Non-dipolar magnetic field at the polar cap of neutron stars and the physics of pulsar radiation
UUniversity of Zielona G´oraFaculty of Physics and Astronomy
Subject matter: Physical ScienceDiscipline: Astronomy
Andrzej Szary
Non-dipolar magnetic field at the polar cap ofneutron stars and the physics of pulsar radiation (Niedipolowe pole magnetyczne nad czap (cid:44) a polarn (cid:44) a gwiazdyneutronowej a fizyka promieniowania pulsar´ow).
Acceptance of supervisor: PhD thesiswritten under the supervision of:prof. dr hab. Giorgi MelikidzeZielona G´ora 2013 a r X i v : . [ a s t r o - ph . H E ] A p r o Natalia, my daughter, and Beata, my wife, for being there... ontents Contents iAbstract vIntroduction vii1 X-ray emission from Radio Pulsars 1
Partially Screened Gap 53
Conclusions 147Acknowledgements 151List of tables 153List of figures 158Bibliography 170 iiiv bstract
Despite the fact that pulsars have been observed for almost half a century, until now manyquestions have remained unanswered. One of the fundamental problems is describing thephysics of pulsar radiation. By trying to find an answer to this fundamental questionwe use the analysis of X-ray observations in order to study the polar cap region of radiopulsars. The size of the hot spots implies that the magnetic field configuration just abovethe stellar surface differs significantly from a purely dipole one. By using the conservationof the magnetic flux we can estimate the surface magnetic field as of the order of 10 G.On the other hand, the temperature of the hot spots is about a few million Kelvins. Basedon these two facts the Partially Screened Gap (PSG) model was proposed to describe theInner Acceleration Region (IAR). The PSG model assumes that the temperature of theactual polar cap is equal to the so-called critical value, i.e. the temperature at which theoutflow of thermal ions from the surface screens the gap completely.We have found that, depending on the conditions above the polar cap, the generationof high energetic photons in IAR can be caused either by Curvature Radiation (CR) orby Inverse Compton Scattering (ICS). Completely different properties of both processesresult in two different scenarios of breaking the acceleration gap: the so-called PSG-offmode for the gap dominated by CR and the PSG-on mode for the gap dominated by ICS.The existence of two different mechanisms of gap breakdown naturally explains the mode-changing phenomenon. Different characteristics of plasma generated in the accelerationregion for both processes also explain the pulse nulling phenomenon. Furthermore, themode changes of the IAR may explain the anti-correlation of radio and X-ray emission invery recent observations of PSR B0943+10 (Hermsen et al., 2013).Simultaneous analysis of X-ray and radio properties have allowed to develop a modelwhich explains the drifting subpulse phenomenon. According to this model the drift takesplace when the charge density in IAR differs from the Goldreich-Julian co-rotationaldensity. The proposed model allows to verify both the radio drift parameters and X-rayefficiency of the observed pulsars. v treszczenie
Pomimo, ˙ze pulsary s (cid:44) a badane ju˙z od prawie p´o(cid:32)l wieku, do dzisiaj nie uda(cid:32)lo si (cid:44) e znale´z´codpowiedzi na wiele pyta´n. Jednym z fundamentalnych problem´ow jest opis fizyki promi-eniowania pulsar´ow. Pr´obuj (cid:44) ac znale´z´c odpowied´z na to fundamentalne pytanie, wykorzys-tujemy analiz (cid:44) e obserwacji rentgenowskich w celu badania obszaru czapy polarnej pulsar´ow.Rozmiar obserwowanych gor (cid:44) acych plam wskazuje, ˙ze konfiguracja pola magnetycznego napowierzchni gwiazdy r´o˙zni si (cid:44) e znacznie od pola czysto dipolowego. Wykorzystuj (cid:44) ac prawozachowania strumienia magnetycznego mo˙zemy oszacowa´c si(cid:32)l (cid:44) e pola magnetycznego w ob-szarze czapy polarnej, kt´ore dla obserwowanych pulsar´ow jest rz (cid:44) edu 10 G. Z drugiejstrony obserwowana temperatura gor (cid:44) acej plamy jest rz (cid:44) edu kilku milion´ow kelwin´ow. Opi-eraj (cid:44) ac si (cid:44) e na tych dw´och faktach wykorzystujemy model cz (cid:44) e´sciowo-ekranowanej przerwyakceleracyjnej (z ang. Partially Screened Gap - PSG), aby opisa´c wewn (cid:44) etrzn (cid:44) a przerw (cid:44) eakceleracyjn (cid:44) a (z ang. Inner Acceleration Region - IAR). Model PSG zak(cid:32)lada, ˙ze temper-atura czapy polarnej jest bliska do tak zwanej warto´sci krytycznej tzn. takiej przy, kt´orejtermiczny odp(cid:32)lyw jon´ow z powierzchni w pe(cid:32)lni ekranuje przerw (cid:44) e akceleracyjn (cid:44) a.W zale˙zno´sci od warunk´ow jakie panuj (cid:44) a w obszarze czapy polarnej, mechanizmemodpowiedzialnym za generowanie wysokoenergetycznych foton´ow w IAR mo˙ze by´c pro-mieniowanie krzywiznowe (z ang. Curvature Radiation - CR) lub odwrotne rozpraszanieComptona (z ang. Inverse Compoton Scattering - ICS). Ca(cid:32)lkowicie r´o˙zne w(cid:32)la´sciwo´sciobu tych proces´ow prowadz (cid:44) a do sytuacji, w kt´orej mo˙zemy wyr´o˙zni´c dwa scenariuszezamkni (cid:44) ecia przerwy akceleracyjnej: tzw. PSG-off dla przerwy zdominowanej przez pro-mieniowanie CR, oraz tzw. PSG-on dla przerwy zdominowanej przez ICS. Istnienie dw´ochr´o˙znych mechanizm´ow zamkni (cid:44) ecia przerwy w naturalny spos´ob t(cid:32)lumaczy zjawisko zmi-any trybu promieniowania pulsar´ow (z ang. mode-changing). R´o˙zna charakterystykaplazmy generowanej w obszarze akceleracyjnym dla obu tych tryb´ow t(cid:32)lumaczy zjawiskosporadycznego braku pojedynczych puls´ow (z ang. pulse nulling) w obserwacjach radiow-ych. Co wi (cid:44) ecej zmiana trybu w jakim pracuje przerwa akceleracyjna mo˙ze zosta´c powi (cid:44) az-ana z antykorelacj (cid:44) a promieniowania radiowego i rentgenowskiego wykazan (cid:44) a w ostatnichobserwacjach PSR B0943+10 (Hermsen et al., 2013).Jednoczesna analiza w(cid:32)la´sciwo´sci promieniowania rentgenowskiego i radiowego poz-woli(cid:32)la na opracowanie modelu dryfuj (cid:44) acych sk(cid:32)ladowych pulsu pojedynczego (z ang. sub-pulses). Model ten zak(cid:32)lada, ˙ze dryf jest wynikiem r´o˙znicy g (cid:44) esto´sci (cid:32)ladunku w IAR wstosunku do g (cid:44) esto´sci korotacji. Proponowany model pozwala zar´owno na weryfikacj (cid:44) ewyznaczonych parametr´ow dryfu oraz na weryfikacj (cid:44) e np. efektywno´sci promieniowaniarentgenowskiego. vi ntroduction
The history of neutron stars began in the early 1930s when Subrahmanyan Chandrasekharcalculated the critical mass for a white dwarf. As soon as the mass of a white dwarf exceedsthe critical value (e.g. due to accretion of matter from a companion star) it collapses and aneutron star is formed. Chandrasekhar estimated that the critical mass was approximately1 . (cid:12) ). Even before James Chadwick’s discovery of neutrons (1932), LevLandau anticipated the existence of neutron stars by writing about stars in which “atomicnuclei come in close contact, forming one gigantic nucleus”. In 1934 Baade and Zwickyproposed that the “supernova process represents the transition of an ordinary star into aneutron star”. Five years later Oppenheimer and Volkoff (1939), using the work of Tolman(1939), computed an upper bound on the mass of a star composed of neutron-degeneratematter. They assumed that the neutrons in a neutron star form a cold degenerate Fermigas which leads to an upper bound of approximately 0 . (cid:12) . Modern estimates of thecritical mass for neutron stars range from approximately 1 . (cid:12) to 3 M (cid:12) (Bombaci, 1996).This uncertainty reflects the fact that the equation of state for extremely dense matter isnot well known. Let us note that the radius of a neutron star should be R ≈
10 km. Onthe other hand nobody expected to detect any emission from neutron stars due to theirsmall size and the lack of theoretical predictions about any radiation processes, exceptfor thermal radiation. Thus, it took almost forty years to detect emission from a neutronstar.The breakthrough came on 28 November 1967 with the radio observations that wereperformed by Jocelyn Bell-Burnell and Anthony Hewish. They observed radio pulsesseparated by 1 .
33 seconds. The world “pulsar” was adopted to reflect the specific propertyof these celestial objects. The suggestion that pulsars were rotating neutron stars was putforth independently by Gold (1968) and Pacini (1968), and was soon proved beyond areasonable doubt by the discovery of a pulsar with a very short (33-millisecond) pulseperiod in the Crab nebula. It was suggested that this pulsar powers the activity of thenebula (Pacini, 1968). Nearly 2000 pulsars have been found so far. Observations of pulsarsprovide valuable information about neutron star physics, general relativity, the interstellarmedium, celestial mechanics, planetary physics, the Galactic gravitational potential, themagnetic field and even cosmology. Studying neutron stars is therefore a very broad issueand it is beyond the scope of this thesis to describe the current status of the theory ofneutron stars or pulsar population studies in detail. We rather refer the reader to theviiiterature (Michel, 1991; M´esz´aros, 1992; Glendenning, 1996; Weber, 1999) and provideonly a basic theoretical background that is relevant to the subject of this thesis.Following the ideas of Pacini (1968) and Gold (1968) radio pulsars can be interpretedas rapidly spinning, strongly magnetised neutron stars radiating at the expense of theirrotational energy. Neutron stars consist of compressed matter with density in its coreexceeding nuclear density ρ nuc = 2 . × g cm − . Direct and accurate mass meas-urements come from timing observations of binary pulsars and are consistent with atypically assumed neutron star mass M ≈ . (cid:12) . Most models predict a radius of R ∼
10 km, which is consistent with the theoretical upper and lower limits. However, themeasurements of neutron star radii are much less reliable than the mass measurements.Therefore, the moment of inertia for these canonical values ( M = 1 . (cid:12) , R = 10 km) I ≈ (2 / M R ≈ g cm may be uncertain by ∼ P = d P/ d t , is related to the rate of rotational kinetic energy loss (spin-downluminosity) ˙ E = L SD = 4 π I ˙ P P − . In most cases only a tiny fraction of ˙ E can beconverted into radio emission. The efficiency, χ radio = L radio / ˙ E , in the radio bands istypically in the range of ∼ − − − . It is assumed that the bulk of the rotationalenergy is converted into magnetic dipole radiation. The expected evolution of the angularvelocity (Ω = 2 π/P ) of a rotating magnetic dipole can be described as ˙Ω ∼ Ω n , and thebreaking index is n = 3 for the pure dipole radiation. Indeed, the observed values ofthe breaking index (e.g. Becker, 2009) confirm the above statement, e.g.: for the Crab n = 2 . ± . n = 2 . ± .
2, for PSR B0540-69 n = 2 . ± .
02, forPSR J1911-6127 n = 2 . ± .
05, for PSR J1846-0258 n = 2 . ± .
01, and for the Velapulsar n = 1 . ± .
2. On the other hand the observations of pulsar wind nebulae suggestthat a significant fraction of the pulsar rotational energy is carried away by a pulsar wind.Furthermore, recent observations of high energy radiation from pulsars show that signific-antly more energy is radiated in the form of X-rays and γ -rays than in the form of radioemission (e.g. Abdo et al., 2010). Thus, pure magnetic breaking does not provide fullinformation about the physical processes that take place in the pulsar magnetosphere.Despite the fact that pulsars have been observed for almost half a century, manyquestions still remain unanswered. One of the fundamental problems concerns the physicsof pulsar radiation. Radio observations alone cannot point to the model (e.g. vacuum gap,slot gap, outer gap, free outflow, etc.) that correctly describes the source of pulsar activity.Observations carried out by relatively new high-energy instruments, e.g. Chandr a and
XMM-Newton , significantly extended the spectra over which we can study pulsars andtheir environments. There is no consensus about the origin of pulsar X-ray emission(Michel, 1991). We can distinguish two main types of models: the polar gap and the outergap. The polar gap models suggest that the emission region is located in the vicinity ofthe neutron star polar caps, while the outer gap models assume that particle accelerationviiind X-ray emission take place close to the pulsar light cylinder . In both types of modelshigh energy radiation is generated by relativistic particles accelerated in charge-depletedregions, while the high energy photons are emitted by means of Curvature Radiation(CR), Synchrotron Radiation (SR) and Inverse Compton Scattering (ICS). Both modelsare able to interpret existing observational data.In this thesis we will use the Partially Screened Gap (PSG) model (Gil et al., 2007a).The PSG model assumes the existence of the Inner Acceleration Region (IAR) abovethe polar cap (a region penetrated by the open field lines) where the electric field hasa component along the magnetic field. In this region particles (electrons and positrons)are accelerated in both directions: outward and toward the stellar surface. Consequently,outflowing particles are responsible for generation of magnetospheric emission (radio andhigh-frequency) while the backflowing particles heat the surface and provide the requiredenergy for thermal emission. The PSG model is an extension of the Standard Modeldeveloped by Ruderman and Sutherland (1975) and takes into account the thermionicion flow from the stellar surface heated up to a high temperature (a few million Kelvins)by the backstreaming particles. In such a scenario an analysis of X-ray radiation is anexcellent method of obtaining insight into the most intriguing region of the neutron star. The light cylinder with radius R LC = cP/ π is defined as a place where the azimuthal velocity of theco-rotating magnetic field lines is equal to the speed of light ( c ) ix Chapter 1X-ray emission from Radio Pulsars
X-ray photons can only be detected by telescopes operating at high altitudes or above theEarth’s atmosphere, thus detectors should be mounted on high-flying balloons, rockets orsatellites. The first (i.e. carried out from space) X-ray observations were performed bya team led by Herbert Friedman in 1948. The team estimated the luminosity of X-rayradiation from the solar corona. They found that X-ray luminosity is weaker by a factorof 10 than luminosity in the optical wave range. Up until the early 1960s it was widelybelieved that all other stars should be so faint in the X-rays that their observationswould be hopeless. The situation changed in 1962 when a team led by Bruno Rossiand Riccardo Giacconi, when trying to find fluorescent X-ray photons from the moon,accidentally detected X-rays from Sco X-1. Subsequent flights launched to confirm thesefirst results detected Tau X-1, a source in the constellation Taurus which coincided withthe Crab supernova remnant (Bowyer et al., 1964). The search for similar sources becamea source of strong motivation for the further development of X-ray astronomy.Before the first direct detection of a neutron star by Hewish et al. (1968), it waspredicted that neutron stars could be powerful sources of thermal X-ray emission dueto a high surface temperature ( T s ). The expected value of the surface temperature wasestimated as T s ∼ were initiated by the Einstein Observatory, whichwas launched by NASA in 1978. Using a high-resolution imaging camera sensitive in the0 . − . Einstein detected X-ray emission froma number of neutron stars (mainly as compact sources in supernova remnants) such asthe middle-aged radio pulsars B0656+14, B1055-52 and the old pulsar B0950+08. The
Einstein observatory re-entered the Earth’s atmosphere and burned up on 25 March 1982. The term ”isolated” is omitted hereafter in the text however all X-ray observations presented in thisthesis concern isolated neutron stars
Chapter 1. X-ray emission from Radio Pulsars
The next ”decade of space science” was opened in the 1990s with the launch of the
ROSAT mission that was sensitive in the 0 . − . ROSAT was the identification of the γ -ray source Geminga as a pulsar,hence a neutron star (Halpern and Holt, 1992).The current era of X-ray observations of neutron stars was begun with the launchof two satellites: the XMM-Newton owned by the European Space Agency and the
Chandra owned by the National Aeronautics and Space Administration. These twograzing-incidence X-ray telescopes were placed in orbit in 1999. They were equippedwith cameras and high-resolution spectrometers sensitive to low-energy X-rays: from 0 . Chandra and from 0 . XMM-Newton . While the twoobservatories have similar designs, they are not identical. The
XMM-Newton observat-ory has three X-ray telescopes that provide six times the collecting area and a broaderspectral range in images than the
Chandra , while the
Chandra has a much finer spatialresolution and a broader spectral range in its high-resolution spectroscopy than does the
XMM-Newton . Both observatories are in a highly-elliptical orbit that permits continuousobservations of up to 40 hours. The
Chandra and
XMM-Newton have greatly increasedthe quality and availability of observations of X-ray thermal radiation from neutron starsurfaces. The total number of isolated neutron stars of different types detected in X-raysis hard to find since not all data have been published. Some authors estimate that aboutone hundred rotation-powered pulsars were detected in the X-rays (Zavlin, 2007a; Becker,2009).
X-ray emission is a common feature of all kinds of neutron stars. Furthermore, X-rayobservations have led to the discovery of other types of neutron stars that for variousreasons were missed in the standard searches for radio pulsars. These new classes, suchas X-ray Dim Isolated Neutron Stars, Central Compact Objects in supernovae remnants,Anomalous X-ray Pulsars, and Soft Gamma-ray Repeaters, are only a small fraction ofthe whole number of observed pulsars but provide valuable information on the diversityof the neutron star population.X-ray radiation from an isolated neutron star can in general consist of two distinguish-able components: thermal and nonthermal emissions. The thermal emission can originateeither from the entire surface of a cooling neutron star or from spots around the magneticpoles on the stellar surface (polar caps and adjacent areas). The temperature of a neut-ron star at the moment of its formation is extremely high - its value is even as high as10 − K. Such a high initial temperature leads to very fast cooling, and after severalminutes the temperature of the star interior falls to 10 − K. After 10 −
100 yr theneutron star will cool down to a few times 10 K. At this point, depending on the stillpoorly known properties of super-dense matter, the temperature evolution can follow two .2. X-ray emission from isolated neutron stars ∼ (0 . − × K by the end of the neutrino cooling era and then fallsexponentially to temperatures lower than ∼ K in ∼ yr. In the accelerated coolingscenario, which implies higher central densities (up to 10 g cm − ) and/or exotic interiorcomposition (e.g. quark plasma), at the age of ∼ −
100 yr the temperature decreasesrapidly down to ∼ (0 . − . × K and is followed by a more gradual decrease down tothe same ∼ K in ∼ yr (Becker, 2009). The thermal evolution of neutron stars isvery sensitive to the composition (and structure) of their interiors, therefore, measuringsurface temperatures is an important tool in studying super-dense matter. In additionto a thermal component emitted from the entire surface, other thermal components canalso be seen. One of these additional components could be related to the reheating of thepolar cap region by relativistic backflowing particles (electron and/or positrons) createdand accelerated in the so-called polar gaps (see Chapter 3). The temperature of these hotspots does not obey the same age dependence as the thermal evolution of neutron stars.Thus, depending on the pulsar age the thermal radiation may be dominated by either theentire surface (for younger neutron stars) or the hot spot components (for older neutronstars). The nonthermal component is usually attributed to the emission produced by Syn-chrotron Radiation (SR) and/or Inverse Compton Scattering (ICS) of charged relativisticparticles accelerated in the pulsar magnetosphere. As the energy of these particles followsa power-law distribution, nonthermal emission is also characterised by power-law spectra.The X-ray spectrum of a neutron star (thermal and nonthermal) depends on manyfactors, e.g. the age of the star ( τ ), inclination angle, strength and geometry of themagnetic field, etc. In most of the very young pulsars ( τ ∼ L SD . A spin-down luminosity generally decreases with the increasingstar age, as L SD ∝ τ − m , where m (cid:39) − τ ∼
100 kyr) and some younger ( τ ∼
10 kyr) pulsars. For the old neutronstars ( τ > T s < . T s (cid:63) R dp = (cid:112) πR /cP .Since the spin-down luminosity L SD is the source for both nonthermal (magneto-spheric) and thermal (polar cap) components, it is hard to predict which one wouldprevail in the X-ray flux of old neutron stars. Figure 1.1a shows the ratio of a thermal Chapter 1. X-ray emission from Radio Pulsars luminosity to a nonthermal one as a function of the pulsar age. Since calculating thisratio is possible only for pulsars with blackbody plus power-law fit, only these pulsarsare included in the Figure. There is also a significant number of pulsars (16) with thespectra dominated by nonthermal components. Let us note that it is impossible to de-termine the thermal components for these pulsars. Most of them are young neutron stars ∼ − yr, but there are also much older ones ( ∼ yr). In addition, there is agroup of 4 pulsars with the spectra dominated by thermal components (without a visiblenonthermal component). Their age also varies in quite a wide range 10 − yr. Figure 1.1:
Ratio of X-ray luminosities (thermal and nonthermal components) as a function of τ (panel a) and B d (panel b). The plots contain only those pulsars for which the BB+PL (Black-Body plus Power-Law) spectral fit exists. The number labels at the points correspond to the pulsarnumbers in Table 1.1. As it follows from the left panel of Figure 1.1, there is no obvious relation betweenpulsar age and the ratio of luminosities. The spectra of pulsars with a similar age may bedominated either by nonthermal (e.g. PSR B1951+32, PSR B1046-58) or thermal (e.g.PSR B0656+14, PSR J0538+2817) components. It is difficult to provide a more detailedanalysis because, on the one hand, the observational errors are large and, on the otherhand, a separation of thermal and nonthermal components is often not possible. Theratio of luminosities also does not show any correlation with the strength of the dipolarmagnetic field (see the right panel of Figure 1.1). Let us note that the value of the dipolarmagnetic field is conventionally calculated by adopting that the spin-down luminosity isequal to the power of magneto-dipole radiation (neglecting the influence of a pulsar wind).Then, assuming a dipolar structure of the neutron star magnetic field down to the stellarsurface, we estimate its strength (measured in Gauss) at the pole as B d = 2 . × (cid:16) P ˙ P − (cid:17) . . (1.1)Here P is a period in seconds and ˙ P − = ˙ P × . The actual strength of the surface .2. X-ray emission from isolated neutron stars Chapter 1. X-ray emission from Radio PulsarsTable 1.1:
Parameters of rotation powered normal pulsars with detected X-ray radiation. Theindividual columns are as follows: (1) Pulsar name, (2) Barycentric period P of the pulsar, (3) Timederivative of barycentric period ˙ P , (4) Canonical value of the dipolar magnetic field B d at the poles,(5) Spin-down energy loss rate L SD (spin-down luminosity) , (6) Dispersion measure DM , (7) Bestestimate of pulsar distance D (used in all calculations), (8) Best estimate of pulsar age or spin-downage τ = P/ (cid:16) P (cid:17) , (9) Pulsar number (used in the Figures). Parameters of the radio pulsar havebeen taken from the ATNF catalogue. Name P ˙ P B d log L SD DM D τ
No. (s) (cid:0) − (cid:1) (cid:0) G (cid:1) (cid:0) erg s − (cid:1) (cid:0) cm − pc (cid:1) (kpc) J0108–1431 0 .
808 0 .
077 0 .
504 30 .
76 2 .
38 0 .
18 166 Myr 1J0205+6449 0 .
066 193 . .
210 37 .
43 141 3 .
20 5 .
37 kyr 2B0355+54 0 .
156 4 .
397 1 .
675 34 .
65 57 . .
04 564 kyr 3B0531+21 0 .
033 422 . .
555 38 .
66 56 . .
00 1 .
24 kyr 4J0537–6910 0 .
016 51 .
78 1 .
846 38 .
69 – 47 . .
93 kyr 5J0538+2817 0 .
143 3 .
669 1 .
464 34 .
69 39 . .
20 30 . .
050 478 . .
934 38 .
18 146 55 . .
67 kyr 7B0628–28 1 .
244 7 .
123 6 .
014 32 .
18 34 . .
45 2 .
77 Myr 8J0633+1746 0 .
237 10 .
97 3 .
258 34 .
51 – 0 .
16 342 kyr 9B0656+14 0 .
385 55 .
00 9 .
294 34 .
58 14 . .
29 111 kyr 10J0821–4300 0 .
113 1 .
200 0 .
743 34 .
52 – 2 .
20 3 . .
531 1 .
709 1 .
924 32 .
65 19 . .
34 4 .
92 Myr 12B0833–45 0 .
089 125 . .
750 36 .
84 68 . .
21 11 . .
274 6 .
799 5 .
945 32 .
11 12 . .
64 2 .
97 Myr 14B0943+10 1 .
098 3 .
493 3 .
956 32 .
00 15 . .
63 4 .
98 Myr 15B0950+08 0 .
253 0 .
230 0 .
487 32 .
75 2 .
96 0 .
26 17 . .
124 96 .
32 6 .
972 36 .
30 129 2 .
70 20 . .
197 5 .
833 2 .
166 34 .
48 30 . .
75 535 kyr 18J1105–6107 0 .
063 15 .
83 2 .
020 36 .
40 271 7 .
00 63 . .
408 4022 81 .
80 36 .
36 707 8 .
40 1 .
61 kyr 20
Continued on next page .2. X-ray emission from isolated neutron stars P ˙ P B d log L SD DM D τ
No. (s) (cid:0) − (cid:1) (cid:0) G (cid:1) (cid:0) erg s − (cid:1) (cid:0) cm − pc (cid:1) (kpc) J1124–5916 0 .
135 747 . .
31 37 .
08 330 6 .
00 2 .
87 kyr 21B1133+16 1 .
188 3 .
734 4 .
254 31 .
94 4 .
86 0 .
36 5 .
04 Myr 22J1210–5226 0 .
424 0 .
066 0 .
338 31 .
53 – 2 .
45 102 Myr 23B1259–63 0 .
048 2 .
276 0 .
666 35 .
91 147 2 .
00 332 kyr 24J1357–6429 0 .
166 360 . .
62 36 .
49 128 2 .
50 7 .
31 kyr 25J1420–6048 0 .
068 83 .
17 4 .
810 37 .
00 360 8 .
00 13 . .
263 0 .
098 0 .
325 32 .
32 8 .
60 0 .
48 42 . .
089 9 .
170 1 .
824 35 .
71 138 2 .
56 154 kyr 28B1509–58 0 .
151 1537 30 .
73 37 .
26 252 4 .
18 1 .
55 kyr 29J1617–5055 0 .
069 135 . .
183 37 .
20 467 6 .
50 8 .
13 kyr 30B1706–44 0 .
102 92 .
98 6 .
235 36 .
53 75 . .
50 17 . .
236 10 .
85 3 .
234 34 .
52 99 . .
84 345 kyr 32J1747–2958 0 .
099 61 .
32 4 .
972 36 .
40 102 5 .
00 25 . .
125 127 . .
075 36 .
41 289 5 .
00 15 . .
134 134 . .
551 36 .
34 234 4 .
00 15 . .
083 25 .
54 2 .
936 36 .
26 197 3 .
50 51 . .
065 44 .
00 3 .
407 36 .
81 – 5 .
00 23 . .
101 75 .
06 5 .
575 36 .
45 231 4 .
00 21 . .
326 7083 97 .
02 36 .
91 – 6 .
00 0 .
73 kyr 39B1853+01 0 .
267 208 . .
08 35 .
63 96 . .
60 20 . .
181 212 . .
99 33 .
71 27 . .
10 88 . .
137 750 . .
47 37 .
08 308 5 .
00 2 .
89 kyr 42B1929+10 0 .
227 1 .
157 1 .
034 33 .
59 3 .
18 0 .
36 3 .
10 Myr 43B1951+32 0 .
040 5 .
845 0 .
971 36 .
57 45 . .
00 107 kyr 44J2021+3651 0 .
104 95 .
60 6 .
361 36 .
53 371 10 . . .
096 1 .
270 0 .
706 34 .
75 21 . .
80 1 .
20 Myr 46B2224+65 0 .
683 9 .
659 5 .
187 33 .
08 36 . .
00 1 .
12 Myr 47B2334+61 0 .
495 191 . .
69 34 .
79 58 . .
10 40 . Chapter 1. X-ray emission from Radio Pulsars
The nonthermal emission, which is generally observed from radio to γ -ray frequencies,should be generated by charged particles accelerated at the expense of rotational energyin the magnetosphere of the neutron star. Nonthermal X-ray radiation is characterisedby highly anisotropic emission patterns, which give rise to large pulsed fractions. Thepulse profiles often show narrow (often double) peaks, however, in many cases nearlysinusoidal profiles are observed. As the X-ray efficiency is strongly correlated with L SD ,the most X-ray luminous sources (among rotationally powered pulsars) are the Crab pulsarand two young pulsars in the Large Magellanic Cloud, which are the only pulsars with L SD > erg s − (Mereghetti, 2011).Becker and Truemper (1997) suggested that in the 0 . . ROSAT sourcesthat are identified as rotation-powered pulsars exhibit an X-ray efficiency which can be ap-proximated as a linear function L X = ξL SD , where the total X-ray efficiency ξ = ξ BB + ξ NT ≈ − , here ξ BB and ξ NT are efficiencies of the thermal (without thecooling component) and nonthermal X-ray emission, respectively. The higher sensitivityof both the Chandra and
XMM-Newton allows detection of less efficient ( ξ < − ) X-raypulsars (see Figure 1.2). Becker (2009) suggested that for these faint pulsars the orient-ation of the magnetic/rotation axes to the observer’s line of sight might not be optimal.We believe that the efficiency of spin-down energy conversion processes is mostly affectedby the strength and structure of the surface magnetic field. The variation of ξ is ratherdue to the nature of physical processes than the geometrical effects. Let us note thatthe nonthermal X-ray luminosities presented in Figure 1.2 are calculated assuming anisotropic radiation pattern. In general, the X-ray emission pattern differs quite essentiallyfrom the isotropic one. Thus, one should introduce a beaming factor as the ratio of theopening angle of the radiation cone to the full solid angle 4 π . Since a beaming factor isgenerally unknown, the actual X-ray efficiency may differ by up to an order of magnitude(or even more) than we have presented.Various fitting parameters and efficiencies of nonthermal X-ray radiation suggest thatthe efficiency of processes responsible for the generation of nonthermal X-ray radiationshould highly depend on the pulsar parameters (see Figure 1.2). The fitting parametersfor the data of all pulsars show a linear trend with ξ ≈ − , however, if we divide theminto two groups of less and more luminous pulsars, we can see that the fitting parametersfor these two groups differ from one another. The efficiency of less luminous X-ray pulsarsdepends on L SD to a lesser extent than is the case for more luminous pulsars.As we mentioned in the Introduction, there are two main types of models: the polarcap models and the outer gap models. The outer gap model was proposed to explain thebright γ -ray emission from the Crab and Vela pulsars (Cheng et al., 1986a,b). Placing a γ -ray emission zone at the light cylinder, where the magnetic field strength is considerablyreduced to B LC = B d ( R/R LC ) , provides higher γ -ray emissivities that are in somewhat .3. Nonthermal X-ray radiation Figure 1.2:
Nonthermal luminosity within the . −
10 keV band ( L NT ) vs spin-down luminosity( L SD ). The black solid line corresponds to the linear fitting for all pulsars, while the blue dottedand red dashed lines correspond to the linear fit for less luminous ( L SD < erg s − ) and moreluminous ( L SD > erg s − ) pulsars, respectively. better agreement with the observations. The observational data can be interpreted withany of the two models, although under completely different assumptions about pulsarparameters. Generally, the X-ray spectrum of relatively young ( τ <
10 kyr) and middle-aged( τ <
10 kyr) pulsars is dominated by the nonthermal component. However, it is notpossible to find an exact correlation between τ and the type of spectra, i.e. which com-ponent, thermal or nonthermal, dominates the spectrum (see the left panel of Figure 1.1).As we mentioned above, it is quite often impossible to resolve the components. The Crabpulsar ( τ = 958 yrs) is the most characteristic example of a young pulsar. The upper limitfor X-ray luminosity of the Crab pulsar (one of the strongest known X-ray radio pulsars)is about L max NT = 8 . × erg s − . This value is calculated assuming an isotropic radiationpattern, however, even if we assume an angular anisotropy of the radiation (beamingfactor ≈ / π ), the lower limit of its luminosity L min NT = 7 . × erg s − continues to bevery high. The luminosities calculated above correspond to the following X-ray efficien-cies: ξ maxNT = 10 − . (isotropic radiation pattern) and ξ minNT = 10 − . (anisotropic radiationpattern). Although ξ NT is quite small, the nonthermal component still obscures all thethermal ones. To obtain a similar efficiency of the thermal radiation from the entire stellarsurface, its temperature should be T s = 5 . × K (assuming R = 10 km), which vastlyexceeds the upper limit ( T s < . × K). Furthermore, the temperature of the polar0
Chapter 1. X-ray emission from Radio Pulsars caps should be about 2 . × K to obtain a comparable luminosity.The Vela-like pulsars compose another characteristic group of pulsars. This groupconsists of pulsars with high spin-down luminosities but considerably low X-ray efficiencies ξ NT (cid:62) − . A characteristic age of the Vela is about 1 . × yrs (10 times older thanthe Crab), but it can still be classified as a very young pulsar. The nonthermal luminosityof a Vela pulsar is L max NT = 4 . × and efficiency ξ maxNT = 10 − . . Some of the Vela-like pulsars (like the Vela itself) also exhibit a thermal component, which in some casescan be comparable to the nonthermal component. The thermal efficiency of the Vela ξ BB = 10 − . is quite similar to ξ maxNT = 10 − . , but if we assume an anisotropic radiationpattern of the nonthermal component than ξ minNT = 10 − . , thus even less than ξ BB .The third group includes pulsars with low spin-down luminosity L SD (cid:62) . Inmost cases, the X-ray spectra of such pulsars (e.g. PSR 9050+08, PSR B1929+10) haveboth thermal and nonthermal components, with similar efficiencies. Thus, the spectrumfitting procedure is more complicated. The nonthermal X-ray efficiencies of these pulsars, ξ NT ∼ − , are considerably higher than those of the Vela-like pulsars. Note that evenwhen the observed spectra are dominated by nonthermal radiation, we cannot rule outa situation that the thermal component is stronger than the nonthermal one, but due tounfavourable geometry we cannot observe it.Even with the improved quality of X-ray observations performed by both the Chandra and
XMM-Newton , the available data do not allow us to fully discriminate between the dif-ferent emission scenarios. However, these data can be used to verify whether the proposedmodel of X-ray emission meets all the requirements. Table 1.2 presents the observed spec-tral properties of pulsars showing nonthermal components. . . N o n t h e r m a l X - r a y r a d i a t i o n Table 1.2:
Observed spectral properties of rotation-powered pulsars with X-ray spectrum showing the nonthermal (power-law) component. The individualcolumns are as follows: (1) Pulsar name, (2) Additional information, (3) Spectral components required to fit the observed spectra, PL: power law, BB:blackbody, (4) Pulse phase average photon index, (5) Maximum nonthermal luminosity L NT , (6) Maximum nonthermal X-ray efficiency ξ maxNT , (7) Minimumnonthermal X-ray efficiency ξ minNT , (8) Total thermal luminosity L BB , (9) Thermal efficiency ξ BB , (10) References, (11) Number of the pulsar. Both nonthermalluminosities and efficiencies were calculated in the . −
10 keV band. The maximum value was calculated with the assumption that the X-ray radiation isisotropic while the minimum value was calculated assuming strong angular anisotropy of the radiation ( ξ minNT ≈ / (4 π ) · ξ maxNT ). Pulsars are sorted by nonthermalX-ray luminosity (5). Name Comment Spectrum Photon-Index log L NT log ξ maxNT log ξ minNT log L BB log ξ BB Ref. No. (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) B0540–69 N158A, LMC PL 1 . +0 . − . . − . − .
37 – – Ka01, Ca08 7B0531+21 Crab PL 1 . +0 . − . . − . − .
81 – – Be09 4J0537–6910 N157B, LMC PL 1 . +0 . − . . − . − .
84 – – Mi05 5B1509–58 Crab-like pulsar PL 1 . +0 . − . . − . − .
10 – – Cu01, De06, Be09 29J1846–0258 Kes 75 BB + PL 1 . +0 . − . . − . − .
88 34 . − .
85 Ng08, He03 39J1420–6048 PL 1 . +0 . − . . − . − .
35 – – Ro01 26J2021+3651 PL, BB 1 . +0 . − . . − . − .
27 33 . − .
75 Va08,He04 45J1617–5055 Crab-like pulsar PL 1 . +0 . − . . − . − .
05 – – Ka09, Be02 30J1747–2958 Mouse PL, BB 1 . +0 . − . . − . − .
41 – – Ga04 33J1811–1925 G11.2-0.3 PL 0 . +0 . − . . − . − .
94 – – Ro03, Ro04 37J1930+1852 Crab-like pulsar PL 1 . +0 . − . . − . − .
25 – – Lu07, Ca02 42
Continued on next page C h a p t e r . X - r a y e m i ss i o n f r o m R a d i o P u l s a r s Table 1.2 - continued from previous pageName Comment Spectrum Photon-Index log L NT log ξ max NT log ξ min NT log L BB log ξ BB Ref. No. (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) J1105–6107 PL 1 . +0 . − . . − . − .
58 – – Go98 19B1757–24 Duck PL 1 . +0 . − . . − . − .
05 – – Ka01 34B1951+32 CTB 80 BB + PL 1 . +0 . − . . − . − .
45 31 . − .
62 Li05 44J0205+6449 3C58 BB + PL 1 . +0 . − . . − . − .
43 33 . − .
83 Sl04 2J1119–6127 G292.2-0.5 BB + PL 1 . +0 . − . . − . − .
51 33 . − .
00 Go07, Ng12 20J1124–5916 Vela-like pulsar PL 1 . +0 . − . . − . − .
27 – – Hu03,Go03 21B1259–63 Be-star bin PL 1 . +0 . − . . − . − .
15 – – Ch09, Ch06 24B0833–45 Vela BB + PL 2 . +0 . − . . − . − .
32 32 . − .
72 Za07b 13B1706–44 G343.1-02.3 BB + PL 2 . +0 . − . . − . − .
47 32 . − .
76 Go02 31J1357–6429 BB + PL 1 . +0 . − . . − . − .
44 32 . − .
99 Za07 25B1853+01 W44 PL 1 . +0 . − . . − . − .
66 – – Pe02 40B1046–58 Vela-like pulsar PL 1 . +0 . − . . − . − .
36 – – Go06 17B1916+14 BB, PL 3 . +1 . − . . − . − .
81 31 . − .
63 Zh09 41J1509–5850 MSH 15-52 PL 1 . +0 . − . . − . − .
02 – – Hu07 28B1823–13 Vela-like BB + PL 1 . +0 . − . . − . − .
77 32 . − .
27 Pa08 38B1800–21 Vela-like pulsar PL + BB 1 . +0 . − . . − . − .
84 – – Ka07 35
Continued on next page . . N o n t h e r m a l X - r a y r a d i a t i o n Table 1.2 - continued from previous pageName Comment Spectrum Photon-Index log L NT log ξ max NT log ξ min NT log L BB log ξ BB Ref. No. (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) J1809–1917 BB + PL 1 . +0 . − . . − . − .
78 31 . − .
56 Ka07 36B2334+61 BB + PL 2 . +3 . − . . − . − .
34 32 . − .
73 Mc06 48J2043+2740 BB + PL 2 . +1 . − . . − . − .
44 30 . − .
98 Be04 46B2224+65 Guitar PL, BB 2 . +0 . − . . − . − .
97 30 . − .
57 Hu12, Hu07b 47B0355+54 BB + PL 1 . +0 . − . . − . − .
83 30 . − .
25 Mc07,Sl94 3B1055–52 BB+BB+PL 1 . +0 . − . . − . − .
67 32 . − .
85 De05 18B0656+14 BB+BB+PL 2 . +0 . − . . − . − .
42 32 . − .
81 De05 10J0633+1746 Geminga BB+BB+PL 1 . +0 . − . . − . − .
37 31 . − .
84 Ja05 9B1929+10 BB + PL 1 . +0 . − . . − . − .
46 30 . − .
53 Mi08 43B0628–28 BB + PL 2 . +0 . − . . − . − .
04 30 . − .
94 Te05 , Be05 8B0950+08 BB + PL 1 . +0 . − . . − . − .
86 28 . − .
82 Za04 16B1451–68 BB + PL 1 . +0 . − . . − . − .
66 29 . − .
06 Po12 27B1133+16 BB, PL 2 . +0 . − . . − . − .
52 28 . − .
38 Ka06 22B0823+26 PL 1 . +0 . − . . − . − .
33 – – Be04 12B0943+10 Chameleon BB, PL 2 . +0 . − . . − . − .
74 28 . − .
62 Zh05,Ka06 15B0834+06 BB + PL – 28 . − . − .
51 28 . − .
41 Gi08 14J0108–1431 BB + PL 3 . +0 . − . . − . − .
29 27 . − .
82 Po12, Pa09 14
Chapter 1. X-ray emission from Radio Pulsars
Thermal X-ray emission seems to be quite a common feature of radio pulsars. The black-body fit to the observed thermal spectrum of a neutron star allows us to obtain theredshifted effective temperature T ∞ and redshifted total bolometric flux F ∞ (measuredby a distant observer). To estimate the actual (unredshifted) parameters, one should takeinto account the gravitational redshift, g r = (cid:112) − GM/Rc , determined by the neutronstar mass M and radius R , here G is the gravitational constant. Then the actual effectivetemperature and actual total bolometric flux can be written as: T = g − T ∞ ,F = g − F ∞ . (1.2)Knowing the distance to the neutron star, D , we can use the effective temperature andtotal bolometric flux to calculate the size of the radiating region. If we assume that theradiation is isotropic (same in all directions, e.g. radiation from the entire stellar surface)then the radius of the radiating sphere (star) can be calculated as (Zavlin, 2007a) R ∞⊥ = D (cid:114) F ∞ σT ∞ = g − R ⊥ , (1.3)where σ ≈ . × − erg cm − s − K − is the Stefan-Boltzmann constant.Knowing that L BB = 4 πD F and using Equations 1.2 and 1.3, we can write that L BB = g − L ∞ BB . (1.4)The modelling of thermal radiation is more complicated if we assume that it comesfrom the hot spot on the stellar surface. One should take into account such factors as:time-averaged cosine of the angle between the magnetic axis and the line of sight (cid:104) cos i (cid:105) ,gravitational bending of light, as well as whether the radiation comes from two oppositepoles of the star or from one hot spot only. In general, the observed luminosity of the hotspot can be written as: L ∞ hs = A ∞ hs σT ∞ , (1.5)where A ∞ hs = πR ∞ is the observed area of the radiating region.The observed area of the radiating spot is also influenced by the geometrical factor f .This geometrical factor depends on following angles: ζ between the line of sight and thespin axis, and α between the spin and magnetic axes, as well as on g r and whether the .4. Thermal X-ray radiation A ∞ hs = g − f A hs ,R hs = g r f − / R ∞ hs . (1.6)Finally, the hot spot luminosity can be calculated as L hs = g − f − L ∞ hs . (1.7)The luminosity of a radiating sphere with radius R ⊥ can be calculated as L sp = 4 A ⊥ σT = 4 πR ⊥ σT . On the other hand, if we assume that the radiation ori-ginates only from one hot spot we can calculate the luminosity as L hs = A hs σT . If thehot spot size is small compared to the star radius ( R hs (cid:28) R ) then the area of the spotcan be calculated as A hs ≈ πR ⊥ . Thus, we have to remember that the luminosity calcu-lated assuming a spherical source will be four times higher than the actual luminosity ofa radiating hot spot L hs = 1 / · L sp (see the next section for details). Figure 1.3:
Coordinate system co-rotating with a star. The system was chosen so that the z-axisis along Ω (the angular velocity) and o lies in the x-z plane (fiducial plane, i.e. at longitude zero).Here, ˆ µ is a unit vector in the direction of the magnetic axis and α is the angle between Ω and ˆ µ , β is the impact parameter. Chapter 1. X-ray emission from Radio Pulsars
Let us consider a neutron star with two antipodal hot spots associated with polar capsof a stellar magnetic field. For simplicity’s sake we assume that the spot size is smallcompared to the star radius R . If the magnetic axis ˆ µ is inclined to the spin axis by anangle α ≤ ◦ , the spots periodically change their position and inclination with respectto a distant observer. To compute the radiation fluxes from the primary (closer to theobserver) as well as the antipodal spot, we need to know their inclinations: cos i = n · o and cos i = ¯n · o = − cos i , where n and ¯n = − n are normal vectors to spots surfaces,and o is the unit vector pointing toward the observer. In the calculations we use acoordinate system co-rotating with a star. The z-axis is along Ω (the angular velocity)and o lies in the x-z plane (see Figure 1.3).In the chosen coordinate system we can write that the spherical coordinates of vectorshave the following components: Ω = (Ω , ,
0) ; o = (1 , α + β,
0) ; ˆ µ = (1 , α, Ω t ) . (1.8)Here the impact parameter β represents the closest approach of the line of sight tothe magnetic axis. Note that ˆ µ = n and ¯n = − ˆ µ ; thus, we can write the followingcomponents of Cartesian coordinates: o = (sin ( α + β ) , , cos ( α + β )) ; n = (sin α cos Ω t, sin Ω t sin α, cos α ) . (1.9)Finally, the inclination angle for both primary and antipodal hot spots can be calcu-lated ascos i = sin α · cos Ω t · sin ( α + β ) + cos α · cos ( α + β ) ;cos i = − cos i = − sin α · cos Ω t · sin ( α + β ) − cos α · cos ( α + β ) . (1.10)We can estimate the contributions of the primary and antipodal spots to the observedX-ray flux by calculating the time-averaged cosine of the angle between the magnetic axisand the line of sight. Note that we should take into account only positive values of cos i since for larger angles ( i > ◦ ) the spot is not visible (at least in this approximation,see Section 1.4.3 for more details). Thus, the contribution of the primary spot can becalculated as follows: (cid:104) cos i (cid:105) = ˆ P cos ( i ) d t if α tan( α + β ) < − α tan( α + β ) > , ˆ t − cos ( i ) d t + ˆ πt + cos ( i ) d t if − < α tan( α + β ) < , (1.11) .4. Thermal X-ray radiation t ± = P ± P π arccos (cid:20) α tan ( α + β ) (cid:21) . (1.12)On the other hand, the contribution of the antipodal spot can be calculated as (cid:104) cos i (cid:105) = α tan( α + β ) < − α tan( α + β ) > , ˆ t + t − cos ( i ) d t if − < α tan( α + β ) < . (1.13)Depending on the orientation of Ω , o and ˆ µ , the thermal radiation may originate from:(1) both the primary and antipodal hot spots (see Figure 1.4); (2) mainly the primary spotbut with a small contribution from the antipodal spot (see Figure 1.5); (3) the primaryspot only (see Figure 1.6). Figure 1.4:
Cosine of the hot spots’ inclination angle as a function of the pulsar phase forPSR B0950+08. The following parameters were used: α = 105 . ◦ , β = 21 . ◦ . For this geo-metry the thermal radiation of both primary and antipodal spots has a significant influence on theobserved thermal flux. Chapter 1. X-ray emission from Radio PulsarsFigure 1.5:
Cosine of the hot spots’ inclination angle as a function of the pulsar phase forPSR B1929+10. The following parameters were used: α = 35 . , β = 25 . . For this geometrythere is only a small contribution from the antipodal spot. Figure 1.6:
Cosine of the hot spots’ inclination angle as a function of the pulsar phase forPSR B0943+10. The following parameters were used: α = 11 . ◦ , β = − . ◦ . For this geo-metry only the primary hot spot is visible. .4. Thermal X-ray radiation The radius of a neutron star is only a few times larger than the Schwarzschild radius. Theapproach presented in the previous section does not include the gravitational bendingeffect, which is very strong in neutron stars. A strong gravitational field just above thestellar surface causes the bending of light. A photon emitted near a neutron star surfaceat an angle δ with respect to the radial direction escapes to infinity at a different angle δ (cid:48) > δ . As a consequence, even when the spot inclination angle to the line of sight is i (cid:38) ◦ we can still observe thermal radiation from this spot. For a Schwarzschild metric we cancalculate an observed flux fraction from the primary f = F /F and antipodal f = F /F spots. Here F is the maximum possible flux that is observed when the primary spot isviewed face-on. The primary and antipodal fluxes are given by (Beloborodov, 2002) f = cos ( i ) (cid:0) − r g R (cid:1) + r g R ,f = − cos ( i ) (cid:0) − r g R (cid:1) + r g R , (1.14)here r g = 2 GM/c is the Schwarzschild radius. The primary spot is visible whencos i > − r g / ( R − r g ) and the antipodal spot when cos i > − r g / ( R − r g ). Consequently,both spots are seen when − r g / ( R − r g ) < cos i < r g / ( R − r g ), and then the observedflux fraction is f min = f + f = 2 r g R . (1.15)Hence, the blackbody pulse of primary and antipodal spots must display a plateauwhenever both spots are in sight. Depending on the geometry of a pulsar we can distin-guish four classes (Beloborodov, 2002). Class I: when the antipodal spot is never seenand the primary spot is visible all the time (see the bottom right panel of Figure 1.7).For such pulsars the blackbody pulse has a perfect sinusoidal shape. Class II: when theprimary spot is seen all the time and the antipodal spot is also in the visible zone for sometime (see panels a, b and c of Figure 1.7). For these pulsars the sinusoidal pulse shapeis interrupted by the plateau. Class III: the primary spot is not visible for a fraction ofthe period and during this time only the antipodal spot is seen. The primary sinusoidalprofile of such pulsars is interrupted by the plateau, and the plateau is interrupted by aweaker sinusoidal subpulse from the antipodal spot. Class IV: both spots are seen at anytime. The observed blackbody flux of such pulsars is constant.The gravitational bending of light can significantly increase the visibility of a pulsar(i.e. the observed flux, compare Figures 1.4 and 1.8). For some specific geometry thegravitational effects can also drastically change primary to the antipodal flux ratio (com-pare Figures 1.9 and 1.5). Our calculations show that for canonical values M = 1 . (cid:12) and R = 10 km the gravitational effect is quite strong and the observed flux fraction isin the range of 0 . −
1, while the geometric approach results in the 0 . − Chapter 1. X-ray emission from Radio PulsarsTable 1.3:
Viewing geometry of pulsars. The individual columns are as follows: (1) Pulsar name,(2) Inclination angle with respect to the rotation axis α , (3) Opening angle ρ , (4) Impact parameter β , (5) Total flux correction factor (including gravitational bending of light) (cid:104) f (cid:105) , (6) Flux correctionfactor of the primary spot (cid:104) f (cid:105) , (7) Flux correction factor of the antipodal spot (cid:104) f (cid:105) , (8, 9, 10)Time-averaged cosine of the angle between the magnetic axis and the line of sight: (cid:104) cos i (cid:105) (the totalvalue), (cid:104) cos i (cid:105) (the primary spot), (cid:104) cos i (cid:105) (the antipodal spot), (10) Number of the pulsar. Thegravitational bending effect was calculated using M = 1 . (cid:12) and R = 10 km . Name α β ρ (cid:104) f (cid:105) (cid:104) f (cid:105) (cid:104) f (cid:105) (cid:104) cos i (cid:105) (cid:104) cos i (cid:105) (cid:104) cos i (cid:105) No. (deg) (deg) (deg)
B0628–28 70 . − . . .
86 0 .
52 0 .
34 0 .
52 0 .
35 0 .
17 8B0834+06 60 . . . .
86 0 .
53 0 .
32 0 .
52 0 .
36 0 .
16 14B0943+10 11 . − . . .
98 0 .
98 0 .
00 0 .
97 0 .
97 0 .
00 15B0950+08 105 . . . .
85 0 .
51 0 .
34 0 .
50 0 .
33 0 .
17 16B1133+16 52 . . . .
86 0 .
61 0 .
25 0 .
48 0 .
40 0 .
07 22B1451–68 37 . − . .
88 0 .
81 0 .
06 0 .
68 0 .
68 0 .
00 27B1929+10 36 . . . .
85 0 .
64 0 .
21 0 .
43 0 .
41 0 .
02 43
Figure 1.7:
Comparison of the observed flux fraction for geometric effect only (blue dotted line) andfor geometric effect with the inclusion of a gravitational bending of light (red solid line). Individualpanels correspond to the following pulsars: (a) PSR B1133+16, (b) PSR B1929+10, (c) PSRB0834+06 (d) PSR B0943+10. Parameters used in the calculations are presented in Table 1.3. .4. Thermal X-ray radiation Figure 1.8:
Observed flux fraction f as a function of the rotation phase for PSR B0950+08.The following parameters were used: α = 105 . ◦ , β = 21 . ◦ , M = 1 . (cid:12) , R = 10 km . Thegravitational bending of light increases the flux ratio of the antipodal to primary spots almost twotimes ( . / . . ) and also increases the antipodal to the primary flux ratio ( ∼ . ). Figure 1.9:
Observed flux fraction f as a function of the rotation phase for PSR B1929+10.The following parameters were used: α = 35 . , β = 25 . ◦ , M = 1 . (cid:12) , R = 10 km . Thegravitational bending of light increases the observed flux fraction two times ( (cid:104) f (cid:105) / (cid:104) cos i (cid:105) = 1 . )and also increases the flux ratio of the antipodal to primary spots almost seven times ( ∼ . ). Chapter 1. X-ray emission from Radio Pulsars
As we have shown in the previous sections, the blackbody fit to the X-ray observationsallows us to directly obtain the surface temperature T s . Using the distance to pulsar D and the luminosity of thermal emission L BB we can estimate the area of spot A bb . Inmost cases, A bb differs from the conventional polar cap area A dp ≈ . × P − m . Weuse parameter b = A dp /A bb to describe the difference between A dp and A bb . Entire surface radiation and warm spot component (b < In most cases the observed spot area A bb is larger than the conventional polar cap area(see Table 1.4). We can distinguish two types of pulsars in this group, with b (cid:28) b (cid:46) τ (cid:46)
10 kyr ), observation of this radiation is verydifficult due to the strong nonthermal component. A common practice is to separatelyfit the nonthermal (PL) and thermal (BB) components. However, the temperature ob-tained in such a BB fit (without the PL component) is most likely overestimated (e.g.see PSR J2021+3651 in Table 1.4). The nonthermal luminosity of an aging neutron stardecreases proportionally to its spin-down luminosity L SD , which is thought to drop withthe star age as L SD ∝ τ − m , where m (cid:39) − τ ∼
100 kyr) and some younger ( τ ∼
10 kyr) pulsars the thermal radiation from the en-tire stellar surface dominates the radiation at soft X-ray energies (e.g. PSR J0633+1746,PSR B1055-52, PSR J0821-4300, PSR B0656+14, PSR J0205+6449, PSR J2021+3651).However, the sample of pulsars is not sufficient to unambiguously identify the coolingscenario.The second type is associated with observations of the warm spot area that is largerthan the conventional polar cap area but still significantly less than the area of the star( b (cid:46) τ ∼ τ ∼
100 kyr) neutron stars. There is one exception, namely PSR J1210-5226, whichis very old ( τ = 105 Myr) and can still be classified as a pulsar with the large warm spotcomponent. Note, however, that the age of this pulsar is estimated using a characteristicvalue and if the pulsar period at birth is comparable with the current period then theage is highly overestimated (see, e.g. PSR J0821-4300). Furthermore, the fit to the X-rayspectrum was performed using only one thermal component and assuming no nonthermalradiation (PL). We believe that in many cases the size of the warm spot component andits temperature are overestimated by neglecting other sources of X-ray radiation, i.e. the .4. Thermal X-ray radiation Figure 1.10:
Cartoon of the magnetic field lines in the polar cap region. Red lines are open fieldlines and green dashed lines correspond to the dipole field. The blue arrows show the direction ofthe curvature photon emission.
The pairs move along the closed magnetic field lines and heat the surface beyondthe polar cap on the opposite side of the star. In such a scenario the heating energyis generated in IAR, and hence the luminosity of such a warm spot is limited by thepower of the outflowing particles (for more details see Section 5.1.2.3). In most casesthe large size of the emitting area and its high temperature make it unlikely that the4
Chapter 1. X-ray emission from Radio Pulsars warm spot is related to the particles accelerated in IAR and is rather connected with thenon-isothermality of the crust (e.g. PSR J1210-5226, PSR J1119-6127).
The hot spot component (b > In many cases the observed hot spot area A bb is less than the conventional polar cap area( b > b < b = A dp /A bb = B s /B d . Thus, if b (cid:29) B s (cid:29) B d .In neutron stars with positively charged polar caps ( Ω · B < T crit the ions can tightly bind tothe condensed surface and a polar gap can form (see Chapter 3 for details). Medin and Lai(2008) calculated the dependence of the critical temperature (for a vacuum gap formation)on the strength of the surface magnetic field. In Figure 1.11 we present the positions ofpulsars with derived surface temperature T s and hot spot area A bb on the B s − T s diagram,where B s is estimated as B s = bB d . The red line represents the dependence of the criticaltemperature T crit on B s . We can see that in most cases the pulsars’ positions follow the B s − T crit theoretical curve. Note that the Figure includes only pulsars with a visible hotspot component (old pulsars). For younger pulsars (with warm spot components) it isnot possible to estimate the surface magnetic field. There are a few cases which do notcoincide with the theoretical curve. We believe that they correspond to the observationsof warm spot component but with the area of radiation smaller than the conventionalpolar cap area (e.g. due to reheating of the surface beyond the polar cap, see Section1.4.4).According to our model the actual surface temperature is almost equal to the criticalvalue T s ≈ T crit , which leads to the formation of the Partially Screened Gap (PSG) abovethe polar caps of a neutron star (Gil et al., 2003). The hot spot parameters derived fromX-ray observations of isolated neutron stars are presented in Table 1.4. .4. Thermal X-ray radiation Figure 1.11:
Diagram of the surface temperature ( T = T s / (cid:0) K (cid:1) ) vs. the surface magnetic field( B = B s / (cid:0) G (cid:1) ). The red line represents the dependence of T crit on B according to Medinand Lai (2008) and the dashed lines correspond to uncertainties in the calculations. The diagramincludes all pulsars with b > with the exception of PSR J2043+2740, for which the blackbody fitwas performed using a fixed radius (estimation of the surface magnetic field is not possible). Errorbars correspond to σ . C h a p t e r . X - r a y e m i ss i o n f r o m R a d i o P u l s a r s Table 1.4:
Spectral properties of rotation-powered pulsars with detected blackbody X-ray components. The individual columns are as follows: (1) Pulsarname, (2) Spectral components required to fit the observed spectra, PL: power law, BB: blackbody, (3) Radius of the spot obtained from the blackbody fit R bb , (4) Surface temperature T s , (5) Surface magnetic field strength B s , (6) b = A dp /A bb = B s /B d , A dp - conventional polar cap area, A bb - actual polarcap area, (7) Bolometric luminosity of blackbody component L BB , (8) Bolometric efficiency ξ BB , (9) Maximum nonthermal luminosity L max NT , (10) Maximumnonthermal X-ray efficiency ξ maxNT , (11) Best estimate of pulsar age or spin down age, (12) References, (13) Number of the pulsar. Nonthermal luminosity andefficiency were calculated in the . −
10 keV band. The maximum value was calculated with the assumption that the X-ray nonthermal radiation is isotropic.Pulsars are sorted by b parameter (6). Name Spectrum R bb T s B s b log L BB log ξ BB log L X log ξ maxNT τ Ref. No. (cid:0) K (cid:1) (cid:0) G (cid:1) (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) B1451–68
BB + PL +24 . − . m 4 . +1 . − . . +114 − .
418 29 . − .
06 29 . − .
56 42 . BB, PL +41 . − . m 3 . +1 . − . . +30 . − .
126 28 . − .
62 29 . − .
64 4 .
98 Myr Zh05,Ka06 15B1929+10
BB + PL +4 . − . m 4 . +0 . − . . +0 . − .
122 30 . − .
53 30 . − .
36 3 .
10 Myr Mi08 43B1133+16
BB, PL +10 . − . m 3 . +0 . − . . +31 . − . . . − .
38 29 . − .
42 5 .
04 Myr Ka06 22B0950+08
BB + PL +26 . − . m 2 . +0 . − . . +1 . − . . . − .
82 29 . − .
76 17 . PL, BB +5 . − . m 5 . +1 . − . . +13 . − . . . − .
57 31 . − .
87 1 .
12 Myr Hu12, Hu07b 47J0633+1746
BB+BB+PL +34 . − . m 1 . +0 . − . . +2 . − . . . − .
44 30 . − .
27 342 kyr Ja05 9—— 11 . +1 − km 0 . +0 . − . . − . BB + PL +56 . − . m 2 . +0 . − . . +3 . − . . . − .
41 28 . − .
41 2 .
97 Myr Gi08 14B0355+54
BB + PL +122 . − . m 3 . +1 . − . . +1 . − . . . − .
25 30 . − .
73 564 kyr Mc07,Sl94 3J0108–1431
BB + PL +24 . − . m 1 . +0 . − . . +0 . − . . . − .
82 28 . − .
19 166 Myr Po12, Pa09 1
Continued on next page . . T h e r m a l X - r a y r a d i a t i o n Table 1.4 - continued from previous pageName Spectrum R bb T s B s b log L BB log ξ BB log L X log ξ maxNT τ Ref. No. (cid:0) K (cid:1) (cid:0) G (cid:1) (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) B0628–28
BB + PL +70 . − . m 3 . +1 . − . . +4 . − . .
14 30 . − .
94 30 . − .
94 2 .
77 Myr Te05, Be05 8J2043+2740
BB + PL +153 . − . m 1 . +0 . − . . +0 . − . .
70 30 . − .
98 31 . − .
34 1 .
20 Myr Be04 46B1719–37 BB +390 . − . m 3 . +0 . − . . +0 . − . .
57 31 . − .
32 – – 345 kyr Oo04 32J1846–0258
BB + PL +153 . − . m 13 . +3 . − . – 0 .
686 34 . − .
85 35 . − .
78 0 .
73 kyr Ng08, He03 39B1055–52
BB+BB+PL +60 . − . m 1 . +0 . − . – 0 .
503 30 . − .
59 30 . − .
57 535 kyr De05 1812 . +2 − km 0 . +0 . − . . − . BB +38 . − . m 2 . +0 . − . – 0 .
330 31 . − .
73 – – 30 . BB + PL +920 . − . m 2 . +0 . − . – 0 .
280 31 . − .
56 31 . − .
68 51 . BB + BB . +0 . − . km 6 . +0 . − . – 0 .
125 33 . − .
91 – – 3 .
70 kyr Go10 11—— 6 . +0 . − . km 3 . +0 . − . . − . BB + PL . +1 . − . km 1 . +0 . − . – 0 .
110 31 . − .
62 33 . − .
35 107 kyr Li05 44B0833–45
BB + PL . +0 . − . km 1 . +0 . − . – 0 .
091 32 . − .
72 32 . − .
22 11 . BB + PL . +0 . − . km 2 . +0 . − . – 0 .
034 32 . − .
99 32 . − .
35 7 .
31 kyr Za07 25J1210–5226 BB .
23 km 3 . .
033 33 .
04 1 .
51 – – 102 Myr Pa02 23B1823–13
BB + PL . +0 . − . km 1 . +0 . − . – 0 .
032 32 . − .
27 31 . − .
67 21 . BB, PL +100 . − . m 1 . +0 . − . – 0 .
028 31 . − .
63 32 . − .
71 88 . BB + PL . +0 . − . km 2 . +0 . − . – 0 .
027 32 . − .
76 32 . − .
37 17 . C h a p t e r . X - r a y e m i ss i o n f r o m R a d i o P u l s a r s Table 1.4 - continued from previous pageName Spectrum R bb T s B s b log L BB log ξ BB log L X log ξ maxNT τ Ref. No. (cid:0) K (cid:1) (cid:0) G (cid:1) (cid:0) erg s − (cid:1) (cid:0) erg s − (cid:1) B2334+61
BB + PL . +0 . − . km 2 . +0 . − . – 0 .
026 32 . − .
73 31 . − .
24 40 . BB+BB+PL . +0 . − . km 1 . +0 . − . – 0 .
017 31 . − .
13 30 . − .
33 111 kyr De05 10—— 20 . +3 − km 0 . +0 . − . . − . BB + PL . +1 . − . km 3 . +0 . − . – 0 .
008 33 . − .
00 32 . − .
42 1 .
61 kyr Go07, Ng12 20J0205+6449
BB + PL . . .
005 33 . − .
83 33 . − .
33 5 .
37 kyr Sl04 2J2021+3651
PL, BB . +4 . − . km 2 . +0 . − . – 0 .
004 33 . − .
75 34 . − .
17 17 . Chapter 2Model of a non-dipolar surfacemagnetic field
Generally, the properties of pulsar radio emission support the assumption that the mag-netic field of pulsars is purely dipolar at least in the radio emission region (Radhakrish-nan and Cooke, 1969). However, radio emission is generated at altitudes R em of morethan several stellar radii (e.g. Kijak and Gil (1997), Kijak and Gil (1998), Krzeszowskiet al. (2009) and references therein). Thus, radio observations do not provide informationabout the structure of the magnetic field at the surface of the neutron star. On the otherhand, strong non-dipolar surface magnetic fields have long been thought to be a necessarycondition for pulsar activities, e.g. the vacuum gap model proposed by Ruderman andSutherland (1975) implicitly assumes that the radius of curvature of field lines above thepolar cap should be about 10 cm in order to sustain pair production. This curvature isapproximately 100 times higher than that expected from a global dipolar magnetic field.Furthermore, to explain radiation from the Crab Nebula, the Crab pulsar should providequite a dense stellar wind, as such a high particle multiplicity is not possible in a purelydipolar magnetic field.There are several theoretical studies concerning the formation and evolution of the non-dipolar magnetic fields of neutron stars (e.g. Blandford et al. 1983; Krolik 1991; Ruderman1991; Arons 1993; Chen and Ruderman 1993; Geppert and Urpin 1994; Mitra et al. 1999;Page et al. 2006). According to Woltjer (1964), the magnetic field in neutron stars resultsfrom the fossil field of the progenitor stars which is amplified during the collapse andremains anchored in the superfluid core of the neutron star. Several authors also notedthat during the collapse (or shortly after) there is possible magnetic field generation in theexternal crust, for instance, by a mechanism like thermomagnetic instabilities (Blandfordet al., 1983). Urpin et al. (1986) also showed that it is possible to form small-scale magneticfield anomalies in the neutron star crust with a typical size of the order of 100 meters.0 Chapter 2. Model of a non-dipolar surface magnetic field
The soft X-ray observations of pulsars presented in Chapter 1 show non-uniform sur-face temperatures which can be attributed to small-scale magnetic anomalies in the crust.Further observational arguments in favour of the non-dipolar nature of the surface mag-netic field can be found in many articles (e.g. Bulik et al. 1992; Thompson and Duncan1995; Bulik et al. 1995; Page and Sarmiento 1996; Thompson and Duncan 1996; Beckerand Truemper 1997; Cheng et al. 1998; Rudak and Dyks 1999; Cheng and Zhang 1999;Murakami et al. 1999; Tauris and Konar 2001; Maciesiak et al. 2012).
In order to model a surface magnetic field we used the scenario proposed by Gil et al.(2002). In this scenario the magnetic field at the neutron star’s surface is non-dipolarin nature, which is due to superposition of the fossil field in the core and crustal fieldstructures. To calculate the actual surface magnetic field described by superposition ofthe star-centred global dipole d and the crust-anchored dipole moment m , let us considerthe general situation presented in Figure 2.1 Figure 2.1:
Superposition of the star-centred global magnetic dipole d and crust-anchored localdipole anomaly m located at r s = ( r s ∼ R, θ = θ r ) and inclined to the z -axis by an angle θ m . Theactual surface magnetic field at radius vector r = ( r, θ ) is B s = B d + B m , where B d = 2 d/r , B m = 2 m/ | rr s | , r is the radius and θ - is the polar angle. R is the radius of the neutron star andL is the external crust thickness. Gil et al. (2002) .2. Modelling of the surface magnetic field B s = B d + B m + ... (2.1)Using the star-centred spherical coordinates with the z -axis directed along the globalmagnetic dipole moment we obtain: B d = (cid:18) d cos θr , d sin θr , (cid:19) , (2.2) B m = 3( r − r s )( m · ( r − r s )) − m | r − r s | | r − r s | . (2.3)Here r s = ( r s , θ r , φ r ), m = ( m, θ m , φ m ) and the spherical components of B m are explicitlygiven in Equation 2.7.The global magnetic moment can be written as d = 12 B p R , (2.4)where B p = 6 . × (cid:16) P ˙ P (cid:17) / G is the dipole component at the pole derived from pulsarspin-down energy loss.The crust-anchored local dipole moment is m = 12 B m ∆ R , (2.5)where ∆ R ∼ . R < L and L ∼ cm is the characteristic crust dimension (for R = 10 cm). For these values a local anomaly can significantly influence the surfacemagnetic field ( B m > B d ) if m/d > − .The system of differential equations for a field line of the vector field B = ( B r , B θ , B φ )in spherical coordinates can be written as d θ d r = B θ rB r d φ d r = B φ r sin( θ ) B r . (2.6)The solution of these equations, with the initial conditions θ = θ ( r = R ) and φ = φ (r=R)determining a given field line at the stellar surface, describes the parametric equation ofthe magnetic field lines. The spherical components of B m can be written in the followingform2 Chapter 2. Model of a non-dipolar surface magnetic field B mr = − D . (3 T r sr − T r + Dm r ) ,B mθ = − D . (3 T r sθ + Dm θ ) ,B mφ = − D . (cid:0) T r sφ + Dm φ (cid:1) . (2.7)Here D = r s + r − r s r (sin θ r sin θ cos ( φ − φ r ) + cos θ r cos θ ) , (2.8)and T = m r r − ( m r r sr + m θ r sθ ) . (2.9)According to the geometry presented in Figure 2.1, the components of the radius vectorof the origin of the crust-anchored local dipole anomaly can be written as r sr = r s (sin θ r sin θ cos ( φ − φ r ) + cos θ r cos θ ) ,r sθ = r s (sin θ r cos θ cos ( φ − φ r ) + cos θ r sin θ ) ,r sφ − r s sin θ r sin ( φ − φ r ) . (2.10)The components of the local dipole anomaly are m r = m (sin θ m sin θ cos ( φ − φ m ) + cos θ m cos θ ) ,m θ = m (sin θ m cos θ cos ( φ − φ m ) + cos θ m sin θ ) ,m φ = − m sin θ m sin ( φ − φ m ) . (2.11)Finally, we obtain the system of differential equations from Equation 2.6 by substitu-tions B r = B dr + B mr , B θ = B dθ + B mθ and B φ = B dφ + B mφ (Equations 2.2 and 2.7)d θ d r = B dθ + B mθ r ( B dr + B mr ) ≡ Θ , (2.12)d φ d r = B mφ r ( B dr + B − r m ) sin θ ≡ Φ . (2.13) As Curvature Radiation (CR) may play a decisive role in radiation processes, it is im-portant to calculate the curvature (or curvature radius) for each field line. The curvature ρ c = 1 / (cid:60) of field lines (where (cid:60) is the radius of curvature) is calculated as (Gil et al.,2002) ρ c = (cid:18) d s d r (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) d r d r d s d r − d r d r d s d r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (2.14) .3. Curvature of magnetic field lines s d r = (cid:113)(cid:2) r Θ + r Φ sin ( θ ) (cid:3) . (2.15)Thus, the curvature can be written in the form ρ c = ( S ) − (cid:0) J + J + J (cid:1) / , (2.16)where J = X S − X S ,J = Y S − Y S ,J = Z S − Z S ,X = sin θ cos φ + r Θ cos θ cos φ − r Φ sin θ sin φ,Y = sin θ sin φ + r Θ cos θ sin φ − r Φ sin θ cos φ,Z = cos θ − r Θ sin θ,X = (2Θ + r Θ ) cos θ cos φ − (2Φ + r Φ ) sin θ sin φ − r (cid:0) Θ + Φ (cid:1) sin θ cos φ + 2 r Θ Φ cos θ sin φ,Y = (2Θ + r Θ ) cos θ sin φ + (2Φ + r Φ ) sin θ cos φ − r (cid:0) Θ + Φ (cid:1) sin θ sin φ + 2 r Θ Φ cos θ cos φ,Z = − Θ sin θ − Θ sin θ − r Θ sin θ − r Θ cos θ,S = (cid:113) r Θ + r Φ sin θ,S = S − (cid:0) t Θ + r Θ Θ + r Φ sin θ + r Φ Φ sin θ + r Θ Φ sin θ cos θ (cid:1) , Θ ≡ dΘ d r , Φ ≡ dΦ d r . (2.17) Let us note that when evaluating Equation 2.6 it was assumed that sin θ (cid:54) = 0. Thus Φ and Φ are undefined for θ = 0. Figure 2.2 presents the first and second derivative (Φ ,Φ ) of the φ -coordinate of a magnetic field line with respect to the r -coordinate for amagnetic field structure with B φ (cid:54) = 0.The singularity in Equation 2.6 may result in an overestimation of curvature of fieldlines that cross the θ = 0 plane for a complex structure of the surface magnetic field. Tosolve this problem, and in addition to the analytical approach, the numerical calculationof the curvature of magnetic field lines was implemented.Let us consider three consecutive points of the given magnetic field line A , B , C (seeFigure 2.3).4 Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.2:
Plot of the first and second derivative of the φ -coordinate of the magnetic field linewith respect to the r -coordinate vs. the distance from the stellar surface. Values were calculatedusing the approach described in Section 2.3. Panels (a) and (b) show the first and second derivativeof the φ -coordinate while panel (c) shows the θ -coordinate of the magnetic field line. line of nodes magnetic field line Figure 2.3:
For any given three points ( A , B , C ) we can always find a common plane. We use thefollowing transformations to achieve this: (I) shift the origin of the system to point A (prime), (II)rotate the shifted system by an angle ς y around the y (cid:48) –axis and by an angle ς x around the x (cid:48)(cid:48) -axis(double prime). After these transformations the z (cid:48)(cid:48) -axis will be aligned with normal vector ˆ N and allpoints will lie in the x (cid:48)(cid:48) y (cid:48)(cid:48) -plane of such a system of coordinates. .3. Curvature of magnetic field lines B we can use the followingprocedure: • simplify the 3-D problem to 2-D by finding a common plane for all three points – move the origin of the coordinate system to point A – rotate the coordinate system to align the z-axis with the normal vector to thecommon plane of all three points ( A , B , C ) • calculate the radius of the circle passing through all three points (in a 2-D coordinatesystem) To simplify the calculations we shift the origin of the coordinate system so that point A will be the origin of the new system: A (cid:48) = (0 , ,
0) ; B (cid:48) = ( B − A , B − A , B − A ) ; (2.18) C (cid:48) = ( C − A , C − A , C − A ) . The unit normal vector to the plane enclosing all three points ( A (cid:48) , B (cid:48) , C (cid:48) ) can becalculated as ˆ N = b × c | b × c | = ( N , N , N ) , (2.19)where b = ( B (cid:48) , B (cid:48) , B (cid:48) ) and c = ( C (cid:48) , C (cid:48) , C (cid:48) ).The next step is to rotate the shifted coordinate system to align the z (cid:48) -axis with normalvector ˆ N . In the new system all three points will lie in the x (cid:48)(cid:48) y (cid:48)(cid:48) -plane. In our calculationswe rotate the shifted system by an angle ς y around the y (cid:48) -axis, R y ( ς y ), and a rotation byan angle ς x around the x (cid:48)(cid:48) -axis, R x ( ς x ). The final rotation matrix can be written as R yx = R y ( ς y ) R x ( ς x ) = cos ς y sin ς x sin ς y sin ς y cos ς x ς x − sin ς x − sin ς y cos ς y sin ς x cos ς y cos ς x (2.20)The Euler angles for these rotations can be calculated as ς x = atan2 ( N , N ) . where atan2 ( y, x ) equals: (1) arctan ( y/x ) if x >
0; (2) arctan ( y/x ) + π if y ≥ x <
0; (3)arctan ( y/x ) − π if y < x <
0; (4) π/ y > x = 0; (5) − π/ y < x = 0; (6) isundefined if y = 0 and x = 0. This function is available in many programming languages. Chapter 2. Model of a non-dipolar surface magnetic field ς y = arctan (cid:16) − N N cos ς x (cid:17) if N (cid:54) = 0arctan (cid:16) − N N sin ς x (cid:17) if N (cid:54) = 0 π if N = 0 and N = 0 (2.21)Finally, we can write the components of all three points in our new (shifted anddouble-rotated) system of coordinates as follows A (cid:48)(cid:48) = R yx A (cid:48) = (0 , ,
0) ; B (cid:48)(cid:48) = R yx B (cid:48) = ( B (cid:48)(cid:48) , B (cid:48)(cid:48) ,
0) ; (2.22) C (cid:48)(cid:48) = R yx C (cid:48) = ( C (cid:48)(cid:48) , C (cid:48)(cid:48) , . Circle passing through 3 points (Bourne, 2012)
Finding the radius of the circle passing through three consecutive points of a given mag-netic field line ( A = (0 , B = ( B , B ), C = ( C , C )) is an exact method for findingthe radius of curvature (cid:60) and hence the curvature ρ = 1 / (cid:60) of this line. Note thatfor simplicity’s sake we hereafter describe points without double prime notation butthey refer to coordinates in the shifted and double-rotated system of coordinates (e.g. B = ( B , B ) = ( B (cid:48)(cid:48) , B (cid:48)(cid:48) )).Slope m of the line joining A to B and slope m of the line joining B to C (see Figure2.4) are given by m = ∆ y ∆ x = B B ,m = ∆ y ∆ x = C − B C − B . (2.23)In general, the centre of the circle passing through our points is given by x c = m m ( A − C ) + m ( A − B ) − m ( B − C )2 ( m − m ) ,y c = 1 m (cid:18) x c − A + B (cid:19) + A + B . (2.24) .3. Curvature of magnetic field lines Figure 2.4:
The radius of curvature (cid:60) of the magnetic field line at a given point B can becalculated as the radius of the circle passing through this and the neighbouring two points ( A , C ).The system of coordinates was moved and double-rotated so that A is in its origin and all points liein the x (cid:48)(cid:48) y (cid:48)(cid:48) -plane. The slopes of the lines joining A to B and B to C are described by Equation2.23. Since point A is in the centre of the coordinate system we can simplify these formulasas follows x c = 2 B B − B B C + B C − B C − ( B − B ) C B C − B C ) ,y c = B − ( B − x c ) B B . (2.25)Finally, we can calculate the radius of curvature simply by finding the distance betweenthe centre of the circle and any of the points on the circle (we have chosen point A ) (cid:60) = 1 ρ = (cid:112) x c + y c . (2.26)In this thesis we consider complex structures of the surface magnetic field, thus thenumerical method presented above was used in all the calculations of curvature. Theanalytical approach may result in an overestimation of curvature for points with θ ≈ Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.5:
Curvature of the magnetic field lines vs. the height above the stellar surface cal-culated using the analytical approach described in Section [2.3] (green lines) and the numericalapproach presented above (red lines). Panel (a) corresponds to the magnetic field line which hasno φ component, while panel (b) corresponds to a more general scenario i.e. the nonzero φ com-ponent of the magnetic field line. As can be seen, the analytical approach is not valid for everycase. This is caused by the undefined value of the φ derivative for θ = 0 (see Equation [2.6]). Here ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) and (cid:60) = (cid:60) / (cid:0) cm (cid:1) . In this section we model the surface non-dipolar magnetic field structure for some pulsars.Note that we can estimate the size of the polar cap and the strength of the surface magneticfield only for pulsars with an observed hot spot (see Section 1.4.4). Here we present onlypulsars listed in Table 3.1.We use spherical coordinates ( r, θ, φ ) to describe the location and orientation of crust-anchored local anomalies. The parameters of anomalies are as follows: r a = ( r a , θ a , φ a )is a radius vector which points to the location of the anomaly and m a = ( m a , θ a , φ a ) isits dipole moment. The value of m a is measured in units of the global dipole moment d ,i.e. the moment which corresponds to the pulsar’s global magnetic field. In the Figuresshowing a possible non-dipolar structure (e.g. Figure 2.6, 2.9, 2.12) the dashed linescorrespond to the dipolar configuration of the magnetic field lines, while the solid linescorrespond to the actual magnetic field lines (taking into account the crust-anchoredanomalies). Green and red lines represent the open magnetic field lines for dipolar andnon-dipolar structures, respectively. .4. Simulation results Pulsar B0628-28, a bright radio pulsar, was discovered by Large et al. (1969) during apulsar search at 408 MHz. The pulsar period P ≈ .
24 s and its first derivative ˙ P − ≈ . B d = 6 × G and a characteristic age τ c ≈ . D = 1 .
44 kpc (evaluated using the Galactic free electron density model of Cordes andLazio, 2002) makes it impossible to use the parallax method to determine the distancewith better accuracy.PSR B0628-28 is one of the longest period pulsars among those detected in X-rays.The pulsar was first detected in the X-ray band by
ROSAT and then later observedwith both the
Chandra and
XMM-Newton . Observations with the
Chandra revealedno pulsations, while the
XMM-Newton observations revealed pulsations with a periodconsistent with the period of radio emission (Tepedelenlıoˇglu and ¨Ogelman, 2005). Theinconsistency of the observations is a reflection of the fact that the pulsar is detectable justat the threshold of sensitivity of both the observatories. The two-component spectral fit(BB+PL) shows that both the nonthermal and thermal components have a comparableluminosity (at least if we assume that the nonthermal radiation is isotropic, see Table1.2). PSR B0628-28 is characterised by one of the largest X-ray efficiencies among theobserved pulsars ξ BB ≈ ξ max NT ≈ − . Figure 2.6:
Possible non-dipolar structure of the magnetic field lines of PSR B0628-28.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) . × − d, ◦ , ◦ (cid:1) , m = (cid:0) . × − d, ◦ , ◦ (cid:1) respectively (blue arrows). Theinfluence of the anomalies is negligible at distances D (cid:38) R , where B m /B d ≈ m/d = 4 . × − (top panel). For more details on the polar cap region see Figure 2.7. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.7:
Zoom of the polar cap region of PSR B0628-28. See Figure 2.6 for a description.
Figure 2.8:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B0628-28. The distance is in units of the stellar radius z = z/R and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results Geminga was discovered in 1972 as a γ -ray source by Fichtel et al. (1975). The visualmagnitude of the pulsar was estimated by Bignami et al. (1987) to be of the order of ∼ . mag . The pulse modulation was discovered in X-rays (Halpern and Holt, 1992), in γ -rays, and at optical wavelengths (Shearer et al., 1998). Geminga has been determinedto be a relatively old ( τ = 342 kyr) radio-quiet pulsar with a period P = 237 ms. Thedistance to the pulsar D = 0 .
16 kpc, evaluated using the parallax method, makes it theclosest pulsar with available X-ray data.The pulsar exhibits one of the weakest radio luminosities known and a cutoff at fre-quencies higher than about 100 MHz. The model presented by Gil et al. (1998) explainsthis weak radio emission with absorption by the magnetised relativistic plasma inside thelight cylinder. As the exact model of radio emission is still unknown (see Section 5.2.3),it is difficult to verify if this weak radio emission is a result of absorption or the absenceof coherent radio emission.The three-component fit to the X-ray spectrum (PL+BB+BB, see Table 1.4) revealsthe hot spot component with a size that is considerably smaller than the conventionalpolar cap size ( b ≈ T s = 0 . Figure 2.9:
Possible non-dipolar structure of the magnetic field lines of PSR J0633+1746.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) . × − d, ◦ , ◦ (cid:1) , m = (cid:0) . × − d, ◦ , ◦ (cid:1) respectively (blue arrows). Theinfluence of the anomalies is negligible at distances D (cid:38) . R , where B m /B d ≈ m/d = 5 . × − (top panel). For more details on the polar cap region see Figure 2.10. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.10:
Zoom of the polar cap region of PSR J0633+1746. See Figure 2.9 for a description.
Figure 2.11:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR J0633+1746. The distance is in units of the stellar radius z = z/R and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results The bright radio emission of PSR B0834+06 shows frequent nulls (nearly 9% of thepulses is absent, see Rankin and Wright, 2008) . With a relatively long rotational period P = 1 .
27 s and ˙ P − ≈ . B d = 3 × G, are close to the average. The characteristic age τ c = 2 .
97 Myr implies thatthe pulsar should be categorised as an old pulsar. The distance to the pulsar, estimatedas D = 0 .
64 kc, was derived from its dispersion measure using the Galactic free-electrondensity model of Cordes and Lazio, 2002. Weltevrede et al. (2006) suggest a drift ofsubpulses, but the estimated value of a subpulse separation is larger than the pulse width.Despite the fact that the geometry based on the carousel model could be fitted to theobservations, there is no clear evidence for a drift of emission between the components ofthe pulsar (Rankin and Wright, 2007).The pulsar was detected in X-ray by Gil et al. (2008) with a total of 70 counts fromover 50 ks exposure time. Because of the low statistical quality of the X-ray data, it wasnot possible to constrain the absorbing column density N H . The two-component spectralfit (BB + PL), as presented in this thesis, was performed using the assumption that boththe thermal and nonthermal fluxes are of the same order. Figure 2.12:
Possible non-dipolar structure of the magnetic field lines of PSR B0834+06.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) × − d, ◦ , ◦ (cid:1) , m = (cid:0) × − d, ◦ , ◦ (cid:1) respectively (blue arrows). Theinfluence of the anomalies is negligible at distances D (cid:38) . R , where B m /B d ≈ m/d = 3 × − (top panel). For more details on the polar cap region see Figure 2.13. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.13:
Zoom of the polar cap region of PSR B0834+06. See Figure 2.12 for a description.
Figure 2.14:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B0834+06. The distance is in units of stellar radius ( z = z/R ) and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results Pulsar B0943+10 is a relatively old pulsar with a characteristic age of τ c = 4 .
98 Myr. Thepulsar period P = 1 . P − ≈ . B d = 4 . × G. Using the Galactic free-electron density model ofCordes and Lazio, 2002, we can estimate the distance to the pulsar D = 0 .
63 kpc.PSR B0943+10 is a well-known example of a pulsar exhibiting both the mode changingand subpulse drifting phenomenon. Strong, regular subpulse drifting is observed only inradio-bright mode, and only hints of the modulation feature have been found in the radio-quiescent mode. Very recent results presented by Hermsen et al. (2013) show synchronousswitching in the radio and X-ray emission properties. When the pulsar is in a radio-brightmode, the X-rays exhibit only an unpulsed component. On the other hand, when thepulsar is in a radio-quiet mode, the flux of X-rays is doubled and a pulsed component isalso visible.
Figure 2.15:
Possible non-dipolar structure of the magnetic field lines of PSR B0943+10.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) . × − d, ◦ , ◦ (cid:1) , m = (cid:0) × − d, ◦ , ◦ (cid:1) respectively (blue arrows). Theinfluence of the anomalies is negligible at distances D (cid:38) . R , where B m /B d ≈ m/d = 2 × − (top panel). For more details on the polar cap region see Figure 2.16. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.16:
Zoom of the polar cap region of PSR B0943+10. See Figure 2.15 for a description.
Figure 2.17:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B0943+10. The distance is in units of stellar radius ( z = z/R ) and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results Pulsar B0950+08 is one of the strongest pulsed radio sources in the metre wavelengthrange. The pulsar radiation also exhibits an interpulse located at 152 ◦ from the mainpulse (Smirnova and Shabanova, 1992). Based on the period P = 1 . P − ≈ .
5, we can estimate the pulsar’s characteristic age τ c = 17 . B d = 0 . × . Forthis pulsar the distance D = 0 .
26 kpc was estimated using the parallax method.PSR B0950+08 was detected in the ultraviolet-optical range (2400 − Hubble Space Telescope . Further observations suggest that theoptical radiation of the pulsar is most likely of a nonthermal origin (Mignani et al., 2002;Zharikov et al., 2004).X-rays from PSR B0950+08 were first detected with the
ROSAT by Manning andWillmore (1994) ( ∼
55 source counts). Further X-ray observations revealed pulsationsof the X-ray flux at the radio period of the pulsar (Zavlin and Pavlov, 2004). The X-ray spectrum manifests two components (thermal and nonthermal). Which of the twocomponents dominates the spectrum depends on the radiation pattern of the nonthermalcomponent (isotropic or anisotropic). Due to the poor quality of the X-ray data, theconnection of the optical and X-ray spectra remained unclear.
Figure 2.18:
Possible non-dipolar structure of the magnetic field lines of PSR B0950+08. Thestructure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) . × − d, ◦ , ◦ (cid:1) , m = (cid:0) . × − d, ◦ , ◦ (cid:1) respectively (blue arrows). The influence of the anomalies is negligibleat distances D (cid:38) . R , where B m /B d ≈ m/d = 5 . × − (top panel). For more details on thepolar cap region see Figure 2.19. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.19:
Zoom of the polar cap region of PSR B0950+08. See Figure 2.18 for a description.
Figure 2.20:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B0950+08. The distance is in units of stellar radius ( z = z/R ) and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results Pulsar B1133+16 is one of the brightest pulsating radio sources in the Northern hemi-sphere (Maron et al., 2000). The relatively long pulse period P = 1 .
19 s and its first de-rivative ˙ P − ≈ . B d = 4 . × G, τ c = 5 .
04 Myr. The pulsar profile exhibits a classic double peak along with the usualS-shaped polarisation-angle traverse. The pulsar also shows the phenomenon of driftingsubpulses but only for some finite time-spans, outside of which the behaviour of individualpulses is chaotic (Honnappa et al., 2012).PSR B1133+16 is located at a high galactic latitude, thus implying a low interstellarextinction (Schlegel et al., 1998). Zharikov et al. (2008) suggested a possible opticalcounterpart with brightness B = 28 mag .X-ray observations performed by Kargaltsev et al. (2005) with the Chandra result ina small number of counts (33 counts from over 17 ks), thus the X-ray spectrum can bedescribed by various models. The photon statistics are so low that they allowed onlyseparate fits for the thermal (BB) and nonthermal (PL) components.
Figure 2.21:
Possible non-dipolar structure of the magnetic field lines of PSR B1133+16.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) × − d, ◦ , ◦ (cid:1) , m = (cid:0) × − d, ◦ , ◦ (cid:1) respectively (blue arrows). Theinfluence of the anomalies is negligible at distances D (cid:38) . R , where B m /B d ≈ m/d = 5 × − (top panel). For more details on the polar cap region see Figure 2.22. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.22:
Zoom of the polar cap region of PSR B1133+16. See Figure 2.21 for a description.
Figure 2.23:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B1133+16. The distance is in units of stellar radius ( z = z/R ) and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . .4. Simulation results With a pulse period of P = 0 .
23 s and a period derivative of ˙ P − ≈ .
2, the pulsar’scharacteristic age is determined to be τ c = 3 . B d = 1 . × . Thedistance to the pulsar D = 0 .
36 kpc was estimated using the parallax.Pavlov et al. (1996) identified a candidate optical counterpart of PSR B1929+10 withbrightness U ∼ . mag , which was later confirmed by proper motion measurements per-formed by Mignani et al. (2002).The X-ray pulse profile of PSR B1929+10 consists of a single, broad peak which isin contrast with the sharp radio one of Misanovic et al. (2008). The two-componentspectral fit (BB+PL) suggests that both the thermal and nonthermal luminosities are ofthe same order. The derived surface temperature T s = 4 . B s = 1 . × G do not coincide with the theoretical curve T s − B s of the criticaltemperature calculated by Medin and Lai (2008). We believe that this inconsistency canbe removed by adding an additional blackbody component (the whole surface or the warmspot radiation). Figure 2.24:
Possible non-dipolar structure of the magnetic field lines of PSR B1929+10.The structure was obtained using two crust anchored anomalies located at: r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , r = (0 . R, ◦ , ◦ ) , with the dipole moments m = (cid:0) × − d, ◦ , ◦ (cid:1) , m = (cid:0) × − d, ◦ , ◦ (cid:1) , m = (cid:0) × − d, ◦ , ◦ (cid:1) respect-ively (blue arrows). The influence of the anomalies is negligible at distances D (cid:38) . R , where B m /B d ≈ m/d = 3 × − (top panel). For more details on the polar cap region see Figure 2.25. Chapter 2. Model of a non-dipolar surface magnetic fieldFigure 2.25:
Zoom of the polar cap region of PSR B1929+10. See Figure 2.24 for a description.
Figure 2.26:
Dependence of a curvature of the open magnetic field lines on the distance from thestellar surface for PSR B1929+10. The distance is in units of stellar radius ( z = z/R ) and thecurvature of the magnetic field lines is ρ − = 1 / (cid:60) = ρ/ (cid:0) − cm − (cid:1) . Chapter 3Partially Screened Gap
The charge-depleted inner acceleration region above the polar cap can be formed if a localcharge density differs from the co-rotational charge density (Goldreich and Julian, 1969).We assume that the crust of the neutron stars mainly consists of iron (
Fe) formed at theneutron star’s birth (e.g. Lai, 2001). Depending on the mutual orientation of Ω and µ , thestellar surface at the polar caps is either positively ( Ω · µ <
0) or negatively ( Ω · µ > Fe ions or electrons. In this thesis we consider the case ofpositively charged polar caps ( Ω · µ < T s is below the critical value T crit . Since the number density of the iron ions in the neutron star crust is many ordersof magnitude larger than the co-rotational charge density (the so-called Goldreich-Juliandensity) ρ GJ = Ω · B / (2 πc ), then a thermionic emission from the polar cap surface is notsimply described by the usual condition (cid:15) i ≈ kT s , where (cid:15) i is the cohesive energy and/orwork function, T s is the actual surface temperature, and k is the Boltzman constant. Theoutflow of iron ions can be described in the form (Gil et al. 2003 and references therein) ρ i ρ GJ ≈ (cid:18) C i − (cid:15) i kT s (cid:19) , (3.1)where ρ i ≤ ρ GJ is the charge density of the outflowing ions. As soon as the surfacetemperature T s reaches the critical value T crit = (cid:15) i C i k , (3.2)the ion outflow reaches the maximum value ρ i = ρ GJ . The numerical coefficient C i = 30 ± T crit is also estimated within an accuracy of about 10%. The cohesive energy is mainly definedby the strength of the magnetic field and was calculated by Medin and Lai (2006, 2007).4 Chapter 3. Partially Screened Gap
As it follows from the X-ray observations (see Section 1.4), the temperature of the hotspot (which is associated with the actual polar cap) is more than 10 K. As we mentionedabove, in order to sustain such a high temperature bombardment by the backstreamingparticles is required. But particle acceleration (and therefore the surface heating) ispossible only if T s < T crit . Gil et al. (2003) introduced the model of the Partially ScreenedGap to describe the polar gap sparking discharge specifically under such circumstances.The PSG model assumes the existence of heavy iron ions ( Fe) with a density near butstill below the co-rotational charge density ( ρ GJ ), thus the actual charge density causespartial screening of the potential drop just above the polar cap. The degree of screeningcan be described by screening factor η = 1 − ρ i /ρ GJ . (3.3)where ρ i is the charge density of the heavy ions in the gap. The thermal ejection of ionsfrom the surface causes partial screening of the acceleration potential drop∆ V = η ∆ V max , (3.4)where ∆ V max is the potential drop in a vacuum gap. We can express the dependence ofthe critical temperature on the pulsar parameters by fitting to the numerical calculationsof Medin and Lai (2007) T crit = 1 . × (cid:40)(cid:20)(cid:16) P ˙ P − (cid:17) . b (cid:21) . + 17 . (cid:41) , (3.5)or T crit = 1 . × ( B . + 0 . B = B s / (10 G) , B s = bB d is a surface magneticfield (applicable only if hot spot components are observed, i.e. b > V should be thermostatically regulated and a quasi-equilibrium state should be established in which heating due to the electron/positronbombardment is balanced by cooling due to thermal radiation (see Gil et al. 2003 formore details). The necessary condition for this quasi-equilibrium state is σT = ηe ∆ V cn GJ , (3.6)where σ is the Stefan-Boltzmann constant, e - the electron charge, and n GJ = ρ GJ /e = 1 . × b ˙ P . − P − . is the co-rotational number density. The Goldreich-Julian co-rotational number density can be expressed in terms of B as n GJ = 6 . × B P − . (3.7) .1. The Model ηn GJ .By using Equations 3.6, 3.5 and 3.7 we can express the acceleration potential dropthat satisfies the heating condition (Equation 3.6) as follows∆ V = 7 . × ( B . + 0 . PηB . (3.8)The above equation may suggest that the acceleration potential drop is inversely pro-portional to the screening factor. In fact, it is just the opposite (see Equations 3.4 and3.26).Knowing that ∆ V = γ max mc /e , where m is the mass of a particle (electron orpositron), we can calculate the maximum Lorentz factor of the primary particles in PSGas γ max = 450 ( B . + 0 . PηB . (3.9) As the actual polar cap is much smaller than the conventional polar cap (see section1.4.4), we cannot use the approximation proposed by Ruderman and Sutherland (1975)that the gap height is of the same order as the gap width ( h ≈ h ⊥ ). On the contrary, thesmall polar cap size and subpulse phenomenon suggest that in the PSG model the sparkhalf-width is considerably smaller than the gap height ( h ⊥ < h ). For such a regime weneed to recalculate a formula for the acceleration potential drop ∆ V .Let us consider a reference frame co-rotating with a star and with the z-axis alignedwith the star’s angular velocity Ω (see Figure 3.1). Figure 3.1:
Co-rotating frame of reference with the z-axis aligned with the angular velocity Ω .The magnetic dipole moment µ is constant in this frame of reference, thus ∂ B /∂t = 0 . Chapter 3. Partially Screened Gap
Let us underline that we will neglect the effects of non-inertiality of the co-rotatingsystem. Thus, we assume that in any given moment we have a system moving with aconstant velocity.In this co-rotating frame of reference we can write the spherical components of anangular velocity as follows Ω = (Ω cos θ, − Ω sin θ, . (3.10)Gauss’s law in the co-rotating frame (after Lorentz transformations) takes the form ∇ · E = 4 πρ ( r ) − π (cid:18) Ω · B πc (cid:19) . (3.11)While Faraday’s law of induction can be written as ∇ × E = 0 . (3.12)Note that if we consider a drift of plasma in the Inner Acceleration Region (IAR), weshould expect temporal variations of the magnetic field ( ∇ × E = − ∂ B / ( c∂t )) (Schiff,1939), but as was shown by van Leeuwen and Timokhin (2012), even if we considerfluctuations of the electric current of the order of the Goldreich-Julian current ρ GJ c , theresulting variation of the magnetic field is so small that ∇ × E = 0 with a high accuracy,and circulation of the non-co-rotational electric field along a closed path is zero.Equation 3.11 in the spherical system of coordinates has the following form2 r E r + ∂E r ∂r + cos θr sin θ E θ + 1 r ∂E θ ∂θ + 1 r sin θ ∂E φ ∂φ = 4 πρ ( r, θ, φ ) − π (cid:18) Ω · B πc (cid:19) . (3.13)The PSG model assumes the existence of ions in the IAR region that affects the chargedensity. Using the screening factor, η , we can write that ρ ( r, θ, φ ) = (1 − η ) ρ GJ ( r, θ, φ ) = (1 − η ) Ω · B πc . In general, η depends on the curvature and strength of the magnetic field, thus itvaries across the polar cap, but we can still assume that η is approximately constant atleast for a given spark. Then2 r E r + ∂E r ∂r + cos θr sin θ E θ + 1 r ∂E θ ∂θ + 1 r sin θ ∂E φ ∂φ = − πη (cid:18) B r Ω cos θ − B θ Ω sin θ πc (cid:19) . (3.14)Let us change the variables as follows: r = R + z and θ = α + ϑ . Here R is the stellarradius and α is the inclination angle between the rotation and the magnetic axis. .1. The Model R + z E r + ∂E r ∂z + cos ( α + ϑ )( R + z ) sin ( α + ϑ ) E θ + 1 R + z ∂E θ ∂ϑ + 1( R + z ) sin ( α + ϑ ) ∂E φ ∂φ == − πη (cid:18) ( B r Ω cos θ − B θ Ω sin θ )2 πc (cid:19) . (3.15)Assuming that R (cid:29) z , which is correct as the gap height is less than the stellar radius( h (cid:28) R ), α (cid:29) ϑ , and B r (cid:29) B θ , which is correct for the polar cap region, we can writeEquation 3.15 in the first approximation ( R → ∞ ) as follows ∂E r ∂z + 1 R ∂E θ ∂ϑ = − πη (cid:18) B r Ω cos θ πc (cid:19) . (3.16)Note that for spark widths considerably smaller than the stellar radius h ⊥ (cid:28) R (∆ ϑ ≈ h ⊥ /R ) we can write that R ∂E θ ∂ϑ (cid:29) cot( α + ϑ ) R E θ .Let us now consider Faraday’s law (Equation 3.12). The curl of an electric field inspherical coordinates can be written as( ∇ × E ) r = 1 r sin θ (cid:18) ∂∂θ ( E φ sin θ ) − ∂E φ ∂φ (cid:19) = 0 , ( ∇ × E ) θ = 1 r (cid:18) θ ∂E r ∂φ − ∂∂r ( rE φ ) (cid:19) = 0 , ( ∇ × E ) φ = 1 r (cid:18) ∂∂r ( rE θ ) − ∂E r ∂θ (cid:19) = 0 . (3.17)Using the same change of variables we performed above ( r = R + z and θ = α + ϑ ),the third equation of System 3.17 can be written as R ∂E θ ∂z = ∂E r ∂ϑ . (3.18)From this equation in the zeroth approximation we can estimate the variations of theelectric field components as R ∆ E θ ∆ ϑ ≈ ∆ E r ∆ z. (3.19)Since h ⊥ (cid:28) R we can write that (cid:104) h ⊥ E θ (cid:105) = (cid:104) hE r (cid:105) = ∆ V. (3.20)From Equation 3.16 we can also briefly estimate that∆ E r h + ∆ E θ h ⊥ = − πη (cid:18) B r Ω cos θ πc (cid:19) . (3.21)Using Equations 3.16 and 3.20 we can write that8 Chapter 3. Partially Screened Gap (cid:104) hE r (cid:105) h + (cid:104) h ⊥ E θ (cid:105) h ⊥ = ∆ Vh + ∆ Vh ⊥ . (3.22)Finally, we can estimate the potential drop in a spark region∆ Vh + ∆ Vh ⊥ = 2 ηB r Ω cos ( α + ϑ ) c . (3.23)If we use the same assumption as Ruderman and Sutherland (1975), i.e.: (1) the sparkhalf-width is of the same order as the gap height h ⊥ = h , (2) there is no ion extractionfrom the stellar surface ( η = 1), and (3) the pulsar magnetic and rotation axes are aligned( α = 0 ◦ ), we get: ∆ V RS = B r Ω c h . (3.24)Note that the potential drop defined by Equation 3.23 differs from that used in theStandard Model by the screening factor (as the presence of ions screens the gap) and bythe factor of cos ( α + ϑ ) which also takes into account non-aligned pulsars. In our case thepolar cap size is much smaller than the conventional polar cap size. It seems reasonableto also consider sparks with widths much smaller than the gap height ( h ⊥ (cid:28) h ), in thatcase the potential drop can be calculated as∆ V = 2 ηB r Ω cos ( α + ϑ ) c h ⊥ . (3.25)Even for a relatively small inclination angle between the rotation and magnetic axis,we can still write ϑ (cid:28) α , thus ∆ V = 4 πηB r cos αcP h ⊥ . (3.26) Since the exact dependence of the electric field on z is unknown we use the same linearapproximation that Ruderman and Sutherland (1975) used. In the frame of the PSGmodel as h ⊥ < h or even h ⊥ (cid:28) h , we can use Equations 3.20 and 3.26 to describe thecomponent of the electric field along the magnetic field line: E ≈ πηB s cos αcP h ⊥ h ( h − z ) , (3.27)which vanishes at the top z = h . The Lorentz factor of particles after passing distance l acc can be calculated as follows γ acc = emc ˆ z z Edz ≈ πηB s e cos αmc P h ⊥ h ( z − z ) (cid:18) h − z + z (cid:19) , (3.28) .1. The Model m is the mass of a particle (electron or positron) and z − z = l acc . Then we canapproximate z + z ≈ h , thus l acc , ap = γ acc mc P πηB s e cos α hh ⊥ . (3.29)Assuming that a non-relativistic particle is accelerated from the stellar surface ( z = 0, γ = 1) we can calculate the distance l acc which it should pass to gain a Lorentz factor γ acc : l acc = h (cid:32) − (cid:114) − γ(cid:96) (cid:33) , (3.30)where (cid:96) = 8 πηB s eh ⊥ cos ( α ) / ( P c m ). Although the approximate formula 3.29 is muchmore readable, in the calculations we use the exact value (see Equation 3.30) as for Lorentzfactors that are considerably smaller than the maximum value, the discrepancy is abouta factor of two, l acc , ap ≈ l acc . The mean free path of a particle (electron and/or positron) l p can be defined as the meanlength that a particle passes until a γ -photon is emitted. In the case of the CR particle,mean free path can be estimated as a distance that a particle with a Lorentz factor γ travels during the time which is necessary to emit a curvature photon (see Zhang et al.1997) l CR ∼ c (cid:18) P CR E γ, CR (cid:19) − = 94 (cid:126) (cid:60) cγe , (3.31)where P CR = 2 γ e c/ (cid:60) is the power of CR, E γ, CR = 3 (cid:126) γ c/ (cid:60) is the photon character-istic energy, and (cid:60) is the curvature radius of the magnetic field lines.For the ICS process calculation of the particle mean free path l ICS is not as simple asthat of the CR process. Although we can define l ICS in the same way that we defined l CR ,it is difficult to estimate the characteristic frequency of emitted photons. We have to takeinto account photons of various frequencies with various incident angles. An estimationof the mean free path of an electron (or positron) to produce a photon is in Xia et al.(1985) l ICS ∼ (cid:20) ˆ µ µ ˆ ∞ σ (cid:48) ( (cid:15), µ ) (1 − βµ i ) n ph ( (cid:15) ) d(cid:15)dµ (cid:21) − . (3.32)Here (cid:15) is the incident photon energy in units of mc , µ = cos ψ is the cosine of the photonincident angle, β = v/c is the velocity in terms of speed of light, σ (cid:48) is the cross section ofICS in the particle rest frame, n ph ( (cid:15), T ) d(cid:15) = 4 πλ c (cid:15) exp ( (cid:15)/ (cid:48) ) − d(cid:15) (3.33)0 Chapter 3. Partially Screened Gap represents the photon number density distribution of semi-isotropic blackbody radiation, (cid:48) = kT /mc , k is the Boltzmann constant, and λ c = h/mc = 2 . × − cm is theelectron Compton wavelength. A detailed description of how to calculate σ (cid:48) can be foundin Section 4.4.1.We should expect two modes of ICS: resonant and thermal-peak (see Section 4.4.3for more details). The Resonant ICS (RICS) takes place if the photon frequency in theparticle rest frame is equal to the electron cyclotron frequency. As shown in Section 4.4.4,the particle mean free path strongly depends on the distance from the polar cap. Boththe photon density and incident angles ( µ and µ ) change with increasing altitude. Inour calculations we take into account both of those effects, thus we replace n ph ( (cid:15), T ), µ and µ with n sp ( (cid:15), T, L ), µ min ( L ) and µ max ( L ), respectively (for more details see Section4.4.4). Here, L is the location of the particle, n sp ( (cid:15), T, L ) is the photon density atlocation L , and µ min ( L ) and µ max ( L ) correspond to the highest and lowest angle betweenthe photons and particle at a given location L . Thus, just above the polar cap for RICSthe mean free path of outflowing positrons is: l RICS ≈ (cid:34) ˆ µ max ( L ) µ min ( L ) ˆ (cid:15) maxres (cid:15) minres (1 − βµ ) σ (cid:48) ( (cid:15), µ ) n sp ( (cid:15), T, L ) d(cid:15)dµ (cid:35) − , (3.34)where the limits of integration over energy, (cid:15) minres and (cid:15) maxres , are chosen to cover the resonantenergy (for more details see Section 4.4.3).The thermal-peak ICS (TICS) includes all scattering processes of photons with fre-quencies around the maximum of the thermal spectrum. As an example we adopt (cid:15) minth ≈ . (cid:15) th , and (cid:15) maxth ≈ (cid:15) th where (cid:15) th = 2 . kT / ( mc ) is the energy, in units of mc , at which blackbody radiation with temperature T has the largest photon numberdensity. The electron/positron mean free path for the TICS process is l TICS ≈ (cid:34) ˆ µ max ( L ) µ min ( L ) ˆ (cid:15) maxth (cid:15) minth (1 − βµ ) σ (cid:48) ( (cid:15), µ ) n ph ( (cid:15), T, L ) d(cid:15)dµ (cid:35) − . (3.35) The photons with energy E γ > mc propagating obliquely to the magnetic field lines canbe absorbed by the field, and as a result, an electron-positron pair is created. To describethe strength of the magnetic field we use β q = B/B q , where B q = m c /e (cid:126) = 4 . × Gis the critical magnetic field strength.For strong magnetic fields ( β q (cid:38) .
2, see Section 4.2.3) the photon mean free path canbe calculated as (see Section 4.2.4 for more details) l ph ≈ (cid:60) mc E γ , (3.36) .2. Gap height β q (cid:46) .
2) we can use an asymptotic approximationderived by Erber (1966) l ph = 4 . e / (cid:126) c ) (cid:126) mc B q B sin Ψ exp (cid:18) χ (cid:19) , (3.37) χ ≡ E γ mc B sin Ψ B q ( χ (cid:28) , (3.38)where Ψ is the angle of intersection between the photon and the local magnetic field. By knowing the acceleration potential drop in PSG ∆ V we can evaluate the gap height h and the screening factor η , which actually depends on the details of the avalanche pairproduction in the gap. First, we need to determine which process, Curvature Radiation(CR) or Inverse Compton Scattering (ICS), is responsible for the γ -photon generation inthe gap region. In order to identify the proper process we need the following parameters: l acc - the distance which a particle should pass to gain the Lorentz factor γ acc , l p - themean length a particle (electron and/or positron) travels before a γ -photon is emitted,and l ph - the mean free path of the γ -photon before being absorbed by the magnetic field.As mentioned above, PSG can exist if Equation 3.6 is satisfied. On the other hand,in order to heat the polar cap surface to high enough temperatures the high enough fluxof back-streaming particles is required. By using Equations 3.8 and 3.26 we can find therelationship between the screening factor, the spark half-width and pulsar parameters ηh ⊥ = 4 .
17 ( B . + 0 . PB (cid:112) | cos α | . (3.39)Thus, for specific pulsar parameters we can define a product of the two main para-meters of PSG, namely the screening factor η and the spark half-width h ⊥ . The Figure 3.2 shows the dependence of particle mean free paths on the Lorentz factor γ for some pulsar parameters (the dependence on pulsar parameters will be discussedin Section 3.3). Let us note that these free paths do not depend on the gap height h (see Equations 3.31, 3.32 and 3.36). The results presented in the Figure do not allow todefine the gap height unambiguously. However, we can find which process is responsiblefor generation of the γ -photon in PSG. For narrow sparks the acceleration potential dropdecreases, and as a result the Lorentz factor of the primary particles is about γ ∼ − .In this regime l ICS (cid:28) l CR , so the gap will be dominated by ICS. Thus, ICS will dominatethe gap if deceleration due to Inverse Compton Scattering prevents further acceleration2 Chapter 3. Partially Screened Gap by the electric field. Let us remember that ICS is not efficient for particles with theLorentz factor γ (cid:38) . If the sparks are wider or η ≈
1, the acceleration potential dropincreases and the Lorentz factors of primary particles reach values about γ ∼ − .In this regime the γ -photon emission is dominated by CR. Let us note that the condition l ICS (cid:28) l CR is satisfied for particles with γ ∼ − , as one can see from Figure 3.2(panel a), but this does not mean that the ICS event happens. Since l acc (cid:28) l ICS , theparticles will be accelerated to higher energies ( γ ∼ − ) before they upscatterthe X-ray photons. Thus, the particles start emission of γ -photons (via CR) as soon ascondition l acc ≈ l CR is met. Figure 3.2:
Dependence of the mean free path of the primary particle on Lorentz factor γ for boththe CR and ICS processes. Panel (a) corresponds to calculations for a relatively higher potential drop(e.g. a wider spark with h ⊥ = 3 m and η = 1 ), while panel (b) corresponds to calculations for arelatively lower potential drop (e.g. a narrow spark with h ⊥ = 1 m and η = 0 . ). The accelerationpaths on both panels were calculated for the same pulsar parameters ( B = 3 . , T = 4 . , (cid:60) = 1 , P = 1 , α = 10 ◦ ). Note that for the RICS process the particle mean free paths were calculated foroptimal conditions (just above the polar cap). As is seen from Figures 3.3 and 3.4, in the CR-dominated gap the primary particle shouldtravel a distance comparable with gap height l acc ≈ h/ γ CR c . On the other hand, the primaryparticles in the ICS-dominated gap reach a characteristic value γ ICS c at altitudes thatare considerably smaller than gap height l acc (cid:28) h (see Figures 3.3 and 3.5). Thus, γ CR c ≈ ≈ γ max is about three orders of magnitude higher than γ ICS c ≈ (cid:28) γ max , here γ max is the value of the Lorentz factor after the particle travels a distance h . Furthermore,the characteristic energy of CR photons is considerably smaller than the energy of emitting(primary) particles, e.g. for γ = 10 , (cid:60) = 1, γ sec ≈ . On the other hand, RICS photons .2. Gap height B > B crit ) magnetic field gain a significant part of theenergy of the scattering (primary) particle. Therefore, the electron/positron pair createdby the RICS photon has energies comparable with the energy of the scattering (primary)particle. This will essentially influence the multiplicity M pr in the ICS gap, as all thenewly created particles will participate in further cascade pair-production. Additionally,RICS in ultrastrong magnetic fields produces approximately the same amount of photonswith (cid:107) and ⊥ polarisation (see Section 4.4.2), while most of the photons produced by CRare (cid:107) -polarised (see Section 4.1). Splitting of the ⊥ -polarised photons will increase thephoton mean free path, but it will also increase the multiplicity in the ICS gaps.Figure 3.3 presents a sketch of a cascade formation for CR- and ICS-dominated gaps.The CR photons are emitted in the upper half of the gap. Most of these photons producepairs at about the same height, in the region where the acceleration potential is almostequal to zero, hereinafter we will call this region the Zero-Potential Front (ZPF). Thenewly created particles have much lower Lorentz factors as compared with the primaryparticle, thus they are not able to emit CR photons. CR ICS
Figure 3.3:
Sketch of differences in a cascade formation for the CR-dominated gap (left panel)and the ICS-dominated gap (right panel). In order to increase readability, only a few points (filledcircles) are shown which correspond to altitudes where γ -photons are emitted. The unfilled circlescorrespond to places where γ -photons are also emitted, but those photons (and their evolution) arenot included in the diagram. Note that for the ICS-dominated gap we plot only the bottom (active)part of the gap ( z (cid:28) h ICS ), furthermore, points of radiation are tracked only for the first populationof newly created particles. The avalanche nature of the ICS-dominated gap will result in a muchhigher multiplicity and continuous backflow of relativistic particles. Chapter 3. Partially Screened Gap
Figure 3.4 presents the primary particle evolution and photon mean free paths of γ -rays produced in the CR-dominated gap. As can be seen, the first γ -photon producesa pair approximately at the same time (and same place) as the primary particle reachesZPF. Thus, the multiplicity in a gap region (the number of particles created by a singleprimary particle) in the CR scenario is strictly related to the number of photons producedby the primary particle M CR ≈ × N CR ph . Figure 3.4:
Cascade formation for a CR-dominated gap. Blue lines represent the mean free pathof γ -photons. The filled circles correspond to places of γ -photon emission. Panel (a) includes thefree paths of γ -photons which produce pairs below ZPF (red circles) while panel (b) includes thefree paths of γ -photons which produce pairs above the acceleration gap (blue circles). The resultswere obtained using the following parameters: N CR ph = 50 , B s = 2 . × G , B d = 2 . × G , T = 3 MK , P = 1 . , (cid:60) = 0 . , and α = 60 . ◦ . The energy of γ -photons produced by ICS depends on the Lorentz factor of the primaryparticles and on the strength of the magnetic field. In ultrastrong magnetic fields theenergy of newly created particles is comparable with the energy of the scattering particle γ new ≈ γ c /
2. Figure 3.5 shows schematically the locations at which γ -photons are emittedby ICS. The first γ -photon is produced already at altitudes of about a few metres and thenconverted to an electron-positron pair well below ZPF. Note that already at relativelylow altitudes ( z (cid:38)
100 m) the photon density decreases rapidly (see Section 4.4.4.1),furthermore, the small size of the polar cap entails a rapid change of the particle-photonincident angles (see Section 4.4.4.2). Those two effects make the ICS process significantonly in the lower parts of the gap ( z (cid:46)
20 m). On the other hand, the multiplicity in theICS-dominated gap is enhanced by all newly created particles which are created in thelower part of the gap. Furthermore, the ICS is more effective for backstreaming particles(see Figure 3.2), thus most of the γ -photons in the gap region will be created by scatteringson electrons. For the ICS scenario it is not possible to evaluate a simple expression forthe multiplicity produced by a single primary particle in a gap region. Furthermore, it is .2. Gap height N ph required to break the gap (both for CRand ICS) without a full cascade simulation. Figure 3.5:
Cascade formation for an ICS-dominated gap. Blue lines represent the mean free pathof γ -photons. The filled circles correspond to places of γ -photon emission. The results were obtainedusing the following parameters: N ICS ph = 15 , B s = 2 . × G , B d = 2 . × G , T s = 3 MK , P = 1 . , (cid:60) = 0 . , and α = 60 . ◦ . The differences between the CR and ICS gaps that we mention above have drastic con-sequences on the cascade formation process. Since the cooling time of the hot spot is veryshort ( τ cool (cid:46) − s , see Gil et al., 2003), to sustain the hot spot temperature just belowthe critical temperature a continuous backflow of relativistic particles is required. Anenergetic enough flux of backstreaming particles can be produced only in ICS-dominatedgaps. The heating of the surface will sustain the outflow of iron ions from the crust,maintaining η <
1, hence we call this mode the PSG-on mode. As the temperature ofthe polar cap is in quasi equilibrium with the backstreaming particles (temperature isclose to the critical value) the gap can break only due to production of a dense enoughplasma n p (cid:29) ηn GJ in the gap region. The multiplicity in the PSG-on mode is much higherthan the multiplicity of CR-dominated gaps. Moreover, in the gap dominated by CR theparticles are created in a cloud-like fashion (see Figure 3.4). The successive clouds heat upthe surface once per τ ≈ h/c , which for a typical gap height h ≈
100 m is much longerthan the time needed for the surface to cool down τ ≈ × − (cid:29) τ cool . Therefore, in theCR-dominated gaps the backstreaming particles cannot sustain the temperature that isclose to the critical value during τ (cid:29) τ (cid:29) τ cool , thus for most of the time the screeningfactor is η ≈ τ heat (cid:29) τ . Let us note6 Chapter 3. Partially Screened Gap that the primary particles in the PSG-off mode are very energetic γ ≈ , and hence thedensity of particles required to close gap ρ c is much lower than the Goldreich-Julian dens-ity. To describe this difference we use the overheating parameter κ = ρ c /ρ GJ . Knowingthat in the PSG-off mode η ≈
1, we use Equation 3.6 and the relation ∆ V = γ acc mc /e to calculate the overheating parameter: κ = σ T n GJ γ max mc . (3.40) Curvature emission by a primary particle is effective for Lorentz factors γ (cid:38) (when l CR ≤ l acc ). An equilibrium between acceleration and deceleration (by reaction force)would be established if the CR power were equal to the ”electric power”. In our case( (cid:60) ≈ γ c ≈ ), the reaction force is not high enough to stop acceleration by theelectric field. In the PSG-off mode the spark region is free from ions ( η ≈ h min ⊥ = (cid:60) − √(cid:60) − h . (3.41)Figure 3.7 presents the minimum spark half-width for three different radii of curvature: (cid:60) = 0 . (cid:60) = 0 . (cid:60) = 1. Note that as long as the gap height does not exceed somespecific value ( h ≈
40 m, h ≈
100 m, h ≈
140 m, respectively for the given curvature radii)the minimum spark half-width is well below 1 m.
Figure 3.6:
Diagram of the minimum spark half-width h min ⊥ for a given gap height h and a radiusof curvature (cid:60) . .2. Gap height Figure 3.7:
Minimum spark half-width vs. gap height calculated for three different radii ofcurvature: (cid:60) = 0 . - red solid line, (cid:60) = 0 . - green dashed line, and (cid:60) = 1 - blue dottedline. On the other hand, we can estimate the acceleration potential ∆ V (and thus the sparkhalf-width h Nph ⊥ ) required to produce a specified number of photons N CR ph within a gap.Figure 3.8 presents the dependence of both h min ⊥ and h Nph ⊥ on the gap height. As resultsfrom the Figure, the gap height in PSG-off does not change drastically with N CR ph , andfor these specific parameters of a pulsar it is h ≈
240 m. For historical reasons, hereafterunless stated otherwise, we will use N CR ph = 50 to calculate the gap parameters of thePSG-off mode. Note that in order to find the gap height, we assume h min ⊥ = h N ph ⊥ , whichresults in a gap that allows both overheating of the entire spark surface by backstreamingparticles and the creation of the required number of photons N CR ph . Figure 3.8:
Dependence of a spark half-width on the gap height for the PSG-off mode. The resultswere obtained using the following pulsar parameters: B = 2 . , T = 3 . , P = 1 . , (cid:60) = 1 . , α = 60 . ◦ . Chapter 3. Partially Screened Gap
In our calculations we use the algorithm presented in Figure 3.9 to find the gap heightin the PSG-off mode for given pulsar parameters: a pulsar period P , a pulsar inclinationangle α , a surface magnetic field strength B s , and a curvature radius of field lines (cid:60) . Figure 3.9:
Flowchart of the algorithm used to estimate the gap height in the PSG-off mode. Theinitial gap height from which we begin our calculations is an arbitrary set to h init . = 10 m , while thestep ∆ h depends on the required accuracy. The number of γ -ray photons created in a spark by asingle primary particle is set to N CR ph = 50 (see text for more details). Figure 3.10 presents the result of finding the gap height in the PSG-off mode for PSRB0943+10. The presented solution corresponds to the magnetic field structure presentedin Section 2.4.4. The average radius of curvature in the gap region is relatively high, (cid:60) = 0 .
7, hence the inclination of the gap region. The polar gap conditions, the strengthof magnetic field B = 2 . R bb = 17 m) and the polar cap temperature T = 3 . h = 166 m, spark half-width h ⊥ = 1 . η = 1 (fixed), κ = 7 × − , γ c = 1 . × . Note that the primary particleswill gain γ max = 1 . × as the CR efficiency is not high enough to stop the acceleration. .2. Gap height Figure 3.10:
Gap structure in the PSG-off mode for PSR B0943+10. Filled columns represent thelocations and sizes of the active regions of sparks. Here we assumed that the active region of a spark(the place where acceleration is high enough to produce a cascade) has a size comparable with thespark half-width. The iron ions extracted from the surface (due to a high surface temperature) arerepresented by circle-plus symbols.
In the PSG-on mode, radiation of the surface just below the spark is in quasi-equilibriumwith the flux of backstreaming particles. When the surface temperature rises, the densityof iron ions increases, thus resulting in a decrease in the potential drop, which in turn,reduces the flux of backstreaming particles. On the other hand, when the surface tem-perature decreases it entails the drop of iron ion density and, consequently, an increasein the flux of backstreaming particles. Thus the polar cap temperature is maintainedslightly below the critical value. This quasi-equilibrium state prevents the gap breakdowndue to surface overheating. However, a high multiplicity in the PSG-on mode leads to aproduction of dense plasma. When the density of the plasma n p (cid:29) ηn GJ , the accelerationpotential drop will be completely screened due to charge separation.Alongside the pulsar parameters the gap height in the PSG-on mode also dependson the spark half-width h ⊥ and on the number of scatterings by the first population ofnewly created particles N ICS ph . For a sample of pulsars we can use drift information to putconstraints on the spark half-width (see Section 3.4). Figure 3.11 presents the procedure offinding the gap height in the PSG-on mode for the following pulsar parameters: a pulsarperiod P , a pulsar inclination angle α , a surface magnetic field strength B s , a surfacetemperature T s , a curvature radius of magnetic field lines (cid:60) , and a spark half-width h ⊥ .First we use Equation 3.6 to estimate the screening factor η which defines the electricfield, and thus the particle acceleration. Then we estimate the number of scatterings for asingle outflowing particle N pr ph for the initial gap height. The initial gap height from which0 Chapter 3. Partially Screened Gap we begin our calculations is an arbitrary set to h init . = 10 m. We track the propagation of γ -photons produced by ICS on a primary particle to find the location L new where pairsare created. Then we calculate their propagation through the acceleration region and weestimate the number of scatterings by every newly created particle of the first population N new ph . If the total number of scatterings by the first population (including the primaryparticle) is N ph < N ICS ph , we resume our calculations assuming a higher gap until the N ph ≥ N ICS ph is met. Figure 3.11:
Flowchart of algorithm used to estimate the gap height in PSG-on mode for a givenspark half-width (see text for more details). .2. Gap height h , the screening factor η ,the characteristic Lorentz factor of a particle at the moment of ICS photon emission γ c ,the maximum value of the Lorentz factor γ max , and the characteristic Lorentz factor ofiron ions γ i . In our calculations, if not stated otherwise, we use N ICS ph = 25 to calculatethe gap parameters of the PSG-on mode. Note that in this approximation we take intoaccount only the first population of newly created particles. In fact, the avalanche natureof the ICS-dominated gap will result in a much higher multiplicity than in the PSG-offmode M ICS (cid:29) M CR . For details of particle/photon propagation, see Chapter 4. Subpulsedrift observations are available only for a few X-ray pulsars with the hot spot component.Thus, to find the approximate gap parameters for pulsars without the predicted sparkhalf-width we use h ⊥ = 2 m.Figure 3.12 presents the result of finding the gap height in the PSG-on mode for PSRB0943+10. In this model the gap parameters, such as the magnetic field strength B s and the surface temperature T s , were restrained to follow the observed values (see Table1.4). The result was obtained for the non-dipolar structure of a surface magnetic fieldpresented in Section 2.4.4 and for the predicted value of a spark half-width h ⊥ ≈ N ICS ph = 25 photons by the first population ofparticles was estimated as h ≈
92 m. Other gap parameters for this solution can be foundin Table 3.1.
Figure 3.12:
Gap structure in the PSG-on mode for PSR B0943+10. Filled columns represent thelocations and sizes of the active regions of sparks. Here we assumed that the active region of a spark(the place where acceleration is high enough to produce a cascade) has a size comparable with thespark half-width. The iron ions extracted from the surface (due to a high surface temperature) arerepresented by circle-plus symbols. Note that iron ions are present in both the non-active and activeregions. The density of ions in the non-active regions is so high that it prevents cascade formationof pairs. Chapter 3. Partially Screened Gap
In Table 3.1 we present the results of finding the gap height for the sample of pulsars.For the PSG-on mode we show the estimated PSG parameters found using the predictedspark half-width and the spark half-width h ⊥ = 2 m. The only exception is Geminga(PSR J0633+1746), for which drift information is not available and we can only presentcalculations for h ⊥ = 2 m. For PSR B0628-28 the predicted spark half-width is large( h ⊥ = 3 . N ICS ph . Webelieve that for this specific pulsar the predicted spark half-width is overestimated. Ac-tually, if a spark is narrower ( h ⊥ = 2 m), it can operate in the PSG-on mode (see Table3.1). This result may suggest that for this specific pulsar the parameters of the sub-pulse phenomenon could be overestimated (e.g. due to aliasing). On the other hand,X-ray observations of Geminga suggest a relatively low temperature of the hot spot( T s ≈ . h ⊥ = 1 m) to allow the gap to operate in the PSG-on mode. We believethat the relatively large hot spot ( R pc = 44 . Table 3.1:
Estimated parameters of PSG for the sample of pulsars. The conditions in the polarcap region: surface temperature, magnetic field strength, polar cap radius, and curvature radiusof the field lines are given the in headers next to the pulsar name. The individual columns are asfollows: (1) PSG mode (see Section 3.2.2), (2) Gap height, (3) Spark half-width, (4) Screening factor,(5) Overheating parameter, (6) Characteristic Lorentz factor of scattering particles , (7) MaximumLorentz factor of primary particles, (8) Lorentz factor of iron ions (if they are relativistic), (9) Particlemean free path, and (10) Photon mean free path. The results are presented for two different gapbreakdown scenarios: the PSG-off and PSG-on modes (see Section 3.2 for more details). a The modes correspond to calculations using the predicted spark half-width (see Table 3.3) b The modes correspond to calculations with a spark half-width h ⊥ = 2 m mode h h ⊥ η κ γ c γ max γ i l p l ph ( N ph ) (m) (m) (m) (m) PSR B0628-28 T = 2 . B = 2 . R pc = 21 . (cid:60) = 0 . . . .
007 1 . × . × – 1 . . a – 3 . b . . .
15 – 6 . × . ×
23 1 . . Continued on next page .2. Gap height h h ⊥ η κ γ c γ max γ i l p l ph (m) (m) (m) (m) PSR B0628-28 T = 2 . B = 2 . R pc = 21 . (cid:60) = 0 . . . .
007 1 . × . × – 1 . . a – 3 . b . . .
15 – 6 . × . ×
23 1 . . PSR J0633+1746 T = 1 . B = 1 . R pc = 44 . (cid:60) = 2 . . . . . × . × – 2 . . b – 2 . PSR B0834+06 T = 2 . B = 1 . R pc = 22 . (cid:60) = 0 . . . . . × . × – 1 . . a . . .
12 – 5 . × . ×
18 2 . . b . . .
11 – 4 . × . ×
20 2 . . PSR B0943+10 T = 3 . B = 2 . R pc = 17 . (cid:60) = 0 . . . . . × . × – 1 . . a,b . . . . × . ×
63 1 . . Continued on next page Chapter 3. Partially Screened Gap
Table 3.1 - continued from previous pagemode h h ⊥ η κ γ c γ max γ i l p l ph (m) (m) (m) (m) PSR B0950+08 T = 2 . B = 2 . R pc = 14 . (cid:60) = 0 . . . . . × . × – 1 . . a . . .
09 – 3 . × . × . . b . . .
03 – 5 . × . ×
17 1 . . PSR B1133+16 T = 2 . B = 2 . R pc = 17 . (cid:60) = 0 . . . . . × . × – 1 . . a . . .
08 – 7 . × . ×
47 1 . . b . . .
11 – 7 . × . ×
33 1 . . PSR B1929+10 T = 3 . B = 2 . R pc = 20 m (cid:60) = 0 . . . . . × . × – 1 . . a . . .
02 – 9 . × . ×
31 1 . . b . . .
02 – 8 . × . ×
39 1 . . .3. PSG model parameters We can distinguish two types of PSG parameters: observed and derived. As we havementioned above, in some cases when X-ray observations are available we can directlyestimate the surface magnetic field B s . On the one hand, B s can be calculated using thesize of the hot spot A bb , and on the other hand we can find B s by using the estimation ofthe critical temperature and the assumption that T s = T crit . One of the most importantrequirements for the PSG model is that these two estimations should coincide with eachother. As is clear from Figure 1.11, in most cases when the hot spot parameters areavailable this requirement is fulfilled. Thus, we can assume that the characteristic valuesof B s vary in the range of (1 − × G, which corresponds to the critical surfacetemperature in the range of (1 . − × K (see Table 1.4). By using these values wecan estimate the derived parameters of PSG, such as the gap height h , the screeningfactor η (or the overheating parameter κ in the PSG-off mode) and the characteristicLorentz factor of primary particles γ c . Let us note that these parameters also depend onthe curvature radius of the magnetic field lines (cid:60) . The curvature can be neither observednor derived, but modelling of the surface magnetic field (see Chapter 2) indicates thatthe curvature radius varies in the range of (0 . − × cm. Below we will discuss theinfluence of pulsar parameters, such as the magnetic field, the curvature of field lines andthe period on derived PSG parameters. The conditions in PSG are mainly defined by the surface magnetic field. In Figure 3.13,panel (a) we present the dependence of the gap height on the surface magnetic fieldcalculated according to the approach described in Section 3.2. It is clear that in the PSG-off mode the gap height decreases as the surface magnetic field increases. In the PSG-onmode, on the other hand, the gap height shows a minimum at a specific value of themagnetic field strength (for a given pulsar’s parameters it is B ≈ κ in the PSG-off mode) on the surface magnetic field. Wecan see that for stronger magnetic fields both η and κ increase, which means that: (1)the density of heavy ions above the polar cap in the PSG-on mode decreases, (2) thedensity of particles required to overheat (and thus to close) the polar cap increases. Letus note that the surface temperature T s stays very near to the critical temperature T crit ,which is shown on the top axis of the Figures. In panel (c) the red, solid and dottedlines correspond to characteristic and maximum Lorentz factors ( γ c , γ max ) in the PSG-on6 Chapter 3. Partially Screened Gap mode, while the blue, dashed and dashed-dotted lines correspond to γ c and γ max in thePSG-off mode. We see that especially for the PSG-off mode γ c does not depend on themagnetic field strength. Note also that in the PSG-off mode (CR-dominated gap), thecharacteristic Lorentz factor (the Lorentz factor for which most of the gamma photonsare produced) slightly differs from the maximum value, γ c ≈ γ max . On the other hand,in the PSG-on mode γ c (cid:28) γ max , which reflects the fact that most of the scatterings takeplace in the bottom part of the gap. Figure 3.13:
Dependence of the gap height (panel a), the screening factor or the overheatingparameter (panel b), and the particle Lorentz factor (panel c) on the surface magnetic field. Solidred lines correspond to the PSG-on mode (ICS-dominated gaps) while dashed blue lines correspond tothe PSG-off mode (CR-dominated gaps). Calculations were performed using the following parameters: P = 0 . , (cid:60) = 0 . , B d = 1 . × G , and α = 36 ◦ . The actual polar cap radius was calculatedseparately for a given surface magnetic field as R pc = R dp (cid:112) B d /B s . In panel (c) the red solid anddotted lines correspond to characteristic and maximum Lorentz factors ( γ c , γ max ) in the PSG-onmode while blue dashed and dashed-dotted lines correspond to γ c and γ max in the PSG-off mode.Corresponding critical temperature is shown on top axis of the figures. The curvature of the magnetic field lines significantly affects the gap height in the PSG-offmode (see Figure 3.14, panel a). In the case of the CR-dominated gap, the curvature ofthe magnetic field lines affects not only the photons’ mean free path (for higher curvaturethe magnetic field will absorb photons faster), but also the particle mean free path and,more importantly, the energy of photons generated in the gap region. The higher energy ofphotons further reduces the photon mean free path, thus resulting in lower heights of thePSG. In contrast, the gap height in the PSG-on mode is only slightly affected by changesin the curvature of the magnetic field lines. In this case the most important parameterwhich determines the cascade properties is the primary particle mean free path whichdoes not depend on the curvature of the magnetic field lines.The overheating parameter in the PSG-off mode inversely depends on the radius ofcurvature of the magnetic field lines (see Figure 3.14, panel b). The higher the curvature, .3. PSG model parameters γ c and γ max ) required to close the gap in the PSG-off mode also increases. This reflects thefact that in order to produce a sufficient number of photons in the gap region, the primaryparticles should be accelerated to higher energies (if the curvature is lower). Higherenergies of the primary particles will increase the emitted γ -photon energy, thereby theywill partly inhibit the growth of the photon mean free path due to the lower curvature. Asmentioned above, the gap height in the PSG-on mode very weakly depends on the photonmean free path, thus both γ c and γ max are not affected by the increase in the radius ofcurvature. Figure 3.14:
Dependence of the gap height (panel a), the screening factor or the overheatingparameter (panel b), and the particle Lorentz factor (panel c) on the curvature radius of magneticfield lines. Calculations were performed using the following parameters: P = 0 . , B d = 1 . × G , B s = 2 . × G , α = 36 ◦ . For a more detailed description see Figure 3.13. As we can see from Figure 3.15, panel (a) and panel (c), in the PSG-on mode the gapheight and the Lorentz factor of primary particles do not depend on the pulsar period.The increase in the screening factor (see Figure 3.15b) compensates the increase in theacceleration potential drop (see Equation 3.8). Thus the particles in the gap region areaccelerated in the same way independently of the pulsar period. On the other hand,the gap height in the PSG-off mode increases with the increasing pulsar period. Thisreflects the fact that in the PSG-off mode the acceleration potential, and hence γ c and γ max , decreases with longer periods (see Equation 3.26). Longer pulsar periods entailan increase in the screening factor (in the PSG-on mode, see Equation 3.39) and in theoverheating parameter (in the PSG-off mode). Note that for periods longer than some8 Chapter 3. Partially Screened Gap specific value (for a given pulsar’s parameters it is P max ≈ Figure 3.15:
Dependence of the gap height (panel a), the screening factor or a overheatingparameter (panel b), and the particle Lorentz factor (panel c) on the pulsar period. Calculationswere performed using the following parameters: P = 0 . , B d = 1 . × G , B s = 2 . × G , α = 36 ◦ , (cid:60) = 0 . . The actual polar cap radius was calculated separately for a given pulsar periodas R pc = R dp (cid:112) B d /B s , where R dp = (cid:112) πR / ( cP ) . For a more detailed description see Figure3.13. The existence of IAR in general causes a rotation of the plasma relative to the NS, asthe charge density differs from the Goldreich-Julian co-rotational density. The powerspectrum of radio emission must have a feature due to this plasma rotation. This featureis indeed observed and is called the drifting subpulse phenomenon.
An explanation for drifting subpulses was offered by Ruderman and Sutherland (1975) asbeing due to a rotating carousel of sub-beams within a hollow emission cone. Accordingto this model a pair cascades may not occur simultaneously across the whole polar capbut is localised in the form of discharges of small regions in the polar gap. Such sparksmay produce plasma columns that stream into the magnetosphere to produce the observedradio emission. The location of the discharges on the polar cap determines the geometricalpattern of instantaneous subpulses within a pulsar’s integrated pulse profile.In the PSG model the stable pattern of subpulses is due to heating of the inactivepart of the spark (the place where no cascade forms due to a low acceleration potential)by all the neighbouring discharges. The lifetime of a single spark is very short. On theother hand, an inactive region is continuously heated by all the neighbouring sparks. Evenwhen one of them dies, the temperature is still high enough (high ion density) to prevent .4. Drift model v dr ≈ πR pc P P βρ P ◦ ◦ , (3.42)where R pc is the actual polar cap size, P ◦ is the characteristic spacing between subpulsesin the pulse longitude, P is the period at which a pattern of subpulses crosses the pulsewindow (in units of the pulsar period), β is the impact angle, and ρ is the opening angle. Figure 3.16:
Top view of a polar cap region of an aligned pulsar. Small circles represent sparks,while the red line corresponds to the line of sight. If we neglect the transition from a non-dipolarstructure of the magnetic field on the stellar surface to a dipolar structure in the region where radioemission is produced, we can assume that the observed subpulse separation P ◦ also describes sparkseparation (cid:37) s (angular separation between the adjacent sparks on the polar cap). In such an approximation the assumption that only half of the spark is active can bewritten as P ◦ ◦ ≈ h ⊥ πR pc βρ . (3.43)Finally, we can define the observed drift velocity of aligned pulsars as v dr ≈ h ⊥ P P . (3.44)0 Chapter 3. Partially Screened Gap
Most observed pulsars are non-aligned rotators. It is very common to apply the carouselmodel to interpret observations of the drifting subpulses of non-aligned pulsars (Ruder-man and Sutherland, 1975). Despite the fact that the carousel model can explain someproperties of subpulses (for example the change in intensity), we believe that this modelis not suitable for describing the spark’s behaviour on the polar cap. There is no phys-ical reason for a spark to circulate around the magnetic axis. The circulation in alignedpulsars is caused by a lack of coronation with respect to the rotation axis. For non-alignedpulsars, the co-rotation velocity in the polar cap region has more or less the same direc-tion: what is more, if we assume circulation around the magnetic axis we will get plasmawith a velocity that is higher than the co-rotational velocity, which is difficult to explainin a region where the charge density is lower than the co-rotational density.As in our model, the drift is caused by a lack of charge in IAR, thus the plasma shoulddrift in approximately the same direction, i.e. in the direction opposite to the co-rotationvelocity. We believe that the change in subpulse intensity is caused by the observationof a different part of a spark and/or different conditions across the polar cap at whichthe spark is formed (e.g. magnetic field strength, curvature of the magnetic field lines,background photon flux).For pulsars with a relatively high inclination angle α we can calculate the drift velocityusing the following approximation v dr ≈ R pc WW β P P P ◦ W ≈ R pc P P P ◦ W β , (3.45)where W is the profile width and W β ≈ W/ (cid:114) − (cid:16) βρ (cid:17) is the profile width calculatedassuming β = 0 (see Figure 3.17). Using the assumption that only half of the spark isactive, we can write that P ◦ W ≈ h ⊥ R pc WW β −→ P ◦ W β ≈ h ⊥ R pc , (3.46)and the drift velocity v dr = 2 h ⊥ P P . (3.47)The spark half-width can be calculated using Equation 3.46 as follows h ⊥ = R pc P ◦ W β . (3.48) .4. Drift model Figure 3.17:
Top view of the polar cap region in the case of a non-aligned pulsar. Small circlesrepresent sparks, the red line corresponds to the line of sight. In general, the observed subpulseseparation P ◦ does not describe the actual spark separation (cid:37) s (the angular separation between theadjacent sparks on the polar cap). In order to calculate the distance between the sparks we use anapproximation from Equation 3.45. In our model the drift is caused by a lack of charge in IAR, thus we can write the equationfor the drift velocity as follows v ⊥ = v dr = c ∆E × B B , (3.49)where ∆ E is the electric field caused by the difference of an actual charge density fromthe Goldreich-Julian co-rotational density. We can a use calculation of the circulation ofan electric field, Equations 3.20 and 3.26, to find the dependence of the drift velocity onthe screening factor: v dr = c E θ B r B r = 4 πηh ⊥ cos αP . (3.50)Finally, by using Equations 3.47 and 3.50 we can find the dependence of the screeningfactor on the observed drift parameters η = 12 πP cos α . (3.51) The key parameters in the above calculations are the pulse width W (or W β ), the char-acteristic spacing between subpulses P ◦ , and the period at which a pattern of subpulses2 Chapter 3. Partially Screened Gap crosses the pulse window P . Of these three only P is easy to apply, both W and P ◦ need serious study before they can be used.In general, the profile width depends on the frequency at which we observe the pulsar,and most normal pulsars show a systematic increase in pulse width and the separation ofprofile components when observed at lower frequencies. The model known as radius-to-frequency mapping explains this effect as a direct consequence of the emission at higherfrequencies being produced closer to the neutron star surface than at lower frequencies. Forthis reason both the pulse width and the spacing between subpulses should be measuredat the same frequency. Note that P is not affected by this effect since its determinationinvolves analyses of many pulses and does not depend on the pulse width. The observedpulse width W , measured in longitude of rotation, can be calculated by applying simplespherical geometry (Gil et al., 1984):sin W ( ρ/ − sin ( β/ α sin ( α + β ) . (3.52)In the above calculations we are using the W β ≈ W/ (cid:114) − (cid:16) βρ (cid:17) approximation, where W β is the pulse width calculated assuming β = 0. In the first approximation we canassume that W β corresponds to the distance 2 R pc at the polar cap which, is valid fornon-aligned pulsars with a relatively high inclination angle. A more accurate value can becalculated using formulas presented in Gil et al. (1984). The running polar coordinatesalong the line of sight trajectory can be expressed in the form ρ ( ϕ ) = 2 arcsin (cid:32)(cid:114) sin ϕ α sin ( α + β ) + sin β (cid:33) , (3.53) σ ( ϕ ) = arctan (cid:18) sin ϕ sin α sin ( α + β )cos ( α + β ) − cos α cos ρ ( ϕ ) (cid:19) . (3.54)In numerical calculations of σ ( ϕ ) it is convenient to use the “atan2” function whichtakes into account the signs of both components and places the angle in the correct quad-rant (see the footnote on page 35). Figure 3.18 presents the geometry of the emissionregion for pulsars with available radio observations of the subpulse drift and X-ray obser-vations of the hot spot. By knowing the actual polar cap radius R pc we can determinethe transverse size of the region responsible for the generation of plasma clouds in IAR(the spark half-width). .4. Drift model Figure 3.18:
Top view of the polar cap region of pulsars with radio drift observations and X-rayhot spot radiation. Red lines correspond to the line of sight while green dashed lines correspond tothe theoretical lines of sight calculated with an assumption that β = 0 ◦ . The geometry of pulsarscan be found in Table 1.3. In our model the motion of sparks and the progressively different positions of theassociated plasma columns are responsible for the observed drift of subpulses. For somepulsars it is possible to measure directly the subpulse separation using a single pulse. Inmost calculations it is assumed that the observed subpulses correspond to the adjacent4
Chapter 3. Partially Screened Gap sparks. In general, this is not necessarily true. The distribution of sparks on the polar capis unknown and it is very likely that the line of sight does not cross the adjacent sparksbut it omits some sparks in between. Therefore, the observed value of P ◦ should beconsidered rather as an upper limit for spark separation. Furthermore, for many pulsarsthe observed value P ◦ > W , which means that it is not related to any structure at thepolar cap but that it corresponds to some other periodicity. We can use Equations 3.39and 3.51 to calculate the spark half-width as follows h ⊥ = 26 . B . + 0 . P P (cid:112) | cos α | B . (3.55)Finally, using Equation 3.46 we can determine the predicted value of the subpulseseparation ˜ P ◦ ≈ . B . + 0 . P P (cid:112) | cos α | B R pc W β . (3.56) The spin-down energy loss is L SD = 3 . × ˙ P − P . (3.57)We can use Equations 3.26, 3.48 and 3.51 to calculate the dependence of the acceler-ation potential drop on the parameters of drifting subpulses:∆ V ≈ . × (cid:32) ˙ P − P (cid:33) . P (cid:18) P ◦ W β (cid:19) . (3.58)The power of heating by backstreaming particles can be calculated as follows L heat = ηn GJ c (∆ V e ) πR . (3.59)The number density of the Goldreich-Julian co-rotational charge can be calculatedusing n GJ = Ω · B s πce , (3.60)where Ω = 2 π/P is an angular velocity, B s = bB d is the surface magnetic field, b = R dp /R pc , B d = 2 . × (cid:113) P ˙ P − G, and R dp = (cid:112) πR / ( cP ) ≈ . × P − . .The Goldreich-Julian density in terms of observed parameters can be written as n GJ = 2 . × (cid:16) P − ˙ P − (cid:17) / cos αR . (3.61)Finally, using Equations 3.51, 3.58 and 3.61 we can estimate the dependence of the .4. Drift model L heat = L X = 6 × (cid:32) ˙ P − P (cid:33) (cid:18) P P ◦ W β (cid:19) . (3.62)The heating efficiency by the backstreaming particles can be calculated as ξ heat = L heat L SD = 0 . (cid:18) P P ◦ W β (cid:19) . (3.63) In the PSG-on mode the bulk of energy is transferred to the iron ions which shield theacceleration potential drop. Similar as for the backstreaming particles, we can estimatethe power of ion acceleration as L ion = (1 − η ) n GJ c (∆ V q ion ) πR , (3.64)where q ion = 26 e = 1 . × − erg . cm . is the ion charge. Using the same approachas for electron, we can calculate the dependence of energy transformed to the ions persecond on the parameters of the radio observations as follows L ion = 9 . × (1 − η ) ˙ P − P P (cid:18) P ◦ W β (cid:19) cos α. (3.65)It is clearly visible that if the screening factor is low η (cid:28)
1, most of the energy in IARis transferred to the iron ions. Using Equations 3.59 and 3.64 we can show that L ion L heat = 26 (1 − η ) η ≈ η . (3.66)Finally, the ion acceleration efficiency can be calculated as ξ ion = L ion L SD ≈
25 1 P (cid:18) P ◦ W β (cid:19) cos α. (3.67)It may seem that ion luminosity exceeds the spin-down luminosity, but note that both P > P ◦ < W β . The predicted values of heating efficiency ξ heat and ion accelerationefficiency ξ ion are presented in the next section. In this section we confront the values of the subpulse drift and X-ray radiation as es-timated by other authors with predicted values estimated using the approach presentedabove. In Table 3.2, alongside our predicted value of ˜ P ◦ we present two other estimates:(1) the subpulse separation estimated using the carousel model developed by Ruderman6 Chapter 3. Partially Screened Gap and Sutherland (1975), P ◦ , RS ; (2) the subpulse separation found using the analysis ofthe Longitude-Resolved Fluctuation Spectrum (Backer, 1970) and the integrated Two-Dimensional Fluctuation Spectrum (Edwards and Stappers, 2003), performed by Wel-tevrede et al. (2006), P ◦ , W . We have found that the subpulse separation estimated usingthe fluctuations spectra is overestimated (in most cases P ◦ , W > W ). By definition P ◦ should correspond to the structure within a single pulse, thus if the geometry is not ex-treme ( (cid:37) > ◦ ) it should comply with P ◦ ≤ W . For this specific sample of pulsars P ◦ , W should not be interpreted as the actual subpulse separation. On the other hand, ˜ P ◦ is ingood agreement with P ◦ , RS . The predicted values ˜ P ◦ for B0834+06 and B0943+10 sug-gest that P ◦ , RS for those pulsars could be overestimated due to the aliasing phenomenon( P ◦ , RS ≈ P ◦ ). For B1929+10 we do not list P ◦ , RS as its value presented in Gil et al.(2008) does not comply with the P ◦ , RS ≤ W condition. We believe that the overestim-ated value of P ◦ , RS for B1929+10 is a result of using the fluctuations spectra presented inWeltevrede et al. (2006) to calculate the number of sparks in the carousel model. Notethat the coincidence of ˜ P ◦ and P ◦ , RS is yet to be clarified, as in our model there is nophysical reason for sparks to circulate around the magnetic axis. In fact, the PSG modelassumes the non-dipolar structure of the magnetic field lines in the gap region and theactual position of the polar cap is not necessarily coincident with the global dipole (e.g.see Figures 2.6, 2.18, 2.24). Table 3.2:
Details of a subpulse drift for pulsars with X-ray hot spot radiation. The individualcolumns are as follows: (1) Pulsar name, (2) Predicted characteristic spacing between subpulses inthe pulse longitude, ˜ P ◦ ; (3) Spacing between subpulses, found in the literature, estimated using thecarousel model, P ◦ , RS ; (4) Spacing between subpulses estimated using fluctuations spectra, P ◦ , W ;(5) Period at which a pattern of subpulses crosses the pulse window (in units of the pulsar period), P ; (6) Number of sparks estimated using the carousel model, N ; (7) Profile width at 10%, W ; (8)Profile width calculated assuming β = 0 , W β ; (9) Angular width of the observed region on the polarcap (cid:37) (see Figure 3.18); (10) References; (11) Number of the pulsar. Name ˜ P ◦ P ◦ , RS P ◦ , W P N W W β (cid:37) Ref. No. (deg) (deg) (deg) ( P ) (deg) (deg) (deg) B0628–28 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. Drift model P ◦ . Note that we consider only pulsars with a visible hot spot com-ponent since only for these pulsars we can estimate the size of the polar cap. The lowvalue of the estimated screening factor ( η (cid:28)
1) suggests that when the drift is visible, thepulsar operates in the PSG-on mode. If the pulsar operates in the PSG-off mode, η ≈ Table 3.3:
Derived parameters of PSG for pulsars with available radio observations of the subpulsedrift and X-ray hot spot radiation. The individual columns are as follows: (1) Pulsar name, (2)Screening factor, η ; (3) Predicted heating efficiency, ξ heat ; (4) Observed bolometric efficiency, ξ BB ;(5) Predicted ion acceleration efficiency, ξ ion ; (6) Surface temperature, T s ; (7) Strength of the surfacemagnetic field, B s ; (8) Observed polar cap radius, R pc ; (9) Estimated spark half-width, h ⊥ ; (10)Number of the pulsar. T s , R pc , b were chosen to fit σ uncertainty. Note that in the calculations ˜ P ◦ was used. Name η log ξ heat log ξ BB log ξ ion T s B s R pc h ⊥ No. (radio) (x − ray) (ions) (cid:0) K (cid:1) (cid:0) G (cid:1) (m) (m) B0628–28 0 . − . − . − .
43 2 . . . . − . − . − .
35 3 . . . . − . − . − .
75 3 . . . . − . − . − .
62 2 . . . . − . − . − .
81 3 . . . . − . − . − .
98 4 . . . Chapter 3. Partially Screened Gap Chapter 4Cascade simulation
In this chapter we present the approach of calculating the pair cascades developed by Medinand Lai (2010) which has been applied to cases with non-dipolar structure of magneticfield. The original approach was adapted to perform full three-dimensional calculationsand extended with effects that may have a greater importance for non-dipolar configurationof surface magnetic fields (e.g. aberration). Additionally, to perform a thorough analysisof the Inverse Compton Scattering we present the detailed description of calculating theICS cross section originally developed by Gonthier et al. (2000).
Following the approach presented by
Medin and Lai (2010) we can divide the cascadesimulation into three parts: • propagation of the primary particle (including photon emission), • photon propagation in strong magnetic field (pair production, photon splitting), • propagation and photon emission of the secondary particles.We use the ”co-rotating” frame of reference (the frame which rotates with the star) totrack both photons and particles. In calculations we consider regions far inside the lightcylinder. Thus, following Medin and Lai (2010), we ignore any bending of the photonpath due to rotation of the star. Furthermore, we also ignore effects of general relativityon trajectories of photons and particles.Figure 4.1 presents a summary flowchart of the algorithm used to calculate the proper-ties of secondary plasma and the spectrum of radiation for a given structure of a neutron’sstar magnetic field and gap parameters. In this thesis the term ”secondary” refers to any newly created particle except for the primary particlesaccelerated in IAR, e.g. the third generation of electrons and positrons are all considered as ”secondary”particles. Chapter 4. Cascade simulationFigure 4.1:
Flowchart of algorithm used to calculate a cascade simulation. .1. Curvature Radiation As we have shown in Chapter 2, an ultrastrong surface magnetic field ( B s > G) isaccompanied by high curvature (curvature radius (cid:60) ≈ . − ∼ /γ ,where γ is a Lorentz factor of an emitting particle (for more details see Rybicki andLightman, 1979).We track the primary particle above the acceleration zone (the gap region) as it movesalong the magnetic field line. The length of the step ∆ s is chosen so as to achieve sufficientaccuracy even for large curvature of the magnetic field line, ∆ s ≈ . (cid:60) min , where (cid:60) min isthe minimum radius of curvature. The distribution of CR photon energy can be writtenas (see Equation 14.93 in Jackson, 1998)d N d (cid:15) = E(cid:15) CR √ π ˆ ∞ (cid:15)/(cid:15) CR K / ( t )d t, (4.1)where E = 4 πe γ / (cid:60) is the total energy radiated per revolution, (cid:15) CR = 3 γ (cid:126) c/ (2 (cid:60) ) isthe characteristic energy of curvature photons, and K / is the n = 5 / s , E ∆ s , can be written as E ∆ s = E ∆ s π (cid:60) . (4.2)Thus, by using Equations 4.1 and 4.2 we can write the formula for the distribution onCR photon energy after a particle passes length ∆ s d N d (cid:15) = ∆ s π (cid:60) √ e γ (cid:126) c ˆ ∞ (cid:15)/(cid:15) CR K / ( t )d t. (4.3)It is convenient to divide the spectrum of photon energy into discrete bins. Then, thenumber of photons in each energy bin can be calculated as N (cid:15) = ˆ (cid:15) i +∆ (cid:15)(cid:15) i dNd(cid:15) d (cid:15), (4.4)where (cid:15) i is the lowest energy for the i -th bin and ∆ (cid:15) is the energy bin width. Oursimulation uses 50 bins with an energy range of (cid:15) = 4 × − keV (soft X-ray) to (cid:15) = 4 × MeV (hard γ -rays).2 Chapter 4. Cascade simulation
Depending on the photon frequency the polarisation fraction of CR photons is between50% and 100% polarised parallel to the magnetic field (see Jackson, 1998, Rybicki andLightman, 1979). Therefore, using similar approach as Medin and Lai (2010) we randomlyassign the polarisation in the ratio of one photon ⊥ -polarised to every seven (cid:107) -polarisedphotons, which corresponds to 75% parallel polarisation. To explain some of the properties of pulsars and their surroundings (e.g. nebulae radi-ation), large magnetospheric plasma densities exceeding the Goldreich-Julian density (seeEquation 3.7) by many orders of magnitude are required. In order to simulate the processof generation of such a dense plasma it is necessary to check the conditions of photondecay into electron-positron pairs.A photon with energy E γ > mc and propagating with a nonzero angle Ψ with respectto an external magnetic field can be absorbed by the field and, as a result an electron-positron pair is created. The concurrent process is photon splitting γ → γγ , which mayoccur even if the photon energy is below the pair creation threshold ( E γ < mc ).In the cascade simulation the photon is emitted (or scattered in the case of ICS) frompoint P ph in a direction tangent to the magnetic field line ∆ s (cid:107) . The direction vector iscalculated as the value of the magnetic field at the point of photon creation (see Equations2.1, 2.2 and 2.7) normalised so that its length is equal to the desired step ∆ s (cid:107) = B ∆ s/B .However, the direction of the magnetic field at the point of photon emission does not takeinto account the randomness of the emission direction due to the relativistic beamingeffect. In Section 4.2.1 we describe a procedure to include the beaming effect in theemission process which alters ∆ s (cid:107) → ∆s ph . Finally, we can write that at the point ofcurvature emission photons are created with energy (cid:15) ph , polarisation (cid:107) or ⊥ , weightingfactor N (cid:15) (number of photons), and with both optical depths (for pair production τ andfor photon splitting τ sp ) set to zero. Since we neglect any banding of the photon path weassume that from the point of emission it travels in a straight line. In each following stepthe photon travels a distance ∆s ph . In the co-rotating frame of reference in every step weneed to take into the account aberration due to pulsar rotation. In order to do so, in everystep we alter the photon position according to the procedure described in Section 4.2.2.As stated by Medin and Lai (2010) we can calculate the change in the pair productionoptical depth , ∆ τ , and in the photon splitting optical depth ∆ τ sp , at the new positionas: ∆ τ (cid:39) ∆ s ph R (cid:107) , ⊥ , (4.5)∆ τ sp (cid:39) ∆ s ph R sp (cid:107) , ⊥ , (4.6)where R (cid:107) , ⊥ and R sp (cid:107) , ⊥ are the attenuation coefficients for (cid:107) or ⊥ polarised photons for pair .2. Photon propagation Due to relativistic beaming the emission direction should be modified by an additionalemission angle of order ∼ /γ . We use the following steps to include the beaming effectin our simulation (see Figure 4.2).(I) The first step is rotation of the xyz frame of reference in order to align the z -axiswith ∆s (cid:107) . In our calculations we used rotation by angle ς y around the y -axis, R y ( ς y ), androtation by angle ς x around the x -axis, R x ( ς x ). The final rotation matrix can be writtenas R yx = R y ( ς y ) R x ( ς x ) = cos ς y sin ς x sin ς y sin ς y cos ς x α − sin α − sin ς y cos ς y sin ς x cos ς y cos ς x . (4.7) magnetic field lineline of nodes Figure 4.2:
Relativistic beaming effect of photon emission (for both CR and ICS). In the simulationwe include the beaming effect by performing three steps: (I) rotation of the xyz frame of referencein order to align the z -axis with ∆s (cid:107) , (II) transformation of the step vector from a Cartesian toa spherical system of coordinates and alteration of the θ and φ components with random values /γ cos Λ and Π , respectively, (III) transformation of the step vector from a spherical to a Cartesiansystem of coordinates and rotation back to the original system of reference. Note that after thesesteps we get a new vector ∆s ph inclined to the primary one, ∆s ph , at an angle ranging from to /γ . The Euler angles for rotations can be calculated as4
Chapter 4. Cascade simulation ς x = atan2 ( s y , s z ) ,ς y = arctan (cid:16) − s x s z cos ς x (cid:17) if s z (cid:54) = 0arctan (cid:16) − s x s y sin ς x (cid:17) if s y (cid:54) = 0 π if s x = 0 and s y = 0 . (4.8)Note that in order to increase readability, the ∆ symbol and (cid:107) index were discarded(e.g. s x = ∆ s (cid:107) ,x ).(II) The second step is the transformation of the step vector’s coordinates in thedouble rotated frame of reference ∆s (cid:48)(cid:48) ph = (cid:0) s (cid:48)(cid:48) x , s (cid:48)(cid:48) y , s (cid:48)(cid:48) z (cid:1) to spherical system of coordinatesand alteration of the θ and φ components as follows s (cid:48)(cid:48) r = (cid:113) s (cid:48)(cid:48) x + s (cid:48)(cid:48) y + s (cid:48)(cid:48) z ,s (cid:48)(cid:48) θ = arccos (cid:32) s (cid:48)(cid:48) z (cid:112) s (cid:48)(cid:48) x + s (cid:48)(cid:48) y + s (cid:48)(cid:48) z (cid:33) + 1 γ cos Λ ,s (cid:48)(cid:48) φ = arctan (cid:18) s (cid:48)(cid:48) y s (cid:48)(cid:48) x (cid:19) + Π , (4.9)where Λ and Π are random angles between 0 and 2 π . The inverse tangent denoted in the φ -coordinate must be suitably defined by taking into account the correct quadrant (seethe “atan2” description in the footnote on page 35).(III) The last step is the transformation of vector components to the Cartesian sys-tem of coordinates, s (cid:48)(cid:48) ph = (cid:2) s (cid:48)(cid:48) r sin ( s (cid:48)(cid:48) θ ) cos (cid:0) s (cid:48)(cid:48) φ (cid:1) , s (cid:48)(cid:48) r sin ( s (cid:48)(cid:48) θ ) sin (cid:0) s (cid:48)(cid:48) φ (cid:1) , s (cid:48)(cid:48) r cos ( s (cid:48)(cid:48) θ ) (cid:3) and rotationback to the original coordinate system ∆s ph = ( R yx ) − s (cid:48)(cid:48) ph .The rotation matrix of this transformation can be written as( R yx ) − = ( R yx ) T = cos ς y − sin ς y sin ς x sin ς y cos ς x sin ς x cos ς y sin ς y cos ς x − sin ς x cos ς x cos ς y . (4.10) Note that in our frame of reference (co-rotating with a star) the path of the photonshould be curved (see Harding et al. 1978). In the dipolar case the angular deviationincreases approximately as s ph Ω /c = s ph /R LC . When the configuration of magnetic fieldin non-dipolar inclusion of an aberration is even more important. Therefore, the locationof photon decay should be modified to include the growth of the photon-magnetic fieldintersection angle.In our simulation we include the aberration effect by alteration of photon position P ph in every step ∆s ph (see Figure 4.3). .2. Photon propagation magnetic field line Figure 4.3:
Aberration due to pulsar rotation. We use the following procedure to includethe aberration effect: (I) rotation around the y -axis to align Ω with µ , (II) rotation by angle ω = 2 π ∆ s bm / ( cP ) around the z -axis (which reflects the pulsar rotation), (III) rotation back tothe original frame of reference (in which µ is aligned with the z -axis). We use the three-step procedure to alter the photon position.(I) Rotation of the xyz frame of reference around the y -axis by angle α , P (cid:48) ph = R y ( α ) P ph . Note that here α refers to the inclination of the magnetic axis withrespect to the rotation axis and we assume that the pulsar’s angular velocity vector Ω lies in the xz -plane. The rotation matrix of this transformation can be written as R y ( α ) = cos α α − sin α α . (4.11)(II) After step (I) the z -axis is aligned to Ω, and in order to include the rotation of thepulsar we need to again rotate the frame of reverence by angle ω = 2 π ∆ s bm / ( cP ) aroundthe z -axis, P (cid:48)(cid:48) ph = R z ( ω ) P (cid:48) ph . We use the following rotation matrix R z ( ω ) = cos ω − sin ω ω cos ω
00 0 1 . (4.12)6 Chapter 4. Cascade simulation (III) The final step is a rotation back to the original frame of reference, P (cid:48)(cid:48)(cid:48) ph = ( R y ( α )) − P (cid:48)(cid:48) ph , using the following rotation matrix( R y ( α )) − = ( R y ( α )) T = cos α − sin α α α . (4.13) The pair production attenuation coefficient can be written as (Medin and Lai, 2010) R (cid:107) , ⊥ = R (cid:48) sin Ψ , (4.14)where R (cid:48) is the attenuation coefficient in the frame where the photon propagates per-pendicular to the local magnetic field (the so-called ”perpendicular” frame), Ψ is theintersection angle between the propagation direction of the photon and the local mag-netic field. To increase readability we suppress the subscripts (cid:107) and ⊥ , but R (cid:48) has to becalculated for both polarisations separately.As stated by Medin and Lai (2010) the total attenuation coefficient for pair productioncan be calculated as R (cid:48) = (cid:80) jk R (cid:48) j,k , where R (cid:48) j,k is the attenuation coefficient for theprocess producing an electron in Landau level j and a positron in Landau level k . Forthe electron-positron pair the sum is taken over all possible states ( j and k ). Note thatproduction of electron-positron pairs is symmetric R (cid:48) jk = R (cid:48) kj . Thus, to represent the paircreation probability in either the ( jk ) or ( kj ) state we will use R (cid:48) jk . For a given Landaulevels j and k , the pair production threshold condition is (Medin and Lai, 2010) E (cid:48) γ > E (cid:48) j + E (cid:48) k , (4.15)where E (cid:48) γ = E γ sin Ψ is the photon energy in the perpendicular frame and E (cid:48) n = mc √ (cid:15) B n is the minimum energy of a particle (electron or positron) in LandauLevel n . This condition can be written in a dimensionless form as x = E (cid:48) γ mc = E γ mc sin Ψ > (cid:104)(cid:112) (cid:15) B j + (cid:112) (cid:15) B k (cid:105) , (4.16)where (cid:15) B = (cid:126) eB/ ( mc ) is the cyclotron energy of a particle (electron or positron) inmagnetic field B in units of mc .The first nonzero pair production attenuation coefficients for both polarisations ( ⊥ and (cid:107) ) are (Daugherty and Harding, 1983; Medin and Lai, 2010) R (cid:48)(cid:107) , = 12 a (cid:15) B x √ x − e − x /(cid:15) B , x > ( x = 1) , (4.17) .2. Photon propagation R (cid:48)⊥ , = 2 × a (cid:15) B x x − (cid:15) B (cid:113) x − − (cid:15) B + (cid:15) B x e − x /(cid:15) B , x > (cid:16) x = (cid:16) (cid:112) (cid:15) B (cid:17) / (cid:17) , (4.18) R (cid:48)(cid:107) , = 2 × a (cid:15) B − (cid:15) B x (cid:113) x − − (cid:15) B + (cid:15) B x e − x /(cid:15) B , x > x , (4.19) R (cid:48)(cid:107) , = 2 × a x (cid:15) B (cid:15) B − (cid:15) B x (cid:113) x − − (cid:15) B + (cid:15) B x e − x /(cid:15) B , x > (cid:16) x = (cid:16) (cid:112) (cid:15) B (cid:17) / (cid:17) , (4.20) R (cid:48)⊥ , = 2 × a x − (cid:15) B (cid:113) x − − (cid:15) B + (cid:15) B x e − x /(cid:15) B , x > x , (4.21)where a is the Bohr radius (let us note that R (cid:48)⊥ , = 0). In the above equations for allchannels except 00 the pair production attenuation coefficients are multiplied by a factorof two (see the text above Equation 4.15).The pair production optical depth is defined as (Medin and Lai, 2010): τ = ˆ s ph R ( s )d s = ˆ s ph R (cid:48) ( s ) sin Ψd s. (4.22)We can assume Ψ (cid:28)
1, because all high-energy photons ( x >
1) will produce pairsmuch earlier than Ψ reaches a value near unity. In this limit sin Ψ (cid:39) s ph / (cid:60) , so the relationbetween x and s ph can be expressed by x (cid:39) s ph (cid:60) E γ mc . (4.23)Equation 4.22 can be rewritten as τ = τ + τ (cid:107) , + τ ⊥ , + ... ; τ = ˆ s s R (cid:107) , d s, τ (cid:107) , = ˆ s s (cid:0) R (cid:107) , + R (cid:107) , (cid:1) d s, τ ⊥ , = ˆ s s R ⊥ , d s, (4.24)where s and s are distances which the photon should pass in order to have energy x and x , respectively (in the perpendicular frame of reference). Let us note that s , s and s are of the same order, and if s < s the attenuation coefficient is zero.8 Chapter 4. Cascade simulation
The pair production optical depth to reach the second threshold is ˆ s s d sR (cid:107) , ( s ) = (cid:15) B a (cid:18) mc E γ (cid:19) (cid:60) ˆ x x d xx √ x − e − x /(cid:15) B , (4.25)where s is the distance travelled by the photon to reach the threshold x ≡
1, and s isthe distance travelled by the photon to reach the second threshold x ≡ (1 + √ (cid:15) B ) / Figure 4.4:
Panel (a) presents the dependence of the pair production optical depth on the magneticfield strength ( β q = B/B q ). Panel (b) presents the dependence of the optical depth on photon energyin the perpendicular frame of reference ( x = (cid:15) sin Ψ / (cid:0) mc (cid:1) ). On both panels the photon energy is (cid:15) = 500 MeV , while panel (b) was obtained for magnetic field strength β q = 1 . As was shown in the previous section (see Figure 4.4) for strong magnetic fields (e.g. β q (cid:38) . τ , τ (cid:107) , , and τ ⊥ , are much larger than one. Therefore, the pair productionprocess takes place according to two scenarios (see also Medin and Lai, 2010). If β q (cid:38) . n (cid:46) β q (cid:46) .
2, the photons will travellonger distances to be absorbed and the created pairs will be in the higher Landau levels.Thus, for strong magnetic fields ( β q (cid:38) .
2) the photon mean free path can be approx-imated as l ph ≈ s = (cid:60) mc E γ , (4.26)while for relatively weak magnetic fields ( β q (cid:46) .
2) we can use the asymptotic approxim- .2. Photon propagation l ph ≈ . e / (cid:126) c ) (cid:126) mc B q B sin Ψ exp (cid:18) χ (cid:19) , (4.27) χ ≡ E γ mc B sin Ψ B q ( χ (cid:28) . (4.28) In our calculations we include photon splitting by following the approach presented byMedin and Lai (2010). Since only the ⊥→(cid:107)(cid:107) process is allowed, for (cid:107) -polarised photons thephoton splitting attenuation coefficient is zero R sp (cid:107) = 0 (Adler, 1971, Usov, 2002, Baringand Harding, 2001). To calculate the splitting attenuation coefficient in the perpendicularframe for ⊥ -polarised photons we use the formula adopted from the numerical calculationof Baring and Harding (1997) : R (cid:48) sp ⊥ (cid:39) α f π (cid:0) (cid:1) (2 x ) (cid:15) B (cid:2) (cid:15) B exp ( − . x ) + 0 . (cid:3) (cid:2) . (cid:15) B exp ( − . x ) + 20 (cid:3) . (4.29)For photon energies x ≤ β q = 1 by less than 30%, while at both β q ≤ . β q (cid:29) R (cid:48) sp ⊥ dropsrapidly with the magnetic field strength for β q <
1, thus photon splitting is unimportantfor β q (cid:46) . Figure 4.5:
Dependence of the photon-splitting attenuation coefficient on the energy of the photonin the perpendicular frame ( x = (cid:15) sin Ψ / (cid:0) mc (cid:1) , vertical axis) and on the strength of the magneticfield ( β q = B/B q , horizontal axis). Chapter 4. Cascade simulation
As noted by Medin and Lai (2010), even though the photon splitting attenuation coef-ficient above the first threshold ( x > x ) is much smaller than for pair production (seeFigure 4.6), in ultrastrong magnetic fields ( β q (cid:38) .
5) the ⊥ -polarised photons split beforereaching the first threshold (see Figure 4.7). On the other hand, the (cid:107) -polarised photonsproduce pairs in the zeroth Landau level. Figure 4.6:
Attenuation coefficients of pair production and photon splitting in the perpendicularframe of reference. Panel (a) was obtained using photon energy E γ = 10 MeV and magnetic fieldstrength B = B q = 4 . × G ( β q = 1 ). Panel (b) presents calculations for photon energy E γ = 10 MeV and magnetic field strength B = 2 . × G ( β q = 5 . ). Figure 4.7:
Optical depth for pair production and photon splitting for ⊥ -polarised photons. Panel(a) presents results for E γ = 10 MeV and B = B q = 4 . × G ( β q = 1 ), while panel (b) wasobtained using the same photon energy but a stronger magnetic field B = 2 . × G ( β q = 5 . ). If β q = 1 the photon creates an electron-positron pair, while in an ultrastrong magnetic field ( β q = 5 . )the photon splits before it reaches the first threshold, x = x . .2. Photon propagation Following the approach presented by Medin and Lai (2010) whenever τ ≥ x = x for (cid:107) -polarised photons and x = x for ⊥ -polarised photons), the photon is turned into an electron-positron pair. Whereasif τ sp ≥ (cid:107) -polarised photon is created with an energy 0 . (cid:15) ph and weighting factor 2∆ N (cid:15) . We assume that the newly created photon travels in the samedirection as the parent photon, ∆s ph . Note that the photon should split with probability1 − e − τ , but as shown by Medin and Lai (2010) for cascade results this effect is negligible.For β q (cid:46) .
1, the particles are produced in high Landau levels with energy equal tohalf of the photon energy each (see Daugherty and Harding, 1983). In our calculationswe assume that the newly created particles (electron-positron pairs) travel in the samedirection as the photon. When β q (cid:38) .
1, on the other hand, we choose the maximumallowed values of j and k for the newly created electron and positron. Note that for β q (cid:38) . Figure 4.8:
Distribution of CR photons produced by a single primary particle for a dipolar (blueline) and non-dipolar (red line) structure of the magnetic field. The minimum radius of curvature inthee dipolar case is (cid:60) min ≈ , while in the non-dipolar case (cid:60) min ≈ . In both cases the radiationwas calculated up to a distance of D = 100 R , and with an initial Lorentz factor of the primaryparticle γ c = 3 . × . Chapter 4. Cascade simulation
Formation of the peaks is caused by the fact that the particle passes regions with threedifferent values of curvature: (I) just above the stellar surface, z ≈ z ≈ . z (cid:38) . R , with approximately dipolar curvature (seeFigure 2.11). Hence, the spectrum is a sum of radiation generated in a highly non-dipolarmagnetic field (high energetic and soft γ -rays) and with radiation at higher altitudeswhere the magnetic field is dipolar (X-rays). The primary particle loses about 63% and1% of its initial energy in the non-dipolar and dipolar case, respectively. As can be seenfrom the Figure, to get high emission of CR photons and, thus, a significant density ofsecondary plasma, a non-dipolar structure of the magnetic field is required.The high energetic photons produced in a strongly non-dipolar magnetic field willeither split or create electron-positron pairs. Figure 4.9 presents the distribution of particleenergy created by CR photons. Note that for β q (cid:46) . Figure 4.9:
Distribution of particle energy created by CR photons calculated for a non-dipolarstructure of the magnetic field. For this specific magnetic field configuration and initial parameters(see the caption of Figure 4.8) the secondary plasma multiplicity is M sec ≈ × . Note that thisresult does not include Synchrotron Radiation and the actual energies of the created pairs are loweras they lose their transverse momenta (see Section 4.3). When pairs (electrons and positrons) are created in high Landau Levels they radiateaway their transverse momentum through Synchrotron Radiation (SR). The secondarypositron (or electron) is created with energy γmc and pitch angle Ψ, which corresponds .3. Synchrotron Radiation n . Following Medin and Lai (2010) we choose theframe in which the particle has no momentum along the direction of external magneticfield. In such a frame of reference the particle propagates in a circular motion transverseto the magnetic field (the so-called ”circular” frame). The relation of the energy of thenewly created particle in the circular frame of reference ( E ⊥ = γ ⊥ mc ) with the particleenergy in the co-rotating frame can be written as (Medin and Lai, 2010) γ ⊥ = (cid:113) γ sin Ψ + cos Ψ = (cid:112) (cid:15) B n. (4.30)The power of synchrotron emission, P SR , can calculated as follows P SR = 2 e c (cid:0) γ ⊥ − (cid:1) c (cid:15) B , (4.31)In the circular frame E ⊥ , is radiated away through synchrotron emission after a particletravels a distance l SRp ≈ (cid:12)(cid:12)(cid:12)(cid:12) E ⊥ P SR c (cid:12)(cid:12)(cid:12)(cid:12) = γ ⊥ mc e c ( γ ⊥ − c (cid:15) B . (4.32)The particle (electron or positron) mean free path for SR is much shorter than forother relevant cascade processes (see Section 4.1 for Curvature Radiation, and Section4.4 for ICS). In fact, it is so short that in our calculations we assume that before movingfrom its initial position the particle loses all of its perpendicular momentum p ⊥ due to SR(see Daugherty and Harding, 1982; Medin and Lai, 2010). Once the particle reaches theground Landau level ( n = 0, p ⊥ = 0) its final energy can be calculate as γ (cid:107) = (cid:0) − β cos Ψ (cid:1) − / = γ/γ ⊥ , (4.33)here β = v/c = (cid:112) − /γ is the particle velocity in units of speed of light.Following the approach presented by Medin and Lai (2010), to simplify the simulationwe assume that in the circular frame synchrotron photons are emitted isotropically in theplane of motion such that there is no perpendicular velocity change of the particle (theLorentz factors γ and γ ⊥ decrease but γ (cid:107) is constant). Thus, the Equation 4.33 remainsvalid until the particle reaches the ground state. In order to simulate the full SR processthe following procedure was adopted: the particle Lorentz factor in the circular frame γ ⊥ drops from its initial value to γ ⊥ = 1 (i.e., n = 0) in a series of steps. Each step entailsemission of one synchrotron photon, with energy (cid:15) ⊥ depending on the current value of γ ⊥ .After the photon emission the energy of the particle is reduced by (cid:15) ⊥ , ∆ γ ⊥ = (cid:15) ⊥ /mc .Subsequently, the particle with reduced energy emits a photon with a new value of (cid:15) ⊥ .This process continues until the particle is at n = 0 Landau level. Depending on theparticle’s Landau level n , the SR photon energy (cid:15) ⊥ is chosen in one of three ways.04 Chapter 4. Cascade simulation (I) When the particle is created in a high Landau Level ( n ≥ N d t d (cid:15) ⊥ = √ π α f (cid:15) B (cid:15) ⊥ × (cid:34) f · F (cid:18) (cid:15) ⊥ f (cid:15) SR (cid:19) + (cid:18) (cid:15) ⊥ γ ⊥ mc (cid:19) G (cid:18) (cid:15) ⊥ f (cid:15) SR (cid:19)(cid:35) , (4.34)where (cid:15) SR = 32 γ ⊥ (cid:126) (cid:15) B (4.35)is the characteristic energy of the synchrotron photons, f = 1 − (cid:15) ⊥ / ( γ ⊥ mc ) is thefraction of the electron’s energy after photon emission, F ( x ) = x ´ ∞ x K / ( t ) d t , and G ( x ) = xK / ( x ). The functions K / and K / correspond to modified Bessel functionsof the second kind. The expression in Equation 4.34 differs from the classical synchrotronspectrum (e.g. Rybicki and Lightman, 1979) by a factor of f = 1 − (cid:15) ⊥ / ( γ ⊥ mc ) whichappears in several places in Equation 4.34 and by a term with the function G ( x ). Notethat in the classical expressions for the total radiation spectra these terms cancel out.However, as noted by Medin and Lai (2010) when the quantum effects are consideredthere is asymmetry between the perpendicular and parallel polarisations such that term G ( x ) remain.(II) If n = 2, the photon’s energy is either that required to lower the particle energyto its first excited state ( n = 1) or to the ground state ( n = 0). The probability of eachprocess depends on the local magnetic field strength. We use the simplified prescriptionbased on the results of Herold et al. (1982) to calculate the transition rates (see alsoHarding and Preece, 1987). If β q < (cid:15) ⊥ = mc (cid:0)(cid:112) β q − (cid:112) β q (cid:1) . If β q (cid:38) (cid:15) ⊥ = mc (cid:0)(cid:112) β q − (cid:1) ), with probability50% each.(III) When n = 1, the photon’s energy is chosen to lower the particle’s energy to itsground state, (cid:15) ⊥ = mc (cid:0)(cid:112) β q − (cid:1) . If after emission of SR photon the particle is notin the ground state, γ ⊥ is recalculated and a new energy of photon is chosen.The photon energy in the co-rotating frame can be calculated as (cid:15) = γ (cid:107) (cid:15) ⊥ . (4.36)The weighting factor of the emitted photon is the same as the secondary particle thatemitted it (∆ N (cid:15) ). In the circular frame the photon is emitted in a random directionperpendicular to the magnetic field. Hence, in the co-rotating frame the emission angle .3. Synchrotron Radiation (cid:115) γ ⊥ − γ ⊥ γ (cid:107) − , (4.37)where Π is a random number from 0 to 2 π . In our simulation we include this emissionangle by using the same approach as presented in Section 4.2.1, but as the maximumvalue we use Ψ instead of 1 /γ .The polarisation fraction of SR photons is the exact opposite of the CR case and itranges from 50% to 100% polarised perpendicular to the magnetic field. Following theapproach presented by Medin and Lai (2010) in our calculations the photon polarisationis randomly assign in the ratio of one (cid:107) to every seven ⊥ photons, which corresponds toa 75% perpendicular polarisation.Figure 4.10 presents the distribution of SR produced by a single secondary particle.To show the nature of the distribution, a relatively high pitch angle was used. Note thatwhen a particle is created at a distance where the magnetic field is relatively weak (e.g. β q = 10 − for γ = 10 ) then most of the energy is radiated in the range of 1 −
10 keV.Thus, we believe that if a strong enough instability forms (that increases the particle’spitch angle), the SR process could be responsible for the production of a non-thermalcomponent of the X-ray spectrum.
Figure 4.10:
Distribution of SR produced by a single secondary particle with Lorentz factor γ = 10 . We have assumed that the particle was created in a region where the magnetic fieldstrength was B = 4 . × ( β q = 10 − ) and with a pitch angle Ψ = 7 ◦ . For such a relatively highpitch angle the particle loses most of its energy ending with Lorentz factor γ end ≈ . Chapter 4. Cascade simulation
Figure 4.11 presents the final spectrum produced by a single primary particle withan initial Lorentz factor of γ c = 3 . × for a non-dipolar configuration of the surfacemagnetic field of PSR J0633+1746 (see Section 2.4.2). Due to CR the particle loses about68% of its initial energy (∆ (cid:15) = 2 . × mc ), which is radiated mainly in close vicinityof a neutron star, where curvature of the magnetic field is the highest. As the γ -photonspropagate they will split (only if the magnetic field is strong enough) and eventually mostphotons will be absorbed by the magnetic field - as a result electron-positron pairs emerge.These pairs radiate away their transverse momenta through SR, producing mainly X-rayphotons (at larger distances) and only a few γ -photons (in a strong magnetic field justabove the stellar surface). Note that at the end (after pair production) only 14% of theprimary particle’s energy (∆ (cid:15) ph = 4 × mc ) is converted into photons and the bulkof its energy, 54% (∆ (cid:15) pairs = 1 . × mc ), is allocated into secondary plasma. Themultiplicity for this specific simulation is of the order M sec = 10 . Note that we use M sec to describe the multiplicity of secondary plasma in contrast to M pr which describesparticle multiplicity in the IAR. Figure 4.11:
Final photon distribution produced by a single primary particle. The blue line corres-ponds to the initial CR photons distribution for a non-dipolar structure of the magnetic field, whilethe red line presents the final distribution with the inclusion of photon splitting, pair production andSR.
Figure 4.12 presents the distribution of particle energy created by CR photons butwith the inclusion of SR emission (red line). Note that synchrotron emission both lowersthe particle energy (after SR maximum at γ ≈ −
8, while without SR at γ ≈ − M sec ≈ . .4. Inverse Compton Scattering Figure 4.12:
Distribution of particle energy created by CR photons calculated for a non-dipolarstructure of the magnetic field. For this specific magnetic field configuration and initial parameters(see the caption of Figure 4.8) the secondary plasma multiplicity is M sec ≈ . Note that this resultdoes not include Synchrotron Radiation and the actual energies of the created pairs are lower as theylose their transverse momenta (see Section 4.3). The Inverse Compton Scattering (hereafter ICS) process in the neutron star vicinity hasbeen studied extensively by Xia (1982); Kardash¨ev et al. (1984); Xia et al. (1985); Daugh-erty and Harding (1989); Dermer (1989, 1990); Bednarek et al. (1992); Chang (1995);Sturner (1995); Zhang and Qiao (1996); Zhang et al. (1997); Zhang and Harding (2000);Harding et al. (2002), etc. According to these studies, the ICS process may play a signific-ant role in the physics of a neutron star’s magnetosphere. Relativistic particles (positronsand electrons) can Compton-scatter thermal radiation from the neutron star surface. As aparticle with a certain relativistic velocity scatters the thermal photons with a blackbodydistribution, it will produce radiation in quite a wide energy range. However, we can dis-tinguish two characteristic frequencies of upscattered photons: one is the frequency dueto resonant scattering, another is the range of frequencies contributed by the scatteringof photons with frequencies around the ”thermal-peak”. The Resonant Inverse ComptonScattering (RICS) corresponds to a scenario when the scattering cross section is largest.On the other hand, Thermal-peak Inverse Compton Scattering (TICS) corresponds tointeractions with photons with the maximum number density. These two modes are verydifferent when it comes to the nature of the process. The photons’ energy in RICS de-pends on the strength of the magnetic field, thus at low altitudes (where the field is verystrong), it can power pair cascades, while TICS can be responsible for magnetosphericradiation at much higher altitudes. Note that for some specific combinations of magneticstrength and distribution of background photons, RICS and TICS are indistinguishableas the resonance frequency falls into the thermal peak range.08
Chapter 4. Cascade simulation
Due to the rapid time scale for synchrotron emission (see section 4.3), a particle in anexcited Landau level almost instantaneously de-excites to the ground level. The particlemotion is therefore strongly confined to the magnetic field direction. In our calculations weconsider the geometry illustrated in Figure 4.13. In the observer’s frame of reference (OF),a particle with Lorentz factor γ travelling along the magnetic field line scatters a photon.Let ψ = arccos µ be the angle between the magnetic field line (particle propagation) andthe direction of photon propagation in OF and ψ (cid:48) = arccos µ (cid:48) in the particle rest frame(PRF). The energy of the photon in PRF is given by (cid:15) (cid:48) = γ(cid:15) (1 − βµ ) . (4.38)After scattering, the photon energy is denoted by (cid:15) (cid:48) s in PRF and (cid:15) s in OF. The anglebetween the direction of propagation of the scattered photon and B (which describes thedirection of particle propagation) is denoted by ψ s = arccos µ s in OF and ψ (cid:48) s = arccos µ (cid:48) s ,where µ (cid:48) s = ( µ s − β ) / (1 − βµ s ) in PRF (Dermer, 1990). Figure 4.13:
Reproduction of the Figure from Dermer (1990). Geometry of the ICS event in theobserver’s frame (left) and the particle rest frame (right). A particle with Lorentz factor γ , beamedalong the direction of the magnetic field, scatters a photon with energy (cid:15) directed at angle ψ withrespect to the magnetic field line. After scattering, the energy and angle of the photon are denotedby (cid:15) s and ψ s , respectively. Quantities in the particle rest frame are denoted by a prime. Restriction to the Thomson regime requires that γ(cid:15) (1 − µ ) (cid:28)
1. In the particle restframe, the angle ψ (cid:48) = arcsin { γ − [sin ψ/ (1 − β cos ψ )] } , and when γ (cid:29) | µ (cid:48) | →
1. Inthe Thomson regime the only important Compton scattering process involves transitionsbetween ground-state Landau levels. Daugherty and Harding (1989) and Dermer (1989)calculated the differential cross section (after summing over polarisation modes and in-tegrating over azimuth) for a photon scattered from ψ (cid:48) = 0 ◦ into angle ψ (cid:48) s = arccos µ (cid:48) s asfollows .4. Inverse Compton Scattering σ (cid:48) d µ (cid:48) s = 3 σ T (cid:0) µ (cid:48) s (cid:1) (cid:20) (cid:15) (cid:48) ( (cid:15) (cid:48) + (cid:15) B ) + (cid:15) (cid:48) ( (cid:15) (cid:48) − (cid:15) B ) + (Γ / (cid:21) , (4.39)where Γ = 4 α f (cid:15) B / σ T is the Thomson cross section, and α f = e / (cid:126) c is the fine-structure constant. In theThomson limit (cid:15) (cid:48) (cid:28)
1, and thus the scattered photon energy in PRF can be approximatedas (cid:15) (cid:48) s (cid:39) (cid:15) (cid:48) + (cid:15) (cid:48) ( µ (cid:48) − µ (cid:48) s ) / ≈ (cid:15) (cid:48) . (4.40)Equations 4.39 and 4.40 show that a differential magnetic Compton scattering crosssection when γ (cid:29) (cid:15) (cid:48) ap-proaches (cid:15) B and is depressed at energies (cid:15) (cid:48) < (cid:15) B . The total cross section for magneticCompton scattering, obtained by integrating Equation 4.39 over µ (cid:48) s , was calculated byDermer (1989); Zhang et al. (1997) and is given by σ (cid:48) = σ IC (cid:20) u ( u + 1) + u ( u − + a (cid:21) , (4.41)where σ IC = σ T , σ T is the Thomson cross section, u = (cid:15) (cid:48) /(cid:15) B , a = α f (cid:15) B . The Klein-Nishina regime includes quantum effects due to the relativistic nature of scatter-ing, and it requires that γ(cid:15) (1 − µ ) (cid:38)
1. The principal effect is to reduce the cross sectionfrom its classical value as the photon energy in PRF becomes large. In the Klein-Nishinaregime instead of σ IC = σ T we can use the following relationship σ IC = σ KN = 34 σ T (cid:26) (cid:15) (cid:48) (cid:15) (cid:48) (cid:20) (cid:15) (cid:48) (1 + (cid:15) (cid:48) )1 + 2 (cid:15) (cid:48) − ln (1 + 2 (cid:15) (cid:48) ) (cid:21) + 12 (cid:15) (cid:48) ln (1 + 2 (cid:15) (cid:48) ) − (cid:15) (cid:48) (1 + 2 (cid:15) (cid:48) ) (cid:27) . (4.42)In an extreme relativistic regime (cid:15) (cid:48) (cid:29) σ KN ≈ σ T (cid:15) (cid:48)− (cid:20) ln (2 (cid:15) (cid:48) ) + 12 (cid:21) . (4.43)The above formula clearly shows that Inverse Compton Scattering is less efficient forphotons with energy in PRF significantly exceeding particle rest energy. Previous studies on upscattering and energy loss by relativistic particles have used thenon-relativistic, magnetic Thomson cross section for resonant scattering or the Klein-Nishina cross section for thermal-peak scattering. As noted by Gonthier et al. (2000),this approach does not account for the relativistic quantum effects of strong magnetic10
Chapter 4. Cascade simulation fields (
B > G). When the photon energy exceeds mc in the particle rest frame,the strong magnetic field significantly lowers the Compton scattering cross section belowand at the resonance. Gonthier et al. (2000) developed expressions for the scattering ofultrarelativistic electrons with γ (cid:29) γ , the photon incident angle ψ gets Lorentz concentratedto ψ (cid:48) ≈ ψ/ γ ≈ ◦ in the PRF. The differential cross section in the rest frame of theparticle can be written asd σ (cid:48)(cid:107) , ⊥ d cos ψ (cid:48) s = 3 σ T π (cid:15) (cid:48) s e − (cid:15) (cid:48) s sin ( ψ (cid:48) s / (cid:15) B ) (cid:15) (cid:48) (2 + (cid:15) (cid:48) − (cid:15) (cid:48) s ) (cid:2) (cid:15) (cid:48) s + (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:3) l ! (cid:18) (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) B (cid:19) G (cid:107) , ⊥ , (4.44)where G (cid:107) = ˆ G (cid:107) no − flip + ˆ G (cid:107) flip , G ⊥ = ˆ G ⊥ no − flip + ˆ G ⊥ flip (4.45)andˆ G (cid:107) no − flip = ˆ π (cid:12)(cid:12)(cid:12) G (cid:107) , (cid:107) no − flip (cid:12)(cid:12)(cid:12) d φ (cid:48) = ˆ π (cid:12)(cid:12)(cid:12) G ⊥ , (cid:107) no − flip (cid:12)(cid:12)(cid:12) d φ (cid:48) ==2 π (cid:110) [( B + B + B ) cos ψ (cid:48) s − ( B + B ) sin ψ (cid:48) s ] + ( B cos ψ (cid:48) s − B sin ψ (cid:48) s ) (cid:111) , ˆ G ⊥ no − flip = ˆ π (cid:12)(cid:12)(cid:12) G (cid:107) , ⊥ no − flip (cid:12)(cid:12)(cid:12) d φ (cid:48) = ˆ π (cid:12)(cid:12)(cid:12) G ⊥ , ⊥ no − flip (cid:12)(cid:12)(cid:12) d φ (cid:48) ==2 π (cid:2) ( B − B − B ) + B (cid:3) , ˆ G (cid:107) flip = ˆ π (cid:12)(cid:12)(cid:12) G (cid:107) , (cid:107) flip (cid:12)(cid:12)(cid:12) d φ (cid:48) = ˆ π (cid:12)(cid:12)(cid:12) G ⊥ , (cid:107) flip (cid:12)(cid:12)(cid:12) d φ (cid:48) ==2 π (cid:110) [( C + C + C ) cos ψ (cid:48) s − ( C + C ) sin ψ (cid:48) s ] + ( C cos ψ (cid:48) s − C sin ψ (cid:48) s ) (cid:111) , ˆ G ⊥ flip = ˆ π (cid:12)(cid:12)(cid:12) G (cid:107) , ⊥ flip (cid:12)(cid:12)(cid:12) d φ (cid:48) = ˆ π (cid:12)(cid:12)(cid:12) G ⊥ , ⊥ flip (cid:12)(cid:12)(cid:12) d φ (cid:48) ==2 π (cid:2) ( C − C − C ) + C (cid:3) . (4.46)The imaginary terms and the φ (cid:48) dependence are isolated in the polarisation componentsand in the phase exponentials, leading to elementary integrations over the azimuthal angle, φ (cid:48) (Gonthier et al., 2000).The differential cross section depends on the final Landau state l , thus a sum must becalculated over all the contributing Landau states. The energy of the scattered photon isgiven by (Gonthier et al., 2000) (cid:15) (cid:48) s = 2 ( (cid:15) (cid:48) − l(cid:15) B )1 + (cid:15) (cid:48) (1 − cos ψ (cid:48) s ) + (cid:8) [1 + (cid:15) (cid:48) (1 − cos ψ (cid:48) s )] − (cid:15) (cid:48) − l(cid:15) B ) sin ψ (cid:48) s (cid:9) , (4.47)where l is the final Landau level of the scattered particle. Each final state has an energy .4. Inverse Compton Scattering l(cid:15) B , thus the maximum contributing Landau state l max can be expressed as: (cid:15) (cid:48) /(cid:15) B − < l max < (cid:15) (cid:48) /(cid:15) B . To obtain the energy-dependent cross section, the Romberg’smethod can be used to numerically integrate the differential cross section over ψ (cid:48) s . For thisparticular case (scattering of relativistic particles) there is only one resonance appearingat the fundamental cyclotron frequency (cid:15) B = β q = eB/ ( mc ).The values of B and C can be expressed as: B = 2 (cid:15) (cid:48) − (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s )2 ( (cid:15) (cid:48) − (cid:15) B ) ,B = − ( (cid:15) (cid:48) − (cid:15) (cid:48) s cos ψ (cid:48) s ) (cid:0) l(cid:15) B − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:1) + 2 l(cid:15) B (cid:15) (cid:48) (cid:15) (cid:48) s sin ψ (cid:48) s ( (cid:15) (cid:48) − (cid:15) B ) ,B = l(cid:15) B (cid:0) l(cid:15) B − (cid:15) B − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:1) (cid:15) (cid:48) s sin [ ψ (cid:48) s ( (cid:15) (cid:48) − (cid:15) B )] ,B = − (cid:15) (cid:48) s + (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) − (cid:15) B ] ,B = − ( (cid:15) (cid:48) − (cid:15) (cid:48) s cos ψ (cid:48) s ) (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) − (cid:15) B ] ,B = l(cid:15) B cos ψ (cid:48) s sin ψ (cid:48) s [ (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) + (cid:15) B ] ,B = 2 l ( l − (cid:15) B (cid:15) (cid:48) s sin ψ (cid:48) s [ (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) + (cid:15) B ] ,C = (cid:112) l(cid:15) B (cid:15) (cid:48) (cid:15) (cid:48) − (cid:15) B ) ,C = − (cid:112) l(cid:15) B (cid:15) (cid:48) + 2 (cid:15) (cid:48) − (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − l(cid:15) B + (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) (cid:48) s sin ψ (cid:48) s ( (cid:15) (cid:48) − (cid:15) B ) ,C = (cid:112) l(cid:15) B ( (cid:15) (cid:48) − (cid:15) (cid:48) s cos ψ (cid:48) s ) (cid:0) l(cid:15) B − (cid:15) B − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:1) (cid:15) (cid:48) s sin ψ (cid:48) s ( (cid:15) (cid:48) − (cid:15) B ) ,C = − (cid:112) l(cid:15) B (cid:15) (cid:48) s cos ψ (cid:48) s (cid:15) (cid:48) s (cid:15) (cid:48) (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) − (cid:15) B ] ,C = (cid:112) l(cid:15) B (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) (cid:48) s (cid:15) (cid:48) (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) − (cid:15) B ] ,C = − (cid:112) l(cid:15) B (cid:15) (cid:48) s + (cid:15) (cid:48) (cid:15) (cid:48) s (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) s sin ψ (cid:48) s (cid:15) (cid:48) s sin ψ (cid:48) s [ (cid:15) (cid:48) s (cid:15) (cid:48) (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) + (cid:15) B ] ,C = (cid:112) l(cid:15) B ( l − (cid:15) B ( (cid:15) (cid:48) − (cid:15) (cid:48) s cos ψ (cid:48) s ) (cid:15) (cid:48) s sin ψ (cid:48) s [ (cid:15) (cid:48) s (cid:15) (cid:48) (1 − cos ψ (cid:48) s ) − (cid:15) (cid:48) + (cid:15) B ] . (4.48)Although the expressions presented above describe the exact cross section for ICS instrong magnetic fields, due to their complexity their usage in cascade simulation is limited.12 Chapter 4. Cascade simulation
An approximation to the exact l = 0 differential cross section can be given by assumingthat the scattering is significantly below the resonance, where (cid:15) (cid:48) < (cid:15) B and also (cid:15) (cid:48) < (cid:15) (cid:48) and (cid:15) (cid:48) s in theregion of validity, it agrees very well with the exact l = 0 cross section. The approximationoverestimates the exact l = 0 cross section above the region of validity (cid:15) (cid:48) > (cid:15) B . However,the approximation is close to the total cross section for both energy regions ( (cid:15) (cid:48) < (cid:15) B and (cid:15) (cid:48) > (cid:15) B ), even for high magnetic field strengths (see Figure 4.14).According to Gonthier et al. (2000), the polarisation-dependent and averaged, approx-imate cross section can be calculated as: σ (cid:48)(cid:107)→(cid:107) = σ (cid:48)⊥→(cid:107) = 3 σ T
16 [ g ( (cid:15) (cid:48) ) − h ( (cid:15) (cid:48) )] (cid:20) (cid:15) (cid:48) − (cid:15) B ) + 1( (cid:15) (cid:48) + (cid:15) B ) (cid:21) , (4.49) σ (cid:48)(cid:107)→⊥ = σ (cid:48)⊥→⊥ = 3 σ T
16 [ f ( (cid:15) (cid:48) ) − (cid:15) (cid:48) h ( (cid:15) (cid:48) )] (cid:20) (cid:15) (cid:48) − (cid:15) B ) + 1( (cid:15) (cid:48) + (cid:15) B ) (cid:21) , (4.50) σ (cid:48) avg = 3 σ T
16 [ g ( (cid:15) (cid:48) ) + f ( (cid:15) (cid:48) ) − (1 + 2 (cid:15) (cid:48) ) h ( (cid:15) (cid:48) )] (cid:20) (cid:15) (cid:48) − (cid:15) B ) + 1( (cid:15) (cid:48) + (cid:15) B ) (cid:21) , (4.51) g ( (cid:15) (cid:48) ) = (cid:15) (cid:48) (3+2 (cid:15) (cid:48) )+2 (cid:15) (cid:48) √ (cid:15) (cid:48) (2+ (cid:15) (cid:48) ) ln (cid:104) (cid:15) (cid:48) − (cid:112) (cid:15) (cid:48) (2 + (cid:15) (cid:48) ) (cid:105) + (cid:15) (cid:48) ln (1 + 4 (cid:15) (cid:48) ) ++ (cid:15) (cid:48) (1 + 2 (cid:15) (cid:48) ) ln (1 + 2 (cid:15) (cid:48) ) + 2 (cid:15) (cid:48) , (4.52) f ( (cid:15) (cid:48) ) = − (cid:15) (cid:48) ln (1 + 4 (cid:15) (cid:48) ) + (cid:15) (cid:48) (1 + 2 (cid:15) (cid:48) ) ln (1 + 2 (cid:15) (cid:48) ) , (4.53) h ( (cid:15) (cid:48) ) = (cid:15) (cid:48) √ (cid:15) (cid:48) (2 − (cid:15) (cid:48) ) arctan (cid:20) √ (cid:15) (cid:48) (2 − (cid:15) (cid:48) )1+ (cid:15) (cid:48) (cid:21) for (cid:15) (cid:48) < , (cid:15) (cid:48) √ (cid:15) (cid:48) ( (cid:15) (cid:48) − ln (cid:34) (cid:16) (cid:15) (cid:48) + √ (cid:15) (cid:48) ( (cid:15) (cid:48) − (cid:17) (cid:15) (cid:48) (cid:35) for (cid:15) (cid:48) > . (4.54)Figure 4.14 presents the total approximate cross section of Compton scattering, theexact QED cross section (summed over all contributing final electron/positron Landaustates) and the exact cross section for final Landau state l = 0 as a function of energyof the incident photon in PRF (in units of cyclotron energy, (cid:15) (cid:48) /(cid:15) B ). As mentioned above,the approximation is valid in the region below the resonance, (cid:15) (cid:48) < (cid:15) B . Although theapproximation overestimates the cross section for l = 0 final Landau state in the regimeof high energetic photons ( (cid:15) (cid:48) > (cid:15) B ), it can be used in this regime as the approximation ofthe total cross section. In our simulation we use this approach to calculate the total ICScross section in both regimes, (cid:15) (cid:48) < (cid:15) B and (cid:15) (cid:48) > (cid:15) B . Calculation of the cross section for theresonance frequency ( (cid:15) (cid:48) = (cid:15) B ) is presented in the next section. .4. Inverse Compton Scattering Figure 4.14:
Total cross section of Compton scattering (in Thomson units) as a function of anincident photon energy in PRF (in units of the cyclotron energy) calculated for a magnetic fieldstrength B = 3 . . The exact QED scattering cross section, summed over all contributing finalelectron/positron Landau states, is indicated as the red dotted curve. The cross section for finalLandau states l = 0 is plotted as a blue dashed line. This section describes an approach used to calculate the RICS cross section for ultrastrongmagnetic fields (
B > G). For weaker fields the calculations are much simpler andresonance is already included in Equation 4.41.The trend as β q increases is for the magnitude of the cross section to drop at allenergies. For weaker magnetic fields ( β q <
1) the width of the resonance increases withincreasing β q , but for β q ≥ (cid:15) (cid:48) = (cid:15) B by introducing a finitewidth Γ. The procedure is to replace the resonant ( (cid:15) (cid:48) − (cid:15) B ) denominator (see Equations4.44 and 4.51) by (cid:2) ( (cid:15) (cid:48) − (cid:15) B ) + Γ / (cid:3) . In the β q (cid:28) ≈ α f (cid:15) B / β q (cid:29) ≈ α f (cid:15) B (1 − / ˜ e ) where ˜ e is Euler’s number (e.g.see Baring et al., 2011). These widths lead to areas under the resonance being independentof (cid:15) B in the magnetic Thomson regime of β q (cid:28) (cid:15) / B when β q (cid:29)
1. Theseresults can be deduced using the l = 0 approximation derived in Equation 4.51. By usingthis approach the averaged, approximate cross section can be written as σ (cid:48) avg = 3 σ T
16 [ g ( (cid:15) (cid:48) ) + f ( (cid:15) (cid:48) ) − (1 + 2 (cid:15) (cid:48) ) h ( (cid:15) (cid:48) )] (cid:20) (cid:15) (cid:48) − (cid:15) B ) + Γ / (cid:15) (cid:48) + (cid:15) B ) (cid:21) . (4.55)14 Chapter 4. Cascade simulation
The common practice to calculate a resonant cross section in an ultrastrong magneticfields is to use the Dirac delta function as follows (e.g. Medin and Lai, 2010) σ (cid:48) res (cid:39) π e (cid:126) mc δ ( (cid:15) (cid:48) s − (cid:15) B ) (4.56)This simplified approach, however, does not include scatterings of photons whose en-ergy in a particle rest frame is not equal but very close to the resonance frequency. Therelativistic quantum effects of strong magnetic fields that are included in the approximatesolution increase the cross section, and thus the efficiency of the ICS process in previousestimates could be underestimated.According to Medin and Lai (2010) in ultrastrong magnetic fields the ICS polarisationfraction is about 50% (approximately 50% of the photons are slightly above resonance and50% are slightly below). Therefore, the polarisation of ICS photons is randomly assignedin the ratio of one ⊥ (perpendicular to the field) to every (cid:107) photon. For the ICS process the calculation of the particle mean free path l ICS is not as simple asthat of the CR process. Although we can define l ICS in the same way as we defined l CR ,it is difficult to estimate a characteristic frequency of emitted photons. We have to takeinto account photons of various frequencies with various incident angles. An estimationof the mean free path of a positron (or electron) to produce a photon is (Xia et al., 1985) l ICS ≈ (cid:20) ˆ µ µ ˆ ∞ (1 − βµ ) σ (cid:48) ( (cid:15), µ ) n ph ( (cid:15) ) d (cid:15) d µ (cid:21) − , (4.57)where (as before) β = v/c is the velocity in terms of speed of light, n ph represents thephoton number density distribution of semi-isotropic blackbody radiation (see Equation3.33). Here σ (cid:48) is the average cross section of scattering in the particle rest frame (seeEquation 4.55). We should expect two modes of the ICS process, i.e. Resonant ICS andThermal-peak ICS. The RICS takes place if the photon frequency in the particle rest frame is equal to thecyclotron electron frequency. Using Equation 4.38 we can write that the incident photonenergy is (cid:15) = (cid:15) B / [ γ (1 − βµ )]. For altitudes of the same order as the polar cap sizewe use µ = 1, µ = 0 as incident angle limits for outflowing particles, and µ = 0, µ = − .4. Inverse Compton Scattering l RICS ≈ (cid:34) ˆ ˆ (cid:15) maxres (cid:15) minres (1 − βµ ) σ (cid:48) ( (cid:15), µ ) n ph ( (cid:15) ) d (cid:15) d µ (cid:35) − , (4.58)where limits of integration, (cid:15) minres and (cid:15) maxres , are chosen to cover the resonance. In oursimulation we use such limits to include the region where the integrated function decreasesup to about two orders of magnitude from its maximum: (cid:15) min / maxres = (cid:15) B ± √ γ (1 − βµ ) . (4.59)Here Γ is the finite width introduced in Section 4.4.2 to describe the decay of an excitedintermediate particle state.Figure 4.15 presents the dependence of the integrand from Equation 4.58 on the in-cident photon energy for a given incident angle. The maximum of the integrand shows asignificant decline for stronger magnetic fields. This is due to both the drop of the crosssection at all energies with an increasing magnetic field (see Section 4.4.2) and due to thefact that for this specific incident angle resonance is in a different range of photon energy.In stronger magnetic fields resonance occurs not only for higher energetic photons butalso the width of the resonance is wider (see the right panel of Figure 4.15). Figure 4.15:
Dependence of the integrand from Equation 4.58 on the energy of the incident photon.Both panels were calculated for surface temperature T = 3 × K , cosine of the incident angle µ = 0 . and Lorentz factor of particle γ = 10 . The left panel corresponds to resonance in magneticfield B = 10 G , while the right panel was obtained using B = 3 × G . Note that both plots do not include the dependence of the photon density on distancefrom the stellar surface. Depending on whether the radiation originates from the wholestellar surface or from the polar cap only, the dependence of the photon number densityon the height above the surface can differ significantly (see Section 4.4.4).16
Chapter 4. Cascade simulation
TICS includes all scattering processes of photons with frequencies around the maximumof the thermal spectrum. In our simulation we adopt (cid:15) minth ≈ . (cid:15) th , and (cid:15) maxth ≈ (cid:15) th where (cid:15) th = 2 . kT / ( mc ) is the energy, in units of mc , at which blackbody radiation withtemperature T has the largest photon number density. The electron/positron mean freepath for the TICS process can be calculated as l TICS ≈ (cid:34) ˆ µ µ ˆ (cid:15) maxth (cid:15) minth (1 − βµ ) σ (cid:48) ( (cid:15), µ ) n ph ( (cid:15) ) d (cid:15) d µ (cid:35) − . (4.60)Figure 4.16 presents the dependence of the integrand from Equation 4.60 on photonenergy for two different incident angles of background photons. As the number densitydepends exponentially on the photon energy, TICS is important only for small incidentangles ( µ ≈ Figure 4.16:
Comparison of the integrand from Equation 4.60 with photon number density. Thebottom panels present the dependence of the photon number density on photon energy in OF. Thered dashed lines correspond to limits used to calculate the particle mean free path for TICS. The toppanels present the dependence of the integrand on photon energy in PRF. Both panels were obtainedusing surface temperature T = 3 × K , Lorentz factor of the particle γ = 10 and magnetic fieldstrength B = 10 G . The cosine of the incident angle, µ = 0 . and µ = 0 . , was used for theleft and right panel, respectively. .4. Inverse Compton Scattering For ultrastrong magnetic fields quite a wide range of the particle Lorentz factor falls intothe peak of background photons (see Figure 4.17). In such a case RICS is enhanced bythe fact that it involves photons with very high density. Furthermore, the RICS processfor such particles is indistinguishable from the TICS (see Figure 4.16). For particles withLorentz Factor γ (cid:38) , the dominant process of radiation is CR. The exact value ofthis limit depends on conditions such as: density of background photons, incident anglesbetween particles and photons, and curvature of magnetic field lines (1 / (cid:60) ). Figure 4.17:
Dependence of a particle mean free path on its Lorentz factor for three differentprocesses: CR, RICS and TICS. The calculations were performed for magnetic field strength B = 2 ,radius of curvature of magnetic field lines (cid:60) = 1 (for the CR process) and hot spot temperature T = 3 (for RICS and TICS). Both RICS and TICS were calculated for a full range of incident angles( µ = 0 , µ = 1 ). Note that for a Lorentz factor in the range of γ ≈ × − the particlemean free paths of RICS and TICS are equal as the resonance falls into the peak of the backgroundphotons. Figure 4.18 presents the dependence of a particle mean free path on the magneticfield strength and the particle Lorentz factor for RICS. The minimum of the mean freepath for relatively weak magnetic fields ( B = 0 .
5) is for particles with Lorentz factor γ ≈ × , while for relatively stronger magnetic fields ( B = 3 .
5) the RICS is mostefficient for particles with energy an order of magnitude larger ( γ ≈ × ). This isa natural consequence of the fact that resonance takes place when the photon energy inPRF is equal to the electron cyclotron energy, which in stronger fields is higher. As canbe seen from the Figure, the particle mean free paths for RICS in stronger magnetic fieldsincrease. This is due to the decreasing resonant cross section with increasing magneticfield strength (see Figure 4.15). Note, however, that this behaviour does not include thefact that photon density in regions with weaker magnetic fields is considerably smaller.In fact, the results of the cascade simulation presented in Chapter 5 show that RICS is18 Chapter 4. Cascade simulation efficient only in the immediate vicinity of a neutron star since photon density at higheraltitudes drops rapidly.
Figure 4.18:
Dependence of a particle mean free path on magnetic field strength ( B ) and theLorentz factor of a particle ( γ ) for the RICS process. The particle mean free path was calculated forsemi-isotropic blackbody radiation ( µ = 0 , µ = 1 ) with temperature T = 2 . . One of the main parameters affecting ICS above the stellar surface is photon density.The initial photon density (at altitude z = 0) highly depends on the temperature of theradiating surface. As shown in Chapter 1 (e.g. see Table 1.4), the entire surface has thelowest temperature ( T (cid:46) . T (cid:46)
3) and up to about three ordersmagnitude lower than hot spot radiation ( T (cid:46) L = ( r, θ, φ ). Then the relative density of photons originatingfrom the entire surface can be calculated as n st ( (cid:15), T st , L ) n ( (cid:15), T st ) = sin (cid:18) ∆ θ st (cid:19) = (cid:18) Rr (cid:19) , (4.61)where n st , ( (cid:15), T st ) is the density of photons with energy (cid:15) at the stellar surface withtemperature T st , and ∆ θ st is the angular diameter of the star at a distance from the star .4. Inverse Compton Scattering r .Likewise, we can write a formula for the relative density of photons originating froma spot (warm or hot) as n sp ( (cid:15), T sp , L ) n sp , ( (cid:15), T sp ) = sin (cid:18) ∆ θ (cid:19) , (4.62)where n sp , ( (cid:15), T sp ) is the density of photons with energy (cid:15) at the spot surface (either hotor warm) with temperature T sp . The angular diameter of the spot can be calculated as∆ θ = arccos (cid:18) r + r − R sp r r (cid:19) , (4.63)here R sp is the spot radius and r , r are the distances to the outer edges of the spot (seeFigure 4.19). Figure 4.19:
Simplified method used for calculation of a photon density originating from an entirestellar surface (blue lines) and from a hot/warm spot (red lines). Here R sp is a spot radius (either hotor warm). Let us note that the simplified method is valid for the entire surface component regardlessof the φ component of location L , while for the spot component it can be used only for small valuesof φ . In a more general case the spot should be projected on the surface perpendicular to the radiusvector r and passing through point L . Figure 4.20 presents the dependence of the relative photon density ( n ( z ) /n ) onthe distance from the stellar surface. Due to the small size of a polar cap (hot spot, R hs = 50 m) the density of the photons drops rapidly and already at a distance of about z = 150 m it is one order of magnitude lower than at the polar cap surface. On the otherhand, for a larger size of the warm spot ( R hs = 1 km) the photon density is reduced byan order of magnitude at a distance of about z = 3 km. From Equation 4.61 it can easilybe seen that the photon density of radiation from the entire stellar surface decreases byan order of magnitude at a distance of about z ≈ R ≈
30 km.20
Chapter 4. Cascade simulationFigure 4.20:
Dependence of the relative photon density on the distance from the stellar surfacefor three different thermal components (the entire stellar surface, the warm spot and the hot spot).The following parameters were used for the calculations: star radius R = 10 km , warm spot radius R ws = 1 km and hot spot radius R hs = 50 m . The very small size of the polar cap also has an additional implication to the back-ground photons’ density. Namely, the density of the background photons just above thepolar cap highly depends not only on the distance from the surface, but also on theposition relative to the cap centre.Figure 4.21 presents the dependence of the relative photon density originating from apolar cap (the hot spot) on the distance from the stellar surface for three different startingpoints on the polar cap. The distance was calculated for points which follow the magneticfield structure of PSR B0656+14. Note that for the extreme magnetic line (which starts atthe cap edge) already at a distance of about z ≈ θ ) and middle line ( θ ) the distances are respectively z ≈
45 m and z ≈
30 m. This result is important as the background photon density directly translates tothe particle mean free path in ICS (see Section 4.4.3). This means that for ICS-dominatedgaps the sparks’ height will vary depending on their location. The breakdown of the gap(spark) in the central region of a polar cap is easier to develop as the particle mean freepath is lower, and eventually it will result in lower heights of the central sparks. Thiswill influence the properties of plasma produced in the central region of open magneticfield lines, and depending on the conditions may result in the formation of plasma eithersuitable to produce radio emission (core emission) or unsuitable to produce radio emission(conal emission but with the line of sight crossing the centre of the beam).To find the dominant component of thermal radiation at a given altitude we needto take into account the initial flux of radiation and how it changes with the distance.Below we present the calculations of a radiation flux (Figure 4.22) for PSR B0656+14.The parameters of an entire surface and warm spot components are in agreement with .4. Inverse Compton Scattering
Figure 4.21:
Dependence of the relative photon density on the distance from the stellar surfacefor a hot spot component of PSR B0656+14. The relative photon density was calculated for threedifferent starting positions: θ (central), θ (at the half distance to the edge), and θ (the capedge). The altitude ( z ) was calculated for points which follow the magnetic field structure ofPSR B0656+14. Figure 4.22:
Dependence of the radiation flux for three different components (the entire stel-lar surface, the warm spot and the hot spot) on the distance from the stellar surface for PSRB0656+14. The following parameters were used for the calculations: entire stellar surface radiation, T st = 0 . , R st = 20 km ; warm spot, T ws = 1 . , R ws = 1 . ; and hot spot, T hs = 2 . , R hs = 50 m . Already at a distance of 240 m the flux of the warm spot radiation becomes higher22
Chapter 4. Cascade simulation than the flux of the hot spot radiation. Furthermore, already at a height of 750 m fluxthe radiation originating from the polar cap (hot spot) becomes lower than the flux ofradiation from the entire stellar surface. With an increasing distance the flux of the warmspot decreases faster than the flux of the entire surface radiation and at a distance of 6 . Another parameter that significantly affects the ICS is the incident angle between thebackground photons and the relativistic particles. Especially for Resonant Inverse ComptonScattering is the incident angle of great importance. Figure 4.23 presents the dependenceof a particle mean free path for ICS on a maximum value of the incident angle ψ crit . Ifincident angles are low, the resonance is outside of the photon spectrum and results invery high values of particle mean free paths. The lower the energy of the particle (lowerLorentz factor), the incident angles should be larger to ensure that the resonance fallsinto an energy range with high photon density. Figure 4.23:
Dependence of the particle mean free path on the maximum value of the incidentangle ψ crit . The particle mean free path l p was calculated for magnetic field strength B = 10 G assuming background blackbody radiation with a temperature T = 3 MK . Two different particleLorentz factors were used for the calculations: γ = 10 (dashed lines) and γ = 10 (solid lines).The red lines correspond to Resonant Inverse Compton Scattering, while the blue lines correspondto Thermal-peak Inverse Compton Scattering. .4. Inverse Compton Scattering ψ crit ≈ ◦ (for γ = 10 ) and ψ crit ≈ ◦ (for γ = 10 ). For such high incident angles theresonance takes place at the thermal peak of the background photons. Therefore, TICSand RICS are indistinguishable, which results in an almost equal particle mean free path(see the text above Figure 4.16 for more details).Due to the very small size of the polar cap the influence of the hot spot componentwill by lower not only because of the change of photon density, but also because of therapid change of the incident angle between the photons and particles. Figure 4.24 presentsthe dependence of the maximum incident angle on the altitude above the stellar surfacefor three thermal components (the entire surface, the warm spot and the hot spot). Asfollows from the Figure, already at an altitude of z ≈
90 m does the maximum valueof the incident angle between the photons from the hot spot and the particles drop to ψ crit = 30 ◦ , which significantly lowers the efficiency of ICS for this source of backgroundphotons (see Figure 4.23). Since the size of the warm spot component is larger, the warmspot radiation will be significant for up to higher altitudes, but already at a distance of z ≈ . ψ crit = 30 ◦ . Figure 4.24:
Dependence of the maximum incident angle on the altitude above the stellar surfacefor three thermal components (the entire surface, the warm spot and the hot spot radiation).
Note that in the Figure we have calculated the maximum value of the intersectionangle at altitudes which correspond to radial progression from the stellar surface. In fact,the actual maximum value of the incident angle also depends on the structure of themagnetic field. Figure 4.25 presents the actual maximum value of the incident angle ofphotons originating from the hot spot for three different magnetic field lines calculated forPSR B0656+14. The actual values of the maximum incident angle just above the surfaceexceed 90 ◦ , but its rapid decline (especially for extreme lines) causes the radiation of the24 Chapter 4. Cascade simulation hot spot component to become insignificant for ICS at relatively low altitudes z ≈
20 m.
Figure 4.25:
Dependence of the maximum incident angle on the altitude above the stellar surfacefor the hot spot component of PSR B0656+14. The maximum incident angle was calculated forthree different starting positions: θ (central), θ (at the half distance to the edge), and θ (the capedge). Both the decrease of photon density and the decrease of the maximum inclination anglecause the parameters of plasma produced by RICS to highly depend on the properties(size and temperature) of the background photons source. The hot spot component willbe the dominant source of background photons for ICS in the gap region ( z (cid:46)
20 m),while the radiation of the warm spot and the entire surface will be the main source of thebackground photons for ICS at higher altitudes.25
Chapter 5Physics of pulsar radiation
In our model most of the γ -photons are produced in the Inner Acceleration Region orin close vicinity of a neutron star. Due to an ultrastrong surface magnetic field, themost energetic γ -photons are produced by Inverse Compton Scattering in the PSG-onmode. If a pulsar is in the PSG-off mode, Curvature Radiation produces fewer energeticphotons than ICS in the PSG-on mode. Photons produced in IAR (both the ICS andCR) are absorbed by strong magnetic fields creating positron-electron plasma in the gapregion, thereby enhancing a cascade, or just above the gap enhancing a secondary plasmapopulation. The absorption of γ -photons in close vicinity of NS makes it impossible todirectly observe the radiation produced in IAR. However, a characteristic of this emissiondefines the parameters of the gap (e.g. multiplicity in the gap region, gap height, etc.),and thus the parameters of secondary plasma. In general, the existence of high potential in IAR (e.g. wide sparks or η ≈
1) results insolutions for which CR is responsible for the emission of γ -photons. The energy of suchradiation depends on the Lorentz factor of primary particles and curvature of the magneticfield lines. Figures 5.1 and 5.2 present the histogram of photons produced in IAR by CRfor PSR B0628-28 and Geminga, respectively. The curvature in IAR of Geminga is lower( (cid:60) ≈ .
1, see Section 2.4.2), thus the primary particle should be accelerated to higherenergies in order to produce the required number of photons in the gap region. Eventuallythe higher Lorentz factor of primary particles will result in the emission of γ -photons withenergy up to 10 GeV for Geminga. On the other hand, the curvature magnetic lines forPSR B0628-28 ( (cid:60) = 0 .
6, see Section 2.4.1) is higher, which reduces the photon meanfree path and it is possible to produce the required number of photons in the gap region26
Chapter 5. Physics of pulsar radiation N CRph for lower the Lorentz factor of primary particles.In CR-dominated gaps we can distinguish three types of photons: (I) radiation withenergy below 1 MeV which is unaffected by the magnetic field (except the splitting) andcan be detected by a distant observer, (II) soft γ -ray photons which create pairs aboveZPF, (III) and high energetic γ -photons responsible for pair production below ZPF. Inan ultrastrong magnetic field the photons from the third group will produce particles justafter reaching the first threshold. Due to the fact that most CR photons are (cid:107) -polarised,photon splitting is insignificant in cascade pair production in the PSG-off mode. Figure 5.1:
Distribution of photons produced in IAR by a single particle for PSR B0628-28. In thecalculations we used parameters of the gap in the PSG-off mode as presented in Table 3.1. We alsoassumed a linear change in the acceleration electric field (see Equation 3.27).
Figure 5.2:
Distribution of photons produced in IAR by a single particle for PSR J0633-1746. Inthe calculations we used parameters of the gap in the PSG-off mode as presented in Table 3.1. .1. Inner Acceleration Region
When the acceleration potential is low enough (narrow sparks with η <
1) to satisfythe condition for effective ICS ( l ICS (cid:46) l acc ), the gap will operate in the PSG-on mode.The energy of ICS radiation in the gap region (RICS) depends on the Lorentz factor ofprimary particles and the strength of magnetic field. In an ultrastrong magnetic field ofIAR implied by the PSG model, the primary particle loses most of its energy during thescattering of background photons. Such extremely energetic photons produce pairs on thezero-th Landau level ( (cid:107) -polarised photons) or split to less energetic photons before reachingthe first threshold (see Section 4.2.6). After the photons split the resulting photons are stillvery energetic and create an electron-positron pair enhancing the avalanche productionof particles. In contrast to the PSG-off, most of the electron-positron pairs in the PSG-on mode are created well below ZPF. Furthermore, there is no additional radiation atlower energies ( (cid:15) < γ -photons ranges from 1 GeV to ≈
20 GeV. The narrow predicted sparkhalf-width of PSR B0950+08 results in a lower potential in IAR, thus increasing theefficiency of ICS (more photons produced by the first population of particles). The particlemean free path for ICS is smaller for backstreaming particles (see Section 3.2.1 for moredetails), thus most photons in the PSG-on mode are produced in the direction towards thestellar surface. Note that not all photons will produce electron-positron pairs since some γ -photons are produced so close to the stellar surface that they reach its surface beforethey manage to reach the first threshold for pair production. Figure 5.3:
Distribution of photons produced in IAR by the first population of newly createdparticles for PSR B0950+08. In the calculations we used the parameters of the gap in the PSG-onmode as presented in Table 3.1. Chapter 5. Physics of pulsar radiationFigure 5.4:
Distribution of photons produced in IAR by the first population of newly createdparticles for PSR B1929+10. In the calculations we used the parameters of the gap in the PSG-onmode as presented in Table 3.1.
An negligible fraction of energy radiated by a primary particle in the PSG-off mode fallsin the X-ray band. What is more, in the PSG-on mode all photons produced by ICS haveenergy which exceeds an electron’s rest energy by many orders of magnitude. Thus, IARmay be responsible only for generating the thermal component of the X-ray spectrum inthe process of heating the stellar surface.
As shown in Section 1.4, thermal emission is a common feature of neutron stars. Due to thelarge uncertainties in X-ray observations, it is not possible to distinguish all three thermalcomponents (entire surface radiation, warm spot component and hot spot radiation) forone specific pulsar. Furthermore, only for a few pulsars (e.g. Geminga, PSR B0656+14)was it possible to distinguish two thermal components alongside the nonthermal one. Inthis thesis we focus on an analysis of pulsars with a visible hot spot component ( b > .1. Inner Acceleration Region
Figure 5.5:
Observed flux of radiation for PSR J0633+1746. In the figure we present threecomponents of radiation: the nonthermal one (green line), the entire surface radiation (blue line),and the hot spot component (red line). The dashed lines correspond to uncertainties in observations(see Table 1.4).
Figure 5.6 presents the X-ray spectrum of PSR B1133+16. The small number of countsdetected resulted in the fact that only separate fits for the BB and PL components wereperformed. Both the BB and PL fits describe the observed spectrum with similar accuracy.In the Figure we present additional thermal components (the entire surface radiation andthe warm spot) which have not been determined by the observations. The Figure showsthat the overlapping thermal components can mimic the power-law dependence of thespectrum at frequencies below 2 keV.
Figure 5.6:
X-ray spectrum of PSR B1133+16. In addition to the observed thermal radiation (redsolid line), two other thermal components are presented: the warm spot radiation (green dashed line)and the entire surface radiation (blue dotted line). Chapter 5. Physics of pulsar radiation
Although this specific combination of thermal components for PSR B1133+16 wouldresult in a photon index greater than the observed one Γ = 2 .
51, the spectral fits for allpulsars should be extended to include more BB components in order to examine the effectof thermal components overlapping at lower frequencies. The results of our calculationssuggest that the nonthermal X-ray radiation should dominate the spectrum at higherfrequencies ≈ −
10 keV, but the power-law-like behaviour at lower frequencies could bethe result of the overlapping of thermal components anticipated in the PSG scenario (seeSection 5.1.2.3).
The polarisation of ICS radiation in an ultrastrong magnetic field is 50% (one (cid:107) to every ⊥ -polarised photon). Synchrotron Radiation of secondary particles created by ⊥ -polarisedphotons would generate hard X-ray photons, however, as was mentioned in Section 4.2.6,these photons will split before they reach the first threshold to produce pairs. Therefore,regardless of whether the gap is dominated by CR or by ICS, Synchrotron Radiation inIAR is not significant. Apart from the obvious X-ray component corresponding to the whole surface radiation, thePSG model can explain both the hot and warm spot radiation. The hot spot radiation is anatural consequence of heating the actual polar cap region by the backstreaming particles(see Section 1.4.4). As was mentioned in Section 1.4.4, the warm spot component can havetwo different sources: (I) the drastic difference of the crustal transport process due to thenon-dipolar structure of the surface magnetic field (for young and middle-aged pulsars),(II) and a mechanism of heating the surface adjacent to the polar cap. In this sectionwe present the second mechanism, i.e. heating of the surface adjacent to the polar cap,which can be applied to both young and old pulsars.Figure 5.7 presents the mechanism of heating the area adjacent to the polar cap forPSR B0950+08. When the gap operates in the PSG-off mode the primary plasma (seeSection 5.1.3) will lose a significant part of its energy via CR as the particles propagatethrough the region of high curvature. For this particular magnetic line’s configurationthe region of high CR extends up to an altitude about 4 km above the stellar surface.The most energetic CR photons emitted in this region have a relatively short mean freepath and they produce electron-positron pairs in the region of open magnetic field lines.However, both the less energetic CR photons and γ -photons produced by SR have a largeenough photon mean free path to produce pairs in the region of the closed magnetic fieldlines. All newly created pairs move along the closed magnetic field lines and heat thesurface beyond the polar cap on the opposite side of the star. .1. Inner Acceleration Region x [10 cm] z [ c m ] NS Figure 5.7:
Global structure of magnetic field lines for PSR B0950+08. The structure was obtainedusing two crust-anchored anomalies (see Section 2.4.5). Green lines correspond to the outer openmagnetic field lines, while the red lines correspond to the closed magnetic field lines at which secondarypairs are produced. Blue, yellow and red dots represent the locations of secondary pair productionfor the outer left, the middle and the outer right open field lines, respectively.
The fraction of energy transferred to the region of the closed field lines highly dependson the region of open magnetic field lines considered in CR/SR emission. In the Figurewe use three different colours (blue, yellow and red) to show the positions of pair creationfor three characteristic open magnetic field lines (the outer left, the middle and the outerright). The simulation results in the following fractions of energy transferred to theregion of the closed field lines are: 0 . . Chapter 5. Physics of pulsar radiation move towards the region with the highest curvature. We can roughly estimate that forthe proposed magnetic field configuration of PSR B0950+08, about 1% of the outflowingenergy is responsible for heating of the surface beyond the polar cap on the opposite sideof the star. Note that due to strong anisotropy of the outflowing and backflowing streamof particles (see Section 5.1.3), this fraction could be enough to obtain the warm spotcomponent with a luminosity equal or in some cases even higher than the luminosity ofthe hot spot component. x [10 cm] z [ c m ] NS
900 m1 km
Figure 5.8:
Global structure of magnetic field lines for PSR B0943+10. The structure was obtainedusing two crust-anchored anomalies (see Section 2.4.4). Green lines correspond to the outer openmagnetic field lines, while the red lines correspond to the closed magnetic field lines at which secondarypairs are produced. Blue, yellow and red dots represent the locations of secondary pair creation forthe outer left, the middle and the outer right open field lines, respectively. .1. Inner Acceleration Region −
5% of the outflowing energy is responsible for the heating. Notethat the magnetic field structure of PSR B0950+08 results in the heating of only one sidebeyond the polar cap, while in the case of PSR B0943+10 the whole surface around thepolar cap is heated. The actual size of the warm spot also depends on the magnetic fieldconfiguration in the heating zone, and can either be decreased or increased.
As we mentioned in Section 3.2.2, PSG-off and PSG-on modes differ essentially by theLorentz factor of primary particles produced in the gap region. Furthermore, differentscenarios of the gap breakdown (due to surface overheating or due to production of denseenough plasma) cause the evolution of primary particles in the two modes to completelydifferent.We assume that in the PSG-off mode the gap breakdown is due to surface overheating;hence the plasma cloud moving away from the stellar surface is a mixture of ions andelectron-positron plasma. In this scenario the ions are the main source of charge densityrequired to screen the gap (see Equation 3.40). As the plasma cloud moves away fromthe stellar surface both the spark height and the spark width increase, which resultsin an increase of the acceleration potential drop. When the particles gain the Lorentzfactors γ (cid:38) , CR begins to produce γ -photons. In the PSG-off mode most of the γ -photons are created near ZPF (see Figure 3.4). All particles created by γ -photons abovethe ZPF do not contribute to the heating of the surface. Furthermore, the accelerationin the upper parts of the gap is relatively weak, and electrons produced in this regionwill also escape from the gap, thus not contributing to the surface heating. Dependingon the details of the cascade formation, the process described above may result in thecreation of strong streaming anisotropies, where the flux of the backstreaming particlesis considerably smaller than the flux of the outstreaming particles. Note that the densityof the backstreaming particles required to overheat the surface is significantly lower thanthe co-rotational density n CR (cid:28) n GJ (see Table 3.1).In the PSG-on mode the quasi-equilibrium of the flux of backstreaming particles andthe flux of the polar cap radiation can cause the gap to break only due to the produc-tion of dense enough plasma. Thus, the surplus of positrons is the main source of thecharge in the plasma cloud moving away from the stellar surface. The ICS process re-sponsible for the cascade production of particles is effective only in the bottom part of34 Chapter 5. Physics of pulsar radiation the gap. Hence, the backstreaming electrons will hit the surface with a Lorentz factor γ c well below the γ max . As there is no strong pair production near (or above) ZPF, the back-streaming/outstreaming anisotropy arises only due to the difference of the Lorentz factorof electrons hitting the stellar surface and the Lorentz factor of positrons accelerated in thegap γ max /γ c ≈
10. The actual density of newly created plasma to completely screen thegap can be calculated only in the full cascade simulation. However, as shown by Timokhin(2010), this density should significantly exceed the co-rotational Goldreich-Julian density n ICS (cid:29) n GJ . We describe the difference between the co-rotational density and the actualdensity of primary plasma required to completely screen the ICS-dominated gap by factor N ICS = n ICS /n GJ (cid:29) In general there are three processes which can produce γ -ray emission in the inner mag-netosphere of a pulsar ( R pc (cid:28) z (cid:28) R LC ): CR, ICS and SR. Which of them producesthe majority of γ -photons depends on the parameters of the primary particles, and thusmainly depends on the mode in which the gap operates. Additionally, the efficiency ofthe ICS process strongly depends on the source of the background photons. When the gap operates in the PSG-off mode, high-energetic particles are produced γ c (cid:38) . As they pass the region with high curvature ( (cid:60) ≈
1) they radiate a signi-ficant part of their energy through CR (see Section 4.1).Figure 5.9 presents the distribution of CR photons produced by a single primaryparticle moving along the open magnetic field line of PSR B1133+16 (see Section 2.4.5for the details of the magnetic field configuration). The initial Lorentz factor of theparticle γ max = 1 . × was set according to the value presented in Table 3.1. As theparticle advanced through the region with high curvature, it lost about 46% of its initialenergy, which was mainly converted to high-energetic γ -photons with an energy up toabout 2 GeV. The γ -photons are produced in a region of a strong magnetic field, thusafter passing a relatively short distance the most energetic photons are absorbed by themagnetic field and electron-positron pairs emerge. The red colour in the Figure corres-ponds to the final spectrum (after photon splitting, pair production and SR) producedby a single primary particle in the PSG-off mode. Most of the energy radiated by theprimary particle was converted into the secondary plasma (see Section 5.2.3) and onlyabout 5% of the particle’s initial energy ended in the form of radiation with a cut-off atabout 30 MeV. .2. Inner magnetosphere of a pulsar Figure 5.9:
Final photon distribution produced by a single primary particle for PSR B1133+16. Theblue line corresponds to the initial CR distribution, while the red line presents the final distributionwith the inclusion of photon splitting, pair production and SR.
To increase the amount of photons reaching the observer, the emission zone, i.e. theregion with the highest curvature, should by located in the area with a weaker magneticfield. Such a configuration allows a photon to travel a longer distance before it is absorbedby the magnetic field. As a result the electron-positron pairs are created at higher Landaulevels, which enhances SR. Figure 5.10 presents the distribution of CR photons for PSRB0950+08. The calculations were performed for the initial Lorentz factor of the particle γ max = 2 . × (see Table 3.1). Figure 5.10:
Final photon distribution produced by a single primary particle for PSR B0950+08.The blue line corresponds to the initial CR distribution, while the red line presents the final distributionwith the inclusion of photon splitting, pair production and SR.
Due to CR the primary particle lost about 40% of its initial energy. In this case36
Chapter 5. Physics of pulsar radiation about a half of the energy radiated by the primary particle was converted into the second-ary plasma and the same amount of energy (about 20% of the particle’s initial energy)ended in the form of radiation. For both PSR B1133+16 and PSR B0950+08, the max-imum of the curvature is of the same order. However, the maximum of curvature forPSR B1133+16 is located at an altitude of about 800 m, while for PSR B0950+08 it islocated at an altitude of about 1 .
75 km (compare Figures 2.23 and 2.20).
In the PSG-on mode the maximum Lorentz factor of primary particles is in the range of10 − (see Table 3.1). As it follows from Figures 4.17 and 4.18, the ICS process ismost effective for particles with a Lorentz factor in the range of 10 − . Particles withhigh energies ( γ (cid:38) ) will upscatter thermal photons only just above the stellar surface,where the density of the background photons is very high (see Section 5.1.1.2). Thus, ifthere is no additional source of background photons, the most energetic particles ( γ (cid:38) )will escape from the inner magnetosphere without losing their energy by ICS. However,the plasma cloud produced by the ICS-dominated gap has a density exceeding the co-rotational Goldreich-Julian density even by a few orders of magnitude (see Section 5.1.3).Such a high charge density reduces the acceleration (Timokhin, 2010) and, consequently,the bulk of particles will escape from the IAR with lower Lorentz factors. It is not possibleto estimate the actual Lorentz factor of particles in the plasma cloud at the moment of gapbreakdown without performing a full cascade simulation. Thus, in this thesis we assumethat at the moment of gap breakdown most of the particles will have an energy that isabout the characteristic value at which the acceleration is stopped by ICS in the bottomparts of the IAR γ c . To increase readability for cascade simulations with very low surfacetemperature, in all the Figures of the ICS distribution we present γ -photons produced by50 primary particles with Lorentz factors in the range of 0 . γ c − γ c .In Figure 5.11 we present the distribution of ICS photons produced by the upscatteringof surface thermal radiation with temperature T s = 0 . γ -photons with energy up to 1 GeV. Since the γ -photons are very energetic and areproduced in a region with a strong magnetic field, they will be absorbed by the magneticfield, thus giving rise to the secondary plasma population (see Section 5.2.3). All pairsin the inner magnetosphere of a pulsar are created in the nonzero Landau level, thus thepair production process is also accompanied by strong SR (see the next section). .2. Inner magnetosphere of a pulsar Figure 5.11:
Distribution of ICS photons produced by an upscattering of surface radiation( T s = 0 . ) for PSR B0834+06. The plot includes all γ -photons upscatterd by primaryparticles with Lorentz factors in the range of . × − . A natural way of increasing the number of γ -photons produced by ICS in the innermagnetosphere is to increase the number of background photons. Figure 5.12 presents thedistribution of ICS photons produced by an upscattering of the surface thermal radiationwith temperature T s = 0 . γ -photons up to higher altitudes (about two stellar radii), thus photons withlower energy emerge (cid:15) min ≈ γ -photons. Figure 5.12:
Distribution of ICS photons produced by an upscattering of the surface radiation( T s = 0 . ) for PSR B0834+06. The plot includes all γ -photons upscatterd by primaryparticles with Lorentz factors in the range of . × − . Chapter 5. Physics of pulsar radiation
Note that although for PSR B0834+06 the X-ray spectral fit was performed withonly one BB component, the surface temperatures used in the calculations (0 . . T s = 1 . R ws = 1 km for PSRB0834+06. When the warm spot is the main source of background photons, the ICSprocess starts at lower altitudes. As a consequence, the scattering produces photonswith higher energy and the primary particles lose up to 90% of their initial energy. Allthese high energetic γ -photons are absorbed by the magnetic field producing electron-positron pairs. Note that for this specific pulsar the existence of such a strong warm spotcomponent is unlikely, but as mentioned in Section 5.1.2.1 the X-ray spectral fits shouldbe extended to include more thermal components to put better constraints on the X-rayemission of pulsars. Figure 5.13:
Distribution of ICS photons produced by an upscattering of warm spot radiation( T s = 1 . , R ws = 1 km ) for PSR B0834+06. The plot includes all γ -photons upscatterd by primary particles with Lorentz factors in the range of . × − . In both PSG-off and PSG-on modes SR plays a significant role in the generation of soft γ -ray photons. Figure 5.14 presents the places at which SR-photons are generated (left .2. Inner magnetosphere of a pulsar z ≈
500 m), while the lessenergetic ones are produced at altitudes z > γ -photons produce electron-positron pairs in a strong magneticfield, thus its observation is not possible. Figure 5.14:
Synchrotron Radiation in the PSG-off mode for PSR J0633+1746. The left panelpresents the places at which SR-photons are generated, while the right panel presents the SR-photonsdistribution. Plots were obtained in a cascade simulation calculated for a single primary particlemoving along the extreme left open magnetic field line.
In Figure 5.15 we present the places of SR-photon generation (left panel) and theenergy distribution of photons (right panel) in the PSG-on mode for Geminga.
Figure 5.15:
Synchrotron Radiation in the PSG-on mode for PSR J0633+1746. The left panelpresents places at which SR-photons are generated, while the right panel presents the SR-photonsdistribution. Plots were obtained in a cascade simulation calculated for a single primary particlemoving along the extreme left open magnetic field line. The ICS process was calculated using thewhole surface radiation with temperature T s = 0 . (see Table 1.4). Chapter 5. Physics of pulsar radiation
The production of SR-photons in the PSG-off mode starts at altitudes about z ≈ z ≈ . (cid:15) min ≈
40 keV and (cid:15) max ≈
50 MeV,respectively. Note the significant difference in the number of photons produced by SR inthe PSG-off and PSG-on modes. The difference is a direct consequence of low secondaryplasma multiplicity in the PSG-on mode (see Section 5.2.3).
The main source of X-ray photons produced in the inner magnetosphere is SR. As men-tioned in Section 5.2.1.1, to increase the amount of photons reaching the observer theemission zone should by located in the area with a weaker magnetic field. In this sec-tion we focus on the results of PSR B0943+10 and PSR 1929+10 for which the proposedconfiguration of a magnetic field satisfies this requirement (see Sections 2.4.4 and 2.4.7).In the PSG-off mode most of the X-ray photons are produced by the SR of newly cre-ated electron-positron pairs. Figure 5.16 presents the final photon distribution producedby a single primary particle of PSR B1929+10 in the PSG-off mode. For a single primaryparticle we can estimate that only about 0 .
7% of the total photon energy is in the rangeof 1 −
10 keV. The bulk of the energy is carried away by newly created particles (73%)and high energetic photons (27%).
Figure 5.16:
Final photon distribution produced by a single primary particle for PSR 1929+10 inthe PSG-off mode. The blue line corresponds to the initial CR photons distribution, while the redline presents the final distribution with the inclusion of photon splitting, pair production and SR.
In Figure 5.17 we present the locations and the photon distribution of SR for PSR1929+10 in the PSG-off mode. All SR-photons produced closer to the stellar surfacewill contribute to γ -ray emission, while the SR-photons produced at higher altitudes willproduce photons in the X-ray band. As it results from Figure 2.26, the curvature at an .2. Inner magnetosphere of a pulsar z ≈ . z ≈ z ≈ . Figure 5.17:
Synchrotron Radiation in the PSG-off mode for PSR B1929+10. The left panelpresents the places at which SR-photons are generated, while the right panel presents the SR-photons distribution. Plots were obtained in a cascade simulation calculated for a single primaryparticle moving along the extreme left open magnetic field line.
To increase radiation in the 1 −
10 keV energy band we should apply the magneticfield structure with considerably higher curvature at altitudes where X-ray photons aregenerated. Although the curvature will not directly affect the SR, it will enhance CR,and thus it will increase the number of pairs produced in the region of a relatively weakmagnetic field. In Figure 5.18 we present the final photon distribution produced by a singleprimary particle for PSR B0943+10 calculated using the magnetic field configuration aspresented in Section 2.4.4.
Figure 5.18:
Final photon distribution produced by a single primary particle for PSR B0943+10in the PSG-off mode. The blue line corresponds to the initial CR photons distribution, while the redline presents the final distribution with the inclusion of photon splitting, pair production and SR. Chapter 5. Physics of pulsar radiation
For this magnetic field structure about 3% of the total photon energy is in the rangeof 1 −
10 keV. The newly created particles carry away about 63% of the energy radiatedby the primary particle, while about 37% of the energy remains in the form of photons.The structure of the magnetic field of PSR B0943+10 allows enhanced pair productionin a region of a weaker magnetic field (see Figure 5.19). The SR that accompanies pairproduction at higher altitudes ( z > −
10 keV energy band. Note, however, that the fraction of energyradiated in this band is still relatively low (3%), and in order to be a substantial part ofthe observed X-ray spectrum the strong anisotropy of backstreaming and outstreamingplasma is required (see Section 5.1.3).
Figure 5.19:
Synchrotron Radiation in the PSG-off mode for PSR B0943+10. The left panelpresents the places at which SR-photons are generated, while the right panel presents the SR-photons distribution. Plots were obtained in a cascade simulation calculated for a single primaryparticle moving along the extreme left open magnetic field line.
In the PSG-on mode even for a complicated structure of the magnetic field most ofthe outflowing energy is converted to secondary plasma. Figure 5.20 presents the ICS-photons distribution produced in the PSG-on mode for PSR B0628-28. The bulk ofenergy is radiated in the form of high energetic γ -photons which are responsible for pairproduction, and thus the formation of secondary plasma. Taking into account not sohigh backstreaming/outstreaming anisotropy in the PSG-on mode, the ICS process is notrelevant for the production of X-ray photons. .2. Inner magnetosphere of a pulsar Figure 5.20:
Distribution of ICS photons produced by an upscattering of the whole surface radiation( T s = 0 . ) for PSR B0628-28. The plot includes all photons upscatterd by primary particleswith Lorentz factors in the range of × − . × . The SR which accompanies the pair creation process in the PSG-on mode mostlyproduces soft γ -photons (see the right panel of Figure 5.21). Although the secondarypairs are produced at similar altitudes in both modes, (compare the left panels of Figures5.19 and 5.21), the higher Lorentz factor of secondary plasma produced in the PSG-on mode results in higher energy of the SR-photons. The results suggest that whenthe gap operates in the PSG-on mode we should expect lower efficiencies of nonthermalX-ray emission than in the PSG-off mode. Note, however, that the final efficiency ofX-ray radiation in the PSG-off mode highly depends on the backstreaming/outstreaminganisotropy and the structure of magnetic field lines. Figure 5.21:
Synchrotron Radiation in the PSG-on mode for PSR B0628-28. The left panelpresents places at which SR-photons are generated, while the right panel presents the SR-photonsdistribution. Plots were obtained in a cascade simulation calculated for primary particles withLorentz factors in the range of × − . × moving along the extreme left open magneticfield line. Chapter 5. Physics of pulsar radiation
The multiplicity of secondary particles in the PSG-off mode is much higher than in thePSG-on mode. However, the primary plasma produced in the IAR of CR-dominatedgaps has a density considerably lower than the Goldreich-Julian co-rotational density(see Equation 3.40). Figure 5.22 presents the energy histogram of secondary plasma forGeminga (left panel) and PSR B1133+16 (right panel). Despite major differences in themagnetic field structure and conditions in the IAR for both pulsars, the secondary plasmadistribution shows many similarities. The only significant difference is the maximumLorentz factor of secondary plasma, which for Geminga is about γ maxsec ≈ , while forPSR B1133+16 is is a few times smaller γ maxsec ≈ × .By using the overheating parameters presented in Table 3.1 we can roughly estimatethat the final multiplicity of particles in the plasma cloud in the PSG-off mode rangesfrom M = κ · M sec ≈ M = κ · M sec ≈
100 (for PSR B1133+16).Note, however, that these values do not take into account the anticipated anisotropy ofbackstreaming and outstreaming particles. The existence of such an anisotropy couldfurther increase the final multiplicity of particles in the plasma cloud leaving the innermagnetosphere. Despite the fact that without a full cascade simulation in the IAR wecannot unambiguously determine the final multiplicity in the plasma cloud, it can beclearly seen that depending on the details of the gap operating in the PSG-off mode, theproduced plasma may be suitable (e.g. PSR B1133+16) or unsuitable (e.g. Geminga) togenerate radio emission (see Section 5.2.3). The main factor determining the parameters ofthe CR-dominated gap, and thus determining whether it is possible to effectively produceradio emission, is the radius of curvature of the magnetic field lines (see Section 3.3.2).
Figure 5.22:
Energy histogram of secondary plasma in the PSG-off mode. The left panel wasobtained in a cascade simulation calculated for a single primary particle moving along the extremeleft open magnetic field line of PSR J0633+1746, while the right panel corresponds to a cascadesimulation for PSR B1133+16.
In ICS-dominated gaps, on the other hand, the density of primary plasma produced .2. Inner magnetosphere of a pulsar M = N ICS × M sec . As mentioned in Section 5.1.3, theexact value of N ICS can be found only by performing the full cascade simulation in IARbut, as shown by Timokhin (2010), we should expect a full screening of the accelerationregion when N ICS reaches a value as high as 20 − T s = 0 . M ≈ Figure 5.23:
Secondary plasma produced in the PSG-on mode for PSR B0628-28. The left panelpresents places at which pairs are produced, while the right panel presents the histogram of particleenergy. Plots were obtained in a cascade simulation calculated for primary particles with Lorentzfactors in the range of × − . × moving along the extreme left open magnetic field line.The ICS process was calculated using the whole surface radiation with temperature T s = 0 . . In the PSG-on mode the main factor which determines the final multiplicity of second-ary plasma is the source of the background photons. As shown in Section 4.4.4, the polarcap radiation (the hot spot component) has a negligible impact on the ICS process abovethe IAR. Figure 5.23 presents the location of pair production and energy distributionof secondary plasma for PSR B0628-28 calculated assuming the whole surface radiationwith temperature T s = 0 . M sec ≈
60. For such conditions the final multiplicity of secondary plasma in the PSG-onmode is of the order of M ≈ − .46 Chapter 5. Physics of pulsar radiationFigure 5.24:
Secondary plasma produced in the PSG-on mode for PSR B0628-28. The left panelpresents the places at which pairs are produced, while the right panel presents the histogram of particleenergy. Plots were obtained in a cascade simulation calculated for primary particles with Lorentzfactors in the range of × − . × moving along the extreme left open magnetic field line.The ICS process was calculated using the whole surface radiation with temperature T s = 0 . . onclusions The hot spot component identified in X-ray observations implies the non-dipolar structureof surface magnetic field. We used the Partially Screened Gap model to explain both theX-ray radiation of radio pulsars and production of secondary plasma suitable for genera-tion of radio emission.
A special case (PSR B0943+10)
Our model predicts two additional sources of X-ray emission: (I) the warm spot compon-ent and (II) enhanced SR radiation in the PSG-off mode. The warm spot component isassociated with particles originating from the antipodal polar cap, while the high lumin-osity of X-ray photons produced in the PSG-off mode is a result of strong anisotropy ofbackstreaming and outstreaming particles.Very recent results presented by Hermsen et al. (2013) show the anti-correlation of ra-dio and X-ray emission of PSR B0943+10. The authors suggest an unpulsed, non-thermalcomponent in radio-bright mode and a 100%-pulsed thermal component along with a non-thermal component in a radio-quiet mode. In our model it is not possible to produce anunpulsed, nonthermal X-ray component without the accompanying blackbody radiationof the polar cap. Although it is possible to produce nonthermal X-ray radiation whichobscures the thermal component (strong SR in the PSG-off mode with a high predom-inance of outstreaming particles), the resulting radiation should be pulsed. We believethat the X-ray radiation of PSR B0943+10 in the radio bright mode was misinterpretedas the nonthermal one. As shown in Figure 1.7 (panel d), for a derived geometry of PSRB0943+10 the polar cap produces unpulsed, thermal radiation. Furthermore, as repor-ted by the authors, in the radio-bright mode both the absorbed blackbody (BB) and theabsorbed power-law (PL) models fit the spectrum equally well (see Table S4 in Hermsenet al., 2013). We believe that the observed radiation modes of PSR B0943+10 correspondto a mode switch between the PSG-on (radio-bright) and the PSG-off mode (radio-quiet).When pulsar is in the PSG-on we observe both the radio emission and thermal radiationwhich originates from the polar cap. In the PSG-off mode the secondary plasma is notsuitable to produce so strong radio emission as in the PSG-on mode, but the polar capradiation is accompanied by pulsed, nonthermal emission produced by SR (see Sec. 5.2.2).14748
Chapter 5. Physics of pulsar radiation
Gamma-ray pulsars
As was shown in Sections 5.1.1 and 5.2.1, γ -rays produced in IAR and the inner mag-netosphere cannot reach the observer due to efficient pair production in those regions.Current models of γ -ray emission propose that the emission comes from outer magneto-spheric gaps. The non-dipolar structure of a magnetic field has two key implicationson γ -ray emission models: (I) the formation of slot gaps is not possible as pairs are pro-duced along all open magnetic field lines, (II) the high density of electron-positron plasma( n p (cid:29) n GJ ) produced in the inner magnetosphere prevents the outer gap formation. Thehigh-density plasma which crosses the null line will screen the outer magnetospheric regiondue to plasma separation (acceleration of electrons and deceleration of positrons). Thus,the formation of outer gaps is possible only in special cases when the pulsar operates inthe PSG-off mode and produces secondary plasma with low density n p ≈ n GJ .As recently reported by Arka and Dubus (2013): “It is possible for relativistic popu-lations of electrons and positrons in the current sheet of a pulsar’s wind right outside thelight cylinder to emit synchrotron radiation that peaks in the sub − GeV to GeV regime,with γ -ray efficiencies similar to those observed for the Fermi/LAT pulsars.” We believethat the observed high-energetic γ -rays are produced in the not yet well explored regionright outside the light cylinder. Radio emission
Pulsed radio emission remains one of the most intriguing puzzles of astrophysics. It is re-markable that despite the large ranges in P , B d , the variations in the pulse profile betweendifferent classes of neutron stars (young, old, millisecond, magnetars) are similar to thosewithin classes (Melrose, 2004). The radio emission of most pulsars can be characterisedby: a relatively narrow frequency range, ∼
100 MHz to ∼
10 GHz, and a high degree ofpolarisation with a characteristic sweep of the position angle. The extremely high bright-ness temperature of pulsar radio emission (typically T b > K) implies that a coherentemission mechanism is involved. Many radio emission mechanisms have been proposed,but no consensus on a specific emission mechanism has emerged. The radio observationsalone cannot identify the emission mechanism and, hence, a model of the magnetosphereis needed to put constraints on the radio emission model. An acceptable emission mech-anism must involve some form of instability to produce coherent radiation. The maindifficulty in finding a specific emission mechanism is that many of the predicted featuresare common all proposed models. Furthermore, the polarisation can also be regarded asgeneric rather than associated with a specific emission mechanism (Melrose, 2006).The X-ray observations have allowed us to put constraints on the polar cap region ofpulsars. The non-dipolar structure of the surface magnetic field causes plasma to formunder similar conditions regardless of the global configuration of the magnetic field. We .2. Inner magnetosphere of a pulsar
The mixed mode
Although in the thesis we consider the PSG-on and PSG-off mode separately, in a realcase both of these modes can coexist either on two separate polar caps or on the samepolar cap occupying its different parts. In the latter case the change of modes is associatedwith varying degrees of intensity of the two modes. Furthermore, if specific conditionsare met, the ICS process can be a main source of γ -photons in the lower parts of the gap,while the CR process can produce γ -photons in the upper parts of the acceleration region.In such a case distinguishing between the two modes is even more difficult. Summary
The main propositions associated with this thesis are as follows:1. The size of the hot spots implies that the magnetic field configuration just abovethe stellar surface differs significantly from a purely dipole one.2. The analysis of X-ray observations shows that the temperature of the actual polarcap is equal to the so-called critical value, i.e. the temperature at which the outflowof thermal ions from the surface screens the gap completely.3. The non-dipolar structure of a surface magnetic field and the high multiplicity ofparticles produced in IAR prevents the formation of slot and outer gaps.4. The PSG model predicts the existence of two scenarios of gap breakdown: the PSG-off mode for CR-dominated gaps and the PSG-on mode for ICS-dominated gaps.5. The two different scenarios of gap breakdown can in a natural way explain the mode-changing phenomenon when both modes produce plasma suitable to generate radioemission, and pulse nulling when the radio emission is not generated in one of themodes.6. The mode changes of the IAR may explain the anti-correlation of radio and X-rayemission in very recent observations of PSR B0943+10 (Hermsen et al., 2013).7. The regular drift of subpulses can be expected only when the gap operates in thePSG-on mode. The proposed model of drift allows to connect the drift informationobtained by radio observations with the X-ray data of rotation-powered pulsars.50
Chapter 5. Physics of pulsar radiation cknowledgements
I would like to express my deep gratitude to Professor Giorgi Melikidze, my researchsupervisor, for his patient guidance, enthusiastic encouragement and useful critique ofthis research work. I would also like to thank Professor Janusz Gil for his advice andsupport which allowed me to complete this thesis. This research project would not havebeen possible without the support of many people. I would like to thank all my col-leagues at the Institute of Astronomy who taught me a lot and never refused to help:Professor Ulrich Geppert, Professor Dorota Gondek-Rosi´nska, Professor Jaros(cid:32)law Kijak,Dr. Krzysztof Krzeszowki, Dr. Wojciech Lewandowski, Professor Andrzej Maciejewski,Dr. Krzysztof Maciesiak, Dr. Olaf Maron, Dr. Roberto Mignani, Dr. Marek Sendyk, andDr. Agnieszka S(cid:32)lowikowska. And a special thanks to Mrs Emilia Gil for her assistance inall the administrative issues.I would also like to extend my thanks to friends and family for their support, sacrifice,patience and wisdom. My special thanks are extended to my parents for their supportand encouragement throughout my studies.
Thank you.
Image by Karolina Ro ˙zko
Chapter 5. Physics of pulsar radiation ist of Tables
List of Tables ist of Figures . −
10 keV band vs spin-down luminosity 91.3 Coordinate system co-rotating with a star . . . . . . . . . . . . . . . . . . 151.4 Cosine of the hot spots’ inclination angle [PSR B0950+08] . . . . . . . . . 171.5 Cosine of the hot spots’ inclination angle [PSR B1929+10] . . . . . . . . . 181.6 Cosine of the hot spots’ inclination angle [PSR B0943+10] . . . . . . . . . 181.7 Comparison of the observed flux fractions for geometric effect only and forgeometric effect with the inclusion of a gravitational bending of light . . . 201.8 Observed flux fraction as a function of the rotation phase [PSR B0950+08] 211.9 Observed flux fraction as a function of the rotation phase [PSR B1929+10] 211.10 Cartoon of the magnetic field lines in the polar cap region . . . . . . . . . 231.11 Diagram of the surface temperature vs. the surface magnetic field . . . . . 252.1 Model of a non-dipolar surface magnetic field . . . . . . . . . . . . . . . . 302.2 First and second derivative of the φ -coordinate of the magnetic field line . 342.3 Curvature of magnetic field lines (numerical approach) . . . . . . . . . . . 342.4 The radius of curvature of the magnetic field line . . . . . . . . . . . . . . 372.5 Curvature of the magnetic field lines vs. the height above the stellar surface 382.6 Possible non-dipolar structure of the magnetic field lines [PSR B0628-28] . 392.7 Zoom of the polar cap region [PSR B0628-28] . . . . . . . . . . . . . . . . 402.8 Curvature of the open magnetic field lines [PSR B0628-28] . . . . . . . . . 402.9 Possible non-dipolar structure of the magnetic field lines [PSR J0633+1746] 412.10 Zoom of the polar cap region [PSR J0633+1746] . . . . . . . . . . . . . . . 422.11 Curvature of the open magnetic field lines [PSR J0633+1746] . . . . . . . . 422.12 Possible non-dipolar structure of the magnetic field lines [PSR B0834+06] . 432.13 Zoom of the polar cap region [PSR B0834+06] . . . . . . . . . . . . . . . . 442.14 Curvature of the open magnetic field lines [PSR B0834+06] . . . . . . . . . 442.15 Possible non-dipolar structure of the magnetic field lines [PSR B0943+10] . 452.16 Zoom of the polar cap region [PSR B0943+10] . . . . . . . . . . . . . . . . 462.17 Curvature of the open magnetic field lines [PSR B0943+10] . . . . . . . . . 462.18 Possible non-dipolar structure of the magnetic field lines [PSR B0950+08] . 472.19 Zoom of the polar cap region [PSR B0950+08] . . . . . . . . . . . . . . . . 4815556 List of Figures ist of Figures T s = 0 . T s = 0 . T s = 1 . R ws = 1 km] 1385.14 Synchrotron Radiation in the PSG-off mode [PSR J0633+1746] . . . . . . 1395.15 Synchrotron Radiation in the PSG-on mode [PSR J0633+1746] . . . . . . 1395.16 Final photon distribution produced by a single primary particle [PSR 1929+10]1405.17 Synchrotron Radiation in the PSG-off mode [PSR B1929+10] . . . . . . . 14158 List of Figures T s = 0 . T s = 0 . T s = 0 . ibliography Abdo, A. A., Ackermann, M., Ajello, M., Atwood, W. B., Axelsson, M., Baldini, L.,Ballet, J., Barbiellini, G., Baring, M. G., Bastieri, D., and et al. (2010). The FirstFermi Large Area Telescope Catalog of Gamma-ray Pulsars. ApJS, 187:460–494.Abrahams, A. M. and Shapiro, S. L. (1991). Molecules and chains in a strong magneticfield - Statistical treatment. ApJ, 382:233–241.Adler, S. L. (1971). Photon splitting and photon dispersion in a strong magnetic field.
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