On the mystery of the interpulse shift in the Crab pulsar
OOn the mystery of the interpulse shift in the Crab pulsar
V.M. Kontorovich • S.V. Trofymenko
Accepted for publication in JPSA, Vol.7, no. 4, 2017
Abstract
A new mechanism of radiation emission inthe polar gap of a pulsar is proposed. It is the curvatureradiation which is emitted by returning positrons mov-ing toward the surface of the neutron star along fieldlines of the inclined magnetic field and reflects from thesurface. Such radiation interferes with transition radi-ation emitted from the neutron star when positrons hitthe surface. It is shown that the proposed mechanismmay be applicable for explanation of the mystery of theinterpulse shift in the Crab pulsar at high frequenciesdiscovered by Moffett and Hankins twenty years ago.
Keywords neutron stars, pulsar PSR B0531+21, non-thermal radiation mechanisms, shift of interpulse, co-herent emission, radiation energy spectrum
It is commonly accepted that a pulsar is a neutron starwith the radius of about ten kilometers which createsintense magnetic field around itself. In the vicinity ofthe surface of the star such field is often supposed to re-semble the field of a magnetic dipole. The star quicklyrotates around the axis which does not coincide withthe direction of its magnetic moment. Such rotationcauses the division of the space around the star into
V.M. KontorovichS.V. Trofymenko Karazin Kharkov National University, 4 Svobody Sq., Kharkov61022, Ukraine, Institute of Radio Astronomy of National Academy of Sciencesof Ukraine, 4 Art Str., Kharkov 61002, Akhiezer Institute for Theoretical Physics of NSC KIPT, 1 Aka-demicheskaya Str., Kharkov 61108, Ukraine. two regions (see e.g. Smith (1977); Manchester & Tay-lor (1977)). The first one is situated within the cylinder(light-cylinder) of radius R c = c/ Ω (where c is the speedof light and Ω is the pulsar rotation angular frequency)with the axis coinciding with the pulsar rotation one.The second region lies outside this cylinder. It leadsto the division of the magnetosphere into the regionsof closed and opened magnetic field lines. The firstones are situated completely within the light-cylinderin the region filled with co-rotating plasma of electron-positron pairs. The second ones emerge from the sur-face of the star in the vicinity of the magnetic poles andintersect the light-cylinder.Quick rotation of the highly magnetized star leadsto generation of intense electric field around it. In thevicinity of the magnetic poles this field has a longitudi-nal component parallel to the magnetic field. It acceler-ates the electrons from the surface of the star along thecurved opened magnetic field lines up to the value ofthe Lorentz-factor of the order of 10 . The hard part ofcurvature radiation emitted by such electrons generatesthe cascade of electron-positron pairs, which form themagnetospheric plasma (see e.g. Beskin (2010)). Dur-ing the acceleration process the electrons move insidea polar gap (Sturrock 1971; Ruderman & Sutherland1975; Arons 2009), between the surface of the star andthe pair plasma.We accept that the polar gaps are also the sources ofelectromagnetic radiation which has relatively narrowangular distribution around the magnetic axis but occu-pies a wide range of frequencies: from radio to gamma.Its origin is attributed to radiation produced by elec-trons accelerated in the polar gap by the electric fieldand penetrating into the magnetospheric plasma. Itseems that a whole lot of mechanisms of radiation pro-duction by such electrons take place in this case replac-ing each other in different frequency ranges. Such mech-anisms have been studied in a large number of papers a r X i v : . [ a s t r o - ph . H E ] A ug (see, e.g. books and reviews Smith (1977); Manchester& Taylor (1977); Malov (2004); Arons (2009); Harding(2009); Beskin (2010); Kaspi & Kramer (2016); Eilek &Hankins (2016) and references therein).Twenty years ago a thorough study of the averageshapes of radiation pulses (average light curves) arriv-ing from the pulsar in Crab nebula was made in (Mof-fett & Hankins 1996). During such investigation theobservation data was collected in wide range of frequen-cies: from hundreds MHz to a hundred keV. Recentlythese results were confirmed and supplemented (Hank-ins, Jones & Eilek 2015). Here we reproduce (with someadditional marks) a part of the figure from (Moffett &Hankins 1996) which represents the results of measure-ments of the phase dependence of the average registeredradiation intensity during one period of the pulsar ro-tation (fig.1). The average light curves are presentedfor different radiation frequencies. Fig. 1
Average light curves obtained at multi-frequencyobservations in (Moffett & Hankins 1996). The shifted (toabout 7 ◦ ) interpulse and high-frequency components aremarked. With gratitude to the authors At frequencies lower than 1.5 GHz two distinctpulses, registered during one period of rotation of thestar, are seen on fig.1. Such pulses, the main pulse (MP)and interpulse (IP), are believed to originate from dif-ferent magnetic poles of the Crab pulsar. Here we willassume that the magnetic axis of the star is nearly or-thogonal to its rotation axis.At frequencies around 3 GHz IP disappears. It ap-pears again at higher frequencies having phase sift δ of about 7 ◦ comparing to the initial IP at lower fre-quencies. Moreover, two more distinct pulses, knownas high-frequency components (HFCs), appear at thesame frequencies. At some higher frequency HFCs dis-appear and IP restores its previous position. Let usnote that MP disappears as well at the frequency ofabout 8 GHz appearing again at higher frequencieswithout any phase shift (Moffett & Hankins 1996). Since the discovery of these peculiar features of radioemission by the Crab pulsar no theoretical explanationof them has been presented. In the present work weconsider a mechanism of radio emission which may beresponsible for the shift of IP in the Crab pulsar. At theheart of this mechanism lies the reflected from the pul-sar surface curvature radiation by returning positrons moving in the polar gap toward the surface of the star.We show that such mechanism provides the possibilityof explaining the IP phase shift. The version of the ori-gin of HFCs is discussed in (Kontorovich 2016) also in-volving the reflection from the star surface and applyingthe idea of nonlinear Raman scattering of the positronradiation on the excitations of the pulsar surface. Theidea of reflection from the pulsar surface is applied herefor consideration of HFCs since they appear at the samefrequencies as the IP shift, which presently does nothave other explanation. For independent treatment ofthe HFCs, not connected with the IP shift, see (Petrova2009). The explanation of MP disappearance was proposed in(Kontorovich & Flanchik 2013). Such proposal wasmade on the basis of consideration of non-relativisticradiation mechanism during longitudinal accelerationof electrons in the polar gap as the mechanism of low-frequency radiation emission. When the acceleratedelectron reaches relativistic velocities this mechanismweakens and turns off at rather high frequency. As wasshown there, the value of such frequency is proportionalto (cid:112) B || , where B || is the magnetic field component par-allel to the rotation axis in the vicinity of the magneticpole of the star. The disappearance of IP may be at-tributed to the same physical reasons. The lower valueof the frequency at which it occurs, comparing to theone in the case of MP, can be caused by the lower valueof B || on the magnetic pole responsible for IP produc-tion (for details see Kontorovich & Trofymenko (2017)).Anyway, the radiation mechanism responsible forproduction of the shifted IP at high frequencies shoulddiffer from the one dominating in the case of unshiftedIP at lower frequencies (such idea correlates with theone expressed in Hankins, Jones & Eilek (2015)).As the origin of the shifted high-frequency IP weconsider the radiation by positrons moving toward the In the regions with the opposite direction of the acceleratingelectric field in the gap the same may refer to electrons movingto the star surface surface of the pulsar in the polar gap. Such positronscan be returned from the lower layers of magnetosphericplasma (just after the corresponding e + e − pairs are cre-ated) by the same electric field which accelerates theelectrons outward the star. The detailed considerationof such return motion of positrons in a pulsar mag-netosphere is presented in (Barsukov et al. 2016) andreferences therein.We assume that in the considered range of frequen-cies of the order of several GHz the surface of the staracts as ideal conductor reflecting all the incident radia-tion. Moving along curved magnetic field lines towardsthe surface of the star the positrons emit curvature ra-diation (CR), just like the electrons moving in the op-posite direction (Komesaroff 1970). Since the positronsare ultra relativistic, such radiation is emitted predom-inantly in the direction close to their velocities. Atsufficiently high frequencies it reflects from the surfaceand propagates outward the star. Moreover, when thepositrons hit the surface the so-called transition radia-tion is generated in the direction outwards the surfaceas well. The total radiation by positrons is the resultof interference of such transition radiation with the re-flected curvature radiation .In order to obtain the angular shift between the pre-dominant direction of the direct radiation by electrons,which is believed to be the origin of the unshifted low-frequency IP, and the corresponding direction of thereflected radiation by positrons (shifted IP) one moreessential assumption is needed. It is the requirementthat the magnetic axis on the surface of the star shouldbe inclined at some angle δ/ δ from the magnetic axis. It hap-pens due to the fact that both the average directions oftransition radiation and reflected CR in this case are Transition radiation is the radiation which is generated when amoving charged particle traverses the border between media withdifferent dielectric properties. It has wide and nearly constantspectrum covering the range from the long-wavelength part of theradio band and up to X-ray band for ultra relativistic particles.For details see (Ginzburg 1984; Ter-Mikaelyan 1972) According to (Dyks et al. 2006) and (Wright 2003), the reflec-tion from the neutron star surface was discussed in the report(Michel 1992) on the conference in Zielona Gora, presumably, inconnection with the problem of the drifting subpulses structure.Let us also remark that an interesting question concerning the so-called reversible radio emission (see i.e. Melikidze & Gil (2006))does not have direct analogy with our problem In the case of a relativistic charged particle oblique incidenceupon a conducting surface the major part of backward TR pulse
Fig. 2
Schematic picture of motion and radiation by elec-trons and positrons in the polar gap of the pulsar in thecase of a tilted magnetic axis. The directions of radiationby electrons and reflected radiation by positrons are shiftedat the angle δ of mirror reflection the directions of mirror reflection of the magnetic axiswith respect to the surface of the star. In order to investigate the properties of radiation pro-duced by the positrons moving toward the surface ofthe star and hitting it, it is necessary to adopt a cer-tain (naturally, simplified) picture of their motion. Fol-lowing the ideas elaborated in (Radhakrishnan 1969;Komesaroff 1970) for radiation by electrons, for cer-tainty, we will mainly consider radiation by positronsmoving along outer open magnetic field lines. Such linesare situated in the vicinity of the boundary between theregion of open and closed lines and have significant cur-vature. However, they are supposed not to belong tothe immediate vicinity of this boundary and still havesignificant accelerating electric field component alongthem. CR produced by such positrons is more inten-sive than the one emitted by the particles moving alongmore straight (with larger curvature radius) field linessituated closer to the magnetic axis (see discussion insec. 6 involving consideration of radiation coherence).Nevertheless, for estimation of the total positron radi-ation flux in sec. 5 and 6 we will take into accountthe contribution from the positrons on all the openedfield lines. In this section the important effect of radi-ation coherence will be included into our considerationas well.Firstly, let us note that CR emitted by a positroncan reflect from the surface of the star and propagate is concentrated close to the direction of mirror reflection withrespect to the direction of the particle velocity to the observer only if the emission occurs within someeffective part of the positron trajectory (further we willcall it effective path). On fig.3 this part leans on theangle α . The value of the positron Lorenz-factor γ within the effective path is restricted by its minimum γ min and maximum γ max values. The first one de-pends on the frequency and the curvature radius of thepositron trajectory (see formula (6) with the substi-tution γ → γ min ). The second one is defined by therelative position of the observation direction and themagnetic axis (for details see sec. 5). Since CR emis-sion by a relativistic positron occurs in the directionclose to its velocity, CR emitted prior to this regiondoes not reflect from the surface and, hence, is not ob-served. By α we denote here the angular coordinateof the positron which is counted from the point of thebeginning of its motion along the effective path.For the sake of simplicity of calculations we assumethat the positron moves along an arc of a circle of radius R ∼ cm . In this case effective path corresponds tothe angle α (cid:39) arccos(1 − R ∗ /R ) − R ∗ /R , where R ∗ ∼ cm is the radius of the pulsar, and has the value ofabout 3 . R ∗ . In fact, it is just the highest estimationfor the effective path since the radiation emitted in thedirection close to the ‘limit’ one (see fig.3) may notescape from the pulsar at all. In order to reach theobserver the reflected radiation should propagate in thedirection closer to the magnetic axis, which requires theemission to occur at lower altitudes (at larger α ). Fig. 3
Schematic picture of a return positron motion andradiation along the curved magnetic field line. α is theangle corresponding to the effective path, α is the currentpositron angular coordinate. The solid red line shows thedirection of the reflected radiation emitted by the positronat its current position, while the dashed one shows the direc-tion of radiation (emitted at α = 0) which just begins beingreflected from the surface (Kontorovich & Trofymenko 2017) In the region corresponding to the effective path weuse the following simplified model of accelerating field: E ( α ) = E ( α/α − α ) θ ( α − α )+ (1) (cid:0) E + E ( α − α ) (cid:1) (1 − α ) θ ( α − α ) , where θ ( x ) is the step function, which is equal to zerofor x < x > α < α ),which lies in the vicinity of the magnetospheric plasma,and then changes parabolically reaching some maxi-mum value and turning to zero on the surface of thestar ( α = 1). In (1) α and α are measured in the unitsof α . The linear section corresponds to smooth pen-etration of the field into the lower magnetosphere, cf.(Shibata et al. 2002), the parabolic – to a simplified ver-sion of known models, see (Arons 2009; Harding 2009). E and E are some constants defining the magnitudeof the electric field.The equation defining the dependence of the positronLorenz-factor γ on α is dγ/dα = eE ( α ) R/mc , (2)which can be derived from the well-known equation fora particle energy gain in external electric field: dε/dt = e Ev . The solution of this equation with the field given by (1)leads to the following expression for the Lorenz-factor: γ ( α ) = θ ( α − α ) { γ (0) + f (1 − α ) α / α } + (3)+ θ ( α − α ) { γ (0) + f (1 − α ) / δγ ( α ) } ,δγ = ( f /α − f )( α − α )++ (cid:0) f (1 + α ) − f (cid:1)(cid:0) α − α (cid:1) / α − f ( α − α ) / α , where f , = eα E , Rα /mc .In the next section it will be shown that it is possibleto choose the values of E and E in such way that ra-diation emitted by a positron in the region of the lineargrowth of the field belongs to radio band. Moreover, theappearance of the shifted IP in the vicinity of a certainfrequency (about 5 GHz, see fig.1) can be realized inthe adopted model. In order to calculate the spectral-angular density of ra-diation emitted by a positron moving along a curved magnetic field line in the polar gap and falling on thesurface of the star we use the method of images. Suchmethod is applicable even in ultra relativistic case if thesurface of the star is approximately considered as flat.In this method the radiation emitted by the positronis considered as radiation emitted by its image movinginside the star the mirror symmetrically to the positronwith respect to the surface (fig.4). When the positronenters the surface of the star it disappears from thepoint of view of the external observer. It happens dueto the fact that the positron’s charge in this case iscompletely screened by polarization charges and cur-rents inside the star surface (which we consider here asa perfect conductor). In the terms of the method ofimages such disappearance of the positron is describedas abrupt stop of the particle and its image at the samepoint on the surface (Bolotovsky 1982; Ginzburg 1984).The charges of the positron and its image screen eachother in this case which is analogous to the particlesdisappearance.
Fig. 4
The geometry of a positron and its image motionused for calculations. The radiation direction angle ψ ap-proximately corresponds to a certain value of the radiatingparticle Lorentz-factor and radiated frequency (Kontorovich& Trofymenko 2017) In the considered method the radiation emittedby the image during its motion is analogous to thereflected curvature radiation of the positron. Thebremsstrahlung generated by the image at its abruptstop on the surface is analogous to transition radiationwhich occurs when the positron hits the surface.By χ we define the value of the angle between theline tangent to the magnetic field line on the surface ofthe star and the direction of the magnetic axis. Theangle ψ is the angle between the radiation observationdirection and the magnetic axis. The spectral-angular density of radiation emitted bythe positron’s image can be calculated with the useof the well-known expression (e.g. see Jackson (1999);Landau & Lifshitz (1987)) for distribution of the energyradiated by a particle with the known law of motion r = r ( t ) : d Wdωdo = e ω π c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) −∞ dt [ n , v ( t )] exp (cid:26) iω ( t − nr ( t ) c ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)Here v ( t ) is the image velocity and n is a unit vectorin the direction of observation.We do not take into account the radiation emittedwhen the positron changes its direction of motion (af-ter the pair creation) under the impact of the electricfield and starts its motion towards the surface. This isdue to the fact that at the beginning of such motionthe positron Lorenz-factor is close to unit. The radia-tion emitted in this region at relatively high consideredfrequencies (several GHz) is less intense than the curva-ture radiation generated during further positron motiontowards the surface.Thus we consider radiation emitted by the positron(or rather its image) which approaches the effectivepath having some moderately relativistic initial velocityv . Then it gains energy accelerating along the curvedmagnetic field line up to standard values of particlesLorenz-factors in this case γ ∼ − and hits thesurface.The radiation is predominantly concentrated in thevicinity of the plane of the positron motion. It is thisplane which we will further consider radiation charac-teristics in. For the vector product in (4) we can write:[ n , v ( t )] = e ⊥ v · sin ( α − α − χ − δ + ψ ) , where e ⊥ is a unit vector perpendicular to the planeof the positron motion. The integration over time in(3) can be substituted by the one over α with the useof the relation v dt = Rdα . The dependence of t on α during the positron motion within the effective path(0 < α < α ) can be expressed with the use of (2) asfollows: t ( α ) = Rc α (cid:90) dα (cid:48) β ( α (cid:48) ) , in which β ( α (cid:48) ) = v( α (cid:48) ) /c = (cid:112) − /γ ( α (cid:48) ). This quantity can be expressed in terms of the positron accel-eration ˙v as well through taking the integral with respect to t inthe expression (4) by parts, see, e.g. (Jackson 1999) Thus we obtain the following expression for spectral-angular density of radiation by a single positron in theconsidered case: d Wdωdo = e w π c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i β sin( η ( ψ )) exp[ i w sin( η ( ψ ))]w[1 − β cos( η ( ψ ))] + (5) α (cid:90) dα sin( α − η ( ψ )) exp (cid:26) i w (cid:18) ctR − sin( α − η ( ψ )) (cid:19)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where η ( ψ ) = α + χ + δ − ψ , w = ωR/c and β = v /c .Fig.5 shows the frequency dependence of the radi-ated energy at different observation angles ψ calculatedwith the use of expression (5). The figure shows that inthe direction defined by a certain angle ψ the consid-ered radiation is emitted in the vicinity of a certain fre-quency ω ( ψ ). This frequency corresponds to the maxi-mum of radiation spectral distribution in the considereddirection. Due to narrow angular distribution of theemitted radiation around the direction of the positroninstantaneous velocity, it is a small part of the positrontrajectory which is responsible for radiation at certainangle ψ . On this small part of the effective path, whichcorresponds to a certain value α = α ψ , the positron hasa certain Lorenz-factor γ ( α ψ ). Thus the characteristicfrequency ω ( ψ ) of radiation reflected from the surfacein the considered direction can be estimated as ω ( ψ ) ∼ cγ ( α ψ ) /R (6)as a frequency of curvature (or synchrotron) radiationspectrum maximum. From fig.4 it is possible to derivethe relation between ψ and α ψ as ψ = α + δ + χ − α ψ .The radiation emitted by positron at the very be-ginning of the effective path also has its characteristicfrequency ω ( α + δ + χ ) (since α ψ = 0 in this case). How-ever, this wave is emitted tangentially to the surface ofthe star (see fig.3) and propagates at large angle to themagnetic axis. Such wave, even if it escapes from themagnetosphere, does not hit the telescope. The samesituation takes place for the waves emitted at lower al-titudes (0 < α (cid:28) α ) at the beginning of the effectivepath. Such waves, though being reflected from the sur-face of the star, still have rather large angle between themagnetic axis and the wave vector direction. The wavesemitted at even lower altitudes after reflection from thesurface propagate in the direction close enough to theone of the magnetic axis to be able to escape the polargap through relatively rarefied layer of plasma.The maximum value ψ max of the angle ψ at whichthe reflected radiation still can escape from the pul-sar and hit the telescope, certainly, depends on a series Fig. 5
Radiation spectral distribution (smoothed) for dif-ferent values of ϕ = ψ − δ − χ . The angle ψ (see Fig.4)corresponds to the value (6) of radiation frequency ω ( ψ )and, hence, to a definite value of radiating positron Lorenz-factor γ . The values f = 4 · , f = 3 · and α = 1 / γ ( α ) are used. The figure showsthat at frequencies exceeding the one of the spectrum max-imum, the radiation intensity rapidly decreases. The spec-trum with the maximum at frequency ω min , at which theshifted IP appears, corresponds to the value γ = γ min (see(7)). of parameters characterizing the magnetosphere and,probably, can not be defined from pure theoretical spec-ulations. Fig.5, illustrating the expression (4), showsthat in the model under discussion the characteristicfrequencies of the waves propagating in the directionscorresponding to smaller values of ψ are higher than ω ( ψ max ) ≡ ω min . Therefore ω min can be approxi-mately considered as the minimum frequency at whichthe pulse of radiation from the positrons can be ob-served. According to (6), the value of ω min is relatedto characteristic minimum value γ min of the positronLorenz-factor, at which the emitted radiation beginshitting the telescope, as γ min ∼ ( ω min R/c ) / (7)Let us take, for example, γ min ∼ f = 4 · and f = 3 · for the corresponding quantities in (3) gives the value ω min / (2 π ) ∼ GHz for the minimum characteristic fre-quency at which the positron radiation pulse appears.As fig.1 shows, such result is in accordance with theobserved value of the frequency at which the shifted IPappears.It is necessary to note that fig.5 shows the frequencydependence of the averaged (over small frequency inter-vals) value of radiation spectral-angular density definedby (4). In fact there are small-amplitude rapid oscilla-tions of the considered quantity around the presentedaverage value. Such oscillations are caused by interfer-ence of the reflected curvature radiation with the tran-
Fig. 6
The radiation spectral distribution (not smoothed)for ϕ = 15 ◦ . Rapid oscillations originate from the interfer-ence of the reflected curvature radiation with the transitionradiation. The figure is presented for the frequencies lowerthan the ones we are interested in so that the oscillationswere distinguishable (Kontorovich & Trofymenko 2017). sition one. We decided not to present such oscillationson fig.5 since they do not play important role for ourestimations. Moreover they will, probably, disappearafter averaging over large number of positrons radiat-ing simultaneously within the polar gap. Nevertheless,the example of the real positron radiation distributionis presented on fig.6. Such distribution corresponds toa larger value of the angle ϕ (and, hence, smaller fre-quencies) for which the time of numerical calculationson the basis of the expression (4) is much shorter.Let us also note that we have chosen the region ofthe linear growth of the electric field to occupy the halfof the effective path. In this case the positron Lorenz-factor reaches the value γ ( α ) ∼ in this region. Let us now estimate the flux of the reflected radiationproduced by positrons moving along magnetic field linestowards the surface of the star above the entire area ofthe polar cap. The observations indicate the necessityof coherent character of radiation emission by chargedparticles in the pulsar magnetosphere. We will makeestimation taking into account this fact and use a se-ries of results concerning coherent radiation emissionby a large number of particles discussed in detail inAppendix. For such estimation we will, as previously,consider the positron trajectories in a simplified wayas arcs of circles of radius R varying with the distancefrom the magnetic axis.For the estimation of CR intensity and derivationof the radiation energy spectrum at first it is enoughto accept the very fact of the existence of inhomoge-neous (clumped) positron flux in the gap. In this case the main contribution to radiation is made by coher-ently radiating volumes, which we will estimate further.The number of such volumes can be rather roughly es-timated through “division” of the total radiating vol-ume by the coherence volume. It is also necessary totake into account the dependence of such volume onthe wavelength, Lorenz-factor and the curvature of thepositrons trajectories.The radiation spectral density at frequency ω pro-duced by a single positron, which moves along such anarc, per unit path can be estimated as follows: W ω ∼ e πRc (cid:16) ωRc (cid:17) / . (8)It directly follows from the expression for spectral den-sity of radiative energy loss by a particle moving alonga circular trajectory (see, e. g. Jackson (1999)).The size of a volume of space which encloses amountof particles radiating coherently we will take as V coh = r || r ⊥ ∼ γ λ / (4 π ) . (9)Here the distances r || ∼ λ/π (10)and r ⊥ ∼ γλ/ (2 π ) (11)define the linear dimensions of such coherently radi-ating volume in the direction of the particles motionand in the one perpendicular to it respectively (seeAppendix). It is the square of the average number N coh ∼ κ · n GJ V coh of positrons in such volume that ra-diation flux produced by these particles is proportionalto. Here n GJ ∼ cm − is the Goldreich-Julian den-sity and κ < V coh which lie withinthe surface element 2 πrdr of the polar cap at certainaltitude is then d N ⊥ ∼ πrdrr ⊥ = 2 πrdrγ λ (2 π ) . (12)The upper estimation for the number of such volumesfalling on the polar cap surface segment of the order of r ⊥ per unit of time is then d N || dt ∼ cλ . (13)The effective solid angle which encloses the consideredradiation emitted by positrons with the Lorentz-factor γ can be estimated as Ω eff ∼ π/γ . This leads to thefollowing expression for the effective area which suchnarrow pulse of the reflected radiation traverses on dis-tance d from the star: S eff ∼ πd /γ .Summarizing the previous considerations we canpresent the expression for the flux of the reflected radi-ation by positrons as follows: J ( ω ) ∼ (cid:90) dzd N ⊥ d N || dt W ω N coh S eff . (14)After substitution of the explicit expressions for thequantities presented here it transforms to J ( ω ) ∼ e κ n GJ λ − (2 π ) π d R PC (cid:90) drrR ( r ) (cid:90) dzγ ( r, z ) , (15)where z = Rα (see Fig.4) is the positron coordinatealong the magnetic field line and R P C is the polar capradius.Let us assume, as previously, the linear growth ofaccelerating electric field strength in the region of thepositrons trajectories contributing to the observed ra-diation flux. In this case with the use of (2) we can ex-press the positron coordinate through its Lorenz-factorin the following form: dz = (cid:115) mc ¯ h eE ( r ) dγ √ γ , (16)where ¯ h is some averaged (with respect to r ) lengthof the positron trajectory interval in the considered re-gion. The quantity E ( r ) is the accelerating electric fieldstrength itself, which varies with the distance r from themagnetic axis. We will take the explicit dependence ofthe electric field on this distance in the following form(cf. Kontorovich & Flanchik (2013)): E ( r ) = E (1 − r /R P C ) , (17)which implies vanishing of the field on the polar capboundary.The dependence of the curvature radius of the mag-netic field lines on r in the vicinity of the surface ofthe star in the case of a dipole field is described by theexpression: R ( r ) = 4 R ∗ r , (18)in which R ∗ ∼ cm is the radius of the pulsar.Substituting the expressions (16)-(18) into (15) weobtain: J ( ω ) ∼ κ n GJ e λ − / / π / d R / ∗ (cid:115) mc ¯ h eE × × R PC (cid:90) drr / (cid:112) − r /R P C γ max (cid:90) γ min ( r,ω ) dγγ / , (19)where the value γ max , which we roughly consider asindependent on r , is the effective value of the positronLorenz-factor at which its radiation reflected from thesurface of the star ceases to hit the telescope (Fig.7).Here γ min is some minimal value of the positron Lorenz-factor for the corresponding magnetic field line. Forinstance, on the outer field lines it coincides with thevalue defined by the expression (7).Here we assume that the initial angular width of thepositron radiation diagram ∼ /γ min , when the parti-cle moves at the beginning of the effective path, exceedsthe value of the minimal angle θ min between the mag-netic axis and the direction to the telescope . Withthe increase of the positron Lorenz-factor at lower alti-tudes the characteristic angle of its radiation diagrambecomes less than θ min (at γ ∼ γ max ) and radiationceases to be caught by the telescope.As the expression (19) shows, the contribution to theradiation flux in our case grows with the increase of γ .Due to this fact the value of the integral in (19) withrespect to γ is mostly defined by the upper limit γ max while the exact value of γ min is not very significant.Let us note that such situation is different from thewell known case of synchrotron radiation of the electroncomponent of cosmic rays with the decreasing energyspectrum. Due to such spectrum of the electron ener-gies the integral with respect to γ in this case is merelydefined by the lower limit and the contribution to theradiation flux is associated only with γ min . It leadsto the well known relation between spectral indices ofsuch cosmic radio emission sources as radio galaxies,quasars, supernova remnants etc.The final estimation of the total reflected positronradiation flux, obtained from (19), is the following: J ( ω ) ∼ κ n GJ e λ − / π / d R / ∗ (cid:115) mc ¯ h eE γ / max R / P C , (20)Here we neglected the value γ min comparing to γ max .The electric field strength E on the magnetic axiscan be estimated as: E ∼ a Ω R ∗ c B ∼ CGSE , where Ω = 2 π/T is the angular frequency of the pulsarrotation, B ∼ Gs is the magnetic field strengthand a ∼ − ÷ − is some small parameter taking As was noted previously, we assume nearly orthogonal mutualdirection of the magnetic and the rotation axes.
Fig. 7
Schematic picture of angular regions (cones) of con-centration of the reflected radiation by positrons at two ex-treme values of the considered Lorenz-factors. At γ = γ max radiation ceases to hit the telescope. The magnetic axisdoes not coincide with the ones of the cones, among otherreasons, due to its assumed inclination with respect to thesurface normal into account the effect of geometrical and gravitationalfactors (for details see Beskin (2010)).Then for the values R P C = 10 cm, ¯ h = 10 cm, d = 6 · cm (which is the distance from the Crabpulsar to the Earth (Manchester et al. (2004))) of thecorresponding quantities in (20) this estimation reducesto the following: J ( ω ) ∼ κ λ − / γ / max · − WHz · m . (21)Such estimation may, probably, be improved if applythe self-consistent calculation of the accelerating field(see ref. in Arons (2009); Muslimov & Harding (2004))and the returning positron flow (Shibata 2002).As the expression (21) shows, the radiation flux inour model depends on a certain parameter γ max defin-ing some average effective value of positron Lorenz-factor at which its radiation ceases to hit the telescope.Its value can be approximately estimated by the com-parison of the calculated value of flux (21) with the oneobtained from the observations of the Crab pulsar ra-dio emission. For the latter, for instance, the resultspresented in (Sieber 1973) and (Smith 1977) can be ap-plied. Namely, quite nice agreement by the order ofmagnitude between the results of our model and obser-vations in the range of frequencies, in which the inter-pusle shift takes place, is achieved if choose γ max ∼ (if take κ ∼ − as well).Expression (21) also shows that in the consideredhere interval of wavelengths the radiation energy spec-trum is defined by the term λ − / ≈ λ . , which is inquite good qualitative agreement with observation re-sults (Sieber (1973); Smith (1977), for observations atlower frequencies see Ellingson et al. (2013)).Let us note that the upper estimation (13) for thenumber of coherence volumes crossing the square r ⊥ perunit of time is supposed to work well for large values of the ratio λ/L . Here L is some characteristic spatialsize of longitudinal inhomogeneities (bunches) of thepositron flow (for details see sec. I and II of Appendixdevoted to partially coherent radiation). For the caseof shorter waves with λ/L < λ to L in (13). In this case we have to multiply (21)by λ/L and radiation spectrum becomes proportionalto λ − / , which even seems to be in better accordancewith observations than (21). In the framework of our model the value γ min of thepositron Lorenz-factor at the beginning of the effectivepath defines the minimal frequency ω of its radiationwhich can reflect from the surface and hit the tele-scope. The corresponding relation between γ min and ω , following from (6), can be presented as: ω ∼ cR γ min . (22)Here the radius R of the positron trajectory (magneticfield line) is defined by (18). Thus the value of γ min canbe expressed through radiated frequency ν = ω/ (2 π )and the distance r from the magnetic field line as γ min ( ν, r ) ∼ (cid:18) πνR ∗ cr (cid:19) / . (23)Here we see that γ min grows with the increase of ν anddecrease of r . Therefore, at some value of r = r ( ν )the magnitude of γ min reaches the one of γ max and theinternal integral in (19) tends to zero. In the region ofpolar cap r < r in this case the considered radiationmechanism does not work. According to (23), the valueof r can be estimated as r ( ν ) ∼ πνR ∗ cγ max . (24)The quantity r is some characteristic radius of themagnetic axis vicinity, in which the moving positronsdo not contribute to the observed radiation. Thus inthe considered model the coherence leads to formationof something like a hollow cone (in accordance withRadhakrishnan (1969)), restricted from the inner andouter sides by the field lines situated respectively ondistances r and R P C (in the vicinity of the star surface)from the magnetic axis (fig.8). The particles movingin the specified region make the main contribution toradiation, which corresponds to classical conceptions ofpulsar radiation mechanisms. Fig. 8
Region of space (orange hollow cone) contributingto positron coherent radiation flux at ν < ν max
With the increase of the frequency ν the “wall thick-ness” of the considered cone decreases ( r → R P C ) andthe contribution to the coherent radiation is made justby more and more contracting belt in the vicinity of theouter magnetic field lines.The frequency ν = ν max , at which r reaches the ra-dius R P C of the polar cap, is the one at which the con-sidered radiation mechanism providing the interpulseshift disappears. With the use of (24) it can be esti-mated as ν max ∼ cR P C γ max πR ∗ . (25)Let us note that more accurate estimation of this fre-quency requires application of self-consistent models ofthe positron flow motion in the polar gap.Taking into account the considered here effect of re-duction of the region above the polar cap, which makescontribution to coherent radiation emission, with theincrease of the frequency, the expression (20) for theradiation flux may be slightly modified. It is associ-ated with the fact that the integration with respect to r in (19) in this case should be made in the interval r < r < R P C . It leads to the following expression forradiation flux: J (cid:48) ( ν ) ∼ J (cid:20) − x / + 97 x / (cid:21) , (26)where x = r R P C = 8 πνR ∗ cR P C γ max . and the quantity J is defined by the expression (20).From (26) it follows that significant modification of the flux due to the considered effect takes place only atfrequencies in the vicinity of ν max (when r ≈ R P C ). Atlower frequencies the values of the modified expression(26) and of the one, not taking into account the effectof radiation region reduction (20), are nearly the same.
In the present work a new mechanism of radio emissionin the polar gap of a pulsar is proposed. It is the ra-diation by positrons accelerated toward the surface ofthe star along curved magnetic field lines by the sameelectric field which accelerates electrons in the oppo-site direction. The total radiation by positrons in thiscase consists of curvature radiation reflected from thesurface of the neutron star as well as of transition ra-diation emitted when the particles hit this surface. Itis shown that the considered radiation mechanism canbe applied for explanation of such unusual fact of theCrab pulsar radio emission as the shift of its interpulseat several GHz frequency. For this the assumption ofthe inclined magnetic field is used. The total flux of thecoherent positron radiation is estimated. The obtainedvalue (as well as the radiation spectrum) quite nicelyagrees with the results of observation. In the frameworkof the proposed model the highest frequency at whichthe interpulse shift takes place is roughly estimated.
Acknowledgments
We are grateful to E.Yu. Bannikova, I. Semenkina,N.F. Shul’ga, D.P. Barsukov, O.M. Ulianov, V.V. Za-kharenko and D.M. Vavriv for help and discussion.
APPENDIX. On the coherence of radiation bypositrons.I. General considerations
The high intensity of radiation coming from the Crabpulsar (as well as from the other pulsars) claims thecoherent (at least partly) emission of such radiationby a large number of particles in the pulsar magneto-sphere (see, e.g. the latest review Melrose (2017)). Thepossibility of coherent radiation emission by a flow ofpositrons moving along curved magnetic field line arisesif the positron flux becomes inhomogeneous.The inhomogeneities, which lead to (partially) co-herent radiation, are associated with evolution of insta-bilities of the bunches in magnetospheric plasma (Stur-rock 1971). Initial inhomogeneities already appear atthe electrons emission from the surface of the star orat the beginning of the positrons return motion in the lower magnetosphere (Ruderman & Sutherland (1975);Cheng & Ruderman (1980); Al’ber et al. (1975); Asseoet al. (1983); Ursov & Usov (1988); for the analogies inthe electron beams ejected from the Sun see e.g. Kras-noselskikh & Voshchepynets (2015)). Further we ap-ply just the very fact of the inhomogeneities existence,which makes the coherent mechanisms possible .Further we estimate the possible spatial size of theregions responsible for coherent radiation emission bypositrons in the framework of the model proposed inthe present paper. Essential role in this case is playedby the large transversal size of the coherently radiatingarea in a relativistic beam. II. Longitudinal coherence
Presently we will consider the situation when thedensity of the positron flux is inhomogeneous (the re-gions of the increased particle density alternate withthe regions of the decreased one) in the direction ofthe positrons motion. Let us denote the characteris-tic size of the inhomogeneity as L . In such case it isusually expected that the radiation is emitted coher-ently if the wavelength exceeds the value of L . However,this value depends on numerous factors and, generallyspeaking, may significantly exceed the wavelengths λ we are presently interested in (which are just about sev-eral centimeter). Is coherent radiation emission possiblein this case? The simple calculations presented furthershow that partial coherence of the emitted radiationmay still take place even in the case λ (cid:28) L . By thepartial coherence we mean here the case when just acertain part (probably, small) of the positrons from theregion of the increased density (let us call it a bunch)are involved in coherent radiation emission.The circular trajectory is a simplified version of thetrajectory of positrons which we use for estimations inour model. Let us consider the spectral power of syn-chrotron (or curvature) radiation emitted in this caseby a bunch of length L consisting of N particles. Ac-cording to Ternov (1995), this quantity is defined bythe following expression: W ( ω ) = w( ω ) S N , (27) The instabilities of the positron flow, probably, may not havetime to increase up to necessary magnitude during a single pas-sage of the gap by a positron. Nevertheless, the positrons maytransfer the excitations to the electron flow moving towards them,which in its turn transfers the excitations to the positrons. Thusthe positive feedback might take place, which transforms the con-vective instability in the system of interacting colliding beamsinto the absolute one, at which the excitations grow with time ineach point of the beam. Such growth brings the bunches to thestationary non-linear regime, in which the maximum possible am-plitudes of the inhomogeneities are established and the particleflows become clumped. where w( ω ) is a spectral power for a single particle and S N is the so-called coherence factor. The explicit ex-pression for it has the following form: S N = N + N ( N − g ω , in which the form factor g ω = (cid:32) π (cid:90) − π dϕσ ( ϕ ) cos( ωϕ/ω ) (cid:33) . Here
N σ ( ϕ ) is the density of the bunch (at the certainmoment of time) as a function of the angular coordinate ϕ on the circular trajectory of the bunch. The function σ ( ϕ ) is supposed to be symmetric with respect to thepoint ϕ = 0.Let us consider the simplest case of a uniform dis-tribution of particles within the bunch. Let the bunchbe situated in the region − ϕ / < ϕ < ϕ /
2. In thiscase σ ( ϕ ) = 1 /ϕ in this region while it equals zeroelsewhere. In this case the coherence factor is definedby the following expression: S N ≈ (cid:18) N λπL (cid:19) sin (cid:18) πLλ (cid:19) . (28)Here we applied the relations ω = c/R , L = Rϕ and ω = 2 πc/λ and took into account the condition N (cid:29) L (cid:29) λ is fulfilled, the sine argument in (28) is a quicklyoscillating function. It can be substituted by 1 /
2, whichis analogous to averaging of this expression over thelengths of the bunches. Finally the coherence factortakes the form: (cid:104) S N (cid:105) ≈ (cid:18) N λ √ πL (cid:19) . (29)Expression (29) shows that in average the spectralpower of radiation by a bunch is proportional to thesquared number of particles from the bunch which aresituated within the distance of about λ/ √ π . In thissense the radiation by a bunch can be considered aspartially coherent if the condition1 N (cid:28) (cid:18) λ √ πL (cid:19) (cid:28) N (cid:28) (cid:104) S N (cid:105) (cid:28) N . Let us also note that under the considered condition( L (cid:29) λ ) the interference of radiation emitted by differ-ent bunches can be neglected. III. Transversal coherence
Previously we considered the possibility of coherentradiation emission by positrons due to inhomogeneity oftheir density in the direction along the positron veloc-ity. However it is essential (see, for example, Potylitsynet al. (2008) and ref. in Shul’ga & Tyutyunnik (2003))that in ultrarelativistic case the size of the spatial re-gion in the direction orthogonal to the particle veloc-ity, which is responsible for coherent radiation emission,can significantly exceed the size of the correspondingregion in longitudinal direction, which, as noted previ-ously, is of the order of radiation wavelength. Indeed,let us consider radiation emitted by two particles mov-ing along the z -axis separated by distance a in the di-rection orthogonal to it, defined by a unit vector e ⊥ . Ifwe present the wave emitted by the first particle in theform e i ( kr − ωt ) , the wave emitted by the second parti-cle in the same moment of time in the point with thesame coordinate z will be e i ( k ( r − a ) − ωt ) . Here we used adenomination a = a e ⊥ . The phase difference betweensuch waves in this case is totally attributed to the dif-ference a of the transverse (with respect to the z -axis)coordinates of the particles and equals δφ = ka = 2 πa sin ϑ/λ. (31)Here for simplicity we consider radiation in the plane ofthe particles motion. ϑ denotes the angle between theradiation direction and the z -axis. In ultrarelativisticcase most part of the emitted radiation is concentratedin the vicinity of the value ϑ ∼ /γ of this angle. From(31) we see that at such angles the condition δφ (cid:28) a (cid:28) γλ or, in our notation, a (cid:28) r ⊥ , where r ⊥ is defined by the expression (11).The condition δφ (cid:28) γ can be fulfilled even for a (cid:29) λ . In the directionswhich correspond to larger values of ϑ , as follows from(31), the size of the transversal region responsible forcoherent radiation emission decreases and for ϑ ∼ ∼ λ .If the particles move not parallel to each other andthe directions of their velocities form a certain angle α ,the coherence of the particles radiation partially brakes.From the consideration presented above it follows thatfor qualitative estimations it can be accepted that thecoherence exists if α (cid:45) /γ , provided the transversaldistance between the particles is a (cid:28) r ⊥ , while for α > /γ the radiation emission by the particles is inco-herent.The positrons moving along different magnetic fieldlines move not parallel to each other. As noted, this fact can modify the transversal dimensions of the region re-sponsible for coherent radiation emission making themdifferent from r ⊥ . It is the case if the angular differencebetween the directions of the positrons motion within r ⊥ exceeds the value 1 /γ . Let us define whether suchsituation takes place in the considered model. We willmake estimation for the region where the influence ofthe magnetic field lines curvature upon the radiationprocess is the most significant. It is the vicinity of theouter open magnetic field lines. We will also considerthe case of rather high altitudes above the pulsar sur-face (around the beginning of the effective path) wherethe field lines curvature is the largest. We consider themagnetic field as close to the dipole one.According to the model of positron motion, acceptedin the present paper, the considered region is situatedon distance of the order of ten kilometers from the cen-ter of the pulsar. The positrons Lorenz-factors here wetake to be of the order of γ min . In this case for centime-ter wavelengths (frequencies of the order of 10 GHz) thedistance r ⊥ ∼ γλ exceeds ten meters in magnitude.If we choose the z -axis parallel to the magnetic axisand y -axis perpendicular to it, the equation of the linetangent to a magnetic field line in the point ( y , z ) willhave the following form (see e.g. Malov (2004)):(3 sin θ − A sin θ /r )( y − y ) + 3 cos θ ( z − z ) = 0 , (32)where r = (cid:112) y + z and θ = arctan( y /z ) are thespherical coordinates of the considered point in yz planeand A is the parameter of the magnetic line (its equa-torial diameter). From (32) we can derive the angle β between the magnetic field line and the y -axis in thispoint: β = arctan(2 A sin θ /r − θ ) . (33)Substituting here the expression for A from the equa-tion of the magnetic field line in spherical coordinates A = r / sin θ , we obtain the expression for β as afunction of r and θ .As was noted, the values of r which we considerare of the order of ten kilometers. The value of A significantly exceeds this value and, for instance, forthe line which divides the region of opened and closedmagnetic field lines reaches the value of about severalthousand kilometers. The corresponding value θ is θ = arcsin (cid:112) r /A ∼ .
1. For such value of θ theexpression (33) can be simplified and we obtain: β ≈ arctan(2 / θ ) . (34) Using (34) we can define the range of θ on which thevalue of β changes by the value ∼ /γ :∆ θ ∼ /γ. (35)Such result is stipulated by the fact that at small valuesof θ which we consider the angle β is close to the angle π/ − θ and the changes of β and θ are the same bythe order of magnitude.The distance in the direction approximately orthog-onal to the direction of magnetic field lines which cor-responds to the value ∆ θ from (35) is ∆ r θ = r ∆ θ which is about ten meters. It is the distance in theplane yz in the direction perpendicular to the magneticlines, within which the moving positrons radiate coher-ently. We see that in the considered case it is of theorder of γλ and we can conclude that the curvature ofthe magnetic field lines does not play important role inthis respect.In order to define all three characteristic dimensionsof the spatial volume in which the moving positrons ra-diate coherently it is still necessary to estimate its sizein the direction orthogonal to yz plane, which is associ-ated with azimuthal coordinate ϕ around the magneticaxis. This size can be found from the analogous condi-tion that the angle between the velocities of positronshaving different coordinates ϕ , which radiate coher-ently, should not exceed the value 1 /γ . Simple geomet-rical considerations give the condition for the maximumvalue ∆ ϕ of the difference of the azimuthal coordinatesof such positrons:∆ ϕ ∼ arccos { (cos γ − − sin β ) / cos β } . (36)The distance associated with this angular difference is∆ r ϕ = r sin θ ∆ ϕ , which for the accepted values of r and θ is about ten meters and also fits by the order ofmagnitude the value γλ .As the positrons move further towards the surfaceof the star and accelerate the value of 1 /γ decreases.The curvature of magnetic field lines and the angles be-tween the velocities of positrons moving along differentlines decrease as well. Therefore for our estimations weassume that the relations ∆ r ϕ ∼ ∆ r θ ∼ r ⊥ are approx-imately applicable within the whole effective path. Forestimation of the total positron radiation flux we applysame assumption also for the positrons moving alongthe field lines situated nearer to the magnetic axis.Summarizing the presented considerations we con-clude that the spatial volume V coh , which is responsiblefor coherent radiation emission by positrons in the con-sidered model, can be rather large. It is estimated as V coh ∼ λ √ π ∆ r θ ∆ r ϕ ∼ λ γ / (4 π ) and contain quite huge number of positrons N coh . Thedenominator 4 π here comes from the more accurateapplication of the equation (31), while √ N coh . Thetotal positron radiation flux will be also proportional tothe number of such volumes V coh which emit radiationwithin a unit of time. Such number is approximatelyestimated in sec. 5.Thus, the transversal coherence, which leads to sig-nificant dependence of the coherence volume V coh onthe Lorenz-factor, plays a key role in the consideredproblem. References
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