High Energy Neutrinos and Photons from Curvature Pions in Magnetars
aa r X i v : . [ a s t r o - ph ] J u l High Energy Neutrinos and Photons from Curvature Pions in Magnetars
T. Herpay , , S. Razzaque , , , , A. Patk´os , , and P. M´esz´aros , , ABSTRACT
We discuss the relevance of the curvature radiation of pions in strongly magnetizedpulsars or magnetars, and their implications for the production of TeV energy neutrinosdetectable by cubic kilometer scale detectors, as well as high energy photons.
Subject headings: neutrinos — cosmic rays — neutron stars — pulsars — magnetars
1. Introduction
In pulsar and magnetar polar caps one expects the formation of vacuum or space-charge limitedgaps (e.g. Harding and Lai, 2006), in which an electric field component parallel to the magneticfield accelerates particles to relativistic velocities. The particles move away from the caps along themagnetic field lines, producing photons by curvature radiation, which at some height h produce an e ± pair front that determines the upper extent of the gap (e.g. Ruderman and Sutherland, 1975,Harding and Muslimov, 2002). The caps with Ω · B < E > ∼ eV) neutrinos through photomeson interactions, pγ → π + → µ + ν µ → e + ν e ¯ ν µ (Zhang et al, 2003).Here we explore the consequences for neutrino production of a different mechanism, the curvatureradiation of pions by protons.Relativistic protons interacting with a strong magnetic field can produce pions, a quantumtreatment of this process being given by Zharkov (1964). Ginzburg and Zharkov (1965) used asemi-classical approach to emphasize the analogy between this pion radiation process and the usualsynchrotron radiation of photons by protons. This semi-classical method was further exploredfor π and ρ mesons by Tokuhisa and Kajino (1999) in the context of pulsars. The pion radiationmechanism is characterized by a parameter χ = γ ( B/B Q ) similar to that used for electron processes Institute of Physics, E¨otv¨os University, H-1117 Budapest, Hungary Research Group for Statistical and Biological Physics of the Hungarian Academy of Sciences, H-1117 Budapest,Hungary Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA Department of Physics, Pennsylvania State University, University Park, PA 16802, USA Center for Particle Astrophysics, Pennsylvania State University, University Park, PA 16802, USA Space Science Division, Code 7653, U.S. Naval Research Laboratory, Washington DC 20375, USA B Q = m p c /e ~ = 1 . × G. Theyshowed that the synchrotron-like π radiation process becomes competitive with (proton) photonsynchrotron radiation when the above parameter χ > ∼ .
1. The related synchrotron-like π + radiationby protons in a magnetic field requires, however, a different semi-classical treatment, since theprotons become neutrons in the process, and this was calculated by Herpay and Patk´os (2008).In the astrophysical literature, particle acceleration in pulsars or magnetars is considered bothnear the polar caps (“inner” or polar gap models), and further out near or beyond the light cylinder(“outer” gap or wind models); e.g. see review by Harding and Lai (2006). Examples of the formerare e.g. Blasi, Epstein and Olinto (2000), Arons (2003), etc.; while examples of the latter are, e.g.Sutherland (1979), Harding and Muslimov (2001, 2002, 2005), Thompson (2008), etc.. Here, weconsider specifically the case of protons or ions accelerated in the inner regions, just above the polarcaps of magnetars. In the inner acceleration region discussed here, the accelerated particles areexpected to radiate away their transverse energy so efficiently that they are effectively constrainedto the ground Landau level as they move along the curved field lines. The principal photon radiationloss is then curvature radiation (Ruderman and Sutherland, 1975; Harding and Lai, 2006). In thissituation, the synchrotron-like pion radiation (which depends on the proton transverse energy) willbe similarly suppressed. However, as for photon curvature radiation, the motion along the curvedfield lines produces an acceleration equivalent to a Larmor motion in a fictitious magnetic field,which will lead to curvature pion radiation (Berezinsky, Dolgov and Kachelriess, 1996). Unlikeprevious authors, in this paper we concentrate on π ± production by the curvature-like mechanismof both protons and heavy nuclei, occurring in the immediate vicinity of the polar caps, and weexplore its consequences for the production of VHE neutrinos, as well as their detectability bycubic kilometer Cherenkov detectors such as IceCube and KM3NeT. We compare the resultingneutrino fluxes to those from the photomeson process, and discuss the associated cosmic ray as wellas GeV-TeV photon production.In § § § § §
2. Ion Acceleration and Energy Losses
The maximum potential drop in a magnetar of radius a = 10 a cm, rotation frequencyΩ = 2 π/P (where P = 10 − P − s is the period) and surface magnetic field at the pole B = 10 B Ba / c ) ≃ (6 . × V) B a P − − . However, the self-consistent gap structure ofmagnetars is not well known, e.g. for the ~ Ω · ~B < ∼ r pc =(2 πa /cP ) / ≈ . × a / P − / − cm. Beyond a height h ∼ r pc , ions may further be acceleratedby an electric field in the saturated regime, given by E k ≃ × B P − − in c.g.s. units (see, e.g.,eq. 1 in Harding and Muslimov 2002). For heights much smaller than the stellar radius, h ≪ a forwhich the saturated field is constant, the energy ǫ reached by the particle is ǫ = ZeE k h , during anacceleration time t a = ǫ/cZeE k ≃ . × − ǫ B − P − Z − s , (1)where ǫ = ( ǫ/ eV). Charged particles accelerated from the polar cap move following closelythe field lines, the main photon radiation loss being curvature radiation, whose power is P c =(2 e cγ / R c ). Curvature radiation generally dominates over (transverse) synchrotron radiation atlow heights in the polar caps. For the limiting open field line of a neutron star rotating with period P = 10 − P − s defining a light cylinder R L = cP/ π the curvature radius is R c = (4 / aR L ) / = 3 × P / − a / cm . (2)and the photon curvature radiation loss time for protons of energy ǫ is t c = (3 ǫR c / e cγ ) = 2 . × − ǫ − P − a s . (3)Another energy loss mechanism for protons in a strong magnetic field is the synchrotron-like pionradiation losses (Ginzburg and Zharkov, 1964). In terms of the parameter χ defined as χ = γ ( B/B Q ) , (4)where B Q = m p c /e ~ = 1 . × G, a semi-classical calculation in the χ ≫ P p,π = dE p /dt of a proton due to π emission as P π ( χ ≫ ≃ ( g / m p c ~ − , (5)which in quantum mechanical calculation acquires a further χ / factor (Ginzburg and Zharkov,1964). Here g / ~ c ≃
14 is the strong coupling constant. The corresponding low χ limit is P π ( χ ≪
1) = ( g / √ m π m p c ~ − χ exp[ − ( √ /χ )( m π /m p )] (6)(Ginzburg and Zharkov, 1964; Tokuhisa and Kajino, 1999). For χ > ∼ . π + meson radiation, the proton 4 –becomes a neutron during the emission process, and a different semi-classical calculation methodis needed, as done by Herpay and Patk´os (2008). A quantum calculation of this process is that ofZharkov (1964). In the limit of χ ≪ π + radiation is P π + = (cid:18) g ~ c (cid:19) m p c ~ ! (cid:18) √ (cid:19) " ( m p /m π ) (2 √ − m π /m p ) (cid:2) √ m p /m π ) (cid:3) χ exp " − √ χ (cid:18) m π m p (cid:19) ≃ . × χ exp[ − . /χ ] erg s − . (7)However, for the same reason that photon curvature radiation dominates over photon synchrotronradiation, the pion radiation due to the proton transverse momentum will be dominated by pionradiation due to the longitudinal component along the curved field lines (Berezinsky et al, 1996).We define an equivalent ‘curvature” B ⊥ field perpendicular to the particle trajectory, which wouldgive a radius of curvature R c equal to the gyro-radius of a proton of energy ǫ and Lorentz factor γ , B ⊥ = ǫ/eR c = 1 . × ǫ P − / − a − / G . (8)The corresponding ‘curvature” χ parameter is χ ⊥ = γ ( B ⊥ /B Q ) ≡ γ ( ~ /R c m p c ) ≃ . × − ǫ P − / − a − / , (9)and in terms of this parameter the photon curvature power is P γ,c = (2 e cγ / R c ) ≡ (2 e / ~ c )( m p c / ~ ) χ . (10)From now on we will refer to χ ≡ χ ⊥ , and in our case we will be interested mainly in the χ ≪ χ ≪ ǫ , t π,c = ǫ/ P π,c , using eq. (7) with eq.(9), is t π,c ∼ ( . × − ǫ − P / − a / exp[31 . ǫ − P / − a / ] s for π ;4 . × − ǫ − P / − a / exp[4 . ǫ − P / − a / ] s for π + (11)where ǫ = ( ǫ/ GeV) is the particle (longitudinal) energy.For the case of heavy ions (
A, Z ), e.g. Fe (56,26), for an ion of energy ǫ the ion Lorenztfactor is γ A = ǫ/ ( Am p c ), and each constituent proton moves with γ A . We can use the previousequations for the photon curvature losses P γ,c , and in principle also the curvature pion losses P π,c of the individual constituent protons moving with γ A , and find the corresponding losses for the 5 –ion as P γ,c,A = Z P γ,c ( γ A ). The proportionality factor Z assumes incoherent radiation by theconstituent nucleons, since at high energies the dominant wavelength becomes smaller than theaverage separation of the constituents. Thus P γ,c,A = Z P γ,c = Z (2 / e / ~ c )( m p c / ~ ) χ . For anion energy ǫ A = 10 ǫ GeV, the acceleration and incoherent photon curvature times are t a,A = 8 . × − ǫ B − P − Z − s , (12) t γ,c,A = 1 . × − ǫ − P − a A Z − s , (13)where we used the same curvature radius R c of equation (2) as for protons, since the nuclei movealong the same field lines. The fictitious ‘curvature” magnetic field B ⊥ and χ ⊥ for ions is the sameeq. (8) and eq. (9) as for protons, except for using γ A , χ A = γ A ( B ⊥ /B Q ) = 2 . × − ǫ A − ( a P − ) − / . (14)If the ions did not fragment before reaching the appropriate energy threshold, the pion curvatureradiation of heavy ions ( A, Z ) moving along the field lines would, in principle, be due to theindividual nucleons in the nucleus moving with Lorentz factor γ A and with the χ parameter of eq.(14). The pion emission power would then be P π,c,A = A P π,c for the above χ A , the protons inthe nucleus radiating π + and the neutrons radiating π − . However, as discussed in § § π energy loss timescale with theacceleration time occurs generally at higher energies than the intersection of the π + losses and theacceleration. The shorter π + time is mainly due to the lower energy cutoff in the radiation power P π + caused by an ( m π + /m p ) factor in the exponent, versus an m π /m p in the exponent of the π power. Thus, the proton energy is determined by a balance between acceleration energy gains anda combination of the π + radiation and photon curvature losses. Assuming that the particles canreach the corresponding height h (which is plausible in the saturated field regime r pc ≪ h ≪ a ), themaximum particle energy for a given value of B and P is approximately given by the intersectionof the appropriate acceleration time with either the π + or the curvature loss time curve, whicheveroccurs at the lowest energy. Thus, from Fig. 1, for B = 10 G and P = 10 − s the curvaturepion losses are beginning to get competitive with curvature photon losses, with t π + /t c ∼ . ǫ max ∼ . × GeV. The pion losses dominate for B P − − > ∼
2. However for lower fields or longerperiods B P − − < ∼
1, it is the photon curvature losses that determine ǫ max . 6 –
3. Trajectories, Acceleration and Energy Losses from the Solution of theProton/Ion Equations of Motion
In the previous section, we determined the maximum energy of the protons/ions by comparingthe characteristic timescales of different radiations with that of the acceleration calculated withouttaking into account the back-reaction of the radiation. Moreover, in estimating the characteristictimes we made the further assumption that the perpendicular momentum of the accelerated particleto the magnetic field lines is much smaller than the parallel component. The simplicity of thisphysical picture greatly helps our intuition, still its fundation needs some elaboration. Therefore itis interesting to compute the trajectory and the saturation energy of the charged particles directlyfrom the full equation of motion which also takes into account the backreaction of the radiation.In this section we discuss in detail the proton/ion trajectory through the acceleration region nearthe polar cap.
In a given electromagnetic field, the relativistic equation of motion of a proton is˙ ~γ = em p c ~E + 1 p ~γ ~γ × ~B ! − P m p c p ~γ + 1 | ~γ | ~γ | ~γ | , ~γ = ~pm p c , (15)where we added on the right hand side the last term to the usual Lorentz force to take into accountthe energy loss of the particle ( P = d E/ d t ) due to radiation processes (photon, π , π + ). Thedirection of this back reaction force is chosen opposite to its momentum, which is true for large | ~γ | (a similar description of the back reaction of the radiation was proposed in Takata et al (2008)and Harding, Usov and Muslimov (2005)). Here, P results from the curvature of the trajectory.It is not split up into separate synchrotron and curvature (along ~B ) radiation parts correspondingto transverse and parallel momentum components, respectively. This approach enables us a morecomplete description, because the presence of an accelerating electric field continously modifies thecurvature even if ~E || ~B . Therefore in eq. (15) P depends on the total curvature radius ( R c ), whichdefines a curvature parameter χ (Berezinsky et al, 1995), χ = ~γ R c ℏ m p c , (16)where R c can be expressed with ~γ and ˙ ~γ ,1 R c = 1 c p ~γ | ~γ | (cid:12)(cid:12)(cid:12) ˙ ~γ × ~γ (cid:12)(cid:12)(cid:12) = 1 c p ~γ | ~γ | q ~γ ˙ ~γ − ( ~γ ˙ ~γ ) . (17)Here P is the sum of the power of γ and π radiations given in section 2: P = P c + P π + P π + ≡ P (cid:16) | ~γ | , χ ( ~γ, ˙ ~γ ) (cid:17) , (18) 7 –The formulae are applicable only for χ < m π /m p . It can be seen from the above expressions that ˙ ~γ appears on the right hand side of the equation of motion (15) in a very complicated way. However ifthe ratio of the power of the energy loss to the energy is small, one can solve the equation iteratively,e.g. one substitutes ˙ ~γ into χ with its value calculated from the equation of motion with only theLorentz term (without P ).In this approximation one considers six coupled first order differential equations for ~r and ~γ wherein time derivatives appear only on the left hand sides˙ ~r = c~γ p ~γ , ˙ ~γ = em p c ~E ( ~r ) + 1 p ~γ ~γ × ~B ( ~r ) ! − P ( | ~γ | , ¯ χ ( ~γ, ~r )) m p c p ~γ + 1 | ~γ | ~γ | ~γ | , (19)where ¯ χ is calculated from (15) for P = 0,¯ χ = e ℏ m p c | ~γ | (cid:12)(cid:12)(cid:12) ~γ × (cid:16)p ~γ ~E ( ~r ) + ~γ × ~B ( ~r ) (cid:17)(cid:12)(cid:12)(cid:12) . (20)One can easily check that this approximation gives back the usual formulas for the intensity ofthe synchrotron or curvature radiation as long as ~E = 0 or ~B = 0, respectively. Note thatequations in (19) would be further simplified if ~E || ~B (see Takata et al, 2008) using appropriatelocal coordinates to treat the motion parallel and perpendicular to the magnetic field lines. In thegeneral case ( ~E × ~B = 0) one cannot avoid to calculate all three independent components of ~γ .Then the use of a coordinate system with one axis fixed along the magnetic field lines does notbring any extra advantage over the use of a unique coordinate system for the whole motion.Unfortunately the exact expression of the electromagnetic field to be used in (15) is not knownfully since the field-charge, the motion and emission of any charged particle modifies the acceleratingfield. The full selfconsistent description of the inner gap is very complicated as was already shortlydiscussed in section 2. In our actual calculations we considered a dipole field for the magneticpart. In addition a so-called unsaturated expression was used for the electric field close to thesurface of the magnetar and saturated field form is adjusted to it far away from the surface. Theexplicit expressions of the electric field strength in the two regimes are given by Harding andMuslimov (2001, 2002). These electric fields take into account the influence of the acceleratedelectrons/positrons and their pair production, which may be the most important effect for theformation of a self consistent electric field. The corresponding unsaturated and saturated potentialscan be found in equations (20) and (24) of Harding and Muslimov (2001).In the course of the numerical solutions of (19) we switched from the unsaturated to thesaturated field at the altitude where the component of the unsaturated electric field parallel to themagnetic field equals to the corresponding component of the saturated field.By consulting the literature it appears, that our choice of the explicit form for the electric fieldis somewhat arbitrary, many other expressions having been proposed earlier (see e.g. Goldreichand Julian, 1969; Sutherland, 1979; Medin and Lai, 2008 and their references). We checked thatthe proton/ion saturation energy reflecting a balance between the acceleration and back reaction, 8 –and also the power of the different radiations are of the same order of magnitude when a simpleelectric quadrupole field is considered instead of the combination of the unsaturated and saturatedfields as described above.The equations discussed above are valid only for protons. In case of accelerated ions, therelevant equation of motion can be arrived at by the replacements e −→ Ze and m −→ Am in(15). We study in this section the case when the radiation intensity by the ions is the incoherentsum of individual nucleonic contributions provided that both photon and pion emission processesof the constituents become incoherent when the ions attain very large energies. Then one has tomake the following modifications in the expression of P in (18), • P γ −→ Z P γ because only protons can emit photons; • P π −→ A P π because both proton and neutron emits π with the same probability andneutrons move together with the protons; • P π + −→ A P π + due to the radiation of π + by protons and the radiation of π − by neutronshave same probability (from the point of view of neutrino radiation by charged pions we donot distinguish π − and π + ).The equation of motion for heavy ions as detailed above is valid as long as the ion doesn’tfragment. Fragmentation can occur as a result of the radiation recoil of the constituent nucleons.The numerical solution of the equation of motion shows that the transverse momentum transferfrom the radiation k ⊥ ≈ χ / m exceeds the nucleon’s binding energy ( ≈
10 MeV for Fe ions) beforethe pion radiation occurs (see the right hand side of Figs. 4). This means that the main part of thepion radiation arises from the acceleration of the individual proton constituents of the dissociatednucleus, as described by equation (15). Therefore, one doesn’t need to deal with the details of ionfragmentation from the point of view of pion radiation, which would be given in this treatment byan incoherent sum of the radiation from the individual nucleon fragments.In the next subsection we shall present the numerical solution of the equation of motion forprotons and the corresponding intensity of each type of radiations along the trajectory.
One of the main goals of this calculation is to decide if one can find such values for the magneticinduction B and the rotation frequency P of the neutron star for which the power of charged pionradiation becomes comparable to the photon radiation. Therefore, based on the comparison of thetimescales as presented in section 2, we focus our study on ( B, P ) pairs for which the intersectionsof the pion and photon cooling time curves with the acceleration time curve happen approximatelyat the same time. Based on the findings of the previous section (see Fig. 1), we examined protonacceleration the following realistic magnetar configurations: ( B , P − ), ( B , P − ), ( B , P − ). 9 –The path of a proton is shown for ( B , P − ) and different initial angles (Θ) measured relativeto axis of the magnetic dipole on the left hand side of Fig. 2. It can be seen that the proton movesinitially nearly along the magnetic field lines when the acceleration process begins, then shoves offthe field lines at about z = 2–3 R and finally moves on a helical trajectory along the z-axis faraway from the surface.On the top of the left hand side of Fig. 3, one can see that the maximum proton energy isalmost independent of the initial angle when ( B , P − ). This is also true in case of other valuesof B and P and also valid for the intensity maximum of the radiations as well. Therefore weonly plotted the time dependence of the momenta and the power of radiations corresponding to θ = 0 . θ pc (except the top left of Fig. 3).It’s interesting to observe time lags in the growth of both momentum components paral-lel/perpendicular ( γ k /γ ⊥ ) to the field lines as can be seen on the top of Fig. 3 and on the righthand side of Fig. 2, in which the time dependence of the relative altitude (( r − R ) /R ) of pro-tons is shown. In this time region, the proton path is closer to circular ( γ ⊥ ≫ γ k ) and remainsroughly constant because synchrotron radiation loss keeps pace with the transverse energy gaindue to acceleration along the field. Then, the path changes to being more elongated due to theincreasing electric field ( | ~E unsat | ∼ ( r − R ) /R ), resulting in a change of the effective radius ofcurvature. This change of the path is indicated by a jump of the value of χ = γ/R c (at ≈ − s)which can be seen in Fig. 4, where we plotted the transverse momentum carried by the transfers ofradiations ( k ⊥ ≈ mχ / ). Between 10 − − − s γ ⊥ is approximately constant and the effect fromthe proton energy gain is practically compensated by a decrease in the effective curvature radius,so the radiation rate is approximately constant (see the bottom parts of Fig. 3 and Fig. 4). Theincrease in radiation rate after 10 − s is due to the effective radius of curvature starting to decreasemore slowly than the proton gains energy, until the saturated regime is reached where energy gainsbalance energy losses.From the bottom parts of Fig. 3, one can see that the π radiation is irrelevant, moreover thecharged pion radiation intensity becomes comparable to that of the photon radiation only in caseof ( B , P − ).In Fig. 4, one can see that the transverse momentum transfer of the radiation is small com-paring to the total energy both for protons and for constituent protons of Fe ions. It supportsthe selfconsistency of our softness assumption for the radiation process implicit in our choice ofthe effective pion–nucleon interacton. However, k ⊥ is large enough to fragment the ions at thebeginning of the trajectory as we noticed in the previous sections.The results of the numerical solution tell us that the protons can reach energies up to ≈ /10 GeV in the examined range of B and P. This is in good agreement with the results of theprevious section where we compared the cooling time curves to the acceleration time curve. Thisconclusion is apparently not sensitive to the fact that the value of the momentum perpendicularto the magnetic field lines is almost the same order of magnitude as the parallel component due 10 –to the essential presence of perpendicular components of the electric field. We also saw that for( B , P − ) or higher B and/or lower P it is the charged pion radiation which prompts the energysaturation of the protons (this is consistent with the conclusions obtained from the comparison ofthe characteristic timescales).
4. Curvature Pion Neutrino Luminosity
In terms of the Goldreich-Julian charged particle density n GJ = ( B/ecP ) = 7 × B P − cm − for a pulsar or magnetar of radius a and period P s, with a polar cap solid angle Ω p = πθ p ∼ (2 π a/cP ) = 6 . × − a P − sr, the outflow rate of ions with charge Z is˙ N i = a Ω p n GJ Z − c ≃ . × B a Z − P − s − . (21)For a π + radiation efficiency η π at the ion terminal Lorentz factor, the resulting number of pion-decay ν µ and subsequent muon-decay ¯ ν µ is 2 per proton, giving for a pulsar at distance D = 10Kpc a ν µ or ¯ ν µ flux at Earth ofΦ ν,c = Z ˙ N i / πD ≃ . × − η π B P − a D − cm − s − , (22)where the label c denotes the curvature pion origin. The corresponding arrival rate in a detectorof area A = 1 A km is ˙ N ν = Φ ν A ≃ . × η π B a P − A D − km − yr − .The typical energy of the muon neutrinos is determined by the energy losses incurred by thepions and muons before the decay. As the pions move out to radii comparable to the light cylinder,the π + Klein-Nishina energy losses are less important than the π + photon curvature losses, whichoccur on a timescale t c,π ∼ . × − ǫ − π a P − s, where ǫ π = ( ǫ π / GeV) is the pion energynormalized to 10 GeV, while the π + decay time is t d,π ∼ . × ǫ π s. The pions cool by curvatureradiation until reaching the light cylinder at t L ∼ . × − P − s, where their energy has droppedto ǫ πL ∼ . × − a / . Thereafter the pions move in the wind zone in essentially ballistic paths,cooling adiabatically on a timescale t ad,π = t L ( ǫ πL /ǫ π ) / = 1 . × − P − a / ǫ − / π s. The pionsdecay when their energy reaches ǫ πd ≃ . × − P / − a / ∼ GeV. The corresponding ν µ hasan energy ǫ ν µ ∼ ǫ ,π,d / ∼ P / − a / TeV. The associated µ + starts with (2/3) of the decaying pionenergy and is subject to adiabatic losses. It decays on a timescale t µd ∼ . × ǫ µ s, where ǫ µ isthe muon energy normalized to 10 GeV, reaching at decay an energy ǫ µd ≃ . × − P / − a / ∼ GeV. The associated muon neutrino has an energy ǫ ν µ ∼ ( ǫ ,µ,d / ∼ . ν e : ν µ : ν τ =0:1:0, forpion decay ν µ , the observed ratio would be 0.2:0.4:0.4. For 3.3 TeV muon decay neutrino fluxes,with a production ratio of 1:0:0 and 0:1:0, respectively for ν e and ¯ ν µ , the respective observed 11 –ratios would be 0.6:0.2:0.2 and 0.2:0.4:0.4. Since the neutrino detectors cannot distinguish betweena muon neutrino and an anti-neutrino, the observed flux of 3.3 TeV muon neutrino would be afactor 0.6 times the production flux. The probability per ν µ of resulting in an upward muon is P ν → µ ∼ . × − ( ǫ ν µ / TeV), which for the 33 and 3.3 TeV neutrino cases lead to upward muoncount rates from curvature pions, after taking into account oscillation effects, of˙ N µ,c ≃ ( . × η π B P − / − a / A D − ( ǫ ν µ / − yr − , . × η π B P − / − a / A D − ( ǫ ν µ / . − yr − , (23)for an on-beam magnetar at 10 Kpc.In the case of proton acceleration, from Fig. 1 these would reach a maximum energy determinedby π + curvature losses for fields B > ∼ G and P ∼ − s, or B P − − > ∼
1. In these cases t π + < ∼ t c andthe π + radiation efficiency η π ∼
1, so the upward muon event rate should be detectable in the 1-100TeV range with cubic kilometer detectors, if the pulsars is on-beam and B P − / − A D − > ∼ − ,e.g. for sources in the local group. On the other hand, for values B P − − <
1, the pion efficiency islow, η π ∼ t c /t π <
1, which would allow potentially detectable neutrino fluxes for sources in our owngalaxy provided η π > ∼ − − − . For B P − − ≪ − , the steep behavior of the pion radiationefficiency implies an unobservably small number of events, even for galactic sources. Thus, thereis a small range of magnetar field strengths B > ∼ P − ∼
5. Photomeson Neutrinos, Cosmic Rays and Photons
Another mechanism for neutrino production in the polar caps is the photomeson process pγ → π + → µ + ν µ → e + ν e ¯ ν µ , acting on the magnetar surface X-rays, typically L x ∼ L35 erg s − (Zhang et al, 2003). The photon number density near the surface is n γ,a = 1 . × L35 / cm − , and for protons of energy much above the threshold ǫ ≫ . − . σ pγ ∼ − cm implies a mean interaction time t pγ ≃ . × − L35 − / s, a mean free path ℓ pγ ∼ × L35 − / cm, or an optical depth τ pγ ∼ .
18 L35 / a . For L35 > ∼ R L = 5 × P − cm, but much larger than the height h where acceleration balancespion or photon curvature losses.If the pion curvature process is inefficient, η π ≪
1, most of the protons survive beyond h , andinside the light cylinder they cool due to curvature radiation (or outside the light cylinder due toadiabatic cooling), until the photomeson process leads to a neutron and a π + which takes 0 . R L ) and adiabatic (when outside R L ) losses in a similar way as in the previous section. Fortypical parameters, the average neutrino energy is < ǫ ν µ > ≃ P / − L35 − / a / TeV, not muchdifferent from the value ∼ P / − a / TeV in the pion curvature initiated case of last section, since 12 –both the protons (before undergoing pγ ) and the charged pions cool by photon curvature radiationinside the light cylinder, and adiabatically outside. The neutrino flux resulting from pγ are larger,for τ pγ ∼ . ≫ η π , and since the typical neutrino energies from pγ are typically only ∼ η π ≪ pγ flux.The situation is different for large η π →
1. In this case it is mainly neutrons that propagateoutwards beyond h . Being neutral, they do not suffer adiabatic losses, and undergo photopioninteractions nγ → π − on a timescale similar to protons, t nγ ≃ . × − L35 − / s. Typically thisoccurs inside or at the light cylinder. We take as a limiting example the case where this occursnear the light cylinder. The negative pions and muons then undergo adiabatic cooling outside R L similarly to their positive counterparts, as discussed previously, resulting in ¯ ν µ . Ignoring thedistinction between ¯ ν µ and ν µ , which cannot be discriminated in Cherenkov detectors, the resultingaverage neutrino energy is now < ǫ ν µ > ∼
280 L35 − / ǫ / ni TeV, where ǫ ni = ( ǫ n / GeV) is theinitial neutron energy. The flux is Φ νnγ ∼ × − P − − B a L35 / D − cm − s − around ǫ ν ∼ nγ interactions, and the upward muon event rate, modulo the factors due to oscillationeffect as previously discussed, is˙ N µ,nγ ≃ . × B P − − a L35 / A D − ( ǫ ν µ / − yr − , (24)for an on-beam magnetar at 10 Kpc, from nγ interactions. The corresponding pion curvatureneutrinos would lead to an upward muon event rate given by equation (23), which is not toodifferent. If observed, the difference in the average neutrino energies from pion curvature ( ∼ nγ interactions ( ∼
280 TeV) could help discriminate between the pion curvatureand the photomeson production mechanisms. If there were such milisecond magnetars with η π → D < ∼
10 Kpc) this would imply, from eqs. (23, 24) upward muons event rates whichwould be strong in a completed cubic kilometer detector, and probably even in partially completedinstallations, so the current presence of such objects in our galaxy is questionable.These ~ Ω · ~B < π + radiation by protonsfrom magnetars with B P − − = 1, for which η π ≃ t π + /t c ∼ .
2, the flux of neutrons at Earth andtheir energies are Φ n ≃ − B P − − D − m − s − sr − ; ǫ n ≃ × eV . (25)A neutron of 6 × eV energy decays after traveling ∼ ≈ × − m − s − sr − (Nagano & Watson, 2000).However, the very small degree of observed cosmic ray anisotropy at these energies again suggeststhat the probability of finding such η π → v = 10 v cm/s andreaches a radius R snr ≃ . × v t d cm in t d days. With a typical shell mass of M snr =2 × m snr g, the column density of target atoms is Σ A ≃ . × m snr t − d v − cm − . The pp cross-section for ∼ eV energy incident neutrons is 100 mb and the corresponding optical depthis τ pp ∼ m snr t − d v − . The energy of the neutrinos produced by a 7 × eV neutron via pp interactions ranges from ∼ m π c γ cm / ≃ . ∼ ǫ n / ǫ − ν distribution normalizedto a multiplicity of 10 − at this energy (Razzaque, M´esz´aros & Waxman, 2003). Here γ cm is theLorentz factor of the center-of-mass of the pp interaction in the Lab frame.There may also be photo-hadronic interactions with photons in the SNR shell created fromthe SN explosion. The peak energy of these blackbody photons at creation is ≈ . E / r / keVfor ∼ E erg SN explosion energy and ∼ r cm progenitor star’s radius. In the SNRshell, however, they cool down to a peak energy of ǫ γ,sn ≃ . E / r / v − t − d eV. Thus neutrons of & eV energy may produce pions by interacting with them. Assuming that the total numberof SN photons do not change after their creation, the column density of them in the SNR shell isΣ γ,sn ≃ . × E / r / v − t − d cm and the opacity for photo-hadronic interactions is . t < ∼ few days after a magnetar is born.Ultra-high energy photons will also be produced, both through curvature radiation of chargedpions and muons, and through decay of the associated neutral pions. E.g. pions produced by ∼ GeV protons would lead to pion-related UHE photons of luminosity L γ ( π ) ∼ B P − − a erg s − , decaying as the field drops and the period lengthens. The curvature photon energy is ǫ γ ∼ P − − a − / ( ǫ π / GeV) PeV. These, as well as the muon curvature and neutral pion decayphotons are all well above the 2 m e c / sin θ threshold for one-photon pair production γB → e ± (e.g.Harding and Lai, 2006), where θ is the angle of propagation relative to field direction. The opticaldepth above threshold for a path length 10 R cm is τ γB ∼ × ( B / sin θ ) / R ( ǫ γ / PeV) − / .(Another UHE photon opacity is photon splitting, a higher order mechanism which above thresholdis less important than one-photon pair formation). The UHE photons will thus be degraded toenergies below the threshold, ǫ γ < ∼ θ − − MeV, for cap angles θ ∼ − θ − .
6. Discussion
Our discussion has assumed that the properties of the inner gaps near the polar caps of pulsarsor magnetars allow protons to be accelerated to energies > ∼ × GeV. This is an open question,since the extent and properties of polar gaps (with ~ Ω · ~B ≤
0) including self-consistently pionradiation effects is unknown. Such studies, if undertaken, would have to take into account notonly the effects of leptons acceleration from charged pion decays, but also high energy photons 14 –from neutral pion decay and their subsequent cascades. Studies of space charge limited gaps withelectron acceleration and pair production (e.g. Harding and Muslimov, 2002; also Baring andHarding, 2002) suggest that Lorentz factors of order 10 may be possible in fast-rotating, high fieldobjects, in which case the effects discussed here may become important.The curvature pion radiation mechanism is likely to be of interest for fast rotating magnetarswhich accelerate protons. This is because, from Fig. 1, we see that for magnetic field and periodvalues B P − − > ∼ B > ∼ G and millisecond periods, themuon event rate is so large that they might have been detected by now, if the magnetar wereon-beam and at a distance D < ∼ Mpc. The likeliest explanation is that such millisecond magnetarsdo not exist in our galaxy (or else they might be off-beam). At the same time, it is reassuring thatfor periods P > ∼ η π = t c /t π ≪
1, and the correspondingmuon event rate in cubic kilometer detectors given by equation (23) is negligible for known galacticmagnetars.An interesting possibility is that some fraction of core collapse supernovae lead, at least ini-tially, to a millisecond magnetar. There is currently no observational evidence for this, but itis a plausible hypothesis, among others from fast convective overturn dynamo arguments, e.g.Thompson and Duncan (1996). In such objects, the envelope optical thickness to νN interac-tions is τ νN ∼ − ( ǫ ν /
10 TeV)( M env / M ⊙ )( v env / . c ) − ( t/ day) − , and as long as the magneticfield and the rotation rate remain high, neutrinos produced by pion curvature and pγ interactionscan escape without further reprocessing. Thus, core collapse supernovae which result initially ina millisecond magnetar with B > ∼ G are expected to undergo a brief period of intense pionand ultra-high energy neutrino production, which significantly exceeds any electromagnetic energylosses. If the fraction of core collapse supernovae leading to such objects were 0 . − .
1, given thefrequency of core collapse SNe in the LMC and in the local group, one could in principle expectsome millisecond magnetars detectable in cubic kilometer detectors within timescales of years. Inthis scenario one would also expect ( §
5) a significant < ∼
10 MeV photon luminosity, which may bedetectable by the GLAST GBM.In summary, we have pointed out that pions produced by protons interacting with the curvedstrong magnetic fields of magnetars may be an important energy loss mechanism for the acceleratedparticles, leading to secondary leptons and photons which can affect the properties of the innergaps. Our model for the strong interaction processes involved here relies on the softness of thepion radiation process. Although the correctness of this assumption needs further, more detailed 15 –investigation, this level of approximation is justifiable here, given the larger uncertainties in theastrophysical model. Our results indicate that the effects analysed in this paper could lead tocopious neutrino production in the TeV-nPeV range, which would provide interesting targets forcubic kilometer scale detectors.This work is supported in part by NSF AST 0307376 and the Hungarian Science FoundationOTKA No. T046129 and T68108. SR is presently a National Research Council Research Associateat the Naval Research Laboratory. We are grateful to A.K. Harding and to the referee for usefulcomments.
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17 –
Proton t a B P - t a B P - t a B P - t a B P - t c P - t c P - t Π + P - t Π + P - t Π P - t Π P - ´ ´ ´ ´ ´ ´ - - - - - - Energy @ GeV D T i m e @ s D Fig. 1.— Energy loss timescales t π + and t π for proton curvature emission of π + and π , comparedto photon curvature radiation loss timescale t c and acceleration time t a , as a function of protonenergy (in GeV), for different field strengths B = 10 B (G) and periods P = 10 − P − (s). 18 – z / R ρ /R magnetic field linesparticle paths y / R x/R P -3 , θ =0.5 θ pc B P -3 , θ =0.5 θ pc B P -2 , θ =0.5 θ pc Fig. 2.— On the left hand side: the proton’s trajectory for initial angles relative the rotation axis θ/θ pc = 0 . , . , . B , P − ( θ pc = 27 . ◦ ). θ pc is the angular size of the polar cap. On theright hand side: the time dependence of the relative altitude of protons above the surface of theneutron star for different values of B and P . γ (a)B , P -3 ;parallel: θ =0.3 θ pc θ =0.4 θ pc θ =0.6 θ pc perpendicular: θ =0.3 θ pc θ =0.4 θ pc θ =0.6 θ pc γ (b)B ,P -2 : parallelperpendicularB ,P -3 : parallelperpendicular1e-201e-101e+001e+101e+20 1e-14 1e-12 1e-10 1e-08 1e-06 1e-04 d E / d t [ M e v / s ] t[s](c) photon π + π d E / d t [ M e v / s ] t[s](d) B P -2 : photon π + B P -3 : photon π + Fig. 3.— Acceleration of protons. The parallel and perpendicular momenta for B , P − (a), andfor B , P − ( θ pc = 8 . ◦ ) and B , P − ( θ pc = 27 . ◦ ) where θ = 0 . θ pc (b). The intensity of eachtype of radiation for B , P − (c), for B , P − and B , P − (d) where θ = 0 . θ pc . The verticallines indicate the switch between the unsaturated and saturated fields. 19 – k ⊥ [ M e V ] t[s]B P -3 B P -3 B P -2 k ⊥ [ M e V ] t[s]B P -3 B P -3 B P -2 Fig. 4.— Transverse momentum transfer the radiations from protons (left hand side) and con-stituent protons of Fe ions (right hand side) along the path. The photon radiation dominates incase of ( B , P − ) and ( B , P − ), and in case of ( B , P − ) up to t ≈ −6