Extinct radio pulsars as a source of subrelativistic positrons
aa r X i v : . [ a s t r o - ph . H E ] J u l MNRAS , 1–7 (2020) Preprint 17 July 2020 Compiled using MNRAS L A TEX style file v3.0
Extinct radio pulsars as a source of subrelativistic positrons
Ya. N. Istomin, ⋆ D. O. Chernyshov, † and D. N. Sob’yanin ‡ P. N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskii Prospekt 53, Moscow 119991, Russia
Accepted 2020 July 16. Received 2020 July 16; in original form 2020 April 08
ABSTRACT
Extinct radio pulsars, in which stationary, self-sustaining generation of a relativis-tic electron-positron plasma becomes impossible when rotation brakes down, can besources of a subrelativistic flux of positrons and electrons. We assume that the observedexcess of positrons in the bulge and the disc of the Galaxy is associated with theseold neutron stars. The production of pairs in their magnetospheres occurs due to one-photon absorption of gamma quanta of the Galactic and extragalactic backgrounds.The cascade process of plasma production leads to the flux of positrons escapingfrom the open magnetosphere ≃ × s − . The total flux of positrons from all oldGalactic neutron stars with rotational periods 1 . < P <
35 s is ≃ × s − . Theenergy of positrons is less than ≃
10 MeV. The estimated characteristics satisfy therequirements for the positron source responsible for the 511-keV Galactic annihilationline.
Key words: stars: neutron – cosmic rays
One of the most famous puzzles related to the prob-lem of the acceleration and propagation of Galactic lep-tons is the origin of positrons responsible for the anni-hilation emission from the Galactic centre. This emissionconsists of the line of electron-positron annihilation witha characteristic energy of 511 keV and also of continuumemission due to 3-photon annihilation. The emission fromthe Galactic centre was the first gamma-ray line discov-ered outside the Solar system. It was detected using theballoon-borne detectors (Johnson et al. 1972) and after-ward was excessively studied by other experiments (see,e.g., the review of Prantzos et al. 2011). The most detailedobservations were made using the SPI spectrometer lo-cated on the International Gamma-Ray Observatory (
IN-TEGRAL ) (Churazov et al. 2005; Kn¨odlseder et al. 2005;Weidenspointner et al. 2006).A recent analysis of
INTEGRAL /SPI data performedby Siegert et al. (2016b) demonstrated that the actualbulge-to-disc ratio for the annihilation emission may be sig-nificantly lower than it was stated earlier. The reason forthis drastic change was taking into account an additionalspatial component of the annihilation emission which coin-cides with the thick disc of the Galaxy. Therefore, it wasconcluded that sources of positrons should somehow be re-lated to old stars, which are abundant both in the Galactic ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] bulge and in the thick disc. The annihilation rates were de-termined as 2 × s − in the bulge and 3 × s − inthe thick disc (Siegert et al. 2016b).It is worth noting that independent observations by theballoon-borne Compton Spectrometer and Imager (COSI)showed a different spatial structure of the annihilation emis-sion (Kierans et al. 2020). The annihilation flux from theGalactic bulge is, however, consistent with the SPI mea-surements (Siegert et al. 2020). As pointed by Siegert et al.(2020), the discrepancy in the spatial structure may resultfrom an inability of a coded-mask instrument such as SPI toextract large-scale emission, e.g., coming from the Galactichalo.A complete model of the annihilation emission from theGalaxy should explain both the spatial morphology and thespectral properties of the emission. Since the spatial profileof the emission suggested by Siegert et al. (2016b) is more orless consistent with the stellar population, it is more naturalto assume that positrons do not travel far away from theirproduction sites and to search for the sources with the spa-tial distribution similar to that of the annihilation emission.Indeed, the simulations of Jean et al. (2009); Martin et al.(2012); Alexis et al. (2014) show that it is very unlikely forpositrons to travel more than ∼
200 pc away from theirsources (see the review of Panther 2018, for more details).Spectral properties of the emission require positronsto annihilate in a warm moderately ionized medium(Churazov et al. 2005; Jean et al. 2006; Churazov et al.2011) being cooled down to the energies in the order of ther-mal energy. Since positrons in most cases are born almost c (cid:13) Ya. N. Istomin et al. relativistic, this energy excess should somehow be dissipatedduring their lifetime.Mildly relativistic positrons lose energy mainly dueto collisions with free or bound electrons, i.e., due toCoulomb or ionization losses (Prantzos et al. 2011). How-ever, exactly the same collisions also lead to annihilationand bremsstrahlung producing gamma-ray emission above511 keV with hard spectrum. Since the annihilation emis-sion does not correlate with the Galactic background, it ispossible to extract the annihilation component and use it torestrict the initial energy of positrons (Aharonian & Atoyan1981). A comparison between theoretical predictions and
IN-TERGAL and COMPTEL data (Beacom & Y¨uksel 2006)as well as a combination of COMPTEL, IBIS, and EGRETdata (Sizun et al. 2006) showed that the maximum energy ofannihilating positrons cannot exceed 1 − −
10 MeV for fully ionized medium.This restriction can be relaxed if there is a colli-sionless mechanism of energy dissipation that dominatesover collisional losses for the energies above 3 −
10 MeV,such as adiabatic (Panther et al. 2018) or synchrotron(Chernyshov et al. 2010) losses. The first model imposes cer-tain requirements on the background medium, while the sec-ond requires the magnetic field in the order of several mG,i.e., unrealistically strong.Taking into account the above arguments, we sum-marise the requirements for the sources of positrons in thefollowing way: • The sources should be able to produce more than 2 × positrons per second in the Galactic bulge and morethan 3 × positrons per second in the thick disc • The maximum energy of positrons produced by thesources should not exceed 3 −
10 MeV • It is likely that the spatial distribution of the sourcesfollows that of the old stellar populationTwo classes of sources are presently considered as themain candidates. The first one involves β + decay of unstablenuclei produced during supernova explosions. In the recentpaper by Crocker et al. (2017) it was shown that the spe-cial subclass of SNe Ia can produce enough Ti to almostcompletely explain the origin of the Galactic annihilationemission.Another class of sources involves leptonic pairs pro-duced by jets in low-mass X-ray binaries (LMXBs) and mi-croquasars. This model is supported by the observed an-nihilation emission from V404 Cygni (Siegert et al. 2016a).Bartels et al. (2018) showed that LMXBs can be responsiblenot only for the annihilating positrons in the Milky Way butalso for the gamma-ray excess observed from the inner partof the Galaxy. However, the amount of positrons LXMBscan produce is still uncertain.In this paper, we show that old neutron stars, whichwere radio pulsars, can be a source of Galactic positrons.Usual radio pulsars are magnetized, rapidly rotating neutronstars with radius R ≃ cm, and magnetization means thatthe star has an intrinsic magnetic moment µ ≃ G cm corresponding to the typical magnetic field at the surface B ≃ G. The rotational period P of neutron starsvaries from milliseconds to about ten seconds, and everypulsar can conveniently be presented by a point on the P − B diagram (Fig. 1). High magnetic fields of neutron stars -3 -2 -1 s u r f a c e m agne t i c f i e l d , B ( G ) pulsar period, P (s) dea t h li ne pulsarextinctionpulsars Figure 1.
The magnetic field at the stellar surface, B , againstthe period of rotation, P , for the observed pulsars ( P − B diagram,double logarithmic scale). Arrows schematically show trajectoriesof motion on the P − B diagram for old neutron stars, extinct radiopulsars. and their rotation make them efficient generators of denserelativistic plasmas consisting of electrons and positronsand determining the observed activity (Istomin & Sob’yanin2007, 2011b). The fact that neutron stars can emit lep-tons was confirmed by the observation of gamma-ray ha-los around the pulsars Geminga and B0656+14 (Abdo et al.2009; Abeysekara et al. 2017).The existence of both the magnetic field and the rota-tion of a neutron star is important for the plasma gener-ation. In strong magnetic fields ( B & G, see Fig. 1),one-photon pair production occurs (i.e. γ → e + e − ), and thecreated electron and positron are subsequently acceleratedby the longitudinal electric field directed along the magneticfield and induced by rotation. The latter acceleration resultsin the radiation of new photons, and their absorption givesnew pairs, thereby forming a cascade process of plasma mul-tiplication. The stationary plasma generation near the stel-lar surface in the polar magnetosphere is maintained underthe condition of a sufficiently strong magnetic field and arather fast rotation, which defines the so-called death lineon the P − B diagram, P ∝ B / .The possible contribution of regular and millisecondpulsars to the positron flux in the Galaxy was discussed pre-viously (Wang et al. 2006), but the energy of positrons wasfound to be too high to explain the Galactic annihilationemission (de Jager & Djannati-Ata¨ı 2009; Prantzos et al.2011). Meanwhile, electrons and positrons will have lowerenergies than those going from the usual pulsars if these areproduced in the magnetospheres of less energetic neutronstars. Such neutron stars may have larger rotational peri-ods but must have the magnetic fields of a typical pulsarbecause such fields are necessary for one-photon pair pro-duction to be effective. When rotation of the neutron star MNRAS , 1–7 (2020) xtinct radio pulsars brakes and the energy of rotation is transmitted mainly tothe plasma generation, the star approaches the boundaryof ‘death’ on the P − B diagram and ceases to be a radiopulsar, now becoming an ‘extinct pulsar’ (see the Fig. 1).We show that extinct pulsars are able to produce enough e + e − pairs to account for the annihilation emission. Be-sides, these sources, unlike active pulsars, produce particleswith energy low enough to satisfy the requirements on thepositron source responsible for the 511-keV Galactic annihi-lation line.The paper is structured as follows. In section 2 we willdiscuss how an electron-positron plasma is generated in themagnetospheres of slowly rotating neutron stars, extinctradio pulsars. Section 3 determines the parameters of theplasma escaping from the open magnetosphere. Section 4 isdevoted to conclusions that follow from the presented modelof the origin of subrelativistic positrons in the Galaxy. After the neutron star has slowed down, intersected thedeath line on the P − B diagram, and become an extinctpulsar, production of e + e − pairs cannot continue near thestellar surface. Instead, pair production occurs in the mag-netosphere. The problem of appearance of the neutron starmagnetosphere from a vacuum with a strong magnetic fieldis very complex (Istomin & Sob’yanin 2009, 2010a,b). Thediffuse Galactic and isotropic extragalactic gamma radia-tion illuminates it and causes one-photon pair production(Istomin & Sob’yanin 2011b). Estimates show that about2 × gamma-ray photons with energy higher than thethreshold energy 2 m e c can be absorbed in the magneto-sphere per second (Istomin & Sob’yanin 2011c). The pro-duced electrons and positrons are accelerated by the longi-tudinal electric field present in the rotating magnetosphereand emit gamma quanta that in turn produce pairs.Thus, the plasma multiplication cascade develops, andthe dense electron-positron plasma, filling a narrow tube ofa size of about 100 m in the transverse direction, expandswith the speed of light both in the direction towards thestellar surface and outwards, so that after the absorptionof one external gamma-ray photon a ‘lightning’ forms, alengthening and simultaneously expanding plasma tube de-veloping on a ms time-scale. The lightning going towardsthe surface enters the region where the magnetic and longi-tudinal electric fields increase and produces a huge numberof electron-positron pairs up to 10 per external photon(Istomin & Sob’yanin 2011c). On the basis of this effect,Istomin & Sob’yanin (2011a) constructed a model for theformation of rotating radio transients (RRATs) from slowrotating neutron stars with P ≃ −
10 s. The other part ofthe lightning spreads outwards, and if it is in the open polarregion of the magnetosphere, consisting of the magnetic fieldlines going to infinity, the plasma flows into the surroundingspace.In the outer magnetosphere, the strength of the mag-netic and longitudinal electric fields rapidly falls, the plasmamultiplication cascade becomes less effective than in theinner magnetosphere, and the energies of electrons andpositrons become not as large as in the case of the part
Figure 2.
Production of electrons and positrons in the openmagnetosphere of a neutron star by high-energy photons fromthe cosmic gamma-ray background. The figure shows the neu-tron star (golden sphere), the closed magnetosphere (grey-shadedarea), where magnetic field lines (blue) begin and end at the stel-lar surface, and the open magnetosphere, where magnetic fieldlines (green) emanate from the polar cap (yellow) and go throughthe light cylinder (blue dashed lines) to infinity. The gamma-rayphotons (wavy arrows) are absorbed in the magnetosphere andconverted to e + e − pairs (orange circles). The produced plasmaflows along open magnetic field lines out of the magnetosphere(wide orange arrows). of the lightning propagating towards the surface. Here, theproduction of pairs involved in the cascade of plasma multi-plication is strongly non-local in nature. The mean free pathof photons with respect to pair production, the length of par-ticle acceleration, and the radius of curvature of magneticfield lines become comparable to the size of the magneto-sphere. The scheme of the electron-positron pair productionis shown in Fig. 2. We will try to determine from generalconsiderations the characteristics of the electron-positronplasma escaping from the magnetosphere, its flow, and char-acteristic energy. The characteristic size r L = c/ Ω ∼ cm of the magne-tosphere of a rotating neutron star, where c is the speed oflight and Ω = 2 π/P ∼ − is the rotation frequency (pe-riod P ∼
10 s), exceeds greatly the stellar radius R ≃ cmand corresponds to the distance at which plasma corotationbreaks down and its magnetic confinement becomes impos-sible. It determines a cylindrical light surface r = r L whichcontains the closed magnetosphere and at which the corota-tion velocity Ω r formally equals c (see Fig. 2). The typicalvertical size of the magnetosphere is in the order of its radius,and the energy flux for the outflow of an electron-positron MNRAS000
10 s), exceeds greatly the stellar radius R ≃ cmand corresponds to the distance at which plasma corotationbreaks down and its magnetic confinement becomes impos-sible. It determines a cylindrical light surface r = r L whichcontains the closed magnetosphere and at which the corota-tion velocity Ω r formally equals c (see Fig. 2). The typicalvertical size of the magnetosphere is in the order of its radius,and the energy flux for the outflow of an electron-positron MNRAS000 , 1–7 (2020)
Ya. N. Istomin et al. plasma escaping from the magnetosphere is W = 4 πr nγm e c , (1)where n is the plasma number density and γ is theLorentz factor. In what follows, it is convenient to char-acterize the plasma density by a dimensionless multiplic-ity λ = n/n GJ , where n GJ is the Goldreich-Julian den-sity (Goldreich & Julian 1969), which approximately corre-sponds to the minimum particle density necessary for anequilibrium magnetospheric charge density when the mag-netic field is not significantly twisted (Sob’yanin 2016). Atthe magnetospheric boundary we have n GJ = Ω B πce (cid:18) Ω Rc (cid:19) . (2)In order to determine the energy flux W (1) throughthe parameters of the neutron star, we first turn to the di-mensions of the quantities under consideration. The energyflux eventually goes from the rotational energy of the starand owes its existence to the rotating magnetic field frozen-in into the star, i.e., to the stellar magnetic moment µ . Since µ is a rotating vector, W cannot be proportional to the firstdegree of µ ; we have W ∝ µ because the correspondingdimensions are [ W ] = ML T − and [ µ ] = M / L / T − ,where M , L , and T are dimensions of mass, length, andtime, so that [ W ] = [ µ ] L − T − . Since R ≪ r L , the charac-teristic length is r L = c/ Ω, while the characteristic time isΩ − . Thus, W = a Ω µ c = 5 . × aB P − R erg s − , (3)where a ≃ a =(2 /
3) sin χ for vacuum magnetic dipole losses and a = i cos χ for current losses in a plasma-filled case, where χ is the inclination angle and i is a dimensionless currentflowing in the magnetosphere (Beskin et al. 1984), so that a ≃ c , and µ entering (3) and containingthe necessary three dimensions T , L , and M are not justarbitrary quantities but are the specific physical quantitiesthat characterize the real parameters of a rotating magne-tized neutron star – the rotation frequency (Ω), the magne-tospheric size ( r L = c/ Ω), and the magnetic moment ( µ ) –but not their fractions or their arbitrary combinations. Fromthese considerations it follows that the dimensionless factor a is generally of order unity, a ≃ λγ = aω c (cid:18) Ω Rc (cid:19) = 1 . × aB P − R , (4)where ω c = eB /m e c is the non-relativistic cyclotron elec-tron frequency, B is the magnetic field at the stellar surface, B = B / G and R = R/ cm are the dimensionlesssurface field and stellar radius. Equation (4) gives reasonableestimations for radio pulsars, λ ∼ − and γ ∼ − (Istomin & Sob’yanin 2007; Timokhin & Harding 2019).Another way to estimate the multiplicity is as follows:When the magnetosphere is filled with a plasma, the brak-ing of stellar rotation can be brought about only by elec-tric currents flowing in the magnetosphere and being closed at the stellar surface. The currents create the torque ofelectromagnetic forces braking the rotation. Since the mag-netic field in the closed magnetosphere rests on a highly-conducting surface of the neutron star, every magnetic tubehas the same electric potential equal to the stellar poten-tial and the electric current can flow in the open magneto-sphere only. Let us estimate the electric current flowing outof the magnetosphere. When magnetic field lines in the openpart of the magnetosphere go beyond the light surface, elec-trons and positrons move with slightly different velocities v e,p = c (1 − γ − e,p /
2) because the longitudinal electric fieldaccelerates one type of particles and decelerates the other,so the resulting current is I = λ B Ω R c | γ − e − γ − p | . (5)We have | γ − e − γ − p | = bγ − with a factor b <
1. The workof electric field on current I per unit time is the power beinglost by the star, W = UI , where U is the potential differ-ence generated by the rotating magnetic field in the openmagnetosphere; the relation is analogous to that takes placein relativistic jets, where the observed intensity correspondsto the power released in the jet via unipolar operation ofthe central engine, a supermassive black hole with a sur-rounding accretion disc (Sob’yanin 2017). Using Faraday’slaw U = − ( ∂ Φ /∂t ) /c , where Φ = 2 πr B (Ω R/c ) is themagnetic flux through the light surface, we arrive at U = 2 πB Ω R c , (6)and from UI = W we get πbλ = aγ . Combining this rela-tion with (4), we may separately find γ and λ , γ = (cid:18) πb (cid:19) / (cid:16) ω c Ω (cid:17) / (cid:18) Ω Rc (cid:19) = 3 . × b / B / P − / R , (7) λ = (cid:18) a πb (cid:19) / (cid:16) ω c Ω (cid:17) / (cid:18) Ω Rc (cid:19) = 4 × (cid:16) ab / (cid:17) B / P − / R . (8)Interestingly, the surface magnetic field B appears inrelations (7) and (8) in a combination B (Ω R/c ) , whichis the magnetic field at the light surface, B L = µ/r . Thelatter field does not depend separately on the surface field B or the stellar radius R but is determined by the stellarmagnetic moment µ and angular velocity Ω only. If we definethe cyclotron frequency for electrons and positrons at thelight surface, ω L c = eB L /m e c , relations (7) and (8) take ona simple form γ ≃ (cid:18) ω Lc Ω (cid:19) / , (9) λ ≃ (cid:18) ω Lc Ω (cid:19) / . (10)The ratio ω Lc / Ω is the particle magnetization at the lightsurface. The sole dependence of particle Lorentz factor onmagnetization presented in (9) is a universal property of thesystems in which the acceleration of charged particles in a
MNRAS , 1–7 (2020) xtinct radio pulsars magnetic field occurs. Particularly, it is impossible to accel-erate a particle whose cyclotron frequency ω c /γ becomes lessthan the characteristic frequency of a system under study(Hillas’ criterion) (Hillas 1984), so the maximum Lorentzfactor is γ max = ω c / Ω. In magnetospheres of rotating mas-sive black holes γ = ( ω c / Ω) β , where the index β < / /
3. In the works by Michel (1969, 1974) β appears to be 1 /
3. Istomin & Gunya (2020) have foundthat direct acceleration of protons by an electric field trans-mitting rotation from the black hole to the magnetosphericplasma is more efficient, β = 1 / β = 2 / E becomes comparable to the magnetic field B L . Aparticle during its gyration departs from the initial mag-netic field line by a distance l , and the electric field doesa work on it, so that γmc = eEl . Since E ≃ B L , wehave l ≃ l = γmc /eB L and then should use the criterion l = r c . The cyclotron radius r c = v/ ( ω c /γ ) is not identi-cally equal to l : firstly, the particle velocity v is slightlyless than the velocity of light, v = c (1 − / γ ), and sec-ondly, the effective magnetic field on the particle trajectoryis slightly less than B L because of its decrease with distance, B ≃ B L (1 − r c /r L ) ≃ B L [1 − γ (Ω /ω Lc )]. We drop off insignif-icant factors because the magnetic field at the light surfacedeviates from the dipole structure; besides, the electric fieldmay also differ from B L by the terms of order γ − , and thesame is true for the difference between l and l . With thesereservations, we get from l = r c l = l (cid:18) − γ (cid:19) (cid:18) γ Ω ω Lc (cid:19) . (11)Equating the small terms gives γ ≃ ω Lc / Ω, which corre-sponds to (9).Now let us find the positron flux L = W/ γm e c createdby the neutron star, L = 1 . × ab − / B / P − / R s − . (12)The factor b < /
3, so we may put b / ≃ P < P ≃ . ≃
10 s, these are few (note aunique ultra-slow pulsar PSR J0250+5854 with P = 23 . J Ω ˙Ω = − W , where J ≃ g cm is the stellar moment of inertia; therefore, P = P (cid:18) tτ (cid:19) / , (13) where τ = c P J π aB R = 1 . × P a J B R yr , (14)with J = J/ g cm being the dimensionless stellar mo-ment of inertia.The magnetic field of isolated neutron stars vir-tually does not decay (Bhattacharya & Srinivasan 1991;Hartman et al. 1997; Johnston & Karastergiou 2017) andexists in superconducting vortices in the stellar core, so whenestimating the maximum rotational period P max with thehelp of (13), we put the time t equal to the Galactic age t G ≃ . × yr, P max = P (cid:18) . × aP B R J (cid:19) / ≃ a / B R J / s . (15)We see that the maximum period of extinct pulsars is vir-tually independent of the period P with which intersectionof the death line has occurred. All the neutron stars thatbecame radio pulsars and were born at the moment of theGalaxy formation, now have period P = P max ≃
35 s.Let us define a pulsar birthrate ν ( t ) via dN = ν ( t ) dt , sothat the total number of neutron stars is N = R t G ν ( t ′ ) dt ′ ≃ (Arnett et al. 1989; Perna et al. 2003). If one assumesthat every supernova explosion leads to the birth of a ra-dio pulsar, then the rate ν of supernova explosions in theGalaxy corresponds to the birthrate of neutron stars. Cur-rent estimations give ν ≃ × − yr − (Tammann et al.1994; Diehl et al. 2006; Adams et al. 2013), and were thisquantity constant in time, the number of neutron stars wouldbe 4 × . It can be, however, that in the past, when theGalaxy was young, the rate of supernova explosions washigher, and the number of old neutron stars is greater thanthat of young (Crocker et al. 2017). An alternative is theearly star formation in the Galactic bulge up to 6 × (Ofek2009). In any case, it is difficult to determine the present pe-riod distribution of old neutron stars without knowledge ofthe history of pulsar formation. The period lies in the rangefrom P ≃ . P max ≃
35 s and depends on the mag-netic field B (note the pulsar death line and equation (15)).To find the average value of the Lorentz factor of outflowingparticles, we put P = P max because γ ∝ P − / correspondsto an integral dependence with positive exponent 1 / γ ≃ . × B / P − / R = 33 (cid:18) J aB R (cid:19) / . (16)It is worth noting that estimate (16) refers to the en-ergy of electrons and positrons flowing directly out of theneutron star magnetosphere. Expanding into the interstel-lar medium, a subrelativistic plasma cools down, and itsLorentz factor drops when the plasma enters the surround-ing gas. Let us estimate this value γ f . The plasma expandsfreely up to a distance l where its pressure becomes equalto the pressure P g of interstellar gas, P g = 2 γ f nm e c . Forthe relativistic gas, the plasma density n at the boundary ofthe cavity of size l is related to the density at the boundaryof the magnetosphere by n = n ( r L /l ) . On the other hand,the work done by the plasma for supplantation of the gasfrom the cavity is 4 πP g l / π (¯ γ − γ f ) m e c n r l . As a result, ¯ γ − γ f = 2 γ f / MNRAS000
35 s and depends on the mag-netic field B (note the pulsar death line and equation (15)).To find the average value of the Lorentz factor of outflowingparticles, we put P = P max because γ ∝ P − / correspondsto an integral dependence with positive exponent 1 / γ ≃ . × B / P − / R = 33 (cid:18) J aB R (cid:19) / . (16)It is worth noting that estimate (16) refers to the en-ergy of electrons and positrons flowing directly out of theneutron star magnetosphere. Expanding into the interstel-lar medium, a subrelativistic plasma cools down, and itsLorentz factor drops when the plasma enters the surround-ing gas. Let us estimate this value γ f . The plasma expandsfreely up to a distance l where its pressure becomes equalto the pressure P g of interstellar gas, P g = 2 γ f nm e c . Forthe relativistic gas, the plasma density n at the boundary ofthe cavity of size l is related to the density at the boundaryof the magnetosphere by n = n ( r L /l ) . On the other hand,the work done by the plasma for supplantation of the gasfrom the cavity is 4 πP g l / π (¯ γ − γ f ) m e c n r l . As a result, ¯ γ − γ f = 2 γ f / MNRAS000 , 1–7 (2020)
Ya. N. Istomin et al. γ f = 3¯ γ/
5. Thus, the energy of positrons interacting with theinterstellar gas is E p = γ f m e c ≃ m e c = 10 MeV . (17)With account of (16), ¯ γ ∝ B − / , the value (17) may beconsidered an upper limit; e.g., for B ≃
10 we have E p ≃ P = P in (12) because thestrong dependence P − / , with integral dependence being P − / , implies the main contribution at the lower limit ofperiods, P ≃ P , L tot ≃ . × aNB / P − / R s − = 3 . × aN B / R s − . (18)Here, N = 10 N is the number of old neutron stars, N ≃ α r ≃ − (Beskin et al. 1993). Inour case of old neutron stars, P ≃
15 s and B ≃
1, the en-ergy flux of the particles emanating from the magnetosphereis W ≃ erg s − . Thus, the power of radio emission froman old neutron star is W r ≃ erg s − .However, the emitted radio waves have significantlylower frequencies, ω = cγ /ρ c , where ρ c ≃ r L = cP/ π is the radius of curvature of the magnetic field lines in thefar open magnetosphere. For the characteristic value of theLorentz factor of electrons and positrons in the magneto-sphere, γ ≃
30, we have the value of the frequency of radi-ated radio waves, f = P − γ ≃ ≃
10 pc, then the observedenergy flux density should be ≃ f ≃ × (10 − ) Hz, the power is equal to10 − erg s − . Radiation is observed on remote space-crafts, even on the orbit of the Moon. Similar radiation wasdetected from other planets of the Solar system which havetheir own magnetic field. Then the Jupiter emits decametricradio emission. Old neutron stars considered in this paper generatepositrons with much lower energy not exceeding 10 MeV.The total flux of positrons produced by such stars, ≃ × s − , corresponds to the necessary flux in the bulge andin the thick disc of the Galaxy. As for the spatial distribu-tion of extinct radio pulsars, it may seem that they, as wellas active radio pulsars, should be widely distributed overthe distance z to the Galactic plane, | z | ≃
300 pc. However,over a long period of time comparable to the age of theGalaxy, neutron stars with high vertical speeds begin to os-cillate in the vertical direction in the gravitational potentialof the Galaxy. Calculations show that half period of oscil-lations is in the order of 4 × yr (Bienaym´e et al. 2006),which on the one hand is greater than the characteristic ageof radio pulsars, ≃ − yr, and on the other hand ismuch less than the age of the Galaxy, ≃ yr, so oldneutron stars cross many times the Galactic plane, wherethey lose their velocity due to friction (Zelnikov & Kuskov2016; Webb et al. 2018). Thus, their distribution is close tothe distribution of old stars, and they satisfy all the require-ments for the positron source responsible for the 511-keVGalactic annihilation line. REFERENCES
Abdo A. A., et al., 2009, ApJ, 700, L127Abeysekara A. U., et al., 2017, Science, 358, 911Adams S. M., Kochanek C. S., Beacom J. F., Vagins M. R., StanekK. Z., 2013, ApJ, 778, 164Aharonian F. A., Atoyan A. M., 1981, Astron. Lett., 7, 395Alexis A., Jean P., Martin P., Ferri`ere K., 2014, A&A, 564, A108Arnett W. D., Schramm D. N., Truran J. W., 1989, ApJ, 339,L25Bartels R., Calore F., Storm E., Weniger C., 2018, MNRAS,480, 3826Beacom J. F., Y¨uksel H., 2006, Phys. Rev. Lett., 97, 071102Beskin V. S., Gurevich A. V., Istomin Ya. N., 1984, Ap&SS, 102,301Beskin V. S., Gurevich A. V., Istomin Ya. N., 1993, Physics ofthe Pulsar Magnetosphere. Cambridge University Press, Cam-bridgeBhattacharya D., Srinivasan G., 1991, The Evolution of NeutronStar Magnetic Field. Springer Science, Dordrecht, p. 219Bienaym´e O., Soubiran C., Mishenina T. V., Kovtyukh V. V.,Siebert A., 2006, A&A, 446, 933Chernyshov D. O., Cheng K. S., Dogiel V. A., Ko C. M., Ip W. H.,2010, MNRAS, 403, 817Churazov E., Sunyaev R., Sazonov S., Revnivtsev M., Var-shalovich D., 2005, MNRAS, 357, 1377Churazov E., Sazonov S., Tsygankov S., Sunyaev R., VarshalovichD., 2011, MNRAS, 411, 1727Crocker R. M., et al., 2017, Nat. Astron., 1, 0135Diehl R., et al., 2006, Nature, 439, 45Goldreich P., Julian W. H., 1969, ApJ, 157, 869Gruzinov A., 2005, Phys. Rev. Lett., 94, 021101MNRAS , 1–7 (2020) xtinct radio pulsars Gruzinov A., 2006, ApJ, 647, L119Hartman J. W., Bhattacharya D., Wijers R., Verbunt F., 1997,A&A, 322, 477Hillas A. M., 1984, ARA&A, 22, 425Istomin Ya. N., Gunya A. A., 2020, MNRAS, 492, 4884Istomin Ya. N., Sob’yanin D. N., 2007, Astron. Lett., 33, 660Istomin Ya. N., Sob’yanin D. N., 2009, J. Exp. Theor. Phys., 109,393Istomin Ya. N., Sob’yanin D. N., 2010a, Astron. Rep., 54, 338Istomin Ya. N., Sob’yanin D. N., 2010b, Astron. Rep., 54, 355Istomin Ya. N., Sob’yanin D. N., 2011a, Astron. Lett., 37, 468Istomin Ya. N., Sob’yanin D. N., 2011b, J. Exp. Theor. Phys.,113, 592Istomin Ya. N., Sob’yanin D. N., 2011c, J. Exp. Theor. Phys.,113, 605Istomin Ya. N., Philippov A. A., Beskin V. S., 2012, MNRAS,422, 232Jean P., Kn¨odlseder J., Gillard W., Guessoum N., Ferri`ere K.,Marcowith A., Lonjou V., Roques J. P., 2006, A&A, 445, 579Jean P., Gillard W., Marcowith A., Ferri`ere K., 2009, A&A,508, 1099Johnson W. N. I., Harnden F. R. J., Haymes R. C., 1972, ApJ,172, L1Johnston S., Karastergiou A., 2017, MNRAS, 467, 3493Kierans C. A., et al., 2020, ApJ, 895, 44Kn¨odlseder J., et al., 2005, A&A, 441, 513Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, AJ,129, 1993Martin P., Strong A. W., Jean P., Alexis A., Diehl R., 2012, A&A,543, A3Michel F. C., 1969, ApJ, 157, 1183Michel F. C., 1974, ApJ, 192, 713Ofek E. O., 2009, PASP, 121, 814Panther F., 2018, Galaxies, 6, 39Panther F. H., Crocker R. M., Birnboim Y., Seitenzahl I. R.,Ruiter A. J., 2018, MNRAS, 474, L17Perna R., Narayan R., Rybicki G., Stella L., Treves A., 2003, ApJ,594, 936Prantzos N., et al., 2011, Rev. Mod. Phys., 83, 1001Siegert T., et al., 2016a, Nature, 531, 341Siegert T., Diehl R., Khachatryan G., Krause M. G. H., Gugliel-metti F., Greiner J., Strong A. W., Zhang X., 2016b, A&A,586, A84Siegert T., et al., 2020, arXiv e-prints, p. arXiv:2005.10950Sizun P., Cass´e M., Schanne S., 2006, Phys. Rev. D, 74, 063514Sob’yanin D. N., 2016, Astron. Lett., 42, 745Sob’yanin D. N., 2017, MNRAS, 471, 4121Spitkovsky A., 2006, ApJ, 648, L51Tammann G. A., L¨offler W., Schr¨oder A., 1994, ApJS, 92, 487Tan C. M., et al., 2018, ApJ, 866, 54Timokhin A. N., Harding A. K., 2019, ApJ, 871, 12Wang W., Pun C. S. J., Cheng K. S., 2006, A&A, 446, 943Webb J. J., Leigh N. W. C., Singh A., Ford K. E. S., McKernanB., Bellovary J., 2018, MNRAS, 474, 3835Weidenspointner G., et al., 2006, A&A, 450, 1013Wu C. S., Lee L. C., 1979, ApJ, 230, 621Zelnikov M. I., Kuskov D. S., 2016, MNRAS, 455, 3597de Jager O. C., Djannati-Ata¨ı A., 2009, Implications of HESSObservations of Pulsar Wind Nebulae. Springer, Berlin, p. 451This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000