THE ASTROPHYSICAL JOURNAL, 498:327È341, 1998 May 1 ( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE STATISTICS OF GAMMA-RAY K. S. CHENG AND L. ZHANG Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong; hrspksc=hkucc.hku.hk, ZhangL=hkuphy1.hku.hk Received 1997 July 15; accepted 1997 December 9

ABSTRACT We use Monte Carlo methods to simulate the properties of a Galactic population of rotation-powered pulsars in the self-consistent outer gap model proposed by Zhang & Cheng, where the initial magnetic Ðeld and spatial and velocity distributions of the neutron at birth are obtained from the statistical results of radio pulsars. We obtain the distance, period, age, magnetic Ðeld, and c-ray Ñux distributions of the c-ray pulsars whose c-ray and radio Ñuxes are above the detectable threshold Ñuxes. Furthermore, we simulate the properties of Geminga-like pulsars and obtain the possible parametric region and the number of the Geminga-like pulsars. In our simulations, the di†erent beaming e†ects of radio and c-ray beams are taken into account. We predict that there may be D11 c-ray pulsars that are detectable at both radio and c-ray energies and D37 Geminga-like pulsars in the Galactic plane ( o b o \ 5¡) if the c-ray beaming fraction is 0.67 and the birthrate of the neutron stars is 1 per century. Compared to the Ñux distribution, individual spectrum, and distance distribution obtained by assuming OB associations of our simulated Geminga-like pulsars and the unidentiÐed EGRET sources at o b o \ 5¡, we conclude that the majority of the unidentiÐed c-ray point sources near the Galactic plane may be Geminga-like pulsars. We also suggest that the most possible radio pulsars likely to be conÐrmed as c-ray pulsars in future are those with L /E0 D 10~3. X sd Subject headings: gamma rays: theory È pulsars: general È stars: neutron È stars: statistics

INTRODUCTION 1. gap models. In polar gap models, charged particles are A database of 706 radio pulsars can be obtained from the accelerated in charge-depleted zones near the pulsarÏs polar archive data of Princeton University (also seeTaylor, Man- cap and c-rays are produced through curvature-radiationÈ chester, & Lyne1993). Furthermore, the upper limits on induced c-B pair cascade (e.g.,Harding 1981) or through pulsed c-ray emission for 350 pulsars have been given (Nel Compton-induced pair cascades(Dermer & Sturner 1994). et al.1996), which include two kinds of pulsars: canonical In the outer gap models, large regions of the magneto- pulsars and millisecond pulsars. Of these pulsars, only six spheric charge depletion (gaps) are assumed to result from a have been conÐrmed to emit high-energy c-rays in the global current Ñow pattern through the magnetosphere, energy range of EGRET (PSR B1509[58 was detected up and charged particles that are likely / are to D1 MeV but has not been detected by EGRET). accelerated to extreme relativistic energies because of large Ramanamurthyet al. (1996) have reported possible evi- electric Ðelds along the magnetic lines in these gaps. The dence for pulsed c-ray emission above 50 MeV from PSR observed c-rays are produced by the synchrotron radiation B0656]14. In addition, Geminga was not detected as a (Vela type) and synchrotron self-Compton mechanism radio . This information allows us to perform useful (Crab type) of secondary eB pairs(Cheng, Ho, & Ruderman statistical studies of the observed c-ray pulsar parameters, 1986a, 1986b,hereafter respectively CHR I, CHR II; Ho such as the distance, period, age, magnetic Ðeld, and c-ray 1989; Chiang& Romani 1992; Cheng & Ding 1994; Cheng Ñux distributions and may also provide clues about how to & Wei1995; Romani 1996) or by synchro-curvature radi- predict possible candidates for c-ray pulsars. In fact, based ation of primary eB pairs(Zhang & Cheng 1997, hereafter on the data observed by EGRET, many authors have ZC97). Based on known c-ray pulsars, studies have been already made some predictions of possible candidates for made of the luminosity and conversion efficiency of c-rays c-ray pulsars (e.g.,Cheng & Ding 1994; Brazier et al. 1994; in various models.Harding (1981) predicted that the c-ray Thompsonet al. 1994; Fierro et al. 1995). In particular, luminosity from a pulsar can be expressed by L th([100 c Cheng& Ding (1994) predicted that PSR B1951]32 should MeV) B 1.2 ] 1035B0.95P~1.7 photons s~1, whereB is be the pulsar conÐrmed to be a c-ray pulsar after PSR the magnetic Ðeld in12 units of 1012 G and P is the pulsar12 B1055[52.Fierro (1995) also pointed out that pulsars PSR period in seconds.Dermer & Sturner (1994) predicted that J0034[0534, J0613[0200, PSR B1046[58, B1832[06, the c-ray luminosity from the polar cap is given by L th B c and B1853]01 are the most probable candidates for c-ray 1032B3@2P~2 ergs s~1. Comparing the observed c-ray lumi- pulsars. This information can also be used to test the valid- nosities12 (or upper limits) with those predicted by the models ity of the various c-ray pulsar models. Moreover, the second ofHarding (1981) and Dermer & Sturner (1994), Nel et al. EGRET catalog and its supplement (Thompson et al. 1995, (1996) suggested that both polar gap models overestimate 1996) lists 96 unidentiÐed point sources, where 30 unidenti- L at high luminosity and underestimateL at low lumi- Ðed point sources are in low Galactic latitudes ( o b o \ 5¡). It nosity.c A recent phenomenological polar gapc model (Rudak is believed that the unidentiÐed point sources at the Galac- & Dyks1997) can give a more satisfactory explanation for tic plane probably consist of young pulsars (e.g.,Kaaret & L from pulsars. For the outer gap models, the predicted Cottam1996; Yadigaroglu & Romani 1997). We will use c-rayc luminosity depends on the solid angle of c-ray these observation data to compare with our analysis. Gen- beaming.Yadigaroglu & Romani (1995) suggested that the erally, c-ray models can be divided into polar gap and outer conversion efficiency of c-rays(g 4 L /E0 , whereE0 is the c c sd sd 327 328 CHENG & ZHANG Vol. 498 spin-down power) should increase with pulsar age to derive will be reÑected back to the stellar surface due to the cyclo- a phenomenological scaling law of the form g \ 3.2 tron resonance scattering if there is a large density of mag- c ] 10~5q0.76, where q is the pulsar age in units of years. netically produced eB pairs near the surface Recently we have proposed a new self-consistent outer gap (Halpern& Ruderman 1993), and will eventually reemit model, called the thick outer gap model, to describe the softer thermal X-rays with characteristic temperature high-energy c-ray radiations from mature pulsars (ZC97). (ZC97) In this paper, we will use theZC97 model to determine T B 3.8 ] 105f 1@4P~5@12B1@4R K (1) general formulae for c-ray Ñux and the luminosity and con- s 12 6 version efficiency of c-rays from canonical pulsars that are and power functions of pulsar parameters, e.g., period and magnetic Ðeld strength. Using as a basis models for c-ray production L soft B 1.4 ] 1031fB P~5@3 ergs s~1 . (2) X 12 from pulsars, many authors have studied detailed statistical Calculations show that the optical depth of the soft X-ray properties of c-ray pulsars by using Monte Carlo methods. photons in the outer gap is only DL p /4nrkT c D For example,Bailes & Kni†en (1992) investigated the polar soft ?eB 10 . However, since the potential dropX ofX thec outer gap is gap model(Harding 1981) and outer gap model (CHR I, ~4 D10 V(CHR I, CHR II), each primary e`/e passing CHR II)to explain the Galactic di†use c-rays. Yadigaroglu 15 ~ through the gap can emit more than 10 multi-GeV curva- & Romani(1995) have considered the properties of c-ray 5 ture photons. Such huge multiplicity can produce a suffi- pulsars in their outer gap model.Sturner & Dermer (1996) cient number of eB pairs to sustain the gap as long as the simulated the properties of the Galactic population of center-of-mass energy of X-ray and curvature photonsis rotation-powered pulsars on the basis of their model for higher than the threshold energy of the / c-ray production from pulsars. Here we will present the pair production, i.e.,E E º (m c ) . From the condition statistical analysis of c-ray properties of rotation-powered e 2 2 for the photon-photonX pairc production, the fractional size pulsars based on our model. In° 2 we will review our outer of the outer gap limited by the soft thermal X-rays from the gap model results and use the database (including only neutron star surface can be determined as those pulsars with upper limits) given byNel et al. (1996) to test our model. In° 3 we describe the Monte Carlo simula- f \ 5.5P26@21B~4@7 . (3) tion of the Galactic pulsar population and present the simu- 12 lation results. In° 4 we simulate the population of It should be emphasized that f ¹ 1 and any Ñuctuation of Geminga-like pulsars and compare that with the unidenti- the gap can be stabilized by the processes described above. Ðed point sources of EGRET. In° 5 we further analyze the In fact, the increase in gap size will increase the X-ray Ñux database of radio pulsars and give some predictions about and the overproduced pairs can reduce the gap size. Simi- the most possible candidates for c-ray pulsars based on our larly, a decrease in the gap size will produce insufficient model. Finally, a brief conclusion and discussion will be pairs and result in an increase in the gap size. Furthermore, given in ° 6. it has been argued(Halpern & Ruderman 1993; ZC97) that the primary electrons/positrons leaving the outer gap emit 2. REVIEW OF THE THICK OUTER GAP MODEL AND curvature photons, which will be converted into the second- COMPARISON WITH THE OBSERVED DATA ary eB pairs by the neutron star magnetic Ðeld if these We have proposed a self-consistent model of a thick outer photons come close to the star, where the pair production gap to describe the high-energy c-ray emission from mature condition in strong magnetic Ðeld is satisÐed (Erber 1966; pulsars(ZC97). In our model, the characteristic photon Ruderman& Sutherland 1975). In ZC97 we have estimated energyE ( f ) is completely determined by the size of the that the magnetic pair production will start at about 10È20 outer gapc ( f ), which is the ratio between the outer gap stellar radii. Since the primary photon energies are of the order of a few GeV for all known c-ray pulsars, the synchro- volume andRL3, whereRL is the light cylinder radius. Half of the primary eB pairs in the outer gap will move toward the tron photons will have a typical energy of the order of a few star and lose their energy via the curvature radiation. The hundred MeV. Most of these synchrotron photons are moving toward the star because the local Ðeld lines are return particle Ñux can be approximated by N0 eB B 1 fN0 , where f is the fractional size of the outer gap andN0 2 is theGJ converging toward the star.Cheng, Gil, & Zhang (1998) Goldreich-Julian particle Ñux(Goldreich & JulianGJ 1969). estimated that when these photons move in toward the star Although most of the energy of the primary particles will be by a factor of 0.37, the local magnetic Ðeld will become lost on the way to the star via curvature radiation, about strong enough to convert these synchrotron photons to 10.6P1@3 ergs per particle will still remain and will be Ðnally pairs which subsequently lose their energies via synchrotron deposited on the stellar surface. This is because the curva- radiation with a characteristic synchrotron energy of the ture radiation loss depends on the 4th power of the Lorentz order of a few tens of MeV. These processes can repeat at factor of the charged particles. When the Lorentz factor is least once more before the synchrotron photon energy below D107, the timescale for the curvature energy loss will reduces to DMeV and the spectral index steepens from be longer than the time needed for the particle traveling to D[1.5 to D[2. They estimated that the luminosity of this the star. This energy will then be emitted in the form of nonthermal X-ray component is X-rays from the stellar surface (such a process has been L syn B 8.8 ] 10~6(tan4 s) f 1@2B5@12P~17@12L , (4) described byHalpern & Ruderman 1993). The character- X 12 sd istic energy of X-rays is given by Eh B 3kT B 1.2 where s is the inclination angle between the magnetic axis ] 103f 1@4P~1@6B1@4 eV, where P is the pulsarX period in and the rotation axis.Cheng et al. (1998) have pointed out 12 units of seconds,B is the dipolar magnetic Ðeld in units of thatL /E0 D 10~3 for pulsars with f ¹ 1 and s D 55¡, and 1012 G and the radius12 of the neutron star is assumed to be this resultX sd seems quite consistent with the observed data 106 cm. Furthermore, the keV X-rays from a hot polar cap (Becker& TruŽ mper 1997). We want to point out that X-ray No. 1, 1998 STATISTICS OF GAMMA-RAY PULSARS 329 luminosity is predicted by the outer gap model, and it plays and an important role in our later analysis for identifying which unidentiÐed EGRET sources and radio pulsars are possible 1 CA r 1 R B R Q \ B ] 1 [ 3 L x cos4 a ] 3 L x cos2 a c-ray pulsars. The predicted X-ray photon spectra for E ¹ 2 xRL RL x rB rB 1 MeV are given by X AR B2 D1@2 ] L x2 sin4 a , (13) dN0 r X \ F (T , E ) ] AE~2 , (5) B dE bb s X X X where rB \ cmc2 sin a/eB; a is the pitch angle of a charged whereF is the blackbody spectrum with a characteristic particle in the curved magnetic Ðeld (B) and aB 0.79f B P x ; q is the pulsarÏs age in units of temperaturebb kT \ Es , which satisÐes 1@2 ~3@4 7@4 17@4 s X years;F(12y) \ /= K (z)dz; K andK are modiÐed y 5@3 5@3 2@3 P Bessel functions of order 5/3 and 2/3; andy \ E /Ec. The F E dE \ L soft , (6) characteristic energy of the radiated photons is givenc by bb X X X 3 A P B3@28 and E \ +cc(x)3Q B 640.5 x~13@4(R xQ ) MeV . c 2 2 B L 2 12 tan4 s A B 1.7 ] 1035f 1@2P~65@12B29@12 , (14) 12 ln MMeV/[+eB(r)/mc]N Therefore, the integrated Ñux withE [ 100 MeV at the (7) Earth is given by c where r is the distance to the star at which the magnetic Ðeld 1 P 10 GeV d2N F(E [ 100 MeV) \ c dE , (15) becomes strong enough to convert the curvature photons c *) d2 dE dt c into pairs. Here we would like to remark that our thick c 100 MeV c where*) is the solid angle of c-ray beaming and d is the outer gap model does not work very well for very young c pulsars, e.g., the Crab pulsar, because we have ignored the distance to the pulsar. In our model, there are three param- eters, i.e.,*) , x andx . These three parameters Compton cooling of the return current. In very young c min max pulsars, the surface temperature is determined by cooling depend on the detailed structure of the outer gap and the instead of polar cap heating and is sufficiently high that the inclination angle of the pulsar. We have calculated the dif- inverse Compton scattering by the surface soft X-ray ferential Ñux of c-rays for the known c-ray pulsars and foundx D 2, and*) andx vary between 0.5 and 3.9 photons can reduce the energy of the return current (CHR max c min II). ZC97 have also shown that the Lorentz factor of accel- and between 0.55 and 0.7, respectively, for known c-ray erated particles inside the outer gap is given by pulsars(ZC97). In Figure 1 we compare our model spectra from 0.1 keV to 10 GeV with the observed data. The dashed c(x) B 2 ] 107f 1@2B1@2P~1@4x~3@4 , (8) curve is the best-Ðt model curve with the Ðtting parameters 12 shown inTable 1. The solid curve is the consistent Ðt in where x \ s/RL, s is the curvature radius, andRL is the which we use one set of parameters for all c-ray pulsars, i.e., radius of the light cylinder. The energy distribution of the *) \ 2.0, x \ 0.55, andx \ 2.0 for all c-ray pulsars. accelerated particles is given by Wec can seemin that the consistent-Ðtmax curves generally agree with the data quite well. In particular, the c-ray Ñux above dN dN A x B9@4 100 MeV predicted by the consistent-Ðt curve agrees with dEe \ dx B 1.4 ] 1030fB P~1 , (9) dEe dx 12 x the observed data to within a factor of 3. This set of consis- max tent Ðtting parameters plays two very important roles in whereEe is the energy of the accelerated particles and x our later analysis. First, it allows us to predict the model is the maximum curvature radius of the magnetic Ðeld linesmax c-ray spectrum and Ñux for unidentiÐed c-ray sources if they in units of the radius of the light cylinder. The radiation are Geminga-like pulsars and radio pulsars. Second, in the spectrum produced by the accelerated particles with a outer gap model, radio and c-rays do not in general come power-law distribution in the outer gap can be expressed as from the same place; we need to estimate the c-ray beaming (Cheng& Zhang 1996; ZC97) factor before we do our statistical analysis. In° 3.1 we will show how to estimate the c-ray beaming factor from *) . d2N N0 P xmax R The energy Ñux of c-rays between 100 MeV and 10 GeV canc c B 0 x3@2 L dE dt E r expressed as c c xmin c CA 1 B A 1 B D 1 P 10 GeV d2N ] 1 ] F(y) [ 1 [ yK dx (10) Sth(E [ 100 MeV) \ E c dE , (16) r2 Q2 r2 Q2 2@3 c c *) d2 c dE dt c c 2 c 2 c 100 MeV c while the corresponding luminosity is with L (E [ 100 MeV) \ *) d2S (E [ 100 MeV) . (17) J3e2c N c c c c c N0 \ 0 0 B 6.2 ] 1037q~11@56X~9@4 s~1 , (11) We can calculateequation (17) numerically for given 0 2hR max L parameters of pulsars. Furthermore, the total c-ray lumi- NCA r 1B R D nosity is given by (ZC97) r \ xR 1 ] B cos2 a ] L x sin2 a , (12) c L R x r L B 3.8 ] 1031f 3P~4B2 ergs s~1 . (18) L B c 12 330 CHENG & ZHANG Vol. 498

FIG. 1.ÈComparisons of our model spectra from 0.1 keV to 10 GeV with the observed data for Ðve known c-ray pulsars detected by EGRET. The dashed curves are the best-Ðt model curves, and the solid curves are consistent-Ðt curves in which we use one set of parameters for all c-ray pulsars.

FromFigure 1 we can see that most of the c-ray luminosity E0 B 3.8 ] 1031P~4B2 ergs s~1, the conversion efficiency is contributed byE [ 100 MeV, so that the c-ray lumi- withsd E [ 100 MeV can12 be approximated by nosity withE [ 100c MeV can be approximated by c c g (E [ 100 MeV) B 10~4q6@7P2 . (19) equation (18).From equation (18), we can estimate the con- c c version efficiency of spin-down power into c-rays. Using The comparisons between the theoretical integrated Ñux,

TABLE 1 BEST-FIT MODELPARAMETERSa OF c-RAY PULSARS

PB d sc Pulsar (s) (G)12 (kpc) x b x b *) b (deg) min max c Vela ...... 0.089 3.4 0.5 ^ 0.1 0.70 2.0 1.0 55.0 B1951]32...... 0.04 0.49 2.5`0.6 0.55 2.0 2.4 55.0 ~1.0 B1706[44...... 0.102 3.1 1.82`0.7 0.62 2.0 0.6 55.0 Geminga ...... 0.237 1.6 0.25 ^~0.50.1 0.7 2.1 3.9 55.0 B1055[52...... 0.197 1.0 1.5 ^ 0.4 0.55 2.1 2.6 77.0

a Values of the consistent-Ðt parameters are x \ 0.55, x \ 2.0, *) \ 2.0, and s \ 55¡ for all c-ray pulsars which are used for calculatingmin the solidmax curves in Fig. 1. b Values of the best-Ðt parameters for c-rays. c Values of the best-Ðt parameters for X-rays. No. 1, 1998 STATISTICS OF GAMMA-RAY PULSARS 331 energy Ñux, luminosity, and conversion efficiency and the where u \ V /350 km s~1 and V is the random velocity of observed data of pulsars are presented inZhang & Cheng the pulsar, which is chosen randomly and added to the (1998). circular velocity. The surface magnetic Ðelds of the pulsars are distributed as a Gaussian in log B with mean log B \ 0 3. GALACTIC PULSAR POPULATION 12.4 and dispersionpB \ 0.3, namely, In order to consider the luminosity and spatial evolution 1 C 1 Alog B [ log B B2D oB(log B) \ exp [ 0 . (26) of a Galactic population of the canonical pulsars, the initial J2np 2 pB values of parameters of pulsars at birth, which include the B initial position, velocity, period, and magnetic Ðeld strength, Here we assume that the magnetic Ðelds of pulsars do not are needed. Furthermore, the beaming and selection e†ects decay. Furthermore, one of the pulsar parameters for at radio and c-ray energies for the canonical pulsars have to Monte Carlo simulation is the initial period for each pulsar. be considered. In this section we will describe our Monte Here we use the assumption that the initial rotation period Carlo simulations of the Galactic pulsar population. distribution of the pulsars is a d-function at some period P (Sturner& Dermer 1996). After giving the initial properties0 of a newly born pulsar, we can follow its trajectory and 3.1. Basic Parameters and Detectability of Pulsars evolve its period until the present time. At any time t, the The conventional assumption for the Galactic pulsar pulsarÏs location is determined by following its motion in population is that pulsars are born at a rate (say D1/100 yr) the Galactic gravitational potential. Using the equations at spin periods ofP , which vary exponentially with dis- given byPaczyn ski (1990), the orbit integrations are per- tance from the Galactic0 center and vary exponentially with formed using the fourth-order Runge-Kutta method with distance from the Galactic plane. The birth location for variable time step(Press et al. 1992) on the variables R, VR, each pulsar is estimated from spatial distributions in z and z,Vz, and /. Once the coordinates [R(t), z(t), /] are deter- R (Paczyn ski 1990; Sturner& Dermer 1996), i.e., mined, we can calculate the Galactic longitude and latitude and the distance of the pulsar, namely, 1 o (z) \ e~@z@@zexp , (20) z z G R(t) H exp l \ 57¡.3 arcsin sin / , [R(t)2]R_2[2R(t)R_cos /]1@2 aR o (R) \ Re~R@Rexp , (21) (27) R R2 exp G z(t) H where z is the distance from the Galactic plane, R is the b \ 57¡.3 arctan , distance from the Galactic center,z \ 75 pc, a \ [R(t)2]R_2[2R(t)R_cos /]1@2 R R exp R [1 [ e~ max@ exp(1 ] R /R )]~1, R \ 4.5 kpc, and (28) R \ 20 kpc. The initialmax exp velocity ofexp each pulsar is the vectormax sum of the circular rotation velocity at the birth and location and a random velocity from the explo- d \ J[R(t)2]R2[2R(t)R cos /)]1@2 ] z(t)2 , (29) sion. The circular rotation velocity is determined by _ _ (Sturner& Dermer 1996) whereR_ \ 8.0 kpc is the Galactocentric distance of the . Furthermore, the evolution of the pulsarÏs period can C Ad' d' d' BD1@2 be represented by v \ R sph ] disk ] halo , (22) circ dR dR dR C A16n2R6 B2BD1@2 P(t) \ P2] NS t (30) where' , ' , and' are spheroidal, disk, and halo 0 3Ic3 componentssph ofdisk Galactichalo gravitational potential, given by Paczyn ski (1990) as if the period evolution is due to the e†ects of magnetic dipole radiation, whereR is the neutron star radius, I is NS GMi the neutron star moment of inertia, and c is the speed of 'i(R, z) \ , (23) light. The 400 MHz radio luminosity,L , of each model R [a (z b ) ] J 2] i] 2] i21@2 2 pulsar is determined from the following400 distribution wherea \ 0.0 kpc,b \ 0.277 kpc, and M \ 1.12 (Narayan& Ostriker 1990): sph sph sph ] 1010 M_ for the spheroidal component,a \ 3.7 kpc, disk oL (P, P0 ) \ 0.5j2e~j , (31) b \ 0.200 kpc, andM \ 8.07 ] 1010 M_ for the disk 400 component.disk The halo componentdisk is given by where j \ 3.6[log (L /SL T) ] 1.8], log SL T \ 6.64 ] 1 log (P0 /P3), and400L is400 in units of mJy kpc4002.In GM C1 A r2B2 r A r BD 3 400 ' (r) \ halo ln 1 ] ] c arctan , order to generate a sample of detectable canonical pulsars, halo rc 2 rc2 r rc we need to consider the beaming e†ect of the radio beam. (24) The radio beaming fraction can be expressed as (Emmering & Chevalier 1989) where r \ (R2]z2)1@2 is the distance to the Galactic center, fr(u) \ (1 [ cos u) ] (n/2 [ u) sin u , (32) rc \ 6.0 kpc, andM \ 5.00 ] 1010 M_. For the random velocity, their three-dimensionalhalo velocity distribution is where a random distribution of magnetic inclination angles accurately given by(Lyne & Lorimer 1994) is assumed and u is the half-angle of the radio emission cone. Earlier studies have assumed thatfr D 0.2 (i.e., 4 u h D 10¡) andf is independent of pulsar period (e.g., o (u) \ , (25) r V n (1 ] u4) Vivekanand& Narayan 1981). However, recent studies 332 CHENG & ZHANG indicate thatfr depends on the pulsar period because the observed photons and can be estimated by(Johnston et al. half-angle varies with period. Di†erent models have di†er- 1992; Sturner& Dermer 1996) ent relations of half-angle and period (e.g.,Narayan & Vive- A400 MHzB4.4 kanand1983; Lyne & Manchester 1988; Biggs 1990; Gil, q (l) \ q (37) Kijak, & Seiradakis1993; Gil & Han 1996). Here we will scat scat,400 l use the model ofLyne & Manchester (1988) corrected by with q 10 ` 10 ` . P \ ~4.61 1.14 log DM ] ~9.22 4.46 log DM Biggs (1990), i.e.,u \ 6¡.2] ~1@2. Then, following The dispersionscat,400 measure of the pulsar is calculated by using Emmering& Chevalier (1989), a sample pulsar with a given DM / n (R, Z)ds, where the electron density is given by P f P \ e period is chosen in one out ofr( )~1 cases using the (Lyne,Manchester, & Taylor 1985; Bhattacharya et al. Monte Carlo method. For the c-ray beaming e†ect, follow- 1992; Sturner& Dermer 1996) ing the same consideration as the radio beaming e†ect given byEmmering & Chevalier (1989), the beaming fraction of 1 @z@ H @z@ H the solid angle swept out by c-ray pulsar beams is given by n (R, z) \ (n e~ @ 1 ] n e~ @ 2) , (38) e 1 ] R/R 1 2 f \ (1 [ cos h ) ] (n/2 [ h ) sin h , whereh is the half- 0 anglec of the c-rayc emissionc cone. Herec we assumec that the whereR \ 8.0 kpc is the Galactocentric radius of the Sun, c-ray beaming fraction is independent of the pulsarÏs period n \ 0.01880 cm~3,H \ 70 pc,n \ 0.0313 cm~3, and 1 1 2 and useS*) T \ 2.0; then we haveh D 37¡.5 and f D0.67. H \ 639 pc. Furthermore, we use the assumption of After the position,c period, magnetic Ðeldc strength, andc lumi- Sturner2 & Dermer that the Arecibo receiver covers the nosity have been determined and the pulsar has been Galactic plane with 30¡ \ l \ 75¡ and o b o \ 5¡, and the rest accepted according to the criteria mentioned above, we of the sky is covered by the Parkes receiver. Therefore, from need to consider the detectability of the pulsar, namely, we the above equations, we can estimateS for each pulsar at determine whether the pulsar will be detectable at both either 1520 or 430 MHz. Then we convertmin S at 1520 or radio and c-ray energies. Therefore, the radio and c-ray 430 MHz into that at 400 MHz by assumingmin a power-law selection e†ects must be considered. For the radio selection energy spectrum with spectral index of [2 for the pulsar e†ects, the minimum detectable average Ñux density,S , of radio emission(Biggs & Lyne 1992) and compare it with the a pulsarÏs radio survey can be estimated by (e.g.,Biggsmin & model pulsar Ñux at 400 MHz. Those pulsars which satisfy Lyne1992; Sturner & Dermer 1996) L /d2 º S (39) 400 min CT ] T (l, b)DA W B1@2 S \ bS rec sky , (33) and min 0 T P[W 0 W \ P (40) wherebS is the Ñux limit when pulse broadening and back- ground emission0 is negligible,T is the receiver tem- are considered to be radio-detectable pulsars, where d is the perature,T is the sky temperaturerec at the survey distance to the pulsar. Now we consider the c-ray selection sky e†ect. In our model, the luminosity of c-rays with E [ 100 frequency,T is a normalization temperature, P is the pulsar c rotation period,0 and W is the broadened radio-pulse width. MeV can be approximated byequation (18). The corre- bS , T , andT are radio receiver parameters, which are sponding energy Ñux can be written as 0 rec 0 3.32 mJy, 20 K, and 45 K for the Parkes 1520 MHz receiver L ([100 MeV) (l \ 1520 MHz is the observing frequency, and dl \ 5.0 S ([100 MeV) \ c , (41) c *) d2 MHz is the channel bandwidth) and 2.46 mJy, 10 K, and 90 c K for the Arecibo 430 MHz receiver (l \ 430 MHz and For the c-ray detection,Yadigaroglu & Romani (1995) used dl \ 0.25 MHz).T is given by(Johnston et al. 1992; the valueS@ \ 3.0 ] 10~10 ergs cm~2 s~1, which can Sturner& Dermer 1996)sky compare toc,min the faintest 5 p sources in the Ðrst EGRET Galactic plane source catalog(Fichtel et al. 1994), but G 275 HA408 MHzB2.6 T (l) \ 25 ] , Sturner& Dermer (1996) adopted 2 ] 10~10 ergs cm~2 sky [1 ] (l/42)2][1 ] (b/3)2] l s~1. In this paper, we will use the value suggested by (34) Yadigaroglu& Romani (1995). The pulsars which satisfy S ([100 MeV)*) º 3 ] 10~10 ergs cm~2 s~1 (42) where l and b are the Galactic longitude and latitude, c c respectively. The broadened pulse width is given by will be detectable as c-ray pulsars. W 2\W2](b q )2]q2]q2 , (35) 0 1 samp DM scat 3.2. Monte Carlo Simulations whereW B 0.1P is the intrinsic radio pulse width; b \ 2; After the initial properties of pulsars at birth are given, we q is the0 receiver sampling time, which is 1.2 ms for1 the can follow the luminosity and spatial evolution of the Parkessamp receiver and 0.25 ms for the Arecibo receiver;q is Galactic pulsar population by means of Monte Carlo simu- the frequency dispersion of the radio pulse as it passesDM lations. The aim of these simulations is to produce the dis- through the and is given by tributions of distance, period, age, magnetic Ðeld, and c-ray Ñux of the c-ray pulsars. Our simulations can be described A DM B q \ q , (36) by a number of sequential steps. DM DM samp 0 1. Generate pulsars with random ages less than 1 ] 106 whereDM \ q l3/8299 dl pc cm~3; DM is the disper- yr, which minimizes computing time. sion measure0 ofsamp the pulsar;q represents the broadening 2. Generate the initial position of a pulsar as random scat of the radio pulse due to scattering of the pulse and the deviates with the probability density functionsoz(z) and subsequent path-length di†erences experienced by the oR(R) given by equations (20) and (21). FIG. 2.ÈNormalized cumulative distributions of distance, period, age, magnetic Ðeld, and c-ray energy Ñux of Galactic high-energy ([100 MeV) c-ray pulsar population for our model with radio Ñux selection e†ects (solid curves) and without radio selection e†ects (dot-dashed curves). For comparison, corresponding distributions of the six observed c-ray pulsars by EGRET are also shown (dashed histograms). 334 CHENG & ZHANG Vol. 498

3. Generate the magnetic Ðeld of the pulsar with a the pulsars with longer period if their magnetic Ðelds are random Gaussian deviate with probability density function within 1012È1013 G. Therefore, the radio selection e†ects in described by equation (26). our model are less important than those predicted by 4. Calculate the initial velocity of the pulsar, in which the Sturner& Dermer (1996). We have also simulated the nor- random velocity is created as a random deviate with prob- malized cumulative distributions of the pulsars with P \ ability density function given byequation (25) and the circu- 10 ms. There is only a negligible di†erence between their0 lar velocity is determined byequation (22). The initial space period distributions. In subsequent simulations, we will velocity of a pulsar is the vector sum of the random velocity only useP \ 30 ms. We performed a Kolmogorov- and circular velocity. Smirnov (KS)0 test to compare the cumulative model dis- 5. Calculate the pulsar period at time t by using equation tributions with the distributions for the six EGRET pulsars. (30) and determine the position at time t by following the One-sample KS statistics is used, since the model samples pulsarÏs motion in the Galactic gravitational potential for a have more than 150 pulsars. The hypothesis that model and given initial position (R, z) (step 2) and initial velocity (vR, observed pulsars were drawn from the same parent popu- v , vz) (step 4). lation can be rejected at better than the 80% level if the Õ6. Calculate the Galactic longitude and latitude by using maximum deviation between the distributions is greater equations (27) and (28), and the distance to the pulsar by than 0.41. In our model, the maximum deviations of dis- using equation (29). tance, period, age, magnetic Ðeld, and Ñux distributions 7. Calculate the radio beaming fraction given by from the observed distributions are 0.30, 0.37, 0.50, 0.20, equation (32) and generate a sample pulsar of period P in and 0.22, respectively. It can be seen that four out of Ðve of one out offr(P)~1 cases by using the Monte Carlo method. the cumulative distributions cannot be rejected at better Then, generate the radio luminosity at 400 MHz as a than the 80% conÐdence level, and the Ðfth cannot be random deviate with probability density function given by rejected at better than the 95% conÐdence level.Sturner & equation (31) and calculate the minimum radio energy Ñux Dermer(1996) examined their model using the KS test. In given byequation (33) and broadened pulse width given by their model 4X-P30, which best Ðtted the observed data equation (35). Those pulsars which satisfy equations (39) (seven pulsars were used for the distance, period, age, and and (40) are considered to be radio-detectable pulsars. magnetic Ðeld distributions but six pulsars for the Ñux 8. Calculate the c-ray luminosity withE [ 100 MeV distribution), the corresponding maximum deviations for and compare it with the Ñux threshold. Thosec pulsars which the Ðve parameters are 0.20, 0.39, 0.25, 0.35, and 0.40, satisfyequation (42) are considered to be both radio- and respectively. In our model, for the cumulative distance dis- c-rayÈdetectable pulsars. All the relevant information is tribution, the model results predict a larger number of c-ray recorded. pulsars between 1 and 2 kpc. The discrepancies between the 9. The c-ray beaming factor,f D 0.67, obtained by com- model results and observed data can be understood by con- paring with known c-ray pulsars,c is used. Since in the outer sidering uncertainties of the observed distances for the six gap model the c-ray beam and radio beam are not corre- known c-ray pulsars (at least 25% uncertainty for each lated in general, these two beaming fractions will be used pulsar). Furthermore, our model distributions of period, independently. age, magnetic Ðeld, and energy Ñux will become more con- sistent with the data if more c-ray pulsar candidates are Following the procedure described above, we perform conÐrmed. In fact, based on the analysis of the EGRET Monte Carlo simulations of 106 pulsars born during the data in the vicinity of 14 radio-bright ([1Jyat1GHz) past million years. Of course, we need to normalize to either supernova remnants,Esposito et al. (1996) suggested that the pulsar birthrate of D10~2 yr~1 or the observed number the c-ray emission near W44 may be due to the pulsar PSR of radio pulsars. We choose an initial periodP \ 10 ms or B1853[01. 30 ms to perform our simulations. Furthermore,0 we con- Finally, we would like to estimate the number of possible sider two kinds of pulsars: one consists of the pulsars that c-ray pulsars from our statistical analysis. We Ðnd that the satisfy both radio selection and c-ray selection e†ects; the ratio of the c-ray pulsars to radio pulsars from our samples other consists of the pulsars for which only the c-ray selec- is about 0.16, i.e., there are about 16 c-ray pulsars in 100 tion e†ect is taken into account. Therefore, radio-quiet radio pulsars. At present, 706 radio pulsars have been c-ray pulsars can be estimated. InFigure 2 the normalized observed, but the ages of only 540 radio pulsars have been cumulative distributions of distance, period, age, magnetic determined, and 105 radio pulsars have ages less than 106 Ðeld, and c-ray energy Ñux of the pulsars with an initial yr. Therefore, we have about 11 c-ray pulsars that are also period of 30 ms in our model are shown (solid curves). radio pulsars, where the c-ray beaming fraction (0.67) is Model distributions without taking the radio-Ñux selection used. So we expect that about 11 high-energy c-ray pulsars e†ects into account are also given (dot-dashed curves). For can be found out of D100 detected radio pulsars with age comparison, the corresponding observed distributions for less than 106 yr. six known c-ray pulsars (seeTable 1) are shown. It should 4. GEMINGA-LIKE PULSARS AND UNIDENTIFIED POINT be noted that the c-ray pulsar PSR B1509[58 and the SOURCES OF EGRET possible c-ray pulsar PSR B0656]14 have not been included because the former has not been detected in the In this section, we consider the properties of Geminga- energy rangeE [ 100 MeV and the latter has not been like pulsars. The Geminga-like pulsars are either those conÐrmed. Generally,c the inclusion of the radio detection pulsars whose radio Ñuxes are lower than the radio survey criterion preferentially eliminates the distant, long-period, Ñux threshold or those whose radio beams miss us. Hence, and dim pulsars. In our model, however, the pulsars with the constraints for the Geminga-like pulsars are (1) f ¹ 1 f ¹ 1 are considered to be c-ray pulsars which emit high- and (2)S º 3 ] 10~10 ergs cm~2 s~1. We obtain the energy c-rays. Fromequation (3), this condition excludes sample ofc the Geminga-like pulsars by means of the Monte No. 1, 1998 STATISTICS OF GAMMA-RAY PULSARS 335

Carlo method as follows. First, we follow steps 1È6 of our fore, the number of the Geminga-like pulsars can be simulation method outlined in° 3.2. Next, prior to step 7, approximated by(N [ N ). Using the method men- step 8 is performed to generate the sample of c-ray pulsars. tioned above, we canc obtain1 c a2 Geminga-like pulsar sample In this sample (the number of pulsars is labeled as N ), and analyze the properties of the pulsars statistically. In some of the pulsars may satisfy the radio survey Ñux thresh-c1 Figure 3 we show the normalized distributions of distance, old. Hence, step 7 is performed to generate those pulsars period, magnetic Ðeld, age, and the size of the outer gap, f,of which are detectable at both radio and c-ray energies, where the Geminga-like pulsars. It can be seen that the Geminga- the number of these pulsars is represented byN . There- like pulsars are distributed mainly in the distance region c2

FIG. 3.ÈNormalized distributions of distance, period, magnetic Ðeld, age, and the outer gap size of Geminga-like pulsars 336 CHENG & ZHANG Vol. 498 from 1 to 2 kpc, and most of the Geminga-like pulsars have model results with the distributions of high-energy ([100 periods between D0.1 and 0.3 s, with magnetic Ðelds from MeV) c-rays from these unidentiÐed point sources at di†er- D8 ] 1011 to D5 ] 1012 G, ages between 105 and 106 yr, ent latitude regions are shown. Our result indicates that the and the size of the outer gap from 0.3 to 0.9. In fact, from Ñuxes of Geminga-like pulsars at o b o \ 5¡ are concentrated these simulated distributions, we should perhaps not be sur- mainly in the range from D3 ] 10~7 to D3 ] 10~6 cm~2 prised that the Geminga is the Ðrst Geminga-like pulsar s~1, which is consistent with the observed data of 30 detected. Except for distance, the other parametersÈthe unidentiÐed point sources detected by EGRET at o b o \ 5¡. period (P \ 0.237 s), the magnetic Ðeld(B \ 1.6), and the Compared to the Ñuxes of all unidentiÐed point sources, age (q \ 3.4 ] 105 yr) of the GemingaÈare12 coincident with there are some di†erence between our model result and the the peak positions of the distributions. So it is the most observed data. This indicates that only some of the typical Geminga-like pulsar. But its distance is only D200 unidentiÐed point sources at o b o º 5¡ are Geminga-like pc, so that is why we Ðnd it Ðrst. Next we consider the pulsars. Furthermore, we can compare the Galactic dis- high-energy c-ray Ñux distribution from these pulsars. At tributions in l and b of the unidentiÐed EGRET sources and present, EGRET has observed 96 unidentiÐed point sources modeled Geminga-like pulsars at o b o \ 5¡. From our sta- (Thompson et al.1995, 1996). Kaaret & Cottam (1996) tistical analysis, the number of Geminga-like pulsars is analyzed the unidentiÐed EGRET sources at o b o \ 5¡ and D55N0 , whereN0 is the birthrate of neutron stars in found signiÐcant correlation between the positions of the units of100 100 yr. Considering100 the beaming e†ect of c-ray unidentiÐed EGRET sources and OB associations. They pulsars( f D 0.67), the number of Geminga-like pulsars is estimated distances to the unidentiÐed EGRET sources D37N0 cin our . InFigure 5 we show the compari- from the OB associations and luminosities, and veriÐed that son of100 the observed normalized distributions with the the distribution of luminosities for the unidentiÐed EGRET modeled results. For the longitude distribution, we divide sources is consistent with that of known c-ray pulsars. the longitude into 18 bins. Assuming the observed unidenti- Therefore, Kaaret & Cottam concluded that a majority of Ðed EGRET sources and model pulsars in each bin are Ni the unidentiÐed EGRET sources may be pulsars. Further- andni, then we have s2/lB0.84 using s2\;i (Ni more, based on their outer gap model,Yadigaroglu & [ ni)2/ni, where l is the number of degrees of freedom. For Romani(1997) found that the spatial distribution and lumi- the latitude distribution (each bin width is 2¡), we have nosities of the unidentiÐed EGRET sources are consistent s2/lB1.42 at o b o \ 5¡. Therefore, the model results are with the proposition that essentially all Galactic unidenti- consistent with the observed distributions in both l and b. Ðed EGRET sources are pulsars. Whether or not these Now we would like to describe how to discriminate the point sources are c-ray pulsars has not yet been conÐrmed, Geminga-like pulsars from other point sources. We have although these data can be used as a possible test for our shown the properties of Geminga-like pulsars in Figure 3. model predictions. InFigure 4 the comparisons of our Using these simulation results and the constraints men-

FIG. 4.ÈComparison of the Ñux distribution of Geminga-like pulsars with that of the unidentiÐed point sources detected by EGRET (Thompson et al. 1995, 1996), where model results are indicated by solid histograms and observed data are represented by dashed histograms. No. 1, 1998 STATISTICS OF GAMMA-RAY PULSARS 337

period varies from D0.1 to D0.3 s and the magnetic Ðeld from D8 ] 1011 to D5 ] 1012 GinD70% probability. In the next section we will use this parameter range. As men- tioned above, based on the argument that EGRET source positions are related with SNRs and OB associations, Kaaret& Cottam (1996) and Yadigaroglu & Romani (1997) have estimated the distances of some EGRET unidentiÐed point sources and concluded that young pulsars can account for essentially all of the excess low-latitude EGRET sources. Moreover, the spectra of some EGRET unidenti- Ðed point sources have been obtained(Fierro 1995). Here we assume that these unidentiÐed point sources are Geminga-like pulsars. We compare the cumulative model distance distribution with the distribution for both Geminga and the 12 unidentiÐed sources given byKaaret & Cottam(1996) in Figure 7. The KS test indicates that the maximum deviation from the observed is about 0.280. Because the hypothesis that the model pulsars and the observed unidentiÐed sources were drawn from the same parent population can be rejected at greater than the 80% conÐdence level if the distribution is greater than 0.285, so the cumulative distribution cannot be rejected at this level. It should be pointed out that our model distance distribu- tion is not consistent with the distribution estimated by Yadigaroglu& Romani (1997). Therefore, we use the parameter range of Geminga-like pulsars given above and the distances obtained byKaaret & Cottam (1996) to calcu- late the spectra of Ðve unidentiÐed point sources that lie in o b o ¹ 5¡, and their distances are less than 2 kpc. We choose FIG. 5.ÈNormalized distributions of the Geminga-like pulsars in di†erent values of period and magnetic Ðeld from the Galactic longitude l and latitude b. The distributions of unidentiÐed parameter range shown inFigure 6 and calculate the EGRET sources at o b o \ 5¡ are represented by the solid histograms, and spectra by using the consistent Ðtting parameters and the those of the modeled Geminga-like pulsars are represented by dashed histograms. distances for each unidentiÐed point source. Finally, the values of period and magnetic Ðeld are determined by com- tioned above, we propose to deduce the parameter range for the Geminga-like pulsars. InFigure 6 the contour in the P-B plane is shown, where the contour lines from inner to outer represent about 23%, 47%, 70%, and 95% probabil- ities, respectively. From this Ðgure it can be seen that the parameter range of period and magnetic Ðeld for the Geminga-like pulsars is located in the area in which the

13.0

12.5

Log B (G)

12.0

11.5

-2.0 -1.5 -1.0 -0.5

Log P (s)

FIG. 6.ÈMagnetic Ðeld strength vs. period for the sample of Geminga- FIG. 7.ÈComparison of the cumulative model distance distribution for like pulsars (see text for details). The contour lines from inner to outer Geminga-like pulsars with that for both Geminga and the 12 unidentiÐed represent about 23%, 47%, 70%, and 95% probabilities. sources given byKaaret & Cottam (1996). FIG. 8.ÈComparison of our model results with observed spectra for Ðve EGRET unidentiÐed point sources, where dashed curves represent our results with consistent-Ðt parameters and solid curves represent our results with best-Ðt parameters. STATISTICS OF GAMMA-RAY PULSARS 339

TABLE 2 BEST-FIT MODEL PARAMETERS AND EXPECTED X-RAY LUMINOSITIES OF FIVE UNIDENTIFIED POINT SOURCES

PB d Name (s) (G)12 (kpc) x x *) T a log L soft b log L syn b min max c s6 X X 2EG J0618]2234 ...... 0.1 0.8 1.34 0.72 2.1 1.4 0.72 32.28 30.81 2EG J1746-2852 ...... 0.3 5.0 1.50 0.68 2.1 0.6 0.78 32.41 30.28 2EG J2020]4026 ...... 0.3 5.0 1.37 0.80 2.1 0.9 0.78 32.41 30.28 2EG J2019]3719 ...... 0.2 2.0 1.37 0.72 2.1 1.2 0.74 32.32 30.30 2EG J1801[2312 ...... 0.1 5.0 1.44 0.70 2.1 1.8 0.88 32.61 32.61

a Ts \ Ts/106. b L soft6 andL syn are in units of ergs s~1. X X paring model results with the observed data. InFigure 8 the have same Ñuxes, so the c-ray from the pulsar at high lati- comparisons of our results with observed spectra for Ðve tude can be easier to detect. EGRET unidentiÐed point sources are shown, where We have estimated the c-ray Ñux withE [ 100 MeV by dashed curves represent the results with consistent-Ðt using the parameters of 52 canonical pulsars.c We assume parameters and solid curves represent the results with best- that the average solid angle is 2 sr. The results are shown in Ðt parameters shown inTable 2. In Table 2 the expected Figure 9, where the Ñuxes from six c-ray pulsars are rep- X-ray luminosities are also given for these sources. In order resented by Ðlled circles; the open circles and Ðlled dia- to account for the fact that our consistent-Ðt parameters monds represent the pulsars with F (E [ 100 MeV) º 1.6 c c produce spectra that are consistent with the observed ] 10~7 cm~2 s~1 and F (E [ 100 MeV) \ 1.6 ] 10~7 c c spectra, we estimated the s2 values for the best-Ðt versus cm~2 s~1 of the Galactic plane group, respectively. The consistent-Ðt parameters for each pulsar. For each pulsar, empty box represents the pulsar located out of the Galactic the Ñuxes for the consistent-Ðt and best-Ðt parameters at the plane, while plus signs represent the pulsars with X-ray observed energy bands are calculated. The s2 values for emission. It can be seen that there are 14 canonical pulsars 2EG J0618]2234, 2EG J1746[2852, 2EG J1801[2312, withF (E [ 100 MeV) º 1.6 ] 10~7 cm~2 s~1 (including c c 2EG J2019]3719, and 2EG J2020]4026 are 2.9, 3.7, 1.6, six known c-ray pulsars). This number is very close to our 2.2, and 3.0, respectively. It can be seen that our results are Monte Carlo results. According to our model, each of these consistent with the observed spectra. Therefore, these pulsars has 67% probability of being detected as a c-ray sources may be Geminga-like pulsars, especially 2EG pulsar. It should be pointed out that in our model there are J2019]3719. six pulsars whose Ñuxes are between 10~7 and 1.6 ] 10~7 cm~2 s~1: PSR B1046[58, PSR B1221[63, PSR 5. POSSIBLE GAMMA-RAY PULSARS B1509[58, PSR B1719[37, PSR B1737[30, and PSR In our Monte Carlo simulations, we predict that there B1853]01. Based on the data analysis of EGRET, Fierro should be 11 pulsars which can be detected in both radio (1995) suggested that PSR B1509[58, B0656]14, and c-ray energy bands. Six radio pulsars have already been B1046[58, and B1853]01 are the strongest candidates for identiÐed as EGRET sources. In this section we attempt to c-ray pulsars. He also pointed out that PSR B1823[13 and determine which radio pulsars should be the most possible B1832[06 are possible candidates of c-ray pulsars, but our c-ray pulsars. Generally, the EGRET threshold for point model results giveF (E [ 100 MeV) D 7.4 ] 10~8 and c c source detection above 100 MeV cannot be lower than D2.3 ] 10~8 cm~2 s~1 for these two pulsars, respectively. F º 10~7 photons cm~2 s~1. However, the actual As to PSR B0656]14, our model result is greater than the thresholdmin depends on the local c-ray background and instrument exposure and accuracy of pulsar parameters. Moreover, the minimum pulsed Ñux withE º 100 MeV from known c-ray pulsars detected by EGRETc is that from PSR B1951]32, which is (1.6 ^ 0.2) ] 10~7 cm~2 s~1 (Ramanamurthyet al. 1995). Here we will use the Ñux as threshold Ñux for detecting a pulsar as a point source in the EGRET range. Based on the data analysis, Fierro (1995) pointed out that the strongest candidates for c-ray emission from radio pulsars are binary millisecond pulsars PSR J0034[0534 and J0613[0200, and the young pulsars PSR B1046[58, B1832[06, and B1853]01. Here we would like to present some predictions of our model. We use the database of radio pulsars provided by Princeton University, where the parameters of a total of 706 pulsars are given. From this database, we select 52 canonical pulsars that FIG. 9.ÈPredicted Ñux of high-energy(E [ 100 MeV) c-rays from the satisfy the condition of f ¹ 1. We divide these pulsars into pulsars as a function of pulsar characteristicc age. Filled circles represent two groups based on their Galactic latitudes: one is the the six detected EGRET pulsars. Open circles and Ðlled diamonds rep- resent the pulsars withF (E [ 100 MeV) º 1.6 ] 10~7 cm~2 s~1 and Galactic plane group with o b o ¹ 10¡, and the other is the c c F (E [ 100 MeV) \ 1.6 ] 10~7 cm~2 s~1 within Galactic latitude nonÈGalactic plane group with o b o [ 10¡. In general, the o bco ¹c 10¡; open squares represent the pulsars in high Galactic latitude local c-ray background at high latitude is lower than that in o b o [ 10¡, and plus signs represent pulsars with X-rays, where*) \ 2.0 sr the Galactic plane, compared to that at low latitude if they is used. c 340 CHENG & ZHANG Vol. 498

TABLE 3 EXPECTED PULSARS WITH F ([100 MeV) [ 1.6 ] 10~7 cm~2 s~1 IN OUR MODEL (EXCLUDINGc SIX KNOWN c-RAY PULSARS)

Age PB d Pulsar (yr) (s) (G)12 (kpc) fF([100 MeV) c B0114]58...... 0.275E]06 0.101 0.780E]00 2.140 0.373 0.1778E[06 B0355]54a ...... 0.564E]06 0.156 0.839E]00 2.070 0.611 0.1661E[06 J0538]2817a ...... 0.619E]06 0.143 0.733E]00 1.770 0.592 0.2239E[06 B0656]14a ...... 0.111E]06 0.385 0.466E]01 0.760 0.700 0.1608E[05 B0740[28...... 0.157E]06 0.167 0.169E]01 1.890 0.443 0.2505E[06 B1449[64...... 0.104E]07 0.179 0.711E]00 1.840 0.797 0.1849E[06 J2043]2740 ...... 0.121E]07 0.096 0.352E]00 1.130 0.550 0.4907E[06 B2334]61a ...... 0.409E]05 0.495 0.986E]01 2.470 0.623 0.1821E[06

a Pulsars with X-rays(Becker & TruŽ mper 1997). upper limit by an order of magnitude and is about 39 times at birth (obtained by the radio pulsar statistical studies) the value deduced byRamanamurthy et al. (1996). In fact, have been used in our simulations, where the radio and the expected Ñuxes in other models are also greater than the c-ray beaming fractions of the pulsars have been taken into observed data for this pulsar(Ramanamurthy et al. 1996). It account. We have modeled the distance, period, age, mag- is likely that we have missed the major c-ray beam of PSR netic Ðeld, and high-energy ([100 MeV) c-ray energy Ñux B0656]14. InTable 3 we show the model results for distributions of the Galactic c-ray pulsar population and pulsars withF ([100 MeV) º 1.6 ] 10~7 cm~2 s~1. Ðnd them consistent with those of the six observed c-ray Therefore, addingc the six known c-ray pulsars, there are pulsars by EGRET. From our statistical analysis, we expect about 14 pulsars with high-energy ([100 MeV) c-rays. that about 11 c-ray pulsars whose Ñuxes are greater than Since in our outer gap model, X-rays and c-rays are strong- the EGRET threshold, and which also are radio pulsars, ly correlated (cf.° 2), the strongest candidates of c-ray may exist in our Galaxy. Furthermore, we have obtained pulsars may therefore be canonical pulsars with strong the distributions of period, magnetic Ðeld, distance, age, and X-ray, e.g., PSR J0538]2817, PSR B0355]54, and PSR energy Ñux for the Geminga-like pulsars. Comparing the B2334]61. More speciÐcally, we suggest that the signa- high-energy ([100 MeV) c-ray Ñux and the Galactic dis- tures for c-ray pulsars are (i) power-law behavior of the tributions in Galactic longitude and latitude with those of c-ray spectrum witha D [1.7 to [2.2 from 100 MeV to a unidentiÐed EGRET point sources in the Galactic plane, we few GeV, (ii) power-lawc behavior of the hard X-ray spec- conclude that a majority of the unidentiÐed EGRET point trum withax D [2 from keV to MeV, (iii) a soft thermal sources are probably Geminga-like pulsars. We have also X-ray component with kT D 102 eV, and (iv) L /E0 D predicted that there may beD55f N0 in our Galaxy, c 100 10~2È10~3. X sd wheref is the beaming fraction of c-ray pulsars, and we c 6. CONCLUSIONS have used an average value (0.67) to estimate the number of the Geminga-like pulsars.Helfand (1994) pointed out that Using the six detected EGRET c-ray pulsars and the 0.2 ¹ f \ 1 (see Fig. 3 of his paper). By Ðtting the observed database for c-ray emission from all known spin-powered data ofc the six known c-ray pulsars, we estimate that f D pulsars withE0 /d2 º 0.5 given by Nel et al. (1996), we c 33 kpc 0.67. Furthermore, we have shown the possible range of have tested the model and deduced a set of consistent-Ðt parameters for the Geminga-like pulsars: their periods are parameters which is very important (i) to predict the spec- mainly from 0.1 to 0.3 s, the magnetic Ðelds are mainly from trum of unidentiÐed c-ray sources and not-yet-conÐrmed 8 ] 1011 to 5 ] 1012 G, and they must satisfy B º 20P2.2 c-ray pulsars and (ii) to carry out our statistical analysis of G. Finally, we have estimated the high-energy12c-ray Ñux the c-ray pulsars. Our results indicate that our model can withE [ 100 MeV by using the database of 706 pulsars successfully explain the c-ray emission from spin-powered given byc Princeton University, and we suggest that the pulsars. Our model predicts that the conversion efficiency of strongest candidates of c-ray pulsars may be canonical high-energy c-rays depends on the characteristic age and pulsars with strong X-ray emission satisfying L /E0 D the period of pulsars, i.e.,g P q6@7P2. Furthermore, we 10 (Chenget al. 1998). X sd have also reviewed the X-rayth emission from pulsars. This is ~3 not only important in calculating the size of the outer gap, which plays an important role in our statistical analysis, but also the X-ray properties of pulsars allow us to predict We thank T. Bulik, A. da Costa, J. Gil, D. R. Lorimer, which EGRET sources and radio pulsars are likely c-ray G. Melikidze, and B. Rudak for useful discussion and sug- pulsars. Based on our model, the properties of the Galactic gestions. We are grateful to T. C. Boyce for a critical high-energy c-ray pulsar population have been simulated reading of our manuscript and the anonymous referee for by means of Monte Carlo methods. The initial magnetic his/her useful comments. This work is partially supported Ðeld, spatial, and velocity distributions of the neutron star by a RGC grant of the Hong Kong Government. No. 1, 1998 STATISTICS OF GAMMA-RAY PULSARS 341

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