张力 张力 2003 年 10 月 21 日于北京 2003 年 10 月 21 日于北京 Gamma-ray Luminosity and Death Lines of Pulsars with Outer Gaps

Outline  Introduction  The Outer Gap Model  Magnetospheric Geometry  X-ray Field in the Magnetosphere  The Fractional Size of an Outer Gap  High-Energy Gamma-ray Luminosity  The Death lines of Pulsars with Outer Gaps  Conclusion and Discussion

1.Introduction 1) Observations of High-Energy Emission from Rotation-Powered Pulsars History: SAS 2, COSB(1970’s-1980’s). EGRET (1990’s) Coming satellite: GLAST

The Observations of Gamma-ray Pulsars The OSSE/EGRET experiments have detected pulsed signals from eight rotation-powered pulsars. Nolan et al. 1993, ApJ 409, 697 Crab Matz et al. 1994, ApJ 434, 288 B * Kanbach et al. 1994, A&A 289, 855 Vela Mayer-Hasselwander et al. 1994, ApJ 421, 276 Geminga Ramanamurthy et al. 1995, ApJ 447, L109 B Thompson et al. 1996, ApJ 465, 385 B Fierro et al. 1998, ApJ 494, 734 Crab, Vela, Geminga Thompson et al. 1999, ApJ 516, 297 B Kaspi et al. 2000, ApJ 528, 445 B

Pulses at multi-wavebands

Energy spectra of gamma-ray pulsars

2) Models for High-Energy Emission from Pulsar There are two kinds of models for high-energy emission from pulsars: (1)polar cap models (Harding et al. 1978, ApJ 225, 226; Daugherty & Harding 1982 ApJ 252, 337;1996 ApJ 458, 278; Zhang & Harding, 2000, ApJ; Sterner et al. 1995, ApJ 445, 736). (2) Outer gap models(Cheng, Ho, Ruderman,1986, ApJ 300, 500; 300, 522; Chiang & Romani 1994, ApJ 436, 754; Romani, 1996, ApJ 470, 469, Zhang & Cheng 1997, ApJ ; Hirotani & Shibata 2001, MNRAS 325, 1228 )

Schematic geometry of polar and outer gaps. Dark solid regions are thin gaps of younger pulsars. Hatched regions are thick gaps of older pulsars

3) Outer gaps: charge density surrounds a rotating NS with B and . The plasma is corotating with the NS within R L =c/ . In the corotating region, E || = E  B/B ~0. But, the flows of the plasma along open field lines will results in some plasma void regions (where    0 ) in the vicinity of null charge surfaces (   B=0). In such charge deficient regions, which are called outer gaps, E ||  0 is sustained,e-/e+ can be accelerated to relativistic energies and the subsequent high-energy  -ray emission and   pair production can maintain the current flow in the magnetosphere.

5)Motivations: Observation indicates (Thompson et al. 2001): L   L ½ sd. New polar gap models (e.g. Zhang &Harding 2000; Harding et al. 2002): L   L ½ sd for Lsd > L break sd, L   L sd for Lsd < L break sd, where L break sd, = 5 x P -1/2 erg/s. For previous outer gap models: L  ~ f 3 L sd f  B -13/20 P 33/20 (CHR II) f  B -4/7 P 26/21 (ZC97) ZC97 model predicts L   (B/P) 0.3, e.g, the ratio of (B/P) 0.3 for PSR B to that for Geminga is ~ 0.9. But, the observed ratio ~8 (Kaspi et al. 2000). Therefore, other physics quantities should be taken into account in order to explain the observed data.

We re-study the gamma-ray emission from the outer gaps of the rotation-powered pulsars. We take the magnetosphere geometry into account and show that the fractional size of the outer gap is a function of period, magnetic field, radial distance to the neutron star and magnetic inclination angle.

2. The Outer Gap Model e + and e - are accelerated by E || Relativistic e + /e - emit gamma-rays via Synchro-curvature, and IC processes Soft X-rays from stellar surface are produced by the collision of return current with NS. Gamma-rays collide with these soft photons to materialize as pairs in the accelerator to maintain the outer gap.

Based on ZC97 model, 2D and 3D models were developed Crab The spectra of Crab

Emission projection and pulse profile for Geminga parameter. Panel A: Photon emission from the pulsar as a function of phase. Panel B: Pulse profile. Phase-resolved and phase- averaged gamma-ray spectra of the Geminga pulsar Geminga Geminga

Magnetospheric Geometry For an oblique magnetic dipole rotator with an angular velocity  and the magnetic moment vector , let its spin axis be along the Oz axis,  be in the plane xOz and  be the angle between  and .  = B p R 3 /2. Effects of Inclination angle is not included in ZC97 model New model New model:

It is believed that the outer gap is extended from its inner boundary to the light cylinder (CHR I). For the oblique magnetic dipole rotator, outer boundary is determined by (Kapoor & Shukre 1998) Inner boundary: The Goldreich-Julian current

X-ray Field in the Magnetosphere Observed X-ray spectra: soft (thermal) + hard (non-thermal) X-rays thermal X-rays. Thermal X-ray emission from neutron star cooling Zhang & Harding (2000) Hibschman & Arons (2001) X-ray emission from polar cap heating Harding and Muslimov (2001)

X-ray emission from outer gap heating In the outer gap models, part of the relativistic particles from the outer gap will collide the stellar surface, producing the thermal X-rays. These relativistic inflowing particles from the outer gap radiate away much of their energy before reaching the polar cap (Zhang & Cheng 1997, Zhu et al. 1997; Wang et al. 1998; Cheng & Zhang 1999).

Average energy of X-rays Two possible cases:NS cooling+ pc heating; og heating Assuming these X-ray can be approximated as the black-body, their spectrum can be expressed as NS cooling + pc heating: Outer gap heating

The Fractional Size of an Outer Gap In two dimensional geometry, the parallel electric field in the outer gap can be approximated as (ZC97) Lesch et al. 1998

Typical energy of the gamma-rays in the outer gap : Inside the outer gap, pair production condition :

X-rays are produced by the NS cooling and pc heating:

X-rays are produced by the outer gap heating:

In order to explain the average properties of high-energy photon emission from the outer gap, we assume that high-energy emission at a average radius represents the typical emission of high-energy photons from a pulsar. The average radius is given by Corresponding fractional size is where f 0 (P, B) = 5.5 P 26/21 B -4/7 12 is the fractional size of outer gap by ignoring the effect of inclination angle (Zhang & Cheng 1997)

Fig.1 Variation of the fractional size of the outer gap with the magnetic inclination angle for some typical pulsar parameters.

3. High-Energy Gamma-ray Luminosity In our model, L  for each gamma-ray pulsar depends on P, B and . However,  values are not known well. Once f(,P,B) for a pulsar is estimated, L  is We find out statistically the relation between L  and Lsd for the canonical pulsars using Monte Carlo method. The details of this Monte Carlo method is given by Cheng & Zhang (1998) and Zhang et al. (2000).

Fig.2 The change of L  with Lsd in the simulated  -ray pulsar. In our simulation, we have used the EGRET threshold as the minimum detectable  -ray energy flux. Open circles and shaped circles are the model radio-quiet and radio-loud  -ray pulsars respectively. Radio-loud : L   L 0.30 sd Radio-quiet : L   L 0.38 sd

Fig.3 L  /L  0 versus L  in the  -ray pulsar population predicted by our outer gap model. We have used the EGRET threshold as the minimum detectable  -ray energy flux. Open circles are the expected data and solid line is the best fit. For comparison, we show the result given by Zhang & Cheng (1997) as a dashed line.

Fig. 4 L  versus Lsd. Solid circles with error bar are the observed data given by Thompson et al. (2001). Solid line:lg L   lg Lsd Dashed line: lg L   lg Lsd

4. The Death lines of Pulsars with Outer Gaps We now consider the condition which the outer gap of a pulsar exists. If the fractional size of the outer gap at rin is larger than unity, then the outer gap would not exist, I.e. f(rin)=1 give the death lines. NS cooling+pc heating (Zhang & Harding 2000) A1= lg Gpc (  ) A2= lg Gpc (  ) Outer gap heating:

It was generally believed that the parent distribution of the magnetic inclinations satisfy an uniform distribution (Gunn & Ostriker 1970; Gil & Han 1996). But recent study by using polarization data of the radio pulsars indicate that the parent distribution of the magnetic inclinations satisfies a cosine-like distribution (Tauris & Manchester 1998). Therefore, we estimate the average value of f(rin) in these two possible parent distributions of the magnetic inclinations, i.e.

For the uniform distribution, ~ 0.43 and <G(  )=0.38. For cosine distribution, NS+pc OG

Fig.5 Death lines of the pulsars with outer gaps. It is assumed that X-rays are produced by the neutron star cooling and polar cap heating. Two cases of the inclination angle distributions are considered: (i)an uniform distribution and (ii)a cosine distribution.

Fig.6 Death lines of the pulsars with self-sustained outer gaps. The observed data are taken from see website

4. Conclusion and Discussion We studied the outer gap size, gamma-ray luminosity and death lines of gamma-ray emission of the pulsars with outer gaps when the geometry of dipole magnetic field is taken into account. f(P, B, (  )) is not only the function of P and B, but also the function of, which depends on . The outer gap will not exist if f(r,  ) > 1. For all gamma-ray pulsars with self-sustained outer gaps simulated by using Monte Carlo method, L   L  sd, where the value of  ~0.38 for EGRET sensitivity. The death lines of the pulsars with outer gaps in the two possible X-ray fields are given and the comparison with the observed data are considered.

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