**The Net Advance of Physics: Gamma-Ray Pulsar Emission Models, by Alice K. Harding -- Section 3B.**

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The -ray emission from pair cascades induced by curvature

radiation of accelerated particles has been investigated by

Daugherty & Harding (1982, 1994, 1996) and Chiang & Romani

(1992), assuming a space-charge limited electron acceleration

model (e.g. Arons 1983) is operating. DH96 found that if the

primary electron acceleration is assumed to occur over several

stellar radii, then the cascade -ray emission shows agreement

with several key features of the observed emission, including the

double-peaked light curves with bridge emission (as discussed

above) as well as phase-averaged and phase-resolved spectra. In

particular, the CRPC models predict high-energy spectral cutoffs,

due to magnetic pair production, at several GeV. The pair

production attenuation cutoffs are much sharper than exponential

cutoffs due to a cutoff in the particle distribution, and can fit the

very steep cutoffs in observed -ray pulsar spectra. Some

correlation of surface magnetic field with pair production or photon

splitting (in the case of PSR1509-58, see Harding & Baring 1996)

cutoff
energy is predicted, with higher *B* producing lower cutoff

energies.

The CRPC models can account for the systematic

soft-hard-soft hardness variation of the pulse profile of Vela

(Kanbach et al. 1994). The hardest emission is predicted to occur

near the pole, where the emission is pure curvature radiation that is

not softened by cascading. Near the polar cap rim, where the peak

emission originates, the cascades are most extended, producing a

softer spectrum with a lower cutoff energy. The emission outside the

rim, and outside the peaks in the pulse profile, comes from

high-altitude curvature radiation from primary particles that have

lost most of their energy and has the lowest cutoff, but not as sharp

as that of the peak spectra. Very importantly, the CRPC/SPC model

predicts pulsed emission at all phases, as has been observed (Fierro

et al. 1996).

The observed increase in hardness of the phase-averaged spectrum

with age can also be understood in the CRPC model. The hardness

of a CRPC cascade spectrum primarily depends on the number of

photon generations in the cascade. As noted by Lu et al. (1994),

there is a rough correlation between the spectral indices of the

observed -ray pulsars and a theoretical estimate of the number of

polar cap cascade generations. The primary curvature radiation

spectrum has a photon index of 5/3 if the particles have significant

energy loss and 2/3 if they do not. Significant energy loss of the

primary particles will occur for short period ( )

pulsars, where the polar cap half-angle, , is larger and

thus the field line radius of curvature at the polar cap rim, is

smaller. The input curvature spectrum is then softened on each

generation of pair production and synchrotron emission. The

spectral index of the escaping -ray spectrum will be (Harding &

Daugherty 1996)

where the number of cascade generations, , is allowed to be a

non-integer and is the spectral index of the primary curvature

spectrum. The number of generations will generally increase with

decreasing pulsar period and increasing surface magnetic field

strength , because the mean free path for pair production will

decrease. The generation number will also decrease with the height

above the surface at which the bulk of the cascade occurs. Figure 3

shows the predicted dependence of spectral index with the

parameter for different
radii *r* of the cascade. Since the

pulsar characteristic age , a correlation of spectral

index with age is predicted. Figure 3 also shows a comparison of

predicted and observed for the observed -ray pulsars. Several

pulsars with ages larger than yr and low magnetic fields,

PSR1929+10 and PSR0950+08, that should be detectable by

EGRET based on their predicted fluxes, have extremely low

predicted spectral indices, indicating that the bulk of their -ray

emission could lie above the EGRET range.

**Figure 3:** Predicted -ray spectral index vs. .