Limits on particle energies by resonant inverse Compton scattering

For a hot polar cap which emits soft photons, RICS can constrain the maximum energies of electrons (or positrons) accelerated in the polar gap in the following two ways: the energy loss competes with acceleration, or by limiting the acceleration length through pair production. For a given accelerating potential, , the Lorentz factor of an electron (or positron) at a distance, , to the polar cap is given by (Dermer 1990; Luo 1996)

where is the initial Lorentz factor, is the Compton wavelength, is the effective temperature (in ), is the star's radius, is the Thomson cross section, is the fine structure constant, , and is the maximum angle between a photon and the particle motion. For thermal emission from the entire neutron star's surface we have , which corresponds to propagation of photons at directions tangential to the surface. For a hot polar cap, this is valid only for (where and P is the pulsar period); for , the angle is given by , corresponding to photons from the polar cap rim on which the field lines have colatitude angle relative to the magnetic pole. The energy loss due to RICS is described by the third term on the right hand side of equation (1).

The maximum Lorentz factor corresponds to the value at which is the gap length determined by the pair production, e.g. due to RICS or due to thermal photons interacting with heavy ions, whichever comes first. Equation (1) can be solved numerically with the boundary condition .

The model for the accelerating potential strongly depends on the physical conditions on the polar cap. If polar caps are sufficiently hot, we may have free emission of charges. The induced electric field causes charge flow, which tends to screen out the field. However, the charge screening is always insufficient, e.g. due to field line curvature (Arons & Scharlemann 1979), and hence charge depletion occurs (the charge density deviates from Goldreich-Julian (G-J) density). An electric potential exists in the charge-depletion region (often called the gap). As an example, we consider the potential (e.g. Arons & Scharlemann 1979; Arons 1983)

where is the angle of the field lines relative to the magnetic pole, is the azimuthal angle of the open field line flux tube, x is the distance to the polar cap (in ), i is the angle between the rotation axis and the magnetic pole, and g(x) is given by , . According to Arons (1983), the angle is in the ranges, for (if the outflowing charges are ions), and for (if the outflowing charges are electrons). The potential given by (2) is due to charge density decrease along field lines (i.e. it becomes less than the G-J density) because of field line curvature. Plots of the Lorentz factor as a function of distance from the polar cap are shown in Figure 1. We assume at . As shown in the figure, energy loss due to RICS can compete with acceleration only for high polar cap temperature . For , the particle energy is constrained mainly through pair production by RICS.

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Figure 1: Lorentz factor versus distance to the polar cap. The electrons (or positrons) are accelerated by the potential (2) with (or ), , . The dashed curve is obtained without RICS. The solid curves (from top to bottom) correspond to , , and . We assume that thermal emission is from polar cap. The dotted curve is the threshold condition for pair production through RICS (i.e. pair production can occur only above the curve).

There are other models of the space-charge-limited potential such as the model discussed by Michel (1974), also by Fawley, Arons & Scharlemann (1977) and Cheng & Ruderman (1977), which do not rely on field line curvature and in general gives a lower voltage. The potential discussed by Michel (1974) arises from the deviation of the charge density from the G-J value because of particle inertia, i.e. the velocity v(x) and the magnetic field B(x) have different scaling with x such that the charge density cannot be maintained at the G-J density all the way along open field lines. The corresponding potential can be described by (Fawley, Arons & Scharlemann 1977)

with , and . For , the electric field drops off faster than . The plot of the Lorentz factor as a function of is shown in Figure 2 for and . The electric potential for is assumed to be constant, . For , the energy loss is so severe that it continues beyond the distance , resulting in a sudden decrease in at . As in Figure 1, for moderately hot polar caps (), the particle energy is constrained by electron-positron cascades started by RICS.

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Figure 2: As Figure 1 but the particles are accelerated by the potential (3) with i=0 ( for outflowing positrons or ions), , , (the upper solid curve), (the lower solid curve). The result is similar to that derived by Sturner (1995). The two dashed plots (upper and lower) correspond respectively to acceleration of electron (positron) and ion without RICS. The dotted curve is the threshold condition for pair production through RICS.

The main mechanism for a high energy photon (emitted by an electron or positron either through curvature or inverse Compton scattering) to produce a pair is single photon decay in a strong magnetic field. To produce a pair, the photon must satisfy the threshold condition where is the photon energy (in ), is the angle between the photon propagation and the magnetic field. When this condition is satisfied, the number of pairs produced depends on the opacity, , which is an integration of the absorption coefficient (for a photon being converted into a pair) along the distance that photon travels. We assume that a photon produces at least one pair if . With this condition, we can write down the threshold condition for producing at least one pair through RICS (Luo 1996) as

The right-hand side of (4) strongly depends on B. If the above condition is satisfied, pair cascade can be initiated by RICS. The threshold condition is represented by the dotted curves in Figures 1 and 2. Note that for each curve of vs. , the location of an intersect (with the dotted curves) point defines a gap length, represented by . (If the curve has two intersecting points, the one that is the nearest to the polar cap is relevant.)

á© Copyright Astronomical Society of Australia 1997