Acceleration efficiency

The efficiency of the polar gap is described by the ratio of the potential drop across the gap, , to the maximum potential across the polar cap with an empty magnetosphere, , viz.

where , , and is the length of the polar gap in units of . Let and be the accelerated particle and spin-down luminosities, respectively. If the energy loss of accelerated particles in the gap is not important, we have since we can write and where is the injection rate calculated from the G-J density . Note that the efficiency defined by (8) is similar to that used by Arons (1996). For the presently available models for accelerating potentials, we always have .

Using (8), we may estimate the efficiency for a given model potential. For the potential described by Eq. (2) and assuming , we have the maximum efficiency

Thus, even when we use , the maximum efficiency is . When a pair cascade occurs, we usually have (for the polar gap) and hence .

One may estimate the efficiency for the potential given by Eq. (3) from

with . Although we have a higher for the Arons & Scharlemann model, the acceleration by the potential given by Eq. (3) is the more effective for .

The evaluation of depends on the specific mechanism for initiating a pair cascade and whether the pair plasma produced through the cascade is dense enough to short out part of the electric field. We consider three mechanisms for starting a pair cascade: RICS, curvature radiation, and inelastic scattering of ions by thermal photons from the polar cap. The corresponding lengths are , and . In general, we have (e.g. Luo 1996). For moderately hot polar caps with effective temperature , the energy loss due to RICS is not important compared to acceleration but the photons produced through RICS can start a pair cascade at the distance less than , . Thus, the effect of RICS is to reduce the gap acceleration efficiency. For space-charge-limited flow, the composition of outflowing charges at the poles with and can be different. For , the primary particles consist mainly of electrons, and for , the main components are heavy ions or positrons. In the ion zone, the gap length is constrained by the pair production by positrons through RICS. The possible source of positrons was discussed by Cheng & Ruderman (1977). For moderately hot polar caps, the gaps at both types of pole have a similar efficiency.

For sufficiently hot polar caps and a superstrong magnetic field (), the energy loss of electrons or positrons due to RICS can be important, and may prevent them starting a pair cascade in the region close to the polar cap. Since the cross section is for iron nuclei (), the energy loss due to RICS for ions is negligibly small, and they can be continuously accelerated until distance . Thus, the acceleration efficiency increases significantly. As an example, for the potential (2) with and , the gap length is about and the maximum energy of electrons (or positrons) is . This gives the particle luminosity where is the injection rate of primary electrons (or positrons). For hot polar caps with , from Figures 1, we obtain . From Figure 4, pair production by ions interacting with thermal photons is important only for the case where the whole star's surface has (the thick solid curve), and this occurs at the distance . Thus, the gap length is controlled by RICS. From Figure 1, we have for positrons (or electrons) and for ions; positrons (or electrons) and ions can be accelerated to the maximum energy and , respectively. Hence, we have much a higher particle luminosity and most of are carried by ions.

© Copyright Astronomical Society of Australia 1997