Heating of the Real Polar Cap of Radio Pulsars

Abstract

The heating of the real polar cap surface of radio pulsars by the bombardment of ultra-relativistic charges is studied. The real polar cap is a significantly smaller area within or close by the conventional polar cap which is encircled by the last open field lines of the dipolar field $\vec{B}_d$. It is surrounded by those field lines of the small scale local surface field $\vec{B}_s$ that join the last open field lines of $\vec{B}_d$ in a height of $\sim 10^5$ cm above the cap. As the ratio of radii of the conventional and real polar cap $R_{dip}/R_{pc}\sim 10$, flux conservation requires $B_s/B_d\sim 100$. For rotational periods $P\sim 0.5$ s, $B_s\sim 10^{14}$ G creates a strong electric potential gap that forms the inner accelerating region (IAR) in which charges gain kinetic energies $\sim 3\times 10^{14}$ eV. This sets an upper limit for the energy that back flowing charges can release as heat in the surface layers of the real polar cap. Within the IAR, which is flown through with a dense stream of extremely energetic charges, no stable atmosphere of hydrogen can survive. Therefore, we consider the polar cap as a solidified "naked" surface consisting of fully ionized iron ions. We discuss the physical situation at the real polar cap, calculate its surface temperatures $T_s$ as functions of $B_s$ and $P$, and compare the results with X-ray observations of radio pulsars.