Charge Densities above Pulsar Polar Caps
A simplified model provided the framework for our investigation into the distribution of energy and charge densities above the polar caps of a rotating neutron star. We assumed a neutron star withm= 1.4M⊙,r= 10km, dipolar field |B0| = 1012G,B||Ω and Ω = 2Π · (0.5s)−1. The effects of general relativity were disregarded. The induced accelerating electric fieldE||reachesE0= 2.5 · 1013V m−1at the surface near the magnetic poles. The current density along the field lines has an upper limitnGJ, when the electric field of the charged particle flow cancels the induced electric field: At the polesnGJ(r=rns,θ= 0) = 1.4 · 1017m−3.The work function(surface potential barrier)EWis approximated by the Fermi energyEFof magnetised matter. Following Abrahams and Shapiro (1992) one needs to revise the surface density from the canonical 1.4 · 108kg m−3down toρFe = 2.9 · 107kg m−3. Withwe obtain a value ofEF=Ew= 417eV. There are two relevant particle emission processes:Field (cold cathode) emissionby quantum-mechanical tunneling of charges through the surface potentialandthermal emissionwhich is a purely classical process. In strong electric fields it is enhanced by the lowering of the potential barrier due to the Schottky effect. The combined Dushman-Schottky equationwithtells us, thatat temperatures> 2 · 105K the the Goldreich-Julian current can be supplied thermal emission alone. The surface temperature however has a lower limit in the order of 105K due to the rotational braking. Therefore, in most cases a sufficient supply of charges for the Goldreich-Julian current is available and the electrical field accelerating the particles will be quenched as a result of their abundance. Otherwise a residual equilibrium electric field Eeqremains with:and hence the equilibrium density is:n=nfieid(Eeq,EW) +nDS(Eeq,EW,T) For a temperature just below the onset of thermal emission (T= 1.85 · 105K) the charge density is found to vary almost linearly with the work functionEWfor values ofEWbetween 0.3 and 2 keV. At the chosen value forEWof 417 eVthe residual electric field amounts to only 8.5% of the vacuum value. Even in the residual electric field the particles are rapidly accelerated to relativistic energies balanced by inverse Compton and curvature radiation losses.