Past exam of the astrophysics course of the University of Oxford / 2015 / 2 / e / Solution

The accreting-pulsar luminosity and material-torque consistency test shows a concrete tension between the simplest isotropic luminosity estimate and the material-torque estimate. Making their rates equal requiresAn exterior accretion Alfvén radius cannot lie inside a 10-km star. Even taking its smallest exterior value, $r_{\mathrm{Alf}}=R$, the observed torque permits only about $32{\dot{M}}_{\mathrm{Edd}}$ in this model, below the $49{\dot{M}}_{\mathrm{Edd}}$ isotropic estimate. The discrepancy is modest enough to be sensitive to the approximate radius and moment of inertia, but it prevents claiming exact simultaneous agreement under the stated central values.

Near corotation, the torque gives only $2.23{\dot{M}}_{\mathrm{Edd}}$. If its radiative efficiency is unchanged, beaming into a fraction about $2.23/48.6\simeq 0.046$ of the sky would reconcile that intrinsic luminosity with the apparent isotropic value. The given truncation formula would then require $B\sim 2.6\times 10^8$ T for a magnetosphere near corotation. Other possibilities include substantial opposing magnetic torque, outflows separating the disk supply rate from the surface rate, a different radiative efficiency and magnetically altered opacity. These are changes to the elementary model that need independent evidence, rather than a license to use an unphysical truncation radius. Coherent pulsations already identify the accretor as a neutron star; they do not by themselves choose between these accretion-flow refinements.