A steady, geometrically thin accretion disc surrounds a neutron star of mass and…

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar death and compact objects · Astrophysical fluid dynamics and plasmas

Stars from the Inside, Ch Accretion power: mass transfer, discs and the weighing of black holes

Problem

A steady, geometrically thin accretion disc surrounds a neutron star of mass

M = 1.4\,M_\odot

and radius

R_* = 10\,\mathrm{km}

, fed at a steady rate

\dot M = 10^{-9}\,M_\odot\,\mathrm{yr^{-1}}

. The disc is Keplerian and radiates locally as a blackbody. Its viscous dissipation rate per unit area of one face, obtained from angular-momentum conservation in the disc with a torque-free inner edge, is

D(R) = \frac{3GM\dot M}{8\pi R^3}\left[1-\left(\frac{R_*}{R}\right)^{1/2}\right],\qquad R\ge R_*.

((a)) By integrating

D(R)

over both faces of the disc from

R_*

to infinity, show that the total disc luminosity is

L_{\rm disc} = GM\dot M/2R_*

. ((b)) Evaluate

L_{\rm disc}

numerically, and compare it with the total accretion luminosity

GM\dot M/R_*

released between infinity and the stellar surface, to say where the other half is radiated.

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