A white dwarf is non-rotating and spherically symmetric

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Gravitational field, potential and Gauss's law for gravity

Problem

A white dwarf is non-rotating and spherically symmetric, of mass

M = 1.2\,M_\odot

with

M_\odot = 1.989 \times 10^{30}\ \mathrm{kg}

, and radius

R = 6.0 \times 10^{6}\ \mathrm{m}

; outside it the field is that of a point mass at its centre. A dust grain of mass

m = 1.0 \times 10^{-6}\ \mathrm{kg}

starts at rest at a very large distance and falls freely and radially to the surface. There is no atmosphere and no drag, the grain's own gravity is ignored, and since

GM/Rc^{2} \ll 1

energy conservation with the Newtonian potential is used together with the relativistic kinetic energy, to first order in

\Phi/c^{2}

. Take

G = 6.674 \times 10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}

, and the gravitational potential zero at infinity. ((a)) Find the gravitational potential

\Phi

at the surface of the white dwarf. ((b)) Find the kinetic energy of the grain on arrival, in joules, and express it as a fraction of the grain's rest energy

mc^{2}

. ((c)) Using

K = (\gamma - 1)mc^{2}

, find the speed of the grain on arrival. ((d)) Treating the white dwarf as a cold uniform sphere, evaluate the uniform-sphere estimate

3GM/5Rc^{2}

of its own gravitational mass defect, and give the ratio of the fraction of rest mass released by fusing hydrogen into helium,

0.0071

, to your fraction from part (b).

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