A red dwarf of mass and radius has a brown dwarf companion of mass and radius

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic stars and compact objects · Relativistic stars and compact objects

Relativity and Gravitation, Ch Escape velocity, the Schwarzschild radius and the idea of a black hole

Problem

A red dwarf of mass

M = 2.446\times10^{29}\ \mathrm{kg}

and radius

R = 1.07\times10^{8}\ \mathrm{m}

has a brown dwarf companion of mass

M = 1.193\times10^{29}\ \mathrm{kg}

and radius

R = 7.00\times10^{7}\ \mathrm{m}

. Treat both as non-rotating spheres whose external potential is

\Phi = -GM/r

, zero at infinity, and ignore the field of each on the other. Take

G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

and

M_\odot = 1.989\times10^{30}\ \mathrm{kg}

. ((a)) Compute the compactness parameter

GM/Rc^2

of each star and rank them. ((b)) Compute the escape speed from the surface of the red dwarf. ((c)) Light leaves the red dwarf's photosphere and is received by a distant observer at rest. Give the redshift

z

to first order in

\Phi/c^2

, and estimate the fractional amount by which this falls short of the exact factor

1 + z = (1 - 2GM/Rc^2)^{-1/2}

, quoted in this chapter from the full theory and not derived here. Does the first-order result apply to this star? ((d)) An astronaut of length

1.80\ \mathrm{m}

, oriented radially, falls freely through the event horizon of a

60\,M_\odot

black hole. Compute the tidal stretch (the relative acceleration of head and feet) at the horizon.

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