Treat Mars and Mercury as non-rotating spheres of uniform composition

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

Treat Mars and Mercury as non-rotating spheres of uniform composition, each with the external potential

\Phi = -GM/r

taken to be zero at infinity. Mars has

M = 6.417\times10^{23}\ \mathrm{kg}

and

R = 3.390\times10^{6}\ \mathrm{m}

; Mercury has

M = 3.301\times10^{23}\ \mathrm{kg}

and

R = 2.440\times10^{6}\ \mathrm{m}

. 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}

. Ignore the fields of the Sun and of all other bodies. ((a)) Compute the compactness parameter

GM/Rc^2

of each planet and say which is the more compact. ((b)) A probe stands at rest on the surface of Mars. What does an accelerometer carried by the probe read, and in which direction does the reading point? ((c)) A radio transmitter on the surface of Mars sends a signal to a receiver at rest very far away. Give the gravitational redshift

z

to first order in

\Phi/c^2

, and estimate the fractional amount by which this first-order value falls short of the exact factor

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

, which is quoted in this chapter from the full theory of gravity and not derived here. Is the first-order formula adequate for Mars? ((d)) An astronaut of length

1.75\ \mathrm{m}

, oriented radially, falls freely through the event horizon of a black hole of mass

15\,M_\odot

. Compute the relative acceleration (the tidal stretch) between the astronaut's head and feet at the horizon.

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