A heavy neutron star has mass and radius

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Relativistic stars and compact objects

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

Problem

A heavy neutron star has mass

M = 2.0\,M_\odot

and radius

R = 11\ \mathrm{km}

. Treat it as spherically symmetric and non-rotating, and suppose a clock on its surface emits light of a known frequency which is received very far away, effectively at infinity. ((a)) Estimate the Schwarzschild radius

r_s = 2GM/c^{2}

of this star, in km. ((b)) Estimate its compactness parameter

GM/Rc^{2}

. ((c)) Estimate the redshift of the received light in two ways: with the first-order formula

z \approx \Delta\Phi/c^{2}

of Chapter 9, where

\Delta\Phi = \Phi_{\rm rec} - \Phi_{\rm em}

, and with the exact factor quoted in this chapter from the full theory of gravity,

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

. ((d)) By what fraction of the exact value does the first-order estimate fall short? Hence decide whether the first-order formula may be used for this star. Use

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

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

and

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

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