A magnetar is modelled as a static spherical star of 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 magnetar is modelled as a static spherical star of mass

M = 1.8\,M_\odot

and radius

R = 10.5\ \mathrm{km}

, with

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

. Take

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

and

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

. Rotation is ignored. ((a)) A probe stands at rest on the surface. Compute the Newtonian estimate

GM/R^{2}

of the accelerometer reading it carries. ((b)) Compute the compactness parameter

GM/Rc^{2}

of the star. ((c)) Compute the surface redshift seen by a distant observer two ways: from the first-order formula

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

with

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

\Phi = -GM/r

and

\Phi = 0

at the telescope, and from the factor

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

quoted in this chapter from the full theory. State whether the first-order formula is adequate here. ((d)) Using the Newtonian tidal field

\Delta a = 2GM\xi/r^{3}

, compute the stretch across a probe of radial extent

\xi = 1.8\ \mathrm{m}

resting at the surface.

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