A near-Earth asteroid's orbit is fitted with a semi-major axis quoted in the…

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

The Geometry of Gravity, Ch Solar-system tests, the PPN framework and alternative theories

Problem

A near-Earth asteroid's orbit is fitted with a semi-major axis quoted in the isotropic radial coordinate of the linearised weak-field metric,

\tilde a=2.20\times10^{11}\,\text{m}

, and eccentricity

e=0.35

. Take

GM_\odot/c^2=1.48\times10^{3}\,\text{m}

and

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

. A scalar-tensor theory under test has

\gamma_{\rm PPN}=0.95

and

\beta_{\rm PPN}=1.00

, giving the perihelion advance per orbit

\Delta\varpi=\frac{2+2\gamma_{\rm PPN}-\beta_{\rm PPN}}{3}\cdot\frac{6\pi GM_\odot}{a(1-e^2)c^2},

where

a

is the areal semi-major axis. ((a)) Using

r\approx\tilde r+GM_\odot/c^2

from the weak-field metric, decide whether converting

\tilde a

to the areal

a

changes the value to be used in the formula above, at the precision given, and state the value of

a

you use. ((b)) Find the orbital period from Kepler's third law,

T_{\rm orb}=2\pi\sqrt{a^3/GM_\odot}

, and hence the number of orbits per century. ((c)) Compute the perihelion advance predicted by this theory, in arcseconds per century.

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