Radio ranging to a spacecraft beyond the Sun measures the excess round-trip Shapiro…

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

Radio ranging to a spacecraft beyond the Sun measures the excess round-trip Shapiro delay when the radar path grazes the solar limb. General relativity (

\gamma_{\rm PPN}=1

) predicts an excess delay

\Delta t_{\rm S}^{\rm GR} = 240.0\ \mu\mathrm{s}

for this geometry. Redoing the delay integral with the PPN metric multiplies the whole logarithm by the same factor that appears in the deflection, so that

\Delta t_{\rm S} = \dfrac{1+\gamma_{\rm PPN}}{2}\,\Delta t_{\rm S}^{\rm GR}

. A measurement finds

\Delta t_{\rm S} = 239.95\pm0.05\ \mu\mathrm{s}

. ((a)) Find

\gamma_{\rm PPN}-1

with its

1\sigma

uncertainty. ((b)) State the resulting bound on

|\gamma_{\rm PPN}-1|

at two standard deviations. ((c)) Use Brans-Dicke theory's relation

\gamma_{\rm PPN}=(1+\omega_{\rm BD})/(2+\omega_{\rm BD})

to convert this bound into a lower bound on

\omega_{\rm BD}

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