Deflection of spacecraft trajectories as a new test of general relativity: Determining the parametrized post-Newtonian parametersβandγ

Abstract

In a previous work, we proposed a new test of general relativity (GR) based on a general deflection formula which applies to all values of asymptotic speed ${V}_{\ensuremath{\infty}}(0<~{V}_{\ensuremath{\infty}}<~1).$ The formula simplifies to Einstein's light deflection result when ${V}_{\ensuremath{\infty}}=1.$ At low velocity, the general deflection equation reduces to the classical Newtonian contribution along with additional terms which contain the GR effect. A spacecraft, such as the proposed interstellar mission which involves a close pass of the Sun, can be used to exaggerate the GR effect so that it can be accurately measured. In this paper we provide a detailed derivation of the general deflection equation, expressed in terms of the parametrized post-Newtonian constants $\ensuremath{\beta}$ and $\ensuremath{\gamma}.$ The resulting formula demonstrates that by measuring spacecraft trajectories we can determine $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ independently. We show via a detailed covariance analysis that $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ may be determined to a precision of $\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $\ensuremath{\sim}8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6},$ respectively, using foreseeable improvements in spacecraft tracking.