Deflection of spacecraft trajectories as a new test of general relativity: Determining the parametrized post-Newtonian parameters $\beta$ and $\gamma$

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 $V_{\infty }(0\mathrm{}<~V_{\infty }<~1).$ The formula simplifies to Einstein’s light deflection result when $V_{\infty }=1.\mathrm{}$ 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 $\beta$ and $\gamma .$ The resulting formula demonstrates that by measuring spacecraft trajectories we can determine $\beta$ and $\gamma$ independently. We show via a detailed covariance analysis that $\beta$ and $\gamma$ may be determined to a precision of $\sim 4\times {10}^{-5}$ and $\sim 8\times {10}^{-6},$ respectively, using foreseeable improvements in spacecraft tracking.