Title: The complete exterior spacetime of spherical Brans-Dicke stars

Abstract: We derive the complete expression for the Brans Class I exterior spacetime explicitly in terms of the energy and pressures profiles of a stationary spherisymmetric gravity source. This novel and generic expression is achieved in a $\textit{parsimonious}$ manner, requiring only a subset of the Brans-Dicke field equation and the scalar equation. For distant orbiting test particles, this expression promptly provides a simple, closed and exact formula of the $\gamma$ Eddington parameter, which reads $\gamma_{\,\text{exact}}=\frac{\omega+1+(\omega+2)\,\varTheta}{\omega+2+(\omega+1)\,\varTheta}$, where $\varTheta$ is the ratio of the star's "total pressure" integral over its energy integral. This $\textit{non-perturbative}$ result reproduces the usual Post-Newtonian $\frac{\omega+1}{\omega+2}$ expression in the case of a "Newtonian star", in which the pressure is negligible with respect to the energy density. Furthermore, it converges to the General Relativity value $\gamma_{\,\text{GR}}=1$ as the star's equation of state approaches that of ultra-relativistic matter (in which case $\varTheta$ approaches 1), a behavior consistent with broader studies on scalar-tensor gravity. Our derivation underscores the essence of these results involving (1) the key relevant portion of the Brans-Dicke field equations, (2) the uniqueness of the Brans Class I vacuum solution for the non-phantom action, viz. $\omega>-3/2$, and (3) the involvement of only two free parameters in this solution. From a practical standpoint, it elucidates how a given stellar interior structure model determines the star's exterior gravitational field and impacts the motions of light objects (such as planets and accretion disks) orbiting it.