Spherically symmetric solution of $f(R,\mathcal{G})$ gravity at low energy

The weak-field and slow-motion limit of $f(R,\mathcal{G})$ gravity is developed up to $(v/c)^{4}$ order in a spherically symmetric background. Considering the Taylor expansion of a general function $f$ around vanishing values of $R$ and $\mathcal{G}$, we present general vacuum solutions up to $(v/c)^{4}$ order for the gravitational field generated by a ball-like source. The spatial behaviors at $(v/c)^{2}$ order are the same for $f(R,\mathcal{G})$ gravity and $f(R)$ gravity, and their corresponding real valued static behaviors are presented and compared with the one in general relativity. The static Yukawa-like behavior is proved to be compatible with the previous result of the most general fourth-order theory. At $(v/c)^{4}$ order, the static corrections to the Yukawa-like behavior for $f(R,\mathcal{G})$ gravity, $f(R)$ gravity, and the Starobinsky gravity are presented and compared with the one in general relativity.