Orbits in bootstrapped Newtonian gravity

Bootstrapped Newtonian gravity is a non-linear version of Newton's law which can be lifted to a fully geometric theory of gravity starting from a modified potential. Here, we study geodesics in the bootstrapped Newtonian effective metric in vacuum and obtain bounds on a free parameter from Solar System data and $S$-star orbits near our Galaxy centre. These bounds make {\em vacuum} bootstrapped Newtonian gravity experimentally indistinguishable from General Relativity.