Scalar-tensor theories with screening mechanisms come with non-linearities
that make it difficult to study setups of complex geometry without resorting to
numerical simulations. In this article, we use the femtoscope code
that we introduced in a previous work in order to compute the fifth force
arising in the chameleon model in the Earth orbit. We go beyond published works
by introducing a departure from spherical symmetry \unicode{x2014} embodied
by a mountain on an otherwise spherical Earth \unicode{x2014} as well as by
implementing several atmospheric models, and quantify their combined effect on
the chameleon field. Building on the numerical results thus obtained, we
address the question of the detectability of a putative chameleon fifth force
by means of space geodesy techniques and, for the first time, quantitatively
assess the back-reaction created by the screening of a satellite itself. We
find that although the fifth force has a supposedly measurable effect on the
dynamics of an orbiting spacecraft, the imprecise knowledge of the mass
distribution inside the Earth greatly curtails the constraining power of such
space missions. Finally, we show how this degeneracy can be lifted when several
measurements are performed at different altitudes.Comment: 32 pages, 19 figure