Interpretation of geodesy experiments in non-Newtonian theories of
  gravity
  The tests of the deviations from Newton's or Einstein's gravity in the Earth
neighbourhood are tied to our knowledge of the shape and mass distribution of
our planet. On the one hand estimators of these "modified" theories of gravity
may be explicitly Earth-model-dependent whilst on the other hand the Earth
gravitational field would act as a systematic error. We revisit deviations from
Newtonian gravity described by a Yukawa interaction that can arise from the
existence of a finite range fifth force. We show that the standard multipolar
expansion of the Earth gravitational potential can be generalised. In
particular, the multipolar coefficients depend on the distance to the centre of
the Earth and are therefore not universal to the Earth system anymore. This
offers new ways of constraining such Yukawa interactions and demonstrates
explicitly the limits of the Newton-based interpretation of geodesy
experiments. In turn, limitations from geodesy data restrict the possibility of
testing gravity in space. The gravitational acceleration is described in terms
of spin-weighted spherical harmonics allowing us to obtain the perturbing force
entering the Lagrange-Gauss secular equations. This is then used to discuss the
correlation between geodesy and modified gravity experiments and the
possibility to break their degeneracy. Finally we show that, given the existing
constraints, a Yukawa fifth force is expected to be sub-dominant in satellite
dynamics and space geodesy experiments, as long as they are performed at
altitudes greater than a few hundred kilometres. Gravity surveys will have to
gain at least two orders of magnitude in instrumental precision before
satellite geodesy could be used to improve the current constraints on modified
gravity.
