Testing the Strong Equivalence Principle with spacecraft ranging towards
  the nearby Lagrangian points
  General relativity is supported by great experimental evidence. Yet there is
a lot of interest in precisely setting its limits with on going and future
experiments. A question to answer is about the validity of the Strong
Equivalence Principle. Ground experiments and Lunar Laser Ranging have provided
the best upper limit on the Nordtvedt parameter $\sigma[\eta]=4.4\times
10^{-4}$. With the future planetary mission BepiColombo, this parameter will be
further improved by at least an order of magnitude. In this paper we envisage
yet another possible testing environment with spacecraft ranging towards the
nearby Sun-Earth collinear Lagrangian points. Neglecting errors in planetary
masses and ephemerides, we forecast $\sigma[\eta]=6.4\,(2.0)\times10^{-4}$ (5
yr integration time) via ranging towards $L_1$ in a realistic (optimistic)
scenario depending on current (future) range capabilities and knowledge of the
Earth's ephemerides. A combined measurement, $L_1$+$L_2$, gives instead
$4.8\,(1.7)\times10^{-4}$. In the optimistic scenario a single measurement of
one year would be enough to reach $\approx3\times10^{-4}$. All figures are
comparable with Lunar Laser Ranging, but worse than BepiColombo. Performances
could be much improved if data were integrated over time and over the number of
satellites flying around either of the two Lagrangian points. We point out that
some systematics (gravitational perturbations of other planets or figure
effects) are much more in control compared to other experiments. We do not
advocate a specific mission to constrain the Strong Equivalence Principle, but
we do suggest analysing ranging data of present and future spacecrafts flying
around $L_1$/$L_2$ (one key mission is, for instance, LISA Pathfinder). This
spacecraft ranging would be a new and complementary probe to constrain the
Strong Equivalence Principle in space.
