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 $\sigma [\eta ]=6.4(2.0)\times {10}^{-4}$ (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 $4.8(1.7)\times {10}^{-4}$. In the optimistic scenario a single measurement of one year would be enough to reach $\approx 3\times {10}^{-4}$. 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 analyzing 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.

Figure 1

Spacecraft ranging towards $L_1$ or $L_2$ as a means by which to test the SEP (not in scale). We calculate the SEP signature as a perturbation on the Earth’s orbit around the Sun (${\mathbf{r}}_{03}$) as well as on the spacecraft ranging (${\mathbf{r}}_{3p}$). We also include perturbations from other planets.

Figure 2

Expected perturbation to the range signal towards a spacecraft in orbit around $L_1$ (solid line) and $L_2$ (dashed line) due to the SEP violation. The two signals have opposite sign.