Searching for new physics in the Solar System with tetrahedral spacecraft formations

Tetrahedral configurations of spacecraft on unperturbed heliocentric orbits allow for highly precise observations of small spatial changes in the gravitational field, especially those affecting the gravity gradient tensor (GGT). The resulting high sensitivity may be used to search for new physics that could manifest itself via deviations from general relativistic behavior yielding a nonvanishing trace of the GGT. We study the feasibility of recovering the trace[GGT] with the sensitivity of $\mathcal{O}({10}^{-24}{\mathrm{s}}^{-2})$—the level where some of the recently proposed cosmological models may have observable effects in the Solar System. Specifically, we consider how a set of local measurements provided by precision laser ranging (to measure the intersatellite ranges) and atom-wave interferometry (to correct for any local nongravitational disturbances) can be used for that purpose. We report on a preliminary study of such an experiment and on the precision that may be reached in measuring the trace[GGT], with the assumption of drag-compensated spacecraft by atom interferometer measurements. For that, we study the dynamical behavior of a tetrahedral formation established by four spacecraft placed on nearby elliptical orbits around the Sun. We develop analytical models for the relevant observables and study the conditions for setting up an optimal tetrahedral configuration. We formulate the observational equations to measure the trace[GGT] relying only on the observables that are available within the formation, such as those based on the laser ranging and the Sagnac interferometry. We demonstrate that the Sagnac observable is a mission-enabling capability that allows us to measure the angular frequency of the tetrahedral rotation with respect to an inertial reference frame with an accuracy that is much higher than that available from any other modern navigational techniques. We show that the quality of the science measurements is affected by the tetrahedron evolution, as its orientation and the shape change while the spacecraft follow their orbits. We present the preliminary mission and instrument requirements needed to measure the trace[GGT] to the required accuracy and thus demonstrate the feasibility of satisfying the stated science objective.

Figure 1

The RIC coordinate frame for a satellite or constellation in heliocentric orbit.

Figure 2

The TCS frame, constructed with satellite 4 as the origin. Note that the TCS frame is rotated by ${\mathit{\omega }}_{\mathtt{TCS}}$ with respect to the RIC frame, as shown by (25).

Figure 3

Left: evolution of the tetrahedron volume as a function of true anomaly $\nu$ and eccentricity, $e$, as given by (99). Vertical axis shows the volume compared to its initial value at the perigee, $V(\nu )/V(0)$, where formation was initiated. Right: changes in the normalized volume $({\mathbf{n}}_{41}·[{\mathbf{n}}_{42}\times {\mathbf{n}}_{43}])=V(\nu )/r_{41}{r}_{42}{r}_{43}$. Horizontal axis: $\nu \in [0,2\pi ]$. Dashed: $e=0$; solid: $e=0.6$.

Figure 4

Left: distance $r_{41}$ as a function of $\nu$ and eccentricity, $e$, as given by (58). Vertical axis shows the distance compared to its initial value at perigee, $r_{41}(\nu )/r_{41}(0)$. Right: pointing angle ${\theta }_{41}(\nu )$ as a function of $\nu$ and eccentricity, $e$, as given by (102). Vertical axis: angle in degrees. Horizontal axis is $\nu \in [0,2\pi ]$. For both plots: dashed: $e=0$, solid: $e=0.6$.

Figure 5

Simulation of the trace of the GGT. Left: results for a specific reference orbit with a perihelion of 0.6 AU and eccentricity $e\sim 0.5909$, with the satellites initially forming a tetrahedron with $\sim 1000\mathrm{km}$ edges. Right: shows the values for the trace of the GGT that are derived from the actual ranges between satellites and the generalized Sagnac observables (see Table 3). The constellation volume is also shown (orange solid line). Dashed purple line shows the heliocentric distance, referencing the secondary vertical axis. Horizontal axis: days since perihelion passage.

Figure 6

Details from the simulation shown in Fig. 5 from a region of maximum sensitivity.

Figure 7

The six intersatellite ranges of the configuration introduced in Fig. 5.

Figure 8

The 12 tetrahedron angles of the configuration introduced in Fig. 5, grouped, using color, by vertex (left) and tetrahedron face (right).