Determination of the rotation of Mercury from satellite gravimetry

Abstract

Space missions can have as a goal the determination of the interior structure of a planet: this is the case for the ESA BepiColombo mission to Mercury. Very precise range and range‐rate tracking from the Earth and onboard accelerometry will provide a huge amount of data, from which it will be possible to study the gravity field of Mercury and other parameters of interest. Gravity can be used to constrain the interior structure, but cannot uniquely determine the interior mass distribution. A much stronger constraint on the interior can be given by also determining the rotation state of the planet. If the planet is asymmetric enough, the gravity field as measured by an orbiting probe tracked from the Earth contains signatures from the rotation. Are these enough to solve for the rotation state, to the required accuracy, from tracking data alone, without measurements of the surface? In order to reach some result analytically, a simplified analytical model is developed, and the symmetry breaking, occurring when the shape of the planet deviates from spherical symmetry, is characterized by explicit formulae. Moreover, a full cycle numerical simulation of the Radio Science Experiment is performed, including the generation of simulated tracking and accelerometer data and the determination, by least‐squares fit, of the Mercury‐centric initial conditions of the probe, of Mercury's gravity field and its rotation state, together with other parameters affecting the dynamics. The conclusion is that there is no reason of principle prohibiting the determination of the rotation from gravimetry, and the sensitivity of the measurements and the coverage are good enough to perform the experiment at the required level of accuracy. This will be important also in ensuring independent terms of comparison for the rotation experiment performed with a high‐resolution camera. The mission is currently under development and much care has to be taken in guaranteeing the scientific goals even if there is some change in scenario.

The relativity experiment of MORE: Global full-cycle simulation and results

“…As a consequence, the parameters to solve for are the following: the PPN and related parameters, expressing the violations of GR and other effects (γ, β, η, α 1 , α 2 , J 2 , μ, ζ) together with the initial conditions for Mercury barycenter (x M , y M , z M ,ẋ M ,ẏ M ,ż M ) and Earth-Moon barycenter -EMB (x EMB , y EMB , z EMB ,ẋ EMB ,ẏ EMB ,ż EMB ) with respect to the Solar System Barycenter (SSB) in the Ecliptic J2000 reference frame 4 Together with these parameters, we need to determine also the initial conditions of the probe for each extended arc (as explained in Section II-A), which correspond to 6 parameters per arc (3 position and 3 velocity components). Moreover, the solve for parameters for the gravimetry and rotation experiments are: the gravity field harmonic coefficients up to degree l = 25, Love number k 2 , two obliquity angles δ 1 and δ 2 and the amplitude ε 1 of Mercury librations in longitude at 88 days.…”

Addressing some critical aspects of the BepiColombo MORE relativity experiment

“…• strategy I (descoping) 5 : we remove 4 out of the 13 parameters from the solve-for list (the three position components of the EMB and the z-component of the velocity of the EMB); • strategy II: we solve simultaneously for the 13 parameters by adding 4 a priori constraint equations in the LS fit.…”

Testing General Relativity with the Radio Science Experiment of the BepiColombo mission to Mercury

“…If we define the space-fixed Mercurycentric frame, Ψ MC , in which writing the equation of motion of the spacecraft, then we need to compute the rotation matrix R to convert the probe coordinates from Ψ BF to Ψ MC . To this aim, we adopt the semi-empirical model defined in [9]. Referring to that paper for an exhaustive discussion, we recall that the rotation matrix can be decomposed as R = R 3 (φ)R 1 (δ 2 )R 2 (δ 1 ), where R i (α) is the matrix associated with the rotation by an angle α about the i-th axis (i = 1, 2, 3), (δ 1 , δ 2 ) define the space-fixed direction of the rotation axis in the Ψ MC frame and φ is the rotation angle around the rotation axis, assuming the unit vector along the longest axis of the equator of Mercury (minimum momentum of inertia) as the rotational reference meridian.…”

“…The main goals of the MORE radio science experiment are concerned with the gravity of Mercury [4][5][6][7][8], the rotation of Mercury [9][10][11] and General Relativity (GR) tests [12][13][14][15][16]. The global experiment consists in determining the value and the formal uncertainty (as defined in Section 2.2) of a number of parameters of general interest, with the addition of further parameters characterizing each specific goal of the experiment.…”