Revisiting the computation of the critical points of the Keplerian distance, Mars orientation and rotation angles
Authors: Gronchi Giovanni F. , Yseboodt Marie 
Published: January 2024

Revisiting the computation of the critical points of the Keplerian distance

Authors: Giovanni F. Gronchi, Giulio Baù, Clara Grassi
We consider the Keplerian distance d in the case of two elliptic orbits, i.e., the distance between one point on the first ellipse and one point on the second one, assuming they have a common focus. The absolute minimum d_min of this function, called MOID or orbit distance in the literature, is relevant to detect possible impacts between two objects following approximately these elliptic trajectories. We revisit and compare two different approaches to compute the critical points of d^2, where we squared the distance d to include crossing points among the critical ones. One approach uses trigonometric polynomials, and the other uses ordinary polynomials. A new way to test the reliability of the computation of d_min is introduced, based on optimal estimates that can be found in the literature. The planar case is also discussed: in this case, we present an estimate for the maximal number of critical points of d^2, together with a conjecture supported by numerical tests.

Mars orientation and rotation angles

Authors: Marie Yseboodt, Rose-Marie Baland, Sébastien Le Maistre
The rotation and orientation of Mars is commonly described using two different sets of angles, namely (1) the Euler angles with respect to the Mars orbit plane and (2) the right ascension, declination, and prime meridian location angles with respect to the Earth equator at J2000 (as adopted by the IAU). We propose a formulation for both these sets of angles, which consists of the sum of a second degree polynomial and of periodic and Poisson series. Such a formulation is shown here to enable accurate (and physically sound) transformation from one set of angles to the other. The transformation formulas are provided and discussed in this paper. In particular, we point that the quadratic and Poisson terms are key ingredients to reach a transformation precision of 0.1 mas, even 30 years away from the reference epoch of the rotation model (e.g., J2000). Such a precision is required to accurately determine the smaller and smaller geophysical signals observed in the high-accuracy data acquired from the surface of Mars. In addition, we present good practices to build an accurate Martian rotation model over a long time span (30 years around J2000) or over a shorter one (e.g., lifetime of a space mission). We recommend to consider the J2000 mean orbit of Mars as the reference plane for Euler angles.
An accurate rotation model should make use of up-to-date models for the rigid (this study) and liquid (Le Maistre et al., 2023) nutations, relativistic corrections in rotation (Baland et al. 2023), and polar motion induced by the external torque (this study). Our transformation model and recommendations can be used to define the future IAU solution for the rotation and orientation of Mars using right ascension, declination, and prime meridian location. In particular, thanks to its quadratic terms, our transformation model does not introduce arbitrary and non-physical terms of very long period and large amplitudes, thus providing unbiased values of the rates and epoch values of the angles.