Mars rotational elements: how to explain the long period terms in the IAU standard?

In order to describe the orientation of the spin axis of Mars, the IAU (International Astronomical Union) Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE) uses the right ascension and declination angles, the equatorial coordinates orienting the planet with respect to the Earth equator. This solution is useful to make cartographic products.
Differently from IAU, the radioscience community commonly uses Euler angles, with obliquity and node’s longitude defined with respect to the planet mean orbit at epoch. In both sets of angles, a third angle, the rotation angle, is used to position the prime meridian. Its trend is the diurnal rotation.

The most usual way to transform Euler angles into IAU angles is to numerically evaluate the IAU angles over a given time interval with the help of spherical geometry, then to perform a frequency analysis on the so-obtained time series (e.g. Jacobson, 2010, Kuchynka et al., 2014 and Jacobson et al., 2018 for Mars). Unfortunately, such a method does not take into account the physical meaning of the planet’s rotational dynamics, which relies on well-known periodicities governed by the celestial mechanics.

In the present study, we provide analytical expressions to precisely transform a set of angles in the other in the case of Mars. The targeted precision of the transformation is smaller than 0.1 mas for each angle on an interval of about 30 years before and after J2000. Each angle is modeled by the sum of a quadratic polynomial, a periodic series (nutation or rotation variations) and a Poisson series (a periodic series multiplied by the time).

The analytical precise transformation shows that even when the Euler angles do not include quadratic terms (as commonly assumed by the radioscience community), the IAU angles do. As a result, since no quadratic term is considered in the IAU-like solutions as inferred by Jacobson (2010), Kuchynka et al. (2014), and Jacobson et al. (2018), an artificial very long period signal with well-chosen values for the amplitude, phase and frequency is needed to mimic the quadratic behavior on an interval of a few tens of years around J2000. Adding a long period modulation instead of a quadratic term largely and artificially alters the angle values at J2000 as well as their rates. For example, the correction was as high as 9° for the initial value of the angle or 100 mas/year for the rate.

This work is done in the frame of the analysis of two Martian radioscience experiments (RISE on InSight, and LaRa on ExoMars), and is financially supported by the Belgian PRODEX program managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office.