Data Processing and Analysis

The GEODYN orbit determination program[8] was used to process the ramped Doppler and range observations for MGS. The modelling included use of the DE403 planetary ephemerides[9], in both the measurement and force models. In addition to the third body perturbations caused by the planets and the Sun, the third body perturbations due to Phobos and Deimos were also included, using a GM of 720,000 m³/s² for Phobos[10], and 120,000 m³/s² for Deimos[11] and the ephemerides of Jacobson[12]. The Mars radiation pressure was modelled using spherical harmonic models, as described by Lemoine[13], based on analysis of the Viking Infrared Thermal Mapper data. The formulation of Knocke et al.[14] was used to define the surface spots and the location of the source vectors for the Mars radiation pressure. The Mars radiation pressure is computed separately for 43 surface spots in three concentric rings and a central cap below the subspacecraft point that are in view of each spacecraft surface. The Stewart atmosphere model was used to compute the atmospheric density at the MGS altitude[15].

The Mars geopotential is modeled in spherical harmonics using the expression[16],

where the expansion is defined using the spherical coordinates of radius, r, latitude, , and longitude, ; and represent the normalized geopotential coefficients; are the normalized associated Legendre functions of degree l and order m; a_e is the reference equatorial radius; G is the universal constant of gravitation, and M is the planet mass. The 1991 IAU system of constants was used to define the orientation of Mars[17].

The a priori model was the MGM0890 solution, a model complete to 70x70, based on the historic Viking Orbiter and Mariner 9 S Band tracking, and the X Band tracking of MGS in the Hiatus and the Science Phasing Orbits [6]. Updated solutions to 70x70 were soon developed that incorporated the tracking from MGS in February and March 1999, and these improved models were applied in the production of orbits for analysis of the MOLA altimetry.

MGS periodically fires its thrusters to desaturate the reaction wheels, which absorb angular momentum from disturbance torques acting on the spacecraft. Three to four angular momentum desaturations occurred per day after entry into the low-altitude mapping orbit. The thruster firings can impart some velocity impulse effectively altering the orbit. Since telemetry informs us of the time and duration of these events, the angular momentum desaturations are modelled using empirical three-axis accelerations. Constant radial, along-track, and cross-track accelerations are applied over the duration of each event (typically two to there minutes), and are estimated as part of each orbit determination solution.

In the first pass through the February and March mapping orbit data, the tracking data were processed in one-day arcs. Later, as the modelling of the Mars geopotential improved, longer arcs (three to five days) were used. Each arc adjusted the spacecraft state, a solar radiation reflectivity coefficient (C_r), a drag coefficient (C_d) per day, as well as the empirical accelerations that modelled each AMD event.

Two distinct sets of orbit analyses were carried out. First, orbits that did not overlap were computed using the orbit modelling just described. These arcs in February and March formed the basis for the improvement in the modelling of the Mars geopotential. Second, quasi-reduced dynamic orbits that overlaped in time were computed to support the analysis of the MOLA altimeter data. These orbits adjusted additional empirical accelerations daily in the form of along-track once per revolution parameters. The adjustment of these extra empirical accelerations helps to remove residual orbit mismodelling. The orbit overlaps between adjacent arcs (except when orbit trim maneuvers occurred), provide a running check on the orbit quality. Since the frequent AMD events (three or four per day) would "interrupt" the period of each once per rev. empirical acceleration, constraint equations in GEODYN "tied" together the once-per-rev accelerations each day, effectively resulting in a single once-per-rev adjustment per day (See Rowlands et al.[18] for a description of the application of these constraint equations).