Mars' time-variable gravity and its determination: Simulated geodesy experiments

Authors


Abstract

[1] The seasonal carbon dioxide (CO2) cycle on Mars results in a time-variable global redistribution of mass. These large-scale variations are associated with changes in the gravity field, mainly in the two zonal gravity coefficients equation image and equation image, which have been recently evaluated from Doppler tracking data of the Mars Global Surveyor (MGS) spacecraft. In the present study, we calculated these variations from the mass redistribution obtained from outputs of two general circulation models (GCM) as well as from CO2 thickness measurements by the High Energy Neutron Detector (HEND) instrument on board the Mars Odyssey spacecraft and compared them to the observations. Tracking observations provide one of the most direct measures of the global-scale atmospheric mass cycle. However, the associated uncertainties are relatively large, partly because the low-degree zonals obtained from a single orbiter tracking analysis are contaminated by higher-degree harmonics which are shown to have nonnegligible seasonal variations. Thus we investigated possibilities to improve the determination of the time-variable gravity field by means of simulated geodesy experiments. Additional radio tracking of a second spacecraft with suitable orbital characteristics was shown to be able to separate the higher-degree geodetic signatures. Radio links between landers on the Martian surface and a near-polar orbiter can further better estimate especially the even zonals.

1. Introduction

[2] Mars' surface mass distribution varies at seasonal timescales due to condensation, sublimation and precipitation of the atmospheric CO2. The size of both polar caps and the atmospheric pressure change, resulting in seasonal variations of the Martian gravity field. Theoretically, geodetic signatures of this CO2 cycle can be found in the variations of both the zonal gravity parameters, equation image, and the rotation rate (i.e., length of day (LOD)).

[3] An orbiting spacecraft can be considered as a gravity sensor through the dynamical effect of the mass redistributions on the spacecraft motion. Perturbations of the satellite velocity induced by the time-variable gravity can be obtained from the analysis of the spacecraft Doppler tracking data. Tracking of Earth orbiting satellites has allowed the measurement of global-scale mass transport of geophysical fluids which is of interest not only to geodesy but also to other Earth sciences such as hydrology, oceanography, glaciology and meteorology [Chao et al., 2000; Chao, 2003].

[4] The influence of time-variable gravity harmonics on a spacecraft's orbit has been recognized since the very early days of the space age. However, extraction of seasonal and secular variations of odd and even zonals requires accurate tracking data [Yoder et al., 1983]. In the course of time, accuracies of spacecraft tracking systems have been increased considerably [Barlier and Lefebvre, 2001]. The recent GRACE mission with additional spacecraft-to-spacecraft tracking, monitors oceanic/atmospheric mass variations with much higher precisions compared to their predecessors [Tapley et al., 2004].

[5] The possibility of detecting seasonal gravity variations of Mars by spacecraft tracking has first been studied by Chao and Rubincam [1990]. Yoder and Standish [1997] and Smith et al. [1999a] confirmed that seasonal mass variations on Mars have large enough signatures in the gravity field so that from tracking data of an orbiting spacecraft one could estimate odd and even zonal harmonics. Recently, the variations of the first three lowest-degree zonal gravity harmonics were extracted from MGS spacecraft tracking data [Smith et al., 2001; Smith and Zuber, 2003]. By using the same data set, and adding Mars Pathfinder/Viking lander data to constrain the rotation, Yoder et al. [2003] estimated annual and subannual components of the normalized zonal harmonic coefficients of degree two, equation image, and three, equation image (see below for their definitions). The resulting even zonal coefficients show a semiannual signal while odd zonal coefficients have principally an annual signal. Those gravitational variations are explained by phase effects associated with the competing influences of north and south pole mass variations.

[6] These studies provided a direct measure of the global cycle of CO2. However, the noise level of the X band Doppler tracking data is only slightly below the time-variable part of the gravity signals, leading to high uncertainties in the reported coefficients. Atmospheric drag affects especially equation image making its extraction more difficult [Yoder et al., 2003]. Moreover, as pointed out by these authors, their recovered signals contain also errors due to contributions of higher-degree zonals. From the theory of spacecraft dynamics, it is known that the tracking data from a single spacecraft can be used to separate odd from even zonal coefficients but is not sufficient to separate each individual degree [see Kaula, 1966]. Consequently, the contamination of the higher-degree zonals in the current equation image and equation image estimates remains unknown and it is difficult to precisely estimate the seasonal CO2 mass distribution.

[7] The gravity variations can be modeled using surface mass distribution variations from theoretical models and other types of measurement. On the basis of surface pressure data from Viking landers, [Chao and Rubincam, 1990] first estimated these seasonal variations. Improved estimates were made by Smith et al. [1999a], who used the output of a global circulation model (GCM). However, their GCM results were approximate in the absence of accurate altimetry data in the polar regions. We use the surface pressure and CO2 ice mass as provided by the GCM of the Laboratoire de Météorologie Dynamique (LMD) [Forget et al., 1999] and by the GCM of NASA Ames [Haberle et al., 1999]. Both of these recent models adopt the MOLA topography [Smith et al., 1999b].

[8] Another estimate of the CO2 surface mass distribution comes from the High Energy Neutron Detector (HEND) on board the Mars Odyssey spacecraft (part of the Gamma Ray Spectrometer). It has been measuring neutron fluxes since 18 February 2002 [Mitrofanov et al., 2003; Litvak et al., 2004]. The presence of hydrogen on the surface or subsurface of Mars strongly affects the flux of fast neutrons due to the high moderation effect of the hydrogen nuclei, with which the neutrons collide. The summer measurements, when no CO2 deposit is present on the surface, are used to determine the amount of hydrogen in the subsurface, and the changes in the neutron flux with seasons are interpreted in terms of CO2 deposits. The neutron flux scales with the thickness of the surface CO2 layer such that a higher flux of neutrons is expected for a thicker atmospheric deposit. The local CO2 deposit variations over one Martian year were calculated from HEND data by Litvak et al. [2004].

[9] The seasonal variations of the zonal harmonic of degree 2, equation image, are linked to changes in the rotation rate about the spin axis (LOD) [see Chao and Rubincam, 1990]. In the calculation of LOD, besides the surface mass redistribution, the atmospheric (wind) angular momentum variations have to be taken into account. Several authors [Cazenave and Balmino, 1981; Defraigne et al., 2000; Van den Acker et al., 2002; Sanchez et al., 2003] estimated the seasonal variations of LOD from GCMs as well as from Viking lander surface pressure measurements. Duron et al. [2003] showed that, with the help of a dedicated geodesy experiment including a network of landers, it is possible to measure equation image and the rotation rate of the planet independently.

[10] The present paper has two main objectives. The first is to investigate the seasonal variations in the global CO2 mass redistribution and in the Martian gravity field by using outputs from atmospheric models, local CO2 measurements, and global tracking of orbiting spacecraft. The second is to assess the precision of the time-variable zonal gravity coefficients from tracking data and to present alternative strategies to improve their determination. The utility of spacecraft tracking data for estimating the global-scale CO2 cycle is also discussed.

[11] The comparison of the time-variable gravity fields is presented in section 2. Section 3 is dedicated to the recovery of time-variable gravity field from simulation experiments with detailed discussions on the solutions from single and dual orbiters as well as on the role of a lander network.

2. Time-Variable Gravity Field

2.1. Theory

[12] The external gravitational potential of a planet is customarily expressed in a spherical harmonics expansion [e.g., Chao et al., 1987]. The normalized Stokes coefficients of degree n and order 0, equation image are given by

equation image

where Pn is Legendre's polynomial of degree n, and R and M are the radius and mass of the planet. Colatitude and longitude are given by equation image and λ, respectively. The mass density is ρ, which is also a function of r, the distance from the center of the planet. The integration is over the whole planet including the atmosphere. equation image are related to dimensionless zonal harmonic coefficients of degree n and order 0, equation image. The elastic yielding of the planet under the surface loading is neglected; its effect is less than 1% [see Van Hoolst et al., 2003].

[13] The seasonal variation of the coefficients of this expansion is directly related to the seasonal mass redistribution on the planet. Mass distribution inside the planet varies on much longer timescales compared to that on the surface except for the small tidal contributions which occurs on shorter timescales. Therefore, in the present analysis associated with the seasonal timescales, any mass variation inside the planet is neglected and seasonal gravity variation is entirely attributed to mass exchanges between atmosphere and surface.

[14] For a mass distribution in the atmosphere, we can take rR and equation (1) can be rewritten as a surface integral by introducing a surface mass distribution σ given by the following integral over the atmosphere:

equation image

[15] Accordingly, any variation in surface density of mass, given by Δσ, changes the gravity field acting on the spacecraft as [Chao et al., 1987]

equation image

[16] In our case, Δσ is due to the seasonal variation in both atmospheric pressure, Δσatm, and polar cap masses, Δσpole; both are used in the calculation of zonal coefficients. Figure 1 shows the contributions of polar cap mass and atmospheric pressure to equation image and equation image variations, calculated from the LMD GCM (see below). In agreement with Smith et al. [1999a], Δσpole makes the major contribution. In the next section, we neglect the atmospheric contribution in the calculation of equation image variations from HEND data. From the results in Figure 1, the errors are less than 6% and 2% for equation image and equation image, respectively.

Figure 1.

Contributions of polar caps and atmospheric pressure to (a) equation image and (b) equation image variations according to the LMD GCM.

2.2. Seasonal Variation of equation image Coefficients

[17] The temporal variations of the surface pressure and CO2 ice mass are provided by the GCMs from the Laboratoire de Météorologie Dynamique (LMD) [Forget et al., 1999] and the NASA Ames research center [Haberle et al., 1999]. In both simulations of the Martian atmosphere dynamics, the surface resolution is 7.5° in latitude and 9.0° in longitude. The dust scenario is consistent with the observations of a moderately dusty planet. Since the models do not handle exactly in the same way the physical parameterization, the sublimation/condensation rates of CO2 are not identical.

[18] The seasonal changes in CO2 deposits, which cover hydrogen rich surface layers can cause variations in neutron flux observed from the orbit between summer and winter up to several times. These variations have been measured by HEND and used to estimate the thickness of CO2 deposits by Litvak et al. [2004]. The CO2 snow depth at different latitudes was calculated by using a model-dependent technique. The CO2 surface density was obtained in kg/m2 over the latitude bands of 60°–70°, 70°–80° and 80°–90° for both hemispheres as a function of time. Maximum masses in north and south seasonal caps were estimated to be 3.6 × 1015 kg and 6.7 × 1015 kg, respectively.

[19] Figure 2 shows the temporal variation in the zonal coefficients of the gravity field from degree 2 to 5 as a function of areocentric longitude of the Sun (Ls), calculated from the LMD GCM, the NASA Ames GCM and the HEND data. Although the general behavior of the three data sets is similar, the differences in the amplitudes at a given time can be larger than 30% (see below for a more detailed comparison). In all three data sets, the higher-degree gravity coefficients (degrees 4 and 5) show also significantly large seasonal variations. This result suggests that the higher-degree terms should not be omitted in the determination procedure of seasonal gravity field variations from tracking data. On the other hand, tracking data often do not allow the adjustment of all the gravity coefficients, hence the studies remain restricted to lower-degree coefficients equation image and equation image, although they might be contaminated from large higher-degree coefficients.

Figure 2.

Seasonal variation of the normalized zonal gravity field coefficients (a) equation image, (b) equation image, (c) equation image, and (d) equation image as predicted by LMD GCM, NASA Ames GCM, and the measurements of CO2 deposit thickness of HEND.

[20] If north and south polar caps would have identical seasonal deposits and be 180° out of phase, the even and odd zonal harmonics would show zero and “double” annual amplitudes, respectively. In reality, due to north-south asymmetry the contributions from the two hemispheres not only do not cancel but also yield semiannual variability for the even zonal harmonics. There are several reasons for this north-south asymmetry. It is due to mass differences between the polar caps as well as to the relative altitude differences between the polar regions, the high eccentricity of the Martian orbit and the different albedos of the polar caps. The southern polar cap is lying about 6 km higher in altitude and therefore has lower pressure and temperature. Solar insolation is lower during southern winter compared to northern winter period. In addition, the northern cap is known to be darker [Paige and Ingersoll, 1985]. As a result of all these factors, the southern pole has lower general temperature. Its winter also lasts longer while its summer is shorter due to the high eccentricity of the orbit. Seasonal CO2 deposits do not sublimate completely on the southern pole as it is the case in the northern hemisphere.

[21] A more detailed analysis of equation image and equation image is presented in Figure 3, where Doppler tracking results of the MGS orbiter are overlaid. From the reported annual and semiannual amplitudes and phases by Smith et al. [2001] and Yoder et al. [2003], the two signals are reconstructed and plotted. The results show a fairly good overall agreement with the GCM and HEND solutions. Both the GCM and the HEND solutions on the CO2 mass variations are model-dependent and hence it is difficult to associate errors to them. From local neutron flux measurements of HEND, the masses of seasonal deposits were estimated using a model for the subsurface layers [Litvak et al., 2004]. Similarly, the GCM results depend on the parametrization of several physical parameters. Some of those parameters such as the albedo and the emissivity of polar caps, are assumed different than their actual values in order to match the CO2 cycle predicted by Viking landers [Hourdin et al., 1995]. The discrepancies between these model-dependent solutions in Figure 3 hence reflect the incomplete physical understanding of the CO2 cycle on planetary scale. Tracking data provide a more direct measure of global CO2 cycle. However, differences in solution procedures to extract the time variable coefficients combined with the low signal-to-noise ratio led to large (up to 40% for equation image) variations in the amplitudes. These differences between the tracking observations are about the same as those between the GCMs and HEND data and hence tracking data do not favor in particular any of these models. It can, however, be expected that gravity observations will better constrain such models in near future, as longer tracking data and more spacecrafts orbiting Mars will be available. The assessment of the precision of the time-variable zonal gravity coefficients from tracking data and alternative strategies to improve their determination are discussed in the following section.

Figure 3.

Seasonal variation of equation image and equation image as extracted from MGS tracking data and predicted by two GCMs as well as by the measurements of CO2 deposit thickness of HEND.

3. Recovery of Time-Variable Gravity Field

[22] In the present section, the recovery of time-variable gravity field is simulated with respect to several geodesy experiments. The objective is to explore possibilities to improve the direct determination of time-variable zonal gravity coefficients from one or more spacecrafts, and to quantify the influence of higher-degree zonals in the case of a single orbiter.

3.1. Theory and Methodology

[23] The spacecraft orbit plays the role of a gravity filter of which the transfer function is fixed by its orbital elements. The orbital velocity or position can be observed from either the Earth or Mars, or both, by Doppler shift and range measurements. The even zonal harmonics are determined principally from the observation of changes in the orbital node, Ω, whereas the odd harmonics affect mainly the spacecraft orbit eccentricity, e, and pericenter angle, ω. The dynamical equations for Ω, e and ω appropriate for a high inclination, I, and small e can be expressed, following Yoder et al. [2003], as

equation image
equation image

where p = e exp(−iω), and ω0 = 3n(R/a)2f3C20. Other parameters are the mean motion, n, the semimajor axis of the orbiter, a = 3796 km, and the radius of Mars, R = 3390 km. The lumped coefficients equation image and equation image are linear combinations of even and odd zonal harmonics, respectively:

equation image

[24] The fn coefficients depend on inclination: f2 = 1, f3 = 1 − 1.25x, f4 = 0.625(R/a)2(7x − 4), f5 = −0.3125(R/a)2(8 − 28x + 21x2), where x = sin2I.

[25] The orbit determination program used is GINS (Géodésie par Intégrations Numériques Simultanées), developed primarily at GRGS/CNES (Toulouse, France). It models orbiter dynamics by taking into account both gravitational and nongravitational forces acting on the spacecraft and adjusts several parameters of these models such as the zonal harmonics of the time-variable gravity field.

[26] For the simulations, we used the spherical harmonic expansion of the static gravity field GMM-2B [Lemoine et al., 2001], to which we added the time series of the zonal gravity coefficients, equation image determined from the NASA Ames GCM. Given these modeled (initial) values, we simulated Doppler shifts of radio signals over 100 arcs of one week each. The orbitography software GINS can derive from the simulated Doppler tracking data, the spacecraft cartesian state vector and the parameters of the gravity model. Through a least-squares procedure, we evaluated the spacecraft positions and velocities on a weekly basis together with the zonal gravity parameters, equation image to equation image, at 10-days intervals.

[27] Doppler shifts of radio signals between spacecrafts and three DSN stations on Earth (X/S band link) as well as between Martian landers and an orbiter (UHF/S band link) are considered for simulation experiments. DSN stations are located at Canberra (Australia), Madrid (Spain) and Goldstone (USA). This study assumes that X/S and UHF/S radio tracking observations are integrated over 60-s and 20-s intervals respectively with an instrumental error budget of 0.1 mm/s [Tyler et al., 1992; Barriot et al., 2001]. The rate of the UHF measurements is fixed to one pass per lander per week. The positions of the Sun and the other planets are given by the JPL DE403 ephemeris. The nongravitational forces acting on the orbiters (desaturation maneuvers, atmospheric drag, solar radiation pressure, albedo, etc.) and the lander locations are assumed to be perfectly known. An accurate determination of the lander locations is feasible within the first 5 weeks of the mission as demonstrated by Vienne et al. [2004]. Duron et al. [2003] showed that the simultaneous adjustment of the rotation rate and the time-variable gravity harmonics can be performed successfully with known lander positions. Accordingly, the rotation of Mars, the rotation rate in particular, is assumed to be perfectly known or determined accurately enough from a Martian network science experiment so that the seasonal variation of even zonals would not be contaminated. With these assumptions, the simulations provide the best estimate of the zonal gravity variations and correspond to an upper limit for the sensitivity of the orbits to the zonal terms.

[28] The results can be analyzed in several ways. First, one can consider the true errors which correspond to the differences between the synthetic and the recovered values. Obviously, the true error cannot be determined within a planetary mission, in which formal errors are estimated from the statistical evaluation of the least-squares procedure and given as standard deviations. Another output of the solution process is the covariance matrix. Covariances between each estimate of the zonal terms in a covariance matrix represent the ability to separate the adjusted parameters with the solution procedure.

3.2. Single Orbiter Analysis

[29] A single orbiter is considered in a Sun-synchronous, near-circular (e = 0.001), near-polar orbit (I = 93.2°) with an altitude of 500 km. This orbit is very close to that of MGS.

[30] The orbit perturbations are related to specific combinations of gravity coefficients (see equation (6)), and the resolved coefficients from tracking data of a single orbiter contain contributions from all degrees. For the chosen inclination, we have: f2 = 1, f3 = −0.25, f4 = 1.48 and f5 = −0.24. As the amplitudes of the time-variable higher-degree zonals are of the same order of magnitude (see Figure 2), the higher-degree zonals have a nonnegligible contribution to equation image and equation image.

[31] In order to investigate how the contributions of higher-degree harmonics affect the solution, we varied the number of zonal coefficients that are taken into consideration in our simulated Doppler shifts. In the first simulation, equation image and equation image were jointly adjusted for a variable gravity field composed of only equation image and equation image. In the second simulation, we also adjusted only equation image and equation image, but the time-variable gravity field included the equation image and equation image temporal variations. In the third simulation, we considered the same gravity field as in the second simulation, but we jointly adjusted all of the zonal parameters, equation image to equation image, simultaneously (see also Table 1).

Table 1. Description of Simulations
SimulationRadio LinkComponents of Gravity FieldAdjusted Parameters
11 orbiterequation image, equation imageequation image, equation image
21 orbiterequation image to equation imageequation image, equation image
31 orbiterequation image to equation imageequation image to equation image
42 orbitersequation image to equation imageequation image to equation image
51 orbiter + 4 landersequation image to equation imageequation image to equation image
62 orbiters + 4 landersequation image to equation imageequation image to equation image

[32] The covariance matrix for the first two simulations, in which only equation image and equation image are jointly adjusted, are shown in Figures 4a and 4b. These results correspond to the first five weeks of data, for which six solutions (on overlapping intervals of 10 days) are obtained for both zonals. The correlations between the adjusted parameters of simulation 1 remain low, that is, less than 0.5. When the higher-degree coefficients are included in the modeled time-variable gravity field, the adjustment results in high correlations. In particular, the interdate correlations of equation image and the correlations between equation image and equation image are significant. The adjustment of all of the coefficients (equation image to equation image) does not provide completely uncorrelated set of solutions (Figure 4c) since the individual components of the lumped coefficients (equation (6)) cannot be resolved with a single spacecraft.

Figure 4.

Correlation matrix (a) for which equation image and equation image are adjusted for a time-variable gravity field defined by the zonals of degree 2 to degree 3, (b) for a simulation where only equation image and equation image are adjusted for a time-variable gravity field defined by the zonals of degree 2 to degree 5, and (c) for a simulation where equation image to equation image are adjusted for a time-variable gravity field defined by the zonals of degree 2 to degree 5. The matrix elements represent the covariances between the adjusted zonal coefficients for the first 5 weeks of the mission (6 solutions are shown for overlapped periods of 10 days).

[33] The true errors of equation image and equation image estimates are shown in Figures 5a and 5b, respectively, for the cases where equation image and equation image are considered in the model and are (simulation 3) or are not (simulation 2) adjusted. The synthetic variations of equation image and equation image are also plotted. The good agreement between the true errors of simulation 2 and the modeled variations of equation image and equation image confirms that the true error in the adjusted terms equation image and equation image of simulation 2, is principally due to the nonadjusted higher-degree terms equation image and equation image, respectively.

Figure 5.

True errors in (a) equation image and (b) equation image compared to the time variation of the nonadjusted equation image and equation image. Simulations 2 and 3.

[34] An attempt to resolve all of the zonal coefficients (simulation 3) removes largely the signature of higher-degree zonals in equation image and equation image, but the errors still remain very large, as expected. The errors in equation image are slightly lower than those associated with equation image. This can also be noted from Figures 6 and 7, where the variations of equation image to equation image and their standard deviations with respect to other zonals are given. The even zonals are better determined than the odd zonals with equation image being the best adjusted parameter. The equation image formal uncertainty in the simulations is about one order of magnitude better than that obtained from MGS tracking data [Yoder et al., 2003], while the formal uncertainty of equation image is similar. This is probably because nongravitational forces are perfectly known in the simulations. For example, uncertainties in the drag force would strongly perturb the determination of the nodal rate, and hence the even coefficients.

Figure 6.

Recovered values regarding the initial variations of (a)equation image, (b)equation image, (c)equation image, and (d)equation image. Modeled variations (solid line), solutions for single orbiter (circle), for a single orbiter with landers (triangle), for two orbiters (x), and for two orbiters with landers (plus symbols) are plotted.

Figure 7.

Same as Figure 6, but for standard deviation (see text for explanation).

[35] We calculated the annual and semiannual amplitudes of the adjusted equation image and equation image variations from a Fourier series representation of the adjusted temporal series (see Table 2). When the seasonal variations of the higher-degree terms equation image and equation image contribute to the gravity field acting on the orbiter, and only the two lowest-degree coefficients are adjusted, the true errors for the annual amplitudes are larger than 50% and 25% for the semiannual amplitudes.

Table 2. Annual and Semiannual Amplitudes of Recovered equation image and equation image
 Δequation imageΔequation image
AnnualSemiannualAnnualSemiannual
AmplitudePhaseAmplitudePhaseAmplitudePhaseAmplitudePhase
Initial8.05E-10−44.47.49E-10−44.51.92E-9−67.52.57E-1037.8
Simulation 16.78E-10−42.06.57E-10−41.02.43E-9−69.32.93E-1036.8
Simulation 21.22E-9−37.81.01E-9−57.62.86E-9−70.63.34E-1040.0
Simulation 38.64E-10−45.55.38E-10−25.92.08E-9−69.56.09E-1026.9

[36] The time series of equation image and equation image extracted from the MGS tracking data will therefore undoubtedly also show large errors. Since those gravity variations have been used by several authors to estimate the seasonal variation of polar masses [Karatekin et al., 2003; Smith and Zuber, 2003; Yoder et al., 2003] and the density of seasonal mass deposit [Aharonson et al., 2004; Smith et al., 2001], we therefore expect that significant errors occur on these estimates. We now explore geodetic experiments to reduce such errors.

3.3. Two Orbiters

[37] A solution to overcome the difficult estimation of time-variable zonal harmonics from a single orbiter tracking is to add another spacecraft with a different orbit which is perturbed by a different combination of zonal terms. At present three spacecrafts are orbiting Mars (Mars Express, MGS and Mars Odyssey), and in the coming years several spacecrafts are expected to orbit Mars. Some of these orbiters could have nonpolar orbits as a part of their mission strategy or could be moved to lower inclinations in their extended missions. We now investigate the simultaneous tracking of such a spacecraft with a polar orbiter. The first spacecraft orbit is kept identical to the one described in section 3.2, while the second has an altitude of 400 km, an inclination of 50°, and a small eccentricity of 0.0206. These new orbital elements are chosen so that the geodetic signature of even zonal harmonics increases (see below).

[38] With the additional orbiter, the correlations between each equation image and equation image estimates are reduced considerably, to below 0.5 (see Figure 9). Moreover, the equation image variations and, to a lesser degree, the equation image variations match almost exactly the modeled input variations (Figure 6). The equation image and equation image variations are also very well recovered due to the additional information brought by the second radio link.

[39] Correlations between the equation image estimates appear in Figure 9a, despite the fact that the mean a posteriori standard deviations on the adjustment of equation image and equation image are one order of magnitude smaller than in the single orbiter case (Figure 7). The perturbation rate of the ascending node due to equation image drops down as the orbit inclination decreases from 90°, reaching zero at 49.1° (for the given inclination I = 50°, the coefficients in equation (6) have the values f2 = 1, f3 = 0.27, f4 = 0.05 and f5 = 0.30). It is then difficult to extract the equation image signal from the data whereas equation image is less perturbed by the existence of equation image. This leads to decorrelation between both estimates. On the other hand, the adjustments of the odd coefficients result in high equation image correlations, they remain lumped according to equation (6). Nevertheless, their a posteriori standard deviations are one order of magnitude smaller than those of the single orbiter.

[40] The mean a posteriori standard deviations on the adjustment of odd and even zonals are one order of magnitude smaller than in the single orbiter case (Figure 7). Their amplitudes vary as much as by a factor of 3 during the adjustment period. The odd terms are better recovered in the first 200 days (especially for a single orbiter) where the orbit is nearly edge-on as seen from the Earth (Figure 8). It can be shown from Kaula's [1966] classical expression for the disturbing potential that the determination of orbit eccentricity and pericenter angle, hence odd zonals, are best when the spacecraft is in edge-on geometry (the odd zonals affect primarily the radial component of the spacecraft velocity whereas the along-track and cross-track perturbations are driven by mainly the even zonal coefficients). With the additional orbiter, the a posteriori standard deviations of odd zonals do not vary much in time as in a near-polar orbit because the lower inclined orbit provides a viewing angle (whose mean is close to edge-on geometry) which favors determination of odd zonals (Figure 8).

Figure 8.

Variation of viewing geometry for the two orbiters; 0° and 90° correspond to face-on and edge-on geometries.

[41] The sensitivity of the orbits to the even or odd harmonics depends on their inclination (see equations (4) and (5)), while the relative amplitudes of the inclination functions, fn (equation (6)) characterize the separability of each zonal coefficients. Next to an MGS-like orbiter, the use of a second spacecraft with an inclination around 50° has principally two benefits; first, lower inclination increases the sensitivity of even zonals, and secondly, the relative amplitudes of the inclination functions of even terms (f4 ≈ 0) provides a better separability of equation image and equation image. The odd relative amplitudes of f3 and f5 remain nearly equal even though it was possible to find another inclination which would also help to separate properly the odd zonals such as I ≈ 40°, I ≈ 63° or I ≈ 73°. Obviously, two orbiters with inclinations favorable to the separation of odd and even degree zonals respectively, would improve further the solution.

3.4. Additional Lander Network

[42] Finally, we consider a Martian lander network, for example three landers around Tharsis and one near the Hellas basin, at antipodal location as suggested by Barriot et al. [2001]. In simulation 5, we consider an additional radio link between the near-polar orbiter and these four Martian landers. In simulation 6, we consider two orbiters as in simulation 4 and a radio link between the lander network and the first orbiter (see Table 1). In the present simulations, we assume the rotation to be perfectly known. If this would not be the case, the lander network would not only improve the ability to detect the changes in the zonal harmonics of the gravity field but would also enable an accurate evaluation of the planetary rotation [Duron et al., 2003; Yoder and Standish, 1997].

[43] In the case of a single orbiter, the use of landers improves significantly only the even zonals, as shown in Figures 6 and 7. The estimated even zonals are much closer to their synthetic variations and their standard deviations are reduced approximately by a factor of two compared to the single orbiter case. This is consistent with Duron et al. [2003], who showed that the resulting adjustment of equation image is improved due to a better orbit determination related to the additional information brought by the lander-orbiter radio signal. On the other hand, the effect of landers on odd zonals is limited only to reduce slightly the standard deviations especially during the face-on geometry (last 200 days) thanks to an alternative viewing geometry of the orbit plan. Although the lander network improves the solution, it is not sufficient to significantly resolve the time-variable gravity field.

[44] In the previous section it was shown that two orbiters may be used to improve the determination of the time-variable gravity field. The presence of a lander network enhances further the solution. When the simulated Doppler shift measurements of the Mars-based tracking of the near-polar orbiter are added, the correlations between the equation image estimates decreases significantly (Figure 9c) compared to the cases where only twin Earth-based tracking data (Figure 9a) and where a lander network with a single orbiter (Figure 9b) are considered. Consequently, a more accurate adjustment of the equation image variations is obtained as shown in Figure 6. Accordingly, the mean a posteriori standard deviations on this adjustment is twice smaller (Figure 7). The equation image correlations remain about the same.

Figure 9.

Correlation matrix for (a) two orbiters, (b) a single orbiter with the lander network, and (c) two orbiters with the lander network. The matrix elements represent the covariances between the adjusted zonal coefficients for the first 25 weeks of the mission (solutions given for overlapped periods of 10 days).

4. Discussion and Conclusions

[45] Seasonal zonal harmonics have been estimated from two GCMs for the Martian atmosphere as well as from HEND observations of the surface CO2 deposits over one Martian year. Although the general behavior of the three data sets is similar, the differences in the amplitudes at a given time can be larger than 30%. These results reflect an incomplete physical understanding of the CO2 cycle on planetary scale. Tracking data provide a more direct measure of global CO2 cycle. We have therefore compared the estimated equation image and equation image with reported annual and semiannual amplitudes and phases by Smith et al. [2001] and Yoder et al. [2003] obtained from Doppler tracking results of the MGS orbiter. Differences in solution procedures to extract the time variable coefficients combined with the low signal-to-noise ratio leads to large (up to 40% for equation image) differences between these tracking data solutions. These discrepancies are of about the same order of amplitude as those given for the GCM and HEND data. Hence, despite showing fairly good overall agreement, tracking data do not favor in particular any of these models. It can, however, be expected that gravity observations will better constrain such models in near future, as longer tracking data and more spacecrafts orbiting Mars will be available.

[46] An inherent problem of the determination of zonal gravity coefficients from tracking of a single orbiter is that the orbit perturbations depend on linear combinations of either odd or even zonal coefficients. The equation image and equation image solutions from MGS tracking therefore contain information from all even and odd zonal terms, respectively. Higher-degree gravity coefficients are shown to have large seasonal variations as well, comparable to the degree 2 and 3 variations, suggesting that the higher-degree terms should not be neglected in the determination procedure of seasonal gravity field variations from spacecraft tracking data. Our tracking data simulations show that the neglect of the higher-degree terms equation image and equation image has an impact of about 50% on the adjusted annual amplitudes of both equation image and equation image. Another way to assess the influence of higher-degree zonals is to calculate the lumped coefficients, equation image and equation image according to equation (6) by using the outputs of a GCM. Figure 10 shows the two MGS tracking solutions in comparison with the modeled lumped coefficients obtained from the output of NASA Ames GCM. The amplitudes of the lumped coefficients increase and their phases vary as the number of zonals in the summation are increased. The influence of coefficients of higher degree is more significant for equation image for the given orbit inclination. When the higher-degree odd zonals are taken into account, the agreement between equation image and the tracking solutions equation image is improved, in particular for the annual amplitude. However, this is not the case for equation image whose signal is much more difficult to detect [Yoder et al., 2003; Smith et al., 2001].

Figure 10.

Contribution of higher-degree zonals on (a) even and (b) odd zonal gravity field coefficients. Following equation (6), contributions of higher degrees are calculated from NASA Ames GCM outputs. The results are shown in comparison with MGS tracking data.

[47] In order to improve zonal gravity coefficient recovery, we investigated the possibility of combining different sets of tracking data. Two spacecrafts with circular orbits having two different inclinations can be used to recover more accurately the time-variable gravity field. In the present study, next to an MGS-like orbiter, we used a second spacecraft with an inclination around 50°. The a posteriori standard deviations of the recovered zonals are about one order of magnitude smaller than those of the single orbiter. Orbital characteristics of the second spacecraft allow the accurate extraction of the geodetic signatures of the even zonal harmonics in particular, thanks to the relative amplitudes of the inclination functions entering the lumped coefficients. The resulting correlations between each equation image and equation image estimates are reduced considerably compared to the single orbiter case. Next to the MGS-like orbiter, it was possible to find another inclination for the second orbiter which would help to separate properly the odd zonals. Two orbiters with inclinations favorable to the separation of odd and even degree zonals, respectively, would improve further the solution.

[48] As a further extension of the present study, a more realistic gravity field (including the omitted degrees, n > 5) could be considered. In this case, the recovery of degree 2 and degree 3 zonal harmonics with two orbiters would depend on the relative amplitudes of ∑ fnequation image for n > 4 and for n > 5 with respect to the f2equation image and f3equation image, respectively (see equation (6)). The recovery of zonal coefficients equation image and equation image could be significantly improved with respect to a single orbiter case, provided that appropriate orbits are chosen. On the other hand, determination of equation image and equation image, would not anymore be possible. To determine more harmonics of the time-variable gravity field, several different orbits and varied inclinations will be necessary since they allow separation of the contributions from each zonal individually [Balmino, 1986]. As for the inclination, a different eccentricity might be helpful to improve time-variable gravity. The influence of a higher eccentricity for the second orbiter such as for Mars Express was studied preliminarily by Rosenblatt et al. [2004]. They showed that the determination of the variable gravity field can be improved by up to a factor 2 by additionally using simulated Mars Express to MGS data, although both have near-polar orbits.

[49] Additional lander-orbiter radio links are shown to improve further the determination of the low-degree zonal gravity coefficients, in particular equation image, both for single and twin orbiters. Although it has been proposed several times, the realization of a lander network mission in the near future remains uncertain. Additional radio links between landers on the surface of Mars and the orbiter yield a decorrelation of the interdate estimates of even degree zonals by mainly providing an alternative view angle for Doppler shift measurements. Moreover, the lander network can be used to obtain an accurate LOD evaluation. As the LOD signal contains both mass and wind effects and the even zonal part of the gravity field contains only the mass effect, such an experiment could also be used to separate the contributions of the mass redistribution and of the atmospheric winds, and to estimate the zonal winds [Duron et al., 2003].

Acknowledgments

[50] We are grateful to Y. Wanherdrick and F. Forget from the LMD and R. Haberle from NASA Ames for sharing their GCM results and to M. Litvak from SRI for providing the HEND data. We wish to thank A. Somerhausen for help with computation and J.-C. Marty, J.-M. Lemoine, and S. Loyer for their careful and constructive comments. We also thank Benjamin Chao and the anonymous reviewer for their careful and constructive remarks. This study was funded by a ESA/Prodex contract and supported by the European Community's Improving Human Potential Programme, under contract RTN2-2001-00414, MAGE.

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