# Mars precession rate determined from radiometric tracking of the InSight Lander

Daniel S. Kahan, William M. Folkner, Dustin R. Buccino, Véronique Dehant, Sébastien Le Maistre, Attilio Rivoldini, Tim van Hoolst, Marie Yseboodt, J. C. Marty

► To cite this version:

Daniel S. Kahan, William M. Folkner, Dustin R. Buccino, Véronique Dehant, Sébastien Le Maistre, et al.. Mars precession rate determined from radiometric tracking of the InSight Lander. Planetary and Space Science, 2021, 199, 10.1016/j.pss.2021.105208. insu-04836297

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# Mars Precession Rate Determined from Radiometric Tracking of the InSight Lander

Daniel S. Kahan<fnref n="1" />, William M. Folkner<fnref n="1" />, Dustin R. Buccino<fnref n="1" />, Véronique Dehant<fnref n="2" />, Sébastien Le Maistre<fnref n="2" />, Attilio Rivoldini<fnref n="2" />, Tim Van Hoolst<fnref n="2" />, Marie Yseboodt<fnref n="2" />, J. C. Marty<fnref n="3" />

March 2021

<footnote n="1">¹ Jet Propulsion Laboratory, 4800 Oak Grove Dr., Pasadena, CA 91109, USA</footnote>

<footnote n="2">² Royal Observatory of Belgium, av. Circulaire 3, B1180 Brussels, Belgium</footnote>

<footnote n="3">³ CNES/GRGS, OMP 14 avenue Edouard Belin 31400 Toulouse, France</footnote>
46 **Abstract**

47 The Interior Exploration using Seismic Investigations, Geodesy, and Heat Transport (InSight) 48 mission has provided an opportunity to improve our knowledge of Mars' interior via the Rotation 49 and Interior Structure Experiment (RISE). RISE provides information on the rotation of Mars by 50 measuring the Doppler shift of radio transmissions from InSight to NASA's Deep Space 51 Network (DSN) on Earth. Through the combination of one year of Doppler data from InSight 52 with previous data from the Viking-1 lander, Mars Pathfinder, and Opportunity missions, data 53 spanning 43 years from 1976 to 2019 are used to estimate the Mars precession rate as -7605 ±3 54 milliarcseconds per year. This result is consistent with the precession rate estimated from 55 Doppler tracking of previous landers and orbiters alone.

56 **Keywords: Mars, interior structure, moment of inertia, precession, InSight, Rotation and**
57 **Interior Structure Experiment**

1 Introduction

59 Landing on Mars on November 26, 2018, the Interior Exploration using Seismic Investigations,
60 Geodesy, and Heat Transport (InSight) mission communicates with Earth via relays with Mars
61 orbiters and by direct radio transmissions to and from NASA's Deep Space Network (DSN).
62 InSight's Rotation and Interior Structure Experiment (RISE) makes use of the InSight/DSN
63 transmissions by measuring the range rate of the round-trip signal, which is proportional to the
64 velocity of the lander along the line of sight (LOS) with respect to Earth. These Doppler
65 measurements provide information on the rotation of Mars (e.g., Folkner et al., 1997; Kuchynka
66 et al., 2014). This paper discusses the methodology for determination of the Mars precession rate
67 based on data collected during the first year of the InSight mission along with previous
68 measurements from the Viking-1 lander, Pathfinder, and Mars Exploration Rover Opportunity.
An updated computation of that value is provided along with a computation of Moment of Inertia (MOI) based on recently estimated parameters such as the degree-2 order-0 gravity field $J_2$.

The precession of Mars' axis of rotation is caused by the Sun's gravitational pull on the equator of the Mars ellipsoid as it rotates. It takes approximately 170,000 years for the spin axis to rotate once about the normal to the orbit plane, while nutations of the rotation axis occur on a period of 1 Martian year or less due to the relative Mars-Sun orientation as well as to the gravitational torques of Phobos and Deimos (Reasenberg & King, 1979; Baland et al., 2020). The MOI determined from the precession rate can help determine interior properties of Mars (Folkner et al., 1997). Additionally, together with the degree-two gravity field, the precession rate determines Mars' global moment of inertia, which is a key element for the calculation of obliquity and insolation history and therefore also for the modeling of the Martian climate over millions of years (Bouquillon, 2000; Ward, 1974; Laskar, 1988; Tourma and Wisdom, 1993; Francois et al., 1990). The data set is not yet long enough to separate the nutation contributions from precession or obliquity rates. The nutations during the InSight time frame are taken into account but fixed to the values from a rigid Mars model (Reasenberg & King, 1979; Baland et al., 2020).

Measurement of the radio range from communications with the Viking landers provided the first estimation of Mars' precession from landers on Mars (Yoder & Standish, 1997). For those missions, there was no programmatic impetus for the archival of tracking data; however, two years of Doppler measurements from Viking-1 were recovered (a small amount of Viking-2 data were recovered but are not employed in this study). Those two years of data from Viking-1 were later combined with data from Mars Pathfinder, yielding an observed precession of -7576 ± 35 milliarcseconds per year (Folkner et al., 1997). In 2014, Kuchynka et al. combined 4 months of
tracking data from a stationary period of the Opportunity rover with the Viking-1 and Pathfinder data to improve the landers/rover precession estimate to -7619.5 ± 6.4 mas/yr. Doppler tracking data of the same landers plus spacecraft in orbit about Mars (Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance Orbiter) have also been used to estimate the Martian precession rate, with a most recent estimate of -7608.3 ± 2.1 mas/yr (Konopliv et al., 2016).

This study builds upon the work of Kuchynka et al. (2014) by adding one year of data from the recently landed InSight mission. The large displacement of the Martian pole between missions allows for more accurate determination of the linear drift of the pole. Now spanning 43 years from 1976 to 2019, the combined data set yields updated constraints on the mean precession rate and other rotation parameters. Section 2 of this paper describes the nature and source of the measurements involved; Section 3 discusses the treatment of range-rate noise; Section 4 discusses the analysis of the data; Section 5 interprets the results; and Section 6 provides conclusions.

## 2 Measurements

The measurements in this analysis are the range rates of the radio signals transmitted from the DSN to the lander and phase coherently transmitted back to the DSN on Earth. The measured Doppler shift approximates the rate of change of distance $ρ$ along the LOS between the tracking station and the lander. Yoder and Standish (1997) provide a description of the range rate as:

$$ \frac{\partial \rho}{\partial t} = \frac{\partial \rho_{EM}}{\partial t} + \frac{\partial \rho_{DSN}}{\partial t} - \frac{\partial}{\partial t} [R_z \sin \delta_E + R_S \cos \delta_E \cos(\varphi + \lambda - \alpha_E)] \quad (1) $$

where $ρ_{EM}$ is the Earth-Mars distance, $ρ_{DSN}$ is the norm of the component of the center of Earth to DSN antenna vector projected to the Earth-Mars direction, $R_z$ is the distance of the lander
from the Martian equatorial plane (geocentric elevation), $R_s$ is the distance from the Martian spin axis to the lander (geocentric latitude), $\phi$ is the angle of rotation of Mars about its spin axis (between the prime meridian direction at $t$ and its direction at the J2000 reference epoch), $\lambda$ is the lander longitude, $\alpha_E$ is the right ascension (the angle in the Martian equatorial plane measured from where the Sun crosses that plane from south to north [Martian vernal equinox]), and $\delta_E$ is the declination (the angle from the Martian mean equatorial plane) of Earth as viewed from Mars. See Figure 1.
<figure><img src="image_2.png" /></figure>

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<caption>Figure 1. Geometry for computation of the Doppler shift of a landed asset on Mars</caption>

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The direction of Mars' spin axis, and therefore precession, can be gleaned through changes in the Earth declination $\delta_E$ and right ascension $a_E$ observed via the Doppler measurements. These changes impact the amplitude and phase of the diurnal effect in the last term of Equation 1. For
example, when Earth declination approaches zero, the Mars rotation axis nears perpendicularity to the LOS, lessening this signature (Yseboodt et al, 2017).

To estimate Mars' precession, the InSight tracking data covering November 27, 2018, to November 30, 2019, are joined with previous data from the Viking-1 lander, encompassing over 1 Martian year from 1976 to 1979, from Mars Pathfinder, covering roughly 90 days in 1997, and from Opportunity, which was stationary for four months in 2012. Table 1 summarizes the lander data sets. A brief description of the parameters for each mission follows, along with a summary of the calibrations applied to account for the tropospheres and ionospheres of Earth and Mars.

<table><caption>**Table 1. Lander/rover tracking data at 60-second integration time considered in this study. The estimated accuracy corresponds to the root mean square of the frequency residuals determined in a fit of the model.**</caption><thead><tr><td></td><td>Number of Passes</td><td>Mean Pass Duration (minutes)</td><td>Number of Data Points</td><td>Time Range</td><td>Sun-Earth-Probe Angles (degrees)</td><td>Approximate Accuracy (mm/s)</td></tr></thead><tbody><tr><td>Viking 1</td><td>208</td><td>73</td><td>15230</td><td>21-Jul-1976 to 20-Dec-1978</td><td>1 to 176</td><td>0.42</td></tr><tr><td>Pathfinder</td><td>107</td><td>75</td><td>7995</td><td>04-Jul-1997 to 25-Sep-1997</td><td>55 to 85</td><td>0.037</td></tr><tr><td>Opportunity</td><td>59</td><td>23</td><td>1373</td><td>10-Jan-2012 to 04-May-2012</td><td>110 to 176</td><td>0.017</td></tr><tr><td>InSight</td><td>241</td><td>49</td><td>11755</td><td>27-Nov-2018 to 30-Nov-2019</td><td>2 to 92</td><td>0.037</td></tr></tbody></table>

## 2.1 Viking

The first of the two-part Viking mission, Viking-1 was the first spacecraft to successfully land and operate from the surface of Mars. The mission consisted of both an orbiter and a lander, and the lander was designed to take high-resolution images and study the surface and atmosphere of Mars. The Viking-1 lander landed in Chryse Planitia (22.49°N, 48.04°W) on July 20, 1976, and the mission lasted until November 11, 1982 (Viking Mission to Mars, 1988).
The Viking-1 lander communicated with its orbiter through an ultra-high frequency (UHF) relay system and directly with Earth via an S-Band communication system (2.1 GHz uplink, 2.3 GHz downlink). The S-Band system consisted of two receivers, one using a low gain antenna (LGA) to receive commands from Earth and one using a high gain antenna (HGA). The HGA, a 76-centimeter diameter parabolic reflector dish, was steerable and could be pointed directly at Earth (Soffen and Snyder, 1976). Over two years of S-Band Doppler data, while Viking-1 was transmitting with the HGA, were retrieved by Folkner et al. (1997) and are similarly used in this study (neither Viking-1 nor Viking-2 Doppler data were archived). The data used in the fitting process consist of 15,230 60-second data points over 208 passes between July 21, 1976, and December 20, 1978 (Table 1).

## 2.2 Pathfinder

The Mars Pathfinder mission was the first mission to deliver a wheeled rover (Sojourner) to another planet. Pathfinder landed in Ares Vallis (19.17°N, 33.21°W) on July 4, 1997 (Golombek et al., 1997), and the mission lasted until September 27, 1997.

Similar to Viking, Pathfinder's radio system included a LGA and a steerable HGA (Golombek, 1997). While the Sojourner rover communicated directly with the base station (formally named the Carl Sagan Memorial Station after landing), this station communicated with Earth via its X-Band communication system (7.2 GHz uplink, 8.4 GHz downlink). Radio tracking of Pathfinder was used to attain a precise measure of Mars' pole of rotation, constraining the radius of the planet's core to between 1300 and 2000 kilometers (Golombek et al., 1997).

The Pathfinder data contribute 7995 60-second data points over 107 passes from July 4, 1997, to September 25, 1997, to the fitting process (Table 1). Although the span of the Pathfinder data is
relatively short, the combination with the Viking data shows the motion of the Martian pole due to precession over a period of twenty years (Folkner et al., 1997).

## 2.3 Opportunity

On January 24, 2004, the Mars Exploration Rover Opportunity landed in Eagle crater on Meridiani Planum (2.2°S, 5.4°W), 21 days after the landing of Spirit at Gusev crater. The primary mission of both rovers was to explore locations on the Martian surface where water may have once been present and to assess past environmental conditions and their capability to support life (Squyres et al., 2004). Opportunity continued operations until June 2018 when it was engulfed by a planet-wide dust storm and lost communications with Earth. Although both rovers were in transit most of the time, between January and May 2012 Opportunity remained in one spot, saving energy during Martian winter. During that time, the MER Radio Science Tracking Experiment (Kuchynka et al., 2014) was conducted by the DSN, with the express purpose of collecting additional Doppler data for improved calculation of Mars rotation parameters.

Similar to Pathfinder, Opportunity transmitted at X-Band and had both a LGA and a steerable HGA. The HGA was used for the bulk of the experiment, with approximately 20 tracks of durations less than 5 minutes acquired on the LGA. Because the HGA passes occurred with Earth at a more sensitive geometry (lower elevation angles), and because they tended to be longer (up to 30 minutes) (Kuchynka et al., 2014), they are the only ones used in this study.

The Opportunity 60-second data applied in this study from the MER Radio Science Tracking Experiment consist of 1373 points over 59 passes acquired between January 10, 2012, and May 4, 2012 (Table 1). The addition of Doppler data from the MER Radio Science Tracking Experiment to the Viking and Pathfinder sets increased the range of direct radiometric tracking
187 of the Martian surface from about 20 years to more than 35 years, and the updated precession
188 rate based on these three missions is discussed by Kuchynka et al. (2014).

189 2.4 InSight

190 InSight landed as intended in Elysium Planitia on Mars (4.50°N, 135.62°E) on November 26,
191 2018 (Golombek et al., 2020), and has recently completed its primary science phase as of
192 December, 2020. A thermally controlled enclosure below the lander's main platform contains the
193 lander's Small Deep-Space Transponder (SDST) and solid-state power amplifier (SSPA). As
194 with Pathfinder and Opportunity, an X-Band (7.2 GHz) carrier signal is sent from a DSN
195 antenna, received by the SDST, amplified by the SSPA, and phase coherently transmitted back to
196 Earth (at 8.4 GHz) to the same DSN antenna (with a turnaround ratio of 880/749).

197 Unlike the other three missions, the SDST and SSPA for InSight RISE are linked to two fixed-
198 pointing medium-gain antennas (MGAs), which are located on the main platform (Folkner et al.,
199 2018). InSight azimuthal orientation was controlled during landing in order to ensure good
200 viewing geometry for RISE via one antenna or the other, while also allowing commanding from
201 Earth throughout the mission. One antenna points 5.06 degrees northward from due west, and the
202 other points 16.8 degrees southward from due east. With this orientation, Earth is usually
203 viewable when rising over the eastward antenna or when setting over the westward antenna; the
204 typical amount of tracking time for one day is 30 minutes to an hour. The tilt at the landing site is
205 4° (Agle et al., 2018), well within the MGA patterns, which were designed to accommodate
206 angles up to 15° (Golombek et al., 2017). Each MGA points 30° upwards from the platform's
207 main deck (Folkner et al., 2018).
The InSight data set consists of 11,755 data points over 241 passes from November 27, 2018, to November 30, 2019, and provides a robust addition to the existing data from the previous three missions, extending the data arc from 35 to 43 years. The location of InSight based on RISE estimates is 4.49751°N by 135.6178693°E (Golombek et al., 2020). Note that in this paper we formally distinguish between the two unique locations of the east and west antennas, which are separated by a constant 1.55 meters.

## 2.5 Media Calibrations

Radio waves bend as they travel through the atmosphere. The refractive index (the ratio of the propagation speed in a vacuum to the speed of propagation in another medium) in Earth's troposphere imparts a significant signature to the propagation of radio waves and thus to the Doppler data. This refractive index is mainly related to common atmospheric quantities of pressure and temperature. After calibration of the data, the noise due to remaining water vapor is the dominant component and tends to be greater during Earth's daylight hours when atmospheric warming increases water vapor capacity (Asmar et al., 2005). The contribution from the dry troposphere is of a larger amount but is well determined from the DSN local pressure and temperature reporting. Another effect, one tenth in scale to that of the troposphere, is caused by Earth's ionosphere. Most of the ionosphere is ionized due to the large amount of solar radiation, which increases the amount of free electrons during the day. As radio waves enter Earth's ionosphere from space, some of the waves are absorbed by the electrons in the ionosphere by an amount inversely proportional to the frequency of the waves. Signals from global positioning system satellites are recorded each day at the DSN and are used to calibrate these effects at X-band frequency (Bar-Sever et al., 2007; Mannucci et al., 1998).
The thin troposphere of Mars has its own small effect on the Doppler measurements, well below the noise caused by water vapor fluctuations on Earth, but comparable to Earth ionosphere noise. For this analysis, we calibrate tropospheric noise at Mars with a simple model provided by Le Maistre (2020). The delay caused by the Mars troposphere is defined as 5 centimeters divided by sin (E) where E is the elevation angle of Earth as viewed by the lander. Although the effects are negligible with respect to the Mars precession (<1 mas/yr), they are nonetheless applied in this analysis. In the coming months with additional data, this calibration may be essential for extracting the liquid core contribution in the nutation signal.

Approximately one fifth as noisy as the Martian troposphere, the Martian ionosphere also has a small effect on the Doppler data (Lillis et al., 2010; Bergeot et al., 2019). Because this amount is close to the level of the Doppler noise, the Martian ionosphere is not calibrated for this study.

See Folkner et al. (2018) or Dehant et al. (2020) for other noise sources considered to be below the contribution from tropospheric water vapor which are disregarded for this study. There is however one exception, the plasma effects that can be quite high when the Sun-Earth-Probe (SEP) angle is close to zero (see below).

### 3. Doppler Noise

Prior to analysis of the InSight data, the first step is to average the data over a convenient time interval in a way that properly accounts for the various noise sources. The DSN receiver records the Doppler shift every 0.1 seconds. At short integration times, the dominant noise source is thermal noise due to limited signal-to-noise ratio in the radio link (Armstrong, 1998). At longer integration times, the dominant noise source is phase scintillation caused by wave propagation through media, especially Earth's troposphere. Based upon the data collected so far for InSight,
the average integration time where the transition from thermal noise to troposphere noise occurs
is around 20 seconds (for example, see Figure 2). Thus, the Doppler data have been integrated to
give phase versus time and the resulting phase is averaged over 20 seconds. Differences in
average phase divided by 20 seconds then gives Doppler data with negligible thermal noise.
These Doppler data are then averaged over 60 seconds to reduce the noise from troposphere
fluctuations. Thanks to their HGA steerable antennas, the previous three mission measurements
had higher signal-to-noise ratio than InSight, so the thermal noise was negligible and the 0.1-
second Doppler data were directly averaged over 60 seconds. Both noise sources are considered
white, having nearly Gaussian noise distributions. To fit Doppler data dominated by white noise,
the data are weighted by the root-mean-square (RMS) of the residuals (the difference between
the observed and expected Doppler shift).
<figure><img src="image_4.png" /><figcaption>Figure 2. Allan Deviation versus correlation time (TAU) based on Doppler residuals over a tracking pass</figcaption></figure>

In addition to the thermal and troposphere noise, the speed of radio propagation, and hence the Doppler measurement, is affected and sometimes dominated by phase scintillation from continual variations in the total electron content (solar plasma) between Mars and Earth. Because the amount of solar plasma is greatest near the Sun, closer position of the Mars-Earth ray path to the Sun (typically denoted as the SEP angle) is correlated with greater plasma noise. For Viking-1, because solar plasma noise is inversely proportional to the square of the radio frequency, the S-Band Doppler dataset is dominated by solar plasma noise, unlike the other landers that used X-band. The Pathfinder and Opportunity data noise were taken at large enough SEP angle so that the solar plasma noise was less than the troposphere noise. For InSight, most Doppler data noise was dominated by troposphere noise, except for ~90 days around September 2, 2019, (solar
conjunction) when the data noise was dominated by solar plasma. We define the cut-off point at SEP = 15°, where the solar plasma noise is greater than the thermal noise, which happens between July 16th and Oct. 18th 2019. During this period the noise becomes colored (Kolmogorov noise, see Folkner, 2017), meaning that subsequent data points become correlated, which is not suitable for proper application of the least-squares fitting method used by the orbit determination and analysis software. To account for charged particle content, Viking data and InSight data near solar conjunction are de-weighted from the RMS by a factor of 5.3 to account for the non-white power spectral density of the solar plasma noise for estimation of signatures with near-diurnal frequency (see Appendix A).

## 4 Analysis

Once compressed, the 60-second tracking measurements from all four missions are analyzed to solve for Mars rotation using the Jet Propulsion Laboratory (JPL) Mission analysis, Operations and Navigation Toolkit Environment (MONTE) program (Evans et al., 2018) and the Centre National d'Etudes Spatiales (CNES) Géodésie par Intégrations Numériques Simultanées (GINS) software (Marty et al., 2011), adapted and used by the Royal Observatory of Belgium (ROB) for planetary applications. Both programs are designed for orbital determination and trajectory optimization and analysis.

In order to compute the predicted Doppler (Eq. 1) to be fitted to the measurements, both Earth and Mars antenna positions must be expressed in the same reference frame, chosen to be the Solar System Barycenter J2000 for the sake of simplicity. Conversion to J2000 from Mars-fixed coordinates is accomplished by modeling of rotation, precession and obliquity rate, and nutation about the spin axis (Konopliv, 2006). See Yoder & Standish (1997) and Sears & Brehme (1968)
for discussion on the correction for relativistic effects. Rotation about the spin axis is expressed by angle $\varphi$ (from Eq. 1) and variations $\delta\varphi$ developed as a sum of four harmonics

$$ \delta\varphi = \sum_{j=1}^{4} (C_j \cos jl + S_j \sin jl) \qquad (2) $$

where $l$ is the orbital mean anomaly (Folkner et al, 1997). Its derivative $d\delta\varphi/dt = \omega$ provides the rate of rotation. The rotation about the pole $\varphi$ and rotation rate $\omega$, Mars' obliquity $\epsilon$ (the tilt of the spin axis relative to the orbital plane) and its rate $d\epsilon/dt$, the mean longitude of the node of intersection between the Mars' orbital and equatorial planes $\psi_0$ and the precession rate $d\psi/dt$, and the rotation constants from Equation 2 are calculated. See Folkner et al. (1997) for more information on determination of Mars' obliquity and mean orbit.

For a specified set of model parameters, considering what we know at present from theory or complementary observations (called the nominal model), MONTE and GINS determine a computed frequency based on the Doppler shift accounting for the known uplink reference frequency and turnaround ratio of the SDST, the motion of both Mars and Earth provided by the DE430 planetary ephemeris (Folkner et al., 2014), Earth orientation parameters provided by JPL's Time and Earth Motion Precision Observation group (for MONTE) and the International Earth Rotation and Reference Systems Service (IERS) Service of Earth Rotation (for GINS), and all of the tropospheric and ionospheric calibrations mentioned in Section 3. The difference between the computed frequency and the actual observed frequency (the residual frequency) is attributed to the difference between the nominal model and the actual Martian spin angle amplitude and spin axis direction. The determination of Mars' rotation and orientation then consists of adjustment to the model parameters in order to match the model predictions with the
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actual tracking observations. MONTE and GINS solve for new parameter values which will
drive residuals to zero mean for all four missions.

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As explained above, the current data set is presently too limited to determine new values for the periodic changes in Mars' orientation, and only precession and obliquity rates are estimated together with mean seasonal changes in Mars' rotation and with the lander positions (except for the longitude of Viking). For this paper, the longitude of Viking-1 was held fixed at -47.95137° as in Kuchynka et al. (2014) to define the Mars-fixed coordinate system. In doing so, the precision of the Viking lander longitude replaces the less certain position of the prime meridian defined as the center of Airy-O crater. The longitude and distance from the Martian axis of rotation (cylindrical radius) for Pathfinder, Opportunity, and each medium-gain antenna of InSight were estimated from the Doppler data, with the distance from the equator (z coordinate) determined by comparing the longitude and radius of each antenna with the topography determined by the Mars Observer Laser Altimeter (Le Maistre, 2016).

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Figure 3 shows the residual frequencies for each of the four missions, having been determined according to the above procedure. For Viking-1 (Figure 3A), the entirety of the observations are solar-plasma dominated, as indicated in Section 3. Thus, residuals are higher than for the subsequent three missions, which is reflected in the higher root-mean square error (RMS). The two shaded regions near solar conjunction are especially affected due to the low SEP angle. Both Pathfinder (Figure 3B) and Opportunity (Figure 3C) data were collected at high SEP angles, resulting in low residuals. Opportunity data have low noise, as evidenced by the standard deviation, since the 4 months of data were acquired near Mars opposition and limited by nighttime troposphere fluctuation. The InSight (Figure 3D) data taken shortly after landing were also near Mars opposition with noise similar to that of Opportunity data, but with noise increasing as
Mars moved into the daytime sky and then near solar conjunction. Only during solar conjunction (i.e. SEP < 15°) do the data become solar plasma limited, resulting in higher residuals. In all cases, the residuals have been brought to zero mean as described above.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Frequency residuals, root mean square error (RMS), and standard deviation (σ) in millimeters per second for (A) Viking-1 (July 21, 1976, to December 20, 1978); (B) Pathfinder (July 4, 1997, to September 25, 1997); (C) Opportunity (January 2, 2012, to May 4, 2012); (D) Insight (November 27,</figcaption></figure>
2018, to November 30, 2019) based on a model fit for all four landers. Shaded regions indicate SEP angle < 15° (not included in computation of RMS or σ).

Table 2 provides updated estimations from GINS of the model parameters described above, along with the conventional Mars mean orbit inclination and longitude of ascending node of 1980 (Konopliv et al., 2006) and the estimated position of the center point between the two InSight antennas. From the combination of lander data, we estimate a precession rate of -7605 ± 3 mas/yr. The results computed with MONTE (-7610 ± 3 mas/yr) and with GINS are within each other's error limits. The difference in these two values is expected from use of slightly different data weighting calibrations at the two institutions as opposed to a difference in the software.
<table><caption>**Table 2.** Mars rotation and orientation parameters adjusted to InSight, Opportunity, Pathfinder, and Viking-1 data. Uncertainties are 1-sigma uncertainty bounds.</caption><thead><tr><td>Mars Rotation Parameter</td><td>Value</td><td>Uncertainty</td></tr></thead><tbody><tr><td>Mars orbit inclination of 1980 J (°)</td><td>24.67682669</td><td>Fixed</td></tr><tr><td>Longitude of Mars orbit ascending node of 1980 N (°)</td><td>3.37919183</td><td>Fixed</td></tr><tr><td>Rotation about pole φ (°)</td><td>133.3861997</td><td>0.00001</td></tr><tr><td>Rotation rate ω (°/day)</td><td>350.8919825</td><td>0.0000002</td></tr><tr><td>Longitude of node at J2000 ψ<sub>0</sub> (°)</td><td>81.9683988</td><td>0.0000003</td></tr><tr><td>Precession rate dψ/dt (mas/yr)</td><td>-7605</td><td>3.0</td></tr><tr><td>Obliquity ε (°)</td><td>25.18938228</td><td>0.000001</td></tr><tr><td>Obliquity rate dε/dt (mas/yr)</td><td>-5.4</td><td>1</td></tr><tr><td>Annual term C1 in δφ (mas)</td><td>481</td><td>6</td></tr><tr><td>Semiannual term C2 in δφ (mas)</td><td>-117</td><td>5</td></tr><tr><td>Triannual term C3 in δφ (mas)</td><td>-19</td><td>4</td></tr><tr><td>Quarterly term C4 in δφ (mas)</td><td>-5</td><td>3.5</td></tr><tr><td>Annual term S1 in δφ (mas)</td><td>-144</td><td>6</td></tr><tr><td>Semiannual term S2 in δφ (mas)</td><td>-94</td><td>4.5</td></tr><tr><td>Triannual term S3 in δφ (mas)</td><td>-8</td><td>4.5</td></tr><tr><td>Quarterly term S4 in δφ (mas)</td><td>-3.5</td><td>4</td></tr><tr><td>InSight longitude (°)</td><td>135.6178</td><td>0.0003</td></tr><tr><td>InSight cylindrical radius (km)</td><td>3382.6129</td><td>0.0001</td></tr><tr><td>InSight z (km)</td><td>266.07</td><td>0.02</td></tr></tbody></table>

**5 Interpretation**

The precession rate $d\psi/dt$ is related to the polar moment of inertia $C$ by (e.g., Konopliv et al., 2011)

$$ \frac{d\psi}{dt} = -\frac{3}{2C} \frac{n_0^2}{\omega} (1 - e^2)^{-\frac{3}{2}} J_2 \cos(\varepsilon) + \dot{\psi}_g + \dot{\psi}_p \quad (3) $$

where $n_0 = 191.408°/yr$ is the mean motion of Mars (Konopliv et al., 2011), $\omega$ is the computed spin rate (Table 2), $e = 0.09341$ is the orbit eccentricity determined from radio range measurements to spacecraft orbiting Mars (Yoder, 1995), $J_2 = 0.00195660888 \pm 2.82e-10$ is the degree-2 unnormalized gravity coefficient estimated from Doppler tracking of spacecraft orbiting Mars (Konopliv et al., 2016), $\varepsilon$ is the computed obliquity (Table 2), $\dot{\psi}_g = 6.75$ mas/yr is the
geodetic precession (Baland et al., 2020), and $\dot{\psi}_p = -0.77$ mas/yr is the precession resulting from the torque of the other planets as well as from Phobos and Deimos (Baland et al., 2020). The obliquity $\epsilon$ is also varying in time but the changes are very small as shown in Table 2 and are of the second order in Equation 3.

The value of the normalized polar moment of inertia to one sigma is then

$$ \frac{c}{Mr_e^2} = 0.363615 \pm 0.000143 \qquad (4) $$

where $M = 6.41712 \times 10^{23}$ kg is the mass of Mars (GM = 42,828.4 km$^3$/s$^2$, Konopliv et al., 2016) and $r_e = 3396$ km is the equatorial radius derived from the Mars Observer Laser Altimeter (Smith et al., 2001).

By combining the polar moment of inertia $C$ with the accurately known constant part of the degree-2 order-0 gravity field $J_2$, the mean moment of inertia $I$ of Mars, which is invariant to all deformations lacking a spheroidal component of degree 0, can be determined from (Rochester and Smylie, 1974)

$$ \frac{I}{Mr_a^2} = \frac{A+B+C}{3Mr_a^2} = \left(\frac{C}{Mr_e^2} - \frac{2}{3}J_2\right) \frac{r_e^2}{r_a^2} = MOI = 0.363701 \pm 0.000143 \qquad (5) $$
where **A** and **B** are the two principal equatorial moments of inertia of Mars and $r_a = 3389.5$ km is its mean radius. The error bars associated with **C** (Eq. 4) and I (Eq. 5) have been computed by propagating uncertainties in Equation 3 parameters using a Monte Carlo procedure. Because of the high accuracy of Mars orbital motion (few meters accuracy), $n_o$, **e**, and $\dot{\psi}_g$ have been considered well-known as well as $\dot{\psi}_p$. Konopliv et al. (2011) computed normalized MOI as 0.3644, and Konopliv et al. (2016) updated the estimate to 0.3637, as displayed in Figure 4.

The MOI quantifies the radial mass distribution within the planet. It is mainly determined by the mass of the mantle and crust because the core has a smaller volume and is closer to the center of Mars. To constrain the radius and density of the core from the moment of inertia, prior assumptions about the composition and thermal state of the mantle as well as about the density and thickness of the crust are required. Figure 4 shows a particular range of interior models that fit the bulk parameters of Mars (e.g., mass and radius) while varying the radius of the core (Rivoldini et al., 2011). The models shown are further characterized by an assumed thickness and density of the crust, composition of the mantle, and mantle temperature profile (case 16, Plesa et al., 2016). The current value and uncertainty of the MOI are indicated. Improvements in the MOI determination narrow the range of possible core radii, but the range is strongly dependent on the assumed crust and mantle structure as is illustrated in the figure, which shows how it is affected by a crust thickness variation of 10 km and by the mantle temperature profile (case 22, Plesa et al., 2016).
<figure><img src="image_6.png" /></figure>

**Figure 4.** Relation between core radius and normalized moment of inertia for a set of plausible Mars mantle compositions assuming a fixed crust density ($\rho$) and thickness (d) and mantle temperature (case 16, Plesa et al., 2016). The acronyms stand for the different mantle mineralogy models (TA: Taylor (2013), d =95 km, $\rho$ = 2865 kg/m$^3$; EH45: Sanloup et al. (1999), d =70 km, $\rho$ = 2834 kg/m$^3$; LF: Lodders and Fegley (1997), d =70 km, $\rho$ = 2834 kg/m$^3$; MM: Mohapatra and Murty (2003), d =65 km, $\rho$ = 2826 kg/m$^3$. The red shaded area indicates the effect of a crust thickness variation of ±10 km on the crust thickness of the MM models and the dashed red line illustrates the effect of the mantle temperature (case 22, Plesa et al., 2016) on the MM models. The hatched area indicates the normalized moment of inertia value (0.3637) at 1 sigma of Konopliv et al. (2016) and the blue shaded area represents the normalized MOI value of this study.
416 **6 Conclusions**

417 With the combination of data from InSight and the three previous Mars lander missions, we have
418 estimated a Mars precession rate of -7605 ± 3 mas/yr. This estimate is in close agreement with
419 Kuchynka et al.'s (2014) results from combined lander and orbiter data of -7606.1 ± 3.5 mas/yr.
420 While our solution is significantly different from Kuchynka et al.'s (2014) lander-only estimate
421 of -7619 ± 6.4 mas/yr, it is consistent with the more recent results from Konopliv et al. (2016)
422 with lander tracking only of -7611.4 ± 7.3 mas/yr and with lander and orbiter tracking of -7608.3
423 ± 2.1 mas/yr (see Figure 5).

424 The extra years of coverage from orbiters in addition to landers in Konopliv et al. (2016) gives
425 slightly smaller uncertainty but has not been done here since inter-annual variations in seasonal
426 rotation effects are difficult to quantify. For example, seasonal atmospheric changes related to
427 sublimation and condensation of Mars' polar ice caps cause seasonal changes in the Martian
428 rotation rate observed in the Viking-1 and Mars orbiter Doppler data (Konopliv et al., 2011).
429 Therefore, while seasonal rotation is driven primarily by the Viking-1 data (over ~1 Martian
430 year) for our lander-only estimate, it is also affected by the 2002-2016 time span for Konopliv et
431 al.'s (2016) solution with orbiters. In addition, General Circulation Models of the atmosphere of
432 Mars show that seasonal changes in rotation rate depend significantly on the presence or not of
433 dust storms (Van den Acker et al., 2002), indicating a large interannual variability (at the 10%
434 level). With an additional year of InSight data, the inter-annual variation will be much better
435 characterized.
<figure><img src="image_7.png" /><figcaption>Figure 5. Mars precession rate and 1-sigma uncertainty based on Viking-1 and Pathfinder (“2 Landers (1997)”, Folkner et al., 1997); Viking-1, Pathfinder, and Opportunity (“3 Landers (2014)”, Kuchynka et al., 2014); Viking-1, Pathfinder, Opportunity, Mars Global Surveyor orbiter, Odyssey orbiter, and Mars Reconnaissance Orbiter (“3 Landers + 3 Orbiters (2014)”, Kuchynka et al., 2014); Viking-1, Pathfinder, and Opportunity (“3 Landers (2016)”, Konopliv et al., 2016); Viking-1, Pathfinder, Opportunity, Mars Global Surveyor orbiter, Odyssey orbiter, and Mars Reconnaissance Orbiter (“3 Landers + 3 Orbiters (2016)”, Konopliv et al., 2016); and Viking-1, Pathfinder, Opportunity, and InSight combined (“4 Landers (2020)”).</figcaption></figure>

Our estimate of precession is used to compute the mean MOI as 0.363701±0.000143. This value
constrains the radius of the core, dependent on models for the crust and mantle thickness.

Additional measurements are needed beyond improvements in precession rate to increase our
knowledge about the interior structure. With the addition of a second year of InSight radio

Doppler data, the RISE dataset will cover a sufficiently long time span to separate the seasonal
rotation rate variations from the nutation of the spin axis direction, allowing for determination of
the free-core nutation period and the core moment of inertia. In addition, the continuous
sounding of seismic events by the InSight SEIS instrument (Lognonné et al., 2019) will
accurately determine the crust thickness and constrain the elastic and thermal properties of the
mantle. By combining this knowledge with the accurate moment of inertia from precession,
measured tides, and the nutation measurements of RISE, our knowledge about the core radius
and density will significantly improve.

## Appendix A

Most orbit-determination software such as used here for estimation of Martian rotation assumes
that the data points are uncorrelated with time. This is correct for Doppler measurements for
which the Earth troposphere fluctuations are the dominant noise; in this case the optimal data
weighting is the inverse of the root-mean-square (RMS) of the measurement residuals.

For Doppler data with dominant noise limited by solar plasma, the Doppler measurements are
not uncorrelated. The optimum least-squares estimate would use a data weighting matrix that
includes the correlated measurement noise. Since the signature of Martian rotation is largely at
the time scale of the Martian day, a near-optimal estimate can be derived by treating the
measurements as uncorrelated but reducing the data weight by a multiple of the RMS to give the
correct power spectral index of the data noise at the diurnal time scale.

For Doppler data with noise dominated by solar plasma fluctuations, the noise power spectral
density $S_{s,p}$ has the form

$$S_{s,p}(f) = h_{s,p} f^{2/3} \qquad (6)$$
where $f$ is the signal frequency and $h_{s,p}$ is amplitude. For uncorrelated data the noise power spectral density $S_w$ is white (independent of frequency) and has the form

$$S_w(f) = h_w f^0 \qquad (7)$$

For a given power spectrum $S$, the variance of the measurement residuals at time interval $\tau$ is given by

$$\sigma_{rms}^2(\tau) = \int_0^\infty S(f)[\sin(\pi \int \tau) / (\pi \int \tau)]^2 df \qquad (8)$$

For the white spectrum this gives

$$\sigma_{rms}^2(\tau) = h_w / (2\tau) \qquad (9)$$

For the solar plasma Doppler spectrum this gives

$$\sigma_{rms}^2(\tau) = h_{s,p} (\pi \tau)^{-1/3} \quad 3.35 \qquad (10)$$

From Equations 9 and 10, we see that the white spectrum would underestimate the solar plasma noise at time scales longer than the interval between data points. Assuming that to fit a spectral signature at $f = 1/(T)$, e.g. $T = 88620$ seconds, the inferred white power spectrum should be scaled up to match the solar power spectrum at that frequency, we define a scaled white power spectrum amplitude by effectively assuming an increase in RMS residuals by a factor $F$:

$$h'_{w} = F^2 \sigma_{rms}^2(\tau) 2\tau = h_{s,p} f^{2/3} \qquad (11)$$

with the scale factor $F$ given by
$$F^2 = \frac{\pi^3}{2 \times 3.35} \left(\frac{T}{\tau}\right)^{2/3} = \left[0.468 \left(\frac{T}{\tau}\right)^{1/3}\right]^2 \quad (12)$$

Applying $T = 88620$ seconds per Martian day and $\tau = 60$-seconds count time,

$$F = [0.468(88620/60)^{1/3}] = 5.33. \quad (13)$$

## Data Availability

ASCII versions of the Doppler data for Viking and Pathfinder are in the REDUCED directories at https://pds-geosciences.wustl.edu/missions/mpf/radioscience.html. Opportunity raw tracking data can be found at: https://pds-geosciences.wustl.edu/mer/mer1-m-rss-1-edr-v1/mer1rs_0002/. InSight tracking data are archived at https://pds-geosciences.wustl.edu/missions/insight/rise.htm.

## Acknowledgements

This paper is InSight Contribution Number 165. This work was partially carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. © 2019. All rights reserved.

For the Belgian authors, this work was financially supported by the Belgian PRODEX program managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office.

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