Deriving Mercury Geodetic Parameters With Altimetric Crossovers From the Mercury Laser Altimeter (MLA)

Abstract

Based on the previous applications of laser altimetry to planetary geodesy at GSFC, we use the recently developed PyXover software package to analyze altimetric crossovers from the Mercury Laser Altimeter (MLA). Using PyXover, we place new constraints on Mercury's geodetic parameters via least squares minimization of crossover discrepancies. We simultaneously solve for orbital corrections for each MLA ground track, for the geodetic parameters of the International Astronomical Union-recommended orientation model for Mercury (pole right-ascension and declination coordinates, prime meridian rotation rate, and librations), and for the Mercury's Love number h₂. We calibrate the formal errors of our solution based on closed-loop simulations and on the level of robustness against a priori values, data selection, and parametrization. Our solution of the Mercury's rotational parameters is consistent with published values. In particular, our new estimate for the orientation of the pole places Mercury in a Cassini state, with an obliquity ϵ = 2.031 ± 0.03 arcmin compatible with previous “surface” related measurements. Moreover, we provide a first data-based estimate of the Love number h₂ = 1.55 ± 0.65. The latter is consistent with expectations from models of Mercury's interior, although its precision does not enable their refinement.

Key Points

• We provide an independent solution for Mercury's orientation parameters based on the analysis of the Mercury Laser Altimeter crossovers

• Our solution places Mercury in a Cassini state with an obliquity ε = 2.03 ± 0.03, larger than the recent gravity-based estimate

• We provide a first estimate of Mercury's tidal Love number h₂ to be in the range from 0.9 to 2.2

Plain Language Summary

Measuring the orientation of bodies in space is one of the few means we have to learn about their internal structure. We analyze Mercury's orientation from distance measurements between the planet's surface and the MESSENGER probe, acquired with laser pulses from orbit around Mercury between 2011 and 2015. In particular, we use observations of the same surface locations at different times, called crossovers. Any difference in the measured elevation at these crossover points results either from an error in MESSENGER's estimated position in space or from an error in the assumed orientation of Mercury. Based on these differences, we make corrections to both MESSENGER's trajectory and to the pole position, rotation rate, and oscillations of Mercury. Tides raised on Mercury by the Sun are also expected to periodically vary the surface elevation by more than 2 m. Since these tidal effects are also expressed as elevation differences at the crossovers, our analysis provides a first measurement of their amplitude. Our updates to Mercury's orientation and tidal response bring important information about its internal structure, such as the size of its core and its internal level of differentiation.

1 Introduction

Mercury is one of the most interesting objects in the Solar System, still challenging our understanding of planetary formation and evolution with its high density, unexpected magnetic field, and the 3:2 resonance between its rotational and orbital periods.

After the early flybys by Mariner 10 in the 1970s, the MESSENGER spacecraft (MErcury Surface, Space ENvironment, GEochemistry, and Ranging; Solomon et al., 2008) executed three equatorial flybys of Mercury in 2008–2009 before entering a highly elliptical, near-polar orbit from March 2011 to April 2015. Mercury's orientation and rotation have been studied by a variety of techniques, as they have implications for the moment of inertia of the outer solid shell and thus its mass distribution, internal structure, and thermal evolution of Mercury (e.g., Genova et al., 2019; Margot et al., 2012; Phillips et al., 2018). Already before MESSENGER, Margot (2009) used ground-based radar observations to develop early orientation models. Several independent confirmations and refinements of Mercury's rotational parameters followed, based on a variety of techniques using multiple MESSENGER data sets. In particular, Mazarico et al. (2014), Verma and Margot (2016), Genova et al. (2019), and Konopliv et al. (2020) all analyzed the radio tracking data of the MESSENGER spacecraft, with different approaches; Stark et al. (2015) co-registered altimetry from the Mercury Laser Altimeter (MLA, Cavanaugh et al., 2007) and shape models derived from the Mercury Dual Imaging System camera images. Solutions for most rotational parameters agree within provided error bars (with a wide range of magnitudes), yet significant differences are present between recent estimates of both the orientation of the pole and Mercury's spin rate.

While several orbit determination (OD)-based studies have provided estimates of Mercury's tidal Love number k₂ (Genova et al., 2019; Mazarico et al., 2014; Verma & Margot, 2016), no data-based solution for the vertical Love number h₂ has been produced to date, mainly because of the small expected signal (a maximum vertical deformation of <2.5 m at the equator and 50 cm at the poles for h₂ = 1), because of the poor knowledge of small scale topography required to use direct altimetry analysis (Thor et al., 2020), and because of MESSENGER's orbital configuration, which strongly limits the density of altimetry crossovers at latitudes <30°N.

MLA collected measurements of surface height during ∼3,200 periapsis passes over Mercury's northern hemisphere. Where two MLA groundtracks intersect, we get a so-called crossover point. A crossover is thus a differential measurement between two distinct observations of the same surface location at two different times. Any difference in height at the crossover point, referred to as its discrepancy v, is thus mainly due to the following effects: (1) errors in the spacecraft orbit and attitude, or small variations (due to thermal deformations or other environmental conditions) to MLA fixed boresight orientation, (2) interpolation errors of the surface topography between MLA footprints, and (3) geophysical signal, for example, due to mismodeled time-varying planetary rotation or to tidal vertical motions. Although crossovers require a complex processing pipeline, they are a powerful tool to explore the state of planetary bodies (Mazarico et al., 2014; Rosat et al., 2008; Rowlands et al., 1999) and provide an opportunity to measure Mercury's orientation and rotation. In our study, we provide an independent solution based on the application of this technique to MLA crossovers with the in-house PyXover code (Bertone et al., 2020), that we recently developed for this analysis. The resulting discrepancies v constitute the observation residuals to be minimized in the least squares (LS) procedure, involving the simultaneous adjustments of MESSENGER orbit corrections and Mercury's geodetic parameters. Within an iterative procedure, we solve for four of the orientation parameters of the model recommended by the International Astronomical Union (IAU, Archinal et al., 2018), that is, right ascension (RA) and declination (DEC) of the spin pole at J2000, spin rate (ω), and a scale factor for the librations amplitude (L) of Mercury, as well as for the degree-2 tidal radial response h₂.

This study is structured as follows. In Section 2, we present our reference data set and the auxiliary data used for this study. Details about the data weighting and solution strategy are provided in Section 3. Finally, our solution and error calibration for Mercury's orientation and tidal parameters based on MLA crossover analysis are presented in Section 4 and discussed in Section 5. Throughout the text, we use small bold letters to denote vector quantities, and capital bold letters for matrices.

2 Data Description, Modeling, and Parametrization

The MESSENGER spacecraft orbited Mercury between 2011 and 2015 in a highly elliptical, near-polar orbit with a periapsis of ∼200–400 km, an apoapsis between ∼15,000 and 20,000 km, and an orbital period of 12 h initially and reduced to 8 h after 1 year. The spacecraft was within ranging distance for the onboard MLA instrument over 15–45 min periods near periapsis, typically at Northern latitudes. MLA collected over 22 million measurements of surface height with a vertical precision of ∼1 m and an accuracy of ∼10 m (Zuber et al., 2012).

Because of the elliptical orbit, the laser spot diameter on the surface varies between ∼10 and 100 m. The inter-spot distance is then ∼350–450 m, so that the average distance between each crossover and its bracketing spots is usually ∼200 m (Zuber et al., 2012). The total MLA data set contains ∼3,200 tracks and ∼3 million crossovers, geographically distributed as shown in Figure 1 (bottom-right). These crossovers represent repeated measurements of the same surface locations, such that any difference between the elevation measured along the two profiles results from an error either in the orbit and attitude reconstruction, or in the a priori knowledge of the planetary rotation and tidal response. Figure 1 shows the partial derivatives of MLA crossovers, and hence their sensitivity to the parameters of interest as a function of their geographical location on the surface of Mercury.

Figure 1

North-polar stereographic map of MLA crossovers sensitivity to several rotational and tidal Mercury's parameters. Bottom-right: pre-fit crossovers discrepancies v on Mercury surface.

From the MLA data set available on the NASA Planetary Data System, we extract the laser pulse emission time in Barycentric Dynamical Time (TDB, Soffel et al., 2003) and the Time of Flight (TOF) of the signal, along with the channel associated with each measurement. Data with a “channel” value >4 include an elevated level of noise and are thus excluded from our analysis. If multiple data points within the nominal 10 Hz sampling rate are available, we only include the one with the lowest channel value, that is, the most reliable.

Our processing, detailed in Section 3, also requires a reference orbit and attitude for the spacecraft carrying the altimeter. We mostly refer to the MESSENGER orbits reconstructed by KinetX based on radio tracking by the Deep Space Network antennas and on the spacecraft attitude provided by on-board star-trackers. Both are available as NAIF/Spice (Acton et al., 2018) kernels on the NASA PDS, where the telemetered attitude has already been corrected for aberration effects. We process these kernels via the SpiceyPy wrapper for Python (Annex et al., 2020). In Section 4.4, we also perform our analysis on MESSENGER orbits based on the Genova et al. (2019) processing baseline, in order to quantify the independence of our solution from a priori orbits and to a more robust estimate.

We model the resulting crossover discrepancies v as a function of errors in the a priori orbit, as well as of deviations from the IAU rotational model (Archinal et al., 2018) and as mismodeling of tidal deformations. These constitute our estimated parameters vector q. Orbital parameters include corrections to the a priori orbit which can be modeled as a constant offset estimated once for each track in every direction of the orbital frame: along-track A, cross-track C, and radial R. In addition, attitude (roll and pitch) biases and time-dependent corrections (e.g., linear or quadratic) could be estimated for each track within PyXover, but we do not want to over-parametrize our solution. Additional geodetic parameters characterize the tidal deformation of Mercury and its orientation in space, and enter the geolocation of the MLA groundtracks via the transformation from the inertial frame (in which the MESSENGER orbits are provided) to the Mercury-fixed frame (where MLA groundtracks need to be rotated to form crossovers). Following the IAU formalism (Archinal et al., 2018), we parameterize the orientation of Mercury by the right ascension (RA or α) and declination (DEC or δ) of Mercury's pole at J2000 (their secular trends are fixed to their nominal IAU values). The planetary prime meridian (PM) direction is also modeled as a quadratic function of time (since J2000). In the following, we indicate by ω and estimate exclusively the linear term of this series, that is, the spin rate. On top of this, we consider the longitudinal libration (L), that is, the sum of all the terms at different periods from Margot (2009). We then estimate corrections to the pole orientation at J2000, to Mercury's spin rate, and a scaling factor for Mercury's librations.

To model the solid tidal displacement u_r at Mercury surface, we use a solid tide model based on the degree 2 potential terms exerted by the Sun (e.g., Van Hoolst & Jacobs, 2003), so that

(1)

where h₂ is the Love number of degree 2, g is the gravitational acceleration at the surface, and

(2)

is the tidal potential caused by the Sun at a point on Mercury surface with coordinates r, with G the universal gravity constant, M the mass of the Sun, R the distance between the centers of mass of the Sun and Mercury, and ψ the angle between the Mercury-centric directions of the Sun and of the point considered.

3 Processing and Solution Strategy

We perform the analysis of MLA crossovers within the recently developed PyXover python package (Bertone et al., 2020), whose modular structure is sketched in Figure 2.

Figure 2

Workflow of the PyXover code: (1) geolocation of altimetry data, (2) crossovers location, and (3) setup of observation equations, QR-filter solution for a chosen set of parameters, with given weights and constraints.

The crossover analysis can be divided into three main steps. First, laser altimetry ranges are geolocated to the planetary surface (i.e., we assign a set of latitude, longitude, and elevation in the planet frame to each MLA shot) and partial derivatives of the groundtracks are computed with the chosen set of parameters q by finite differencing. Initial geolocation is based on a set of reference orbit solutions for MESSENGER (see Section 4) and on a priori knowledge of Mercury's orientation (e.g., Archinal et al., 2018). Tidal deformations are modeled according to Van Hoolst and Jacobs (2003) with the a priori value for h₂ set to 0.

Second, intersections between the tracks are identified and characterized. The horizontal coordinates of the crossover points are first computed in a local North polar stereographic projection, to provide an approximate location of all possible intersections. For computational reasons, the tracks are subsampled to a ratio of 1:4, that is, looking for intersections between straight lines connecting MLA measurements ∼ 1,200 m from each other. This (time consuming) step is performed only once to locate the horizontal coordinates of potential crossovers. Subsequent iterations only refine crossover coordinates based on results from previous iterations. Short track segments constituted of the (fully sampled) four MLA observations involved in a potential crossover are thus analyzed. We reproject the coordinates of each subtrack around the preliminary crossover coordinates and fine-tune them. To finally compute the elevation discrepancy

(3)

for all confirmed crossovers, we interpolate MLA-derived elevations along each track A and B using cubic splines to determine elevations η_A and η_B at the refined crossover coordinates (x₀, y₀). The discrepancies vector v constitutes the residuals to be minimized in the LS optimization process. Moreover, we associate each measurement v with a weight according to its reliability, following criteria detailed in Section 3.2.

The corrections δq resulting from the LS inversion detailed in Section 3.3 are then applied to parameters values from previous iteration q_i, so that q_i+1 = q_i + δq. Updated orbits and geodetic parameters constitute the basis for the following iteration, including a new geolocation of the MLA data, the fine determination of new crossovers tri-dimensional coordinates and of a new residual vector v and of the associated partial derivatives. We set the following criteria for convergence: first, when the root mean square error (RMSE) of residuals stabilizes within 5%, we fix the weighting of observations to the latest evaluation and we start estimating h₂, which is initially held fixed to 0 because of its correlation with orbital errors; then, we further iterate with fixed observation weights until the relative improvement of residuals RMSE falls below 1% and corrections for global parameters are lower than their formal errors at 3σ. This usually happens within <10 iterations. The choice of different convergence criteria would impact the rate of convergence rather than the final solution.

3.1 Computation of the Crossover Partial Derivatives

From Equation 3, we obtain the partial derivatives of each discrepancy v at intersection of tracks A and B with respect to a parameter q belonging to q as

(4)

By expanding Equation 3, we also obtain

(5)

with λ_X the longitude, ϕ_X the co-latitude, and η_X the elevation of a measurement from track X, while q is the vector of the solved-for orbital and geodetic parameters, so that

(6)

During the geolocation phase, we compute the partial derivatives , and numerically, by finite differencing of the ground location of individual MLA shots. Derivatives with respect to h₂ are an exception. Indeed, based on Equation 1, and considering that η(r, t) = η₀(r) + u_r(r, t), the analytical expression of is straightforward. We also obtain updated epochs for the intersection of the laser pulses with the surface for tracks perturbed with respect to each parameter, in order to compute the accurate planetary state at bounce. We get “perturbed groundtracks” (λ,ϕ)_q by linear extrapolation as

(7)

from the nominal groundtrack (λ,ϕ)₀ using an appropriate increment Δq. Based on these perturbed tracks, we locate perturbed crossover coordinates and further correct the elevation by . For each track X, we finally compute Equation 6 numerically by

(8)

where ± indicate the elevation at crossover points of “positively and negatively” perturbed tracks. We are then able to fully compute Equation 4 and thus populate the partials (or first design) matrix A.

3.2 Data Weighting

The quality of the crossovers included in the analysis can affect the estimation of our results. Instead of removing poor pseudo-measurements, we associate a weight to each crossover based on several factors determining its reliability: belonging to a “well behaved” MLA track, being close to neighboring MLA measurements, belonging to close-to-nadir measurements, and not having an unreasonably large a priori discrepancy. This helps the stability of the LS solution by maintaining a uniform data set among iterations.

First, we evaluate the quality of OD for each MLA track involved in our analysis. For each of the N_τ = 3,200 tracks, we analyze the residuals of all N_w(i_τ) crossovers resulting from intersections of track i_τ with the remaining tracks over the whole MESSENGER mission. The average bias of the resulting time series (preemptively screened for large outliers) enable the evaluation of the quality of each track. Figure 3 shows examples of a “good” track (left), where the rather noisy residuals are centered around 0, and of a “bad” track (right), where residuals are globally biased. The resulting error σ_τ associated to each track is then propagated to a full covariance matrix at the crossover level, by setting

(9)

where Σ_τ is a (N_τ × N_τ) covariance matrix containing σ_τ for each track on the main diagonal and zeros elsewhere, while A_τ is the (N_w × N_τ) transfer matrix between tracks and crossovers, where for each observation column j, A_τj = ±1 if the track τ_j intervenes in the crossover, else A_τj = 0. We obtain the related weight matrix by taking the element-wise inverse of Σ_wτ. The resulting weight matrix is clearly nondiagonal. This procedure identifies and down-weights crossovers carrying erroneous information from one of the parent tracks. As shown in Figure 3 (right), an unreliable track also includes crossovers with v∼0, which could degrade the solution if included in the analysis by a less sophisticated screening.

Figure 3

Examples of (a) good and (b) bad tracks (RMSE in meters vs. time in seconds along the track, red dashed lines at ±RMSE). Using this criterion, also crossovers with v∼0 but involving track (b) will be significantly down-weighted.

The second main source of error in v is the interpolation noise. The use of altimetry crossovers considerably reduces the reliance of our analysis on the Digital Terrain Model (DTM) quality. Still, because of the finite MLA sampling of 10 Hz and the high orbital velocity of the spacecraft around periapsis (∼4.3 km/s), chances that the crossover location coincides with an altimetric measurement are low. Because of the limited knowledge of Mercury topography at baselines relevant for our analysis (i.e., from the 20 m laser footprint to the 400 m of average separation, see Zuber et al., 2012), we use a cubic spline to interpolate elevation profiles from the bracketing track points to the crossover location. This operation introduces an additional error σ_wι, which in principle depends on both the separation and the terrain roughness (i.e., the interpolation error will be lower on a smooth plain than in a rough area). In this study, however, we use an average terrain roughness of 100 m/km² based on Kreslavsky et al. (2014), as detailed roughness maps are not available at latitudes <65°N. For each crossover, we compute the average of the minimal separation of each profile and use it as reference baseline for the observation. We extrapolate the regional roughness at this baseline using the spectral power of Mercury's surface as derived from the stereo DTM data provided by Steinbrügge, Stark, et al. (2018). We consider this roughness at separation as an indicator of the relative interpolation error σ_wι between crossovers, and the associated weight matrix to have a value of 1/σ_wι on the main diagonal.

On top of this, Huber weighting (defined as if , wt = 1 otherwise) is then applied to each crossover according to its off-nadir angle (, q = 1), while crossovers with abnormally large residuals (v > 50 m, q = 1) are strongly down-weighted.

All weight components are then multiplied for each crossover observation to get the final weight matrix W to be associated with residuals v and partial derivatives in A used in computing the LS solution.

3.3 Solution Strategy

Given the partial derivatives in A, we estimate corrections δq to the parameter vector q by minimizing the RMSE of the measurement residuals vector v so that

(10)

where values for all quantities have to be intended at iteration i, except for q₀ indicating the a priori value of the estimated parameters.

Weak constraints P are applied, mainly to contain the impact of correlations between orbit and geodetic parameters. We use ridge regression (Tikhonov et al., 1998) to penalize statistically large variations for orbit parameters and deviations of the average orbital corrections in each direction (from an expectation of 0). This helps improve correlations between, for example, an offset in the determination of the spin and a solid rotation of all (quasi-) polar MESSENGER orbits in the cross-track direction. We use variance component estimation (VCE, Kusche, 2003) to determine the optimal weights between observations and constraints and to both stabilize the solution and get more realistic error estimates. Following Lemoine et al. (2013), we define the VCE determined weight by

(11)

for observations and

(12)

for constraints, and with the constrained normal matrix N = A^TWA + P. In particular, we compute two separate weights for parameter constraints and for constraints acting on average values of orbit corrections, so that

(13)

where Λ is a diagonal matrix having as elements

(14)

λ is a vector the size of q, manually set to constrain parameters with respect to each other (based on the reliability of prior knowledge and on preliminary simulations), P_q is a diagonal square matrix with the size of the total number of parameters and values 0 or 1, depending on the corresponding parameter in the solution vector p, and

(15)

with the identity and the unary matrix, respectively. This procedure stabilizes our solution, by providing an optimal balance between amplitude of the solution vector δq and the minimization of the residuals vector v.

By introducing the Cholesky square root of P on both sides of the observation equations, we finally set-up the square-root information filter (Bierman, 1977) solution algorithm, as depicted in Figure 2. Given the size of the A matrix (i.e., up to 3 × 10⁶ observations/rows per ∼8,000 parameters/columns) and its low density, we use sparse algebra operations provided by the SciPy library (Virtanen et al., 2020) to efficiently perform the required matrix manipulations.

4 Iterative Solution and Error Assessment

We perform an iterative weighted LS solution of orbit corrections and geodetic parameters based on the processing setup presented in Section 3. We base our solution on a set of 10⁶ crossovers selected according to their computed weight (i.e., their quality) and to their balanced geographical distribution. The quality threshold is thus higher above 60°N latitude, while at low latitudes only the worst 20% of the crossovers are excluded, given the latter are more sensitive to parameters of interest. We show in Figure 4 the distribution of the weights as a function of discrepancies v and of the separation to the bracketing points.

Figure 4

Log-log representation of the weights assigned to each crossover point as function of its discrepancy. The color scale shows the average minimal separation between the crossover and the neighboring observations. Huber weighting ensures a sharp cutoff for crossovers with v > 100 m. One can see that small values of v do not ensure a high weight. Also, most observations with high separation show large residuals and low weights.

We use KinetX-recovered MESSENGER orbits and the IAU orientation models (Archinal et al., 2018) recommended for Mercury as a priori. Following the procedure sketched in Section 3, we estimate corrections for orbit and geodetic parameters until convergence is reached (see Figure 5). Typical orbital corrections are in the order of 50–100 m for “per-track” biases in along- and cross-track directions and 20 m in the radial direction, as shown in Figure 6.

Figure 5

Convergence of residuals RMSE and geodetic parameter corrections for a reference iterated solution. The y-scales indicate, for each iteration, the parameter improvements in units of the associated formal errors (red, 3σ) and, bottom-right, the percentage RMSE change (blue-red dashed lines at 5%–1%, respectively).

Figure 6

Orbit corrections at convergence, parametrized as biases in MESSENGER orbital frame (radial, along-, and cross-track) estimated for each MLA track (i.e., once per orbit). Larger corrections in the along- and cross-track indicate a lower sensitivity of both radio-science and crossovers to these components. A few larger outliers up to several hundred meters have been removed to enhance visualization.

In order to assess the quality of the obtained solution, we check several factors. As shown in Figure 7, the distribution and RMSE of post-fit crossover discrepancies significantly improves, as expected. Also, we check that individual MESSENGER tracks benefit from our estimated corrections, by comparing pre- and postfit distributions. Our iterated solution results in significant improvements on the base of all the above criteria.

Figure 7

Pre- (FWMH = 35 m) and post-fit (FWMH = 24 m) assessment of discrepancies residuals (left) and improvement in the distribution of tracks quality, evaluated by the average bias of their crossovers (right).

Formal errors resulting from LS and VCE notoriously neglect systematic errors intervening in the solution. In Sections 4.1-4.4, we thus analyze several possible sources of systematic errors, that is, the a priori chosen for MESSENGER trajectory and the Mercury's rotational state, data selection and other intrinsic biases in our crossover analysis (which we evaluate by processing a simulated MLA data set). The resulting error budget is summarized in Table 1, while our final solution including calibrated error bars is shown in Figure 11. Correlations between these parameters are <0.3 when using the whole MLA crossovers data set, while only ∼3% of all orbit parameters have correlations >0.9, mainly between along-track and radial corrections estimated for the same track.

Table 1. Summary of Solutions, Statistical, and Systematic Sources of Error for Our Crossovers Analysis

Parameter	Solution	Formal	Systematic	A priori	Subset	Intrinsic
RA (°)	281.0093	5.40 × 10^−5	5.8 × 10^−4	5.30 × 10^−4	5 × 10^−5	1.50 × 10^−5
DEC (°)	61.4153	2.80 × 10^−5	4.60 × 10^−4	3.80 × 10^−4	8.50 × 10^−5	7 × 10^−6
ω (°/d)	6.138510	1.50 × 10^−7	2.70 × 10^−6	2.70 × 10^−6	4 × 10^−8	8. × 10^−8
L (as)	39.03	0.2	0.9	0.7	0.15	0.04
h2	1.55	0.3	0.35	0.2	0.08	0.1

• Note. Least squares provided formal errors are scaled by a factor of 3 to provide a more robust range of parameter values, while the “systematic” column is the sum of: influence of a priori values, data selection, and other intrinsic biases in our analysis.

4.1 Influence of A Priori MESSENGER Orbit and Rotational Parameters

We compute solutions based on different Doppler orbit reconstructions (KinetX and Genova et al., 2018) and from both IAU (Archinal et al., 2018) and Genova et al. (2019) values for Mercury's rotational parameters. We analyze the four possible combinations and compare parameter solutions at convergence in Figure 8. Clustering is visible for most solutions. The solution shown in Table 1 and in Figure 11 is the weighted average of these solutions according to their respective formal errors. We use the statistical dispersion of these solutions to evaluate the systematic error introduced by the choice of a priori values and to update formal error bars, as summarized in column “a priori” of Table 1.

Figure 8

Comparison of our solutions for [RA, DEC, ω, L, h₂], based on a set of 10⁶ crossovers, and using the same parametrization and data selection criteria, but on different combinations of MESSENGER orbit reconstructions (KinetX and Genova et al., 2018) and rotational parameters, that is, IAU, Archinal et al. (2018) and Genova et al. (2019), as a priori values (green and red triangles). The dispersion of the converged solutions (colored dots) is used to evaluate the systematic error introduced by the choice of a priori.

Orbits derived from Genova et al. (2019) show a lower consistency with MLA crossovers than KinetX orbits, possibly due to the chosen minimal parametrization with empirical terms. Hence, we first estimate a priori offsets for the spacecraft positions to get a refined a priori geolocated track for the iterative crossover analysis. In particular, we use the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) numerical optimization method (Jorge Nocedal, 2006) to minimize differences of MLA measured elevations to Mercury's DTM, which reduces crossover residuals to a level close to the one obtained from KinetX orbits.

4.2 Influence of Data Sampling

To analyze the impact of data sampling on our solution, we construct 10 different random subsets of 5 × 10⁵ crossovers out of the full MLA data set (after removing 10% of data with the lowest quality). We choose a stratified resampling without replacement, in order to retain the latitudinal distribution of the original data set. Common data among any pair of subsets do not exceed 20%. We compute a fully iterated solution for each subset and measure their dispersion.

The dispersion of most solutions falls well within the formal error bars provided by the LS (the dispersion of L and h₂ are comparable with formal error bars), and we conclude that our solution is robust with respect to an arbitrary selection of MLA measurements and resulting crossovers. Statistical results of this analysis are summarized in column “subset” of Table 1.

4.3 Influence of Orbit Constraints

As discussed in Section 3.3, we apply VCE to identify optimal relative weights for data (i.e., crossovers discrepancies) and parameter constraints. We get and relative to the unweighted data. By choice, constraints are only acting on orbital ACR corrections (full value and average over the whole mission), while geodetic parameters are freely estimated.

We found that constraint , acting on the global average of estimated ACR corrections (which is expected to be close to 0 as the a priori dynamic solution is known to be unbiased on the whole), is the main factor to consider and explored the impact of a wide range of values. Figure 9 shows the variation of the global three-dimensional mean (green) and RMS (red) of orbital corrections over the whole mission, as a function of and of the crossovers RMSE. As expected, the crossover fits (x-axis) degrade with tighter constraints. Different mean values correspond to a global shift of ACR corrections, rather than to isolated outliers (as verified with median values and visual inspection). We use this representation to perform an L-curve analysis (Hansen, 1999) based on the (green) means versus crossovers RMSE curve, and get a weight , close to the one suggested by VCE. The total RMS of corrections (red) shows the orbit variations allowed by our current parametrization and weighting, consistently with Figure 6.

Figure 9

Influence of the relative weighting of data and constraints on estimated orbit corrections (top) and tidal parameter h₂ (bottom). Each point corresponds to the solution of a subset of 500,000 crossover discrepancies with different constraint applied on the averages of ACR orbit corrections computed over the whole mission. Resulting averages of 3-D corrections (green) are used to validate the VCE-based weighting via an L-curve analysis; the resulting total RMS (red) of corrections is also shown for reference. Relative constraints favored by VCE and L-curve (gray area indicating ) also define a range of favored h₂ values.

Concerning our solution for Mercury's orientation and tidal parameters, we only found a significant impact on the estimate of h₂, while estimates for other parameters are stable within their error bars. Figure 9 (bottom) illustrates the range of possible h₂ estimates for a set of (IAU, KinetX)-based solutions only differing by . We highlight in gray the range of “optimal estimates” favored by VCE and L-curve analysis. The average value of this range, together with formal and systematic errors shown in Table 1, result in a best estimate of h₂ = 1.55 ± 0.65. Solutions based on Genova et al. (2019) a priori orbits generally require a stronger orbit regularization to converge on consistent results.

4.4 Validation on Simulated Data

To fully characterize the behavior of the solutions, and in order to choose an appropriate parametrization and weighting scheme, we conduct extensive simulations with time-of-flight ranges consistently generated from a realistic topography.

To model small scale Mercury topography, we compute a fractal noise map composed by five superposed levels: the main noise level has an amplitude of 30 m on a 600 m baseline, while for each of the following ones the amplitude is divided by and the baseline is halved, consistently with the structure function of Mercury topography estimated by, for example, Susorney et al. (2017), Steinbrügge, Stark, et al. (2018). Instead of a map for the full surface, we generate a limited size “stamp” (Mazarico et al., 2015) of 0.25° × 0.25° with a periodic pattern in both latitude and longitude, such as the one shown in Figure 10. For each set of coordinates on Mercury surface, we define the local elevation as the sum of MLA derived topography and of the simulated small-scale noise.

Figure 10

Left: Simulated small-scale topography of Mercury surface. Right: Crossover discrepancies histogram (meters) of perturbed simulation setup (blue) and “zero-test” (only including interpolation noise, red).

We simulate the full MLA data set and repeat the selection procedure outlined in Section 4 to select the “best” 10⁶ crossovers. We first check the impact of the interpolation error on crossovers residuals and on the recovery of the geodetic parameters, by considering a perfect knowledge of MESSENGER trajectory and Mercury's orientation and tides. We show the distribution of the resulting discrepancies (FWMH < 10 m) in Figure 10, while estimated parameter corrections (expected to be 0) have amplitudes of 10⁻⁵ deg for RA, 10⁻⁶ deg for DEC, 5 × 10⁻⁹ deg/day for ω, 10⁻³ arcsec for L, and 0.075 for h₂.

Then, we analyze a more realistic situation, where both the orbits and the geodetic parameters are perturbed. To simulate a realistic mismodeling occurring in the processing of real data, we degrade our a priori knowledge by applying both a bias and linear drift in ACR and a bias to pointing parameters, but only estimate a set of ACR biases per each track. Perturbations have been set to an RMSE of 50 m (+40 m/day) in AC and 20 m (+10 m/day) in R, 0.5 arcsec for the pointing, five arcsec for right-ascension and declination of the pole, 3 as/y for the spin rate, and 1.5 arcsec for librations, according to the value of current uncertitudes (Archinal et al., 2018). Orbit and pointing perturbations are randomly chosen for each track, according to the selected RMSE. An histogram of discrepancies for each experiment is shown in Figure 10: pre-fit discrepancies of the perturbed solution are comparable with the ones of real MLA data. We report in Table 1 (column “intrinsic”) the parameter residuals at convergence, that is, the difference between the applied perturbations and the solution. We consider these as intrinsic errors from the processing pipeline (e.g., interpolation noise, numerical errors, and imperfections in our modeling and parametrization), also contributing to the systematic errors budget of our analysis.

5 Discussion

Our solution for the Mercury's rotational parameters, based on the full MLA data set and an average of solutions with different a priori orbits and values (see Section 4.1), is shown in Figure 11 along with previous solutions provided by other groups using various techniques (camera and altimetry, Doppler, Earth-based radar). Our updated values and calibrated error bars (based on the analysis presented in Section 4) are consistent with most recent solutions and provide an independent validation.

Figure 11

Our solutions for Mercury's orientation (RA, DEC, ω, L) and tidal Love number h₂ based on MLA crossovers analysis (red, 3σ errors), compared with Margot et al. (2012) (blue), Mazarico et al. (2014) (black), Stark et al. (2015) (yellow), Verma and Margot (2016) (mauve), Genova et al. (2019) (green), and Konopliv et al. (2020) (cyan), all using different data sets and techniques. As ours is the first data-based estimate of Mercury's h₂, we compare our solution to theoretical predictions from Steinbrügge, Padovan, et al. (2018) (brown) and Goossens et al. (2019) (orange). Black dashed lines indicate either the Cassini plane (RA/DEC) or Mercury's resonant spin rate (ω).

Our solution puts Mercury in a precise Cassini state, as predicted by dynamical models (Peale, 1988), without any explicit constraints to place it in this state. While deviations from the Cassini state of the order of a few arc-seconds are expected (e.g., due to the precession of perihelion or to tidal dissipation, see Baland et al., 2017), these are of the order of our error bars and significantly smaller than offsets presented by most previous solutions (also see, e.g., Dumberry, 2020). Compared to the gravity measurements provided by Genova et al. (2019) (also in agreement with a Cassini state), we get a higher obliquity ϵ = 2.031 ± 0.03 arcmin, consistent with a normalized polar moment of inertia C/MR² = 0.343 ± 0.006 (with C, M, and R the polar moment of inertia, mass, and radius of Mercury, respectively). Explicit equations for these quantities are given in Genova et al. (2019) (supplementary material), where we note that Equation 5 contains a typo and should read

while Baland et al. (2017) provides useful numerical values and a detailed discussion of the underlying dynamical theory. Recent estimates for Mercury's rotational parameters, obliquity, and polar moment of inertia (and associated errors) are summarized and compared in Table 2, updating a similar table from Baland et al. (2017). Our crossover-based solution is hence closer to previous estimates from Earth-based radar (Margot et al., 2012) and from imagery and altimetry (Stark et al., 2015), but with smaller error bars. Since these techniques are tied to and sensitive to the rotation of the crust only, while gravity measurements by Verma and Margot (2016) and Genova et al. (2019) sense the whole planet, the discrepancy between these values might be interpreted as a different state for different layers of the planet. Geophysical implications of our results are presented later in this section.

Table 2. Values of Mercury's Orientation Parameters, ϵ, and C/MR²: Updated Version Based on Baland et al. (2017)

	RA (°)	DEC (°)	ω (°/day)	L (m)	ϵ (arcmin)	C/MR^2
Margot (2009)	281.0103 ± 1.4 × 10^−3	61.4155 ± 1.4 × 10^−3	6.1385025	38.5  ±  1.6	2.04 ± 0.08	0.346 ± 0.014
Mazarico et al. (2014)	281.00480 ± 0.0054	61.41436 ± 0.0021	6.138511 ± 1.15 × 10^−6	–	2.06 ± 0.16	0.349 ± 0.014
Stark et al. (2015)	281.00980 ± 8.8 × 10^−4	61.4156 ± 1.6 × 10^−3	6.13851804 ± 9.4 × 10^−7	38.9 ± 1.3	2.029 ± 0.085	0.3437 ± 0.011
Verma and Margot (2016)	281.00975 ± 4.8 × 10^−3	61.41828 ± 2.8 × 10^−3	–	–	1.88 ± 0.16	0.318 ± 0.028
Genova et al. (2019)	281.0082 ± 9.4 × 10^−4	61.4164 ± 3 × 10^−4	6.1385054 ± 1.3 × 10^−6	40.0 ± 8.7	1.968 ± 0.027	0.333 ± 0.005
Konopliv et al. (2020 ^a	281.0138 ± 2.5 × 10^−3	61.4161 ± 1.7 × 10^−3	6.138514 ± 6 × 10^−6	–	2.04 ± 0.12^a	0.345 ± 0.020^a
This study	281.0093 ± 6.3 × 10^−4	61.4153 ± 4.8 × 10^−4	6.138510 ± 2.8 × 10^−6	39.03 ± 1.1	2.031 ± 0.03	0.343 ± 0.006

• Note. In bold, the values currently adopted by the IAU (Archinal et al., 2018).
• ^a The obliquity ϵ given by Konopliv et al. (2020) is inconsistent with the pole axis orientation they report, as already noted by Steinbrügge et al. (2020): we derived values for ϵ and C/MR².

Concerning Mercury's spin rate, our solution favors Mazarico et al. (2014), rather than the other analysis which used MLA data (Stark et al., 2015). Error bars values in Table 1 are the result of a thorough evaluation (e.g., already reflect the sensitivity of the solution to a priori values and parametrization) and can thus be used as such. Regarding the amplitude of Mercury's longitudinal librations, our solution is consistent with the literature, with error bars comparable with previous “surface measurements” by Margot et al. (2012), Stark et al. (2015). Based on the polar moment of inertia and estimate for longitudinal librations, we compute the ratio C_cr+m/C = 0.423 ± 0.012, where C_cr+m is the fractional polar moment of inertia of the solid crust plus mantle and values ∼0.5 indicate a fluid outer core.

We then use a Markov Chain Monte Carlo (MCMC) process to generate an ensemble of interior models of Mercury. Our models follow earlier works: pressure variations with depth are computed using hydrostatic assumptions, and we numerically integrate the differential equations for pressure, gravity, and temperature (Hauck et al., 2013; Knibbe & van Westrenen, 2015; Sohl & Spohn, 1997). Based on these values, we determine the local density from equations of state. Our approach is entirely based on our earlier work as reported in Genova et al. (2019) (using the same parameters for the equations of state). We use our newly derived values for C/MR² and C_cr+m/C as measurements, together with a constraint of 0.2% on the bulk density of Mercury (the same as in Genova et al., 2019). Our MCMC results thus satisfy Mercury's mass constraint. As was the case before, the outer core radius is the parameter that is best determined (Genova et al., 2019; Hauck et al., 2013). As shown in Figure 12, our best estimate for the outer core radius r_oc = 2020 ± 50 km (at 3σ) is significantly larger than what estimated in gravity analysis by Genova et al. (2019), but close to the value estimated by Hauck et al. (2013). Because our polar moment value is close to that used by the latter, our MCMC results are also very similar. This is the case for the outer core radius, but also for other parameters such as the mantle density and weight fraction of Si in the core (not shown here). An important difference with the results used in Hauck et al. (2013), however, is that our rotation state is exactly in the Cassini state. This allows us to directly apply the procedure outlined by Peale et al. (2002) to derive Mercury's internal structure from our estimates. In addition, our error bars are smaller, which results in smaller error bars on the outer core radius. Because our polar moment value is larger than that of Genova et al. (2019), our outer core radius is also larger. Because the estimate of Genova et al. (2019) was based on gravity, indicating a sensitivity to the whole planet, and ours on measurements related to the crust, this further illustrates a possible difference between these measurements. While it is unclear which measurement type (if any) would yield the correct answer on its own, one has to be aware of these differences, because they have consequences for the resolved interior models: a higher polar moment as resolved from crustal measurements results in a larger outer core radius, and might not be able to constrain a solid inner core (Hauck et al., 2013; Margot et al., 2018). We note the latter in our results as well. Despite a smaller error, our current MCMC results do not make a distinction between the solid and liquid core as we find that their density is often close to one another.

Figure 12

Outer core radius, r_oc, resulting from Markov chain Monte Carlo solutions consistent with values of Mercury's moment of inertia C/MR² and C_cr+m/C based on our analysis of MLA altimetry crossovers. As for previous solutions based on the tracking of surface features, our best estimate for r_oc is significantly larger than gravity estimates by Genova et al. (2019) (also shown, for comparison).

As differential measurements of Mercury's surface elevation, altimetry crossovers are sensitive to vertical displacements due to the tidal influence of other bodies (mainly the Sun), and hence to the Love number h₂. However, the geographical distribution of MLA crossovers and of tidal deformations at Mercury surface (see Figure 13), along with their amplitude (>2 m only at limited longitudes and close to Mercury equator) makes tidal variations particularly challenging to measure with currently available measurements from orbit.

Figure 13

The geographical distribution of MLA crossovers (darker areas indicate more crossovers per sq. km) is superposed to a map of the total Mercury's tidal deformations (for h₂ = 1) integrated over a Mercury year. The comparison shows that although most MLA crossovers lie in regions where deformations are below 1 m, a large variety of tidal patterns are covered.

When combined with measurements of the gravitational Love number k₂, h₂ provides important constraints on the deep interior of a body, such as its inner core size (Steinbrügge, Padovan, et al., 2018; Van Hoolst & Jacobs, 2003). Up to now, the value of Mercury's h₂ has only been predicted based on Mercury's mean density and moment of inertia inferred from the MESSENGER mission data analysis (0.77 < h₂ < 0.93, Steinbrügge, Padovan, et al., 2018, based on C/MR² = 0.34 and k₂ = 0.46) and on Markov-Chain Monte-Carlo (MCMC) analysis of Mercury's interior taking into account experimental measurements of its moment of inertia and gravitational tidal response (h₂ = 1.02 ± 0.06, by Goossens et al., 2019, based on estimates by Genova et al., 2019).

In Section 4, we provided our solution for h₂ and highlighted how its estimate from MLA data is a delicate matter, sensitive to orbital errors and to the choice of constraints and parametrization. We discussed in Section 4.3 how to partially mitigate these factors, and set our error bars accordingly. Our error budget includes the main error sources identified in previous studies (e.g., elevation interpolation, orbital and pointing errors, see Steinbrügge, Padovan, et al., 2018) but also systematic errors due to sampling, LS constraints, and the choice of a priori orbits and rotational parameters. Figure 14 shows that larger h₂ values would correspond to larger k₂ values, according to MCMC-derived correlations. As such, only the lower part of our solution range for h₂ = 1.55 ± 0.65 would map on current best estimates of k₂ by Genova et al., 2019 at 3σ level. We could in principle apply tighter constraints toward 0 on either our h₂ estimate or MESSENGER orbit corrections to “regularize” our solution toward values closer to the “expected” h₂∼1, but we rather choose to provide a loosely constrained solution. While the upper and central part of our solution range is hardly compatible with, for example, recent measurements of Mercury's k₂, it's important to remind that modeling predictions are sensitive to a wide range of highly correlated parameters. As an example, Figure 14 shows the dependency of h₂ on Mercury's mantle viscosity, whose range of values is currently mainly controlled by k₂ estimates. For such reasons, we opt to use MCMC predictions as “guidance” to interpret our results, rather than as prior constraints to force our solution to fall within a given range.

Figure 14

h₂ and k₂ have a positive correlation when using current reference values in simulations of Mercury interior: Results from the MCMC analysis shown in Figure 12 indicate that h₂ < 1.1 is expected given current k₂ estimates (left). However, this behavior is sensitive to a multitude of model parameters such as mantle viscosity (right) or the rheology model (here, we adopt Andrade rheology, see Andrade, 1910; Jackson, 1993). For these reasons, we opt not to force our h₂ solution to be close to model predictions.

While acknowledging these limitations, we analyze the implications of our data-driven h₂ estimate. The ratio of currently available k₂ = 0.5690 ± 0.025 (Genova et al., 2019) with our new estimate of h₂ = 1.55 ± 0.65 (taking a robust 3σ range for formal errors, or ± 0.45 at 1σ) yields h₂/k₂ = 2.7 ± 1.2 (at 3σ, or ± 0.9 at 1σ), which can be compared with Steinbrügge, Padovan, et al. (2018) to predict the size of Mercury's inner core. Even if the error associated with our solution does not allow to finely discriminate between different interior models, it constitutes a first experimental confirmation from h₂ of the range obtained by Genova et al. (2019), that is, a solid inner core with a radius of 590–1,400 km, marginally favoring Mercury's inner core radius to be >1,000 km (following the relations given in Steinbrügge, Padovan, et al., 2018) although this would result in a lower density, approaching the one of the outer core.

6 Conclusion

In this study, we presented new solutions for Mercury's rotational state based on crossover analysis of the altimetry data set collected by MLA over the full mission (including two equatorial flybys in 2008 and the 2011–2015 orbital phase). Crossover analysis has several advantages, including a lower dependence on the knowledge of small scale topography, and is a powerful tool to determine the orientation and tidal deformations of a celestial body (Mazarico et al., 2014).

In particular, we analyze the MLA crossovers “data set” with an original procedure, including a detailed light-propagation model and optimized procedures to locate the three-dimensional coordinates of MLA crossovers within the newly developed in-house software package PyXover. We apply an extensive error modeling based on a set of factors, as detailed in Sectiodn 3.2, and VCE to ensure an optimal weighting of data and constraints to 0 applied on orbital corrections. These result in a solution based on a refined data set and covariance information. We present the first data-based solution for Mercury's tidal Love number h₂, which is consistent with the presence of a solid inner core predicted by previous studies. Our results point to a complex scenario, and they highlight the great interest to improve h₂ and k₂ determination with future analysis of existing and upcoming data. Moreover, our solution for Mercury's orientation places it in a precise Cassini state, while the corresponding moment of inertia C/MR² and C_cm+r/C are consistently computed within our solution. This results in values for the radius of the outer core that are larger than what measured by gravity (Genova et al., 2019) and consistent with other analysis based on methods sensitive to the rotation of Mercury's crust only. We interpret the apparent inconsistency between results based on gravity and “crust-related” analysis as possible evidence of different states for different layers of the planet, as discussed in Section 5.

While limits posed by MESSENGER observation geometry and accuracy exist, the quality of Doppler-based orbit reconstruction can be improved by MLA contribution, as we showed crossovers to be sensitive to inconsistencies in MESSENGER orbit. While the parametrization employed in this study can only partially correct these imperfections, a combined reconstruction of MESSENGER orbits based on both Doppler and altimetry data, for example, as crossovers constraints, could potentially benefit the estimate of both orbital and empirical parameters included in the reconstruction of science orbits. In turn, such improvements would benefit the interpretation of many products and observations by the MESSENGER mission. Future observations of Mercury by the ESA mission BepiColombo (Benkhoff et al., 2010), expected to reach its orbital phase in 2025, will further constrain these parameters by extending measurements from low orbit to the Southern hemisphere of the planet. In particular, gravity estimates will profit from a less elliptical orbit and the refined X/Ka-band transponder (MORE, Iess et al., 2009) on-board the Mercury Planetary Orbiter (MPO), allowing to remove a large part of plasma noise from tracking data. The Italian Spring Accelerometer (ISA, Iafolla et al., 2010) will also contribute to a refined calibration of non-gravitational forces, for example, solar radiation pressure, acting on the spacecraft. Beside the positive impact on Mercury's gravity field estimation, these factors will likely results in an improved knowledge of its Love number k₂ and orientation. Altimetry measurements by the on-board BepiColombo Laser Altimeter (BELA, HosseiniArani et al., 2020; Thomas et al., 2007) could then be combined with MLA measurements to extend and refine the present analysis, either in form of crossovers or as individual measurements of surface elevation (Steinbrügge, Stark, et al., 2018; Thor et al., 2020).

Acknowledgments

S. Bertone acknowledges support by the Swiss National Science Foundation within the Advanced Postdoc Mobility grant P300P2_177776. The material presented here is based upon work supported by NASA's Planetary Science Division Research Program under the CRESST II cooperative agreement with award number 80GSFC17M0002 (S. Bertone and S. Goossens).

Open Research

Data Availability Statement

The MLA data were processed on the GSFC NCCS ADAPT cluster (https://www.nccs.nasa.gov/systems/ADAPT) using the PyXover software (Bertone et al., 2020). MESSENGER orbit and attitude information used in this paper are available on the Navigation and Ancillary Information Facility (NAIF, https://naif.jpl.nasa.gov/pub/naif/pds/data/mess-e_v_h-spice-6-v1.0/messsp_1000/data/). In particular, unless differently specified, we refer to the spk/msgr_040803_150430_150430_od431sc_2.bsp orbit kernel. Other MESSENGER and Mercury's ephemeris used in this work are available on the NASA GSFC Planetary Geodynamics Data Archive (https://pgda.gsfc.nasa.gov/products/71). MLA observations are stored on the Geosciences Node of NASA's Planetary Data System (Neumann, 2018, see rdr_radr/section).