# Determination of Venus's Rotation State Using Radio-tracking Data from the Venus Express Spacecraft

Maëva Lévesque, Pascal Rosenblatt, Jean-Charles Marty, Caroline Dumoulin

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Maëva Lévesque, Pascal Rosenblatt, Jean-Charles Marty, Caroline Dumoulin. Determination of Venus's Rotation State Using Radio-tracking Data from the Venus Express Spacecraft. The Planetary Science Journal, 2026, 7, pp.21. 10.3847/psj/ae2dfe. hal-05478093

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# Determination of Venus's Rotation State Using Radio-tracking Data from the Venus Express Spacecraft

Maëva Lévesque<fnref n="1" />, Pascal Rosenblatt<fnref n="1" />, Jean-Charles Marty<fnref n="2" />, and Caroline Dumoulin<fnref n="1" />

Received 2025 March 24; revised 2025 December 5; accepted 2025 December 10; published 2026 January 23

## Abstract

Despite its similarities to Earth, our understanding of Venus remains limited, particularly regarding its long retrograde rotation period. This has been measured using both Earth-based and in-orbit methods. However, the values obtained exhibit significant variations that cannot be explained by models of various physical mechanisms that could affect the rotation period of Venus. In this study, we present a new determination of the rotation period of Venus based on the precise orbit determination method applied to the radio-tracking data of the Venus Express (VEX) spacecraft. This method consists of a least-squares adjustment of the difference between Doppler data collected by ground-based stations and simulated Doppler data from a force model governing the spacecraft's motion. We find a rotation period of 243.02064 ± 0.0010 days, which is consistent with results from studies using tracking data from previous orbiters, and with Earth-based radar techniques for the same period covered by the VEX tracking dataset. An attempt was also made to adjust other parameters, including the precession rate, the gravitational field, and the Love number $k_2$. Nevertheless, no reliable results for these parameters could be obtained due to the lack of tracking data in the low altitude part of the highly eccentric orbit of VEX. The upcoming Venus missions, such as VERITAS (NASA) and EnVision (ESA) are expected to provide accurate Doppler tracking measurements that will improve our understanding of the planet.

Unified Astronomy Thesaurus concepts: Planetary science (1255); Venus (1763); Orbit determination (1175); Planetary probes (1252)

## 1. Introduction

Although Venus is similar to Earth in terms of size and mass, our knowledge of this planet, particularly regarding its internal structure, remains limited (A. Mocquet et al. 2011). One characteristic of Venus is its long retrograde rotation period of approximately 243 Earth days, which sets it apart from other planets in the solar system.

Measurements of the rotation period of Venus have been conducted since the 1960s using various approaches. The first relied on Earth-based radar, which involves sending radar waves from ground antennas and receiving at these antennas the reflected echoes from different regions of the planet's surface (I. I. Shapiro et al. 1990; M. A. Slade et al. 1990; B. A. Campbell et al. 2019) or measuring the displacement of radar speckles (J.-L. Margot et al. 2021). These methods have generally consisted of short, irregularly spaced campaigns obtained over decades, resulting in sparse time-series data. Other observations have been carried out using orbiting spacecraft, such as Pioneer Venus Orbiter (PVO), Magellan, and Venus Express (VEX), employing different techniques. One approach is radar imaging, which follows the same principle as Earth-based measurements but operates from orbit (M. Davies et al. 1992). Another method combines thermal images from the VIRTIS instrument on VEX with altimetry data from Magellan (N. Mueller et al. 2012). The comparison of the two datasets revealed a longitude deviation, allowing for a new determination of Venus's rotation period over 16 yr.

The last method is based on the study of the effect of the planet's rotation on the precise reconstruction of the spacecraft's orbital motion (A. Konopliv et al. 1999). These satellite-based methods provide nearly continuous datasets, but are limited over the satellite's operating period.

These different solutions of the rotation period of Venus extend over a range of at least 7 minutes (20 ppm), of which only an amplitude of 3 minutes (8.6 ppm) can be explained by modeling various phenomena that could influence the length of day (LOD; L. Cottereau et al. 2011). Indeed, the effect of the solar torque on Venus, considered as a single solid body, has been studied and induces variations of the LOD with an amplitude of the order of 2 minutes. This is the most significant influence on the LOD for Venus (L. Cottereau et al. 2011). The second effect influencing the LOD comes from the angular momentum induced by the winds in the atmosphere: this effect can induce LOD variations between 25 and 50 s depending on the knowledge of atmospheric dynamics (L. Cottereau et al. 2011). Finally, the presence of a liquid core causes LOD variations between 3.5 and 20.4 s, depending on its size (L. Cottereau et al. 2011). Moreover, while the characteristic periods of these phenomena are shorter than one Earth year, measurements of Venus's rotation period are typically based on estimates over a period of a year or more. Thus, an accurate monitoring of Venus's rotation period or LOD could provide insights into these processes.

In this study, we investigate navigation tracking data of the VEX spacecraft with the specific aim of improving the solution for the rotation period of Venus. We follow the methodology employed by A. Konopliv et al. (1999) with the Magellan tracking data. Although these VEX Doppler data

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<footnote n="1">¹ Nantes Université, Univ Angers, Le Mans Université, CNRS, Laboratoire de Planétologie et Géosciences, LPG UMR 6112, 44000 Nantes, France; maeva.levesque@univ-nantes.fr</footnote>

<footnote n="2">² CNES—Space Geodesy Office, Toulouse, France</footnote>
<table><caption>**Table 1**
Orbital Parameters of VEX</caption><thead><tr><td>Parameters</td><td>Values</td></tr></thead><tbody><tr><td>Eccentricity</td><td>0.84</td></tr><tr><td>Altitude at pericenter (km)</td><td>250</td></tr><tr><td>Altitude at apocenter (km)</td><td>66,000</td></tr><tr><td>Orbital period (hour)</td><td>24</td></tr><tr><td>Inclination in IAU body-fixed frame (deg)</td><td>~ 90</td></tr></tbody></table>

have mostly been used for navigation purposes, they can be used for scientific investigation (P. Rosenblatt et al. 2012).

This manuscript is structured as follows: In Section 2, we introduce the type of data and the method of precise orbit determination (POD) that we used to perform our study. The results of the POD are presented in Section 3, including an estimate of the rotation period of Venus. Finally, in Section 4, the solution is compared with previous studies, and its implications for future work are discussed.

## 2. Method

In this study, we use navigation tracking data of the VEX spacecraft. This mission of the European Space Agency (ESA) was launched in 2005 and orbited Venus for approximately 8 yr from 2006 to 2014 with the aim of studying the atmosphere of the planet. VEX was in a polar and highly elliptical orbit, with an altitude of 250 km at the pericenter and 66,000 km at the apocenter, corresponding to an orbital period of approximately 24 hr. Table 1 gives a brief summary of the orbital parameters of VEX.

### 2.1. Doppler Dataset

In this study, the tracking data collected by the ESA Tracking Network (ESTRACK) is examined. ESTRACK comprises ground stations strategically located at different longitudes around the Earth: Cebreros (Spain), New Norcia (Australia), and Malargüe (Argentina). For the VEX mission, the three stations have collected tracking data for the purpose of navigation. In addition, a part of the tracking data at New Norcia has been used for the radio-science experiment (H. Svedhem et al. 2007). The tracking data consist of Doppler shift measurements on the carrier frequency of the radio link between the spacecraft and the ground antennas. For VEX, these data include two-way Doppler measurements in the X band with an uplink frequency of 7.2 GHz and a downlink frequency of 8.4 GHz, using a 10 s averaging interval. As the radio signal propagates through the Earth's atmosphere, its velocity decreases, and the heterogeneity of the medium distorts its optical path, increasing its travel time. A correction is then applied to each station using models based on the general formulation of the IERS 2010 conventions (G. Petit & B. Luzum 2010; J. C. Marty 2013). This approach allows the precise monitoring of changes in the velocity of the spacecraft. During each 24 hr orbit of VEX, approximately 8 hr of tracking were performed over approximately the same part of the orbit (i.e., between mean anomalies of about 40° and 170°, corresponding to altitudes between 32,000 and 62,000 km) throughout the studied period.

### 2.2. Precise Orbit Determination (POD)

The present study analyzes VEX's Doppler tracking data from mid-2006 to mid-2014 using the POD method (T. Moyer 1971; B. Tapley et al. 2004; J. C. Marty 2013). This approach involves performing a least-squares adjustment to minimize the difference between Doppler measurements collected by ground-based tracking stations and simulated Doppler data. The simulated Doppler data are derived from the trajectory computed by numerical integration of the forces governing the spacecraft's motion, including both gravitational and non-gravitational forces, respectively, acting on the center of mass of the spacecraft and the faces of the spacecraft.

For gravitational forces, we use the MGNP180U solution of Venus's gravitational field, developed to degree and order 180, as well as its associated rotational state, derived from Doppler tracking data of two NASA missions: PVO and Magellan (A. Konopliv et al. 1999). In addition, the gravitational influences of external bodies such as the Sun and the other planets are accounted for using a point mass representation, based on data from the JPL DE440 ephemeris (R. S. Park et al. 2021).

For nongravitational forces that also drive the spacecraft's motion, we employ a macro model that represents the spacecraft as a collection of flat plates, each defined by its surface area and optical properties (J. C. Marty 2013). The VEX macro model, provided by the European Space Operations Center (ESOC), consists of six flat plates for the bus, two for each solar array, and one for the High Gain Antenna (P. Rosenblatt et al. 2012). The orientation of each surface in inertial space is expressed by quaternions, also provided by ESOC. Some of this information is available as SPICE kernels on the NAIF website.<fnref n="3" />

The nongravitational force model takes into account the solar radiation pressure resulting from the transfer of momentum from photons impacting the spacecraft surface, as well as atmospheric drag, which depends on the atmospheric density. The Hedin-VTS3 model of Venus's upper atmosphere density (A. E. Hedin et al. 1983) has been used to compute the drag force. However, the impact of using the upper atmosphere density provided by the Venus Global Reference Atmospheric Model (Venus-GRAM; Venus-GRAM 2005 Rel. October 2009) is investigated in the next section. We also consider residual accelerations caused by imbalanced thruster firings during the wheel-off-loading (WoL) events. These firings cause changes in the satellite's velocity (ΔV) that typically occur once per day and generally near the apocenter. Using the ΔV and the duration of each event provided by ESOC, we compute the residual accelerations along the X-, Y-, and Z-axis components of the reference frame tied to the spacecraft.

### 2.3. Fitting the Force Model to the Doppler Tracking Data

The iterative least-squares adjustment is performed on successive arcs of several days using the Géodésie par Intégrations Numériques Simultanées (GINS) software developed by the Centre National d'Études Spatiales (J. C. Marty 2013). In this study, we select a total of 399 arcs with a duration of 7 days for data analysis. They are segmented to ensure that the arc begins and ends with tracking

<footnote n="3">³ http://naif.jpl.nasa.gov/pub/naif/VEX/kernels</footnote>
<table><caption>**Table 2**
List of Local and Global Parameters Adjusted for VEX</caption><thead><tr><th>Local Parameters</th><th>Global Parameters</th></tr></thead><tbody><tr><td>Solar radiation pressure scale factor (one parameter)</td><td>Rotation period</td></tr><tr><td>Atmospheric drag scale factor (one parameter per revolution)</td><td>Right ascension of the spin axis</td></tr><tr><td>Correction to the initial state vector (six parameters)</td><td>Decl. of the spin axis</td></tr><tr><td>Acceleration due to Wheel-off-Loading (three parameters)</td><td>Coefficients of degree 2 of the gravity field</td></tr></tbody></table>

data, thus avoiding gaps that could complicate the adjustment of some parameters. We also tested arcs of 3 and 12 days; however, these trials did not significantly improve the results obtained with 7 day data arcs due to a lack of information or dynamical discontinuities caused by WoL events. The adjustment provides estimates of a set of parameters that define the spacecraft's force model. These parameters are adjusted for each arc and are referred to as local parameters in order to mitigate the inaccuracy of the force model (see Table 2). For the VEX mission, the adjusted parameter set includes the initial state vector at the beginning of the arc, a solar radiation pressure and drag force scale factors, as well as a residual acceleration at each WoL event. One value of the solar pressure factor is adjusted for the entire arc, while values of the drag scale factor are adjusted on a per-revolution basis (or 7 per arc). Since the spacecraft is not tracked from Earth during most of the pericenter passes, this approach yields more information than a single adjustment per arc and is implemented by applying a continuity constraint of 0.01 to each successive drag scale factor. As the spacecraft's position and velocity from the ESOC navigation orbit are used as the initial state vector of each arc, they are refined during the adjustment process in order to compute a precise orbit. Finally, the three components of the WoL accelerations are adjusted at each of the daily WoL events (or 21 per arc). We used a priori values given by ESOC and applied an a priori constraint of 100%. These accelerations are adjusted along the X-, Y-, and Z-axes of the reference frame tied to the spacecraft, the origin of which is located at the center of mass of the spacecraft. In this frame, the high-gain antenna is aligned with the X-axis, and the solar panels are aligned with the Y-axis.

These various parameters are estimated using an iterative least-squares adjustment process. In the first iteration, the residuals and partial derivatives of each measurement are computed. The parameters are initialized using either selected models or user-defined a priori values (e.g., WoL accelerations and continuity constraints for atmospheric drag). Based on predefined criteria, measurements are excluded if they have a minimum elevation angle of 20° or residuals exceeding 100 Hz. Then, partial derivatives with respect to local parameters are calculated, allowing for the correction of their initial values. The process continues with subsequent iterations. Doppler residuals are recomputed using the updated parameters, and individual measurements with residuals greater than 3 times the rms of the arc are discarded. These Doppler measurements are equally weighted with a constant weight of 0.05 mm s⁻¹. Iterations continue until convergence is reached, which is defined as a change in the rms of the postfit residuals that is smaller than 10% between successive iterations. Once convergence is reached, the partial derivatives are recalculated, and local parameter solutions are obtained for each arc, after which selection criteria are applied. First, we select only data arcs with a solar radiation pressure scale factor around 1, since this is the value expected if the macro model is

accurate. However, we allow a large range around this value (100%) to avoid removing too many arcs that can then be used for the global parameter estimation. Second, we select data arcs with rms of the postfit Doppler residuals below 10 mHz (0.176 mm s⁻¹ in the case of two-way range rate) to minimize the effect of noisy Doppler data around solar superior conjunction periods. The last selection criterion ensures that only arcs for which at least 60% of the Doppler measurements are retained after the fitting process are included in the analysis. During this process, individual low-quality measurements with large residuals are automatically discarded. If more than 40% of the Doppler measurements are rejected, the arc is considered unreliable and excluded entirely. This helps to avoid inaccurate local parameter adjustments caused by insufficient data coverage. To retain the maximum number of arcs, we chose not to apply any selection criteria based on the atmospheric drag scale factor.

The selection criteria are employed to ensure that only arcs with well-adjusted local parameters corresponding to a precise, reconstructed orbit are included in the next stage of the analysis: the adjustment of the global parameters (see Table 2). Indeed, it is challenging to adjust the geophysical parameters, such as the rotation period of the planet, in each data arc, because the geophysical signatures in the spacecraft orbital perturbations are weak. Consequently, global parameters are derived for all selected arcs covering the period from mid-2006 to mid-2014 for VEX.

# 3. Results

## 3.1. Precise Orbit Determination of the Venus Express Spacecraft

After completing the least-squares adjustment using the GINS software, 330 arcs out of a total of 399 arcs successfully converged. Following the application of our selection criteria, 245 arcs remained for analysis, representing 74% of the total number of converging arcs.

Figure 1 shows the rms values of the postfit Doppler residuals for each arc throughout the VEX mission period. The selected arcs are evenly distributed over the analyzed period (2006–2014), indicating the homogeneous quality of the orbit determination over this period. The mean value of the rms of these residuals, 2.81 ± 1.46 mHz, is in agreement with a modern spacecraft orbiting Venus and tracked in the X band, and of the same order of magnitude as for the MESSENGER mission, which was also tracked in the X band (A. K. Verma & J.-L. Margot 2016). Peaks observed in the residuals correspond to the periods close to the superior solar conjunction, when the Sun is positioned between Venus and the Earth. These events are identified by a minimum Sun–Probe–Earth angle, which is defined as the angle between the lines connecting the spacecraft to the Sun and Earth. During such periods, the radio signal encounters significant perturbations as it passes
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Top: the rms of the postfit residuals (in mHz and in mm s⁻¹) is presented as a function of time (years) for the 330 converging arcs. Among these arcs, the 245 selected arcs are shown in black, while the remaining excluded arcs by the selection criteria are highlighted in red. The mean value of the residuals is indicated by a red dashed line. Peaks of higher values correspond to periods of superior solar conjunction. Bottom: evolution of the Sun–Probe–Earth angle. Minimum values indicate moments of superior conjunction, when the Sun is between the spacecraft and the Earth.</figcaption></figure>

close to the Sun, leading to elevated residual values (A. Konopliv et al. 1999).

A solar radiation pressure scale factor is estimated for each arc, with a mean value of approximately 1 (1.05 ± 0.18 (1σ)), indicating that our solar radiation pressure model is, on the whole, accurate enough. The mean atmospheric drag scale factor per revolution is 1.00 ± 1.27. The associated error is large, despite the fact that the mean value is close to 1. This result is consistent, given the lack of VEX tracking near periapsis. The initial position correction is approximately 611.23 ± 924.94 m, while the initial velocity correction is 22.77 ± 40.05 mm s⁻¹. Although these values are relatively high, they are reasonable given that the VEX navigation orbit was reconstructed using a force model probably less accurate than ours.

Figure 2 shows the adjusted accelerations due to the daily desaturation event of the reaction wheels for each arc. The adjustments are of the order of approximately 10⁻⁵ m s⁻² along the X- and Y-axes, corresponding to mean values of 3.48 × 10⁻⁴ ± 1.43 × 10⁻⁴ m s⁻² and 9.05 × 10⁻⁵ ± 1.15 × 10⁻⁴ m s⁻², respectively. In contrast, the mean adjustment along the Z-axis is 3.52 × 10⁻⁴ ± 1.64 × 10⁻³

m s⁻², showing an increase compared to the X- and Y-axes. The higher values of residual accelerations along the Z-axis are explained by its orientation perpendicular to the spacecraft plate, which supports the pairs of thrusters performing the WoL maneuvers. This observation is consistent with that of the ESA's Mars Express platform (P. Rosenblatt et al. 2008).

The correct level of our Doppler residuals and the estimated values of the local parameters indicate that we have computed a physically realistic orbit. To provide reliable estimates of the orbit accuracy, we employ orbit overlaps. This approach involves two pairs of successive data arcs. The orbit of the first data arc is extrapolated from the result of the Doppler fit. The extrapolated spacecraft position is then compared with the position derived from the Doppler fit of the second data arc. An orbit overlap period of 12 hr has been chosen to avoid WoL events, since only one was usually planned per 24 hr. Figure 3 shows the rms values of these overlaps in the along-track, cross-track (perpendicular to the orbital plane), and radial directions in the orbital plane and oriented toward the planet. The median value obtained for this study was 20 and 13 m, along the along-track and cross-track directions, respectively, and 4 m in the radial direction, which is comparable
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Accelerations due to desaturation events of reaction wheels measured along the X-, Y-, and Z-axes (m s⁻²) as a function of time (years). Red dashed lines indicate the median value for each axis of the reference frame tied to the spacecraft.</figcaption></figure>

to the results obtained with the Mars Express spacecraft (P. Rosenblatt et al. 2008).

## 3.2. Venus's Rotation Period Determination

Following the selection of arcs based on the outcomes of local parameter fits, which resulted in 245 arcs, the rotation period of Venus was determined by combining the normal matrices from each data arc. These matrices were stacked to form a single global matrix, which was then inverted to estimate the rotation period. In addition, the local parameters were readjusted to ensure consistency with the determined rotation period and the orbit determination process. To stabilize the global inversion, a constraint is applied on the readjustment of the X-, Y-, and Z-components of the WoL event accelerations. The constraint is an a priori constraint of the form: initial value ± sigma. The initial value comes from the adjustment of the acceleration from WoL events as a local parameter. The sigma defines the allowed uncertainty and is set to the acceleration values provided by ESOC, divided by a

factor of 10. In addition, we apply a relatively loose constraint of 0.1 on the atmospheric drag scale factor. These two parameters are subject to constraints due to the limited available information, and the least restrictive constraints are adopted.

The estimation of the rotation period depends on the atmospheric density model (Hedin or Venus-GRAM) used in the adjustment process. However, this dependence is reduced by introducing a per-revolution scale factor for atmospheric drag. Therefore, it must be taken into account in the final range of solutions. Additionally, we checked the dependency of the final rotation period on the initial rotation value by deriving solutions using the initial periods reported by M. Davies et al. (1992), A. Konopliv et al. (1999), and J.-L. Margot et al. (2021). Figure 4 summarizes the resulting solutions as a function of the atmospheric density model and initial rotation period. Blue, green, and red correspond to the initial values reported by M. Davies et al. (1992), A. Konopliv et al. (1999), and J.-L. Margot et al. (2021), respectively. The error bars
<figure><img src="image_6.png" /><figcaption>**Figure 3.** The rms of the overlap differences along the along-track, cross-track, and radial directions. The median value for each direction is indicated by the red dashed lines.</figcaption></figure>

associated with each estimate correspond to the formal errors derived from the least-squares adjustment.

Taking all these dependencies into account, we determined a solution for the rotation period that encompasses all those presented in Figure 4. The shaded area in the figure shows the range of values that account for the different parameters influencing the solution, which are the atmospheric density model used and the rotation state. From the radio tracking data of VEX, we obtained a Venus rotation period of 243.0206 ± 0.0010 days. This adopted error corresponds to a variation of approximately 1.44 minutes (4.115 ppm).

4. Discussion

Figure 5 compares the result of the present study for the rotation period of Venus with different solutions from the literature, expressed in days, with respect to the periods covered by the respective datasets and the method used to derive these values. The 1σ errors are also given for each solution. Detailed references and methods are listed in Table 3.

Our estimate for the rotation period of Venus is within the range of previous solutions. Our result is found to be in good agreement with the estimate of A. Konopliv et al. (1999), which also uses the POD method, but with Doppler data from Magellan and PVO. Additionally, our range of solutions overlaps the estimate from B. A. Campbell et al. (2019), which is based on radar images from Earth's ground stations. Moreover, our solution is also aligned with the result obtained by S. Shaffer & S. Hensley (1991) derived from Magellan radar images. In their study, they analyzed the location of features visible in images taken from two different orbits. A total of 300 features were selected, ranging from 60°north to 44°south latitude. The offset in their positions allows for the calculation of the planet's rotation period.

As illustrated in Figure 5, the oldest measurements have large error bars, while around the 1980s these become smaller. The variability between studies may be related to the methods used and associated biases, as well as the quality of the datasets. We can notice that the solution of 243.0185 ± 0.0001 days from M. Davies et al. (1992) is lower than the others.
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Estimates of the rotation period of Venus (in days) as a function of the assumed atmospheric density model (Hedin and Venus-GRAM) and the initial rotation period (Davies, Konopliv, and Margot). The blue, green, and red markers correspond to initial values from M. Davies et al. (1992), A. Konopliv et al. (1999), and J.-L. Margot et al. (2021), respectively. The dashed horizontal lines indicate the reference values from each study. The error bars for each estimate represent the formal error from the least-squares adjustment. The gray shaded area covers all solutions, representing the range of values used to obtain the Venus rotation period.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Estimates of Venus's rotation period are presented alongside their measurement time baselines and corresponding $1\sigma$ error bars. Numbers refer to Table 3, in which references and methodologies are listed. The result of this study is the number 9.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Time series of Venus's rotation period, with one value reported for every 12" (initial rotation: J.-L. Margot et al. 2021, density model: Venusgram). The results of our study are shown in black, while the findings from J.-L. Margot et al. (2021) covering the period from mid-2006 to 2014 are displayed in red. The median values are indicated by dashed lines in their respective colors. The error bars for each estimate represent the formal error.</figcaption></figure>

<table><caption>**Table 3**
References for Estimates of Venus's Rotation Period (1σ Error Bars) Displayed in Figure 5</caption><thead><tr><th>References</th><th>Period of Rotation (days)</th><th>Measurement Time Baseline</th><th>Method</th></tr></thead><tbody><tr><td>(1) M. A. Slade et al. (1990)</td><td>243.022 ± 0.003</td><td>1972–1982</td><td>Earth-based radar observations</td></tr><tr><td>(2) I. I. Shapiro et al. (1990)</td><td>243.026 ± 0.006</td><td>1975–1983</td><td>Earth-based radar observations</td></tr><tr><td>(3) M. Davies et al. (1992)</td><td>243.0185 ± 0.0001</td><td>1990–1991</td><td>Magellan radar images</td></tr><tr><td>(4) S. Shaffer & S. Hensley (1991)</td><td>243.0200 ± 0.0024</td><td>1990–1991</td><td>Magellan radar images</td></tr><tr><td>(5) A. Konopliv et al. (1999)</td><td>243.0200 ± 0.0002</td><td>1978–1982 (PVO)<br>1992–1994 (Magellan)</td><td>Spacecraft POD<br>(PVO + Magellan)</td></tr><tr><td>(6) N. Mueller et al. (2012)</td><td>243.0230 ± 0.002</td><td>1990–1992 (Magellan)<br>2006–2008 (VEX)</td><td>Space-borne altimetry (Magellan)<br>+ thermal images (VEX)</td></tr><tr><td>(7) B. A. Campbell et al. (2019)</td><td>243.0212 ± 0.0006</td><td>1988–2017</td><td>Earth-based radar observations</td></tr><tr><td>(8) J.-L. Margot et al. (2021)</td><td>243.0226 ± 0.0013</td><td>2006–2020</td><td>Earth-based radar observations</td></tr><tr><td>(9) This study (2025)</td><td>243.0206 ± 0.0010</td><td>2006–2014</td><td>Spacecraft POD (VEX)</td></tr></tbody></table>

This is probably because the features selected in the radar images are mainly located near the north pole of Venus, at a latitude of approximately 80°. Its use as a reference in the IAU is therefore not very representative of the most recent studies, which could have implications for future missions to Venus.

The solutions for the rotation period of Venus shown in Figure 5 are generally averages over the study period, which means that they do not give access to possible variations in the LOD over that period. We try here to identify possible variations by estimating the rotation period every 12 arcs, representing a solution every 3 months. These estimates are shown in black in Figure 6. We observe a high degree of variability. Indeed, performing the adjustments over a small number of arcs leads to a less stable solution. We also tried to evaluate one solution every 4 arcs and 8 arcs, but the dispersion increased even more. In Figure 6, the measurements of the instantaneous periods obtained by J.-L. Margot et al. (2021) relative to the operating period of VEX are also displayed (in red). Their solutions are also scattered by 21 minutes (60 ppm). Therefore, both studies show a similar amount of dispersion, although the data from J.-L. Margot et al. (2021) include significant discontinuities in their time

series. The median rotation periods from both studies are close over the VEX operating period: 243.0219 days for J.-L. Margot et al. (2021) and 243.0214 days for this study. This shows that two different methods give consistent rotation periods over the same period. In both studies, the dispersion is too large to detect reliable periodic variations in the LOD, and thus to detect such an expected signal with an amplitude of about 3 minutes (L. Cottereau et al. 2011). Despite an estimate approximately every 3 months or so, there are significant variations between individual solutions that make it difficult to validate models of LOD variations.

The rotation period estimate derived from this study was obtained simultaneously with the R.A. and decl. of Venus's pole. The strong correlation between rotation period and the R. A. indicates that they must be adjusted at the same time during the fitting process. We obtained a value of 272°.741 ± 0.024 for the R.A. and 67°.148 ± 0.014 for the decl., taking into account both atmospheric models as well as the initial rotation period. These values correspond to the mid-period of VEX, i.e., 06/2010. To compare our values with those in the literature, we used the approach of J.-L. Margot et al. (2021). Using their estimate of the precession rate, we have precessed
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Spin axis orientation of Venus with $1\sigma$ uncertainties, i.e., 2D confidence intervals at 68.3% levels. Solutions from J.-L. Margot et al. (2021), M. Davies et al. (1992), and A. Konopliv et al. (1999) are represented in black, blue, and red, respectively. The results from this study are shown in green.</figcaption></figure>

<table><caption>**Table 4**
Estimate of Venus's R.A. and Decl. Precessed to J2000.0 with the Precession Rate from J.-L. Margot et al. (2021)</caption><thead><tr><td>References</td><td>R.A.</td><td>Decl.</td></tr></thead><tbody><tr><td>M. Davies et al. (1992)</td><td>272.753 ± 0.020</td><td>67.156 ± 0.010</td></tr><tr><td>A. Konopliv et al. (1999)</td><td>272.737 ± 0.006</td><td>67.153 ± 0.003</td></tr><tr><td>J.-L. Margot et al. (2021)</td><td>272.739 ± 0.0008</td><td>67.151 ± 0.0007</td></tr><tr><td>This study (2025)</td><td>272.750 ± 0.024</td><td>67.153 ± 0.014</td></tr></tbody></table>

the values of the R.A. and decl. published by M. Davies et al. (1992) and A. Konopliv et al. (1999) from the mid-date of the corresponding observations, i.e., 1991 January and 1993 September, respectively, to the epoch J2000.0. In addition, we have precessed our solutions to J2000.0 from 2010 June. The values obtained are summarized in Table 4 with uncertainties of $1\sigma$.

Figure 7 displays the spin axis orientation of Venus from Table 4 with $1\sigma$ uncertainties, i.e., 2D confidence intervals at 68.3% level as in J.-L. Margot et al. (2021). We have a large uncertainty, which can be explained by taking into account the two atmospheric models, Hedin and Venus-GRAM, as well as the initial rotation period. All other solutions are within our

uncertainties. Here we also show that we cannot determine a precession rate because our estimate of the pole position is not accurate enough.

We estimated Venus's degree-2 gravity field coefficients simultaneously with its rotation period and spin-axis orientation. The results are presented in Table 5, with error bars accounting for the dependence of the atmospheric density model and the initial rotation period, as was done for the other parameters. Table 5 also compares our estimates with those of A. Konopliv et al. (1999), showing very good agreement. Due to the limited sensitivity to higher-degree harmonics provided by the highly elliptical orbit of VEX, and the fact that this sensitivity scales with both semimajor axis and eccentricity, this analysis focused on degree-2 terms in order to avoid biases linked to poorly resolved higher degrees. In orbit determination, spacecraft motion is driven by linear combinations of several gravity coefficients rather than by each coefficient independently. This leads to lumped responses in the tracking data. Tests that included degree-3 terms confirmed this behavior, showing that, while the rotation parameters are not significantly affected, variations in degree-2 estimates increase. In contrast, Konopliv's solution extended up to degree 180.
<table><caption>**Table 5**
Estimates of the Degree-2 Gravity Field Coefficients of Venus from A. Konopliv et al. (1999) and This Study</caption><thead><tr><th>Normalized Harmonic Coefficients of Degree-2</th><th>A. Konopliv et al. (1999) (3σ)</th><th>This Study</th></tr></thead><tbody><tr><td>C<sub>20</sub></td><td>-1.9697e-06 ± 2.1000e-09</td><td>-1.9674e-06 ± 5.0659e-09</td></tr><tr><td>C<sub>21</sub></td><td>2.68e-08 ± 1.90e-09</td><td>2.57e-08 ± 5.47e-09</td></tr><tr><td>S<sub>21</sub></td><td>1.32e-08 ± 1.90e-09</td><td>1.17e-08 ± 4.73e-09</td></tr><tr><td>C<sub>22</sub></td><td>8.578e-07 ± 3.000e-09</td><td>8.551e-07 ± 1.467e-09</td></tr><tr><td>S<sub>22</sub></td><td>-9.55e-08 ± 3.00e-09</td><td>-9.50e-08 ± 3.64e-09</td></tr></tbody></table>

The Love number $k_2$ characterizes variations of the gravity field induced by the viscoelastic deformation of the body under the influence of the tidal forces. To date, only one value of $k_2$ has been estimated from the tracking data of the PVO and Magellan spacecraft (A. S. Konopliv & C. F. Yoder 1996). This value, $k_2 = 0.295 \pm 0.033$, has an uncertainty too large to effectively constrain internal structure models and distinguish between a liquid or solid core (C. Dumoulin et al. 2017). We have adjusted the rotation period, R.A., decl., and degree-2 coefficients of the gravity field by fixing the $k_2$ value to that of A. S. Konopliv & C. F. Yoder (1996).

To overcome the current limitations in estimating the various geophysical parameters or to improve the quality of the data, further investigations are needed, which should be provided by upcoming missions such as VERITAS (NASA) and EnVision (ESA). Indeed, these missions are expected to improve our understanding of Venus's internal structure by providing more frequent and precise observations to study LOD variations, the precession rate, and the $k_2$ Love number (G. Cascioli et al. 2021; P. Rosenblatt et al. 2021).

In addition, our results using VEX tracking data demonstrate the limitations of POD in accurately retrieving the global parameters that generate small orbital perturbations over a short period. The usual method of accumulation over a much longer period is limited by the inaccurate estimation of local parameters. Mitigating this effect requires using more pertinent local parameters, as described in A. M. Gargiulo et al. (2024). Such improvements should improve the expected results of upcoming missions (VERITAS and EnVision), which will provide more precise and continuous Doppler coverage.

## 5. Conclusion

In this study, we used navigation tracking data from the VEX spacecraft to refine the measurements of Venus's rotation period. We used the so-called POD method to obtain a new solution of $243.0206 \pm 0.0010$ days. This estimate is consistent with previous estimates obtained with different methods and datasets. The precise orbit of the VEX spacecraft reconstructed in this study is available as SPICE kernels at the NAIF website.<fnref n="4" /> These results also show that data that are not classified as scientific data can be used for further scientific investigations and therefore are worthy of being archived.

Comparison with previous studies supports the validity of our results. However, estimating other geophysical parameters was challenging, and only the estimate of the Venus rotation period and the degree-2 of the gravity field coefficients are reliable. Although we have determined rotation periods at regular 3 month intervals, the results exhibit significant

dispersion. Future missions such as EnVision (ESA) and VERITAS (NASA) will provide more accurate Doppler tracking data acquired at lower orbital altitudes than previous missions, enabling the estimation of more reliable and better-resolved time series of the rotation period, improving our understanding of processes affecting Venus's rotation rate.

## Acknowledgments

The reviews by two anonymous reviewers have been helpful in clarifying some points in the paper. This work acknowledges the financial support from CNES and the Région Pays de la Loire, as well as the CNRS. The authors would like to thank the VEX Radio Science Experiment (VeRa) team, in particular Martin Pätzold and Bernd Häusler, for their help in collecting the Doppler navigation dataset. The authors also thank the ESOC Flight Dynamics team for providing all the necessary information on the VEX spacecraft. This research used resources of the GLiCID Computing Facility (Ligerien Group for Intensive Distributed Computing, doi:10.60487/glicid, Pays de la Loire, France).

## ORCID iDs

Maëva Lévesque https://orcid.org/0009-0008-5643-8665

Pascal Rosenblatt https://orcid.org/0000-0002-2688-0726

Caroline Dumoulin https://orcid.org/0000-0001-8670-8410

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