https://www.nature.com/articles/s41586-023-06150-0 Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Advertisement Advertisement Nature * View all journals * Search * Log in * Explore content * About the journal * Publish with us * Subscribe * Sign up for alerts * RSS feed 1. nature 2. articles 3. article * Article * Published: 14 June 2023 Spin state and deep interior structure of Mars from InSight radio tracking * Sebastien Le Maistre ORCID: orcid.org/0000-0002-9524-9479^1,2, * Attilio Rivoldini ORCID: orcid.org/0000-0002-8626-9283^1, * Alfonso Caldiero ORCID: orcid.org/0000-0002-1624-0581^1,2, * Marie Yseboodt^1, * Rose-Marie Baland ORCID: orcid.org/0000-0002-5907-0033^1, * Mikael Beuthe ORCID: orcid.org/0000-0002-9339-5894^1, * Tim Van Hoolst ORCID: orcid.org/0000-0002-9820-8584^1,3, * Veronique Dehant ORCID: orcid.org/0000-0002-9516-8572^1,2, * William M. Folkner ORCID: orcid.org/0000-0001-5133-9934^4, * Dustin Buccino^4, * Daniel Kahan ORCID: orcid.org/0000-0003-4211-536X^4, * Jean-Charles Marty^5, * Daniele Antonangeli ORCID: orcid.org/0000-0002-4952-5700^6, * James Badro ORCID: orcid.org/0000-0001-9337-4789^7, * Melanie Drilleau ORCID: orcid.org/0000-0001-5625-9706^8, * Alex Konopliv ORCID: orcid.org/0000-0001-8669-1866^4, * Marie-Julie Peters^1, * Ana-Catalina Plesa ORCID: orcid.org/0000-0003-3366-7621^9, * Henri Samuel ORCID: orcid.org/0000-0002-3740-9235^7, * Nicola Tosi ORCID: orcid.org/0000-0002-4912-2848^9, * Mark Wieczorek ORCID: orcid.org/0000-0001-7007-4222^10, * Philippe Lognonne ORCID: orcid.org/0000-0002-1014-920X^7, * Mark Panning ORCID: orcid.org/0000-0002-2041-3190^4, * Suzanne Smrekar^4 & * ... * W. Bruce Banerdt ORCID: orcid.org/0000-0003-3125-1542^4 Show authors Nature volume 619, pages 733-737 (2023)Cite this article * 1861 Accesses * 425 Altmetric * Metrics details Subjects * Atmospheric dynamics * Mineralogy Abstract Knowledge of the interior structure and atmosphere of Mars is essential to understanding how the planet has formed and evolved. A major obstacle to investigations of planetary interiors, however, is that they are not directly accessible. Most of the geophysical data provide global information that cannot be separated into contributions from the core, the mantle and the crust. The NASA InSight mission changed this situation by providing high-quality seismic and lander radio science data^1,2. Here we use the InSight's radio science data to determine fundamental properties of the core, mantle and atmosphere of Mars. By precisely measuring the rotation of the planet, we detected a resonance with a normal mode that allowed us to characterize the core and mantle separately. For an entirely solid mantle, we found that the liquid core has a radius of 1,835 +- 55 km and a mean density of 5,955-6,290 kg m^-3, and that the increase in density at the core-mantle boundary is 1,690-2,110 kg m^ -3. Our analysis of InSight's radio tracking data argues against the existence of a solid inner core and reveals the shape of the core, indicating that there are internal mass anomalies deep within the mantle. We also find evidence of a slow acceleration in the Martian rotation rate, which could be the result of a long-term trend either in the internal dynamics of Mars or in its atmosphere and ice caps. Access through your institution Buy or subscribe This is a preview of subscription content, access via your institution Access options Access through your institution Access through your institution Change institution Buy or subscribe Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $29.99 / 30 days cancel any time Learn more Subscribe to this journal Receive 51 print issues and online access $199.00 per year only $3.90 per issue Learn more Rent or buy this article Prices vary by article type from$1.95 to$39.95 Learn more Prices may be subject to local taxes which are calculated during checkout Additional access options: * Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support Fig. 1: Estimates of the main rotation parameters. [41586_2023_6150_Fig1_HTML] Fig. 2: Interpretation of nutation parameters in terms of interior structure. [41586_2023_6150_Fig2_HTML] Fig. 3: Density jump at the core-mantle boundary as a function of FCN period. [41586_2023_6150_Fig3_HTML] Fig. 4: Fraction of light elements in the core of Mars. [41586_2023_6150_Fig4_HTML] Data availability The RISE data that support the findings of this study are available on the Planetary Data system: https://pds-geosciences.wustl.edu/ missions/insight/rise.htm. Doppler data for Viking are in the REDUCED directories: https://pds-geosciences.wustl.edu/missions/mpf/ radioscience.html. DSN media calibration files are given in the RISE PDS archives: https://pds-geosciences.wustl.edu/insight/ urn-nasa-pds-insight_rise_raw/data_tro/. The full correlation matrix is available as source data for Extended Data Fig. 3b. Source data are provided with this paper. Code availability Distribution of the MONTE navigation code is restricted by the Export Administration Regulations of the US Department of Commerce. Eligible readers may request a copy of MONTE, under a licence that does not permit redistribution, at https://montepy.jpl.nasa.gov/. GINS software is the property of CNES. It can be used for research only; any other commercial or non-commercial uses are strictly prohibited. CNES grants to the GINS Licensee (natural person) a free non-exclusive licence. References 1. Banerdt, W. B. et al. Initial results from the InSight mission on Mars. Nat. Geosci. 13, 183-189 (2020). Article ADS CAS Google Scholar 2. Folkner, W. M. et al. The rotation and interior structure experiment on the InSight mission to Mars. Space Sci. Rev. 214, 100 (2018). Article ADS Google Scholar 3. Dehant, V. & Mathews, P. M. Precession, Nutation and Wobble of the Earth (Cambridge Univ. Press, 2015). 4. Sasao, T., Okubo, S. & Saito, M. A simple theory on dynamical effects of stratified fluid core upon nutational motion of the Earth. Proc. IAU Symp. 78, 165-183 (1980). ADS Google Scholar 5. Folkner, W. M. et al. Interior structure and seasonal mass redistribution of Mars from radio tracking of Mars Pathfinder. Science 278, 1749-1752 (1997). Article ADS CAS PubMed Google Scholar 6. Yoder, C. F. & Standish, E. M. Martian precession and rotation from Viking lander range data. J. Geophys. Res. 102, 4065-4080 (1997). Article ADS Google Scholar 7. Evans, S. et al. MONTE: the next generation of mission design and navigation software. CEAS Space J. 10, 79-86 (2018). Article ADS Google Scholar 8. Marty, J. C. et al. GINS: the CNES/GRGS GNSS scientific software. 3rd Int. Coll. Sci. Fundam. Asp. Galileo Program. ESA Proc. WPP326 31, 8-10 (2011). Google Scholar 9. Le Maistre, S. et al. Lander radio science experiment with a direct link between Mars and the Earth. Planet. Space Sci. 68, 105-122 (2012). Article ADS Google Scholar 10. Konopliv, A. S. et al. Detection of the Chandler wobble of Mars from orbiting spacecraft. Geophys. Res. Lett. 47, e2020GL090568 (2020). Article ADS Google Scholar 11. Baland, R.-M. et al. The precession and nutations of a rigid Mars. Celest. Mech. Dyn. Astron. 132, 47 (2020). Article ADS MathSciNet MATH Google Scholar 12. Banfield, D. et al. InSight Auxiliary Payload Sensor Suite (APSS). Space Sci. Rev. 215, 4 (2019). Article ADS Google Scholar 13. Sanloup, C. et al. Density measurements of liquid Fe-S alloys at high-pressure. Geophys. Res. Lett. 27, 811-814 (1999). Article ADS Google Scholar 14. Yoshizaki, T. & McDonough, W. F. The composition of Mars. Geochim. Cosmochim. Acta 273, 137-162 (2020). Article ADS CAS Google Scholar 15. Smrekar, S. E. et al. Pre-mission InSights on the interior of Mars. Space Sci. Rev. 215, 3 (2019). Article ADS Google Scholar 16. Yoder, C. F., Konopliv, A. S., Yuan, D. N., Standish, E. M. & Folkner, W. M. Fluid core size of Mars from detection of the solar tide. Science 300, 299-303 (2003). Article ADS CAS PubMed Google Scholar 17. Stahler, S. C. et al. Seismic detection of the martian core. Science 373, 443-448 (2021). Article ADS PubMed Google Scholar 18. Rivoldini, A., Van Hoolst, T., Verhoeven, O., Mocquet, A. & Dehant, V Geodesy constraints on the interior structure and composition of Mars. Icarus 213, 451-472 (2011). Article ADS CAS Google Scholar 19. Khan, A. et al. A geophysical perspective on the bulk composition of Mars. J. Geophys. Res. 123, 575-611 (2018). Article CAS Google Scholar 20. Wieczorek, M. A., Beuthe, M., Rivoldini, A. & Van Hoolst, T. Hydrostatic interfaces in bodies with nonhydrostatic lithospheres. J. Geophys. Res. Planets 124, 1410-1432 (2019). ADS Google Scholar 21. Kiefer, W. S., Bills, B. G. & Nerem, R. S. An inversion of gravity and topography for mantle and crustal structure on Mars. J. Geophys. Res. Planets 101, 9239-9252 (1996). Article ADS CAS Google Scholar 22. Defraigne, P., Dehant, V. & Van Hoolst, T. Steady-state convection in Mars' mantle. Planet. Space Sci. 49, 501-509 (2001). Article ADS CAS Google Scholar 23. Samuel, H. et al. The thermo-chemical evolution of Mars with a strongly stratified mantle. J. Geophys. Res. Planets 126, e2020JE006613 (2021). Article ADS CAS Google Scholar 24. McNamara, A. K. A review of large low shear velocity provinces and ultra low velocity zones. Tectonophysics 760, 199-220 (2019). Article ADS Google Scholar 25. Steenstra, E. S. & van Westrenen, W. A synthesis of geochemical constraints on the inventory of light elements in the core of Mars. Icarus 315, 69-78 (2018). Article ADS CAS Google Scholar 26. Gendre, H., Badro, J., Wehr, N. & Borensztajn, S. Martian core composition from experimental high-pressure metal-silicate phase equilibria. Geochem. Perspect. Lett. 21, 42-46 (2022). Article Google Scholar 27. Shibazaki, Y. et al. Hydrogen partitioning between iron and ringwoodite: implications for water transport into the Martian core. Earth Planet. Sci. Lett. 287, 463-470 (2009). Article ADS CAS Google Scholar 28. Zharkov, V. N. The internal structure of Mars: a key to understanding the origin of terrestrial planets. Sol. Syst. Res. 30, 456-465 (1996). ADS Google Scholar 29. Tsuno, K., Frost, D. J. & Rubie, D. C. The effects of nickel and sulphur on the core-mantle partitioning of oxygen in Earth and Mars. Phys. Earth Planet. Inter. 185, 1-12 (2011). Article ADS CAS Google Scholar 30. Defraigne, P., Rivoldini, A., Van Hoolst, T. & Dehant, V. Mars nutation resonance due to free inner core nutation. J. Geophys. Res. Planets 108, 5128 (2003). Article ADS Google Scholar 31. Mittelholz, A. et al. Timing of the martian dynamo: new constraints for a core field 4.5 and 3.7 Ga ago. Sci. Adv. 6, eaba0513 (2020). Article ADS CAS PubMed PubMed Central Google Scholar 32. Lodders, K. Relative atomic Solar System abundances, mass fractions, and atomic masses of the elements and their isotopes, composition of the solar photosphere, and compositions of the major chondritic meteorite groups. Space Sci. Rev. 217, 44 (2021). Article ADS CAS Google Scholar 33. Estefan, J. A. & Sovers, O. J. A Comparative Survey of Current and Proposed Tropospheric Refraction-Delay Models for DSN Radio Metric Data Calibration. JPL Publication 94-24 (NASA 1994). 34. Le Maistre, S. Martian lander radio science data calibration for Mars troposphere. Radio Sci. 55, e2020RS007155 (2020). Article ADS Google Scholar 35. Buccino, D., Border, J. S., Folkner, W. M., Kahan, K. & Le Maistre, S. Low-SNR Doppler data processing for the InSight radio science experiment. Remote Sens. 14, 1924 (2022). Article ADS Google Scholar 36. Le Maistre, S., Rosenblatt, P., Dehant, V., Marty, J.-C. & Yseboodt, M. Mars rotation determination from a moving rover using Doppler tracking data: what could be done? Planet. Space Sci. 159, 17-27 (2018). Article ADS Google Scholar 37. Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. S. & Kuchynka, P. The planetary and lunar ephemerides DE430 and DE431. IPN Progr. Rep. 42, 196 (2014). Google Scholar 38. Jacobson, R. A. & Lainey, V. Martian satellite orbits and ephemerides. Planet. Space Sci. 102, 35-44 (2014). Article ADS Google Scholar 39. Dehant, V., Defraigne, P. & Van Hoolst, T. Computation of Mars' transfer functions for nutations, tides and surface loading. Phys. Earth Planet. Inter. 117, 385-395 (2000). Article ADS Google Scholar 40. Van Hoolst, T., Dehant, V., Roosbeek, F. & Lognonne, P. Tidally induced surface displacements, external potential variations, and gravity variations on Mars. Icarus 161, 281-296 (2003). Article ADS Google Scholar 41. Archinal, B. A. et al. Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2015. Celest. Mech. Dyn. Astron. 130, 22 (2018). Article ADS MathSciNet Google Scholar 42. Konopliv, A. S., Yoder, C. F., Standish, E. M., Yuan, D.-N. & Sjogren, W. L. A global solution for the Mars static and seasonal gravity, Mars orientation, Phobos and Deimos masses, and Mars ephemeris. Icarus 182, 23-50 (2006). Article ADS Google Scholar 43. Dehant, V. et al. The radioscience LaRa instrument onboard ExoMars 2020 to investigate the rotation and interior of Mars. Planet. Space Sci. 180, 104776 (2020). Article Google Scholar 44. Kahan, D. S. et al. Mars precession rate determined from radiometric tracking of the InSight lander. Planet. Space Sci. 199, 105208 (2021). Article Google Scholar 45. Baland, R.-M., Hees, A., Yseboodt, M., Bourgoin, A. & Le Maistre, S. Relativistic contributions to the rotation of Mars. Astron. Astrophys. 670, A29 (2023). Article Google Scholar 46. Lange, L. et al. InSight pressure data recalibration, and its application to the study of long-term pressure changes on Mars. J. Geophys. Res. Planets 127, e2022JE007190 (2022). Article ADS CAS PubMed PubMed Central Google Scholar 47. de la Torre Juarez, M., Piqueux, S., Kass, D. M., Newman, C. & Guzewich, S. D. Pressure deficit in Gale crater and a larger northern polar cap after the Mars year 34 global dust storm. AGU Fall Meeting Abstr. P51C-02 (2019). 48. Kuchynka, P. et al. New constraints on Mars rotation determined from radiometric tracking of the Opportunity Mars Exploration Rover. Icarus 229, 340-347 (2014). Article ADS Google Scholar 49. Panning, M. P. et al. Planned products of the Mars structure service for the InSight mission to Mars. Space Sci. Rev. 211, 611-650 (2017). Article ADS Google Scholar 50. Connolly, J. A. D. Computation of phase equilibria by linear programming: a tool for geodynamic modeling and its application to subduction zone decarbonation. Earth Planet. Sci. Lett. 236, 524-541 (2005). Article ADS CAS Google Scholar 51. Stixrude, L. & Lithgow-Bertelloni, C. Thermodynamics of mantle minerals - I. Physical properties. Geophys. J. Int. 162, 610-632 (2005). Article ADS Google Scholar 52. Greenwood, S., Davies, C. J. & Pommier, A. Influence of thermal stratification on the structure and evolution of the Martian core. Geophys. Res. Lett. 48, e2021GL095198 (2021). Article ADS Google Scholar 53. Terasaki, H. et al. Pressure and composition effects on sound velocity and density of core-forming liquids: implication to core compositions of terrestrial planets. J. Geophys. Res. Planets 124 , 2272-2293 (2019). Article ADS Google Scholar 54. Tsuno, K., Grewal, D. S. & Dasgupta, R. Core-mantle fractionation of carbon in Earth and Mars: the effects of sulfur. Geochim. Cosmochim. Acta 238, 477-495 (2018). Article ADS CAS Google Scholar 55. Okuchi, T. Hydrogen partitioning into molten iron at high pressure: implications for Earth's core. Science 278, 1781-1784 (1997). Article ADS CAS PubMed Google Scholar 56. Clesi, V. et al. Low hydrogen contents in the cores of terrestrial planets. Sci. Adv. 4, e1701876 (2018). Article ADS PubMed PubMed Central Google Scholar 57. Malavergne, V. et al. Experimental constraints on the fate of H and C during planetary core-mantle differentiation. Implications for the Earth. Icarus 321, 473-485 (2019). Article ADS CAS Google Scholar 58. Yuan, L. & Steinle-Neumann, G. Strong sequestration of hydrogen into the Earth's core during planetary differentiation. Geophys. Res. Lett. 47, e2020GL088303 (2020). Article ADS CAS Google Scholar 59. Anderson, D. L. & Minster, J. B. The frequency dependence of Q in the Earth and implications for mantle rheology and Chandler wobble. Geophys. J. R. Astron. Soc. 58, 431-440 (1979). Article ADS Google Scholar 60. Yseboodt, M., Dehant, V. & Peters, M.-J. Signatures of the Martian rotation parameters in the Doppler and range observables. Planet. Space Sci. 144, 74-88 (2017). Article ADS Google Scholar Download references Acknowledgements This is InSight Contribution Number 211. The research done at the Royal Observatory of Belgium was supported by the Belgian PRODEX programme managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office, contract numbers PEA4000129361 and PEA4000140326. A.C. was supported by the French Community of Belgium within the frame of a FRIA grant. The work performed at the Jet Propulsion Laboratory, California Institute of Technology, was under contract with NASA. J.-C.M., D.A., J.B., M.D., H.S., M.W., P.L. acknowledge support from CNES and ANR (ANR-19-CE31-0008-08), and J.B., H.S. and P.L. thank IdEx Universite de Paris (ANR-18-IDEX-0001). D.A. received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement 724690). Those authors computing BSL models were given access to the HPC resources of IDRIS under the allocation A011041317 made by GENCI. J.B. acknowledges support from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement 101019965--ERC advanced grant SEPtiM). Author information Authors and Affiliations 1. Royal Observatory of Belgium, Brussels, Belgium Sebastien Le Maistre, Attilio Rivoldini, Alfonso Caldiero, Marie Yseboodt, Rose-Marie Baland, Mikael Beuthe, Tim Van Hoolst, Veronique Dehant & Marie-Julie Peters 2. UC Louvain, Louvain-la-Neuve, Belgium Sebastien Le Maistre, Alfonso Caldiero & Veronique Dehant 3. Institute of Astronomy, KU Leuven, Leuven, Belgium Tim Van Hoolst 4. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA William M. Folkner, Dustin Buccino, Daniel Kahan, Alex Konopliv, Mark Panning, Suzanne Smrekar & W. Bruce Banerdt 5. Centre National d'Etudes Spatiales, Toulouse, France Jean-Charles Marty 6. IMPMC, Sorbonne Universite, MNHN, CNRS, Paris, France Daniele Antonangeli 7. Universite de Paris, Institut de Physique du Globe de Paris, CNRS, Paris, France James Badro, Henri Samuel & Philippe Lognonne 8. Institut Superieur de l'Aeronautique et de l'Espace SUPAERO, Toulouse, France Melanie Drilleau 9. DLR Institute of Planetary Research, Berlin, Germany Ana-Catalina Plesa & Nicola Tosi 10. Laboratoire Lagrange, Universite Cote d'Azur, Observatoire de la Cote d'Azur, CNRS, Nice, France Mark Wieczorek Authors 1. Sebastien Le Maistre View author publications You can also search for this author in PubMed Google Scholar 2. Attilio Rivoldini View author publications You can also search for this author in PubMed Google Scholar 3. Alfonso Caldiero View author publications You can also search for this author in PubMed Google Scholar 4. Marie Yseboodt View author publications You can also search for this author in PubMed Google Scholar 5. Rose-Marie Baland View author publications You can also search for this author in PubMed Google Scholar 6. Mikael Beuthe View author publications You can also search for this author in PubMed Google Scholar 7. Tim Van Hoolst View author publications You can also search for this author in PubMed Google Scholar 8. Veronique Dehant View author publications You can also search for this author in PubMed Google Scholar 9. William M. Folkner View author publications You can also search for this author in PubMed Google Scholar 10. Dustin Buccino View author publications You can also search for this author in PubMed Google Scholar 11. Daniel Kahan View author publications You can also search for this author in PubMed Google Scholar 12. Jean-Charles Marty View author publications You can also search for this author in PubMed Google Scholar 13. Daniele Antonangeli View author publications You can also search for this author in PubMed Google Scholar 14. James Badro View author publications You can also search for this author in PubMed Google Scholar 15. Melanie Drilleau View author publications You can also search for this author in PubMed Google Scholar 16. Alex Konopliv View author publications You can also search for this author in PubMed Google Scholar 17. Marie-Julie Peters View author publications You can also search for this author in PubMed Google Scholar 18. Ana-Catalina Plesa View author publications You can also search for this author in PubMed Google Scholar 19. Henri Samuel View author publications You can also search for this author in PubMed Google Scholar 20. Nicola Tosi View author publications You can also search for this author in PubMed Google Scholar 21. Mark Wieczorek View author publications You can also search for this author in PubMed Google Scholar 22. Philippe Lognonne View author publications You can also search for this author in PubMed Google Scholar 23. Mark Panning View author publications You can also search for this author in PubMed Google Scholar 24. Suzanne Smrekar View author publications You can also search for this author in PubMed Google Scholar 25. W. Bruce Banerdt View author publications You can also search for this author in PubMed Google Scholar Contributions Writing the original draft: S.L.M., A.R., M.B., A.C., M.Y., R.-M.B., T.V.H., M.D. and H.S. Writing, review and editing: M.W. and D.A. Formal analysis: S.L.M. and A.C. Validation: D.B., A.K. and M.-J.P. Software: J.-C.M., S.L.M. and A.C. Data curation: D.K., D.B., S.L.M. and A.C. Methodology: A.R., M.Y., R.-M.B., M.B., T.V.H., D.A., J.B., M.D., A.-C.P., H.S. and N.T. Investigation: S.L.M., A.R., W.M.F., D.K. and D.B. Conceptualization: W.M.F., V.D. and P.L. Project administration: W.B.B., S.S. and M.P. Corresponding author Correspondence to Sebastien Le Maistre. Ethics declarations Competing interests The authors declare no competing interests. Peer review Peer review information Nature thanks Michael Efroimsky and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Additional information Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Extended data figures and tables Extended Data Fig. 1 RISE Doppler residuals and signature of the liquid core. Post-fit Doppler residuals and their histogram distributions for RISE (a) and Viking 1 lander (b) as a function of the mission time after landing and of the DSN facilities location. (c) Pre-fit Doppler residuals of RISE data as a function of time computed with the latest rotation models of Konopliv et al. (2020)^10 (in blue) and Kahan et al. (2021) ^44 (in orange). (d) Pre-fit residuals with nominal values of the classical rotation model parameters coming from converged RISE solution using only the first 24 months of data (blue) and only the last 24 months of data (orange). This shows that the classical model fails at matching RISE full arc data since a clear trend is observed in the regions not covered by the corresponding nominal solution. (e) Theoretical signature of the liquid core for the RISE timing, separated in red for the East antenna and in green for the West antenna tracking. The FCN parameters are \(F=0.06\) and FCN period of -243 days. The orange envelope shows the signature when the FCN period is slightly different (between -238 and -248 days). The pink box is the interval where the SEP angle is smaller than 15deg (conjunction) while the grey boxes are the intervals where the Earth declination is close to 0deg. The signature of a parameter in the Doppler observable is the difference between the observable computed using a nominal/non-zero value for this parameter and that obtained when the parameter is set to 0^60. Extended Data Fig. 2 RISE data calibrations. (a) Correction for media delays as applied to the RISE data points. On the left, the corrections due to the Earth's atmosphere as a function of the elevation at the DSN, on the right those due to Mars' troposphere, as a function of their elevation at Mars. (b) RISE Doppler residuals: only the red points are processed, i.e. used in our analysis. Points acquired at low elevation above the DSN station (orange diamonds) are affected by large noise from the Earth's atmosphere. They are still part of our analysis thanks to our accurate Earth noise calibration. The rest of the points are eliminated due to low SEP (blue squares), or high residual value (green crosses). (c) Estimated wet troposphere bias parameter per pass, classified by DSN station identifier. Shaded area is the a priori uncertainty. Extended Data Fig. 3 Mars rotation and orientation angles and their correlations. (a) Reference frames and Mars orientation angles (orange and green) for conversion between the Earth mean equator of J2000 (in blue) and Mars body-fixed coordinates (in red). (b) Correlation matrix between MONTE solved parameters (GINS correlation matrix is equivalent) using the full set of RISE data (see Supplementary Table S1 for symbol definition). Values smaller than 0.3 are set to 0 for readability. Source Data Extended Data Fig. 4 Comparison between the classical model of rotation of Mars and the one proposed in this study. Temporal evolution of the 30-months solutions for the FCN period (a), the core amplification factor (b), and the precession rate (c), with the classical spin model (orange) and with the model with corrections on the rotation rate for the post-dust-storm period (blue). Shaded envelopes are \(1\sigma \) uncertainty bounds. Extended Data Table 1 2-way Doppler data at 60 s of integration time considered in this study Full size table Extended Data Table 2 Best solution of the main Mars rotation parameter estimates corresponding to an average of the GINS and MONTE sets of full arc solutions reported in Supplementary Table S1 Full size table Supplementary information Supplementary Information This file contains sections 1-13, Supplementary Figs. 1-16, Supplementary Tables 1-5 and references. Source data Source Data Extended Data Fig. 3. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and Permissions About this article Check for updates. Verify currency and authenticity via CrossMark Cite this article Le Maistre, S., Rivoldini, A., Caldiero, A. et al. Spin state and deep interior structure of Mars from InSight radio tracking. Nature 619, 733-737 (2023). https://doi.org/10.1038/s41586-023-06150-0 Download citation * Received: 28 November 2022 * Accepted: 27 April 2023 * Published: 14 June 2023 * Issue Date: 27 July 2023 * DOI: https://doi.org/10.1038/s41586-023-06150-0 Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Comments By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Access through your institution Buy or subscribe Access through your institution Change institution Buy or subscribe Behind the Paper The real story behind the research, from conception to publication, the highs and the lows Advertisement Advertisement Explore content * Research articles * News * Opinion * Research Analysis * Careers * Books & Culture * Podcasts * Videos * Current issue * Browse issues * Collections * Subjects * Follow us on Facebook * Follow us on Twitter * Subscribe * Sign up for alerts * RSS feed About the journal * Journal Staff * About the Editors * Journal Information * Our publishing models * Editorial Values Statement * Journal Metrics * Awards * Contact * Editorial policies * History of Nature * Send a news tip Publish with us * For Authors * For Referees * Language editing services * Submit manuscript Search Search articles by subject, keyword or author [ ] Show results from [All journals] Search Advanced search Quick links * Explore articles by subject * Find a job * Guide to authors * Editorial policies Nature (Nature) ISSN 1476-4687 (online) ISSN 0028-0836 (print) nature.com sitemap About Nature Portfolio * About us * Press releases * Press office * Contact us Discover content * Journals A-Z * Articles by subject * Nano * Protocol Exchange * Nature Index Publishing policies * Nature portfolio policies * Open access Author & Researcher services * Reprints & permissions * Research data * Language editing * Scientific editing * Nature Masterclasses * Live Expert Trainer-led workshops * Research Solutions Libraries & institutions * Librarian service & tools * Librarian portal * Open research * Recommend to library Advertising & partnerships * Advertising * Partnerships & Services * Media kits * Branded content Career development * Nature Careers * Nature Conferences * Nature events Regional websites * Nature Africa * Nature China * Nature India * Nature Italy * Nature Japan * Nature Korea * Nature Middle East * Privacy Policy * Use of cookies * Your privacy choices/Manage cookies * Legal notice * Accessibility statement * Terms & Conditions * Your US state privacy rights Springer Nature (c) 2023 Springer Nature Limited Close Nature Briefing Sign up for the Nature Briefing newsletter -- what matters in science, free to your inbox daily. Email address [ ] Sign up [ ] I agree my information will be processed in accordance with the Nature and Springer Nature Limited Privacy Policy. Close Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing * *