Articles | Volume 40, issue 1
https://doi.org/10.5194/angeo-40-91-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/angeo-40-91-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Regular paper
| 
17 Feb 2022
Regular paper |
| 17 Feb 2022
| 17 Feb 2022
Reconstruction of Mercury's internal magnetic field beyond the octupole
Simon Toepfer
CORRESPONDING AUTHOR
Institut für Theoretische Physik,
Technische Universität Braunschweig, Braunschweig, Germany
Ida Oertel
Institut für Theoretische Physik,
Technische Universität Braunschweig, Braunschweig, Germany
Vanita Schiron
Institut für Theoretische Physik,
Technische Universität Braunschweig, Braunschweig, Germany
Yasuhito Narita
Space Research Institute, Austrian Academy of Sciences, Graz, Austria
Institut für Geophysik und extraterrestrische Physik,
Technische Universität Braunschweig,
Braunschweig, Germany
Karl-Heinz Glassmeier
Institut für Geophysik und extraterrestrische Physik,
Technische Universität Braunschweig,
Braunschweig, Germany
Max-Planck-Institut für Sonnensystemforschung,
Göttingen, Germany
Daniel Heyner
Institut für Geophysik und extraterrestrische Physik,
Technische Universität Braunschweig,
Braunschweig, Germany
Patrick Kolhey
Institut für Geophysik und extraterrestrische Physik,
Technische Universität Braunschweig,
Braunschweig, Germany
Uwe Motschmann
Institut für Theoretische Physik,
Technische Universität Braunschweig, Braunschweig, Germany
DLR Institute of Planetary Research, Berlin,
Germany
Related authors
Yasuhito Narita, Daniel Schmid, and Simon Toepfer
Ann. Geophys., 42, 79–89, https://doi.org/10.5194/angeo-42-79-2024, https://doi.org/10.5194/angeo-42-79-2024, 2024
Short summary
Short summary
The magnetosheath is a transition layer surrounding the planetary magnetosphere. We develop an algorithm to compute the plasma flow velocity and magnetic field for a more general shape of magnetosheath using the concept of potential field and suitable coordinate transformation. Application to the empirical Earth magnetosheath region is shown in the paper. The developed algorithm is useful when interpreting the spacecraft data or simulation of the planetary magnetosheath region.
Leonard Schulz, Karl-Heinz Glassmeier, Ferdinand Plaschke, Simon Toepfer, and Uwe Motschmann
Ann. Geophys., 41, 449–463, https://doi.org/10.5194/angeo-41-449-2023, https://doi.org/10.5194/angeo-41-449-2023, 2023
Short summary
Short summary
The upper detection limit in reciprocal space, the spatial Nyquist limit, is derived for arbitrary spatial dimensions for the wave telescope analysis technique. This is important as future space plasma missions will incorporate larger numbers of spacecraft (>4). Our findings are a key element in planning the spatial distribution of future multi-point spacecraft missions. The wave telescope is a multi-dimensional power spectrum estimator; hence, this can be applied to other fields of research.
Simon Toepfer, Karl-Heinz Glassmeier, and Uwe Motschmann
Ann. Geophys., 41, 253–267, https://doi.org/10.5194/angeo-41-253-2023, https://doi.org/10.5194/angeo-41-253-2023, 2023
Short summary
Short summary
The present study discusses the modeling and interpretation of magnetospheric structures via electromagnetic knots for the first time. The mathematical foundations of electromagnetic knots are presented, and the formalism is reformulated in terms of the classical wave telescope technique. The method is tested against synthetically generated magnetic field data describing a plasmoid as a two-dimensional magnetic ring structure.
Yasuhito Narita, Simon Toepfer, and Daniel Schmid
Ann. Geophys., 41, 87–91, https://doi.org/10.5194/angeo-41-87-2023, https://doi.org/10.5194/angeo-41-87-2023, 2023
Short summary
Short summary
Magnetopause is a shielding boundary of planetary magnetic field. Many mathematical models have been proposed to describe or to reproduce the magnetopause location, but they are restricted to the real-number functions. In this work, we analytically develop a magnetopause model in the complex-number domain, which is advantageous in deforming the magnetopause shape in a conformal (angle-preserving) way, and is suited to compare different models or map one model onto another.
Simon Toepfer, Yasuhito Narita, Daniel Heyner, Patrick Kolhey, and Uwe Motschmann
Geosci. Instrum. Method. Data Syst., 9, 471–481, https://doi.org/10.5194/gi-9-471-2020, https://doi.org/10.5194/gi-9-471-2020, 2020
Short summary
Short summary
The Capon method serves as a powerful and robust data analysis tool when working on various kinds of ill-posed inverse problems. Besides the analysis of waves, the method can be used in a generalized way to compare actual measurements with theoretical models, such as Mercury's magnetic field analysis. In view to the BepiColombo mission this work establishes a mathematical basis for the application of Capon's method to analyze Mercury's internal magnetic field in a robust and manageable way.
Bruce T. Tsurutani, Gurbax S. Lakhina, Rajkumar Hajra, Richard B. Horne, Masatomi Iizawa, Yasuhito Narita, Ingo von Borstel, Karl-Heinz Glassmeier, Volker Bothmer, Klaus Reinsch, Philipp Schulz, and Sami Solanki
EGUsphere, https://doi.org/10.5194/egusphere-2025-5536, https://doi.org/10.5194/egusphere-2025-5536, 2025
This preprint is open for discussion and under review for Annales Geophysicae (ANGEO).
Short summary
Short summary
During the 10–11 May 2024 geomagnetic storm, the red auroral rays appear at higher altitudes and connect to green rays lower down. The effect is linked to energetic electrons precipitating into the atmosphere during the storm. As the electrons continue downward, they hit oxygen below 200 km altitude and produce green light (5577 Å), named Stable Auroral Green (SAG) arcs. These observations mark the first reported sightings of such detailed, combined features.
Yasuhito Narita, Daniel Schmid, and Uwe Motschmann
Ann. Geophys., 43, 417–425, https://doi.org/10.5194/angeo-43-417-2025, https://doi.org/10.5194/angeo-43-417-2025, 2025
Short summary
Short summary
It is often the case that only magnetic field data are available for in situ planetary studies using spacecraft. Either plasma data are not available or the data resolution is limited. Nevertheless, the theory of plasma instability tells us how to interpret the magnetic field data (wave frequency) in terms of flow speed and beam velocity, generating the instability. We invent an analysis tool for Mercury's upstream waves as an example.
Yasuhito Narita, Daniel Schmid, and Simon Toepfer
Ann. Geophys., 42, 79–89, https://doi.org/10.5194/angeo-42-79-2024, https://doi.org/10.5194/angeo-42-79-2024, 2024
Short summary
Short summary
The magnetosheath is a transition layer surrounding the planetary magnetosphere. We develop an algorithm to compute the plasma flow velocity and magnetic field for a more general shape of magnetosheath using the concept of potential field and suitable coordinate transformation. Application to the empirical Earth magnetosheath region is shown in the paper. The developed algorithm is useful when interpreting the spacecraft data or simulation of the planetary magnetosheath region.
Leonard Schulz, Karl-Heinz Glassmeier, Ferdinand Plaschke, Simon Toepfer, and Uwe Motschmann
Ann. Geophys., 41, 449–463, https://doi.org/10.5194/angeo-41-449-2023, https://doi.org/10.5194/angeo-41-449-2023, 2023
Short summary
Short summary
The upper detection limit in reciprocal space, the spatial Nyquist limit, is derived for arbitrary spatial dimensions for the wave telescope analysis technique. This is important as future space plasma missions will incorporate larger numbers of spacecraft (>4). Our findings are a key element in planning the spatial distribution of future multi-point spacecraft missions. The wave telescope is a multi-dimensional power spectrum estimator; hence, this can be applied to other fields of research.
Simon Toepfer, Karl-Heinz Glassmeier, and Uwe Motschmann
Ann. Geophys., 41, 253–267, https://doi.org/10.5194/angeo-41-253-2023, https://doi.org/10.5194/angeo-41-253-2023, 2023
Short summary
Short summary
The present study discusses the modeling and interpretation of magnetospheric structures via electromagnetic knots for the first time. The mathematical foundations of electromagnetic knots are presented, and the formalism is reformulated in terms of the classical wave telescope technique. The method is tested against synthetically generated magnetic field data describing a plasmoid as a two-dimensional magnetic ring structure.
Yasuhito Narita, Simon Toepfer, and Daniel Schmid
Ann. Geophys., 41, 87–91, https://doi.org/10.5194/angeo-41-87-2023, https://doi.org/10.5194/angeo-41-87-2023, 2023
Short summary
Short summary
Magnetopause is a shielding boundary of planetary magnetic field. Many mathematical models have been proposed to describe or to reproduce the magnetopause location, but they are restricted to the real-number functions. In this work, we analytically develop a magnetopause model in the complex-number domain, which is advantageous in deforming the magnetopause shape in a conformal (angle-preserving) way, and is suited to compare different models or map one model onto another.
Daniel Schmid and Yasuhito Narita
Ann. Geophys. Discuss., https://doi.org/10.5194/angeo-2022-30, https://doi.org/10.5194/angeo-2022-30, 2023
Revised manuscript not accepted
Short summary
Short summary
Here we present a useful tool to diagnose the bow shock condition around planets on basis of magnetic field observations. From the upstream and downstream shock normal angle of the magnetic field, it is possible to approximate the relation between compression ratio, Alfvenic Mach number and the solar wind plasma beta. The tool is particularly helpful to study the solar wind conditions and bow shock characteristics during the planetary flybys of the ongoing BepiColombo mission.
Yasuhito Narita
Ann. Geophys., 39, 759–768, https://doi.org/10.5194/angeo-39-759-2021, https://doi.org/10.5194/angeo-39-759-2021, 2021
Short summary
Short summary
The concept of electromotive force appears in various electromagnetic applications in geophysical and astrophysical fluid studies. The electromotive force is being recognized as a useful tool to construct a more complete picture of turbulent space plasma and has the potential to test for the fundamental processes of dynamo mechanism in space.
Daniel Schmid, Yasuhito Narita, Ferdinand Plaschke, Martin Volwerk, Rumi Nakamura, and Wolfgang Baumjohann
Ann. Geophys., 39, 563–570, https://doi.org/10.5194/angeo-39-563-2021, https://doi.org/10.5194/angeo-39-563-2021, 2021
Short summary
Short summary
In this work we present the first analytical magnetosheath plasma flow model for the space environment around Mercury. The proposed model is relatively simple to implement and provides the possibility to trace the flow lines inside the Hermean magnetosheath. It can help to determine the the local plasma conditions of a spacecraft in the magnetosheath exclusively on the basis of the upstream solar wind parameters.
Horia Comişel, Yasuhito Narita, and Uwe Motschmann
Ann. Geophys., 39, 165–170, https://doi.org/10.5194/angeo-39-165-2021, https://doi.org/10.5194/angeo-39-165-2021, 2021
Short summary
Short summary
Identification of a large-amplitude Alfvén wave decaying into a pair of
ion-acoustic and daughter Alfvén waves is one of the major goals in the
observational studies of space plasma nonlinearity.
Growth-rate maps
may serve as a useful tool for predictions of the wavevector spectrum of density
or magnetic field fluctuations in various scenarios for the
wave–wave coupling processes developing at different stages in
space plasma turbulence.
Yasuhito Narita, Ferdinand Plaschke, Werner Magnes, David Fischer, and Daniel Schmid
Geosci. Instrum. Method. Data Syst., 10, 13–24, https://doi.org/10.5194/gi-10-13-2021, https://doi.org/10.5194/gi-10-13-2021, 2021
Short summary
Short summary
The systematic error of calibrated fluxgate magnetometer data is studied for a spinning spacecraft. The major error comes from the offset uncertainty when the ambient magnetic field is low, while the error represents the combination of non-orthogonality, misalignment to spacecraft reference direction, and gain when the ambient field is high. The results are useful in developing future high-precision magnetometers and an error estimate in scientific studies using magnetometer data.
Simon Toepfer, Yasuhito Narita, Daniel Heyner, Patrick Kolhey, and Uwe Motschmann
Geosci. Instrum. Method. Data Syst., 9, 471–481, https://doi.org/10.5194/gi-9-471-2020, https://doi.org/10.5194/gi-9-471-2020, 2020
Short summary
Short summary
The Capon method serves as a powerful and robust data analysis tool when working on various kinds of ill-posed inverse problems. Besides the analysis of waves, the method can be used in a generalized way to compare actual measurements with theoretical models, such as Mercury's magnetic field analysis. In view to the BepiColombo mission this work establishes a mathematical basis for the application of Capon's method to analyze Mercury's internal magnetic field in a robust and manageable way.
Cited articles
Abramowitz, M. and Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover Publications, ISBN-10 0486612724, 1972. a
Anderson, B. J., Johnson, C. L., Korth, H., Winslow, R. M., Borovsky, J. E., Purucker, M. E., Slavin, J. A., Solomon, S. C., Zuber, M. T., and McNutt Jr. R. L.: Low-degree structure in Mercury's
planetary magnetic field, J. Geophys. Res., 117, E00L12, https://doi.org/10.1029/2012JE004159, 2012. a, b, c, d, e, f, g, h
Backus, G.:
Poloidal and toroidal fields in geomagnetic field modeling,
Rev. Geophys., 24, 75–109, https://doi.org/10.1029/RG024i001p00075, 1986. a, b, c
Backus, G., Parker, R., and Constable, C.:
Foundations of Geomagnetism,
Cambridge University Press, Cambridge, https://doi.org/10.1017/S0016756897386464, 1996. a, b, c
Benkhoff, J., van Casteren, J., Hayakawa, H., Fujimoto, M., Laakso, H., Novara, M., Ferri, P., Middleton, H. R., and Ziethe, R.: BepiColombo–Comprehensive exploration of Mercury: Mission overview and science goals, Planet. Space Sci., 85, 2–20, https://doi.org/10.1016/j.pss.2009.09.020, 2010. a, b
Benkhoff, J., Murakami, G., Baumjohann, W., Besse, S., Bunce, E., Casale, M., Cremosese, G., Glassmeier, K.-H., Hayakawa, H., Heyner, D., Hiesinger, H., Huovelin, J., Hussmann, H., Iafolla, V., Iess, L., Kasaba, Y., Kobayashi, M., Milillo, A., Mitrofanov, I. G., Montagnon, E., Novara, M., Orsini, S., Quemerais, E., Reininghaus, U., Saito, Y., Santoli, F., Stramaccioni, D., Sutherland, O., Thomas, N., Yoshikawa, I., and Zender, J.: BepiColombo – Mission Overview and Science Goals, Space Sci. Rev., 217, 90, https://doi.org/10.1007/s11214-021-00861-4, 2021. a, b
Capon, J.: High resolution frequency-wavenumber spectrum analysis,
Proc. IEEE, 57, 1408–1418, https://doi.org/10.1109/PROC.1969.7278, 1969. a, b
Connerney, J. E. P.: The magnetic field of Jupiter: A generalized inverse approach, J. Geophys. Res., 86, 7679–7693, https://doi.org/10.1029/JA086iA09p07679, 1981. a, b, c
Connerney, J. E. P., Kotsiaros, S., Oliversen, R. J., Espley, J. R., Jørgensen, J. L., Joergensen, P. S., Merayo, J. M. G., Herceg, M., Bloxham, J., Moore, K. M., Bolton, S. J., and Levin, S. M.: A new model of Jupiter's magnetic field from Juno's first nine orbits, Geophys. Res. Lett., 45, 2590–2596, https://doi.org/10.1002/2018GL077312, 2018. a, b
Eckart, C. and Young, G.: The approximation of one matrix by another of lower rank, Psychometrika, 1, 211–218, https://doi.org/10.1007/BF02288367, 1936. a
Exner, W., Heyner, D., Liuzzo, L., Motschmann, U., Shiota, D., Kusano, K., and Shibayama, T.: Coronal mass ejection hits mercury: A.I.K.E.F. hybrid-code results compared to Messenger data,
Planet. Space Sci, 153, 89–99, https://doi.org/10.1016/j.pss.2017.12.016, 2018. a
Exner, W., Simon, S., Heyner, D., and Motschmann, U.: Influence of Mercury's exosphere on the structure of the magnetosphere,
J. Geophys. Res., 125, e27691, https://doi.org/10.1029/2019JA027691, 2020. a, b
Glassmeier, K.-H., Auster, H.-U., Heyner, D., Okrafka, K., Carr, C., Berghofer, G., Anderson, B. J., et al.: The fluxgate magnetometer of the BepiColombo Mercury Planetary Orbiter, Planet. Space Sci., 58, 287–299, https://doi.org/10.1016/j.pss.2008.06.018, 2010. a
Glassmeier, K.-H. and Tsurutani, B. T.: Carl Friedrich Gauss – General theory of terrestrial magnetism – a revised translation of the German text, Hist. Geo Space. Sci., 5, 11–62, https://doi.org/10.5194/hgss-5-11-2014, 2014. a, b, c
Glassmeier, K.-H. and Heyner, D.: Planetary magnetic fields, Space Physics and Aeronomy Collection Volume 2: Magnetospheres in the Solar System, Geophysical Monograph 259, edited by: Maggiolo, R., André, N., Hasegawa, H., and Welling, D. T., John Wiley & Sons, Inc, 367–389, https://doi.org/10.1002/9781119815624.ch24, 2021. a, b
Heyner, D., Auster, H.-U., Fornacon, K.-H., Carr, C., Richter, I., Mieth, J. Z. D., Kolhey, P. et al.:
The BepiColombo planetary magnetometer MPO-MAG: What can we learn from the Hermean magnetic field?, Space Sci. Rev., 217, 52, https://doi.org/10.1007/s11214-021-00822-x, 2021. a, b, c, d, e, f, g, h, i
Holme, R. and Bloxham, J.: The magnetic fields of Uranus and Neptune: Methods and models, J. Geophys. Res., 101, 2177–2200, https://doi.org/10.1029/95JE03437, 1996. a
Katsura, T., Shimizu, H., Momoki, N., and Toh, H.:
Electromagnetic induction revealed by Messenger's vector magnetic data: The size of Mercury's core, Icarus, 354, 114112, https://doi.org/10.1016/j.icarus.2020.114112, 2021. a
Lowes, F. J.: Mean-square values on sphere of spherical
harmonic vector fields, J. Geophys. Res., 71, 2179–2179, https://doi.org/10.1029/JZ071i008p02179, 1966. a
Mauersberger, P.: Das Mittel der Energiedichte des geomagnetischen Hauptfeldes an der Erdoberfläche und seine säkulare Änderung, Gerlands Beitr. Geophys, 65, 207–215, 1956. a
Milillo, A., Fujimoto, M., Murakami, G., Benkhoff, J., Zender, J., Aizawa, S., Dósa, M. et al.: Investigating Mercury's environment with the two-spacecraft BepiColombo mission, Space Sci. Rev., 216, 93, https://doi.org/10.1007/s11214-020-00712-8, 2020. a, b
Motschmann, U., Woodward, T. I., Glassmeier, K.-H.,
Southwood, D. J., and Pinçon, J.-L.:
Wavelength and direction filtering by magnetic measurements
at satellite arrays: Generalized minimum variance analysis,
J. Geophys. Res., 101, 4961–4966, https://doi.org/10.1029/95JA03471, 1996. a
Müller, J., Simon, S., Motschmann, U., Schüle, J., Glassmeier, K.-H., and Pringle, G. J.: A.I.K.E.F.: Adaptive hybrid model for space plasma simulations,
Comp. Phys. Comm, 182, 946–966, https://doi.org/10.1016/j.cpc.2010.12.033, 2011. a, b
Narita, Y.: Plasma turbulence in the solar system, Springer-Verlag Berlin Heidelberg, https://doi.org/10.1007/978-3-642-25667-7, 2012. a
Narita Y., Plaschke F., Magnes W., Fischer D., and Schmid, D.: Error estimate for
fluxgate magnetometer in-flight calibration on a spinning spacecraft, Geosci.
Instrum. Method Data Syst., 10, 13–24, https://doi.org/10.5194/gi-10-13-202, 2021. a
Oliveira, J. S., Langlais, B., Pais, M. A., and Amit, H.: A modified equivalent source dipole method to model partially distributed magnetic field measurements, with application to Mercury, J. Geophys. Res., 120, 1075–1094, https://doi.org/10.1002/2014JE004734, 2015. a
Oliveira, J. S., Hood, L. L., and Langlais, B.:
Constraining the early history of Mercury and its core dynamo by studying the crustal magnetic field, J. Geophys. Res., 124, 2382–2396, https://doi.org/10.1029/2019JE005938, 2019. a
Olsen, N.:
Ionospheric F currents at middle and low latitudes estimated from Magsat data,
J. Geophys. Res., 102, 4569–4576, https://doi.org/10.1029/96JA02949, 1997. a, b, c
Philpott, L. C., Johnson, C. L., Winslow, R. M., Anderson, B. J., Korth, H., Purucker, M. E., and Solomon, S. C.: Constraints on the secular variation of Mercury’s magnetic field from the
combined analysis of Messenger and Mariner 10 data, Geophys. Res. Lett., 41, 6627–6634,
https://doi.org/10.1002/2014GL061401, 2014. a
Plattner, A. M. and Johnson, C. L.: Mercury's northern rise core-field magnetic anomaly, Geophys. Res. Lett., 48, e2021GL094695, https://doi.org/10.1029/2021GL094695, 2021. a
Slavin, J. A., Imber, S. M., and Raines J. M.: A Dungey cycle in the life of Mercury's magnetosphere, in: Magnetospheres in the Solar System, edited by: Maggiolo, R., André, N., Hasegawa, H., Welling, D. T., Zhang, Y., Paxton, L. J., Geophys. Monogr. Ser., https://doi.org/10.1002/9781119815624.ch34, 2021.
a
Tikhonov, A. N., Goncharsky, A., Stepanov, V. V., and Yagola, A. G.: Numerical methods for the solution of ill-posed problems, Springer Netherlands, Netherlands, https://doi.org/10.1007/978-94-015-8480-7, 1995. a, b
Toepfer, S., Narita, Y., Heyner, D., and Motschmann, U.:
Error propagation of Capon's minimum variance estimator,
Front. Phys., 9, 684410, https://doi.org/10.3389/fphy.2021.684410, 2021b. a, b, c, d
Wang, J., Huo, Z., and Zhang, L.: Reconstructing Mercury's magnetic field in magnetosphere using radial basis functions, Planet. Space Sci., 210, 105379, https://doi.org/10.1016/j.pss.2021.105379, 2021. a, b
Short summary
Revealing the nature of Mercury’s internal magnetic field is one of the primary goals of the BepiColombo mission. Besides the parametrization of the magnetic field contributions, the application of a robust inversion method is of major importance. The present work provides an overview of the most commonly used inversion methods and shows that Capon’s method as well as the Tikhonov regularization enable a high-precision determination of Mercury’s internal magnetic field up to the fifth degree.
Revealing the nature of Mercury’s internal magnetic field is one of the primary goals of the...
