Articles | Volume 41, issue 2
https://doi.org/10.5194/angeo-41-449-2023
© Author(s) 2023. 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-41-449-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Regular paper
| 
07 Nov 2023
Regular paper |
| 07 Nov 2023
| 07 Nov 2023
The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Karl-Heinz Glassmeier
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Max Planck Institute of Solar System Research, Göttingen, Germany
Ferdinand Plaschke
Institute of Geophysics and Extraterrestrial Physics, Technische Universität Braunschweig, Braunschweig, Germany
Simon Toepfer
Institute for Theoretical Physics, Technische Universität Braunschweig, Braunschweig, Germany
Uwe Motschmann
Institute for Theoretical Physics, Technische Universität Braunschweig, Braunschweig, Germany
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Tomas Karlsson, Ferdinand Plaschke, Austin N. Glass, and Jim M. Raines
Ann. Geophys., 42, 117–130, https://doi.org/10.5194/angeo-42-117-2024, https://doi.org/10.5194/angeo-42-117-2024, 2024
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The solar wind interacts with the planets in the solar system and creates a supersonic shock in front of them. The upstream region of this shock contains many complicated phenomena. One such phenomenon is small-scale structures of strong magnetic fields (SLAMS). These SLAMS have been observed at Earth and are important in determining the properties of space around the planet. Until now, SLAMS have not been observed at Mercury, but we show for the first time that SLAMS also exist there.
Niklas Grimmich, Ferdinand Plaschke, Benjamin Grison, Fabio Prencipe, Christophe Philippe Escoubet, Martin Owain Archer, Ovidiu Dragos Constantinescu, Stein Haaland, Rumi Nakamura, David Gary Sibeck, Fabien Darrouzet, Mykhaylo Hayosh, and Romain Maggiolo
EGUsphere, https://doi.org/10.5194/egusphere-2024-1087, https://doi.org/10.5194/egusphere-2024-1087, 2024
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In our study, we looked at the boundary between the Earth's magnetic field and the interplanetary magnetic field emitted by the Sun, called the magnetopause. While other studies focus on the magnetopause motion near Earth's equator, we have studied it in polar regions. The motion of the magnetopause is faster towards the Earth than towards the Sun. We also found that the occurrence of unusual magnetopause locations is due to similar solar influences in the equatorial and in the polar regions.
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
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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.
Adrian Pöppelwerth, Georg Glebe, Johannes Z. D. Mieth, Florian Koller, Tomas Karlsson, Zoltan Vörös, and Ferdinand Plaschke
Ann. Geophys. Discuss., https://doi.org/10.5194/angeo-2023-31, https://doi.org/10.5194/angeo-2023-31, 2023
Revised manuscript accepted for ANGEO
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In the magnetosheath, a near-Earth region of space, we observe increases in plasma velocity and density called jets. As they propagate, the jets disturb the ambient plasma. We study this interaction with three spacecraft simultaneously. Using the plasma motion, we reconstruct the propagation axis and calculate a simple model of the jet shape. We find that the cross section perpendicular to the propagation and the dynamic pressure decrease from the center of the jet to the rear.
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
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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
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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, Ida Oertel, Vanita Schiron, Yasuhito Narita, Karl-Heinz Glassmeier, Daniel Heyner, Patrick Kolhey, and Uwe Motschmann
Ann. Geophys., 40, 91–105, https://doi.org/10.5194/angeo-40-91-2022, https://doi.org/10.5194/angeo-40-91-2022, 2022
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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.
Martin Volwerk, Beatriz Sánchez-Cano, Daniel Heyner, Sae Aizawa, Nicolas André, Ali Varsani, Johannes Mieth, Stefano Orsini, Wolfgang Baumjohann, David Fischer, Yoshifumi Futaana, Richard Harrison, Harald Jeszenszky, Iwai Kazumasa, Gunter Laky, Herbert Lichtenegger, Anna Milillo, Yoshizumi Miyoshi, Rumi Nakamura, Ferdinand Plaschke, Ingo Richter, Sebastián Rojas Mata, Yoshifumi Saito, Daniel Schmid, Daikou Shiota, and Cyril Simon Wedlund
Ann. Geophys., 39, 811–831, https://doi.org/10.5194/angeo-39-811-2021, https://doi.org/10.5194/angeo-39-811-2021, 2021
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On 15 October 2020, BepiColombo used Venus as a gravity assist to change its orbit to reach Mercury in late 2021. During this passage of Venus, the spacecraft entered into Venus's magnetotail at a distance of 70 Venus radii from the planet. We have studied the magnetic field and plasma data and find that Venus's magnetotail is highly active. This is caused by strong activity in the solar wind, where just before the flyby a coronal mass ejection interacted with the magnetophere of Venus.
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
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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.
Martin Volwerk, David Mautner, Cyril Simon Wedlund, Charlotte Goetz, Ferdinand Plaschke, Tomas Karlsson, Daniel Schmid, Diana Rojas-Castillo, Owen W. Roberts, and Ali Varsani
Ann. Geophys., 39, 239–253, https://doi.org/10.5194/angeo-39-239-2021, https://doi.org/10.5194/angeo-39-239-2021, 2021
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The magnetic field in the solar wind is not constant but varies in direction and strength. One of these variations shows a strong local reduction of the magnetic field strength and is called a magnetic hole. These holes are usually an indication that there is, or has been, a temperature difference in the plasma of the solar wind, with the temperature along the magnetic field lower than perpendicular. The MMS spacecraft data have been used to study the characteristics of these holes near Earth.
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
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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
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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
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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.
Ovidiu Dragoş Constantinescu, Hans-Ulrich Auster, Magda Delva, Olaf Hillenmaier, Werner Magnes, and Ferdinand Plaschke
Geosci. Instrum. Method. Data Syst., 9, 451–469, https://doi.org/10.5194/gi-9-451-2020, https://doi.org/10.5194/gi-9-451-2020, 2020
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We propose a gradiometer-based technique for cleaning multi-sensor magnetic field data acquired on board spacecraft. The technique takes advantage on the fact that the maximum-variance direction of many AC disturbances on board spacecraft does not change over time. We apply the proposed technique to the SOSMAG instrument on board GeoKompsat-2A. We analyse the performance and limitations of the technique and discuss in detail how various disturbances are removed.
Daniel Schmid, Ferdinand Plaschke, Yasuhito Narita, Daniel Heyner, Johannes Z. D. Mieth, Brian J. Anderson, Martin Volwerk, Ayako Matsuoka, and Wolfgang Baumjohann
Ann. Geophys., 38, 823–832, https://doi.org/10.5194/angeo-38-823-2020, https://doi.org/10.5194/angeo-38-823-2020, 2020
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Recently, the two-spacecraft mission BepiColombo was launched to explore Mercury. To measure the magnetic field precisely, in-flight calibration of the magnetometer offset is needed. Usually, the offset is evaluated from magnetic field observations in the solar wind. Since one of the spacecraft will remain within Mercury's magnetic environment, we examine an alternative calibration method. We show that this method is applicable and may be a valuable tool to determine the offset accurately.
Ferdinand Plaschke, Maria Jernej, Heli Hietala, and Laura Vuorinen
Ann. Geophys., 38, 287–296, https://doi.org/10.5194/angeo-38-287-2020, https://doi.org/10.5194/angeo-38-287-2020, 2020
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Jets of solar-wind plasma commonly hit the Earth's magnetosphere. Using data from the four Magnetospheric Multiscale (MMS) spacecraft, we show statistically that within jets the magnetic field is more aligned with the plasma flow direction than outside of these jets. Our study confirms prior simulation results, but it also shows that the average effect is moderate. The jets' magnetic field is important with respect to their impact on space weather.
Martin Volwerk, Charlotte Goetz, Ferdinand Plaschke, Tomas Karlsson, Daniel Heyner, and Brian Anderson
Ann. Geophys., 38, 51–60, https://doi.org/10.5194/angeo-38-51-2020, https://doi.org/10.5194/angeo-38-51-2020, 2020
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The magnetic field that is carried by the solar wind slowly decreases in strength as it moves further from the Sun. However, there are sometimes localized decreases in the magnetic field strength, called magnetic holes. These are small structures where the magnetic field strength decreases to less than 50 % of the surroundings and the plasma density increases. This paper presents a statistical study of the behaviour of these holes between Mercury and Venus using MESSENGER data.
Ferdinand Plaschke
Geosci. Instrum. Method. Data Syst., 8, 285–291, https://doi.org/10.5194/gi-8-285-2019, https://doi.org/10.5194/gi-8-285-2019, 2019
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Measuring the magnetic field onboard spacecraft requires regular in-flight calibration activities. Among those, determining the output of magnetometers under vanishing ambient magnetic fields, the so-called magnetometer offsets, is essential. Typically, characteristic rotations in solar wind magnetic fields are used to obtain these offsets. This paper addresses the question of how many solar wind data are needed to reach certain accuracy levels in offset determination.
Horia Comişel, Yasuhito Narita, and Uwe Motschmann
Ann. Geophys., 37, 835–842, https://doi.org/10.5194/angeo-37-835-2019, https://doi.org/10.5194/angeo-37-835-2019, 2019
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Here we present a scenario that the decay of a field-aligned Alfvén wave can occur simultaneously at various angles to the mean magnetic field, generating a number of second-order fluctuations or waves (after the pump wave as the first-order fluctuation). We refer to the simultaneous decay as
multi-channel couplingsfollowing the notion in scattering theory. Our goal is to study the hypothesis of the multi-channel coupling by running a three-dimensional hybrid plasma simulation.
Laura Vuorinen, Heli Hietala, and Ferdinand Plaschke
Ann. Geophys., 37, 689–697, https://doi.org/10.5194/angeo-37-689-2019, https://doi.org/10.5194/angeo-37-689-2019, 2019
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Before the solar wind encounters the Earth's magnetic field, it is first slowed down and deflected by the Earth's bow shock. We find that downstream of the bow shock regions where the shock normal and the solar wind magnetic field are almost parallel and the shock is more rippled, plasma jets with high earthward velocities are observed significantly more often than elsewhere downstream of the shock. Our results help us forecast the occurrence of these jets and their effects on Earth.
Johannes Z. D. Mieth, Dennis Frühauff, and Karl-Heinz Glassmeier
Ann. Geophys., 37, 163–169, https://doi.org/10.5194/angeo-37-163-2019, https://doi.org/10.5194/angeo-37-163-2019, 2019
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The magnetopause (MP) is the primary interaction region between solar wind and the magnetic field of planet Earth and understanding of its behaviour also helps to better understand space weather. One famous model of the MP is the Shue et al. model, designed for the dayside and near-Earth situation. We take data of the ARTEMIS mission orbiting the moon and compare the MP position and shape to the model. We find differences in the location prediction but good agreement for the MP normal direction.
Ferdinand Plaschke, Hans-Ulrich Auster, David Fischer, Karl-Heinz Fornaçon, Werner Magnes, Ingo Richter, Dragos Constantinescu, and Yasuhito Narita
Geosci. Instrum. Method. Data Syst., 8, 63–76, https://doi.org/10.5194/gi-8-63-2019, https://doi.org/10.5194/gi-8-63-2019, 2019
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Raw output of spacecraft magnetometers has to be converted into meaningful units and coordinate systems before it is usable for scientific applications. This conversion is defined by 12 calibration parameters, 8 of which are more easily determined in flight if the spacecraft is spinning. We present theory and advanced algorithms to determine these eight parameters. They take into account the physical magnetometer and spacecraft behavior, making them superior to previously published algorithms.
Yasuhito Narita and Uwe Motschmann
Ann. Geophys., 36, 1537–1543, https://doi.org/10.5194/angeo-36-1537-2018, https://doi.org/10.5194/angeo-36-1537-2018, 2018
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Venus has no intrinsic magnetic field. On the other hand, we discover that an interplanetary magnetic field may nevertheless penetrate the planetary ionosphere by the diffusion process and reach the planetary surface when the solar wind condition remains for a sufficiently long time, between 12 and 54 h, depending on the condition of ionosphere.
Minna Palmroth, Heli Hietala, Ferdinand Plaschke, Martin Archer, Tomas Karlsson, Xóchitl Blanco-Cano, David Sibeck, Primož Kajdič, Urs Ganse, Yann Pfau-Kempf, Markus Battarbee, and Lucile Turc
Ann. Geophys., 36, 1171–1182, https://doi.org/10.5194/angeo-36-1171-2018, https://doi.org/10.5194/angeo-36-1171-2018, 2018
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Magnetosheath jets are high-velocity plasma structures that are commonly observed within the Earth's magnetosheath. Previously, they have mainly been investigated with spacecraft observations, which do not allow us to infer their spatial sizes, temporal evolution, or origin. This paper shows for the first time their dimensions, evolution, and origins within a simulation whose dimensions are directly comparable to the Earth's magnetosphere. The results are compared to previous observations.
Ferdinand Plaschke and Heli Hietala
Ann. Geophys., 36, 695–703, https://doi.org/10.5194/angeo-36-695-2018, https://doi.org/10.5194/angeo-36-695-2018, 2018
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Fast jets of solar wind particles drive through a slower environment in the magnetosheath, located sunward of the region dominated by the Earth’s magnetic field. THEMIS multi-spacecraft observations show that jets push ambient particles out of their way. These particles flow around the faster jets into the jets’ wake. Thereby, jets stir the magnetosheath, changing the properties of this key region whose particles and magnetic fields can directly interact with the Earth’s magnetic field.
Tomas Karlsson, Ferdinand Plaschke, Heli Hietala, Martin Archer, Xóchitl Blanco-Cano, Primož Kajdič, Per-Arne Lindqvist, Göran Marklund, and Daniel J. Gershman
Ann. Geophys., 36, 655–677, https://doi.org/10.5194/angeo-36-655-2018, https://doi.org/10.5194/angeo-36-655-2018, 2018
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We have studied fast plasma jets outside of Earth’s magnetic environment. Such jets are small-scale structures with a limited lifetime, which may be important in determining the properties of the near-Earth space environment, due to their concentrated kinetic energy. We have used data from the NASA Magnetospheric MultiScale (MMS) satellites to study their properties in detail, to understand how these jets are formed. We have found evidence that there are at least two different types of jets.
Sudong Xiao, Tielong Zhang, Guoqiang Wang, Martin Volwerk, Yasong Ge, Daniel Schmid, Rumi Nakamura, Wolfgang Baumjohann, and Ferdinand Plaschke
Ann. Geophys., 35, 1015–1022, https://doi.org/10.5194/angeo-35-1015-2017, https://doi.org/10.5194/angeo-35-1015-2017, 2017
Dennis Frühauff, Ferdinand Plaschke, and Karl-Heinz Glassmeier
Ann. Geophys., 35, 117–121, https://doi.org/10.5194/angeo-35-117-2017, https://doi.org/10.5194/angeo-35-117-2017, 2017
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Vector magnetic field instruments mounted on spacecraft require precise in-flight calibration of the offsets of all three axes, i.e., the output in vanishing ambient field. While calibration of the spin plane offsets is trivial, we apply a new technique for determining the spin axis offset, not relying on solar wind data but on magnetosheath encounters. This technique is successfully applied to the satellites of the THEMIS mission to update the calibration parameters of the complete mission.
Martin Volwerk, Daniel Schmid, Bruce T. Tsurutani, Magda Delva, Ferdinand Plaschke, Yasuhito Narita, Tielong Zhang, and Karl-Heinz Glassmeier
Ann. Geophys., 34, 1099–1108, https://doi.org/10.5194/angeo-34-1099-2016, https://doi.org/10.5194/angeo-34-1099-2016, 2016
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The behaviour of mirror mode waves in Venus's magnetosheath is investigated for solar minimum and maximum conditions. It is shown that the total observational rate of these waves does not change much; however, the distribution over the magnetosheath is significantly different, as well as the growth and decay of the waves during these different solar activity conditions.
David Fischer, Werner Magnes, Christian Hagen, Ivan Dors, Mark W. Chutter, Jerry Needell, Roy B. Torbert, Olivier Le Contel, Robert J. Strangeway, Gernot Kubin, Aris Valavanoglou, Ferdinand Plaschke, Rumi Nakamura, Laurent Mirioni, Christopher T. Russell, Hannes K. Leinweber, Kenneth R. Bromund, Guan Le, Lawrence Kepko, Brian J. Anderson, James A. Slavin, and Wolfgang Baumjohann
Geosci. Instrum. Method. Data Syst., 5, 521–530, https://doi.org/10.5194/gi-5-521-2016, https://doi.org/10.5194/gi-5-521-2016, 2016
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This paper describes frequency and timing calibration, modeling and data processing and calibration for MMS magnetometers, resulting in a merged search choil and fluxgate data product.
Ferdinand Plaschke and Yasuhito Narita
Ann. Geophys., 34, 759–766, https://doi.org/10.5194/angeo-34-759-2016, https://doi.org/10.5194/angeo-34-759-2016, 2016
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Spacecraft-mounted magnetic field instruments (magnetometers) need to be routinely calibrated. This involves determining the magnetometer outputs in vanishing ambient magnetic fields, the so-called offsets. We introduce and test a new method to determine these offsets with high accuracy, the mirror mode method, which is complementary to existing methods. The mirror mode method should be highly beneficial to current and future magnetic field observations near Earth, other planets, and comets.
Y. Narita, H. Comişel, and U. Motschmann
Ann. Geophys., 34, 591–593, https://doi.org/10.5194/angeo-34-591-2016, https://doi.org/10.5194/angeo-34-591-2016, 2016
D. Schmid, M. Volwerk, F. Plaschke, Z. Vörös, T. L. Zhang, W. Baumjohann, and Y. Narita
Ann. Geophys., 32, 651–657, https://doi.org/10.5194/angeo-32-651-2014, https://doi.org/10.5194/angeo-32-651-2014, 2014
R. Nakamura, F. Plaschke, R. Teubenbacher, L. Giner, W. Baumjohann, W. Magnes, M. Steller, R. B. Torbert, H. Vaith, M. Chutter, K.-H. Fornaçon, K.-H. Glassmeier, and C. Carr
Geosci. Instrum. Method. Data Syst., 3, 1–11, https://doi.org/10.5194/gi-3-1-2014, https://doi.org/10.5194/gi-3-1-2014, 2014
F. Plaschke, H. Hietala, and V. Angelopoulos
Ann. Geophys., 31, 1877–1889, https://doi.org/10.5194/angeo-31-1877-2013, https://doi.org/10.5194/angeo-31-1877-2013, 2013
C. Nabert, K.-H. Glassmeier, and F. Plaschke
Ann. Geophys., 31, 419–437, https://doi.org/10.5194/angeo-31-419-2013, https://doi.org/10.5194/angeo-31-419-2013, 2013
Related subject area
Subject: Magnetosphere & space plasma physics | Keywords: Experimental and mathematical techniques
Concerning the detection of electromagnetic knot structures in space plasmas using the wave telescope technique
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
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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.
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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.
The upper detection limit in reciprocal space, the spatial Nyquist limit, is derived for...
