Magnetosheath Thickness Research Articles

Mercury’s Bow Shock and Magnetopause Variations According to MESSENGER Data

Using data from the MESSENGER spacecraft magnetometer that describes the magnetopause and the bow shock crossing points of the Mercury’s magnetosphere, we have calculated the parameters of the paraboloids of revolution approximating the obtained points. For each spacecraft orbit, the subsolar magnetopause and bow shock standoff distances were obtained, based on the paraboloid parameters for each crossing point. The dependences of the magnetopause and bow shock subsolar standoff distances on the Mercury’s position relative to the Sun have been obtained. These profiles agree with decreases of the solar wind plasma dynamic pressure and the interplanetary magnetic field strength with heliocentric distance. The variations of the interplanetary and magnetosheath magnetic field were investigated. The average subsolar magnetosheath thickness and the value of the magnetic field jump at the bow shock during the transition from the upstream interplanetary magnetic field region to the magnetosheath were obtained.

Cross-comparison of global simulation models applied to Mercury’s dayside magnetosphere

We present the first comparison of multiple global simulations of the solar wind interaction with Mercury’s dayside magnetosphere, conducted in the framework of the international collaborative project SHOTS - Studies on Hermean magnetosphere Oriented Theories and Simulations. Two 3D magnetohydrodynamic and two 3D hybrid simulation codes are used to investigate the global response of the Hermean magnetosphere without its exosphere to a northward-oriented interplanetary magnetic field. We cross-compare the results of the four codes for a theoretical case and a MESSENGER orbit with similar upstream plasma conditions. The models agree on bowshock and magnetopause locations at 2.1 ​± ​0.11 and 1.4 ​± ​0.1 Mercury planetary radii, respectively. The latter locations may be influenced by subtle differences in the treatment of the plasma boundary at the planetary surface. The predicted magnetosheath thickness varies less between the codes. Finally, we also sample the plasma data along virtual trajectories of BepiColombo’s Magnetospheric and Planetary Orbiter. Our ability to accurately predict the structure of the Hermean magnetosphere aids the analysis of the onboard plasma measurements of past and future magnetospheric missions.

Magnetosheath Propagation Time of Solar Wind Directional Discontinuities

AbstractObserved delays in the ground response to solar wind directional discontinuities have been explained as the result of larger than expected magnetosheath propagation times. Recently, Samsonov et al. (2017, https://doi.org/10.1002/2017GL075020) showed that the typical time for a southward interplanetary magnetic field (IMF) turning to propagate across the magnetosheath is 14 min. Here by using a combination of magnetohydrodynamic simulations, spacecraft observations, and analytic calculations, we study the dependence of the propagation time on solar wind parameters and near‐magnetopause cutoff speed. Increases in the solar wind speed result in greater magnetosheath plasma flow velocities, decreases in the magnetosheath thickness and, as a result, decreases in the propagation time. Increases in the IMF strength result in increases in the magnetosheath thickness and increases in the propagation time. Both magnetohydrodynamic simulations and observations suggest that propagation times are slightly smaller for northward IMF turnings. Magnetosheath flow deceleration must be taken into account when predicting the arrival times of solar wind structures at the dayside magnetopause.

A new method for solving the MHD equations in the magnetosheath

Abstract. We present a new analytical method to derive steady-state magnetohydrodynamic (MHD) solutions of the magnetosheath in different levels of approximation. With this method, we calculate the magnetosheath's density, velocity, and magnetic field distribution as well as its geometry. Thereby, the solution depends on the geomagnetic dipole moment and solar wind conditions only. To simplify the representation, we restrict our model to northward IMF with the solar wind flow along the stagnation streamline. The sheath's geometry, with its boundaries, bow shock and magnetopause, is determined self-consistently. Our model is stationary and time relaxation has not to be considered as in global MHD simulations. Our method uses series expansion to transfer the MHD equations into a new set of ordinary differential equations. The number of equations is related to the level of approximation considered including different physical processes. These equations can be solved numerically; however, an analytical approach for the lowest-order approximation is also presented. This yields explicit expressions, not only for the flow and field variations but also for the magnetosheath thickness, depending on the solar wind parameters. Results are compared to THEMIS data and offer a detailed explanation of, e.g., the pile-up process and the corresponding plasma depletion layer, the bow shock and magnetopause geometry, the magnetosheath thickness, and the flow deceleration.

Thin magnetosheath as a consequence of the magnetopause deformation: THEMIS observations

This paper presents a simultaneous observation of the bow shock and magnetopause by THEMIS probes that allows determination of the actual magnetosheath thickness at the subsolar point. Moreover, Geotail located at the dusk dayside magnetosheath registered a brief excursion to the magnetosphere in this time. The spacecraft configuration reveals a significant deformation of the magnetopause surface that locally decreases its curvature radius. The highly curved magnetopause results in the decrease of the magnetosheath thickness to about half of its standard value in a particular observation point. The observed phenomenon is attributed to a rotation of the interplanetary magnetic field (IMF). Although it is generally expected that the bow shock and magnetopause move in accord, being driven mainly by the solar wind dynamic pressure, we suggest that the local and transient thinning of the magnetosheath can result from different responses of its boundaries to a sudden change of the pressure and/or IMF orientation.

Average Thickness of Magnetosheath Upstream of Magnetic Clouds at 1 AU versus Solar Longitude of Source

Starting with a large number (N = 100) of Wind magnetic clouds (MCs) and applying necessary restrictions, we find a proper set of N = 29 to investigate the average ecliptic plane projection of the upstream magnetosheath thickness as a function of the longi- tude of the solar source of the MCs, for those cases of MCs having upstream shock waves. A few of the obvious restrictions on the full set of MCs are the need for there to exist a driven upstream shock wave, knowledge of the MC's solar source, and restriction to only MCs of low axial latitudes. The analysis required splitting this set into two subsets according to av- erage magnetosheath speed: slow/average (300 - 500 km s −1 ) and fast (500 - 1100 km s −1 ) speeds. Only the fast set gives plausible results, where the estimated magnetosheath thick- ness (�S) goes from 0.042 to 0.079 AU (at 1 AU) over the longitude sector of 0° (adjusted source-center longitude of the average magnetic cloud) to 40° off center (East or West), based on N = 11 appropriate cases. These estimates are well determined with a sigma ( σ) for the fit of 0.0055 AU, where σ is effectively the same as √ (chi-squared) for the appro- priate quadratic fit. The associated linear correlation coefficient forS versus |Longitude| was very good (c.c. = 0.93) for the fast range, andS at 60° longitude is extrapolated to be 2.7 times the value at 0°. For the slower speeds we obtain the surprising result thatS is typically more-or-less constant at 0.040 ± 0.013 AU at all longitudes, indicating that the MC as a driver, when moving close to the normal solar wind speed, has little influence on magnetosheath thickness. In some cases, the correct choice between two candidate solar- source longitudes for a fast MC might be made by noting the value of the observedS just

Simulation of the dawn-dusk magnetosheath asymmetry under quasi-steady states

The dawn-dusk asymmetry of the magnetosheath under quasi-steady states has been studied by using a newly developed 3D MHD magnetosphere simulation model. The results show that the dawn-dusk asymmetry is substantial because of the Parker spiral IMF. It is found that the dawn-dusk magnetosheath thickness asymmetry is the effect of different shock conditions. The plasma density and flux asymmetry are mainly caused by the different thickness of the dawn-dusk magnetosheath, and the magnetic reconnection on the magnetopause has no significant effects. It is also showed that the Plasma Depletion Layer in front of the dayside magnetopause can cause duskward plasma flow, and the total plasma flux on the dusk side will be higher.

Comparative study of bow shock models using Wind and Geotail observations

Wind and Geotail observed bow shock (BS) crossings were selected from the 1998 to 2001 ISTP database. We analyzed 625 case events containing 4381 Geotail BS crossings and 130 case events containing 917 Wind BS crossings. The location of the BS crossings, in the aberrated GSE coordinate system, varied over a wide range from −85 Re to 45 Re along the X‐GSE axis, up to 90 Re in the perpendicular direction. ACE, Wind, and Geotail measurements were used to determine the upstream solar wind conditions in the interplanetary medium. These conditions were determined for the BS crossings in each case event by using the delay time of direct solar wind propagation from an upstream monitor to the probe satellite (Wind or Geotail). The solar wind conditions for the BS crossings varied over a wide range of dynamic pressures Pd (from 0.02 nPa to 49 nPa), IMF Bzs (from −26 nT to 23 nT), thermal/magnetic pressure ratios β (from 0.002 to 50), and magnetosonic Mach numbers Mms (from 1.02 to 29). Such a wide spatial and dynamic range of BS crossings permits us to consider the different parameters that control the BS size and shape, such as the radius of curvature of the magnetopause which depends on Pd and Bz, the Alfven, sonic, and magnetosonic Mach numbers, and the IMF orientation. To study the dependence on these parameters, we compared the accuracy of the BS models formulated by Peredo et al. [1995], Russell and Petrinec [1996], Verigin et al. [2001b], and Chao et al. [2002] for the prediction of selected BS crossings observed in different bow shock regions and with various upstream solar wind conditions. It was found that the Chao et al. [2002] model had the best capability for predicting the BS crossings. The solar wind dynamic pressure and magnetosonic Mach number were determined to be the most important parameters controlling the BS size and shape. The important role of the dawn‐dusk asymmetry of the bow shock tail region is emphasized. The effect of the southward IMF influence on the dayside magnetosheath thickness is revealed and discussed.

On the magnetosheath thicknesses of interplanetary coronal mass ejections

Interplanetary coronal mass ejections (ICMEs) often move relative to the ambient solar wind plasma at a speed that exceeds the fast magnetosonic velocity, producing a standing shock wave in the frame of the ICME and a magnetosheath whose plasma flows around the obstacle much as the terrestrial magnetosheath flows around the Earth's magnetosphere. The half-thickness of an ICME is typically 0.1 AU at 1 AU . If this dimension represented the characteristic scale size of the obstacle, then the thickness of the magnetosheath should be about 0.025 AU but it is typically closer to 0.1 AU . In order to treat this problem we convert the original Spreiter et al. (Planet. Space Sci. 14 (1966) 223) formula for the terrestrial magnetosheath thickness to one that is appropriate for the ICME magnetosheath. This treatment allows us to conclude that the characteristic radius of curvature of an ICME at 1 AU is about 0.4 AU . This radius of curvature is provided by both the bend of the axis of the rope and by an elongation of the ICME in the direction perpendicular to both the rope axis and the solar wind flow. Thus near 1 AU ICMEs have a radial thickness that is smaller than their other two characteristic dimensions.

Observations of the radial magnetosheath profile and a comparison with gasdynamic model predictions

The gasdynamic model of the magnetosheath flow predicts changes of plasma parameters across the magnetosheath. According to this model, the ion flux peaks at the bow shock and decreases toward the magnetopause. MHD models predict a further plasma depletion near the magnetopause for specific orientations of the interplanetary magnetic field. We present a statistical study of ion flux measurements on the dusk flank of the magnetosheath and its comparison with present magnetosheath models. The study is based on three years of INTERBALL‐1 measurements supported by simultaneous WIND solar wind observations. Our results reveal a much flatter magnetosheath radial profile than was expected from the gasdynamic prediction. The profile becomes steeper for higher upstream Mach number or plasma beta. The depletions of the ion flux behind the bow shock and at the magnetopause are attributed to the reflection and acceleration of particles in the bow shock region and magnetopause reconnection, respectively. A comparison of the measured and predicted ion fluxes suggests that the magnetosheath thickness is a rising function of the IMF magnitude.

Low Mach number predictions in an extended axially symmetric MHD theory of the magnetosheath

A recent axisymmetric magnetohydrodynamic (MHD) model for the Earth's magnetosheath near the Sun‐Earth line, with boundary conditions given by Rankine‐Hugoniot (RH) equations at the bow shock and the Chapman‐Ferraro (CF) condition at the magnetopause, is extended by including smaller terms in the CF condition that become important at low Mach numbers. Three conclusions are predicted for the magnetosheath. (1) The magnetosheath thickness Δms decreases with decreasing Alfven Mach number MA for MA > M* (in which only the “gasdynamic” root of the RH equations for the bow shock boundary is physical), where M* ∼ 2 is the root‐transition value for the RH equations at which all 3 roots coalesce. When MA < M*switch‐on shock roots (purely MHD) are also physical, and Δms associated with those roots shows a rapid increase to a large value, followed by a rapid decrease back to a very small value as the decreasing MA approaches 1. Thus Δms associated with these switch‐on roots may give more distant bow shocks, and may rapidly vary with small MA changes. (2) Δms is very sensitive to the value for the CF constant kCF at the magnetopause. For larger MA and sonic Mach numbers Ms the ratio of Δms to the geocentric radial distance amp of the magnetopause varies from 0.44 for 90% coupling efficiency of the solar wind momentum density to the magnetopause down to 0.18 for 60% coupling efficiency. This suggests a balance between the solar wind coupling efficiency to the magnetopause and the thickness of the magnetosheath, in which an increase (decrease) in the coupling efficiency increases (decreases) the thickness of the magnetosheath to counter that efficiency by moving the bow shock further upstream (downstream), hence minimizing the variation in time of that coupling efficiency. (3) The linear relation between Δms and the density ratio ρsw/ρbs observed in simulations breaks down for MA or Ms ≲ 2. The slope steadily drops as Ms → 1, but increases as MA → 1.

Magnetic field orientation effects on the standoff distance of Earth's bow shock

Three‐dimensional, global MHD simulations of solar wind flow onto a prescribed magnetopause obstacle are used to show that a bow shock's nose location as and the relative subsolar magnetosheath thickness Δms/amp are strong functions of the IMF cone angle θ (between vsw and Bsw) and the Alfven Mach number MA For a given MA the shock is more distant for higher θ (restricted to the interval 0–90° by symmetries), while as/amp and Δms/amp increase with decreasing MA for θ ≳ 20° but decrease with decreasing MA for θ ∼ 0°. Large differences in Δms/amp are predicted between θ = 0° and 90° at low MA, with smaller differences remaining even at MA ∼ 10. The θ = 0° results confirm and extend the previous work of Spreiter and Rizzi [1974]. The simulations show that successful models for the subsolar shock location cannot subsume the dependences on MA and θ into a sole dependence on Mms. Instead, they confirm a recent prediction [Cairns and Grabbe, 1994] that as/amp and Δms/amp should depend strongly on θ and MA for MA ≲ 10 (as well as on other MHD variables). Detailed comparisons between theory and data remain to be done. However, preliminary comparisons show good agreement, with distant shock locations found for low MA and large θ ≳ 45° and closer locations found for θ ≲ 20° even at MA ∼ 8.

Near‐simultaneous bow shock crossings by WIND and IMP 8 on December 1, 1994

Near‐simultaneous dawn‐side bow shock crossings by WIND and IMP 8 on December 1, 1994 are analyzed to determine shock location and shape and to examine the changes in shock structure and the foreshock MHD wave properties with increasing downstream distance. The WIND and IMP 8 crossings took place at sun‐Earth‐spacecraft angles of 64.7° and 115.3°, respectively. The solar wind speed and interplanetary magnetic field magnitude were near their long‐term average values. However, the orientation of the IMF was unusual in that it rotated from an angle of ∼50–60° to the sun‐Earth line at the beginning of the interval of shock crossings to less than 20° just after the final crossings. The ratio of the downstream to upstream components of the magnetic field tangential to the shock decreases from 4.1 at WIND to 3.1 at IMP 8 in general agreement with theory. In addition, the overshoot in the shock magnetic ramp observed at WIND is greatly diminished by the downstream distance of IMP 8. In the foreshock, MHD waves with periods of 10–20 s and amplitudes of 3–6 nT were observed at both spacecraft. However, at WIND they have a strong compressional component which is much weaker farther downstream at IMP 8. Unexpectedly, the radial distance of the shock at both spacecraft is only ∼80–85% of that predicted by recent models. Motivated by this event, we have statistically analyzed a larger data set of bow shock crossings which took place under quasi‐field‐aligned flow conditions. On this basis it is suggested that magnetosheath thickness may decrease by ∼10% as the IMF becomes increasingly flow aligned.

The polytropic index for the solar wind at Earth's bow shock

Analysis of spacecraft shock‐crossing data on the magnetosheath thickness to determine the polytropic index γ of the solar wind is shown to overestimate the value for γ when the gasdynamic (GD) equations are used for the shock jump. This is done by comparing GD estimates for γ extrapolated from observed data points to values obtained when more appropriate MHD equations are used. As a result of this bias a shock‐crossing data analysis carried out recently using the GD equations appears to have obtained a value for γ that is over 0.5 too large. This is in part associated with the breakdown of GD with respect to nonzero angle between the bow‐shock normal and the magnetic field for finite Alfven Mach numbers. The magnetosonic phenomenological model (sonic Mach number replaced with the magnetosonic Mach number in the GD equations), which was also recently used to determine γ, is found to reduce this bias by 70–90%. This analysis also strongly suggests that the phenomenological model inadequately treats variations in the angle between the bow‐shock normal and the magnetic field, which may also produce systematic errors in the experimental determination of γ using this model.

MHD theory of Earth's magnetosheath for an axisymmetric model

An analysis is made of the Earth's magnetosheath along the Sun‐Earth line under conditions that the IMF (Interplanetary Magnetic Field) is nearly parallel or anti‐parallel to the solar wind flow. MHD conservation equations in temporally‐averaged steady‐state form for the mass, momentum and energy density are combined with the magnetic divergence and induction equations, a hard conducting‐sphere model for the magnetopause, and an adiabatic equation of state in the magnetosheath. The equations are integrated from the nose of the bow shock to the magnetopause and reduced to a set of nonlinear‐coupled equations for the magnetosheath thickness and average magnetosheath parameters, which are then used to obtain a new equation for the thickness of the magnetosheath. This is the first analytical equation for the magnetosheath thickness that has been derived, and it exhibits an interesting functional dependence on Alfven and sonic Mach number MA and Ms,the angle θ0 between the bow shock normal and the IMF, the Chapman‐Ferraro constant k0 at the magnetopause, the polytropic index γ, and the thermal conductivity Qr at the bow shock. The thickness is found to decrease for decreasing Ms, but increase with decreasing MA It exhibits the qualitative feature found in both gasdynamic and θ0 ≥ 45° MHD simulations of an approximate linear variation of the magnetosheath thickness with the density jump ratio X across the bow shock, but it also exhibits a unique negative slope and an offset that is a function both of MA and of K0.

MHD simulations of Earth's bow shock at low Mach numbers: Standoff distances

Global, three‐dimensional, ideal MHD simulations of Earth's bow shock are reported for low Alfven Mach numbers MA and quasi‐perpendicular magnetic field orientations. The simulations use a hard, infinitely conducting magnetopause obstacle, with axisymmetric three‐dimensional location given by a scaled standard model, to directly address previous gasdynamic (GD) and field‐aligned MHD (FA‐MHD) work. Tests of the simulated shocks’ density jumps X for 1.4 ≲ MA ≲ 10 and the high MA shock location, and reproduction of the GD relation between magnetosheath thickness and X for quasi‐gasdynamic MHD runs with MA ≫ MS, confirm that the MHD code is working correctly. The MHD simulations show the standoff distance as increasing monotonically with decreasing MA. Significantly larger as are found at low MA than predicted by GD and phenomenological MHD models and FA‐MHD simulations, as required qualitatively by observations. The GD and FA‐MHD predictions err qualitatively, predicting either constant or decreasing as with decreasing MA. This qualitative difference between quasi‐perpendicular MHD and FA‐MHD simulations is direct evidence for as depending on the magnetic field orientation θ. The enhancement factor over the phenomenological MHD predictions at MA ∼ 2.4 agrees quantitatively with one observational estimate. A linear relationship is found between the magnetosheath thickness and X, modified both quantitatively and intrinsically by MHD effects from the GD result. The MHD and GD results agree in the high MA limit. An MHD theory is developed for as, restricted to sufficiently perpendicular θ and high sonic Mach numbers MS. It explains the simulation results with excellent accuracy. Observational and further simulation testing of this MHD theory, and of its predicted MA, θ, and MS effects, is desirable.

Three‐dimensional position and shape of the bow shock and their variation with Alfvénic, sonic and magnetosonic Mach numbers and interplanetary magnetic field orientation

A large set of bow shock crossings (i.e., 1392) observed by 17 spacecraft has been used to explore the three‐dimensional shape and location of the Earth's bow shock and its dependence on solar wind and interplanetary magnetic field (IMF) conditions. This study investigates deviations from gas dynamic flow models associated with the magnetic terms in the magnetohydrodynamic (MHD) equations. Empirical models predicting the statistical position and shape of the bow shock for arbitrary values of the solar wind pressure, IMF, and Alfvénic Mach number (MA) have been derived. Individual crossings have been rotated into aberrated GSE coordinates to remove asymmetries associated with the earth's orbital motion. Variations due to changes in solar wind dynamic pressure have been taken into consideration by normalizing the observed crossings to the average value 〈p〉 = 3.1 nPa. The resulting data set has been used to fit three‐dimensional bow shock surfaces and to explore the variations in these surfaces with sonic (MS), Alfvénic (MA) and magnetosonic (MMS) Mach numbers. Analysis reveals that among the three Mach numbers, MA provides the best ordering of the least square bow shock curves. The subsolar shock is observed to move Earthward while the flanks flare outward in response to decreasing MA; the net change represents a 6‐10% effect. Variations due to changes in the IMF orientation were investigated by rotating the crossings into geocentric interplanetary medium coordinates. Past studies have suggested that the north‐south extent of the bow shock surface exceeds the east‐west dimension due to asymmetries in the fast mode Mach cone. This study confirms such a north‐south versus east‐west asymmetry and quantifies its variation with MS, MA, MMS, and IMF orientation. A 2‐7% effect is measured, with the asymmetry being more pronounced at low Mach numbers. Combining the bow shock models with the magnetopause model of Roelof and Sibeck (1993), variations in the magnetosheath thickness at different local times are explored. The ratio of the bow shock size to the magnetopause size at the subsolar point is found to be 1.46; at dawn and dusk, the ratios are found to be 1.89 and 1.93, respectively. The subsolar magnetosheath thickness is used to derive the polytropic index γ according to the empirical relation of Spreiter et al. (1966). The resulting γ = 2.3 suggests the empirical formula is inadequate to describe the MHD interaction between the solar wind and the magnetosphere.

The magnetosphere as a sufficient source for upstream ions on November 1, 1984

We consider 10 tests which distinguish the source of energetic particles in upstream ion events. Some tests are definitive, and others inadequate. In particular, we note that the Fermi model requires a finite time of magnetic field line connection to the bow shock before energetic ions appear and inverse velocity dispersion when the interplanetary magnetic field rotates to a more radial direction but that these features are also expected for the leakage model under certain conditions. We present simultaneous magnetospheric, magnetosheath, and upstream energetic particle observations during two well‐documented upstream events on November 1, 1984, and apply the 10 tests. The events are unusual in that they occurred during a period when the magnetosphere was greatly compressed, providing for an increase in the energetic magnetospheric particle population and a decrease in the magnetosheath thickness. Magnetospheric leakage satisfactorily explains our observations on this day, but in situ Fermi acceleration does not. Consequently, we consider magnetospheric leakage a sufficient source for upstream particles during these two events.

An analytic treatment of the structure of the bow shock and magnetosheath

A detailed theoretical examination of the jump conditions of the bow shock in an ideal magneto‐hydrodynamic flow pattern is used to obtain two sets of approximate analytic representations of the parameters behind the bow shock. From these analytic representations contour maps can be drawn to show clearly the influence of the solar wind magnetic fields, especially their directions on these parameters. We next use these results to examine the expected thickness of the magnetosheath. In order to obtain an analytical solution that is tractable we assume that the average values of the parameters along the radial direction in the magnetosheath are simply equal to their values just behind the bow shock. We emphasize that this assumption allows variations of these parameters along the radial direction. A very simple approximate formula for the thickness of the magnetosheath is obtained which satisfies the boundary conditions and conservation laws of mass and momentum flux. Two years magnetometer data from the Isee satellite crossing the magnetosheath during the period between October 1977 to October 1979 are used to determine empirically the thickness of the magnetosheath and to compare these observed thickness with theoretical estimates. Because of the sensitive dependence of the formula for the magnetosheath thickness on the value of γ, the polytropic exponent of the plasma gas, we can use this comparison to determine the optimum value of γ. A value of γ=2 gives much better agreement than 3, 5/3 or 7/5.

Orientation and shape of the Earth's bow shock in three dimensions

Nearly 2500 shock crossings from HEOS-1, HEOS-2 and 5 IMP spacecraft, covering most of the northern and part of the southern bow shock surface for X values X > − 20 R E , have been used to carry out a detailed study of the three-dimensional shape and location of the bow shock. The influence of the different solar wind conditions has been reduced by normalising the observed crossings to an average solar wind dynamical pressure ( N 0 = 9.4 cm −3, V 0 = 450 km s −1). It has been shown that the shock surface is symmetric with respect to the ecliptic plane and intersects the coordinate axes at 11.9 R E ( X), + 27.0 and − 22.9 R E ( Y), + 23.9 and − 24.5 R E ( Z) for the average dynamical pressure ( N 0 = 9.4 cm −3, V 0 = 450 km s −1, with M A = 9.3 , M MS = 6.1) . The observed aberration of the shock surface is 8.9° ± 1°, i.e. 5.1° larger than the aberration predicted from the Earth's motion. This asymmetry around the solar wind apparent direction is described by equation (6) for different Mach numbers M A and confirms the predictions of Walters [ J. geophys. Res. 71, 1319 (1964)] and Michel [ J. geophys. Res. 70, 1 (1965)]. The magnetosheath thickness is 3.3 R E along the X-axis, 11.4 R E (+ Y), 8.7 R E (− Y), 9.9 R E (+ Z) and 10.9 R E along the negative Z axis.