On the Temperatures of Planetary Magnetosheaths at the Subsolar Points

This research explores the relationship between the temperatures of the solar corona and planetary magnetosheaths. Based on the second law of thermodynamics, the maximum temperature of the planetary magnetosheaths cannot exceed that of the solar corona. A theoretical investigation is presented into the expansion of the solar corona, the propagation of solar wind, and the compression of planetary magnetosheaths by bow shocks. The method used is general and fits the dynamics of multiple components, thermal anisotropy, and non‐Maxwellian plasmas in the steady state, and approximate formulas are obtained. The results indicate that, for the steady state, planetary magnetosheaths at the subsolar points in the solar system have approach peak mean temperatures. Second, a systematic statistical survey of the average temperature of the planetary magnetosheaths is presented and shows that the average plasma temperature of the subsolar point magnetosheaths of Earth and Saturn are 206 eV (2.39 MK) and 171 eV (1.98 MK), respectively, which are close to that of the corona. The statistical results are consistent with the theoretical estimations. These results are of significant use for estimating the thermal properties of the planetary magnetospheres.

The extremely hot corona expands outward and generates outflowing solar wind into interplanetary space (Barnes, 1992;Cranmer et al., 2017;Marsch, 1999;Parker, 1958). Near the Sun, the solar wind accelerates rapidly (McComas et al., 2007) and, with increasing distance from the Sun, the velocity of the solar wind increases whereas its density and temperature gradually decrease (Barnes et al., 1992;Cranmer et al., 2017;Koet et al., 1999;Marsch, 1999;McComas et al., 2007McComas et al., , 2008Parker, 1958). As per Parker's model (Parker, 1958), the velocity V of the solar wind far from the Sun varies with the heliocentric distance r and approximately follows the formula = 2 √ ln ( ∕ ) , where is the Parker critical heliocentric distance and is the critical velocity, and the density n of the solar wind is ∝ ( 2 ) −1 due to conservation of matter. The temperature of the solar wind decreases with distance from the Sun as ∝ − , where the factor ranges from 2/7 to 4/3 (Barnes, 1992;Cranmer et al., 2009;McComas et al., 2008;Scudder, 2015). Weber and Davis (1967) established a steady-solarwind model that considers the azimuthal component of the solar wind and indicates that the magnetic field in the solar wind may apply a torque to the Sun and lead to the loss of the angular momentum of the Sun. As the solar wind reaches the Earth at a heliocentric distance of 1 AU, measurements (Burlaga and Szabo, 1999) reveal that its mean velocity ≈ 400 km∕s , its mean density ≈ 5 cm −3 , its average electron temperature ≈ 0.1-0.2 MK (12-25 eV), its average proton temperature ≈ 0.01-0.40 MK (1.2-50 eV), and its average plasma temperature is ⟨ ⟩ ≈ 1 2 ( + ) ≈ 0.13 MK (15 eV), if helium ions are omitted (Burlaga and Szabo, 1999;Cranmer et al., 2017).
On the other hand, the solar wind expands with the magnetic field frozen in. According to the Parker spiral field model, the radial component of the interplanetary magnetic field (IMF) ∝ 1∕ 2 and its azimuthal component ∝ ( − 0) ∕ 2 (Parker, 1958). Near Earth at 1 AU, the magnetic strength of the IMF is ≈6 nT with the IMF spiral angle being ≈45°, and the ratio of thermal to magnetic energy in the solar wind is ≈ 0.1-6.0 (Burlaga and Szabo, 1999).
If the expansion of corona and solar wind can be regarded as a heat engine process, its thermal efficiency at 1 AU is = (⟨ ⟩ − ⟨ ⟩ ) ∕⟨ ⟩ ≈ (30 − 1.3) × 10 5 ∕ ( 3 × 10 6 ) = 96% . Thus, the solar corona heat engine rather effectively transfers heat into kinetic energy. When the solar wind passes by the planets, such as Mercury, Earth, Jupiter, or Saturn, it impacts their magnetic field to produce planetary magnetospheres and bow shocks. The upstream solar wind traverses the bow shocks and is compressed to form the denser and hotter plasmas of the magnetosheaths (Chapman et al., 2004;Masters et al., 2011;Petrinec & Russell, 1997). The physical parameters of the upstream solar wind and downstream magnetosheath plasmas approximately obey the Rankine-Hugoniot jump conditions (Hudson, 1970;Liu et al., 2007). The observations by the Mercury Surface, Space Environment, Geochemistry, and Ranging (MESSENGER) craft show that, the proton temperature in Mercury's magnetosheath at the subsolar point is ≈ 1.  with the most probable proton temperature being ≈ 3 MK (350 eV) (Gershman et al., 2013). Under extreme solar-wind conditions, the proton temperature in Mercury's magnetosheath can reach up to 6.0 MK (700 eV) (Slavin et al., 2014). According to observations by the Double Star Project from 2004 to 2005  the time-averaged electron temperature of Earth's magnetosheath on the dayside is ≈ 50 eV, while the time-averaged ion temperature is ≈ 200 eV (Shen et al., 2008). Therefore, the average plasma temperature of Earth's magnetosheath on the dayside is about ⟨ ⟩ ≈ 1 2 ( + ) ≈ 125 eV . A statistical analysis of the THEMIS observations (Wang et al., 2012) indicates that, at the subsolar point of Earth's dayside magnetosheath, the mean electron and ion temperatures are ≈ 40 eV and ≈ 210 eV, respectively; thus, the average plasma temperature at the subsolar point of Earth's magnetosheath is ⟨ ⟩ ≈ 1 2 ( + ) ≈ 125 eV . Given a fast solar wind, the mean electron and ion temperatures at the subsolar point of Earth's magnetosheath are ≈ 53 eV and ≈ 400 eV , respectively, with the average plasma temperature being ⟨ ⟩ = 1 2 ( + ) ≈ 227 eV . Based on the measurements by Voyager 1 and 2 of Jupiter and Saturn, Richardson (1987Richardson ( , 2002 revealed that protons in their SHEN ET AL.

10.1029/2022JA030782
3 of 15 magnetosheaths have a double-Maxwellian distribution and are composed of both cold and hot components with temperatures ≈ 100 eV and ≈ 600 eV , respectively. These two proton components have comparable densities, so the average proton temperature in the magnetosheaths of Jupiter and Saturn is estimated as ≈ ( + ) ∕2 ≈ 350 eV . The explorations of Saturn by Cassini find that the average ion temperature of Saturn's magnetosheath is ≈ 210-370 eV (Sergis et al., 2013). Thomsen et al. (2018) surveyed the features of Saturn's magnetosheath in detail based on Cassini measurements and showed that the mean temperatures of the electrons and protons at the subsolar point of Saturn's magnetosheath are ≈ 34 eV and ≈ 340 eV , respectively, with the average temperature of Saturn's magnetosheath being ⟨ ⟩ = 1 2 ( + ) ≈ 187eV . The temperature of both electrons and protons from Saturn's magnetosheath gradually decreases as we move away from the noon (Thomsen et al., 2018). Therefore, the observations indicate that the plasma temperatures of the planetary magnetosheaths are comparable to each other to within several MK or several hundred eV and are very close to the temperature of the solar corona.
Satellite observations of velocity distribution functions (VDFs) from solar wind and the magnetosheaths of planets (such as Earth, Mercury, Saturn, and Uranus) frequently exhibit non-Maxwellian features, which feature suprathermal tails at high energies and quasi-Maxwellian behavior at low energies (Christon et al., 1988;Maksimovic et al., 1997;Pierrard et al., 2004). Non-Maxwellian VDFs have also been found in the solar wind and around Earth's bow shock with two different temperatures and densities: a dense "core" thermal population superimposed on a hot "halo" superthermal population (Feldman et al., 1975(Feldman et al., , 1983a1983b;Gaelzer et al., 2008;Lin, 1998;Marsch, 2006). Distributions of superthermal tails are well modeled by the kappa distribution since it fits both the Maxwellian (thermal) and non-Maxwellian (superthermal) high-energy part of the distribution (Pierrard & Lemaire, 1996;Schippers et al., 2008). Electron VDFs in Earth's magnetosheath and magnetosphere have also been observed with flat tops instead of a quasi-Maxwellian low-energy part that can be fit by neither Maxwellian nor kappa distribution functions. Such flat-top VDFs often have one or two distinct components and are well fit by a generalized (r,q) distribution function (Qureshi et al., 2004(Qureshi et al., , 2019. The magnetosheaths are also rather turbulent and filled with various waves, such as fast and slow magnetosonic waves, Alfven waves, mirror modes, whistler waves, and even solitary waves (Sckopke et al., 1990;Song et al., 1990Song et al., , 1992Song et al., , 1994Southwood & Kivelson, 1995).
As for the simplified cases when the magnetic field, solar gravity, and coronal heat conduction may be omitted, the expansion of the solar corona may be regarded as an adiabatic process, with the plasma entropy conserved. The coronal plasmas with an extremely high temperature expand outward from a stationary state, accelerate into interplanetary space, and decelerate at bow shocks in the vicinity of planets to form the hot magnetosheath plasmas with exceedingly small bulk velocities. According to the second law of thermodynamics, the plasma temperatures of the planetary magnetosheaths cannot exceed the maximum temperature of the source region plasmas-the solar corona (i.e., ≤ ). However, in actual situations, some of the thermal energy of the corona is spent to overcome the pull of solar gravity, and the magnetic field in the corona may also accelerate the solar wind. Furthermore, the electrons and ions of the solar wind or the magnetosheaths have different temperatures, making this a complex problem with many factors. Therefore, a detailed theoretical investigation is required for the whole process: outward expansion of the solar corona, propagation of the solar wind, and compression of the planetary magnetosheaths by the bow shocks. The result should lead to a quantitative relationship between the temperatures of the solar corona and that of the planetary magnetosheaths.
This research analyzes theoretically in Section 2 the motion of the solar wind and presents approximations relating the mean temperatures of the solar corona to that of the planetary magnetosheaths for the steady-state situations. Section 3 pursues statistical investigations into features of the plasma temperatures in planetary magnetosheaths, and Section 4 presents the discussion and conclusions.

Theoretical Analysis of Physical Processes
We investigate the propagation of the solar wind from the corona to the planetary magnetosheaths, considering its multiple components, thermal anisotropy, and non-Maxwellian features. Multicomponent magnetohydrodynamics (MHD) describes approximately the coronal expansion and the propagation of the solar wind (Echim et al., 2011;Parker, 1958).

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In the solar system, the planets orbit the Sun in the ecliptic plane, so we investigate the steady propagation of the solar wind in this plane, as illustrated in Figure 1. To make the physics explicit and facilitate the analysis, we first present a short derivation of Bernoulli's equation that is applicable to the solar wind with its multiple components, thermal anisotropy, and non-Maxwellian features.
The steady coronal solar wind and magnetosheath plasmas obey the continuity equation: where and are the bulk velocity and mass density of the plasmas, respectively. For simplicity, we assume that the plasma electrons and ions have the same bulk velocities. Generally, the notation = V̂ is used, where ̂ is the unit radial vector with respect to the heliocenter. The mass density of the plasmas can be expressed as = ∑ a a , where m a and n a are the mass and number densities of species a, respectively.
The steady expansion of the corona, the propagation of the solar wind, and the compression of the magnetosheath plasmas also obey the following equation of energy (Rossi & Olbert, 1970;Echim et al., 2011): where the total energy of the solar wind is composed of kinetic, thermal, electromagnetic, and gravitational potential energy. The thermal energy density of the solar wind is where a T is the thermal energy density of species a, a ‖ and a ⟂ are the temperature of the species a parallel and perpendicular to the magnetic field, respectively, k is Boltzmann's constant, and em is the electromagnetic energy density. The gravitational potential = − S∕ ，where G is the gravitational constant, M S is the mass of the Sun, and r is the heliocentric distance. In contrast with the solar gravity, the gravity due to planets is rather weak so any effects on the solar wind are omitted. The thermal pressure tensor of species a is defined as P a ij = ∫ vipjfa( , )d 3 in the frame of reference of the plasma bulk velocity , where fa( , ) is the phase-space density of species a at position and momentum (Rossi & Olbert, 1970). The phase-space density of the solar wind and of the planetary magnetosheath plasmas can be non-Maxwellian (Maksimovic et al., 1997;Richardson, 2002;Qureshi et al., 2014;Qureshi et al., 2019;Vasyliunas et al., 1968). The components of the total thermal pressure tensor of the magnetized plasmas in the coordinates of the magnetic field are where the pressure of species a parallel and perpendicular to the magnetic field is P a ⟂ = nakT a ⟂ and P a ‖ = nakT a ‖ , respectively. The flux density of electromagnetic energy is Generally, the magnetic flux is frozen in the plasmas and = − × , so that . In Equation 2, is the total heat flux in the solar wind. Presently, we still do not understand the heating mechanism of the solar corona and it is still under investigation (Cranmer et al., 2017;Klimchuk, 2015;McComas et al., 2007;Parnell & De Moortel, 2012). Given that the heating and radiative losses are at equilibrium in the core region of the solar corona, the temperature of the coronal plasmas is extremely high. This investigation assumes that the coronal plasma expands under the thermodynamic force or pressure and thereby produces the outward-flowing solar wind. Therefore, any heating and radiative losses in the coronal region are not considered here.
Applying the equation of conservation of energy (2) in the steady state to a tube with cross-sectional area A enclosing the solar wind (see Figure 1) yields Here, Bt is the component of the magnetic field perpendicular to the radial direction.
On the other hand, the continuity Equation 1 reduces to Comparing the Equations 6 and 7 yields Denote the angle between and as . In the steady state, ≈ 0 • in the outer corona, whereas ≈ 45 • at 1 AU. In Cartesian coordinates with respect to the magnetic field, as illustrated in Figure 2, Here the transverse plasma beta is defined as = T∕ , that is, the ratio of the total thermal energy T to the transverse magnetic field energy B 2 t ∕2 0 . Thus, Bernoulli's Equation 8 becomes where the term  The z axis is along the direction of the magnetic field , and the radial direction ̂ is in the x-z coordinate plane.

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Commonly, the ion ratio He 2+ ∕H + in the solar wind is less than 6% and averages 4.5% (Cranmer et al., 2017;Song, Russell, Zhang, et al., 1999;McComas et al., 2008). In the plasma of the solar wind, heavy elements are exceedingly rare, so we just consider electrons, protons and 4 He 2+ (i.e., particles) and neglect other ions, which is a reasonable approximation. We further assume the ion ratio He 2+ ∕H + in the solar wind is < 0.06 and that the number densities of protons, He 2+ and electrons are np , n = np , and ne = np + 2n = (1 + 2 )np , respectively. The total number density in the plasmas is then n = ne + np + n = (1 + 2 )np + np + np = (2 + 3 )np , and the mass density of the solar wind is where me , mp , and m are the electron, proton, and He 2+ masses. Given me ≪ mp and m ≈ 4mp , the average atomic weight in the solar plasma is ≈ (1 + 4 )∕(2 + 3 ) . Therefore, Equation 9 yields where the radial temperature of species a {e, p, } is defined as T a n = cos 2 T a ‖ + sin 2 T a ⟂ .
The total thermal energy density is The omnidirectional temperature of a species can be defined as The average temperature of the plasmas is then When the helium abundance is very small, the contribution of He 2+ ions can be neglected, and the average temperature of the plasmas takes the form ⟨T⟩ = ( T e + T p ) ∕2 Therefore, Bernoulli's Equation 11 reduces to (1 + 2 )T e n + T p n + T n 2 + 3 Relating the corona to the solar wind and planetary magnetosheaths yields where the subscript "cor", "sw," and "sh" denote the corona, solar wind and magnetosheaths, respectively. Here, we neglect the heat fluxes of the solar wind and the planetary magnetosheath collisionless plasmas. The influence of the gravity on both the solar wind and manetosheaths are neglectable. At the subsolar point of the planetary magnetosheaths, the temperatures reach maximal values, whereas the bulk velocities of the downstream plasmas are small. Obviously, Equation 21 contains the matter continuity and energy conservation parts of the shock jump conditions constrain the upstream and downstream plasmas in the vicinity of the planetary bow shocks.
As shown in the Introduction, the temperature of the solar wind has decreased considerably, and ⟨ ⟩ ≈ 15eV , and (⟨ ⟩ − ⟨ ⟩ ) ∕⟨ ⟩ ≈ 96% . So that the part of the thermal energy for the solar wind in Equation 21 is rather small.
For the steady corona, the magnetic field B is almost in the radial direction due to the drawing of the outflowing streamers, so ≈ 0 and the transverse plasma beta = T∕ B 2 t 2 0 ≫ 1 . Usually, all the species in the solar corona are thermally isotropic due to the rapid Coulomb collisions (Cranmer et al., 2017). Therefore, the perpendicular temperature and the parallel temperature for the species a in the corona are the same, that is, ). So that we get T a n ≈ T a from Equation 15, indicating that the transverse temperature and the average temperature are approximately equal. According to Equation 19, we have in the corona ( (1 + 2 )T e n + T p n + T n ) ∕(2 + 3 ) ≈ ( (1 + 2 )T e + T p + T ) ∕(2 + 3 ) = ⟨T⟩ . Furthermore, in the coronal region with the highest temperature T max cor , the temperature gradient is almost zero (i.e., ∇Tcor ≈ 0 ), so the heat flux is neglected in these zones. Therefore, Equation 21 becomes (1 + 2 )T e n + T p n + T n 2 + 3 The above formula shows the relationship between the temperatures of the solar corona, solar wind and the planetary magnetosheaths at the subsolar points in the steady state. The maximum temperature of the planetary magnetosheaths at the subsolar points is determined by the highest temperature of the solar corona. The transverse plasma beta t of the magnetosheaths on the right-hand side of Equation 22 is still not fixed because the equation of momentum was not included in this investigation. The transverse plasma beta t of the magnetosheaths can be determined observationally.
We now consider the statistically averaged situations. Although the temperatures of protons and electrons in the planetary plasmasheaths have anisotropy generally, there exist a trend of isotropization caused by various instabilities stimulated, thus the velocity distributions of the particles do not deviate from isotropy too much, especially for high beta plasmas. Here we make the approximation that the perpendicular temperature and the parallel temperature for the speci a in the magnetosheaths are equal statistically, that is,  Thomsen et al. (2018) have statistically obtained an empirical formula for the relationship between the speed of the solar wind and the velocity and temperature of the Saturn's magnetosheath, that is, 8 of 15 1 2 mpV 2 sw = 1.687 × ⟨ 1 2 mpV 2 sh + kTp + B 2 ∕2 0 n ⟩ sh , where n is the number density of protons, which has certain similarity to the above equation as omitting the contribution of the thermal energy of the solar wind.
According to the observations of Wang et al. (2012) of Earth's magnetosheath and on the observations of Thomsen et al. (2018) of Saturn's magnetosheath, Vsh ≤ 100 km∕s, Vsh∕Vsw ≤ 1∕4, and V 2 sh ∕V 2 sw ≤ 0.060 in general. Upon approaching the nose (the stagnant point) of the magnetopause along the Sun-planetary line, the velocity of the magnetosheath plasma decreases to about zero, so the kinetic energy is much less than the total energy in the planetary magnetosheaths at the subsolar points and thus can be neglected. Generally, in the planetary magnetosheaths, β ≈ 0.1-20 (Gershman et al., 2013;Richardson, 2002;Sergis et al., 2013;Thomsen et al., 2018). As shown in Figure S1-S3 in Supporting Information S1, the average plasma beta of the magnetosheaths of Mercury, Earth and Saturn are 2.4, 7.3 and 9.6, respectively, that is, 2.4 ≤ ⟨ ⟩sh ≤ 9.6 . Then the corresponding the average transverse plasma beta obey 4.8 ≤ ⟨ t ⟩ sh ≈ 2⟨ ⟩sh ≤ 19.2 . So that we can make an approximation that ⟨ t ⟩ sh ≫ 1 , that is, the magnetic field is assumed to have a neglect role on the evolution of the solar wind.
which is the expected temperature of the corona with no heat source and gravitation that can expand to the same solar wind. Finally we can obtain where the contribution of the kinetic energy of magnetosheath plasmas is omitted on considering V 2 sh ∕V 2 sw ≤ 0.060 . This implies the average temperatures of the planetary magnetosheaths are about equal to the effective temperature of the solar corona. The observational investigations in the following Section 3 shows that the average plasma temperatures of the magnetosheaths of Earth and Saturn at the subsolar points are 184 eV (2.13 MK) and 171 eV (1.98 MK), respectively. On the other hand, there is still no sufficient observational information about the corona temperature. As shown in the next Section, based on Kohl et al. (2006), the average temperature in the polar coronal holes at solar minimum are about ⟨T⟩cor ≈ 146eV ≈ 1.7MK . This implies that the average temperatures of the planetary magnetosheaths are possibly very close to that of the solar corona. Therefore, it is most likely that the first term in the right side hand of Equation 25 are actually in balance and cancel each other, and thus ⟨T⟩cor ∼ ⟨T⟩sh . This result is consistent with the second law of thermodynamics if we neglect the plasma bulk velocity in the planetary magnetosheath at the subsolar point and assume the magnetic field have an omissible role in the propagation of the solar wind with very high plasma beta.
Therefore, in the present work, it is expected that, in the steady state, the temperatures of all planetary magnetosheaths in the solar system at their subsolar points are comparable and possibly close to the temperature of the solar corona.

Statistical Investigations
We now present a statistical investigation of the plasma temperature in the magnetosheaths of Mercury, Earth, Jupiter, and Saturn and compare the results with the theoretical results from the previous section.
The Supporting Information (jgra55009-sup-0001-2021JA029108-si) gives the plots (Figures S1-S3 in Supporting Information S1) of the distributions of plasma betas in the magnetosheaths of Mercury, Earth and Saturn as well as the plots ( Figures S4-S9 in Supporting Information S1) of the distributions of ion and electron temperatures in the magnetosheaths of Mercury, Earth, Jupiter and Saturn. Figure 3 shows the distributions of ion temperature in the magnetosheaths of Mercury, Earth, Jupiter, and Saturn, respectively.
For Mercury, we use the data from the Messenger satellite from 2012 to 2013 (Slavin et al., 2007). It is found that the Messenger satellite was in Mercury's magnetosheath region for 94 days during 2012 and 2013. Based on the proton measurements during these periods, we obtain the distribution of proton temperatures in Mercury's magnetosheath. The proton temperature is drawn from the NTP data in FIPS-DDR with a time resolution of 1 min (Andrews et al., 2007). We are still in lack of the data for the temperatures of other ions, for example, He + , He 2+ and O + . Considering that the protons make up most of the ions in the magnetosheath, we would use the proton temperature to approximate the ion temperature in Mercury's magnetosheath. Figure 3 and also Figure  S4 in Supporting Information S1 (jgra55009-sup-0001-2021JA029108-si) show that the maximum ion temperature in Mercury's magnetosheath can reach as high as 1,450 eV, with the most probable ion temperature being ≈ 275 eV , and the average ion temperature being ≈ 414 eV with its standard deviation ≈ 232 .
As for Earth, we analyze the data from the MMS1 satellite Pollock et al., 2016;Torbert et al., 2015) acquired from 2015 to 2021. We obtain 169 days when the detector was in the subsolar region (with the zenith angles from the X axis being <30°) of Earth's magnetosheath. Using the data within this interval, we obtain the distribution of ion and electron temperatures in Earth's magnetosheath. The ion temperature is derived from the ion vertical temperature Ti⟂ and the ion parallel temperature Ti‖ in FPI_FAST_L2_DIS-MOMS of MMS1 (Pollock et al., 2016), and the electron temperature is obtained from the electron vertical temperature Te⟂ and the electron parallel temperature Te‖ in FPI_FAST_L2_DES-MOMS of MMS1 (Pollock et al., 2016). The time resolution of the data is 4.5 s. The calculation formulas for the total temperatures of ions and electrons are Ti = 1 3 Ti‖ + 2 3 Ti⟂ and Te = 1 3 Te‖ + 2 3 Te⟂ , respectively, as shown in Equation 17. Figure 3 as well as Figure S5 in Supporting Information S1 show the distribution of ion temperature in Earth's magnetosheath. These results show that the maximum ion temperature in Earth's magnetosheath can reach up to 1,200 eV, the average ion temperature is i ≈ 362eV , and the most probable ion temperature in Earth's magnetosheath is ≈ 185eV . The standard deviation of the ion temperature is ≈ 221 eV . The average ion temperature of the subsolar magnetosheath of Earth obtained here based on MMS measurements is consistent with that of Wang et al. (2012) based on THEMIS observations within the range of error. Figure 4 as well as Figure S6 in Supporting Information S1 present the distribution of electron temperature in Earth's subsolar magnetosheath. These results show that the maximum electron temperature can reach 140 eV, the average electron temperature is ≈ 49 eV , and the most probable electron temperature is ≈ 35 eV , with a standard deviation ≈ 18 eV . The mean electron temperature of the subsolar magnetosheath of Earth based on MMS measurements is in agreement with the statistical result of Wang et al. (2012) from THEMIS observations within the range of error.
For Jupiter, we have used the data of Voyager 2 satellite (Kohlhase & Penzo, 1977) from July 2 to 5 July 1979. Ion temperatures were derived from the ION-L-MODE data from the Plasma Subsystem (PLS) which has a time resolution of 96 s (Bridge et al., 1977). Figure 3 and also Figure S7 in Supporting Information S1 show the distribution of ion temperature in Jupiter's magnetosheath. The maximum ion temperature in Jupiter's magnetosheath can reach 560 eV, the average ion temperature is ≈ 309 eV , and the most probable ion temperature is ≈ 330 eV , with a standard deviation ≈ 75 eV .
For Saturn, we apply the data of the Cassini satellite from 2007 to 2008 (Matson et al., 2002). Counting the data from 2007 to 2008, we find 66 days during which the detector was located in the subsolar region of Saturn's magnetosheath (with the zenith angles from the X axis being <30°). Using the data during these 66 days, we obtain the distribution of plasma temperature in Saturn's subsolar magnetosheath. The proton temperature is derived from the DDR-ION-MOMENTS data from the Cassini Plasma Spectrometer (CAPS) with a time resolution of approximately 7 min (Young et al., 2004), and the electron temperature is generated from the DDR-ELE-MOMENTS data from the CAPS with a time resolution of approximately 32s (Young et al., 2004). Similar to the situation of Mercury previously, we use the proton temperature to approximate the ion temperature in the magnetosheath. Figure 3 and also Figure S8 in Supporting Information S1 show the distribution of ion temperature in Saturn's magnetosheath. The ion temperature can reach 800 eV, the average ion temperature is ≈ 304 eV , and the most likely ion temperature is ≈ 270 eV ,with a standard deviation ≈ 104 eV . Figure 4 and also Figure S9 in Supporting Information S1 show the distribution of electron temperature in the magnetosheath of Saturn. These results show that the electron temperature can reach 100 eV, the average electron temperature is ≈ 37 eV , and the most probable electron temperature is ≈ 33 eV , with a standard deviation ≈ 12 eV . These results are ≈ 1 2 (104 + 12)eV ≈ 58 eV . And also, based on the proton and electron measurements from the Jovian Auroral Distributions Experiment (JADE) (McComas, Alexander, et al., 2017) on board Juno, recently Ranquist et al. (2019) have statistically found that the average temperature of the Jupiter's magnetosheath at the region with the zenith angle between 15° and 35° (a little dawnside) is 197 eV, which is in consistence with the results here. As shown in Table 1 and Figures 3 and 4, S1-S6 in Supporting Information S1, the deviations from the average temperatures generally become smaller and the distributions of the temperatures grow more Gaussian with the distances of the planets away from the Sun. Kohl et al. (2006) presented that the electron and proton temperatures in the polar coronal holes at solar minimum are about ⟨T e ⟩ ≈ 1.5MK and ⟨T p ⟩ ≈ 2.0MK , respectively, with the average temperature being approximately ⟨T⟩ ≈ (⟨T e ⟩ + ⟨T p ⟩) ∕2 ≈ 1.7MK ≈ 146eV . (Considering the high temperature of Alpha particles, it can be enhanced even further.) Cranmer et al. (2017) have given a slightly smaller value. It is expected that the average temperature of the plasmas in equatorial corana is larger than that in the polar coronal holes (Kohl et al., 2006). Kohl et al. (2006) have also shown that the average proton temperature of the corona streamers is 2.0∼3.2 MK, or 170∼280 eV. So that the average temperatures of the planetary magnetosheaths are very close to that of the solar corona within the statistical deviations. These statistical results support the theoretical results presented in the previous section.
In addition, for both Earth's and Saturn's magnetosheaths, the electron temperatures are much less than the proton temperatures. The ratio of ion-to-electron temperature in the magnetosheaths of Earth and Saturn are 7.4 and 8.2, respectively, which indicates that, as the solar wind streams out from the solar corona, the electrons cool significantly. It is possibly caused by the ambipolar diffusion process (Lemaire & Pierrard, 2001;Parks, 2018).

Discussion and Conclusions
The magnetosheaths supply matter and energy to planetary magnetospheres and plays a critical role in the evolution of the magnetospheres (Axford & Hines, 1961;Dungey, 1961;Phan et al., 2000;Fujimoto et al., 2008;Wang et al., 2012). The upstream solar wind plasmas, which originate from the solar corona, are compressed by the bow shocks and form the downstream magnetosheath plasmas. The thermal properties of the planetary magnetosheaths are closely related to the features of the solar corona. Numerous observational investigations indicate that the temperature of planetary magnetosheaths at subsolar points is on the order of several hundred eV or several MK (Gershman et al., 2013;Richardson, 1987Richardson, , 2002 (Delaboudinière et al., 1995;Laming et al., 1995;Schrijver et al., 1999;Schmelz & Winebarger, 2015;Tu et al., 1999). This research seeks to find a quantitative relationship between the temperature of the solar corona and that of the planetary magnetosheaths.
The thermal energy of the solar corona that is converted into kinetic energy to accelerate the solar wind is almost entirely converted back to thermal energy when the plasma crosses a planetary bow shock. As viewed from the second law of thermodynamics, the maximum temperature of a planetary magnetosheath generally cannot exceed that of the solar corona if we omit the role of magnetic field for high plasma beta situations. This investigation includes a detailed theoretical analysis of the steady expansion of the solar corona, the propagation of the solar wind, and the compression of planetary magnetosheaths by the bow shocks with the general Bernoulli's equation. The approach is universal and considers the dynamics of multiple components, thermal anisotropy, and non-Maxwellian plasmas in the steady state. In the core region of the solar corona, the heating input and the radiative loss reach thermal equilibrium, thereby maintaining the extremely high temperature of the corona plasmas. At present, no clear understanding exists of the real heating mechanism of the solar corona (Cranmer et al., 2017;Klimchuk, 2015;McComas et al., 2007;Parnell & De Moortel, 2012). In this research, we only study the outward expansion of the outer corona under the thermodynamic driving and evade possible heating and radiation losses. This approximation is reasonable and does not seriously affect the results obtained herein. It is somewhat odd that the Rankine-Hugoniot shock jump conditions have not been directly applied in this investigation related to planetary bow shocks. Actually, Bernoulli's equation and the matter conservation and energy conservation parts of Rankine-Hugoniot shock jump conditions are equivalent for describing the bow shocks. In this study, we have not omitted the Rankine-Hugoniot shock jump conditions, but applied Bernoulli's equation instead to describe the variations of the quantities at the two sides of the bow shock.
We propose an approximate formula relating the temperature of the solar corona to that of the planetary magnetosheaths. The quantitative results indicate that the average temperatures of all planetary magnetosheaths at the subsolar points are comparable. In general, the peak temperatures of the planetary magnetosheaths at the subsolar regions are very close to that of the solar corona. These theoretical results are consistent with measurements of planetary magnetosheaths (Gershman et al., 2013;Richardson, 1987Richardson, , 2002Slavin et al., 2014;Shen et al., 2008;Wang et al., 2012;Sergis et al., 2013).
We also provide a systematic statistical investigation into the average temperatures of the magnetosheaths of Mercury, Earth, Jupiter, and Saturn. The results indicate that the average proton/ion temperatures of the magnetosheaths of Mercury, Earth, Jupiter, and Saturn are 414, 362, 309, and 304 eV, respectively, whereas the average electron temperatures of the magnetosheaths of Earth and Saturn are 49 and 37 eV, respectively (no electron data are available for Mercury and Jupiter at present). The average plasma temperatures of the magnetosheaths of Earth and Saturn are 206 and 171 eV (or 2.39 and 1.98 MK), respectively, which are very close to the average temperature of the solar corona. The statistical results are consistent with the theoretical results. However, the electrons cool considerably as they travel away from the Sun in the solar wind.
The quantitative relationship obtained herein regarding the temperature of planetary magnetosheaths can be applied to the steady solar wind propagation and planetary magnetosheaths. These results can also be meaningful for investigating the magnetosheaths of Venus and Mars without involving the intrinsic magnetic field (Øieroset et al., 2004). Observations indicate that the heliosheath temperature is approximately 2 MK , which can also be explained by the theoretical results of Section 2. The planetary magnetosheaths, interplanetary coronal mass ejection sheaths, and heliosheath are similar in terms of hot protons, which should also be the case for the steady-state fast solar wind originating from the coronal holes. However, the results obtained in this investigation do not hold for the explosive processes of the coronal mass ejections, during which the coronal magnetic energy contributes to the outward acceleration of the solar wind.
The relationship obtained herein between the temperature of the solar corona and that of planetary magnetosheaths is useful for evaluating the thermal features of the planetary magnetospheres based on the conditions of the solar corona. The plasmas in the tail plasma sheet originate mainly from the magnetosheath. The ratio of proton to electron temperature is approximately seven in Earth's plasma sheet, which is about the same as that in the magnetosheath. The higher the temperature of the magnetosheath, the higher that of the plasma sheet.