What Density of Magnetosheath Sodium Ions Can Provide the Observed Decrease in the Magnetic Field of the “Double Magnetopause” during the First MESSENGER Flyby?

: On 14 January 2008, the MESSENGER spacecraft, during its ﬁrst ﬂyby around Mercury, recorded the magnetic ﬁeld structure, which was later called the “double magnetopause”. The role of sodium ions penetrating into the Hermean magnetosphere from the magnetosheath in generation of this structure has been discussed since then. The violation of the symmetry of the plasma parameters at the magnetopause is the cause of the magnetizing current generation. Here, we consider whether the change in the density of sodium ions on both sides of the Hermean magnetopause could be the cause of a wide diamagnetic current in the magnetosphere at its dawn-side boundary observed during the ﬁrst MESSENGER ﬂyby. In the present paper, we propose an analytical approach that made it possible to determine the magnetosheath Na + density excess providing the best agreement between the calculation results and the observed magnetic ﬁeld in the double magnetopause.


Introduction
Sodium Fraunhofer lines at 5890 and 5896 angstroms were found by the on-ground telescope during investigation of Mercury in 1985 (e.g., [1], and references therein). Later it occurred that the sodium exosphere is time-dependent and controlled, in particular, by the solar wind. During two first MESSENGER flights the so-called "double magnetopause" was found at the dawn-side magnetospheric boundary [2,3]. For the first flyby (M1) the interplanetary magnetic field (IMF) was northward when the spacecraft exited into the dawn-side magnetosheath, while for the second one (M2) it was southward. In the both cases the magnetic field decrease between two narrow current sheets was observed and called "dayside boundary layer (D-BL)". From this fact M¨Uller et al. [3] concluded that the D-BL and two current sheets at its boundaries do not depend on IMF direction. Belenkaya [4] showed for the terrestrial magnetosphere that the magnetopause currents generated by the magnetosheath ions and caused by the violation of the symmetry of the plasma parameters are directed from dawn to dusk regardless of the north-south IMF orientation. Thus, the magnetosheath ions of different sorts can be responsible, in principle, for these qualitatively similar magnetic field and plasma structures.
Slavin et al. [2] mentioned that at~19:10:35 and at 19:14:15 on 14 January 2008, current sheets with similar small thicknesses and orientations were observed when MESSENGER exited magnetosphere at the dawn-side during M1. At 19:14:15, the MESSENGER spacecraft intersected the magnetopause, and the outer current sheet was interpreted by the authors as the "real" magnetopause, while the inner current sheet was supposed to be created by the pick-up ions of the planetary origin (predominantly Na + ), which entered the magnetosphere from the solar wind side. It was emphasized by Slavin et al. [2] that the outer current is twice as strong than the inner one. The authors mentioned that both Symmetry 2021, 13, 1168 2 of 6 current layers were separated by the wide D-BL region~1000 km. This length is almost a half of the Mercury radius (R M = 2440 km (e.g., [5]). However, it should be noted that the inner current sheet (at~19:10:35) was as narrow as the outer one, and if it were generated by sodium ions, then it should have been much wider. Figure 1 illustrates a rough diagram of the Hermean magnetosphere. The magnetopause bounds the cavity filled with a magnetic field (solid lines with arrows). Parts of the Larmor circles of sodium pick-up ions from the magnetosheath located out of the magnetosphere are marked with dashed curves, and parts placed in the magnetosphere that create the boundary current are shown by solid curves. Trajectory M1 is shown.
Symmetry 2021, 13, x FOR PEER REVIEW 2 of 6 the outer current is twice as strong than the inner one. The authors mentioned that both current layers were separated by the wide D-BL region~1000 km. This length is almost a half of the Mercury radius (RM = 2440 km (e.g., [5]). However, it should be noted that the inner current sheet (at ~19:10:35) was as narrow as the outer one, and if it were generated by sodium ions, then it should have been much wider. Figure 1 illustrates a rough diagram of the Hermean magnetosphere. The magnetopause bounds the cavity filled with a magnetic field (solid lines with arrows). Parts of the Larmor circles of sodium pick-up ions from the magnetosheath located out of the magnetosphere are marked with dashed curves, and parts placed in the magnetosphere that create the boundary current are shown by solid curves. Trajectory M1 is shown. Mϋller et al. [3] showed that the thin inner current sheet of a double magnetopause can be created by proton pressure gradients. The authors used 3D hybrid numerical simulations for investigation of such double magnetopause observed during first MESSENGER flyby. The planet's magnetic field was taken from the Alexeev et al. [6] model, but some simplifications were used later by the authors. Mϋller et al. [3] concluded that although the inner current sheet can be explained without sodium ions, their influence should nevertheless be taken into account. They noted that for working of the mechanism suggested by Slavin et al. [2], most of the sodium ions should be produced out of the Mercury's magnetosphere, but not inside it. Sodium pickup ions density in the magnetosheath during M1 was estimated by Sarantos et al. [7] as 0.1-3 cm −3 . Here, we consider what can be expected if the situation suggested by Slavin et al. [2] was implemented.

Magnetopause Current Generated by Sodium Ions Penetrated from the Magnetosheath. Methods of Investigation
The magnetizing current arises if at least one plasma parameter (for example, density) or magnetic field changes at the boundary between two media. If sodium ions are present in the magnetosheath with a density higher than their density in the magnetosphere, they participate in the creation of an additional magnitizing current at the magnetopause. Sodium ions and neutrals are of planetary origin. Since the Mercury's atmosphere is very weak, energetic ions, neutrals, meteors, and the thermal or photon stimulated desorption produce the sodium exosphere around Mercury. Besides sodium, other substances are formed from the planet's surface, but sodium probably predominates ( [8,9], and references therein). Ionisation is produced by UV radiation and the solar wind charge particles precipitation. Sometimes this process is more intense outside the magnetosphere, where Na + ions are picked-up and accelerated, while solar wind charged particles are slowed down. MÜller et al. [3] showed that the thin inner current sheet of a double magnetopause can be created by proton pressure gradients. The authors used 3D hybrid numerical simulations for investigation of such double magnetopause observed during first MESSENGER flyby. The planet's magnetic field was taken from the Alexeev et al. [6] model, but some simplifications were used later by the authors. MÜller et al. [3] concluded that although the inner current sheet can be explained without sodium ions, their influence should nevertheless be taken into account. They noted that for working of the mechanism suggested by Slavin et al. [2], most of the sodium ions should be produced out of the Mercury's magnetosphere, but not inside it. Sodium pickup ions density in the magnetosheath during M1 was estimated by Sarantos et al. [7] as 0.1-3 cm −3 . Here, we consider what can be expected if the situation suggested by Slavin et al. [2] was implemented.

Magnetopause Current Generated by Sodium Ions Penetrated from the Magnetosheath. Methods of Investigation
The magnetizing current arises if at least one plasma parameter (for example, density) or magnetic field changes at the boundary between two media. If sodium ions are present in the magnetosheath with a density higher than their density in the magnetosphere, they participate in the creation of an additional magnitizing current at the magnetopause. Sodium ions and neutrals are of planetary origin. Since the Mercury's atmosphere is very weak, energetic ions, neutrals, meteors, and the thermal or photon stimulated desorption produce the sodium exosphere around Mercury. Besides sodium, other substances are formed from the planet's surface, but sodium probably predominates ( [8,9], and references therein). Ionisation is produced by UV radiation and the solar wind charge particles precipitation. Sometimes this process is more intense outside the magnetosphere, where Na + ions are picked-up and accelerated, while solar wind charged particles are slowed down.
Slavin et al. [2] showed the MESSENGER magnetic field data, including its exit from the dawn magnetosphere in the first flyby. The interplanetary magnetic field was northward. This situation is similar with the northward IMF for the Earth, since the dipole magnetic axes for both planets are directed southward. Belenkaya [10] considered the structure of the terrestrial magnetopause current for the northward IMF. It was shown how solar wind protons create a dawn-dusk magnetizing current at the equatorial dayside Symmetry 2021, 13, 1168 3 of 6 magnetospheric boundary. A kinetic approach was applied. Using the obtained distribution function, we derived an equation for the current density of the magnetosheath ions [10] as a function of the distance from the magnetopause x (x positive sunward). Here, we rewrite it for the sodium ions as Equation (1): where n msh0 and T msh are the Na + ion's magnetosheath density excess relative to the magnetospheric density, and temperature in the magnetosheath, respectively; m msh and ρ msh are the mass and gyro-radius of the magnetosheath sodium ions penetrating the magnetosphere, respectively. Larmor radius of the sodium ion penetrating from the magnetosheath is: where m msh = 3.8 × 10 −26 kg, z Na+ = 1, e is the electron charge, and B msph is the average field in the magnetosphere near the dawn magnetopause (which should be in the absence of Na + ). If we assume, following Belenkaya [10], that the width of the magnetosheath sodium ions' current sheet inside the magnetosphere is L ≈ 2ρ msh , then from the observed D-BL thickness we can find ρ msh . From the Slavin et al. [2] data it follows that L~1032 km (MÜller et al. [3] estimated L as 1000-1100 km), and consequently ρ msh = 516 km. Sarantos et al. [7] mentioned that the slow-down of the magnetosheath flow may be due to the loading by the sodium ions.
In Figure 2, one can see that the current sheet arises at the Mercury's magnetopause inside the magnetosphere, if the density of Na + ions outside the magnetosphere at the time under consideration was higher than inside it. Solar wind protons have much smaller gyroradii than sodium ions. For this reason, the diamagnetic current sheet created by Na + penetrating the Hermean magnetosphere is much thicker than the current carried by protons. All these currents are directed from dawn to dusk independent of the IMF B mshz direction [10]. solar wind protons create a dawn-dusk magnetizing current at the equatorial magnetospheric boundary. A kinetic approach was applied. Using the obtained tion function, we derived an equation for the current density of the magnetoshe [10] as a function of the distance from the magnetopause x (x positive sunward). H rewrite it for the sodium ions as Equation (1) where nmsh0 and Tmsh are the Na + ion's magnetosheath density excess relative to t netospheric density, and temperature in the magnetosheath, respectively; mmsh are the mass and gyro-radius of the magnetosheath sodium ions penetrating the tosphere, respectively. Larmor radius of the sodium ion penetrating from the tosheath is: msh = msh Na + Na + msph where mmsh = 3.8 × 10 −26 kg, zNa+ = 1, e is the electron charge, and Bmsph is the averag the magnetosphere near the dawn magnetopause (which should be in the absence If we assume, following Belenkaya [10], that the width of the magnetosheath sodi current sheet inside the magnetosphere is L  2ρmsh, then from the observed D-B ness we can find ρmsh. From the Slavin et al. [2] data it follows that L~1032 km (M al. [3] estimated L as 1000-1100 km), and consequently ρmsh = 516 km. Sarantos mentioned that the slow-down of the magnetosheath flow may be due to the loa the sodium ions.
In Figure 2, one can see that the current sheet arises at the Mercury's magne inside the magnetosphere, if the density of Na + ions outside the magnetosphere at under consideration was higher than inside it. Solar wind protons have much sm roradii than sodium ions. For this reason, the diamagnetic current sheet created penetrating the Hermean magnetosphere is much thicker than the current carried tons. All these currents are directed from dawn to dusk independent of the IMF rection [10].  Linear integral current density carried by the magnetosheath sodium ions is [10]: The magnetic field generated by the sodium ion current density j mshy (Equation (1)) is determined by the equation: curl (B z (x)/µ 0 ) = j mshy.
B z (x) for −2ρ msh < x < 0 generated by this current sheet according to [10] is: where is the background magnetic field strength. Here, B msh0 and B msph0 are magnetic fields in the magnetosheath and magnetosphere, respectively, outside the considered current sheet of Na + ions. From the magnetic field data presented by Slavin et al. [2], rough estimations give B msh0~3 0 nT and B msph0~9 0 nT, thus, B 0 = −60 nT.

Results
For the sodium ion density outside the magnetosphere and inside it, close to the magnetopause, Exner et al. [11] give a modeled result of 1-100 cm −3 . MÜller et al. [3], considering the possibility that Na + ions play a significant role in creation of a double boundary layer, noted that the density of these ions can reach~10 cm −3 in the inner dawn magnetosphere. Thus, the estimated value of n msh0 varies considerably. Moreover, this value is time dependent and therefore subject to change.

Results
For the sodium ion density outside the magnetosphere and inside it, close to th netopause, Exner et al. [11] give a modeled result of 1-100 cm −3 . Mϋller et al. [3], co ing the possibility that Na + ions play a significant role in creation of a double bo layer, noted that the density of these ions can reach ~10 cm −3 in the inner dawn m sphere. Thus, the estimated value of nmsh0 varies considerably. Moreover, this value dependent and therefore subject to change.
In Figure 3 we present calculations by Equation (4), which show that nmsh0 = provides the best coincidence with observations.    Figure 3 shows that in the considered case, under the assumption that the sodium magnetosheath pick-up ions penetrate into the Mercury's magnetosphere, creating the observed boundary layer, and in the approach used by Belenkaya [4,10], the density excess of these Na + magnetosheath ions relative to the magnetospheric density of sodium ions should be n msh0~2 cm −3 .

Discussion and Conclusions
MÜller et al. [3] stated that the phenomena leading to the formation of a double magnetopause observed at the dawn magnetospheric boundary during the M1 and M2 flybys are not well understood. The authors used the hybrid simulation code A.I.K.E.F. (Adaptive Ion-Kinetic Electron-Fluid) [12]. The asymmetry of the plasma and magnetic field parameters leads to special processes that can be described using the kinetic approach. The A.I.K.E.F. code was used, in particular, for investigation of the interaction of the solar wind with Mercury [12]. The hibrid approach includes kinetic analysis of ions, since their Larmor radii are comparabel to the scale of the obstacle, and electrons are considered to be neutralizing liquid. This code was also used, for example in [13], to study the interaction of the Kronian magnetosphere with Dione.
For the M1 flight around Mercury, MÜller et al. [3] received good agreement with observations even in the absence of sodium ions in the solar wind and in the Hermean exosphere. They found that the inner narrow current sheet can be generated by proton pressure gradients caused by trapped protons. However, MÜller et al. [3] noted that, nevertheless, heavy ions may be important for the Hermean magnetospheric current systems generation.
Here, we estimate the sodium ions density excess in the magnetosheath required to generate a wide boundary layer in the Hermean dawn magnetosphere during the M1 flyby, under the assumptions that the inner narrow current sheet is caused by proton pressure gradients [3], and the outer current sheet is a typical magnetopause shielding current created by solar wind protons [2].
In order to do this, we used the analytical expressions obtained by Belenkaya [10] for the magnetopause diamagnetic currents carried by the magnetosheath ions. Comparison with the observed magnetic field data in the D-BL region [2] during the M1 flyby shows that the magnetosheath Na + density excess should be n msh0~2 cm −3 . In this case, the violation of the symmetry of the plasma parameters at the magnetopause could explain the observations.