4.1 Magnetic field control of the surface charging
In contrast with the non-magnetized solar wind case, in which the solar wind electrons are able to access the nightside surface of the obstacle freely from any direction (Kimura and Nakagawa, 2008), the motion of the electrons in the magnetized solar wind case is controlled by the solar wind magnetic field. The electrons are confined within a Larmour radius to the magnetic field line and we can more-or-less approximate electron flow to be along the magnetic field. Only the electrons on the magnetic field lines that connect with the obstacle can contribute to surface charging.
Figure 6(a) illustrates an example of the magnetic field with the angle θB = -45° measured from the flow direction of the solar wind. The thermal electrons come from the upper left, or the lower right, direction along the field lines. The electron flux arriving at the obstacle per unit area on the surface should be largest at around -45° and 135°, where the magnetic field is perpendicular to the surface. (Precisely speaking, the positions shift slightly due to the solar wind bulk velocity.) As we assume the body to be non-conducting, the electric charge accumulates at the position of impact and does not migrate on the surface. The electric charge on the upstream-side surface can be easily neutralized by the incoming solar wind ions and only the electrons that reach the downstream-side surface contribute to the charge accumulation. On the positive-y side, a small area in the vicinity of the terminator (45° < θ < 90°) collects electrons from the upstream side. A small number of field lines are connected with the positive-y side and they intersect the surface at an oblique angle. On the other hand, a large area of the negative-y side and a part of the positive-y side extending from -90° to 45° collects a larger number of electrons, with the maximum near θ ~ -45°. Thus, the surface charging and the potential drop are expected to be large on the negative-y side. On the positive-y side, the potential drop should be smaller and shifted to the terminator. The negative charging would vanish at around θ ~ 45° where the magnetic field lines do not intersect the surface. These features are consistent with the electric potential shown in Fig. 4.
Figure 6(b) shows an example of the magnetic field with θB = -15°. In this case, a wide range of the downstream-side surface -90° < θ < 75° is covered by the electron flux coming from downstream, and only a small area near the terminator of the positive-y side is hit by the electrons from upstream. Figure 7 shows the result of simulation run #2 for the magnetic field direction θB = -15°. Compared with Fig. 4, the position of the maximum potential drop has shifted to the center of the void, consistent with Fig. 6(b). The asymmetry is less significant because a wide area of the downstream side is exposed to the electron flux from downstream. The negative potential of the central wake extends far beyond 7RO, as in the non-magnetized solar wind case (figure 2 of Kimura and Nakagawa, 2008).
The asymmetry of the electric field structure vanishes when the magnetic field is parallel, or perpendicular to the solar wind flow. The asymmetry is caused by the oblique magnetic field; it would be most significant at the heliospheric distance of 1 AU, where θ ~ -45° on average. On the Moon, for example, the negative charging and the maximum potential drop are expected to be shifted to the dusk-side of the nightside surface. Since the solar wind magnetic field is variable, there would be sudden changes of charge and discharge as observed by Apollo missions (Colwell et al., 2007, and references therein) at abrupt changes of the magnetic field direction.
4.2 Streaks of the enhanced electron density
In the absence of photoelectrons, the electrically-neutral surface on the upstream side of the non-magnetized body is not an obstacle as seen from the electrons of the upstream solar wind. Only the negatively-charged surface is the obstacle that expels the electrons. Figure 8 shows the electric potential as seen from the electrons (that is, reversed in sign) plotted against the distance along a magnetic field line. Initially, the magnetic field line upstream of the non-magnetized body (which crosses the x-axis at x = -2 in Fig. 8) is nearly equipotential. A potential difference as large as 2ϕ0 appears on the field line at the terminator (which crosses the x-axis at x = -1 in Fig. 8). The electrons that cannot climb up the potential difference are accelerated away from the terminator by the electric field component parallel to the magnetic field, and flow down along the field line. The electrons, once accelerated, keep going along the field line although the potential gap is restricted to a small area near the obstacle.
We can estimate the speed of the electron flow υ|| along the magnetic field line to be υ|| ~ 2υe. The electrons flow down along the magnetic field line, while the field line is convected down at the solar wind speed υsw = 0.25υe. Combining the thermal velocity (υ|| cos θB,υ|| sin θB, 0) with the convection velocity (υsw, 0, 0) as illustrated in Fig. 9, we obtain the flow direction -40° measured from the direction of the solar wind flow. On the positive-y side, the electrons flow along the magnetic field line against the solar wind bulk flow with the velocity (- υ|| cos θB, -υ|| sin θB, 0) and the flow direction as seen from the obstacle is 129° from the solar wind direction. These are consistent with the direction of the streaks of the electron enhancements, -37° and 131°, as has been observed in Fig. 1.
In the case of the magnetic field whose direction is θB = -15° from the x-axis, the flow directions of the accelerated electrons are calculated to be -13° and 163°. Figure 10 shows the electron density obtained from the simulation run #2 with θB =-15°. The electron enhancements streak in the direction -14° measured from the x-axis on the negative-y side and in the direction 160° on the positive-y side, although this is rather faint due to the very small area of negative charge on the surface near the terminator on the positive-y side. They agree with the above expectations.
4.3 Dependence on the Debye length
In general, the spatial extent of the electric field caused by the surface charging is of the order of the Debye length. In this paper, a Debye length λD as large as 0.25RO has been employed. There might be a concern that the effect of the surface charging is limited for an object whose radius RO is much larger with respect to the Debye length.
Figure 11 shows the result of the simulation run #3, for which the Debye length is reduced to be 0.125RO by slowing down the electron thermal speed. The solar wind bulk speed and the ion thermal speed are also reduced in the same proportion. As expected, the spatial extent of the electron void around the terminator is smaller in Fig. 11(a) than in Fig. 1. On the other hand, the void in the central wake is essentially the same.
The asymmetry of the potential structure is clearer in Fig. 11(b) than in Fig. 4. The largest potential drop is ϕ ~ -3ϕ0. (Note that ϕ0 is also reduced by slowing down the electron thermal speed υe.) As the ratio of the potential drop to the electron thermal energy is nearly the same as before, the electrons gain as much flow speed as before and the streaks of electron enhancement appear in Fig. 11(a).
The potential drop in the downstream wake extends far beyond 7 RO in Fig. 11(b), differently from the larger Debye length case in Fig. 4. The relative importance of the wake potential to the surface charging increases for a larger scale obstacle.
4.4 Weaker magnetic field case
We have employed an intense magnetic field Ωe = 12ωp and there might be a concern that the control by the magnetic field is too strong in these simulations. So we carried out another simulation run (#4) in which the magnitude of the magnetic field is as small as Ωe = 0.75ωp. In this case, the ion Larmour radius riL is as large as 7.54RO, i.e. the ions are almost non-magnetized, and the electron Larmour radius reL is 0.23RO, larger than the Debye length λD = 0.125RO. Figure 12 shows the electric potential obtained from run #4. The asymmetry of the potential structure is recognized, although it is not as clear as in Fig. 11(b), due to the large electron Larmour radius. It shows that the magnetic field control of the surface charging of the non-magnetized obstacle is significant, as long as the electron Larmour radius is smaller than the size of the obstacle.
4.5 Comparison with observations at the Moon
Limitation of the scale size of the obstacle with respect to the Debye length, together with the absence of photoemission, prohibits us from making a direct comparison of the simulation result with the observations made at the Moon. Too small a ratio of the obstacle size to the Debye length magnifies the effect of surface charging with respect to the potential drop at the wake boundary. Nevertheless, some aspects of the model can be compared with the lunar data. Such a comparison would help elucidate what aspects of the model are appropriate for all scale sizes of objects and which are more limited to smaller objects.
As we have seen in Section 4.2, the potential drop at the terminator is of the order of 2ϕ0, which corresponds to 60–80 V for the typical solar wind electrons having a thermal energy of 15–20 eV. This is consistent with the Apollo SIDE observation of 70 eV ions accelerated by the negative lunar surface potential (Freeman and Ibrahim, 1975) and a surface potential as low as -100 V on some terminator crossings (Lindeman et al., 1973).
The largest potential drop on the nightside surface of the obstacle, 3ϕ0, which corresponds to 90–120 V, is consistent with a lunar surface potential of -120 V inferred from the Lunar Prospector observation of the electrons at an altitude of 20-40 km (Halekas et al., 2002), but somewhat smaller than the newly found potential drop of -200 V near the edge of the wake (Halekas et al., 2008). Halekas et al. (2008) also reported that the surface potential drop with respect to the local plasma is smaller in the central wake than near the wake boundary. No such signature is found in this simulation. Halekas et al. (2008) attributed this to secondary electrons, which are not included in the present simulation.
The minimum electron density obtained by Lunar Prospector in the lunar wake (figure 6 of Halekas et al., 2005) appears to be shifted slightly to the duskside, consistent with the result of the present simulations. This is likely, because the magnetic field lines of the average IMF at 1 AU are perpendicular to the dusk-to-night surface of the moon.
It is difficult to apply the wake potential obtained from the simulation with a large Debye length to the lunar observations. In the classical theory of a plasma expansion into a vacuum (Samir et al., 1983), the electrons were thought to rush into the void faster than the ions due to the faster thermal speed. However, in the present simulation with nightside surface charging, the electrons are retarded by the negative charging of the downstream-side surface, and cannot precede the ions. Figure 13(a) shows the ion and electron densities for several distances from the obstacle, obtained from simulation run #3. In the vicinity of the obstacle (x = 1 RO), the ions enter the void faster than the electrons, producing a positive excess of charge in the vicinity of the wake boundary which affects the potential structure as observed in the top panel of Fig. 13(b). Such an effect of the surface charging should be more limited within a small area.
A negative excess of charge is found in the central wake at x = 2 – 3RO in Fig. 13 and disappears at 4RO.At x = 3RO, well beyond the Debye length from the obstacle (although the Debye length becomes large in a low density plasma), the potential drop in the central wake with respect to the ambient solar wind is about 0.5ϕ0-1ϕ0. If we assume that this is the wake potential and that the wake potential is essentially independent of the Debye length, as long as the ratio of the thermal speeds to the solar wind speed is kept constant, it is not necessary to evaluate it in terms of the electron thermal energy
, but rather, we can convert it directly into volts using υe = 0.05c for the simulation run #3. It is calculated to be 0.64–1.3 kV. This is much stronger than it appears in Fig. 11 in which the surface charging effect is magnified. Although this is a very rough estimation, it is of the same order as the potential of -442 V estimated from the WIND observation of backstreaming electrons (Farrell et al., 1996) and -480 V estimated from NOZOMI observation of counterstreaming electrons (Futaana et al., 2001).
This simulation reproduced the streaks of the electron enhancement along the magnetic field line on the same hemisphere as the electron enhancement detected by Kaguya LRS/WFC-H, but with a slight difference in the location. This might be due to the effect of surface charging, or the absence of photoelectrons in this simulation. Photoelectrons might affect the position of the first contact of the magnetic field line with the negative surface density. At present, we cannot conclude that they are the same phenomena or not.
The horizontal ion entry along the magnetic field lines as reported by Futaana et al. (2010) is not observed in the present simulation.