The interaction between the Moon and the solar wind
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We study the interaction between the Moon and the solar wind using a three-dimensional hybrid plasma solver. The proton fluxes and electromagnetical fields are presented for typical solar wind conditions with different magnetic field directions. We find two different wake structures for an interplanetary magnetic field that is perpendicular to the solar wind flow, and for one that is parallell to the flow. The wake for intermediate magnetic field directions will be a mix of these two extreme conditions. Several features are consistent with a fluid interaction, e.g., the presence of a rarefaction cone, and an increased magnetic field in the wake. There are however several kinetic features of the interaction. We find kinks in the magnetic field at the wake boundary. There are also density and magnetic field variations in the far wake, maybe from an ion beam instability related to the wake refill. The results are compared to observations by the WIND spacecraft during a wake crossing. The model magnetic field and ion velocities are in agreement with the measurements. The density and the electron temperature in the central wake are not as well captured by the model, probably from the lack of electron physics in the hybrid model.
Key wordsMoon solar wind hybrid model plasma
Bodies that lack a significant atmosphere and internal magnetic fields, such as the Moon and asteroids, can to a first approximation be considered passive absorbers of the solar wind. The solar wind ions and electrons directly impact the surface of these bodies due to the lack of atmosphere, and the interplanetary magnetic field passes through the obstacle relatively undisturbed because the bodies are assumed to be non-conductive. Since the solar wind is absorbed by the body, a wake is created behind the object. This wake is gradually filled by solar wind plasma downstream of the body, through thermal expansion and the resulting ambipolar electric field, along the magnetic field lines. For a review of the Moon-solar wind interaction, including recent findings, see Halekas et al. (2011), who also point out the fact that we can view the near lunar space as a plasma laboratory. Although the Moon lacks any significant atmosphere and ionosphere there are many interesting features to this seemingly simple solar wind interaction. To understand the global interaction between the solar wind and the moon we need computer models that can give us a three-dimensional picture that complements the in-situ plasma observations.
Magnetohydrodynamic (MHD) modeling of the Moon-solar wind interaction has a long history, starting with the work of Spreiter et al. (1970), who analytically treated the special case of an interplanetary magnetic field (IMF) aligned with the solar wind flow direction by finding a transformation of the MHD equations into a hydrodynamic problem. However, there are many kinetic processes that cannot be described by fluid models, e.g., the non-Maxwellian particle populations in the wake region, so particle models should capture more of the physical processes. An approximation of the refill of the lunar wake is the general one-dimensional problem where plasma expands into a vacuum (Widner et al., 1971; Samir et al., 1983; Mora, 2003), and such models have specifically been applied to the refill of the lunar wake (Farrell et al., 1998; Birch and Chapman, 2001). A nice property of such models is that even if they are one-dimensional, they can to some degree approximate a two-dimensional model of the lunar wake, where time corresponds to distance downstream the wake, i.e., the one-dimensional model is applied perpendicular to the solar wind flow and convects with the flow. A disadvantage of one-dimensional models, e.g., along x, is that the magnetic field component along the x-axis is forced to be constant, from the requirement that ∇ ∙ B = 0. This is a severe restriction for modeling the lunar wake, where the field component across the wake can increase. Due to the computational complexity, particle in cell (PIC) models, that include electrons as particles, are for the foreseeable future limited to at most two spatial dimensions, so what has been investigated using such models is the interaction of an infinite cylinder with the solar wind (Harnett and Winglee, 2003; Kimura and Nakagawa, 2008).
Hybrid models, where ions are particles and electrons a fluid, are less computational demanding that full PIC models and are suitable for studying processes of size comparable to the ion inertial length and gyro radius. A two-dimensional hybrid model by Trávníček et al. (2005) focused mostly on wave activity in the lunar wake. The M. Holmström (e-mail: ), S. Fatemi, Y. Futaana, and H. Nilsson Moon-solar wind interaction is however a fully three-dimensional problem, since the IMF component perpendicular to the solar wind flow introduces asymmetry into an otherwise cylindrical symmetric problem. A three dimensional hybrid model by Kallio (2005) had a fairly coarse grid on a small region, and did not handle the wake region in a self consistent way (the electric field was specified in that region). Recently another three dimensional hybrid model has been presented in Wiehle et al. (2011). They compare observations by one of the ARTEMIS spacecrafts to a model run with time dependent inflow conditions, and find a good agreement. The time dependence enables comparison with observations even during changing solar wind conditions, but makes it more difficult to see how the wake forms differently depending on the solar wind conditions.
Here we also apply a three dimensional hybrid model to the Moon-solar wind interaction, but for steady inflow conditions. This allow us to study how the wake depends on the direction of the IMF, and in what follows we describe the method and present a comprehensive global overview of the different plasma quantities in the wake.
We use a cell-centered representation of the magnetic field on a uniform grid. All spatial derivatives are discretized using standard second order finite difference stencils. Time advancement is done by a predictor-corrector leapfrog method with subcycling of the field update, denoted cyclic leapfrog (CL) by Matthews (1994). An advantage of the discretization is that the divergence of the magnetic field is zero, down to round off errors. Also, the discretization conserves energy very well (Holmstrom, 2011). The ion macroparticles (each representing a large number of real particles) are deposited on the grid by a cloud-in-cell method (linear weighting), and interpolation of the fields to the particle positions are done by the corresponding linear interpolation. Initial particle positions are drawn from a uniform distribution, and initial particle velocities from a drifting Maxwellian distribution. Further details on the hybrid model can be found in Holmstrom (2010).
The implementation of the algorithm was done in the FLASH software framework, developed at the University of Chicago (Fryxell et al., 2000), that implements a block-structured adaptive (or uniform) Cartesian grid and is parallelized using the Message-Passing Interface (MPI) library for communication. Further details on the FLASH hybrid solver can be found in Holmstrom (2011). The general solver will be included in future releases of FLASH.
A crucial point in modeling the interaction between the Moon and the solar wind is how to choose the inner boundary conditions, at the surface of the moon. Here we model the Moon as an absorber of the solar wind. Ideally we should then remove all ions that hit the surface of the Moon, but that would introduce a discontinuity in the charge density, from solar wind values outside the sphere, to zero inside the sphere. However, numerical differentiation of discontinuous functions on a grid will lead to oscillations and instabilities in the fields. Therefore we choose to keep also ions inside the Moon, but we gradually reduce their weight by a factor fobs after each time step while they are inside the Moon (the mass and charge of each particle are reduced). Thus, the Moon acts as a gradual sink for ions, and this approach also avoids the problem of ρI =0 in Eq. (1). However, we do not plot these reduced ions in what follows, only protons that have never hit the Moon are used when computing the various statistics. This can be seen as a form of smoothing, done in a self consistent way. Often smoothing of fields are used in hybrid solvers, but it is then done in an ad-hoc way, as an extra step. The downside of our approach is that we introduce a numerical parameter, but experience shows that the solution will not change by much when reducing fobs, indicating that the solution is converging. The reduction parameter cannot be chosen too small however, since a steep gradient in ion density will cause oscillations in the magnetic field. We have verified numerically that this approach keeps the IMF constant inside the Moon, and thus that no spurious currents are present there. There are no inner boundary conditions on the electro-magnetic fields.
In what follows, we have used typical solar wind conditions at 1 AU (Kivelson and Russell, 1995), unless otherwise noted. The solar wind velocity is 450 km/s, the number density is 7.1 cm−3, and the ion temperature is 1.2 ∙ 105 K. The electron temperature is 1.4 ∙ 105 K, and the IMF is in the xy-plane with a magnitude of 7 nT. For these plasma parameters, the ion inertial length is 85 km, the Alfvén velocity is 58 km/s, the thermal proton gyro radius is 66 km (the gyro time is 9 s), and the ion plasma beta is 0.6.
We now examine the ion velocities, computed from the ion current and the charge density as JI/ρI, as is usual for particle in cell models, since the ion current density and the ion charge density are quantities that are available on the computational grid. The small white region downstream of the Moon is a region without any particles that has not hit the Moon, thus no velocities can be computed there. The x-velocity is shown as the deviation from the solar wind velocity, when the IMF is perpendicular to the flow, in Fig. 2(4). We see a slight slow down of the solar wind in the rarefaction cone. There is also a slow down behind the +z hemisphere (and a speed up behind the −z hemisphere) seen in Fig. 2(4b). This is probably a geometric effect, where part of the velocity distribution function is emptied by collisions with the lunar surface. The y-velocity (Fig. 2(5)) shows how the wake symmetrically refills from both sides of the wake. The ions move inward not only in the central part of the wake, but also in the rarefaction cone. For the case of an aligned IMF, the z-velocity (Fig. 2(6)) also shows a flow in the rarefaction cone toward the wake center. There we also see a flow of ions at the wake boundary in the IMF plane, and there seem to be a small kinetic effect of alternating upward and downward going ions in the upper and lower part of the wake (Fig. 2(6b)). This seems associated with the wave pattern in density inside the rarefaction cone (Fig. 2(1–3)), possibly caused by oscillations at the wake boundary.
The magnetic field magnitude (three first rows) is shown for three different IMF directions in Fig. 3, where also each of the vector components (rows 4–9) are shown for two different IMF directions (perpendicular and aligned to the solar wind flow).
The Mach cone of decreased density and magnetic field makes an angle to the solar wind flow direction of about 10 degrees, as seen in Fig. 2 and Fig. 3. Since the outer edge of the cone is diffuse, it is impossible to give a more exact angle. The ion-acoustic wave velocity is Open image in new window , and the Alfvén velocity is also vA ≈ 58 km/s for these solar wind conditions, corresponding to angles of 7 degrees. We see that the density depletion travel perpendicular to the magnetic field. This together with the fact that the propagation velocity is larger than both the ion-acoustic and the Alfvén velocity suggests that the fast magnetosonic wave, with a phase velocity of Open image in new window leading to an angle of 10 degrees, is responsible for the creation of the Mach cone in the hybrid model. This is consistent with the observation of Wiehle et al. (2011) that all three MHD modes (fast, Alfvén and slow) will be manifested in the lunar wake.
3.2 Comparison with WIND observations
We here compare the hybrid model results with observations by the WIND spacecraft on December 27, 1994, when it traversed the lunar wake approximately perpendicular to the solar wind flow, in the plane of the IMF, at a distance of x = −6.5Rl ≈ −11300 km. Ogilvie et al. (figure 1, 1996) plots several plasma parameters during the wake crossing. Apparently the solar wind conditions changed during the observation. This is most apparent in the plasma number density that decreased by more than a factor of two from the undisturbed solar wind on the inbound trajectory to the solar wind on the outbound trajectory. Therefore we make two runs of the hybrid model for the two sets of solar wind conditions, here denoted before and after. The solar wind conditions used for the before (after) case are as follows. A solar wind velocity of 470 (500) km/s, a number density of 5 (2) cm−3, an ion temperature of 0.9 (0.5) ∙ 105 K, and an electron temperature of 1.5 (2) ∙ 105 K. The IMF was (6.39, 2.33, 0) nT for the before case, and (5.57, 5.02, 0) nT for the after case. The time step was 0.08 (0.05) s and fobs = 0.98(0.985). Since the plasma parameters are different in the two cases, these numerical parameters have different values to ensure the stability of the solutions.
Similarly, the lack of electron dynamics in the hybrid model will affect the electron temperature, Te, in the wake. Observations show an increased electron temperature in the wake (figure 1, Ogilvie et al., 1996). But from Eq. (2), and the ideal gas law, Open image in new window for γ = 5/3, so Te in the hybrid model is then lower in the wake.
Since the only deviation from axial symmetry is the component of the IMF that is perpendicular to the solar wind flow, we can see that the two extreme cases are when the IMF is perpendicular to the flow, and when the IMF is aligned with the flow. For a constant IMF magnitude, the flow and field for cases when the IMF is at an angle to the flow will be a mixture of these extreme cases, as seen in Fig. 2(2) and Fig. 3(2).
There are many processes related to electron dynamics that are not captured by a hybrid model. The process of plasma expansion into a vacuum has a lot of details, and instabilities that can form (Birch and Chapman, 2001). Also, the region near the lunar surface probably has strong electric fields, and charge separation effects (Farrell et al., 2008). However using the hybrid approximation it is possible to model the global three-dimensional interaction between the Moon and the solar wind, so full PIC models and hybrid models are complementary tools in the study of this interaction. Also, the Moon-solar wind interaction, although seemingly simple, is complex enough to be a good test of the applicability of different models; fluid, hybrid, and PIC models.
We have here considered the simplest possible model of the Moon in the solar wind—a spherical sink of solar wind protons. There are many physical processes that make the real interaction more complex than that. Two examples are magnetic anomalies that deflect the solar wind and will cause downstream disturbances (Lue et al., 2011), and the reflection of solar wind protons by the dayside lunar surface (Holmstrom et al., 2010).
5. Summary and Conclusions
We have presented a fully self consistent three-dimensional hybrid plasma model of the interaction between the Moon and the solar wind and shown the resulting global ion fluxes and fields for typical solar wind conditions and different IMF directions. Magnetic field enhancement in the wake, and a rarefaction wave in density and magnetic field are seen in the model. The two extreme cases are that of an IMF perpendicular to the solar wind flow, and that of an IMF that is parallel to the solar wind flow. We find that the wake for intermediate IMF directions can be seen as a superposition of these two states. Some kinetic effects were also observed in the model. There are density and field variation in the far wake, maybe caused by an ion beam instability from the refill of the wake. Visible are also kinks in the magnetic field at the inner boundary of the rarefaction cone for an IMF perpendicular to the solar wind flow direction.
Observations by the WIND spacecraft were fairly well reproduced by the hybrid model, showing that a hybrid model with the moon as a spherical sink of ions is able to represent many global features of the interaction. However, the model central wake density and electron temperature were different from observations, probably due to the lack of electron dynamics in the hybrid model.
This work was supported by the Swedish National Space Board, and conducted using the resources of the High Performance Computing Center North (HPC2N), Umeå University, Sweden. The software used in this work was in part developed by the DOE-supported ASC/Alliance Center for Astrophysi-cal Thermonuclear Flashes at the University of Chicago. The work of S. Fatemi was supported by the National Graduate School of Space Technology, Luleå University of Technology
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