**Radio Atmosphere Science Center
Kyoto University at Kyoko 611, Japan
Now at SRI International, Menlo Park, California 94025, U. S. A.
Adv. Space Res., vol 15, (8/9)29-(8/9)42, 1995
Copyright 1995 by COSPAR
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The energetic electrons responsible for the generation of VLF waves in the electron foreshock are mainly of solar wind origin. While energetic electrons could possibly be of magnetospheric origin at the Earth, the presence of electron foreshock VLF waves upstream of the bow shock of Venus, for example, implies that the electrons are indeed accelerated at the shock. Leroy and Mangeney /15/, amongst others, discussed the acceleration of electrons at a bow shock, using the de Hoffman-Teller (HT) transformation /11/ to facilitate their discussion of acceleration, transmission and reflection. The HT transformation is carried out by moving along the shock surface with the velocity that aligns the upstream flow velocity and magnetic field in the transformed frame. In this frame the motional electric feld has been transformed away. Thus there can be no energization of reflected electrons in the HT frame, assuming a time stationary shock structure.
The HT frame is a particularly useful frame for discussing the processes involved in electron transmission and reflection at the shock, as can be seen in Figure 1. The left of this figure, after /15/, shows phase space contours of the incident and reflected solar wind electrons. The solar wind has been transformed into the HT frame, and the incident flow is field-aligned, with positive velocities corresponding to flow into the shock from upstream. The dashed hyperbola in the figure gives the boundary between transmitted and reflected electrons. In the HT frame only the jump in the magnetic field causes electron reflection at the shock. In the absence of a field-aligned electric field within the shock, the boundary between transmitted and reflected electrons is a cone. The shock electric field, which retards the ions, accelerates the electrons through the shock, widening the transmission cone into the hyperbolic shape sketched in panel a) of Figure 1 The portion of the incident solar wind that lies outside of this boundary is reflected at the shock and travels back upstream. It is this reflected population that is believed to generate electron plasma oscillations in the electron foreshock.
Fig. 1. a) Phase space contours of the incident and reflected solar wind electron population, in the de Hoffman-Teller frame (after /15/). b) Reflected electron flux and energy as a function of bn (after /15/).
The de Hoffman-Teller velocity in the observer's frame (which is assumed to be at rest with respect to the shock surface) is given by
where n is the shock normal, v0 is the solar wind velocity in the observer's frame, b is the unit vector along the magnetic field,vn is the angle between the solar wind velocity vector and the shock normal, and bn is the angle between the magnetic field and the shock normal. Taking primed vectors to be measured in the HT frame, the transformation from shock frame to HT frame is given by v' = v - vHT.
In the HT frame the solar wind moves towards the shock with a speed given by v0' = v0 cos vn/cos bn along the magnetic field. Since the magnetic field increase at the shock acts as a mirror, the solar wind sees a magnetic mirror moving with the speed v0', and particles reflected by this moving mirror gain twice this speed in the solar wind frame. Thus this process has been named Fast Fermi acceleration /16/. At first sight, it appears that electrons can gain an arbitrary amount of energy, depending on bn. As bn 90°, v0' , ignoring relativistic effects. However, as pointed out in /15/, the mirror efficiency 0 as bn 90°. This can be seen on consideration of Figure 1 As v0' increases, the whole solar wind population moves to higher parallel velocity, and more and more of the solar wind population lies within the transmission cone. This is further demonstrated in panel b) of Figure 1.
Another point alluded to in /15/ and /16/, but not discussed in detail, concerns the effect of shock curvature. In carrying out the HT frame transformation we assume an infinite planar shock. In reality, planetary bow shocks are curved. This affects the reflection process in two ways. First the area of the curved bow shock over which reflection occurs becomes vanishingly small as bn 90°, and so the net flux of reflected particles must 0. Second, the amount of energy gain will be restricted by the curvature of the shock. In the HT frame the reflected particles gain no energy, but in the shock frame the particles have gained energy, corresponding to the moving mirror speed. This energy gain can only come from the motional electric field -v0 B. Furthermore, the electrons gain this energy by drifting along the shock, i.e., shock-drift acceleration /4, 5/. Although shock-drift acceleration is usually discussed in the context of ion acceleration at the shock, the same principles apply to electron acceleration. Taking a solar wind flow velocity of 400 kms-1, and an upstream magnetic field strength of 10 nT, the motional electric field is 4 mV/m. The tangential electric field is continuous across the shock, and so a particle gains an energy of the order 25 keV/Re in drifting along the shock surface. For a shock, such as the terrestrial bow shock, whose radius of curvature is of the order 20 Re, electrons can gain moderately high energies before shock curvature effects become significant. However, for smaller shocks, such as the Venus bow shock, whose radius of curvature 2 Rv (1 Rv = 6052 km, 1 Re = 6371 km), curvature effects are likely to limit the energy gain of the electrons. At Mars, with an even smaller radius of curvature, and weaker magnetic field, we would expect the effect of shock curvature to be even more significant.
Since shock curvature changes bn along the bow shock, we expect electrons which encounter the shock further downstream to in general gain less energy than those electrons that encounter the bow shock near the point of where the IMF is a tangent to the shock surface. This tangent field line also marks the limit of accessibility for reflected particles, since they are convected downstream by the solar wind motional electric field. The lower energy particles take longer to travel any given distance along the field, and so they are convected further downstream by the solar wind electric field. This time-of-flight effect was first used by Filbert and Kellogg /17/ to explain the presence of electron plasma oscillations in the electron foreshock. In Figure 1 all the particles outside of the transmission cone are reflected. However, at any particular location in the foreshock, there exists a critical parallel velocity, below which the particles are convected downstream before they can propagate from the shock to the point of observation. The resultant low energy cut-off in the reflected population introduces a positive slope in the reduced distribution function (i.e., the distribution function integrated over v ), which can generate plasma oscillations through the bump-on-tail instability.
Fitzenreiter et al. /18/ developed a model for electron distributions in the foreshock. In their work they took into account the curvature of the shock through changes in the HT speed as a function of bn, and through kinematic restrictions on the allowed trajectories. Figure 2 shows some results of their modeling. Two effects are clear from the figure. The first is that the reflected distribution lies at oblique angles with respect to the magnetic field. The boundaries of the "earlike" structure are given by the transmission cone at high energy, and the time-of-flight cut-off velocity at low energy. As discussed in /18/, the distribution is built up from reflected electrons from different portions of the shock. The net effect of this is to steepen the time-of-flight gradient in the distribution. The second effect in Figure 2 is that the instrument sampling smears any gradients in the phase space distribution. Thus, because of instrument sampling, and also the smoothing that occurs in response to the wave generation, the typical signature of the reflected electrons tends to be a high energy tail. The break in the distribution marks the cut-off velocity.
Fig. 2. Foreshock electron distributions (after /18/). The left panels (a) show theoretical phase space density contours, together with the reduced distribution function. The right panels (b) show the same phase distributions, but sampled in a manner similar to a particle instrument. In this case the reduced distribution functions (solid curves) are compared with actual observations (dotted curves).
The basic concepts of "Fast Fermi" acceleration at the bow shock and time-of-flight cut-offs appear to be sufficient to explain many of the features of VLF waves in the electron foreshock, as we shall see below. However, there are still some effects not yet incorporated into theory, primarily the restriction on electron energization imposed by shock curvature, as briefly discussed above. This can best be seen through studies of the electron foreshock at Venus.
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