Pitch angle distributions of energetic ions in the lobes of the distant geomagnetic tail

0032-0633/90$3.00+0.00 Pergmon Press plc

Pluner. SpuceS~,..Vol.3X.No.7,pp 851482,1990 PnntedI"GreatBntam.

PITCH ANGLE DISTRIBUTIONS OF ENERGETIC IONS IN THE LOBES OF THE DISTANT GEOMAGNETIC TAIL C. J. OWEN, S. W. H. COWLEY, I. G. RICHARDSON* and A. BALOGH Blackett Laboratory, Imperial College, London SW7 2BZ, U.K.

(Received in finalform 16February 1990) Abstract-Analysis

of energetic (> 35 keV) ion data from the ISEE- spacecraft obtained during 19821983. when the spacecraft made a series of traversals of the distant geomagnetic tail (Xo,, z -230 RE), indicates that the pitch angle distribution of energetic ions in the distant tail lobes is usually highly anisotropic, being peaked closely perpendicular to the magnetic field direction, but with a small net flow in the antisunward direction. In this paper we present a model, based on the motion of single particles into and within the tail lobes, which accounts for these observed distributions. This model assumes that the lobe ions originate in the magnetosheath, where the energetic ion population consists of two components ; a spatially uniform “solar” population, and a population of “terrestrial” origin, which decreases in strength with downtail distance. The density of energetic ions in the magnetosheath is therefore high close to the Earth, and falls with increasing distance down the tail. The ions are assumed to enter the lobes across the open magnetopause boundary, and the speed and point of entry as a function of pitch angle are determined from consideration of the subsequent motion of the ions within the lobe, including mirroring close to the Earth. The pitch angle distribution at any point within the lobe may then be constructed, assuming that the value of the distribution function along the particle trajectory is conserved. In general, those ions with a large field-aligned component to their motion enter the lobes in the deep tail, where the “terrestrial” source is weak, whilst those moving closely perpendicular to the field enter the lobes at positions much closer to the Earth, where the source is strong. The fluxes of these latter ions are therefore much enhanced above the rest of the pitch angle distribution, and are shown to account for the form of the observed distributions. It is further shown that at a given point in the tail, the degree to which the ion distribution function is localized to large pitch angles depends on the spatial scale with which the magnetosheath source population decreases away from the Earth. The distributions broaden as the spatial scale gets larger, and show only modest variations with pitch angle in the limit that the source is uniform. The model thus also accounts for the more isotropic ion population observed in the lobe during solar particle events, when the “terrestrial” component of the magnetosheath source may be considered negligible in comparison to the enhanced “solar” component.

1. INTRODUCTION

Between October 1982 and December 1983 the ISEE3 spacecraft was engaged in a series of orbital manoeuvres which resulted in four passes into the nightside region of the Earth’s magnetosphere (Tsurutani and von Rosenvinge, 1984). Results from this mission revealed that the distant geomagnetic tail has a coherent structure, similar to that in the nearEarth tail, out to at least 240 R, from the Earth (e.g. Tsurutani et al., 1987). This structure consists of two lobes of strong magnetic field (- 10 nT), directed towards the Earth in the northern lobe, and away from the Earth in the southern lobe. These two regions are separated by the equatorial plasma sheet. which usually contains fast streaming plasma and weak disordered fields. It is in the latter region, together with

* Present address Code 661, Greenbelt.

: NASA/Goddard

Space Flight MD 20771. U.S.A.

Center,

the associated plasma sheet boundary layers, that the highest energetic (E > 35 keV) ion intensities found in the deep tail are observed (Cowley et al., 1984). The proton flux in this region usually lies in the range dJ/dE - 3 x 10’ to 3 x lo3 (s cm2 sr keV)- ’ for protons of 35-56 keV (Richardson and Cowley. 1987). This central region has been the subject of numerous publications concerned with the structure of the plasma sheet and its variations during substorms (e.g. Hones et al., 1984; Zwickl et al., 1984; Slavin et al., 1985; Scholer et al., 1986; Richardson et al., 1987a,b, 1989; Baker et al., 1988; Fairfield et al., 1989; Schindler et al., 1989), and thus has received a high level of attention and interest during recent years. In contrast, the tail lobes remain a relatively neglected area of study in both near-Earth and deep-tail data sets, largely due to the less exotic and less varied nature of phenomena observed in these regions. The nature of the thermal plasma populations in the lobes and the trajectories of the constituent particles have 851

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been discussed from a theoretical viewpoint by Pillip and Morfill (1978) and by Cowley (198 1). The former paper concerns the role played by the lobe plasma in the formation of the plasma sheet, whilst the latter discusses the effect of a significant IMF By component on the lobe field and plasma populations. In contrast, discussion of the energetic particle component of the lobe plasma populations has remained almost totally absent from the literature, except by way of passing reference (e.g. Cowley et al., 1984). Since there exists almost no description of the nature of the energetic lobe ions within the literature, we will now briefly outline the basic features of the observed distributions in order to form a basis for the subsequent theoretical development. A detailed discussion of the observed energetic ion component in the tail lobes is, however, beyond the scope of the present paper. In the outline presented here we will use data obtained from the Energetic Particle Anisotropy Spectrometer (EPAS) instrument on board ZSEE-3 (Balogh ef al., 1978 ; van Rooijen ef al., 1979). This instrument measures ion fluxes in three dimensions in velocity space by use of a system of three identical semiconductor particle telescopes, Tl-T3, inclined at 30., 60’, and 135’ to the spacecraft spin axis (which points North, perpendicular to the ecliptic plane), as shown on the left-hand side of Fig. la. Each of the

(a) Spin Axis 4 North (Z)

telescopes has a full viewing cone of 32’ centred on its look direction, as shown in the figure, and a geometrical factor of 0.05 cm* sr. As the spacecraft spins about its axis, the ion events detected in each telescope are binned into one of eight 45” azimuthal sectors, SlLS8, shown on the right-hand side of Fig. la, giving a total of 24 directional flux measurements for each energy channel in each full data sample. The energy range of the instrument for protons is 3551600 keV. divided into eight logarithmically spaced energy channels. One full energy-angle distribution is obtained every 16 s. The direction-averaged fluxes from the EPAS instrument show that the energetic ion intensities within the lobes are much reduced compared with those in the plasma sheet [dJ/dE < 100 (s cm* sr keV))’ for ions of 35-56 keV], and also compared with the magnetosheath adjacent to the tail lobes, where intensities are comparable to the plasma sheet, but vary on time scales of a few minutes. The deep tail lobes are therefore regions of depressed energetic ion intensity compared with both the adjacent plasma sheet and magnetosheath regions, although the EPAS data indicate that the lobe ion fluxes are larger close to the tail magnetopause. In addition, the capability of the EPAS instrument to make directional measurements allows us to infer the angular distributions of

Sun(X) t

East WI

Telescope

Elevation

Angles

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FIG. I (a). THE GEOMETRY OF THE EPAS VIEWINGDIRECTIONS. The left-hand figure shows the fields of view of the three EPAS telescopes, TILT3, relative to the spacecraft spin axis. The full viewing cone is 32’. The right-hand figure shows the viewing directions of the azimuthal sectors SlLS.8, looking down on the ecliptic plane from the North. The Sun (Earth) is at the top of the figure.

Energetic ions in the distant tail lobes the energetic ions in the lobe. In particular, since the lobe magnetic fields are so highly ordered, pointing in a closely earthward (and sunward) direction in the northern lobe, and a closely tailward (and antisunward) direction in the southern lobe, it is a relatively straightforward exercise to relate each of the EPAS look-directions to a particular range of pitch angles. If we consider telescope T2 of the EPAS instrument, which has a look-direction lying closest to (3Ok 16’ above) the ecliptic plane (see Fig. la). the number of counts in the 45- azimuthal sector centred on the sunward look-direction (Sl) gives an indication of the flux of ions flowing along the field in an antisunward direction (i.e. particles with pitch angles within a few tens of degrees of 0’ when the spacecraft is located in the southern lobe or 180’ if the spacecraft is in the northern lobe). Conversely, the particle counts in the 45’ sector centred on the antisunward look-direction (S5) indicate the flux of particles moving in a sunward direction (0’ in the northern lobe, 180’ in the southern). Further, the two 45’ sectors with look-directions perpendicular to the Sunspacecraft line (S3 and S7) must also be directed approximately perpendicular to the magnetic field direction in both lobes, and the counts in these sectors therefore indicate the flux of particles with pitch angles close to -90’. The remaining four sectors of course indicate the fluxes of particles with pitch angles intermediate between those mentioned above. Consideration of the relative number of counts in the eight azimuthal sectors of telescope T2 therefore gives a broad indication of the form of the pitch angle distribution in the lobes. Some T2 sector plots for the lowest EPAS energy channel (35-56 keV) are shown in Fig. lb. In this figure the ion counts are plotted linearly vs viewing direction, normalized to the maximum sector count rate (which is indicated beneath each plot). and with the sunward viewing direction towards the top of the page, as on the right-hand side of Fig. la. Each plot shows an average count rate over some 320 s. and the average direction of the magnetic field projected onto the ecliptic (GSE .x-j,) plane during each period is indicated by the adjacent small black arrows. The deviation of the magnetic field out of the ecliptic can be seen in the sub-panel beneath each plot. which shows a projection of the field direction onto the GSE X-Z plane. Figure 1b shows the sectored count rates from four typical periods when the spacecraft was located in the lobes (but at varying distance down the tail) during normal solar wind conditions (upper panel), and four periods during a solar particle event, when the solar wind energetic ion fluxes are considerably enhanced

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by the presence of flare-associated particles (second panel). The lower two half panels show, for comparison, the sectored count rates observed when the spacecraft is located in the plasma sheet and magnetosheath, respectively. As can be seen in each of the plots in the upper panel of Fig. 1. the pitch angle distribution in the lobes under normal circumstances has a characteristic “pancake”-like nature (see also Cowley et al.. 1984, and Richardson and Cowley, 1985). The distributions have the highest count rates in the two look-directions which are most perpendicular to the field direction, and the lowest counts in the field-aligned direction. The ion distribution is therefore peaked at pitch angles close to 90‘ to the magnetic field direction, and falls off at both higher and lower pitch angles. This is in contrast to the plasma sheet ions under normal conditions (first plot. lower-left half panel) which show a clear streaming in the tailward direction in the deep tail (as indicated by the high count rate in the single sector Sl representing the most sunward look-direction; see also Cowley et al., 1984; Daly et al., 1984) and also to the tailward convecting distributions observed in the magnetosheath (e.g. Williams et al., 1988). This latter case is illustrated in the first plot of the lower-right half panel, which shows higher count rates in the more sunward lookdirections. The plots in the top panel do, however, indicate a net antisunward streaming in the lobe energetic ion distribution, resulting in generally higher count rates in the three sectors with sunward components to their look-directions (Sl. S2 and S8) than in those with antisunward look-directions (S4S6). The flux of tailward streaming particles is thus usually somewhat larger than the earthward-streaming flux, an impression which is confirmed by more detailed analysis. The fundamental properties of the lobe energetic ion distribution function are therefore that it is quite highly peaked perpendicular to the field direction, whilst maintaining a moderate net antisunward anisotropy in the field-aligned components of the flow. The above characteristics of the energetic ion lobe distribution hold except during times when the solar wind and magnetosheath fluxes are enhanced by the presence of flare-associated solar particles. One such solar particle event commenced at 06:03 U.T. on 3 February (day 34) 1983 when BEE-3 was located in the deep tail, and lasted for 5-6 days. An associated interplanetary shock, accompanied by intense low energy ion fluxes passed the Earth at l6:15 U.T. the following day. During the period of this event, the ion fluxes in the magnetosheath were one or two orders of magnitude above those normally observed, as can

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Energetic ions in the distant tail lobes be seen in the lower-right half panel of Fig. lb. The first plot in this half pane1 is for the day preceding the solar particle event onset, and the second plot for the day following the passage of the shock. Although the form of the distribution does not dramatically alter between the two days (both plots are typical of those observed for an antisunward convecting distribution), the count rate recorded during the solar particle event is much higher than that for more normal times. Further analysis of the EPAS data also indicates that the spectrum of the ions is much harder during this event than under norma conditions. Under these circumstances, the lobe energetic ion distribution is dramatically altered, becoming much more nearly isotropic in nature, as can be seen from the four plots in the middle panel of Fig. 1. These show the counts recorded by T2 on each of the 4 days following the onset of the above solar particle event. In each case the count rate is approximately independent of lookdirection (and much higher than those recorded during normal conditions). The lobe energetic ion fluxes during solar particle events are therefore approximately isotropic. Any model attempting to explain the observations in the lobe should be able to account for the differences in the ion characteristics which occur between those observed during solar particle events and those appearing under more usual conditions. The aim of this paper is to present a simple explanation of the properties of the energetic ion populations in the tail lobes, deduced from the properties of the source distribution for these ions, and their motion into and within the lobes. We shall argue that the lobe ion population originates predominantly in the magnetosheath, and we shall assume that these ions gain entry to the lobes across an open tail magnetopause boundary. The energetic ion distribution function in the lobe then depends on the characteristics of the distribution of the magnetosheath source, the nature of the magnetopause boundary between the lobe and the source, the physics

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of particle transmission across this boundary, as well as the subsequent motion of the ions within the lobe. The next section, therefore, discusses the properties of the distribution of energetic ions in the magnetosheath, which is argued to be the source of the lobe ions, and the interaction of the particles with the tail magnetopause. Section 3 then deals with the particle motion in a simple model lobe, in which the magnetic field is assumed to be constant in both strength and direction. The pitch angle dist~butions produced by the model are also discussed in the latter section, including their dependence on both position in the lobe and on the properties of the magnetosheath ion distribution. Finally, in Section 4 the mode1 is revised to include some variation of field strength along the tail in order to make the model slightly more realistic. It is shown that the results do not differ qualitatively in form from the simple modei, which may therefore be considered adequate to explain the main observed characteristics of the lobe energetic ions.

2. THE SOURCE OF ENERGETIC PARTICLES IN THE DEEP TAIL LOBES

Simple considerations indicate that the only reasonable source of the energetic ion distribution observed in the tail lobes of the geomagnetic tail lies in the tailward-convecting population found in the adjacent magnetosheath. In principle, the lobe ions could also originate by direct injection along lobe field lines from the ionosphere, but the ionospheric source is typically confined to energies below a few tens of electron volts. Even in the presence of discrete aurora1 phenomena, the injected ionospheric ions generally only reach energies up to a few kiloelectron volts, which is still an order of magnitude or more below those considered here. In addition, due to the magnetic mirror effect, such ions would be expected to possess a highly collimated meld-aligned dist~bution in the distant tail,

FIG. I(b). SOME EXAMPLESOFTYPICAL EPAS T2 SECTOR PLOTS WITHIN THE EARTH'S NIGHTSIDE REGION. Each plot in this figure shows the count rates observed by each of the eight T2 look-directions, indicated by the radius of the corresponding 45’ sector. The azimuth of the magnetic field and its projection on the X-Z plane are indicated by the arrows in each plot. The top panel shows plots of data taken when the spacecraft was located in the lobes of the tail during normal conditions, for a range of distances downtail from the Earth. Each of these plots shows high count rates in the lookdirections perpendicular to the field direction, indicating that the distribution is peaked at pitch angles close to 90 The generally higher counts in the sunward look-directions than the antisunward directions also indicate a small down&ii anisotropy in the flow. The second panel also shows plots of data taken whilst the spacecraft was in the lobes. but during the 4 days following the occurrence of a solar particle event. and shows a smaller angular variation in the count rates for each panel, indicating that the distribution is more isotropic under these conditions. The lower two half panels show plots of data from the plasma sheet and magnetosheath regions for comparison (one each for normal and solar particle event conditions). In these cases the distributions are strongly peaked in the antisunward direction, due to the rapid taibvard streaming in these regions.

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as opposed to the field-perpendicular distributions which are actually observed. In this paper we shall show, however, that a magnetosheath source can give rise to lobe ion populations with the observed characteristics outlined in the previous section. It is argued that under normal conditions the ions in the magnetosheath (and lobes) are dominated by particles having a “terrestrial” origin, from either the bow shock or the Earth’s radiation belts. Either of these sources will give rise to a magnetosheath population whose intensity decreases with increasing distance from the Earth, as will be discussed further below. It will be shown that in these circumstances the lobe ion population will be peaked perpendicular to the field, with a small tailward anisotropy, as observed. On the other hand, during solar particle events, the magnetosheath population will be dominated by solar particles whose intensity varies only slowly with distance from the Earth. It will be shown that in this case the lobe population should be nearly isotropic, again as observed. In this paper we shall therefore assume that some portion of the magnetosheath ion distribution effects entry into the lobe through an open, “leaky” tail magnetopause, and that these particles then constitute the source of the lobe population. Under these circumstances the nature of the lobe energetic ion distribution is governed by three factors : (a) the physics of particle transmission across the magnetopause into the lobe; (b) the form of the magnetosheath energetic ion distribution just outside the magnetopause ; and (c) the propagation of the ions from the magnetopause to the point ofobservation within the lobes, and the conservation of the particle distribution along the trajectories from Liouville’s theorem. Topics (a) and (b) will be discussed in this section. whilst topic (c) is discussed in Section 3. Using these results, the distribution function of energetic ions of given velocity 11and pitch angle CIat any given point within the lobe can be calculated by mapping back the particle trajectories to the magnetopause using the results of (c), and relating to the magnetosheath distribution (a) using (b) above. Here, we shall first consider the transmission of energetic ions across the magnetopause current sheet, i.e. topic (a) above. We first note that the Larmor radius of the energetic protons considered in this paper (E > 35 keV) within the lobe and the adjacent magnetosheath, where the magnetic field strength is usually -10 nT, is -0.5& (I R, = I Earth radius z 6400 km). This is large compared with the likelvi thickness of the maenetonause current sheet.

which is of order of a few thermal proton gyroradii thermal protons (Rg 2 l/30 RE for magnetosheath with temperature z IO6 K; e.g. Lui, 1987). For particles of this energy it therefore suffices to model the field system as two distinct regions (lobe and magnetosheath) of differing strength and direction, separated by a current sheet of zero thickness. The configuration of the magnetic fields at the magnetopause envisaged here, is shown in Fig. 2a. In this figure, a section of the magnetopause is shown lying in the X-Y plane separating the magnetosheath (regions of positive Z) from the lobe (regions of negative Z). The field in the magnetosheath Bs, is orientated at an angle OS, to the plane of the magnetopause, and at an angle &, to the positive X-axis, and is frozen into the solar wind plasma flow, which moves tailward (negative X-direction) at a speed vsu (assumed constant along the length of the tail in this model). Across the magnetopause the field changes sharply, becoming stronger in the lobe. In this region, the field direction is aligned closely along the X-axis, but has a small component in the negative --direction, thus making a small angle H, to the plane of the magnetopause. Note that the normal (Z) component of the field is continuous across the boundary (i.e. it is a rotational discontinuity), and that the field in the lobe moves slowly towards the tail centre plane with the ExB drift of the lobe plasma, in conjunction with the tailward convection of the magnetosheath field at the magnetosheath flow speed. The speed of the inward drift in the tail lobe is thus vsH sin f& transverse to the lobe field. or csH tan 0, along the r-direction. In order to estimate the typical normal component B, of the field at the tail magnetopause, we note that the field strength (B ( in the lobe is of the order 10 nT (see, for example, the review by Tsurutani et al., 1987), whilst the tail radius R, is of order 30 R, (e.g. Sibeck ct al., 1986). and the length L, of the tail is about 1000 RF (Dungey, 1965). Since the total flux content of the tail lobe (Q, = ]B( . nR,‘/2 Wb) enters the lobe through the tail magnetopause. the average normal component of the field through the magnetopause current sheet is given by B, = @,IxR,L, = IBI . R,/2L, = O.l5nT,

(1)

i.e. two orders of magnitude less than the transverse components of the field. Note, however, that if the open flux occurs in patches at the magnetopause [as would be the case if flux is added to the tail principally by flux transfer events (FTEs). or sporadic dayside reconnection; e.g. Russell and Elphic, 19781 then B, in these patches would be larger than this value. However, for our purposes. the lobe field is best represented by a large field component transverse to the mag-

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netopause current sheet (B1 x 10 nT), with a much smaller normal component (B, z 0.15 nT). The transSHEATH verse component in the magnetosheath is generally \ smaller than that in the lobe, and may be orientated in essentially any direction. Results of numerical calculations of the particle motion between two regions separated by a rotational discontinuity in the magnetic field (such as at the tail magnetopause) have been presented by Hudson (1974) for a variety of field strengths and orientations. This work demonstrates that although in general each particle crosses the current sheet many times, a large majority of ions incident on the magnetosheath side of the current sheet are usually transmitted through FIELD b) LINE EARTH’S into the lobe. The only significant departures from REST REST FRAME FRAME this result occur when the transverse component of the magnetosheath field is highly antiparallel to that in the lobe. and is much larger than the normal component threading the boundary (i.e. there is an extremely sharp change of magnetic field direction between the two regions), when some particles of certain pitch angles may be reflected back into the magnetosheath. The worst case, with the magnetosheath transverse field component exactly antiparallel to that in the lobe, with ~Blobe~/~BEhealh~ = 1, results in approx. 50 % of particles reflected back into the magnetosheath. However, this figure decreases rapidly as the magnetosheath field strength drops (-40% for ~B,obc~/lBsheath~ = 1.5), and as the fields FIG. 2. DIAGRAM ILLUSTRATING THE ASSUMED FIELD become less antiparallel (30% with an angle of 135” STRUCTURE OF, AND PARTICLE BEHAVIOUR IN, THE TAIL LOBE between the transverse field components, 0% at 80”, MAGNETOPAUSEREGION. The upper diagram shows the coordinate system and field for 140bclll&cathl = 1). H owever, the particular part structure on either side of the boundary. The magnetic field of the transmitted pitch angle distribution which is (represented by the heavy arrowed line) changes direction missing due to reflection If(v,,, uL) = 0 in the lobe] is and strength abruptly at the point that it crosses the magfound to be very highly sensitive to small variations netopause. In the magnetosheath it has strength BsH, and is in the angle between the fields in the two regions. oriented at an angle 4s” to the x-axis in the plane of the boundary, and elevated by an angle HSHout of that plane. Since, in reality, the magnetosheath field is usually In practise each of these three parameters generally varies observed to vary rapidly in both strength and direcrapidly in time, and the field is continually convected tailtion, this effect is likely to be smoothed out and so wards with the magnetosheath flow at the speed cSH. By does not need to be a major consideration here. Also, contrast, the field in the lobe is steady, in both strength (BL) and direction (tilted by a small angle fIL out of the plane of Hudson (1974) considered the case of an infinitesithe boundary from the x-axis). The field in the lobe convects mally thin current sheet ; if we relax this assumption, towards the tail centre plane with the ExB drift speed such that the current sheet has finite thickness, rsII sin 0,. The lower diagram shows the transformation of and the field direction does not change so rapidly, we velocities between the field line rest frame (de Hoffmanwould expect that even more particles would gain Teller frame) and the Earth’s frame. The field line (magnetosheath) frame is represented by the axes (L’;. vi), and is access to the lobe. Hence, for the purposes of this direction with shifted by an amount cS,, in the antisunward work, it is satisfactory to assume for simplicity that all respect to the Earth’s frame (I‘,. c,). The lobe field direction IS represented by the solid arrowed line inclined at angle GIL particles incident on the current sheet are transmitted through it. to the t’, axis. In the Earth’s rest frame the particle motion consists of an E x B drift towards the tail centre plane at In the Earth’s rest frame the velocity of the ions speed rF.XB, a component of velocity L’,!along the field direcchanges in both magnitude and direction as they cross tion, and a component rL perpendicular to the field direction. the magnetopause boundary. However, in the rest The geometry of the diagram can be used to determine the frame of the magnetic field lines the speed of the velocity r* of the particle in the magnetosheath (field line) frame [see equation (6a,b)]. particles is conserved since the accelerating electric a)

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fields are transformed away (deHoffman and Teller, 1950). The transformation of energetic ion velocities from the field line to the Earth’s frame is illustrated in the velocity space diagram shown in Fig. 2b. The axes in this figure represent the X- and z-components of the particle velocities in both the Earth’s (o,,~,) and the field line (v:,ui) rest frames. Since the magnetosheath field lines are convected tailward with the magnetosheath flow, it follows that the vz and 0: axes are separated by speed us”. The direction of the magnetic field in the lobe is represented by the heavy arrowed line marked B, which is drawn passing through the origin of the field line rest frame. The motion of the particles in the lobe in the Earth’s frame can be considered to be made up of three contributions. Firstly, the particles are convected with the field lines towards the centre plane of the tail lobe. This is the E x B drift in the Earth’s rest frame, and is represented by the arrow labelled vex Bin Fig. 2b. The remaining contributions are the components of the ion motion parallel and perpendicular to the field line direction, labelled u,, and uI respectively (note, however, that this latter component gyrates about the field direction at the ion gyrofrequency). The corresponding velocity in the field line rest frame is represented by the arrow labelled u* connecting the origin of that frame with the tip of the combined velocity vector in the Earth’s rest frame. Consideration of the geometry of this diagram allows the speed u* to be expressed in term of the other parameters as follows u* =
(24 (for

(2b)

In this paper we shall use this approximation to relate the speeds u,, and uI of particles just inside the magnetopause to their speed in the magnetosheath field line rest frame, u*. Since there is no change of speed of the particles as they cross the magnetopause in this latter frame, the speed u* is also the speed of the particle (in the field line rest frame) when it was located in the magnetosheath. However, in order to map the ion distribution function from the magnetosheath into the tail lobe using Liouville’s theorem, we must in general also consider the possibility that the particles may undergo pitch angle scattering on crossing the magnetopause. A general model for this process would depend implicitly on the strength and orientation of the fields on both sides of the magnetopause (i.e. on B,, OL, BSH, OS,, and $sn). However, considerable simplification results if we

assume that the ion distribution in the field line rest frame in the magnetosheath is isotropic. In this case, the value of the distribution function associated with a particular particle depends only on the speed of the particle in that frame, and not on its pitch angle, i.e. the results are then independent of the change in pitch angle. Williams et al. (1988) have recently shown that in the highly variable magnetic fields of the magnetosheath, the energetic ions (E > 5 keV) are indeed isotropic in the flow rest frame, and are thus, to a good approximation, simply convected tailward with the thermal magnetosheath plasma to produce the anisotropies observed in the satellite frame. The assumption of an isotropic magnetosheath source plasma is thus well justified. We have already indicated above that the speed of the particles in the field line rest frame is conserved as they cross the magnetopause, and the value of fis also conserved along the trajectories by Liouville’s theorem. The value off(u,,,v,) for a particle just inside the magnetopause will then simply be related to the value of f(~*) of the isotropic source plasma by use of equation (2) and pitch angle scattering at the magnetopause does not affect this result. Having thus considered how the ion distributions are mapped across the magnetopause [topic (a) above], we now turn to consider the properties of the observed magnetosheath energetic ion distribution functions [topic (b)]. Analysis of EPAS data (not reported in detail here) shows that the spectrum of the energetic particles in the magnetosheath plasma frame may be conveniently described by a power law in energy such that

where dJ/dE dR is the ion differential flux (number of particles per unit area, per steradian, per unit energy, per unit time), E is the energy of the particles in the magnetosheath frame, and k is a constant. The parameter f is an empirical exponent which, for typical conditions in the magnetosheath, has a value of ~3.5, although the spectra tend to harden during solar particle events to values of F that may even be less than unity. This is important when considering the difference in the observed lobe distributions between normal and solar particle event conditions, as will be explained later. The distribution function S is related to the differential energy flux by the relationship _=-dJ dEdQ

fve2 m, ’

(4)

where II* is the speed of the particle in the mag-

Energetic

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netosheath frame, and mi is the ion mass. Eliminating dJ/dE dR between equations (3) and (4) gives the distribution function in terms of the particle speed t’* in the magnetosheath rest frame as f(v*)

= k’/v*2(r+ 11,

(5)

The diswhere k’ = 2’k/m, w ‘) is a new constant. tribution function of the ions in the magnetosheath is illustrated schematically in Fig. 3a, showing contours of constant f in velocity space. As in Fig. 2b, the magnetosheath frame is represented by the axes r:: and tll_which are shifted by an amount JrSHI in the

a)MAGNETOSHEATH DISTRIBUTION

v

Vz'

b) TAIL SOURCE DISTRIBUTION

PARTICLES

FIG. 3. SCHEMATIC ILLUSTRATION OF THE DISTRIBUTION OF ENERGETIC IONS IN THE MAGNETOSHEATH. AND THE PORTION OF THAT DISTRIBUTION WHICH ACTS AS A SOURCE FOR THE LOBE PARTICLES. The axes are the same as employed in Fig. 2b. The magnetosheath distribution is taken to be isotropic in the field line frame, and is thus represented by a set of concentric circles (contours of constant distribution function) about the origin of this frame. The lower panel includes a lobe field line (solid arrowed line at an angle I), to the r, axis). which is at rest in the magnetosheath frame. The portion of the magnetosheath distribution which has access to the lobe is that portion whose field-aligned velocity in the field line rest frame has a component directed towards the tail centre. away from the magnetopause as shown. In the Earth’s rest frame this population includes particles with both earthward and tailward components of velocity. The particles in the hatched region move tailwards in the Earth’s rest frame.

859

negative vi direction relative to the Earth’s rest frame (L),, v,), corresponding to the motion of the field lines away from the Earth with the magnetosheath flow. The contours of constant f are represented by the concentric circles about the origin of the solar wind frame (t’:, vi), i.e. the ion distribution function is isotropic in this frame, in conformity with the above discussion. Figure 3b illustrates the corresponding ion distribution function just inside the tail lobe (i.e. the source of the lobe population) assuming perfect transmission of particles across the magnetopause, as indicated above. This diagram has the same format as Fig. 3a above, but we now include the direction of the lobe magnetic field, indicated by the solid line and arrow, making an angle 0r with the u, axis. As described above, the component of the field threading from the magnetosheath into the lobe is small compared with the transverse components, such that 8, is also small (and shown exaggerated here for clarity). Since there is no acceleration of the particles as they cross the tail magnetopause, the form of the distribution is unaltered in this frame. However, only the portion of the distribution which has a positive parallel component of motion in the solar wind frame may move away from the magnetopause and into the lobes, i.e. only half of the magnetosheath distribution acts as a source for the lobes, as shown in the figure. It should be noted that in the Earth’s rest frame this population includes particles which have parallel velocity components in both directions along the field line. Those particles with positions in velocity space in the hatched region have field-aligned velocities in a tailward direction, whilst those to the left of this region move earthward along the field. However, those particles in the hatched region have field-aligned speeds which are less then 1usH1cos O,_z 1vSH1,and hence have a net motion into the tail lobe with the E x B drift. rather than moving back out into the magnetosheath. Having established the general form of the magnetosheath distribution at a particular point on the magnetopause, and which parts of the ion distribution act as a source of the lobe energetic ions, we now consider the variation of the magnetosheath distribution with distance down the tail. As we shall see in the next section, particles arriving at a particular point within the lobe have trajectories that intersect the magnetopause over a large range of distances along the tail length, such that the variation of the source distribution with distance down the tail is a critical factor in determining the lobe distributions. As indicated in the Introduction, the omnidirectional intensity of energetic ions in the mag- 200 RE downtail from the Earth is 1-2 netosheath

860

C. J. OWEN ef al.

orders of magnitude higher than those fluxes measured within the lobes, although there are considerable variations in the flux levels within the magnetosheath. Further, analysis of EPAS data from the solar wind upstream of the bow shock during normal conditions (non-solar particle events) shows that the magnetosheath flux is generally enhanced by a similar factor above the solar wind energetic ion intensity. It is thus evident that some process must occur earthward of the ISEE- location in the deep tail (i.e. at X > - 100 I&) which results in enhanced energetic ion fluxes in the magnetosheath. Several processes have been suggested as causes of such fluxes. First, particles may be energized at the bow shock (Asbridge et al., 1978). Crooker et al. (1981) showed that energetic ions were preferentially present on magnetosheath flux tubes when the angle between the bow shock surface normal and the magnetosheath field, at the point where the’plasma streamlines intercept the shock, is less than a certain critical value. This finding is consistent with acceleration of particles by the Fermi mechanism in the turbulent field regions associated with quasi-parallel bow shocks, as described by Lee (1982). This process may, in part at least, be responsible for the variable nature of the magnetosheath ion flux. The second process that may account for these observations is the escape of energetic particles from the ring current or plasma sheet across the near-Earth magnetopause, either during dayside reconnection, or by some other “leakage” process. Clearly, as ring current flux tubes become open as a consequence of dayside reconnection, energetic particles will rapidly escape from the magnetosphere into the magnetosheath, and have been observed to do so by e.g. Scholer ef al. (1981). However, particles may also escape (or “leak”) from the magnetosphere by grad-B loss from closed flux tubes (Anagnostopoulos et al., 1986; Sibeck et al., 1987). Both of these processes may contribute to the formation of the tailward streaming energetic particle layer observed adjacent to and outside the tail lobe magnetopause (e.g. Baker and Stone. 1977, 1978; Lanzerotti et al., 1979). The highly variable nature of the shocked field in the magnetosheath will result in the scattering of energetic particles in this region. As a consequence, ions from either source discussed above will move approximately with the flow (ic. a convecting ion population. as observed), while also diffusing along the mean ficld. In particular, for particles with a magnetospheric source, this implies the possibility of a back-scattered population along the open flux tubes, rather than a freely escaping population. As a result of scattermg. it is possible for ions from a ring current source to re-

encounter the magnetopause at some point further tailwards. The relative importance of the magnetosphere and the bow shock as sources of energetic particles in the magnetosheath and solar wind is the subject of much current debate (e.g. Sarris et al., 1987; Sibeck et al., 1987) which is beyond the scope of the discussion here. However, it should be noted that, for our purposes, the outcome of this debate is largely irrelevant since both sources lead to similar effects within the magnetosheath, specifically that the intensity of the energetic ions is greatly enhanced over solar wind levels close to the Earth (other than during solar particle events), and then drops off with increasing distance downstream, eventually returning to the solar wind levels at sufficiently large distances, as illustrated in Fig. 4. In the case of the remnant of the escaped ring current distribution, this decline in fluxes is due to the continual diffusion of particles away from the magnetosphere as discussed above. Although diffusion is also important in the case of ions produced

FK. 4. A

NOON~MIVNIGHT

CROSS-SECTION

THROUGH

EARTHS'

MAGNETOSPHERIC

SYSTEM SCHEMATICALLY

TRATING

THE

EQUIVALENCE

IN THE

MECHANISMS

FOR

PKOVUCTION

DEEP

TAIL

ENERGETIC

THE

ILLUS-

OF THE IONS

IN

TWO THE

MAGNETOSHEATH.

The hatched region in the upper part of the diagram indicates the spatial extent occupied by ions produced at the point S on the bow shock. whilst the cross-hatched area in the lower part of the diagram indicates the region occupied by particles escaping across the dayside magnetopause at point M. Due to convectton and diffusion processes in the magnetosheath, both sources result in a magnetosheath population at the magnctopause which falls in intensity with increasing dislance from the Earth.

Energetic

ions in the distant tail lobes

by energization at the bow shock, it is also evident that, in general, magnetosheath particles detected in the nearer Earth magnetosheath have streamlines that intersect the bow shock closer to the nose of the shock, where the acceleration process is likely to be most efficient. The trajectories of those energetic ions which reach the magnetopause in the deep tail map back to the bow shock at locations further downwind where the shock and the acceleration process may be weaker, resulting in lower fluxes than at locations closer to the Earth. The magnetosheath energetic ion population can therefore be thought of as consisting of two components, a component from the terrestrial source which declines in strength with increasing distance from the Earth. and a component from the ambient solar wind energetic ion population which is essentially independent of distance down the tail. and which. under normal conditions, is much weaker than the terrestrial source near the Earth. For the purposes of this study we shall. therefore, assume a simple variation of the magnetosheath source strength with distance s (negative) downstream, given by s(s) = S,, exp (X/L) + I.

(6)

whcrc S, is the factor by which the intensity of the “terrestrial” ions close to the Earth (X = 0) exceed the background intensity of ions of similar energy in the solar wind, and L is the scale length of the variation. Note that the EPAS data indicate that S, should be of order IO’-IO3 under typical conditions. reflecting the 2-3 orders of magnitude increase in the near-Earth magnetosheath flux over that observed in the solar wind. However, under solar particle event conditions when the solar wind fluxes are considerably enhanced. and are the dominant component of the magnetosheath distribution. S,, may be neglected in equation (I). giving a constant source strength along the entire length of the tail. In this analysis, the factor L is essentially an unknown parameter. and we shall give results for a range of values of L in subsequent sections. We shall also neglect effects due to temporal flux variations in the magnctosheath. so that the source function s(.Y) depends only on distance down the tail. A general formula for the value of the distribution function for particles entering the lobes at a position X. with velocity (c, , 1‘ L) may now be obtained by combining equations (2b) and (5). and by including the effect of the spatial source strength function s(.Y). such that (for cos 0, = 1)

861

Since the value of the distribution functionfis conserved along a trajectory, the value off associated with a particular particle energy and pitch angle at a particular point within the lobe may be determined by tracing back the trajectories of such particles to the lobe magnetopause, to find the position x and velocity (c,,, I’~) at entry. From equation (7) it follows that under normal conditions particles which enter the lobe close to the Earth will have a higher value of .f than those entering further tailward. Also, equation (7) gives a larger value of,ffor those particles which enter the lobes with negative (tailward) velocities than those which enter at the same point, and with the same energy in the Earth’s frame, and then move earthward, since the former have smaller values of c* in the magnetosheath frame. This is shown by the dashed line in Fig. 3b, which is a circle centred on the origin of the Earth’s frame. which represents particles of equal energy in that frame. In the small 0, approximation. the maximum tailward field-aligned speed a particle entering the lobe can possess is the speed of the magnetosheath flow. t’s,, , as there must exist a net earthward motion of particles in the field line rest frame for particle entry. On the other hand. particles entering with earthward motion in the Earth’s rest frame may have any field-aligned velocity, up to the total speed of the particle in the small OL limit. For F = 3.5, typical of the magnetosheath energetic ion distribution, the ratio offfor particles entering with the maximum tailward field-aligned speed (those with velocity vector marked Tin Fig. 3b) to those entering the lobe at the same point s, and then moving along the field lines in the earthward direction with l’I = 0 (the vector marked E in Fig. 3b), is approx. 3 for particles of energy 35 keV (the lowest EPAS energy channel), and is less than this for higher energies. Hence the source distribution at a particular point on the magnetopause is anisotropic in the Earth’s frame. but not strongly so. However. the spatial variation of the magnetosheath flux [equation (6)]. depends on the value of So, and the scale length L for the variation of the source strength function. As explained above. the data indicate that the near-Earth magnetosheath flux exceeds the background solar wind flux by 2 or 3 orders of magnitude under normal (non-solar particle event) conditions. Hence the magnctoshcath flux along the magnctopause also varies over a few orders of magnitude. The value of,/‘at the magnetopause is therefore more strongly dependent on _Ythan it is on the values oft’, and r, The latter are still important. however, especially when the scale length for magnetosheath density variations L is large. e.g. during solar particle events, when the spatially varying “terrestrial” component of the magnetosheath source

862

c. J. OWEN

population may be considered negligible with respect to the highly enhanced spatially uniform “solar” component. Having thus discussed the nature of the magnetosheath source population for the tail lobes, in the next section we shall go on to study the motion of particles within the tail lobe with a view to establishing the point at which each particle in the pitch angle distribution at a given point within the lobe crosses the tail magnetopause, and the velocity with which it does so. Using the above analysis for the distribution function of particles at entry, and invoking Liouvilles’ theorem, we can then construct the pitch angle distribution of the particle population at any point within the model lobe. 3. MOTION OF PARTICLES IN A UNIFORM-FIELD LORE WITH A PERFECT MIRROR AT ONE END

In the last section we discussed the properties of the distribution functionf(x, u,,, uI) of the ions just inside the tail magnetopause based on physical arguments relating to the sources of these particles and their transmission across the magnetopause. In order to find the distribution functionf(r,,v,) of particles of velocity v0 at a general point r,, within the lobe, we now need only to track the particle orbit back to the magnetopause, deduce the point xh and velocity vb at the point of entry of the particle and, using the properties of the magnetosheath source distribution discussed above, thus find f(x,, vJ. From Liouville’s theorem we then havef(r,, v,) =f(x,, vi,), such that it

69 01

is possible to deduce the distribution function at any point in the lobe. The results can then be compared with the observations described in Section I above. In this section we shall therefore investigate the motion of particles within a simple model lobe, in order to determine the downtail position xi, and velocity vb of the particles at the magnetopause, given that they are later located at some point (x,, zO)in the lobe and are moving with velocity v”. We shall, at first, assume the simplest possible model lobe to illustrate the main principles involved in the formation of the pancake energetic ion pitch angle distributions. The coordinate system and magnetic field structure assumed for this model lobe are illustrated in Fig. 5. The x-direction points towards the Sun with the Earth located at x = 0 as before. All points within the lobe ConsequentIy have negative xcoordinates. The z-direction represents the direction normal to the tail current sheet and magnetopause; note that there is no J’, or crosstail dependence in the model. The model is bounded in the r-direction by the tail centre plane at z = 0, and by the magnetopause at z = zmp, such that the lobe is represented as an elongated box. The magnetic field is assumed to be everywhere constant, with the exception of the earthward boundary, at x = 0, where a perfect magnetic mirror is applied, corresponding to an infinite step increase of the field strength at this plane. This is a simple method of introducing the effect of mirroring in the high held region near the Earth, which is an important component of the model. Any particle incident on this mirror is assumed to be perfectly reflected,

z t

MAGNETOSHEATH

FIG.~.THE~OORD~NATESYSTEMANDMAGNETICFIELDSTK~C~URF.ASSUMEUIN~F~ES~MPLEMODELLOBE. In this model the s-direction is taken to be aligned with the Sun-Earth line with the Ear&h located at x = 0. Points within the lobe therefore have negatrve .x-coordinates. In the z-direction the lobe is bounded by the tail centre plane at : = 0. and by the magnetopause at I: = zmp. The field in the lobes has a large component in the positive s-direction. and a smaller component in the negative z-direction, such that the field line is tilted through a small angle fIL with respect to the s-direction. Each field line thus threads into the magnetosheath at some point along its length. At the earthward end of the lobe we apply a perfect mirror, which we assume to reflect all incident particles. instantaneously reversing their field-aligned component of motion.

Energetic ions in the distant tail lobes i.e. its field-aligned component

of velocity is instantaneously reversed. The x-component of the field, B,, is assumed positive and to be considerably larger in magnitude than the z-component, - 1B, 1,so that the angle ff, between the x-axis and the field direction is small [in this section we shall be using the values for B,Y,B_calculated previously (B, = 10 nT, B.. = -0.15 nT), giving eL z 1")To a good approximation, therefore, the field in this lobe thus points directly towards the Earth. (& reverses sign in the other lobe so the field points away from the Earth ; however, the sign of & is of no importance to the particle kinematics within the lobes.) Each field line therefore threads into the magnetosheath at some point along its length, where the component of the field threading the boundary B, = - ]&I_ The magnetosheath plasma is again assumed to move tailwards with the speed osH (taken to be positive in the negative x-direction) at all points along the tail magnetopause, and hence the point at which any particular field line crosses the magnetopause also moves tailward at this speed. This tailward motion of the field at the magnetopause results in a convection of the field tines in the lobe towards the tail centre plane at a speed c‘, = t+B,/B., ( =oEXr, in the limit of small OL,i.e. when IBY;z B), a considerably smaller speed than that of the magnetosheath flow (for example, if vsn = 500 km s- ‘, and BJB, = 0.015, as above, then L’;z 7.5 km s-‘). The motion of the field lines within the tail is associ: ated with a convection electric field across the tail lobe in the y-direction given by E = vSHBn (z 0.1mV rn- ’ with the above numbers), representing the magnetosheath electric field mapped along the equipotential open field lines into the tail lobes. Energetic particles (with speeds much greater than that of the field lines) within the lobe move. for the most part, towards or away from the Earth along the field direction, but also have a small E x B drift towards the tail centre plane, at the speed c; given above. The guiding centres of the particles therefore move towards the tail centre plane with a given field line. Further, there are no grad-B. or curvature drifts within this model, nor do the guiding centres experience any form of acceleration within the lobes. or at the magnetic mirror, other than that which reverses the field-aligned velocity of the particles at the latter. We will now consider a particie of energy E and pitch angle SL,whose guiding centre arrives at a point (xO,zO) within the lobe at a time I= 0 (tx = 0 corresponds to a particle moving directly along the field line, towards the Earth in this case). As mentioned above, all x-positions within the lobe are negative. and we shall take particle velocities to be positive in the positive x-direction. From the geometry of the

863

field described above, it follows that at I = 0 this particle has its guiding centre located on the field line that crosses the magnetopause at a position

Since this point is being convected tailwards by the magnetosheath flow at a constant speed ~‘s”, at any other time f this field line cuts the magnetopause at the point x,(r) = xm -r&i 1. (9) Considering now the motion of the particle in the xdirection, the x-position of the particle in the lobe at a time t (provided that it has not interacted with the mirror) is simply given by the equation x,(r) = xQf (v,i cos

e,)t = xo+v,if,

00)

since 0, is small and cos 0,_is approximately unity, an approximation that we shall use throughout the rest of this paper. In terms of E and a, u,, is given by I:,, = &2E/m,)

cos a.

(II)

Since the guiding centre of the particle remains on a particular field line as a consequence of its E x B drift, the motion of the particle in the z-direction does not need to be explicitly considered in order to find the point at which it crosses the magnetopause. When the particle x-coordinate matches that of the magnetopause intersection point of the held line on which it is moving at some time t, the particle must be located at the magnetopause, leaving the lobe if t is positive, or entering if t is negative. It is this latter case in which we are interested. One further complication arises from the possibility that the particle has undergone a reversal of its motion at the perfect mirror at x = 0 at some earlier time t, (negative). This has happened to particles arriving at (x,,z,,) which are moving tailwards with velocities that are comparable with and larger than the magnetosheath flow speed. The limiting value of fiefd-aligned velocity I+_separating those particles which have mirrored from those which have not is given by the field-aligned velocity of the particle which enters the lobe at x = 0 and then drifts directly to the point (xO,zO). i.e. the particle which is located at x = 0 at the same time as its field line crosses the magnetopause at x = 0. Putting xr(t) = s(t) = 0 in equations (9) and (10) and eliminating t [i.e. equating the time taken by the particle to travel from x = 0 to xc, at speed nL to the time taken for the field line to travel to xf(l = 0) at speed as& we obtain an expression for the limiting field-aligned velocity

C. J. OWEN et al.

864

Note that uL is always smaller in magnitude than us”. For particles of energy E this corresponds to a limiting pitch angle aL given by c$ = cos- ’ {t&?l,/2E)},

(13)

which we note will be a little larger than 90” for the particles of interest here, whose speed corresponding to energy E is large compared with nsu. All particles with pitch angles greater than this value must have interacted with the perfect mirror before reaching the point (x~,z,,), while those with smaller pitch angle have yet to do so. Since we have assumed that the mirror simply reverses the particles’ field-aligned motion, the source points for particles which have mirrored can be calculated by noting that the time t, at which each particle interacts with the mirror can be deduced by putting x,(t,) = 0 in equation (I 0), giving

t, = -x()/a,, and its motion prior to interacting (t < tJ is thus given by .x,(t)

=

-t+,(t-t,).

(14) with the mirror

(15)

We therefore have a set of simple equations giving the position x within the lobe at any time t. Using the fact, noted above, that particles located at the same s-position as the magnetopause crossing point of their field line must be entering the lobe (if t is negative), it is possible to calculate the entry points of particles for any value of pitch angle and energy. Since the field in this model lobe is uniform, the field-aligned velocity (and thus the pitch angle a) of each particle is unchanged between its point of entry and its point of detection, except for those particles that have mirrored at x = 0. In the latter case the parallel velocity is simply reversed in sign (such that the pitch angle of the particles at the magnetopause, c(,,,~= 7-r- tc, where tl is the measured pitch angle). The value of the distribution functionffor each particle can then be deduced from the calculated values of x and z’,,. as explained in the previous section. We shall first discuss the simplest case, corresponding to solar particle event conditions, in which we have a uniform magnetosheath source distribution [i.e. S, = 0 in equations (6) and (7)J such that the dependence on .Y is removed from the problem. The value off associated with each pitch angle therefore depends only on the pitch angle amp with which the particles cross the magnetopause. This is illustrated in Fig. 6a, which shows the relationship in velocity space between particles of a given speed V in the Earth’s frame (with origin 0), and contours of constant .f in the source distribution function just inside the magnetopause. As in previous figures, the Earth and mag-

netosheath flow frames are separated by a “distance” vsn, and the contours of constantfare represented by the concentric circles about the origin 0’ of the latter frame. For a typical distribution (I- > 1) the contours closer to the origin of this frame represent the higher values off The particles of a given speed V in the Earth’s frame are represented in the figure by the heavy circle centred on the origin of this frame. Particles of 0” pitch angle are located at the point where this circle intersects the positive c’,,axis (point A), whilst particles of 90” pitch angle are located at the points where the circle cuts the t’l axis (points B). As indicated above, as c( at the observing point (sO,rO) varies from 0’ to l80”, the corresponding pitch angle ~~~ at the magnetopause entry point varies from 0” to rL and then back to O”, with a discontinuity at CI~from t[L itself to n--cc,, due to reflection from the perfect mirror at x = 0. In the absence of any dependence of f on the point of entry of the particles, the resulting pitch angle distribution due to this effect is shown schematically in Fig. 6b. The value off is a minimum at a = 0’ and l80’, since the entry pitch angle cl,,,,, is 0” at both points (particles at point A in Fig. 6b), while f is a maximum at zL (point C), with a discontinuity at that point due to the discontinuity in CI,_to rr-uL. Except for the narrow region rr - c(~ < c( < aL, the lobe distribution is symmetric about 90”, such that f(a) =

f(n-r). The resulting lobe distribution shown schematically in Fig. 6b indicates that for a spatially uniform source there is an anisotropy favouring particles perpendicular to the field direction and a weak net tailward flow. However, for energetic particles whose thermal speed considerably exceeds rsu, the degree of anisotropy is small unless the gradient in the spectrum is large. As pointed out in Section 2, for typical magnetosheath source spectra under “normal” conditions the distribution only varies by a factor of -3 from maximum to minimum, and for the “hard” spectra observed during solar particle events, the variation will be even less than this (- I .6). These results thus seem compatible with the essentially isotropic populations observed in the lobe during solar particle events. However. in order to explain the highly anisotropic lobe distributions observed during “normal” conditions, we must also consider the effect of a spatially varying magnetosheath source, which results from the presence of ions of “terrestrial” origin, as postulated in the previous section. Some of the trajectories of the guiding centres of energetic particles within the model lobe are illustrated in Fig. 7, using the same coordinate system as employed in Fig. 5. Figure 7 shows a single held line (solid arrowed line) within the lobe, passing through

865

Energetic ions in the distant tail lobes

PARTICLES OF GIVEN SPEED V IN EARTH’S FRAME

FIG. 6. (a) .f ON

PITCH

VELOCITY ANGLE,

SPACE DIAGRAM AND

(b)

I MAGNETO SHEATH FLOW

ILLUSTKATING

A SCHEMATIC

ASSUMING

I EARTH’S FRAME

THE

ILLUSTKATION

A SPATIALLY

UNIFORM

DEPENDENCE OF THE

-

OF THE RFXJLTlNG

MAGNETOSHEATH

LOBE DISTRIBUTION PITCH

ANGLE

FUNCTlON DISTRIBUTION

SOURCE.

The upper diagram shows the velocity space conhguration in the Earth and magnetosheath flow rest frames described in Fig. 2b. but now neglecting the angle ULsuch that the field is directed very nearly along the V, direction. The concentric circular arcs centred on the origin of the magnetosheath how frame are contours of constant,/‘of the magnetosheath source (highestJclosest to the origin), whilst the solid circle centred on the origin of the Earth’s frame represents particles of given speed in that frame. As the measured pitch angle of the particles increases from 0” to the limiting pitch angle aL, the locus of particle velocities at the magnetopause moves along the solid circle from point A through points B and D to point C, cutting contours of continually Increasing.6 At the limiting pitch angle a,_ the magnetopause pitch angle jumps back to point D due to mirroring near the Earth. and then returns to point A with continually increasing pitch angle at the lobe measurmg point. The resulting pitch angle distribution is shown in (b). The distribution has a peak and a discontinuity at the limiting pitch angle q_, and falls away smoothly at both higher and lower pitch angles. The letters on this figure correspond to the source points in veloctty space marked on the upper diagram.

the measuring point (x.~,=,&at t = 0. The solid horizontal line crossing the z-axis at c,,,r represents the tail magnetopause (note that the angle between the field line and the magnetopause has been greatly exaggerated in this diagram for purposes of clarity). whilst the thin arrowed lines represent the trajectories of particles of a given speed C: but of varying pitch angle a, which reach (sO,zO),The first trajectory, which intersects the magnetopause at z = z,,,r at the point x,, represents a particle with zero pitch angle. This particle enters the lobe when the field line shown crosses the magnetopause at the point .Y,, and subsequently moves along the field line towards the Earth.

and is convected inwards with the field line as the magnetopause end of the latter is convected tailward by the magnetosheath flow. The trajectory is a straight line since all velocity components arc constant in this model. Since this particle has the maximum heldaligned speed for ions of this energy, the point s, is the furthest downtail source point contributing particles to the distribution at (x0,:,). In the case of a magnetosheath source flux that falls with increasing downtail distance, particles of this pitch angle would thus be expected to have the smallest value of,f(r?,,, vi) at the measuring point (note, however, this is also true for a uniform source, since such particles have the

866

C. J.

OWEN

et al.

MAGNETOSHEATH

PERFECT MIRROR

Rc.7. DIAGRAM SHOWING ENERGETIC ~O~TRAJECTORIESWITHI~T~SIMPLE MODEL LOBE. In this figure we show. for clarity, only a single field line (heavy arrowed fine). which passes through the

point of observation in the lobe. The lighter lines indicate the guiding centre trajectories of a number of particles, of the same speed c but differing pitch angles, which pass through the measuring point (so, z,,). The first trajectory, that of a field-aligned particle, enters the lobe from the magnetosheath at xl, and then moves earthward to (x~,-_~).A particle of 90; pitch angle crosses at .x! and E x B drifts towards the tail centre plane. The trajectory of the particle of limiting pitch angle CL~ whtch separates those particles which have mirrored from those which have not starts at x4 and drifts tailwards to the measuring point. All particles with pitch angles at the measuring point which are larger than this value, for example the antifieldaligned particle which originally enters the lobe at x5, have mirrored before reaching (xO,zO).Jf the source distribution function has an earthward gradient, then the value ofJis highest for the particles entering closest to the Earth, such that the particle entering at ,yr with pitch angle a,. represents the peak in the distribution function. Further, since the antifield-aligned particle enters the lobe closer to the Earth than the field-aligned particle. the distribution contains a consistent downtail anisotropy.

highest speed (and thus lowestfl in the magnetosheath flow frame, as discussed above]. The second trajectory shown is that of the 90” pitch angle particle. This particle has no field-aligned component of velocity, and simply drifts inwards with the E x B drift, as indicated by the trajectory crossing the magnetopause at x = .y2 (note that in the cos 0,. = 1 approximation x2 =: x,,). Particles with pitch angles between 0’ and 90” enter the lobe between the points .Y, and x2. and the value of .f generally increases with pitch angle between these limits, due to both the increasing pitch angle at the magnetopause and the increasing source strength (although the latter dominates if S,, is large and L is small). The third trajectory, crossing the magnetopause at the point x3 represents the trajectory of a particle with pitch angle a few degrees larger than 90.. This particle moves slowly tailward as the field convects toward the centre plane, at a speed somewhat less than the magnetosheath flow speed. The fourth trajectory connects the point (0, z,,,,,)to the measuring point (x0, zO),and represents the limiting trajectory of the particle whose pitch angle x,_is given by equation (13). This particle has the maximum pitch angle possible for a particle to reach (.I+,,2”) without having

first interacted with the mirror, and entered the lobe at the same time as the field line first entered at the point (O,Z,., ) For a source function S(x) which falls monotonically with downtail distance,fis maximum for this pitch angle, again due to the form of both S(X) andf‘(v) at the magnetopause. Note that all the particles so far described reach the point (.x0.ZJ with the same field-aligned speed u,, with which they first entered the lobe. This is not true of the remaining trajectory, which represents the course taken by a particle with a pitch angle of 180”. Like all particles with pitch angles greater than aL, this particle has had its motion reversed at the perfect mirror, which it hit at the point (0, z,) at a time t, (negative) given by equation (14). Since its field-aligned velocity was reversed at this point, it follows that previous to the interaction with the mirror this particle had the same components of velocity as the partide first described, which entered the lobes at x = x1 (i.e. the particle with zero pitch angle). Wence the part of the trajectory undertaken by this particle before the interaction with the perfect mirror is parallel to the trajectory of the corresponding field-aligned particle described above. Since the 180” particle has a longer

Energetic

ions in the distant

tail lobes

867

FIG. 8. SKETCH 0~ THE PITCH ANGLE DISTRIBUTION (SOLID CURVE) RESULTING FROM THE INCLUSION OF A SPATIALLY VARYING “TERRESTRIAL"SOURCE,COMPARED WITH THE DISTRIBUTION OBTAINED FROM A UNIFORM SOLAR SOURCE(DASHED LINE,cf. FIG.~~). The peak in fand the discontinuity are located at q in both cases, but the “terrestrial” source gives rise to a distribution function which is much more strongly peaked perpendicular to the field than for the spatially uniform “solar” source. In addition, for the “terrestrial” source the value off for pitch angles larger than 90’ is, always larger than that at the corresponding supplementary pitch angle (less than 90’), giving a net tailward ion flux.

path

to

particle,

the

point

(x,,,zJ

than

the

field-aligned

but the same speed, it must have spent longer in the lobe, as is evident from the geometry of Fig. 7, and consequently crossed the magnetopause at a position closer to the Earth. Note that if the source flux falls with distance down the tail, the value offfor particles with pitch angles between a,_ and 180” will fall with increasing pitch angle. This decline is due to the form off at the magnetopause (as in Fig. 6b), and to the fall in S(x), but is not as steep as the corresponding decline inJfor particles between OLand 90’, and the value offat 180’ is larger than that at 0”. A sketch of the pitch angle distribution resulting from the inclusion of a spatially varying source distribution is shown in Fig. 8. Here we assume that the source strength at the Earth exceeds that of the background magnetosheath source by a factor So. and falls exponentially with increasing distance from the Earth with a scale length L [cf. equation (6)]. However, for simplicity of discussion we assume that both So and L are sufficiently large to maintain the terrestrial magnetosheath source fluxes well above the background level [i.e. So exp (- I.K,,,,,~/L)x 1 in equation (6)] for all lobe entry points .Y~,,. In this figure the dashed line represents the distribution sketched in Fig. 6b due solely to the variation of entry velocity

with pitch angle at (xO,z,,), whilst the solid line represents the typical distribution expected when the effects due to a spatially varying source are also included. As explained above, the particles of limiting pitch angle aL enter the lobe at the point x = 0, so that in the presence of the spatially varying source, the value offis increased fromf(a,) to (So+ l).f(a,) at this pitch angle. The discontinuity infstill exists at a,_ due to the discontinuity in entry velocity at this point. Away from aL the distribution falls away to lower values offmuch more rapidly than in the spatially uniform case, since particles of higher and lower pitch angles enter the lobes at greater distances downtail, where the source strength factor is progressively smaller. Thus in this case the ion distribution is much more strongly peaked perpendicular to the field. In addition. since particles of 180” pitch angle enter the lobe closer to the Earth than those of 0”, these particles come from a region of higher source strength. and thus have a higher value off. Particles of 180 pitch angle (entering the lobe at the point x5 in Fig. 7) increase their value of f by a factor S,exp (-lsJ/L)+ I with the inclusion of the spatially varying source. whilst those of 0” increase by the smaller factor So exp (-1x,1/L) + 1. Consequently, the symmetry of the distribution for pitch angles away from 90“ noted in the uniform source case (Fig. 6b) is broken, such

868

C. J. OWEN et al.

that particles with tl > aL have higher values offthan their counterparts with pitch angle rc- IX, giving a small net tailward flux of energetic ions. Having established the basic form of the lobe pitch angle distribution function, we now wish to consider the variation of the distribution function which will result from different measuring positions within the lobe. We first consider the effect of distance of the measuring point from the magnetopause. In the case of a uniform magnetosheath source the value off depends only on the pitch angle of the particle at the magnetopause, which for most particles is independent of the distance of the measuring point from the magnetopause. However, the particle which enters the lobe at x = 0 and moves to the measuring point now has a different trajectory, and hence different pitch angle. The value of the limiting pitch angle crL (locating the peak of the distribution) therefore changes with distance from the magnetopause. As can be seen directly from equations (12) and (13) if the value of (zmp-zO), which is the distance between the measuring point and the magnetopause, is increased, the value of CI~decreases. At very large distances from the magnetopause, c(~ z 90”, and the distribution becomes completely symmetric about the 90” pitch angle. Conversely, at very short distances from the magnetopause, the limiting pitch angle for a particle of speed V is CL‘% cos- ’ (us,,/V). For particles whose speed V >> vSH, a,. at the magnetopause is only a few degrees above 90”, as previously noted. The distribution resulting from a uniform source in the mag-

I

/ x+

netosheath thus does not vary greatly across the lobe. In order to study the effect on the results of distance from the magnetopause when the magnetosheath source varies along the length of the tail, we need first to consider the change in position of the particle entry points, which we do with the aid of Fig. 9. This figure is of the same format as Fig. 7, but the measuring point is located at a greater distance from the magnetopause than before. The relative position of the magnetopause in the previous figure to the measuring point is shown as a dotted line at position :I, for reference. The trajectories shown are for the same pitch angles considered in the previous figure, and it is important to note that these trajectories are unchanged with respect to the measuring point (so, z,J. The new magnetopause source points are thus obtained by projecting back along each trajectory (via the mirror if necessary) to the point of intersection with the new magnetopause location. For those pitch angles that are well away from the peak (i.e. those particles which are largely field aligned, or anti-field aligned) the source points are therefore much further away from the Earth than in the previous case (see trajectories ending at s, and x5). For the spatially varying source, particles of these pitch angles consequently have lower values of f(v,, , VJ than the case described above. The 90” pitch angle particle has only a small change in its source position (x2), which is negligible in the cos tIL = 1 approximation, such that the value ofSfor this particle is unchanged. The third trajectory, which represented a pitch angle between

MAGNETOSHEATH

LOBE

’

FIG. 9. As FOR FIG. 7. BUT FOR A MEASURING POINT FURTHER FROM THE MAGNETOPAUSE. The relative position of the magnetopause in Fig. 7 relative to the measuring point is denoted by the dotted line. The five trajectories shown are for the same particle pitch angles as shown in Fig. 7. The points of entry for the field- and antifield-aligned particles have moved further downtail, and the limiting pitch angle is now associated with the particle starting from .Y, whose pitch angle is closer to 90” than in the previous case.

Energetic

ions in the distant

90” and the limiting pitch angle c(,_in the previous case, now enters the lobe at the point (x3 = 0, z,,), and as such represents the new limiting pitch angle. This particle will now have the peak value off(r,!, cl), whilst the particle that had the peak value in the previous case is now a mirroring particle which enters at .xq, with a reduced value off. As explained above, the value of pitch angle corresponding to the limiting trajectory is thus slightly decreased towards 90” from the previous case. Note that the peak value off(now located at a shghtly reduced pitch angle) is, to a close approximation, unchanged from the previous case, whilst the values offat pitch angles away from the peak are depressed due to the increased distance of the source point from the Earth. With increasing distance from the magnetopause, therefore, the distribution resulting from the inclusion of a spatially varying source population becomes progressively narrower and peaked near 90”, whilst the peak value remains approximately constant. These changes are illustrated schematically in Fig. 10. Distribution 1 is that described in Fig. 8, whilst distribution 2 illustrates the changes anticipated for a measuring point further from the tail magnetopause. Again we assume that all fluxes at the entry points are well above the uniform background level. As described above, the peak at the

I

I:\ ‘\

DISTRIBUTION2 DISTRISUTION

1

bf bf

I 0” FIG.

45’

I

t

I

90’

135”

180”

n

10. SKETCH OF THE DISTRIBUTIONFUNCTION fvs PITCH

ANGLE,

ILLUSTRATING TANCE

THE

VARIATIONS

EXPECTED

WITH

DIS-

FROM THE TAIL MAGNETOPAUSE.

Distribution 1 (dashed line) is that discussed in Fig. 8 (soiid line in that figure), whilst distribution 1 (solid line) is that expected at a position further from the tail magnetopause. In the latter case, the peak in the distribution moves closer to 90”, and the discontinuity becomes smaller. The height of the peak is, to a first approximation, unchanged. The “flanks” of the distribution, however, are reduced in value compared with the distribution closer to the magnetopause, such that the pitch angle distribution becomes narrower. and the omnidirectional fiux becomes smaller.

tail lobes

869

limiting pitch angle moves closer to 90’, and the size of the discontinuity in f is reduced, since the discontinuity of pitch angle at the magnetopause (from ttL to n-tt,) is also reduced. The size of the peak is only marginally affected. however. Since the 90” pitch angle particle enters the lobe at approximately the same point, the value off is unchanged for this particle. Both the 0’ and 180’ particles enter the lobe at positions further tailward, and have their values of f‘reduced by a factor Sf, which is the same for both by virtue of the fact that both entry points move tailwards by the same amount 6x. Note also that the area under the new distribution is less than that of the previously discussed distribution This results in a reduced direction-averaged flux of ions at the new measuring point. Hence the lobe omnidirectional energetic ion flux should also fall with increasing distance from the tail magnetopause in this case. Furthermore, the new distribution is somewhat more symmetrical, indicating a fall in the net tailward flow in the lobe with increasing distance from the magnetopause. We shall finatly consider how the pitch angle distribution functions depend on the distance .Xof the measuring point along the tail. Considering first the case of a uniform source in the magnetosheath. there is again no change in most of the distribution from that shown in Fig. 6b, since this is only a function of entry pitch angle in this case. However, the limiting pitch angle between particles which have and have not mirrored moves to larger pitch angles (i.e. closer to 1807 with increasing distance x away from the Earth, at a given distance from the magnetopause. This can be seen from equation (12) where, dividing top and bottom by x0, the denominator becomes smaller with decreasing x0 (i.e. further from the Earth), leading to higher values of pitch angle for the particles mapping back to the magnetopause at x = 0. The peak in the pitch angle distribution consequently moves to larger pitch angles for decreasing x* In the limiting case for IX/ >>(+-Z), t(L z cos- ’ (Z&/V)* To include the effects of a spatially varying source, we again need to consider the change of points of entry of particles for the new x--position. For particles that have not mirrored, the trajectories relative to the measuring point are exactly the same as the previous case described above. The point of entry for the fieldaligned particle is a constant distance downtail from the measuring point independent of x (for constant -_). and this holds for all particles which have not mirrored. Consequently, while the absolute fluxes of these particles will drop in this case, the profile of the pitch angle distribution remains unchanged over the pitch angle range W-90”, providing the source fluxes

870

C. J. OWN et al.

for all particles remain well above the uniform background level. Those particles which have mirrored remain to be considered in this context. From Fig. 7 it can be seen that as the measuring point moves tailward ( lx01 increasing) at fixed z0 the point at which the field-aligned particle (180”) encounters the mirror moves closer to the tail centre plane (z, reduces). This results in the source point of this particle also moving tailward, however the geometry of the system results in this tailward movement being less than that of the measuring point. Since we have already shown that the source points of the non-mirrored particles lie at a fixed distance from the measuring point for constant zO, it follows that the distance between the source points of the 0’ pitch angle particle and the 180” particle increases as x0 moves downtail. Providing the scale length for magnetosheath flux variations is sufficiently large, such that the source flux at the entry point of the 180” particle is still above the background level, this results in an increased “front to back” ratio f( 18O”)/f(O”) at the measuring point. However, the source points of most particles move further from the Earth with the measuring point, so that the omnidirectional flux (proportional to the area under the pitch angle distribution) decreases with increasing distance from the Earth. These effects are illustrated in Fig. 11, in the same format as Figs 8 and 10. Again, the curve marked distribution 1 is that discussed in Fig. 8, whilst that marked distribution 2 is the distribution expected for a tailwards displacement 6.x, of the measuring point. As in the case of varying distance from the magnetopause, the peak in the distribution is, to a good approximation, unchanged. However, as the measuring point moves tailwards, the peak in the distribution moves to larger pitch angles. and the discontinuity infis slightly larger. All pitch angles ,<90” reduced by the constant amount Sf are (zexp (- ~cSS~~/L)), since all entry points for these particles are shifted tailwards by the same amount. The value off at 180” is, however, reduced by a smaller amount due to the smaller shift in entry positions for mirrored particles as discussed above. The degree of anisotropy of the distribution is thus enhanced, whilst the omnidirectional flux of particles is reduced as the measuring point moves tailwards. Some numerical results illustrating the effects of measuring position in the lobe on the pitch angle distribution are shown in Figs 12 and 13. In producing each of these figures we have assumed the following parameters in our mode1 : B, = 10 nT, B, = -0.15 nT, aSH = 400 km s-‘, L = 100 R,,So = 500, f = 3.5, and the particle speed V = 2600 km s ’ (corresponding to a proton of energy E,,c 35 keV or an O+ ion of z 560 keV). In Fig. 12 we show the variation

DISTRIBUTION

2

Af

I

0"

45’

90’

135’

180’

FIG. 11. SKETCH OF THE DISTRIBUTION FUNCTIONJVS ANGLE, ILLUSTRATING THE VARIATIONS EXPECTED INCREASING DISTANCE DOWN THE TAIL.

a PITCH WITH

Distribution 1 (dashed line) is that discussed in Fie. 8 (solid line in that figure), whilst distribution 2 (solid link) is that expected at a position further down the tail. In the latter case, the peak in the distribution moves to larger pitch angles (towards 180’). and the discontinuity is larger. As in Fig. 10, the height of the peak is essentially unchanged. The flanks of the distribution are, however, reduced compared with the distribution expected closer to the Earth, such that the pitch angle distribution becomes narrower, and the omnidirectional flux smaller. The f(180@)~~(0”) ratio is also increased.

of the distribution with distance from the tail magnetopause, with pitch angle distributions plotted for (z,,,,-20) = 1 R,,5R,,lSR,,and25R,,ataconstant distance of 100 RE downtail. In each of these plots the dotted line indicates the distribution expected in the absence of the terrestrial source (i.e. S, = 0), shown for reference. As can be seen from Fig. 12, and as explained above, the peak in the distribution is approximately the same in each plot, although the position of the peak (at aL) moves to smaller pitch angles with increasing distance from the magnetopause. Also, the flanks of the distribution show that the fluxes of the more field-aligned particles reduce with distance from the magnetopause, as the points of entry move further tailwards (note, however, that the points of entry of the 0’ and 180” remain a constant distance from each other). Note also that in the plots C and D the value off has fallen to the background level at pitch angles away from 90”, reflecting the fact that these particles enter the lobe at positions where the terrestrial source strength has declined to levels which are small compared with the assumed uniform background flux. Hence the distributions become more anisotropic, peaked near 90”, with increasing distance from the tail magnetopause, and the omni-

871

Energetic ions in the distant tail lobes

=-

100R~

Bx=lOnT Bz=- O.lSnT

L= 100R~ so=500

VSH=mkms-’ f-3.5

E=fSkeV

______--______;____

----C-.-_-L

_-~_c-_--~-_..-c----L

..-c.----l

FIG. 12.

PLOTS

OF DlSTRIBUTlON

FUNCTION PERFECT

f

VS PITCH

MIRROR

ANGLE

AT A POSITION

100

RE

DOWNTAlL

FROM THE

IN THE MODEL LOBE.

The model parameters assumed in producing these plots are shown at the top of the figure. The peak “terrestrial” source strength S, is taken to be 500 times the solar wind background density, and falls off exponentially with distance down the tail with a scale length of 100 R,. The solid line in each plot represents the distribution from the combmed exponential and uniform sources, whilst the dashed line represents the distribution for the spatially uniform source alone. The various panels then show the expected ion pitch angle distributions at distances of I REq 5 RE. 15 Rs and 25 RE from the magnetopause. As the measuring point moves away from the magnetopause, the peak in the distributions moves to pitch angles closer to 9@, the distributions become more highly peaked near to 90 ‘, and the omnidirectional flux is reduced. At sufficiently large distances from the magnetopause, the “uniform” population dominates, except very close to 90’.

C.

872 :mp-~)=5R~

i)

L=IOORE so=500

J.

OWEN

et

Bx=lOnT Bz=-0.15nT

x,,=-25R~ xmp(OO) = -314 RE xmp(180°) = -307 RE

aL = 90.6’

al.

VSH = 400 km s-’ r=3.5

E = 35 keV

xo=-lcIclR~

B)

xmp(OO) = -388 RE xmp(1804 = -362 RR

aL = 92’

____-_-__--________

0

0 pit=; Q

o”CJYcs

(bJeJcj>

x0=-400R~ xmP(Oo) = -688 RE ~~~(1800) = -582 RE

pit=?

on$e

xo=-80OR~

D)

aL = 94.8’

xmp(OO)= -1088 RE xmp(180’) = -875 RE 3.00 _------_-----

QL = 96.2’ ____ -__

2.50 :_____:____:____:____! 2.00 -_____:____:____:-___:

FIG. 13.

PLOTS

OF DISTRIBUTION

FUNCTlON,/VS DOWNTAIL.

PITCH

ANGLE

FOR MEASURING

IN THE SAME FORMAT

AS FIG.

POINTS AT VARIOUS

DISTANCES

12.

The model parameters are the same as those employed in that figure. and each plot is for a measuring point 5 RE from the magnetopause. The various panels show results for distances of 25 RE, 100 RE. 400 R, and 800 R, downtaii from the perfect mirror. The peak in the distribution is located closest to 90” for positions close to the perfect mirror, while the pitch angle anisotropy increases and the omnidirectional flux decreases with Increasing distance from the mirror.

directional flux (a function of the area under each curve) also falls. In addition, each plot in the series indicates a slightly higher mean value of.ffor particles with a > 90” than those with LX< 90 , which lcads to a small net downtail flux of particles at all positions, although the effect is largest at the magnetopause. and falls with increasing distance towards the tail centre

plane. Note also that the results indicate that deep within the lobe one part of the ion distribution function may be dominated by the terrestrial source (near 90’ pitch angle), while the other may be dominated by the solar source (near 0’ and 180‘). Finally, Fig. 13 illustrates the effect of moving the measuring point in the x-direction in this model. The

Energetic ions in the distant tail lobes

same parameters have been used as in the previous figure, but we now fix the distance of the measuring point at 5 R, from the magnetopause, and show the distributions at distances of 25 RE, 100 RE, 400 R, and 800Rr downtaif from the Earth. Again the peak in the distribution is approximately constant in each plot, but the position of the peak moves to slightly larger pitch angles with increasing distance from the Earth. The flanks of the distribution fall with increasing distance as the entry points for particles moving predominantly along the field move tailward with the measuring point, so that the distribution becomes more peaked perpendicular to the field, and the omnidirectional flux is reduced. Note that the distance between the entry points of the 0” and 180” pitch angle particles grows with increasing distance down the tail as explained above, leading to a larger ratio ,f( 1SO”)/~(O’)in regions where f is well above background levels (represented by the dashed line). Apart from the differences due to varying observing positions in the lobes. we also expect that the scale length. L, of the variation of fluxes in the magnetosheath and the hardness of the spectrum, F, will have an effect on the pitch angle distribution observed in the lobes. A typical value of the scale length in the magnetosheath is not well established at present. So far we have assumed a scale length L = 100 R,, which gives a variation of several orders of magnitude in the fluxes over the expected spread of source points (of order 1000 R,, depending on the position in the lobes, see Figs I2 and 13). However. due to the uncertainty in scale length, it is instructive to consider the extreme cases that could apply, namely very short scale lengths, and very long scale lengths. The first case, for a short scale length. is illustrated in Fig. 14. The parameters used in each of these panels are identical to those used in Fig, 12, but the scale length has been reduced by an order of magnitude to L = IO R,. The plots are for distances of I R,. 5 RF, I5 R,, and 25 R, from the magnetopause at a constant IOOR, from the Earth. The trajectories. source points and entry velocities of particles within the lobe are unaltered from the corresponding cases discussed in Fig. 12. so that differences between the two figures arise solely from the variation in source scale length. Consequently the peaks in the distribution are at the same pitch angles, and have the same values off: However, due to the small scale length of the terrestrial component, the magnetosheath fluxes fall to the uniform background much closer to the Earth than before. so that, correspondingly. the pitch angle distributions in the lobe contain only a spike of terrestrial particles near to the limiting pitch angle, and then fall to the uniform background at smaller and larger pitch

873

angles. The spike becomes narrower with increasing distance from the magnetopause, while the remainder of the distribution function is essentially independent of position within the lobe. The second case, of the very long scale lengths, is equivalent to the uniform source case (S, = 0) discussed above, and is important in its application to solar particle events. As explained in the Introduction, the energetic particle fluxes in the solar wind increase by several orders of magnitude during such periods, and are typically much higher than those fluxes which are produced by the terrestrial source (other than within the plasma sheet itself). Under these circumstances the “terrestrial” source is negligible in the magnetosheath compared to the solar source. At the same time the spectra of the energetic magnetosheath particles become much harder during these events, such that the exponent F may at times be less than unity (i.e. the fluxes of more energetic particles are much enhanced relative to the lower energy particles). Although the trajectories of the particles within the model lobe, and consequently the source points and entry velocities, remain the same as above, the dependence offon the particle entry point is zero (as discussed for a uniform source above), and the dependence on entry velocity is also now much reduced due to the hardness of the spectrum (F = 1 gives a ratio of I.6 between the minimum and maximum value of *fin the lobe). The source fluxes during solar particle events are, however, much higher than during more normal periods, so that under the former conditions this model predicts a more nearly isotropic distribution. almost independent of position within the lobe and with a higher omnidirectional flux. which is indeed what is observed by the EPAS instrument. In summary, we have shown how a very simple model lobe with a perfect mirror at its earthward end can reproduce, semi-quantitatively, the properties of the lobe energetic ion pitch angle distribution observed in the distant tail by ISEE-3, both under normal, and solar particle event conditions. The model produces pitch angle distributions which are highly peaked close to 90’ with a small downtail anisotropy under “normal” conditions, when a near-Earth source is present, which dominant, declines with increasing distance downtail. In this case the ion fluxes in the lobe generally decrease with increasing distance from the magnetopausc, and with increasing distance from the Earth. At a given downtail distance the omnidirectional flux will also generally be much lower than those observed in the adjacent magnetosheath, since most of the lobe particles have source points much further down the tail. where the ion flux is reduced. Under soiar particle

C. J. OWEN et al.

874

L= 10R~ so=500

=-100R~

0

z,,,~-zo= 1 RE ~~~(0’) = -157 RE

Bx= 1OnT Bz=-0.15nT B)

aL = 95.3’

VSH = 400 km s-l r=3.5

E = 35 keV

zrnp-~=5R~

xmp(O’) = -388 RE xmp(l 80’) = -362 RE

xmp(1800) = -131 RE

aL = 92’

_________----------

2)

zmp-zo=15R~

-966 RE xmp(180’) = -940 RE

xmp(OO)

=

D)

aL = 90.8’

The parameters the scale length the “uniform” lobes near the sharp

A VERY

SMALL SCALE

xmp(OO) = -1543 RE

% = 90.5O

~,,,~(180~)=-1517R~

FIG. 14. PLOTSOF DISTRIBUTION FUNCTION,fVS PITCH ANGLE WITH

z,,,~-z.,=~~RE

LENGTH

FOR THE CASE OF A “TERRESTRIAL”ION

FOR SPATIAL

SOURCE

VARIATIONS.

used in each of the plots are identical to the corresponding plots in Fig. 12, except that L has been reduced from 100 to 10 R,. The enhancement in magnetosheath density above background is thus confined to the near-Earth region, so that only particles entering the Earth have enhanced values of,L resulting in distribution functions that have extremely peaks. Over most of the pitch angle range the uniform “solar” source dominates.

conditions, however, the variation inf’with pitch angle is expected to be much smaller, such that the spacecraft would observe a nearly isotropic distribution. In addition, the fluxes would be nearly independent of position in the lobe. and comparable in intensity to that in the magnetosheath.

The model, by its very simplicity, obviously makes several assumptions which are quite large departures from reality. In the next section we shall discuss in greater detail the more severe of these assumptions, and how relaxing them would affect the results discussed here.

Energetic

ions in the distant

4. MOTION OF PARTICLES IN A MODEL LOBE

is unchanged from that described in the last section, but the motion closer to the Earth will be substantially altered, with particles of different pitch angles mirroring at different positions within the near-Earth region. In the rest of this section we shall discuss this motion, and the effect on the pitch angle distributions that we predict in the lobes, in more detail. The observed variation of the field strength in the near-Earth tail is a complicated function of distance. A full treatment is inappropriate here, however, since we wish only to investigate possible major departures from the previous results that might arise from a more gradual “mirror” configuration of the field. Here we shall therefore model the tail lobe by retaining our perfect mirror at x = 0, but including a gentle slope in the field strength between 0 and the point x,, which in this case we shall take to be - 100 RE. The field strength variation thus assumed is shown as a function of downtail distance in Fig. 15. The field strength at distances greater than x, is a constant 10 nT as observed, and we shall assume that the x- and zcomponents of the field are the same as those employed in the last section (i.e. B, = 10 nT. B, = - 0.15 nT). Between 0 and x, we assume a linear ramp in the field strength, with a gradient such that the field strength doubles across this region, reaching 20 nT at x = 0, where we again apply a perfect mirror (i.e. infinite field strength). We assume that the component of the field normal to the tail axis, B:, remains small compared with that parallel to the tail axis, B,, such that to a first approximation the field remains

WITH INCREASING FIELD STRENGTH CLOSET0

THE EARTH

In the previous

section the basic properties of the in a simple, uniform field lobe model were discussed in order to determine the points at which particles of particular pitch angles enter the lobes. The only complications arose from particles interacting with the perfect mirror at x = 0, representing the strong field region near the Earth. Since this simply reverses the x-component of the particles’ velocity, simple equations for the points of entry for each particle were obtained. However, this field configuration, whilst representing the distant tail lobe quite reasonably is, at first sight, a gross oversimplification to the field in the near-Earth tail (x > - lOOR,). As the distance from the Earth increases the observed field strength in the tail drops, rapidly at first, but then more slowly, settling to an approximately constant value of 8-12 nT at distances beyond 100-200 R, downtail (Slavin et al., 1985). The initial field decrease is largely due to the flaring of the tail which results from the balance between the solar wind ram pressure and the magnetic pressure of the lobe flux tubes, as discussed by Coroniti and Kennel (1972). Consequently, in order to make our model lobe more realistic, we need to replace the perfect mirror at x = 0 with some more gradual field strength increase in the near-Earth region (x > - 100 Rr). The motion of particles at distances greater than - 100 R, particle

875

tail lobes

motion

lBl*

tI 20 nT

e k lu-l n rr!

A--

-100

-200

RE

DISTANCE FIG. 15.

SKETCH

SHOWING

THE VARIATION

OF FIELD

THE “REVISED”

RE

DOWNTAIL

STRENGTH

WITH

DISTANCE

DOWN

THE TAIL

ASSUMED

IN

LOBE MODEL.

The field is unaltered at distances greater than 100 R, downtail, and we retain the perfect mirror at x = 0. Between these two points the field is assumed to rise linearly from IO nT at x = - 100 R, to 20 nT immediately outside the perfect mirror. This field variation therefore includes, in an approximate way, the effect of the observed near-Earth increase in the lohe field strength.

C. J. OWEN

876

pointing directly towards or away from the Earth. The field strength in the near-Earth region (0 > x > x,) is thus (recalling values of x in the lobe are all negative)

d&4

B(x) = B,,,+xJy,

(16)

where B,,, = 20 nT is the peak field in the ramp region and dB(x)/dx = 0.1 nT R; ’ in this case. We wish to employ the same technique for determining the source points of the lobe energetic particles as we did in the last section. For this we need to know explicitly where the field line at the measuring point (xO,zO) threads through the tail magnetopause [cf. equation (S)]. An exact field structure within the ramp region needs to be assumed for measuring points with x0 > x,, in order to identify the points at which each field line intersects the magnetopause. However, for source points further downtail the field structure is identical to that employed in the previous section, so the field line-magnetopause intersection point can be calculated in exactly the same way as before. For this reason, we shall confine ourselves to studying the source points for particles measured in the deeper tail, such that, with the exception of B(x), the near-Earth field does not need to be explicitly defined. Assuming that the field line is convected tailward with the magnetosheath flow at a constant, speed us,, along the entire length of the tail, the field line located at (x0, zO)

et al.

(where x0 < x,) at r = 0 cuts the magnetopause at positions given by equation (9) at any other time t. as described in the last section. It therefore suffices to redefine the motion of the particle in the x-direction along the field to include the effects of the linear increase in field strength earthward of x,, in order to locate its source point, using the technique employed in the last section. As before, particle trajectories fall into distinct types, i.e. those which have no part of their trajectory in the field ramp region and have yet to mirror, those which have some part of their trajectory in the ramp region (either due to the particles having entry points within this region, or due to particles having entry points in the uniform field region, but which mirror in the ramp field), and those which hit the perfect mirror. The new trajectories are illustrated in Fig. 16, in the same format as employed previously in Figs 7 and 9. Obviously, those particles that do not enter the mirror region behave in exactly the same way as described in the last section, as they move in exactly the same field configuration, and therefore cross the magnetopause at the same positions x,, x2, and x3. The value offassociated with these particles is thus unchanged for the same source function. These particles are bounded by the trajectory of the particle which enters the lobe at x = x, (the boundary of the ramp region) and then moves tailward to the point (x,,z,). The pitch angle cc; of this particle can be obtained in an analogous way to

Z t

MAGNETOSHEATH

FIG. 16. SKETCH SHOWING PARTICLE TRAJECTORIES OF DIFFEKING TYPES IN THE"REVISED" MODEL LOBE. The format employed is the same as in Fig. 7. The trajectories shown are for the same pitch angles used in Fig. 7. The limiting pitch angle between unmirrored particles and those which have undergone mirroring within the near-Earth region or at the perfect mirror is given by that particle entering the lobe at x3. The unmirrored particles have the same trajectories as in the previous model, and hence the same value offfor each source. Particles with pitch angles shghtly larger than the limiting pitch angle are quickly mirrored, and have source points tallward of - 100 R,. Those which are mirrored at the perfect mirror spend a short time in the near-Earth region, and also have source points tailward of - 100 R,. The particle entering at x, therefore represents the peak in the resulting distribution in this case.

Energetic ions in the distant tail lobes the limiting pitch angle particle in the last section. Putting xr(t) = xp(Q = x, in equations (9) and (lo), respectively, and then eliminating 1, we find

I

(x0-A) rJ ((x,--x,)-_(~,p-;o)~,~Il~.I) .

VW - --

a; = cos-’ i

(17) This again gives a pitch angle a few degrees above 90”, depending on position in the lobe, and is represented in Fig. 16 by the trajectory crossing the magnetopause at xj. All particles with larger pitch angles than this value must have spent some part of their trajectories within the near-Earth region, where their motion is reversed either by gradual deceleration in the field strength ramp, or by interaction with the perfect mirror at x = 0. We shall now study the motion of these particles in some detail. The increase in magnetic field strength required to reverse the field-aligned motion of a gyrating particle can be obtained from the conservation of magnetic moment ,u. This holds provided the field strength as “seen” by the particle varies slowly in time compared with the gyroperiod. This is likely to be the case within the real lobes so we shall invoke conservation of g at all points within our model lobe (the perfect mirror is, of course, an artifice representing the Earth, which also has the property of conserving p). Note also that we shall ignore gradient-B and curvature-B drifts which are also negligible in the real lobe. For a particle detected at some point (.x0, zO) in the lobe. with pitch angle a,, the magnetic moment is given by p = E sin’ cr,/B(x,),

(18)

where E is the energy of the particle, and B(x,) is the field strength at the measuring point. If the particle enters a region of increasing field strength, the pitch angle moves closer to 90’, such that it eventually mirrors and its motion is reversed. The field strength at the mirror point X, (i.e. where the particle pitch angle is 90’) is thus given by B(x,)

= B(x,)/sin’

ao.

(19)

For measuring points outside of the near-Earth region we have B(s,) = 10 nT. whilst the maximum field strength before the perfect mirror is encountered is 20 nT. It therefore follows that. in this case, all particles appearing at (so, z0 ) with pitch angles between the limiting pitch angle and 135< have some part of their trajectory in the near-Earth region, but have not hit the perfect mirror, whilst those with pitch angles larger than 135” must have interacted with the perfect mirror. An exception to this concerns particles that may have their source points within the field ramp

817

region, but these represent a special case which we shall deal with separately later. Note that those particles with pitch angles close to al (i.e. those close to 90 , such that sin’s is close to unity) need only a small increase in field strength to cause mirroring, and consequently these particles do not penetrate very far into the near-Earth region before having their motion reversed. This is illustrated schematically in Fig. 16 by the trajectory which crosses the magnetopause at position .x4. The trajectory is quickly reversed in the increasing field strength region, and the particle reemerges after a very short time. Note, however, that this trajectory is associated with the same particle pitch angle as that which originated at x = 0 in the simpler model in the last section, and as such then represented the peakfin the pitch angle distribution. The source point for this particle is now tailward of x,, and it enters the lobe with an earthward motion. This particle thus has a reduced value of,f, which is now less than the value offat aL. For particles which have very large (i.e. close to 180’) pitch angles at the measuring point, the increase in the field strength between .Y= s, and x = 0 is insufficient to reverse their motion, and so they hit the perfect mirror. The trajectory of such a particle is illustrated in Fig. 16 by the trajectory crossing the magnetopause at position x5. Note that the portions of the latter two trajectories and the field line within the field ramp region are shown dashed since the exact form of these quantities is not explicitly defined in this region. The parts of the pitch angle distribution which are made up by particles which execute the various types of trajectory are illustrated in Fig. 17. This shows a plot of v,, against z:, divided into sectors according to which type of trajectory the particles have executed. Those particles whose velocities fall into the sector marked “A” are those with u < a;, which have not yet entered the ramp region. Those in the other three sectors have all moved into the field ramp region at some point prior to their arrival at the measuring point; those in sector “B” with pitch angles above 135’ have interacted with the perfect mirror. whilst those in sector “C” have had their motion reversed in the field ramp. The remaining sector. “D”, contains those particles which have their source points within the near-Earth region. and are a special case that we shall discuss more fully below. The source points for particles that enter the ramp region in this model (sectors “B” and “C” in Fig. 17) can be calculated by considering the acceleration of particles in this region. The force acting on the particles which reduces their field-aligned component of velocity (taken again to be approximately the .r-direc-

C. J. OWEN et al.

878

actual mirror point is then a solution (x,-x,)

of the equation

= t’,,t, - (FX/2mi)r:.

(22)

Since the acceleration is constant, 1, represents half the total time the particle spends in the ramp region. In addition to this, the particle takes a time I, = (x.0 +X&.

FIG. 17. DIAGRAMINDICATINGTHETYPE~FTRAIECTORYTHAT PARTICLESOFEACH PITCHANGLEHAVEUNDERGONETOREACH THE MEASURING POINT IN THE-REVISED" LOBE MODEL.

Those particles whose velocity components fall within the sector marked “A” are the unmirrored particles, bounded by the pitch angle ar. Particles with pitch angles larger than 135” (those with velocity components in sector B) have had their motion reversed at the perfect mirror, whilst those in sector C have mirrored within the near-Earth field ramp region. All of these particles have source points tailward of - 100 R,, outside the ramp. The remaining sector (D) represents those particles which may have entry points within the near-Earth region. which are not covered by our theory. In practise the size of this sector is negligibly small providing the measuring point is not close to both the ramp region boundary and the magnetopause.

tion)

is given by

Since we have assumed a linear field increase in this region, the force on each particle is constant within the ramp region and depends only on the particles’ ~1, i.e. on the particle pitch angle at the measuring point. Consequently the field-aligned acceleration of each particle is constant within the mirror region. The position s, at which each particle mirrors (i.e. the point at which its field-aligned motion reverses) may also be deduced from equations (16) and (19), [II,,, x,

= -

- B(.Y,)/sin’

dB(.u)/ds

r,]

(21)

for particles that do not hit the perfect mirror (x0 < 135”, sector “C”), whilst x, = 0 for all other mirrored particles. The time fm taken for the particle to move from the edge of the mirror region to the

(23)

to move from the mirror region to the measuring point where it is located at t = 0. The particle thus first entered the ramp region at a time t, = -(21t,J+ lt,l). Equation (9) can be used to locate the intersection of the field line, on which the particle is located, with the magnetopause at this time. Provided this point is further downtail than x,, the particle has a source point in the distant tail which, using the values of fr, xr(tE), and the fact that the particle is located at x = x,, and has speed u,, , can then be calculated exphcitly using the methods employed in Section 3. The possibility that ~~(2~) > x,, such that the particle has a source point within the near-Earth region will be discussed below. The value off for both these types of particle (i.e. those mirrored in the ramp region and those mirrored at the perfect mirror) can thus be deduced from these source points and by noting that, since they have all had their motion reversed at some point within the near-Earth region, the entry velocity t’,, is the subsequently measured velocity reversed in sign. The theory so far described in this section covers all particles that are both detected and have their source points in the distant lobes of this model (i.e. all the particles whose velocity components fall in sectors “A”, “B”, and “C” of Fig. 17). As described above we will not consider measuring points within the near-Earth region, however it is possible that some particles detected in the distant lobe have their source points in this region (sector “D” of Fig. 17). If, at the time t when the particle is located at x = x,. the field line intersects the magnetopause at a point closer to the Earth (such that the field line is located entirely within the near-Earth region), it follows that the particle must be located in the magnetosheath, and therefore must enter the lobe at some later time tE+dtE, at some position within the near-Earth region. However, we note that for particles with fieldaligned velocities which are large compared with the magnetosheath flow speed usH, the intersection point of the field line at the magnetopause moves only a small distance tailwards during the time taken for the particle to move down the field line to the mirror point and back to the magnetopause. Hence particles of this type detected at (x0, zO) (tailward of x,) can only have sources within the near-Earth region for x0 very close

Energetic ions in the distant tail lobes

to x, and z,, very close to the magnetopause location z,,,,,, i.e. a very restricted region of the lobe in this model. Those particles with field-aligned speeds comparable to the magnetosheath flow speed [i.e. those within a few degrees of the limiting pitch angle described by equation (17)] may also have sources within the ramp region. However, we note that the distance they may penetrate into this region is very small since they mirror at a field strength only slightly higher than that found in the deep tail. Hence their point of entry into the lobe must also be within a few Earth radii of x = x,. These particles may be observed at a general measuring point within the lobe (unlike the “fast” particles with sources in this region which can only be detected at positions close to the magnetopause and close to x,). However, the trajectory of these particles to the measuring point is very close to that of the limiting pitch angle particle, which crosses the magnetopause at x = x,. The field-aligned velocity of these particles within the constant field section of the lobe must therefore be very close to that of the limiting pitch angle particle, such that the velocity components of these “ramp” particles occupy a small section (section “D”) in Fig. 17, just above the limiting pitch angle. In addition, the particles are accelerated within the near-Earth region, and may have had their direction of motion reversed, such that although they are detected with a larger tailward fieldaligned speed than that of the limiting pitch angle particle, they entered the lobe with a speed that is of order or less than that of the limiting particle. It is therefore likely that any increase in the value off for these particles above the value off(a,) due to their source point being slightly closer to the Earth (for either type of source), will be more than cancelled by a decrease in ,f due to the entry velocity being less tailward than that of the limiting pitch angle particle. The exact balance of these two effects depends on the exact nature of the assumed magnetosheath flux variation, but the result is that the value off for the limiting pitch angle particle is, to a very close approximation, the maximum value of .f observed at that measuring point. Since the source point for this new maximum is further tailwards than that for the constant field strength lobe model of the last section, we expect the peak produced by this new model to be smaller, the decrease depending principally on the value of L for the exponentially falling source. It should be noted that although the motion of particles described in the last few paragraphs is somewhat complicated, the trajectories discussed correspond to a very small fraction of the total range of pitch angles at a “typical” measuring point.

879

3~00T--__---____-__--_-_

I

I

I

I

2.50

2.00

‘;; 1.50 .%

5 ’

“\_i ‘..,

‘.

_-_

----

-0.00

r - - _,_+__ - - - r - - - - ; ,’

I

,’

-0.50

-1.00

“.I...,. _+. ___,

“.,c

I

*--. I !_-___-z-_ --I _ _ _ - - ‘- - - - - -,--__ _ - --_.; : : ’

f

/

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I

I

I

I

I

I

,

135

180

,,,,,,,,,~,,,,/,,,~,,,,,,,,/,,,,,,,,~ 0

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90

Pitch angle (deg) FIG. 18. DISTRIBUTION FUNCTION~VS PITCH ANGLE FOR THE UNIFORM AND RAMP LOBE MODELS, FOR A MEASURING POINT 150R, DOWNTAIL FROM THE EARTH AND 10 R, FROM THE MAGNETOPAUSE. The magnetosheath source parameters are the same as those employed in Fig. 12, and the dashed line again represents the distribution expected if no near-Earth enhancement of the source flux is present. The solid line represents the distribution obtained from the ramp model. the dotted line that from the uniform field model. Although there are differences in detail between the two distributions, they both show the same qualitative behaviour. This figure illustrates that the simple lobe model is adequate to explain the observed energetic ion distributions in the tail lobes.

Figure 18 shows the pitch angle plots obtained using the uniform field lobe model described in the last section (dotted line, marked A) and by using the revised ramp model (solid line, marked B) at a measuring point given by x0 = - 150 R,, and -zO = IOR,, with x, = - lOOR,. and the same -P izput parameters (B, = 10 nT, B_ = -0.15 nT, vs,=400kms-‘,L=100R,,S0=500,~=3.5,and the particle speed ~2600 km s- ‘) used in the last section in conjunction with Figs 12 and 13. The dashed line (marked C) again shows the distribution expected for a uniform source (i.e. with no enhancement of flux due to a terrestrial source). In this case, as explained in the previous section, the variation infdepends only on the entry velocity of the particles. This parameter is the same for all particles in both models, except in the region al < tl < tlL, where particles which were

C. J. OWEN er al.

880

unmirrored in the uniform model become mirrored in the ramp model, and thus reverse the sign of their entry velocity. This results in a small shift in the peak of the distribution towards lower pitch angles, and a slight reduction in the height of the peak, since the maximum tailwards speed of entry possible is reduced. The distributions produced by the two models thus differ only over a very small range of pitch angles for a uniform source. For a spatially varying source, in which the position of particle entry to the lobes has to be explicitly considered, we note that since the particles around the peak in the ramp model have source positions much further down the tail than those in the uniform model. the peak represented by the solid line is shifted slightly, and is not as high (reduced by a factor e- ’ for the parameters used here). The low pitch angle particles (2 < al), which do not enter the ramp region, have exactly the same form as before, whilst the results for very large pitch angle particles (those above 150’) are not significantly altered as the near-Earth acceleration results in only a small change in their velocities in this region. Particles with pitch angles between 120. and 150’ are slightly enhanced as their source points are closer to the Earth due to the fact that they are significantly slowed in the near-Earth region. whilst the field line speed is unaltered. The results presented in this section therefore show that the qualitative form of the pitch angle distribution is not significantly altered by the inclusion of a near-Earth ramp in the field strength, for either a uniform or spatially varying source. We therefore conclude that the results presented in the previous section of this paper show all the essential features of the observed distribution. despite the simplicity of the model assumed. The exact form of the pitch angle distribution in the lobe does depend on the exact form of the field strength variation in the near-Earth region, but we have shown that the general form is unaffected. The peak in the distribution and the associated asymmetries are constant features of the distribution. which are only altered in detail by varying the field within the near-Earth region of the model lobe. The major considerations in predicting the form of the pitch angle distribution in the lobes are thus just the variation of the source distribution in the magnetosheath and the mechanism for entry of the particles into the lobes. as discussed in previous sections.

5.

SUMMARY

In this paper WChave considered, from a theoretical viewpoint, the form of the distributions of energetic ions in the lobes of the deep geomagnetic tail. and

have formulated a simple model which is found to account for the observed features of the distributions. As explained in the Introduction, there has been very little work done in this area in the past, in terms either of theory or observation, since most work on the tail has concentrated on the plasma sheet and associated phenomena. Pillip and Morfill (1978) and Cowley (1981) have studied the thermal plasma in the lobe (i.e. the plasma mantle), but these works concentrated on the low energy plasma component for which the speed V of the particles is comparable with, or lower than, the magnetosheath flow speed t)sH. In this paper we have concentrated on energetic ions for which the particle speed V z+ vSH. Observations (e.g. Cowley et al., 1984, and those presented here in Fig. 1) indicate that under usual conditions the lobe pitch angle distribution is highly peaked at pitch angles closely perpendicular to the magnetic field direction (“pancake” distributions), while also showing a small downtail anisotropy. During periods of solar particle enhancements however, the lobe distributions become much more closely isotropic. We have proposed a model which successfully reproduces these features of the data. In this model, the magnetosheath energetic ions, which are taken to act as the source population for the lobes, consist of two components, a “solar” source which is spatially uniform along the tail length, and a “terrestrial” source which decreases in strength with downtail distance. In general, therefore, enhanced fluxes exist at the earthward end of the tail which gradually fall to the solar wind background level some distance downtail. The near-Earth enhancement may be due either to production of energetic ions at the bow shock, or escape of ions across the magnetopause from the near-Earth ring current or plasma sheet. Diffusion of ions in the turbulent magnetosheath field creates distributions which are isotropic in the flow rest frame. In our model we have assumed that these ions gain entry to the tail lobes across the magnetically open magnetopause. Consideration of the motion of the particles within a simple model lobe. in which the field strength is constant everywhere but at the earthward end, where we apply a perfect mirror, allows the determination of the points at which particles of a given pitch angle and speed which reach a specified measuring point, crossed the magnetopause. The relative value of f, the distribution function, associated with that pitch angle and speed can then be determined from the source distribution at the point of entry, since the value offis conserved along the particles’ trajectory from the magnetopause to the point of measurement. The resulting theoretical pitch

Energetic ions in the distant tail lobes

881

G. and Sckopke, N. (1978) Energetic plasma idns in the Earth’s magnetosheath. Geophys. Res. Lett. 5,953. Baker, D. N., Craven, J. D., Elphic, R. C., Fairfield, D. H., Frank, L. A., Singer, H. J., Slavin, J. A., Richardson, I. G., Owen, C. 3. and Zwickl, R. D. (1988) The CDA W-8 substorm event of 28th January 1983: a detailed global study. Adtt. Space Res. 8(9), 113. Baker, D. N. and Stone, E. C. (1977) The magnetopause electron layer along the distant magnetotail. Geophys. Res. L&t. 4, 133. Baker, D. N. and Stone, E. C. (1978) The magnetopause energetic electron layer. 1. Observations along the distant magnetotail. .I. geophvs. Res. %3,4327. Balogh, A., Dijen. G. van. Genechten, J. van, Henrion. J., Hynds, R. J., Korfmann. G., Iversen. T., Rooijen, J. van, Sanderson, T., Stevens, G. and Wenzel, K.-P. (1978) The low energy proton experiment on ISEE-C. IEEE Trans. Geosci. Electronics GE-16 No. 3, 176. Coroniti, F. V. and Kennel, C. F. (1972) Changes in the ma~etospheric con~guration during the substorm growth phase. .I. geophys. Res. 77, 3361. Cowley, S. W. H. (1981) Magnetospheric asymmetries associated with the y-component of the IMF. Planet. Space Sci. 29, 79. Cowley, S. W. H., Hynds, R. J.. Richardson, I. G., Daly, P. W., Sanderson, T. R., Wenzel. K.-P., Slavin, J. A. and Tsurutani, B. T. (1984) Energetic ion regimes in the deep geomagnetic tail : lSEE 3. Geoohvs. Res. Let?. 11,275 Crooker:N. U., Eastman, T. E.: Frank, L. A., Smith,E. J. and Russell, C. T. (1981) Energetic magnetosheath ions and the interplanetary magnetic field orientation. .I. geophys. Res. 86,455. Daly, P. W., Sanderson, T. R. and Wenzel, K.-P. (19841 -_Skvey of energetic (E > 35 keV) ion anisotropies in the deen eeomaenetic tail. J. ueonhvs. Res. 89. 10733. deHo&an, Frand Teller, E:(l&) Ma~eto-hydrodynamic shocks. Phys. Rev. 80,692. Dungey, J. W. (1965) The length of the magnetospheric tail. J. geophys. Res. 70, 1753. Fairfield, D. H., Baker, D. N., Craven, J. D., Elphic, R. C., Fennell, J. F., Frank, L. A., Richardson. I. G., Singer, H. J., Slavin, J. A., Tsurutani, B. T. and Zwickl, R. D. (1989) Substorms, plasmoids. flux ropes and magnetotail flux loss on March 23, 1983: CDAU’-8. J. geophys. Res. 94,15135. Hones, E. W., Jr., Baker, D. N., Bame, S. J., Feldman, W. C., Gosling, J. T., McComas, D. J.. Zwickl, R. D., Slavin, J. A., Smith, E. J. and Tsurutani, B. T. (1984) Structure of the magnetotail and its response to geomagnetic activity. Geophys. Res. Lett. 11, 5. Hudson, P. D. (1974) The reflections of charged particles by rotational discontinuities. Plmet. Space Sci. 22, 1571. Ackno~leugemenl-During the course of this work two of Lanzerotti, L. J., Krimigis, S. M., Bostrom, C. O., Axford, the authors (C.J.O. and I.G.R.) were supported by U.K. W. I., Lepping, R. P. and Ness, N. F. (1979) Measurements SERC Research Assistantships. The ISEE- EPAS instruof plasma flow at the dawn magnetopause by Voyager 1. ment is a joint project of the Blackett Laboratory, Imperial J. oeoohvs. Res. 84. 6483. College, London. the Space Science Dept of ESA. ESTEC, Lee.“M’.zA.I (1982) Coupled hydromagnetic wave excitation Noordwijk. and the Space Research Laboratory, Utrecht. and ion acceleration upstream of the Earth’s bow shock ___ ____ J. aeoohvs. Res. 87. 5063. Lui, A. T.‘Y. (1987) Roadmap to the magnetotail domains, REFERENCES in Magnetotail Physics (Edited by Lui, A. T. Y.), p. 3. Johns Hookins Universitv Press. London. Anagnostopoulos, G. C., Sarris. E. T. and Krimigis, S. M. Pillip, W. 6. and Morfill, G. (1978) The formation of the (1986) Magnetospheric origin of energetic (E > 50 keV) plasma sheet resulting from plasma mantle dynamics. J. ions upstream of the bow shock: the October 31, 1977, geaphys. Res. 83, 5670. evjent. I. geo$_ss. Res. 91, 3020. Richardson, I. G. and Cowley, S. W. H. (1985) PlasmoidAsbridge, J. R.. Bame. S. J.. Gosling, J. T., Paschmann.

angle plots show a peak in the distribution at a few degrees above 90” to the field direction, and an anisotropy, in which particles moving tailwards have generally higher values of f than those moving earthwards, which is in agreement with the observed distributions. The spatial variation of these distributions within the lobe was also studied. For a purely uniform source (i.e. the background fluxes from the solar wind), we find only limited variations in the form of the lobe energetic particle pitch angle distributions. However, if the spatially varying terrestrial source is inciuded, we find that the energetic particle density, or omnidirectional flux (which is a function of the area under thefvs pitch angle curve for a given speed) decreases, the distribution narrows, and the net downtail flow becomes smaller as the measuring point moves further from the magnetopause. The omnidirectional flux, and the width of the peak also become smaller with increasing distance down the tail, although in this case, the net downtail flux becomes larger. For the spatially varying source, however, the parameter having the most affect on the pitch angle distributions is the scale length of the source flux variation in the magnetosheath. For very small scale lengths the distributions are peaked sharply at the maximum, whilst for very long scale lengths the distribution is broader, and is more nearly isotropic. The latter case explains the observations made during solar particle events, when the spatially varying flux in the magnetosheath is negligible compared with the high background fluxes existing in the solar wind, when the scale length effectively becomes infinite. Finally, we have refined our lobe model to include a (more realistic) field increase in the near-Earth region. Although the details of the results differ from those produced by the simple model, the qualitative difference between the models is small, and we conclude that our simple model is thus sufficient to explain the main features of the energetic particle populations observed within the tail lobes.

‘I

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