Energy and pitch-angle dispersions of LLBL/cusp ions seen at middle altitudes: predictions by the open magnetosphere model

. Numerical simulations are presented of the ion distribution functions seen by middle-altitude space-craft in the low-latitude boundary layer (LLBL) and cusp regions when reconnection is, or has recently been, taking place at the equatorial magnetopause. From the evolution of the distribution function with time elapsed since the ﬁeld line was opened, both the observed energy/observation-time and pitch-angle/energy dispersions are well reproduced. Distribution functions showing a mixture of magnetosheath and magnetospheric ions, often thought to be a signature of the LLBL, are found on newly opened ﬁeld lines as a natural consequence of the magnetopause e(cid:128)ects on the ions and their ﬂight times. In addition, it is shown that the extent of the source region of the magnetosheath ions that are detected by a satellite is a function of the sensitivity of the ion instrument . If the instrument one-count level is high (and/or solar-wind densities are low), the cusp ion precipitation detected comes from a localised region of the mid-latitude magnetopause (around the magnetic cusp), even though the reconnection takes place at the equatorial magnetopause. However, if the instrument sensitivity is high enough, then ions injected from a large segment of the dayside magnetosphere (in the relevant hemisphere) will be detected in the cusp. Ion precipitation classed as LLBL is shown to arise from the low-latitude magnetopause, irrespective of the instrument sensitivity. Adoption of threshold ﬂux deﬁnitions has the same e(cid:128)ect as instrument sensitivity in artiﬁcially restricting the apparent source region.


Introduction
Recently.two complementary models of the precipitation of magnetosheath ions into the dayside auroral ionosphere have been developed and have successfully reproduced the energy-latitude dispersion plume of ions spanning the precipitation regions classed cusp, mantle and polar cap.These models are based on the open magnetosphere theory in which the plasma gains entry to the magnetosphere by ¯owing along newly-opened ®eld lines produced by magnetopause reconnection.The model of Onsager et al. (1993) employs ®xed, steadystate distributions of magnetic and electric ®eld throughout the dayside magnetosphere to evaluate the trajectories of the ions.The model developed by Lockwood and Smith (1994), Lockwood (1995a) and Lockwood and Davis (1996) is based on the same general principles but is dierent in a number of details.One of the more fundamental dierences from the Onsager et al. model is that the spatial distribution of electric and magnetic ®elds does not need to be speci®ed.Instead, it is assumed that the frozen-in theorem and E á B = 0 apply everywhere away from the reconnection site and the precipitation is studied as a function of the time elapsed since the reconnection of the ®eld line onto which it is frozen.This is done by solving the tangential stress balance on newly opened ®eld lines to evaluate how they evolve over the magnetopause.A major advantage of the model by Lockwood and co-workers is that it can readily be used to study time-varying situations like pulsed reconnection: an important disadvantage is that non-ideal-MHD eects (such as gradient-B and curvature drifts of the more energetic particles) cannot easily be included.
A key part of the identi®cation of the low-latitude boundary layer (LLBL) precipitation is the presence of both magnetosheath-like and magnetosphere-like ion populations.In this respect, the presence of energetic magnetosphere-like ions, at energies above those of the ions from the magnetosheath, is a crucial de®ning feature of the LLBL (Woch and Lundin, 1993) and is usually interpreted as showing that the LLBL ®eld lines are closed.However, Lockwood (1997) has recently argued that the open magnetosphere theory provides an explanation not only of the LLBL ion precipitation, but Ann.Geophysicae 15, 1501±1514 (1997) Ó EGS ± Springer-Verlag 1997 also of any dayside boundary plasma sheet (BPS), with both being on open ®eld lines.The open-closed ®eld line boundary is inferred to be equatorwards of the BPS (or void if ¯uxes are low or dicult to classify), close to the poleward edge of the central plasma sheet (CPS).The locations where all these classes of precipitation are typically found have been surveyed statistically by Newell and Meng (1992).
The velocity ®lter eect on injected magnetosheath ions was ®rst discussed by Rosenbauer et al. (1975).These authors considered the trajectories of ions of dierent energies and pitch angles from a single point on the magnetopause in the presence of the large-scale dawn-to-dusk electric ®eld which gives convection of plasma and magnetic ¯ux from the dayside magnetosphere into the tail lobe.In particular, they provided an explanation of upgoing magnetosheath ions, seen by mid-altitude satellites in the mantle region of the magnetosphere, in terms of ions entering through the cusp region and mirroring at low altitudes in the converging geomagnetic ®eld lines.The lower-energy ions are swept further towards the tail lobe by the ®eldperpendicular convection because they have lower ®eldaligned velocities and thus longer ¯ight times (from the magnetopause down to the mirror point and then back up to the satellite).There are two competing pitch-angle eects on the ¯ight time: higher pitch-angle ions have lower ®eld-parallel velocities than more ®eld-aligned ions of the same energy; however, the mirror point for smaller pitch angles is further below the satellite, making the path travelled longer.Rei et al. (1977) used the velocity ®lter concept to show that the energy-latitude dispersion of magnetosheath ions seen at low altitudes in the topside ionosphere was also due to the eect of ion ¯ight times in a convecting magnetosphere.Hill and Rei (1977) also showed that the ion ¯uxes at the high-energy end of this dispersion ramp were at higher energies than in the magnetosheath, providing evidence that they were associated with the accelerated ion ¯ows seen at the dayside magnetopause (see review by Cowley, 1982).The highest-energy ions are found at the equatorward edge of the dispersion ramp for southward IMF, but at the poleward edge for northward IMF (Woch and Lundin, 1992).
The fact that the velocity ®lter eect is seen in the ion dispersion is consistent with the ions, in the main, undergoing adiabatic, scatter-free motion from the magnetopause to the satellite.This being the case, an ion's magnetic moment is conserved so that its pitch angles at its magnetopause source, at a distance z along the ®eld line from the magnetopause, and at the satellite (respectively, a m , a(z) and a s ) are related by: sin 2 m af m sin 2 zafz sin 2 s af s Y 1 where B m , B(z) and B s are the corresponding magnetic ®eld intensities at these three locations.Using order-ofmagnitude estimates of f s 5 Â 10 À5 T (for a lowaltitude satellite, geocentric distance, r $1 R E ) and B m = 50 nT, and noting that ions with mirror points at or below the satellite have a s < 90°, Eq. ( 1) shows that only the ions close to ®eld-aligned at the magnetopause (a m < 2°) will reach satellites in the topside ionosphere.However, at higher altitudes (and we here use a typical r of 4 R E ), the lower B s ($ 200 nT) means that a larger angular slice of the distribution function at the dayside magnetopause (a m up to about 30°) can be detected.The ion velocity along the ®eld line is (2E/m) 1/2 ´cos a(z), where E is the (conserved) ion energy and m is its mass.From Eq. ( 1) the time of ¯ight T, along the ®eld-aligned distance of s from the magnetopause to the satellite, is therefore (Burch et al., 1986): Note that in general, the solution of Eq. ( 2) requires knowledge of the ®eld B(z) at all z from the magnetopause between zero and s: however, this is not true in the special case of purely ®eld-aligned ions for which a s is zero, and Eq. ( 2) reduces to T = s(m/2E) 1/2 .Equation (2) shows that ions observed at larger pitch angles, a s , have a longer time-of-¯ight at a given energy; conversely, for ions to have the same time of ¯ight, higher energies are required at larger pitch angles.If we consider a point magnetopause source, the ions which are swept the same amount downstream by the convection electric ®eld will all share the same ¯ight time, T, and so those seen together at the satellite must have higher energies E at higher pitch angles, a s .On an ion spectrogram, in which ion ¯ux is contoured as a function of energy and observation time, the ions will form a characteristic V-shape as the satellite spins and repetitively samples the full range of a s .These ``ion Vs'' are observed by mid-altitude satellites in the LLBL/cusp region (e.g.Burch et al., 1982Burch et al., , 1985;;Woch and Lundin, 1992;Kremser et al., 1995).This has therefore been interpreted as showing that the LLBL/ cusp ions come from a relatively narrow source region.
A spread in source locations would allow a range of T to be sampled simultaneously and so introduce a range of ion energies at any one pitch angle (which is equivalent to a spread of pitch angles at any one energy), i.e. it gives width to the ion Vs.From these ideas, Menietti and Burch (1988) used the width of the ion Vs to deduce that the source region was only about 1 R E in extent.
On the other hand, Lockwood and Smith (1993) have argued that the spread of the ion energies seen in cusp at low altitudes reveals that the magnetopause source region of these magnetosheath ions must be of order 10± 20 R E in extent.This is consistent with the open magnetosphere theory in which plasma streams continuously across the magnetopause at all times for which the ®eld line is open.At times when the ®eld line threads the dayside magnetopause, ions crossing the boundary are accelerated such that they acquire large ®eld-aligned velocities towards the Earth as the ®eld lines contract Earthwards.The contrasts with later times, when the ®eld lines thread the boundary in the mantle region (the magnetopause edge of the tail lobe) where the ions crossing the boundary are decelerated and ¯ow towards the tail as the ®eld lines are stretched in the tail lobe [see discussion by Cowley et al. (1991) and Lockwood (1995a, b)].Thus, at larger elapsed times since reconnection, most of the injected sheath ions do not precipitate down to the inner magnetosphere (and so, in the main, they reach neither low nor middle altitudes); rather, they mainly ¯ow antisunwards into the tail lobe.In addition, the density and temperature of the source magnetosheath population is highest on the dayside, but decreases with distance away from the nose of the magnetosphere (Spreiter et al., 1966).As a result, the high-¯ux cusp precipitation evolves into the lower-¯ux, lower-energy mantle precipitation and, later, into the almost non-existent polar-cap ion precipitation.These eects mean that, although the ions continue to ¯ow across the magnetopause as the newly opened ®eld line evolves into the tail lobe, they cease to be a source of high ¯uxes in the inner magnetosphere.These eects can act to restrict the eective source region for the ions in the ionospheric cusp region to 10±20 R E , though they would not be consistent with the smaller ($1 R E ) extents derived from the ion Vs.
In this paper, we investigate whether the open magnetosphere theory is indeed consistent with the observed ion Vs and discuss the size of the source region.In Sect. 2 we extend the modelling of Lockwood (1995a) to cover all pitch angles sampled by satellites at mid-altitudes.In addition, we include the acceleration of magnetospheric ions at the magnetopause, via the mechanism proposed by Lockwood et al. (1996).The results are discussed in Sect. 3 and compared with observations.In Sect. 4 we show that the geometric factor and sensitivity of the ion detector determines how extensive the source region is inferred to be and also discuss the role of ¯ux thresholds employed in de®nitions of the precipitation class.

The model
The model used is a development of that described and employed by Lockwood (1995a) to investigate the dispersion of injected sheath ions in the (open) LLBL/ cusp/mantle/polar-cap dispersion plume.This model was developed from the theory presented by Lockwood and Smith (1994) and has been used by Lockwood and Davis (1996) successfully to predict the observed ion precipitation signatures (in spectrogram format) for various low-altitude satellite passes with pulsed magnetopause reconnection.The model combines ®ve key elements: (1) gas dynamic predictions of the morphology of the magnetosheath near the magnetopause boundary (Spreiter et al., 1966); (2) the method for determining the evolution of newly opened ®eld lines over the magnetopause of Cowley and Owen (1989); (3) the theory of the interactions of ions with the magnetopause current sheet by Cowley (1982), recently extended to cover another magnetopause AlfveÂ n wave by Lockwood et al. (1996); (4) the velocity ®lter eect of ion ¯ight times, originally invoked by Rosenbauer et al. (1975); and (5) Liouville's theorem of the conservation of phase space density for a dynamical processes (e.g.Elliott, 1993).
The starting point of the model is the method to compute how newly opened ®eld lines evolve over the dayside magnetopause, away from the reconnection site, using the stress-balance concepts introduced by Cowley and Owen (1989).This is achieved using a gas-dynamic model of the boundary-tangential magnetosheath bulk ¯ow over the boundary (Spreiter et al., 1966) and an input draped magnetosheath ®eld orientation (as a function of position), as speci®ed by Lockwood (1995a).(Note that this draped sheath ®eld determines the spatial distribution of the boundary-normal ®eld, B n , at the magnetopause, which is therefore eectively an input to the model and not self-consistently computed).Application of the algorithm of Cowley and Owen (1989) yields not only the location where each newly opened ®eld line threads the boundary (here referred to generically as the point P n ) as a function of time elapsed since reconnection, but also gives the ®eld-line speed over the boundary (the de Homan-Teller velocity) at each P n (de Homan and Teller, 1950).We de®ne the distance along the magnetopause (along the locus of the points where the convecting ®eld line threads the boundary) from the reconnection site to the point P n to be d n and the time taken for the open ®eld line to reach P n to be t n .The Spreiter et al. gas-dynamic model is used to give the ion gas density and temperature at each P n , and the theory of Cowley (1982) is used with the de Homan-Teller velocity to compute the distribution function of the ions injected by ¯owing along the newly opened ®eld lines across the magnetopause at each P n .This theory of the ion acceleration is a vital part of both this model and that by Onsager et al. (1993): its application and its veri®cation in ion observations from the magnetopause have been discussed by Lockwood (1995a).In order to apply the theory, the angle /(t n ) that the magnetospheric part of the open ®eld line makes with the boundary must be known.This was determined from the boundary normal ®eld at P n , B n (as just discussed), and the magnitude of the interior ®eld B sp , taken from a Tsyganenko T87 model in a way described in the following (/ = sin A1 [B n /B sp ]).
The next element of the model is to compute the time of ¯ight, T, of the ions from each P n to the altitude of the satellite (in this paper we consider a satellite at a geocentric distance of r = 4 R E ).In the previous work, this was restricted to ®eld-parallel ions for ease of computation (one major complication introduced in the following is to consider ions at other pitch angles): to compute T, Lockwood (1995a) employed the Tsyganenko T87 magnetic ®eld model to give an estimate of the ®eld-aligned distance s from each P n to the altitude of the satellite.The T87 model contains no systematic open ¯ux and thus ®eld lines do not thread the magnetopause in the way required to de®ne the P n .In order to simulate the open magnetosphere, an arti®cial magnetopause was introduced just inside the main magnetopause current sheet implied by the model.This gave not only an estimate the required distance s, but also the estimate of the ®eld strength just inside the boundary, B sp , which was used to compute the angle /.With knowledge of their time of ¯ight we can, for every time elapsed since reconnection, evaluate when and where any ions observed by the satellite were injected across the magnetopause.We here use the same notation as Lockwood (1995a), in which the time elapsed since reconnection is t s À t o (t s is the time of observation at the satellite and t o is the time that the ®eld line was opened).The point of entry of every particle is known because and so for a known (t s À t o ) and T, the time t n can be evaluated and, as already described, d n is known.From the distribution functions computed by the Cowley theory at each P n , we know the phase space density of the ions of the energy E corresponding to the ¯ight time T. Because the ion motion considered is adiabatic (E á B = 0 in ideal MHD so ®eld-parallel electric ®elds are neglected) and scatter free, both the ion energy and, by Liouville's theorem, the phase space density are conserved in travelling from P n to the satellite.Thus the distribution function f (E, a s ) of the ions at the satellite can be computed, as a function of time elapsed since reconnection (t s À t o ).The location of the source point on the magnetopause P n (quanti®ed by the distance d n ) of the ions seen at any one position in phase space (E, a s ) at that (t s À t o ) is also known.
As already mentioned, we here extend the modelling of Lockwood (1995a) to allow for pitch angles other than zero.Equation (2) shows that in order to evaluate the time of ¯ight T for a given ion energy E we need not only the distance s, but also the variation of the magnetic ®eld strength B along the ®eld line along the distance s.This is also taken from the Tsyganenko T87 magnetic ®eld model.
The model used here also includes the eect of acceleration of magnetospheric ions by re¯ection o both of the two AlfveÂ n waves launched by the reconnection site.This concept is a generalisation of the Cowley (1982) theory and was used by Lockwood et al. (1996) to model the highest-energy ions of an observed ion dispersion plume reported by Newell et al. (1991a).A second observed example, reported by Moen et al. (1996), was also successfully explained using this theory (Lockwood and Moen, 1996).The basic concept is illustrated in Fig. 1.This schematic shows three newly opened ®eld lines emanating from a reconnection site, X; the ®eld line passing through X makes up the two magnetic separatrices (s) and is for a (t s À t o ) of zero; the other two ®eld lines are for larger (t s À t o ).The reconnection site, X, launches two AlfveÂ n waves (also called intermediate mode or rotational discontinuity, RD) shown by the dashed lines: the interior wave, i, stands in the in¯ow to the reconnecting boundary on the magnetospheric side, the exterior wave, e, stands in the in¯ow from the magnetosheath side.In the sense that the majority of the ®eld rotation takes place at it, e is the magnetopause.In this simpli®ed view of the reconnecting magnetopause, all of the ®eld rotation takes place at these two AlfveÂ n waves and, because the AlfveÂ n speed is much greater on the magnetospheric side of the boun-dary (owing to the lower plasma density), the ®eld kink at the interior wave moves at a speed V i which is considerably greater than that of the kink at the exterior wave, V e .It should be noted that MHD theories suggest a more complex structure than is proposed here (e.g.Biernat et al., 1989), including a fast wave, an intermediate shock and a slow expansion wave on the magnetosheath side of the reconnection layer, and a fast wave, AlfveÂ n wave and slow shock on the magnetospheric side: the two sides being separated by a contact discontinuity.Although such structure is reproduced in MHD simulations, it is not found in hybrid simulations, which yield the simpler picture of a mixing of the two plasmas in an open LLBL which is bounded by two AlfveÂ n waves (Lin and Lee, 1993).The attraction of these hybrid simulations (for the work presented here) is that they also reproduce the ion distribution functions seen near the magnetopause, as predicted by Cowley (1982) and observed by Gosling et al. (1990), Fuselier et al. (1991) and Smith and Rodgers (1991).An important part of Cowley's predictions is that ions incident on an RD are either transmitted through it or re¯ected o it.The hybrid simulations by Chapman and Moukis (1996) show that this occurs because ions fall into a potential well at the centre of the RD and emerge (with the same energy and pitch angle) on one or other side, depending on their initial gyrophase.Thus a fraction r of incident ions at a certain point in (V ^, V i ) velocity space are re¯ected while the remaining fraction (1Ar) are transmitted.Chapman and Moukis also show that this does not occur in ideal MHD simulations.This behaviour is almost exactly as assumed by Cowley (1982) when he made his predictions.
Figure 1 shows the bulk motion of the relevant ion populations in the frame of the Earth.The magnetosheath population adjacent to the boundary is shown ¯owing away from the X-line, but because its speed is considerably slower than V e , it is eectively being overtaken by the ®eld-line kink and will be incident upon the boundary e.This results in the magnetosheath plasma motion in the ®eld line rest frame (the de Homan-Teller frame) being purely ®eld aligned and towards the RD at the local AlfveÂ n speed, V Ash , i.e. the WhaleÂ n relation holds (Cowley and Owen, 1989).This population is therefore labelled i-sh (i for incident, sh for sheath origin).Some of this i-sh population will pass through the magnetopause to form the t-sh population (t for transmitted) which near the X-line moves Earthwards with a bulk ¯ow speed of about (V e +V Ash ) in the Earth's frame.Lockwood (1995a, b) has discussed how this velocity and the t-sh ion population subsequently evolves with the distance d n .The changes occur for three reasons: ®rstly, the ®eld line accelerates away from the reconnection site towards the tail lobe; secondly the ion acceleration decays as the ®eld line straightens (specifically, the angle / that the magnetospheric arm of the ®eld line makes with the magnetopause increases); thirdly, the characteristics of the magnetosheath source plasma change with d n .On the magnetospheric side of the boundary, the ion population, sp, is at rest in the Earth's frame and thus the interior kink of the newlyopened ®eld lines will engulf it at speed V i , which will be the local (interior) AlfveÂ n speed, V Asp (i.e, as at the exterior RD, the WhaleÂ n relation holds).On interaction with the wave i, any sp ions re¯ected move away from the boundary and towards Earth with bulk ¯ow speed of about 2V Asp (these re¯ected ions are referred to ri-sp: ri for re¯ected o the interior wave and sp for magnetosphere in origin).Note however, that the ri-sp population can only be seen from the magnetospheric side of i , so at low altitudes they should only found equatorward of the interior wave.Most sp ions will pass through i and some of these will be re¯ected o e to give a population termed re-sp, which near the X-line has an Earthward bulk ¯ow speed of about 2V e .The re-sp population will evolve with (t s À t o ) for three reasons.The ®rst two reasons are the same as for the t-sh population, namely the de Homan-Teller velocity changes as the ®eld line accelerates, and acceleration decays as the ®eld line straightens.The third reason is that the source sp ions are progressively lost by interaction with the magnetopause.Once the ®eld line is opened a loss of sp ions commences, either because they are re¯ected o the magnetopause to become re-sp or ri-sp ions, or because they are transmitted through it into the magnetosheath.At any one (t s À t o ) > 0, the sp ions which remain are those which have not had time to interact with the open magnetopause: they have a halfbounce time-of-¯ight (from the reconnection site down to a mirror point and back up to the magnetopause) which is greater than (t s À t o ).
Applying the theory of Cowley (1982) to each of these populations, the model generates the total ion distribution functions at each P n in the open LLBL: in the example shown in Fig. 2, P n is in the immediate vicinity of a subsolar reconnection site (i.e.d n and t n are nearly zero in this example).These plots show the phase space density, f, colour coded as a function of the ®eldperpendicular and ®eld-parallel ion velocities (in the Earth's frame of reference and positive towards the Earth).The upper plot is for the part of the open LLBL that is between the interior wave i and the ion edge (which is closer to the separatrix s in Fig. 1).On the other hand, the lower plot is for within the open LLBL and between the two waves i and e.The input parameters to the model are derived to give the same conditions at the reconnection site (at the nose of the magnetosphere) as were deduced from observations in the low-altitude cusp region by Lockwood et al. (1994Lockwood et al. ( , 1996)): they are listed in full in Table 1.Note that many of the parameters are not ®xed by the theory or simulations discussed here (for example the re¯ection coecients r i and r e ) and values used are simply those that give the best ®ts to the data.The ri-sp ions can only be seen in the upper plot and appear at the highest Earthward ®eld-aligned velocities.The main peak in f shows the characteristic D-shaped distribution of the tsh population, as predicted by Cowley (1982) and as has been observed in a number of studies (Gosling et al., 1990;Fuselier et al., 1991;Smith and Rodgers, 1991) and produced in the hybrid simulations of Lin and Lee (1993).The population centred on the origin is the initial, isotropic sp population (usually called the central plasma sheet, CPS). Figure 3 gives the spectra of the ®eld-aligned ions (a m = 0) of the various components of the total distribution function shown in Fig. 2a. Figure 3 is in the form of a log-log plot of the dierential energy ¯ux, J E , as a function of the energy, E. The t-sh, sp, resp and ri-sp populations are labelled.Note that in this paper, we make predictions of the phase space density, f, the dierential number ¯ux (sometimes referred to as the intensity), J, and the dierential energy ¯ux, J E : at any one energy, E, these are related by: Figure 4a shows an example set of ¯ight times of all the ions injected/accelerated at this P n , in the immediate vicinity of the X-line (d n » 0), to a mid-altitude satellite at a geocentric distance of r = 4 R E .These are computed from the magnetic ®eld model, using Eq. ( 2).The time of ¯ight, T, is colour coded on a logarithmic scale as a function of the ®eld-perpendicular and ®eld-parallel ion velocities of the ions when they reach the satellite.The ¯ight times can be used in the way described to determine the distribution function f(E,a s ) at a given elapsed time since reconnection (t s À t o ).

Distribution functions
Parts b-f of Fig. 4 show a set distribution functions generated in the manner described in the previous section and using the model inputs listed in Table 1.The sequence shows how the distribution function evolves with increasing elapsed time since reconnection, (t s À t o ).
From the preceding discussion of the model, it should be remembered that this evolution is caused by a convolution of three factors: the spatial variation of the ion gas in the magnetosheath, the eects of the motion of the ®eld line tailward and the time of ¯ight of the ions from the magnetopause to the satellite.In Fig. 4b, (t s À t o ) = 75 s and two ion populations can be seen.The lower-energy population is a loss-cone distribution of the sp ions (i.e.CPS) which were present on closed ®eld lines (t s À t o `0) and have yet to be in¯uenced by the fact that the ®eld line has been opened.Note that it has been assumed here that equatorial scattering has ®lled the loss cone corresponding to mirror points in the opposite hemisphere; however, this assumption was not necessary and a double loss cone distribution could equally well have been used.At the satellite, the loss of sp ions is ®rst noted at the highest energies as the lowest ¯ight time sp ions fail to arrive: for (t s À t o ) as low as 75 s, only loss of ions with T < (t s À t o ) = 75 s could be noted and the ¯uxes of such ions in the sp population is negligibly small.The boundary in (V ^, V i ) phase space de®ned by T = (t s À t o ) is here called the time-of-¯ight cut-o.Below this cut-o only the sp ions from the closed ®eldline region can be seen, above it only the populations generated by the open magnetopause (t-sh, re-sp and risp) can be found.For reference with Fig. 3, the lower cut-o energy of ®eld-aligned ions is given by: where s x is the distance along the ®eld line from the Xline to the altitude of the satellite, which is here 23.5 R E .For Fig. 4b, E ic is 2.08 ´10 4 eV (in Fig. 3, log 10 (E ic ) is 4.32).
In addition to the sp population in Fig. 4b, a second higher-energy population forming a ``halo'' has appeared which is not seen on closed ®eld lines.These are only seen above the time-of-¯ight cut-o and Fig. 3 shows that for log 10 (E ic ) > 4.32 the dominant ions are ri-sp.Thus the halo is caused by ri-sp ions reaching the satellite.Note that somewhat lower energies have reached the satellite at a s = 0 than for larger pitch angles, in accordance with Eq. ( 2).This population has developed further in Fig. 4c which is for (t s À t o ) = 100 s (E ic = 1.17 ´10 4 eV; log 10 (E ic ) = 4.07), as lower energy ri-sp and re-sp ions have had time to reach the satellite.At a (t s À t o ) shortly after this, the (apparent) gap between the t-sh and re-sp/ri-sp populations disappears.Lockwood (1997) argues that the ion populations in Fig. 4b and c would qualify them as a BPS precipitation.
Figure 4d is for (t s À t o ) = 200 s (E ic = 2.93 ´10 3 eV; log 10 (E ic ) = 3.47), when substantial changes have taken place.Firstly, the interior AlfveÂ n wave has passed over the satellite, which can therefore no longer detect any ri-sp ions.This means that the re-sp population is revealed at the highest energies.In addition, the time-of-¯ight cut-o has moved down to still lower values and Fig. 3 shows that for this E ic , t-sh ions will dominate just above the time-of-¯ight cut-o (giving phase-space densities of over 10 11 m A6 s 3 , shown in red in Fig. 4d).The sp ions at energies above the time-of-¯ight cut-o (ions with b t s À t o ) have been lost, but this loss is masked by the presence of the t-sh and re-sp ions.The dashed line in Fig. 5 shows the spectrum of ®eld-aligned ions at the satellite at this (t s À t o ), shown here as the dierential number ¯ux, J, and as a function of energy on a log-log scale.The presence of the re-sp ions at high energies means that the satellite observing a ®eld line of this (t s À t o ) would classed as being in the LLBL (Woch and Lundin, 1993).At (t s À t o ) = 300 s (Fig. 4e), the time-of-¯ight cuto has moved to yet lower velocities (E ic = 1.30´10 3 eV; log 10 (E ic ) = 3.11), below which sp ions are seen, but above which t-sh ions and a few remnant re-sp ions are seen.The population is now like that seen in the magnetosheath so that it is classed as cusp or cusp proper.Lastly, for (t s À t o ) = 500 s (E ic = 468 eV; log 10 (E ic ) = 2.67), Fig. 4f shows that the re-sp ions are almost all gone and the lower cut-o energy of the tsh population has moved to yet lower values.This precipitation is also classed as cusp.This is con®rmed by the dierential number ¯ux spectrum of ®eld-parallel ions, shown by the solid line in Fig. 5. [Compare with, for example, the cusp spectra in Figs. 1 and 2 of Woch and Lundin (1992)].As the ¯ux is approaching the lower limit for the cusp classi®cation, this case is quite close to the boundary of the cusp and mantle precipitation regions (Newell et al., 1991b).
Comparison of Fig. 5 with the cusp and LLBL spectra presented in Fig. 1 of Woch and Lundin (1992) shows that the model is able to reproduce the important features of both the cusp and the LLBL precipitations.Note that the model predictions presented here dier from the observed examples at low energies (10 eV±1 keV); this is because the observed CPS population contains a low-energy ionospheric component (cf.Fig. 2 of Woch and Lundin, 1993), which is not included in Fig. 5: for simplicity, the sp population used in the model was a single Maxwellian.The model also explains the evolution of the LLBL spectrum into that of the  The various component ion populations, t-sh, sp, ri-sp and re-sp are discussed in the text.This plot gives the dierential energy ¯ux J E as a function of energy E on a log-log scale cusp, for example as revealed by the sequence shown in Fig. 2 of Woch and Lundin (1992).
In cases such as that for (t s À t o ) = 200 s, the magnetospheric (sp) ions below the cut-o appear to belong to the same population as those magnetospheric ions that have been re¯ected o the exterior wave (re-sp) and that are seen at energies above where transmitted sheath (t-sh) ions dominate: these two populations (sp and re-sp) could therefore be ®tted with a single Maxwellian (of higher temperature and density than the sp population ± cf.Fig. 3).This point is demonstrated in Fig. 6, in which the J(E) spectrum for The only indication that this ®t may not be valid is the fact that the observed ¯uxes are a bit lower at energies just below the time-of-¯ight cut-o, E ic .This situation can indeed be seen in the boundary-layer spectra presented in Fig. 1 of Woch and Lundin, (1992).Note that the re-sp ions can appear as a continuation of the sp ions because although only some of the sp ions are lost at the magnetopause, those that are re¯ected are accelerated.The slope of the sp distribution function f(E) is steep at higher energies and so the rise in ion energy E on re¯ection acts to increase f and so counteracts the loss of ions by transmission through the magnetopause.
Therefore, an important point about Fig. 4d and 6 is that this distribution function [for (t s À t o ) = 200 s] appears to be a mixture of an sp population, with magnetosheath ions added, whereas, in fact, the higher energy ions are re-sp and not sp at all.This situation is that expected in the LLBL, with a mixture of magnetosphere-like and magnetosheath-like populations seen at the same location.The apparent presence of a full magnetospheric population is one reason why this LLBL is thought to be on closed ®eld lines; however, the modelled distributions are on open ®eld lines (t s À t o b 0), and the population which mimics the CPS is in fact made up of the remnant sp population at low energies (ions with `ts À t o ) with the re-sp population at higher energies.These eects therefore oer a good explanation of sheath plasma injected onto seemingly closed magnetospheric ®eld lines in the LLBL: they are on ®eld lines which have been open for a time of order 200 s.

Pitch-angle dispersion
The distribution functions like those shown in Fig. 4 can be re-plotted in a pitch angle-energy format, as often used for satellite data.This is done here in Fig. 7.The distribution function has been converted to dierential energy ¯ux using Eq. 4 because either this or count rates are usually plotted (count rates being proportional to dierential energy ¯ux for a dierential ion instrument).In Fig. 7, the spin-angle distributions are plotted for eight dierent (t s À t o ) values, as given at the top of each panel.The dierential energy ¯ux J E is colour coded on a logarithmic scale as a function of the logarithm of the ion energy, E, and the pitch angle a s : in each panel the pitch angle varies linearly from a s = 180°on the left, to a s = 0 (the downward ®eld-aligned direction) at the centre and then returns linearly to a s = 180°on the right.
In the ®rst panel of Fig. 7 (t s À t o 40 s) we see mainly the sp population which cannot be dierentiated from that on closed ®eld lines as the ion loss is only at such high energies that the ¯uxes could not be detected anyway.However, ions belonging to the ``halo'' of ri-sp ions, as seen in Fig. 4b, could be seen at low pitch angles by detectors if their one-count level was below (an unrealistically low) 10 A2 cm A2 s A1 sr A1 .This higherenergy population grows in ¯ux and descends in minimum energy as (t s At o ) increases giving a ``bowllike'' appearance on the spectrogram.By (t s À t o ) = 200 s, the ®rst magnetosheath ions can be seen at the lower pitch angles and lower energies of this bowl feature, after which the characteristic V-shaped feature of the cusp precipitation becomes clear.This  development of the form from a bowl to a V was noted by Menietti and Burch (1988).Note that at the lower edge of the ion Vs, all ions have come from the reconnection site because they have the lowest energy and thus longest ¯ight time T at that pitch angle a s .As a result, this edge is not dependent on the contour level, but this is certainly not true of the upper edge of the V. Thus the width of the V (in energy at any one pitch angle) depends on the sensitivity of the instrument, speci®cally the geometric factor and the one-count level.As a result, the extent of the source region inferred from the V will depend on the instrument.This point is addressed further in the next section.
Comparison of Fig. 7 with observations shows that the model reproduces well the observed ion Vs in this spectrogram format.For example, Fig. 7 can be compared with the second panel of Fig. 2 of Kremser et al. (1995).In making this comparison, it must be remembered that Fig. 7 has not been convolved with any instrument response characteristics and many of the features shown will be below the one-count level.In particular, note that the J E scale in Fig. 7 covers 12 decades, whereas the data presentation given by Kremser et al. (1995) covers only 3.7.For this reason, the low-¯ux features at the highest energies are not as clear in the data as they are in this model.Nevertheless, the data clearly reveal ions at higher energies than are seen equatorwards of the cusp on closed ®eld lines.The bottom panel of Fig. 2 of Kremser et al: (1995) is from a high-energy-ion instrument which detects ions of energy up to about 100 keV, as in Fig. 7.These high-energy ions are observed to share the same energy/time-ofobservation dispersion ramp as the cusp ions, as is also predicted in Fig. 7. Furthermore, from the ratios of the ¯uxes of dierent species, Kremser et al. (1995) ®nd that these higher-energy ions are of magnetospheric origin and suggest that they are generated by interaction with the magnetopause.This is con®rmed to be the mechanism in the modelling presented here.
Looking closely at the spin-angle distribution for (t s À t o ) = 500 s, it can be seen that a minimum is starting to form at zero pitch angle, with peaks at larger values.This is also seen in the data presented by Kremser et al. (1995) and represents the evolution towards upgoing, mirrored mantle ions, as discussed by Rosenbauer et al. (1975).
Note that in Fig. 4, 5 and 7, the magnetospheric CPS ions (sp) are always seen at energies below the time-of-¯ight cut-o energy (which is de®ned by t s À t o and so depends on the pitch angle and the time elapsed since reconnection), whereas the injected sheath ions and energised magnetospheric ions are simultaneously present above this cut-o energy.This predicted continuation of sp ions at energies below the injected magnetosheath ions is a feature of all observations of dispersed LLBL and cusp ions, at both middle and low altitudes.

Injection locations of observed cusp and LLBL ions
It is instructive to return to the debate about where the precipitating ions seen in the cusp region were injected across the magnetopause.As was discussed in the introduction, Menietti and Burch (1988) used the ion Vs modelled in Fig. 7 to derive a spread of source locations of about 1 R E , whereas Lockwood and Smith (1993) argued that the spread of ion energies seen in low-  altitude data reveal ions injected over regions 10±20 R E wide.The modelling by Onsager et al. (1993) gave source extents which were somewhere between the two.Lockwood and Smith (1994) and Lockwood (1997) have discussed several reasons for these discrepancies.Menietti and Burch argued from the similarity of their estimate to the known dimensions of magnetopause ¯ux transfer events (FTEs) that the cusp precipitation originated in bursts of reconnection; however, Lockwood and Smith (1994) point out that because the ions are largely frozen onto each newly opened ®eld line, the distribution function seen on any one ®eld line (and thus the width of the cusp ion V) depends on the sequence of ions injected onto that ®eld line as a function of time, not on what happens to other ®eld lines around it.Thus the precipitation is determined by the evolution of each newly opened ®eld line after reconnection and is independent of the reconnection rate with which it was opened.Lockwood and Smith also show that the reconnection-rate variations, rather than restricting the width of the dispersion signature, cause one of a number of step-like features, as recently modelled in spectrogram format by Lockwood and Davis (1996).
Figure 8 shows the dierential energy ¯ux spectrum J E (E) at three (t s À t o ) for ®eld-aligned ions (a s = 0).From the preceding discussion, we class that for (t s À t o ) = 200 s as LLBL, that for (t s À t o ) = 300 s as in the cusp near the cusp/LLBL boundary and that for (t s À t o ) = 500 s as also being in the cusp but near the cusp/mantle border.In all three cases, the time-of-¯ight cut-o can be clearly de®ned: below this cut-o, the remnant sp ions are seen.The dotted line shows one possible one-count level of the instrument (J E = 10 7 cm A2 s A1 sr A1 ), below which no ions would be detected.One can see that the width (energy range) of the population detected depends on not only on (t s À t o ), but also on the one-count level: the width will decrease/ increase if the threshold J E is higher/lower.A corresponding decrease/increase in the energy width of the magnetosheath-like ion precipitation feature, as detected by any one instrument, would also be caused by a fall/rise in the solar-wind density, relative to that input into the model to generate the results shown here (see Table 1).
The velocity ®lter eect of the convection electric ®eld means that a spread in ion energies at the satellite (at any one pitch angle) corresponds to a range of source locations (Lockwood and Smith, 1993).The range of detectable energies will depend on the instrument geometric factor, which sets the one-count level (note that because counts are proportional to dierential energy ¯ux J E for any one dierential ion detector, the threshold (one-count level) J E is independent of energy).The spread of source locations of detected ions will increase with higher instrument sensitivity (lower onecount levels).This eect is investigated here in Fig. 9.The ®gure shows the distances d n from the reconnection site to the magnetopause source points (P n ) of those ions in Fig. 8 whose ¯ux equals the one-count level.These distances are given as a function of the value of that onecount level.For comparison, the dotted lines in Fig. 8  and 9 give one example of a threshold J E (one-count level) of 10 7 cm A2 s A1 sr A1 .The three curves shown are for the same three values of (t s À t o ) as in Fig. 8 and are plotted using the same types of line.The curves in Fig. 9 mark the boundaries of the source regions of the detected ions and so de®ne their apparent extent, as a function of the instrument one-count level.
If we consider the LLBL spectrum in Fig. 8 (t s À t o 200 s), we see that the largest J E (E, a s = 0) is about 10 8.5 cm A2 s A1 sr A1 and that this is at the lower cut-o energy, E ic = 2.93 ´10 3 eV (log 10 (E ic ) = 3.47).These ions have the largest T which, at a given (t s À t o ), means that they have the minimum t n [Eq.( 3)] which is zero, i.e. these ions were injected at the reconnection site (at d n = 0).[As discussed by Lockwood (1995a, b), there is a caveat which needs placing on this argument when the time-of-¯ight cut-o energy falls below the minimum energy injected at the X-line, which is mV ex 2 / 2, where V ex is the value of V e in the immediate vicinity of the X-line; however, this only occurs at larger (t s À t o ) than in Fig. 8 and 9)].As we lower the threshold in J E , Fig. 8 shows that the low-energy limit of the detected ions does not change and these ions still all come from the X-line.However, ions are detected at increasingly higher energies and these ions thus have lower T and higher t n and d n .This can be seen in Fig. 9.For the lowest J E threshold shown (10 5.7 cm A2 s A1 sr A1 ), the upper d n has grown to 4 R E , i.e. the source region has expanded to cover a region on the low-latitude magnetopause 4 R E wide from the reconnection site.At this (t s À t o ) of 200 s, the ®eld line that is being observed actually threads the boundary at a distance of 8.7 R E from the X-line: thus if we could detect sheath ions of zero ¯ight time (in®nite energy) this would be the limit of the source region.This means that at this elapsed time since reconnection, the satellite is magnetically connected to the magnetic cusp region of the mid-latitude magnetopause (8.7 R E from the X-line) but the precipitation at the satellite is classed as LLBL and was injected by ¯owing along open ®eld lines across the lowlatitude magnetopause, within a few R E of the reconnection site.For (t s À t o ) = 300 s, a very similar situation applies, but the source region is wider.The relatively ¯at top to the spectrum shown in Fig. 8 means that as we lower the threshold, the source region widens very rapidly after the ®rst detection of the ions.This can be seen in Fig. 9, such that at J E of 10 8.5 cm A2 s A1 sr A1 , the source region is 3 R E wide, and this grows to over 8 R E for 10 5.7 cm A2 s A1 sr A1 .At this (t s À t o ), the ®eld line threads the boundary at 15.2 R E from the X-line.
However, for the case of (t s À t o ) = 500 s, the behaviour is signi®cantly dierent.Fig. 8 shows that the peak J E (E, a s = 0) is not at the lower cut-o, but is at a higher E of around 1 keV.Thus, as we lower the detection threshold, the ®rst ions we see are not from the reconnection site in this case, rather they come from near d n = 7 R E (Fig. 9).The extent of this source region spreads rapidly about this d n until at a threshold J E of 10 7.9 cm A2 s A1 sr A1 , the lower cut-o ions are seen for the ®rst time: the source then extends from the X-line to d n of 14 R E .At the left of the plot, this source range has increased to 0±18 R E .These values are similar to those modelled by Lockwood (1995a) and are comparable to those inferred from observations by Lockwood and Smith (1993).At this (t s À t o ), the ®eld line threads the boundary 39.5 R E .Notice, therefore, that the precipitation is classed as cusp on a ®eld line that actually threads the magnetopause so far down the tail that it would be called part of the mantle.
The preceding discussion has been in terms of the eects of instrument sensitivity on the location and extent of the source of the ions seen.The same sort of eects will be present if the dierential energy ¯ux of the ions is used to de®ne the precipitation classi®cation.In other words, if one de®nes cusp as requiring ¯ux values over a suciently high threshold, one will inevitably select those ions that come from a restricted region around the magnetic cusp.

Summary and conclusions
The open magnetosphere model can explain the energy/ pitch-angle dispersion of injected magnetosheath ions seen at middle and low altitudes, as well as the energy/ observation-time dispersion.
The ion distribution function evolves as a function of time elapsed since reconnection in a way that explains the BPS and LLBL precipitations as being on open ®eld lines, in addition to the cusp, mantle and polar cap.The implications of this have been discussed by Lockwood (1997); speci®cally, some outstanding anomalies concerning the location of the precipitations with respect to the pattern of convection can be resolved.
The distribution seen in the LLBL is a mixture of magnetosheath ions and magnetospheric ions.However, at energies below the time-of-¯ight cut-o the magnetospheric ions are that part of the pre-existing closed-®eld-line (CPS) population which has not yet had time to be in¯uenced by the fact that the ®eld line has been opened.At energies above the magnetosheath-like ions, ions of magnetospheric origin are seen.However, these are not part of the same population as is seen simultaneously at lower energies.Instead, these ions have been re¯ected o the magnetopause.The re¯ection coecient employed here has a constant value of 0.4: this may be rather large, although it is consistent with the magnetopause observations of Fuselier et al. (1991).However, these ions would be detected even if this re¯ection is much weaker, because the ions are accelerated when they are re¯ected.The slope of the distribution function f(E) is very steep at the hot tail of the distribution and thus the rise in energy E tends to increase the phase space densities of the re¯ected ions: they can be detected at high energies even if the re¯ection coecient is very low.There is, however, no reason why these two opposing eects should exactly counteract each other and make the re-sp tail an exact continuation of the remnant sp population.This can be seen to be true for many examples of boundary-layer spectra (e.g.Fig. 1 of Woch and Lundin, 1992).
The higher-energy, magnetosphere-like population which de®nes the LLBL (the re-sp) decays away as the ®eld line evolves.This is because the ion acceleration turns to a deceleration as the ®eld line straightens (plus, in reality, the re¯ection coecient may decrease).In addition, the source sp population is progressively lost by interaction with the boundary.At the same time, the time-of-¯ight cut-o decreases with time elapsed since reconnection, allowing the lower-energy sheath ions to arrive.In this way the LLBL precipitation evolves into cusp as is often observed (e.g.Newell et al., 1991a;Woch and Lundin, 1992).Lockwood (1997) has pointed out that time-of-¯ight dispersion gives several severe conceptual diculties if one thinks of the LLBL/cusp boundary as at the closed/open ®eld line boundary.
The precipitation classed as LLBL at middle and low altitudes is on ®eld lines that are magnetically connected to the mid-latitude magnetopause, i.e. which thread the magnetic cusp.Similarly, the precipitation classed as cusp is on ®eld lines that thread the magnetopause in the high-latitude boundary layer (or mantle) and that classed as BPS is magnetically connected to the lowlatitude magnetopause.This is to be expected because the ®eld lines evolve towards the tail during the ¯ight time of the ions.
The extent of the source region of ions of a given classi®cation depends on the threshold set to de®ne that precipitation class and, in some cases, may be set by the instrument's sensitivity.A suciently high threshold de®nition of cusp particles restricts the source region to the vicinity of the magnetic cusp.However, a lower threshold shows that the particles in fact originate from the entire dayside magnetopause, at all latitudes down to the reconnection site.The threshold also in¯uences the extent of the inferred source of LLBL ions.However, LLBL ions always arise from adjacent to the reconnection site.

Fig. 1 .
Fig.1.Schematic of the open LLBL emanating from a magnetopause reconnection site, X.The magnetic separatrices s are the locations of the ®eld lines at time elapsed since reconnection (t s À t o ) = 0: newly opened ®eld lines are also shown for two later (t s À t o ).The exterior and interior AlfveÂ n waves (launched by the reconnection site into the magnetosheath and magnetosphere in¯ow regions) are e and i, along which the kinks in the newly opened ®eld lines evolve at speeds V e and V i .The bulk motions (in the Earth's frame of reference) of various ion populations discussed in the text are shown with broad arrows.

Fig. 2 .
Fig. 2. Ion distribution functions in the openLLBL in the immediate vicinity of the reconnection X-line (d n » 0, t n » 0).The parallel velocity is positive towards the Earth and is in the Earth's frame of reference: (top) between the interior wave (i in Fig.1) and the ion edge which is closer to the separatrix s; (bottom) between the exterior and interior waves (e and i)

Fig. 3 .
Fig.3.The spectra of ®eld-aligned ions (pitch angle, a m = 0) in the LLBL at the magnetopause, corresponding to the top part of Fig.2.The various component ion populations, t-sh, sp, ri-sp and re-sp are discussed in the text.This plot gives the dierential energy ¯ux J E as a function of energy E on a log-log scale

Fig. 4a .
Fig.4a.Ion ¯ight time from a point P n on the magnetopause to the altitude of satellite, T, colour coded (on a logarithmic scale) as a function of the ®eld-parallel and ®eld-perpendicular ion velocities at the satellite.In this example, P n is at the (subsolar) reconnection site

Fig. 5 .
Fig.5.Dierential number ¯ux (intensity) ion spectra at two elapsed times since reconnection.The dashed line is for (t s À t o ) of 200 s and is typical of spectra in the LLBL; the solid line is for (t s À t o ) of 500 s and is typical of spectra in the cusp.The dierential number ¯ux J is plotted as a function of energy E on a log-log scale.

Fig. 6 .
Fig. 6.The intensity spectrum for (t s À t o ) = 200 s, as shown in Fig. 5 (dashed line), compared with two Maxwellian sp populations: A is the input to the model with temperature T sp = 1.4 ´10 7 K and density, N sp = 4 ´10 5 m A3 ; B is a ®t to the visible parts of the sp and re-sp populations with temperature T sp = 2.5 ´10 7 K and density, N sp = 8 ´10 5 m A3 .The dominant ion population in the three parts of the total spectrum are marked (sp at low energies, t-sh at middle energies and re-sp at the high energies).
Fig.7.Pitch-angle/energy spectra obtained by spins of a mid-altitude satellite at eight elapsed times since reconnection (t s À t o ), as given at the top of each case.The dierential energy ¯ux is colour contoured (on a log scale) as a function of pitch angle and the logarithm of energy: in each panel, the pitch angle a s varies from 180°to 0 (downward) and back to 180°.

Fig. 8 .
Fig.8.Dierential energy ¯ux spectra of ®eld-aligned ions (a s = 0) detected at the mid-altitude satellite at three elapsed times since reconnection, (t s À t o ): 200 s (LLBL), 300 s (near the LLBL-cusp border) and 500 s (cusp).The dotted line shows an example threshold (imposed as a classi®cation criterion or by the instrument one-count level) of J E = 10 7 cm A2 s A1 sr A1

Fig. 9 .
Fig.9.The distance d n (of the magnetopause source point from the Xline) for ions whose ¯ux J E is at a threshold level, as a function of that level and for the three elapsed times since reconnection (t s À t o ) used in Fig.8

Table 1 .
Input parameters to the model