A model of the Earth's distant bow shock

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. A12, PAGES 26,927-26,941, DECEMBER 1, 1997 A model of the Earth's distant bow shock L. Bennett, M. G. Kivelson, and K. K. Khurana Institute of Geophysics and Planetary Physics, University of California, Los Angeles L. A. Frank and W. R. Paterson Department of Physics and Astronomy, University of Iowa, Iowa City Abstract. We present a new global model of the Earth's bow shock that is parametrized by the solar wind conditions. We begin with a conic section base model taken to be correct for average solar wind conditions. Then we apply modifications to the base model, based on physical arguments, to account for the changes in the size and shape of the bow shock caused by changes in the prevailing solar wind dynamic pressure, Alfv n and sonic Mach numbers, and interplanetary magnetic field orientation. We show that our model matches the location and timing of shock crossings observed, at large downfall distances, by the Galileo spacecraft during December 1990 and December 1992 and by the Pioneer 7 spacecraft during September Magnetic field and plasma moments in a shock normal coordinate system change across the model shock surface as required by conservation laws. 1. Introduction Models of the large-scale shape and structure of the bow shock have been based on observations made within 100 Rs of the Earth, primarily on data taken within 50 Rs, and heavily biased toward the dayside. These studies have characterized the shock surface and established the effects of changing solar wind conditions on the near-earth portion of the shock. The second Galileo flyby of Earth provided an opportunity to study the more distant Earth bow shock and to understand the effects of changing solar wind plasma parameters on this portion of the shock. This study presents a new model of the bow shock which accounts for these effects and matches the location of the crossings observed by Galileo. The study of Earth's bow shock has a long history. We briefly review the work pertinent to this study. Fairfield [1971] reviewed the earliest studies of the position of the bow shock. He also modeled the bow shock, finding the hyperbohc model that provided the best fit to 389 near-equatorial crossings measured by Imp 1-4, Explorer 33, and Explorer 35 which extended from the nose of the shock to 60 Rs downtail. This model Although Dryer and Heckman [1967] and Slavin et al. [1984] predicted the position of the distant, downallowed for a dawn-dusk asymmetry of the bow shock. t.ail bow shock, all these studies emphasized the fit of The Fairfield curve has remained a good representation the model to the near-earth region. Greenstadt et al. of the shock position as additional spacecraft obser- [1990] identified shock crossings data taken by the vations have been added to the database [Slavin and ISEE 3/ICE spacecraft in 1982 and 1983 as the space- Holzer, 1981; Slavin et al., 1984; Greenstadt et al., 1990]. Gasdynamic simulations [$preiter et al., 1966; Copyright 1997 by the American Geophysical Union. Paper number 97JA /97/97JA ,927 Dryer and Heckman, 1967; Spreiter and Stahara, 1980, and references therein] have been shown to agree well with the observations from the nose to about 60 Rs downtail [Fairfield, 1971; Slavin et al., 1984] but, beyond that, the observations place the shock farther out than the simulations predict. Recently, the shape and position of the shock in front of the dayside terminator have received additional attention [e.g., Farris et al., 1991; Farris and Russell 1994; Cairns et al., 1995; Cairns and Lyon, 1995; Peredo et al., 1995]. Farris and Russell [1994] and Cairns et al. [1995] showed that the standoff distance of the shock is dramatically changed for low magnetosonic Mach numbers. Additionally Cairns and Lyon [1996] have shown that the nose position of the shock is influenced by the interplanetary magnetic field direction (IMF), particularly at low Mach numbers. Peredo et al. [1995] added observations to those used by Fairfield [1971] and Slavin and Holzer [1981] and used the larger database to model the shock with a three-dimensional, ß second-order equation whose coefficients are functions of the Alfv n Mach number. craft's trajectory was being altered for its encounter with comet Giacobini-Zinner. These observations were taken as far as 110 Rs downtail and were shown to be consistent with a version of the Fairfield [1971] model that was taken to be symmetric about the tail axis. we shall refer to the symmetric model as the Fair-

2 26,928 BENNETT ET AL.: MODEL OF THE EARTH'S I)ISTANT BOW SHOCK field/greenstadt model or the F/G model. This model is a best fit to a large number of observations and is, therefore, a representation of the shock surface for average solar wind conditions. However, it does not describe how the shock changes for changing solar wind conditions. Our new model describes the shock surface for any solar wind conditions by starting with the F/G model as a base and modifying it to account for the prevailing solar wind dynamic pressure, Mach number, and IMF direction. The new model accurately represents the locations of the bow shock observed by the Galileo spacecraft at a range of distances from 355 to 120 Rz antisunward of Earth. We also relate the model to shock crossings observed by Pioneer 7. During much of the inbound portion of Galileo's second flyby of Earth, its trajectory passed near the bow shock. Multiple crossings occurred as the shock moved in and out across the spacecraft. Our model describes how the structure of the shock responds to changes in the solar wind properties, and this enables us to match the timing of each crossing seen by Galileo. A similar model was developed by Khurana and Kivel- son [1994] for analysis of the bow shock of Venus. The first models of the Cytherean bow shock were based on data from Venera 9 and 10 [Verigin et al., 1978] and the Pioneer Venus Orbiter [e.g., Russell et al., 1979; Slavin et al., 1980] taken 2Rv downtail. Smirnov et al. [1980] and Slavin et al. [1984] added 11 Venera 9 and 10 crossings, between 9 and 15 Rv downtail, and obtained fits that were consistent with those based only on the data taken closer to the planet, but the scatter of the data about the curve was quite large. These fits to the data did not explicitly depend on the solar wind conditions. However, by binning data according to the value of the solar wind plasma parameters, the effects of dynamic pressure, Mach numbers, and solar cycle on the bow shock have been quantified [e.g., Slavin et al., 1980; Russell et al., 1988; Zhang et al., 1990]. Khurana and Kivelson [1994] developed a shock surface model that depended on the Alfv n Mach number, the sonic Mach number, and the direction of the IMF. During the Galileo flyby of Venus Pioneer Venus observations showed that the solar wind plasma conditions were constant while Galileo measurements showed that the IMF rotated at constant magnitude. Khurana and Kivelson [1994] used their model to interpret the timings of 12 shock crossings observed by the Galileo spacecraft between 5 Rv and 10 Rv downtail as it skimmed along the Cytherean bow shock. En route to Jupiter, the Galileo spacecraft encountered the Earth twice. On the second encounter the spacecraft was returning to Earth from the asteroid belt. It approached Earth from the antisolar, downtail direction northward of the ecliptic plane on the duskward side of the magnetotail. Closest approach was at 1509 UT on December 8, Shortly after 0800 UT on December 5, 1992, Galileo began encoun- tering the Earth bow shock at a distance of over 350 Rz downtail of Earth. As Galileo moved earthward, it encountered the bow shock several times, with the last unambiguous crossing at 1150 UT on December 7, 1992 at x _ 117 Rz downtail. Initial reports on the shock encounters were based on the magnetometer data [Kivelson et al., 1995]. This study reexamines the Galileo shock encounters using data from both the magnetometer [Kivelson et al., 1992] and the plasma investigation [Frank eta!., 1992], which allow us to evaluate the relevant solar wind parameters. Using this information, we develop a new model of the bow shock that depends on the solar wind conditions. We show that our model provides the timing of the observed shock crossings quite well. Our model also accounts for the locations at which the distant bow shock was encountered by Galileo on its first pass by Earth [Kivelson et al., 1993] and by Pioneer 7 on its outbound trajectory[villante, 1976]. 2. Shock Data In this section we use magnetic field and plasma data to identify the shock crossings in the data acquired by Galileo during its second pass of Earth in December, In Figure i we have plotted the portions of the data containing shock encounters. The upper panels show the total electron velocity, the electron density, and the total magnetic field for the period from 0730 UT to 1800 UT December 5, The lower panels plot the same quantities for the period from 1900 UT December 6, 1992 to 1500 UT December 7, The lighter shaded regions mark times when the spacecraft was in the magnetosheath. There were no shock crossings during the interval not plotted. Disturbances in the solar wind made it difficult to identify shock crossings during several periods which are marked in Figure i by darker shading. For example, between 1545 UT and 1650 UT December 5, 1992, the magnetic field strength decreased in average value and showed rapid fluctuations of considerable amplitude before resuming the slow changes associated with a relatively steady flow. The density also fluctuated about an average value higher than in the solar wind prior to this period. We have interpreted these signatures as evidence of a change of solar wind conditions, most notably an increase in density, which displaced the bow shock inward of the spacecraft. The density slowly increased throughout December 7, 1992, reaching a nomi- nal value of over 50 particles/cm 3 with several intervals of much larger values. It is likely that there were several shock encounters between 1200 UT and 1500 UT on December 7, However, the extreme solar wind conditions and large fluctuations in all quantities make it very difficult to identify the shock crossings. The position of Galileo at the time of each identified shock encounter is shown in Figure 2. The coordinate system is cylindrical abetrated GSE. The values

3 BENNETT ET AL.: MODEL OF THE EARTH'S DISTANT BOW SHOCK 26, ' : i. ;sui "i::.? Z::;... i..?!?' :.;.!'.?.. :!:'"' : V i! i i... ' ;: : ::::"' " ' ": ' 1992-Dec-5 DOY: O.0 t :30 09:30 11:30 13:30 15: Time 17: Dec-6 DOY: :00 21:00 23:00 01:00 03:00 05:00 07:00 09:00 11:00 13:00 Time Figure 1. Data for the portions of Galileo'second flyby of Earth containing clear shock encounters. The top panelshow the total electron velocity (kin/s), electron density (particles/cm3), and total magnetic field (nanoteslas) for the interval 0730 to 1800 UT on December 5, The bottom panel show the same quantities for the period from 1900 UT December 6, 1992, to 1500 UT December 7, The unshaded regions of the graph are times when the spacecraft was in the solar wind. The lighter shaded portions are times when Galileo was in the magnetosheath. The darkest shaded regions are periods when disturbed solar wind conditions made the accurate identification of shocks difficult. No shock crossings were identified during the unplotted intervals ,--, 200 ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I of the components of the solar wind velocity used in the transformation into the abetrated coordinate system are given in Table 1. In the transformation we are including the effects of the motion of the Earth around the Sun and of the transverse components of the solar wind velocity. Fairfield [1971] and Gree stadt et al. [1990] specified a 4 ø rotation, which is the aberration corresponding to the orbital velocity of the Earth if the solar wind velocity is purely radial at 430 km/s. The shock crossings observed by Galileo are well organized into three groups (see Figure I and discussion below). 1 o0 i i i I i i i i i i i i i O0 0 x [R d Figure 2. Positions of Galileo shock encounters compared with the F/G shock model. The solid circles are the position of Galileo in a cylindrical, abetrated GSE coordinate system at each shock crossing. The solid line is the F/G model. The data used to rotate into the abetrated coordinate system are given in Table 1.

4 26,930 BENNETT ET AL.: MODEL OF THE EARTH'S DISTANT BOW SHOCK Table 1. Data Used to Transform Observations and Models Into the Abetrated GSE Coordinate System F/G model 430 Galileo, Dec. 5, Galileo, Dec. 6-7, Galileo, Dec. 7, Galileo, Dec. 5, Pioneer 7, Sept. 3, Pioneer 7, Sept. 5, Pioneer 7, Sept. 6, NA indicates the data were unavailable. V, km/s V, km/s Vz, km/s Abberation Angle 30 NA 4.0 ø ø ø ø ø 30 NA 3.0 ø 3O NA 3.6 ø -10 NA -1.2 ø The figure also shows the F/G model of the bow shock, which is described by p2 = ax 2 _ bx + c (1) where a = 0.04, b = 45.3, and c = for x in Rz. The F/G model may also be represented as L r e cos 0 (2) lar wind dynamic pressure Psw = 2.5 npa, sonic Mach nmnber Ms = 7.2, Alfv n Mach nmnber MA = 9.4, and an IMF spiral angle of 45 ø. We seek a model that can predict the structure of the bow shock for any solar wind conditions. By starting with a model that is correct for average solar wind conditions and modifying it for the prevailing parameters we obtain a new model using only a few observations of the distant bow shock. Note that while the base model is symmetric about the tail axis, the physically motivated modifications to the model described below introduce asymmetry about the tail axis. The dynamic pressure in the solar wind is the primary quantity governing the scale size of the magnetosphere. As the solar wind dynamic pressure increases, where = = - + both the distance to the nose of the magnetopause and the diameter of the magnetosphere decrease, with both quantities varying as p, /6 sw [e.g., $preiter et al., 1966; Fairfield, 1971; Holzer and Slavin, 1978]. Since the magis the distance from the focus, at (x0, 0, 0), to a point, netosphere is the obstacle that deflects the solar wind, (x, y, z), on the curve, e is the eccentricity, L is the semithe characteristic lengths in models of the bow shock latus rectum, and 0 is the angle, at the focus, from the must also scale like P, /6 SW ß major axis to the point on the hyperbola. Por the P/G model, x0 = 3.5 Rz, e = 1.02, and L = The posi- $ibeck et al. [1991] and Petrinec and Russell [1993] showed that the position of the magnetopause is also tions of the bow shock encounters observed by Galileo dependent on IMP Bz with the subsolar point moving are not well estimated by this model. The crossings of 1 Rz earthward for each 7.4 nt of southward IMP. December 6 and 7 are all inside the P/G model, and the earlier, very distant crossings of December 5 are Petrinec and Russell [1993] found no dependence of magnetopause position on northward IMP, while Sibeck all nearly 30 Rz outside the curve. However, the P/G model was a best fit to a large number of observations et al. [1991] found that the subsolar point moved outward for increasing northward IMP. The scale size of and is therefore a model of the shock position for averthe magnetopause in the dawn meridian was shown to age solar wind conditions and should not be expected be relatively insensitive to IMP B z. The dependence to fit individual crossings. of magnetopause position on the IMP Bz has been explained by erosion of the dayside magnetopause due to 3. New Shock Surface Model: Physical reconnection [Aubry et al., 1970; Sibeck et al., 1991; Background Petrinec and Russell, 1993] and by the reduction of the dayside magnetic field caused by the increase of region 1 Birkeland currents during periods of southward IMP In this section, we describe a new model that extends the F/G model and is consistent with the Galileo [Hill and Rassbach, 1975; Maltsev and Lyatsky, 1975; observations. We start with a base model, the P/G Sibeck et al., 1991]. model, of the bow shock which is assumed to apply to Spreiter et al. [1066], Slavin et al. [1084], Khurana nominal, average solar wind conditions. Por average and Kivelson [1994], and others have shown that the solar wind conditions, Slavin and Holzer [1981] give to- position of the bow shock in the asymptotic limit is tal solar wind velocity Vsw = 430 km/s, proton density governed by the magnetosonic Mach number, Mms = np= 7 cm -3, total magnetic field Btot = 6 nt, so- Vsw/Vms, where Vsw is the solar wind velocity and Vms is the fast magnetosonic speed. In the asymptotic limit, the shock surface is a cone, called the Mach cone, with angular half width -- sin- (l/ms/vsw) = sin- (1/Mms) (4)

5 The fast magnetosonic speed is given by BENNETT ET AL.: MODEL OF THE EARTH'S DISTANT BOW SHOCK 26, l[(v 2 v"(v - 1 where va is the Aftvan speed, cs is the sound speed, and 8s. is the angle between the magnetic field vector and the shock normal in the asymptotic limit. In this limit, the direction of propagation of the fast magnetosonic wave, which steepens to become the shock, will lie along the local shock normal at the point where the wave intersects the shock. In the asymptotic limit, is constant along the shock surface in the x direction; however, it changes with the angle about the tail axis. Thus, Vms and Mms vary with the angle c about the tail axis in the plane perpendicular to the tail axis and the Mach cone is asymmetric, with the half width p being a function of c. Here c is defined such that Z - y where Y and Z are the coordinates in the aberrated GSE system. We refer to the magnetosonic velocity or Mach number at a particular angle as the "local" velocity or Mach number. So, as VA, cs, or the IMF direction changes, the shape of the shock will change and the shock will move in or out from the tail axis. Another Mach number effect that has received con- siderable interest of late is the sunward motion of the nose of the bow shock as the solar wind Mach num- ber decreases. This topic has been discussed extensively in the literature and remains somewhat controversial [e.g., Farris and Russell, 1994; Cairns et al., 1995; Cairns and Lyon 1995; Cairns et al., 1996; Russell and Petrinec, 1996]. For aerodynamics Seiff [1962] and for gasdynamics Spreiter et al. [1966] have shown that the distance of the shock from the nose of the ob- stacle A normalized by the radius of the obstacle D (the distance from the center of the Earth to the nose of the magnetopause for the Earth bow shock) is related to the density jump across the shock by A D = (?) for large (> 5) Mach numbers. The hydro dynamic shock jump conditions imply where M is the upstream sonic Mach number and is the adiabatic index. Then the distance to the nose p2 This relation predicts a maximum shock standoff distance of 2.1 times the obstacle size. However, observations at Venus by Russell and Zhan# [1992] and at Earth by Cairns et al. [1995] have shown that the bow shock must stand farther out for very low Mach numbers. Farris and Russell [1994] have proposed a form of (9) that allows the bow shock to move to infinity as the Mach number goes to 1: 3 n -- 3 nm (7-1) M + 2 ) p (7 + 1)(M - 1) (10) where M, in this case, is the upstream magnetosonic Mach number. An alternative formula explaining the behavior of the nose of the bow shock at low Mach numbers has been proposed by Cairns and Lyon [1995] based on three-dimensional, ideal MHD simulations. Their form is x, - x= p (3.4 \ P2 p + 0.4) (11) where the density jump is calculated directly from their MHD simulation. This theory predicts a maximum standoff distance for the shock of 3.8 x, p as the Mach number goes to 1. As there have been few observations of the bow shock at Mach numbers smaller than 3, the actual behavior of the shock in these situations is unclear. Below we show how either of these theories, or any theory that gives x as a multiple (n) of xn, p, where n is any function of the upstream and downstream plasma parameters, can be incorporated into our model. 4. New Shock Surface Model: Implementation We now describe how each of the above physical effects is incorporated into our model. Taking 8-0 in (2), y = z = 0, and L x,-x0+ l+e (12) The standoff distance of the nose of the bow shock from the Earth is x and the transverse scale size of the shock is given by the semi-latus rectum. We assume that the bow shock responds to changes in the solar wind dynamic pressure as the magnetopause does, so that all length quantities in (12) vary as D /6 sw, and X, l -_ x0,a,e xo1 = La e L1 -_ ( p ppvp,, V,, ) 1/ ( x.,.. (13) where p is the mass density, V is the total solar wind velocity, the subscript ave indicates the average conditions given above, the subscript 1 indicates the updated values, and the subscript p indicates the current, pre- of the bow shock from the center of the Earth, xn, is related to the distance to the nose of the magnetopause from the center of the Earth, Xnmp, by vailing conditions. We assume e is unchanged and the shape of the shock surface is unaffected by changes in (v + the solar wind pressure.

6 26,932 BENNETT ET AL.: MODEL OF THE EAP, TIt'S DISTANT BOW SHOCK Increasing southward IMF Bz causes the magnetopause to move earthward (see above). We assume the bow shock moves earthward through the same distance as the nose of the magnetopause without changing shape or size. This is done in the model by decreasing xn by I Rz for each 7.5 nt of southward IMF so that Xn2 = Xn -- Bz/7.5 Bz <0 x 2 = x Bz _ 0 (14) keeping L and e constant which implies that x0 will also decrease by the same amount (see equation (12)). This effect is negligible far downstream of Earth, with a I RE earthward shift of the nose corresponding to a shift in cylindrical radius of 0.3 Rz at x = -100 Rz and 0.25 RE at x RE. The sunward motion of the nose of the bow shock at low Mach numbers can be incorporated into our model by taking X n ---- I Xnm p (15) where n is a function of the plasma parameters. In the gasdynamic solution [e.g., Spreiter et al., 1966] n is n, -- I + 1.1(h'- + 1)M where M is the upstream sonic Mach number. For the Farris and Russell [1994] formulation ny (7 + 1)(M - 1) (17) with M the upstream magnetosonic Mach number, and for the Cairns and Lyon [1995] formulation n p (18) Then assuming that the shape of the magnetopause remains unchanged, we may take P Xn 3 -- Xn 2 np (19) I ave for whichever form of n is desired. In order to displace the bow shock without changing its shape, we recalculate the focus position keeping the semi-latus rectum and the eccentricity constant. The IMF dependence of the nose position of the shock described in Cairns and Lyon [1996] had not been included in our model but could be incorporated in a similar fashion to the modifications applied in this step. The equation for the updated cylindrical radius of the bow shock, p is now V/ Z, -( ) ( 0) P -- I + e cos( ) - x03 Calculating the effect on the bow shock of changes in the local magnetosonic Mach number requires knowl- Figure 3. Construction of the asymptotic bow shock adapted from Spreiter et al. [1966]. The dashed circle represents the distance that waves, emanating from a point source at the vertex of the cone, would be convected downstream by the solar wind at velocity Vsw. The solid curve is a Friedrichs diagram of phase speed Vms, which is the solution of (5), and represents the distance that waves move through the solar wind in its rest frame. The intersection of the two curves defines points on the shock surface. edge of the magnetosonic velocity as a function of which can be calculated by following the prescription of Khurana and Kivelson [1994](see also Spreiter et al. [1966]). In the asymptotic limit it is assumed that the waves generated by the interaction of a supersonic flow with a point obstacle are confined to a conic region which is the Mach cone of the shock. The wave vec- tor of the waves which steepen to form the shock will be parallel to the local shock normal and so the Mach cone can be specified by finding two perpendicular lines such that one (representing a line on the shock surface) passes through the point obstacle and the other (representing the wave vector) passes through the center of the wave front surface. Such a construction is shown in Figure 3. The diameter of the dashed circle represents the distance the center of the wave front surface is convected downstream by the solar wind. The solid curve is the wave front surface which is the solution of (5). Since lines drawn from a point on a circle to the two ends of a diameter of the circle are normal to each other, the intersection of the two curves satisfies the above criteria and represents points on the shock surface. Extending this construction into three dimensions we obtain the cross section of the asymptotic bow shock and the Mach cone. For a given c, the local at a point on the shock can be calculated by from the system of equations, Vz 2-1[ + 4) + v + c]) - 4v c]

7 BENNETT ET AI..' MODIiL OI: TI I1' I.œARTI I'S DISTANT BOW S}IOCK 26,933 tan(a)- v_ (22) (23) The coordinate system used is abetrated GSE. With the local magnetosonic velocity known the model can be modified further to include the dependence of the flaring of the shock on the local Mach number using a method taken from Khurana and Kivelson [1994] which we will call the KK94 method. (An alternative method is presented in the appendix.) Since the solution for Vms depends on angle around the tail axis, it is this modification that introduces the asymmetries in the tail cross section. We begin by calculating the local Mach cone angle for the prevailing conditions, pp, and for the average conditions we have assumed for the base model, Pb. Then, the cylindrical radius of the Mach cone for the base model (see Figure 4) can be expressed as = - ) + p0 where P0 is the cylindrical radius of the asymptote at the nose of the shock determined by the detailed physics of the interaction of the solar wind with the nose of the magnetopause. Here base model refers to the model already modified by the previously discussed effects. We 8O 6O p 40 2o o - 12o Model for prevailing conditions Asymptotic model for prevailing conditions Ap Figure 4. Illustration of how the KK94 method is used to modify the base model. The height of the asymptote at the nose of the shock is P0. The thinner lines are for the base model. The thicker lines are for the model for the prevailing Math number, which in this case is lower than the base model's. The distance between the asymptotic models (dashed lines) is added to the base model (thinner solid line) at each x x assume that the change in the flaring of the shock will not change P0 or xn. The cylindrical radius of the Mach cone for the prevailing conditions is and the displacement of the shock surface from the base model, for any a:, is Ap -- pp -- pb = - - where % and po are found from (4) for prevailing and average Mms, respectively. The position of the shock in the modified model is then Pshock -- Pl q- Ap (28) 5. New Shock Surface Model' Comparisons With Observations The parameters of the model are fully defined in terms of the solar wind properties. Figure 1 suggests that the shocks observed on the Galileo flyby of Earth in December 1992 can be split into three groups based on relatively constant prevailing solar wind plasma conditions: from 0730 to 1800 UT December 5, 1992, from 1900 UT December 6, 1992, to 0200 UT December 7, 1992, and from 0500 to 1200 UT December 7, The magnetometer and plasma instruments on the Galileo spacecraft provided all the parameters necessary to specify the solar wind conditions except the proton temperature and the abundance and temperature of helium. For the proton temperature we have adopted the results of Newbury [1996] and J. A. Newbury et al. (submitted to J. Geophys. Res., 1997) who investigated Te and T at 1 AU in the solar wind by analyzing 18 months of continuous ISEE 3 data. They give Tv = 4.5 x x 104 K for low-speed (< 350 kin/s) solar wind flows and T = 9.5 x X 104 K for intermediate (350 < Vsw < 475 kin/s) flows. We have also assumed 4% helium with THe = 3.5T [Slavin and Holzer, 1981]. Solar wind plasma conditions for each period are obtained by averaging over the intervals when the spacecraft was in the solar wind (unshaded regions in Figure 1). Using the median values for all parameters, we obtain a solar wind pressure Psw = 3.2 npa and Mms = 3.3 for the shocks on December 5, Psw = 7.8 npa and Mms = 5.3 for the shocks on December 6-7, and Psw = 30.5 npa and Mms = 4.8 for the shocks on December 7. The values of Mms given here are calculated for 0Bn = 90 ø. The value used in the model is the local value calculated for the actual 0B as described above. We transform into the aberrated G SE coordi- nate system using the data given in Table 1 and modify the base model using the method described above to obtain a model of the bow shock locally correct for the prevailing solar wind conditions. We have neglected the

8 26,934 BENNETT Erl ' AL.: MODEL OF Tt-II,; EARTH'S 1)ISTANT BOW SHOCK 3OO n,' eee Dec. 5 P=w = 3.2 Mms = Dec. 6-7 Psw = 7.8 Mrn s = === Dec. 7 Psw = 30.5 Mms = 4.8' oooooooooooooo 0,,, I,,,, I,,,, I,,,, I' O0 0 x [R E] Figure 5. Comparisons between the measured shock locations and models modified using the KK94 method. The circles are the observations from December 5, 1992, and the dotted line is our model for that day. The triangles and the dashed line are for the period from late December 6 to early December 7. The squares and the dot-dashed line are for late December 7. The solid line is the F/G model. The solar wind dynamic pressure, Psw, has units of nanopascals. change in the position of the nose of the shock given in and solar wind plasma data from the Massachusetts (19) because the Mach numbers for these observations Institute of Technology plasma instrument on Imp 8. are all greater than 3. The results are shown in Figure 5. The fits of the Inbound toward Earth, Galileo crossed the shock only once on December 5, The spacecraft re-entered model to the data are quite good. The December 5 the solar wind near the nose of the shock. The elecshocks lie within 4 RE of the model. The cylindri- tron temperature was unavailable, and so we assume cal radii of shocks from the second period (Decem- Te = 1.41 x 105K. The observation is well represented ber 6-7) depart from the model by only RE, and by the model. the December 7 shocks lie within 2 RE of the model. The departures are smaller than the uncertainty of the data. Achieving these fits required a change of the base We also calculated models for observations made by Pioneer 7 during September 1966 [Villante, 1976]. These shock crossings occurred between 350 and 420 RE anmodel from the F/G model, which has an eccentricity tisunward of Earth, farther out than the Galileo obof 1.02, to a model that has an eccentricity of servations. No data were available for the transverse The difference between these two curves is 15 RE at x = -300 RE, 6 RE at x = RE, and i RE at x = 0 and well within the uncertainty of the fit to the components of the solar wind velocity and so we assumed V u = 30 km/s, which is the orbital velocity of the Earth, and Vz = O. The model represents the pair data used to develop the F/G model. The models plotted in Figure 5 were calculated using the Mach number and the IMF direction averaged over extended intervals. In order to improve the predictions of our models we have carried out the procedures described above for the shocks on December 5, 1992, using the tail axis of a model with 0B, = 90ø; and P0, the distance from the tail axis of a model with 0s, = 0 ø. The shaded portions of the graph are times when the spacecraft is in the magnetosheath, inferred from the total magnetic field. When Pmod > PGa] the model predicts that the spacecraft is in the magnetosheath. From this plot we see that the spacecraft remained close to the bow shock with small excursions of the bow shock causing the observed crossings. We predict each of the crossings quite well. The density, total magnetic field, and other plasma parameters are roughly constant in the solar wind during the interval shown in Figure 6, so the major cause of the motion of the bow shock is rotations of the IMF. The parameter P90 is the maximum possible radius of the bow shock at a given x, for a given set of solar wind conditions, and p0 is the minimum. At downtail distances near 350 RE the maximum displacement of the shock due solely to IMF rotations is nearly 25 RE. For the Galileo observations, 0Bn was nearly always above 50 ø and so the excursions of the bow shock were small (< 5 We have also used our model to represent distant shock encounters observed by Galileo during its first pass by Earth and by Pioneer 7. For each encounter, or series of encounters, we calculate a model of the bow shock for the prevailing conditions. Figure 7 compares the model and the observed shock crossing for the Galileo Earth i pass. Model parameters were determined from Galileo magnetometer data of crossings near 0500 UT September 3, 1966 (Figure 8a) and the single crossing at 1755 UT September 5, 1966 (Figure 8b) well, but predicts the bow shock to be well inside the spacecraft (Ap/p 0.3) for the event at 0750 UT September 6, 1966 (Figure 8c). The data during this period had many gaps, and it is impossible to construct an accurate time line. However, the observation would match a model bow shock for Vu = s averages for the IMF direction and 5 min averages for M t and Ms (linearly interpolated across the intervals inside the magnetosheath.) Figure 6 shows plots of km/s in the solar wind frame, equivalent to Vy = -10 the total magnetic field; the local Osn; PGat, the actual km/s in the Earth frame. This is unusually large but position of Galileo; Pmod, the distance from the tail axis fails within the range of values that have been reported of the model using the local 0Bn; p90, the distance from by Feldman et al. [1977].

9 BENNETT ET AL.: MODEL OF 'ri- EARTH'S DISTANT BOW SHOCK 26, I ' I, :00 09:00 10:00 11:00 Time 12:00 13:00 14:00 15:00 Figure 6. Difference between the cylindrical radius at Galileo's position, PGal, and the model for December 5, 1992, Pmod. The location of the shock encounters are identified from jumps in the total magnetic field (shown in the top panel) and shading shows when the spacecraft was in the magnetosheath. The parameter 0Bn is the angle between the model shock normal and the instantaneous magnetic field vector. The straight solid line in the bottom panel is the trajectory of Galileo. The fluctuating solid line is the cylindrical radius of the model bow shock using 20 s magnetic field data. When Pmod > PGal the model predicts that the spacecraft is in the magnetosheath. The upper dotted line, P90, is the cylindrical radius of a model with 0B, = 90 ø and the lower dotted line, P0, is the cylindrical radius of a model with 0B, = Shock Normal Direction and Compression Ratios The shock normal direction can be calculated from the model. Using the model for December 5, 1992, obtained with the KK94 method (see Figure 5), we have rotated the magnetic field and velocity data from 0730 UT to 1530 UT December 5, 1992, into a shock normal coordinate system. The normal and tangential components of the data are shown in Figure 9. The shock crossings are identified using the total field and density, and the shaded regions are times when the spacecraft is in the magnetosheath. The tangential component of the magnetic field should show a jump at each shock encounter, which it does. The normal component of the magnetic field should be constant across a shock. In most of the cases shown here, it is; however, in three cases we see large rotations of the normal component' associated with the shock encounter. Such rotations, if not produced by temporal changes, would indicate that we have identified the wrong normal direction or that we have misidentified the bow shock. However, we inter- pret the field rotations as temporal changes of the IMF orientation that have caused the shock to move over the spacecraft. Consequently, we expect that in some fraction of crossings, significant rotations may follow entry into the magnetosheath. The velocity data in Figure 9 show the expected results. The tangential component, which is large owing to the highly oblique nature of these distant shocks, is on average constant throughout the period shown. The normal component decreases when the spacecraft enters the magnetosheath. That we see these small changes in the minor component of the velocity data, using the

10 . ß 26,936 BENNETT ET AL.' MODEL OF TIIE EARTH'S DISTANT BOW SHOCK 400 3OO 200 loo 0,,,. I, X [RE] Figure 7. Model of the bow shock at the time of Galileo's first flyby of Earth. The solid circle is the position of Galileo at the time of the inbound shock crossing on December 5, The solid line is the model using the solar wind velocity data given in Table 1. shock normal calculated from the model, indicates that we have obtained a good representation of the shock surface. The compression ratio n2/nl and a magnetic field compression ratio B2/B1 can be calculated by solving the Rankine-Hugoniot equations (see Decker [1988] for an example) using the shock normal direction calculated from the model. The calculated and observed compression ratios for the December 5, 1992, observations from Galileo are shown in Figure 10, plotted versus distance down the tail to organize the data. The calculated values compare reasonably well with the observed data, although the values of the magnetic compression are consistently low. The largest differences are 5%. 7. Response of Shock Model to Variation of Parameters To understand the relative importance of the effects described in the last two sections, we have varied one of the parameters in the solar wind (Psw, keeping the others constant at solar wind values. For each varying parameter we have calculated a model using the average value and low and high values representing the 5 and 95% probablility values given by Feldman et al. [1977]. The models are calculated in the equatorial plane on the duskside. The IMF direction is taken to be the nominal spiral angle. Figure 11a shows the effects of dynamic pressure with models for average pressure (2.5 npa) and pressure a factor of 5 above and below average. This effect is large at the nose of the shock due to the change in the position of the nose of the mag- netopause and in the distant tail due to the change in the diameter of the magnetosphere. Figure 11b shows the changes in the bow shock caused by changes in the sonic Mach number where the high and low values of M. are 5.6 and 10. The flaring of the bow shock increases as M. decreases, as expected. Figure 11c shows models for changing MA where the extreme values are 4.4 and 20. The magnetosonic Mach number for 0Bn 90 ø is (1 1 ) -1/2 mms r- 2s2 (29 Mms is dominated by the lower of the Alfv n and sonic Mach numbers. So since M is on average larger than Ms in the solar wind, increasing it does not dramatically change the magnetosonic Mach number and the change in flaring is small. However, since M can become very small the shock can be flared out considerably. Figure 11f shows cros sections in the y-z plane for the average and low M models taken at x = -350 R and at x = 0 RE. This further illustrates the change in the bow shock as a function of downtail distance. The change is dramatic in the distant tail but is negligible in the terminator plane. The IMF direction also influences the shape of the bow shock. Figure 11d shows models where 0Sn is taken to be 0 ø, 45 ø, and 90 ø along the entire shock surface. This is unrealistic, but it indicates the magnitude of the effect quite well. The dispacement of the shock forward of the terminator is negligible but becomes larger ' ' I... I... I... I... I ' ' I... I... I... I... I X [RE] Figure 8. Shock crossings observed by Pioneer 7 during September, 1966 [Viiiante, 1976] compared with the predictions of our models for conditions at the time of each crossing. The data and the model are shown (a) for a pair of crossings observed near 0500 UT September 3, 1966, (b) for a single crossing at 1755 UT September 5, 1966, and (c) for 0750 UT September 6, The dotted line in Figure 8c is for a model with solar wind V u = -10 km/s in the Earth frame. (a).

11 BENNETT ET AL.: MODEL OF THE EARTH'S DISTANT BOW SHOCK 26, i,! Dec-5 DOY: :00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 Time Figure 9. Magnetic field and velocity data for 0730 UT to 1530 UT December 5, 1992, plotted in a shock normal coordinate system. The shock normal was calculated using the model of the shock on December 5, 1992, obtained using the KK94 method. The shock crossings are identified using the total magnetic field and density, and shading shows when the spacecraft was in the magnetosheath. 1.5 I ' [ ' 1 I I, I -300 as distance down the tail increases. While this effect is not large, it is important because the IMF direction fluctuates even when the other plasma paramaters are constant. Also the asyrnmetry of the bow shock about the tail axis changes with local 0sn. Figure 11e illustrates this by showing cross sections of the tail in the y- z plane taken at x = RE for the field completely in the negative x direction, completely in the y direction, and at the nominal spiral angle of 45 ø in the x-y plane. When the magnetic field is completely in the x direction, 0Bn is constant around the tail and the cross section becomes symmetric. When the field is completely in the y direction, the cross section of the tail is symmetric about the y or z axes but extends further in y than in z. To illustrate the displacement of the nose of the shock at low Mach nu_mbers, we have calculated models for x Alfv n Mach numbers from 1.1 to 10 with other solar Figure 10. Density compression ratio (n2/n ) wind parameters taken to be average. Figure 12 shows and magnetic compression ratio (B /B ) observed by Galileo (solid circles) on December 5, 1992, and calcua series of models using the theory of Farris and Ruslated (open circles) by solving the Rankine-Hugoniot sell [1994], and Figure 13 shows models with the same equations using a shock normal from the model. The input conditions using the Cairns and Lyon [1995] thedata are plotted versus downtail distance. ory. The curves are very similar for MA > 3. Between

12 ß 26,938 BENNETT ET AL.' MODEL OF THE EARTH'S DISTANT BOW SHOCK 3O0 '--'2OO ' ' [ ] I ' " ] ' I ' ' ' ' I ' '-- ' ' ' ' I... I... I ' '. ' ' ' I... I ' ' -(a) Solar Wind Pressure "(c) Alfven Math Number.'(e) IMF Direction" _--.- -_. - : ::?' :' -. "--.. :: f I' '!/' - _"'"--...!: '" '%, loo '(b) Sonic Mach Number "(d) 0Sn '(f) M -' 200 loo 0 --!, I,,,,,,,,,,I, / i,, i i i i i i i [ i I! x [.3 x [Rd Y [,3 Figure 11. Cuts through the model bow shock surface showing changes when one solar wind parameter is varied keeping the others constant. Solid lines represent average solar wind values. (a) Pressure dependence with 12.5 npa (dashed), 2.5 npa (solid), and 0.5 npa (dotted). (b) Sonic Mach n,,mber dependence with Ms = 10 (dashed), Ms = 7.2 (solid), and Ms = 5.6 (dotted). (c) Affv n mach number dependence with MA = 20 (dashed), MA = 9.4 (solid), and MA = 4.4 (dotted). (d) The Bn dependence assuming Bn constant along the entire shock surface for Bn ø (dashed), Bn = 45 ø (solid), and Bn = 0ø (dotted).(e) Cros sections of the bow shock taken at z = -350 RE for three orientations of the IMF using an average solar wind value for the magnitude of B. The dashed curve is for the IMF in the negative z direction. The solid curve is for the IMF in the y direction. The dotted curve is for the IMF at a 45 ø angle in the z - y plane. (f) Crossections of the bow shock for the average MA case (solid curve) and the low Mn case (dashed curve) shown in Figure 11c. The inner curves are cuts at z = 0 and the outer at z = -350 RE. \ 100 -lo0 2OO and 3 the curves differ at the nose but differ little in the distant tail, as expected. Below Mach number 1.1 the sunward displacement of the Farris and Russell [1994] model produces very large effects. 8. Discussion ß. oo 50 We have developed a model of the bow shock surface that depends on the solar wind conditions. The shock surface equations are obtained using the following prescription. Using solar wind velocity data, rotate observed shock locations into an aberrated GSE coordinate system. The shock model is defined in this coor x [ ] Figure 12. Series of models using the Farris and Russell [1994] theory of the sunward motion of the bow shock for low Mach n,,mbers. The Alfv n Mach re,mher is varied with M t - 10, 5, 3, 2, 1.5, and 1.1. All other solar wind parameters are nominal.

13 ß BI NNI:œTT El' AL.: MODI;I. OF T111 IœARTI t's I)ISTANT 13()W SI lock 26,939 ß oo 50 of December 5, 1992, the normal and tangential components of the magnetic field and velocity change across the shock as expected from theory. The normal component of the velocity is only 1/3 the tangential velocity and changes by only 30% across the shock at 350 Rz downtail. That we see the predicted drop in the normal component of the flow when the spacecraft is in the magneotsheath shows that our model represents the shock normal direction well and reveals the quality of the plasma data taken by Galileo. Finally, this new model which can predict the shape of the distant bow shock for any solar wind conditions may prove useful in interpreting anomalous bow shock encounters observed by other spacecraft at large distances downtail. 0-1 O O0 x dinate system. Calculate xn3, x03, and L1 using (13), (14), and (19) for the prevailing P,, IMF B,, and solar wind Much number. Then calculate Pl from (20). At a given x and c (equation(6)), for known IMF, Urn, œom calculate Ap from (27) and the final Pshock from (28). The KK94 method modifies a base model to re- flect the prevailing solar wind Mms. The full asymmerry about the magnetic field vector is present at all distances in this method. To obtain our fits we de- creased the eccentricity of the hyperbolic base model from 1.02, for the F/G model, to This corresponds to a cylindrical radial displacement of 15 RE at x = -300 RE and 1 RE at x = 0. This change in the base model was necessary because, even though the F/G model is an excellent representation of the near- Earth portion of the bow shock, it does not include the asymmetry about the magnetic field direction which is an important effect for distant observations. The model shocks represent the observations to within uncertainties in the data. The ability to adjust the base model to reflect different, or more detailed, physics in the region near the obstacle is an advantage of the KK94 method. Also, this method is capable of handling a base model that is not a conic section. This will make the KK94 Appendix: e Adjustment Method A second method, which we refer to as the ½ adjustment method, for calculating the changes in the bow Figure 13. Series of models using the Cairns and Lyon shock flaring caused by changes in Mms uses the fact [1995] theory of the sunward motion of the bow shock that the asymptote of a hyperbolic model of the shock for low Much numbers. The Alfv n Much number is lies on the Much cone of the shock [e.g., Slavin et al., varied with MA = 10, 5, 3, 2, 1.5, and 1.1. All other 1984]. The asymptotes are determined by the magnesolar wind parameters are nominal. tosonic Much number, with the angle of the asymp- tote given by b = sin-l(1/mms). This angie is a function only of the eccentricity of the hyperbola with p-- tan -1(v/½ ), so ½ is a function of Mms. Thus, when the local Much number changes, the eccentricity of the model changes from ½ to ½1. Once again, the procedure of (13)- (19) is used to determine x0a and L. Then the hyperbolic model for the prevailing conditions is completely defined. There are two primary differences between this method and the KK94 method. Here the flaring of the shock, since it is calculated directly from the prevailing local Much number, is independent of the flaring of the base model. In the KK94 method the eccentricity of the base model is a parameter. Also in the ½ adjustment method the asymmetry of the shock, resuiting from the variation of Vms with the magnetic field orientation, reaches a maximum only in the asymptotic limit. In the KK94 method the full asymmetry about the magnetic field vector is present at all distances. In Figure 14 we compare data and models calculated using this method for the shock crossings observed by Galileo during December We used the same solar wind plasma properties as for Figure 5 based on the KK94 method. The shocks on December 5 lie about 3% outside the fit. The shocks for the period on December 5-6 lie about 6% inside the fit and the shocks for December 7 lie 3% inside the fit. Uncertainties in proton preferable for future, more detailed models. temperature and other physical parameters can easily account for variations of this magnitude. On the other hand, the difference from the F/G model is as large 9. Summary as 30 RE, or 16%. While the models using the KK94 The new model of the large-scale bow shock surface method provide a better fit, the models using the ½ adgives excellent correspondence between the calculated justment method provide an adequate description of the shock locations and compression ratios and observa- observations made by Galileo, within reasonable uncertions from several spacecraft. For the shock crossings tainties in the physical parameters. Therefore the ½ad-

14 26,940 BENNETT ET AI_,.: MODEL OF 'FILE EARTtt'S I)ISTANT BOW SHOCK 3OO 200 loo.. Dec. 5 Ps Mms Dec. 6-7 Ps. = 7.8 Mms = III Dec. 7 Psw = 30.5 Mms = 4-.8'.. e o X [RE] Figure 14. Comparisons between the measured shock locations and models modified using the e adjustment Cairns, I. H., and J. G. Lyon, Magnetic field orientation effects on the standoff distance of Earth's bow shock, Geophys. Res. Left, œ$, 2883, Cairns, I. H., D. H. Fairfield, R. R. Anderson, V. E. H. Carlton, K. I. Paularena, and A. J. Lazarus, Unusual locations of Earth's bow shock on September 24-25, 1987: Mach number effects, J. Geophys. Res., 100, 47, Cairns, I. H., D. H. Fairfield, R. R. Anderson, K. I. Paularena, and A. J. Lazarus, Reply to comment on "Unusual locations of Earth's bow shock on September 24-25, 1987: Mach number effects," J. Geophys. Res., 101, 7679, Decker, R. B., Computer modeling of test particle acceleration at oblique shocks, Space Sci. Rev.,. 8, 195, Dryer, M., and G. R. Heckman, On the hypersonic analogue as applied to planetary interaction with the solar plasma, Planet. Space Sci., 15, 515, Fairfield, D. H., Average and unusual locations of the Earth's magnetopause and bow shock, J. Geophys. Res., 76, 6700, Farris, M. H., and C. T. Russell, Determining the standoff distance of the bow shock: Mach number dependence and use of models, J. Geophys. Res., 99, 17681, Farris, M. H., S. M. Petrinec, and C. T. Russell, The thickness of the magnetosheath: Constraints on the poly- method. The circles are the observations from Decem- tropic index, Geophys. Res. Left, 18, 1821, bet 5, The dotted line is the model of the bow Feldman, W. C., J. R. Asbridge, S. J. Bame, and J. T. shock for that day. The triangles and the dashed line Gosling, Plasma and magnetic fields from the Sun, in are for the period from late December 6, 1992, to early December 7, The squares and the dot-dashed line are for late December 7, The solid line is the F/G model. The solar wind dynamic pressure, Psw, has units of nanopascals. justment method is a viable method for modeling the changes in the bow shock shape resulting from changes The Solar Output and its Variation, edited by O. R. White, pp , Colorado Univ. Press, Boulder, Frank, L. A., K. L. Ackerson, J. A. Lee, M. R. English, and G. L. Pickett, The plasma instrumentation for the Galileo mission, Space Sci. Rev., 60, 283, Greenstadt, E. W., D. P. Traver, F. V. Coroniti, E. J. Smith, and J. A. Slavin, Observations of the flank of Earth's bow shock to -110 Rs by ISEE3/ICE3, Geophys. Res. in the solar wind magnetosonic Mach number. Left., 17, 753, Computer codes necessary to utilize our bow shock Hill, T. W., and M. E. Rassbach, Interplanetary magnetic model will be available via world wide web at edu/galileo/newmodel.htm. Acknowledgments. We thank Steve Joy of UCLA for his assistance in obtaining and preparing the data. L.B. also thanks F. Bagenal, C. T. Russell, and R. J. Walker for many useful discussions. Imp 8 plasma data courtesy of A. J. Lazarus and K. Paularena. This work was supported by the Jet Propulsion Laboratory, Pasadena, under contracts JPL (L.B., M.G.K., and K.K.K.) and JPL (L.A.F. and W.R.P.). K.K.K. also wishes to acknowledge support by the National Aeronautics and Space Administration under contract NAGW UCLA-IGPP publication The editor thanks D. Sibek and S. A. Curtis for their assistance in evaluating this paper. References Aubry, M.P., C. T. Russell, and M. G. Kivelson, Inward motion of the magnetopause before a substorm, J. Geophys. Res., 75, 7018, Cairns, I. H., and J. G. Lyon, MHD simulations of Earth's bow shock at low Mach numbers: Standoff distances, J. Geophys. Res., 100, 17173, field direction and the configuration of the dayside magnetosheath, J. Geophys. Res., 80, 1, Holzer, R. E., and J. A. Slavin, Magnetic flux transfer associated with expansions and contractions of the dayside magnetopause, J. Geophys. Res., 83, 3831, Khurana, K. K., and M. G. Kivelson, A variable crosssection model of the bow shock of Venus, J. Geophys. Res., 99, 8505, Kivelson, M. G., K. K. Khurana, J. D. Means, C. T. Russell, and R. C. Snare, The Galileo magnetic field investigation, Space Sci. Rev., 60, 357, Kivelson, M. G., et al., The Galileo Earth encounter: Magnetometer and allied measurements, J. Geophys. Res., 98, 11,299, Kivelson, M. G., A. Prevost, F. V. Coroniti, K. K. Khurana, and D. J. Southwood, Galileo fiybys of Earth: The nature of the distant shock, Adv. Space Res., 16, 197, Maltsev, Y. P., and W. B. Lyatsky, Field-aligned currents and erosion of the dayside magnetopause, Planet. Space Sci., 23, 1257, Newbury, J. A., Rule of thumb: Electron temperature in the solar wind at I AU, Eos Trans. A GU, 77, 471, Peredo, M., J. A. Slavin, E. Mazur, and S. A. Curtis, Threedimensional position and shape of the bow shock and their variation with Alfv nic, sonic and magnetosonic Mach numbers and interplanetary magnetic field orientation, J. Geophys. Res., 100, 7907, Petrinec, S. M., and C. T. Russell, External and internal

15 BENNETT ET AL.: MODEL OF THE EARTH'S DISTANT BOW SHOCK 26,941 influences on the size of the dayside terrestrial magnetosphere, (7eophys. Res. Lett, 20, 339, Russell, C. T., and S. M. Petrinec, Comment on "Unusual locations of Earth's bow shock on September 24-25, 1987: Mach number effects," J. Geophys. Res., 101, 7677, Russell, C. T., R. C. Elphic, and J. A. Slavin, Pioneer magnetometer observations of the Venus bow shock, Nature, 282, 815, Russell, C. T., E. Chou, J. G. Luhman, P. Gazis, L. H. Brace, and W. R. Hoegy, Solar and interplanetary control of the location of the Venus bow shock, J. Geophys. Res., 93, 5461, Russell, C.T., and T. L. Zhang, Unusually distant bow shock encounters at Venus, Geophys. Res. Left., 19, 833, Seiff, A., Gasdynamics in space exploration, NASA Spec. Publ. 2., Sibeck, D. G., R. E. Lopez, and E. C. Roelof, Solar wind control of the magnetopaase shape, location, and motion, J. Geophys. Res., 96, 5489, Slavin, J. A., and R. E. Holzer, Solar wind flow about the terrestrial planets, 1, Modeling bow shock position and shape, J. Geophys. Res., 86, 11401, Slavin, J. A., R. C. Elphic, C. T. Russell, F. L. Scarf, J. H. Wolfe, J. D. Mihalov, D. S. Intriligator, L. H. Brace, H. A. Taylor, J., and R. E. Daniel, J., The solar wind interaction with Venus: Pioneer Venus observations of bow shock location and structure, J. Geophys. Res., 85, 7625, Slavin, J. A., R. E. Holzer, J. R. Spreiter, and S.S. Sta- hara, Planetary Mach cones: Theory and observation, J. Geophys. Res., 89, 2708, Stairnov, V. N., O. L. Vaisberg, and D. S. Intriligator, An empirical model of the Venusian outer environment, 2, Shape and location of the bow shock, J. Geophys. Res., 85, 7651, Spreiter, J. R., and S.S. Stahara, A new predictive model for determining solar wind-terrestrial planet interactions, J. Geophys. Res., 85, 6769, Spreiter, J. R., A. L. Summers, and A. Y. Alksne, Hydromagnetic flow around the magnetosphere, Planet. Space Sci., 1., 223, Verigin, M. J., K. I. Gringauz, T. Gombosi, T. K. Breus, V. V. Bezrukikh, A. P. Remizov, and G.I. Volkov, Plasma near Venus from the Venera 9 and 10 wideangle analyzer data, J. Geophys. Res., 83, 3721, Villante, U., Evidence for a bow shock at 400 RZ: Pioneer 7, J. Geophys. Res., 81, 1441, Zhang, T.-L., J. D. Luhmann, and C. T. Russell, The solar cycle dependence of the location and shape of the Venus bow shock, J. Geophys. Res., 95, 14961, L. Bennett, K. K. Khurana, and M. G. Kivelson, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA ( lbennett@igpp.ucla.edu) L. A. Frank and W. R. Paterson, Department of Physics and Astronomy, University of Iowa, Iowa City, IA (Received March 27, 1997; revised June 24, 1997; accepted June 27, 1997.)