AN EMPIRICAL MODEL OF THE SIZE AND SHAPE OF THE NEAR-EARTH MAGNETOTAIL S.M. Petrinec and C.T. Russell Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Ca. Abstract. A model of the size and shape of the near-earth magnetotail which depends upon distance downtail, pv2sw, and IMF Bz has been developed. This empirical model was developed by first using observations of the magnetic field in the lobe by ISEE-1 and simultaneous observations of the magnetic field and plasma properties of the solar wind by IMP-8 to determine the flaring angle of the near-earth magnetotail in the region -22.5 Re < ~r < -10 Re. These studies show that the flaring angle depends on x. pv2sw, and IMF Bz. In particular, the median flare angle increases linearly with decreasing B~ when the IMF is southward, while the flare angle is constant when the IMF is northward. The empirical function derived for the flaring angle of the magnetotail is used to determine a relation for the radius of the tail Introduction An accurate model of the size and shape of the near-earth magnetotail is an essential element for developing an understanding of the dynamic processes that occur therein, since the near tail appears to be the source region for substorms (Russell and McPherron [1973]). Early studies of the magnetotail concentrated on magnetopause crossings by spacecraft (Behannon [1968], Spreiter and Alksne [1969], Howe and Binsack [1972]), and the field line configuration and field magnitude as a function of distance downtail (Behannon [1970], Mihalov and Sonett [1968], Mihalov et al. [1968], Spreiter and Alksne [1969]). Later works examined changes in the lobe magnetic field with solar wind and substorm conditions, using spacecraft observations (Caan et al. [1973, 1975]), while Coroniti and Kennel [1972] developed a theoretical approach for estimating the magnetotail shape and the magnetospheric configuration during quiet and substorm growth phase times. Unfortunately, available models for the size of the magnetotail are not satisfactory. The radius of the magnetotail varies with distance downtail and solar wind pressure, as the magnetic pressure of the tail lobe (which decreases with increasing distance from the earth) balances the component of solar wind dynamic pressure normal to the magnetotail boundary, with some (small) assistance from the solar wind thermal and magnetic pressures. Howe and Binsack [1972] used magnetopause crossings by Explorer 33 and Explorer 35 to empirically determine that the nightside magnetopause boundary varied approximately as RT = 23.9~. The constant {23.9 } can be varied as the inverse one-sixth root of the solar wind dynamic pressure, although the median dynamic pressure for the crossings is not known. Consistent with the observed effect of dynamic pressure on the tail cross section, Russell et al. [1981] found that the solar wind dynamic pressure was a good predictor of the field strength in the lobes at the orbit of the moon. However, the size of the magnetotail also varies with the direction of the z-component (GSM) of the IMF. When the IMF turns southward, magnetic flux is eroded from the dayside magnetosphere and deposited in the nightside [Aubry et al., 1971]. The tail cross section must then increase until such time as a substorm occurs. The same neglect of any Bz effects is found in the model of Spreiter and Alksne [1969]. A recent model that attempts to redress these problems has been developed hy Ohtani and Kokubun [1990]. Here, magnetopause crossings of the IMP-8 spacecraft were used to determine the location of the magnetotail boundary as a multiple linear regression of Lxl, ~. BIMF. and BZIMF- Though this model attempts to incorporate all expected effects. the data base is limited and the model employs a flaring angle that is constant, regardless of solar wind parameters or distance downtail. As will be shown, this is clearly not the case. In A complementary study, Nakai et al. [1991] used the ISEE-1 measurements of magnetic field in the lobe to determine how the magnetic field varies with distance downtail. Using this relation (BT = (1.03+0.14)X103 L~ 20~00s)~ magnetic field measurements were mapped to a distance of -20 Re. The magnetic field at x = -20 Re was then determined as a function of solar wind properties (pv2sw, Ps (=nk(Ti+Te) + BT2/2~O), and BZV (or IALI, a magnetospheric index)). If one assumes the magnetic flux to be constant, the tail radius can be determined, being proportional to BT-1/2 (~BX-1/2). This approach does not appear to work in the near tail because the magnetic field fall-off with distance is too steep, resulting in a tail radius of zero at the terminator. Nakai et al. also determined how the flaring angle of the magnetotail varies as a function of pv2sw and IALI~ at a fixed distance of -20 Re~ but did not make any attempt to use this information to estimate the size of the magnetotail. In short, the available models do not allow us to accurately calculate an instantaneous estimate of the size, shape, and hence magnetic flux content of the tail. We therefore found it necessary to develpoe our own magnetotail boundary model. The Data The spacecraft measurements used in this work include magnetometer and plasma (from the Fast Plasma Experiment) observations by ISEE- 1 and -2, as well as plasma and magnetometer observations by IMP-8. The solar wind data were corrected according to the procedure of Petrinec and Russell (1993). Fifty orbits of the ISEE spacecraft were examined for times when these spacecraft were in the lobes of the magnetotail. The criterion used for magnetotail lobe determination was that the ion density measured by the FPE instrument on board ISEE-2 be less than 0.1 cm~3. In addition. observations were only used at times when the ISEE spacecraft were greater than 10 Re downtail. This data set was further reduced by considering times when there were IMP-8 solar wind data available. Time delays between IMP-8 and the ISEE spacecraft were not taken into account, but since the temporal resolution of the data used in this work is 5 minute averages, time delays are at most on the order of _5 minutes. Calculation of the Flaring Angle oc It is assumed in this study that the total pressure in the lobes of the magnetotail is balanced by the total pressure of the solar wind, as described by )lob~ =pv25~sin2~x+ ( )sw +(nk(T, +Te))sW (1) where a is the angle between the solar wind flow direction and the tangent to the magnetopause surface in the meridian plane. By using Equation (1), we assume that the lobe field is a vacuum field, and is uniform across the lobe for a given distance x. The angle a is then easily calculated from the solar wind and magnetotail lobe observations. We thus have compiled a data set of 3383 points from fifty orbits of the ISEE spacecraft. The Model The flaring angle is found to vary most strongly with the solar wind dynamic pressure. This indicates that although the shape of the dayside magnetosphere is self-similar with respect to changes in dynamic pressure, the magnetotail is not. Dynamic pressure values are binned as ~0-.5, .5-1, 1-2, 2-4, 4-8, 8-16, and 16-32 nPa}, as shown in Figure 1. Earlier studies have determined that when the IMF is southward, magnetic flux is transferred from the dayside magnetosphere into the magnetotail. Regressions between flare angle and pv2 and x are determined only for those times during which the IMF was northward (1680 points). Figure 1 shows the regression calculated between medians of log2(sin2a) and log2(pv2). Silimar to the result obtained by Nakai et al. [1991], we find that sin2oc (pv2)-0 487+0 024. Using the median dynamic pressure for this data set, the solar wind dynamic pressure dependence is then removed from the flare angle data, so that sin2o~ 76x(pv2)-o 487+o.o24 f[x B ] The functional dependence between the flare angle and distance downtail is weaker than the previous relation between flare angle and dynamic pressure. However, this relation is crucial since it will be used to determine the general magnetotail shape. Observations are binned according to distance downtail, with an equal number of points per bin. There are several factors which must be considered when deciding the best function to use. For a general function, the flare angle is defined as dy/dx-- tana. The shape can thus be determined (with a constant to be estimated later) by a relation between a and x. The chosen function must give an expression for the radius of the tail that is physically reasonable. In addition, the function must provide a good fit to the data. Lastly. the derived function should be valid over a large range of dynamic pressures and IMF Bz values. Table 1 shows conelation coefficients for linear fits between various functions of oc and x (polynomial fits were attempted, but either could not be integrated or did not give a reasonable shape for the the magnetotail boundary). The correlation between various functions of a versus x results in the highest correlation. Differences in the correlation coefficients are quite small. The best correlations are between 1/sin(2)a and x, and between 1/sina and x, but they also have a limited range of validity in pv(2)Bz space (for each relation, the dependence of flare angle on IMF Bz was also determined, and the area in pv(2)-Bz space where the magnetotail solution became complex was mapped). The best relation between a and x was determined to be: sin(2)a=g(pv(2),Bz)x(ax+b)(2) (2) where a=.00819_.00175 and b=.549_.030, as shown in Figure 2. Defining A = a~(pl~2, B ) and B = b~g(pv2,Bz), we are able to solve for the shape of the magnetotail as: y = A~ B)2 +C1 (3) In order to solve for Cl, we use a result from Petrinec et al. [1991] The terminator distance is found to be 14.6 Re for a median solar wind dynamic pressure of 2.52 nPa, and is the approximate intersection point for dayside magnetopause fits of the ISEE data for both northward and southward IMF subsets. This distance is assumed to vary as the inverse one-sixth power of the solar wind dynamic pressure. Therefore, Equation (2) can be rewritten as: y = 1 (~1- (Ax + B)2 - ~) + Rterm j 2 52 ) (4) With Equation (2), we are able to normalize the calculated flare angles to a common distance of -17.56 Re (The median distance of the ISEE spacecraft in the data set). With a set of flare angles normalized by solar wind dynamic pressure and distance, we can investigate how the flare angle, and thus the size of the magnetosphere vary with IMF Bz. As with the regression with x, we bin the IMF Bz values such that there is an equal number of points within each bin. Medians of sin2o~ (normalized by pv2 and downtail distance) are calculated for each bin of IMF Bz, and plotted in Figure 3 as a function of Bz. There is a clear difference in behavior between the values of a when the IMF is southward and when it is northward. Sin2o~ increases linearly as the magnitude of the southward IMF increases, indicating that the size of the magnetotail becomes larger, as expected if flux is being transferred into the magnetotail from the dayside magnetosphere. However, when the IMF is northward, the flare angle is constant. In summary, then, the flare angle of the magnetotail as determined from solar wind and magnetotail lobe data can be best described as: sin2 a = 7 o44(pl~2)- 487+ 024 ((8.19 + 1.75) x 10 3 x+.549+.030) x f(-7.99 + 1.09) x 10 3Bs+.164+.005) where Bs equals Bz when the IMF is southward, and zero otherwise (pv2, x, and Bs have units of nPa, Re, and nT, respectively). The magnetotail radius is described by Equation (4). Figures 4 a-e illustrate sample magnetotail functions for a range of dynamic pressures, with each panel displaying three different IMF Bz values: 0.0 nT (or greater), -5.0 nT, and - 10.0 nT. Figures 5 a-c show sample magnetotail shapes for a range of IMF Bz values, with each panel displaying five different pv2 values: 0.5 nPa, 1.0 nPa, 2.0 nPa, 4.0 nPa. and 8.0 nPa. Discussion and Conclusions A magnetotail model has been developed using simultaneous, multipoint observations of the magnetic field and plasma within the magnetotail lobes and the solar wind. By approaching the problem in this manner, instead of using the more standard procedure of using actual crossings of the magnetopause by spacecraft, we have managed to avoid some of the problems that plague the more straightforward method (e.g., spacecraft orbital biases, multiple magnetopause crossings on a single pass or questionable crossings, consideration of aberration angle, etc.). However, there are some fundamental assumptions that have been made in this work as well. We have implicitly assumed that we are far enough from the earth that the magnetotail field lines are straight. In addition, we assume that the tail lobe is a vacuum field. We also assume that the magnetotail is circular in cross-section, and does not change as the dipole tilt angle changes. Lastly, all of the empirical solutions had, to some degree, a region of pv2-Bz space where there was no solution for the magnetotail shape. When B in equation (4) is greater than 1, the calculated tail radius becomes complex. The region of pv2-Bz space for which this occurs is illustrated in Figure 6. When the solar wind dynamic pressure is below 0.115 nPa, there is no solution. For larger dynamic pressures, there is no solution only for very large, negative values of IMF Bz. (e.g., pv2 = 1.0 nPa, Bz < -38.4 nT). Finally, it should be noted that this and other models of the magnetotail boundary that depend on the IMF only use the instantaneous value of the IMF, both in their derivation and their use. The amount of flux in the magnetotail and the size of the nightside magnetosphere are the results of a time-integrated effect of the IMF, particularly when there is a substantial southward component to the IMF. This model was developed from magnetotail lobe data in the range -22.rRe 1, and the magnetopause shape cannot be determined. The contour plot shows where observations used in the development of this tail model lie in this pauameter space (bin size = 0.2 nPax2.0 nT).