The position and shape of the post-terminator magnetospheric boundary is studied as a function of downtail distance, solar wind dynamic pressure, and the z-component of the IMF. We have used 360 orbits of ISEE-2 for which plasma data are available to determine when the spacecraft was in the lobe of the magnetotail. The lobe pressure has then been calculated from the total magnetic field measured by ISEE-2, and simultaneous calculations of the solar wind pressure components (dynamic, thermal, and magnetic) are made from IMP-8 measurements. Perpendicular pressure balance between the magnetotail lobes and the solar wind is used to calculate a set of flare angles for the magnetotail boundary. The flare angle of the magnetotail is found to increase for southward IMF, but the variation for northward IMF is much weaker, indicating that the magnetotail size increases appreciably only for southward IMF. The flaring angle has been integrated to get the average shape and size of the magnetotail, matching the boundary to the average dayside magnetopause shape at the terminator. The magnetotail shape and size has been determined as a function of the solar wind condition. We have compared this model with other empirical models of the magnetotail size and shape, as well as a regression to IMP- 6 magnetotail boundary crossings. Comparisons are made for nominal as well as extreme solar wind conditions. We find that the tail size depends more strongly on the solar wind dynamic pressure than predicted by the Roelof and Sibeck  model, but is in excellent agreement with the dependence obtained by Lui . This tail model has been used to examine several sudden impulse events, comparing the tail magnetic flux content for those events which trigger substorms and for those that do not.
An important consequence of the solar wind interaction with the near-Earth environment is the existence of a magnetotail. Much of the magnetic flux and energy of the magnetosphere is transported and stored within this region, either tied to the Earth on the closed field lines of the plasma sheet or the open field lines of the magnetotail lobes. Models of the magnetotail shape and size are extremely useful for estimating in a dynamic sense the magnetic flux content of the magnetotail. This is beneficial for studies of dynamic processes which occur within the magnetotail, and their relation to sudden changes in the solar wind. In particular, this method of analysis may provide insight into the state of the magnetotail just prior to the triggering of the substorm expansion phase. Knowledge of the conditions which lead to substorms is a crucial element for improving our understanding of energy storage and release within the terrestrial magnetosphere.
Studies of the magnetic field profiles of the magnetotail and boundary crossings were performed soon after the discovery of the magnetotail /1, 2/. A statistical study of the magnetotail position and shape soon followed, using plasma observations to identify the boundary /3/. In addition, theoretical estimates of the size of the magnetotail and the magnetic flux contained within were made /4, 5, 6/. However, the statistical model only gives an average shape for the magnetotail boundary, and does not examine the effect of the solar wind condition on the shape and position. The theoretical models are expressed in terms of the level of magnetic flux within the tail, and are not easily recast according to solar wind parameters.
More recently, other empirical studies of the magnetotail boundary derived from spacecraft observations have been performed /7, 8, 9, 10, 11, 12/. Most of these studies are based on direct observations of the magnetopause by spacecraft which flew through the magnetotail region. However, the works of Petrinec and Russell /11, 12/, are based on the assumption of pressure balance between the solar wind and magnetotail lobes, perpendicular to the boundary. The latter study /12/, the result of which is shown below, uses a much larger data set and derives a more robust tail shape than was determined in the former work /11/. In this work, calculated flare angles are found to vary with the solar wind dynamic pressure, IMF Bz, and downtail distance at which the lobe magnetic field values are measured. The flare angle () is then integrated down the tail from the terminator, to give the following expression:
The first term is an empirical model describing the dayside magnetopause /13/, where this nightside model is attached. The stand-off distance of the magnetopause is , and is the Heaviside step function linking the dayside and nightside solutions, bn and bs are the value of the IMF Bz in nT (e.g., bn = 5 nT, bs = -3 nT), and the argument of the Arcsine function is expressed in radians. The dynamic pressure is in nPa, and x and y are the distance along and perpendicular to the Sun-Earth line respectively, in Re.
It is of interest to compare and contrast this model with several earlier empirical models. A direct comparison is made in Figure 1 between our results /12/ and those of Howe and Binsack /3/, Ohtani and Kokubun /8/, and Roelof and Sibeck /10/. This comparison is performed for a solar wind dynamic pressure of 2.0 nPa (a typical value), and an IMF Bz of 0 nT. Although there are large discrepancies between our model and those of Howe and Binsack /3/, Ohtani and Kokubun /8/, it is easily seen that this model and that of Roelof and Sibeck model are very similar; agreement within 1.2% out to a distance of 50 Re. However, extreme values of solar wind dynamic pressure find significant differences between these two models, as shown in Figure 2. These variations indicate a distinctly different dependence on solar wind dynamic pressure in the two models. Since these extreme conditions of solar wind dynamic pressure are not very common, it is difficult to judge which model is better. A separate study performed several years earlier /7/ used IMP-6 crossings of the magnetotail between -15 Re and -20 Re to estimate the relation between the tail radius and the solar wind number flux (which is directly related to the dynamic pressure through the ion mass). The dependence found was RT ~ (nv2)-0.227±0.026. This agrees very well with the pressure dependence at this distance obtained in our model /12/ (RT ~ (nv2)-0.226), but the agreement with Roelof and Sibeck /10/ is less favorable (RT ~ (nv2)-0.077). It should be noted, however, there were only 12 crossings used in that earlier study /7/, with only one crossing occurring at a dynamic pressure greater than 5 nPa.
Fig. 1. Magnetotail traces for a nominal solar wind dynamic pressure of 2.0 nPa and an IMF Bz value of 0 nT.
Fig. 2. Magnetotail traces from two empirical models /10, 12/, for extreme values of solar wind dynamic pressure.
Another indication that this model /12/ includes the correct dynamic pressure dependence is an examination of sudden impulse events, when ISEE-2 is in the magnetotail lobe (shown in /12/). When the sudden impulse passes by the Earth, the magnetic field in the tail is increased as the tail is compressed. Using the measured solar wind parameters, we can estimate the tail radius as a function of time. For those events for which no substorm onset is triggered, the calculated tail radius (and hence, tail cross-sectional area) exactly counteracts the increase in lobe magnetic field strength, such that the calculated magnetic flux remains constant. This would not be the case if the tail radius dependence on solar wind dynamic pressure was incorrect.
In summary, our near-Earth magnetotail model disagrees with those of Howe and Binsack /3/ and Ohtani and Kokubun /8/, but agrees remarkably well with the model of Roelof and Sibeck during nominal conditions. Under extreme conditions, however, these two models diverge. Although we feel that our model reflects the correct dynamic pressure dependence, additional studies of the magnetotail are needed in order to confirm this assertion.
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