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).