A method is reported for calculating the shapes of ionic comet tails that use the magnetic field along the tail to channel the tail plasma. The local momentum field in the solar wind determines magnetic field line locations along the tail. Thus, the magnetic field acts as a transparent wind sock in trapping field lines in the cometary plasma around the nucleus long enough to be effectively fastened to the comet's head

Brandt, J. C.

Rothe, E. D.

English

NASA Technical Reports Server

N76-21088
THE WIND-SOCK THEORY OF COMET TAILS
John C. Brandt and Edward D. Rothe
I. Introduction
Type I or ionic comet tails on the average make an angle of a few
degrees at the nucleus with the prolonged radius vector in the direction
opposite to the comet's orbital motion. This fact was explained by
•^
Biermann (1951) as the aberration angle caused by the comet's motion
in the outflowing solar wind plasma, and, as is well known, led to the
discovery of the solar wind itself. Mathematically, the direction of
—»
the tail T is given by the vector equation
T = w - V (1)
—» —»
where w is the solar wind velocity and V is the comet's orbital velocity.
Equation (1) or simplified forms of it have been used extensively to
derive properties of the solar wind (Belton and Brandt 1966; Brandt
1967; Brandt, Harrington and Roosen 1973). The solar wind properties
derived from ionic comet tails agree with directly determined properties
in all cases where comparison is possible and, hence, the validity of
equation (1) has been established. If the solar wind determines the
gross shape of the entire plasma tail, what is this shape and how can
it be calculated?
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There are at least three conceptually distinct approaches to
calculating the shapes on ionic comet tails.
(1) The Mechanical or Bessel-Bredichin Theory. The details of
this approach first treated by Bessel (1836) are widely known. A
constant acceleration is assumed to act on the tail material, and in
the calculations, this is included by using a reduced solar gravity.
Bredichin defined Type I comet tails as syndynes with extra repulsive
force (1 - |j) » 18. Unfortunately, syndynes are tangent to the prolonged
radius vector at the nucleus which is contrary to the observations. The
tail curvature given by a syndyne with (1 - p.) « 18 is probably not
correct either and we return to this point below.
(2) The Smoke Theory. Here, the force on the tail material is
given by a momentum transfer depending on the relative velocity of the
solar wind with respect^to the tail material (see Belton 1965, Appendix 1)
Hence, if r' is the velocity of the tail material, we would need to
include a force of the form
F (w - ?') . (2)
—*f
and calculate the speed of the tail material f at all points. Our
understanding of the solar wind interaction with plasma tails is
insufficient to permit accurate calculation of the forces required on
the smoke theory,, This difficulty obviously applies to an entire class
of theories requiring specific forces. Fortunately, knowledge of
specific forces may not be necessary.
879
(3) Wind-Sock Theory. Here we do not need the forces accelerating
the material along the tail. The magnetic field along the tail is used
to channel the tail plasma and the location of the magnetic field lines
in the tail is determined by the local momentum field in the solar wind.
The magnetic field acts as a transparent wind sock. This viewpoint
implies that the field lines are trapped in the cometary plasma around
the nucleus long enough for them effectively to be fastened to the comet's
head. The first explicit statement of this concept known to us was by
Alfven (1957) who wrote:
"The tail should no more be considered as gas moving
freely in space. Instead the tail is a real part of the
0
 comet, fastened to the head by magnetic field lines."
II. Theory
The gross shape of an ionic comet tail on the wind-sock theory
assuming constant solar wind speed can be calculated by applying the
—* —» —*
equation T = w - v pointwise along the tail. The,basic geometry in
the plane of the comet's orbit is shown in Figure 1. By projecting the
components of solar wind velocity into the cometocentric coordinate
system, we obtain the basic equation for the wind-sock theory, viz.,
, -V sin y - w sin Q1 + w , cos & cos i'/cos bdy _ r , a
w cos a - V cos Y + w sin Oi cos i /cos b
Many of the quantities used are illustrated in Figure 1. In addition,
w and w, are the radial and azimuthai'components of the solar wind
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velocity. The angles i' and b are the inclination of the orbit
with respect to the solar equator and the heliographic latitude,
respectively. If necessary, this equation could easily be generalized
to three dimensions; in this case, the direction cosines at each point
(dx, dy, dz) would be similarly determined.
A simple analytical result can be obtained on the basis of some
reasonable approximations. For a comet away from the sun (i.e., non-sungrazers)
and near perihelion, a « 1 and w » cos Y, respectively, are good
approximations. Then, equation (3) can be written
. -V sin Y - w at + w, cos i'/cos b
dx ~ w ' '
Equation (4) can be used to evaluate the coefficients in a Taylor's
series. If we let
-y = A+Bx +Cx2 +Dx3 + ... , (5)
we find
A = 0
fv sin v - w.cos i'/cos b~l
_ £
L w - V cos Y J wv. ~ v '-v  j
(6)
B
882
If we write
X = x/r
(7)
Y = y/r
where r is the comet's heliocentric distance, our final equation becomes
-Y=BX +f x2+ ... (8)
^
for X ^ 0.3, the cubic term is 1.3% or less compared to the sum of the
first two terms.
The Taylor's series solution can be obtained without the approximations
used to write equation (4) and is
r o /WACOS i'\ /w^sin Y cos i7 \
TJ 2 4- ( 0 ) V I ^ 4- u r- ic ' 1
r \ cos b / \ cos b r /
(w - V cos Y)
Ex:2
2-Y=BX+ —
 v
 ' > ^^ '- =^ -+... (9)
w   Y  J
For most cases, the term in brackets in equation (9) is close to 1 and
equation (9) reduces to equation (8). In doubtful cases, equation (9)
provides a check on the applicability of the simple solution.
Our approximate (but rather accurate solution) for steady solar
wind conditions depends only on the quantity B which is the tangent of
the aberration angle at the nucleus used in the earlier work. The
calculated tails are nearly straight near the head, but show curvature
well away from the head. The curvature arises from the geometrical
divergence of the radial direction in a spherical coordinate system.
883
III. Applications
First, we briefly reexamine the historical question of Bredichin's
identification of the Type I tails with syndynes from the mechanical
theory for (1 - jj.) « 18. We have checked back to some of Bredichin's
(1884) original work in which the observations (for the Great Comet of
1882) were given. They show little or no tail curvature and we have no
difficulty fitting the wind-sock theory with modern solar wind parameters
(Figure 2). Bredichin's fit with (1 - |_i) « 18 would not be bad in an
rms sense, even though the aberration angle at the nucleus was in
error on the average by w 5 and the curvature was too large. Visual
observations of very bright comets in the 19th century may be a valuable
untapped source of solar wind data.
The wind-sock theory can also be applied to the geomagnetic tail
(Figure 3). Behannon's (1970) observations gave an aberration angle
of 3.1 and this is closely approximated by the solar wind parameters
chosen. Figure 3 shows that observations would be required at tenths
of A.U. from Earth to detect the effects of the tail curvature; such
f
observations are unlikely in the near.future. A comparison of an
accurate computer integration of equation (3) with the Taylor's Series
result of equation (8) is also shown.
Figure 4 shows a photograph of Comet Kohoutek taken at the Joint
Observatory for Cometary Research (JOCR) on January 19, 1974. We would
anticipate no difficulty in explaining the gross shape of the main ion
tail on the basis of steady solar wind conditions. However, it is
important to note that our model may require comparison with averages
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of observations if sufficiently steady solar wind conditions are not
found. For example, Comet Kohoutek on January 20, 1974 (Figure 5)
showed a large disturbance in the shape of the main tail. Our solution
with B = 0.36 would be a reasonable fit to the tail shape except for
the disturbance as is shown schematically in Figure 6. The disturbance
could be caused by a high-speed solar wind stream. Note that the quiet
conditions for B= 0.36 are somewhat unusual; a large negative value of
w . is necessary to produce a reasonable w value. We suspect that
changes in solar wind conditions produce changes in gross plasma tail
shapes and that study of tail shapes may provide information on
velocity structures in the solar wind.
IV. Conclusions
We have presented a simple version of the wind-sock theory of
ionic comet tails. The simple model is consistent with all facts
known to us. There are straightforward improvements that can be made
for the case of steady solar wind conditions (e.g., inclusion of
effects due to the tail's magnetic field). Consideration of the
non-steady case are also of considerable interest.
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References
Alfven, H. 1957, Tellus. 9, 92.
Behannon, K. W. 1970, J. Geophys. Res.. 75^ 743.
Belton, M. J. S. 1965, A. J., 70, 451.
Belton, M. J. S., and Brandt, J. C. 1966, A£. J. Suppl.. 13, 125 (No. 117),
Bessel, W. 1836, A.N., 13, 185.
Biermann, L. 1951, £s. £_. .Ajg., 29j 274.
Brandt, J. C. 1967, Ap. J., 147., 201.
Brandt, J. C., Harrington, R. S., and Roosen, R. G. 1973, Aj>. J., 184, 27.
Bredichin, Th. 1884, Ann. Moscou Obs.. !£, 7.
891
DISCUSSION
B. Jambor; If I understand you correctly your Bredikhin approach to the
problem fails because you do not get enough curvature in the tail; I would be
curious to know what a more precise approach like the Finson-Probstein, not
relying on approximations of series expansion, would yield.
J. C. Brandt; Yes. You know, I took that solely because for years that
has been the definition of a Type 1 tail. These Type 1 and Type 2 appear in
Bredikhin1 s papers, and I was curious as to how this got started.
D. J. Malaise: The windsock model has the nice feature that you can
compute the shape of the tail. Has it not the drawback that you have to drop the
assumption that the tail lies in the orbital plane of the comet. Even small de-
partures from the orbital plane makes the computation of the true direction of
the tail quite indeterminate in some projection situations.
J. C. Brandt; Now clearly, you can create such a comet. The Comet
Mrkos was one such thing. It was at 90 degrees inclination, and that's going to be
a problem. But with any care at all, it's not a problem.
K. Jockers: Your windsock model is the model of a tail which can with-
stand any tension along its axis but has to be in lateral momentum equilibrium.
The small curvature of the tails is caused by a diverging but stationary solar
wind flow field. How can this model be applied to an evidently non-stationary
situation as on Jan. 20?
J. C. Brandt; I think, if you stop and think about it, that you can make
qualitative statements about what happened.
K. Jockers; You know, that windsock has to respond to the changing
non-stationary situation and that is completely different than that line you have
calculated.
J. C. Brandt; It is not necessarily completely different from the line,
but it may be. But if you know how to calculate that, why don't you let me know
and we'll do it.
892


The wind-sock theory of comet tails

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