* Originally Published In: Journal of Geophysical Research, 101*, 7677-7678, 1996.

Copyright © 1996 American Geophysical Union

S. M. Petrinec

*
STELAB, Nagoya University, Toyokawa 442, JAPAN*

In their recent paper *Cairns et al.*
[1995*a*] conclude that when the magnetosonic Mach
number is low there can be substantial changes in the
standoff distance and/or shock shape. The purpose of this
note is not to disagree with that statement which is in
accord with our earlier work [*Russell and
Zhang*, 1992] but to suggest a method whereby these
two changes can be separated from each other so that their
analysis can be made less ambiguous.

Over the forward part of the magnetosheath the flow is
subsonic and much of the shock remains in communication with
much of the obstacle. Thus the shape of the obstacle plays an
important role in determining the shape of the shock and
hence the forward part of the terrestrial shock is found to
be elliptical in contrast to the hyperbolic shape it is
expected to have far behind the Earth. The shape of the
obstacle also plays a role in determining the standoff
distance. A pointed obstacle would have a much smaller
standoff distance than a sphere. A sphere would have a
smaller standoff distance than an ellipse (with the Earth at
the focus) etc. Physically, the standoff distance depends on
the radius of curvature of the obstacle at the stagnation
point in the flow and thus is insensitive to the choice of
the origin of the coordinate system. *Spreiter et al.* [1966] gave a standoff
formula as a function of Mach number for a specific shape, an
elliptical, magnetosphere-like obstacle, with distance
measured from the center of the Earth. All later work
including that of *Cairns et al.*
[1995*a*] has followed this procedure of measuring
the standoff distance from the center of the Earth, rather
than comparing with the radius of curvature of the obstacle
thus making the standoff ratio coordinate-system dependent.

The reason the shock front sits anywhere is to allow all
the shocked plasma to pass between the shock front and the
obstacle. If the plasma is compressed less as when the Mach
number drops to low values, then the shock must move away
from the obstacle. The failing of the *Spreiter et al.* [1966] formula is that
it does not predict this motion correctly at very low Mach
numbers. Since the shock should move to infinity when there
is an infinitesimal compression of the flow in order that the
(slightly) shocked flow can move around the obstacle, and
since this statement is true for a gas or a plasma, we need
not look to an MHD effect to explain the failing of Spreiter
et al.'s formula. *Farris and Russell*
[1994] proposed an alternate formula equally valid for
gas and magnetohydrodynamics that appears to redress this
failing. More tests of this formula are certainly in order.
However, in order to carry out such tests it is important to
separate dynamic pressure effects from Mach number effects,
and shape changes from size changes. The methodology of *Cairns et al.* [1995*a*] does not do
this adequately.

The equation of a conic section is:

where *R* is the distance from the origin to the
curve; *K* is the semilatus rectum, the point on the
curve at right angles to the line joining the origin (at the
focus) and the "nose" of the conic section;
is the eccentricity
of the ellipse; and
is the angle between the
line joining the nose and the origin and the line joining the
origin and any point on the curve. The parameters *R*
and *K* have dimensions of length;
has dimensions of angle;
and
is dimensionless
as befits a shape parameter. The eccentricity,
, is greater than 1
for a hyperbola; equals 1 for a parabola; and is less than 1
for an ellipse. *Cairns et al.*
[1995*a*] assume
=1, despite the
evidence from previous work that
is significantly
smaller than 1. [*Holzer et al.*,
1966; *Slavin and Holzer*, 1981;
*Farris et al.* 1991]. This formula
may be rewritten for
=1:

If we equate this formula with that of *Cairns et al.* [1995*a*] with its
focus at the Earth we find that *a _{s}* =

In order to separate the effects of shape changes, we
recommend the use of the eccentricity as used by many
previous authors [*Holzer et al.*,
1966; *Slavin and Holzer*, 1981;
*Russell*, 1985]. Since it is
dimensionless it does not vary as the scale size as the
parameter *b _{s}* does, and will be much less
sensitive to changes in dynamic pressure than

In the following reply *Cairns et al.*
[1995*b*] present new simulation results [*Cairns and Lyon,* 1995]. We note first
that the shock location moves to very large standoff
distances at low Mach numbers as expected [*Russell and Petrinec*, 1995] showing that
the treatment of *Cairns and Grabbe*
[1994] is an incorrect approach. *Cairns
et al.* [1995*b*] plot the shock position versus
Alfvenic Mach number and not the magnetosonic Mach number.
When the more physical comparison is made with the
magnetosonic Mach number, the simulation result is
qualitatively the same as Farris and Russell's model but
still on the high side. In contrast the low Mach number gas
dynamic simulation of *Spreiter and
Stahara* [1995] is qualitatively similar to the Farris
and Russell model but on the low side. Since the new MHD
model does not approach the asymptotic, and non
controversial, limit expected at high Mach numbers, we think
the problem may be with the MHD model. Inspection of Figure 2
of *Cairns and Lyon* [1995] suggests
the problem is in the simulation itself. For a perpendicular
shock, the fractional jump in density, in magnetic field
strength, and the drop in velocity should all be equal, but
they are not. The density has the correct jump for the stated
conditions but the magnetic field jump and velocity drop are
incorrect. On a more minor note, we point out that the reason
*Farris et al.* [1991] did not consider
shape changes is that they restricted their attention to
*M _{ms}* > 4.5.

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