^{1}
Institute of Space and Astronautical Science, 3-1-1 Yoshinodai,
Sagamihara, Kanagawa 229, Japan

^{2}
Institute of Geophysics and Planetary Physics,
University of California, Los Angeles, CA 90095-1567

Understanding the interaction of planetary obstacles with the solar wind is fundamental to the entire field of space physics. However, some details of magnetohydrodynamics and even hydrodynamics lead to confusion and the use of incorrect assumptions. One area of confusion involves the position of a bow shock in front of a planetary obstacle; especially as the upstream magnetosonic Mach number approaches unity. Another area concerns the Newtonian approximation along the surface of an obstacle to the solar wind. In particular, confusion arises when the obstacle itself is defined as a boundary across which pressure must be balanced. Again, problems with the Newtonian approximation at the magnetopause are most evident for low upstream Mach number. We investigate the analytic hydrodynamic and MHD formulations across the bow shock and along the boundary of obstacles to the solar wind flow, and offer better analytic approximations.

The standoff distance of a detached bow shock from a blunt obstacle has been of interest from the earliest studies of supersonic aerodynamic flow (Laitone and Pardee, 1947, Nagamatsu, 1949, and many others). The position and shape of the bow shock is such as to allow all of the shocked fluid to flow between the shock and the obstacle. Since these relations contain no inherent length scales, however, the position and shape of the shock is extremely difficult to derive from gasdynamic or hydrodynamic relations. Instead, assumptions and approximations are used to estimate the curvature of the shock front, the form of the stream functions, and the vorticity and distribution of pressure downstream from the shock. Thus, progress in the theoretical understanding of the shape and size of a detached shock wave in front of a blunt obstacle is closely linked to the success of computer simulations and laboratory experiments.

It has long been known (Hayes, 1955) that the thickness of the sheath along the stagnation streamline divided by the obstacle standoff distance is proportional to the density ratio across the shock at high upstream Mach numbers. Seiff (1962) deduced a constant of proportionality for the case of a shock wave in front of a sphere , based on previous experimental results. Increased scatter in the experimental results for low density ratios (low upstream Mach numbers), however, suggested that this relation may not always be valid.

Spreiter *et al.* (1966) used a
Rankine-Hugoniot relation to rewrite the relation of Seiff (1962) in terms of the upstream sonic Mach
number. The coefficient 0.78 was also replaced with 1.1, based
on results of wind tunnel experiments using an ellipsoidal model
with the approximate shape of the magnetopause. The resulting
relation agrees well with spacecraft observations at high Mach
numbers, but it was cautioned that this formulation only be used
for upstream sonic Mach numbers () greater than 5. An extrapolation to = 1 reveals a ratio of shock to
obstacle standoff distances of 2.1; a value with no physical
basis, and is contrary to the expectation that the gasdynamic
shock retreats infinitely far from the obstacle as the Mach
number approaches unity (Landau and Lifshitz,
1959). Farris and Russell (1994)
approached this shortcoming from a different perspective. It was
noted that a simple relation involving the downstream sonic Mach
number resulted in the same asymptotic value as the density
ratio across the shock at high upstream Mach numbers. This
relation also causes the ratio of bow shock to obstacle standoff
distances to retreat to infinity as the Mach number decreases
towards unity, and is written as:

Eq. 1.

where the magnetosonic Mach number has replaced the sonic
Mach number, *D _{OB}* is the distance from the
focus of the ellipsoid to its nose, and

The position of the shock subsolar point from the obstacle standoff position can also be slightly influenced by the upstream magnetic field. The solution for the density ratio across the bow shock was solved for arbitrary by Russell and Petrinec (1996), and couples the 'switch-on' shock solution to the third real solution when = in the appropriate regions. Russell and Petrinec then used the conjecture of Farris and Russell (1994) to determine the shock subsolar position (Figure 1). This work was done to correct the claim by Cairns and Grabbe (1994) that there exists a factor of ~4 change in the shock standoff distance from the ob-stacle at low Alfven Mach numbers between = and other angles. The error with this result is that specific values of were inserted into the cubic equation derived from the Rankine-Hugoniot relations before they were solved. This produces 3 real roots at = , as this procedure decouples the 2 equivalent 'switch-on' shock solutions from the third solution. The 'switch-on' shock solution was discarded by Cairns and Grabbe (1994), leading to an incorrect estimate of the shock distance from the obstacle at low Alfven Mach numbers. While it is true that the upstream magnetic field can influence the shock subsolar position, the differences are not as large as had been claimed for low Alfven Mach numbers by Cairns and Grabbe.

Fig. 1. The ratio of the bow shock to magnetopause subsolar
positions, for *a*) low plasma . *b*) high plasma . (adapted from Russell and
Petrinec (1996))

It should be noted that when the upstream magnetic field is neither aligned with nor perpendicular to the upstream velocity, the velocity downstream from the subsolar bow shock obtains a tangential component (this also occurs for the 'switch-on' shock). Under these conditions the streamline which passes through the shock subsolar point is not the streamline that stagnates at the magnetopause nose (Walters, 1964). This does not change the calculated density ratio across the shock, but could cause small changes in any relations used to determine the distance of the shock from an obstacle.

As noted by Grabbe and Cairns (1995),
several contradictory studies exist regarding the dependence of
the distance between the bow shock and obstacle standoff
distances on the upstream Alfven Mach number. Computational
solutions by Spreiter and Rizzi (1974)
found a decrease in the standoff distance ratio between the bow
shock and obstacle with decreasing Alfven Mach number (for = ). This claim has received observational support
from an empirical study by Peredo *et al.*
(1995). However, this result contrasts with the
computational study of Cairns and Lyon
(1995) ( = and ) and theoretical studies by Cairns and Grabbe (1994), Grabbe and
Cairns (1995), Farris and Russell (1994),
and Russell and Petrinec (1996) (the latter
two using magnetosonic Mach number). Additional studies are
needed to better understand the exact role of the magnetic field
on the bow shock position.

The magnetopause is at its most basic definition the boundary
across which the pressure of the magnetosheath is balanced by
the pressure produced by the Earth's intrinsic magnetic field
(and a very small contribution from the interior thermal plasma
pressure of the magnetosphere). While total pressure is balanced
across the bow shock and across the magnetopause in the
equilibrium state, the total pressure at the magnetopause is
*not* equal to the total pressure across the bow shock,
since the flow parameters within the magnetosheath must change
to satisfy the hydrodynamic relations and deflect the plasma
flow around the obstacle.

In general, the magnetic field of the plasma precludes us from fully understanding analytically the plasma flow parameters in the magnetosheath region in terms of the solar wind parameters. However, much can be understood from consideration of the stagnation streamline in the hydrodynamic regime, since it lies closest to the magnetopause. The Rankine-Hugoniot relations across the shock are written as:

Eq. 2.

Eq. 3.

where is the solar wind
sonic Mach number and *M _{s}* is the downstream
Mach number. Bernoulli's equation and the adiabatic flow condition are used between the downstream
side of the bow shock and the obstacle stagnation position to
determine the stagnation thermal pressure. By replacing Eq. 2
for

Eq. 4.

The thermal pressure along the obstacle surface is then determined from the Newtonian approximation, and the density and velocity are determined from the adiabatic condition and Bernoulli's equation, respectively:

Eq. 5.

Eq. 6.

Eq. 7.

where *Q* is used to indicate the uncertainty in the
Newtonian approximation. In Eqs. 5-7, defines the angle between the upstream flow
velocity vector and the normal to the obstacle. We next examine
these relations, using different functions in place of *Q*.

The formulation is simple
and useful for pressure balance at the high Mach number dayside
magnetopause. It has been used by numerous authors, with various
values for *k*. A proper understanding of the value of
*k* was first considered in magnetospheric calculations by
Spreiter *et al.* (1966) (see Eq. 4),
and subsequently used in later studies (though not always). This
relation breaks down as
approaches , however, because
the exterior pressure approaches zero. This implies that either
the magnetotail radius never reaches an asymptotic value far
downtail or the total pressure interior to the magnetopause
becomes zero far downtail; both of which are contrary to
observations. In addition, the magnetosheath velocity along the
magnetopause surface from Eq. 7 exceeds the solar wind velocity
as approaches (Figure 2). This cannot happen in
a hydrodynamic flow, as no source exists for the additional
kinetic energy (Spreiter *et al.*,
1966). Thus, this relation clearly is a poor approximation
for the far downtail region.

The relation has been
used by several authors, and provides reasonably accurate
solutions for large upstream Mach numbers. Often the solar wind
magnetic field pressure is also added to the right hand side,
and is replaced with
*P*_{static}. This equation again is an
approximation. The solar wind pressure contributions are added
to provide a finite external pressure to the magnetosphere as
approaches . Here we consider the case without a solar wind
magnetic field. While the magnetosheath thermal pressure and
density are now non-zero at =
, the stagnation thermal
pressure now does not agree with that derived in Eq. 4. Even
worse, while the speed remains less than the solar wind speed
far downtail for = 1.1, it
is also imaginary in the subsolar region and over much of the
dayside magnetosphere.

We propose that the relation is the simplest formulation which satisfies the hydrodynamic requirements both at the stagnation position and in the far downtail region. This also appears to be the only solution for which parameters vary monotonically along the obstacle. Thus at a minimum it should be used for boundaries which are defined by a balance of pressure. This has been used in earlier aerodynamic studies (Linnell, 1957; Daskin and Feldman, 1957), but not in magnetospheric applications.

Fig. 2. Thermal pressure, mass density, and velocity along an obstacle surface for an upstream sonic Mach number of 1.1 and = 5/3, and is normalized to the values upstream from the shock.

Hayes, W. D., Some aspects of hypersonic flow, The Ramo-Wooldridge Corp., (1955).

Landau, L. O., and E. M. Lifshitz, *Fluid
Mechanics*, Pergamon, New York, (1959).

Linnell, R. D., Hypersonic flow around a
sphere, *J. Aero. Sci.*, 65-66, (1957).

Nagamatsu, H. T., Theoretical investigations
of detached shock waves, *GALCIT Pub.*, (1949).

Seiff, A., Recent information on hypersonic
flow fields, *NASA SP-24*, 19-32, (1962).

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