07-05-2013, 04:04 PM
Analytical solutions for anisotropic MHD shocks
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Abstract.
A new method to analytically solve the anisotropic
MHD system of equations describing shock transitions is
presented. As this system is known to be under-determined
(there is more unknown parameters than available equations)
free parameters must be chosen. From observational contraints
it appears that the magnetic amplitude jump is a good
candidate as it is generally available more frequently and
more precisely than other jump variables. With this approach
we obtain an explicit expression for the density compression
ratio for arbitrary upstream parameters and shock geometry.
Downstream anisotropy and pressure are also calculated.
The results are tested against an other approach and compared
with observations from the Earth’s bow shock and the
solar wind termination shock.
Introduction
The MHD formalism describing transitions across shocks
has been employed successfully in many astrophysical situations.
The general goal is to predict downstream conditions
from the knowledge of upstream conditions and shock
geometry. The latter is characterized by the shock angle
Bn between the upstream magnetic field and the shock normal.
From this prediction it is possible, for instance, to
get insight on the wave generation processes at work in
the downstream regions of planetary bow shocks or solar
wind termination shock, namely magnetosheaths or the heliosheath.
Temperature anisotropy instabilities are among
the most common means to generate waves. Consequently
the formalism adopted must account for pressure variations
in directions parallel and perpendicular to the ambiant magnetic
field.
Full analytical resolution
The main challenge in solving the system of Eqs. (1–6) is
to eliminate the right unknown at each step. To get a full
analytical solution in the end one should look for simple expressions
(first or second order) of each variable.
Observational tests
Comparison with Earth’s bow shock data
The applicability of RH jump conditions to observed shocks
has been verified (for instance Winterhalter et al., 1984). To
validate the present approach we use six bow shocks crossings
referenced in Chao et al. (1995) (see Table 1). They
all correspond to low Mach number solar wind conditions.
Alfv´en Mach numbers in Table 1 are computed from Eq. (12)
in Chao et al. (1995). The last three columns of Table 1 display
the density compression ratio as it is observed, from
our Eq. (18) and from equations of Chao et al. (1995) respectively.
For a given shock, differences between the three
values are very small. First, our (direct) method gives results
very close to those obtained by the method of Chao et
al. (1995). Slight discrepancies may come from our use of
calculated MA (round values instead of exact). Second, our
calculated ratios agree very well with observed values.
Conclusion
The analysis developed in this work is intended to complete
general studies on anisotropic MHD shocks by giving, for
the first time, a full analytic expression of the density compression
ratio as a function of the upstream parameters and
shock angle and strength. It has been validated by comparison
with another method and observations in different astrophysical
contexts. Such compact formula may be used to
easily compute downstream parameters when only magnetic
measurements are available and when upstream parameters
can be inferred (when plasma data are absent, in the case of
Voyager 1 for instance). It is also possible to analyze the
sensitivity of the results to uncertainties in the inputs and to
propose error bars.