Remote Sensing of Saturn’s Bow Shock
Chrystal Moser & Charles Stine
UNIVERSITY OF NEW HAMPSHIRE
To state the problem mathematically we
consider the general formula for a parabolic
shock in terms of two parameters, and the
equation for the path of the Langmuir wave,
which is a tangent line to the shock:
For our first approach in solving this
problem we use an numerical method.
Combining the above equations results in a
quadratic equation in terms of t:
PRAM [Pa]
In order to ensure that we have a tangent
point, t can only have one solution, which
occurs when the value under the square root in
the quadratic formula is equal to zero:
The numerical program takes this
equation, which contains both As and Bs, and
starts by picking a small As value and
evaluating the equation for many different Bs
values; then it increases As by a small amount
and repeats the process. Thus it tests every
combination of the two over a plausible range
for each, and it writes down those combinations
which make the equation close to zero.
Our second approach is analytical, when
the above equation is fully expanded and the
known quantities are grouped it looks like this:
Where:
The analytical approach leaves us with
two solutions, however we see that one of the
two solutions is trivial to discount. The
remaining solution matches the with the
solution from the numerical method precisely. In
the end, by comparing the sets of possible
answers each method produced, we are able to
find a single common answer, one that makes
physical sense. By doing this for each pairing,
we hope to find correlations between the rampressure at a certain time, and the shape of the
bow shock at the same time.
Average RamP
2.183
2.642
2.418
2.45
3.004
2.91
2.772
2.423
2.263
2.604
2.959
2.246
2.774
1.165
As
30.9581
31.0688
31.0269
28.7128
28.9833
28.8526
43.231
31.2587
23.5219
28.1011
24.756
23.4566
25.8247
23.6177
Bs
0.0343
0.039
0.0372
0.0332
0.043
0.0378
0.0403
0.0344
0.0399
0.0297
0.0324
0.0319
0.0245
0.0518
BS [RS-1]
As the planet Saturn orbits the sun, it's
magnetic field carves a wake through the
solar wind. This wake is similar to that of a
speed boat across a calm lake: the solar
wind particles collide with the magnetic field
of the planet and pile up until they fall to the
side. This is the planet's bow shock.
Changes in speed and density of the solar
wind cause the shape of the shock to
change over the course of time. When a
certain set of solar wind conditions happen
to occur, a Langmuir wave is released along
a tangential trajectory to the shock. These
waves can be intercepted by satellites such
as the Voyager 1 and 2 probes. From that
information, and knowing the position of the
satellite when it picked up the Langmuir
wave, it is possible to calculate the exact
shape of the shock. We seek to use the data
from multiple Langmuir waves to find the
shape of the bow shock for different sets of
solar wind conditions, and find correlations
between the two. The result will make it
possible to predict the shape of Saturn's
bow shock from the characteristics of the
solar wind, which is what we are looking for.
Results
AS [RS]
Methods
Introduction
That is a single equation with two
unknowns, which cannot be solved. We deal
with this by pairing up lines in the data table
that have the same solar wind conditions.
Each line gives us a version of that equation,
and because the conditions are the same in
each line of the pair, we know that the As and
Bs values are the same as well. That leaves us
with two equations and two unknowns. We
then perform elimination, and use the quadratic
formula to get a pair of solutions:
PRAM [Pa]
Having produced a table of average ram
pressure across each pair, we proceed to plot
our results. It becomes apparent that there is
quite a poor correlation. We think this is
because our tolerance for choosing similar ram
pressures, while pairing our events, was too
high. We intend to continue working on finding
different approaches to our calculations, and
how we represent our data, in the hope that a
useful correlation may yet arise.