QUESTION:
How were the density, pressure and temperature profiles of Mars'
atmosphere
deduced from the deceleration recording during the EDL?

ANSWER from Tim Schofield on November 30, 1998:
The derivation of density, pressure and temperature profiles from
deceleration measurements is summarized very briefly in a paper in
Science
(Schofield et al. 1998. Science, 278, 1752-1757). A much more detailed
paper will appear in the Journal of Geophysical Reasearch in a few
months.
The raw acceleration data and the profile of the results will also be
made
available in a few weeks through the NASA planetary data system (PDS)
(http://pds.jpl.nasa.gov/index.html).

The atmospheric density profile is derived from the following pieces of
information supplied by Pathfinder.

1. Deceleration measured as a function of time as Pathfinder enters the
atmosphere of Mars.
2. The trajectory of the entry vehicle (position and velocity as a
function of time).

The trajectory is calculated from initial navigation data, just before
Pathfinder enters the atmosphere, and the deceleration measurements
themselves. It is modelled from the initial conditions, assuming that
the
entry vehicle is influenced only by the gravity of Mars and atmospheric
forces. The acceleration measurements detect the atmospheric forces,
which,
to a good approximation, are directed along the axis of symmetry of the
entry vehicle and along its direction of motion.

Deceleration and density are related by the following expression.

              rho = acc * 2m / (Cd A V**2)

where rho is atmospheric density, acc is deceleration, and, m, A, V and
Cd
are entry vehicle mass, cross-section area, velocity, and drag
coefficient.
acc is measured, m & A are known constants, V is derived from the
modelled
trajectory, and Cd is a function of V and atmospheric pressure, and is
derived from Viking-era tests and fluid dynamical models of the entry
vehicle.

The equation above gives density as a function of time, which can be
converted to density as a function of altitude using the trajectory
data.
Pressure as a function of altitude can be derived from density as a
function of altitude using the hydrostaic relation:

             dP(z)  =  - rho(z) g(z) dz

where P(z) is pressure, g(z) is the acceleration due to gravity, and z
is
altitude. Given pressure and density at a given level, temperature can
be
derived from the equation of state (gas equation), given the mean
molecular
weight of the atmosphere.

There are several details which influence the analysis. These include
the
rotation of Mars, the shape of the Mars gravity field, the variation of
atmospheric molecular weight with altitude at the highest atmospheric
levels, and the choice of a first guess temperature (pressure) at the
top
of the atmosphere for the integration of the hydrostatic relation. All
of
these have been considered in the latest analyses of the data.