The motion of the Huygens Probe in Titan's atmosphere is determined by the action of two forces, gravity and wind drag. The acceleration due to wind drag is taken from Flury (1986) to be:
where is the atmospheric density, is the relative velocity of the Probe with respect to the atmospheric wind, and is the ballistic coefficient (units: kg/mē) defined by:
with m = probe mass, drag coefficient, and A = effective probe area to the flow.
The wind velocities in the (x,y,z) directions of a Cartesian
coordinate system on Titan's surface are given by:
u = zonal wind (x-axis: positive toward east)The Probe's equation of motion during the Titan descent phase can thus be written:
v = meridional wind (y-axis: positive toward north)
w = vertical wind (z-axis: positive upwards)
where is Titan's gravitational acceleration, and
The meridional and vertical winds, v and w, are assumed to be considerably weaker than the zonal wind u. In this case, the latitude of the Probe (y component) remains approximately constant.
Knowing the Probe's velocity on Titan, it is not difficult to determine the Doppler shift projected onto the PRL ray path back to the Orbiter:
The vector defines the line-of-sight of the PRL. is negative during the Titan descent, so that the received frequency is increased (blue shifted). The PRL Doppler shift (9) is 37.6 kHz for the nominal orbiter starting approach velocity of -5.53 km/s. The velocity of the Probe projected onto the line-of-sight can be written:
= probe velocity w.r.t. Titan center = orbiter velocity w.r.t Titan center = unit vector pointing from Orbiter to Probe
Furthermore, we define the projection of the orbiter velocity onto the line-of-sight by:
The angles and define the direction from the Probe to the Orbiter in a local Cartesian coordinate system on the Probe oriented along a natural Titan coordinate grid (x-axis positive toward east; y-axis positive toward north; z-axis positive upward). The angle , basically the zenith angle of the Orbiter as seen from the Probe (Ott 1991), is also sometimes designated as the "probe aspect angle" (PAA). The azimuthal angle of the Probe-to-Orbiter line-of-sight is labeled the "line-of-sight azimuth" (LOSA) by some authors (Atkinson 1989, Pollack et al. 1992).
The term in (12), where the mean value of u, the drift velocity due to zonal winds, is the quantity to be determined by DWE. This velocity is co-aligned with the contribution in (13) due to Titan rotation. It is assumed for simulation purposes that the rotation is synchronous with the Titan orbital period ( 11.7 m/s). As noted in the previous section, the relative importance of , and thus the quality of the reconstructed wind profile, is strongly dependent on the values of and .
The term in (14), arising from meridional drift, should be small since is not expected to be important. The term in (15) contains the vertical descent velocity . This quantity can be obtained to a high degree of accuracy either from the range rates deduced from the Probe's proximity sensor data, or using measurements of temperature T and pressure P from the Huygens Atmospheric Structure Instrument (HASI). In the latter case, based on the assumption of hydrostatic equilibrium and ideal gas behavior, the descent velocity is determined from
where variations in H = , the atmospheric scale height, are assumed to be negligible over the altitude range of the Huygens descent. The final error associated with the determination of the reconstructed probe descent velocity is estimated to be of the order of 1%.
Knowing from the proximity sensor or from HASI
measurements, it is possible to reconstruct
from the Doppler shift of the PRL (9).
Using in situ measurements of the density ,
we can then extract the exact height profile from the
motion equations (5) and (7).
Comparitive wind measurements are expected near the surface from two
independent sources on board the Huygens Probe:
In order to provide a rough estimate of the Doppler shifts involved, Fig. 4 shows the results of a simplified calculation of the five line-of-sight velocity contributions to the PRL frequency shift, through . The geometry for the calculations was taken from the nominal Huygens descent profile on Titan (duration: 135 min). The upper and lower panels show the same curves on different ordinate scales.
Figure 4: Velocity contributions to PRL Doppler shift for the nominal Huygens descent profile.
Only the change in the projected orbiter velocity from its initial value, i.e. , is plotted in Fig. 4. The total projected velocity is plotted for the other contributions, all of which happen to be negative (blue shift in PRL frequency) in this example. The 100% nominal zonal wind model was used for computing the zonal wind contribution . The Titan rotation term is nearly constant during the entire descent. The meridional winds in were taken to be a constant +1 cm/s (blowing toward north). The change of parachutes at t = 15 min is clearly seen in the curve for , which is the second largest contributer to the changing Doppler shift.
In addition to the derivation of a zonal wind height profile from
the large-scale drift, DWE may be able to provide valuable science
and navigation services from the small-scale variations in frequency
as a 'by-product' of the analysis.
Two such possibilities are: