General approach

The observable quantities which we use in this analysis are the relative position of the two stars in detector coordinates and the ratio of their flux, both as a function of time. The relative position, in particular, can be measured with extraordinary precision. The angular deflection and flux of starlight which interacts with Titan’s atmosphere will be a function of the geometry of the occultation and atmospheric properties of Titan such as composition, scale height, zonal wind field, etc. In addition, since we are not in fact measuring the angular deflection or flux of the starlight directly but rather the relative separation and recorded flux of two stars on a detector, several other instrumental parameters must also be considered. We will ignore these instrumental effects for now, and present first a model for the propagation of starlight through Titan’s atmosphere and to an observer on Earth. Instrumental effects are addressed in Section 4.4.7.

Our primary interest in this chapter is to determine the zonal winds in Titan’s strato- sphere, whose presence is reflected both in the position of refracted images on Titan’s limb and, to a far lesser extent in the flux of these images. We also consider the effects of tem- perature and haze. Other observed phenomena, such as scintillation due to kilometer-scale pressure variations in Titan’s stratosphere, are ignored and treated only as a source of noise. The model proves far too complex to allow a direct inversion of the measured stellar positions and fluxes to retrieve atmosphere parameters. Instead, we compute predicted observables, effectively creating a synthetic dataset based on a small set of parameters, and optimize the parameters to minimize deviations between the synthetic and actual observations.

Figure 4.7: Relative position and flux of stars during successive occultations by Titan, in detector coordinates xdet and ydet.

We use a numerical technique based on previous analytic models of stellar occultations by Pluto, Saturn, and Titan. nocite1992AJ....103..991E Elliot and Young (1992, hereafter EY92), derived equations for the expected angular deflection and flux of light refracted through a weakly bound atmosphere, one in which the scale height is a non-negligible frac- tion of the body’s radius. They include the effects of gradients in atmospheric temperature, composition, and opacity and assume the planet and its atmosphere are spherically sym- metric. This model successfully reproduces most of the phenomena which we observe (with the notable exception of the scintillation), but significant deviations from the predicted stellar positions occur near deepest occultation of both stars, which we will demonstrate are caused by strong winds in Titan’s stratosphere.

The effect of a non-spherically symmetric atmosphere on the propagation of starlight was addressed by both H93 and N95 in their analyses of the central flash produced by the Titan and Saturn occultations of 28 Sgr. Both authors, however, used the simplifying approximation that the central flash is produced by refraction at a single surface of constant refractivity in the atmosphere. The models of H93 and N95 therefore predict only the direction of the refraction of starlight, equivalent to predicting only the position angle of refracted images along the limb. We initially attempted to combine the techniques of EY92, H93, and N95 to develop an analytic model for both the direction and magnitude of the angular deflection of starlight incident on an axially symmetric (but non-spherical) atmosphere. However, the tilt of Titan’s polar axis (the presumed axis of symmetry of the atmosphere) with respect to the direction of the incoming starlight renders integrals of atmospheric parameters along the path of the ray prohibitively complex. It proved simpler and more intuitive to calculate such integrals numerically.

Though the geometry of the current problem is complex, we can make several simplifying assumptions regarding Titan’s atmosphere which could not be made in the aforementioned studies. First, Titan’s atmosphere can be considered isothermal over the region of interest (240–600 km above Titan’s surface). As shown in Fig. 4.4.1, radiative transfer models of Titan’s atmospheric temperature structure based on Voyager observations (Lellouch, 1990; Yelle et al., 1997) predict at most a 30 K difference in temperature between 10−7 and 3×10−3 bar. These models are supported by retrievals of the temperature profile from lightcurves measured during the occultation of 28 Sgr (H93), though the scintillations seen in both those lightcurves and the present observations demonstrate that fine scale temper-

Table 4.1: Variables used in the occultation model Coordinate systems

R, φ, λ Planetocentric radius, co-latitude, longitude

ρ, θ, z Cylindrical coordinates in planet plane

ρ0, θ0 Polar coordinates in observer plane

x0, y0 Cartesian coordinates in observer plane

α, δ Relative right ascension and declination

xdet, ydet Relative detector coordinates

Atmospheric parameters

T, P, N Temperature, pressure, number density

ν Refractivity

Other important variables

Θ,Φ Deflection angle, normalized flux of starlight

ature structure does exist (Sicardy et al., 1999). We furthermore assume the atmospheric composition to be unchanging over this region, as reflected in the mean molecular weight. Titan’s homopause lies well above the region of interest, near 1000 km (M¨uller-Wodarg and Yelle, 2002).

Figure 4.8: Radiative transfer models of the temperature structure in Titan’s stratosphere by Yelle et al. (1997) (solid lines) and Lellouch (1990) (dashed line).