Surface albedo map

The purpose of our analysis is to measure the position and flux of the transient clouds recorded in the 24 nightly mean images included in this study. To do this, we must estimate

the PSF of each mean image, and the light contributed by scattering of sunlight at Titan’s surface in the stratospheric haze. The approach which we will take is to first determine a rough map of Titan’s surface albedo at 2.0µm by combining the 16 Palomar images. Using this surface map and the mean haze opacity structure determined in Ch. 3, we compute model images of Titan and solve for the PSF which best describes the transformation between modeled and observed images. Subtracting psf-convolved model images from those observed, we can accurately measure the position and flux of the transient clouds.

We compute a map of Titan’s relative 2.0 µm surface albedo using a technique similar to that described in Ch. 2 for the analysis of 940-nm images. The first step is to correct the K0 images for the light scattered off stratospheric haze. This is done in an ad hoc

fashion by subtracting a scaled Brackett-γ (Brγ, 2.16–2.18 µm) filtered image, taken on 23 September 2002. The brightness of the eastern and western limbs is strongly asymmetric in this haze-only image, due to the 6◦.4 phase angle of Titan at the time of the observations. We radially averaged this image, then subtracted an appropriate linear combination of the radial average and the actual Brγ image (or its mirror-image) to match the observed limb asymmetry of the K0 images. The difference images record only sunlight scattered below the altitude at whichτCH₄ ≈1 for the Brγ filter, approximately 50 km (Roe et al., 2002).

Using a modification of the technique of Smith et al. (1996), we solve for the relative albedo at each location on a latitude-longitude grid, while simultaneously determining a relative photometric scaling factor for each haze-subtracted image. We model the recorded radiance I_i,j at each location i on Titan’s surface in haze-subtracted image j in terms of the albedo A_i at zero phase, a photometric scaling factorα_j, and a simple Minnaert-type phase function with exponentk, as

α_jI_i,j =A_i(µ_ijµ0,ij)k, (6.1)

whereµis the cosine of the observer’s zenith angle, andµ0 is the cosine of the solar zenith angle. We find that k = 1 adequately describes the phase behavior of Titan’s surface in these haze-subtracted images. For Gaussian-distributed uncertainties in the measured radiancesσ_ij, the best estimate of the surface albedoA_i at each location will be that which

minimizes the χ2 function

(χ2)_i=X j

(A_iµ_ijµ0,ij−αjIij)2

(α_jσ_ij)2 . (6.2)

Solving for A_i at the minimum, we find that

A_i= P j I ijµijµ0,ij αjσ2_ij 2P_j µ2ijµ20,ij α2jσij2 , (6.3)

where the sums are taken only over the images in which the location i is viewed at less than a maximum emission angle of 65◦. For any set of photometric scaling factors α_j, the optimal distribution of surface albedoA_i can be directly calculated with Eq. 6.3.

However, both the absolute and relative photometric calibration of the Palomar images are unknown a priori. Though we cannot determine the true surface albedo distribution from these data alone, we can use the above model to solve for a map of the relative surface albedo A0_i, while simultaneously determining the relative photometric calibration of the images α0_j. We used the Levenberg-Marquardt algorithm (Mor´e et al., 1980) implemented in IDL by Markwardt (2003) to numerically optimize the 16 photometric scaling factorsα0_j

by minimizing theχ2 function

χ2 =X i X j (A0_iµ_ijµ0,ij −α0jIij)2 (α0_jσ_ij)2 , (6.4)

where the sum over j is again taken over the images which view each location i, and the relative albedo map A0_i is recalculated at each iteration using Eq. 6.3.

In an important final step, we normalize the relative surface albedo map A0_j to the 2.0µm albedo values determined for selected regions of Titan’s surface (with lower spatial resolution) by fitting a radiative transfer model to resolved spectra of Titan (Ch. 3). The mean surface albedo of the central region of Titan’s sub-earth hemisphere on 26 September 1999 (central longitude 57.4◦W, emission angle <65◦) was found to beAmean= 0.09, with a range from 0.05±0.01 to 0.14±0.02. The calibrated surface albedo map is displayed in Fig. 6.5.

It was necessary to correct this surface map for the continual presence of clouds south of 60◦S before using it to generated model images of a cloud-free Titan. This was done

Figure 6.4: Crude map of Titan’s surface albedo at 2.0µm, derived by fitting a radiative transfer model to resolved 2.0–2.3µm spectra of Titan taken on 26 September 1999 (Ch. 3, see Fig. 3.8). Only those locations viewed at an incidence angle of <65◦ are mapped.

Figure 6.5: High resolution map of Titan’s surface albedo at 2.0 µm, uncorrected for tro- pospheric clouds at the south pole. A relative albedo map calculated from 16 Palomar AO haze-corrected K0 images was normalized to spectrally-derived albedos (Fig. 6.4) north of 55◦S. The highest and lowest albedos detected north of the polar clouds are 0.17 at (115◦W, 13◦S) and 0.03 at (259◦W, 5◦S). The region north of 40◦N was not observed.

Figure 6.6: Titan’s rotational lightcurve. The integrated flux in 16 Palomar K0 images, corrected for the changing distance between Titan and the Earth and Sun, is displayed with black dots, with 1-σerror bars. A 3rd-order Fourier series fit to these data is shown by a gray line. A compilation of K0 photometric observations made over the period 1989–1995 is shown in diamonds, reproduced from Griffith et al. (1998). The amplitude of Titan’s lightcurve has decreased as our view has become dominated by Titan’s south polar region. The photometric effect of Titan’s daily transient clouds is insignificant at this scale.

simply by asserting that the surface albedo everywhere south of 60◦S is equal to 0.07, the median value over the region 60◦S–40◦N. Similarly, the region north of 40◦N, which was never sampled in Palomar images at an incidence angle of less than 65◦, is also assumed to have an albedo of 0.07.