Saturn 1991–1993: Clouds and Hazes

ICARUS

119, 53–66 (1996) 0002

ARTICLE NO.

Saturn 1991–1993: Clouds and Hazes1 J. L. ORTIZ, F. MORENO, AND A. MOLINA2 Instituto de Astrofı´sica de Andalucı´a, CSIC, Granada, Spain E-mail: [email protected] Received March 24, 1995; revised August 7, 1995

of further investigation is the temporal changes in the saturnian hazes and clouds. In a recent paper, Ortiz et al. (1993) analyzed 1991 Saturn images at methane bands and nearby continuum wavelengths between 0.6 and 0.96 em. We then pointed out temporal changes in Saturn’s clouds by comparing our results with those of West et al. (1982), and stressed the need for a more frequent and systematic study in order to find out the timescale of these changes and also to identify the physical processes which lead to them. In a subsequent work, Ortiz et al. (1995) presented and compared some results from four different data sets over a 3-year period using the same instrumentation and reduction techniques. The cited work showed unambiguous saturnian cloud changes. In addition, some of the results presented in these papers (such as the differences in continuum reflectivity) suggested the possible existence of a large amount of particles with radii close to the red wavelengths, since the largest changes in extinction and scattering efficiency occur at size parameters of 6–4 for real refractive indices in the range 1.33–1.44 (see Fig. 1). Recently, Karkoschka and Tomasko (1993) modeled the upper saturnian troposphere and stratosphere using Hubble Space Telescope 1991 images. In the upper troposphere they found that spherical particles with radii of 1.5 em provided good fits to their data and could explain some of the Ortiz et al. (1993) results at other wavelengths, which were obtained very close in time to the HST observations. They also found that the stratospheric particles which gave the best fits had radii of 0.15 em. Here, we interpret the most remarkable temporal variations presented in Ortiz et al. (1995) in terms of radiative multiple scattering models in which we incorporate Mie phase functions for particles with several radii. Thus, we include the suggestions on particle sizes. The models presented here also include recent data on the tempera˚ absorption coefficients ture dependence of the 6190 A (Mickelson and Larson, 1992) and some other improvements. A summary of the observing dates and the studied wavelengths is shown in Tables I and II.

Using the data set presented in Ortiz et al. (Icarus 117, 328–344, 1995) we further analyze some of the most outstanding changes from 1991 to 1993 in the saturnian atmosphere. For this purpose, limb-to-limb reflectivity scans at planetographic latitudes 08, 108, 208, 408, and 708 have been studied in terms of multiple scattering radiative transfer models. The large changes in reflectivity and methane absorption at equatorial latitudes from 1991 to 1993 are attributed to a considerable increase in the aerosol abundance at different levels within the extended tropospheric haze.  1996 Academic Press, Inc.

I. INTRODUCTION

Clouds and hazes in Saturn can be studied in a variety of ways, one of which is the observation of the planet in reflected sunlight at continuum and molecular bands. The different absorption coefficients at several molecular wavelengths allow us to probe distinct atmospheric levels, and methane absorption is especially suitable since methane is one of the few well-mixed species in Saturn’s atmosphere. Hence, a number of papers have been published in the past dealing with spectroscopic measurements which covered methane and ammonia bands and H2 quadrupole lines from the Earth (see, e.g., Cochran and Cochran 1981, Baines 1983, Killen 1988, Moreno et al. 1991, Karkoschka and Tomasko 1992) or dealing with direct images obtained through narrow band filters (West et al. 1982). From these Earth-based studies together with photometric and polarimetric measurements from spacecraft (e.g., Tomasko and Doose 1984) among other techniques, a considerable amount of information on the saturnian aerosols has been obtained. However, one of the subjects worthy 1 Based on data obtained at the 1.5-m telescope of the Estacio´n de Observacio´n de Calar Alto, Instituto Geogra´fico Nacional, which is jointly operated by the Instituto Geogra´fico Nacional and the Consejo Superior de Investigaciones Cientı´ficas through the Instituto de Astrofı´sica de Andalucı´a. 2 Also at Departamento de Fı´sica Aplicada, Universidad de Granada, Granada, Spain.

53 0019-1035/96 $12.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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TABLE I Observing Runs

Note. CA stands for Calar Alto Observatory (Spain), and NOT stands for Nordic Optical Telescope (La Palma, Spain).

II. RADIATIVE MODELS

The models used in the present work are based on the vertical structure depicted in Fig. 2. These models are characterized by a number of parameters, functions, and assumptions. For a given set of parameters and functions, one can calculate the amount of reflected sunlight by the model for a specific geometry, compare the results with the observations at the same geometry, and alter the values of the adjustable parameters (by using some iterative scheme) until the model results fit the observational data. The detailed fitting procedure is outlined in the next paragraph. The parameters and functions which describe our atmospheric model were classified in two categories, adjustable and fixed parameters. The adjustable parameters in our fitting algorithms were: g˜ h , th , g˜ cl , tcl , g˜ y , P1 , P2 , P3 , P4 , P5 , where g˜ h , g˜ cl and g˜ y are the single scattering albedos of the stratospheric haze, the tropospheric haze, and the semiinfinite cloud respectively. th and tcl , are the aerosol optical depths for the stratospheric and tropospheric hazes, respectively. P1 , P2 , P3 , P4 , and P5 are the pressure levels of the interfaces (see Fig. 2). Our fixed parameters (parameters not altered during the fitting) were latitude, wavelength, CH4 mixing ratio, H2 molar fraction, P(T) profile, gravity acceleration, integrated absorption coefficient, FWHM of the filter, and the height increment for layering procedures. The P(T) profile was adopted from Lindal et al. (1985). The gravity was set to 9.05 m/sec2 at the equator and corrected for latitude. The methane mixing ratio was fixed to 3.0 1023 (Karkoschka and Tomasko 1992). The height ˚ CH4 increment was 5 km. Recent data on the 6190 A

absorption as a function of temperature by Mickelson and Larson (1992) were used. The rest of parameters had the same values as those used in Moreno et al. (1991). Phase Functions Two groups of models were generated according to the phase functions used. For group A, we used the Tomasko and Doose (1984) phase function for the bottom tropospheric cloud, and Mie phase functions for Hansen (1971) particle distributions with a 5 1.5 em and b 5 0.1 for the extended tropospheric haze. The refractive indices were those of ammonia ice. For the stratospheric haze we used the resulting Mie phase function for a particle distribution with a 5 0.15 em and b 5 0.1. In the clear gas regions we used Rayleigh’s phase function. For group B, we used a Mie phase function obtained from a particle distribution with a 5 0.75 em and b 5 0.1, which described the scattering within the extended tropospheric haze and the bottom cloud. The stratospheric particles were assumed to be spherical with a 5 0.15 em and b 5 0.1. In the clear gas regions we used Rayleigh’s phase function. Once all the values of the parameters are given, we can calculate gas and Rayleigh optical depths for each of the layers by assuming that the gas density follows a scale height distribution (strictly valid for hydrostatic equilibrium) and some other basic laws. For the aerosol distributions, we assumed a constant number of scatterers per unit volume. After the optical depths are calculated one can compute the effective single scattering albedos according to g˜ 5 tsca / ttot , where ttot is the total optical depth (including gas absorption) and tsca is just the optical depth due to scattering. The next step is to solve the radiative transfer problem. For the radiative transfer computations, an ‘‘adding’’ technique was used, and in some cases the results were

FIG. 1. Mie computations on exctinction efficiency (upper graphs), scattering efficiency (middle graphs), and single scattering albedo (lowergraphs) as a function of size parameter, for the refractive indices shown in the labels.

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TABLE II Characteristics of the Narrow Band Filters

checked with a discrete ordinates method. The number of zenith angles for integration and the number of Fourier terms in the azimuthal expansion of the phase functions were fixed to 30 and 12, respectively, for some of these Mie phase functions which have narrow angular features (strong forward scattering peak). It should be remembered that the use of adding algorithms is based on the assump-

tion of plane-parallel atmospheres, which implies that the results for points close to the limbs are not reliable (depending on the penetration depth at each wavelength) and should be rejected in any attempt to model experimental results. In the present work we rejected points with e, e0 lower than 0.3. Concerning other limitations, the use of an effective single scattering albedo is not valid for aerosol distributions which are not proportional to the gas abundance. Hence, our extended clouds were subdivided into sufficiently thin layers so that an effective single scattering albedo holds for each layer. In the present work, the extended tropospheric cloud was segregated into 10 layers. We used Mie theory to generate phase functions, but we did not derive optical depths and single scattering albedos as a function of refractive indices and number of particles. In the case of aggregate particles whose radii is close to the wavelength, particle extinction and single scattering albedo are not coincident with Mie scattering computations (West 1991). That is the reason why we preferred to fit optical depths and single scattering albedos instead of imaginary refractive indices and particle abundance. III. FITTING STRATEGY

The iterative scheme consisted of the minimization of a deviation function. For a given latitude and a given filter, a deviation function was defined as: deviation 5

F

G

oi51 Wi ((I/F )i,model 2 (I/F )i,obser )2 100 Npts (I/F )max oi51 Wi Npts

1/2

(i 5 1, ..., Npts), (1)

FIG. 2. Model structure used in this paper.

where (I/F )i,obser is the measured reflectivity at pixel i, (I/ F)i,model is the model reflectivity for the same scattering geometry as in pixel i, (I/F )max is the maximum reflectivity, Npts is the number of observational points in a scan, and Wi is a weighting factor for (I/F )i,obser .

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TABLE IIIA Latitude 108, Models A (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

A ‘‘supersigma’’ function was obtained by adding the six deviations (one per filter). A ‘‘methanesigma’’ was also defined by summing only the deviations for the three methane wavelengths. Additionally, we used another function which we called ‘‘LTLVsigma.’’ This function measured the suitability of the limb-to-limb behavior of the band/ continuum ratios computed from the models. The best fits were searched for by minimizing any of the ‘‘supersigma,’’ ‘‘methanesigma,’’ and ‘‘LTLVsigma’’ functions. This was achieved by means of a downhill simplex method (Press et al. 1992). Other approaches of modeling consist in the fit of both the limb-darkening coefficient and absolute reflectivity at some geometry, but we chose to retrieve the whole limb-to-limb scans since this is a more general approach for a variety of atmospheres, not only in the case of Saturn. Thus, our strategy is a simultaneous multi-wavelength fit. Only on a few occasions did we attempt to model the atmosphere using a single scan, as will be discussed later. Another difference with other published strategies is related to the weighting factors, which were chosen in order

to minimize the relative error instead of the absolute error. This had the advantage of a faster convergence. The reflectivity data were permitted to vary within their expected uncertainties (10%). For this purpose, a ‘‘recalibration factor’’ in the interval (0.9 1.1) was fitted for each filter. However, as methane absorption coefficients are also somewhat uncertain, we allowed for ‘‘recalibration’’ factors within the interval (0.8 1.2). The last improvement of the strategy was to allow for a wavelength dependence of the albedos and optical depth of the tropospheric haze. This dependence has often been neglected in previous methane band modeling efforts despite the fact that it may be important if a large part of the scatterers are micrometer sized. Furthermore, our modeling strategy was partially focused on trying to derive optical depths as a function of wavelength in order to derive conclusions on the scatterers’ sizes regardless of their shapes. A potential problem with this strategy is the large number of adjustable parameters. We first approached the best solutions by lowering the number of adjustable parameters.

TABLE IIIB Models B (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

FIG. 3. Observational reflectivities (stars) and model results (line) along with Saturnian scans at the six wavelengths shown in the labels. The planetographic latitude was 108 and the corresponding date is July 1991.

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FIG. 4.

Same as for Fig. 3 but for May 1993 data.

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TABLE IVA Aerosol Optical Depth per km at Latitude 108 for Different Dates, Using models A

th , g˜ cl6350 , tcl6350 , g˜ cl7500 , tcl7500 , g˜ cl9500 , tcl9500 , g˜ y , P5, P4 , P3 , P2 , and P1 . IV. MODELING RESULTS AND DISCUSSION

TABLE IVB Aerosol Optical Depth per km at Latitude 108 for Different Dates, Using models B

Thus, we began by fitting the continuum wavelengths while keeping the pressure levels fixed to some arbitrary values since this does not affect the continuum reflectivity. We assumed g˜ cl 5 g˜ y and determined the single scattering albedo as a function of wavelength using the continuum scans. Afterward, we assumed that the aerosol optical depths and aerosol single scattering albedos within the extended haze were the same for each pair of band-continuum wavelengths. The rest of parameters were fitted and the albedos refined, using all the data. Hence, the fitted parameters for each latitude were: g˜ h ,

Due to the fact that we could not fit the low continuum limb darkening at 408 in 1991 using the Tomasko and Doose (1984) phase function for the tropospheric haze (we had to use g˜ less than 0.95, a value which gives much lower reflectivities than what we observed, even considering larger errors than estimated), we decided to use Mie scattering phase functions. As mentioned earlier, we used particles whose radii were close to the wavelength based on the large changes of continuum reflectivity at several locations, which might suggest the existence of a considerable number of particles whose radius is in the range of our observing wavelengths, since the largest changes in particle extinction and scattering efficiency occur at size parameters of 4–6 for real refractive indices in the range 1.44–1.33. Given that the size of the backscattering peak is larger, a lowcontinuum limb-darkening fit was obtained without lowering g˜ too much. Results for Latitude 108 The best fits for the different dates are shown in Table III for models A and B. Reflectivities from the July 1991 and May 1993 models in Table III are shown in Figs. 3 and 4, respectively, along with the observations. From the supersigma minimization results, the main difference in the atmospheric structure is related to the optical depth

FIG. 5. Optical depth per km as obtained from the models for latitude 108, vs the different observing runs (from 1 to 4: July 1991, August ˚ , circles 7500 A ˚ , and filled circles 6350 A ˚. 1992, May 1993, and September 1993, respectively). Stars correspond to 9500 A

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of the extended tropospheric haze, which increased from 1991 to 1992. The aerosol content per km can be computed from the tables, yielding the results shown in Table IV and Fig. 5. Assuming a penetration level of 200 mbar for the 8900˚ channel in 1991 (the level where the transmission factor A is e21) and an upper haze level located at 70 mbar for a 108 latitude, the penetration depth in the tropospheric haze is around 41 km. Using the t /Dz values for the 1991 data we get a total aerosol optical depth of 7.4, which corresponds to t 5 0.46 per km-am, since there is 16 km-am of gas within the 70- and 200-mbar boundaries. The value t 5 0.46 per km-am is consistent with the value for the same latitude plotted in Fig. 8 of Karkoschka and Tomasko (1993). From 1991 to 1992 there is a remarkable increase in the optical depth per km, therefore, the temporal changes may be interpreted as an increase in the aerosol specific abundance. The wavelength dependence of the optical depth seen in the graphs may be related to different aerosol contents in the different levels which are probed by the three methane bands (less aerosol content at deeper levels); however, it may also be related to the variation in aerosol extinction as a function of wavelength, which might indicate particle size. Although the uncertainty in t /Dz is still under assessment (see the paragraph on the model sensitivity to the parameters), there seems to be an increase of extinction with wavelength, which suggests particles with radii larger than at least 0.5 em for nr 5 1.44 and even larger for lower nr (Fig. 1) if we assume spherical particles. From another point of view, we can explain the temporal increase in reflectivity at the deep methane wavelengths by increasing the level of the tropospheric haze (that is, lowering P3 ) in test models, but the resulting limb-darkening coefficient was lower, contrary to the observations. ˚ reflectivity was upon Another means of raising the 8900-A increasing the optical depth within the extended haze; this resulted in a small increase in the limb-darkening coefficient as well. On the other hand, the increase in the continuum limb darkening may be explained as an increase in the single scattering albedo. Then, the temporal changes seem to be related to a simultaneous increase in optical depth and albedo, which may be caused by material coming from the bottom clouds or even by a change in the particle mean radius. However, the change in the continuum limb darkening may have occurred due to a change in the particle phase function, especially in the backscattering peak, which is the most important part of the phase function for studies from the Earth. As West (1991) pointed out, this part of the phase function is very sensitive to particle shape when its diameter is comparable to the wavelength. This effect of non-sphericity may be responsible for at least part of the temporal changes in continuum limb darkening and

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reflectivity, since ammonia ices are more likely non-spherical particles. When minimizing methanesigma we also obtained an increase in the tropospheric optical depth per km, from 1991 to 1992, but the stratospheric optical depth was much larger, and has increased since 1991. By minimizing LTLVsigma we obtained similar results to those obtained with supersigma. Latitudes 08 and 208 Results for latitudes 08 and 208 are shown in Tables V and VI, respectively. From the supersigma minimization results we computed t /Dz ratios, which are shown in Figs. 6 and 7. We again found a considerable increase in t /Dz from 1991 to 1992 at all the wavelengths although we found a smaller absolute change at 208 as well as a different t /Dz wavelength dependence. We further investigated the possibility of getting better results by minimizing individual methane scans at latitude 208 (see Table VII). Some of the fits improved, but showed very large stratospheric optical depths and different cloud levels depending on the wavelength used. These single fits locate the upper tropospheric layer at different altitudes and give very low single scattering albedos which may be unrealistic. In addition, the best fits obtained using only ˚ scans show large stratospheric optical 8900- and 7250-A depths but different haze tops, while the large stratospheric ˚ scans. The only optical depths give bad fits to the 6190-A way to reconcile this result is by assuming a much lower ˚ . If the stratospheric stratospheric optical depth at 6190 A optical depth is lower at this shorter wavelength, the aerosols might have radii larger than 0.15 em. Therefore, from these fits, some large aerosol particles may have reached the stratosphere. Concerning the kind of models, models B seem to give slightly better fits at all the ‘‘equatorial’’ latitudes analyzed here. Other Latitudes We also modeled latitudes 408 and 708, which are representative of ‘‘clearer’’ regions. The results for the four observational runs are shown in Tables VIII and IX. We did not find remarkable temporal changes. The changes in the models are related basically to the single scattering albedos, which may be caused either by temporal changes or by the different calibrations. Models A seem to give slightly better fits at these latitudes. For latitude 408 the wavelength dependence seen in the tables may imply less extinction toward longer wavelengths (lower size parameters). This might suggest particles with radii smaller than 0.5 em for nr 5 1.44 and smaller than 0.7 em for nr 5 1.33. However, the wavelength dependence may suggest a

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TABLE VA Latitude 08, Models A (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE VB Models B (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE VIA Latitude 208, Models A (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE VIB Models B (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

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FIG. 6. Same as for Fig. 5 for latitudes 08, 208, 408, and 708, using models A.

FIG. 7. Same as for Fig. 6, using models B.

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TABLE VIIA Models A: Results from Fittings to Single Methane Band Scans at 208 (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE VIIB Models B (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE VIII Latitude 408, Models A (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

larger aerosol concentration at deeper levels. The optical depth per km is shown in Figs. 6 and 7. Comparing the fits at different latitudes we notice the lower aerosol content toward higher latitudes (in the tropospheric haze) and the deeper pressure levels of the tropospheric haze top. Sensitivity to the Model Parameters We carried out computations in order to estimate the sensitivity of the models to the different adjustable parame-

ters Table X. This is just a way of exploring the level of uncertainty in the parameters within our model, but may be different for other kinds of models. From Table X we can see that the sensitivity is highest for P3 , while the bottom cloud single scattering albedos have little effect on the results (and were not altered in Table X), especially when the tropospheric haze is optically thick. In these cases of high optical depths of the tropospheric haze, g˜ y is not constrained at all, and the values presented in the tables should be considered arbitrary ‘‘initial values.’’ Concerning

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TABLE VIIIB Models B (The v ˜ ` Values Used Were in the Range 0.98 to 0.99)

TABLE IXA Latitude 708, Models A (The v ˜ ` Values Used Were in the Range 0.980 to 0.985)

TABLE IXB Models B (The v ˜ ` Values Used Were in the Range 0.980 to 0.985)

TABLE X Sensitivity to the Model Parameters

Note. The altered parameter is in bold face. The change in supersigma gives an estimate to the parameter uncertainty.

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the P1 level, it is not accurately determined since it only affects the weak absorption, provided that tcl is not very large. Hence, P1 should be considered as another ‘‘initial value’’ in most of the tables. We also found that there are families of models with the same optical depth per km but with different P2 and tcls, and hence, the sensitivity of the models to P2 and tcls is low. However, it is good for tcl /Dz, since changes in both P3 and tcl make large perturbations in supersigma. We estimate a 20% uncertainty in the tcl /Dz ratio. This estimate may be too low if we recall that the molecular absorption coefficients are somewhat uncertain. For this reason, the numbers in the tables were not rounded off according to any estimate from model perturbations.

the Comisio´n Nacional de Ciencia y Tecnologı´a under Contracts ESP940719, ESP94-0803, and ESP93-0338.

V. CONCLUSIONS

KARKOSCHKA, E., AND M. G. TOMASKO 1993. Saturn’s upper atmospheric hazes observed by the Hubble Space Telescope. Icarus 126, 428–441.

The best multi-wavelength simultaneous fits which were obtained using our model structure show that the most remarkable equatorial changes are a result of an increase in tropospheric aerosol abundance. Although we favor the multi-wavelength fits, some good fits performed using single methane band scans show large changes in the stratospheric aerosol content too (strictly speaking, in the aerosol optical depth). In any case, both kinds of fits (multi-wavelength and single-wavelength) require larger aerosol contents at the equatorial latitudes from 1991 to 1992 and 1993.

KILLEN, R. M. 1988. Longitudinal variations in the saturnian atmosphere. Icarus 73, 227–247.

VI. CAVEATS FOR THE FUTURE

We are presently working on algorithms which may be applied to four latitudes simultaneously. This strategy has the advantage of giving the same recalibration factors for all the latitudes. In addition, we are trying to minimize the uncertainty in the absolute calibrations at continuum wavelengths since an accurate knowledge of absolute reflectivity and limb darkening may give reliable dependences of single scattering albedos and optical depths as a function of wavelength. ACKNOWLEDGMENTS We are grateful to Drs. Robert A. West, Eric Karkoschka, and Kathy Rages for helping us improve this work. This research was supported by

REFERENCES ˚ methane feature BAINES, K. H. 1983. Interpretation of the 6818.9 A observed on Jupiter, Saturn, and Uranus. Icarus 56, 543–559. COCHRAN, A. L., AND W. D. COCHRAN 1981. Longitudinal variability of methane and ammonia bands on Saturn. Icarus 48, 488–495. HANSEN, J. E. 1971. Multiple scattering of polarized light in planetary atmospheres. II. Sunlight reflected by terrestrial water clouds. J. Atmos. Sci. 28, 1400–1426. KARKOSCHKA, E., AND M. G. TOMASKO 1992. Saturn’s upper troposphere 1986–1989. Icarus 97, 161–181.

LINDAL, G. F., D. N. SWEETMAN, AND V. R. ESHLEMAN 1985. The atmosphere of Saturn: An analysis of the Voyager radio ocultation measurements. Astron. J. 90, 1136–1146. MICKELSON, M. E., AND L. E. LARSON 1992. The temperature dependent absorption coefficient for methane at 619 nm. Bull. Am. Astron. Soc. 24, 990. MORENO, F., A. MOLINA, AND J. L. ORTIZ 1991. CCD spectroscopic observations of Saturn, Uranus, Neptune, and Titan during the 1990 apparitions. Icarus 93, 88–95. ORTIZ, J. L., F. MORENO, AND A. MOLINA 1993. Absolutely calibrated CCD Images of Saturn at Methane Band and Continuum Wavelengths During its 1991 Opposition. J. Geophys. Res. 98, 3053-3063. ORTIZ, J. L., F. MORENO, AND A. MOLINA 1995. Saturn 1991–1993: Reflectivities and Limb-darkening coefficients at methane bands and nearby continua. Temporal changes. Icarus 117, 328–344. PRESS, W. H., S. A. TEUKOLSKY, W. T. VETTERLING AND B. P. FLANNERY 1992. ‘‘Numerical Recipes in FORTRAN.’’ Cambridge Univ. Press, Cambridge, UK. TOMASKO, M. G. AND L. R. DOOSE 1984. Polarimetry and photometry of Saturn from Pioneer 11: Observations and constraints on the distribution and properties of cloud aerosol particles. Icarus 57, 1–34. WEST, R. A., M. G. TOMASKO, M. P. WIJENSINGHE, L. R. DOOSE, H. J. REITSEMA, AND S. M. LARSON 1982. Spatially resolved methane band photometry of Saturn. I. Absolute reflectivity and center to limb varia˚ Bands. Icarus 51, 51–64. tions in the 6190-, 7250-, and 8900-A WEST, R. A. 1991. Optical properties of aggregate particles whose outer diameter is comparable to the wavelength. Appl. Opt., 30, 5316–5324.