Saturn's rings: Particle composition and size distribution as constrained by observations at microwave wavelengths

ICARUS 44, 683--705 (1980)

Saturn's Rings: Particle Composition and Size Distribution as Constrained by Observations at Microwave Wavelengths II. Radio Interferometric Observations

J E F F R E Y N. CUZZI, JAMES B. POLLACK, AND AUDREY L. SUMMERS Theoretical and Planetary Studies Branch, Space Science Division, NASA, Ames Research Center, Moffett Field, California 94035 Received July 25, 1980; revised October 23, 1980 The sizes, composition, and number of particles comprising the rings of Saturn may be meaningfully constrained by a combination of radar- and radio-astronomical observations. In a previous paper, we have discussed constraints obtained from radar observations. In this paper, we discuss the constraints imposed by complementary "passive" radio observations at similar wavelengths. First, we present theoretical models of the brightness of Saturn's rings at microwave wavelengths (0.34-21.0 cm), including both intrinsic ring emission and diffuse scattering by the rings of the planetary emission. The models are accurate simulations of the behavior of realistic ring particles and are parameterized only by particle composition and size distribution, and ring optical depth. Second, we have reanalyzed several previously existing sets of interferometric observations of the Saturn system at 0.83-, 3.71-, 6.0-, 11.1-, and 21.0-cm wavelengths. These observations all have spatial resolution sufficient to resolve the rings and planetary disk, and most have resolution sufficient to resolve the ring-occulted region of the disk as well. Using our ring models and a realistic model of the planetary brightness distribution, we are able to establish improved constraints on the properties of the rings. In particular, we find that: (a) the maximum optical depth in the rings is - 1 . 5 _+ 0.3 referred to visible wavelengths; (b) a significant decrease in ring optical depth from h3.7 to h21.0 cm allows us to rule out the possibility that more than - 3 0 % of the cross section of the rings is composed of particles larger than a meter or so; this assertion is essentially independent of uncertainties in particle adsorption coefficient; and (c) the ring particles cannot be primarily of silicate composition, independently of particle size, and the particles cannot be primarily smaller than -0.1 cm, independently of composition.

1975; Jaffe, 1977; Janssen and Olsen, 1978; Schloerb et al., 1979, 1980; Epstein et al., Studies of Saturn's rings at microwave 1980) and their analysis (see, e.g., Pollack, frequencies over the last several years have 1975; Epstein et al., 1980) may be used to contributed much to our understanding of further constrain the ring particles as to the composition and size distribution of the composition and size distribution. For inconstituent particles. Part of the strength of stance, it is unlikely that the rings are these studies lies in the ability to employ composed primarily of silicates (Pollack, both active (i.e., radar-) and passive (radio-) 1975) and, if the particle composition is astronomical tools at similar or identical primarily of ices, the typical particle size wavelengths. Radar observations at 3.5- may not exceed a few meters or so (Janssen and 12.6-cm wavelengths (Goldstein and and Olsen, 1978; Epstein et al., 1980). The Morris, 1973; Goldstein et al., 1977; Ostro latter results, however, rely on direct obet al., 1980) and their analysis (Cuzzi and servation of thermal emission by the partiPollack, 1978; Ostro et al., 1980) require cles and depend on the choice of absorption the ring particles to be wavelength-sized or coefficient of ice, a function of temperature larger. Passive radio observations at similar and lattice properties of the ice. Current wavelengths (Briggs, 1974; Cuzzi and Dent, knowledge of the ice adsorption coefficient I. INTRODUCTION

683 0019-1035/80/120683-23502.00/0

684

CUZZI, POLLACK, AND SUMMERS

is uncertain to plus or minus at least a factor of 3 (Whalley and Labb6, 1979; Whalley, private communication, 1979), as will be discussed later in greater detail. Previous analysis and interpretation of both radar and radio observations of the rings has been hampered by lack of good information on the optical depth of the rings at microwave wavelengths, which is a function of both the optical depth of the rings at visible wavelengths and ring particle scattering and extinction efficiencies at m i c r o w a v e wavelengths. Both are incompletely understood (Cuzzi and Pollack, 1978). Besides being important contributing parameters to inferences of particle size and composition based on radar and radio observations, the ring optical depth itself and its variation with wavelength are significant independent diagnostics of ring particle size distribution. Several of the previously cited passive radio observations over a significant range of wavelengths (1-21 cm) have been interferometric observations with spatial resolution sufficiently high to be sensitive to the region on the planet where the planetary disk is occulted by the rings. These observations have thus obtained values of ring optical depth as well as values of ring brightness temperature. The values of optical depth have been quoted as generally "comparable" to those at visible wavelengths, with perhaps some decrease indicated at the longer wavelengths (Briggs, 1974). Values of brightness temperature have been given as - 10°K (Cuzzi and Dent, 1975; Jaffe, 1977; Schloerb et a l . , 1979, 1980) or merely as upper limits (Briggs, 1974). We have constructed a model for the brightness of the rings due both to their own thermal emission and their diffuse scattering of the planetary emission. The model follows the approach of Cuzzi and van Blerkom (1974, henceforth CVB), but improves on and corrects certain aspects of the earlier calculations of angular scattering

of the diffuse component and allows incorporation of the properties of realistic particles on arbitrary size and composition into the scattering calculations, as discussed by Cuzzi and Pollack (1978, henceforth CP). In this paper, typical results of the model calculations are presented; several interesting features are noted, dealing with total brightness and variation of brightness with radius within the rings, with azimuth, with ring tilt angle, and with wavelength, as a function of ring particle composition and size distribution. The model is then used to reanalyze and interpret previously published interferometric observations of the Saturn system. It was decided to completely reanalyze the original data, as opposed to merely interpreting the previously published results. It was desirable that the results should be derived with respect to a more realistic model and should share a more uniform mode of derivation (minimization of residuals) than previously used. Planetary atmospheric limb darkening and its variation with wavelength, for instance, were generally not treated in previously reported results. Further, several of the previously published results have not considered the changing apparent oblateness of the planetary disk with changing sub-Earth latitude, an important effect. We also wished to investigate possible correlations between parameters of interest. This reanalysis has, in most cases, resulted in reduced error bars and improved knowledge of both ring brightness and optical depth and their variation with wavelength from 0.86 to 21 cm. The newly obtained results are used to constrain specific theoretical models of ring particle size and composition. In Section II of this paper, we describe our model and present some typical results. In Section III, we discuss the observations and their reanalysis and give results. In Sections IV and V, we discuss the results and their implications and present our conlcusions. In an appendix, we consider mutual particle shadowing effects.

SATURN'S RINGS II. MODEL DESCRIPTION AND TYPICAL RESULTS Two different models are c o m p a r e d with the data: a simple, uniform-brightness ring with crudely varying optical depth and a more realistic and detailed numerical model which predicts ring brightness and microwave optical depth using radiative transfer calculations. In this section, we discuss the makeup o f the detailed model and present some typical results. The philosophy of the detailed model is basically that of CVB, who calculated the total brightness of the rings due to both the scattering o f planetary thermal emission and the intrinsic ring particle emission. H o w e v e r , our method o f determining the scattering properties o f the rings is an improvement over the approach of CVB.

A. Scattering Properties o f the Ring Layer Certain aspects o f the model are treated in detail by CP, and we present only a brief description of these aspects below. We consider two alternate structural models for the rings: a m o n o l a y e r model and the classical " m a n y - p a r t i c l e - t h i c k " model. In the next section, we describe our m o n o l a y e r models. In the " c l a s s i c a l " plane-parallel

685

model, the rings are modeled as a manyparticle-thick, vertically homogeneous layer of local normal optical depth %, composed of particles o f specified composition. From the scattering particle composition, which determines the material refractive indices (see Table I), and size distribution, we determine using Mie theory, the average particle single-scattering albedo 030, extinction efficiency Qe, and phase function for scattering P(®). The extinction efficiency is the sum of the scattering and absorption efficiencies, Qe = Qs + Q a - A semiempirical theory of scattering by nonspherical particles (CP) is applied to the Mie calculations in determining the average particle phase function, extinction efficiency, and albedo. The size distributions used were those which are best suited to fit radar observations. T w o different types of distributions are used: a narrow size distribution of the form n(r) = n0r(1-av)/~e-(r/~v) with areaweighted mean radius f (Hansen and Travis, 1974), and a powerlaw size distribution o f the form n(r) = nor -3, with smallest particle size rmin r I and largest particle size rmax = r 2 ( C P ) , typical of meteoritic size distributions and particle ensembles characterized by comminution processes in general. The diffuse reflection and transmission functions for the ring layer S(030, To, ~---

TABLE I REFRACTIVE INDICES USED AND SOURCES

Material

nr

ni

Source

Water icea

1.78

7.5 x 10-5/~,(cm)b

Silicates Metal

2.32 1000

0.0290 1000

Theoreticalcurve for T = 100°K Whalley and Labb6 (1969) Campbell and Ulrichs (1969) Cuzzi and Pollack (1978)

a In our use of these values of nr and ni, solid ice is implicitly assumed. Of course, the particles are very possibly granular and porous to some extent. However, an exceedingly low density appears to be ruled out observationally by the large radar reflectivity, which requires a substantial value of n r, and also theoretically by the likelihood of constant bumping and jostling of the particles. This assumption is not inconsistent with the fine surface observed at visible and near-infrared wavelengths. b We note a misprint in Eq. (42) of Pollack (1975). Also, the value of nl used by Janssen and Olsen 0978) is normalized to a data point at 1.2 cm and is -3x lower than ours at all wavelengths. This difference is within the experimental uncertainty.

686

CUZZI, POLLACK, AND SUMMERS

/z0, ~b0,/z, ~b) and T(o~0, to, tZo, 60, tz, ~b) for scattered radiation into direction (/x, ~b) from incident direction (/~0, ~b0) (Chandrasekhar, 1950; CP), are determined using a doubling technique (e.g., Hansen, 1969, CP). The scattering geometry is shown in Fig. 6 of CP. The optical depth at the wavelength of interest, r0(h) is obtained from the optical depth at visible wavelengths by r0(h) = Qe(X)r0(vis)/Qe(vis),

(1)

where Qe(vis) = 1 as discussed by CP and ~-0(vis) is a function of radial distance from the planet as shown in Fig. 1, adapted from Cook et al. (1973). A discussion of the impact of possible shadowing effects on treatment of radiative transfer in the layer is given in Appendix A, and shows them to be negligible. In addition to scaling the nominal optical depths at visible wavelengths (Fig. 1) to relevant ones at wavelength h using Eq. (1), we also scaled the visible optical depths themselves by factors of 0.5, 1.0, 1.5, and 2.0, a range of variation not grossly inconsistent with uncertainties in r0(vis) (see, e.g., CP). It is of central importance to perform accurate calculations of ring scattering functions S and T because the typical particle phase functions for centimeter-to-meter-sized particles, believed to typify the ring particles (Pollack et 1.0

x

0.5

0.5

0.125 0.031 [ - i

c ~NG

8~G RADIUS

AR.NG J=

FIG. 1. Ring optical depth model. The individual ringlets are a s s u m e d to maintain their relative values. T h e value of 7max is varied b e t w e e n - 1 and 2 for purposes of c o m p a r i s o n of models with data. Such an uncertainty is within current uncertainty in B ring optical depths.

1973, CP), are strongly forward-directed at centimeter wavelengths and especially so at millimeter wavelengths. The S and T functions inherit the same behavior, as manifested in the ring brightnesses, as demonstrated in Section II.D. The use of a Henyey-Greenstein phase function by CVB was intended to demonstrate this behavior, and a value o f g = (cos O) =

al.,

2rr

cos ® ' p ( c o s O)d(cos O) = 0.6 was

used. The phase functions we calculate here are often even more strongly forward scattering (g - 0.7-0.8). The above aspects of the ring scattering model are treated in detail by CP. One aspect that is relevant in the case at hand, and not discussed by CP, is the truncation of the forward peak of the particle phase function. For the shorter wavelengths (-0.3 to 3 cm) and larger particle sizes (meters) which we attempted to treat, the particle phase function exhibits an extremely intense "diffraction" spike in the forward direction. Exact treatment of such a strongly concentrated forward peak entails great computational expense. Fortunately, in essentially all cases of interest, this extremely narrow forward peak may be treated as unscattered and the phase function, albedo, and layer optical depth adjusted accordingly (e.g., Hansen and Pollack, 1969; Irvine, 1975). We chose to use a less extreme truncation algorithm than that of Hansen and Pollack (1969) and truncated only the most extremely forward-directed radiation (at angles less than 5-10 ° from the incident direction). Thus, we retained the bulk of the "forward lobe" (see Fig. 2), thereby only minimally affecting our calculations of the (primarily forward-scattered) diffuse reflection by the rings of the planetary emission. Test calculations showed that the truncation never affected the distribution and magnitude of the diffusely reflected brightness by more than - 5 % . In cases where thermal emission is the dominant contribution to ring brightness, as at the shorter wavelengths, an even more ex-

SATURN'S RINGS

687

already extremely narrow forward "diffraction" peak. The accuracy of the approximation was further tested by comparison with more exact calculations. (a) ~"

i

-1

B. Calculated Brightness o f the Rings

(c)

20'

4'0

100 120 140 160 180 6'0 8'0 ' ' ' ' ' SCATTERINGANGLE,(9, deg

FIG. 2. A typical p h a s e function (for an r ~ powerlaw with effective m e a n x = 2 z r r / h - 15). (a) True p h a s e function. (b) P h a s e function after truncation, as described in text. Part of the highly forward-scattered radiation is a s s u m e d to be unscattered. (c) Renormalization final phase function after truncation.

treme truncation may be used. For instance, Epstein et al. (1980) employed the model discussed in this paper, but used a "similarity"-type transformation (Irvine, 1975) which essentially treats all the forward-scattered radiation as unscattered and assumes isotropic scattering for the remainder. This approach would not be appropriate at wavelengths of 3-10 cm, where the scattering is both more significant and not so strongly concentrated in the forward direction. One other approximation, for the largest values of x = 2rrr/h, is our use of scaling rules to obtain particle single-scattering properties. For transparent particles (2rn~ ~ 1) for which x -> 1, little change in either particle albedo or phase function is incurred by simultaneously decreasing r and increasing n~ such that nir = constant. This "scaling" approach to albedos (e.g., Hansen and Pollack, 1969) and phase functions is quite accurate for albedoes, and satisfactory even for phase functions, ifx ~500. For such large particles, the major effect of increasing r is to further narrow an

The brightness of the rings arises from t w o components: intrinisic emission from the particles themselves, and diffuse reflection and transmission of the planetary thermal emission. Both the ring properties and the planetary thermal emision are functions of wavelength. Our treatment of the diffuse reflection and transmission is the same as that used in CVB; the planet is taken as an extended source of known brightness, and the S and T functions are used to calculate the brightness diffusely reflected and transmitted into the direction of Earth by a set of points distributed both in azimuth around the ring and in radius from the planet. The scattering geometry is shown in Fig. 1 of CVB. However, where CVB obtained their S and T functions using a Monte Carlo method, ours were obtained, as discussed in Section II.A, using the doubling method. The diffusely transmitted brightnesses obtained by CVB were significantly in error (too large) due to the coarseness of the Monte Carlo grid used and numerical difficulties at angles close to straight-through diffuse transmission where, in fact, our more exact approach is quite well behaved. Further, we include here calculations of brightness radially across the entire ring system, from the C ring to the A ring. For the brightness of the planet we use an atmospheric model which provides a good fit to disk-average observations of Saturn across the wavelength range of interest. Our model atmosphere program produces the emergent microwave flux, or brightness temperature, by numerically integrating the source function along the line of sight. The opacity is provided exclusively by gaseous ammonia (Berge and Gulkis, 1976) which is assumed to be uniformly mixed with mixing ratio (Klein et al., 1978) c~ = (NH:~/H2) = 3

688

CUZZI, POLLACK, AND SUMMERS

x 10-4 at temperatures higher than the local saturation temperature and in vapor pressure equilibrium at lower temperatures. Other sources of opacity (e.g., far-infrared transitions, clouds) are not likely to be significant at wavelengths of 1-21 cm. The models begin with the "nominal" temperature-pressure relation of Klein et al. (1978), and the program extrapolates the T-P relation to greater depths. Slight modifications are made to correct the extrapolated T - P relation to the local adiabat, which is in function of temperature through the variation of specific heat with ortho/para-hydrogen ratio. Integration proceeds to a depth of -1000 bars; from below this level little radiation escapes even at the longest wavelengths. Typical 100 limb-darkening curves for several wavelengths are shown in Fig. 3, and prePLANET dicted disk-average brightness temperatures as a function of wavelength are compared with observations in Fig. 4. These values are in good agreement with the results obtained by the (basically i i i i I i i i 50 I .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 identical) model of Gulkis and Klein (see, sin 0 e.g., Klein et al., 1978). FIG. 4. Typical limb-darkening profiles from o u r The difference between the models prob- model a t m o s p h e r e for a range of w a v e l e n g t h s and a = ably reflects numerical effects in one or (NHa/H2) = 3 × 10 -4 (solid curve) and 6 x 10-4 both models, and is at present unexplained. (dashed curve). However, as we are not obtaining a value for a, but merely obtaining a limb-darken- that value of a for which our model best ing curve, we anticipate no error from using suits the disk-temperature data. Although wavelength dependent and obviously limb 700 darkened, the brightness of the planet is assumed to be spatially constant for the g 6o0 purpose of providing a source function for ~H31 ~ 3 × 1 0 ~ 1 (CURRENT P A P E R ) \ i ~ ring scattering. We have tested this as~_ 4oo sumption by integrating over the limb-darkened profile obtained from the model and obtained good agreement. However, the full limb-darked planetary brightness distri~ 100 bution is used across the planet itself for the 0 I I I I ] I I I I I I l I I I ~ o I purpose of comparison with observations, .1 1 10 100 as discussed in Section III. WAVELENGTH, cm We obtain the thermal emission accuFIG. 3. C o m p a r i s o n of disk-average model temperarately for each anisotropically scattering tures from Klein et al. (1978) (solid curve) and from ringlet using a detailed balance approach the model used in this paper (dashed curve), with observations recalibrated by Klein et al. analogous to that of Goody (1964), who

SATURN' S RINGS treated the semi-infinite case. In this approach, the ring layer is treated as if within a blackbody enclosure at the temperature TR of the ring. The emission from the layer at wavelength X and arbitrary angle of view, when combined with the hemispherically integrated diffuse scattering and transmission of the ambient radiation and the direct transmission through the layer, must be equal to the Planck function at temperature Tn and wavelength h. In this way the emission at arbitrary wavelength and view angle may be determined. The ring layer is assumed to be at a uniform physical temperature of 94°K, based on thermal infrared observations suitable for the large ring tilt angles characteristic of all the observations with which we shall be dealing (see, e.g., Kawata and Irvine, 1975). The thermal emission calculations have been checked against analytical solutions for both isotropic and anisotropic scattering cases for a range of optical depths (Horak, 1952). C. T r e a t m e n t o f " M o n o l a y e r " M o d e l s

Two different types of "monolayer" models were distinguished by CP. One was a large multiply internally scattering monolayer of "particles" similar in their radar properties to the Galilean satellites (Campbell et al., 1977), and one was a monolayer of wavelength-sized particles such as constitute the many-particle-thick models described above. We first describe calculations of the scattered and emitted brightness of the multiply internally reflecting "particles." Each large "megaparticle" is assumed to be composed of individual scatterers, of the sizes and compositions previously discussed, dispersed within a low density matrix of ice. The situation is then analogous to that in a scattering and absorbing/emitting atmosphere; the average albedo of the individual embedded scattering particles ~0', is defined as follows to account for absorption along the photon path between successive scattering:

689

(Oo' = Qs/[Qs + Qa

+ (4~'ni/X) (p/ps)l*

(2)

In the above equation, n i is the imaginary index of refraction of the solid material of density Ps, of which the "matrix" is composed, and p is the density of the real matrix in which the scatterers (of density Ps) are embedded. Also, l* is the mean free photon path length. For particles in free space, p = 0 and ~0' reduces to the usual definition of t~0 such as calculated from Mie theory (Section II.A and CP). It is not possible to directly evaluate I * for this case, as the observed ring volume densityD 0 0 -2 > D > l0 -a) would refer to the surface microstructure of the matrix material and not to the spacing of the microwave-scattering particles. However, for such particles to be viable radar scatterers, their mean separation I must be greater than a few times the mean scattering particle size r; since I* ~ ID -213 ~ l(I/r) 2 (see Appendix), then l* > (l/r)ar. Therefore, we assumed the lower bound for I* of/* ~ 40r; a larger value would increase absorption and emission and require even lower matrix densities for the megaparticles. For the Mie monolayer, no special treatment is required beyond including only single scattering in the calculations of the S and T functions for intermediate ring tilt angles. If the particles are in a monolayer, they will be more closely packed than if in a many-particle-thick layer of the same optical depth. Adjacent particles subtend an angle of sin-l(r/I) ~ sin-t(0.2-0.4), where r/l is obtained from the observed depth. For most calculations of scattering of relevance to the observations we discuss here and previously (CP), the tilt angles are greater than sin-l(0.4) - sin-l(~); therefore, the radiation is primarily single scattered. We do not have an accurate method for calculating the S and T functions at other angles and therefore are unable to rigorously calculate the thermal emission using the "detailed balance" approach described earlier. However, we do not feel

690

CUZZI, POLLACK, AND SUMMERS

that the layer or angle-integrated albedo of a monolayer ring, including small-angle effects (/~ < 0.4) will differ significantly from the corresponding many-particle-thick case of equal optical depth. Consequently, neither would we expect the thermal emission to differ substantially from the corresponding many-particle-thick case. Thus, radio brightness studies at a single tilt angle such as we analyze here are not able to distinguish between monolayer and many-particle-thick layers of wavelength-sized ("Mie") particles.

D. Typical Results for Ring Brightness In this section, we present examples of calculated total ring brightness as a function of azimuth and of radial distance from the planet and given an example of the diffusely reflected and diffusely transmitted contributions to the total brightness. We also illustrate the overall variation of the results with wavelength.

1. Brightness due only to scattering; classical ring model. In Table II, we show the calculated ring brightnesses for the manyparticle-thick model at 3.71 cm wavelength due to diffuse reflection (a) and diffuse transmission (b), as well as the total (c). We note that the rings are significantly brighter in the forward direction, even when not directly in front of the planet (azimuths 30°) than near the ansae (azimuth - 90°), essentially confirming the earlier results of CVB. In general, however, our scattered brightnesses are significantly lower than those of CVB. The discrepancy arises from the details of the numerical approach used by CVB, which produced inaccuracies in the calculation of diffuse transmission when integrating over certain locations on the planet. The values we present here are calculated on a finer grid and are free of this source of error. As a further check on the current technique, integration over a

T A B L E II DIFFUSE SCATTERING BY A RING LAYER OF WATER ICE PARTICLESa Ring elem ent

Radial distan ce

Ring a z i mut h (deg)

to(h) °

(R /R~ ) 0.

20.

40.

60.

80.

(a) Diffuse reflection 2. 2. 1. 4. 3. 3. 7. 6. 5. 6. 5. 4. 3. 3. 2.

100.

120.

140.

160.

180.

1. 2. 5. 4. 2.

1. 2. 4. 4. 2.

1. 2. 4. 4. 2.

1. 2. 4. 4. 2.

1. 2. 4. 4. 2.

0.058 0.232 0.928 1.856 0.928

I nner C Outer C I nner B Outer B Middle A

1.21 1.43 1.60 1.81 2.14

2. 5. 8. 7. 4.

2. 5. 7. 6. 3.

I nner C Outer C I nner B Outer B Middle A

1.21 1.43 1.60 1.81 2.14

8. 18. 17. 5. 7.

8. 17. 14. 4. 5.

(b) Diffuse t r a n s m i s s i o n 6. 3. 2. 1. 10. 5. 4. 3. 8. 5. 4. 4. 3. 3. 2. 2. 3. 2. 2. 2.

1. 2. 4. 2. 1.

1. 2. 3. 2. 1.

1. 2. 3. 2. 1.

1. 2. 3. 2. 1.

0.058 0.232 0.928 1.856 0.928

I nner C Outer C Inner B Outer B Middle A

1.21 1.43 1.60 1.81 2.14

10. 23. 25. 12. 11.

10. 22. 21. 10. 8.

(c) Total diffuse scattering 8. 5. 3. 2. 14. 8. 7. 5. 15. 11. 9. 9. 9. 8. 6. 6. 6. 5. 4. 4.

2. 5. 8. 6. 3.

2. 4. 7. 6. 3.

2. 4. 7. 6. 3.

2. 4. 7. 6. 3.

0.058 0.232 0.928 1.856 0.928

a Size distribution n(r) = nor-a; rmin = 1 cm, rmax = 100 cm; at a w a v e l e n g t h of 3.71 cm. The m a x i m u m optical depth, at visible w a v e l e n g t h s , ro(vis) = 1.

SATURN'S RINGS source function at all angles has been performed in both transmission and reflection to ascertain that energy is conserved for layers of finite optical depth. Thus, we are confident that the discrepancy is understood and that the values we present here are correct to within - 10%. It is of interest that diffuse transmission still dominates the ring brightness in front of the planet relative to diffuse reflection. Also o f significance in Table II is the comparison between the brightness of the C ring and that of the A ring. The A ring (R/R~ = 2.14) has a larger optical depth, but the outer C ring (R/R~ -- 1.43) is brighter in scattered radiation, especially in front of the planet. Even the very optically thin inner C ring (R/R~ = 1.21) is equally as bright as the A ring. This is due to the fact that the planet or source function, as seen from the inner rings, subtends a much larger solid angle than as seen from the outer rings and thus produces a greater scattered brightness for the inner rings in spite of their lower optical depth.

60

691 ~.-I(A) \

40

1~'-2(A)

A:

PL3,rMAX = 1000

|

B:

PL3,rMAx = 100

X-

o

~ 3o

,-',k",, \ 1 WAVELENGTH, c m

lo

FIG. 5. T h e r m a l e m i s s i o n f r o m S a t u r n ' s B r i n g a s a function of wavelength. Two model calculations are s h o w n , b o t h f o r w a t e r i c e p a r t i c l e s . (a) P o w e r l a w n ( r ) = no r-a, rmi n = 1 c m , rma x = 1000 c m . (b) P o w e r l a w n ( r ) = nor -3, rm~n = 1 c m , r . . . . = 100 c m . A l s o s h o w n a r e s u i t a b l y n o r m a l i z e d c u r v e s w i t h h -1 (solid) a n d h -2 (dashed) wavelength dependence for each case. The b e t t e r fit t o t h e h -1 c u r v e is e x p l a i n e d in t h e t e x t .

2. Brightness due to intrinsic emission; classical ring model. Figure 5 shows typical results for the thermal emission component o f the ring brightness, for two choices of particle size distribution, as a function of wavelength. Also shown for illustration are two normalized curves with different wavelength dependence. The function with h -2 d e p e n d e n c e o f brightness is characteristic o f the emission expected from a thin, low-loss slab (Janssen and Olsen, 1978; Whalley and L a b b r , 1969), much thicker than any of the wavelengths in question. Clearly, the emission by the rings does not increase with decreasing wavelength as fast as h -2. This is due to the fact that a large n u m b e r of the particles are nearly conservative scatterers. For lowloss, dielectric particles, this is a consequence o f the size of the particle being comparable to the wavelength. As noted by Irvine and Pollack (1968), o30 - 1 - Kxr for (1 - o30) "~ 1, where K x is the absorption coefficient, Kx = 47rni/h. Because the

emission from a slab of scatterers of albedo ~50 is proportional to (1 - ~0) ~/2 (e.g., Horak, 1952) and Kxa(1/h2), one obtains an approximate relation for the emitted intensity that is proportional to h -1 for (1 - ~0) 1. This wavelength dependence is a much more suitable approximation to the exact calculations than one characterized by h -2, as may be seen from Fig. 5.

3. Total ring brightness; scattering plus thermal emission; classical ring model. The total calculated brightness of the rings, due to diffuse scattering and transmission as well as thermal emission, is shown for a range of relevant wavelengths in Table III. The enhanced brightness in front of the planet, due to the dominance of forward scattering in the phase functions of the individual particles, is evident at all wavelengths. At the shorter wavelengths, the total brightness increases and b e c o m e s

692

CUZZI, POLLACK, AND SUMMERS TABLE III

CALCULATED RING BRIGHTNESS TEMPERATURES, INCLUDING BOTH DIFFUSE SCATTERING AND THERMAL EMISSION, FOR A RING LAYER COMPOSED OF ICE PARTICLES a

Radial distance

Ring azimuth (deg)

r0(~.)b

R/Rb 0.

20.

40.

60.

1.21 1.43 1.60 1.81 2.14

6. 15. 30. 32. 2I.

6. 15. 29. 31. 18.

5. 11. 23. 29. 17.

4. 9. 21. 28. 16.

1.21 1.43 1.60 1.81 2.14

7. 18. 26. 19. 15.

7. 18. 24. 17. 11.

6. 11. 15. 14. 8.

1.21 1.43 1.60 1.81 2.14

10. 23. 25. 12. I 1.

10. 22. 21. 10. 8.

1.21 1.43 1.60 1.81 2.14

12. 27. 27. 13. 12.

1.21 1.43 1.60 1.81 2.14

15. 36. 42. 22. 19.

80.

100.

120.

140.

160.

180.

h = 0.34 cm 3. 3. 9. 8. 21. 20. 28. 27. 16. 16.

3. 8. 19. 27. 15.

2. 7. 19. 27. 15.

2. 7. 19. 26. 15.

2. 7. 19, 26. 15.

0.038 0.152 0.609 1.219 0.609

3. 7. 13. 14. 8.

h = 0.83 cm 3. 2. 7. 6. 12. 12. 13. 13. 7. 7.

2. 5. 11. 12. 7.

2. 5. 11. 12. 7.

2. 5. 10. 12. 6.

2. 5. 10. 12. 6.

0.045 0.183 0.734 1.467 0.734

8. 14. 15. 9. 5.

5. 8. 11. 8. 5.

h = 3.71 cm 3. 2. 7. 5. 9. 9. 6. 6. 4. 4.

2. 5. 8. 6. 3.

2. 4. 7. 6. 3.

2. 4. 7. 6. 3.

2. 4. 7. 6. 3.

0.058 0.232 0.928 1.856 0.928

11. 25. 25. 11. 9.

10. 16. 16. 9. 6.

5. 9. 12. 8. 5.

,k = 6.0 cm 4. 3. 7. 6. 10. 9. 7. 6. 4. 4.

2. 5. 8. 6. 3.

2. 5. 8. 6. 3.

2. 5. 8. 6. 3.

2. 5. 8. 6. 3.

0.060 0.243 0.973 1.946 0.973

15. 33. 37. 20. 15.

12. 23. 26. 16. 10.

7. 14. 19. 13. 8.

X = 21cm 5. 4. 10. 8. 16. 14. 12. 11. 7. 6.

3. 7. 12. 10. 5.

3. 6. 12. 10. 5.

3. 6. 12. 10. 5.

3. 6. 12. 10. 5.

0.050 0.203 0.814 1.628 0.814

a Size distribution n(r) = nor-3, with rmin 0.1 cm and rmax bMaximum optical depth r0(vis) = 1, as in Table 11. =

more uniform, due to the increasing import a n c e o f t h e r m a l e m i s s i o n ( F i g . 5). The increased importance and azimuthal uniformity of thermal emission at shorter wavelengths makes possible certain simplifying assumptions, such as use of similarity relations to approximate the scattering problem by isotropic scattering (Irvine, 1975; E p s t e i n e t a l . , 1980). T h e r e s u l t s i n T a b l e III f o r h = 0.34 c m s h o w little varia-

=

100 cm.

t i o n o f t o t a l b r i g h t n e s s a t a z i m u t h s ( ~ 3 0 °, a f u n c t i o n of R / R O t h a t a r e n o t o v e r l y i n g the p l a n e t itself. T h a t is, the f o r w a r d s c a t t e r i n g is h i g h l y c o n c e n t r a t e d in t h e forward direction. This extra brightness i n f r o n t o f t h e p l a n e t is w e l l a c c o u n t e d f o r in t h i s c a s e b y t h e s i m i l a r i t y r e l a t i o n s , which treat the forward-scattered radiation as unscattered and reduce the optical d e p t h a c c o r d i n g l y . T h i s b e h a v i o r facili-

SATURN'S RINGS tates calculations at the shortest wavelengths (Epstein et al., 1980). Here, however, we use the exact calculations, as centimeter-wavelength brightnesses are more complex functions of azimuth. In particular, the difference between the value of the diffusely scattered brightness directly in front of the planet and its value near the ansae is crucial to inferences about ring optical depth, and these relative brightnesses would not be well reproduced by an "isotropic" approximation at centimeter wavelengths, as may be clearly seen by inspection of Table III. One final point of interest demonstrated in Table III is that the total ring brightness reaches a minimum value between h - 3.7 and 11.1 cm. This is due to (a) the increasing contribution of thermal emission toward still shorter wavelengths and (b) the increasing brightness of the planet (see, e.g., Fig. 1), the main source of the ring brightness, toward larger wavelengths. For the model shown, with a maximum particle size of - 1 m, the optical depth remains essentially constant over the range of wavelengths shown. For wavelengths longer than - 1 cm, the particle albedos are all very close to unity; therefore, the ring brightness variation with wavelength, to

693

first order, merely "reflects" that of the planet. III. COMPARISON OF MODELS WITH DATA

A. Data Sets Used and Their Analysis We have limited the data sets for this study to interferometric data, which are of sufficiently high resolution to resolve the rings, disk, and in some cases, the region of the disk occulted by the rings. The data sets, which are interferometric visibility amplitudes, are described and summarized in Table IV. All are previously published with the exception of the observations of Jaffe (1977), which are obtained at the Westerbork array (HiSgbom and Brouw, 1974). These data are reproduced in Table V. The errors used for these data were the quadrature sum of the electrical noise quoted by Jaffe (1977) and a noise due to gain variation equal to 5% of the observed visibility amplitude. Model visibility amplitudes are calculated for the date, instrument, and wavelength of interest by numerical integration over the brightness distribution of the planet-ring system. The instrinsic optical oblateness of the planet is taken from Cook et al. (1973); uncertainty in this parameter is not significant for our results. The appar-

TABLE IV DATA SETS USED Wavelength (cm)

Instrument

0.83

Table Mt. interferometer N.R.A.O. interferometer N.R.A.O. interferometer Westerbork synthesis array N.R.A.O. interferometer N.R.A.O. interferometer

3.71 3.71 6.0

11.13 21.0

Spacing used

Dates

Ring tilt (deg)

Reference

-25 -21 -25.6

Janssen and Olsen (1978)

1900, 2400 m

May 1975 Jan. 1976 April 1972

100, 1800, 1900 rn

Dec. 1973

-26.2

Cuzzi and Dent (1975)

See Table V

Dec. 1972

-26.4

Jaffe (1977)

1900, 2400 m

April 1972

-25.6

Briggs (1974)

900, 1800, 2700 m

Aug. 1971

-25.2

Briggs (1973)

62 m (EW)

Briggs (1974)

694

CUZZI, POLLACK, AND SUMMERS TABLE V 6-cm VISIBILITYAMPLITUDES(mJy)" Houran~e -45

U/B (wavelengths

11.785

0

+45

+76

16.667

I 1.785

4.032

+4.246

5.827

m ')

V/B

-4.246

0.0

Baseline (m) 198 270 342 414 486 558 630 702 774 846 918 990 1062 1134 1206 I278 1350 1422 1494 1566

920 795 735 595 540 445 385 265 225 130 80 30 25 65 105 95 90 90 115 - 75

820 675 590 445 345 225 125 40 - 35 80 120 130 -120 90 -50 -55 + 15 +60 +60 +75

900 805 765 615 555 485 375 285 225 125 70 30 30 65 - 105 -175 - 120 -115 80 75

900 915 895 860 830 770 720 655 640 615 540 490 460 400 350 330 290 240 190 185

" Dam of Jaffe (1977). Jy - 10 ~s W m z str-' H z '. U and V are the north and east components of the baseline vector projected on the sky. Calibration sources: 3C147, S~c,)~, - 8.18 Jy: 3C309.1, S~00 3.69 Jy.

ent oblateness of the planet is less than the intrinsic oblateness for nonzero ring tilt, and this effect is much more important than current uncertainty in optical oblateness The actual oblateness of Saturn as measured by Pioneer XI (Gehrels et al., 1980) is smaller than our adopted value by an amount which is 2 to 3 times less than the "tilt effect" on the oblateness. Sample calculations show that the Pioneer value decreases our residuals somewhat, but does not alter our results significantly as compared with our quoted errors. The planettilt effect was neglected in several previous analyses, leading to erroneously low ring optical depths. The computed model visibility amplitudes for each parameter set D(ro(X), TA, TB, Tc . . . . ) are used to generate reduced X2 sums by N

X2(fi) = ~ [(ml(fi) - dO/o'i] 2, i=l

where m~ are the model-generated and d~ the N observed visibility amplitudes with associated relative errors o-~. In all cases the absolute scaling of the data was taken as a parameter, obviating concerns about absolute calibration. The minimum X z = X2. was determined, giving the best-fit parameter set fi* and optimum scaling factor, and the associated formal errors in the parameters, Aft*, were obtained from the parameter values at w h i c h x 2 = [(N - r + 1)/(N r ) ] ' X 2., where r is the number of free parameters used in the fit (Meyer, 1975). In all cases, the best-fit parameters were determined at the minimum X2 obtained while varying all parameters of interest simultaneously. T w o approaches were used to obtain constraints on ring properties. We (a) fit uniformly bright rings (A, B, and C where permitted by instrumental resolution, otherwise an overall average ring temperature) to each data set and constrained subsequent model calculations by these results; and (b) compared our detailed model-calculated ring brightness models with the data in two ways as described in Section IIIB2. Our optical depths were obtained primarily using the former approach, with relative radial variation of optical depth at visible wavelengths as shown in Fig. 1, and different values for rmax. The values of optical depth we derive are microwave optical depths to(h) with relative radial variation as shown in Fig. I and are denoted by their maximum value in the B ring, rmax. They implicitly assume uniformly bright rings because of the models used (method (a) above). Therefore, they are very likely to be systematic underestimates of the true microwave optical depth r0(h); this is because the best fits for ring temperature are most sensitive to brightness at the ansae, and the rings are very likely to be brighter in front of the planet due to preferential forward scattering (see Table III). Therefore, in a fit to a uniform ring temperature model this excess will be attributed to an anomalously low optical depth. In compar-

SATURN'S

ing with specific ring models, we account for this effect as described later in this section.

B. Results of Data Reanalysis 1. Comparison with "uniformly bright rings" model. The results of the analysis using the simple, "uniformly bright rings" model are given in Table VI. Where only one value of ring temperature is shown, the instrumental resolution was insufficient to obtain uncorrelated values for separate ring components. The errors quoted are formal, ltr errors, which include the " i n t e r n a l " fitting error as well as uncertainty due to the model atmosphere, as discussed below. In cases where insufficient resolution existed to obtain independent estimates of T A, TB, and T c, it was generally obvious from two-parameter fits that the results were highly correlated. The most significant reason that our results are better determined than they were previously, as shown by the formal detection of the rings in nearly all cases, is probably the previous neglect of the geometrical effect of the tilted disk of the

695

RINGS

planet. The sensitivity of the results to the ammonia mixing ratio, which determines the limb darkening, was tested at the (most sensitive) 21-cm point by varying a from 10-4 to 6 × 10-4 , a reasonable range of uncertainty (e.g., Klein et al., 1978); there is no effect on our derived optical depths, but there is an effect on derived ring temperature which is comparable to the formal (internal) error; uncertainties given in Table VI include the corresponding uncertainty due to the above uncertainty in ~. There is less effect of uncertainty in a on results at the shorter wavelengths. We do not attempt, at this time, to actually determine o~ from these data; when more sensitive observations near ring plane crossing are obtained with the ring contribution absent, our estimates of ring brightness will be improved and the uncertainties decreased.

2. Comparison with Detailed Models of the Rings. The ring brightness results in Table VI have all been converted into average ring brightnesses for uniformity and are shown in Fig. 6 along with selected model calculations, also suitably averaged over

T A B L E VI RING

BRIGHTNESS

Wavelength

AND OPTICAL

TA

TR

DEPTH

FROM

FITTING

MODELS

WITH

Tc

to(h) ( m a x . v a l u e , R i n g B)

0

--

UNIFORM

RING

BRIGHTNESS

Comments

(cm) 0.83

11 _+ 3

3.71

11 ÷_ 2

10 _+ 2

8 _÷ 4

2.2_0.~+~'°

3.71

6 - 3

15 _+ 4

8 ___ 4

1.4_+°:26

10 _+ 4 7 -+- 3

--

6.0

--

1 • 2 --0.35 +°'r

11.1

< 12.5 °

--

1.8+_~:~

21.0

6 -+ 3

--

0.4_+~:~

Average of results obtained s e p a r a t e l y for M a y 1975 a n d Jan. 1976; n o i n f o r m a t i o n o b t a i n e d o n C ring or o p t i c a l d e p t h s , T A = TB a s s u m e d ; B r i g g s (1974). No information obtainable from 500-m s p a c i n g . 1800, 1900 m; C u z z i a n d D e n t (1975). 100 m , TA = TB = Tc a s s u m e d . N o i n f o r m a t i o n o n C ring; TA = TB = Tc a s s u m e d . N o i n f o r m a t i o n o n C ring; TA = TB = Tc a s s u m e d . N o i n f o r m a t i o n o n C ring; TA = TB = Tc a s s u m e d .

696

CUZZI, POLLACK, AND SUMMERS 60

~¢

H-H __ T= 300 \J MONOLAYER

\\/

=0,0

\~/PL3. 1-1ooo \ ~ •P,31-3oo MONOLAYER p = 0.01 PL3, 1-100

j_

/

H-H,T= 6 '

,~ 1°0 I

•1

-

J 1

1tO ~..

cm

FIG. 6. C o m p a r i s o n o f average ring temperature as a function o f wavelength with model calculations for a range o f typical models o f water ice composition. Filled circles: results o f this paper. Open circle: from Schloerb e t al. (1979). O p e n triangle: from Epstein e t al. (1980). Model curves: all solid lines are for "/'max 1; d a s h e d line for Zmax= 1.5. C u r v e s labeled PL3 are for a powerlaw o f the form n ( r ) = n o r -~ a n d are further labeled by their range rmin - rmax (cm). C u r v e s labeled " m o n o l a y e r " are for large, multiply internally scattering particles with matrix density p, as described in Section II.C. C u r v e s labeled H - H are for H a n s e n H o v e n i e r size distributions with variance 0.3 (see, e.g., CP) and are labeled by their m e a n radius f. =

the rings. Also shown are the ring brightness results o f Schloerb et al. (1979) at h3.71 cm and Epstein et al. (1980) at h0.34 cm. Clearly, little discrimination between models b a s e d on ring brightness is possible at wavelengths b e t w e e n 3 and 11 cm. Values o f ring brightness at shorter wavelengths, h o w e v e r , are m o r e diagnostic. F o r instance, we also show in Fig. 6 the result of Epstein et al. (1980) at h3.4 m m , which has been interpreted by those authors as restricting the largest particle size to no m o r e than 3 m, including a large uncertainty in the ice absorption coefficient. The model results are given for the nominal refractive indices o f Table I and support these previous results. Further, by c o m p a r ison o f the model with results at both 0.34 and 0.83 cm, we can apparently rule out the powerlaw distribution of index 3 for maxim u m radii larger than a few meters, as well as " m e g a p a r t i c l e " models with matrix density greater than a few × l0 -2 g cm -a, an

unlikely state for particles undergoing constant jostling. The o b s e r v a t i o n s seem to present a brightness t e m p e r a t u r e trend that is systematically less wavelength-dependent and possibly higher than the model predictions. T h e possibility of a small, wavelength-independent absorption has been suggested previously (Janssen and Olsen, 1978) and is a distinct possibility in the rings. H o w e v e r , the formal errors in the results are still too large to allow us to decide this question. Although these results are dependent upon refractive indices of the particles, uncertain to about a factor of 3, fair agreement will be d e m o n s t r a t e d between the implications o f these brightness t e m p e r a t u r e s and the implications o f the optical d e p t h results. In Fig. 7 we c o m p a r e our derived ring optical depths, plotted by the m a x i m u m value in the B ring, with predictions of specific model cases. We have included the result o f Schloerb et al. (1979) at 3.71-cm wavelength, which was derived with consideration of the tilt o f the oblate planet, as

.

a

H-H. F= 300

|

~

"-o / 1 J-I "rMAX = I

~ ~

/

J

/

.L

~,, ~,jfPL3.1-10

.,_..4..~T

~

j ~,.~PLS. 1-200 I

H

P

-

L

H

~

,

3

¥

,

/

=

1-10

1~.. P,5.1-2oo

±

0-

H-H, ¥ = 300 3

-H..~=3

1'o X, c m

FIG. 7. C o m p a r i s o n of o b s e r v e d " a p p a r e n t " microw a v e normal optical depth of the B ring with model

calculations. Data and models as described in Fig. 6 caption; in addition, model results are shown here for a steeper powerlaw distribution of the form n = n lr-s (labeled PL5). Upper set of model curves is for ~'max= 2, and lower set is for 7"max = 1.

SATURN'S RINGS modified for the effects of the Cassini Division by Schloerb et al. (1980). It is not likely that more detailed reanalysis will change this value appreciably, as planetary limb darkening is not large at 3.7-cm wavelength. The model optical depths shown were obtained from detailed model calculations such as were shown in Tables II and III from T0(h)

= - Isin B I In [ T°cc - T(90°)] [ Tpl J'

where B is the ring tilt angle, Tocc is the average brightness temperature of the B ring-occulted region of the planet, T(90°) is the average B-ring brightness at the ansae, and Tp~ is the disk-average temperature of the planet. For reasons discussed earlier, the values are lower than the true microwave optical depth r0(h). However, they provide a meaningful comparison with what the observations actually measure. For the tilt angles of these observations, blockage of the planet by the A ring was negligible. The models shown in Figs. 6 and 7 are

TABLE

697

typical cases only. There is a large range of parameter space to explore, and both the accuracy of the data and computational resources discourage more extensive model calculations at this time. We show here primarily a range of typical models that might be acceptable. As mentioned in Section IIA, we have already ruled out at the 2o- level a large range of models, which, for clarity, are not shown in the figures. These models are listed in Table VII. They include essentially all models with primarily silicate particles, all ice-particle models with primarily meter-or-larger-size particles, all models with particles smaller than 0.1 cm, and all "multiply internally scattering" monolayer models with matrix density >0.03 g cm -3. Certain sizes of silicate particles have been judged unacceptable by previous authors as well (Pollack, 1975; CP; Janssen and Olsen, 1978; Schloerb et a l . , 1979, 1980), and ice particles larger than several meters have been judged unacceptable by Briggs (1974), Janssen and Olsen (1978), and Epstein et al. (1980) for a range of reasonable values of the absorption

VII

M O D E L S R U L E D O U T AT T H E 2 ~ L E V E L

Constituent

Dataset source Ices J a n s s e n a n d O l s e n (1978)

B r i g g s (1974)

1. t: _< 0.1 c m 2. t:-~ 1 0 0 c m ('/'max = 1) 3. P o w e r l a w : ( r -3) rmax ~ 1000 c m ('i'max = 1--2) 4. M o n o l a y e r ; m a t r i x d e n s i t y > 0.03 g c m -3 Less restrictive

Silicate

Metal

=0.3to?= 30 "/'max = 0.5 t o 2 . 0

i : = 0.3 t o J r = 30 rmax = 0.5 t o 2 . 0

~ 0.2 c m f o r "/'max ~ 1 f o r ~ ~ 2.0 c m rmax > 2

Cuzzi and D e n t (1975)

Less restrictive

t : = 0.3 t o f = 30 rm.x = 0.5 t o 2 . 0

tr -- 0 . 2 c r n f o r "rmax 1 f o r r - 2.0 c m rma x > 2 ~

698

CUZZI, POLLACK, AND SUMMERS

coefficient of water ice at IO0°K. From our brightness temperature data, and the results of CP, we can now definitely rule out all particles of primarily silicate composition. For ice particles, we cannot improve significantly on previous estimates of particle size; however, we have demonstrated good consistency over a wide range of wavelengths between observed ring brightness temperature and scattering model calculations for allowable particle sizes (ibid) in models such as have been used to explain radar observations. We discuss the ring brightness implications further in the next section. Our results for ring optical depth contain a greater amount of new information. Consider first values near 3.7-cm wavelength, where the most diagnostic (smallest error) results are in good agreement and where the model results are not strongly dependent on choice of particle size distribution. It appears that the true maximum optical depth in the B ring, referred to visible wavelengths, must be greater than unity and is probably less than 2. The observed optical depth at 6 cm is also consistent with this result for a variety of size distributions. Further, we also obtain an apparent decrease of optical depth with increasing wavelength. Such an effect has been qualitatively noted by Briggs (1974) and Jaffe (1977), but, using our improved values, we are now able to more adequately constrain the ring particle size distribution. We have employed a simple approach to extracting the essential information from the results in Fig. 7. We note that all of the model calculations of ring optical depth shown in Fig. 7 may be closely approximated by straight lines over the wavelength region 3.7-21 cm. We then determine from the observed optical depths, by the analysis of variance approach discussed in Section IIIA, the best weighted-fit straight line and its ___20limits on slope and intercept. We then note characteristic of particle size distributions that are consistent with this result and from this draw general conclusions as

to the ring particle size distribution. For this exercise, we have averaged the positive and negative error bars on the observed optical depths. The resultant best-fit straight line and its _+20- limits are shown with the observed optical depths in Fig. 8. There appears to be a statistically significant decrease in optical depth between 3.7 and 21 cm. This behavior has significant implications as to the relative fraction of large (~meter-sized) particles in the rings. It implies that there is significantly less total cross-sectional area represented by meter-sized (and larger) particles than by centimeter-sized particles, independent of assumptions as to particle absorption coefficient and therefore of its poorly known composition (ices, clathrates, etc.) or temperature dependence. In the next section, we discuss our results and their implications. IV. DISCUSSION OF RESULTS

In this paper, we present results of two kinds: theoretical calculations of the diffuse scattering and thermal emission properties of Saturn's rings for a variety of realistic choices of particle size distribution and composition and constraints on the proper3

A2 ¢.u

2a LIMIT LIMIT "~' \ \ 1 '~'~'~-. 2a LIMIT \~EAST WEIGHTEDx2 \2a

LIMIT

I 100

~., cm

FIG. 8. Observed apparent optical depths, as described in Fig. 7, with best-fitting straight line and 95% (2or) confidence limits. The large error bars on certain values at 3.7 and 11 cm (see Table VI) essentially remove these points from the fitting process.

SATURN'S RINGS ties of the ring particles derived from reanalyzed interferometric observations over a range of wavelengths. First, we summarize and discuss our theoretical results. We have shown that the brightness of Saturn's rings at centimeter wavelengths is indeed dominated by diffuse scattering and transmission of the planetary thermal emission and that the ensuing azimuthal brightness distribution is asymmetric, or enhanced in the forward (subearth) direction. These results qualitatively confirm and quantitatively correct the results of earlier work (CVB). Further, we have shown that, in spite of its quite small optical depth, the C ring exhibits a brightness comparable to or greater than that of the A ring at wavelengths (~ l cm) where the ring brightness is dominated by scattering. This is because the size of the source function provided by the planet increases as seen from locations closer to the planet. Further, our theoretical calculations demonstrate that the thermal emission component of ring brightness increases more like h -1 than like h -2 with decreasing wavelength. This is a direct consequence of the primarily scattering nature of the ring layer. We have also demonstrated that shadowing effects, which probably dominate nonideal conditions for standard radiative-transfer calculations, are small at all wavelengths of interest for volume densities characteristic of the "classical" manyparticle-thick ring model. We have also reanalyzed several sets of good quality interferometric observations of the Saturn system. By employing realistic models which represent our current knowledge of the properties of the planet, we have increased the accuracy of the results (ring brightness and optical depth) to a point where more meaningful inferences may be drawn as to ring particle properties than was possible previously. Our chief improvements have been to include more realistic ring properties (variation of optical depth with radius, including the Cassini Division), planet properties (re-

699

alistic limb darkening) and geometry (smaller oblateness for the tilted spheroidal planet). Our new results for brightness temperature are not overly sensitive to variation, within current uncertainty, of ammonia mixing ratio, which determines the degree of planetary limb darkening. Our optical-depth results appear to be even less sensitive to this source of uncertainty. Surely, future interferometric observations over a wide range of wavelengths will be of great value in improving on the results we present here. There is obviously a large range of parameters which may be adjusted to "fit" the ring brightness and optical depth data; "/'max, rmax, and powerlaw index are three we have considered. However, certain specific constraints may be set on these parameters, individually or jointly. First, from the ring optical depth results at 3.7-cm wavelength (Fig. 7), it appears to be well established that the maximum optical depth in the B ring is at least unity and very likely on the order of 1.5 ___ 0.3, with the inner B ring scaled accordingly. Because, for the observations we used (see Table IV), the ring tilt angle was fairly constant a n d fairly large, no complications arise from tilt-effects or effects of ring A, which occulted a negligible amount of the disk at these times. However, for this reason we derive little information as to the microwave optical depth of ring A. That ring A has a smaller, but nontrivial, optical depth may be inferred from the results of Schloerb et al. (1979), as well as radially resolved radar observations (Pettengill et al., 1977). An optical depth as large as -1.5 for the thickest part of the B ring facilitates theoretical understanding of ground-based radar observations in terms of a multiplescattering, many-particle-thick model (CP). The presence of unresolved "holes" in the B ring, as inferred from Pioneer XI observations (Esposito et al., 1980) may affect our determination of the mean optical depth of the B ring slightly. Because of the rather large tilt angles characterizing these obser-

700

CUZZI, POLLACK, AND SUMMERS

vations, the change induced due to the wave (and, by implication, visible) wavepossibility of such " h o l e s " is only compa- lengths is provided by particles smaller rable to our quoted error in optical depth. than a meter or so. This result also rules out However, if a few percent of the rings have all "multiply internally scattering" particle optical depth of only a few percent, as models, even for matrix densities less than suggested by Esposito et al., then the an unrealistic 0.03 g cm -3, because their "typical" optical depth of the majority of optical depth would be wavelength indethe rings, referred to visible wavelengths, pendent. This latter result is in agreement is increased from - 1 . 5 to -1.8. We with the implications of the radar results of therefore have increased our upper error Ostro et al. (1980). bar accordingly. From the fact that the powerlaw size Our results as to variation of optical distribution with index 3 provides equal depth with wavelength may be interpreted cross-sectional area per decade of size, and in a variety of ways. There is an obvious from the fact that such distributions may be tradeoff between f o r m of size distribution ruled out for maximum sizes as small as assumed and size of the largest particle in lm, we feel that we can state confidently that distribution (e.g., Fig. 7). We offer this that less than - 3 0 % of the cross-sectional result as a cautionary note which must be area of the rings is presented by meter-andapplied to any interpretation of the size of larger-size particles. the "largest particle" in Saturn's rings. However, it must not be inferred that However, bearing this in mind, we still particles larger than a meter do not exist believe that certain constraints may now be within the rings; that this is true is shown applied to the particle size distribution. For by the optical depth curves for the powinstance, it has been pointed out that any erlaw with index 5, which easily satisfies size estimate based on thermal emission by the observational constraints with a 2-m an essentially transparent, low-loss particle maximum size. In fact, for such a size is subject to the uncertainty in the particle distribution, no upper limit may be placed composition and absorption coefficient. We on the size of the "largest" particle. observe a statistically significant decrease Our inferences from comparison of ringin ring optical depth with 'increasing wave- brightness observations with ice-particle length (Fig. 8) which appears to rule out, at models are consistent with the above the several-o- level, any particle size distri- results. It appears to be unlikely that the bution with cross section dominated by powerlaw of index 3, with maximum partimeter-and-larger-size particles, with very cle radii greater than a few meters, will fit little dependence on their refractive in- the data. However, as with optical-depth dices. For instance, the curve in Fig. 7 for a results, one must bear in mind the connecHansen-Hovenier distribution with mean tion between powerlaw index and size of radius of 300 cm is ruled out by the optical- largest particle. Powerlaws of larger index (i.e. PL5) would easily permit maximum depth data. A powerlaw of index 3 [n(r) = n0 r-3] was radii of several meters. The effects of a chosen by Cuzzi and Pollack (1978) for somewhat larger value of rmax (1.50), as comparison with radar observations as an shown in Fig. 6, are not major and would be example of a very broad particle size distri- easily offset by a small change in maximum bution. It has, in fact, the property that radius or powerlaw slope. It might be areach decade of radius provides the same gued that the form of the data favors a more net cross section. Clearly, powerlaws of gradual increase with decreasing wavethis index, with maximum size greater than length than any of the models, if a small, lm, are ruled out by the data. This implies wavelength-independent emission is added. that most of the ring cross section at micro- However, we feel that in the light of the

SATURN' S RINGS uncertainties, such arguments would be premature at this time. In addition, by comparison of detailed model calculations directly with specific diagnostic data sets, we can rule out all models with particles primarily smaller than 0.1 cm, independent of composition and, in conjunction with the results of CP, all models with particles composed primarily of silicates. It is not possible to rule out wavelength-size metallic particles (with a thin cover of water ice) from these observations. It will be extremely important to observe the rings at as many wavelengths as possible in the range hl00/zm to hl cm during the next few years as the rings open up. Interferometric observations at all wavelengths will also be extremely useful for improving our knowledge of the behavior of the ring optical depth. V. C O N C L U S I O N S

Comparison of theoretical calculations of ring layer scattering and emission behavior, for realistic models of ring layer, ring particle, and planet properties with interferometric observations of the Saturn system at microwave wavelengths has been made. We conclude that the maximum optical depth of the B ring, referred to visible wavelengths, is 1.5 --0.3 +0.s and that less than - 3 0 % of the B-ring cross-sectional area can be composed of meter-and-larger-size ice particles. There is no way, however, of ruling out the existence, within this constraint, of some quite particles in the rings from current data. The rings cannot be composed of primarily silicate particles. APPENDIX A

On shadowing by ring particles: An assumption which is implicit in most studies of scattering processes in radiative transfer is that the particles are sufficiently well separated that both (a) coherent scattering and (b) shadowing may be neglected. The usual criteria for (a) to be satisfied are that,

701

first, the particles should not be closer to each other than about two wavelengths and, second, that their spacings should be more or less random (van de Hulst, 1957). This situation is believed to prevail throughout the ring layer, under the "classical" many-particle-thick hypothesis in which the particles have typical separation I 10 times the typical particle size r (Kawata and Irvine, 1974). For typical particle sizes greater than a few centimeters, the above criteria for (a) are met for all wavelengths considered here. For wavelengths greater than lm, the problem would need to be addressed in more detail. The case of shadowing (b) is, however, somewhat more troublesome. The existence of a shadow near the forward-scattering direction is essentially due to the scattered wave contributions from different parts on the perimeter of the particle which have not yet been able to fully interact; that is, the "far-field" zone has not been reached. The extent of this near-field, or shadow zone, Is (see Fig. A1), is a function of both wavelength and particle size and may be estimated in a variety of ways as being Is ~ x r

(AI)

where x = 27rr/h (see, e.g., Brillouin, 1949). In cases where Is is greater than the line-of-sight distance to the next particle, l*, interesting effects may come into play. We must distinguish here between l and I*, as the angular distribution of the scattered radiation in a s p e c i f i c d i r e c t i o n (i.e., the direction defined by the incident radiation) is in question. The difference is nontrivial when regarded in three dimensions, as 1 r D -113 and l* - r / D , where D is the volume density; for the rings 10-2 > D > 10-3 (Kawata and Irvine, 1974). Thus, the condition o f " negligible shadowing" occurs for Is < l* = r / D , or r i D > xr; giving h > 21rDr. For particle sizes in the range of centimeters to meters, wavelengths longer than a few centimeters will be relatively free of shadowing effects. However, shadowing at wavelengths in the important millimeter

702

CUZZI, POLLACK, AND SUMMERS INCIDENT LIGHT I OF I W A V E L E N G T H , ?~

bt 0 = C 0 $ 0 0

I I

Fro. AI. Sketch of s h a d o w s in a ring layer o f thickness z0. The length of the s h a d o w cast by a particle of radius r is l~(r), and the volume-weighted m e a n s h a d o w length is is. An arbitrary "incident b e a m " is s h o w n at angle 0 = cos-t(/Zo) to the ring normal. It should be recalled that the sketch is a projection of a three-dimensional situation and s h a d o w s m a y not significantly overlap.

range (l~ < I*) must be considered in more detail. There are several possible effects of shadowing on scattering processes; shadowing could conceivably affect the typical average particle albedo, phase function, or extinction efficiency, and it can effectively " r e m o v e " particles from interaction with the radiation field. We claim that all three possible effects may be neglected. First, consider the particle albedo. Clearly, the total scattered energy crossing a sphere o f radius R centered on the particle may not change depending on whether R is greater than or less than Is. Therefore, the albedo o f the particle is not affected. The latter two effects may be considered together, and from two different standpoints. We note that essentially the same amount of diffracted energy lies in a narrow ring or cone at the edge of the geometrical shadow (Brillouin, 1949) as is contained in the more familiar forward-directed diffraction peak seen in the far field (beyond the shadow zone). We note that essentially the same amount o f diffracted energy lies in a narrow ring or

cone at the edge of the geometrical shadow (Brillouin, 1949) as is contained in the more familiar forward-directed diffraction peak seen in the far field (beyond the shadow zone). This situation may be regarded from the standpoint of the radiation field as a small shift of the peak of the phase function in the near-field to slightly larger scattering angles, with reduction of the directly forward-scattered intensity to zero. Because the change in effective scattering angle is so small (AO -- l/x) little net effect on the net transfer of radiation is expected. Certainly, the use o f far-field phase functions in the near-field is an approximation, however, for situations of interest here, (substantial multiple scattering, random and uniform particle spatial distribution, polydisperse particle size distribution, low spatial and angular resolution) it appears to be acceptable. The near field effect may also be regarded from the standpoint of other particles within the near field. If the volume density is sufficiently high relative to the extent of the near field (ls > l*) to cause some particles to be within the shadow zone and not to interact with the scattered radiation, it is also high enough to cause particles to lie within the Brillouin zone of enhanced radiation surrounding the shadow zone. It may be easily shown that the volume of the shadow in the near field is comparable to the volume of the "Brillouin z o n e . " Therefore, for a uniform distribution of particles, the situation Is > l*, which removes some particles from interaction with scattered radiation simultaneously enhances (by an equal amount) the n u m b e r of particles interacting with scattered radiation. Therefore the opacity o f the layer is unaffected. On these grounds, we maintain that the overall effects of shadowing, and of nearfield phenomena in general, are not significant to the problem at hand. This argument is considerably strengthened when it is realized that the total fractional volume of particles so affected (particles lying in and near the shadow zone) is small ( - 1 0 % ) at the wavelengths of interest.

SATURN'S RINGS Below, we demonstrate the calculation of this fraction. The question of mutual shadowing of particles has been theoretically attacked in great detail by Irvine (1966) in the geometrical optics limit and by Franklin and Cook (1965) in a more diffraction-limited regime. However, these studies have been more directed toward the problem of the intersection of shadowed volume with observed volume and do not bear directly on our problem here. These latter authors have integrated the shadow volume throughout the ring layer, considering (as in our case) conical shadows and have truncated the conical shadows when they extend beyond the ring layer, as no particles exist outside the layer to be shadowed. We have adopted a much simpler order-of-magnitude approach which is sufficiently accurate for our purposes and greatly clarifies the problem. Consider the shadowed ring layer shown in Fig. AI, in which a typical size distribution is represented showing numerical dominance by small particles and mass dominance by large particles. It will be assumed that the shadows fill a slab of thickness Zo + [slxo, whereas the particles only fill a slab of thickness z0. We may then integrate over shadows cast by all particles in the ring layer, of thickness z0, without involving complex geometry for truncation of conical shadows at the lower ring layer boundary. The value of this approach will be seen when the short wavelength limit is considered. LettingdN(r) be the incremental number of particles of radius r within (z, z + dz), the fraction of particles shadowed = c(h) = net fractional volume in shadow within the the ring layer is

c(X)=c= (Zz~ ~ )

fo°

where V~(r)/V is the fractional shadow volume due to particles of radius r, and

is = f~2 ls (~Zrr21s)n(r)dr/ f:~2 ½7rr21~n(r)dr. (13) If n(r, z) = nor-3 (CP), and ls - (2zrr/h) • r, we obtain simply

c(X)

--

Vs(r) dN(r, z, z + dz) 1 ---V-(A2)

f:o f]~½ ~rZl~(r)n(r, z) dr dz, Zo

c ~- [ Z o / ( Z o +

~0is)] (TrZnor22/3X). (A4)

Further, it is easily shown from Eq. (A5) that, for the same size distribution, is = (Tr/h)r22.

(A5)

That is, the shadowing process is dominated by the largest particles (radius r2). Some insight is obtained by taking the limiting cases of (a) is "~ z0 and (b) is ~ z0, corresponding to the long and short-wavelength limits. It has been shown by Cuzzi et al. (1979) that, due to gravitational scattering in a size distribution of the form n (r) = n 0r-a,

°°/ 10-11

10 2

10-3

/x0

703

i

.3

110

310 10.0~ ~., c m

30.0l

100.0

FIG. A2. The fraction of shadow volume, c, is plotted against wavelength h for a maximum particle size of 1 m and a range of possible ring volume densities. The solid curve shows the result for a nominal volume density D of 5 × 10-a. The crosshatched region shows a reasonable range of uncertainty in volume density (10 -z > D > 10-a).

704

CUZZI, POLLACK, AND SUMMERS

that z0 -

10r2. Therefore, from Eq. (AS),

is/So ~ 7r sol 100h ~ 7rrz/1 Oh.

If rz = 1 m (this paper and Epstein et al., 1980), then is~So < ( < ) 1 (case a) implies h > ( > ) 3 cm; and, conversely, [ J z o > ( > ) 1 (case b) implies h < ( < ) 3 cm. In case (a), we obtain c(~,) ~ D r z / X and in case (b) we obtain c(h) ~ D z o / r z . It is interesting to note for case (b), the short-wavelength limit, that c(h) is independent of wavelength, as one might expect. Further, from the results of Cuzzi e t al. (1979) that Zo/rz ~ constant, we realize that the shadowing effect at short wavelengths depends only on the volume densityD, and, for typical values of D and zo/rz ~ 10, is quite small. Typical values ofc(h) as a function of wavelength for 10 -2 > D > 10 -3 andr~ - 1 m are shown in Fig. A2. The only effect of varying rz is to vary the wavelength which separates cases (a) and (b). A final check on this approach is to compare the result with the observed opposition effect at visible wavelengths, believed to be due to mutual particle shadowing. The increase in brightness of the rings in the sharp "swing" within - 1.5 ° of opposition is - 0 . 2 5 magnitudes (Franklin and Cook, 1965), or a factor of - 2 6 % . Considering that this increase includes contributions from partially as well as totally shadowed particles, and probably also a certain component due to the intrinsic phase variation of the particles themselves, the agreement with the above simple estimate is satisfactory. In conclusion, the net effect of all nonideal properties of the ring layer regarding particle spacing appears to be negligible at even the shorter wavelengths under consideration here. ACKNOWLEDGMENTS We are extremely grateful to F. Briggs, W. Jail'e, and M. Janssen for their helpful cooperation and willingness to allow us to reanalyze their data. We are also grateful to S. Gulkis for his help with the atmospheric model and to F. P. Schloerb and D. Muhleman for providing results in advance of publication. We

thank O. B. Toon for helpful discussions on mutual shadowing by particles and near-field effects and J. Ledden and W. A. Dent for pointing out the geometrical effect of tilting the planet. We thank Larry Esposito for a careful reading and critical evaluation of the manuscript. We also thank Gary Veum and David Comstock for valuable assistance with reduction and analysis of the data. Part of this research was performed while J. N. Cuzzi was an NAS-NRC Resident Research Associate at Ames Research Center.

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JAFFE, W. J. (1977). 6 cm radio observation of Saturn. PreptinL JANSSEN, M. A., AND OLSEN, E. T. (1978). A measurement of the brightness temperature of Saturn's rings at 8-mm wavelength. Icarus 33, 263-278. KAWATA, Y., AND IRVlNE, W. M. (1974). Models of Saturn's rings which satisfy the optical observations. In The Exploration o f the Planetary System (Woszczyk and Iwaniszewska, Eds.), pp. 441-464. KAWATA, Y., AND IRVINE, W. M. (1975). Thermal emission from a multiply-scattering model of Saturn's rings. Icarus 24, 472-482. KLEIN, M. J., JANSSEN, M. A., GULKIS, S., AND OLSEN, E. T. (1978). Saturn's microwave spectrum: Implications for the atmosphere and the rings. In The Saturn System (D. Hunten and D. Morrison, Eds.). NASA Conference Publication 2068. MEYER, S. L. (1975). Data Analysis for Scientists and Engineers, Chaps. 31, 32. Wiley, New York: OSTRO, S., PETTENGILL, G. H., AND CAMPBELL, D. B. (1980). Radar observations of Saturn's rings at intermediate tilt angels. Icarus 41, 381-388. PETTENGILL, G. H., OSTRO, S., AND CAMPBELL, D. B. (1977). Saturn's rings: Radial distribution of radar scatterers. Bull. Amer. Astron. Soc. 9, 462. POLLACK, J. B. (1975). The tings of Saturn. Space Sci. Rev. 18, 3-93. POLLACK, J. B., SUMMERS, A. L., AND BALDWIN, B. Estimates of the size of the particles in the rings of Saturn and their cosmogonic implications. Icarus 20, 263-278. SCHLOERB, F. P., MUHLEMAN,D. O., AND BERGE, G. L. (1979). Interferometric observations of Saturn and its rings at a wavelength of 3.71 cm. Icarus 39, 214-231. SCHLOERB, F. P., MUHLEMAN, D. O., AND BERGE, G. L. (1980). Interferometry of Saturn and its rings at 1.30 cm wavelength. Icarus 42, 125-135. VAN DE HULST, H. C. (1957). Light Scattering by Small Particles, Wiley, New York. WHALLEY, E., AND LABB/~, H. J. (1969). Optical spectra of orientationally disordered crystals. III. Infrared spectra of the sound waves. J. Chem. Phys. 51, 3120.