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

~CARVS33, 233--262 (1978)

Saturn's Rings: Particle Composition and Size Distribution as Constrained by Microwave Observations I. Radar Observations JEFFREY

N . C U Z Z I ~ AND J A M E S

B. P O L L A C K

Theoretical and Planetary Studies Branch, Space Science Division, Ames Research Center, NASA, Moffett Field, California 94035 :Received June 30, 1977; revised August 29, 1977 We have calculated the radar backscattering characteristics of a variety of compositional and structural models of Saturn's rings and compared them with observations of the absolute value, wavelength dependence, and degree of depolarization of the rings' radar cross section (reflectivity). In the treatment of particles of size comparable to the wavelength of observation, allowance is made for the nonspherical shape of the particles by use of a new semiempirical theory based on laboratory experiments and simple physical principles to describe the particles' single scattering behavior. The doubling method is used to calculate reflectivities for systems that are many particles thick using optical depths derived from observations at visible wavelengths. If the rings are many particles thick, irregular centimeter- to meter-sized particles composed primarily of water ice attain sufficiently high albedos and scattering efficiencies to explain the radar observations. In that case, the wavelength independence of radar reflectivity implies the existence of a broad particle size distribution that is well characterized over the range 1 cm ~ r ~ i m by n(r)dr = nor-3dr. A narrower size distribution with d ~ 6 cm is also a possibility. Particles of primarily silicate composition are ruled out by the radar observations. Purely metallic particles, either in the above size range and distributed within a many-particlethick layer or very much larger in size and restricted to a monolayer, may not he ruled out on the basis of existing radar observations. A monolayer of very large ice "particles" that exhibit multiple internal scattering may not yet be ruled out. Observations of the variation of radar reflectivity with the opening angle of the rings will permit further discrimination between ring models that are many particles thick and ring models that are one "particle" thick. I. INTRODUCTION C u r r e n t c o n c e p t s of t h e n a t u r e of Saturn's rings have been greatly influenced b y t h e r e s u l t s of r e c e n t r a d a r i n v e s t i g a t i o n s w h i c h s h o w t h e r i n g s t o b e v e r y effective r a d a r reflectors. I t h a s b e e n f o u n d t h a t t h e r a d a r r e f l e c t i v i t y of t h e r i n g s is g r e a t e r , b o t h a t ),3.5- a n d X l 2 . 6 - c m w a v e l e n g t h s , t h a n t h a t of a n y o t h e r o b j e c t in t h e s o l a r system. The reflected power at both wavel e n g t h s is, w i t h i n o b s e r v a t i o n a l error, corn1 NAS-NRC Resident Research Associate.

pletely depolarized (Goldstein and Morris, 1973; G o l d s t e i n et al., 1976). Subsequent passive interferometric observations at centimeter and decimeter wavelengths have shown that the rings e x h i b i t s i g n i f i c a n t b l o c k a g e of e m i s s i o n f r o m t h e p l a n e t a r y disk, i n d i c a t i v e of a n o p t i c a l d e p t h in t h e B r i n g of t h e o r d e r of unity, comparable to that at visible wavel e n g t h s (Briggs, 1974; C u z z i a n d D e n t , 1975; M u h l e m a n et al., 1976). T h e s e s a m e r a d i o o b s e r v a t i o n s , a n d t h o s e of J a n s s e n a n d Olsen (1976), also c o n f i r m e d p r e v i o u s 233

oo19-1035/78/o332-02335o2.oo/0 Copyright O 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

234

CUZZI AND POLLACK

observations (Berge and Read, 1968 ; Berge and Muhleman, 1973) that the brightness temperature of the. rings at centimeter and decimeter wavelengths is very much lower than the physical temperature of the ring particles themselves as given by infrared observations (Morrison, 1976). Currently, two distinct conceptual models of the ring system have been advanced to explain all known observations. It seems proper to keep in mind that each concept represents one extreme of a continuum of possible ring states. On the one hand, the "classical" ring model presumes the ring particles to be distributed within a slab or layer that is extremely thin relative to its horizontal extent, yet much thicker than the size of a typical constituent particle. On the other hand, inelastic collisions between particles may have damped random relative velocities to such an extent that the rings are only one particle thick (Jeffreys, 1947; Cook et al., 1973). Although the dynamical arguments for fattening are quite strong, photometric observations seem to favor the classical concept (Kawata and Irvine, 1974 ; H~tmeen-Anttila and Vaaraniemi, 1975). These concepts are dealt with in great detail in review papers by Bobrov (1970) and Pollack (1975). Each concept offers a different explanation for the microwave behavior of the rings. Here, we investigate the constraints placed on both conceptual models of the rings by radar observations. In a subsequent paper we will deal with additional constraints imposed by microwave brightness observations. The concept of a many-particle-thick ring attributes the high radar reflectivity and depolarization to multiple scattering by high-albedo particles in a layer of significant optical depth. It was pointed out by Pollack et al. (1973) that wavelengthsized particles of composition plausible for the outer solar system (water ice or silicate) could attain sufficiently high albedos and scattering efficiencies to satisfy these obser-

vations. Pure metallic particles of size larger than a wavelength would have similar properties. Tile monolayer concept may be applied to constituent particles of a variety of sizes, structures, and compositions. Although multiple scattering between constituent particles will not contribute significantly to the net radar reflectivity or depolarization, this may be compensated for by high backscatter gain or by multiple scattering within the interior of constituent particles. Here, we calculate the radar reflectivities characteristic of monolayer and manyparticle-thick ring models, assuming a wide range in particle size and composition, and compare the results of these calculations with radar observations. In Section II, we describe the treatment of radar scattering by a many-particle-thick ring model. In Section III, we describe the radar observations. In Sections IV and V, we compare the behavior of many-particle-thick and single-particle-thick ring models, respectively, with the observations. In Section VI, we present our summary and conclusions. II. SCATTERING BEHAVIOR OF A MANYPARTICLE-THICK RING MODEL We now describe the calculations leading to the determination of the radar reflectivity of ring systems that are many particles thick. As suggested by Pollack et al. (1973), the particle sizes of greatest interest for such a ring system are coinparable to the radar wavelengths. Therefore, we use Mie theory to calculate the scattering properties (efficiencies and phase function) for spherical particles of these sizes. We use realistic material properties (refractive indices) in the Mie calculations for three candidate compositions : water ice, a representative silicate, and pure metal. The refractive indices of these candidate materials effectively span the entire spectrum of reasonable possibilities. Using a new algorithm, we correct the above

SATURN'S RINGS: RADAR OBSERVATIONS

235

circumference to wavelength, x = 2~a/h, is the size parameter that determines the domain of theory within which a given problem falls. The domain of Mie theory is approximately the range of 0.01 ~ x ~ 100. The cross sections and phase function are obtained by solving the wave equation rigorously on a spherical boundary (the particle) and evaluating the angular variation of the scattered wave (the phase function) in the far-field limit. The implications of these aspects of the theory are discussed further below. The ring particles are not likely to be A. Single-Scattering Properties Including spherical, however, and some of the characParticle Irregularity teristic behavior of Mie scattering is due A particle of geometrical cross section exclusively to spherical geometry. We have 7ra2 presents cross sections for absorption developed a new algorithm based on labora(zA) or scattering (as) of radiation of tory experiments and simple physical wavelength ~ given by principles to allow for effects of nonsphericity in treating ensembles of ranzA = QA(a, ~)lra 2, (1) domly oriented particles in the Mie domain as = Qs(a, h)~ra2, (Pollack and Cuzzi, 1978). Existing laboratory data show several where QA(a, ~) and Qs(a, ~) are efficiencies for absorption and scattering. Also, the clearly evident effects of nonsphericity on cross section for total extinction is ZE scattering behavior. Parameters integrated = aA + as, and thus QE(a, ~) = QA(a, ~) over angle, such as albedo and efficiency, are + Qs(a, ~). The single-scattering albedo, not significantly affected by nonsphericity &0, is defined as &0 = ZS/aE, and is the (e.g., Greenberg et al., 1971; Purcell and fraction of energy removed from an incident Pennypacker, 1973; Proctor and Barker, beam that is not absorbed by the particle 1974). In particular, irregular particles as but is scattered into another direction. The well as spherical particles may have phase function p(O) gives the angular extinction efficiencies in excess of the distribution of scattered energy, where geometrical optics value of 2 (see Fig. 1). The main effect of nonsphericity on ~1 f 0 p(O) sin OdO = 1. For a planeparallel, homogeneous layer of thickness z0 scattering behavior is in the phase function. containing such particles, the normal A sufficient amount of experimental work optical depth is r0(~,) = naEzo, where n is has been done to permit satisfactory working hypotheses to be formulated. For particle volume density. Mie theory [-see, e.g., van de Hulst example, Fig. 2a compares the phase (1957) or Kerker (1969)-] allows calculation function for scattering by a cube, averaged of the cross sections for absorption and over many orientations, to that calculated scattering, and the angular dependence of for a sphere of equal volume and refractive the scattered radiation, for particles of index using Mie theory. The good agreegiven composition whose radius is neither ment indicates that even a cube of the sufficiently small nor sufficiently large for given size (x ~ 4) scatters like a sphere. Rayleigh or geometrical optics approxi- Figure 2b shows a similar comparison for a mations to apply. The ratio of oartic]e "rough" sphere, with a corrugated surface, calculated scattering properties for the effects of particle irregularity or nonsphericity. We average these single-scattering properties over different choices of particle size distribution. We use the doubling method to solve the equation of transfer for the net reflectivity, including the component due to multiple interparticle scattering. To obtain the microwave optical depth of the ring layer, we use observed optical depths at visible wavelengths and our calculated particle efficiencies.

236

CUZZI AND POLLACK 5.0

.....

•

i

0

iO

2D

50

4.0

5.0

6.0

p : 4~ra(nr-l)IX

FIG. 1. Data show the extinction efficiency of "rough" spheres, composed of stacked cylinders, as obtained using analog techniques. Efficiency is plotted against the phase-shift parameter p. Data refer to different orientations of the electric vector to the cylinder axes. The solid curve is the theoretical (Mie) result for equal-volume spheres of the same index of refraction (m = 1.37 + 0i) (from Greenberg et al., 1971).

which shows good agreement with theory even for x ~ 14. I t appears that, for m a n y irregular shapes, there will be some maxim u m value of x = x0 t h a t varies with shape

and possibly refractive index, below which the particle "looks" spherical. The phase function for scattering by an ensemble of large (x > x0) cubes is shown ]05

.

.

.

.

.

x

104

I03

,o41

. . . . . . . . . . . . . . . . . .

x = 575 • • • CUBE - SPHERE (MiE)

A

• • •

•

= 14 02 ROUGH SPHERE SMOOTH SPHERE

I03

A

]

P (~) IO~ II.

p(®)

i01 °°°° i

i00 L

, 90

e

-

I 180

I00 0

i

90

~

18o

Fie,. 2. Comparison of the angular distribution of the intensity scaitered by an irregular particle obtained by analog technique with the the(,retical (Mie) prediction for all equal-volume sphere (,f the same refractive index. The size parameter x = 2~ra/X. The intensity is given in arbitrary unils (from Zerull and Giese, 1974). (a) Intensity scattered t)y a small cube. (b) Similar comparison for a large corrugated sphere.

SATURN'S RINGS: RADAR OBSERVATIONS in Fig. 3. The data show an essentially linear relationship between 0 and log p (O) for large scattering angles. There is no trace of the characteristic Mie backscatter enhancement (rainbow" and glory) associated with spherical geometry. This behavior is also seen for other irregular-particle shapes (e.g., Holland and Gagne, 1970). Characterization of particles with x > x0 was accomplished by the construction of parameterized phase functions. We separated the scattered radiation into a diffracted component, a reflected component, and an internally refracted and transmitted component. This has been attempted before with good results for spheres of x ~- 15 and scattering angles less than ~ 6 0 ° (Hodkinson, 1963; Hodkinson and Greenleaves, 1963), and for all scattering angles for very large spheres (Hansen and Travis, 1974). Application of this technique to irregular particles is based partly on the equivalence of the components that are diffracted and externally reflected by a collection of randomly oriented irregular particles, with components diffracted and reflected by a collection of spheres of equal cross-sectional area and refractive index (van de Hulst, 1957; Hodkinson and Greenleaves, 1963; Hansen and Travis, 1974). We do not attempt an exact solution for the transmitted part, but instead use a simple parameterization of this component which is suggested by experimental data. The most significant effect of nonsphericity is that irregular particles permit more energy to be totally reflected internally than do spheres, and thus scatter more energy at large scattering angles than do spheres. Spheres allow no total internal reflection, and scatter at large angles ( > 9 0 °) only the small fraction of incident energy that enters the particle after striking the sphere nearly tangentially. In this sense, spheres are the least effective of all l)artMe shapes in a multiple-scattering c¢ m t e x t .

Our algorithm assumes that the refracted

237

104

5.87 <- x <- IZ79 I0 s

l

102

p (@)

I0 ~

I0 o

r 90

0

0--

~

180

FIG. 3. Comparison of the angular distribution of the intensity scattered by a distribution of large cubes, obtained using analog techniques, with theoretical (Mie) predictions for an identical distribution of equal-volume spheres. The size parameter x = 2~a/X. The intensity is given in arbitrary units (from Zerull and Giese, 1974). (transmitted) component R (0) is linear in log R(O) vs O, with slope determined by a parameter F T B which we define as FTB =

R(O)dO

/00

//

R(O)dO

(2)

0

for reasons of mathematical simplicity. This linearity was suggested by data such as those shown in Fig. 3. By normalizing and combining the diffracted, reflected, and refracted components, and combining the x < x0 (Mie) and x > x0 (parameterized) phase functions, we obtain a net mean phase function. By comparing such phase functions to the data of Zerull (1976), Zerull and Giese (1974), and Holland and Gagne (1970), we have determined a range of values of x0 and F T B which best characterizes scattering by irregular particles of different shapes, sizes, and refractive indices. The agreement with experimental data is quite good, justifying the use of the algorithm. Two of the many comparisons we have made of our calculated irregular-particle phase fune-

238

CUZZI AND POLLACK I

I

]

log P (0)

I

I

I

I

I

SEMI-EMPIRICAL THEORY o

~

~ ] ; ....,/

/ /

\ ' ~ .

-I

{ ZERULL AND GtESE DATA ^ II CUBES, 1.9 _
120

--

f//

20

I

40

I

I

I

I

I

60 80 I00 120 140 SCATTERING ANGLE, deg

160

180

Fro. 4. C o m p a r i s o n of the phase f u n c t i o n for scattering b y ~ d i s t r i b u t i o n of large and small

cubes, obtained using analog techniques, with theoretical predictions. Dashed curve, Mie theory; solid curve, semiempirical theory (as described in text).

tions with corresponding data are shown in Figs. 4 and 5. In the scattering calculations discussed in the next section, we I

log P (O)

I

have used representative values of x0 and F T B in constructing irregular-particle phase functions similar to those shown in

I

-I

I

I

I

I

--

..

0

"'-.~.~

SEM I-EMPIR

L

EORY

.

-I { ZERULL DATA, CONVEX A N D CONCAVE PARTICLES, 5.9 _
/ L

r~:l.5 + o.oo5,

|

SEMI-EMPIRICAL THEORY: Xo = I0, FTB = 4.0, | SAR = 1.3 _~

_1 __.1

-20

m

20

1

40

__

I .

I

~ _ _

•

60 80 I00 120 140 SCATTERING ANGLE, deg

I

160

180

FIG. 5. Comparison of the pha.~e function for scattering by a distribution of irregularly shaped particles described by Zerull (1974, 1976) as "convex" and "concave" with theoretical predictions. Dashed curve, Mie theory; solid curve, semiempirical theory (as described in text).

SATURN'S RINGS: RADAR OBSERVATIONS

239

Figs. 4 and 5. We have chosen x0 = 3, where SAR is the ratio of surface areas of F T B = 2, and x0 = 8, F T B = 2 to repre- an irregular particle and a sphere of the sent fairly common particles with different same volume. For ,50s ~ 1, ~0 i ~- ~50s. All degrees of irregularity. As shown in Figs. 4 of these matters are treated in greater and 5, x0 = 3 and x0 = 8 characterize detail b y Pollack and Cuzzi (1978). Irregular metallic particles with a distrifairly rough (cubes) and fairly smooth particles, respectively. A value of F T B = 2 bution of sizes between 0.1 ~< x ~< 100 m a y was fairly typical of several particle shapes be modeled quite closely b y their spherical t h a t we investigated. For completeness, we counterparts, as there is no transmitted have also included one case of a fairly component. Such particles scatter only b y rough particle (x0 = 3) t h a t scatters its diffraction and reflection, and the reflected refracted (transmitted) component iso- component is isotropic unless a >> X, the tropically ( F T B = 1), although such be- limit in which a baekseatter direetivity havior was not found in comparisons we m a y be produced. This limiting ease is made with existing data (Pollack and discussed below as part of the monolayer model. Cuzzi, 1978). The use of phase functions either calcuOne further correction is applied to the irregular-particle efficiencies. The initial lated b y Mie t h e o r y or due to Fraunhofer calculation gives efficiencies for spheres of diffraction presupposes the particles to be equal volume. For a particle with x > x0, in the far field of each other, which requires the total scattered intensity will be propor- a particle separation of m a n y wavelengths (Kerker, 1969). Also, the particles must not tional to the total surface area. T h e surface-to-volume ratio is lower for spheres shadow each other. T h e shadow cone t h a n for arbitrarily shaped particles. Cubes extends to a distance a2/X ,,~ ax (Brillouin, have 30% more surface area t h a n spheres 1949). Thus, the particles will not shadow of the same volume and are t h e r e b y more each other if t h e y have a mean separation effective scatterers. The enhancement of ~ a x 11~, which is on the order of several scattering efficiency of irregulars over t h a t particle radii for centimeter-sized particles of equal-volume spheres is well illustrated at microwave wavelengths. Volume densib y Greenberg et al. (1971). Having calcu- ties implied b y the opposition effect lated a scattering efficiency for a given (Kawata and Irvine, 1974) indicate t h a t distribution of spheres with x > x0, we then these criteria are satisfied in the context of must increase it to allow for this effect. the many-particle-thick ring. The polarization characteristics of seatUnless the particles are metallic, the absorption cross section is proportional to tering b y irregular particles are less well volume and is not affected. Thus, b o t h the known t h a n are the effieieneies or phase scattering cross section and the albedo are functions. The directly baekseattered (oncehigher for irregular particles (superscript i) scattered) radiation is not depolarized b y t h a n for spherical particles (superscript s) spheres. We define the depolarization ratio for single scattering, DR, as of the same volume : D R = I2/I1,

Qs i -= SAR. Qs S, Qs i ~0 i =

- -

QEi

Qs' =

(3)

Qs ~ -[- (QA~/SAR) SAR SAR -t- (1/&o ' -- 1)

where I2 and I1 are directly backseattered intensities in orthogonal polarizations, and the reflected radiation is expected to be in polarization (1) if the particles are spherical and the incident beam is completely polarized. Then, for spheres, D R = 0.

240

CUZZI AND POLLACK

Observed values of D R for irregular particles of x "-~ 10 are typically of the order of ½ to ½ (Sassen, 1974; Zerull, 1974, 1976 ; McNeill and Carswell, 1975). Larger values are observed for larger particles (Schotland et al., 1971) or particles of high dielectric constant (Sassen, 1975). The relevance of such depolarization of the directly backscattered component to the radar reflectivity is discussed next.

B. Calculation of the Diffuse Reflectivity of a Layer, Including Multiple Scattering Consider the plane-parallel layer in Fig. 6, illuminated by a plane wave (radar transmission) incident from direction ( - ~ , q~). The intensity of radiation diffusely reflected or transmitted by the layer into direction (~', ¢') is given by

I(u', 4,') = (~/4u')S(~o; u, 4~, u', 4;) (u' <_ 0), I (u',4~') = (~Y/4u')T(ro; u, rb, u', ¢') (u' > 0),

(4)

where the functions S and T are the diffuse reflectivity and transmissivity of the layer (Chandrasekhar, 1960). The total intensity reflected by a layer of very small optical depth r*<< 1 is ~/~:Cos8

I'~-.(

1 /

1

-

i -

i PLANEZ:Z i

i i,

.i . . . L ~

Fla. 6. G e o m e t r y of the plane-parallel layer. T h e incident radiation is shown as a plane wave of flux ~r.~ normal to itself. T h e seat lering angle O is a fttnclion of tL, U', 4', and ~'; therefore p(tt, q~, it', qT) = p(O).

(Chandrasekhar, 1960) I(•', ¢') = - 4~' \ u + ~'/

× I , °xE x p(u, ¢, u', ¢')

(5)

for the geometry in Fig. 6. In (5), p(u, ¢, u', ¢') is the phase function, normalized by

p(O) 4~ This expression gives the singly scattered intensity that is diffusely reflected from a many-particle-thick layer of arbitrary optical depth. A similar expression holds for the diffusely transmitted intensity. For r* << 1, there is essentially no multiple scattering so (5) is exact. For the case of interest, in which ~0 ~ 1 and r0(X) >~ 1, the contribution of multiple scattering is significant. We have used the doubling method (Hansen, 1969a, 1970; Hansen and Travis, 1974) to determine the diffuse reflectivity of such a layer. In this method, which assumes the layer to be homogeneous, two identical layers of optical depth r, for which the reflectivity and transmissivity are known exactly (starting with r = r*<< 1), are numerically combined to give the behavior of a single layer of optical depth 2r. Final values for arbitrary r0(X) are quickly attained even for quite large r0. In performing these calculations, irregular particle phase functions are used in the initial reflectivity given by (5). The calculated reflectivities have a fractional accuracy better than 10-~. Our version of the doubling method has been checked against theoretical results for layers of both large and small optical depths using both isotropic and anisotropic phase functions (Carlstadt and Mullikin, li)(;G; ttorak, 1952).

241

SATURN'S RINGS: RADAR OBSERVATIONS TABLE I CANONICAL I{ING OPTICAL DEPTHS (AFTER COOK et Inner radius

Outer radius

1.09 R ~ 1.21 1.39 1.53 1.64 1.88 2.03

1.21 1.39 1.53 l . 64 1.88 1.95 2.29

r (v)

Rb"

0.02 0.08 0.15 0.30 0.60 1.20 0.30

al.,

1973)

l{inglet area '~ Area of A and B r i n g s /

Colmnents

C inner

0.08 0.18 0.10 0.12 0.31 0.10 0.46

C middle C outer B inner B nfiddle B outer A middle

R~ = 8'.'2463 at 10 AU.

As shown in Table I, our ring layer consists of seven radial ringlets of optical depths t h a t differ b y a factor of 2. This makes it poisible to compute reflectivities for each ringlet successively using the doubling program. The reflectivities are then averaged as weighted b y the ratio of the visible area of each ringlet to the area of A and B rings:

S = E S,(rOw6

(6)

where S~ is the diffuse reflectivity of ringlet i of optical depth r~(X), and w~ is the ratio of the visible area of ringlet i to the area of the A and B rings. In addition, we calculate the singly scattered return for each ringlet using (5) with the proper optical depth, and obtain a weighted average (I~) in the same way. The depolarization of the net diffusely reflected return (IT) is defined in the same m a n n e r as was D R : IT(2)

IT(l) 0.5I~ + [ D R / ( D R

+ 1)7/~

=

,

(7)

0.5I~ + [ 1 / ( D R + 1)-]I~ where [M = IT -- I~ is tered component. T h e component is assumed depolarized. Liou and

the multiply scatmultiply scattered to be completely Schotland (1971)

have shown t h a t even the twice-scattered radiation is ~ 8 0 % depolarized b y spheres. One would expect irregular particles in r a n d o m orientations to be at least as effective at depolarizing multiply scattered radiation.

C. Distributions of Particle Size T h e existence of a distribution of particle sizes is to be expected in realistic cases. We define mean parameters characteristic of a given particle size distribution as follows:

f QE(a, k)Tra2n(a)da/ f n (a) Tra2da, (s) where n(a) is the n u m b e r of particles with radii between a and a + da and QA(X), 0s(X), and /3(k;~, ¢, #', ¢') are similarly defined. Because it enters into our calculations primarily in the above integral form, the exact form of the size distribution is not well determined b y existing data. The calculations are sensitive only to the mean radius and variance or width of the size distribution. Thus, any distribution simply characterized b y mean size and variance will suffice for these calculations. One emmnonly used size distril}uti,m is (Hansen and Travis, 1974)

n(a)da = no(a)a-a")l"e-(~/~)da,

(9)

242

CUZZI AND POLLACK

where d is the mean radius weighted b y geometric cross section,

b y a size distribution n (a) = noa -~, fa "f n 0a-'y (x) Q (a, X)7ra2da

a= f an(a)~a~da/f n(a)~a:da,

i

-~2~r)

and v is the variance (fractional width) of the distribution, also weighted b y geometric cross section. We consider two representative size distributions, each with physical significance as to the nature of the rings. A relatively narrow distribution might be expected to result from condensation of particles from a protoplanetary nebula (Goldreich and Ward, 1973). Alternatively, a power law with index near 3 is commonly found to prevail in numerous fragmentation processes (Hartmann, 1969). We return to the significance of the size distribution in Section VI. For the fairly narrow size distribution we have used, given b y (9) with v = 0.3, the mean size is well defined and thus the mean size parameter 2 = 2~rd/X is also well defined. Examples of the shapes of such size distributions and their effects on scattering parameters are given b y Hansen and Travis (1974). The power-law distribution of interest is given b y (9) in the limit of v --+ o0 ' n (a) da = noa-3da.

(lO)

This v e r y broad size distribution yields essentially wavelength-independent mean parameters. The " m e a n size" for this size distribution is determined b y the range of radii over which it is valid, t h a t is, =

fa af noa -3

• a. ~-a 2. da

i

/fa af noa-37ra2da i

= (at-

ai)/ln (af/ai).

(11)

Also, consider the wavelength dependence of the mean of a parameter y (x) as weighted

no

i

x-'y(x)q(x)~x2dx.

(12)

Given equivalent values of xl and xf for each wavelength (xi << 1 and xf >> 1 are sufficient), the mean value of the parameter is independent of wavelength if s = 3. The dependence of ~(X) on af is not significant, being logarithmic so long as 2~raf/h >> 1;

[-Q(27raf/~) ~ constJ. (13)

In this case, we chose af as large as could be conveniently treated b y the Mie calculations ( ~ 1 m) and varied al = ami, to investigate the effect of cutoffs in the ring particle-size distribution at different minim u m radii due, for instance, to the Poynt i n g - R o b e r t s o n effect. D. Particle Composition and Complex Reractive Indices We consider three particle compositions of interest: a typical silicate, pure metal, and water ice. Only water ice has been positively identified in the rings (Kuiper et al., 1970; Pilcher et al., 1970), although b r o a d b a n d visible spectra indicate the presence of other material t h a t is strongly absorbing at short visible wavelengths (Lebovsky et al., 1970). This reddening phenomenon is also exhibited b y the Galilean satellites and could arise from trace surface constituents (Johnson, private communication, 1976). Of course, these observations refer only to the very surface of the ring particles and thus are not sensitive to bulk composition. The microwave observations are significant in t h a t they enable conclusions to be drawn as to the bulk composition of the particles. For similicity, we have not considered clathrates of methane or ammonia with water ice, about which very little is known.

SATURN'S RINGS: RADAR. OBSERVATIONS

243

TABLE II [{EFRACTIVE INDICES FOR P A R T I C L E COMPOSITIONS U S E D

Material

Wavelength (cm)

n~

Ice (water) Ice (water) Rock (acidic) Metal (iron)

3.5 12.6 3.5, 12.6 3.5 12.6 3.5 12.6 3.5, 12.6

1.78 1.78 2.32 6500 12300 1000 1000 7.8

Stony-iron

nl

The dielectric properties of such materials are not likely to differ significantly from those of water ice (Miller, 1973). Particle composition enters into our calculations t h r o u g h the use of the corresponding complex indices of refraction. The complex index of refraction, no, is related to the dielectric constant, co, as no

=

e : II2 =

( e - - i ~ ) ~/2 =

n~ --

ini,

(14)

where nr and n~ are the real and imaginary parts of n~, and e and n are similarly related to co. A measure of the absorption of radiation b y a particle is provided b y ni or ~. The values of and sources for the indices used here are given in Table II. These values were derived from l a b o r a t o r y measurements for the materials of interest (as described below). The value for silicates is an average over values for acidic rocks given b y Campbell and Ulrichs (1969) for room-temperature conditions. H a d we t a k e n an average over all rock types, the value of ni would have been greater due to the contribution of the more absorbing basic rock types. We chose the lower-loss case to obtain a conservative upper limit on the reflectivity of silicate particles. Values of the refractive indices for metallic particles were obtained using the H a g e n - R u b e n s t h e o r y (van de Hulst,

Source

0. 0 0 0 0 2 0. 000006 0.029 6500 12300 1000 1000 O.62

Whalley and Labbd (1969) Whalley and Labbd (1969) Campbell and Ulrichs (1969) Theoretical (see text) Theoretical (see text) Used (see text) Used Campbell and Ulrichs (1969)

1957). We have used the relations nc = ~¢'/~ = (~ - iv) '/2

= E~-

(4.~/~)iJ-~,

2nrni

4~o/~,

(15)

which give ~lr 2 - -

=

ni 2 ~

~,

(16)

where a here is conductivity at angular frequency w. At sufficiently long wavelengths (X >> 2 ~m), E is roughly constant (see, e.g., Wickramasinghe and Guillaume, 1965) so t h a t ~ >> e and nc2 = - i ~ (the H a g e n - R u b e n s regime). Thus, 7~¢--~ (7/2) I/2 i(~/2) 1/2 (van de Hulst, p. 288). I n the microwave region, ,7 = 4rz0/~, where z0 is the dc conductivity (van de Hulst, p. 268). I n calculating no for metallic particles, we have included thermal effects (Gerritsen, 1956) t h a t increase the conductivity of pure iron by a factor of 4 relative to the value at 20°C (International Critical Tables, Vol. I, p. 104). Correcting the value at 20°C to a temperature of 100°K, we obtain a value of z0 = 4 × 107 mho m - ' . This value was checked against the value of a0 derived from the H a g e n - R u b e n s relations using a value of nc measured for iron at X = 11.4 ~m (Leksina and Penkina, 1967). Good agreement was obtained. Using the above value for ~0 and the H a g e n - R u b e n s relations for n~, we obtain the values at microwave wavelengths given in Table II. Note t h a t the theoretical values refer to pure -

-

244

CUZZI AND POLLACK

met-d. The lower values actually used were dictated b y computational limitations ( ~ = ni = 1000), but so long as n~ and n~ >> 1, their exact values are not critical for these calculations. As shown in Table II, we also considered meteoritic material of the stony-iron composition investigated b y Campbell and Ulrichs (1969), which is only 20% free iron. Meteoritic metal, b y comparison, is 9 5 - 9 7 % pure iron-nickel (Wood, 1963). The refractive indices of water ice were obtained from Whalley and Labb~ (1969). These authors have data extending longward in wavelength to about X = 600 #m, which show b o t h a significant t e m p e r a t u r e dependence and an inverse-square wavelength variation of the absorption coefficient for wavelengths greater t h a n several hundred microns. T h e y show t h a t extrapolation of their results leads to crude agreement with existing, poorly determined absorption data at microwave wavelengths. The existing factor of 2 or 3 uncertainty in these values does not affect our results for the radar case, because the values of n~ are all so small at these wavelengths. We have used their value for the absorption coefficient KA at T = 100°K, which gives

fti

=

KAX/47r = (9.9 X 10-4/X2)X/47r = 7.9 × 10-5/X (cm).

E. Optical Depths Used We assume t h a t the particles responsible for visible-wavelength behavior are the same particles responsible for microwave behavior. This assumption allows us to relate the microwave optical depth to the observed optical depth at visible wavelengths b y a scaling relationship. We obtain r0(X) = n~E(X)Z0 from the discussion of Section I I A as the normal optical depth at a given wavelength of a layer of physical thickness z0. The measured optical depth at visible wavelengths is r(v) = nSE(V)Zo. In b o t h cases, mean parameters are understood. Then the optical depth at wave-

length X is

= r(V)(dE(X)/f2E(V).

(17)

Strictly, for a particle of size much larger t h a n a wavelength, QE(X) = 2, because in addition to light being scattered and absorbed on a particle's geometric cross section, there is an additional and, by Babinet's principle, equal a m o u n t of energy diffracted and thus rigorously removed from the incident direction (see, e.g., van de Hulst, 1957; Born and Wolf, 1964). In a variety of relevant circumstances, however, this diffracted energy is concentrated within a v e r y small solid angle near the incident direction, and is indistinguishable from the incident beam. In such cases, oifly the energy geometrically intercepted by the particle is seen to be extracted from the beam, and thus the apparent extinction efficiency is unity. Indeed, most geometrical optics (a >> X) treatments of scattering in Saturn's rings use Q~.(x) = 1 (e.g., Irvine, 1966; K a w a t a and Irvine, 1974). We have considered existing observations of extinction of light at visible wavelengths b y the rings with regard to whether optical depths implied b y these observations are characterized b y QE(v) = 1 or QE(v) = 2. This is obviously an important distinction to clarify when scaling r(v) b y calculated values of QE (X) to obtain microwave optical depths. T h a t is, in using (17) we must know the proper value of QE(v) to associate with each observed value of r(v). We will show t h a t QE(v) = 1 characterizes all existing observations at visible wavelengths. In the discussion below, we defer momentarily the question of multiple (diffuse) scattering. Consider first a plane-parallel beam of light of unit flux (ergs sec -1 cm -~) of wavelength X falling on a layer of particles of radius a, where a>> X (see Fig. 7). Consider the reduction in intensity of the beam as seen by an observer at P

SATURN'S RINGS: RADAR OBSERVATIONS

245

j

i.

m

•

i

FIG. 7. Viewing geometry that illustrates situations giving rise to QE = 1 or QE = 2. Point P is the observer or detector of area d~. An equal area &r on the layer attenuates the incident wave. The energy diffracted by dz out of the beam is equal to the energy diffracted into P, d~ by the annulus A. The resolution half-angle of the detector is ~. behind the layer. T h e flux at P will be reduced b y the a m o u n t physically intereepted (Tra2) b y each particle in the line of sight to the source. E a c h particle will also diffract an equal a m o u n t of light (ira 2) which will fall primarily within a narrow region a p p r o x i m a t i n g the surface of a cone of half-angle 0D. This m a y be t h o u g h t of as the first bright ring of the diffraction p a t t e r n . First, we consider a detector with an infinitely narrow field of view. F r o m Fig. 7, it is evident t h a t if 0D is greater t h a n the angle subtended b y the detector, m o s t of the diffracted energy will not reach the detector. Thus, for each particle, 27ra2 is r e m o v e d f r o m the b e a m a n d 0~(X) = 2 (ease 1). If the diffracted energy does reach the detector (0D less t h a n the angle subtended b y the detector), it will be indistinguishable f r o m the incident b e a m and the effective value of QE(X) will be u n i t y (ease 2). These eases are discussed b y Sinclair (1947). N o w suppose t h a t the detector is sensitive to all radiation arriving within a cone of half-angle ~b; t h a t is, the observer has a resolution element of solid angle ~r~2 on the layer. We assume t h a t the layer has angular extent exceeding ¢ as seen f r o m P, t h a t is, it is seen as " e x t e n d e d . " E v e r y w h e r e on the layer, particles scatter the incident

beam, and there are particles whose "diffraction cone" passes t h r o u g h P (Fig. 7). I n fact, there is an annulus (A) of particles t h a t scatters exactly as m u c h light in the direction of P b y diffraction as particles in the line of sight from P to the source r e m o v e d b y diffraction (see Fig. 7). If ~ > 0D, this light is indistinguishable from the "incident b e a m " a n d again only 7ra2 per particle is seen to be removed, giving QE(X) = 1 (case 3). If q~ < 0D, the annulus is observed as a bright ring, and the intensity of the source is seen to be reduced b y 2 r a 2 per particle in the line of sight (case 4). A similar discussion is given b y v a n de H u l s t (1957, p. 107). A s o m e w h a t analogous situation is presented b y an extended source of light behind the layer. B y extended, we m e a n having extent 08, as seen from the layer, larger t h a n 0D. I n this ease, light reaches a scattering particle from a range of angles, and the light diffracted out of the line-of-sight direction f r o m source to observer is replaced b y light diffracted into the line-of-sight direction which would not otherwise h a v e reached the detector, giving (~E(X) = 1 for this ease as well. There are three direct q u a n t i t a t i v e observations of the optical d e p t h at some point in the rings b y extinction of transm i t t e d radiation at visible wavelengths.

246

CUZZI AND POLLACK

The most direct arc the reduction by Cook and Franklin (1958) of observations by Barnard (1890) of an eclipse of Iapetus by the C and D rings, and a polarimetric determination of the transparency of the B ring by Kemp and Murphy (1973). Each of these is characterized by QE(V)----1. The Iapetus eclipse is of the nature of case 3 above, even if we consider the Sun to be a point source, unless the particles are smaller than 10-4 m. In this case, ¢ is the angle subtended by the ring as seen from Iapetus. The polarimetric measurement involves two situations--the transmission of solar radiation (case 3, as above) and the extended source case (Saturn seen through the rings). This measurement is also characterized by QE(V) = 1, unless the particle size is less than 5 X 10-7 m. The photographic determination of the optical depths of the A and B rings (Ferrin, 1974) by their apparent transmission of light from the disk of the planet is in agreement with the above measurements. It is also of the "extended source" variety (QE(v) = 1), unless the particle size is less than 5 X 10-7 m. The other techniques that have been used to obtain ring optical depths involve theoretical calculation of their brightness (Cook et al., 1973 ; Lumme, 1970 ; H~tmeenAnttila and Pyykko, 1972), qualitative estimates using stellar extinction (Bobrov, 1970), and observed infrared brightness (Murphy, 1973). These techniques and the previously discussed observations have led to a "canonical" set of ring optical depths (Cook et al., 1973) which are discussed in more detail by Pollack (1975). We shall adopt this canonical set as our standard case, but we wish to point out that most existing values of ring optical depth are subject to systematic effects, a seeming exception being the Iapetus eclipse that refers only to the C and D rings. In all other cases of extinction measurements, neglect of diffusely transmitted (multiply scattered) light leads to an underestimate

of the true optical depth. In all of the above cases of reflected brightness calculations, isotropic scattering has been assumed. This assumption results in overestimates of the multiply scattered contribution to the reflected brightness (Pollack, 1975 ; Esposito and Lumme, 1976; Lumme and Irvine, 1976b), leading to an underestimate of the optical depths. Thus, although we adopt the canonical set of ring optical depths at visible wavelengths shown in Table I, we wish to emphasize the possibly systematically low nature of these values. For comparison with Table I, Esposito and Lumme obtain optical depths for the brightest parts of the A and B rings rA(v) = 0.4-0.6 and rB(v) 0.9 from a more realistic modeling of the rings' brightness at visible wavelengths. Their calculations did not establish an upper limit on rB(V). The values given in Table I are modified slightly from those given by Cook et al. (1973) to facilitate use of the doubling methodi A similar set was used by Briggs (1974). Because extinction at visible wavelengths, as pointed out above, refers only to physical blockage, we must apply the irregular-particle surface-area correction to these optical depths; that is, the effective efficiencies relative to spheres of equal volume are equal to SAR. Thus, ro(~) = QE i (~,) ~ ( v ) / Q ~ (v)

= [SAR. QsS(h) + Q~(X)J~(v)/SAR [-QsS(~) -[-QAs(~)/SARJr(v). (18) =

Therefore, our total optical depths are slightly lower than if we had assumed the particles to be spherical (SAR = 1). We have used the fact that QEi(V) = 1 in (18). Because the true extinction efficiency of wavelength- or larger-sized particles is --~2, microwave optical depths used in the calculations are typically about twice the values shown.

SATURN'S RINGS: RADAR OBSERVATIONS III. RADAR OBSERVATIONS R a d a r observations of the rings provide a determination of their absolute backscattering cross section, the wavelength dependence of the cross section, and the ability of the rings to depolarize an incident polarized wave. A typical radar observation consists of transmission of a totally polarized signal and reception of the return in one or both of the orthogonally polarized components. According to Goldstein et al. (1976), the sum of the radiation reflected from the rings in b o t h components of polarization implies a radar cross section a - - 1 . 3 7 ± 0.16. Here, ~ is the ratio of the observed power to the power which would be received from an isotropically scattering, perfectly reflecting sphere or spheres, at the ring's distance from Earth, subtending the same solid angle as the A and B rings. The reflected intensities in the two orthogonal polarizations are comparable to within the scatter between individual observations ( ~ 3 0 % ) (Goldstein et al., 1976). Thus, = IT (2)/IT (1) ~ 0.70, where polarization state 1 is the polarization state expected for radiation reflected from a large, smooth sphere. To within calibration error ( ~ 1 5 % ) , the rings have the same reflectivity and depolarization at b o t h ~3.5- and h12.6-cm wavelengths (Goldstein et al., 1976). For comparison of these observations with the data, we take the quadrature sum of quoted internal crror (12%) and estimated calibration or systematic error (---15%), giving a total standard error of 19%, or a reflectivity of 1.37±0.26. The relationship between the above standard definition of a and the standard diffuse reflectivity of a layer of optical depth r, which is parallel to the ring plane, is given b y = ( 1 / ~ o ) S ( r ; ~¢, 0, ~e, ~),

(19)

where tL~ = Isin Do [ and Do is the elevation angle of the E a r t h above the plane of the

247

rings (Cuzzi and van Blerkom, 1974). We m a y thus compare our model calculations for extended layers directly with the observations. We have three aspects of the observations to compare with our calculations : the absolute value of the reflectivity, the wavelength independence of the reflectivity, and the depolarization. Other standard comparisons m a y also be made. The albedo of an equally reflective L a m b e r t surface parallel to the ring plane would be S/4t~t~' = 0.78 for the Do ( ~ 2 5 °) of observation. The geometrical albedo would be S/4tt = 0.34. Note that tt = tL' for the backscattering measurements under discussion. IV. COMPARISON OF MANY-PARTICLETHICK RING MODELS WITH RADAR OBSERVATIONS In these calculations, we cover a reasonably complete parameter space of ring particle characteristics. We have made calculations for ring models composed of particles of three compositions, three degrees of irregularity, and two forms of size distribution. We first consider the total reflectivity. Figures 8 to 10 show computed cross sections ~ for model rings containing water ice particles with different size distributions and shape parameters. The size distributions represented in Figs. 8 and 9 are fairly narrow, using v = 0.3 in (9). In Fig. 8, we show t h a t relatively smooth particles (x0 = 8, curves labeled A) yield quite substantial reflectivities over limited ranges of mean radius d. These high reflectivities are due to the combination of a large backscatter gain (the glory) characteristic of spherical particles with the high efficiencies (Q~ >~ 2) characteristic of particles with x >~ 1. Larger particles (x > x0) look less spherical and have relatively lower backscatter gain, and smaller particles have low scattering efficiency. Thus, different mean radii are needed to produce the high reflectivities at the different radar wavelengths. The range of particle size within

248

CUZZI AND POLLACK WATER

2.2 2.1

l

ICE

PARIICLES

I REFLECTIVITY

- ....

g-2.0 ~-

J VS. MEAN P A R T I C L E

1.8

UPPER C U R V E S ( A ) ; -

~

qTOTAL

°"" ~L 1.4

/\A/

SIZE AT

3 . 5 0 c m WAVELENGTH 1 2 . 6 0 c m WAVELENGTH ×o=8,

FTB:2~

P L O W E RCURVES (B): x o = 3, FTB = 2

',

i

>_ 1.0 ~u .S uJ du_ .6 w

0

O. I

1.0 40.0 I00.0 MEAN RADIUS OF SIZE D I S T R I B U T I O N , c m VARIANCE OF DISTRIBUTION = 0 . 3

1000.0

FIG. 8. Reflectivity of a many-particle-thick ring layer composed of ice particles with a relatively narrow size distribution, as a function of mean particle radius (as described in text). The short dashed curve is the reflectivity at X3.5 em of a layer of particles of x0 = 3, FTB = 2, with 50% enhanced optical depth.

w h i c h t h e dual criteria of w a v e l e n g t h i n d e p e n d e n c e and net r e f l e e t i v i t y are m e t is quite narrow, perhaps 5 ~< d ~< 7 cm. For s m o o t h e r particles, s a y x0 ~> 10, t h e reflect i v i t y is higher. H o w e v e r , the range of p e r m i t t e d c~ is no larger. Thus, w e regard such a narrow size d i s t r i b u t i o n as improbable. T h e r e are m o r e likely s i t u a t i o n s t h a t WATER

2.2.

]

t~" 2.0 c~ u_ 1 . 8 -

,.,,)

1.6-

PARTICLES

I

REFLECTIVITY

2.1-

e

ICE

exhibit w a v e l e n g t h independence. For particles w i t h x > x0, the o n l y c h a n g e s in scattering b e h a v i o r w i t h w a v e l e n g t h are a c o n t i n u o u s narrowing of the diffraction p e a k and a gradual decrease in albedo for dielectric particles. For the narrow size distributions in Fig. 8, the range d > 50 e m for the s m o o t h particles (curves A) exhibits wavelength-independent reflee-

....

SIZE AT

3 . 5 0 c r n WAVELENGTH 11~.60 c m WAVELENGTH

UPPER CURVES (A): x o = 3, F T B = I LOWER CURVES (B): x o = 3, F T B = 2

T

1.4'

i

v s . MEAN P A R T I C L E

TOTAL

1.21.0-

.8 L

.6L

-

E F r E C T O r SO°/.,,' :'" . . . . . . . . ~--._. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INCREASE OF ," ,/_ ~ ~ . A ........... OPT IC AL ,/ ~ . ~ DEPTH 0 N ~

.I

.4 2 .2o 0.1

,

I.O IO.O IOO.O MEAN RADIUS OF S I Z E D I S T R I B U T I O N , c m VARIANCE OF D I S T R I B U T I O N = O . 3 0

IOO0.0

FIG. 9. Reflectivity of a many-particle-thick ring layer composed of less forward scattering ice particles with a relatively narrow size distribution (as described in text). The short dashed curve is the reflcctivity at k3.5 cm of a layer of particles of x0 = 3, FTB = 1, with 50% enhanced optical depth.

249

SATURN'S RINGS: RADAR OBSERVATIONS WATER ICE PARTICLES 24 2.2 ~2.0 ~_ ] J8 ~-I-~/

_~ k2 ~•

~ .4 .2 0 0.1

. . . . . . . l l REFLECTIVITY VS. MINIMUM PARTICLE RADIUS AT 3.50 cm WAVELENGTH . . . . . 12,60 cm WAVELENGTH UPPER CURVES (A): Xo= S, FTB = 2 T LOWER CURVE (S): x o = 5, FTB = 2 I TOTAL ...... -• ..~-""

.................... "~.~_ ~ / ~ - ~ ~"-.

~ POWERLAW SIZE

~--~.~ x

_J I -~ 4 i

ENHANCED OPTICAL

J

......................::.L-.-~.... I DISTRIBUTION, n(a)= % 0 3 a>amin :OELSE !

l

1

1.0

I0.0

I000

Qmin (ore)

FIG. 10. Reflectivity of a many-particle-thick ring layer composed of ice particles with a powerlaw size distribution, as a function of m i n i m u m particle size (as described in text). The short dashed curve is the reflectivity at X3.5 cm of a layer of particles with x0 = 8, F T B = 2, with 50% enhanced optical depth.

tivity, but in this case the reflectivities are much lower t h a n observed. The rough particle cases shown in Fig. 8 (x0 = 3, curve B) exhibit wavelength independence over a larger range of x (mean radii ~> 4 cm) because of the lack of the strongly-wavelength-dependent " g l o r y " for such particles. However, rough (x0 = 3) particles do not produce the observed reflectivity either with the nominal or the 50%-enhanced ring model optical depths. For x>>x0, the two cases (x0 = 3 and x0 = 8) have identical behaviors, as all particles in the size distribution have X>Xo.

In Fig. 9, we show for comparison with the above rough particle case the less forward scattering, less probable case x0 = 3, F T B = 1. With a 50% enhancement in optical depth, such a model has a reflectivity t h a t approaches the observed reflectivity for mean particle radii in the range 10 ~< d < 100 cm. Again, we feel such a combination of favorable enhancements (in F T B and r) m a y be improbable. The results for particles characterized b y a power-law distribution of radii n(a) = noa -3 for amin ~< a < 100 cm are auto-

matically independent of wavelength to the extent discussed in Section I I FEq. (12)]. These results (Fig. 10) are most sensitive to the value of the lower b o u n d a r y of the x-integration, where particle efficiencies are changing most rapidly. F r o m Fig. 10, we infer t h a t such a size distribution of relatively smooth (x0 = 8) ice particles gives good agreement with observed reflectivities over a range 1 ~ amln ~ 3 cm. Again, however, it does seem t h a t optical depths about 50% higher t h a n our nominal values are required. Very rough (x0 = 3) particles with a power-law size distribution do not give sufficiently high reflectivities in models of reasonable ring optical depth to m a t c h the observed values. We have investigated power-law size distributions b o t h with n(a) = const for a < amin, and with n(a) = 0 for a < amin, and have found the variation between these cases to be less t h a n t h a t due to reasonable variation of amin. For the power-law distribution, the range of permitted amin is significantly larger for larger b u t still reasonable values of x0. For x0 = 10, the range is about a decade, or 0.4 ~ amin ~< 5 era. This is not the case for the narrow size distribution in

250

CUZZI AND POLLACK

Fig. 8. For this reason, we feel t h a t the broad power-law size distribution is somewhat more probable. For the silicate particles shown in Fig. 11, we see t h a t no combination of reasonable parameters even begins to approach the dual criteria of net reflectivity and wavelength independence. Power-law distributions of such particles have lower reflectivities t h a n curve B in Fig. 10. The albedos of silicate particles are typically <~0.8 and thus multiple scattering does not increase the net return significantly for a 50% enhancement in optical depth. This difference in the radar reflectivity characteristics of silicate and water ice particles is due to the silicates having m u c h larger values of n~ t h a n the ices at radar wavelengths. Assuming t h a t the average refractive index of a mixture of components is given b y the average of the refractive indices of the individual components as weighted b y their mass fractions, the observed reflectivities would not be obtained even with a 50% optical d e p t h enhancement if the ring particles contained more t h a n 10% b y mass of such silicates mixed in with pure water ice.

Figure 12 shows reflectivities for metallic particles of different mean radii with a size distribution given by (9) for v = 0.3. As such particles are high-albedo, high-efficiency scattcrcrs, it is not surprising t h a t they produce high reflectivities. Because metallic particles scatter only b y diffraction and external reflection, the properties of spheres and randomly oriented irregular metallic particles are likely to be equivalent (van de Hulst, 1957; Hodkinson and Greenleaves, 1963). Therefore, the reflectivities shown are calculated directly for spheres. A power-law distribution of such particles in a layer of 50% enhanced optical depth would produce a cross section (not shown) of a -- 0.99. Thus, for • >~ 1 cm, 50% enhanced optical depths, and either size distribution, a layer of metallic particles approaches the observed reflectivity and wavelength independence. Interestingly, because metallic particles never exhibit scattering efficiencies much in excess of the geometrical optics value of 2 (Kerker, 1969; Hansen and Travis, 1974), the net reflectivity of a given distribution of metallic particles is slightly lower t h a n t h a t of an equivalent distribution of water ice particles. This occurs despite the higher

SILICATE P A R T I C L E S

2.2

•

T

REFLECTIVITY VS. MEAN PARTICLE SIZE AT - 5.50 cm WAVELENGTH 12.60 crn WAVELENGTH

2.1 #'" 2.0

.

~

1.8 ~

L6

~ m

1.4

"

1.2

~_>

1.0

o .8 ~ dW .6

.

.

.

BOTH CURVES FOR xo = 8, FTB = 2

T

(VARIATION OF x o AND FTB HAS NEGLIGIBLE EFFECT)

; TOTAL

_

^ ; 2.52 +0.029 i) (m

~

,,/~,, ~ ,

/

1,'-~ ~

/EFFECT OF 50% INCREASE IN OPTICAL DEPTH ON

.2

J 1

....................

0.I

1.0 I0.0 I00.0 MEAN RADIUS OF SlZE DISTRIBUTION, crn

I000.0

VARIANCE OF DISTRIBUTION : 0.50

FIG. 11. Reflectivity of a many-particle-thick ring layer composed of silicate particles with a relatively narrow size distribution, as a function of mean particle radius. The short dashed curve is the reflectivity at X3.5 cm of a layer of 50% enhanced optical depth.

251

SATURN’S RINGS : RADAR OBSERVATIONS REFLECTIVITY 2.27

I

VS. MEAN SIZE METALLIC nr = ni = 1000 ~ ----12.60

2.1

3.50

PARTICLES I cm cm

WAVELENGTH WAVELENGTH

c- 2.0 t

$-

1.4

“OOI

i TOTAL

01 1.0 MEAN RADIUS OF SIZE DlSTRlBUTlON. VARIANCE OF DISTRIBUTION = 0.30

100 cm

100.0

FIG. 12. Reflectivity of a many-particle-thick ring layer composedof pure metal particles with a relatively narrow size distribution, as a function of mean particle radius. ratio of backward-to-forward scattering characteristic of metallic particles. The difference between pure metal and ice particle behavior is not significant at the present level of uncertainty in optical depths and observed radar reflectivities. However, material of stony-iron composition (72, = 7.8 + 0.62i) has a radar cross section u = 0.42 for the above power-law size distribution and 50% enhanced optical depths. Particles composed of such material would not satisfy the radar observations. Very small metal particles (X 5 0.1 in Fig. 12) are inefficient scatterers and, for very small values of Z, become absorbing as well. However, for values of refractive index In., 1 - 103-104, such particles maintain high albedos until 5 < In, 1--2/3(see, e.g., van de Hulst, Chaps. 6, 10). The efficiency of a small, highly irregular particle of high refractive index may be significantly enhanced (by a factor - In, 1”) over the low efficiencies typical of spheres of the same volume (van de Hulst, 1957; Greenberg, 1972; Atlas et al., 1953). Dielectric materials (ice, rock) do not permit such large enhancements because of their lower value of n,. This combination of effects would seem to allow metallic needles of the proper size (-1 mm in

length) to attain sufficiently high albedos and efficiencies to satisfy the observations. However, for such small particles, the scat’tering efficiency has the form characteristic of Rayleigh scattering (Qs ccx4), and a strong wavelength variation of radar reflectivity would ensue in contradiction to the observat’ions. In addition, the random orientation of the particles required by the depolarization would reduce both their average albedo and efficiency below the potentially high values that characterize only orientations parallel to the electric vector of t’he incident wave. Thus, a preponderance of small (a < 1 cm) particles of any composition or configuration is apparently ruled out by the radar observations. The general form of the scattering behavior of the above ring models is of some interest. Figure 13 shows the angular variation of the diffusely scattered and transmitted intensity in the plane of incidence for two cases. We compare a semi-infinite layer of conservative (ij, = 1) isotropic scatterers (curve a) with a ring model having optical depths 50% larger than the nominal values in Table I (curve b). The scatters in the ring model are pure ice particles with 20 = 8 and FTB = 2 and

252

CUZZI AND POLLACK SCATTERED INTENSITIES( S ergsec°lcm-2str -I) 4~ FOR UNIT INCIDENT FLUX (~.~'=[erg sec-lcm -2) V A R I A T I O N WITH ZENITH ANGLE

e = Cos i#

IN THE PLANE @ = 0

INCIDENT DIRECTION ~o : 0,4 (8 ~ 66.5 °)

RING P~ANE 47r

o'

b

4~r

FIG. 13. The ~angular distribution of diffusely scuttered and transmitted radiation in the plane of incidence. The incident plane wave has unit flux normal to itself. Curves a and a': total and singly scattered intensities reflected by a semi-infinite layer of isotropic scatterers with &0 = 1. Curves b and b': total and singly scattered intensities diffusely reflected and transmitted by a typical ring model composed of particles of centimeter to meter size in a many-particle-thick ring layer. h a v e size distribution n ( a ) = noa -3 for 1 < a < 100 cm. The illumination in b o t h cases is a plane w a v e incident f r o m the direction (--#e, 0) where ~e = 0.4. The scattered radiation fields are quite different. T h e p r e d o m i n a n t forward p e a k in the phase function of the wavelength-sized ice particles (e.g., Figs. 4 and 5) manifests itself b o t h in the diffusely t r a n s m i t t e d and in the diffusely scattered radiation. T h e t o t a l scattered flux contained within the large forward peaks in the scattered and t r a n s m i t t e d components is not as large as it m i g h t a p p e a r because of the small solid angles subtended b y these lobes. We h a v e verified numerically t h a t the t o t a l flux diffusely scattered and diffusely t r a n s m i t t e d b y the model into all angles is less, as it m u s t be, t h a n the total flux scattered b y the semi-infinite layer. Figure 13 also shows the c o m p o n e n t of the t o t a l scattered intensity which has undergone only one scattering (curves a' or b'), as calculated using (5). I t is clear t h a t multiple scattering increases the net

return in m o s t directions b y several times relative to the singly scattered component. We t u r n next to the third aspect of the observations--depolarization. T h e large multiply scattered c o m p o n e n t characteristic of m o s t m a n y - p a r t i c l e - t h i c k ring models t h a t exhibit the observed reflectivity provides a n a t u r a l explanation for the observed depolarization of the reflected intensity. The dependence of the depolarization of the net reflected intensity (5) on the depolarization of the singly scattered intensity (DR), as discussed in Section I I A , is shown ill Fig. 14. The observed depolarization is produced in essentially all cases b y m a n y - p a r t i c l e - t h i c k rings composed of particles with D R ~ 0.5, a typical value for irregular particles with x --~ 10. Results for metallic particles (not shown in Fig. 14) lie in the same range as curves A to D. T h e results shown in Fig. 14 refer to the normal values of optical depth. A 5 0 % increase in optical d e p t h would increase the multiply scattered component, t h e r e b y increasing b y a b o u t 5 % the values of

SATURN'S RINGS: RADAI~ OBSERVATIONS shown. A slight variation is seen among different extended ring models--those that have the largest singly scattered component (curves A and B) show the lowest depolarization. At present levels of uncertainty, it seems fair to say that the many-particlethick ring models that satisfy observed reflectivity criteria also satisfy the observed depolarization criterion without additional assumptions. We now summarize the results for the many-particle-thick case. Pure ice particles of radius > 1 cm which are fairly, but not extremely, irregular exhibit the observed radar reflectivity if the true optical depths are ~ 5 0 % greater than the nominal, currently accepted values given in Table I. Such particles exhibit the observed wavelength independence of reflectivity, either if the distribution of their radii follows a power-law distribution n(a) = noa -3 with a,,~i. "~ 1 cm or if their radii are predominantly very near 6 cm. If the ring particles were to contain more than 10% silicates by mass, their albedos would be too low to satisfy the radar observations unless the ring optical depths are more than 50% greater than currently accepted values. Pure metallic particles of mean radius ~ 1 cm are marginally capable of satisfying

DEPOLARIZATION OF TOTAL REFLECTED RADIATION

IORT~L

the observed reflectivity and wavelength independence either in a power-law or relatively narrow size distribution if the true optical depths are ~ 5 0 % greater than currently accepted values. Tiny ( ~ 1 mm) metallic needles do not exhibit the observed wavelength independence of reflectivity. All of the many-particle-thick models that exhibit the observed reflectivity and wavelength independence also produce the observed depolarization. Only a moderate amount of depolarization by single scattering is required since multiple scattering makes a significant contribution to the net observed depolarization. IV. COMPARISON OF MONOLAYER MODELS WITH OBSERVATIONS A . General Properties

We have considered a variety of particle sizes, shapes, and compositions that might satisfy the observational constraints within the context of a monolayer. Each is treated somewhat differently. The expression for the singly scattered intensity from a many-particle-thick layer [-Eq. (5)] is not generally valid for a monolayer. In fact, the singly scattered

VALUES

.8

p

C .6 /

.4 .2

/

/ 1

0

253

: SIO 2 PARTICLES, 0 = 1.0, X o = 3, F T B = I, ),3.5 Cm B: ICE PARTICLES, g = 6.O,X o = 8, FTB = 2,). 12.6 c m C: ICE PARTICLES, G = 3 . 0 , X o = 3 , F T B = I , k S . 5 C m D: ICE PARTICLES, POWERLAW, rmin =LO, X o = 8 , F T B = 2 E: MONOLAYER OF PARTICLES SUCH AS ABOVE ] I I I

.2 .4 .6 .8 1.0 DEPOLARIZATION OF SINGLY SCATTERED RADIATION OORTROGONAL / 1EXPECTED) DR

FIG. 14. The dependence of the depolarization of the net reflected intensity (~) on the dep0larization of the singly scattered radiation (DR) for various ring models. Shaded area is the range of permitted values as constrained b y the observations. Case E shows the behavior of any of the particle types shown (A to D) if the particles are restricted to a monolayer.

254

CUZZI AND POLLACK

intensity from a monolayer is greater t h a n t h a t from an extended layer of the same optical depth unless r/t~ <~ 1. This difference results from the lack of blockage of the incident or scattered radiation b y particles in front of a given particle in a monolayer (see, e.g., Irvine, 1966). Our approach to obtaining the reflect i v i t y of such a monolayer follows the definition of o- = F g ~ , (20) where F = Fx(#0) is the fraction of the area of the rings seen to be filled b y scatterers at wavelength X and at ring-tilt angle sin -1 (~0), g is the "gain" for scattering in the backscattering direction over t h a t for an isotropic scatterer; and v is particle reflectivity or, equivalently, singlescattering albedo. We obtain the fraction filled at visible wavelengths F~(#0) from our nominal values of r(v), which are inferred from extinction observations discussed in Section II. Thus, F ~ ( # o ) = 1 -- e -'~(~)/~o,

simulates the truncation of the diffraction peak for x>> 1 (Hansen, 1969b) and truncates less, or not at all, for x ~ 1 where the scattering is less sharply concent r a t e d in the forward direction. Let f = Q D / Q s be the fraction of scattered (including diffracted) energy t h a t will be regarded as unscattered, where f o r Q E < 1,

QD = Q E - - 1

for 1 _~ QE < 2,

QD = 1

f o r 2 _ ~ QE,

(22)

and QD is a "diffraction efficiency," which attains the proper values in limiting cases and varies smoothly in between. Then, in standard fashion (Hansen and Pollack, 1969; Hansen, private communication, 1975), we obtain various quantities with the "diffracted" component removed, denoted b y primes, from their values with diffraction included : Q'E ---- (1 -- fS~0)QE, [

(21)

where subscript i indicates values for each ringlet and t~0 = 0.44. As shown below, (21) does not give the correct dependence of extinction on #0 for a monolayer. However, most quoted optical depths have been obtained from extinction measurements at values of t~0 near the above value (0.44). Therefore, negligible error is introduced b y use of (21) as described. As discussed earlier, the scattering and extinction efficiencies and thus cross sections of particles t h a t constitute the monolayer m a y be quite different at microwave wavelengths from their (geometric optics) values at visible wavelengths. However, we do not wish to include in Fx(tL0) the component of extinction t h a t results from diffraction b y particles with x >> 1, which surely contributes very little to the reflectivity of a monolayer. Therefore, we adopt the following simple algorithm for obtaining Fx(#o), which

QD = 0

~'0 =

(1 -- &°) 1-1 1 ~- ~0(1----f-)J ~ •'

(23)

p'(Tr) = p ( r ) / ( 1 -- f) ---- g, and F~x(,o) = Q'EF,~(~o) _< 1. Following previous notation (20), (23), and (6)1,

[-i.e., Eqs.

ax(p0) = p'(Tr)~'0 ~ F~x(#0)w~. (24) i

Values of ax(t~0) calculated in the above way are actually somewhat larger t h a n t h e y would be if we had not excluded the diffracted energy from the extinction efficiency, because we require t h a t F~x (~0) _~ 1. This m a y be seen by substituting (23) into (24). In normal use, truncation does not alter backscattered intensities. This treatment is less t h a n rigorous, and there are other difficulties such as the fact t h a t the particles in ringlets where Fix(#0) ~ 1 are not widely separated from each other, and

SATURN'S RINGS: RADAR OBSERVATIONS near-field and coherence effects would be significant. These calculations are useful, however, for a rough comparison. The variation of ~(~) with ~ for monolayer models is qualitatively different from that shown by many-particle-thick models. For a monolayer of finite "optical depth," not all of the layer area is filled with particles [-F(~0) _< 17. As the layer is viewed at angles increasingly farther from the layer normal, the fraction seen to be "filled" increases. This is due to the constant total projected area of particles (assuining them to be equidimensionM) and the smaller total projected area of empty regions. As the angle of view from the ring normal increases, the total projected ring area decreases proportional to #, but the projected area of particles remains constant until the rings appear filled. Thus, a(u) = F(t~)gn = F(go)(tto/tt)g~l

(25)

and F(u)~< 1. The importance of this variation is discussed later. B. Monolayers Composed of Centimeter- to Meter-Sized Particles of Ice, Silicate, or Metal

We first consider particles for which the physical properties are known, that is, centimeter- to meter-sized ice, silicate, and metal particles. Reflectivitics for monolayers composed of such particles ~re calculated using these properties and nominal values of ring optical depth. We then describe the probable behavior of systems composed of two hypothetical "particle" types: large, jagged, metallic chunks, and low-loss particles that exhibit multiple internal scattering. For these cases, the scattering characteristics of individual particles are not well established. We have calculated ax(#0) in the above way for monolayer ring systems identical in optical depth and constituent particle properties to the extended layer models of Section IIIB. These values of ~x are

255

TABLE III REFLECTIWTY(~ = Fgn) OFA MONOLAYERCOMPOSEDOF P A R T I C L E S OF S I Z E C O M P A R A B L E TO A W A V E L E N G T H : F O R D I F F E R E N T CONSTITUENT PARTICLES Model

d

x0

FTB ~(3.5cm)

a(12.6cm)

0.5 2.0 4.0 8.0 20.0 P o w e r la w 0.5 4.0 6.0 8.0 50.0 P o w e r la w

8 8 8 8 8 8 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2

0.55 -2.54 1.23 0.72 1.43 0.54 0.46 0.45 0.45 -0.43

-0,14 0,75 2,47 1,69 1,43 -0,39 0,50 0.50 0.50 0,43

8 8 8 8 8 8

2 2 2 2 2 2

0,003 0.35 1.08 0.93 -0.27

-0,02 -0,91 1,34 0.98

0.002 0.15 0.59 1.12 0.92 0.71 0.67 0.71

0.000012 0. 0009 0. 002 0.15 1.04 0.93 0.73 0.71

Ice particles 1 2 3 4 5 6 7 8 9 10 11 12

Silicate p a r tic le s 13 14 15 16 17 18

0.1 0.5 1.0 3.0 5.0 10.0

M e t a l p a r tic le s 19 20 21 22 23 24 25 26

0.056 0.17 0.28 0.56 1.67 5.57 16.70 P o w e r la w

given in Table III. Generally, the lack of a reflectivity component due to multiple scattering is offset by larger singly scattered intensities. Increases similar to those seen in extended-layer models result from 50% enhancements in optical depth. Although model 6 in Table III satisfies the reflectivity- and wavelength-independence criteria, these monolayer models would require perhaps unrealistically large single-scattering depolarization values to match the observed depolarization (see Fig. 14). Measured values of singlescattering depolarization for particles of these sizes range from 0.3 to 0.5 (as discussed earlier). More irregularly shaped particles, which might be expected to depolarize more effectively (x0 = 3, FTB = 2), do not exhibit a sufficiently

256

CUZZI AND POLLACK

high net refleetivity. This objection applies even more strongly to spherical particles, which may attain extremely large values of p(~r) = g (Pettengill and Hagfors, 1974), but which do not depolarize at all. On this basis, perfectly spherical particles may be ruled out as constituents of either a monolayer or an extended layer. Since present knowledge of the depolarization properties of irregular particles may be incomplete, it may be premature to rule out model 6 in Table I I I only on the basis of the observed depolarization. The variation of reflectivity with ring tilt angle is another constraint that is discussed later for this model and other possible models. Available data imply that particles must be large (x > 100) to depolarize the directly backscattered radiation effectively upon single scattering. Observations of backscattering by such very large particles have shown depolarizations greater than 80~0 (Schotland et al., 1971). For such particles to satisfy the microwave brightness observations discussed in Section I, the intrinsic emissivity of the surface material itself must be quite small (<0.1). Large jagged metallic particles (Goldstein and Morris, 1973) are possible candidates, but solid water ice or silicate particles of such size are not (Pollack, 1975). Another possibility is that the large "particles" are composed of internal scattering elements of high albedo embedded within an essentially transparent matrix, providing high reflectivity and depolarization by multiple internal scattering (Irvine, 1973; Cook and Franklin, 1976). We now deal with these two possibilities. C. Large Metallic Particles

Little is known about the backscatter gain g of large metallic particles. For the nominal optical depths we have used at visible wavelengths, ~ i Fix(~o)wi ~ 0.7, so g could be as small as 2 and the observed net reflectivity (o = 1.37) would result.

We have adopted the Schonberg sphere (van de Hulst, 1957; Hapke, 1966) as a typical large, rough, opaque particle. This is a multifaceted irregular particle with each facet randomly oriented and reflecting by the Lambert law. For such a particle, g = 8/3. We assume that there would be no wavelength dependence of g, because each facet is much larger than either radar wavelength. For perfectly reflecting (7 = 1) Schonberg spheres, the observed reflectivity will result if the rings are only 52% filled EFx(#0) = 0.52]. On the other hand, one may invoke laboratory studies of bright, opaque surfaces that show opposition brightening (Oetking, 1966). Thus, the Lambert law may give only a lower limit to g [-upper limit to Fx (~0)]. The major difficulty with such a model is in explaining the depolarization using external reflection only. The large particles for which Sehotland et al. (1971) observed 80~0 depolarizations were probably ice (transparent), for which ease scattering from subsurface irregularities dominates the depolarization (Wilhelmi et al., 1975). The least ad hoe mechanism for producing depolarization from large, jagged metallic particles is to model the surface as a collection of randomly oriented dipoles (Hagfors, 1967); in which ease only 33% depolarization is expected. Thus, it would seem that such constituent particles would not satisfy the observed depolarization.

D. Particles with Multiple Internal Reflections or Scatterings

Recent observations of the Galilean satellites (Campbell et al., 1976; Campbell, private communication, 1976) show them to be very bright objects and highly depolarized. In particular, Europa has a cross section a = 1.7 4- 0.4 (est.) and the return is 4 2 ~ depol.'~rized; Ganymede has a cross section ~ = 0.8 4-0.2 (est.) and

257

SATURN'S RINGS: RADAR OBSERVATIONS

the return is 62% depolarized. In addition, these objects appear almost uniformly bright, with v~lues of g ~--4 (Campbell, private communication, 1976). These results may be understood in the light of multiple scattering by subsurface inhomogeneities (Wilhelmi et al.) 1975 ; Pollack and Whitehill, 1972) within a low-loss "regolith" of ice or frost. For the rings, observed microwave brightness temperatures will place severe limits on the matrix holding any such "particles" together (to be discussed in a future paper). On the basis of radar data alone, our nominal values of Fix(#o)~ 0.7 require such individual "particles" to have a cross section ~ ~ 2.0, which is still somewhat higher than that of Europa, the brightest of the Galilean satellites. However, with g = 4 for Europa, one obtains an albedo of only 42% (Campbell, private communication, 1976), which suggests that higher reflectivities could conceivably be achieved by "particles" of this sort. Such particles would not need to be the same size as the Galilean satellites, but merely to have similar surface properties.

Very large, low-loss "particles" with multiple internal scattering may provide the requisite reflectivity, wavelength dependence, and depolarization, but they would need to be of material of significantly lower loss, due either to composition or mass density, than the outer Galilean satellites. Future observations will soon be capable of distinguishing between the two classes of models, using the variation of ~ (it) with # discussed in Section IVA. This variation has a qualitatively different form for monolayer models than for models that are many particles thick (Fig. 15). A monolayer composed of low-loss particles with multiple internal scattering would behave in a way similar to that shown for the large metallic particles. Figure 15 shows the variation in the reflectivity of monolayer models as the Earth approaches the plane of the rings. The error bar shows the relative error in 2.2

q

--

:"oF .....

~

~[ . . . .

I

F

.

1.9 t1.8 1.7

E. Discussion of Monolayer Models and Future Discrimination to Be Made Three distinct possibilities have been seen to provide explanations for different aspects of the radar observations in the context of a monolayer, but, on the basis of current knowledge, none provides a completely satisfactory explanation for all radar observations. Small (centimetersized) ice particles with a power-law size distribution provide the net reflectivity and wavelength independence required, but might not provide the depolarization required. A similar situation holds for very large (many-meter) metal particles. Our current understanding of single-scattering depolarization may not yet be sufficiently complete for us to rule out these hypotheses, however.

/ 1

\\ \ LARGE M E T A L L I C

~RTICLES(R>>lm)

1°6 1.5 I.O R (present)

ICE PARTICLE MONO- \ LAYER (MODEL G, \

1.3

LO .9 .8

A: ICE PARTICLES, MEAN SIZE ~ I0 c m , x o = 5, F T B = I B: ICE PARTICLES, POWERLAW, MINIMUM SIZE = 2.5 crn, Xo= 8,

.7 "c°

4 /

FTB = 2

0

.I

I

i

.2 .5 .4 ILLe = sin (D e) I

.5 __

i

0 6 112 117 24 50 D e = E A R T H INCLINATION ABOVE RING P L A N E , deg

FIG. 15. Variation of ring reflectivit,y with ring opening angle fox' different ring models. Edge-on aspect eorresp(mds to De = 0. The error bar shows the relative uncertainty in the existing observation (about 10%).

258

CUZZI AND POLLACK

the existing measurement (~10%). The increase in the reflectivity of monolayer models results from the increase discussed previously in the fraction of projected ring area filled with particles. Such an increase in reflectivity would be detectable with existing sensitivity in the near future. By comparison, the reflectivity of manyparticle-thick models decreases slightly as the rings close up because of the increasing importance near grazing incidence of single scattering, which is strongly forwarddirected. Also, as may be seen from Fig. 13, bistatic radar observations using spacecraft of the angular variation of the diffusely scattered intensity will be extremely important. v. SUMMARY AND CONCLUSIONS We have investigated in detail the radar refleetivity and depolarization of plausible ring models, considering particles composed of ice, silicate, or metal both in manyparticle-thick and single-particle-thick layers. We have presented a model for the refleetivity of a many-particle-thick ring which includes the effects of particle irregularity, realistic particle refractive indices, and finite ring optical depth. The optical depths required of models of this type are about 50% higher than those obtained by directly scaling currently accepted values at visible wavelengths. This difference is comparable to the uncertainty in existing values (of. Briggs, 1974). Systematic effects in the observations and analyses leading to current values of ring optical depths allow them to be underestimated, so the 50(~0 larger values are not unreasonable. Several such extended ring models provide a straightforward explanation for the mean value, wavelength independence, and depolarization of the radar reflectivity. The ring particles must typically be larger than a centimeter in mean radius to exhibit tile

observed wavelength independence. We find that they must be of very high albedo : (1 ~0) < 10-2. Spherical particles may be ruled out on the basis of the observed depolarization. Particles composed either of water ice (conceivably mixed with other ices) or pure metal could satisfy these constraints. Similar conclusions have been reached by Goldstein and Morris (1973), Pollack et al. (1973), Briggs (1974), Pollack (1975), Cuzzi and Dent (1975), and Goldstein et al. (1976). If the particles are pure metal, neither the mean size nor the form of the size distribution is well determined. Metal particles of radius ~>1 cm and a wide variety of size distributions approach the observed behavior, but it does seem that larger optical depths are required for metal particles than for ice particles. Tiny metallic needles do not satisfy the observed wavelength independence of radar refleetivity. Material of stony-iron composition (20% free iron) does not give a sufficiently high refleetivity. Water ice is, however, a more likely candidate material than metal on eosmochemical grounds (Lewis, 1973). For water ice particles that are not too irregular, the absolute value and wavelength independence of the observed refleetivity is most naturally satisfied if the radii of the particles follow a power-law size distribution of the form n ( a ) ~ noa -~. Such a model lends itself to interpretation as a manifestation of a fragmentation process (Hartmann, 1969), operative at least over the size range 1 e m _ < a _ < 100 em. The smallest particle in such a distribution must be no larger than a few centimeters and no smaller than a few millimeters in radius. This range agrees well with expected values of particle size stable to PoyntingRobertson drag over the lifetime of the solar system (Pollack, 1975). The concept that the ring particles follow •~ power-law size distribution of the form n ( a ) = noa -~ is consistent with the exis-

259

S A T U R N ' S R I N G S : RADAR OBSERVATIONS T A B L E IV A('CEPTABIIATY OF RING MODELS OF ])IFFERENT STRUCTURE~ COMPOSITION, AND PARTICLE SIZE

Particle composition

Model structure

(Metal)

(Ice)

(Rock)

A : a < < X (a < l c m )

No, due to low net reflectivity and strong h-dependence

No, due to low net reflectivity and strong h-dependence

No, due to strong },-dependence

B:a~X (narrow size dist.)

Possible ((i = 6 :t: 1 cm only)

No, due to low net reflectivity

Possible

C: a>>X (a > l m ) (narrow size distribution)

No, due to low net reflectivity

No, due to low net reflectivity

Possible

D : Power law (a > 1 cm)

Possible

No, due to low net reflectivity

Possible

A : a < < X (a < l c m )

No, due to low net reflectivity and strong h-dependence

No, due to low net reflectivity and strong X-dependence

No, due to strong },-dependence

B : a ~ X (a > 1 cm)

Unlikely, due to low depolarization

Unlikely, due to low depolarization

Unlikely, due to low depolarization

C: a>>X (a > 1 m) narrow size distribution)

Possible (multiple internal scattering)

No, due to low net reflectivity

Unlikely, due to low expected depolarization

D Power law (a > 1 cm)

Unlikely, due to low depolarization

No, due to low net reflectivity

No, due to low net reflectivity

Extended layer

n(a)

=

noa

3

Monolayer

n(a)

= noa -a

tence of large (many-meter) particles in the ring system. Such large particles would be fewer in number and would provide less total area than the small particles. Thus, they would contribute little to the overall radar, radio, infrared, or visible properties of the rings. However, they could contain most of the mass of the ring system and could provide interesting dynamical effects. For instance, the observed brightness asymmetry of the A ring (Lumme and Irvine, 1976a; Reitsema et al., 1976) has been used to infer the existence of large particles in the ring system. Greenberg et al. (1977) infer a size distribution n ( a ) noa -33 if the brightness asymmetry is due to synchronously rotating particles larger than 50 m. Colombo et al. (1976) suggest that a few large particles could =

create density disturbances which would produce the observed brightness asymmetry. Another possibility is a narrower size distribution of ice particles with mean radius d ~-- 6 cm. The range of permitted values of ~ depends on the width of the distribution used. For our choice (variance = 0.3), d ~ 6 4- 1 cm. Such a size distribution might be expected to be typical of particles that condensed from a protoplanetary nebula, and were prevented from accreting further by tidal influences within Saturn's Roche limit. Both of the above forms of size distribution of ice particles adequately explain the observed radar depolarization on the basis of multiple scattering between nonspherical particles. To distinguish further

260

CUZZI AND POLLACK

a m o n g t h e d i f f e r e n t f o r m s of size d i s t r i b u t i o n , e i t h e r o b s e r v a t i o n s ( r a d a r or radio) a t o~her w a v e l e n g t h s or b i s t a t i c obserw~t i o n s ( u s i n g s p a c e c r a f t ) of t h e a n g u l a r d i s t r i b u t i o n of t h e s c a t t e r e d i n t e n s i t y will be n e c e s s a r y . I n t h e c o n t e x t of a m o n o l a y e r , v e r y l a r g e (x > 100), j a g g e d m e t a l l i c p a r t i c l e s m a y produce a high and wavelength-independent r e f l e e t i v i t y , as d o e e n t i m e r - t o m e t e r - s i z e d ice p a r t i c l e s w i t h a p o w e r - l a w size d i s t r i b u t i o n . O n t h e b a s i s of c u r r e n t k n o w l e d g e , i t seems t h a t n e i t h e r of t h e s e t w o possibilities would produce the observed dep o l a r i z a t i o n . H o w e v e r , t h e y m a y n o t as y e t b e r i g o r o u s l y r u l e d out. P a r t i c l e s w i t h m u l t i p l e i n t e r n a l s c a t t e r i n g m a y be a b l e t o s a t i s f y all o b s e r v a t i o n s , b u t " p a r t i c l e " albedos ~50% higher than the brightest of t h e G a l i l e a n s a t e l l i t e s w o u l d be n e c e s s a r y . S u c h a m o d e l is t h e m o s t a t t r a c t i v e of t h e monolayer variety. In the near future, o b s e r v a t i o n s of t h e v a r i a t i o n of z ( v ) w i t h will d i s c r i m i n a t e f u r t h e r b e t w e e n m o n o layer models and models that are many particles thick. Our conclusions are briefly s u m m a r i z e d in T a b l e IV. ACKNOWLEDGMENTS We thank Drs. R. Goldstein and G. Pettengill for several discussions of the radar observations in advance of publication. We thank O. B. Toon, D. Campbell, F. Drake, and W. Irvine for helpful conversations and revelations, and L. Esposito, R. Greenberg, W. Irvine, and K. Lumme for permission to quote as yet unpublished work. We thank A. Summers, M. Covert, and B. Baldwin for computing assistance. REFERENCES ATLAS, D., KERKER, M., AND HITSCHFELD, W.

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