Smithsonian

Contributions to Astrophysics

VOLUME 2, NUMBER 13

OPTICAL PROPERTIES OF SATURN'S RINGS:

I. TRANSMISSION

by ALLAN F. COOK, II, and FRED A. FRANKLIN

mtt

SMITHSONIAN INSTITUTION

Washington, D. C.

1958
Publications of the Astrophysical Observatory

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Optical Properties of Saturn's Rings:

I. Transmission

By Allan F. Cook, II,1 and Fred A. Franklin

In the present paper, the first in a projected

series on the rings of Saturn, we propose to

discuss observations bearing upon the transmission properties of the three rings of Saturn.

Aspects of this problem have been treated by

Bobrov (1952) in one of a series of six interesting papers (Bobrov, 1940, 1951, 1952, 1954,

1956a, 1956b) on the ring system. Forthcoming papers will consider other physical aspects

of the rings and their interaction with the

interplanetary medium.

Rough estimates of the optical thickness of

Saturn's rings may be obtained from material

available from occultations of stars by the rings.

The most important of these occurred on March

14, 1920, when observers in the Union of South

Africa (Reid et al., 1920) and in India (Bhaskaran, 1920) followed the passage of BD+11°

2269 through rings A and B. If we assume that

the diffraction disk of the star could be seen

if its brightness were 1 percent that of the

surrounding ring, we can show that the optical

thickness, T, of ring B must be 0.59. Such a

low value of the contrast is valid only because

the star was visibly very orange, having a

spectral class of K3.

Ainslie (1917a, 1917b) and Knight (in

Ainslie and Knight, 1917) observed the occultation of the star 212 B. Geminorum by ring A

on February 9, 1917. Since both observers

followed the star (whose visual magnitude was

6.68 and spectral class Ml) through ring A,

its transparency is amply demonstrated. Any

attempt to estimate the optical thickness of

ring A by an argument similar to that used for

the 1920 occultation leads to a lower limit

greater than that for ring B. One final occultation was reported (Leonard, 1940) of BD+6°

259 on November 24, 1939. This star was not

visible through the ring system, probably because of its faintness and solar spectral class,

and the generally poor seeing and transparency.

Bobrov (1952) has used this observation to

determine a lower limiting value of 0.83 for

the optical thickness of ring B.

In his 1952 paper Bobrov claims that, on a

reproduction in L'Astronomie of a photograph

taken by Camichel (1946), the ball of Saturn

is clearly discernible as a bright area through

ring B. Bobrov's argument is greatly weakened

by the presence of other bright spots on the

ring. It seems exceedingly unlikely that with

his value of 0.7 for the average optical thickness

of ring B, corresponding to a contrast of 3

percent, the ball of the planet should be at all

visible in such a reproduction. From a detailed

study of the original photographs, Camichel

(private communication) is of the same opinion.

We object not to Bobrov's value of T, but to

this particular means of inferring it.

1

 Harvard College Observatory.

1

 Harvard College Observatory, and Smithsonian Astrophysical Observatory.

453660—58 377
378 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The eclipse of Iapetus by the shadow of the

rings, as observed by Barnard (1890) on Noember 1-2, 1889, provides probably the best

transmission data concerning the optical thickness of both rings B and C. Only for the latter, however, do we have measures of the

optical thickness as a function of distance

from the planet, because oncoming daylight

prevented Barnard from following the satellite

more than just into the penumbra of ring B.

Buchholz (1895) discussed Barnard's observations, using several simplifying approximations;

he assumed the sun to be a uniformly illuminated disk, and Iapetus to be a point reflector.

In the present report we have removed these

two assumptions by considering a limbdarkened sun and by trying to represent the

distribution of light on the finite surface of

Iapetus. Our first problem then is to discuss

the light curve of Iapetus and to find a way of

representing it. Photometric measurements

of Iapetus have been carried out by Pickering

(1879) in 1878-79, Wendell (1909) from 1896

to 1900, Guthnick (1910) from 1905 to 1908,

Graff (1939) in 1922, and Widorn (1952) in

1949, who summarized the data. In general,

the light variation ranges from about magnitude 10.0 to magnitude 12.0 while the period

of the light curve is virtually the same as that

of revolution about Saturn, with maximum

brightness occurring at western elongation.

From this fact one may infer that Iapetus

always presents the same face toward Saturn;

hence, the inclination of the axis of rotation

of Iapetus to its orbit is very probably near

90°, and the normal to the plane of the orbit

is inclined to the plane of the sky by 18° or less.

This implies that mean intensities of meridianal zones may be determined at least approximately.

As a first attempt we compared the following model with the above observations. Let

A(\) be the average brightness as a function

of the longitude X, At and A2 be average relative

brightnesses based upon a two-hemisphere

model for the bright and dark hemispheres

respectively, and A3 be the brightness of a

periodic term in cos 2X; then:

Then for the range 0<XO<TT we have essentially a Fourier series representation of the

average brightness:

* 2X) cos (X—

-T/2<X<T/ 2 A(\)=AI+A3 cos 2X

A(X) =A2-\-A3 cos 2X.

J

'X0+r/2 -1

(A2-\-A3 cos 2X) cos (X—Xo)<& •

Til J

This leads to the result:

-A2)-\-\j2{Ai—Aa) cos Xo

COS 2XQ,

which gives a negative brightness for a small

region just on the dark side of the boundary

between the bright and dark hemispheres.

In model II we tried:

0<|X|<TT/4

*74<|X|<ir/2 _

x/2<|X|<3x/4 A(X)=A2-A3

3*-/4<|X|<ir Ai\)=As+A3.

We can now carry out an integration similar

to the one above and then determine the best

set of values for Au A2 and A3 from the observed light curves. Finally, one must ask

how well these three can represent the entire

light curve. Such a model gives quite satisfactory agreement with the observed results of

Guthnick (1910) and Wendell (1909), but requires an extremely small brightness for two

octants. More explicitly, for Guthnick's data

we have: ^=0.765, ^2=0.125 and A=0.110,

while for Wendell's results: ^4t=0.780, A2—

0.115 and ^13=O.1O5. For each case the derived curve deviates from the observed one by

less than ±0.06 magnitudes.

A model such that

0<|X|<a A(\)=A1

«<W< T A(\)=A2

where from Wendell's data Ax was determined to

be 0.855, A2 to be 0.145 and a to be 76°, fits

about as well. Finally a more elaborate model

was tried on the data of both Guthnick and

Wendell:

0<|X|<X! ^(X)=^Lx

Xi<|x|<x2 2(x)=^i2

A2<|X|<ir A(\)=A3
OPTICAL PROPERTIES OF SATURN'S RINGS 379

with the added restriction that XI+X2=T.

For Wendell's curve: A=0.679, ^2=0.205,

43=0.116 and X!=58°, while for Guthnick's

data: ^4i=0.691, A2=0.212, ^43=0.097 and

Xi=53°. These results fit the respective observed curves to within ±0.03 magnitudes.

No reasonable model can represent the light

curves of either Graff or Pickering. The former

is considerably asymmetric while in the latter

case the satellite was compared directly with

an image of Saturn introduced by low power

optics. Furthermore, the considerable color

difference between Saturn and Iapetus should

have made Pickering's observations subject to

the Purkinje effect.

A general superposition of all curves except

Pickering's shows agreement to within ±0.10

magnitudes. In view of this fact and the relatively acceptable agreement given by all

models, it seems most reasonable to adopt the

simplest, i. e., a two-hemisphere model which

agrees to within ±0.10 magnitudes for the results of Guthnick and Wendell when the values

-Ai=0.879 and ^42=0.121 are employed.

This assumption is supported by the work of

Widorn (1952) who showed that his observations may be represented extremely accurately

on the basis of a two-hemisphere model. With

this problem settled we can proceed to consider the eclipse of Iapetus.

Adopting the orbital elements of Iapetus

given by G. Struve (1933), we can determine

the position of the satellite during the eclipse.

We can now rectify the observed light curve of

Barnard in order to obtain a continuous curve

of the optical thickness of ring C. We shall

first assume that the light intensity, I(r,s),

transmitted to the satellite can be represented

by a quadratic, whence,

J(r,«)=/(r)+6 cos 0 jrs+c cos2 6(4rY (1)

where r is the radial distance in the ring plane

of the point through which passes the straight

line joining the centers of the sun and Iapetus,

s is measured in the direction of motion of the

satellite, and 0 is the angle between a line perpendicular to the s-direction and the direction

defined by the isolines of light intensity transmitted by the rings, assumed to be straight

lines over the satellite disk. This assumption

introduces an error of less than 1 percent. R is

the radius of the solar disk of confusion, which

at the time of the eclipse was O'/2775. The

quantity Ar is the change in r as one moves a

distance R along the direction perpendicular

to the isolines. The displacement in this latter

direction we shall denote by H, so that 8=H

sec 0.

Using the familiar form of the equation of an

ellipse, we have

Ar

(2)

where e is the eccentricity of the ellipses formed

by projection of circles in the ring plane on the

plane of the sky as seen from the sun, and x, y

are the rectangular coordinates of Iapetus relative to Saturn and the principal axes of the

projection of the rings on the plane of the sky.

The coefficients b and c are given by the expressions :

= /(rm»x)+/(rmin)-2/(r0)

 m '

(3)

where /(TWJ) and I(rmln) are the intensities at

a distance of one radius of the solar disk of confusion within and without r in the direction

perpendicular to the isolines.

We shall first introduce the effect of a limbdarkened sun on equation (1) by the following

integration:

J-R

where F, a normalizing divisor, equals / ;

L(S,H), the solar limb darkening, is %+%R

X^R2—H2—S2, and S is a coordinate lying in

the direction of the isolines. This integration

yields:

(5)
380 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

In order to consider the effect of the disk of

confusion of the satellite itself, we must carry

out another integration over the satellite disk:

M dH +V I'MdS

J

'R'siat fHcotB "I

dH I'(r)dS

-R'sta$ J-jR'*-H> J

+A2\

I'(r)dS

'BinB

(6)

In equation (6), F'=(A,+4a ) (T#' 2/2), R' is

the angular radius of Iapetus which was taken

as O'/O9O. Ax and At are the brightnesses of

the trailing and the leading halves of the satellite

respectively and are quoted above. Since

maximum and minimum light of Iapetus occur

at western and eastern elongations respectively,

it seems reasonable to assume that during the

eclipse the bright and dark regions possessed

the same area.

This expression reduces upon integration

to:

% cos0[sin20-cos20]

Barnard's measures, which in our notation are

denoted by /"(r), are eye estimates relative to

the two satellites Tethys and Enceladus, whose

difference in brightness he takes equal to 1.0

magnitudes. More recent photometry (Guthnick, 1914) suggests that this difference is

more nearly 1.1 magnitudes. Barnard's curve

was therefore corrected for this difference.

In applying equation (7) we can use values

of /"(r) to approximate the 6's and c's and

hence obtain a first approximation to /(r). If

necessary one can repeat the process until no

change in I(r) occurs. For the crape ring,

since the corrections to I(r) are small, one

iteration is sufficient. Final values of I(r) are

shown in figure 1.

As mentioned previously, Barnard carried

his observations just into the shadow of ring B

before on-coming dawn made further estimates

impossible.

We can reduce his few measures in a fashion

similar to the one employed for ring C. We

consider the effect of the sun by the integration:

where /3 is the perpendicular distance from the

center of the solar disk of confusion to the as0.15 -

10.0 10.5 11.0 11.5 12.0 12.5 13.0

DISTANCE FROM CENTER OF SATURN IN SECONDS OF ARC AT MEAN DISTANCE

FIGURE 1.—The optical thickness of ring C plotted against distance from the planet's center.
OPTICAL PROPERTIES OF SATURN'S RINGS 381

sumed sharp straight-line boundary of ring B,

and a is the transmission of this ring. Upon

integration, equation (8) yields:

Y+ir/2]

Ar

-jk (l-7!)

3 / 2

 [46+3cTAr]

bAr

(9)

The quantity y in this last equation equals

&/R. It is impossible to integrate equation

(9) over the satellite disk, so the following procedure will be adopted. We can obtain values

of b and c by extrapolating the observed curve

for ring C; that is, by ignoring the effect of

ring B. Equation (9) then yields a set of

values for I'(r). We shall now assume that

the effect of passing over the ring B boundary

can be treated by applying an equation similar

to (7), viz.:

7"(rW(r)

,4&'Ar Rr

+

c o s [s m e

~

cos e]

(10)

The quantities b' and c' are found from the

computed curve for I'if) given by equation (9)

in accordance with the relation

I'(r,s)=r{r)+br cos cos2

 *(g (11)

The use of equations (9), (10), and (11) permits

us to calculate a family of curves of /"(r) for

differing values of the ring B transmission, a,

and inner boundary, p. This latter quantity

enters into the solution through its dependence

upon 7, 7=^r^- The best value of p, 13!22±

0T06 at the mean distance of Saturn, is somewhat higher than those commonly found in the

literature which are usually the result of nighttime micrometer measures. However, Lowell

and E. C. Slipher (see Lowell, 1915) obtained

13T21, a value corrected for irradiation by means

of daylight measures. We suspect that values

much lower than this are incorrect due to the

presence of irradiation.

The second quantity obtainable is a probable

lower limit to the transmission of ring B. The

best value of the transmission, a, as viewed

during the eclipse is 0.05. As the elevation of

the sun above the ring plane at the same time

was —11° 11', this value of the transmission

corresponds to an optical thickness of 0.58. It

may be seen that the extreme limits on the

transmission are 0<a<0.10 whence those on

the optical thickness are « >T>0.45 . For further details the reader is referred to figure 2.

As a final result, the measures of the optical

thickness of the crape ring can be used to determine an expression proportional to the pressure

of the gas, probably originating from the solar

corpuscular stream, in this ring. The argument

runs as follows: Let the rate of introduction of

particles of radius I across the inner edge of

ring B be W(l) per unit length of the circumference of the ring, and let the number of

particles in a column one square centimeter in

cross section in the ring be n(r,l) where r is the

distance from Saturn. Let a steady state of

inward flow of particles due to loss of angular

momentum to the gas be assumed. Then the

rate of loss of angular momentum by the

particles gives us the equation:

^ P,l*n(r,l) J* (nO— n(r,l)l*, (12)

where v is the velocity of the ring particles and

pp their density. The quantity ki(r) is a factor

measuring the effective drag on a particle's

cross section, xZ2. Using the fact that the

period of revolution equals &2r

3/2

, we may write

equation (12) as:

4ir2 ppl

3

 n( dr

Ji

In the steady state we must have:

(14)
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

0.00

1.50

Legend:

- Curve for inner radius of ring B. 13.22 .

at mean distance and transmission of

incident sunlight through ring B, 0.05.

Curves for changes of ±0.06" in radius.

Curves for changes of ±0.05 in transmission

I -,—I V • L- 1 r~~\ , i

4 0 " 23h 20m

40TM u" 20 m

UNIVERSAL TIME 1889 NOVEMBER 1 2

4 0 m 2 Q m

FIGURE 2.—The filled circles are Barnard's magnitude estimates which have been used to deduce limiting and most probable

values for the transmission of ring B and the location of its inner boundary.

where p is, as before, the inner radius of ring B.

Rearranging this last equation and combining

it with equation (13), we obtain:

(15)

PW(l)dl.

(16)

By definition the optical thickness is

Letting kt be the unknown constant,

we have

(17)

(18)

Note that kx(r) increases sharply inward. If

Saturn possesses a magnetic field which is operative at a distance equal to that of the crape

ring, then this field would divert ionized particles from the corpuscular stream away from

the ring plane. If on the other hand Saturn

possesses no effective magnetic field at such

distances, then for strong gravitational focussing (i. e., very slow particles) ki(r) should be proportional to r~

%

. The increase will be less steep

for faster particles. The fact that the increase

is much steeper than this seems to imply that

the solar corpuscular stream cannot be directly

responsible for the shape of kt(r). We are

therefore led to postulate the existence of a

cloud of neutral gas, the shape of whose pressure curve we are in a position to determine.

The factor ki(r) has the form:

&i(r)=fc4p<(»<—v) (19)

where p, is the gas density, ve the circular velocity and kt

 a constant of proportionality. If

P is the gas pressure, then in equilibrium:

dP

where

a—-

(20)

(21)

If we combine this with equation (19) we obtain:

dr~ T4 r sT T

 (22)
OPTICAL PROPERTIES OF SATURN'S RINGS 383

We must now introduce equation (18) into this

last result in order to obtain the equation:

4x

dr or

(23)

Equation (23) was integrated graphically

from the inner boundary of ring B (r=p=

13T22), where the integrand is practically zero, to

r=10?3, beyond which point the crape ring

becomes invisible. The result of this integration, that is, the shape of the gas pressure curve

as a function of the distance from Saturn, is

exhibited in figure 3.

We are pleased to thank Dr. Fred L. Whipple

for directing our attention to this problem and

for his continued aid and encouragement.

Miss Frances W. Wright also deserves our gratiDISTANCE FROM CENTER OF SATURN IN SF.CONCS OF ARC AT MEAN DISTANCE

P-PCP)

FIGURE 3.—The variation of —; (sec equation (23))

with distance from the planet's center.

tude for aid in the translation of Russian

articles.

References

AINSLIE, M. A.

1917a. Monthly Notices Roy. Astron. Soo. London,

vol. 77, p. 456.

1917b. Observatory, vol. 40, p. 157.

AINSLIB, M. A., AND KNIGHT, J.

1917. Journ. British Astron. Assoc, vol. 27, p.

212.

BABNABD, E. E.

1890. Monthly Notices Roy. Astron. Soc. London, vol. 50, p. 107.

BHASKARAN, T. P.

1920. Monthly Notices Roy. Astron. Soc. London,

vol. 80, p. 756.

BOBROV, M. S.

1940. Russian Astron. Journ., vol. 17, No. 6, p. 1.

1951. Proc. Acad. Sci. U. S. S. R., vol. 77, p. 581.

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CAMICHEL, H.

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GUTHNICK, P.

1910. Beob. K6nigl. Sternw. Berlin, No. 14.

1914. Astron. Nachr., vol. 198, p. 233.

LEONARD, F. C.

1940. Publ. Astron. Soc. Pacific, vol. 52, p. 39

LOWELL, P.

1915. Lowell Memoirs, vol. 1, No. 2.

PICKERING, E. C.

1879. Harvard Annals, vol. 11, Part 2, p. 247.

REID, W.; DUTTON, C. L. O'B.; AND MCINTYBB, W. G.

1920. Journ. British Astron. Assoc., vol. 30, pp.

230, 306; vol. 31, p. 37.

STBUVE, G.

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WENDELL, O. C.

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WIDORN, T.

1952. Mitt. Univ. Sternw. Wien, vol. 5, p. 19.

Abstract

All existing observational data relating to the transmission properties of Saturn's rings are discussed. From

such data the optical thickness of the crape ring is determined as a function of its distance from the planet.

Limiting values for ring B and a new value for the inner radius of ring B are established. The shape of the

pressure curve of a gas cloud evidently associated with ring C can be computed from its optical thickness.

m. a. imunn miiiH emu.