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Saturn's rings and Pioneer 11
ICARUS
24,492-498(1975)
Saturn’s
Rings and Pioneer
MICHAEL Planetary
Science
Institute,
11
J. PRICE Tucson,
Arizona
Received September 9, 1974; revised November
85704 22, 1974
Quantitative predictions of the diffuse reflection and transmission properties of Saturn’s rings, relevant to the September 1979 Pioneer 11 flyby, are presented. Predictions are based on an elementary anisotropic scattering model. Interparticle separations are considered to bc sufficiently large that mutual shadowing is negligible. Likely ranges in both the single scattering albedo and perpendicular optical thickness of the ring are considered. Situations of pronounced back-scattering and of isotropic scattering are treated individually. Spacecraft measurement of the radiation suffering diffuse scattering by the ring can provide a useful test of the basic ring model.
I. INTRODUCTION
scattering models for the ring. Whatever the actual spacecraft flyby trajectory at Saturn, the relevant diffuse reflection and transmission characteristics of the ring are predicted in this paper on the basis of the optimum models derived by Price (1973, 1974). Emphasis is placed on predicting the visual appearance of the ring system as a function of viewpoint.
Recently, Price (1973, 1974) rediscussed the optical scattering properties of Saturn’s ring using Earth-based photometric function data at visual wavelengths. Evidence indicating both that primary scattering dominates and that mutual shadowing is an irrelevant concept, was reviewed. Simple anisotropic scattering radiative transfer models were used to define the probable ranges both in the single scattering albedo, and in the general shape of the scattering phase function. Limitations on the mean perpendicular optical thickness of the ring were also obtained. Results indicated that the ring particles are highly efficient back-scatterers of visual radiation. Macroscopic particles account for the basic shape of the scattering phase function. Based on an infinite optical thickness for the ring, a minimum single scattering albedo -0.75 was found. Use of conservative scattering led to a minimum optical thickness -0.7. The analysis was consistent with the ring particles being centimeter-size pieces of ice. Currently, the NASA Pioneer 11 spacecraft is enroute to Saturn via Jupiter. On board the spacecraft is an Imaging Photopolarimeter Experiment (Gehrels, et al. 1974), providing a unique opportunity for investigating the validity of current optical Copyright 0 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
2. DIFFUSE SCATTERING BY THE RING The basic ring model (Price, 1974) is a homogeneous, plane-parallel, layer of anisotropically scattering particles illuminated by a parallel beam of solar radiation. Interparticle distances are sufficiently large that each particle may be considered to be in the f&r field of its neighbors. The optical scattering properties of the ring are completely described by three basic radiative transfer parameters : single scattering albedo (&), anisotropy parameter (x), and optical thickness (T). The shape of the scattering phase function will be described by the elementary form p(0) = b(1 + xcos O),
(1)
where 0 is the scattering angle. Kate that Eq. (1) is not entirely general since it does not encompass very elongated phase functions. In the general case of arbitrary 7, the theory of multiple scattering must be 492
SATURN’S
RINGS AND PIONEER
applied. Relevant analytical solutions and approximate computational techniques developed by Chandrasekhar (1960) have been used cf. Price (1974). Our quantitative predictions based on the optimum Price (1974) ring model must encompass the probable ranges in the single scattering albedo, in the optical thickness, and in the anisotropy parameter. Specifically, two extreme values for the single scattering albedo have been selected, i. e., 6 = 0.5 and f = 1.0. In order to examine the relative photometric properties of the major ring components,
493
11
two distinct values for the optical thickness have been adopted. For investigations of ring C, the region near Cassini’s division, and the outer extremity of ring A, T = 0.1 will be used. For ring B, T = 1 has been chosen. For ring A, an intermediate value for T would be suitable. Dependence of the predictions on choice of the anisotropy parameter will be illustrated by considering the two cases of maximum backscattering efficiency (x = -1) and of isotropic scattering (x = 0). Since Pioneer 11 will encounter Saturn only one year before the ring system .I0
.08
.08
.06
.06
I
L
F
F .04
.04
.02
.02
0
0
.2
.4
.6
.8
0
1.0
JO,
,
I
I
I
$ /
.4
.6
.8
1.0
I
I
I
I
.I0
I
TR*NSMISSION ALL
.08 f
AZIMUTH
ANGLES
_
.08
X’O
(
p.
/
ii=
.06 3 $
.2
P
P
= 0.05 0.5
.06 L F
i:
.04
.04 .p. /
0
*
T’.,
.2
.4
.6
.8
1.0
P
FIG. 1. Theoretical predictions of the diffuse reflection and transmission properties of Saturn’s ring. Isotropic scattering (x = 0) is assumed. Two values of the single scattering albedo (G) are considered, 0.5 and unity. Calculations are presented for two values of the optical thickness (r), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (CL,,)is taken as 0.05. Each diagram refers to all azimuth angles. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units F, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, P.
494
MICHAELJ.PRICE
presents an “edge-on” aspect to the sun (in late 1980), the chosen solar illumination angle must necessarily be low. All predictions will therefore correspond to p,, equal to 0.05 ; p,, (or p) is the cosine of the angle of incidence (or emergence) with respect to the outward normal. Because the radiative transfer theory tacitly assumes the ring to be laterally both homogeneous and infinite in extent, it can be expected to break down for small II, p0 values. All predictions will therefore correspond to p 2 0.05 only. REFLECTION:
x = -I,
3. THE VIEW FROM THE SPACECRAFT Bearing in mind that the flight time of Pioneer 11 from the Earth to Saturn (6.5 years) greatly exceeds the design lifetime of the spacecraft (900 days), our predictions must consider the possibility of partial or even complete failure of the imaging photometric experiment. Should calibration difficulties prohibit absolute photometry, relative photometry between the individual ring components may still be possible. Both absolute and relative /1” = 0.05, ii= 0.5
/ .06 I T 04
FIG. 2. Theoretical predictions of the diffuse reflection properties of Saturn’s ring. Maximum back-scattering (Z = -1) is assumed. The single scattering albedo (&) is taken as 0.5. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (pc) is taken as 0.05. Four values for the azimuth angle (Z/J)are considered: O”, 45”, 90”, and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, TV.
SATURN’S
RING9 AND PIONEER
similar to the situation when Pioneer 11 flew by. Absolute dependence of the diffuse reflection and transmission properties of the ring on both the single scattering albedo and optical thickness can be most easily examined by considering the simple case of isotropic scattering (x = 0). Complications introduced by anisotropic scattering will be discussed later. Our quantitative predictions are contained in Fig. 1. In the case of diffuse reflection, the specific intensity decreases monotonically as the
photometric predictions of the appearance of Saturn’s ring will therefore be made. Should the spacecraft be no longer functioning at Saturn encounter, the predictions will yet be relevant to the imaging experiment on board the pair of 1977 Mariner Jupiter-Saturn missions. Both Mariner spacecraft are expected to encounter Saturn during the first half of 1981. By then, the ring will have passed through its “edge-on” aspect, presenting its other face to the Sun. Even so, the geometry of solar illumination will be very TRANSMISSION:
495
11
x
““M
-0
.2
.6
.4 P
.e
I.0
0
.2
.4
.6
6
1.0
P
FlG. 3. Theoretical predictions of the diffuse transmission properties of Saturn’s ring. Maximum back-scattering (z = -1) is assumed. The single scattering albedo (G) is taken as 0.5. Calculations are presented for two values of the optical thickness (r), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (p,,) is taken as 0.05. Four values for the azimuth angle (9) are considered: O”, 45’, 90”, and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, TV.
MICHAEL
496
angle of reflection becomes more nearly normal to the ring plane. From all viewpoints, the absolute specific intensity is sensitive to G for constant r. Increasing i;, from 0.5 to unity changes the specific intensity by a factor 2-3. By comparison, the specific intensity is relatively insensitive to T for constant G. Diffuse reflection measurements can therefore provide useful information concerning the single scattering albedo. In the case of diffuse transmission, the specific intensity is sensitive both to the single scattering REFLECTION:
J. PRICE
albedo and to the optical thickness. For small optical thicknesses (T = O.l), the specific intensity decreases monotonically as the angle of transmission becomes more nearly normal to the ring plane. For large optical thicknesses (T = l), the specific intensity remains nearly constant. From all viewpoints, the specific intensity decreases with increasing optical thickness for constant single scattering albedo. For small TV,the absolute specific intensity is very sensitive to 7 for constant G. For p equals 0.1, increasing T from 0.1 to x = -I,
pLo= 0.05,
“w = I
0 0
.2
.4
6
8
1.0
P
P
P
FIG. 4. Theoretical predictions of the diffuse reflection properties of Saturn’s ring. Maximum conservative ((;, = 1) back-scattering (2 = -1) is assumed. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of sola,r radiation with respect to the outward normal to the ring plane (pO) is taken as 0.05. Four values for the azimuth angle ($) are considered: 0”, 45”, 90”, and 180’. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where rrP is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, ~1.
SATURN’S
RINGS AND PIONEER
unity results in changing the specific intensity by a factor 5-15, depending on the 6 value. For p near unity, the absolute specific intensity is not sensitive the specific to T. From all viewpoints, intensity is dependent on the single scattering albedo for constant 7. Increasing B from 0.5 to unity changes the specific intensity by a factor 2-3. Diffuse transmission measurements can therefore provide useful information on optical thickness, if the single scattering albedo is known and the spacecraft elevation angle is small. TRANSMISSION
:
497
11
Expected dependences of the diffuse reflection and transmission properties of the ring on azimuth angle are illustrated in Figs. 2, 3, 4 and 5. Each figure corresponds to our preferred radiative transfer model in which the anisotropy parameter, 5, is set equal to -1. Two individual single scattering albedos are again used, i;, = 0.5 and 1. For c = 0.5, Figs. 2 and 3 illustrate the diffuse reflection and transmission characteristics, respectively. For ij = 1, Figs. 4 and 5 illustrate the diffuse reflection and transmission characteristics, respectively. Each diagram considers four x
-I,
q
/Lo = 0.05,
ii = I
.I0
.06 I
F .04
.06
0
.2
.4
.6 P
.6
1.0
0
0
.2
.4
.6
.6
1.0
P
FIG. 5. Theoretical predictions of the diffuse transmission properties of Saturn’s ring. Maximum conservative (6 = 1) back-scattering (Z = - I) is assumed. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (p,,) is taken as 0.05. Four values for the azimuth angle (#) are considered: O”, 45”, 90” and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where xP is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, I*.
MICHAELJ. PRICE
498
azimuth angles # = O”, 45”, 90”, and 180’. For #< 90”, the diffuse reflection and transmission properties of the ring remain qualitatively very similar to the isotropic scattering case. Not surprisingly, highest values for the specific intensity occur for $ = 0” (i.e., the backscattering direction). In the forward-scattering direction (4 = ISO’), the reflection and transmission characteristics are very alike ; the specific intensity is insensitive to p and 7. Evidently, discrimination between scattering phase functions will require observations with $ > 90”. For T --f 0, the amounts of radiation suffering diffuse reflection and transmission become predictable on the basis of primary scattering alone. By integrating the releva’nt contributions from the entire range in optical thickness, the diffuse reflection characteristics of the ring can be readily shown (Chandrasekhar, 1960) to take the form
I/F = a~(@) --&
xP-
exln
[-(l/p + ~/P&I>. (2)
Similarly, the diffuse transmission teristics are given by I/F = ip(O)&
primary scattering) contributes to the determination of the diffuse reflection characteristics of the ring. 4. CONCLUDINGREMARKS Several major qualitative predictions may be made. When the ring is viewed from the spacecraft by reflected sunlight, its visual appearance will be similar to that observed from the Earth. In surface brightness, ring A will always be fainter than ring B. Contrast between rings A and B will gra,dually increase with the spacecmft elevation angle. By comparison, when the ring is viewed from the spacecraft in transmitted sunlight, its visual appearance will differ widely from that observed from the Earth. For small spaoecraft elevation angles, ring A will exhibit a surface brightness much greater bhan that of ring B. However, as the spacecraft elevation angle increases, contrast between t’he major ring components will entirely disappear. When viewed from a direction perpendicular to the ring plane, the sy&em should appear uniformly bright.
charac-
(e-r’fi - e-rluo).
ACKNOIVLEDGMENT
(3)
Equations (2) and (3) become applicable for 720.1. For p, p,, --f 0, predictions of the diffuse reflection cha’racteristics for arbitrary T converge to the prima,ry scattering case vertical pene(i.e., 7 + 0). Physically, tration of the ring by solar radiation is inhibited at shallow angles of incidence by correspondingly long optica’ paths. The escape of radiation suffering diffuse reflectionis similarly affected. In the “edgeon” limit, only the upper boundary (i.e.,
This research was supported by the National Aeronautics and Space Administration under contract NrZSW-2521.
REFERENCES PRICX, M. J. (1973). Optical scatterinfi properties of Saturn’s hug. ~IstYon..J. 78, 11X-120. Pnrc~, M. J. (1974). Optical scattering properties of Saturn’s ring. 11. Icarus 23, 388-398. GEHRELS, T. et al. (1974) The imaging photopolarimeter experiment on Pionwr 10. &ience 183, 318. CHANDRASEKI-IAR, 8. (1960). Radiutioe Tramfer. Dover, New Work.
24,492-498(1975)
Saturn’s
Rings and Pioneer
MICHAEL Planetary
Science
Institute,
11
J. PRICE Tucson,
Arizona
Received September 9, 1974; revised November
85704 22, 1974
Quantitative predictions of the diffuse reflection and transmission properties of Saturn’s rings, relevant to the September 1979 Pioneer 11 flyby, are presented. Predictions are based on an elementary anisotropic scattering model. Interparticle separations are considered to bc sufficiently large that mutual shadowing is negligible. Likely ranges in both the single scattering albedo and perpendicular optical thickness of the ring are considered. Situations of pronounced back-scattering and of isotropic scattering are treated individually. Spacecraft measurement of the radiation suffering diffuse scattering by the ring can provide a useful test of the basic ring model.
I. INTRODUCTION
scattering models for the ring. Whatever the actual spacecraft flyby trajectory at Saturn, the relevant diffuse reflection and transmission characteristics of the ring are predicted in this paper on the basis of the optimum models derived by Price (1973, 1974). Emphasis is placed on predicting the visual appearance of the ring system as a function of viewpoint.
Recently, Price (1973, 1974) rediscussed the optical scattering properties of Saturn’s ring using Earth-based photometric function data at visual wavelengths. Evidence indicating both that primary scattering dominates and that mutual shadowing is an irrelevant concept, was reviewed. Simple anisotropic scattering radiative transfer models were used to define the probable ranges both in the single scattering albedo, and in the general shape of the scattering phase function. Limitations on the mean perpendicular optical thickness of the ring were also obtained. Results indicated that the ring particles are highly efficient back-scatterers of visual radiation. Macroscopic particles account for the basic shape of the scattering phase function. Based on an infinite optical thickness for the ring, a minimum single scattering albedo -0.75 was found. Use of conservative scattering led to a minimum optical thickness -0.7. The analysis was consistent with the ring particles being centimeter-size pieces of ice. Currently, the NASA Pioneer 11 spacecraft is enroute to Saturn via Jupiter. On board the spacecraft is an Imaging Photopolarimeter Experiment (Gehrels, et al. 1974), providing a unique opportunity for investigating the validity of current optical Copyright 0 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
2. DIFFUSE SCATTERING BY THE RING The basic ring model (Price, 1974) is a homogeneous, plane-parallel, layer of anisotropically scattering particles illuminated by a parallel beam of solar radiation. Interparticle distances are sufficiently large that each particle may be considered to be in the f&r field of its neighbors. The optical scattering properties of the ring are completely described by three basic radiative transfer parameters : single scattering albedo (&), anisotropy parameter (x), and optical thickness (T). The shape of the scattering phase function will be described by the elementary form p(0) = b(1 + xcos O),
(1)
where 0 is the scattering angle. Kate that Eq. (1) is not entirely general since it does not encompass very elongated phase functions. In the general case of arbitrary 7, the theory of multiple scattering must be 492
SATURN’S
RINGS AND PIONEER
applied. Relevant analytical solutions and approximate computational techniques developed by Chandrasekhar (1960) have been used cf. Price (1974). Our quantitative predictions based on the optimum Price (1974) ring model must encompass the probable ranges in the single scattering albedo, in the optical thickness, and in the anisotropy parameter. Specifically, two extreme values for the single scattering albedo have been selected, i. e., 6 = 0.5 and f = 1.0. In order to examine the relative photometric properties of the major ring components,
493
11
two distinct values for the optical thickness have been adopted. For investigations of ring C, the region near Cassini’s division, and the outer extremity of ring A, T = 0.1 will be used. For ring B, T = 1 has been chosen. For ring A, an intermediate value for T would be suitable. Dependence of the predictions on choice of the anisotropy parameter will be illustrated by considering the two cases of maximum backscattering efficiency (x = -1) and of isotropic scattering (x = 0). Since Pioneer 11 will encounter Saturn only one year before the ring system .I0
.08
.08
.06
.06
I
L
F
F .04
.04
.02
.02
0
0
.2
.4
.6
.8
0
1.0
JO,
,
I
I
I
$ /
.4
.6
.8
1.0
I
I
I
I
.I0
I
TR*NSMISSION ALL
.08 f
AZIMUTH
ANGLES
_
.08
X’O
(
p.
/
ii=
.06 3 $
.2
P
P
= 0.05 0.5
.06 L F
i:
.04
.04 .p. /
0
*
T’.,
.2
.4
.6
.8
1.0
P
FIG. 1. Theoretical predictions of the diffuse reflection and transmission properties of Saturn’s ring. Isotropic scattering (x = 0) is assumed. Two values of the single scattering albedo (G) are considered, 0.5 and unity. Calculations are presented for two values of the optical thickness (r), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (CL,,)is taken as 0.05. Each diagram refers to all azimuth angles. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units F, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, P.
494
MICHAELJ.PRICE
presents an “edge-on” aspect to the sun (in late 1980), the chosen solar illumination angle must necessarily be low. All predictions will therefore correspond to p,, equal to 0.05 ; p,, (or p) is the cosine of the angle of incidence (or emergence) with respect to the outward normal. Because the radiative transfer theory tacitly assumes the ring to be laterally both homogeneous and infinite in extent, it can be expected to break down for small II, p0 values. All predictions will therefore correspond to p 2 0.05 only. REFLECTION:
x = -I,
3. THE VIEW FROM THE SPACECRAFT Bearing in mind that the flight time of Pioneer 11 from the Earth to Saturn (6.5 years) greatly exceeds the design lifetime of the spacecraft (900 days), our predictions must consider the possibility of partial or even complete failure of the imaging photometric experiment. Should calibration difficulties prohibit absolute photometry, relative photometry between the individual ring components may still be possible. Both absolute and relative /1” = 0.05, ii= 0.5
/ .06 I T 04
FIG. 2. Theoretical predictions of the diffuse reflection properties of Saturn’s ring. Maximum back-scattering (Z = -1) is assumed. The single scattering albedo (&) is taken as 0.5. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (pc) is taken as 0.05. Four values for the azimuth angle (Z/J)are considered: O”, 45”, 90”, and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, TV.
SATURN’S
RING9 AND PIONEER
similar to the situation when Pioneer 11 flew by. Absolute dependence of the diffuse reflection and transmission properties of the ring on both the single scattering albedo and optical thickness can be most easily examined by considering the simple case of isotropic scattering (x = 0). Complications introduced by anisotropic scattering will be discussed later. Our quantitative predictions are contained in Fig. 1. In the case of diffuse reflection, the specific intensity decreases monotonically as the
photometric predictions of the appearance of Saturn’s ring will therefore be made. Should the spacecraft be no longer functioning at Saturn encounter, the predictions will yet be relevant to the imaging experiment on board the pair of 1977 Mariner Jupiter-Saturn missions. Both Mariner spacecraft are expected to encounter Saturn during the first half of 1981. By then, the ring will have passed through its “edge-on” aspect, presenting its other face to the Sun. Even so, the geometry of solar illumination will be very TRANSMISSION:
495
11
x
““M
-0
.2
.6
.4 P
.e
I.0
0
.2
.4
.6
6
1.0
P
FlG. 3. Theoretical predictions of the diffuse transmission properties of Saturn’s ring. Maximum back-scattering (z = -1) is assumed. The single scattering albedo (G) is taken as 0.5. Calculations are presented for two values of the optical thickness (r), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (p,,) is taken as 0.05. Four values for the azimuth angle (9) are considered: O”, 45’, 90”, and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where nF is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, TV.
MICHAEL
496
angle of reflection becomes more nearly normal to the ring plane. From all viewpoints, the absolute specific intensity is sensitive to G for constant r. Increasing i;, from 0.5 to unity changes the specific intensity by a factor 2-3. By comparison, the specific intensity is relatively insensitive to T for constant G. Diffuse reflection measurements can therefore provide useful information concerning the single scattering albedo. In the case of diffuse transmission, the specific intensity is sensitive both to the single scattering REFLECTION:
J. PRICE
albedo and to the optical thickness. For small optical thicknesses (T = O.l), the specific intensity decreases monotonically as the angle of transmission becomes more nearly normal to the ring plane. For large optical thicknesses (T = l), the specific intensity remains nearly constant. From all viewpoints, the specific intensity decreases with increasing optical thickness for constant single scattering albedo. For small TV,the absolute specific intensity is very sensitive to 7 for constant G. For p equals 0.1, increasing T from 0.1 to x = -I,
pLo= 0.05,
“w = I
0 0
.2
.4
6
8
1.0
P
P
P
FIG. 4. Theoretical predictions of the diffuse reflection properties of Saturn’s ring. Maximum conservative ((;, = 1) back-scattering (2 = -1) is assumed. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of sola,r radiation with respect to the outward normal to the ring plane (pO) is taken as 0.05. Four values for the azimuth angle ($) are considered: 0”, 45”, 90”, and 180’. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where rrP is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, ~1.
SATURN’S
RINGS AND PIONEER
unity results in changing the specific intensity by a factor 5-15, depending on the 6 value. For p near unity, the absolute specific intensity is not sensitive the specific to T. From all viewpoints, intensity is dependent on the single scattering albedo for constant 7. Increasing B from 0.5 to unity changes the specific intensity by a factor 2-3. Diffuse transmission measurements can therefore provide useful information on optical thickness, if the single scattering albedo is known and the spacecraft elevation angle is small. TRANSMISSION
:
497
11
Expected dependences of the diffuse reflection and transmission properties of the ring on azimuth angle are illustrated in Figs. 2, 3, 4 and 5. Each figure corresponds to our preferred radiative transfer model in which the anisotropy parameter, 5, is set equal to -1. Two individual single scattering albedos are again used, i;, = 0.5 and 1. For c = 0.5, Figs. 2 and 3 illustrate the diffuse reflection and transmission characteristics, respectively. For ij = 1, Figs. 4 and 5 illustrate the diffuse reflection and transmission characteristics, respectively. Each diagram considers four x
-I,
q
/Lo = 0.05,
ii = I
.I0
.06 I
F .04
.06
0
.2
.4
.6 P
.6
1.0
0
0
.2
.4
.6
.6
1.0
P
FIG. 5. Theoretical predictions of the diffuse transmission properties of Saturn’s ring. Maximum conservative (6 = 1) back-scattering (Z = - I) is assumed. Calculations are presented for two values of the optical thickness (T), 0.1 and unity. The cosine of the angle of incidence of solar radiation with respect to the outward normal to the ring plane (p,,) is taken as 0.05. Four values for the azimuth angle (#) are considered: O”, 45”, 90” and 180”. Azimuth angle is measured in the ring plane, the zero point corresponding to the projected Saturn-Sun line. Ordinates are the specific intensity, I, measured in units P, where xP is the solar flux at Saturn. Abscissae are the cosines of the angles of emergence with respect to the outward normal to the ring plane, I*.
MICHAELJ. PRICE
498
azimuth angles # = O”, 45”, 90”, and 180’. For #< 90”, the diffuse reflection and transmission properties of the ring remain qualitatively very similar to the isotropic scattering case. Not surprisingly, highest values for the specific intensity occur for $ = 0” (i.e., the backscattering direction). In the forward-scattering direction (4 = ISO’), the reflection and transmission characteristics are very alike ; the specific intensity is insensitive to p and 7. Evidently, discrimination between scattering phase functions will require observations with $ > 90”. For T --f 0, the amounts of radiation suffering diffuse reflection and transmission become predictable on the basis of primary scattering alone. By integrating the releva’nt contributions from the entire range in optical thickness, the diffuse reflection characteristics of the ring can be readily shown (Chandrasekhar, 1960) to take the form
I/F = a~(@) --&
xP-
exln
[-(l/p + ~/P&I>. (2)
Similarly, the diffuse transmission teristics are given by I/F = ip(O)&
primary scattering) contributes to the determination of the diffuse reflection characteristics of the ring. 4. CONCLUDINGREMARKS Several major qualitative predictions may be made. When the ring is viewed from the spacecraft by reflected sunlight, its visual appearance will be similar to that observed from the Earth. In surface brightness, ring A will always be fainter than ring B. Contrast between rings A and B will gra,dually increase with the spacecmft elevation angle. By comparison, when the ring is viewed from the spacecraft in transmitted sunlight, its visual appearance will differ widely from that observed from the Earth. For small spaoecraft elevation angles, ring A will exhibit a surface brightness much greater bhan that of ring B. However, as the spacecraft elevation angle increases, contrast between t’he major ring components will entirely disappear. When viewed from a direction perpendicular to the ring plane, the sy&em should appear uniformly bright.
charac-
(e-r’fi - e-rluo).
ACKNOIVLEDGMENT
(3)
Equations (2) and (3) become applicable for 720.1. For p, p,, --f 0, predictions of the diffuse reflection cha’racteristics for arbitrary T converge to the prima,ry scattering case vertical pene(i.e., 7 + 0). Physically, tration of the ring by solar radiation is inhibited at shallow angles of incidence by correspondingly long optica’ paths. The escape of radiation suffering diffuse reflectionis similarly affected. In the “edgeon” limit, only the upper boundary (i.e.,
This research was supported by the National Aeronautics and Space Administration under contract NrZSW-2521.
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