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Icarus 170 (2004) 70–90www.elsevier.com/locate/icaru

Photometric modeling of Saturn’s rings.II. Azimuthal asymmetry in reflected and transmitted light

H. Salo,a,∗ R. Karjalainen,a and R.G. Frenchb

a Department of Physical Sciences, Astronomy Division, PO BOX 3000, FI-90014 University of Oulu, Finlandb Astronomy Department, Wellesley College, Wellesley, MA 02481, USA

Received 12 September 2003; revised 10 March 2004

Available online 18 May 2004

Abstract

DynamicalN-body simulations (Salo, 1992, Nature 359, 619) suggest the formation of trailing density enhancements in the outer portioof Saturn’s rings, due to local gravitational instabilities. These Julian–Toomre type wakes, having a pitch angle of about 20–25 withrespect to the local tangential direction, seem to provide a plausible explanation for the observed quadrupole brightness variationA ring (Salo and Karjalainen, 1999, Bull. Am. Astron. Soc. 31, 1160; French et al., 2000, Bull. Am. Astron. Soc. 32, 806; Porc2001, Bull. Am. Astron. Soc. 33, 1091). We have carried out systematic photometric modeling of gravitational wake structuredynamical simulations, performed for the parameter values of the A ring, using the Monte Carlo radiative transfer code describeand Karjalainen (2003, Icarus 164, 428). Comparisons to the observed asymmetry in various cases are presented (asymmetryand transmitted light, ring longitude and opening angle dependence), in all cases confirming the applicabilityof the wake model. Typicallyminimum brightness corresponds to viewing/illumination alongthe long axis of wakes; however, the sense of modeled asymmetry revat small tilt angles in diffuse transmission. Implications of wakes on the occultation optical depth profiles and the A ring overall brbehavior are also discussed: it is shown that the wake structure needs to be taken into account when the Cassini occultation proA ring are interpreted in terms of variations in surface density. Also, the presence of wakes offers a plausible explanation for the ieffect seen in the mid A-ring. 2004 Elsevier Inc. All rights reserved.

Keywords:Planetary rings; Saturn; Radiative transfer; Collisional physics; Computer techniques

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1. Introduction

The outer parts of Saturn’s rings exhibit bisymmeazimuthal brightness variations, first noticed byCamichel(1958), and later confirmed byFerrin (1975) and Reitsemet al. (1976). According to these observations the brigness of the A ring has two minima, before the westerneastern ansae, in the sense of rotation of the rings. Suquent detailed studies(Lumme and Irvine, 1976a, 1979Lumme et al., 1977; Thompson et al., 1981)confirmed the180 symmetry of the brightness variations and showeda similar asymmetry is seen in all visual wavelengths,that the amplitude of the asymmetry peaks when theopening angle is around 12. In these ground-based observ

* Corresponding author. Fax: +358-8-553-1934.E-mail address:[email protected] (H. Salo).

0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2004.03.012

-

tions the minimum brightness is seen at longitudes of a245–250 and 65–70, measured with respect to the suobserver point in the direction of the orbital motion. Hoever, due to the small range of phase angles accessibleEarth, it is difficult to distinguish whether the minimumpredominantly related to the observing or illuminationrection. The detailed analysis of low-phase Voyager ima(Dones et al., 1993)showed that the amplitude of variatiopeaks strongly in the mid-A ring, reaching a full amplitud((Imax − Imin)/〈I 〉) of about 35% in reflected light at thsaturnocentric distance of 128,000 km. Also, the longitof the minimum seemed to be mainly determined byviewing direction. The A ring asymmetry is also seentransmitted light(Franklin et al., 1987), whereas for the Bring no asymmetry was detected. Note that Saturn’s rhave often been reported to show a difference betweenvisual brightnesses of the East and West ansae, the a

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Photometric modeling of a ring azimuthal asymmetry 71

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tude and sign of which varies with time (“east–west asymetry,” see, e.g.,Piironen and Lukkari (1980)and detailedreferences therein). This phenomenon, if real, is likelybe distinct from the permanently present quadrupole asmetry addressed here: it has been suggested to rise dreflection by debris clouds caused by meteoroidal impon the rings(Hämeen-Anttila and Itävuo, 1976).

A likely cause of the azimuthal quadrupole brightneasymmetry is small-scale particle inhomogeneities, traiby about 20–25 with respect to the local tangential drection (Colombo et al., 1976). As discussed inFranklinet al. (1987), such systematically oriented inhomogeneitcould arise due to dynamical wakes excited around msive particle aggregates(Franklin and Colombo, 1978Karttunen, 1983), or from the superposition of numeroJulian–Toomre (1966)gravity wakes excited about the rinparticles themselves (seeFranklin et al., 1987; Dones anPorco, 1989). In both cases the expected orientation ofwakes is determined by the differential rotation, whicha Keplerian velocity field yieldsφwake∼ 20. Other modelsin terms of elongated swarms of particle debris resulfrom collisions have also been discussed (Gorkavyi andTaidakova, 1989, seeFridman and Gorkavyi, 1999, p. 122),but these require additional assumptions about the lifetimthe debris swarms, in order to yield the desired orientatThe large maximum amplitude of the asymmetry suggthat, whatever structures are responsible for the brightvariations, they need to cover a large fraction of the rarea, at least in the mid-A ring, and have a very large intrinbrightness contrast. This seems to favor Julian–Toomrespontaneous wakes over those forced by embedded mlets, unless the latter are much more abundant than usassumed. Moreover, the formation of trailing Julian–Toomwakes via local gravitational instabilities has been confirmby numericalN -body simulations(Salo, 1992), with pa-rameter values consistent with Voyager measuremensize distribution and surface density, using theBridges etal. (1984)laboratory measurements of the elasticity ofparticles.

The ability of density wakes to cause brightness vations follows from the fact that the fractional reflectisurface area will be direction-dependent (seeFig. 1). In par-ticular, when the wakes are viewed more or less along tlong axis, the rarefied regions between the dense wakebetter visible, reducing the overall reflection, in comparisto the perpendicular viewing direction when the rarefiedgions are partially hidden (see, e.g.,Franklin et al., 1987Thompson, 1982; Dones and Porco, 1989). The directionof illumination is likewise expected to have importance,well as the multiple scattering occurring predominantlyside wakes. Somewhat different models, in terms of mtiple scattering in optically thick ellipsoidal particle blob(Lumme and Irvine, 1979a)yield the same expected depedence of longitude minima and maxima with respect toassumed orientation of the blobs. A-ring wakes also manthemselves in the asymmetry of Saturn’s microwave ther

o

-

e

Fig. 1. Schematic explanation for the relation of wake structure to azimuasymmetry. At low elevation angles the wakes trailing by about 21 withrespect to the local tangential direction are seen roughly along theiraxis at ring longitudes of 249 and 69 , and perpendicular to their lonaxis at longitudes of 339 and 159 . In the former case rarefied regionbetween wakes are visible, reducing thereflecting surface area: this shoucorrespond to minimum brightness. In the latter case rarefied regionhidden by the wakes in low tilt angle images, maximizing the reflecarea and thus corresponding to maximum brightness. For a more reaillustration, seeFig. 3.

radiation transmitted through the rings(Dunn et al., 2004).Wake structure has also been suggested to explain the ametry in the thermal radiation scattered by the rings(van derTak et al., 1999; Dunn et al., 2002); due to Saturn being thsource of radiation this asymmetry manifests as a differebetween East and West ansae (not to be confused witvisual ‘east–west asymmetry’ mentioned above).

We have carried out photometric modeling of JuliaToomre wake structures seen in dynamical simulations,have made systematic comparisons to the observed ametry at various geometries. The photometric calculamethod, based on Monte Carlo modeling of light rays stered by a system of discrete simulation particles, asas its application to azimuthally homogeneous ring modhas been described inSalo and Karjalainen (2003)(hereafterPaper I). Some preliminary results for the asymmetry halready been reported inSalo and Karjalainen (1999), Saet al. (2000), and French et al. (2000), supporting the wakeexplanation for the asymmetry (see alsoPorco et al., 2001).In the present study we demonstrate that wakes obtavia dynamical simulations can account naturally for sevbasic properties of the asymmetry, including the tilt-andependence found in ground-based observations, as wthe longitude dependence in Voyager observations ofreflected and transmitted light. Implications for the occution optical depth are also briefly discussed, as well asconsequences of wakes on the elevation angle depend(tilt effect) of the A ring brightness. Two different dynamicsimulation runs are studied, for the distances corresping to the maximum of the observed asymmetry amplituusing models (1) with identical particles and (2) with a trucated power-law size distribution, for a fixed value of surfdensity, corresponding to that typically assumed for the mA ring. Here, we are primarily interested in exploring tdifferences between these two models, rather than tryinachieve an exact match with observations. The next pap

72 H. Salo et al. / Icarus 170 (2004) 70–90

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this series will study the dependence of the simulated asmetry on various dynamical parameters: the saturnocedistance, the internal density and elasticity of particles,ring surface density, and the size distribution of particles

2. Gravitational wakes and the asymmetry

2.1. Gravitational instability in planetary rings

The local equilibrium state in planetary rings is governby the balance between collisional dissipation and thecous gain of energy from the systematic velocity field.the absence of self-gravity the resulting equilibrium velocdispersion is determined by the elastic properties of pcles and by their size distribution. Due to mutual gravparticles tend to form densityenhancements via local graitational instability. This tendency for gravitational collapis opposed by the velocity dispersion of particles and byferential rotation, which stabilize disturbances on smalllarge scales, respectively(Toomre, 1964). The condition foraxisymmetric gravitational stability is that the Toomre pameterQ > 1, where

(1)Q = σrκ

3.36GΣ

measures the velocity dispersion. Hereσr andΣ representhe radial velocity dispersion and the surface density,spectively, and the epicyclic frequencyκ equals the angulafrequencyΩ in the case of a Keplerian velocity field.practice, even ifQ exceeds unity but remains below 2–significant non-axisymmetric local condensations may fo(Julian and Toomre, 1966; Salo, 1992, 1995; DaisakaIda, 1999). Due to differential rotation, these condensatioappear in the form of trailing wakes, inclined typically∼ 20 with respect to the tangential direction. These foring condensations heat the system, so that in the absendissipative collisions the ring would soon become featureagain asQ rises to 1. In the presence of impacts a stattical steady-state is reached, with new wakes replacingold ones which disperse in timescales of order of the orbperiod. The typical radial scale of wakes in simulations coresponds, at least roughly, to Toomre’s critical wavelen(Toomre, 1964),

(2)λcr = 4π2GΣ/κ2,

which for the parameter values of Saturn’s rings correspoto

(3)λcr = 70 m

(a

108 m

)3(Σ

1000 kg m−2

).

The rough requirement for the formation of wakes,Q < 2,indicates for identical particles with radiusr and internaldensityρ (in which caseΣ = 4

3ρrτ ) that

(4)σr

rΩ< 14

(a

108 m

)3(ρ

−3

)τ.

900 kg m

f

Since the minimum velocity dispersion maintained bypacts alone may be estimated to be about 2–3rΩ (in the caseof identical particles and a constant coefficient of restitutseeBrahic, 1977; Wisdom and Tremaine, 1988; Salo, 19),it is clear that rarefied ring regions withτ < τmin are stableagainst the formation of wakes: the above minimum velodispersion 2–3rΩ , combined withEq. (4)indicates

(5)τmin ≈ 0.2

(a

108 m

)−3(ρ

900 kg m−3

)−1

,

or about 0.3–0.1, from the inner C ring to the outer A ringthe density of solid ice is assumed. In regions withτ > τminwakesmay form, depending on the actual elasticity law, tinternal density of particles, and the particles’ size distution. Namely, in the presence of a size distribution, smparticles achieve a larger velocity dispersion than the laones, which acts as a stabilizing factor. The importancthe elasticity law stems from the fact that in the case ovelocity dependent coefficient of restitutionεn = ε(v), thesystem can attain, depending on the exact form of the veity dependence, a steady-state with a finite velocity dissionσr rΩ , in contrast to the case of constantεn, leadingto σr a few timesrΩ whenever the system is thermally sble: εn < εcr, whereεcr ≈ 0.65 for τ → 0 (Hämeen-Anttila,1978; Goldreich and Tremaine, 1978). The internal densityis also important, since for a smallρ the maximum spacdensity is limited even in the case of close packing of pticles. Taking into account these uncertainties, dynamN -body simulations seem to offer the most reliable wayaddress the formation and properties of the gravity wakeSaturn’s rings.

2.2. Dynamical simulations

In the present study we want to address the genapplicability of gravity wakes to explain the observedimuthal asymmetry, as well to illustrate some other possphotometric consequences of wake structure. For thispose just a few typical simulation examples are studied,dynamical parameters chosen to represent the mid Awhere the asymmetry is most pronounced. The simulatare performed with a local method described inSalo (1995).The basic idea in the method is to restrict all collisioand gravitational calculations to a small region insiderings, co-moving with the local mean angular speed of pacles. Linearized dynamical equations are employed, andparticles leaving the simulation system due to differenrotation are treated by using periodic boundary conditionsfirst introduced inWisdom and Tremaine (1988) and Toomand Kalnajs (1991). As discussed in Paper I, these perioboundary conditions are straightforward to incorporate intophotometric calculations.

An example of the time evolution in a dynamical simution is shown inFig. 2, for internal densityρ = 450 kg m−3,dynamical optical depthτdyn = 0.5 and saturnocentric distancea = 130,000 km. Identical particles are used, with t


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,lationon

Fig. 2. Time evolution of our standard dynamical simulation model with identical particles. The upper two rows show snapshots from the simulation sytem,as projected onto the equatorial plane. The planet is to the left and the direction of orbital motion is upward. The run with dynamical optical depthτdyn = 0.5

is performed for a saturnocentric distance ofa = 130,000 km. Identical particles with radiir = 1.667 m, and internal densityρ = 450 kg m−3 are assumedyielding a surface densityΣ = 500 kg m−2. The Bridges et al. (1984)velocity dependent coefficient of restitution is assumed. The size of the simusystem corresponds toLx × Ly = 4λcr × 4λcr = 305 m× 305 m, and the number of particles isN = 5300. The bottom row displays 2D autocorrelatifunctions for the snapshots corresponding to the middle row.

-e

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particle radius set tor = 1.667 m, to yield a surface density Σ = 500 kg m−2. For the coefficient of restitution wuse the standard velocity-dependent model ofBridges et al.(1984),

(6)εn(vn) = min[1, (vn/vc)

−0.234],wherevn is the normal component of the relative velocof the impacting bodies and the scale parametervc equalsvB = 0.0077 cm s−1 in the Bridges et al. (1984)measure-ments. For meter-sized particles this elasticity model yieresults which are fairly close to using a constantεn ∼ 0.5.In what follows, this run, as well as its counterpart withpower-law size distribution, will be used as our standardnamical model for producing particle distributions for tphotometric calculation of the brightness asymmetry. Nhowever, that the strength of the wake structure and thesulting amplitude of asymmetry are sensitive to all thmodel parameters. For example, using more elastic impwould weaken the wake structure, which could be compsated by assuming a largerτdyn, or a somewhat largerρ.On the other hand, a significantly larger internal denssay ρ = 900 kg m−3 corresponding to the solid ice de

sity, would increase the tendency of wakes to degradegravitational aggregates due to pairwise sticking of pacles(Salo, 1992, 1995; Karjalainen and Salo, 2004),1 whichwould reduce the resulting brightness asymmetry. The abchosen parameter values thus represent one possiblebination which leads to a strong asymmetry at the A-rdistance.

The system followed inFig. 2 starts with a uniform dis-tribution of particles, with velocity dispersion correspondto Q ≈ 1. After one orbital period the system has alreadivided into small local condensations, with a radial scof roughlyλcr/2. These condensations rapidly heat the stem, so thatQ reaches about 2 after five orbital periodSimultaneously the scale of the wakes increases. Duringlater evolution a statistical steady-state between collisiocooling and viscous and gravitational heating is achiev

1 In terms of the often employed gravity parameterrp (see, e.g.,Ohtsuki,1993; Salo, 1995), the summed radius of a particle pair normalizedtheir mutual Hill radius, the adoptedsimulation parameters correspondsrp = 1.18. Sincerp exceeds unity, no gravitational sticking of individuparticles is expected, even in the case of zero relative velocity.


ug of

Fig. 3. Dynamical wakes viewed from different directions, providing more quantitative comparison of how the rings might appear locally at the longitudesof expected minimum (left) and maximum (right) brightnesses. Three different elevation angles (B) are illustrated. For a high elevation angle (upper frames),the rarefied regions are roughly equally visible at both longitudes. However, when B is reduced (middle row), a larger portion of rarefied regions is visible inthe left-hand frame than in the right-hand frame, indicating azimuthal asymmetry. For very small B, the reflecting areas again become nearly equal, de toimperfect alignment of wakes. Particle positions are taken from the run ofFig. 2 at 42 orbits after the beginning of the simulation. Note that the shadinparticles in this plot is not related to their apparent brightness.

dis-ex-

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so that the average properties of the system (velocitypersion, wake structure) remain constant, although theyhibit large fluctuations due to the emergence and dissoluof individual strong wakes. More examples are provideSalo (1995), where several simulations of wake structuwere displayed for various surface densities and saturnotric distances (see alsoDaisaka and Ida, 1999; Ohtsuki aEmori, 2000). These examples illustrate quite clearly tstrong dependence of wake amplitudes on distance, wwill be addressed in more detail in our future study.

As seen inFig. 2, the wakes represent considerablehancements in density, often having almost empty gapstween them. In the vertical direction the thickness ofwakes is comparable to their radial width. In order tolustrate how such formations may affect the brightnesthe ring,Fig. 3displays one snapshot of the simulation stem as it would appear from different viewing directioEspecially for intermediate elevation angles, the fractionrarefied gaps that are visible depends strongly on the ving longitude, supporting the simple sketch inFig. 1.

In the case of a size distribution, the density contrasttween wake/inter-wake regions is reduced, as the smallticles tend to have a much more uniform distribution thanlargest particles. This is illustrated inFig. 4, for a power-lawsize distribution of the form

(7)dN/dr ∝ r−q, rmin < r < rmax, q = 3.

-

-

-

We usermax/rmin = 10 and choose the upper size lim(rmax = 4.25 m) in a way which yields the same surfadensity as the model for identical particles. This distributis considerably narrower than the size distribution inferfrom Voyager radio-occultation measurements(Marouf etal., 1983), with rmax/rmin ∼ 500 andrmin ∼ 1 cm. How-ever, the observational size range is fairly model-dependfor exampleFrench and Nicholson (2000)derived a ranger ∼ 30 cm–20 m, based on the amount of forward diffraclight estimated from comparison of Voyager and groubased occultation experiments. Our truncated model shreveal the qualitative effects of a size distribution, whkeeping the required number of particles computationmanageable.

The simulation snapshots in the upper rows ofFig. 2 in-dicate that the instantaneous direction of wakes is ravariable, making it difficult to estimate very accurately thaverage orientation of wakes. The same is true for the 2Dtocorrelation functions for individual snapshots, calculaas in Salo (1995), displayed in the bottom row ofFig. 2for the same instants as for which the snapshots of parpositions are displayed in the middle row. However, whaveraged over longer time spans (Fig. 5), the characteristic structure of the wakes becomes evident: in the innermdense part, the average angle is about 25–30 with respectto local tangential direction, but decreases toward 15–20when the tails of the autocorrelation function are conside


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s theates

Fig. 4. A typical snapshot from our standard size distribution simulation model. The parametersτdyn, ρ, a, andεn(vn) are the same as for the model of identicparticles inFig. 2, while here the power-law size distribution is used, with the indexq = 3, extending fromrmin = 0.425 m tormax = 4.25 m, yielding thesameΣ as before. In the left-hand frame all particles projected to the central plane are shown, while the other two frames display the particles withr 2rminandr rmax/2, respectively. The size of the simulation system corresponds again toLx × Ly = 4λcr × 4λcr = 305 m× 305 m, and the number of particleis N = 17,500.

Fig. 5. On the left: the time-averaged 2D autocorrelation function for the run ofFig. 2. The contours display densities at 1.1, 1.25, 1.5, and 2.0 timeambient density. Solid lines mark directions corresponding to 15 , 20 , 25, and 30 with respect to the tangential direction, while the dashed curve indicthe loci of maximum density. On the right: the same is shown for the dynamicalsimulation with a size distribution, using only the locations of the particleswith r > 1.34 m, comprising one half of totalτdyn.

e ofsedid-

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urey, as

irec-esrin-truc-the

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In the case of a size distribution, the average pitch anglwakes is larger than that for identical particles. As discusbelow, the 2D-autocorrelation function is useful in proving qualitative interpretation for the longitude of minimumbrightness found in actual photometric modeling. Howeno quantitative results can be derived, as the time-averautocorrelation function contains no information of the vtical extent nor the mutual spacing of the adjacent wake

2.3. Photometric modeling of wakes

As described in the Introduction, the local wake structoffers a possible explanation for the observed asymmetr

the effective scattering surface area depends on the dtion of viewing with respect to the orientation of the wak(Figs. 1 and 3). The observed asymmetry can, at least in pciple, be used to probe the density contrast due to wake sture, and therefore also the particle properties affectingformation and strength of wakes (internal density and eticity of particles). However, detailed photometric modeliis needed for quantitative analysis, addressing the effecmultiple scattering, large volume density, and various obvation geometries on the brightness variations corresponto density variations. Also, the picture is complicated byfact that the wake structure is most pronounced in the la


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particles, whereas the surfacearea is dominated by the smaparticles.

In Paper I a Monte Carlo method for determiningphotometric properties of arbitrary particle fields wasscribed. The method is based on following a large numof photon paths through successive scatterings on thdividual particles until their final escape from the partisystem. When combined with an indirect Monte Carlo tenique (adding the contributions of individual scatteringsthe brightness at a given viewing direction), very goodcuracy can be obtained with a computationally managenumber of photons. In order to apply the method to meling the asymmetry, we assume that the particle positobtained from dynamical wake simulations sample the tcal particle distributions in eachring longitude, in a coordi-nate system aligned with the local radius vector. For a gviewing geometry, each longitude of the rings has its ownlumination and viewing direction for which the brightnesscalculated.

Figure 6illustrates the coordinate system used to moasymmetry. We denote byθ0 andθs the ring-observer anring-Sun longitude differences, measured in the ring pin the direction of orbital motion, from the sub-observer asub-solar points, respectively. Correspondingly,B0 andBs

stand for the elevation of the observer and Sun with res

Fig. 6. Coordinate system used in modeling of the asymmetry.Rv and Rrepresent the saturnocentric radiusvectors of the observer and the studiring region while D = Rv − R. θ0 and θs stand for the longitude of thring region with respect to the sub-observer and sub-solar directions,counterclockwise direction (the direction of orbital motion), whileφ0 andφs denote the observer and solar azimuths with respect to the simux-axis, aligned with the local saturnocentric radius vector.φwake stands forthe average angle of the wakes with respect to the local tangential dire(the simulationy-axis), andα0 andαs denote the azimuthal viewing anillumination angle with respect to the wakes. For clarity, the elevation anof the observer and Sun,B0 andB ′, are not included in the graph.

-

t

to the ring plane in a saturnocentric coordinate system. Insimulation coordinate system, with thex-axis aligned alongthe local radius vector, the illumination and viewing diretions are specified by (φs,B

′) and (φ0,B), respectively. Foillumination by the Sun,B ′ = Bs andφs = −θs to a goodaccuracy. Similarly, in the case of large observing dista(Earth-based observations)B = B0 andφ0 = −θ0. However,in some Voyager observations made from a close rangetance less than a few million kilometers), there is a slidifference. LetR denote the saturnocentric radius vectorthe studied ring region andRv that of Voyager (in a saturnocentric coordinate system aligned with the coordinaxis of the dynamical simulation),

R = a,0,0,(8)Rv = cosθ0 cosB0,−sinθ0 cosB0,sinB0∆,

where∆ = | Rv| is the distance of the spacecraft from Surn’s center. The spacecraft direction vector with respect tthe ring region is

(9)D = Rv − R ≡ cosφ0 cosB,sinφ0 cosB,sinB| D|,from which φ0 and B can be calculated as a functionB0, θ0, anda/∆. Whena/∆ is small, we have

B = B0 + ∆B ≈ B0 + (a/∆)sinB0 cosθ0,

(10)φ0 = −(θ0 + ∆θ0) ≈ −(

θ0 + a

∆

sinθ0

cosB0

).

In some of the close-range Voyager images analyzeDones et al. (1993), ∆/a ≈ 10, so that the maximum shi|∆θ0| reaches 5, large enough to be significant in compason of simulations with observations.

The azimuthal viewing directionα0 with respect to themajor axis of wakes is related to the average pitch aφwake of wakes with respect to the local tangential directby

α0 = φ0 − 90 − φwake

(11)≈ −(θ0 + ∆θ0 + 90 + φwake).

The wakes are viewed most closely along their long axα0 ≈ 0 or 180, corresponding to

θ0 = 270 − φwake− ∆θ0 or

(12)θ0 = 90 − φwake− ∆θ0.

For a distant observer,∆θ0 = 0, so that wakes appear edgon for the longitudesφwakebefore the ansae, as was assumin Fig. 1. For a nearby observer these longitudes movewardθ0 = 270 and away from 90, as indicated by the sigof ∆θ0 in Eq. (10). The azimuthal illumination direction owakes is similarly

αs = φs − 90 − φwake

(13)= −(θ0 + ∆θs + 90 + φwake),

where∆θs = θs − θ0 is the difference in sub-solar and suobserver longitudes. Illumination roughly parallel to wak


show

how

Fig. 7. The upper frame shows the individual brightness vs. ring longitude curves calculated for 40 simulation snapshots of the identical particle run ofFig. 2. Monte Carlo modeling assumes a Lambert phase function and particle albedoA = 0.5, observing elevationB = 12.6 and Sun elevation ofB ′ = 8,and ∆θs = θs − θ0 = 0. Infinite viewing distance is assumed. Altogether 25 different ring longitudes were studied, withNphot = 2 × 104 photons foreach longitude and each snapshot. The thick solid curve is the averageI/F curve for all snapshots, while the dashed curves display the±1 − σ standarddeviation of the individual curves. In the lower frames the symbols connected with lines indicate the modeled longitude of the brightness minimum (left) andthe brightness asymmetry amplitude (right) 2(Imax− Imin)/(Imax+ Imin), for each snapshot separately. For comparison, the curves without symbolsquantities characterizing surface number density autocorrelation functions: thelongitude of minimum is compared with 270 − φauto, whereφautodenotes themajor axis direction of the autocorrelation function at the contour level of 1.5 times the ambient density. Similarly, the brightness amplitude is compared to themaximum value of the autocorrelation function aty/λcr = 1, multiplied by an arbitrary factor of 0.3. Note that the purpose of these comparisons is to sthe qualitative dependence between the density structure and the photometric behavior, not to yield any quantitative estimates.

sedring

tlyunc-opri-py

denthus

urveed atre,

loyedimel pe-ex-sely

rtieskes

el,ap-

atored-ctureughatterverylednedns,

thus occurs at

θ0 = 270 − φwake− ∆θs or

(14)θ0 = 90 − φwake− ∆θs.

Two different single scattering phase functions are uin the photometric calculations, determining the scatteprobability as a function of phase angleα,

(1) a Lambert phase function

(15)PL(α) = 8

3π

[sinα + (π − α)cosα

],

(2) a power-law phase function

(16)Ppower(α) = cn(π − α)n,

defined in Dones et al. (1993): in particular, we adopn = 3.092, cn = 0.153, in which case this function closeapproximates the phase curve of Callisto. Both phase ftions describe strongly backscattering particles, as apprate for macroscopic ring particles, the implied anisotroparameter (see, e.g.,Dones et al. (1993)or Paper I) beingg = −〈cosα〉 = −0.44 and−0.55, respectively.

2.4. Preliminary tests

Since the simulated wakes are transient, time-depenfeatures, the strength and direction of the wakes and

t

also the shape of the modeled brightness vs. longitude cmay be expected to show a large scatter when evaluatdifferent time steps of the dynamical simulation. Therefoaverages over snapshots from many time steps are empin the photometric calculations. As long as the sampled tsteps are separated by at least the order of one orbitariod, no correlation with the previous particle positions ispected. In this sense the time-averaging corresponds cloto the spatial averaging taking place when global propeof ring patches much larger than the size of individual waare observed.

Figure 7 displays a representative photometric modshowing the brightness vs. longitude for individual snshots taken from the dynamical run ofFig. 2. The specificgeometry studied corresponds to theDones et al. (1993)ob-servations of reflected light nearthe eastern ansa, except ththe difference in Sun and Voyager azimuths has been ign(∆θs = 0; also∆θ0 = 0). Only those portions of the dynamical run are used when the steady-state wake struhas been achieved (after about 10 orbital periods). Althothe brightness vs. longitude curve shows considerable scbetween individual time steps, the average curve has asmooth form. Also shown are the longitude of the modebrightness minimum, and the asymmetry amplitude, defiin this paper as the full amplitude of brightness variationormalized by the mean brightness,

(17)Aasy= 2(Imax− Imin)/(Imax+ Imin),


-ngthingtherela

rec-the

sitye

ofsing

ex-al

theri-

Pa-sing

emtsam-me

lated, astheugh

rveddy-

ori-endsedions is

loren-

into

eded,i-try,

heesnfor-ls.

-all-

etry

lyofve-incer-

we,r-

the, inex-

undeldpar-gis isint ofd inho-idesar-

d in

measured for each time separately. These photometric quantities are also compared with the wake direction and streestimated from the autocorrelation function correspondto each snapshot. Clearly, there is a similar trend inphotometric values and those characterizing the autocortion function. The minimum of the averageI/F curve is atθo ≈ 249, corresponding to a wake orientationφwake≈ 21with respect to the tangential direction. The average dition of the autocorrelation function depends naturally ondensity level studied: comparison toFig. 5 indicates that thephotometric behavior seems to be related to the low dentails of the autocorrelation function (21 corresponds to thdirection of contours of the autocorrelation plot at levelsabout 1.2 times the average density), which is not surprifor the low viewing elevation studied here (B = 12). Forlarger viewing elevations the brightness minimum can bepected to move further away from the ansa, since the centrparts of the wakes with largerφwake will then be more easilyvisible.

Tests with different numbers of photon,Nphot, indicatethat the main uncertainty of the derived amplitude andlongitude of minimum brightness results from the time vaation in wake structure, rather than from the 1/

√Nphot un-

certainty intrinsic to the Monte Carlo calculation (seeper I). For example, the average curves obtained when uonly 103 instead of 104 photons for the sameNpos = 40particle snapshots as inFig. 7 yield practically the samamplitude (within one percent) and longitude of minimu(within 0.1). The error limits shown in all subsequent ploare therefore calculated from the standard deviation ofplitudes and minimum longitudes obtained for different tisteps, divided by

√Npos, ignoring any formal Monte Carlo

errors. The amplitudes themselves, however, are calcufrom the maximum and minimum of the combined curveswell as the minimum longitudes (not from the mean ofamplitudes and minima for individual snapshots, althothe difference is small).

In order to compare the simulated wakes to the obseasymmetry, we need to make sure that the size of thenamical calculation region is large enough to yield anentation and amplitude of wakes which does not deptoo sensitively on the periodic boundary conditions impoin dynamical simulations, i.e., the size of the calculatregion. Since the range of included gravitational forcelimited to a region with a radiusrgrav = minLx/2,Ly/2around each particle (seeSalo, 1995), a larger dynamicasimulation region takes into account the interaction of mdistant wakes (Lx andLy denote the full radial and tangetial extent of the calculation region, respectively). InSalo(1995) this calculation region dependence was studiedterms of the autocorrelation function, which was foundbe nearly constant once the size of the region exceaboutLx × Ly = 4λcr × 4λcr (see alsoToomre and Kalnajs1991). Figure 8shows this directly in terms of the minmum longitude and amplitude of the calculated asymmefor dynamical runs similar to those inFig. 2, except for a

-

Fig. 8. The dependence of modeled brightness asymmetry on the size of tdynamical simulation region. Different runs with the same parameter valuas inFig. 2 are compared, for different sizesof the square-size calculatioregion (and correspondingly different numbers of particles). Parametersthe photometric calculations are the same as inFig. 7. The error bars correspond to±1 standard deviation of the values calculated for individuasnapshots, divided by

√Npos, whereNpos= 40 is the number of snapshot

different size of the calculation region. Indeed the direction of the minimum is found to be rather constant forruns withL = Lx = Ly 4λcr, in agreement with the autocorrelation function being almost constant. The asymmamplitude, however, seems to decline slightly withL/λcr

even beyond this. The effect is fairly small and is liketo result from the fact that the total gravitational effectwakes is slightly overestimated when only a few first walengths are included in the force calculation. However, sfor a fixedτdyn andΣ the particle number grows in propotion to L2, the CPU time consumption in such largeL/λcr

dynamical runs becomes very large. As a compromiseperform all our models with 4λcr by 4λcr simulation regionswhich are likely to yield correct amplitudes within a few pecent.

Finally, we also performed some additional checks ofphotometric method when applied to modeling of wakesaddition to the tests already described in Paper I. Forample, the results of photometric calculations were foto be unaffected if the size of the modeled particle fiwas doubled, by including the appropriate copies of theticles implied by the periodic boundary conditions amonthe actual simulation particles themselves (note that thdifferent from changing the size of the original regiondynamical simulations). Since any error in the treatmenboundaries is likely to become much more pronouncethe case of inclined wakes than in the case of spatiallymogeneous particle systems studied in Paper I, this provan additional confirmation of the correct treatment of pticle replicas in terms of image photons, as describePaper I.


mi-oy-alsoetry

sim-onsforing

hengehasli-ingminismor

ownfht-odelfor

ve).at-ter-e towithe

outen

ount

t-des

av-

c-ym-

3. Comparison to observations

In this section we compare the predictions of dynacal simulations to the azimuthal asymmetry seen in Vager observations of reflected and transmitted light. Wedemonstrate that the tilt-angle dependence of the asymmamplitude found in ground-based observations has aple interpretation in terms of wakes. For these comparisthe two standard dynamical runs of the previous sectionidentical and size distribution model are used for producthe photometric models.

3.1. Brightness asymmetry in reflected and transmittedradiation

In Dones et al. (1993)the azimuthal dependence of tA ring brightness was analyzed from a series of close-raVoyager 2 images of the eastern ansa. Due to the small pangle (α = 13) the contribution from Saturn shine is neggible for strongly backscattering ring particles. Accordto these observations the brightness has a well-definedimum nearθ0 = 249. In addition the observed minimumasymmetric, in the sense that the brightness increasesrapidly for θ0 < 249 than for θ0 > 249. Figure 9com-pares the modeled asymmetry with the observations shin Fig. 19 inDones et al. (1993). Except for the difference oa few degrees in the minimum longitude, the overall brigness level and its variations are well reproduced in the mwith identical particles, when using reasonable values

e

-

e

particle albedo and phase function (A ≈ 0.43 for a power-law phase function approximating Callisto’s phase curAlso shown in the plot is the contribution due to single sctering alone, indicating that the shape of the curve is demined by single scattering, and is thus not very sensitivthe assumed phase function: a Lambert phase functionA ≈ 0.65 would yield practically identical results. On thother hand, our model with a size distribution yields aba factor of two smaller amplitude, and the minimum is evfurther away from the ansa than is observed.

The modeled minimum brightness longitude inFig. 9dif-fers from that inFig. 7, which yielded 249 in the case ofequal illumination and viewing azimuths, for the sameB

andB ′. This shift follows from the∆θs = 10.2 difference inthe Sun and observer longitudes,which is taken into accin the model curve ofFig. 9. Also the shift∆θ0 ≈ −2.2 isincluded, seeEq. (10). In fact, the modeled minimum brighness seems to fall very close to the mean of the longitucorresponding to the viewing and illumination along theerage long axis of wakes, which according toEqs. (12) and(14) indicates

θ0 = 270 − φwake− 1

2∆θs − 1

2∆θ0

(18)≈ 270 − 21 − 5.2 − (−1.1) ≈ 245,

for an intrinsic wake pitch angleφwake ≈ 21. This differ-ence in illumination and viewing directions may also acount for the asymmetric shape of the minimum. This asmetry is present in the model curve ofFig. 9, although it is

te

thwith

Fig. 9. Observed shape of the asymmetry at the distance of 128,000 km, as seen in Voyager 2 images at phase angleα = 13 , with Voyager elevationB = 12.6and Sun elevation of 8 (Dones et al., 1993). Also shown is the modeled curve for the dynamical simulation ofFig. 2. We have taken into account the finidistance of the spacecraft from the rings (∆ = 3.3×106 km), and usedB0 = 12.8 as the Voyager latitude with respect to the ring plane whileθs − θ0 = 10.2 ,yielding the same values forB andα as listed for variousθ0 values in Table III inDones et al. (1993). For this case the maximum shift in viewing azimudue to the proximity of the spacecraft to the rings amounts to∆θ0 ≈ 2.2 . A power-law phase function approximating Callisto’s phase curve is used,A = 0.43. The single-scattered contribution is shown separately. Both the standard dynamical models for identical particles and for a size distribution are used.


orhtness

nd

.

ther

Fig. 10. The modeled brightness vs. ring longitude for different solar directions with respect to the observer,θs − θ0, using the standard dynamical model fidentical particles. The solar elevation angle is fixed atB ′ = 8 and infinite viewing distance is assumed. The upper solid curves show the modeled brigfor observing elevationB = 12.6 , while the lower solid curves correspond to transmitted radiation atB = −29. A Callisto phase function withA = 0.43is used. The dash-dotted curves stand for the contribution of Saturn-shine, calculated with Barkstrom’s law(Barkstrom, 1973), using numerical factorsinterpolated from the values tabulated in Table V inDones et al. (1993)(average over zones and belts in blue, Saturn’s oblateness is ignored). The upper alower curves again correspond toB = 12.6 andB = −29, respectively. The dashed curves (above the solid curves, as seen in frames forθs − θ0 = −90 ,−140 and 60) indicate the combined solar and Saturn-shine. The long (short) arrows extending from the upperx-axis indicate the ring longitudes wherethe wakes are viewed (illuminated) roughly perpendicularly,α0 = 0 (αs = 0), while the arrows extended from the lowerx-axis indicate the solar longitudeThe points in the frame forθs − θ0 = 10 denote observations byDones et al. (1993)(labeled by “D”) and those in the frame forθs − θ0 = −90 standfor observations ofFranklin et al. (1987)(labeled by “F”), both with and withoutSaturn-shine (data points fromFranklin et al. (1987), Fig. 5d, scaled by anarbitrary factor). The geometry is also depicted in the lower frames: the filled circles stand for the local minima in modeled brightness, while long and shortticks mark the longitudes corresponding to viewing (α0 = 0) and illumination (αs = 0) along the wakes, respectively. Open symbols stand for the osymmetrically located minima.

note

mtlyd on

nthe

e in

ger

bellu-ar-

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ve, in

fonal

a,the

el

hane aforht

somewhat weaker than in the observations. By contrast,that the brightness profile inFig. 7 for ∆θs = 0 is com-pletely symmetric. The larger shift of the minimum frothe ring ansa for the size distribution model follows direcfrom the larger pitch angle of wakes, as anticipated basethe autocorrelation plots inFig. 5.

The effect of the illumination direction on the locatioof the minimum, as well as on the asymmetric shape ofminimum, becomes more evident inFig. 10, where the up-per curves display the modeled asymmetry vs. longituda series of runs again withB ′ = 8 andB = 12.6 as in theDones et al., Fig. 19, observations, but with varied∆θs . Ineach frame the arrows indicate the directions correspondinto α0 = 0 or 180 (long arrows extending from the uppx-axis) and toαs = 0 or 180 (short arrows extending fromthe upperx-axis). Clearly, both these directions seem torelated to a local minimum brightness, the influence of imination being stronger on the combined minimum. In p

ticular, comparing the frames with∆θs = 60 and−40, onesees that the brightness gradient is in both cases steepthe side corresponding more closely to parallel illumination.Most likely this follows from the fact that observations haB ′ < B (Dones, 2003, personal communication). Indeedthe case of larger solar elevation,B ′ > B, the asymmetry othe minimum should be reversed, as confirmed in additiexperiments withB ′ = 20 instead of 8. For well-separatedillumination and viewing,∆θs = −90, the minimum be-comes very wide and even divides into two local minimone being close to the parallel illumination of wakes andother close to the parallel viewing.

Also shown inFig. 10 (lower curves) is a set of modcurves for transmitted light withB ′ = 8, B = −29. Al-though the overall brightness level is now much lower tin reflected light, the relative brightness variations havsimilar longitude dependence. Also shown in the plot,∆θs = −90, is the observed curve in transmitted lig


eed,hal

Fig. 11. A more detailed comparison with the observed asymmetry in transmitted light. Symbols denote theFranklin et al. (1987)measurements of thVoyager 2 image FDS 44169.26, fora = 127,000 km,B = −28.8 , B ′ = 8.1 , θs − θ0 = −90 , traced from their Fig. 5d. The SLS longitude system is uswhere SLS= 92.4 − θ0. Besides the total observedI/F , the amount of Saturn-shine estimated inFranklin et al. (1987)is displayed. Model curves (for botSaturn-shine and Saturn-shine+ transmitted solar radiation) are calculated using a Callisto phase function withA = 0.43, both with the standard dynamicrun for identical particles and for a size distribution. Original observations given in digital units are scaled by multiplication with an arbitraryfactor.

lsoini-

me-in-

alwith

eenrge-inlyra-

rsalandyareong

of

atgi-s

,

no(for

,

m-

m-seeanstonfec-forrom

ms

(from Fig. 5d in Franklin et al. (1987), with an arbitraryintensity scale). The contribution from Saturn-shine is ashown (dashed lines), since it is important near the mmum analyzed byFranklin et al. (1987). A more detailedcomparison to their observations is shown inFig. 11. Alto-gether, although the modeled fractional amplitude is sowhat smaller than the observed one, the location of the mimum is well accounted for. Again, the model with identicparticles agrees better with observations than the modela size distribution.

The fact that a very similar longitude dependence is sin both reflected and transmitted light is related to the laelevation angles studied inFig. 10. In this case the brightness in transmitted light seems also be determined maby the fraction of scattering area, just as for the reflecteddiation. However, as discussed inFranklin et al. (1987), forsmallα and low elevation angles one would expect a reveof brightness minima and maxima between transmittedreflected radiation, basically because only the low-densitinter-wake regions transmit any light at all, and theseshadowed by the wakes except for viewing very nearly althe long axis of the wakes. Indeed, when the modelsFig. 10are extended to smallerB and |B ′|, such a reversebehavior is obtained (Fig. 12). For example, whenB = 8andB ′ is slightly negative andθs ∼ θ0, there is amaximumin transmitted light nearθ0 = 250, which changes into alocal minimum when|B ′| > 4. For still lower viewing el-evation (B = 3) the local maximum persists for somewhlarger|B ′|, andI/F becomes almost independent of lontude forB ′ = −16 if θs − θ0 = 0. In this case the effect

of even very small non-zeroθs − θ0 are very pronouncedthe minimum and maximumI/F being very sensitive tothe exact geometry. For comparison,Fig. 12 also displaysthe reflected radiation at similar low elevations, showingqualitative changes in comparison to larger elevationsB = 3, B ′ = 2 the largerI/F for θs − θ0 = 0 in compari-son to−10 or 10 reflects the opposition brightening).

3.2. Brightness asymmetry at different elevation angles

Ground-based observations(Lumme and Irvine, 1976a1979b; Lumme et al., 1977; Thompson et al., 1981)indi-cate that the asymmetry amplitude increases whenB ≈ B ′decreases from 26 to 12, while for still smaller elevationsthe amplitude again decreases(Thompson, 1982). Also thewidth of the asymmetry minimum shows signs of becoing more narrow for smaller elevations(Lumme and Irvine,1976a). This qualitative tilt-angle dependence of the asymetry is naturally accounted for by the wake model (Fig. 3). Since the vertical extent of wakes is smaller ththeir radial separation, for largeB the scattering area varieonly little with the azimuthal viewing angle with respectthe wakes’ major axis direction. With decreasing elevatiothe shielding of rarefied regions becomes increasingly eftive for nearly perpendicular viewing azimuths. However,very small elevations the gaps are completely hidden fany direction, since the wakes are not perfectly straight.Fig-ure 13illustrates these effects more quantitatively, in terof the modeledI/F vs.θ0 for different elevations.


s

d.

e,

how

Fig. 12. Modeled asymmetry in transmitted and reflected light for low elevations. In the upper frames the brightness vs. ring longitude is shown forB = 8 andB = 3, for different solar elevations. A Lambert phase function withA = 0.5 is used. For sufficiently small|B ′| and|B| the ring longitude where viewing ialong the wakes may appear as a brightness maximum in transmitted light, contrary to the case of transmitted light for larger elevation angles (seeFig. 10), orfor the case of reflected light (lower frames). Also indicated is the influence of small changes inθs − θ0. The dynamical model for identical particles is use

Fig. 13. The dependence of asymmetry on tilt angle. On the left, brightness vs. elevation is shown for photometric simulations performed for various elevationangles. The dynamical simulation data correspond to the standard model with identical particles. Photometric runs are performed forα = 0 andB = B ′,using a Lambert phase function andA = 0.5. In the middle frame the asymmetry amplitude (Eq. (17)) is shown, both for the runs shown on the left framas well as for corresponding models using the standard size-distribution simulation. Large symbols with error bars show observations byThompson (1982)(Table 3, blue light forα < 1). The right frame shows the longitude of modeled minimum brightness with respect to ansa; the axis labels in the right sthe same in terms of ring longitudeθ0.


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In general, comparison to ground-based observat(Thompson, 1982)shows a good qualitative agreementthe asymmetry amplitude, the observations falling betwthat implied by photometric models for our standard dynaical run with identical particles and with a size distributioIn addition, in agreement with observations the width ofmodeled minimum is reduced for small elevations, as mbe expected since the wakes must be viewed more andprecisely along their long axis for the gaps to be visible. Tfigure also indicates the large reduction of the amplituobtained in the case of a size distribution, in agreementthe models shown inFig. 9, resulting from the tendency osmall particles to fill the rarefied gaps to some degree.direction of modeled minimum longitude shows also a cltendency of being shifted away from ansa for the largerevations. A likely explanation is that for higher elevatiothe brightness is becoming increasingly dominated byinner parts of the wakes: the autocorrelation plots ofFig. 5indicate that the central parts of wakes make on averalarger angle with respect to the tangential direction thantails which dominate for more shallow viewing directionAlso, there is a clear difference in the longitude of minimbrightness between models with identical particles and wa size distribution, the latter models yielding a minimufurther away from the ansa. This is also consistent withdifferences in the autocorrelation function.

3.3. Role of multiple scattering

As discussed in relation toFig. 9, the modeled asymmetry is dominated by single scattering. Nevertheless,higher orders of scattering exhibit a rather similar brightnvs. longitude dependence, with even higher fractionalplitude. This is illustrated inFig. 14, for the elevation angleof 10 yielding near-maximal asymmetry amplitude; pratically identical behavior is obtained for both Lambert aCallisto phase functions. Since the fractional contributof multiple scattering increases with albedo, the expeasymmetry amplitude will also increase. A similar concsion results fromLumme and Irvine (1979a)model based onmultiple scattering in optically thick blobs. However, sinthe overall contribution of multiple scattering is fairly smin our models (seeFig. 9), the expected magnitude of theffect is not very large. This is shown inFig. 15, indicatingthat for the geometry ofFig. 14 the amplitude varies by amost about 15% for different values of albedo. For a Clisto phase function the maximal difference is smaller thanfor a Lambert phase function,because its more backscatteing character leads to a larger relative contribution of sinscattering at small phase angles. This model result, thcrease of asymmetry amplitude with albedo, is qualitativconsistent with the observations reported inLumme et al.(1977). According to their observations for an elevation agle of about 16, the amplitude of asymmetry in red ligh(effective wavelengthλeff = 590 nm) appeared even 50larger that in blue light (λeff = 415 nm). Such a dependen

e

Fig. 14. Comparison of asymmetry profile in first (k = 1) and sec-ond (k = 2) order scattering. In each case, the curve is normalizedImean= 1

2 (Imax+ Imin), calculated for the order studied. For clarity, thigher orders are not shown: ink = 3 the amplitude is slightly smallethan that fork = 2, while for k 4, the amplitude is even bigger thafor k = 2. Photometric simulations for the identical particle model, wB = B ′ = 10, zero phase angle are shown; a similar stronger amplitudhigher order scatterings is also seen for other elevation angles.

Fig. 15. Dependence of asymmetry amplitude on albedo and phasfunction. Photometric simulations for the identical particle model, witB = B ′ = 10, are shown, both forα = 6 and α = 0 . The increasedoverall amplitude of a Lambert phase function in comparison to a Calphase function follows from the larger contribution of multiple scatterin the former case, which, according toFig. 14, exhibits a stronger relativevariation than the singly-scattered component. On the other hand, theduced opposition amplitude reflects the decrease in the variations osingly-scattered flux, taking place ina similar fashion regardless of thphase function.

on wavelength was confirmed intheir later analysis including smearing corrections(Thompson et al., 1981), althoughits exact magnitude was difficult to estimate due to obsetional uncertainties.

Figure 15also compares the amplitudes for phase anα = 6 and α = 0, indicating that for both studied phas


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functions the asymmetry amplitude is reduced by about 1at opposition. Since this reduction is not sensitive to thesumed albedo, it must be a single scattering effect. Nopposition, the singly scattered flux increases, due toduced mutual shadowing between particles (see Paper I areferences therein). The reduced asymmetry amplitude indicates that this brightening is more important at brightnminima, compared to that in longitudes yielding a brightnmaximum. A likely reason is that the opposition brightenbecomes stronger with the path optical depth (see Paphaving thus the largest importance in viewing along the laxis of the dense wakes (corresponding to minimum briness). The reduction of opposition amplitude is also qitatively consistent with the observations inLumme et al.(1977), who found significantly reduced asymmetry amptude in their observations forα = 0.11 in comparison toα ∼ 1. Again, this result was at least marginally confirmin the later analysis inThompson et al. (1981).

3.4. Optical depth

Our dynamical models use fixedτdyn = 0.5, defined asthe total surface area of particles divided by the area ocalculation region,

(19)τdyn =rmax∫

rmin

πr2 dN(r),

whereN(r) is the number density of various-sized particlOn the other hand, the photometric optical depth alongiven direction is defined by

(20)τpath= − log(fpath),

wherefpath denotes the fraction of light passing throughlayer in that direction. In the case of a geometrically thi

,

spatially homogeneous, low-volume-density ring (D → 0),τpath is proportional to the geometric path length, so thatperpendicular photometric optical depth,

(21)τphot= τpath(B)sinB,

is independent of elevationB, and equalsτdyn. In Paper I wefound that generallyτphot> τdyn for spatially homogeneousystems, the enhancement increasing with the volumeing factorD, consistent withEsposito (1979) and Peltoniemand Lumme (1992). On the other hand,τphot was found to bepractically independent ofB as long as the vertical thickneof the layer was at least a few particle diameters.

In the case of spatially inhomogeneous rings,τphot de-rived viaEq. (21)is expected to depend onB, as suggestebyHämeen-Anttila and Vaaraniemi (1975)in their ring mod-els including narrow ringlets separated by low density gIn such a case, the amount of light passing through thedepends mainly on the fractional area and optical deptrarefied gaps. The same is also true in the case of grwakes. In addition, since the visibility of gaps depends ondirection of the light ray with respect to the local wake dirtion, the derivedτphot will depend on bothB andθ0. This isillustrated inFig. 16, showing theτpath(B, θ0)sin(B) profilesfor the same models whose brightness was shown inFig. 13.Indeed, the ring longitudes which correspond most cloto the minimum brightness (direction of wakes) appearas minima forτphot if defined as above inEq. (21). More-over, this minimum value increases monotonically withB,due to the sinB factor, asτpath mainly measures the fractioof gaps throughfpath and varies only little withB. How-ever, τpath(B, θ0)sin(B) does not go to zero forB → 0,due to the finite vertical extent and the imperfect alignmof the wakes, and the non-zero particle density in the gSimilarly, the maximumτphot corresponds to perpendiculviewing with respect to wakes. The maximumτphot varies

al

Fig. 16. The dependence of photometric optical depth on the viewing direction. At left the optical depth profiles as a function of ring longitude are shownfor different elevations, using the standard dynamical run for identical particles. In each case the path optical depth is converted to normal optical depth inthe manner typically assumed for a spatially homogeneous low volume density ring (τphot = τpathsinB). In the middle, the maximum and minimum opticdepths are shown as a function of elevation, as well as the optical depth corresponding to the ansa. At right the same is shown for the standard size distributionmodel. For comparison the optical depths are also shown for similar modelsusing dynamical data from runs where self-gravity is not included (‘uniform’). Ineach caseτdyn = 0.5.


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less withB, except that it starts to rise for smallB. This in-dicates thatτpathacross the wakes increases with longer plength, although not in the same manner as for a unifring. At largeB the minimum and maximumτphot approacha common value, which, however, is clearly smaller thτdyn. In the case of a size distribution, the overallτphot islarger than for identical particles (but still less thanτdyn),due to the small particles having a more uniform spatialtribution, reducing the effect of rarefied regions. Also,sensitivity toB andθ0 is reduced, but remains still significant.

For comparison,Fig. 16also shows theτphot vs.B depen-dence in the case of dynamical runs similar to our identparticle and size distribution models, except that the sgravity of particles has been ignored (labeled as ‘uniformIn these modelsτphot exceedsτdyn by about 10%, consistent with their significantly non-zeroD (the central planefilling factors are about 0.125 and 0.145 for the model widentical particles and the model with a size distribution,spectively) (see Paper I).

3.5. Inverse tilt-effect

Ground-based observations indicate that the brightof the B ring increases (“tilt-effect”) with elevation anglby about 25% betweenB = B ′ = 6 and 26, whereasthe brightness of the A ring is constant within about 1(Lumme and Irvine, 1976b). In Lumme et al. (1983)the Bring tilt effect was interpreted in terms of increased mtiple scattering with increasing elevation, this effect beless significant for the A ring because of its smaller ocal depth. Moreover, in Paper I we showed that the B rtilt effect could also be explained by its large central plavolume density, combined with the size distribution and vtical distribution of particles: for increasing elevations tdense central portions, appearing relatively brighter becausof their opposition brightening,would become increasinglvisible, leading to enhanced brightness. Again, this brigness effect would be less significant for the A-ring. Hoever, due to limited radial resolution, these ground-bameasurements(Lumme and Irvine, 1976b)for the A ringreferred just to the brightest, innermost A ring. The recenanalysis of HST images(Cuzzi et al., 2002)shows the fullbrightness profiles of rings vs. distance and confirmsgeneral trend, i.e., the strong increase of brightness withvation for the B ring and a smaller increase for the innermA ring. However, their analysis also indicates that forA ring at a = 125,000–130,000 km, the brightness in fadecreaseswith elevation angle, suggesting that this regioneffectively optically thin. Most interestingly, this decreaseeven stronger than the slight reduction expected in the siscattered component, calculated for the A ring optical dein these regions (τphot∼ 0.45). Since this region with exceptional behavior coincides with the region where the strongasymmetry is seen, it is of interest to check whether“inverse tilt effect” could be explained by the presence

gravity wakes. Indeed, signs of such an effect were seeFig. 13, suggesting that in the presence of strong wakesoverall brightness decreases with elevation.

A more detailed study is shown inFig. 17, in terms ofthe same quantities shown inCuzzi et al. (2002)’s Fig. 8,namely the brightness at the ring ansa, for a phase aα = 6. Photometric models for dynamical simulations wiout self-gravity are also shown (labeled by “ng”), to empsize the difference as compared to the case where wakenot present. In the left-hand frame the brightnessI/F vs.elevation angle is shown, as well as the theoretical sinscattering brightness for uniform rings with volume densD → 0,

(22)

(Iss

F

)refl

= AP(α)µ0

4(µ + µ0)

(1− exp

[−τ (1/µ + 1/µ0)])

,

whereµ0 = |sinB ′| and µ = |sinB|. In the constructionof theoretical single-scattering curves the photometricτ ’sdisplayed inFig. 16have been used, accounting for the dcreasing trend of the curve (Iss would be more weakly decreasing for a fixedτphot ∼ 0.45). In the right-hand framethe scaled intensityI/Iss is shown, normalized to unity aB = B ′ = 24, so it is directly comparable to theCuzzi etal. (2002)figure. Since for small elevations and small phaangles the brightness is expected to be dominated by siscattering, this scaled intensity should be nearly consfor the case of classical low-volume-density rings. Moover, when the non-zero volume density is taken into acco(the spatially uniform models; ‘ng’) the scaled intensityfact increases withB = B ′. However,Fig. 17also indicatesthat the trend reverses in the presence of gravity wadue to increased visibility of gaps for larger elevations. Tamplitude of this inverse tilt effect, about−10% betweenB = 4–24 in the gravitating case with identical particleis in qualitative agreement with Fig. 8 ofCuzzi et al., 2002.In the size distribution model the trend is still presentthough considerably weaker. According to our models,type of contribution to the tilt effect is expected whenethe wakes are strong enough to cause significant asymm

4. Discussion and summary

Photometric modeling of gravity wakes obtained in dnamical N -body simulations has been carried out instraightforward manner by the Monte Carlo (MC) techniqdescribed in Paper I. We assumed that the layer of simulaparticles is illuminated by a large number of photons, aring either from the Sun or from Saturn, and followed the pof each photon through its successive scatterings until itable to escape from the layer. The direct MC-methodaugmented with an indirect MC method, significantly reding the variance of the final result. The periodic boundconditions of the dynamical simulation were taken expitly into account, which is very important in modelinglow illumination and viewing elevations.


lyed

Fig. 17. The inverse tilt effect in the presence of wakes. In the left frames the ansa brightnesses are shown for photometric models utilizing 4 differentdynamical simulations: standard identical particle and size distribution models (“sg-models”) and corresponding models without self-gravity (marked bysymbols; “ng-models”). Also shown are theoretical single-scattering contributions, calculated with the classical formula using the appropriateτpath for eachelevation. At right the scaledI/Iss, explained in the text, is shown, with the curves normalized by values atB = B ′ = 24 . Whereas in the case of spatialuniform non-gravitating models, the scaledI/Iss increases slightly withB = B ′, the opposite trend occurs for the case of wakes. The figure can be comparto Fig. 8 inCuzzi et al. (2002), showing the same type of observed reversed behavior for the portion of the A ring where the asymmetry is strongest.

essinedis-

mngillu-t-nde

aresenlvedoto-her

tetricdiedvs.viorser-

e ofto

se o

medara-ical

indi-

e of

esslue

,forthetionwhatthatofonons.thever-htor-his

In order to apply the method to modeling of the brightnasymmetry, we assume that the particle positions obtafrom dynamical simulations sample the typical particle dtributions at each ring longitude, in a coordinate systealigned with the local radius vector. For a given viewigeometry, each longitude of the rings has its specificmination and viewing direction, for which the ring brighness is calculated. Since wakes are transient, time-depefeatures, simulation snapshots from several time stepsemployed, corresponding to the spatial averaging prein any observations covering a large number of unresowakes. Two different dynamical models were used in phmetric modeling, one with identical particles and the otwith a truncated power-law size distribution (rmax/rmin =10, q = 3). In both cases the saturnocentric distancea =130,000 km, dynamical optical depthτdyn = 0.5, surfacedensityΣ = 500 kg m−2, internal densityρ = 450 kg m−3

and theBridges et al. (1984)velocity-dependent coefficienof restitution were assumed. The effects of various geomfactors on the obtained brightness curves were then stu

For both dynamical models, the overall brightnesslongitude curve was in rough agreement with the behaof reflected and transmitted light as seen in Voyager obvations. For the model of identical particles the amplitudthe asymmetry in reflected light was in fact rather closeobservations, but the match was clearly worse in the ca

nt

t

.

f

the size distribution. This clearly indicates that the assudynamical parameters need refinements. An obvious pmeter to reconsider is the adopted value of the dynamoptical depth, especially since the models themselvescate that the resultingτphot ≈ 0.3–0.4 of the models is infact below the observed value∼ 0.5. Indeed, increasingτdynwould increase the asymmetry amplitude also in the cassize distribution models.

Even though the longitude of the modeled brightnminimum differed by a few degrees from the observed va(245 vs. 249, for the geometry studied inDones et al.1993), the models were able to provide an explanationthe asymmetric shape of the minimum, in terms ofbrightness depending on both the viewing and illuminalongitudes. The model brightness seems to be somemore sensitive to the illumination direction, in the sensethe gradient ofI/F in the models is larger on the sidethe minimum corresponding more closely to illuminatialong the wakes, in qualitative agreement with observatiThis follows from the particular geometry studied, wheresolar elevation is smaller than the viewing elevation. Netheless, for lowα the modeled minimum in reflected ligfell very nearly at the average of the longitudes which crespond to parallel viewing and illumination of wakes. Tis not very different from what was reported inDones et al.(1993): for the images FDS 43916.44–43917.12, withB =


-43-

a

luete

om-

m-wing

also

un-toe

on

erningbut

irly

entrror

ef-haninghepthkesac-

12.6, B ′ = 8.0, and∆θs = 10.3, they measured a minimum at θ0 = 249, whereas for the image FDS 43974.with B = 16.5, B ′ = 8.0, ∆θs = 18.1 the observed minimum was shifted toθ0 = 246, or 3 further away from theansa. Photometric modeling withthese parameters yieldsminimum at 242, similarly shifted by 3 in comparison to245 obtained for the previous values. This modeled vafor ∆θs = 18.1 is also roughly consistent with the estimaof Eq. (18), which gives

θ0 = 270 − φwake− 1

2∆θs − 1

2∆θ0

(23)≈ 270 − 21 − 9.0 − (−2.5) ≈ 242.5.

(For this close range image∆/a = 10.3, giving∆θ0 ≈ −5near the eastern ansa.) For a wider range ofθs − θ0 thebrightness profile depends on solar longitude in a more cplicated way.

In order to further illustrate how the modeledI/F vs. θ0

curves are modified for close-range geometry,Fig. 18showsthe full 360 range plots of the models corresponding to iages FDS 43916.44 and FDS 43974.43 (ignoring shadoof the rings by the planet). As seen, the modeledI/F ’s aredifferent for the two ansae, and the modeled minima are

at different distances from the ansae, even though thederlying dynamical model is symmetrical with respecta 180 shift in ring longitude. The difference is due to thdifferent local directions between viewing and illuminatilongitudes. For the eastern ansa (θ0 ≈ 270) the longitudedifference

(24)φs − φ0 = −∆θs + ∆θ0

is for this geometry significantly larger than for the westansa, reducing the overall brightness level. The local viewelevations and phase angles are also slightly different,their effect is negligible.

Altogether, it seems that our modeling accounts fawell for the shift of the brightness minimum with∆θs , atleast for small-α observations, when the distance-dependgeometric factors are taken into account. The slight ein the minimum longitudes themselves suggests that thefective pitch angle of wakes is a few degrees smaller timplied by our present dynamical model. Again, increasthe modeledτphot might help, even if the dynamical pitcangle would remain the same, since the larger optical dmight be expected to make the low density tails of wamore efficient in determining the photometric behavior:

e
Fig. 18. Models of full 360 brightness vs longitude curves for twoclose-range Voyager images analyzed inDones et al. (1993). The upper frames show thI/F curves, while the lower ones display the difference in the viewing and illumination longitudes,φs − φ0. For infinite distance this would equal−∆θs ,shown with the dashed line (the lower left/right image corresponds to theupper left/right frame). In the upper frames the arrows indicate the minima near thetwo ansae, while the dashed (solid) ticks indicate the longitudes where illumination (viewing) is closest to being parallel to wakes, with an averagepitch angleof 21 .


a

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-tly.isolu-d/or

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nt ines,ons.

ex-our

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cording to the autocorrelation plot (Fig. 5) these tails havesmaller angle with respect to the tangential direction.

For transmitted light the minimum location was also qusuccessfully accounted for, although the only observatidata analyzed is for geometries where the Saturn-shinalso large, causing some uncertainty in the extraction odirect solar contribution. Somewhat counterintuitively,same general longitude dependence was seen as in thflected light. Nevertheless, when very small elevations anphase angles were modeled, the situation became revas compared to reflected light, as envisioned inFranklin etal. (1987).

The trend of the asymmetry amplitude with elevationgle we find matches fairly closely that found in the groubased observations from the 1970’s(Lumme and Irvine,1976a, 1979b; Lumme et al., 1977; Thompson et al., 19,the amplitude peaking at intermediate elevations. The mels also predict a change in the width and the location ofminimum with increasing elevation (the minimum wideniand moving slightly away from the ansa), but the limitedcuracy of these early observations does not allow even qitative comparisons. Basically, we suggest the asymmto be dominated by single scattering, but as the multiscattered contribution exhibits a similar and even stronlongitude variation, the asymmetry amplitude may depon wavelength via the single-scattering albedo. This isconsistent with observations(Lumme et al., 1977), althoughthe observed effect might be even larger than implied bysimulation models we studied. If so, this problem could adue to the too small optical depth in our present modwhich reduces the role of multiple scattering. The mels also predict a reduction in the asymmetry amplitnear opposition, due to reduced intrinsic variation of singscattered flux. Again, this is at least qualitatively consiswith Lumme et al. (1977)’s observations. It will be very interesting to compare all these aspects of the modeled ametry with the recent Hubble Space Telescope observatdescribed inCuzzi et al. (2002), covering a wide range oring elevations and phase angles, with high accuracy obsevations in several wavelengths (French et al., in preparat

The strong dependence of the modeled optical don the viewing direction with respect to wakes is a facthat is likely to be present wherever the presence of wamay be suspected based on azimuthal brightness variaClearly, in the presence of wakes the normal optical dededuced from occultation measurements will depend setively on both the elevation and ring longitude of the scif Eq. (21)for τphot is used. In particular, this dependenshould be taken into account when interpreting occultatiooptical depth profiles of the A ring, to be obtained durCassini’s orbital tour.

An important new result is the natural explanationwakes offer for the inverse tilt effect(Cuzzi et al., 2002seen in those regions of the A ring where the asymmis strongest. According to our models (seeFig. 17) this fol-lows from the reduced brightness of the ring as more

-

d

-

-,

.

.

more of the rarefied inter-wake regions become visible wincreasing elevation. Altogether, this observational phenenon may be considered as additional very strong evidof wakes, besides the asymmetry itself.

Even if the wakes are likely to exist in A-ring, it is uncertain whether Cassini will be able to resolve them direcAccording toEq. (4), the expected radial scale of wakesat most about 100 meters, still below even the best restion of Cassini images. Nevertheless, Fourier analysis anwavelet analysis of the UVIS high-resolution occultatprofiles should be able to show if excess signal is prein these wavelengths. If resolved, the radial scale of wstructure will give accurate constraints for the local surfdensity of rings, complementing those derived from salite excited density waves. Also, the mere confirmationwakes, resolved or not, would already give an estimatthe local velocity dispersion in terms of Toomre paramebeing close to unity.

5. Conclusions

According to our models:

• Gravitational wakes account in detail for the obserbrightness-longitude dependence seen in Voyager ovations of the A ring, both for the reflected and transmted radiation. The minimum brightness correspondviewing roughly along the wakes, minimizing the sctering area.

• Basically similar behavior is predicted for both reflecand transmitted radiation, except in the caseB andB ′close to zero, when the minimum and maximum lontudes are reversed in transmitted light (i.e., the brigness maximum corresponds to viewing and illuminatalong wakes).

• The variations of asymmetry amplitude with ring opeing angle, and its peak for intermediate elevations∼ 12is well accounted for. Also, the suggested trend of aplitude increasing with increased wavelength, as wethe reduction of amplitude at opposition, seem to bqualitative agreement with observations.

• Wakes provide a natural explanation for the inverseeffect of the mid A ring, the brightness being reducedthe rarefied region between wakes become better viat larger elevations.

• The presence of wakes needs to be taken into accouthe interpretation of Cassini stellar occultation profilas well as Voyager and Earth-based stellar occultati

On the quantitative level several problems remain inplaining the observed asymmetry. In particular, althoughmodels with identical particles yielded a roughly corrasymmetry amplitude, the amplitude obtained in the,sumably more realistic, size distribution model was smathan observed. Also, the photometric optical depth of


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models was somewhat smaller than the actual A-ring valThis clearly indicates that the dynamical parameters adohere are not optimal. Our future study will address in a stematic manner the dependence of azimuthal asymmetrvarious dynamical parameters, like the internal densityelasticity of particles, and the underlaying dynamical optdepth. An important constrain for this survey, as well asany quantitative model of Saturn ring’s structure, is providby the strong peaking of the observed asymmetry ampliin the mid A-ring.

Acknowledgments

This work was supported by the Academy of Finland,Väisälä Foundation, and by the NASA Planetary Geoloand Geophysics Program. We thank Luke Dones and DDunn for their very detailed and helpful reviews.

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