The dark side of Saturn’s B ring: Seasons as clues to its structure

Icarus 223 (2013) 28–39

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The dark side of Saturn’s B ring: Seasons as clues to its structure C. Ferrari ⇑, E. Reffet AIM, UMR 7158, Université Paris Diderot, Irfu/DSM/CEA Saclay, CNRS/INSU Bat. 709, Orme des Merisiers, 91191 Gif sur Yvette cedex, France

a r t i c l e

i n f o

Article history: Received 8 June 2012 Revised 13 October 2012 Accepted 3 December 2012 Available online 19 December 2012 Keywords: Planetary rings Saturn, Rings Thermal histories Infrared observations

a b s t r a c t The unlit side of the dense B ring of Saturn does not receive direct sunlight. Yet it cools down as the Sun sets on the ring from solstice to equinox. A multi-scale thermal model that treats the heat transfer through a packed ensemble of particles is built to study the orbital and seasonal temperature variations of both lit and unlit sides of this opaque ring. Heat transfer by radiation, conduction or through contacts are considered both at the ring and the particles scale. A statistical approach shows that three simple constraints, such as the heat diffusion time through the ring, the current estimates on its thermal inertia and its optical depth, yield important constraints on its thickness, its filling factor and on particle properties such as their thermal inertia, porosity or size. The B ring is probably a few-meters-thick, with a filling factor D below 0.25, made of a population of porous but still conductive particles. Its thickness is compatible with a surface mass density ranging between 50 and 75 g/cm2 in its densest part. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The bright Saturn’s B ring remains poorly characterized. Its lack of transparency due to a very large optical depth has indeed been preventing deep sounding of its vertical structure and particles properties. The ability of the CASSINI spacecraft to observe it from a multitude of viewing angles, especially at high elevation above the ring plane, has dramatically changed our knowledge of its radial structure. Series of stellar and radio occultations observed by the UVIS, VIMS and RSS instruments have provided a new picture. Composed of four sub-rings, B1–B4, the optical depth of this main ring exhibits large variations with distance to Saturn: it undulates in the B1 ring, is bimodal in the B2 and B3 rings and highly variable, and shows no specific radial structure in B4 (Colwell et al., 2009). The derivation of the normal optical depth from the transmitted starlight has revealed its correlation with the spacecraft elevation BSC. The latter induces the idea of a ring almost entirely dominated by self-gravity wakes of very large optical depth alternating with gaps almost empty of material. These high spatial resolution data in the radial direction have shown that the ratio of the gap width Sw to the wakes width Ww is decreasing with increasing normal optical depth. Their relative thickness Hw/Ww was shown to be below 0.1, decreasing with increasing optical depth and width Ww. Given their expected width, the wakes might be quasi-monolayers of the largest particles (Colwell et al., 2007). The authors recommend to use Hw/Sw = 0.2, i.e. a thickness that decreases with spacing as optical depth increases. However diminishing wake spacing appears as the only way for their model to reproduce high optical ⇑ Corresponding author. E-mail address: [email protected] (C. Ferrari). 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2012.12.006

depths and this behavior might be an artifact (Colwell et al., 2007). In VIMS observations, the optical depth is similarly dependent on the gap width and appears rather insensitive to the ring thickness H (Nicholson and Hedman, 2010). On the other hand, a strong correlation appears in the VIMS near-infrared spectra between the redness and the optical depth greater than 2 which may indicate changes in the vertical structure of the sub-B-rings (Nicholson et al., 2008). The particle size distribution in this opaque ring is unknown but in its thinnest parts, as derived from both the Voyager photopolarimeter PPS occultation profiles and the ground-based occultation of 28 Sgr (French and Nicholson, 2000). These observations suggest a distribution with minimum and maximum sizes of 30 cm and 20 m respectively, and a slope index q = 2.75. Numerical simulations show that the density contrast of the wakes relatively to gaps is enhanced for the population of the largest particles compared to the small ones. The larger the volume density of particles, the larger the contrast of the wakes relatively to the gaps (Robbins et al., 2010). Derived from CIRS infrared spectrometer observations, temperatures of both the lit and unlit sides of Saturn’s rings, together with their seasonal variations with the solar elevation B0, are expected to provide constraints on the vertical structure and particles properties of this opaque region (Spilker et al., 2003). Early in the CASSINI mission, the vertical thermal gradient between both sides ranges between 10 and 25 K for s P 1.5, depending on phase angle. This gradient is shown to endure with decreasing B0, but with always lower contrast. Temperatures homogenize vertically at equinox, as expected for a ring warmed up only by a central heating source, here Saturn (Flandes et al., 2010; Spilker et al., 2011). First measurements in 2005, at B0  22°, also show a positive correlation between the temperature of the lit side and the optical

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

depth s whereas a perfect anticorrelation is observed on the unlit side (Spilker et al., 2006). These correlations can be understood if the vertical mixing of particles is getting more limited as the ring becomes optically thicker. Being structured in self-gravity wakes, such a ring offers a substantial barrier to vertical motion. The idea of a limited vertical mixing and heat transport by convection is reinforced by the comparison of Morishima’s model to some observations of the B ring. It treats the heat transport by slow (and large) or fast (and small) spinning particles of finite thermal inertia through a classical multi-layered ring assuming particles have slightly inclined orbits that allow vertical motion from the lit to the unlit side. This model is reconciled with B ring data by assuming the particles to bounce in the mid-plane of the ring instead of crossing it (Morishima et al., 2010). The same authors also derived a particle thermal inertia from a dataset which spanned from July 2004 to early 2007 when the solar elevation was 24.5° 6 B0 6 13.7°. Their model was adjusted on lit side temperatures only and took advantage of the thermal transient in the planetary shadow where temperature variations are most sensitive to particles thermal inertia. They determine an average thermal inertia of 13 ± 4 J/m2/K/s1/2, being on first order larger for fast rotators, i.e. 27 J/m2/K/s1/2, than for the slowest ones, at 8 J/m2/K/s1/2. These latter values stay compatible with values derived earlier with a slab model of slow rotators from ground-based 2 1=2 data at low phase, i.e. 5þ18 (Ferrari et al., 2005). The 2 J=m =K=s variation of thermal inertia with spin rate was interpreted as a lack of fluffy regolith on the fastest rotators due to a larger centrifugal force at their surface (Morishima et al., 2011). In parallel, CIRS observations of the unlit side show no cooling within the planetary shadow (Ferrari et al., 2009; Reffet and Ferrari, 2011). This confirms the weak proportion of sunlight directly transmitted to this side and the reduced vertical heat transport by vertical motion of particles at the time-scale of the orbital period. Yet the unlit side temperature was revealed to change by more than 1 K on a time-scale of about 2 weeks in March 2009, when B0 = 2°, providing a preliminary constraint on the diffusion time of the solar heat wave through the B ring as Sun sets on the rings (Ferrari et al., 2009; Reffet and Ferrari, 2011). In this dense ring, particles are most probably continuously in close contact, piling up in self-gravity wakes or packed in layers. Modelling the seasonal evolution of this ring of non-zero filling factor D is beyond the limits of models which follow the formalism of classical radiative transfer. We address in this paper the questions of how heat transfer may happen through such packed ensembles of particles and of what few constraints on the observed diffusion time, the thermal inertia and the optical depth, can tell us about the ring and particles structures. In Section 2, a new multi-scale thermal model is presented. It neglects vertical motion and treats the B ring as a porous packed ensemble of particles. Heat transfer is computed both at the scale of the ring slab and at the scale of particles. It considers radiation through interstices, conduction within the solid phase and through areas of contact between ring particles, or between grains that form them. Given the ring and particles structural properties, an effective conductivity of the slab that takes into account all the processes listed above is calculated. The heat diffusion equation is solved for a discretized ring slab and its thermal evolution is integrated to study the temporal variations of the temperatures of both its lit and unlit sides. Presented in Section 3, a statistical analysis examines diffusion time, thermal inertia and optical depth as a function of ring and particles properties such as porosities, conductive or insulating structures, ring thickness, particle or grains effective sizes. It shows that current constraints on these three observables can provide important clues on these properties. How heat transfer may happen in Saturn’s B ring in these conditions and what its thermal evolution would then look like, both at the orbital and seasonal time-scales, are studied

29

Table 1 Table of symbols for the thermal model. Symbol

Meaning

i 2 {0, 1} A a

Index for (ring, regolith) medium Ring bolometric Bond albedo Distance from Saturn center Diffusivities, a = KEFF/qMCS Conducting, Insulating, Bruggeman EMA structure of the effective medium Specific heat of solid water ice (J/kg/K) Non-shadowed fractional area Ring volume filling factor Sun–Saturn distance in AU Ring emissivity Radiative exchange factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thermal inertias, C ¼ K EFF qM C S Ring thickness Thermal conductivity of solid water ice Thermal conductivities of contacts Effective conductivities Radiative conductivities Medium conductivities Porosities Sizes Volume density of water ice (918 kg/m3) Volume densities of media Solar constant (1370 W/m2) Stefan constant Temperature Saturn year, orbital period at distance a

ai B, C, I CS(T) C(s, l0) D = 1  p0 DAU

 FE

Ci H K0(T) KCi KEFFi KRi KMi pi Ri

q0 qMi S

r T TS, Ta

Bold: Parameters of the model.

in depth. When the confrontation of the model to an extensive thermal dataset of Saturn’s B ring is beyond the scope of this paper, it is nevertheless kept in mind. A distance from Saturn of a = 105,000 km has been chosen as a reference point. The latter correspond to the wealthiest dataset available from CIRS on this ring. It also corresponds to one of the regions with the highest optical depth. Finally, the derived preliminary constraints on ring and particles porosities, structures and thermal inertias, together with the implications on the ring thickness or its surface density are discussed in Section 4. The notations used throughout the paper and the parameters of the model are listed in Table 1 for reference. 2. Modelling heat transfer in the opaque B ring A ring is a structured ensemble of centimeter-to-meter-sized particles which intimate structure is unknown. These particles may be nearly spheres, possibly covered by regolith grains, which size may be of the order of hundred microns (Poulet and Cuzzi, 2002), or they may be fluffy aggregates. As the solid phase, they constitute the ring volume capacity at storing heat. This capacity is controlled by their porosity, or that of their regolith if any. Their effective thermal conductivity is also driven by their structure, which favor conduction or insulation (for instance with cracks or isolated layers), by contacts between grains or more generally by the continuous solid paths to conduct heat in depth. Heat is also transferred by radiation through voids. At the scale of the ring slab, Sun heats up particles which warm up, store and re-radiate energy towards nearby particles through ring voids. When particles collide, some heat may also be exchanged during contacts. This multi-scale medium may be treated as continuous with effective thermal properties that depend on properties at the various scales. In an optically thick medium like the B ring, extinction of sunlight along the optical path is important and energy deposit occurs mainly on the most superficial layers. As the surface of ring particles is covered with water ice (Poulet and Cuzzi, 2002), the absorption of particles is important at infrared wavelengths below 200 lm where most of the ring thermal emission takes place. Heat

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transfer by scattering of the infrared light among particles or among grains is therefore expected to be limited. At these wavelengths, the extinction is dominated by absorption on a crowded optical path. Heat transfer in the B ring can therefore be reasonably treated under the Rosseland approximation which is valid in highly absorptive media. Radiative and conductive heat transfer can be combined when they are independent by adding conductivities in the diffusion equation. For multi-scale problems and dependent heat transfers, the case is more complex as described in Section 2.2.1 and the use of an effective theory is required. 2.1. Heat transfer In this context, the heat diffusion through the ring layer along the vertical x direction is regulated by the ring diffusivity a0 = KEFF0/ qM0CS following the classical one-dimensional equation in cartesian coordinates for a plane-parallel geometry:

@T @2T ðx; tÞ ¼ a0 2 @t @x

ð1Þ

where KEFF0 is the ring effective conductivity, qM0 its volume density and CS the specific heat of the solid phase within particles. The boundary conditions on the unlit (x = 0) and lit (x = H) ring sides are:

F 0;H  rT 40;H ¼ K EFF0

  @T @x 0;H

ð2Þ

r being the Stefan constant and  the emissivity. The ring is assumed to be optically thick and absorptive enough for the energy deposit to happen on the first layer of both sides. The input fluxes on the lit and unlit sides, FH and F0 respectively, are: FH ¼

ð1  AÞSCðs; l0 Þ D2AU

þ FP

and F 0 ¼ F P

ð3Þ

FP being the actual ring emitted flux at distance a from the planet in absence of solar warming (B0 = 0°), DAU the Sun–Saturn distance in astronomical units and A the bolometric Bond albedo of the ring. FP has been indeed measured by the CIRS instrument at equinox in 2009 and is shown to be equal on both lit and unlit sides (Spilker et al., 2011), FP  0.30 W/m2 at distance a = 105,000 km. C(s, l0) is the ring non-shadowed fractional area, which takes into account mutual shadowing between particles within the ring. Its value is close to l0 = sin B0 for very dense rings. The maximum shadowing approximation as proposed by Froidevaux (1981) is used here. The solar input varies on an orbital period Ta = 2p/X(a) as it is temporarily eclipsed by the planet. The duration of the eclipse at distance a = 150,000 km largely varies with solar elevation, from about 2 h at equinox to only 30 mn near solstice. The solar elevation B0 varies from 0° at equinox to ±26.7° at summer or winter on a seasonal timescale of about 7.5 years. The austral summer at Saturn started on October, 12th, 2002 and ended on August, 11th 2009. The CIRS dataset that will be used to constrain the B ring structure covers a significant fraction, starting close after the orbit insertion in July 2004, right until the equinox. Within this time interval, DAU has increased from 9.04 to 9.43 AU. The annual timescale is TS = 29.4571 years, the orbital is here Ta  9.5 h. The associffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ated seasonal skin depth dS ¼ a0 T S =p is therefore ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pabout ffiffiffiffiffiffiffiffiffiffiffiffi 170 times larger than the orbital one, da ¼ a0 T a =p ¼ dS T a =T S . The ring is vertically sampled so as to correctly resolve the ring orbital skin depth, the shortest one. A Crank–Nicholson scheme is used to solve the diffusion equation with a linearization in temperature at the radiative boundaries. The propagation of the thermal wave through the ring is calculated every 500 s, corresponding to an azimuthal sampling of about 5°. This provides sufficient time sampling to accurately describe the transient regime and insure reasonable calculation time over a kronian season.

2.2. Multi-scale effective thermal conductivity A huge literature in planetary sciences describes the complexity in modelling heat transfer in porous media such as icy surfaces of comets and satellites, lunar or martian regoliths. The topic is also largely developed beyond this context for packed beds. Modelling heat transfer in dense rings requires convolving transfer at different scales, the ring and the particle, within which main transfer processes can be of different magnitudes and nature. An analytical model that provides consistent expressions for thermal conductivities by contacts, conduction or radiation for a given medium structure cannot be found. Experimental approaches have only recently provided effective thermal conductivities for highly porous material at low pressure and explore their dependencies on a controlled medium structure (Krause et al., 2011). Given the complexity of the modelling, our purpose here is to set a range of reasonable values for these conductivities in a medium of porosity p = 1  D that may significantly differ from unity. The magnitudes of the effective thermal conductivities of the ring or particle/regolith media and their connections with their structural properties have to be estimated to understand how heat transfer may happen. 2.2.1. Effective thermal conductivities An important aspect of research in porous media is to find analytical formulae that lump all relevant transfer mechanisms into one single representative value such as an effective thermal conductivity KEFF. Wiener (1912) proposed optimal bounds for the effective conductivity of a mixture of two phases, assumed here to be solid and void, distributed in alternate layers either parallel to the heat transfer direction (conductive) or perpendicular to it (insulating case). The Maxwell–Garnett formulation of the effective thermal conductivity of spheres dispersed in a continuous matrix made of a different material is often used as an intermediate case (Maxwell-Garnett, 1904). It is derived from the Lorentz– Lorenz formula which gives the effective conductivity for a threephases system, two phases and the environment. Maxwell–Garnett chose one of the phase as the environment to treat the two-phases case. His formulation has the drawback of not being symmetrical, i.e. the phases cannot be interchanged, which leads to physical issues at intermediate porosities. The Bruggeman effective medium approximation (EMA) sets the effective medium as the environment. It has the advantage of considering both phases symmetrically (Bruggeman, 1935). The formula is then valid for any porosity and the two phases can be interchanged. It is obviously an intermediate case between the Wiener bounds. More recent works have also explored this aspect via Monte-Carlo simulation, classical radiative transfer approach or experimental measurements (Van Antwerpen et al., 2010) but at this stage a simple analytical formulation appears sufficient to explore the question of the effective conductivity of the ring. The conductivity KM of a porous medium, as derived from the effective medium approximation, can be related to the thermal conductivity of its solid phase KS and the radiative conductivity of voids KR. It can be expressed as KM = /(rk)KS, with / a function of rk = KR/KS that depends of the structure type retained (insulating, conductive or intermediate, Eq. (4)). The Wiener and Bruggeman EMA formulations are implemented as a function of rk and medium porosity p according to:

Insulating : / ¼

1 p=rk þ ð1  pÞ

Conductive : / ¼ ð1  pÞ þ prk Bruggeman : /2 þ b/ þ 0:5r k ¼ 0 b ¼ 0:5ð2  3p þ r k ð3p  1ÞÞ

ð4Þ

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31

For our multi-scale heat transfer problem, this is to be applied both at the scale of the particles and of the ring slab. To discriminate one from the other, a subscript i respectively equal to 1 and 0 is used in our notations when necessary. Therefore, rk1 = KR1/KS is defined for the particle regolith. The solid thermal conductivity of icy regolith grains KS is about 10 W/m/K and slightly decreases with temperature according to KS = K0(T) = 567/T (Klinger, 1981). For the ring, the solid thermal conductivity has to be replaced by the effective conductivity of the particles (KS0 = KEFF1) which gives rk0 = KR0/KEFF1. Finally, the effective conductivities KEFF sum up the conduction through contacts KCi to the effective medium conductivity KMi and their multi-scale implementation takes the following form:

K EFF1 ¼ K C1 þ K M ðK R1 ; K S ; p1 Þ K EFF0 ¼ K CO þ K M ðK R0 ; K EFF1 ; p0 Þ

ð5Þ

K Mi ðK Ri ; K Si ; pi Þ ¼ /K Si Nonetheless, one has to keep in mind that KR depends on KS as explained in the following section. 2.2.2. Radiative conductivity In dense rings/regoliths, heat exchange by radiation between particles/grains through the voids may not be negligible. As discussed above, the radiative transfer equation governing infrared intensity through the layers can be expressed as a diffusion equation with a radiative conductivity KR that varies in T3, i.e. strongly dependent on temperature. This radiative conductivity is often assumed to be independent of the solid thermal conductivity KS. Most recent studies on heat transfer in discontinuous media like packed beds have intended to express this radiative conductivity in the form of KR = 8RFErT3, where FE is a dimensionless radiation exchange factor, R the effective size in the medium and T the average bed temperature (for a review see Van Antwerpen et al., 2010; Piqueux and Christensen, 2009). The most difficult step in this method is to determine the radiative exchange factor FE and its dependency on the structure of the medium. Some of these works showed the dependence of KR on KS. The approaches of Breitbach and Barthels (1980) or Singh and Kaviany (1994) appear to be the most relevant to our case (Van Antwerpen et al., 2010). Breitbach and Barthels used the unit cell approach which describes the layers as an ensemble of cells that exchange heat. It allows to treat the layers in a continuous approach, even in the case of closely packed particles and when the scattering by one particle dependent on its neighbors. Singh and Kaviany (1994), using Monte-Carlo method for radiation transport and a finite difference scheme for temperature distribution, have demonstrated that the effective radiative conductivity KR is strongly influenced by the emissivity  and the solid thermal conductivity KS of the particles/grains. Recent modelling of heat transfer in planetary soils (Piqueux and Christensen, 2009) uses an exchange factor FE which does not depend on the thermal conduction of the solid phase but only on the emissivity of the grains as proposed by Kasparek and Vortmeyer (1978). As these two last models have been determined only at few porosities, the Breitbach and Bartels’ model, hereafter BB, is chosen in the current study. The latter can be used for any range of porosity, solid conductivity and emissivity and is equivalent to the others when available. The expression used for the radiative exchange factor FE is given below (Eq. (6)), plotted and compared to the two other models for self-consistency in Fig. 1.

pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi 1p Bþ1 1 FE ¼ 1  1  p p þ 2=  1 B 1 þ 2=K1

ð6Þ

f

The dimensionless conductivity Kf is the ratio of conduction-toradiative conductivities Kf = KS/8RrT3 and B = 1.25((1  )/)10/9.

Fig. 1. Radiative exchange factor FE as a function of the porosity (p = 0.48 or p = 0.95) in a packed bed of meter-sized particles (R = 1 m) at mean temperature T = 80 K, with emissivity  and solid thermal conductivity KS. Curves correspond to Breitbach and Barthels (BB), Singh and Kaviany diffused (SKD) and Kasparek and Vortmeyer (V78) models.

Fig. 1 shows that for a porosity p = 0.48 the exchange factor FE is similar among authors when KS P 0.3 W/m/K. For such large solid conductivities KS, heat transfer via conduction through the grain/ particle is relatively large and they become isothermal. The absorbed radiation is therefore re-emitted isotropically, which increases the exchange factor through the layer. The exchange factor also increases with increasing emissivity as more radiation is absorbed and re-emitted, favoring an efficient conduction of heat through particles (Singh and Kaviany, 1994). As the solid thermal conductivity gets smaller, heat transfer by conduction is reduced and the radiation flux has to be reflected or re-emitted by the particle surface through the voids. For a given medium structure, the exchange factor is then lower than its value at larger KS. At small KS, FE is also less dependent on the solid phase properties and in particular on its emissivity (Singh and Kaviany, 1994; Rubiolo and Gatt, 2002). For a given solid conductivity, FE increases as the porosity gets larger because a higher fraction of voids favors heat transfer by radiation. It varies by an order of magnitude when the porosity roughly doubles from 0.5 to 0.95 and may be as high as 10 for high porosities, emissivities and solid conductivities. For the same reason, considering a given porosity, optical depth of the layer and solid conductivity, the radiative exchange factor increases with a decreasing size. In addition, the radiative conductivity KR = 8RFErT3 changes by orders of magnitude when either regolith or ring are considered, due to the relative range of sizes considered. Moreover, Saturn’s B ring temperatures range between 50 K and 90 K over a season, which may induce variation of KR by another order of magnitude. The radiative conductivity is then highly dependent on the structure of the medium and its thermal properties. 2.2.3. Thermal conductivity of contacts Heat flux crosses contacts between ring particles or, at the regolith scale, between grains within the particle. Contacts between particles may be brief but frequent in a dense ring. Grains are continuously in contact and may be compacted by inter-particle collisions. The thermal resistance of contacts, from which their

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conductivity KCi is derived, is the subject of a huge research in multiple fields. This is also a complex problem as it highly depends on the mechanical properties and rugosities of the surfaces in contact. Modelling the conductivity through contacts rapidly becomes tedious if the grain/particles have an irregular surface (Bahrami et al., 2006). Analytical works provide some derivation in case of spheres that have been confirmed by experimental work (Buonanno and Carotenuto, 1996). Regolith grains are modelled here as solid spheres in contact. The thermal conductivity of contact proposed by Watson (1964) is driven by the grain size R. Its validity is limited to R P 10 lm, it gives KC = 1.5  108R1 6 1.5  103 W/m/K. Piqueux and Christensen (2009) also report numerous references attesting a contact thermal conductivity KC 6 103 W/m/K. According to the JRT theory (Johnson, 1985), in the case of spheres, the area of contact and the deformation induced by the compression can be calculated as a function of the mechanical properties of the material (here solid water ice), as a function the Young modulus E (J/m3), the Poisson ratio m and the surface tension c (J/m2). Johnson proposes to write the thermal conductivity of contact as KC = SCKS:



KC ¼

p 9pcð1  m2 Þ g

8RE

2=3 KS

ð7Þ

with g = 4 for a cubic lattice of grains and g  1 in general. For solid water ice E = 9 GPa (109 J/m3) at T = 273 K, m = 0.33 and c = 0.076 J/m2 (Sirono and Yamamoto, 1997). Lindgren (1970) has shown that the Young modulus decreases with increasing temperature, along E(T) = 6.6  109 (4.276  0.012T), i.e. 24.3 GPa P E P 18.8 GPa for 50 K 6 T 6 120 K. For R = 10 lm and E = 9 GPa, SC = 3.3  103/g, which gives a maximum value for this conductivity relative to the solid water ice one, i.e. K0(T) = 567/T, which is about 7.1 W/m/K at T = 80 K (Klinger, 1981). Depending on the ice structure, E might also be lower, between 3 and 5.7 GPa, which would increase the thermal conductivity as the area of contact would be larger for the same force (Zamankhan, 2010). 2.2.4. Heat transfer in particles and ring According to the above equations, ring and particle effective thermal conductivities are controlled by sizes, porosities, structure (insulating or conducting layers and mixed voids and spheres) and contacts. Their relative magnitudes in the extreme cases of ring and regolith structures being either both conducting or both

insulating are displayed in Fig. 2. Ten-cm-sized particles (R0 = 10 cm) in a ring of porosity p0 = 0.9 are chosen for this example. They would form a ring of optical depth s = 3(1  p0)H/4R0 = 3 if the ring thickness were H = 4 m. In Saturn’s B ring, regolith grains may be as small 10 lm (60% in volume fraction) with still a significant population of 100-lm-sized grains (30%, Poulet et al., 2003). Given that wavelengths of maximum thermal emission are in this range, only the larger grains, typically of a few hundred microns, can be considered here. For this example, R1 = 500 lm is assumed. To emphasize the effect of size and temperature on effective conductivities, p1 = p0 = 0.9. Contacts between particles are supposed to be continuous in time, which maximizes the thermal conductivity of contacts between particles. The Johnson model (1985) for contact conductivity KC is assumed with g = 1. At the regolith scale, the radiative conductivity is set by the grain size, since FE  4 regardless of KS for the assumed porosity p1 = 0.9 and grain size. At T = 80 K, KR1  5  104 W/m/K. For an insulating regolith, the rk1 ratio /  105, given p1 = 0.9. The porosity is high, the structure is insulating, so that the solid conductivity of the regolith is KM1  104  103 W/m/K. The contact conductivity is comparable but decreases with temperature due to the Young modulus variation with temperature. It results in a regolith effective conductivity KEFF1  103 W/m/K, dominated by conduction through contacts between small spheres KC1 at low temperature and by radiation KR1 above 85 K. In the case of a conducting regolith, even at high porosity, the effective thermal conductivity is dominated by the solid conductivity of the grains with rk1  104  103 and /  1  p = 0.1, which leads to KEFF1  1 W/m/K. At the scale of the ring slab, the radiative conductivity of the ring KR0 is R0/R1 (200) times larger than KR1 and of the order of 102  101 W/m/K. It is much larger than the conductivity of contacts KC0 which is of the order of a few 105 W/m/K, due to decreasing E(T). In the case of insulating ring and regolith structures, the ring effective conductivity KEFF0 is about 102 W/m/K, as rk0  10  102 and /  1/(1  p) = 10 (KEFF1  103 W/m/K). Heat transfer through the ring is here driven by its radiative conductivity, i.e. by the particle size and the volume filling factor D = 1  p0. If both ring and regolith conduct heat (KEFF1  1 W/m/K), rk0  102  101 and /  (1  p) = 0.1  1, the ring effective conductivity is KEFF0  101  1 W/m/K, an order of magnitude smaller or similar to the regolith effective conductivity KEFF1. Heat transfer is driven in this case by both radiative and solid conductivities, with an

Fig. 2. Conductivities of particle regolith (left) and ring (right), i.e. KRi, KCi, KEFFi versus temperature T. Regolith grains and ring particles sizes are R1 = 500 lm and R0 = 10 cm respectively. The ring volume filling factor D = 1  p0 = 0.1 and its emissivity is  = 0.9. The regolith porosity is p1 = p0 = 0.9 and the JRT theory (Johnson, 1985) with g = 1 is assumed for contacts in both media. Only Wiener bounding values for conduction are considered here, ring and regolith media either both conducting or both insulating.

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increasing influence of radiation as temperature increases. Considering Bruggeman theory of effective medium allows describing intermediate combinations between these two extreme cases. Calculated from Eqs. (1)–(3) and using the multi-scale effective conductivity described in Eqs. (4) and (5), the thermal history of a ring of thickness H = 4 m while the solar elevation B0 decreases above the ring plane from summer to equinox is shown in Fig. 3. The ring filling factor is D = 1  p0 = 0.1 and the Bruggeman effective medium formulation is used to model its structure. Particles size is R0 = 10 cm. The regolith structure is assumed conductive,

(a)

its porosity is p1 = 0.9 and the grains size is R1 = 500 lm. Fig. 3a displays seasonal temperature variations of its lit side both at ingress and egress of the planetary shadow (TLI and TLE), and of its unlit side (TUL). Fig. 3b displays the orbital variations of the temperature through the ring during about two orbits when B0 = 15°. A 8.5 Kwide transient regime due to the planetary shadow crossing can be observed on its top layer). At one or two orbital thermal skin depths da  0.35 m, the thermal wave due to the transient is delayed and attenuated. It rapidly vanishes with depth. For this modelled ring, the transient regime is hardly detectable on the unlit side as the orbital skin depth da is much lower than the ring thickness. Nonetheless, the temperature of the unlit side exhibits seasonal variations as the seasonal ring thermal skin depth dS  58 m is much larger than the ring thickness (Fig. 3a). The effective diffusivity of the ring in this specific case is a0  1.15105 m2/s and corresponds to a diffusion time tD  16 days or 2.4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi weeks. The effective ring thermal inertia C0 ¼ K EFF0 qM0 C S calculated by the model in this example slightly decreases from about 24 to 10 J/m2/K/s1/2 from winter/summer to equinox, as conductivity is partially radiative. The effective thermal inertia of particles is about constant, i.e. C1 = 214 J/m2/K/s1/2. Particles are isothermal as both orbital and seasonal skin depths are larger than their size R0. 3. Overall constraints on the B ring structure

(b)

The model presented here depends on 10 parameters which are A, , pi, Ri, H, how ring and regolith layers conduct heat (insulating I, conducting C or intermediate B structure) and how the conductivity by contacts is modelled (Table 1). It is destined to derive ring and regolith thermal and structural properties of the B ring from both its orbital and seasonal thermal histories, as observed by the CASSINI-CIRS instrument during the nominal and equinox missions (2004–2009). A tenable fit to these data requires to restrain the domains of parameter values as far as possible. A priori knowledge on the B ring is used in that purpose. The most probable ring and particles thermal or structural properties are identified here. How the heat transfer happens at each scale in these cases is scrutinized. 3.1. Preliminary constraints The a priori knowledge on the B ring considered here concerns seasonal temperature variations, the diffusion time through the ring tD, its optical depth s, its thermal inertia C0 but also the range of model parameters such as its thickness H, the size of particles R0 or regolith grains R1 and chemical composition. The B ring temperature ranges between 50 and 90 K within the period considered (Flandes et al., 2010). The change in temperature of the unlit side of the B ring attributed to heat transfer from the lit side is supposed

Table 2 Median ring and regolith properties.

Fig. 3. (a) Seasonal temperature variations of the lit side of a ring at ingress (TLI) and egress (TLE) of the planetary shadow and of its unlit side (TUL) as a function of solar elevation B0. (b) Orbital temperature variation of the upper layer of the lit side (x = H), the unlit side (x = 0) and layers located at one and two orbital skin depths da = 0.35 m. Particles and grains sizes are R0 = 10 cm and R1 = 500 lm respectively, ring and regolith properties are (Bruggeman EMA with p0 = 0.9) and (Conducting with p1 = 0.9) respectively. The ring distance from Saturn is a = 105,000 km, its thickness H = 4 m, the layer thickness is R0. The ring emissivity  = 0.9 and its Bond albedo A = 0.5. The JRT theory with g = 1 is assumed for contacts. The resulting ring optical depth is s = 3.

Struct.a

Nb(%)

p0

p1

R0 (cm)

R1 (mm)

H (m)

C–C C–B C–I B–C B–B B–I I–C I–B I–I

0.40 0.08 0.05 0.29 0.05 0.05 0.23 0.04 0.03

0.88 0.88 0.76 0.77 0.76 0.75 0.77 0.66 0.65

0.94 0.98 0.99 0.96 0.99 0.99 0.97 0.99 0.99

7.3 5.4 9.9 8.2 9.9 11.9 8.7 24.7 26.2

0.7 0.8 1.0 0.8 1.0 1.3 0.8 4.9 5.7

4.0 3.0 3.5 2.8 3.4 3.5 2.9 3.9 4.3

a Ring-regolith conducting properties: C = Conductive, B = Bruggeman, I = Insulating. Johnson et al. contact theory (Johnson, 1985) with g = 1 is chosen. b Percentage of draws within criterion (tD 6 20 days) and (2 6 C0 6 50 J m2 K1 s1/2) and (3 6 s 6 6). The initial number of draws is M = 450,000.

34

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

20

of the regolith grains is set at 100 lm and they cannot be larger than 1 cm. The chemical composition of grains is assumed to be pure water ice. This affects the ring Bond albedo and emissivity and all conductivities, solid, radiative or by contacts (through mechanical properties). The emissivity is assumed to be between 0.5 and 1. Random sets of parameters {T, , pi, Ri, H} are drawn within these ranges to calculate the diffusion time tD = H2/a0, the ring thermal

Ring thermal inertia

Diffusion time (days)

to happen in less than tD = 20 days as a conservative boundary. The B ring optical depth is in the range [3–6]. Its thermal inertia C0 is of the order of 15 J/m2/K/s1/2. A conservative range is fixed to [2–50] J/m2/K/s1/2, somewhat extended compared to measured values as the effective conductivity in this model can be radiative and vary significantly with temperature. The ring particle size R0 has to be within the range (1 cm–3 m), for a ring thickness H which is supposed to be between 1 and 100 m. The lower limit for the size

15 10 5 0 1

10

10

1

10

H (m)

20

Ring thermal inertia

Diffusion time (days)

H (m)

15 10 5 0 1

10

10

100

1

10

20 15 10 5 0 0.1

1.0

10.0

10

0.1

1.0

20 15 10 5

0.2

0.4

0.6

10.0

R1 (mm) Ring thermal inertia

Diffusion time (days)

R1 (mm)

0 0.0

0.8

1.0

10

0.0

0.2

0.4

20 15 10 5

0.2

0.4

0.6

p1

0.6

0.8

1.0

0.6

0.8

1.0

p0 Ring thermal inertia

Diffusion time (days)

p0

0 0.0

100

R0 (cm) Ring thermal inertia

Diffusion time (days)

R0 (cm)

0.8

1.0

10

0.0

0.2

0.4

p1

Fig. 4. Diffusion time tD and ring thermal inertia C0 versus ring and regolith conducting properties. Draws obeying the cumulative criterion are represented here: (green open circle) for C–C ring-regolith conducting properties, (cyan open squares) for C–B combination, (magenta open star) for B–C and I–C (orange open triangles) combinations. Filled black symbols represent the median values as given in Table 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

35

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi inertia C0 ¼ K EFF0 ð1  p0 Þð1  p1 Þq0 C S ðTÞ and the optical depth s = 3(1  p0)H/4R0, for any combination of ring and regolith conducting properties (conduction, insulating or intermediate). Density distribution of parameters are uniform within the specified ranges, either with linear binning (for pi, T, ) or logarithmic one (for Ri and H). The percentage of draws N that fits in the cumulative constraints ranges tD 6 20 days, 2 6 C0 6 50 J/m2/K/s1/2 and 3 6 s 6 6 are given in Table 2 for each ring-regolith combination. The number of tested draws, M = 450,000, insures a reliable estimation. The median values of the parameter sets that obey the selection criterion are also given as reference.

Four combinations of ring-regolith structures emerge from this study as most capable to follow these constraints, given the a priori knowledge considered by the model (higher N values in Table 2). Three of them correspond to a porous but still conductive regolith structure (C) coupled with a ring of any structure type (C, B or I). The last one, may be less probable, corresponds to a conductive but porous ring with an insulating porous regolith (C–B combination). At this stage, insulating regolith structures are clearly rejected as less probable combinations. Fig. 4 displays the diffusion time and ring thermal inertia as a function of model parameters for the selected draws and these four combinations.

2.0

% of samples in bin

% of samples in bin

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

1.5

1.0

0.5

0.0 1

10

100

1

10

R0 (cm)

100

H (m)

3.0

% of samples in bin

% of samples in bin

1.4 2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.5

1.0

0.6

0.7

p0

0.8

0.9

1.0

emissivity 5

% of samples in bin

% of samples in bin

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.1

1.0

4 3 2 1 0 −6 10

10.0

−5

10

5

3.0

4

2.5

3 2 1 0 0.0

−3

10

Diffusivity

% of samples in bin

% of samples in bin

R1 (mm)

−4

10

2.0 1.5 1.0 0.5 0.0

0.2

0.4

0.6

p1

0.8

1.0

1

10

100

Regolith thermal inertia

Fig. 5. Density functions of model parameters and other thermal properties as a function of conducting properties at ring and regolith scales. They are estimated from histograms of draws that obey the cumulative criterion (green for C–C ring-regolith properties, cyan for C–B combination, magenta for B–C combination and orange for the I– C one). Dashed lines report the initial number of samples per bin multiplied by the number of bins and divided by M = 450,000. This number has to be uniformly unity if the distribution is uniform along bins. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

36

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

diffusion times between 3 and 20 days and a ring thickness of about 3–4 m, in case of conductive structures at both scales (C–C). As a0 C20 ¼ K 2EFF0 , the effective thermal conductivity of the ring KEFF0 is expected to be about 0.06 W/m/K, given C0 = 15 J m2 K1 s1/2 and a0 = 1.5 times 105 m2/s.

The optical depth exhibits clouds of points very similar to the diffusion time. The distribution of the kept draws in the parameters space is indicative of further refinement in the constraints on the probable ring and particles properties. At the ring scale, the thickness H is found to be most probably below 10 m. Thick rings are clearly rejected. Median values of 2.8–4 m are suggested (Table 2). The histograms of selected draws displayed in Fig. 5 confirm this and show that the density function of the thickness H peaks between 3 and 4 for a conductive structure (both ring and regolith, i.e. C–C). For the other structures a thinner B ring is favored, most probably because a lower conductivity requires a thinner ring to fit the time diffusion constraint. The median values for the ring porosity p0 are above 0.7 (Fig. 4 and Table 2). Histograms of p0 show that the density functions peak between 0.85 and 0.95, i.e. a filling factor D in the range 0.05–0.15. Most conductive ring beds (C–C or C–B) favor generous ring porosities when more insulating ones (B–C or I–C) adopt lower ones. The median values for the effective size of particles R0 are about decimeter-sized, in between 5.4 and 8.7 cm (Table 2). Plots show that R0 is most probably below 1 m (Fig. 4) and the density functions of selected draws peak about 7 cm (Fig. 5). These peaks are most pronounced for less conductive ring structures, i.e. B–C or I–C ring-regolith combinations. The H/R0 ratio ranges between 33 and 55. At the scale of a particle, the median values for the regolith grain size R1 are similar for the four cases. The regolith grain size R1 is below 10 mm, most probably below 1 mm. The regolith, having to be conductive (C), is not dominated by radiation or contacts, which are sensitive to grain size. In consequence the constraint on the regolith grain size R1 is weak. If the regolith structure has to be conductive, it still can be significantly porous (p1 P 0.94 in Table 2). Plot of draws show that most selected draws are found for p1 P 0.8. The thermal inertia C1 of the regolith is most probably in between 100 and 300 J/m2/K/s1/2 for a conductive regolith structure. In case of an intermediate Bruggeman structure the regolith thermal inertia might be much lower, about a few J/m2/K/s1/2 (Fig. 5). The criterion puts no constraints on the emissivity which shows a flat density function (Fig. 5). The density functions of diffusion time, ring thermal inertia and optical depth of selected draws show no significant asymmetry within the constrained ranges. As far as the ring diffusivity a0 is concerned, the density functions of the selected draws show a large peak between 9  106 and 3  105 m2/s (Fig. 5). These limits correspond to

How heat transfer may happen in the presence of conducting particles within a ring of variable conducting structure is studied here, i.e. for the C–C, B–C and I–C ring-regolith combinations. The median values of the ring and regolith parameters pi, Ri, H are in these cases representative enough of their density functions as constrained by the diffusion time, the ring thermal inertia and the optical depth. When representative, median values can be used to evaluate the magnitudes of conductivities in each case and provide insights on the heat transfer at each scale (Fig. 6). In the case of a conductive ring-regolith combination (C–C), the regolith conductivity through contacts is below 103 W/m/K (Fig. 6a). It is comparable to the radiative conductivity between small grains. Even if the regolith porosity is large, i.e. 0.94, the regolith conductivity KEFF1 stays as high as 0.5–1 W/m/K and is dominated by solid conduction. At the scale of the ring medium, the contact conductivity is negligible, the radiative conductivity in the ring voids is about 0.01–0.1 W/m/K. The heat transfer through the ring is efficient here thanks to a conductive structure of particles (C), even if they remain porous. The largest fraction of heat is therefore mainly transferred by conduction through the solid phase, despite the ring porosity. When the temperature increases, a significant fraction is radiated. As the conductivity by contacts stays below 103 W/m/K, the above results do not depend on the chosen contact theory. In the case of a less conductive ring structure (B or I), with comparable regolith properties (conductive with p1 = 0.96 or 0.97), the way heat is transferred is different (Fig. 6b and c). The conductivity through the solid phase in the ring is lower due to a less conductive structure and it is comparable to the radiative conductivity of the ring (Fig. 6b). The effective ring conductivity increases with temperature, from 0.02 to 0.1 W/m/K for the Bruggeman case (B–C) and from 0.01 to 0.1 W/m/K for the insulating ring bed structure (I–C) (Fig. 6c). Here the ring thermal inertia is expected to depend more on temperature and on the ring particle size R0 which controls the radiative conductivity. That is why the particle size R0 is

(b)

10−2

10−4

10−6 50

KR 60

KC 70

80

KM 90

100

Temperature T(K)

KEFF 110

(c) B−C

100

10−2

10−4

KR

10−6 120

Thermal conductivity (W/m/K)

C−C

100

Thermal conductivity (W/m/K)

Thermal conductivity (W/m/K)

(a)

3.2. How heat transfer may happen in the B ring

50

60

KC 70

80

KM 90

100

Temperature T(K)

KEFF 110

I−C

100

10−2

10−4

KR

10−6 120

50

60

KC 70

80

KM 90

100

KEFF 110

120

Temperature T(K)

Fig. 6. Conductivities KRi (- -), KCi (- ) and KEFFi versus temperature T for ring (black) and regolith (red) media as structured according to the most probable ring-regolith conducting properties C–C (a), B–C (b) and I–C (c). Median values as given in (Table 2) are used with  = 0.9. The JRT theory with g = 1 is assumed for contacts. The red dotted line represents the solid ice conductivity of grains K0(T). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

better constrained in these latter cases than in the C–C conductive combination (Fig. 5). The magnitudes of these conductivities are compatible with KEFF0 about 0.06 W/m/K. 3.3. Seasonal and orbital thermal cycles of the B ring What the thermal history of the B ring might be at a distance a = 105,000 km from Saturn, both at the time-scales of a season and of an orbit, with the planetary shadow crossing as a transient periodical event, is displayed in Fig. 7. Median values of the ring and regolith parameters (Table 2) are used for these cases. At equinox, the ring temperature is about 50 K. At summer, temperatures peak at 90–91 K on the lit side and at about 61–65 K on the unlit one. These temperatures are very close to the observed estimates (90–95 K) on the lit side but significantly different on the unlit side (72 K, Flandes et al., 2010). In the case of the C–C conductive structures, the effective conductivity is constant versus temperature, about 0.1 W/m/K. The energy received on the lit side is easily transferred to the unlit side whatever the season is. As the Sun raises from the ring plane, it warms up faster than the other two

TLI TLE TUL

90

Temperature (K)

combinations, which conductivity is more radiative and comparatively lower at low temperature (Fig. 6). The lit side tends thus to be relatively colder as Sun raises. The more insulating ring-regolith B–C and I–C combinations exhibit warmer lit sides and larger thermal gradients despite a smaller thickness H due to this temperature-dependent conductivity. The difference in seasonal variations between these two families is most significant on the unlit side at solar elevations ranging between 5° and 20°. At the orbital time-scale, typically 9.5 h at that distance of Saturn, effects of the regolith porosity on the ring thermal inertia can also be observed where particles heat up and cool down at the planetary shadow ingress and egress (Fig. 7). The higher the regolith porosity, the smaller the ring thermal inertia and the larger the azimuthal gradient in the shadow. Time-dependent observations of the lit and unlit sides temperatures may indeed discriminate these structures that can exhibit vertical temperature gradient differences as high as 6 or 7 K. Also data around the shadow zone may provide complementary constraints to evaluate the ring thermal inertia and porosities.

4. Discussion and conclusions

100

80

70

60

50

40 −25

−20

−15

−10

0

−5

solar elevation B0 (deg) 85

80

Temperature (K)

37

75

70

65 0

5

10

15

Time (h) Fig. 7. Seasonal and orbital thermal histories versus ring and regolith conducting properties at a = 105,000 km. (Top) Ring temperatures versus solar elevation (seasonal variations) for probable combinations, i.e. C–C (green), B–C (magenta) and I–C (orange). Ring lit side temperatures at ingress (TLI, full line) and egress (TLE, dotted line) of the planetary shadow are plotted, together with the unlit side temperature TUL. The Bond albedo is A = 0.5, the emissivity  = 0.9. (Bottom) Temperature variations at the orbit time-scale (typically 9.5 h) for a solar elevation of B0 = 15° (same color coding). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

This new multi-scale thermal model of the B ring assumes heat transport by conduction through the solid phase of particles or through contacts and by radiation between particles or their constitutive grains. It treats very dense rings as packed ensembles of spheres, similar to an atmosphereless planetary surface covered with a regolith, but with finite thickness. It also separates particles from ring thermal properties with a multi-scale approach. It is aimed at modelling the thermal history of the B ring at both orbital and seasonal timescales. A tenable fit to seasonal temperatures variations of the B ring over 5 years requires restraining as far as possible the validity domains for the 10 model parameters. With only three primary constraints on the heat diffusion time from the lit to the unlit side, on the ring thermal inertia and its optical depth, a statistical approach already provides very interesting insights on the multi-scale properties and structure of the ring and how heat transfer may happen along seasons. As most improbable, insulating regolith structures (I and B) are rejected. Even if porous, a conductive structure of the particles is required for the ring to diffuse heat efficiently. Their effective conductivity may have to be about 0.1–1 W/m/K (Fig. 6). Conduction by radiation or contacts appears negligible in the particle regolith relatively to solid conduction, so that constraints can hardly be set on the regolith grain size R1 via this thermal model (Fig. 5). As a consequence, the solutions are not sensitive to the contact theory used and the grain size R1 may be fixed to any value below 10 mm. The B ring exhibits a slight thermal phase effect of about 10 K between 0° and 130° of phase angle, with an almost constant temperature above 50° (Altobelli et al., 2008). The present model can hardly reproduce such an anisotropy in emission as it assumes that particles within layers are isothermal. This approximation seems to be valid at least on the unlit side layer where no phase effect is observed (Altobelli et al., 2008). Introducing an anisotropy in thermal emission on the upper layer of the lit side is possible by adding a dependence of the non-shadowed fractional area on the local hour angle, i.e. C(s, l0, a). This will be considered and discussed in more details as the model will be compared to an extensive thermal dataset of the B ring. The thermal inertia of particles may be as high as a few hundred J/m2/K/s1/2, assuming they are made of solid icy grains. Lower thermal inertia C1 may also be found with less probability (Fig. 5). Considering the observed thermal inertia (15 J/m2/K/s1/2) as the particle one has long been a problem for former thermal models

38

C. Ferrari, E. Reffet / Icarus 223 (2013) 28–39

since it remains very difficult to explain how it can be two orders of magnitude lower than the thermal inertia of solid water ice (2100 J/m2/K/s1/2), even by accounting for both very high porosity and very low solid conductivity (Ferrari et al., 2005). Considering the observed thermal inertia as the ring one, like our present multi-scale model does, drives to the opposite conclusion of conducting particles with a moderate thermal inertia, easily explained by a regolith porosity p1 P 0.95. The low thermal inertia of the B ring with such conducting particles may be due to its porosity p0 P 0.75 (or filling factor D = 1  p0 6 0.25) and its effective conductivity, 0.01 6 KEFF0 6 0.1 W/m/K. Numerical simulations of ring dynamics suggest very flat rings with D  0.1  0.3 (Salo and Karjalainen, 2003; Salo and French, 2010), in perfect match with filling factors proposed here. At this point, the statistical approach cannot yet conclude for a specific ring vertical structure and how heat transfer may happen in the B ring, either driven by solid conduction of by radiation between conducting particles. It shows however that the ring is most probably only a few meters thick, certainly below 10 m. This is constrained by the diffusion time. The ring particle effective size R0 of 7–8 cm is best constrained when part of the heat transfer happens through radiation, as the radiative conductivity highly depends on it. The height-over-size ratio H/R0 might be as high as 33–55. Stellar occultation studies have indicated that the B ring appears is composed of self-gravity wakes in its densest part. The height-over-wavelength ratio Hw/kw is of the order of 1/15, given the height-over-width ratio Hw/Ww = 0.1 and the height-overspacing ration Hw/Sw = 0.2 (Colwell et al., 2007). Strong constraints on the ring thickness that may be provided by this thermal model coupled with the accurate determination of the aspect ratios of self-gravity wakes, yield direct constraints on their wavelength kw and surface density r0 at distance a as

r0 ¼ k w

X2 4p2 G

ð8Þ

where X is the orbital mean motion at distance a and G the gravitational constant, according to the theory. Ring surface densities are usually estimated from the damping of waves created by satellites at resonant locations. The study of the Mimas 4:2 bending wave in the very outer B ring set r0 = 54 ± 10 g/cm2 (Lissauer, 1985), while the Janus 2:1 density wave at a = 96,246 km yielded r0 = 70 ± 10 g/cm2 (Esposito et al., 1983), an increase which is compatible with the outer B ring probing if wavelength is constant with distance. Assuming an aspect ratio Hw/kw = 1/15 and a range of ring thickness H of 2.8–4 m, as supported here, yield surface densities between 53 and 76 g/cm2. This derived range is fully compatible with the above measurements. Temperature measurements of both lit and unlit sides of the B ring now exist as a function of the solar elevation but also as a function of distance to Saturn. Similar thermal sounding of the ring thickness may then be driven all along the B ring whereas very few waves can be detected, and used to estimate the surface density as a function of distance. This multi-scale thermal model appears promising at revealing the inner structure of the thick B ring of Saturn. A confrontation of the model to an extended seasonal dataset of temporal temperature variations is now required to refine the constraints on the structural and thermal properties of this ring (Reffet and Ferrari, in preparation). References Altobelli, N. et al., 2008. Thermal phase curves observed in Saturn’s main rings by Cassini-CIRS: detection of an opposition effect ? Geophys. Res. Lett. 36, L10105. Bahrami, M., Yovanovich, M.M., Culham, J.R., 2006. Effective thermal conductivity of rough spherical packed beds. Int. J. Heat Mass Transfer 49, 3691–3701.

Breitbach, G., Barthels, H., 1980. The radiant heat transfer in the high temperature reactor core after failure of the after heat removal systems. Nucl. Technol. 49, 392–399. Bruggeman, D.A.G., 1935. Berechnung Verschiederner Physikalischer Konstanten von Hetrogenen Substanzen. Annalen der physik 24, 636–664. Buonanno, G., Carotenuto, A., 1996. The effective thermal conductivity of a porous medium with interconnected particles. Int. J. Heat Mass Transfer 40, 393– 405. Colwell, J.E., Esposito, L.W., Sremcevic, M., Stewart, G.R., McClintock, W.E., 2007. Self-gravity wakes and radial structure of Saturn’s B ring. Icarus 190, 127– 144. Colwell, J.E., Nicholson, P.D., Tiscarino, M.S., Murray, C.D., French, R.G., Marouf, E.A., 2009. The structure of Saturn’s rings. In: Dougherty, M.K., Esposito, L.W., Krimigis, S.M. (Eds.), Saturn from Cassini–Huygens. Springer, pp. 375–412. Esposito, L.W., O’Callaghan, M., West, R.A., 1983. 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