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Icarus 218 (2012) 60-68

Contents lists available at SciVerse ScienceDirect

Icarus
j o u r n a l hom ep ag e: www. el se vi er .com /l ocate / i c aru s

Adhesion and collisional release of particles in dense planetary rings
Anna Bodrova
a b c

a,b,

, J?rgen Schmidt a, Frank Spahn a, Nikolay Brilliantov

c

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany Department of Physics, Moscow State University, Leninskie Gory, 119991 Moscow, Russia Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom

article

info

abstract
We propose a simple theoretical model for aggregative and fragmentative collisions in Saturn's dense rings. In this model the ring matter consists of a bimodal size distribution: large (meter sized) boulders and a population of smaller particles (tens of centimeters down to dust). The small particles can adhesively stick to the boulders and can be released as debris in binary collisions of their carriers. To quantify the adhesion force we use the JKR theory (Johnson, K., Kendall, K., Roberts, A. [1971]. Proc. R. Soc. Lond. A 324, 301-313). The rates of release and adsorption of particles are calculated, depending on material parameters, sizes, and plausible velocity dispersions of carriers and debris particles. In steady state we obtain an expression for the amount of free debris relative to the fraction still attached to the carriers. In terms of this conceptually simple model a paucity of subcentimeter particles in Saturn's rings (French, R.G., Nicholson, P.D. [2000]. Icarus 145, 502-523; Marouf, E. et al. [2008]. Abstracts for ``Saturn after Cassini-Huygens'' Symposium, Imperial College London, UK, July 28 to August 1, p. 113) can be understood as a consequence of the increasing strength of adhesion (relative to inertial forces) for decreasing particle size. In this case particles smaller than a certain critical radius remain tightly attached to the surfaces of larger boulders, even when the boulders collide at their typical speed. Furthermore, we find that already a mildly increased velocity dispersion of the carrier-particles may significantly enhance the fraction of free debris particles, in this way increasing the optical depth of the system. ã 2011 Elsevier Inc. All rights reserved.

Article history: Received 31 May 2011 Revised 28 October 2011 Accepted 5 November 2011 Available online 28 November 2011 Keywords: Planetary rings Saturn, Rings Collisional physics

1. Introduction Saturn's dense ring system consists of a large number of water ice particles, which, owing to frequent mutual dissipative collisions, flattened into a thin disc (Colwell et al., 2009; Cuzzi et al., 2009; Schmidt et al., 2009). Light scattering properties of the rings are consistent with a power-law size distribution n(r) $ rÐb with a slope near b = 3, ranging from centimeters to tens of meters, varying with the radial distance from the planet (French and Nicholson, 2000; Showalter and Nicholson, 1990; Zebker et al., 1985 see also Cuzzi et al., 2009; Schmidt et al., 2009). In particular, a lack of particles smaller than, roughly, 1 cm (French and Nicholson, 2000) is implied by comparison of the optical depth of the main rings at different wavelength (from microwaves to UV), and detailed photometry of the A ring (Dones et al., 1993). Such a size distribution may arise from a balance of aggregation between particles and fragmentative collisions of aggregates (Davis et al., 1984; Weidenschilling et al., 1984). The adhesive forces, leading to aggregation, are strongest for small particles (Chokshi et al., 1993). Although small debris should be steadily created in hypervelocity impacts of micrometeor Corresponding author at: Department of Physics, Moscow State University, Leninskie Gory, 119991 Moscow, Russia. E-mail address: bodrova@polly.phys.msu.ru (A. Bodrova).
0019-1035/$ - see front matter ã 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2011.11.011

oids on the rings (Cuzzi and Durisen, 1990; Durisen, 1984), it seems plausible that it is rapidly and permanently adsorbed on the surfaces of the larger particles, which would explain the small contribution of this particle size to the optical depth of the rings. This scenario was qualitatively discussed by Dones et al. (1993), who estimated the critical radius for sticking using the adhesion model proposed by Chokshi et al. (1993). In this paper we investigate this idea quantitatively in terms of a kinetic model with a bimodal distribution of particle sizes, for binary adhesive and dissipative particle collisions. We consider consistently the kinetics of coagulation and fragmentation of particles. The balance of fragmentation of particles and their gravitational aggregation, leading to the establishment of the power-law size distribution in Saturn's rings was studied by Longaretti (1989). A simple model for ballistic aggregation and fragmentation was introduced recently by Brilliantov et al. (2009), however, without specifying a detailed mechanism of fragmentation. We discuss the possibility that an increased velocity dispersion of the carrier-particles may significantly enhance the fraction of free debris particles, in this way increasing the optical depth of the system. Such a process may be important to understand quantitatively the observed brightness variations in perturbed ring regions, like satellite resonances (Dones et al., 1993; Nicholson Ä et al., 2008) and the lobes of the propellers (Spahn and Sremcevic,


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A. Bodrova et al. / Icarus 218 (2012) 60-68

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Ä 2000) observed in the A ring (Sremcevic et al., 2007; Tiscareno et al., 2006). The plan of the paper is as follows. In Section 2 we present the theoretical model, and calculate release and sticking rates of debris particles on the surfaces of carriers. The balance of these processes is analyzed in Section 3, establishing a steady distribution of free debris, depending on the parameters of the model. We discuss our results and their applications to Saturn's rings in Section 4, and give our conclusions in Section 5.

! ! ! velocity of debris and carriers as V cd Ì V c Ð V d . Let us also denote rffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E V 2;d . the velocity dispersions of carriers and debris as v c;d Ì c Although the velocity distribution function of particles in Saturn's rings is anisotropic (Goldreich and Tremaine, 1978), we do not expect a significant influence of this anisotropy on the average outcome of aggregation and fragmentation. For sake of simplicity we assume here an isotropic Maxwellian distribution of random (thermal) speeds of particles

2. Model We consider a bimodal system consisting of identical large boulders (carriers) of radius R and smaller particles with a range of radii r ( R (Fig. 1). Due to adhesion small particles can attach to the surface of bigger ones. We assume that boulders and debris particles are indestructible, while the fragmentation of the aggregates formed by carriers and attached debris is possible. Thus the total number of small particles of size r per unit volume Nd(r) is comprised of free debris with number density nv(r) and a population covering the surface of boulders with surface density ns(r). Then the mass conservation implies:

! f V c ;d Ì

3 2

!
2 c;d

3=2

pv

3V exp Ð 2v

2 c;d 2 c ;d

! : ?3î

Nd ?r î Ì nv ?rî? 4pR2 ns ?rîNc ;

? 1î

where Nc is the number density of carriers. We neglect the change of carrier size due to the attached debris particles. The equilibrium number density of small particles nv(r) (debris) arises from the balance of their aggregation with boulders and detachment in collisions of carriers. The kinetic equation, describing these processes reads

dnv ?r î Ì I? ?r îÐ IÐ ?r î; dt

? 2î

where I+ is debris production rate (the number density of debris of radius r, released per unit time in unit volume), IÐ - their adsorption rate (the number density of smaller particles, adsorbed at the surface of carriers per unit time in unit volume). I+ and IÐ are calculated below. We neglect the interaction of debris particles with each other. ! ! Let V c and V d be the velocities of carriers and debris particles measured in a frame that co-rotates with the local Keplerian speed at given distance from the central planet. We define the relative

For simplicity we assume, that the velocity dispersion vd is equal for debris of all radii (Salo, 1992b). We also do not take into account the deviation of velocity distribution function from the Maxwellian (which corresponds to the Rayleigh distribution in orbit elements), found by Ohtsuki and Emori (2000) in the computer simulation of dense self-gravitating systems. We assume, that all particles are composed of ice with Young modulus Y =7 Ò 109 Pa and Poisson ratio m = 0.25 (Chokshi et al., 1993). It is also common to introduce the elastic constant 2 D Ì 3 ?1Ðm î Ì 2 ÒÐ10 PaÐ1 . The material densities of carriers and Y 2 smaller particles are also assumed to be equal: qc = qd = 900 kg/m3. The adhesive interactions are characterized by the adhesion coefficient c, which is twice the surface tension coefficient. The adhesion coefficient was estimated theoretically for pure ice surfaces as c = 0.74 N/m (Chokshi et al., 1993). Hatzes et al. (1991) have shown experimentally, that adhesive interactions of ice particles significantly depend on their surface structure. In the recent experiments of Gundlach et al. (2011) the adhesion coefficient of micrometer-sized ice particles was found to be c = 0.19 N/m. Adhesive interactions become important only in the final phase of a collision, while generally the impact phase is governed by elastic and dissipative forces. The latter tend to damp the relative motion. These dissipative losses are usually quantified by the restitution coefficient, defined as the ratio of the normal components of the relative velocity after and before an impact: ! ! ! e Ì V 0cd Ñ ~ = V cd Ñ ~ , where V 0cd is the relative velocity after e e the collision and ~ is a unit vector, along the line, joining centres e of particles during their impact. In general, the restitution coefficient is velocity-dependent (Bridges et al., 1984; Brilliantov et al., 1996; Ramirez et al., 1999), but in the present study we assume it to be constant. We take here e ' 0.3 as a typical value for the restitution coefficient for ice particles, colliding with relative velocities $1 cm/s and effective radii $1cm (Bridges et al., 1984). 2.1. Debris adsorption rate ! If V cd is small enough, carriers and debris can form aggregates due to adhesion. The adhesive interaction can be described in the framework of JKR theory (Johnson et al., 1971). The force, acting during the collision of a boulder with a smaller particle, consists of two parts (Johnson et al., 1971):

Fa Ì
Fig. 1. A bimodal system consisting of carriers of radius R $ 1 m and debris of radii r < 10 cm. The debris can either move freely or be attached to the surface of carriers due to adhesive forces. Debris can be released during the collision of carriers, if the normal component of inertial force, acting on debris Fin cos h (Eq. (14)) exceeds the adhesion force Fsep (Eq. (7)). (Here h is the angle between the line, connecting the centres of colliding carriers and the line, connecting the centres of the carrier and the debris particle, resting on its surface.) Debris particles can be re-adsorbed in subsequent collisions.

a3 Ð DRcd

rffiffiffiffiffiffiffiffiffi 6pc 3=2 a: D

?4î

The first term corresponds to the Herzian elastic force, the second one - to adhesion. Here a is the contact radius of the colliding particles, Rcd = Rr/(R + r) - their effective radius. The work, required to release a bound debris particle from a carrier surface, can be written in the form (Brilliantov et al., 2007; Brilliantov and Spahn, 2006):


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Wa Ì

Z

a

sep

F

a

a0

1=3 dn da Ì q0 p5 c5 D2 ?Rcd î4=3 ; da

q0 ' 1:457:

? 5î

Here the deformation n is related to the contact radius according to (Johnson et al., 1971):

ratio of the kinetic coefficients IÐ and I+, the effect is expected to cancel out. Performing the integration with the velocity distribution function (3), we obtain:

a2 nÌ Ð Rcd

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8pcDa : 3

IÐ ?r î Ì A?r înv ?r î;
where

?10î

? 6î

The contact radius and the deformation in the equilibrium (Fa(a0)= 0) can be written in the following way: a0 Ì ?6pcDR2 î3 ; n?a0 î Ì eff 1 21 3 1 ?6pcDî3 R3 . The contact radius of separation asep Ì 3 pcDR2 can eff 3 2 be found from the condition: dFa/da = 0. It corresponds to the minimal value of adhesive force (Eq. (4)) or to maximal external pulling Fsep, which the adhesive interactions between carriers and debris can resist (Johnson et al., 1971):
1

A?r î Ì

Nc ?Rî?R ? r î2 ?v 2 ? v 2 î1=2 c d 3W a ? r î 1 3W a ?rî exp Ð 2 Ò 1Ð 1? 2 e mcd v 2 ? v 2 e mcd c d
2 c



8p 3

1

=2

v

2 c

1 ?

!

v

2 d

: ?11î

Expanding the exponent for r ( R or W a ( mcd ?v ? the following expression for the kinetic coefficient A:

v

2 d

î, one gets

F

sep

Ì ÐF a ?asep î Ì

3 pcRcd : 2

? 7î

pffiffiffiffiffiffiffi W 2 ?r îNc R2 1 A?r î Ì 3 6p a 4 2 e md Ðv 2 ? v 2 Ñ c d
2.2. Collisional production of debris

3 =2

$r

Ð10=3

:

?12î

If the kinetic energy of colliding boulders and debris with effective mass mcd = mcmd/(mc + md) (here mc and mc are respectively the mass of carrier and of a debris particle) in the final stage of ;2 the collision W kin Ì 1 mcd V 0cd is too small to overcome the adhesion 2 barrier Wa P Wkin, then

V

cd

sffiffiffiffiffiffiffiffiffiffiffiffi 2W a < ga Ì $r mcd e2

Ð5=6

;

? 8î

and the particles stick (Brilliantov and Spahn, 2006). The dependence of ga on the radius of debris r is depicted in Fig. 2. The smaller the particles, the easier they stick to the surface of larger ones. The amount of debris particles per unit volume, attaching to a carrier per unit time (debris adsorption rate) reads

Let us consider a collision of two boulders, covered by small ! ! particles. Let ~1 and ~2 denote their positions and V c1 ; V c2 their r r velocities. The vector connecting their centres at the collision instant is ~12 Ì ~1 Ð ~2 Ì r 12 Ñ ~, their relative velocity is r r r e ! ! ! V 12 Ì V c1 Ð V c2 , and the normal component of their relative ! e velocity reads: g Ì V 12 Ñ ~ . During the collision particles undergo the deformation n =2R Ð r12. For the purpose of calculation of debris production, we neglect the dissipative forces. Then the elastic force acting between the boulders during the collision obeys the Hertzian law:

Z ! ! !! IÐ ?r î Ì ?R ? r î2 Nc ?Rînv ?r î dV d dV c d~ V d f V c ef ! ! Ò V cd Ñ ~H Ð V cd Ñ ~ H?g a Ð V cd î e e

FH Ì ? 9î

pffiffiffiffiffiffiffiffi Reff 3=2 n: D

?13î

Here the Heaviside step-functions select particles which approach with the relative normal velocity, smaller than ga. Note that in Eq. (9) we do not account for effects induced by the flattening of the ring in the equatorial plane of the central planet. This implies the dependence of the particles density on the respective velocity dispersion. However, it will affect in the same manner the corresponding gain term I+ (Section 2.2). Because the steady state balance of free and adsorbed debris depends solely on the

Here Reff = R/2 is the effective radius of the carriers. We also denote the effective mass of colliding carriers as meff = mc/2. In the collision the maximal inertial force acting on debris particles sitting on the surface of carriers occurs in the moment, when the compression of the boulders is maximal:

F in Ì

1 1 F H ?nmax î md12 Ì md : r 2 2 meff

?14î

This maximal compression of boulders nmax is achieved, when the Rn elastic energy W el Ì 0 max F H dn equals the initial kinetic energy 1 meff g 2 of the normal component of the relative motion: 2

nmax Ì

5 Dmeff 4 R1=2 eff

!

2 =5

g

4 =5

:

?15î

We assume, that the debris particle leaves the surface of the carrier, if the projection of the maximal inertial force (Eq. (14)) on the carrier's surface normal (see Fig. 1) exceeds the maximal external pulling (Eq. (7)), which the adhesive interactions between carriers and debris can resist:

F in cos h P F

sep

?16î

Fig. 2. Maximal relative velocity of debris and carriers, leading to adhesion of debris ga (Eq. (8)) (dashed line) and minimal relative velocity of carriers gc, leading to the detachment of debris of size r from their surface (Eq. (17)) (solid line). The adhesion coefficient c = 0.74 N/m, the radius of carriers R = 6 m. In the upper corner of the diagram all collisions between carriers lead to detachment of debris, while the lower corner corresponds to sticky collisions between carriers and debris.

The debris can release from both boulders, from the side facing the collision partner. We ignore the process of immediate re-attachment of a debris particle on the collision partner. Small particles that are sitting on the contact surfaces of the colliding carriers are compressed during collision and do not release. However, the radius of a contact surface is much smaller, than radii of colliding icy boulders, therefore, this effect can be neglected. For simplicity we also neglect possible release of debris due to lateral motion of debris


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over the boulders' surface (rolling or sliding) induced by the action of inertial forces and shearing away of debris during grazing collisions of carriers. Substituting Eqs. (13)-(15) in Eq. (16) and setting cos h =1, we obtain the minimal relative velocity of carriers, leading to the detachment of debris of size r from their surface:

The solution of the equation has the form:

nv ?r; tî Ì nv ?r; 1î ? ?nv ?r ; 0îÐ nv ?r ; 1îî exp?Ðt=sî;
that is, nv(r) evolves exponentially to the steady-state value:

?29î

nv ?r; 1î Ì

g c ?r î Ì

?324pî pffiffiffi 5

1=3

Ð

c5=2 D

Ñ

1=3

qc q5=2 d

!

1=3



Reff 1?

5
r R

N d ?r î 1 ? A?r î=B?r î

?30î

=6

r

Ð5=3

?17î

with the relaxation time:

s Ì 1=?A ? Bî $ 1=Nc ?Rî:

?31î

The dependence of gc on the radius of debris r is depicted in Fig. 2. For smaller particles a larger impact velocity is needed to release them in an impact. The total number of small debris particles produced in a collision of two carriers is

Dns ?r î Ì 4pR2 ns ?r î

Zp
0

=2

Ð dh sin hH F in cos h Ð F

sep

Ñ

:

?18î

Evaluating the integral yields

6=5 ! g Dns ?r î Ì 4pR ns ?r îH?g Ð g c îÒ 1 Ð c : g
2

?19î

The relaxation time is inversely proportional to Nc(R), while the number density of free debris in the steady state nv(r, 1) does not depend on the number density of carriers Nc(R) (since both A $ Nc(R) and B $ Nc(R)). The fraction of free debris particles, calculated from Eq. (30) is shown in Fig. 3. Above a certain radius rcr ? 1 Dr all small particles 2 remain free, while below rcr Ð 1 Dr they are all attached to the sur2 face of carriers. The transition between regimes of completely adsorbed and completely free debris is very sharp. The critical radius of transition rcr can be roughly determined as the point where nv/Nd = 1/2 or A(rcr) = B(rcr) in the following way:

The total number density of debris particles of size r, released per unit time and unit volume in boulders collision, is given by:

! ! !! I? ?r î Ì ?2Rî2 N2 ?Rî dV c1 dV c2 d~ V c1 f V c2 ef c ! ! Ò H Ð V 12 Ñ ~ V 12 Ñ ~Dns ?r î: e e
Performing the integration, we obtain:

Z

Ð B1 ?r cr î rcr Ì ~cr ln r A?r cr î
where

3=10

;

?32î

?20î

~cr Ì r



3 5v

3
2 c

=10

1=2 1=5 9cReff 16pqc D : 3 4qd

?33î

I? Ì B?Nd Ð nv î rffiffiffiffiffiffiffiffiffi ! pv 2 5 53 7 2 c 1 ? a exp?ÐaîÐ a5 C ; a B Ì 16R Nc 2 2 5 3
with the incomplete Gamma-function:

?21î ?22î

Taking into account the fact that the last factor in Eq. (32) only weakly depends on rcr, we can approximate rcr with a sufficient accuracy as

Ð B1 ?~cr î r rcr ' ~cr ln r A?~cr î r

3=10

:

?34î

C?x; aî Ì
where

Z
a

1

dyy

xÐ1 Ðy

e;

?23î

a?rî Ì

3g 2 ?r î c $r 4v 2 c

The transition becomes smoother with increasing rcr. This is seen in Fig. 3 and may be easily shown analytically. Let us assume, that the fraction d of debris of radius r cr Ð 1 Dr , as well as the fraction 1 Ð d of 2 debris of radius r cr ? 1 Dr, remains free, where d is some small num2 ber. The transition width Dr has then the following form:

Ð10=3

:

?24î

The expression (22) for B may be further simplified if we estimate the Gamma-function, using the steepest descend (Laplace) method, which yields:

3 ln ?1=d Ð 1î Dr ' rcr : 5 ln ?B1 ?r cr î=A?rcr îî

?35î

B ' B1 exp?Ðaî;
where

?25î

rffiffiffiffiffiffiffiffiffi B1 Ì 16R2 N
and
c

pv 2 3 2 c ; if a P 5 3 5a pv
3
2 c

?26î

rffiffiffiffiffiffiffiffiffi" B1 Ì 16R N Ò if a 6
2 c

3=5 # 5 5 29 a 3 2 ; ? a 1? aÐ Ða 2 2 50 10 4 ?27î

2 : 5

3. Steady state abundance of small particles (debris) The kinetic equation for the time evolution of the number density of free debris particles of size r reads:

dnv ?r ; t î Ì B?r î?Nd ?r îÐ nv ?r ; t îî Ð A?r înv ?r; tî: dt

?28î

Fig. 3. Dependence of the fraction of free debris nv/Nd in the steady state on debris radius r for different values of velocity dispersion of carriers: vc = 0.001 m/s (dashed line), 0.005 m/s (solid line) and 0.01 m/s (dotted line). The velocity dispersion of debris is vd = 0.01 m/s. The radius of boulders is taken as R = 6 m. Two regions are observed: if r < rcr, where rcr is some critical radius, practically all debris are adsorbed, while all particles of radius r > rcr are free. The critical radius of transition rcr can be roughly determined as the point where nv/Nd = 1/2. The transition between two regimes is very sharp, moreover, the width of the transition Dr decreases with the decreasing rcr.


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Note, that if no fragmentation occurs in the system (B(r) = 0), nv(r, 1) becomes equal to zero: all debris will eventually stick to large particles even if vd ) ga. The critical radius rcr tends to infinity in this case. The magnitude of the critical radius rcr is primarily governed by the inertial forces, acting on debris during collision of carriers and leading to the detachment of debris. Therefore, according to Eqs. (32) and (33), the critical radius depends mostly on quantities, which are essential in the detachment processes (radii R and the velocity dispersion vc of carriers) and has much weaker (logarithmic) dependence on quantities, associated with aggregation (the velocity dispersion vd of debris). This statement is illustrated at Fig. 4. The larger the thermal velocity of colliding carriers, the smaller particles are released, therefore the critical radius significantly decreases with increasing of vc (left panel). In contrast, the dependence of rcr on the debris velocity dispersion is very week: Fig. 4, right panel shows no significant variations of rcr, when vd varies over several orders of magnitude. If the carriers become larger and more massive, their acceleration during the collision decreases and less debris leaves their surface. Therefore, the critical radius increases with increasing carrier size (Fig. 5, Eqs. (33) and (34)). The value of adhesion coefficient c = 0.74 N/m was estimated theoretically for collisions of completely smooth particles (Chokshi et al., 1993). The adhesion coefficient of realistic ice particles can be few times smaller (Gundlach et al., 2011), which leads to a decreasing rcr, due to reduction of the adhesion strength (see Figs. 4 and 5).

Fig. 5. Contour graph, showing the dependence of the critical radius for debris release rcr on velocity dispersion vc and radii R of carriers. Solid contour lines are for the adhesion coefficient c = 0.74 N/m (Chokshi et al., 1993) and the dashed lines for c = 0.19 N/m (Gundlach et al., 2011). Parameter ranges plausible for Saturn's rings are highlighted in grey.

4. Application to planetary rings Before we consider the application of our model to real systems it is worth to discuss some idealizations of the model and its impact on the adequate description of the planetary rings. 4.1. Justification of some model's idealizations In the present study we propose the following simplified model: The system of interest is comprised of a population of large particles, the carriers of the same size, and smaller particles, debris, with some size distribution. Aggregation between carriers and debris is driven by surface adhesion. For given velocity dispersions of the two populations, their sizes, and material properties, we quantify the amount of the steady-state fraction of free debris particles, i.e. those not attached to the carrier particles. The balance depends sensitively on the size of the debris particles. We obtain a critical radius, such that all smaller debris are practically

absorbed by the carriers, while debris larger than the critical radius are all free. In our model all debris particles are assumed to stick on the surfaces of carriers directly. There is no interaction between debris particles themselves, i.e. we neglect the process of particles attaching to debris that already sticks on a carrier, as well as the collisional aggregation of small particles. Effectively, these processes should tend to make the absorption of debris more efficient, since the relative strength of adhesion (compared to inertial forces) increases for smaller particles. Thus, qualitatively the inclusion of these processes should result in a somewhat larger critical radius than the one given directly by the model. We also neglect the release of debris in collisions of carriers by lateral, shearing forces, and by the direct action of the carriers, shrugging off debris from their collision partner during a collision. These processes would make the collisional release more efficient and should work in the direction of a smaller value of the critical radius. One may expect that the two neglected effects compensate each other. We can qualitatively estimate the influence of tangential motion of carriers in the following way. The maximal inertial force, acting on debris particles, cannot exceed

F in Ì

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the total force F total Ì F 2 ? F 2 consists of Herzian force FH H T and tangential force FT. The tangential force FT 6 lFH, where l is

1 F total ?nmax î md ; 2 meff

?36î

Fig. 4. Dependence of the panel). The critical radius r vc. The radius of carriers R c = 0.19 N/m (dashed line)

cr

critical radius rcr on velocity dispersion of carriers vc for vd = 0.01 m/s (left panel) and on velocity dispersion of debris vd for vc = 0.005 m/s (right does not change significantly even if vd varies in the range of several orders, however, it sensitively depends on the velocity dispersion of carriers = 6 m. The adhesion coefficient c = 0.74 N/m (solid line) corresponds to theoretical value, calculated for a pure ice surface (Chokshi et al., 1993), was found in experiments of Gundlach et al. (2011).


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the friction coefficient. So if we take into account tangential interactions, the maximal inertial force, acting on debris, will be only ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ? l2 times larger. Taking into account that the friction coefficient of ice is estimated to be of the order of l = 0.03 according to the experimental data (Kietzig et al., 2010; Oksanen and Keinonen, 1982; Serway, 1995), it may be concluded that the effect of tangential interactions is negligible. One should note, however, that the friction coefficient of ice depends on temperature, velocity between surfaces and normal load (Kietzig et al., 2010; Oksanen and Keinonen, 1982), and the effective friction coefficient of actual ring particles is unknown. An impact of the rotational degrees of freedom may be also easily estimated. Indeed, the centrifugal force Fcen = 2Rmeffx2, where x % g/2R is the characteristic rotational velocity of carriers, and g - their normal velocity, is significantly smaller than a typical Herzffi pffiffiffiffiffiffiffi ian force Reff n3=2 =D for carriers of size up to 20 m. Therefore we do not take the centrifugal force into account in the present study. Finally, we neglect any tidal forces and self-gravity in the system. Tidal forces should work against adhesion, leading to a lower critical radius, while self-gravity would enhance the tendency for clumping. Realistically, in Saturn's rings one expects the formation of self-gravity wakes (Salo, 1992a), i.e. clusters of ring particles, formed by gravitational instability, temporally bound but destroyed by shear when revolving under differential rotation around the planet. From the viewpoint of our model, one would expect the carrier particles to be arranged in such a wake pattern. In the wakes the population of debris particles, smaller than rcr, should be absorbed on the carriers, while there still exists free debris in the rarefied regions between the wakes. The simplified kinetics of our model then describes average properties of the rings, averaged over a spatial domain larger than the typical length scale of the wake pattern. Note that we assume that only adhesive forces are responsible for the adsorption of small particles on the large ones, that is we, ignore gravitational forces. This is justified, since a simple analysis shows, that for the sizes of carriers (less than $10 m) and debris (less than $0.1 m) and for the values of adhesion coefficient, adopted in the present work, the gravitational forces indeed may be neglected, see also Fig. 1 of Albers and Spahn (2006). Strictly, our mathematical approach applies to a system with a bimodal size distribution. However, if we neglect any debris-debris interaction, the results can be interpreted for a system with a continuous size distribution of each of the sub-populations consisting of debris particles or carriers. On the one hand, if we vary the size of the debris particles, we observe a very abrupt transition from a state where practically all debris is attached to the carriers to a state where all debris is free (Fig. 3). This leads us to the definition of the critical radius, where we expect for a system with a distribution of debris particle sizes that all debris smaller than rcr is absorbed. On the other hand, we observe a monotonic increase of the critical radius with increasing carrier size (Eqs. (33) and (34)) so that for two carrier sizes R(1) < R(2) in the system we expect ?1î ?2î ?1î ?2î r cr < r cr . Thus, in steady state all particles rcr < r < rcr , which (1) cannot be absorbed by carriers of size R , will eventually be absorbed by the larger carriers of size R(2). Therefore, the effective ?2î critical radius for the total system is r cr . More generally, for a system with continuous distribution of carrier sizes we expect that the effective rcr depends mostly on the largest carrier size Rmax and we expect that the results presented in Fig. 3-5 hold, if we replace R by Rmax. In this case, the velocity dispersion of the largest carriers is on the order of a few times XRmax and the velocity dispersion of smaller carriers increases mildly with decreasing radius (Salo, 1992b). In such a way, the arguments developed in this paper for a bimodal system can be extended to the description of debris adsorption in systems with a continuous power-law size distribution n(r) = n0rÐb.

The maximal size of the main population of Saturn's ring particles is constrained to be on the order of Rmax % 1 ÑÑÑ 10 m (French and Nicholson, 2000; Zebker et al., 1985). The ring also contains a population of larger embedded moonlets, tens to hundreds of me Ä ters in size (Sremcevic et al., 2007; Tiscareno et al., 2006, 2008). However, these moonlets are rare and their distribution falls of much steeper towards larger sizes than for the population of main ring particles. Thus, for the moonlets the timescale to establish a collisional balance of debris absorption and release, Eq. (31) is huge and their effect on the amount of free debris particles can be neglected. 4.2. Absorption of dust in dense rings Photometry of Saturn's A ring implies a very small fraction of free dust, if any. This conclusion seems to hold more generally for parts of the ring system with optical depth s > 0.1 but also for the dense rings of Uranus and Neptune (see discussion in Dones et al. (1993)). However, if not present in the system, dust should be rapidly replenished as the result of micrometeoroid bombardment on the rings (Cuzzi and Durisen, 1990; Durisen, 1984; Ip, 1983). One possibility is that non-gravitational forces rapidly remove dust from the system, such as electromagnetic and radiation forces (Burns et al., 1979; Burns et al., 1984; Rubincam, 2006; Ä Vokrouhlicky et al., 2007). Although these forces can provide significant perturbations for micron sized particles, however, the unavoidable collisional interaction of the dust and macroscopic ring particles should enforce a locked dynamical evolution of both species and a preferential removal of dust by these processes appears unlikely. Alternatively, it has been suggested that sweep up of dust particles and their sticking on larger ring particles is responsible for the paucity of dust (Dones et al., 1993). This conclusion is supported by our results. In Fig. 6 we show the (maximal) carrier size R and velocity dispersions consistent with a micron sized critical radius rcr. In fact, the release of micron sized dust from ring particles would require either implausibly large collision speeds or implausibly small radii of the carrier population. This large distance of the boundary for dust release from the domain of realistic ring parameters may explain why no free dust component is seen even in the highly perturbed regions of Saturn's main rings, such as the outer A ring beyond the Keeler gap, or the density waves at the strongest resonances in the rings (Dones et al., 1993). The situation is different for the F ring of Saturn. Here, the confinement of the ring by Prometheus and Pandora, as well as the gravitational stirring of embedded moonlets and clumps (Cuzzi and Burns, 1988; Dermott, 1981; Kolvoord et al., 1990; Kolvoord

Fig. 6. The case of micron sized dust: carrier radius and velocity dispersion for fixed critical radius rcr = 10 lm. The adhesion coefficient is c = 0.19 N/m (Gundlach et al., 2011). The abundant release of microscopic particles may be relevant in perturbed regions of the F ring. In this case fairly large collision velocities for the carriers are required and dominant carrier sizes not larger than few millimeters.


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and Burns, 1992; Lissauer and Peale, 1986; Showalter and Burns, 1982; Showalter et al., 1992; Winter et al., 2007) can lead to a fairly large velocity dispersion as high as several meters per second. The size of small particles in F ring is estimated to be in the order of several microns (Hedman et al., 2011), which corresponds according to our model to the dominant carrier size not exceeding few millimeters (Fig. 6). The local or temporal perturbations may lead to detectable changes in the size distribution of the F ring dust (Hedman et al., 2010, 2011). Significant perturbations in the F ring, such as transitions of clumps of material through the core ring, are implied by Cassini data (Esposito et al., 2008; Charnoz, 2009), and they may occur even daily (Murray et al., 2008). 4.3. Lower size cut-off in Saturn's dense rings The lower size cut-off of the particles in Saturn's dense rings is to date probably best constrained (French and Nicholson, 2000) by a ring occultation of the star 28Sgr, observed from Earth (Nicholson et al., 2000), combined with Voyager photopolarimeter (PPS) stellar occultation data (Esposito et al., 1983). Assuming a power law size distribution, modelling of the scattered signal implies 1 cm as the lower size cut-off in the C ring and in the outer A ring, but 30 cm in the B ring and the inner to mid A ring (French and Nicholson, 2000). This is consistent within a factor of two with the analysis of radio occultations performed with the CASSINI Radio Science Subsystem (Marouf et al., 2008 see also Cuzzi et al., 2009), giving 5 mm in C and outer A, and 50 cm in the inner A ring. If adhesion is the origin for the observed lack of small particles in the rings, then our expression for the critical radius should give a prediction for the lower size cut-off. In Fig. 5 we show the critical radius for various values of the velocity dispersion and (largest) size of carrier particles. The dependence on other parameters is weak, and the values used are given in the figure caption. From the identification of gravitational wakes in large parts of Saturn's rings (Colwell et al., 2006, 2007; French et al., 2007; Hedman et al., 2007), it follows that the Toomre parameter for gravitational instability

might be lower. For comparison, we also show critical radii for c = 0.19 N/m (Gundlach et al., 2011). In this case we obtain somewhat smaller values for rcr around one to a few centimeters. In conclusion, our results show that the observed lower size cut-off in Saturn's rings is consistent with the process of adhesional sticking of small particles on larger ring particles, taking into account a kinetic balance of sticking and release. 4.4. Increase of optical depth in perturbed regions A local change of the steady state velocity dispersion of the ring matter can change the balance of adhesion and release of debris from the main ring particles. As a consequence, the input of energy and angular momentum in perturbed ring regions can lead to the collisional release of small debris particles, previously attached to larger ring particles, which could result in an observable increase of optical depth. This process was suggested as an explanation for the bright halos observed around the radial location of strong density waves in the A ring (Dones et al., 1993; Nicholson et al., 2008; Salo and Karjalainen, 2003). For example, the brightening measured in Dones et al. (1993) for the Mimas 5:3 resonance, was well fitted by Salo and Karjalainen (2003) with a photometric model. Salo and Karjalainen (2003) assumed the power-law size distribution with the slope b = 3 in the range from rcr = 50 cm to Rmax = 5 m in the background region, while the lower size cut-off was reduced to rcr = 15 cm in the resonance zone due to release of small debris. Another example are the propeller structures, forming as a response to the gravitational action of moonlets embedded in the Ä rings (Spahn and Sremcevic, 2000), which were observed in the A ring (Tiscareno et al., 2006). These structures appear brighter in the images, than the surrounding background, which indicates the increasing effective optical depth in the vicinity of propellers (Tiscareno et al., 2008). Tiscareno et al. (2008, 2010) suggested, that this phenomena may be attributed to the disruption of the self-gravity wakes by the moonlet. In the present manuscript we Ä consider another hypothesis, proposed by Sremcevic et al. (2007), which assumes, that the additional material can be released in those regions where the perturbation of the moonlet induces collisions with a somewhat larger speed, when compared to the unperturbed ring. An increase of the velocity of carriers by the factor of 10 can decrease the critical radius by a factor of several (Eqs. (33) and (34)). If the critical radius decreases from rcr,2 to rcr,1 then material in the size range rcr,1 < r < rcr,2 adds to the system the (dynamical) optical depth

QÌ

v

c

X

3:36GR

?37î

should assume values around 2, or slightly smaller values. Here, R is the surface mass density of the rings, G is the gravitational constant, and X the Kepler frequency. This gives a rough constraint for the velocity dispersion of the carriers

v

c

% 1:1 Ò 10

Ð3

mQ s2

!

R
500 kg=m2

# !" 2 Ò 10Ð4 sÐ1

X

?38î

On the other hand, dense, collision dominated rings should maintain a velocity dispersion on the order of X times particle diameter (Araki and Tremaine, 1986; Shukhman, 1984; Wisdom and Tremaine, 1988) so that

Ds Ì

Z
r

r

cr;2

pr2 n?rîdr

?40î

cr;1

v

c

% 4 Ò 10Ð3

m X s 2 Ò 10Ð4 s

!
Ð1

! R 10m

?39î

if n(r) is the size distribution of this material. If all ring material, including debris and carriers, follows the same size distribution in form of a power law n(r) $ rÐb, then the relative enhancement of the optical depth can be expressed as

D

Shu et al. (1985) have estimated the collision velocity in the region of density waves, excited by of Mimas's 5:3 resonance in the A ring, as 0.4-3 cm/s. In the figure we indicate plausible ranges for the velocity dispersion (0.1-1 cm/s) and largest particle sizes (5-15 m, (French and Nicholson, 2000; Marouf et al., 2008; Tyler et al., 1983) in grey. In this parameter range an adhesion coefficient of c = 0.74 N/m gives critical radii from a few centimeters to a few tens of centimeters. This value of the adhesion coefficient follows from calculation of the strength of hydrogen bonding of clean ice surfaces in contact (Chokshi et al., 1993). For more realistic surfaces the effective c

s r3Ð2b Ð r3Ð1b cr; cr; Ì s R3Ðb Ð r3Ð1b cr;

?41î

where R is the size of the largest carrier and b the slope of the size distribution. Examples for the so obtained estimate of the optical depth enhancements in perturbed ring regions are shown in Fig. 7. One expects Ds/s on the order of a few percent up to tens of a percent if the velocity dispersion is enhanced by a factor of five. For comparison we also show numbers derived from dynamical and photometric simulations of a propeller structure (Salo and Schmidt, Ä 2007) analyzed and compared to Cassini images by Sremcevic


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5. Conclusion and future work We have presented a simple theoretical model to describe processes of adsorption and detachment of small debris particles on the surface of larger carrier particles. We have shown, that below a certain critical radius all debris particles are resting on the surface of the carriers. For parameters plausible for Saturn's rings, the value of this critical radius is consistent with an absence of subcentimeter particles in the dense rings. Particles with dimension greater than the critical radius may move freely in space. The critical radius decreases with decreasing effective adhesion coefficient, carrier size, and especially with increasing relative velocity of colliding boulders. In such a way, a perturbed velocity dispersion in the vicinity of moonlets or at the locations of satellite resonances in the rings may lead to the release of additional debris particles, enhancing the optical depth and affecting the brightness of satellite-generated structures in the images. In future work we plan to improve the model of debris production, taking into account release of debris due to rolling (Dominik and Tielens, 1997), induced by the action of inertial forces. The theory could be extended by taking into account the dependence of velocity dispersion on the masses of particles. As for the comparison to data, it would be useful to evaluate the actual magnitude of the enhancement of velocity dispersions in perturbed regions of Saturn's rings. As a next step it would be worth to quantify the influence of released debris particles on the brightness of perturbed regions of the main rings and in certain regions of the Fring. Acknowledgments
Fig. 7. Upper panel: critical radius rcr(vc) and rcr for a 50% enhanced carrier velocity dispersion vc. Also shown are the expected relative changes of optical depth (from Eq. (41)) in the case of increasing of carrier velocity from vc to 1.5vc, for two values of the size slope b. Lower panel: relative change in optical depth depending on relative enhancement of vc for fixed vc = 2 mm/s. The size of the largest carrier R and the adhesion coefficient c have a small influence on the optical depth variation. For comparison we also show symbols (diamonds) from dynamical and photometric simulations of a propeller structure (Salo and Schmidt, 2007) analyzed and Ä compared to Cassini images by Sremcevic et al. (2007) (see text for details).

We thank Martin Seiï and Heikki Salo for fruitful discussions. We also thank Keiji Ohtsuki and an anonymous reviewer for the constructive reports that led to a significant improvement of the paper. This work was supported by the Deutscher Akademischer Austauschdienst (DAAD, Grant A/09/84185) and Deutsches Luft und Raumfahrtzentrum (DLR, Grant 50OH0901). References

et al. (2007). Here, debris particles were added to a ring background (dynamical optical depth s = 0.5, particle albedo - = 0.5, steady state velocity dispersion 2-3 mm/s, including self-gravity wakes) at those locations perturbed by an embedded moonlet where the perturbed impact velocity of ring particles exceeded 1 cm/s. The debris followed Keplerian trajectories, with probabilities for re-absorption given by the particle positions obtained from the dynamical simulations. The radii of debris particles then were scaled, to adjust for the optical depth necessary to match the Ä brightness of the propeller structure in images. Sremcevic et al. (2007) obtained for the debris an optical depth of Ds = 0.025, using an albedo of - = 0.9 for the debris particles. This value is shown in the lower panel of Fig. 7 as a diamond symbol. A roughly doubled optical depth of the debris is needed if debris particles have albedo - = 0.5 (upper diamond symbol). We note that the debris optical Ä depth from the simulations of Sremcevic et al. (2007) represents an average over the whole simulations box. Locally, the optical depth enhancement is higher or lower. Also, the criterion for debris release vc > 1 cm/s does not strictly correspond to the perturbed velocity dispersion, which was used to derive the theoretical curves in Fig. 7. Nevertheless, the comparison demonstrates that the optical depth perturbations predicted by the theory developed in this paper are at least in reasonable agreement with available preliminary analysis of perturbed ring regions in Cassini data.

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