Regolith depth growth on an icy body orbiting Saturn and evolution of bidirectional reflectance due to surface composition changes

Icarus 212 (2011) 268–274

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Regolith depth growth on an icy body orbiting Saturn and evolution of bidirectional reflectance due to surface composition changes Joshua P. Elliott a,1, Larry W. Esposito b,⇑ a b

ITT Visual Information Solutions, 4990 Pearl East Circle, Boulder, CO 80301, United States Laboratory for Atmospheric and Space Physics, University of Colorado, 392 UCB, Boulder, CO 80309, United States

a r t i c l e

i n f o

Article history: Received 30 October 2009 Revised 4 June 2010 Accepted 30 October 2010 Available online 11 December 2010 Keywords: Regoliths Saturn, Rings Ultraviolet observations

a b s t r a c t Using a Markov chain model, we consider the regolith growth on a small body in orbit around Saturn, subject to meteoritic bombardment, and assuming all impact ejecta are re-collected. We calculate the growth of regolith and the fractional pollution, assuming an initial pure ice body and amorphous carbon as a pollutant. We extend the meteorite flux of Cuzzi and Estrada (Cuzzi, J., Estrada, P. [1998]. Icarus 132, 1–35) to larger sizes to consider the effect of disruption of the moonlet on other moonlets in the ensemble. This is a relatively small effect, completely negligible for moonlets of 1 m radius. For the given impact model, fractional pollution reaches 22% for 1 m bodies, but only 3% for 10 m bodies, 1.7% for 20 m bodies, and 1% for 30 m bodies after 4 byr. By considering an ensemble of moonlets, which have identical cross-sections for releasing and capturing ejecta, this analysis can be extended to a model of particles in Saturn’s rings, where the calculated spectra can be compared to observed ring spectra. The measured spectral reflectance of Saturn’s rings from Cassini observations therefore constrains the size and age of the ring particles. The comparison between 1 m, 10 m, 20 m, and 30 m particles confirms that for larger ring mass, the current rings would be less polluted; for the largest particles, we expect negligible changes in the UV spectrum after 4 byr of meteoritic bombardment. We consider two end members for mixing of the meteoritic material: areal and intimate. Given the uncertainties in the actual mixing of the meteoritic infall and in its composition (as a worst case, we assume the meteoritic material is 100% amorphous carbon, intimately mixed) initially pure ice 30 m ring particles would darken after 4 byr of exposure by 15%. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction We examine how the bidirectional reflectance of an icy body, for example, a saturnian moonlet or ring particle, changes due to surface composition changes from meteoritic bombardment. Our model assumes that as a meteorite impact on the surface of the body, composed initially of water ice, excavates a hemisphere of icy material. This volume of ice (as well as the volume of the impactor) are then evenly distributed over the surface of the body. We assume that all ejected material effectively falls back to the surface because we assume the body exists within a ring of similarly sized bodies and that all material will recollect onto one or more of these bodies. Over large timescales the result is an average regolith depth on each body in the ring. The model defines the

⇑ Corresponding author. Address: Laboratory for Atmospheric and Space Physics, University of Colorado, 1234 Innovation Drive, Boulder, CO 80303, United States. Fax: +1 303 492 1132. E-mail addresses: [email protected] (J.P. Elliott), larry.esposito@lasp. colorado.edu (L.W. Esposito). 1 Fax: +1 303 402 4644. 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.10.031

thickness of this layer of ejecta to be the regolith depth, and tracks its time evolution. The respective volumes of the ice and the meteoritic material give the fractional pollution of the regolith as it changes over time. This fractional pollution evolution is input to Hapke’s (2002) model for bidirectional reflectance to calculate the evolution of the reflectance spectrum. 2. The Markov chain approach We model the regolith evolution as a stochastic process, using a Markov chain approach. A similar formalism has been applied for radiative transfer (Esposito and House, 1978; Esposito, 1979a,b) ring dynamics (Brophy and Esposito, 1989) and ring particle size evolution (Canup and Esposito, 1995, 1996, 1997; Colwell and Esposito, 1990a,b, 1992, 1993; Throop and Esposito, 1998; Barbara and Esposito, 2002). The general Markov chain is described in Kemeny and Snell (1960) and Krishnan (2006). We define a transition matrix P where the jth entry of the ith row represents the probability of transitioning from the ith state to the jth state after one time step, where the states are small ranges in regolith depth (we use both cm and mm intervals).

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We fit a power-law distribution to each segment of the impact flux distribution (see Table 1):

2.1. Markov chain We multiply the initial state vector x(0) (whose zeroth entry is unity and all others are zero to represent the initial state of having no regolith) by the matrix P, to obtain a vector of transition probabilities whose entries are the probabilities of being in state j (the jth height interval) after one time step. Successive multiplications yield distributions for further time steps, as indicated below:

xð0ÞP ¼ xð1Þ xð1ÞP ¼ xð2Þ

ð1Þ

xðt  1ÞP ¼ xðtÞ where the vector x(t) is the distribution of likelihood of being in each of the height intervals at time, t, i.e., after t steps. The following identity drastically reduces the number of matrix multiplications:

xð0ÞP t ¼ xðtÞ

2.2. Expectation value

jmax X

8 q1 > < b1 a ; amin 6 a 6 a1 _nðaÞ ¼ n0 b2 aq2 ; a1 6 a 6 a2 > : b3 aq3 ; a2 6 a

xj j

ð3Þ

j¼0

_ ¼ Fg  m

Z

3 cm

amin

4p 3 _ qa nðaÞda 3

ð6Þ

where Fg = 3 is the gravitational focusing factor for Saturn (Cuzzi and Estrada, 1998). Cuzzi and Estrada note that their mass flux may be uncertain by a factor of 3. The matrix elements for the transition matrix P are calculated as follows:

Pij ¼

A  Dt

R a;Dh¼jiþ1 a;Dh¼ji

_ nðaÞda; j > i; a 6 amax

0;

ð7Þ

j
where A is the surface area of the moonlet, Dt is the time step in which one impact is expected (defined below), and Dh is the change in regolith depth given below:

Dh ¼

1 12R2

ðH1 a  h0 Þ2 ð2H1 a þ h0 Þ

ð8Þ

from the formula for the volume of a hemispherical cap, where the cap is the volume of new excavated icy material:

3. Transition matrix

V¼

The crux of this model is to determine the transition matrix P. This matrix P incorporates the physics of our simulation and the time evolution. In our simulation, the transitions probabilities depend only on the current depth, and are independent of time: this defines the stochastic process as a Markov chain (Kemeny and Snell, 1960). The construction of its elements will now be discussed.

1 2 pb ð3r  bÞ 3

ð9Þ

where r is the radius of the crater and b is the thickness of the cap. In our case b = r  h0, where h0 is the current depth of the regolith so:

V¼

1 pðr  h0 Þ2 ð2r þ h0 Þ 3

3.1. Off-diagonal elements

Table 1 Coefficients.

Number Flux [cm-2sec-1]

10-10

Extrapolation

10-20 CE98 Fig 17

10-30

10-40 10-6 b

q

2.52  1019 9.85  1014 9.61  1020

2.26 1.04 3.91

ð10Þ

Meteoroid Number Flux

100

3.1.1. Without disruptions The basic driving force for the evolution is meteoroid bombardment (see Cuzzi and Estrada, 1998), who give the number flux distribution of incoming meteoroids as a function of radius a as a discrete distribution in units of impacts/cm2/s. Cuzzi and Estrada define this flux in Section 4.2 of their 1998 paper. We fit a broken power law function to their distribution in order to accurately integrate their distribution when calculating the transition matrix elements for our Markov chain.

1 2 3

ð5Þ

With the breakpoints a1 = 2.48  105 cm and a2 = 7.97  103 cm. See Fig. 1. The normalization, n0 can be found using the following equation and the value for the mass flux given in Cuzzi and Estrada (1998) which is 4.5  1017 g/cm2/s (the upper limit is 3 cm because this is the largest size in Cuzzi and Estrada (1998) and corresponds to that mass flux):

(

At each time step, the state distribution vector yields the probability distribution of the states in the Markov chain, that is, the probabilities of the system being found in each particular state j. The expectation value of the state vector gives the system mean as a function of time. The mean depth is given by

ð4Þ

with

ð2Þ

where t is the number time steps, P is the transition matrix, and x(0) is the initial state vector, and x(t) is the vector that represents the probability distribution after t time steps. By squaring the matrix repeatedly instead of successive multiplications we obtain a probability distribution for exponentially increasing times.

hji ¼

q _ nðaÞ ¼ ba

10-4

10-2

100

102

104

Meteoroid Radius [cm] Fig. 1. Cuzzi and Estrada (1998) meteoroid imparting flux distribution including our extension, and the new continuous power-law distribution, offset for clarity. The extension allows us to consider disruptive impacts (see text).

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We use a simple diameter to impactor scaling to define r in terms of a: r = H1a. We calculate H1 from the mass-yield ratio Y of excavated mass to impactor mass given by Cuzzi and Estrada (1998), Section 6.2.2. That is, we calculate H1 so that the volume of the mass in the hemispherical crater of radius, r, is Y times the mass of the impactor, assuming the same mass density for impactor, target and regolith.

Y ¼ 3  104 ¼

me 23 pr 3 qe 1 3 ¼ ¼ H mi 43 pa3 qi 2 1

ð11Þ

ffiffiffiffiffiffi r p 3 H1 ¼ ¼ 2Y  39 a

where me and mi are the masses of the excavated material and the impactor material and qe and qi are their respective densities, r is the radius of the excavated crater and a is the radius of the impactor. Since the surface area of the moonlet is:

A ¼ 4pR2

ð12Þ V , A

then the increase in regolith depth is Dh ¼ assuming the ejected material is spread uniformly over the surface of the moonlet. For the small objects considered (1–30 m radius) the ejecta will initially escape the moonlet, but since they cannot escape the planet’s gravity, they are soon recaptured. This justifies the assumption of the uniform coverage which gives Eq. (8). Dt is given by:

Dt ¼

A

1 R amax amin

_ nðaÞda

ð13Þ

where amin is the smallest size given in the Cuzzi and Estrada (1998) distribution (106 cm), and amax = R/H1 which is the size of impactor capable of disrupting the moonlet. Thus, the average number of non-disruptive impacts is one each time step.

since, all elements to the left of the diagonal vanish. The resultant transition matrix is an upper triangular matrix, with row sums of unity to machine accuracy. 3.1.4. Condensing the matrix by summing over depth bins One last item on the transition matrix P. As stated earlier, the jth entry in the ith row represents the transition probability of going from the ith state to the jth state. Each of these states covers a small range in regolith depth. For accuracy reasons we use 1 mm bins (1 cm bins yielded a slightly different result and was deemed less accurate), however this proved problematic when looking at the 10 m moonlet case, because the matrix has 10,000  10,000 elements, which proved too computer intensive and was time prohibitive. The solution was to create the initial 1 mm bin matrix and then condense it down to one with 1 cm bins, with a center-of-thebin summing approach. This was done by summing the elements 0–4 of the mm matrix into the zeroth bin of the cm matrix (this represents no increase in h), then elements 5–14 to the next bin of the cm matrix (this represents a depth increase of 1 cm), then 15–24, 25–34, etc., giving us our cm bin matrix, with the benefit of the initial transition probabilities calculated at 1 mm resolution. Testing shows good correspondence between the results of the uncondensed and condensed matrices. 3.2. Regolith depth Now we use P to examine the regolith growth rate. Using Eq. (2) we repeatedly square the matrix, and upon each squaring, we compute the vector of state probabilities (the probability distribution of regolith depth) and its expectation value. The expectation value of the depth of the regolith at each time step is given by:

hðtÞ ¼

jmax X

_ q xj ðtÞj þ F g mt=

ð17Þ

j¼0

3.1.2. Disruptions The disruption of the moonlet occurs if an impactor larger than amax strikes the moonlet where amax = R/H1, that is, where the predicted crater radius is as large as that of the icy body target. To account for this, we consider our test body to be one of an ensemble of N moonlets, each of radius R. When one moonlet is destroyed we assume that its volume is deposited uniformly onto the surfaces of the other moonlets. This produces a small change in regolith depth on the other moonlets. To calculate the probability of this occurring, we define:

P d ¼ N  A  Dt

Z

1

_ nðaÞda

where the summation is just Eq. (3) and the first term on the right hand side represents the regolith contributed by the new icy excavated material, and the second term is the small change in depth due to the meteoritic volume. Fg is the gravitational focusing. Figs. 2 and 3 illustrate how regolith depth changes over time. Four cases are examined. Each case is calculated as described in this paper, but with different moonlet radii. For illustrative purposes the second plot, Fig. 3, examines how the first two cases, 1 m and 10 m, change with and without the additional probability given by disruptions as in Eq. (14) changes the outcome. Fig. 3 plots

ð14Þ

amax

This probability is then added to the appropriate matrix element so that:

V R ¼ N  A 3N

ð15Þ

This quantity, when rounded to the nearest integer gives the index j = (i + Dhd). In the case of a disruption, this will be the index of the final state for the regolith depth after disruption. To this element of each row, Pd is added to include the effect of disruption on regolith growth. 3.1.3. Diagonal elements The diagonal elements represent the probability that no change occurs in a given time step. (When in state i, you remain in state i.) These elements are calculated:

Pi¼j ¼ 1 

jmax X j¼iþ1

Regolith Depth over Time

102

Depth [cm]

Dhd ¼

104

100

10-2

10-4 102

QO75 1m moonlet w/ Disruptions 10m moonlet w/ Disruptions 20m moonlet w/ Disruptions 30m moonlet w/ Disruptions

104

106

108

1010

Time [years]

Pij

ð16Þ

Fig. 2. Evolution of the regolith depth over time. The dashed line is from Quaide and Oberbeck (1975) for lunar regolith growth.

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Regolith Depth over Time

104

10-1

102

Depth [cm]

Fractional Pollution

100

1m moonlet 1m moonlet w/ Disruptions 10m moonlet 10m moonlet w/ Disruptions

10-2

10

0

10-3

10-2

QO75 PowerLaw mm-to-cm bins, 1m moonlet PowerLaw 1m moonlet w/ Disruptions PowerLaw mm-to-cm bins, 10m moonlet PowerLaw 10m moonlet w/ Disruptions

10-4 102

104

106

108

10-4

10-5 102

1010

104

Time [years] Fig. 3. To Pd or not to Pd? The effect of including the probability for disruption of the moonlet. Dashed: from Quaide and Oberbeck (1975).

two cases for each radius of moonlet, one with disruptions, and one without disruptions. The later plot illustrates an interesting point: For the 1 m moonlet case, no noticeable change occurs when adding in Pd, where as in the 10 m case, there is. This can be understood as due to the fact that for the 1 m case, Dhd is only a fraction of a cm, and for the 10 m case Dhd is around 3 cm. The larger particle size provides a significant source of mass through disruptions that increases the regolith depth. 4. Fractional pollution The above calculations indicate the composition of the regolith changes over time. This will allow us to calculate the bidirectional reflectance evolution (covered in the next section). We define the fractional pollution as the ratio of meteoritic material in the regolith to the total amount of regolith:

fp ¼ Pj max

_ q F g mt=

ð18Þ

_ j¼0 xj ðtÞj þ F g mt=q

Figs. 4 and 5 give the evolution of the fractional pollution for each size moonlet examined. As before, the second plot shows the difference in the fractional pollution associated with the use of Eq. (14), including disruptions. 10

10-1

1m moonlet w/ Disruptions 10m moonlet w/ Disruptions 20m moonlet w/ Disruptions 30m moonlet w/ Disruptions

10-2

10-3

10-4

10-5 102

104

106

108

1010

Fig. 5. Two moonlet sizes are plotted illustrating scenarios with and without disruptions to show the change in fractional pollution for the larger moonlet size.

5. Bidirectional reflectance The fractional pollution is now used to calculate how the bidirectional reflectance spectrum changes over time. The approach is taken from Hapke (1993, 2002), who calculates the bidirectional reflectance:

r¼

w 4p

l0 ½pðgÞBSH ðgÞ þ Mðl0 ; lÞBCB ðgÞ l0 þ l

ð19Þ

This equation has multiple parameters. The single scatter albedo is given by w. l0 and l are the cosines of the angles of incidence and emergence respectively. p(g) is the single term Henyey–Greenstein phase function, and g is the phase angle. BSH and BCB are the Shadow Hiding Opposition Effect and the Coherent Backscatter Opposition Effect respectively. And finally M(l0, l) is Hapke’s model from his 2002 paper for multiple scattering. This model is slightly different from his 1993 book, but it contains many of the same elements. One notable difference is the approximation to the H-function, which as incorporated into M(l0, l) in his 2002 paper is:

  1 1  2r0 x 1 þ x ln HðxÞ  1  wx r0 þ 2 x

ð20Þ

where x is the dummy variable for l0 or l. To exploit this model, we follow a similar approach to that used by Hendrix and Hansen (2007). Like Hendrix and Hansen we use spectral data from Warren (1984) for the real and imaginary indices of refraction of H2O and data from Zubko et al. (1996) for amorphous carbon, which is taken as a typical ‘‘pollutant’’, namely the meteoritic material impacting the surface of the moonlet. For simplicity, we assume the pollutant is 100% amorphous carbon. Realistic estimates of outer Solar System material may be only 1/3 carbon (Crukshank and Dallel Ore (2003) match the spectral reflectance of KBO’s and Centaurs with 1–60% amorphous carbon). Our results thus represent a more extreme case. Hapke’s model is dependant on the indices of refraction and the wavelength of light. In particular the absorption coefficient:

Fractional Pollution

0

106

Time [years]

108

1010

Time [years] Fig. 4. Fractional pollution from 1 m, 10 m, 20 m, and 30 m moonlet simulations. Bold points are calculated for 4 byr and are found by cubic-spline interpolation, these are for later comparison when examining the bidirectional reflectance in the next section.

a¼

4pk k

ð21Þ

where k is the imaginary index of refraction and k is the wavelength that corresponds to that index. When calculating the single scatter albedo, a becomes quite important.

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We have:

H2O and C mix, 1m moonlet w/ Disruptions, Areal Mixture

ð1  Si Þ w ¼ Q s ¼ Se þ ð1  Se Þ H ð1  Si HÞ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r i þ exp  aða þ sÞhDi H¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ r i exp  aða þ sÞhDi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  a=ða þ sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ri ¼ 1 þ a=ða þ sÞ Se ¼

ðn  1Þ2 þ k 2

ð22Þ

2 2

ðn þ 1Þ þ k 4 Si ¼ 1  nðn þ 1Þ2

where s is that internal path length, taken to be s  1  10–17 after Roush (1994), and hDi is taken to be the expectation value for the average regolith particle diameter. Tests with s = 0.3 show no noticeable difference. In this model, the modeled fractional pollution from above gives an estimate of how the bidirectional reflectance changes over time on our simulated icy body. Two mixing models are shown in Sections 5.1 and 5.2 for areal and intimate mixtures respectively. Fig. 6 shows the unpolluted ice spectrum in the UV range for various regolith grain sizes. This plot is essentially identical to one in Hendrix and Hansen (2007). 5.1. Bidirectional reflectance – areal mixture model To get a sense of the evolution over time, we first use a simple areal style mixing model, assuming regions of only H2O and regions of only carbon. Using this model it is fairly easy to mix the two spectra for H2O and carbon: It is just a simple sum of the two. Fig. 7 shows the spectral evolution due to meteoritic bombardment over long timescales for two different grain sizes, assuming the areal style mix. The series of plots below illustrates the evolution of the spectrum as the fractional pollution changes over time for different moonlet sizes. For each plot, the thick solid lines represent the spectrum at t = 0, while the dashed lines represent the spectrum at t = 4  109 years. The thinner dotted lines are incremental changes in the spectrum at given times in exponentially increasing time steps. As indicated above, the value of the fractional pollution at 4 byr has been calculated, and this is the value used in the calculation of the final spectrum line on each plot, this allows an easy comparison of the different cases. For simplicity, only two regolith grain sizes have been taken into account in these plots, 1 lm (red) and 15 lm (blue). For

Bidirectional Reflectance

1.000

0.100

0.010

0.001 0.10

0.12

0.14

0.16

0.18

0.20

Wavelength [µm] Fig. 7. Bidirectional reflectance spectrum from Hapke (2002) for bombardment by amorphous carbon at the Cuzzi and Estrada (1998) rate using an areal mixture model. Red: 1 lm grains. Blue: 15 lm grains. Successive curves show time n evolution from 0 to 4 byr. The time steps used are T ¼ Dtð2 Þ . The last time step at 4 byr is calculated based on the fractional pollution at 4 byr, which is calculated by cubic-spline interpolation as shown in Fig. 4.

each calculation we assume the same grain size for both the water ice and carbon constituents. Each plot shows this evolution for a particular moonlet size on logarithmic axes. Note that for larger moonlet sizes, the change is negligible. One cannot easily discern visually the change in reflectance over time by looking at the plots for moonlets of 10 m in size or greater. See Figs. 7–10. 5.2. Bidirectional reflectance – intimate mixture model If the regolith grains are intimately mixed, then an areal model can no longer be used. Hapke (1993) and others (Clark, 1983; Clark and Lucey, 1984) have shown that a dark pollutant intimately mixed with lighter materials can reduce the observed reflectance by a much greater amount than that which results from an areal mixture. To examine the bidirectional reflectance of an intimate regolith mixture we use the intimate mixing model given by Hapke (1993):

P

Mj wj j qj Dj

w¼ P

Mj j qj Dj

P

j

ð23Þ

M j wj pj ðgÞ qj Dj

pðgÞ ¼ P

Water Ice

0.4

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp=22.6% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp=22.6%

M j wj j qj Dj

H2O and C mix, 10m moonlet w/ Disruptions, Areal Mixture

0.3

Bidirectional Reflectance

Bidirectional Reflectance

1.000

0.2 1µm

15µm

0.1

0.0 0.10

0.12

0.14

0.16

0.18

0.20

Wavelength [µm]

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 3.4% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 3.4%

0.100

0.010

0.001 0.10

0.12

0.14

0.16

Wavelength [µm] Fig. 6. Water ice spectrum profile for varying grain sizes from 1 lm to 15 lm. From Warren (1984).

Fig. 8. Same as Fig. 7 for 10 m bodies.

0.18

0.20

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H2O and C mix, 20m moonlet w/ Disruptions, Areal Mixture

H2O and C mix, 10m moonlet w/ Disruptions, Intimate Mixture

1.000

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 1.7% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 1.7%

Bidirectional Reflectance

Bidirectional Reflectance

1.000

0.100

0.010

0.001 0.10

0.12

0.14

0.16

0.18

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 3.4% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 3.4%

0.100

0.010

0.001 0.10

0.20

0.12

0.14

0.16

0.18

0.20

Wavelength [µm]

Wavelength [µm]

Fig. 12. Same as Fig. 11 for 10 m bodies.

Fig. 9. Same as Fig. 7 for 20 m bodies.

H2O and C mix, 30m moonlet w/ Disruptions, Areal Mixture H2O and C mix, 20m moonlet w/ Disruptions, Intimate Mixture 1.000

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 1.0% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 1.0%

Bidirectional Reflectance

Bidirectional Reflectance

1.000

0.100

0.010

0.001 0.10

0.12

0.14

0.16

0.18

0.20

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 1.7% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 1.7%

0.100

0.010

0.001 0.10

Wavelength [µm]

0.12

0.14

0.16

0.18

0.20

Wavelength [µm] Fig. 10. Same as Fig. 7 for 30 m bodies. Fig. 13. Same as Fig. 11 for 20 m bodies.

where M is the mass fraction, q is the particle density, and D is the particle diameter. Using this model, we again examine the cases of 1 lm and 15 lm regolith grain sizes on 1 m, 10 m, 20 m, and 30 m moonlets. For this intimate mixture model, one notices that the

bidirectional reflectance still has a noticeable change at longer wavelengths at large timescales. For the larger moonlet sizes, this change is fairly small even at 4 byr. See Figs. 11–14.

H2O and C mix, 1m moonlet w/ Disruptions, Intimate Mixture H2O and C mix, 30m moonlet w/ Disruptions, Intimate Mixture

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp=22.6% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp=22.6%

1.000

Bidirectional Reflectance

Bidirectional Reflectance

1.000

0.100

0.010

0.001 0.10

0.12

0.14

0.16

0.18

t = 0 years ; 1µ grains ; Fp=0% t = 4e9 years ; 1µ grains ; Fp= 1.0% t = 0 years ; 15µ grains ; Fp=0% t = 4e9 years ; 15µ grains ; Fp= 1.0%

0.100

0.010

0.20

Wavelength [µm] Fig. 11. Bidirectional reflectance spectrum from Hapke (2002) for bombardment by amorphous carbon at the Cuzzi and Estrada (1998) rate using an intimate mixture model.

0.001 0.10

0.12

0.14

0.16

0.18

Wavelength [µm] Fig. 14. Same as Fig. 11 for 30 m bodies.

0.20

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6. Conclusions

References

There is an obvious difference between the 1 m, 10 m, 20 m, and 30 m moonlet case. The bidirectional reflectance in the 1 m case has a large change over the course of time, whereas for the later cases less and less change is observed as the moonlet size increases. This is not surprising because the incoming flux of impactors is proportional to R2 but the volume of the body in which the impactors are mixed is proportional to R3. This means less fractional pollution for larger body sizes. Although the development so far has been for a single moonlet, this analysis is equally applicable to any number of ring particles of a given size, that mutually exchange ejecta. The production and capture areas are equal for each body: We can interpret the probability distribution for an individual body (x(t) in Eq. (1)) also as the fraction of bodies with regolith depth h. Then, the expectation value (Eq. (2)) is just the average regolith depth in the ensemble. This allows us to apply the analysis to the ensemble of ring particles in a planetary ring. Thus, it might be possible to determine the average particle size for particular regions of Saturn’s rings. For a given optical depth, darker rings would be composed of smaller particles and brighter rings would be composed of larger particles. Conversely, this can provide an estimate of the exposure age of the ring region. The two spectral mixing models shown in this paper, areal and intimate, do yield substantially different results. Intimate mixing results in a greater change, or darkening, of the spectrum than does the areal mixing model. If the rings are massive this would suggest that it is indeed possible for the rings to look ‘‘young’’ for very long time periods. In the 10 m case, for example, the fractional pollution only reaches about 3% after 4 byr, which would amount to less of a change in the observed spectrum than for the 1 m case (22% fractional pollution), leaving a brighter ‘‘youthful’’ looking ring. In contrast, a ring of 1 m moonlets would be measurably darker since the fractional pollution there after 4 byr is about 22%. Likewise for the larger 20 m and 30 m sizes, the fractional pollution only reaches 1.7% and 1% respectively. This naturally confirms the expectation that more massive rings would be less polluted (Esposito et al., 2008).

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Acknowledgments This research was supported by the Cassini project. We appreciate helpful reviews from G. Fillachione and R. Clark.