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Author's personal copy
Icarus 212 (2011) 268–274
Contents lists available at ScienceDirect
Icarus
journal homepage: www.elsevier.com/locate/icarus
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: joshua.elliott@colorado.edu (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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J.P. Elliott, L.W. Esposito / Icarus 212 (2011) 268–274
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<i
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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J.P. Elliott, L.W. Esposito / Icarus 212 (2011) 268–274
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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J.P. Elliott, L.W. Esposito / Icarus 212 (2011) 268–274
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
Author's personal copy
274
J.P. Elliott, L.W. Esposito / Icarus 212 (2011) 268–274
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
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R2 but the volume of the body in which the impactors are mixed
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Although the development so far has been for a single moonlet,
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Acknowledgments
This research was supported by the Cassini project. We appreciate helpful reviews from G. Fillachione and R. Clark.