Mon. Not. R. Astron. Soc. 399, L79–L83 (2009)
doi:10.1111/j.1745-3933.2009.00727.x
Can ice survive in main-belt comets? Long-term evolution models
of comet 133P/Elst-Pizarro
Dina Prialnik and Eric D. Rosenberg
Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Accepted 2009 July 28. Received 2009 July 20; in original form 2009 June 11
ABSTRACT
We present detailed evolutionary calculations spanning 4.6 × 109 yr for a model representing
main-belt comet 133P/Elst-Pizarro, considering different initial compositions: mixtures of ices
and dust. We find that only crystalline H2 O ice may survive in the interior of the nucleus, and
may be found at depths ranging from ∼50 to 150 m. Other volatiles will be completely lost.
We further show, by means of fully three-dimensional evolutionary calculations, that if the ice
is exposed, say as a result of a collision, the resulting activity is similar to the observed pattern
of the behaviour of the comet for a large spin axis inclination, even if only a patch of ice is
exposed.
Key words: comets: general – comets: individual: 133P/Elst-Pizarro.
1 I N T RO D U C T I O N
In recent years, a new class of comets – known as ‘main-belt’ comets
(MBCs) – has been identified, the prototype being 133P/Elst-Pizarro
(EP), which was first classified as a main-belt asteroid. Cometary
activity for EP was first detected in 1996 as a dust tail, and at the
time it was thought to be the result of a recent impact. The dust
could be either a trail of debris (Toth 2000) or the manifestation
of volatile-driven activity caused by the exposure of deep-lying ice
(Boehnhardt et al. 1998). The latter interpretation gained support
when the comet became active again one orbital period later, in 2002
(Hsieh, Jewitt & Fernández 2004; Ferrı́n 2006). This apparition also
enabled the determination of its albedo of 0.044, typical of comet
nuclei, and its rotation period of 3.471 h. The third episode of activity, detected another orbital period later (Jewitt, Lacerda & Peixinho
2007), as well as the seasonal modulation of the activity (all three
episodes of mass loss occurred near perihelion), leave little doubt
that mass ejection is triggered by volatile sublimation. Two more
objects belonging to the MBC class soon joined EP: 176P/LINEAR
and P/2005 U1 (Read), which showed vigorous activity between
2005 October and December, ejecting about 10 times as much dust
as EP (Hsieh, Jewitt & Fernández 2009; Hsieh, Jewitt & Ishiguro
2009). Very recently, a fourth object belonging to this class was
detected, P/2008 R1 (Jewitt, Yang & Haghighipour 2009).
The orbits of these objects are stable; hypotheses regarding their
origin are divided between formation in place (Haghighipour 2008)
and capture of comets from distant regions of the Solar system
into the outer asteroid belt during the late heavy-bombardment era
(Levison et al. 2008, based on the Nice model). Either way, it seems
that MBCs have spent between 3.9 and 4.6 Byr in the main belt.
E-mail: [email protected]
C
C 2009 RAS
2009 The Authors. Journal compilation A plausible scenario for the cometary activity of MBCs in general,
and EP in particular, is that there was indeed a recent impact on EP,
but the prolonged activity is driven by sublimation of ices exposed
by the impact, rather than by the impact itself. This would explain
both the prolonged nature of the dust emission and its recurrence
at successive perihelia. It implies, however, that MBCs have ices
buried below the surface. May ices survive 4.6 Byr of evolution in
an orbit that is relatively close to the Sun? This is the question that
we propose to address in the present Letter. Simplified analytical
estimates (Schorghofer 2008) indicate that this may be possible, but
such estimates do not consider gas flow through the porous nucleus,
nor sublimation in the deep porous interior.
The calculation is divided into two parts. In the first (described
in Section 2), the evolution of an initially homogeneous body composed of ices and dust is evolved for 4.6 Byr by means of a onedimensional (1D) evolution code. This yields a configuration which
is no longer homogeneous, but is still spherically symmetric, that is
temperature, density and composition change with radial distance.
At this point, the result of a hypothetical collision is simulated by
exposing the inner layers, that is transferring the composition of the
inner layers to the outer part of the spherical object. The resulting
configuration is then fed into a 3D code for detailed evolutionary
calculations (described in Section 3), spanning several orbits, which
simulate the present activity pattern of the object.
2 L O N G - T E R M E VO L U T I O N – 4 . 6 B Y R
The code used for numerical modelling is a 1D code described in
detail by Prialnik (1992) and Sarid et al. (2005); a discussion of the
input physics may be found in Prialnik, Benkhoff & Podolak (2004).
Only a very brief description follows. The most general composition
of a comet-like object is assumed to consist of water ice, water
vapour, dust (silicates and minerals) and other volatiles, which may
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D. Prialnik and E. D. Rosenberg
Table 1. Initial composition and physical parameters.
be frozen, free or trapped in the amorphous water ice. The water ice
may be either amorphous or crystalline; the former will crystallize
at a temperature-dependent rate. The mass conservation equations
for vapour are of the form:
∂ρv
+ ∇ · J v = qv ,
∂t
(1)
where ρ v is the density, J v the flux and q v the rate of gas release
or absorption by sublimation/condensation. The number of such
equations is equal to the number of volatiles considered. The heattransfer equation has the form:
∂uα
(2)
− ∇ · (K∇T ) +
ρα
cα J α · ∇T = S,
∂t
α
α
L
cos ξ,
4πd(t)2
Value
Radius (R)
Albedo (A)
Emissivity ()
Dust : ice (by mass)
Temperature
Hertz factor
Dust thermal conductivity
Dust heat capacity
2.5 km
0.04
0.90
1:1
60 K
0.01
0.2 W m−1 K−1
1.3 × 103 J kg−1
Table 2. Model parameters.
Model
where u is the internal energy, K is the thermal conductivity and S
represents all the available energy sources and sinks, such as latent
heat of phase transitions, or heat release by crystallization.
This results in a set of coupled, time-dependent equations,
which require additional assumptions about the physical processes
involved: heat conduction and gas flow in a porous medium,
amorphous–crystalline transition, phase transitions, pore growth
and shrinkage and properties of the medium: porosity, tensile
strength, pore-size distribution, ice to dust ratio and volatile
abundances. The evolution equations are of the second order in
space, and hence each requires two boundary conditions: vanishing
heat and mass fluxes at the centre, vanishing gas pressures at the
surface and the requirement of energy balance at the surface:
F (R) = σ T (R, t)4 + Q(T )H − (1 − A)
Parameter
H2 O ice
Density (g cm−3 )
Porosity
Volatiles
A
B
C
D
a
0.5
0.66
–
c
0.5
0.66
–
c
1.3
0.09
–
c
0.5
0.66
0.05
(3)
where F(R) is the conduction heat flux, σ is the Stefan–Boltzmann
constant, Q(T) and H are the rate and latent heat of sublimation,
respectively, d is the heliocentric distance and ξ is the local solar zenith angle. The corresponding set of difference equations is
linearized and solved fully implicitly by an iterative process.
The orbital period of EP is ∼5.5 yr, and in the long-term calculations we aim to cover 4.6 Byr of evolution. Clearly, therefore, orbital
changes cannot be resolved, as this would require a prohibitive number of time-steps and would introduce huge cumulative numerical
errors that would render the results meaningless. To circumvent this
difficulty, but still keep track of the correct overall amount of energy
absorbed by the comet, we replace the elliptic orbit by a circular one,
corresponding to the same total orbital insolation. The total amount
of solar energy intercepted by a spherical comet nucleus of radius
R, revolving the Sun in an orbit of eccentricity e and semimajor axis
a (corresponding to a period P orb ) is
t+Porb
πL
L
R2
dt = Eorb = πR 2
,
(4)
2
4πd(t)
2 GM a(1 − e2 )
t
Figure 1. Time-scales of relevant processes as a function of temperature:
sublimation (marked by the volatile species), crystallization, diffusion of
heat (through dust, amorphous ice and crystalline ice) and of gas for a depth
of 2.5 km.
tion time-scales are obtained by
ρice
√
τ=
,
SP m/2πkT
(5)
where ρ ice is the solid ice density, P is the saturated vapour pressure, m is the molecular mass of the species and S is the surface to
volume ratio. The diffusion time-scales are calculated for the entire
radius of the body (2.5 km); for lesser depths r, they scale as
(r/R)2 . We note that the heat diffusion time-scale is sufficiently
low compared to the evolution time (0.001–0.1) to allow heating
down to the centre. For sublimation to occur in the deep interior
on a time-scale of the order of Byr, the internal temperature must
reach at least ∼90 K. At this temperature, crystallization will be
completed in only ∼106 yr (1 Myr). The extent of sublimation cannot be straightforwardly estimated, however, since phase equilibrium may be eventually established. Hence detailed calculations are
necessary.
where L and M are the solar luminosity and mass, respectively,
and G is the gravitational constant. The radius a c of a circular orbit
with the same E orb is thus given by a c = a(1 − e2 ). Needless to say,
the nucleus spin has to be neglected and a fast-rotator approximation
is assumed. Thus, here d(t) = constant = a c and cos ξ = 0.25. The
number of time-steps for a complete run lies between a few 107 to
108 . Time-steps, determined automatically by the implicit evolution
code, vary between a few hundred seconds to a few thousand years.
We consider different initial configurations, having in common
the characteristics listed in Table 1 and differing in properties listed
in Table 2. As significant insight may be gained by the time-scale
analysis, the relevant time-scales are shown in Fig. 1. The sublima
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C 2009 RAS, MNRAS 399, L79–L83
2009 The Authors. Journal compilation Evolution models of comet 133P/Elst-Pizarro
For the first model that we consider (Model A), the initial composition is a mixture of pure amorphous water ice (no occluded gases)
and dust. It may be taken to represent a captured body, formed in
more distant and cold regions of the Solar system. The ice crystallizes completely during the first 6 × 105 yr of evolution, with
the crystallization front – initiated by absorption of solar energy –
advancing from the surface all the way to the centre, feeding mainly
on the released latent heat. If gases were trapped in the amorphous
ice, they would have escaped to the surface and out of the nucleus; the gas diffusion time-scale is only a few thousand years and
temperatures are too high for gases to refreeze. While the crystallization front is receding towards the centre, the crystalline ice starts
sublimating at the surface, with the sublimation front very slowly
advancing inwards, leaving behind an ice-depleted dust mantle. The
weak vapour flux escaping from the surface is incapable of dragging with it significant amounts of dust. By the time crystallization
is completed, the dust mantle thickness reaches about 5 m. Some of
the vapour flows inwards and refreezes ahead of the crystallization
front. Thus, the ice fraction profile peaks at some depth below the
dust mantle, as shown in Fig. 3. This peak is also fed by vapour flowing outwards from the deep interior, where the temperature has risen
to about 125 K, allowing for bulk sublimation. We note that phase
equilibrium may be obtained only when the gas diffusion time-scale
is higher than the sublimation one; otherwise the vapour escapes to
the surface, preventing vapour-pressure buildup. Hence, according
to the intersection of the sublimation and gas diffusion time-scales
in Fig. 1, steady state requires T 120 K. The steady state is not
perfect, but rather a quasi-steady-state, since some vapour does escape to the surface; hence the ice is slowly depleted, but far more
slowly than the rate of free sublimation would imply. As a result of
internal sublimation, after 4.6 × 109 yr of evolution, the ice density
throughout most of the interior is significantly lower than the initial
density, as seen in Fig. 3. Overall, the total ice content of the body at
present is slightly less than half of the initial. The depth at which ice
may be found at present, assuming no collapse of the dust mantle,
is about 90 m. The depth of ice is shown as a function of time in
Fig. 2.
If the body formed in place in the asteroid belt, it is more likely
that the ice was crystalline to start with. Therefore, the second
model we consider (Model B) is identical to the first, except that
the ice is initially crystalline throughout. In this case, the internal
temperature attained is slightly lower, and less ice sublimates both
in the interior and below the surface, so that ice may be found at
present at a depth of ∼45 m (see Fig. 3). The difference between the
Figure 2. Comparison of the evolution of the four models: A (blue dot),
B (red short dash), C (green long dash) and D (magenta dash–dotted), for
central temperature (left) and depth of H2 O ice (right).
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C 2009 RAS, MNRAS 399, L79–L83
2009 The Authors. Journal compilation L81
Figure 3. Comparison of the present structure of the four models: A (blue
dot), B (red short dash), C (green long dash) and D (magenta dash–dotted),
for temperature profile (left) and H2 O density (right).
amorphous and crystalline ice bodies is due to the additional heat
source in the former, provided by the exothermic crystallization.
Since for crystalline ice the only source of heat is solar energy
and since the absorbed energy in a given orbit depends only on
the cross-sectional area (the radius, for a spherical body), the next
question we ask is whether an initially denser body, that is a body
that contains more ice for the same ice/dust ratio and size (input
energy), will also preserve a larger quantity of ice. Hence, the third
model (Model C) is identical to the second, except that the bulk
initial density is about 2.5 times larger. The porosity is, of course,
correspondingly lower. This causes the internal vapour pressure to
rise and hence the temperatures attained are higher. The relatively
high pressure may be maintained, because the flow of gas is impeded
by the lower porosity. In fact, the outflow of vapour is so weak that
only a very small fraction of the ice sublimates in the interior. The
H2 O-depleted layer is about 50 m thick, similar to the previous case,
but the internal density of ice remains almost unchanged.
Having seen that water ice may be preserved in the deep interior,
although the internal temperatures attained are sufficiently high to
allow complete sublimation on the time-scales of Byr, the question
arises whether ices of higher volatility than H2 O may also survive.
The next model (Model D) includes, therefore, a mixture of four
volatile ices – CO2 , HCN, NH3 and C2 H2 – besides crystalline water
ice and dust. The ice/dust ratio and the bulk density are the canonical
ones (1 and 0.5 g cm−3 , respectively), but water ice is only 0.45 of
the mass, the remaining 0.05 being divided equally among the other
volatiles. We note that the corresponding sublimation time-scales
are a great many orders of magnitude lower than that of H2 O for a
given temperature and differ by several orders of magnitudes among
them. We do not consider supervolatile ices, such as CO, since the
temperature in the region where the comet was formed could not
have been so low as to allow condensation of CO (see Fig. 1).
We find that – in the long run – all the volatiles except water are
depleted, even at the centre of the body. The pattern is very similar to that of water ice sublimation, except that the fronts advance
more rapidly. This is shown in Fig. 4, where the accumulation of
ice ahead of the front is conspicuous. Not surprisingly, the various
fronts advance at different rates, according to volatility, as indicated
by the corresponding time-scales of Fig. 1. The advance is not selfsimilar; volatiles interfere with each other, competing for the same
heat source, and all processes are extremely sensitive to temperature. The significant result is that after about 108 yr, no volatile
besides H2 O is left, not even a trace. The succession of volatile
depletion at the centre is illustrated by the evolution of the central
temperature shown in Fig. 2: the temperature is almost constant during the sublimation of a species and rises until the sublimation of the
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D. Prialnik and E. D. Rosenberg
rotation. Briefly, to solve the heat equation in three dimensions, we
use the following set of implicit difference equations:
U (t + δt) − U (t)
V ,
Fx (t + δt)Sx + q(t + δt)V =
δt
(6)
where x represents the direction (i + , i − , j + , j − , k + and k − ), Fx is the
heat flux vector component in the x direction, Sx is the element’s
area perpendicular to the x direction, q is the heat generation rate
inside the element, V is the element’s volume, δt is the timestep and U is the thermal energy per unit volume. The system of
non-linear equations is linearized and solved iteratively. Boundary
conditions at the surface consist of energy balance as in equation (3),
but for each surface element separately; thus not only is d a function
of time, but ξ is a function of local co-latitude and hour-angle.
Here, too, as in the long-term calculation, water ice may sublimate into the internal pores, but as sublimation is expected to occur
very close to the surface, we assume vanishing pressure in the pores,
meaning that the vapour escapes to the surface with no time delay.
Essentially, this approximation is ∇ · J = −∂ρ s /∂t, where ρ s is
the solid ice density (Choi et al. 2002). The mass and energy equations are thus decoupled and significantly easier to solve, yielding
an upper limit for the effect of internal sublimation. In this approximation, the vapour flux at each depth (radial surface of a volume
element) is given by the sum of vapour masses produced in the elements directly below this depth (see Rosenberg & Prialnik 2009).
Internal sublimation affects both porosity and pore sizes.
Given the observational data and the results of long-term evolution, summarized in Table 3, we are left, essentially, with only one
free parameter: the spin axis inclination. The orientation of the spin
axis is another free parameter, but its significance decreases with
decreasing eccentricity, and the eccentricity of MBCs is relatively
low. In order to delimit the possible range of results, we consider
a range of obliquities between 10◦ and 80◦ . Additional models are
calculated by assuming that only a patch of the surface – amounting
to 1/8 of the area – has ice exposed down to a depth of several tens of
metres, while the rest is composed of dust (rock). This configuration
is meant to simulate the result of a collision (within the limitations
of a spherical configuration). Here, a crucial factor is the location of
the patch with respect to the spin axis, and hence for the tilt angle of
80◦ we have calculated two cases, with the patch at the north pole
and at the south pole.
The H2 O production rate is obtained by integrating the fluxes over
the entire surface of the nucleus. Since crystalline H2 O ice is the
only volatile and any activity will be driven by its sublimation, we
use the empirical correlation derived by Jorda, Crovisier & Green
(1992) to represent the activity in terms of visual magnitude, which
Figure 4. Profiles of ice mass fractions for four different volatiles (HCN –
red; NH3 – blue; CO2 – magenta; C2 H2 – green) at several times during evolution, illustrating the recess of sublimation fronts and, eventually, complete
depletion of ices.
next – less volatile – species sets in. As for the H2 O ice, the behaviour is similar to the other cases, but sublimation is strongest in
this case, with the ice boundary receding down to ∼150 m. Over
three times more H2 O ice is lost as compared to Model B, in addition
to the other volatiles. Energetically, this does not pose a problem
since the energy required for sublimating the entire ice content of
the body, 4πρ ice H sub /3 ≈ 4 × 1019 J, amounts to only about 10−7
(!) of the absorbed solar energy during the lifetime of the Solar
system, (1 − A) L πR 2 τ /(4πd 2 ) ≈ 4 × 1026 J. An increase
of the fraction of heat conducted into the interior by even a factor
of 10 would hardly affect the surface heat balance (equation 3).
Such an increase is made possible for Model D by the low internal
temperature maintained during the sublimation of the more volatile
species (see Fig. 2), which results in a larger temperature gradient.
This effect illustrates the intricate ways of mutual influence for a
group of volatiles.
In conclusion, we have shown that H2 O ice may be retained in
the interior of a main-belt body, despite its proximity to the Sun
and the long evolution time in its orbit, but to a lesser extent than
indicated by simplified models (Schorghofer 2008). Ice is expected
to be found at a depth of ∼50–150 m, depending on initial structure
and composition. No other volatile species would be able to survive,
not even by assuming an extremely low thermal conductivity, as we
have done in the present study. Therefore, any activity resulting
from exposure of the ice-rich layer by an impact will be driven by
water sublimation. Since water has its own characteristic vapourpressure dependence on temperature, this leads us to the prediction
that the rise and decline in activity along an eccentric orbit should
be compatible with the H2 O sublimation pattern.
Table 3. Initial composition and physical parameters.
3 L AT E ( P R E S E N T ) E VO L U T I O N
For the short-term evolution, we use the 3D thermal evolution code
developed by Rosenberg & Prialnik (2007), which is ideally suited
for a composition of water ice and dust, where the key factor is
C
Parameter
Value
Radius (R)
Albedo
Emissivity
Semimajor axis
Eccentricity
Spin period (P rot )
Spin axis inclination
Dust: ice (by mass)
Ice
Temperature
Density (ρ)
2.5 km
0.04
0.90
3.156 au
0.165
3.471 hr
10◦ –80◦
1:1
Crystalline H2 O
125 K
500 kg m−3
C 2009 RAS, MNRAS 399, L79–L83
2009 The Authors. Journal compilation Evolution models of comet 133P/Elst-Pizarro
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of the highly tilted spin axis. There remains the problem of the low
amplitude obtained in our calculations, but this is largely due to the
fact that dust ejection is not taken into account here. Future models
will have to examine various mixtures of ice and dust and various
dust grain size distributions.
In conclusion, our two strong predictions regarding EP are as
follows: (a) based on the long-term calculations, the source of activity is sublimation of water ice; (b) based on the 3D calculations
of present behaviour, the spin axis of EP has a very high tilt angle.
AC K N OW L E D G M E N T
DP acknowledges support for this work provided by the GIF grant
859-25.7/2005.
REFERENCES
Figure 5. The variation of water production rate and derived magnitude
with time along five orbits, for various inclinations of the spin axis, for
models with a homogeneous initial composition and models with an icerich patch. Perihelia and aphelia are marked by vertical lines, solid and dash,
respectively. The enclosed sketch illustrates the extent of the ice-rich patch
on the surface of the nucleus.
is easier to compare with observations. The results of all the models
are shown in Fig. 5. In this calculation, the dust is left behind when
the ice sublimates, and thus a dust mantle forms and grows very
slowly with repeated perihelion passages, although its thickness is
not uniform over the surface. This is the reason for the decline in
water production rate with repeated revolutions. We note that the
peak production rate decreases with increasing tilt.
Ferrı́n (2006) analysed in detail the light curve of EP. Regardless
of details, the main features that emerge are as follows: (a) an
amplitude of about 2 mag for peak activity, (b) a time lag of about
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surge of activity at aphelion – all three quite unusual in comparison
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The aphelion activity is greatly emphasized by the inhomogeneous
model, where the patch of exposed ice is located near the south pole
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