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Influence of porosity on the thermal behaviour of comet surfaces
Adv. Space Res. Vol. 29, No. 5, pp. 691-704, 2002 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273- 1177/02 $22.00 + 0.00 PII: SO273- I177(02)00003-O
Pergamon www.elsevier.com/locate/asr
INFLUENCE OF POROSITY ON THE THERMAL BEHAVIOUR OF COMET SURFACES
D.MGhlmann, DLR Institut fiir Raumsimulation,
51170 Kdn, Germany
ABSTRACT Comet nuclei are assumed
to be of porous structure.
A consequence
sublimating matter is that sublimation will take place also at Jntemal“ at the outer surface. This considerably increase.
The resulting
to a non-porous
of
surfaces and not only
modifies the thermal and sublimation
surface material will be cooler if compared gradients
of the porosity
behaviour. The
surface while the temperature
comet diurnal and depth variations
of temperature
and
outgassing properties are described in some detail in their dependence on heliocentric distance for different porosities, pore radii and heat conductivities
for a model nucleus on the orbit of
comet 46PTWirtanen. The results are compared to astronomically comet. Based on that, ,,most appropriated“
observed properties of this
numerical values or ranges of parameters which
can be used to characterize the properties of comet 46P/Wirtanen 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
are identified.
1. INTRODUCTION The results comparatively Porosity
of the Halley-missions low mass density
in 1986 indicate
that cometary
p. This implies a high porosity
nuclei
p of cometary
are of matter.
is defined by the ratio of ,,empty space“ and volume filled with matter. A further
parameter
is the pore size. This can be modeled by cylindrical
tubes with a corresponding
,,pore radius“ rp. In real matter tubes do not extend unrestrictedly through the body into large depths. This property is proposed here to be described by a percolativity K which is assumed to decay exponentially
with depth z. This percolativity
limits the ranges of interconnected
tubes and thus of the gas flow in porous matter. The temperature
in porous matter is governed by heat conduction
transport by gas. A characteristic
Jhermal
penetration length“ L can be determined in case of
a periodic heating, as it can be assumed for rotating and periodically At the outer surface conduction
insolation
energy
and infrared re-radiation.
and partly by energy
is transformed
Furthermore,
691
orbiting cometary nuclei.
into energies
of sublimation,
heat
it can be assumed that there is no heating
692
D. MBhlmann
at depths which are large if compared to the orbital thermal penetration
depth. Therefore,
a
constant temperature can be assumed as a boundary condition at a sufficient depth. A model of a porous surface of pure water ice will be used in the following numerically
thermal and sublimation
profiles in their dependence
on heliocentric
The restriction to water ice only can be justified as a first approximation
to derive distance.
by the fact that water
ice is the dominating cometary matter. Other volatiles are not expected to have any essential influence
on the thermal
Furthermore,
balance
in the upper surface
the low gravity of 46P/Wirtanen
layers
of comet
46P/Wirtanen.
and the observed gas production rates indicate
that this comet is not covered by a mantle. It is expected that this comet nucleus has a surface with ,,free sublimation“. Thermal and sublimation properties of cometary matter will be computed numerically for different porosity properties
and heat conductivities
within the frame of the model as it is
described in the following. 2. THE MODEL Porous and volatile matter sublimate
at sufficient temperatures
inner or ,,internal“ surfaces. Internal sublimation outer surface if the porous matter is ,,percolative“,
will contribute
at the outer surface and at to the total outflow at the
i.e. if the gas which is released at the
internal surfaces can reach the outer surface. The internal energy balance is determined
by
heat conduction and energy transport due to gas flow. Here, and as a first approximation,
the
energy transport by the gas will be taken into account only by assuming that half of the isotropically
released gas will propagate
upwards and leave the comet without effectively
cooling the upper layers. This sublimation caused outflow is related to a corresponding term in the energy balance. downward
The other half of the released
into cooler regions where it recondensates
gas is assumed
to propagate
at the cooler internal surfaces. The
related energy transport can be neglected if it is assumed that this recondensation
takes place
near to the depth of release. Thus, the ,,sink term“ in the equation of heat conductivity, is proportional
sink
to 2p/r,, has to be halved in this approximation
which
to describe the internal energy
transformation. The finally used equations are
PC%
- -&A%
= - % H(T)z(T)
(1)
for the temperature field T(z,t), and 1-A -y-'h
SCf(s,‘P,t) =
A aT(z,t,<,cp) -L
aZ
+ &oT4(z = O,t,s,q)
+ (I- p)Hz(z =
o,t,S,q) (2)
2=0
and dT(z,t) ___ dZ
ZZ 0 ;>>I
(3)
Thermal Behaviour
of Comet Surfaces
693
for the boundary conditions. The cometary energy balance at the surface is governed by solar heating, infrared re-radiation,
sublimation caused cooling, and by heat flow into the nucleus.
This balance has to be described for the surface and for the pores. Here it is assumed that the temperature
of the upper surface and ,,of the pores“ is equal, that there is no heat conduction
through the ,,empty“ pores, and that the sublimating (and also the heat conducting)
surface is
reduced by (l-p). This limits the applicability of Eq. (2) to spatial resolutions of or larger than the pore size. The relation
Z(T(z)) =
describes
the temperature
normalized
dependence
&FiiEg eTFz) of the sublimation
rate Z. Here, z is the depth
to L, i.e. the real depth x is given by x=Lz. Analogously,
the rotation
period P. The function
f(S,q,t)
describes
time t is normalized to
the local cometographic
and time-
dependence of (the angle of incident dependent) solar irradiation at the cometary surface. It is for equatorial latitudes given between sunrise (at n/2) and sunset (at 3n/2) by f=-cos(;?&P). Furthermore, heliocentric
SC=1368 W rn-’ is the Solar distance,
c is the specific
constant“,
i.e. the flux of solar energy at 1 AU
heat, A is the albedo of the surface,
and E is the
emissivity of the surface. The diurnal penetration depth L is then given by L = &&c coefficients
in Eq. (4) are according to Fanale and Salvail (1984) determined by a = 3.56 10”
N me’, b = 6141.667 K. Furthermore, molecule,
. The
and k is Boltzmanns’
m = 3 1O-‘6kg is the mass of the sublimated
constant. H=8 10.” Ws per molecule
water
is the latent heat for
sublimation of one water molecule, This value correspond to the value of H at a temperature of 198 K in the formulation
of Delsemme and Miller (1971). Note that the precision of these
numbers came into question
fitting of measurements
without taking into
account the Clausius-Clapeyron
equation (Huebner, 1998). Furthermore,
the energy necessary
to release
from a water ice surface
microscopic
a water molecule
due to incorrect
structure of this surface. Therefore,
depends
uncertainties
strongly
on state and
of the order of about lo%, as
they are found in different values in the literature (Cowan and A’Heam, 1979, and Delsemme and Miller, 1971) have to be accepted. A more precise determination
of these numbers seems
to be an actual task. Equation (4) describes free sublimation into vacuum. This approximation applied as long as kinetic pressure of the released gas can be neglected
can be
if compared to the
saturation pressure. This is at 200 K of about 0.16 Pa while the kinetic pressure in the pores is at 1 AU conditions of the order of or less than 10-j Pa. It has to be noted that it is assumed in Eq. (1) that horizontal heat flow can be neglected
if compared
to vertical heat propagation.
This approximation
may fail at local
shadows and at the day-night boundary. The above given or an equivalent and sublimation
set of equations is usually taken to model thermal
properties of cometary surfaces (cf. Capria et al., 1996, Julian et al., 2000).
This set of equations is simplified here insofar as only pure water will be taken into account.
694
D.M~hlmann
Porosity, outgassing
pore radius
through
appropriatedly
by
reduction of connectivity energetically changes
K
When
can be modeled
takes into account that there may happen a
of tubes with increasing depth. Here, a value of x=1/L can be used
example. This value forx
is used to take into account that most of the
relevant processes will happen within a depth of the order of L. Thus, structural
are expected
plausible
are the yet ,,free“ parameters.
also a ,,percolativity“
exp(-Xx). This percolativity
K =
as a characteristic
and heat conductivity
the pores is considered,
to be mainly within depths of this order of magnitude.
suggestion
,,percolativity“
only.
qualitatively
influence essentially
Experiments
are necessary
and quantitatively.
to validate
this
This is a concept
It can be expected that percolativity
the outgassing behaviour if the percolativity
of
does not
depth is much larger than
the thermal penetration depth L. This is due to the fact that most of outgassing happens in the hottest,
i.e. in the uppermost
following. Heat conductivity
h is influenced
section for heat conduction between neighbouring proportional
parts of the surface.
This is shown more in detail in the
by two effects. This is at first the reduced cross
of a solid continuum,
which is due only to the small bridges
connected particles. This is described by the Hertz factor h, which is
to the square of the ratio of bridge size to particle size. The heat conductivity
the grain material is then given by h*=hhc,,,,, where hcrainis the heat conductivity material. This material is water ice in the models of this paper. Furthermore, will be reduced by macroscopic bridge-connected
porosity
heat conduction
p with respect to that of a solid continuum
grains) with heat conductivity
of
of the grain (of
3L*of the material outside the pores according
to h = h* (l-p). This is caused by the reduced
surface that heat flow must cross. In the following
calculations,
to cover a wide range of possible
h*=h,,,
W rn-’ K“ and of h*=0.02
2.5
conductivity
k *. This
(5)
corresponds
conductivities,
conductivity
model
values
of
W mm’K-’ are used for the ,,solid state“ heat
for the smaller conductivity
to a Hertz factor of l/125,
which is well compatible to the value used by Huebner et al. (1999). As can be seen in the following Table I, which gives a summarizing
overview of the
used model parameters, pore radii of rp = 0 m, rp = 1O-3m and of rp = 10m5m are used. Model I is a ,,reference“
model with respect to pure and non-porous
water ice. The other models in
Table I are chosen to fit the expected range of physical parameters (Mohlmann,
2001) and to demonstrate
for comet 46PAVirtanen
effects due to changes of heat conductivity,
and pore radius. The largest value of the heat conductivity,
i.e. h*=&
porosity
=2.5 W me’ Km’ is
chosen as an upper limit and following Klinger’s (1980) formula for the heat conductivity
of
crystalline water ice in the range of about 200 K. This large value might not be unrealistic for upper surface layers which could be hardened and densified due to recondensation as they have been observed mentioned heat conductivity
in the lab (Kochan
processes
et al., 1989). The lower of the above
values is suggested as to be typical for a lower limit (cf. Seiferlin
et al. (1996), Steiner and Komle (1991)) for porous ice with pore sizes of r, =10e3 m and temperatures around 200 K and less.
Thermal Behaviour of Comet Surfaces
695
Model
h*[W mm’K*‘]
Uml
2.5
p 0
rAmI
I
0
0.245
II
2.5
0.5
1o-3
0.245
III
0.02
0
0
0.022
IV
0.02
0.5
1o-3
0.022
V
0.02
0.5
1o-5
0.022
VI
0.02
0.8
1o-3
0.022
Table I
Model parameters used in the numerical computations. Furthermore,
the assumption
has been made throughout
nucleus is rotating around a fixed axis perpendicular precessional motions. The computations The numerical computations initial conditions
this paper that the comet
to the orbital plane and that there are no
refer to an equatorial location at the nucleus.
of the temperature evolution from the relatively arbitrary
will be stopped after reaching an ,,equilibrium
defined here by a temperature
distribution
fulfilling
state“. This equilibrium
the numerical
is
surface energy balance
better than 1% (i.e. for relative numerical errors smaller than 0.01 of the integral over one rotation
of the difference
surface sublimation
insolation
power and powers of surface
and inward heat flow which are normalized
energy). Therefore, ,,equilibrium
between
the temperatures
temperatures“
re-radiation,
by the daily insolation
derived and used within this paper are to be seen as
as they will evolve in and at a cometary
surface as long as the
rotation period is very small if compared to the orbit period. The computations
are performed
for different heliocentric distances. 3. RESULTS Results of the above characterized
numerical computations
will be described in detail
in the following. There is often a reference to heliocentric distances of 1.06 AU or 3 AU. The distance of 1.06 AU ist the perihelion distance of 46P/Wirtanen,
and the distance of 3 AU is
of special interest for the Lander of the ROSETTA mission. This Lander is planned to reach the surface of 46P/Wirtanen
at a heliocentric distance of about 3 AU.
3.1 Surface EnerPv Balance and Relevant Processes The diurnal variation of the different energy fluxes at a freely sublimating icy surface of a comet nucleus at 1.06 AU heliocentric
distance is described
Figure 1. It can be seen that the contributions
to this energy balance depend strongly on model
parameters.
In case of zero or very small porosity,
heliocentric
distance the dominating
minor importance.
surface
for equatorial sublimation
energy sink. Heat conduction
latitudes in
is at 1.6 AU
and re-radiation
are of
This is different for a porosity around p=O.5, where heat conduction
related sublimation and related outflow.
become essential. The internal heat is converted Surface sublimation,
comet, is not important
i.e. the sublimation
in this case of the larger conductivity,
and
into sublimation energy
at the ,,outer“ surface of the but it becomes
dominating
696
D. Mahlmann
again for a comparatively
smaller
heat conductivity.
Generally
spoken, sublimation
is the
dominant loss process at about 1.06 AU heliocentric distance. Comparison of heat conduction in parts A and B, and also in parts C and D in Figure 1 illustrate the porosity caused change in the physical situation at otherwise identical properties for the ,,sublimation conductivity,
dominated“
part of the orbit. In spite of decreasing
effective
heat
heat conduction increases for increasing porosity. This is caused by the internal
cooling due to sublimation
and the related outflow of released gas. Internal cooling causes
stronger temperature gradients and an increase in the related heat flow. Comparison sublimation
of parts B and D teaches that the ratio of surface sublimation to internal
increases
for decreasing
conductivity.
This is due to the higher
surface
temperatures in case of weak conductivities.
z -200
Fig. 1. Surface energy balance at equatorial latitudes at 1.06 AU heliocentric distance for models I-IV. All profiles are given from midnight to midnight over one rotation period z after reaching
,,numerical
Abbreviations
equilibrium
conditions“
(over
a different
number
of rotations).
are: In for insolation, Ir for infrared re-radiation, Hc for heat conduction, Ss for
surface sublimation, and Si for that part of internal sublimation which reaches the surface.
Thermal
Behaviour
of Comet
Surfaces
697
The relations in the surface energy balances change with heliocentric 2 gives the corresponding Figures
energy balances for 3 AU. It can be seen by comparing parts A in
I and 2 that infrared re-radiation
larger heliocentric
distance. Figure
and heat conduction
become more important for
distances while sublimation becomes less effective. Comparison of parts B
and D in Figure 2 teaches that changes in porosity (or density) are not followed
by strong
changes in the surface energy balance. Contrary to this, and as it is described by parts D and B in Figure 2, internal sublimation Si and the related outflow at the surface increase remarkably for smaller pore radii. This behaviour is a consequence
of the structure of the rhs-term in Eq.
(1) which takes into account the increase of the ,,area of internal surfaces per volume“ for decreasing pore size. Thus it is the size structure of the pores which is an essential parameter, additional to the porosity itself. Exactly, it is the ratio p/r, which governs this thermal and
W
l-ii-l
F[Wrril
n-?l
150rh*=o.02
l-7
qw f-n-*]
lso fh*=0.02 W 125
In
III
D
100 75 50 25
5.8
Z
6
-25
-25
outgassing behaviour. Fig.2. Surface energy balance at equatorial latitudes at 3 AU heliocentric
distance for models
I,IV,V,VI. All profiles are given from midnight to midnight over one rotation period r after reaching
,,numerical
Abbreviations
equilibrium
conditions“
(over
a different
number
of rotations).
are: In for insolation, Ir for infrared re-radiation, Hc for heat conduction, Ss for
surface sublimation, and Si for that part of internal sublimation which reaches the surface.
698
D. MGhlmann
3.2 Diurnal Death Profiles of the TemDerature
model
TN
IV
model V p=o.5
8
model
J-WI
Z
10
VI
p=O.8 rP= 1 Oe3m
180
A
Fig. 3 Diurnal depth profiles of the temperature at equatorial latitudes at 3 AU for models IV, V and VI.
Thermd
The temperature penetration
Hehaviour
of Comet Surfaces
699
in cometary surfaces depends strongly on depth. The diurnal thermal
length L = dw
is the characteristic
measure for this depth dependence.
can be seen in Figure 3 that most of the diurnal temperature
changes happen within depths
down to only a few L. The depth is described in Figure 3 by the dimensionless Furthermore, Jimiting“
It
depth 2=x/L.
it is use to describe the depth dependence of periodic temperature profiles by the
curves of the depth profiles at the time of the minimum and of maximum surface
temperature. These are given for a heliocentric distance of 3 AU in Figure 3. Comparison effective
of models IV and V indicates,
as it has to be expected,
that cooling is more
for smaller pore sizes (for unchanged porosity). This is due to the increase of the
area of internal surfaces per volume, as it is described by p/r,. It can be seen by comparing models IV and VI that the small changes in porosity
values, as they are only possible for
realistic porosities, are not followed by significant changes in the thermophysical
behaviour.
Thus, the most critical porosity parameter in current comet modelling is the pore size. 3.3 Diurnal Heat Flow and Temuerature
Gradients at the Surface
m-21
j[ 40
20
!I“
1
1.
i 5.2
5.4
5.6
._ 5.8
‘I:
6
Fig. 4 Diurnal depth profiles of the heat flow and surface heat flow at 3 AU for equatorial latitudes for models IV andV.
700
D. MBhlmann
Thermal evolution
properties
of the cometary
of the surface temperature
surface
have an essential
and related sublimation.
influence
on the
Thus, the heat flow at the
surface is an important surface property. It can easily be seen in Figure 4 that the heat flow is inward directed at the dayside, but there is a flow from inside towards the surface during the night. The difference
between the dayside inward flow and the smaller night side flow is
caused by transformation
of heat into internal sublimation. This is a specific characteristic
of
porous bodies of volatile matter. As it has shown above, inward heat flow is in porous bodies larger for smaller pore sizes. Cooling
due to internal
sublimation
behaviour. It can be derived from Figure 4 that the temperature
is the cause of this
gradients are strongest in the
uppermost parts of the surface. They can reach a few 10” K m-l within the upper millimeter. The related thermal stresses can overcome there the cohesive strengths of the surface material. This can be followed by a thermal cracking triggered separation of millimeter-sized particles which can be lifted-off into the coma by the outstreaming contribution
to the gas production
or smaller
gas. There they will give a
rate additional to that of the cometary
surface (including
the outflow due to internal sublimation). The importance of this ,,second source“ is expected to increase for decreasing heliocentric distance. 3.4 Dependence of Surface TemDeratures on Heliocentric Distance Typical diurnal temperature follow
for other models,
profiles have been shown in Figure 3. Similar profiles
but they differ remarkably
in their maximum
and minimum
temperatures as they are given in Figure 5.
180
160
160.
4’
‘.“...,.
“,....
2
3
s
qpJl
5
2
3
Fig. 5 Maximum and minimum equatorial surface temperatures in dependence on heliocentric distance. Models are described by Table I. Model IV A is Model IV with P=24h. It can be seen in Figure 5 that maximum temperatures conductivity
(cf. models II and IV). Furthermore,
increase for decreasing
heat
decreasing pore size will be followed by
stronger cooling (cf. models IV and V), while the maximum surface temperature
increases for
increasing porosity, as can be seen by comparing models IV and VI. Minimum temperatures, which occur around sunrise (and not at midnight) are highest for larger heat conductivity models I and II with respect to IV, V and VI). Lower minimum temperatures
(cf.
are shown to
evolve for decreasing
pore radii (cf. models IV and V) and with increasing
models IV and VI). Note that the model with the highest porosity,
porosity
(cf.
i.e. model VI, has the
highest maximum and the lowest minumum temperatures. The derived
minimum-
Figure 6) agree for comparable
and maximum
values of temperature
and sublimation
models well with values derived by Benkhoff
(1996) Capria et al. (1996) by the ISSI team (Huebner
(cf.
and Boice
et a1.(1999)), and Julian et al., (2000).
Model IV A was chosen especially for comparison with ISSI results. 3.5 Denendence of the Sublimation Rate on Heliocentric Distance Figure 6 gives the dependence heliocentric
of total sublimation
me’ s-‘1 on
distance at equatorial latitudes for different models. The total sublimation rate is
defined here as to consist of the sublimation rate through
rates Z[molecules
the pores due to internal
maximum total sublimation
rate of the surface material and of the outflow
sublimation.
Models
with a comparatively
rate are of interest when compared to astronomically
water production rates of 46P/Wirtanen,
large
observed
which are surprisingly large in view of the small size
of this comet nucleus (cf. Schulz and Schwehm, 1999).
‘1 4 mol. rn* 2sece 1.
1.
Fig. 6 Diurnal profile of the maximum total sublimation
rates for different
models and in
dependence on heliocentric distance. Model IV A is Model IV with P=24h. 3.6 Deoth Denendence
of Internal Sublimation
Sublimation in porous matter has to be described by a ,,surface sublimation rate“ [mol. mm’s-‘1 of a volume with a ,,sublimation density“ S[molecules mm3s-‘1 with an outflow through a defined surface, i.e. here through the cometary surface. This sublimation
density S can be
described by
s =
!pz. rP
(6)
702
D. MGhlmann
Note that S in Figure 7 is a sublimation
density,
i.e. the number
of sublimated
molecules per volume. This should not be confused with the sublimation rate, i.e. the number of released molecules per surface.
S[mol. m-3s-‘] 3.5x10zr
S[mol. m-3sM’] 1.06 AU
3x1o26
r
1.06 AU
=10w5m
2.5~10~~ 2x1o26 1.5x101b~ 1x1026 5x1 o= 0.02
0.04
0.06
0.08
2
0.1
0.02
0.04
0.06
0.08 z
0.1
S[mol. m‘3s-1]
S[mol. m-3s“] 3 AU
r = 1G5m
0.02
0.04
Fig. 7 Depth dependence the time of maximum
0.06
0.08
z
0.02
0.1
0.04
0.06
0.08
z 0.1
within 0.1 z of the sublimation density S for models IV and V at
total sublimation.
The depth z is measured
in units of the diurnal
thermal penetration depth L, p=O.5, and h*=0.02. In case of the above mentioned
model assumption
that only half of the internally
released molecules will reach the surface, a factor of 0.5, as it has been taken into account in the rhs-term of Eq. (1) has to be used additionally rate. The depth dependence
in Eq. (6) to describe the real gas release
of the sublimation density S according to Equ. (6) is described in
Figure 7. It can be seen that most of the internal sublimation takes place directly beneath the surface within depths less than half a millimeter in the case of model IV. The thickness of the layer of highest sublimation density decreases with decreasing pore size. 4. CONCLUSIONS The results
of modelling
summarized in the following:
cometary
surface
and outgassing
related
properties
will be
Thrrmnl
l
As shown with Figure
Behaviour
of Comet Surfxes
703
4, most of the water relevant
energetic
processes
in freely
sublimating cometary surfaces happen in the upper parts of the surface within depths of a few centimeters.
This importance
increases for decreasing sublimation
of the uppermost
surface parts for cometary
activity
heliocentric
distance insofar as the erosion depth due to surface
approaches increasingly
the thermal penetration depth. It reaches about 40%
of the thermal penetration
depth at 1.06 AU for model IV. This steady ,,reduction of the
surface height“ was not taken into account in the above described cause something
of the thermal gradients
like a ..steepening“
computations.
It will
in the uppermost
active
layers. l
It is shown in Figures 1 and 2 that the energetic regime at cometary surfaces is different at 1.06 AU and 3 AU heliocentric
distance. Sublimation
is the dominating
1.06 AU. Infrared re-emission and heat conduction are characteristic l
loss process at
for 3 AU.
The porosity related parameters which governs the thermal and sublimational porous cometary
surfaces are given by p/rp. This is illustrated
behaviour of
by Figures
1 to 3. The
currently most unknown porosity parameter of cometary matter is the pore size. l
Comparatively conductivities
small pore sizes in the range of IO-’ m to 10e5 m and weak heat in the range of 10.’ W mm’K-’ seem to be appropriated
behaviour of a porous water-ice surface of 46P/Wirtanen possibly
larger). This result is based on computed
to describe better the
with porosities around p=O.5 (or
total sublimation
rates which are
highest for these parameter ranges. High total sublimation rates are required to make the astronomically
determined size of 46P/Wirtanen
(Lamy et al., 1996) and the resulting total
gas production rates compatible. l
Thermal
gradients
are strongest
directly
at and below the cometary
surface.
These
gradients increase for decreasing heliocentric distance. l
It should be noted that differences water
production
thermocracking
between observationally
rates of 46P/Wirtanen
triggered and lifted-off
could be explained millimeter-sized
determined
by a ,,second source“ of
or smaller particles which will
sublimate in the coma. Note that the above mentioned ,,steepening“ around perihelion
may additionally
and modelled
enhance the thermomechanical
of thermal gradients stresses and cracking
processes in the uppermost surface parts.
REFERENCES Benkhoff, J. and D.C. Boice, Modeling of the thermal properties of the gas flux from a porous, ice-dust body in the orbit of P/Wirtanen, 1996, Planet.Space Sci., 44, No 7, 665-673 Capria, M.T., F. Capaccioni,
A. Coradini,
M.C. De Sanctis, S. Espinasse, C. Federico, R.
Orosei, and M. Salomone, A P/Wirtanen evolution model, Plurzet. Spuce Sci., 44, No.9, 987- 1000, 1996 Cowan, J.J. and M.F. A’Heam, Vaporization of cometary nuclei:Light curves and lifetimes, Moor1 and Plarzrts, 21, 155-171, 1979
704
D. Miihlmann
Delsemme, A.H. and D.C. Miller, The continuum of comet Bumham (1960 II), PZanet.Space S&19,1229-1257,
1971
Fanale, F.P., and J.R. Salvail, An idealized short period comet model : surface insolation, H,O flux, dust flux, and mantle evolution, ICARUS, 60,476, 1984 Huebner, W.F., private communication,
1998
Huebner, W.F., J.Benkhoff, M.T.Capria, A.Coradini, M.C.De Sanctis, A.Enzian, R.Orosei, and D. Prialnik, Results from the Comet Nucleus Model Team at the International Space Science Institute, Bern, Switzerland, Adv. Space Res., Vo1.23, No.7, 1283-1298,1999 Julian, W.H., Samarasinha, N.H., and M.J.S. Belton, Thermal Structure of Cometary Active Regions:Comet
lP&lalley, Icarus, 144, 160-171,200O
Klinger, J., Influence of a phase transition ice on the heat and mass balance of comets, Science, 209,27 l-27 1, 1980 Kochan, H., K. Roessler, L. Ratke, M. Heyl, H. Hellmann and G. Schwehm, Crustal strength of different model comet materials, ESA SP-302, 115-l 19, 1989 Lamy, P.L., M.F. A’Heam, I.Thoth, and H.A. Weaver, HST observations of the nuclei of comets 4.5 P/Honda-Mrkos-Pajdusakova,
22 P/Kopff, and 46 PWirtanen,
Bull. Amer.
Astrcm. Sot., 28, abstract # 08.04, p. 1083,1998, see also Comet 46 P/Wirtanen, ZAU Circ. 6478,1996 Mohlmann, D., ESA-SP-116.5, Physical properties of cometary surfaces, ESA, 2001 Schulz, R. and G. Schwehm, Coma Composition and Evolution of Rosetta Target Comet 46PfWirtanen
Space Science Rev., 90, 321-328,1999
Seiferlin, K., N.I. Kiimle, G. Kargl, and T. Spohn, Line heat-source measurements of the thermal conductivity of porous HZ0 ice, CO* ice and mineral powders under space conditions, Planet. Space Sci., 44, 691-704-1996 Steiner, G., and N.I. Kijmle, Thermal Budget of Multicomponent Res.,No.
E3, 96, l&897-18,903,
1991
Porous Ice, J. Geophys.
Pergamon www.elsevier.com/locate/asr
INFLUENCE OF POROSITY ON THE THERMAL BEHAVIOUR OF COMET SURFACES
D.MGhlmann, DLR Institut fiir Raumsimulation,
51170 Kdn, Germany
ABSTRACT Comet nuclei are assumed
to be of porous structure.
A consequence
sublimating matter is that sublimation will take place also at Jntemal“ at the outer surface. This considerably increase.
The resulting
to a non-porous
of
surfaces and not only
modifies the thermal and sublimation
surface material will be cooler if compared gradients
of the porosity
behaviour. The
surface while the temperature
comet diurnal and depth variations
of temperature
and
outgassing properties are described in some detail in their dependence on heliocentric distance for different porosities, pore radii and heat conductivities
for a model nucleus on the orbit of
comet 46PTWirtanen. The results are compared to astronomically comet. Based on that, ,,most appropriated“
observed properties of this
numerical values or ranges of parameters which
can be used to characterize the properties of comet 46P/Wirtanen 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
are identified.
1. INTRODUCTION The results comparatively Porosity
of the Halley-missions low mass density
in 1986 indicate
that cometary
p. This implies a high porosity
nuclei
p of cometary
are of matter.
is defined by the ratio of ,,empty space“ and volume filled with matter. A further
parameter
is the pore size. This can be modeled by cylindrical
tubes with a corresponding
,,pore radius“ rp. In real matter tubes do not extend unrestrictedly through the body into large depths. This property is proposed here to be described by a percolativity K which is assumed to decay exponentially
with depth z. This percolativity
limits the ranges of interconnected
tubes and thus of the gas flow in porous matter. The temperature
in porous matter is governed by heat conduction
transport by gas. A characteristic
Jhermal
penetration length“ L can be determined in case of
a periodic heating, as it can be assumed for rotating and periodically At the outer surface conduction
insolation
energy
and infrared re-radiation.
and partly by energy
is transformed
Furthermore,
691
orbiting cometary nuclei.
into energies
of sublimation,
heat
it can be assumed that there is no heating
692
D. MBhlmann
at depths which are large if compared to the orbital thermal penetration
depth. Therefore,
a
constant temperature can be assumed as a boundary condition at a sufficient depth. A model of a porous surface of pure water ice will be used in the following numerically
thermal and sublimation
profiles in their dependence
on heliocentric
The restriction to water ice only can be justified as a first approximation
to derive distance.
by the fact that water
ice is the dominating cometary matter. Other volatiles are not expected to have any essential influence
on the thermal
Furthermore,
balance
in the upper surface
the low gravity of 46P/Wirtanen
layers
of comet
46P/Wirtanen.
and the observed gas production rates indicate
that this comet is not covered by a mantle. It is expected that this comet nucleus has a surface with ,,free sublimation“. Thermal and sublimation properties of cometary matter will be computed numerically for different porosity properties
and heat conductivities
within the frame of the model as it is
described in the following. 2. THE MODEL Porous and volatile matter sublimate
at sufficient temperatures
inner or ,,internal“ surfaces. Internal sublimation outer surface if the porous matter is ,,percolative“,
will contribute
at the outer surface and at to the total outflow at the
i.e. if the gas which is released at the
internal surfaces can reach the outer surface. The internal energy balance is determined
by
heat conduction and energy transport due to gas flow. Here, and as a first approximation,
the
energy transport by the gas will be taken into account only by assuming that half of the isotropically
released gas will propagate
upwards and leave the comet without effectively
cooling the upper layers. This sublimation caused outflow is related to a corresponding term in the energy balance. downward
The other half of the released
into cooler regions where it recondensates
gas is assumed
to propagate
at the cooler internal surfaces. The
related energy transport can be neglected if it is assumed that this recondensation
takes place
near to the depth of release. Thus, the ,,sink term“ in the equation of heat conductivity, is proportional
sink
to 2p/r,, has to be halved in this approximation
which
to describe the internal energy
transformation. The finally used equations are
PC%
- -&A%
= - % H(T)z(T)
(1)
for the temperature field T(z,t), and 1-A -y-'h
SCf(s,‘P,t) =
A aT(z,t,<,cp) -L
aZ
+ &oT4(z = O,t,s,q)
+ (I- p)Hz(z =
o,t,S,q) (2)
2=0
and dT(z,t) ___ dZ
ZZ 0 ;>>I
(3)
Thermal Behaviour
of Comet Surfaces
693
for the boundary conditions. The cometary energy balance at the surface is governed by solar heating, infrared re-radiation,
sublimation caused cooling, and by heat flow into the nucleus.
This balance has to be described for the surface and for the pores. Here it is assumed that the temperature
of the upper surface and ,,of the pores“ is equal, that there is no heat conduction
through the ,,empty“ pores, and that the sublimating (and also the heat conducting)
surface is
reduced by (l-p). This limits the applicability of Eq. (2) to spatial resolutions of or larger than the pore size. The relation
Z(T(z)) =
describes
the temperature
normalized
dependence
&FiiEg eTFz) of the sublimation
rate Z. Here, z is the depth
to L, i.e. the real depth x is given by x=Lz. Analogously,
the rotation
period P. The function
f(S,q,t)
describes
time t is normalized to
the local cometographic
and time-
dependence of (the angle of incident dependent) solar irradiation at the cometary surface. It is for equatorial latitudes given between sunrise (at n/2) and sunset (at 3n/2) by f=-cos(;?&P). Furthermore, heliocentric
SC=1368 W rn-’ is the Solar distance,
c is the specific
constant“,
i.e. the flux of solar energy at 1 AU
heat, A is the albedo of the surface,
and E is the
emissivity of the surface. The diurnal penetration depth L is then given by L = &&c coefficients
in Eq. (4) are according to Fanale and Salvail (1984) determined by a = 3.56 10”
N me’, b = 6141.667 K. Furthermore, molecule,
. The
and k is Boltzmanns’
m = 3 1O-‘6kg is the mass of the sublimated
constant. H=8 10.” Ws per molecule
water
is the latent heat for
sublimation of one water molecule, This value correspond to the value of H at a temperature of 198 K in the formulation
of Delsemme and Miller (1971). Note that the precision of these
numbers came into question
fitting of measurements
without taking into
account the Clausius-Clapeyron
equation (Huebner, 1998). Furthermore,
the energy necessary
to release
from a water ice surface
microscopic
a water molecule
due to incorrect
structure of this surface. Therefore,
depends
uncertainties
strongly
on state and
of the order of about lo%, as
they are found in different values in the literature (Cowan and A’Heam, 1979, and Delsemme and Miller, 1971) have to be accepted. A more precise determination
of these numbers seems
to be an actual task. Equation (4) describes free sublimation into vacuum. This approximation applied as long as kinetic pressure of the released gas can be neglected
can be
if compared to the
saturation pressure. This is at 200 K of about 0.16 Pa while the kinetic pressure in the pores is at 1 AU conditions of the order of or less than 10-j Pa. It has to be noted that it is assumed in Eq. (1) that horizontal heat flow can be neglected
if compared
to vertical heat propagation.
This approximation
may fail at local
shadows and at the day-night boundary. The above given or an equivalent and sublimation
set of equations is usually taken to model thermal
properties of cometary surfaces (cf. Capria et al., 1996, Julian et al., 2000).
This set of equations is simplified here insofar as only pure water will be taken into account.
694
D.M~hlmann
Porosity, outgassing
pore radius
through
appropriatedly
by
reduction of connectivity energetically changes
K
When
can be modeled
takes into account that there may happen a
of tubes with increasing depth. Here, a value of x=1/L can be used
example. This value forx
is used to take into account that most of the
relevant processes will happen within a depth of the order of L. Thus, structural
are expected
plausible
are the yet ,,free“ parameters.
also a ,,percolativity“
exp(-Xx). This percolativity
K =
as a characteristic
and heat conductivity
the pores is considered,
to be mainly within depths of this order of magnitude.
suggestion
,,percolativity“
only.
qualitatively
influence essentially
Experiments
are necessary
and quantitatively.
to validate
this
This is a concept
It can be expected that percolativity
the outgassing behaviour if the percolativity
of
does not
depth is much larger than
the thermal penetration depth L. This is due to the fact that most of outgassing happens in the hottest,
i.e. in the uppermost
following. Heat conductivity
h is influenced
section for heat conduction between neighbouring proportional
parts of the surface.
This is shown more in detail in the
by two effects. This is at first the reduced cross
of a solid continuum,
which is due only to the small bridges
connected particles. This is described by the Hertz factor h, which is
to the square of the ratio of bridge size to particle size. The heat conductivity
the grain material is then given by h*=hhc,,,,, where hcrainis the heat conductivity material. This material is water ice in the models of this paper. Furthermore, will be reduced by macroscopic bridge-connected
porosity
heat conduction
p with respect to that of a solid continuum
grains) with heat conductivity
of
of the grain (of
3L*of the material outside the pores according
to h = h* (l-p). This is caused by the reduced
surface that heat flow must cross. In the following
calculations,
to cover a wide range of possible
h*=h,,,
W rn-’ K“ and of h*=0.02
2.5
conductivity
k *. This
(5)
corresponds
conductivities,
conductivity
model
values
of
W mm’K-’ are used for the ,,solid state“ heat
for the smaller conductivity
to a Hertz factor of l/125,
which is well compatible to the value used by Huebner et al. (1999). As can be seen in the following Table I, which gives a summarizing
overview of the
used model parameters, pore radii of rp = 0 m, rp = 1O-3m and of rp = 10m5m are used. Model I is a ,,reference“
model with respect to pure and non-porous
water ice. The other models in
Table I are chosen to fit the expected range of physical parameters (Mohlmann,
2001) and to demonstrate
for comet 46PAVirtanen
effects due to changes of heat conductivity,
and pore radius. The largest value of the heat conductivity,
i.e. h*=&
porosity
=2.5 W me’ Km’ is
chosen as an upper limit and following Klinger’s (1980) formula for the heat conductivity
of
crystalline water ice in the range of about 200 K. This large value might not be unrealistic for upper surface layers which could be hardened and densified due to recondensation as they have been observed mentioned heat conductivity
in the lab (Kochan
processes
et al., 1989). The lower of the above
values is suggested as to be typical for a lower limit (cf. Seiferlin
et al. (1996), Steiner and Komle (1991)) for porous ice with pore sizes of r, =10e3 m and temperatures around 200 K and less.
Thermal Behaviour of Comet Surfaces
695
Model
h*[W mm’K*‘]
Uml
2.5
p 0
rAmI
I
0
0.245
II
2.5
0.5
1o-3
0.245
III
0.02
0
0
0.022
IV
0.02
0.5
1o-3
0.022
V
0.02
0.5
1o-5
0.022
VI
0.02
0.8
1o-3
0.022
Table I
Model parameters used in the numerical computations. Furthermore,
the assumption
has been made throughout
nucleus is rotating around a fixed axis perpendicular precessional motions. The computations The numerical computations initial conditions
this paper that the comet
to the orbital plane and that there are no
refer to an equatorial location at the nucleus.
of the temperature evolution from the relatively arbitrary
will be stopped after reaching an ,,equilibrium
defined here by a temperature
distribution
fulfilling
state“. This equilibrium
the numerical
is
surface energy balance
better than 1% (i.e. for relative numerical errors smaller than 0.01 of the integral over one rotation
of the difference
surface sublimation
insolation
power and powers of surface
and inward heat flow which are normalized
energy). Therefore, ,,equilibrium
between
the temperatures
temperatures“
re-radiation,
by the daily insolation
derived and used within this paper are to be seen as
as they will evolve in and at a cometary
surface as long as the
rotation period is very small if compared to the orbit period. The computations
are performed
for different heliocentric distances. 3. RESULTS Results of the above characterized
numerical computations
will be described in detail
in the following. There is often a reference to heliocentric distances of 1.06 AU or 3 AU. The distance of 1.06 AU ist the perihelion distance of 46P/Wirtanen,
and the distance of 3 AU is
of special interest for the Lander of the ROSETTA mission. This Lander is planned to reach the surface of 46P/Wirtanen
at a heliocentric distance of about 3 AU.
3.1 Surface EnerPv Balance and Relevant Processes The diurnal variation of the different energy fluxes at a freely sublimating icy surface of a comet nucleus at 1.06 AU heliocentric
distance is described
Figure 1. It can be seen that the contributions
to this energy balance depend strongly on model
parameters.
In case of zero or very small porosity,
heliocentric
distance the dominating
minor importance.
surface
for equatorial sublimation
energy sink. Heat conduction
latitudes in
is at 1.6 AU
and re-radiation
are of
This is different for a porosity around p=O.5, where heat conduction
related sublimation and related outflow.
become essential. The internal heat is converted Surface sublimation,
comet, is not important
i.e. the sublimation
in this case of the larger conductivity,
and
into sublimation energy
at the ,,outer“ surface of the but it becomes
dominating
696
D. Mahlmann
again for a comparatively
smaller
heat conductivity.
Generally
spoken, sublimation
is the
dominant loss process at about 1.06 AU heliocentric distance. Comparison of heat conduction in parts A and B, and also in parts C and D in Figure 1 illustrate the porosity caused change in the physical situation at otherwise identical properties for the ,,sublimation conductivity,
dominated“
part of the orbit. In spite of decreasing
effective
heat
heat conduction increases for increasing porosity. This is caused by the internal
cooling due to sublimation
and the related outflow of released gas. Internal cooling causes
stronger temperature gradients and an increase in the related heat flow. Comparison sublimation
of parts B and D teaches that the ratio of surface sublimation to internal
increases
for decreasing
conductivity.
This is due to the higher
surface
temperatures in case of weak conductivities.
z -200
Fig. 1. Surface energy balance at equatorial latitudes at 1.06 AU heliocentric distance for models I-IV. All profiles are given from midnight to midnight over one rotation period z after reaching
,,numerical
Abbreviations
equilibrium
conditions“
(over
a different
number
of rotations).
are: In for insolation, Ir for infrared re-radiation, Hc for heat conduction, Ss for
surface sublimation, and Si for that part of internal sublimation which reaches the surface.
Thermal
Behaviour
of Comet
Surfaces
697
The relations in the surface energy balances change with heliocentric 2 gives the corresponding Figures
energy balances for 3 AU. It can be seen by comparing parts A in
I and 2 that infrared re-radiation
larger heliocentric
distance. Figure
and heat conduction
become more important for
distances while sublimation becomes less effective. Comparison of parts B
and D in Figure 2 teaches that changes in porosity (or density) are not followed
by strong
changes in the surface energy balance. Contrary to this, and as it is described by parts D and B in Figure 2, internal sublimation Si and the related outflow at the surface increase remarkably for smaller pore radii. This behaviour is a consequence
of the structure of the rhs-term in Eq.
(1) which takes into account the increase of the ,,area of internal surfaces per volume“ for decreasing pore size. Thus it is the size structure of the pores which is an essential parameter, additional to the porosity itself. Exactly, it is the ratio p/r, which governs this thermal and
W
l-ii-l
F[Wrril
n-?l
150rh*=o.02
l-7
qw f-n-*]
lso fh*=0.02 W 125
In
III
D
100 75 50 25
5.8
Z
6
-25
-25
outgassing behaviour. Fig.2. Surface energy balance at equatorial latitudes at 3 AU heliocentric
distance for models
I,IV,V,VI. All profiles are given from midnight to midnight over one rotation period r after reaching
,,numerical
Abbreviations
equilibrium
conditions“
(over
a different
number
of rotations).
are: In for insolation, Ir for infrared re-radiation, Hc for heat conduction, Ss for
surface sublimation, and Si for that part of internal sublimation which reaches the surface.
698
D. MGhlmann
3.2 Diurnal Death Profiles of the TemDerature
model
TN
IV
model V p=o.5
8
model
J-WI
Z
10
VI
p=O.8 rP= 1 Oe3m
180
A
Fig. 3 Diurnal depth profiles of the temperature at equatorial latitudes at 3 AU for models IV, V and VI.
Thermd
The temperature penetration
Hehaviour
of Comet Surfaces
699
in cometary surfaces depends strongly on depth. The diurnal thermal
length L = dw
is the characteristic
measure for this depth dependence.
can be seen in Figure 3 that most of the diurnal temperature
changes happen within depths
down to only a few L. The depth is described in Figure 3 by the dimensionless Furthermore, Jimiting“
It
depth 2=x/L.
it is use to describe the depth dependence of periodic temperature profiles by the
curves of the depth profiles at the time of the minimum and of maximum surface
temperature. These are given for a heliocentric distance of 3 AU in Figure 3. Comparison effective
of models IV and V indicates,
as it has to be expected,
that cooling is more
for smaller pore sizes (for unchanged porosity). This is due to the increase of the
area of internal surfaces per volume, as it is described by p/r,. It can be seen by comparing models IV and VI that the small changes in porosity
values, as they are only possible for
realistic porosities, are not followed by significant changes in the thermophysical
behaviour.
Thus, the most critical porosity parameter in current comet modelling is the pore size. 3.3 Diurnal Heat Flow and Temuerature
Gradients at the Surface
m-21
j[ 40
20
!I“
1
1.
i 5.2
5.4
5.6
._ 5.8
‘I:
6
Fig. 4 Diurnal depth profiles of the heat flow and surface heat flow at 3 AU for equatorial latitudes for models IV andV.
700
D. MBhlmann
Thermal evolution
properties
of the cometary
of the surface temperature
surface
have an essential
and related sublimation.
influence
on the
Thus, the heat flow at the
surface is an important surface property. It can easily be seen in Figure 4 that the heat flow is inward directed at the dayside, but there is a flow from inside towards the surface during the night. The difference
between the dayside inward flow and the smaller night side flow is
caused by transformation
of heat into internal sublimation. This is a specific characteristic
of
porous bodies of volatile matter. As it has shown above, inward heat flow is in porous bodies larger for smaller pore sizes. Cooling
due to internal
sublimation
behaviour. It can be derived from Figure 4 that the temperature
is the cause of this
gradients are strongest in the
uppermost parts of the surface. They can reach a few 10” K m-l within the upper millimeter. The related thermal stresses can overcome there the cohesive strengths of the surface material. This can be followed by a thermal cracking triggered separation of millimeter-sized particles which can be lifted-off into the coma by the outstreaming contribution
to the gas production
or smaller
gas. There they will give a
rate additional to that of the cometary
surface (including
the outflow due to internal sublimation). The importance of this ,,second source“ is expected to increase for decreasing heliocentric distance. 3.4 Dependence of Surface TemDeratures on Heliocentric Distance Typical diurnal temperature follow
for other models,
profiles have been shown in Figure 3. Similar profiles
but they differ remarkably
in their maximum
and minimum
temperatures as they are given in Figure 5.
180
160
160.
4’
‘.“...,.
“,....
2
3
s
qpJl
5
2
3
Fig. 5 Maximum and minimum equatorial surface temperatures in dependence on heliocentric distance. Models are described by Table I. Model IV A is Model IV with P=24h. It can be seen in Figure 5 that maximum temperatures conductivity
(cf. models II and IV). Furthermore,
increase for decreasing
heat
decreasing pore size will be followed by
stronger cooling (cf. models IV and V), while the maximum surface temperature
increases for
increasing porosity, as can be seen by comparing models IV and VI. Minimum temperatures, which occur around sunrise (and not at midnight) are highest for larger heat conductivity models I and II with respect to IV, V and VI). Lower minimum temperatures
(cf.
are shown to
evolve for decreasing
pore radii (cf. models IV and V) and with increasing
models IV and VI). Note that the model with the highest porosity,
porosity
(cf.
i.e. model VI, has the
highest maximum and the lowest minumum temperatures. The derived
minimum-
Figure 6) agree for comparable
and maximum
values of temperature
and sublimation
models well with values derived by Benkhoff
(1996) Capria et al. (1996) by the ISSI team (Huebner
(cf.
and Boice
et a1.(1999)), and Julian et al., (2000).
Model IV A was chosen especially for comparison with ISSI results. 3.5 Denendence of the Sublimation Rate on Heliocentric Distance Figure 6 gives the dependence heliocentric
of total sublimation
me’ s-‘1 on
distance at equatorial latitudes for different models. The total sublimation rate is
defined here as to consist of the sublimation rate through
rates Z[molecules
the pores due to internal
maximum total sublimation
rate of the surface material and of the outflow
sublimation.
Models
with a comparatively
rate are of interest when compared to astronomically
water production rates of 46P/Wirtanen,
large
observed
which are surprisingly large in view of the small size
of this comet nucleus (cf. Schulz and Schwehm, 1999).
‘1 4 mol. rn* 2sece 1.
1.
Fig. 6 Diurnal profile of the maximum total sublimation
rates for different
models and in
dependence on heliocentric distance. Model IV A is Model IV with P=24h. 3.6 Deoth Denendence
of Internal Sublimation
Sublimation in porous matter has to be described by a ,,surface sublimation rate“ [mol. mm’s-‘1 of a volume with a ,,sublimation density“ S[molecules mm3s-‘1 with an outflow through a defined surface, i.e. here through the cometary surface. This sublimation
density S can be
described by
s =
!pz. rP
(6)
702
D. MGhlmann
Note that S in Figure 7 is a sublimation
density,
i.e. the number
of sublimated
molecules per volume. This should not be confused with the sublimation rate, i.e. the number of released molecules per surface.
S[mol. m-3s-‘] 3.5x10zr
S[mol. m-3sM’] 1.06 AU
3x1o26
r
1.06 AU
=10w5m
2.5~10~~ 2x1o26 1.5x101b~ 1x1026 5x1 o= 0.02
0.04
0.06
0.08
2
0.1
0.02
0.04
0.06
0.08 z
0.1
S[mol. m‘3s-1]
S[mol. m-3s“] 3 AU
r = 1G5m
0.02
0.04
Fig. 7 Depth dependence the time of maximum
0.06
0.08
z
0.02
0.1
0.04
0.06
0.08
z 0.1
within 0.1 z of the sublimation density S for models IV and V at
total sublimation.
The depth z is measured
in units of the diurnal
thermal penetration depth L, p=O.5, and h*=0.02. In case of the above mentioned
model assumption
that only half of the internally
released molecules will reach the surface, a factor of 0.5, as it has been taken into account in the rhs-term of Eq. (1) has to be used additionally rate. The depth dependence
in Eq. (6) to describe the real gas release
of the sublimation density S according to Equ. (6) is described in
Figure 7. It can be seen that most of the internal sublimation takes place directly beneath the surface within depths less than half a millimeter in the case of model IV. The thickness of the layer of highest sublimation density decreases with decreasing pore size. 4. CONCLUSIONS The results
of modelling
summarized in the following:
cometary
surface
and outgassing
related
properties
will be
Thrrmnl
l
As shown with Figure
Behaviour
of Comet Surfxes
703
4, most of the water relevant
energetic
processes
in freely
sublimating cometary surfaces happen in the upper parts of the surface within depths of a few centimeters.
This importance
increases for decreasing sublimation
of the uppermost
surface parts for cometary
activity
heliocentric
distance insofar as the erosion depth due to surface
approaches increasingly
the thermal penetration depth. It reaches about 40%
of the thermal penetration
depth at 1.06 AU for model IV. This steady ,,reduction of the
surface height“ was not taken into account in the above described cause something
of the thermal gradients
like a ..steepening“
computations.
It will
in the uppermost
active
layers. l
It is shown in Figures 1 and 2 that the energetic regime at cometary surfaces is different at 1.06 AU and 3 AU heliocentric
distance. Sublimation
is the dominating
1.06 AU. Infrared re-emission and heat conduction are characteristic l
loss process at
for 3 AU.
The porosity related parameters which governs the thermal and sublimational porous cometary
surfaces are given by p/rp. This is illustrated
behaviour of
by Figures
1 to 3. The
currently most unknown porosity parameter of cometary matter is the pore size. l
Comparatively conductivities
small pore sizes in the range of IO-’ m to 10e5 m and weak heat in the range of 10.’ W mm’K-’ seem to be appropriated
behaviour of a porous water-ice surface of 46P/Wirtanen possibly
larger). This result is based on computed
to describe better the
with porosities around p=O.5 (or
total sublimation
rates which are
highest for these parameter ranges. High total sublimation rates are required to make the astronomically
determined size of 46P/Wirtanen
(Lamy et al., 1996) and the resulting total
gas production rates compatible. l
Thermal
gradients
are strongest
directly
at and below the cometary
surface.
These
gradients increase for decreasing heliocentric distance. l
It should be noted that differences water
production
thermocracking
between observationally
rates of 46P/Wirtanen
triggered and lifted-off
could be explained millimeter-sized
determined
by a ,,second source“ of
or smaller particles which will
sublimate in the coma. Note that the above mentioned ,,steepening“ around perihelion
may additionally
and modelled
enhance the thermomechanical
of thermal gradients stresses and cracking
processes in the uppermost surface parts.
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