Towards a model of cometary nuclei for engineering studies for future space missions to comets

Planet. Space Sci., Vol. 44, No. I, pp. 637-653, 1996

Pergamon

Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/96 $15.00+0.00 PII: SOO32-0633(96)000674

Towards a model of cometary nuclei for engineering studies for future space missions to comets Jiirgen Klinger,’ Anny-Chantal

Levasseur-Regourd,’

Naceur Bouziani’ and Achim Enzian’

‘Laboratoire de Glaciologie et GCophysique de YEnvironnement, CNRS, Universite Joseph Fourier de Grenoble, France, B.P. 96, F38402 St. Martin d’Heres, France *Universiti Paris 6, Akronomie CNRS. B.P. 3, F91371 Verrikres, France Received 20 November 1995; revised 16 April 1996; accepted 18 April 1996

brief review is given of the present state ledge on comets. Existfmg comet models are critically analyzed in view of these observational facts. This analysis leads to the conclusion that the most promising model approach is one where the comet nucleus is considered as a porous medium containing an intimate mixture of dust particles and different ices. Compact nuclei can be treated as limiting cases. An outline of a p~~o-t~~~o~l modei based on this approach is given. The bypotbesis is made that comets initially contain a mixture of amorphous water ice, solid carbon monoxide and silicate dust. As examples, lo-1 production rates of CO and integrated production rates of H@ and 60 for comets PjSchwassmannWachmann 1 and P~S~hwassm~-Wa~hrn~n 3 are calculated, making the extreme assumption that all the devolatilized dust remains on the surface. The depletion of CO in the near surface layers is determined after ten revolutions. Under the assumptions that all the dust remains on the surface or that all the dust is lifted off, thermal profiles at the equator of a nucleus on the orbit of comet P/Wirtanen with the rotation axis perpendicular to the orbital plane are determined when the surface temperature is maximum or minimum, The evolution of the maximum and minimum surface temperature is computed over ten orbital periods for two values of the bulk thermal conductivity of the dust coverage. For the same comet, the relative I&O and CO content as a function of depth are calculated for a dust covered and a non-dust covered nucleus. Improvements of the present day model are sugklested and a strategy is prop0 for adapting this model for complex and thus more realistic situations. Copyright 0 I996 Elsevier Science Ltd A

Correspondence

to: J. Klinger

Introduction In order to maximize the chance of success of future space missions to comets such as the Rosetta mission, reliable comet models are needed. Indeed, during the preparation

and the development phase of the experiments supposed to fly on board a mission to a comet, the techniques used must be tested using models of comet nuclei which reflect the present state of knowledge and which include also unexpected features having a reasonable likeliness. The development of engineering models are particularly important for the thermal sounders, the radar wave transmission experiment and the gamma ray spectrometer. Such kind of models are also needed during the approach phase of the nucleus. During the data transmission phase, reliable comet models are vital for a real time interpretation of the data and decision making for the following mission phases. The purpose of this paper is to establish the basis of such a model approach using the presently available observational data and theoretical considerations. In this paper, we model temperature profiles as well as the evolution of the composition (presently the H,O and CO content) as a function of depth, of cometocentric longitude, latitude, of the orientation of the rotation axis, and of the local and global dust ejection and dust accumulation on the surface. Making specific assumptions on the texture of cometary materials, it will be possible to model also the tensile strengths of comet nuclei. In order to assess the model parameters, special attention is given to two prominent objects: comet P/Halley and comet P/Shoemaker-Levy 9. As only Jupiter family comets are accessible to space missions with presently available technologies we show examples of model calculations on two Jupiter family comets : P/SchwassmannWachmann 3 (SW3) and P/Wirtanen. In order to show the influence of the orbital parameters, comparisons are made with comet Schwassmann-Wachmann 1 (SWl),

J. Klinger 62’1 (11.:Towards

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which has an orbital period of about 15years, roughly speaking twice that of typical Jupiter family comets. The propositions made here are revisable at any moment, depending on the understanding of comets. A light diffusion model which hopefully will be able to establish the link between the model approach exposed here and the coma observations is in progress and will be presented in a separate paper. The development of a true 2-D nucleus model with meridional transport is also in progress.

Observational

facts

The rmcleus of comet PIHalleJ. The overall shape as determined by the cameras of Giotto and Vega is approximately that of an ellipsoid with main axes of 8 x 8 x 16 km. The volume of the nucleus is (550& 160) km3 and the surface area is (400$80) km3 (Keller, 1990). In view of the large scale inhomogeneities found in the surface topography and the elongated shape, several authors suggest that the nucleus had been built up from smaller bodies of a size ranging from 1 to 5 km (Keller et al., 1987, 1988; Keller, 1990; Sagdeev and Szegb, 1990). An emissive centre in the inner coma, a few kilometres in size with an integrated temperature significantly in excess of 300K has been detected by the infrared spectrometer IKS on board Vega 1 (Combes et al., 1986). This result means that at least the major part and eventually the total visible surface of the nucleus is covered by refractory material (silicates and organics of low volatility for example). The visible albedo of the surface is < 0.04 (Keller et al., 1986). No direct determination of the mass of comet P/Halley was possible. Using. the non-gravitational perturbations of the orbit due to mass lost by the nucleus, Rickman (1989) estimated the mass of comet P/Halley as 1.5 x 10” kg. Combined with the volume determined from the camera data, this leads to an average density of 300 kgm-‘. Due to the uncertainties over the shape of the gas production curve, the density range may eventually extend up to 700 kg rnp3 (Festou et al., 1993). Sagdeev et al. (1988) gave a density range for Halley between 200 and 1500 kgme3 with an average of 600 kg mm3 and Peale ( 1989) a range of 304900 kg m-‘. The results of Rickman lead to the conclusion, which has now found a large consensus in the community. that comets are fluffy aggregates of refractories and ices. Several groups of investigators have found that the nucleus rotates with a period of approximately 2.2days. However, several observation series showed evidence for a periodicity near 7.4days. There is still little consensus on the actual mode of rotation of the nucleus. It has been suggested that the nucleus spins with a period of 2.2 days about an axis which is oblique to the principal axis of inertia and rolls on its long axis with a period of 7.4days (or equivalently, has a rotation period of 7.4days along the major axis and a precession period of 2.2days), although the lack of resonance between these two periods is puzzling (Sekanina, 1990 ; Belton et al., 1991). More recently, Belton et al. (1995) found a precession of the

a model of cometary nuclei

long axis around the angular momentum vector with a period of 3.69 days compounded with a rotational period of 7.1 days. The ratio of these two periods is close to 2, but the precessional state (long axis or short axis mode) is still discussed. The images from Giotto show several bright zones in the near coma essentially on the sunlit side, thus indicating that the dust and gas production mainly comes from restricted areas, even on the day side. The total gas production rate has been evaluated as 6 x 10” molecules s- ’ (Krankowsky et al., 1986). This is the amount of water vapour that would sublimate from a surface area of 36 km’ covered with water ice and exposed perpendicularly to the Sun under vacuum at the heliocentric distance of the Giotto encounter. Such a surface area represents 3040% of the projected area of Halley’s nucleus. From these findings it has been concluded that the activity of comet P/Halley is restricted to 10% of the surface area (20% of the day side). If we exclude the presence of important internal heat sources, such a conclusion is compatible with the heat balance of the nucleus only if the active areas are located at sub-solar points, and if the sublimation occurs directly on the surface without any dust coverage. It is not clear yet if these conditions were fulfilled during the Giotto encounter. Further, it is generally considered that the footprints of the dust jets are located at the same places as the maximum gas production. Recent hydrodynamical calculations of Crifo et al. (1995) showed that, depending on the size distribution of the dust, a confinement of the dust by the gas may take place. Thus the footprints of the dust “jets” which diffuse the solar light and have been detected by the Giotto camera are not necessarily identical with the source regions of the gas which propels the dust. Majolet et al. (1994) on the other hand were able to show that jet-like features in the near coma can be obtained around an entirely active nucleus, with an activity being simply modulated by the local solar incidence angle. Almost one hundred different species have been identified (Huebner et al., 1991; Geiss et al., 1991 : Crovisier. 1994). Water vapour was by far the most abundant constituent. Its abundance was about 80% of the gas in the coma. The second gas in rank of abundance was carbon monoxide, about two thirds of which were released by an extended source in the coma. Formaldehyde was also found to be released from an extended source. An important fraction may indeed be trapped on dust grains in complex organic molecules. Due to various reactions, the chemical composition in the coma is different from that of the volatiles in the nucleus. Apart from water vapour and carbon monoxide, the other parent gases such as carbon dioxide, methane, ammonia, molecular nitrogen, formaldehyde, and hydrogen cyanide, contributed at most a few per cent to the coma gas (Krankowsky. 1991). Taking into account the fact that some elements are found both in volatiles and in refractory molecules, the elemental abundances are estimated to fit broadly those of the Sun, apart from the expected deficiency in hydrogen and helium and the apparent coma depletion in carbon and nitrogen. There is a good agreement in elemental abundance between type I carbonaceous chondrites and cometary dust (Jessberger et al., 1988). A high variability was found

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J. Klinger et al.: Towards a mode1 of cometary nuclei between individual grains. About one third of the dust consisted of CHON and silicates, whereas the last third contained no low atomic number elements except oxygen (Langevin et al., 1987; Kissel and Krueger, 1987). An unexpected amount of submicrometre size particles was found, together with large grains, up to about 100pm. Submicrometre dust particles, originally glued together by water ice or organic compounds are fragmented in flight as a result of collisions and of sublimation due to heating.

Comet P/Shoemaker-Levy 9 During July, 1992, comet P/Shoemaker-Levy 9 passed at less than 40,000 km from the upper atmospheric layers of Jupiter and split into more than 10, possibly 25 pieces. It is reasonable to think that the disruption of this comet is due to tidal forces. Prior to the impacts, some fragments disappeared. Others suffered secondary splitting, with separation velocities of about 1 m s-‘. Models describing the parent nucleus as a strengthless agglomerate of subnuclei only held by gravity, or as a discrete nucleus with low tensile strength in which fissures propagate. have been proposed (Scotti and Melosh, 1993 ; Asphaug and Benz, 1994; Solem, 1994; Sekanina, 1995). asphaug and Benz (1994) found a bulk density of the parent comet in the range of 300-700 kgmp3, Solem (1994) gives as density a value of 500 kgrn--j. Using the classical Roche theory of tidal disruption of a viscous body without internal cohesion, Boss (1994) estimated upper limits of the density of P/Shoemaker-Levy 9 between 1140 and 2430 kg rn-‘. At least the first value is compatible with a porous body, but the real density value may be substantially lower. especially if the comet was held together by cohesive forces. From Galileo measurements, the size of the nuclei has been estimated to be in a range between 0.3 and 1 km. From Hubble Space Telescope data, the smallest fragments could be 0.5 km across, while the largest could have a diameter of 2.9 km (Weaver rt al., 1995). Sekanina (1995) found values between 0.6 and 4 km. The size distribution seems to be better assessed than the size of the individual fragments.

Other cornet nuclei Comet P/Halley is the only comet whose nucleus has been imaged. However, the radar cross-sections of some other comets have been measured (Goldstein et al., 1984) and numerous photometric measurements of light curves are available. The minimum and maximum size of comet nuclei remain unknown. Published values, derived from infrared observations (once the visual magnitude and its normalization to 0” phase angle are known) and radar observations (making reasonable assumptions on the radio reflectivity and the rotational characteristics), are in the range of l-10 km. Outer Solar System objects, such as SW1 and Chiron, are considered by several authors as large comets with nuclei of about 40 and 200 km in diam-

eter, respectively. Nevertheless, according to a recent work by Meech et al. (1993) the size of comet SW1 could be much smaller than estimated earlier. Although some huge comets (e.g. the great comet of 1729) have been suggested to have masses up to 10” kg, most masses, as estimated from indirect arguments based on perturbations of cometary orbits by jet-reaction forces due to outgassing, are within 10” and 1015kg. Considering comets as low-density objects and making reasonable assumptions on their size (several km up to 10 km) we are to deduce from the survival of Sun grazing comets, a tensile strength of approximately 1O”Pa (Klinger et al., 1989a).

Origin of’ comets Based on the studies of Sinding (1948) and van Woerkom (1948), Oort (1950) deduced that a comet reservoir should exist between 20,000 and 200,000 AU. The statistical study of Oort has been improved by Marsden et al. (1978). Using a sample of 200 long period comets with well determined orbits. these authors found an average aphelion distance of Oort cloud objects as 45,OOOAU. Thus, comets are without any doubt members of the Solar System. The question is now where, under what conditions and at what time scale comets formed. A formation directly in the Oort cloud is difficult due to the low density of matter there, leading to unreasonably long formation times. Kuiper (1949, 1951) proposed that comets could have formed outside the orbit of Neptune and that these objects have been ejected to the Oort cloud by the cooperative action of Pluto’s and Neptune’s gravity field. Even if we know now that at Kuiper’s time the mass of Pluto has been overestimated, this approach is still viable. In the meantime, several possible “Kuiper belt” objects. this means trans-Neptunian objects have indeed been detected (see Jewitt and Luu (1993) and references therein). Recently. Cochran et al. (1995) presented a statistical study of a Hubble Space Telescope survey of potential Kuiper belt objects. They estimated that in the range of orbits they studied, the number of comets brighter than their limiting magnitude is at least 10”. Some of these trans-Neptunian objects may eventually be injected into the inner solar system, thus forming short period comets, Jupiter family comets for example. What is interesting in the present context is that comets have probably been formed directly from interstellar grains in a cold environment (Greenberg, 1982). The presence of very small grains in the coma of comet P/Halley and the chemical similarity between interstellar molecules and molecules in cometary comae are in favour of this idea. Far from the Sun, the relative velocity of accreting particles should be small ( ~0.1 km s-l). As Donn (1963) already pointed out, such relative velocities are a necessary condition for the preservation of ices more volatile than H1O in comets. But this author also showed that the bodies accreted with such small velocities should be fluffy. As previously stated, there are strong indications that comets are indeed low density bodies. It should be mentioned also that it can be concluded from dynamical deter-

640 minations of the density of several small Saturn satellites that these bodies have porosities 2 30% (Dones. 1995). If we take as a hypothesis that comets formed by accumulation of basic elements of submicrometre size we can consider different scenarios to end up with kilometre size objects: (i) growth of objects by accumulation of individual grains (Ballistic Particle Cluster Aggregation, BPCA), (ii) growth of objects by accumulation of individual grains, small agglomerates of grains and agglomerates of agglomerates... (Balistic Cluster Cluster Aggregation, BCCA), (iii) growth of agglomerates up to a given limiting size of agglomerates and subsequent accretion of these agglomerates. In a great number of natural systems similar to the accumulation of comets, the second growth mechanism is preferred to the first one. But the main feature of the fractal objects obtained in this way is that their average density diminishes with growing size. This means that as long as no compaction mechanism becomes efficient macroscopic agglomerates should have extremely low average densities. A model of comet aggregation based a BCCA process has been published by Donn (1991). According to Weidenschilling (1994) the formation of comets occurred in a two-stage process (corresponding to the third scenario) : in the protostellar nebula small grains form clusters up to a certain size (tenths of metres). When this size is reached the grains decouple from the gas phase of the nebula and sediment to the equator plane where they form bodies of several kilometres across. As already mentioned, the break up of comet P/Shoemaker-Levy 9 favours this concept. Let us mention here that the idea that comets should be composed of “blocks” of hundreds of metres, up to kilometres in size had already been published during the 1940s (see Minnaert (1948) and references therein). A similar idea has been published more recently by Weissman (1986) under the name of “rubble pile model”. An alternative idea has been proposed by Gombosi and Houpis (1986) and more recently by Miihlmann (1995). In this concept the comet nucleus is considered as composed of “boulders” glued together by an icy substance. The surface regions where these icy zones emerge are considered as the “active” zones. But this concept still needs to be explained in terms of an aggregation mechanism.

Comet models Whipple (1950) showed in a convincing manner that comets cannot be a swarm of dust grains carrying adsorbed gas molecules but must have a solid nucleus. This so-called “icy conglomerate” model is still the basis of virtually all comet models.

Clathrate hydrates and molecular hydrates in comets In order to explain the simultaneous sublimation of ices with very different vapour pressures, Delsemme and Swings (1952) proposed that comets contain clathrate hydrates. Clathrate hydrates are non-stoichiometric com-

J. Klinger rt al.: Towards a model of cometary nuclei pounds formed by a lattice of water molecules. These water molecules engage gas atoms or molecules such as CO. CO,, CHJ. Nz, Ar, Kr, Xe ,... The atoms or molecules stabilize the clathrate structure. The “guest” atoms or molecules are weakly bound to the host lattice by Van der Waals forces and can easily migrate from one cage to another (see Miller, 1985). Delsemme and Swings suggest that when the comet approaches the Sun, these hydrates decompose and liberate simultaneously the water and the guest molecules. It has been shown in the meantime that this idea is thermodynamically questionable (see Klinger et al., 1986). Indeed when clathrate hydrates loose their guest atoms or molecules the clathrate structure collapses to form an ordinary ice structure. Thus the sublimation of water ice is conditioned by the saturation pressure of ordinary water ice. This means that the decomposition of clathrate hydrates does not explain the simultaneous outgassing of water and substances much more volatile than water. These considerations obviously do not exclude the existence of clathrate hydrates in comets. Nevertheless the formation of clathrate hydrates in comets sets very severe limits to the pressure and temperature conditions during comet formation and/or cometary evolution. More recent laboratory work seems not to be in contradiction with this view (see Blake et nl., 1991). The question of simultaneous outgassing of substances with different volatility finds a natural explanation if we consider that comet nuclei are porous (see below). The case of molecular hydrates may be very different. Here the hydrate forming molecules are bound to the water molecules by strong hydrogen bonds. For this reason molecular hydrates are thought to be stable up to temperatures where water ice sublimates. A prominent hydrate forming molecule is NH,.

Exothermic processes in comets Already in the 1950s it had been proposed that cometary matter may contain free radicals. The (exothermic) recombination of these free radicals could be an energy source for comet outbursts. These reactions could be triggered by solar wind particles or simply by solar heating. The radicals may have been formed by UV or energetic cosmic ray particle bombardment of interstellar grains which formed the comet nucleus. An alternative possibility of radical formation is that during the stay of the comet in the Oort cloud the outer layers of the nucleus are chemically transformed by cosmic rays (Donn and Urey, 1956 ; Shul’man, 1972). These ideas still merit consideration. Another possibility of such a “chemical fuelling” of comet outbursts has been proposed by Rettig et al. (1992) : outbursts could be due to the polymerization of HCN. Patashnick et al. (1974) proposed that the exothermic crystallization of amorphous ice could be an efficient energy source for comet outbursts. If comets have been formed from unaltered interstellar grains and taking into account the fact that interstellar molecular clouds have typical temperatures on the order of 1OK. we have to consider that cometary ices should be initially in their low temperature form. Knowing that the dominant com-

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J. Klinger et al.: Towards a model of cometary nuclei ponent of the volatile phase of comets is water ice, we can conclude that this water ice is initially amorphous. The properties of amorphous water ice have been studied by Dowel1 and Rinfret (1960) Ghormley (1968) and Mayer and Pletzer (1985). Due to these authors, the main properties of amorphous water ice are : The surface structure is extremely porous with a great number of deep adsorption sites allowing an efficient trapping of other volatile molecules during the condensation phase of the ice. When heated to temperatures close to 100 K, the surface smooths progressively. The diminution of the surface free energy associated with this process is about 20 kJ kg-‘. The smoothing of the surface means that volatile molecules such as CO or CO, adsorbed in deep sites are efficiently trapped. This means that cometary ices may contain volatiles such as CO or CO? even at conditions of pressure and temperature under which those volatiles should have sublimated. During heating the amorphous ice crystallizes irreversibly and exothermally first into the cubic form and after further heating into the stable hexagonal form. The enthalpy change during the phase transition from amorphous to cubic is about 67 kJ kg-‘. According to Schmitt et al. (1989), the crystallization of amorphous ice follows an activation law as

(1) During the crystallization process the volatiles trapped in the amorphous matrix (CO, C02) are expelled. The survival of amorphous ice in comet nuclei is conditioned by the temperature within the comet throughout the history since the formation of the body. The problem of interior temperatures of comets had already been considered in early work (see Minnaert (1948) and references therein). Klinger (198 1, 1983) evaluated the temperature limit against which the centre of the comet should tend making the following assumptions : The comet contains only water ice. The nucleus is a “good integrator”, this means that the orbital period is much smaller than the characteristic time for the penetration to the centre of a thermal perturbation applied at the surface. The final deep temperature is obtained from the heat balance of the nucleus integrated over the orbital period. The heat balance at the surface is that of a “fast rotator”. This concept has been used to evaluate the chance of survival of amorphous ice in comets. The idea was that, taking into account the fact that the crystallization of amorphous ice is exothermic, the ice should crystallize in a “runaway” process in all comets where this “orbital mean temperature” is higher than the temperature under which amorphous ice can survive. From these considerations Klinger (1981, 1983) concluded that typical Jupiter family comets like P/Encke, P/Tempel 2 and P/d’Arrest should contain crystalline ice. In comets like

P/Halley, the surface erosion close to perihelion should progress faster than the penetration of the crystallization front. Thus, such type of comets is likely to contain still amorphous ice in the central parts. Comet SW1 which sometimes shows unpredictable changes in its lightcurve, so-called “outbursts”, although the average level of activity is correlated with heliocentric distance, the correlation is somewhat different for the outbursts. In particular, no characteristic frequencies could be found in the variations of the visual magnitudes (Cabot et al., 1995, 1996). The approach of Klinger (198 1, 1983) has been criticized by McKay et al. (1986) and by Herman and Weissman (1987). A detailed analysis of these last two papers shows that these criticisms are due to a misinterpretation of Klinger’s original work. In the case of the paper by McKay et al. the discrepancies are also due to differences in the boundary conditions. Klinger (1983) computed the final central temperature from the energy balance integrated over one orbit whereas McKay et al. used the instantaneous surface temperature in radiative equilibrium with the Sun. In any case, Herman and Weissman (1987) confirm the main conclusion of Klinger that Jupiter family comets should contain crystalline ice. It should be mentioned here that in the mode1 published by Espinasse et al. (1989, 1991) (see below), the crystallization of amorphous ice in the nucleus of a Jupiter family comet (P/Churyumov-Gerasimenko) occurs indeed as a runaway process. But obviously the time scale of this process depends on the physical parameters used in the model. Recently, Kouchi et al. (1994) tried to answer the question if amorphous water ice is able to survive the formation process of comets. Based on their own measurements of the heat conduction coefficient of amorphous ice, Kouchi et al. found out that the survival of amorphous ice is only possible in a rather narrow range of conditions. This is due to the fact that the measured value of the heat conduction coefficient of amorphous ice is several orders of magnitude lower than that estimated by Klinger (1980). What has not been taken into account in the work by Kouchi et al. is the heat transfer due to the gas phase which should be very important at temperatures much lower than 100K when volatiles like CO or CH4 are present. In other words, Kouchi et al. consider the most pessimistic case. The amorphous to crystalline phase transition of water ice has been included in a great number of models since. Without claiming completeness let us mention the models by Herman and Podolak (1985) Prialnik and Bar-Nun (1987), Espinasse et al. (1989, 1991) Prialnik (1992), and Tancredi et al. (1994).

Dust covers on comet nuclei

The development of a dust coverage on the surface of the nucleus and the stability of such a coverage has been modelled by Brin and Mendis (1979). These authors considered that cometary dust is composed of spherical particles of a given size distribution and a constant density. Depending on the gas flow from the cometary surface dust

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particles up to a critical radius can be lifted off. In this model the surface dust coverage is periodically blown off as the comet approaches perihelion. Recently, Ktihrt and Keller (1994) demonstrated that under the assumptions of the Brin and Mendis model a stable surface coverage cannot develop. As already mentioned, there is no doubt that most of the surface and perhaps the whole surface is free of ice. Thus these authors propose that the surface should not be covered by loose dust but that this surface cover should be held together by cohesive forces. It is indeed plausible that such cohesive forces exist on the surface of the comet nucleus either due to polymerization or simply due to sintering of organic molecules. This could occur by cosmic ray bombardment of the comet during its stay in the Oort cloud or by thermal processing in the inner solar system. The main consequence of such “baking together” of cometary grains would be an increase of the effective heat conduction coefficient of the surface material and of its mechanical strength. What should be clearly stated here is that a baking together of the dust on a comet surface does not mean that the dust coverage on a cometary surface becomes gas tight. It is much more likely that cometary dust grains will form sinter bridges. If the surface coverage is initially porous it will remain porous simply because no compaction mechanism such as gravity is available. Besides : if the result of the sintering of the surface material were a compaction of this material we had to expect a rather important thermal inertia which should be detectable in the activity of the comet. Up to now there is no evidence for such kind of thermal inertia.

Transport phenomena

in a porous cornet nucleus

As already mentioned, it was suspected since the 1960s that comet nuclei where porous bodies. Horanyi et al. (1983) studied the diffusion of H,O vapour through a porous dust cover. These authors considered that water ice evaporates under a dust layer and then diffuses through this layer. It is considered that the gas is thermalized during this diffusion. As the top layers in this model are “dry” dust, the surface temperatures can take values much higher than the sublimation temperature of water ice ( - 200 K). Let us recall that the measurements of the IKS instrument on board Vega 1 indicate indeed temperatures higher than 300 K on the surface of comet P/Halley. This problem has been reconsidered by Fanale and Salvail (1987, 1990). In their model they include a dust mantle and sublimation of HZ0 as well as CO2 or CO, respectively. They considered that HZ0 and the second volatile sublimate at a certain depth and that the gas produced in this way diffuses to the surface. Smoluchowski (1982) was the first to recognize that in a porous mixture containing ices the vapour phase produced by the sublimation of these ices contributes to the heat transfer within the nucleus. In this model the contribution of the gas phase is taken into account using an “effective” heat conduction coefficient. This coefficient strongly depends on the pore size, on temperature and on the chemical nature of the gas. As an example, according to this work the contribution of H,O vapour becomes

nuclei

significant at 200K in a medium with a relative pore volume of 0.5 and an average pore size of 10~“m. Under the same conditions the contribution of a CO, gas phase exceeds that of the solid matrix by more than one order of magnitude. It was also Smoluchowski ( 1986) who pointed out that in a pore system. volatiles are able to diffuse to the outer part of the nucleus as well as to the inner parts where they may recondense due to the lower temperatures there, an effect which has been disregarded in the models by Fanale and Salvail ( 1987. 1990). Espinasse rt al. ( 1989, 1991) developed a model of the heat and mass transport in a porous comet nucleus in which the heat and mass exchanges are formulated in a physically more relevant manner than in Smoluchowski’s work. The heat conduction equation and the transport equation of the vapour phase are solved simultaneously :

1 ?P, = V(DV(P,)) + Q,: RT ?t where Q is the average density, c the specific heat capacity, T the temperature, I the time, i.,, the effective heat conduction coefficient of the solid matrix, R the universal gas constant, P, the partial pressure of constituent number i, and D the diffusion coefficient. Equation (2) can be easily recognized as the classical energy equation for the solid matrix. The source term Qi takes into account the heat gain or loss of the solid matrix due to the exchange of latent and sensible heat with the gas phase, but also the heat gain and loss due to other processes such as phase transitions within the solid matrix. In other words, with the assumption of constant values for g and c, this equation translates the enthalpy change of the solid matrix. Equation (3) is one of the standard formulations of Fick’s second law. The source term Q,;, describes the gain or loss of matter in the gas phase due to sublimation and recondensation processes. As sublimation and condensation processes are associated with a gain or loss of energy, the source terms insure a coupling between the heat conduction equation and the transport equations of the gas phase. The matter and heat exchange between the solid and the gas phase are considered as instantaneous processes. The expressions of the transport coefficients are given in detail in Espinasse et a/. (1991) and are not repeated here. It has been pointed out by Steiner et al. (1991) that the model by Espinasse et al. (1989) and thus implicitly Espinasse et al. (1991) is based on the same conservation equation as their model and as the model by Mekler et (II. (1990). The boundary conditions of equation (3) at the surface of the nucleus are given by the classical heat balance equation : F= toT’fH.

B+;.,,AVT),=.

where F is the solar flux absorbed emissivity, g the Stefan-Boltzmann

(4)

per unit surface, c the constant, H the latent

J. Klinger et al.: Towards a model of cometary nuclei heat of sublimation, E the sublimation rate per surface unit, ,& the effective heat conduction coefficient, and R the radius of the nucleus. In the centre of the nucleus it is considered that the heat flux is zero. In the model by Espinasse et al. (1989, 1991) it is considered that the nucleus is initially composed of amorphous water ice containing a molar fraction of 0.0550.3 of a second volatile species (CO or CO,). Up to a molar fraction of 0.1 this second volatile is considered to be trapped in the amorphous water ice, while the remainder forms an independent solid phase. The crystallization of the amorphous ice is exothermic and follows the activation law given in equation (1). It is considered that during the crystallization the volatiles trapped in the amorphous matrix are released. The main results obtained in these model calculations are : The gas phase contributes in a substantial manner to the heat transfer to deeper layers. This finding is confirmed independently by the analysis of the thermal profiles in the samples of the KOSI comet simulation experiments (Spohn et al., 1989). Volatiles are able to diffuse to deeper layers where they recondense. This also has been confirmed by the KOSI experiments (Hsiung and Roessler, 1989). Depending on the volatility of the different ices in a comet, sublimation can occur near the surface or in a substantial depth under the surface. The relative abundances in the coma are variable from one point of the orbit to another and eventually between subsequent orbits. In any case the coma composition does not reflect the composition of the solid nucleus. The gas production is asymmetric with respect to perihelion, the production being more important after perihelion. The water ice in model nuclei on Jupiter family orbits containing initially amorphous ice will crystallize in a runaway process. The time scale of this process evidently depends on the model parameters. Espinasse et al. (1991) found a typical time of a few centuries. Several groups developed models based on similar concepts using more or less equivalent mathematical formulations. As far as the model parameters are comparable the results of these calculations are similar. Let us mention in particular : l

l

l

l

l

The model by Mekler et al. (1990) where the comet nucleus is treated as a porous medium containing crystalline water ice only. The model by Prialnik (1992) in which amorphous water ice, CO and dust are included. The model by Orosei et al. (1995) which is based directly on that of Espinasse et al. In this model dust is added. The volatile phase is composed of HZ0 and COz. The model of Benkhoff and Huebner (1995) in which up to three minor volatile components are included (CO, COZ, CH3, OH). This model in particular confirms the result of Espinasse et al. (1991) that the relative abundances of species in the coma are in general not the same as those in the nucleus. The model of Tancredi et al. (1994) in which the authors run a model of a typical Jupiter family comet over

643 500years. For the heat conduction coefficient of amorphous ice they use an intermediate value between the theoretical estimate of Klinger (1980) and the experimental value of Kouchi et al. (1992). Further they consider that within the nucleus the gas phase is saturated, this means in equilibrium with the corresponding solid phase. Compared to the other models, this can be considered as a limiting case, where the vapour transport is sufficiently slow. But this evidently depends on the model parameters, in particular on the pore size. They find that a Jupiter family comet may conserve amorphous ice in the interior up to 10”years after its capture. Let us mention here that using a result of the original work of Espinasse (1989) Schmitt et al. (1991) were able to show that the phase transition of amorphous to crystalline ice in a porous comet nucleus or a reaction having similar thermodynamical characteristics is a possible explanation of the outburst of comet P/Halley at 14 AU from the Sun. Prialnik and Bar-Nun (1992) come independently to the same conclusion. As already mentioned, the KOSI experiments have demonstrated the importance of heat transport by the vapour phase in a porous medium which contains ices (see Spohn and Benkhoff, 1990 ; Benkhoff and Spohn, 1991; Steiner et al., 1991).

Towards a working model of comet nuclei for future space missions Taking into account the above-mentioned observational constraints and having critically analysed the model concepts published up to now, it seems adequate to consider that comets are more or less porous bodies containing dust mixed with more or less important proportions of ices of different volatility, water ice being the major constituent. The formation mechanism is not clear yet but it is likely that comets initially were agglomerates of interstellar grain-mantle particles (Greenberg, 1982, 1986), containing a silicate core covered by a partially photolysed ice mantle. In order to allow the preservation of ices more volatile than HzO, comets should have formed by low velocity accretion of such interstellar grains, eventually by a two-step process, thus favouring the appearance of subnuclei (cometesimals). Nevertheless, at present we cannot completely exclude alternative scenarios of formation, for example a formation of comets from rather compact “boulders” of meteoritic material glued together by ices. Alternatively, we have to consider models in which more or less dense “cometesimals” containing a certain amount of ices are glued together more or less tightly. The “active zones” in this picture could be the lines where the borders between cometesimals emerge on the surface. The strategy we have to use is the-following : Develop a model based on a given concept, trying to include the rotational state, the shape of the nucleus and the topography in an appropriate way. Deduce from this model a maximum of data which can be checked by observations.

J. Klinger ef al.: Towards a model of cometary nuclei

644 Table 1. Likely initial composition of the volatile component of comet nuclei (in mol %)

H,O co

80% 5-20%

co> CH, NH,

<5%
l

l

initially amorphous trapped in amorphous H,O or as independent phase trapped or as independent phase trapped or as independent phase trapped or as an independent phase, forms ammonia hydrates when heated to temperatures over 130 K

Develop an alternative model based on a different concept and compare again with observational data. Reactualize the model concepts in view of the results and new observational facts.

The model approach which seems the most appropriate and flexible to perform such type of work is that of a porous nucleus as developed by Espinasse et al. (1989, 1991), Prialnik (1992), Orosei et al. (1995), Enzian et al. (1995), Benkhoff and Huebner (1995), or Tancredi et al. (1994). If such type of models are correctly parametrized, situations in which the heat transfer by the vapour phase is negligible can be included as extreme cases. The advantage of the formulation of Espinasse et al. (1989, 1991) is that it gives a simultaneous description of the thermal evolution and the matter loss and redistribution by taking into account the coupling between matter and energy conservation. Further, this model is extremely flexible. For this reason we based the calculations which follow on the formulation by Espinasse et al.

Baseline model For the chemical composition of the baseline model it is reasonable to take, as a first approach, the abundance values as measured during the Giotto encounter with comet P/Halley even if we must be aware that the coma composition does not necessarily reflect the composition of the nucleus. These abundances are given in Table 1. For our first model approach we merely use a mixture of solid Hz0 and solid CO. Based on cosmochemical considerations, Greenberg (1986) proposed a dust to ice mass ratio of 1. After the encounter of Giotto with comet Halley, several authors suggest dust to ice ratios between 1 and 3. Once again it is not possible to conclude from the coma composition to the composition of the nucleus and, even less, to deduce the initial composition of the nucleus. But in the present situation let us take as a baseline model a dust to ice ratio of one in mass. Following the results of Rickman (1989) we consider average densities of comet nuclei between 260 and 650 kg rnp3, 500 kg me3 being a reasonable average. Thus it is not unreasonable to consider that the relative pore volume is 0.5-0.8, depending on the dust to ice ratio. When the near surface layers are devolatilized we should expect an increase of the relative pore volume (a decrease in average density). It is evident from the Giotto camera

data that comets are more or less irregularly shaped bodies. Nevertheless, for reasons of simplicity we first consider a spherical nucleus. From the previous considerations we can now present a simplistic pseudo-3-D working model able to be used as a basis for engineering studies. We proceed in the following manner : The nucleus is divided into ten latitude strips receiving equal solar flux in the case when the rotation axis is perpendicular to the orbital plan. This division is conserved even if the rotation axis is oriented differently. Within every latitude strip 30 zones with equal surface areas are defined, every area covering 12 deg in cometocentric longitude. Within every area, a point is chosen in such a way that the locally received solar radiation is equal to the average solar radiation per surface unit received by the area. In this way 300 surface grid points are defined over the nucleus. The transport equations (2) and (3) are solved for every surface grid point. For every surface point we define an exponential depth profile with 130 points. In this way the whole nucleus is defined by a 3-D grid with approximately 40,000 points. Heat transport parallel to the surface is neglected. The locally absorbed solar flux is expressed as : F(Y~#,~) = S. r;‘( 1 - A)(cos 0 cos 4 cos 6 + sin $ sin 6) (5) where S is the solar constant, Ye the heliocentric distance, A the albedo, 0 the hour angle, 4 the latitude, and 6 the declination. The coma is considered as optically thin. The time step of the orbital movement of the comet is taken as l/30 of the rotation period. This means that between two time steps every grid point has taken the place of its immediate neighbour on the same latitude strip in the sense of the rotation. In the present work we only considered the extreme cases of 0” and 90” of obliquity of the rotation axis but the same principle can obviously be applied to any orientation of the rotation axis as long as it is stable in space. The choice of an adequate equivalent geometry for the effective heat conduction coefficient of the solid phase is a delicate problem. Several expressions are in use in different models. In our model we use Russel’s formula (see Espinasse et al., 1993) :

Lff=(lj -

&did[(

ip+

1- 1C12’3)&old + $2'3Ahvl l)/lsolld + $2i3(@‘3- 1)3+,

(6)

where rc/is the relative pore volume. Other authors prefer a formula due to Zehner and Schliinder (see Steiner et al., 1991). One of the arguments for this is that this formula is symmetric for solid and void space. As at present it is very difficult to decide what is the physically more relevant expression, we used Russel’s formula as in the original work by Espinasse (1989) with a further term taking into account the radiative heat transfer. for crystalline and For &olld the classical expressions

J. Klinger et al.: Towards a model of cometary nuclei

645

Table 2. Parameters used in our baseline model Mass ratio of ice to dust Mass ratio of CO to Hz0 Relative pore volume Pore radius Bulk density of dust Bulk thermal conductivity of dust Surface emissivity Surface albedo Nucleus radius Nucleus rotation period

1 0.05-0.2 0.8 lpm 3200 kgmm3 1 or4.2Wmm’K-’ 0.9 0.1 5000 m lOdays (5 days for P/Wirtanen)

orbit of Schwassmann-Wachmann with obliquity 0"

1

lE-3

lE-4

1 E-5

amorphous ice are used (see Klinger, 1980, 1981). &,,,is approximated as :

I.,,? = 4mT3r,.

1 E-6

(7) lE-7

The total gas production rate is computed by adding up the gas productions of all 300 surface areas. Concerning the development of the dust cover we considered two extreme situations : 1. All the dust accumulates on the surface. In other words,

the dust coverage grows thicker and thicker. This case can be considered as an accelerated aging of the comet. 2. All the dust contained in the nucleus is dragged away by the gas. A physically more relevant model taking into account the momentum transfer between ice and dust is in progress and will be published later. The model parameters are defined in Table 2.

-r 0

’

I

’

I

’

I

’

I

’

I

’

’

2

’

4

’

6

’

8

*

10

”

In Fig. 1 the CO production (in mol m-‘) at the equator of SW1 is shown, the obliquity being 0”. Figure 2 shows the CO production at one pole of SW1 for an obliquity of 90”. At the beginning of the run we remark a strong CO “burst” in both cases. This erratic activity disappears during the fourth revolution in the equatorial model but persists (less violently) in the polar model. Let us mention in this context that a study of Whipple (1980) indeed

Results

The model presented here has been applied to comets SW3 and P/Wirtanen. The reason for this choice was that comet SW3 and (more recently) P/Wirtanen have been considered as possible targets for the Rosetta mission. For comparison we also studied comet SWl. This last object is interesting for several reasons: the orbit of SW1 is nearly circular and confined between the orbits of Jupiter and Saturn. As already mentioned, in spite of the fact that at such heliocentric distances ordinary water ice sublimation is negligible, the comet shows a rather important irregular activity. It has been suspected that the crystallization of amorphous ice may have some importance in this behaviour (see Klinger, 1983; FroeschlC et al., 1983). Further an important release of CO has been discovered recently by Senay and Jewitt (1994), and Crovisier et al. (1995). In the studies concerning SW1 and SW3 we only consider the case where the dust remains on the surface. In the case of SW1 the water production is expected to be small. Thus the present study was focused on the production of CO. In these calculations, the bulk thermal conductivity is taken as 4.2 W m-’ K-l.

12

number of revolutions Fig. 1. CO production (in molm-‘) at the equator of a model comet on the orbit of SWl. Rotation axis perpendicular to the orbital plane. Bulk thermal conductivity of the dust: 4.2 W m-’ Km’. Rotation period : 10 days. It is considered that all the dust accumulates on the surface

1 lE.3

T

iE-4

T

lE-5

m=

1 E-6

1 lE-7

0

4

8

number

12

16

of revolutions

Fig. 2. CO production (in molmm2) on the pole for SWl. Obliquity 90”. All other parameters as in Fig. 1

646 suggested that the obliquity of the rotation axis of SW1 is high. Qualitatively we can explain this behaviour as follows: when one pole is turned to the Sun the surface temperature is obviously much higher over an important part of the orbit compared to the equatorial model. This favours the crystallization of the amorphous subsurface layers. The heat of crystallization released during this process will be partially used for sublimating the CO trapped in the amorphous layers and partially for heating the deeper layers. The CO will partially diffuse to the surface and form the coma gas which has been detected. When the heat wave reaches very cold layers the crystallization heat is efficiently dissipated and CO production decreases rapidly. In other words: the interplay between an exothermic process depending on temperature in a non-linear manner (crystallization of amorphous ice) and an endothermic process (CO sublimation), having also a nonlinear temperature dependence, is likely to produce a chaotic behaviour. The CO production rate of SW 1 measured by Crovisier et al. (1995) was 5 x lOI molecules s-‘. Let us estimate if these values are compatible with our results. The height of the spikes of CO in the polar model is close to 10~3molm-” s-r during revolution 9 and falls to 5 x 10p6mol mm2 ssl during revolution 16. Let us make the very crude assumption that the production rate is constant and equal to the production rate at the sub-solar point over the projected surface of the comet. In order to produce the measured amount of CO we need an equivalent disc having a radius between 5 and 75 km. The last value seems important but let us remember that we consider in this model that all the dust remains on the surface. As already pointed out we can consider that the aging of the comet has been accelerated. In any case a simple evaluation of the available mass of a comet shows that if the measured production rate of SW1 is maintained, this comet must be either extremely big or is supposed to be extinct within a few centuries. In other words, in the present case the predictions of our model are compatible with observation. The influence of the development of a dust coverage on the CO production will be checked in future work. Figure 3 shows the CO production of SW3 at the equator for 0’ obliquity. The behaviour is similar to that of SW1 but a study over a much longer time scale is needed. In Fig. 4 integrated production rates for H,O and CO are shown for SW3 and SW1 for two different CO contents (5% left, and 20% right). We remark in this short-term study a rather irregular behaviour in the CO production. Runs over longer time spans are needed to follow the further evolution. In the case of comet SW3 the small Hz0 production is most likely due to the extreme assumption of dust accumulation on the surface. What is interesting is to compare the CO production of a CO rich (right) to a CO poor (left) model comet on the orbit of SW3 (top). We see a much more irregular CO production for the poor nucleus compared to the rich one. In this last case the irregularities in CO production seem to disappear after 15 revolutions. This point has to be confirmed by integrations over a longer period. In the case of SW1 (bottom) we see the appearance of sporadic phases of high activity for the poor nucleus during revolution 19 and for the rich nucleus

J. Klinger et al.: Towards a model of cometary nuclei lE-2 1

lE-3

IE-4

1 E-5

1 E-6 ~.

lE-7

-,,__

0

1

I 5

IO

IS

n urn ber of revolutions

25

20

i 30

111

Fig. 3. CO production

(in molm-‘) at the equator or SW3. Obliquity 0 ‘. All other parameters as in Fig. 1

during revolution 16. Once again a study over a longer time span will be necessary. It should be noted that the integrated CO production rate shows maximum values of the same order of magnitude as the measured values. The average value over 20 revolutions is about two orders of magnitude lower than the measured value for the poor nucleus and about one order of magnitude lower for the rich one. The three most likely explanations for this are : (i) In this model all the dust accumulates on the surface (accelerated aging). (ii) As the obliquity is zero in this case, the maximum surface temperatures are lower compared to the case where one pole faces the Sun during a part of the orbit. (iii) The size of the model comet is probably smaller than that of the real comet. The depletion of CO in the sub-surface layers of SW1 after ten revolutions is visualized as a function of latitude in Fig. 5. In Fig. 6 the CO depletion in the interior of SW1 is compared to that in SW3. What is remarkable is the progressive depletion of CO with depth and the recondensation of CO at a certain depth. Figure 7 visualizes the temperature profiles in SW 1 as a function of latitude at noon slightly after perihelion. In Fig. 8 the equatorial temperature profiles of SW1 and SW3 are compared under the same conditions as in Fig. 7. The accidents in the thermal profiles are related to the concentration plateaus and the enrichment zones in the CO concentration profiles. In Fig. 9, the relative ice content (HZ0 and CO) with respect to the initial concentration is plotted as a function of depth for a point at the equator of a model comet on the orbit of P/Wirtanen. In the case shown on Fig. 9a the assumption is made that the dust remains on the surface. In Fig. 9b, the same concentration profiles are shown for the case where all the dust is lifted off. In this last case, the HZ0 content remains constant. The fact that the starting point is slightly displaced to the right with respect to the border of the drawing has no physical relevance (please

641

J. Klinger et al.: Towards a model of cometary nuclei 6 -5 -7

4

u!3 g2

-2g -1

1

l0 u

-1

cn -2 -0 -3 -4

0

5

10

15

20

0

5

6

IO

20

15

6

,.,..,a.,.,,..,

-5 -7

4

v!3 22

-2,g -1

1

-0

:0 0

-1

j\ I! I iIf\ tlii I\ iIi\ fl ii f\(\ i\ I\ r'i i\ g L' )$a h Q $ $ ;i L; ','$ G ; L b J 4 i

cn -2 0 -3 -4

t

- -3 9 0

5

10

15

20

I 5

0

revolution number

.,

.

I 10

.,

, ,

I,,

15

,

,

-4 20

revolution number

Fig. 4. Integrated CO and Hz0 production rates over 20 orbits for SW1 and SW3. Model parameters as Fig. 1. Top: SW3. Bottom: SWI. Left: CO/H,0 ratio 0.05. Right: CO/H,0 ratio 0.2. The continuous lines indicate the CO production, the dotted lines the H,O production

l.-...

1600

0

Isurface /

;

, ,

0.01

,/I,

i,,

0.10

nucleus center

0-0°- /I .#$_300 /

,

1,111,

,//

1 .oo

,,I

/

10.00

,

,,///,(

100.00

I

I

,,l”l,

I

1

1000.00

depth r Iml

Fig. 5. CO content (in mol mm3) in the sub-surface layers of SW1 at different latitudes after ten revolutions. Model parameters as in Fig. 1

note that the x-axis is graduated in a logarithmic scale). In the case of the dust covered nucleus, the near surface layers are progressively depleted in HzO. This is easy to understand, knowing that on an uncovered nucleus a substantial amount of heat is used for surface sublimation and thus the produced water vapour has not to diffuse through a surface coverage. For CO we find in both cases

a complete disappearance close to the surface (as expected) and an enrichment at a certain depth. In Fig. 10 temperature profiles at the equator are shown for comet P/Wirtanen for the highest daytime temperature and the lowest night-time temperature at perihelion and at aphelion ,for the dust covered and for the non-dust covered case. As expected (and observed during the Vega encounter with comet P/Halley), the temperature of the ice free dust cover is able to get values much higher than the sublimation temperature of water ice. Finally, we checked the influence of the bulk thermal conductivity. Under the hypothesis that the dust remains on the surface, the influence of the bulk thermal conductivity of the dust on the maximum daytime temperatures and on the minimum night-time temperatures. For 4.2 W m-‘K-’ the maximum daytime temperature at perihelion is only slightly higher than 300 K, whereas the minimum night-time temperatures at perihelion exceed 250 K. For a bulk heat conduction coefficient of the dust of 1 W m-‘K-‘, the maximum daytime temperature at perihelion exceeds 350K, whereas the minimum temperatures at perihelion go only slightly over 200K. The minimum and maximum temperatures at aphelion are rather insensitive to the bulk heat conduction coefficient.

Discussion We consider that comets are porous media with a spongelike topology (the pores communicate with each other). This has the following consequences : l

The contribution of the solid matrix to the heat conduction to deeper layers is considerably reduced.

J. Klinger et al.: Towards a model of cometary nuclei

648 2000

1600

1200

culating in the pore system contribute in an important manner to the heat transport in the nucleus without introducing a significant thermal inertia.

r

-4

Following this concept we come to a more subtle picture of such phenomena like “active zones” and “mantle development” as generally found in the literature :

i

600

/I

400

SWI-

surface 0

,

, / ,,,I,,

0.10

I

(

, / l,,,,,

.oo

j

,

SW3

, / ,,,,,,

,

100.00

10.00

/ , ,111,

,

! /

1000.00

depth [m]

Fig. 6. Comparison of the CO content (in molm-‘) between SW1 and SW3 at the equator. Model parameters as Fig. 1

Sublimation processes not only occur at the surface but also in deeper layers. The more volatile the ice, the deeper the zone where the ice is able to sublimate. The gas freed by sublimation in the porous medium is able to diffuse to the surface or to deeper and/or colder layers within the nucleus where they are able to recondense. The migration as well as sublimation and recondensation processes of the different gas phases cir-

Volatiles are continuously redistributed during the daily, annual and secular variation of the insulation undergone by the comet surface. This means that a dust coverage on the comet surface does not grow in a monotonous manner. As long as the thermal gradient between the surface and the subsurface layers is negative (the surface hotter than the interior), the dust progressively devolatilizes, the surface being in most cases ice free. When the thermal gradient is reversed, for example during cometary night, volatiles may recondense in the near surface dust layers. These volatiles are freed quickly at sunrise, contributing efficiently to the heat transfer to deeper layers and thus trigger the onset of cometary activity without contributing substantially to the thermal inertia of the nucleus. Such effects have been demonstrated by laboratory simulations (see Klinger et al., 1989b). Such kind of effects are not evidenced in most comet models, simply because the daynight variations are averaged out! The redistribution of volatile matter may lead to a stratification of the nucleus characterized by variations of the chemical composition and local increase of hardness due to sintering. Such type of effects have been found in qualitative simulation experiments. The redistribution of cometary volatiles could also produce some local variations in the isotope ratios. Studies on the isotope ratios of cometary material have been done but

\ 160

160

120

80

80

40

40

surface

o-

1

0.01

l~nmT/

1

0.10

rmm

,

1 .oo

I111111,

I

r711111,

10.00 100.00 depth [m]

,

/I,,“‘,

-T

1000.00

-14 10000.00

Fig. 7. Temperature profiles after ten revolutions for SW1 for different latitudes, slightly after perihelion. Model parameters as Fig. 1

O-+,-, 0.01

FITl[ilrii?TT

0.10

1.oo

10.00

T--l-

100.00

I / ,r,Tir

-7

I

1000.00

depth [m] Fig. 8. Comparison

between temperature profiles of SW1 and SW3 at the equator after ten revolutions, slightly after perihelion. Model parameters as Fig. 1

J. Klinger et al.: Towards a model of cometary nuclei

649

(a) 1.4 c

1.2-

.-0 $

l.O-

(a)

,OOj

E a> 0.8 E 0 0 $

0.6 0.4 -

L 0.2 0.0

W ”

I

-3

‘.

-2

I. -1

”

I.

0

'. I ’ 1

”

log,, Depth

’

2

I

0

1

3

c

400 ',

1.2

L-

H,O

”

”

“,

1

2

Depth

[ml

”

““““‘I

”

”

3

4

1.0

3

4

g3007 ii! 3 3iE 200:

0.6 ? p

-2

-1

0

1

log,,

/ -4 c

t

-3

2

3

4

[ml

Depth of HZ0 and CO normalized to the initial concentration as a function of depth at the equator of a model comet on the orbit of P/Wirtanen. Rotation period : 5 days. (a) All the dust accumulates on the surface. (b) No dust accumulates on the surface Fig. 9. Concentration

l

”

0

co

E a> 0.8 E 0 0

-1 bl,,

(b)

1.4

.-0 $

-2

[ml L

(b)

-3

4

did not bring conclusive results (Klinger et al., 1989~; Roessler et al., 1991). Depending on the dynamical age, the topography, the orientation of the rotation axis and on the rotation period, the devolatilization of the near surface layer will show local variations. The more the devolatilization progresses, the smaller the gas leakage through the dust coverage. The dust lift-off from the surface depends on the local gas flow and on the gas velocity (eventually on centrifugal forces) on one side and on the cohesive forces between dust grains (eventually on gravity) on the other. The formation of dust “jets” as seen on the images of the Giotto camera depends on the hydrodynamical interaction between gas and dust, but also on the scattering properties of the dust particles and thus on the particle distribution. The development of a model of these effects is under progress.

Let us mention

that the model

presented

here has been

100:

O,~~~,~~,,~~~r...,...,...,... -3 -2 -1 0 1 @I,,

Depth

2

[ml

Fig. 10. Temperature profiles for maximum and minimum surface temperatures at the equator at perihelion (top curves) and aphelion (bottom curves) of a model comet on the orbit of P/Wirtanen. (a) With all dust remaining on the surface. (b) With no dust remaining on the surface. Model parameters as in Fig. 9

used for the preparation of the radar wave transmission experiment CONSERT which has been selected for the payload of the landers “Champollion” and “RoLand” of the Rosetta mission. The present model seems to describe adequately the heat and mass transfer within a porous comet nucleus. Nevertheless, several weak points have been discovered recently (Bouziani, 1995) : The different gas phases are treated independently. This is certainly justified as long as the gas transport occurs in the Knudsen regime (the mean free path of the gas molecules being much longer than the pore size). This is certainly not adequate for the diffusion of a minor component when molecule-molecule collisions become important. Thus the coupling between the different gas phases must be taken into account. In order to describe the gas transport in a wide pressure

650

range, Fanale and Salvail (1987) introduced three different expressions of the diffusion coefficient. These expressions are now widely used. As it is difficult to set a limit to the validity of the different expressions it is more adequate to express the diffusion coefficients by a unique equation valid over the whole pressure range. A study is now under progress in order to check the influence of the above-mentioned points on the results of the model. Further, the exchange between the gas phase is considered as an instantaneous process compared to the time scales of the gas transport in the pore system. This should in general be true but it should be wise to confirm this assumption by laboratory experiments on appropriate samples. The combination of the model presented here with a true bi-dimensional model including meridional transport is under progress. This work will be presented separately. In order to take into account the possibility that comets are composed of subnuclei, we propose, as a first approach, to build a nucleus from spherical subnuclei. By doing this we have then the possibility to vary independently the porosity and the composition in the subnuclei and in the transition zones. Thus we can for example construct model nuclei containing subnuclei from ice free (porous or non-porous) material held together by a porous or non-porous icy glue. As already stated, comets are rather irregular than spherical bodies. Thus, we can consider the possibility to generate an irregular object having the physical characteristics of the previously developed models. With such an approach we can test the influence of the shape and the topography on the evolution of comet nuclei. In view of our present experience it is realistic to achieve these two last points using a parallel computing technique, as far as we do not claim a too important spatial resolution. More generally speaking we are aware of the fact that several groups are now developing models of comet nuclei. In order to make the results of different models comparable it seems necessary to us to standardize the model parameters used. The parameters of Table 2 could be a starting point for this.

Conclusion An analysis of the available observational data show that we are far from having a coherent overall picture of the composition and structure of comet nuclei and their evolution with time. Thus an important effort of well coordinated observational programmes is necessary. Even if none of the existing models finds a consensus in the community of specialists, a critical analysis of the available work shows that the “porous icy nucleus” approach is the most credible and flexible one. Parametrized in an adequate manner, this approach is able to describe cometary processes in a physically relevant manner and to take into account limiting cases such as compact nuclei or boulders glued together by ices. Quantitative present day comet models together with available observational data lead to a concept of comet

J. Klinger ef 01.: Towards a model of cometary nuclei

nuclei which is more subtle than the classical picture of an ice containing zone overlaid by tight, more or less thick dust “mantle” having some holes which manifest themselves as “active” zones. Taking into account the combined gas and heat transport in a porous medium containing ices we have to expect a continuous redistribution of volatiles at time scales ranging from the rotation period to the whole lifetime of the comet (see Figs 5, 6 and 9). The presently developed hydrodynamic coma models and the light diffusion models developed in parallel will hopefully allow to establish a link between comet models and comet observations. In order to make different comet nucleus models comparable we propose to standardize the model parameters. Aclinoltlengemerlts. We acknowledge support from the French national planetary programme PNP of INSUjCNRS and from the French space agency CNES. Many thanks to all members of the group of planetology of “Laboratoire de Glaciologie et Geophysique de 1’Environnement” for helpful discussions, and particular thanks to Mr H. Cabot for critically reading of the manuscript.

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