Dust and Neutral Gas Modeling of the Inner Atmospheres of Comets

REVIEWS OF GEOPHYSICS, VOL. 24, NO. 3, PAGES 667-700, AUGUST 1986

Dust and Neutral Gas Modeling of the Inner Atmospheres of Comets

T. I. GOMBOSI, 1 A. F. NAGY, AND T. E. CRAVENS

Space Physics Research Laboratory, University of Michigan, Ann Arbor

This paper summarizes our present, preencounter understanding of the physical and chemical processes controling the inner (r < 1000 km) region of cometary atmospheres. Special emphasis was attached to compiling a self-consistent set of governing equations. We are aiming this review at readers who want to understand the present status of the mantle and coma regions and/or who want to develop new, next generation models which will be needed as the large volume of new observational data will become available in the near future.

CONTENTS

Introduction ............................................. 667

Gas and dust production of cometary nuclei ................ 668 Structure and composition of cometary surface layers ...... 668 Vaporization and gas production models .................. 670 Heat transfer in the mantle: governing equations .......... 672 Heat transfer in the mantle: approximate solutions ........ 674

Theory of atmospheric processes .......................... 679 Transport equations and physical processes ............... 679 Photochemistry ........................................ 681

Dusty gas flow in the near-nucleus region ................... 685 Governing equations ................................... 685 Gas-dust momentum and energy transfer ................. 686 Steady state solutions without radiative transfer ........... 686 Approximate supersonic steady state solutions ............ 689 Radiative transfer ...................................... 692 Time-dependent models ................................ 695

Summary discussion ..................................... 697

1. INTRODUCTION

The cometary atmosphere is a unique phenomenon in the solar system. Owing to its negligible gravity the tiny nucleus of a few kilometers in diameter produces a highly variable, extensive dusty atmosphere of dimensions ranging from 10 '• to 105 km. In contrast to planetary atmospheres, the dimensions of the nucleus are much smaller than the scale length of the coma. In spite of the apparent differences, many of the physical and chemical processes dominating planetary and cometary atmospheres are similar. The large spatial extent and continual expansion of a cometary atmosphere provide a con- venient tool for studying the time history of atmospheric processes, and this could be useful when compared with similar processes in a gravitationally bound atmosphere.

Following Whipple's [1950] pioneering work, cometary nuclei are now thought to be chunks of ice, rock, and dust with negligible surface gravity. As these "dirty iceballs" approach the sun, water vapor and other volatile gases sublimate from the surface generating a rapidly expanding huge cloud of dust and gas. The radiation field in the inner cometary atmosphere is expected to be quite different from the unattenuated solar radiation as molecular absorption and emission, as well as multiple scattering and thermal emission by dust particles, modify the original spectrum.

•Also at Central Research Institute for Physics, Budapest, Hungary.

Copyright 1986 by the American Geophysical Union.

Paper number 6R0128. 8755-1209/86/006R-0128515.00

667

One of the most important features influencing cometary dynamics is the "retarded" nature of gas and dust production. The radiation reaching the surface and supplying energy for sublimation must first penetrate an extensive, absorbing dusty atmosphere. Any change in the gas and dust production alters the optical characteristics of the atmosphere, thus causing a delayed (or "retarded") effect on the production rates them- selves. The delay is caused by the finite radial transport time of the outflowing dust and gas.

Since the mid-1970s, when primitive bodies became potential targets of deep-space missions, numerous review papers dealing with various aspects of comets have been published. The latest and most comprehensive example of such a review is that of Mendis et al. [1985]. During the last few years, two extensive collections of papers dealing with comets were also published (Comets, edited by L. L. Wilkening, and Cornetary Exploration, edited by T. I. Gombosi, were published in 1982 and 1983, respectively). The recently published reviews and monographs provide a detailed source of information about our present (preencounter) understanding of cometary physics and chemistry.

We decided to write a nonconventional and more tutorial

review using the presently available material; instead of reviewing individual papers and calculations, we try to concentrate on the most recent developments and compile a more or less self-consistent picture of the inner cometary region, giving due credit to the original papers but at the same time trying to provide an interrelated description of this region. This approach will inevitably result in overemphasizing some aspects of modeling activities, especially those which are important for interfacing various regions in the near-nucleus region, such as mantle thermal balance, gas outflow, and gas- dust interaction. We are aiming this review at readers who want to understand the present state of models of the nucleus- coma interface regioh and/or who want to develop new, next generation models which will be needed as the large volume of new observational data will become available in the next year or two. We are also providing many of the input parameters which are necessary to carry out these model calculations. Finally, this paper does not attempt to review the presently available (mostly indirect) observations in any detail, especially since so much new information is expected to become available soon; it only refers to observations in terms of justi- fying and/or checking inputs to and results of model calculations. For an excellent "conventional review" the readers

should turn to the Mendis et al. [1985] paper. This paper concentrates on inner coma processes and

models. This is the region where dust and gas are produced and accelerated to their terminal velocities. In the inner coma

668 GOMBOSI ET AL.: DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES

BOW SHOCK

pu 2 pu 2 + nk T

INNER CONTACT SHOCK(?) SURFACE (?)

/ /

nkTIi pu •

SONIC

LINE II EUS

io o (kin)

Fig. 1.

PLASMA-NEUTRAL DECOUPLING

HYDROGEN COLLISIONLESS CORONA GAS

EXPANSION

SONIC

LINE EUS

Schematic representation of the cometary plasma (upper panel) and neutral gas (lower panel) environment.

the neutral gas can be considered to be collision dominated, and consequently the application of a hydrodynamic approach is justified. A schematic view of the cometary atmosphere is shown in Figure 1. It can be seen that the region of interest for this paper is spatially limited (r < 1000 km) and has a great influence on cometary dynamics.

In section 2, physical and chemical processes in the upper layers of the nucleus will be summarized. Section 3 overviews the basic aeronomical phenomena in cometary atmospheres with a number of useful tables, providing information about solar spectra, reaction rates, etc. In section 4 the dusty gas dynamics in inner cometary atmospheres is reviewed. In this section, tables summarizing calculated gas and dust terminal velocities and densities for comets Halley, Giacobini-Zinner, Kopff, and Wild-2 at different heliocentric distances are also presented.

2. GAS AND DUST PRODUCTION OF

COMETARY NUCLEI

2.1. Structure and Composition of Cometary Surface Layers

Our present understanding of cometary nuclei is based on Whipple's [1950] pioneering "dirty snowball" idea according to which the nucleus consists of a mixture of frozen volatiles

and nonvolatile dust. Whipple's hypothesis quickly replaced

the century-long series of "sandbank" models, wherein the nucleus was thought of as a diffuse cloud of small particles trav- eling together. In a detailed comparison of the two competing models [Whipple, 1964], it was pointed out that even the simplest icy conglomerate model is able to explain basic cometary features, such as (1) the repeated nature of coma formation and gas loss for many revolutions of some short-period comets, (2) the survival of some sun-grazing comets with perihelion distances of less than 0.005 AU, (3) the splitting of cometary nuclei, and (4) the occurrence of nongravitational forces. Today there seems to be little doubt that comet nuclei are monolithic dust and ice conglomerates with a radius of a few kilometers [cf. Wyckoff, 1982; Delsemme, 1982; Donn and Rahe, 1982; Mendis et al., 1985]. The available observational evidence also indicates that the interior of the nucleus of a

comet is relatively homogeneous, although the surface is probably differentiated and nonuniform [cf. Delsemme, 1982].

The chemical composition and physical structure of the surface layers of a cometary nucleus are very important factors affecting the mass, momentum, and energy of the outflowing gas-dust mixture, as well as the relative abundances of various gas molecules. The sublimated gas molecules (often called parent molecules) undergo frequent collisions and various fast photochemical processes in the near-nucleus region, thus pro- ducing a whole chain of daughter atoms and molecules [cf. Huebner, 1985]. As there are no direct observations available

GOMBOSI ET AL.: DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES 669

about the structure and composition of cometary nuclei, or even the near-nucleus region, we have only indirect evidence to help us to delineate the main physical and chemical processes which control these continuously expanding and dy- namically evolving complex dusty atmospheres. Most of our observations come from spectrophotometry of cometary exo- spheres and tails. Cometary spectroscopy has developed spec- tacularly during the last decade, but many of the parent molecules and the secondary atoms, radicals, and ions still remain unobserved: we mostly detect those species that can be excited by solar radiation through a variety of fluorescence mechanisms [Delsemrne, 1985a]. New and increasingly sophisticated observing techniques recently made it possible to obtain high- resolution images of molecules hardly detectable with conventional methods (such as H20 or S2).

In Table 1 (taken from Delsernrne [1985a]) we summarize the chemical species identified in cometary spectra. It should be noted that this is a composite list (i.e., not every molecule was detected in each comet). Inspection of Table 1 shows that the volatile components of comets are mainly composed of elements, such as H, C, N, O, and S. Delsernrne [1985a] has also compiled the mean relative abundances of these elements and obtained the following values: H/O = 1.5-2.5; C/O - 0.2; N/O = 0.1; S/O = 0.003. The most interesting feature of these results is the depletion of cometary hydrogen by some 3 orders of magnitude compared to solar system abundances.

Delsentnte's [1985a] results are in good agreement with the generally accepted view that cometary volatiles are mainly composed of water ice, which controls the sublimation of the icy conglomerate of most comets [cf. Wyckoff, 1982; Del- sentme, 1982, 1985a; Donn and Rahe, 1982; Feldrnann, 1982; Keller, 1983; Mendis et al., 1985. Delsernrne's [1985a] abun- dance values are also consistent with Shulntan's [1983a] studies, which are predicting that during the accretion process only those volatile species which are thermodynamically com- patible with water can condensate into the ice mixture of cometary nuclei.

Reviewing the available, mostly circumstantial, evidence, Delsentnte [1985b] concluded that water ice controls the sublimation process in the vast majority of comets, including some whose behavior was earlier suspected to be determined by the vaporization of ices more volatile than water (comets More- house and Kohoutek are two typical examples). It also seems very likely that clathrate hydrates are quite common in comets (this possibility was first suggested by Delsernrne and Swings [1952]). Clathrate hydrates have two very important features: (1) a maximum of one "guest" molecule (such as CO) can be trapped within a H20 lattice of five to seven molecules,

TABLE 1. Observed Species in Cometary Spectra [Delsemme, 1985a]

Organic Inorganic Metals Ions Dust

C H Na C +

C 2 NH K CO + C 3 NH 2 Ca CO2 + CH NH 3 V CH + CN O Mn H:O + CO OH Fe OH +

HCO H20 Co Ca + H2CO S Ni N2 + CS S2 Cu CN + HCN Cr H:S + CH3CN

silicates

and (2) their latent heat of vaporization is almost the same as the latent heat of pure water ice. In the presence of clathrate hydrates, H20 and more volatile CO molecules are evapor- ated simultaneously in agreement with some observations; however, the simple clathrate hydrate model cannot explain H20/CO production rate ratios smaller than 5. On the other hand, it was presently pointed out by R. Prinn (private communication, 1985) that not all volatile molecules can be trapped into the H:O lattice; for example, a CO: molecule is simply too big to "fit" into the ice structure. This means that in addition to clathrate hydrates other volatile ice components might also be present in some comets. This question will be discussed in greater detail in a later section.

The presently prevailing view is that the solid components of cometary nuclei form an extremely porous, low-density, weak structure, rather than a coherent mass of rocky solids penetrated by gas or liquids that froze [cf. Whipple and Hueb- net, 1976; Donn and Rahe, 1982; Mendis et al., 1985]. Com- etary structures seem to be associated with the fragile fireballs observed by the Prairie Network [Ceplecha, 1977] and the chondritic aggregate micrometeorites collected in the stratosphere [Fraundorf et al., 1982]. Fraundorf et al. [1982] concluded that cometary particles were probably formed by two episodes of aggregation: the first involved assembly of basic building blocks of cometary solids, typically ranging in size from 0.1 to 1 #m, while the second aggregation process produced the "bunch of grapes" type observed morphology where the observed void spaces have previously been occupied by ice. Applying percolation theory, Hordnyi and Kecskem•ty 1-1983] have estimated the size distribution of aggregates composed of elementary building blocks and found it to be n(a)• a -4'4 (a being the particle size). Similar predictions were made earlier by Monte Carlo calculations [Daniels and Hughes, 1980, 1981]. These distributions are in reasonable agreement with dust observations indicating a spectral index of 4.2 at the nucleus [cf. Hanner, 1983].

When the nucleus made up from icy conglomerates approaches the sun, it absorbs an increasingly larger flux of solar radiation, and the vaporization rate of volatile molecules at the surface increases. Gravitational forces are negligible; therefore the vaporized gases leave the surface and form an expanding exosphere. In this process the gas drags away some of those dust grains which have already been evacuated of their ice component. The surface escape velocity is small but finite, so there is a critical dust size, a .... characterizing the largest solid particle which can be dragged away by the outflowing gas (see section 4.2). In his original presentation of the icy conglomerate model, Whipple [1950] predicted that an inert layer of large dust particles, evacuated of the volatile component, would form an insulating crust on the surface (mantle), causing a significant postperihelion decrease in lu- minosity. The development of such a mantle was discussed later by Shulman [1972], Mendis and Brin [1977], and Brin and Mendis [1979] and subsequently by Hordnyi et al. [1984], Fanale and Salvail [1984], Podolak and Hermann [1985], and Houpis et al. [1985]. A schematic representation of such a mantle is shown in Figure 2.

The thickness of the mantle varies with time because the

continuous vaporization increases the thickness of the evacuated layer, and the "erosion" due to the drag of the outflowing gas decreases it. Mendis and Brin [1977] assumed that the erosion mechanism takes place throughout the mantle with all particles smaller than ama x being dragged away by the outflowing vapor. Uncertainties, such as how a dust grain can be

670 GOMBOSI ET AL..' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES

Fig. 2.

SOLAR BLACK BODY

RADIATION RADIATION

TTTTTTTT i•. SUR FACE }?.•.':•.':','.•.•.•.?:':':•'1 ....................

0•:•:,:•:•:•:,:.':•: C O N DU C T I O N / / / / / / / /,:::.,,.:::,:::.,,-::: - HEATN OF DFFUSN II G A S f'•'?;:;:;';:•'•'•:•'•'•'•'•'•'•'•

ß ;,:.: $ U B I I M AT I 0 N .x.:.:.:,:,:,:.:.:,:.x,:.:.x.:.:,:

MANTLE

CORE

Schematic representation of energy transfer in the upper layer of a cometary nucleus.

forced through the mantle and how the gas loss is related to the dust loss, were among major unclarified questions of the Mendis-Brin model. Recently, several papers were published improving the original Mendis and Brin [1977] model, while keeping their original idea of mantle evolution, i.e., that the mantle thickness varies with time because of the interplay between vaporization which increases the thickness of the evacuated layer and the "erosion" due to the drag of the outflowing gas which decreases it. Hordnyi et al. [1984] introduced a "friable sponge" model of cometary nuclei, which makes it possible to express quantitative relations between gas and dust productions. Fanale and Salvail [1984] took into account the diffusive flow of gas through the mantle, Podolak and Hermann [1985] included the effects of cracks and pores in the mantle, and Houpis et al. [1985] introduced a chemically differentiated, multilayered mantle.

2.2. Vaporization and Gas Production Models

As a first approximation, vaporization models neglect the differentiated thin surface layer and consider the whole nucleus to be a homogeneous mixture of volatiles and dust. These homogeneous models implicitly assume that the uppermost dust layer, freshly evacuated of its ice component, is instantaneously blown away by the outflowing gas so that the slowly shrinking nucleus retains its undifferentiated pristine nature at all instances. Another consequence of the homoge- neity assumption is that basic physical parameters, such as the bolometric albedo (A•), surface emissivity (e•), and thermal conductivity 0c) do not depend explicitly on time or spatial location; however, quantities like thermal conductivity, latent heat of vaporization, etc., are usually temperature dependent, and thus may show temporal and spatial variations. As hori- zontal temperature gradients are typically much smaller than the ones measured in the radial direction, homogeneous models usually neglect transverse heat flows. These explicit and implicit assumptions, almost without exception, result in models describing a one-dimensional radial heat flow problem, where for any given angular position the local external radiation field (which might be time and angular position dependent) controls the temperature distribution along the radial direction in the nucleus.

Homogeneous models assume that the absorbed radiation flux is balanced by a combination of blackbody reradiation, vaporization of surface volatiles and the maintenance of the thermal structure of the nucleus; in other words, they apply a version of the following energy balance equation at the cometary surface [cf. Squires and Beard, 1961]:

(1 -- ,ZlB)Jrad(O , (])) -t- (1 -- ,Zlm)ltr(O, (])) --- 8so'Ts 4

+ • rIizj(T,)Lj(T,)/[Narnj] + tc(T•) grad (T,) (1) where t is time; r is nucleocentric distance; O and •b are the local hour angle and latitude, respectively; J,,a(t, O, fib) is the total direct and multiple scattered solar radiation energy flux reaching the nucleus surface; ltr is thermal radiation energy density received at the surface, from radiation emitted by dust particles in the coma; A•R is infrared albedo of the surface, T•(t, O, •b) is the surface temperature; a is the Stefan-Boltzmann constant (a = 5.67 x 10 -5 erg cm -2 s -• K-'•); N,• is the Avo- gadro number; Lj(T) is the latent heat of vaporization per mole for the jth volatile species (erg/mol); zj(T) is the outflowing mass flux of the jth species (g/cm2/s); to(T) is thermal conductivity; rnj is mass of the jth particle species; and r/j is the fraction of the surface area covered by the jth ice component. Inside the nucleus the following heat conduction equation can be applied:

pcCc Ot r 2 Or r2K • = Qint (2) where Pc, Cc, and Qint are the mass density of the nucleus, specific heat, and internal energy source, respectively. Several authors have considered various possible internal energy sources, such as decay of radioactive isotopes [Whipple and Stefanik, 1966; Wallis, 1980] or transition of amorphous ice to a crystalline state [Pataschnik et al., 1974; Smoluchowski, 1981a, b; Klinger, 1981].

The simplest models of cometary gas production at the surface of a nucleus assume an optically thin coma and neglect the heat conduction term in equation (1), thus reducing the problem to a simple transcendental algebraic equation I'Wei- gert, 1959; Watson et al., 1961; Huebner, 1965, 1967; Del- semrne, 1966; Shulman, 1969; Delsemrne and Miller, 1971]. More sophisticated models assume the presence of an insulating dust mantle [Mendis and Brin, 1977; Hordnyi et al., 1984; Fanale and Salvail, 1984; Podolak and Hermann, 1985], take into consideration light scattering caused by dust particles in the coma [Hellrnich and Keller, 1981; Weissman and Kieffer, 1981; Marconi and Mendis, 1982, 1983, 1984, 1986], include nuclear rotation [Dobrovolskii and Markovich, 1972; Srnolu- chowski, 1981b; Rickman and Froeschl•, 1983; Hordnyi et al., 1984-1, or use a combination of these' effects..

One of the central questions addressed by all these models is how to calculate the production rate of outflowing gas particles. In a pioneering work, Delsemrne and Swings [1952] considered a cometary nucleus covered by a homogeneous surface of volatile snow. They assumed that the surface did not contain macroscopic irregularities; i.e., surface irregularities were much smaller than the mean free path of vaporized particles. As the pressure of the cometary atmosphere was much smaller at the nucleus surface than the critical pressure of the phase transition triple point, the liquid phase was unstable and sublimation of frozen volatiles was responsible for gas production. Assuming that the sublimated gas was in equilibrium with the surface, Delsemrne and Swings [1952] applied the Clausius- Clapeyron equation to determine the steady state saturated gas pressure:

( ' p,(T) = p, exp LkN, (3) where p• is the vapor pressure, p, is the saturated vapor pressure at a reference temperature T,, L(T) is the latent heat of

GOMBOSI ET AL.'. DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES 671

vaporization, and k is Boltzmann's constant (k = 1.38 x 10-•6 erg/K). Delsemme and Swings [1952] also assumed that the sublimated molecules behave as a perfect gas (Ps = nkT), thus defining the gas number density n. In order to determine the bulk velocity of the outflowing gas, Delsemme and Swings [1952] considered a low-pressure kinetic model. They started by deriving the flux of scattered gas particles reaching the snow surface and condensing back to solid phase:

Z- = 0.25mnuth (4)

where m is the mass of a gas molecule and Uth = (8kT/r•m) ø'5, the mean thermal speed of the gas molecules. Under steady state conditions the condensing flux z- is equal to the vaporizing flux z +. In this kinetic outflow model, Delsemme and Swings [1952] assumed that cometary sublimation was not an equilibrium process and neglected the condensing flux altoge- ther (z- = 0). At the same time they kept the vaporizing flux at the steady state level, thus finding a net gas production rate of

ps(T)(m) ø'• Zkin(T) = (2r•kT)O.5 (5)

For more than three decades, equations (3) and (5) have been widely used for calculating cometary gas production rates (compare earlier review papers by Whipple and Huebner [1976], Delsemme [1982], and Mendis et al. [1985]). During the years, however, it drew criticism from several authors, who made some modifications (usually resulting in a change of the gas production rate within about a factor of 2) to improve the kinetic production model [Marconi and Mendis, 1984; Hordnyi et al., 1984].

Expressing more fundamental criticism, Shulman [1972] pointed out that equation (5) would have had to have been significantly modified if one had taken into account the he- terogenous chemical nature of a cometary surface (i.e., that cometary surfaces are a mixture of various volatiles and dust; consequently, complex intermolecular forces modify the molecular constants in the Clausius-Clapeyron equation).

The whole idea of applying a kinetic approach was criticized by Markovich [1963] and Mendis et al. [1972] and later by Wallis [1982], Gombosi et al. [1985], and Mendis et al. [1985]. These authors pointed out that at the cometary surface the gas mean free path was typically 10-10 2 cm; consequently, the neglect of collisions is inappropriate and gas dy- namical methods have to be applied at the nucleus-coma interface. Wallis [1982] specifically noted that the kinetic sublimation curves published by Delsemme and Miller [1971] imply that collisionless effusion from an H20-dominated comet could only be appropriate outside 3 AU.

Self-consistent dusty hydrodynamic calculations [Probstein, 1968; Hellmich, 1979; Marconi and Mendis, 1982, 1983, 1984, 1986; Gombosi et al., 1983] employ a surface gas density value obtained from equation (3) and determine the interrelated production rate and outflow velocity (z = nUout) by solving the coupled steady state dust and gas equations with the appropriate boundary conditions at infinity (Po• = 0). These calculations will be discussed in detail in section 4. In attempting to develop a realistic model of the gas flow through the evacuated porous dust layer (mantle), one needs to approximate the inhibited gas flow and the dust pickup in a manner which is both realistic and still permits meaningful solutions to be obtained. Two such models have been published recently [Fanale and Salvail, 1984; Gombosi et al., 1985] which have

certain basic similarities but also have a number of different

fundamental assumptions. Extending the hydrodynamic approach to the vaporization

process, Gombosi et al. [1985] proposed a gas dynamic "reservoir outflow" model to describe cometary gas production. In their model the sublimating surface below the mantle is replaced by a reservoir containing a stationary perfect gas with temperature T• and pressure Ps. It is assumed that the gas flow is so inhibited through the mantle that the gas is practically stationary and there is no significant pressure drop within the mantle. At the top of the evacuated dust layer the gas dis- charges to the low-pressure external medium dragging away dust particles from the top of the mantle. This approximation is applicable even in the case of a "bald" nucleus, because the sublimation has to take place below a microscopic layer of freshly evacuated dust (else the outflowing gas could not drag away dust grains). The gas from the reservoir is discharged to the low-pressure external medium either directly or through a thin layer of porous dust covering the nuclear surface. The gas production rate and outflow velocity are affected by the dust loading and by the gas parameters immediately outside the nucleus. Combining the results of a series of numerical solutions with the time-dependent dusty hydrodynamic equations describing gas outflow from such a reservoir and using the predictions of steady state gas dynamics, Gombosi et al. [1985] concluded that in a first approximation the gas production rate was

Fps(m) ø'5 Zo = 2(7kT•)O.• (6)

where 7 is the gas specific heat ratio and the numerical constant F can be expressed as

( 2 ) ø-•(•'+•)/(•'-•) r = 7 (7) 7+1

Comparing equations (5) and (6), it can be easily seen that for a typical 7 value of 4/3 (water vapor) the difference between the kinetic outflow [Delsemme and Swings, 1952] and the reservoir outflow [Gombosi et al., 1985] models is only about 20%. However, this difference becomes much larger when one takes into account the choking effect of dust loading. A new set of time-dependent numerical calculations, similar to those reported by Gombosi et al. [1985] has recently been carried out for this review paper. The dependence of gas production rates and outflow velocities on surface temperatures and dust/gas mass production rate ratios (25) was determined, on the basis of this new and extended numerical modeling of transonic dusty hydrodynamic reservoir outflow. (It should be noted that, in general, 25 is not necessarily equal to the solid/volatile mass ratio inside the nucleus. This question will be addressed in section 2.3.) The following analytic expressions approximate these new results to within a few percent for 25 in the range 0-5:

z(T, 25)= zo(T)(1.17- 0.07325) (8)

0.62Uth(T) Uout(T, 25)= (9)

1 + 0.2825

Numerical calculations leading to these expressions will be described in detail in section 4.4.

Equations (8) and (9) have several advantages compared with the predictions of the kinetic model. First, they were obtained from hydrodynamic calculations which describe the


collision-dominated outflow region more properly than a collisionless effusion model. Second, these new results take into consideration the effects of dust loading, which was neglected by the kinetic approach.

A diffusive model of gas production was published by Fanale and Salvail [1984], who considered the gas flow through the evacuated upper dust layer. They visualized this layer as a coherent solid with pores and capillaries and assumed that the sublimated gas flowing through the tubes is in the Knudsen regime. In this case the local gas velocity is [Fanale and Salvail, 1984]

4 (2k T• •ø'5 r0 grad (p) (10) Um= • \-•m ,/ t'• p where T• is the sublimating temperature and ro and t,, are the average capillary radius and tortuosity, respectively. The pressure drop through the mantle was expressed as

Ap = Psi1 - f (Tou•)/(ht•o)] (11)

where To is the surface temperature and f is the mantle porosity, while the gas outflow velocity from the mantle (Uo) was assumed to be 0.6 times the mean thermal velocity Uth. As- suming steady state conditions, Fanale and Salvail [1984] derived the gas production rate as

raps

zFs = • u•N½•:ro 2 (12) where u• is the gas diffusion velocity at the sublimating interface, while the number of capillaries per unit area (No) can be expressed as

N½ = f/(3•ro2tm) (13)

Combining equations (10), (11), and (13), one can obtain an alternate expression for the Fanale and Salvail [1984] diffusive gas production rate:

A

ZFS -- Zki n A/Ac n t- (To/TOO. 5 (14) where A is the thickness of the evacuated dust layer and

A = 0.8/tin

A½ = (10/9)(fro/tm)

Fanale and Salvail [1984] adopted f= 0.5, tm= 5.0 values and assumed that r 0 was approximately equal to half of the average grain size ((a)• 1.5 #m). Using these values, one obtains A = 0.16 and Ac • 0.08 #m. It is interesting to note that for A--• 0 the Fanale and Salvail [1984] model (with their parameter selection) gives a gas production rate which is about 6 times smaller than the kinetic production rate calculated for the same surface temperature.

It should be noted again that the Fanale and Salvail [1984] diffusive production model and the reservoir outflow model [Gornbosi et al., 1985] are both based on the same basic process: pressure difference drives out gas from a stationary (or almost stationary) gas reservoir. In the case of the Fanale and Salvail [1984] model the outflow velocity is constant (0.6Uth), while in the reservoir outflow model the pressure in the mantle is assumed to be constant, but variations in the external pressure value, as well as the dust mass loading, are taken into account.

Before discussing more complex cometary surface models, it should be noted that the latent heat of vaporization, L(T),

needs to be treated with some caution. It was pointed out by Delsernrne and Miller [1971] that for most volatiles L strongly varies with temperature; the notable exceptions are water and clathrate hydrate ices, whose latent heats hardly vary between 125 K and 273 K. Some authors have used a so-called "mean

latent heat" for describing the sublimation of multicomponent ice mixtures [e.g., Huebner and Giguere, 1980; Huebner and Keady, 1983]. This method was recently criticized by Mendis et al. [1985], who concluded that there was no physical justi- fication for this practice and it grossly overestimates the production rate of the less volatile species, while underestimating the release of the more volatile component.

2.3. Heat Transfer in the Mantle: Governing Equations

Most recent mantle energy balance calculations have considered two components (one solid and one volatile) and have applied assumptions with different levels of sophistication [Mendis and Brin, 1977, 1978; Brin and Mendis, 1979; Weiss- man and Kieffer, 1981, 1984; Hordnyi et al., 1984; Fanale and Salvail, 1984; Podolak and Hermann, 1985; Houpis et al., 1985]. As the gas production of almost all known comets seems to be primarily controlled by the vaporization of water [Delsernrne, 1985b], the following sections in this paper will mainly be devoted to the discussion of the temperature distribution, gas and dust production rates in the surface layer of a water, or clathrate hydrate, dominated nucleus. A recently published and more complicated three-component multilayered mantle model [Houpis et al., 1985] explaining a poten- tially high CO/CO2 production rate of comet West (1975n) [Feldrnan and Brune, 1976] will also be discussed later in the paper.

A schematic representation of the differentiated top layers is shown in Figure 2. A core consisting of the original dust-ice mixture is covered by an evacuated dust layer which forms the mantle. The vaporization process takes place at the mantle- core interface and not at the top of the nucleus. The absorbed radiation energy has to penetrate the insulating mantle in order to reach the sublimating surface, and consequently the surface temperature (T O in what follows) and the sublimation temperature (T•) cannot be represented any more with a single temperature T s. At the nucleus surface the energy balance equation can be expressed as

aT0 (1 -- AB)Jra d + (1 - AiR)!tr = esaTo ½ + *era(To) -•r (15)

where ,c m represents the heat conductivity in the mantle. At the core-mantle interface there is a jump in the heat flux due to the energy used up by sublimation. The heat conductivity function is different in the mantle and in the core, because the physical parameters are different in the evacuated mantle and in the ice-dust mixture core. At this interface the energy balance equation can be written in the form

•T ] L(T•) •T ] = q•, z(T,) + %(T,) (16)

here A is mantle thickness, R• is nuclear radius, % is thermal conductivity in the core, while e represents an infinitesimally small but positive number. The area fraction, r/,•, covered by ice can be expressed as [cf. Hordnyi et al., 1984; Fanale and Salvail, 1984]

qw __ rl2/3 = (pa p a •2/3 (17)

GOMBOSI ET AL..' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES 673

where r/ is volume ratio of ice in the core, Pa is dust bulk density, Pi is ice mass density, and Zc is dust/ice mass ratio in the core. The energy balance equations in the mantle and in the core are (assuming only radial variations, heat conduction as the dominant energy transport mechanism, and the heating of the outward diffusing gas as the main energy loss process)

Mantle

pmCm r3t r2•r r2tcm • = - •m qwz(Ti) • (18) Core

pcCc r3t r2r3r r2tcc = 0 (19) where Pm is average mass density of the mantle, Pc is average mass density of the core, Cm is specific heat of the mantle, and Cc is specific heat of the core. The energy loss term in equation (18) is not zero, since it is assumed that the penetrating vapor is in thermal equilibrium with the dust throughout the mantle and so represents a sink of heat. Hordnyi et al. [1984] have argued that it is reasonable to expect the gas to accommodate to the mantle temperature at each location because the mean free path of a gas molecule is much larger than the average grain separation in the mantle [cfi Whipple and $tefa- nik, 1966].

Equations (15)-(19) contain some basic parameters describing such fundamental properties of the mantle and core as density, heat conductivity, and specific heat. As nobody has ever examined the structure of a nucleus, these basic parameters have mainly been obtained by guesswork, so that the values chosen are uncertain. On the other hand, one has the right to expect nucleus models to be internally self-consistent: two interrelated parameters should not be derived using con- tradictory assumptions. Here an attempt is made to compile a set of consistent functions from the various mantle/core models.

The ice mass density is assumed to be Pi = 0.9 g/cm 3. The average dust density is derived from the average density of dust particles observed in the coma. When deriving Pd, the dust density function used by Divine for comet Halley calculations [Divine and Newburn, 1983] and a Hanner-type dust size distribution [Hanner, 1983] is used (this distribution is a modified version of the $ekanina and Miller [1973] and $ekanina [1980] distribution functions):

p(a) = Po- p•a/(a + a•) (20)

n(a) = [1 - ao/a]M(ao/a) N (21)

where M=12, N=4.2, a1=2 #m, ao=0.1 #m, P o=3 g/cm 3, and P l = 2.2 g/cm 3 [Divine et al., 1986]. These values result in an average dust bulk density value of Pa - 0.84 g/cm 3 and an average dust size of (a) = 0.69 #m. Now the average core and mantle densities can be obtained as

Pm = q)•cPi (22)

p• = q(1 + Zc)p,

Following Hordnyi et al. [1984], a dust specific heat value of Ca - 8 x 106 erg/g/K was adopted by the Halley environment working group of the Interagency Consultative Group [Divine et al., 1986]. As there is no specific information available about the thermal properties of cometary nuclei, this value seems to be as good as anything else and was also adopted in

this work. The situation is considerably better with respect to water ice. Klinger [1981] has derived an empirical temperature law for the ice heat capacity from the measurements of Giaque and Stout [1936]:

C,(T) = 7.49 x 104T + 9.00 x 105 (erg/g/K) (23)

In the core there is an ice-dust mixture and the effective specific heat can be expressed as

Cc -- (Zc Ca + C,)/(1 + Zc) (24)

The specific heat in the evacuated mantle is taken to be identical with the dust specific heat, Cm----Ca. Klinger [1981] also published an expression for the crystalline water ice thermal conductivity in the form of

tc,(T) = 5.67 x 107/T (erg/cm/s/K) (25)

Following Mendis and Brin [1977], recent calculations [Hordnyi et al., 1984; Fanale and $alvail, 1984; Podolak and Hermann, 1985; Houpis et al., 1985] represent the thermal conductivity in the mantle with an expression taking into account contact and radiative conduction and neglect the contribution of gas conduction:

tcm(T ) = tco + 4rrgalT 3 (26)

where tco is the contact conduction coefficient, ea is dust infrared emissivity, and I is average intergain distance. Brin and Mendis [1977] assumed that I was equal to the maximum grain size in the mantle. This approximation overestimates the role of radiative conduction: an alternative approach published by Hordnyi et al. [1984], expressing I as a function of the dust/ice mass ratio, seems to be somewhat more realistic:

Mendis and Brin [1977] used a too = 60 erg/cm/s/K value, which is in the range of values derived for lunar materials [Linsky, 1966]. There is no better value at the present time, and one can, for the time being, also use this numerical value. The emissivity % is not known either, but a very approximate value of % = 0.9 used by Halley numerical models [Divine et al., 1986] is adopted. Comparing equations (25) and (26), one can see that at reasonable cometary core temperatures (T < 300) the dust conductivity (•60 erg/cm/s/K) is much smaller than the conductivity of ice; consequently the core thermal conductivity can be well approximated by the following expression:

lVc -- flKi (28)

In the mantle-core thermal calculation one also has to

know the sublimation parameters for water ice (the parameters for clathrate hydrates are very similar). It was pointed out by Delsemme and Miller [1971] that for water the latent heat is fairly insensitive to temperature and its value can reason- ably be approximated by Lw = 4.80 x 10 TM erg/mol. The sublimation vapor pressure of boiling water, p, (see equation (2.3)), is 106 dyn/cm 2 at the T• - 373 K reference temperature.

In order to be able to solve equations (18) and (19) with the appropriate boundary condition, one has to define the relation between the mass ratio of outflowing dust and gas (Z) and the solid/volatile ratio in the core (Zc)- The original Mendis and Brin [1977] model and some of the recent calculations [Fanale and $alvail, 1984; Podolak and Hermann, 1985] as-


sumed that at any orbital position all the evacuated grains smaller than a critical size (the largest grain which can be blown off by the outflowing gas; for details, see section 4.2) will escape, while larger grains will be retained as part of the mantle. This type of model (sometimes implicitly) assumes flui- dization of the mantle; this way the newly evacuated small grains can be forced through the existing mantle.

An alternative model was suggested by Hordnyi et al. [1984] and later adopted by Houpis et al. [1985]. This "friable sponge" model assumes that (1) the dust grains in the mantle have the same spatial configuration (spongelike so as to permit the outflow of gas of the vaporizing ice at the core- mantle interface) as they have in the core; (2) the destruction time of particles larger than the critical size, amax, is short; these big grains are extremely friable and break into smaller pieces before their accumulation results in a violation of assumption 1; (3) the mass loss rate of the mantle (or dust production rate) is proportional to the momentum flux of the outflowing gas (the proportionality factor,/•a, is a characteristic parameter of each comet).

Z d = ]•dUout•wZ (29)

Combining equations (9) and (29), one obtains a relation between/•a and Z:

Z = 1.79[(1 + 0.69flariwUth) •/2-- 1] (30)

It should be noted that a Z > Zc value usually means a decreasing mantle thickness, while Z < Zc leads to a growing mantle. As Uth , and consequently Z, varies with the T O surface temperature, comets may have different mantle evolution pat- terns along their orbit depending on the friability of their nuclear material. In the case when A = 0 and equation (30) yields a Z > Zc value, it should be replaced by Z = Zc, because one can only blow away the freshly evacuated dust layer.

The sublimation process increases the mantle thickness, as the upper layer of the core is evacuated. The mass loss rate per unit area of the ice is

dt - rlwz = •lPi '• s (31) where (dA/dt)s is the change in A due to sublimation. The rate of decrease of the mantle thickness due to dust production is

dM• dMi (dA) dt- Z '-• = •m • e (32) where (dA/dt)e is the change in A due to erosion. The time rate of change of A can be obtained by subtracting (dA/dt)e from (dA/dt)s [cfi Hordnyi et al., 1984]:

d• (pa+piz•l/3(l_ Z)• (33) In general, if one knew the radiation flux at the surface of

the nucleus at all instances after the comet started its journey into the solar system, equations (18), (19), and (33) could be solved simultaneously with the boundary conditions defined by equations (8), (9), (15), and (16) for the •ntir• life of the comet and one could determine the A(t, •, ½) and T(t, •, •) functions. Unfortunately, the J•a(t, •, ½) Junction is dependent on the optical characteristics o[ the coma, controlled by •arli•r gas and dust production. It is obvious that self- consistent treatment of this problem is not an easy task and such a solution is still a f•w y•ars down the road. On the other hand, various groups made signiacant progress in solving

parts of this very complex problem. In the following sections the present status of these efforts will be discussed.

2.4. Heat Transfer in the Mantle: Approximate Solutions

Before discussing various approximate solutions of equations (18), (19), and (33), it is worthwhile to estimate the time constants characterizing heat transport and mantle growth processes. The thermal time constants of the mantle and core can be defined as [Klinger, 1981]

Pm Cm A2 Pc CcRn 2 '•m -- '•c = (34)

TC2Km TC2Kc

where Zm and Zc represent the thermal time constants of the mantle and core, respectively. Substituting estimates of comet parameters into expressions (34), one obtains z m and z c values of the orders of magnitude of an hour and hundreds of years, respectively. On the basis of these time constants, nucleus models can be characterized by the following classes of models:

Rotating nucleus. Two groups have published calculations taking into consideration the angular dependence of external radiation at the surface. Weissman and Kieffer [1981, 1984] considered a nucleus with no mantle and took into consider-

ation the heat conducted into the core. They considered the core to be a sink of energy rather than a very large heat reservoir, which may increase the surface temperature at larger heliocentric distances (especially in the outbound part of the orbit). In a semiempirical way, Weissman and Kieffer [1981, 1984] also took into consideration the optical characteristics of the coma, thus decreasing the diurnal variation of the radiative flux reaching the surface. Hordnyi et al. [1984] adopted a more sophisticated mantle-core nucleus model but used the unattenuated solar radiation profile to study the diurnal variation of the surface and sublimating temperatures.

Both calculations concluded that (1) for active comets the amount of energy conducted into core was much smaller than the energy used for sublimation, (2) the temperature distribution in the uppermost layer of the nucleus reaches a steady state diurnal pattern after less than 10 rotations, and (3) the longitudinal temperature distribution reaches its daily maximum in the afternoon and its daily minimum in the predawn hours.

It seems to be very reasonable to expect a diurnal variation in the surface illumination at larger heliocentric distances where the coma is not well developed. However, closer to the sun, multiple light scattering on dust grains in the coma dramatically modifies the angular distribution of the absorbed radiation. Various radiative transfer calculations [Hellreich and Keller, 1981; Marconi and Mendis, 1982, 1983, 1984, 1986; Weissman and Kieffer, 1981, 1984] have indicated that as a result of the large collecting area represented by the thick dust coma the radiation energy density at the nucleus is probably somewhat larger than that of the unattenuated solar radiation and its distribution is fairly isotropic. This means that the more active a comet is, the less diurnal variation one can expect. Figure 3 (taken from Weissman and Kieffer [1984]) shows the extreme differences between surface temperature distributions obtained for Comet Halley at perihelion (0.5871 AU) in the presence and absence of a dense coma. In the calculations a rotation axis obliquity of 20 ø was assumed. The temperature distribution on a bare ice nucleus without coma is shown in Figure 3a, and the same nucleus with coma is

GOMBOSI ET AL.' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES 675

NUCLEUS SURFACE TEMPERATURE R = 0.587 AU

75.

60.

45.

30.

15.

O,

-15.

-30.

-45.

-60.

-75.

205

90.

• a.) 160

170

180. 270. 360.

LOCAL HOUR ANGLE

NUCLEUS SURFACE TEMPERATURE R -- 0.587 AU

75.

60.

45.

30.

15.

-15.

-30.

-45.

-60.

-75

I I '1

20 04

I

b.)

O. 90. 180. 270. 360.

LOCAL HOUR ANGLE

Fig. 3. Temperature distributions on the nucleus of Comet Halley at perihelion (0.5871 AU) (a) with no dust coma and (b) with dust coma. The rotation pole obliquity, estimated to be approximately perpendicular to the orbit plane, was chosen to be 20 ø (taken from Weissman and Kieffer [1984]).

shown in Figure 3b. For the case of the nucleus without coma there is a strong diurnal variation and a noticeable thermal lag in the surface response, despite the very low value adopted by Weissman and Kieffer [1984] for the thermal inertia. How- ever, when the dust coma is added to the model, the nucleus becomes more isothermal.

Core thermal hysteresis. These calculations neglect the mantle layer and concentrate on the effects caused by a nucleus which has a large but finite heat reservoir [Smolu- chowski, 1981a, b; Klinger, 1981; Hermann and Podolak, 1985]. These authors also considered an internal heat source caused

by amorphous to hexagonal water ice phase transition, togeth-

er with heat conduction in a finite core. As has already been mentioned earlier, this model has been criticized by Shulman [1983b], who thinks that it is hard to argue for an evolution- ary scenario which could prevent amorphous ice from crys- tallization at the beginning of the formation of the nucleus. On the other hand, the amorphous to crystalline transition model [Smoluchowski, 1981a, b; Klinger, 1981] enjoys con- siderable popularity in the cometary community, in part because it very naturally explains the flaring phenomena observed in comet Schwassman-Wachmann 1. This question is still open, and the presently available observational evidence is insufficient to resolve it.


T(K) 210 ...... , , ,

•90

170 •

150 ' .

1:50

I10

90

70 •'

30 I I I I I I I I ' .5• •.0 m2.0•.0 5.0 •0 20•0

R(Au)

Fig. 4. Variation of surface temperature with heliocentric distance for a hexagonal water ice ball moving along the orbit of Comet Halley (taken from Hermann and Podotak [1985]).

Another important question regarding this type of calculation is the question of how quickly heat can be conducted within the core. The smaller the surface heat conductivity is, the longer it will take to conduct heat deep into the nucleus. Using an appropriately small value for the heat conductivity of the top layer, the surface layer can supply additional energy for sublimation after perihelion, thus shifting the maximum production rate up to • 100 days postperihelion. The problem is that this "appropriate" conductivity value represents a low porosity, an almost "concrete" like nucleus. It is difficult to visualize how such a low-porosity nucleus can produce dust particles at all. On the other hand, when the calculations take into consideration a pure hexagonal ice nucleus, the predicted light curve has its maximum at perihelion. Figure 4 (taken from Hermann and Podolak [1985]) shows the surface temperature variation of a pure hexagonal ice nucleus in the orbit of Comet Halley after many revolutions. It can be seen that close to the sun where the temperature is above • 185 K, sublimation of water ice controls the surface temperature. The cooling due to sublimation is so strong and the heat conducted to the core is so small that the surface temperature rises only slightly as the comet passes the perihelion portion of the orbit. On the outbound leg between about 1.3 and 3.5 AU a portion of the thermal energy stored in the surface layer is being conducted into the nucleus to continue warming the deeper layers. Between 3.5 and 7.0 AU this inward heat flow continuously decreases, and beyond 7 AU the surface is partially heated by the interior. This process continues as the comet passes its aphelion and starts its journey inward. On the other hand, the heat content of the nucleus has already been depleted by this time; consequently, internal heating contributes less and less to the maintenance of the surface temperature.

Orbital evolution of the mantle thickness. It was first suggested by Whipple [1950] that an evacuated dust layer may cover the surface of cometary nuclei, although the first quantitative model of the mantle thickness variation along a comet orbit was published by Mendis and Brin [1977] and Brin and Mendis [1979]. Using an evacuation-erosion model, they predicted different mantle thicknesses for the inbound and out-

bound portions of comet orbits, which resulted in a hysteresis of the light curve. The original Mendis and Brin [1977] model was recently further elaborated by several authors. Hortinyi et

al. [1984] introduced the friability concept (to connect gas and dust production rates), suggested that the outward diffusing gas is continuously heated in the mantle, and assumed that erosion primarily takes place at the surface of the mantle. Fanale and Salvail [1984] proposed a model for gas diffusion through the mantle, Podolak and Hermann [1985] combined the original Mendis and Brin [1977] mantle thickness variation calculation with a finite core heat capacity model, while Houpis et al. [1985] allowed for chemical differentiation of the upper, partially evacuated layers. It is important to note that all these models predict similar production rate hysteresis curves: brightness variation is simply a result of the growth and destruction of the mantle, so all modified Mendis and Brin [1977] type models predict very similar production rate curves, even though the physical processes responsible for the mantle distruction might be different.

Various models predict somewhat different mantle thickness variation along the cometary orbit. Mendis and Brin [1977], Brin and Mendis [1979], Hortinyi et al. [1984]. Podolak and Hermann [1985], and Houpis et al. [1985] obtained similar mantle thickness curves, which gave typical mantle thickness values of 0.1-1 cm for active periodic comets. Fanale and Sal- vail [1984] used their diffusive mantle model to calculate the growth and distruction of the mantle, which resulted in expression (14) for the gas mass production rate. Inspection of equation (14) reveals that the gas production rate starts to drop drastically as A becomes larger than the critical thickness, A c. Using the relations given by Fanale and Salvail [1984], one can express A• as a function of basic morpho- logical parameters of the evacuated mantle. As a result of the parameter values adopted by Fanale and Salvail [1984] their critical mantle thickness is approximately equal to the average capillary radius in the mantle, which was assumed to be half of the average grain size (< 1 #m). As the gas production rate drastically decreases when A >> A•, the typical mantle thickness for active periodic comets is the same order of magnitude as A•; consequently, Fanale and Salvail [1984] obtained 10-100 times smaller mantle thickness values as did the other

groups. Had they adopted a different set of material constants (for instance, the average intergrain distance (equation (27)) for the average capillary radius (as suggested by Hordnyi et al. [1984]) and a smaller tortuosity value), they would have obtained an order of magnitude larger mantle thickness, in agreement with the other calculations. This different set of constants could also result in a better agreement between the diffusive and kinetic gas production rates for the A--} 0 limiting case (which presently differ by a factor of 6). However, at this point this discrepancy between competing models cannot be resolved, because we have no observations about surface morphology of comets.

The governing equations of the mantle thickness variation discussed earlier can be simplified further by neglecting the radiative term in the mantle heat conductivity (with the adopted material constant values, this term contributes less than 5% to •Cm below about 500 K) and assuming that the temperature distribution in the mantle can always be considered to be steady state (this assumption seems justifiable because the thermal time constant in the mantle is about an hour, which is much smaller than the time constants of orbital motion or

mantle growth). In this case, one still has to solve equation (33) to obtain the orbital variation of the mantle thickness, but the surface and sublimating temperatures can be obtained by solving the following transcendental algebraic equations:

GOMBOSI ET AL.' DUST, NEUTRAL GAS MODELING OF COMETAllY INNER ATMOSPHERES 677

iO :>9

w

w

j 0 :>7

:>6

)•C = 0.5 /•d = 1.5 10-5s/crn

<3

IO

I.O

O.I

i

i

. I i

0.1 1.0 I0.0 0.1 1.0 I0.0

d (AU) d (AU)

Fig. 5. Variation of gas production rate (Qg) and mantle thickness (A) for a dirty iceball nucleus (with a friability of 1.5 x 10- • s/cm) moving along the orbit of Comet Halley.

(1 - AB)Jra d + (1 - AiR)ltr- esrrro 4

+ 5--m (To - (35)

3k A ( 3kN,t ) 2-• r/wZ- = In 1 + (T O - T•) (36) •c o 2L

F (•) •/2 ( L )1.17-0.073Z z = 2 Pr exp kNA T• Toø'5

The total gas mass production rate is

Mg = 4rcRn2rlw 2

-L

exP(kN,tT1) (37)

(38)

Figures 5 and 6 (calculated using the friable sponge model of Hor•inyi et al. [1984]) show the orbital variation of mantle thickness (A) and that of the total gas production rate (Qg = Maim) for a dirty iceball nucleus moving along the orbit of Comet Halley. In this calculation the external radiation field (Jrad) was assumed to be equal to the unattenuated solar radi- aton energy flux density, the heat flux conducted to the core was neglected, and the friability parameter was chosen to be either/9d = !.5 x 10 -5 s/cm (Figure 5)or//d = 2 x 10 -5 s/cm (Figure 6). The integration started at aphelion with a "bald" nucleus (A = 0) and was continued through several successive revolutions. Inspection of Figure 5 shows that for a friability parameter smaller than a critical value (~ 1.8 x 10-5 s/cm in this case) the mantle thickness increases monotonically with

10 29

• 102e

Xc=0.5 I0 sø i

_

I0 z? 0.01 0.1 I I0 0.1

d (AU)

io

O.I

/•d: 2.1CF5 s/cm i

d (AU)

Fig. 6. Variation of gas production rate (Qg) and mantle thickness (A) for a dirty iceball nucleus (with a friability of 2 x 10-5 s/cm) moving along the orbit of Comet Halley.

678 GOMBOSI ET AL.' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES

4

0.5 I 1.5 2 2.5 3 :5.5

d(AU)

Fig. 7. Variation of the H20/CO2 production rate ratio for a chemically differentiaated dirty iceball nucleus (with a friability of 2.25 x 10 -5 s/cm) moving along the orbit of Comet Halley (taken from Houpis et al. [1985]).

time, while the gas and dust production rates continuously decrease. During consecutive revolutions this comet becomes fainter and fainter (i.e., the gas production rate decreases) and is finally suffocated by dust. The surface temperature of such a comet increases continuously and finally approaches a limiting value, T O ..... = (Jrad/t;str) 0'25 (i.e., practically all absorbed energy is reradiated as blackbody radiation). At 1 AU, this typically means a surface temperature of about 400 K. It is interesting to note that the faint earth-grazing Comet Iras- Araki-Alcock (1983d) is probably such a dying object. Hanner et al. [1985] have concluded on the basis of a series of infrared observations that this comet had an approximate radius of about 5 km, a surface albedo of 0.9, and a subsolar surface temperature of about 400 K. The average gas production rate was • 2 x 10 -7 g/cm2/s, while the dust/gas mass ratio in the coma was •0.25. Applying the friable sponge model (most evacuation/erosion models would give qualitatively similar results) to Iras-Araki-Alcock (1983d), one obtains the following parameters: sublimating temperature of •190 K, mantle thickness of •0.5 cm, and friability parameter of • 10-5 s/cm. These parameters clearly describe a dying comet, where the accumulating dust layer ultimately quenches gas and dust production.

The mantle thickness does not increase monotonically any more when the friability is larger than a critical value, but insteatt a repetitive cycle appears. Apart from the first approach to the vicinity of the sun, A and Qg follow the same curve during subsequent revolutions. Inspection of Figure 6 shows that at about 1.5 AU preperihelion the increase of A stops and then the mantle thickness starts to decrease. By the time the comet reaches ,-• 1 AU postperihelion, the mantle is practically blown off all at once and the gas production jumps by more than 1 order of magnitude. When the comet again leaves the vicinity of the sun, a new mantle is developed; this new mantle is blown off during the next perihelion passage. This process is repeated during subsequent revolutions.

When the friability parameter is further increased, the mantle thickness starts to decrease while still further away from the sun, and it will eventually be blown off before perihelion, rather than postperihelion. This early blow off results in fairly high perihelion production rates (,-• 3 x 1030 molecules/s for a 3-km nucleus radius) and in a symmetric light curve around perihelion. The friable sponge orbital calcula-

tions predict either a large hysteresis or practically no hysteresis for the near-perihelion part of a Halley type comet depending on the adopted friability parameter value. For Comet Halley the relatively modest hysteresis is probably related to the variation of the coma optical parameters, while the nucleus itself remains practically "bald" for the perihelion part of the orbit [Brin and Mendis, 1979' Weissman and Kieffer, 1984]. It should be remembered that these friable sponge calculations did not include any radiative transfer consideration of the flux incident on the nucleus, and therefore this question has to be further investigated using a combined mantle-coma radiative transfer model.

Multilayer mantle. In order to explain the unusually high CO/OH ratio reported for Comet West (1975n), Houpis et al. [1985] have recently proposed a three-component mantle/core model. In this model the pristine nucleus consists of an icy conglomerate containing dust, frozen clathrate hydrate (with one CO molecule trapped between six water molecules), and free CO2 ice. As this conglomerate approaches the sun, the more volatile CO2 which is not trapped in clathrate will escape first leaving behind a steadily growing mantle of dust and clathrate ice. This clathrate mantle partially insulates the core containing more volatile species. As the comet moves closer to the sun, its surface gradually heats up, and the clathrate hydrate also starts to sublimate creating an uppermost dust mantle completely evacuated of its volatile component.

In their calculation, l-loupis et al. [1985] applied a friable sponge type model [Hor•inyi et al., 1984] to describe the temporal evolution of the dust and clathrate mantles. The model neglected the heat flux into the core, as well as the heating of outflowing gas.

Figure 7 shows the orbital variation of the H20/(CO2 + CO) production rate ratio Rv after 10 revolutions. For this

particular calculation, Houpis et al. [1985] adopted a fid = 2.25 x 10 -5 s/cm value for the friability parameter. The production rate ratio was calculated using the relation

6•clZcl Rv = (39)

gI½lZ½l -[- 7r/,,z,,

where •/½1 and r/• represent the fractions of the sublimating surfaces covered by clathrate and CO2, respectively; Zcl and z• are the clathrate and CO2 production rates. It is obvious from equation (32) that Rv < 6, and consequently this model will always predict a CO and CO2 rich comet. Inspection of Figure 7 reveals that at large heliocentric distances this comet will behave as a cO 2 (and CO) rich comet (R• < 1), while closer to the sun the CO + CO2 content in the coma decreases to about 20-25%.

The multilayer mantle model is very attractive, because it is able to explain the high initial activity of new comets and can predict a changing H20/(CO 2 + CO) ratio along the comet orbit, On the other hand, some authors [cfi Shulman, 1983a] have difficulties visualizing how this type of nucleus could originally condense out of the presolar nebula. Shulman's arguments seem to be somewhat simplistic, especially if one takes into account the widely used concept of "condensation in sequence" [cf. Safranov, 1972; Alfv•n and Arrhenius, 1976] and that larger volatile molecules (for example, CO2) simply do not "fit" into the water ice structure (R. Prinn, private communication, 1985). This "size incompatibility" raises the possibility that the clathrate hydrate contains mainly smaller "guest" molecules (such as CO), while molecules like CO2 form the free volatile ice component.


3. THEORY OF ATMOSPHERIC PROCESSES

3.1. Transport Equations and Physical Processes

As has already been discussed in the preceding section, presently there are two models describing gas production of a cometary nucleus: the kinetic outflow model of Delsemrne and Swings [1952] (see equation (5)) and the recently published hydrodynamic reservoir outflow model proposed by Gornbosi et al. [1985] (see equation (8) for a "bald" nucleus and equation (37) for a nucleus with an insulating mantle). The next question to be addressed is what happens to the outflowing gas once it leaves the surface of the nucleus.

The atmospheres of comets, commonly referred to as c½•mas, are different in a number of important ways from planetary atmospheres. The most important distinguishing characteristics of comas are (1) the lack of any significant gravitational force, (2) relatively fast radial outflow velocities, and (3) the time-dependent nature of their physical properties. A direct consequence of these features is the expanding nature of cometary atmospheres.

The first in situ measurement of a cometary neutral atmosphere will not be made until later this year; nevertheless, remote optical observations have provided some clues about the general nature of comas. The limited information presently available makes discussions of general cometary atmospheres meaningful, even though it is known that there are significant differences in the physical and chemical makeup of the different comets. Studies of cometary atmospheres generally assume that water vapor is the major parent molecule, with only a minor amount of other volatiles present in the nucleus. In the following subsections the main physical and chemical processes, which are believed to control the gross behavior of comas, are outlined.

In order to carry out quantitative studies of the gases flowing away from the surface of the nucleus, the appropriate coupled set of conservation equations have to be solved. A comprehensive summary of these transport equations and their relative applicability was given by Schunk [1975, 1977] in a couple of review papers; other authors who have extensively discussed these equations, applicable to atmospheric and plasma studies, include Holt and Haskell [1965] and Ta- nenbaurn [1967]. All these equations are obtained by taking moments of the Boltzmann equation. Some differences do exist between the equations derived by different authors, because some authors obtain the moment equations with respect to the random velocity, while others use the actual (total) velocity.

As mentioned earlier, the most comprehensive presentation and studies of these transport equations, relevant to aeronomy, are those of Schunk [1975, 1977]; so this brief review will follow his approach, with one important difference. In a cometary atmosphere, unlike that of earth, the mean flow velocities can be comparable to the thermal velocities; therefore certain approximations adopted for studies of the terrestrial environment are no longer appropriate.

Schunk [1975] presented a general system of transport equations for flowing neutral gases and plasmas, which were derived by using Grad's [1958] formulation and Burgers' [1969] collision terms. These systems of equations, sometimes referred to as the "13 moment equations," include continuity, momentum, internal energy, pressure tensor, and heat flow equations for each species under consideration. However, considering our very limited present-day understanding of com-

etary atmospheres, only the first three of these equations, corresponding to a "five-moment approximation," will suffice to characterize the neutral gas behavior within the collision- dominated region of comas.

In this five-moment approximation the properties of the gas are expressed in terms of just the species density, flow velocity, and temperature. The conservation equations for the density, velocity, and temperature, neglecting Coriolis, viscous stress, internal energy, and certain heat flow effects [see Holt and Haskell, 1965; Burgers, 1969; Schunk, 1975], are

C9ns •ns • + V(nsUs) - (40a) •t •t

Dsus •Ms (40b) rnsns -•- + Vps - rnsnsGs - cSt

3 Dsp s 5 cSE s (40c) 2 Dt +•ps(V.us)+V.qs- •t where Ds/Dt is the convective derivative, ns is the number density of neutral gas species s, u s is the velocity of neutral gas species s, ms is the mass of neutral gas species s, Ps is the kinetic pressure of neutral gas species s (=nskTs), T s is the temperature of neutral gas species s, Gs is the external volume force (e.g., gravity), CSns/& is the density source term due to collisions, cSMs/& is the momentum source term due to collisions, and •Es/•t is the energy source term due to collisions.

As stated earlier, the energy equation (40c) was obtained neglecting the internal energy of the molecules. It is commonly assumed that the average energy per particle with internal degrees of freedom can be written as [cf. Burgers, 1969]

Ws = « VskOs (41 a)

where v s is the number of internal degrees of freedom and Os is the effective temperature associated with the internal mo- tions. When the gas is in thermodynamical equilibrium, Os has the same value as T•, which we defined as the temperature associated with the translational motion. Under these con-

ditions the average total energy, Us, is

= + 1 V s)kTs It follows that the specific heat at constant volume, Cv, is

(4lb)

(• 1 )k (41c) C,,= +• vs m The ratio of specific heats, commonly denoted as 7, can thus be written as

C v 5 + v s 7 = -- - (41d)

C,, 3 + Vs

If the moment of the Boltzmann equation is taken with respect to not only the kinetic (translational) energy but the sum of kinetic and internal energies, the resulting energy equation (neglecting conduction of internal energy) for the total energy is

) Dt ps + nsUs + ps + nsUs (V.us)+V-qs- •t (41e)

Now using the definition of the polytropic index, 7, as given


TABLE 2. Hard Sphere Radii of Typical Cometary Species [Allen, 1973]

Radius, Species nm

H20 0.175 OH 0.145

H 2 0.11 CO 2 0.19 CO 0.17

CH½ 0.175 NH 3 0.15

in (41a), the above equation can be transformed into the form commonly used in hydrodynamics [cf. Zucrow and Hoffman, 1976]'

1 Dsp s 7 6Es + ps(V' Us) + V'qs - (41f)

•- 1 Dt •- 1 cSt

There is a whole hierarchy of approximations for the collisional source terms. Relatively low order approximations can be adopted for comets given our very limited understanding of cometary atmospheres. The density source term is taken to be simply the chemical production minus loss rate of a given neutral species'

cSn s - p/-- l•' (42a)

where Ps' is the production rate of neutral species s and ls' is the loss rate of neutral species s. Although we mentioned only chemical production and loss rates, processes such as sublimation of icy grains and ionization of neutral gases can easily be included in the production and loss rates, respectively.

Rigorous derivations of the momentum and energy source terms have, in general, been carried out considering elastic collision processes only, with inelastic processes introduced only at the end in a heuristic manner. The momentum source term, considering only elastic hard sphere interactions [Schunk, 1975], is

s(_psis) (St - • nsmsVs,(Us- u,)- 0.2 Vs, •-•s • qs Pt t

(42b)

The momentum transfer collision frequency between neutral species s and t (rs,), the reduced mass (#s,), and the reduced temperature (Tst) can be expressed as

msmt #st = • (43a)

ms +mt

mits + msT• Tst = (43b)

mt+ ms

8r• ø's ntmt 2(2k Tst• ø'5 - rs, (43c) Vst 3 ms +mr \-•s• / where rst is the sum of the radii of the colliding particles (see Table 2 for a few representative values).

Estimates of equivalent hard sphere radii are given in Table 2; note that they are all in the range of 0.1 to 0.2 nm. It is important to point out that the few actually measured total scattering cross sections are significantly greater than the hard sphere cross sections calculated from the given radii. For example, the measured value of the H20-H20 cross section is

about 3.35 x 10 -•'• cm 2 [Snow et al., 1973] compared with the hard sphere cross section of 9.62 x 10- x6 cm,•.

The second term on the right-hand side of equation (42b) accounts for the thermal diffusion and thermoelectric effects.

In neutral atmospheric applications the latter effect is generally neglected and even thermal diffusion is negligible, unless very large temperature gradients are present (R. W. Schunk, private communication, 1985).

The energy source term for neutral gas mixtures was given by Schunk [ 1975], for hard sphere interactions, as

tSEs --• nsmsVs• 3k(Ts- Tt) (44) tSt ms +mt

The above energy source term accounts for energy transfer between neutral gas species s and t via collisions; collisions between the neutral gas, s, and ions and electrons can also be accounted for by the above expression if the appropriate collision frequencies are used. There is another source of energy which is important in cometary atmospheres and which needs to be added, in a somewhat heuristic manner, to the above equation. Radiative energy absorption, scattering and emission processes (Qrad) by the radiatively "active" molecules found in cometary atmospheres (e.g., H:O and CO:), along with heating due to chemical processes (Qch) represent the energy transfer mechanism, which has to be considered; therefore an approximation of the total energy source term, appropriate for cometary atmospheres, is

6Es • nsmsVs, 3k(T s Tt) q- Qcxt (45) fit ms + rnt

where Qcxt--Qraa q- Qeh is the rate of net external heating/cooling. The main contribution to radiative cooling is the infrared radiation from the H20 molecules. In order to obtain an appropriate quantitative value for this net heating term, very complex radiative transfer calculations are necessary. The following semiempirical equation for H20 radiative cooling was given by Shimizu [1976]'

Qrad( H 2 O) 8.5 X 10- • 9 Tw 2rtw2 =- ergcm-3s-• (46) nw + 2.7 x 107Tw

where nw and Tw are the H20 number density and temperature, respectively. However, this expression does not take into account the above mentioned radiative transfer effects

inside the dense coma. Huebner [1985] has indicated that radiative cooling is not important in the inner coma because of radiative trapping. He obtained the following estimate for the infrared optical depth •IR (see section 3.2 for the definition of optical depth) at a distance r from the nucleus'

0.4nsaaR,• 2 •IR --' (47)

r

where n s is number density of absorbing gas at the nucleus and a a is mean infrared absorption cross section (an approximate value of aa= 4 x 10 -•5 cm a was used by Huebner [1985]). Using this approach, Huebner [1985] gave the following modified expression for Shimizu's [1976] radiative cooling formula'

Qrad(H20 ) 8.5 x 10-19Tw2nw 2 = - exp (-- IIR) erg crn- 3 s- • n w + 2.7 x 107Tw

(48a)

Recently, Crovisier [1984] has published an improved


TABLE 3. Incident Solar UV Flux Data (5-105 nm) for Solar Minimum (SC 21REFW) and Solar Maximum (79314) Together With Photoabsorption and Photoionization Cross Sections

•, (•abs (•ion (•abs (•ion (•abs (•ion (•abs (•ion (•ion nm Omi n Oma x H 20 H 20 CO 2 CO 2 CO CO N 2 N 2 O

5.0-10.0 0.38 1.60 1.25 1.25 4.42 4.42 1.92 1.92 0.60 0.60 1.92 10.0-15.0 0.13 0.45 2.33 2.33 7.51 7.51 3.53 3.53 2.32 2.32 1.63 15.0-20.0 1.84 7.18 3.47 3.47 11.03 11.03 5.48 5.48 5.40 5.40 3.00 20.0-25.0 0.92 5.75 4.97 4.97 14.98 14.98 8.02 8.02 8.15 8.15 5.05 25.630 0.27 0.73 6.00 6.00 17.88 17.88 10.02 10.02 9.65 9.65 6.03 28.415 0.10 5.68 7.20 7.20 21:21 21.21 11.70 11.70 10.60 10.60 7.12 25.0-30.0 0.84 6.69 6.82 6.82 20.00 20.00 11.01 11.01 10.08 10.08 6.78 30.331 0.24 3.67 7.80 7.80 23.44 23.44 12.52 12.52 11.58 11.58 7.64 30.378 6.00 12.65 7.80 7.80 23.44 23.44 12.47 12.47 11.60 11.60 7.64 30.0-35.0 0.87 9.47 8.60 8.60 23.88 23.88 13.61 13.61 14.60 14.60 8.05 36.807 0.74 1.93 10.20 10.20 25.70 25.70 15.43 15.43 18.00 18.00 9.82 35.0-40.0 0.21 3.77 10.41 10.41 25.81 25.81 15.69 15.69 17.51 17.51 9.91 40.0-45.0 0.39 1.40 12.32 12.32 27.52 27.52 18.01 18.01 21.07 21.07 11.22 46.522 0.18 0.51 13.80 13.80 28.48 28.48 19.92 19.92 21.80 21.80 11.00 45.0-50.0 0.31 2.60 14.52 14.52 29.27 29.27 20.09 20.09 21.85 21.85 12.65 '50.0-55.0 0.51 2.11 16.63 16.44 31.61 31.61 21.61 21.44 24.53 24.53 12.35 55.437 0.80 1.84 18.00 18.00 33.20 33.20 22.28 22.31 24.69 24.69 12.47 58.433 1.58 5.92 21.00 20.30 34.21 34.21 22.52 21.38 23.20 23.20 13.21 55.0-60.0 0.48 1.19 19.20 18.50 34.00 34.00 22.41 21.62 22.38 22.38 12.85 60.976 0.45 2.29 21.40 20.40 25.31 20.16 18.42 16.93 23.10 23.10 13.22 62.973 1.50 3.50 21.40 19.80 25.86 21.27 18.60 16.75 23.20 23.20 13.18 60.0-65.0 0.17 0.69 21.40 19.82 25.88 21.14 19.78 17.01 23.22 23.22 13.12 65.0-70.0 0.22 0.55 21.40 16.93 25.96 21.72 25.59 17.04 29.75 25.06 10.31 70.331 0.39 0.82 21.50 15.20 21.76 17.71 24.45 16.70 26.30 23.00 8.32 70.0-75.0 0.17 0.51 21.01 15.15 22.48 17.02 25.98 17.02 30.94 23.20 6.85 76.515 0.20 0.51 19.30 15.50 53.96 50.39 26.28 12.17 35.46 23.77 4.38 77.041 0.24 1.02 18.50 14.20 26.48 20.00 15.26 9.20 26.88 18.39 4.25 79.015 0.48 1.10 16.60 13.90 21.79 17.07 33.22 15.44 19.26 10.18 3.70 75.0-80.0 1.16 3.46 18.25 14.22 31.83 21.53 21.35 11.38 30.71 16.75 4.12 80.0-85.0 1.93 16.00 17.16 12.76 12.84 10.67 22.59 17.13 15.05 0.00 3.95 85.0-90.0 4.43 16.13 19.34 10.52 49.06 19.66 37.64 11.70 46.63 0.00 0.79 90.0-95.0 4.22 14.54 20.02 7.86 70.89 0.00 49.44 0.00 16.99 0.00 0.02 97.702 5.96 15.49 15.80 5.20 29.91 0.00 28.50 0.00 0.70 0.00 0.00 95.0-100.0 1.79 5.26 16.12 5.60 34.41 0.00 52.90 0.00 36.16 0.00 0.00

102.572 4.38 15.93 8.80 0.00 15.10 0.00 0.01 0.00 0.00 0.00 0.00 103.191 3.18 10.92 7.60 0.00 14.90 0.00 0.00 0.00 0.00 0.00 0.00 100.0-105.0 3.63 10.30 9.04 0.00 18.18 0.00 0.00 0.00 0.00 0.00 0.00

The fluxes are given in 10 9 cm -2 s -x units, while the cross-section values have to be multiplied by 10 -x8 crn 2. The information presented in this table, except the H20 and O cross-section values, was obtained from Torr et al. [1979] and Tort and Tort [1985]; the H20 cross sections were taken from Cairns et al. [1971] and Katayama et al. [1973], and the O cross sections were taken from Samson and Pareek [1985].

model of the water vapor radiative cooling effect assuming no equilibrium between the ortho and para states. Crovisier [1984] presented a cooling rate curve computed using the GEISA spectroscopic data bank. Crovisier's [1984] results indicate that in the dense part of the coma the radiative cooling is much less important than $himizu [1976] estimated (by a factor of 10 at T = 30 K, a factor of 3 at T = 100 K). The following expression is an analytic approximation to Croois- let's [1984] results (combined with Huebner's [1985] optical depth correction):

Qrad(H20) = -4.4 x 10-22Tw3'35nw exp (--'CiR) T<52K

(48b)

Qrac!(H20) = --2.0 x 10-2øYw2'47J% exp (--'•'I10 T>_52K

Very recently, Marconi and Mendis [1986] have proposed a pioneering idea to consider the trapping of infrared thermal radiation by collision-dominated water vapor. They took into consideration the excitation of rotational/vibrational levels of water molecules by the dust-generated thermal radiation. A large fraction of this elevated intei'nal energy is then transformed via collisions into translational energy. This additional

energy source results in significantly increased gas temperatures in the region of adiabatic expansion and also in larger gas terminal velocities (however, dust terminal velocities are hardly affected). These effects will be discussed in more detail in section 4.

3.2. Photochemistry • The evaporating volatiles leaving the nucleus undergo nu-

merous physical and chemical processes. The physical processes were briefly reviewed in section 3.1, and in this section the photochemical processes are discussed. The freshly evapor- ated molecules, called parent molecules, are rapidly photo- dissociated or photoionized, and therefore most of the chemical kinetics of cometary atmospheres involve the resulting highly reactive radicals and ions.

Model calculations of the photochemistry need to start with an assumed parent molecule composition at the surface, followed by a self-consistent calculation of the densities of all absorbing species and solar flux intensity as a function of the radial distances from the surface. Most of the absorption of the solar radiation is believed to take place at the surface or in the dense region near the surface, where the parent molecules


TABLE 4. Incident Solar UV Flux Data (105 nm< 2 < 200 nm) and Photoabsorption Cross Sectio•ns

(1)rain, (1) .... rr(H20), rr(CO2), 109 cm -2 s-1 109 cm -2 s-1 10-x8 cm 2 l0 -18 cm 2

105.0-110.0 3.07 7.08 5.76 110.0-115.0 0.70 1.77 8.80

115.0-120.0 2.78 7.24 4.58 121.567 300.00 927.27 5.90 120.0-125.0 5.20 13.152 7.44

125.0-130.0 4.10 12.52 7.18

130.217 1.16 2.42 7.30 130.486 1.19 2.49 6.80

130.603 1.29 2.71 6.80 133.453 1.96 4.53 4.80

133.566 2.68 6.2{) 4.70 130.0-135.0 6.18 14.37 5.20 139.376 1.40 3.54 0.90 135.0-140.0 6.18 18.86 1.75 140.277 0.98 2.48 0.80 140.0-145.0 9.53 25.17 0.60 145.0-150.0 15.44 35.18 0.95 154.820 4.24 10.75 2.50 150.0-155.0 25.84 51.17 1.80

155.077 2.19 55.59 2.50 156.100 1.10 2.55 2.60

155.0-160.0 37.83 68.13 3.10 160.0-165.0 55.85 104.76 4.40

165.720 4.09 9,46 4.90 165.0-170.0 162.22 209.63 4.80 170.0-175.0 230.73 319.92 3.80 175.0-180.0 374.24 461.90 2.00 180.0-185.0 609.76 776.33 0.30

185.0-190.0 779.67 955.95 0.013 190.0-195.0 1078.23 1332.81 0.0018 195.0-200.0 1662.21 2026.47 0.00

20.08 30.58

0.92

0.07

0.11 2.99 0.70 0.75

0.75 ...

...

...

...

...

,..

...

...

. .

The information presented in this table was obtained from H. E. Hinteregger (private communication, 1985), Hudson [1971], and Nic- olet [ 1984].

dominate. Therefore it appears at first that it should be relatively easy to calculate the spectral intensity of the solar flux as a function of the distance from the comet surface once a

parent gas composition is selected. However, multiple scattering of the solar radiation and thermal reradiation by the outflowing dust grains means that complex and self-consistent radiative transfer calculations are necessary, in general, to obtain reliable flux intensities [l-lellmich, 1979, 1981; Weiss- man and Kieffer, 1981, 1984; Marconi and Mendis, 1982, 1983, 1984, 1986]. Such detailed radiative transfer calculations are especially important in calculating the entergy budget of the nucleus itself and regions near the surface. The net result of these processes is to increase the amount of radiative energy reaching the comet and thus leading to increased gas and dust production rates and outflow velocities. However, for photochemical calculations only the ultraviolet radiation with wavelength of less than about 200 nm is important, and in this wavelength region it is reasonable to assume that the only effect of the dust is that of an absorber [cf. Mendis et al., 1985].

The calculation of the dissociating and ionizing solar flux (2 < 200 nm) requires a knowledge of the number densities of the neutral constituents, ns(r ), as a function of radial distance r, the absorption cross section of these constituents as a function of wavelength, asa(2), and the spectrum of the unattenuated solar radiation 1©(2). In terms of those quantities, the solar flux at any distance r is given by

1(2, r) = I o•(2) exp [-,(2, r)] (49)

where

,(2, r)= • o'sa(.•) dr' ns(F) (50)

here dr' is the incremental distance along the path from the radial position in the direction toward the sun.

The photodissociation, PdS(r), and photoionization, PiS(r), rates for species s can be calculated at a given radial position if the solar flux at that position and the dissociation asd(,•) and ionization asi(2) cross sections are known'

© PaS(r) = ns(r ) d2 1(2, r)asa(2) (51)

© ,PiS(r) = ns(r) d2 1(2, r)asi(2) (52)

There is quantitative information available on the spectrum of the unattenuated EUV solar radiation at wavelengths below 185 nm; most of these data were obtained by spec- trophotometers carried aboard the Atmosphere Explorer (AE) satellites C, D, and E [Hinteiegger et al., 1973; Hinteregger, 1977]. Hinteregger [1981] selected the period of July 13-28, 1976, as the reference period representative of solar conditions at minimum activity for cycle 21 (R z = 0; F•0.7 = 68), The detailed reference spectrum of EUV irradiance for wavelengths below 200 nm, corresponding to solar minimum conditions, is given in terms of 1659 wavelength increments and is available on a magnetic tape from the National Space Science Data Center as file number SC# 21REFW. H. E. Hinteregger (per- sonal communication, 1985) also provided a reference spectrum corresponding to a period of highest solar activity, No- vember 1979 (R: = 302; F•0.7 = 367); this spectrum is referred to as F79314. For most aeronomic calculations, such spectral resolution, represented by 1659 wavelength increments, is not necessary; therefore Torr et al. [1979] and Tort and Torr [1985] merged the data sets, for wavelengths below 105 mm, into 37 wavelength intervals. Torr et al. [1979] also provided, appropriately averaged, absorption and ionization cross sections for N2, 02, O, CO2, and CO corresponding to these 37 wavelength increments; the ones relevant for cometary atmospheres are presented in Table 3. The solar flux data for wavelengths of less than 105 nm for both solar cycle maximum and minimum are also given in Table 3, along with H20 cross sections, which were compiled from various sets of experi- mental data [cfi Hudson, 1971]. Fluxes in the wavelength region 105-200 nm for the July 1976 and November 1979 conditions are given in Table 4, along with corresponding H20 and CO2 absorption cross sections.

The discussion, up to this point, has been limited to the reference spectra of July 1976 and February 1979; however, what one needs in most cases is the solar irradiance for a

given date, which means that some appropriate scaling of the reference flux is necessary. The most straightforward approach to such scaling is based on certain measured variables associated with key EUV emissions. Hinteregger [1981] recom- mended that two classes of emissions should be considered:

K - 1 for which the key index is the emission at 2 = 102.6 nm (H Lye) and K = 2 for which the emission at 2 = 33.5 nm (Fe XVI) is used as the appropriate index. Thus an estimate of the solar flux at wavelength 2 is given by the following expression:

I(•) = lrcf(•)[1 + Cx(R K -- 1)] (53)

where RK is the ratio of flUX intensity at the desired time to that of the reference value (• = 102.6 nm or 33.5 nm for K = 1


or 2, respectively) and C• is the effective contrast ratio provided with the reference flux. Data on these key emission features, as well as a few more selected lines and intervals, covering the time period of July 2, 1977, to December 31, 1980, are also available from the Nationai Space Science Data Center under file number SC # 210BS.

This scaling approach, of course, only works for time periods for which these key EUV parameters are provided. For periods when such direct information is not available, some other easily available parameter indicative, at least in- directly, of EUV activity needs to be used. The solar full disk flux at 2800 MHz (10.7 cm), given routinely in the form of daily values, F•o.?, has been the most widely accepted and used as an index of solar EUV activity. The following relation for the ratio of the solar flux on a given day to that of the reference value was suggested by Hinteregger [1981]:

l(•)/lrer(I[ ) = B o + Bi{(F,o.?)8,a- 71.5)

+ B2{F,o.?- (F,o.?)8,a + 3.9) (54)

where F•o.7 is the daily value of the 10.7-cm flux (10 -22 W m -2 Hz -a) and (Fao. 7)8a a is the 81-day average value of the 10.7-cm flux. Hinteregger [1981] provided best estimates for the parameters Bo, B•, and B 2 for 15 selected lines and intervals. Two sets of these B parameters were obtained by fitting the available data base. Bo was set to unity for one set, which means that the calculated solar flux reduces to the reference value for July 1976 when the appropriate Fao.7 values are

TABLE 5. The B Parameters Associated With Equation (57) for Solar UV Flux Variation Adjustments

it, nm B o B• B 2

16.8-19.0 1.0 0.01122 0.00256 1.110 0.01026 0.00251

19.0-20.6 1.10 0.03123 0.00590 1.509 0.02678 0.00567

20.6-•5,5 1.0 0.02391 0.00711 ,

1.184 0.02228 0.00708 25.5-30.0 1.0 0.03830 0.01259

1.196 0.03656 0.01254 "30.4" 1.0 0.00605 0.00325

0.952 0.00647 0.00326 51.0- 58.0 1.0 0.00964 0.00255

1.286 0.00714 0.00242 58.4 1.0 0.01130 0.00438

1.290 0.00876 0.00425 59.0-60.6 1.0 0.01101 0.00244

1.311 0.00830 0.00230 102.6 1.0 0.01380 0.00500

1.310 0.01106 0.00492 33.5 1.0 0.59425 0.38i 10

-6.618 0.66159 0.38319 121.6 1.0 0.01176 0.00306

1.046 0.01136 0.00605 28.4 1.0 0.22811 0.11638

0.731 0.23050 0.11643 20.0-20.4 1.0 0.04 562 0.00918

1.726 0.03928 0.00885

17.8-18.3 1.0 0.01272 0.00290 1.169 0.01225 0.00282

16.9-17.3 1.0 0.00616 0.00178 1.018 0.00600 0.00177

The information in this table was provided by H. E. Hinteregger (private communication, 1985).

TABLE 6. Solar Cycle Minimum Photodissociation Frequencies (Rates) at 1 AU

Mean

Dissociation Excess

Parent Dissociation Threshold, Frequency, Energy, Molecule Products nm 10 -6 s- • eV

H 2 H(ls) + H(2s, 2p) 84.48 0.034 0.44 CH C + H 358.99 1.2 x 104 0.48 OH O + H 282.30 18. 4.6

C2 C + C 203.0 0.17 12. CN C + N 160.0 4.0 6.9 CO(X•; +) C + O 111.78 0.28 2.6

C(xO) + O(xO) 86.34 0.035 2.3 CO(a3H) C q- O 243.18 72. 2.2 N2 N + N 127.04 0.66 3.4 NO N q- O 191.0 2.2 1.8 02 O(3p) q- O(3p) 242.37 0.060 8.3

O(3p) q- O(•D) 175.9 4.2 1.3 O(•S) + O(•S) 92.3 0.04 1 0.82

SO S + O 231.4 620. 0.62 NH 2 NH + H 300.0 2.1 6.4 H20 H + OH 242.46 10. 3.4

H 2 + O(t D) 177.0 1.4 3.5 HCN H + CN(A2IIi) 192.0 13. 4.3 H2S HS + H 317.0 3.3 x 103 0.77 CO2 CO(XXZ +) + O(3p) 227.5 0.017 1.7

CO(XtZ +) + O(•D) 167.1 0.92 4.3 CO(a3I'I) + O 108.2 0.28 0.20

N20 N 2 + O(XD) 340.7 1.0 2.8 N2 + O(X $) 211.5 4.9 6.8

OCS CO + S(3p) 399.0 17. 2.7 CO + S(•D) 291.0 54. 1.9 CO + S(X$) 212.0 30. 2.1 CS + O(3p) 182.0 0.069 0.13 CS + O(XD) 141.0 6,3 0.85

SO 2 SO + O 221.0 190. 0.49 S + O 2 207.0 58. 0.68

CS 2 CS(XIZ +) + S(3p) 277.8 320. 0.77 CS(X•E +) + S(•D) 221.1 22. 0.28 CS(a3H) + S(3p) 157.1 4.7 0.69 CS(AXIl) + S(3p) 133.7 52. 0.92

NH 3 NH2 + H 279.8 170. 1.8 NH(a xA) + H2 224.0 4.0 1.7 NH + H + H 147.0 2.0 2.1

C2H 2 H + C2H 230.6 10. 3.2 H 2 q- C2 200.6 2.7 3.1

H2CO H 2 + CO > 700. 160. > 2.1 H + HCO 334. 84. 0.37 H + H + CO 275.0 32. 3.

HNCO NH(c•II) + CO 354. 15. 5.1 H + NCO(A2•:) 253. 14. 4.1

CH,• CH3 + H 277. 1.2 5.9 CH2(a•A 0 + H 2 237.3 5.5 5.2 CH + H 2 + H 137. 0.5 1.9

HNO 3 OH + NO 2 598. 210. 4.3 C2H,, C2H 2 q- H 2 720. 24. 6.2

C2H2 + H + H 196. 23. 1.7 CH3OH H2CO + H 2 •700. 250. 6.5

CH 3 q- OH • 315. 13. 4.4 HO2NO 2 HO 2 + NO 2 1340. •330 4.2

OH + NO 3 730. •330. 3.4 CH3CHO CH 4 + CO -,• 700. •9.8 3.7

CH 3 + HCO 350.1 24. 0.87 CH3CO + H 337.5 22. 0.43

C2H 6 C2H,, q- H 2 874.3 3.7 9.0 CH 3 + CH3 322. 0.88 7.4 C2H 5 q- H 290. 3.3 6.8 CH 4 + CH 2 272.6 2.2 6.1

From W. F. Huebner (work in preparation, 1986) and Mendis et al. [1985].

substituted. Bo was not forced to unity for the other set, which resulted in a somewhat better overall fit (higher correlation coefficients) to the data base, but in this case the reference flux is not recovered exactly for the July 1976 period. Both sets of


TABLE 7. Some Assumed Compositions for the Frozen Gases in the Nucleus (%)

Composition Number

Species 1 2 3 4 5 6 7 8 9 10

H20 CH4 NH 3 CO2 CO

H2CO N2 HCN

CH3CN NH2CH 3 H2C3H2 C2H2 HCOOH

CH3OH NO

02 C2H,, HCN

H2N2H2 HC•H

61.110 30.450 8.440

55.560 ...

11.110

33.330 ...

53.340 48.890

2.220 11.110 11.110 11.110

33.330 28.890

48.890

11.110

11.110 ...

28.890 ß

43.004 33.004 61.127 74.036 62.433 13.464 1.650 5.094 ..- 0.001 0.080 0.094 0.007 ... 0.002

12.057 ..- 1.698 7.404 24.886 2.8!4 ... 13.583 4.970 12.428

22.105 25.931 0.082 4.462 .-.

5.225 6.129 15.282 ." 0.039 0.482 0.566 0.102 2.941 0.012 0.402 0.471 ..- 1.420 0.012 0.201 0.236 ......... 0.005 0.236 0.020 ... 0.125 0.161 0.094 0.071 2.231 0.031 ß .- 17.445 ......... ß .. 14.144 0.815 ...... ...... 1.019 ......

...... 1.019 ......

...... 0.061 .-. 0.031

...... 0.020 ......

......... 1.826 --.

......... 0.710 -.-

Reproduced from Huebner [1985]' for details, refer to the original reference.

B parameters are given in Table 5. It should be noted that if an estimate of the solar flux over wavelength intervals not covered by the B parameters is desired, one needs to estimate the two key wavelength intensities at • = 33.5 nm and • = 102.6 nm, using equation (54), and then use equation (53) to scale the reference flux as necessary.

A useful quantity for many aeronomic calculations is the frequency at which a given process (e.g., photodissociation) takes place owing to the unattenuated solar flux outside the atmosphere. Such photodissociation frequencies can be calculated from the available solar flux and cross-section infor-

mation and are independent of the atmospheric densities. W. F. Huebner (work in preparation, 1986) calculated a wide variety of these frequencies for solar cycle minimum conditions; the ones most relevant for cometary atmospheres are given in Table 6. Table 6 also gives the energy threshold for the given dissociation and ionization processes in terms of wavelength. (The following simple relationship allows this information to be transformed from wavelength, /•, to eV if so desired: E(eV) = 1239.8/;t(nm).)

There have been no direct in situ measurements of the neu-

tral gas composition and abundances in the coma. Estimates of the total densities and composition have been based on remote optical observations and guesses have been based on present ideas on how and where comet nuclei were formed. Table 7, reproduced from Huebner [1985], summarizes the variety of suggested composition ratios in the nucleus. In order to get an accurate description of the parent molecule densities in the atmosphere, one needs to correctly model the sublimation/evaporation processes followed by a self- consistent solution of the coupled continuity, momentum, and energy equations, which is such a complex undertaking that only limited studies of this type have been carried out to date (see sections 2 and 4). A zero-order approximation to the parent molecule distribution in the inner coma can be obtained by (1) selecting an initial composition ratio from Table 7, (2) assuming a 1/r 2 density variation, and (3) selecting an appropriate total gas production rate. Estimates of gas pro-

duction rates as a function of heliocentric distance, for a number of comets of interest, are given in Tables 11-14.

The dissociated and ionized molecules and atoms partici- pate in a wide variety of chemical processes; some of these reactions lead to certain important constituents, which are not immediately apparent from the parent molecules (e.g., H30 + as the major ionic component). We give here some representative examples of these various processes, and in Table 8 the best available rate coefficients of interest, for calculations of cometary atmospheres, are summarized.

Up to this point the discussions only dealt with photodissociation and photoionization; respective examples of these are

H20 + hv(/l _< 242.46 nm)=• OH(X2I-[) + H(2S) (55)

H20 q- hv(/• < 98.3 nm)=• H20 + q- t; (56)

Photodissociative ionization, an example of which is shown below, is also important in a number of circumstances'

H20 + hv(,• _< 66.5 nm)=,, H: + O* + e (57)

Three-body recombination is another process which is important in certain cases, specifically when the total density is high'

O + O + M =,, 02 + M q- 5.12 eV (58)

Dissociative recombination of molecular ions generate dissociation products, which often have relatively high kinetic energies and/or are in an excited state:

O2 + + e =• O(•D) + O(3P) + 4.98 eV (59)

An example of atom-atom (ion) interchange, which often leads to the formation of a vibrationally excited molecule, 02(v < 3), is

O(3p) + OH(X2II)=:- H + O 2 + 0.72 eV (60)

One important aspect of processes such as (55)-(59) is that the energy carried by the dissociation or reaction products can


largely go into heating the neutral and/or ionized gas, at least in the collision-dominated region of a cometary atmosphere.

4. DUSTY GAS FLOW IN THE NEAR-NUCLEUS REGION

4.1. Governing Equations

It was recognized as early as the mid-1930s that gas outflow plays an important role in cometary dust production [Orlov,

TABLE 8. Chemical Reaction Rates of Potential Importance in Cometary Atmospheres

Reaction

Rate

Coefficient, cm-3 s-1

H20 + O(1D)• OH + OH H20 + H2 O+ --• H3 O+ + OH H20 + OH + --• H30 + + O

--• H20+ + OH H20+H +--•H2 O+ +H H20+O +--•H2 O+ +O

H20 + CO2 + --• He O+ + CO 2 --• HCO2 + + OH

H20 + CO +--• H20 + + CO --• HCO + + OH

H20 + N2 + --• H2 O+ + N2 OH + OH--, H20 + O

OH + O--, 0 2 + H OH + CO--, CO 2 + H

OH + H2 O+ --• H3 O+ + O OH + OH + --, H20 + + O

OH+O +--•OH + +O 402 + +H

OH+CO +-•OH + +CO --• HCO + + O

OH+N2 +•OH + +N 2 H + + O--• O + + H

H 2+H20+--•H3 O+ +H H 2 + N 2 --• N2H + H

H 2+OH +--•H2 O+ +H H 2 + O+--• OH + + H O+H+--•H+O +

O+H2 +--•OH + +H O+H3 +--•OH + +H 2 O+N2 +-•NO + +N

---.0 + +N 2 CO 2 -•' OH + --• HCO2 + + O

--, HCO + + 02 CO 2+H +--•HCO + +O CO 2 -•- O + '•O2 + -•- CO

CO 2 + CO + --• CO 2 + + CO CO 2 + N2 + --• CO2 + -¾ N 2

CO + OH-, CO 2 + H CO + H2 +'-*cO + + H 2

--• Clio + +H

CO + Ha +-•cHO + + H 2 N 2+OH +•N2 H+ +O

N 2 + O + • NO + + N N 2+H 2+4N2 H+ +H N2 + H3+-•N2 H+ + H2

CH,• + H2 O+ --• Ha O+ + CH 3 CH,• + H + --, CH3 + + H 2

•CH,, + +H CH•, + He + --* CH• + + H

--• CH,• + + H 2 --• CHa + + H 2 + H

CHa, + Ha +--•CH• + + H I CH,• + O + -• CH,• + + O

CH,• + OH + --• H30 + + CH 2

2.2 x 10 -1ø 2.05 x 10 -9

1.3 x 10 -9 1.6 x 10 -9 8.2 x 10 -9 2.3 x 10 -9 1.7 x 10 -9

5 x 10 -1ø 1.7 x 10 -9

9 x 10 -1ø 1.6 x 10 -9 4.2 x 10-12

2.2 x 10 -11 exp (117/T) 9.30 x 10 -13

6.9 x 10 -1ø 7 x 10 -1ø

3.6 x 10 -1ø 3.6 x 10 -1ø 3.1 x 10 -1ø 3.1 x 10 -1ø 6.3 x 10 -1ø

7.96 x 10 -1ø 1.4 x 10 -9

1.73 x 10 -9 1.05 x 10 -9 1.58 x 10 -9 7.96 x 10-1ø 1.00 x 10 -9 8.00 x 10-1ø 1.40 x 10-1ø 1.00 x 10 -11

1.6 x 10 -9 5.4 x 10 -1ø

3 x 10 -9 1.1 x 10 -9 9.6 x 10 -1ø 5.5 x 10 -1ø 9.3 x 10-13

6.44 x 10-1ø 2.16 x 10 -9 1.70 x 10 -9

1.1 x 10 -9 1.20 x 10-12 2.00 x 10 -9 1.70 x 10 -9 1.30 x 10 -9 2.28 x 10 -9 1.52 x 10 -9 1.14 x 10 -1ø 1.41 x 10 -9 2.28 x 10 -9 2.40 x 10 -9 1.00 x 10 -9 3.00 x 10 -11

The information in this table was taken from Giguere and Huebner [1978], Huebner and Giguere [1980], Mendis et al. [1985], and W. F. ,Huebner (work in preparation, 1986).

1935]. In early treatments of the gas-dust interaction it was assumed that the dust drag coefficient was independent of the gas parameters and that the gas velocity was constant in the dust acceleration region [Whipple, 1951; Weigert, 1959; Do- brovolskii, 1966; Huebner and Weigert, 1966]. This constant velocity assumption was dropped in the late 1960s. A two- component treatment of the gas-dust interaction was published almost simultaneously by Probstein [1968], Brunner and Michel [1968], and Shulman [1969], assuming that the heavy dust grains have no thermal motion and collide only with gas molecules. These authors pointed out that the gas mean free paths were much larger than the dust particle dimensions, and consequently, the gas flow could be considered to be free molecular relative to the dust component. In Probstein's [1968] dusty gas dynamic treatment, the traditional gas energy conservation equation was replaced by a combined dust-gas energy integral. He assumed a single characteristic dust size (a = 0.5 #m) and neglected any external dust or gas heating in the interaction region. It was later pointed out by Shulman [1972], and Hellreich and Keller [1980] that the dust grains are significantly heated in the inner coma by multiple scattered solar radiation. Furthermore, calculations of Hellreich [1981] and Gombosi et al. [1983] have also demonstrated that using realistic dust size distributions rather than Probstein's single-size approximation produces significantly different results.

The radiative transfer problem in a dusty cometary coma is far from simple and has been investigated by a number of groups using different methods [Hellreich, 1979; Hellreich and Keller, 1981; Marconi and Mendis, 1982, 1983, 1984]. Most recently, Marconi and Mendis [1986] have investigated the effects of the dust thermal radiation on the energetics of neutral gas and found that this radiation can significantly increase H20 internal energies. Due to neutral-neutral collisions, this elevated internal energy can also increase the random energy of the neutral gas, thus resulting in much higher minimum neutral gas temperatures in the gas-dust interaction region than predicted by previous calculations.

Modeling efforts have shown that the spatial extent of the gas-dust interaction region is limited to less than ,-• 30 Rn [cf. Keller, 1983; Mendis et al., 1985]. Gas particles typically spend about 100 s in this region; this time scale is too short for any significant change in the gross chemical composition of the gas. There seems to be a general concensus in the litera- ture that a single-fluid dusty hydrodynamical approach is ade- quate for describing the dynamics of the gas-dust interaction region (cf. the most recent review of Mendis et al. [1985]). In this section the equations that govern the evolution of gas and dust parameters in a spherically symmetric water vapor dominated inner coma are summarized.

Using this approximation, equations (40) can be simplified, giving the following equations which describe the mass, momentum, and energy conservation of the neutral gas:

O(Ap) O(Apu) + - 0 (61a)

Ot

O(Apu) Ot

c•(Apu 2) } a-•-• + A •rr = -AF,d (6lb)

a(1 a-• • Apu: + 1

7-1 Ap + •rr Apu3 + Apu 7--1

= A(Qext- Qgd) (61c)


where p is gas mass density, p is gas pressure, u is gas velocity, A is area function (for spherical geometry A = r2), Fga is momentum transfer rate from the gas to the dust, Qga is energy transfer rate from the gas to the dust, Qcxt is external heat source rate (photochemical heating and infrared cooling). In the innermost coma, where the gas-dust interaction takes place, Qext can be approximated with relatively simple expressions; however, at larger cometocentric distances this term becomes fairly complicated.

By neglecting collisions between dust particles one can obtain the following equation of motion for an individual grain in a radially expanding cometary atmosphere [cf. Wei- •]ert, 1959]:

4• dVa 1 4• G '•' paa3 '• = a2•t • CDP(U -- Va)2 --•- paa3 7 Mcøm (62)

Here Va is the velocity of a spherical dust particle with size a and bulk density Pa, CD is the drag coefficient, and u and p represent the gas radial expansion velocity and density, respectively. The second term on the right-hand side describes gravitational attraction from a cometary nucleus of mass Mco m (G is the gravitational constant).

In the presence of an external radiation field the energy balance equation for a dust particle is [Probstein, 1968]

4•t paa3Ca dTa '•' • = 4•ra2qa + /ra2t•absJr -- 4/ra2t•½maTa 4 (63) where Ta is dust particle temperature, qa is the gas-dust heat transfer rate per unit surface area, Jr is the radiation energy flux, Ca is the dust specific heat, and eabs and /•em are the dust absorption and infrared reemission emissivities, respectively. Finally, the dust size distribution function fa must obey the following continuity equation:

d(Afa) %(0

dt - _•ta3pa 5(r- Rn) (64) where % is the production rate (per unit surface area) of dust particles with radius a. Equation (64) assumes that once a dust particle is released from the surface, it will be conserved in the coma: we neglect icy grain sublimation or particle fragmenta- tion processes.

4.2. Gas-Dust Momentum and Ener•7y Transfer

In Probstein's [1968] formulation of drag and heat transfer to a spherical dust particle in its free molecular environment, grains are assumed to reflect the gas molecules diffusively around the surface, to have a perfect thermal accommodation, and to have a constant surface temperature. In this approximation the free molecular drag coefficient based on the pro- jected area for a sphere with diffuse reflection is given by

2;zø'STaø'5 2Sa 2 + 1 CD = 3saTO. 5 + •- exp (--sa 2)

4Sa 4 + 4Sa 2-- 1 + 2Sa4 err (Sa) (65)

Here T is the gas temperature; sa is the gas-dust relative Mach number:

u-- V a mO.5 (66) sa = (2kT) ø'5 Using the free molecular approximation, Probstein [1968]

also calculated the gas-dust heat transfer rate per unit dust grain surface area'

qa = p(U- VaXrrec- ra)C H (67)

where the "recovery" temperature, Tre c, and the heat transfer function, Cn, are the following (a correction made by Kita- mura [1986] is incorporated into Tree)'

Tre•- 27 + 2(7- 1)sa 2

_1 -øY3 exp (--sa 2) erf -1 0.5 + Sa 2 -{" San

CH--

(68)

7+1 k

7-1 8ms a 2 [n-ø'Ssa exp (--Sa2)+(O.5+Sa 2) err (sa) ]

(69)

The gas to dust momentum and energy transfer rates can be obtained by integrating over all dust sizes'

mdafa 4• dVa (70) Fga = •- Paa3 dt

Qgd famdafa (4• Paa3Va alva ) = Jao [ 3 • + 4•a2qa (71) Here ao represents the minimum dust size in the mantle, assumed to be identical to the size of elementary building blocks found in chondritic aggregate micrometeorites collected in the stratosphere [Fraundorff et al., 1982]. The maximal liftable dust particle size, am, can be expressed from equation (62):

3CDZUoutRn 2 am = (72)

8GM•omPam

This formula can be determined from the equations published by Wallis 1-1982], but the corresponding formula (72) obtained by Wallis 1-1982] was incorrect by a factor of 2.

4.3. Steady State Solutions Without Radiative Transfer

In early treatments of the gas-dust interaction [Whipple, 1951; Weigert, 1959; Dobrovolskii, 1966; Huebner and Wei- gert, 1966] it was assumed that gas particles had mean free paths much larger than the grain size and collided elastically with the dust grains. These assumptions resulted in a constant Co = 2 value. Furthermore, these calculations assumed steady state and a constant gas flow velocity u. Using these assumptions and neglecting cometary gravity, one can obtain a transcendental expression for the dust velocities (which is a simplified version of Dobrovolskii's [1966] solution):

Va +'n(1--•)=Xa(1---•) where the constant Xa can be expressed as

(73)

Assuming perfect thermal accommodation and free molecular drag, Probstein [1968] published new and more sophisticated expressions for the CD drag coefficient and qa heat transfer functions (see equations (65)-(69) in this review). He also solved the coupled gas-dust equations using a single characteristic grain size. This solution represented a major step for- ward, since the traditional gas energy conservation equation

3CDzR n Xa - (74)

16apau


was replaced by a combined gas-dust energy integral. It should be noted that more recent radiative transfer calcula-

tions have shown that the situation is much more complicated and that the interaction with the external radiation field significantly violates the original Probstein [1968] assumption [Hellreich, 1979; Hellreich and Keller, 1981; Weissman and Kieffer, 1981, 1984; Marconi and Mendis, 1982, 1983, 1984, 1986].

In a steady state self-consistent dusty gas dynamic treatment the gas continuity, momentum, and energy equations can be combined to yield the following first-order differential equation for the gas velocity:

du u (•--' Fgd Y -- 1 Qgd -- Qext.) (75) dr 1 -- M 2 P 7 pu

where M is the gas Mach number and A' represents the spatial derivative of the area function. Probstein [1968] neglected Qext and examined the flow behavior at various cometocentric dis-

tances. He pointed out that the gas pressure must tend to zero at infinity and at the same time the velocity must remain finite, so

lim M = oo (76)

At large distances from the nucleus there is no gas-dust interaction; so Fed--• 0 and Qgd--• 0 and

du u A'

lim - M2 > 0 (77) r-.o• dr A

Probstein [1968] also proved that the numerator of equation (75) is negative near the nucleus and has its maximum absolute value at the surface. A limiting case of this dusty gas flow model occurs if the dust/gas production rate ratio (Z) tends to zero. In this case, the presence of the dust does not affect the gas motion. This is the equivalent of seeding at the source an already established steady, radial, isentropic inviscid com- pressible source flow. In this case, the gas flow properties can be obtained from the well-known solution for a supersonic source in which the Mach number at the nucleus (source) surface is taken to be 1 in order to satisfy the appropriate boundary conditions at infinity and at the surface. On the other hand, a Z > 0 value indicates strong gas-dust interaction at the source resulting in a lower gas outflow velocity, Prob- stein [1968] concluded that the gas flow had to be subsonic at the surface and supersonic at large cometocentric distances.

It is evident that M - 1 is a singular (sonic or critical) point of equation (75). This means that an additional constraint needs to be imposed: the solution must properly traverse the sonic point, or in other words the numerator has to vanish at M = 1. The equation has usually been solved numerically by employing a time-consuming "shooting" method, which forces the gas velocity function through the sonic point on a trial and error basis [Hellreich, 1979; Hellreich and Keller, 1981; Gombosi et al., 1983; Marconi and Mendis, 1982, 1983, 1984]. Recently, Gombosi et al. [1985] have published a new method to determine the gas and dust parameters at the sonic point, thus making the numerical solution much simpler. This method will be discussed in the next section.

As has already been mentioned, the physical solution of equation (75) is a transonic accelerating gas flow. At first it is not obvious which external effect "accelerates" the gas and enables it to become supersonic. In order to visualize this effect, Gombosi et al. [1985] have applied their reservoir out-

3 ' I • I ' I ' I ' I LU•- n• • x=o •:::) Z -

LU •- •

0 • I • I • I • I :>.5 :>.7 :>.9 3.1 3.3 3.õ

R {KM)

Fig. 8. Variation of the effective area function with cometocentric distance for various dust to gas mass production rate ratios, X (taken from Gombosi et al. [1985]).

flow model (cf. section 2.2) and introduced an effective area function defined by

= (78) Aef f dr P 7 pu

By adopting the effective area function, the steady state gas equation can be written in the following simple form:

dtt tt Aef f' dr - 1 - M 2 Aef f (79)

This equation is just the steady state equation describing the free, unrestricted discharge of gas into vacuum through a nozzle having an area function of Aeff(r) [see Zucrow and Hoffman, 1976]. When no external sources influence the gas discharge, the effective area function is identical with the geo- metrical area function, A. In this case, the gas flows out of the stationary reservoir starting at the sonic velocity. The situation changes dramatically when the outflowing gas has had a chance to drag away dust. Figure 8 (taken from Gombosi et al. [ 1985]) shows the variation of the effective area function with cometocentric distance for various dust to gas ratios (Z). In this particular calculation, To = T• = 200 K, Rn = 2.5 km, and 7 = 4/3 values were used. It can be seen that for Z = 0, Aef f has its minimum value at the surface, and hence the gas velocity will start at the sonic velocity at the surface. For larger values of Z, Aeff first decreases, then reaches its minimum value at the sonic point, and then increases. This shows that the dust interaction {the dominant external effect near the nucleus) can in effect be visualized by using the concept of a Laval nozzle. First, the outflow geometry "narrows," and then it "opens up." In short, the presence of dust allows the gas to start at the nucleus with subsonic velocity and then go through a sonic point and reach supersonic velocities. This transition would have been impossible without the presence of dust. The situation strongly resembles the steady state solar wind equation, first solved by Parker [1958]. The gravitational effect of the sun for the solar wind corresponds to frictional forces between the dust and gas for the outflowing cometary gas.

Using a single characteristic dust size and conserving the combined gas-dust energy integral, Probstein [1968] produced numerical solutions to the steady state coupled gas-dust equations. More sophisticated calculations later confirmed the basic features of his results [I-lellmich and Keller, 1980; Gom- bosi et al., 1983; Marconi and Mendis, 1982, 1983, 1984, 1986].

In his calculations, Probstein [1968] introduced a dimen-


0.4 - 1.5

f 4.0 02 ' IO

O.OI O.I I IO IO0

1.6

1.2

1.0

0.8

0.6

0.4

-0.3

.8

X=O

0.2

0.01 0.1 I I0 I00

Fig. 9. The variation of outflow Mach number (Mi) and normalized dust terminal velocity with Probstein's dimensionless similarity parameter,/•a (taken from Probstein [1968]).

sionless similarity parameter characterizing the ability of a dust particle to adjust to the local gas velocity'

8 (C•,To) ø'5 aPa (80) J•a = 5 jR n where Cp is the gas specific heat at constant pressure and To is the surface temperature (Probstein assumed that T O was equal to the outflow temperature T•). The length l•aRn is analogous to a mean free path: for JlaRn << r the dust adjusts quickly to the gas velocity, while for JlaRn >> r the particle remains much slower than the gas.

Assuming that T•(Rn) iS identical to To, using y = 1.4 and C a = 5.8 X 10 6 erg/K/g values and making the Pa • 0, r a---} T--} 0 and V•--} u approximations, Probstein [1968] obtained an analytic expression for the upper limit of the velocity of small dust grains:

Va'lim--Uth y- 1 2(1 + Z) ø'5 1 + 2 Mi 2 + Z Cl•J (81)

where M i is the gas outflow Mach number. Probstein [1968] also published a set of normalized numeri-

cal solutions to the coupled gas-dust equations. His results are reproduced in Figure 9, where outflow Mach numbers (Figure 9a) and normalized dust terminal velocities (Figure 9b) are presented as a function of Jla. The various curves are marked with the values of the corresponding dust/gas mass production rate ratio, Z. $ekanina [1981] determined a semiempirical fit to these curves and obtained the following expression for dust terminal velocities:

, u 1 0.65 \ I • •a2/3') (1 + 0'38Zø'6) + 0'225/•aø'5 (82)

Modifying an earlier Sekanina [1979] fit to the Probstein [1968] solution, Wallis [1982] published a different expression for the dust terminal velocities'

V•.oo = u o o(0.9 + 0.4Z ø'5 + 0.5J1•ø'5) -• (83)

These semiempirical expressions are useful only when order- of-magnitude estimates are needed. They are only approximations to rescaled numerical results of a single dust size calculation, and their application to a description of a range of particle sizes has not been justified. Wallis [1982] pointed out an additional problem with equations (82) and (83): they give nonphysical results when both the dust mass loading and the /• similarity parameter are small (in this case V•,oo will be larger than Uoo).

A piecewise approximate solution for the dust terminal velocity distribution can be obtained by solving for the J?•-, 0 and the Jla >> 1 cases and constructing a function smoothly connecting the two limiting solutions. The result is

where

= 3Uoo 2 C•, ø'5 raø'5 + _ reffO.5 (85) Tmi n q- tcP a

Yef f = (86)

Here rmi n and Tc represent the minimum gas temperature in the gas-dust interaction region and the gas temperature at the sonic point, respectively. We note that expression (84) does

I.O

UcD

O. OI

-- ß - ß • o...,. o,..... o.... % -

\\ 2 '

-- ..... •• I SEKANINA

2 PRESENT WORK

:3 SEKANINA-WALLIS

4 DOBROVOLSKll ß NUMERICAL SOLUTION

I I I

O. I 1.0 I0.0 I00.0 I000.0

•Q

Fig. 10. Comparison of various analytic approximations of dust terminal velocities with steady state numerical solutions obtained with a Hanner type dust size distribution function.


not contain the dust/gas mixing ratio, Z, in an explicit form. On the other hand, u oo and z depend on Z; so the dust loading does have an influence on the solution. In a first approximation, however, numerical results indicate that in the presence of external gas heating (but neglecting infrared trapping) uoo can be approximated as u• • 1.1Uth.

Figure 10 shows a comparison between the various terminal velocity approximations and the results of a numerical solution of the coupled gas and dust equations. When evaluating expressions (73), (82), and (83), numerical constants listed in Table 9 were used' T O was taken to be 200 K. From the numerical solution, uoo = 0.635 km/s, Uo,,t = 0.281 km/s, u½ = 0.300 km/s, T c= 146 K, z= 1.05 x 10 -5 g/cm2/s, Z=0-27, Tmi , = 5 K, and T a = 277.8 K values were obtained at 1 AU heliocentric distance. When calculating the Dobrovolskii [1966] solution, we used a CD = 2 value in equation (73).

It can be easily shown (and seen in Figure 10) that the asymptotic behavior of all four solutions is similar; Va,• fla -ø'5 as fla--• oo. At this point it should be noted again that equations (82) and (83) were obtained using 7 = 1.4, C a = 5.8 x 106 erg/K/g values and a single characteristic dust size

(a = 0.5 #m)' so the comparison has to be done with appropriate caution. It should be pointed out that equation (86) gave a better than 5% approximation to a set of numerical solutions over a wide range of Z values (Z was varied between Z = 0 and z= 5).

Hellmich [1979], Hellmich and Keller [1981], and Gombosi et al. [1983] investigated the effect of a realistic dust size distribution on dust terminal velocities. In their comparative study, Gombosi et al. [1983] first solved the coupled gas-dust equation system with a realistic dust source distribution function, assuming a spectral index N - 4.2 (cf. equation (16)) and using 30 dust size classes. In the next step they solved the same equation system 30 times using single dust sizes and renormalized the solutions to give a dust terminal velocity distribution. The results are shown in Figure 11, which shows that by using a realistic dust size distribution the dust terminal velocities decrease about 20%.

4.4. Approximate Supersonic Steady State Solutions

The coupled gas-dust differential equation system was solved during the last several decades by many authors using different approximations. Early calculations using mainly ana-

TABLE 9. Adopted Comet Nucleus Parameters

Value

Solar energy flux at 1 AU, erg/cm2/s Solar UV flux at 1 AU, photon/cm2/s Gas mean molecular mass

Gas specific heat ratio Latent heat of vaporization, erg/mol Dust grain albedo Dust grain IR emissivity Dust grain specific heat, erg/g/K Minimum dust size, •m Maximum dust size, cm P0, g/cm3 p•, g/cm 3 a 0,/•m a•,/•m M

N

1.35 x 10 6 3.50 x 10 •2 18

4/3 5 x 10 TM 0.1

0.9

8X 10 6 0.1 1

3.0 2.2

0.1 2.0 12

4.2

1.5

1.0

qC.,o 0.5

0.0

•.. SINGLE DUST

-- -'"%- SIZE .,•LUTIONS

- SIZE DISTRIBUTION i i i

o. i.o io.o ioo.o

a(•m) Fig. 11. Comparison of dust terminal velocities obtained with

single dust size solutions and using a realistic dust size distribution (taken from Gornbosi et al. [1983]).

lytic or semiempirical methods were summarized in the preceding section. Following Probstein's [1968] approach, more sophisticated numerical models were developed by several groups. Hellmich [1979], Hellmich and Keller [1981], Marconi and Mendis [1982, 1983, 1984, 1986], and Gombosi et al. [1983] published steady state solutions, using "trial and error" methods to get through the 0/0 type singularity at the sonic point. In order to obtain a "physical" transonic solution, these authors had to "prescribe" the smooth behavior at the critical point. These calculations assumed a given surface gas density, temperature, and dust mass loading ratio and varied the surface Mach number until a smooth transonic solution was ob-

tained to equation (75). The surface Mach number turned out to be a function of the dust mass loading ratio, Z: for instance, in the case of a Halley type comet the Mach number was close to 1 when there was only a negligible amount of dust in the coma, while for a Z = 1 value, the outflow Mach number was about 0.5. On the other hand, this time-consuming shooting method was too expensive to be used extensively with a large number of dust sizes and realistic external energy sources.

Recently, Gombosi et al. [1985] have published a time- dependent treatment of inner coma dusty hydrodynamics, in which steady state results are a natural byproduct. A time- dependent treatment of the gas-dust interaction process does not result in singular differential equations; therefore transonic solutions evolve naturally with time. Applying the reservoir outflow model (section 2.2), these authors benchmarked their steady state solutions against the results of a "shooting method" type numerical code and found a reasonable agreement between the two solutions.

Based on fluid dynamics considerations, Gombosi et al. [1985] suggested a simple approximate recipe for the flow parameters at the critical point. They pointed out that between the nucleus surface and the sonic point (typically located at a few tens of meters from the surface) the gas flow is isentropic within a few percent. In this case, the following relations hold between the gas quantities in the reservoir and those at the sonic point for an isentropic discharge of perfect gas from a stationary reservoir through a converging- diverging (Laval) nozzle:

..... ½ (87) \po/ Here w is sound velocity, while the subscripts "0" and "c" refer to conditions in the reservoir and at the sonic point, respectively. The dimensionless quantity ½ can be expressed in terms


of the gas production rate z'

(Aozwo•'•- •'/'•+ •' = (88)

A new set of dusty gas dynamics calculations was carried out for this review paper using the time-dependent code developed by Gornbosi et al. [19853. The sublimating and surface temperatures were both taken to be 200 K; there were 12 logarithmically spaced dust sizes between 0.1 and 100/•m; and the gas molecular mass, gas specific heat ratio, and dust specific heat were taken to be 18, 4/3, and 8 x 106 ergs/g/K, respectively. For the dust bulk density and source distribution functions, expressions (20) and (21) were used. Ten calculations were performed using different Z values distributed between 0 and 5. The initial condition was an empty coma in each calculation. At t = 0 the gas started to flow out of the reservoir and began to carry away dust grains, then the system was allowed to evolve until steady state was reached. In the next step, analytic approximations were fitted to the numerical values of gas production rate (z), outflow velocity (11out) , sonic point radius (Re) and gas terminal velocity (uo•), as a function of the dust to gas mass production rate ratio, Z. It was found that the following expressions describe the numerical results within a couple of percent in the Z = 0-5 interval'

z = z0(1.17 - 0.073Z) (89)

0.62Uth Uout = (90)

(1 + 0.28Z)

R• = R,(1 + 0.0246Z •'s44) (91)

where z0 has already been defined (expression (6)). Substituting expressions (89) and (9•1)into equation (88), one obtains

F 1.17 - 0.073Z '¾•-•)/{•+ • ½ = •77 (1 + •-.•4•-•ZF'•4)2'J (92) In order to check the validity of the isentropy assumption in the subsonic region, one can calculate the sonic point gas parameters from equations (87) using expression (92) for ½ and compare the results with the numerical solutions. The two sets of sonic point parameters agreed to within a few percent giving confidence in these fast steady state calculations which start from the sonic point (one can call this the "supersonic approximation").

Previous numerical calculations have shown that the tem-

perature of the dust particles adapts very quickly to the local equilibrium temperature defined by

(4% + CabsJr• ø'25 Ta = •' •'e• '? (93) where Jr is the local external radiation field and qa is the gas to dust heat transfer rate (see equation (67)). As the thermal time constant of a dust grain is ra • 10 4 a 2 (where % is measured in seconds and "a" is given in centimeters), most dust particles reach their equilibrium temperature by the sonic point. In these new supersonic approximation calculations the dust temperature at the sonic point was defined by expression (93) for all dust sizes.

The total dust mass production rate is z a = •z; the source dust size distribution function (Na) is assumed to follow a Hanner type distribution [cf. Hanner, 1983] given by Na = Non(a). The normalization factor No can be obtained from the

following equation:

;•z = da z a = -•- N O da a3pan(a)Va (94) o o

The last quantity to be defined at the sonic point is the dust velocity distribution, Va. Dust particles are practically stationary at the surface and have Va << u velocities at the sonic point, typically located • 100 m from the nucleus. In a steady state, low dust velocity approximation, where cometary gravity is neglected, equation (62) can be expressed as

dVa2 3 u2( _•)2 dr -- 4• PCa •pa 1-- (95) In this approximation, Ca is independent of the dust size (but varies with r). Assuming that the gas velocity does not change drastically in the small subsonic region, one can make the u • u c approximation and expand about Va/uc, which leads to the following equation:

2Q_•) 3 3(Va• 4 C* 1 (Va•2._}_ -[- -[- .... (96) \Uc/ \Uc/ where the C* constant can be determined from the

dAeff(r) I = 0 (97) dr r = R•

condition, thus ensuring the appearance of the 0/0 type singularity at the sonic point. Equations (87)-(97) uniquely define the gas and dust parameters at the sonic point. The du/dr value at the sonic point can be obtained using the L'Hospital relation:

du (d2Aeff/dr2)r=Rc = (98)

dr r= • (d/dr)(1 - M 2 r=Rc

Equation (75) can now be integrated outward from the sonic point using the initial gas and dust parameter values obtained from equations (89)-(97) and the (98) boundary condition.

The results of a series of calculations starting from the sonic point were compared with the solutions obtained by using other steady state methods. Several test cases were calculated with the well-established shooting method, as well as with the time-dependent code. The results of the three independent methods were compared at various cometocentric distances between the sonic point and 3000 km. The typical deviation between the various results was a few percent' it never exceed- ed 10%. This comparison has proved that the supersonic approximation solution is an accurate enough approximation for describing the steady state radial evolution of the gas and dust parameters in the acceleration region. This new technique has a big advantage' it is our experience that on the average it is 3 to 10 times faster than the more traditional shooting method (depending on the complexity of the photochemical and radiative energy source terms). The big saving is that it is not necessary to solve the dust equation of motion in the subsonic region, where the radial grid size is very small owing to the large acceleration of small grains.

Some results of a supersonic approximation calculation are shown in Figure 12. In this calculation, T O = T• = 200 K, Ca=8 x 106 ergs/g/K, 7=4/3, Z=0.27, d=l AU values were used, and there were 12 dust sizes logarithmically spaced between 0.1 /•m and 100/•m. Photodissociation heating and $himizu [1976] cooling were the external energy sources con-

o .J i,i

IOOO

IO0

I0


0.42/z 1.3/a

4.2/2-

42•z

T I -- 200K 0.27

0.43

T (K) •To M

loo

lO

-1

0.1

- 0.01

10

I • 0.1 •

I I0 r I00 1 10 r 100 1 10 I

R-• Rn a b c

Fig. 12. (a) Steady state gas and dust radial velocity, (b) gas temperature, and (c) Mach number profiles obtained with a Hanner type dust size distribution function.

r lOO

Rn

sidered. Inspection of Figure 12 shows that within about 2 Rn the gas approaches 80% of its terminal velocity value, while the dust accelerates somewhat more slowly. Within a cometocentric distance of 10 R n, both the gas and dust almost reach their terminal velocities. The dust temperature (not shown) rapidly converges to its asymptotic value, which is

( l•absJr • 0'2 5 T, = •,4aemff/ (99) Adiabatic cooling decreases the gas temperature to a few degrees at •-, 30 R,; then heating due to photodissociation re- verses the trend. The gas Mach number (naturally) has an opposite behavior:it first increases to about 11 and then starts to decrease.

It was recently shown by Marconi and Mendis [1986] that the trapping of infrared thermal radiation in the collision- dominated inner coma significantly influences the gas temperature profile. According to their new results the gas temperature decreases less than a factor of 2 from its surface value in

the gas-dust interaction region, and the revised temperature minimum is about 100 K. Marconi and Mendis [1986] have also demonstrated that IR radiation trapping increases the gas terminal velocity by 30-40%, while the dust terminal velocities remain practically unchanged.

The new code incorporating the supersonic approximation was applied to calculate new sets of inner coma gas and dust parameters for four comets of special interest: Halley, Giacobini-Zinner (G-Z), Kopff, and Wild-2. Comets Halley and G-Z are being visited by spacecraft, while Kopff and Wild-2 have been discussed as potential targets of a comet rendezvous mission. Table 9 summarizes those model parameter values which were assumed to be the same for all four

TABLE 10. Main Parameters of Some Selected Comets

Halley G-Z Kopff Wild-2

Orbital period, years 76.10 9.600 6.460 6.390 Semimajor axis, AU 17.95 3.516 3.467 3.441 Eccentricity 0.9673 0.7075 0.5441 0.5402 Perihelion distance, AU 0.5871 1.028 1.580 1.582 Aphelion distance, AU 35.31 6.004 5.353 5.299 Nucleus radius, km 3.00 1.00 1.45 1.45 Dust/gas mass ratio 0.2 1.0 0.3 0.3

comets, while parameters varying from comet to comet are shown in Table 10. The parameter values were taken from N. Divine (private communication, 1985) and Divine et al. [1986]. Gas production rates were calculated at each heliocentric distance from the light curves using Newburn's [1981] empirical method. The cometary light curves were taken from Divine et al. [1986] (Halley), Brandt and Yeomans [1985] (G-Z), Seka- nina [1984] (Kopff), and Divine [1985] (Wild-2). There is observational evidence that both the Comet Halley and Giacobini-Zinner production rate predictions were too low (P. D. Feldman, private communication, 1985). Tables 11 and 12 might still be useful by providing "intelligent guesses," which can be appropriately scaled. Next the gas production rate per unit surface area was derived using equation (38), and a single surface temperature was obtained using equation (37) (assuming a A = 0 mantle thickness). After all these preliminary steps the gas and dust parameters were calculated at the sonic point, and the coupled differential equation system was solved in the supersonic region. The external energy sources were taken from Gornbosi et al. [1985]. The calculation was stopped at 300-km cometocentric distance, where the gas and dust became practically decoupled and both components ap- proached their terminal velocities. The results obtained for the four comets under consideration are summarized in Tables

11-14. The listed parameters are heliocentric distance (AU), surface temperature (K), total gas production rate (molecules/s), average gas mass production rate per unit area, dust production rate, maximum liftable dust size, gas terminal velocity, gas number density at 300 km, gas temperature at 300 km, and dust terminal velocities for different grain sizes.

Inspection of Tables 11-14 reveals that calculated gas terminal velocities are practically independent of heliocentric distance. This result is in clear contradiction with Whipple's [1978] empirical law, which was derived from Bobrovnikov's [1954] paper summarizing observations of 57 different comets between 0.66 and 6.74 AU:

uoo = 0.535d -ø'6 (100)

where d is the heliocentric distance (measured in AUs). Similar results were obtained by Malaise [1970], who analyzed the Greenstein effect on the rotational levels of the CN (O-O) band for three different comets. Malaise [1970] carried out this very difficult analysis for four spectra obtained while the


TABLE 11. Comet Halley Parameters at r = 300 km

d, AU

-2.00 - 1.5000 - 1.0000 -0.7500 0.5871 0.7500 1.0000 1.5000 2.0000

Tsurf 183.20 188.49 195.32 199.98 203.97 206.11 202.50 196.93 192.36 Q•as 2.28E + 28 5.68E + 28 1.71E + 29 3.47E + 29 6.21E + 29 8.40E + 29 5.03E + 29 2.20E + 29 1.07E + 29 2gas 1.07E - 07 1.51E - 06 4.55E - 06 9.23E - 06 1.65E - 05 2.23E - 05 1.34E - 05 5.83E - 06 2.85E - 06 Zaust 1.21E - 07 3.02E - 07 9.10E - 07 1.85E - 06 3.30E - 06 4.47E - 06 2.67E - 06 1.17E - 06 5.70E - 07 ama x 0.56 1.42 4.35 8.93 16.14 21.94 13.01 5.60 2.70 u•a • 0.621 0.631 0.643 0.653 0.663 0.657 0.648 0.639 0.633 rtgas 3.70E + 07 9.06E + 07 2.68E + 08 5.36E + 08 9.44E + 08 1.29E + 09 7.81E + 08 3.46E + 08 1.70E + 08 T•a • 3.4 4.4 6.4 8.3 10.4 7.7 5.7 3.7 2.7 Va(0.13) 0.210 0.299 0.424 0.505 0.562 0.580 0.536 0.446 0.362 Va(0.24) 0.169 0.245 0.364 0.448 0.515 0.540 0.484 0.386 0.305 V,(0.42 ) 0.135 0.200 0.307 0.390 0.461 0.490 0.428 0.329 0.253 V,(0.75) 0.109 0.164 0.258 0.335 0.406 0.436 0.373 0.279 0.210 V,(1.33) 0.089 0.135 0.218 0.288 0.354 0.385 0.323 0.236 0.175 V,(2.37) 0.073 0.112 0.184 0.247 0.309 0.339 0.280 0.201 0.147 Va(4.22) 0.061 0.094 0.156 0.212 0.268 0.296 0.242 0.171 0.124 V,(7.50) 0.050 0.078 0.132 0.180 0.231 0.257 0.207 0.1 44 0.104 V,(13.34) 0.041 0.064 0.109 0.151 0.195 0.218 0.174 0.120 0.086 V,(23.71) 0.033 0.052 0.088 0.123 0.161 0.181 0.143 0.097 0.069 V,(42.17) 0.026 0.041 0.070 0.099 0.130 0.146 0.115 0.077 0.055 V,(74.99) 0.020 0.032 0.055 0.077 0.102 0.116 0.091 0.061 0.042 V,(133.35) 0.015 0.024 0.042 0.060 0.080 0.091 0.071 0.047 0.033 Va(237.14) 0.012 0.018 0.032 0.046 0.061 0.070 0.054 0.036 0.025 V,(421.70) 0.009 0.014 0.024 0.035 0.047 0.054 0.041 0.027 0.019 V,(749.90) 0.007 0.011 0.018 0.027 0.036 0.04 1 0.031 0.021 0.014 V,(1333.52) 0.008 0.014 0.020 0.027 0.031 0.024 0.016 0.011 V,(2371.38) 0.006 0.011 0.015 0.020 0.023 0.018 0.012 0.008 V,(4216.97) 0.005 0.008 0.011 0.015 0.018 0.014 0.009 0.006 V,(7498.95) 0.003 0.006 0.009 0.012 0.013 0.010 0.007 0.005

Read, for example, "2.28E + 28" as "2.28 x 1028."

comets were between 0.1 and 1 AU. The four data points were best fitted with a spectral index of 0.5. It was later pointed out by Delsermne [1982] that Whipple's [1978] empirical formula was also consistent with the d -ø'5 law and interpreted the data in terms of rapid thermalization of the CN radical in the inner coma, concluding that the gas temperature followed a T • d- • relation. Delsemme's [1982] final conclusion was that this radial dependence can be explained if one assumes that the gas heating is proportional to the solar radiation flux, while the cooling is mainly caused by infrared radiation cooling [Shintizu, 1976].

There is clearly a contradiction between the results of dusty gas dynamic calculations and the Whipple [1978]-Delsentnte [1982] empirical gas terminal velocity relation. One possible explanation is that the present dusty model of gas-dust interaction (which assumes perfectly spherical particles, total accommodation, etc.) needs major revisions. Another possibility is that the early photographic observations summarized by Bobrovnikov [1954] were biased by the dust component, while the uncertainty of Malaise's [1970] measurements is not known very well (these possibilities were mentioned by Del- sentme [1982] and Whipple [1982, private communication, 1985]). Calculated dust terminal velocities exhibit an approximately d -• dependence on heliocentric distance. The main reason behind this behavior of calculated dust terminal veloci-

ties is the variation of Prob}tein's [1968] accommodation parameter, fla-It is obvious from equation (80) that fia decreases with an increase in the gas production rate. On the other hand, a small fi• value predicts that the dust grain will quickly accommodate to the gas flow; consequently its terminal velocity will be higher than that of a particle characterized by a large fi• value. The conclusion is that the same dust grain will

have a larger terminal velocity when the comet is more active than in the case of a lower gas production rate.

The gas and dust parameters shown in Tables 11-14 can serve as initial values for gas and dust calculations in the outer parts of the coma, where gas-dust interaction is not important any more. On the other hand, chemistry, radiative transfer, ionization, solar wind interaction, etc., make the situation complicated in the decoupled region.

4.5. Radiative Transfer

In most of the earlier dusty gas dynamics calculations, interaction of the solar radiation field with dust grains and gas particles was neglected. Studying a large number of potential neutral gas and ion reactions and their contribution to the coma optical depth, Huebner and Giguere [1978] concluded that gas emission and absorption are unimportant outside the resonance bands and lines of the most dominant molecules

and atoms. They concluded that only the ultraviolet wavelength region might be optically thick owing to photolyric losses. On the other hand, there is recent indication [Marconi and Mendis, 1986] that infrared radiation trapping by the H20 molecules may also contribute significantly to radiation transfer in the coma.

The inner coma contains a large number of dust particles. A simple estimate based on an r -2 number density distribution of a more or less typical grain size (-• 1 #m) predicts that the optical thickness of the column above the surface is less than 0.1, causing only a marginal insulation effect. On the other hand, in the immediate vicinity of the nucleus the dust velocity is very small (see Figure 11a); consequently just above the surface the dust particle density varies much faster than r-2. This "pileup" of slow dust grains may cause important effects,


TABLE 12. Comet Giacobini-Zinner Parameters at r = 300 km

d, AU

-2.00 - 1.5000 - 1.2500 1.0300 1.2500 1.5000 2.0000

Tsurf 189.26 194.88 198.03 199.32 196.20 192.96 187.63 Qgas 4.77E + 27 1.18E + 28 1.92E + 28 2.33E + 28 1.45E + 28 8.72E + 27 3.63E + 27 Zgas 1.140E - 06 2.820E - 06 4.580E - 06 5.570E - 06 3.470E - 06 2.090E - 06 8.690E - 07 Zdust 1.140E -- 06 2.820E - 06 4.580E - 06 5.570E - 06 3.470E - 06 2.090E - 06 8.690E - 07 ama x 3.08 7.74 12.66 15.42 9.53 5.69 3.34 Ugas 0.672 0.678 0.682 0.687 0.682 0.677 0.671 ngas !.02E + 07 2.50E + 07 4.04E + 07 4.87E + 07 3.15E + 07 1.85E + 07 7.78E + 06 Tgas 4.0 5.6 6.9 9.1 7.4 6.0 4.2 Va(0.13) 0.203 0.289 0.344 0.368 0.313 0.259 0.181 Va(0.24 ) 0.162 0.237 0.286 0.309 0.258 0.210 0.144 Va(0.42) 0.130 0.193 0.236 0.256 0.211 0.170 0.115 Va(0.75 ) 0.104 0.158 0.194 0.212 0.173 0.138 0.092 Va(1.33) 0.085 0.130 0.161 0.176 0.143 0.113 0.075 Va(2.37 ) 0.070 0.108 0.135 0.148 0.119 0.094 0.062 Va(4.22) 0.058 0.090 0.113 0.125 0.100 0.078 0.051 Va(7.50) 0.048 0.075 0.095 0.104 0.083 0.065 0.042 Va(13.34) 0.039 0.061 0.078 0.086 0.068 0.053 0.034 Va(23.71) 0.031 0.049 0.063 0.069 0.055 0.043 0.027 Va(42.17) 0.025 0.039 0.049 0.055 0.043 0.034 0.022 Va(74.99) 0.019 0.030 0.038 0.043 0.034 0.026 0.017 Va(133.35) 0.015 0.023 0.029 0.033 0.026 0.020 0.013 Va(237.14) 0.011 0.018 0.022 0.025 0.020 0.015 0.010 V•(421.70) 0.008 0.013 0.017 0.019 0.015 0.011 0.007 V•(749.90) 0.006 0.010 0.013 0.014 0.011 0.009 0.006 Va(1333.52 ) 0.005 0.008 0.010 0.011 0.008 0.007 0.004 Va(2371.38) 0.004 0.006 0.007 0.008 0.006 0.005 0.003 Va(42!6.97) 0.003 0.004 0.006 0.006 0.005 0.004 0.003 V•(7498.95) 0.002 0.003 0.004 0.005 0.004 0.003 0.002

Read, for example, "4.77E + 27" as "4.77 x 1027."

leading to significant modifications of nuclear outgassing. This potential effect was first investigated by Keller and Hellmich [Hellreich, 1979; Hellreich and Keller, 1980, 1981] and later by Weissman and Kieffer [1981, 1984] and Marconi and Mendis [1982, 1983, 1984, 1986].

Any steady state solution to this problem has to satisfy boundary conditions at two separate surfaces' at large cometocentric distances the radiation field is the unattenuated

solar radiation, while at the nucleus the net absorbed energy supports gas and dust production, which in turn determines the optical properties of the coma. The retarded nature of the problem makes self-consistent solutions rather difficult to obtain. Based on a spherically symmetric dust distribution produced by an isothermal nucleus, Hellreich [1979] developed an iterative model to calculate steady state radiation transfer in the inner coma and to determine the energy input to the nucleus. The calculation was started with an optically transparent coma. In the first step, gas and dust production rates and the resulting dusty gas flow parameters were calculated. Next, Hellreich [1979] recalculated the radiation field in the coma and at the nucleus, thus obtaining a new surface temperature. In this step, direct absorption and thermal emission as well as multiple scattering effects caused by dust particles were taken into account. The process was repeated until a converged solution was obtained. A similar technique was later employed by the San Diego group, too [cf. Marconi and Mendis, 1982, 1983, 1984, 1986].

Recently, a comprehensive two-band, three-stream radiative transfer model was developed by Marconi and Mendis [1984, 1986]. They separately considered the variation of the UV flux, which is responsible for the major photolytic processes and the visual and near-infrared radiation, which is mainly

responsible for heating the dust grains and the nucleus. In their model, Marconi and Mendis [1984] considered six constituents' nonthermal hydrogen (produced in photolyric processes), thermalized hydrogen, heavy neutrals (heavier than H), ions, electrons, and a single-size dust population. Photolytic reaction rates determine the mass source rates of various

species and contribute to the momen.tum and energy source terms. The nonthermal hydrogen was .treated in a semikinetic way, the dust motion was approximated by the usual kinetic model, while the behavior of the remaining components was described by hydrodynamic equations. An ambipolar electric field was determined from the electron momentum equation assuming charge neutrality and the absence of net electric current. It was also assumed that ion and neutral velocities

were equal' in effect this assumption grossly overestimates the ion-neutral drag effect at larger radial distances [Cravens et al., 1984' Kb'r&mezey, 1984]. Electron heat conduction and, neutral molecule radiative cooling [Shimizu, 1976] were also taken into account, contributing to the coma energetics.

Figure 13 (taken from Marconi and Mendis [1984]) shows the radial profile of the normalized radiation energy flux. The calculation was carried out for a Halley class comet at 1-AU heliocentric distance. In the figure the contributions of the direct radiation, JmR, the multiple scattered flux, JMS, and the diffuse radiation, JBB (produced by the thermal radiation of dust grains), are shown together with the total flug, Jxox. In- spection of Figure 13 shows that the direct solar radiation decreases monotonically toward the nucleus, and practically no direct radiation reaches the surface. The multiple scattered flux peaks right above the surface, somewhat closer to the nucleus than the dust thermal radiation. The radiation flux

scattered within the inner dust coma is partially trapped, and

694 GOMBOSI ET AL.' DUST, NEUTRAL GAS MODELING OF COMETARy INNER ATMOSPHERES

TABLE 13. Comet Kopff Parameters at •' = 300 km

d, AU

-1.75 1.58 1.75 2.00

T•urf 189.40 193.39 195.98 194.00 190.37 Qg, s 1.45E + 28 2.77E + 28 4.16E + 28 3.05E + 28 1.70E + 28 zg,s 1.65E - 06 3.15E - 06 4.73E - 06 3.47E - 06 1.94E - 06 Zaus, 4.95E - 07 9.46E - 07 1.42E - 06 1.04E - 06 5.81E - 07 am,,, 3.19 6.16 9.31 6.80 3.76 ug, s 0.667 0.673 0.676 0.674 0.669 ngas 2.31E + 07 4.38E + 07 6.55E + 07 4.82E + 07 2.71E + 07 Tg,, 3.5 3.9 4.2 3.8 3.4 Va(0.13) 0.244 0.310 0.356 0.320 0.259 Va(O. 24 ) O• 197 0.255 0.297 0.264 0.210 Va(0.42 ) 0.159 0.209 0.246 0.217 0.170 Va(0.75) 0.128 0.171 0.203 0• 178 0.138 Va(1.33) 0.105 0.141 0.168 0.147 0.113 Va(2.37) 0.087 0.117 0.141 0.123 0.094 V•(4.22) 0.072 0.098 0.119 0.103 0.078 V•(7.50) 0.060 0.082 0.099 0.086 0.065 Va(13.34) 0.049 0.067 0.082 0.070 0.053 Va(23.71) 0.039 0.054 0.066 0.057 0.042 Va(42.17) 0.031 0.043 0.052 0.045 0.033 Va(74.99) 0.024 0.033 0.040 0.035 0.026 Va(133.35) 0.018 0.025 0.031 0.027 0.020 Va(237.14) 0.014 0.019 0.024 0.020 0.015 V•(421.70) 0.011 0.015 0.018 0.015 0.011 Va(749.90) 0.008 0.011 0.014 0.012 0.009 V•(1333.52) 0.006 0.008 0.010 0.009 0.007 Va(2371.38) 0.005 0.006 0.008 0.007 0.005 Va(4216.97) 0.003 0.005 0.006 0.005 0.004 V•(7498.95) 0.003 0.004 0.004 0.004 0.003

Read, for example, "1.45E + 28" as "1,45 x 10 TM"

it overcompensates the screening of the direct solar radiation. The total radiation flux reaching the nucleus is about 25% higher than the unattenuated solar radiation; consequently, the gas and dust production rates are somewhat larger than they would have been if the optical properties of the coma were neglected. This effect, however, strongly depends on the optical characteristics Of the dust grains. It was pointed out by Keller [1983] that the trapping of multiple scattered photons leads to flux enhancements of about 25% if pure olivine is considered for the material of dust particles with a high scattering albedo and negligible absorption (dielectric material). On the other hand, dust grains with a large absorption coefficient (such as magnetiLe) suppress multiple scattering. In this case, the grains are strongly heated and the emitted infrared diffuse radiation further enhances the energy input to the nucleus. The physical structure of the dust particles plays a very important role in the radiative transfer process: direct observations are needed to resolve uncertainties of present calculations.

The radial variation of gas and dust parameters obtained by the comprehensive model of Marconi and Mendis [1984] ex- hibits a very similar behavior to the profiles shown in Figure 12. This is understandable, because in that calculation the radiation fieldswithin t-he inner coma was assumed to be con-

stant, which turns out to be true within about 50%. This simplifying assumption does not cause significant changes in the global behavior of the solutions.

The newest development of the radiative transfer calculations is the first theoretical model describing infrared radiation trapping [Marconi and Mendis, !986]. They pointed out that the trapped infrared radiation represents only a negligible fraction of the total radiation energy density but can stlll sig-

nificantly influence the gas parameter profiles. The main idea behind the IR trapping is the recognition that most of the dust thermal radiation is emitted in the 1- to 20-#m wavelength range, where several rotational and vibrational transitions exist for the highly polar water molecules and have very large resonance absorption cross sections (~4 x 10-x,• cm 2 [Hueb- ner, 1985]). In a collisionless gas the resonant radiation is continuously absorbed and reemitted by the water molecules; in other words, it is trapped by the gas. Marconi and Mendis [1986] were the first to recognize that in the collision- dominated inner coma a large fraction of the rotational/vibrational excitation energy of water molecules can be transformed via collisions into translational energy, thus increasing the gas temperature. This new heating is the opposite of the radiative cooling [Shimizu, 1976; CrOvisier, 1984] in Which it is assumed that molecular collisions excite the water molecules

and the excitation energy is lost by the emission of infrared radiation. Marconi and Mendis [1986] derived an approximate formula to describe the combined effect of the Crovisier

[1984] cooling and the gas heating due to infrared radiation trapping:

QIR -' -- Co Tw•'nw exp (- Zw)

am + 9 x 10-3qe[1 - exp (-Zw)] da a2f,•T• 4 o

(101)

where QIR is given in units of erg cm-3 s-x. In expression (101), nw and Tw represent the water vapor concentration and temperature, respectively' qe is an efficiency factor (its value is around unity); and fa and T• are the differential dust number density and dust temperature, respectively. The infrared opti-

GOMBOSI ET AL.' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOgPHERES 695

TABLE 14. Comet Wild-2 Parameters at r = 300 km , .

d, AU

-2.00 -1.75 1.58 1.75

Tsurf 186.66 189.84 192.25 189.84 186.66 Qgas 9.15E + 27 1.56E + 28 2.31E + 28 1.56E + 28 9.15E + 27 Zga s 1.04E - 06 1.78E - 06 2.63E - 06 1.78E - 06 1.04E - 06 Zau•t 3.12E - 07 5.33E - 07 7.88E - 07 5.33E - 07 3.12E - 07 ama x 2.00 3.44 5.12 3.44 2.00 /•/gas 0.664 0.669 0.673 0.669 0.664 ngas 1.47E + 07 2.48E + 07 3.65E + 07 2.48E + 07 1.47E + 07 Tgas 3.8 4.3 4.8 4.3 3.8 Va(0.13) 0.203 0.252 0.291 0.252 0.203 Va(0.24) 0.163 0.204 0.239 0.204 0.163 V•(0.42) 0.130 0.164 0.194 0.164 0.130 V•(0.75) 0.104 0.133 0.159 0.133 0.104 Va(1.33) 0.085 0.109 0.130 0.109 0.085 Va(2.37 ) 0.070 0.090 0.108 0.090 0.070 V•(4.22) 0.058 0.075 0.091 0.075 0.058 V•(7.50) 0.048 0.062 0.075 0.062 0.048 Va(13.34) 0.039 0.051 0.062 0.051 0.039 V•(23.71) 0.031 0.04 1 0.050 0.04 1 0.031 V•(42.17) 0.025 0.032 0.039 0.032 0.025 V•(74.99) 0.019 0.025 0.030 0.025 0.019 V•(133.35) 0.015 0.019 0.023 0.019 0.015 Va(237.14) 0.011 0.014 0.018 0.014 0.011 V•(421.70) 0.008 0.011 0.013 0.01 ! 0.008 v•( 749.90) 0.006 0.008 0.010 0.008 0.006 Va(1333.52) 0.005 0.006 0.008 0.006 0.005 Vd2371.38) 0.004 0.005 0.006 0.005 0.004 V•(4216.97) 0.003 0.004 0.004 0.004 0.003 Va(7498.95) 0.002 0.003 0.003 0.003 0.002

Read, for example, "9.15E + 27" as "9.15 x 1027"

cal depth of water vapor, %, is defined by expression (50). The Co and b constants are the following:

C0 = 4.4 x 10 -22 T w < 52 K

C0=2.0x 10-2ø Tw>_52K

b = 3.35 T• < 52 K

b = 2.47 T w >_ 52 K

Inspection of expression (101) shows that when % >> 1, QiR

Z

z

z

z

N

o z

2.0

1.5

1.0

,,

i i i i i i i i i i i

I JBB •

0.5 l -• JDIR 0 20 40 60 80 IO0 120 140 160 180 200 2ZO

DISTANCE (Km)

Fig. 13. The radial profile of the normalized radiation energy flux. The various curves represent the direct solar radiation (JmR), the multiple scattered flux (JMS), the diffuse radiation (JBB), and the total flux (Jxox) (taken from Marconi and Mendis [ 1984]).

describes the infrared heating by the trapped radiation, while in the Zw << 1 case, QiR becomes the Shimizu [1976] cooling term.

Marconi and Mendis [1986] incor•porated the new infrared heating term into their dusty gas dynamics model. Their results indicate that infrared radiation trapping significantly modifies the radial gas parameter profiles. Figure 14a (taken from Marconi and Mendis [1986]) shows the radial variation of gas temperature. One can see that owing to the additional dust heating the temperature does not decrease below • 100 K, while in previous calculations the minimum temperature was only a few degrees. The additional energy source also significantly increases the gas terminal velocity (typically by about 50%). On the other hand, the infrared heating deposits s{gnifi- cant amounts of "new" energy at distances where only a rela- •tively small fractio•n of dust acceleration takes place; consequently the dust terminal velocities are only Slightly affected. Figure 14b shows two dust terminal veloci[y profiles.: one was obtained by neglecting the IR heating, and the other curve was calculated with the new gas heating term. Inspection of Figure 14b reveals that the velocity difference for particles larger than 0.1/•m is less than 10%.

4.6. Time-Dependent Models

Comets are highly variable celestial objects which exhibit spectacular changes with different time scales. The orbital motion results in variations with a characteristic time scale of 10 days or so' this effect usually can be treated in a q•asi steady state manner. In some comets, violent flareups jets, expanding halos, tail discontinuities, etc., may develop within minutes or hours, although their decay phase usually lasts considerably longer. These phenomena require a time-


250 , ,

T R

200 T K

150

I00 TK

50

o •o RADIAL DISTANCE (km)

a

0.7

0.5

,e, 0.:• E

>8 0.2.

o.I

0.07

i i _ _

- R n = :3 km - _ .

_ DUST IR HEATING _

_ • _ INCLUDED X = 0.3 -

- -

__ NO IR HEATING _

BY DUST _ -

z _

_

i I

I IO IOO

a (/•m) b

Fig. 14. (a) Radial dependence of rotational (TR) and translational (T s:) gas temperatures calculated with infrared radiation trapping (taken from Marconi and Mendis [1986]). (b) Dust terminal velocity profiles obtained with and without the trapped radiation heating.

dependent, multidimensional treatment in order to describe the temporal and spatial evolution in the near-cometary environment. Recently, Gombosi et al. [1985] have published the first time-dependent dusty hydrodynamical description of a cometary inner coma. The model describes a spherically symmetric transonic flow in the immediate vicinity of the nucleus where the gas-dust interaction dominates the dynamics. A more sophisticated two-dimensional calculation was published by Kitamura [1986], who solved the dusty gas dynamic equations for a long-lasting jet. Gombosi and Hordnyi [1986] con- structed a dusty hydi'odynb. mic-kinetic hybrid model to describe the evolution of dust halos.

Time-dependent inner coma calculations have several advantages over steady state models. One important technical advantage is that a time-dependent treatment of the gas and dust interaction process does not result in singular differential equations; consequently, transonic solutions evolve naturally with time.

In their calculations, Gombosi et al. [1985], Kitamura [1986], and Gombosi and Hortinyi [1986] simultaneously solved the partial differential equation system (61) describing the gas behavior with the dust equations (62), (63), and (64). The dust-gas interaction was described by Probstein's [1968] model, while the gas was heated by H20 photodissociation and cooled by the Shimizu [1976] infrared cooling term.

As a result of the time-dependent dusty flow calculations, Gombosi et al. [1985] found that in addition to the similarity type expanding gas halo lip, 1981], another type of distur- bance will also propogate outward in a cometary coma following an outburst on the nucleus. Next, Gombosi and Hortinyi [1986] calculated the evolution of gas and dust distributions following a spatially and temporally localized comet outburst using a dusty hydrodynamic--kinetic hybrid method. In the inner coma the time-dependent continuity, momentum, and energy equations of a dusty gas flow were solved assuming spherical symmetry within the 30 ø wide jet using 12 logarithmically spaced dust sizes following a Hanner type size distribution (with a surface spectral index of 4.2). Beyond 300 km a three-dimensional kinetic method [cfi Hordnyi and Mendis, 1985] was used to calculate dust grain trajectories. it was found that following the onset of the simulated comet outburst, a gas-dust blast wave propagates outward in the inner coma. About 60 min after the increased gas and dust production .was initiated at the nucleus a new equilibrium was reached in the inner coma. The most important feature of this new steady state is the significant increase of the gas pressure (Figure 15). These higher terminal velocity values resulted in increased apex distances for dust particles emitted during the outburst. Gombosi and Hordnyi [1986] concluded that comet outbursts may generate long-lasting distinct dust envelopes in front of the regular dust coma, which later propagate tailward and finally dissolve (Figure 16).. These type of dust envelopes were observed at several comets (cf. comet Donati).

Recently, a couple of groups started to work on multidimensional modeling of cometary jets. Sagdeer et al. [1985] have calculated the steady state shape of a long-lasting, localized jet assuming that after the passage of the initial blast wave there is no significant gas pressure gradient across the jet boundary. On the basis of this assumption, Sagdeer et al. [1985] concluded that the angular extent of the dust jet varies with cometocentric distance as r {•- •)/2.

Kitamura [1986] recently published the first time-dependent axisymmetric dusty gas jet calculation describing inner coma gas and dust distributions following a long-lasting, localized surface outburst. The jet profile at the surface was approximated by a Gaussian function with a half width of 10 ø. The

z 0.2

O.O

• ' ' •3600s ß t=O$

I , I

0.1 1.0 I0 I00

o (/.t.m)

Fig. 15. Dust terminal velocity distributions before and 60 min after the onset of the comet outburst (taken from Gombosi and Hordnyi [ 1986]).

GOMBOSI ET AL..' DUST, NEUTRAL GAS MODELING OF COMETARY INNER ATMOSPHERES 697

DUST SIZE = 0.42/•m

t:6h t=lSh

UNIT LENGTH =104km

Fig. 16. Snapshots of 0.42-#m dust particle distributions following a 30 ø wide subsolar outburst occurring at t = 0 (taken from Gom- bosi and Hordnyi [1986]).

main feature of Kitamura's [1986] numerical results is that the narrow outburst results in a conical jet. The physical reason for this conical jet formation is twofold: first, the surface gas pressure gradient initiates a lateral gas transport which quickly pushes the gas density peak to • 30 ø, and second, near the nucleus the dust grains attain a tangential velocity which is comparable to their radial velocities, thus depleting the dust population along the • = 0 ø line. Further away from the nucleus the dust particles lost most of their tangential velocities; consequently the modified dust structure "freezes" at a cometocentric distance of about 10 km. The result of these two

processes is a laterally varying dust/gas mass ratio, resulting in different loading effects. Figure 17 (taken from Kitamura [1986]) shows the gas velocity field. One can see that the sonic distance is further inside the jet than in the undisturbed region. It also can be seen that there is a dust loading peak near 30 ø , due to the lateral transport of dust grains in the immediate vicinity of the nucleus. An additional factor influencing the radial gas and dust velocity profiles is that the gas in the enhanced density region (around • = 30 ø) can adia- batically expand not only radially, but also tangentially. This means that only a smaller fraction of the gas internal energy will be transferred to the dust population than in the radially expanding gas regions. As a result of the combination of these processes, a slow, high-density dust jet is formed around • = 30 ø. Inside the jet the dust is faster than the ambient population; consequently the dust number density is smaller. Figure 18 (taken from Kitamura [1986]) shows steady state dust equidensity contours in the (r, •) plane for a 10 ø wide outburst occurring at • = 0 ø. One can see the formation of the dust jet wings at around 30 ø . On the other hand, owing to the increased dust velocities inside the jet cone, in this region the dust density is smaller than the ambient value.

Kitamura's [1986] results represent a significant improve- ment in modeling dusty gas jets. We expect the development of further interesting jet models following the Halley flyby projects, when (hopefully) more detailed observations will be available about the structure of inner coma dust and gas features.

5. SUMMARY DISCUSSION

In the next few months and years, we will learn a great deal more about both the nucleus and the dust/gas environment of comets through in situ measurements (ICE, VEGA, Giotto, Sakigake, and Suisei missions) and intense remote observations from the ground (International Halley Watch) as well as earth orbit (Astro 1, Space Telescope, etc.). These measurements will significantly advance our understanding of the physical and chemical processes controlling comets and their environment. Therefore the purpose of this review is not to be an encyclopedia of knowledge but to provide a description and summary of some of the most relevant basic physical and chemical processes along with the best present-day models which can be used as benchmarks to compare the new results with and then facilitate data interpretation and the resulting

THETA [DEG)

R

Fig. 17. Steady state gas velocity field following a long-lasting 10 ø wide subsolar comet outburst. The arrows represent the direction of the gas flow, while the solid line marked "sonic" represents the M = 1 curve (taken from Kitamura [1986]).


THETA {DEG)

BO

5O •o

o

7o

8o

9o

1oc oo

11o 1o

120 2O

140

15

3O

4O

50

i i i 0 50 100

R IKM)

Fig. 18. Steady state inner coma equidensity contours in the (r, •) plane for a Gaussian-shaped surface jet profile with a half width of 10 ø occurring at ½I) = 0 ø (taken from Kitamura [1986]).

advances in our knowledge. The history of solar system studies has continuously demonstrated the importance of the sym- biotic relation between theoretical studies and observations in

achieving progress. We hope that these reviews will make some contribution to the dramatic advances expected in the next few years.

Acknowledgments. We wish to thank H. L. F. Houpis, A. K6r6smezey, M. L. Marconi, and D. A. Mendis for numerous helpful comments and suggestions. This work was supported by NASA grants NAGW-15 and NGR-23-005-015 and NSF grants ATM-85- 08753 and INT-83-19732. Acknowledgment is also made to the Na- tional Center for Atmospheric Research, sponsored by the National Science Foundation, for computing time used in this research.

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(Received October 2, 1985; accepted February 28, 1986.)

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Dust and Neutral Gas Modeling of the Inner Atmospheres of Comets