REVIEWS
OF GEOPHYSICS,
VOL. 24, NO. 3, PAGES 667-700, AUGUST
1986
Dust and Neutral Gas Modeling of the Inner Atmospheresof Comets
T. I. GOMBOSI,
1 A. F. NAGY,ANDT. E. CRAVENS
SpacePhysicsResearchLaboratory, Universityof Michigan, Ann Arbor
This paper summarizesour present,preencounterunderstandingof the physical and chemical processescontrolingthe inner (r < 1000 km) region of cometary atmospheres.Specialemphasiswas attached
to compilinga self-consistent
set of governingequations.We are aiming this review at readerswho want
to understand
thepresentstatusof themantleandcomaregionsand/orwhowantto developnew,next
generation models which will be needed as the large volume of new observational data will become
available in the near future.
CONTENTS
Introduction
.............................................
Gas and dust production of cometary nuclei ................
Structure and compositionof cometary surfacelayers ......
Vaporization and gas production models ..................
Heat transfer in the mantle: governing equations ..........
Heat transfer in the mantle: approximate solutions ........
Theory of atmospheric processes ..........................
Transport equations and physical processes...............
Photochemistry ........................................
Dusty gas flow in the near-nucleusregion ...................
Governing equations ...................................
Gas-dust momentum and energy transfer .................
Steady state solutionswithout radiative transfer ...........
Approximate supersonicsteady state solutions ............
Radiative transfer
......................................
Time-dependent models ................................
Summary discussion .....................................
1.
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689
692
695
697
INTRODUCTION
The cometary atmosphereis a unique phenomenonin the
solar system.Owing to its negligiblegravity the tiny nucleus
of a few kilometers in diameter produces a highly variable,
One of the most important features influencing cometary
dynamicsis the "retarded"nature of gas and dust production.
The radiation reaching the surfaceand supplying energy for
sublimationmust first penetratean extensive,absorbingdusty
atmosphere.Any changein the gas and dust production alters
the optical characteristicsof the atmosphere,thus causing a
delayed (or "retarded") effect on the production rates themselves.The delay is causedby the finite radial transport time
of the outflowing dust and gas.
Since the mid-1970s,when primitive bodies became potential targets of deep-spacemissions,numerous review papers
dealing with various aspectsof comets have been published.
The latest and most comprehensiveexample of such a review
is that of Mendis et al. [1985]. During the last few years, two
extensivecollectionsof 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)understandingof cometary physics
and chemistry.
We decided to write
a nonconventional
and more tutorial
extensive
dustyatmosphere
of dimensions
rangingfrom 10'• to review using the presently available material; instead of re105km. In contrastto planetaryatmospheres,
the dimensions viewing individual papers and calculations, we try to conof the nucleusare much smaller than the scalelength of the centrate on the most recentdevelopmentsand compile a more
coma. In spite of the apparent differences,many of the physi- or less self-consistentpicture of the inner cometary region,
cal and chemical processesdominating planetary and com- giving due credit to the original papers but at the same time
etary atmospheresare similar. The large spatial extent and trying to provide an interrelated description of this region.
continual expansionof a cometaryatmosphereprovide a con- This approach will inevitably result in overemphasizingsome
venient tool for studyingthe time history of atmosphericproaspectsof modeling activities,especiallythose which are imcesses,and this could be useful when compared with similar portant for interfacing various regions in the near-nucleus
region, such as mantle thermal balance,gas outflow, and gasprocesses
in a gravitationallybound atmosphere.
Following Whipple's [1950] pioneering work, cometary dust interaction. We are aiming this review at readers who
nuclei are now thought to be chunks of ice, rock, and dust want to understandthe presentstate of modelsof the nucleuswith negligible surface gravity. As these "dirty iceballs" ap- coma interfaceregioh and/or who want to developnew, next
proach the sun,water vapor and other volatile gasessublimate generationmodelswhich will be neededas the large volume of
from the surfacegeneratinga rapidly expandinghuge cloud of new observationaldata will becomeavailable in the next year
dust and gas. The radiation field in the inner cometary atmo- or two. We are also providing many of the input parameters
sphereis expectedto be quite differentfrom the unattenuated which are necessaryto carry out these model calculations.
solar radiation as molecular absorption and emission,as well Finally, this paper does not attempt to review the presently
as multiple scatteringand thermal emissionby dust particles, available (mostly indirect) observations in any detail, especially sinceso much new information is expectedto become
modify the original spectrum.
available soon; it only refersto observationsin terms of justifying and/or checkinginputs to and resultsof model calcula•Also at Central ResearchInstitute for Physics,Budapest, tions. For an excellent "conventional
review" the readers
Hungary.
shouldturn to the Mendis et al. [1985] paper.
This paper concentrates on inner coma processesand
models. This is the region where dust and gas are produced
Copyright 1986by the AmericanGeophysicalUnion.
Paper number 6R0128.
8755-1209/86/006R-0128515.00
and accelerated
667
to their terminal
velocities.
In the inner coma
668
GOMBOSI
ETAL.: DUST,NEUTRALGASMODELINGOFCOMETARY
INNERATMOSPHERES
BOW
SHOCK
INNER
CONTACT SHOCK(?)
SURFACE(?)
/
/
SONIC
LINE
pu2
nkTIipu
•
pu2+nk
T
II
EUS
ioo
(kin)
PLASMA-NEUTRAL
DECOUPLING
HYDROGEN
CORONA
COLLISIONLESS
GAS
EXPANSION
SONIC
LINE
EUS
representation
ofthecometary
plasma(upperpanel)andneutralgas(lowerpanel)environment.
Fig. 1. Schematic
seriesof "sandbank"
models,whereinthe nuthe neutralgascan be considered
to be collisiondominated, the century-long
and consequentlythe applicationof a hydrodynamicap- cleuswasthoughtof as a diffusecloudof smallparticlestravof the two competing
proachis justified.A schematicview of the cometaryatmo- elingtogether.In a detailedcomparison
models
[Whipple,
1964],
it
was
pointed
out that even the
sphereis shownin Figure1. It canbe seenthat theregionof
icy conglomerate
modelis ableto explainbasiccominterestfor this paperis spatiallylimited(r < 1000km) and simplest
etaryfeatures,suchas (1) the repeatednatureof comaformahasa greatinfluenceon cometarydynamics.
In section2, physicaland chemicalprocesses
in the upper tion and gaslossfor many revolutionsof someshort-period
cometswith perilayersof the nucleuswill be summarized.
Section3 overviews comets,(2) the survivalof somesun-grazing
the basic aeronomicalphenomenain cometaryatmospheres helion distancesof less than 0.005 AU, (3) the splitting of
with a number of usefultables,providing information about cometarynuclei,and (4) the occurrenceof nongravitational
solar spectra,reactionrates,etc. In section4 the dustygas forces.Today thereseemsto be little doubt that cometnuclei
dynamics
in innercometaryatmospheres
is reviewed.
In this are monolithicdust and ice conglomerateswith a radius of a
1982;Donnand
section,tablessummarizingcalculatedgas and dust terminal few kilometers[cf. Wyckoff,1982;Delsemme,
velocitiesand densitiesfor cometsHalley, Giacobini-Zinner, Rahe, 1982; Mendis et al., 1985]. The availableobservational
Kopff, and Wild-2 at differentheliocentric
distances
are also evidence also indicates that the interior of the nucleus of a
cometis relativelyhomogeneous,
althoughthe surfaceis probpresented.
ablydifferentiated
andnonuniform[cf.Delsemme,
1982].
2. GAS AND DUST PRODUCTION OF
The chemicalcompositionand physicalstructureof the surCOMETARY NUCLEI
facelayersof a cometarynucleusare very importantfactors
affectingthe mass,momentum,and energyof the outflowing
2.1. Structureand Compositionof Cometary
gas-dust
mixture,aswellastherelativeabundances
of various
SurfaceLayers
gas molecules.The sublimatedgas molecules(often called
undergofrequentcollisionsand variousfast
Our presentunderstanding
of cometarynucleiis basedon parentmolecules)
processes
in the near-nucleus
region,thusproWhipple's[1950] pioneering"dirty snowball"idea according photochemical
to which the nucleus consists of a mixture of frozen volatiles
ducinga whole chain of daughteratomsand molecules[cf.
available
and nonvolatiledust. Whipple'shypothesisquickly replaced Huebner,1985]. As there are no directobservations
GOMBOSIET AL.: DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
about the structure and composition of cometary nuclei, or
even the near-nucleusregion, we have only indirect evidence
to help us to delineatethe main physicaland chemicalprocesseswhich control these continuouslyexpanding and dynamically evolving complex dusty atmospheres.Most of our
observationscome from spectrophotometryof cometary exospheresand tails. Cometary spectroscopyhas developedspectacularly during the last decade,but many of the parent moleculesand the secondaryatoms, radicals, and ions still remain
unobserved:we mostly detectthosespeciesthat can be excited
by solar radiation through a variety of fluorescencemechanisms [Delsemrne,1985a]. New and increasinglysophisticated
observingtechniquesrecentlymade it possibleto obtain highresolutionimagesof moleculeshardly detectablewith conventional methods(suchas H20 or S2).
In Table 1 (taken from Delsernrne[1985a]) we summarize
the chemicalspeciesidentified in cometary spectra.It should
be noted that this is a composite list (i.e., not every molecule
was detectedin each comet).Inspectionof Table 1 showsthat
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 abundancesof theseelements
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 interestingfeature of these
results is the depletion of cometary hydrogen by some 3
ordersof magnitudecomparedto solar systemabundances.
Delsentnte's[1985a] results are in good agreementwith the
generally accepted view that cometary volatiles are mainly
composedof water ice, which controls the sublimation of the
icy conglomerate of most comets [cf. Wyckoff, 1982; Delsentme,1982, 1985a; Donn and Rahe, 1982; Feldrnann, 1982;
Keller, 1983; Mendis et al., 1985. Delsernrne's[1985a] abundance values are also consistentwith Shulntan's[1983a] studies, which are predicting that during the accretion process
only thosevolatile specieswhich are thermodynamicallycompatible with water can condensateinto the ice mixture of cometary nuclei.
Reviewing the available, mostly circumstantial, evidence,
Delsentnte[1985b] concludedthat water ice controls the sublimation processin the vast majority of comets,including some
whosebehavior was earlier suspectedto be determinedby the
vaporization of ices more volatile than water (comets Morehouseand Kohoutek are two typical examples).It also seems
very likely that clathrate hydrates are quite common in
comets (this possibility was first suggestedby Delsernrneand
Swings [1952]). Clathrate hydrates have two very important
features:(1) a maximum of one "guest"molecule(suchas CO)
can be trapped within a H20 lattice of five to sevenmolecules,
669
and (2) their latent heat of vaporization is almost the same as
the latent heat of pure water ice. In the presenceof clathrate
hydrates, H20 and more volatile CO moleculesare evaporated 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 presentlypointed out by R. Prinn (private communication, 1985) that not all volatile molecules can be
trapped into the H:O lattice; for example,a CO: moleculeis
simply too big to "fit" into the ice structure.This means that
in addition to clathratehydratesother volatile ice components
might also be presentin some comets.This question will be
discussedin greater detail in a later section.
The presentlyprevailing 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
penetratedby gas or liquids that froze [cf. Whipple and Huebnet, 1976; Donn and Rahe, 1982; Mendis et al., 1985]. Cometary structuresseemto be associatedwith the fragile fireballs
observedby 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 particleswere probably formed by two
episodesof aggregation: the first involved assemblyof basic
building blocks of cometary solids, typically ranging in size
from 0.1 to 1 #m, while the secondaggregationprocessproduced the "bunch of grapes"type observedmorphology where
the observed void spaceshave previously been occupied by
ice. Applying percolation theory, Hordnyi and Kecskem•ty
1-1983]have estimatedthe sizedistributionof aggregatescomposed of elementary building blocks and found it to be n(a)•
a-4'4 (a beingthe particlesize).Similarpredictions
weremade
earlier by Monte Carlo calculations [Daniels and Hughes,
1980, 1981]. These distributionsare in reasonableagreement
with dust observationsindicating a spectralindex of 4.2 at the
nucleus[cf. Hanner, 1983].
When the nucleus made up from icy conglomeratesapproachesthe sun,it absorbsan increasinglylarger flux of solar
radiation, and the vaporization rate of volatile moleculesat
the surface increases. Gravitational forces are negligible;
therefore the vaporized gases leave the surface and form an
expanding exosphere.In this processthe gas drags away some
of those dust grains which have already been evacuated of
their ice component. The surfaceescapevelocity is small but
finite, so there is a critical dust size, a.... characterizingthe
largest solid particle which can be dragged away by the outflowing gas(seesection4.2). In his original presentationof 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
TABLE 1. Observed Species in Cometary Spectra [Delsemme, (mantle), causing a significant postperiheliondecreasein lu1985a]
minosity. The development of such a mantle was discussed
later by Shulman [1972], Mendis and Brin [1977], and Brin
Organic
Inorganic
Metals
Ions
Dust
and Mendis [1979] and subsequentlyby Hordnyi et al. [1984],
Fanale and Salvail [1984], Podolak and Hermann [1985], and
C
H
Na
C+
silicates
Houpis et al. [1985]. A schematic representation of such a
C2
NH
K
CO +
C3
NH 2
Ca
CO2+
mantle is shownin Figure 2.
CH
CN
NH 3
O
V
Mn
CH +
H:O +
CO
OH
Fe
OH +
HCO
H2CO
CS
HCN
CH3CN
H20
S
S2
Co
Ni
Cu
Cr
Ca +
N2 +
CN +
H:S +
The
thickness
of the mantle
varies
with
time
because
the
continuousvaporization increasesthe thicknessof the evacuated layer, and the "erosion"due to the drag of the outflowing
gas decreasesit. Mendis and Brin [1977] assumedthat the
erosion mechanismtakes place throughout the mantle with all
particles smaller than ama
x being dragged away by the outflowing vapor. Uncertainties,suchas how a dust grain can be
670
GOMBOSIET AL..'DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
SOLAR
BLACK
RADIATION
BODY
(1 -- ,ZlB)Jrad(O
, (]))-t-(1 -- ,Zlm)ltr(O,
(]))---8so'Ts
4
RADIATION
+ • rIizj(T,)Lj(T,)/[Narnj]
+ tc(T•)
grad(T,)
TTTTTTTT
i•.
SURFACE}?.•.':•.':','.•.•.•.?:':':•'1
....................
0•:•:,:•:•:•:,:.':•:CONDUCT I ON
/ / / / / / / /,:::.,,.:::,:::.,,-:::
-
HEATN OF
DFFUSNII
GA S
f'•'?;:;:;';:•'•'•:•'•'•'•'•'•'•'•
ß;,:.:$ U B I I M AT I 0 N
MANTLE
.x.:.:.:,:,:,:.:.:,:.x,:.:.x.:.:,:
CORE
Fig. 2. Schematic representationof 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 questionsof 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
evacuatedlayer and the "erosion"due to the drag of the outflowing gas which decreasesit. Hordnyi et al. [1984] introduced a "friable sponge" model of cometary nuclei, which
makes it possibleto expressquantitative relations betweengas
and dust productions. Fanale and Salvail [1984] took into
account the diffusiveflow of gas through the mantle, Podolak
and Hermann [1985] included the effectsof cracks and pores
in the mantle, and Houpis et al. [1985] introduced a chemically differentiated,multilayered mantle.
2.2.
Vaporization and Gas ProductionModels
As a first approximation, vaporization models neglect the
differentiated thin surface layer and consider the whole nucleus to be a homogeneousmixture of volatiles and dust.
These homogeneousmodelsimplicitly assumethat the uppermost dust layer, freshly evacuated of its ice component, is
instantaneouslyblown away by the outflowing gas so that the
slowly shrinking nucleus retains its undifferentiated pristine
nature at all instances.Another consequenceof the homogeneity assumptionis that basicphysicalparameters,such as the
bolometric albedo (A•), surfaceemissivity(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 horizontal temperaturegradientsare 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 describinga one-dimensionalradial 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 structureof the nucleus;in other words, they apply a
version of the following energy balance equation at the cometary surface[cf. Squiresand Beard, 1961]:
(1)
where t is time; r is nucleocentricdistance; O and •b are the
local hour angle and latitude, respectively;J,,a(t, O, fib)is the
total direct and multiple scatteredsolar radiation energy flux
reaching the nucleussurface;ltr is thermal radiation energy
densityreceivedat the surface,from radiation emitted by dust
particlesin the coma; A•Ris infrared albedoof the surface,T•(t,
O, •b) is the surface temperature; a is the Stefan-Boltzmann
constant(a = 5.67x 10-5 ergcm-2 s-• K-'•); N,• is theAvogadro number;Lj(T) is the latent heat of vaporizationper
mole for the jth volatilespecies
(erg/mol);zj(T) is the outflowingmassflux of thejth species
(g/cm2/s);to(T)is thermal
conductivity;
rnjis massof thejth particlespecies;
and r/j is
the fraction of the surfacearea coveredby the jth ice component. Inside the nucleusthe following heat conductionequation can be applied:
pcCc
Ot r2Orr2K
• = Qint
(2)
where Pc, Cc, and Qint are the mass density of the nucleus,
specificheat, 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 amorphousice to
a crystalline state [Pataschnik et al., 1974; Smoluchowski,
1981a, b; Klinger, 1981].
The simplestmodelsof cometary gas production at the surface of a nucleusassumean optically thin coma and neglect
the heat conduction term in equation (1), thus reducing the
problem to a simple transcendentalalgebraic equation I'Weigert, 1959; Watson et al., 1961; Huebner, 1965, 1967; Delsemrne,1966; Shulman, 1969; Delsemrneand Miller, 1971].
More sophisticatedmodels assumethe presenceof an insulating dust mantle [Mendis and Brin, 1977; Hordnyi et al., 1984;
Fanale and Salvail, 1984; Podolak and Hermann, 1985], take
into considerationlight scatteringcausedby dust particlesin
the coma [Hellrnich and Keller, 1981; Weissmanand Kieffer,
1981; Marconi and Mendis, 1982, 1983, 1984, 1986], include
nuclear rotation [Dobrovolskii and Markovich, 1972; Srnoluchowski,1981b; Rickmanand Froeschl•, 1983; Hordnyi et al.,
1984-1,
or usea combination
of these'
effects..
One of the central questionsaddressedby all thesemodels
is how to calculatethe productionrate of outflowing gas particles.In a pioneeringwork, Delsemrne
and Swings[1952] considereda cometarynucleuscoveredby a homogeneoussurface
of volatile snow. They assumedthat the surfacedid not contain macroscopicirregularities;i.e., surfaceirregularitieswere
much smaller than the mean free path of vaporized particles.
As the pressureof the cometary atmospherewas much smaller
at the nucleussurfacethan the critical pressureof the phase
transition triple point, the liquid phasewas unstableand sublimation of frozen volatileswas responsiblefor gas production.
Assumingthat the sublimatedgas was in equilibrium with the
surface, Delsemrneand Swings [1952] applied the ClausiusClapeyron equation to determine the steady state saturated
gas pressure:
p,(T)
=p,exp
LkN,
('
(3)
where p• is the vapor pressure,p, is the saturatedvapor pressure at a referencetemperature T,, L(T) is the latent heat of
GOMBOSIET AL.'. DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
vaporization,and k is Boltzmann'sconstant(k = 1.38 x 10-•6
certain
erg/K). Delsemmeand Swings [1952] also assumed that the
sublimatedmoleculesbehaveas a perfectgas (Ps= nkT), thus
defining the gas number density n. In order to determine the
bulk velocity of the outflowing gas, Delsemmeand Swings
[1952] considereda low-pressurekinetic model. They started
by deriving the flux of scatteredgas particles reaching the
snowsurfaceand condensingback to solidphase:
fundamentalassumptions.
basic similarities
but also have a number
671
of different
Extendingthe hydrodynamicapproachto the vaporization
process,Gombosiet al. [1985] proposeda gasdynamic"reservoir outflow" model to describecometarygas production.In
their model the sublimating surfacebelow the mantle is replacedby a reservoircontaininga stationaryperfectgas with
temperatureT• and pressurePs.It is assumedthat the gas flow
is
so inhibited through the mantle that the gas is practically
Z- = 0.25mnuth
(4)
stationary and there is no significantpressuredrop within the
wherem is the massof a gasmoleculeand Uth= (8kT/r•m)
ø'5, mantle. At the top of the evacuateddust layer the gas disthe mean thermal speed of the gas molecules.Under steady chargesto the low-pressureexternal medium draggingaway
state conditionsthe condensingflux z- is equal to the vapo- dust particlesfrom the top of the mantle. This approximation
rizing flux z +. In this kinetic outflow model, Delsemmeand is applicable even in the caseof a "bald" nucleus,becausethe
Swings[1952] assumedthat cometary sublimation was not an sublimation has to take place below a microscopiclayer of
equilibrium processand neglectedthe condensingflux altoge- freshlyevacuateddust (elsethe outflowinggascould not drag
ther (z- = 0). At the same time they kept the vaporizing flux away dust grains).The gasfrom the reservoiris dischargedto
at the steadystatelevel, thus finding a net gas production rate the low-pressureexternalmedium either directly or through a
of
thin layer of porousdustcoveringthe nuclearsurface.The gas
productionrate and outflow velocity are affectedby the dust
ps(T)(m)
ø'•
loading and by the gas parametersimmediately outside the
nucleus. Combining the results of a seriesof numerical soluFor more than three decades,equations (3) and (5) have tions with the time-dependentdusty hydrodynamicequations
been widely used for calculating cometary gas production describinggas outflow from such a reservoir and using the
rates (compareearlier review papersby Whipple and Huebner predictionsof steadystategasdynamics,Gombosiet al. [1985]
[1976], Delsemme[1982], and Mendis et al. [1985]). During concludedthat in a first approximation the gas production
Zkin(T)
= (2r•kT)O.5
(5)
the years,however, it drew criticism from severalauthors, who
made somemodifications(usuallyresultingin a changeof 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
significantlymodified if one had taken into account the he-
terogenouschemicalnature of a cometary surface(i.e., that
cometary surfacesare a mixture of various volatiles and dust;
consequently,complex intermolecularforcesmodify the molecularconstantsin 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], Gombosiet al. [1985], and Mendis et al.
[1985]. These authors pointed out that at the cometary sur-
rate
was
Fps(m)
ø'5
Zo= 2(7kT•)O.•
(6)
where 7 is the gas specificheat ratio and the numerical constant F can be expressedas
( 2)ø-•(•'+•)/(•'-•)
r = 7
7+1
(7)
Comparing equations (5) and (6), it can be easily seen that
for a typical 7 value of 4/3 (water vapor) the differencebetween the kinetic outflow [Delsemmeand Swings,1952] and
the reservoir outflow [Gombosiet al., 1985] models is only
about 20%. However, this difference becomesmuch larger
when one takes into account the choking effect of dust loading. A new set of time-dependentnumerical calculations,simifacethe gasmeanfree path was typically10-102 cm; conse- lar to those reported by Gombosiet al. [1985] has recently
quently,the neglectof collisionsis inappropriateand gas dy- been carried out for this review paper. The dependenceof gas
namical methods have to be applied at the nucleus-coma production rates and outflow velocities on surface temperinterface.Wallis [1982] specificallynoted that the kinetic sub- atures and dust/gasmassproduction rate ratios (25)was deterlimation curves published by Delsemmeand Miller [1971]
mined, on the basis of this new and extended numerical modimply that collisionlesseffusion from an H20-dominated eling of transonic dusty hydrodynamicreservoir outflow. (It
comet could only be appropriateoutside3 AU.
should be noted that, in general,25is not necessarilyequal to
Self-consistent
dusty hydrodynamiccalculations[Probstein, the solid/volatilemassratio inside the nucleus.This question
1968; Hellmich, 1979; Marconi and Mendis, 1982, 1983, 1984, will be addressedin section 2.3.) The following analytic ex1986; Gombosiet al., 1983] employ a surfacegas densityvalue pressionsapproximate these new results to within a few perobtainedfrom equation(3) and determinethe interrelatedpro- cent for 25in the range 0-5:
duction rate and outflow velocity (z = nUout)
by solving the
z(T, 25)= zo(T)(1.17- 0.07325)
(8)
coupledsteadystate dust and gas equationswith the appropriate boundaryconditionsat infinity (Po•= 0). Thesecalcula0.62Uth(T)
Uout(T,25)=
(9)
tions will be discussedin detail in section4. In attemptingto
1 + 0.2825
develop a realistic model of the gas flow through the evacuated porous dust layer (mantle),one needsto approximatethe Numerical calculations leading to these expressionswill be
inhibited gas flow and the dust pickup in a manner which is described in detail in section 4.4.
both realistic and still permits meaningfulsolutionsto be obEquations (8) and (9) have several advantagescompared
tained. Two such models have been published recently with the predictionsof the kinetic model. First, they were
[Fanale and Salvail, 1984; Gombosiet al., 1985] which have obtained from hydrodynamic calculationswhich describethe
672
GOMBOSIET AL.: DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
collision-dominatedoutflow region more properly than a col-
needsto be treated with somecaution. It was pointed out by
Delsernrne
and Miller [1971] that for most volatilesL strongly
varieswith temperature;the notable exceptionsare water and
clathrate hydrate ices,whoselatent heatshardly vary between
lisionless effusion model. Second, these new results take into
consideration the effectsof 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 assumedthat the sublimatedgas flowing through the tubesis in
the Knudsen regime. In this case the local gas velocity is
[Fanale and Salvail, 1984]
4(2k
T••ø'5
r0grad
(p)
Um=
• \-•m,/ t'• p
125 K and 273 K. Some authors have used a so-called "mean
latent heat" for describingthe sublimationof multicomponent
ice mixtures [e.g., Huebner and Giguere, 1980; Huebner and
Keady, 1983]. This method was recently criticized by Mendis
et al. [1985], who concludedthat there was no physicaljustification for this practiceand it grosslyoverestimatesthe production rate of the lessvolatile species,while underestimating
the releaseof the more volatile component.
(10)2.3.
Heat Transfer in the Mantle:
GoverningEquations
where T• is the sublimatingtemperatureand ro and t,, are the
averagecapillary radius and tortuosity, respectively.The pressure drop through the mantle was expressedas
Ap = Psi1 - f (Tou•)/(ht•o)]
Most recent mantle energy balance calculationshave consideredtwo components(one solid and one volatile) and have
applied assumptionswith different levels of sophistication
[Mendis and Brin, 1977, 1978; Brin and Mendis, 1979; Weissman 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
seemsto be primarily controlled by the vaporization of water
[Delsernrne,1985b], the following sectionsin this paper will
mainly be devotedto the discussionof the temperaturedistribution, gas and dust productionratesin the surfacelayer of a
water, or clathrate hydrate, dominated nucleus.A recently
publishedand more complicatedthree-componentmultilayered mantle model [Houpis et al., 1985] explaining a potentially high CO/CO2 production rate of comet West (1975n)
[Feldrnan and Brune, 1976] will also be discussedlater in the
(11)
where Tois the surfacetemperatureandf is the mantle porosity, while the gas outflow velocity from the mantle (Uo)was
assumedto be 0.6 times the mean thermal velocity Uth.Assuming 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 sublimatinginterface, while the number of capillariesper unit area (No) can be
expressedas
N½= f/(3•ro2tm)
paper.
(13)
Combining equations (10), (11), and (13), one can obtain an
alternate expressionfor the Fanale and Salvail [1984] diffusive
gas production rate:
A
ZFS
--Zki
nA/Ac
nt-(To/TOO.
5
(14)
where A is the thicknessof the evacuateddust layer and
A = 0.8/tin
A½= (10/9)(fro/tm)
A schematicrepresentationof the differentiatedtop layersis
shown in Figure 2. A core consistingof the original dust-ice
mixture is coveredby an evacuateddust layer which forms the
mantle. The vaporization processtakes place at the mantlecore interfaceand not at the top of the nucleus.The absorbed
radiation energy has to penetrate the insulating mantle in
order to reach the sublimatingsurface,and consequentlythe
surfacetemperature(TO in what follows) and the sublimation
temperature(T•) cannot be representedany more with a single
temperature Ts. At the nucleus surface the energy balance
equationcan be expressedas
aT0
Fanale and Salvail [1984] adoptedf= 0.5, tm= 5.0 valuesand
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 interestingto 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 samesurfacetemperature.
It should be noted again that the Fanale and Salvail [1984]
diffusive production model and the reservoir outflow model
[Gornbosiet al., 1985] are both basedon the samebasicprocess:pressuredifferencedrives out gas from a stationary (or
almost stationary) gas reservoir.In the caseof the Fanale and
Salvail [1984] model the outflow velocityis constant(0.6Uth),
while in the reservoiroutflow model the pressurein the mantle
is assumedto be constant,but variationsin the external pressure value, as well as the dust mass loading, are taken into
(1--AB)Jra
d+ (1- AiR)!tr
= esaTo
½+ *era(To)
-•r (15)
where ,cm representsthe 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, becausethe
physicalparametersare different in the evacuatedmantle and
in the ice-dustmixture core. At this interface the energy balanceequationcan be written in the form
•T]
= q•,
L(T•)
z(T,) + %(T,)
•T]
(16)
here A is mantle thickness,R• is nuclear radius, % is thermal
conductivity in the core, while e representsan infinitesimally
small but positive number. The area fraction, r/,•, coveredby
ice can be expressedas [cf. Hordnyi et al., 1984; Fanale and
Salvail, 1984]
account.
Beforediscussing
more complexcometarysurfacemodels,it
should be noted that the latent heat of vaporization, L(T),
qw
__
rl2/3
=(pa
pa •2/3
(17)
GOMBOSIET AL..' DUST, NEUTRAL GAS MODELINGOF COMETARYINNER ATMOSPHERES
where r/ is volume ratio of ice in the core, Pa is dust bulk
density,Pi is ice massdensity, and Zc is dust/ice mass ratio in
the core. The energy balance equations in the mantle and in
the core are (assumingonly radial variations, heat conduction
as the dominant energy transport mechanism,and the heating
of the outward diffusing gas as the main energylossprocess)
673
this work. The situation is considerablybetter with respectto
water ice. Klinger [1981] has derived an empirical temperature law for the ice heat capacity from the measurementsof
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-dustmixture and the effectivespecific heat can be expressedas
Mantle
Cc -- (ZcCa + C,)/(1 + Zc)
pmCm
r3t r2•rr2tcm
• = - •mqwz(Ti)
• (18)
Core
pcCc
r3t r2r3r
r2tcc = 0
(19)
where Pmis averagemassdensityof the mantle, Pc is average
massdensityof the core, Cmis specificheat of the mantle, and
Cc is specificheat of the core. The energy loss term in equation (18) is not zero, sinceit is assumedthat the penetrating
vapor is in thermal equilibrium with the dust throughout the
mantle and so representsa sink of heat. Hordnyi et al. [1984]
have argued that it is reasonableto expect the gas to accommodate to the mantle temperature at each location because
the mean free path of a gas moleculeis much larger than the
averagegrain separationin the mantle [cfi Whipple and $tefanik, 1966].
Equations (15)-(19) contain some basic parametersdescrib-
ing such fundamental properties of the mantle and core as
density, heat conductivity, and specificheat. As nobody has
ever examined the structure of a nucleus,these basic parameters have mainly been obtained by guesswork, so that the
(24)
The specificheat in the evacuatedmantle is taken to be identical with the dust specificheat, Cm----Ca.Klinger [1981] also
publishedan expressionfor the crystallinewater ice thermal
conductivityin 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
conductivityin the mantle with an expressiontaking into account contact and radiative conductionand neglectthe contribution of gas conduction:
tcm(T
) = tco+ 4rrgalT
3
(26)
where tcois the contactconductioncoefficient,eais dust infrared emissivity,and I is average intergain distance. Brin and
Mendis [1977] assumed that I was equal to the maximum
grain sizein the mantle. This approximation overestimatesthe
role of radiative conduction: an alternative approach published by Hordnyi et al. [1984], expressingI as a function of
the dust/ice massratio, seemsto be somewhatmore realistic:
values chosen are uncertain. On the other hand, one has the
right to expect nucleusmodels to be internally self-consistent:
two interrelated parametersshould not be derived using contradictory assumptions.Here an attempt is made to compile a
set of consistent functions from the various mantle/core
models.
The ice massdensityis assumed
to be Pi = 0.9 g/cm3. The
average dust density is derived from the average density of
dust particles observedin the coma. When deriving Pd, the
dust densityfunction usedby 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):
where M=12,
p(a) = Po- p•a/(a + a•)
(20)
n(a)= [1 - ao/a]M(ao/a)
N
(21)
N=4.2,
a1=2
#m, ao=0.1
and an averagedust size of (a) = 0.69 #m. Now the average
densities can be obtained
lVc-- flKi
#m, Po=3
g/cm3, and Pl = 2.2 g/cm3 [Divineet al., 1986].Thesevalues
resultin an averagedustbulkdensityvalueof Pa- 0.84g/cm3
core and mantle
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:
as
In
the
mantle-core
thermal
calculation
(28)
one also
has to
know the sublimationparametersfor water ice (the parameters for clathrate hydratesare very similar).It was pointed out
by Delsemmeand Miller [1971] that for water the latent heat
is fairly insensitive to temperature and its value can reason-
Pm = q)•cPi
(22)
p• = q(1 + Zc)p,
ably be approximated
by Lw= 4.80 x 10TMerg/mol.The sublimation vapor pressure of boiling water, p, (see equation
(2.3)),is 106dyn/cm2 at the T• - 373K reference
temperature.
Following Hordnyi et al. [1984], a dust specificheat value of
In order to be able to solveequations(18) and (19) with the
Ca- 8 x 106erg/g/Kwasadoptedby the Halleyenvironment appropriate boundary condition, one has to define the relation
working group of the InteragencyConsultativeGroup [Divine
et al., 1986]. As there is no specific information available
about the thermal properties of cometary nuclei, this value
seemsto be as good as anything elseand was also adopted in
betweenthe massratio 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-
674
GOMBOSIET AL.: DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
parts of this very complex problem. In the following sections
the presentstatusof theseefforts will be discussed.
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 section4.2)
will escape,while larger grains will be retained as part of the
mantle. This type of model (sometimesimplicitly) assumesfluidization of the mantle; this way the newly evacuated small
grainscan be forcedthrough the existingmantle.
An alternative model was suggestedby Hordnyi et al.
[1984] and later adopted by Houpis et al. [1985]. This "friable
sponge"model assumesthat (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 coremantle 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 extremelyfriable and break into smaller
piecesbefore 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 eachcomet).
2.4. Heat Transfer in the Mantle:
ApproximateSolutions
Before discussingvarious approximatesolutionsof equations (18), (19), and (33), it is worthwhile to estimatethe time
constantscharacterizingheat transport and mantle growth
processes.The thermal time constantsof the mantle and core
can be definedas [Klinger, 1981]
'•m--
Pm
Cm
A2
PcCcRn
2
'•c=
TC2Km
TC2Kc
(34)
where Zmand Zc represent the thermal time constantsof the
mantle and core, respectively.Substitutingestimatesof comet
parametersinto expressions
(34), one obtainszmand zc values
of the ordersof magnitudeof an hour and hundredsof years,
respectively.On the basis of these time constants,nucleus
models can be characterized by the following classesof
models:
Zd = ]•dUout•wZ
(29)
Rotating nucleus. Two groupshave publishedcalculations
taking into considerationthe angular dependenceof external
radiation at the surface.Weissmanand Kieffer [1981, 1984]
Combining equations (9) and (29), one obtains a relation between/•a and Z:
considered a nucleus with no mantle and took into consider-
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 consequentlyZ, varies with the TO surface
temperature,cometsmay have differentmantle evolutionpatterns 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 replacedby Z = Zc,because
one can only blow away the freshlyevacuateddust layer.
The sublimation processincreasesthe mantle thickness,as
the upper layer of the core is evacuated.The masslossrate per
unit area of the ice is
dt- rlwz
= •lPi'• s
(31)
where (dA/dt)sis the changein A due to sublimation.The rate
of decreaseof the mantle thicknessdue to dust productionis
dt- Z'-•
= •m
• e
dM•
dMi
(dA)
ation the heat conductedinto the core. They consideredthe
core to be a sink of energy rather than a very large heat
reservoir, which may increase the surface temperature at
larger heliocentricdistances(especiallyin the outbound part
of the orbit). In a semiempiricalway, Weissmanand Kieffer
[1981, 1984] also took into considerationthe optical characteristicsof the coma, thus decreasingthe diurnal variation of
the radiative flux reachingthe surface.Hordnyi et al. [1984]
adopted a more sophisticatedmantle-corenucleusmodel but
used the unattenuated solar radiation profile to study the
diurnal variation of the surfaceand sublimatingtemperatures.
Both calculations concluded that (1) for active comets the
amount of energyconductedinto core was much smaller than
the energy used for sublimation,(2) the temperaturedistribution in the uppermostlayer of the nucleusreachesa steady
state diurnal pattern after less than 10 rotations, and (3) the
longitudinaltemperaturedistribution reachesits daily maximum in the afternoonand its daily minimum in the predawn
hours.
(32)
where(dA/dt)eis the changein A due to erosion.The time rate
of change of A can be obtained by subtracting(dA/dt)efrom
(dA/dt)s[cfi Hordnyi et al., 1984]:
It seemsto be very reasonableto expecta diurnal variation
in the surface illumination at larger heliocentricdistances
where the coma is not well developed.However, closerto the
sun, multiple light scatteringon 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;
Weissmanand Kieffer, 1981, 1984] have indicated that as a
resultof the large collectingarea representedby the thick dust
coma the radiation energydensityat the nucleusis probably
somewhatlarger than that of the unattenuatedsolar radiation
and its distribution is fairly isotropic. This means that the
d•(pa+piz•l/3(l_
Z)•
(33)
In general, if one knew the radiation flux at the surface of
the nucleusat all instancesafter the comet started its journey
into the solar system,equations (18), (19), and (33) could be
solved simultaneouslywith the boundary conditionsdefined
by equations (8), (9), (15), and (16) for the •ntir• life of the
comet and one could determinethe A(t, •, ½) and T(t, •, •)
functions.Unfortunately, the J•a(t, •, ½) Junctionis dependent on the optical characteristicso[ the coma, controlled by
•arli•r gas and dust production. It is obvious that selfconsistenttreatment of this problem is not an easy task and
sucha solutionis still a f•w y•ars down the road. On the other
hand, various groups made signiacant progress in solving
more active a comet is, the less diurnal variation one can
expect. Figure 3 (taken from Weissmanand Kieffer [1984])
shows the extreme differencesbetween surfacetemperature
distributionsobtained for Comet Halley at perihelion(0.5871
AU) in the presenceand absenceof a dense coma. In the
calculationsa rotation axis obliquity of 20ø was assumed.The
temperature distribution on a bare ice nucleus without coma
is shown in Figure 3a, and the same nucleus with coma is
GOMBOSIET AL.' DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
NUCLEUS SURFACE TEMPERATURE
R = 0.587
675
AU
75.
•
60.
a.)
160
45.
170
30.
15.
205
O,
-15.
-30.
-45.
-60.
-75.
90.
180.
LOCAL
NUCLEUS
75.
I
SURFACE
HOUR
270.
360.
ANGLE
TEMPERATURE
I
R -- 0.587
AU
'1
b.)
60.
20
45.
04
30.
15.
-15.
-30.
-45.
-60.
I
-75
O.
90.
180.
LOCAL
HOUR
270.
360.
ANGLE
Fig. 3. Temperaturedistributions
on the nucleusof CometHalley at perihelion(0.5871AU) (a) with no dustcomaand
(b) with dust coma.The rotation pole obliquity,estimatedto be approximatelyperpendicularto the orbit plane, was
chosento be 20ø(takenfrom Weissman
andKieffer [1984]).
shownin Figure 3b. For the caseof the nucleuswithout coma
there is a strong diurnal variation and a noticeablethermal
lag in the surfaceresponse,despitethe very low value adopted
by Weissmanand Kieffer [1984] for the thermal inertia. However, when the dust coma is added to the model, the nucleus
becomes more isothermal.
Core thermal hysteresis.These calculationsneglect the
mantle layer and concentrateon the effectscausedby a nucleus which has a large but finite heat reservoir [Smoluchowski,1981a,b; Klinger, 1981; Hermannand Podolak,1985].
These authors also considered an internal
heat source caused
by amorphousto hexagonalwater ice phasetransition,togeth-
er with heat conductionin 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 evolutionary scenariowhich could prevent amorphousice from crystallization 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 considerablepopularity in the cometary community, in part because it very naturally explains the flaring phenomena observed in comet Schwassman-Wachmann1. This question is
still open, and the presently available observational evidence
is insufficient
to resolve it.
676
GOMBOSIET AL.: DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
T(K)
210
......
,
,
al. [1984] introducedthe friability concept(to connectgasand
dust production rates), suggestedthat the outward diffusing
gas is continuously heated in the mantle, and assumedthat
erosion primarily takes place at the surface of the mantle.
Fanale and Salvail [1984] proposeda model for gas diffusion
through the mantle, Podolak and Hermann [1985] combined
the original Mendis and Brin [1977] mantle thicknessvariation calculationwith a finite core heat capacitymodel, while
Houpis et al. [1985] allowed for chemicaldifferentiation of the
upper, partially evacuatedlayers. It is important to note that
all these models predict similar production rate hysteresis
curves:brightnessvariation is simply a result of the growth
,
•90
170
•
150
'
.
1:50
I10
90
70 •'
30
I
.5•
I
I I
I
•.0 m2.0•.0
I
5.0
I
•0
and destruction of the mantle, so all modified Mendis and Brin
I
'
20•0
R(Au)
Fig. 4. Variation of surfacetemperaturewith heliocentricdistance
for a hexagonal water ice ball moving along the orbit of Comet
Halley (taken from Hermannand Podotak[1985]).
Another important questionregardingthis type of calculation is the question of how quickly heat can be conducted
within the core. The smaller the surfaceheat conductivity is,
the longer it will take to conductheat deep into the nucleus.
Using an appropriatelysmall value for the heat conductivity
of the top layer, the surfacelayer can supplyadditional energy
for sublimation after perihelion, thus shifting the maximum
productionrate up to • 100 dayspostperihelion.The problem
is that this "appropriate"conductivityvalue representsa low
porosity, an almost "concrete"like nucleus.It is difficult to
visualize how such a low-porosity nucleuscan produce dust
particlesat all. On the other hand, when the calculationstake
into considerationa pure hexagonalice nucleus,the predicted
light curve has its maximum at perihelion. Figure 4 (taken
from Hermann and Podolak [1985]) showsthe surfacetemperature variation of a pure hexagonalice nucleusin the orbit of
Comet Halley after many revolutions.It can be seenthat close
to the sun where the temperature is above • 185 K, sublimation of water ice controls the surface temperature. The
cooling due to sublimationis so strongand the heat conducted to the core is so small that the surfacetemperature rises
only slightlyas the comet passesthe perihelionportion of the
orbit. On the outboundleg betweenabout 1.3 and 3.5 AU a
portion of the thermal energy stored in the surfacelayer is
being conductedinto the nucleusto continue warming the
deeperlayers. Between3.5 and 7.0 AU this inward heat flow
continuouslydecreases,
and beyond 7 AU the surfaceis partially heated by the interior. This processcontinuesas the
comet passesits aphelion and starts its journey inward. On
the other hand, the heat content of the nucleus has already
beendepletedby this time; consequently,
internalheatingcontributes lessand lessto the maintenanceof the surfacetemperature.
Orbital evolutionof the mantle thickness. It was first suggestedby Whipple [1950] that an evacuateddust layer may
cover the surfaceof cometary nuclei, although the first quantitative model of the mantle thicknessvariation along a comet
orbit was publishedby Mendis and Brin [1977] and Brin and
Mendis [1979]. Using an evacuation-erosionmodel, they predicted
different
mantle
thicknesses
for the inbound
and out-
[1977] type models predict very similar production rate
curves,even though the physicalprocessesresponsiblefor the
mantle distruction might be different.
Various modelspredict somewhatdifferentmantle 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 thicknesscurves,which gave typical mantle thickness
valuesof 0.1-1 cm for active periodic comets.Fanale and Salvail [1984] used their diffusive mantle model to calculate the
growth and distruction of the mantle, which resultedin expression(14) for the gas mass production rate. Inspection of
equation (14) reveals that the gas production rate starts to
drop drastically as A becomeslarger than the critical thickness,Ac. Using the relations given by Fanale and Salvail
[1984], one can expressA• as a function of basic morphological parametersof the evacuatedmantle. As a result of the
parametervaluesadopted by Fanale and Salvail [1984] their
critical mantle thicknessis approximately equal to the average
capillary radius in the mantle, which was assumedto be half
of the averagegrain size (< 1 #m). As the gas production rate
drasticallydecreaseswhen A >>A•, the typical mantle thicknessfor active periodic cometsis the sameorder 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 averageintergrain distance(equation (27)) for
the average capillary radius (as suggestedby 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
constantscould also result in a better agreementbetween the
diffusiveand kinetic gas production rates for the A--} 0 limiting case(which presentlydiffer by a factor of 6). However, at
this point this discrepancybetweencompetingmodelscannot
be resolved, because we have no observations about surface
morphologyof comets.
The governingequationsof the mantle thicknessvariation
discussedearlier can be simplified further by neglectingthe
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 assumingthat the temperature distribution in the mantle can always be consideredto
be steady state (this assumptionseemsjustifiable becausethe
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
bound portions of comet orbits, which resultedin a hysteresis (33) to obtain the orbital variation of the mantle thickness,but
of the light curve.The original Mendis and Brin [1977] model the surfaceand sublimatingtemperaturescan be obtained by
was recentlyfurther elaboratedby severalauthors.Hortinyi et
solvingthe followingtranscendentalalgebraicequations:
GOMBOSIET AL.' DUST, NEUTRAL GAS MODELING OF COMETAllYINNER ATMOSPHERES
)•C=0.5
677
/•d= 1.5 10-5s/crn
iO:>9
IO
i
i
I.O
w
<3
w
j 0:>7
O.I
:>6
.
I
0.1
1.0
i
I0.0
0.1
d (AU)
1.0
I0.0
d (AU)
Fig. 5. Variationof gasproductionrate (Qg)and mantlethickness
(A) for a dirty iceballnucleus(with a friabilityof
1.5 x 10- • s/cm)movingalongthe orbit of CometHalley.
(1 - AB)Jra
d + (1 - AiR)ltr- esrrro
4
Figures 5 and 6 (calculatedusing the friable spongemodel
of Hor•inyi et al. [1984]) show the orbital variation of mantle
(35)
+ 5--m
(To-
thickness
(A) and that of the total gasproductionrate (Qg=
Maim)for a dirty iceballnucleusmovingalongthe orbit of
Comet Halley. In this calculationthe external radiation field
(Jrad)was assumedto be equal to the unattenuatedsolar radiaton energy flux density, the heat flux conducted to the core
(TO- T•))
(36)
3k A ( 3kN,t
wasneglected,
and the friabilityparameterwaschosento be
either/9d
=
!.5
z= F
2 Pr
exp(kNA
Toø'5 exP(kN,tT1) x 10-5 s/cm(Figure5)or//d = 2 x 10-5 s/cm
(•)•/2
LT•)1.17-0.073Z
2-•r/wZ•co = In 1+
2L
-L
(37)
The total gas massproduction rate is
(Figure 6). The integration started at aphelion with a "bald"
nucleus(A = 0) and was continuedthrough severalsuccessive
revolutions.Inspectionof Figure 5 showsthat for a friability
parametersmallerthan a criticalvalue(~ 1.8 x 10-5 s/cmin
Mg= 4rcRn2rlw
2
(38)
this case)the mantle thicknessincreasesmonotonically with
s/cm
/•d:2.1CF5
Xc=0.5
I0 sø
io
i
1029
_
• 102e
O.I
I0 z?
0.1
i
0.01
I
d (AU)
I0
0.1
d (AU)
Fig. 6. Variationof gasproductionrate (Qg)and mantlethickness
(A) for a dirty iceballnucleus(with a friabilityof
2 x 10-5 s/cm)movingalongtheorbit of CometHalley.
678
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARY
INNER ATMOSPHERES
4
tions predict either a large hysteresisor practically no hysteresis for the near-perihelion part of a Halley type comet dependingon the adoptedfriability parametervalue. For Comet
Halley the relativelymodesthysteresis
is probablyrelatedto
the variation of the coma optical parameters, while the nucleusitself remainspractically"bald" for the perihelionpart of
the orbit [Brin and Mendis, 1979' Weissman and Kieffer,
1984]. It should be rememberedthat thesefriable spongecalculations did not include any radiative transfer consideration
of thefluxincidenton the nucleus,
andthereforethisquestion
has to be further investigatedusinga combinedmantle-coma
0.5
I
1.5
2
2.5
3
:5.5
d(AU)
Fig. 7. Variation of the H20/CO2 productionrate ratio for a
chemicallydifferentiaateddirty iceball nucleus(with a friability of
2.25 x 10-5 s/cm) moving along the orbit of Comet Halley (taken
from Houpiset al. [1985]).
radiative
transfer model.
Multilayer mantle. In order to explain the unusuallyhigh
CO/OH ratio reportedfor Comet West (1975n),Houpis et al.
[1985] have recentlyproposeda three-componentmantle/core
model. In this model the pristine nucleusconsistsof an icy
conglomeratecontaining dust, frozen clathrate hydrate (with
one CO moleculetrappedbetweensix watermolecules),
and
free CO2 ice. As this conglomerateapproachesthe sun, the
more volatile CO2 which is not trapped in clathrate will
time, while the gas and dust production rates continuously escapefirst leavingbehinda steadilygrowingmantle of dust
decrease.During consecutiverevolutionsthis comet becomes and clathrate ice. This clathrate mantle partially insulatesthe
fainter and fainter (i.e.,the gas productionrate decreases)
and core containing more volatile species.As the comet moves
is finally suffocatedby dust.The surfacetemperatureof sucha closerto the sun, its surfacegradually heats up, and the clathrate hydratealsostartsto sublimatecreatingan uppermost
comet increasescontinuouslyand finally approachesa limitdust
mantle completelyevacuatedof its volatile component.
ing value,TO..... = (Jrad/t;str)
0'25(i.e.,practically
all absorbed
In their calculation, l-loupis et al. [1985] applied a friable
energyis reradiatedas blackbodyradiation).At 1 AU, this
typically meansa surfacetemperatureof about 400 K. It is spongetype model [Hor•inyi et al., 1984] to describethe teminterestingto note that the faint earth-grazingComet Iras- poral evolutionof the dust and clathratemantles.The model
Araki-Alcock (1983d)is probably sucha dying object.Hanner
et al. [1985] have concludedon the basisof a seriesof infrared
observationsthat this comet had an approximate radius of
about 5 km, a surface albedo of 0.9, and a subsolar surface
temperatureof about 400 K. The averagegasproductionrate
was • 2 x 10-7 g/cm2/s,whilethe dust/gasmassratio in the
coma was •0.25. Applying the friable spongemodel (most
evacuation/erosion
modelswould give qualitativelysimilar results)to Iras-Araki-Alcock(1983d),one obtainsthe following
parameters: sublimating temperature of •190 K, mantle
neglected
the heatflux into the core,as well as the heatingof
outflowing gas.
Figure 7 shows the orbital variation of the H20/(CO2
+ CO) productionrate ratio Rv after 10 revolutions.For this
particular calculation,Houpis et al. [1985] adopted a fid=
2.25 x 10-5 s/cmvaluefor the friabilityparameter.The production rate ratio was calculatedusingthe relation
Rv =
6•clZcl
gI½lZ½l
-[- 7r/,,z,,
(39)
thickness
of •0.5 cm,andfriabilityparameterof • 10-5 s/cm. where •/½1and r/• representthe fractionsof the sublimating
These parametersclearly describea dying comet, where the
accumulatingdustlayer ultimatelyquenchesgasand dustpro-
surfacescoveredby clathrateand CO2, respectively;Zcland z•
are the clathrateand CO2 productionrates.It is obviousfrom
duction.
equation(32) that Rv< 6, and consequently
this model will
The mantle thicknessdoes not increasemonotonically any
more when the friability is larger than a critical value, but
always predict a CO and CO2 rich comet. Inspection of
Figure 7 revealsthat at large heliocentricdistancesthis comet
will behaveas a cO 2 (and CO) rich comet(R• < 1), while
proachto the vicinityof the sun,A and Qgfollowthe same closer to the sun the CO + CO2 content in the coma de-
insteatt a repetitive cycle appears.Apart from the first ap-
curve during subsequentrevolutions.Inspectionof Figure 6
showsthat at about 1.5 AU preperihelionthe increaseof A
stopsand then the mantle thicknessstartsto decrease.By the
time the comet reaches,-•1 AU postperihelion,the mantle is
practicallyblown off all at onceand the gasproductionjumps
by more than 1 order of magnitude.When the comet again
leavesthe vicinity of the sun,a new mantle is developed;this
new mantle is blown off during the next perihelionpassage.
This processis repeatedduringsubsequent
revolutions.
When the friability parameter is further increased, the
mantle thickness starts to decreasewhile still further away
from the sun, and it will eventually be blown off before perihelion, rather than postperihelion.This early blow off results
creases to about 20-25%.
The multilayermantle model is very attractive,becauseit is
ableto explainthe highinitial activityof newcometsand can
predicta changingH20/(CO 2 + CO) ratio along the comet
orbit, On the other hand, some authors [cfi Shulman,1983a]
have difficultiesvisualizing how this type of nucleuscould
originally condenseout of the presolar nebula. Shulman's
argumentsseemto be somewhatsimplistic,especiallyif 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" raisesthe
in fairly high perihelionproduction
rates(,-•3 x 1030mole- possibilitythat the clathrate hydrate containsmainly smaller
cules/sfor a 3-km nucleusradius)and in a symmetriclight "guest" molecules(such as CO), while moleculeslike CO2
curve around perihelion.The friable spongeorbital calcula- form the free volatile ice component.
GOMBOSIET AL.: DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
3.
679
THEORY OF ATMOSPHERIC PROCESSES
etary atmospheres,only the first three of theseequations,correspondingto a "five-momentapproximation," will sufficeto
3.1. Transport Equationsand Physical Processes
characterize the neutral gas behavior within the collisionAs has already been discussedin the precedingsection,pres- dominated region of comas.
In this five-moment approximation the propertiesof the gas
ently there are two models describinggas production of a
cometary nucleus:the kinetic outflow model of Delsemrneand are expressedin terms of just the speciesdensity,flow velocity,
Swings [1952] (see equation (5)) and the recently published and temperature.The conservationequationsfor the density,
hydrodynamic reservoir outflow model proposed by Gornbosi velocity, and temperature, neglectingCoriolis, viscousstress,
et al. [1985] (seeequation (8) for a "bald" nucleusand equa- internal energy, and certain heat flow effects [see Holt and
tion (37) for a nucleuswith an insulating mantle). The next Haskell, 1965; Burgers,1969; Schunk,1975], are
questionto be addressedis what happens to the outflowing
C9ns
•ns
• + V(nsUs)
(40a)
gas once it leavesthe surfaceof the nucleus.
•t
•t
The atmospheresof comets,commonly referred to as c½•mas,
are different in a number of important ways from planetary
atmospheres.The most important distinguishingcharacterrnsns
-•- + Vps
- rnsnsGs
- cSt
isticsof comas are (1) the lack of any significantgravitational
force, (2) relatively fast radial outflow velocities,and (3) the
2 Dt +•ps(V.us)+V.qs•t
time-dependentnature of their physical properties.A direct
consequenceof thesefeaturesis the expanding nature of comwhere Ds/Dt is the convectivederivative, ns is the number
etary atmospheres.
densityof neutral gasspeciess, us is the velocityof neutral gas
The first in situ measurementof a cometary neutral atmospeciess, ms is the mass of neutral gas speciess, Ps is the
sphere will not be made until later this year; nevertheless,
kinetic pressure of neutral gas speciess (=nskTs), Ts is the
remote optical observationshave provided some clues about
temperatureof neutral gas speciess, Gs is the external volume
the generalnature of comas.The limited information presently
force (e.g., gravity), CSns/&
is the density source term due to
available makes discussionsof generalcometary atmospheres
collisions,cSMs/&is the momentum source term due to colmeaningful,even though it is known that there are significant
lisions,and •Es/•t is the energysourceterm due to collisions.
differencesin the physicaland chemicalmakeup of the differAs stated earlier, the energy equation (40c) was obtained
ent comets.Studiesof cometary atmospheresgenerallyassume
neglectingthe internal energyof the molecules.It is commonly
that water vapor is the major parent molecule, with only a
assumedthat the average energy per particle with internal
minor amount of other volatilespresentin the nucleus.In the
degreesof freedomcan be written as [cf. Burgers,1969]
following subsectionsthe main physical and chemical processes,which are believed to control the gross behavior of
Ws= «VskOs
(41a)
Dsus
•Ms
3Dsp
s 5
comas, are outlined.
In order to carry out quantitative studiesof the gasesflowing away from the surface of the nucleus, the appropriate
coupled set of conservationequations have to be solved. A
comprehensivesummary 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 discussedthese equations, applicable to atmospheric
and plasma studies,include Holt and Haskell [1965] and Tanenbaurn[1967]. All these equations are obtained by taking
moments of the Boltzmann equation. Some differencesdo
exist between the equationsderived by different authors, becausesomeauthorsobtain the moment equationswith respect
to the random velocity, while others use the actual (total)
velocity.
As mentionedearlier, the most comprehensivepresentation
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
environmentare no longer appropriate.
Schunk [1975] presented a general system of transport
equationsfor flowing neutral gasesand plasmas,which were
derived by using Grad's [1958] formulation and Burgers'
[1969] collisionterms.Thesesystemsof equations,sometimes
referredto as the "13 moment equations,"includecontinuity,
momentum, internal energy, pressure tensor, and heat flow
equationsfor each speciesunder consideration.However, consideringour very limited present-dayunderstandingof com-
(40b)
cSE
s
(40c)
where vs is the number of internal degreesof freedomand Os
is the effectivetemperatureassociatedwith the internal motions.When the gasis in thermodynamicalequilibrium,Os has
the same value as T•, which we defined as the temperature
associated
with
the translational
motion.
Under
these con-
ditionsthe averagetotal energy,Us,is
= +1V
s)kTs
(4lb)
It follows that the specificheat at constantvolume, Cv,is
(• 1)k
C,,= +• vsm
(41c)
The ratio of specificheats,commonly denoted as 7, can thus
be written
as
Cv
5 + vs
7 = -- C,, 3 + Vs
(41d)
If the moment of the Boltzmann equation is taken with
respectto not only the kinetic (translational)energy but the
sumof kineticand internalenergies,the resultingenergyequation (neglectingconductionof internal energy)for the total
energy is
Dt ps+ nsUs
+
)
ps+ nsUs
(V.us)+V-qs-•t
(41e)
Now usingthe definitionof the polytropicindex, 7, as given
680
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
TABLE 2. Hard SphereRadii of Typical Cometary Species[Allen,
1973]
The secondterm on the right-hand side of equation (42b)
Radius,
nm
Species
H20
0.175
OH
0.145
H2
CO 2
0.11
0.19
CO
0.17
CH½
NH 3
0.175
0.15
about 3.35x 10-•'• cm2 [Snowet al., 1973] comparedwith
thehardspherecrosssection
of 9.62x 10- x6cm,•.
accounts for the thermal
diffusion
and thermoelectric
tSEs
--•nsmsVs•
3k(TsTt)
tSt
in (41a), the aboveequation can be transformedinto the form
commonly used in hydrodynamics[cf. Zucrow and Hoffman,
1976]'
1
•-
Dsps
7
6Es
+
ps(V' Us)+ V'qs 1 Dt
•- 1
cSt
(41f)
There is a whole hierarchyof approximationsfor the collisional sourceterms.Relativelylow order approximationscan
be adoptedfor cometsgivenour very limited understanding
of
cometaryatmospheres.
The densitysourceterm is taken to be
simply the chemicalproduction minus loss rate of a given
neutral species'
cSn
s
- p/-- l•'
effects.
In neutral atmosphericapplicationsthe latter effectis generally neglectedand even thermal diffusionis negligible,unless
very large temperaturegradientsare present(R. W. Schunk,
private communication,1985).
The energysourceterm for neutral gas mixtureswas given
by Schunk[ 1975], for hard sphereinteractions,as
(42a)
where Ps'is the productionrate of neutral speciess and ls' is
the lossrate of neutral speciess. Although we mentionedonly
chemical production and loss rates, processessuch as sublimation of icy grainsand ionizationof neutral gasescan easily
be includedin the productionand lossrates,respectively.
Rigorous derivationsof the momentum and energysource
terms have, in general, been carried out consideringelastic
collision processesonly, with inelastic processesintroduced
only at the end in a heuristicmanner. The momentum source
term, considering only elastic hard sphere interactions
[Schunk,1975], is
ms+mt
(44)
The above energy sourceterm accountsfor energy transfer
betweenneutral gas speciess and t via collisions;collisions
betweenthe neutral gas,s, and ions and electronscan also be
accountedfor by the aboveexpressionif the appropriatecollision frequenciesare used.There is another sourceof energy
which is important in cometaryatmospheresand which needs
to be added, in a somewhat heuristic manner, to the above
equation. Radiative energy absorption,scatteringand 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)representthe
energytransfermechanism,which has to be considered;therefore an approximationof the total energysourceterm, appropriate for cometaryatmospheres,
is
6Es• nsmsVs,
3k(T
s Tt)
qQcxt(45)
fit
ms+ rnt
where Qcxt--Qraaq- Qeh is the rate of net external heating/cooling.The main contributionto radiative coolingis the
infrared radiation from the H20 molecules.In order to obtain
an appropriate quantitative value for this net heating term,
very complexradiative transfercalculationsare necessary.The
following semiempiricalequation for H20 radiative cooling
was given by Shimizu[1976]'
Qrad(
H2O)=- 8.5
X10•9Tw
2rtw2
ergcm-3s-•
nw+ 2.7 x 107Tw
(46)
where nw and Tw are the H20 number density and temper(St- • nsmsVs,(Usu,)-0.2 Vs,
•-•s
• qsPt
s(_psis)
ature, respectively.However, this expressiondoes not take
t
(42b)
The momentum transfer collision frequency between neutral
speciess and t (rs,),the reducedmass (#s,),and the reduced
temperature(Tst)can be expressedas
#st= •
msmt
ms +mt
Tst=
-
mits + msT•
mt+ ms
8r•
ø'sntmt
rs,2(2k
Tst•
ø'5
Vst 3 ms
+mr \-•s•/
(43a)
into
account
the above
mentioned
radiative
transfer
effects
inside the dense coma. Huebner [1985] has indicated that
radiative coolingis not important in the inner coma because
of radiative trapping. He obtained the following estimatefor
the infrared optical depth •IR (seesection3.2 for the definition
of optical depth)at a distancer from the nucleus'
•IR --'
0.4nsaaR,•
2
r
(47)
(43b)
(43c)
where rstis the sum of the radii of the collidingparticles(see
Table 2 for a few representativevalues).
Estimatesof equivalenthard sphereradii 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 measuredtotal
scatteringcrosssectionsare significantlygreaterthan the hard
spherecrosssectionscalculatedfrom the given radii. For example, the measuredvalue of the H20-H20 crosssectionis
where ns is number density of absorbinggas at the nucleus
and aa is mean infrared absorptioncrosssection(an approxi-
mate value of aa= 4 x 10-•5 cma was used by Huebner
[1985]). Using this approach,Huebner[1985] gavethe following modifiedexpressionfor Shimizu's[1976] radiative cooling
formula'
Qrad(H20
)= - 8.5
x10-19Tw2nw
2exp(-- IIR)ergcrn-3 s- •
nw+ 2.7 x 107Tw
(48a)
Recently, Crovisier [1984] has published an improved
GOMBOSI
ETAL.' DUST,NEUTRALGASMODELINGOFCOMETARY
INNERATMOSPHERES
681
TABLE 3. Incident Solar UV Flux Data (5-105 nm) for Solar Minimum (SC 21REFW) and Solar
Maximum(79314)TogetherWith Photoabsorption
and PhotoionizationCrossSections
(•abs
(•ion
(•abs
(•ion
(•abs
(•ion
(•ion
nm
•,
Omi
n
Oma
x
H 20
(•abs
H 20
(•ion
CO2
CO2
CO
CO
N2
N2
O
5.0-10.0
10.0-15.0
15.0-20.0
20.0-25.0
1.60
0.45
7.18
5.75
1.25
2.33
3.47
4.97
1.25
2.33
3.47
4.97
6.00
7.20
6.82
7.80
7.80
8.60
10.20
10.41
12.32
13.80
14.52
6.00
7.20
6.82
7.80
7.80
8.60
10.20
10.41
12.32
13.80
14.52
4.42
7.51
11.03
14.98
17.88
21.21
20.00
23.44
23.44
23.88
25.70
25.81
27.52
28.48
29.27
1.92
3.53
5.48
8.02
0.73
5.68
6.69
3.67
12.65
9.47
1.93
3.77
1.40
0.51
2.60
4.42
7.51
11.03
14.98
17.88
21:21
20.00
23.44
23.44
23.88
25.70
25.81
27.52
28.48
29.27
1.92
3.53
5.48
8.02
25.630
28.415
25.0-30.0
30.331
30.378
30.0-35.0
36.807
35.0-40.0
40.0-45.0
46.522
45.0-50.0
0.38
0.13
1.84
0.92
0.27
0.10
0.84
0.24
6.00
0.87
0.74
0.21
0.39
0.18
0.31
10.02
11.70
11.01
12.52
12.47
13.61
15.43
15.69
18.01
19.92
20.09
10.02
11.70
11.01
12.52
12.47
13.61
15.43
15.69
18.01
19.92
20.09
0.60
2.32
5.40
8.15
9.65
10.60
10.08
11.58
11.60
14.60
18.00
17.51
21.07
21.80
21.85
0.60
2.32
5.40
8.15
9.65
10.60
10.08
11.58
11.60
14.60
18.00
17.51
21.07
21.80
21.85
6.03
7.12
6.78
7.64
7.64
8.05
9.82
9.91
11.22
11.00
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
58.433
55.0-60.0
60.976
62.973
60.0-65.0
65.0-70.0
70.331
70.0-75.0
76.515
77.041
79.015
75.0-80.0
80.0-85.0
85.0-90.0
90.0-95.0
97.702
95.0-100.0
102.572
103.191
100.0-105.0
0.80
1.58
0.48
0.45
1.50
0.17
0.22
0.39
0.17
0.20
0.24
0.48
1.16
1.93
4.43
4.22
5.96
1.79
4.38
3.18
3.63
1.84
5.92
1.19
2.29
3.50
0.69
0.55
0.82
0.51
0.51
1.02
1.10
3.46
16.00
16.13
14.54
15.49
5.26
15.93
10.92
10.30
18.00
21.00
19.20
21.40
21.40
21.40
21.40
21.50
21.01
19.30
18.50
16.60
18.25
17.16
19.34
20.02
15.80
16.12
8.80
7.60
9.04
18.00
20.30
18.50
20.40
19.80
19.82
16.93
15.20
15.15
15.50
14.20
13.90
14.22
12.76
10.52
7.86
5.20
5.60
0.00
0.00
0.00
33.20
34.21
34.00
25.31
25.86
25.88
25.96
21.76
22.48
53.96
26.48
21.79
31.83
12.84
49.06
70.89
29.91
34.41
15.10
14.90
18.18
33.20
34.21
34.00
20.16
21.27
21.14
21.72
17.71
17.02
50.39
20.00
17.07
21.53
10.67
19.66
0.00
0.00
0.00
0.00
0.00
0.00
22.28
22.52
22.41
18.42
18.60
19.78
25.59
24.45
25.98
26.28
15.26
33.22
21.35
22.59
37.64
49.44
28.50
52.90
0.01
0.00
0.00
22.31
21.38
21.62
16.93
16.75
17.01
17.04
16.70
17.02
12.17
9.20
15.44
11.38
17.13
11.70
0.00
0.00
0.00
0.00
0.00
0.00
24.69
23.20
22.38
23.10
23.20
23.22
29.75
26.30
30.94
35.46
26.88
19.26
30.71
15.05
46.63
16.99
0.70
36.16
0.00
0.00
0.00
24.69
23.20
22.38
23.10
23.20
23.22
25.06
23.00
23.20
23.77
18.39
10.18
16.75
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
12.47
13.21
12.85
13.22
13.18
13.12
10.31
8.32
6.85
4.38
4.25
3.70
4.12
3.95
0.79
0.02
0.00
0.00
0.00
0.00
0.00
1.92
1.63
3.00
5.05
The fluxesare givenin 109 cm-2 s-x units,whilethe cross-section
valueshaveto be multipliedby
10-x8 crn2. The informationpresented
in this table,exceptthe H20 and O cross-section
values,was
obtainedfrom Torr et al. [1979] and Tort and Tort [1985]; the H20 crosssectionswere taken from
Cairns et al. [1971] and Katayama et al. [1973], and the O crosssectionswere taken from Samsonand
Pareek [1985].
model of the water vapor radiativecoolingeffectassumingno
equilibrium between the ortho and para states. Crovisier
[1984] presenteda cooling rate curve computed using the
GEISA spectroscopic
data bank. Crovisier's[1984] resultsindicatethat in the densepart of the coma the radiativecooling
is much lessimportantthan $himizu[1976] estimated(by a
factor of 10 at T = 30 K, a factor of 3 at T = 100 K). The
followingexpressionis an analytic approximationto Crooislet's [1984] results(combinedwith Huebner's[1985] optical
depth correction):
Qrad(H20)
= -4.4 x 10-22Tw3'35nw
exp(--'CiR)
T<52K
Qrac!(H20)
= --2.0 x 10-2øYw2'47J%
exp(--'•'I10
T>_52K
(48b)
Very recently,Marconi and Mendis [1986] have proposeda
pioneeringidea to considerthe trappingof infraredthermal
radiation by collision-dominatedwater vapor. They took into
considerationthe excitation of rotational/vibrational levelsof
water moleculesby the dust-generatedthermal radiation. A
large fraction of this elevated intei'nal energy is then transformedvia collisionsinto translationalenergy.This additional
energy source results in significantlyincreasedgas temperaturesin the regionof adiabaticexpansionand also in larger
gas terminal velocities(however,dust terminal velocitiesare
hardlyaffected).Theseeffectswill be discussed
in moredetail
in section 4.
3.2. Photochemistry
•
The evaporatingvolatilesleavingthe nucleusundergonumerousphysicaland chemicalprocesses.
The physicalprocesseswere briefly reviewedin section3.1, and in this section
the photochemicalprocesses
are discussed.
The freshlyevaporated molecules,called parent molecules,are rapidly photo-
dissociated
or photoionized,
andtherefore
mostof thechemical kinetics of cometary atmospheresinvolve the resulting
highlyreactiveradicalsand ions.
Model calculationsof the photochemistryneedto start with
an assumedparent moleculecompositionat the surface,followed by a self-consistent
calculationof the densitiesof all
absorbingspeciesand solarflux intensityas a functionof the
radial distancesfrom the surface.Most of the absorption of
the solar radiation is believedto take placeat the surfaceor in
the denseregion near the surface,wherethe parent molecules
682
TABLE 4.
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARY
INNER ATMOSPHERES
Incident Solar UV Flux Data (105 nm< 2 < 200 nm)
and PhotoabsorptionCross Sectio•ns
(1)rain,
109cm-2 s-1
105.0-110.0
110.0-115.0
(1)....
109cm-2 s-1
rr(H20),
rr(CO2),
10-x8 cm2 l0 -18 cm2
3.07
0.70
7.08
1.77
2.78
300.00
7.24
927.27
4.58
5.90
120.0-125.0
125.0-130.0
5.20
4.10
13.152
12.52
7.44
7.18
130.217
1.16
2.42
7.30
130.486
130.603
133.453
1.19
1.29
1.96
2.49
2.71
4.53
6.80
6.80
4.80
133.566
2.68
6.2{)
4.70
115.0-120.0
121.567
5.76
8.80
130.0-135.0
139.376
135.0-140.0
140.277
140.0-145.0
145.0-150.0
154.820
150.0-155.0
155.077
156.100
6.18
1.40
6.18
0.98
9.53
15.44
4.24
25.84
2.19
1.10
14.37
3.54
18.86
2.48
25.17
35.18
10.75
51.17
55.59
2.55
5.20
0.90
1.75
0.80
0.60
0.95
2.50
1.80
2.50
2.60
155.0-160.0
160.0-165.0
165.720
165.0-170.0
170.0-175.0
175.0-180.0
37.83
55.85
4.09
162.22
230.73
374.24
68.13
104.76
9,46
209.63
319.92
461.90
3.10
4.40
4.90
4.80
3.80
2.00
180.0-185.0
185.0-190.0
190.0-195.0
609.76
779.67
1078.23
776.33
955.95
1332.81
0.30
0.013
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 presentedin this table was obtained from H. E.
Hinteregger (private communication, 1985), Hudson [1971], and Nicolet [ 1984].
where
,(2, r)= • o'sa(.•) dr'ns(F)
(50)
here dr' is the incremental distancealong the path from the
radial position in the direction toward the sun.
The photodissociation,PdS(r),and photoionization, PiS(r),
rates for speciess can be calculatedat a given radial position
if thesolarfluxat that positionandthedissociation
asd(,•)
and
ionizationasi(2)crosssections
areknown'
d2 1(2,r)asa(2)
©
,PiS(r)
= ns(r) d2 1(2,r)asi(2)
©
PaS(r)
= ns(r
)
(51)
(52)
There is quantitativeinformationavailableon the spectrum
of the unattenuated EUV solar radiation at wavelengths
below 185 nm; most of these data were obtained by spectrophotometerscarried aboard the AtmosphereExplorer (AE)
satellites
C, D, andE [Hinteiegger
et al., 1973;Hinteregger,
1977]. Hinteregger[1981] selectedthe period of July 13-28,
1976, as the referenceperiod representativeof solar conditions
at minimum activity for cycle 21 (Rz = 0; F•0.7 = 68), The
detailedreferencespectrumof EUV irradiancefor wavelengths
below 200 nm, correspondingto solar minimum conditions,is
given in terms of 1659 wavelengthincrementsand is available
on a magnetic tape from the National Space ScienceData
Center as file number SC# 21REFW. H. E. Hinteregger(personal communication,1985) also provided a referencespectrum correspondingto a period of highestsolar activity, No-
vember1979(R: = 302; F•0.7 = 367);this spectrumis referred
to as F79314. For most aeronomiccalculations,such spectral
resolution, representedby 1659 wavelength increments,is not
necessary;therefore Torr et al. [1979] and Tort and Torr
[1985] mergedthe data sets,for wavelengthsbelow 105 mm,
dominate. Therefore it appearsat first that it should be relainto
37 wavelengthintervals.Torr et al. [1979] also provided,
tively easy to calculatethe spectralintensityof the solar flux
appropriately averaged,absorption and ionization crosssecas a function of the distance from the comet surface once a
tions for N2, 02, O, CO2, and CO correspondingto these37
parent gas compositionis selected.However, multiple scatterwavelength
increments;the onesrelevant for cometaryatmoing of the solar radiation and thermal reradiation by the outspheresare presentedin Table 3. The solar flux data for waveflowing dust grains means that complex and self-consistent
lengthsof lessthan 105 nm for both solar cyclemaximum and
radiative transfer calculations are necessary,in general, to
obtain reliable flux intensities [l-lellmich, 1979, 1981; Weiss- minimum are also given in Table 3, along with H20 cross
sections,which were compiled from various sets of experiman and Kieffer, 1981, 1984; Marconi and Mendis, 1982, 1983,
1984, 1986]. Such detailed radiative transfer calculationsare mental data [cfi Hudson, 1971]. Fluxes in the wavelength
region 105-200 nm for the July 1976 and November 1979
especiallyimportant in calculatingthe entergybudget of the
conditionsare given in Table 4, along with corresponding
nucleus itself and regions near the surface.The net result of
H20 and CO2 absorptioncrosssections.
theseprocessesis to increasethe amount of radiative energy
The discussion,up to this point, has been limited to the
reachingthe comet and thus leading to increasedgas and dust
referencespectra of July 1976 and February 1979; however,
production rates and outflow velocities.However, for photowhat one needs in most cases is the solar irradiance for a
chemicalcalculationsonly the ultraviolet radiation with wavegiven date, which means that some appropriate scalingof the
length of less than about 200 nm is important, and in this
referenceflux is necessary.
The moststraightforwardapproach
wavelength region it is reasonable to assumethat the only
to suchscalingis basedon certain measuredvariablesassocieffect of the dust is that of an absorber [cf. Mendis et al.,
ated with key EUV emissions.Hinteregger [1981] recom1985].
mended that two classes of emissions should be considered:
The calculation of the dissociatingand ionizing solar flux
K - 1 for which the key index is the emissionat 2 = 102.6nm
(2 < 200 nm) requiresa knowledgeof the number densitiesof
(H Lye) and K = 2 for which the emissionat 2 = 33.5 nm (Fe
the neutral constituents,ns(r), as a functionof radial distancer,
XVI) is usedas the appropriateindex. Thus an estimateof the
the absorptioncrosssectionof theseconstituentsas a function
solarflux at wavelength2 is givenby the followingexpression:
of wavelength,asa(2),and the spectrumof the unattenuated
solar radiation 1©(2). In terms of those quantities,the solar
(53)
I(•) = lrcf(•)[1 + Cx(RK -- 1)]
flux at any distancer is given by
where RK is the ratio of flUX intensity at the desired time to
(49)
1(2, r) = I o•(2)exp [-,(2, r)]
that of the referencevalue (• = 102.6 nm or 33.5 nm for K = 1
GOMBOSI
ETAL.:DUST,NEUTRALGASMODELINGOFCOMETARY
INNERATMOSPHERES
683
or 2, respectively)and C• is the effectivecontrastratio provid- TABLE 6. Solar Cycle Minimum PhotodissociationFrequencies
(Rates) at 1 AU
ed with the referenceflux. Data on thesekey emissionfeatures,
as well as a few more selectedlinesand intervals,coveringthe
Mean
time period of July 2, 1977, to December 31, 1980, are also
Dissociation
Excess
available
fromtheNationai
Space
Science
DataCenter
under
file number SC # 210BS.
This scaling approach, of course, only works for time
periodsfor which thesekey EUV parametersare provided.
For periods when such direct information is not available,
some other easily available parameter indicative, at least indirectly,of EUV activity needsto 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 acceptedand
usedas an index of solar EUV activity.The followingrelation
for the ratio of the solar flux on a given day to that of the
referencevalue was suggested
by Hinteregger[1981]:
l(•)/lrer(I[) = Bo + Bi{(F,o.?)8,a- 71.5)
+ B2{F,o.?- (F,o.?)8,a + 3.9)
(54)
whereF•o.7is the dailyvalueof the 10.7-cmflux (10-22 W
m-2 Hz -a) and (Fao.7)8aa is the 81-dayaveragevalueof the
Parent
Dissociation
Threshold,
Molecule
Products
nm
H(ls) + H(2s,2p)
CH
C+ H
358.99
OH
O + H
282.30
C2
C+ C
203.0
0.17
CN
C + N
160.0
4.0
1.0
1.110
1.10
1.509
0.01122
0.01026
0.03123
0.02678
0.00256
0.00251
0.00590
0.00567
20.6-•5,5
1.0
0.02391
0.00711
25.5-30.0
1.184
1.0
0.00708
0.01259
0.952
1.0
1.286
1.0
1.290
1.0
1.311
1.0
1.310
0.02228
0.03830
0.03656
0.00605
0.00647
0.00964
0.00714
0.01130
0.00876
0.01101
0.00830
0.01380
0.01106
1.0
19.0-20.6
"30.4"
51.0- 58.0
58.4
59.0-60.6
102.6
33.5
1.196
1.0
0.00326
0.00255
0.00242
0.00438
0.00425
0.00244
0.00230
0.00500
0.00492
0.59425
0.38i 10
-6.618
0.66159
0.38319
121.6
1.0
0.01176
0.00306
28.4
1.046
1.0
0.01136
0.22811
0.00605
0.11638
0.731
1.0
1.726
1.0
1.169
1.0
0.23050
0.04 562
0.03928
0.01272
0.01225
0.00616
0.11643
0.00918
0.00885
0.00290
0.00282
0.00178
1.018
0.00600
0.00177
20.0-20.4
17.8-18.3
16.9-17.3
The information in this table was provided by H. E. Hinteregger
(private communication,1985).
0.28
0.035
72.
0.66
4.6
12.
6.9
2.6
2.3
2.2
3.4
191.0
2.2
1.8
0.060
8.3
O(3p) q- O(•D)
O(•S) + O(•S)
175.9
92.3
4.2
0.041
1.3
0.82
SO
S+ O
231.4
NH 2
H20
HCN
NH + H
H + OH
H 2 + O(tD)
H + CN(A2IIi)
300.0
242.46
177.0
192.0
H2S
CO2
HS + H
CO(XXZ+) + O(3p)
317.0
227.5
3.3x 103
0.017
0.77
1.7
CO(XtZ +) + O(•D)
CO(a3I'I)+ O
167.1
108.2
0.92
0.28
4.3
0.20
N 2 + O(XD)
N2 + O(X$)
CO + S(3p)
340.7
211.5
399.0
1.0
4.9
17.
2.8
6.8
2.7
CO + S(•D)
CO + S(X$)
CS + O(3p)
CS + O(XD)
SO + O
S + O2
291.0
212.0
182.0
141.0
221.0
207.0
54.
30.
0.069
6,3
190.
58.
1.9
2.1
0.13
0.85
0.49
0.68
CS(XIZ+) + S(3p)
277.8
320.
0.77
CS(X•E +) + S(•D)
CS(a3H)+ S(3p)
CS(AXIl)+ S(3p)
NH2 + H
NH(a xA)+ H2
221.1
157.1
133.7
279.8
224.0
22.
4.7
52.
170.
4.0
0.28
0.69
0.92
1.8
1.7
NH 3
C2H2
H2CO
H + C2H
H 2 q- C2
H 2 + CO
H + HCO
H + H + CO
,
0.01254
0.00325
18.
242.37
NH + H + H
16.8-19.0
0.48
O(3p)q-O(3p)
TABLE 5. The B Parameters AssociatedWith Equation (57) for
Solar UV Flux Variation Adjustments
B2
0.44
1.2 x 104
N q- O
CS2
B•
0.034
02
SO2
Bo
111.78
86.34
243.18
127.04
eV
NO
10.7-cm flux. Hinteregger [1981] provided best estimatesfor
N20
the parametersBo, B•, and B2 for 15 selectedlines and intervals. Two setsof theseB parameterswere obtained by fitting OCS
the availabledata base.Bo was set to unity for one set,which
means that the calculatedsolar flux reducesto the reference
value for July 1976 when the appropriate Fao.7 values are
it, nm
10 -6 s- •
H2
CO(X•; +) C + O
C(xO)+ O(xO)
CO(a3H) C q- O
N2
N +N
84.48
Frequency, Energy,
HNCO
NH(c•II) + CO
H + NCO(A2•:)
CH,•
CH3 + H
CH2(a•A0 + H 2
CH + H 2 + H
HNO 3
OH + NO 2
C2H,,
C2H2 q-H 2
C2H2 + H + H
CH3OH H2CO + H 2
CH 3 q-OH
HO2NO2 HO 2 + NO 2
OH + NO 3
CH3CHO CH 4 + CO
CH 3 + HCO
CH3CO + H
C2H6
C2H,, q-H 2
CH 3 + CH3
C2H5 q-H
CH 4 + CH 2
147.0
620.
2.1
10.
1.4
13.
2.0
0.62
6.4
3.4
3.5
4.3
2.1
230.6
200.6
> 700.
10.
2.7
160.
334.
275.0
84.
32.
0.37
3.
354.
253.
277.
237.3
137.
598.
720.
196.
•700.
• 315.
1340.
730.
-,•700.
350.1
337.5
874.3
322.
15.
14.
1.2
5.5
0.5
210.
24.
23.
250.
13.
•330
•330.
•9.8
24.
22.
3.7
0.88
5.1
4.1
5.9
5.2
1.9
4.3
6.2
1.7
6.5
4.4
4.2
3.4
3.7
0.87
0.43
9.0
7.4
290.
272.6
3.3
2.2
3.2
3.1
> 2.1
6.8
6.1
From W. F. Huebner(work in preparation,1986)and Mendiset al.
[1985].
substituted.Bo wasnot forcedto unity for the other set,which
resulted in a somewhat better overall fit (higher correlation
coefficients)
to the data base,but in this casethe referenceflux
is not recoveredexactlyfor the July 1976 period.Both setsof
684
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
TABLE 7. SomeAssumedCompositionsfor the Frozen Gasesin the Nucleus(%)
Composition Number
Species
1
61.110
30.450
8.440
H20
CH4
NH 3
CO2
CO
2
55.560
3
4
5
53.340
48.890
48.890
11.110
11.110
11.110
11.110
2.220
11.110
33.330
33.330
28.890
...
...
11.110
7
8
9
43.004
13.464
0.080
33.004
1.650
0.094
61.127
5.094
0.007
74.036
.....
12.057
...
28.890
H2CO
N2
6
ß
HCN
CH3CN
NH2CH 3
H2C3H2
C2H2
HCOOH
2.8!4
22.105
.....
25.931
1.698
13.583
0.082
5.225
6.129
15.282
0.482
0.402
0.201
0.005
0.161
ß.ß..
0.566
0.471
0.236
0.236
0.094
17.445
14.144
0.102
...........
0.020
0.071
.........
0.815
7.404
24.886
12.428
.-.
."
0.039
2.941
1.420
0.012
0.012
...
2.231
0.125
0.031
......
02
C2H,,
......
1.019
......
......
......
......
0.061
.-.
HCN
......
0.020
......
H2N2H2
HC•H
.........
.........
1.019
62.433
0.001
0.002
4.970
4.462
NO
CH3OH
10
1.826
0.710
0.031
--.
-.-
Reproducedfrom Huebner[1985]' for details,referto the originalreference.
B parametersare given in Table 5. It should be noted that if
an estimate of the solar flux over wavelengthintervals not
coveredby the B parametersis desired,one needsto estimate
the two key wavelength intensities at • = 33.5 nm and
• = 102.6 nm, usingequation(54), and then useequation(53)
to scalethe referenceflux as necessary.
A useful quantity for many aeronomic calculations is the
frequencyat which a given process(e.g., photodissociation)
takes place owing to the unattenuated solar flux outside the
atmosphere.Such photodissociationfrequenciescan be calcu-
duction rates as a function of heliocentric distance, for a
numberof cometsof interest,are givenin Tables 11-14.
The dissociatedand ionized moleculesand atoms participate in a wide variety of chemical processes;some of these
reactionslead to certainimportant constituents,which are not
lated from the available
Up to this point the discussions
only dealt with photodissociationand photoionization;respectiveexamplesof these
solar flux and cross-section infor-
mation and are independentof the atmosphericdensities.W.
F. Huebner (work in preparation, 1986)calculateda wide variety of these frequenciesfor solar cycle minimum conditions;
the onesmost relevantfor cometaryatmospheresare givenin
Table 6. Table 6 also givesthe energythresholdfor the given
immediatelyapparentfrom the parentmolecules
(e.g.,H30 +
as the major ionic component).We give here some representativeexamplesof thesevarious processes,
and in Table 8
the best available rate coefficientsof interest, for calculations
of cometaryatmospheres,are summarized.
are
H20 + hv(/l_<242.46nm)=•OH(X2I-[)+ H(2S) (55)
H20 q-hv(/•< 98.3nm)=• H20 + q-t;
(56)
dissociationand ionizationprocesses
in termsof wavelength.
Photodissociative
ionization,an exampleof whichis shown
(The following simplerelationshipallows this information to
be transformed from wavelength, /•, to eV if so desired: below,is alsoimportant in a numberof circumstances'
E(eV) = 1239.8/;t(nm).)
H20 + hv(,•_<66.5nm)=,,H: + O* + e
(57)
There have been no direct in situ measurements of the neu-
tral gas composition and abundancesin the coma. Estimates
of the total densitiesand compositionhave been based on
remote optical observationsand guesseshave been basedon
present ideas on how and where comet nuclei were formed.
Table 7, reproduced from Huebner [1985], summarizesthe
variety of suggestedcompositionratios in the nucleus.In
Three-bodyrecombinationis anotherprocesswhich is important in certain cases,specificallywhen the total densityis
high'
O + O + M =,,02 + M q- 5.12eV
(58)
Dissociativerecombinationof molecularions generatedisorder to get an accuratedescriptionof the parent molecule sociationproducts,which often have relatively high kinetic
densitiesin the atmosphere,
one needsto correctlymodelthe energiesand/or are in an excitedstate:
sublimation/evaporation processes followed by a selfO2+ + e =• O(•D) + O(3P)+ 4.98eV
(59)
consistentsolutionof the coupledcontinuity,momentum,and
An exampleof atom-atom (ion) interchange,which often
energy equations,which is such a complex undertaking that
only limited studiesof this type have beencarriedout to date leads to the formation of a vibrationally excited molecule,
(seesections2 and 4). A zero-orderapproximationto the 02(v < 3), is
parent molecule distribution in the inner coma can be obtained by (1) selectingan initial compositionratio from Table
7, (2) assuming
a 1/r2 densityvariation,and (3) selecting
an
appropriate total gas productionrate. Estimatesof gas pro-
O(3p)+ OH(X2II)=:-H + O2 + 0.72eV
(60)
One important aspectof processes
suchas (55)-(59) is that
the energycarriedby the dissociation
or reactionproductscan
GOMBOSI
ET AL.' DUST, NEUTRALGAS MODELINGOF COMETARY
INNERATMOSPHERES
largelygo into heatingthe neutraland/or ionizedgas,at least
in the collision-dominated
regionof a cometaryatmosphere.
685
1935]. In early treatmentsof the gas-dustinteractionit was
assumedthat the dust drag coefficientwas independentof the
gasparametersand that the gasvelocitywasconstantin the
4.
4.1.
DUSTY GAS FLOW IN THE NEAR-NUCLEUS
REGION
GoverningEquations
It wasrecognizedas early as the mid-1930sthat gasoutflow
playsan important role in cometarydust production[Orlov,
TABLE 8. Chemical Reaction Rates of Potential Importance in
Cometary Atmospheres
Rate
Coefficient,
cm-3 s-1
Reaction
H20 + O(1D)• OH + OH
H20 + H2O+ --• H3O+ + OH
H20 + OH + --• H30 + + O
--• H20+ + OH
H20+H +--•H2O+ +H
H20+O +--•H2O+ +O
H20 + CO2+ --• HeO+ + CO2
--• HCO2 + + OH
H20 + CO +--• H20 + + CO
--• HCO + + OH
H20 + N2 + --• H2O+ + N2
OH + OH--, H20 + O
OH + O--, 0 2 + H
OH + CO--, CO 2 + H
OH + H2O+ --• H3O+ + 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
CO2 -•' OH + --• HCO2+ + O
--, HCO + + 02
CO2+H +--•HCO + +O
CO2 -•-O + '•O2 + -•-CO
CO2 + CO + --• CO2+ + CO
CO2 + N2 + --• CO2+ -¾N 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,• + H2O+ --• HaO+ + 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
2.05
1.3
1.6
8.2
2.3
1.7
5
1.7
9
1.6
4.2
x
x
x
x
x
x
x
x
x
x
x
x
10 -1ø
10 -9
10 -9
10 -9
10 -9
10 -9
10 -9
10 -1ø
10 -9
10 -1ø
10 -9
10-12
2.2 x 10-11 exp (117/T)
9.30 x 10-13
6.9 x 10 -1ø
7 x
3.6 x
3.6 x
3.1 x
3.1 x
6.3 x
7.96 x
1.4 x
1.73 x
1.05 x
1.58 x
7.96 x
1.00 x
8.00 x
1.40 x
1.00 x
1.6 x
5.4 x
3 x
1.1 x
9.6 x
5.5 x
9.3 x
6.44 x
2.16 x
1.70 x
1.1 x
1.20 x
2.00 x
1.70 x
1.30 x
2.28 x
1.52 x
1.14 x
1.41 x
2.28 x
2.40 x
1.00 x
3.00 x
10 -1ø
10 -1ø
10 -1ø
10 -1ø
10 -1ø
10 -1ø
10 -1ø
10 -9
10 -9
10 -9
10 -9
10-1ø
10 -9
10-1ø
10-1ø
10 -11
10 -9
10 -1ø
10 -9
10 -9
10 -1ø
10 -1ø
10-13
10-1ø
10 -9
10 -9
10 -9
10-12
10 -9
10 -9
10 -9
10 -9
10 -9
10 -1ø
10 -9
10 -9
10 -9
dustgrainshaveno thermalmotionand collideonly with gas
molecules.Theseauthorspointed out that the gas mean free
pathsweremuchlargerthan the dustparticledimensions,
and
consequently,
the gasflow couldbe consideredto be freemolecular relative to the dust component.In Probstein's[1968]
dusty gas dynamictreatment,the traditional gas energyconservation equation was replaced by a combined dust-gas
energyintegral.He assumeda singlecharacteristic
dust size
(a = 0.5 #m) and neglectedany externaldustor gasheatingin
the interaction region. It was later pointed out by Shulman
[1972], and Hellreichand Keller [1980] that the dust grains
are significantlyheatedin the inner coma by multiple scattered solar radiation. Furthermore, calculations of Hellreich
[1981] and Gombosi
et al. [1983] havealsodemonstrated
that
usingrealisticdust sizedistributionsrather than Probstein's
single-sizeapproximationproducessignificantlydifferentresults.
The radiative transferproblemin a dustycometarycoma is
far from simple and has been investigatedby a number of
groupsusingdifferentmethods[Hellreich,1979;Hellreichand
Keller, 1981; Marconi and Mendis, 1982, 1983, 1984]. Most
recently,Marconi and Mendis [1986] have investigatedthe
effectsof the dust thermal radiation on the energeticsof neu-
tral gasand foundthat thisradiationcan significantly
increase
H20 internalenergies.Due to neutral-neutralcollisions,this
elevatedinternalenergycan also increasethe random energy
of the neutral gas, thus resultingin much higher minimum
neutral gas temperaturesin the gas-dustinteractionregion
than predictedby previouscalculations.
Modelingeffortshaveshownthat the spatialextentof the
gas-dustinteractionregionis limitedto lessthan ,-•30 Rn [cf.
Keller, 1983; Mendis et al., 1985]. Gas particles typically
spendabout 100 s in this region;this time scaleis too short
for any significantchangein the grosschemicalcomposition
of the gas.Thereseemsto be a generalconcensus
in the literature that a single-fluiddustyhydrodynamicalapproachis adequatefor describingthe dynamicsof the gas-dustinteraction
region(cf. the mostrecentreviewof Mendiset al. [1985]). In
thissectionthe equationsthat governthe evolutionof gasand
dustparameters
in a spherically
symmetricwatervapordominated inner coma are summarized.
Using this approximation,equations(40) can be simplified,
givingthe followingequationswhichdescribethe mass,momentum,and energyconservation
of the neutralgas:
O(Ap) O(Apu)
+
- 0
Ot
2)
O(Apu) c•(Apu
Ot
10 -9
10 -11
The information in this table was taken from Giguereand Huebner
[1978], Huebnerand Giguere[1980], Mendiset al. [1985], and W. F.
,Huebner(work in preparation,1986).
dust accelerationregion [Whipple, 1951; Weigert, 1959; Dobrovolskii,1966; Huebner and Weigert, 1966]. This constant
velocityassumptionwas droppedin the late 1960s.A twocomponenttreatmentof the gas-dustinteractionwas publishedalmostsimultaneouslyby Probstein[1968], Brunnerand
Michel [1968], and Shulman[1969], assumingthat the heavy
a(1
} a-•-•+ A•rr= -AF,d
(61a)
(6lb)
1
+ 7--1 Apu
a-•• Apu:
+ 7-1 Ap+ •rr Apu3
= A(Qext- Qgd)
(61c)
686
GOMBOSI
ET AL.' DUST,NEUTRALGAS MODELINGOFCOMETARY
INNERATMOSPHERES
where p is gasmassdensity,p is gaspressure,u is gasvelocity,
also calculatedthe gas-dustheat transfer rate per unit dust
A is areafunction
(forspherical
geometry
A = r2),Fgais mo- grain surfacearea'
mentumtransferrate fromthe gasto the dust,Qgais energy
qa = p(U- VaXrrec- ra)CH
transfer rate from the gas to the dust, Qcxtis external heat
source rate (photochemicalheating and infrared cooling). In
the innermost coma, where the gas-dust interaction takes
place, Qextcan be approximated with relatively simple expressions;however,at larger cometocentricdistancesthis term
becomesfairly complicated.
By neglecting collisions between dust particles one can
obtain the following equation of motion for an individual
grain in a radially expandingcometary atmosphere[cf. Wei•]ert, 1959]:
4•
dVa
1
4•
(67)
where the "recovery"temperature,Trec, and the heat transfer
function,Cn, are the following(a correctionmade by Kitamura[1986] is incorporatedinto Tree)'
Tre•-
27 + 2(7- 1)sa
2
-øY3
exp
(--sa
2)erf
-1
_1
0.5 + Sa2 -{"San
G
CH--
'•' paa3
'• = a2•t
• CDP(U
-- Va)2
--•- paa3
7 Mcøm
(62)
7+1
k
(68)
2 [n-ø'Ssa
exp(--Sa2)+(O.5+Sa
2)err(sa)
]
7-1 8msa
(69)
Here Vais the velocity of a sphericaldust particle with size a
and bulk density Pa, CD is the drag coefficient,and u and p
The gasto dust momentumand energytransferrates can be
represent the gas radial expansion velocity and density, reobtainedby integratingover all dust sizes'
spectively.The secondterm on the right-hand side describes
gravitational attraction from a cometary nucleus of mass
Mcom(G is the gravitationalconstant).
In the presenceof an external radiation field the energy
balanceequation for a dust particle is [Probstein,1968]
•dt
mdafa
4•Paa3
dVa
(70)
=Jao
[(4•
3Paa3Va
•alva
+4•a2qa
Qgd
famdafa
) (71)
Fga
=
Here ao representsthe minimum dust size in the mantle, as4•tpaa3Ca
'•'
•dTa
= 4•ra2qa
+ /ra2t•absJr
--4/ra2t•½maTa
4 (63) sumed
to be identicalto the sizeof elementarybuildingblocks
where Ta is dust particle temperature,qa is the gas-dustheat found in chondritic aggregatemicrometeoritescollectedin the
transfer rate per unit surfacearea, Jr is the radiation energy stratosphere[Fraundorff et al., 1982]. The maximal liftable
flux, Ca is the dust specificheat, and eabs
and /•emare the dust dust particle size,am,can be expressedfrom equation(62):
absorption and infrared reemissionemissivities,respectively.
3CDZUoutRn
2
am=
(72)
Finally, the dust size distribution functionfa must obey the
8GM•omPam
followingcontinuityequation:
This formula can be determinedfrom the equationspublished
d(Afa)
%(0
by Wallis 1-1982],but the correspondingformula (72) obtained
by Wallis 1-1982]was incorrectby a factor of 2.
dt - _•ta3pa
5(r-Rn)
(64)
where % is the productionrate (per unit surfacearea) of dust
particleswith radius a. Equation (64) assumesthat oncea dust
particle is releasedfrom the surface,it will be conservedin the
coma: we neglecticy grain sublimationor particle fragmentation processes.
4.3. Steady State SolutionsWithout Radiative Transfer
In early treatments of the gas-dustinteraction [Whipple,
1951; Weigert, 1959; Dobrovolskii, 1966; Huebner and Weigert, 1966] it was assumedthat gas particles had mean free
paths much larger than the grain size and collided elastically
4.2. Gas-DustMomentumand Ener•7yTransfer
with the dust grains.These assumptionsresultedin a constant
In Probstein's[1968] formulation of drag and heat transfer Co = 2 value. Furthermore, thesecalculationsassumedsteady
to a sphericaldust particle in its free molecular environment, state and a constantgas flow velocity u. Using theseassumpgrains are assumedto reflect the gas molecules diffusively tions and neglectingcometarygravity, one can obtain a transaround the surface,to have a perfectthermal accommodation, cendental expressionfor the dust velocities(which is a simand to have a constant surfacetemperature.In this approxi- plified versionof Dobrovolskii's[1966] solution):
mation the free molecular drag coefficientbased on the projected area for a spherewith diffusereflectionis givenby
(73)
2;zø'STaø'52Sa
2+ 1
CD
= 3saTO.
5 + •-
Va+'n(1--•)=Xa(1---•)
wherethe constantXa can be expressedas
exp
(--sa
2)
+
4Sa
4 + 4Sa
2-- 1
2Sa4 err(Sa) (65)
Here T is the gastemperature;sais the gas-dustrelative Mach
number:
u--Va mO.5
sa= (2kT)
ø'5
(66)
Using the free molecular approximation, Probstein [1968]
Xa -
3CDzRn
16apau
(74)
Assumingperfectthermal accommodationand free molecular drag, Probstein[1968] publishednew and more sophisticatedexpressions
for the CD drag coefficientand qaheat transfer functions(seeequations(65)-(69) in this review). He also
solved the coupled gas-dustequations using a single characteristic grain size. This solution representeda major step forward, since the traditional gas energy conservationequation
GOMBOSIET AL.: DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
was replaced by a combined gas-dust energy integral. It
should be noted
that more
recent radiative
transfer
3
calcula-
' I • I ' I ' I ' I
LU•-
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-consistentdusty gas dynamic treatment the gas continuity, momentum, and energy equations
can be combinedto yield the following first-order differential
equationfor the gasvelocity:
n••
•:::)
Z
687
x=o
-
LU•- •
0
•
I
:>.5
:>.7
•
I
•
:>.9
I
•
3.1
I
3.3
3.õ
R {KM)
Fig. 8. Variation of the effectivearea function with cometocentric
distancefor variousdust to gas massproduction rate ratios, X (taken
from Gombosiet al. [1985]).
dr
1 --u
M 2 (•--'
P Y
71
pu
du
Fgd
-Qgd
--Qext.)
(75)
where M is the gasMach number and A' representsthe spatial
derivativeof the area function.Probstein[1968] neglectedQext
and examined
the flow behavior
at various cometocentric
dis-
tances.He pointed out that the gaspressuremust tend to zero
at infinity and at the same time the velocity must remain
finite, so
flow model (cf. section 2.2) and introduced an effective area
function definedby
Aeff dr
lim M = oo
(76)
dtt
action;soFed--•0 andQgd--•0 and
limdr r-.o•
u
dr-
A'
M2A > 0
P
7
pu
(78)
By adopting the effectivearea function, the steady state gas
equation can be written in the following simpleform:
At large distancesfrom the nucleusthere is no gas-dustinterdu
=
tt
Aeff'
1- M2Aef
f
(79)
(77) This equation is just the steady state equation describingthe
Probstein[1968] also proved that the numerator of equation
(75) is negative near the nucleusand has its maximum absolute value at the surface.A limiting caseof this dusty gas flow
model occursif the dust/gasproductionrate ratio (Z) tendsto
zero. In this case,the presenceof the dust does not affect the
gas motion. This is the equivalentof seedingat the sourcean
already establishedsteady, radial, isentropic inviscid compressiblesourceflow. In this case,the gas flow propertiescan
be obtained from the well-known solution for a supersonic
source in which the Mach number at the nucleus (source)
surfaceis taken to be 1 in order to satisfythe appropriate
boundary conditions at infinity and at the surface. On the
other hand, a Z > 0 value indicatesstronggas-dustinteraction
at the sourceresultingin a lower gas outflow velocity,Probstein [1968] concludedthat the gas flow had to be subsonicat
the surfaceand supersonicat large cometocentricdistances.
It is evidentthat M - 1 is a singular(sonicor critical)point
of equation (75). This means that an additional constraint
needsto be imposed:the solution must properly traversethe
sonic point, or in other words the numerator has to vanish at
M = 1. The equation has usually been solvednumericallyby
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;
Gombosiet al., 1983; Marconi and Mendis, 1982, 1983, 1984].
Recently,Gombosiet al. [1985] have publisheda new method
to determinethe gas and dust parametersat the sonicpoint,
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 transonicacceleratinggas 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,Gombosiet al. [1985] have applied their reservoirout-
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 sourcesinfluence the gas
discharge,the effectivearea function is identical with the geometrical area function,A. In this case,the gas flows out of the
stationary reservoir starting at the sonic velocity. The situation changesdramatically when the outflowing gas has had a
chanceto drag away dust. Figure 8 (taken from Gombosiet al.
[ 1985]) showsthe variation of the effectivearea function with
cometocentricdistancefor various dust to gas ratios (Z). In
this particularcalculation,To= T• = 200 K, Rn = 2.5 km, and
7 = 4/3 valueswere used.It can be seenthat for Z = 0, Aeff
hasits minimumvalueat the surface,and hencethe gasvelocity will start at the sonic velocity at the surface.For larger
values of Z, Aeff first decreases,then reachesits minimum
value at the sonic point, and then increases.This shows that
the dust interaction {the dominant external effect near the
nucleus)can in effectbe visualizedby usingthe conceptof a
Laval nozzle.First, the outflow geometry"narrows,"and then
it "opensup." In short, the presenceof dust allows the gas to
start at the nucleus with subsonic velocity and then go
through a sonic point and reach supersonicvelocities.This
transitionwould have beenimpossiblewithout the presenceof
dust. The situation strongly resemblesthe steady state solar
wind equation, first solved by Parker [1958]. The gravitational effect of the sun for the solar wind correspondsto
frictional forcesbetweenthe dust and gas for the outflowing
cometary gas.
Using a single characteristicdust size and conservingthe
combinedgas-dustenergyintegral,Probstein[1968] produced
numericalsolutionsto the steadystatecoupledgas-dustequations. More sophisticatedcalculations later confirmed the
basicfeaturesof his results[I-lellmichand Keller, 1980; Gombosiet al., 1983; Marconi and Mendis, 1982, 1983, 1984, 1986].
In his calculations,Probstein [1968] introduced a dimen-
688
GOMBOSI
ET AL.' DUST,NEUTRALGAS MODELINGOF COMETARY
INNERATMOSPHERES
,
u 1
I •0.65
•a2/3')
+0'225/•aø'5
\(1+0'38Zø'6)
(82)
0.4
02
-
Modifying an earlier Sekanina [1979] fit to the Probstein
[1968] solution, Wallis [1982] published a different expressionfor the dust terminal velocities'
1.5
f
'
O.OI
O.I
4.0
IO
I
V•.oo
= uoo(0.9+ 0.4Zø'5+ 0.5J1•ø'5)
-•
IO
IO0
1.6
X=O
-0.3
1.2
(83)
Thesesemiempiricalexpressions
are usefulonly when orderof-magnitude estimatesare needed.They are only approximations to rescaled numerical results of a single dust size
calculation,and their applicationto a descriptionof a range of
particle sizeshas not beenjustified. Wallis [1982] pointed out
an additional problem with equations(82) and (83): they give
nonphysicalresultswhen both the dust massloading and the
/• similarityparameterare small (in this caseV•,oowill be
.8
larger than Uoo).
A piecewiseapproximate solution for the dust terminal velocity distributioncan be obtainedby solvingfor the J?•-, 0
and the Jla>>1 casesand constructinga function smoothly
connectingthe two limiting solutions.The result is
1.0
0.8
0.6
0.4
0.2
0.01
0.1
I
I0
I00
where
=3Uoo
2C•,
ø'5raø'5
+_reffO.5 (85)
Fig. 9. The variation of outflow Mach number (Mi) and normalized dust terminal velocity with Probstein's dimensionlesssimilarity
parameter,/•a (taken from Probstein[1968]).
sionlesssimilarity parameter characterizing the ability of a
dust particle to adjust to the local gas velocity'
8(C•,To)
aPa
J•a
=5
jRnø'5
Yeff =
Tmin q- tcPa
(86)
Here rmin and Tc representthe minimum gas temperaturein
the gas-dustinteractionregion and the gas temperatureat the
sonic point, respectively.We note that expression(84) does
(80)
whereCpis thegasspecific
heatat constant
pressure
and Tois
the surfacetemperature(Probsteinassumedthat TOwas equal
to the outflow temperature T•). The length l•aRnis analogous
to a meanfreepath: for JlaRn
<<r the dustadjustsquicklyto
I.O
--ß- ß• o...,.
o,.....
o....
%
-
the gas velocity, while for JlaRn>>r the particle remainsmuch
slower than the gas.
Assumingthat T•(Rn)iS identical to To, usingy = 1.4 and
Ca = 5.8 X 106 erg/K/g valuesand makingthe Pa• 0, r a---}
T--} 0 and V•--} u approximations,Probstein[1968] obtained
an analytic expressionfor the upper limit of the velocity of
small dust grains:
UcD
--.....
Va'lim--Uth
y- 1 2(1
+Z)ø'51+ 2 Mi2+ZCl•J
(81)
whereM i is the gasoutflow Mach number.
Probstein[1968] also publisheda set of normalizednumerical solutionsto the coupledgas-dustequations.His resultsare
reproducedin Figure 9, where outflow Mach numbers(Figure
9a) and normalized dust terminal velocities (Figure 9b) are
presentedas a function of Jla.The various curvesare marked
with the valuesof the correspondingdust/gasmassproduction
rate ratio, Z. $ekanina[1981] determineda semiempiricalfit to
these curves and obtained the following expressionfor dust
terminal
velocities:
\\
O. OI
••
I
SEKANINA
2
PRESENT
:3
SEKANINA-WALLIS
4
DOBROVOLSKll
ß
NUMERICAL
I
O. I
1.0
2'
WORK
SOLUTION
I
I0.0
I
I00.0
I000.0
•Q
Fig. 10. Comparison of various analytic approximationsof dust
terminal velocities with steady state numerical solutions obtained
with a Hanner type dust sizedistribution function.
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
not contain the dust/gasmixing ratio, Z, in an explicit form.
On the otherhand,uooand z dependon Z; so the dustloading
does have an influence on the solution. In a first approximation, however, numerical results indicate that in the presenceof externalgas heating(but neglectinginfrared trapping)
uoocan be approximatedas u• • 1.1Uth.
Figure 10 showsa comparisonbetweenthe variousterminal
velocityapproximationsand the resultsof a numericalsolution of the coupledgas and dust equations.When evaluating
expressions
(73), (82), and (83), numericalconstantslisted in
Table 9 were used' TO was taken to be 200 K. From the
numericalsolution, uoo= 0.635 km/s, Uo,,t= 0.281 km/s, u½=
0.300 km/s, Tc= 146 K, z= 1.05x 10-5 g/cm2/s,Z=0-27,
Tmi
, = 5 K, and Ta = 277.8 K valueswere obtained at 1 AU
heliocentric distance. When calculating the Dobrovolskii
[1966] solution,we useda CD = 2 value in equation (73).
It can be easily shown (and seen in Figure 10) that the
asymptoticbehaviorof all four solutionsis similar; Va,•
689
1.5
1.0
•..
--
qC.,o 0.5
0.0
o.
SINGLE
DUST
-'"%SIZE
.,•LUTIONS
- SIZE
DISTRIBUTIO
i
i
i.o
io.o
i
ioo.o
a(•m)
Fig. 11. Comparison of dust terminal velocities obtained with
singledust size solutionsand usinga realisticdust sizedistribution
(takenfrom Gornbosi
et al. [1983]).
lytic or semiempiricalmethodswere summarizedin the precedingsection.Following Probstein's[1968] approach,more
fla-ø'5 as fla--•oo.At thispointit shouldbe notedagainthat
sophisticatednumerical models were developedby several
equations(82) and (83) were obtained using 7 = 1.4, Ca = 5.8
groups.Hellmich[1979], Hellmichand Keller [1981], Marconi
x 106 erg/K/g valuesand a singlecharacteristicdust size
and Mendis [1982, 1983, 1984, 1986], and Gombosi et al.
(a = 0.5 #m)' so the comparisonhas to be done with appropri[1983] publishedsteadystatesolutions,using"trial and error"
ate caution.It shouldbe pointed out that equation(86) gave a
methods to get through the 0/0 type singularity at the sonic
better than 5% approximationto a set of numericalsolutions
point. In order to obtain a "physical"transonicsolution,these
over a wide range of Z values(Z was varied betweenZ = 0 and
authorshad to "prescribe"the smoothbehaviorat the critical
z= 5).
point. These calculationsassumeda given surfacegas density,
Hellmich [1979], Hellmich and Keller [1981], and Gombosi
temperature,and dust massloading ratio and varied the suret al. [1983] investigatedthe effect of a realistic dust size face Mach number until a smooth transonic solution was obdistribution on dust terminal velocities.In their comparative
tained to equation(75). The surfaceMach number turned out
study, Gombosiet al. [1983] first solvedthe coupledgas-dust
to be a function of the dust massloading ratio, Z: for instance,
equation systemwith a realisticdust sourcedistribution funcin thecaseof a HalleytypecomettheMachnumberwasclose
tion, assuminga spectralindex N - 4.2 (cf. equation(16)) and
to 1 when there was only a negligible amount of dust in the
using 30 dust size classes.In the next step they solved the
coma, while for a Z = 1 value, the outflow Mach number was
same equation system30 times using single dust sizesand
about 0.5. On the other hand, this time-consumingshooting
renormalized the solutions to give a dust terminal velocity
method was too expensiveto be usedextensivelywith a large
distribution. The resultsare shown in Figure 11, which shows
number of dust sizesand realisticexternal energysources.
that by usinga realisticdust sizedistributionthe dust terminal
Recently, Gombosiet al. [1985] have published a timevelocities decrease about 20%.
dependenttreatment of inner coma dusty hydrodynamics,in
which steady state results are a natural byproduct. A timedependenttreatment of the gas-dustinteractionprocessdoes
4.4. ApproximateSupersonicSteadyState Solutions
not result in singular differential equations; therefore tranThe coupled gas-dust differential equation system was sonicsolutionsevolvenaturally with time. Applying the resersolvedduring the last severaldecadesby many authors using voir outflow model (section2.2), these authors benchmarked
differentapproximations.Early calculationsusingmainly ana- their steady state solutionsagainst the resultsof a "shooting
method"type numericalcode and found a reasonableagreement between the two solutions.
TABLE 9. Adopted Comet Nucleus Parameters
Value
Solarenergyflux at 1 AU, erg/cm2/s
SolarUV flux at 1 AU, photon/cm2/s
Gas mean molecular
mass
Gas specificheat ratio
Latent heat of vaporization, erg/mol
Dust grain albedo
Dust grain IR emissivity
Dust grain specificheat, erg/g/K
Minimum dust size,•m
Maximum dust size, cm
P0,g/cm3
p•, g/cm3
a 0,/•m
a•,/•m
M
N
1.35 x 106
3.50 x 10 •2
18
4/3
5 x 10 TM
0.1
0.9
8X 106
0.1
1
3.0
2.2
0.1
2.0
12
4.2
Based on fluid dynamics considerations,Gombosi et al.
[1985] suggesteda simple approximate recipe for the flow
parameters at the critical point. They pointed out that between the nucleussurfaceand the sonic point (typically located at a few tens of meters from the surface)the gas flow is
isentropicwithin a few percent. In this case, the following
relations hold betweenthe gas quantitiesin the reservoir and
thoseat the sonicpoint for an isentropicdischargeof perfect
gas from a stationary reservoir through a convergingdiverging(Laval) nozzle:
.....
\po/
½
(87)
Here w is soundvelocity,while the subscripts"0" and "c" refer
to conditionsin the reservoir and at the sonic point, respectively. The dimensionless
quantity ½ can be expressedin terms
690
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
following equation:
of the gas production rate z'
=
(Aozwo•'••'/'•+
•'
;•z=
(88)
A new set of dusty gas dynamics calculations was carried
out for this review paper usingthe time-dependentcodedeveloped by Gornbosiet al. [19853. The sublimatingand surface
temperatureswere both taken to be 200 K; there were 12
logarithmicallyspaceddust sizesbetween0.1 and 100/•m; and
the gasmolecularmass,gasspecificheat ratio, and dust specif-
Uout=
0.62Uth
(1 + 0.28Z)
R• = R,(1 + 0.0246Z
•'s44)
(91)
(95)
In this approximation, Ca is independentof the dust size (but
varies with r). Assumingthat the gas velocity doesnot change
drastically in the small subsonic region, one can make the
u • uc approximationand expand about Va/uc,which leadsto
the following equation:
1(Va•2._}_
-[-....
3-[-3(Va•
\Uc/ 2Q_•)
\Uc/4
where the C* constant can be determined
C*
dr
(96)
from the
dAeff(r)
I =0
(89)
(90)
o
dr --4•
1--_•)2
dVa2
3PCa•pa
u2(
dust to gas massproductionrate ratio, Z. It was found that
the followingexpressions
describethe numericalresultswithin
a coupleof percentin the Z = 0-5 interval'
z = z0(1.17- 0.073Z)
daza= -•-NO daa3pan(a)Va
(94)
The last quantity to be defined at the sonic point is the dust
velocitydistribution, Va.Dust particlesare practicallystationary at the surfaceand have Va<<u velocitiesat the sonicpoint,
typically located • 100 m from the nucleus.In a steady state,
low dust velocity approximation, where cometary gravity is
neglected,equation (62) can be expressedas
ic heatweretakento be 18,4/3, and 8 x 106ergs/g/K,respectively. For the dust bulk density and sourcedistribution functions, expressions(20) and (21) were used. Ten calculations
were performed usingdifferent Z valuesdistributedbetween0
and 5. The initial condition was an empty coma in each calculation. At t = 0 the gas startedto flow out of the reservoirand
beganto carry away dust grains,then the systemwas allowed
to evolve until steady state was reached. In the next step,
analytic approximationswere fitted to the numericalvaluesof
gas production rate (z), outflow velocity (11out)
, sonic point
radius (Re)and gasterminal velocity(uo•),as a functionof the
o
r=R•
(97)
condition,thus ensuringthe appearanceof the 0/0 type singularity at the sonic point. Equations (87)-(97) uniquely define
the gas and dust parametersat the sonic point. The du/dr
value at the sonicpoint can be obtained usingthe L'Hospital
where z0 has already beendefined(expression(6)). Substituting
expressions
(89) and (9•1)intoequation(88),oneobtains
relation:
du
=
(d2Aeff/dr2)r=Rc
(98)
drr=• (d/dr)(1
- M2
½= •77
F(1+1.17
•-.•4•-•ZF'•4)2'J
- 0.073Z
'¾•-•)/{•+
• (92)Equation (75) can now be integratedoutward from the sonic
r=Rc
point usingthe initial gas and dust parameter valuesobtained
from equations(89)-(97) and the (98) boundary condition.
The resultsof a seriesof calculationsstartingfrom the sonic
point were compared with the solutions obtained by using
other steady state methods. Several test caseswere calculated
with the well-establishedshooting method, as well as with the
time-dependentcode. The results of the three independent
methods were compared at various cometocentricdistances
Previous numerical calculations have shown that the tembetweenthe sonic point and 3000 km. The typical deviation
perature of the dust particlesadaptsvery quickly to the local betweenthe various resultswas a few percent' it never exceeded 10%. This comparisonhas proved that the supersonicapequilibrium temperaturedefinedby
proximationsolutionis an accurateenoughapproximationfor
describingthe steadystateradial evolution of the gasand dust
parametersin the accelerationregion.This new techniquehas
a big advantage'it is our experiencethat on the averageit is 3
where Jr is the local externalradiation field and qa is the gas to 10 times faster than the more traditional shootingmethod
to dust heat transfer rate (see equation (67)). As the thermal (dependingon the complexityof the photochemicaland raditime constantof a dustgrainis ra • 104 a2 (where% is mea- ative energy sourceterms). The big saving is that it is not
sured in secondsand "a" is given in centimeters),most dust necessaryto solvethe dust equation of motion in the subsonic
particles reach their equilibrium temperature by the sonic region, where the radial grid size is very small owing to the
point. In thesenew supersonicapproximationcalculationsthe largeaccelerationof smallgrains.
Someresultsof a supersonicapproximationcalculationare
dust temperatureat the sonicpoint was definedby expression
(93) for all dust sizes.
shown in Figure 12. In this calculation, TO= T• = 200 K,
The total dust massproductionrate is za = •z; the source Ca=8 x 106 ergs/g/K,7=4/3, Z=0.27, d=l AU values
were used,and there were 12 dust sizeslogarithmicallyspaced
dust size distribution function (Na) is assumedto follow a
between 0.1 /•m and 100/•m. Photodissociationheating and
Hanner type distribution [cf. Hanner, 1983] given by Na =
Non(a).The normalizationfactor No can be obtainedfrom the $himizu [1976] cooling were the external energy sourcesconIn order to check the validity of the isentropy assumptionin
the subsonicregion, one can calculate the sonic point gas
parametersfrom equations(87) usingexpression(92) for ½ and
comparethe resultswith the numericalsolutions.The two sets
of sonic point parameters agreed to within a few percent
giving confidencein thesefast steadystate calculationswhich
start from the sonic point (one can call this the "supersonic
approximation").
+CabsJr•
Ta
=(4%
•' •'e•
'?ø'25
(93)
691
GOMBOSI
ET AL.' DUST, NEUTRALGAS MODELINGOF COMETARY
INNERATMOSPHERES
T(K)
•To
M
IOOO
-1
0.42/z
1.3/a
loo
10
4.2/20.1
42•z
IO0
lO
o
TI -- 200K
.J
i,i
0.27
0.43
I0
- 0.01
I
I
I0
r
I00
•
1
10
R-•
a
b
0.1
I
r
100
•
1
10
r
lOO
Rn
Rn
c
Fig. 12. (a) Steadystategasand dust radial velocity,(b) gastemperature,and (c) Mach numberprofilesobtainedwith a
Hanner type dust sizedistributionfunction.
comets, while parameters varying from comet to comet are
shown in Table 10. The parameter valueswere taken from N.
the dust acceleratessomewhat more slowly. Within a com- Divine (private communication,1985)and Divine et al. [1986].
etocentricdistanceof 10 Rn, both the gas and dust almost Gas production rates were calculatedat each heliocentricdisreach their terminal velocities. The dust temperature (not tance from the light curvesusing Newburn's[1981] empirical
method. The cometarylight curveswere taken from Divine et
shown)rapidly converges
to its asymptoticvalue,whichis
al. [1986] (Halley), Brandt and Yeomans[1985] (G-Z), Sekanina [1984] (Kopff), and Divine [1985] (Wild-2). There is observational evidence that both the Comet Halley and
Adiabatic coolingdecreases
the gas temperatureto a few de- Giacobini-Zinner productionrate predictionswere too low (P.
greesat •-,30 R,; then heating due to photodissociationre- D. Feldman, private communication,1985).Tables 11 and 12
which
versesthe trend. The gas Mach number (naturally) has an might still be usefulby providing "intelligentguesses,"
oppositebehavior:it firstincreases
to about 11 and then starts can be appropriately scaled.Next the gas production rate per
unit surfacearea was derived usingequation (38), and a single
to decrease.
It was recentlyshownby Marconi and Mendis [1986] that surface temperature was obtained using equation (37) (asthe trapping of infrared thermal radiation in the collision- suminga A = 0 mantle thickness).After all thesepreliminary
dominated inner coma significantlyinfluencesthe gas temper- stepsthe gasand dust parameterswere calculatedat the sonic
ature profile. Accordingto their new resultsthe gas temper- point,and the coupleddifferentialequationsystemwassolved
in the supersonicregion. The external energy sourceswere
ature decreases less than a factor of 2 from its surface value in
et al. [1985]. The calculationwas stopped
the gas-dustinteraction region, and the revised temperature taken from Gornbosi
at
300-km
cometocentric
distance, where the gas and dust
minimum is about 100 K. Marconi and Mendis [1986] have
sidered.Inspectionof Figure 12 showsthat within about 2 Rn
the gas approaches80% of its terminal velocityvalue, while
(l•absJr
•0'2
5
T,=•,4aemff/
(99)
also demonstratedthat IR radiation trapping increasesthe gas became practically decoupled and both components apThe resultsobtainedfor the
terminal velocity by 30-40%, while the dust terminal veloci- proachedtheir terminalvelocities.
four comets under consideration are summarized in Tables
ties remain practically unchanged.
The new code incorporatingthe supersonicapproximation 11-14. The listed parametersare heliocentricdistance(AU),
was appliedto calculatenew setsof inner coma gasand 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 discussedas potential targets of a comet
rendezvousmission.Table 9 summarizesthosemodel parameter values which were assumed to be the same for all four
surface temperature (K), total gas production rate (molecules/s),averagegasmassproductionrate per unit area, dust
productionrate, maximumliftable dust size,gasterminal velocity, gasnumberdensityat 300 km, gastemperatureat 300
km, and dust terminal velocitiesfor different grain sizes.
Inspectionof Tables 11-14 revealsthat calculatedgas terminal velocitiesare practicallyindependentof heliocentricdistance. This result is in clear contradiction with Whipple's
TABLE
10.
Main Parameters of Some Selected Comets
Orbital period,years
Semimajoraxis,AU
Eccentricity
Perihelion distance,AU
Apheliondistance,AU
Nucleus radius,km
Dust/gasmassratio
Halley
G-Z
Kopff
Wild-2
76.10
17.95
0.9673
9.600
3.516
0.7075
6.460
3.467
0.5441
6.390
3.441
0.5402
1.028
1.580
1.582
35.31
6.004
5.353
5.299
3.00
0.2
1.00
1.0
1.45
0.3
1.45
0.3
0.5871
[1978] empiricallaw, which was derivedfrom Bobrovnikov's
[1954] papersummarizingobservations
of 57 differentcomets
between 0.66 and 6.74 AU:
uoo= 0.535d
-ø'6
(100)
whered is the heliocentricdistance(measuredin AUs). Similar
resultswere 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 analysisfor four spectraobtained while the
692
GOMBOSIET AL.' DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
TABLE 11. Comet Halley Parametersat 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
2gas
2.28E+ 28
1.07E- 07
5.68E+ 28
1.51E- 06
1.71E+ 29
4.55E- 06
3.47E+ 29
9.23E- 06
6.21E+ 29
1.65E- 05
8.40E+ 29
2.23E- 05
5.03E+ 29
1.34E- 05
2.20E+ 29
5.83E- 06
1.07E+ 29
2.85E- 06
Zaust
ama
x
1.21E- 07
0.56
3.02E - 07
1.42
9.10E - 07
4.35
1.85E- 06
8.93
3.30E - 06
16.14
4.47E - 06
21.94
2.67E - 06
13.01
1.17E- 06
5.60
5.70E - 07
2.70
u•a•
rtgas
T•a•
0.621
3.70E+ 07
3.4
0.631
9.06E+ 07
4.4
0.643
2.68E+ 08
6.4
0.653
5.36E+ 08
8.3
0.663
9.44E+ 08
10.4
0.657
1.29E+ 09
7.7
0.648
7.81E+ 08
5.7
0.639
3.46E+ 08
3.7
0.633
1.70E+ 08
2.7
Va(0.13)
Va(0.24)
V,(0.42)
V,(0.75)
V,(1.33)
V,(2.37)
Va(4.22)
V,(7.50)
V,(13.34)
V,(23.71)
V,(42.17)
V,(74.99)
V,(133.35)
Va(237.14)
V,(421.70)
V,(749.90)
V,(1333.52)
V,(2371.38)
V,(4216.97)
V,(7498.95)
0.210
0.169
0.135
0.109
0.089
0.073
0.061
0.050
0.041
0.033
0.026
0.020
0.015
0.012
0.009
0.007
0.299
0.245
0.200
0.164
0.135
0.112
0.094
0.078
0.064
0.052
0.041
0.032
0.024
0.018
0.014
0.011
0.008
0.006
0.005
0.003
0.424
0.364
0.307
0.258
0.218
0.184
0.156
0.132
0.109
0.088
0.070
0.055
0.042
0.032
0.024
0.018
0.014
0.011
0.008
0.006
0.505
0.448
0.390
0.335
0.288
0.247
0.212
0.180
0.151
0.123
0.099
0.077
0.060
0.046
0.035
0.027
0.020
0.015
0.011
0.009
0.562
0.515
0.461
0.406
0.354
0.309
0.268
0.231
0.195
0.161
0.130
0.102
0.080
0.061
0.047
0.036
0.027
0.020
0.015
0.012
0.580
0.540
0.490
0.436
0.385
0.339
0.296
0.257
0.218
0.181
0.146
0.116
0.091
0.070
0.054
0.041
0.031
0.023
0.018
0.013
0.536
0.484
0.428
0.373
0.323
0.280
0.242
0.207
0.174
0.143
0.115
0.091
0.071
0.054
0.041
0.031
0.024
0.018
0.014
0.010
0.446
0.386
0.329
0.279
0.236
0.201
0.171
0.144
0.120
0.097
0.077
0.061
0.047
0.036
0.027
0.021
0.016
0.012
0.009
0.007
0.362
0.305
0.253
0.210
0.175
0.147
0.124
0.104
0.086
0.069
0.055
0.042
0.033
0.025
0.019
0.014
0.011
0.008
0.006
0.005
Read,for example,"2.28E+ 28" as "2.28x 1028."
cometswere between0.1 and 1 AU. The four data points were
bestfitted with a spectralindex of 0.5. It was later pointed out
by Delsermne
[1982] that Whipple's[1978] empiricalformula
was also consistentwith the d-ø'5 law and interpretedthe
data in termsof rapid thermalizationof the CN radicalin the
inner coma, concludingthat the gas temperaturefollowed a
T • d- • relation.Delsemme's
[1982] final conclusion
wasthat
this radial dependencecan be explainedif one assumesthat
the gas heating is proportional to the solar radiation flux,
while the coolingis mainly causedby infraredradiation cooling [Shintizu,1976].
There is clearly a contradictionbetweenthe resultsof dusty
gas dynamiccalculationsand the Whipple [1978]-Delsentnte
[1982] empirical gas terminal velocity relation. One possible
explanationis that the presentdustymodel of gas-dustinteraction (which assumesperfectly sphericalparticles,total accommodation,etc.) needsmajor revisions.Another possibility
is that the early photographicobservationssummarizedby
Bobrovnikov[1954] were biasedby the dust component,while
the uncertainty of Malaise's [1970] measurementsis not
known very well (thesepossibilitieswere mentionedby Delsentme[1982] and Whipple [1982, private communication,
1985]). Calculateddust terminal velocitiesexhibit an approxi-
mately d-• dependence
on heliocentricdistance.The main
reason behind this behavior of calculated dust terminal veloci-
have a larger terminal velocity when the comet is more active
than in the caseof a lower gasproductionrate.
The gas and dust parametersshown in Tables 11-14 can
serveas initial valuesfor gasand dustcalculationsin the outer
parts of the coma,wheregas-dustinteractionis not important
any more. On the other hand, chemistry,radiative transfer,
ionization, solar wind interaction, etc., make the situation
complicatedin the decoupledregion.
4.5. Radiative Transfer
In most of the earlier dustygasdynamicscalculations,interaction of the solar radiation field with dust grains and gas
particleswas 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 emissionand absorptionare unimportant outsidethe
resonance bands and lines of the most dominant
molecules
and atoms. They concludedthat 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 moleculesmay also contributesignificantlyto radiation
transfer in the coma.
The inner coma containsa large number of dust particles.A
simpleestimatebasedon an r -2 numberdensitydistribution
tiesis thevariation
of Prob}tein's
[1968]accommodation
pa- of a more or lesstypical grain size (-• 1 #m) predictsthat the
rameter,fla-It is obviousfrom equation(80) that fia decreases optical thicknessof the column above the surfaceis lessthan
with an increasein the gas production rate. On the other 0.1, causingonly a marginal insulation effect. On the other
hand,a smallfi• valuepredictsthat the dustgrainwill quickly hand, in the immediatevicinity of the nucleusthe dust velocity
accommodateto the gasflow; consequentlyits terminal veloc- is very small (see Figure 11a); consequentlyjust above the
the dustparticledensityvariesmuchfasterthanr-2.
ity will be higher than that of a particle characterizedby a surface
large fi• value.The conclusionis that the samedustgrain will This "pileup"of slow dustgrainsmay causeimportant effects,
GOMBOSIET AL.' DUST, NEUTRALGAS MODELINGOF COMETARYINNER ATMOSPHERES
TABLE
12.
Comet Giacobini-Zinner
Parameters
693
at r = 300 km
d, AU
-2.00
Tsurf
189.26
- 1.5000
- 1.2500
1.0300
1.2500
1.5000
2.0000
194.88
198.03
199.32
196.20
192.96
187.63
Qgas
Zgas
4.77E+ 27 1.18E+ 28 1.92E+ 28 2.33E+ 28 1.45E+ 28 8.72E+ 27 3.63E+ 27
1.140E- 06 2.820E- 06 4.580E- 06 5.570E- 06 3.470E- 06 2.090E- 06 8.690E- 07
Zdust
ama
x
1.140E -- 06 2.820E - 06 4.580E - 06 5.570E - 06 3.470E - 06 2.090E - 06 8.690E - 07
3.08
7.74
12.66
15.42
9.53
5.69
3.34
Ugas
ngas
Tgas
0.672
0.678
0.682
0.687
0.682
0.677
0.671
!.02E + 07 2.50E+ 07 4.04E+ 07 4.87E+ 07 3.15E+ 07 1.85E+ 07 7.78E+ 06
4.0
5.6
6.9
9.1
7.4
6.0
4.2
Va(0.13)
Va(0.24
)
Va(0.42)
Va(0.75
)
Va(1.33)
Va(2.37
)
Va(4.22)
Va(7.50)
Va(13.34)
Va(23.71)
Va(42.17)
Va(74.99)
Va(133.35)
0.203
0.162
0.130
0.104
0.085
0.070
0.058
0.048
0.039
0.031
0.025
0.019
0.015
0.289
0.237
0.193
0.158
0.130
0.108
0.090
0.075
0.061
0.049
0.039
0.030
0.023
0.344
0.286
0.236
0.194
0.161
0.135
0.113
0.095
0.078
0.063
0.049
0.038
0.029
0.368
0.309
0.256
0.212
0.176
0.148
0.125
0.104
0.086
0.069
0.055
0.043
0.033
0.313
0.258
0.211
0.173
0.143
0.119
0.100
0.083
0.068
0.055
0.043
0.034
0.026
0.259
0.210
0.170
0.138
0.113
0.094
0.078
0.065
0.053
0.043
0.034
0.026
0.020
Va(237.14)
0.011
V•(421.70)
0.008
V•(749.90)
Va(1333.52
)
Va(2371.38)
0.006
0.005
0.004
Va(42!6.97)
V•(7498.95)
0.181
0.144
0.115
0.092
0.075
0.062
0.051
0.042
0.034
0.027
0.022
0.017
0.013
0.018
0.022
0.025
0.020
0.015
0.010
0.013
0.017
0.019
0.015
0.011
0.007
0.010
0.008
0.006
0.013
0.010
0.007
0.014
0.011
0.008
0.011
0.008
0.006
0.009
0.007
0.005
0.006
0.004
0.003
0.003
0.004
0.006
0.006
0.005
0.004
0.003
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."
leadingto significantmodificationsof nuclearoutgassing.This
potential effect was first investigatedby Keller and Hellmich
[Hellreich, 1979; Hellreichand Keller, 1980, 1981] and later by
Weissmanand Kieffer [1981, 1984] and Marconi and Mendis
[1982, 1983, 1984, 1986].
Any steady state solution to this problem has to satisfy
boundary conditionsat two separatesurfaces'at large com-
responsiblefor heating the dust grains and the nucleus.In
their model, Marconi and Mendis [1984] consideredsix constituents' nonthermal hydrogen (producedin photolyric processes),
thermalizedhydrogen,heavy neutrals(heavierthan H),
ions, electrons,and a single-sizedust population. Photolytic
producedby an isothermalnucleus,Hellreich[1979] devel-
current. It was also assumed that ion and neutral velocities
reaction
rates determine
the mass source rates of various
speciesand contribute to the momen.tumand energy source
etocentric distances the radiation
field is the unattenuated
terms.The nonthermalhydrogenwas .treatedin a semikinetic
solar radiation, while at the nucleusthe net absorbedenergy way, the dust motion was approximated by the usual kinetic
supportsgas and dust production, which in turn determines model, while the behavior of the remaining componentswas
the optical propertiesof the coma.The retardednature of the describedby hydrodynamicequations.An ambipolar electric
problem makes self-consistentsolutions rather difficult to field was determined from the electron momentum equation
obtain. Based on a sphericallysymmetric dust distribution assumingcharge neutrality and the absenceof net electric
oped an iterative model to calculatesteady state radiation
transferin the inner coma and to determinethe energyinput
were equal' in effectthis assumptiongrosslyoverestimatesthe
ion-neutral drag effect at larger radial distances[Cravens et
to the nucleus.The calculation was started with an optically al., 1984' Kb'r&mezey, 1984]. Electron heat conduction and,
transparent coma. In the first step, gas and dust production neutral molecule radiative cooling [Shimizu, 1976] were also
rates and the resultingdusty gas flow parameterswere calcu- taken into account,contributing to the coma energetics.
lated. Next, Hellreich [1979] recalculatedthe radiation field in
Figure 13 (taken from Marconi and Mendis [1984]) shows
the coma and at the nucleus,thus obtaininga new surface the radial profile of the normalizedradiation energyflux. The
temperature.In this step,direct absorptionand thermal emis- calculation was carried out for a Halley classcomet at 1-AU
sion as well as multiple scatteringeffectscausedby dust parti- heliocentric distance. In the figure the contributions of the
cleswere taken into account.The processwas repeateduntil a directradiation,JmR,the multiplescatteredflux, JMS,and the
converged solution was obtained. A similar technique was diffuseradiation,JBB(producedby the thermalradiationof
later employed by the San Diego group, too [cf. Marconi and dust grains),are showntogetherwith the total flug, Jxox.InMendis, 1982, 1983, 1984, 1986].
spectionof Figure 13 shows that the direct solar radiation
Recently,a comprehensive
two-band,three-streamradiative decreasesmonotonically toward the nucleus,and practically
transfer model was developedby Marconi and Mendis [1984, no direct radiation reachesthe surface.The multiple scattered
1986]. They separately consideredthe variation of the UV
flux peaks right above the surface,somewhat closer to the
flux, which is responsiblefor the major photolytic processes nucleus than the dust thermal radiation. The radiation flux
and the visual and near-infrared radiation, which is mainly
scatteredwithin the inner dust coma is partially trapped,and
694
GOMBOSI
ETAL.'DUST,NEUTRAL
GASMODELING
OFCOMETARy
INNER
ATMOSPHERES
TABLE13. Comet
KopffParameters
at•'= 300km
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
4.73E
- 06
3.05E+ 28
3.47E
- 06
1.70E+ 28
Zaus,
4.95E - 07
9.46E - 07
1.42E - 06
1.04E - 06
5.81E - 07
zg,s
1.65E
- 06
3.15E
- 06
1.94E
- 06
am,,,
3.19
6.16
9.31
6.80
3.76
ug,
s
0.667
0.673
0.676
0.674
0.669
2.31E+ 07
3.5
4.38E+ 07
3.9
6.55E+ 07
4.2
4.82E+ 07
3.8
2.71E+ 07
3.4
ngas
Tg,,
Va(0.13)
Va(O.
24)
0.244
O•197
0.356
0.297
0.246
0.320
0.264
0.159
0.310
0.255
0.209
0.217
0.259
0.210
0.170
Va(0.42
)
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)
V•(4.22)
0.087
0.072
0.117
0.098
0.141
0.119
0.123
0.103
0.094
0.078
V•(7.50)
0.060
0.082
0.099
0.086
0.065
Va(13.34)
Va(23.71)
Va(42.17)
Va(74.99)
Va(133.35)
Va(237.14)
V•(421.70)
Va(749.90)
0.049
0.039
0.031
0.024
0.018
0.014
0.011
0.008
0.067
0.054
0.043
0.033
0.025
0.019
0.015
0.011
0.082
0.066
0.052
0.040
0.031
0.024
0.018
0.014
0.070
0.057
0.045
0.035
0.027
0.020
0.015
0.012
0.053
0.042
0.033
0.026
0.020
0.015
0.011
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 "1,45x 10TM"
it overcompensatesthe screeningof the direct solar radiation.
The total radiation flux reaching the nucleusis about 25%
higher than the unattenuated solar radiation; consequently,
the gas and dust production rates are somewhatlarger than
nificantly influencethe gas parameter profiles.The main idea
behind the IR trapping is the recognitionthat most of the dust
thermal radiation is emitted in the 1- to 20-#m wavelength
range, where several rotational and vibrational transitions
they wouldhavebeenif the opticalpropertiesof the coma existfor the highlypolarwatermolecules
andhaveverylarge
were neglected.This effect, however,strongly dependson the resonance
absorptioncrosssections
(~4 x 10-x,•cm2 [Hueboptical characteristics
Ofthe dust grains.It waspointed out by
ner, 1985]). In a collisionlessgas the resonant radiation is
Keller [1983] that the trappingof multiplescatteredphotons continuouslyabsorbedand reemitted by the water molecules;
leads to flux enhancementsof about 25% if pure olivine is in other words,it is trapped by the gas. Marconi and Mendis
consideredfor the material of dust particleswith a high scat- [1986] were the first to recognize that in the collisiontering albedo and negligibleabsorption(dielectricmaterial). dominated inner coma a large fraction of the rotational/vibraOn the other hand, dust grains with a large absorption coef- tional excitation energy of water molecules can be transficient (suchas magnetiLe)suppressmultiple scattering.In this formed via collisionsinto translationalenergy,thus increasing
case,the grains are strongly heated and the emitted infrared the gas temperature.This new heating is the oppositeof the
diffuse radiation further enhancesthe energy input to the nu- radiative cooling[Shimizu,1976; CrOvisier,1984] in Whichit is
cleus.The physicalstructureof the dust particlesplays a very assumedthat molecular collisions excite the water molecules
important role in the radiativetransferprocess:direct obser- and the excitation energy is lost by the emissionof infrared
vations are neededto resolveuncertaintiesof presentcalcula- radiation.Marconiand Mendis[1986] derivedan approxitions.
mate formula to describe the combined effect of the Crovisier
The radial variationof gasand dustparametersobtainedby
the comprehensivemodel of Marconi and Mendis [1984] ex-
[1984] cooling and the gas heating due to infrared radiation
trapping:
hibitsa verysimilarbehaviorto the profilesshownin Figure
12. This is understandable, because in that calculation the
radiation
fieldswithin
QIR-' -- CoTw•'nw
exp(- Zw)
t-he inner coma was assumed to be con-
stant, which turns out to be true within about 50%. This
am
(101)
+ 9 x 10-3qe[1- exp(-Zw)]
da a2f,•T•
4
simplifying assumptiondoes not causesignificantchangesin
the global behavior of the solutions.
The newest developmentof the radiative transfer calcula- whereQIRis givenin units of erg cm-3 s-x. In expression
tions is the first theoreticalmodeldescribinginfraredradiation (101), nwand Twrepresentthe water vapor concentrationand
trapping [Marconi and Mendis, !986]. They pointed out that temperature,respectively'qeis an efficiencyfactor(its valueis
the trapped infrared radiation representsonly a negligible around unity); and fa and T• are the differential dust number
fraction
of thetotalradiation
energy
density
butcanstlllsig- density and dust temperature,respectively.The infrared optio
GOMBOSIET AL.' DUST, NEUTRAL GAS MODELINGOF COMETARYINNER ATMOgPHERES
TABLE
14.
Comet
Wild-2
Parameters
695
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
Zga
s
9.15E+ 27
1.04E- 06
1.56E+ 28
1.78E- 06
2.31E+ 28
2.63E- 06
1.56E+ 28
1.78E- 06
9.15E+ 27
1.04E- 06
Zau•t
ama
x
3.12E - 07
2.00
5.33E - 07
3.44
7.88E - 07
5.12
5.33E - 07
3.44
3.12E - 07
2.00
/•/gas
0.664
ngas
0.669
1.47E+ 07
2.48E+ 07
0.673
3.65E+ 07
0.669
2.48E+ 07
0.664
1.47E+ 07
Tgas
3.8
4.3
4.8
4.3
3.8
Va(0.13)
Va(0.24)
0.203
0.163
0.252
0.204
0.291
0.239
0.252
0.204
0.203
0.163
V•(0.42)
0.130
0.164
0.194
0.164
0.130
V•(0.75)
Va(1.33)
Va(2.37
)
V•(4.22)
0.104
0.085
0.070
0.058
0.133
0.109
0.090
0.075
0.159
0.130
0.108
0.091
0.133
0.109
0.090
0.075
0.104
0.085
0.070
0.058
V•(7.50)
0.048
0.062
0.075
0.062
0.048
Va(13.34)
V•(23.71)
0.039
0.031
0.051
0.041
0.062
0.050
0.051
0.041
0.039
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)
V•(421.70)
v•(749.90)
0.011
0.008
0.006
0.014
0.011
0.008
0.018
0.013
0.010
0.014
0.01!
0.008
0.011
0.008
0.006
Va(1333.52)
Vd2371.38)
0.005
0.004
0.006
0.005
0.008
0.006
0.006
0.005
0.005
0.004
V•(4216.97)
Va(7498.95)
0.003
0.002
0.004
0.003
0.004
0.003
0.004
0.003
0.003
0.002
Read,for example,"9.15E+ 27"as"9.15x 1027"
cal depth of water vapor, %, is definedby expression(50). The
Co and b constantsare the following:
describesthe infrared heating by the trapped radiation, while
in the Zw<<1 case,QiRbecomes
the Shimizu[1976] cooling
term.
C0 = 4.4 x 10-22
Tw< 52 K
C0=2.0x 10-2ø
Tw>_52K
heating term into their dusty gas dynamics model. Their resultsindicatethat infrared radiation trapping significantlymo-
b = 3.35
T• < 52 K
b = 2.47
Tw >_52 K
difiesthe radial gasparameterprofiles.Figure 14a (takenfrom
Marconi and Mendis [1986]) showsthe radial variation of gas
MarconiandMendis[1986] incor•porated
the newinfrared
temperature.
Onecanseethatowingto theadditional
dust
Inspectionof expression(101) showsthat when % >>1, QiR heating the temperature does not decreasebelow • 100 K,
while in previouscalculationsthe minimum temperature was
Z
only a few degrees.The additional energysourcealso signifi2.0
,,
i
i
i
i
i
i
i
i
i
i
i
cantlyincreases
the gasterminalvelocity(typicallyby about
50%).On theotherhand,theinfraredheatingdeposits
s{gnificant amountsof "new" energyat distanceswhere only a rela-
•tivelysmall fractio•nof dust accelerationtakesplace;conse-
quentlythedustterminalvelocities
areonlySlightlyaffected.
Figure14bshows
twodustterminalveloci[y
profiles.:
onewas
1.5
z
z
z
obtained by neglectingthe IR heating, and the other curve
was calculated with the new gas heating term. Inspection of
Figure 14b reveals that the velocity differencefor particles
larger than 0.1/•m is lessthan 10%.
1.0
I JBB
•
0.5
l-• JDIR
4.6. Time-Dependent
Models
N
0
o
z
20
40
60
80
IO0
120
DISTANCE
140
160 180 200
2ZO
(Km)
Fig. 13. The radial profile of the normalized radiation energy
flux.Thevariouscurves
represent
thedirectsolarradiation(JmR),
the
multiple scatteredflux (JMS),the diffuseradiation (JBB),and the total
flux (Jxox)(takenfrom MarconiandMendis[ 1984]).
Cometsare highlyvariablecelestialobjectswhichexhibit
spectacularchangeswith different time scales.The orbital
motion results in variations with a characteristic time scale of
10daysor so'thiseffectusually
canbetreated
in a q•asi
steady state manner. In some comets, violent flareups jets,
expandinghalos,tail discontinuities,
etc.,may developwithin
minutes or hours, although their decay phase usually lasts
considerably longer. These phenomena require a time-
696
GOMBOSIET AL.' DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
250
,
,
TR
200
TK
150
I00
TK
50
o
•o
RADIAL DISTANCE
(km)
Mendis,1985]wasusedto calculatedustgraintrajectories.
it
wasfoundthat followingthe onsetof the simulated
comet
a
i
i
_
E
Rn = :3 km
_
.
_
DUST
_• _
0.5
,e,
_
-
0.7
IR HEATING
INCLUDED
_
X =0.3
-
-
0.:•
-
__
>8 0.2.
NO IR HEATING
In their calculations, Gombosi et al. [1985], Kitamura
[1986], and Gombosiand Hortinyi [1986] simultaneously
solved the partial differential equation system(61) describing
the gas behavior with the dust equations(62), (63), and (64).
The dust-gasinteraction was describedby Probstein's[1968]
model,while the gas was heatedby H20 photodissociation
and cooledby the Shimizu[1976] infraredcoolingterm.
As a result of the time-dependentdusty flow calculations,
Gombosiet al. [1985] found that in addition to the similarity
type expandinggas halo lip, 1981], another type of disturbance will also propogate outward in a cometary coma following an outburst on the nucleus. Next, Gombosi and
Hortinyi [1986] calculatedthe evolutionof gasand dust distributions following a spatially and temporally localized comet
outburstusinga dustyhydrodynamic--kinetichybrid method.
In the inner coma the time-dependent
continuity,momentum,
and energy equationsof a dusty gas flow were solved assuming sphericalsymmetry within the 30ø wide jet using 12
logarithmicallyspaceddust sizesfollowing a Hanner type size
distribution (with a surfacespectralindex of 4.2). Beyond 300
km a three-dimensional kinetic method [cfi Hordnyi and
_
outburst, a gas-dustblast wave propagatesoutward in the
inner coma. About 60 min after the increasedgas and dust
production .wasinitiated at the nucleusa new equilibrium was
reachedin the inner coma. The most important feature of this
new steadystate is the significantincreaseof the gas pressure
(Figure 15). These higher terminal velocity values resultedin
increasedapex distancesfor dust particlesemitted during the
outburst. Gombosiand Hordnyi [1986] concludedthat comet
outbursts
maygenerate
long-lasting
distinctdustenvelopes
in
BY
DUST
front of the regular dust coma,which later propagatetailward
and finally dissolve(Figure 16)..These type of dust envelopes
o.I
z
were observedat severalcomets(cf.cometDonati).
0.07
Recently, a couple of groups started to work on multidii
I
mensional
modeling of cometary jets. Sagdeer et al. [1985]
I
IO
IOO
have calculatedthe steadystate shapeof a long-lasting,locala (/•m)
ized jet assumingthat after the passageof the initial blast
b
wave there is no significantgaspressuregradient acrossthe jet
Fig. 14. (a) Radial dependence
of rotational(TR)and translational boundary. On the basis of this assumption, Sagdeer et al.
(Ts:) gas temperaturescalculatedwith infrared radiation trapping [1985] concludedthat the angularextentof the dustjet varies
(takenfromMarconiandMendis[1986]).(b) Dustterminalvelocity with cometocentric distance as r {•- •)/2.
_
-
_
_
profilesobtainedwith and withoutthe trappedradiationheating.
Kitamura [1986] recentlypublishedthe first time-dependent
axisymmetricdusty gasjet calculation describinginner coma
gas and dust distributionsfollowing a long-lasting,localized
dependent,multidimensionaltreatment in order to describe surfaceoutburst. The jet profile at the surfacewas approxithe temporal and spatial evolution in the near-cometaryenvi- mated by a Gaussianfunctionwith a half width of 10ø. The
ronment. Recently,Gombosiet al. [1985] have publishedthe
first time-dependentdusty hydrodynamicaldescriptionof a
cometary inner coma. The model describesa sphericallysym-
metrictransonicflow in the immediatevicinityof the nucleus
where the gas-dust interaction dominates the dynamics. A
more sophisticatedtwo-dimensionalcalculationwas published
by Kitamura [1986], who solvedthe dusty gasdynamicequations for a long-lastingjet. Gombosiand Hordnyi [1986] con-
structeda dustyhydi'odynb.
mic-kinetic
hybridmodelto describe the evolution
•
ß
'
' •3600s
t=O$
z 0.2
of dust halos.
Time-dependentinner coma calculationshave severaladI
,
I
O.O
vantagesover steady state models. One important technical
0.1
1.0
I0
I00
advantageis that a time-dependenttreatment of the gas and
o (/.t.m)
dust interactionprocessdoesnot resultin singulardifferential
Fig. 15. Dust terminal velocity distributionsbefore and 60 min
equations; consequently,transonic solutionsevolve naturally after the onset of the comet outburst (taken from Gombosi and
with time.
Hordnyi [ 1986]).
GOMBOSIET AL..' DUST, NEUTRAL GAS MODELING OF COMETARYINNER ATMOSPHERES
DUSTSIZE=0.42/•m
t:6h
697
processesis a laterally varying dust/gasmassratio, resultingin
different loading effects. Figure 17 (taken from Kitamura
[1986]) showsthe gas velocity field. One can seethat 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 30ø, 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 profilesis that the gas
in the enhanced density region (around • = 30ø) can adiabatically expand not only radially, but also tangentially.This
means that only a smaller fraction of the gas internal energy
will be transferredto the dust population than in the radially
expandinggas regions.As a result of the combinationof these
processes,a slow, high-density dust jet is formed around
• = 30ø. Inside the jet the dust is faster than the ambient
t=lSh
population;consequently
the dust numberdensityis smaller.
Figure 18 (taken from Kitamura [1986]) showssteadystate
dust equidensitycontoursin the (r, •) plane for a 10ø wide
outburst occurring at • = 0ø. One can see the formation of
the dustjet wingsat around30ø. On the otherhand,owingto
the increaseddust velocitiesinsidethejet cone,in this region
the dust densityis smallerthan the ambient value.
Kitamura's[1986] resultsrepresenta significantimprove-
ment in modelingdusty gasjets. We expectthe development
of further interestingjet models following the Halley flyby
projects, when (hopefully) more detailed observationswill be
availableabout the structureof inner coma dust and gasfeatures.
UNIT LENGTH=104km
Fig. 16. Snapshotsof 0.42-#m dust particle distributions following a 30ø wide subsolaroutburst occurringat t = 0 (taken from Gombosiand Hordnyi [1986]).
main feature of Kitamura's [1986] numerical resultsis that the
narrow outburst resultsin a conicaljet. The physical reason
for this conicaljet formation is twofold: first, the surfacegas
pressure gradient initiates a lateral gas transport which
quickly pushesthe gasdensitypeak to • 30ø,and second,near
the nucleusthe dust grains attain a tangential velocity which
is comparableto their radial velocities,thus depletingthe dust
population along the • = 0ø line. Further away from the nucleusthe dust particleslost most of their tangential velocities;
consequentlythe modified dust structure "freezes"at a cometocentric
distance
of about
10 km. The result of these two
THETA
5.
SUMMARY DISCUSSION
In the next few months and years,we will learn a great deal
more about both the nucleusand the dust/gasenvironment 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 understandingof the
physicaland chemicalprocesses
controllingcometsand their
environment.Thereforethe purposeof this reviewis not to be
an encyclopediaof knowledge but to provide a description
and summaryof someof the most relevantbasicphysicaland
chemical processesalong with the best present-daymodels
which can be used as benchmarksto compare the new results
with and then facilitate data interpretation and the resulting
[DEG)
R
Fig. 17. Steadystate gas velocityfield following a long-lasting10ø wide subsolarcomet outburst.The arrows represent
the directionof the gasflow, while the solidline marked "sonic"representsthe M = 1 curve(taken from Kitamura [1986]).
698
GOMBOSI
ETAL.' DUST,NEUTRALGASMODELING
OFCOMETARY
INNERATMOSPHERES
and thermodynamicsof the inner coma, Astron.Astrophys.,130,
THETA {DEG)
361, 1984.
Daniels, P. A., and D. W. Hughes, Monte Carlo simulationon the
massdistributionin an accretingsystemof dust particlesin Solid
Particlesin the Solar System,editedby J. Halliday and B. A. Mcintosh,p. 325,D. Reidel,Hingham,Mass.,1980.
Daniels, P. A., and D. W. Hughes,Accretionof cosmicdust--A com-
•o
5O
putersimulation,
Mon.Not.R.Astron
Soc.,
195,1001,1981.
o
BO
Delsemme,A. H., Vers un module physico-chemique
du noyau
com6taire,Natureet OriginedesCom6tes,Mere.Soc.R. Sci.Liege,
7o
Ser. 5, 12, 77, 1966.
Delsemme,A. H. Chemical compositionof cometarynuclei, in
Comets,
editedby L. L. Wilkening,p. 85, Universityof Arizona
8o
Press,Tucson, 1982.
9o
oo
1oc
1o
11o
2O
120
4O
i
0
505,D. Reidel,Hingham,Mass.,1985a.
Delsemme,
A. H., The sublimationtemperature
of the cometarynucleus:Observational
evidencefor H20 snows,in Ices in the Solar
System,
editedby J. Klinger,D. Benest,
A. Dollfus,andR. Swolu-
50
15
editedby J. Klinger,D. Benest,
A. Dollfus,andR. Swoluchowski,
p.
chowski,
p. 367,D. Reidel,Hingham,Mass.,1985b.
Delsemme,
A. H., and D.C. Miller, Physico-chemical
phenomena
in
comets,III, The continuumof CometBurnham(1960II), Planet
3O
140
Delsemme,A. H., What we do not know about cometaryices:A
reviewof the incompleteevidence,in Ices in the Solar System,
i
i
50
100
R IKM)
Fig. 18. Steadystateinnercomaequidensity
contours
in the(r, •)
planefor a Gaussian-shaped
surface
jet profilewitha halfwidthof
10øoccurringat ½I)= 0ø(takenfromKitamura[1986]).
SpaceSci.,19, 1229,1971.
Delsemme,A. H., and P. Swings,Hydratesde gaz dansles noyaux
com6taires
et lesgrainsinterstellaires,
Ann.Asrophys.,
15, 1, 1952.
Divine,N., and R. L. Newburn,Jr., Numericalmodelsfor cometary
dustenvironments,
in CometaryExploration,vol. 2, editedby T. I.
Gombosi,
p. 81,Hungarian
Academy
of Sciences,
Budapest,
1983.
Divine,N., H. Fechtig,T. I. Gombosi,M. S. Hanner,H. U. Keller,S.
M. Larson,D. A. Mendis,R. L. Newburn,R. Reinhard,Z. Sekanina,andD. K. Yeomans,The CometHalleydustand gasenvironment,SpaceSci.Rev.,43, 1, 1986.
advancesin our knowledge.
The historyof solarsystemstud- Dobrovolskii,O. V., Komety,Nauka Press,Moscow,1966.
Dobrovolskii,
O. V., and M. Z. Markovich,On nongravitational
efieshascontinuously
demonstrated
the importanceof the symfectsin two classesof modelsfor cometarynuclei,in The Motion,
biotic relation between theoretical studies and observationsin
Evolutionof Orbitsand Origin of Comets,editedby G. A.
achieving
progress.
We hopethat thesereviewswill make
some contribution to the dramatic advancesexpectedin the
next few years.
Chebotarev,E. I. Kazimirchak-Polonskaya,
and B. G. Marsden,p.
287,D. Reidel,Hingham,Mass.,1972.
Donn,B.,andJ. Rahe,The structure
andoriginof cometary
nuclei,in
Comets,
editedby L. L. Wilkening,p. 203, Universityof Arizona
Press,Tucson, 1982.
Acknowledgments.
We wish to thank H. L. F. Houpis,A. Feldman,P. D., Ultravioletspectroscopy
of comae,in Comets,
edited
K6r6smezey,
M. L. Marconi,andD. A. Mendisfornumerous
helpful
by L. L. Wilkening,p. 461,Universityof ArizonaPress,Tucson,
commentsand suggestions.
This work was supportedby NASA
grantsNAGW-15and NGR-23-005-015
and NSF grantsATM-8508753 and INT-83-19732. Acknowledgment
is also made to the Na-
tionalCenterfor Atmospheric
Research,
sponsored
by the National
Science
Foundation,for computingtimeusedin thisresearch.
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