Cometary Atmospheres : 

Modeling the Spatial Distribution 

of Observed Neutral Radicals 

Michael R. Combi 

Atmospheric and Environment a1 Research, Inc . 840 Memorial Drive 

Cambridge, MA 02139-3794 

December 1986 

Report for the Period 

June 1, 1985 to September 30, 1986 

( NASA -CE- 1862 17) CC !4 El A& Y A1 BC2EBEBES : 

RCCELIBG !IHE SPaIXAL IISZPIEClICL CI 

CESEGVED NEUXEAL EADICALS Accual Beport, 1 

3un. 19E5 - 51 Aug. 1Stb (Atrfcsyheric and 

Envirc~mental fescarch, Inc-) 21 p CSCL 03B G3/90 43655 

Nk7-19318 

Unclas 
I. Int roduct ion 

The primary goals of the research program are the inclusion and 

evaluat ion of the following processes which are ususally neglected in modeling 

cometary atmospheres: (1) dependence on heliocentric distance of initial 

comet expansion velocities and nucleus vaporization rates, (2) dependence on 

heliocentric velocity of photochemical lifetimes and photon emission rates 

where applicable, (3) the correct isotropic ejection of daughter radicals upon 

photodissociation of parent molecules, (4) solar radiation pressure, and 

(5) collisional effects on radical kinematics. 

modeling effort would be to generate the information necessary to reduce 

observed radical column abundances to radical production rates for CN, C2, C3, 

and OH. 

quantitative understanding of the physical processes which are important in 

cometary atmospheres. In all instances, model development is constrained by 

the observed spatial distributions of these radicals already phblished in the 

literature, as well as by the best currently available photochemical data. In 

addition, the same models developed here could also be easily applied to 

analyze the new observed spatial distributions of cometary radicals. 

The principal results of the 

Secondary goals which would naturally follow would be a better 

A fairly large amount of the filter photometric data can be expected to 

be acquired during the 1985-1986 apparition of Comet P/Halley from the members 

of the Photometry/Polarimetry network organized under the International Halley 

Watch. The large amount of high quality data requires much more sophisticated 

models for analysis than are currently available. 

The type of model developed is a many particle-trajectory Monte Carlo 

model. The facility and economy of using this type of model for treating the 

complex geometry and dependencies mentioned above has been discussed and 

demonstrated in the published literature (Combi and Delsemme 1980a,b; Combi 

1980). The traditional approach initiated by Eddington's (1910) fountain 

model and continued in such later works by Haser (1957, 19661, Wallace and 

Miller (1958), Keller and Meier (1976) and Festou (1981) becomes 

computationally unmanageable for multi-dimensional time-dependent models. 

This is also true for the one-dimensional fluid models which have begun to 

model the physics correctly (Marconi and Mendis 1982, Huebner and Keady 1983). 

Progress during the second year in the following areas will be discussed: (1) C2 and CN spatial distributions and their heliocentric distance 

1 
dependences, (2) C3 and OH scale lengths, (3) incorporation of coma temperature into the Monte Carlo particle-trajectory model, and (4) collaboration in 

the analysis of Comet Halley CCD images and long slit spectra. 

2 
11. CL and CN in Comets 

Attached in an appendix to this report is a reprint of the paper 

published in the Astrophysical Journal which deals with the spatial 

distributions and inferred production rates of the C2 and CN radicals in 

comet 6. 

The principal contents of the paper briefly enumerated are: 

new observations of C2 brightness profiles, 

compilation of C2 and CN scale lengths from previously published data, as 

well as their heliocentric distance dependences, 

re-evaluation of published filter photometry with the new scale length 

data which resolves the previously held anomalous C2-to-CN production 

ratio's behavior with increasing heliocentric distance, and 

an analysis of sunward and antisunward C2 profiles with the Monte Carlo 

particle-trajectory model which implies a total source (parent) lifetime 

0.5 km s-l for C2 radicals. 

(1) 

(2) 

(3) 

(4) 

of 3.1 x 10 4 s at 1 AU and an excess photolysis-energy ejection speed of 

3 
111. C3 and OH in Comets 

+ Although the (lnU-lC transition of the C3 radical at - 4040 A has been 

g 

observed in cometary spectra for many years (see Swings and Haser 19561, and 

OH is believed to be the photodissociation product of water, surprisingly 

little has been done in the area of observing the spatial distributions of 

these two important species. 

Other than qualitative descriptions of the apparent spatial distribution 

from long-slit and objective prism spectra (see A'Hearn 1982) that the C3 

radical was less spatially extended from the nucleus than C2 and CN, little 

quantitative study of the C3 coma had been done until the very recent paper 

by Cochran (1985) which presents six brightness profiles of C3 in 5 comets 

covering a range of heliocentric distances from 0.74 to 1.58 AU. 

involved fitting Haser models simultaneously to all six profiles where the 

parent and daughter scale lengths were assumed to vary with heliocentric 

distance according to r , where n N 2. These power laws were compared with 

those adopted by A'Hearn (1982) for photometric data reduction and that found 

by Newburn and Spinrad (1984) for only the Cg parent. Cochran concluded on 

the basis of her analyses and comparisons that data for two of the profiles 

2 (Comets Crommelin at 0.74 AU and Austin at 0.78 AU) were best fit by the r 

law of Newburn and Spinrad but that data for the other four profiles (Comets 

Tuttle at 1.02 AU, Stefan-Oterma at 1.58 AU, and Kopff at 1.58 and 1.81 AU) 

were best fit by a different r2 law. 

Her analysis 

n 

An alternative way of approaching this type of analysis, as has been 

adopted in this project, is to analyze each brightness profile separately and 

compare the resulting heliocentric distance dependences of best fit model 

parameters. 

profiles are listed in Table 1. The dependences on heliocentric distance for 

the parent and daughter scale lengths are shown in Figure 1 a and b. For 

reference, the best-fit power laws are also plotted. The variations are not 

at all consistent with r2 dependences, as might be expected, but are r 4.9 

the parent and r3.4 for the daughter scale lengths. 

inconsistent with straight forward photochemical product ion and decay as are 

appropriate for C2 and CN. 

when assuming a simple r2 law for both parent and daughter scale lengths, 

A'Hearn (1982) finds C3 production rates in proportion to those for dust 

(continuum), CN and OH. 

The best-fit Haser model scale lengths for each of the six C3 

for 

Such variations are 

However, one must also keep in mind the fact that, 

4 
The spatial distribution of OH has been observed by various satellitebased W instruments, but sufficiently high quality observations are unfortunately not available over a wide enough range of heliocentric distance to 

enable a quantitative study of the heliocentric distance dependence to be 

made. Table 2 lists the results of fitting Haser's model to each of the 

available brightness profiles. 

daughter scale lengths (which may or may not be true), we find the following 

values at 1 AU: 

If an r2 law is assumed for both parent and 

y (parent) = (6.3 f 1.9) x lo4 km 

y (daughter) = (2.5 f 0.9) x 10 km. 5 

These values are somewhat larger than those adopted by photometric observers, 

4.1 x 10 km and 1.16 x 10 km for the parent and daughter, respectively 

(A'Hearn 1982). 

4 5 

5 
IV. Coma Temperatures 

Gas dynamic and dusty gas dynamic models of the cometary coma predict 

that the vaporized gas can in general be described as having a bulk outflow 

speed and a temperature (Marconi and Mendis 1982, 1984; Huebner and Keady 

1983, Gombosi et al. 1985). The outflow speed and temperature are in general 

a variable function of the distance (r) from the nucleus. In previous Monte 

Carlo models, including those developed for this project, the outflow has 

simply been described as a single constant speed. 

As part of another NASA project with AER (NASW-3966: Principal 

Investigator, William H. Smyth; Co-Investigator, Michael R. Combi), we have 

incorporated the Monte Carlo model developed under this project in another 

model which describes the production of cometary atoms H, C and 0 through the 

photodissociation of likely parent molecules (H, 0, CO, and C02). One of the 

objects of that work is to see if the spatial distribution of the observed 

extended hydrogen coma of comets can be modeled by a physically based 

model which includes (1) time dependent lifetimes and production rates, 

(2) realistic photodissociation kinematics, (3) three-dimensional heliocentric 

atom trajectories with radiation pressure, and (4) collisional thermalization 

of hot H atoms. 

Early attempts at modeling observed 2-dimensional Lyman-a isophotes show 

that the H atoms which are thermalized by a simple radially flowing parent gas 

introduce a very low speed component to the velocity distribution which is not 

present in the data. However, for reasonable outer coma gas temperatures of 

several hundred degrees, the thermal speed component of the thermalized H atom 

can be comparable to, or even larger than, the outflow speed. 

This thermalization of H atoms, as well as the effect of the radial 

dependence of the coma temperature on the thermalization of all species is 

important for gaining insight into the physical conditions in the inner coma 

as well as determining the observed spatial distributions of neutral gases. 

Therefore, the outflow of parent molecules from the nucleus is more 

realistically described by a convected Maxwell-Boltzmann distribution rather 

than a simple single speed radial outflow. At some point in the coma, the gas 

flow of a parent molecule can be characterized by a bulk outflow speed, u, and 

a temperature, T. The speed distribution, f(v) dv, of molecules of mass, m, 

at temperature T is given by the Maxwell-Boltzmann distribution, 

6 
4 

1/2 f(v) dv = 

ll 

where vm is the most probable speed, 

1 /2 v = (2kT/m) m 

In order to simulate this speed distribution on a microscopic level with 

the particle-trajectory model, we can use the principle of Monte Carlo. Since 

the Boltzmann distribution is a normalized probability density function, a 

simulated distribution can be generated from a random number sequence, Ri on 

the interval from 0 to 1 by solving for vi in the equation, 

v2 exp(-v2/vm2) dv = Ri . ll V 

m 

The integral in this equation is not analytical, but can be transformed into a 

single parameterized function by making the substitutions: u v/vm, 

ui z vi/vm and du =- dv. 1 The equation then becomes 

V 

m 

which has been numerically solved and included as an interpolated table of the 

form, ui = u(R;), in the model. 

The effects of a thermal coma in the model will be investigated more 

fully in the third year of this project, and coma temperatures through the 

photochemical heating produced by the photodissociation of water molecules 

will be determined. 
V. Distribution of Neutral Radicals in Comet P/Ealley 

A collaboration had been initiated last year with Dr. Uwe Fink of the 

University of Arizona to analyze both CCD filtered images and long slit 

spectra of Comet P/Halley. Late in this project year, the principal 

investigator traveled to Tucson for an extended stay to assess the entire data 

set and to begin the job of data reduction. Tables 3 and 4 show summaries of 

the radical images and the long slit spectra. 

In Tucson, we went through the procedure of processing one pair of images 

including flat fielding, background subtract ion and continuum subtraction. 

the images processed were those taken on May 11, 1986 with the filters at 

9180 A, which includes the (1-0) band of the CN red system, and at 8600 A, 

which is a well-placed continuum filter. The relative weights associated with 

the two images were computed roughly from the filter transmission parameters 

and the system spectral response estimated from some of the long-slit 

spectroscopy data. 

Figure 2 showns the two reduced but unsubtracted filter images and the 

resulting subtracted CN image. It is apparent that, On May 11, we are seeing 

one of the CN jets which have been reported by other observes (A'Hearn et al. 

1986) and that, like other reports, there is no coincident continuum jet. 

Because of the large amount of high quality data available from this 

observing program, only a selected set of the images can be reduced and 

analyzed under the limited scope of the project. This work will continue 

during the third project year. The emphasis will be placed upon the remainder 

of the May 11 data to test for the existence of jets in the emissions of the 

other radicals, and possibly on data from one or two other nights spanning a 

range of heliocentric distance. The existence of gas jets could have farranging implications as to the production mechanism(s) of cometary radicals, 

as evidenced by the suggestion of A'Hearn et al. (1986) that the CN and C2 in 

the jets (-15% of the total) may be produced by the newly discovered CHON 

grains which are likely to be rich in hydrocarbons. 

The flexibility of the Monte Carlo particle-trajectory model will enable 

us to begin to model localized time-dependent multiple sources for cometary 

radicals for comparison with image data. 

8 
Table 1 

Haser Model Scale Lengths for C3a 

Comet 

(f;ff> yd 

(km) 

C r omme 1 in 0.74 1.4 103 3.3 104 

Austin 0.78 1.5 lo3 3.7 lo4 

Tut t le 1.02 3.2 104 1.5 105 

S t ephan-0 t erma 1 .58 7.7 104 9.9 105 

Kopf f 1.58 7.9 104 3.6 105 

Kopf f 1.81 1.0 105 4.8 105 

aCochran ( 1985) 

9 
Table 2 

Haser Model Scale Lengths for OH 

Kohouteka (1973x11) 0.62 5.1 104 8.1 104 

XBMC (19751x1 0.82 4.8 104 1.2 105 

Bennet td ( 197011) 0.83 < 7 104 1.4 lo5 

Enckee ( 1980x1) 0.83 5.8 104 8.2 104 

0.87 2.7 104 4.8 105 

Bradf ieldb (1979x1 1.03 2.29 104 > 3 105 

4 Bradfieldb (1979X) 0.71 1.6 104 8.2 x 10 

KBMC (19751x1 

aFestou (1981) 

bWeaver et al. (1981) 

'Kobayashi-Berger-Milan, Festou (1981) 

dKeller and Lillie (1974) 

eFeldman et al. (1984) 
I 

I 

d 

I 

I 

4 

I 

0 

m 

N 

dNhlrc) N 

N 

I 

N 

1 

I 

I 

m 

N 

,-I 

1 

I 

I 

m 

N 

e4 

cc) 

Im 

4 

N 

hl 

I 

0 

I 

I 

1 

\D 

d 

d 

.... 

OOdd 

11 
Table 4 

Summary of Long Slit Spectroscopy of Comet P/Halley 

Date r A Number of 

1985-86 AU AU Spectra 

Aug 23 2.84 3.24 3 

Aug 24 

Sep 17 

Sep 24 

Oct 20 

Oct 21 

Nov 15 

Dec 8 

Dec 9 

Jan 10 

Jan 11 

Jan 12 

Mar 1 

Mar 2 

Mar 3 

2.83 

2.53 

2.43 

2.09 

2.07 

1.72 

1.38 

1.36 

0.87 

0.86 

0.84 

0.72 

0.74 

0.75 

3.21 

2.49 

2.27 

1.43 

1.40 

0.74 

0.70 

0.72 

1.32 

1.34 

1.36 

1.27 

1.25 

1.22 

1 

3 

2 

5 

2 

9 

10 

5 

13 

11 

9 

15 

11 

6 

Mar 4 0.76 1.20 6 

Mar 5 0.77 1.18 9 

Apr 14 1.38 0.43 22 

Apr 15 1.39 0.44 19 

May 9 1.75 1.05 14 

Jun 5 2.13 1.93 6 

Jun 6 2.14 1.96 5 

12 
n 

P 

v 

eo 

0 

P 

LD 

0 - 

IZ 

w 

Q 2 

I 

0" 

n 

lu 

W 

LD 

0 - m 

0 

.- 

n 

3 

< 

u 5 

M 

C 

u 

cd 

0 

c 

a> 

0 

.- 

L 

w 

.- 0 

13 
ORIGINAL PAGE IS 

OF POOR QUALm e 

- 

'(. 

. ... . , 

. .' ', >;i ?r .; :, , 

., 1.;: ,*: . .. . _. , ;'" 

CN + Continuum 

(9 180& 

Continuum 

(8600A) 

Figure 2 

CCD Images of Comet 

P/Halley on May 11, 1986. 

Shown are the two direct 

images (above) and the 

subtracted image (below) 

wh ich shows the d is t r ibution of CN alone. An 

apparent CN jet can be seen 

in the cont rast-enhanced 

reduced image below. 

CN 

14 
Ref e renc e s 

A'Hearn, M. F. (1982) In Comets (ed. L. Wilkening), The University of Arizona 

Press, Tucson, AZ, pp. 433-460. 

A'Hearn, M.F., S. Hoban, P.V. Birch, C. Bowers, R. Martin, D.A. Klinglesmith 

(1986) Nature (in press). 

Cochran, A. L. (1985) Astron. J. 90, 2609. 

Combi, M.R., and A.H. Delsemme (1980a) Ap. J. - 237, 633. 

Combi, M.R., and A.H. Delsemme (1980b) Ap. J. 237, 641. 

Eddington, A.S. (1910) M. N. R. A. S. 79, 442. 

- 

- 

- 

Feldman, P. D., H. A. Weaver and M. C. Festou (1984) Icarus 60, 455. 

Festou, M.C. (1981a) Astron. Astrophys. 95, 69. 

- 

Festou, M.C. (1981b) Astron. Astrophys. 96, 52. 

Haser, L. (1957) Bull. Acad. Roy. SOC. Belgique - 43, 740. 

Haser, L. (1966) 

Huebner, W.F., and J.J. Keady (1983) International Conference on Cometary 

Mem. SOC. Roy. Sci. Liege - 12, series 5, 233. 

Exploration, Hungarian Academy of Sciences. 

Johnson, J.R., U. Fink and S.M. Larson (1984) Icarus 60, 351. 

Keller, H. U., and C. F. Lillie (1974) Astron. Astrophys. 34, 187. 

Keller, H.U., and R.R. Meier (1976) 

- 

- 

Astron. Astrophys.52, 273. 

Marconi, M.L., and D.A. Mendis (1982) Ap. J. 260, 386. 

Newburn, R. L., and H. Spinrad (1984) Astron. J. 89, 289. 

- 

- 

Swings, P., and L. Haser (1956) Univ. Liege and U.S. Air Force Cambridge Res. 

Tech. Report 56-267, Louvain Centerick Press. 

Weaver, H. A., P. D. Feldman, M. C. Festou and M. F. A'Hearn (1981) 

Astrophys. J. 251 809. -' 

15 
APPENDIX 
THE ASTROPHYSICAL JOURNAL 308:472-484,1986 September I 

e 1986. The American Astronomical Society. All rights ruened. Prinled in U.S.A. 

NEUTRAL COMETARY ATMOSPHERES. V. C, and CN IN COMETS 

M. R. COMBI' 

Atmospheric and Environmental Research, Inc., Cambridge, Massachusetts 

AND 

A. H. DELSEMME 

Department of Physics and Astronomy, The University of Toledo 

Received 1985 December 9; accepted 1986 February 20 

ABSTRACT 

Brightness profiles of C2 in comets Bennett 1970 I1 and Kohoutek 1973 XI1 are presented. Model analysis 

of these profiles yields radial Haser scale lengths for production and destruction of C2 which, when combined 

with other scale length determinations in the literature, are shown to vary as the square of the heliocentric 

distance. This is consistent with photochemical production and destruction. Also presented is an updated 

compilation of CN scale lengths, which shows that the mean parent scale length law varies as r".'. A reanalysis of published cometary photometry, using the new scale length laws, yields a C, to CN ratio which is 

independent of heliocentric distance. The previously documented drop-off in C, production rate relative to 

other neutral species for heliocentric distances larger than -1.5 AU was a simple artifact of the previously 

assumed scale length variations. 

Analysis of the sunward-tailward distortion of the brightness profiles with a Monte Carlo particle-trajectory 

model shows that C2 is released from its parent molecule with an ejection speed of -0.5 km s-', owing to the 

excess photolysis energy. This result also implies that the photochemical lifetimes for the C, parent and C, 

respectively are 3.1 x 104 s and 1.2 x lo5 s at 1 AU. 

Subject headings: comets - molecular processes 

1. INTRODUCTION 

The Swan system (d %,-a of C, dominates the ernission spectra of most comets at visible wavelengths. Furthermore, the detection of C, along with C3 has suggested the 

possible presence of hydrocarbons in comets. Thus, although it 

is really one of the minor components of the comae of comets, 

C, has been the subject of considerable study. Despite this 

effort, though, there are still many unanswered questions 

regarding both its production mechanism(s) and its detailed 

excitation mechanism. 

Although it has been long accepted that the excitation of C, 

emission bands is through resonance fluorescence with solar 

light (Swings 1941), and much progress has been made in modeling the observed Swan band intensity distribution (Arpigny 

1966; Krishna Swamy and ODell 1977, 1979; A'Hearn 1978; 

Lambert and Danks 1983), fundamental unknowns still exist. 

The ground state of C, is x 'X,'; therefore, intercombination 

transitions must play an important role. Krishna Swamy and 

ODell (1979) had assumed an intercombination transition of 

a %,-x 'X,+ with a rate, A x s-', which was later 

found to be consistent with the Mulliken (d 'Z,+-x 'X + 

system intensity discovered by A'Hearn and Feldman (1980). 

However, more recent studies by Johnson, Fink, and Larson 

(1983) and Lambert and Danks (1983) do not favor this ad hoc 

assumption and point to the likely importance of intercombination transitions involving excited triplet states. 

A principal method for studying the production and destruction mechanisms for observed cometary radicals has been the 

observation and model analysis of their spatial distributions. 

Modeling has generally been done with Haser's (1957) model, 

0) 

' Work was begun while a Postdoctoral Research Associate at the University ofToledo. 

which provides radial scale lengths for production and decay. 

These can be related to true photochemical lifetimes if the 

ejection velocity of the daughter radical and the expansion 

velocity of the parent molecule are known (Combi and Delsemme 198Oa, hereafter Paper I). 

We present here revised sunward and antisunward brightness profiles of the C, (0-0) band of the Swan system in Comet 

Bennett 1970 I1 and a pair of profiles in Comet Kohoutek 

1973 XII. Haser model scale lengths determined for these profiles are then compared with those found by other investigators 

to examine their heliocentric distance dependence. The 

separate sunward and antisunward profiles are then analyzed 

with the Monte Carlo particle-trajectory model, as described 

in Paper I and used similarly by Combi (1980, hereafter Paper 

111) for pairs of CN profiles. Since the solar radiation pressure 

on C, radicals can be calculated, the sunward-antisunward 

distortion can be modeled to deduce photochemical lifetimes 

from the spatial distribution. The newly revised C, scale length 

laws and those determined from an updated compilation of 

CN observations are then used to reexamine the production 

rates as determined from filtered photometry. Finally, possible 

sources of C2 are discussed in the light of the other results 

presented in this paper. 

11. MODELING THE SPATIAL DISTRIBUTIONS OF 

NEUTRAL RADICALS 

There are active today three major, broadly defined 

approaches to modeling the spatial distributions of neutral 

cometary species. The traditional method is, of course, that put 

forth in the model of Haser (1957). which considers observed 

daughter radicals to be produced at a constant rate from a 

source of exponentially decaying parent molecules streaming 

radially from a point source nucleus. The radicals continue to 

472 
NEUTRAL COMETARY ATMOSPHERES. V. 473 

move radially and decay exponentially themselves. The two 

decay times are related to scale lengths by the assumed radial 

velocity. A closed form expression for the column density can 

be written in terms of simple integrals of modified Bessel functions. The two scale lengths give the model enough parameter 

flexibility such that it can be fitted to almost any observed 

radial brightness profile. Unfortunately, early attempts to identify suspected parent molecules by comparing observed scale 

lengths with photochemical lifetimes (Potter and DelDuca 

1964; Delsemme and Moreau 1973) generally found cometary 

scale lengths to be too short to be explainable by photochemistry alone. 

At this point, two other possible sources of dissociation or 

ionization of parent molecules had been suggested. These were 

gas phase chemistry (Aitken 1974; Oppenheimer 1975), particularly fast ion-molecule reactions, and an internal ionization 

source created by the interaction between solar wind flow 

through the cometary plasma (Ip and Mendis 1975, 1976~1, 

1977) in combination with chemistry. These ideas evolved 

through large, complicated chemistry models having more 

than 100 species and more than loo0 gas phase and photochemical reactions (see Huebner, Giguere, and Slattery 1982; 

and Mitchell, Prasad, and Huntress 1981) finally into multifluid chemical-dynamic models which address the feedback of 

photochemistry and radiative transfer on the energy balance of 

the inner coma (Marconi and Mendis 1982, 1983; Huebner 

and Keady 1983). The former of these two schemes, a onedimensional chemistry model, has been adopted by Cochran 

(1982, 19851) to analyze observations of cometary radicals. 

This model calculates the nonequilibrium chemistry occurring 

in a single fluid parcel of cometary gas moving radially away 

from the nucleus's surface at a constant velocity and expanding, of course, as r2. Results for CN (Cochran 1982) and C, 

(Cochran 19851) are apparently consistent with photochemical 

production of these two species. In fact, the earlier results of 

this type of chemical model generally underproduced radicals 

which could not be produced by a one- or two-step photodissociation (Giguere and Huebner 1978). 

A third general approach to the question of modeling the 

spatial distribution of cometary radicals was developed simultaneously with the chemical models and is based on the principal weakness of the simple Haser model. That is, when a 

radical or atom is produced during the photodissociation of its 

immediate parent molecule or radical, there is in general some 

excess energy which is divided between the internal energy of 

the fragments (i.e., rotationally, vibrationally, or electronically 

excited states) and translational energy. This translational 

energy can impart extra nonradial velocities to the newly 

created fragments which are of the same order or even much 

larger than the typical outflow velocities of the parent molecules of 0.3 to 1.0 km s-' (Whipple 1980; Delsemme 1982), 

e.g., 1.2-1.5 km s-' for OH from H,O, -20 km s-' for H 

from H,O, -8 km s-' for H from OH, - 1 km s-' for CN 

from HCN, 3-7 km s-' for C and 0 from CO and CO, 

(Huebner and Carpenter 1979; Festou 19816). 

The vectorial model (Festou 1978; Festou et al. 1979; Festou 

1981a, 6) addressed the nonradial radical velocities in a way 

similar to Haser's model, i.e., in terms of a closed-form 

multiple-integral expression for the space or column density of 

an observed radical. Festou successfully applied the vectorial 

model to the distributions of OH as well as H in the inner 

(I lo5 cm) coma. The vectorial model requires at least a triple 

numerical integration to calculate the column density for 

single-parent outflow and daughter ejection velocities. 

Work by the authors (Combi 1979; Papers I, 111; Combi and 

Delsemme 19806 hereafter Paper 11) addressed the same question of nonradial radical velocities in two different ways. One 

was the Average Random Walk Model (ARWM), which was 

based on a simple geometric reinterpretation of Haser model 

scale lengths as radial projections of true nonradial scale 

lengths. The second was the introduction of a Monte Carlo 

model, which simulates the actual photochemical kinetics by 

calculating explicit particle trajectories for many individual 

radicals. We successfully applied the ARWM to show that 16 

symmetric brightness profiles (Le., average of sunwardantisunward pairs) of CN in comets Bennett 1970 I1 and West 

1976 VI could be explained by the photodissociation of cometary HCN but not CH,CN. Paper 111 went on to explain the 

observed distortion in the sunward-antisunward pairs of CN 

brightness profiles in terms of the expected excess velocity 

imparted to CN during the photodissociation of HCN combined with the radiation pressure force using the Monte Carlo 

model. 

Whereas the optical data (see above, and Cochran 1982) are 

consistent with HCN as the parent of observed CN, recent 

radio observations provide conflicting evidence. Irvine et ai. 

(1984) presented the results of radio observations of two 

comets (1983d and 1983e) from 14 observatories. Observations 

of the 1-0 transition of HCN in comet ZRAS-Araki-Alcock 

1983d were made at the National Radio Astronomy Observatory (NRAO) and the Five College Radio Astronomy Observatory (FCRAO) on 1983 UT May 8.6 and 10.1, respectively. The 

NRAO observations yielded a tentative detection indicating a 

production rate for HCN of -3 x 10" s-', which is roughly 

equivalent to the optical production rate for CN. On the other 

hand, the FCRAO observation only two days later provided 

only an upper limit which was an order of magnitude lower. 

The principal qualitative results found in both studies of 

non-radial radical motion were that the use of the Haser model 

resulted in a measurement, in fact, of a radial projection of the 

true scale length and thus would always give smaller values 

than one might suspect from photochemical lifetimes, and that 

the effect of nonradial motion is both measurable and important. Furthermore, the cases of H and OH from HzO and CN 

from HCN can in fact be understood in terms of purely (but 

geometrically correct) photochemical models, for generally 

large comets with supposedly well developed collision zones 

and ionospheres. 

The single-fluid-one-dimensional chemistry models (e.g.. 

Giguere and Huebner 1978; Mitchell, Prasad, and Huntress 

1981; Cochran 1982, 1985a), of course, cannot easily address 

the problem of nonradial motion. Huebner and Keady (1983) 

have used our ARWM to approximate the transition from 

collision-dominated flow to free flow in vacuum in their multifluid chemical dynamic models. They are currently working on 

a more fundamental treatment of the transition region 

(Huebner 1985). Finally, a two-dimensional multi-fluid model 

(one fluid for each observed species) that treats this transition 

zone in a proper way would be required to attack the problem 

of the radiation pressure distortion, which is quite apparent for 

C, and CN for heliocentric distances less than 1 AU. 

Let us enumerate a few points regarding these various modeling efforts: 

1. That chemical reactions (especially the fast ion-neutral 

.reactions) occur in the inner regions (r < loo0 km) of comets 

with large gas production rates for heliocentric distances 1 AU 

or less is not questioned. The rates for the reactions involving 

particular species may well dominate photochemical rates in 
474 COMB1 AND DELSEMME 

the inner coma regions. A major reshuming of ion species may 

in fact occur, and the special case of observed C('D) may be 

produced by dissociative recombination. However, the situation is complicated by the fact that a gas parcel only spends 

on the order of IO00 s in this region, and all the gas phase 

chemistry must occur in that time scale. 

2. Photochemical reaction rates vary from to 

s-l. However, since the rates depend only on the square 

of the heliocentric distance and in some cases on the heliocentric velocity, they have no time constraint and should generally dominate in the long run. 

3. The impressively large base of information gathered by 

A'Hearn and his collaborators over the last 12 yr in the form of 

photometric observations of the major visible radicals and 

dust has demonstrated that there are no systematic compositional differences between the gas composition of the comae of 

large and small gas-producing comets. This corresponds to gas 

production rates (and collision zone radii) varying by well over 

two orders of magnitude. The only consistently anomalous 

behavior has been that of the apparent fall-off in the C, abundance relative to all other species for heliocentric distances 

larger than 1.2 AU, and even this behavior seems to be independent of the comets' gas production rates (A'Hearn and 

Cowan 1980; A'Hearn 1982; see 9 IV below). The uniformity 

seems also to be evident in the growing base of ultraviolet data, 

especially from IUE [again with the exception of q'D), 

Weaver et al. 1981 ; Feldman 19823. 

4. Even the results of Cochran (1982, 1985a), which used a 

single-fluid, one-dimensional, constant outflow velocity, nonequilibrium chemistry model, point to photodissociation of 

parent molecules as by far the dominant source of the observed 

radical species. 

5. The simple single-fluid, constant outflow velocity chemistry models are clearly inappropriate outside the collision 

zone. Here, exothermic nonthermal nonradial velocities and 

radiation pressure dominate the kinematics and either photodissociation, photoionization, or charge exchange impact with 

the solar wind dominate the radical production and decay 

processes. 

6. In the case of negligible impact by gas phase chemistry in 

this type of chemistry model, the density distribution of a 

radical should be virtually identically reproduced by a sum of 

either simple Haser models for the case of several single-step 

photodissociations or, at worst, some type of multiple-step 

grandparent-parent model, such as was introduced by Malaise 

(1966). For example, Cochran (19856) has recently fitted Haser 

models to observed C,, C,, and CN profiles which had been 

previously analyzed with a one-dimensional chemistry model 

(Cochran 1982,1985~). 

7. A major weakness of almost all modeling efforts is the 

fact that the principal shaping factor of the outer radical coma 

(Le., the daughter scale length region - 1-2 x lo5 km) may in 

fact be the sporadic activity of the nucleus's vaporization. This 

was suggested some time ago by Malaise (1 970) and was borne 

out by the measured Haser scale lengths for CN decay determined for comets Bennett 1970 I1 and West 1976 VI (Paper 

11). Figure 1 shows the variation of the two Haser scale lengths 

for observed CN brightness profiles. We have added points 

from Combi (1978); Cochran (1982); Delsemme and Combi 

(1983, hereafter Paper IV); and Johnson, Fink, and Larson 

(1983) to our original data. Although the variation of the 

parent scale lengths is not inconsistent with a simple rz law 

(owing to the large amount of scatter), the best power-law fit is 

E 

actually proportional to r'.'. On the other hand, we found no 

measurable trend in the Haser scale length for CN decay as a 

function of heliocentric distance, as we had for the Haser scale 

length for decay of the CN parent (HCN). Rather, we found a 

fairly random distribution of values generally greater than 

lo5 s. Simply put, this means that whereas a steady state 

vaporization rate for time scales lo4 s or less may make it 

possible to identify parent molecule decay rates (or at least 

Haser scale lengths or source region sizes), either sporadic or 

periodic variations in the vaporization rate on time scales of 

1-5 x lo5 s may make it at least very difficult to model full 

observed brightness distributions. It should also be mentioned 

here that Cucchiaro and Malaise (1982) and Keller and Meier 

(1980) have attempted some models with time-dependent outbursts in production rate to explain observed spatial distributions. 

8. Finally, the results of a chemical model run depend critically on many rate constants, nearly half of them poorly 

known, and on the initial assumed composition of the 

nucleus's volatile mix (which is exactly what is unknown). The 

problem, of course, cannot be uniquely inverted even if all the 

rate constants are precisely known. The agreement of the 

model profiles with observations is a necessary but not a sufficient condition for having the correct nuclear mix. Furthermore, some of the dominant photolytic rates used in these 

models are approximations derived from Haser model fits to 

observed brightness profiles. 

In conclusion, since there has yet to be demonstrated a reasonable case for the production of any of the principally 

observed neutral cometary species-with the possible exception of C('D)-by gas phase chemical reactions, it seems reasonable to choose some combination (or sum) of the 

exospheric photochemical models with which to attempt to 

understand the observations of the spatial distributions of 

neutral cometary species. A method we (Papers I, 11, 111) have 

used still remains quite viable. This is to use the Haser model, 

realizing that the scale lengths may have no direct physical 

meaning (even for pure photochemistry) to characterize the 

spatial extent and heliocentric distance dependence of the 

source region. From this point, one can then move on to the 

ARWM interpretation of the scale lengths; or to explicit modeling with the Monte Carlo particle-trajectory models (PTM) 

which can handle radiation pressure, collisions, velocity distributions, and time dependencies as necessary; or to both. 

Note that the simplest version of the particle-trajectory 

model, i.e., with nonradial ejection of photodissociated radicals, isfundamentally equivalent to the vectorial model (Festou 

1981~). Also, even though neither the vectorial nor the particletrajectory model is as directly invertible as Haser's we are still 

left with only a handful of parameters, such as lifetimes and 

velocities, to specify, and we do not have to prespecify the 

identity of a proposed parent but simply find the best-fit lifetimes and velocities for a given data set. 

111. BRIGHTNESS PROFILES OF c, 

The brightness profiles of C, (0-0) presented here were 

determined from microdensitometer scans of spectrograms of 

comet Bennett 1970 I1 and comet Kohoutek 1973 XII. The 

details regarding the observations and the principal data 

reduction have been discussed at length in earlier papers for 

both the Toledo plates of comet Bennett (Delsemme and 

Combi 1979; Paper 11) and the Lick plates of comet Kohoutek 

(Paper IV), and will not be pursued further in this paper. See 
I I I1 1 I I11 

0 

I I I 11 1 1 Ill 1111 

0.5 10 1.5 2.0 

Heliocentric Distance (AU) 

Rc. la 

t 

3 2 

t 

1 .o 1.5 2.0 

Heliocentric Distance (AU) 

FIG. Ib 

FIG. 1.-Variation of CN Haser (a) parent and (h) daughter scale length with heliocentric distance. Circles, values which result from Haser model fits to whole 

brightness profiles. In (a) the line shows the best-fit power law of 1.6 x lo4 ris km. In (h) the daughter scale lengths exhibit no obvious trend but likely reflect only 

coma activity (Combi and Delsemme 19806). 
476 COMB1 AND DELSEMME 

I I I I Illll I 1 I I1111 

TABLE I 

OBSERVATlONAL DATA 

TABLE 2 

c, PARENT AND DAUGHTER SCALE LENGTHS FROM HASER'S MODEL 

Comet 

Kohoutek 1973 XI1 ..... 1974 Jan 9 0260 0.465 0.846 855 

Bennett 1970 I1 .......... 1970 Apr 18 0960 0.841 1.054 62.5 

1970 Apr 26 0960 0.970 1.247 51.8 

1970 Apr 27 0960 0.986 1.271 50.7 

Heliocentric distance. 

Geocentric distance. 

E Sun-comet-Earth angle. 

also Delsemme and Combi 1979) for a discussion of the earlier 

C, profiles deduced from these plates of comet Bennett by 

Delsemme and Moreau (1973). A systematic error in the original data reduction by Delsemme and Moreau, due to unsuspected vignetting in the spectrograph, was the reason for 

reanalyzing the CN data in Paper I1 and the C, data in this 

paper. We had cautioned in Paper I1 that a revision for the 

original C, scale lengths (similar to that for CN) was also 

likely. Relevant information concerning comet parameters for 

each observation are given in Table 1. The brightness profiles 

are shown in Figures 2-5. As noted in the figure legends, the 

profiles are exactly sunward and antisunward for the Bennett 

plates but are 51" from the true projected radius vector for the 

Kohoutek plate. 

Parent and daughter scale lengths were determined with 

Haser's (1957) model using the same non-linear least-squares 

method as in Paper I1 fsee also Combi 1979) for the average 

symmetric profiles and are listed as part of Table 2. Also in 

Table 2 is a compilation of other measured Haser scale lengths 

Observer 

ODell and 

Osterbrock 1962 . . . . Delsemme and 

Miller 1971 ......... Kumar and 

Southall 1976.. . . . . . Malaise 1976' . .. . . . . . . 

Cochran 1985' ..... . . . 

This work ............. 

Burnham 1960 11 

Tago-Sato-Kosaka 

Bennett 1970 I1 

Bennett 1970 11 

1980 XI11 P/Tuttle 

1980 X P/Stefan-Otei 

Meier 1980 XI1 

Kohoutek 1973 XI1 

Bennett 1970 11 

Bennett 1970 I1 

Bennett 1970 11 

1969 IX 

1 .OO 

1.25 

0.664 

0.7 13 

1.019 

1.755 

0.465 

0.84 1 

0.970 

0.986 

ma 1.58 

4.10b 4.95 

4.40 5.09 

3.61 4.83 

3.95 5.08 

4.25b 5.11b 

4.62b 5.6Sb 

4.72b 5.59b 

3.69 4.29 

4.05 4.78 

4.08 4.97 

4.23 4.92 

Scale length was computed by Combi 1978 from the original data. 

Determined from the data for this paper. 

determined for C, profiles in different comets from whole 

brightness profile observations only. Newburn and Spinrad 

(1984) have computed Haser model parent scale lengths for C,, 

C,, and CN from two-point spectrophotometry of five different 

comets. They determined column densities within a 4 diameter aperture centered on the nucleus and displaced 1715 and 

35" from the nucleus. From the nucleus value and one displaced point, they determined a parent scale length and a production rate. In order to do this, they had to assume the value 

for the daughter scale lengths as adopted by A'Hearn (1982) 

t 

1970 II 

r = .841 

4 1 
1970 II Y 

r = .970 

I I I I I Ill1 1 I I I Ill 

104 io5 106 

lo’ 

103 

Distance from the Nucleus (km) 

FIG. 3.-Samc as Fig. 2, for 1970 April 26 

10’ I I I I I I Ill1 I 1 1 I11111l 1 I 1 I Ill 

io3 lo4 lo5 lo6 

Distance from the Nucleus (km) 

FIG. 4.-Same as Fig. 2, for 1970 April 27 
478 

Distance from the Nucleus (km) 

FIG. 5.-Brightness profiles of C, in comet Kohoutek 1973 XII; symbols as in Fig. 2. The points marked sunward and antisunward are actually 51” from the true 

directions. Because of the limited spatial extension, especially sunward, no asymmetric profile information could be extracted. However, the corresponding Monte 

Cario PTM (solid lines) were computed for this observational geometry with model parameters scaled appropriately from the other pairs of protiles. 

from our earlier papers (Delsemme and Moreau 1973; Paper 

11). In a few cases, they determined scale lengths using each 

displaced point in combination with the nucleus values, and 

differences up to a factor of 2 resulted. When using entire 

brightness profiles, both scale lengths can be determined with 

uncertainties of only 10%-25%, so we limit our study to this 

type of data (see Table 2 for references). Figures 6a and 6b 

show the variation with the comet’s heliocentric distance of the 

C, parent and the C, radial Haser length respectively. Simple 

power-law fits to the data imply the following expressions for 

the C, parent and C, scale lengths: 

yH(C, parent) = 1.6 x 104ri0*0.3 km , 

yH(C,) = 1.1 x 105r$0*0.3 km , 

where rH is the comet’s heliocentric distance. The scatter of 

values in Figures 6a and 66 likely results from both the 

10%-25% uncertainties in the Haser model fitting procedure 

as well as temporal fluctuations in the vaporization rate of the 

cometary nucleus. 

It is worth noting here that if we include the C, parent scale 

lengths determined by Newburn and Spinrad (1984), the 

overall power-law fit is not changed much, although their data 

do deviate much more from the mean power law. This is also 

true when comparing their CN results with the results from 

whole profiles (Fig. 1). 

IV. THE PRODUCTION RATES OF c, AND CN IN COMETS 

A’Hearn, Thurber, and Millis (I 977, hereafter ATM) have 

published an extensive set of filter photometric observations of 

C,, C,, and CN over a wide range of heliocentric distance for 

comet West 1976 VI. In Paper I1 we rereduced their CN data 

with the updated scale lengths and scale length laws and found 

a CN production rate law consistent with that of C, as determined by ATM. Both of these pointed to a dependence on the 

heliocentric distance consistent with an inverse square law out 

to rH z 2.5 AU. The case for C,, on the other hand, has been 

different. 

ATM assumed both C, and CN parent scale lengths varying 

as rH+l. Subsequent papers (A’Hearn, Millis, and Birch 1979; 

A’Hearn and Cowan 1980; A’Hearn and Millis 1980) used this 

same scale length law for C, but now use our revised values for 

CN. In Table 3 we present the rereduced C, and CN production rates, assuming the new Haser scale length laws as 

well as the corrected g-factor for C, (0-0) (i.e., 

4.5 x lo-’, ergs s-’ per radical; A’Hearn 1985). 

The production rates for both C, and CN in comet West 

now vary almost exactly as rHz. The steep drop-off in C2 production rate from an rHz law for rH > 1.5 AU has been eliminated. A plot of the CJCN ratio versus rH for comet West is 

shown in Figure 7. A very slight downward trend remains 

apparent for most points where rH = 2.5 AU. Since it appears 

that some irregularity occurred in the production rate curves 

for C,, C,, CN, and solids in comet West near rH 2 1.5 AU 

(A’Hearn and Cowan 1980), we should examine the behavior of 

the C,/CN ratio in other comets as a function of heliocentric 

distance. A’Hearn and Millis (1980) found that this apparent 

relative depletion of C, at larger heliocentric distances 

occurred in many comets, both periodic and new. A’Hearn and 

Cowan (1980) attributed this to the hypothesis that the C2 
1 I 1 111 I I I I I I I1 1111 

0.5 1 .o 1.5 2.0 

Heliocentric Distance (AU) 

FIG. 60 

Heliocentric Distance (AU) 

FIG. 66 

FIG. 6.-Variation of C, Haser (a) parent and (b) daughter scale lengths with heliocentric distance. Circles, values in Table 2; lines, best power-law fit of rfi" 
480 COMB1 AND DELSEMME Vol. 308 

0.468.. .... 27.675 27.707 

0.522.. .... 27.662 27.790 

0.522. ..... 28.757 27.782 

0.522.. .... 21.664 27.733 

0.600.. .... 27.483 27.495 

0.651 ...... 27.560 27.567 

0.773.. .... 27.295 27.340 

0.820 ...... 27.432 27.426 

0.899. ..... 27.206 27.208 

1.044.. .... 27.203 27.129 

1.406.. .... 26.665 26.640 

1.443.. .... 26.535 26.645 

1.443.. .... 26.544 26.625 

1.554 ...... 26.688 26.722 

1.767.. .... 26.738 26.814 

1.852 ...... 26.683 26.837 

1.852 ...... 26.666 26.844 

2.002.. .... 26.439 26.604 

2.146 ...... 26.41 7 26.554 

2.209.. .... 26.157 26.403 

2.497.. .... 26.255 26.182 

2.540 ...... 26.097 26.243 

2.550 ...... 26.281 26.388 

with the new revised Haser scale length laws. 

Recalculated from the photometry of ATM 

parent is embedded in grains of a less volatile component than 

water, whose vaporization turns off at a higher temperature 

than the bulk vaporization of the nucleus. They constructed a 

model which quantitatively explains the data. This explanation 

now no longer seems necessary. 

We have also rereduced all the photometry of six comets, 

observed by A'Hearn and Millis (1980) for C, and CN, and 

find similar results, as shown in Table 4. The values of the 

ratios of the production rates have been categorized in a way 

similar to that done by A'Hearn and Millis, in two groups: 

those with a heliocentric distance less than 1.5 AU, and those 

greater than 1.5 AU. These results are summarized in Table 5. 

Using the old scale length laws, a sizeable depletion (45%) of 

C, relative to CN for r > 1.5 AU was found as in the case for 

comet West, but with the revised scale lengths laws this depletion again disappears. 

Thus we have explained the apparent drop of C, production 

rate for larger heliocentric distance to be simply an artifact of 

an inappropriate Haser scale length law. 

. 

V. RADIATION PRESSURE ON c2 

Paper I11 showed that the difference between the sunward 

and antisunward brightness profiles of a cometary radical due 

to solar radiation pressure could be quantitatively modeled 

and used to disentangle the velocity and lifetime from the 

decay scale length. The same many-particle Monte Carlo coma 

model has been used to analyze the profiles of C, as was used 

in Paper I11 for CN. 

The model discussed in detail in Paper I correctly models 

photochemical kinematics by taking into account the isotropic 

0.2 I I I I I I I I I 0 0 0 

DO 0 

on 

0 

0.0 

I 

- 

0 

- 0 

-0.4 

-0.6 

- 0 

e 

0 

0 

0 

0 

0 

E 

e e ** ** 0 

- e e e e 

**o e 

0 

0 

eo 

1 I I I I I I I I 

-0.4 -0.2 0.0 0.2 0.4 

log r (AU) 

FIG. 7.-Ratio of the production rates of C, to CN in comet West 1976 VI. Open squares, reduction of photometric band fluxes determined by ATM and reduced 

with the Haser scale lengths adopted by A'Hearn and Cowan (1980). Filled circles, our reduction of the photometric band fluxes reduced with the ne\\ Haser scale 

length laws as presented in this paper. 
No. 1, 1986 NEUTRAL COMETARY ATMOSPHERES. V. 48 1 

TABLE 4 

PRODUCTION RATES' OF c, AND CN IN SIX COMETS 

1978 XIV P/Ashbrook-Jackson 

2.349. ..... 25.31 ... 2.347.. .... 24.92 25.05 

2.344.. .... 25.02 24.79 

2.288.. .... 24.95 24.74 

1978 XI P/Wild 2 

1.803.. .... 25.19 25.18 

1.497.. .... 25.62 25.44 

1.495.. .... 25.63 25.49 

1.495.. .... 25.64 25.43 

1.493.. .... 25.62 25.42 

1.493.. .... 25.68 25.48 

Meier 1978 XXI 

2.858.. .... 26.72 25.57 

2.587.. .... 26.93 26.69 

2.553 ...... 27.00 26.73 

2.553.. .... 27.14 26.73 

2.222 ...... <25.15 26.79 

2.211 ...... ... 26.82 

1978 XX P/Haneda-Campos ~~ 

1.177.. .... 24.66 23.88 

Bradfield 1979c 

0.960.. .... 24.83 24.84 

0.978.. .... 24.84. 24.88 

Meier 197% ~ 

1.479.. .... 25.19 24.87 

a Recalculated from the photometry of 

A'Hearn and Millis 1980 with the new revised 

Haser scale length laws. 

ejection of the daughter radicals. In the model, the trajectories 

of many radicals (usually lo5) are actually calculated, and the 

column and space densities are found by counting the number 

of radicals in grids and bins of known area and volume respectively. This is in contrast to the usual cometary free-flow 

models which compute densities by numerical integration of 

emission flux functions. 

Haser's model can be fitted to data in terms of three independent parameters: the ratio of Q (production rate) to u 

(radial outflow speed), and the two scale lengths (one for production and one for decay of the daughter radical). In contrast, 

TABLE 5 

c, TO CN PRODUCTION RATIO mR Six COMETS 

RANGE OF log (QcJQcNY 

DISTANCE Currentb New 

r< 1.5 ...... +0.072 +0.176 

r > 1.5 ...... - 0.088 +0.178 

HELIOCENTRIC 

(AU) Scale lengths Scale Lengths 

Qc2 and QCN are the respective C, and CN proAs adopted by A'Hearn and Millis 1980. 

duction rates. 

the Monte Carlo PTM without radiation pressure (and the 

vectorial model), owing to the isotropic ejection directions of 

daughter radicals, introduces two extra free parameters, which 

are the parent outflow and the radical ejection speeds. 

However, if a sizeable radiation pressure acceleration is present 

and can be independently calculated, and the distorted coma 

can be observed, the two additional parameters become constrained. Therefore, if one can determine the radiation pressure 

acceleration from the total solar fluorescence efficiency (gfactor) and can measure sunward and antisunward brightness 

profiles (or a 2-dimensional mapping), then the velocities and 

lifetimes can be completely deconvolved. 

The response of modeled sunward and antisunward profiles 

to the relative speeds of the radical and the parent is illustrated 

in Figure 8. The location of the sunward limit is set approximately by the maximum radial speed u,,, (i.e., the scalar sum 

loo II 1 I1111111 I Ill,1111 I Ill 

Projected Distance (km 

103 104 io5 

FIG. I.-Effects of radical ejection velocity on the modeled radiation pressure asymmetry. Three pairs of profiles are shown as modeled with the Monte 

Carlo particle-trajectory model. All three exhibit the same symmetric radial 

scale lengths, as determined with the ARWM, but have different velocity ratios 

of radical ejection to parent outflow. As the radical ejection velocity is 

increased, the location where the sunward and antisunward profiles diverge 

from one another moves backward toward the nucleus. This response yields an 

effective method for disentangling velocities and lifetimes from observed scale 

lengths. 
482 COMB1 AND DELSEMME Vol. 308 

of the parent and radical ejection speeds) and the radiation 

pressure acceleration a as a viaJ2a envelope. The distance 

from the nucleus where the sunward and antisunward profiles 

diverge is set by the relative parent outflow and the radical 

ejection speeds. All three sets of model profiles in Figure 8 have 

the same radial (Haser) scale lengths, with the appropriate 

lifetimes being set using the ARWM. They also have the same 

sunward limit set by the maximum velocity and the acceleration. As radical speed increases, the location where the two 

profiles diverge moves back toward the nucleus. 

The radiation pressure on a C2 radical at 1 AU can be calculated from the fluorescence efkiency of the (0-0) band of the 

Swan System (4.5 x lo-'' ergs s-'; A'Hearn 1975) and the 

observed relative band ratios for the rest of the system 

("earn 1985). We find a value of 0.81 cm s-'. There are other 

spectroscopic systems which may contribute to the radiation 

pressure, principally the singlet Phillips (A 'lT,,-X 'Ea+) and 

Mulliken (d 'Ey+-x 'E,,') systems. This raises the question of 

whether intercombination transitions are important. It 

appears that two alternate scenarios can be envisioned: either 

intercombination transitions are not important and we have 

two populations of C2 radicals, triplets and singlets, coexisting 

in the ratio of -20 to 1 (Johnson, Fink, and Larson 1983; 

Davis et a/. 1984); or intercombination transitions are important and we have one population of C, radicals spending 

-95% of their time in triplet states and -5% in singlet states. 

In either case, total C2 abundances determined from the Swan 

band observations would be underestimated by -5%. In the 

former case, contributions to the radiation pressure on the 

triplet population of C2 radicals would result from triplet 

(mainly Swan) transitions only. Here also, singlet and triplet 

populations of C, would be accelerated at different rates. In 

the latter case, the radiation pressure would be determined by 

the triplet transitions 95% of the time and by the singlet transitions only 5% of the time, and thus the effective acceleration 

on all C, radicals would be the same and a few percent less 

than that determined from the triplet g-factor alone. Such a 

difference could be tested observationally from the comparison 

of the spatial distributions of triplet and singlet C, emissions. 

One should expect significantly less distortion from circularity 

for the isophotes of a separate population of singlet C, radicals 

than for triplet C, radicals (the former case), and identically 

distorted isophotes from a mixed population (the latter case). 

However, we are concerned with Swan band (triplet) observations, and there is a large uncertainty in the absolute g-factor 

which may be as large as 40% ("earn 1985). Therefore, a 

value near 0.81 cm s-' is quite reasonable for our purposes. 

An alternate approach to constraining the model parameters 

would be to adopt the parent outflow speed law of Delsemme 

(1982). From the compilation of halo expansion velocities measured by Bobrovnikoff (1954) and Beyer (1961), Whipple (1980) 

has suggested that the initial radial velocity of parent molecules expanding from the nucleus is given approximately as 

0.535r,0.6 in km s- '. Delsemme (1982) has added to this data 

set the result of Malaise (1970), who deduced outflow velocities 

near the nucleus using the Swings-Greenstein effect on CN 

rotational lines. Furthermore, Delsemme has provided a 

simple theoretical argument that, since a temperature law of 

T = To rH is expected, the true radial velocity expansion law 

should now be given as I u = (0.58 & 0.03)r,0.5 . 

The change in the power-law exponent is due to the fact that 

Bobrovnikoff's and Beyer's data beyond 3 AU must clearly 

reflect vaporization of gases more vblatile than water; the constant shift corresponds to a factor of 1.25, yielding an average 

molecular weight of 28 for these gases. Either CO or a mixture 

of CO, with lighter gases could explain this shift. 

If we adopt this parent speed law, then the best-fit models to 

the observed profiles will yield both lifetimes, the radical 

outflow speed and the radiation pressure acceleration. It was 

found that the best fits to all three of the pairs of C, profiles 

(shown as the solid lines in Figs. 2, 3, and 4) in comet Bennett 

were obtained with an ejection velocity of -0.5 km s-l. The 

corresponding profiles for the limited comet Kohoutek profiles 

have also been calculated and are shown in Figure 5. The 

numerical results of the Monte Carlo PTM analysis of the 

three pairs of C, profiles in comet Bennett are summarized as 

follows: 

' 

Assumptions: 

1. v (parent) = 0.58/r1" km s-' 

2. C,X + hv-rC, + X +energy 

0.5 km s-' 

 parent) 3.1 x lo4 s 

 daughter) 1.2 x lo5 s 

a(rad. pr.) 0.70 cm s-' 

Resultsat 1 AU: 

VAC,) 

A value for the radiation pressure acceleration of 0.70 cm s-', 

found from the model analysis, compares rather favorably with 

the value calculated from the g-factor (0.81 cm s-'). In fact, 

this is well within the 40% uncertainty expected for the gfactor. The self-consistency in the results lends confidence in 

the model, the g-factor, and a parent outflow speed of 

-0.6 km s-' near 1 AU. Of course, since we have observations over only a narrow range of heliocentric distance (0.84- 

1.0 AU), we really can neither confirm nor deny the validity of 

the Delsemme velocity law. 

VI. POSSIBLE SOURCES OF c, 

We now turn to the question of examining the possible C, 

source(s) in comets in terms of the deduced production lifetime 

(i.e., destruction of the parent), found to be -3.1 x lo4 s at 

rH = 1 AU and the ejection speed of 0.5 km s-'. The uncertainty in this lifetime can be crudely estimated from the uncertainties in the (0-0) band g-factor (40%), the parent velocity 

law (6%), and the effective scale length fitting procedure 

( 5 15%). Since the spatial distortion is in effect set by a u2/2a 

paraboloid envelope, uncertainties in the inferred velocities 

enter as the square root of those in the radiation pressure 

acceleration (the g-factor). Therefore, a total uncertainty of 

-20% in the determined parent lifetime is expected if one 

relies on the parent velocity law and of - 35% if one relies on 

the g-factor. 

C, once seemed like a viable candidate for at least a major 

source of cometary C,, but the g-factor for the 14040 band has 

been revised upward by a factor of 40 (A'Hearn 1982). Thus, 

the C, production rate is now believed to be nearly two orders 

of magnitude less than those of C, and CN. 

In their extensive work on solar photochemical radiation 

rates, Huebner and Carpenter (1979) had calculated the photochemical lifetimes of C,H, and C,H4. The following reactions 

which may ultimately produce C,, shown with the lifetime of 

the original reactant, are: 

1. CzH4 + hv- C,H, + H, , T = 2.1 x IO4 s 

(Huebner and Carpenter 1979); 

(Huebner and Carpenter 1979); 

theoretically unknown. 

2. C,H, + hv+ C,H , = 3.1 x 104 s 

3. C2H + hv- C, + H , 
No. 1, 1986 NEUTRAL COMETARY ATMOSPHERES. V. 483 

TABLE 6 

C,H, PHOTOCHEMISTRY 

Rate’ E,,,,, uhcarY Branching 

Reaction (s-’) (eV) (km s-I) Ratio 

C,H, + hv-rC,H + H ... 1 x lo-’ 3.2 0.97 0.79 

+ C, + H, ..... 2.1 x 3.1 1.4 0.21 

Total C,H, .............. 1.27 x 1 .o 

Old C,H, .............. 3.1 x IO-’ 1 .o 

C,H + hv-r C, + H Unknown ...... ... ... ... 

New reaction rates from Huebner (1985); old reaction rate from Huebner 

and Carpenter 1979. 

It seems that with a total parent lifetime of 3.1 x lo4 s found 

in this paper (i.e., “total ” meaning all steps of a multiple-step 

process added up to yield a simple effective parent lifetime), 

CzH4 can probably be eliminated as a principal parent. Furthermore, even C,H, would only be possible if the lifetime for 

reaction (3) were very short. This would be precisely the same 

conclusion reached with the chemical model by Cochran 

(19854, who placed a value of 2000 s on the lifetime for C,H in 

order for acetylene to be the primary parent frozen in the 

cometary nucleus. However, new information regarding the 

C,H, photodissociation has become available (Huebner 1985), 

which revises the lifetime of C,H, upward (or the reaction rate 

downward) by more than factor of 2. These new data are given 

in Table 6. The reaction rate for photodissociation of the C,H 

radical is still unknown. A photochemical lifetime of 

7.9 x lo5 s and the large ejection speeds would appear to 

eliminate C,H, as the only or principal primary parent for C,. 

However, the determination of photodissociation rates from 

solar spectral fluxes and laboratory-measured photoabsorption cross sections is not a simple problem. Even in cases where 

a complete absorption spectrum has been measured, the validity of this approach is complicated by (1) the multiline nature 

of the solar UV spectrum, (2) the relatively low wavelength 

resolution of the experiments (- 10 A) and, most importantly, 

(3) the high density and high internal temperature (-270 K) of 

the target gas. As cometary molecules leave the collisiondominated inner region, they naturally undergo rapid radiative 

cooling, leaving them in their lowest rotational and vibrational 

levels (T < 100 K). Actual photodissociation cross sections 

(and the major branching ratios) can be more than two orders 

of magnitude different from values measured in the laboratory 

(Jackson 1982). 

These difficulties can be best illustrated by the widely 

varying total solar photodissociation rates quoted for C2H2 

over the years in the literature: 1.6 x lo5 s by Potter and 

DelDuca (1964); 5.7 x lo3 s by Jackson (1976); 3.1 x lo4 s by 

Huebner and Carpenter (1979); and 7.9 x lo4 s by Huebner 

(1985). Finally, Huebner (1985) has suggested that C, may be 

produced by a combination of many sources, possibly including C,H,. In any event, the effective lifetime of the C, source(s) 

has been determined empirically here to be - 3.1 x IO4 s. 

The case for attempting to identify the parent(s) of cometary 

C, is a perfect example of the dangers in trying to use complicated chemical models for routine analyses of observed spatial 

distributions of cometary radicals. Revisions by factors of 2-5 

are to be expected as better laboratory measurements and ab 

initio calculations of gas phase chemical and photochemical 

reactions are made. In this case, a revision by a factor of 2.5 in 

only one out ofmore than lo00 reactions completely reversed an 

interpretation. (Furthermore, the same interpretation would 

have been drawn using the old reaction rate and the Haser and 

Monte Carlo model approach adopted in this paper.) 

VII. SUMMARY 

1. Brightness profiles of C, in comets Bennett 1970 I1 and 

Kohoutek 1973 XI1 have been presented. 

2. Radial (Haser) scale lengths for production and destruction of C, have been determined from these profiles, and when 

compared with those determined from other brightness profiles 

in the literature are found to be consistent with both scale 

lengths varying as the square of the heliocentric distance r. An 

updated compilation of the CN Haser scale lengths yields an 

rl.’ for the CN parent. 

3. The production rates of C, and CN as determined from 

the photometry of ATM and A’Hearn and Millis (1980) have 

been recalculated using Haser’s model and the new scale length 

r-dependences. The drop in production rate of C, relative to 

CN (and presumably other species) for heliocentric distances 

greater than 1.5 AU is now eliminated. Furthermore, the production rates of both C, and CN is comet West vary almost 

precisely as the inverse square of the heliocentric distance. 

4. Monte Carlo particle-trajectory models which take into 

account the radiation pressure acceleration on C, were fitted 

to the sunward and antisunward pairs of profiles for comet 

Bennett. These results yield a photochemical lifetime of 

3.1 x lo4 s at 1 AU for the parent@) of C, and an ejection 

velocity of -0.5 km s-l for C, upon dissociation and a radiation pressure acceleration consistent with that expected for 

solar fluorescence, if one assumes a parent outflow of 

-0.6 km s-’ (Delsemme 1982). A photochemical lifetime of 

1.2 x lo5 s for C, was found. This interpretation is consistent 

with the radiation pressure acceleration as determined by the 

total C, solar fluorescence rate. 

5. We have examined the possible molecular sources for 

cometary C, in terms of the parent lifetime of 3.1 x lo4 s at 

1 AU. C,H4 in a multiple-step process seems unlikely as an 

ultimate source. C,H, as the sole or even primary source is 

also inconsistent with the recently updated photodissociation 

rates for C,H, (Huebner 1985). 

The authors would like to thank Dr W. F. Huebner and Dr. 

A. L. Cochran for helpful discussions. Work at AER was supported under NASA contract NASW-3950. Work at the University of Toledo was supported in part by grants ASY 

82-07435 from NSF and NSG-7301 from NASA. 

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-. 1978. Ap. J.. 219,768. 

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484 COMB1 AND DELSEMME 

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Delsemme, A. H. 1982. in Comets, ed. L. L. Wilkenina (Tucson: Universitv of 

J., 281,455. 

- Arizona Press), p.85. 

Delsemme, A. H., and Combi, M. R. 1979, Ap. J., 288,330. -. 1983. Ap. J.. 271,388 (Paper IV). 

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Giguere, P. T., and Huebner, W. F. 1978, Ap. J., 223,638. 

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Note added in prooC-It has been pointed out by M. F. A’Hearn that, whereas a C, triplet-to-singlet ratio of 20 to 1 was implied 

by the results of Johnson, Fink, and Larson (1983), as discussed in 0 V, A’Hearn and Feldman (1980) found a value closer to 6 to 1. 

Nonetheless, the value is large and as such has no effect on the content of the discussion in 0 V. 

M. R. COMBI: Atmospheric and Environmental Research, Inc., 840 Memorial Drive, Cambridge, MA 021 39-3758 

A. H. DELSEMME: Department of Physics and Astronomy, University of Toledo, Toledo, OH 43506 
t 

I. Report No. 2. &rernment Accession No. 

6. Title and Subtitle 

Cometary Atmospheres: Modeling the Spatial 

Distribution of Observed Neutral Radicals 

3. Recipient's Catalog No. 

5. Report Dote 

December 1986 

6. Performing Organixotion Code 

1. Author(s 

Michael R. Combi 

8. Performing Orgonixation Report No. 

Cambridge, MA 02139-3794 

?. Performing Orgonixation Name and Address 

Atmospheric and Environmental Research, Inc. 

840 Memorial Drive 

NASW-3950 

13. Type of Report and Period Covered 

10. Work Unit No. 

11. Contract or Grant NO/ 

2. Sponsorin9 Agency Name and Address 

NASA Headquarters 

Headquarters Contract Division 

Washington, DC 20546 

Annual Rep0 r t 

6/1/85 - 8/31/86 

14. Sponsoring Agency Code 

9. kcurity Classif. (of this toport) 20. kcurity Classif. (of this page) 21. No. of Pages 

Unclassified Unclassified 15 

5. Supplementary Notes 

22. Price. 

6.Abstroct 

Progress during the second year of a program of research on 

the modeling of the spatial distributions of cometary radicals is 

discussed herein in several major areas. New scale length laws for 

cometary C2 and CN have been determined which explain that the 

previously-held apparent drop of the C2/CN ratio for large heliocentric distances does not exist and that there is no systematic 

variation; Monte Carlo particle trajectory model (MCPTM) analysis 

of sunward and anti-sunward brightness profiles of cometary C2 has 

been completed. This analysis implies a lifetime of 3.1 x lo4 

seconds for the C2 parent and an ejection speed for C2 of -0.5 km/s 

upon dissociation from the parent. A systematic reanalysis of 

published C3 and OH data was begun. Preliminary results find a 

heliocentric distance dependence for C3 scale lengths with a much 

larger variation than for C2 and CN. Scale lengths for OH are 

generally somewhat larger than currently accepted values. The MCPTM 

has been updated to include the coma temperature. Finally, the 

collaborative effort with the University of Arizona programs has 

yielded some preliminary CCD images of Comet P/Halley. 

7. Key Words (Salrcted by Author(.)) 18. Distribution Statement 

comets