Axisymmetric dusty gas jet in the inner coma of a comet

ICARUS (1~, 241--257 (1986)

Axisymmetric Dusty Gas Jet in the Inner Coma of a Comet YOSHIMI KITAMURA Institute of Space and Astronautical Science, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan Received August 19, 1985; revised November 19, 1985 The behavior of expanding pure and dusty gas jets is investigated in the inner coma of an HzOdominated comet by numerically solving the axisymmetric, time-dependent, coupled hydrodynamic equations for H20 gas and single-sized dust (0.65/zm) in polar coordinates (r, 0, ~). The jet profile is assumed to be Gaussian on the surface of a nucleus. The viscosity of the gas is taken into account. Two-dimensional distributions of the densities, velocities in the r and 0 directions, and temperatures for the gas and dust have been obtained. For the dusty jet, the axisymmetric transonic solution for the gas has been calculated time-dependently. For a narrow dusty gas jet (i.e., o f breadth 10°), the gas density peaks shift from the central axis of the jet (0 = 0°) to its wings (0 - 30 °) with the gas flowing away from the cometary nucleus, owing to a steep density gradient in the 0 direction. Dragged by this laterally expanding gas outflow, the dust particles are swept away from the central axis and are also concentrated more sharply at 0 - 35 ° than the gas particles. This lateral expansion o f the jet is overwhelming only within the innermost region (r _-< 10 km). The jet feature for the gas becomes indiscernible by the time the flow reaches the outer boundary (r = 100 kin), while the corresponding dust feature remains even at the outer boundary. The radial velocities of the gas and dust are enhanced inside the jet, compared with those in the background. For a broad pure gas jet (i.e., of breadth 30°), on the other hand, the gas density peaks do not shift to the wings and the jet feature can still be seen at the outer boundary, in contrast to the narrow case. © 1986 Academic Press, Inc.

1. I N T R O D U C T I O N

Many observations of comets show that they have spherically symmetric comae. However, there are observations which indicate the existence of asymmetry in cometary comae. This asymmetry is attributed mainly to the anisotropic ejection of gas and dust from the nucleus, as suggested by the following observational evidence. (i) Spiral jets in Halley's dust coma. Larson and Sekanina (1984) discovered spiral dust jets in the photographs of Comet Halley taken at Mount Wilson in 1910, by a new image-processing algorithm. They found that the dust jets evolved into expanding envelopes on a time scale of days. They showed that the dust density in the jet was several times higher than that in the background, if the breadth of the jet was large (-90°), but that its density exceeded tens of times the ambient density, if narrow (-10°). The expansion velocity of the dust

in the jet was lower than the theoretical value (Sekanina and Larson, 1984). (ii) A s y m m e t r y in O H coma. In the radio observations of OH in Comets Meier (1978 XXI), Bradfield (1979 X), and Austin (1982g), Bockel6e-Morvan and Gerard (1984) observed asymmetry both in the velocity profile and in the East-West brightness distribution. They interpreted this asymmetry in terms of the anisotropic outgassing from the nucleus. However, the anomalous outflow velocity of the parent molecules of OH was not detected even in the case of large anisotropy. (iii) Fan-shaped coma and jet force. Several comets (Periodic Comets Encke, Tempel 2, Borrelly, Schwassmann-Wachmann 3, etc.) displayed broad fan-shaped comae in the direction of the Sun, probably as a result of the inhomogeneous surface structure (Sekanina, 1979). The jet force from the anisotropic sublimation causes a precession of the spin axis and perturbs the

241 0019-1035/86 $3.00 Copyright © 1986by AcademicPress. Inc. All rights of reproductionin any form reserved.

242

YOSHIMI KITAMURA

orbital motion of a comet (Whipple and Sekanina, 1979). These observations support that the anisotropic ejection from the surface is the general nature of cometary nuclei. The anisotropy will probably be due to the volatile pockets in the dust mantle on the surface. The models of cometary atmospheres presented so far are all limited to spherical geometry. In view of the observations indicating asymmetric comae, two- or three-dimensional models should be constructed in the next step. Being interested in the jet phenomena in cometary comae such as observed in Halley's comet, we have investigated the expansion of pure and dusty gas jets by solving the axisymmetric, time-dependent, coupled hydrodynamic equations for the gas and dust. In the next section we shall describe the basic equations and model parameters. In the third section the boundary conditions and the numerical schemes used in solving the hydrodynamic equations are described. In the fourth section we shall discuss our results, and the summary is presented in the last section.

the continuity equation, 0 o~ (r3pge'/~) =

0 (rZpget/~Vr) Ox

0 O0 (r2pget/rv°) - cot 0 r2pget/'~Vo, (2.1) the momentum equation in the r direction, 0

0

Ot (r3pget/~Vr) -

Ox (rZpget/'v~ + rZpe'/') 0 ~0 (r2pget/~VrV°) -- c o t 0 r2pget/rVrVO ,)

-)

+

r-pget/~vb +

-

r3e t/~ (Fdrag,r

2rZpe 'IT + Fgrav,g)

+ r3et/~Vr,

(2.2)

the momentum equation in the 0 direction, 0

0

Ot (r3pget/~v°) -

Ox (r2pgemVrV°) 0 ", 2 O0 (r"pge'/Tv° +

r2pet/r)

- cot 0 r2pget/'~Vo 2. D U S T Y H Y D R O D Y N A M I C M O D E L

-- r2pget/rOrVo

2.1. Basic Equations --

We study axisymmetric jets by adopting the polar coordinates (r, 0, 4)), with the central axis of the jet as the z axis. Since the typical spin period is about 10 hr, the nucleus rotates only for 10° during 103 sec, by which time the system can reach steady state in the inner coma (r ~ 100 km). Therefore the effect of the spin rotation can be neglected. We treat dust as one of the components of the fluid, and solve the time-dependent hydrodynamic equations representing the mass, momentum (in both r and 0 directions), and energy conservations for the gas and dust. We introduce a variable x defined by In(r/ r.,0, where r,u~ is the radius of the nucleus. For H20 gas, the four basic equations in the conservative form are written as

r3 =,t/rls" ~

~drag,O

(2.3)

+ r3et/TVo,

the energy equation,

Ot 0 (r2pget#(1 v2 + eg+ P) Vr) -

ox

E

0 { r2pget/¢ (12 v Z + e g + -~g p ) Vo} O0 - cot 0 r2pge t/r -~

Vo

+ r3e'/T(Qh~ - LH20 -- Q~x) - r3et/~{Or(Fd~ag.~+ Fg~av,g) + VoFd~ag.O} + r3et/r(dPcond +

(1)diss

+

urV r + voVo) ,

(2.4)

JET IN A COMETARY ATMOSPHERE where

the energy equation,

pg(r,O,t) = mass density of the gas, or(r,O,t) = velocity of the gas in the r direc-

O 0"t (r3pded) --

tion,

c30 (r2pdedUo) -- COt 0 r2paeaua

unit mass.

~- r3(Qsun -- Lra d d- Qex), (2.8)

~"is the lifetime of an H 2 0 molecule for photodissociation and is given by I × 105 (rh/ AU) 2 sec ( H u e b n e r and Carpenter, 1979), where r h is the S u n - c o m e t distance. The pressure of the gas p is given by pgkBTg/mg, where mg is the m a s s of an H~O molecule, kB the B o l t z m a n n constant, and Tg the gas t e m p e r a t u r e . Tg is related to eg as Tg = ((y 1) mg/kB) eg, where y is the ratio of specific heats for gas (4/3 for H 2 0 gas). We can neglect the gravitational force b e t w e e n the gas and nucleus, Fgrav,g. F o r the dust, the four basic equations in the c o n s e r v a t i v e f o r m are the continuity equation, 0

0

o-7 (r3pd) =

0x (r2mu3 0

O0 (r2pdUO) - cot 0 &OdUo,

(2.5)

the m o m e n t u m equation in the r direction, 0

O

c~X (r2pdU2) 0

,:30 (r2pdUrUo) -- ( c o t 0 r2pdtlrllO -- r2pdtl 2) + r3(Fdrag,r -- Fgrav,d),

(2.6)

the m o m e n t u m equation in the 0 direction, 0 O-'~ (r3pdU°) ----"

0

Ox 0

O 0X (r2pdedUr) O

oo(r,O,t) = velocity of the gas in the 0 direction, (v 2 = v~ + v~) p(r,O,t) = pressure of the gas, eg(r,O,t) = internal energy of the gas per

0-~ (r3pdllr) =

243

pa(r,O,t) = m a s s density of the dust, Ur(r,O,t) = velocity of the dust in the r direction,

uo(r,O,t) = velocity of the dust in the 0 direction,

ed(r,O,t) -= internal energy of the dust per unit mass.

ea is related to the dust t e m p e r a t u r e Td as ed = CdTd, where Co is the specific heat capacity of the dust. We take the value of magnetite, 6 × 106 erg g-t K - t for Co ( J A N A F , 1971), where we ignore the temperature d e p e n d e n c e of Cd, because the dust t e m p e r a t u r e is found to be almost constant all o v e r the c o m e t a r y a t m o s p h e r e , as discussed in Section 4.2. F o r dust with a radius of - 1 /xm as usually considered, the gravitational force between the dust and nucleus, Fgrav.d, can be neglected. The cooling t e r m due to the infrared emission of an H 2 0 molecule, LH2O, is given by Shimizu (1976). F o r the p h o t o c h e m i c a l heating term, Qhv, we select five important reactions in determining the gas t e m p e r a t u r e , referring to Marconi and Mendis (1983),

H 2 0 + hv -~ O H + H R1 = 1.02 × 10 -5 (rh/AU) -2 sec -l,

C)

AEI = 1.9 eV, H 2 0 + hv--~ H2 + O(ID) R2 = 1.35 x 10 -6 (rh/AU) -2 sec -1,

(r2pdltrlgO)

(~)

AE2 = 1.9 eV,

oo (r2pdu2°) - (cot 0 r2pd u2 + r2paUrUo)

+ r3Fdrag,o,

where

(2.7)

H 2 0 + hv--* HzO+ + e R3 = 3.34 x 10 -7 (rh/AU) 2 sec-l, AE3 = 12.3 eV,

(~)

244

YOSHIMI KITAMURA

H20 + O(ID)---* OH + OH

0 5 = {(I --

(~)

R 4 = 3 X 10 - 1 ° c m 3 s e c - ' ,

® AEH3o+}nn2onH2o+Rs, (2.15)

1.26 eV,

/~E 4 = H20

+ (1

OIo~H'H20)nH2or)AEo(~) H -- OI°-(H~'O+'H20)nH20r~ H30+ !

+ H20 + ---> H 3 0 + + OH R5 = 2.05 × 10-9 cm 3 sec -I,

(~)

and

Qh~ = Qi + Q2 + Q3 + Q4 + Qs,

AE5 = 1.1 eV,

(2.16)

where o-(p,H20) is the cross section for the where R is the reaction rate and AE the elastic collision between_ a species p and an energy release (Marconi and Mendis, 1983). H20 molecule, and ~ the energy that the The number density of H20 is not affected species p has obtained by the ith reaction. by these reactions, because its density is We do not take into account the attenuation much larger than those of its products in the of the solar UV flux, because the photoinner region (3 _-< r -< 100 km). Since O(1D) chemical heating no longer plays a major and H20 ÷ are lost mainly by the reactions role in the region where the optical depth of of (~) and (~), respectively, the densities of the UV is greater than unity (r ~< I00 km) these species in a steady state are given by (Giguere and Huebner, 1978). The heating term due to the absorption of nO(ID) = R2/R4 (2.9) the solar radiation by the dust, Qsun is writand ten as, nH20+ = R3/Rs.

(2.10)

Qsun =

qabs'tra2FQnd,

(2.17)

In calculating the photochemical heating where qabs is the absorption efficiency of the rate contributed by each reaction, we as- dust, a the dust radius, Fo the solar energy sume that the excess energy is converted flux, and nd the number density of the dust. completely to the kinetic energy of each The cooling term due to the thermal radiaproduct. The collision process transfers tion by the dust, Lrad, is this converted energy into the energy of Lrad = qemit47ra2o-T4nd, (2.18) H20 gas. We simplify this transfer process by assuming that a species p having the ini- where qemit is the emission efficiency of the tial energy AEp loses the energy (1 - o~,)AEp dust, and o- the Stefan-Boltzmann conafter the nth collision with H20 molecules, stant. where O~pis defined by (m 2 + m220)/(mp + The drag force, Farag, and heat exchange mH20)2 (Kitamura et al., 1985). Then the function, Qex, in the free-molecular approxcontribution of the ith reaction to the pho- imation have been calculated by Probstein tochemical heating term, Qi, can be written (1969), assuming a drifted Maxwellian veas locity distribution and total accommodation on the dust surface for the gas: Q! = {(1 -- ot~H'H20)nH2Or)AE? + (1 --

Ot~)~H'H20)nHZOr)AEOoH}nH2oRI' (2.11)

Q2 = (1

-

O/H~(H2'H20)nH2Or)AEH(~)2/"/H2oR2,

Fdrag = 0 . 5 C D T ' / ' a 2 1 v - U l n d P g ( V -

Q4

=

(2.13)

2(1 - C~'H20)nH2 o~)

AEo~HnH2ono(ID)R4, (2.14)

(2.19)

and Qex = 47ra21 v -

ulndSt y

(2.12) Q3 = 0 ,

U)

y-

kB pg(Trling

Td).

(2.20)

The drag coefficient, CD, the Stanton number, St, and the recovery temperature, Tr are given as functions of to by Probstein

JET IN A COMETARY ATMOSPHERE (1969), where to is defined by Iv - u]/ V~aTg/mg. F o r Tr, we correct a minor mistake in his paper as

1

245

a (

+ r 3 s i n ~ O0 rh sin 0 - ~ - / ;

T,

Tr(rO) = V + 1 {2y + 2(y

_

-

1)to 2

y--I

]

0.5 + ~o2 + (~o/V~ erf(co)) exp(-002) " (2.21) where erf(t0) is the error function. Finally, we will describe the viscosity terms for H20 gas. An H20 molecule being assumed to be a rigid spherical particle, the viscosity coefficient,/~, is given (Chapman and Cowling, 1970) by /x = ~-~

D---~

(2.22)

and the thermal conduction coefficient, h, is 25 1 k3B~/~-T-T 1 h = 32 3' - 1 ~ / ~ D---~,

(2.23)

where D is the diameter of an H20 molecule. For the axisymmetric flow, the viscosity terms, V~ and Vo, the heat conduction term, @co.d, and the dissipation term, qbdis~, are as follows: 2 0 2r31x Vr - 3r 4 Ox Ox r2//, 0 (sin 0 vo)} sin 0 a0

{

1

°(9)

0

sin 0 Ix

+ r 2 sin 0 O0

0-0

+rsin0~xx

l o{

;

(2.24)

o(? t + r2~ -OOr ~J

Vo - r4 ax r3lz ~x + ~

~

~-~ (sin 0 vo)

- r~ ~x

r 2 sin 0 0

Vo-~ (l~ cos 0);

(2.25)

(2.26)

2/x{l 3 3 ~ ~XX (r2Ur)

(IDdiss -+

1

O

r sin 0 O0

(sin 0 vo)

}z

+~{2(OVr] 2 (Vr+OVO] 2 \3x/ + 2 30/ + 2(v~+ c o t O v o ) 2+ \O0 °

+ r~xx

.

(2.27)

2.2. Model Parameters Our calculation has been performed for rh = 1 AU. We assume that the cometary nucleus is composed of H20 ice and dust particles, and put r,,c = 3 km. We set the outer boundary of our calculation at r = 100 kin. In this inner region (r N 100 kin), the hydrodynamic description is permitted, because the ratio of mean free path ( - 10 .3 r 2 kin) to scale length of the flow ( - r kin) is less than unity. The dust particles are assumed to be single sized because of the limitation of computational time. We take 0.65 txm as the radius, a, which corresponds to the peak value of the dust-size distribution for Comet Halley (Divine et al., 1985) weighted by the geometrical cross section of a dust particle. In view of the observed results that the temperature of the dust exhibiting the black-body-like spectrum in the IR is higher than that of the black body (Ney, 1982), we take T~s) = 348 K at r = r , , o 25% higher than the black-body temperature; this value corresponds to the efficiency of q ~ = 1 and qemit = 0.41. N e x t we shall describe the jet parameters. The gas density in the background, n~ ) is put to 2.5 x 1012 cm -3 on the surface of a nucleus, which corresponds to the total gas production rate o f QH2o = 1.0 x 1029 s e c - 1 , provided that the initial expansion velocity of the gas, v~~) is equal to the sound veloc-

246

YOSHIMI KITAMURA

ity, c~s) (= 0.35 km sec-~). The 0 profile of the gas density, n~)(O) is assumed to be a Gaussian distribution as n~gS)(0) = {(~ - 1) exp(-(0/orj)2) + 1}n~), (2.28) where 33 is the density enhancement factor inside the jet and o-j the breadth of the jet. We put33 = 10 and o-j = 10° for a narrow jet (Larson and Sekanina, 1984) and 30 ° for a broad one. We assume that the gas is ejected radially from the nucleus and its temperature is constant all over the surface. The value of the initial Mach number of the gas, M0, is important in determining the pure gas flow (Shul'man, 1970a,b; Probstein, 1969; Wallis, 1982). When the gas expands spherically into a vacuum, the subsonic solution cannot occur as far as the cooling term in the energy equation dominates near the nucleus (Shul'man, 1970a). We take M0 = 1 all over the surface, referring to the observations (Bockel6e-Morvan and Gerard, 1984; Sekanina and Larson, 1984) indicating an absence of anomalous expansion velocity inside the jet. In the dusty coma, however, the initial Mach number becomes less than unity because of the interaction between the gas and dust near the nucleus (Probstein, 1969). This initial Mach number is automatically determined by a time-dependent scheme (Gombosi et al., 1985). Across a sonic point, the gas flow changes from subsonic into supersonic. The highly supersonic solution can o c c u r as well even in this dusty case, but the high initial Mach number seems to be unrealistic. The ratio of dust to gas mass production rate, X, is put to zero all over the surface for the pure gas jet, and unity for the dusty jet. The dust particles are assumed to be at rest on the surface. (In practice, the velocity of the dust is put to a small positive value on the surface, as described in Section 3.1, to avoid the divergence of its density.)

3. M E T H O D

OF THE COMPUTATION

3.1. Initial a n d B o u n d a r y Conditions In our calculation gas is set to start sublimating from the nucleus into a vacuum at t = 0. Dust begins to be dragged out by this expanding gas. We continue the time-dependent calculation until the steady-state solution is obtained. At the inner boundary on the nucleus surface, we put n~SJ(O) as given by Eq. (2.28), v¢o~(O) = O, TCgS~(0) = 200 K, u~S~(0) = Uo, where u0 is an arbitrarily chosen small velocity, u~(O) = 0, and Th~)(O) -- 348 K. v~s) is time-dependently determined for the dusty gas jet, as stated in Section 2.2. We put v~)(O) = c~~) at t = 0. To compute the time evolution of v~s), the partial derivative, OV~/Or[,:rn,c must be given. This derivative can be derived from Eqs. (2.1) to (2.4) by putting O/Ot = 0. After some manipulations we have 1 OVr + ( M ~ -

I)lOv~ r O0 Ovo\ 2 cot 0 + MrMo (OUr = - - V r -t- - o0 ) r ~ - ~ + Ox r r

(M~ - I ) r ~ - x

+ 1

3/-1 - TP

(Qh~ - Ln2o - Qex)

1 __ _ _ {(Fdrag,r + Fgrav,g)U r TP

+ Forag.oVo}, (3.1) where M , and Mo stand for Vr/Cs and vo/cs, respectively. Since the flow in the 0 direction is assumed to vanish on the surface, this relation reduces to 1_ 0v___Z~_ r 0x

!

-

Fdrag,r -- Fgrav,g ) "yp

T - 1 (Qhv "yp

'/

- LH2O -- Qex) + ~

(3.2)

as already derived by Gombosi et al. (1985). The interaction terms of Farag,, and Qex can be determined by putting O/Ot = 0 in Eqs. (2.6) and (2.8):

JET IN A COMETARY ATMOSPHERE 1 OUr Fdrag'r ---- ~ (r2pdttr) OX

(3.3)

1 Oed Qex = ~ (rZpdur) OX"

(3.4)

and

For the pure gas jet, on the other hand, _(s) • v~S)(O) is always put to ts F r o m the assumption that the ratio of dust to gas mass production rate is constant, the density of the dust on the surface, n~)(O) is given by n~S)(O) = X

mgn~S)(O)v~S)(O) mdu~s)(o)

(3.5)

H e r e we have derived the inner boundary conditions by assuming the establishment of steady state on the nucleus surface. This assumption is valid as far as the steadystate solution is concerned. To discuss the time evolution, however, we must consider the time-dependent model of a cometary nucleus such as the reservoir outflow model by Gombosi et al. (1985). At the outer boundary of r = 100 km, we can solve the time-dependent equations by using the one-sided upwind difference in both pure and dusty cases. This is because the gas flow is supersonic at r = 100 km and a signal propagates only from the upper flow. When we take into account the physical viscosity for the pure gas jet, we extend the outer boundary to r = 1000 km and put Ov,/Ox = O, OVo/OX = - Vo, and OTg/Ox = 0, conditions which correspond to a free-molecular flow of the gas. The flow becomes free-molecular at about 1000 km as known from Section 2.2. At 0 = 0 and zr, the physical quantities must vary smoothly across these two lines, as far as the axisymmetric flow is concerned. This is equivalent to the boundary conditions that Opg/O0 = .Opd/O0 = O, Ovr/O0 = CgUr/O0 = O, VO = UO = 0, and OTg/,90 = OTo/O0 =- O.

3.2. N u m e r i c a l S c h e m e a n d C o n v e r g e n c e of Solution

We have solved the equations for the gas

247

by M a c C o r m a c k ' s (1969) explicit method and those for the dust particles by the second-order upwind differencing method (Roache, 1982). To reduce the computational time, we put a time step as large as possible in the stable region of the numerical schemes. Namely, the Courant number, defined by c = At(lv,.I + cs)/Ai in the ith direction, is put to unity. We have introduced an artificial viscosity term of the order of magnitude comparable to the truncation error of the finite differences, in order to avoid the spatial wriggles of the numerical solutions in the subsonic region. F o r the radial direction, we have divided the region o f 3 _-< r -< 100 km into 40 logarithmically equal bins in the case of the pure gas jet. On the other hand, in the dusty case, the region 3 _-< r -< 3.5 km is divided into 25 logarithmically equal bins and the region of 3.5 =< r -< 100 km into 40 bins to increase the accuracy o f the numerical solution for the transonic flow. At first, we obtain the transonic solution in the region o f 3 -< r =< 3.5 km. Using the numerical values at r = 3.5 km thus obtained as new inner boundary conditions, we calculate the supersonic flow in the region 3.5 =< r =< 100 km. F o r the 0 direction, the region 0 -< 0 =< zr is divided equally into 90 bins in both pure and dusty cases. F o r the pure gas jet, we have computed the jet expansion not only by M a c C o r m a c k ' s explicit method, but also by Beam and Warming's (1978) implicit method and confirmed that the two solutions calculated by these different methods agree well with each other. We judged whether the numerical solution reached steady state or not by monitoring the time-sequential values for each quantity. In practice, we continued our calculation until the numerical values remained unchanged down to two decimal places. 4. R E S U L T S A N D D I S C U S S I O N

4.1. P u r e H 2 0 G a s J e t

Figure 1 shows the density profile of the

248

YOSHIMI KITAMURA

i,] 0.2013. R (KM) 3.0 3,9 5.6 8.1 ii,6 16.6 23.7 34.0 - - 4 8 . 7 ~ 6 9 , 8

~K 12,

11,

z

10.

0.15-

~

R: IOKM

g~o. io

~ l O 0 , O

9.

/

R ~ qKM

R=IOlY~M

8

. 180.150.120. 90. 60. 3 0 .

~ O. 3 0 - 80. 90.120.150.180.

0.05

THETA (DEG)

FIG. 1. G a s d e n s i t y vs 0 at v a r i o u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for o-j = 10° ( n a r r o w j e t ) a n d X = 0 ( p u r e g a s jet).

gas in the 0 direction for the inviscid narrow jet (o-j = 10°). T w o features are noticeable from Fig. 1. One is the rapid lateral expansion of the jet with increasing r, strong only within the innermost region of r -_< 10 km (see also Fig. 2). The other is the shift of the density peaks from the central axis to the wings (0 - 30°). The steep density gradient inside the jet produces the large velocity of the gas in the 0 direction near the nucleus. Since this significant acceleration is limited to the region around 0 = 15°, the gas molecules expanding fast from 0 ~< 30 ° are decelerated by the gas in the background, and as 0.20 2 A.

o. ]5 -

= o. l o

o. 00. o

\

.A

"-% %".

":::-

i. bo

2. oo

LOG R (KM)

FIG. 2. G a s v e l o c i t y in the 0 d i r e c t i o n v s r at 0 = 12 a n d 40 ° for o-j = 10° a n d × = 0. T h e s o u n d v e l o c i t y p r o f i l e s for t h e s e 0 v a l u e s are r e p r e s e n t e d b y t r i a n g l e s . B e c a u s e o f the p o l a r c o o r d i n a t e s , vo d e c r e a s e s w i t h i n c r e a s i n g r ( e v e n for the m o l e c u l a r flow).

0.0

073b. 6b. 9b.12b.15b.18b. THETA (DEG)

FIG. 3. G a s v e l o c i t y in the 0 d i r e c t i o n vs 0 at v a r i o u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for o-j = 10° a n d X = 0.

a result the new peaks are formed. In spite of the negative pressure gradient in the 0 direction, the jet continues to expand laterally because of the effect of inertia. However, oo finally becomes slightly negative at about the outer boundary (Fig. 3), and as a result the backward flow onto the central axis appears (Fig. 1). In the density contour map (Fig. 4), the expanded jet is easily seen at angles between 0 = l0 ° and 50 °, For a broad jet (o-j = 30 °) as shown in Fig. 5, on the contrary, its characteristic feature can still be seen at the outer boundary: Disintegration of the density peaks does not occur in this case, because of the gentle slope of the density profile inside the jet. The lateral expansion affects the radial flow. F o r the narrow jet the gas velocity in the r direction is enhanced in the center (0 - 0°), while diminished slightly in the wings (0 - 30°) from the level in the background (Fig. 6). This is because the lateral expansion causes different radial pressure gradients at different angles. In the center, where the lateral expansion is strong near the nucleus, the density decreases significantly with increasing r (Fig. 1) and the cooling of the gas by expansion is the strongest (Fig. 7). On the other hand, in the wings, where the gas is relatively com-

JET IN A COMETARY A T M O S P H E R E

249

THETA(DEG}

8(

;0

90

~}0

10[

O0 0

1:

I

I

0

I

50

100

B (KM)

FIG. 4. D e n s i t y c o n t o u r m a p o f the g a s in the r-O p l a n e for o'j = 10° a n d X = 0, w h e r e R m e a n s the r a d i a l d i s t a n c e f r o m t h e n u c l e u s c e n t e r . V a l u e s o f 1010 a n d 10 H c m -3 are a s s o c i a t e d w i t h t h e s e c o n t o u r

lines.

pressed, the decrease in density and cooling by expansion are milder (Fig. 1 and 7). The temperature decreases in a manner similar to the above density decrease. Therefore, the pressure gradient in the r direction is more enhanced in the center and

less in the wings. The enhancement of or inside the jet as well as the lateral expansion decreases the gas density in the center, i. O01 0.90 O. BO-

14.

~ z .m

1I.

~ ~ - ~ / / ~ ~ ~ ~ / / ~ ~ ~ ~ _

10. 9. -

IO0 10

~ 0.60-

13-

~ 12.

R (KM)

O- 70-

-

3.9

O- 4 0

5. 6 8. 1 11,6

0.30

23.7 34.o 100,0

i

SOUND INITIAL

0,20 0.~0 0.

o. 3b. sb. 9b.12b.lsb.18b. THETA (DEG)

eieo.isb.]2b. 9b. 6b. 3b. b. 3b. sb. 9b.12b.lsb.]eb. THETA(DEG) FiG. 5. G a s d e n s i t y v s 0 a t v a r i o u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for ~rj = 30 ° ( b r o a d j e t ) a n d X=0.

Fit;. 6. G a s v e l o c i t y in t h e r d i r e c t i o n vs 0 at v a r i o u s r a d i a l d i s t a n c e s R f r o m t h e n u c l e u s c e n t e r for trj = 10 ° a n d X = 0. T h e s o u n d v e l o c i t y o n the s u r f a c e is r e p r e s e n t e d b y t r i a n g l e s . T h e initial M a c h n u m b e r is set to u n i t y for t h e c a s e o f X = 0.

250

YOSHIMI D (KM) 3

200.

150.

~Ioo.

50. 10

O.

1oo U. 3u. {SU. 90.120.150.180.

F1G. 7. G a s t e m p e r a t u r e vs 0 at v a r i o u s radial dist a n c e s R f r o m the n u c l e u s c e n t e r for o-j = 10° a n d X=0.

but the former effect is smaller than the latter one. Figure 8 shows that the radial acceleration is effective only within the innermost region (r =< 10 km), where the lateral expansion of the jet is remarkable. The above results are not changed by the effect of viscosity in the inner coma (r < I00 km). This can be confirmed by the fact that the result for the viscous gas jet agrees well with that for the inviscid jet. In the outer region of r > 100 km, however, the viscosity slightly flattens the expanded 1.00-

o. 80 0.70 ~0.60

~0.50 ~0.40

~ ~

o

broad peak at 0 - 30 °. After all, the jet would not be changed seriously at larger distances (r > 100 km) because of the high Mach number. Disintegration of density peaks is not peculiar to Gaussian distribution. This is confirmed by simulating time-dependently the one-dimensional jet expansion for various shape functions, such as Gaussian, parabolic, triangular, and rectangular distributions. The disintegration commonly appeared.

4.2. Dusty H20 Gas Jet

THETA (DEG)

0.90

KITAMURA

F o r the dusty jet, a slight decrease in the gas density appears at 0 - 35 ° (Fig. 9), superimposed on the disintegration explained in Section 4.1. This decrease can be more clearly seen in Fig. 10, and is understood in terms of the g a s - d u s t interaction. As can be seen in Fig. 17, the concentration of dust occurs at just the same position. This concentration makes it somewhat harder for a gas particle to flow across this 0line. As a result the gas flow becomes rapidly radial in the region 0 ~< 35 °, while in the outer region the lateral expansion of the jet still remains. In terms of re, vo decreases with increasing r more rapidly in 0 ~< 35 ° than in 0 > 35 ° (Fig. 11). Therefore, the number of gas particles which can reach at 0 - 35 ° from the interior diminishes. Compared with the pure case, backward flow onto the central

9( ° ) 14.

180 30

13.

:g

.

6

~

R (KM) 3.0 3,5 4.9

6.8 11.

G

~ o. s o -

--18:7 ~ ~ - ~ ~ ~ . - ~ ~

0.20

--26.2 - - 36.6 51,1 71,5

"-J

o. ioO. 0

9.

100,0

SOUND

o.

l.bo

2. bo

LOG R (KM)

EIG. 8. G a s v e l o c i t y in the r d i r e c t i o n vs r at 0 = 0, 30, a n d 180 ° for trj = 10° a n d X = 0. T h e s o u n d v e l o c i t y profiles for t h e s e 0 v a l u e s a r e r e p r e s e n t e d b y t r i a n g l e s .

8.

~80.tsb.l~b. 9b. sb. 3b, b. 3b. sb. 9b,12b.lsb.18b, IHETA (DEG)

FIG. 9. G a s d e n s i t y v s 0 at v a r i o u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for o'j = 10 ° a n d X = 1 ( d u s t y gas jet).

JET IN A COMETARY ATMOSPHERE

251

THETA (DEG)

8C

0

90

~0

IOC

O0 0

11

I

0

I

I

50 FI (KM)

tO0

FIG. 10. D e n s i t y c o n t o u r m a p o f the gas in the r-O p l a n e for crj = 10° a n d X = 1, w h e r e R m e a n s the r a d i a l d i s t a n c e f r o m the n u c l e u s c e n t e r . V a l u e s o f 10 m a n d 10" c m -3 are associated with these c o n t o u r lines.

axis occurs at smaller distances (compare Fig. 12 with Fig. 2), owing to the drag force by the dust. Moreover, the jet expansion reaches as much as 0 - 75 °, broader than 0 0.20

a - ~KM

.~ O. 15

60 ° for the pure gas jet. This is partly because the heating by the dust slightly enhances the lateral expansion (compare Fig. 11 with Fig. 3), and partly because the gas 0.20

0.15

~o. lo

~0.10

%

A

~

%

',.

R= IOKM IOOKM

~0.05

0. 05

o.o

.~g. 6b. 9b.12645b.18b. THETA (BE6)

FIG. 11. G a s v e l o c i t y in the 0 d i r e c t i o n vs 0 at various radial d i s t a n c e s R from the n u c l e u s c e n t e r for trj = 10 ° a n d X = 1.

0.

00.

1. O0 --'-----'~2"7"00 L06 R (KM)

FIG. 12. G a s v e l o c i t y in the 0 d i r e c t i o n v s r at 0 = 12 a n d 40 ° f o r o,j = 10° a n d X = 1. T h e s o u n d v e l o c i t y profiles for t h e s e 0 v a l u e s a r e r e p r e s e n t e d b y t r i a n g l e s .

252

YOSHIMI KITAMURA

I.00] 0.90 0.80 0.70 v 0.60 0.50 0.40 O. ~0 1 _

1. OO 0.90 0.80 0.70

/

-

0.60 ,:3

-

0

~ 1 8 0

0.500.40-

I~ITIAL

I . I

0.30-

3

O. 2 0 y

0.20-

O. 10 1

O. lO-

o. o Io. 3b. 6b. 9b.12b.lsb.18b.

0.0

SOUND

~.bo

O.

/HETA (DEG)

2. bo

LOG R (KM)

FIG. 13. Gas velocity in the r direction v s 0 a t v a r i radial distances R from the nucleus center for o-j = 10 ° and X = 1. The sound velocity profile on the surface is represented by triangles. Note that the initial Mach number is less than unity in this dusty case.

Gas velocity in the r direction v s r a t 0 = 0, 10 ° and X = 1. The sound velocity profiles for these 0 values are represented by triangles. F I G . 14.

ous

3 0 , a n d 180 ° f o r trj =

is relatively more accelerated in the 0 direction in a longer time interval than the pure case because of the smaller radial velocity. Different from the lateral expansion, the radial flow becomes transonic owing to the drag force by the dust. This solution can be obtained by the time-dependent scheme. The calculated initial velocity of the gas is shown with the sound velocity in Fig. 13. In the central part of the jet, the stronger drag force decreases the initial velocity, corn-

pared with that in the background. Sublimating with M0 < 1, the gas smoothly goes across the sonic point near the nucleus (Fig. 14). The sonic line is shown in Fig. 15. It can be seen that the sonic point is more distant from the surface in the central part of the jet than in the background, owing to the small initial Mach number. The line is fairly prolonged in the direction of 0 = 20 °, because the radial acceleration by the pressure gradient is weak in this direction. Not only does the dust decrease the velocity of the gas, but it also heats the gas.

THETA [OEG]

0

4~/SON

R (KM}

IC

o 3.3

F I G . 15. Velocity field of the gas for o-j = 10° and × = 1. Each velocity vector is plotted in the r-O half plane. The line assigned b y " S O N I C " indicates the sonic line.

JET IN A COMETARY ATMOSPHERE

253

R (KM) 3

200.

150.

0.05-

0.04IOO.

7., 0.03-

50. ~ o. 020.

O. 30. 50. 90,120.150.180.

O. Ol-

THETA (DEG)

FIG. 16. G a s t e m p e r a t u r e vs 0 at v a r i o u s r a d i a l dist a n c e s R f r o m the n u c l e u s c e n t e r for oq = 10° a n d X=I.

This heating enhances the gas velocity inside the jet more strongly, compared with the enhancement of the pure case (see Figs. 13 and 6). A temperature increase in the center is seen in Fig. 16. The other local heating at 0 - 35°, however, is not effective in the acceleration of the gas. For the dust, dragged by the gas flow, the density peaks also shift to the wings and fairly sharp peaks are formed at 0 - 35°, peaks which remain even at the outer boundary (Fig. 17). The sharpness arises from the absence of the pressure of the 4-

~,

),o

§2.

1I00.0

2. 180.150.120. 90. 60. 30.

O. 30- 60. 90.120.150.180.

THETA (BEG)

FIG. 17. D u s t d e n s i t y v s 0 at v a r i o u s r a d i a l d i s t a n c e s R f r o m t h e n u c l e u s c e n t e r for o~j = 10° a n d X = 1. N o t e t h a t the profile a s s i g n e d to R = 3 k m is n o t the profile at the p o s i t i o n o f t h e s u r f a c e b u t t h a t at the c l o s e s t m e s h p o i n t to t h e s u r f a c e .

O. 0 q ~ - - - ~ O. 30. 60. 90.120.150.180. THETA (DE8)

FIG. 18. D u s t v e l o c i t y in the 0 d i r e c t i o n v s 0 at vario u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for o-j = l 0 ° a n d X = 1.

dust. From inside these peaks (0 ~< 35°), dust particles are dragged along the O line by the lateral expansion of the gas, and form the peaks at 0 - 35°, while the gas particles gradually lose the momentum in the 0 direction and can no longer drag the dust into the outside of 0 ~> 35°. In consequence the outside edges become sharper. This corresponds to the steep decrease in the right sides of the peaks in Fig. 18. Additional small peaks at 0 - 45 ° in Fig. 17 are made up by the gas particles in the outside (0 -> 35°), which still have considerable velocities in the 0 direction (see Fig. 11). Another feature seen in Fig. 17 is that the density of the dust in the central part of the jet remains decreased. This is because the pressure gradient is negligibly small and as a result the backward flow does not occur. Figure 19 clearly shows these features. The radial velocity of the dust is produced by the drag force. Its velocity is enhanced inside the jet (Fig. 20) as is that of the gas. In the central part of the jet, where the density and radial velocity of the gas are large near the nucleus, the drag force by the gas becomes significant. The dust velocity

254

YOSHIMI

KITAMURA

THETA(OEG)

8C

0

90

)0

I0[

O0

1:

0

I

o

I

I

50 R (KM)

tO0

FIG. 19. D e n s i t y c o n t o u r m a p o f the d u s t in the r-O p l a n e for o-j = 10° a n d X = 1, w h e r e R is the r a d i a l d i s t a n c e f r o m the n u c l e u s c e n t e r . V a l u e s o f 10-' a n d 10° c m 3 are a s s o c i a t e d w i t h t h e s e c o n t o u r lines.

at 0 - 35 °, however, is not accelerated so strongly (which can be seen as a hollow in Fig. 20), because the radial velocity of the gas is small at this position (Fig. 13) and, moreover, because the dust concentration at 0 - 35 ° decreases the gas density. Finally, we shall briefly discuss the temperature distributions for the gas and dust. The gas temperature in the innermost region (r _-< 10 km) is an important factor in determining the lateral expansion of the jet as well as the density gradient in the 0 direction. In this region the cooling by both expansion and the IR emission of H20 molecules is dominant. The heating by the dust is not so effective, as known from the result that the lateral expansion is not so enhanced in the dusty case. In addition, the heat conduction and the dissipation terms can be neglected, like the viscosity terms. Outside the innermost region the photochemical heating gradually increases, but

the lateral expansion of the jet essentially ceases there. In our model, the radiative transfer for H20-IR emission has not been taken into account. H o w e v e r , Crovisier 0.40

0.35 ~ . 0.30

R(I~) 100 10

~ 0.20~0.15 0.10 0.05

O.

~ 0

,

,

,

INITIAL ,

,l

O- 30. 60- 90.120.150.180THETA (DE6)

FIG. 20. D u s t v e l o c i t y in the r d i r e c t i o n vs 0 at vario u s r a d i a l d i s t a n c e s R f r o m the n u c l e u s c e n t e r for tTj = 10° a n d X = 1. T h e v e l o c i t y o n the s u r f a c e is zero.

JET IN A COMETARY ATMOSPHERE (1984) concluded that the inner coma was optically thick for most of the rotational lines of an H20 molecule. Marconi and Mendis (1984) pointed out that the infrared emission from the hot dust in the inner coma was trapped by the rotational levels of an HzO molecule and this increased the heat input onto the nucleus surface. If we take into account these effects, the lateral expansion of the dusty jet may be more rapid. This is to be studied in the future. The dust temperature is found to be almost constant all over the region, because in the energy equation both the heating by the solar radiation and the cooling by the thermal radiation are overwhelming, compared with the energy exchange between the gas and dust.

4.3. Dependence of the Jet Expansion on the Initial Profile and the Dust Size To see the dependence of the solution on the initial profile of the jet (at r = r,uc), we have calculated the expansion of the dusty gas jet in the innermost region (r =< 10 km) by applying the Fermi-type distribution instead of the Gaussian distribution. The former profile is given by

~-1 ng~)(O) =

exp((O - Oj)/o-j) + 1 + 1) n~),

(4.1)

where 3'] = 10, 0j = 10°, and o-j = 1°. In almost all features, we found qualitative agreement with those for the Gaussian distribution, for example, the shifts of the gasand dust-density peaks toward the wingside, the dust concentration and the decrease of the gas density at 0 - 35°, the enhancement of the gas and dust velocities inside the jet, the local increase of the gas temperature at 0 - 35°. To see the dependence of the jet expansion on the dust size, we have calculated the jet expansion for two different radii of

255

a = 0.39 and 1.5 tzm (from 0.65 /xm). The former (0.39) corresponds to the peak value of the dust-size distribution (Divine et al., 1985), and at about this value the size distribution weighted by the cross section of dust takes a half-maximum. The latter (1.5) corresponds to the other half-maximum of the weighted distribution. These calculations show that the smaller the dust radius is, the larger the gas-dust interaction becomes, if we fix X = 1. However, the position of the concentration of the dust particles in the 0 space does not change for radii of both 0.39 and 1.5/xm. Hence it is expected that the 0 profiles of the gas and dust densities would be essentially unchanged, if allowing for a dust-size distribution. For gas velocities, the deviation from the result for 0.65/zm is about 10%. For dust velocities, however, the deviation is slightly larger and amounts to scores of percents. Recently, Yamamoto and Ashihara (1985) have proposed that the H20 gas sublimating from the nucleus surface condenses into ice particles in the vicinity of the nucleus owing to the cooling by expansion. Under the condition that the depletion of the gas is largest near the nucleus, an anisotropy born on the surface will be kept up to a considerable distance (r - 100 km) by these ice particles. It is because the mutual collisions among the ice particles are so rare that the anisotropy is not easily homogenized, different from the situation for the gas. The condensation of H20 gas is to be taken into account in our calculation in the next step. 5. SUMMARY

We have studied the axisymmetric expansion of pure (X = 0) and dusty (X -~ 1) gas jets in the inner coma (r =< 100 km) of an HzO-dominated comet by solving time-dependently the coupled hydrodynamic equations for the gas and dust. We have adopted the jet parameters, referring to the observation of Halley's spiral jets (Larson and Sekanina, 1984). The jet profile on the nucleus surface is assumed to be Gaussian distribu-

256

YOSHIMI KITAMURA

tion with breadth trj. The results are summarized as follows: (1) For narrow (trj = 10°) pure and dusty gas jets, their features are already indefinite at r - 100 km. For the broad (o-j = 30 °) pure gas jet, on the other hand, its feature is still discernible at r - 100 km. (2) For narrow pure and dusty gas jets, the jets expand rapidly in the 0 direction, and as a result the gas density peaks shift from the central axis (0 = 0 °) to the wings (0 30 °) in contrast with the broad case. This lateral expansion of the jets occurs effectively within the innermost region (r _-< 10 km). In addition, backward flow onto the central axis is found in the central part of the jet at large distances from the nucleus. (3) For the narrow dusty gas jet, the density peaks of the dust also shift to the wings and the dust particles concentrate sharply at 0 - 35 °. Owing to this concentration, the density of the gas decreases locally at the same position. The lateral expansion of the dust essentially ceases within the innermost region. Different from the gas, the sharp peaks of the dust concentration remain even at the outer boundary. (4) For narrow pure and dusty gas jets, the radial velocity of the gas is enhanced in the central part of the jet (0 - 0 °) and diminished in the wings (0 - 30°), compared with that in the background. The radial velocity of the dust is also enhanced inside the jet. (5) The axisymmetric transonic flow near the nucleus has been obtained for the dusty jet by the time-dependent scheme. (6) The effect of viscosity on the lateral expansion of the jet can be neglected in the inner coma (r _-< 100 km). (7) The Fermi-shaped jet does not qualitatively change the results for the Gaussian distribution except some minor details. The position of the sharp peaks of dust (0 - 35 °) does not strongly depend on dust size. ACKNOWLEDGMENTS The author thanks Professor M. Sbimizu for valuable comments and continuous encouragement. He also acknowledges helpful discussions with Dr. T.

Y a m a m o t o and Dr. O. Ashihara. For numerical schemes, he is indebted to Dr. T. Terasawa and Mr. M. Hoshino for useful suggestions. Numerical computations were performed by F A C O M M380 of the Institute o f Space and Astronautical Science. REFERENCES BEAM, R. M., AND R. F. WARMING (1978). An implicit factored scheme for the compressible NavierStokes equations. AIAA J. 16, 393-402. BOCKELI~E-MORVAN, D., AND E. GERARD (1984). Radio observations of the hydroxyl radical in comets with high spectral resolution. Kinematics and a s y m metries of the OH c o m a in C/Meier (1978 XXI), C/Bradfield (1979 Xt, and C/Austin (1982g). Astron. Astrophys. 131, 111-122. CHAPMAN, S., AND T. G. COWL1NG (1970). The Mathematical Theory of Non-uniform Gases, 3rd ed., Ch. 10. Cambridge Univ. Press, New York. CROVISiER, J. (1984). The water molecule in comets: Fluorescence mechanisms and thermodynamics of the inner coma. Astron. Astrophys. 130, 361-372. DIVINE, N., H. FECHTIG, T. 1. GOMBOSI, M. S. HANNER, H. U. KELLER, S. M. LARSON, D. A. MENDIS, RAY L. NEWBURN, JR., R. REINHARD, Z. SEKANINA, AND D. K. YEOMANS (1985). The C o m e t Halley dust and gas environment. Cometary science team, preprint series, No. 72, JPL. Space Sci. Rev., in press. GIGUERE, P. T., AND W. F. HUEBNER (1978). A model of c o m e t comae. I. Gas-phase chemistry in one dimension. Astrophys. J. 223, 638-654. GOMBOSI, T. I., Z. E. CRAVENS, AND A. F. NAGY (1985). Time-dependent dusty gasdynamical flow near cometary nuclei. Astrophys. J. 293, 328-341. HUEBNER, W. F., AND C. W. CARPENTER (1979). Solar Photo Rate Coefficients. LA-8085-MS, Informal Report, L o s A l a m o s Scientific Laboratory. J A N A F (1971). Thermochemical Tables, 2nd ed., p. 792. Natl. Bur. Stand., Washington, D.C. KITAMURA, Y., O. ASHIHARA, AND T. YAMAMOTO (1985). A model for the hydrogen c o m a of a comet. Icarus 61, 278-295. LARSON, S. M., AND Z. SEKANINA (1984). C o m a morphology and dust-emission pattern of Periodic C o m e t Halley. I. High-resolution images taken at Mount Wilson in 1910. Astron. J. 89, 571-578. MACCORMACK, R. W. (1969). The Effect of Viscosity in Hypervelocity Impact Cratering. A I A A Paper 69354. MARCONI, M. L., AND D. A. MENDIS (1983). The atmosphere of a dirty-clathrate c o m e t a r y nucleus: A two-phase, multifluid model. Astrophys. J. 273, 381-396. MARCONI, M. L., AND D. A. MENDIS (1984). The effects of the diffuse radiation fields due to multiple scattering and thermal reradiation by dust on the dynamics and thermodynamics o f a d u s t y c o m e t a r y atmosphere. Astrophys. J. 287, 445-454.

JET IN A COMETARY ATMOSPHERE NEY, E. P. (1982). Optical and infrared observations of bright comets in the range of 0.5 ~ m to 20 p.m. In Comets (L. L. Wilkening, Ed.), pp. 323-340. Univ. of Arizona Press, Tucson. PROBSrEIN, R. F. (1969). The dusty gasdynamics of comet heads. In Problems of Hydrodynamics and Continuum Mechanics (M. A. Lavrent'ev, Ed.). pp. 568-583. SIAM, Philadelphia. ROACHE, P. J. (1982). Computational Fluid Dynamics, revd. printing, pp. 64-74. Hermosa, Albuquerque, New Mexico. SEKnNINA, Z. (1979). Fan-shaped coma, orientation of rotation axis, and surface structure of a cometary nucleus. I. Test of a model on four comets. Icarus 37, 420-442. SEKANINA, Z., AND S. M. LARSON (1984). Coma morphology and dust-emission pattern of Periodic Comet Halley. II. Nucleus spin vector and modeling of major dust features in 1910. Astron. J. 89, 14081425. SHIMIZU, M. (1976). The structure of cometary atmo-

257

spheres. I. Temperature distribution. Astrophys. Space Sci. 40, 149-155. SHUL'MAN, L. M. (1970a). Hydrodynamics of the circure-nuclear region of a comet. In Astrometry and Astrophysics, No. 4, Physics of Comets (V. P. Konopleva, Ed.), pp. 85-99. NASA TT F-599. SHUL'MAN, L. M. (1970b). The physical conditions in the boundary layer of a cometary nucleus. In Astrometry and Astrophysics, No. 4, Physics of Comets (V. P. Konopleva, Ed.), pp. 100-109. NASA TT F-599. WALLIS, M. K. (1982). Dusty gas-dynamics in real comets. In Comets (L. L. Wilkening, Ed.), pp. 357369. Univ. of Arizona Press, Tucson. WHIPPLE, F. L., AND Z. SEKANINA (1979). Comet Encke: Precession of the spin axis, nongravitational motion, and sublimation. Astron. J. 84, 1894-1909. YAMAMOTO, T., AND O. ASHIHARA (1985). Condensation of ice particles in the vicinity of a cometary nucleus. Astron. Astrophys. 152, LI7-L20.