A growing amount of observational and theoretical evidence suggests that most main sequence stars are surrounded by disks of cometary material. The dust production by comets in such disks is investigated when the central stars evolve up the red giant and asymptotic giant branch (AGB). Once released, the dust is ablated and accelerated by the gas outflow and the fragments become the seeds necessary for condensation of the gas. The origin of the requisite seeds has presented a well known problem for classical nucleation theory. This model is consistent with the dust production observed in M giants and supergiants (which have increasing luminosities) and the fact that earlier supergiants and most WR stars (whose luminosities are unchanging) do not have significant dust clouds even though they have significant stellar winds. Another consequence of the model is that the spatial distribution of the dust does not, in general, coincide with that of the gas outflow, in contrast to the conventional condensation model. A further prediction is that the condensation radius is greater that that predicted by conventional theory which is in agreement with IR interferometry measurements of alpha-Ori

Matese, John J.

Reynolds, R. T.

Whitmire, D. P.

English

NASA Technical Reports Server

r/
N91-15018
SUBLIMATING COMETS AS THE SOURCE OF NUCLEATION SEEDS FOR GRAIN
CONDENSATION IN THE GAS OUTFLOW FROM AGB STARS
D.P. Whitmire*, J.J. Matese*, R.T. Reynolds**
*Physics Department, Univ. of Southwestern
Louisiana, Lafayette, LA 70504-4210
**NASA Ames Research Center, Moffett Field, CA 94035
ABSTRACT. A growing amount of observational and theoretical evidence
suggests that most main sequence stars are surrounded by disks of
cometary material. In this paper we investigate the dust production by
comets in such disks when the central stars evolve up the red giant and
asymptotic giant branch (AGB). Once released, the dust will be ablated
and accelerated by the gas outflow and the fragments will become the
seeds necessary for condensation of the gas. The origin of the
requisite seeds has presented a well known problem for classical
nucleation theory. This model is consistent with the dust production
observed in M giants and supergiants (which have increasing
luminosities) and the fact that earlier supergiants and most WR stars
(whose luminosities are unchanging) do not have significant dust clouds
even though they have significant stellar winds. Another consequence of
the model is that the spatial distribution of the dust will not in
general coincide with that of the gas outflow, in contrast to the
conventional condensation model. A further prediction is that the
condensation radius is greater than that predicted by conventional
theory, in agreement with IR interferometry measurements of a-Ori.
I. INTRODUCTION
The sources of the small dust grains observed by IRAS around Vega,
B-Pic and numerous other old main sequence stars (Aumann, 1985; Backman
and Gillett, 1987; Walker and Wolstencroft, 1988) are believed to be
disks of cometary material (Harper et al., 1984; Weissman, 1984; Matese
et al., 1987). Modeling of the IR emission yields dust disk radii
extending to several hundred AU (Gillett, 1986). The disk in B-Pic has
been resolved optically and it extends to _I000 AU (Smith and Terrile,
1987). The mass of the small (_-10um) grains directly observed is
> O.01M_ (e.g. Gillett, 1986). However, the lifetime of these small
grainS is less than the probable ages of the stars (Whitmire et al.,
1988) and therefore more massive unobserved sources must also be
present. Estimates of the total disk masses are highly uncertain since
it must be assumed that a standard mass distribution index holds over a
range from the observed small grains up to an assumed maximum mass. If
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the maximum mass is taken to be comet size, total mass estimates range
from _10-300M_(Harper et al., 1984; Weisman, 1984; Gillett, 1986).
Taking into account observational bias, it is likely that most main
sequence stars (including the sun) possess disks of cool orbiting grains
(Aumann, 1985; Backman and Gillett, 1987), implying that the phenomenon
is not indicative of young systems as originally conjectured. Further.
several F-type stars with 60_n excesses have age estimates of _2 X 109
yr Backman and Gillett, 1987).
These observations are consistent with the theoretical expectation
(Kuiper, 1951; Cameron, 1962) that a primordial disk of residual
unaccreted planetesimals should exist just beyond the planetary region
of the Solar System. This disk has previously been invoked as the
source of the observed steady state short period comets (Fernandez,
1980; Matese and Whitmire, 1986a,b) whose flux (Fernandez, 1980) and
inclinations (Duncan et al., 1988) are inconsistent with their origin as
captured long period comets. In addition to this residual flattened
planetesimal disk there is also theoretical evidence suggesting the
existence of an isotropic massive (_I00 M_ ) inner Oort cloud extending
inward to just outside the planetary region (Hills, 1981; Weissman,
1985; Bailey, 1983). However, since the dust clouds around
B-Pic and a-Psa are known to be flattened we assume that it is the
residual planetesimal disk that dominates within several hundred AU of
the star.
In this paper we investigate the dust production by disk comets
around intermediate mass stars (_I-IOH O ) as they evolve into red giants
and especially red supergiants (AGB p_ase_. During the AGB phase the
stellar luminosity can increase to IOq-IO_LQ , depending on mass. We
will argue that if a modest population of comets (_IM_) exists0within
100 AU of these stars they can readily supply the seeds (2 10 A)
necessary for dust condensation in the gas outflow, thereby mitigating
the classical nucleation problem of statistically producing sufficient
critical clusters in this environment (Donn and Nuth, 1985; Draine,
1985).
II. ANALYSIS
Iben and Renzini (1983) have modeled the evolution of intermediate
mass stars in the AGB phase. The time dependence of the stellar
luminosity can be written as
L(t) = L(0) exp(t/_ L) (1)
where
L(O) = 26.5(Mi/M )2"68L O (2)
and
_L = 6.5 X I07(M/M )-3"64yr. (3)i
496
In these formulae t = 0 at the beginning of the AGB phase, and M i is the
initial zero-age main sequence stellar mass. These formulae are
applicable for masses in the range IMO _ M _ IOM® , however for masses
< 2MQ_M i should be set = 2M@ . The stella_ luminosity, radial distance
R and black body temperature T are related by
L(t) = (R/AU)2(T(t)/280K)4L o (4)
while the time-dependent black body temperature can be written
T(t) = T(O) exp(t/_) where the temperature time scale _T = 4_ L .
Since, in a standard cometary mass distribution, most of the mass
residesli_ithe largest bodies, we consider comets of a single massM --- 0 g. The total mass loss rate for observed comets can be fit
tom_e form (Ney, 1982)
dM/dt = -kM2/3exp(-1552K/T) (5)
where the Boltzmann factor is appropriate for the activation energy of
water released from clathrates and the coefficient k depends on specific
surface absorptivity and activity, being smaller for older comets like
Encke. This semiempirical formula is applicable for black body
temperatures T up to _700K , which encompasses the temperatures of
relevance in the present analysis.
Equation (5) can be integrated to give
.1552K. ,1552K, 640 yr (I - fi/3)
EI_'T----_ - EI_T0_ _L
(6)
where E I is the Error function, f - M/M = fraction of original massax
surviving. Th_ n_rical factor was evaTuated by taking
k = 3.7 X 10-- g--_/s (comet Encke). We shall assume that the second
term is negligiOle compared to the first, an adequate approximation
after several luminosity timescales have elapsed. Eq. (6) gives the
value of T when M has been reduced to fMma x.
We now locate that position in the disk where sublimation is
maximal. Setting M = 0 we find
exp (-1552K/T)
P
fi/3
640 yr p
1552K/Tp _L 2
(7)
where Tp is the black body temperature where the comet is undergoing
peak sublimation and f_ is the surviving mass fraction at peak
sublimation. To is also a rough lower limit to the black body
temperature at the inner edge where f = O. An adequate approximate
solution to this equation is found to be
497
Tp = 1860K/ln(_L/400 yr). (8)
Having obtained the temperature at which comets are undergoing peak
sublimation rates we next determine the disk location R_ where this
P
occurs and take this as an adequate (upper bound) approximation to the
inner edge. Rp is found fr_n
Rp(t) = (L(t /L o )_/(Tp/280K)2AU (9)
or
Rp(t)/R,(t) = _(T,/Tp) 2 (10)
where * refers to stellar parameters. In Table I Tp, f and R /R, areP P
tabulated for several stellar masses Rp/R, is given for two stellar
temperatures corresponding to _ -Ori (oxygen supergiant) and IRC 10216
(carbon supergiant).
Table I
Rp/R,
Mi/M O Tp (K) fp T,=3300K T,=2300K
2 194 0.33 145 70
4 270 0.34 75 36
6 341 O. 35 47 23
8 423 0.36 30 15
I0 509 O. 37 21 I0
Finally, if one approximates that sublimation occurs only at the
location of the peak sublimation
dMnet/dt = (dM/dR) (dR/dt)IR (t)= (2_L)
P
-I (RdM/dR) i
IR (t)
P
(11).
For any standard radial power law RdM/dR = AM({2R _ 2R) in order of
magnitude. Therefore, if there is,_IM_)of comets available we should be
releasing it at a rate
-7M_ /yr M i = 2M.ofor
,_ AM/_L_ %" (I2)dMnet/dt
I0-4M@ /yr for M. = tOM
/
k_ z
To order of magnitude this result is of general applicability and can
also be used, for example, to estimate the dust production rate around
red giants as well as red supergiants.
498
III. DISCUSSION
Although thesedust production rates are incapable by themselves of
explaining the observed dust clouds around most M giants and supergiants
they can readily supply the seeds (> I0_) necessary for dust
condensation in the gas outflows. Seeds of radii_10A which
subsequently grow to_O.Ium represent a dust mass enhancement factor
of_10 • The required seeds should be readily produced by ablation in
the dense gas outflow, sputtering, grain-grain collisions, or they could
simply correspond to the lower end of a standard mass distribution of
the sublimating particles. The seeds will simultaneously be accelerated
until there is little relative velocity between the seeds and the gas
outflow, at which point condensation may occur rapidly. We note that
the temperature Tp in Table I is the black body temperature at the
location of those comets which are undergoing peak sublimation, but the
temperature of the small condensing grains (_ 0.1_m) that are actually
observed can be significantly higher (e.g. IO00K).
The inner radius of the best studied oxygen supergiant _-Ori has
been resolved with IR interferometry and found to be = 35 - 60 R.
(Bloemhof, 1984; Howell et al., 1981), consistent with the proposed
model but much larger than expected by conventional condensation theory
(Draine, 1985). In other cases it is often assumed that T = 1000K which
implies an inner edge --5R, for black body radiators (Tielens, 1988).
The proposed model is consistent with the fact that M giants and
supergiants have significant dust clouds while earlier supergiants and
most WR stars do not, even though they have significant mass loss via
stellar winds. It is only the cool giants and supergiants that have
rapidly increasing luminosities, and therefore short time scales _. , as
they evolve. Most of the comets within a few hundred AU should be L
concentrated in a disk. Thus, we expect the sublimated seeds and
condensed dust to be preferentially concentrated near the disk. Dust
disks are compatible with the observed asymmetries in the IR emission in
IRC 10216 and other cool giants (e.g. Zuckerman, 1980; Beckwith,
1985). A prediction of the model is that the spatial distribution of
the dust will not in general coincide with that of the gas outflow, in
contrast to conventional condensation theory. Although Rp is uncertain
by a factor of = 2, the model predicts that the inner disk edges will
tend to be further out than suggested by conventional condensation
theory, especially for the lower mass stars.
ACKNOWLEDGEMENTS
This work was supported in part by the NASA Ames/Stanford ASEE
Summer Faculty Fellowship Program and by a NASA Ames University
Consortium Grant.
499
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Sublimating comets as the source of nucleation seeds for grain condensation in the gas outflow from AGB stars

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