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Numerical Two-Dimensional Calculations
of the Formation of the Solar Nebula
PETER H. BODENHEIMER
Lick Observatory
ABSTRACT
The protostellar phase of stellar evolution is of considerable impor-
tance with regard to the formation of planetary systems. The initial mass
distribution and angular momentum distribution in the core of a molecular
cloud determine whether a binary system or a single star is formed. A rel-
atively slower rotating and centrally condensed cloud is likely to collapse to
a disk-like structure out of which planets can form. The above parameters
then determine the temperature and density structure of the disk and the
characteristics of the resulting planetary system.
There has been considerable recent interest in two-dimensional nu-
merical hydrodynamical calculations with radiative transfer, applied to the
inner regions of collapsing, rotating protostellar clouds of about 1 Me.
The calculations start at a density that is high enough so that the gas
is decoupled from the magnetic field. During the collapse, mechanisms
for angular momentum transport are too slow to be effective, so that an
axisymmetric approximation is sufficiently accurate to give useful results.
Until the disk has formed, the calculations can be performed under the
assumption of conservation of angular momentum of each mass element.
In a numerical calculation, a detailed study of the region of disk formation
can be performed only if the central protostar is left unresolved.
With a suitable choice of initial angular momentum, the size of the
disk is similar to that of our planetary system. The disk forms as a relatively
thick, warm equilibrium structure, with a shock wave separating it from the
surrounding infalling gas. The calculations give temperature and density
17
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distributions throughout the infalling cloud as a function of time. From
these, frequency-dependent radiative transfer calculations produce infrared
spectra and isophote maps at selected viewing angles. The theoretical spec-
tra may be compared with observations of suspected protostellar sources.
These disks correspond to the initial conditions for the solar nebula, whose
evolution is then driven by processes that transport angular momentum.
OBSERVATIONAL CONSTRAINTS ON THE PROPERTIES OF
THE INITIAL SOLAR NEBULA
From observations of physical and cosmochemical properties of the
solar system and from astronomical observations of star-forming regions
and young stars, certain constraints can be placed on the processes of
formation and evolution of the solar nebula.
a) Low-mass stars form by the collapse of initially cold (10 K),
dense (105 particles cm~3) cores of molecular clouds. The close physical
proximity of such cores with T Tauri stars, with imbedded infrared sources
which presumably are protostars, and with sources with bipolar outflows,
presumably coming from stars in a very early stage of their evolution, lends
support to this hypothesis (Myers 1987~.
by The specific angular momenta 0) of the cores, where observed,
fall in the range 102° - 102i cm2 s~i (Goldsmith and Arquilla 1985;
Heyer 1988~. In the lower end of this range, the angular momenta are
consistent with the properties of our solar system: for example, Jupiter's
orbital motion has j ~ 102° cm2 sol. In the upper end of the range,
collapse with conservation of angular momentum would lead to a halt of
the collapse as a consequence of rotational effects at a characteristic size
of ~ 2000 AU, far too large to account for the planetary orbits. In fact,
hydrodynamical calculations suggest that collapse in this case would in fact
lead to fragmentation into a binary or multiple system.
c) The infrared radiation detected in young stars indicates the pres-
ence of disks around these objects (Hartmann and Kenyon 1988~. A
particularly good example, where orbital motions have been observed, is
HL Tau (Sargent and Beck~rith 1987~. The deduced masses of the disk and
star are 0.1 Me and 1.0 M, respectively. The radius of the disk is about
2000 AU. Roughly half of all young pre-main-sequence stars are deduced
to have dislo;, mostly unresolved, with masses in the range 0.01-0.1 M and
sizes from 10 to 100 AU (Strom et at 1989~.
d) The rotational velocities of T Mauri stars are small, typically 20 km
s~i or less (Hartmann et aL 1986~. The distribution of angular momentum
in the system consisting of such a star and disk is quite different from
that in the core of a molecular cloud, which is generally assumed to be
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uniformly rotating with a power-law density distribution. Substantial angular
momentum transport, from the central regions to the outer regions, must
take place early in the evolution. The required transport is unlikely to
occur during collapse; therefore it must occur during the disk evolution
phase before the star emerges as a visible object.
e) The lifetime of the pre-main-sequence disks is difficult to deter-
mine from observation, but it probably does not exceed 107 years (Strom
et al. 1989~. The mechanisms for angular momentum transport, which
deplete disk mass by allowing it to fall into the star, must have time scales
consistent with these observations, as well as with time scales necessary to
form gaseous giant planets.
f) The temperature conditions in the early solar nebula can be
roughly estimated from the distribution of the planets' and satellites' mass
and chemical composition (Lewis 1974~. The general requirements are that
the temperature be high enough in the inner regions to vaporize most solid
material, and that it be low enough at the orbit of Jupiter and beyond to
allow the condensation of ices. Theoretical models of viscous disks produce
the correct temperature range, as do collapse models of disk formation with
shock heating.
g) The evidence from meteorites is difficult to interpret in terms
of standard nebular models. First, there is evidence for the presence of
magnetic fields, and second, the condensates indicate the occurrence of
rapid and substantial thermal fluctuations. Suggestions for explaining this
latter effect include turbulent transport of material and non-axisymmetric
structure (density waves) in the dish
h) The classical argument, of course, is that the coplanarity and
circularity of the planets' orbits imply that they were produced in a disk.
i) A large fraction of stars are observed to be in binary and mul-
tiple systems (Abt 1983~; the orbital values of j in the closer systems are
comparable to those in our planetary system. It has been suggested (Boss
1987; Safronov and Ruzmaikina 1985) that if the initial cloud is slowly
rotating and centrally condensed, it is likely to form a single star rather
than a binary. Pringle (1989) has pointed out that if a star begins collapse
after having undergone slow diffusion across the magnetic field, it will be
centrally condensed and will therefore form a single star. If, however, the
collapse is induced by external pressure disturbances, the outcome is likely
to be a binary. On the other hand, Miyama (1989) suggests that single star
formation occurs in initial clouds with j ~ 102i cm2 set. After reaching a
rotationally supported equilibrium that is stable to fragmentation, the cloud
becomes unstable to nonaxisymmetric perturbations, resulting in angular
momentum transport and collapse of the central regions.
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THE PHYSICAL PROBLEM
The above considerations illustrate several of the important questions
relating to the formation of the solar nebula: What are the initial conditions
for collapse of a protostar? At what density does the magnetic field
decouple from the gas? What conditions lead to the formation of a single
star with a disk rather than a double star? Can the embedded IRAS sources
be identified with the stage of evolution just after disk formation? What is
the dominant mechanism for angular momentum transport that produces
the present distribution of angular momentum in the solar system? The
goal of numerical calculations is to investigate these questions by tracing
the evolution of a protostar from its initial state as an ammonia core in a
molecular cloud to the final quasi equilibrium state of a central star, which is
supported against gravity by the pressure gradient, and a circumstellar disk,
which is supported in the radial direction primarily by centrifugal effects.
A further goal is to predict the observational properties of the system at
various times during the collapse. A full treatment would include a large
number of physical effects: the hydrodynamics, in three space dimensions,
of a collapsing rotating cloud tenth a magnetic field; the equation of state of
a dissociating and ionizing gas of solar composition, cooling from molecules
and grains in optically thin regions; frequency-dependent radiative transfer
in optically thick regions; molecular chemistry; the generation of turbulent
motions as the disk and star approach hydrostatic equilibrium; and the
properties of the radiating accretion shock which forms at the edge of the
central star and on the surfaces of the disk (Shu et al. 1987~.
The complexity of this problem makes a general solution intractable
even on the fastest available computing machinery. For example, the length
scales range from 10~7 centimeters, the typical dimension of the core of
a molecular cloud, to 10~i centimeters, the size of the central star. The
density of the material that reaches the star undergoes an increase of about
15 orders of magnitude from its original value of ~10-~9 g cm~3. Also,
the numerical treatment of the shock front must be done very carefully.
The number of grid points required to resolve the entire structure is very
large in two space dimensions; in three dimensions it is prohibitively large.
Even if the detailed structure of the central object is neglected and the
system is resolved down to a scale of 0.1 AU, the Courant-Friedrichs-
Lewy condition in an explicit calculation requires that the time step be
less than one-millionth of the collapse time of the cloud. Therefore,
a number of physical approximations and restrictions have been made
in all recent numerical calculations of nebular formation. For example,
magnetic fields have not been included, on the grounds that the collapse
starts only when the gas has become almost completely decoupled from
the field because of the negligible degree of ionization at the relatively
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high densities and cold temperatures involved. Also, in most calculations,
turbulence has been neglected during the collapse. Even if it is present, the
time scale for transport of angular momentum by this process is expected
to be much longer than the dynamical time. It turns out that angular
momentum transport can be neglected during the collapse, and therefore
an axlsymmetnc (two-olmenslonal) approximation IS adequate ourmg this
phase. Three-dimensional effects, such as angular momentum transport by
gravitational torques become important later, during the phase of nebular
evolution. A further approximation involves isolating and resolving only
specific regions of the protostar. In one-dimensional calculations (stabler
et al. 1980), it has been possible to resolve the high4ensity core as well
as the low-densi~r envelope of the protostar. However, in two space
dimensions, proper resolution of the region where the nebular disk forms
cannot be accomplished simultanously with the resolution of the central
star. In several calculations the outer regions of the protostar are also not
included, so the best possible resolution can be obtained on the length scale
1-50 AU. Thus the goal outlined above, the calculation of the evolution of
a rotating protostar all the way to its final stellar state, has not yet been
fully realized.
The stages of evolution of a slowly rotating protostar of about 1 Me
can be outlined as follows:
a) The frozen-in magnetic field transfers much of the angular mo-
mentum out of the core of the molecular cloud, on a time scale of 107
years.
b) The gradual decoupling between the magnetic field and the matter
allows the gas to begin to collapse, with conservation of angular momentum.
c) The initial configuration is centrally condensed. During collapse,
the outer regions, with densities less than about 10-~3 g cm~3 remain
optically thin and collapse isothermally at 10 K The gas that reaches
higher densities becomes optically thick, most of the released energy is
trapped, and heating occurs.
d) The dust grains, which provide most of the opacity in the pro-
tostellar envelope, evaporate when the temperature exceeds 1500 K An
optically thin region is created interior to about 1 AU. Further, at temper-
atures above 2000 K, the molecular hydrogen dissociates, causing renewed
instability to collapse.
e) The stellar core and disk form from the inner part of the cloud.
The remaining infalling material passes though accretion shocks at the
boundaries of the core and disk; most of the infall kinetic energy is con-
verted into radiation behind the shock The surrounding infalling material
is optically thick, and the object radiates in the infrared, with a peak at
around 6() 100 am.
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f) A stellar wind is generated in the stellar core, by a process that
is not well understood. The wind breaks through the infalling gas at the
rotational poles, where the density gradient is steepest and where most of
the material has already fallen onto the core. This bipolar outflow phase
lasts about 105 years.
g) Infall stops because of the effects of the wind, or simply because
the material is exhausted. The stellar core emerges onto the Hertzsprung-
Russell diagram as a T Tauri star, still with considerable infrared radiation
coming from the disk.
h) The disk evolves, driven by processes that transfer angular momen-
tum, on a timescale of 106 to 107 years. Angular momentum is transferred
outwards through the disk while mass is transferred inwards. The rotation
of the central object slows, possibly through magnetic braking in the stellar
wind. Possible transport processes in the disk include turbulent (convec-
tively driven) viscosity, magnetic fields, and gravitational torques driven by
gravitational instability in the disk or by non-axisymmetric instabilities in
the initially rapidly rotating central star.
The following sections describe numerical calculations of phases b
through e, from the time when magnetic effects become unimportant to the
time when at least part of the infalling material is approaching equilibrium
in a disk.
REVIEW OF TWO-DIMENSIONAL CALCULATIONS OF
THE FORMATION PHASE
Modern theoretical work on this problem goes back to the work of
Cameron (1962, 1963), who discuss in an approximate way the collapse
of a protostar to form a disk In a later work, Cameron (1978) solved
numerically the one-dimensional (radial) equations for the growth of a
viscous accretion disk, taking into account the accretion of mass from an
infalling protostellar cloud. The initial cloud was assumed to be uniformly
rotating with uniform density. The hydrodynamics of the inflow was not
calculated in detail; rather, infalling matter was assumed to join the disk
at the location where its angular momentum matched that of the disk A
similar approach was taken by Cassen and Summers (1983) and Ruzmaikina
and Maeva (1986), who, however, took into account the drag caused by
the infalling material, which has angular momentum different from that
of the disk at the arrival point. The latter authors discuss the turbulence
that develops for the same reason (see also Safronov and Ruzmaildna
1985~. This section concentrates on full two-dimensional calculations of the
collapsing cloud during the initial formation of the disk.
One approach to this problem (,I§charnuter 1981; Regev and Shaviv
1981; Morfill et al. 198S; ~charnuter 1987) is based on the assumption that
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1985; Durisen et al. 19863. Angular momentum would be transported from
the central object to the disk, and the value of ,B for the core would be
reduced below the critical value. However, its remaining total angular
momentum would be still too large to allow it to become a normal star.
A further important feature of the calculation was its prediction of the
temperatures that would be generated in the planet-forming region. Over
a time scale of 3 x 104 years the temperature of material with the same
specific angular momentum as that of the orbit of Mercury ranged from
400 600 K The predicted temperature for Jupiter remained fairly constant
at 100 K, while that for Pluto approximated 15 K In the inner region of the
nebula these temperatures are slightly cooler than those generally thought
to exist during planetary formation or those calculated in evolving models
of a viscous solar nebula (Ruden and Lin 1986~.
A further calculation was made by Tscharnuter (1987) with similar
physics but a different initial condition. A somewhat centrally condensed
and non-spherical cloud of 1.2 Me starts collapse from a radius of 4 x 10~5
cm, a mean density of 8 x 10-~5 g cm~3, and j ~ 102° cm2 sol. A major
improvement was a refined equation of state. The use of this equation of
state to calculate the collapse of a spherically symmetric protostar starting
from a density of 10-~9 g cm~3 produces violent oscillations in central
density and temperature after the stellar core has formed. The instability is
triggered when the adiabatic exponent P~ = (bin P/Bln pits falls below 4/3,
and the source of the energy for the reexpansion is association of hydrogen
atoms into molecules.
In the two-dimensional case, the much higher starting density and
the correspondingly higher mass inflow rate onto the core, as well as the
effects of rotation, are sufficient to suppress the instability. A few relatively
minor oscillations, primarily in the direction of the rotational pole, dampen
quickly. The numerical procedure uses a grid that moves in the (spherical)
radial direction and thus is able to resolve the central regions well, down
to a scale of 10~° cm. This calculation is carried to the point where a
fairly well-defined core of 0.07 Me has formed, which is still stable to
non-axisymmetric perturbations (,B = 0.08~. A surrounding disk structure
is beginning to form, out to a radius of about 1 AU. The density and
temperature in the equatorial plane at that distance are about 3 x 10-9
g cm~3 and 2500 K, respectively. Further accretion of material into the
core region is likelier to increase the value offs. The calculation was not
continued because of the large amount of computer time required.
A different approach to the problem of the two-dimensional collapse
of the protostar has been considered by Adams and Shu (1986) and Adams,
et al. (1987~. The aim is to obtain emergent spectra through frequency-
dependent radiative transfer calculations. In order to bypass the difficulties
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of a full two-dimensional numerical calculation, they made several appro~-
mations. The initial condition is a "singular" isothermal sphere, in unstable
equilibrium, with sound speed cat, uniformly rotating with angular velocity
Q. In the initial state the density distribution is given by p oc R-2, where
R is the distance to the origin, and the free-fall accretion rate onto a
central object of mass M is given by M = 0.975 c,3/G. The hydrodynamical
solution for the infalling envelope is taken to be that given by Terebey et al.
(1984), a semianalytic solution under the approximation of slow rotation.
The thermal structure and radiation transport through the envelope can
be decoupled from the hydrodynamics (Stabler et al. 1980~. The model at
a given time consists of an (unresolved) core, a circumstellar disk, and a
surrounding infalling, dusty, and optically thick envelope. The radiation
produced at the accretion shocks at the core and disk is reprocessed in
the envelope, and emerges at the dust photosphere, primarily in the mid-
infrared. The thermal emission of the dust in the envelope is obtained by
approximating the rotating structure as an equivalent spherical structure;
however, the absorption in the equation of transfer is calculated taking
the full two-dimensional structure into account. The model is used to fit
the observed infrared radiation from a number of suspected protostars,
by variation of the parameters M, cat, Q. ~D, and q*, where the last two
quantities are the efficiencies with which the disk transfers matter onto
the central star and with which it converts rotational energy into heat and
radiation, respectively. These models provide good fits to the spectra of
the observed sources for typical parameters M = 0.2 -1.0 M<;, c, = 0.2 -
0.35 km s~t, Q = 2 x 10-~4 - 5 x 10-~3 red s~i, ED = 1 and y* = 0.5. Of
particular interest is the fact that in many cases the deduced values of Q
fall in the range j ~ 102° cm2 sol, which is appropriate for "solar nebula"
disks. The contribution from the disk broadens the spectral energy distri-
bution and brings it into better agreement with the observations than does
the non-rotating model. More recent observational studies of protostellar
sources (Myers et al. 1987; Cohen et al. 1989) also are consistent with the
hypothesis that disks have formed within them.
RECENT MODELS W1MI HYI)RODYNAMICS AND
RAI)IATIVE TRANSPORT
Full hydrodynamic calculations of the collapse, including frequency-
dependent radiative transport, have recently been reported by Bodenheimer
e! al. ˘~19883. The purpose of the calculations was to obtain the detailed
structure of the solar nebula at a time just after its formation and to obtain
spectra and isophotal contours of the system as a [unction of viewing
angle and time. Because of the numerical difficulties discussed above, the
protostar was resolved only on scales of 10~3 - 10~5 cm. These calculations
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have now been redone with the extension of the outer boundary of the
grid to 5 x 10~5 cm, with improvements in the radiative transport, and
with a somewhat better spatial resolution, about 1 AU in the disk region
(Bodenheimer et al. 1990~.
The initial state, a cloud of 1 M<3 with a mean density of 4 x 1O-is g
cm~3, can be justified on the grounds that only above this value does the
magnetic field decouple from the gas and allow a free-fall collapse, with
conservation of angular momentum of each mass element, to start (Nakano
1984; Tscharnuter 1987~. The initial density distribution is assumed to be a
power law, the temperature is assumed to be isothermal at 20 K, and the
angular velocity is taken to be uniform with a total angular momentum of
1053 g cm2 sol. Because the cloud is already optically thick at the initial
state, the temperature increases rapidly once the collapse starts. The inner
region with R < 1 AU is unresolved; the mass and angular momentum
that flow into this core are calculated. At any given time, a crude model of
this material is constructed under the assumption that it forms a Maclaurin
spheroid. From a calculation of its equatorial radius Re, the accretion
luminosity L = GMM/Re is obtained. For each timestep At the accretion
energy LAt is deposited in the inner zone as internal energy and is used
as an inner boundary condition for the radiative transfer. Most of the
energy radiated by the protostar is provided by this central source. During
the hydrodynamic calculations, radiative transfer is calculated according
to the diffusion approximation, which is a satisfactory approximation for
an optically thick system. Rosseland mean opacities were taken from
the work of Pollack e! al. (1985~. After the hydrodynamic calculations
were completed, frequency-dependent radiative transfer was calculated for
particular models according to the approach of Bertout and Yorke (1978),
with their grain opacities which include graphite, ice, and silicates.
The results of the calculations show the formation of a rather thick disk,
with increasing thickness as a function of distance from the central object.
As a function of time the outer edge of the disk spreads from 1 AU to
60 AU, because of the accretion of material of higher angular momentum.
The shock wave on the surface is evident, and the internal motions in the
disk are relatively small compared with the collapse velocities. At the end
of the calculation the mass of the disk is comparable to that of the central
object, and it is not gravitationally unstable according to the axisymmetric
local criterion of lbomre (19643. The central core of the protostar, inside
10~3 cm, contains about 0.6 M and sufficient angular momentum so that
,B x 0.4. This region is almost certainly unstable to bar-like perturbations.
Theoretical spectra show a peak in the infrared at about 40~; when viewed
from the equator the wavelength of peak intensity shifts redward from that
at the pole. A notable difference between equator and pole is evident
in the isophotal contours. At 40 ~m, for example, the peak intensity
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27
shifts spatially to points above and below the equatorial plane because of
heavy obscuration there. This effect becomes more pronounced at shorter
wavelengths. Maximum temperatures in the midplane of the disk reached
1500 K in the distance range 1-10 AU. At the end of the calculation, after
an elapsed time of 2500 years, these temperatures ranged from 700 K at 2
AU to 500 K at 10 AU and were decreasing with time.
FURTHER EVOLUTION OF THE SYSTEM
In the preceding example most of the infalling material joined the disk
or central object on a short time scale, because of the high initial density.
For a lower initial density, processes of angular momentum transport in
the disk would begin before accretion was completed. The problem of
the rapidly spinning central regions is apparently not solved by including
angular momentum transport by turbulent viscosity during the collapse
phase. Furthermore, no plausible physical mechanism for generating tur-
bulence on the appropriate scale has been demonstrated. The angular
momentum transport resulting from gravitational torques arising from the
non-axisymmetric structure of the central regions is likely to leave them
with values of ,B near 0.2 (Durisen et al. 1986~. Therefore, even further
transport is required. A related mechanism has been explored by Boss
(1985, 1989~. He has made calculations of protostar collapse, starting from
uniform density and uniform angular velocity, with a three-dimensional
hydrodynamic code, including radiation transport in optically thick regions.
Small, initial non-axisymmetric perturbations grow during the collapse, so
that the central regions, on a scale of 10 AU, become significantly non-
axisymmetric even before a quasi-equilibrium configuration is reached. The
deduced time scales for angular momentum transport depend on the initial
conditions but range from 103 to 106 years for systems with a total mass
of 1 Me. However, since the evolution has not actually been calculated
over this time scale, it is not clear how long the non-axisymmetry will last
or how it will affect the angular momentum of the central core.
It is likely that some additional process is required to reduce j of the
central object down to the value of 10~7 characteristic of T Tauri stars.
The approach of Safronov and Ruzmaikina (1985) is to assume that the
initial cloud had an even smaller angular momentum (i ~ 10~9 cm2 sol)
than that assumed in most other calculations discussed here. The cloud
would then collapse and form a disk with an equilibrium radius much less
than that of Jupiter's orbit. Outward transport of angular momentum into
a relatively small amount of mass is then required to produce the solar
nebula. Magnetic transport could be important in the inner regions, which
are warm and at least partially ionized (Ruzmaildna 1981~. However, out-
side about 1 AU (Hayashi 1981) the magnetic field decays faster than it
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PIANETARY SCIENCES
amplifies, and magnetic transport is ineffective. A supplementary mech-
anism must be available to continue the process. One possibility is the
turbulence generated in the surface layers of the disk caused by the shear
between disk matter and infalling matter. Another possibility is that the
initial cloud had higher j, and the rapidly spinning central object is braked
through a centrifugally driven magnetic wind which can remove the angular
momentum relatively quickly (Shu et al. 1988~.
As far as the evolution of the disk itself is concerned, other important
mechanisms that have been suggested include (a) gravitational instability;
(b) turbulent viscosity induced by convection, and (c) sound waves and shock
dissipation. The former can occur if the disk is relatively massive compared
with the central star or if the disk is relatively cold. Although it is still an
open question whether this instability can result in the formation of a binary
or preplaneta~y condensations, the most likely outcome is the spreading
out of such condensations, because of the shear, into spiral density waves
(Larson 1983~. Lin and Pringle (1987) have estimated the transport time
to be about 10 times the dynamical time. Processes (b) and (c) have time
scales more in line with the probable lifetimes of nebular disks. Convective
instability in the vertical direction (tin and Papaloizou 1980), induced by
the temperature dependence of the grain opacities, gives disk evolutionary
times of about 106 years (Ruden and Lin 1986~. An alternate treatment of
the convection (Cabot et al. l9g7a,b) gives a time scale longer roughly by
a factor of 10. Sound waves induced by various external perturbations give
transport times in the range 106 to 107 years (Larson 1989~. A complete
theory of how the disk evolves after the immediate formation phase may
involve several of the mechanisms just mentioned, and its development will
require a considerable investment of thought and numerical calculation.
ACKNOWLEDGEMENTS
This work was supported in part by a special NASA theory program
which provides funding for a joint Center for Star Formation Studies at
NASA-Ames Research Center, University of California, Berkeley, and
University of California, Santa Crud Further support was obtained from
National Science Foundation grant AST-8521636.
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OCR for page 30
Representative terms from entire chapter:
momentum transport
30
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Toomre, A. 19
