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Three-Dimensional Evolution of
Early Solar Nebula
ALAN P. BOSS
Carnegie Institution of Washington
in.
INITIAL CONDITIONS FOR PROTOSTELLAR COLLAPSE
Mathematically speaking, solar nebula formation is an initial value
problem. That is, it is believed that given the proper initial conditions and
knowledge of the dominant physical processes at each phase, it should be
possible to calculate the evolution of a dense molecular cloud core as it
collapses to form a protosun and solar nebula. Also, through calculating
the evolution of the dust grains in the nebula, it should be possible to
learn how the initial phases of planetary accumulation occurred. While
this extraordinarily ambitious goal has not yet been achieved, considerable
progress has been made in the last two decades, with the assistance of
current computational resources. This paper reviews the progress toward
the goal of a complete theory of solar nebula formation, with an emphasis
on three spatial dimension (3D) models of solar nebula formation and
evolution.
In principle, astronomical observations should provide the initial con-
ditions required for theoretical calculations of solar nebula formation. This
assumes that physical conditions in contemporary regions of solar-type star
formation in our galaxy are similar to the conditions ~ 4.56 x 109 years
ago. Whether or not this is a reasonable assumption, considering that the
age of the solar system is a fair fraction of the Hubble time, there really
is no other means of constraining the initial conditions for solar system
formation.
Millimeter wave and infrared telescopes have revealed that low-mass
(solar-type) stars are presently forming preferentially in groups distributed
throughout large molecular cloud complexes in the disk of our galaxy. These
31
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molecular clouds have complex, often filamentary structures on the largest
scales (~100pc). On the smallest scales (~0.lpc), where finite telescope
resolution begins to limit the observations, molecular clouds are composed
of centrally condensed cloud cores surrounded by cloud envelopes. The
cloud cores are gravitationally bound and, should they begin to collapse
upon themselves because of self-gravity, are quite likely to form stars
(Myers and Benson 1983~. Indeed, when infrared observations of embedded
protostellar objects (presumed to be embedded pre-main-sequence T Tauri
stars or very young protostars) are combined with millimeter wave maps
of molecular cloud cores, it appears that roughly one half of the cloud
cores contain embedded protostars (Beichman et al. 19863. The physical
conditions in molecular cloud cores, or their predecessors, should then
provide the best indication of the initial conditions that are appropriate for
solar nebula formation.
Practically speaking, there are several reasons why astronomical obser-
vations can only provide us with a range of possible initial conditions for
solar nebula formation. First, since even the short time scales associated
with low-mass star formation (~105 - 106 years) are quite long compared
to human lifetimes, one can never be sure what a particular collapsing
cloud core will produce. Second, although interferometric arrays have the
potential to greatly increase our understanding of cloud core properties,
the length scales appropriate for the initial phases of collapse are only
marginally resolved by current millimeter wave telescopes. Third, because
many (if not most) cloud cores have already collapsed to form protostars,
their properties may not be appropriate for constraining the earliest phases
of collapse. If the immediate predecessors of cloud cores could be identi-
fied, then the constraints on the initial conditions for protostellar collapse
would be improved considerably.
Given these limitations, observations of cloud cores suggest the fol-
lowing initial conditions for solar nebula formation: 1) Cloud cores have
masses in the range of 0.1 to 10Me, implying that evolution within the
molecular cloud complex has already reduced the mass of sub-structures by
several orders of magnitude, from masses characteristic of giant molecular
clouds (~105 - 106M<~), to masses in the stellar range; 2) Dense cloud
cores have maximum densities on the order of 10-~9 - 10-~8 g cm~3,
sizes less than 0.1 pc, and temperatures close to 10K These are basically
the same initial conditions that have long been used to model protosolar
collapse (Larson 1969~.
One of the remaining great uncertainties is the amount of rotation
present in cloud cores, because even the most rapidly rotating clouds have
Doppler shifts comparable to other sources of line broadening, such as
thermal broadening, translational cloud motions, and turbulence. While
there is strong evidence that at least some dense clouds rotate close to
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33
centrifugal equilibrium (specific angular momentum JIM ~ 1021 cm2s~1
for solar mass clouds), it is unknown how common such rapid rotation is,
or how slowly clouds can rotate. Three-dimensional calculations (described
in the next sub-section) indicate that the initial amount of rotation is
critical for determining whether clouds collapse to form single or multiple
protostars.
Finally, most three-dimensional calculations of protostellar collapse
have ignored the possible importance of magnetic fields, in no small part
because of the computational difficulties associated with their inclusion in
an already formidable problem. OH Zeeman measurements of magnetic
field strengths in molecular clouds yield values (~30pG) in cool clouds
implying that magnetic fields dominate the dynamics on the largest scales
(~10—10ppc). However, there is evidence from the lack of correlations
between magnetic field directions and dense cloud minor and rotational
axes that, on the smaller scales (and higher densities) of dense cloud cores,
magnetic fields no longer dominate the dynamics (Heyer 1988~. Loss of
magnetic field support is probably caused by ambipolar diffusion of the ions
and magnetic field lines during contraction of the neutral bunk of the cloud.
The evidence for decreased importance of magnetic fields at densities
greater than ~10-2° g cm-3 suggests that nonmagnetic models may be
adequate for representing the gross dynamics of solar nebula formation.
SINGLE VERSUS BINARY STAR FORMATION
The first numerical models of the collapse of interstellar clouds to
form solar-lype stars disregarded the effects of rotation, thereby reducing
the problem to spherical symmetry and the mathematics to one-dimensional
(ID) equations (Bodenheimer 1968~. The assumption of spherical symme-
try ensures that a single star will result, but unfortunately such calculations
can say nothing about binary star or planet formation. The first major dy-
namical problem in solar nebula formation is avoiding fragmentation into
a binary protostar.
Orson (1969) found that ID clouds collapse non-homologously, form-
ing a protostellar core onto which the remainder of the cloud envelope
accretes. Once the envelope is accreted, the protostar becomes visible as
a low-mass, pre-main-sequence star (T Mauri star). Models based on these
assumptions have done remarkably well at predicting the luminosities of T
Tauri stars (Stabler 1983~. However, there is a recent suggestion (I§char-
nuter 1987a) that the protostellar core may not become well established
until much later in the overall collapse than was previously thought. The
ID models also showed that the first six or so orders of magnitude increase
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35
fragmentation during its own collapse. Hence 3D models of solar nebula
formation have also concentrated on low J/M clouds, in a search for clouds
that do not undergo rotational fragmentation during their isothermal or
nonisothermal collapse phases. Three-dimensional calculations, including
radiative transfer in the Eddington approximation, have shown that solar
mass clouds with J/M ~ 102° cm2 s- ~ are indeed required in order to avoid
binary formation (Boss 1985; Boss 1986~.
There are two other ways of suppressing binary fragmentation other
than starting with insufficient J/M to form and maintain a binary protostel-
lar system. First, when the initial mass of a collapsing cloud is lowered
sufficiently, fragmentation is halted, yielding a lower limit on the mass of
protostars formed by the fragmentation of molecular clouds of around 0.01
Me (Boss 1986~. The minimum mass arises from the increased importance
of thermal pressure as the cloud mass is decreased; thermal pressure re-
sists fragmentation. This limit implies that there may be a gap between
the smallest mass protostars (C`brown dwarfs") and the most massive plan-
ets (the mass of Jupiter is ~ 0.001 Magi. Second, clouds that are initially
strongly centrally condensed can resist binary fragmentation simply because
of their initial geometric prejudice toward forming a single object (Boss
1987~. While initially uniform density and initially moderately condensed
clouds readily fragment, given large J/M and/or low thermal pressure, it
appears to be impossible to fragment a cloud starting from an initial power
law density profile (see Figure 1~. Considering that the majority of stars are
found in binary or multiple systems, it does not appear likely that power
law initial density profiles are widespread in regions of star formation, but
such a profile could have led to solar nebula formation. Thus, the 3D
calculations have shown that formation of the Sun requires the collapse of
either a very slowly rotating, high-thermal energy cloud, or else the collapse
of a cloud starting from a power law initial density profile. In contrast to
ID calculations, however, no 3D calculation has been able to collapse a
cloud core all the way to the pre-main-sequence. In part, this is because
of the greatly increased computational effort necessary to evolve a mul-
tidimensional cloud through the intermediate phases. Equally important
though, is the problem that when rotation is included, accumulation of
the central protosun requires an efficient mechanism for outward angular
momentum transport. Identifying this mechanism and its effects is one of
the major remaining uncertainties in solar nebula models.
ANGULAR MOMENTUM TRANSPORT MECHANISMS
Given the formation of a rotationally flattened, presolar nebula through
the collapse of a cloud core that has avoided binary fragmentation, the next
major dynamical problem is accumulating the protosun out of the disk
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u
FIGURE 1 Density contours in the midplane of three models of protostellar collapse with
varied initial density profiles (Boss 1987~. The rotation axis falls in the center of each plot;
counterclockwise rotation is assumed. Each contour represents a factor of two change in
density; contours are labelled with densities in g cm~3. (a) Initially uniform density profile,
(by initially Gaussian density profile, and (c) initially power law profile (r i). As the initial
degree of central concentration increases, the amount of nonaxisymmet~y produced during
collapse decreases. Qualititativeh~r similar results hold when the initial cloud mass or initial
angular velocity is decreased; binary formation is stifled. Diameter of region shown: (a)
580 AU, (b) 300 AU, (c) 110 AU.
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matter. Angular momentum must flow outward, if mass is to accrete onto
the protosun, and also if the angular momentum structure of the solar
system is to result from a cloud with more or less uniform Jim. The
physical process responsible for this dynamical differentiation is thought to
have operated within the solar nebula itself, rather than in the material
collapsing to form the nebula.
Three different processes have been proposed for transporting mass
and angular momentum in the solar nebula: viscous shear, magnetic
stresses, and gravitational torques (see also Bodenheimer, this volume).
Molecular viscosity is far too small to be important, so turbulent viscosity
must be invoked if viscous stresses are to dominate. The most promising
means for driving turbulence in the solar nebula appears to be through
convective instability in the vertical direction, perpendicular to the neb-
ula midplane (tin and Papaloizou 1980~. The main uncertainty associated
with connectively driven viscous evolution, aside from the effective strength
of the turbulent stresses, is the possibility that such a nebula is unstable
to a diffusive instability that would break up the nebula into a series of
concentric rings (Cabot et al. 1987~.
As previously mentioned, magnetic fields need not be dominant during
the early phases of presolar collapse, and frozen-in magnetic fluxes scale
in such a way that they never become important, if they are not important
initially. While some meteorites show evidence for remanent magnetic
fields requiring solar nebula field strengths on the order of 30,uT (Sugiura
and Strangway 1988), the magnetic pressure (B2/8,r) corresponding to such
field strengths is still considerably less than even thermal pressures in hot
solar nebula models (Boss 1988), implying the negligibility of magnetic
fields for the gross dynamics of the solar nebula.
The remaining candidate for angular momentum transport is gravi-
tational torques between nonaxisymmetric structures in the solar nebula.
Possible sources of nonaxisymmetry include intrinsic spiral density waves
(Larson 1984), large-scale bars (Boss 1985), and biaxial central protostars
(Yuan and Cassen 1985~. Early estimates of the efficiency of gravitational
torques (Boss 1984b) implied that a moderately nonaxisymmetric nebula
can have a time scale for angular momentum transport just as short as
a strongly turbulent accretion disk Three-dimensional calculations of the
aborted fission instability in rapidly rotating polytropes (Durisen et al. 1986)
were perhaps the first to demonstrate the remarkable ability of gravitational
torques to remove orbital angular momentum from quasi~quilibrium, non-
axisymmetric structures similar to the solar nebula.
At later phases of nebula evolution, nonaxisymmetry and spiral density
waves can also be driven by massive protoplanets. The possible effects range
from gap clearing about the protoplanet, in which case the protoplanet must
evolve along with the nebula (tin and Papaloizou 1986), to rapid orbital
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decay of the protoplanet onto the protosun (Ward 1986~. While the effects
of viscous or magnetic stresses can be studied with 2D (axisymmetric) solar
nebula models, in order to model the effects of gravitational torques, a
nonaxisymmetric (generally 3D) solar nebula model must be constructed.
THREE-DIMENSIONAL SOLAR NEBULA MODELS
Only a few attempts have been made at studying the nonaxisymmetric
structure of the early solar nebula. Cassen et al. (1981) used a type of
N-body code to study the growth of nonaxisymmetry in an infinitely thin,
isothermal model of the solar nebula. Cassen et al. (1981) found that
when the nebula is relatively cool (~1OOK) and more massive than the
central protosun, nonaxisymmetry grows within a few rotational periods,
resulting either in spiral arm formation, or even fragmentation into giant
gaseous protoplanets in the particularly extreme case of a nebula 10 times
more massive than the protosun. Cassen and Manley (1988) are presently
engaged in using this code to study the onset of gravitational instability in
nebula models with simulated thermal gradients.
Boss (1985) used a 3D hydrodynamics code to model the early phases
of solar nebula formation through collapse of a dense cloud core, and
found that formation of a strong bar-like structure resulted. However,
because the explicit nature of the code limited Boss (1985) from evolving
the model very far in time, these results are only suggestive of the amount of
nonaxisymmetry that could arise in the solar nebula. Recently, Boss (1989)
has tried to circumvent this computational problem by calculating a suite of
3D models starting from densities high enough to bypass the intermediate,
quasi-equilibrium evolution phases that obstruct explicit codes. While these
initial densities for collapse (~lo-~3 - 1o-~2 g cm-3) are clearly not
realistic given the present understanding of star formation, it can be argued
(Boss 1989) that starting from these high densities should not greatly distort
the results.
The 3D models of Boss (1989) show that gravitational torques can be
quite efficient at transporting angular momentum in the early solar neb-
ula. The models show that collapsing presolar clouds become appreciably
nonaxisymmetric (as a result of a combination of nonlinear coupling with
the infall motions, rotational instability, and/or self-gravitation), and that
trailing spiral arm patterns often form spontaneously; trailing spiral arms
lead to the desired outward transport of angular momentum. The most
nonaxisymmetric models tend to be massive nebulae surrounding low mass
protosuns in agreement with the results of Cassen et al. (1981~. Extrap-
olated time scales for angular momentum transport, and hence nebula
evolution, can be as short as ~103 years for strongly nonaxisymmetric mod-
els, or about ~106 - 107 years for less nonaxisymmetric models. Because
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AMERICAN AND SOVIET RESEARCH
39
these time scales are comparable to or less than model ages for naked
T Tauri stars (Walter 1988), solar-type, pre-main-sequence stars that show
no evidence for circumstellar matter, it appears that gravitational torques
can indeed be strong enough to account for the transport of the bulk of
nebula gas onto the protosun on the desired time scales. While these initial
estimates are encouraging, it remains to be learned exactly how a solar
nebula evolves due to gravitational torques.
IMPLICATIONS FOR PLANETARY FORMATION
The 3D solar nebula models of Boss (1989) show little tendency for
breaking up directly into small numbers of giant gaseous protoplanets,
contrary to one of the models of Cassen et al. (1981~. This difference
is probably a result of several features of the Boss (1989) models. The
inclusion of 3D radiative transfer means that the compressional heating
accompanying nebula formation can be included, leading to considerably
higher temperatures than assumed in Cassen et al. (1981), and hence
greater stability against break-up. Also, the gradual buildup of the nebula
through collapse in the Boss (1989) models means that incipient regions of
gravitational instability can be sheared away into trailing spiral arms by the
differential rotation of the nebula before the regions become well-defined.
These models thus suggest that planet formation must occur through the
accumulation of dust grains (Safronov 1969; Wetherill 1980~.
Considering that dust grain evolution is not yet included in 3D codes,
detailed remarks about the earliest phases of dust grain accumulation are
not possible. However, the models of Boss (1989) can be used to pre-
dict surface densities of dust grains in the solar nebula, and these surface
densities are quite important for theories of planetary accumulation. For
example, Goldreich and Ward (1973) suggested that a dust surface density
at 1 AU of a~ ~ 7.5 g cm~2 would be sufficient to result in a gravitational
instability of a dust sub-disk (Safronov 1969) that could speed up the inter-
mediate stages of planetary accumulation. More recently, Lissauer (1987)
has proposed the rapid formation of Jupiter through runaway accretion of
icy-rock planetesimals in a nebula with ad > 15 g am~2 at 5 AU. Rapid
formation is required in order to complete giant planet formation prior to
dispersal of the solar nebula. Using a gas to dust ratio of 200:1 at 1 AU and
50:1 at 5 AU, these critical surface densities correspond to gas surface den-
sities of 1500 g cm~2 at 1 AU and 750 g cm-2 at 5 AU. Similar minimum
densities are inferred from reconstituting the planets to solar composition
(Weidenschilling 1977~. The models of Boss (1989) have surface densities
in the inner solar nebula that are nearly always sufficient to account for
terrestrial planet formation. However, surface densities in the outer solar
nebula are less than the critical amount, unless the nebula is quite massive
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40
i, I I T r ! r-r T · ~ I r -F l]
'\\ V~///
~,,,,,,~,,,,''~ ',''',,,~/,,,a
~7TT~ ~T I ~ I I I T~ T I I I I I I I I I I I I I I I I I I I I I I I I I 5~
'9,
PLANETARY SCIENCES
\\—~
~ /'.
1 1 1 1 1 1/
- //~
Tr~r~TfTTT r r r r ~ I I I I ~ I ~ I ~ I I ~ r~~ ~
~ ~ ~.
~/~\
~\\` ~/
\ ~
\ ,~_
I I I i~1 1 '~.L ~_~LL I ' I I I ~ I I I I I ~ ~ ~ I I I ' I I ~ I I I I ~ I ~ ~ I ~
FIGURE 2 Density contou~s in the midplane of three solar nebula models formed by
collapse onto protosuns with varied initial masses Ms (Boss 1989), plotted as in F;gure
1. (a) M,' = 1 M~, (b) M5 = 0.01 M~, (c) M5 = 0M~. The initial nebula mass was
1M for each model, and the initial specific angular momentum was J/M = 6.2 X 10~9
cm2s~l. The resulting nebulae become increasingly nonaxisymmetnc as the initial protosun
mass is decreased; (b) forms trailing spiral arms that result in efficient transport of angular
momentum, while (c) actually fragments into a transient binaty system. These models also
illustrate the ability of a massive central object to stabilize a protostellar disk. Region
shown is 21) AU across for each model.
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AMERICAN AND SHEET RESEARCH
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(~1M<~. Any protoJupiter that is formed rapidly in a massive nebula is
likely to be lost during subsequent evolution, either because gap clearing
will force the protoplanet to be transported onto the protosun with the rest
of the gas (tin and Papaloizou 1986), or because motion of the protoplanet
relative to the gas will result in orbital decay onto the protosun (Ward
1986~. The planetary system is the debris leftover from formation of the
voracious Sun, and so prematurely formed protoplanets are at peril.
Variations in the initial density and angular velocity profiles do not
appear to be able to produce sufficiently high surface densities at 5 AU in
low-mass solar nebula (Boss 1989), so it does not appear that the surface
densities required for planet formation can be accounted for simply by
collapse onto a nebula. Diffusive redistribution of water vapor could pref-
erentially accumulate ices wherever temperatures drop to 160K (Stevenson
and Lunine 1988), but this mechanism can only be invoked to explain the
formation of one of the outer planets. The most promising means for
enhancing surface densities of the outer solar nebula appears to be through
nebula evolution subsequent to formation. Viscous accretion disks can
increase the surface density in the outer regions where the angular momen-
tum is being deposited (tin and Bodenheimer 1982; Lissauer 1987~. While
the long-term evolution of a 3D nebula subject to gravitational torques
is as yet unknown, gravitational torques should produce a similar result
(tin and Pringle 1987~. Determining the evolution of a nona~symmetric
solar nebula thus appears to be a central issue in finding a solution to the
problem of rapid Jupiter formation.
_ ~ ~
Finally, the 3D models of Boss (1988, 1989) have important implica-
tions for the thermal structure of the solar nebula. Provided the artifice of
starting from high initial densities does not severely overestimate nebula
temperatures, it appears that the compressional energy released by infall
into the gravitational well of a solar-mass object can heat the midplane of
the inner solar nebula to temperatures on the order of 1500 K for times on
the order of 105 years. Such temperatures are high enough to vaporize all
but the most refractory components of dust grains. In particular, because
the vaporization of iron grains around 1420 K removes the dominant source
of opacity, temperatures may be regulated to values close to ~1500 K by
the thermostatic effect of the opacity. A hot inner solar nebula can account
for the gross depletion of volatiles on the terrestrial planets (relative to
solar) by allowing the volatiles to be removed along with the H and He of
the nebula. Hot solar nebula models were introduced by Cameron (1962)
in one of the first solar nebula investigations, but have since fallen into
disfavor (Wood 1988), so it will be interesting to see whether high tempera-
tures in the inner nebula can be successfully resurrected, and whether they
will prove to be useful in explaining planetary and asteroidal formation.
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ACKNOWLEDGMENTS
This research was partially supported by U.S. National Aeronautics
and Space Administration grant NAGW-1410 and by U. S. National Science
Foundation grant AST-8515644.
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Representative terms from entire chapter:
cloud cores
