Astronomy&
Astrophysics

A&A 504, 109–113 (2009)
DOI: 10.1051/0004-6361/200912120
c ESO 2009Protostellar collapse: rotation and disk formation
W. M. Tscharnuter1 , J. Schönke1 , H.-P. Gail1 , M. Trieloﬀ2 , and E. Lüttjohann1
1

Zentrum für Astronomie (ZAH), Institut für Theoretische Astrophysik (ITA), Universität Heidelberg, Albert-Ueberle-Str. 2,69120 Heidelberg, Germany
e-mail: wmt@ita.uni-heidelberg.de2
Institut für Geowissenschaften, Universität Heidelberg, Im Neuenheimer Feld 236, 69120 Heidelberg, Germany
Received 20 March 2009 / Accepted 8 June 2009
ABSTRACT

We present some important conclusions from models of the collapse of rotating molecular cloud cores with axial symmetry, corresponding to the evolution of young stellar objects from class 0 to the beginning of class I. There are three main findings of the calculations: (1) the typical timescale for building up a preplanetary disk, which was found to be of the order of one free-fall time decisively
shorter than the widely assumed timescale related to the so-called “inside-out collapse”; (2) redistribution of angular momentum
and the accompanying dissipation of kinetic (rotational) energy causing the growing disk to become more stable and strengthening
the intrinsic meridional circulation pattern of the accretion flow; and (3) the origin of calcium-aluminium-rich inclusions (CAIs).
Because of the persistent equatorial outflow, material that has undergone substantial chemical and mineralogical modifications in thehot (>∼900 K) interior of the protostellar core may have a good chance of being advectively transported outward into the cooler remoteparts (>
∼4 AU, say) of the growing disk and surviving there until it is incorporated into a meteoritic parent body.
Key words. stars: formation – accretion, accretion disks – planetary systems: formation

1. Introduction
Stars are understood to form from regions within molecular
clouds that for some reason became gravitationally unstable andstarted to collapse under the influence of their own gravitational
attraction (see, e.g., Larson 2003; McKee & Ostriker 2007). It isclear that angular momentum (e. g., Bodenheimer 1995; Tohline
2002) and magnetic fields (e.g., Pudritz et al. 2008) play an important role during many stages of protostellar collapse. Angularmomentum is particular important because it is responsible forthe formation of accretion disks that are the birthplaces of planetary systems. Therefore, the study of rotating collapse is inevitable if one intends to understand the formation of the Solar
System and other planetary systems.
The earliest of such studies were a number of analytic approaches to rotating collapse without (Terebey et al. 1984) andwith (Allen et al. 2003) magnetic fields and analytic studies of
disk formation (see Cassen 1994, and references therein) andsimple models for the build-up and evolution of accretion disks(e.g., Lin & Pringle 1990; Nakamoto & Nakagawa 1994). These
introduce a number of approximations about the nature of thecollapse process that appear plausible but lack justification.
In the past two decades, various attempts have been madeto follow the 2-D and 3-D collapse of rotating protostellar
fragments by means of extended numerical calculations (e. g.,
Boss 1989; Bodenheimer et al. 1990; Yorke et al. 1993; Saigo
et al. 2008) based on grid-methods, by a solution method
based on a series development into orthogonal polynomials(Tscharnuter 1987), and by studies based on the SPH-method(e.g., Stamatellos et al. 2007).
However, almost all of these studies covered only the collapse up to the formation of the first core. Only the early 2-D
model of Tscharnuter (1987), the 3-D study of Bate (1998) andthe models of Stamatellos et al. (2007) and Saigo et al. (2008)

achieved suﬃcient resolution to follow the second collapse. Asa rule, 3-D models neglect most of the basic physics and consider only an oversimplified equation of state. Promising ideas
to overcome this unfortunate situation have been put forward by
Stamatellos et al. (2007).
In this paper, we discuss some important cosmochemicalconsequences of a new 2-D model calculation by following
simultaneously the evolution of a rotating Bonnor-Ebert-spherelike initial state with ∼1 M initial mass through an axiallysymmetric first and second collapse to nearly stellar central
densities and the early build-up and evolution of the associatedaccretion disk. The model takes all of the essential physics ofthe problem into account and the simulation covers the evolutionary phases of young stellar objects corresponding to class 0
and early class I. It clearly demonstrates the rapid co-formationof a compact stellar object and a very extended accretion disk
within a period of no more than ∼1.1 free-fall times.

2. Methods
Appropriate implicit methods have been developed only for
1-D and 2-D collapse problems with spherical and axial symmetry, respectively (see, e. g., Wuchterl & Tscharnuter 2003;
Tscharnuter 1987). Of course, implicit methods (being notoriously expensive) are useful only if the physical processes tobe considered exhibit a hierarchy of timescales and the system
as a whole evolves into a quasi-stationary state. If the longest
timescale exceeds the smallest one by a large amount, the use of
implicit methods is mandatory, as in our case, where the accretion time is several orders of magnitude longer than the oscillation period of the stellar core.
For a detailed description of the method upon which the implicit 2-D hydrodynamical code is based, we refer to Tscharnuter(1987). For the equation of state, both the dissociation of H2 andArticle published by EDP Sciences

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W. M. Tscharnuter et al.: Protostellar collapse: rotation and disk formation

3.2. Formation and growth of the disk-like core

Table 1. Starting parameters.
Quantitymass
angular momentum
radiusspin angul. veloc.
centrifugal barriermean density

SymbolValue

Dimension

M
L
R
Ω
Rcfb
ρ̄

1.037
2.682×1053
1.200×1017
3.160×10−14
1.504×1015
2.850×10−19M
g cm2 s−1cm
s−1cm
g cm−33π
32Gρ̄1.247×105
5.326×10−18yr
g cm−33π
32Gρc2.885×104
10
2.437×10−3

yrK–mean free-fall time
central densityt¯ﬀ =central free-fall time
temperature
ratio rot./grav. energytﬀ,c =

ρcTrotθ = |EEgrav|

ionization of H and He is taken into account, and the opacities
at low temperatures are dominated by dust with and without ice
mantles. In a spherical polar coordinate system, we discretize thevariables within a staggered radial grid (256 gridpoints) and represent the dependence on the polar angle by choosing a Legendre
expansion (up to 27 coeﬃcients) for each primary variable. The
discretized equations are written in a conservation form on aself-adaptive radial grid according to Dorfi & Drury (1987), andthe shock fronts are smoothed by artificial (tensor-) viscosity.
The maximum resolution achieved amounts to about 0.03 R inthe central parts. The adaptive grid is chosen in a way that the
innermost rigidly rotating homogeneous sphere always contains
a fraction of 10−6 of the total mass.
Concerning the coeﬃcient of turbulent viscosity, νtur , wehave adopted the so-called β-viscosity prescription of Duschl
et al. (2000), an extension of the well-known, but likewise
heuristic, α-viscosity for application to self-gravitating disks. To
be specific, νtur := βr2 Ω, where r is the radial coordinate, Ω the(local) angular velocity, and β = 10−4 . . . 10−2 the inverse of thecritical Reynolds number indicating the onset of turbulence. In
our calculation, we have chosen β = 10−2 .

3. Results and conclusions
3.1. Initial configuration and the first collapse phase

Table 1 lists a set of appropriate starting parameters leading
to the collapse of a rotating protostellar cloud fragment. The
Bonnor-Ebert-sphere-like initial configuration exhibits alreadya moderate density concentration toward the center and is assumed to rotate like a rigid body. The revolution period, 2π/Ω =6.3 × 106 yr, adopted in our calculation is long compared to themean free-fall time. Hence, the centrifugal barrier is situated at
a distance from the center of only about 1% of the cloud radius,
and the collapse flow develops almost perfectly with sphericalsymmetry for most parts of the cloud.
It is convenient to reset the clock to zero when the optical depth of the collapsing fragment, counted from the outeredge to the center, exceeds 2/3 for the first time (e.g., Wuchterl& Tscharnuter 2003). This event may be interpreted to be thebeginning of the protostellar evolution. All ages given subsequently are relative to this moment.
Shortly after age zero, an accretion shock forms marking thenatural boundary of a flattened quasi-hydrostatic rotating structure, which is pressure-supported, parallel, and centrifugallysupported perpendicular to the axis of rotation.

Figure 2 shows (a) the density-temperature-diagram for the very
center of the collapsing cloud, (b) the stability parameter θc as
a function of the relative core mass Mc /Mtot , and (c) the timedependence of the relative core mass. The labels A through E
indicate characteristic stages of the evolution:
A: the core starts to grow in mass;
B: continuation of the accretion after thermal relaxation causedby opacity eﬀects (sublimation of ice mantels);
C: sublimation of the dust grains commences oﬀ-center near theaxis of rotation as a “hot polar cap”, leaving a growing opacity gap and violent thermal relaxation eﬀects;
D: start of the second collapse due to dissociation of H2 ;E: formation of the “stellar” core.
Our first important result is that we found a typical rise time,
Mc / Ṁc , of the core’s mass to be only a small fraction (1–2%)of the mean free-fall time, i. e., only a few thousand years.
Figure 2c shows an almost constant accretion rate of 3 ×
10−5 M yr−1 lasting for about 2600 yr, from slightly after “A”
until “D”, where the second collapse sets in. The dots between“D” and “E” represent the individual models tracing the dynamical transition (the “second collapse”) from the disk-forming firstcore to the second “stellar” core.
Figure 1 demonstrates the increasing geometrical dimensions and the changing shape of the first core, which occurs
within a few thousand years; the bulge in the polar direction,
indicated by the route of the accretion shock in the vicinity of
the rotational axis, is the result of a rather intricate interplay between the redistribution of angular momentum, the accompanying generation of entropy (viz. heat), and the transport of energy.
The net eﬀects are rising temperatures and lower rotation rates
in the inner parts of the core, which are necessary to trigger thesecond collapse. Test calculations demonstrated that, without acertain amount of angular momentum transport, the core would
stay too cool and, hence, become prone to the onset of destructive gravitational instabilities that trigger binary formation.
A glimpse at Fig. 2b indicates that the first core cannot escape from existing in the regime of secular instability. The ratio, θ = Erot /|Egrav |1 , quickly rises above the critical numberof 0.14, so that a small amount of dissipation leads to symmetry
breaking: the originally axisymmetric core will assume a triaxial
shape and presumably evolve into a distinctive bar/spiral configuration exerting gravitational torques, which, in turn, cause
an enhanced redistribution of angular momentum (see, e. g.,
Bodenheimer & Ostriker 1973; Ostriker & Bodenheimer 1973;
Pickett et al. 1997; Toman et al. 1998; Imamura et al. 2000). Thisfinding may motivate our choice of rather eﬃcient turbulent (β-)
viscosity with β = 0.01.
3.3. Formation of the “stellar” core

The second collapse is a result of the interplay between thermodynamics, redistribution of angular momentum as a dissipative process, and energy transport. Their combined eﬀect is illustrated by Fig. 1: particularly in the central parts, angular momentum transport tends to prevent extreme flattening, while theheat input by the accompanying dissipation of rotational energy
creates increasing pressure forces. As a consequence, the pressure distribution becomes more spherical, and after some “incubation” period of slow contraction (between “C” and “D” in
1
Usually written β := T/ |W|, but we wish to avoid confusion with T ,the temperature, and β, the scaling parameter of the turbulent viscosity.

W. M. Tscharnuter et al.: Protostellar collapse: rotation and disk formation

111

Fig. 1. Meridional cross sections. Displayed are equi-density contours (annotated numbers are logarithms of the density in g cm−3 ), the spacing
being 0.5 dex, and arrows representing the velocity field. Red “spots” indicate expansion (cf. Fig. 3a). The closed heavy line marks the accretion
shock, and the arrow at the lower right corner of the four panels represents the respective maximum infall velocities. Upper row: snapshots slightly
after B and before C, respectively. Lower row: snapshots for D and E (cf. Fig. 2). At E (right panel), two shocks, an outer and an inner one,bounding the growing “pre-planetary” disk and the “stellar” core, respectively, coexist (the two shocks are not displayed here, but cf. Fig. 4 for theformation of the stellar core).

Fig. 2), dissociation of H2 eventually leads to dynamical collapse(“D”–“E”). Figure 2b shows that the stability parameter, θc , approaches the critical value of 0.27, which is indicative of dynamical instability. Interestingly enough, our axisymmetric model
suggests the formation of a self-gravitating doughnut-like structure of the density distribution (cf. Fig. 4).
3.4. An intermediate hot solar nebula?

The model provides for the first time a realistic initial temperature and density structure of the accretion disk and the flow pattern in the disk and its surroundings, including well resolved accretion shocks, on scales as small as well below 1 AU. Figure 3b
shows a rather hot inner disk region extending out to about 4 AU.
This structure was not seen in earlier calculations because of alack of resolution, use of inadequate equations of state, and theomission of radiative transfer; it lasts for a couple of 1000 yr.
This inner portion is hot enough for formation of materials suchas those found as calcium-aluminium rich inclusions (CAIs)
in meteorites; it might be the “hot solar nebula” that cosmochemists have always advocated.
CAIs are a fundamental – though rare (few permille to percent) – ingredient of chondrites. Although modified by a secondary processing, they consist of mineral assemblages which
are expected for high temperature (>1600 K) solar nebula condensates, and are considered to be oldest solar system material(Amelin et al. 2002; Bouvier et al. 2007) and apparently formed
over a relatively short time interval, but their detailed formation
setting is basically unknown.

The CAI formation time interval has been vividly debated,based on short-lived nuclide chronometry, mainly the 26Al -26 Mg
system (τ1/2 = 0.7 Ma). While most studies yielded a so-called“canonical” initial 26Al /27Al in the range 5 × 10−5 , Thrane et al.
(2006), Young et al. (2005), and Bizzarro et al. (2004) argued fora “supracanonical” 26Al /27Al ratio of 5.8 . . . 7 × 10−5 , implying
formation and processing of CAI material over a few 100 000
years. Jacobsen et al. (2008) expended a comprehensive eﬀort
to demonstrate that CAIs have an identical (canonical) 26 Al/27 Al
ratio of 5.23 ± 0.13 × 10−5, implying a formation time interval of
only 40 ka.
After residence in the solar nebula the CAIs became incorporated into chondritic parent bodies, which are strongly related tocarbonaceous chondrites. This must have happened up to a few
Ma later, as indicated by chondrule ages. Chondrules formed before chondrite parent-body accretion and are 1–4 Ma younger
than CAIs (e.g., Scott 2007). It has always been unclear how
they could be dynamically transported from their apparent formation regions in the inner solar system outwards to chondrite
forming regions, and survive a few Ma before becoming incorporated into carbonaceous chondrite parent bodies. Our modelshows that from the onset of disk formation on, there is a complex accretion flow structure, where the average inflow is superimposed on circulation currents that result in a net outward
directed flow close to the midplane and inward directed flow inhigher layers of the disk (cf. Fig. 1). This flow results in bothaccretion and a large-scale transport of matter (with velocities
10–50 m s−1 , cf. Fig. 3a) close to the disk’s midplane from the
hot part outward to distances of several AU from the center.

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W. M. Tscharnuter et al.: Protostellar collapse: rotation and disk formation
(a)

(b)

2 nd collapse(c)Fig. 2. Evolution of the core. a) central density vs. central temperature; b) ratio of rotational over gravitational energy vs. core mass; c)core mass vs. time. Plotted are the respective quantities for each of the
104 timesteps individually. Because of the rapid evolution during thesecond collapse the path from D to E is covered only by a few points.
For the meaning of labels A through E see Sect. 3.2.

(a)

(b)

Fig. 3. Evolution of the core from its quasi-stationary stage to the end of
the second collapse. Spatial distribution of a) the radial velocity component, ur , b) the temperature, T , in the equatorial plane for three instantsof time indicated by dash-dotted (2301 yr), dashed (2948 yr), and solidlines (3070 yr), respectively; crosses mark the gridpoints.

Fig. 4. Meridional cross sections (cf. Fig. 1). Formation of the “stellar”
core on a very short timescale. Plotted are three snapshots of the final
shock evolution at (from above) t0 −10 d, t0 , and t0 + 7 d, respectively,
with t0 = 3070 yr. The contour lines and arrows have the same meaning
as in Fig. 1, numbers are again logarithms of the density in g cm−3 , thespacing is 0.5 dex, and the arrows indicate the velocity field. Note that
now the geometrical dimensions are only several solar radii (R ).

This type of global accretion flow structure is generally
found in hydrodynamic calculations (e. g., Kley & Lin 1992;
Różyczka et al. 1994; Keller & Gail 2004). Material from the
hot region may be mixed by this across the inner ∼5 AU andpart of this material may survive until the onset of planetesimalformation since outflow close to the midplane also continues
to exist in later phases of disk evolution, though with reduced
velocity (e.g., Keller & Gail 2004; Ciesla 2009). In the full3-D picture, the flow will presumably exhibit non-axisymmetricstructures (e.g., cyclones). However, since they are only local
phenomena, we do not expect them to impede the global meridional circulation considerably.
According to these perceptions, our model provides a mostfeasible setting for CAI formation comprising a very short timeinterval at the starting point of stellar accretion, which implies
both high temperature processing out to relatively large innerdisk radii, and a possible outward transport mechanism to ensure survival and later incorporation into chondritic planetesimals. Although other disk activities may also provide hightemperature energetic events (e.g., FU Orionis outbursts), they
were mostly active on significantly longer timescales, and appear

W. M. Tscharnuter et al.: Protostellar collapse: rotation and disk formation

unsuitable to explain why CAIs formed over such a restricted
time interval of 40 ka only.
The model therefore seems to oﬀer an explanation of how
the short CAI-forming period in the Solar Nebula is related tothe earliest evolutionary phase of the accretion disk. If true, CAI
formation follows the second collapse immediately and is an accurate indicator of the formation time of the protoplanetary disk.
Acknowledgements. This work has been supported by the Forschergruppe 759,
“The Formation of Planets: The Critical First Growth Phase” of the Deutsche
Forschungsgemeinschaft (DFG).

References
Allen, A., Li, Z.-Y., & Shu, F. H. 2003, ApJ, 599, 363
Amelin, Y., Krot, A. N., Hutcheon, I. D., et al. 2002, Science, 297, 1678
Bate, M. R. 1998, ApJ, 508, L95
Bizzarro, M., Baker, J. A., & Haack, H. 2004, Nature, 431, 275, Erratum: 2005,
Nature, 435, 1280
Bodenheimer, P. 1995, ARA&A, 33, 199
Bodenheimer, P., & Ostriker, J. P. 1973, ApJ, 180, 159
Bodenheimer, P., Yorke, H. W., Różyczka, M., et al. 1990, ApJ, 355, 651
Boss, A. P. 1989, ApJ, 345, 554
Bouvier, A., Blichert-Toft, J., Moynier, F., et al. 2007, Geochim. Cosmochim.
Acta, 71, 1583

113

Cassen, P. 1994, Icarus, 112, 405
Ciesla, F. J. 2009, Icarus, 200, 655
Dorfi, E. A., & Drury, L. O. 1987, J. Comp. Phys., 69, 175
Duschl, W. J., Strittmatter, P. A., & Biermann, P. L. 2000, A&A, 357, 1123
Imamura, J. N., Durisen, R. H., & Pickett, B. K. 2000, ApJ, 528, 946
Jacobsen, B., Yin, Q., Moynier, F., et al. 2008, Earth Plan. Sci. Lett., 272, 353
Keller, C., & Gail, H.-P. 2004, A&A, 415, 1177
Kley, W., & Lin, D. N. C. 1992, ApJ, 397, 600
Larson, R. B. 2003, Reports on Progress in Physics, 66, 1651
Lin, D. N. C., & Pringle, J. E. 1990, ApJ, 358, 515
McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565
Nakamoto, T., & Nakagawa, Y. 1994, ApJ, 421, 640
Ostriker, J. P., & Bodenheimer, P. 1973, ApJ, 180, 171
Pickett, B. K., Durisen, R. H., & Link, R. 1997, Icarus, 126, 243
Pudritz, R. E., Banerjee, R., & Ouyed, R. 2008, ArXiv e-prints
Różyczka, M., Bodenheimer, P., & Bell, K. R. 1994, ApJ, 423, 736
Saigo, K., Tomisaka, K., & Matsumoto, T. 2008, ApJ, 674, 997
Scott, E. R. D. 2007, Ann. Rev. Earth Planet. Sci., 35, 577
Stamatellos, D., Whitworth, A. P., Bisbas, T., & Goodwin, S. 2007, A&A, 475,
37
Terebey, S., Shu, F. H., & Cassen, P. 1984, ApJ, 286, 529
Thrane, K., Bizzarro, M., & Baker, J. A. 2006, ApJ, 646, L159
Tohline, J. E. 2002, ARA&A, 40, 349
Toman, J., Imamura, J. N., & Pickett, B. K. 1998, ApJ, 497, 370
Tscharnuter, W. M. 1987, A&A, 188, 55
Wuchterl, G., & Tscharnuter, W. M. 2003, A&A, 398, 1081
Yorke, H. W., Bodenheimer, P., & Laughlin, G. 1993, ApJ, 411, 274
Young, E. D., Simon, J. I., Galy, A., et al. 2005, Science, 308, 223