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abstract:

ABSTRACT

We examine the criterion for selfgravitational fragmentation in the protostellar disks through which massive stars accrete. Rapid accretion and high angular momentum push such disks toward fragmentation, whereas viscous heating and the high protostellar luminosity stabilize them. Using a simple analytic model, we establish the conditions under which disks may become unstable and investigate the consequences on both the accreting star and the fragmenting disk. We find that for a broad range of protostar masses and for reasonable accretion durations, massive disks larger than approximately 100 AU are prone to fragmentation. Concentrating on the turbulent core model, we predict that disks typically cross this threshold for stars that accrete more than about 15 solar masses. This critical mass depends sensitively on the core angular momentum, however, which we can only approximate. If high angular momentum is the norm, then disk fragmentation may starve accretion in massive stars and create a subpopulation within massive clusters.

Fragmentation of Massive Protostellar Disks
K. M. Kratter & C. D. Matzner Dept. of Astronomy and Astrophysics, University of Toronto
STABLE

UNSTABLE

CRITERION FOR FRAGMENTATION:

Q=

cs G

1

Gammie, 2001

M

c

3 s

steady accn.

G

We obtain criteria for disk fragmentation via the Toomre instability (Toomre, 1964). We define a disk as unstable when the Toomre Q parameter drops below one, and following the shearing sheet gravitational instability simulations of Gammie (2001), we additionally require that the disk cool quickly in order to avoid the feedback loop of gravitoturbulence. The cooling criterion, together with the assumption that the disk is in a steady state--where the accretion rate from the mass reservoir is equal to the accretion rate through the disk--sets a maximal value of =0.23 (Gammie, 2001), where is the Shakura Sunyaev viscosity parameter Making these assumptions, we define a critical sound speed and thus critical temperature at which Q=1; If the disk drops below this critical temperature, it will be prone to fragmentation. To determine the temperature structure of the disk, we consider two primary sources of heating: viscous diffusion at the midplane and irradiation from the central protostar.

Evolution of critical fragmentation radius over the accretion history of a 30 solar mass star in the McKee & Tan turbulent core model. Also shown are the expected disk radii and intersection of the innermost infall streamline with the disk

Matzner & Levin 2005

Patel et al. 2005

The figure above illustrates our model for disk irradiation by the central source. The top right shows dust continuum emission from the Cepheus A HW2 protostar. The bottom right shows position velocity plots for disk source, IRAS 20126+4104.
Cesaroni et al. 2005

The disk is heated at the midplane due to viscous dissipation of energy. The accretion rate, efficiency of angular momentum transport, and opacity determine the temperature:

Using the outflow cavity of model of Matzner & Levin (2005), we approximate the flux received at the disk surface by a geometric factor, f, based on the width of the outflow cavity.

The angular momentum is computed by imposing a line width size scaling and an isotropic, homogeneous, Gaussian velocity field throughout the core.
2

UNSTABLE

We show the predicted critical fragmentation radius from the McKee &Tan turbulent core collapse model as well as expected disk radii from our angular momentum model. Numbered symbols represent observed disks and toroids.

STABLE

F v=

3 M 8

2

=

8 3

T

4 v

T =

4 irr

fL 4R
2

[ vi r 1 - v j r 2 ] = k r 1-r j= f j R

2



2



ij

criteria for fragmenation:

DISCUSSION

Recent observations of regions of massive star formation reveal the presence of both disks and outflows. Although uncertainties remain regarding the exact formation scenario for these stars, our model relies primarily on assumptions which are constrained by observations. Disks appear to play a role in formation, and outflows likely excavate cavities and allow irradiation to reach the disk. Our determination of the typical fragmentation radius depends primarily on the above assumptions, and the typical accretion rates. If massive protostellar disks are typically larger than this fiducial value of ~100 AU, we predict the formation of lowmass stellar companions of spectral types GM within the disk of the parent star. Their final mass depends on precisely how accretion proceeds once fragmentation occurs. Subsequent migration of fragments close to the parent star might permit them to interact in the later stages of stellar evolution.

Various estimates for masses of diskborne fragments. The isolation mass (Goodman & Tan, 2004) represents an upper constraint on growth via accretion. That the gap opening mass (Rafikov, 2002) is smaller than the initial mass scale of fragments suggests that gap opening is rapid, which might inhibit accretion.

Lines of constant critical radius for a given mass and accretion rate. It is clear that for expected formation times, the critical radius is nearly constant at ~100AU. (The highlighted region illustrates formation times within a factor of ~2 of the predictions of McKee & Tan 2003)

ACKNOWLEDGEMENTS: We thank Roman Rafikov, Yanqin Wu, and Ray Jayawardhana for helpful discussions. We would
also like to thank Susana Lizano for insightful suggestions regarding disk observations. C.D.M.'s research is funded by NSERC and the Canada Research Chairs program. K.M.K. is supported by a U. Toronto fellowship.

REFERENCES
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Goodman J., Tan J. C., 2004, ApJ 608, 108 Matzner C. D., Levin Y., 2005, ApJ, 628, 817 McKee C. F., Tan J. C., 2003, Apj, 585 850 Olmi L. et al., 2003 A&A, 407, 255 Patel N. A. et al., 2005, Nature, 437, 109 Rafikov, R. R., 2002 ApJ, 572, 566

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