Angular momentum transport within young massive protoplanetary discs may be dominated by self-gravity at radii where the disk is too weakly ionized to allow the development of the magneto-rotational instability. We use time-dependent one-dimensional disc models, based on a local cooling time calculation of the efficiency of transport, to study the radial structure and stability (against fragmentation) of protoplanetary discs in which self-gravity is the sole transport mechanism. We find that self-gravitating discs rapidly attain a quasi-steady state in which the surface density in the inner disc is high and the strength of turbulence very low (alpha ~ 10^{-3} or less inside 5 au). Temperatures high enough to form crystalline silicates may extend out to several au at early times within these discs. None of our discs spontaneously develop regions that would be unambiguously unstable to fragmentation into substellar objects, though the outer regions (beyond 20 au) of the most massive discs are close enough to the threshold that fragmentation cannot be ruled out. We discuss how the mass accretion rates through such discs may vary with disc mass and with mass of the central star, and note that a determination of the \dot{M}-M_* relation for very young systems may allow a test of the model.
Deep Dive into Time-dependent models of the structure and evolution of self-gravitating protoplanetary discs.
Angular momentum transport within young massive protoplanetary discs may be dominated by self-gravity at radii where the disk is too weakly ionized to allow the development of the magneto-rotational instability. We use time-dependent one-dimensional disc models, based on a local cooling time calculation of the efficiency of transport, to study the radial structure and stability (against fragmentation) of protoplanetary discs in which self-gravity is the sole transport mechanism. We find that self-gravitating discs rapidly attain a quasi-steady state in which the surface density in the inner disc is high and the strength of turbulence very low (alpha ~ 10^{-3} or less inside 5 au). Temperatures high enough to form crystalline silicates may extend out to several au at early times within these discs. None of our discs spontaneously develop regions that would be unambiguously unstable to fragmentation into substellar objects, though the outer regions (beyond 20 au) of the most massive disc
Low mass stars form from the collapse of cold, dense molecular cloud cores (Terebey, Shu & Cassen 1984). Although the rotation rates of such cores are generally quite small (Caselli et al. 2002) they nonetheless contain amounts of angular momentum far in excess of the rotational angular momentum of a single star. Most of the mass must therefore pass through a protostellar disc, and the answers to many open problems in star and planet formation hinge on the nature of the angular momentum transport that is needed for disc accretion.
In most astrophysical discs the fundamental question of what mechanism dominates angular momentum transport is widely considered to have been solved -MHD turbulence initiated by the magneto-rotational instability (MRI) can provide the necessary viscosity (Balbus & Hawley 1991;Papaloizou & Nelson 2003). Protoplanetary disks, however, ⋆ E-mail: wkmr@roe.ac.uk † Scottish Universities Physics Alliance are so cold and dense that thermal processes probably fail to yield even the very small degree of ionization needed to sustain MHD turbulence (Blaes & Balbus 1994). Under these conditions disc self-gravity may provide an alternate and possibly dominant mechanism for transporting angular momentum through the growth of the gravitational instability (Toomre 1964;Lin & Pringle 1987;Laughlin & Bodenheimer 1994).
Study of the development of gravitational instability in protostellar discs has often been motivated largely by the possibility that the instability will lead to fragmentation of the disc and the formation of gas giant planets (Boss 1998). The conditions for fragmentation are, however, quite difficult to achieve, especially in the inner, planet-forming regions (Matzner & Levin 2005;Rafikov 2005; Boley et al. 2006;Whitworth & Stamatellos 2006;Stamatellos & Whitworth 2008;Forgan et al. 2009). What seems more likely is that discs will evolve towards quasisteady states in which the instability acts to transport angular momentum outwards (Gammie 2001;Rice et al. 2003;Lodato & Rice 2004;Vorobyov & Basu 2007). This is of in-terest in its own right, as it implies that the conditions within young protoplanetary discs -at precisely the epoch when planetesimals and perhaps larger bodies are forming -may largely be set by the physics of angular momentum transport via gravitational instability. The recent progress in understanding the conditions for fragmentation has yielded a much clearer understanding of this physics, which we use here to construct realistic time-dependent models for the structure of self-gravitating protoplanetary discs. Our models rely on two specific properties of transport via gravitational instability. First, that transport can be approximated as a local viscous process for all except the most massive discs (Lodato & Rice 2004, 2005). Second, under conditions of thermal equilibrium the strength of angular momentum transport is set by the cooling rate of the disc (Gammie 2001;Rice et al. 2003). Our results show that self-gravitating discs will settle into quasi-steady states that appear independent of the initial conditions, but that their properties are quite unlike those that result from angular momentum transport via generic turbulent processes. Specifically, we find that the surface density profile is reasonably steep and in the cases considered here, ∼ 80 % of the mass within 50 au is located inside 10 -20 au. The quasi-steady mass accretion rate depends strongly on the disc mass and on the mass of the central star. For a constant star to disc mass ratio, however, the relationship between mass accretion rate and central star mass is similar to that found observationally.We further show that the secular evolution of the disc does not typically result in any regions that would be unstable to fragmentation, with the region inside 10 -20 au being particularly stable unless some mechanism -such as convection (Boss 2004) -can significantly increase the cooling rate.
Our focus in this paper is on the outer cool regions of protoplanetary discs, and for this reason and for simplicity we consider disk models in which self-gravity provides the sole source of angular momentum transport. Of course both the extreme inner region (inside 0.1 au) and the upper layers of the disc further out could become sufficiently ionised for the MRI to operate (Gammie 1996), and this would modify our results and admit new physical effects. In particular, a pile-up of material brought to within 1 -2 au by self-gravity could eventually trigger the onset of the MRI and episodic outbursts (Armitage, Livio & Pringle 2001;Zhu, Hartmann & Gammie 2009). These might be related to the FU Orionis phenomenon (Hartmann & Kenyon 1996). Ultimately the evolution of discs around very young stars could be driven by both MRI and the gravitational instability (Terquem 2008).
The paper is organised as follows. In Section 2 we describe how we can self-consistently model protostellar discs evolving through self-gravity
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