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The Spectral Energy Distribution of Self-Gravitating Protostellar Disks

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The Spectral Energy Distribution of Self-Gravitating Protostellar Disks A&A 375, 455–468 (2001) Astronomy DOI: 10.1051/0004-6361:20010868 & c ESO 2001 Astrophysics The spectral energy distribution of self-gravitating protostellar disks G. Lodato1 and G. Bertin1,2 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy 2 Universit`a degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, 20133 Milano, Italy Received 12 February 2001 / Accepted 12 June 2001 Abstract. The long wavelength emission of protostellar objects is commonly attributed to a disk of gas and dust around the central protostar. In the first stages of disk accretion or in the case of high mass protostars, the disk mass is likely to be sufficiently large, so that the disk self-gravity may have an impact on the dynamics and the emission properties of the disk. In this paper we describe the spectral energy distribution (SED) produced by a simple, non-flaring, self-gravitating accretion disk model. Self-gravity is included in the calculation of the rotation curve of the disk and in the energy balance equation, as a term of effective heating related to Jeans instability. In order to quantify in detail the requirements on the mass of the disk and on the accretion rate posed on the models by realistic situations, we compare the SEDs produced by these models with the observed SEDs of a small sample of well-studied protostellar objects. We find that relatively modest disks – even lighter than the central star – can lead to an interesting fit to the infrared SED of the FU Orionis objects considered, while in the case of T Tauri stars the required parameters fall outside the range suggested as acceptable by the general theoretical and observational scenario. On the basis of the present results, we may conclude that the contribution of a self- gravitating disk is plausible in several cases (in particular, for FU Orionis objects) and that, in the standard irradiation, dominated disk scenario, it would help soften the requirements encountered by Keplerian accretion models. Key words. accretion, accretion disks – gravitation – stars: pre-main sequence 1. Introduction “passive”. In contrast, during FU Orionis outbursts, the accretion rate appears to be relatively high, so that here It now seems well established that pre-main-sequence the disk can dissipate sufficient amounts of energy to pro- stars, such as T Tauri and FU Orionis objects, are gen- duce the observed luminosity. In the construction of dy- erally surrounded by a disk of gas and dust. The tradi- namical models, a related issue is that of the total disk tional evidence for such circumstellar disks comes from the mass. Clues have indeed been gathered that the disk mass excess luminosity found at long wavelengths, far-infrared may be large in some protostellar systems (see Lay et al. (Rydgren et al. 1976) and millimetric (Beckwith et al. 1994), especially in the earlier phases of the process of 1990), indicating that this radiation comes from a region star formation (Yorke & Bodenheimer 1999, for example, of relatively low temperature, far from the central star. conclude from hydrodynamical simulations that the disk The first attempt to try explain such excess infrared mass at the beginning of the accretion phase is compa- luminosity dates back to the 70s (Lynden-Bell & Pringle rable to that of the central object) and in the context of 1974), when it was argued that it might be due to the high-mass star formation, where observations point to the release of gravitational binding energy from an accretion presence of massive disks (Cesaroni et al. 1994). disk. In recent years, the general picture that has emerged is that, in most T Tauri systems, the mass accretion rate is Standard accretion disk theory (Shakura & Sunyaev −8 −7 too low (M˙ ∼ 10 −10 M /yr) to account for the ob- 1973) is based on the assumption that the disk is optically served luminosity and that the observed long wavelength thick and geometrically thin and that the power viscously 0 2 excess is likely to result from the reprocessing of the star- dissipated in the disk D = νασ(rΩ ) /2 is all radiated light from the outer disk (Hartmann 1998). Therefore, the away. In the optically thick case, it can be shown that the accretion disk of T Tauri stars is generally thought to be spectral index n =dLog(νFν )/dLog ν in the infrared part of the spectrum is related to the exponent q in the surface q Send offprint requests to: G. Lodato, temperature profile Ts = T0(r0/r) ,sothatn ' 4 − 2/q e-mail: [email protected] (Adams et al. 1988). For a Keplerian rotation curve one Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20010868 456 G. Lodato and G. Bertin: The SED of protostellar disks −3/4 4/3 has Ts(r) ∝ r , leading to Lν = νFν ∝ ν , while for rotation occur even for disk masses Mdisk . M?,which 2/3 T Tauri stars one typically finds Lν ∝ ν (Kenyon & could be a plausible range even for some T Tauri disks. Hartmann 1995). The difficulty becomes even more evi- Based on these models of self-regulated accretion, it is dent for the so-called “flat-spectrum T Tauri” (such as T now possible to quantify in detail what would be the pa- Tau itself and HL Tau), which show a spectral index n ≈ 0. rameter requirements needed to fit the infrared spectra of In turn, if the rotation curve is flat, the surface tempera- protostellar objects. In this respect, our results will turn −1/2 ture profile decreases more slowly, as Ts(r) ∝ r .Inthis out to be quite different (see discussion in Sect. 3.5) from latter case, the spectral index would be n = 0, resembling earlier estimates (e.g., see Kenyon & Hartmann 1987; Shu that of flat-spectrum objects. Adams et al. (1988) indeed et al. 1987) that did not include the important ingredient proposed that the flat spectrum could arise from a disk of self-regulation, which is likely to change significantly with such a non-standard surface temperature profile, but the energy balance equations in the self-gravitating part argued that the observed disk mass in these systems (al- of the disk (see also discussion in Sect. 2.2). though difficult to measure) is probably too low to produce In this paper we describe the spectral energy distribu- substantial changes in the rotation curve and then sug- tion produced by a class of self-regulated, self-gravitating gested that the non-standard temperature profile should accretion disks and discuss its dependence on the various be ascribed to collective processes of angular momentum parameters involved and the differences from the non-self- and energy transport associated with self-gravity-induced gravitating case. In addition, we determine the physical instabilities. parameters needed to fit the SEDs of realistic protostel- In the absence of a convincing physical justification lar objects, by referring to a small sample of well-studied for non-standard descriptions of accretion disks, the at- FU Orionis systems and T Tauri stars. A general theory tention has been turned to other ways of explaining the of FU Orionis and T Tauri systems, or a comprehensive far-infrared spectra of T Tauri stars. As noted above, it is picture of the individual objects picked here to test the now commonly thought that in most of these systems the pure self-gravitating model against observed SEDs, is be- accretion rate is low and that the heating of the disk is pri- yond the goals of this paper. The models considered are marily due to irradiation from the central star. The effect extremely simple and do not address the issue of the phys- of irradiation is enhanced at large radii by the disk flaring, ical ingredients that determine the long-term evolution of when the thickness of the disk increases rapidly with ra- the system (in particular, the physical ingredients respon- dius. Models of the vertical structure of these flared disks sible for the outburst phase of the FU Orionis systems): have been constructed (Kenyon & Hartmann 1987; Chiang they are taken to match current “snapshots” of such evolv- & Goldreich 1997; but concerns about stability have been ing systems, assuming that the steady-state equations are expressed by Dullemond 2000). On the other hand, in the temporarily adequate to describe the observed situation. case of flat spectrum sources and of most FU Orionis ob- The quality of the fit to the available data for the long jects, flaring alone is ineffective and the far-infrared emis- wavelength spectral energy distribution is very good. In sion has been attributed to an infalling envelope of dust the case of FU Orionis systems the required accretion rate −5 −4 (Kenyon & Hartmann 1991; Calvet et al. 1994). is of the order of M˙ ≈ 10 −10 M /yr and the im- The hypothesis that the flat long wavelength spec- plied disk mass is of the order of one solar mass. The trum of protostellar objects arises from a self-gravitating implied mass accretion rate for T Tauri stars is found to −7 −6 disk with flat rotation curve actually remains attractive be M˙ ≈ 10 −10 M /yr. The choice of the outer ra- from a physical point of view, especially for its simplicity. dius of the disk determines the mass of the disk relative Unfortunately, only scattered efforts have been made in to that of the central star. The estimates, although uncer- the direction of constructing self-gravitating models of ac- tain, point to rather massive disks (Mdisk ≈ 0.3 M ).
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