Thermal and hydrostatic structure of the protoplanetary disks: Influences of wind strengths, mass distributions, and stellar wind velocity laws
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 28 September 2010
Accepted: 18 December 2012
Published: 23 August 2013
The structures of the protoplanetary disk have been examined under various conditions of the stellar wind and the mass distribution of the disk, by assuming that the disk is steady and geometrically thick. T Tauri stars are commonly accompanied by disks as well as the stellar wind. Therefore, the disk around a T Tauri star should be influenced by the stellar wind. The height of the geometrical surface of the disk is suppressed by the dynamical pressure of the wind but depends very weakly on the wind strength. The surface becomes slightly higher when the wind strength becomes weaker. Furthermore, the dependency on the mass distribution of the disk is also weak. As a natural result, the temperature distribution in the disk is insensitive to the wind strength and also the mass distribution of the disk. Thus, we can conclude that the temperature and the geometrical surface height of the disk under the stellar wind does not depend on either the wind properties or the mass distribution of the disk.
It is commonly believed that planets form in a disk that passively receives and reflects radiation from a central star. Therefore, in the study of planetary formation, it is necessary to obtain a good grasp of the physical properties which provide information about the initial conditions of subsequent planetary formation processes. In particular, it is most important to know the temperature distribution over the entire disk, because this governs the distribution of the solid mass in the disk at the beginning of planetary formation.
During the last two or three decades, astronomical observations have revealed a sequence of stellar formation processes, from protostars to pre-main-sequence stars (e.g., Shu et al., 1987). Among young stellar objects, such as the protostars and T Tauri stars, we are particularly interested in T Tauri stars, because their ages almost correspond to the characteristic planetary formation time (e.g., Hayashi et al., 1985). The development of observational studies on disks surrounding young stellar objects has stimulated theoreticians who have investigated various aspects of protoplane-tary disks: emission features from young stellar objects at various stages (e.g., Beckwith et al., 1990; Hartmann and Kenyon, 1990; Calvet et al., 1992; Edwards et al., 1994; Chiang et al., 2001); the dynamical and thermal structure of a protoplanetary disk (Kusaka et al., 1970, which is referred to as K70 hereafter, and Chiang and Goldreich, 1997, which is referred to as CG97 hereafter); the gas temperature and the chemical structure of a protoplanetary disk (e.g., Kamp and Dullemond, 2004; Jonkheid et al., 2004); the thermal instability due to flaring of a passive disk (e.g., D’Alessio et al., 1999; Watanabe and Lin, 2008) and due to self-shadowing (e.g., Dullemond, 2000; Dullemond et al., 2001; Dullemond and Domink, 2004); and the mass distribution in the disk (Hayashi, 1981).
Among these subjects, we put a priority on studies of the temperature structure in the passive disk in which the first physical (and also chemical) processes of planetary formation would begin. The temperature in the disk essentially determines the mass and the chemical composition of solid materials in the disk. If the disk temperature is higher than 170 K (which is the freezing temperature of ice under the typical gas pressure; e.g., Hayashi et al., 1985), only rocky and metallic materials can condense to form solid particles. On the other hand, if the temperature is below the freezing point of ice, solid particles containing a great amount of ice can exist. The mass of solid particles in the region where solid ice exists would be about four times larger than that in the region where ice sublimes (Hayashi et al., 1985). It is very important to specify the place where ice condenses, namely, the position of the snow line in a protoplanetary disk, because the snow line governs the boundary between regions of the terrestrial-type planets and the Jovian-type planets (e.g., Sasselov and Lecar, 2000; Garaud and Lin, 2007). Furthermore, it should also be pointed out that the temperature distribution of a proto-planetary disk governs the thermal and dynamical instability of the disk (e.g., D’Alessio et al., 1999; Dullemond and Domink, 2004; Watanabe and Lin, 2008).
Despite its importance, however, there are few theoretical works, to date, on the temperature distribution of the passive disk. K70 is a pioneering work on the temperature distribution of the disk determined by solar radiation onto its surface. Another temperature model of the passive disk around T Tauri stars was presented by CG97. They took into account an optically thin, slightly high-temperature region on the top of the disk and evaluated the radiation reprocessed by this surface layer to heat the interior disk. Contrary to these highly-elaborate calculations applied to consider the spectral feature of T Tauri stars, there is, unfortunately, an omission in the works mentioned above. Namely, they constructed their thermal models for passive protoplanetary disks without regard to the fact that T Tauri stars would be accompanied by a stellar wind.
Kuhi (1964) recognized the presence of winds from T Tauri stars by analyzing in detail their emission profiles. Nowadays, it is widely accepted that most of the classical T Tauri stars have an extremely strong wind (e.g., Carr, 1989; Strom et al., 1989; Greene and Mayer, 1995; McCaugh-rean and O’Dell, 1996). From extensive observations it has become clear that the strong wind from a T Tauri star is closely related to disk accretion because of the strong correlation between the mass loss rates and the mass accretion rates (Cabrit et al., 1990; Calvet et al., 1992; Hartigan et al., 1995). As a protoplanetary disk evolves from an active phase (from classical T Tauri stars) to a passive phase (to weak-line T Tauri stars), the stellar wind from the central star may change from a strong accretion-driven wind to a weak wind (e.g., Kenyon and Hartmann, 1995; Calvet, 1997; Wood et al., 2002). On the mass loss rates of the stellar wind in the passive phase, some observational evidence shows that these are under 1 × 10−7M⊙ yr−1 (Hartmann et al., 1990; Hartigan et al., 1995; Calvet, 1997), or as weak as 1 × 10−8M⊙ yr−1 (Cabrit et al., 1990; Hartigan et al., 1995; White and Hillenbrand, 2004). It can readily be imagined that if the disk around the central star has a hydrostatic state the stellar wind from the star might blow the stellar wind with mass loss rates nearly equal to, or weaker than, 1 × 10−8M⊙yr−1.
They limited themselves to the case where the wind velocity is constant.
The work has a problem concerning the basic equations (Yun et al., 2010).
As will be seen in the next section, the basic equations contain an important parameter that characterize the radial distribution of the disk mass. So, as a natural extension of Y07, we will make clear the effect of this parameter on the disk temperature. The aim of the present study is to investigate the thermal behavior of the protoplanetary disk exposed to stellar radiation and the stellar wind over a wide range of three principal parameters: the strength of the stellar wind, the parameters describing the mass distribution of the disk, and the velocity law of the stellar wind.
In Section 2, we will describe the assumptions adopted as well as the basic equations. The contents of Section 2 are nearly the same as those of Y70 (note that a basic equation is revised following Yun et al., 2010).
In Section 3, we describe the thermal and the dynamical structure of a disk. First, we will see in detail the effect of the wind strength under the fixed parameters describing the disk mass distribution and the velocity distribution of the stellar wind. Next, we will comment briefly on the similarity of the disk models, taking into account the difference in mass distribution of the disk and of the wind velocity variation. From the descriptions in Section 3, we will see that, surprisingly, the disk models are very similar to each other irrespective of the wide variety in the considered parameters. Finally, conclusions and remarks are presented in Section 4.
2. Assumptions and Basic Equations
We consider a protoplanetary disk in the passive phase, exposed to the wind and radiation from a central T Tauri star. As a preparation for obtaining the geometrical figure and the temperature distribution of the disk, we will describe briefly the assumptions made and the basic equations, since the contents are very similar to those of Y07. In this study, the mass distribution of the disk and the velocity law of the stellar wind are somewhat extended compared with those of Y07, and the dynamical balance equation is revised because the equation used in Y07 is incorrect (see Yun etal., 2010).
2.1 Adopted assumptions
2.2 Basic equations
2.3 Non-dimensional forms of the basic equations
As mentioned above, one of the basic equations of our previous study Y07 is revised here. Nevertheless, the topological behavior of the solutions to Eq. (18) is very similar to that in Y07 as long as we are concerned with an inner disk of less than 50 AU. Thus, we can obtain a suitable disk structure by the same method as in Y07.
In this section, we present the results which are obtained by solving numerically the set of basic equations described in Section 2 over a wide range of wind strength, as well as the radial mass distribution parameter γ (see Eq. (1)). We also mention briefly the results for the case of the wind model with a radially-dependent wind velocity. For the central star, we choose T* = 4000 K, R* = 2R⊙, and M* = 1M⊙. As for the inner and the outer boundaries, we take these to be 0.03 AU ( η = 3) and 50 AU ( η = 5 × 103).
3.1 Disk models with different wind strengths
In Fig. 1, we illustrate the flaring index, f, as a function of η for the five cases of the wind parameters t*/ β between 1 × 108 yr and 1 × 1012 yr. As seen in Y07, the profile of the flaring index f changes for each case of the adopted value of t*/ β. Although, we can say that it is very small in all cases of the wind parameter at any distance from the central star. Furthermore, it becomes small with an increase in wind strength because the stronger wind suppresses the surface of the disk more strongly by the dynamic pressure of the wind.
The disk temperatures and their logarithmic derivatives (i.e., −d ln T/d ln r) at r = 0.1 AU and 1 AU for the wind parameters from 1 × 108 yrto 1 × 1012 yr.
t* / β [yr]
1 × 108
1 × 109
1 × 1010
1 × 1011
1 × 1012
The logarithmic derivatives of the temperature, p, defined by −d ln T/d ln r, are shown in Fig. 2(b) for the wind parameters, and the values of p at the distances 0.1 AU and 1 AU are tabulated in Table 1. From the results under the five conditions of the wind, we can confirm that the index p becomes larger for the stronger wind case in most regions of the disk.
3.2 Different mass distributions in the disk
The height of the disk surface, zs, is shown in Fig. 5(b) for four cases of γ. zs increases with an increase in γ in the inner region, whereas it decreases with γ in the outer region. But we can safely say that the height behaves similarly for all the case of γ taken here. Note that the height of the disk surface is determined by a balance between the dynamical pressure of the wind and the gas pressure in the disk which is proportional to the local matter density. As mentioned earlier, the surface density in the outer region (r ≥ 1 AU) behaves in an inverse way to that in the inner region (r ≤ 1 AU) when the power index γ changes. As a result, in the inner region the gas pressure increases and, hence, the height zs becomes higher when γ is larger. In the outer region the surface density and the gas pressure decrease. So the height of the disk becomes lower with an increment in the power index γ.
As seen from Fig. 5(a), the flaring index, f, does not change in the inner region (r ≤ 1 AU) but in the outer region, where r ≥ 1 AU, there is a notable discrepancy in f for the case of γ = −2. As mentioned in the previous section, f is approximately proportional to zs/r in the outer region where the first term of the right hand side of Eq. (16) becomes important (see Fig. 1). Thus, we can say that the decline in f is caused by that in zs.
Anyway, the dependency on γ of the flaring index f, and the height zs of the disk, is very weak. Furthermore, the temperature does not depend on the radial mass distribution of the disk. This is also true for the other cases of t*/ β, as long as we are concerned with cases where t*/ β is greater than 1 × 108 yr.
3.3 Influence of the adopted stellar wind model
In Figs. 6(b) and 6(c), we illustrate the temperature and the height of the disk, respectively, which are calculated numerically by the use of Parker’s solution. For comparison, we also give the results based on the constant velocity wind model with vc as shown by dashed curves. Obviously the two models give almost the same results (as presented by Y07), though the height of the disk is very slightly suppressed in the outer region (r ≥ 30 AU), where (vp/vc)2 is larger than 2. Anyway, we can say that the results described in Subsection 3.1 are not changed even under the stellar wind expressed by Parker’s solution, since we can also confirm that we have obtained similar results for the other cases of the mass loss rate.
In any case, we can surely say that the difference in the wind velocity law does not appreciably affect the structure of the disk.
4. Conclusions and Discussion
The temperature and the height of the disk surface are very insensitive to the adopted parameters describing the wind model (i.e., the wind strength and the velocity law of the wind) and the mass distribution model of the disk. In particular, in the inner region (r ≤ 0.1 AU) they are determined almost uniquely.
The flaring index of the disk, f, is suppressed by the dynamical pressure of the stellar wind and, as a result, is limited to a low level smaller than 0.1. Furthermore, f does not depend very much in general on the adopted wind model or on the mass distribution model of the disk. For example, even if the wind strength changes over four orders of magnitude, f remains in a narrow variance of factor 2.
As long as we are concerned with the wind coexisting with the passive disk, both the temperature and the height of the disk depend only very weakly on the adopted parameters and are written approximately in the form of a power-law function of r. We obtain the temperature and height of the disk given by Eq. (25) and (26), respectively. To our regret, they are not expressed by a simple power-law function in the region 0.1 AU < r < 1 AU.
Finally, we will add a few other remarks. As already pointed out in Y07, in addition to the direct radiative flux from the central star, the disk would receive radiative energy emitted from the surrounding diffuse gas, as well as heating energy due to cosmic rays. In the outer region (r ≥ 30 AU) of the disk, where the temperature is very low, such an additional energy flux cannot be ignored. Of course, the degree of the temperature increment in the outer region depends entirely on the physical configuration around the protoplanetary disk. In this sense, the temperature of the disk obtained in this study should be regarded as a minimum temperature in the outer region.
The strength of the stellar wind is assumed to be a constant temporarily in the present study. As mentioned in the first section, however, the stellar wind must change its strength with time. In Section 3, we found that the height of the disk changes very slightly with the adopted wind parameter t*, as in the case of the temperature not being affected appreciably by t*. Regarding the height of the surface, we can propose another kind of mechanism that is potentially important. Besides the shear region between the stellar wind and the disk, it is possible to generate a kind of instability, such as the Kelvin-Helmholtz instability. The wave produced on the contact surface will change the height of the surface and, as a result, change the heating efficiencies of the stellar radiation on the disk, too. It will be important to study the stability of the disk when the stellar wind changes its strength with time. This is a matter for further studies.
Authors have appreciated fruitful discussions with members in the Department of Earth and Planetary Sciences, Tokyo Institute of Technology. An advise given by Reviewer A helped authors to reconsider and reconfirm their conclusions. Authors have greatly appreciated to him or her, too.
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