Effect of Dust Evaporation and Thermal Instability on Temperature Distribution in a Protoplanetary Disk

Abstract

The thermal instability of accretion disks is widely used to explain the activity of cataclysmic variables, but its manifestation in gas and dust disks in young stars has been studied in less detail. A semi-analytical stationary model is presented for calculating the equatorial temperature of a gas and dust disk around a young star. The model considers the opacity caused by dust and gas, as well as the evaporation of dust at temperatures above 1000 K. Using this model, the distributions of the equatorial temperature of the gas and dust disk are calculated under various assumptions on the source of opacity and the presence of dust. It is shown that when all the above processes are considered, the thermal balance equation in the region $r < 1$ AU has multiple temperature solutions. Thus, the conditions for thermal instability are satisfied in this region. As an illustration of the possible influence of instability on the nature of accretion in a protoplanetary disk, we consider a viscous disk model with $\alpha$-parametrization of turbulent viscosity. It is shown that in such a model a non-stationary mode of disk evolution is realized with alternating phases of accumulation of matter in the inner disk and phases of its rapid accretion onto the star, which leads to a burst character of accretion. The results obtained indicate the need to take this instability into account when modeling the evolution of protoplanetary disks.

Appendix A

1.1 6. DERIVATION OF THE FORMULA FOR THE EQUATORIAL TEMPERATURE OF THE CIRCUMSTAR DISC

Consider a circumstellar disk in a state of thermal equilibrium. In the plane-parallel approximation, the thermal structure of such a disk in the vertical direction can be described by a system of moment transfer equations for thermal radiation:

$$\frac{{dF}}{{dz}} = c\rho {{\kappa }_{{\text{P}}}}(B - E),$$

(A.1)

$$\frac{c}{3}{\kern 1pt} \frac{{dE}}{{dz}} = - \rho {{\kappa }_{{\text{R}}}}F,$$

(A.2)

where $F$ is the thermal radiation flux, $E$ is the radiant energy density, $B = a{\kern 1pt} {{T}^{4}}$ is the radiant energy density at thermodynamic equilibrium, $a$ is the radiation density constant, $T$ is the medium temperature, $c$ is the speed of light, $z$ is the vertical coordinate measured from the equator, $\rho$ is the medium density, ${{\kappa }_{{\text{P}}}}$ and ${{\kappa }_{{\text{R}}}}$ are the opacity coefficients, averaged according to Planck and Rosseland. Equation (A.1) describes the change in flux due to the difference between the emission and absorption of radiant energy. Equation (A.2) relates the radiation flux to the energy density in the Eddington approximation. The system of equations (А.1), (А.2) is closed by the equation

$$\frac{{dF}}{{dz}} = \rho S,$$

(A.3)

according to which the thermal radiation flux is generated by some heating source $\rho S$, where $S$ [erg s^–1 g^–1] is defined as the heating power per unit mass of matter. Let’s rewrite these equations using the surface density $\Sigma = \int_0^z {\rho (z{\kern 1pt} ')dz{\kern 1pt} '}$ as a variable:

$$c{{\kappa }_{{\text{P}}}}(B - E) = S,$$

(A.4)

$$\frac{c}{3}{\kern 1pt} \frac{{dE}}{{d\Sigma }} = - {{\kappa }_{{\text{R}}}}F,$$

(A.5)

$$\frac{{dF}}{{d\Sigma }} = S.$$

(A.6)

We will assume that the heating power $S$ is due to two processes: absorption of stellar radiation and viscous dissipation,

$$S = {{S}_{{{\text{star}}}}} + {{S}_{{{\text{vis}}}}}.$$

(A.7)

The heating power due to the viscous dissipation of the gas in the stationarity approximation can be found by the formula:

$${{S}_{{{\text{vis}}}}} = \frac{{{{\Gamma }_{{{\text{vis}}}}}}}{{{{\Sigma }_{0}}}} = \frac{3}{{8\pi }}{\kern 1pt} \frac{{GM\dot {M}}}{{{{R}^{3}}}}{\text{/}}{{\Sigma }_{0}},$$

(A.8)

where $M$ is the mass of the star, $\dot {M}$ is the rate of accretion of matter through the disk, $R$ is the distance from the star to the element of the disk under consideration, ${{\Sigma }_{0}}$ is the surface density from the equator to the upper boundary of the disk, and $G$ is the gravitational constant. The use of (A.8) is also based on the assumption that the rate of viscous dissipation per unit volume is proportional to the density of the medium. The heating of the disk by stellar radiation is found by us using the formula:

$${{S}_{{{\text{star}}}}} = {{\kappa }_{{\text{F}}}}{{F}_{0}}{\kern 1pt} \exp \left( { - {\kern 1pt} \frac{{{{\kappa }_{{\text{F}}}}({{\Sigma }_{0}} - \Sigma )}}{\mu }} \right){\kern 1pt} ,$$

(A.9)

where ${{\kappa }_{{\text{F}}}}$ is the absorption coefficient averaged over the spectrum of the star, ${{F}_{0}} = \frac{L}{{4\pi {{R}^{2}}}}$ is the radiation flux from the star reaching the disk surface, $L$ is the luminosity of the star, and $\mu$ is the cosine of the angle between the direction to the star and the normal to the disk surface. Formula (A.9) is derived from the formal solution of the radiative transfer equation under the assumption that the absorption coefficient ${{\kappa }_{{\text{F}}}}$ is constant along the vertical direction. In this case, we also neglect stellar radiation from the opposite surface of the disk. Accounting for disk heating by stellar radiation by introducing the function ${{S}_{{{\text{star}}}}}$ into system (А.4)–(А.6) is based on the assumption that the disk radiates weakly in the visible range, i.e., this range weakly intersects with the range of thermal radiation of the disk itself. We introduce the notation

$${{\tau }_{{{\text{uv}}}}} = \frac{{{{\kappa }_{{\text{F}}}}{{\Sigma }_{0}}}}{\mu }{\kern 1pt} ,$$

(A.10)

which is the optical depth of the medium to stellar radiation up to the current position in the disk. Integration of equation (A.6), taking into account expressions (A.7)–(A.9) and the condition of equality of the thermal radiation flux at the equator (due to the symmetry of the problem), gives

$$F = \mu {\kern 1pt} {{F}_{0}}{\kern 1pt} {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}(\exp \left( {\frac{{{{\tau }_{{{\text{uv}}}}}\Sigma }}{{{{\Sigma }_{0}}}}} \right) - 1) + {{S}_{{{\text{vis}}}}}\Sigma {\kern 1pt} .$$

(A.11)

In particular, on the disk surface, the thermal radiation flux is equal to

$$F({{\Sigma }_{0}}) = \mu {{F}_{0}}{\kern 1pt} \left( {1 - {{e}^{{{{\tau }_{{{\text{uv}}}}}}}}} \right) + {{S}_{{{\text{vis}}}}}{{\Sigma }_{0}}{\kern 1pt} .$$

(A.12)

Substituting equation (A.11) into equation (A.5) and integrating the resulting equation from the equator to the upper boundary of the disk, one can obtain the relationship between the radiant energy density at the surface $E({{\Sigma }_{0}})$ and at the disk equator $E(0)$:

$$\begin{gathered} E({{\Sigma }_{0}}) - E(0) \\ = - \frac{{3{{\kappa }_{{\text{R}}}}{{\mu }^{2}}{{F}_{0}}}}{{c{{\kappa }_{{\text{F}}}}}}\left( {1 - {{\tau }_{{{\text{uv}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}} - {{\tau }_{{{\text{uv}}}}}} \right) - \frac{{3{{\kappa }_{{\text{R}}}}{{S}_{{{\text{vis}}}}}}}{{2c}}{\kern 1pt} \Sigma _{0}^{2}{\kern 1pt} . \\ \end{gathered} $$

(A.13)

When obtaining relation (A.13), it was assumed that ${{\kappa }_{{\text{R}}}}$ is constant along the vertical direction. As a boundary condition on the disk surface, one can use the relation

$$F({{\Sigma }_{0}}) = \eta {\kern 1pt} cE({{\Sigma }_{0}}),$$

(A.14)

where the coefficient $\eta$ depends on the assumed anisotropy of the outgoing thermal radiation. The value $\eta = \frac{1}{2}$ corresponds to isotropy over the upper hemisphere, while $\eta = 1$ describes the case of a strictly vertical radiation output. In what follows, we will assume that $\eta = \frac{1}{2}$. Combining equations (A.12), (A.13), and (A.14), one can obtain an expression for the radiant energy of thermal radiation at the equator:

$$\begin{gathered} E(0) = \frac{{{{F}_{0}}}}{c} \\ \times \;\left[ {2\mu \left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right) + 3{{\mu }^{2}}\frac{{{{\kappa }_{{\text{R}}}}}}{{{{\kappa }_{{\text{F}}}}}}\left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}} - {{\tau }_{{{\text{uv}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right)} \right] \\ + \;\frac{{2{{S}_{{{\text{vis}}}}}{{\Sigma }_{0}}}}{c}\left( {1 + \frac{3}{4}{{\tau }_{{\text{R}}}}} \right){\kern 1pt} , \\ \end{gathered} $$

(A.15)

where the Rosseland optical thickness to thermal radiation is introduced:

$${{\tau }_{{\text{R}}}} = {{\kappa }_{{\text{R}}}}{{\Sigma }_{0}}{\kern 1pt} .$$

(A.16)

The desired equatorial temperature ${{T}_{{{\text{mid}}}}}$ is found from the relation

$$B(0) = a{\kern 1pt} T_{{{\text{mid}}}}^{4}{\kern 1pt} .$$

(A.17)

The value $B(0)$, in turn, is expressed through equation (A.4), which, taking into account the value of the source function at the equator (A.7)–(A.9), takes the form

$$B(0) = E(0) + \frac{{{{\kappa }_{{\text{F}}}}{{F}_{0}}{\kern 1pt} {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}}}{{c{{\kappa }_{{\text{P}}}}}} + \frac{{{{S}_{{{\text{vis}}}}}}}{{c{{\kappa }_{{\text{P}}}}}}{\kern 1pt} .$$

(A.18)

Combining equations (A.15), (A.17), and (A.18), one obtains:

$$\begin{gathered} a{\kern 1pt} T_{{{\text{mid}}}}^{4} = \frac{{\mu {{F}_{0}}}}{c}\left[ {2\left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right)\mathop {}\limits_{}^{} } \right. \\ + \;\left. {3\mu \frac{{{{\tau }_{{\text{R}}}}}}{{{{\tau }_{{{\text{uv}}}}}}}\left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}} - {{\tau }_{{{\text{uv}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right) + \frac{{{{\tau }_{{{\text{uv}}}}}}}{{{{\tau }_{{\text{P}}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right] \\ + \;\frac{{{{\Gamma }_{{{\text{vis}}}}}}}{c}\frac{1}{{{{\tau }_{{\text{P}}}}}}\left[ {1 + 2{{\tau }_{{\text{P}}}}\left( {1 + \frac{3}{4}{{\tau }_{{\text{R}}}}} \right)} \right], \\ \end{gathered} $$

(A.19)

where the Planck optical depth with respect to thermal radiation is introduced

$${{\tau }_{{\text{P}}}} = {{\kappa }_{{\text{P}}}}{{\Sigma }_{0}}{\kern 1pt} .$$

(А.20)

Equation (A.19) can also be usefully presented in the following form:

$${{\Lambda }_{{{\text{IR}}}}} = {{\Gamma }_{{{\text{star}}}}} + {{\Gamma }_{{{\text{vis}}}}}{\kern 1pt} ,$$

(А.21)

where ${{\Lambda }_{{{\text{IR}}}}}$ [erg s^–1 cm^–2] is the rate of cooling of the equatorial layers of the disk, ${{\Gamma }_{{{\text{star}}}}}$ [erg s^–1 cm^–2] is the rate of heating of the equatorial layers by stellar radiation, ${{\Gamma }_{{{\text{vis}}}}}$ [erg s^–1 cm^–2] is the rate of heating due to viscous dissipation:

$${{\Lambda }_{{{\text{IR}}}}} = \frac{{4{{\tau }_{{\text{P}}}}\sigma T_{{{\text{mid}}}}^{4}}}{{1 + 2{{\tau }_{{\text{P}}}}\left( {1 + \frac{3}{4}{{\tau }_{{\text{R}}}}} \right)}}{\kern 1pt} ,$$

(А.22)

$${{\Gamma }_{{{\text{star}}}}} = \frac{{\mu {{F}_{0}}{{\tau }_{{\text{P}}}}\left[ {2\left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right) + 3\mu \frac{{{{\tau }_{{\text{R}}}}}}{{{{\tau }_{{{\text{uv}}}}}}}\left( {1 - {{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}} - {{\tau }_{{{\text{uv}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right) + \frac{{{{\tau }_{{{\text{uv}}}}}}}{{{{\tau }_{{\text{P}}}}}}{{e}^{{ - {{\tau }_{{{\text{uv}}}}}}}}} \right]}}{{1 + 2{{\tau }_{{\text{P}}}}\left( {1 + \frac{3}{4}{{\tau }_{{\text{R}}}}} \right)}}{\kern 1pt} ,$$

(А.23)

$${{\Gamma }_{{{\text{vis}}}}} = \frac{3}{{8\pi }}{\kern 1pt} \frac{{GM\dot {M}}}{{{{R}^{3}}}}{\kern 1pt} .$$

(А.24)

Let us analyze the behavior (A.19) in the absence of viscous heating ${{S}_{{{\text{vis}}}}} = 0$. For small optical thicknesses with respect to stellar radiation ${{\tau }_{{{\text{uv}}}}} \ll 1$, we obtain

$$a{\kern 1pt} T_{{{\text{mid}}}}^{4} = \frac{{{{\kappa }_{{\text{F}}}}}}{{{{\kappa }_{{\text{P}}}}}}{\kern 1pt} \frac{{{{F}_{0}}}}{c}{\kern 1pt} ,$$

(А.25)

i.e., the temperature of the medium is determined by the ratio of the opacity of the medium to the stellar and thermal radiation. If the disk is optically thick to stellar radiation ${{\tau }_{{{\text{uv}}}}} \gg 1$ and the ratio ${{\kappa }_{{\text{R}}}}{\text{/}}{{\kappa }_{{\text{F}}}}$ is small (which is usually the case), then

$$a{\kern 1pt} T_{{{\text{mid}}}}^{4} = \frac{{2\mu {{F}_{0}}}}{c}{\kern 1pt} ,$$

(А.26)

i.e., equatorial temperature depends only on the total flux of stellar radiation entering the disk. If the disk is optically thick to stellar radiation and viscous heating is present ${{S}_{{{\text{vis}}}}} \ne 0$, then

$$a{\kern 1pt} T_{{{\text{mid}}}}^{4} = \frac{{2\mu {{F}_{0}}}}{c} + \frac{{{{\Gamma }_{{{\text{vis}}}}}}}{c}{\kern 1pt} \frac{1}{{{{\tau }_{{\text{P}}}}}}{\kern 1pt} \left[ {1 + 2{{\tau }_{{\text{P}}}}\left( {1 + \frac{3}{4}{{\tau }_{{\text{R}}}}} \right)} \right]{\kern 1pt} .$$

(А.27)

As an example, Fig. 5 shows the equatorial temperature distributions obtained using formula (A.19) as a function of the surface density of the disk. The following parameters were used to construct these distributions: $M = 1{\kern 1pt} {{M}_{ \odot }}$, $L = 1{\kern 1pt} {{L}_{ \odot }}$, $R = 1$ a.u., $\mu = 0.05$, ${{\kappa }_{{\text{P}}}} = {{\kappa }_{{\text{R}}}} = 1$ cm²/g, ${{\kappa }_{{\text{F}}}} = 100$ cm²/g. The constructed dependencies illustrate the limits obtained in (A.25) and (A.26) and also show the effect of viscous heating.

Fig. 5.

Dependences of the equatorial temperature on the surface density of the disk for various accretion rates $\dot {M} = 0$, ${{10}^{{ - 8}}}$ and ${{10}^{{ - 6}}} {{M}_{ \odot }}$/year.