Simulations of Dynamical Gas–Dust Circumstellar Disks: Going Beyond the Epstein Regime

Abstract—In circumstellar disks, the size of dust particles varies from submicron to several centimeters, while planetesimals have sizes of hundreds of kilometers. Therefore, various regimes for the aerodynamic drag between solid bodies and gas can be realized in these disks, depending on the grain sizes and velocities: Epstein, Stokes, and Newton, as well as transitional regimes between them. This means that simulations of the dynamics of gas–dust disks require the use of a drag coefficient that is applicable for a wide range for sizes and velocities for the bodies. Furthermore, the need to compute the dynamics of bodies of different sizes in the same way imposes high demands on the numerical method used to find the solution. For example, in the Epstein and Stokes regimes, the force of friction depends linearly on the relative velocity between the gas and bodies, while this dependence is non-linear in the transitional and Newton regimes. On the other hand, for small bodies moving in the Epstein regime, the time required to establish the constant relative velocity between the gas and bodies can be much less than the dynamical time scale for the problem—the time for the rotation of the disk about the central body. In addition, the dust may be concentrated in individual regions of the disk, making it necessary to take into account the transfer of momentum between the dust and gas. It is shown that, for a system of equations for gas and monodisperse dust, a semi-implicit first-order approximation scheme in time in which the interphase interaction is calculated implicitly, while other forces, such as the pressure gradient and gravity are calculated explicitly, is suitable for stiff problems with intense interphase interactions and for computations of the drag in non-linear regimes. The piecewise drag coefficient widely used in astrophysical simulations has a discontinuity at some values of the Mach and Knudsen numbers that are realized in a circumstellar disk. A continuous drag coefficient is presented, which corresponds to experimental dependences obtained for various drag regimes.

Appendices

APPROACH TO DESCRIBING THE DYNAMICS OF DUST IN A CIRCUMSTELLAR DISK

The suitability of a hydrodynamical approach to describing a medium is usually determined using two conditions:

• The mean free path of the particles making up the medium should be much shorter than the length scale of the system. We will take the disk scale height H as a length scale of the system.

Further, some element of the medium is distinguished whose size is much less than the length scale of the medium, but much larger than the mean free path of the particles. In our case, this is the size of a computational grid cell, $\delta R$.

• The number of particles in a given volume of the medium $\delta R$ should be sufficiently large.

The first condition is required so that the loss/gain of particles by an element of the medium as a result of chaotic motions of the particles does not appreciably influence the mean characteristics of the element. The second condition is required for computations using the particle distribution function in the six-dimensional space (coordinates and velocities) of the mean characteristics, such as the mean density in a cell, the mean velocity, and the mean energy.

There is no doubt that these conditions are satisfied by the gas in a circumstellar disk. However, we must investigate this question for the dust component. We estimated the mean free path of a solid particle assuming that the dust subdisk consists of monodisperse, spherical particles. In this case,

$${{\lambda }_{{\text{d}}}} = \frac{1}{{\sqrt 2 \pi {{a}^{2}}{{n}_{{\text{d}}}}}} = \frac{{4a{{\rho }_{{\text{s}}}}}}{{2\sqrt 2 {{\rho }_{{\text{d}}}}}},$$

(A.1)

since, by virtue of their monodisperse nature,

$${{n}_{{\text{d}}}}(a) = \frac{{3{{\rho }_{{\text{d}}}}}}{{4\pi {{\rho }_{{\text{s}}}}{{a}^{3}}}}.$$

(A.2)

We assumed that the volume density of the dust in the equatorial plane of the disk was given by

$${{\rho }_{{\text{d}}}} = \frac{{{{\Sigma }_{{\text{d}}}}}}{H},$$

(A.3)

where the height of the disk is

$$H = \sqrt {\frac{{2{{k}_{{\text{B}}}}T}}{{{{\mu }_{{\text{H}}}}}}\frac{{{{r}^{3}}}}{{GM}}} {\kern 1pt} {\kern 1pt} .$$

(A.4)

Let the surface density of the dust and the temperature be power-law functions of the radius,

${{\Sigma }_{{\text{d}}}}(r) = {{\sigma }_{0}}{{(r{\text{/}}1\;{\text{AU}})}^{p}}$, $T = {{T}_{0}}{{(r{\text{/}}1\;{\text{AU}})}^{q}}$. Adopting the characteristic values for a circumstellar disk

${{\sigma }_{0}} = 1$ g cm^–2, ${{T}_{0}}$ = 100 K, ${{\mu }_{{\text{H}}}} = 2.3$, $M = 1{{M}_{ \odot }}$, p = –1, $q = - 0.5$, we obtain

$${{\rho }_{{\text{d}}}} = 4.3 \times {{10}^{{-13}}}{{(r{\text{/}}1\;{\text{AU}})}^{{-2.25}}}\;\mathop {{\text{g}}\;{\text{cm}}}\nolimits^{-3} .$$

(A.5)

It follows from (A.1) and (A.5) that grains smaller than 1 cm in the outer parts of a disk have mean free paths much shorter than $\delta R$. It is clear that the dust subdisk can be modeled as a continuous medium in this case. On the other hand, these estimates indicate that ${{\lambda }_{{\text{d}}}}$ is close to $H$ in inner regions of the disk. In this case, instead of the mean velocity over the volume, we must consider the velocity of the solid phase as a vector plus a dispersion. A possible development of the model could be the use of the Vlasov–Boltzmann equation [8] or moments of the Boltzmann equation [46]. The solution of the Vlasov–Boltzmann equation in Lagrange coordinates implies the use of discrete particles when modeling the dynamics of the solid phase [1, 33, 47]. Solution for the moment Boltzmann equation can be found in an Euler approach using grid methods [48].

The moment Boltzmann equation take into account the possible anisotropic behaviour of the velocity dispersion of the medium, but computing the components of the velocity dispersion requires the introduction of additional equations. When solving the Vlasov–Boltzmann equation using a particle method, the velocity dispersion varies in a self-consistent way [49], but the difficulty is correctly computing the collision integral. In gas dynamics, this integral is zero for frequent elastic collisions. This may not be the case for the grains.

We have concluded that dust in the main part of the disk can be represented as a continuous medium, while there may be regimes in the central region of the disk in which solid particles have a velocity dispersion, based on estimates of the mean free paths of particles, neglecting their interactions with the gas. However, the same conclusion is supported by estimates of the spatial scale on which exchanges of momentum between the gas and dust occur:

$${{\lambda }_{{{\text{stop}}}}} = \left\| {u - {v}} \right\|{{t}_{{{\text{stop}}}}},$$

(A.6)

where ${{t}_{{{\text{stop}}}}} = {\text{St/}}{{\Omega }_{{\text{K}}}}$. According to [50] Eq. (8), the azimuthal velocity of a particle can be estimated as

$$u = {{V}_{{\text{K}}}}\left( {1 - \frac{1}{{1 + \mathop {{\text{St}}}\nolimits^2 }}\eta } \right).$$

(A.7)

Here, $\eta$ characterizes the degree of deviation of the velocity of the gas from the Keplerian value, and, for typical disk parameters, $\eta \sim 0.01$. We then find that

$${{\lambda }_{{{\text{stop}}}}} = r{\text{St}}\left( {1 - \frac{1}{{1 + \mathop {{\text{St}}}\nolimits^2 }}\eta } \right) \sim r{\text{St}}.$$

(A.8)

It is clear that, when ${\text{St}} < 0.01$, the velocities of grains within a single cell will differ little, while they will differ appreciably when ${\text{St}} \sim 1$. The simulation results [7] indicate that dust that has grown is located in the inner part of the disk.

MODEL OF A STATIONARY GASEOUS DISK

We calculated the radial distributions of parameters of an axially symmetric, gaseous disk with inner boundary ${{R}_{{{\text{min}}}}}$ and outer boundary ${{R}_{{{\text{max}}}}}$ with mass ${{M}_{{{\text{disk}}}}}$ around a star with mass $M$ using the model of [4], which is based on the following assumptions:

• the surface density of the matter in the disk displays a power-law dependence on the radius with power-law index $\xi$:

$$\Sigma (r) = {{\Sigma }_{0}}\mathop {\left( {\frac{r}{{{{R}_{0}}}}} \right)}\nolimits^{ - \xi } ,$$

(B.1)

• where ${{\Sigma }_{0}}R_{0}^{\xi }$ is given by

$$\int\limits_0^{2\pi } {\int\limits_{{{R}_{{{\text{min}}}}}}^{{{R}_{{{\text{max}}}}}} {\Sigma (r)rdrd\varphi } } = {{M}_{{{\text{disk}}}}},$$

(B.2)

• the density and temperature of the gas in the disk are such that the Toomre parameter varies as

$$Q(r) = \frac{{{{c}_{{\text{s}}}}(r)\Omega (r)}}{{\pi G\Sigma (r)}} = 2\mathop {\left( {\frac{{{{R}_{{\max}}}}}{r}} \right)}\nolimits^{0.75} ,$$

(B.3)

• the gaseous disk is in equilibrium, so that

$${{u}_{r}} = 0,\quad {{u}_{\varphi }}(r) = \sqrt {\frac{{GM}}{r} + \frac{r}{{{{\rho }_{{\text{g}}}}(r)}}\frac{{dp(r)}}{{dr}}} ,$$

(B.4)

• the height H and radius r of the disk are related as

$$\frac{{{{c}_{{\text{s}}}}(r)}}{{{{V}_{{\text{K}}}}(r)}} = \frac{{H(r)}}{r},$$

(B.5)

• the gas density at each radius does not vary in the vertical direction, so

$${{\rho }_{{\text{g}}}}(r) = \frac{{\Sigma (r)}}{{H(r)}}.$$

(B.6)

We assumed $\xi = 1$ and used the above assumptions to find the distributions ${{c}_{{\text{s}}}}(r)$, ${{\rho }_{{\text{g}}}}(r)$, ${{u}_{\varphi }}(r)$ in the disk, which are required for the solution of (26).

It follows from (B.2) that

$${{\Sigma }_{0}} = \frac{{{{M}_{{{\text{disk}}}}}(2 - \xi )}}{{2\pi R_{0}^{\xi }(R_{{{\text{max}}}}^{{2 - \xi }} - R_{{{\text{min}}}}^{{2 - \xi }})}}.$$

(B.7)

It follows from (B.3) that

$${{c}_{{\text{s}}}}(r) = \frac{{Q(r)\pi G\Sigma (r)}}{{\Omega (r)}} = \frac{{\pi rQ(r)G\Sigma (r)}}{{{{V}_{{\text{K}}}}(r)}},$$

(B.8)

and therefore

$$\frac{{{{c}_{{\text{s}}}}(r)}}{{{{V}_{{\text{K}}}}(r)}} = \frac{{\pi Q(r)G\Sigma (r)r}}{{V_{{\text{K}}}^{2}}} = \frac{{\pi Q(r)\Sigma (r){{r}^{2}}}}{M}.$$

(B.9)

We then obtain from (B.9) and (B.5)

$$H(r) = \frac{{\pi Q(r)\Sigma (r){{r}^{3}}}}{M},$$

(B.10)

whence

$${{\rho }_{{\text{g}}}}(r) = \frac{{\Sigma (r)}}{{H(r)}} = \frac{M}{{\pi Q(r){{r}^{3}}}}.$$

(B.11)

Note that, by virtue of (B.3),

$${{\rho }_{{\text{g}}}}(r) = {{\rho }_{{{\text{g}}0}}}\mathop {\left( {\frac{r}{{{{R}_{0}}}}} \right)}\nolimits^{ - 2.25} .$$

(B.12)

We took ${{u}_{\varphi }}$ to be related to ${{V}_{{\text{K}}}}$ as

$$u_{\varphi }^{2} = V_{{\text{K}}}^{2} + \left( {\frac{{dln{{\rho }_{{\text{g}}}}}}{{dlnr}}} \right)c_{{\text{s}}}^{2} = V_{{\text{K}}}^{2}(1 - \eta ),$$

(B.13)

where

$$\eta \, = \frac{{c_{{\text{s}}}^{2}}}{{V_{{\text{K}}}^{2}}}\left| {\frac{{dln{{\rho }_{{\text{g}}}}}}{{dlnr}}} \right|\, = \,2.25\frac{{c_{{\text{s}}}^{2}}}{{V_{{\text{K}}}^{2}}}\, = \frac{{2.25{{\pi }^{2}}{{Q}^{2}}(r){{\Sigma }^{2}}(r){{r}^{4}}}}{{{{M}^{2}}}},$$

(B.14)

so that

$$\begin{gathered} {{u}_{\varphi }} = {{V}_{{\text{K}}}}\sqrt {1 - \frac{{2.25{{\pi }^{2}}{{Q}^{2}}(r){{\Sigma }^{2}}(r){{r}^{4}}}}{{{{M}^{2}}}}} \\ \, = {{V}_{{\text{K}}}}\sqrt {1 - \frac{{2.25{{\Sigma }^{2}}(r)}}{{\rho _{{\text{g}}}^{2}(r){{r}^{2}}}}} {\kern 1pt} {\kern 1pt} . \\ \end{gathered} $$

(B.15)

The distribution of the surface density and temperature for the disk in Toy Models 1 and 2 is shown in Fig. 13.

Fig. 13.

(Left panel) Radial distributions of the surface density and temperature in a circumstellar disk for Toy Models 1 and 2. (Right panel) Radial distribution of η and Toomre parameter for Toy Models 1 and 2.