The dust-to-gas ratio in the gravitational collapse of filamentary clouds

Abstract

The aim of this study is to investigate the evolution of the dust-to-gas ratio in the self-similar collapse of a filamentary molecular cloud. For this purpose, we use single fluid dusty-gas model in our study, which describes a single fluid moving with the barycentric velocity of the mixture instead of two-fluid method. The self-similar technique is used to consider the problem in two phases of the collapse, i.e. isothermal and polytropic collapse phases. Regarding the analytical methods, we obtain a semi-analytical solution for the dust-to-gas ratio as a function of the barycentric velocity at large radii of the filament. Furthermore, the polytropic collapse is solved numerically at large radii of a collapsing filamentary cloud. The results show that the profile of dust-to-gas ratio is very different for the isothermal and polytropic cases. This issue suggests that thermodynamic processes during the collapse have the most significant effect on the evolution of the dust-to-gas ratio. It was also proved that the intrinsic density and the grain size play important roles on the values of the dust-to-gas ratio during the self-similar collapse. Finally, the results address some outstanding issues about the dust distribution during the gravitational collapse (either the isothermal and adiabatic collapse) which has not received much attention before.

Appendices

Appendix A: Self-similar variables and observational parameters

Here, we study the relationship between the self-similar variables and the observational parameters. It is certainly established that there are three phases for a collapsing cloud, i.e. the isothermal collapse phase, the adiabatic collapse phase, and the accretion phase (Bodenheimer 2011). Thus, it is assumed that the collapsing cloud undergoes into the adiabatic collapse phase after passing the isothermal collapse phase. This means that the final boundary condition for the isothermal case is being consistent with the initial boundary condition for the polytropic case. At this stage, the radius of outer edge is (approximately) “0.01” pc at a time $t=10^{5}$ yr (e.g. Bodenheimer 2011). For a better comparison between the self-similar variables and observational parameters, we should know the asymptotic behaviors of physical variables at large radii in the self-similar approach. The asymptotic behavior of these variables can be estimated by expanding them at $|\:x\:|\gg 1$ as the following form

$$\begin{aligned} \rho \approx \frac{\rho _{0}}{x^{2}},\quad m\approx m_{0}, \quad u_{r} \approx u_{0} x,\quad u_{z}\approx 0, \quad f\approx f_{0}x^{-\alpha }, \end{aligned}$$

(34)

where $\rho _{0}$, $m_{0}$, $u_{0}$ and $f_{0}$ are constants. Also, we have

$$ x=\frac{r}{\sqrt{K}}\:t^{n}, $$

(35)

where $K$ is

$$ K=\frac{c_{s}^{2}}{(n+2)\varrho ^{n+1}}. $$

(36)

The important key to connection between the self-similar variables and observational parameters is the equation (35). If the RHS of the equation (35) is chosen regarding the observational data, the LHS of this equation (that is the most important parameter of the self-similar approach) will be so realistic.

Substituting the temperature

$$ T=10 \mbox{ K} , $$

(37)

and the sound speed (e.g., Bodenheimer 2011)

$$ c_{s}=0.2 \times 10^{5} \mbox{ cm}/\mbox{s}\quad \left ( \frac{T}{10\mbox{ K}}\right )^{1/2}, $$

(38)

into the equation (36), we have $K=2.5\times 10^{20}$ for the case of “$n=-1/3$”. Now, if one substitutes the values of “$K$” and $t=10^{5}\mbox{ yr}$ (at $r=0.01\mbox{ pc}$) into the equation (35), the result is $x \approx 100$. Thus, $\varrho _{0}$ becomes

$$\begin{aligned} \varrho _{0} =&2\pi G t^{2} x^{2} \rho , \\ =&4, \end{aligned}$$

(39)

where the values of “$G$” and “$\varrho$” are “$6.67\times 10^{-8}$ cm³ g⁻¹ s⁻²” and “$\varrho =10^{-18} \mbox{ g}\,\mbox{cm}^{-3}$”, respectively.

According to the Eqs. (11) and (18), we can write

$$\begin{aligned} \sigma =&2\pi G t \sigma _{0}, \\ =& 200. \end{aligned}$$

(40)

In order to get the values of $u_{0}$ and $m_{0}$, the below equations are useful

$$ v_{r}=\sqrt{K}u_{r}\left (\frac{r}{x\sqrt{K}}\right )^{\frac{n+1}{n}}, $$

(41)

and

$$ m_{0}=\frac{\rho _{0}}{2}\left (\frac{n+u_{0}}{n+1}\right ), $$

(42)

which result in $u_{0}=0.98$ and $m_{0}=2$. Furthermore, we set $t_{0}=10^{5}\mbox{ yr}$. Now it is useful to consider the stopping time according to the Eq. (11) as follows

$$ t_{s}=\frac{\sigma _{0}}{\varrho }=5\times 10^{5} \mbox{ yr}. $$

(43)

Since this time is very close to the dynamical timescale (the free-fall time is $3\times 10^{5} \mbox{ yr}$), the dust should be mainly coupled to the gas.

Appendix B: Stokes number and stopping time

In the previous sections, we considered the variation of dust to gas ratio by choosing different values of $\sigma$ in the dynamic model. However, it is crucial to discuss the corresponding stopping time $t_{s}$, as well as the Stokes number by referring to the eddy turnover time $t_{ed}$. In the Appendix A, we found $t_{s}=5\times 10^{5}\mbox{ yr}$. This time is very close to the dynamical timescale (the free-fall time is $3\times 10^{5} \mbox{ yr}$), the dust should be mainly coupled to the gas or $\textrm{St} \rightarrow 0$.

As we know, the case of $\textrm{St}\ll 1$ leads to $t_{s}\ll t_{ed}$. In this case, the dust is weakly coupled to the gas. In addition, the case of $\textrm{St}\gg 1$ leads to $t_{s}\gg t_{ed}$. In this case, the dust is mainly decoupled to the gas. It is helpful to rewrite the Eq. (9) with more details. Regarding the values of $\sigma$ in the Appendix A, the stopping time is

$$\begin{aligned} t_{s}&=7.5\times 10^{3} \mbox{ yr}\left ( \frac{\varrho _{gr}}{3\mbox{ g}\,\mbox{cm}^{-3}}\right )\left ( \frac{s_{gr}}{0.1\mbox{ }\upmu \mbox{m}}\right )\left ( \frac{c_{s}}{0.2\mbox{ km}\,\mbox{s}^{-2}}\right )^{-1} \\ &\quad{}\times \left ( \frac{\varrho }{10^{-20}\mbox{ g}\,\mbox{cm}^{-3}}\right )^{-1}. \end{aligned}$$

(44)

In the above equation, the density is substituted regarding the initial stage of the collapse (i.e. isothermal case). At this time, the eddy turnover time can be obtained by $t_{ed}=L/(2\mathcal{M} c_{s})$ where ℳ is the turbulent Mach number and $L$ is the size of the cloud (Tricco et al. 2017). Accordingly, we have

$$ t_{ed}=5\times 10^{5} \mbox{ yr}\left (\frac{L}{1 \mbox{ PC}} \right )\left (\frac{2\mathcal{M}}{10}\right )^{-1}\left ( \frac{c_{s}}{0.2\mbox{ km}\,\mbox{s}^{-2}}\right )^{-1}. $$

(45)

Regarding the equation (44) and (45), we can write

$$\begin{aligned} \textrm{St}&=1.5\times 10^{-2} \: \left ( \frac{\varrho _{gr}}{3\mbox{ g}\,\mbox{cm}^{-3}}\right )\left ( \frac{s_{gr}}{0.1\mbox{ }\upmu \mbox{m}}\right )\left ( \frac{2\mathcal{M}}{10}\right ) \\ &\quad{}\times\left ( \frac{\varrho }{10^{-20}\mbox{ g}\,\mbox{cm}^{-3}}\right )^{-1} \left (\frac{L}{1\mbox{ PC}}\right )^{-1}. \end{aligned}$$

(46)

As we see, the values of $\textrm{St}$ depend on some physical properties of the dust grains and the clouds, as well as the nature of the turbulence. For instance, Tricco et al. (2017) assumed that the turbulence is initiated and sustained at $\mathcal{M}=10$. In this case, the dusty shocks are expected to be of “J-type” at this Mach number, with a sharp jump in the gas properties (Lehmann and Wardle 2016). Obviously, if we substitute $10^{-18}\mbox{ g}\,\mbox{cm}^{-3}$ and $L=0.01 \mbox{ PC}$ in the equation (B3), the value of the Stokes number remains unchanged.