Accretion Rate

Using equation (2.26), the necessary time for a mass-shell at $R$ to reach the center (free-fall time) is expressed as

$$T(R)\equiv\left(\frac{R^3}{2GM(R)}\right)^{1/2}\frac{\pi}{2}$$ (4.93)

(for detail of this section see Ogino, Tomisaka, & Nakamura 1999).

Consider two shells whose initial radii are $R$ and $R+\Delta R$. The time difference for these two shells to reach the center $\Delta T(R)$ can be written down using equation (4.93) as

$$\Delta T(R) =\frac{\pi{R}^{1/2}}{2^{3/2}(GM(R))^{1/2}} \left[\frac{3}{2}-\frac{{R}}{2M(R)}\frac{dM(R)}{dR}\right]\Delta R.$$ (4.94)

Mass in the shell between $R$ and $R+\Delta R$, $\Delta M\equiv M(R+\Delta R)-M(R)$ $=(dM/dR)\Delta R$, accretes on the central object in $\Delta T(R)$. Thus, mass accretion rate for a pressure-free cloud is expressed as $\Delta M/\Delta T$. This leads to the expression as

$$\frac{d M}{d T}(R)=\frac{2^{3/2}}{\pi}\frac{G^{1/2}M(R)^{3/2... ... M(R)}{d R}} {\frac{3}{2}-\frac{{R}}{2M(R)}\frac{dM(R)}{d R}}.$$ (4.95)

Figure 4.11: Mass accretion rate against the typical density of the cloud.

This gives time variation of the accretion rate. Consider two clouds with the same density distribution $\partial\log\rho/\partial r$ but different absolute value. Since these two clouds have the same $\partial \log M(R)/\partial \log R$, the mass accretion rate depends only on $M(R)/R$, and is expressed as

$$\frac{d M}{d T}(R)\propto M(R)^{3/2}.$$ (4.96)

This indicates that the accretion rate is proportional to $\rho^{3/2}$, while the time scale is to $\rho^{-1/2}$. This is confirmed by hydrodynamical simulations of spherical symmetric isothermal clouds (Ogino et al.1999). When the initial density distribution is the SIS as $\rho \propto r^{-2}$, the mass included inside $R$ is proportional to radius $M(R) \propto R$. In this case, equation (4.95) gives a constant accretion rate in time. In Figure 4.11 we plot the mass accretion rate against the cloud density. $\alpha$ represents the cloud density relative to that of a hydrostatic Bonnor-Ebert sphere. This shows clearly that the mass accretion rate is proportional to $\alpha^{3/2}$ for massive clouds $\alpha > 4$. This is natural since the assumption of pressure-less is valid only for a massive cloud in which the gravity force is predominant against the pressure force.

Similar discussion has been done by Henriksen, André, & Bontemps (1997) to explain a decline in the accretion rate from Class 0 to Class I IR objects. They assumed initial density distribution of

$$\rho\left\{\begin{array}{l}= \rho_0 (r \le r_N),\\ =\rho_0\left(\frac{r}{r_N}\right)^{-2/D_1} (r > r_N), \end{array}\right.$$ (4.97)

as shown in Figure 4.12. Since the free-fall-time of the gas contained in the inner core $r \le r_N$ is the same, such gas reaches the center once. It makes a very large accretion rate at $t=(3\pi / 32 G \rho_N)^{1/2}$ as $\dot{M}=(4\pi/3) \rho_0 r_N^3\delta(t-(3\pi / 32 G \rho_N)^{1/2})$. If $D_1=1$, $\rho \propto r^{-2}$ for $r \mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} r_N$. Since $M\propto R^1$ and $t_{\rm ff}\propto R$, equation (4.95) predicts $\dot{M}\propto R^{0}\propto t^0$. A constant accretion rate is expected for this power-law and the accretion rate is converged to a constant value after the stellar mass is much larger than than that was containd in $r_N$, $M_* \gg M(r_N)$. If $D_1=2$, $\rho \propto r^{-1}$ for $r \mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} r_N$. Since for this power $M\propto R^2$ and $t_{\rm ff}\propto R^{1/2}$, equation(4.95) predicts $\dot{M}\propto R^{3/2}\propto t^3$. They gave $\dot{M}\propto t^{3D_1-3}$ for $D_1> 2/3$.

Figure 4.12: A model proposed to explain time variation in accretion rate by Henriksen, André, & Bontemps (1997). The density distribution $\rho (r)$ at $t=0$ (left) and expected accretion rate (right).