The instability of viscous self-gravitating protostellar disk affected by density bump

Abstract

In this work, we study the instability of viscous self-gravitating protostellar disk affected by infalling Low-mass condensations (LMCs) from the envelope of collapsing molecular cloud cores. The infalling low-mass-condensations (LMCs) are considered as density bumps through the nearly Keplerian viscous accretion disk, and their evolutions are analyzed by using the linear perturbation approximation. We investigate occurrence of instability in the evolution of these density bumps. We find the unstable regions of the bumped accretion disk and evaluate the growth time scale (GTS) of the instability. We also study the radial accretion and azimuthal rotation in these unstable regions. The results show that the GTS will be minimum at a special radius so that the unstable regions can be divided in two parts (inner and outer regions). The perturbed radial and azimuthal velocities in the inner unstable regions are strengthened, while in the outer unstable regions are weakened. Decreasing the radial and azimuthal velocities in the outer unstable regions may lead to coagulation of matters. This effect can help the fragmentation of the disk and formation of the self-gravitating bound objects.

Appendix:  The coefficients

The coefficients of Eqs. (19)–(21) are: $a_{1}=\frac{u_{R_{0}}}{R}+\frac{du_{R_{0}}}{dR}$, $a_{2}= m\frac{u_{\varphi_{0}}}{R}$, $a_{3}=u_{R_{0}}$, $a_{4}=\frac{\varSigma _{0}}{R}+\frac{d\varSigma_{0}}{dR}$, $a_{5}=\varSigma_{0}$, $a_{6}=m\frac {\varSigma_{0}}{R}$, $b_{1}=\frac{du_{R_{0}}}{dR}+\frac{2}{3R\varSigma _{0}}\frac{d}{dR}(\nu_{0}\varSigma_{0})$, $b_{2}=m\frac{u_{\varphi _{0}}}{R}$, $b_{3}=u_{R_{0}}-\frac{4}{3R\varSigma_{0}}\frac{d}{dR}(R\nu _{0}\varSigma_{0})$, $b_{4}=-\frac{4}{3}\nu_{0}$, $b_{5}=-\frac {k_{B}}{\mu m_{H}}\frac{T_{0}}{\varSigma_{0}}\frac{d\varSigma _{0}}{dR}+\frac{4}{3R \varSigma_{0}^{2}}\frac{d}{dR}(R\nu_{0}\varSigma _{0}\frac{du_{R_{0}}}{dR}) -\frac{2}{3R\varSigma_{0}^{2}}\frac {d}{dR}(\nu_{0}\varSigma_{0}u_{R_{0}})+\frac{2\nu_{0}}{3R^{2}\varSigma _{0}}\frac{d}{dR}(Ru_{R_{0}}) -\frac{2}{3R^{2}}\frac{\nu _{0}u_{R_{0}}}{\varSigma_{0}}$, $b_{6}=\frac{k_{B}}{\mu m_{H}}\frac {T_{0}}{\varSigma_{0}}$, $b_{7}=-2\frac{u_{\varphi _{0}}}{R}$, $b_{8}=1$, $c_{1}=\frac{u_{R_{0}}}{R}+\frac {1}{R^{2}\varSigma_{0}}\frac{d}{dR}(R\nu_{0}\varSigma_{0})$, $c_{2}=m\frac{u_{\varphi_{0}}}{R}$, $c_{3}=u_{R_{0}}-\frac {1}{R^{2}\varSigma_{0}}\frac{d}{dR}(R^{2}\nu_{0}\varSigma_{0})+\frac{\nu _{0}}{R}$, $c_{4}=-\nu_{0}$, $c_{5}=m\frac{k_{B}}{\mu m_{H}}\frac {T_{0}}{R\varSigma_{0}}$, $c_{6}=\frac{1}{R^{2}\varSigma_{0}^{2}}\frac {d}{dR}(R^{3}\nu_{0}\varSigma_{0}\frac{d}{dR}(\frac{u_{\varphi _{0}}}{R}))$, $c_{7}=0$, $c_{8}=\frac{u_{\varphi_{0}}}{R}+\frac {du_{\varphi_{0}}}{dR}$, and $c_{9}=\frac{m}{R}$.