The ability of a protostellar disc to fragment and the properties of molecular cloud cores

We explore the ability of a protostellar disc to fragment using an evolutionary disc model. Our disc model includes the mass influx from a molecular cloud core, the irradiation from the central star, the magnetorotational instability (MRI), and the gravitational instability (Kratter et al. in Astrophys. J. 681:375, 2008). We use the fragmentation criterion of Gammie (Astrophys. J. 553:174, 2001) and Rafikov (Astrophys. J. 621:L69, 2005) to judge whether or not a protostellar disc can fragment. We find that there is a link between whether a protostellar disc can fragment and the properties of the molecular cloud cores (angular velocity ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega$\end{document}, temperature Tcore\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\mathrm{core}}$\end{document}, and mass Mcore\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\mathrm{core}}$\end{document}). In the parameter space ω−Mcore\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega-M_{\mathrm{core}}$\end{document}, there is a critical value ωcrit\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega_{\mathrm{crit}}$\end{document}, which divides the parameter space into two regions: one is the fragmentation region, the other is the non-fragmentation region. The protostellar disc can only fragment when ω>ωcrit\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega> \omega_{\mathrm{crit}}$\end{document}. The reason can be understood as follows. The protostellar disc is formed from the gravitational collapse of a molecular cloud core, the properties of the molecular cloud core determine the properties of the protostellar disc. Thus the two categories of protostellar discs correspond to two categories of molecular cloud cores. Moreover, we find that ωcrit\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega_{\mathrm{crit}}$\end{document} is approximately a linear function of Mcore\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\mathrm{core}}$\end{document} in log-scale coordinates.