Infall onto the Protoplanetary Disk during the Gravitational Collapse of a Molecular Cloud

Abstract

The development of models for matter infall from a collapsing molecular cloud is an essential part of numerical studies on the formation and evolution of protoplanetary disks. In this article, the widely used Nakamoto and Nakagawa (1994) model is analyzed and modifications are proposed to complement the initial model. These improvements include calculation for the outer boundary of a molecular cloud and refinement of the initial density distribution within. Also, due to the finite size of a cloud, the approach for computing the rate of mass infall onto the protoplanetary disk during collapse is modified. The proposed enhancements are aimed at eliminating the key limitations of the initial model, but do not affect its advantages, such as simplicity in numerical implementation. Using the modified model, we investigate the duration of the embedded phase of the evolution of young stellar objects and compare the modeling results with observational data. The results indicate a shorter duration of the embedded phase of the evolution of young stellar objects, especially in star-forming regions with a high amplitude of the initial density perturbation of prestellar condensation.

Figures (5)

• Figure 1: Distribution of mass (shown in color) of matter infall across disk in radial direction (along horizontal axis) and its change over time (along vertical axis). Boundary of colored region shows change in centrifugal radius of disk over time. Black and red dashed lines delineate boundaries of regions outside of which $90\%$ and $50\%$ of mass that fell onto disk is contained, respectively. Calculations were performed using NN-0 model.
• Figure 2: Gas concentration profiles in molecular cloud at $T_{\rm c}$ = 10 K. Dots indicate profiles constructed for different moments of time based on three-dimensional modeling data Vorobyov2024. Constant concentration profile constructed using distribution (\ref{['rho_c_LP69']}) at $T_{\rm c}$ = 10 K and $A$ = 5 is shown by red solid line. Profiles corresponding to singular isothermal sphere with distribution (\ref{['rho_c_Shu77']}) for $A =$ 2 and 5 are plotted as dotted and dashed lines, respectively.
• Figure 3: Changes in rate of matter infall over time using various modifications. Solid red line is M1 modification; dashed green line, M2 modification. Dashed black line shows infall rate as calculated by V2010. Same cloud parameters were used in all cases: $M_{\rm c}$ = 1 $M_{\rm \odot}$, $r_{\rm out}$ = 8337 au, $T_{\rm c}$ = 15 K и $\omega_{\rm c}$ = 2.8$\times$10$^{-14}$ s$^{-1}$. To enable comparison with V2010 calculations, density distribution (\ref{['rho_c_Shu77']}) characteristic of SIS for $A$ is used.
• Figure 4: Density distribution $\rho(r)$ in molecular cloud (\ref{['rho_c_LP69']}) for models NN-1, NN-2, NN-3, and NN-7. For same values of central concentration, mass, and temperature of cloud, it is amplitude of density disturbance $A$ that determines width of central plateau and density at large distances from center of cloud.
• Figure 5: Changes in mass infall rate over time in models NN-1 -- NN-7. Circles show times corresponding to infall of $50\%$ of initial cloud mass ($t_{\rm C0}$); diamonds show times corresponding to infall of $90\%$ of mass ($t_{\rm CI}$). Vertical dashed lines indicate boundaries of interval corresponding to most probable total duration of Classes 0 and I according to estimates Dunham2015.