Gravitational collapse at low to moderate Mach numbers: The relationship between star formation efficiency and the fraction of mass in the massive object

TL;DR

This paper investigates how the formation of massive objects through gravitational collapse depends on initial conditions, environmental factors, and collision dynamics, providing insights into star formation efficiency and mass distribution.

Contribution

It introduces a comprehensive analysis of simulations linking star formation efficiency, mass fractions, and collision parameters across different environments.

Findings

f_tot increases with star formation efficiency

f_* relation depends on initial protostar count

Collision parameter helps predict the importance of collisions

Abstract

The formation of massive objects via gravitational collapse is relevant both for explaining the origin of the first supermassive black holes and in the context of massive star formation. Here, we analyze simulations of the formation of massive objects pursued by different groups and in various environments, concerning the formation of supermassive black holes, primordial stars, as well as present-day massive stars. We focus particularly on the regime of small virial parameters, i.e., low ratios of the initial kinetic to gravitational energy, low to moderate Mach numbers, and the phase before feedback is very efficient. We compare the outcomes of collapse under different conditions using dimensionless parameters, particularly the star formation efficiency \epsilon_*, the fraction f_* of mass in the most massive object relative to the total stellar mass, and the fraction f_{\rm tot} of mass of the most massive object as a function of the total mass. We find that in all simulations analyzed here, f_{\rm tot} increases as a function of \epsilon_*, although the steepness of the increase depends on the environment. The relation between f_* and \epsilon_* is found to be more complex and also strongly depends on the number of protostars present at the beginning of the simulations. We show that a collision parameter, estimated as the ratio of the system size divided by the typical collision length, allows us to approximately characterize whether collisions will be important. We analyze the statistical correlation between the dimensionless quantities using the Spearman coefficient and confirm via a machine learning analysis that good predictions of f_* can be obtained from \epsilon_* together with a rough estimate of the collision parameter. This suggests that a good estimate of the mass of the most massive object can be obtained once the maximum efficiency for a given environment is known.