Letter to the Editor

Maximum accretion rate of supermassive stars

¹ Département d’Astronomie, Université de Genève, Chemin des Maillettes 51, 1290 Versoix, Switzerland
e-mail: lionel.haemmerle@unige.ch
² Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
³ Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
⁴ Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

Received: 25 May 2021
Accepted: 20 July 2021

Abstract

Context. The formation of the most massive quasars observed at high redshifts requires extreme inflows of gas down to the length scales of the central compact object.

Aims. Here we estimate the maximum inflow rate allowed by gravity down to the surface of supermassive stars, the possible progenitors of these supermassive black holes.

Methods. We use the continuity equation and the assumption of spherical symmetry and free fall to derive the maximum allowed inflow rates for various density profiles. We apply our approach to the mass–radius relation of rapidly accreting supermassive stars to estimate an upper limit to the accretion rates allowed during the formation of these objects.

Results. We find that, as long as the density of the accreted gas is smaller than or equal to the average density of the accretor, the maximum allowed rate, Ṁ_max, is given uniquely by the compactness of the accretor. We argue that a density inversion between accreting matter and the accretor is inconsistent with gravitational collapse. For the compactness of rapidly accreting supermassive stars, Ṁ_max is related to the stellar mass, M, by a power law, Ṁ_max ∝ M^3/4. The rates of atomically cooled halos (0.1−10 M_⊙ yr⁻¹) are allowed as soon as M ≳ 1 M_⊙. The largest rates expected in galaxy mergers (10⁴ − 10⁵ M_⊙ yr⁻¹) become accessible once the accretor is supermassive (M ≳ 10⁴ M_⊙).

Conclusions. These results suggest that supermassive stars can accrete up to masses > 10⁶ M_⊙ before they collapse via the general-relativistic instability. At such masses, the collapse is expected to lead to the direct formation of a supermassive black hole, even within metal-rich gas, resulting in a black hole seed that is significantly heavier than in conventional direct collapse models for atomic cooling halos.