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Abstract:

The observations of high-redshifts quasars at z ≳ 6 have revealed that supermassive black holes (SMBHs) of mass ∼109M⊙∼109M⊙ were already in place within the first ∼Gyr after the big bang. Supermassive stars (SMSs) with masses 103−5M⊙103−5M⊙ are potential seeds for these observed SMBHs. A possible formation channel of these SMSs is the interplay of gas accretion and runaway stellar collisions inside dense nuclear star clusters (NSCs). However, mass-loss due to stellar winds could be an important limitation for the formation of the SMSs and affect the final mass. In this paper, we study the effect of mass-loss driven by stellar winds on the formation and evolution of SMSs in dense NSCs using idealized N-body simulations. Considering different accretion scenarios, we have studied the effect of the mass-loss rates over a wide range of metallicities Z* = [.001–1]Z⊙ and Eddington factors fEdd=L∗/LEdd=0.5,0.7,and0.9fEdd=L∗/LEdd=0.5,0.7,and0.9⁠. For a high accretion rate of 10−4M⊙yr−110−4M⊙yr−1⁠, SMSs with masses ≳103M⊙yr−1≳103M⊙yr−1 could be formed even in a high metallicity environment. For a lower accretion rate of 10−5M⊙yr−110−5M⊙yr−1⁠, SMSs of masses ∼103−4M⊙∼103−4M⊙ can be formed for all adopted values of Z* and fEdd, except for Z* = Z⊙ and fEdd = 0.7 or 0.9. For Eddington accretion, SMSs of masses ∼103M⊙∼103M⊙ can be formed in low metallicity environments with Z* ≲ 0.01 Z⊙. The most massive SMSs of masses ∼105M⊙∼105M⊙ can be formed for Bondi–Hoyle accretion in environments with Z* ≲ 0.5 Z⊙. An intermediate regime is likely to exist where the mass-loss from the winds might no longer be relevant, while the kinetic energy deposition from the wind could still inhibit the formation of a very massive object.

Abstract:

More than 200 supermassive black holes (SMBHs) of masses ≳109M⊙≳109M⊙ have been discovered at z ≳ 6. One promising pathway for the formation of SMBHs is through the collapse of supermassive stars (SMSs) with masses ∼103−105M⊙∼103−105M⊙ into seed black holes which could grow upto few times 109M⊙109M⊙ SMBHs observed at z ∼ 7. In this paper, we explore how SMSs with masses ∼103−105M⊙∼103−105M⊙ could be formed via gas accretion and runaway stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs) using idealized N-body simulations. We explore physically motivated accretion scenarios, e.g. Bondi–Hoyle–Lyttleton accretion and Eddington accretion, as well as simplified scenarios such as constant accretions. While gas is present, the accretion time-scale remains considerably shorter than the time-scale for collisions with the most massive object (MMO). However, overall the time-scale for collisions between any two stars in the cluster can become comparable or shorter than the accretion time-scale, hence collisions still play a crucial role in determining the final mass of the SMSs. We find that the problem is highly sensitive to the initial conditions and our assumed recipe for the accretion, due to the highly chaotic nature of the problem. The key variables that determine the mass growth mechanism are the mass of the MMO and the gas reservoir that is available for the accretion. Depending on different conditions, SMSs of masses ∼103−105M⊙∼103−105M⊙ can form for all three accretion scenarios considered in this work.

Abstract:

Since its formulation by Sir Isaac Newton, the problem of solving the equations of motion for three bodies under their own gravitational force has remained practically unsolved. Currently, the solution for a given initialization can only be found by performing laborious iterative calculations that have unpredictable and potentially infinite computational cost, due to the system's chaotic nature. We show that an ensemble of converged solutions for the planar chaotic three-body problem obtained using an arbitrarily precise numerical integrator can be used to train a deep artificial neural network (ANN) that, over a bounded time interval, provides accurate solutions at a fixed computational cost and up to 100 million times faster than the numerical integrator. In addition, we demonstrate the importance of training an ANN using converged solutions from an arbitrary precise integrator, relative to solutions computed by a conventional fixed precision integrator, which can introduce errors in the training data, due to numerical round-off and time discretization, that are learned by the ANN. Our results provide evidence that, for computationally challenging regions of phase space, a trained ANN can replace existing numerical solvers, enabling fast and scalable simulations of many-body systems to shed light on outstanding phenomena such as the formation of black hole binary systems or the origin of the core collapse in dense star clusters.

Abstract:

Runaway collisions in dense clusters may lead to the formation of supermassive black hole (SMBH) seeds, and this process can be further enhanced by accretion, as recent models of SMBH seed formation in Population III star clusters have shown. This may explain the presence of SMBHs already at high redshift, z > 6. However, in this context, mass loss during collisions was not considered and could play an important role for the formation of the SMBH seed. Here, we study the effect of mass loss, due to collisions of protostars, in the formation and evolution of a massive object in a dense primordial cluster. We consider both constant mass-loss fractions as well as analytic models based on the stellar structure of the collision components. Our calculations indicate that mass loss can significantly affect the final mass of the possible SMBH seed. Considering a constant mass loss of 5 per cent for every collision, we can lose between 60–80 per cent of the total mass that is obtained if mass loss were not considered. Using instead analytical prescriptions for mass loss, the mass of the final object is reduced by 15–40 per cent, depending on the accretion model for the cluster we study. Altogether, we obtain masses of the order of 104M⊙104M⊙⁠, which are still massive enough to be SMBH seeds.

Abstract:

Collisions were suggested to potentially play a role in the formation of massive stars in present day clusters, and have likely been relevant during the formation of massive stars and intermediate mass black holes within the first star clusters. In the early Universe, the first stellar clusters were particularly dense, as fragmentation typically only occurred at densities above 109 cm−3, and the radii of the protostars were enhanced as a result of larger accretion rates, suggesting a potentially more relevant role of stellar collisions. We present here a detailed parameter study to assess how the number of collisions and the mass growth of the most massive object depend on the properties of the cluster. We also characterize the time evolution with three effective parameters: the time when most collisions occur, the duration of the collisions period, and the normalization required to obtain the total number of collisions. We apply our results to typical Population III (Pop. III) clusters of about 1000 M⊙, finding that a moderate enhancement of the mass of the most massive star by a factor of a few can be expected. For more massive Pop. III clusters as expected in the first atomic cooling halos, we expect a more significant enhancement by a factor of 15–32. We therefore conclude that collisions in massive Pop. III clusters were likely relevant to form the first intermediate mass black holes.