Realistic Accretion Research Articles

Supermassive star formation via episodic accretion: protostellar disc instability and radiative feedback efficiency

The formation of SMSs is a potential pathway to seed SMBHs in the early universe. A critical issue for forming SMSs is stellar UV feedback, which may limit the stellar mass growth via accretion. In this paper we study the evolution of an accreting SMS and its UV emissivity under conditions of realistic variable accretion from a self-gravitating circumstellar disc. First we conduct a 2D hydrodynamical simulation to follow the long-term protostellar accretion until the stellar mass exceeds $10^4~M_\odot$. The disc fragments due to gravitational instability, creating a number of small clumps that rapidly migrate inward to fall onto the star. The resulting accretion history is thus highly time-dependent: short episodic accretion bursts are followed by longer, relative quiescent phases. We show that the circumstellar disc for the so-called direct collapse model is more unstable and generates greater variability over shorter timescales than normal Pop III cases. We conduct a post-process stellar evolution calculation using the obtained accretion history. Our results show that, regardless of the strong variability of the accretion rates, the stellar radius monotonically increases with almost constant effective temperature at $T_{\rm eff} \simeq 5000$ K as the stellar mass increases. The resulting UV feedback is too weak to hinder mass accretion due to the low flux of stellar UV photons, thus verifying our implicit assumption of no stellar feedback during the hydrodynamic simulations. The insensitivity of stellar evolution to variable accretion is attributed to the fact that typical timescales of variability, $\lesssim 10^3$ years, are too short to affect the stellar structure. We argue that this evolution will continue until the SMS eventually collapses to produce a massive black hole by the general relativistic instability after the stellar mass reaches $\gtrsim 10^5~M_\odot$.

The effect of black hole spin by different viscosities on the stability of relativistic accretion disk

When luminosity of black hole accretion disks is greater than 0.06 Eddington luminosity ( $L \geq0.06 L_{E} $ ), according to standard disk model, the inner region of these disks is radiation-pressure dominated (RPD) and unstable in thermal and viscous modes. Whereas this object conflicts with the observations, if $0.01 L_{E}\leq L \leq0.5 L_{E}$ . Thus, the change of viscosity or considering relativistic corrections that are essential in realistic black hole accretion disks may solve this problem between theory and observation. An explanation by taking into account the effect of black hole spin with the presence of different viscosities on the secular and thermal instabilities in geometrically thin optically thick accretion disks is presented. In order to consider the effects of spin and general relativity, kerr space-time is used. A diffusive form of viscous stress tensor, a new form of fully relativistic shear stress tensor that is not used so far, power law, and parametric viscosity are considered. Investigations have been done in some parameters of viscosity and some values of spin. We conclude that the type of viscosity is effective in stabilization of the disk, whereas spin has no effect. Increasing stability or instability in the disk can be caused by spin, which this behavior is related to the type of viscous parameter. Thereby, spin can be effective on the temperature, accretion rate, and in the luminosity of the system.

Origin of nonlinearity and plausible turbulence by hydromagnetic transient growth in accretion disks: Faster growth rate than magnetorotational instability.

We investigate the evolution of hydromagnetic perturbations in a small section of accretion disks. It is known that molecular viscosity is negligible in accretion disks. Hence, it has been argued that a mechanism, known as magnetorotational instability (MRI), is responsible for transporting matter in the presence of a weak magnetic field. However, there are some shortcomings, which question the effectiveness of MRI. Now the question arises, whether other hydromagnetic effects, e.g., transient growth (TG), can play an important role in bringing nonlinearity into the system, even at weak magnetic fields. In addition, it should be determined whether MRI or TG is primarily responsible for revealing nonlinearity in order to make the flow turbulent. Our results prove explicitly that the flows with a high Reynolds number (Re), which is the case for realistic astrophysical accretion disks, exhibit nonlinearity via TG of perturbation modes faster than that by modes producing MRI. For a fixed wave vector, MRI dominates over transient effects only at low Re, lower than the value expected to be in astrophysical accretion disks, and low magnetic fields. This calls into serious question the (overall) persuasiveness of MRI in astrophysical accretion disks.

SMOOTHING ROTATION CURVES AND MASS PROFILES

We show that spiral activity can erase pronounced features in disk galaxy rotation curves. We present simulations of growing disks, in which the added material has a physically motivated distribution, as well as other examples of physically less realistic accretion. In all cases, attempts to create unrealistic rotation curves were unsuccessful because spiral activity rapidly smoothed away features in the disk mass profile. The added material was redistributed radially by the spiral activity, which was itself provoked by the density feature. In the case of a ridge-like feature in the surface density profile, we show that two unstable spiral modes develop, and the associated angular momentum changes in horseshoe orbits remove particles from the ridge and spread them both inwards and outwards. This process rapidly erases the density feature from the disk. We also find that the lack of a feature when transitioning from disk to halo dominance in the rotation curves of disk galaxies, the so called "disk-halo conspiracy", could also be accounted for by this mechanism. We do not create perfectly exponential mass profiles in the disk, but suggest that this mechanism contributes to their creation.

Cosmic background radiation in the vicinity of a Schwarzschild black hole: No classic firewall

The Cosmic Blackbody Background Radiation pervades the entire Universe, and so falls into every astrophysical black hole. The blueshift of the infalling photons, measured by a static observer, is infinite at the event horizon. This raises a question as to whether a "firewall" of high energy density may form just outside the horizon, or whether the effect can be attributed exclusively to a singular behavior of the static observer's frame at the horizon. In principle, the presence of such firewall may alter the motion of the infalling matter, influence the black hole evolution, or even invalidate the {\it vacuum} Einstein field equation solution as a realistic approximation for black holes. In this paper we show by means of analytic calculations that all these effects indeed exist, but their magnitude is typically negligibly small, even though the matter stress tensor is divergent in the static frame at $r=2M$. That is not surprising because of the divergent relation of that frame to a freely falling frame as $r \rightarrow 2M$; however it represents a kind of classical analogue for the Black Hole Complementarity principle that has been proposed for quantum effects near a black hole. What is perhaps more surprising is the divergence of the radiation stress tensor for massive particles moving on circular geodesic orbits for values of $r$ approaching $r = 3M$. However such orbits will not occur for infalling matter in realistic accretion discs.

Exploring the outer limits of numerical relativity

We perform several black-hole binary evolutions using fully nonlinear numerical relativity techniques at separations large enough that low-order post-Newtonian expansions are expected to be accurate. As a case study, we evolve an equal-mass nonspinning black-hole binary from a quasicircular orbit at an initial coordinate separation of $D=100M$ for three different resolutions. We find that the orbital period of this binary (in the numerical coordinates) is $T=6422M$. The orbital motion agrees with post-Newtonian predictions to within 1%. However, we find that the time derivative of the coordinate separation is dominated by a purely gauge effect leading to an apparent contraction and expansion of the orbit at twice the orbital frequency. Based on these results, we improved our evolution techniques and studied a set of black hole binaries in quasicircular orbits starting at $D=20M$, $D=50M$, and $D=100M$ for $\ensuremath{\sim}5$, 3, and 2 orbits, respectively. We then find good agreement between the numerical results and post-Newtonian predictions for the orbital frequency and radial decay rate, radiated energy and angular momentum, and waveform amplitude and phases. The results are relevant for the future computation of long-term waveforms to assist in the detection and analysis of gravitational waves by the next generation of detectors as well as the long-term simulations of black-hole binaries required to accurately model astrophysically realistic circumbinary accretion disks.

THE INFLUENCE OF OUTER SOLAR SYSTEM ARCHITECTURE ON THE STRUCTURE AND EVOLUTION OF THE OORT CLOUD

We study the influence of outer Solar System architecture on the structural evolution of the Oort Cloud (OC) and the flux of Earth-crossing comets. In particular, we seek to quantify the role of the giant planets as "planetary protectors". To do so, we have run simulations in each of four different planetary mass configurations to understand the significance of each of the giant planets. Because the outer planets modify the structure of the OC throughout its formation, we integrate each simulation over the full age of the Solar System. Over this time, we follow the evolution of cometary orbits from their starting point in the protoplanetary disk to their injection into the OC to their possible re-entry into the inner planetary region. We find that the overall structure of the OC, including the location of boundaries and the relative number of comets in the inner and outer parts, does not change significantly between configurations; however, as planetary mass decreases, the trapping efficiency (TE) of comets into the OC and the flux of comets into the observable region increases. We determine that those comets that evolve onto Earth-crossing orbits come primarily from the inner OC but show no preference for initial protoplanetary disk location. We also find that systems that have at least a Saturn-mass object are effective at deflecting possible Earth-crossing comets but the difference in flux between systems with and without such a planet is less than an order of magnitude. We conclude by discussing the individual roles of the planets and the implications of incorporating more realistic planetary accretion and migration scenarios into simulations, particularly on existing discrepancies between low TE and the mass of the protoplanetary disk and on determining the structural boundaries of the OC.

Simulation studies of dark energy clustering induced by the formation of dark matter halos

In this paper, we present a simulation method within the two-component spherical collapse model to investigate dark energy perturbations associated with the formation of dark matter halos. The realistic mass accretion history of a dark matter halo taking into account its fast and slow growth is considered by imposing suitable initial conditions and isotropized virializations for the spherical collapse process. The dark energy component is treated as a perfect fluid described by two important parameters, the equation of state parameter $w$ and the sound speed $c_s$. Quintessence models with $w>-1$ are analyzed. We adopt the Newtonian gauge to describe the spacetime which is perturbed mainly by the formation of a dark matter halo. It is found that the dark energy density perturbation $\delta_{DE}$ depends on $w$ and $c_s$, and its behavior follows closely the gravitational potential $\Phi$ of the dark matter halo with $\delta_{DE}\approx -(1+w)\Phi/c_s^2$. For $w>-1$, the dark energy perturbation presents a clustering behavior with $\delta_{DE}>0$ during the entire formation of the dark matter halo, from linear to nonlinear and virialized stages. The value of $\delta_{DE}$ increases with the increase of the halo mass. For a cluster of mass $M\sim 10^{15} M_{\odot}$, $\delta_{DE}\sim 10^{-5}$ within the virialized region for $c_s^2 \in [0.5, 1]$, and it can reach $\delta_{DE}=O(1)$ with $c_s^2=0.00001$. For a scalar-field dark energy model, we find that with suitably modeled $w$ and $c_s$, its perturbation behavior associated with the nonlinear formation of dark matter halos can well be analyzed using the fluid approach, demonstrating the validity of the fluid description for dark energy even considering its perturbation in the stage of nonlinear dark matter structure formation.

FORMATION OF TERRESTRIAL PLANETS FROM PROTOPLANETS UNDER A REALISTIC ACCRETION CONDITION

The final stage of terrestrial planet formation is known as the giant impact stage where protoplanets collide with one another to form planets. So far this stage has been mainly investigated by N-body simulations with an assumption of perfect accretion in which all collisions lead to accretion. However, this assumption breaks for collisions with high velocity and/or a large impact parameter. We derive an accretion condition for protoplanet collisions in terms of impact velocity and angle and masses of colliding bodies, from the results of numerical collision experiments. For the first time, we adopt this realistic accretion condition in N-body simulations of terrestrial planet formation from protoplanets. We compare the results with those with perfect accretion and show how the accretion condition affects terrestrial planet formation. We find that in the realistic accretion model, about half of collisions do not lead to accretion. However, the final number, mass, orbital elements, and even growth timescale of planets are barely affected by the accretion condition. For the standard protoplanetary disk model, typically two Earth-sized planets form in the terrestrial planet region over about 100 M years in both realistic and perfect accretion models. We also find that for the realistic accretion model, the spin angular velocity is about 30% smaller than that for the perfect accretion model that is as large as the critical spin angular velocity for rotational instability. The spin angular velocity and obliquity obey Gaussian and isotropic distributions, respectively, independently of the accretion condition.

The influence of the monomer shape in the first stage of dust growth in the protoplanetary disk

<i>Context. <i/>Laboratory experiments have shown that the initial stage of the dust growth in the protoplanetary disk starts with low-velocity hit-and-stick collisions between sub-micron grains, before the relative impact energy becomes large enough to cause the morphological restructuring of the forming aggregates. The results of these collisions are loose aggregates characterized by fractal dimensions <i>≤<i/>2. Numerical studies generally model this early stage with colliding clusters made of monodisperse spherical monomers.<i>Aims. <i/>In this paper we aimed to investigate how a more complex representation of the monodisperse monomers structure influences the morphology of the final aggregate particles in terms of fractal dimension, porosity, cross-section, and friction time. This study had also the purpose of testing the validity of the current fractal models in representing irregular particles.<i>Methods. <i/>We used three kinds of hit-and-stick aggregation methods: two particle-cluster aggregations producing compact and extremely loose aggregates, respectively, and simplified cluster-cluster aggregation as intermediate class in terms of fluffyness and porosity. In our measures, we used two kinds of monomer shapes: spheres and elongated prolate ellipsoids with axis ratio 3:1 resembling the ones used to model interstellar dust grains.<i>Results. <i/>We found that the monomer shape has little influence (<i>≲<i/>2%) on the fractal dimension of the final aggregate, once large particles are taken into account, independently on the morphological class. Elongated monomers produce structures with larger fractal dimension in the limit of small particles. For loose objects the influence on the porosity is even smaller, whereas elongated monomers give particles with ~5% higher porosity in the case of large compact fractals. Regardless the ellipsoidal monomer used has, by construction, a cross-section ~26% larger than the one of the volume-equivalent sphere, the difference in cross-section per unit mass is decreasing when aggregates are considered. Clusters made by elliptical monomers result much coupled to the surrounding gas than objects made by spherical monomers belonging to the same morphological class. Differences in the friction time are estimated to be below 15%. We conclude that the monomer shape influence on the morphology of the final particles during the hit-and-stick aggregation step is much lower than the differences due to considering different morphological classes in modeling and the uncertainties produced in a realistic accretion scenario, where a wide distribution of morphologies is expected. Present models of irregular particles as fractals made by spherical monomers can therefore also represent well more complex shapes, before the restructuring phase is taken into account.

EFFECTS OF MAGNETIC FIELDS ON THE DISKOSEISMIC MODES OF ACCRETING BLACK HOLES

The origin of the rapid quasi-periodic variabilities observed in a number of accreting black hole X-ray binaries is not understood. It has been suggested that these variabilities are associated with diskoseismic oscillation modes of the black hole accretion disk. In particular, in a disk with no magnetic field, the so-called g-modes (inertial oscillations) can be self-trapped at the inner region of the disk due to general relativistic effects. Real accretion disks, however, are expected to be turbulent and contain appreciable magnetic fields. We show in this paper that even a weak magnetic field (with the magnetic energy much less than the thermal energy) can modify or "destroy" the self-trapping zone of disk g-modes, rendering their existence questionable in realistic black hole accretion disks. The so-called corrugation modes (c-modes) are also strongly affected when the poloidal field approaches equal-partition. On the other hand, acoustic oscillations (p-modes), which do not have vertical structure, are not affected qualitatively by the magnetic field, and therefore may survive in a turbulent, magnetic disk.

Cosmological Simulations of the Preheating Scenario for Galaxy Cluster Formation: Comparison to Analytic Models and Observations

We perform a set of nonradiative cosmological simulations of a preheated intracluster medium in which the entropy of the gas was uniformly boosted at high redshift. The results of these simulations are used first to test the current analytic techniques of preheating via entropy input in the smooth accretion limit. When the unmodified profile is taken directly from simulations, we find that this model is in excellent agreement with the results of our simulations. This suggests that preheated efficiently smooths the accreted gas, and therefore a shift in the unmodified profile is a good approximation even with a realistic accretion history. When we examine the simulation results in detail, we do not find strong evidence for entropy amplification, at least for the high-redshift preheating model adopted here. In the second section of the paper, we compare the results of the preheating simulations to recent observations. We show—in agreement with previous work—that for a reasonable amount of preheating, a satisfactory match can be found to the mass-temperature and luminosity-temperature relations. However, as noted by previous authors, we find that the entropy profiles of the simulated groups are much too flat compared to observations. In particular, while rich clusters converge on the adiabatic self-similar scaling at large radius, no single value of the entropy input during preheating can simultaneously reproduce both the core and outer entropy levels. As a result, we confirm that the simple preheating scenario for galaxy cluster formation, in which entropy is injected universally at high redshift, is inconsistent with observations.

Atmospheres of Protoplanetary Cores: Critical Mass for Nucleated Instability

We systematically study quasi-static atmospheres of accreting protoplanetary cores for different opacity behaviors and realistic planetesimal accretion rates in various parts of the protoplanetary nebula. We demonstrate that there are two important classes of atmospheres: (1) those having an outer convective zone that smoothly merges with the surrounding nebular gas, and (2) those possessing an almost isothermal outer radiative region that effectively decouples the atmospheric interior from the nebula. The type of atmosphere accumulating around a given core depends on the core mass, nebular parameters, and accretion luminosity of the core. Cores in the inner parts of the protoplanetary disk (within roughly 0.3 AU from the Sun) have large luminosities resulting in atmospheres of the first type, while cores in the giant planet region (beyond several AU) have small accretion luminosities and always accumulate massive atmospheres of the second type. The critical core mass needed for the nucleated instability to commence is found to vary considerably as a function of distance from the Sun. This mass is 5-20 M⊕ at 0.1-1 AU, which is too large to permit the formation of hot Jupiters by nucleated instability around the cores that have grown in situ. In the region of giant planets the critical core mass depends on the gas opacity and planetesimal accretion rate but is insensitive to the nebular temperature or density provided that the opacity in the outer radiative region does not depend on the gas density (e.g., dust opacity). The critical mass in the region of giant planets can be as high as 20-60 M⊕ (for an opacity of 0.1 cm2 g-1) if planetesimal accretion is fast enough for protoplanetary cores to form prior to the nebular gas dissipation.

The Nonlinear Behavior of the Black Hole System GRS 1915+105

Using nonlinear time series analysis, along with surrogate data analysis, it is shown that the various types of long-term variability exhibited by the black hole system GRS 1915+105 can be explained in terms of a deterministic nonlinear system with some inherent stochastic noise. Evidence is provided for a nonlinear limit cycle origin of one of the low-frequency QPOs detected in the source, while some other types of variability could be due to an underlying low-dimensional chaotic system. These results imply that the partial differential equations that govern the magnetohydrodynamic flow of the inner accretion disk can be approximated by a small number (≈3-5) of nonlinear but ordinary differential equations. While this analysis does not reveal the exact nature of these approximate equations, they may be obtained in the future, after results of magnetohydrodynamic simulation of realistic accretion disks become available.

X-ray observations of black-hole accretion disks

Our current theoretical and observational understandings of the accretion disks around Galactic black-holes are reviewed. Historically, a simple phenomenological accretion disk model has been used to interpret X-ray observations. Although such a phenomenological interpretation is still useful, high quality X-ray data from contemporary instruments allow us to test more realistic accretion disk models. In a simple and ideal case, the standard optically thick accretion disk model is successful to explain observations, such that the inner disk radius is constant at three times the Schwarzschild radius over large luminosity variations. However, when disk luminosity is close to or exceeds the Eddington luminosity, the standard disk model breaks, and we have to consider the “slim disk” solution in which radial energy advection is dominant. Recent observations of Ultra-luminous X-ray sources (ULXs), which may not be explained by the standard disk model, strongly suggest the slim disk solution. We compare theoretical X-ray spectra from the slim disk with observed X-ray spectra of ULXs. We have found that the slim disk model is successful to explain ULX spectra, in terms of the massive stellar black-holes with several tens of solar mass and the super-Eddington mass accretion rates. In order to explain the large luminosities (>10 40 ergs s −1) of ULXs, “intermediate black-holes” (>100 M ⊙) are not required. Slim disks around massive stellar black-holes of up to several tens of solar mass would naturally explain the observed properties of ULXs.

Dynamics of a classical spinning particle in Vaidya space-time

Based on the Mathisson-Papapetrou-Dixon (MPD) equations and the Vaidya metric, the motion of a spinning point particle orbiting a non-rotating star while undergoing radiation-induced gravitational collapse is studied in detail. A comprehensive analysis of the orbital dynamics is performed assuming distinct central mass functions which satisfy the weak energy condition, in order to determine a correspondence between the choice of mass function and the spinning particle's orbital response, as reflected in the gravitational waves emitted by the particle. The analysis presented here is likely most beneficial for the observation of rotating solar mass black holes or neutron stars in orbit around intermediate-sized Schwarzschild black holes undergoing radiation collapse. The possibility of detecting the effects of realistic mass accretion based on this approach is considered. While it seems unlikely to observe such effects based on present technology, they may perhaps become observable with the advent of future detectors.

New multi-zoom method for N-body simulations: application to galaxy growth by accretion

The growth of galaxies is driven by two processes: mergers with other galaxies and smooth accretion of intergalactic gas. The relative share of this two processes depends on the environment (rich cluster or field), and determines the morphological evolution of the galaxy. In this work we focus on the properties of accretion onto galaxies. Through numerical simulations we investigate the geometrical properties of accretion. To span the scale range required in these simulations we have developed a new numerical technique: the multi-zoom method. We run a series of Tree-SPH simulations in smaller and smaller boxes at higher and higher mass resolution, using data recorded at the previous level to account for the matter inflow and the tidal field from outside matter. The code is parallelized using OpenMP. We present a validation test to evaluate the robustness of the method: the pancake collapse. We apply this new multizoom method to study the accretion properties. Zooming in onto galaxies from a cosmological simula- tion, we select a sample of 10 well resolved galaxies (5000 baryonic particles or more). We sum up their basic properties and plot a Tully-Fisher relation. We find that smooth accretion of intergalactic cold gas dominates mergers for the mass growth of galaxies at z < 2. Next we study the baryonic accretion rate which shows different behaviours depending on the galaxy mass. The bias is also computed at different radii and epochs. Then we present galactocentric angular maps for the accretion integrated between z = 2a ndz = 0, which reveal that accretion is highly anisotropic. Average accretion rates plotted against galactocentric latitude show a variety of behaviours. In general, accretion in the galactic plane is favored, even more for baryonic matter than for dark matter. Our results form a basis for prescribing realistic accretion in simulations of isolated galaxies.

Total and Jet Blandford-Znajek Power in the Presence of an Accretion Disk

A rotating black hole probably powers ultrarelativistic jets in gamma-ray bursts, relativistic jets from some active galactic nuclei, and jets from some black hole x-ray binaries. Prior estimates of the power output of a black hole have assumed an infinitely thin disk, a magnetic field based upon a slowly rotating black hole, and have not self-consistently determined the geometry or magnitude of the magnetic field for a realistic accretion disk. We provide useful formulae for the total and jet Blandford-Znajek (BZ) power and efficiency as determined self-consistently from general relativistic magnetohydrodynamic numerical models. Of all jet mechanisms, we suggest that only the BZ mechanism is able to produce an ultrarelativistic jet.

Protostellar Disk Dynamos and Hydromagnetic Outflows in Primordial Star Formation

Are magnetic fields important in primordial star formation? Assuming that star formation occurs via an accretion disk that is turbulent, initially because of local gravitational instability, we calculate the disk structure for realistic accretion rates. We predict that local gravitational viscosity is able to drive accretion, without the disk fragmenting. We then estimate the rate of dynamo amplification of seed magnetic field. Turbulence in a stratified disk can be helical, with different signs of the helicity in each hemisphere. This provides a key ingredient for production of global-scale magnetic fields whose sign of flux is sustained over many orbit times. The resulting fields can drive collimated protostellar outflows that reduce the star formation efficiency from the initial gas cloud, especially once the protostar has contracted to the main sequence, at ~100 M☉. We estimate that the outflows are powerful enough to eject some material from the host dark matter halo and to initiate relatively strong magnetization of the local intergalactic medium. Close to the protostar, the outflow acts to shield the disk and equatorial regions from radiative feedback, such as ionizing photons, and this may enable accretion up to relatively large stellar masses. We conclude that magnetic fields cannot be ignored in models of primordial star formation.

Formation of the First Stars by Accretion

The process of star formation from metal-free gas is investigated by following the evolution of accreting protostars with emphasis on the properties of massive objects. The main aim is to establish the physical processes that determine the upper mass limit of the first stars. Although the consensus is that massive stars were commonly formed in the first cosmic structures, our calculations show that their actual formation depends sensitively on the mass accretion rate and its time variation. Even in the rather idealized case in which star formation is mainly determined by acc, the characteristic mass scale of the first stars is rather uncertain. We find that there is a critical mass accretion rate crit 4 × 10-3 M☉ yr-1 that separates solutions with acc crit) where the maximum mass limit decreases as acc increases. In the latter case, the protostellar luminosity reaches the Eddington limit before the onset of hydrogen burning at the center via the CN cycle. This phase is followed by a rapid and dramatic expansion of the radius, possibly leading to reversal of the accretion flow when the stellar mass is about 100 M☉. Under a realistic time-dependent accretion rate that starts at high values (~10-2 M☉ yr-1) and decreases rapidly in the high-mass regime (M* 90 M☉), the evolution follows the case of acc < crit and accretion can continue unimpeded by radiation forces. Thus, the maximum mass is set by consideration of stellar lifetimes rather than by protostellar evolution. In this case, the upper limit can be as high as ~600 M☉. We consider also the sensitivity of the results to the presence of heavy elements with abundances in the range Z = 5 × 10-5 to 5 × 10-3 Z☉. The main evolutionary features of protostars are similar to those of metal-free objects, except that the value of crit increases for metal-enriched protostars. Since the accretion rate is lower in a slightly polluted environment, the condition acc < crit is expected to be more easily met. We find that for metallicities below ~10-2 Z☉, where radiation forces onto dust grains in the flow are negligible, a slightly metal-rich gas favors continued accretion and the formation of very massive stars.