Although fundamental for astrophysics, the processes that produce massive stars are not well understood. Large distances, high extinction, and short timescales of critical evolutionary phases make observations of these processes challenging. Lacking good observational guidance, theoretical models have remained controversial. This review offers a basic description of the collapse of a massive molecular core and a critical discussion of the three competing concepts of massive star formation:• monolithic collapse in isolated cores• competitive accretion in a protocluster environment• stellar collisions and mergers in very dense systems We also review the observed outflows, multiplicity, and clustering properties of massive stars, the upper initial mass function and the upper mass limit. We conclude that high-mass star formation is not merely a scaledup version of low-mass star formation with higher accretion rates, but partly a mechanism of its own, primarily owing to the role of stellar mass and radiation pressure in controlling the dynamics.

We perform a large set of radiation hydrodynamics simulations of primordial star formation in a fully cosmological context. a Our statistical sample of 100 First Stars show that the first generation of stars have a wide mass distribution M popIII = 10 ∼ 1000 M ⊙ . We first run cosmological simulations to generate a set of primordial star-forming gas clouds. We then follow protostar formation in each gas cloud and the subsequent protostellar evolution until the gas mass accretion onto the protostar is halted by stellar radiative feedback. The accretion rates differ significantly among the primordial gas clouds which largely determine the final stellar masses. For low accretion rates the growth of a protostar is self-regulated by radiative feedback effects and the final mass is limited to several tens of solar masses. At high accretion rates the protostar's outer envelope continues to expand and the effective surface temperature remains low; such protostars do not exert strong radiative feedback and can grow in excess to one hundred solar masses. The obtained wide mass range suggests that the first stars play a variety of roles in the early universe, by triggering both core-collapse supernovae and pair-instability supernovae as well as by leaving stellar mass black holes. We find certain correlations between the final stellar mass and the physical properties of the star-forming cloud. These correlations can be used to estimate the mass of the first star from the properties of the parent cloud or of the host halo, without following the detailed protostellar evolution.

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The first stars fundamentally transformed the early universe by emitting the first light and by producing the first heavy elements. These effects were predetermined by the mass distribution of the first stars, which is thought to have been fixed by a complex interplay of gas accretion and protostellar radiation. We performed radiation-hydrodynamics simulations that followed the growth of a primordial protostar through to the early stages as a star with thermonuclear burning. The circumstellar accretion disk was evaporated by ultraviolet radiation from the star when its mass was 43 times that of the Sun. Such massive primordial stars, in contrast to the often-postulated extremely massive stars, may help explain the fact that there are no signatures of the pair-instability supernovae in abundance patterns of metal-poor stars in our galaxy.

Supermassive stars (SMSs) forming via very rapid mass accretion (Ṁ * 0.1 M ⊙ yr −1 ) could be precursors of supermassive black holes observed beyond redshift of about 6. Extending our previous work, we here study the evolution of primordial stars growing under such rapid mass accretion until the stellar mass reaches 10 4−5 M ⊙ . Our stellar evolution calculations show that a star becomes supermassive while passing through the "supergiant protostar" stage, whereby the star has a very bloated envelope and a contracting inner core. The stellar radius increases monotonically with the stellar mass, until ≃ 100 AU for M * 10 4 M ⊙ , after which the star begins to slowly contract. Because of the large radius the effective temperature is always less than 10 4 K during rapid accretion. The accreting material is thus almost completely transparent to the stellar radiation. Only for M * 10 5 M ⊙ can stellar UV feedback operate and disturb the mass accretion flow. We also examine the pulsation stability of accreting SMSs, showing that the pulsation-driven mass loss does not prevent stellar mass growth. Observational signatures of bloated SMSs should be detectable with future observational facilities such as the James Webb Space Telescope. Our results predict that an inner core of the accreting SMS should suffer from the general relativistic instability soon after the stellar mass exceeds 10 5 M ⊙ . An extremely massive black hole should form after the collapse of the inner core.