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For adiabatic collapse, $T{\rho }^{1-\gamma }$ is constant. With $\gamma =5/3$,The rising Jeans mass produces adiabatic suppression of fragmentation: smaller subregions become more pressure-supported as density increases.

For isothermal fragmentation, $T$ stays approximately constant, andThe instability scale then falls during collapse, allowing hierarchical fragmentation until cooling fails, opacity rises, or another source of support intervenes.

Primordial metal-free gas cools inefficiently, principally through molecular hydrogen, and remains relatively hot. It therefore has a larger Jeans mass and tends toward a top-heavy initial mass function of massive Population III stars. Metal lines and dust let enriched gas remain cool to higher density, so Population II stars extend to much lower birth masses.

These alternatives map directly onto black-hole seed channels. Massive Population III remnants produce light Population III remnant black-hole seeds. If cooling and fragmentation are strongly suppressed while a primordial halo supplies rapid inflow, near-monolithic collapse can produce a heavy direct-collapse black-hole seed. Intermediate cooling and fragmentation in a dense cluster can instead permit a runaway stellar-collision black-hole seed. The Jeans argument selects plausible mass scales; angular momentum, feedback, chemistry, and accretion determine which channel actually operates.

Star formation is the conversion of interstellar gas into self-gravitating protostars and eventually stars. A region can contract when gravity overcomes support from gas pressure and other motions or fields. Cooling matters because it removes compression energy: during approximately isothermal fragmentation, rising mass density lowers the Jeans mass, permitting smaller regions to collapse independently.

When the gas becomes opaque and cannot cool efficiently, adiabatic suppression of fragmentation changes that behaviour. A first hydrostatic core and second protostellar collapse describe successive physical stages rather than two arbitrary labels for the same contraction. Accretion, rotation and magnetic fields affect the resulting objects, while stellar feedback limits how much gas becomes stellar mass. The initial mass function summarizes the resulting birth-mass distribution.