Star formation

Star formation is the gravitational collapse and fragmentation of cold interstellar gas into protostars and eventually stars. Cooling, turbulence, magnetic fields, rotation, and feedback determine its efficiency and characteristic masses.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

Gravitational fragmentation divides a self-gravitating gas distribution into condensations that can undergo separate collapse. Pressure support, cooling and rotation influence the scales and outcome; it is not equivalent to an instantaneous equal-mass fission of a rigidly rotating sphere. Isothermal fragmentation is a useful limiting case in which thermal adjustment keeps the gas temperature approximately fixed.

The Kennicutt–Schmidt law relates star-formation-rate surface density to gas surface density, commonly as ${\Sigma }_{\mathrm{SFR}}=A{\Sigma }_g^N$. Disk-averaged total-gas measurements give $N$ near $1.4$, while resolved molecular-gas relations are often closer to linear.

The gas-depletion time is the time required to consume a gas reservoir at its current star-formation rate, $t_{\mathrm{dep}}=M_g/{\dot{M}}_{\ast }$. It ignores replenishment, returned stellar mass, and changes in the rate.

The star-formation efficiency per free-fall time is the fraction of gas converted into stars during one gravitational free-fall time, ${ϵ}_{\mathrm{ff}}=t_{\mathrm{ff}}/t_{\mathrm{dep}}$. Molecular clouds commonly have values of order one per cent.

The Jeans mass is the characteristic gas mass above which self-gravity overcomes pressure support. Up to the convention used to define the collapsing region,for an ideal gas of fixed composition.

For a uniform ideal gas sphere, set $\rho =mn$, $p=n{k}_{B}T$ and $M=4\pi \rho R^3/3$. Building the sphere by concentric shells gives gravitational potential energy $W=-3G{M}^2/(5R)$. During a small homologous contraction, released gravitational energy balances the work against gas pressure when $3G{M}^2|dR|/(5R^2)=4\pi R^{2}p|dR|$, or $GMm/R=5k_{B}T$. Eliminating $R$ gives the displayed mass. At the same initial mass density and temperature, larger masses have stronger gravitational contraction than pressure support in this estimate. Its numerical coefficient belongs to the uniform-sphere pressure-work model; the infinite-medium Jeans instability and a pressure-confined isothermal sphere use different equilibrium and boundary assumptions.

During isothermal collapse, $T$ is constant and the Jeans mass decreases as ${\rho }^{-1/2}$. Successively denser regions can therefore become independently unstable and fragment to smaller masses.

For adiabatic monatomic gas, $T\propto {\rho }^{2/3}$ and the Jeans mass increases as ${\rho }^{1/2}$. Once cooling fails and collapse becomes adiabatic, further fragmentation is suppressed.

The initial mass function is the distribution of stellar birth masses. It records the mass scales selected by cloud fragmentation, accretion, feedback, and stellar multiplicity.

For an initial mass function proportional to $M^{-p}$ on $M\ge M_{\min }$ with $p>1$, integration and normalization give the displayed upper-tail fraction. The probability integral transform makes $X$ uniform on $[0,1]$ for a randomly sampled birth mass. Evolutionary lifetime regions can then be measured as ordinary areas when the independent age coordinate is also uniform.