Star formation

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.

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.

During continued compression of a first hydrostatic core, molecular hydrogen dissociation absorbs energy and softens the effective equation of state. This can trigger a second rapid contraction before a compact stellar core becomes pressure-supported. It is distinct from the initial nearly isothermal collapse and from later Kelvin-Helmholtz contraction at roughly fixed mass.

An accelerated contraction phase associated with molecular hydrogen dissociation absorbing compression energy and reducing effective pressure support. A compact stellar core forms when the gas can again oppose contraction. Its timing and geometry depend on the accretion and thermodynamic history.

A transient pressure-supported stage of a collapsing cloud after rising optical depth inhibits cooling. Continued accretion and changes of the molecular equation of state can lead to further collapse. It is distinct from the later compact stellar core.

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.

Replacing the sound speed in a Jeans mass estimate by the Alfvén speed $B/\sqrt{{\mu }_0\rho }$ gives the displayed magnetic-support mass scale. A sphere whose diameter is the corresponding wavelength gives $C={\pi }^{5/2}/6$; the crossing-time cloud support mass instead gives $C=(4\pi /3)(3\pi /32{)}^{3/2}$. The coefficient is model dependent, and a real magnetic field supplies anisotropic stresses rather than isotropic gas pressure.

During isotropic magnetic flux freezing, $B\propto {\rho }^{2/3}$, so $M_B$ and the number of such masses in a fixed material cloud remain constant. This is another expression of a conserved mass-to-flux ratio. It contrasts with the decreasing thermal Jeans mass of isothermal fragmentation, and does not rule out motion or fragmentation along field lines.

Equate the free-fall time of a uniform sphere $\sqrt{3\pi /(32G\rho )}$ to a radius-based sound-crossing time $R/c_s$, then substitute $R$ in $M=4\pi \rho R^3/3$. This gives the displayed mass. It has the scaling of a Jeans mass, but its coefficient differs from the convention using a sphere of diameter equal to the Jeans length. Equality of these two times is an approximate support comparison, not the exact marginal mode of an infinite-medium Jeans instability.

For a cloud supported by an effective speed of sound $v$ and confined by a surface pressure $P$, dimensional analysis fixes the gravitational instability mass scale to $v^4/(G^{3/2}{P}^{1/2})$. The dimensionless coefficient depends on the equilibrium model and the convention used for the instability threshold.

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 specifies the distribution of stellar masses at birth. Writing $\xi (M)=dN/dM$ means that $\xi (M)dM$ counts the expected number born in a small mass interval. Counts per logarithmic interval differ: $dN/d\ln M=M\xi (M)$. Confusing these conventions changes the reported slope even for the same population.

For a chosen finite birth-mass range, $\int \xi (M)dM$ gives the number of stars and $\int M\xi (M)dM$ their total mass. The mass and luminosity weighting of a stellar initial mass function explains why a small number of luminous massive stars can dominate the light without dominating the count. A present-day distribution also reflects deaths, continued star formation, unresolved companions and observational selection, so it is not automatically the birth distribution.

If birth counts obey $dN/dm\propto m^{-s}$ and individual luminosity scales as $m^q$, multiply by mass or luminosity and by $dm=md\ln m$ to obtain the displayed logarithmic weights. With $s=2.35$ and $q=4$, birth mass has weight $m^{-0.35}$ while light has weight $m^{2.65}$. Finite mass cutoffs are essential; stellar lifetimes also alter the luminosity weighting of an evolved population.

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.