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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.

Galactic star formation is limited by how much gas becomes cold, shielded and self-gravitating, and by how long it remains in that state. A useful local estimate isThe star-formation efficiency per free-fall time ${ϵ}_{\mathrm{ff}}$ absorbs the fact that turbulence, magnetic support and stellar feedback prevent all eligible gas from collapsing in one free-fall time of a uniform sphere. Rotation and disk stability influence the supply of dense clouds, while gravitational collapse and gravitational fragmentation operate inside them.

For galaxy-averaged measurements, the Kennicutt–Schmidt law has ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^N$, with the classic mixed spiral/starburst sample giving $N\simeq 1.4$. This empirical relation was obtained using H-alpha and far-infrared star-formation tracers together with atomic-hydrogen and CO gas measurements.^[ref] Dust attenuation, contamination by older stellar heating, the stellar initial mass function and the CO-to-molecular-gas conversion must be controlled. A disk-averaged total-gas exponent is not automatically a small-scale cloud law.

Young stars then photoionize gas, drive winds and eventually inject supernova energy and momentum. These processes can disrupt clouds, sustain random motions or expel gas; radiative cooling and fresh inflow rebuild the cold supply. Large-scale star formation is a regulated conversion of gas, rather than unimpeded free fall of the entire disk. A long gas-depletion time ${\Sigma }_g/{\Sigma }_{\mathrm{SFR}}$ is compatible with short galactic dynamical times in the dense fraction.

A dense cloud core first collapses when self-gravity overwhelms its effective support. While radiation escapes readily, compression can remain approximately isothermal and the Jeans mass decreases as density rises, encouraging gravitational fragmentation. Angular momentum usually leads to a disk rather than unrestricted spherical infall.

As the central optical depth increases, cooling becomes slower than compression and pressure produces a first hydrostatic core. molecular hydrogen dissociation near $2000\mathrm{K}$ absorbs compressional energy and lowers the effective adiabatic response, permitting a second collapse of a protostar. After dissociation the gas stiffens and a much more compact stellar core forms. An accretion shock and surrounding disk then convert infall energy into radiation. A small central core forms before most of the envelope arrives, as demonstrated in radiation-hydrodynamic collapse calculations.^[ref]

The isothermal inside-out collapse of a singular isothermal sphere model provides one idealized supply law, $\dot{M}\sim a_T^3/G$, but rotation, magnetic fields and an unsteady reservoir modify real accretion. Its luminosity scale is $L_{\mathrm{acc}}\sim GM\dot{M}/r_{\ast }$. Dust can absorb the emerging short-wavelength radiation and re-emit it in the infrared, so an embedded source need not have a directly visible stellar photosphere.

As envelope accretion declines, gravitational contraction becomes important on the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(r_{\ast }L)$. deuterium-burning energy-generation rate can temporarily influence the radius; sustained core hydrogen burning eventually establishes the main sequence if the final mass is sufficient. A low-mass convective object approaches along a Hayashi-like track, while more massive objects need not share that track. accretion, contraction and nuclear burning are distinct energy sources, and a protostar is not generally a fixed-mass object.

Let $M=M_1+M_2$, and let $\mu =M_1{M}_2/M$ be the reduced mass. The stellar distances from the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. For a circular Kepler orbit, ${\Omega }^2=GM/a^3$, so summing the two orbital contributions gives the circular-binary orbital angular momentumThe separation $a$ is inside the square root. This also has the required dimensions of angular momentum.

For the homologous rotating collapse, label a fluid element by $s=r/R$ and its polar angle $\theta$. Its enclosed mass $m(s)$ is constant during stellar homology, as are the dimensionless density profile and the inertia coefficient $\alpha$. Conservation of angular momentum and solid-body rotation giveThe magnitude of the centrifugal acceleration is ${\Omega }^{2}r\sin \theta$, while the Newtonian gravitational field has magnitude $Gm(s)/r^2$. ThereforeUsing only the radial component of centrifugal force replaces $\sin \theta$ by ${\sin }^2\theta$ and leaves the scaling unchanged. The centrifugal force vanishes on the rotation axis; the central point is understood through the limiting field rather than a ratio of two zero forces.

At the equator of the cloud, critical rotation of a spherical cloud means ${\Omega }_c^2{R}_c=GM/R_c^2$, where $R_c=R_{\mathrm{crit}}$. ConsequentlyImmediately after the first fission, each daughter has mass $M/2$ and radius $a/2$ because the spheres touch. Their orbital moment of inertia is $M{a}^2/4$, while their combined spin moment of inertia is $\alpha M{a}^2/4$. With a common spin and orbital frequency ${\Omega }_f=\sqrt{GM/a^3}$, the synchronous fission model for binary formation givesThe common frequency can change during fission; corotation requires equal instantaneous frequencies, not an unchanged frequency from before the rearrangement. Equating the two values of $J$ yieldsFor a nonzero positive inertia coefficient, the necessary and sufficient condition in this algebraic model for $a<R_c$ isFor example, a moment of inertia of a uniform solid sphere has $\alpha =2/5$ and gives $a/R_c=64/49>1$; it fails the required compact-fission condition. A sufficiently centrally concentrated cloud can have smaller $\alpha$.

For the next collapse, neglect exchange of spin with the unchanged outer orbit, so each daughter's spin angular momentum is separately conserved. Immediately after the first fission that spin isFor a daughter of mass $m=M/2$ to reach critical rotation of a spherical cloud at radius $R_{c,d}$, the same relation gives $S_d=\alpha m\sqrt{Gm{R}_{c,d}}$. HenceEquivalently, its centrifugal-to-gravity ratio starts at $1/4$ when its radius is $a/2$ and reaches one after contraction by a factor four. Applying the same fission rule to this daughter produces an inner pair with separationCorotation at this second fission is local to each newly formed inner pair; it is not a single common frequency for the entire four-star hierarchy.

This spin assumption matters. If tidal locking instead kept each shrinking daughter synchronized to the fixed outer orbital period throughout the intervening collapse, its angular velocity would remain fixed. Its centrifugal-to-gravity ratio would then scale as $R_d^3$ and decrease, so it would never reach the proposed second fission. This is a counterexample to extending the corotation assumption through the whole contraction. The printed ratio describes the separately spin-conserving interpretation of the repeated fission hierarchy.

The model can generate a compact hierarchy algebraically, but is not a general physical account of multiple-star formation. In the allowed range $\alpha <1/3$, one has $a^{\prime }/a<1/8$, suggesting well-separated inner and outer scales. However, rapidly rotating gas deforms, touching daughters are tidally distorted, and pressure, gas flows, dissipation and spin-orbit torques cannot generally be ignored. The assumed identical profiles, equal mass splits, instantaneous corotation and later torque-free contractions are restrictive. Star formation can involve gravitational fragmentation and redistribution of angular momentum; the toy budget neither proves that fission occurs nor predicts the population of real multiple systems.