 Solution

A protostar forms when a self-gravitating cloud fragment can contract faster than pressure support and energy loss can arrest it. A useful local scale is the Jeans length, ${\lambda }_J\sim c_s/\sqrt{G\rho }$, with characteristic Jeans mass $M_J\sim c_s^3/(G^{3/2}{\rho }^{1/2})$ and free-fall timefor an initially uniform sphere. Geometry, turbulence and magnetic support change the detailed threshold; none of these estimates asserts that every cloud contracts as a uniform sphere.

While cooling is efficient, compression is nearly isothermal. Then $M_J$ decreases as the density rises, allowing fragmentation. Rotation becomes increasingly important during contraction because a parcel conserving specific angular momentum $j$ encounters a centrifugal scale $r_c\sim j^2/(GM)$. The resulting accretion disc must transport angular momentum before matter can reach the central object. Infall, a dense accreting interior and an outer envelope can therefore coexist; a protostar is not initially an isolated fixed-mass star.

Increasing optical depth eventually slows radiative cooling. The central gas then becomes less compressible and can form a pressure-supported first core. Molecular hydrogen dissociation absorbs compressional energy and can trigger a second collapse before a compact stellar core is established. Subsequent accretion supplies a luminosity of orderThe retained fraction depends on shock and disc radiation, so this expression is an energy scale rather than a guaranteed luminosity all emerging from the stellar surface.

A central accreting core can be nearly hydrostatic on its short sound-crossing time while its mass, entropy and radius evolve on longer times. A low-mass object is often largely convective because of its opacity and steep radiative temperature gradient; deuterium burning can temporarily contribute to support and moderate contraction. Neither convection nor the timing of deuterium ignition is universal across all masses and accretion histories.

When the main accretion phase ends, gravitational contraction remains an energy source. The virial theorem explains how contraction both heats the interior and supplies radiation: for an ideal-gas-supported object with negligible surface-pressure correction, $2U+W\simeq 0$, so only part of the released binding energy appears as increased thermal energy. The corresponding Kelvin-Helmholtz time isFully convective pre-main-sequence evolution approximately follows a Hayashi track, with decreasing radius and luminosity at comparatively weakly changing effective temperature. Objects developing radiative interiors can follow Henyey tracks. Eventually sustained core hydrogen burning balances the luminosity and the star reaches the main sequence. Massive objects can ignite hydrogen while substantial accretion continues, so “accretion ends, then hydrogen ignites” is not a universal ordering.

A useful structural distinction is therefore between an accretion-powered protostar and a largely fixed-mass pre-main-sequence contraction model. An infrared-bright envelope or disc traces obscured formation and reprocessing of radiation; a schematic evolutionary track should not replace the actual mass- and entropy-dependent stellar-structure problem.