= Solution
\b[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 time
$$
t_{\rm ff}=\sqrt{\frac{3\pi}{32G\rho}}
$$
for 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 order
$$
L_{\rm acc}\sim\frac{GM\dot M}{R}.
$$
The 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, $2U+W\simeq0$, so only part of the released <binding energy> appears as increased <thermal energy>. The corresponding Kelvin-Helmholtz time is
$$
t_{\rm KH}\sim\frac{GM^2}{RL}.
$$
Fully 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.
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