The Jeans criterion defines the critical conditions required for an interstellar cloud to undergo spontaneous gravitational collapse. According to the virial theorem, collapse occurs when twice the internal kinetic energy ($2K$) is less than the absolute value of the gravitational potential energy ($|U|$). For a spherical cloud of constant density $\rho_0$ and temperature $T$, the Jeans mass ($M_J$) represents the minimum mass required to initiate collapse: $$M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$$ where $k$ is Boltzmann’s constant, $\mu$ is the mean molecular weight, and $m_H$ is the mass of a hydrogen atom. Similarly, the Jeans length ($R_J$) is the minimum radius required for a cloud of a given density to collapse: $$R_J \approx \left(\frac{15kT}{4\pi G\mu m_H \rho_0}\right)^{1/2}$$ Diffuse HI clouds are generally stable against collapse because their masses are much lower than their Jeans mass, whereas the dense cores of giant molecular clouds (GMCs) often satisfy this criterion.

Homologous Collapse: Once the Jeans criterion is satisfied, if pressure gradients are initially negligible, the cloud undergoes free-fall collapse. This collapse is isothermal as long as the cloud remains optically thin and can radiate away the released gravitational potential energy. The equation of motion for a mass shell at radius $r$ is: $$\frac{d^2r}{dt^2} = -G \frac{M_r}{r^2}$$ The free-fall timescale ($t_{ff}$) is the time required for a uniform cloud to collapse to a point (assuming $r_{final} \approx 0$): $$t_{ff} = \left(\frac{3\pi}{32} \frac{1}{G\rho_0}\right)^{1/2}$$ Because $t_{ff}$ is independent of the initial radius $r_0$, all parts of a uniform cloud collapse in the same amount of time, a process known as homologous collapse. If the cloud is centrally condensed, it undergoes an inside-out collapse where the core collapses faster than the outer layers.

Fragmentation of Collapsing Clouds: As an isothermal collapse proceeds, the density $\rho$ increases while $T$ remains constant, which causes the Jeans mass to decrease ($M_J \propto \rho^{-1/2}$). Initial density inhomogeneities then allow smaller sections of the cloud to satisfy the Jeans criterion independently and collapse locally, leading to fragmentation. This cascading process segments the cloud into many smaller objects, explaining why stars often form in groups. Fragmentation ceases when the cloud becomes optically thick and the collapse becomes adiabatic. In an adiabatic collapse, the temperature rises ($T \propto \rho^{\gamma-1}$), causing $M_J$ to increase with density ($M_J \propto \rho^{1/2}$ for atomic hydrogen), which establishes a minimum fragment mass: $$M_{J,min} \approx 0.03 \left( \frac{T^{1/4}}{e^{1/2}\mu^{9/4}} \right) M_\odot$$ For typical parameters, this limit is approximately $0.01$ to $0.5$ $M_\odot$.

A protostar forms when the central region of a collapsing fragment becomes optically thick, causing internal pressure to rise and slowing the collapse into a state of near-hydrostatic equilibrium. This hydrostatic core typically begins with a radius of about $5$ AU. The protostar is initially powered by a shock wave at its surface where infalling material from the envelope arrives at supersonic speeds, converting kinetic energy into heat. A second, more rapid collapse occurs when central temperatures reach ~2000 K, causing molecular hydrogen to dissociate, which removes the pressure support needed for equilibrium. Hydrostatic equilibrium is re-established once the core radius shrinks to about 1.3 times the size of the Sun.

Ambipolar Diffusion: Magnetic fields can inhibit cloud collapse by providing magnetic pressure. While charged ions and electrons are “frozen-in” to magnetic field lines, neutral particles (atoms and molecules) are not. Under the influence of gravity, neutrals slowly migrate toward the center of the cloud by drifting across magnetic field lines, a process called ambipolar diffusion. The neutrals collide with ions during this drift, which transfers some gravitational force to the magnetic field. The characteristic timescale for this diffusion is: $$t_{AD} \approx \frac{2R}{v_{drift}} \approx 10 \text{ Gyr} \left(\frac{n_{H_2}}{10^{10} \text{ m}^{-3}}\right) \left(\frac{B}{1 \text{ nT}}\right)^{-2} \left(\frac{R}{1 \text{ pc}}\right)^2$$ This long timescale explains why dense cores can remain stable for millions of years before free-fall collapse begins.

Hayashi Track: The Hayashi track is a nearly vertical evolutionary path on the Hertzsprung-Russell (H-R) diagram followed by a protostar with a deeply convective envelope. Because convection is highly efficient at transporting luminosity, the star’s luminosity drops rapidly while its effective temperature increases only slightly as it contracts. The Hayashi track serves as a boundary; no stable hydrostatic stars can exist to its right (lower temperatures) because no mechanism can transport the required luminosity at those temperatures.

Pre-Main-Sequence Evolution: The pre-main-sequence (PMS) phase begins once the protostar settles into a quasi-static contraction. The evolution is now controlled by the Kelvin-Helmholtz timescale ($t_{KH}$), as the star radiates away gravitational energy released by its slow collapse. For a $1 M_\odot$ star, this takes approximately 40 Myr. During this phase, deuterium burning may occur, temporarily slowing the contraction. As the core heats up, a radiative zone develops and expands, causing the evolutionary track to turn away from the Hayashi track and move horizontally toward higher temperatures.

Brown Dwarfs: Objects with masses below approximately $0.072 M_\odot$ (for solar composition) are known as brown dwarfs. Their cores never reach temperatures high enough to sustain stable hydrogen fusion to counteract gravitational collapse. They may temporarily burn deuterium (if $M > 0.013 M_\odot$) or lithium (if $M > 0.06 M_\odot$), but they eventually cool and fade. They are characterized by cool L and T spectral types.

Birth of Massive Stars: Massive stars ($M > 10 M_\odot$) evolve much faster than low-mass stars, reaching the main sequence in as little as 28,000 years for a $60 M_\odot$ star. Because their central temperatures are high, they ignite hydrogen via the CNO cycle, which is highly temperature-dependent and maintains a convective core even after reaching the main sequence. Their PMS tracks are nearly horizontal, as they leave the Hayashi track almost immediately. Their intense ionizing radiation may disperse the surrounding cloud, potentially limiting the formation of nearby low-mass stars.

Zero-Age Main-Sequence (ZAMS): The Zero-Age Main Sequence (ZAMS) is the diagonal line on the H-R diagram where stars of various masses first reach a state of equilibrium hydrogen burning. At this point, nuclear energy production exactly balances the star’s luminosity, and gravitational contraction stops.

Initial Mass Function (IMF): The initial mass function (IMF), denoted as $\xi$, describes the relative number of stars that form in different mass intervals from a fragmented cloud. Fragmentation typically produces a large abundance of low-mass stars and very few massive stars. While the function is well-modeled for higher masses, it is less certain for objects below $0.1 M_\odot$, where it may become relatively flat.

HII regions are large clouds of ionized hydrogen surrounding hot, young O and B stars. These stars emit intense ultraviolet radiation ($\lambda < 91.2$ nm) that ionizes the surrounding gas. Recombination of electrons and protons leads to a cascade of photons, including the red $H\alpha$ Balmer line, which causes these regions to fluoresce. The Strömgren radius ($r_S$) defines the equilibrium size of an HII region, where the ionization rate equals the recombination rate: $$r_S \approx \left(\frac{3N}{4\pi\alpha}\right)^{1/3} n_H^{-2/3}$$ where $N$ is the rate of ionizing photons and $\alpha$ is the recombination coefficient.

OB Associations: OB associations are loose groups of young, massive O and B stars. These clusters are typically gravitationally unbound because the intense radiation and stellar winds from the massive stars quickly disperse the remaining gas cloud that provided the binding mass. Consequently, the member stars tend to drift apart over time.

T Tauri Stars: T Tauri stars are low-mass ($0.5$ to $2 M_\odot$) pre-main-sequence stars that have emerged from their dust cocoons but have not yet reached the ZAMS. They are characterized by irregular luminosity variations, strong lithium absorption, and emission lines from hydrogen, calcium, and iron. Many exhibit a P Cygni profile in their $H\alpha$ lines—a broad emission peak with a blueshifted absorption trough—indicating significant mass loss via strong stellar winds.

Herbig-Haro Objects: Herbig-Haro (HH) objects are small, bright nebulae found near young stars, created by high-speed jets of gas ejected from the protostar or its accretion disk. As these jets collide with the interstellar medium at supersonic speeds, the resulting shocks excite and ionize the gas, producing characteristic emission-line spectra.

Circumstellar Disk Formation: As a protostellar cloud collapses, it spins up to conserve angular momentum ($L = I\omega = \text{constant}$). The resulting centripetal acceleration halts the collapse in the plane perpendicular to the rotation axis, while collapse along the axis continues. This leads to the formation of a flattened circumstellar accretion disk. The Hill radius ($R_H$) defines the region around a growing protoplanet within the disk where its gravity dominates: $$R_H = a \left(\frac{M}{M_\odot}\right)^{1/3}$$ where $a$ is the orbital radius and $M$ is the protoplanet mass. Most main-sequence stars rotate much slower than expected from simple collapse, implying that angular momentum is transferred away, likely by magnetic braking and stellar winds.