# The Formation of Protostars

The earliest stage of stellar evolution is the formation of protostars, pre-nuclear burning objects.  However, our concept of how globules and cores in molecular clouds collapse to form protostars is far from complete.  Sir James Jeans defined the ‘Jeans Criterion’ in the early 20th Century for the conditions required for collapse to occur.  In order to initiate spontaneous collapse, the mass of a cloud must exceed the Jeans mass defined as:

The Jeans Criterion can also be phrased in terms of a corresponding Jeans radius:

In both of these equations, $G$ is the gravitational constant, $k$ is the Boltzmann constant, $m_H$ is the mass of a hydrogen atom, $\mu$ is the mean molecular weight, $T$ is the temperature of the cloud, and $\rho_0$ is the initial density of the cloud.  The Jeans radius defines the minimum radius necessary for a cloud to collapse.  However, this criterion does not take the external pressure on the cloud due to the surrounding interstellar medium into account.  If that external pressure is accounted for, the critical mass required for gravitational collapse of a cloud is instead given by the Bonnor-Ebert mass:

Here, $P_0$ is the external gas pressure, $v_T$ is the isothermal sound speed, and $c_\text{BE}$ is a dimensionless constant ($c_\text{BE}$ ≈ 1.18).

If the criterion for gravitational collapse defined above is met and we ignore the effect of rotation, turbulence, or magnetic fields, the subsequent collapse of the cloud is essentially described as an isothermal free-fall.  This stage of protostellar evolution is termed ‘Homologous Collapse.’  In reality, a single cloud does not collapse to form a single star.  Instead, clouds fragment as they collapse, resulting in the formation of a group of stars.  Finally, this analysis still ignores many additional features of a collapsing cloud that should be accounted for including the initial velocity of the cloud’s outer layers, radiation transport through the cloud, vaporization of dust grains, dissociation of molecules, ionization of atoms, rotation of the cloud, and magnetic fields.  However, this simple analysis does illustrate many important features of protostar formation.  For a complete discussion of this simplified analysis and the additional complications required for a more sophisticated analysis, check out Caroll and Ostlie’s book An Introduction to Modern Astrophysics.

# Describing and Classifying the Evolution of a Protostar

The evolution of a protostar towards the Zero Age Main Sequence (ZAMS) is often defined by the evolution of the system geometry and the corresponding evolution of the protostar’s Spectral Energy Distribution (SED).  A SED is essentially a graph of how the flux of emission from an object depends on the wavelength the object is observed at– flux plotted versus wavelength (for a more detailed discussion of SEDs, see this other WordPress post).  A graphical overview of the four stages of protostar evolution is shown below (Andrea Isella’s thesis, 2006). Class 0 objects are characterized by a very embedded central core in a much larger accreting envelope. The mass of the central core grows in Class I objects and a flattened circumstellar accretion disk develops. For Class II objects, the majority of circumstellar material is now found in a disk of gas and dust. Finally, for Class III objects, the emission from the disk becomes negligible and the SED resembles a pure stellar photosphere.