Equations for Stellar Formation

Most school-level courses treat the formation of stars as an entirely qualitative topic: lots of description and no equations, except for those already on other parts of the syllabus like gravitational ones. If you're looking to make your astrophysics teaching a little more mathematical, the Jeans Instability (named after Sir James Jeans, not the denim fabric) can be slotted in quite nicely. It was the first analysis done of the conditions that must exist in a cloud of interstellar gas if it is to undergo gravitational collapse and form a protostar. Most treatments of this online or in textbooks are aimed at an undergraduate audience, but the necessary expressions can be derived fairly simply from knowledge that may well be on your syllabus under other topics.

We start with the idea of a single gas particle of mass m on the outer edge of a spherical cloud of radius R. The particle is at temperature T (in Kelvin) and the total mass of the cloud is M. In this situation, the cloud will undergo collapse if the particle's vibrational KE is less than its gravitational PE:


Solving for yields an equation for the Jeans Mass, which is the minimum mass needed for a cloud of given T and to collapse:


You can now calculate (for example) the maximum possible radius that a cloud of Hydrogen gas at 10 Kelvin could have and still collapse to form a 1-Solar Mass star.

The next stage can be to solve the expression for Jeans mass for R, then substitute it into a few other (hopefully) familiar equations to yield an expression for the minimum density that a cloud would need in order to undergo gravitational collapse: