The Jeans Mass The Jeans mass is determined by asking when the magnitude of the gravitational potential energy exceeds the magnitude of the gas kinetic energy. It is given by MJ = 3kTR / 2Gm where k is the Boltzmann constant, T the temperature, R the radius of the cloud, G the gravitational constant, and m the average mass of a gas particle. If the cloud's mass exceeds this value, gravity can overcome the gas kinetic energy and initiate collapse. The corresponding Jeans density is DJ = constant x (T 3 / m 3M 2) where constant = 81k3 / 32πG3 and M is the mass of the cloud. If the cloud is compressed so that its density exceeds this value, it can begin to collapse.

#### The Jeans Collapse Criterion

We believe that stars form when a portion of a molecular cloud collapses gravitationally. Since we have seen that this collapse is resisted by various things, and since we see evidence for many molecular clouds that have not collapsed, it is clear that the collapse initiating star formation occurs only under some circumstances. Can we find a simple condition that tells us when a cloud becomes unstable to collapse?

##### The Jeans Mass
This question was answered in a very simple model by James Jeans (see the right panel), who showed that for a cloud of a given radius and temperature, there is a critical mass that is now called the Jeans mass. If it is exceeded, the cloud becomes unstable to collapse. The Jeans mass depends on the radius of the cloud, its temperature, and the average mass of the particles in the cloud, as shown in the box.
##### The Jeans Density
The average density of a region is its mass divided by its volume. Using this fact, the Jeans mass can also be converted into a corresponding critical density called the Jeans density. If a cloud is compressed to a point where the Jeans density is exceeded, it becomes unstable to gravitational collapse. From the form of the equation for the critical density (see the box above), collapse is favored by a low temperature and a large mass, as we would expect based on our discussion. The following table gives the critical Jeans density for three different cloud masses at the same temperature. Clearly more massive clouds have lower critical densities and thus are more susceptible to gravitational collapse.

Jeans Density for a Cloud at T = 20K
Solar Masses Critical Density (g/cm3)
1 1 x 10-19
1000 1 x 10-25
1,000,000 1 x 10-31
##### Collapse of the Sun
The solar density is about 1 gram per cubic centimeter. From the adjacent table, we may conclude that a 1 solar mass star like our Sun probably formed from a region of a molecular cloud that was about 19 powers of 10 less dense than the present Sun. If we assume for an estimate that the initial density was uniformly distributed, this suggests that the present Sun collapsed from a cloud some 10,000 AU in radius, which is about 250 times larger than the present Solar System and 2 million times larger than the present Sun. Here is a calculator that lets you explore the role of temperature and mass in determining the Jeans critical density.