## Star formation

In Chapter 14, we discussed the contents of the interstellar medium, the material out of which new stars must be formed. In this chapter, we will identify those parts of the interstellar medium that are involved in star formation, and see what we know, and what we have to learn, about the star formation process.

### 15.1 I Gravitational binding

In Chapter 13, we talked about gravitational binding for clusters of stars. The same concepts apply to interstellar clouds, with the stars in the cluster being replaced by the particles that make up the cloud (either H or H2 ). The gravitational potential energy is now due to the interaction among all of the particles in a cloud. For a uniform spherical cloud, the gravitational potential energy is-(3/5)GM2/R. The kinetic energy is still related to the rms velocity dispersion, but with a large number of particles, which can easily be related to the cloud temperature, so the kinetic energy is (3/2)(M/m)kT, where M is the total mass of the cloud and m is the mass per particle.

The clouds are kept together by the gravitational attraction amongst all of the particles in the cloud. If the gravitational forces that hold the cloud together are greater than the forces driving it apart, we say the cloud is gravitationally bound. We can think of the random thermal motions in the gas as resisting the collapse.

The condition for gravitational binding (total energy negative) is then

Dividing both sides by GM and multiplying by (5/3) gives

The mass and radius of a cloud are not independent, since they are related to the density p = M/(4-n-/3)R3. We might therefore like to use equation (15.1) to estimate the smallest size cloud of a given p, m and T for which the cloud is grav-itationally bound. This quantity is called the Jeans length, Rj. James Jeans obtained essentially the same result with a more sophisticated analysis. We therefore eliminate M in equation (15.1), and change the inequality to an equality, since we are looking for the value of R that is on the boundary between bound and unbound. This gives

Note that (15/8^)1/2 = 0.77, which is close to unity. As the geometry of the cloud changes, the exact value of the constant will change, but it will still be close to unity. We then write

We can rewrite this in terms of n, the number of particles per unit volume (n = p/m), as

We can also use equation (15.1) to give us the minimum mass for which a cloud of given p, T and m will be bound. This minimum mass is called the Jeans mass. It is the mass of an object whose radius is Rj, so

Example 15.1 Jeans length and mass Find the Jeans length and mass in a cloud with 105 H atoms per centimeter cubed and a temperature of 50 K.

solution

We use equation (15.4) to find Rj:

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