The Jeans Limit

We shall later study the birth of stars and galaxies. The initial stage is, roughly speaking, a gas cloud that begins to collapse due to its own gravitation. If the mass of the cloud is high enough, its potential energy exceeds the kinetic energy and the cloud collapses. From the virial theorem we can deduce that the potential energy must be at least twice the kinetic energy. This provides a criterion for the critical mass necessary for the cloud of collapse. This criterion was first suggested by Sir James Jeans in 1902.

The critical mass will obviously depend on the pressure P and density p. Since gravitation is the compressing force, the gravitational constant G will probably also enter our expression. Thus the critical mass is of the form b jc

M = CPaGbp

i = 1 j = i + 1 ij where the latter form is obtained by rearranging the double sum, combining the terms where C is a dimensionless constant, and the constants a, b and c are determined so that the right-hand side has the dimension of mass. The dimension of pressure is kg m—1 s—2, of gravitational constant kg-1 m3 s—2 and of density kg m—3. Thus the dimension of the right-hand side is kg

s where U is the potential energy of the system. Thus, the virial theorem becomes simply

Since this must be kilograms ultimately, we get the following set of equations:

a — b + c = 1, —a + 3b — 3c = 0 — 2a — 2b = 0 .

The solution of this is a = 3/2, b = —3/2 and c = —2. Hence the critical mass is p3/2

This is called the Jeans mass. In order to determine the constant C, we naturally must calculate both kinetic and potential energy. Another method based on the propagation of waves determines the diameter of the cloud, the Jeans length A.j, by requiring that a disturbance of size Aj grow unbounded. The value of the constant C depends on the exact form of the perturbation, but its typical values are in the range [1/n, 2n]. We can take C = 1 as well, in which case (6.52) gives a correct order of magnitude for the critical mass. If the mass of ri — rj n r mimj r

a cloud is much higher than Mj, it will collapse by its own gravitation.

In (6.52) the pressure can be replaced by the kinetic temperature 7k of the gas (see Sect. 5.8 for a definition). According to the kinetic gas theory, the pressure is

where n is the number density (particles per unit volume) and k is Boltzmann's constant. The number density is obtained by dividing the density of the gas p by the average molecular weight n = p/x, whence

By substituting this into (6.52) we get

vmw 4P

* Newton's Laws

1. In the absence of external forces, a particle will remain at rest or move along a straight line with constant speed.

2. The rate of change of the momentum of a particle is equal to the applied force F:

3. If particle A exerts a force F on another particle B, B will exert an equal but opposite force —F on A.

If several forces F1, F2,... are applied on a particle, the effect is equal to that caused by one force F which is the vector sum of the individual forces (F = F1 + F2 +...).

Law of gravitation: If the masses of particles A and B are m A and m B and their mutual distance r, the force exerted on A by B is directed towards B and has the magnitude GmA mB/r2, where G is a constant depending on the units chosen.

Newton denoted the derivative of a function f by f and the integral function by f '. The corresponding notations used by Leibniz were d f/dt and f f dx. Of

Newton's notations, only the dot is still used, always signifying the time derivative: f = d f/dt. For example, the velocity r is the time derivative of r, the acceleration r its second derivative, etc.

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