Addendum 4: Stellar Evolution (Chapter 17)

First define some constants and dimensional units needed below

Note: In order to calculate the classic Jean's radius and mass for collapse of an interstellar gas cloud to form stars, we need first to calculate the gravitational potential energy of a sphere and the average kinetic energy of particles in a gas of a given temperature. The former requires a solving a simple integral.

1a. Gravitational potential energy of two masses separated by distance r. The force of gravity between the two masses M₁ and M₂ separated by distance r is

The (gravitational) potential energy is the integral of the force over distance. Bringing one mass from infinity to distance r gives:

Where the minus sign indicates that the system is in a gravitational potential well (bound)

1b. Gravitational potential energy of a uniform sphere of mass M and radius R. To find the total gravitational potential energy of a uniform massive sphere, consider an initial sphere of radius r. Add an annulus (thin spherical shell) to the sphere of density r and thickness dr. The mass contained in the shell is

The differential change in gravitational energy required to bring the shell from infinity to r must be:

Hence, to build the sphere to a radius R, we integrate over the entire sphere:

where we have substituted:

1c. Temperature and mean energy of gas particles. Any gas in thermal equilibrium has a simple reltationship between the mean energy per particle and the temperature of the gas:

where k_b is the Boltzmann constant (given above).

Example: The mean energy (in electron volts, eV) per particle of room temperature gas is

2a. Jeans Radius for cloud collapse. A cloud wtith radius R, mass M, and temperature T will collapse to form a star if the total energy of the cloud is <0, i.e, if the (absolute value) of the potential energy exceeds the thermal energy of the cloud:

N is total number of particles in cloud

where

Assuming an isothermal (constant temperature) and constant density r cloud, we can solve for the critical radius ("Jean's radius") at which the cloud will collapse:

but the number of gas particles can be written:

where m is the average mass per particle in the cloud, assumed to be hydrogen

so

or

Solving for R gives:

or

Assuming that the cloud is mostly hydrogen, m ~ m_H. Then the constants can be collected as

So

Example: Typical interstellar molecular clouds have densities n ~ 10³ - 10⁴ atom cm^-3 and a temperature of 30K. At what radii will the clouds collapse?

This size range is fairly typical of collapsed regions in star formation nebulae as illustrated below.

2b. Jean's Mass to collapse. The corresponding mass required for collapse is easily calculated from the equations for the Jeans radius:

or

or

collecting constants and expressing M_J is:

so

Example: What are Jean's masses (in solar masses) of the clouds described in the previous example?

2c. Characteristic time for a Jean's cloud to collapse. The acceleration (force per u it mass) felt by a particle at the edge of the Jean's cloud (mass M, radius R) is

and

Hence, the characteristic 'free fall' time of the mass to fall to the center is

or

Notes:
1. This result depends only on the mass density, not size!
2. This result ignores both angular momentum (rotation) and magnetic fields, both of which will resist collapse, and hence lengthen the true collapse time. The free-fall collapse time given above should be understood as a lower limit to a more realistic collapse time calculation.

Example: A Jean's cloud has a mass density n = 2 10³ cm^-3. How long will it take the cloud to collapse (ignoring rotation and magnetic fields)?

3. Luminosity of a collapsing cloud. As the cloud collapses, it loses total energy, which is radiated away. To determine the luminosity as a function of time, we use the Virial Theorem, which state that for an isolated system the change in potential and kinetic energies are related by:

this means that 1/2 the change ingraviational potential energy results in increase particle motion. The other halfmust be released as radiation, cuase by heating of the gas. From the first paragraph above we had

As change in E with time is given by the derivative dE/dt:

Since 1/2 of this amount must be lost to radiation, the luminosiy of the cloud is:

calcualtign dR/dt = V(t) isn't easy, but we can estimate an average value by using the free-fall time and the initial Jean's radius:

So

Example: A cloud with density n = 4 10³ cm^-3 and initial temperature 50K collapses for form stars. What is the Jean's mass (solar masses) and radius (pc), collapse time (yr), and average luminosity of the star (solar luminosities) as it collapses?

This is quite faint, which means that the collapsing cloud looks very dark. These regions are sometimes called Bok globules as seen in image below. They are studied using infrared telescopes, since the temperature is low.