Question

Assuming a composition entirely of atomic hydrogen, and a density of
1.67x10^-19 kg/m³, use the formulae for Jeans Mass and Radius (where
appropriate), to determine which of the following gas clouds will collapse:
Cloud A: Mass = 10,000 M_sun, T = 120K
Cloud B: Mass = 7,000 M_sun, T = 80K
Cloud C: Radius = 8 pc T = 60K

Answer 1

First define some constants and dimensional units needed below

  

      

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.

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.

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?

density of hydrogen = 1.67x10^-19 kg/m³,

Number density    =  1.67x10^-19 kg/m³  /

=10^6/ m^3

Cloud A: Mass = 10,000 M_sun, T = 120K

Thus == 1000 sun mass

thus possible

Cloud B: Mass = 7,000 M_sun, T = 80K

= 1000 sun mass

thus possible

Cloud C: Radius = 8 pc T = 60K

=100 pc

Thus acreation is not possible here.