Question

Discuss Jeans criterion for the formation of stars and obtain an expression for Jeans mass
and Jeans length

Answer 1

Answer on Question #54985, Physics / Astronomy | Astrophysics

Star formation begins in Giant Molecular Clouds (GMCs). The Milky Way has several thousand of these objects, each with masses of $10^{4}\mathrm{M}_{\odot} < \mathrm{M} < 10^{7}\mathrm{M}_{\odot}$, and sizes between 10 and 100 pc.

A cloud will collapse if the gravitational potential is stronger than its thermal (and magnetic) support. In other words $|\mathrm{E}_{\mathrm{grav}}| > 2\mathrm{E}_{\mathrm{i}}$. As we just saw, the internal energy is:

$$E_{i} = \int_{0}^{M_{T}} \frac{1}{\gamma - 1} \times \frac{P}{\rho} dM = \int_{0}^{M_{T}} \frac{1}{\gamma - 1} \times \frac{1}{\mu m_{H}} k T dM$$

For simplicity, let's take a uniform density, isothermal cloud. For such a cloud:

$$\left| E_{\text{grav}} \right| = \int_{0}^{R} \frac{GM(r)}{r} dM = \frac{3}{5} \frac{GM^{2}}{R} = 2E_{i} = \frac{2}{\gamma - 1} \times \frac{1}{\mu m_{H}} k TM$$

or:

$$M_{J} = \frac{5}{3} \times \frac{2}{\gamma - 1} \times \frac{1}{G \mu m_{H}} k TR$$

All scales larger than this are unstable to collapse.

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