Astrophysics
Table of Contents
1. Mean Molecular Weight
\[ \mu := \frac{\expval{m}}{m_{\rm H}} \] where \( m \) is the mass of a molecule, and \( m_{\rm H} \) is the mass of a hydrogen atom.
2. Jeans Instability
2.1. Jeans Mass
- \( M_J \)
Giant molecular cloud (GMC) can form a proto star if the mass exceeds Jeans mass.
It can be derived in several ways. The first one involves virial theorem.
If a system deviates from the stability given by the Lagrange's identity, it would expand or shrink.
- \( 2K > |U| \) expand
- \( 2K < |U| \) shrink
For the GMC to collapse into a protostar,
\begin{align*} 2K < |U| \implies &2 \cdot \frac{3}{2} \frac{M}{\mu m_{\rm H}}k_{\rm B}T < \frac{3}{5}\frac{GM^2}{R} \\ \implies & \left( \frac{5k_BT}{G\mu m_{\rm H}} \right)^{\frac{3}{2}} \left( \frac{3}{4\pi \rho} \right)^{\frac{1}{2}} := M_J< M. \end{align*}The second one compares the changes in the potential and thermal energy. If the decrease in gravitational potential energy is equal to the increase of thermal energy, there is the equilibrium.
\begin{align*} \dd{W} =& -nk_{\rm B}T \cdot 4 \pi R^2 \dd{R}, \quad \dd{V} = \frac{3}{5}\frac{GM^2}{R^2}\dd{R} \\ & \dd{W} + \dd{V} = 0. \end{align*}This gives the same result.