====== Jeans instability ======

Stars form from the condensation of gas clouds in the interstellar medium (ISM). A condensing cloud must become dense enough for starting nuclear reactions. As a cloud contracts, it sheds angular momentum and must overcome the pressure of magnetic fields inherent in the ionized gas of the ISM.

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Many small perturbations (regions of higher density) can arise within a single cloud. The condition for such a perturbation to collapse inward (contracting toward its center) is that the magnitude of the gravitational potential energy \( E_p \) of an average atom in the cloud must exceed its kinetic energy \( E_k \):

\begin{equation}
E_k \lesssim |E_p|.
\end{equation}

If this inequality holds, self-gravity dominates over pressure, and the perturbation collapses. Otherwise, the region remains stable or disperses.

---

The **kinetic energy** per particle in a gas is roughly  
$$
E_k \sim m_H v^2,
$$  
where \( m_H \) is the hydrogen atom mass and \( v \) is the characteristic thermal velocity of the gas.

The **gravitational potential energy** per particle in a roughly uniform cloud of mass \( M \) and radius \( R \) is approximately  
$$
E_p \sim -\frac{G M m_H}{R}.
$$

To compare these energies, note that the mass of the cloud can be written in terms of its density \( \rho \) as  
$$
M \approx \frac{4}{3}\pi R^3 \rho.
$$

Substituting into the potential energy gives  
$$
|E_p| \sim \frac{G \left(\frac{4}{3}\pi R^3 \rho\right) m_H}{R} = \frac{4\pi}{3} G \rho R^2 m_H.
$$

Setting the two energies in the inequality gives the approximate **threshold for collapse**:

$$
m_H v^2 \lesssim \frac{4\pi}{3} G \rho R^2 m_H.
$$

Simplifying, we find

\begin{equation}
R \gtrsim \frac{v}{\sqrt{G \rho}}.
\end{equation}

This represents the **critical radius** above which gravity dominates over internal pressure.

---

Since the velocity \( v \) of gas particles is comparable to the **speed of sound** \( v_s \) in that gas, we define the **Jeans length** as:

\begin{equation}
\lambda_J \approx \frac{v_s}{\sqrt{G \rho}}.
\end{equation}

The Jeans length marks the minimum wavelength (or scale size) of a perturbation that will grow under its own gravity.  
Smaller perturbations (with \( \lambda < \lambda_J \)) are stabilized by pressure; larger ones (with \( \lambda > \lambda_J \)) collapse.

---

We can also derive a corresponding **critical mass**, known as the **Jeans mass**, by taking the mass contained within a sphere of radius \( \lambda_J / 2 \):

$$
M_J = \frac{4}{3}\pi \rho \left( \frac{\lambda_J}{2} \right)^3.
$$

Substituting the expression for \( \lambda_J \), we obtain:

$$
M_J = \frac{4\pi}{3} \rho \left( \frac{v_s}{2\sqrt{G\rho}} \right)^3 = \frac{\pi^{3/2}}{6} \frac{v_s^3}{G^{3/2} \rho^{1/2}}.
$$

Replacing \( v_s^2 \approx \frac{kT}{m_{\text{av}}} \), we get:

\begin{equation}
M_J \approx \left( \frac{kT}{G m_{\text{av}}} \right)^{3/2} \frac{1}{\sqrt{\rho}}.
\end{equation}

---

In the **Cold Neutral Medium (CNM)** of the ISM, the Jeans length is typically around 60 ly, and the Jeans mass is about \( 4000 \, M_\odot \).  
However, since CNM clouds are often smaller (around 30 ly), they do not collapse easily. If such a cloud were to shrink by a factor of four in size, its critical mass would drop by \( 4^3 = 64 \), allowing fragmentation into smaller clumps—each potentially giving birth to a star cluster.

In the **Intergalactic Medium (IGM)**, where densities are far lower (\( n_H \sim 100 \, \text{m}^{-3} \)), the Jeans mass can reach \( 10^9 \, M_\odot \), typical of an entire galaxy.

===== Insights =====

The Jeans instability defines the precise point where **self-gravity overcomes internal pressure**.  
It determines whether a gas region will remain stable, oscillate, or collapse to form stars or galaxies.

  * A region larger than \( \lambda_J \) cannot maintain equilibrium — pressure support is insufficient.  
  * A region smaller than \( \lambda_J \) remains stable against collapse.  
  * The Jeans mass provides a natural *mass scale* for star formation in any medium.

Thus, temperature (through sound speed) and density together decide whether a molecular cloud will fragment and give birth to stars or remain diffuse.

===== Inquiries =====

  - Derive the Jeans length starting from the condition \( E_k \lesssim |E_p| \) and explain each physical step.  
  - Why does increasing temperature make it harder for a cloud to collapse?  
  - How does the Jeans mass change with decreasing density, and what does this imply for galaxy-scale structure formation?