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--- courses:ast402:star-formation [2026/06/04 01:08] – shuvo
+++ courses:ast402:star-formation [2026/06/04 01:12] (current) – [Jeans Criterion] shuvo
@@ Line 8: / Line 8: @@
 Diffuse HI clouds are generally stable against collapse because their masses are much lower than their Jeans mass, whereas the **dense cores** of giant molecular clouds (GMCs) often satisfy this criterion.
-The **Jeans criterion** defines the critical conditions of mass and radius required for an interstellar cloud to undergo spontaneous gravitational collapse. The following derivation starts from the **virial theorem** and identifies the point where gravitational forces overwhelm internal gas pressure.
+The following derivation starts from the **virial theorem** and identifies the point where gravitational forces overwhelm internal gas pressure.
-### **1. The Virial Theorem Starting Point**
+**1. The Virial Theorem Starting Point**
 For a stable, gravitationally bound system in equilibrium, the virial theorem states that twice the total internal kinetic energy ($2K$) plus the gravitational potential energy ($U$) must equal zero:
 $$2K + U = 0$$
@@ Line 17: / Line 17: @@
 To determine if a cloud will collapse, we compare these two energy terms. If **$2K < |U|$**, the gravitational force dominates, and the cloud will collapse. The critical boundary for stability occurs when **$2K = |U|$**.
-### **2. Formulating Kinetic and Potential Energy**
+**2. Formulating Kinetic and Potential Energy**
 To apply this to a molecular cloud, we use the following approximations:
-*   **Internal Kinetic Energy ($K$):** Assuming an ideal monatomic gas, the total kinetic energy for a cloud containing $N$ particles at temperature $T$ is:
+**Internal Kinetic Energy ($K$):** Assuming an ideal monatomic gas, the total kinetic energy for a cloud containing $N$ particles at temperature $T$ is:
-    $$K = \frac{3}{2} NkT$$
+$$K = \frac{3}{2} NkT$$
-    where $k$ is Boltzmann’s constant.
+where $k$ is Boltzmann’s constant.
-    The total number of particles is the cloud mass ($M_c$) divided by the average mass per particle ($m = \mu m_H$):
+The total number of particles is the cloud mass ($M_c$) divided by the average mass per particle ($m = \mu m_H$):
-    $$N = \frac{M_c}{\mu m_H}$$
+$$N = \frac{M_c}{\mu m_H}$$
-    where $\mu$ is the mean molecular weight and $m_H$ is the mass of a hydrogen atom. Substituting this gives:
+where $\mu$ is the mean molecular weight and $m_H$ is the mass of a hydrogen atom. Substituting this gives:
-    $$K = \frac{3}{2} \frac{M_c kT}{\mu m_H}$$
+$$K = \frac{3}{2} \frac{M_c kT}{\mu m_H}$$
-*   **Gravitational Potential Energy ($U$):** For a spherical cloud of constant density $\rho_0$, the potential energy is approximately:
+**Gravitational Potential Energy ($U$):** For a spherical cloud of constant density $\rho_0$, the potential energy is approximately:
-    $$U \approx -\frac{3}{5} \frac{GM_c^2}{R_c}$$
+$$U \approx -\frac{3}{5} \frac{GM_c^2}{R_c}$$
-    where $R_c$ is the cloud radius.
+where $R_c$ is the cloud radius.
-### **3. Deriving the Jeans Mass ($M_J$)**
+**3. Deriving the Jeans Mass ($M_J$)**
 Substituting these expressions into the collapse condition ($2K < |U|$):
 $$2 \left( \frac{3}{2} \frac{M_c kT}{\mu m_H} \right) < \frac{3}{5} \frac{GM_c^2}{R_c}$$
@@ Line 52: / Line 52: @@
 **$$R_J \approx \left( \frac{15kT}{4\pi G\mu m_H \rho_0} \right)^{1/2}$$**
-### **Summary of Results**
+**Summary of Results**
-*   If a cloud's mass **$M_c > M_J$**, or its radius **$R_c > R_J$**, it is unstable and will undergo gravitational collapse.
+If a cloud's mass **$M_c > M_J$**, or its radius **$R_c > R_J$**, it is unstable and will undergo gravitational collapse. \\
-*   Diffuse HI clouds are generally stable because their masses are much lower than their Jeans mass, whereas the **dense cores** of giant molecular clouds often satisfy this criterion and collapse to form stars.
+Diffuse HI clouds are generally stable because their masses are much lower than their Jeans mass, whereas the **dense cores** of giant molecular clouds often satisfy this criterion and collapse to form stars.
 =====Homologous Collapse =====