courses:ast402:star-formation
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| courses:ast402:star-formation [2026/06/04 01:10] – shuvo | courses:ast402:star-formation [2026/06/04 01:12] (current) – [Jeans Criterion] shuvo | ||
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| 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. | 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. |
| **1. The Virial Theorem Starting Point** | **1. The Virial Theorem Starting Point** | ||
| Line 21: | Line 21: | ||
| **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: | ||
| - | | + | $$K = \frac{3}{2} NkT$$ |
| - | 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: | ||
| - | | + | $$U \approx -\frac{3}{5} \frac{GM_c^2}{R_c}$$ |
| - | 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 ($M_J$)** | ||
courses/ast402/star-formation.1780557008.txt.gz · Last modified: by shuvo