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--- courses:ast402:star-formation [2026/06/03 01:55] – [Formation of Protostars] 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.
-**Homologous 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**
+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:
+$$2K + U = 0$$
+where $K$ and $U$ are typically time-averaged values.
+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**
+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:
+$$K = \frac{3}{2} NkT$$
+where $k$ is Boltzmann’s constant.
+The total number of particles is the cloud mass ($M_c$) divided by the average mass per particle ($m = \mu m_H$):
+$$N = \frac{M_c}{\mu m_H}$$
+where $\mu$ is the mean molecular weight and $m_H$ is the mass of a hydrogen atom. Substituting this gives:
+$$K = \frac{3}{2} \frac{M_c kT}{\mu m_H}$$
+**Gravitational Potential Energy ($U$):** For a spherical cloud of constant density $\rho_0$, the potential energy is approximately:
+$$U \approx -\frac{3}{5} \frac{GM_c^2}{R_c}$$
+where $R_c$ is the cloud radius.
+**3. Deriving the Jeans Mass ($M_J$)**
+Substituting these expressions into the collapse condition ($2K < |U|$):
+$$2 \left( \frac{3}{2} \frac{M_c kT}{\mu m_H} \right) < \frac{3}{5} \frac{GM_c^2}{R_c}$$
+Simplifying the left side:
+$$\frac{3 M_c kT}{\mu m_H} < \frac{3}{5} \frac{GM_c^2}{R_c}$$
+To express this in terms of mass and density, we replace the radius $R_c$ using the volume formula for a sphere ($M_c = \frac{4}{3} \pi R_c^3 \rho_0$):
+$$R_c = \left( \frac{3M_c}{4\pi\rho_0} \right)^{1/3}$$
+Substituting $R_c$ into the simplified inequality:
+$$\frac{3 M_c kT}{\mu m_H} < \frac{3}{5} GM_c^2 \left( \frac{4\pi\rho_0}{3M_c} \right)^{1/3}$$
+Isolating $M_c$ reveals the minimum mass required to initiate collapse, known as the **Jeans mass ($M_J$)**:
+**$$M_J \approx \left( \frac{5kT}{G\mu m_H} \right)^{3/2} \left( \frac{3}{4\pi\rho_0} \right)^{1/2}$$**
+**4. Deriving the Jeans Length ($R_J$)**
+The Jeans criterion can also be expressed as the minimum radius needed for a cloud of a given density to collapse. By substituting the mass expression ($M_c = \frac{4}{3} \pi R_c^3 \rho_0$) back into the primary inequality:
+$$\frac{3 (\frac{4}{3}\pi R_c^3 \rho_0) kT}{\mu m_H} < \frac{3}{5} \frac{G(\frac{4}{3}\pi R_c^3 \rho_0)^2}{R_c}$$
+Solving for $R_c$ gives the **Jeans length ($R_J$)**:
+**$$R_J \approx \left( \frac{15kT}{4\pi G\mu m_H \rho_0} \right)^{1/2}$$**
+**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. \\
+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 =====
 Once the Jeans criterion is satisfied, if pressure gradients are initially negligible, the cloud undergoes **free-fall collapse**. This collapse is **isothermal** as long as the cloud remains optically thin and can radiate away the released gravitational potential energy. The equation of motion for a mass shell at radius $r$ is:
 $$\frac{d^2r}{dt^2} = -G \frac{M_r}{r^2}$$
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 The Zero-Age Main Sequence (ZAMS) is the diagonal line on the H-R diagram where stars of various masses first reach a state of **equilibrium hydrogen burning**. At this point, nuclear energy production exactly balances the star's luminosity, and gravitational contraction stops.
-**Initial Mass Function (IMF):**
+=====Initial Mass Function (IMF) =====
 The initial mass function (IMF), denoted as $\xi$, describes the **relative number of stars** that form in different mass intervals from a fragmented cloud. Fragmentation typically produces a large abundance of **low-mass stars** and very few massive stars. While the function is well-modeled for higher masses, it is less certain for objects below $0.1 M_\odot$, where it may become relatively flat.
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 where $dN$ represents the number of stars in a specific mass interval. To find the number of stars in a linear mass range ($dN/dM$), the equation can be rewritten using the chain rule:
 $$dN = \xi(\log_{10} M) \cdot d(\log_{10} M) = \frac{\xi(\log_{10} M)}{M \ln 10} \, dM$$
-This shows that the distribution is **strongly mass-dependent**, typically resulting in a large abundance of low-mass stars and very few massive stars.
+This shows that the distribution is **strongly mass-dependent**, typically resulting in a large abundance of low-mass stars and very few massive stars. Key Characteristics of IMF are:\\
-**Key Characteristics of IMF are:\\
 **Mass-Dependency:** Fragmentation segments a cloud into many smaller objects, with the probability of formation heavily favoring lower masses.\\
 **Low-Mass Behavior:** The IMF is considered less certain for objects below approximately **$0.1 \, M_\odot$**. In this regime, the function may become **relatively flat**, suggesting a high population of low-mass stars and brown dwarfs.\\