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--- courses:ast402:interstellar-medium [2026/05/18 20:52] – [Gas and Dust in the Interstellar Medium (ISM)] shuvo
+++ courses:ast402:interstellar-medium [2026/05/18 21:06] (current) – [Jeans instability] shuvo
@@ Line 12: / Line 12: @@
 Assuming a constant scattering cross section $\sigma_\lambda$, the optical depth can also be expressed as $\tau_\lambda = \sigma_\lambda N_d$, where $N_d$ is the column density of the scattering dust particles.
-In Mie theory, the dimensionless size parameter is defined as $x = \frac{2\pi a}{\lambda}$, where $a$ is the spherical dust grain radius and $\lambda$ is the wavelength of the incident radiation. Typical interstellar dust grains have a characteristic radius of $a \approx 0.1\,\mu\text{m}$.
+In Mie theory, the dimensionless size parameter is defined as $$x = \frac{2\pi a}{\lambda}$$ where $a$ is the spherical dust grain radius and $\lambda$ is the wavelength of the incident radiation. Typical interstellar dust grains have a characteristic radius of $a \approx 0.1\,\mu\text{m}$.
 **Color Excess**\\
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 =====Heating and Cooling of the ISM=====
-*   **Heating:** A major source of heating in molecular clouds is **cosmic rays** (high-energy charged particles). Cosmic-ray protons collide with and ionize H and H$_2$, ejecting electrons that distribute their kinetic energy through the cloud via collisions. Other heating sources include photoelectric ejection of electrons from dust grains by UV starlight, X-ray ionization, and supernova shocks.
+*   **Heating:** A major source of heating in molecular clouds is **cosmic rays** (high-energy charged particles). Cosmic-ray protons collide with and ionize H and H$_2$, ejecting electrons that distribute their kinetic energy through the cloud via collisions. Other heating sources include photoelectric ejection of electrons from dust grains by UV starlight, X-ray ionization, and supernova shocks.\\
 *   **Cooling:** The primary cooling mechanism relies on the emission of infrared (IR) photons that escape the cloud. Collisions between atoms, molecules, or dust grains transfer kinetic energy into atomic/molecular excited states. The species then decays to its ground state, emitting an IR photon that carries the energy away. An example of this is the collisional excitation of oxygen:
 O + H $\rightarrow$ O$^*$ + H
 O$^*$ $\rightarrow$ O + $\gamma$.
+In a state of thermal equilibrium, the total energy input per unit volume per second (volumetric heating rate) must exactly equal the total energy loss per unit volume per second (volumetric cooling rate):
+\[ \text{Heating Rate (per unit volume)} = \text{Cooling Rate (per unit volume)} \]
+\[ \Gamma \times n_{\text{H}} = \Lambda \]
+where $\Gamma$ is $\text{ erg s}^{-1}\text{ per H atom}$ and $\Lambda$ is in $\text{ erg cm}^{-3}\text{ s}^{-1}$.
 =====Jeans instability=====
 It describes the critical conditions under which a molecular cloud will spontaneously collapse under its own gravity to form a protostar. This concept, first investigated by Sir James Jeans in 1902, is based on the virial theorem, which assesses stability by balancing the total internal kinetic energy ($K$) and the absolute value of the gravitational potential energy ($|U|$) of a system. If twice the internal kinetic energy is less than the absolute value of the gravitational potential energy ($2K < |U|$), the inward pull of gravity will overwhelm the outward push of gas pressure, initiating a collapse.
-**The Mathematical Condition for Collapse**
+**The Mathematical Condition for Collapse**\\
 For a spherical cloud of constant initial density $\rho_0$, mass $M_c$, and radius $R_c$, the gravitational potential energy is approximately:
 $U \sim -\frac{3}{5}\frac{GM_c^2}{R_c}$.
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 $\frac{3M_c k T}{\mu m_H} < \frac{3}{5}\frac{GM_c^2}{R_c}$.
-**The Jeans Mass and Jeans Length**
+**The Jeans Mass and Jeans Length**\\
 The radius of the cloud can be expressed in terms of its mass and initial density as $R_c = \left( \frac{3M_c}{4\pi\rho_0} \right)^{1/3}$. Substituting this radius into the inequality allows us to solve for the minimum mass necessary to initiate a spontaneous collapse, known as the **Jeans mass** ($M_J$). The Jeans criterion states that collapse will occur if the cloud's mass exceeds the Jeans mass:
-$M_c > M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$.
+$$M_c > M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$$.
 Alternatively, the Jeans criterion can be expressed in terms of the cloud's physical size, meaning the cloud will collapse if its radius $R_c$ is greater than the **Jeans length** ($R_J$). The Jeans length is given by the expression:
 $R_c > R_J \approx \left(\frac{15kT}{4\pi G\mu m_H\rho_0}\right)^{1/2}$.
-**Fragmentation During Collapse**
+**Fragmentation During Collapse**\\
 As a molecular cloud collapses isothermally (where released gravitational energy is efficiently radiated away), its density increases while its temperature remains relatively constant. This increase in density progressively lowers the Jeans mass. As a result, localized regions of enhanced density within the cloud begin to independently satisfy the Jeans criterion and collapse on their own, leading to a cascading fragmentation of the original cloud into smaller structures.
 This fragmentation process eventually halts when the collapsing fragments become opaque to radiation, trapping the heat and causing the collapse to transition from isothermal to adiabatic. During an adiabatic collapse, the temperature of the gas rises following the relation $T = K'\rho^{\gamma-1}$. Under these adiabatic conditions, the Jeans mass begins to increase according to the relation $M_J \propto \rho^{(3\gamma-4)/2}$, which sets a minimum limit on the mass of the fragments produced.