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--- courses:ast402:interstellar-medium [2026/05/18 20:33] – [Gas and Dust in the Interstellar Medium (ISM)] shuvo
+++ courses:ast402:interstellar-medium [2026/05/18 21:06] (current) – [Jeans instability] shuvo
@@ Line 11: / Line 11: @@
 $A_\lambda = 1.086 \tau_\lambda$.
 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 $a \approx 0.1\,\mu\text{m}$.
 **Color Excess**\\
 Astronomers quantify the exact color of a star using a color index, such as $B−V$, which is the difference between the star's apparent magnitudes in the blue (B) and visual (V) wavelength filters. Because of interstellar reddening, the observed color index of a star behind a dust cloud will be larger (redder) than its true, intrinsic color index. The difference between the observed color and the intrinsic color is defined as the color excess, commonly denoted as E(B−V)the mathematical formula for color excess is: $$E(B−V)≡(B−V)−(B−V)_o$$.
+To study the wavelength-dependent nature of extinction, astronomers utilize extinction curves that compare the ratio of extinction in various wavelength bands, often relying on ratios of color excesses such as  $\frac{A_{\lambda-A_V}}{A_B-A_V}$.
-To study the wavelength-dependent nature of extinction, astronomers utilize extinction curves that compare the ratio of extinction in various wavelength bands, often relying on ratios of color excesses such as  $\frac{A_{\lambda/A_V}}{A_B-A_V}$.
+The total-to-selective extinction ratio, conventionally denoted as $R_V$ is the proportionality constant that directly links the total visual extinction ($A_V$) to the color excess $E(B-V)$ (which is also referred to as selective extinction). It is mathematically defined as:
+$$R_V = \frac{A_V}{E(B-V)}$$
+Because $R_V$ is typically found to have a relatively standard average value in the diffuse interstellar medium (commonly around $R_V≈3.1$),this mathematical relationship is a powerful tool. It allows astronomers to estimate the total visual extinction ($A_V$) and accurately correct a star's distance modulus simply by measuring how much the star's light has been reddened by intervening dust.
 **Mie Theory**\\
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 **HI 21-cm Radiation**\\
 Neutral hydrogen (H I) is mapped largely through its 21-cm radio-wavelength emission. This emission is produced when the inherent spin of the atom's electron flips from being aligned with the proton (a higher energy state) to being anti-aligned (a lower energy state). The resulting photon has a wavelength of 21.1 cm and a frequency of 1420 MHz. As long as this emission line is optically thin, the optical depth at the line's center is given by:
-$\tau_H = 5.2 \times 10^{-23} \frac{N_H}{T \Delta v}$
+$$\tau_H = 5.2 \times 10^{-23} \frac{N_H}{T \Delta v}$$
 where $N_H$ is the H I column density, $T$ is the temperature in kelvins, and $\Delta v$ is the full width of the line at half maximum in km s$^{-1}$.
+For an optically thin HI cloud, the column density $N_{\text{HI}}$ is directly proportional to the integrated brightness temperature over the line profile:
+\[ N_{\text{HI}} = 1.822 \times 10^{18} \int T_b(v) \, dv \]
+where $T_b$ is in Kelvin and $v$ is measured in $\text{km s}^{-1}$.
 **Molecular Hydrogen**\\
@@ Line 52: / Line 64: @@
 =====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.