Astrophysics I – Formula Collection & Cheat Sheet (Chapters 1–6)

Based on ETH Zurich Course Notes (Prof. A. Refregier) & Choudhuri's "Astrophysics for Physicists"

0. Fundamental Constants & Astronomical Units

Physical Constants (SI & CGS notes)

Note: $1\text{ erg}={10}^{-7}\text{ J}$, $1\text{ dyne}={10}^{-5}\text{ N}$, $1\text{ W}=1{\text{ J s}}^{-1}={10}^7{\text{ erg s}}^{-1}$.

• Gravitational Constant: $G=6.674\times {10}^{-11}{\text{ m}}^3{\text{ kg}}^{-1}{\text{ s}}^{-2}$

• Speed of Light: $c=2.998\times {10}^8{\text{ m s}}^{-1}$

• Planck Constant: $h=6.626\times {10}^{-34}\text{ J s}$ ($\hslash =h/2\pi =1.055\times {10}^{-34}\text{ J s}$)

• Boltzmann Constant: $k_B=1.381\times {10}^{-23}{\text{ J K}}^{-1}=8.617\times {10}^{-5}{\text{ eV K}}^{-1}$

• Proton / Hydrogen Mass: $m_p\approx m_H=1.673\times {10}^{-27}\text{ kg}$

• Electron Mass: $m_e=9.109\times {10}^{-31}\text{ kg}$

• Stefan-Boltzmann Constant: $\sigma =\frac{2{\pi }^5{k}_B^4}{15c^2{h}^3}=5.670\times {10}^{-8}{\text{ W m}}^{-2}{\text{ K}}^{-4}$

• Radiation Density Constant: $a_B=\frac{4\sigma }{c}=7.566\times {10}^{-16}{\text{ J m}}^{-3}{\text{ K}}^{-4}$

• Thomson Cross-Section: ${\sigma }_T=\frac{8\pi }{3}{\left(\frac{e^2}{4\pi {ϵ}_0{m}_e{c}^2}\right)}^2=6.652\times {10}^{-29}{\text{ m}}^2$

Astronomical Units & Rules of Thumb

• Astronomical Unit (Earth–Sun distance): $1\text{ AU}=1.496\times {10}^{11}\text{ m}\approx 1.5\times {10}^{11}\text{ m}$

• Parsec (Distance at which $1\text{ AU}$ subtends $1^{″}$): $1\text{ pc}=\frac{1\text{ AU}}{\tan (1^{″})}\approx \frac{360\times 3600}{2\pi }\text{ AU}=206265\text{ AU}=3.086\times {10}^{16}\text{ m}=3.26\text{ ly}$

• Year: $1\text{ yr}=3.156\times {10}^7\text{ s}\approx \pi \times {10}^7\text{ s}$

• Solar Mass: $M_{\odot }=1.989\times {10}^{30}\text{ kg}\approx 2\times {10}^{30}\text{ kg}$

• Solar Radius: $R_{\odot }=6.955\times {10}^8\text{ m}\approx 7\times {10}^8\text{ m}$

• Solar Luminosity & Temperature: $L_{\odot }=3.846\times {10}^{26}\text{ W}\approx 4\times {10}^{26}\text{ W}$, $T_{\text{eff},\odot }\approx 5780\text{ K}$

1 & 2. Basic Concepts (Choudhuri Ch. 1)

2.1 Astrophysical Scales

• Distance: $\text{AU}$ (planetary systems) $\to \text{pc}$ (interstellar distances, nearest star $\sim 1.3\text{ pc}$) $\to \text{kpc}$ (galactic scales, Milky Way radius $\sim 10\text{ kpc}$) $\to \text{Mpc}$ (intergalactic, Andromeda $\sim 0.7\text{ Mpc}$) $\to \text{Gpc}$ (observable Universe $\sim 10\text{ Gpc}$).

• Mass: Planets (${10}^{-7}\text{–}{10}^{-3}{M}_{\odot }$) $\to$ Stars ($0.08\text{–}100M_{\odot }$) $\to$ Globular Clusters (${10}^5{M}_{\odot }$) $\to$ Galaxies (${10}^{10}\text{–}{10}^{12}{M}_{\odot }$) $\to$ Galaxy Clusters (${10}^{14}\text{–}{10}^{15}{M}_{\odot }$).

2.3 Parallax & Celestial Coordinates

• Trigonometric Parallax: Apparent angular shift $\theta$ (half the peak-to-peak annual shift) of a star as Earth orbits the Sun:

  $$d\text{ [pc]}=\frac{1}{\theta \text{ [arcsec]}}\text{or}\theta \text{ [rad]}=\frac{1\text{ AU}}{d}$$

• Coordinate Systems on the Celestial Sphere:

  1. Horizontal: Altitude $a$ (or zenith angle $z={90}^{\circ }-a$) and Azimuth $A$. Observer-dependent.

  2. Equatorial: Declination $\delta$ (latitude projected from Earth's equator, $-{90}^{\circ }\le \delta \le +{90}^{\circ }$) and Right Ascension $\alpha$ (longitude measured eastward from the Vernal Equinox $\mathrm{{\rm Y}}$, expressed in $0\text{h}$ to $24\text{h}$, where $1\text{h}={15}^{\circ }$). Note: Precession of Earth's axis shifts $(\alpha ,\delta )$ with a period of $26000\text{ yr}$, requiring a reference epoch (e.g., J2000).

  3. Galactic: Galactic latitude $b$ and longitude $l$, centered on the Sun with $(l=0^{\circ },b=0^{\circ })$ pointing toward the Galactic Center.

2.4 Flux, Luminosity & Magnitude System

• Energy Flux ($F$): Power received per unit area from an isotropic source of luminosity $L$ at distance $d$:

  $$F=\frac{L}{4\pi d^2}$$

• Apparent Magnitude ($m$): Logarithmic measure of observed flux (Pogson's ratio: $\mathrm{\Delta }m=5⟺F_1/F_2=100$):

  $$m_1-m_2=-2.5{\log }_{10}\left(\frac{F_1}{F_2}\right)⟺\frac{F_1}{F_2}={10}^{-0.4(m_1-m_2)}$$

  Smaller (or more negative) $m$ means a brighter object!

• Absolute Magnitude ($M$): Apparent magnitude an object would have if placed at a reference distance of $d=10\text{ pc}$.

• Distance Modulus ($\mu =m-M$):

  $$m-M=5{\log }_{10}\left(\frac{d}{10\text{ pc}}\right)=5{\log }_{10}(d\text{ [pc]})-5$$

• Bolometric vs. Filter Magnitudes ($U,B,V$):

  • Bolometric ($m_{\text{bol}},M_{\text{bol}}$): Integrated over all wavelengths ($M_{\text{bol}}-M_{\text{bol},\odot }=-2.5{\log }_{10}(L/L_{\odot })$, with $M_{\text{bol},\odot }\approx 4.74$).

  • Filter Bands: $U$ (UV, $\approx 365\text{ nm}$), $B$ (Blue, $\approx 440\text{ nm}$), $V$ (Visual, $\approx 550\text{ nm}$).

  • Color Index: $B-V=m_B-m_V=M_B-M_V$ (distance-independent in vacuum). Measures surface temperature: bluer/hotter stars have smaller (more negative) $B-V$; redder/cooler stars have larger $B-V$.

  • Bolometric Correction: $BC=m_{\text{bol}}-V=M_{\text{bol}}-M_V$.

2.5 Gravitation & Relativistic Criterion

• Newtonian vs. General Relativity (GR): Escape velocity $v_{\text{esc}}=\sqrt{2GM/r}$ equals $c$ at the Schwarzschild radius:

  $$R_S=\frac{2GM}{c^2}\approx 2.95\left(\frac{M}{M_{\odot }}\right)\text{ km}$$

• Relativity Parameter ($f$): Determines when GR effects become significant ($f\sim 1$):

  $$f=\frac{2GM}{c^{2}r}=\frac{R_S}{r}=\frac{8\pi G\rho r^2}{3c^2}$$

  • Solar surface: $f\sim 4\times {10}^{-6}$ (Newtonian valid; weak GR perturbations like Mercury's perihelion).

  • White Dwarfs: $f\sim {10}^{-4}$; Neutron Stars: $f\sim 0.1\text{–}0.3$; Black Holes: $f\sim 1$.

  • Observable Universe: Even for tiny density $\rho \sim {10}^{-26}{\text{ kg m}}^{-3}$, huge scale $r\sim 3\text{ Gpc}$ yields $f\sim 1$ $⟹$ GR is mandatory for cosmology!

2.7 Electromagnetic Spectrum & Atmospheric Windows

• Ground-based windows: Only Optical/Near-IR ($\sim 300\text{ nm}\text{–}1\mu \text{m}$), Radio ($\sim 1\text{ cm}\text{–}10\text{ m}$; blocked at $>10\text{ m}$ by the ionosphere), and specific sub-mm bands reach the ground. Gamma-rays, X-rays, UV, and Mid/Far-IR are absorbed by ${\text{O}}_3,{\text{H}}_2\text{O},{\text{CO}}_2$, etc., requiring space telescopes.

• Angular Resolution: ${\theta }_{\text{min}}\simeq 1.22\frac{\lambda }{D}$ (diffraction limit). In optical, atmospheric turbulence (seeing) limits ground telescopes to ${\theta }_{\text{seeing}}\sim {0.5}^{″}\text{–}{1}^{″}$ unless Adaptive Optics (AO) is used. Radio astronomy uses interferometry (baseline $B$ replaces dish diameter $D$) to achieve high resolution.

3. Interaction of Radiation with Matter (Choudhuri Ch. 2)

3.2 Radiative Field & Moments

• Specific Intensity ($I_{\nu }$): Energy $dE$ passing through area $\cos \theta dA$ (projected perpendicular to ray), in solid angle $d\mathrm{\Omega }$, time $dt$, and frequency range $d\nu$:

  $$dE=I_{\nu }\cos \theta dAdtd\mathrm{\Omega }d\nu \left[{\text{W m}}^{-2}{\text{ s}}^{-1}{\text{ Hz}}^{-1}{\text{ sr}}^{-1}\right]$$

  Key property: In free space, $I_{\nu }$ is conserved along a ray ($d{I}_{\nu }/ds=0$), so surface brightness is independent of distance!

• Moments of Specific Intensity:

  1. Mean Intensity (0th moment): $J_{\nu }=\frac{1}{4\pi }\int I_{\nu }d\mathrm{\Omega }$

  2. Specific Energy Density (0th moment): $U_{\nu }=\frac{1}{c}\int I_{\nu }d\mathrm{\Omega }=\frac{4\pi }{c}{J}_{\nu }$ (For isotropic radiation: $U_{\nu }=\frac{4\pi }{c}{I}_{\nu }$)

  3. Specific Net Flux (1st moment): $F_{\nu }=\int I_{\nu }\cos \theta d\mathrm{\Omega }$ (Isotropic radiation has $F_{\nu }=0$)

  4. Radiation Pressure / Momentum Flux (2nd moment): $p_{\nu }=\frac{1}{c}\int I_{\nu }{\cos }^2\theta d\mathrm{\Omega }$ (For isotropic radiation, $⟨{\cos }^2\theta ⟩=\frac{1}{3}⟹p_{\nu }=\frac{1}{3}{U}_{\nu }$, and total radiation pressure $P_{\text{rad}}=\frac{1}{3}U$).

3.2 Radiative Transfer Equation (RTE)

• Microscopic Coefficients:

  • Emission coefficient ($j_{\nu }$): Energy emitted per unit volume, time, frequency, and solid angle: $dE=j_{\nu }dVdtd\nu d\mathrm{\Omega }$.

  • Absorption coefficient (${\alpha }_{\nu }$): Fractional loss of intensity per unit path length: $d{I}_{\nu }=-{\alpha }_{\nu }{I}_{\nu }ds$ (units: ${\text{m}}^{-1}$). Related to mass opacity ${\chi }_{\nu }$ (or ${\kappa }_{\nu }$) and particle cross-section ${\sigma }_{\nu }$:

    $${\alpha }_{\nu }=\rho {\chi }_{\nu }=n{\sigma }_{\nu }$$

  • Photon Mean Free Path: ${\ell }_{\nu }=\frac{1}{{\alpha }_{\nu }}=\frac{1}{\rho {\chi }_{\nu }}=\frac{1}{n{\sigma }_{\nu }}$.

• Radiative Transfer Equation:

  $$\frac{d{I}_{\nu }}{ds}=-{\alpha }_{\nu }{I}_{\nu }+j_{\nu }$$

• Optical Depth (${\tau }_{\nu }$) & Source Function ($S_{\nu }$):

  $$d{\tau }_{\nu }={\alpha }_{\nu }ds⟹{\tau }_{\nu }(s)={\int }_{s_0}^s{\alpha }_{\nu }(s^{\prime })d{s}^{\prime },S_{\nu }\equiv \frac{j_{\nu }}{{\alpha }_{\nu }}$$

  Dividing the RTE by ${\alpha }_{\nu }$ gives the standard form:

  $$\frac{d{I}_{\nu }}{d{\tau }_{\nu }}=-I_{\nu }+S_{\nu }$$

• Formal Solution (for constant source function $S_{\nu }$):

  $$I_{\nu }({\tau }_{\nu })=I_{\nu }(0)e^{-{\tau }_{\nu }}+S_{\nu }(1-e^{-{\tau }_{\nu }})$$

  • Optically Thin (${\tau }_{\nu }\ll 1$): $e^{-{\tau }_{\nu }}\approx 1-{\tau }_{\nu }⟹I_{\nu }({\tau }_{\nu })\approx I_{\nu }(0)+(S_{\nu }-I_{\nu }(0)){\tau }_{\nu }$.

    • If no background source ($I_{\nu }(0)=0$): $I_{\nu }\approx S_{\nu }{\tau }_{\nu }=\int j_{\nu }ds$ (Emission lines where ${\tau }_{\nu }$ is larger).

    • If bright background ($I_{\nu }(0)>S_{\nu }$): $I_{\nu }\approx I_{\nu }(0)(1-{\tau }_{\nu })$ (Absorption lines against continuum).

  • Optically Thick (${\tau }_{\nu }\gg 1$): $e^{-{\tau }_{\nu }}\to 0⟹I_{\nu }({\tau }_{\nu })\approx S_{\nu }$ (In LTE, $S_{\nu }=B_{\nu }(T)⟹$ Blackbody spectrum!).

3.3 Application to Stars: Diffusion & Eddington-Barbier

• Radiative Diffusion in Stellar Interiors: Deep inside a star ($\tau \gg 1$), radiation is nearly isotropic and in Local Thermodynamic Equilibrium ($S_{\nu }=B_{\nu }(T)$). Expanding $B_{\nu }$ to first order along ray path $s$ ($r=r_0+s\cos \theta$) yields:

  $$I_{\nu }(r,\theta )\approx B_{\nu }(T)-\frac{1}{{\alpha }_{\nu }}\frac{d{B}_{\nu }}{dr}\cos \theta$$

  Integrating over solid angle gives the net radiative flux $F_{\nu }=\int I_{\nu }\cos \theta d\mathrm{\Omega }=-\frac{4\pi }{3{\alpha }_{\nu }}\frac{d{B}_{\nu }}{dr}$. Integrating over all frequencies:

  $$F(r)=-\frac{4\pi }{3\rho {\chi }_R}\frac{dT}{dr}{\int }_0^{\mathrm{\infty }}\frac{\partial B_{\nu }}{\partial T}d\nu =-\frac{4a_{B}c{T}^3}{3\rho {\chi }_R}\frac{dT}{dr}$$

• Rosseland Mean Opacity (${\chi }_R$): Harmonic mean of ${\chi }_{\nu }$ weighted by the temperature derivative of the Planck function (favors frequencies where opacity is low and flux escapes easily):

  $$\frac{1}{{\chi }_R}\equiv \frac{{\int }_0^{\mathrm{\infty }}\frac{1}{{\chi }_{\nu }}\frac{\partial B_{\nu }}{\partial T}d\nu }{{\int }_0^{\mathrm{\infty }}\frac{\partial B_{\nu }}{\partial T}d\nu }$$

• Photon Random Walk (Diffusion Time): A photon travels mean free path $\ell$ between scatterings. To cover radial distance $R$ requires $N\sim (R/\ell {)}^2$ steps, taking time $t_{\text{diff}}\sim \frac{N\ell }{c}\sim \frac{R^2}{\ell c}$. For the Sun ($\ell \sim {10}^{-3}\text{ m}$), $t_{\text{diff}}\sim {10}^4\text{–}{10}^5\text{ yr}$ (vs. $R_{\odot }/c\approx 2.3\text{ s}$ for neutrinos!).

• Stellar Atmospheres & Limb Darkening: Using vertical optical depth $\tau$ measured inward from the surface ($\tau =0$), the Eddington approximation yields $T^4(\tau )=\frac{3}{4}{T}_{\text{eff}}^4(\tau +\frac{2}{3})$. Thus $T=T_{\text{eff}}$ at the photosphere ($\tau =2/3$).

• Eddington-Barbier Relation: Looking at angle $\theta$ to the normal, the emergent intensity roughly equals the source function at line-of-sight optical depth unity (${\tau }_s=1⟺\tau =\cos \theta$):

  $$I(\tau =0,\theta )\approx S(\tau =\cos \theta )$$

  Since temperature increases inward with $\tau$, looking toward the stellar limb ($\theta \to {90}^{\circ },\cos \theta \to 0$) probes higher, cooler layers $⟹$ Limb Darkening (and confirms temperature decreases outward in the photosphere, explaining Fraunhofer absorption lines).

3.4 Opacity Sources

1. Thomson Scattering (Free electrons): Frequency-independent: ${\alpha }_T=n_e{\sigma }_T$. Dominates in hot, ionized interiors.

2. Bound-Bound (Line transitions): Discrete excitations between atomic levels.

3. Bound-Free (Photoionization) & Free-Free (Inverse Bremsstrahlung): Follow Kramers' Opacity Law:

   $$\chi \propto \rho T^{-3.5}$$

4. ${\text{H}}^{-}$ Ion (Negative Hydrogen):Neutral $\text{H}$ atom binds a second electron (binding energy $0.75\text{ eV}$). Dominant continuous opacity source in the photospheres of cool, solar-type stars ($T\sim 3000\text{–}6000\text{ K}$).

3.5 Thermodynamic Equilibrium (TE) vs. Local TE (LTE)

• Four Distributions of Complete TE:

  1. Maxwell-Boltzmann (Particle velocities): $f(v)d^{3}v={\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}{d}^{3}v⟹\frac{1}{2}m⟨v^2⟩=\frac{3}{2}{k}_{B}T$.

  2. Planck Law (Radiation field):

     $$B_{\nu }(T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /k_{B}T}-1},U=\frac{4\pi }{c}{\int }_0^{\mathrm{\infty }}{B}_{\nu }(T)d\nu =a_B{T}^4,F=\pi B(T)=\sigma T^4$$

     • Wien Displacement Law: ${\lambda }_{\text{max}}T=2.898\times {10}^{-3}\text{ m K}$.

     • Rayleigh-Jeans Limit ($h\nu \ll k_{B}T$): $B_{\nu }(T)\approx \frac{2{\nu }^2}{c^2}{k}_{B}T$.

     • Wien Limit ($h\nu \gg k_{B}T$): $B_{\nu }(T)\approx \frac{2h{\nu }^3}{c^2}{e}^{-h\nu /k_{B}T}$.

  3. Boltzmann Excitation Equation (Atomic energy levels):

     $$\frac{n_b}{n_a}=\frac{g_b}{g_a}{e}^{-(E_b-E_a)/k_{B}T}$$

  4. Saha Ionization Equation (Ionization states $i\leftrightarrow i+1+e^{-}$):

     $$\frac{n_{i+1}}{n_i}{P}_e=\frac{2g_{i+1}}{g_i}\frac{(2\pi m_e{)}^{3/2}(k_{B}T{)}^{5/2}}{h^3}{e}^{-{\chi }_i/k_{B}T}(\text{where }{P}_e=n_e{k}_{B}T\text{ and }{\chi }_i=\text{ionization energy})$$

     Concept: High $T$ and low $n_e$both favor ionization. Because the prefactor is huge ($\sim {10}^9$ in stars), hydrogen ionizes at $k_{B}T\sim 0.1{\chi }_H\sim 1\text{ eV}$ ($T\sim {10}^4\text{ K}$) rather than $1.6\times {10}^5\text{ K}$!

• Kirchhoff's Law (valid in TE and LTE): $S_{\nu }\equiv \frac{j_{\nu }}{{\alpha }_{\nu }}=B_{\nu }(T)⟺j_{\nu }={\alpha }_{\nu }{B}_{\nu }(T)$.

• Condition for LTE: Matter is locally in thermal equilibrium at temperature $T(\overrightarrow{r})$ if the particle/photon mean free path is much smaller than the characteristic temperature gradient length scale: $\ell \ll L_T\equiv \frac{T}{|\mathrm{\nabla }T|}$.

4. Stars: Structure, Evolution & Nucleosynthesis (Choudhuri Ch. 3, 4 & 5)

4.1 The Four Stellar Structure Equations

For a static, spherically symmetric star of mass $M$ and radius $R$:

1. Mass Conservation:

   $$\frac{d{M}_r}{dr}=4\pi r^2\rho$$

2. Hydrostatic Equilibrium:

   $$\frac{dP}{dr}=-\frac{G{M}_r\rho }{r^2}$$

3. Energy Generation: ($ϵ$ = nuclear power generated per unit mass $[{\text{W kg}}^{-1}]$)

   $$\frac{d{L}_r}{dr}=4\pi r^2\rho ϵ$$

4. Energy Transport:

   $$\frac{dT}{dr}=\{\begin{array}{ll}-{\displaystyle \frac{3\chi \rho }{4a_{B}c{T}^3}}{\displaystyle \frac{L_r}{4\pi r^2}}& \text{(Radiative)}\\ (1-{\displaystyle \frac{1}{\gamma }}){\displaystyle \frac{T}{P}}{\displaystyle \frac{dP}{dr}}& \text{(Convective / Adiabatic)}\end{array}$$

• Constitutive Relations (Closure): Requires Equation of State $P(\rho ,T,X_i)$, Opacity $\chi (\rho ,T,X_i)$, and Energy Generation Rate $ϵ(\rho ,T,X_i)$, where composition mass fractions satisfy $X(\text{H})+Y(\text{He})+Z(\text{Metals})=1$ (Solar: $X_{\odot }\approx 0.70,Y_{\odot }\approx 0.28,Z_{\odot }\approx 0.02$).

• Boundary Conditions: At center ($r=0$): $M_r=0,L_r=0$. At surface ($r=R$): $\rho =0,T=0$ (or $P=0$).

4.1–4.2 Equations of State & Mean Molecular Weight ($\mu$)

• Total Pressure: $P=P_{\text{gas}}+P_{\text{rad}}=\frac{\rho }{\mu m_H}{k}_{B}T+\frac{1}{3}{a}_B{T}^4$.

• Mean Molecular Weight ($\mu$): Average mass per free particle in units of $m_H$ ($\rho =n\mu m_H$). For a fully ionized gas (where element $i$ yields $1$ nucleus + $Z_i$ electrons $\approx A_i/2$ for metals):

  $$\frac{1}{\mu }=2X+\frac{3}{4}Y+\frac{1}{2}Z$$
  • Pure neutral H: $\mu =1$; Pure ionized H: $\mu =0.5$; Pure ionized He: $\mu =4/3$; Ionized solar mix: $\mu \approx 0.61$.

4.1 Convective Instability (Schwarzschild Criterion)

• Displaced fluid blob expands adiabatically ($P\propto {\rho }^{\gamma }$). If it ends up lighter/hotter than its new surroundings, it keeps rising (convection).

• Schwarzschild Criterion for Convection: Convection occurs if the radiative temperature gradient is steeper than the adiabatic gradient:

  $${\left|\frac{dT}{dr}\right|}_{\text{rad}}>{\left|\frac{dT}{dr}\right|}_{\text{ad}}=(1-\frac{1}{\gamma })\frac{T}{P}\left|\frac{dP}{dr}\right|(\text{For monatomic gas, }\gamma =\frac{5}{3}⟹1-\frac{1}{\gamma }=0.4)$$

• Where does convection happen?

  1. High opacity $\chi$ (e.g., cool outer envelopes of low-mass stars $M\lesssim M_{\odot }$ and red giants).

  2. High central energy flux $L_r/4\pi r^2$ (e.g., cores of massive stars $M\gtrsim 1.3M_{\odot }$ where temperature-sensitive CNO cycle concentrates $ϵ$ at the very center).

4.3 Order-of-Magnitude Estimates & Homology Scaling Relations

• Solar Central Pressure & Temperature (from Hydrostatic Equilibrium):
  Approximate $\frac{dP}{dr}\sim -\frac{P_c}{R}$, $M_r\sim \frac{M}{2}$, $r\sim \frac{R}{2}$, $\overline{\rho }=\frac{M}{\frac{4}{3}\pi R^3}$:

  $$P_c\sim \frac{3}{2\pi }\frac{G{M}^2}{R^4}(\text{For Sun: }{P}_{c,\odot }\sim {10}^{14}\text{ Pa}\sim {10}^9\text{ atm; detailed model gives }2.5\times {10}^{16}\text{ Pa})$$

  Using ideal gas $P_c=\frac{{\rho }_c{k}_B{T}_c}{\mu m_H}$ with ${\rho }_c\sim \overline{\rho }\propto M/R^3$:

  $$T_c\sim \frac{2\mu m_H}{k_B}\frac{GM}{R}(\text{For Sun: }{T}_{c,\odot }\sim 1.5\times {10}^7\text{ K})$$

• Stellar Scaling Derivations:

  1. Mass conservation: $\rho \propto \frac{M}{R^3}$

  2. Hydrostatic equilibrium: $\frac{P}{R}\propto \frac{M\rho }{R^2}⟹P\propto \frac{M^2}{R^4}$

  3. Ideal Gas EOS: $P\propto \rho T⟹\frac{M^2}{R^4}\propto \frac{M}{R^3}T⟹T\propto \frac{M}{R}$ (Note: If radiation pressure dominates, $P\propto T^4⟹T\propto M^{1/2}/R$).

  4. Radiative Transport (assuming Thomson opacity $\chi \approx \text{const}$):

     $$\frac{T}{R}\propto \frac{\chi \rho }{T^3}\frac{L}{R^2}⟹L\propto \frac{(TR{)}^4}{\chi M}\propto \frac{M^4}{M}⟹\mathbf{L}\propto {\mathbf{M}}^{\mathbf{3}}\text{(independently of }R\text{!)}$$

     (If Kramers' opacity $\chi \propto \rho T^{-3.5}$ is used: $L\propto M^{5.5}{R}^{-0.5}\approx M^5$).

  5. Energy Generation: Suppose $ϵ\propto \rho T^{\nu }$. Since $L\propto Mϵ\propto M\left(\frac{M}{R^3}\right){\left(\frac{M}{R}\right)}^{\nu }=M^{\nu +2}{R}^{-(\nu +3)}$, equating with $L\propto M^3$ gives:

     $$R\propto M^{\frac{\nu -1}{\nu +3}}\text{Example: For p-p chain }(\nu \approx 4),R\propto M^{3/7}\approx M^{0.43};\text{ frequently simplified to }\mathbf{R}\propto {\mathbf{M}}^{\mathbf{0.5}}\text{–}{\mathbf{M}}^{\mathbf{0.8}}.$$

  6. Main-Sequence Slope on H-R Diagram: Using $L=4\pi R^2\sigma T_{\text{eff}}^4$, $L\propto M^3$, and $R\propto M^{1/2}$:

     $$T_{\text{eff}}^4\propto \frac{L}{R^2}\propto \frac{M^3}{M}=M^2⟹T_{\text{eff}}\propto M^{1/2}⟹\mathbf{L}\propto {\mathbf{T}}_{\text{eff}}^{\mathbf{6}}⟹\log L=6\log T_{\text{eff}}+\text{const}$$

4.4–4.5 Stellar Observations, H-R Diagram & Mass Limits

• Harvard Spectral Classification (Temperature Sequence):

  $$\mathbf{O}\mathbf{-}\mathbf{B}\mathbf{-}\mathbf{A}\mathbf{-}\mathbf{F}\mathbf{-}\mathbf{G}\mathbf{-}\mathbf{K}\mathbf{-}\mathbf{M}(\text{Hot }\sim 40000\text{ K}⟶\text{Cool }\sim 2500\text{ K; Sun is G2V})$$

  Why do Balmer ($\text{H}\alpha ,\text{H}\beta$, $n=2\to 3,4$) lines peak in A-type stars ($T\approx 10000\text{ K}$)?
  Combined Boltzmann $\times$ Saha effect: In cool stars ($T<9000\text{ K}$), almost all $\text{H}$ is in the ground state ($n=1$, Boltzmann suppression of $n=2$). In very hot stars ($T>11000\text{ K}$), $\text{H}$ is mostly ionized ($\text{H II}$, Saha depletion of neutral $\text{H}$). The product $(n_2/n_{\text{total}})$ peaks sharply at $T\approx 9500\text{ K}$!

• Luminosity Classes: $\text{I}$ (Supergiants), $\text{III}$ (Red Giants), $\text{V}$ (Main Sequence Dwarfs), $\text{WD}$ (White Dwarfs).

• Binary Stars (Mass Determination): For a binary system with semi-major axis $a=a_1+a_2$ and period $P$:

  $$M_1+M_2=\frac{4{\pi }^2}{G}\frac{a^3}{P^2},\frac{M_1}{M_2}=\frac{a_2}{a_1}=\frac{v_2}{v_1}$$

• Ends of the Main Sequence:

  • Lower Mass Limit ($M_{\text{min}}\approx 0.08M_{\odot }$): Below $0.08M_{\odot }$, electron degeneracy pressure halts gravitational contraction before $T_c$ reaches the hydrogen ignition threshold ($\sim 4\times {10}^6\text{ K}$) $⟹$ Brown Dwarfs.

  • Upper Mass Limit ($M_{\text{max}}\sim 100M_{\odot }$) & Eddington Luminosity ($L_{\text{Edd}}$): Radiation pressure force outward on an electron ($f_{\text{rad}}=\frac{{\sigma }_{T}F}{c}=\frac{{\sigma }_{T}L}{4\pi r^{2}c}$) cannot exceed inward gravity on an electron-proton pair ($f_{\text{grav}}=\frac{GM{m}_p}{r^2}$):

    $$L_{\text{Edd}}=\frac{4\pi GM{m}_{p}c}{{\sigma }_T}\approx 3.2\times {10}^4\left(\frac{M}{M_{\odot }}\right)L_{\odot }$$

4.6–4.7 Timescales & Nuclear Energy Production

• Three Fundamental Stellar Timescales:

  1. Dynamical (Free-Fall) Timescale ($t_{\text{dyn}}$): Time to collapse if pressure support vanishes:

     $$t_{\text{dyn}}\sim \sqrt{\frac{R^3}{GM}}\sim \frac{1}{\sqrt{G\overline{\rho }}}(\text{Sun: }\sim 1600\text{ s}\approx 27\text{ min})$$

  2. Kelvin-Helmholtz (Thermal) Timescale ($t_{\text{KH}}$): Time a star can shine from gravitational potential energy alone (by Virial theorem, $E_{\text{th}}=-\frac{1}{2}{E}_{\text{grav}}\sim \frac{G{M}^2}{R}$):

     $$t_{\text{KH}}\sim \frac{G{M}^2}{RL}(\text{Sun: }\sim 3\times {10}^7\text{ yr})$$

  3. Nuclear Timescale ($t_{\text{nuc}}$): Main-sequence lifetime burning fraction $f\approx 0.1$ of hydrogen into helium (efficiency $\eta =0.007$ of rest mass $m_p{c}^2$):

     $$t_{\text{nuc}}\sim \frac{0.007fM{c}^2}{L}\propto \frac{M}{L}\propto M^{-2}(\text{Sun: }\sim {10}^{10}\text{ yr})$$

• Quantum Tunneling & Gamow Peak:

  • Classical Coulomb barrier for two protons at $r_0\sim 1\text{ fm}$ is $E_C=\frac{Z_1{Z}_2{e}^2}{4\pi {ϵ}_0{r}_0}\approx 1\text{ MeV}$, whereas solar core thermal energy is $k_B{T}_c\approx 1.3\text{ keV}$!

  • Tunneling probability $P_{\text{tunnel}}\propto e^{-b/\sqrt{E}}$ (where $b=\frac{\sqrt{2m}\pi Z_1{Z}_2{e}^2}{4\pi {ϵ}_{0}h}=\sqrt{E_G}$). Multiplying Maxwell-Boltzmann tail ($e^{-E/k_{B}T}$) by tunneling probability ($e^{-\sqrt{E_G/E}}$) produces the narrow Gamow Peak at energy $E_0={\left(\frac{b{k}_{B}T}{2}\right)}^{2/3}$:

    $$⟨\sigma v⟩\propto T^{-2/3}\exp [-3{\left(\frac{{\pi }^{2}m{Z}_1^2{Z}_2^2{e}^4}{2(4\pi {ϵ}_0{)}^2{h}^2{k}_{B}T}\right)}^{1/3}]$$

• Hydrogen Burning Chains ($4{}^1\text{H}\to {}^4\text{He}+2e^{+}+2{\nu }_e+26.7\text{ MeV}$):

  • p-p Chain: Dominates at $T_c\lesssim 1.8\times {10}^7\text{ K}$ ($M\lesssim 1.3M_{\odot }$, including the Sun). Bottleneck is weak interaction $p+p\to {}^2\text{H}+e^{+}+{\nu }_e$. Temperature dependence near $T_{\odot }$: ${ϵ}_{\text{pp}}\propto \rho X^2{T}^4$.

  • CNO Cycle: Uses ${}^{12}\text{C},{}^{14}\text{N},{}^{16}\text{O}$ as nuclear catalysts. Higher Coulomb barrier ($Z_1{Z}_2=6\text{–}7$) makes it extremely $T$-sensitive: ${ϵ}_{\text{CNO}}\propto \rho X{Z}_{\text{CNO}}{T}^{16\text{–}20}$. Dominates for $T_c\gtrsim 1.8\times {10}^7\text{ K}$ ($M\gtrsim 1.3M_{\odot }$).

• Advanced Burning Stages:

  • Triple-Alpha Process ($3{}^4\text{He}\leftrightarrow {}^8\text{Be}+{}^4\text{He}\to {}^{12}\text{C}+\gamma$): Requires $T\gtrsim {10}^8\text{ K}$, ${ϵ}_{3\alpha }\propto {\rho }^2{Y}^3{T}^{40}$.

  • Massive Stars ($M\gtrsim 8M_{\odot }$): Burn $\text{C}\to \text{Ne/Mg}\to \text{O}\to \text{Si}\to {}^{56}\text{Fe}$ (Onion-skin structure). Fusion stops at ${}^{56}\text{Fe}$ because iron/nickel have the maximum nuclear binding energy per nucleon ($\approx 8.8\text{ MeV}$); fusing heavier elements is endothermic.

4.8–4.10 Stellar Evolution, Compact Remnants & Degeneracy

• Solar Confirmation:

  • Helioseismology: Acoustic pressure modes ($p$-modes) probe sound speed $c_s=\sqrt{\gamma P/\rho }\propto \sqrt{T/\mu }$ throughout the solar interior, showing base of convective zone at $r\approx 0.71R_{\odot }$.

  • Solar Neutrino Problem: Homestake experiment detected only $\sim 1/3$ of predicted ${\nu }_e$ flux; solved by neutrino oscillations (${\nu }_e\leftrightarrow {\nu }_{\mu },{\nu }_{\tau }$ via MSW effect), proving neutrinos have mass!

• Cluster H-R Diagrams & Turn-Off Point: Since $t_{\text{nuc}}\propto M^{-2}$, the mass (or luminosity) at the Main-Sequence Turn-Off (MSTO) directly measures the age of a star cluster!

• Fermi Gas & Electron Degeneracy Pressure (Choudhuri Ch. 5):
  When density is high, Pauli exclusion principle fills momentum states up to the Fermi momentum $p_F$:

  $$n_e={\int }_0^{p_F}\frac{2}{h^3}4\pi p^{2}dp=\frac{8\pi p_F^3}{3h^3}⟹p_F={\left(\frac{3h^3}{8\pi }{n}_e\right)}^{1/3}\propto {\rho }^{1/3}$$

  • Non-Relativistic Degeneracy ($p_F\ll m_{e}c$): Kinetic energy $E_p=\frac{p^2}{2m_e}$:

    $$P_{\text{deg, NR}}=\frac{1}{3}{\int }_0^{p_F}vpf(p)4\pi p^{2}dp=\frac{h^2}{20m_e}{\left(\frac{3}{\pi }\right)}^{2/3}{n}_e^{5/3}\propto {\rho }^{\mathbf{5}\mathbf{/}\mathbf{3}}(\text{Polytrope }\gamma =5/3,n=3/2)$$

  • Ultra-Relativistic Degeneracy ($p_F\gg m_{e}c$): $E_p\approx pc,v\approx c$:

    $$P_{\text{deg, R}}=\frac{hc}{8}{\left(\frac{3}{\pi }\right)}^{1/3}{n}_e^{4/3}\propto {\rho }^{\mathbf{4}\mathbf{/}\mathbf{3}}(\text{Polytrope }\gamma =4/3,n=3)$$

• White Dwarfs & Chandrasekhar Mass Limit:

  • Equating hydrostatic pressure $P_c\propto G{M}^2/R^4$ with non-relativistic degeneracy pressure $P\propto (\rho {)}^{5/3}\propto (M/R^3{)}^{5/3}=M^{5/3}{R}^{-5}$ yields the White Dwarf Mass-Radius Relation:

    $$\mathbf{R}\propto {\mathbf{M}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{3}}(\text{More massive white dwarfs are smaller!})$$

  • As $M$ increases, $R$ shrinks, $\rho$ rises, and electrons become relativistic ($P\propto {\rho }^{4/3}\propto M^{4/3}{R}^{-4}$). Equating $P_c\propto G{M}^2/R^4$ with $P_{\text{deg, R}}\propto (hc)M^{4/3}{R}^{-4}$ causes $R$ to cancel out completely, yielding a maximum stable mass (Chandrasekhar Limit):

    $$M_{\text{Ch}}\sim {\left(\frac{hc}{G}\right)}^{3/2}\frac{1}{m_p^2}⟹M_{\text{Ch}}\approx 1.44{\left(\frac{2}{{\mu }_e}\right)}^2{M}_{\odot }\approx 1.4M_{\odot }(\text{for He/C/O where }{\mu }_e\approx 2)$$

• End States & Supernovae:

  • $M\lesssim 8M_{\odot }$: Red Giant $\to$ AGB $\to$ Planetary Nebula + White Dwarf ($R\sim {10}^4\text{ km}\sim R_{\oplus }$).

  • $M\gtrsim 8M_{\odot }$: Iron core exceeds $M_{\text{Ch}}$ $\to$ photodisintegration & inverse beta decay ($p+e^{-}\to n+{\nu }_e$) $\to$ Core-Collapse Supernova (Type II, Ib/c) ($99\mathrm{\%}$ of gravitational binding energy $\mathrm{\Delta }E\sim \frac{G{M}^2}{R_{\text{NS}}}\sim 3\times {10}^{46}\text{ J}$ is released as neutrinos!) $\to$ leaves a Neutron Star ($R\sim 10\text{ km}$, supported by neutron degeneracy & strong force up to $M_{\text{OV}}\sim 2\text{–}3M_{\odot }$; observed as Pulsars) or a Black Hole.

  • Type Ia Supernova: Thermonuclear explosion of a C-O White Dwarf in a binary system accreting matter near $M_{\text{Ch}}\approx 1.4M_{\odot }$. No hydrogen lines; uniform peak luminosity makes it a premier Standard Candle.

5. Our Galaxy & The Interstellar Medium (Choudhuri Ch. 6)

5.1–5.2 Star Counts, Structure & Stellar Populations

• Uniform Star Count Relation (Euclidean, no extinction): Stars brighter than apparent magnitude $m$ lie within distance $d(m)={10}^{1+0.2(m-M)}\text{ pc}$. Since volume $V\propto d^3$:

  $$N(<m)\propto d^3\propto {\mathbf{10}}^{\mathbf{0.6}\mathbf{m}},A(m)\equiv \frac{dN}{dm}\propto {10}^{0.6m}$$

  (Observed counts grow slower than ${10}^{0.6m}$ due to the finite disk thickness and interstellar dust extinction).

• Olbers' Paradox: In an infinite, static, uniform Universe, total flux $\mathcal{L}={\int }_{-\mathrm{\infty }}^{m}l(m^{\prime })A(m^{\prime })d{m}^{\prime }\propto \int {10}^{-0.4m^{\prime }}{10}^{0.6m^{\prime }}d{m}^{\prime }\propto {10}^{0.2m}\to \mathrm{\infty }$ as $m\to \mathrm{\infty }$. Resolved by finite age of the Universe and cosmic expansion.

• Milky Way Geometry:

  • Disk: Radius $\sim 15\text{ kpc}$; Thin disk scale height $\sim 300\text{ pc}$, Thick disk $\sim 1\text{ kpc}$. Sun is at $R_0\approx 8\text{ kpc}$ from the Galactic Center, orbiting at ${\mathrm{\Theta }}_0\approx 220{\text{ km s}}^{-1}$ (period $T_{\text{gal}}=\frac{2\pi R_0}{{\mathrm{\Theta }}_0}\approx 2.3\times {10}^8\text{ yr}$).

  • Bulge: Radius $\sim 1\text{–}2\text{ kpc}$ around a central Supermassive Black Hole (${\text{Sgr A}}^{\ast }$, $M_{\text{BH}}\approx 4\times {10}^6{M}_{\odot }$).

  • Stellar Halo: Spherical ($R\sim 30\text{–}50\text{ kpc}$), contains old stars and ~150 Globular Clusters.

  • Dark Matter Halo: Extends to $>100\text{ kpc}$, total mass $M_{\text{MW}}\sim {10}^{12}{M}_{\odot }$.

• Differential Galactic Rotation & Oort Constants (Solar Neighborhood $d\ll R_0$):
  For stars on circular orbits with angular speed $\omega (R)=\mathrm{\Theta }(R)/R$ at galactic longitude $l$:

  $$v_R=Ad\sin (2l),v_T=Ad\cos (2l)+Bd$$

  where the Oort Constants $A$ (shear) and $B$ (vorticity) are:

Notice that $A-B=\frac{{\mathrm{\Theta }}_0}{R_0}={\omega }_0$ and $-(A+B)={\left(\frac{d\mathrm{\Theta }}{dR}\right)}_{R_0}$.

• Flat Rotation Curve & Dark Matter: At large radii ($R\gtrsim R_0$), $\mathrm{\Theta }(R)\approx \text{const}$. Since $\frac{{\mathrm{\Theta }}^2}{R}=\frac{GM(R)}{R^2}⟹\mathbf{M}\mathbf{(}\mathbf{R}\mathbf{)}\mathbf{=}\frac{{\mathbf{\Theta }}^{\mathbf{2}}\mathbf{R}}{\mathbf{G}}\propto \mathbf{R}$, mass continues to grow linearly well beyond the visible stellar disk $⟹$ Dark Matter Halo with density profile ${\rho }_{\text{DM}}(R)\propto R^{-2}$.

• Stellar Populations:

  Property	Population I (Disk)	Population II (Halo & Globular Clusters)
  Age	Young to intermediate ($<10\text{ Gyr}$)	Old ($\sim 12\text{–}13\text{ Gyr}$)
  Metallicity ($Z$)	Metal-rich ($Z\sim 0.01\text{–}0.03$)	Metal-poor ($Z\sim {10}^{-4}\text{–}{10}^{-3}$)
  Kinematics	Ordered circular rotation in plane ($v_{\text{rot}}\gg {\sigma }_v$)	Random, high-eccentricity 3D orbits (${\sigma }_v\gg v_{\text{rot}}$)
  Environment	Gas/dust rich, ongoing star formation	Gas-free, no ongoing star formation

5.3 Interstellar Extinction & Reddening

• Dust grains ($a\sim 0.01\text{–}0.2\mu \text{m}$) scatter and absorb starlight ($F=\frac{L}{4\pi d^2}{e}^{-{\tau }_{\lambda }}$).

• Distance Modulus with Extinction ($A_{\lambda }$):

  $$m_{\lambda }-M_{\lambda }=5{\log }_{10}\left(\frac{d}{10\text{ pc}}\right)+A_{\lambda },\text{where }{A}_{\lambda }=2.5{\log }_{10}(e){\tau }_{\lambda }\approx 1.086{\tau }_{\lambda }$$

  In the Galactic plane, average visual extinction is $\frac{A_V}{d}\approx 1.5{\text{ mag kpc}}^{-1}$.

• Interstellar Reddening & Color Excess $E(B-V)$: Because dust cross-section increases at shorter wavelengths (${\sigma }_{\lambda }\propto {\lambda }^{-1}$ to ${\lambda }^{-4}$), blue light is attenuated more than visual light ($A_B>A_V$):

  $$E(B-V)\equiv (B-V{)}_{\text{obs}}-(B-V{)}_{\text{intrinsic}}=A_B-A_V,R_V\equiv \frac{A_V}{E(B-V)}\approx 3.1(\text{Milky Way average})$$

5.4 Spectral Lines, Einstein Coefficients & 21-cm Hydrogen Line

• Einstein Relations (between levels $l$ and $u$ with $E_u-E_l=h{\nu }_0$):

  • $A_{ul}$: Spontaneous emission rate $[{\text{s}}^{-1}]$; $B_{lu},B_{ul}$: Stimulated absorption/emission coefficients.

  • Detailed balance in TE ($I_{\nu }=B_{\nu }(T)$) requires:

    $$g_l{B}_{lu}=g_u{B}_{ul},A_{ul}=\frac{8\pi h{\nu }_0^3}{c^3}{B}_{ul}$$

  • Microscopic Emission & Absorption Coefficients (with normalized line profile $\int \varphi (\mathrm{\Delta }\nu )d\nu =1$):

    $$j_{\nu }=\frac{h{\nu }_0}{4\pi }{n}_u{A}_{ul}\varphi (\mathrm{\Delta }\nu ),{\alpha }_{\nu }=\frac{h{\nu }_0}{c}(n_l{B}_{lu}-n_u{B}_{ul})\varphi (\mathrm{\Delta }\nu )=\frac{h{\nu }_0}{c}{n}_l{B}_{lu}(1-e^{-h{\nu }_0/k_{B}T})\varphi (\mathrm{\Delta }\nu )$$

• Line Broadening & Curve of Growth:

  1. Natural / Collisional Broadening: Lorentzian profile $\varphi (\mathrm{\Delta }\nu )\propto \frac{\gamma }{(\mathrm{\Delta }\nu {)}^2+(\gamma /4\pi {)}^2}$ (dominates line wings).

  2. Thermal Doppler Broadening: Gaussian profile $\varphi (\mathrm{\Delta }\nu )=\frac{1}{\sqrt{\pi }\mathrm{\Delta }{\nu }_D}{e}^{-(\mathrm{\Delta }\nu /\mathrm{\Delta }{\nu }_D{)}^2}$ with width $\frac{\mathrm{\Delta }{\nu }_D}{{\nu }_0}=\frac{v_{\text{th}}}{c}=\frac{1}{c}\sqrt{\frac{2k_{B}T}{m}}$ (dominates line core).

  • Equivalent Width ($W$): Area of absorption line relative to continuum: $W=\int (1-e^{-{\tau }_{\nu }})d\lambda$.

  • Curve of Growth (3 Regimes of $W$ vs. column density $N=\int nds$):

    1. Linear (${\tau }_0\ll 1$): Optically thin core $⟹\mathbf{W}\propto \mathbf{N}$ (independent of $\mathrm{\Delta }{\nu }_D$).

    2. Flat / Saturated (${\tau }_0\gtrsim 1$): Gaussian core saturates $⟹\mathbf{W}\propto \mathbf{\Delta }{\nu }_{\mathbf{D}}\sqrt{\ln \mathbf{N}}$ (depends on thermal/turbulent velocity $b$).

    3. Damped / Square-Root (${\tau }_0\gg 1$): Broad Lorentzian wings dominate $⟹\mathbf{W}\propto \sqrt{\mathbf{N}\gamma }$.

• The 21-cm Hyperfine Line of Neutral Hydrogen ($\text{H I}$):

  • Transition between parallel ($F=1,g_u=3$) and antiparallel ($F=0,g_l=1$) proton-electron spins in ground-state $\text{H}$: $\lambda =21.1\text{ cm}$, ${\nu }_0=1420.4\text{ MHz}$, $A_{ul}=2.85\times {10}^{-15}{\text{ s}}^{-1}$ ($t_{\text{spont}}\approx 1.1\times {10}^7\text{ yr}$).

  • Since $h{\nu }_0/k_B\approx 0.068\text{ K}\ll T_{\text{ISM}}$, Boltzmann ratio is simply $\frac{n_u}{n_l}\approx \frac{g_u}{g_l}=3⟹n_u=\frac{3}{4}{n}_H,n_l=\frac{1}{4}{n}_H$.

  • 21-cm Emission (Optically thin): Gives neutral hydrogen column density $N_H=\int n_{H}ds$ independent of temperature:

    $$I=\int j_{\nu }dsd\nu =\frac{3}{16\pi }h{\nu }_0{A}_{ul}\int n_{H}ds\propto N_H$$

  • 21-cm Absorption (against radio source): Because of stimulated emission factor $(1-e^{-h{\nu }_0/k_{B}T})\approx \frac{h{\nu }_0}{k_{B}T}$, optical depth depends inversely on temperature:

    $${\tau }_{\nu }=\frac{3c^{2}h{A}_{ul}}{32\pi {\nu }_0{k}_B}\varphi (\mathrm{\Delta }\nu )\int \frac{n_H}{T}ds\propto \int \frac{n_H}{T}ds$$

    Combining 21-cm emission ($\int n_{H}ds$) and absorption ($\int \frac{n_H}{T}ds$) simultaneously measures $N_H$ and the gas temperature $T$!