Essential Equations

The specific intensity $I_\nu$ of radiation is defined by
$$I_\nu \equiv {dP \over (\cos \theta ~ d \sigma) d \nu d \Omega}\rlap{\quad \href{Brightness.html#SpecificIntensity}{\rm {(2A1)}}}$$
where $dP$ is the power received by a detector with projected area $(\cos \theta d \sigma)$ in the solid angle $d \Omega$ and in the frequency range $\nu$ to $\nu + d\nu$.  The mks units of $I_\nu$ are W m$^{-2}$ Hz$^{-1}$ sr$^{-1}$.  Specific intensity is conserved (is constant) along any ray in empty space.  It is a property of the source only, independent of the distance between the source and the observer.

The flux density $S_\nu$ of a source is the spectral power received per unit detector area:
$$S_\nu\equiv \int_{\rm source}I_\nu(\theta,\phi) \cos \theta d\Omega\rlap{\quad \href{Brightness.html#FluxDensity}{\rm {(2A2)}}}$$
If the source is compact enough that $\cos\theta \approx 1$ then

$$S_\nu\approx \int_{\rm source}I_\nu(\theta,\phi) d\Omega \rlap{\quad \href{Brightness.html#FluxDensity}{\rm {(2A3)}}}$$
The mks units of flux density are W m$^{-2}$ Hz$^{-1}$.  Astronomers use $1{\rm ~Jansky} = 1{\rm ~Jy} \equiv 10^{-26} {\rm ~W~m}^{-2}{\rm ~Hz}^{-1}$. 

The spectral luminosity $L_\nu$ of a source is the total power per unit frequency radiated at frequency $\nu$; its mks units are W Hz$^{-1}$.  In free space and at distances $r$ much greater than the source size, the inverse-square law
$$L_\nu = 4 \pi r^2 S_\nu\rlap{\quad \href{Brightness.html#InverseSquareLaw}{\rm {(2A4)}}}$$
relates spectral luminosity and flux density.

The linear absorption coefficient at frequency $\nu$ of an absorber is defined as the infinitesimal probability $dp_\nu$ that a photon will be absorbed in a layer of infinitesimal thickness $ds$.
$$\kappa_\nu \equiv {dp_\nu \over ds}\rlap{\quad \href{Radxfer.html#KappaNu}{\rm {(2B1)}}}$$
The optical depth $\tau_\nu$ at frequency $\nu$ is defined as the sum of those infinitesimal probabilities through the absorber, starting at the source end.
$$\tau_\nu \equiv \int_{s_{\rm out}}^{s_{\rm in}} -\kappa_\nu(s') ds'\rlap{\quad \href{Radxfer.html#Opacity}{\rm {(2B2)}}} $$
The emission coefficient at frequency $\nu$ is the infinitesimal increase $dI_\nu$ in specific intensity per infinitesimal distance $ds$:
$$\epsilon_\nu \equiv {dI_\nu \over ds}\rlap{\quad \href{Radxfer.html#EmissionCoefficient}{\rm {(2B3)}}}$$
The equation of radiative transfer is
$${d I_\nu \over ds} = -\kappa_\nu I_\nu + \epsilon_\nu\rlap{\quad \href{Radxfer.html#RadXferEq}{\rm {(2B4)}}}$$
For any substance in Local Thermodyamic Equilibrium (LTE), Kirchoff's law connects the emission and absorption coefficients via the specific intensity $B_\nu$ of blackbody radiation.
$${\epsilon_\nu \over \kappa_\nu} = B_\nu(T)\rlap{\quad \href{Radxfer.html#KirchoffsLaw}{\rm {(2B5)}}}$$
The brightness temperature of a source with any specific intensity $I_\nu$ is defined as
$$T_{\rm b}(\nu) \equiv {I_\nu c^2 \over 2 k \nu^2}\rlap{\quad \href{Radxfer.html#BrightnessTemp}{\rm {(2B6)}}}$$

For an opaque body in LTE, Kirchoff's law ties together the spectral emissivity $e_\nu$ (the emissivity of the body divided by the emissivity of a black body) to the absorption coefficient $a_\nu$ (fraction of radiation absorbed by the body) and the reflection coefficient $r_\nu$ (fraction of radiation reflected by the body).
$$e_\nu = a_\nu = 1 - r_\nu \rlap{\quad \href{Radxfer.html#KirchoffsLaw2}{\rm {(2B7)}}}$$

The spectral energy density of radiation is
$$u_\nu = {1 \over c} \int I_\nu d \Omega \rlap{\quad \href{BlackBodyRad.html#RadEdensity}{\rm {(2C1)}}}$$

The Rayleigh-Jeans approximation for the specific intensity of blackbody radiation when $h \nu \ll kT$ is
$$B_\nu = {2 k T \nu^2 \over c^2}\rlap{\quad \href{BlackBodyRad.html#RJlaw}{\rm {(2C2)}}}$$

Planck's law for the specific intensity of blackbody radiation is
$$B_\nu = {2 h \nu^3 \over c^2} {1 \over \exp\bigl({h \nu \over k T}\bigr) -1}\rlap{\quad \href{BlackBodyRad.html#PlanckLaw}{\rm {(2C3)}}}$$

The total intensity of blackbody radiation is:
$$B(T) \equiv \int_0^\infty B_\nu(T) d\nu = {\sigma T^4 \over \pi} \rlap{\quad \href{BlackBodyRad.html#IntBright}{\rm {(2C4)}}}$$
where the Stefan-Boltzmann constant $\sigma$ is defined by
$$\sigma \equiv {2 \pi^5 k^4 \over 15 c^2 h^3} \approx 5.67 \times 10^{-5} {{\rm erg} \over {\rm cm}^2 {\rm ~s~K}^4 {\rm ~sr}}$$

The total energy density of blackbody radiation is
$$u =  {4 \sigma T^4\over c} \rlap{\quad \href{BlackBodyRad.html#BBEdensity}{\rm {(2C5)}}}$$

The Nyquist formula approximating the spectral power generated by a warm resistor in the limit $h \nu \ll k T$ is
$$P_\nu = k T \rlap{\quad \href{BlackBodyRad.html#NyquistLaw}{\rm {(2C6)}}}$$
At any frequency, $$P_\nu = {h \nu  \over {\exp\bigl({h \nu \over kT}\bigr) -1}} \rlap{\quad \href{BlackBodyRad.html#QuantumNyquistLaw}{\rm {(2C7)}}}$$

The radiated electric field at distance $r$ from a charge $q$ at angle $\theta$ from the acceleration $\dot{v}$ is:

$$E_\bot = {q \dot{v} \sin\theta \over r c^2}\rlap{\quad \href{LarmorRad.html#LarmorEField}{\rm {(2D1)}}}$$

In a vacuum, the Poynting flux, or power per unit area (erg s$^{-1}$ cm$^{-2}$), is
$$\vert \vec{S} \vert = {c \over 4 \pi}E^2\rlap{\quad \href{LarmorRad.html#PoyntingFlux}{\rm {(2D2)}}}$$

The total power emitted by an accelerated charge is given by Larmor's formula
$$P = {2 \over 3}{q^2 \dot{v}^2 \over c^3}\rlap{\quad \href{LarmorRad.html#LarmorPower}{\rm {(2D3)}}}$$
valid only if $v \ll c$.

The power pattern of a short dipole antenna is
$$P \propto \sin^2 \theta\rlap{\quad
\href{AntennaTheory.html#DipolePattern}{\rm {(3A1)}}}$$

The power emitted by a short ($l \ll \lambda$) dipole driven by a current $I = I_0 e^{-i \omega t}$ is
$$\langle P \rangle = {\pi^2 \over 3 c} \biggl({I_0 l \over \lambda}\biggr)^2 \rlap{\quad\href{AntennaTheory.html#DipolePower}{\rm {(3A2)}}}$$

Radiation resistance is defined by
$$R \equiv {2 \langle P \rangle \over I_0^2}\rlap{\quad
\href{AntennaTheory.html#RadiationResistance}{\rm {(3A3)}}}$$

Energy conservation implies the average power gain of a lossless antenna is
$$ \langle G \rangle = 1\rlap{\quad
\href{AntennaTheory.html#AverageGain}{\rm {(3A4)}}}$$

The effective area of an antenna is defined by
$$ A_{\rm e} \equiv P_\nu / S_{\rm (matched)}\rlap{\quad
\href{AntennaTheory.html#EffectiveArea}{\rm {(3A5)}}}$$

The average effective area of any lossless antenna is $$\langle A_{\rm e} \rangle = {\lambda^2 \over 4 \pi}\rlap{\quad \href{AntennaTheory.html#AverageArea}{\rm {(3A6)}}}$$

Reciprocity implies
$$G(\theta, \phi) \propto A_{\rm e}(\theta, \phi)\rlap{\quad \href{AntennaTheory.html#Reciprocity}{\rm {(3A7)}}}$$
Reciprocity and energy conservation imply:
$$A_{\rm e}(\theta, \phi) = {\lambda^2 G(\theta, \phi) \over 4 \pi}\rlap{\quad \href{AntennaTheory.html#GainArea}{\rm {(3A8)}}}$$
Antenna temperature is defined by
$$T_{\rm A} \equiv {P_\nu \over k}\rlap{\quad \href{AntennaTheory.html#AntennaTemp}{\rm {(3A9)}}}$$

The antenna temperature produced by an unpolarized point source of flux density $S$ is
$$T_{\rm A} = {A_{\rm e} S \over 2 k}\rlap{\quad \href{AntennaTheory.html#AntSens}{\rm {(3A10)}}}$$
An effective collecting area $A_{\rm e} \approx 2761 {\rm ~m}^2$ corresponds to a point-source sensitivity of $1 {\rm ~K~Jy}^{-1}$.

Beam solid angle is defined by
$$\Omega_{\rm A} \equiv \int_{4 \pi} P_{\rm n} (\theta, \phi) d \Omega\rlap{\quad \href{AntennaTheory.html#BeamSolidAngle}{\rm {(3A11)}}}$$
where $P_{\rm n} (\theta, \phi)$ is the power pattern normalized to one at the peak.

For a uniform compact source of brightness temperature $T_{\rm B}$ covering solid angle $\Omega_{\rm s}$,
$$T_{\rm A} \approx T_{\rm B} {\Omega_{\rm s} \over \Omega_{\rm A}}$$

The main beam solid angle is defined by the integral over the main beam to the first zero only:
$$\Omega_{\rm MB} \equiv \int_{\rm MB} P_{\rm n} (\theta, \phi) d \Omega\rlap{\quad \href{AntennaTheory.html#MainBeamSolidAngle}{\rm {(3A12)}}}$$
and is used in the definition of main beam efficiency:
$$\eta_{\rm B} \equiv {\Omega_{\rm MB} \over \Omega_{\rm A}}\rlap{\quad \href{AntennaTheory.html#MainBeamEfficiency}{\rm {(3A13)}}}$$

The equation for the height $z$ at axial distance $r$ of a paraboloidal reflector of focal length $f$ is

$$z = {r^2 \over 4 f}~, ~f = {\rm ~focal~length}\rlap{\quad
\href{ReflectorAntennas.html#Paraboloid}{\rm {(3B1)}}}$$

The far-field distance of an aperture of diameter $D$ used at wavelength $\lambda$ is

$$ R_{\rm ff} \approx {2 D^2 \over \lambda}\rlap{\quad
\href{ReflectorAntennas.html#FarField}{\rm {(3B2)}}}$$

Field pattern:

$$l \equiv \sin\theta\rlap{\quad
\href{ReflectorAntennas.html#DefineL}{\rm {(3B3)}}}$$

$$u \equiv {x \over \lambda}\rlap{\quad
\href{ReflectorAntennas.html#DefineU}{\rm {(3B4)}}}$$

$$f(l) = \int_{\rm aperture} g(u) e^{-i 2 \pi l u} du \rlap{\quad
\href{ReflectorAntennas.html#FieldPattern}{\rm {(3B5)}}}$$

In the far field, the electric-field pattern of an aperture antenna is the Fourier transform of the electric field illuminating the aperture.

Uniformly Illuminated Linear Aperture:

$$P(\theta) \propto {\rm sinc}^2 \biggl( {\theta D \over \lambda} \biggr)\rlap{\quad
\href{ReflectorAntennas.html#PowerPattern}{\rm {(3B6)}}}$$ where ${\rm sinc}(x) \equiv \sin(\pi x)/ (\pi x)$

$$\theta_{\rm HPBW} \approx 0.89 {\lambda \over D}\rlap{\quad
\href{ReflectorAntennas.html#Beamwidth}{\rm {(3B7)}}}$$

Two-dimensional aperture field pattern:

$$f(l,m) \propto \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(u,v) e^{-i 2 \pi(l u + mv)} d u d v \rlap{\quad
\href{2DApertures.html#2DFieldPattern}{\rm {(3C1)}}}$$

where $m$ is the $y$-axis analog of $l$ on the $x$-axis, and $v \equiv y / \lambda$.

The electric field pattern of a two-dimensional aperture is the two-dimensional Fourier transform of the aperture field illumination. Uniformly Illuminated Rectangular Aperture Power Pattern:
$$G = {4 \pi D_{\rm x} D_{\rm y} \over \lambda^2}
{\rm sinc}^2\biggl({ l D_{\rm x} \over \lambda} \biggr) {\rm sinc}^2 \biggl({m D_{\rm y} \over \lambda} \biggr) \approx
{4 \pi D_{\rm x} D_{\rm y} \over \lambda^2}
{\rm sinc}^2\biggl({ \theta_{\rm x} D_{\rm x} \over \lambda} \biggr) {\rm sinc}^2 \biggl({\theta_{\rm y} D_{\rm y} \over \lambda} \biggr)\rlap{\quad
\href{2DApertures.html#2DPowerPattern}{\rm {(3C2)}}}$$

Aperture efficiency is defined by
$$\eta_{\rm A} \equiv { {\rm max}(A_{\rm e}) \over A_{\rm geom}}\rlap{\quad
\href{2DApertures.html#ApertureEfficiency}{\rm {(3C3)}}}$$ 

The half-power beamwidth (HPBW) for a circular Gaussian illumination pattern is

$$\theta_{\rm HPBW} \approx 1.2 {\lambda \over D}\rlap{\quad
\href{2DApertures.html#HPBW}{\rm {(3C4)}}}$$

The surface efficiency $\eta$ of a reflector whose surface errors $\epsilon$ have rms $\sigma$ is given by the Ruze equation:

$$\eta_{\rm s} = \exp \biggl[ - \biggl({4 \pi \sigma \over \lambda} \biggr)^2 \biggr]\rlap{\quad
\href{2DApertures.html#SurfaceEfficiency}{\rm {(3C5)}}}$$

Noise temperature is defined by

$$T_{\rm N} \equiv {P_\nu \over k}\rlap{\quad
\href{Radiometers.html#NoiseTemp}{\rm {(3E1)}}} $$

The system noise temperature is the sum of noise contributions from all sources:

$$T_{\rm sys} = T_{\rm cmb} + \Delta T_{\rm source} + T_{\rm atm} + T_{\rm spillover} + T_{\rm rcvr} + \dots \rlap{\quad
\href{Radiometers.html#SystemNoise}{\rm {(3E2)}}}$$

The ideal total-power radiometer equation is

$$\sigma_{\rm T} \approx T_{\rm sys} \biggl[ {1 \over \Delta \nu_{\rm RF} \tau} \biggr]^{1/2}\rlap{\quad
\href{Radiometers.html#IdealRadiometer}{\rm {(3E3)}}}$$

The practical total-power radiometer equation includes the effects of gain fluctuations:

$$\sigma_{\rm T} \approx T_{\rm sys} \biggl[ {1 \over \Delta \nu_{\rm RF} \tau} + \biggl( {\Delta G \over G} \biggr)^2 \biggr]^{1/2}\rlap{\quad
\href{Radiometers.html#Radiometer}{\rm {(3E4)}}}$$

The Dicke-switching radiometer equation is

$$\sigma_{\rm T} \approx 2 T_{\rm sys}\biggl[ { 1 \over \Delta \nu_{\rm RF} \tau} \biggr]^{1/2}\rlap{\quad
\href{Radiometers.html#DickeRadiometer}{\rm {(3E5)}}} $$

The rms confusion caused by unresolved continuum sources in a Gaussian beam with HPBW $\theta$ at frequency $\nu$ is
$$\biggl({\sigma_{\rm c} \over {\rm mJy~beam}^{-1}}\biggr) \approx 0.2 \biggr({\nu \over {\rm GHz}}\biggr)^{-0.7} \biggl( {\theta \over {\rm arcmin}}\biggr)^2 \rlap{\quad
\href{Radiometers.html#rmsconfusion} {\rm {(3E6)}}} $$ Individual sources weaker than the confusion limit $S \approx 5\sigma_{\rm c}$ cannot be detected reliably.
confusion fluctuations in map of NGP

The complex visibility of a two-element radiometer is related to the source brightness distribution by
$$V_\nu = \int I_\nu(\hat{s})
\exp(-i2\pi \vec{b} \cdot\hat{s} / \lambda) \, d \Omega\rlap{\quad
\href{Interferometers1.html#Complexvis}{\rm {(3F1)}}} $$

To minimize bandwidth smearing in bandwidth $\Delta \nu$, the image angular radius $\Delta \theta$ should satisfy
$$\Delta \theta \Delta \nu \ll \theta_{\rm s} \nu \rlap{\quad
\href{Interferometers1.html#bandsmear}{\rm {(3F2)}}} $$

To minimize time smearing in an image of angular radius $\Delta \theta$ the averaging time should satsify
$$\Delta\theta \Delta t \ll {\theta_{\rm s} P \over 2 \pi} \approx \theta_{\rm s} \times 1.37\times10^4 {\rm ~s} \rlap{\quad
\href{Interferometers1.html#timesmear}{\rm {(3F3)}}} $$

The relation between the source brightness distribution $I_\nu(l,m)$ and the visibilities $V_\nu(u,v,w)$ for an interferometer in three dimensions is
$$V_\nu (u,v,w) = \int \int {I_\nu (l,m) \over (1 -l^2 -m^2)^{1/2}}
\exp[-i 2 \pi(ul+vm+wn)] dl dm \rlap{\quad
\href{Interferometers2.html#3dinterferometer}{\rm {(3G1)}}}$$
For a two-dimensional interferometer confined to $(u,v)$ plane, the source brightness distribution $I_\nu(l,m)$ is the Fourier transform of the fringe visibilities $V_\nu(u,v)$:
$${I_\nu(l,m) \over (1 - l^2 -m^2)^{1/2}} = \int\int V_\nu(u,v,0) \exp[+i 2 \pi (ul +vm)] du dv \rlap{\quad
\href{Interferometers2.html#earthrotation} {\rm {(3G2)}}}$$

The point-source sensitivity (or brightness sensitivity in units of flux density per beam solid angle) for an interferometer with $N$ antennas, each with effective area $A_{\rm eff}$, is
$$\sigma_{\rm S} = {2 k T_{\rm sys} \over A_{\rm eff} [N(N-1)\Delta \nu_{\rm RF} \tau]^{1/2}} \rlap{\quad
\href{Interferometers2.html#fluxsensitivity} {\rm {(3G3)}}}$$

The brightness sensitivity (K) corresponding to a point source sensitivity $\sigma_{\rm S}$ and a beam solid angle $\Omega_{\rm A}$ is
$$\sigma_{\rm T} = \biggl({\sigma_{\rm S} \over \Omega_{\rm A}}\biggr) {\lambda^2 \over 2k}\rlap{\quad
\href{Interferometers2.html#brightsensitivity} {\rm {(3G4)}}}$$ For a Gaussian beam of HPBW $\theta_0$, $$\Omega_{\rm A} = { \pi \theta_0^2 \over 4 \ln 2} \approx 1.133 \theta_0^2$$

The nonrelativistic Maxwellian distribution of particle speeds $v$ is
$$f(v) = {4 v^2 \over \sqrt{\pi}} \biggl( { m \over 2 k T} \biggr)^{3/2} \exp \biggl( - {m v^2 \over 2 k T} \biggr)\rlap{\quad
\href{FreeFreeEmission.html#Maxwellian}{\rm {(4B1)}}}$$
The free-free emission coefficient is
$$\epsilon_\nu = {\pi^2 Z^2 e^6 N_{\rm e} N_{\rm i} \over 4 c^3 m_{\rm e}^2 } \biggl( {2 m_{\rm e} \over \pi k T} \biggr)^{1/2} \ln \biggl( {b_{\rm max} \over b_{\rm min}} \biggr)\rlap{\quad \href{FreeFreeEmission.html#FFemcoefficient}{\rm {(4B2)}}}$$
$$b_{\rm min} \approx {Z e^2 \over m_{\rm e} v^2}\rlap{\quad \href{FreeFreeEmission.html#Bmin}{\rm {(4B3)}}}$$
The free-free absorption coefficient is
$$\kappa_\nu = {1 \over \nu^2 T^{3/2}} \biggl[ {Z^2 e^6 \over c} N_{\rm e} N_{\rm i} {1 \over \sqrt { 2 \pi (m_{\rm e} k)^3 } } \biggr]{\pi^2 \over 4} \ln \biggl( { b_{\rm max} \over b_{\rm min} }\biggr)\rlap{\quad \href{FreeFreeEmission.html#FFabscoefficient}{\rm {(4B4)}}}$$

At frequencies low enough that $\tau_\nu \gg 1$, the HII region becomes opaque, its spectrum approaches that of a black body with temperature $T\ \sim 10^4$ K, and the flux density varies as $S \propto \nu^2$. At very high frequencies, $\tau_\nu \ll 1$, the HII region is nearly transparent, and
$$S_\nu \propto {2 k T \nu^2 \over c^2} \tau_\nu \propto \nu^{-0.1}$$
On a log-log plot, the overall spectrum of a uniform HII region looks like this, with the spectral break corresponding to the frequency at which $\tau_\nu \approx 1$:

The emission measure of a plasma is defined by
$${ EM \over {\rm pc~cm}^{-6}} \equiv \int_{\rm los} \biggl( { N_{\rm e} \over {\rm cm}^{-3} } \biggr)^2 d \biggl( {s \over {\rm pc} } \biggr)\rlap{\quad \href{FreeFreeEmission.html#EmissionMeasure}{\rm {(4B5)}}}$$

The free-free optical depth of a plasma is
$$\tau_\nu \approx 3.28 \times 10^{-7} \biggl( {T_{\rm e} \over 10^4 {\rm ~K} } \biggr)^{-1.35} \biggl( {\nu \over {\rm GHz} } \biggr)^{-2.1} \biggl( { EM \over {\rm pc~cm}^{-6} } \biggr)\rlap{\quad \href{FreeFreeEmission.html#TauFF}{\rm {(4B6)}}}$$

The ionization rate $N_{\rm Ly}$ of Lyman continuum photons produced per second required to maintain an HII region is
$$\biggl( { N_{\rm Ly} \over {\rm s}^{-1} } \biggr) \approx 6.3 \times 10^{52 } \biggl( { T_{\rm e} \over 10^4 {\rm ~K} } \biggr)^{-0.45} \biggl( { \nu \over {\rm GHz } } \biggr)^{0.1} \biggl( { L_\nu \over 10^{20} {\rm ~W~Hz}^{-1} } \biggr)\rlap{\quad \href{FreeFreeEmission.html#LyAlphaRate}{\rm {(4B7)}}}$$
where $L_\nu = 4 \pi D^2 S_\nu$ is the free-free luminosity at any frequency $\nu$ high enough that $\tau_\nu \ll 1$.

The magnetic force on a moving charge is
$$\vec{F} = { e (\vec{v} \times \vec{B}) \over c}\rlap{\quad
\href{Magnetobremsstrahlung.html#MagneticForce}{\rm {(5A1)}}}$$

The gyro frequency is defined by
$$\omega_{\rm G} \equiv { e B \over m c}\rlap{\quad
\href{Magnetobremsstrahlung.html#GyroFrequency}{\rm {(5A2)}}}$$

The nonrelativistic electron gyro frequency in MHz is
$$\biggl({\nu_{\rm G} \over {\rm MHz}}\biggr) = 2.8 \biggl( { B \over {\rm Gauss} }\biggr)\rlap{\quad
\href{Magnetobremsstrahlung.html#ElectronGyro}{\rm {(5A3)}}}$$

The Lorentz Transform is

$$ x = \gamma (x' + vt') \qquad y = y' \qquad z = z' \qquad t = \gamma (t' + \beta x' / c)\rlap{\quad
\href{SynchrotronPower.html#Lorentz}{\rm {(5B1)}}} $$
$$ x' = \gamma (x - vt) \qquad y' = y \qquad z' = z \qquad t' = \gamma (t - \beta x / c)\rlap{\quad
\href{SynchrotronPower.html#LorentzPrime}{\rm {(5B2)}}} $$ where
$$\beta \equiv v/c\rlap{\quad
\href{SynchrotronPower.html#Beta}{\rm {(5B3)}}}$$ and
$$\gamma \equiv (1 - \beta^2)^{-1/2}\rlap{\quad
\href{SynchrotronPower.html#Gamma}{\rm {(5B4)}}}$$
If $(\Delta x', \Delta y', \Delta z', \Delta t')$ and $(\Delta x, \Delta y, \Delta z, \Delta t)$ are the coordinate differences between two events, the differential form of the (linear) Lorentz transform is:
$$\Delta x = \gamma(\Delta x' + v \Delta t') \qquad \Delta y = \Delta y' \qquad \Delta z = \Delta z' \qquad \Delta t = \gamma (\Delta t' + \beta \Delta x' / c)\rlap{\quad
\href{SynchrotronPower.html#DiffLT}{\rm {(5B5)}}} $$
$$\Delta x' = \gamma(\Delta x - v \Delta t) \qquad \Delta y' = \Delta y \qquad \Delta z' = \Delta z \qquad \Delta t' = \gamma (\Delta t - \beta \Delta x / c)\rlap{\quad
\href{SynchrotronPower.html#DiffLTPrime}{\rm {(5B6)}}} $$

The Thomson cross section of an electron is defined by
$$\sigma_{\rm T} \equiv {8 \pi \over 3} \biggl( { e^2 \over m_{\rm e} c^2 } \biggr)^2 \approx 6.65 \times 10^{ -25} {\rm ~cm}^2\rlap{\quad
\href{SynchrotronPower.html#ThomsonArea}{\rm {(5B7)}}}$$
Magnetic energy density:
$$U_{\rm B} = {B^2 \over 8 \pi}\rlap{\quad
\href{SynchrotronPower.html#Umag}{\rm {(5B8)}}}$$
Synchrotron power of one electron:
$$P = 2 \sigma_{\rm T} \beta^2 \gamma^2 c\, U_{\rm B} \sin^2\alpha\rlap{\quad
\href{SynchrotronPower.html#Power}{\rm {(5B9)}}}$$

Synchrotron power averaged over pitch angles:

$$\langle P \rangle = {4 \over 3} \sigma_{\rm T} \beta^2 \gamma^2 c U_{\rm B}\rlap{\quad
\href{SynchrotronPower.html#AveragePower}{\rm {(5B10)}}} $$

The synchrotron spectrum of a single electron is
$$P(\nu) = {\sqrt{3} e^3 B \sin \alpha \over m c^2} \biggl( { \nu \over \nu_{ \rm c}} \biggr) \int_{\nu/\nu_{\rm c}}^\infty K_{5/3} (\eta) d \eta\rlap{\quad
\href{SynchrotronSpectrum.html#Spectrum}{\rm {(5C1)}}}$$ where $K_{ 5/3}$ is a modified Bessel function and the critical frequency is
$$\nu_{\rm c} = {3 \over 2} \gamma^2 \nu_{\rm G} \sin \alpha \approx \gamma^2 \nu_{\rm G} \,
\propto E^2 B_\bot \rlap{\quad
\href{SynchrotronSpectrum.html#CriticalFrequency}{\rm {(5C2)}}}$$

The synchrotron spectrum of a single electron plotted in terms of
$$F(x) \equiv x \int_x^\infty K_{5/3} (\eta) d \eta$$
where $x \equiv \nu / \nu_{\rm c}$.

The observed energy distribution of cosmic-ray electrons in our Galaxy is roughly a power law:
$$N(E) dE \approx K E^{-\delta} dE\rlap{\quad
\href{SynchrotronSpectrum.html#CRSpectrum}{\rm {(5C3)}}}$$
where $N(E) dE$ is the number of electrons per unit volume with energies $E$ to $E + dE$ and $\delta \approx 5/2$.
The corresponding synchrotron emission coefficient is
$$\epsilon_\nu \propto B^{(\delta + 1) /2} \nu^{(1 - \delta)/2}\rlap{\quad
\href{SynchrotronSpectrum.html#EmissionSpectrum}{\rm {(5C4)}}}$$

The [negative sign convention] spectral index is
$$\alpha = {\delta - 1 \over 2}\rlap{\quad
\href{SynchrotronSpectrum.html#SpectralIndex}{\rm {(5C5)}}}$$

For a given synchrotron luminosity, the electron energy density is
$$U_{\rm e} \propto B^{-3 /2} \rlap{\quad
\href{SynchrotronSrcs.html#ElectronU}{\rm {(5D1)}}}$$
The total energy density of both cosmic rays and magnetic fields is
$$U = (1 + \eta) U_{\rm e} + U_{\rm B}\rlap{\quad
\href{SynchrotronSrcs.html#TotalU}{\rm {(5D2)}}}$$
where $\eta$ is the ion/electron energy ratio.

At minimum total energy, the ratio of particle to field energy is nearly unity (equipartition):
$${{\rm particle~energy} \over {\rm field~energy}} = {(1 + \eta) U_{\rm e} \over U_{\rm B}} = {4 \over 3}\rlap{\quad
\href{SynchrotronSrcs.html#MinimumE}{\rm {(5D3)}}}$$

The minimum-energy magnetic field is
$$B_{\min} = [4.5 (1 +\eta) c_{12} L]^{2/7} R^{-6/7} {\rm ~Gauss}\rlap{\quad
\href{SynchrotronSrcs.html#minEB} {\rm {(5D4)}}}$$
The corresponding energy in relativistic particles is
$$E_{\rm min}{\rm (total)} = c_{13} [(1+ \eta) L]^{4/7} R^{9/7} {\rm ~ergs} \rlap{\quad
\href{SynchrotronSrcs.html#minEpart} {\rm {(5D5)}}}$$
The synchrotron lifetime is
$$\tau \approx c_{12} B_\bot^{-3/2} \rlap{\quad
\href{SynchrotronSrcs.html#synclifeEq} {\rm {(5D6)}}}$$ where the functions $c_{12}$ and $c_{13}$ in Gaussian cgs units are plotted below.  Frequency limits $\nu_{\rm min} = 10^7$ Hz and $\nu_{\rm max} = 10^{11}$ Hz are commonly used.

Plots of $c_{12}$ and $c_{13}$ as functions of [negative] spectral index $\alpha \equiv - d \log S / d \log \nu$ for $\nu_{\rm min} = 10^6$ Hz (dashed curves) and $10^7$ Hz (solid curves) and $\nu_{\rm max} = 10^{10}$ Hz, $10^{11}$ Hz, and $10^{12}$ Hz.

The Eddington Limit for luminosity is
$$\biggl( { L_{\rm E} \over L_\odot } \biggr) \approx 3.3 \times 10^4 \biggl( { M \over M_\odot} \biggr)\rlap{\quad
\href{SynchrotronSrcs.html#Eddington}{\rm {(5D7)}}}$$

The effective temperature of a relativistic electron emitting at frequency $\nu$ in magnetic field $B$ is
$$\biggl( { T_{\rm e} \over {\rm K} } \biggr) \approx 1.18 \times 10^6 \biggl( { \nu \over {\rm Hz}} \biggr)^{1/2} \biggl( { B \over {\rm Gauss }} \biggr)^{-1/2}\rlap{\quad
\href{SynchrotronSrcs.html#ElectronTemp}{\rm {(5D8)}}}$$

At a sufficiently low frequency $\nu$, 
$$S_{\rm \nu} \propto \nu^{-5/2}\rlap{\quad
\href{SynchrotronSrcs.html#SSASlope}{\rm {(5D9)}}}$$

$$\biggl( { B \over {\rm Gauss}} \biggr) \approx 1.4 \times 10^{12} \biggl( {\nu \over {\rm Hz}} \biggr) \biggl( { T_{\rm b} \over {\rm K} } \biggr)^{-2}\rlap{\quad
\href{SynchrotronSrcs.html#SSABfield}{\rm {(5D10)}}}$$

Nonrelativistic Thomson-scattering power:
$$P = \sigma_{\rm T} c U_{\rm rad}\rlap{\quad
\href{InverseCompton.html#PScattered}{\rm {(5E1)}}}$$

The relativistic Doppler equation is
$$\nu' = \nu [\gamma(1 + \beta \cos \theta)]\rlap{\quad
\href{InverseCompton.html#Doppler}{\rm {(5E2)}}}$$

The relativistic inverse-Compton power (net) emitted is:
$$P_{\rm IC} = {4 \over 3} \sigma_{\rm T} c \beta^2 \gamma^2 U_{\rm rad}\rlap{\quad
\href{InverseCompton.html#ICPower}{\rm {(5E3)}}}$$

IC/synchrotron power ratio:
$${P_{\rm IC} \over P_{\rm syn} } = { U_{\rm rad} \over U_{\rm B}}\rlap{\quad
\href{InverseCompton.html#PowerRatio}{\rm {(5E4)}}}$$
The average frequency $\langle \nu \rangle$ of upscattered photons having initial frequency $\nu_0$ is
$${\langle \nu \rangle \over \nu_0} = {4\over 3} \gamma^2\rlap{\quad
\href{InverseCompton.html#ICFrequency}{\rm {(5E5)}}}$$
Maximum rest-frame brightness temperature of an incoherent synchrotron source:
$$T_{\rm max} \sim 10^{12}{\rm ~K}\rlap{\quad
\href{InverseCompton.html#Tmax}{\rm {(5E6)}}}$$

The apparent transverse velocity of a moving source component is
$$\beta_\bot{\rm (apparent)} = {\beta \sin \theta \over 1 - \beta \cos\theta}\rlap{\quad
\href{ExtraGalactic.html#ApparentBeta}{\rm {(5F1)}}}$$

For any $\beta$ the angle $\theta_{\rm m}$ that maximizes $\beta_\bot{\rm (apparent)}$ satisfies
$$\cos\theta_{\rm m} = \beta\rlap{\quad
\href{ExtraGalactic.html#BetaThetamax}{\rm {(5F2)}}}$$ and
$$\sin\theta_{\rm m} = \gamma^{-1}\rlap{\quad
\href{ExtraGalactic.html#GammaThetamax}{\rm {(5F3)}}}$$ and the largest apparent
transverse speed is

$${\rm max}[\beta_\bot{\rm (apparent)}] = \beta \gamma\rlap{\quad
\href{ExtraGalactic.html#ApparentBetamax}{\rm {(5F4)}}}$$

The transverse ($\theta= \pi /2$) Doppler shift is
$${\nu \over \nu'} = \gamma^{-1}\rlap{\quad
\href{ExtraGalactic.html#TransverseDoppler}{\rm {(5F5)}}}$$ 

Doppler boosting for Doppler factor $\delta \equiv \nu / \nu'$:
$$\delta^{2 + \alpha} < {S \over S_0} < \delta^{3 + \alpha}\rlap{\quad
\href{ExtraGalactic.html#Boosting}{\rm {(5F6)}}}$$

Thermal and nonthermal luminosities of star-forming galaxies:

$$\biggl({ L_{\rm T} \over {\rm W~Hz}^{-1} } \biggr) \approx 5.5 \times 10^{20} \biggl( {\nu \over {\rm GHz}} \biggr)^{-0.1} \biggl[ {SFR(M > 5 M_\odot) \over M_\odot {\rm ~yr}^{-1} } \biggr]\rlap{\quad
\href{ExtraGalactic.html#LThermal}{\rm {(5F7)}}}$$

$$\biggl({ L_{\rm NT} \over {\rm W~Hz}^{-1} } \biggr) \approx 5.3 \times 10^{21} \biggl( {\nu \over {\rm GHz}} \biggr)^{-0.8} \biggl[ {SFR(M > 5 M_\odot) \over M_\odot {\rm ~yr}^{-1} } \biggr]\rlap{\quad
\href{ExtraGalactic.html#LNonthermal}{\rm {(5F8)}}}$$

The redshift $z$ of a source is defined by
$$(1 + z) \equiv {\lambda_{\rm observed} \over \lambda_{\rm emitted}}\rlap{\quad
\href{ExtraGalactic.html#Redshift}{\rm {(5F9)}}}$$

Minimum mean density of a pulsar with period $P$:
$$\rho > {3 \pi \over G P^2}\rlap{\quad
\href{Pulsars.html#MinDensity}{\rm {(6A1)}}}$$

Magnetic-dipole radiation power:
$$P_{\rm rad} = {2 \over 3} {(\ddot{m}_\bot)^2 \over c^3}\rlap{\quad
\href{Pulsars.html#MagneticLarmor}{\rm {(6A2)}}}$$

Rotational power loss:
$${dE_{\rm rot} \over d t} = {- 4 \pi^2 I \dot{P} \over P^3}\rlap{\quad
\href{Pulsars.html#RotationEnergy}{\rm {(6A3)}}}$$

Minimum magnetic field strength:
$$\biggl( {B \over {\rm Gauss}} \biggr) > 3.2 \times 10^{19} \biggl( { P \dot{P} \over {\rm s} } \biggr)^{1/2}\rlap{\quad
\href{Pulsars.html#Bmin}{\rm {(6A4)}}}$$

The characteristic age of a pulsar is defined by
$$ \tau \equiv { P \over 2 \dot{P}}\rlap{\quad
\href{Pulsars.html#PulsarAge}{\rm {(6A5)}}}$$

The plasma frequency is
$$ \nu_{\rm p} = \biggl({e^2 n_{\rm e} \over
\pi m_{\rm e}}\biggr)^{1/2} \approx 8.97 {\rm ~kHz} \times \biggl({n_{\rm e} \over {\rm cm}^{-3}}\biggr)^{1/2}\rlap{\quad
\href{Pulsars.html#PlasmaFrequency}{\rm {(6A6)}}}$$

The group velocity of pulses is
$$v_{\rm g} \approx c \biggl( 1 - \frac{\nu_{\rm p}^2} {2 \nu^2} \biggr)\rlap{\quad
\href{Pulsars.html#GroupVelocity}{\rm {(6A7)}}}$$

The dispersion delay of a pulsar is
$$\biggl({t \over {\rm sec}} \biggr) \approx 4.149 \times 10^3 \biggl({{\rm DM} \over {\rm pc~cm}^{-3}}\biggr) \biggl( {\nu \over {\rm MHz}} \biggr)^{-2}\rlap{\quad
\href{Pulsars.html#DispersionDelay}{\rm {(6A8)}}}$$
$${\rm DM} \equiv \int_0^d n_{\rm e} dl\rlap{\quad
\href{Pulsars.html#DM}{\rm {(6A9)}}}$$
in units of pc cm$^{-3}$ is the dispersion measure to a pulsar at distance $d$.

The Bohr radius of a hydrogen atom is
$$a_{\rm n} = {n^2 \hbar^2 \over m_{\rm e} e^2}
\approx 0.53 \times 10^{-8} {\rm cm} \times n^2\rlap{\quad
\href{Recombination.html#AtomicRadius}{\rm {(7A1)}}}$$

The frequency of a recombination line is: $$\nu = R_{\rm M} c \biggl[ {1 \over n^2} - {1 \over (n + \Delta n)^2} \biggr] \qquad {\rm where} \qquad R_{\rm M} \equiv R_\infty \biggl( 1 + {m_{\rm e} \over M} \biggr)^{-1}\rlap{\quad
\href{Recombination.html#RecombFrequency}{\rm {(7A2)}}}$$

Approximate line separation frequency $\Delta \nu \equiv \nu(n) - \nu(n+1)$ for $n \gg 1$:
$${\Delta \nu \over \nu} \approx { 3 \over n}
\href{Recombination.html#ApproxFrequency}{\rm {(7A3)}}}$$

The spontaneous emission rate is
$$A_{{\rm n+1},{\rm n}} \approx {64 \pi^6 m_{\rm e}
e^{10} \over 3 c^3 h^6 n^5}
\approx 5.3 \times 10^9 \biggl({1 \over n^5}\biggr) {\rm ~s}^{-1}\rlap{\quad
\href{Recombination.html#SponRate}{\rm {(7A4)}}}$$

Gaussian normalized line profile:

$$\phi(\nu) = {c \over \nu_0} \biggl( { M \over 2 \pi k T}
\biggr)^{1/2} \exp \biggl[ - {M c^2 \over 2 k T}
{(\nu - \nu_0)^2 \over \nu_0^2 } \biggr]
\href{Recombination.html#GaussianProfile}{\rm {(7A6)}}}$$
$$\Delta \nu = \biggl( {8 \ln 2 \, k \over c^2} \biggr)^{1/2}
\biggl( { T \over M} \biggr)^{1/2} \nu_0
\href{Recombination.html#LineFWHM}{\rm {(7A7)}}}$$
$$ \phi(\nu_0) = \biggl( { \ln 2 \over \pi} \biggr)^{1/2}
{2 \over \Delta \nu}\rlap{\quad
\href{Recombination.html#LinePeak}{\rm {(7A8)}}}$$

Einstein coefficients diagram
Rate balance:
$$N_{\rm U} A_{\rm UL} + N_{\rm U} B_{\rm UL} \bar{U} = N_{\rm L} B_{\rm LS} \bar{U}\rlap{\quad
\href{LineRadxfer.html#RateBalance}{\rm {(7B3)}}}$$

Two detailed balance equations connecting Einstein coefficients:
$${g_{\rm L} \over g_{\rm U} } {B_{\rm LU} \over B_{\rm UL} } = 1 \rlap{\quad
\href{LineRadxfer.html#BLUBUL}{\rm {(7B4)}}}$$
$${A_{\rm UL} \over B_{\rm UL}} = {8 \pi h \nu_0^3 \over c^3} \rlap{\quad
\href{LineRadxfer.html#AULBUL}{\rm {(7B5)}}}$$

The Boltzmann equation for a two-level system is:
$${N_{\rm U} \over N_{\rm L}} = {g_{\rm U} \over
g_{\rm L}} \exp \biggl( - {h \nu_0 \over k T} \biggr)
\href{LineRadxfer.html#BoltzmannNUNL}{\rm {(7B6)}}}$$

Line opacity coefficient in LTE:
$$\kappa_\nu = {c^2 \over 8 \pi \nu_0^2} {g_{\rm U} \over g_{\rm L}} N_{\rm L} A_{\rm UL} \biggl[ 1 - \exp \biggl( - {h \nu_0 \over k T} \biggr) \biggr] \phi(\nu)
\href{LineRadxfer.html#LineOpacity}{\rm {(7B7)}}}$$

The excitation temperature $T_{\rm e}$ is defined by
$${N_{\rm U} \over N_{\rm L}} \equiv {g_{\rm U} \over g_{\rm L}} \exp \biggl( - { h \nu_0 \over k T_{\rm e}} \biggr) \rlap{\quad
\href{LineRadxfer.html#ExcitationTemp}{\rm {(7B8)}}}$$

Recombination-line opacity coefficient:
$$\kappa_{\nu_0} \approx \biggl( { N_{\rm e}^2 \over T_{\rm e}^{5/2}  \Delta \nu}\biggr) \biggl( {4 \pi e^6 h \over 3 m_{\rm e}^{3/2} k^{5/2} c} \biggr) \biggl( {\ln 2 \over 2} \biggr)^{1/2}
\href{RecombinationSources.html#RecombOpacityCoefficient}{\rm {(7C1)}}}$$

Recombination line opacity:
$$\tau_{\rm L} \approx 1.92 \times 10^3 \biggl(
{T_{\rm e} \over {\rm K}} \biggr)^{-5/2} \biggl( { {\rm EM} \over {\rm pc}\,{\rm cm}^{-6} } \biggr) \biggl( { \Delta \nu \over {\rm kHz} } \biggr)^{-1}
\href{RecombinationSources.html#LineOpacity}{\rm {(7C2)}}}$$

Recombination line brightness temperature:
$$T_{\rm L} \approx T_{\rm e} \tau_{\rm L} \approx 1.92 \times 10^3 \biggl( { T_{\rm e} \over {\rm K}} \biggr)^{-3/2} \biggl( { {\rm EM} \over {\rm pc}\,{\rm cm}^{-6} } \biggr) \biggl( {\Delta \nu \over {\rm kHz} } \biggr)^{-1}\rlap{\quad
\href{RecombinationSources.html#LineTemp}{\rm {(7C3)}}}$$

Recombination line/continuum ratio:
$${T_{\rm L} \over T_{\rm C}} \approx 7.0 \times 10^3 \biggl( { \Delta v \over {\rm km~s}^{-1} } \biggr)^{-1} \biggl( { \nu \over {\rm GHz}} \biggr)^{1.1} \biggl( {T_{\rm e} \over {\rm K} } \biggr)^{-1.15} \biggl[ 1 + { N({\rm He}^+) \over N({\rm H}^+) } \biggr]^{-1}\rlap{\quad
\href{RecombinationSources.html#LCRatio}{\rm {(7C4)}}}$$ where $[1 + N({\rm He}^+) / N({\rm H}^+)] \approx 1.08$

Electron temperature from line/continuum ratio:
$$\biggl( {T_{\rm e} \over {\rm K}} \biggr)
\approx \biggl[ 7.0 \times 10^3 \biggl( { \nu \over {\rm GHz}} \biggr)^{1.1} \, 1.08^{-1} \, \biggl( { \Delta v \over {\rm km~s}^{-1} } \biggr)^{-1}
\biggl( {T_{\rm C} \over T_{\rm L}} \biggr) \biggr]^{0.87}\rlap{\quad
\href{RecombinationSources.html#ElectronTemp}{\rm {(7C5)}}}$$

Quantization of angular momentum:
$$L = n \hbar\rlap{\quad
\href{MolecularSpectra.html#AngMomQuant}{\rm {(7D1)}}}$$

Angular momentum of a diatomic molecule:
$$L = m r_{\rm e}^2 \omega\rlap{\quad
\href{MolecularSpectra.html#AngMom}{\rm {(7D2)}}}$$ where
$$m \equiv \biggl( {m_{\rm A} m_{\rm B} \over
m_{\rm A} + m_{\rm B}} \biggr)\rlap{\quad
\href{MolecularSpectra.html#ReducedMass}{\rm {(7D3)}}}$$ is the reduced mass and $r_{\rm e}$ is the separation of the atoms with masses $m_{\rm A}$ and $m_{\rm B}$.

Rotational energy levels of a diatomic molecule with moment of inertia $I$:
$$E_{\rm rot} = { J (J + 1) \hbar^2 \over 2 I}~, \qquad
J = {\rm 0,~1,~2,} \dots \rlap{\quad
\href{MolecularSpectra.html#EnergyQuant}{\rm {(7D4)}}}$$

For a transition satisfying the selection rule
$$\Delta J = \pm 1 \rlap{\quad
\href{MolecularSpectra.html#SelectionRule}{\rm {(7D5)}}}$$ the line frequency is:
$$\nu = {h J \over 4 \pi^2 m r_{\rm e}^2}
\rlap{\quad\href{MolecularSpectra.html#Frequency}{\rm {(7D6)}}}$$

Minimum temperature to excite the $J \rightarrow J -1$ transition at frequency $\nu$:
$$T_{\rm min} \approx { \nu h (J + 1) \over 2 k } \rlap{\quad
\href{MolecularSpectra.html#MinimumTemp}{\rm {(7D7)}}}$$

Spontaneous emission coefficient:
$$A_{\rm UL} = {64 \pi^4 \over 3 h c^3} \nu_{\rm UL}^3
\vert \mu_{\rm UL} \vert^2 \rlap{\quad
\href{MolecularSpectra.html#EmissionCoef}{\rm {(7D8)}}}$$ where
$$\vert \mu_{{\rm J}\rightarrow{\rm J}-1} \vert^2 = {\mu^2 J \over 2 J + 1 } \rlap{\quad
\href{MolecularSpectra.html#DipoleMoment}{\rm {(7D9)}}}$$ and $\mu$ is the electric dipole moment of the molecule.

The critical density is defined by
$$ n^* \approx {A_{\rm UL} \over \sigma v} \rlap{\quad
\href{MolecularSpectra.html#CriticalDensity}{\rm {(7D10)}}}$$
where $\sigma \sim 10^{-15}$ cm$^{-2}$ is the collision cross section and $v \sim 10^5$ cm s$^{-1}$ is the typical H$_2$ molecular velocity.

The HI hyperfine line frequency is:
$$\nu_{10} = {8 \over 3} g_{\rm I} \biggl( {m_{\rm e} \over m_{\rm p} } \biggr) \alpha^2 (R c)
\approx 1420.405751{\rm ~MHz}\rlap{\quad \href{HILine.html#Frequency}{\rm {(7E1)}}}$$

The HI hyperfine line emission coefficient is:
$$A_{10} \approx 2.85 \times 10^{-15} {\rm ~s}^{-1}
\href{HILine.html#EmissionCoef}{\rm {(7E2)}}}$$

The HI spin temperature $T_{\rm s}$ is defined by:
$${N_1 \over N_0} \equiv {g_1 \over g_0}
\exp \biggl( - {h \nu_{10} \over k T_{\rm s} } \biggr)
\href{HILine.html#SpinTemp}{\rm {(7E3)}}}$$ where $g_1 / g_0$ = 3.

The HI line opacity coefficient is:
$$\kappa_\nu \approx {3 c^2 \over 32 \pi} {A_{10}
N_{\rm H} \over \nu_{10} } { h \over k T_{\rm s} }
\phi(\nu) \rlap{\quad
\href{HILine.html#OpacityCoef}{\rm {(7E4)}}}$$

The hydrogen column density $\eta_{\rm H}$ is defined as the integral of density along the line of sight:
$$\eta_{\rm H} \equiv \int_{\rm los} N_{\rm H} (s) d s
\href{HILine.html#ColumnDensityDef}{\rm {(7E5)}}}$$

If the HI line is optically thin ($\tau \ll 1$) then:
$$\biggl( { \eta_{\rm H} \over {\rm cm}^{-2} } \biggr)
\approx 1.82 \times 10^{18} \int \biggl[ {T_{\rm b} (v)
\over {\rm K} } \biggr] d \biggl( {v \over {\rm km~s}^{-1} } \biggr) \rlap{\quad
\href{HILine.html#ColumnDensityEq}{\rm {(7E6)}}}$$

The hydrogen mass of a galaxy if $\tau \ll 1$:
$$ \biggl( {M_{\rm H} \over M_\odot} \biggr) \approx
2.36 \times 10^5 \biggl( {D \over {\rm Mpc} } \biggr)^2
\int \biggl[ {S(v)  \over {\rm Jy} } \biggr]
\biggl( {d v \over {\rm km~s}^{-1} } \biggr)
\href{HILine.html#HydrogenMass}{\rm {(7E7)}}}$$

The total mass of a galaxy is
$$ \biggl( {M \over M_\odot} \biggr) \approx
2.33 \times 10^5 \biggl[ { (v_{\rm r} / \sin i) \over
{\rm km~s}^{-1} } \biggr]^2 \biggl( {r \over {\rm kpc}}
\biggr) \rlap{\quad
\href{HILine.html#TotalMass}{\rm {(7E8)}}}$$