# Appendix E Essential Equations

The specific intensity $I_{\nu}$ of radiation is defined by

 $\boxed{I_{\nu}\equiv\frac{dP}{(\cos\theta~{}d\sigma)\,d\nu\,d\Omega},}{}$ (\ref{eqn:SIorSB})

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 $\nu+d\nu$. Likewise $I_{\lambda}$ is the brightness per unit wavelength:

 $\boxed{I_{\lambda}\equiv\frac{dP}{(\cos\theta\,d\sigma)\,d\lambda\,d\Omega}.}{}$ (\ref{eqn:ilambdadef})

These two quantities are related by

 $\boxed{\frac{I_{\lambda}}{I_{\nu}}=\bigg{|}\frac{d\nu}{d\lambda}\bigg{|}=\frac% {c}{\lambda^{2}}=\frac{\nu^{2}}{c}.}{}$ (\ref{eqn:ilambdaoverinu})

The flux density $S_{\nu}$ of a source is the spectral power received per unit detector area:

 $\boxed{S_{\nu}\equiv\int_{\mathrm{source}}I_{\nu}(\theta,\phi)\cos\theta\,d% \Omega.}{}$ (\ref{eqn:fluxdensity})

If the source is compact enough that $\cos\theta\approx 1$ then

 $\boxed{S_{\nu}\approx\int_{\mathrm{source}}I_{\nu}(\theta,\phi)d\Omega.}{}$ (\ref{eqn:simplefluxdensity})

The MKS units of flux density are $\mathrm{W~{}m}^{-2}\mathrm{~{}Hz}^{-1}$; $1\mathrm{~{}jansky~{}(Jy)}\equiv 10^{-26}\mathrm{~{}W~{}m}^{-2}\mathrm{~{}Hz}^% {-1}$.

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 $d$ much greater than the source size, the inverse-square law

 $\boxed{L_{\nu}=4\pi d^{2}S_{\nu}}{}$ (\ref{eqn:speclum})

relates the spectral luminosity of an isotropic source to its flux density.

The linear absorption coefficient at frequency $\nu$ of an absorber is defined as the probability $dP(\nu)$ that a photon will be absorbed in a layer of thickness $ds$:

 $\boxed{\kappa(\nu)\equiv\frac{dP(\nu)}{ds}.}{}$ (\ref{eqn:kappanu})

The opacity or optical depth $\tau$ is defined as the sum of those infinitesimal probabilities through the absorber, starting at the source end:

 $\boxed{\tau\equiv\int_{s_{\mathrm{out}}}^{s_{\mathrm{in}}}-\kappa(s^{\prime})% ds^{\prime}.}{}$ (\ref{eqn:opacity})

The emission coefficient at frequency $\nu$ is the infinitesimal increase $dI_{\nu}$ in specific intensity per infinitesimal distance $ds$:

 $\boxed{j_{\nu}\equiv\frac{dI_{\nu}}{ds}.}{}$ (\ref{eqn:EmissCoeff})

The equation of radiative transfer is

 $\boxed{\frac{dI_{\nu}}{ds}=-\kappa I_{\nu}+j_{\nu}.}{}$ (\ref{eqn:RadXferEq})

For any substance in Local Thermodynamic Equilibrium (LTE), Kirchhoff’s law connects the emission and absorption coefficients via the specific intensity $B_{\nu}$ of blackbody radiation:

 $\boxed{\frac{j_{\nu}}{\kappa}=B_{\nu}(T).}{}$ (\ref{eqn:KirchhoffsLaw})

The brightness temperature of a source with any specific intensity $I_{\nu}$ is defined as

 $\boxed{T_{\mathrm{b}}(\nu)\equiv\frac{I_{\nu}c^{2}}{2k\nu^{2}}.}{}$ (\ref{eqn:BrightnessTemp})

For an opaque body in LTE, Kirchhoff’s law connects the emission coefficient $e_{\nu}$ (the spectral power per unit area emitted by the body divided by the spectral power per unit area emitted by a blackbody) 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):

 $\boxed{e_{\nu}=a_{\nu}=1-r_{\nu}.}{}$ (\ref{eqn:KirchhoffsLaw2})

The spectral energy density of radiation is

 $\boxed{u_{\nu}=\frac{1}{c}\int I_{\nu}\,d\Omega.}{}$ (\ref{eqn:RadEdensity})

The Rayleigh–Jeans approximation for the specific intensity of blackbody radiation when $h\nu\ll kT$ is

 $\boxed{B_{\nu}=\frac{2kT\nu^{2}}{c^{2}}=\frac{2kT}{\lambda^{2}}.}{}$ (\ref{eqn:RJLaw})

The energy of a photon is

 $\boxed{E=h\nu.}{}$ (\ref{eqn:photonenergy})

Planck’s equation for the specific intensity of blackbody radiation at any frequency is

 $\boxed{B_{\nu}=\frac{2h\nu^{3}}{c^{2}}\,\frac{1}{\displaystyle\exp\Bigl{(}% \frac{h\nu}{kT}\Bigr{)}-1}.}{}$ (\ref{eqn:PlanckLaw})

The total intensity of blackbody radiation is

 $\boxed{B(T)\equiv\int_{0}^{\infty}B_{\nu}(T)d\nu=\frac{\sigma T^{4}}{\pi},}{}$ (\ref{eqn:IntBright})

where the Stefan–Boltzmann constant $\sigma$ is defined by

 $\boxed{\sigma\equiv\frac{2\pi^{5}k^{4}}{15c^{2}h^{3}}\approx 5.67\times 10^{-5% }\frac{\mathrm{erg}}{\mathrm{cm}^{2}\mathrm{~{}s~{}K}^{4}\mathrm{~{}sr}}.}{}$ (\ref{eqn:SBConstant})

The total energy density of blackbody radiation is

 $\boxed{u=\frac{4\sigma T^{4}}{c}=aT^{4},}{}$ (\ref{eqn:BBEdensity})

where $a\equiv 4\sigma/c\approx 7.56577\times 10^{-15}\mathrm{~{}erg~{}cm}^{-3}% \mathrm{~{}K}^{-4}$ is the radiation constant.

The photon number density of blackbody radiation is

 $\boxed{\biggl{(}\frac{n_{\gamma}}{\mathrm{cm}^{-3}}\biggr{)}\approx 20.3\biggl% {(}\frac{T}{\mathrm{K}}\biggr{)}^{3}.}{}$ (\ref{eqn:bbphotondensity})

The mean photon energy of blackbody radiation is

 $\boxed{\langle E_{\gamma}\rangle\approx\,2.70\,kT.}{}$ (\ref{eqn:meanphotonenergy})

The frequency of the peak blackbody brightness per unit frequency $B_{\nu}$ is

 $\boxed{\biggl{(}\frac{\nu_{\mathrm{max}}}{\mathrm{GHz}}\biggr{)}\approx 59% \biggl{(}\frac{T}{\mathrm{K}}\biggr{)}.}{}$ (\ref{eqn:bbnupeak})

The wavelength of the peak blackbody brightness per unit wavelength $B_{\lambda}$ is given by Wien’s displacement law:

 $\boxed{\biggl{(}\frac{\lambda_{\mathrm{max}}}{\mathrm{cm}}\biggr{)}\approx 0.2% 9\biggl{(}\frac{T}{\mathrm{K}}\biggr{)}^{-1}.}{}$ (\ref{eqn:WienLaw})

The flux density of isotropic radiation is

 $\boxed{S_{\nu}=\pi I_{\nu}.}{}$ (\ref{eqn:isotfluxdensity})

The Nyquist approximation for the spectral power generated by a warm resistor in the limit $h\nu\ll kT$ is

 $\boxed{P_{\nu}=kT.}{}$ (\ref{eqn:NyquistLaw})

At any frequency, the exact Nyquist formula is

 $\boxed{P_{\nu}=\frac{h\nu}{\displaystyle\exp\Bigl{(}\frac{h\nu}{kT}\Bigr{)}-1}% .}{}$ (\ref{eqn:QuantumNyquistLaw})

The critical density needed to close the universe is

 $\boxed{\rho_{\mathrm{c}}=\frac{3H_{0}^{2}}{8\pi G}\approx 8.6\times 10^{-30}% \mathrm{~{}g~{}cm}^{-3}.}{}$ (\ref{eqn:closuredensity})

Redshift $z$ is defined by

 $\boxed{z\equiv\frac{\lambda_{\mathrm{o}}-\lambda_{\mathrm{e}}}{\lambda_{% \mathrm{e}}}=\frac{\lambda_{\mathrm{o}}}{\lambda_{\mathrm{e}}}-1=\frac{\nu_{% \mathrm{e}}}{\nu_{\mathrm{o}}}-1,}{}$ (\ref{eqn:redshift})

where $\lambda_{\mathrm{e}}$ and $\nu_{\mathrm{e}}$ are the wavelength and frequency emitted by a source at redshift $z$, and $\lambda_{\mathrm{o}}$ and $\nu_{\mathrm{o}}$ are the observed wavelength and frequency at $z=0$.

Redshift $z$ and expansion scale factor $a$ are related by

 $\boxed{(1+z)=a^{-1}.}{}$ (\ref{eqn:scalesize})

The CMB temperature at redshift $z$ is

 $\boxed{T=T_{0}(1+z).}{}$ (\ref{eqn:tcmb})

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

 $\boxed{E_{\bot}=\frac{q\dot{v}\sin\theta}{rc^{2}}.}{}$ (\ref{eqn:LarmorEField})

In a vacuum, the Poynting flux, or power per unit area, is

 $\boxed{|\vec{S}|=\frac{c}{4\pi}E^{2}.}{}$ (\ref{eqn:PoyntingFlux})

The total power emitted by an accelerated charge is given by Larmor’s formula

 $\boxed{P=\frac{2}{3}\frac{q^{2}\dot{v}^{2}}{c^{3}},}{}$ (\ref{eqn:LarmorPower})

which is valid only if $v\ll c$.

Exponential notation for trigonometric functions is

 $\boxed{e^{-i\omega t}=\cos(\omega t)-i\sin(\omega t).}{}$ (\ref{eqn:expnotation})

Electric current is defined as the time derivative of electric charge:

 $\boxed{I\equiv\frac{dq}{dt}.}{}$ (\ref{eqn:currentdef})

The power pattern of a short dipole antenna is

 $\boxed{P\propto\sin^{2}\theta.}{}$ (\ref{eqn:DipolePattern})

The power emitted by a short ($l\ll\lambda$) dipole driven by a current $I=I_{0}e^{-i\omega t}$ is

 $\boxed{\langle P\rangle=\frac{\pi^{2}}{3c}\biggl{(}\frac{I_{0}l}{\lambda}% \biggr{)}^{2}.}{}$ (\ref{eqn:DipolePower})

 $\boxed{R\equiv\frac{2\langle P\rangle}{I_{0}^{2}}.}{}$ (\ref{eqn:RadResist})

Energy conservation implies the average power gain of any lossless antenna is

 $\boxed{\langle G\rangle=1}{}$ (\ref{eqn:AvgGain})

and

 $\boxed{\int_{\mathrm{sphere}}G\,d\Omega=4\pi.}{}$ (\ref{eqn:gainintegral})

The beam solid angle is defined by

 $\boxed{\Omega_{\mathrm{A}}\equiv\frac{4\pi}{G_{\mathrm{max}}}=\frac{1}{G_{% \mathrm{max}}}\int_{4\pi}G(\theta,\phi)\,d\Omega.}{}$ (\ref{eqn:txBSA})

The effective area of an antenna is defined by

 $\boxed{A_{\mathrm{e}}\equiv 2P_{\nu}/S_{\nu},}{}$ (\ref{eqn:EffectiveArea})

where $P_{\nu}$ is the output power density produced by an unpolarized point source of total flux density $S_{\nu}$.

The average effective area of any lossless antenna is

 $\boxed{\langle A_{\mathrm{e}}\rangle=\frac{\lambda^{2}}{4\pi}.}{}$ (\ref{eqn:AvgArea})

Reciprocity implies

 $\boxed{G(\theta,\phi)\propto A_{\mathrm{e}}(\theta,\phi).}{}$ (\ref{eqn:Reciprocity})

Reciprocity and energy conservation imply

 $\boxed{A_{\mathrm{e}}(\theta,\phi)=\frac{\lambda^{2}G(\theta,\phi)}{4\pi}.}{}$ (\ref{eqn:GainArea})

Antenna temperature is defined by

 $\boxed{T_{\mathrm{A}}\equiv\frac{P_{\nu}}{k}.}{}$ (\ref{eqn:AntennaTemp})

The antenna temperature produced by an unpolarized point source of flux density $S$ is

 $\boxed{T_{\mathrm{A}}=\frac{A_{\mathrm{e}}S}{2k}.}{}$ (\ref{eqn:AntSens})

If $A_{\mathrm{e}}\approx 2761\mathrm{~{}m}^{2}$, the point-source sensitivity is $1\mathrm{~{}K~{}Jy}^{-1}$.

For a uniform compact source of brightness temperature $T_{\mathrm{b}}$ covering solid angle $\Omega_{\mathrm{s}}$,

 $\boxed{\frac{T_{\mathrm{A}}}{T_{\mathrm{b}}}=\frac{\Omega_{\mathrm{s}}}{\Omega% _{\mathrm{A}}}.}{}$ (\ref{eqn:TaCompact})

The main beam solid angle is defined by the integral over the main beam to the first zero only:

 $\boxed{\Omega_{\mathrm{MB}}\equiv\frac{1}{G_{\mathrm{max}}}\int_{\mathrm{MB}}G% (\theta,\phi)\,d\Omega}{}$ (\ref{eqn:MainBeamSolidAngle})

and is used in the definition of main beam efficiency:

 $\boxed{\eta_{\mathrm{B}}\equiv\frac{\Omega_{\mathrm{MB}}}{\Omega_{\mathrm{A}}}% .}{}$ (\ref{eqn:MainBeamEfficiency})

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

 $\boxed{z=\frac{r^{2}}{4f}.}{}$ (\ref{eqn:parabola})

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

 $\boxed{R_{\mathrm{ff}}\approx\frac{2D^{2}}{\lambda}.}{}$ (\ref{eqn:farfield})

In the far field, the electric field pattern of an aperture antenna is the Fourier transform of the aperture illumination:

 $\displaystyle\boxed{l\equiv\sin\theta,}{}$ (\ref{eqn:ldeff}) $\displaystyle\boxed{u\equiv\frac{x}{\lambda},}{}$ (\ref{eqn:defineU}) $\displaystyle\boxed{f(l)=\int_{\mathrm{aperture}}g(u)e^{-i2\pi lu}du.}{}$ (\ref{eqn:fieldpattern})

The power pattern of a uniformly illuminated linear aperture is

 $\boxed{P(\theta)\propto\mathrm{sinc}^{2}\biggl{(}\frac{\theta D}{\lambda}% \biggr{)},}{}$ (\ref{eqn:uniformpowerpattern})

where $\mathrm{sinc}(x)\equiv\sin(\pi x)/(\pi x)$, and the half-power beamwidth is

 $\boxed{\theta_{\mathrm{HPBW}}\approx 0.89\frac{\lambda}{D}.}{}$ (\ref{eqn:beamwidth})

The half-power beamwidth (HPBW) of a a typical radio telescope with tapered illumination is

 $\boxed{\theta_{\mathrm{HPBW}}\approx 1.2\frac{\lambda}{D}.}{}$ (\ref{eqn:HPBW})

The two-dimensional aperture field pattern is

 $\boxed{f(l,m)\propto\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(u,v)e^{-i2% \pi(lu+mv)}du\,dv,}{}$ (\ref{eqn:2dfieldpattern})

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.

The power pattern of a uniformly illuminated rectangular aperture is

 $\boxed{G\approx\frac{4\pi D_{x}D_{y}}{\lambda^{2}}\mathrm{sinc}^{2}\biggl{(}% \frac{\theta_{x}D_{x}}{\lambda}\biggr{)}\mathrm{sinc}^{2}\biggl{(}\frac{\theta% _{y}D_{y}}{\lambda}\biggr{)}.}{}$ (\ref{eqn:2DPowerPattern})

Aperture efficiency is defined by

 $\boxed{\eta_{\mathrm{A}}\equiv\frac{\mathrm{max}(A_{\mathrm{e}})}{A_{\mathrm{% geom}}}.}{}$ (\ref{eqn:ApertureEfficiency})

The beam solid angle of a Gaussian beam is

 $\boxed{\Omega_{\mathrm{A}}=\,\biggl{(}\frac{\pi}{4\ln 2}\biggr{)}\theta_{% \mathrm{HPBW}}^{2}\approx 1.133\,\theta_{\mathrm{HPBW}}^{2}.}{}$ (\ref{eqn:gaussbsa})

The surface efficiency $\eta_{\mathrm{s}}$ of a reflector whose surface errors $\epsilon$ have rms $\sigma$ is given by the Ruze equation:

 $\boxed{\eta_{\mathrm{s}}=\exp\biggl{[}-\biggl{(}\frac{4\pi\sigma}{\lambda}% \biggr{)}^{2}\biggr{]}.}{}$ (\ref{eqn:SurfaceEfficiency})

Noise temperature is defined by

 $\boxed{T_{\mathrm{N}}\equiv\frac{P_{\nu}}{k}.}{}$ (\ref{eqn:NoiseTemp})

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

 $\boxed{T_{\mathrm{s}}=T_{\mathrm{cmb}}+T_{\mathrm{rsb}}+\Delta T_{\mathrm{% source}}+[1-\exp(-\tau_{\mathrm{A}})]T_{\mathrm{atm}}+T_{\mathrm{spill}}+T_{% \mathrm{r}}+\cdots.}{}$ (\ref{eqn:SystemNoise})

The ideal total-power radiometer equation is

 $\boxed{\sigma_{T}\approx T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}% \biggr{]}^{1/2}.}{}$ (\ref{eqn:IdealRadiometer})

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

 $\boxed{\sigma_{T}\approx T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}+% \biggl{(}\frac{\Delta G}{G}\biggr{)}^{2}\biggr{]}^{1/2}.}{}$ (\ref{eqn:Radiometer})

 $\boxed{\sigma_{T}\approx 2T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}% \biggr{]}^{1/2}.}{}$ (\ref{eqn:DickeRadiometer})

The rms confusion caused by unresolved continuum sources in a Gaussian beam with HPBW $\theta$ at frequency $\nu$ is

 $\boxed{\biggl{(}\frac{\sigma_{\mathrm{c}}}{\mathrm{mJy~{}beam}^{-1}}\biggr{)}% \approx\begin{cases}0.2\biggr{(}\frac{\nu}{\mathrm{GHz}}\biggr{% )}^{-0.7}\biggl{(}\frac{\theta}{\mathrm{arcmin}}\biggr{)}^{2}}&(\theta>0.17% \mathrm{~{}arcmin}),\\ 2.2\biggr{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.7}\biggl{(}% \frac{\theta}{\mathrm{arcmin}}\biggr{)}^{10/3}}&(\theta<0.17\mathrm{~{}arcmin}% ).\end{cases}{}$ (\ref{eqn:rmsconfusion})

Individual sources fainter than the confusion limit $\approx 5\sigma_{\mathrm{c}}$ cannot be detected reliably.

Radiometer input noise temperature $T_{\mathrm{r}}$ can be measured by the $Y$-factor method; it is

 $\boxed{T_{\mathrm{r}}=\frac{T_{\mathrm{h}}-YT_{\mathrm{c}}}{Y-1}.}{}$ (\ref{eqn:YfactorT})

The response of a two-element interferometer to a source of brightness distribution $I_{\nu}(\hat{s})$ is the complex visibility

 $\boxed{\mathcal{V}_{\nu}=\int I_{\nu}(\hat{s})\exp(-i2\pi\vec{b}\cdot\hat{s}/% \lambda)\,d\Omega.}{}$ (\ref{eqn:Complexvis})

To minimize bandwidth smearing in bandwidth $\Delta\nu$, the image angular radius $\Delta\theta$ should satisfy

 $\boxed{\Delta\theta\,\Delta\nu\ll\theta_{\mathrm{s}}\nu.}{}$ (\ref{eqn:bandsmear})

To minimize time smearing in an image of angular radius $\Delta\theta$ the averaging time should satisfy

 $\boxed{\Delta\theta\,\Delta t\ll\frac{\theta_{\mathrm{s}}P}{2\pi}\approx\theta% _{\mathrm{s}}\cdot 1.37\times 10^{4}\mathrm{~{}s}.}{}$ (\ref{eqn:timesmear})

The source brightness distribution $I_{\nu}(l,m)$ and the visibilities $\mathcal{V}_{\nu}(u,v,w)$ for an interferometer in three dimensions are related by

 $\boxed{\mathcal{V}_{\nu}(u,v,w)=\int\int\frac{I_{\nu}(l,m)}{(1-l^{2}-m^{2})^{1% /2}}\exp[-i2\pi(ul+vm+wn)]dl\,dm.}{}$ (\ref{eqn:3Dinterferometer})

For a two-dimensional interferometer confined to the $(u,v)$ plane, the source brightness distribution $I_{\nu}(l,m)$ is the Fourier transform of the fringe visibilities $\mathcal{V}_{\nu}(u,v)$:

 $\boxed{\frac{I_{\nu}(l,m)}{(1-l^{2}-m^{2})^{1/2}}=\int\int\mathcal{V}_{\nu}(u,% v,0)\exp[+i2\pi(ul+vm)]du\,dv.}{}$ (\ref{eqn:3DSourceBrightness})

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 $A_{\mathrm{e}}$, is

 $\boxed{\sigma_{S}=\frac{2kT_{\mathrm{s}}}{A_{\mathrm{e}}[N(N-1)\Delta\nu\,\tau% ]^{1/2}}.}{}$ (\ref{eqn:fluxsensitivity})

The brightness sensitivity (K) corresponding to a point source sensitivity $\sigma_{S}$ and a beam solid angle $\Omega_{\mathrm{A}}$ is

 $\boxed{\sigma_{T}=\biggl{(}\frac{\sigma_{S}}{\Omega_{\mathrm{A}}}\biggr{)}% \frac{\lambda^{2}}{2k},}{}$ (\ref{eqn:brightsensitivity})

where $\Omega_{\mathrm{A}}=\pi\theta_{\mathrm{HPBW}}^{2}/(4\ln 2)\approx 1.133\theta_% {0}^{2}$ for a Gaussian beam of HPBW $\theta_{\mathrm{HPBW}}$.

The (nonrelativistic) Maxwellian distribution of particle speeds $v$ is

 $\boxed{f(v)=\frac{4v^{2}}{\sqrt{\pi}}\biggl{(}\frac{m}{2kT}\biggr{)}^{3/2}\exp% \biggl{(}-\frac{mv^{2}}{2kT}\biggr{)}.}{}$ (\ref{eqn:MaxwellianDistribution})

The free–free emission coefficient is

 $\boxed{j_{\nu}=\frac{\pi^{2}Z^{2}e^{6}n_{\mathrm{e}}n_{\mathrm{i}}}{4c^{3}m_{% \mathrm{e}}^{2}}\biggl{(}\frac{2m_{\mathrm{e}}}{\pi kT}\biggr{)}^{1/2}\ln% \biggl{(}\frac{b_{\mathrm{max}}}{b_{\mathrm{min}}}\biggr{)},}{}$ (\ref{eqn:FFemcoefficient})

where

 $\boxed{b_{\mathrm{min}}\approx\frac{Ze^{2}}{m_{\mathrm{e}}v^{2}}.}{}$ (\ref{eqn:impactmin})

The free–free absorption coefficient is

 $\boxed{\kappa=\frac{1}{\nu^{2}T^{3/2}}\biggl{[}\frac{Z^{2}e^{6}}{c}n_{\mathrm{% e}}n_{\mathrm{i}}\frac{1}{\sqrt{2\pi(m_{\mathrm{e}}k)^{3}}}\biggr{]}\frac{\pi^% {2}}{4}\ln\biggl{(}\frac{b_{\mathrm{max}}}{b_{\mathrm{min}}}\biggr{)}.}{}$ (\ref{eqn:FFabscoefficient})

At frequencies low enough that $\tau\gg 1$, the Hii region becomes opaque, its spectrum approaches that of a blackbody with temperature $T\ \sim 10^{4}$ K, and the flux density varies as $S\propto\nu^{2}$. At very high frequencies, $\tau\ll 1$, the Hii region is nearly transparent, and

 $\boxed{S_{\nu}\propto\frac{2kT\nu^{2}}{c^{2}}\tau(\nu)\propto\nu^{-0.1}.}{}$ (\ref{eqn:freefreetransparent})

On a log-log plot, the overall spectrum of a uniform Hii region has a break near the frequency at which $\tau\approx 1$.

The emission measure of a plasma is defined by

 $\boxed{\frac{\mathrm{EM}}{\mathrm{pc~{}cm}^{-6}}\equiv\int_{\mathrm{los}}% \biggl{(}\frac{n_{\mathrm{e}}}{\mathrm{cm}^{-3}}\biggr{)}^{2}d\biggl{(}\frac{s% }{\mathrm{pc}}\biggr{)}.}{}$ (\ref{eqn:EmissionMeasure})

The free–free optical depth of a plasma is

 $\boxed{\tau\approx 3.28\times 10^{-7}\biggl{(}\frac{T}{10^{4}\mathrm{~{}K}}% \biggr{)}^{-1.35}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-2.1}\biggl{(}% \frac{\mathrm{EM}}{\mathrm{pc~{}cm}^{-6}}\biggr{)}.}{}$ (\ref{eqn:TauFF})

The ionization rate $Q_{\mathrm{H}}$ of Lyman continuum photons produced per second required to maintain an Hii region is

 $\boxed{\biggl{(}\frac{Q_{\mathrm{H}}}{\mathrm{s}^{-1}}\biggr{)}\approx 6.3% \times 10^{52}\biggl{(}\frac{T}{10^{4}\mathrm{~{}K}}\biggr{)}^{-0.45}\biggl{(}% \frac{\nu}{\mathrm{GHz}}\biggr{)}^{0.1}\biggl{(}\frac{L_{\nu}}{10^{20}\mathrm{% ~{}W~{}Hz}^{-1}}\biggr{)},}{}$ (\ref{eqn:LyAlphaRate})

where $L_{\nu}$ is the free–free luminosity at any frequency $\nu$ high enough that $\tau(\nu)\ll 1$.

The magnetic force on a moving charge is

 $\boxed{\vec{F}=\frac{q(\vec{v}\times\vec{B})}{c}.}{}$ (\ref{eqn:MagneticForce})

The gyro frequency is defined by

 $\boxed{\omega_{\mathrm{G}}\equiv\frac{qB}{mc}.}{}$ (\ref{eqn:GyroFrequency})

The (nonrelativistic) electron gyro frequency in MHz is

 $\boxed{\biggl{(}\frac{\nu_{\mathrm{G}}}{\mathrm{MHz}}\biggr{)}=2.8\biggl{(}% \frac{B}{\mathrm{gauss}}\biggr{)}.}{}$ (\ref{eqn:ElectronGyro})

The Lorentz transform is

 $\boxed{x=\gamma(x^{\prime}+vt^{\prime}),\qquad y=y^{\prime},\qquad z=z^{\prime% },\qquad t=\gamma(t^{\prime}+\beta x^{\prime}/c),}{}$ (\ref{eqn:event1})
 $\boxed{x^{\prime}=\gamma(x-vt),\qquad y^{\prime}=y,\qquad z^{\prime}=z,\qquad t% ^{\prime}=\gamma(t-\beta x/c),}{}$ (\ref{eqn:event2})

where

 $\boxed{\beta\equiv v/c}{}$ (\ref{eqn:Beta})

and

 $\boxed{\gamma\equiv(1-\beta^{2})^{-1/2}}{}$ (\ref{eqn:Gamma})

is called the Lorentz factor. If $(\Delta x^{\prime},\Delta y^{\prime},\Delta z^{\prime},\Delta t^{\prime})$ 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

 $\displaystyle\boxed{\Delta x=\gamma(\Delta x^{\prime}+v\Delta t^{\prime}),% \quad\!\Delta y=\Delta y^{\prime},\quad\!\Delta z=\Delta z^{\prime},\quad\!% \Delta t=\gamma(\Delta t^{\prime}+\beta\Delta x^{\prime}/c),}{}$ (\ref{eqn:DiffLT}) $\displaystyle\boxed{\Delta x^{\prime}=\gamma(\Delta x-v\Delta t),\quad\Delta y% ^{\prime}=\Delta y,\quad\Delta z^{\prime}=\Delta z,\quad\Delta t^{\prime}=% \gamma(\Delta t-\beta\Delta x/c).}{}$ (\ref{eqn:DiffLTPrime})

The Thomson cross section of an electron is defined by

 $\boxed{\sigma_{\mathrm{T}}\equiv\frac{8\pi}{3}\biggl{(}\frac{e^{2}}{m_{\mathrm% {e}}c^{2}}\biggr{)}^{2}.}{}$ (\ref{eqn:ThomsonArea})

Magnetic energy density is given by

 $\boxed{U_{B}=\frac{B^{2}}{8\pi}.}{}$ (\ref{eqn:Umag})

The synchrotron power of one electron is

 $\boxed{P=2\sigma_{\mathrm{T}}\beta^{2}\gamma^{2}c\,U_{B}\sin^{2}\alpha.}{}$ (\ref{eqn:Power})

Synchrotron power averaged over all pitch angles $\alpha$ is

 $\boxed{\langle P\rangle=\frac{4}{3}\sigma_{\mathrm{T}}\beta^{2}\gamma^{2}cU_{B% }.}{}$ (\ref{eqn:AveragePower})

The synchrotron spectrum of a single electron is

 $\boxed{P(\nu)=\frac{\sqrt{3}e^{3}B\sin\alpha}{m_{\mathrm{e}}c^{2}}\biggl{(}% \frac{\nu}{\nu_{\mathrm{c}}}\biggr{)}\int_{\nu/\nu_{\mathrm{c}}}^{\infty}K_{5/% 3}(\eta)d\eta,}{}$ (\ref{eqn:Spectrum})

where $K_{5/3}$ is a modified Bessel function and the critical frequency is

 $\boxed{\nu_{\mathrm{c}}=\frac{3}{2}\gamma^{2}\nu_{\mathrm{G}}\sin\alpha\approx% \gamma^{2}\nu_{\mathrm{G}}\,\propto E^{2}B_{\bot}.}{}$ (\ref{eqn:CriticalFrequency})

The observed energy distribution of cosmic-ray electrons in our Galaxy is roughly a power law:

 $\boxed{n(E)dE\approx KE^{-\delta}dE,}{}$ (\ref{eqn:CRSpectrum})

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

 $\boxed{j_{\nu}\propto B^{(\delta+1)/2}\nu^{(1-\delta)/2}.}{}$ (\ref{eqn:EmissionSpectrum})

The (negative sign convention) spectral index of both synchrotron radiation and inverse-Compton radiation is

 $\boxed{\alpha=\frac{\delta-1}{2}.}{}$ (\ref{eqn:SyncSpectralIndex})

The effective temperature of a relativistic electron emitting at frequency $\nu$ in magnetic field $B$ is

 $\boxed{\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}\approx 1.18\times 1% 0^{6}\biggl{(}\frac{\nu}{\mathrm{Hz}}\biggr{)}^{1/2}\biggl{(}\frac{B}{\mathrm{% gauss}}\biggr{)}^{-1/2}.}{}$ (\ref{eqn:ElectronEffectiveTemp})

At a sufficiently low frequency $\nu$,

 $\boxed{S_{\mathrm{\nu}}\propto\nu^{-5/2}}{}$ (\ref{eqn:SSASlope})

and

 $\boxed{\biggl{(}\frac{B}{\mathrm{gauss}}\biggr{)}\approx 1.4\times 10^{12}% \biggl{(}\frac{\nu}{\mathrm{Hz}}\biggr{)}\biggl{(}\frac{T_{\mathrm{b}}}{% \mathrm{K}}\biggr{)}^{-2}.}{}$ (\ref{eqn:SSABfield})

For a given synchrotron luminosity, the electron energy density is

 $\boxed{U_{\mathrm{e}}\propto B^{-3/2}.}{}$ (\ref{eqn:ElectronU})

The total energy density of both cosmic rays and magnetic fields is

 $\boxed{U=(1+\eta)U_{\mathrm{e}}+U_{B},}{}$ (\ref{eqn:TotalU})

where $\eta$ is the ion/electron energy ratio.

At minimum total energy, the ratio of particle to field energy is $\sim 1$ (equipartition):

 $\boxed{\frac{\mathrm{particle~{}energy}}{\mathrm{field~{}energy}}=\frac{(1+% \eta)U_{\mathrm{e}}}{U_{B}}=\frac{4}{3}.}{}$ (\ref{eqn:MinimumE})

The minimum-energy magnetic field is

 $\boxed{B_{\min}=[4.5(1+\eta)c_{12}L]^{2/7}R^{-6/7}\mathrm{~{}gauss}}{}$ (\ref{eqn:minEB})

and the corresponding total energy is

 $\boxed{E_{\mathrm{min}}\mathrm{(total)}=c_{13}[(1+\eta)L]^{4/7}R^{9/7}\mathrm{% ~{}ergs}.}{}$ (\ref{eqn:minEtot})

 $\boxed{\tau\approx c_{12}B_{\bot}^{-3/2},}{}$ (\ref{eqn:synclifeapprox})

where the functions $c_{12}$ and $c_{13}$ in Gaussian CGS units are plotted in Figures 5.10 and 5.11. Frequency limits $\nu_{\mathrm{min}}=10^{7}$ Hz and $\nu_{\mathrm{max}}=10^{11}$ Hz are commonly used.

The Eddington limit for luminosity is

 $\boxed{\biggl{(}\frac{L_{\mathrm{E}}}{L_{\odot}}\biggr{)}\approx 3.3\times 10^% {4}\biggl{(}\frac{M}{M_{\odot}}\biggr{)}.}{}$ (\ref{eqn:Eddington})

The nonrelativistic Thomson-scattering power is

 $\boxed{P=\sigma_{\mathrm{T}}cU_{\mathrm{rad}}.}{}$ (\ref{eqn:PScattered})

The relativistic Doppler equation is

 $\boxed{\nu^{\prime}=\nu[\gamma(1+\beta\cos\theta)].}{}$ (\ref{eqn:Doppler})

The net inverse-Compton power emitted is

 $\boxed{P_{\mathrm{IC}}=\frac{4}{3}\sigma_{\mathrm{T}}c\beta^{2}\gamma^{2}U_{% \mathrm{rad}}.}{}$ (\ref{eqn:ICPower})

The IC/synchrotron power ratio is

 $\boxed{\frac{P_{\mathrm{IC}}}{P_{\mathrm{syn}}}=\frac{U_{\mathrm{rad}}}{U_{B}}% .}{}$ (\ref{eqn:PowerRatio})

The average frequency $\langle\nu\rangle$ of upscattered photons having initial frequency $\nu_{0}$ is

 $\boxed{\frac{\langle\nu\rangle}{\nu_{0}}=\frac{4}{3}\gamma^{2}.}{}$ (\ref{eqn:ICFrequency})

The maximum rest-frame brightness temperature of an incoherent synchrotron source is limited by inverse-Compton scattering to

 $\boxed{T_{\mathrm{max}}\sim 10^{12}\mathrm{~{}K}.}{}$ (\ref{eqn:Tmax})

The apparent transverse velocity of a moving source component is

 $\boxed{\beta_{\bot}\mathrm{(apparent)}=\frac{\beta\sin\theta}{1-\beta\cos% \theta}.}{}$ (\ref{eqn:ApparentBeta})

For any $\beta$ the angle $\theta_{\mathrm{m}}$ that maximizes $\beta_{\bot}\mathrm{(apparent)}$ satisfies

 $\boxed{\cos\theta_{\mathrm{m}}=\beta}{}$ (\ref{eqn:BetaThetamax})

and

 $\boxed{\sin\theta_{\mathrm{m}}=\gamma^{-1}.}{}$ (\ref{eqn:GammaThetamax})

The largest apparent transverse speed is

 $\boxed{\mathrm{max}[\beta_{\bot}\mathrm{(apparent)}]=\beta\gamma.}{}$ (\ref{eqn:ApparentBetamax})

The transverse Doppler shift (at $\theta=\pi/2$) is

 $\boxed{\frac{\nu}{\nu^{\prime}}=\gamma^{-1}.}{}$ (\ref{eqn:TransverseDoppler})

The Doppler boosting for Doppler factor $\delta\equiv\nu/\nu^{\prime}$ is in the range

 $\boxed{\delta^{2+\alpha}<\frac{S}{S_{0}}<\delta^{3+\alpha}.}{}$ (\ref{eqn:Boosting})

Thermal and nonthermal radio luminosities of star-forming galaxies are

 $\boxed{\biggl{(}\frac{L_{\mathrm{T}}}{\mathrm{W~{}Hz}^{-1}}\biggr{)}\approx 5.% 5\times 10^{20}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.1}\biggl{[}\frac% {\mathrm{SFR}(M>5M_{\odot})}{M_{\odot}\mathrm{~{}yr}^{-1}}\biggr{]}}{}$ (\ref{eqn:LThermal})

and

 $\boxed{\biggl{(}\frac{L_{\mathrm{NT}}}{\mathrm{W~{}Hz}^{-1}}\biggr{)}\approx 5% .3\times 10^{21}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.8}\biggl{[}% \frac{\mathrm{SFR}(M>5M_{\odot})}{M_{\odot}\mathrm{~{}yr}^{-1}}\biggr{]}.}{}$ (\ref{eqn:LNonthermal})

The minimum mean density of a pulsar with period $P$ is

 $\boxed{\rho>\frac{3\pi}{GP^{2}}.}{}$ (\ref{eqn:MinDensity})

A rotating magnetic dipole radiates power

 $\boxed{P_{\mathrm{rad}}=\frac{2}{3}\frac{(\ddot{m}_{\bot})^{2}}{c^{3}}.}{}$ (\ref{eqn:MagneticLarmor})

The spin-down luminosity of a pulsar is

 $\boxed{-\dot{E}\equiv-\frac{dE_{\mathrm{rot}}}{dt}=\frac{-4\pi^{2}I\dot{P}}{P^% {3}}.}{}$ (\ref{eqn:SpindownLuminosity})

The minimum magnetic field strength of a pulsar is

 $\boxed{\biggl{(}\frac{B}{\mathrm{gauss}}\biggr{)}>3.2\times 10^{19}\biggl{(}% \frac{P\dot{P}}{\mathrm{s}}\biggr{)}^{1/2}.}{}$ (\ref{eqn:Bmin})

The characteristic age of a pulsar is defined by

 $\boxed{\tau\equiv\frac{P}{2\dot{P}}\,.}{}$ (\ref{eqn:PulsarAge})

The braking index of a pulsar in terms of its observable period $P$ and the first and second time derivatives is

 $\boxed{n=2-\frac{P\ddot{P}}{\dot{P}^{2}}.}{}$ (\ref{eqn:BI})

At frequency $\nu$ the refractive index of a cold plasma is

 $\boxed{\mu=\biggl{[}{1-\left(\frac{\nu_{\mathrm{p}}}{\nu}\right)^{2}}\biggr{]}% ^{1/2},}{}$ (\ref{eqn:refractiveindex})

where $\nu_{\mathrm{p}}$ is the plasma frequency

 $\boxed{\nu_{\mathrm{p}}=\biggl{(}\frac{e^{2}n_{\mathrm{e}}}{\pi m_{\mathrm{e}}% }\biggr{)}^{1/2}\approx 8.97\mathrm{~{}kHz}\,\biggl{(}\frac{n_{\mathrm{e}}}{% \mathrm{cm}^{-3}}\biggr{)}^{1/2}.}{}$ (\ref{eqn:PlasmaFrequency})

The group velocity of pulses is

 $\boxed{v_{\mathrm{g}}\approx c\biggl{(}1-\frac{\nu_{\mathrm{p}}^{2}}{2\nu^{2}}% \biggr{)}.}{}$ (\ref{eqn:GroupVelocity})

The dispersion delay of a pulsar is

 $\boxed{\biggl{(}\frac{t}{\mathrm{sec}}\biggr{)}\approx 4.149\times 10^{3}% \biggl{(}\frac{\mathrm{DM}}{\mathrm{pc~{}cm}^{-3}}\biggr{)}\biggl{(}\frac{\nu}% {\mathrm{MHz}}\biggr{)}^{-2},}{}$ (\ref{eqn:DispersionDelay})

where

 $\boxed{\mathrm{DM}\equiv\int_{0}^{d}n_{\mathrm{e}}\,dl}{}$ (\ref{eqn:DM})

in units of pc cm${}^{-3}$ is the dispersion measure of a pulsar at distance $d$.

The Bohr radius of a hydrogen atom is

 $\boxed{a_{n}=\frac{n^{2}\hbar^{2}}{m_{\mathrm{e}}e^{2}}\approx 0.53\times 10^{% -8}\mathrm{cm}\cdot n^{2}.}{}$ (\ref{eqn:AtomicRadius})

The frequency of a recombination line is

 $\boxed{\nu=R_{M}c\biggl{[}\frac{1}{n^{2}}-\frac{1}{(n+\Delta n)^{2}}\biggr{]},% \qquad\mathrm{where}\qquad R_{M}\equiv R_{\infty}\biggl{(}1+\frac{m_{\mathrm{e% }}}{M}\biggr{)}^{-1}.}{}$ (\ref{eqn:RecombFrequency})

The approximate recombination line separation frequency $\Delta\nu\equiv\nu(n)-\nu(n+1)$ for $n\gg 1$ is

 $\boxed{\frac{\Delta\nu}{\nu}\approx\frac{3}{n}.}{}$ (\ref{eqn:ApproxFrequency})

The spontaneous emission rate is

 $\boxed{A_{n+1,n}\approx\frac{64\pi^{6}m_{\mathrm{e}}e^{10}}{3c^{3}h^{6}n^{5}}% \approx 5.3\times 10^{9}\biggl{(}\frac{1}{n^{5}}\biggr{)}\mathrm{~{}s}^{-1}.}{}$ (\ref{eqn:SponRate})

The normalized Gaussian line profile is

 $\boxed{\phi(\nu)=\frac{c}{\nu_{0}}\biggl{(}\frac{M}{2\pi kT}\biggr{)}^{1/2}% \exp\biggl{[}-\frac{Mc^{2}}{2kT}\frac{(\nu-\nu_{0})^{2}}{\nu_{0}^{2}}\biggr{]}% ,}{}$ (\ref{eqn:GaussianProfile})

where

 $\boxed{\Delta\nu=\biggl{(}\frac{8\ln 2\,k}{c^{2}}\biggr{)}^{1/2}\biggl{(}\frac% {T}{M}\biggr{)}^{1/2}\nu_{0}}{}$ (\ref{eqn:LineFWHM})

and

 $\boxed{\phi(\nu_{0})=\biggl{(}\frac{\ln 2}{\pi}\biggr{)}^{1/2}\frac{2}{\Delta% \nu}.}{}$ (\ref{eqn:LinePeak})

Rate balance is given by

 $\boxed{n_{\mathrm{U}}A_{\mathrm{UL}}+n_{\mathrm{U}}B_{\mathrm{UL}}\bar{u}=n_{% \mathrm{L}}B_{\mathrm{LU}}\bar{u}.}{}$ (\ref{eqn:RateBalance})

The detailed balance equations connecting Einstein coefficients are

 $\displaystyle\boxed{\frac{g_{\mathrm{L}}}{g_{\mathrm{U}}}\frac{B_{\mathrm{LU}}% }{B_{\mathrm{UL}}}=1,}{}$ (\ref{eqn:BLUBUL}) $\displaystyle\boxed{\frac{A_{\mathrm{UL}}}{B_{\mathrm{UL}}}=\frac{8\pi h\nu_{0% }^{3}}{c^{3}}.}{}$ (\ref{eqn:AULBUL})

The spectral line radiative transfer equation is

 $\boxed{\frac{dI_{\nu}}{ds}=-\biggl{(}\frac{h\nu_{0}}{c}\biggr{)}(n_{\mathrm{L}% }B_{\mathrm{LU}}-n_{\mathrm{U}}B_{\mathrm{UL}})\phi(\nu)I_{\nu}+\biggl{(}\frac% {h\nu_{0}}{4\pi}\biggr{)}n_{\mathrm{U}}A_{\mathrm{UL}}\phi(\nu).}{}$ (\ref{eqn:spectralradxfer})

The Boltzmann equation for a two-level system is

 $\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}=\frac{g_{\mathrm{U}}}{g_{\mathrm{% L}}}\exp\biggl{(}-\frac{h\nu_{0}}{kT}\biggr{)}.}{}$ (\ref{eqn:BoltzmannNUNL})

The line opacity coefficient in LTE is

 $\boxed{\kappa=\frac{c^{2}}{8\pi\nu_{0}^{2}}\frac{g_{\mathrm{U}}}{g_{\mathrm{L}% }}n_{\mathrm{L}}A_{\mathrm{UL}}\biggl{[}1-\exp\biggl{(}-\frac{h\nu_{0}}{kT}% \biggr{)}\biggr{]}\phi(\nu).}{}$ (\ref{eqn:LineOpacity})

The excitation temperature $T_{\mathrm{x}}$ is defined by

 $\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}\equiv\frac{g_{\mathrm{U}}}{g_{% \mathrm{L}}}\exp\biggl{(}-\frac{h\nu_{0}}{kT_{\mathrm{x}}}\biggr{)}.}{}$ (\ref{eqn:ExcitationTemp})

The recombination-line opacity coefficient is

 $\boxed{\kappa(\nu_{0})\approx\biggl{(}\frac{n_{\mathrm{e}}^{2}}{T_{\mathrm{e}}% ^{5/2}\Delta\nu}\biggr{)}\biggl{(}\frac{4\pi e^{6}h}{3m_{\mathrm{e}}^{3/2}k^{5% /2}c}\biggr{)}\biggl{(}\frac{\ln 2}{2}\biggr{)}^{1/2}}{}$ (\ref{eqn:RecombOpacityCoefficient})

and the recombination line opacity is

 $\boxed{\tau_{\mathrm{L}}\approx 1.92\times 10^{3}\biggl{(}\frac{T_{\mathrm{e}}% }{\mathrm{K}}\biggr{)}^{-5/2}\biggl{(}\frac{\mathrm{EM}}{\mathrm{pc}\,\mathrm{% cm}^{-6}}\biggr{)}\biggl{(}\frac{\Delta\nu}{\mathrm{kHz}}\biggr{)}^{-1}.}{}$ (\ref{eqn:LineOpacity2})

The recombination line brightness temperature is given by

 $\boxed{T_{\mathrm{L}}\approx T_{\mathrm{e}}\tau_{\mathrm{L}}\approx 1.92\times 1% 0^{3}\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}^{-3/2}\biggl{(}\frac{% \mathrm{EM}}{\mathrm{pc}\,\mathrm{cm}^{-6}}\biggr{)}\biggl{(}\frac{\Delta\nu}{% \mathrm{kHz}}\biggr{)}^{-1}.}{}$ (\ref{eqn:LineTemp})

The recombination line/continuum ratio is

 $\boxed{\frac{T_{\mathrm{L}}}{T_{\mathrm{C}}}\approx 7.0\times 10^{3}\biggl{(}% \frac{\Delta v}{\mathrm{km~{}s}^{-1}}\biggr{)}^{-1}\biggl{(}\frac{\nu}{\mathrm% {GHz}}\biggr{)}^{1.1}\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}^{-1.1% 5}\biggl{[}1+\frac{N(\mathrm{He}^{+})}{N(\mathrm{H}^{+})}\biggr{]}^{-1},}{}$ (\ref{eqn:LCRatio})

where $[1+N(\mathrm{He}^{+})/N(\mathrm{H}^{+})]\approx 1.08$.

The electron temperature from the line/continuum ratio is

 $\boxed{\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}\approx\biggl{[}7.0% \times 10^{3}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{1.1}\,1.08^{-1}\,% \biggl{(}\frac{\Delta v}{\mathrm{km~{}s}^{-1}}\biggr{)}^{-1}\biggl{(}\frac{T_{% \mathrm{C}}}{T_{\mathrm{L}}}\biggr{)}\biggr{]}^{0.87}.}{}$ (\ref{eqn:ElectronTemp})

Quantization of angular momentum is given by

 $\boxed{L=n\hbar.}{}$ (\ref{eqn:AngMomQuant})

The angular momentum of a diatomic molecule is

 $\boxed{L=mr_{\mathrm{e}}^{2}\omega,}{}$ (\ref{eqn:AngMom})

where

 $\boxed{m\equiv\biggl{(}\frac{m_{\mathrm{A}}m_{\mathrm{B}}}{m_{\mathrm{A}}+m_{% \mathrm{B}}}\biggr{)}}{}$ (\ref{eqn:ReducedMass})

is the reduced mass and $r_{\mathrm{e}}$ is the separation of the atoms with masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$.

The rotational energy levels of a diatomic molecule with moment of inertia $I$ are

 $\boxed{E_{\mathrm{rot}}=\frac{J(J+1)\hbar^{2}}{2I},\qquad J=0,1,2,\ldots.}{}$ (\ref{eqn:EnergyQuant})

For a transition satisfying the selection rule

 $\boxed{\Delta J=\pm 1,}{}$ (\ref{eqn:SelectionRule})

the line frequency is

 $\boxed{\nu=\frac{hJ}{4\pi^{2}mr_{\mathrm{e}}^{2}}.}{}$ (\ref{eqn:Frequency})

The minimum temperature needed to excite the $J\rightarrow J-1$ transition at frequency $\nu$ is

 $\boxed{T_{\mathrm{min}}\approx\frac{\nu h(J+1)}{2k}.}{}$ (\ref{eqn:MinimumTemp})

The spontaneous emission coefficient is

 $\boxed{A_{\mathrm{UL}}=\frac{64\pi^{4}}{3hc^{3}}\nu_{\mathrm{UL}}^{3}|\mu_{% \mathrm{UL}}|^{2},}{}$ (\ref{eqn:EmissionCoef})

where

 $\boxed{|\mu_{\mathrm{J}\rightarrow\mathrm{J}-1}|^{2}=\frac{\mu^{2}J}{2J+1}}{}$ (\ref{eqn:DipoleMoment})

and $\mu$ is the electric dipole moment of the molecule.

The critical density is

 $\boxed{n^{*}\approx\frac{A_{\mathrm{UL}}}{\sigma v},}{}$ (\ref{eqn:CriticalDensity})

where $\sigma\sim 10^{-15}\mathrm{~{}cm}^{-2}$ is the collision cross section and $v\sim 10^{5}\mathrm{~{}cm~{}s}^{-1}$ is the typical H${}_{2}$ molecular velocity.

The CO-to-H${}_{2}$ conversion factor $X_{\mathrm{CO}}$ in our Galaxy is

 $\boxed{X_{\mathrm{CO}}=(2\pm 0.6)\times 10^{20}\mathrm{~{}cm}^{-2}\mathrm{~{}(% K~{}km~{}s}^{-1})^{-1}.}{}$ (\ref{eqn:COconversionfactor})

The Hi hyperfine line frequency is

 $\boxed{\nu_{10}=\frac{8}{3}g_{\mathrm{I}}\biggl{(}\frac{m_{\mathrm{e}}}{m_{% \mathrm{p}}}\biggr{)}\alpha^{2}(R_{M}c)\approx 1420.405751\mathrm{~{}MHz}.}{}$ (\ref{eqn:Frequency2})

The Hi hyperfine line emission coefficient is

 $\boxed{A_{10}\approx 2.85\times 10^{-15}\mathrm{~{}s}^{-1}.}{}$ (\ref{eqn:EmissionCoef2})

The Hi spin temperature $T_{\mathrm{s}}$ is defined by

 $\boxed{\frac{n_{1}}{n_{0}}\equiv\frac{g_{1}}{g_{0}}\exp\biggl{(}-\frac{h\nu_{1% 0}}{kT_{\mathrm{s}}}\biggr{)},}{}$ (\ref{eqn:SpinTemp})

where $g_{1}/g_{0}$ = 3.

The Hi line opacity coefficient is

 $\boxed{\kappa(\nu)\approx\frac{3c^{2}}{32\pi}\frac{A_{10}n_{\mathrm{H}}}{\nu_{% 10}}\frac{h}{kT_{\mathrm{s}}}\phi(\nu).}{}$ (\ref{eqn:OpacityCoef})

The hydrogen column density $\eta_{\mathrm{H}}$ is defined as the integral of density along the line of sight:

 $\boxed{\eta_{\mathrm{H}}\equiv\int_{\mathrm{los}}n_{\mathrm{H}}(s)\,ds.}{}$ (\ref{eqn:ColumnDensityDef})

If the Hi line is optically thin ($\tau\ll 1$) then the Hi column density is

 $\boxed{\biggl{(}\frac{\eta_{\mathrm{H}}}{\mathrm{cm}^{-2}}\biggr{)}\approx 1.8% 2\times 10^{18}\int\biggl{[}\frac{T_{\mathrm{b}}(v)}{\mathrm{K}}\biggr{]}d% \biggl{(}\frac{v}{\mathrm{km~{}s}^{-1}}\biggr{)}.}{}$ (\ref{eqn:ColumnDensityEq})

If $\tau\ll 1$ the hydrogen mass of a galaxy is

 $\boxed{\biggl{(}\frac{M_{\mathrm{H}}}{M_{\odot}}\biggr{)}\approx 2.36\times 10% ^{5}\biggl{(}\frac{D}{\mathrm{Mpc}}\biggr{)}^{2}\int\biggl{[}\frac{S(v)}{% \mathrm{Jy}}\biggr{]}\biggl{(}\frac{dv}{\mathrm{km~{}s}^{-1}}\biggr{)}.}{}$ (\ref{eqn:HydrogenMass})

The total mass of a galaxy is

 $\boxed{\biggl{(}\frac{M}{M_{\odot}}\biggr{)}\approx 2.33\times 10^{5}\biggl{(}% \frac{v_{\mathrm{rot}}}{\mathrm{km~{}s}^{-1}}\biggr{)}^{2}\biggl{(}\frac{r}{% \mathrm{kpc}}\biggr{)}.}{}$ (\ref{eqn:TotalMass})