# Appendix E Essential Equations

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

 $\boxed{I_{\nu}\equiv\frac{dP}{(\cos\theta~{}d\sigma)\,d\nu\,d\Omega},}$ (2.2)

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:

 $\boxed{I_{\lambda}\equiv\frac{dP}{(\cos\theta\,d\sigma)\,d\lambda\,d\Omega}.}$ (2.3)

These two quantities are related by

 $\boxed{\frac{I_{\lambda}}{I_{\nu}}=\bigg{|}\frac{d\nu}{d\lambda}\bigg{|}=\frac% {c}{\lambda^{2}}=\frac{\nu^{2}}{c}.}$ (2.5)

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

 $\boxed{S_{\nu}\equiv\int_{\mathrm{source}}I_{\nu}(\theta,\phi)\cos\theta\,d% \Omega.}$ (2.9)

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

 $\boxed{S_{\nu}\approx\int_{\mathrm{source}}I_{\nu}(\theta,\phi)d\Omega.}$ (2.10)

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

 $\boxed{L_{\nu}=4\pi d^{2}S_{\nu}}$ (2.15)

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 $ds$:

 $\boxed{\kappa(\nu)\equiv\frac{dP(\nu)}{ds}.}$ (2.18)

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

 $\boxed{\tau\equiv\int_{s_{\mathrm{out}}}^{s_{\mathrm{in}}}-\kappa(s^{\prime})% ds^{\prime}.}$ (2.23)

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

 $\boxed{j_{\nu}\equiv\frac{dI_{\nu}}{ds}.}$ (2.26)

The equation of radiative transfer is

 $\boxed{\frac{dI_{\nu}}{ds}=-\kappa I_{\nu}+j_{\nu}.}$ (2.27)

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:

 $\boxed{\frac{j_{\nu}}{\kappa}=B_{\nu}(T).}$ (2.30)

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

 $\boxed{T_{\mathrm{b}}(\nu)\equiv\frac{I_{\nu}c^{2}}{2k\nu^{2}}.}$ (2.33)

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):

 $\boxed{e_{\nu}=a_{\nu}=1-r_{\nu}.}$ (2.47)

The spectral energy density of radiation is

 $\boxed{u_{\nu}=\frac{1}{c}\int I_{\nu}\,d\Omega.}$ (2.76)

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

 $\boxed{B_{\nu}=\frac{2kT\nu^{2}}{c^{2}}=\frac{2kT}{\lambda^{2}}.}$ (2.79)

The energy of a photon is

 $\boxed{E=h\nu.}$ (2.81)

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

 $\boxed{B_{\nu}=\frac{2h\nu^{3}}{c^{2}}\,\frac{1}{\displaystyle\exp\Bigl{(}% \frac{h\nu}{kT}\Bigr{)}-1}.}$ (2.86)

The total intensity of blackbody radiation is

 $\boxed{B(T)\equiv\int_{0}^{\infty}B_{\nu}(T)d\nu=\frac{\sigma T^{4}}{\pi},}$ (2.89)

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

 $\boxed{\sigma\equiv\frac{2\pi^{5}k^{4}}{15c^{2}h^{3}}\approx 5.67\times 10^{-5% }\frac{\mathrm{erg}}{\mathrm{cm}^{2}\mathrm{~{}s~{}K}^{4}\mathrm{~{}sr}}.}$ (2.90)

The total energy density of blackbody radiation is

 $\boxed{u=\frac{4\sigma T^{4}}{c}=aT^{4},}$ (2.93)

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

 $\boxed{\biggl{(}\frac{n_{\gamma}}{\mathrm{cm}^{-3}}\biggr{)}\approx 20.3\biggl% {(}\frac{T}{\mathrm{K}}\biggr{)}^{3}.}$ (2.100)

The mean photon energy of blackbody radiation is

 $\boxed{\langle E_{\gamma}\rangle\approx\,2.70\,kT.}$ (2.101)

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

 $\boxed{\biggl{(}\frac{\nu_{\mathrm{max}}}{\mathrm{GHz}}\biggr{)}\approx 59% \biggl{(}\frac{T}{\mathrm{K}}\biggr{)}.}$ (2.104)

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

 $\boxed{\biggl{(}\frac{\lambda_{\mathrm{max}}}{\mathrm{cm}}\biggr{)}\approx 0.2% 9\biggl{(}\frac{T}{\mathrm{K}}\biggr{)}^{-1}.}$ (2.106)

The flux density of isotropic radiation is

 $\boxed{S_{\nu}=\pi I_{\nu}.}$ (2.109)

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

 $\boxed{P_{\nu}=kT.}$ (2.117)

At any frequency, the exact Nyquist formula is

 $\boxed{P_{\nu}=\frac{h\nu}{\displaystyle\exp\Bigl{(}\frac{h\nu}{kT}\Bigr{)}-1}.}$ (2.119)

The critical density needed to close the universe is

 $\boxed{\rho_{\mathrm{c}}=\frac{3H_{0}^{2}}{8\pi G}\approx 8.6\times 10^{-30}% \mathrm{~{}g~{}cm}^{-3}.}$ (2.126)

Redshift $z$ is defined by

 $\boxed{z\equiv\frac{\lambda_{\mathrm{o}}-\lambda_{\mathrm{e}}}{\lambda_{% \mathrm{e}}}=\frac{\lambda_{\mathrm{o}}}{\lambda_{\mathrm{e}}}-1=\frac{\nu_{% \mathrm{e}}}{\nu_{\mathrm{o}}}-1,}$ (2.127)

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

 $\boxed{(1+z)=a^{-1}.}$ (2.128)

The CMB temperature at redshift $z$ is

 $\boxed{T=T_{0}(1+z).}$ (2.129)

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

 $\boxed{E_{\bot}=\frac{q\dot{v}\sin\theta}{rc^{2}}.}$ (2.136)

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

 $\boxed{|\vec{S}|=\frac{c}{4\pi}E^{2}.}$ (2.139)

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

 $\boxed{P=\frac{2}{3}\frac{q^{2}\dot{v}^{2}}{c^{3}},}$ (2.143)

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

Exponential notation for trigonometric functions is

 $\boxed{e^{-i\omega t}=\cos(\omega t)-i\sin(\omega t).}$ (3.2)

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

 $\boxed{I\equiv\frac{dq}{dt}.}$ (3.4)

The power pattern of a short dipole antenna is

 $\boxed{P\propto\sin^{2}\theta.}$ (3.14)

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

 $\boxed{\langle P\rangle=\frac{\pi^{2}}{3c}\biggl{(}\frac{I_{0}l}{\lambda}% \biggr{)}^{2}.}$ (3.17)

 $\boxed{R\equiv\frac{2\langle P\rangle}{I_{0}^{2}}.}$ (3.25)

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

 $\boxed{\langle G\rangle=1}$ (3.32)

and

 $\boxed{\int_{\mathrm{sphere}}G\,d\Omega=4\pi.}$ (3.33)

The beam solid angle is defined by

 $\boxed{\Omega_{\mathrm{A}}\equiv\frac{4\pi}{G_{\mathrm{max}}}=\frac{1}{G_{% \mathrm{max}}}\int_{4\pi}G(\theta,\phi)\,d\Omega.}$ (3.34)

The effective area of an antenna is defined by

 $\boxed{A_{\mathrm{e}}\equiv 2P_{\nu}/S_{\nu},}$ (3.35)

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

 $\boxed{\langle A_{\mathrm{e}}\rangle=\frac{\lambda^{2}}{4\pi}.}$ (3.41)

Reciprocity implies

 $\boxed{G(\theta,\phi)\propto A_{\mathrm{e}}(\theta,\phi).}$ (3.44)

Reciprocity and energy conservation imply

 $\boxed{A_{\mathrm{e}}(\theta,\phi)=\frac{\lambda^{2}G(\theta,\phi)}{4\pi}.}$ (3.46)

Antenna temperature is defined by

 $\boxed{T_{\mathrm{A}}\equiv\frac{P_{\nu}}{k}.}$ (3.47)

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

 $\boxed{T_{\mathrm{A}}=\frac{A_{\mathrm{e}}S}{2k}.}$ (3.48)

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 $\Omega_{\mathrm{s}}$,

 $\boxed{\frac{T_{\mathrm{A}}}{T_{\mathrm{b}}}=\frac{\Omega_{\mathrm{s}}}{\Omega% _{\mathrm{A}}}.}$ (3.56)

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

 $\boxed{\Omega_{\mathrm{MB}}\equiv\frac{1}{G_{\mathrm{max}}}\int_{\mathrm{MB}}G% (\theta,\phi)\,d\Omega}$ (3.57)

and is used in the definition of main beam efficiency:

 $\boxed{\eta_{\mathrm{B}}\equiv\frac{\Omega_{\mathrm{MB}}}{\Omega_{\mathrm{A}}}.}$ (3.58)

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

 $\boxed{z=\frac{r^{2}}{4f}.}$ (3.60)

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

 $\boxed{R_{\mathrm{ff}}\approx\frac{2D^{2}}{\lambda}.}$ (3.64)

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

 $\displaystyle\boxed{l\equiv\sin\theta,}$ (3.69) $\displaystyle\boxed{u\equiv\frac{x}{\lambda},}$ (3.72) $\displaystyle\boxed{f(l)=\int_{\mathrm{aperture}}g(u)e^{-i2\pi lu}du.}$ (3.73)

The power pattern of a uniformly illuminated linear aperture is

 $\boxed{P(\theta)\propto\mathrm{sinc}^{2}\biggl{(}\frac{\theta D}{\lambda}% \biggr{)},}$ (3.79)

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

 $\boxed{\theta_{\mathrm{HPBW}}\approx 0.89\frac{\lambda}{D}.}$ (3.82)

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

 $\boxed{\theta_{\mathrm{HPBW}}\approx 1.2\frac{\lambda}{D}.}$ (3.96)

The two-dimensional aperture field pattern is

 $\boxed{f(l,m)\propto\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(u,v)e^{-i2% \pi(lu+mv)}du\,dv,}$ (3.97)

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

 $\boxed{G\approx\frac{4\pi D_{x}D_{y}}{\lambda^{2}}\mathrm{sinc}^{2}\biggl{(}% \frac{\theta_{x}D_{x}}{\lambda}\biggr{)}\mathrm{sinc}^{2}\biggl{(}\frac{\theta% _{y}D_{y}}{\lambda}\biggr{)}.}$ (3.107)

Aperture efficiency is defined by

 $\boxed{\eta_{\mathrm{A}}\equiv\frac{\mathrm{max}(A_{\mathrm{e}})}{A_{\mathrm{% geom}}}.}$ (3.111)

The beam solid angle of a Gaussian beam is

 $\boxed{\Omega_{\mathrm{A}}=\,\biggl{(}\frac{\pi}{4\ln 2}\biggr{)}\theta_{% \mathrm{HPBW}}^{2}\approx 1.133\,\theta_{\mathrm{HPBW}}^{2}.}$ (3.118)

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

 $\boxed{\eta_{\mathrm{s}}=\exp\biggl{[}-\biggl{(}\frac{4\pi\sigma}{\lambda}% \biggr{)}^{2}\biggr{]}.}$ (3.129)

Noise temperature is defined by

 $\boxed{T_{\mathrm{N}}\equiv\frac{P_{\nu}}{k}.}$ (3.149)

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

 $\boxed{T_{\mathrm{s}}=T_{\mathrm{cmb}}+T_{\mathrm{rsb}}+\Delta T_{\mathrm{% source}}+[1-\exp(-\tau_{\mathrm{A}})]T_{\mathrm{atm}}+T_{\mathrm{spill}}+T_{% \mathrm{r}}+\cdots.}$ (3.150)

The ideal total-power radiometer equation is

 $\boxed{\sigma_{T}\approx T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}% \biggr{]}^{1/2}.}$ (3.154)

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

 $\boxed{\sigma_{T}\approx T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}+% \biggl{(}\frac{\Delta G}{G}\biggr{)}^{2}\biggr{]}^{1/2}.}$ (3.158)

 $\boxed{\sigma_{T}\approx 2T_{\mathrm{s}}\biggl{[}\frac{1}{\Delta\nu\,\tau}% \biggr{]}^{1/2}.}$ (3.162)

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

 $\boxed{\biggl{(}\frac{\sigma_{\mathrm{c}}}{\mathrm{mJy~{}beam}^{-1}}\biggr{)}% \approx\begin{cases}{\displaystyle 0.2\biggr{(}\frac{\nu}{\mathrm{GHz}}\biggr{% )}^{-0.7}\biggl{(}\frac{\theta}{\mathrm{arcmin}}\biggr{)}^{2}}&(\theta>0.17% \mathrm{~{}arcmin}),\\ {\displaystyle 2.2\biggr{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.7}\biggl{(}% \frac{\theta}{\mathrm{arcmin}}\biggr{)}^{10/3}}&(\theta<0.17\mathrm{~{}arcmin}% ).\end{cases}}$ (3.163)

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

 $\boxed{T_{\mathrm{r}}=\frac{T_{\mathrm{h}}-YT_{\mathrm{c}}}{Y-1}.}$ (3.168)

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

 $\boxed{\mathcal{V}_{\nu}=\int I_{\nu}(\hat{s})\exp(-i2\pi\vec{b}\cdot\hat{s}/% \lambda)\,d\Omega.}$ (3.186)

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

 $\boxed{\Delta\theta\,\Delta\nu\ll\theta_{\mathrm{s}}\nu.}$ (3.192)

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

 $\boxed{\Delta\theta\,\Delta t\ll\frac{\theta_{\mathrm{s}}P}{2\pi}\approx\theta% _{\mathrm{s}}\cdot 1.37\times 10^{4}\mathrm{~{}s}.}$ (3.194)

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

 $\boxed{\mathcal{V}_{\nu}(u,v,w)=\int\int\frac{I_{\nu}(l,m)}{(1-l^{2}-m^{2})^{1% /2}}\exp[-i2\pi(ul+vm+wn)]dl\,dm.}$ (3.197)

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 $\mathcal{V}_{\nu}(u,v)$:

 $\boxed{\frac{I_{\nu}(l,m)}{(1-l^{2}-m^{2})^{1/2}}=\int\int\mathcal{V}_{\nu}(u,% v,0)\exp[+i2\pi(ul+vm)]du\,dv.}$ (3.198)

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

 $\boxed{\sigma_{S}=\frac{2kT_{\mathrm{s}}}{A_{\mathrm{e}}[N(N-1)\Delta\nu\,\tau% ]^{1/2}}.}$ (3.203)

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

 $\boxed{\sigma_{T}=\biggl{(}\frac{\sigma_{S}}{\Omega_{\mathrm{A}}}\biggr{)}% \frac{\lambda^{2}}{2k},}$ (3.204)

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

 $\boxed{f(v)=\frac{4v^{2}}{\sqrt{\pi}}\biggl{(}\frac{m}{2kT}\biggr{)}^{3/2}\exp% \biggl{(}-\frac{mv^{2}}{2kT}\biggr{)}.}$ (4.34)

The free–free emission coefficient is

 $\boxed{j_{\nu}=\frac{\pi^{2}Z^{2}e^{6}n_{\mathrm{e}}n_{\mathrm{i}}}{4c^{3}m_{% \mathrm{e}}^{2}}\biggl{(}\frac{2m_{\mathrm{e}}}{\pi kT}\biggr{)}^{1/2}\ln% \biggl{(}\frac{b_{\mathrm{max}}}{b_{\mathrm{min}}}\biggr{)},}$ (4.39)

where

 $\boxed{b_{\mathrm{min}}\approx\frac{Ze^{2}}{m_{\mathrm{e}}v^{2}}.}$ (4.43)

The free–free absorption coefficient is

 $\boxed{\kappa=\frac{1}{\nu^{2}T^{3/2}}\biggl{[}\frac{Z^{2}e^{6}}{c}n_{\mathrm{% e}}n_{\mathrm{i}}\frac{1}{\sqrt{2\pi(m_{\mathrm{e}}k)^{3}}}\biggr{]}\frac{\pi^% {2}}{4}\ln\biggl{(}\frac{b_{\mathrm{max}}}{b_{\mathrm{min}}}\biggr{)}.}$ (4.52)

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

 $\boxed{S_{\nu}\propto\frac{2kT\nu^{2}}{c^{2}}\tau(\nu)\propto\nu^{-0.1}.}$ (4.54)

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

 $\boxed{\frac{\mathrm{EM}}{\mathrm{pc~{}cm}^{-6}}\equiv\int_{\mathrm{los}}% \biggl{(}\frac{n_{\mathrm{e}}}{\mathrm{cm}^{-3}}\biggr{)}^{2}d\biggl{(}\frac{s% }{\mathrm{pc}}\biggr{)}.}$ (4.57)

The free–free optical depth of a plasma is

 $\boxed{\tau\approx 3.28\times 10^{-7}\biggl{(}\frac{T}{10^{4}\mathrm{~{}K}}% \biggr{)}^{-1.35}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-2.1}\biggl{(}% \frac{\mathrm{EM}}{\mathrm{pc~{}cm}^{-6}}\biggr{)}.}$ (4.60)

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

 $\boxed{\biggl{(}\frac{Q_{\mathrm{H}}}{\mathrm{s}^{-1}}\biggr{)}\approx 6.3% \times 10^{52}\biggl{(}\frac{T}{10^{4}\mathrm{~{}K}}\biggr{)}^{-0.45}\biggl{(}% \frac{\nu}{\mathrm{GHz}}\biggr{)}^{0.1}\biggl{(}\frac{L_{\nu}}{10^{20}\mathrm{% ~{}W~{}Hz}^{-1}}\biggr{)},}$ (4.62)

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

 $\boxed{\vec{F}=\frac{q(\vec{v}\times\vec{B})}{c}.}$ (5.1)

The gyro frequency is defined by

 $\boxed{\omega_{\mathrm{G}}\equiv\frac{qB}{mc}.}$ (5.4)

The (nonrelativistic) electron gyro frequency in MHz is

 $\boxed{\biggl{(}\frac{\nu_{\mathrm{G}}}{\mathrm{MHz}}\biggr{)}=2.8\biggl{(}% \frac{B}{\mathrm{gauss}}\biggr{)}.}$ (5.7)

The Lorentz transform is

 $\boxed{x=\gamma(x^{\prime}+vt^{\prime}),\qquad y=y^{\prime},\qquad z=z^{\prime% },\qquad t=\gamma(t^{\prime}+\beta x^{\prime}/c),}$ (5.12)
 $\boxed{x^{\prime}=\gamma(x-vt),\qquad y^{\prime}=y,\qquad z^{\prime}=z,\qquad t% ^{\prime}=\gamma(t-\beta x/c),}$ (5.13)

where

 $\boxed{\beta\equiv v/c}$ (5.14)

and

 $\boxed{\gamma\equiv(1-\beta^{2})^{-1/2}}$ (5.15)

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

 $\boxed{\Delta x=\gamma(\Delta x^{\prime}+v\Delta t^{\prime}),\quad\!\Delta y=% \Delta y^{\prime},\quad\!\Delta z=\Delta z^{\prime},\quad\!\Delta t=\gamma(% \Delta t^{\prime}+\beta\Delta x^{\prime}/c),}$ (5.16)
 $\boxed{\Delta x^{\prime}=\gamma(\Delta x-v\Delta t),\quad\Delta y^{\prime}=% \Delta y,\quad\Delta z^{\prime}=\Delta z,\quad\Delta t^{\prime}=\gamma(\Delta t% -\beta\Delta x/c).}$ (5.17)

The Thomson cross section of an electron is defined by

 $\boxed{\sigma_{\mathrm{T}}\equiv\frac{8\pi}{3}\biggl{(}\frac{e^{2}}{m_{\mathrm% {e}}c^{2}}\biggr{)}^{2}.}$ (5.33)

Magnetic energy density is given by

 $\boxed{U_{B}=\frac{B^{2}}{8\pi}.}$ (5.35)

The synchrotron power of one electron is

 $\boxed{P=2\sigma_{\mathrm{T}}\beta^{2}\gamma^{2}c\,U_{B}\sin^{2}\alpha.}$ (5.37)

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

 $\boxed{\langle P\rangle=\frac{4}{3}\sigma_{\mathrm{T}}\beta^{2}\gamma^{2}cU_{B% }.}$ (5.42)

The synchrotron spectrum of a single electron is

 $\boxed{P(\nu)=\frac{\sqrt{3}e^{3}B\sin\alpha}{m_{\mathrm{e}}c^{2}}\biggl{(}% \frac{\nu}{\nu_{\mathrm{c}}}\biggr{)}\int_{\nu/\nu_{\mathrm{c}}}^{\infty}K_{5/% 3}(\eta)d\eta,}$ (5.66)

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

 $\boxed{\nu_{\mathrm{c}}=\frac{3}{2}\gamma^{2}\nu_{\mathrm{G}}\sin\alpha\approx% \gamma^{2}\nu_{\mathrm{G}}\,\propto E^{2}B_{\bot}.}$ (5.67)

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

 $\boxed{n(E)dE\approx KE^{-\delta}dE,}$ (5.70)

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

 $\boxed{j_{\nu}\propto B^{(\delta+1)/2}\nu^{(1-\delta)/2}.}$ (5.78)

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

 $\boxed{\alpha=\frac{\delta-1}{2}.}$ (5.79)

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

 $\boxed{\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}\approx 1.18\times 1% 0^{6}\biggl{(}\frac{\nu}{\mathrm{Hz}}\biggr{)}^{1/2}\biggl{(}\frac{B}{\mathrm{% gauss}}\biggr{)}^{-1/2}.}$ (5.85)

At a sufficiently low frequency $\nu$,

 $\boxed{S_{\mathrm{\nu}}\propto\nu^{-5/2}}$ (5.89)

and

 $\boxed{\biggl{(}\frac{B}{\mathrm{gauss}}\biggr{)}\approx 1.4\times 10^{12}% \biggl{(}\frac{\nu}{\mathrm{Hz}}\biggr{)}\biggl{(}\frac{T_{\mathrm{b}}}{% \mathrm{K}}\biggr{)}^{-2}.}$ (5.91)

For a given synchrotron luminosity, the electron energy density is

 $\boxed{U_{\mathrm{e}}\propto B^{-3/2}.}$ (5.98)

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

 $\boxed{U=(1+\eta)U_{\mathrm{e}}+U_{B},}$ (5.100)

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

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

 $\boxed{\frac{\mathrm{particle~{}energy}}{\mathrm{field~{}energy}}=\frac{(1+% \eta)U_{\mathrm{e}}}{U_{B}}=\frac{4}{3}.}$ (5.107)

The minimum-energy magnetic field is

 $\boxed{B_{\min}=[4.5(1+\eta)c_{12}L]^{2/7}R^{-6/7}\mathrm{~{}gauss}}$ (5.109)

and the corresponding total energy is

 $\boxed{E_{\mathrm{min}}\mathrm{(total)}=c_{13}[(1+\eta)L]^{4/7}R^{9/7}\mathrm{% ~{}ergs}.}$ (5.110)

 $\boxed{\tau\approx c_{12}B_{\bot}^{-3/2},}$ (5.112)

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

 $\boxed{\biggl{(}\frac{L_{\mathrm{E}}}{L_{\odot}}\biggr{)}\approx 3.3\times 10^% {4}\biggl{(}\frac{M}{M_{\odot}}\biggr{)}.}$ (5.117)

The nonrelativistic Thomson-scattering power is

 $\boxed{P=\sigma_{\mathrm{T}}cU_{\mathrm{rad}}.}$ (5.132)

The relativistic Doppler equation is

 $\boxed{\nu^{\prime}=\nu[\gamma(1+\beta\cos\theta)].}$ (5.142)

The net inverse-Compton power emitted is

 $\boxed{P_{\mathrm{IC}}=\frac{4}{3}\sigma_{\mathrm{T}}c\beta^{2}\gamma^{2}U_{% \mathrm{rad}}.}$ (5.152)

The IC/synchrotron power ratio is

 $\boxed{\frac{P_{\mathrm{IC}}}{P_{\mathrm{syn}}}=\frac{U_{\mathrm{rad}}}{U_{B}}.}$ (5.154)

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

 $\boxed{\frac{\langle\nu\rangle}{\nu_{0}}=\frac{4}{3}\gamma^{2}.}$ (5.160)

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

 $\boxed{T_{\mathrm{max}}\sim 10^{12}\mathrm{~{}K}.}$ (5.163)

The apparent transverse velocity of a moving source component is

 $\boxed{\beta_{\bot}\mathrm{(apparent)}=\frac{\beta\sin\theta}{1-\beta\cos% \theta}.}$ (5.167)

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

 $\boxed{\cos\theta_{\mathrm{m}}=\beta}$ (5.170)

and

 $\boxed{\sin\theta_{\mathrm{m}}=\gamma^{-1}.}$ (5.171)

The largest apparent transverse speed is

 $\boxed{\mathrm{max}[\beta_{\bot}\mathrm{(apparent)}]=\beta\gamma.}$ (5.172)

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

 $\boxed{\frac{\nu}{\nu^{\prime}}=\gamma^{-1}.}$ (5.180)

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

 $\boxed{\delta^{2+\alpha}<\frac{S}{S_{0}}<\delta^{3+\alpha}.}$ (5.183)

Thermal and nonthermal radio luminosities of star-forming galaxies are

 $\boxed{\biggl{(}\frac{L_{\mathrm{T}}}{\mathrm{W~{}Hz}^{-1}}\biggr{)}\approx 5.% 5\times 10^{20}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.1}\biggl{[}\frac% {\mathrm{SFR}(M>5M_{\odot})}{M_{\odot}\mathrm{~{}yr}^{-1}}\biggr{]}}$ (5.184)

and

 $\boxed{\biggl{(}\frac{L_{\mathrm{NT}}}{\mathrm{W~{}Hz}^{-1}}\biggr{)}\approx 5% .3\times 10^{21}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{-0.8}\biggl{[}% \frac{\mathrm{SFR}(M>5M_{\odot})}{M_{\odot}\mathrm{~{}yr}^{-1}}\biggr{]}.}$ (5.185)

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

 $\boxed{\rho>\frac{3\pi}{GP^{2}}.}$ (6.5)

A rotating magnetic dipole radiates power

 $\boxed{P_{\mathrm{rad}}=\frac{2}{3}\frac{(\ddot{m}_{\bot})^{2}}{c^{3}}.}$ (6.10)

The spin-down luminosity of a pulsar is

 $\boxed{-\dot{E}\equiv-\frac{dE_{\mathrm{rot}}}{dt}=\frac{-4\pi^{2}I\dot{P}}{P^% {3}}.}$ (6.20)

The minimum magnetic field strength of a pulsar is

 $\boxed{\biggl{(}\frac{B}{\mathrm{gauss}}\biggr{)}>3.2\times 10^{19}\biggl{(}% \frac{P\dot{P}}{\mathrm{s}}\biggr{)}^{1/2}.}$ (6.26)

The characteristic age of a pulsar is defined by

 $\boxed{\tau\equiv\frac{P}{2\dot{P}}\,.}$ (6.31)

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

 $\boxed{n=2-\frac{P\ddot{P}}{\dot{P}^{2}}.}$ (6.37)

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

 $\boxed{\mu=\biggl{[}{1-\left(\frac{\nu_{\mathrm{p}}}{\nu}\right)^{2}}\biggr{]}% ^{1/2},}$ (6.39)

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

 $\boxed{\nu_{\mathrm{p}}=\biggl{(}\frac{e^{2}n_{\mathrm{e}}}{\pi m_{\mathrm{e}}% }\biggr{)}^{1/2}\approx 8.97\mathrm{~{}kHz}\,\biggl{(}\frac{n_{\mathrm{e}}}{% \mathrm{cm}^{-3}}\biggr{)}^{1/2}.}$ (6.40)

The group velocity of pulses is

 $\boxed{v_{\mathrm{g}}\approx c\biggl{(}1-\frac{\nu_{\mathrm{p}}^{2}}{2\nu^{2}}% \biggr{)}.}$ (6.42)

The dispersion delay of a pulsar is

 $\boxed{\biggl{(}\frac{t}{\mathrm{sec}}\biggr{)}\approx 4.149\times 10^{3}% \biggl{(}\frac{\mathrm{DM}}{\mathrm{pc~{}cm}^{-3}}\biggr{)}\biggl{(}\frac{\nu}% {\mathrm{MHz}}\biggr{)}^{-2},}$ (6.45)

where

 $\boxed{\mathrm{DM}\equiv\int_{0}^{d}n_{\mathrm{e}}\,dl}$ (6.46)

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

The Bohr radius of a hydrogen atom is

 $\boxed{a_{n}=\frac{n^{2}\hbar^{2}}{m_{\mathrm{e}}e^{2}}\approx 0.53\times 10^{% -8}\mathrm{cm}\cdot n^{2}.}$ (7.6)

The frequency of a recombination line is

 $\boxed{\nu=R_{M}c\biggl{[}\frac{1}{n^{2}}-\frac{1}{(n+\Delta n)^{2}}\biggr{]},% \qquad\mathrm{where}\qquad R_{M}\equiv R_{\infty}\biggl{(}1+\frac{m_{\mathrm{e% }}}{M}\biggr{)}^{-1}.}$ (7.12)

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

 $\boxed{\frac{\Delta\nu}{\nu}\approx\frac{3}{n}.}$ (7.15)

The spontaneous emission rate is

 $\boxed{A_{n+1,n}\approx\frac{64\pi^{6}m_{\mathrm{e}}e^{10}}{3c^{3}h^{6}n^{5}}% \approx 5.3\times 10^{9}\biggl{(}\frac{1}{n^{5}}\biggr{)}\mathrm{~{}s}^{-1}.}$ (7.23)

The normalized Gaussian line profile is

 $\boxed{\phi(\nu)=\frac{c}{\nu_{0}}\biggl{(}\frac{M}{2\pi kT}\biggr{)}^{1/2}% \exp\biggl{[}-\frac{Mc^{2}}{2kT}\frac{(\nu-\nu_{0})^{2}}{\nu_{0}^{2}}\biggr{]},}$ (7.32)

where

 $\boxed{\Delta\nu=\biggl{(}\frac{8\ln 2\,k}{c^{2}}\biggr{)}^{1/2}\biggl{(}\frac% {T}{M}\biggr{)}^{1/2}\nu_{0}}$ (7.35)

and

 $\boxed{\phi(\nu_{0})=\biggl{(}\frac{\ln 2}{\pi}\biggr{)}^{1/2}\frac{2}{\Delta% \nu}.}$ (7.37)

Rate balance is given by

 $\boxed{n_{\mathrm{U}}A_{\mathrm{UL}}+n_{\mathrm{U}}B_{\mathrm{UL}}\bar{u}=n_{% \mathrm{L}}B_{\mathrm{LU}}\bar{u}.}$ (7.42)

The detailed balance equations connecting Einstein coefficients are

 $\displaystyle\boxed{\frac{g_{\mathrm{L}}}{g_{\mathrm{U}}}\frac{B_{\mathrm{LU}}% }{B_{\mathrm{UL}}}=1,}$ (7.50) $\displaystyle\boxed{\frac{A_{\mathrm{UL}}}{B_{\mathrm{UL}}}=\frac{8\pi h\nu_{0% }^{3}}{c^{3}}.}$ (7.51)

The spectral line radiative transfer equation is

 $\boxed{\frac{dI_{\nu}}{ds}=-\biggl{(}\frac{h\nu_{0}}{c}\biggr{)}(n_{\mathrm{L}% }B_{\mathrm{LU}}-n_{\mathrm{U}}B_{\mathrm{UL}})\phi(\nu)I_{\nu}+\biggl{(}\frac% {h\nu_{0}}{4\pi}\biggr{)}n_{\mathrm{U}}A_{\mathrm{UL}}\phi(\nu).}$ (7.57)

The Boltzmann equation for a two-level system is

 $\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}=\frac{g_{\mathrm{U}}}{g_{\mathrm{% L}}}\exp\biggl{(}-\frac{h\nu_{0}}{kT}\biggr{)}.}$ (7.64)

The line opacity coefficient in LTE is

 $\boxed{\kappa=\frac{c^{2}}{8\pi\nu_{0}^{2}}\frac{g_{\mathrm{U}}}{g_{\mathrm{L}% }}n_{\mathrm{L}}A_{\mathrm{UL}}\biggl{[}1-\exp\biggl{(}-\frac{h\nu_{0}}{kT}% \biggr{)}\biggr{]}\phi(\nu).}$ (7.67)

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

 $\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}\equiv\frac{g_{\mathrm{U}}}{g_{% \mathrm{L}}}\exp\biggl{(}-\frac{h\nu_{0}}{kT_{\mathrm{x}}}\biggr{)}.}$ (7.70)

The recombination-line opacity coefficient is

 $\boxed{\kappa(\nu_{0})\approx\biggl{(}\frac{n_{\mathrm{e}}^{2}}{T_{\mathrm{e}}% ^{5/2}\Delta\nu}\biggr{)}\biggl{(}\frac{4\pi e^{6}h}{3m_{\mathrm{e}}^{3/2}k^{5% /2}c}\biggr{)}\biggl{(}\frac{\ln 2}{2}\biggr{)}^{1/2}}$ (7.94)

and the recombination line opacity is

 $\boxed{\tau_{\mathrm{L}}\approx 1.92\times 10^{3}\biggl{(}\frac{T_{\mathrm{e}}% }{\mathrm{K}}\biggr{)}^{-5/2}\biggl{(}\frac{\mathrm{EM}}{\mathrm{pc}\,\mathrm{% cm}^{-6}}\biggr{)}\biggl{(}\frac{\Delta\nu}{\mathrm{kHz}}\biggr{)}^{-1}.}$ (7.96)

The recombination line brightness temperature is given by

 $\boxed{T_{\mathrm{L}}\approx T_{\mathrm{e}}\tau_{\mathrm{L}}\approx 1.92\times 1% 0^{3}\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}^{-3/2}\biggl{(}\frac{% \mathrm{EM}}{\mathrm{pc}\,\mathrm{cm}^{-6}}\biggr{)}\biggl{(}\frac{\Delta\nu}{% \mathrm{kHz}}\biggr{)}^{-1}.}$ (7.97)

The recombination line/continuum ratio is

 $\boxed{\frac{T_{\mathrm{L}}}{T_{\mathrm{C}}}\approx 7.0\times 10^{3}\biggl{(}% \frac{\Delta v}{\mathrm{km~{}s}^{-1}}\biggr{)}^{-1}\biggl{(}\frac{\nu}{\mathrm% {GHz}}\biggr{)}^{1.1}\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}^{-1.1% 5}\biggl{[}1+\frac{N(\mathrm{He}^{+})}{N(\mathrm{H}^{+})}\biggr{]}^{-1},}$ (7.98)

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

The electron temperature from the line/continuum ratio is

 $\boxed{\biggl{(}\frac{T_{\mathrm{e}}}{\mathrm{K}}\biggr{)}\approx\biggl{[}7.0% \times 10^{3}\biggl{(}\frac{\nu}{\mathrm{GHz}}\biggr{)}^{1.1}\,1.08^{-1}\,% \biggl{(}\frac{\Delta v}{\mathrm{km~{}s}^{-1}}\biggr{)}^{-1}\biggl{(}\frac{T_{% \mathrm{C}}}{T_{\mathrm{L}}}\biggr{)}\biggr{]}^{0.87}.}$ (7.99)

Quantization of angular momentum is given by

 $\boxed{L=n\hbar.}$ (7.100)

The angular momentum of a diatomic molecule is

 $\boxed{L=mr_{\mathrm{e}}^{2}\omega,}$ (7.104)

where

 $\boxed{m\equiv\biggl{(}\frac{m_{\mathrm{A}}m_{\mathrm{B}}}{m_{\mathrm{A}}+m_{% \mathrm{B}}}\biggr{)}}$ (7.105)

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

 $\boxed{E_{\mathrm{rot}}=\frac{J(J+1)\hbar^{2}}{2I},\qquad J=0,1,2,\ldots.}$ (7.107)

For a transition satisfying the selection rule

 $\boxed{\Delta J=\pm 1,}$ (7.108)

the line frequency is

 $\boxed{\nu=\frac{hJ}{4\pi^{2}mr_{\mathrm{e}}^{2}}.}$ (7.111)

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

 $\boxed{T_{\mathrm{min}}\approx\frac{\nu h(J+1)}{2k}.}$ (7.119)

The spontaneous emission coefficient is

 $\boxed{A_{\mathrm{UL}}=\frac{64\pi^{4}}{3hc^{3}}\nu_{\mathrm{UL}}^{3}|\mu_{% \mathrm{UL}}|^{2},}$ (7.131)

where

 $\boxed{|\mu_{\mathrm{J}\rightarrow\mathrm{J}-1}|^{2}=\frac{\mu^{2}J}{2J+1}}$ (7.132)

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

The critical density is

 $\boxed{n^{*}\approx\frac{A_{\mathrm{UL}}}{\sigma v},}$ (7.135)

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

 $\boxed{X_{\mathrm{CO}}=(2\pm 0.6)\times 10^{20}\mathrm{~{}cm}^{-2}\mathrm{~{}(% K~{}km~{}s}^{-1})^{-1}.}$ (7.140)

The Hi hyperfine line frequency is

 $\boxed{\nu_{10}=\frac{8}{3}g_{\mathrm{I}}\biggl{(}\frac{m_{\mathrm{e}}}{m_{% \mathrm{p}}}\biggr{)}\alpha^{2}(R_{M}c)\approx 1420.405751\mathrm{~{}MHz}.}$ (7.141)

The Hi hyperfine line emission coefficient is

 $\boxed{A_{10}\approx 2.85\times 10^{-15}\mathrm{~{}s}^{-1}.}$ (7.146)

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

 $\boxed{\frac{n_{1}}{n_{0}}\equiv\frac{g_{1}}{g_{0}}\exp\biggl{(}-\frac{h\nu_{1% 0}}{kT_{\mathrm{s}}}\biggr{)},}$ (7.148)

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

The Hi line opacity coefficient is

 $\boxed{\kappa(\nu)\approx\frac{3c^{2}}{32\pi}\frac{A_{10}n_{\mathrm{H}}}{\nu_{% 10}}\frac{h}{kT_{\mathrm{s}}}\phi(\nu).}$ (7.153)

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

 $\boxed{\eta_{\mathrm{H}}\equiv\int_{\mathrm{los}}n_{\mathrm{H}}(s)\,ds.}$ (7.154)

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

 $\boxed{\biggl{(}\frac{\eta_{\mathrm{H}}}{\mathrm{cm}^{-2}}\biggr{)}\approx 1.8% 2\times 10^{18}\int\biggl{[}\frac{T_{\mathrm{b}}(v)}{\mathrm{K}}\biggr{]}d% \biggl{(}\frac{v}{\mathrm{km~{}s}^{-1}}\biggr{)}.}$ (7.155)

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

 $\boxed{\biggl{(}\frac{M_{\mathrm{H}}}{M_{\odot}}\biggr{)}\approx 2.36\times 10% ^{5}\biggl{(}\frac{D}{\mathrm{Mpc}}\biggr{)}^{2}\int\biggl{[}\frac{S(v)}{% \mathrm{Jy}}\biggr{]}\biggl{(}\frac{dv}{\mathrm{km~{}s}^{-1}}\biggr{)}.}$ (7.166)

The total mass of a galaxy is

 $\boxed{\biggl{(}\frac{M}{M_{\odot}}\biggr{)}\approx 2.33\times 10^{5}\biggl{(}% \frac{v_{\mathrm{rot}}}{\mathrm{km~{}s}^{-1}}\biggr{)}^{2}\biggl{(}\frac{r}{% \mathrm{kpc}}\biggr{)}.}$ (7.172)