Appendix E Essential Equations

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

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

(2.2)

where $dP$ is the power received by a detector with projected area $(\cos \theta d\sigma )$ in the solid angle $d\mathrm{\Omega }$ and in the frequency range $\nu$ to $\nu +d\nu$. Likewise $I_{\lambda }$ is the brightness per unit wavelength:

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

(2.3)

These two quantities are related by

$$\boxed{\frac{I_{\lambda }}{I_{\nu }}=\left|\frac{d\nu }{d\lambda }\right|=\frac{c}{{\lambda }^2}=\frac{{\nu }^2}{c}.}$$

(2.5)

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

$$\boxed{S_{\nu }\equiv {\int }_{\mathrm{source}}{I}_{\nu }(\theta ,\varphi )\cos \theta d\mathrm{\Omega }.}$$

(2.9)

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

$$\boxed{S_{\nu }\approx {\int }_{\mathrm{source}}{I}_{\nu }(\theta ,\varphi )d\mathrm{\Omega }.}$$

(2.10)

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

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:

$$\boxed{\tau \equiv {\int }_{s_{\mathrm{out}}}^{s_{\mathrm{in}}}-\kappa (s^{\prime })d{s}^{\prime }.}$$

(2.23)

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

$$\boxed{j_{\nu }\equiv \frac{d{I}_{\nu }}{ds}.}$$

(2.26)

The equation of radiative transfer is

$$\boxed{\frac{d{I}_{\nu }}{ds}=-\kappa I_{\nu }+j_{\nu }.}$$

(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

$$\boxed{u_{\nu }=\frac{1}{c}\int I_{\nu }d\mathrm{\Omega }.}$$

(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

$$\boxed{B_{\nu }=\frac{2h{\nu }^3}{c^2}\frac{1}{\exp \left({\displaystyle \frac{h\nu }{kT}}\right)-1}.}$$

(2.86)

The total intensity of blackbody radiation is

$$\boxed{B(T)\equiv {\int }_0^{\mathrm{\infty }}{B}_{\nu }(T)d\nu =\frac{\sigma T^4}{\pi },}$$

(2.89)

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

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

(2.90)

The total energy density of blackbody radiation is

$$\boxed{u=\frac{4\sigma T^4}{c}=a{T}^4,}$$

(2.93)

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

The photon number density of blackbody radiation is

$$\boxed{\left(\frac{n_{\gamma }}{{\mathrm{cm}}^{-3}}\right)\approx 20.3{\left(\frac{T}{\mathrm{K}}\right)}^3.}$$

(2.100)

The mean photon energy of blackbody radiation is

$$\boxed{⟨E_{\gamma }⟩\approx \mathrm{\hspace{0.17em}2.70}kT.}$$

(2.101)

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

$$\boxed{\left(\frac{{\nu }_{\max }}{\mathrm{GHz}}\right)\approx 59\left(\frac{T}{\mathrm{K}}\right).}$$

(2.104)

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

$$\boxed{\left(\frac{{\lambda }_{\max }}{\mathrm{cm}}\right)\approx 0.29{\left(\frac{T}{\mathrm{K}}\right)}^{-1}.}$$

(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

$$\boxed{P_{\nu }=\frac{h\nu }{\exp \left({\displaystyle \frac{h\nu }{kT}}\right)-1}.}$$

(2.119)

The critical density needed to close the universe is

$$\boxed{{\rho }_{\mathrm{c}}=\frac{3H_0^2}{8\pi G}\approx 8.6\times {10}^{-30}\mathrm{g}{\mathrm{cm}}^{-3}.}$$

(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

$$\boxed{E_{\perp }=\frac{q\dot{v}\sin \theta }{r{c}^2}.}$$

(2.136)

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

$$\boxed{|\overrightarrow{S}|=\frac{c}{4\pi }{E}^2.}$$

(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

$$\boxed{⟨P⟩=\frac{{\pi }^2}{3c}{\left(\frac{I_{0}l}{\lambda }\right)}^2.}$$

(3.17)

Radiation resistance is defined by

$$\boxed{R\equiv \frac{2⟨P⟩}{I_0^2}.}$$

(3.25)

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

$$\boxed{⟨G⟩=1}$$

(3.32)

and

$$\boxed{{\int }_{\mathrm{sphere}}Gd\mathrm{\Omega }=4\pi .}$$

(3.33)

The beam solid angle is defined by

$$\boxed{{\mathrm{\Omega }}_{\mathrm{A}}\equiv \frac{4\pi }{G_{\max }}=\frac{1}{G_{\max }}{\int }_{4\pi }G(\theta ,\varphi )d\mathrm{\Omega }.}$$

(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

$$\boxed{⟨A_{\mathrm{e}}⟩=\frac{{\lambda }^2}{4\pi }.}$$

(3.41)

Reciprocity implies

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

(3.44)

Reciprocity and energy conservation imply

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

(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 $1\mathrm{K}{\mathrm{Jy}}^{-1}$.

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

$$\boxed{\frac{T_{\mathrm{A}}}{T_{\mathrm{b}}}=\frac{{\mathrm{\Omega }}_{\mathrm{s}}}{{\mathrm{\Omega }}_{\mathrm{A}}}.}$$

(3.56)

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

$$\boxed{{\mathrm{\Omega }}_{\mathrm{MB}}\equiv \frac{1}{G_{\max }}{\int }_{\mathrm{MB}}G(\theta ,\varphi )𝑑\mathrm{\Omega }}$$

(3.57)

and is used in the definition of main beam efficiency:

$$\boxed{{\eta }_{\mathrm{B}}\equiv \frac{{\mathrm{\Omega }}_{\mathrm{MB}}}{{\mathrm{\Omega }}_{\mathrm{A}}}.}$$

(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:

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

The power pattern of a uniformly illuminated linear aperture is

$$\boxed{P(\theta )\propto {\mathrm{sinc}}^2\left(\frac{\theta D}{\lambda }\right),}$$

(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

$$\boxed{f(l,m)\propto {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(u,v)e^{-i2\pi (lu+mv)}dudv,}$$

(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

$$\boxed{G\approx \frac{4\pi D_x{D}_y}{{\lambda }^2}{\mathrm{sinc}}^2\left(\frac{{\theta }_x{D}_x}{\lambda }\right){\mathrm{sinc}}^2\left(\frac{{\theta }_y{D}_y}{\lambda }\right).}$$

(3.107)

Aperture efficiency is defined by

$$\boxed{{\eta }_{\mathrm{A}}\equiv \frac{\max (A_{\mathrm{e}})}{A_{\mathrm{geom}}}.}$$

(3.111)

The beam solid angle of a Gaussian beam is

$$\boxed{{\mathrm{\Omega }}_{\mathrm{A}}=\left(\frac{\pi }{4\ln 2}\right){\theta }_{\mathrm{HPBW}}^2\approx 1.133{\theta }_{\mathrm{HPBW}}^2.}$$

(3.118)

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

$$\boxed{{\eta }_{\mathrm{s}}=\exp [-{\left(\frac{4\pi \sigma }{\lambda }\right)}^2].}$$

(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:

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

(3.150)

The ideal total-power radiometer equation is

$$\boxed{{\sigma }_T\approx T_{\mathrm{s}}{\left[\frac{1}{\mathrm{\Delta }\nu \tau }\right]}^{1/2}.}$$

(3.154)

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

$$\boxed{{\sigma }_T\approx T_{\mathrm{s}}{[\frac{1}{\mathrm{\Delta }\nu \tau }+{\left(\frac{\mathrm{\Delta }G}{G}\right)}^2]}^{1/2}.}$$

(3.158)

The Dicke-switching radiometer equation is

$$\boxed{{\sigma }_T\approx 2T_{\mathrm{s}}{\left[\frac{1}{\mathrm{\Delta }\nu \tau }\right]}^{1/2}.}$$

(3.162)

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

(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

$$\boxed{T_{\mathrm{r}}=\frac{T_{\mathrm{h}}-Y{T}_{\mathrm{c}}}{Y-1}.}$$

(3.168)

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

$$\boxed{{𝒱}_{\nu }=\int I_{\nu }(\widehat{s})\exp (-i2\pi \overrightarrow{b}\cdot \widehat{s}/\lambda )d\mathrm{\Omega }.}$$

(3.186)

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

$$\boxed{\mathrm{\Delta }\theta \mathrm{\Delta }\nu \ll {\theta }_{\mathrm{s}}\nu .}$$

(3.192)

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

$$\boxed{\mathrm{\Delta }\theta \mathrm{\Delta }t\ll \frac{{\theta }_{\mathrm{s}}P}{2\pi }\approx {\theta }_{\mathrm{s}}\cdot 1.37\times {10}^4\mathrm{s}.}$$

(3.194)

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

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

(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 ${𝒱}_{\nu }(u,v)$:

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

(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

$$\boxed{{\sigma }_S=\frac{2k{T}_{\mathrm{s}}}{A_{\mathrm{e}}{[N(N-1)\mathrm{\Delta }\nu \tau ]}^{1/2}}.}$$

(3.203)

The brightness sensitivity (K) corresponding to a point source sensitivity ${\sigma }_S$ and a beam solid angle ${\mathrm{\Omega }}_{\mathrm{A}}$ is

$$\boxed{{\sigma }_T=\left(\frac{{\sigma }_S}{{\mathrm{\Omega }}_{\mathrm{A}}}\right)\frac{{\lambda }^2}{2k},}$$

(3.204)

where ${\mathrm{\Omega }}_{\mathrm{A}}=\pi {\theta }_{\mathrm{HPBW}}^2/(4\ln 2)\approx 1.133{\theta }_0^2$ for a Gaussian beam of HPBW ${\theta }_{\mathrm{HPBW}}$.

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

$$\boxed{f(v)=\frac{4v^2}{\sqrt{\pi }}{\left(\frac{m}{2kT}\right)}^{3/2}\exp (-\frac{m{v}^2}{2kT}).}$$

(4.34)

The free–free emission coefficient is

$$\boxed{j_{\nu }=\frac{{\pi }^2{Z}^2{e}^6{n}_{\mathrm{e}}{n}_{\mathrm{i}}}{4c^3{m}_{\mathrm{e}}^2}{\left(\frac{2m_{\mathrm{e}}}{\pi kT}\right)}^{1/2}\ln \left(\frac{b_{\max }}{b_{\min }}\right),}$$

(4.39)

where

$$\boxed{b_{\min }\approx \frac{Z{e}^2}{m_{\mathrm{e}}{v}^2}.}$$

(4.43)

The free–free absorption coefficient is

$$\boxed{\kappa =\frac{1}{{\nu }^2{T}^{3/2}}\left[\frac{Z^2{e}^6}{c}{n}_{\mathrm{e}}{n}_{\mathrm{i}}\frac{1}{\sqrt{2\pi {(m_{\mathrm{e}}k)}^3}}\right]\frac{{\pi }^2}{4}\ln \left(\frac{b_{\max }}{b_{\min }}\right).}$$

(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

$$\boxed{\frac{\mathrm{EM}}{\mathrm{pc}{\mathrm{cm}}^{-6}}\equiv {\int }_{\mathrm{los}}{\left(\frac{n_{\mathrm{e}}}{{\mathrm{cm}}^{-3}}\right)}^{2}d\left(\frac{s}{\mathrm{pc}}\right).}$$

(4.57)

The free–free optical depth of a plasma is

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

(4.60)

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

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

(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

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

(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

$$\boxed{\left(\frac{{\nu }_{\mathrm{G}}}{\mathrm{MHz}}\right)=2.8\left(\frac{B}{\mathrm{gauss}}\right).}$$

(5.7)

The Lorentz transform is

$$\boxed{x=\gamma (x^{\prime }+v{t}^{\prime }),y=y^{\prime },z=z^{\prime },t=\gamma (t^{\prime }+\beta x^{\prime }/c),}$$

(5.12)

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

(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 $(\mathrm{\Delta }{x}^{\prime },\mathrm{\Delta }{y}^{\prime },\mathrm{\Delta }{z}^{\prime },\mathrm{\Delta }{t}^{\prime })$ and $(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z,\mathrm{\Delta }t)$ are the coordinate differences between two events, the differential form of the (linear) Lorentz transform is

$$\boxed{\mathrm{\Delta }x=\gamma (\mathrm{\Delta }{x}^{\prime }+v\mathrm{\Delta }{t}^{\prime }),\mathrm{\Delta }y=\mathrm{\Delta }{y}^{\prime },\mathrm{\Delta }z=\mathrm{\Delta }{z}^{\prime },\mathrm{\Delta }t=\gamma (\mathrm{\Delta }{t}^{\prime }+\beta \mathrm{\Delta }{x}^{\prime }/c),}$$

(5.16)

$$\boxed{\mathrm{\Delta }{x}^{\prime }=\gamma (\mathrm{\Delta }x-v\mathrm{\Delta }t),\mathrm{\Delta }{y}^{\prime }=\mathrm{\Delta }y,\mathrm{\Delta }{z}^{\prime }=\mathrm{\Delta }z,\mathrm{\Delta }{t}^{\prime }=\gamma (\mathrm{\Delta }t-\beta \mathrm{\Delta }x/c).}$$

(5.17)

The Thomson cross section of an electron is defined by

$$\boxed{{\sigma }_{\mathrm{T}}\equiv \frac{8\pi }{3}{\left(\frac{e^2}{m_{\mathrm{e}}{c}^2}\right)}^2.}$$

(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

$$\boxed{⟨P⟩=\frac{4}{3}{\sigma }_{\mathrm{T}}{\beta }^2{\gamma }^{2}c{U}_B.}$$

(5.42)

The synchrotron spectrum of a single electron is

$$\boxed{P(\nu )=\frac{\sqrt{3}{e}^{3}B\sin \alpha }{m_{\mathrm{e}}{c}^2}\left(\frac{\nu }{{\nu }_{\mathrm{c}}}\right){\int }_{\nu /{\nu }_{\mathrm{c}}}^{\mathrm{\infty }}{K}_{5/3}(\eta )d\eta ,}$$

(5.66)

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

$$\boxed{{\nu }_{\mathrm{c}}=\frac{3}{2}{\gamma }^2{\nu }_{\mathrm{G}}\sin \alpha \approx {\gamma }^2{\nu }_{\mathrm{G}}\propto E^2{B}_{\perp }.}$$

(5.67)

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

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

(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

$$\boxed{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)\approx 1.18\times {10}^6{\left(\frac{\nu }{\mathrm{Hz}}\right)}^{1/2}{\left(\frac{B}{\mathrm{gauss}}\right)}^{-1/2}.}$$

(5.85)

At a sufficiently low frequency $\nu$,

$$\boxed{S_{\nu }\propto {\nu }^{-5/2}}$$

(5.89)

and

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

(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):

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

(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

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

(5.110)

The synchrotron lifetime is approximately

$$\boxed{\tau \approx c_{12}{B}_{\perp }^{-3/2},}$$

(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 ${\nu }_{\min }={10}^7$ Hz and ${\nu }_{\max }={10}^{11}$ Hz are commonly used.

The Eddington limit for luminosity is

$$\boxed{\left(\frac{L_{\mathrm{E}}}{L_{\odot }}\right)\approx 3.3\times {10}^4\left(\frac{M}{M_{\odot }}\right).}$$

(5.117)

The nonrelativistic Thomson-scattering power is

$$\boxed{P={\sigma }_{\mathrm{T}}c{U}_{\mathrm{rad}}.}$$

(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 $⟨\nu ⟩$ of upscattered photons having initial frequency ${\nu }_0$ is

$$\boxed{\frac{⟨\nu ⟩}{{\nu }_0}=\frac{4}{3}{\gamma }^2.}$$

(5.160)

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

$$\boxed{T_{\max }\sim {10}^{12}\mathrm{K}.}$$

(5.163)

The apparent transverse velocity of a moving source component is

$$\boxed{{\beta }_{\perp }(\mathrm{apparent})=\frac{\beta \sin \theta }{1-\beta \cos \theta }.}$$

(5.167)

For any $\beta$ the angle ${\theta }_{\mathrm{m}}$ that maximizes ${\beta }_{\perp }(\mathrm{apparent})$ 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

$$\boxed{\max [{\beta }_{\perp }(\mathrm{apparent})]=\beta \gamma .}$$

(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

(5.183)

Thermal and nonthermal radio luminosities of star-forming galaxies are

$$\boxed{\left(\frac{L_{\mathrm{T}}}{\mathrm{W}{\mathrm{Hz}}^{-1}}\right)\approx 5.5\times {10}^{20}{\left(\frac{\nu }{\mathrm{GHz}}\right)}^{-0.1}\left[\frac{\mathrm{SFR}(M>5M_{\odot })}{M_{\odot }{\mathrm{yr}}^{-1}}\right]}$$

(5.184)

and

$$\boxed{\left(\frac{L_{\mathrm{NT}}}{\mathrm{W}{\mathrm{Hz}}^{-1}}\right)\approx 5.3\times {10}^{21}{\left(\frac{\nu }{\mathrm{GHz}}\right)}^{-0.8}\left[\frac{\mathrm{SFR}(M>5M_{\odot })}{M_{\odot }{\mathrm{yr}}^{-1}}\right].}$$

(5.185)

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

$$\boxed{\rho >\frac{3\pi }{G{P}^2}.}$$

(6.5)

A rotating magnetic dipole radiates power

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

(6.10)

The spin-down luminosity of a pulsar is

$$\boxed{-\dot{E}\equiv -\frac{d{E}_{\mathrm{rot}}}{dt}=\frac{-4{\pi }^{2}I\dot{P}}{P^3}.}$$

(6.20)

The minimum magnetic field strength of a pulsar is

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

(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

$$\boxed{\mu ={[1-{\left(\frac{{\nu }_{\mathrm{p}}}{\nu }\right)}^2]}^{1/2},}$$

(6.39)

where ${\nu }_{\mathrm{p}}$ is the plasma frequency

$$\boxed{{\nu }_{\mathrm{p}}={\left(\frac{e^2{n}_{\mathrm{e}}}{\pi m_{\mathrm{e}}}\right)}^{1/2}\approx 8.97\mathrm{kHz}{\left(\frac{n_{\mathrm{e}}}{{\mathrm{cm}}^{-3}}\right)}^{1/2}.}$$

(6.40)

The group velocity of pulses is

$$\boxed{v_{\mathrm{g}}\approx c(1-\frac{{\nu }_{\mathrm{p}}^2}{2{\nu }^2}).}$$

(6.42)

The dispersion delay of a pulsar is

$$\boxed{\left(\frac{t}{\sec }\right)\approx 4.149\times {10}^3\left(\frac{\mathrm{DM}}{\mathrm{pc}{\mathrm{cm}}^{-3}}\right){\left(\frac{\nu }{\mathrm{MHz}}\right)}^{-2},}$$

(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

$$\boxed{a_n=\frac{n^2{\mathrm{\hslash }}^2}{m_{\mathrm{e}}{e}^2}\approx 0.53\times {10}^{-8}\mathrm{cm}\cdot n^2.}$$

(7.6)

The frequency of a recombination line is

$$\boxed{\nu =R_{M}c[\frac{1}{n^2}-\frac{1}{{(n+\mathrm{\Delta }n)}^2}],\mathrm{where}\mathit{\hspace{1em}\hspace{1em}}{R}_M\equiv R_{\mathrm{\infty }}{(1+\frac{m_{\mathrm{e}}}{M})}^{-1}.}$$

(7.12)

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

$$\boxed{\frac{\mathrm{\Delta }\nu }{\nu }\approx \frac{3}{n}.}$$

(7.15)

The spontaneous emission rate is

$$\boxed{A_{n+1,n}\approx \frac{64{\pi }^6{m}_{\mathrm{e}}{e}^{10}}{3c^3{h}^6{n}^5}\approx 5.3\times {10}^9\left(\frac{1}{n^5}\right){\mathrm{s}}^{-1}.}$$

(7.23)

The normalized Gaussian line profile is

$$\boxed{\varphi (\nu )=\frac{c}{{\nu }_0}{\left(\frac{M}{2\pi kT}\right)}^{1/2}\exp [-\frac{M{c}^2}{2kT}\frac{{(\nu -{\nu }_0)}^2}{{\nu }_0^2}],}$$

(7.32)

where

$$\boxed{\mathrm{\Delta }\nu ={\left(\frac{8\ln 2k}{c^2}\right)}^{1/2}{\left(\frac{T}{M}\right)}^{1/2}{\nu }_0}$$

(7.35)

and

$$\boxed{\varphi ({\nu }_0)={\left(\frac{\ln 2}{\pi }\right)}^{1/2}\frac{2}{\mathrm{\Delta }\nu }.}$$

(7.37)

Rate balance is given by

$$\boxed{n_{\mathrm{U}}{A}_{\mathrm{UL}}+n_{\mathrm{U}}{B}_{\mathrm{UL}}\overline{u}=n_{\mathrm{L}}{B}_{\mathrm{LU}}\overline{u}.}$$

(7.42)

The detailed balance equations connecting Einstein coefficients are

	$$\frac{g_{\mathrm{L}}}{g_{\mathrm{U}}}\frac{B_{\mathrm{LU}}}{B_{\mathrm{UL}}}=1,$$		(7.50)
	$$\frac{A_{\mathrm{UL}}}{B_{\mathrm{UL}}}=\frac{8\pi h{\nu }_0^3}{c^3}.$$		(7.51)

The spectral line radiative transfer equation is

$$\boxed{\frac{d{I}_{\nu }}{ds}=-\left(\frac{h{\nu }_0}{c}\right)(n_{\mathrm{L}}{B}_{\mathrm{LU}}-n_{\mathrm{U}}{B}_{\mathrm{UL}})\varphi (\nu )I_{\nu }+\left(\frac{h{\nu }_0}{4\pi }\right)n_{\mathrm{U}}{A}_{\mathrm{UL}}\varphi (\nu ).}$$

(7.57)

The Boltzmann equation for a two-level system is

$$\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}=\frac{g_{\mathrm{U}}}{g_{\mathrm{L}}}\exp (-\frac{h{\nu }_0}{kT}).}$$

(7.64)

The line opacity coefficient in LTE is

$$\boxed{\kappa =\frac{c^2}{8\pi {\nu }_0^2}\frac{g_{\mathrm{U}}}{g_{\mathrm{L}}}{n}_{\mathrm{L}}{A}_{\mathrm{UL}}[1-\exp (-\frac{h{\nu }_0}{kT})]\varphi (\nu ).}$$

(7.67)

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

$$\boxed{\frac{n_{\mathrm{U}}}{n_{\mathrm{L}}}\equiv \frac{g_{\mathrm{U}}}{g_{\mathrm{L}}}\exp (-\frac{h{\nu }_0}{k{T}_{\mathrm{x}}}).}$$

(7.70)

The recombination-line opacity coefficient is

$$\boxed{\kappa ({\nu }_0)\approx \left(\frac{n_{\mathrm{e}}^2}{T_{\mathrm{e}}^{5/2}\mathrm{\Delta }\nu }\right)\left(\frac{4\pi e^{6}h}{3m_{\mathrm{e}}^{3/2}{k}^{5/2}c}\right){\left(\frac{\ln 2}{2}\right)}^{1/2}}$$

(7.94)

and the recombination line opacity is

$$\boxed{{\tau }_{\mathrm{L}}\approx 1.92\times {10}^3{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)}^{-5/2}\left(\frac{\mathrm{EM}}{\mathrm{pc}{\mathrm{cm}}^{-6}}\right){\left(\frac{\mathrm{\Delta }\nu }{\mathrm{kHz}}\right)}^{-1}.}$$

(7.96)

The recombination line brightness temperature is given by

$$\boxed{T_{\mathrm{L}}\approx T_{\mathrm{e}}{\tau }_{\mathrm{L}}\approx 1.92\times {10}^3{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)}^{-3/2}\left(\frac{\mathrm{EM}}{\mathrm{pc}{\mathrm{cm}}^{-6}}\right){\left(\frac{\mathrm{\Delta }\nu }{\mathrm{kHz}}\right)}^{-1}.}$$

(7.97)

The recombination line/continuum ratio is

$$\boxed{\frac{T_{\mathrm{L}}}{T_{\mathrm{C}}}\approx 7.0\times {10}^3{\left(\frac{\mathrm{\Delta }v}{\mathrm{km}{\mathrm{s}}^{-1}}\right)}^{-1}{\left(\frac{\nu }{\mathrm{GHz}}\right)}^{1.1}{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)}^{-1.15}{[1+\frac{N({\mathrm{He}}^{+})}{N({\mathrm{H}}^{+})}]}^{-1},}$$

(7.98)

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

The electron temperature from the line/continuum ratio is

$$\boxed{\left(\frac{T_{\mathrm{e}}}{\mathrm{K}}\right)\approx {[7.0\times {10}^3{\left(\frac{\nu }{\mathrm{GHz}}\right)}^{1.1}{\mathrm{\hspace{0.17em}1.08}}^{-1}{\left(\frac{\mathrm{\Delta }v}{\mathrm{km}{\mathrm{s}}^{-1}}\right)}^{-1}\left(\frac{T_{\mathrm{C}}}{T_{\mathrm{L}}}\right)]}^{0.87}.}$$

(7.99)

Quantization of angular momentum is given by

$$\boxed{L=n\mathrm{\hslash }.}$$

(7.100)

The angular momentum of a diatomic molecule is

$$\boxed{L=m{r}_{\mathrm{e}}^2\omega ,}$$

(7.104)

where

$$\boxed{m\equiv \left(\frac{m_{\mathrm{A}}{m}_{\mathrm{B}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right)}$$

(7.105)

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