# F.1 Physical Constants

Symbol Name Value Units
${a}$ $\mathrm{radiation~{}constant}$ $7.56577\times 10^{-15}$ $\mathrm{erg~{}cm}^{-3}\mathrm{~{}K}^{-4}$
${a_{0}}$ $\mathrm{Bohr~{}radius}$ $5.29177\times 10^{-9}$ $\mathrm{cm}$
${c}$ $\mathrm{speed~{}of~{}light~{}in~{}vacuum}$ $2.99792\times 10^{10}$ $\mathrm{cm~{}s}^{-1}$
${e}$ $\mathrm{electron~{}charge~{}(magnitude)}$ $4.80325\times 10^{-10}$ $\mathrm{statcoulomb~{}(or~{}esu)}$
$\mathrm{eV}$ $\mathrm{electron~{}volt}$ $1.60218\times 10^{-12}$ $\mathrm{erg}$
${G}$ $\mathrm{gravitational~{}constant}$ $6.67428\times 10^{-8}$ $\mathrm{dyne~{}cm}^{2}\mathrm{~{}g}^{-2}$
${h}$ Planck’s constant $6.62607\times 10^{-27}$ $\mathrm{erg~{}s}$
${k}$ Boltzmann’s constant $1.38065\times 10^{-16}$ $\mathrm{erg~{}K}^{-1}$
${m_{\mathrm{e}}}$ $\mathrm{electron~{}mass}$ $9.10938\times 10^{-28}$ $\mathrm{g}$
${m_{\mathrm{p}}}$ $\mathrm{proton~{}mass}$ $1.67262\times 10^{-24}$ $\mathrm{g}$
${\mu_{\mathrm{B}}}$ $\mathrm{Bohr~{}magneton}$ $9.27401\times 10^{-21}$ $\mathrm{erg~{}gauss}^{-1}$
${R_{\infty}}$ $\mathrm{Rydberg~{}constant}$ $1.09737\times 10^{5}$ $\mathrm{cm}^{-1}$
${R_{\infty}c}$ $\mathrm{Rydberg~{}frequency}$ $3.28984\times 10^{15}$ $\mathrm{s}^{-1}$
${\sigma}$ Stefan–Boltzmann constant $5.67040\times 10^{-5}$ $\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}\,\mathrm{sr}^{-1}\,\mathrm{K}^% {-4}$
${\sigma_{\mathrm{T}}}$ $\mathrm{Thomson~{}cross~{}section}$ $6.65245\times 10^{-25}$ $\mathrm{cm}^{2}$
${u}$ $\mathrm{atomic~{}mass~{}unit}$ $1.66054\times 10^{-24}$ $\mathrm{g}$

# F.2 Astronomical Constants

Symbol Name Value Units
$\mathrm{au}$ $\mathrm{astronomical~{}unit}$ $1.49598\times 10^{13}$ $\mathrm{cm}$
$\approx 500$ $\mathrm{light~{}seconds}$
${H_{0}}$ $\mathrm{Hubble~{}constant}$ $67.8$ $\mathrm{km~{}s}^{-1}\mathrm{~{}Mpc}^{-1}$
$\mathrm{kpc}$ $\mathrm{kiloparsec}$ $10^{3}$ $\mathrm{pc}$
${L_{\odot}}$ $\mathrm{solar~{}luminosity}$ $3.826\hphantom{1~{}}\times 10^{33}$ $\mathrm{erg~{}s}^{-1}$
$\mathrm{ly}$ $\mathrm{light~{}year}$ $9.4606\hphantom{~{}}\times 10^{17}$ $\mathrm{cm}$
${M_{\odot}}$ $\mathrm{solar~{}mass}$ $1.989\hphantom{1~{}}\times 10^{33}$ $\mathrm{g}$
$\mathrm{Mpc}$ $\mathrm{megaparsec}$ $10^{6}$ $\mathrm{pc}$
$\mathrm{pc}$ $\mathrm{parsec}$ $3.0856\hphantom{~{}}\times 10^{18}$ $\mathrm{cm}$
${R_{\odot}}$ $\mathrm{solar~{}radius}$ $6.9558\hphantom{~{}}\times 10^{10}$ $\mathrm{cm}$
$\mathrm{yr}$ $\mathrm{tropical~{}year}$ $3.1557\hphantom{~{}}\times 10^{7}$ $\mathrm{s}$
$\approx 10^{7.5}$ $\mathrm{s}$

# F.3 MKS (SI) and Gaussian CGS Units

The International System of Units, or Système Internationale d’Unités (SI), was adopted in 1960 to succeed the MKS (meter, kilogram, second) system of units dating from 1799 and the CGS (centimeter, gram, second) system introduced in 1874. The various CGS units for the electric and magnetic fields are inconveniently large or small and have different dimensions, so the practical units ampere for electric current, ohm for resistance, and volt for electromotive force were introduced in 1893. SI is the direct descendant of the MKS system, with base units meter, kilogram, and second, plus ampere for current and kelvin for temperature. The kilogram is defined by the mass of the international prototype of the kilogram. The other base units are defined by laboratory measurements: the second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom, the meter is the distance traveled by light in vacuum during a time interval of 1$/$299,792,458 of a second, the ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to $2\times 10^{-7}$ newton per meter of length, and the kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Thus the SI speed of light in a vacuum is exactly 299,792,458 m s${}^{-1}$ by definition. For the purposes of this book, the terms MKS and SI are interchangeable.

$\mathrm{Type}$ $\mathrm{MKS~{}unit}$ $\mathrm{CGS~{}unit}$ $\mathrm{conversion}$
$\mathrm{length}$ $\mathrm{m}$ $\mathrm{cm}$ $10^{2}\mathrm{~{}cm}=1\mathrm{~{}m}$
$2.54\mathrm{~{}cm}\equiv 1\mathrm{~{}inch~{}(exactly)}$
$\mathrm{mass}$ $\mathrm{kg}$ $\mathrm{g}$ $10^{3}\mathrm{~{}g}=1\mathrm{~{}kg}$
$\mathrm{time}$ $\mathrm{s}$ $\mathrm{s}$
$\mathrm{energy}$ $\mathrm{joule}$ $\mathrm{erg}$ $10^{7}\mathrm{~{}erg}=1\mathrm{~{}joule}=1\mathrm{~{}kg~{}m}^{2}\mathrm{~{}s}^% {-2}$
$\mathrm{force}$ $\mathrm{newton}$ $\mathrm{dyne}$ $10^{5}\mathrm{~{}dyne}=1\mathrm{~{}newton}=1\mathrm{~{}kg~{}m~{}s}^{-2}$
$\mathrm{frequency}$ $\mathrm{Hz}$ $\mathrm{Hz}$ $1\mathrm{~{}Hz}=1\mathrm{~{}s}^{-1}$
$\mathrm{power}$ $\mathrm{W}$ $\mathrm{erg~{}s}^{-1}$ $10^{7}\mathrm{~{}erg~{}s}^{-1}=1\mathrm{~{}W}=1\mathrm{~{}kg~{}m}^{2}\mathrm{~% {}s}^{-3}$
$\mathrm{temperature}$ $\mathrm{kelvin}$ $\mathrm{kelvin}$
$\mathrm{charge}$ $\mathrm{coulomb}$ $\mathrm{statcoulomb}$ $3\times 10^{9}\,\mathrm{statcoulomb}\leftrightarrow 1\,\mathrm{coulomb}$
$(1\mathrm{~{}statcoulomb}=1\mathrm{~{}esu})$
$\mathrm{current}$ $\mathrm{ampere}$ $\mathrm{statampere}$ $3\times 10^{9}\mathrm{~{}statamp}\leftrightarrow 1\mathrm{~{}amp}$
$(1\mathrm{~{}amp}=1\mathrm{~{}coulomb~{}s}^{-1})$
$\mathrm{electric~{}field}$ ${\mathrm{volt~{}m}^{-1}}$ ${\mathrm{statvolt~{}cm}^{-1}}$ $(1/3)\times 10^{-4}\mathrm{~{}statvolt~{}cm}^{-1}\leftrightarrow$
$1\mathrm{~{}volt~{}m}^{-1}$
$\mathrm{magnetic~{}field}$ $\mathrm{tesla}$ $\mathrm{gauss}$ $10^{4}\mathrm{~{}gauss}\leftrightarrow 1\mathrm{~{}tesla}$
$\mathrm{resistance}$ $\mathrm{ohm}$ ${\mathrm{sec~{}cm}^{-1}}$ $(1/9)\times 10^{-11}\mathrm{~{}s~{}cm}^{-1}\leftrightarrow 1\mathrm{~{}ohm}$
$\mathrm{voltage}$ $\mathrm{volt}$ $\mathrm{statvolt}$ $(1/3)\times 10^{-2}\mathrm{~{}statvolt}\leftrightarrow 1\mathrm{~{}volt}$
$(1\mathrm{~{}volt}=1\mathrm{~{}joule~{}coulomb}^{-1})$

Engineers and most physicists prefer MKS units, so radio astronomers use MKS units to describe their equipment and the results of their observations. Most astrophysicists prefer Gaussian CGS units to describe astrophysical processes and astronomical sources. Thus both systems of units appear in the literature. Be careful when converting between them because astrophysical quantities can be so large or small that conversion mistakes are not intuitively obvious—everyday experience doesn’t show that the mass of the Sun is $2\times 10^{30}$ kg and not $2\times 10^{30}$ g.

The MKS and CGS units of length, mass, time, and temperature have the same dimensions, so it is possible to state that $10^{2}\mathrm{~{}cm}=1\mathrm{~{}m}$, for example. However, the MKS and CGS units for electromagnetic quantities such as charge, current, voltage, etc. are trickier because they do not have the same dimensions. The Gaussian CGS unit of charge (statcoulomb) is defined so that Coulomb’s law for the magnitude of the electrostatic force $F$ between two charges $q_{1}$ and $q_{2}$ separated by distance $r$ is simply

 $F=\frac{q_{1}q_{2}}{r^{2}}.$ (F.1)

The MKS unit of charge (coulomb) is defined in terms of currents and Ampere’s law, and Coulomb’s law becomes

 $F=\frac{c^{2}}{10^{7}}\frac{q_{1}q_{2}}{r^{2}},$ (F.2)

where $c=2.99729458\times 10^{8}\mathrm{~{}m~{}s}^{-1}$ is the vacuum speed of light in MKS units. The MKS form of Coulomb’s law is usually written as

 $F=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}},$ (F.3)

where $\epsilon_{0}$ is called the permittivity of free space and

 $\epsilon_{0}=\frac{10^{7}}{4\pi c^{2}}\approx 8.854\times 10^{-12}$ (F.4)

has units $\mathrm{m}^{-2}\mathrm{~{}s}^{2}=\mathrm{C}^{2}\mathrm{~{}N}^{-1}\mathrm{~{}m}% ^{-2}$. The corresponding MKS magnetic quantity is the permeability of free space

 $\mu_{0}\equiv 4\pi\times 10^{-7}\mathrm{~{}newton~{}ampere}^{-2}=(\epsilon_{0}% c^{2})^{-1}.$ (F.5)

Comparing forces in Equations F.1 and F.2 shows that

 $1\mathrm{~{}coulomb}=10c\mathrm{~{}statcoulomb}$ (F.6)

Consequently statcoul and coulomb have different dimensions, and the numerical conversion factor factor relating them is

 $1\mathrm{~{}coulomb}\leftrightarrow 2.99729458\times 10^{9}\mathrm{~{}% statcoulomb},$ (F.7)

where the symbol $\leftrightarrow$ indicates numerical conversion relating given amounts of dimensionally incompatible quantities. For example, given that the charge of an electron is $e=-4.80325\times 10^{-10}\mathrm{~{}statcoulomb}$ in CGS units, Equation F.7 can be used to show that the electron charge is $e=-1.60253\times 10^{-19}\mathrm{~{}coulomb}$ in MKS units. The charge conversion factor is simplified to $3\times 10^{9}$ in the table above, where $3$ is the conventional dimensionless shorthand for the exact and dimensionally correct $2.99729458\mathrm{~{}(m~{}s}^{-1})=10^{-8}c$; likewise, $(1/3)$ for $1/[2.99729458\mathrm{~{}(m~{}s}^{-1})]$ and $1/9$ for $1/[2.99729458\mathrm{~{}(m~{}s}^{-1})]^{2}$.

Entire equations can also be converted between CGS and MKS if the dimensions are carefully preserved. For example, the CGS form of Larmor’s formula (Equation 2.143) for the power $P$ radiated by a charge $q$ under acceleration $\dot{v}$ is

 $P=\frac{2}{3}\frac{q^{2}\dot{v}^{2}}{c^{3}},$

so

 $\biggl{(}\frac{P}{\mathrm{erg~{}s}^{-1}}\biggr{)}=\frac{2}{3}\biggl{(}\frac{q}% {\mathrm{statcoul}}\biggr{)}^{2}\biggl{(}\frac{\dot{v}}{\mathrm{cm~{}s}^{-1}}% \biggr{)}^{2}\biggl{(}\frac{c}{\mathrm{cm~{}s}^{-1}}\biggr{)}^{-3}.$

The numerical conversion

 $3\times 10^{9}\mathrm{~{}statcoulomb}\leftrightarrow 1\mathrm{~{}coulomb}$

implies the dimensionally correct equality

 $2.99729458\mathrm{~{}m~{}s}^{-1}\times 10^{9}\mathrm{~{}statcoulomb}=10c% \mathrm{~{}statcoulomb}=1\mathrm{~{}coulomb},$

so

 $\displaystyle\biggl{(}\frac{P}{10^{-7}\mathrm{~{}W}}\biggr{)}$ $\displaystyle=\frac{2}{3}\biggl{(}\frac{q}{(10c)^{-1}\mathrm{~{}coul}}\biggr{)% }^{2}\biggl{(}\frac{\dot{v}}{10^{-2}\mathrm{~{}m~{}s}^{-1}}\biggr{)}^{2}\biggl% {(}\frac{c}{10^{-2}\mathrm{~{}m~{}s}^{-1}}\biggr{)}^{-3},$ $\displaystyle 10^{7}\biggl{(}\frac{P}{\mathrm{W}}\biggr{)}$ $\displaystyle=\frac{2}{3}(10c)^{2}\biggl{(}\frac{q}{\mathrm{coul}}\biggr{)}^{2% }(10^{4})\biggl{(}\frac{\dot{v}}{\mathrm{m~{}s}^{-1}}\biggr{)}^{2}(10^{-6})% \biggl{(}\frac{c}{\mathrm{m~{}s}^{-1}}\biggr{)}^{-3},$ $\displaystyle\biggl{(}\frac{P}{\mathrm{W}}\biggr{)}$ $\displaystyle=\frac{2}{3}\biggl{(}\frac{c^{2}}{10^{7}}\biggr{)}\biggl{(}\frac{% q}{\mathrm{coul}}\biggr{)}^{2}\biggl{(}\frac{\dot{v}}{\mathrm{m~{}s}^{-1}}% \biggr{)}^{2}\biggl{(}\frac{c}{\mathrm{m~{}s}^{-1}}\biggr{)}^{-3}$ $\displaystyle=\frac{2}{3}\biggl{(}\frac{1}{4\pi\epsilon_{0}}\biggr{)}\biggl{(}% \frac{q}{\mathrm{coul}}\biggr{)}^{2}\biggl{(}\frac{\dot{v}}{\mathrm{m~{}s}^{-1% }}\biggr{)}^{2}\biggl{(}\frac{c}{\mathrm{m~{}s}^{-1}}\biggr{)}^{-3}.$

Thus Larmor’s equation in MKS units is

 $P=\frac{1}{6\pi\epsilon_{0}}\frac{q^{2}\dot{v}^{2}}{c^{3}}.$

MKS units are used in practical electromagnetics—voltmeters read MKS volts, ammeters read MKS amperes, a 9 volt battery delivers 9 MKS volts, etc. An important advantage of Gaussian CGS units is that the Lorentz force law becomes

 $\vec{F}=q(\vec{E}+\vec{\beta}\times\vec{B}),$ (F.8)

where $\vec{\beta}\equiv\vec{v}/c$ is dimensionless, so $\vec{E}$ and $\vec{B}$ have the same dimensions (base units $\mathrm{cm}^{-1/2}\mathrm{~{}g}^{1/2}\mathrm{~{}s}^{-1}$) in Gaussian CGS units, and $1\mathrm{~{}statvolt~{}cm}^{-1}=1\mathrm{~{}gauss}$, emphasizing the relativistic result that electric and magnetic fields are equivalent fields viewed in different reference frames. This equivalence is not apparent in the MKS version of Equation F.8:

 $\vec{F}=q(\vec{E}+\vec{v}\times\vec{B}),$ (F.9)

which drops the speed of light and also shows that $\vec{E}$ and $\vec{B}$ have different dimensions in the MKS system. Thus the equality

 $|\vec{E}|=|\vec{B}|$ (F.10)

makes sense in the CGS system but not in MKS.

For a detailed explanation of the common systems of units, see the appendix in Jackson [54].

# F.4 Other Constants and Units

$\mathrm{Symbol}$ $\mathrm{Value}$ $-2.5\log[S(\mathrm{Jy})]+8.90=-2.5\log_{10}[S/(3631\mathrm{~{}Jy})]$ $1/60\mathrm{~{}deg}$ $1/60\mathrm{~{}arcmin}$ $10^{-10}\mathrm{~{}m}$ $1\mathrm{~{}debye}\equiv 10^{-18}\mathrm{~{}statcoulomb~{}cm}$ $10\log_{10}(P_{1}/P_{2})$ $(\pi/180)\mathrm{~{}rad}$ $2.71828\ldots$ $10^{9}\mathrm{~{}Hz}$ $10^{-26}\mathrm{~{}W~{}m}^{-2}\mathrm{~{}Hz}^{-1}=10^{-23}\mathrm{~{}erg~{}s}^% {-1}\mathrm{~{}cm}^{-2}\mathrm{~{}Hz}^{-1}$ $10^{6}\mathrm{~{}Hz}$ $10^{-3}\mathrm{~{}Jy}$ $10^{-6}\mathrm{~{}m}$ $10^{-6}\mathrm{~{}Jy}$ $3.14159\ldots$ $\mathrm{angle~{}subtending~{}unit~{}arc~{}length~{}on~{}a~{}unit~{}circle}$ $=180/\pi\approx 57.296\mathrm{~{}deg,}=640000/\pi\approx 206265\mathrm{~{}arcsec}$ $\mathrm{solid~{}angle~{}subtending~{}unit~{}area~{}on~{}a~{}unit~{}sphere}$ $10^{12}\mathrm{~{}Hz}$

# F.5 Radar and Waveguide Frequency Bands

$\mathrm{Band}$ $\mathrm{Frequency}$ $\mathrm{name}$ $\mathrm{P}$ $0.25<\nu<0.50$ $\mathrm{Previous}$ $\mathrm{L}$ $1<\nu<2$ $\mathrm{Long~{}(wavelength)}$ $\mathrm{S}$ $2<\nu<4$ $\mathrm{Short~{}(wavelength)}$ $\mathrm{C}$ $4<\nu<8$ $\mathrm{Compromise~{}(between~{}S~{}and~{}X)}$ $\mathrm{X}$ $8<\nu<1{\hbox to 0.0pt{2}}$ $\mathrm{X~{}shape~{}of~{}crosshairs}$ $\mathrm{K{\hbox to 0.0pt{u}}}$ $12<\nu<18$ $\mathrm{Kurz-under}$ $\mathrm{K}$ $18<\nu<26{\hbox to 0.0pt{.5}}$ $\mathrm{Kurz~{}(short~{}in~{}German)}$ $\mathrm{K{\hbox to 0.0pt{a}}}$ ${\hbox to 0.0pt{2}}6.5<\nu<40$ $\mathrm{Kurz-above}$ $\mathrm{Q}$ $33<\nu<50$ $\mathrm{U}$ $40<\nu<60$ $\mathrm{U~{}is~{}before~{}V~{}in~{}alphabet}$ $\mathrm{V}$ $50<\nu<75$ $\mathrm{Very~{}absorbed~{}by~{}O}_{2}$ $\mathrm{W}$ $75<\nu<11{\hbox to 0.0pt{0}}$ $\mathrm{W~{}is~{}after~{}V~{}in~{}alphabet}$

# F.6 Dimensional Analysis

The dimensions of any physical quantity can be written as a product of powers of the fundamental physical dimensions of length, mass, and time. The dimensions of velocity can be written as $\mathrm{(length)}\times\mathrm{(time)}^{-1}$, for instance. The derived dimensions temperature and charge are used for convenience.

Both sides of every equation should have the same dimensions because valid laws of physics are independent of the units used. Dimensional analysis provides a useful check for errors in derivations of new equations. For example, the total flux of a blackbody radiator at temperature $T$ is

 $S(T)=\sigma T^{4}.$

The dimensions of flux are power per unit area (e.g., units erg s${}^{-1}$ cm${}^{-2}$), so the dimensions of the Stefan–Boltzmann constant $\sigma$ should be $\mathrm{power}\times\mathrm{area}^{-1}\times\mathrm{(temperature)}^{-4}$ (e.g., units erg s${}^{-1}$ cm${}^{-2}$ K${}^{-4}$). However, the total brightness of a blackbody radiator is

 $B(T)=\frac{\sigma T^{4}}{\pi},$

and the dimensions of brightness are $\mathrm{power}\times\mathrm{area}^{-1}$ per unit solid angle (e.g., units erg s${}^{-1}$ cm${}^{-2}$ sr${}^{-1}$), indicating that $\sigma$ has dimensions of $\mathrm{power}\times\mathrm{area}^{-1}$ per unit solid angle $\times\mathrm{(temperature)}^{-4}$ (e.g., dimensions erg s${}^{-1}$ cm${}^{-2}$ sr${}^{-1}$ K${}^{-4}$). This does not violate the rule that both sides of the equation should have the same dimensions because the extra sr${}^{-1}$ is itself dimensionless: solid angle has dimensions of (angle)${}^{2}=(\mathrm{length/length})^{2}$. For clarity, the extra sr${}^{-1}$ should be written explicitly where appropriate.

Just because angles and solid angles are dimensionless doesn’t mean they don’t have units. The natural unit for angle is the radian, defined as the angle subtended at the center of a circle by an arc whose length equals the radius of the circle. Only this angle has dimensions (length/length) in which both lengths have the same units. Any other unit of angle, the degree for example, does not have that property and must be called out explicitly in equations. Likewise, the only natural unit for solid angle is the steradian, defined as the solid angle subtended by a unit area on the surface of a sphere of unit radius.

The dimensions of some quantities can be written in more than one way, and the simplest is not always the clearest. For example, the noise power per unit frequency generated by a resistor at temperature $T$ is

 $P_{\nu}=kT.$

Both $P_{\nu}$ and $kT$ have dimensions of energy (e.g., erg or joule). However, it is easier to think of $P_{\nu}$ as a power per unit frequency (e.g., erg s${}^{-1}$ Hz${}^{-1}$ or W Hz${}^{-1}$), even though power per unit frequency also has dimensions of energy because the dimension of Hz${}^{-1}$ is time (s) which cancels the s${}^{-1}$.

Finally, the arguments of many functions are dimensionless. Examples include $\sin\theta$, where $\theta$ is an angle (dimensions of length/length), $\cos(\omega t)$, where the dimensions of angular frequency $\omega$ (inverse time) and $t$ (time) cancel, and $\exp[h\nu/(kT)]$, where the numerator $h\nu$ and denominator $(kT)$ both have dimensions of energy.