Essential Radio Astronomy

Appendix F Constants, Units, and Dimensions

F.1 Physical Constants

Symbol Name Value Units
a radiationconstant 7.56577×10-15 ergcm-3K-4
a0 Bohrradius 5.29177×10-9 cm
c speedoflightinvacuum 2.99792×1010 cms-1
e electroncharge(magnitude) 4.80325×10-10 statcoulomb(oresu)
eV electronvolt 1.60218×10-12 erg
G gravitationalconstant 6.67428×10-8 dynecm2g-2
h Planck’s constant 6.62607×10-27 ergs
k Boltzmann’s constant 1.38065×10-16 ergK-1
me electronmass 9.10938×10-28 g
mp protonmass 1.67262×10-24 g
μB Bohrmagneton 9.27401×10-21 erggauss-1
R Rydbergconstant 1.09737×105 cm-1
Rc Rydbergfrequency 3.28984×1015 s-1
σ Stefan–Boltzmann constant 5.67040×10-5 ergs-1cm-2sr-1K-4
σT Thomsoncrosssection 6.65245×10-25 cm2
u atomicmassunit 1.66054×10-24 g

F.2 Astronomical Constants

Symbol Name Value Units
au astronomicalunit 1.49598×1013 cm
500 lightseconds
H0 Hubbleconstant 67.8 kms-1Mpc-1
kpc kiloparsec 103 pc
L solarluminosity 3.826×1033 ergs-1
ly lightyear 9.4606×1017 cm
M solarmass 1.989×1033 g
Mpc megaparsec 106 pc
pc parsec 3.0856×1018 cm
R solarradius 6.9558×1010 cm
yr tropicalyear 3.1557×107 s
107.5 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 2×10-7 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.

Type MKSunit CGSunit conversion
length m cm 102cm=1m
mass kg g 103g=1kg
time s s
energy joule erg 107erg=1joule=1kgm2s-2
force newton dyne 105dyne=1newton=1kgms-2
frequency Hz Hz 1Hz=1s-1
power W ergs-1 107ergs-1=1W=1kgm2s-3
temperature kelvin kelvin
charge coulomb statcoulomb 3×109statcoulomb1coulomb
current ampere statampere 3×109statamp1amp
electricfield voltm-1 statvoltcm-1 (1/3)×10-4statvoltcm-1
magneticfield tesla gauss 104gauss1tesla
resistance ohm seccm-1 (1/9)×10-11scm-11ohm
voltage volt statvolt (1/3)×10-2statvolt1volt

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 2×1030 kg and not 2×1030 g.

The MKS and CGS units of length, mass, time, and temperature have the same dimensions, so it is possible to state that 102cm=1m, 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 q1 and q2 separated by distance r is simply

F=q1q2r2. (F.1)

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

F=c2107q1q2r2, (F.2)

where c=2.99729458×108ms-1 is the vacuum speed of light in MKS units. The MKS form of Coulomb’s law is usually written as

F=14πϵ0q1q2r2, (F.3)

where ϵ0 is called the permittivity of free space and

ϵ0=1074πc28.854×10-12 (F.4)

has units m-2s2=C2N-1m-2. The corresponding MKS magnetic quantity is the permeability of free space

μ04π×10-7newtonampere-2=(ϵ0c2)-1. (F.5)

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

1coulomb=10cstatcoulomb (F.6)

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

1coulomb2.99729458×109statcoulomb, (F.7)

where the symbol indicates numerical conversion relating given amounts of dimensionally incompatible quantities. For example, given that the charge of an electron is e=-4.80325×10-10statcoulomb in CGS units, Equation F.7 can be used to show that the electron charge is e=-1.60253×10-19coulomb in MKS units. The charge conversion factor is simplified to 3×109 in the table above, where 3 is the conventional dimensionless shorthand for the exact and dimensionally correct 2.99729458(ms-1)=10-8c; likewise, (1/3) for 1/[2.99729458(ms-1)] and 1/9 for 1/[2.99729458(ms-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 v˙ is




The numerical conversion


implies the dimensionally correct equality



(P10-7W) =23(q(10c)-1coul)2(v˙10-2ms-1)2(c10-2ms-1)-3,
107(PW) =23(10c)2(qcoul)2(104)(v˙ms-1)2(10-6)(cms-1)-3,
(PW) =23(c2107)(qcoul)2(v˙ms-1)2(cms-1)-3

Thus Larmor’s equation in MKS units is


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

F=q(E+β×B), (F.8)

where βv/c is dimensionless, so E and B have the same dimensions (base units cm-1/2g1/2s-1) in Gaussian CGS units, and 1statvoltcm-1=1gauss, 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:

F=q(E+v×B), (F.9)

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

|E|=|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

Symbol Value
ABmagnitude -2.5log[S(Jy)]+8.90=-2.5log10[S/(3631Jy)]
arcmin 1/60deg
arcsec 1/60arcmin
Å 10-10m
D 1debye10-18statcoulombcm
dB 10log10(P1/P2)
deg (π/180)rad
e 2.71828
GHz 109Hz
Jy 10-26Wm-2Hz-1=10-23ergs-1cm-2Hz-1
MHz 106Hz
mJy 10-3Jy
μm 10-6m
μJy 10-6Jy
π 3.14159
radian(rad) anglesubtendingunitarclengthonaunitcircle
steradian(sr) solidanglesubtendingunitareaonaunitsphere
THz 1012Hz

F.5 Radar and Waveguide Frequency Bands

Band Frequency
name (GHz) Mnemonic
P 0.25<ν<0.50 Previous
L 1<ν<2 Long(wavelength)
S 2<ν<4 Short(wavelength)
C 4<ν<8 Compromise(betweenSandX)
X 8<ν<12 Xshapeofcrosshairs
Ku 12<ν<18 Kurz-under
K 18<ν<26.5 Kurz(shortinGerman)
Ka 26.5<ν<40 Kurz-above
Q 33<ν<50
U 40<ν<60 UisbeforeVinalphabet
V 50<ν<75 VeryabsorbedbyO2
W 75<ν<110 WisafterVinalphabet

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 (length)×(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


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 σ should be power×area-1×(temperature)-4 (e.g., units erg s-1 cm-2 K-4). However, the total brightness of a blackbody radiator is


and the dimensions of brightness are power×area-1 per unit solid angle (e.g., units erg s-1 cm-2 sr-1), indicating that σ has dimensions of power×area-1 per unit solid angle ×(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(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


Both Pν and kT have dimensions of energy (e.g., erg or joule). However, it is easier to think of Pν 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 sinθ, where θ is an angle (dimensions of length/length), cos(ωt), where the dimensions of angular frequency ω (inverse time) and t (time) cancel, and exp[hν/(kT)], where the numerator hν and denominator (kT) both have dimensions of energy.