0.4 Units and Physical Constants

Table 1: Units 10 Angstrom 1 A˚= 10− m 1 Wavenumber 1 cm− = c Hz Charge 1 C = 2.997925 109 (e.s.u.) Electric dipole 1 C m = 2.99792×5 1011 cm (e.s.u.) 18 × 30 Dipole moment 1 Debye = 10− cm (e.s.u.) = 3.33564 10− C m Magnetic ﬁeld 1 T = 104 gauss × Energy 1 J = 107 erg 19 Energy 1 eV = 1.60217733(49) 10− J = 11604.45(10) K 23 2 × 1 1 26 2 1 Flux 1 Jy = 10− erg cm− s− Hz− = 10− W m− Hz− Astronomical unit 1 AU = 1.4959787066 1011 m Parsec 1 pc = 3.0856776 10×16 m Force 1 N = 105 dyn × 2 Pressure 1 Pa = 10 dyn cm−

0.4.1 SI and Gaussian cgs units In the astronomical literature two unit systems are often used: SI and Gaus- sian cgs (plus ‘natural’ units such as parsec, solar mass, etc). Rybicki & Lightman use cgs. For most quantities this means that the units are ex- pressed using centimeters, grams, and seconds rather than the meters, kilo- grams, and seconds of SI. However, for the Maxwell equations the choice of units is important. 7 1 In SI the magnetic permeability is deﬁned, µ0 = 4π 10− H m− . Because 2 in all systems "0µ0 = 1/c , in SI the dielectrical constant becomes "0 = 7 2 1 8 1 10 /(4πc ) F m− with c = 3.0 10 m s− . In SI units the Maxwell equations take the form ×

# D# = ρ # E# = ∂B" ∇ · ∇ × ∂t # B# = 0 # H# = # + ∂D" (1) ∇ · ∇ × ∂t These equations are supplemented by the deﬁnition of the Lorentz force

8 Table 2: Physical Constants 8 1 Speed of light c = 2.99792458 10 m s− × 19 Elementary charge e = 1.60217733(49) 10− C × 10 e = 4.8032068(15) 10− esu 31 × Electron mass me = 9.11 10− kg × 27 Proton mass mp = 1.67 10− kg × 12 1 Dielectric constant "0 = 8.85419 10− F m− × 11 3 1 2 Gravitational constant G = 6.67259(85) 10− m kg− s− × 34 Planck constant h = 6.6260755(40) 10− J s × 27 = 6.6260755(40) 10− erg s ×23 1 Boltzmann constant k = 1.380658(12) 10− J K− × 16 1 = 1.380658(12) 10− erg K− × 5 2 4 1 Stefan-Boltzmann σ = 5.67051(19) 10− erg cm− K− s− × 8 2 4 = 5.67051(19) 10− W m− K− × Solar Luminosity L = 3.845(8) 1026 W " × 33 1 = 3.845(8) 10 erg s− Solar Radius R = 6.995 10×8 m " × Solar Mass M = 1.9891 1030 kg " × and the definitions D# = " E# + P# and H# = B# /µ M# . In vacuum, D# = " D# 0 0 − 0 and H# = B# /µ0. The cgs system comes in two flavors, e.s.u. and e.m.u. The first is defined 2 10 1 as "0 = 1/4π and hence mu0 = 4π/c with c = 3.0 10 cm s− . The second | 2 × 10 1 is defined as µ = 4π and hence " = 1/(4πc ) with c = 3 10 cm s− . The 0 0 × Gaussian cgs system is a combination, and uses e.s.u. for electrical quantities and e.m.u. for magnetic quantities. In Gaussian units the Maxwell equations take the form

1 ∂B" # D# = 4πρ # E# = − ∇ · ∇ × c ∂t # B# = 0 # H# = 4π# + 1 ∂D" (2) ∇ · ∇ × 2 c ∂t

In vacuum, "0 = µ0 = 1 in these units, and D# = E# and H# = B# . Conversion from SI to Gaussian units can be done by the following re- placements

9 Replace by Replace by 2 "0 1/4π µ0 4π/c D# D# /4π H# cH# /4π χe 4πχe χm 4πχm B# B# /c Vm cVm/4π Φ Φ/c m# cm# M# cM# P (pole) cP I# 4πI#/c Source: Duﬃn, Electricity and Magnetism (London: MacGraw-Hill).