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Constants and Conversions
Fundamental constants
Gravitational constant: G = 6. 67408 × 10 −^8 dyn cm^2 g−^2 = 6. 67408 × 10 −^8 g−^1 cm^3 s−^2 = 6. 67408 × 10 −^8 erg cm g−^2 = 2. 960 × 10 −^4 M− ⊙^1 AU^3 days−^2 = 1. 327 × 1011 M− ⊙^1 km^3 s−^2 Speed of light: c = 2. 99792458 × 1010 cm s−^1 Planck’s constant: h = 2 π h¯ = 6. 6261 × 10 −^27 erg s = 4. 136 × 10 −^21 MeV s h¯ = 1. 0546 × 10 −^27 erg s = 6. 5827 × 10 −^22 MeV s hc¯ = 197 .3 MeV fm = 197. 3 × 10 −^13 MeV cm Electrical charge unit: e = 4. 8032068 × 10 −^10 esu = 4 .8032068 erg^1 /^2 cm^1 /^2 = 4 .8032068 g^1 /^2 cm^3 /^2 s−^1 Fine structure constant: α = ( 137. 036 )−^1 = 0. 0073 Weak (Fermi) constant: GF = 8. 958 × 10 −^44 MeV cm^3 = 1. 16637 × 10 −^5 GeV−^2 [GF/(¯hc)^3 ; ¯h = c = 1] Mass of electron: me = 9. 1093898 × 10 −^28 g = 5. 4858 × 10 −^4 amu = 0 .5109991 MeV/c^2 Mass of proton: mp = 1. 6726231 × 10 −^24 g = 1 .00727647 amu = 938 .27231 MeV/c^2 Mass of neutron: mn = 1. 6749286 × 10 −^24 g = 1 .0086649 amu = 939 .56563 MeV/c^2 Atomic mass unit (amu) = 1. 6605390 × 10 −^24 g Avogadro’s constant: NA = 6. 0221409 × 1023 mol−^1 Boltzmann’s constant: k = 1. 38065 × 10 −^16 erg K−^1 = 8. 617389 × 10 −^5 eV K−^1 Ideal gas constant: Rgas ≡ NAk = 8. 314511 × 107 erg K−^1 mole−^1 Stefan–Boltzmann constant: σ = 5. 67051 × 10 −^5 erg cm−^2 K−^4 s−^1 Radiation density constant: a ≡ 4 σ /c = 7. 56591 × 10 −^15 erg cm−^3 K−^4 = 4. 7222 × 10 −^9 MeV cm−^3 K−^4 Planck mass: MP = 1. 2 × 1019 GeV/c^2 Planck length: ℓP = 1. 6 × 10 −^33 cm Planck time: tP = 5. 4 × 10 −^44 s Planck temperature: TP = 1. 4 × 1032 K
Solar quantities
Solar (photon) luminosity: L⊙ = 3. 828 × 1033 erg/s Solar absolute magnitude Mv = 4. 83 Solar bolometric magnitude M⊙ bol = 4. 74 Solar mass: M⊙ = 1. 989 × 1033 g Effective surface temperature: T ⊙ eff = 5780 K Solar radius: R⊙ = 6. 96 × 1010 cm Central density: ρ ⊙core ≃ 160 g/cm^3 Central pressure: P ⊙core ≃ 2. 7 × 1017 dyn cm−^2 Central temperature: T ⊙ core ≃ 1. 6 × 107 K Color indices: B − V = 0. 63 U − B = 0. 13 Solar constant: 1. 36 × 106 erg cm−^2 s−^1
General quantities
1 tropical year (yr) = 3. 1556925 × 107 s = 365 .24219 d 1 parsec (pc) = 3. 0857 × 1018 cm = 206,265 AU= 3.2616 ly 1 lightyear (ly) = 9. 4605 × 1017 cm 1 astronomical unit (AU) = 1. 49598 × 1013 cm Energy per gram from H → He fusion = 6. 3 × 1018 erg/g Thomson scattering cross section: σT = 6. 652 × 10 −^25 cm^2 Mass of Earth M⊕ = 5. 98 × 1027 g Radius of Earth R⊕ = 6. 371 × 108 cm
Useful conversion factors
1 eV = 1. 60217733 × 10 −^12 ergs = 1. 60217733 × 10 −^19 J 1 J = 107 ergs = 6. 242 × 1018 eV 1 amu = 1. 6605390 × 10 −^24 g 1 fm = 10 −^13 cm 0 K = − 273 .16 Celsius 1 atomic unit (a 0 ) = 0. 52918 × 10 −^8 cm 1 atmosphere (atm) = 1. 013250 × 106 dyn cm−^2 1 Pascal (Pa) = 1 N m−^2 = 10 dyn cm−^2 1 arcsec = 1 ′′^ = 4. 848 × 10 −^6 rad = 1 /3600 deg
1 A
◦ = 10 −^8 cm 1 barn (b) = 10 −^24 cm^2 1 Newton (N) = 105 dyn 1 Watt (W) = 1 J s−^1 = 107 erg s−^1 1 Gauss (G) = 10−^4 Tesla (T) 1 g cm−^3 = 1000 kg m−^3 Opacity units: 1 m^2 kg−^1 = 10 cm^2 g−^1
if ¯h = c = 1. These results then imply that [M] may be chosen as the single independent dimension of our set of ¯h = c = 1 natural units. This dimension is commonly measured in MeV (10^6 eV) or GeV (10^9 eV). Useful conversions are
hc¯ = 197 .3 MeV fm 1 fm = (^1971). 3 MeV−^1 = 5 .068 GeV−^1
1 fm−^1 = 197 .3 MeV 1 GeV = 5 .068 fm−^1.
where 1 fm = 10−^13 cm (one fermi or one femtometer).
Natural Units in Cosmology In cosmology we often employ a set of ¯h = c = kB = 1 natural units, where kB is the Boltzmann constant. Then from E = kBT and kB = 8. 617 × 10 −^14 GeV K−^1 = 1,
1 GeV = 1. 2 × 1013 K,
where K denotes kelvins. From §12.6 we then have for the Planck mass, Planck energy, Planck temperature, Planck length, and Planck time in these natural units,
MP = EP = TP = ℓ−P 1 = tP− 1.
To convert to standard units, note that from Eq. (12.16) the gravitational constant may be expressed as
G = (^) M^12 P
where the Planck mass is MP = 1. 2 × 1019 GeV.
From Eqs. (12.17), (B.14), and (B.11), the corresponding Planck length is
ℓP = (^) M^1 P
= 1. 6 × 10 −^33 cm,
multiplying by c−^1 gives the corresponding Planck time,
tP = 5. 4 × 10 −^44 s,
and from Eqs. (B.12) and (B.14) the Planck temperature is
TP = 1. 4 × 1032 K.
Conversions between geometrized (G = c = 1) units and standard units
Quantity Symbol Geometrized Standard Conversion unit unit
Mass M L M GM/c^2 Length L L L L Time t L T ct Spacetime distance s L L s Proper time τ L T c τ Energy E L M (L /T )^2 GE/c^4 Momentum p L M (L /T ) Gp/c^3 Angular momentum J L 2 M (L 2 /T ) GJ/c^3 Luminosity (power) L dimensionless M (L 2 /T 3 ) GL/c^5 Energy density ε L −^2 M /(LT 2 ) G ε/c^4 Momentum density πi L −^2 M /(L 2 T ) G πi/c^3 Pressure P L −^2 M /(LT 2 ) GP/c^4 Energy / unit mass ε dimensionless (L /T )^2 ε/c^2 Ang. mom. / unit mass ℓ L L 2 /T ℓ/c Planck constant h¯ L 2 M (L 2 /T ) Gh¯/c^3
The standard unit of length is L , the standard unit of mass is M , and the standard unit of time is T. Geometrized to standard conversion: Replace quantities in column 2 with quantities in the last column. Standard to geometrized conversion: Multiply by the factor of G and c appearing in the last column.
