CGS units
CGS unit system uses centimeter/gram/seconds as its the base units. This page lists the most common constants of nature in cgs.
# typical CGS constants in python
c = 2.99792e+10 # speed of light; cm/s
me = 9.10939e-28 # electron mass; g
mp = 1.67262e-24 # proton mass; g
e0 = 4.80325e-10 # electron charge; esu
h_pc = 6.62606e-27 # Plank constant; erg*s
sigT = 6.65246e-25 # Thomson cross section; cm^2
sigSB = 5.67040e-5 # S-B constant; erg/(s*cm^2*K^4)
kB = 1.38065e-16 # Boltzmann constant; erg/K
BQ = 4.41396e13 # Schwinger field strength; G
lamC = 2.4240e-10 # Compton wavelength; cm
Msun = 1.989e33 # solar mass; g
pc = 3.086e18 # parsec; cm;
Physical constants
| Quantity | Symbol | Value | Units | Full units |
|---|---|---|---|---|
| Boltzmann’s constant | $$k_\mathrm{B}$$ | $$1.3807 \times 10^{-16}$$ | $$\mathrm{erg}\,\mathrm{K}^{-1}$$ | $$\mathrm{cm}^2 \, \mathrm{g} \,\mathrm{s}^{-2} \, \mathrm{K}^{-1}$$ |
| elementary (electron) charge | $$e$$ | $$ 4.8032 \times 10^{-10}$$ | $$ \mathrm{esu}$$ | $$ \mathrm{cm}^{3/2} \, \mathrm{g}^{1/2} \,\mathrm{s}^{-1}$$ |
| gravitational constant | $$G$$ | $$6.6726 \times 10^{-8}$$ | $$\mathrm{cm}^3\,\mathrm{g}^{-1}\,\mathrm{s}^{-2}$$ | |
| Planck’s constant | $$h$$ | $$6.6261 \times 10^{-27}$$ | $$\mathrm{erg} \, \mathrm{s}$$ | $$\mathrm{cm}^2 \, \mathrm{g} \,\mathrm{s}^{-1}$$ |
| reduced Planck’s constant | $$\hbar \equiv h/2\pi$$ | $$1.0546 \times 10^{-27}$$ | $$\mathrm{erg} \, \mathrm{s}$$ | $$\mathrm{cm}^2 \, \mathrm{g} \,\mathrm{s}^{-1}$$ |
| speed of light in vacuum | $$c$$ | $$2.9979 \times 10^{10}$$ | $$\mathrm{cm}\,\mathrm{s}^{-1}$$ | |
| Stefan-Boltzmann constant | $$\sigma_\mathrm{SB}$$ | $$ 5.6704 \times 10^{-5}$$ | $$\mathrm{g}^2 \, \mathrm{s}^{-3} \,\mathrm{K}^{-4}$$ | |
| electron mass | $$m_e$$ | $$ 9.1094 \times 10^{-28} $$ | $$(511\,\mathrm{keV}/c^2)$$ | $$ \mathrm{g}$$ |
| proton mass | $$m_p$$ | $$ 1.6726 \times 10^{-24} $$ | $$(938.272\,\mathrm{MeV}/c^2)$$ | $$ \mathrm{g}$$ |
| neutron mass | $$m_n$$ | $$ 1.6749 \times 10^{-24} $$ | $$(939.563\,\mathrm{MeV}/c^2)$$ | $$ \mathrm{g}$$ |
| atomic mass unit | $$u$$ | $$ 1.6605 \times 10^{-24} $$ | $$(931.494\,\mathrm{MeV}/c^2)$$ | $$ \mathrm{g}$$ |
| proton/electron mass ratio | $$m_p/m_e$$ | $$ 1836.2 $$ | ||
| dynamic viscosity | $$\mu$$ | $$ \mathrm{g}\,\mathrm{cm}^{-1} \, \mathrm{s}^{-1}$$ | ||
| kinematic viscosity | $$\nu \equiv \frac{\mu}{\rho}$$ | $$ \mathrm{cm}^{2} \, \mathrm{s}^{-1}$$ | ||
| frequency of 1 eV | $$\nu_0 = e/h$$ | $$2.4180 \times 10^{14}$$ | $$ Hz $$ | |
| energy of 1 eV | $$h \nu_0$$ | $$1.6022 \times 10^{-12}$$ | $$ \mathrm{erg} $$ | |
| energy of 1 erg | $$\mathrm{erg}$$ | $$6.62315 \times 10^{11}$$ | $$10^{-7}\,\mathrm{J}$$ | $$ \mathrm{eV} $$ |
| temperature of 1 eV | $$11604 $$ | $$ \mathrm{K} $$ | ||
| Avogadro number | $$N_A$$ | $$6.0221 \times 10^{23} $$ | $$ \mathrm{mol}^{-1} $$ | |
| gas constant | $$R_A \equiv N_A k_{\mathrm{B}}$$ | $$8.3145 \times 10^{7} $$ | $$ \mathrm{erg}\,\mathrm{K}^{-1}\,\mathrm{mol}^{-1} $$ |
Common derived units include:
- $\color{blue}{\mathrm{erg}} = \mathrm{g}\,\mathrm{cm}^2 \, \mathrm{s}^{-2}$ for energy (from $E = m c^2$).
- $\color{blue}{\mathrm{dyne}} = \mathrm{g} \, \mathrm{cm} \, \mathrm{s}^{-2}$ for force,
- $\color{blue}{\mathrm{Gauss}} = 10^{-4} \, \mathrm{Tesla}$ for magnetic field strength,
- $\color{blue}{\mathrm{esu}}$ for charge.
QED constants
Derived constants (especially in the theory of quantum electrodynamics) are
| Quantity | Symbol | Value | Approx. | Full units |
|---|---|---|---|---|
| fine structure constant | $$\alpha \equiv \frac{e^2}{\hbar c}$$ | $$7.2974 \times 10^{-3}$$ | $$\approx \frac{1}{137}$$ | |
| classical electron radius | $$r_0 \equiv \frac{e^2}{m_e c^2}$$ | $$ 2.8179 \times 10^{-13}$$ | $$ \mathrm{cm}$$ | |
| Thomson cross-section | $$\sigma_\mathrm{T} \equiv \frac{8\pi}{3} r_0^2$$ | $$ 6.6524 \times 10^{-25}$$ | $$ \mathrm{cm}^2$$ | |
| electron Compton wavelength | $$\lambda_\mathrm{C} \equiv \frac{h}{m_e c}$$ | $$ 2.4263 \times 10^{-10}$$ | $$ \mathrm{cm}$$ | |
| critical Schwinger field | $$B_\mathrm{Q} \equiv \frac{m_e^2 c^3}{\hbar e}$$ | $$ 4.4139 \times 10^{13}$$ | $$ \mathrm{G}$$ |
Astronomical constants
| Quantity | Symbol | Value | Approx. | Full units |
|---|---|---|---|---|
| Solar mass | $$M_\odot$$ | $$ 1.989 \times 10^{33} $$ | $$ \mathrm{g}$$ | |
| Earth mass | $$M_E$$ | $$ 5.974 \times 10^{27} $$ | $$\approx 3 \times 10^{-6} M_\odot$$ | $$ \mathrm{g}$$ |
| Jupiter mass | $$M_J$$ | $$ 1.899 \times 10^{30} $$ | $$\approx 1 \times 10^{-3} M_\odot$$ | $$ \mathrm{g}$$ |
| Solar radius | $$R_\odot$$ | $$ 6.955 \times 10^{10} $$ | $$ \mathrm{cm}$$ | |
| Earth radius | $$R_E$$ | $$ 6.378 \times 10^{8} $$ | $$\approx 10^{-2} R_\odot$$ | $$ \mathrm{cm}$$ |
| Jupiter radius | $$R_J$$ | $$ 7.149 \times 10^{9} $$ | $$\approx 10^{-1} R_\odot$$ | $$ \mathrm{cm}$$ |
| Solar luminosity | $$L_\odot$$ | $$ 3.839 \times 10^{33} $$ | $$ \mathrm{erg}\,\mathrm{s}^{-1}$$ | |
| astronomical unit (radius of earth’s orbit) | $$\mathrm{AU}$$ | $$ 1.496 \times 10^{13} $$ | $$ \mathrm{cm}$$ | |
| parsec | $$\mathrm{pc}$$ | $$ 3.086 \times 10^{18} $$ | $$ \mathrm{cm}$$ | |
| light-year | $$\mathrm{ly}$$ | $$ 9.461 \times 10^{17} $$ | $$ \mathrm{cm}$$ | |
| year | $$\mathrm{yr}$$ | $$ 3.156 \times 10^{7} $$ | $$\approx \pi \times 10^7$$ | $$ \mathrm{s}$$ |
| gravitational radius | $$r_g \equiv \frac{G M}{c^2}$$ | $$ 1.4822 \left(\frac{M_\bullet}{M_\odot}\right) $$ | $$\mathrm{km}$$ | |
| Schwarzschild radius | $$r_\mathrm{S} \equiv \frac{2G M}{c^2}$$ | $$ 2.95 \left(\frac{M_\bullet}{M_\odot}\right) $$ | $$\mathrm{km}$$ | |
| Eddington luminosity | $$L_\mathrm{Edd} \equiv \frac{4\pi G M c m_p}{\sigma_\mathrm{T}}$$ | $$ 1.26 \times 10^{38} \left(\frac{M_\bullet}{M_\odot}\right) $$ | $$ \mathrm{erg} \, \mathrm{s}^{-1}$$ |
Electrodynamics (CGS-gaussian)
Electrodynamics has multiple unit systems. Commonly used system is the gaussian system with Maxwell’s equations given as:
\[\nabla \times \mathbf{E} = - \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t} \quad\mathrm{Faradays~law}\] \[\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} + \frac{4\pi}{c} \mathbf{J} \quad \mathrm{Amperes~law}\] \[\nabla \cdot \mathbf{E} = 4\pi \rho \quad \mathrm{Poisson~equation}\] \[\nabla \cdot \mathbf{B} = 0 \quad \mathrm{no~monopoles}\]In Gaussian, units the vacuum permeability $\mu \approx 1$ and the vacuum permittivity $\epsilon_0 \approx 1$.
| Quantity | Symbol | Unit | gaussian/SI conversion |
|---|---|---|---|
| Electric charge | $$e$$ | $$ \mathrm{esu}$$ | $$3 \times 10^9 \,\frac{\mathrm{statcoul}}{\mathrm{C}}$$ |
| Electric field | $$\mathbf{E} $$ | $$\mathrm{statvolt}\,\mathrm{cm}^{-1}$$ | $$3.33 \times 10^{-5} \,\frac{\mathrm{statvolt}\,\mathrm{cm}^{-1}}{\mathrm{V}\,\mathrm{m}^{-1}}$$ |
| Magnetic field | $$\mathbf{B}$$ | $$\mathrm{Gauss}$$ | $$10^{4} \,\frac{\mathrm{G}}{\mathrm{T}}$$ |
| Current | $$\mathbf{J}$$ | $$\mathrm{statamp} \equiv \mathrm{esu}\,\mathrm{s}^{-1}$$ | $$3 \times 10^{9} \,\frac{\mathrm{statamp}}{\mathrm{A}}$$ |
| Electric potential (voltage) | $$V_\mathbf{E}$$ | $$\mathrm{statvolt}$$ | $$10^{2} \,\frac{\mathrm{statvolt}}{\mathrm{V}}$$ |
The factors of $3$ come from inclusion of $c \approx 3 \times 10^{10} \,\mathrm{cm}\,\mathrm{s}^{-1}$ and factors of $100$ from $\mathrm{m}/\mathrm{cm} = 100$.
Note on Gaussian unit system
Electromagnetism requires addition of a new unit dimension; this is charge. Charge times electric field has same units in SI and gaussian unit system, $q \mathbf{E} = M L T^{-2}$. However, magnetic field times the charge differs, as it is $q\mathbf{B} = M T^{-1}$ in SI and $q\mathbf{B} = M L T^{-1}$ in gaussian. In addition, in SI unit $B^2$, $E^2$, and $e^2$ require an additional quantity with dimensions, either $\epsilon_0$ or $1/\mu_0 = \epsilon_0/c^2$, to convert the value into quantities involving $M =$ mass, $L=$ length, and $T=$ time. No extra quantities are required in gaussian unit system.
Plasma physics
Electron gyrofrequency
\[\omega_B \equiv \frac{e B}{\gamma m_e c} = 1.76 \times 10^7 ~B\,\mathrm{rad}\,\mathrm{s}^{-1}\]Electron plasma frequency
\[\omega_p \equiv \left(\frac{4\pi n_e e^2}{\gamma m_e}\right)^{1/2} = 5.64 \times 10^4 ~n_e^{1/2} \,\mathrm{rad}\,\mathrm{s}^{-1}\]References
Useful references include:
