Definitions, approximate numerical values, and parameter dependences of common space plasma parameters and characteristic scales, normalized to \( m = m_{\mathrm{species}} \), \( q = |q_{\mathrm{species}}| \), \( [B] = 1\, \mathrm{nT} \), \( [n] = 1\, \mathrm{cm^{-3}} \), \( [T] = 10^5\, \mathrm{K} \), and \( [v] = 1\, \mathrm{km/s} \), with Mach numbers using \( [v] = 100\, \mathrm{km/s} \). The ion and electron temperatures are assumed equal, \( T_\mathrm{i} = T_\mathrm{e} = T \), and the adiabatic index is \( \gamma = 5/3 \).
| Parameter | Definition | Proton | Electron | Dependence |
|---|---|---|---|---|
| Alfvén Mach number | \( M_\mathrm{A} = \frac{v}{v_\mathrm{A}} = \frac{v \sqrt{\mu_0 m n}}{B} \) | \( 4.6 \) | - | \( \frac{v \sqrt{m n}}{B} \) |
| Alfvén speed | \( v_\mathrm{A} = \frac{B}{\sqrt{\mu_0 m n}} \) | \( 22\, \mathrm{km/s} \) | - | \( \frac{B}{\sqrt{m n}} \) |
| Beta | \( \beta = \frac{p_{\mathrm{th}}}{p_B} = \frac{2\mu_0 n k_\mathrm{B} T}{B^2} \) | \( 3.5 \) | \( 3.5 \) | \( \frac{n T}{B^2} \) |
| Cyclotron frequency | \( f_\mathrm{c} = \frac{1}{\tau_\mathrm{c}} = \frac{q B}{2\pi m} \) | \( 15\, \mathrm{mHz} \) | \( 28\, \mathrm{Hz} \) | \( \frac{q B}{m} \) |
| Cyclotron period | \( \tau_\mathrm{c} = \frac{1}{f_\mathrm{c}} = \frac{2\pi m}{q B} \) | \( 66\, \mathrm{s} \) | \( 36\, \mathrm{ms} \) | \( \frac{m}{q B} \) |
| Debye length | \( \lambda_\mathrm{D} = \sqrt{\frac{\epsilon_0 k_\mathrm{B} T}{n q^2}} \) | - | \( 22\, \mathrm{m} \) | \( \sqrt{\frac{T}{n q^2}} \) |
| Debye number | \( N_\mathrm{D} = \frac{4 \pi}{3} n \lambda_\mathrm{D}^3 = \frac{4 \pi}{3} n (\frac{\epsilon_0 k_\mathrm{B} T}{n q^2})^{3/2} \) | - | \( 4.4 \times 10^{10} \) | \( \frac{T^{3/2}}{q^3 \sqrt{n}} \) |
| Gyro radius | \( r_\mathrm{g} = \frac{m v_\perp}{q B} \) | \( 10\, \mathrm{km} \) | \( 5.7\, \mathrm{m} \) | \( \frac{m v}{q B} \) |
| Gyro radius - thermal | \( r_\mathrm{g,th} = \frac{m v_\mathrm{th}}{q B} \) | \( 300\, \mathrm{km} \) | \( 7.0\, \mathrm{km} \) | \( \frac{\sqrt{m T}}{q B} \) |
| Inertial length | \( d = \frac{c}{\omega_\mathrm{p}} = \frac{c}{2 \pi f_\mathrm{p}} = \sqrt{\frac{m}{\mu_0 q^2 n}} \) | \( 230\, \mathrm{km} \) | \( 5.3\, \mathrm{km} \) | \( \sqrt{\frac{m}{n q^2}} \) |
| Ion-acoustic (sound) speed | \( c_\mathrm{s} = \sqrt{\frac{\gamma k_\mathrm{B} (T+T)}{m}} = \sqrt{2 \gamma}\, v_\mathrm{th} \) | \( 53\, \mathrm{km/s} \) | - | \( \sqrt{\frac{\gamma T}{m}} \) |
| Magnetosonic Mach number | \( M_\mathrm{ms} = \frac{v}{v_\mathrm{ms}} = \frac{v}{\sqrt{v_\mathrm{A}^2 + c_\mathrm{s}^2}} = \frac{M_\mathrm{A}}{\sqrt{1 + \gamma \beta}} \) | \( 1.8 \) | - | - |
| Magnetosonic speed | \( v_\mathrm{ms} = \sqrt{v_\mathrm{A}^2 + c_\mathrm{s}^2} = \sqrt{\frac{1}{m} (\frac{B^2}{\mu_0 n} + 2 \gamma k_\mathrm{B} T)} \) | \( 57\, \mathrm{km/s} \) | - | - |
| Plasma frequency | \( f_\mathrm{p} = \frac{1}{\tau_\mathrm{p}} = \frac{\omega_\mathrm{p}}{2\pi} = \frac{1}{2 \pi} \sqrt{\frac{n q^2}{m \epsilon_0}} \) | \( 0.21\, \mathrm{kHz} \) | \( 9.0\, \mathrm{kHz} \) | \( \sqrt{\frac{n q^2}{m}} \) |
| Plasma period | \( \tau_\mathrm{p} = \frac{1}{f_\mathrm{p}} = \frac{2\pi}{\omega_\mathrm{p}} = 2\pi \sqrt{\frac{m \epsilon_0}{n q^2}} \) | \( 4.8\, \mathrm{ms} \) | \( 0.11\, \mathrm{ms} \) | \( \sqrt{\frac{m}{n q^2}} \) |
| Pressure - dynamic | \( p_\mathrm{dyn} = m n v^2 \) | \( 1.7\, \mathrm{fPa} \) | \( 0.91\, \mathrm{aPa} \) | \( m n v^2 \) |
| Pressure - magnetic | \( p_\mathrm{B} = \frac{B^2}{2 \mu_0} \) | \( 0.40\, \mathrm{pPa} \) | - | \( B^2 \) |
| Pressure - thermal | \( p_\mathrm{th} = n k_\mathrm{B} T \) | \( 1.4\, \mathrm{pPa} \) | \( 1.4\, \mathrm{pPa} \) | \( n T \) |
| Thermal speed | \( v_\mathrm{th} = \sqrt{\frac{k_\mathrm{B} T}{m}} \) | \( 29\, \mathrm{km/s} \) | \( 1200\, \mathrm{km/s} \) | \( \sqrt{\frac{T}{m}} \) |