6 Unit Conversion
6.1 Formula Conversion
Part of the materials in this chapter are referenced from the NRL Plasma Formulary and notes by Dana Longcope.
To derive a dimensionally correct SI formula from one expressed in Gaussian (CGS) units, substitute for each quantity according to \(\bar{Q} = \bar{k}Q\), where \(\bar{k}\) is the coefficient in the second column of Table 6.1 corresponding to \(Q\) (overbars denote variables expressed in Gaussian units). To go from SI to natural units in which \(\hbar = c = 1\), use \(Q = \hat{k}^{-1}\hat{Q}\), where \(\hat{k}\) is the coefficient corresponding to the third column.
Here \[ \begin{aligned} \alpha &= 10^2\,\text{cm}\,\text{m}^{-1} \\ \beta &= 10^7\,\text{erg}\,\text{J}^{-1} \\ \epsilon_0 &= 8.8542\times 10^{-12}\,\text{F}\,\text{m}^{-1} \\ \mu_0 &= 4\pi\times 10^7\,\text{H}\,\text{m}^{-1} \\ c &= (\epsilon_0\mu_0)^{-1/2} = 2.9979\times 10^8\,\text{m}\,\text{s}^{-1} \\ \hbar &= 1.0546\times 10^{-34}\,\text{J}\,\text{s} \end{aligned} \]
| Physical Quantity | CGS Units to SI | Natural Units to SI |
|---|---|---|
| Capacitance | \(\alpha/4\pi\epsilon_0\) | \(\epsilon_0^{-1}\) |
| Charge | \((\alpha\beta/4\pi\epsilon_0)^{1/2}\) | \((\epsilon_0 \hbar c)^{-1/2}\) |
| Charge density | \((\beta/4\pi\alpha^5\epsilon_0)^{1/2}\) | \((\epsilon_0 \hbar c)^{-1/2}\) |
| Current | \((\alpha\beta/4\pi\epsilon_0)^{1/2}\) | \((\mu_0/\hbar c)^{1/2}\) |
| Current density | \((\beta/4\pi\alpha^3\epsilon_0)^{1/2}\) | \((\mu_0/\hbar c)^{1/2}\) |
| Electric field | \((4\pi\beta\epsilon_0/\alpha^3)^{1/2}\) | \((\epsilon_0/\hbar c)^{1/2}\) |
| Electric potential | \((4\pi\beta\epsilon_0/\alpha)^{1/2}\) | \((\epsilon_0/\hbar c)^{1/2}\) |
| Electric conductivity | \((4\pi\epsilon_0)^{-1}\) | \(\epsilon_0^{-1}\) |
| Energy | \(\beta\) | \((\hbar c)^{-1}\) |
| Energy density | \(\beta/\alpha^3\) | \((\hbar c)^{-1}\) |
| Force | \(\beta/\alpha\) | \((\hbar c)^{-1}\) |
| Frequency | 1 | \(c^{-1}\) |
| Inductance | \(4\pi\epsilon_0/\alpha\) | \(\mu_0^{-1}\) |
| Length | \(\alpha\) | 1 |
| Magnetic induction | \((4\pi\beta/\alpha^3\mu_0)^{1/2}\) | \((\mu_0\hbar c)^{-1/2}\) |
| Magnetic intensity | \((4\pi\mu_0\beta/\alpha^3)^{1/2}\) | \((\mu_0\hbar c)^{-1/2}\) |
| Mass | \(\beta/\alpha^2\) | \(c/\hbar\) |
| Momentum | \(\beta/\alpha\) | \(\hbar^{-1}\) |
| Power | \(\beta\) | \((\hbar c^2)^{-1}\) |
| Pressure | \(\beta/\alpha^3\) | \((\hbar c)^{-1}\) |
| Resistance | \(4\pi\epsilon_0/\alpha\) | \((\epsilon_0/\mu_0)^{1/2}\) |
| Time | 1 | c |
| Velocity | \(\alpha\) | \(c^{-1}\) |
- Bohr radius in CGS: \(\bar{a}_0 = \bar{\hbar}^2 / \bar{m}\bar{e}^2\)
In SI: \(\alpha a_0 = (\hbar \beta)^2/[(m\beta/\alpha^2)(e^2 \alpha\beta / 4\pi\epsilon_0)]\), or \(a_0 = \epsilon_0 h^2 / \pi m e\).
6.1.1 Maxwell’s Equations
- Faraday’s law
- SI: \(\nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)
- CGS: \(\nabla\times\mathbf{E} = -\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}\)
- Ampère’s law
- SI: \(\nabla\times\mathbf{H} = \frac{\partial\mathbf{D}}{\partial t} + \mathbf{J}\)
- CGS: \(\nabla\times\mathbf{H} = \frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} + \frac{4\pi}{c}\mathbf{J}\)
- Poisson equation
- SI: \(\nabla\cdot\mathbf{D} = \rho\)
- CGS: \(\nabla\cdot\mathbf{D} = 4\pi\rho\)
- Absence of magnetic monopoles
- SI: \(\nabla\cdot\mathbf{B} = 0\)
- CGS: \(\nabla\cdot\mathbf{B} = 0\)
- Lorentz force on charge \(q\)
- SI: \(q(\mathbf{E} + \mathbf{v}\times\mathbf{B})\)
- CGS: \(q\left(\mathbf{E} + \frac{1}{c}\mathbf{v}\times\mathbf{B}\right)\)
- Constitutive relations
- SI & CGS: \(\mathbf{D} = \epsilon\mathbf{E}\), \(\mathbf{B} = \mu\mathbf{H}\)
The electromagnetic energy in volume V is given by \[ \begin{aligned} W =& \frac{1}{2}\int_V\mathrm{d}V(\mathbf{H}\cdot\mathbf{B} + \mathbf{E}\cdot\mathbf{D})\quad\text{SI} \\ =& \frac{1}{8\pi}\int_V\mathrm{d}V(\mathbf{H}\cdot\mathbf{B} + \mathbf{E}\cdot\mathbf{D})\quad\text{CGS} \end{aligned} \tag{6.1}\]
Poynting’s theorem is \[ \frac{\partial W}{\partial t} + \int_S \mathbf{N}\cdot\mathrm{d}\mathbf{S} = -\int_V\mathrm{d}V \mathbf{J}\cdot\mathbf{E} \tag{6.2}\] where S is the closed surface bounding V and the Poynting vector (energy flux across S) is given by \[ \begin{aligned} \mathbf{N} =& \mathbf{E}\times\mathbf{H}\quad\text{(SI)} \\ =& c\mathbf{E}\times\mathbf{H}/4\pi\quad\text{(CGS)} \end{aligned} \tag{6.3}\]
EM formulas can be converted to the corresponding formula from SI to CGS using the substitutions in the left column of Table 6.2. The first three are fundamental relations between the unit systems. The next three are obtained using the equations listed, and the last three are related variables.
| Physical Quantity | SI | CGS |
|---|---|---|
| Magnetic field | \(\mathbf{B}\) | \(\mathbf{B}\) |
| Vacuum permeability | \(\mu_0\) | \(4\pi\) |
| Vacuum permittivity | \(\epsilon_0\) | \(1/4\pi c^2\)1 |
| Electric field | \(\mathbf{E}\) | \(c\mathbf{E}\) |
| Current density | \(\mathbf{J}\) | \(\mathbf{J}/c\) |
| Charge density | \(\rho\) | \(\rho/c\) |
| Charge | \(q\) | \(q/c\) |
| Current | \(I\) | \(I/c\) |
| Resistivity | \(\eta\) | \(c^2\eta\) |
Some common expression conversions are listed in Table 6.3.
| SI | CGS |
|---|---|
| \(q\mathbf{v}\times\mathbf{B}\) | \(q\frac{\mathbf{v}}{c}\times\mathbf{B}\) |
| \(\mathbf{J}\times\mathbf{B}\) | \(\frac{1}{c}\mathbf{J}\times\mathbf{B}\) |
| \(\frac{q_1 q_2}{4\pi\epsilon_0}\) | \(q_1 q_2\) |
| \(\frac{B}{\sqrt{\mu_0 \rho}}\) | \(\frac{B}{\sqrt{4\pi \rho}}\) |
| \(\frac{1}{2\mu_0}B^2\) | \(\frac{1}{8\pi}B^2\) |
| \(\frac{\epsilon_0}{2}E^2\) | \(\frac{1}{8\pi}E^2\) |
| \(\omega_p^2 = \frac{q^2 n}{\epsilon_0 m}\) | \(\omega_p^2 = \frac{4\pi q^2 n}{m}\) |
| \(\Omega_c=\frac{qB}{m}\) | \(\Omega_c = \frac{qB}{mc}\) |
6.2 Unit System Conversion
A rule of thumb is to check the number of constants in the system: making one constant equal to 1 requires to shrink the number of basic variables by 1. In ideal MHD Equation 8.49, there are 4 basic quantities (length, mass, time, and current density) and 1 physical constant \(\mu_0\). If we include temperature, then there will be 5 basic quantities and 2 physical constants \(\mu_0\), \(R\). This indicates that we need 3 reference quantities in order to make the 2 physical constants equal to 1 in the normalized units. For example, in the Earth’s magnetosphere, we select a reference length, e.g. \(l_0=1\,\mathrm{R}_\mathrm{E}\), a reference mass density, e.g. \(\rho_0 = 1.67\times10^{-17}\,\mathrm{kg}\cdot\mathrm{m}^{-3}\), and a reference magnetic field, e.g. \(B_0 = 3.12\times 10^{-5}\,\mathrm{T}\) (Earth’s magnetic field strength at the equator). All the rest conversion factors can be derived from these together with the physical constants \(\mu_0, R, m_i\) expressed SI units. (\(R\) is needed for temperature and \(m_i\) is needed for number density.) Inserting the initially chosen values, we get a full set of conversion factors from variables in normalized units \(n^\prime, B^\prime, u^\prime, p^\prime, T^\prime, \mathcal{E}^\prime\) to SI units \(n, B, u, p, T, \mathcal{E}\): \[ U_\mathrm{SI} = U^\prime * U_0 \] where each conversion factor is summarized in Table 6.4. Note that in the definition of number density, the denominator is \(m_i\) instead of \(m_0\) and \(R=2k_B/m_i\) only appears in temperature. This is because mass does not appear in ideal MHD: for a given \(\rho\), the same results are obtained for a heavy species with small number density and a light species with large number density, e.g. \(\rho=2m_i n_i = m_i 2n_i\). In principle we can choose the reference mass arbitrarily, but here in order to make the values in SI units “look good” we choose \(m_i\). If you need to compare with a hybrid model, an easy way to make self-consistent conversion is to make mass \(m_0=m_i\) as a substitute for any of the basic variables. In this case, the dimensionless \(\rho=n\). Also note that in Table 6.4 the MHD pressure and temperature are the sum of electron’s and ion’s pressure and temperature, which is the reason a factor of “2” pops out in the definition of temperature. If we are comparing to a hybrid model which assumes massless electrons, a proper modification would be \(p_0=\rho_0v_0^2/2\) and \(T_0=p_0*m_i/(k_B\,\rho_0)=m_i v_0^2 / k_B\).
| Basic variable | Notation | Definition | Value |
|---|---|---|---|
| Length | \(l_0\) | \(l_0=\mathrm{R}_\mathrm{E}\) | \(6.371\times10^6\,\mathrm{m}\) |
| Magnetic field | \(B_0\) | \(B_0=B_\mathrm{E}\) | \(3.12\times10^{-5}\,\mathrm{T}\) |
| Mass density | \(\rho_0\) | \(\rho_0=\rho_\mathrm{E}\) | \(1.67\times10^{-17}\,\mathrm{kg}\cdot\mathrm{m}^{-3}\) |
| Derived variable | Notation | Definition | Value |
|---|---|---|---|
| Mass | \(m_0\) | \(m_0=\rho_0 l_0^3\) | \(4.32\times10^3\,\mathrm{kg}\) |
| Velocity | \(v_0\) | \(v_0=B_0/\sqrt{\mu_0\rho_0}\) | \(6.81\times10^6\,\mathrm{m}\cdot\mathrm{s}^{-1}\) |
| Time | \(t_0\) | \(t_0=l_0/v_0\) | \(0.94\,\mathrm{s}\) |
| Number density | \(n_0\) | \(n_0=\rho_0/m_i\) | \(10^{10}\,\mathrm{m}^{-3}\) |
| Pressure | \(p_0\) | \(p_0=\rho_0v_0^2\) | \(7.75\times10^{-4}\,\mathrm{N}\cdot\mathrm{m}^{-2}\) |
| Temperature | \(T_0\) | \(T_0=m_i v_0^2 / (2k_B)\) | \(2.81\times10^9\,\mathrm{K}\) |
| Energy density | \(\mathcal{E}_0\) | \(\mathcal{E}_0=p_0\) | \(7.75\times10^{-4}\,\mathrm{J}\cdot\mathrm{m}^{-3}\) |
6.2.1 Example: \(\rho,\rho\mathbf{u},\mathcal{E},\mathbf{B}\) to \(n,u,T\)
When dealing with velocity distribution functions, one easy way is to write \(f(\mathbf{v})=f(\mathbf{v}; n,\mathbf{u},T)\). For instance, a 3D Maxwellian is given as \[ f(\mathbf{v}) = n\left( \frac{m}{2\pi k_B T} \right)^{3/2} \exp\left(-\frac{m(\mathbf{v}-\mathbf{u})^2}{2k_B T}\right) \] From dimensionless conserved variables \((\rho,\rho\mathbf{u},\mathcal{E},\mathbf{B})\) to SI units \((n,\mathbf{u},T,\mathbf{B})\), we have \[ \begin{aligned} n_\mathrm{SI} &= n_0\,n = \rho_\mathrm{SI} / m_i = \frac{\rho_0}{m_i}\rho \\ u_\mathrm{SI} &= u_0\left( \frac{\rho u}{\rho} \right) \\ T_\mathrm{SI} &= T_0\frac{(\gamma-1)}{\rho}\left[ \mathcal{E} - \frac{1}{2}\rho u^2 - \frac{1}{2}B^2 \right] \\ B_\mathrm{SI} &= B_0\,B \\ \end{aligned} \]
From the unit conversion we can see that modifying the speed of light in numerical simulations only modify \(\epsilon_0\), but not \(\mu_0\).↩︎