Units

Aperture uses a unit system based on the Heaviside-Lorentz units. In the HL units, Maxwell equations look like:

\[ \begin{align} \nabla\cdot\mathbf{E} &= \rho \\ \nabla\cdot\mathbf{B} &= 0 \\ \frac{\partial \mathbf{E}}{\partial t} &= c\nabla\times\mathbf{B} - \mathbf{J} \\ \frac{\partial \mathbf{B}}{\partial t} &= -c\nabla\times\mathbf{E} \end{align} \]

In addition, we need to choose a length scale and time scale to make the equations dimensionless. We demand that \(c = 1\), therefore choosing a length scale automatically determines the appropriate time scale, and vice versa. Energy is measured using units of \(m_ec^2\), while particle momentum is measured in units of \(m_ec\). If we specify a unit of time as \(t_0 = \omega_0^{-1}\), the Lorentz force then takes the dimensionless form:

\[ \frac{d\mathbf{p}}{m_ec\omega_0 dt} = \frac{e}{m_ec\omega_0}\left(\mathbf{E} + \frac{\mathbf{v}}{c}\times\mathbf{B}\right), \]

It is therefore appropriate to use \(eE/m_ec\omega_0\) as the dimensionless form of the electric (or magnetic) field. The dimensionless value of the magnetic field is therefore equal to the dimensionless ratio \(\omega_B/\omega_0\), or the dimensionless electron cyclotron frequency. The dimensionless charge density can be defined using the Gauss's law \(\nabla\cdot\mathbf{E}=\rho\):

\[ \frac{eE}{m_ec\omega_0}\frac{c/\omega_0}{x} = \frac{e\rho}{m_e\omega_0^2}. \]

The right hand side is therefore the natural definition of dimensionless charge density. In the code units, the dimensionless plasma frequency is nicely related to the charge density as follows:

\[ \omega_p \equiv \sqrt{\frac{ne^2}{m_e}} = \sqrt{\frac{e\rho}{m_e\omega_0^2}}\omega_0. \]

In other words, the dimensionless plasma frequency is simply the square root of the dimensionless charge density, \(\tilde{\omega}_p = \sqrt{\tilde{\rho}}\). Tilde here means the dimensionless quantity. This relationship is very convenient to use in plasma simulations.

In HL units, the energy density of electric/magnetic field is \(\mathbf{E}^2/2\) or \(\mathbf{B}^2/2\) respectively, in contrast with the corresponding expressions of \(\mathbf{E}^2/8\pi\) and \(\mathbf{B}^2/8\pi\) in CGS units. The magnetization \(\sigma\) is now defined as:

\[ \sigma \equiv \frac{B^2}{nm_ec^2} = \frac{(eB/m_ec\omega_0)^2}{e^2nm_ec^2}m_e^2c^2\omega_0^2 = \frac{(eB/m_ec\omega_0)^2}{e\rho/m_e\omega_0^2}. \]

Identifying the terms, one can see that \(\sigma = \tilde{B}^2/\tilde{\rho}\).

In an actual simulation, one needs to specify either a length scale \(x_0\) or a time scale \(\omega_0^{-1}\), and it will completely specify the dimensionless unit system. For example, for a global pulsar simulation, the natural length scale is either the stellar radius \(R_*\), or the light cylinder radius \(R_\mathrm{LC}\). For a black hole simulation the natural length scale may be the gravitational radius \(r_g\). For a reconnection simulation it is often okay to choose the upstream \(\omega_0 = \omega_p\). In that case, one simply needs to initialize the upstream plasma density such that \(\tilde{\rho} = 1\).