Main parameters

All temporal and spatial units are normalized correspondingly to the timestep $\Delta t$ and the cell size $\Delta x$; the fields are normalized to the fiducial value $B_{\rm norm}$, and the densities are normalized to $n_{\rm ppc}$. We use the following equations

$$ d_e = \frac{c}{\omega_{\rm p}},~~\omega_{\rm p}^2=\frac{4\pi n_e e^2}{m_e},~~r_L=\gamma\beta \frac{m_e c^2}{|e|B},~~\sigma=\frac{B^2}{4\pi n_e m_e c^2}. $$

For all the details on how the code units are defined see the following section.

$c=\mathrm{CC}\frac{\Delta x}{\Delta t}$

$d_e=\mathrm{COMP}\Delta x$

$n_{\rm ppc}=\mathrm{PPC}\Delta x^{-3}$

$\mathrm{CC}=$

$\mathrm{COMP}=$

$\mathrm{PPC}=$

$\sigma=$

$\omega_{\rm p}^{-1}=~$$~\Delta t$

$\omega_B^{-1}=(\gamma\beta)^{-1}~$$~\Delta t$

$r_L=\gamma\beta~$$~\Delta x$

Pulsar setup

Pulsar setup is initialized with a conducting sphere of radius $R_*$ in the middle of the simulation box. The field is initially dipolar with an enforced strength $B_*$ at the surface. The conductor rotates with a period $P$. .

We use the following equations (all the quantities are in code units)

$$ \Omega = \frac{2\pi}{P},~~R_{\rm LC} = \frac{c}{\Omega},~~B_{\rm LC}\approx B_*\left(\frac{R_{\rm LC}}{R_*}\right)^{-3}, $$
$$ n_{\rm GJ} = \frac{\Omega B}{2\pi c|e|},~~\frac{\Delta V_{\rm pc}}{m_e c^2}=\omega_B^0\frac{B_*}{B_{\rm norm}}\frac{R_*}{c}\left(\frac{R_*}{R_{\rm LC}}\right)^2,~~s_{\rm LC}=\frac{d_e(n_{\rm GJ}^{\rm LC})}{R_{\rm LC}}. $$

$R_*=~$$~{\color{lightgray}\Delta x}$

$B_*=~$$~{\color{lightgray}B_{\rm norm}}$

$\frac{\Delta V_{\rm pc}}{m_e c^2}=~$

$d_e^*=~$${\color{lightgray}\langle\gamma\rangle^{1/2} \left(\frac{n}{n_{\rm GJ}^*}\right)^{-1/2} \Delta x}$

$n^*_{\rm GJ}=~$$~{\color{lightgray}\Delta x^{-3}}$

$\sigma^*=~$${\color{lightgray}\left(\frac{n}{n_{\rm GJ}^*}\right)^{-1}\left(\frac{B}{B_*}\right)^2}$

$P=~$$~{\color{lightgray}\Delta t}$

$R_{\rm LC}=~$$~{\color{lightgray}\Delta x}$

$B_{\rm LC}=~$$~{\color{lightgray}B_{\rm norm}}$

$d_e^{\rm LC}=~$${\color{lightgray}\langle\gamma\rangle^{1/2} \left(\frac{n}{n_{\rm GJ}^*}\right)^{-1/2} \Delta x}$

$r_L^{\rm LC}={\color{lightgray}\gamma\beta}~$${\color{lightgray}\left(\frac{B}{B_{\rm LC}}\right)^{-1} \Delta x}$

$s_{\rm LC}=~$

$n^{\rm LC}_{\rm GJ}=~$$~{\color{lightgray}\Delta x^{-3}}$

$\sigma^{\rm LC}=~$${\color{lightgray}\left(\frac{n}{n^{\rm LC}_{\rm GJ}}\right)^{-1}\left(\frac{B}{B_{\rm LC}}\right)^2}$

Other setup

Some explanation goes here...