# Unit normalisations

All equations in Legolas are in dimensionless form, as is common practice when dealing with (M)HD.

As usual we have three degrees of freedom: the unit magnetic field $B_u$ and unit length $L_u$ should be set, together with either the unit density $\rho_u$ OR unit temperature $T_u$.

## Mean molecular weight

Unit normalisations depend on the molecular weight $\bar{\mu}$, and in Legolas we usually distinguish between two cases:

• Pure proton plasma: $\bar{\mu} = 1$, this is the default case.

$\bar{\mu} = \dfrac{m_i n_i}{n_i} = m_i \rightarrow \bar{\mu} = 1$
• Electron-proton plasma: $\bar{\mu} = 0.5$, in this case the molecular weight should be explicitly set.

$\bar{\mu} = \dfrac{m_e n_e + m_i n_i}{n_e + n_i} \simeq \dfrac{m_i n_i}{n_e + n_i} = \dfrac{1}{2}m_i \rightarrow \bar{\mu} = \dfrac{1}{2}$

## Normalisations

We have two options: either the unit density is set or the unit temperature is set. In both cases a unit magneticfield and a unit length should be set as well, so the unit pressure is given by

$p_u = \dfrac{B_u^2}{\mu_0},$

taking a plasma-beta reference value of 2. The magnetic constant is given by $\mu_0$ ($= 4\pi$ in cgs).

We then have the following options:

• If $\rho_u$ chosen along with $B_u$ and $L_u$:

$T_u = \dfrac{\bar{\mu}m_p p_u}{k_B \rho_u},$
• If $T_u$ chosen along with $B_u$ and $L_u$:

$\rho_u = \dfrac{\bar{\mu}m_p p_u}{k_B T_u},$

All other normalisations follow from these, where the Alfvén speed is assumed as reference velocity:

$\begin{gather} m_u = \rho_u L_u, \\ n_u = \dfrac{\rho_u}{m_p}, \\ v_u = \dfrac{B_u}{\sqrt{\mu_0 \rho_u}}, \\ t_u = \dfrac{L_u}{v_u}, \\ \Lambda_u = \dfrac{p}{t_u n_u^2}, \\ \kappa_u = \dfrac{\rho_u L_u v_u^3}{T_u}, \end{gather}$

which, from top to bottom, read as mass unit, numberdensity unit, velocity unit, time unit, cooling curve unit and conduction unit. The proton mass is given by $m_p$ and $k_B$ denotes the Boltzmann constant.

Note: The unit normalisations are only relevant when radiative cooling, thermal conduction or temperature-dependent resistivity is included. We always set base values though (as one should), which are given in cgs units by $B_u = 10$ G, $L_u = 10^9$ cm and $T_u = 10^6$ K. If no normalisations are specified these are used by default.