# 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.

## Mean molecular weight

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

• Electron-proton plasma: $\bar{\mu} = 0.5$, this is the default case.

$\bar{\mu} = \dfrac{m_e n_e + m_i n_i}{n_e + n_i} \simeq \dfrac{m_i n_i}{n_e + n_i} = \dfrac{1}{2}m_i \rightarrow \bar{\mu} = \dfrac{1}{2}$
• Pure proton plasma: $\bar{\mu} = 1$, in this case the molecular weight should be explicitly set.

$\bar{\mu} = \dfrac{m_i n_i}{n_i} = m_i \rightarrow \bar{\mu} = 1$

## Normalisations

Legolas has three options to specify units, all in cgs. In what follows $m_p$ denotes the proton mass, $k_B$ the Boltzmann constant, and $\mu_0 = 4\pi$ the magnetic constant.

1. Reference unit density, unit magnetic field and unit length $(\rho_u, B_u, L_u)$, then

$p_u = \frac{B_u^2}{\mu_0}, \quad T_u = \frac{\bar{\mu} p_u m_p}{k_B \rho_u}, \quad n_u = \frac{\rho_u}{m_p}, \quad v_u = \frac{B_u}{\sqrt{\mu_0 \rho_u}}.$
2. Reference unit temperature, unit magnetic field and unit length $(T_u, B_u, L_u)$, then

$p_u = \frac{B_u^2}{\mu_0}, \quad \rho_u = \frac{\bar{\mu} p_u m_p}{k_B T_u}, \quad n_u = \frac{\rho_u}{m_p}, \quad v_u = \frac{B_u}{\sqrt{\mu_0 \rho_u}}.$
3. Reference unit numberdensity, unit temperature and unit length $(n_u, T_u, L_u)$, then

$p_u = \bar{\mu} n_u k_B T_u, \quad \rho_u = m_p n_u, \quad v_u = \sqrt{\frac{p_u}{\rho_u}}, \quad B_u = \sqrt{\mu_0 p_u}.$

All other normalisations follow from those above and are given by

• unit mass: $M_u = \rho_u L_u^3$
• unit time: $t_u = \dfrac{L_u}{v_u}$
• unit resistivity: $\eta_u = \dfrac{L_u^2}{t_u}$
• unit cooling curve: $\Lambda_u = \dfrac{p_u}{t_u n_u^2}$
• unit conduction: $\kappa_u = \dfrac{\rho_u L_u v_u^3}{T_u}$

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 set using option 2. with default values $B_u = 10$ G, $L_u = 10^9$ cm and $T_u = 10^6$ K.