Peebles’ parametrisation. This is due to it being more robust to variations in the position of the structure’s outer radius and therefore not as strongly affected by the many minor mergers over a halo’s merging history. Therefore, the two descriptions are not readily interchangeable and results need to be compared using the same parameter.

Peebles Spin Parameter {#sec:peebles}
----------------------

[@peebles_1969] proposed to parameterise the (sub-)halo spin in the following way: $$\lambda = \frac{J \sqrt{{\left|E\right|}}}{GM^{5/2}},
 \label{eqn:peebles}$$ where $J$ is total angular momentum, $E$ the energy and $M$ the mass of the (sub-)structure. With this choice a value of $\lambda \simeq 0.4$ represents a purely rotationally supported object [@frenk_2012].

Applying this parametrisation @bett_2007 determined the spin distribution of [haloes]{} in the Millennium Simulation [@springel_millenium_2005]. The Millennium Simulation has a mass resolution of $m_p= 8.6\times10^8\,h^{-1}{\rm M_\odot}$ and therefore contains very few sub[haloes]{}. The vast majority of the objects in the TREEclean catalogue of @bett_2007 are [haloes]{} rather than sub[haloes]{}. The probability density function of $\log\lambda$ they found to fit the distribution used the following parametrisation; $$P(\log \lambda) = A \left( \frac{\lambda}{\lambda_0} \right) ^3
\exp \left[ -\alpha \left(\frac{\lambda}{\lambda_0}\right)^{3/\alpha} \right]
 \label{eqn:bettfit}$$ where A is given by, $$A = 3 \ln 10 \frac{\alpha^{\alpha-1}}{\Gamma(\alpha)},
\label{eqn:bettA}$$ and $\Gamma(\alpha)$ 