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Let's take velocity dispersion as an example here. There are two relevant velocity dispersions in our studies. The first is the true 3D/2D velocity dispersion $\sigma$ which we cannot measure in the real universe. \cite{2008ApJ...672..122E} measured $\sigma$ for halos in N-body simulations and showed that they relate to the critical mass $M_{200}$ of the host halo on a very tight relation  \begin{equation}  \sigma = 1093 \frac{h(z) M}{1e15\M_{odot}}^{0.34} M}{1e15 M_{odot}}^{0.34}  \end{equation}