Description of the model

(MM, A) LIMEPY: (Multi-Mass, Anisotropic) Lowered Isothermal Model Explorer in Python

Isotropic models

The isotropic distribution functions are defined as (Gomez-Leyton & Velazquez 2014)

\[f(\hat{E}) = \displaystyle A\,E_\gamma(g, \hat{E})\]

where \(\displaystyle \hat{E} = (\phi(r_{\rm t}) - \phi - v^2/2)/s^2\), \(\phi\) is the potential, \(r_{\rm t}\) is the trunction radius, \(s\) is a velocity scale and

\[\begin{split}\displaystyle E_\gamma(g, \hat{E}) = \begin{cases} \exp(\hat{E}), &g=0 \\ \displaystyle \exp(\hat{E})P(g, \hat{E}), &g>1 \end{cases}\end{split}\]

where \(0 < \phi-\phi(r_{\rm t}) <\phi_0/s^2\) is the (positive) potential and \(P(a, x)\) is the regularised lower incomplete gamma function \(P(a, x) = \gamma(a, x)/\Gamma(x)\). For some integer values of g several well known models are found

Anisotropic models

Radial anisotropy as in Michie (1963) can be included as follows

\[f(E, J^2) = \exp(-\hat{J}^2)f(\hat{E}),\]

where \(\hat{J}^2 = (rv_t)^2/(2r_{\rm a}^2s^2)\), here \(r_{\rm a}\) is the user-defined anisotropy radius.

Multi-mass model

Multi-mass models are found by summing the DFs of individual mass components and adopting for each component (following Gunn & Griffin (1979))

\[\begin{split}s_j &\propto \mu_j^{-\delta}\\ r_{{\rm a},j} &\propto \mu_j^{\eta}\end{split}\]

where \(\mu_j = m_j/\bar{m}\) and \(\bar{m}\) is the central density weighted mean mass.