# 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.