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Description of the model
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(MM, A) LIMEPY: (Multi-Mass, Anisotropic) Lowered Isothermal Model Explorer in Python 
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Isotropic models
^^^^^^^^^^^^^^^^


The isotropic distribution functions are defined as `(Gomez-Leyton \&
Velazquez 2014) <http://adsabs.harvard.edu/abs/2014JSMTE..04..006G>`_

.. math::
   f(\hat{E}) = \displaystyle A\,E_\gamma(g, \hat{E})

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

.. math:: \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}

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

*  g = 0 : `Woolley (1954) <http://adsabs.harvard.edu/abs/1954MNRAS.114..191W>`_
*  g = 1 : `King (1966) <http://adsabs.harvard.edu/abs/1966AJ.....71...64K>`_
*  g = 2 : `Wilson (1975) <http://adsabs.harvard.edu/abs/1975AJ.....80..175W>`_

Anisotropic models
^^^^^^^^^^^^^^^^^^

Radial anisotropy as in `Michie (1963)
<http://adsabs.harvard.edu/abs/1963MNRAS.125..127M>`_ can be
included as follows

.. math::
   f(E, J^2) = \exp(-\hat{J}^2)f(\hat{E}),

where :math:`\hat{J}^2 = (rv_t)^2/(2r_{\rm a}^2s^2)`, here :math:`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) <http://adsabs.harvard.edu/abs/1979AJ.....84..752G>`_)

.. math::
   s_j       &\propto  \mu_j^{-\delta}\\
   r_{{\rm a},j}  &\propto  \mu_j^{\eta}

where :math:`\mu_j = m_j/\bar{m}` and :math:`\bar{m}` is the central density weighted mean mass.