# Source code for galpy.potential.PerfectEllipsoidPotential

###############################################################################
#   PerfectEllipsoidPotential.py: Potential of the perfect ellipsoid
#                                 (de Zeeuw 1985):
#
#                                 \rho(x,y,z) ~ 1/(1+m^2)^2
#
#                                 with m^2 = x^2+y^2/b^2+z^2/c^2
#
###############################################################################
import numpy
from ..util import conversion
from .EllipsoidalPotential import EllipsoidalPotential
[docs]class PerfectEllipsoidPotential(EllipsoidalPotential):
"""Potential of the perfect ellipsoid (de Zeeuw 1985):

.. math::

\\rho(x,y,z) = \\frac{\\mathrm{amp\,a}}{\\pi^2\\,bc}\\,\\frac{1}{(m^2+a^2)^2}

where :math:\\mathrm{amp} = GM is the total mass and :math:m^2 = x^2+y^2/b^2+z^2/c^2.
"""
[docs]    def __init__(self,amp=1.,a=5.,b=1.,c=1.,
zvec=None,pa=None,glorder=50,
normalize=False,ro=None,vo=None):
"""
NAME:

__init__

PURPOSE:

initialize a perfect ellipsoid potential

INPUT:

amp - amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass or G x mass

a - scale radius (can be Quantity)

b - y-to-x axis ratio of the density

c - z-to-x axis ratio of the density

zvec= (None) If set, a unit vector that corresponds to the z axis

pa= (None) If set, the position angle of the x axis (rad or Quantity)

glorder= (50) if set, compute the relevant force and potential integrals with Gaussian quadrature of this order

ro=, vo= distance and velocity scales for translation into internal units (default from configuration file)

OUTPUT:

(none)

HISTORY:

2018-08-06 - Started - Bovy (UofT)

"""
EllipsoidalPotential.__init__(self,amp=amp,b=b,c=c,
zvec=zvec,pa=pa,glorder=glorder,
ro=ro,vo=vo,amp_units='mass')
a= conversion.parse_length(a,ro=self._ro)
self.a= a
self.a2= self.a**2
self._scale= self.a
self._amp*= self.a/(numpy.pi**2*self._b*self._c)
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)): #pragma: no cover
self.normalize(normalize)
self.hasC= not self._glorder is None
self.hasC_dxdv= False
self.hasC_dens= self.hasC # works if mdens is defined, necessary for hasC
return None

def _psi(self,m):
"""\psi(m) = -\int_m^\infty d m^2 \rho(m^2)"""
return -1./(self.a2+m**2)

def _mdens(self,m):
"""Density as a function of m"""
return (self.a2+m**2)**-2

def _mdens_deriv(self,m):
"""Derivative of the density as a function of m"""
return -4.*m*(self.a2+m**2)**-3

def _mass(self,R,z=None,t=0.):
"""
NAME:
_mass
PURPOSE:
evaluate the mass within R (and z) for this potential; if z=None, integrate to ellipsoidal boundary
INPUT: