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
# Adjust amp
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:
R - Galactocentric cylindrical radius
z - vertical height
t - time
OUTPUT:
the mass enclosed
HISTORY:
2021-03-08 - Written - Bovy (UofT)
"""
if not z is None: raise AttributeError # Hack to fall back to general
return 2.*numpy.pi*self._b*self._c/self.a\
*(numpy.arctan(R/self.a)-R*self.a/(1.+R**2.))