/
PowerTriaxialPotential.py
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PowerTriaxialPotential.py
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###############################################################################
# PowerTriaxialPotential: Potential of a triaxial power-law
#
# amp
# rho(x,y,z)= ---------
# m^\alpha
#
# with m^2 = x^2+y^2/b^2+z^2/c^2
#
###############################################################################
import numpy
from ..util import conversion
from .EllipsoidalPotential import EllipsoidalPotential
class PowerTriaxialPotential(EllipsoidalPotential):
"""Class that implements triaxial potentials that are derived from power-law density models (including an elliptical power law)
.. math::
\\rho(r) = \\frac{\\mathrm{amp}}{r_1^3}\\,\\left(\\frac{r_1}{m}\\right)^{\\alpha}
where :math:`m^2 = x^2+y^2/b^2+z^2/c^2`.
"""
def __init__(
self,
amp=1.0,
alpha=1.0,
r1=1.0,
b=1.0,
c=1.0,
zvec=None,
pa=None,
glorder=50,
normalize=False,
ro=None,
vo=None,
):
"""
Initialize a triaxial power-law potential.
Parameters
----------
amp : float or Quantity, optional
Amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass or Gxmass.
alpha : float
Power-law exponent.
r1 : float or Quantity, optional
Reference radius for amplitude.
b : float
Y-to-x axis ratio of the density.
c : float
Z-to-x axis ratio of the density.
zvec : numpy.ndarray, optional
If set, a unit vector that corresponds to the z axis.
pa : float or Quantity, optional
If set, the position angle of the x axis (rad or Quantity).
glorder : int, optional
If set, compute the relevant force and potential integrals with Gaussian quadrature of this order.
ro : float, optional
Distance scale for translation into internal units (default from configuration file).
vo : float, optional
Velocity scale for translation into internal units (default from configuration file).
Notes
-----
- 2021-05-07 - 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",
)
r1 = conversion.parse_length(r1, ro=self._ro)
self.alpha = alpha
# Back to old definition
if self.alpha != 3.0:
self._amp *= r1 ** (self.alpha - 3.0) * 4.0 * numpy.pi / (3.0 - self.alpha)
# Multiply in constants
self._amp *= (3.0 - self.alpha) / 4.0 / numpy.pi
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 2.0 / (2.0 - self.alpha) * m ** (2.0 - self.alpha)
def _mdens(self, m):
"""Density as a function of m"""
return m**-self.alpha
def _mdens_deriv(self, m):
"""Derivative of the density as a function of m"""
return -self.alpha * m ** -(1.0 + self.alpha)