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IsochronePotential.py
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IsochronePotential.py
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###############################################################################
# IsochronePotential.py: The isochrone potential
#
# - amp
# Phi(r)= ---------------------
# b + sqrt{b^2+r^2}
###############################################################################
import numpy
from ..util import conversion
from .Potential import Potential
class IsochronePotential(Potential):
"""Class that implements the Isochrone potential
.. math::
\\Phi(r) = -\\frac{\\mathrm{amp}}{b+\\sqrt{b^2+r^2}}
with :math:`\\mathrm{amp} = GM` the total mass.
"""
def __init__(self, amp=1.0, b=1.0, normalize=False, ro=None, vo=None):
"""
Initialize an isochrone potential.
Parameters
----------
amp : float or Quantity, optional
Amplitude to be applied to the potential, the total mass. Can be a Quantity with units of mass or Gxmass.
b : float or Quantity, optional
Scale radius of the isochrone potential.
normalize : bool or float, optional
If True, normalize such that vc(1.,0.)=1., or, if given as a number, such that the force is this fraction of the force necessary to make vc(1.,0.)=1. Default is False.
ro : float or Quantity, optional
Distance scale for translation into internal units (default from configuration file).
vo : float or Quantity, optional
Velocity scale for translation into internal units (default from configuration file).
Notes
-----
- 2013-09-08 - Written - Bovy (IAS)
"""
Potential.__init__(self, amp=amp, ro=ro, vo=vo, amp_units="mass")
b = conversion.parse_length(b, ro=self._ro)
self.b = b
self._scale = self.b
self.b2 = self.b**2.0
if normalize or (
isinstance(normalize, (int, float)) and not isinstance(normalize, bool)
): # pragma: no cover
self.normalize(normalize)
self.hasC = True
self.hasC_dxdv = True
self.hasC_dens = True
def _evaluate(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return -1.0 / (self.b + rb)
def _Rforce(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
dPhidrr = -1.0 / rb / (self.b + rb) ** 2.0
return dPhidrr * R
def _zforce(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
dPhidrr = -1.0 / rb / (self.b + rb) ** 2.0
return dPhidrr * z
def _R2deriv(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return (
-(
-self.b**3.0
- self.b * z**2.0
+ (2.0 * R**2.0 - z**2.0 - self.b**2.0) * rb
)
/ rb**3.0
/ (self.b + rb) ** 3.0
)
def _z2deriv(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return (
-(
-self.b**3.0
- self.b * R**2.0
- (R**2.0 - 2.0 * z**2.0 + self.b**2.0) * rb
)
/ rb**3.0
/ (self.b + rb) ** 3.0
)
def _Rzderiv(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return -R * z * (self.b + 3.0 * rb) / rb**3.0 / (self.b + rb) ** 3.0
def _dens(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return (
(3.0 * (self.b + rb) * rb**2.0 - r2 * (self.b + 3.0 * rb))
/ rb**3.0
/ (self.b + rb) ** 3.0
/ 4.0
/ numpy.pi
)
def _surfdens(self, R, z, phi=0.0, t=0.0):
r2 = R**2.0 + z**2.0
rb = numpy.sqrt(r2 + self.b2)
return (
self.b
* (
(R * z) / r2
- (self.b * R * z * (self.b**2 + 2.0 * R**2 + z**2))
/ ((self.b**2 + R**2) * r2 * rb)
+ numpy.arctan(z / R)
- numpy.arctan(self.b * z / R / rb)
)
/ R**3
/ 2.0
/ numpy.pi
)