###############################################################################
# PowerSphericalPotential.py: General class for potentials derived from
# densities with two power-laws
#
# amp
# rho(r)= ---------
# r^\alpha
###############################################################################
import numpy
from scipy import special
from .Potential import Potential, _APY_LOADED
if _APY_LOADED:
from astropy import units
[docs]class PowerSphericalPotential(Potential):
"""Class that implements spherical potentials that are derived from power-law density models
.. math::
\\rho(r) = \\frac{\\mathrm{amp}}{r_1^3}\\,\\left(\\frac{r_1}{r}\\right)^{\\alpha}
"""
[docs] def __init__(self,amp=1.,alpha=1.,normalize=False,r1=1.,
ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a power-law-density potential
INPUT:
amp - amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass or Gxmass
alpha - inner power
r1= (1.) reference radius for amplitude (can be Quantity)
normalize - 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.
ro=, vo= distance and velocity scales for translation into internal units (default from configuration file)
OUTPUT:
(none)
HISTORY:
2010-07-10 - Written - Bovy (NYU)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass')
if _APY_LOADED and isinstance(r1,units.Quantity):
r1= r1.to(units.kpc).value/self._ro
self.alpha= alpha
# Back to old definition
if self.alpha != 3.:
self._amp*= r1**(self.alpha-3.)*4.*numpy.pi/(3.-self.alpha)
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)):
self.normalize(normalize)
self.hasC= True
self.hasC_dxdv= True
self.hasC_dens= True
def _evaluate(self,R,z,phi=0.,t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at R,z
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
Phi(R,z)
HISTORY:
2010-07-10 - Started - Bovy (NYU)
"""
if self.alpha == 2.:
return numpy.log(R**2.+z**2.)/2.
else:
return -(R**2.+z**2.)**(1.-self.alpha/2.)/(self.alpha-2.)
def _Rforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rforce
PURPOSE:
evaluate the radial force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the radial force
HISTORY:
2010-07-10 - Written - Bovy (NYU)
"""
return -R/(R**2.+z**2.)**(self.alpha/2.)
def _zforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_zforce
PURPOSE:
evaluate the vertical force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the vertical force
HISTORY:
2010-07-10 - Written - Bovy (NYU)
"""
return -z/(R**2.+z**2.)**(self.alpha/2.)
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rderiv
PURPOSE:
evaluate the second radial derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the second radial derivative
HISTORY:
2011-10-09 - Written - Bovy (NYU)
"""
return 1./(R**2.+z**2.)**(self.alpha/2.)\
-self.alpha*R**2./(R**2.+z**2.)**(self.alpha/2.+1.)
def _z2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_z2deriv
PURPOSE:
evaluate the second vertical derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t- time
OUTPUT:
the second vertical derivative
HISTORY:
2012-07-26 - Written - Bovy (IAS@MPIA)
"""
return self._R2deriv(z,R) #Spherical potential
def _Rzderiv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rzderiv
PURPOSE:
evaluate the mixed R,z derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
d2phi/dR/dz
HISTORY:
2013-08-28 - Written - Bovy (IAs)
"""
return -self.alpha*R*z*(R**2.+z**2.)**(-1.-self.alpha/2.)
def _dens(self,R,z,phi=0.,t=0.):
"""
NAME:
_dens
PURPOSE:
evaluate the density for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the density
HISTORY:
2013-01-09 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R**2.+z**2.)
return (3.-self.alpha)/4./numpy.pi/r**self.alpha
def _surfdens(self,R,z,phi=0.,t=0.):
"""
NAME:
_surfdens
PURPOSE:
evaluate the surface density for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the surface density
HISTORY:
2018-08-19 - Written - Bovy (UofT)
"""
return (3.-self.alpha)/2./numpy.pi*z*R**-self.alpha\
*special.hyp2f1(0.5,self.alpha/2.,1.5,-(z/R)**2)
[docs]class KeplerPotential(PowerSphericalPotential):
"""Class that implements the Kepler (point mass) potential
.. math::
\\Phi(r) = -\\frac{\\mathrm{amp}}{r}
with :math:`\\mathrm{amp} = GM` the mass.
"""
[docs] def __init__(self,amp=1.,normalize=False,
ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a Kepler, point-mass potential
INPUT:
amp - amplitude to be applied to the potential, the mass of the point mass (default: 1); can be a Quantity with units of mass density or Gxmass density
alpha - inner power
normalize - 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.
ro=, vo= distance and velocity scales for translation into internal units (default from configuration file)
OUTPUT:
(none)
HISTORY:
2010-07-10 - Written - Bovy (NYU)
"""
PowerSphericalPotential.__init__(self,amp=amp,normalize=normalize,
alpha=3.,ro=ro,vo=vo)
def _mass(self,R,z=0.,t=0.):
"""
NAME:
_mass
PURPOSE:
evaluate the mass within R for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
t - time
OUTPUT:
the mass enclosed
HISTORY:
2014-07-02 - Written - Bovy (IAS)
"""
return 1.
