###############################################################################
# SphericalShellPotential.py: The gravitational potential of a thin,
# spherical shell
###############################################################################
import numpy
from .Potential import Potential, _APY_LOADED
if _APY_LOADED:
from astropy import units
[docs]class SphericalShellPotential(Potential):
"""Class that implements the potential of an infinitesimally-thin, spherical shell
.. math::
\\rho(r) = \\frac{\\mathrm{amp}}{4\pi\,a^2}\\,\\delta(r-a)
with :math:`\\mathrm{amp} = GM` the mass of the shell.
"""
[docs] def __init__(self,amp=1.,a=0.75,normalize=False,ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a spherical shell potential
INPUT:
amp - mass of the shell (default: 1); can be a Quantity with units of mass or Gxmass
a= (0.75) radius of the shell (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.; note that because the force is always zero at r < a, this does not work if a > 1
ro=, vo= distance and velocity scales for translation into internal units (default from configuration file)
OUTPUT:
(none)
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass')
if _APY_LOADED and isinstance(a,units.Quantity):
a= a.to(units.kpc).value/self._ro
self.a= a
self.a2= a**2
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)):
if self.a > 1.:
raise ValueError('SphericalShellPotential with normalize= for a > 1 is not supported (because the force is always 0 at r=1)')
self.normalize(normalize)
self.hasC= False
self.hasC_dxdv= False
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:
2018-08-04 - Written - Bovy (UofT)
"""
r2= R**2+z**2
if r2 <= self.a2:
return -1./self.a
else:
return -1./numpy.sqrt(r2)
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:
2018-08-04 - Written - Bovy (UofT)
"""
r= numpy.sqrt(R**2+z**2)
if r <= self.a:
return 0.
else:
return -R/r**3
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:
2018-08-04 - Written - Bovy (UofT)
"""
r= numpy.sqrt(R**2+z**2)
if r <= self.a:
return 0.
else:
return -z/r**3
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:
2018-08-04 - Written - Bovy (UofT)
"""
r= numpy.sqrt(R**2+z**2)
if r <= self.a:
return 0.
else:
return (z**2-2*R**2)/r**5
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:
2018-08-04 - Written - Bovy (UofT)
"""
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:
2018-08-04 - Written - Bovy (UofT)
"""
r= numpy.sqrt(R**2+z**2)
if r <= self.a:
return 0.
else:
return -3*R*z/r**5
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 surface density
HISTORY:
2018-08-19 - Written - Bovy (UofT)
"""
r2= R**2+z**2
if r2 != self.a2:
return 0.
else: # pragma: no cover
return numpy.infty
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 density
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
if R > self.a: return 0.
h= numpy.sqrt(self.a2-R**2)
if z < h: return 0.
else: return 1./(2.*numpy.pi*self.a*h)
