###############################################################################
# PlummerPotential.py: class that implements the Plummer potential
# GM
# phi(R,z) = - ---------------------------------
# \sqrt(R^2+z^2+b^2)
###############################################################################
import numpy
from .Potential import Potential, kms_to_kpcGyrDecorator, _APY_LOADED
if _APY_LOADED:
from astropy import units
[docs]class PlummerPotential(Potential):
"""Class that implements the Plummer potential
.. math::
\\Phi(R,z) = -\\frac{\\mathrm{amp}}{\\sqrt{R^2+z^2+b^2}}
with :math:`\\mathrm{amp} = GM` the total mass.
"""
[docs] def __init__(self,amp=1.,b=0.8,normalize=False,
ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a Plummer potential
INPUT:
amp - amplitude to be applied to the potential, the total mass (default: 1); can be a Quantity with units of mass or Gxmass
b - scale parameter (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:
2015-06-15 - Written - Bovy (IAS)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass')
if _APY_LOADED and isinstance(b,units.Quantity):
b= b.to(units.kpc).value/self._ro
self._b= b
self._scale= self._b
self._b2= self._b**2.
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
self._nemo_accname= 'Plummer'
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:
2015-06-15 - Started - Bovy (IAS)
"""
return -1./numpy.sqrt(R**2.+z**2.+self._b2)
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:
2015-06-15 - Written - Bovy (IAS)
"""
dPhidrr= -(R**2.+z**2.+self._b2)**-1.5
return dPhidrr*R
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:
2015-06-15 - Written - Bovy (IAS)
"""
dPhidrr= -(R**2.+z**2.+self._b2)**-1.5
return dPhidrr*z
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:
2015-06-15 - Written - Bovy (IAS)
"""
return 3./4./numpy.pi*self._b2*(R**2.+z**2.+self._b2)**-2.5
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:
2018-08-19 - Written - Bovy (UofT)
"""
Rb= R**2.+self._b2
return self._b2*z*(3.*Rb+2.*z**2.)/Rb**2.*(Rb+z**2.)**-1.5/2./numpy.pi
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_R2deriv
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:
2015-06-15 - Written - Bovy (IAS)
"""
return (self._b2-2.*R**2.+z**2.)*(R**2.+z**2.+self._b2)**-2.5
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:
2015-06-15 - Written - Bovy (IAS)
"""
return (self._b2+R**2.-2.*z**2.)*(R**2.+z**2.+self._b2)**-2.5
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:
2015-06-15 - Written - Bovy (IAS)
"""
return -3.*R*z*(R**2.+z**2.+self._b2)**-2.5
@kms_to_kpcGyrDecorator
def _nemo_accpars(self,vo,ro):
"""
NAME:
_nemo_accpars
PURPOSE:
return the accpars potential parameters for use of this potential with NEMO
INPUT:
vo - velocity unit in km/s
ro - length unit in kpc
OUTPUT:
accpars string
HISTORY:
2014-12-18 - Written - Bovy (IAS)
"""
ampl= self._amp*vo**2.*ro
return "0,%s,%s" % (ampl,self._b*ro)