# Source code for galpy.potential.IsochronePotential

###############################################################################
#   IsochronePotential.py: The isochrone potential
#
#                                     - amp
#                          Phi(r)= ---------------------
#                                   b + sqrt{b^2+r^2}
###############################################################################
import numpy
from ..util import conversion
from .Potential import Potential
[docs]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.
"""
[docs]    def __init__(self,amp=1.,b=1.,normalize=False,
ro=None,vo=None):
"""
NAME:

__init__

PURPOSE:

initialize an isochrone 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 radius of the isochrone potential (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:

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.
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.,t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at R,z
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
Phi(R,z)
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return -1./(self.b+rb)

def _Rforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rforce
PURPOSE:
evaluate the radial force for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
dPhidrr= -1./rb/(self.b+rb)**2.
return dPhidrr*R

def _zforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_zforce
PURPOSE:
evaluate the vertical force for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
the vertical force
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
dPhidrr= -1./rb/(self.b+rb)**2.
return dPhidrr*z

def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rderiv
PURPOSE:
evaluate the second radial derivative for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return -(-self.b**3.-self.b*z**2.+(2.*R**2.-z**2.-self.b**2.)*rb)/\
rb**3./(self.b+rb)**3.

def _z2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_z2deriv
PURPOSE:
evaluate the second vertical derivative for this potential
INPUT:
z - vertical height
phi - azimuth
t- time
OUTPUT:
the second vertical derivative
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return -(-self.b**3.-self.b*R**2.-(R**2.-2.*z**2.+self.b**2.)*rb)/\
rb**3./(self.b+rb)**3.

def _Rzderiv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rzderiv
PURPOSE:
evaluate the mixed R,z derivative for this potential
INPUT:
z - vertical height
phi - azimuth
t- time
OUTPUT:
d2phi/dR/dz
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return -R*z*(self.b+3.*rb)/\
rb**3./(self.b+rb)**3.

def _dens(self,R,z,phi=0.,t=0.):
"""
NAME:
_dens
PURPOSE:
evaluate the density for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
the density
HISTORY:
2013-09-08 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return (3.*(self.b+rb)*rb**2.-r2*(self.b+3.*rb))/\
rb**3./(self.b+rb)**3./4./numpy.pi

def _surfdens(self,R,z,phi=0.,t=0.):
"""
NAME:
_surfdens
PURPOSE:
evaluate the surface density for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
the surface density
HISTORY:
2018-08-19 - Written - Bovy (UofT)
"""
r2= R**2.+z**2.
rb= numpy.sqrt(r2+self.b2)
return self.b*((R*z)/r2-(self.b*R*z*(self.b**2+2.*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./numpy.pi