# Source code for galpy.potential.PseudoIsothermalPotential

###############################################################################
#   PseudoIsothermalPotential.py: class that implements the pseudo-isothermal
#                                 halo potential
###############################################################################
import numpy
from ..util import conversion
from .Potential import Potential
[docs]class PseudoIsothermalPotential(Potential):
"""Class that implements the pseudo-isothermal potential

.. math::

\\rho(r) = \\frac{\\mathrm{amp}}{4\\,\pi\\, a^3}\\,\\frac{1}{1+(r/a)^2}

"""
[docs]    def __init__(self,amp=1.,a=1.,normalize=False,
ro=None,vo=None):
"""
NAME:

__init__

PURPOSE:

initialize a pseudo-isothermal potential

INPUT:

amp - amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass or Gxmass

a - core radius (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:

2015-12-04 - Started - Bovy (UofT)

"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass')
a= conversion.parse_length(a,ro=self._ro)
self.hasC= True
self.hasC_dxdv= True
self.hasC_dens= True
self._a= a
self._a2= a**2.
self._a3= a**3.
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)): #pragma: no cover
self.normalize(normalize)
return None

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:
2015-12-04 - Started - Bovy (UofT)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
out= (0.5*numpy.log(1+r2/self._a2)
+self._a/r*numpy.arctan(r/self._a))/self._a
if isinstance(r,(float,int)):
if r == 0:
return 1./self._a
else:
return out
else:
out[r==0]= 1./self._a
return out

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:
2015-12-04 - Started - Bovy (UofT)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
return -(1./r-self._a/r2*numpy.arctan(r/self._a))/self._a*R/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:
2015-12-04 - Started - Bovy (UofT)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
return -(1./r-self._a/r2*numpy.arctan(r/self._a))/self._a*z/r

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:
2015-12-04 - Started - Bovy (UofT)
"""
return 1./(1.+(R**2.+z**2.)/self._a2)/4./numpy.pi/self._a3

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:
2011-10-09 - Written - Bovy (IAS)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
return (1./r2*(1.-R**2./r2*(3.*self._a2+2.*r2)/(self._a2+r2))\
+self._a/r2/r*(3.*R**2./r2-1.)*numpy.arctan(r/self._a))\
/self._a

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:
2012-07-25 - Written - Bovy (IAS@MPIA)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
return (1./r2*(1.-z**2./r2*(3.*self._a2+2.*r2)/(self._a2+r2))\
+self._a/r2/r*(3.*z**2./r2-1.)*numpy.arctan(r/self._a))\
/self._a

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)
"""
r2= R**2.+z**2.
r= numpy.sqrt(r2)
return (3.*self._a/r2/r2*numpy.arctan(r/self._a)\
-1./r2/r*((3.*self._a2+2.*r2)/(r2+self._a2)))*R*z/r\
/self._a