Source code for galpy.potential.FlattenedPowerPotential

###############################################################################
#   FlattenedPowerPotential.py: Power-law potential that is flattened in the
#                               potential (NOT the density)
#
#                                     amp
#                          phi(R,z)= --------- ; m^2 = R^2 + z^2/q^2
#                                   m^\alpha
###############################################################################
import numpy
from astropy import units
_CORE=10**-8
[docs]class FlattenedPowerPotential(Potential):
"""Class that implements a power-law potential that is flattened in the potential (NOT the density)

.. math::

\\Phi(R,z) = -\\frac{\\mathrm{amp}\,r_1^\\alpha}{\\alpha\\,\\left(R^2+(z/q)^2+\\mathrm{core}^2\\right)^{\\alpha/2}}

and the same as LogarithmicHaloPotential for :math:\\alpha=0

See Figure 1 in Evans (1994) <http://adsabs.harvard.edu/abs/1994MNRAS.267..333E>_ for combinations of alpha and q that correspond to positive densities

"""
[docs]    def __init__(self,amp=1.,alpha=0.5,q=0.9,core=_CORE,normalize=False,r1=1.,
ro=None,vo=None):
"""
NAME:

__init__

PURPOSE:

initialize a flattened power-law potential

INPUT:

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

alpha - power

q - flattening

core - core radius (can be Quantity)

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:

2013-01-09 - Written - Bovy (IAS)

"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='velocity2')
core= core.to(units.kpc).value/self._ro
r1= r1.to(units.kpc).value/self._ro
self.alpha= alpha
self.q2= q**2.
self.core2= core**2.
# Back to old definition
self._amp*= r1**self.alpha
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:
2013-01-09 - Started - Bovy (IAS)
"""
if self.alpha == 0.:
return 1./2.*numpy.log(R**2.+z**2./self.q2+self.core2)
else:
m2= self.core2+R**2.+z**2./self.q2
return -m2**(-self.alpha/2.)/self.alpha

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:
2010-07-10 - Written - Bovy (NYU)
"""
if self.alpha == 0.:
return -R/(R**2.+z**2./self.q2+self.core2)
else:
m2= self.core2+R**2.+z**2./self.q2
return -m2**(-self.alpha/2.-1.)*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:
2010-07-10 - Written - Bovy (NYU)
"""
if self.alpha == 0.:
return -z/self.q2/(R**2.+z**2./self.q2+self.core2)
else:
m2= self.core2+R**2.+z**2./self.q2
return -m2**(-self.alpha/2.-1.)*z/self.q2

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:
2011-10-09 - Written - Bovy (NYU)
"""
if self.alpha == 0.:
denom= 1./(R**2.+z**2./self.q2+self.core2)
return denom-2.*R**2.*denom**2.
else:
m2= self.core2+R**2.+z**2./self.q2
return -m2**(-self.alpha/2.-1.)*((self.alpha+2)*R**2./m2-1.)

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:
2012-07-26 - Written - Bovy (IAS@MPIA)
"""
if self.alpha == 0.:
denom= 1./(R**2.+z**2./self.q2+self.core2)
return denom/self.q2-2.*z**2.*denom**2./self.q2**2.
else:
m2= self.core2+R**2.+z**2./self.q2
return -1./self.q2*m2**(-self.alpha/2.-1.)*((self.alpha+2)*z**2./m2/self.q2-1.)

def _dens(self,R,z,phi=0.,t=0.):
"""
NAME:
_dens
PURPOSE:
evaluate the density force for this potential
INPUT:
z - vertical height
phi - azimuth
t - time
OUTPUT:
the density
HISTORY:
2013-01-09 - Written - Bovy (IAS)
"""
if self.alpha == 0.:
return 1./4./numpy.pi/self.q2*((2.*self.q2+1.)*self.core2+R**2.\
+(2.-1./self.q2)*z**2.)/\
(R**2.+z**2./self.q2+self.core2)**2.
else:
m2= self.core2+R**2.+z**2./self.q2
return 1./self.q2*(self.core2*(1.+2.*self.q2)+R**2.*(1.-self.alpha*self.q2)+z**2.*(2.-(1.+self.alpha)/self.q2))*m2**(-self.alpha/2.-2.)/4./numpy.pi