# Source code for galpy.potential.PowerSphericalPotentialwCutoff

###############################################################################
#   PowerSphericalPotentialwCutoff.py: spherical power-law potential w/ cutoff
#
#                                     amp
#                          rho(r)= ---------   e^{-(r/rc)^2}
#                                   r^\alpha
###############################################################################
import numpy
from scipy import special
from ..util import conversion
from .Potential import Potential, kms_to_kpcGyrDecorator
[docs]class PowerSphericalPotentialwCutoff(Potential):
"""Class that implements spherical potentials that are derived from
power-law density models

.. math::

\\rho(r) = \\mathrm{amp}\,\\left(\\frac{r_1}{r}\\right)^\\alpha\\,\\exp\\left(-(r/rc)^2\\right)

"""
[docs]    def __init__(self,amp=1.,alpha=1.,rc=1.,normalize=False,r1=1.,
ro=None,vo=None):
"""
NAME:

__init__

PURPOSE:

initialize a power-law-density potential

INPUT:

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

alpha= inner power

rc= cut-off 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:

2013-06-28 - Written - Bovy (IAS)

"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='density')
r1= conversion.parse_length(r1,ro=self._ro)
rc= conversion.parse_length(rc,ro=self._ro)
self.alpha= alpha
# Back to old definition
self._amp*= r1**self.alpha
self.rc= rc
self._scale= self.rc
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
self._nemo_accname= 'PowSphwCut'

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:
2013-06-28 - Started - Bovy (IAS)
"""
r= numpy.sqrt(R**2.+z**2.)
out= 2.*numpy.pi*self.rc**(3.-self.alpha)*(1/self.rc*special.gamma(1.-self.alpha/2.)*special.gammainc(1.-self.alpha/2.,(r/self.rc)**2.)-special.gamma(1.5-self.alpha/2.)*special.gammainc(1.5-self.alpha/2.,(r/self.rc)**2.)/r)
if isinstance(r,(float,int)):
if r == 0:
return 0.
else:
return out
else:
out[r==0]= 0.
return out

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:
2013-06-26 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R*R+z*z)
return -self._mass(r)*R/r**3.

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:
2013-06-26 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R*R+z*z)
return -self._mass(r)*z/r**3.

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:
2013-06-28 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R*R+z*z)
return 4.*numpy.pi*r**(-2.-self.alpha)*numpy.exp(-(r/self.rc)**2.)*R**2.\
+self._mass(r)/r**5.*(z**2.-2.*R**2.)

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:
2013-06-28 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R*R+z*z)
return 4.*numpy.pi*r**(-2.-self.alpha)*numpy.exp(-(r/self.rc)**2.)*z**2.\
+self._mass(r)/r**5.*(R**2.-2.*z**2.)

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:
2013-08-28 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R*R+z*z)
return R*z*(4.*numpy.pi*r**(-2.-self.alpha)*numpy.exp(-(r/self.rc)**2.)
-3.*self._mass(r)/r**5.)

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:
2013-06-28 - Written - Bovy (IAS)
"""
r= numpy.sqrt(R**2.+z**2.)
return 1./r**self.alpha*numpy.exp(-(r/self.rc)**2.)

def _mass(self,R,z=None,t=0.):
"""
NAME:
_mass
PURPOSE:
evaluate the mass within R for this potential
INPUT:
z - vertical height
t - time
OUTPUT:
the mass enclosed
HISTORY:
2013-XX-XX - Written - Bovy (IAS)
2021-04-07 - Switched to hypergeometric function equivalent to incomplete gamma function for better behavior at alpha < -3 - Bovy (UofT)
"""
if z is not None: raise AttributeError # use general implementation
return 2.*numpy.pi*R**(3.-self.alpha)/(1.5-self.alpha/2.)\
*special.hyp1f1(1.5-self.alpha/2.,2.5-self.alpha/2.,
-(R/self.rc)**2.)

@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**(self.alpha-2.)
return "0,%s,%s,%s" % (ampl,self.alpha,self.rc*ro)