###############################################################################
# LogarithmicHaloPotential.py: class that implements the logarithmic
# potential Phi(r) = vc**2 ln(r)
###############################################################################
import warnings
import numpy
from .Potential import Potential, kms_to_kpcGyrDecorator
from ..util import galpyWarning, conversion
_CORE=10**-8
[docs]class LogarithmicHaloPotential(Potential):
"""Class that implements the logarithmic potential
.. math::
\\Phi(R,z) = \\frac{\\mathrm{amp}}{2}\\,\\ln\\left[R^2+\\left(\\frac{z}{q}\\right)^2+\\mathrm{core}^2\\right]
Alternatively, the potential can be made triaxial by adding a parameter :math:`b`
.. math::
\\Phi(x,y,z) = \\frac{\\mathrm{amp}}{2}\\,\\ln\\left[x^2+\\left(\\frac{y}{b}\\right)^2+\\left(\\frac{z}{q}\\right)^2+\\mathrm{core}^2\\right]
With these definitions, :math:`\\sqrt{\mathrm{amp}}` is the circular velocity at :math:`r \gg \mathrm{core}` at :math:`(y,z) = (0,0)`.
"""
[docs] def __init__(self,amp=1.,core=_CORE,q=1.,b=None,normalize=False,
ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a logarithmic potential
INPUT:
amp - amplitude to be applied to the potential (default: 1); can be a Quantity with units of velocity-squared
core - core radius at which the logarithm is cut (can be Quantity)
q - potential flattening (z/q)**2.
b= (None) if set, shape parameter in y-direction (y --> y/b; see definition)
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:
2010-04-02 - Started - Bovy (NYU)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='velocity2')
core= conversion.parse_length(core,ro=self._ro)
self.hasC= True
self.hasC_dxdv= True
self.hasC_dens= True
self._core2= core**2.
self._q= q
self._b= b
if not self._b is None:
self.isNonAxi= True
self._1m1overb2= 1.-1./self._b**2.
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)): #pragma: no cover
self.normalize(normalize)
self._nemo_accname= 'LogPot'
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:
2010-04-02 - Started - Bovy (NYU)
2010-04-30 - Adapted for R,z - Bovy (NYU)
"""
if self.isNonAxi:
return 1./2.*numpy.log(R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
+(z/self._q)**2.+self._core2)
else:
return 1./2.*numpy.log(R**2.+(z/self._q)**2.+self._core2)
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:
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
return -Rt2/R/(Rt2+(z/self._q)**2.+self._core2)
else:
return -R/(R**2.+(z/self._q)**2.+self._core2)
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:
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
return -z/self._q**2./(Rt2+(z/self._q)**2.+self._core2)
else:
return -z/self._q**2./(R**2.+(z/self._q)**2.+self._core2)
def _phiforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_phiforce
PURPOSE:
evaluate the azimuthal force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the azimuthal force
HISTORY:
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
return R**2./(Rt2+(z/self._q)**2.+self._core2)\
*numpy.sin(2.*phi)*self._1m1overb2/2.
else:
return 0
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:
"""
if self.isNonAxi:
R2= R**2.
Rt2= R2*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
denom2= denom**2.
return 1./4./numpy.pi\
*(2.*Rt2/R2*(denom-Rt2*denom2)\
+denom/self._q**2.-2.*z**2.*denom2/self._q**4.\
-self._1m1overb2\
*(2.*R2*numpy.sin(2.*phi)**2./4.*self._1m1overb2\
*denom2+denom*numpy.cos(2.*phi)))
else:
return 1./4./numpy.pi/self._q**2.*((2.*self._q**2.+1.)*self._core2+R**2.\
+(2.-self._q**-2.)*z**2.)/\
(R**2.+(z/self._q)**2.+self._core2)**2.
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)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
return (denom-2.*Rt2*denom**2.)*Rt2/R**2.
else:
denom= 1./(R**2.+(z/self._q)**2.+self._core2)
return denom-2.*R**2.*denom**2.
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)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
return denom/self._q**2.-2.*z**2.*denom**2./self._q**4.
else:
denom= 1./(R**2.+(z/self._q)**2.+self._core2)
return denom/self._q**2.-2.*z**2.*denom**2./self._q**4.
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)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
return -2.*Rt2/R*z/self._q**2./(Rt2+(z/self._q)**2.+self._core2)**2.
else:
return -2.*R*z/self._q**2./(R**2.+(z/self._q)**2.+self._core2)**2.
def _phi2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_phi2deriv
PURPOSE:
evaluate the second azimuthal derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the second azimuthal derivative
HISTORY:
2017-10-15 - Written - Bovy (UofT)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
return -self._1m1overb2\
*(R**4.*numpy.sin(2.*phi)**2./2.*self._1m1overb2\
*denom**2.
+R**2.*denom*numpy.cos(2.*phi))
else:
return 0.
def _Rphideriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rphideriv
PURPOSE:
evaluate the mixed R,phi derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
d2Phi/dR/dphi
HISTORY:
2017-10-15 - Written - Bovy (UofT)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
return -(denom-Rt2*denom**2.)*R*numpy.sin(2.*phi)*self._1m1overb2
else:
return 0.
def _phizderiv(self,R,z,phi=0.,t=0.):
"""
NAME:
_phizderiv
PURPOSE:
evaluate the mixed phi,z derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
d2Phi/dz/dphi
HISTORY:
2021-04-30 - Written - Bovy (UofT)
"""
if self.isNonAxi:
Rt2= R**2.*(1.-self._1m1overb2*numpy.sin(phi)**2.)
denom= 1./(Rt2+(z/self._q)**2.+self._core2)
return 2*R**2*z*numpy.sin(phi)*numpy.cos(phi)*self._1m1overb2\
*denom**2/self._q**2
else:
return 0.
@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)
"""
warnings.warn("NEMO's LogPot does not allow flattening in z (for some reason); therefore, flip y and z in NEMO wrt galpy; also does not allow the triaxial b parameter",galpyWarning)
ampl= self._amp*vo**2.
return "0,%s,%s,1.0,%s" % (ampl,
self._core2*ro**2.*self._q**(2./3.), #somewhat weird gyrfalcon implementation
self._q)
