###############################################################################
# DoubleExponentialDiskPotential.py: class that implements the double
# exponential disk potential
#
# rho(R,z) = rho_0 e^-R/h_R e^-|z|/h_z
###############################################################################
import numpy
from scipy import special
from .PowerSphericalPotential import KeplerPotential
from .Potential import Potential, _APY_LOADED
if _APY_LOADED:
from astropy import units
_TOL= 1.4899999999999999e-15
_MAXITER= 20
[docs]class DoubleExponentialDiskPotential(Potential):
"""Class that implements the double exponential disk potential
.. math::
\\rho(R,z) = \\mathrm{amp}\\,\\exp\\left(-R/h_R-|z|/h_z\\right)
"""
[docs] def __init__(self,amp=1.,hr=1./3.,hz=1./16.,
maxiter=_MAXITER,tol=0.001,normalize=False,
ro=None,vo=None,
new=True,kmaxFac=2.,glorder=10):
"""
NAME:
__init__
PURPOSE:
initialize a double-exponential disk potential
INPUT:
amp - amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass density or Gxmass density
hr - disk scale-length (can be Quantity)
hz - scale-height (can be Quantity)
tol - relative accuracy of potential-evaluations
maxiter - scipy.integrate keyword
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:
DoubleExponentialDiskPotential object
HISTORY:
2010-04-16 - Written - Bovy (NYU)
2013-01-01 - Re-implemented using faster integration techniques - Bovy (IAS)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='density')
if _APY_LOADED and isinstance(hr,units.Quantity):
hr= hr.to(units.kpc).value/self._ro
if _APY_LOADED and isinstance(hz,units.Quantity):
hz= hz.to(units.kpc).value/self._ro
self.hasC= True
self.hasC_dens= True
self._kmaxFac= kmaxFac
self._glorder= glorder
self._hr= hr
self._scale= self._hr
self._hz= hz
self._alpha= 1./self._hr
self._beta= 1./self._hz
self._gamma= self._alpha/self._beta
self._maxiter= maxiter
self._tol= tol
self._zforceNotSetUp= True #We have not calculated a typical Kz yet
#Setup j0 zeros etc.
self._glx, self._glw= numpy.polynomial.legendre.leggauss(self._glorder)
self._nzeros=100
#j0 for potential and z
self._j0zeros= numpy.zeros(self._nzeros+1)
self._j0zeros[1:self._nzeros+1]= special.jn_zeros(0,self._nzeros)
self._dj0zeros= self._j0zeros-numpy.roll(self._j0zeros,1)
self._dj0zeros[0]= self._j0zeros[0]
#j1 for R
self._j1zeros= numpy.zeros(self._nzeros+1)
self._j1zeros[1:self._nzeros+1]= special.jn_zeros(1,self._nzeros)
self._dj1zeros= self._j1zeros-numpy.roll(self._j1zeros,1)
self._dj1zeros[0]= self._j1zeros[0]
#j2 for R2deriv
self._j2zeros= numpy.zeros(self._nzeros+1)
self._j2zeros[1:self._nzeros+1]= special.jn_zeros(2,self._nzeros)
self._dj2zeros= self._j2zeros-numpy.roll(self._j2zeros,1)
self._dj2zeros[0]= self._j2zeros[0]
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)): #pragma: no cover
self.normalize(normalize)
#Load Kepler potential for large R
self._kp= KeplerPotential(normalize=4.*numpy.pi/self._alpha**2./self._beta)
def _evaluate(self,R,z,phi=0.,t=0.,dR=0,dphi=0):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
potential at (R,z)
HISTORY:
2010-04-16 - Written - Bovy (NYU)
2012-12-26 - New method using Gaussian quadrature between zeros - Bovy (IAS)
DOCTEST:
>>> doubleExpPot= DoubleExponentialDiskPotential()
>>> r= doubleExpPot(1.,0) #doctest: +ELLIPSIS
...
>>> assert( r+1.89595350484)**2.< 10.**-6.
"""
if True:
if isinstance(R,float):
floatIn= True
R= numpy.array([R])
z= numpy.array([z])
else:
if isinstance(z,float):
z= z*numpy.ones_like(R)
floatIn= False
outShape= R.shape # this code can't do arbitrary shapes
R= R.flatten()
z= z.flatten()
out= numpy.empty(len(R))
indx= (R <= 6.)
if numpy.sum(True^indx) > 0:
out[True^indx]= self._kp(R[True^indx],z[True^indx])
R4max= numpy.copy(R)
R4max[(R < 1.)]= 1.
kmax= self._kmaxFac*self._beta
for jj in range(len(R)):
if not indx[jj]: continue
maxj0zeroIndx= numpy.argmin((self._j0zeros-kmax*R4max[jj])**2.) #close enough
ks= numpy.array([0.5*(self._glx+1.)*self._dj0zeros[ii+1] + self._j0zeros[ii] for ii in range(maxj0zeroIndx)]).flatten()
weights= numpy.array([self._glw*self._dj0zeros[ii+1] for ii in range(maxj0zeroIndx)]).flatten()
evalInt= special.jn(0,ks*R[jj])*(self._alpha**2.+ks**2.)**-1.5*(self._beta*numpy.exp(-ks*numpy.fabs(z[jj]))-ks*numpy.exp(-self._beta*numpy.fabs(z[jj])))/(self._beta**2.-ks**2.)
out[jj]= -2.*numpy.pi*self._alpha*numpy.sum(weights*evalInt)
if floatIn: return out[0]
else: return numpy.reshape(out,outShape)
def _Rforce(self,R,z,phi=0.,t=0.):
"""
NAME:
Rforce
PURPOSE:
evaluate radial force K_R (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
K_R (R,z)
HISTORY:
2010-04-16 - Written - Bovy (NYU)
DOCTEST:
"""
if True:
if isinstance(R,numpy.ndarray):
if not isinstance(z,numpy.ndarray): z= numpy.ones_like(R)*z
out= numpy.array([self._Rforce(rr,zz) for rr,zz in zip(R,z)])
return out
if (R > 16.*self._hr or R > 6.) and hasattr(self,'_kp'): return self._kp.Rforce(R,z)
if R < 1.: R4max= 1.
else: R4max= R
kmax= self._kmaxFac*self._beta
kmax= 2.*self._kmaxFac*self._beta
maxj1zeroIndx= numpy.argmin((self._j1zeros-kmax*R4max)**2.) #close enough
ks= numpy.array([0.5*(self._glx+1.)*self._dj1zeros[ii+1] + self._j1zeros[ii] for ii in range(maxj1zeroIndx)]).flatten()
weights= numpy.array([self._glw*self._dj1zeros[ii+1] for ii in range(maxj1zeroIndx)]).flatten()
evalInt= ks*special.jn(1,ks*R)*(self._alpha**2.+ks**2.)**-1.5*(self._beta*numpy.exp(-ks*numpy.fabs(z))-ks*numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks**2.)
return -2.*numpy.pi*self._alpha*numpy.sum(weights*evalInt)
def _zforce(self,R,z,phi=0.,t=0.):
"""
NAME:
zforce
PURPOSE:
evaluate vertical force K_z (R,z)
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
K_z (R,z)
HISTORY:
2010-04-16 - Written - Bovy (NYU)
DOCTEST:
"""
if True:
if isinstance(R,numpy.ndarray):
if not isinstance(z,numpy.ndarray): z= numpy.ones_like(R)*z
out= numpy.array([self._zforce(rr,zz) for rr,zz in zip(R,z)])
return out
if R > 16.*self._hr or R > 6.: return self._kp.zforce(R,z)
if R < 1.: R4max= 1.
else: R4max= R
kmax= self._kmaxFac*self._beta
maxj0zeroIndx= numpy.argmin((self._j0zeros-kmax*R4max)**2.) #close enough
ks= numpy.array([0.5*(self._glx+1.)*self._dj0zeros[ii+1] + self._j0zeros[ii] for ii in range(maxj0zeroIndx)]).flatten()
weights= numpy.array([self._glw*self._dj0zeros[ii+1] for ii in range(maxj0zeroIndx)]).flatten()
evalInt= ks*special.jn(0,ks*R)*(self._alpha**2.+ks**2.)**-1.5*(numpy.exp(-ks*numpy.fabs(z))-numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks**2.)
if z > 0.:
return -2.*numpy.pi*self._alpha*self._beta*numpy.sum(weights*evalInt)
else:
return 2.*numpy.pi*self._alpha*self._beta*numpy.sum(weights*evalInt)
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
R2deriv
PURPOSE:
evaluate R2 derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
-d K_R (R,z) d R
HISTORY:
2012-12-27 - Written - Bovy (IAS)
"""
if True:
if isinstance(R,numpy.ndarray):
if not isinstance(z,numpy.ndarray): z= numpy.ones_like(R)*z
out= numpy.array([self._R2deriv(rr,zz) for rr,zz in zip(R,z)])
return out
if R > 16.*self._hr or R > 6.: return self._kp.R2deriv(R,z)
if R < 1.: R4max= 1.
else: R4max= R
kmax= 2.*self._kmaxFac*self._beta
maxj0zeroIndx= numpy.argmin((self._j0zeros-kmax*R4max)**2.) #close enough
maxj2zeroIndx= numpy.argmin((self._j2zeros-kmax*R4max)**2.) #close enough
ks0= numpy.array([0.5*(self._glx+1.)*self._dj0zeros[ii+1] + self._j0zeros[ii] for ii in range(maxj0zeroIndx)]).flatten()
weights0= numpy.array([self._glw*self._dj0zeros[ii+1] for ii in range(maxj0zeroIndx)]).flatten()
ks2= numpy.array([0.5*(self._glx+1.)*self._dj2zeros[ii+1] + self._j2zeros[ii] for ii in range(maxj2zeroIndx)]).flatten()
weights2= numpy.array([self._glw*self._dj2zeros[ii+1] for ii in range(maxj2zeroIndx)]).flatten()
evalInt0= ks0**2.*special.jn(0,ks0*R)*(self._alpha**2.+ks0**2.)**-1.5*(self._beta*numpy.exp(-ks0*numpy.fabs(z))-ks0*numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks0**2.)
evalInt2= ks2**2.*special.jn(2,ks2*R)*(self._alpha**2.+ks2**2.)**-1.5*(self._beta*numpy.exp(-ks2*numpy.fabs(z))-ks2*numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks2**2.)
return numpy.pi*self._alpha*(numpy.sum(weights0*evalInt0)
-numpy.sum(weights2*evalInt2))
def _z2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
z2deriv
PURPOSE:
evaluate z2 derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
-d K_Z (R,z) d Z
HISTORY:
2012-12-26 - Written - Bovy (IAS)
"""
if True:
if isinstance(R,numpy.ndarray):
if not isinstance(z,numpy.ndarray): z= numpy.ones_like(R)*z
out= numpy.array([self._z2deriv(rr,zz) for rr,zz in zip(R,z)])
return out
if R > 16.*self._hr or R > 6.: return self._kp.z2deriv(R,z)
if R < 1.: R4max= 1.
else: R4max= R
kmax= self._kmaxFac*self._beta
maxj0zeroIndx= numpy.argmin((self._j0zeros-kmax*R4max)**2.) #close enough
ks= numpy.array([0.5*(self._glx+1.)*self._dj0zeros[ii+1] + self._j0zeros[ii] for ii in range(maxj0zeroIndx)]).flatten()
weights= numpy.array([self._glw*self._dj0zeros[ii+1] for ii in range(maxj0zeroIndx)]).flatten()
evalInt= ks*special.jn(0,ks*R)*(self._alpha**2.+ks**2.)**-1.5*(ks*numpy.exp(-ks*numpy.fabs(z))-self._beta*numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks**2.)
return -2.*numpy.pi*self._alpha*self._beta*numpy.sum(weights*evalInt)
def _Rzderiv(self,R,z,phi=0.,t=0.):
"""
NAME:
Rzderiv
PURPOSE:
evaluate the mixed R,z derivative
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
d2phi/dR/dz
HISTORY:
2013-08-28 - Written - Bovy (IAS)
"""
if True:
if isinstance(R,numpy.ndarray):
if not isinstance(z,numpy.ndarray): z= numpy.ones_like(R)*z
out= numpy.array([self._Rzderiv(rr,zz) for rr,zz in zip(R,z)])
return out
if R > 6.: return self._kp.Rzderiv(R,z)
if R < 1.: R4max= 1.
else: R4max= R
kmax= 2.*self._kmaxFac*self._beta
maxj1zeroIndx= numpy.argmin((self._j1zeros-kmax*R4max)**2.) #close enough
ks= numpy.array([0.5*(self._glx+1.)*self._dj1zeros[ii+1] + self._j1zeros[ii] for ii in range(maxj1zeroIndx)]).flatten()
weights= numpy.array([self._glw*self._dj1zeros[ii+1] for ii in range(maxj1zeroIndx)]).flatten()
evalInt= ks**2.*special.jn(1,ks*R)*(self._alpha**2.+ks**2.)**-1.5*(numpy.exp(-ks*numpy.fabs(z))-numpy.exp(-self._beta*numpy.fabs(z)))/(self._beta**2.-ks**2.)
if z >= 0.:
return -2.*numpy.pi*self._alpha*self._beta*numpy.sum(weights*evalInt)
else:
return 2.*numpy.pi*self._alpha*self._beta*numpy.sum(weights*evalInt)
def _dens(self,R,z,phi=0.,t=0.):
"""
NAME:
_dens
PURPOSE:
evaluate the density
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
rho (R,z)
HISTORY:
2010-08-08 - Written - Bovy (NYU)
"""
return numpy.exp(-self._alpha*R-self._beta*numpy.fabs(z))
def _surfdens(self,R,z,phi=0.,t=0.):
"""
NAME:
_surfdens
PURPOSE:
evaluate the surface density
INPUT:
R - Cylindrical Galactocentric radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
Sigma (R,z)
HISTORY:
2018-08-19 - Written - Bovy (UofT)
"""
return 2.*numpy.exp(-self._alpha*R)/self._beta\
*(1.-numpy.exp(-self._beta*numpy.fabs(z)))
