Source code for galpy.potential.RingPotential

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#   RingPotential.py: The gravitational potential of a thin, circular ring
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import numpy
from scipy import special
from ..util import conversion
from .Potential import Potential
[docs]class RingPotential(Potential): """Class that implements the potential of an infinitesimally-thin, circular ring .. math:: \\rho(R,z) = \\frac{\\mathrm{amp}}{2\pi\,R_0}\\,\\delta(R-R_0)\\,\\delta(z) with :math:`\\mathrm{amp} = GM` the mass of the ring. """
[docs] def __init__(self,amp=1.,a=0.75,normalize=False,ro=None,vo=None): """ NAME: __init__ PURPOSE: initialize a circular ring potential INPUT: amp - mass of the ring (default: 1); can be a Quantity with units of mass or Gxmass a= (0.75) radius of the ring (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.; note that because the force is always positive at r < a, this does not work if a > 1 ro=, vo= distance and velocity scales for translation into internal units (default from configuration file) OUTPUT: (none) HISTORY: 2018-08-04 - Written - Bovy (UofT) """ Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass') a= conversion.parse_length(a,ro=self._ro) self.a= a self.a2= self.a**2 self._amp/= 2.*numpy.pi*self.a if normalize or \ (isinstance(normalize,(int,float)) \ and not isinstance(normalize,bool)): if self.a > 1.: raise ValueError('RingPotential with normalize= for a > 1 is not supported (because the force is always positive at r=1)') self.normalize(normalize) self.hasC= False self.hasC_dxdv= False
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: 2018-08-04 - Written - Bovy (UofT) """ # Stable as r -> infty m= 4.*self.a/((numpy.sqrt(R)+self.a/numpy.sqrt(R))**2+z**2/R) return -4.*self.a/numpy.sqrt((R+self.a)**2+z**2)*special.ellipk(m) 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: 2018-08-04 - Written - Bovy (UofT) """ m= 4.*R*self.a/((R+self.a)**2+z**2) return -2.*self.a/R/numpy.sqrt((R+self.a)**2+z**2)\ *(m*(R**2-self.a2-z**2)/4./(1.-m)/self.a/R*special.ellipe(m) +special.ellipk(m)) 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: 2018-08-04 - Written - Bovy (UofT) """ m= 4.*R*self.a/((R+self.a)**2+z**2) return -4.*z*self.a/(1.-m)*((R+self.a)**2+z**2)**-1.5*special.ellipe(m) def _R2deriv(self,R,z,phi=0.,t=0.): """ NAME: _Rderiv 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: 2018-08-04 - Written - Bovy (UofT) """ Raz2= (R+self.a)**2+z**2 Raz= numpy.sqrt(Raz2) m= 4.*R*self.a/Raz2 R2ma2mz2o4aR1m= (R**2-self.a2-z**2)/4./self.a/R/(1.-m) return (2*R**2+self.a2+3*R*self.a+z**2)/R/Raz2*self._Rforce(R,z)\ +2.*self.a/R/Raz*(m*(R**2+self.a2+z**2)/4./(1.-m)/self.a/R**2\ *special.ellipe(m)\ +(R2ma2mz2o4aR1m/(1.-m)*special.ellipe(m) +0.5*R2ma2mz2o4aR1m*(special.ellipe(m)-special.ellipk(m)) +0.5*(special.ellipe(m)/(1.-m)-special.ellipk(m))/m)\ *4*self.a*(self.a2+z**2-R**2)/Raz2**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: 2018-08-04 - Written - Bovy (UofT) """ Raz2= (R+self.a)**2+z**2 m= 4.*R*self.a/Raz2 # Explicitly swapped in zforce here, so the z/z can be cancelled # and z=0 is handled properly return -4.*(3.*z**2/Raz2-1. +4.*((1.+m)/(1.-m)-special.ellipk(m)/special.ellipe(m))\ *self.a*R*z**2/Raz2**2/m)\ *self.a/(1.-m)*((R+self.a)**2+z**2)**-1.5*special.ellipe(m) 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: 2018-08-04 - Written - Bovy (UofT) """ Raz2= (R+self.a)**2+z**2 m= 4.*R*self.a/Raz2 return (3.*(R+self.a)/Raz2 -2.*((1.+m)/(1.-m)-special.ellipk(m)/special.ellipe(m))\ *self.a*(self.a2+z**2-R**2)/Raz2**2/m)*self._zforce(R,z)