###############################################################################
# MiyamotoNagaiPotential.py: class that implements the Miyamoto-Nagai
# potential
# GM
# phi(R,z) = - ---------------------------------
# \sqrt(R^2+(a+\sqrt(z^2+b^2))^2)
###############################################################################
import numpy
from .Potential import Potential, kms_to_kpcGyrDecorator, _APY_LOADED
if _APY_LOADED:
from astropy import units
[docs]class MiyamotoNagaiPotential(Potential):
"""Class that implements the Miyamoto-Nagai potential
.. math::
\\Phi(R,z) = -\\frac{\\mathrm{amp}}{\\sqrt{R^2+(a+\\sqrt{z^2+b^2})^2}}
with :math:`\\mathrm{amp} = GM` the total mass.
"""
[docs] def __init__(self,amp=1.,a=1.,b=0.1,normalize=False,
ro=None,vo=None):
"""
NAME:
__init__
PURPOSE:
initialize a Miyamoto-Nagai potential
INPUT:
amp - amplitude to be applied to the potential, the total mass (default: 1); can be a Quantity with units of mass or Gxmass
a - scale length (can be Quantity)
b - scale height (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:
2010-07-09 - Started - Bovy (NYU)
"""
Potential.__init__(self,amp=amp,ro=ro,vo=vo,amp_units='mass')
if _APY_LOADED and isinstance(a,units.Quantity):
a= a.to(units.kpc).value/self._ro
if _APY_LOADED and isinstance(b,units.Quantity):
b= b.to(units.kpc).value/self._ro
self._a= a
self._scale= self._a
self._b= b
self._b2= self._b**2.
if normalize or \
(isinstance(normalize,(int,float)) \
and not isinstance(normalize,bool)):
self.normalize(normalize)
self.hasC= True
self.hasC_dxdv= True
self.hasC_dens= True
self._nemo_accname= 'MiyamotoNagai'
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:
2010-07-09 - Started - Bovy (NYU)
"""
return -1./numpy.sqrt(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)
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:
2010-07-09 - Written - Bovy (NYU)
"""
return -R/(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)**(3./2.)
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:
2010-07-09 - Written - Bovy (NYU)
"""
sqrtbz= numpy.sqrt(self._b2+z**2.)
asqrtbz= self._a+sqrtbz
if isinstance(R,float) and sqrtbz == asqrtbz:
return (-z/
(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)**(3./2.))
else:
return (-z*asqrtbz/sqrtbz/
(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)**(3./2.))
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:
2010-08-08 - Written - Bovy (NYU)
"""
sqrtbz= numpy.sqrt(self._b2+z**2.)
asqrtbz= self._a+sqrtbz
if isinstance(R,float) and sqrtbz == asqrtbz:
return 3./\
(R**2.+sqrtbz**2.)**2.5/4./numpy.pi*self._b2
else:
return (self._a*R**2.+(self._a+3.*sqrtbz)*asqrtbz**2.)/\
(R**2.+asqrtbz**2.)**2.5/sqrtbz**3./4./numpy.pi*self._b2
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)
"""
return 1./(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)**1.5 \
-3.*R**2./(R**2.+(self._a+numpy.sqrt(z**2.+self._b2))**2.)**2.5
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)
"""
sqrtbz= numpy.sqrt(self._b2+z**2.)
asqrtbz= self._a+sqrtbz
if isinstance(R,float) and sqrtbz == asqrtbz:
return (self._b2+R**2.-2.*z**2.)*(self._b2+R**2.+z**2.)**-2.5
else:
return ((self._a**3.*self._b2 +
self._a**2.*(3.*self._b2 - 2.* z**2.)
*numpy.sqrt(self._b2 + z**2.)
+ (self._b2 + R**2. - 2.*z**2.)*(self._b2 + z**2.)**1.5
+self._a* (3.*self._b2**2. - 4.*z**4. + self._b2*(R**2. - z**2.)))/
((self._b2 + z**2.)**1.5* (R**2. + asqrtbz**2.)**2.5))
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)
"""
sqrtbz= numpy.sqrt(self._b2+z**2.)
asqrtbz= self._a+sqrtbz
if isinstance(R,float) and sqrtbz == asqrtbz:
return -(3.*R*z/(R**2.+asqrtbz**2.)**2.5)
else:
return -(3.*R*z*asqrtbz
/sqrtbz/(R**2.+asqrtbz**2.)**2.5)
@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
return "0,%s,%s,%s" % (ampl,self._a*ro,self._b*ro)