/
DoubleExponentialDiskPotential.py
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/
DoubleExponentialDiskPotential.py
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###############################################################################
# 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 ..util import conversion
from .Potential import Potential, check_potential_inputs_not_arrays
def _de_psi(t):
return t * numpy.tanh(numpy.pi / 2.0 * numpy.sinh(t))
def _de_psiprime(t):
return (numpy.sinh(numpy.pi * numpy.sinh(t)) + numpy.pi * t * numpy.cosh(t)) / (
numpy.cosh(numpy.pi * numpy.sinh(t)) + 1
)
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)
"""
def __init__(
self,
amp=1.0,
hr=1.0 / 3.0,
hz=1.0 / 16.0,
normalize=False,
ro=None,
vo=None,
de_h=1e-3,
de_n=10000,
):
"""
Initialize a double exponential disk potential
Parameters
----------
amp : float or Quantity, optional
Amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass density or Gxmass density.
hr : float or Quantity, optional
Disk scale-length.
hz : float or Quantity, optional
Scale-height.
normalize : bool or float, optional
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.
de_h : float, optional
Step used in numerical integration.
de_n : int, optional
Number of points used in numerical integration (use 1000 for a lower accuracy version that is typically still high accuracy enough, but faster).
ro : float, optional
Distance scale for translation into internal units (default from configuration file).
vo : float, optional
Velocity scale for translation into internal units (default from configuration file).
Notes
-----
- 2010-04-16 - Written - Bovy (NYU)
- 2013-01-01 - Re-implemented using faster integration techniques - Bovy (IAS)
- 2020-12-24 - Re-implemented again using more accurate integration techniques for Bessel integrals - Bovy (UofT)
"""
Potential.__init__(self, amp=amp, ro=ro, vo=vo, amp_units="density")
hr = conversion.parse_length(hr, ro=self._ro)
hz = conversion.parse_length(hz, ro=self._ro)
self.hasC = True
self.hasC_dens = True
self._hr = hr
self._scale = self._hr
self._hz = hz
self._alpha = 1.0 / self._hr
self._beta = 1.0 / self._hz
self._zforceNotSetUp = True # We have not calculated a typical Kz yet
# For double-exponential formula
self._de_h = de_h
self._de_n = de_n
self._de_j0zeros = special.jn_zeros(0, self._de_n) / numpy.pi
self._de_j1zeros = special.jn_zeros(1, self._de_n) / numpy.pi
self._de_j0_xs = numpy.pi / self._de_h * _de_psi(self._de_h * self._de_j0zeros)
self._de_j0_weights = (
2.0
/ (
numpy.pi
* self._de_j0zeros
* special.j1(numpy.pi * self._de_j0zeros) ** 2.0
)
* special.j0(self._de_j0_xs)
* _de_psiprime(self._de_h * self._de_j0zeros)
)
self._de_j1_xs = numpy.pi / self._de_h * _de_psi(self._de_h * self._de_j1zeros)
self._de_j1_weights = (
2.0
/ (
numpy.pi
* self._de_j1zeros
* special.jv(2, numpy.pi * self._de_j1zeros) ** 2.0
)
* special.j1(self._de_j1_xs)
* _de_psiprime(self._de_h * self._de_j1zeros)
)
# Potential at zero in case we want that
_gamma = self._beta / self._alpha
_gamma2 = _gamma**2.0
self._pot_zero = (
2.0 * (_gamma - 1.0) * numpy.sqrt(1.0 + _gamma2)
+ 2.0 * numpy.arctanh(1.0 / numpy.sqrt(1.0 + _gamma2))
- numpy.log(1.0 - _gamma / numpy.sqrt(1.0 + _gamma2))
+ numpy.log(1.0 + _gamma / numpy.sqrt(1.0 + _gamma2))
) / (2.0 * (1.0 + _gamma2) ** 1.5)
self._pot_zero *= -4.0 * numpy.pi / self._alpha**2.0
# Normalize?
if normalize or (
isinstance(normalize, (int, float)) and not isinstance(normalize, bool)
): # pragma: no cover
self.normalize(normalize)
def _evaluate(self, R, z, phi=0.0, t=0.0, dR=0, dphi=0):
"""
Evaluate the potential at (R,z)
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
Potential at (R,z)
Notes
-----
- 2010-04-16 - Written - Bovy (NYU)
- 2012-12-26 - New method using Gaussian quadrature between zeros - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
if isinstance(R, (float, int)):
floatIn = True
R = numpy.atleast_1d(R)
z = numpy.atleast_1d(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()
fun = (
lambda x: (self._alpha**2.0 + (x / R[:, numpy.newaxis]) ** 2.0) ** -1.5
* (
self._beta
* numpy.exp(-x / R[:, numpy.newaxis] * numpy.fabs(z[:, numpy.newaxis]))
- x
/ R[:, numpy.newaxis]
* numpy.exp(-self._beta * numpy.fabs(z[:, numpy.newaxis]))
)
/ (self._beta**2.0 - (x / R[:, numpy.newaxis]) ** 2.0)
)
out = (
-4.0
* numpy.pi
* self._alpha
/ R
* numpy.nansum(fun(self._de_j0_xs) * self._de_j0_weights, axis=1)
)
out[(R == 0) * (z == 0)] = self._pot_zero
if floatIn:
return out[0]
else:
return numpy.reshape(out, outShape)
@check_potential_inputs_not_arrays
def _Rforce(self, R, z, phi=0.0, t=0.0):
"""
Evaluate radial force K_R (R,z)
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
Radial force (R,z)
Notes
-----
- 2010-04-16 - Written - Bovy (NYU)
- 2012-12-26 - New method using Gaussian quadrature between zeros - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
fun = (
lambda x: x
* (self._alpha**2.0 + (x / R) ** 2.0) ** -1.5
* (
self._beta * numpy.exp(-x / R * numpy.fabs(z))
- x / R * numpy.exp(-self._beta * numpy.fabs(z))
)
/ (self._beta**2.0 - (x / R) ** 2.0)
)
return (
-4.0
* numpy.pi
* self._alpha
/ R**2.0
* numpy.nansum(fun(self._de_j1_xs) * self._de_j1_weights)
)
@check_potential_inputs_not_arrays
def _zforce(self, R, z, phi=0.0, t=0.0):
"""
Evaluate vertical force K_z (R,z)
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
Vertical force (R,z)
Notes
-----
- 2010-04-16 - Written - Bovy (NYU)
- 2012-12-26 - New method using Gaussian quadrature between zeros - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
fun = (
lambda x: (self._alpha**2.0 + (x / R) ** 2.0) ** -1.5
* x
/ R
* (
numpy.exp(-x / R * numpy.fabs(z))
- numpy.exp(-self._beta * numpy.fabs(z))
)
/ (self._beta**2.0 - (x / R) ** 2.0)
)
out = (
-4.0
* numpy.pi
* self._alpha
* self._beta
/ R
* numpy.nansum(fun(self._de_j0_xs) * self._de_j0_weights)
)
if z > 0.0:
return out
else:
return -out
@check_potential_inputs_not_arrays
def _R2deriv(self, R, z, phi=0.0, t=0.0):
"""
Evaluate radial force K_R (R,z) R2 derivative
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
-d K_R (R,z) d R
Notes
-----
- 2012-12-27 - Written - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
fun = (
lambda x: x**2
* (self._alpha**2.0 + (x / R) ** 2.0) ** -1.5
* (
self._beta * numpy.exp(-x / R * numpy.fabs(z))
- x / R * numpy.exp(-self._beta * numpy.fabs(z))
)
/ (self._beta**2.0 - (x / R) ** 2.0)
)
return (
4.0
* numpy.pi
* self._alpha
/ R**3.0
* numpy.nansum(
fun(self._de_j0_xs) * self._de_j0_weights
- fun(self._de_j1_xs) / self._de_j1_xs * self._de_j1_weights
)
)
@check_potential_inputs_not_arrays
def _z2deriv(self, R, z, phi=0.0, t=0.0):
"""
Evaluate vertical force K_Z (R,z) Z2 derivative
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
-d K_Z (R,z) d Z
Notes
-----
- 2012-12-26 - Written - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
fun = (
lambda x: (self._alpha**2.0 + (x / R) ** 2.0) ** -1.5
* x
/ R
* (
x / R * numpy.exp(-x / R * numpy.fabs(z))
- self._beta * numpy.exp(-self._beta * numpy.fabs(z))
)
/ (self._beta**2.0 - (x / R) ** 2.0)
)
return (
-4.0
* numpy.pi
* self._alpha
* self._beta
/ R
* numpy.nansum(fun(self._de_j0_xs) * self._de_j0_weights)
)
@check_potential_inputs_not_arrays
def _Rzderiv(self, R, z, phi=0.0, t=0.0):
"""
Evaluate mixed R,z derivative d2phi/dR/dz.
Parameters
----------
R : float
Cylindrical Galactocentric radius
z : float
Vertical height
phi : float, optional
Azimuth (default: 0.0)
t : float, optional
Time (default: 0.0)
Returns
-------
float
d2phi/dR/dz
Notes
-----
- 2013-08-28 - Written - Bovy (IAS)
- 2020-12-24 - New method using Ogata's Bessel integral formula - Bovy (UofT)
"""
fun = (
lambda x: (self._alpha**2.0 + (x / R) ** 2.0) ** -1.5
* (x / R) ** 2.0
* (
numpy.exp(-x / R * numpy.fabs(z))
- numpy.exp(-self._beta * numpy.fabs(z))
)
/ (self._beta**2.0 - (x / R) ** 2.0)
)
out = (
-4.0
* numpy.pi
* self._alpha
* self._beta
/ R
* numpy.nansum(fun(self._de_j1_xs) * self._de_j1_weights)
)
if z > 0.0:
return out
else:
return -out
def _dens(self, R, z, phi=0.0, t=0.0):
return numpy.exp(-self._alpha * R - self._beta * numpy.fabs(z))
def _surfdens(self, R, z, phi=0.0, t=0.0):
return (
2.0
* numpy.exp(-self._alpha * R)
/ self._beta
* (1.0 - numpy.exp(-self._beta * numpy.fabs(z)))
)