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SCFPotential.py
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SCFPotential.py
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import hashlib
import numpy
from numpy.polynomial.legendre import leggauss
from scipy import integrate
from scipy.special import gamma, gammaln, lpmn
from ..util import conversion, coords
from ..util._optional_deps import _APY_LOADED
from .Potential import Potential
if _APY_LOADED:
from astropy import units
from .NumericalPotentialDerivativesMixin import NumericalPotentialDerivativesMixin
class SCFPotential(Potential, NumericalPotentialDerivativesMixin):
"""Class that implements the `Hernquist & Ostriker (1992) <http://adsabs.harvard.edu/abs/1992ApJ...386..375H>`_ Self-Consistent-Field-type potential.
Note that we divide the amplitude by 2 such that :math:`Acos = \\delta_{0n}\\delta_{0l}\\delta_{0m}` and :math:`Asin = 0` corresponds to :ref:`Galpy's Hernquist Potential <hernquist_potential>`.
.. math::
\\rho(r, \\theta, \\phi) = \\frac{amp}{2}\\sum_{n=0}^{\\infty} \\sum_{l=0}^{\\infty} \\sum_{m=0}^l N_{lm} P_{lm}(\\cos(\\theta)) \\tilde{\\rho}_{nl}(r) \\left(A_{cos, nlm} \\cos(m\\phi) + A_{sin, nlm} \\sin(m\\phi)\\right)
where
.. math::
\\tilde{\\rho}_{nl}(r) = \\frac{K_{nl}}{\\sqrt{\\pi}} \\frac{(a r)^l}{(r/a) (a + r)^{2l + 3}} C_{n}^{2l + 3/2}(\\xi)
.. math::
\\Phi(r, \\theta, \\phi) = \\sum_{n=0}^{\\infty} \\sum_{l=0}^{\\infty} \\sum_{m=0}^l N_{lm} P_{lm}(\\cos(\\theta)) \\tilde{\\Phi}_{nl}(r) \\left(A_{cos, nlm} \\cos(m\\phi) + A_{sin, nlm} \\sin(m\\phi)\\right)
where
.. math::
\\tilde{\\Phi}_{nl}(r) = -\\sqrt{4 \\pi}K_{nl} \\frac{(ar)^l}{(a + r)^{2l + 1}} C_{n}^{2l + 3/2}(\\xi)
where
.. math::
\\xi = \\frac{r - a}{r + a} \\qquad
N_{lm} = \\sqrt{\\frac{2l + 1}{4\\pi} \\frac{(l - m)!}{(l + m)!}}(2 - \\delta_{m0}) \\qquad
K_{nl} = \\frac{1}{2} n (n + 4l + 3) + (l + 1)(2l + 1)
and :math:`P_{lm}` is the Associated Legendre Polynomials whereas :math:`C_n^{\\alpha}` is the Gegenbauer polynomial.
"""
def __init__(
self,
amp=1.0,
Acos=numpy.array([[[1]]]),
Asin=None,
a=1.0,
normalize=False,
ro=None,
vo=None,
):
"""
Initialize a SCF Potential from a set of expansion coefficients (use SCFPotential.from_density to directly initialize from a density)
Parameters
----------
amp : float or Quantity, optional
Amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass or Gxmass.
Acos : numpy.ndarray, optional
The real part of the expansion coefficient (NxLxL matrix, or optionally NxLx1 if Asin=None).
Asin : numpy.ndarray, optional
The imaginary part of the expansion coefficient (NxLxL matrix or None).
a : float or Quantity, optional
Scale length.
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.
ro : float or Quantity, optional
Distance scale for translation into internal units (default from configuration file).
vo : float or Quantity, optional
Velocity scale for translation into internal units (default from configuration file).
Notes
-----
- 2016-05-13 - Written - Aladdin Seaifan (UofT)
"""
NumericalPotentialDerivativesMixin.__init__(
self, {}
) # just use default dR etc.
Potential.__init__(self, amp=amp / 2.0, ro=ro, vo=vo, amp_units="mass")
a = conversion.parse_length(a, ro=self._ro)
##Errors
shape = Acos.shape
errorMessage = None
if len(shape) != 3:
errorMessage = "Acos must be a 3 dimensional numpy array"
elif Asin is not None and shape[1] != shape[2]:
errorMessage = "The second and third dimension of the expansion coefficients must have the same length"
elif Asin is None and not (shape[2] == 1 or shape[1] == shape[2]):
errorMessage = "The third dimension must have length=1 or equal to the length of the second dimension"
elif Asin is None and shape[1] > 1 and numpy.any(Acos[:, :, 1:] != 0):
errorMessage = (
"Acos has non-zero elements at indices m>0, which implies a non-axi symmetric potential.\n"
+ "Asin=None which implies an axi symmetric potential.\n"
+ "Contradiction."
)
elif Asin is not None and Asin.shape != shape:
errorMessage = "The shape of Asin does not match the shape of Acos."
if errorMessage is not None:
raise RuntimeError(errorMessage)
##Warnings
warningMessage = None
if numpy.any(numpy.triu(Acos, 1) != 0) or (
Asin is not None and numpy.any(numpy.triu(Asin, 1) != 0)
):
warningMessage = (
"Found non-zero values at expansion coefficients where m > l\n"
+ "The Mth and Lth dimension is expected to make a lower triangular matrix.\n"
+ "All values found above the diagonal will be ignored."
)
if warningMessage is not None:
raise RuntimeWarning(warningMessage)
##Is non axi?
self.isNonAxi = True
if (
Asin is None
or shape[1] == 1
or (numpy.all(Acos[:, :, 1:] == 0) and numpy.all(Asin[:, :, :] == 0))
):
self.isNonAxi = False
self._a = a
NN = self._Nroot(Acos.shape[1], Acos.shape[2])
self._Acos = Acos * NN[numpy.newaxis, :, :]
if Asin is not None:
self._Asin = Asin * NN[numpy.newaxis, :, :]
else:
self._Asin = numpy.zeros_like(Acos)
self._force_hash = None
self.hasC = True
self.hasC_dxdv = True
self.hasC_dens = True
if normalize or (
isinstance(normalize, (int, float)) and not isinstance(normalize, bool)
):
self.normalize(normalize)
return None
@classmethod
def from_density(
cls,
dens,
N,
L=None,
a=1.0,
symmetry=None,
radial_order=None,
costheta_order=None,
phi_order=None,
ro=None,
vo=None,
):
"""
Initialize an SCF Potential from a given density.
Parameters
----------
dens : function
Density function that takes parameters R, z and phi; z and phi are optional for spherical profiles, phi is optional for axisymmetric profiles. The density function must take input positions in internal units (R/ro, z/ro), but can return densities in physical units. You can use the member dens of Potential instances or the density from evaluateDensities.
N : int
Number of radial basis functions.
L : int, optional
Number of costheta basis functions; for non-axisymmetric profiles also sets the number of azimuthal (phi) basis functions to M = 2L+1).
a : float or Quantity, optional
Expansion scale length.
symmetry : {'spherical','axisymmetry',None}, optional
Symmetry of the profile to assume. None is the general, non-axisymmetric case.
radial_order : int, optional
Number of sample points for the radial integral. If None, radial_order=max(20, N + 3/2L + 1).
costheta_order : int, optional
Number of sample points of the costheta integral. If None, If costheta_order=max(20, L + 1).
phi_order : int, optional
Number of sample points of the phi integral. If None, If costheta_order=max(20, L + 1).
ro : float or Quantity, optional
Distance scale for translation into internal units (default from configuration file).
vo : float or Quantity, optional
Velocity scale for translation into internal units (default from configuration file).
Returns
-------
SCFPotential object
Notes
-----
- Written - Jo Bovy (UofT) - 2022-06-20
"""
# Dummy object for ro/vo handling, to ensure consistency
dumm = cls(ro=ro, vo=vo)
internal_ro = dumm._ro
internal_vo = dumm._vo
a = conversion.parse_length(a, ro=internal_ro)
if not symmetry is None and symmetry.startswith("spher"):
Acos, Asin = scf_compute_coeffs_spherical(
dens, N, a=a, radial_order=radial_order
)
elif not symmetry is None and symmetry.startswith("axi"):
Acos, Asin = scf_compute_coeffs_axi(
dens,
N,
L,
a=a,
radial_order=radial_order,
costheta_order=costheta_order,
)
else:
Acos, Asin = scf_compute_coeffs(
dens,
N,
L,
a=a,
radial_order=radial_order,
costheta_order=costheta_order,
phi_order=phi_order,
)
# Turn on physical outputs if input density was physical
if _APY_LOADED:
# First need to determine number of parameters, like in
# scf_compute_coeffs_spherical/axi
numOfParam = 0
try:
dens(0)
numOfParam = 1
except:
try:
dens(0, 0)
numOfParam = 2
except:
numOfParam = 3
param = [1] * numOfParam
try:
dens(*param).to(units.kg / units.m**3)
except (AttributeError, units.UnitConversionError):
# We'll just assume that unit conversion means density
# is scalar Quantity
pass
else:
ro = internal_ro
vo = internal_vo
return cls(Acos=Acos, Asin=Asin, a=a, ro=ro, vo=vo)
def _Nroot(self, L, M):
"""
Evaluate the square root of equation (3.15) with the (2 - del_m,0) term outside the square root.
Parameters
----------
L : int
Evaluate Nroot for 0 <= l <= L.
M : int
Evaluate Nroot for 0 <= m <= M.
Returns
-------
numpy.ndarray
The square root of equation (3.15) with the (2 - del_m,0) outside.
Notes
-----
- Written on 2016-05-16 by Aladdin Seaifan (UofT).
"""
NN = numpy.zeros((L, M), float)
l = numpy.arange(0, L)[:, numpy.newaxis]
m = numpy.arange(0, M)[numpy.newaxis, :]
nLn = gammaln(l - m + 1) - gammaln(l + m + 1)
NN[:, :] = ((2 * l + 1.0) / (4.0 * numpy.pi) * numpy.e**nLn) ** 0.5 * 2
NN[:, 0] /= 2.0
NN = numpy.tril(NN)
return NN
def _calculateXi(self, r):
"""
Calculate xi given r.
Parameters
----------
r : float
Evaluate at radius r.
Returns
-------
xi : float
The calculated xi.
Notes
-----
- 2016-05-18 - Written - Aladdin Seaifan (UofT)
"""
a = self._a
if r == 0:
return -1
else:
return (1.0 - a / r) / (1.0 + a / r)
def _rhoTilde(self, r, N, L):
"""
Evaluate rho_tilde as defined in equation 3.9 and 2.24 for 0 <= n < N and 0 <= l < L
Parameters
----------
r : float
Evaluate at radius r
N : int
size of the N dimension
L : int
size of the L dimension
Returns
-------
numpy.ndarray
The value of rho tilde
Notes
-----
- Written on 2016-05-17 by Aladdin Seaifan (UofT)
"""
xi = self._calculateXi(r)
CC = _C(xi, N, L)
a = self._a
rho = numpy.zeros((N, L), float)
n = numpy.arange(0, N, dtype=float)[:, numpy.newaxis]
l = numpy.arange(0, L, dtype=float)[numpy.newaxis, :]
K = 0.5 * n * (n + 4 * l + 3) + (l + 1.0) * (2 * l + 1)
rho[:, :] = (
K
* ((a * r) ** l)
/ ((r / a) * (a + r) ** (2 * l + 3.0))
* CC[:, :]
* (numpy.pi) ** -0.5
)
return rho
def _phiTilde(self, r, N, L):
"""
Evaluate phi_tilde as defined in equation 3.10 and 2.25 for 0 <= n < N and 0 <= l < L
Parameters
----------
r : float
Evaluate at radius r
N : int
size of the N dimension
L : int
size of the L dimension
Returns
-------
numpy.ndarray
phi tilde
Notes
-----
- Written on 2016-05-17 by Aladdin Seaifan (UofT)
"""
xi = self._calculateXi(r)
CC = _C(xi, N, L)
a = self._a
phi = numpy.zeros((N, L), float)
n = numpy.arange(0, N)[:, numpy.newaxis]
l = numpy.arange(0, L)[numpy.newaxis, :]
if r == 0:
phi[:, :] = -1.0 / a * CC[:, :] * (4 * numpy.pi) ** 0.5
else:
phi[:, :] = (
-(a**l)
* r ** (-l - 1.0)
/ ((1.0 + a / r) ** (2 * l + 1.0))
* CC[:, :]
* (4 * numpy.pi) ** 0.5
)
return phi
def _compute(self, funcTilde, R, z, phi):
"""
Evaluate the NxLxM density or potential
Parameters
----------
funcTilde : function
Must be _rhoTilde or _phiTilde
R : float
Cylindrical Galactocentric radius
z : float
vertical height
phi : float
azimuth
Returns
-------
numpy.ndarray
An NxLxM density or potential at (R,z, phi)
Notes
-----
- Written on 2016-05-18 by Aladdin Seaifan (UofT)
"""
Acos, Asin = self._Acos, self._Asin
N, L, M = Acos.shape
r, theta, phi = coords.cyl_to_spher(R, z, phi)
PP = lpmn(M - 1, L - 1, numpy.cos(theta))[0].T ##Get the Legendre polynomials
func_tilde = funcTilde(r, N, L) ## Tilde of the function of interest
func = numpy.zeros(
(N, L, M), float
) ## The function of interest (density or potential)
m = numpy.arange(0, M)[numpy.newaxis, numpy.newaxis, :]
mcos = numpy.cos(m * phi)
msin = numpy.sin(m * phi)
func = (
func_tilde[:, :, None]
* (Acos[:, :, :] * mcos + Asin[:, :, :] * msin)
* PP[None, :, :]
)
return func
def _computeArray(self, funcTilde, R, z, phi):
"""
Evaluate the density or potential for a given array of coordinates.
Parameters
----------
funcTilde : function
Must be _rhoTilde or _phiTilde.
R : numpy.ndarray
Cylindrical Galactocentric radius.
z : numpy.ndarray
Vertical height.
phi : numpy.ndarray
Azimuth.
Returns
-------
numpy.ndarray
Density or potential evaluated at (R,z, phi).
Notes
-----
- 2016-06-02 - Written - Aladdin Seaifan (UofT)
"""
R = numpy.array(R, dtype=float)
z = numpy.array(z, dtype=float)
phi = numpy.array(phi, dtype=float)
shape = (R * z * phi).shape
if shape == ():
return numpy.sum(self._compute(funcTilde, R, z, phi))
R = R * numpy.ones(shape)
z = z * numpy.ones(shape)
phi = phi * numpy.ones(shape)
func = numpy.zeros(shape, float)
li = _cartesian(shape)
for i in range(li.shape[0]):
j = tuple(numpy.split(li[i], li.shape[1]))
func[j] = numpy.sum(self._compute(funcTilde, R[j][0], z[j][0], phi[j][0]))
return func
def _dens(self, R, z, phi=0.0, t=0.0):
if not self.isNonAxi and phi is None:
phi = 0.0
return self._computeArray(self._rhoTilde, R, z, phi)
def _mass(self, R, z=None, t=0.0):
if not z is None:
raise AttributeError # Hack to fall back to general
# when integrating over spherical volume, all non-zero l,m vanish
N = len(self._Acos)
return R**2.0 * numpy.sum(self._Acos[:, 0, 0] * self._dphiTilde(R, N, 1)[:, 0])
def _evaluate(self, R, z, phi=0.0, t=0.0):
if not self.isNonAxi and phi is None:
phi = 0.0
return self._computeArray(self._phiTilde, R, z, phi)
def _dphiTilde(self, r, N, L):
a = self._a
l = numpy.arange(0, L, dtype=float)[numpy.newaxis, :]
n = numpy.arange(0, N, dtype=float)[:, numpy.newaxis]
xi = self._calculateXi(r)
dC = _dC(xi, N, L)
return -((4 * numpy.pi) ** 0.5) * (
numpy.power(a * r, l)
* (l * (a + r) * numpy.power(r, -1) - (2 * l + 1))
/ ((a + r) ** (2 * l + 2))
* _C(xi, N, L)
+ a**-1 * (1 - xi) ** 2 * (a * r) ** l / (a + r) ** (2 * l + 1) * dC / 2.0
)
def _computeforce(self, R, z, phi=0, t=0):
"""Computes dPhi/dr, dPhi/dtheta, dPhi/dphi"""
Acos, Asin = self._Acos, self._Asin
N, L, M = Acos.shape
r, theta, phi = coords.cyl_to_spher(R, z, phi)
new_hash = hashlib.md5(numpy.array([R, z, phi])).hexdigest()
if new_hash == self._force_hash:
dPhi_dr = self._cached_dPhi_dr
dPhi_dtheta = self._cached_dPhi_dtheta
dPhi_dphi = self._cached_dPhi_dphi
else:
PP, dPP = lpmn(
M - 1, L - 1, numpy.cos(theta)
) ##Get the Legendre polynomials
PP = PP.T[None, :, :]
dPP = dPP.T[None, :, :]
phi_tilde = self._phiTilde(r, N, L)[:, :, numpy.newaxis]
dphi_tilde = self._dphiTilde(r, N, L)[:, :, numpy.newaxis]
m = numpy.arange(0, M)[numpy.newaxis, numpy.newaxis, :]
mcos = numpy.cos(m * phi)
msin = numpy.sin(m * phi)
dPhi_dr = -numpy.sum((Acos * mcos + Asin * msin) * PP * dphi_tilde)
dPhi_dtheta = -numpy.sum(
(Acos * mcos + Asin * msin) * phi_tilde * dPP * (-numpy.sin(theta))
)
dPhi_dphi = -numpy.sum(m * (Asin * mcos - Acos * msin) * phi_tilde * PP)
self._force_hash = new_hash
self._cached_dPhi_dr = dPhi_dr
self._cached_dPhi_dtheta = dPhi_dtheta
self._cached_dPhi_dphi = dPhi_dphi
return dPhi_dr, dPhi_dtheta, dPhi_dphi
def _computeforceArray(self, dr_dx, dtheta_dx, dphi_dx, R, z, phi):
"""
Evaluate the forces in the x direction for a given array of coordinates.
Parameters
----------
dr_dx : numpy.ndarray
The derivative of r with respect to the chosen variable x.
dtheta_dx : numpy.ndarray
The derivative of theta with respect to the chosen variable x.
dphi_dx : numpy.ndarray
The derivative of phi with respect to the chosen variable x.
R : numpy.ndarray
Cylindrical Galactocentric radius.
z : numpy.ndarray
Vertical height.
phi : numpy.ndarray
Azimuth.
Returns
-------
numpy.ndarray
The forces in the x direction.
Notes
-----
- 2016-06-02 - Written - Aladdin Seaifan (UofT)
"""
R = numpy.array(R, dtype=float)
z = numpy.array(z, dtype=float)
phi = numpy.array(phi, dtype=float)
shape = (R * z * phi).shape
if shape == ():
dPhi_dr, dPhi_dtheta, dPhi_dphi = self._computeforce(R, z, phi)
return dr_dx * dPhi_dr + dtheta_dx * dPhi_dtheta + dPhi_dphi * dphi_dx
R = R * numpy.ones(shape)
z = z * numpy.ones(shape)
phi = phi * numpy.ones(shape)
force = numpy.zeros(shape, float)
dr_dx = dr_dx * numpy.ones(shape)
dtheta_dx = dtheta_dx * numpy.ones(shape)
dphi_dx = dphi_dx * numpy.ones(shape)
li = _cartesian(shape)
for i in range(li.shape[0]):
j = tuple(numpy.split(li[i], li.shape[1]))
dPhi_dr, dPhi_dtheta, dPhi_dphi = self._computeforce(
R[j][0], z[j][0], phi[j][0]
)
force[j] = (
dr_dx[j][0] * dPhi_dr
+ dtheta_dx[j][0] * dPhi_dtheta
+ dPhi_dphi * dphi_dx[j][0]
)
return force
def _Rforce(self, R, z, phi=0, t=0):
if not self.isNonAxi and phi is None:
phi = 0.0
r, theta, phi = coords.cyl_to_spher(R, z, phi)
# x = R
dr_dR = numpy.divide(R, r)
dtheta_dR = numpy.divide(z, r**2)
dphi_dR = 0
return self._computeforceArray(dr_dR, dtheta_dR, dphi_dR, R, z, phi)
def _zforce(self, R, z, phi=0.0, t=0.0):
if not self.isNonAxi and phi is None:
phi = 0.0
r, theta, phi = coords.cyl_to_spher(R, z, phi)
# x = z
dr_dz = numpy.divide(z, r)
dtheta_dz = numpy.divide(-R, r**2)
dphi_dz = 0
return self._computeforceArray(dr_dz, dtheta_dz, dphi_dz, R, z, phi)
def _phitorque(self, R, z, phi=0, t=0):
if not self.isNonAxi and phi is None:
phi = 0.0
r, theta, phi = coords.cyl_to_spher(R, z, phi)
# x = phi
dr_dphi = 0
dtheta_dphi = 0
dphi_dphi = 1
return self._computeforceArray(dr_dphi, dtheta_dphi, dphi_dphi, R, z, phi)
def OmegaP(self):
return 0
def _xiToR(xi, a=1):
return a * numpy.divide((1.0 + xi), (1.0 - xi))
def _RToxi(r, a=1):
out = numpy.divide((r / a - 1.0), (r / a + 1.0), where=True ^ numpy.isinf(r))
if numpy.any(numpy.isinf(r)):
if hasattr(r, "__len__"):
out[numpy.isinf(r)] = 1.0
else:
return 1.0
return out
def _C(xi, N, L, alpha=lambda x: 2 * x + 3.0 / 2, singleL=False):
"""
Evaluate the Gegenbauer polynomial for 0 <= l < L and 0<= n < N
Parameters
----------
xi : float
Radial transformed variable
N : int
Size of the N dimension
L : int
Size of the L dimension
alpha : function, optional
A lambda function of l. Default alpha = 2l + 3/2
singleL : bool, optional
If True only compute the L-th polynomial (default: False)
Returns
-------
numpy.ndarray
An LxN Gegenbauer Polynomial
Notes
-----
- 2016-05-16 - Written - Aladdin Seaifan (UofT)
- 2021-02-22 - Upgraded to array xi - Bovy (UofT)
- 2021-02-22 - Added singleL for use in compute...nbody - Bovy (UofT)
"""
floatIn = False
if isinstance(xi, (float, int)):
floatIn = True
xi = numpy.array([xi])
if singleL:
Ls = [L]
else:
Ls = range(L)
CC = numpy.zeros((N, len(Ls), len(xi)))
for l, ll in enumerate(Ls):
for n in range(N):
a = alpha(ll)
if n == 0:
CC[n, l] = 1.0
continue
elif n == 1:
CC[n, l] = 2.0 * a * xi
if n + 1 != N:
CC[n + 1, l] = (
2 * (n + a) * xi * CC[n, l] - (n + 2 * a - 1) * CC[n - 1, l]
) / (n + 1.0)
if floatIn:
return CC[:, :, 0]
else:
return CC
def _dC(xi, N, L):
l = numpy.arange(0, L)[numpy.newaxis, :]
CC = _C(xi, N + 1, L, alpha=lambda x: 2 * x + 5.0 / 2)
CC = numpy.roll(CC, 1, axis=0)[:-1, :]
CC[0, :] = 0
CC *= 2 * (2 * l + 3.0 / 2)
return CC
def scf_compute_coeffs_spherical_nbody(pos, N, mass=1.0, a=1.0):
"""
Numerically compute the expansion coefficients for a spherical expansion for a given $N$-body set of points
Parameters
----------
pos : numpy.ndarray
Positions of particles in rectangular coordinates with shape [3,n]
N : int
Size of the Nth dimension of the expansion coefficients
mass : float or numpy.ndarray, optional
Mass of particles (scalar or array with size n), by default 1.0
a : float, optional
Parameter used to scale the radius, by default 1.0
Returns
-------
tuple
Expansion coefficients for density dens that can be given to SCFPotential.__init__
Notes
-----
- 2020-11-18 - Written - Morgan Bennett (UofT)
- 2021-02-22 - Sped-up - Bovy (UofT)
"""
Acos = numpy.zeros((N, 1, 1), float)
Asin = None
r = numpy.sqrt(pos[0] ** 2 + pos[1] ** 2 + pos[2] ** 2)
RhoSum = numpy.einsum("j,ij", mass / (1.0 + r / a), _C(_RToxi(r, a=a), N, 1)[:, 0])
n = numpy.arange(0, N)
K = 4 * (n + 3.0 / 2) / ((n + 2) * (n + 1) * (1 + n * (n + 3.0) / 2.0))
Acos[n, 0, 0] = 2 * K * RhoSum
return Acos, Asin
def _scf_compute_determine_dens_kwargs(dens, param):
try:
param[0] = 1.0
dens(*param, use_physical=False)
except:
dens_kw = {}
else:
dens_kw = {"use_physical": False}
return dens_kw
def scf_compute_coeffs_spherical(dens, N, a=1.0, radial_order=None):
"""
Numerically compute the expansion coefficients for a given spherical density
Parameters
----------
dens : function
A density function that takes a parameter R
N : int
Size of expansion coefficients
a : float, optional
Parameter used to scale the radius (default is 1.0)
radial_order : int, optional
Number of sample points of the radial integral. If None, radial_order=max(20, N + 1) (default is None)
Returns
-------
tuple
(Acos,Asin) - Expansion coefficients for density dens that can be given to SCFPotential.__init__
Notes
-----
- 2016-05-18 - Written - Aladdin Seaifan (UofT)
"""
numOfParam = 0
try:
dens(0)
numOfParam = 1
except:
try:
dens(0, 0)
numOfParam = 2
except:
numOfParam = 3
param = [0] * numOfParam
dens_kw = _scf_compute_determine_dens_kwargs(dens, param)
def integrand(xi):
r = _xiToR(xi, a)
R = r
param[0] = R
return (
a**3.0
* dens(*param, **dens_kw)
* (1 + xi) ** 2.0
* (1 - xi) ** -3.0
* _C(xi, N, 1)[:, 0]
)
Acos = numpy.zeros((N, 1, 1), float)
Asin = None
Ksample = [max(N + 1, 20)]
if radial_order != None:
Ksample[0] = radial_order
integrated = _gaussianQuadrature(integrand, [[-1.0, 1.0]], Ksample=Ksample)
n = numpy.arange(0, N)
K = 16 * numpy.pi * (n + 3.0 / 2) / ((n + 2) * (n + 1) * (1 + n * (n + 3.0) / 2.0))
Acos[n, 0, 0] = 2 * K * integrated
return Acos, Asin
def scf_compute_coeffs_axi_nbody(pos, N, L, mass=1.0, a=1.0):
"""
Numerically compute the expansion coefficients for a given $N$-body set of points assuming that the density is axisymmetric
Parameters
----------
pos : numpy.ndarray
Positions of particles in rectangular coordinates with shape [3,n]
N : int
Size of the Nth dimension of the expansion coefficients
L : int
Size of the Lth dimension of the expansion coefficients
mass : float or numpy.ndarray, optional
Mass of particles (scalar or array with size n), by default 1.0
a : float, optional
Parameter used to scale the radius, by default 1.0
Returns
-------
tuple
Expansion coefficients for density dens that can be given to SCFPotential.__init__
Notes
-----
- 2021-02-22 - Written based on general code - Bovy (UofT)
"""
r = numpy.sqrt(pos[0] ** 2 + pos[1] ** 2 + pos[2] ** 2)
costheta = pos[2] / r
mass = numpy.atleast_1d(mass)
Acos, Asin = numpy.zeros([N, L, 1]), None
Pll = numpy.ones(len(r)) # Set up Assoc. Legendre recursion
# (n,l) dependent constant
n = numpy.arange(0, N)[:, numpy.newaxis]
l = numpy.arange(0, L)[numpy.newaxis, :]
Knl = 0.5 * n * (n + 4.0 * l + 3.0) + (l + 1) * (2.0 * l + 1.0)
Inl = (
-Knl
* 2.0
* numpy.pi
/ 2.0 ** (8.0 * l + 6.0)
* gamma(n + 4.0 * l + 3.0)
/ gamma(n + 1)
/ (n + 2.0 * l + 1.5)
/ gamma(2.0 * l + 1.5) ** 2
/ numpy.sqrt(2.0 * l + 1)
)
# Set up Assoc. Legendre recursion
Plm = Pll
Plmm1 = 0.0
for ll in range(L):
# Compute Gegenbauer polys for this l
Cn = _C(_RToxi(r, a=a), N, ll, singleL=True)
phinlm = -((r / a) ** ll) / (1.0 + r / a) ** (2.0 * ll + 1) * Cn[:, 0] * Plm
# Acos
Sum = numpy.sum(mass[numpy.newaxis, :] * phinlm, axis=-1)
Acos[:, ll, 0] = Sum / Inl[:, ll]
# Recurse Assoc. Legendre
if ll < L:
tmp = Plm
Plm = ((2 * ll + 1.0) * costheta * Plm - ll * Plmm1) / (ll + 1)
Plmm1 = tmp
return Acos, Asin
def scf_compute_coeffs_axi(dens, N, L, a=1.0, radial_order=None, costheta_order=None):
"""
Numerically compute the expansion coefficients for a given axi-symmetric density
Parameters
----------
dens : function
A density function that takes parameters R and z
N : int
Size of the Nth dimension of the expansion coefficients
L : int
Size of the Lth dimension of the expansion coefficients
a : float, optional
Parameter used to shift the basis functions (default is 1.0)
radial_order : int, optional
Number of sample points of the radial integral. If None, radial_order=max(20, N + 3/2L + 1) (default is None)
costheta_order : int, optional
Number of sample points of the costheta integral. If None, If costheta_order=max(20, L + 1) (default is None)
Returns
-------
tuple
(Acos,Asin) - Expansion coefficients for density dens that can be given to SCFPotential.__init__
Notes
-----
- 2016-05-20 - Written - Aladdin Seaifan (UofT)
"""
numOfParam = 0
try:
dens(0, 0)
numOfParam = 2
except:
numOfParam = 3
param = [0] * numOfParam
dens_kw = _scf_compute_determine_dens_kwargs(dens, param)
def integrand(xi, costheta):
l = numpy.arange(0, L)[numpy.newaxis, :]
r = _xiToR(xi, a)
R = r * numpy.sqrt(1 - costheta**2.0)
z = r * costheta
Legandre = lpmn(0, L - 1, costheta)[0].T[numpy.newaxis, :, 0]
dV = (1.0 + xi) ** 2.0 * numpy.power(1.0 - xi, -4.0)
phi_nl = (
a**3
* (1.0 + xi) ** l
* (1.0 - xi) ** (l + 1.0)
* _C(xi, N, L)[:, :]
* Legandre
)
param[0] = R
param[1] = z
return phi_nl * dV * dens(*param, **dens_kw)
Acos = numpy.zeros((N, L, 1), float)
Asin = None
##This should save us some computation time since we're only taking the double integral once, rather then L times
Ksample = [max(N + 3 * L // 2 + 1, 20), max(L + 1, 20)]
if radial_order != None:
Ksample[0] = radial_order
if costheta_order != None:
Ksample[1] = costheta_order
integrated = _gaussianQuadrature(integrand, [[-1, 1], [-1, 1]], Ksample=Ksample) * (
2 * numpy.pi
)
n = numpy.arange(0, N)[:, numpy.newaxis]
l = numpy.arange(0, L)[numpy.newaxis, :]
K = 0.5 * n * (n + 4 * l + 3) + (l + 1) * (2 * l + 1)
# I = -K*(4*numpy.pi)/(2.**(8*l + 6)) * gamma(n + 4*l + 3)/(gamma(n + 1)*(n + 2*l + 3./2)*gamma(2*l + 3./2)**2)
##Taking the ln of I will allow bigger size coefficients
lnI = (
-(8 * l + 6) * numpy.log(2)
+ gammaln(n + 4 * l + 3)
- gammaln(n + 1)
- numpy.log(n + 2 * l + 3.0 / 2)
- 2 * gammaln(2 * l + 3.0 / 2)
)
I = -K * (4 * numpy.pi) * numpy.e ** (lnI)
constants = -(2.0 ** (-2 * l)) * (2 * l + 1.0) ** 0.5
Acos[:, :, 0] = 2 * I**-1 * integrated * constants
return Acos, Asin
def scf_compute_coeffs_nbody(pos, N, L, mass=1.0, a=1.0):
"""
Numerically compute the expansion coefficients for a given $N$-body set of points
Parameters
----------
pos : numpy.ndarray
Positions of particles in rectangular coordinates with shape [3,n]
N : int
Size of the Nth dimension of the expansion coefficients
L : int
Size of the Lth and Mth dimension of the expansion coefficients
mass : float or numpy.ndarray, optional
Mass of particles (scalar or array with size n), by default 1.0
a : float, optional
Parameter used to scale the radius, by default 1.0
Returns
-------
tuple
Expansion coefficients for density dens that can be given to SCFPotential.__init__
Notes
-----
- 2020-11-18 - Written - Morgan Bennett (UofT)
"""
r = numpy.sqrt(pos[0] ** 2 + pos[1] ** 2 + pos[2] ** 2)
phi = numpy.arctan2(pos[1], pos[0])
costheta = pos[2] / r