# -*- coding: utf-8 -*-
"""This module defines ``class GaussianEllipseKappa`` to compute the lensing properties
of an elliptical Gaussian profile with ellipticity in the convergence using the formulae
from Shajib (2019)."""
__author__ = "ajshajib"
import numpy as np
from scipy.special import wofz
from scipy.integrate import quad
from copy import deepcopy
from lenstronomy.LensModel.Profiles.gaussian_kappa import GaussianKappa
import lenstronomy.Util.param_util as param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["GaussianEllipseKappa"]
[docs]
class GaussianEllipseKappa(LensProfileBase):
"""This class contains functions to evaluate the derivative and hessian matrix of
the deflection potential for an elliptical Gaussian convergence.
The formulae are from Shajib (2019).
"""
param_names = ["amp", "sigma", "e1", "e2", "center_x", "center_y"]
lower_limit_default = {
"amp": 0,
"sigma": 0,
"e1": -0.5,
"e2": -0.5,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"amp": 100,
"sigma": 100,
"e1": 0.5,
"e2": 0.5,
"center_x": 100,
"center_y": 100,
}
[docs]
def __init__(self, use_scipy_wofz=True, min_ellipticity=1e-5):
"""Setup which method to use the Faddeeva function and the ellipticity limit for
spherical approximation.
:param use_scipy_wofz: If ``True``, use ``scipy.special.wofz``.
:type use_scipy_wofz: ``bool``
:param min_ellipticity: Minimum allowed ellipticity. For ``q > 1 - min_ellipticity``, values for spherical case will be returned.
:type min_ellipticity: ``float``
"""
if use_scipy_wofz:
self.w_f = wofz
else:
self.w_f = self.w_f_approx
self.min_ellipticity = min_ellipticity
self.spherical = GaussianKappa()
super(GaussianEllipseKappa, self).__init__()
[docs]
def function(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""Compute the potential function for elliptical Gaussian convergence.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \\pi\\sigma^2) \\exp(-(x^2+y^2/q^2)/2\\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Potential for elliptical Gaussian convergence
:rtype: ``float``, or ``numpy.array`` with shape equal to ``x.shape``
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.function(x, y, amp, sigma, center_x, center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_b = 1.0 / 2.0 / sigma_**2
_p = np.sqrt(_b * q**2 / (1.0 - q**2))
if isinstance(x_, int) or isinstance(x_, float):
return self._num_integral(x_, y_, amp_, sigma_, _p, q)
else:
f_ = []
for i in range(len(x_)):
f_.append(self._num_integral(x_[i], y_[i], amp_, sigma_, _p, q))
return np.array(f_)
def _num_integral(self, x_, y_, amp_, sigma_, _p, q):
"""
:param x_:
:param y_:
:param _p:
:param q:
:return:
"""
def pot_real_line_integrand(_x):
sig_func_re, sig_func_im = self.sigma_function(_p * _x, 0, q)
alpha_x_ = (
amp_
* sigma_
* self.sgn(_x)
* np.sqrt(2 * np.pi / (1.0 - q**2))
* sig_func_re
)
return alpha_x_
def pot_imag_line_integrand(_y):
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * _y, q)
alpha_y_ = (
-amp_
* sigma_
* self.sgn(x_ + 1j * _y)
* np.sqrt(2 * np.pi / (1.0 - q**2))
* sig_func_im
)
return alpha_y_
pot_on_real_line = quad(pot_real_line_integrand, 0, x_)[0]
pot_on_imag_parallel = quad(pot_imag_line_integrand, 0, y_)[0]
return pot_on_real_line + pot_on_imag_parallel
[docs]
def derivatives(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""Compute the derivatives of function angles :math:`\\partial f/\\partial x`,
:math:`\\partial f/\\partial y` at :math:`x,\\ y`.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \\pi\\sigma^2) \\exp(-(x^2+y^2/q^2)/2\\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Deflection angle :math:`\\partial f/\\partial x`, :math:`\\partial f/\\partial y`
for elliptical Gaussian convergence.
:rtype: tuple ``(float, float)`` or ``(numpy.array, numpy.array)`` with each ``numpy.array``'s shape equal
to ``x.shape``.
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.derivatives(x, y, amp, sigma, center_x, center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
# rotated coordinates
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_p = q / sigma_ / np.sqrt(2 * (1.0 - q**2))
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * y_, q)
alpha_x_ = (
amp_
* sigma_
* self.sgn(x_ + 1j * y_)
* np.sqrt(2 * np.pi / (1.0 - q**2))
* sig_func_re
)
alpha_y_ = (
-amp_
* sigma_
* self.sgn(x_ + 1j * y_)
* np.sqrt(2 * np.pi / (1.0 - q**2))
* sig_func_im
)
# rotate back to the original frame
f_x = alpha_x_ * cos_phi - alpha_y_ * sin_phi
f_y = alpha_x_ * sin_phi + alpha_y_ * cos_phi
return f_x, f_y
[docs]
def hessian(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""Compute Hessian matrix of function :math:`\\partial^2f/\\partial x^2`,
:math:`\\partial^2 f/\\partial y^2`, :math:`\\partial^2/\\partial x\\partial y`.
:param x: x coordinate
:type x: ``float`` or ``numpy.array``
:param y: y coordinate
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \\pi\\sigma^2) \\exp(-(x^2+y^2/q^2)/2\\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian
:type sigma: ``float``
:param e1: Ellipticity parameter 1
:type e1: ``float``
:param e2: Ellipticity parameter 2
:type e2: ``float``
:param center_x: x coordinate of centroid
:type center_x: ``float``
:param center_y: y coordianate of centroid
:type center_y: ``float``
:return: Hessian :math:`A/(2 \\pi \\sigma^2) \\exp(-(x^2+y^2/q^2)/2\\sigma^2)` for elliptical Gaussian
convergence.
:rtype: tuple ``(float, float, float)`` , or ``(numpy.array, numpy.array, numpy.array)`` with each ``numpy.array``'s shape equal to ``x.shape``.
"""
phi_g, q = param_util.ellipticity2phi_q(e1, e2)
if q > 1 - self.min_ellipticity:
return self.spherical.hessian(x, y, amp, sigma, center_x, center_y)
# adjusting amplitude to make the notation compatible with the
# formulae given in Shajib (2019).
amp_ = amp / (2 * np.pi * sigma**2)
# converting ellipticity definition from q*x^2 + y^2/q to q^2*x^2 + y^2
sigma_ = sigma * np.sqrt(q) # * q
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_g)
sin_phi = np.sin(phi_g)
# rotated coordinates
x_ = cos_phi * x_shift + sin_phi * y_shift
y_ = -sin_phi * x_shift + cos_phi * y_shift
_p = q / sigma_ / np.sqrt(2 * (1.0 - q**2))
sig_func_re, sig_func_im = self.sigma_function(_p * x_, _p * y_, q)
kappa = amp_ * np.exp(-(q**2 * x_**2 + y_**2) / 2 / sigma_**2)
shear = (
-1
/ (1 - q * q)
* (
(1 + q**2) * kappa
- 2 * q * amp_
+ np.sqrt(2 * np.pi)
* q
* q
* amp_
* (x_ - 1j * y_)
/ sigma_
/ np.sqrt(1 - q * q)
* (sig_func_re - 1j * sig_func_im)
)
)
# in rotated frame
f_xx_ = kappa + shear.real
f_yy_ = kappa - shear.real
f_xy_ = shear.imag
# rotate back to the original frame
f_xx = (
f_xx_ * cos_phi**2 + f_yy_ * sin_phi**2 - 2 * sin_phi * cos_phi * f_xy_
)
f_yy = (
f_xx_ * sin_phi**2 + f_yy_ * cos_phi**2 + 2 * sin_phi * cos_phi * f_xy_
)
f_xy = (
sin_phi * cos_phi * (f_xx_ - f_yy_) + (cos_phi**2 - sin_phi**2) * f_xy_
)
return f_xx, f_xy, f_xy, f_yy
[docs]
def density_2d(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""Compute the density of elliptical Gaussian :math:`A/(2 \\pi \\sigma^2)
\\exp(-(x^2+y^2/q^2)/2\\sigma^2)`.
:param x: x coordinate.
:type x: ``float`` or ``numpy.array``
:param y: y coordinate.
:type y: ``float`` or ``numpy.array``
:param amp: Amplitude of Gaussian, convention: :math:`A/(2 \\pi\\sigma^2) \\exp(-(x^2+y^2/q^2)/2\\sigma^2)`
:type amp: ``float``
:param sigma: Standard deviation of Gaussian.
:type sigma: ``float``
:param e1: Ellipticity parameter 1.
:type e1: ``float``
:param e2: Ellipticity parameter 2.
:type e2: ``float``
:param center_x: x coordinate of centroid.
:type center_x: ``float``
:param center_y: y coordianate of centroid.
:type center_y: ``float``
:return: Density :math:`\\kappa` for elliptical Gaussian convergence.
:rtype: ``float``, or ``numpy.array`` with shape = ``x.shape``.
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(
x, y, amp, sigma, e1, e2, center_x, center_y
)
return (f_xx + f_yy) / 2
[docs]
@staticmethod
def sgn(z):
"""Compute the sign function :math:`\\mathrm{sgn}(z)` factor for deflection as
sugggested by Bray (1984). For current implementation, returning 1 is
sufficient.
:param z: Complex variable :math:`z = x + \\mathrm{i}y`
:type z: ``complex``
:return: :math:`\\mathrm{sgn}(z)`
:rtype: ``float``
"""
return 1.0
# np.sqrt(z*z)/z #np.sign(z.real*z.imag)
# return np.sign(z.real)
# if z.real != 0:
# return np.sign(z.real)
# else:
# return np.sign(z.imag)
# return np.where(z.real == 0, np.sign(z.real), np.sign(z.imag))
[docs]
def sigma_function(self, x, y, q):
r"""Compute the function :math:`\varsigma (z; q)` from equation (4.12) of Shajib
(2019).
:param x: Real part of complex variable, :math:`x = \mathrm{Re}(z)`
:type x: ``float`` or ``numpy.array``
:param y: Imaginary part of complex variable, :math:`y = \mathrm{Im}(z)`
:type y: ``float`` or ``numpy.array``
:param q: Axis ratio
:type q: ``float``
:return: real and imaginary part of :math:`\varsigma(z; q)` function
:rtype: tuple ``(type(x), type(x))``
"""
y_sign = np.sign(y)
y_ = deepcopy(y) * y_sign
z = x + 1j * y_
zq = q * x + 1j * y_ / q
w = self.w_f(z)
wq = self.w_f(zq)
# exponential factor in the 2nd term of eqn. (4.15) of Shajib (2019)
exp_factor = np.exp(-x * x * (1 - q * q) - y_ * y_ * (1 / q / q - 1))
sigma_func_real = w.imag - exp_factor * wq.imag
sigma_func_imag = (-w.real + exp_factor * wq.real) * y_sign
return sigma_func_real, sigma_func_imag
[docs]
@staticmethod
def w_f_approx(z):
"""Compute the Faddeeva function :math:`w_{\\mathrm F}(z)` using the
approximation given in Zaghloul (2017).
:param z: complex number
:type z: ``complex`` or ``numpy.array(dtype=complex)``
:return: :math:`w_\\mathrm{F}(z)`
:rtype: ``complex``
"""
sqrt_pi = 1 / np.sqrt(np.pi)
i_sqrt_pi = 1j * sqrt_pi
wz = np.empty_like(z)
z_imag2 = z.imag**2
abs_z2 = z.real**2 + z_imag2
reg1 = abs_z2 >= 38000.0
if np.any(reg1):
wz[reg1] = i_sqrt_pi / z[reg1]
reg2 = (256.0 <= abs_z2) & (abs_z2 < 38000.0)
if np.any(reg2):
t = z[reg2]
wz[reg2] = i_sqrt_pi * t / (t * t - 0.5)
reg3 = (62.0 <= abs_z2) & (abs_z2 < 256.0)
if np.any(reg3):
t = z[reg3]
wz[reg3] = (i_sqrt_pi / t) * (1 + 0.5 / (t * t - 1.5))
reg4 = (30.0 <= abs_z2) & (abs_z2 < 62.0) & (z_imag2 >= 1e-13)
if np.any(reg4):
t = z[reg4]
tt = t * t
wz[reg4] = (i_sqrt_pi * t) * (tt - 2.5) / (tt * (tt - 3.0) + 0.75)
reg5 = (
(62.0 > abs_z2) & np.logical_not(reg4) & (abs_z2 > 2.5) & (z_imag2 < 0.072)
)
if np.any(reg5):
t = z[reg5]
u = -t * t
f1 = sqrt_pi
f2 = 1
s1 = [1.320522, 35.7668, 219.031, 1540.787, 3321.99, 36183.31]
s2 = [1.841439, 61.57037, 364.2191, 2186.181, 9022.228, 24322.84, 32066.6]
for s in s1:
f1 = s - f1 * u
for s in s2:
f2 = s - f2 * u
wz[reg5] = np.exp(u) + 1j * t * f1 / f2
reg6 = (30.0 > abs_z2) & np.logical_not(reg5)
if np.any(reg6):
t3 = -1j * z[reg6]
f1 = sqrt_pi
f2 = 1
s1 = [5.9126262, 30.180142, 93.15558, 181.92853, 214.38239, 122.60793]
s2 = [
10.479857,
53.992907,
170.35400,
348.70392,
457.33448,
352.73063,
122.60793,
]
for s in s1:
f1 = f1 * t3 + s
for s in s2:
f2 = f2 * t3 + s
wz[reg6] = f1 / f2
return wz