__author__ = "gipagano"
import numpy as np
import lenstronomy.Util.util as util
import lenstronomy.Util.param_util as param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["NIE_POTENTIAL", "NIEPotentialMajorAxis"]
[docs]
class NIE_POTENTIAL(LensProfileBase):
"""This class implements the elliptical potential of Eq.
(67) of `LECTURES ON GRAVITATIONAL LENSING <https://arxiv.org/pdf/astro-ph/9606001.pdf>`_
and Eq. (1) of `Blandford & Kochanek 1987 <http://adsabs.harvard.edu/full/1987ApJ...321..658B>`_,
mapped to Eq. (8) of `Barnaka1998 <https://iopscience.iop.org/article/10.1086/305950/fulltext/37798.text.html>`_
to find the ellipticity bounds
"""
param_names = ["center_x", "center_y", "theta_E", "theta_c", "e1", "e2"]
lower_limit_default = {
"center_x": -100,
"center_y": -100,
"theta_E": 0,
"theta_c": 0,
"e1": 0,
"e2": 0,
}
upper_limit_default = {
"center_x": 100,
"center_y": 100,
"theta_E": 10,
"theta_c": 10,
"e1": 0.2,
"e2": 0.2,
}
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def __init__(self):
self.nie_potential_major_axis = NIEPotentialMajorAxis()
super(NIE_POTENTIAL, self).__init__()
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def param_conv(self, theta_E, theta_c, e1, e2):
if self._static is True:
return (
self._thetaE_transf_static,
self._thetac_static,
self._eps_static,
self._phi_G_static,
)
return self._param_conv(theta_E, theta_c, e1, e2)
def _param_conv(self, theta_E, theta_c, e1, e2):
"""Convert the spherical averaged Einstein radius to an elliptical (major axis)
Einstein radius and the individual eccentricities to the modulus of the
eccentricity.
:param theta_E: Einstein radius
:param theta_c: core radius
:param e1: eccentricity component
:param e2: eccentricity component
:return: transformed Einstein radius, core radius, ellipticity modulus,
orientation angle phi_G
"""
eps = np.sqrt(e1**2 + e2**2)
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
theta_E_conv = self._theta_q_convert(theta_E, q)
theta_c_conv = self._theta_q_convert(theta_c, q)
return theta_E_conv, theta_c_conv, eps, phi_G
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def set_static(self, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius
:param theta_c: core radius
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: profile center
:param center_y: profile center
:return: self variables set
"""
self._static = True
(
self._thetaE_transf_static,
self._thetac_static,
self._eps_static,
self._phi_G_static,
) = self._param_conv(theta_E, theta_c, e1, e2)
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def set_dynamic(self):
"""
:return:
"""
self._static = False
if hasattr(self, "_thetaE_transf_static"):
del self._thetaE_transf_static
if hasattr(self, "_thetac_static"):
del self._thetac_static
if hasattr(self, "_eps_static"):
del self._eps_static
if hasattr(self, "_phi_G_static"):
del self._phi_G_static
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def function(self, x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coord (in angles)
:param y: y-coord (in angles)
:param theta_E: Einstein radius (in angles)
:param theta_c: core radius (in angles)
:param e1: eccentricity component, x direction(dimensionless)
:param e2: eccentricity component, y direction (dimensionless)
:return: lensing potential
"""
theta_E_conv, theta_c_conv, eps, phi_G = self.param_conv(
theta_E, theta_c, e1, e2
)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self.nie_potential_major_axis.function(
x__, y__, theta_E_conv, theta_c_conv, eps
)
# rotate back
return f_
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def derivatives(self, x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coord (in angles)
:param y: y-coord (in angles)
:param theta_E: Einstein radius (in angles)
:param theta_c: core radius (in angles)
:param e1: eccentricity component, x direction(dimensionless)
:param e2: eccentricity component, y direction (dimensionless)
:return: deflection angle (in angles)
"""
theta_E_conv, theta_c_conv, eps, phi_G = self.param_conv(
theta_E, theta_c, e1, e2
)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__x, f__y = self.nie_potential_major_axis.derivatives(
x__, y__, theta_E_conv, theta_c_conv, eps
)
# rotate back
f_x, f_y = util.rotate(f__x, f__y, -phi_G)
return f_x, f_y
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def hessian(self, x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coord (in angles)
:param y: y-coord (in angles)
:param theta_E: Einstein radius (in angles)
:param theta_c: core radius (in angles)
:param e1: eccentricity component, x direction(dimensionless)
:param e2: eccentricity component, y direction (dimensionless)
:return: hessian matrix (in angles)
"""
theta_E_conv, theta_c_conv, eps, phi_G = self.param_conv(
theta_E, theta_c, e1, e2
)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__xx, f__xy, _, f__yy = self.nie_potential_major_axis.hessian(
x__, y__, theta_E_conv, theta_c_conv, eps
)
# rotate back
kappa = 1.0 / 2 * (f__xx + f__yy)
gamma1__ = 1.0 / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
def _theta_q_convert(self, theta_E, q):
"""Converts a spherical averaged Einstein radius/core radius to an elliptical
(major axis) Einstein radius. This then follows the convention of the SPEMD
profile in lenstronomy. (theta_E / theta_E_gravlens) = sqrt[ (1+q^2) / (2 q) ]
:param theta_E: Einstein radius in lenstronomy conventions
:param q: axis ratio minor/major
:return: theta_E in convention of kappa= b *(q2(s2 + x2) + y2)−1/2
"""
theta_E_new = theta_E / (np.sqrt((1.0 + q**2) / (2.0 * q))) # / (1+(1-q)/2.)
return theta_E_new
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class NIEPotentialMajorAxis(LensProfileBase):
"""This class implements the elliptical potential of Eq.
(67) of `LECTURES ON GRAVITATIONAL LENSING <https://arxiv.org/pdf/astro-ph/9606001.pdf>`_
and Eq. (1) of `Blandford & Kochanek 1987 <http://adsabs.harvard.edu/full/1987ApJ...321..658B>`_,
mapped to Eq. (8) of `Barnaka1998 <https://iopscience.iop.org/article/10.1086/305950/fulltext/37798.text.html>`_
to find the ellipticity bounds
"""
param_names = ["theta_E", "theta_c", "eps", "center_x", "center_y"]
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def __init__(self, diff=0.0000000001):
self._diff = diff
super(NIEPotentialMajorAxis, self).__init__()
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def function(self, x, y, theta_E, theta_c, eps):
f_ = theta_E * np.sqrt(theta_c**2 + (1 - eps) * x**2 + (1 + eps) * y**2)
return f_
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def derivatives(self, x, y, theta_E, theta_c, eps):
"""Returns df/dx and df/dy of the function."""
factor = np.sqrt(theta_c**2 + (1 - eps) * x**2 + (1 + eps) * y**2)
f_x = (theta_E / factor) * (1 - eps) * x
f_y = (theta_E / factor) * (1 + eps) * y
return f_x, f_y
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def hessian(self, x, y, theta_E, theta_c, eps):
"""Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx,
d^f/dy^2."""
factor = np.sqrt(theta_c**2 + (1 - eps) * x**2 + (1 + eps) * y**2)
f_xx = (1 - eps) * (theta_E / factor) - (theta_E / factor**3) * (
1 - eps
) ** 2 * x**2
f_yy = (1 + eps) * (theta_E / factor) - (theta_E / factor**3) * (
1 + eps
) ** 2 * y**2
f_xy = -(theta_E / factor**3) * (1 - eps**2) * x * y
return f_xx, f_xy, f_xy, f_yy