__author__ = "sibirrer"
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", "NIEMajorAxis"]
[docs]
class NIE(LensProfileBase):
"""Non-singular isothermal ellipsoid (NIE)
.. math::
\\kappa = \\theta_E/2 \\left[s^2_{scale} + qx^2 + y^2/q]−1/2
"""
param_names = ["theta_E", "e1", "e2", "s_scale", "center_x", "center_y"]
lower_limit_default = {
"theta_E": 0,
"e1": -0.5,
"e2": -0.5,
"s_scale": 0,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"theta_E": 10,
"e1": 0.5,
"e2": 0.5,
"s_scale": 100,
"center_x": 100,
"center_y": 100,
}
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def __init__(self):
self.nie_major_axis = NIEMajorAxis()
super(NIE, self).__init__()
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def function(self, x, y, theta_E, e1, e2, s_scale, 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 e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: lensing potential
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self.nie_major_axis.function(x__, y__, b, s, q)
# rotate back
return f_
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def derivatives(self, x, y, theta_E, e1, e2, s_scale, 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 e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: alpha_x, alpha_y
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# 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_major_axis.derivatives(x__, y__, b, s, q)
# 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, e1, e2, s_scale, 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 e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: f_xx, f_xy, f_yx, f_yy
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# 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_major_axis.hessian(x__, y__, b, s, q)
# 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
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def density_lens(self, r, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""3d mass density at 3d radius r. This function assumes spherical
symmetry/ignoring the eccentricity.
:param r: 3d radius
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: 3d mass density at 3d radius r
"""
# kappa=1/2 at Einstein radius
rho0 = 1 / 2 * theta_E / np.pi
return rho0 / (r**2 + s_scale**2)
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def mass_3d_lens(self, r, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""Mass enclosed a 3d radius r. This function assumes spherical
symmetry/ignoring the eccentricity.
:param r: 3d radius
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: 3d mass density at 3d radius r
"""
rho0 = 1 / 2 * theta_E / np.pi
return rho0 * 4 * np.pi * (r - s_scale * np.arctan(r / s_scale))
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def param_conv(self, theta_E, e1, e2, s_scale):
if self._static is True:
return self._b_static, self._s_static, self._q_static, self._phi_G_static
return self._param_conv(theta_E, e1, e2, s_scale)
def _param_conv(self, theta_E, e1, e2, s_scale):
"""
convert parameters from 2*kappa = bIE [s2IE + r2(1 − e *cos(2*phi)]−1/2 to
2*kappa= b *(q2(s2 + x2) + y2)−1/2
see expressions after Equation 8 in Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:return: critical radius b, smoothing scale s, axis ratio q, orientation angle phi_G
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
theta_E_conv = self._theta_E_prod_average2major_axis(theta_E, q)
b = theta_E_conv * np.sqrt((1 + q**2) / 2)
s = s_scale / np.sqrt(q)
# s = s_scale * np.sqrt((1 + q**2) / (2*q**2))
return b, s, q, phi_G
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def set_static(self, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: self variables set
"""
self._static = True
(
self._b_static,
self._s_static,
self._q_static,
self._phi_G_static,
) = self._param_conv(theta_E, e1, e2, s_scale)
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def set_dynamic(self):
"""
:return:
"""
self._static = False
if hasattr(self, "_b_static"):
del self._b_static
if hasattr(self, "_s_static"):
del self._s_static
if hasattr(self, "_phi_G_static"):
del self._phi_G_static
if hasattr(self, "_q_static"):
del self._q_static
@staticmethod
def _theta_E_prod_average2major_axis(theta_E, q):
"""Converts a product averaged Einstein radius (of semi-minor and semi-major
axis) to a major axis Einstein radius for an Isothermal ellipse. The standard
lenstronomy conventions are product averaged Einstein radii while other codes
(such as e.g. gravlens) use the semi-major axis convention.
.. math::
\\frac{\\theta_{E, prod ave}}{\\theta_{E, major}} = \\sqrt{(1+q^2) / (2 q) }
:param theta_E: Einstein radius in lenstronomy conventions (product average of major and minor axes)
:param q: axis ratio minor/major
:return: theta_E in convention of kappa= b *(q^2(s^2 + x^2) + y^2)^{−1/2} (major axis)
"""
theta_E_major_axis = theta_E / (np.sqrt((1.0 + q**2) / (2.0 * q)))
return theta_E_major_axis
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class NIEMajorAxis(LensProfileBase):
"""
This class contains the function and the derivatives of the non-singular isothermal ellipse.
See Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf
.. math::
\\kappa = b * (q2(s2 + x2) + y2)^{−1/2}`
"""
param_names = ["b", "s", "q", "center_x", "center_y"]
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def __init__(self, diff=0.0000000001):
self._diff = diff
super(NIEMajorAxis, self).__init__()
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def function(self, x, y, b, s, q):
psi = self._psi(x, y, q, s)
alpha_x, alpha_y = self.derivatives(x, y, b, s, q)
f_ = (
x * alpha_x
+ y * alpha_y
- b * s * 1.0 / 2.0 * np.log((psi + s) ** 2 + (1.0 - q**2) * x**2)
)
return f_
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def derivatives(self, x, y, b, s, q):
"""Returns df/dx and df/dy of the function."""
if q >= 1:
q = 0.99999999
psi = self._psi(x, y, q, s)
f_x = (
b / np.sqrt(1.0 - q**2) * np.arctan(np.sqrt(1.0 - q**2) * x / (psi + s))
)
f_y = (
b
/ np.sqrt(1.0 - q**2)
* np.arctanh(np.sqrt(1.0 - q**2) * y / (psi + q**2 * s))
)
return f_x, f_y
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def hessian(self, x, y, b, s, q):
"""Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx,
d^f/dy^2."""
alpha_ra, alpha_dec = self.derivatives(x, y, b, s, q)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, b, s, q)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, b, s, q)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
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@staticmethod
def kappa(x, y, b, s, q):
"""convergence.
:param x: major axis coordinate
:param y: minor axis coordinate
:param b: normalization
:param s: smoothing scale
:param q: axis ratio
:return: convergence
"""
kappa = b / 2.0 * (q**2 * (s**2 + x**2) + y**2) ** (-1.0 / 2)
return kappa
@staticmethod
def _psi(x, y, q, s):
"""Expression after equation (8) in Keeton&Kochanek 1998.
:param x: semi-major axis coordinate
:param y: semi-minor axis coordinate
:param q: axis ratio minor/major
:param s: smoothing scale in major axis direction
:return: phi
"""
return np.sqrt(q**2 * (s**2 + x**2) + y**2)