import numpy as np
from lenstronomy.LensModel.Profiles.sie import SIE
from lenstronomy.LensModel.Profiles.convergence import Convergence
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
from lenstronomy.Util import param_util
__all__ = ['CurvedArcTanDiff']
[docs]class CurvedArcTanDiff(LensProfileBase):
"""
Curved arc model with an additional non-zero tangential stretch differential in tangential direction component
Observables are:
- curvature radius (basically bending relative to the center of the profile)
- radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope)
- tangential stretch (plus sign). Infinity means at critical curve
- direction of curvature
- position of arc
Requirements:
- Should work with other perturbative models without breaking its meaning (say when adding additional shear terms)
- Must best reflect the observables in lensing
- minimal covariances between the parameters, intuitive parameterization.
"""
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'dtan_dtan', 'direction', 'center_x', 'center_y']
lower_limit_default = {'tangential_stretch': -100, 'radial_stretch': -5, 'curvature': 0.000001, 'dtan_dtan': -10, 'direction': -np.pi, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'tangential_stretch': 100, 'radial_stretch': 5, 'curvature': 100, 'dtan_dtab': 10, 'direction': np.pi, 'center_x': 100, 'center_y': 100}
def __init__(self):
self._sie = SIE(NIE=False)
self._mst = Convergence()
super(CurvedArcTanDiff, self).__init__()
[docs] @staticmethod
def stretch2sie_mst(tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y):
"""
:param tangential_stretch: float, stretch of intrinsic source in tangential direction
:param radial_stretch: float, stretch of intrinsic source in radial direction
:param curvature: 1/curvature radius
:param dtan_dtan: d(tangential_stretch) / d(tangential direction) / tangential stretch
:param direction: float, angle in radian
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return: parameters in terms of a spherical SIS + MST resulting in the same observables
"""
center_x_sis, center_y_sis = center_deflector(curvature, direction, center_x, center_y)
r_curvature = 1. / curvature
lambda_mst = 1./radial_stretch
kappa_ext = 1 - lambda_mst
theta_E = r_curvature * (1. - radial_stretch / tangential_stretch)
# translate tangential eigenvalue gradient in lens ellipticity
#TODO this is only an apprximation which is not sufficiently tested
e = abs(2 * dtan_dtan / tangential_stretch / curvature) / lambda_mst
q = 1 - (2 * e)
e = np.minimum(e, 1)
q = np.sqrt((1 - e) / (1 + e))
q = np.maximum(0.001, q)
if dtan_dtan > 0:
phi = direction + np.pi / 4
else:
phi = direction - np.pi / 4
e1_sie, e2_sie = param_util.phi_q2_ellipticity(phi, q)
# ellipticity adopted Einstein radius to match local tangential and radial stretch
factor = np.sqrt(1 + q ** 2) / np.sqrt(2 * q)
theta_E_sie = theta_E * factor
return theta_E_sie, e1_sie, e2_sie, kappa_ext, center_x_sis, center_y_sis
[docs] def function(self, x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y):
"""
ATTENTION: there may not be a global lensing potential!
:param x:
:param y:
:param tangential_stretch: float, stretch of intrinsic source in tangential direction
:param radial_stretch: float, stretch of intrinsic source in radial direction
:param curvature: 1/curvature radius
:param direction: float, angle in radian
:param dtan_dtan: d(tangential_stretch) / d(tangential direction) / tangential stretch
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
lambda_mst = 1. / radial_stretch
theta_E_sie, e1_sie, e2_sie, kappa_ext, center_x_sis, center_y_sis = self.stretch2sie_mst(tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)
f_sis = self._sie.function(x, y, theta_E_sie, e1_sie, e2_sie, center_x_sis, center_y_sis) # - self._sis.function(center_x, center_y, theta_E, center_x_sis, center_y_sis)
alpha_x, alpha_y = self._sie.derivatives(center_x, center_y, theta_E_sie, e1_sie, e2_sie, center_x_sis, center_y_sis)
f_sis_0 = alpha_x * (x - center_x) + alpha_y * (y - center_y)
f_mst = self._mst.function(x, y, kappa_ext, ra_0=center_x, dec_0=center_y)
return lambda_mst * (f_sis - f_sis_0) + f_mst
[docs] def derivatives(self, x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y):
"""
:param x:
:param y:
:param tangential_stretch: float, stretch of intrinsic source in tangential direction
:param radial_stretch: float, stretch of intrinsic source in radial direction
:param curvature: 1/curvature radius
:param direction: float, angle in radian
:param dtan_dtan: d(tangential_stretch) / d(tangential direction) / tangential stretch
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
lambda_mst = 1. / radial_stretch
theta_E_sie, e1_sie, e2_sie, kappa_ext, center_x_sis, center_y_sis = self.stretch2sie_mst(tangential_stretch,
radial_stretch, curvature, dtan_dtan,
direction, center_x, center_y)
f_x_sis, f_y_sis = self._sie.derivatives(x, y, theta_E_sie, e1_sie, e2_sie, center_x_sis, center_y_sis)
f_x0, f_y0 = self._sie.derivatives(center_x, center_y, theta_E_sie, e1_sie, e2_sie, center_x_sis, center_y_sis)
f_x_mst, f_y_mst = self._mst.derivatives(x, y, kappa_ext, ra_0=center_x, dec_0=center_y)
f_x = lambda_mst * (f_x_sis - f_x0) + f_x_mst
f_y = lambda_mst * (f_y_sis - f_y0) + f_y_mst
return f_x, f_y
[docs] def hessian(self, x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y):
"""
:param x:
:param y:
:param tangential_stretch: float, stretch of intrinsic source in tangential direction
:param radial_stretch: float, stretch of intrinsic source in radial direction
:param curvature: 1/curvature radius
:param direction: float, angle in radian
:param dtan_dtan: d(tangential_stretch) / d(tangential direction) / tangential stretch
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
lambda_mst = 1. / radial_stretch
theta_E_sie, e1_sie, e2_sie, kappa_ext, center_x_sis, center_y_sis = self.stretch2sie_mst(tangential_stretch,
radial_stretch, curvature, dtan_dtan,
direction, center_x, center_y)
f_xx_sis, f_xy_sis, f_yx_sis, f_yy_sis = self._sie.hessian(x, y, theta_E_sie, e1_sie, e2_sie, center_x_sis, center_y_sis)
f_xx_mst, f_xy_mst, f_yx_mst, f_yy_mst = self._mst.hessian(x, y, kappa_ext, ra_0=center_x, dec_0=center_y)
return lambda_mst * f_xx_sis + f_xx_mst, lambda_mst * f_xy_sis + f_xy_mst, lambda_mst * f_yx_sis + f_yx_mst, lambda_mst * f_yy_sis + f_yy_mst
def center_deflector(curvature, direction, center_x, center_y):
"""
:param curvature: 1/curvature radius
:param direction: float, angle in radian
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return: center_sis_x, center_sis_y
"""
center_x_sis = center_x - np.cos(direction) / curvature
center_y_sis = center_y - np.sin(direction) / curvature
return center_x_sis, center_y_sis