import numpy as np
from lenstronomy.LensModel.Profiles.spp import SPP
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ['CurvedArcSPP', 'center_deflector']
[docs]class CurvedArcSPP(LensProfileBase):
"""
lens model that describes a section of a highly magnified deflector region.
The parameterization is chosen to describe local observables efficient.
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', 'direction', 'center_x', 'center_y']
lower_limit_default = {'tangential_stretch': -100, 'radial_stretch': -5, 'curvature': 0.000001, 'direction': -np.pi, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'tangential_stretch': 100, 'radial_stretch': 5, 'curvature': 100, 'direction': np.pi, 'center_x': 100, 'center_y': 100}
def __init__(self):
self._spp = SPP()
super(CurvedArcSPP, self).__init__()
[docs] @staticmethod
def stretch2spp(tangential_stretch, radial_stretch, curvature, 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 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 power-law profile resulting in the same observables
"""
center_x_spp, center_y_spp = center_deflector(curvature, direction, center_x, center_y)
r_curvature = 1. / curvature
gamma = (1./radial_stretch - 1) / (1 - 1./tangential_stretch) + 2
theta_E = abs(1 - 1./tangential_stretch)**(1./(gamma - 1)) * r_curvature
return theta_E, gamma, center_x_spp, center_y_spp
[docs] @staticmethod
def spp2stretch(theta_E, gamma, center_x_spp, center_y_spp, center_x, center_y):
"""
turn Singular power-law lens model into stretch parameterization at position (center_x, center_y)
This is the inverse function of stretch2spp()
:param theta_E: Einstein radius of SPP model
:param gamma: power-law slope
:param center_x_spp: center of SPP model
:param center_y_spp: center of SPP model
:param center_x: center of curved model definition
:param center_y: center of curved model definition
:return: tangential_stretch, radial_stretch, curvature, direction
"""
r_curvature = np.sqrt((center_x_spp - center_x)**2 + (center_y_spp - center_y)**2)
direction = np.arctan2(center_y - center_y_spp, center_x - center_x_spp)
tangential_stretch = 1 / (1 - (theta_E/r_curvature) ** (gamma - 1))
radial_stretch = 1 / (1 + (gamma - 2) * (theta_E/r_curvature) ** (gamma - 1))
curvature = 1./r_curvature
return tangential_stretch, radial_stretch, curvature, direction
[docs] def function(self, x, y, tangential_stretch, radial_stretch, curvature, 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 center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
theta_E, gamma, center_x_spp, center_y_spp = self.stretch2spp(tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)
f_ = self._spp.function(x, y, theta_E, gamma, center_x_spp, center_y_spp)
alpha_x, alpha_y = self._spp.derivatives(center_x, center_y, theta_E, gamma, center_x_spp, center_y_spp)
f_0 = alpha_x * (x - center_x) + alpha_y * (y - center_y)
return f_ - f_0
[docs] def derivatives(self, x, y, tangential_stretch, radial_stretch, curvature, 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 center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
theta_E, gamma, center_x_spp, center_y_spp = self.stretch2spp(tangential_stretch,
radial_stretch, curvature,
direction, center_x, center_y)
f_x, f_y = self._spp.derivatives(x, y, theta_E, gamma, center_x_spp, center_y_spp)
f_x0, f_y0 = self._spp.derivatives(center_x, center_y, theta_E, gamma, center_x_spp, center_y_spp)
return f_x - f_x0, f_y - f_y0
[docs] def hessian(self, x, y, tangential_stretch, radial_stretch, curvature, 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 center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
theta_E, gamma, center_x_spp, center_y_spp = self.stretch2spp(tangential_stretch,
radial_stretch, curvature,
direction, center_x, center_y)
return self._spp.hessian(x, y, theta_E, gamma, center_x_spp, center_y_spp)
[docs]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_spp_x, center_spp_y
"""
center_x_spp = center_x - np.cos(direction) / curvature
center_y_spp = center_y - np.sin(direction) / curvature
return center_x_spp, center_y_spp