__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}
def __init__(self):
self.nie_potential_major_axis = NIEPotentialMajorAxis()
super(NIE_POTENTIAL, self).__init__()
[docs] 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
[docs] 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)
[docs] 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
[docs] 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_
[docs] 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
[docs] 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./2 * (f__xx + f__yy)
gamma1__ = 1./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.+q**2) / (2. * q))) #/ (1+(1-q)/2.)
return theta_E_new
[docs]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']
def __init__(self, diff=0.0000000001):
self._diff = diff
super(NIEPotentialMajorAxis, self).__init__()
[docs] 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_
[docs] 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
[docs] 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