__author__ = "sibirrer"
import numpy as np
from lenstronomy.Util import util
from lenstronomy.LensModel.Profiles.nfw import NFW
from lenstronomy.LensModel.Profiles.nfw_ellipse import NFW_ELLIPSE
from lenstronomy.LensModel.Profiles.cored_steep_ellipsoid import CSEProductAvgSet
import lenstronomy.Util.param_util as param_util
__all__ = ["NFW_ELLIPSE_CSE"]
[docs]
class NFW_ELLIPSE_CSE(NFW_ELLIPSE):
"""
this class contains functions concerning the NFW profile with an ellipticity defined in the convergence
parameterization of alpha_Rs and Rs is the same as for the spherical NFW profile
Approximation with CSE profile introduced by Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf
Match to NFW using CSEs is approximate: kappa matches to ~1-2%
relation are: R_200 = c * Rs
"""
profile_name = "NFW_ELLIPSE_CSE"
param_names = ["Rs", "alpha_Rs", "e1", "e2", "center_x", "center_y"]
lower_limit_default = {
"Rs": 0,
"alpha_Rs": 0,
"e1": -0.5,
"e2": -0.5,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"Rs": 100,
"alpha_Rs": 10,
"e1": 0.5,
"e2": 0.5,
"center_x": 100,
"center_y": 100,
}
[docs]
def __init__(self, high_accuracy=True):
"""
:param high_accuracy: if True uses a more accurate larger set of CSE profiles (see Oguri 2021)
:type high_accuracy: boolean
"""
self.cse_major_axis_set = CSEProductAvgSet()
self.nfw = NFW()
if high_accuracy is True:
# Table 1 in Oguri 2021
self._s_list = [
1.082411e-06,
8.786566e-06,
3.292868e-06,
1.860019e-05,
3.274231e-05,
6.232485e-05,
9.256333e-05,
1.546762e-04,
2.097321e-04,
3.391140e-04,
5.178790e-04,
8.636736e-04,
1.405152e-03,
2.193855e-03,
3.179572e-03,
4.970987e-03,
7.631970e-03,
1.119413e-02,
1.827267e-02,
2.945251e-02,
4.562723e-02,
6.782509e-02,
1.596987e-01,
1.127751e-01,
2.169469e-01,
3.423835e-01,
5.194527e-01,
8.623185e-01,
1.382737e00,
2.034929e00,
3.402979e00,
5.594276e00,
8.052345e00,
1.349045e01,
2.603825e01,
4.736823e01,
6.559320e01,
1.087932e02,
1.477673e02,
2.495341e02,
4.305999e02,
7.760206e02,
2.143057e03,
1.935749e03,
]
self._a_list = [
1.648988e-18,
6.274458e-16,
3.646620e-17,
3.459206e-15,
2.457389e-14,
1.059319e-13,
4.211597e-13,
1.142832e-12,
4.391215e-12,
1.556500e-11,
6.951271e-11,
3.147466e-10,
1.379109e-09,
3.829778e-09,
1.384858e-08,
5.370951e-08,
1.804384e-07,
5.788608e-07,
3.205256e-06,
1.102422e-05,
4.093971e-05,
1.282206e-04,
4.575541e-04,
7.995270e-04,
5.013701e-03,
1.403508e-02,
5.230727e-02,
1.898907e-01,
3.643448e-01,
7.203734e-01,
1.717667e00,
2.217566e00,
3.187447e00,
8.194898e00,
1.765210e01,
1.974319e01,
2.783688e01,
4.482311e01,
5.598897e01,
1.426485e02,
2.279833e02,
5.401335e02,
9.743682e02,
1.775124e03,
]
else:
# Table 3 in Oguri 2021
self._a_list = [
1.434960e-16,
5.232413e-14,
2.666660e-12,
7.961761e-11,
2.306895e-09,
6.742968e-08,
1.991691e-06,
5.904388e-05,
1.693069e-03,
4.039850e-02,
5.665072e-01,
3.683242e00,
1.582481e01,
6.340984e01,
2.576763e02,
1.422619e03,
]
self._s_list = [
4.041628e-06,
3.086267e-05,
1.298542e-04,
4.131977e-04,
1.271373e-03,
3.912641e-03,
1.208331e-02,
3.740521e-02,
1.153247e-01,
3.472038e-01,
1.017550e00,
3.253031e00,
1.190315e01,
4.627701e01,
1.842613e02,
8.206569e02,
]
super(NFW_ELLIPSE_CSE, self).__init__()
[docs]
def function(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""Returns elliptically distorted NFW lensing potential.
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: lensing potential
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
# potential calculation
f_ = self.cse_major_axis_set.function(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q
)
const = self._normalization(alpha_Rs, Rs, q)
return const * f_
[docs]
def derivatives(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""Returns df/dx and df/dy of the function, calculated as an elliptically
distorted deflection angle of the spherical NFW profile.
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: deflection in x-direction, deflection in y-direction
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__x, f__y = self.cse_major_axis_set.derivatives(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q
)
# rotate deflections back
f_x, f_y = util.rotate(f__x, f__y, -phi_q)
const = self._normalization(alpha_Rs, Rs, q) / Rs
return const * f_x, const * f_y
[docs]
def hessian(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy the
calculation is performed as a numerical differential from the deflection field.
Analytical relations are possible.
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__xx, f__xy, __, f__yy = self.cse_major_axis_set.hessian(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, 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_q) * gamma1__ - np.sin(2 * phi_q) * gamma2__
gamma2 = +np.sin(2 * phi_q) * gamma1__ + np.cos(2 * phi_q) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
const = self._normalization(alpha_Rs, Rs, q) / Rs**2
return const * f_xx, const * f_xy, const * f_xy, const * f_yy
def _normalization(self, alpha_Rs, Rs, q):
"""Applying to eqn 7 and 8 in Oguri 2021 from phenomenological definition.
:param alpha_Rs: deflection at Rs
:param Rs: scale radius
:param q: axis ratio
:return: normalization (m)
"""
rho0 = self.nfw.alpha2rho0(alpha_Rs, Rs)
rs_ = Rs
const = 4 * rho0 * rs_**3
return const