Source code for lenstronomy.LensModel.Profiles.elliptical_density_slice

__author__ = "lynevdv"

import numpy as np
import cmath as c
from lenstronomy.Util import param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase

__all__ = ['ElliSLICE']


[docs]class ElliSLICE (LensProfileBase): """ This class computes the lensing quantities for an elliptical slice of constant density. Based on Schramm 1994 https://ui.adsabs.harvard.edu/abs/1994A%26A...284...44S/abstract Computes the lensing quantities of an elliptical slice with semi major axis 'a' and semi minor axis 'b', centered on 'center_x' and 'center_y', oriented with an angle 'psi' in radian, and with constant surface mass density 'sigma_0'. In other words, this lens model is characterized by the surface mass density : ..math:: \\kappa(x,y) = \\left{ \\begin{array}{ll} \\sigma_0 & \\mbox{if } \\frac{x_{rot}^2}{a^2} + \\frac{y_{rot}^2}{b^2} \\leq 1 \\\ 0 & \\mbox{else} \\end{array} \\right}. with ..math:: x_{rot} = x_c \\cos \\psi + y_c \\sin \\psi \\ y_{rot} = - x_c \\sin \\psi + y_c \\cos \\psi \\ x_c = x - center_x \\ y_c = y - center_y """ param_names = ['a', 'b', 'psi', 'sigma_0', 'center_x', 'center_y'] lower_limit_default = {'a': 0., 'b': 0., 'psi': -90./180.*np.pi, 'center_x': -100., 'center_y': -100.} upper_limit_default = {'a': 100., 'b': 100., 'psi': 90. / 180. * np.pi, 'center_x': 100., 'center_y': 100.}
[docs] def function(self, x, y, a, b, psi, sigma_0, center_x=0., center_y=0.): """ lensing potential :param a: float, semi-major axis, must be positive :param b: float, semi-minor axis, must be positive :param psi: float, orientation in radian :param sigma_0: float, surface mass density, must be positive :param center_x: float, center on the x axis :param center_y: float, center on the y axis """ kwargs_slice = {'center_x': center_x, 'center_y': center_y, 'a': a, 'b': b, 'psi': psi, 'sigma_0': sigma_0} x_ = x - center_x y_ = y - center_y x_rot = x_*np.cos(psi) + y_*np.sin(psi) y_rot = -x_*np.sin(psi) + y_*np.cos(psi) try: len(x_) except: if (x_rot ** 2 / a ** 2) + (y_rot ** 2 / b ** 2) <= 1: return self.pot_in(x_, y_, kwargs_slice) else: return self.pot_ext(x_, y_, kwargs_slice) else: f = np.array([self.pot_in(x_[i], y_[i], kwargs_slice) if (x_rot[i] ** 2 / a ** 2) + (y_rot[i] ** 2 / b ** 2) <= 1 else self.pot_ext(x_[i], y_[i], kwargs_slice) for i in range(len(x_))]) return f
[docs] def derivatives(self, x, y, a, b, psi, sigma_0, center_x=0., center_y=0.): """ lensing deflection angle :param a: float, semi-major axis, must be positive :param b: float, semi-minor axis, must be positive :param psi: float, orientation in radian :param sigma_0: float, surface mass density, must be positive :param center_x: float, center on the x axis :param center_y: float, center on the y axis """ kwargs_slice = {'center_x': center_x, 'center_y': center_y, 'a': a, 'b': b, 'psi': psi, 'sigma_0': sigma_0} x_ = x - center_x y_ = y - center_y x_rot = x_ * np.cos(psi) + y_ * np.sin(psi) y_rot = -x_ * np.sin(psi) + y_ * np.cos(psi) try: len(x_) except: if (x_rot ** 2 / a ** 2) + (y_rot ** 2 / b ** 2) <= 1: return self.alpha_in(x_, y_, kwargs_slice) else: return self.alpha_ext(x_, y_, kwargs_slice) else: defl = np.array([self.alpha_in(x_[i], y_[i], kwargs_slice) if (x_rot[i] ** 2 / a ** 2) + (y_rot[i] ** 2 / b ** 2) <= 1 else self.alpha_ext(x_[i], y_[i], kwargs_slice) for i in range(len(x_))]) return defl[:, 0], defl[:, 1]
[docs] def hessian(self, x, y, a, b, psi, sigma_0, center_x=0., center_y=0.): """ lensing second derivatives :param a: float, semi-major axis, must be positive :param b: float, semi-minor axis, must be positive :param psi: float, orientation in radian :param sigma_0: float, surface mass density, must be positive :param center_x: float, center on the x axis :param center_y: float, center on the y axis """ diff = 0.000000001 alpha_ra, alpha_dec = self.derivatives(x, y, a, b, psi, sigma_0, center_x, center_y) alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, a, b, psi, sigma_0, center_x, center_y) alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, a, b, psi, sigma_0, center_x, center_y) 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
[docs] @staticmethod def sign(z): """ sign function :param z: complex """ x = z.real y = z.imag if x > 0 or (x == 0 and y >= 0): return 1 else: return -1
[docs] def alpha_in(self, x, y, kwargs_slice): """ deflection angle for (x,y) inside the elliptical slice :param kwargs_slice: dict, dictionary with the slice definition (a,b,psi,sigma_0) """ z = complex(x, y) zb = z.conjugate() psi = kwargs_slice['psi'] e = (kwargs_slice['a'] - kwargs_slice['b']) / (kwargs_slice['a'] + kwargs_slice['b']) sig_0 = kwargs_slice['sigma_0'] e2ipsi = c.exp(2j * psi) I_in = (z - e * zb * e2ipsi) * sig_0 return I_in.real, I_in.imag
[docs] def alpha_ext(self, x, y, kwargs_slice): """ deflection angle for (x,y) outside the elliptical slice :param kwargs_slice: dict, dictionary with the slice definition (a,b,psi,sigma_0) """ z = complex(x, y) r, phi = param_util.cart2polar(x, y) zb = z.conjugate() psi = kwargs_slice['psi'] a = kwargs_slice['a'] b = kwargs_slice['b'] f2 = a ** 2 - b ** 2 sig_0 = kwargs_slice['sigma_0'] median_op = False # when (x,y) is on one of the ellipse axis, there might be an issue when calculating the square root of # zb ** 2 * e2ipsi - f2. When the argument has an imaginary part ==0, having 0. or -0. may return different # answers. Therefore, for points (x,y) close to one axis, we take 3 points (one is x,y ; another one is a delta # away from this position, perpendicularly to the axis ; another one is at -delta perpendicularly away from # x,y). We calculate the function for each point and take the median. This avoids any singularity for points # along the axis but it slows down the function. if np.abs(np.sin(phi - psi)) <= 10 ** -10 \ or np.abs(np.sin(phi - psi - np.pi / 2.)) <= 10 ** -10: # very close to one of the ellipse axis median_op = True e2ipsi = c.exp(2j * psi) eipsi = c.exp(1j * psi) if median_op is True: eps = 10 ** -10 z_minus_eps = complex(r * np.cos(phi - eps), r * np.sin(phi - eps)) zb_minus_eps = z_minus_eps.conjugate() z_plus_eps = complex(r * np.cos(phi + eps), r * np.sin(phi + eps)) zb_plus_eps = z_plus_eps.conjugate() I_out_minus = 2 * a * b / f2 * (zb_minus_eps * e2ipsi - eipsi * self.sign(zb_minus_eps * eipsi) * c.sqrt(zb_minus_eps ** 2 * e2ipsi - f2)) * sig_0 I_out_plus = 2 * a * b / f2 * (zb_plus_eps * e2ipsi - eipsi * self.sign(zb_plus_eps * eipsi) * c.sqrt(zb_plus_eps ** 2 * e2ipsi - f2)) * sig_0 I_out_mid = 2 * a * b / f2 * (zb * e2ipsi - eipsi * self.sign(zb * eipsi) * c.sqrt(zb ** 2 * e2ipsi - f2)) * sig_0 I_out_real = np.median([I_out_minus.real, I_out_plus.real, I_out_mid.real]) I_out_imag = np.median([I_out_minus.imag, I_out_plus.imag, I_out_mid.imag]) else: I_out = 2 * a * b / f2 * ( zb * e2ipsi - eipsi * self.sign(zb * eipsi) * c.sqrt(zb ** 2 * e2ipsi - f2)) * sig_0 I_out_real = I_out.real I_out_imag = I_out.imag # buf = zb ** 2 * e2ipsi - f2 ##problem with 0. and -0. giving different answers # if buf.real == 0: # buf = complex(0, buf.imag) # if buf.imag == 0: # buf = complex(buf.real, 0) return I_out_real, I_out_imag
[docs] @staticmethod def pot_in(x, y, kwargs_slice): """ lensing potential for (x,y) inside the elliptical slice :param kwargs_slice: dict, dictionary with the slice definition (a,b,psi,sigma_0) """ psi = kwargs_slice['psi'] a = kwargs_slice['a'] b = kwargs_slice['b'] sig_0 = kwargs_slice['sigma_0'] e = (a - b) / (a + b) rE = (a + b) / 2. pot_in = 0.5 * ((1 - e) * (x * np.cos(psi) + y * np.sin(psi)) ** 2 + (1 + e) * ( y * np.cos(psi) - x * np.sin(psi)) ** 2) * sig_0 cst = sig_0 * rE ** 2 * (1 - e ** 2) * np.log(rE) return pot_in + cst
[docs] def pot_ext(self, x, y, kwargs_slice): """ lensing potential for (x,y) outside the elliptical slice :param kwargs_slice: dict, dictionary with the slice definition (a,b,psi,sigma_0) """ z = complex(x, y) # zb = z.conjugate() psi = kwargs_slice['psi'] a = kwargs_slice['a'] b = kwargs_slice['b'] sig_0 = kwargs_slice['sigma_0'] r, phi = param_util.cart2polar(x, y) median_op = False # when (x,y) is on one of the ellipse axis, there might be an issue when calculating the square root of # z ** 2 * em2ipsi - f2. When the argument has an imaginary part ==0, having 0. or -0. may return different # answers. Therefore, for points (x,y) close to one axis, we take 3 points (one is x,y ; another one is a delta # away from this position, perpendicularly to the axis ; another one is at -delta perpendicularly away from # x,y). We calculate the function for each point and take the median. This avoids any singularity for points # along the axis but it slows down the function. if np.abs(np.sin(phi - psi)) <= 10 ** -10 \ or np.abs(np.sin(phi - psi - np.pi / 2.)) <= 10 ** -10: # very close to one of the ellipse axis median_op = True e = (a - b) / (a + b) f2 = a ** 2 - b ** 2 emipsi = c.exp(-1j * psi) em2ipsi = c.exp(-2j * psi) if median_op is True: eps = 10 ** -10 z_minus_eps = complex(r * np.cos(phi - eps), r * np.sin(phi - eps)) z_plus_eps = complex(r * np.cos(phi + eps), r * np.sin(phi + eps)) pot_ext_minus = (1 - e ** 2) / (4 * e) * (f2 * c.log( (self.sign(z_minus_eps * emipsi) * z_minus_eps * emipsi + c.sqrt(z_minus_eps ** 2 * em2ipsi - f2)) / 2.) - self.sign(z_minus_eps * emipsi) * z_minus_eps * emipsi * c.sqrt( z_minus_eps ** 2 * em2ipsi - f2) + z_minus_eps ** 2 * em2ipsi) * sig_0 pot_ext_plus = (1 - e ** 2) / (4 * e) * (f2 * c.log( (self.sign(z_plus_eps * emipsi) * z_plus_eps * emipsi + c.sqrt(z_plus_eps ** 2 * em2ipsi - f2)) / 2.) - self.sign(z_plus_eps * emipsi) * z_plus_eps * emipsi * c.sqrt( z_plus_eps ** 2 * em2ipsi - f2) + z_plus_eps ** 2 * em2ipsi) * sig_0 pot_ext_mid = (1 - e ** 2) / (4 * e) * ( f2 * c.log((self.sign(z * emipsi) * z * emipsi + c.sqrt(z ** 2 * em2ipsi - f2)) / 2.) - self.sign(z * emipsi) * z * emipsi * c.sqrt(z ** 2 * em2ipsi - f2) + z ** 2 * em2ipsi) * sig_0 pot_ext = np.median([pot_ext_minus.real, pot_ext_plus.real, pot_ext_mid.real]) else: pot_ext = ((1 - e ** 2) / (4 * e) * ( f2 * c.log((self.sign(z * emipsi) * z * emipsi + c.sqrt(z ** 2 * em2ipsi - f2)) / 2.) - self.sign(z * emipsi) * z * emipsi * c.sqrt(z ** 2 * em2ipsi - f2) + z ** 2 * em2ipsi) * sig_0).real return pot_ext