Source code for lenstronomy.LensModel.Profiles.nie

__author__ = 'sibirrer'

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', 'NIEMajorAxis']


[docs]class NIE(LensProfileBase): """ Non-singular isothermal ellipsoid (NIE) .. math:: \\kappa = \\theta_E/2 \\left[s_{scale} + qx^2 + y^2/q]−1/2 """ param_names = ['theta_E', 'e1', 'e2', 's_scale', 'center_x', 'center_y'] lower_limit_default = {'theta_E': 0, 'e1': -0.5, 'e2': -0.5, 's_scale': 0, 'center_x': -100, 'center_y': -100} upper_limit_default = {'theta_E': 10, 'e1': 0.5, 'e2': 0.5, 's_scale': 100, 'center_x': 100, 'center_y': 100} def __init__(self): self.nie_major_axis = NIEMajorAxis() super(NIE, self).__init__()
[docs] def function(self, x, y, theta_E, e1, e2, s_scale, 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 e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: lensing potential """ b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale) # shift x_ = x - center_x y_ = y - center_y # rotate x__, y__ = util.rotate(x_, y_, phi_G) # evaluate f_ = self.nie_major_axis.function(x__, y__, b, s, q) # rotate back return f_
[docs] def derivatives(self, x, y, theta_E, e1, e2, s_scale, 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 e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: alpha_x, alpha_y """ b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale) # 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_major_axis.derivatives(x__, y__, b, s, q) # 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, e1, e2, s_scale, 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 e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: f_xx, f_xy, f_yx, f_yy """ b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale) # 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_major_axis.hessian(x__, y__, b, s, q) # 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
[docs] def density_lens(self, r, theta_E, e1, e2, s_scale, center_x=0, center_y=0): """ 3d mass density at 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity. :param r: 3d radius :param theta_E: Einstein radius :param e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: 3d mass density at 3d radius r """ # kappa=1/2 at Einstein radius rho0 = 1 / 2 * theta_E / np.pi return rho0 / (r**2 + s_scale**2)
[docs] def mass_3d_lens(self, r, theta_E, e1, e2, s_scale, center_x=0, center_y=0): """ mass enclosed a 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity. :param r: 3d radius :param theta_E: Einstein radius :param e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: 3d mass density at 3d radius r """ rho0 = 1 / 2 * theta_E / np.pi return rho0 * 4 * np.pi * (r - s_scale * np.arctan(r/s_scale))
[docs] def param_conv(self, theta_E, e1, e2, s_scale): if self._static is True: return self._b_static, self._s_static, self._q_static, self._phi_G_static return self._param_conv(theta_E, e1, e2, s_scale)
def _param_conv(self, theta_E, e1, e2, s_scale): """ convert parameters from 2*kappa = bIE [s2IE + r2(1 − e *cos(2*phi)]−1/2 to 2*kappa= b *(q2(s2 + x2) + y2􏰉)−1/2 see expressions after Equation 8 in Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf :param theta_E: Einstein radius :param e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :return: critical radius b, smoothing scale s, axis ratio q, orientation angle phi_G """ phi_G, q = param_util.ellipticity2phi_q(e1, e2) theta_E_conv = self._theta_E_prod_average2major_axis(theta_E, q) b = theta_E_conv * np.sqrt((1 + q**2)/2) s = s_scale / np.sqrt(q) #s = s_scale * np.sqrt((1 + q**2) / (2*q**2)) return b, s, q, phi_G
[docs] def set_static(self, theta_E, e1, e2, s_scale, 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 e1: eccentricity component :param e2: eccentricity component :param s_scale: smoothing scale :param center_x: profile center :param center_y: profile center :return: self variables set """ self._static = True self._b_static, self._s_static, self._q_static, self._phi_G_static = self._param_conv(theta_E, e1, e2, s_scale)
[docs] def set_dynamic(self): """ :return: """ self._static = False if hasattr(self, '_b_static'): del self._b_static if hasattr(self, '_s_static'): del self._s_static if hasattr(self, '_phi_G_static'): del self._phi_G_static if hasattr(self, '_q_static'): del self._q_static
@staticmethod def _theta_E_prod_average2major_axis(theta_E, q): """ Converts a product averaged Einstein radius (of semi-minor and semi-major axis) to a major axis Einstein radius for an Isothermal ellipse. The standard lenstronomy conventions are product averaged Einstein radii while other codes (such as e.g. gravlens) use the semi-major axis convention. .. math:: \\frac{\\theta_{E, prod ave}}{\\theta_{E, major}} = \\sqrt{(1+q^2) / (2 q) } :param theta_E: Einstein radius in lenstronomy conventions (product average of major and minor axes) :param q: axis ratio minor/major :return: theta_E in convention of kappa= b *(q^2(s^2 + x^2) + y^2􏰉)^{−1/2} (major axis) """ theta_E_major_axis = theta_E / (np.sqrt((1.+q**2) / (2. * q))) return theta_E_major_axis
[docs]class NIEMajorAxis(LensProfileBase): """ This class contains the function and the derivatives of the non-singular isothermal ellipse. See Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf .. math:: \\kappa = b * (q2(s2 + x2) + y2􏰉)^{−1/2}` """ param_names = ['b', 's', 'q', 'center_x', 'center_y'] def __init__(self, diff=0.0000000001): self._diff = diff super(NIEMajorAxis, self).__init__()
[docs] def function(self, x, y, b, s, q): psi = self._psi(x, y, q, s) alpha_x, alpha_y = self.derivatives(x, y, b, s, q) f_ = x * alpha_x + y * alpha_y - b * s * 1. / 2. * np.log((psi + s) ** 2 + (1. - q ** 2) * x ** 2) return f_
[docs] def derivatives(self, x, y, b, s, q): """ returns df/dx and df/dy of the function """ if q >= 1: q = 0.99999999 psi = self._psi(x, y, q, s) f_x = b / np.sqrt(1. - q ** 2) * np.arctan(np.sqrt(1. - q ** 2) * x / (psi + s)) f_y = b / np.sqrt(1. - q ** 2) * np.arctanh(np.sqrt(1. - q ** 2) * y / (psi + q ** 2 * s)) return f_x, f_y
[docs] def hessian(self, x, y, b, s, q): """ returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2 """ alpha_ra, alpha_dec = self.derivatives(x, y, b, s, q) diff = self._diff alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, b, s, q) alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, b, s, q) 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 kappa(x, y, b, s, q): """ convergence :param x: major axis coordinate :param y: minor axis coordinate :param b: normalization :param s: smoothing scale :param q: axis ratio :return: convergence """ kappa = b/2. * (q**2 * (s**2 + x**2) + y**2)**(-1./2) return kappa
@staticmethod def _psi(x, y, q, s): """ expression after equation (8) in Keeton&Kochanek 1998 :param x: semi-major axis coordinate :param y: semi-minor axis coordinate :param q: axis ratio minor/major :param s: smoothing scale in major axis direction :return: phi """ return np.sqrt(q**2 * (s**2 + x**2) + y**2)