Source code for lenstronomy.LensModel.Profiles.gaussian_kappa

__author__ = 'sibirrer'
#this file contains a class to make a gaussian

import numpy as np
import scipy.special
import scipy.integrate as integrate
from lenstronomy.LensModel.Profiles.gaussian_potential import Gaussian
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase

__all__ = ['GaussianKappa']


[docs]class GaussianKappa(LensProfileBase): """ this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix """ param_names = ['amp', 'sigma', 'center_x', 'center_y'] lower_limit_default = {'amp': 0, 'sigma': 0, 'center_x': -100, 'center_y': -100} upper_limit_default = {'amp': 100, 'sigma': 100, 'center_x': 100, 'center_y': 100} def __init__(self): self.gaussian = Gaussian() self.ds = 0.00001 super(LensProfileBase, self).__init__()
[docs] def function(self, x, y, amp, sigma, center_x=0, center_y=0): """ returns Gaussian """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) sigma_x, sigma_y = sigma, sigma c = 1. / (2 * sigma_x * sigma_y) if isinstance(x_, int) or isinstance(x_, float): num_int = self._num_integral(r, c) else: num_int = [] for i in range(len(x_)): num_int.append(self._num_integral(r[i], c)) num_int = np.array(num_int) amp_density = self._amp2d_to_3d(amp, sigma_x, sigma_y) amp2d = amp_density / (np.sqrt(np.pi) * np.sqrt(sigma_x * sigma_y * 2)) amp2d *= 2 * 1. / (2 * c) return num_int * amp2d
@staticmethod def _num_integral(r, c): """ numerical integral (1-e^{-c*x^2})/x dx [0..r] :param r: radius :param c: 1/2sigma^2 :return: """ out = integrate.quad(lambda x: (1-np.exp(-c*x**2))/x, 0, r) return out[0]
[docs] def derivatives(self, x, y, amp, sigma, center_x=0, center_y=0): """ returns df/dx and df/dy of the function """ x_ = x - center_x y_ = y - center_y R = np.sqrt(x_**2 + y_**2) sigma_x, sigma_y = sigma, sigma if isinstance(R, int) or isinstance(R, float): R = max(R, self.ds) else: R[R <= self.ds] = self.ds alpha = self.alpha_abs(R, amp, sigma) return alpha / R * x_, alpha / R * y_
[docs] def hessian(self, x, y, amp, sigma, center_x=0, center_y=0): """ returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2 """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) sigma_x, sigma_y = sigma, sigma if isinstance(r, int) or isinstance(r, float): r = max(r, self.ds) else: r[r <= self.ds] = self.ds d_alpha_dr = -self.d_alpha_dr(r, amp, sigma_x, sigma_y) alpha = self.alpha_abs(r, amp, sigma) f_xx = -(d_alpha_dr/r + alpha/r**2) * x_**2/r + alpha/r f_yy = -(d_alpha_dr/r + alpha/r**2) * y_**2/r + alpha/r f_xy = -(d_alpha_dr/r + alpha/r**2) * x_*y_/r return f_xx, f_xy, f_xy, f_yy
[docs] def density(self, r, amp, sigma): """ :param r: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma return self.gaussian.function(r, 0, amp, sigma_x, sigma_y)
[docs] def density_2d(self, x, y, amp, sigma, center_x=0, center_y=0): """ :param x: :param y: :param amp: :param sigma: :param center_x: :param center_y: :return: """ sigma_x, sigma_y = sigma, sigma amp2d = self._amp3d_to_2d(amp, sigma_x, sigma_y) return self.gaussian.function(x, y, amp2d, sigma_x, sigma_y, center_x, center_y)
[docs] def mass_2d(self, R, amp, sigma): """ :param R: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma amp2d = amp / (np.sqrt(np.pi) * np.sqrt(sigma_x * sigma_y * 2)) c = 1./(2 * sigma_x * sigma_y) return amp2d * 2 * np.pi * 1./(2*c) * (1. - np.exp(-c * R**2))
[docs] def mass_2d_lens(self, R, amp, sigma): """ :param R: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma amp_density = self._amp2d_to_3d(amp, sigma_x, sigma_y) return self.mass_2d(R, amp_density, sigma)
[docs] def alpha_abs(self, R, amp, sigma): """ absolute value of the deflection :param R: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma amp_density = self._amp2d_to_3d(amp, sigma_x, sigma_y) alpha = self.mass_2d(R, amp_density, sigma) / np.pi / R return alpha
[docs] def d_alpha_dr(self, R, amp, sigma_x, sigma_y): """ :param R: :param amp: :param sigma_x: :param sigma_y: :return: """ c = 1. / (2 * sigma_x * sigma_y) A = self._amp2d_to_3d(amp, sigma_x, sigma_y) * np.sqrt(2/np.pi*sigma_x*sigma_y) return 1./R**2 * (-1 + (1 + 2*c*R**2) * np.exp(-c*R**2)) * A
[docs] def mass_3d(self, R, amp, sigma): """ :param R: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma A = amp / (2 * np.pi * sigma_x * sigma_y) c = 1. / (2 * sigma_x * sigma_y) result = 1. / (2*c) * (-R * np.exp(-c*R**2) + scipy.special.erf(np.sqrt(c) * R) * np.sqrt(np.pi/(4 * c))) return result*A * 4 * np.pi
[docs] def mass_3d_lens(self, R, amp, sigma): """ :param R: :param amp: :param sigma: :return: """ sigma_x, sigma_y = sigma, sigma amp_density = self._amp2d_to_3d(amp, sigma_x, sigma_y) return self.mass_3d(R, amp_density, sigma)
@staticmethod def _amp3d_to_2d(amp, sigma_x, sigma_y): """ converts 3d density into 2d density parameter :param amp: :param sigma_x: :param sigma_y: :return: """ return amp * np.sqrt(np.pi) * np.sqrt(sigma_x * sigma_y * 2) @staticmethod def _amp2d_to_3d(amp, sigma_x, sigma_y): """ converts 3d density into 2d density parameter :param amp: :param sigma_x: :param sigma_y: :return: """ return amp / (np.sqrt(np.pi) * np.sqrt(sigma_x * sigma_y * 2))