Source code for lenstronomy.LensModel.Profiles.cored_density_2

__author__ = 'sibirrer'

import numpy as np
from scipy.integrate import quad
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
from lenstronomy.Util import derivative_util as calc_util

__all__ = ['CoredDensity2']


[docs]class CoredDensity2(LensProfileBase): """ class for a uniform cored density dropping steep in the outskirts credits for suggesting this profile goes to Kfir Blum .. math:: \\rho(r) = 2/\\pi * \\Sigma_{\\rm crit} R_c^2 * (R_c^2 + r^2)^{-3/2} This profile drops like an NFW profile as math:`\\rho(r)^{-3}`. """ model_name = 'CORED_DENSITY_2' _s = 0.000001 # numerical limit for minimal radius param_names = ['sigma0', 'r_core', 'center_x', 'center_y'] lower_limit_default = {'sigma0': -1, 'r_core': 0, 'center_x': -100, 'center_y': -100} upper_limit_default = {'sigma0': 10, 'r_core': 100, 'center_x': 100, 'center_y': 100}
[docs] def function(self, x, y, sigma0, r_core, center_x=0, center_y=0): """ potential of cored density profile :param x: x-coordinate in angular units :param y: y-coordinate in angular units :param sigma0: convergence in the core :param r_core: core radius :param center_x: center of the profile :param center_y: center of the profile :return: lensing potential at (x, y) """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_ ** 2 + y_ ** 2) r = np.maximum(r, self._s) if isinstance(r, int) or isinstance(r, float): return self._num_integral_potential(r, sigma0, r_core) else: f_ = [] for i in range(len(r)): f_.append(self._num_integral_potential(r[i], sigma0, r_core)) return np.array(f_)
def _num_integral_potential(self, r, sigma0, r_core): """ :param r: :param r_core: :return: """ def _integrand(x): return self.alpha_r(x, sigma0=sigma0, r_core=r_core) f_ = quad(_integrand, 0, r)[0] return f_
[docs] def derivatives(self, x, y, sigma0, r_core, center_x=0, center_y=0): """ deflection angle of cored density profile :param x: x-coordinate in angular units :param y: y-coordinate in angular units :param sigma0: convergence in the core :param r_core: core radius :param center_x: center of the profile :param center_y: center of the profile :return: alpha_x, alpha_y at (x, y) """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) r = np.maximum(r, self._s) alpha_r = self.alpha_r(r, sigma0, r_core) f_x = alpha_r * x_ / r f_y = alpha_r * y_ / r return f_x, f_y
[docs] def hessian(self, x, y, sigma0, r_core, center_x=0, center_y=0): """ :param x: x-coordinate in angular units :param y: y-coordinate in angular units :param sigma0: convergence in the core :param r_core: core radius :param center_x: center of the profile :param center_y: center of the profile :return: Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y) """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_ ** 2 + y_ ** 2) r = np.maximum(r, self._s) d_alpha_dr = self.d_alpha_dr(r, sigma0, r_core) alpha = self.alpha_r(r, sigma0, r_core) dr_dx = calc_util.d_r_dx(x_, y_) dr_dy = calc_util.d_r_dy(x_, y_) f_xx = d_alpha_dr * dr_dx * x_ / r + alpha * calc_util.d_x_diffr_dx(x_, y_) f_yy = d_alpha_dr * dr_dy * y_ / r + alpha * calc_util.d_y_diffr_dy(x_, y_) f_xy = d_alpha_dr * dr_dy * x_ / r + alpha * calc_util.d_x_diffr_dy(x_, y_) return f_xx, f_xy, f_xy, f_yy
[docs] @staticmethod def alpha_r(r, sigma0, r_core): """ radial deflection angle of the cored density profile :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: deflection angle """ return sigma0 * r_core ** 2 * np.log((r_core**2 + r**2) / r_core**2) / r # this is mass_2d / (r * pi)
#return sigma0 * r_core * np.arctan(r/r_core)
[docs] @staticmethod def d_alpha_dr(r, sigma0, r_core): """ radial derivatives of the radial deflection angle :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: dalpha/dr """ return sigma0 * r_core ** 2 * (-1./r**2 * np.log((r_core**2 + r**2) / r_core**2) + 1/r * r_core**2 / (r**2 + r_core**2) * 2 *r/r_core**2)
[docs] @staticmethod def kappa_r(r, sigma0, r_core): """ convergence of the cored density profile. This routine is also for testing :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: convergence at r """ return sigma0 * r_core ** 2 / (r_core ** 2 + r ** 2)
[docs] @staticmethod def density(r, sigma0, r_core): """ rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-3/2) :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: density at radius r """ return 1./2 * sigma0 * r_core**2 * (r_core**2 + r**2) ** (-3./2)
[docs] def density_lens(self, r, sigma0, r_core): """ computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity. :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: density at radius r """ return self.density(r, sigma0, r_core)
[docs] def density_2d(self, x, y, sigma0, r_core, center_x=0, center_y=0): """ projected density at projected radius r :param x: x-coordinate in angular units :param y: y-coordinate in angular units :param sigma0: convergence in the core :param r_core: core radius :param center_x: center of the profile :param center_y: center of the profile :return: projected density """ x_ = x - center_x y_ = y - center_y r = np.sqrt(x_ ** 2 + y_ ** 2) r = np.maximum(r, self._s) return self.kappa_r(r, sigma0, r_core)
[docs] def mass_2d(self, r, sigma0, r_core): """ mass enclosed in cylinder of radius r :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: mass enclosed in cylinder of radius r """ return sigma0 * r_core ** 2 * np.pi * np.log((r_core**2 + r**2) / r_core**2)
[docs] @staticmethod def mass_3d(r, sigma0, r_core): """ mass enclosed 3d radius :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: mass enclosed 3d radius """ r_ = np.sqrt(r**2 + r_core**2) return 2 * np.pi * sigma0 * r_core**2 * (r_ * np.log(r_ + r) - np.log(r_core) * r_ - r) / r_
[docs] def mass_3d_lens(self, r, sigma0, r_core): """ mass enclosed a 3d sphere or radius r given a lens parameterization with angular units For this profile those are identical. :param r: radius (angular scale) :param sigma0: convergence in the core :param r_core: core radius :return: mass enclosed 3d radius """ return self.mass_3d(r, sigma0, r_core)