Source code for lenstronomy.LensModel.Profiles.coreBurkert

__author__ = 'dgilman'

import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase

__all__ = ['CoreBurkert']


[docs]class CoreBurkert(LensProfileBase): """ lensing properties of a modified Burkert profile with variable core size normalized by rho0, the central core density """ param_names = ['Rs', 'alpha_Rs', 'r_core', 'center_x', 'center_y'] lower_limit_default = {'Rs': 1, 'alpha_Rs': 0, 'r_core': 0.5, 'center_x': -100, 'center_y': -100} upper_limit_default = {'Rs': 100, 'alpha_Rs': 100, 'r_core': 50, 'center_x': 100, 'center_y': 100}
[docs] def function(self, x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0): """ :param x: angular position :param y: angular position :param Rs: angular turn over point :param alpha_Rs: deflection angle at Rs :param center_x: center of halo :param center_y: center of halo :return: """ rho0 = self._alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs, r_core=r_core) if Rs < 0.0000001: Rs = 0.0000001 x_ = x - center_x y_ = y - center_y R = np.sqrt(x_ ** 2 + y_ ** 2) f_ = self.cBurkPot(R, Rs, rho0, r_core) return f_
[docs] def derivatives(self, x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0): """ deflection angles :param x: x coordinate :param y: y coordinate :param Rs: scale radius :param rho0: central core density :param r_core: core radius :param center_x: :param center_y: :return: """ rho0 = self._alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs, r_core=r_core) if Rs < 0.0000001: Rs = 0.0000001 x_ = x - center_x y_ = y - center_y R = np.sqrt(x_ ** 2 + y_ ** 2) dx, dy = self.coreBurkAlpha(R, Rs, rho0, r_core, x_, y_) return dx, dy
[docs] def hessian(self, x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0): """ :param x: x coordinate :param y: y coordinate :param Rs: scale radius :param rho0: central core density :param r_core: core radius :param center_x: :param center_y: :return: """ if Rs < 0.0001: Rs = 0.0001 x_ = x - center_x y_ = y - center_y R = np.sqrt(x_ ** 2 + y_ ** 2) rho0 = self._alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs, r_core=r_core) kappa = self.density_2d(x_, y_, Rs, rho0, r_core) gamma1, gamma2 = self.cBurkGamma(R, Rs, rho0, r_core, x_, y_) f_xx = kappa + gamma1 f_yy = kappa - gamma1 f_xy = gamma2 return f_xx, f_xy, f_xy, f_yy
[docs] def mass_2d(self, R, Rs, rho0, r_core): """ analytic solution of the projection integral (convergence) :param R: projected distance :param Rs: scale radius :param rho0: central core density :param r_core: core radius """ x = R * Rs ** -1 p = Rs * r_core ** -1 gx = self._G(x, p) m_2d = 2 * np.pi * rho0 * Rs ** 3 * gx return m_2d
[docs] def coreBurkAlpha(self, R, Rs, rho0, r_core, ax_x, ax_y): """ deflection angle :param R: :param Rs: :param rho0: :param r_core: :param ax_x: :param ax_y: :return: """ x = R * Rs ** -1 p = Rs * r_core ** -1 gx = self._G(x, p) a = 2 * rho0 * Rs ** 2 * gx / x return a * ax_x / R, a * ax_y / R
[docs] def density(self, R, Rs, rho0, r_core): """ three dimensional cored Burkert profile :param R: radius of interest :type R: float/numpy array :param Rs: scale radius :type Rs: float :param rho0: characteristic density :type rho0: float :return: rho(R) density """ M0 = 4*np.pi * Rs ** 3 * rho0 return (M0 / (4*np.pi)) * ((r_core + R) * (Rs ** 2 + R ** 2)) ** -1
[docs] def density_2d(self, x, y, Rs, rho0, r_core, center_x=0, center_y=0): """ projected two dimenstional core Burkert profile (kappa*Sigma_crit) :param x: x coordinate :param y: y coordinate :param Rs: scale radius :param rho0: central core density :param r_core: core radius """ x_ = x - center_x y_ = y - center_y R = np.sqrt(x_ ** 2 + y_ ** 2) x = R * Rs ** -1 p = Rs * r_core ** -1 Fx = self._F(x, p) return 2 * rho0 * Rs * Fx
[docs] def mass_3d(self, R, Rs, rho0, r_core): """ :param R: projected distance :param Rs: scale radius :param rho0: central core density :param r_core: core radius """ Rs = float(Rs) b = r_core * Rs ** -1 c = R * Rs ** -1 M0 = 4*np.pi*Rs**3 * rho0 return M0 * (1+b**2) ** -1 * (0.5*np.log(1+c**2) + b**2*np.log(c*b**-1 + 1) - b*np.arctan(c))
[docs] def cBurkPot(self, R, Rs, rho0, r_core): """ :param R: projected distance :param Rs: scale radius :param rho0: central core density :param r_core: core radius """ x = R * Rs ** -1 p = Rs * r_core ** -1 hx = self._H(x, p) return 2 * rho0 * Rs ** 3 * hx
[docs] def cBurkGamma(self, R, Rs, rho0, r_core, ax_x, ax_y): """ :param R: projected distance :param Rs: scale radius :param rho0: central core density :param r_core: core radius :param ax_x: x coordinate :param ax_y: y coordinate :return: """ c = 0.000001 if isinstance(R, int) or isinstance(R, float): R = max(R, c) else: R[R <= c] = c x = R * Rs ** -1 p = Rs * r_core ** -1 gx = self._G(x, p) fx = self._F(x, p) m_x = 2 * rho0 * Rs ** 3 * gx kappa = 2 * rho0 * Rs * fx a = 2 * (m_x * R ** -2 - kappa) return 0.5 * a * (ax_y ** 2 - ax_x ** 2) / R ** 2, -a * (ax_x * ax_y) / R ** 2
def _u(self, x): return np.sqrt(1 + x ** 2) def _g(self, x, p): return np.sqrt(1 - x ** 2 * p ** 2) def _f(self, x, p): return np.sqrt(x ** 2 * p ** 2 - 1) def _H(self, x, p): prefactor = (p + p ** 3) ** -1 * p if isinstance(x, np.ndarray): inds0 = np.where(x * p == 1) inds1 = np.where(x * p < 1) inds2 = np.where(x * p > 1) func = np.ones_like(x) func[inds1] = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x[inds1]) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x[inds1]) - 0.5 * np.arctanh(self._u(x[inds1]) ** -1)) * np.arctanh( self._u(x[inds1]) ** -1) + \ 2 * (self._g(x[inds1], p) - 0.5 * np.arctanh(self._g(x[inds1], p))) * \ np.arctanh(self._g(x[inds1], p)) + (1 + p ** 2) * np.log(x[inds1]) ** 2 - np.pi * p * \ np.log(1 + self._u(x[inds1])) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x[inds1]) * ( 0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) func[inds2] = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x[inds2]) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x[inds2]) - 0.5 * np.arctanh(self._u(x[inds2]) ** -1)) * np.arctanh( self._u(x[inds2]) ** -1) + \ -2 * (self._f(x[inds2], p) - 0.5 * np.arctan(self._f(x[inds2], p))) * \ np.arctan(self._f(x[inds2], p)) + (1 + p ** 2) * np.log(x[inds2]) ** 2 - np.pi * p * \ np.log(1 + self._u(x[inds2])) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x[inds2]) * ( 0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) func[inds0] = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x[inds0]) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x[inds0]) - 0.5 * np.arctanh(self._u(x[inds0]) ** -1)) * np.arctanh( self._u(x[inds0]) ** -1) \ + (1 + p ** 2) * np.log(x[inds0]) ** 2 - np.pi * p * \ np.log(1 + self._u(x[inds0])) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x[inds0]) * ( 0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) else: if x * p < 1: func = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x) - 0.5 * np.arctanh(self._u(x) ** -1)) * np.arctanh(self._u(x) ** -1) + \ 2 * (self._g(x, p) - 0.5 * np.arctanh(self._g(x, p))) * \ np.arctanh(self._g(x, p)) + (1 + p ** 2) * np.log(x) ** 2 - np.pi * p * \ np.log(1 + self._u(x)) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x) * (0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) elif x * p > 1: func = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x) - 0.5 * np.arctanh(self._u(x) ** -1)) * np.arctanh( self._u(x) ** -1) + \ -2 * (self._f(x, p) - 0.5 * np.arctan(self._f(x, p))) * \ np.arctan(self._f(x, p)) + (1 + p ** 2) * np.log(x) ** 2 - np.pi * p * \ np.log(1 + self._u(x)) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x) * ( 0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) else: func = 0.9058413472016891 + (-0.9640065632861909 + np.pi * self._u(x) - 0.9058413472016892 * p) * p + 2 * p ** 2 * ( self._u(x) - 0.5 * np.arctanh(self._u(x) ** -1)) * np.arctanh(self._u(x) ** -1) \ + (1 + p ** 2) * np.log(x) ** 2 - np.pi * p * \ np.log(1 + self._u(x)) + (0.3068528194400547 + 0.25 * np.log(p ** 2)) * \ np.log(p ** 2) + np.log(x) * (0.6137056388801094 + 0.6137056388801094 * p ** 2 + np.log(p ** 2)) return prefactor * func def _F(self, x, p): """ solution of the projection integal (kappa) arctanh / arctan function :param x: r/Rs :param p: r_core / Rs :return: """ prefactor = 0.5 * (1 + p ** 2) ** -1 * p if isinstance(x, np.ndarray): inds0 = np.where(x * p == 1) inds1 = np.where(x * p < 1) inds2 = np.where(x * p > 1) func = np.ones_like(x) func[inds0] = self._u(x[inds0]) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x[inds0]) ** -1)) func[inds1] = self._u(x[inds1]) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x[inds1]) ** -1)) - \ (2 * p * self._g(x[inds1], p) ** -1 * np.arctanh(self._g(x[inds1], p))) func[inds2] = self._u(x[inds2]) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x[inds2]) ** -1)) - \ (2 * p * self._f(x[inds2], p) ** -1 * np.arctan(self._f(x[inds2], p))) return prefactor * func else: if x * p == 1: func = self._u(x) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x) ** -1)) elif x * p < 1: func = self._u(x) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x) ** -1)) - \ (2 * p * self._g(x, p) ** -1 * np.arctanh(self._g(x, p))) else: func = self._u(x) ** -1 * (np.pi + 2 * p * np.arctanh(self._u(x) ** -1)) - \ (2 * p * self._f(x, p) ** -1 * np.arctan(self._f(x, p))) return prefactor * func def _G(self, x, p): """ analytic solution of the 2d projected mass integral integral: 2 * pi * x * kappa * dx :param x: :param p: :return: """ prefactor = (p + p ** 3) ** -1 * p if isinstance(x, np.ndarray): inds0 = np.where(x * p == 1) inds1 = np.where(x * p < 1) inds2 = np.where(x * p > 1) func = np.ones_like(x) func[inds0] = np.log(0.25 * x[inds0] ** 2 * p ** 2) + np.pi * p * (self._u(x[inds0]) - 1) + \ 2 * p ** 2 * (self._u(x[inds0]) * np.arctanh(self._u(x[inds0]) ** -1) + np.log(0.5 * x[inds0])) func[inds1] = np.log(0.25 * x[inds1] ** 2 * p ** 2) + np.pi * p * (self._u(x[inds1]) - 1) + \ 2 * p ** 2 * (self._u(x[inds1]) * np.arctanh(self._u(x[inds1]) ** -1) + np.log(0.5 * x[inds1])) + 2 * self._g(x[inds1], p) * np.arctanh( self._g(x[inds1], p)) func[inds2] = np.log(0.25 * x[inds2] ** 2 * p ** 2) + np.pi * p * (self._u(x[inds2]) - 1) + \ 2 * p ** 2 * (self._u(x[inds2]) * np.arctanh(self._u(x[inds2]) ** -1) + np.log(0.5 * x[inds2])) - 2 * self._f(x[inds2], p) * np.arctan( self._f(x[inds2], p)) else: if x * p == 1: func = np.log(0.25 * x ** 2 * p ** 2) + np.pi * p * (self._u(x) - 1) + \ 2 * p ** 2 * (self._u(x) * np.arctanh(self._u(x) ** -1) + np.log(0.5 * x)) elif x * p < 1: func = np.log(0.25 * x ** 2 * p ** 2) + np.pi * p * (self._u(x) - 1) + \ 2 * p ** 2 * (self._u(x) * np.arctanh(self._u(x) ** -1) + np.log(0.5 * x)) + 2 * self._g(x, p) * np.arctanh(self._g(x, p)) else: func = np.log(0.25 * x ** 2 * p ** 2) + np.pi * p * (self._u(x) - 1) + \ 2 * p ** 2 * (self._u(x) * np.arctanh(self._u(x) ** -1) + np.log(0.5 * x)) - 2 * self._f(x, p) * np.arctan(self._f(x, p)) return func * prefactor def _alpha2rho0(self, alpha_Rs=None, Rs=None, r_core=None): p = Rs * r_core ** -1 gx = self._G(1, p) rho0 = alpha_Rs / (2 * Rs ** 2 * gx) return rho0 def _rho2alpha(self, rho0=None, Rs=None, r_core=None): p = Rs / r_core gx = self._G(1, p) alpha = 2 * Rs ** 2 * gx * rho0 return alpha