Source code for lenstronomy.LensModel.Profiles.p_jaffe

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

__all__ = ['PJaffe']


[docs]class PJaffe(LensProfileBase): """ class to compute the DUAL PSEUDO ISOTHERMAL ELLIPTICAL MASS DISTRIBUTION based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A Module name: 'PJAFFE'; An alternative name is dPIED. The 3D density distribution is .. math:: \\rho(r) = \\frac{\\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)} with :math:`Rs > Ra`. The projected density is .. math:: \\Sigma(R) = \\Sigma_0 \\frac{Ra Rs}{Rs-Ra}\\left(\\frac{1}{\\sqrt{Ra^2+R^2}} - \\frac{1}{\\sqrt{Rs^2+R^2}} \\right) with .. math:: \\Sigma_0 = \\pi \\rho_0 \\frac{Ra Rs}{Rs + Ra} In the lensing parameterization, .. math:: \\sigma_0 = \\frac{\Sigma_0}{\\Sigma_{\\rm crit}} """ param_names = ['sigma0', 'Ra', 'Rs', 'center_x', 'center_y'] lower_limit_default = {'sigma0': 0, 'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100} upper_limit_default = {'sigma0': 10, 'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100} def __init__(self): """ """ LensProfileBase.__init__(self) self._s = 0.0001
[docs] def density(self, r, rho0, Ra, Rs): """ computes the density :param r: radial distance from the center (in 3D) :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: density at r """ Ra, Rs = self._sort_ra_rs(Ra, Rs) rho = rho0 / ((1 + (r / Ra) ** 2) * (1 + (r / Rs) ** 2)) return rho
[docs] def density_2d(self, x, y, rho0, Ra, Rs, center_x=0, center_y=0): """ projected density :param x: projected coordinate on the sky :param y: projected coordinate on the sky :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :param center_x: center of profile :param center_y: center of profile :return: projected density """ Ra, Rs = self._sort_ra_rs(Ra, Rs) x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) sigma0 = self.rho2sigma(rho0, Ra, Rs) sigma = sigma0 * Ra * Rs / (Rs - Ra) * (1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2)) return sigma
[docs] def mass_3d(self, r, rho0, Ra, Rs): """ mass enclosed a 3d sphere or radius r :param r: radial distance from the center (in 3D) :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: M(<r) """ m_3d = 4 * np.pi * rho0 * Ra ** 2 * Rs ** 2 / (Rs ** 2 - Ra ** 2) * (Rs * np.arctan(r / Rs) - Ra * np.arctan(r / Ra)) return m_3d
[docs] def mass_3d_lens(self, r, sigma0, Ra, Rs): """ mass enclosed a 3d sphere or radius r given a lens parameterization with angular units :param r: radial distance from the center (in 3D) :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: M(<r) in angular units (modulo critical mass density) """ rho0 = self.sigma2rho(sigma0, Ra, Rs) return self.mass_3d(r, rho0, Ra, Rs)
[docs] def mass_2d(self, r, rho0, Ra, Rs): """ mass enclosed projected 2d sphere of radius r :param r: radial distance from the center in projection :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: Sigma(<r) """ Ra, Rs = self._sort_ra_rs(Ra, Rs) sigma0 = self.rho2sigma(rho0, Ra, Rs) m_2d = 2 * np.pi * sigma0 * Ra * Rs / (Rs - Ra) * (np.sqrt(Ra ** 2 + r ** 2) - Ra - np.sqrt(Rs ** 2 + r ** 2) + Rs) return m_2d
[docs] def mass_tot(self, rho0, Ra, Rs): """ total mass within the profile :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: total mass """ Ra, Rs = self._sort_ra_rs(Ra, Rs) sigma0 = self.rho2sigma(rho0, Ra, Rs) m_tot = 2 * np.pi * sigma0 * Ra * Rs return m_tot
[docs] def grav_pot(self, r, rho0, Ra, Rs): """ gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param r: radial distance from the center (in 3D) :param rho0: density normalization (see class documentation above) :param Ra: core radius :param Rs: transition radius from logarithmic slope -2 to -4 :return: gravitational potential (modulo 4 pi G and rho0 in appropriate units) """ Ra, Rs = self._sort_ra_rs(Ra, Rs) pot = 4 * np.pi * rho0 * Ra ** 2 * Rs ** 2 / (Rs ** 2 - Ra ** 2) * (Rs / r * np.arctan(r / Rs) - Ra / r * np.arctan(r / Ra) + 1. / 2 * np.log((Rs ** 2 + r ** 2) / (Ra ** 2 + r ** 2))) return pot
[docs] def function(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0): """ lensing potential :param x: projected coordinate on the sky :param y: projected coordinate on the sky :param sigma0: sigma0/sigma_crit (see class documentation above) :param Ra: core radius (see class documentation above) :param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above) :param center_x: center of profile :param center_y: center of profile :return: lensing potential """ Ra, Rs = self._sort_ra_rs(Ra, Rs) x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) f_ = -2*sigma0 * Ra * Rs / (Rs - Ra) * (np.sqrt(Rs ** 2 + r ** 2) - np.sqrt(Ra ** 2 + r ** 2) + Ra * np.log(Ra + np.sqrt(Ra ** 2 + r ** 2)) - Rs * np.log(Rs + np.sqrt(Rs ** 2 + r ** 2))) return f_
[docs] def derivatives(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0): """ deflection angles :param x: projected coordinate on the sky :param y: projected coordinate on the sky :param sigma0: sigma0/sigma_crit (see class documentation above) :param Ra: core radius (see class documentation above) :param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above) :param center_x: center of profile :param center_y: center of profile :return: f_x, f_y """ Ra, Rs = self._sort_ra_rs(Ra, Rs) x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) if isinstance(r, int) or isinstance(r, float): r = max(self._s, r) else: r[r < self._s] = self._s alpha_r = 2*sigma0 * Ra * Rs / (Rs - Ra) * self._f_A20(r / Ra, r / Rs) f_x = alpha_r * x_/r f_y = alpha_r * y_/r return f_x, f_y
[docs] def hessian(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0): """ Hessian of lensing potential :param x: projected coordinate on the sky :param y: projected coordinate on the sky :param sigma0: sigma0/sigma_crit (see class documentation above) :param Ra: core radius (see class documentation above) :param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above) :param center_x: center of profile :param center_y: center of profile :return: f_xx, f_xy, f_yx, f_yy """ Ra, Rs = self._sort_ra_rs(Ra, Rs) x_ = x - center_x y_ = y - center_y r = np.sqrt(x_**2 + y_**2) if isinstance(r, int) or isinstance(r, float): r = max(self._s, r) else: r[r < self._s] = self._s gamma = sigma0 * Ra * Rs / (Rs - Ra) * (2 * (1. / (Ra + np.sqrt(Ra ** 2 + r ** 2)) - 1. / (Rs + np.sqrt(Rs ** 2 + r ** 2))) - (1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2))) kappa = sigma0 * Ra * Rs / (Rs - Ra) * (1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2)) sin_2phi = -2*x_*y_/r**2 cos_2phi = (y_**2 - x_**2)/r**2 gamma1 = cos_2phi*gamma gamma2 = sin_2phi*gamma f_xx = kappa + gamma1 f_yy = kappa - gamma1 f_xy = gamma2 return f_xx, f_xy, f_xy, f_yy
def _f_A20(self, r_a, r_s): """ equation A20 in Eliasdottir (2007) :param r_a: r/Ra :param r_s: r/Rs :return: f(R/a, R/s) """ return r_a/(1+np.sqrt(1 + r_a**2)) - r_s/(1+np.sqrt(1 + r_s**2))
[docs] def rho2sigma(self, rho0, Ra, Rs): """ converts 3d density into 2d projected density parameter, Equation A4 in Eliasdottir (2007) :param rho0: density normalization :param Ra: core radius (see class documentation above) :param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above) :return: projected density normalization """ return np.pi * rho0 * Ra * Rs / (Rs + Ra)
[docs] def sigma2rho(self, sigma0, Ra, Rs): """ inverse of rho2sigma() :param sigma0: projected density normalization :param Ra: core radius (see class documentation above) :param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above) :return: 3D density normalization """ return (Rs + Ra) / Ra / Rs / np.pi * sigma0
@staticmethod def _sort_ra_rs(Ra, Rs): """ sorts Ra and Rs to make sure Rs > Ra :param Ra: :param Rs: :return: Ra, Rs in conventions used """ if Ra >= Rs: Ra, Rs = Rs, Ra if Ra < 0.0001: Ra = 0.0001 if Rs < Ra + 0.0001: Rs += 0.0001 return Ra, Rs