__author__ = 'ntessore'
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
from lenstronomy.LensModel.Profiles.spp import SPP
from scipy.special import hyp2f1
__all__ = ['EPL', 'EPLMajorAxis']
[docs]class EPL(LensProfileBase):
""""
Elliptical Power Law mass profile
.. math::
\\kappa(x, y) = \\frac{3-\\gamma}{2} \\left(\\frac{\\theta_{E}}{\\sqrt{q x^2 + y^2/q}} \\right)^{\\gamma-1}
with :math:`\\theta_{E}` is the (circularized) Einstein radius,
:math:`\\gamma` is the negative power-law slope of the 3D mass distributions,
:math:`q` is the minor/major axis ratio,
and :math:`x` and :math:`y` are defined in a coordinate sys- tem aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
.. math::
\\kappa(r) = \\frac{3-\\gamma}{2} \\left(\\frac{\\theta'_{E}}{r \\sqrt{1 − e*\\cos(2*\\phi)}} \\right)^{\\gamma-1}
with :math:`\\epsilon` is the ellipticity defined as
.. math::
\\epsilon = \\frac{1-q^2}{1+q^2}
And an Einstein radius :math:`\\theta'_{\\rm E}` related to the definition used is
.. math::
\\left(\\frac{\\theta'_{\\rm E}}{\\theta_{\\rm E}}\\right)^{2} = \\frac{2q}{1+q^2}.
The mathematical form of the calculation is presented by Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819.
The current implementation is using hyperbolic functions. The paper presents an iterative calculation scheme,
converging in few iterations to high precision and accuracy.
A (faster) implementation of the same model using numba is accessible as 'EPL_NUMBA' with the iterative calculation
scheme.
"""
param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {'theta_E': 0, 'gamma': 1.5, 'e1': -0.5, 'e2': -0.5, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'theta_E': 100, 'gamma': 2.5, 'e1': 0.5, 'e2': 0.5, 'center_x': 100, 'center_y': 100}
def __init__(self):
self.epl_major_axis = EPLMajorAxis()
self.spp = SPP()
super(EPL, self).__init__()
[docs] def param_conv(self, theta_E, gamma, e1, e2):
"""
converts parameters as defined in this class to the parameters used in the EPLMajorAxis() class
:param theta_E: Einstein radius as defined in the profile class
:param gamma: negative power-law slope
:param e1: eccentricity modulus
:param e2: eccentricity modulus
:return: b, t, q, phi_G
"""
if self._static is True:
return self._b_static, self._t_static, self._q_static, self._phi_G_static
return self._param_conv(theta_E, gamma, e1, e2)
@staticmethod
def _param_conv(theta_E, gamma, e1, e2):
"""
convert parameters from :math:`R = \\sqrt{q x^2 + y^2/q}` to
:math:`R = \\sqrt{q^2 x^2 + y^2}`
:param gamma: power law slope
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:return: critical radius b, slope t, axis ratio q, orientation angle phi_G
"""
t = gamma - 1
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
b = theta_E * np.sqrt(q)
return b, t, q, phi_G
[docs] def set_static(self, theta_E, gamma, e1, e2, center_x=0, center_y=0):
"""
:param theta_E: Einstein radius
:param gamma: power law slope
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: profile center
:param center_y: profile center
:return: self variables set
"""
self._static = True
self._b_static, self._t_static, self._q_static, self._phi_G_static = self._param_conv(theta_E, gamma, e1, e2)
[docs] def set_dynamic(self):
"""
:return:
"""
self._static = False
if hasattr(self, '_b_static'):
del self._b_static
if hasattr(self, '_t_static'):
del self._t_static
if hasattr(self, '_phi_G_static'):
del self._phi_G_static
if hasattr(self, '_q_static'):
del self._q_static
[docs] def function(self, x, y, theta_E, gamma, e1, e2, 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 gamma: power law slope
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: profile center
:param center_y: profile center
:return: lensing potential
"""
b, t, q, phi_G = self.param_conv(theta_E, gamma, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self.epl_major_axis.function(x__, y__, b, t, q)
# rotate back
return f_
[docs] def derivatives(self, x, y, theta_E, gamma, e1, e2, 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 gamma: power law slope
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: profile center
:param center_y: profile center
:return: alpha_x, alpha_y
"""
b, t, q, phi_G = self.param_conv(theta_E, gamma, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__x, f__y = self.epl_major_axis.derivatives(x__, y__, b, t, 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, gamma, e1, e2, 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 gamma: power law slope
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: profile center
:param center_y: profile center
:return: f_xx, f_xy, f_yx, f_yy
"""
b, t, q, phi_G = self.param_conv(theta_E, gamma, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__xx, f__xy, f__yx, f__yy = self.epl_major_axis.hessian(x__, y__, b, t, 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 mass_3d_lens(self, r, theta_E, gamma, e1=None, e2=None):
"""
computes the spherical power-law mass enclosed (with SPP routine)
:param r: radius within the mass is computed
:param theta_E: Einstein radius
:param gamma: power-law slope
:param e1: eccentricity component (not used)
:param e2: eccentricity component (not used)
:return: mass enclosed a 3D radius r
"""
return self.spp.mass_3d_lens(r, theta_E, gamma)
[docs] def density_lens(self, r, theta_E, gamma, e1=None, e2=None):
"""
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 within the mass is computed
:param theta_E: Einstein radius
:param gamma: power-law slope
:param e1: eccentricity component (not used)
:param e2: eccentricity component (not used)
:return: mass enclosed a 3D radius r
"""
return self.spp.density_lens(r, theta_E, gamma)
[docs]class EPLMajorAxis(LensProfileBase):
"""
This class contains the function and the derivatives of the
elliptical power law.
.. math::
\\kappa = (2-t)/2 * \\left[\\frac{b}{\\sqrt{q^2 x^2 + y^2}}\\right]^t
where with :math:`t = \\gamma - 1` (from EPL class) being the projected power-law slope of the convergence profile,
critical radius b, axis ratio q.
Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819
"""
param_names = ['b', 't', 'q', 'center_x', 'center_y']
def __init__(self):
super(EPLMajorAxis, self).__init__()
[docs] def function(self, x, y, b, t, q):
"""
returns the lensing potential
:param x: x-coordinate in image plane relative to center (major axis)
:param y: y-coordinate in image plane relative to center (minor axis)
:param b: critical radius
:param t: projected power-law slope
:param q: axis ratio
:return: lensing potential
"""
# deflection from method
alpha_x, alpha_y = self.derivatives(x, y, b, t, q)
# deflection potential, eq. (15)
psi = (x*alpha_x + y*alpha_y)/(2 - t)
return psi
[docs] def derivatives(self, x, y, b, t, q):
"""
returns the deflection angles
:param x: x-coordinate in image plane relative to center (major axis)
:param y: y-coordinate in image plane relative to center (minor axis)
:param b: critical radius
:param t: projected power-law slope
:param q: axis ratio
:return: f_x, f_y
"""
# elliptical radius, eq. (5)
Z = np.empty(np.shape(x), dtype=complex)
Z.real = q*x
Z.imag = y
R = np.abs(Z)
R = np.maximum(R, 0.000000001)
# angular dependency with extra factor of R, eq. (23)
R_omega = Z*hyp2f1(1, t/2, 2-t/2, -(1-q)/(1+q)*(Z/Z.conj()))
# deflection, eq. (22)
alpha = 2/(1+q)*(b/R)**t*R_omega
# return real and imaginary part
alpha_real = np.nan_to_num(alpha.real, posinf=10**10, neginf=-10**10)
alpha_imag = np.nan_to_num(alpha.imag, posinf=10**10, neginf=-10**10)
return alpha_real, alpha_imag
[docs] def hessian(self, x, y, b, t, q):
"""
Hessian matrix of the lensing potential
:param x: x-coordinate in image plane relative to center (major axis)
:param y: y-coordinate in image plane relative to center (minor axis)
:param b: critical radius
:param t: projected power-law slope
:param q: axis ratio
:return: f_xx, f_yy, f_xy
"""
R = np.hypot(q*x, y)
R = np.maximum(R, 0.00000001)
r = np.hypot(x, y)
cos, sin = x/r, y/r
cos2, sin2 = cos*cos*2 - 1, sin*cos*2
# convergence, eq. (2)
kappa = (2 - t)/2*(b/R)**t
kappa = np.nan_to_num(kappa, posinf=10**10, neginf=-10**10)
# deflection via method
alpha_x, alpha_y = self.derivatives(x, y, b, t, q)
# shear, eq. (17), corrected version from arXiv/corrigendum
gamma_1 = (1-t)*(alpha_x*cos - alpha_y*sin)/r - kappa*cos2
gamma_2 = (1-t)*(alpha_y*cos + alpha_x*sin)/r - kappa*sin2
gamma_1 = np.nan_to_num(gamma_1, posinf=10**10, neginf=-10**10)
gamma_2 = np.nan_to_num(gamma_2, posinf=10**10, neginf=-10**10)
# second derivatives from convergence and shear
f_xx = kappa + gamma_1
f_yy = kappa - gamma_1
f_xy = gamma_2
return f_xx, f_xy, f_xy, f_yy