from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
import numpy as np
__all__ = ["SIE"]
[docs]
class SIE(LensProfileBase):
"""Class for singular isothermal ellipsoid (SIS with ellipticity)
.. math::
\\kappa(x, y) = \\frac{1}{2} \\left(\\frac{\\theta_{E}}{\\sqrt{q x^2 + y^2/q}} \\right)
with :math:`\\theta_{E}` is the (circularized) Einstein radius,
:math:`q` is the minor/major axis ratio,
and :math:`x` and :math:`y` are defined in a coordinate system aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
.. math::
\\kappa(r) = \\frac{1}{2} \\left(\\frac{\\theta'_{E}}{r \\sqrt{1 − e*\\cos(2*\\phi)}} \\right)
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}.
"""
param_names = ["theta_E", "e1", "e2", "center_x", "center_y"]
lower_limit_default = {
"theta_E": 0,
"e1": -0.5,
"e2": -0.5,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"theta_E": 100,
"e1": 0.5,
"e2": 0.5,
"center_x": 100,
"center_y": 100,
}
[docs]
def __init__(self, NIE=True):
"""
:param NIE: bool, if True, is using the NIE analytic model. Otherwise it uses PEMD with gamma=2 from fastell4py
"""
self._nie = NIE
if NIE:
from lenstronomy.LensModel.Profiles.nie import NIE
self.profile = NIE()
else:
from lenstronomy.LensModel.Profiles.epl import EPL
self.profile = EPL()
self._s_scale = 0.0000000001
self._gamma = 2
super(SIE, self).__init__()
[docs]
def function(self, x, y, theta_E, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coordinate (angular coordinates)
:param y: y-coordinate (angular coordinates)
:param theta_E: Einstein radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: centroid
:param center_y: centroid
:return:
"""
if self._nie:
return self.profile.function(
x, y, theta_E, e1, e2, self._s_scale, center_x, center_y
)
else:
return self.profile.function(
x, y, theta_E, self._gamma, e1, e2, center_x, center_y
)
[docs]
def derivatives(self, x, y, theta_E, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coordinate (angular coordinates)
:param y: y-coordinate (angular coordinates)
:param theta_E: Einstein radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: centroid
:param center_y: centroid
:return:
"""
if self._nie:
return self.profile.derivatives(
x, y, theta_E, e1, e2, self._s_scale, center_x, center_y
)
else:
return self.profile.derivatives(
x, y, theta_E, self._gamma, e1, e2, center_x, center_y
)
[docs]
def hessian(self, x, y, theta_E, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coordinate (angular coordinates)
:param y: y-coordinate (angular coordinates)
:param theta_E: Einstein radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: centroid
:param center_y: centroid
:return:
"""
if self._nie:
return self.profile.hessian(
x, y, theta_E, e1, e2, self._s_scale, center_x, center_y
)
else:
return self.profile.hessian(
x, y, theta_E, self._gamma, e1, e2, center_x, center_y
)
[docs]
@staticmethod
def theta2rho(theta_E):
"""Converts projected density parameter (in units of deflection) into 3d density
parameter.
:param theta_E:
:return:
"""
fac1 = np.pi * 2
rho0 = theta_E / fac1
return rho0
[docs]
@staticmethod
def mass_3d(r, rho0, e1=0, e2=0):
"""Mass enclosed a 3d sphere or radius r.
:param r: radius in angular units
:param rho0: density at angle=1
:return: mass in angular units
"""
mass_3d = 4 * np.pi * rho0 * r
return mass_3d
[docs]
def mass_3d_lens(self, r, theta_E, e1=0, e2=0):
"""Mass enclosed a 3d sphere or radius r given a lens parameterization with
angular units.
:param r: radius in angular units
:param theta_E: Einstein radius
:return: mass in angular units
"""
rho0 = self.theta2rho(theta_E)
return self.mass_3d(r, rho0)
[docs]
def mass_2d(self, r, rho0, e1=0, e2=0):
"""Mass enclosed projected 2d sphere of radius r.
:param r:
:param rho0:
:param e1:
:param e2:
:return:
"""
alpha = 2 * rho0 * np.pi**2
mass_2d = alpha * r
return mass_2d
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def mass_2d_lens(self, r, theta_E, e1=0, e2=0):
"""
:param r:
:param theta_E:
:param e1:
:param e2:
:return:
"""
rho0 = self.theta2rho(theta_E)
return self.mass_2d(r, rho0)
[docs]
def grav_pot(self, x, y, rho0, e1=0, e2=0, center_x=0, center_y=0):
"""Gravitational potential (modulo 4 pi G and rho0 in appropriate units)
:param x:
:param y:
:param rho0:
:param e1:
:param e2:
:param center_x:
:param center_y:
:return:
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
mass_3d = self.mass_3d(r, rho0)
pot = mass_3d / r
return pot
[docs]
def density_lens(self, r, theta_E, e1=0, e2=0):
"""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 in angles
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:return: density
"""
rho0 = self.theta2rho(theta_E)
return self.density(r, rho0)
[docs]
@staticmethod
def density(r, rho0, e1=0, e2=0):
"""Computes the density.
:param r: radius in angles
:param rho0: density at angle=1
:return: density at r
"""
rho = rho0 / r**2
return rho
[docs]
@staticmethod
def density_2d(x, y, rho0, e1=0, e2=0, center_x=0, center_y=0):
"""Projected density.
:param x:
:param y:
:param rho0:
:param e1:
:param e2:
:param center_x:
:param center_y:
:return:
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
sigma = np.pi * rho0 / r
return sigma