__author__ = "sibirrer"
import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["SIS"]
[docs]
class SIS(LensProfileBase):
"""This class contains the function and the derivatives of the Singular Isothermal
Sphere.
.. math::
\\kappa(x, y) = \\frac{1}{2} \\left(\\frac{\\theta_{E}}{\\sqrt{x^2 + y^2}} \\right)
with :math:`\\theta_{E}` is the Einstein radius,
"""
param_names = ["theta_E", "center_x", "center_y"]
lower_limit_default = {"theta_E": 0, "center_x": -100, "center_y": -100}
upper_limit_default = {"theta_E": 100, "center_x": 100, "center_y": 100}
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def function(self, x, y, theta_E, center_x=0, center_y=0):
x_shift = x - center_x
y_shift = y - center_y
f_ = theta_E * np.sqrt(x_shift * x_shift + y_shift * y_shift)
return f_
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def derivatives(self, x, y, theta_E, center_x=0, center_y=0):
"""Returns df/dx and df/dy of the function."""
x_shift = x - center_x
y_shift = y - center_y
R = np.sqrt(x_shift * x_shift + y_shift * y_shift)
if isinstance(R, int) or isinstance(R, float):
a = theta_E / max(0.000001, R)
else:
a = np.empty_like(R)
r = R[R > 0] # in the SIS regime
a[R == 0] = 0
a[R > 0] = theta_E / r
f_x = a * x_shift
f_y = a * y_shift
return f_x, f_y
[docs]
def hessian(self, x, y, theta_E, center_x=0, center_y=0):
"""Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx,
d^f/dy^2."""
x_shift = x - center_x
y_shift = y - center_y
R = (x_shift * x_shift + y_shift * y_shift) ** (3.0 / 2)
if isinstance(R, int) or isinstance(R, float):
prefac = theta_E / max(0.000001, R)
else:
prefac = np.empty_like(R)
r = R[R > 0] # in the SIS regime
prefac[R == 0] = 0.0
prefac[R > 0] = theta_E / r
f_xx = y_shift * y_shift * prefac
f_yy = x_shift * x_shift * prefac
f_xy = -x_shift * y_shift * prefac
return f_xx, f_xy, f_xy, f_yy
[docs]
@staticmethod
def rho2theta(rho0):
"""Converts 3d density into 2d projected density parameter :param rho0:
:return:
"""
theta_E = np.pi * 2 * rho0
return theta_E
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@staticmethod
def theta2rho(theta_E):
"""Converts projected density parameter (in units of deflection) into 3d density
parameter :param theta_E: Einstein radius :return:"""
fac1 = np.pi * 2
rho0 = theta_E / fac1
return rho0
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@staticmethod
def mass_3d(r, rho0):
"""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
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def mass_3d_lens(self, r, theta_E):
"""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)
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@staticmethod
def mass_2d(r, rho0):
"""Mass enclosed projected 2d sphere of radius r :param r:
:param rho0:
: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):
"""
:param r: radius
:param theta_E: Einstein radius
:return: mass within a radius in projection
"""
rho0 = self.theta2rho(theta_E)
return self.mass_2d(r, rho0)
[docs]
def grav_pot(self, x, y, rho0, center_x=0, center_y=0):
"""Gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param
x:
:param y:
:param rho0:
: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
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@staticmethod
def density(r, rho0):
"""Computes the density :param r: radius in angles :param rho0: density at
angle=1 :return: density at r."""
rho = rho0 / r**2
return rho
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def density_lens(self, r, theta_E):
"""Computes the density at 3d radius r given lens model parameterization. The
integral in projected in units of angles (i.e. arc seconds) results in the
convergence quantity.
:param r: 3d radius
:param theta_E: Einstein radius
:return: density(r)
"""
rho0 = self.theta2rho(theta_E)
return self.density(r, rho0)
[docs]
@staticmethod
def density_2d(x, y, rho0, center_x=0, center_y=0):
"""Projected density :param x:
:param y:
:param rho0:
: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