__author__ = "sibirrer"
import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["SIS_truncate"]
[docs]
class SIS_truncate(LensProfileBase):
"""This class contains the function and the derivatives of the truncated Singular Isothermal Sphere potential:
.. math::
\\psi(x, y) =
\\begin{cases}
\\theta_{E} \\, r & r < r_{\\text{trunc}} \\\\
\\theta_{E} \\, r_{\\text{trunc}} + \\frac{1}{2} \\theta_{E} \\left(3 - \\frac{r}{r_{\\text{trunc}}}\\right) (r - r_{\\text{trunc}}) & r_{\\text{trunc}} \\leq r < 2 r_{\\text{trunc}} \\\\
\\frac{3}{2} \\theta_{E} \\, r_{\\text{trunc}} & r \\geq 2 r_{\\text{trunc}}
\\end{cases}
where :math:`\\theta_{E}` is the Einstein radius and :math:`r_{\\text{trunc}}` is the truncation radius
"""
param_names = ["theta_E", "r_trunc", "center_x", "center_y"]
lower_limit_default = {
"theta_E": 0,
"r_trunc": 0,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"theta_E": 100,
"r_trunc": 100,
"center_x": 100,
"center_y": 100,
}
[docs]
def function(self, x, y, theta_E, r_trunc, center_x=0, center_y=0):
"""
:param x: set of x-coordinates
:type x: array of size (n)
:param y: set of y-coordinates
:type y: array of size (n)
:param theta_E: Einstein radius of lens
:type theta_E: float (in arcsec)
:param r_trunc: truncated radius
:type r_trunc: float (in arcsec)
:param center_x: profile center
:param center_y: profile center
:returns: 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):
if r < r_trunc:
f_ = theta_E * r
elif r < 2 * r_trunc:
f_ = theta_E * r_trunc + 1.0 / 2 * theta_E * (3 - r / r_trunc) * (
r - r_trunc
)
else:
f_ = 3.0 / 2 * theta_E * r_trunc
else:
f_ = np.zeros_like(r)
f_[r < r_trunc] = theta_E * r[r < r_trunc]
r_ = r[(r < 2 * r_trunc) & (r > r_trunc)]
f_[(r < 2 * r_trunc) & (r > r_trunc)] = (
theta_E * r_trunc
+ 1.0 / 2 * theta_E * (3 - r_ / r_trunc) * (r_ - r_trunc)
)
f_[r > 2 * r_trunc] = 3.0 / 2 * theta_E * r_trunc
return f_
[docs]
def derivatives(self, x, y, theta_E, r_trunc, center_x=0, center_y=0):
"""Computes the first derivatives df/dx and df/dy.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius of lens
:param r_trunc: truncated radius
:type r_trunc: float (in arcsec)
:param center_x: profile center
:param center_y: profile center
:returns: first derivatives (df/dx, df/dy)
"""
x_shift = x - center_x
y_shift = y - center_y
dphi_dr = self._dphi_dr(x_shift, y_shift, theta_E, r_trunc)
dr_dx, dr_dy = self._dr_dx(x_shift, y_shift)
f_x = dphi_dr * dr_dx
f_y = dphi_dr * dr_dy
return f_x, f_y
[docs]
def hessian(self, x, y, theta_E, r_trunc, center_x=0, center_y=0):
"""Computes the Hessian matrix.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius of lens
:param r_trunc: truncated radius
:type r_trunc: float (in arcsec)
:param center_x: profile center
:param center_y: profile center
:returns: Hessian matrix components (d^2f/dx^2, d^2f/dxdy, d^2f/dydx, d^2f/dy^2)
"""
x_shift = x - center_x
y_shift = y - center_y
dphi_dr = self._dphi_dr(x_shift, y_shift, theta_E, r_trunc)
d2phi_dr2 = self._d2phi_dr2(x_shift, y_shift, theta_E, r_trunc)
dr_dx, dr_dy = self._dr_dx(x, y)
d2r_dx2, d2r_dy2, d2r_dxy = self._d2r_dx2(x_shift, y_shift)
f_xx = d2r_dx2 * dphi_dr + dr_dx**2 * d2phi_dr2
f_yy = d2r_dy2 * dphi_dr + dr_dy**2 * d2phi_dr2
f_xy = d2r_dxy * dphi_dr + dr_dx * dr_dy * d2phi_dr2
return f_xx, f_xy, f_xy, f_yy
def _dphi_dr(self, x, y, theta_E, r_trunc):
"""First derivative of the potential in radial direction.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius of lens
:param r_trunc: truncated radius
:type r_trunc: float (in arcsec)
:returns: derivative dphi/dr
"""
r = np.sqrt(x * x + y * y)
if isinstance(r, int) or isinstance(r, float):
if r == 0:
a = 0
elif r < r_trunc:
a = theta_E
elif r < 2 * r_trunc:
a = theta_E * (2 - r / r_trunc)
else:
a = 0
else:
a = np.zeros_like(r)
a[(r < r_trunc) & (r > 0)] = theta_E
r_ = r[(r < 2 * r_trunc) & (r >= r_trunc)]
a[(r < 2 * r_trunc) & (r >= r_trunc)] = theta_E * (2 - r_ / r_trunc)
a[r >= 2 * r_trunc] = 0
return a
def _d2phi_dr2(self, x, y, theta_E, r_trunc):
"""Second derivative of the potential in radial direction.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius of lens
:param r_trunc: truncated radius
:type r_trunc: float (in arcsec)
:return: second derivative (d^2phi / dr^2)
"""
r = np.sqrt(x * x + y * y)
if isinstance(r, int) or isinstance(r, float):
if r < r_trunc:
a = 0
elif r < 2 * r_trunc:
a = -theta_E / r_trunc
else:
a = 0
else:
a = np.zeros_like(r)
a[r < r_trunc] = 0
a[(r < 2 * r_trunc) & (r > r_trunc)] = -theta_E / r_trunc
a[r > 2 * r_trunc] = 0
return a
def _dr_dx(self, x, y):
"""Derivative dr/dx, dr/dy.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:return: derivatives dr/dx and dr/dy
"""
r = np.sqrt(x**2 + y**2)
if isinstance(r, int) or isinstance(r, float):
if r == 0:
r = 1
else:
r[r == 0] = 1
return x / r, y / r
@staticmethod
def _d2r_dx2(x, y):
"""Second derivatives of dr/dx and dr/dy.
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:return: d^2r/dx^2 & d^2r/dy^2
"""
r = np.sqrt(x**2 + y**2)
if isinstance(r, int) or isinstance(r, float):
if r == 0:
r = 1
else:
r[r == 0] = 1
return y**2 / r**3, x**2 / r**3, -x * y / r**3