__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 Singular Isothermal Sphere
"""
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):
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. / 2 * theta_E * (3 - r / r_trunc) * (r - r_trunc)
else:
f_ = 3./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. / 2 * theta_E * (3 - r_ / r_trunc) * (r_ - r_trunc)
f_[r > 2*r_trunc] = 3./2 * theta_E * r_trunc
return f_
[docs] def derivatives(self, x, y, theta_E, r_trunc, 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
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):
"""
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
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):
"""
:param x:
:param y:
:param r_trunc:
:return:
"""
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:
:param y:
:param theta_E:
:param r_trunc:
:return:
"""
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 of dr/dx, dr/dy
:param x:
:param y:
:return:
"""
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 derivative
:param x:
:param y:
:return:
"""
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