__author__ = "lynevdv"
import numpy as np
import lenstronomy.Util.param_util as param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["Multipole"]
[docs]
class Multipole(LensProfileBase):
"""
This class contains a multipole contribution (for 1 component with m>=2)
This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
Equation B12
m : int, multipole order, m>=2
a_m : float, multipole strength
phi_m : float, multipole orientation in radian
"""
param_names = ["m", "a_m", "phi_m", "center_x", "center_y"]
lower_limit_default = {
"m": 2,
"a_m": 0,
"phi_m": -np.pi,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"m": 100,
"a_m": 100,
"phi_m": np.pi,
"center_x": 100,
"center_y": 100,
}
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def function(self, x, y, m, a_m, phi_m, center_x=0, center_y=0):
"""
Lensing potential of multipole contribution (for 1 component with m>=2)
This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
:param x: x-coordinate to evaluate function
:param y: y-coordinate to evaluate function
:param m: int, multipole order, m>=2
:param a_m: float, multipole strength
:param phi_m: float, multipole orientation in radian
:param center_x: x-position
:param center_y: y-position
:return: lensing potential
"""
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
f_ = r * a_m / (1 - m**2) * np.cos(m * (phi - phi_m))
return f_
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def derivatives(self, x, y, m, a_m, phi_m, center_x=0, center_y=0):
"""
Deflection of a multipole contribution (for 1 component with m>=2)
This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
Equation B12
:param x: x-coordinate to evaluate function
:param y: y-coordinate to evaluate function
:param m: int, multipole order, m>=2
:param a_m: float, multipole strength
:param phi_m: float, multipole orientation in radian
:param center_x: x-position
:param center_y: y-position
:return: deflection angles alpha_x, alpha_y
"""
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
f_x = np.cos(phi) * a_m / (1 - m**2) * np.cos(m * (phi - phi_m)) + np.sin(
phi
) * m * a_m / (1 - m**2) * np.sin(m * (phi - phi_m))
f_y = np.sin(phi) * a_m / (1 - m**2) * np.cos(m * (phi - phi_m)) - np.cos(
phi
) * m * a_m / (1 - m**2) * np.sin(m * (phi - phi_m))
return f_x, f_y
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def hessian(self, x, y, m, a_m, phi_m, center_x=0, center_y=0):
"""
Hessian of a multipole contribution (for 1 component with m>=2)
This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
:param x: x-coordinate to evaluate function
:param y: y-coordinate to evaluate function
:param m: int, multipole order, m>=2
:param a_m: float, multipole strength
:param phi_m: float, multipole orientation in radian
:param center_x: x-position
:param center_y: y-position
:return: f_xx, f_xy, f_yx, f_yy
"""
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
r = np.maximum(r, 0.000001)
f_xx = 1.0 / r * np.sin(phi) ** 2 * a_m * np.cos(m * (phi - phi_m))
f_yy = 1.0 / r * np.cos(phi) ** 2 * a_m * np.cos(m * (phi - phi_m))
f_xy = -1.0 / r * a_m * np.cos(phi) * np.sin(phi) * np.cos(m * (phi - phi_m))
return f_xx, f_xy, f_xy, f_yy