__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
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}
[docs] 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 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: x-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_
[docs] 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
: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: x-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
[docs] 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 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: x-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./r * np.sin(phi)**2 * a_m *np.cos(m*(phi-phi_m))
f_yy = 1./r * np.cos(phi)**2 * a_m *np.cos(m*(phi-phi_m))
f_xy = -1./r * a_m * np.cos(phi) * np.sin(phi) * np.cos(m*(phi-phi_m))
return f_xx, f_xy, f_xy, f_yy