from ...backend_obj import backend
from ...utils import translate_rotate
[docs]
def reduced_deflection_angle_multipole(x0, y0, m, a_m, phi_m, x, y):
"""
Calculates the reduced deflection angle.
Parameters
----------
x: ArrayLike
x-coordinates in the lens plane.
y: ArrayLike
y-coordinates in the lens plane.
x0: ArrayLike
x-coordinate of the center of the lens.
y0: ArrayLike
y-coordinate of the center of the lens.
m: ArrayLike
The multipole order(s).
a_m: ArrayLike
The multipole amplitude(s).
phi_m: ArrayLike
The multipole orientation(s).
Returns
-------
tuple[ArrayLike, ArrayLike]
The reduced deflection angles in the x and y directions.
Equation (B11) and (B12) https://arxiv.org/pdf/1307.4220, Xu et al. 2014
"""
x, y = translate_rotate(x, y, x0, y0)
phi = backend.arctan2(y, x).reshape(1, -1)
m = m.reshape(-1, 1)
a_m = a_m.reshape(-1, 1)
phi_m = phi_m.reshape(-1, 1)
ax = backend.cos(phi) * a_m / (1 - m**2) * backend.cos(
m * (phi - phi_m)
) + backend.sin(phi) * m * a_m / (1 - m**2) * backend.sin(m * (phi - phi_m))
ax = backend.sum(ax, dim=0).reshape(x.shape)
ay = backend.sin(phi) * a_m / (1 - m**2) * backend.cos(
m * (phi - phi_m)
) - backend.cos(phi) * m * a_m / (1 - m**2) * backend.sin(m * (phi - phi_m))
ay = backend.sum(ay, dim=0).reshape(y.shape)
return ax, ay
[docs]
def potential_multipole(x0, y0, m, a_m, phi_m, x, y):
"""
Compute the lensing potential.
Parameters
----------
x: ArrayLike
x-coordinates in the lens plane.
y: ArrayLike
y-coordinates in the lens plane.
x0: ArrayLike
x-coordinate of the center of the lens.
y0: ArrayLike
y-coordinate of the center of the lens.
m: ArrayLike
The multipole order(s).
a_m: ArrayLike
The multipole amplitude(s).
phi_m: ArrayLike
The multipole orientation(s).
Returns
-------
potential: ArrayLike
Lensing potential.
*Unit: arcsec^2*
Equation (B11) and (B3) https://arxiv.org/pdf/1307.4220, Xu et al. 2014
"""
x, y = translate_rotate(x, y, x0, y0)
r = backend.sqrt(x**2 + y**2).reshape(1, -1)
phi = backend.arctan2(y, x).reshape(1, -1)
m = m.reshape(-1, 1)
a_m = a_m.reshape(-1, 1)
phi_m = phi_m.reshape(-1, 1)
potential = r * a_m / (1 - m**2) * backend.cos(m * (phi - phi_m))
potential = backend.sum(potential, dim=0).reshape(x.shape)
return potential
[docs]
def convergence_multipole(x0, y0, m, a_m, phi_m, x, y):
"""
Compute the lensing convergence.
Parameters
----------
x: ArrayLike
x-coordinates in the lens plane.
y: ArrayLike
y-coordinates in the lens plane.
x0: ArrayLike
x-coordinate of the center of the lens.
y0: ArrayLike
y-coordinate of the center of the lens.
m: ArrayLike
The multipole order(s).
a_m: ArrayLike
The multipole amplitude(s).
phi_m: ArrayLike
The multipole orientation(s).
Returns
-------
convergence: ArrayLike
Lensing convergence.
*Unit: unitless*
Equation (B10) and (B3) https://arxiv.org/pdf/1307.4220, Xu et al. 2014
"""
x, y = translate_rotate(x, y, x0, y0)
r = backend.sqrt(x**2 + y**2).reshape(1, -1)
phi = backend.arctan2(y, x).reshape(1, -1)
m = m.reshape(-1, 1)
a_m = a_m.reshape(-1, 1)
phi_m = phi_m.reshape(-1, 1)
convergence = 1 / (2 * r) * a_m * backend.cos(m * (phi - phi_m))
convergence = backend.sum(convergence, dim=0).reshape(x.shape)
return convergence
