Source code for caustics.lenses.func.point
from ...backend_obj import backend
from ...utils import translate_rotate
from ...constants import G_over_c2, rad_to_arcsec
[docs]
def reduced_deflection_angle_point(x0, y0, Rein, x, y, s=0.0):
"""
Compute the reduced deflection angles. See the Meneghetti lecture notes
equation 3.1 for more detail.
Parameters
----------
x0: ArrayLike
x-coordinate of the center of the lens.
*Unit: arcsec*
y0: ArrayLike
y-coordinate of the center of the lens.
*Unit: arcsec*
Rein: ArrayLike
Einstein radius of the lens.
*Unit: arcsec*
x: ArrayLike
x-coordinates in the lens plane.
*Unit: arcsec*
y: ArrayLike
y-coordinates in the lens plane.
*Unit: arcsec*
s: float
Softening parameter to prevent numerical instabilities.
*Unit: arcsec*
Returns
-------
x_component: ArrayLike
Deflection Angle in the x-direction.
*Unit: arcsec*
y_component: ArrayLike
Deflection Angle in the y-direction.
*Unit: arcsec*
"""
x, y = translate_rotate(x, y, x0, y0)
th = backend.sqrt(x**2 + y**2) + s
ax = x * Rein**2 / th**2
ay = y * Rein**2 / th**2
return ax, ay
[docs]
def potential_point(x0, y0, Rein, x, y, s=0.0):
"""
Compute the lensing potential. See the Meneghetti lecture notes
equation 3.3 for more detail.
Parameters
----------
x0: ArrayLike
x-coordinate of the center of the lens.
*Unit: arcsec*
y0: ArrayLike
y-coordinate of the center of the lens.
*Unit: arcsec*
Rein: ArrayLike
Einstein radius of the lens.
*Unit: arcsec*
x: ArrayLike
x-coordinates in the lens plane.
*Unit: arcsec*
y: ArrayLike
y-coordinates in the lens plane.
*Unit: arcsec*
s: float
Softening parameter to prevent numerical instabilities.
*Unit: arcsec*
Returns
-------
ArrayLike
The lensing potential.
*Unit: arcsec^2*
"""
x, y = translate_rotate(x, y, x0, y0)
th = backend.sqrt(x**2 + y**2) + s
return Rein**2 * backend.log(th)
[docs]
def convergence_point(x0, y0, x, y):
"""
Compute the convergence (dimensionless surface mass density). This follows
essenitally by definition.
Parameters
----------
x0: ArrayLike
x-coordinate of the center of the lens.
*Unit: arcsec*
y0: ArrayLike
y-coordinate of the center of the lens.
*Unit: arcsec*
x: ArrayLike
x-coordinates in the lens plane.
*Unit: arcsec*
y: ArrayLike
y-coordinates in the lens plane.
*Unit: arcsec*
Returns
--------
ArrayLike
The convergence (dimensionless surface mass density).
*Unit: unitless*
"""
x, y = translate_rotate(x, y, x0, y0)
return backend.where((x == 0) & (y == 0), backend.inf, 0.0)
[docs]
def mass_to_rein_point(M, d_ls, d_l, d_s):
"""
Compute the Einstein radius of a point mass. See Meneghetti lecture notes equation 1.39
Parameters
----------
M: ArrayLike
Mass of the lens.
*Unit: solar masses*
d_ls: ArrayLike
Distance between the lens and the source.
*Unit: Mpc*
d_l: ArrayLike
Distance between the observer and the lens.
*Unit: Mpc*
d_s: ArrayLike
Distance between the observer and the source.
*Unit: Mpc*
Returns
-------
ArrayLike
The Einstein radius.
*Unit: arcsec*
"""
return rad_to_arcsec * backend.sqrt(4 * G_over_c2 * M * d_ls / (d_l * d_s))
[docs]
def rein_to_mass_point(r, d_ls, d_l, d_s):
"""
Compute the Einstein radius of a point mass. See Meneghetti lecture notes equation 1.39
Parameters
----------
r: ArrayLike
Einstein radius of the lens.
*Unit: arcsec*
d_ls: ArrayLike
Distance between the lens and the source.
*Unit: Mpc*
d_l: ArrayLike
Distance between the observer and the lens.
*Unit: Mpc*
d_s: ArrayLike
Distance between the observer and the source.
*Unit: Mpc*
Returns
-------
ArrayLike
The mass of the lens
*Unit: solar masses*
"""
return (r / rad_to_arcsec) ** 2 * d_l * d_s / (4 * G_over_c2 * d_ls)
