from ...backend_obj import backend
from ...utils import translate_rotate, derotate
from ...constants import c_km_s, rad_to_arcsec
[docs]
def reduced_deflection_angle_sie(x0, y0, q, phi, Rein, x, y, s=0.0):
"""
Calculate the physical deflection angle. For more detail see Keeton 2002
equations 34 and 35, although our ``Rein`` is defined as :math:`b/\\sqrt(q)` in
Keeton's notation.
Parameters
----------
x0: ArrayLike
The x-coordinate of the lens center.
*Unit: arcsec*
y0: ArrayLike
The y-coordinate of the lens center.
*Unit: arcsec*
q: ArrayLike
The axis ratio of the lens.
*Unit: unitless*
phi: ArrayLike
The orientation angle of the lens (position angle).
*Unit: radians*
Rein: ArrayLike
The Einstein radius of the lens.
*Unit: arcsec*
x: ArrayLike
The x-coordinate of the lens.
*Unit: arcsec*
y: ArrayLike
The y-coordinate of the lens.
*Unit: arcsec*
s: float
The core radius of the lens (defaults to 0.0).
*Unit: arcsec*
Returns
--------
x_component: ArrayLike
The x-component of the deflection angle.
*Unit: arcsec*
y_component: ArrayLike
The y-component of the deflection angle.
*Unit: arcsec*
"""
# Handle the case where q = 1.0, numerical instability
q = backend.where(q == 1.0, q - 1e-6, q)
x, y = translate_rotate(x, y, x0, y0, phi)
# intermediary variables
q2_ = q**2
f = backend.sqrt(1 - q2_)
rein_q_sqrt_f_ = Rein * backend.sqrt(q) / f
psi = backend.sqrt(q2_ * (x**2 + s**2) + y**2)
ax = rein_q_sqrt_f_ * backend.atan(f * x / (psi + s)) # fmt: skip
ay = rein_q_sqrt_f_ * backend.atanh(f * y / (psi + q2_ * s)) # fmt: skip
return derotate(ax, ay, phi)
[docs]
def potential_sie(x0, y0, q, phi, Rein, x, y, s=0.0):
"""
Compute the lensing potential. For more detail see Keeton 2002
equation 33, although our ``Rein`` is defined as :math:`b/\\sqrt(q)` in
Keeton's notation.
Parameters
----------
x0: ArrayLike
The x-coordinate of the lens center.
*Unit: arcsec*
y0: ArrayLike
The y-coordinate of the lens center.
*Unit: arcsec*
q: ArrayLike
The axis ratio of the lens.
*Unit: unitless*
phi: ArrayLike
The orientation angle of the lens (position angle).
*Unit: radians*
Rein: ArrayLike
The Einstein radius of the lens.
*Unit: arcsec*
x: ArrayLike
The x-coordinate of the lens.
*Unit: arcsec*
y: ArrayLike
The y-coordinate of the lens.
*Unit: arcsec*
s: float
The core radius of the lens (defaults to 0.0).
*Unit: arcsec*
Returns
-------
ArrayLike
The lensing potential.
*Unit: arcsec^2*
"""
ax, ay = reduced_deflection_angle_sie(x0, y0, q, phi, Rein, x, y, s)
ax, ay = derotate(ax, ay, -phi)
x, y = translate_rotate(x, y, x0, y0, phi)
# intermediary variables
q2_ = q**2
x2_ = x**2
max_s_ = max(s, 1e-6)
rein_q_sqrt_s_ = Rein * backend.sqrt(q) * s
psi = backend.sqrt(q2_ * (x2_ + s**2) + y**2)
return (
x * ax
+ y * ay
- rein_q_sqrt_s_
* backend.log(backend.sqrt((psi + max_s_) ** 2 + (1 - q2_) * x2_))
+ rein_q_sqrt_s_ * backend.log((1 + q) * max_s_)
)
[docs]
def convergence_sie(x0, y0, q, phi, Rein, x, y, s=0.0):
"""
Calculate the projected mass density. This is converted from the SIS
convergence definition.
Parameters
----------
x0: ArrayLike
The x-coordinate of the lens center.
*Unit: arcsec*
y0: ArrayLike
The y-coordinate of the lens center.
*Unit: arcsec*
q: ArrayLike
The axis ratio of the lens.
*Unit: unitless*
phi: ArrayLike
The orientation angle of the lens (position angle).
*Unit: radians*
Rein: ArrayLike
The Einstein radius of the lens.
*Unit: arcsec*
x: ArrayLike
The x-coordinate of the lens.
*Unit: arcsec*
y: ArrayLike
The y-coordinate of the lens.
*Unit: arcsec*
s: float
The core radius of the lens (defaults to 0.0).
*Unit: arcsec*
Returns
-------
ArrayLike
The projected mass density.
*Unit: unitless*
"""
x, y = translate_rotate(x, y, x0, y0, phi)
psi = backend.sqrt(q**2 * (x**2 + s**2) + y**2)
return 0.5 * backend.sqrt(q) * Rein / psi
[docs]
def sigma_v_to_rein_sie(sigma_v, dls, ds):
"""
Convert the velocity dispersion to the Einstein radius. See equation 16.22
in Dynamics and Astrophysics of Galaxies by Jo Bovy
Parameters
----------
sigma_v: ArrayLike
The velocity dispersion of the lens.
*Unit: km/s*
dls: ArrayLike
The angular diameter distance between the lens and the source.
*Unit: Mpc*
ds: ArrayLike
The angular diameter distance between the observer and the source.
*Unit: Mpc*
Returns
-------
ArrayLike
The Einstein radius.
*Unit: arcsec*
"""
return rad_to_arcsec * 4 * backend.pi * (sigma_v / c_km_s) ** 2 * dls / ds
