from ...backend_obj import backend
from ...constants import arcsec_to_rad, G_over_c2
from ...utils import translate_rotate
[docs]
def convergence_0_pseudo_jaffe(mass, Rc, Rs, d_l, critical_surface_density):
"""
Compute the convergence (dimensionless surface mass density). This is
rearranged from Eliasdottir et al 2007 equation A11.
Parameters
----------
mass: ArrayLike
Total mass of the lens (Msun).
*Unit: Msun*
Rc: ArrayLike
Core radius of the lens.
*Unit: arcsec*
Rs: ArrayLike
Scaling radius of the lens.
*Unit: arcsec*
d_l: ArrayLike
Distance to the lens.
*Unit: Mpc*
critical_surface_density: ArrayLike
Critical surface density of the universe at the lens redshift.
*Unit: Msun / Mpc^2*
Returns
--------
ArrayLike
The convergence (dimensionless surface mass density) at the center of
the pseudo jaffe.
*Unit: unitless*
"""
return mass / (2 * backend.pi * critical_surface_density * Rc * Rs * (d_l * arcsec_to_rad) ** 2) # fmt: skip
[docs]
def mass_enclosed_2d_pseudo_jaffe(radius, mass, Rc, Rs, s=0.0):
"""
Compute the mass enclosed within a given radius. See Eliasdottir et al 2007 equation A10.
Parameters
----------
radius: Optional[ArrayLike]
Radius at which to calculate enclosed mass (arcsec).
*Unit: arcsec*
mass: ArrayLike
Total mass of the lens
*Unit: Msun*
Rc: ArrayLike
Core radius of the lens.
*Unit: arcsec*
Rs: ArrayLike
Scaling radius of the lens.
*Unit: arcsec*
"""
theta = radius + s
frac_enclosed_num = (
backend.sqrt(Rc**2 + theta**2) - Rc - backend.sqrt(Rs**2 + theta**2) + Rs
) # arcsec
frac_enclosed_denom = Rs - Rc # arcsec
return mass * frac_enclosed_num / frac_enclosed_denom
[docs]
def reduced_deflection_angle_pseudo_jaffe(
x0, y0, mass, Rc, Rs, x, y, d_l, critical_surface_density, s=0.0
):
"""
Compute the reduced deflection angle. See Eliasdottir et al 2007 equation A19.
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*
mass: ArrayLike
Total mass of the lens
*Unit: Msun*
Rc: ArrayLike
Core radius of the lens.
*Unit: arcsec*
Rs: ArrayLike
Scaling 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*
d_l: ArrayLike
Distance to the lens.
*Unit: Mpc*
critical_surface_density: ArrayLike
Critical surface density of the universe at the lens redshift.
*Unit: Msun / Mpc^2*
s: float
Softening parameter to prevent numerical instabilities.
*Unit: arcsec*
"""
x, y = translate_rotate(x, y, x0, y0)
R = backend.sqrt(x**2 + y**2) + s
f = R / Rc / (1 + backend.sqrt(1 + (R / Rc) ** 2)) - R / (Rs * (1 + backend.sqrt(1 + (R / Rs) ** 2))) # fmt: skip
alpha = 2 * convergence_0_pseudo_jaffe(mass, Rc, Rs, d_l, critical_surface_density) * Rc * Rs / (Rs - Rc) * f # fmt: skip
ax = alpha * x / R
ay = alpha * y / R
return ax, ay
[docs]
def potential_pseudo_jaffe(x0, y0, mass, Rc, Rs, x, y, d_l, d_s, d_ls, s=0.0):
"""
Compute the lensing potential for the pseudo jaffe lens. See Eliasdottir et
al 2007 equation A18.
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*
mass: ArrayLike
Total mass of the lens
*Unit: Msun*
Rc: ArrayLike
Core radius of the lens.
*Unit: arcsec*
Rs: ArrayLike
Scaling 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*
d_l: ArrayLike
Distance to the lens.
*Unit: Mpc*
d_s: ArrayLike
Distance to the source.
*Unit: Mpc*
d_ls: ArrayLike
Distance from the lens to the source.
*Unit: Mpc*
s: float
Softening parameter to prevent numerical instabilities.
*Unit: arcsec*
"""
x, y = translate_rotate(x, y, x0, y0)
R_squared = x**2 + y**2 + s # arcsec^2
surface_density_0 = convergence_0_pseudo_jaffe(
mass, Rc, Rs, d_l, 1.0
) # Msun / Mpc^2
coeff = -(
8
* backend.pi
* G_over_c2
* surface_density_0
* (d_l * d_ls / d_s)
* Rc
* Rs
/ (Rs - Rc)
) # arcsec
scale_a = backend.sqrt(Rs**2 + R_squared) # arcsec
scale_b = backend.sqrt(Rc**2 + R_squared) # arcsec
scale_c = Rc * backend.log(Rc + backend.sqrt(Rc**2 + R_squared)) # arcsec
scale_d = Rs * backend.log(Rs + backend.sqrt(Rs**2 + R_squared)) # arcsec
scale_factor = scale_a - scale_b + scale_c - scale_d # arcsec
return coeff * scale_factor
[docs]
def convergence_pseudo_jaffe(
x0, y0, mass, Rc, Rs, x, y, d_l, critical_surface_density, s=0.0
):
"""
Compute the convergence (dimensionless surface mass density). See Eliasdottir et al 2007 Equation A3.
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*
mass: ArrayLike
Total mass of the lens
*Unit: Msun*
Rc: ArrayLike
Core radius of the lens.
*Unit: arcsec*
Rs: ArrayLike
Scaling 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*
d_l: ArrayLike
Distance to the lens.
*Unit: Mpc*
critical_surface_density: ArrayLike
Critical surface density of the universe at the lens redshift.
*Unit: Msun / Mpc^2*
s: float
Softening parameter to prevent numerical instabilities.
*Unit: arcsec*
"""
x, y = translate_rotate(x, y, x0, y0)
R_squared = x**2 + y**2 + s
coeff = convergence_0_pseudo_jaffe(mass, Rc, Rs, d_l, critical_surface_density) * Rc * Rs / (Rs - Rc) # fmt: skip
return coeff * (1 / backend.sqrt(Rc**2 + R_squared) - 1 / backend.sqrt(Rs**2 + R_squared)) # fmt: skip
