__author__ = "lynevdv"
import numpy as np
import cmath as c
from lenstronomy.Util import param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["ElliSLICE"]
[docs]
class ElliSLICE(LensProfileBase):
"""
This class computes the lensing quantities for an elliptical slice of constant density.
Based on Schramm 1994 https://ui.adsabs.harvard.edu/abs/1994A%26A...284...44S/abstract
Computes the lensing quantities of an elliptical slice with semi major axis 'a' and
semi minor axis 'b', centered on 'center_x' and 'center_y', oriented with an angle 'psi'
in radian, and with constant surface mass density 'sigma_0'.
In other words, this lens model is characterized by the surface mass density :
..math::
\\kappa(x,y) = \\left{
\\begin{array}{ll}
\\sigma_0 & \\mbox{if } \\frac{x_{rot}^2}{a^2} + \\frac{y_{rot}^2}{b^2} \\leq 1 \\\
0 & \\mbox{else}
\\end{array}
\\right}.
with
..math::
x_{rot} = x_c \\cos \\psi + y_c \\sin \\psi \\
y_{rot} = - x_c \\sin \\psi + y_c \\cos \\psi \\
x_c = x - center_x \\
y_c = y - center_y
"""
param_names = ["a", "b", "psi", "sigma_0", "center_x", "center_y"]
lower_limit_default = {
"a": 0.0,
"b": 0.0,
"psi": -90.0 / 180.0 * np.pi,
"center_x": -100.0,
"center_y": -100.0,
}
upper_limit_default = {
"a": 100.0,
"b": 100.0,
"psi": 90.0 / 180.0 * np.pi,
"center_x": 100.0,
"center_y": 100.0,
}
[docs]
def function(self, x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0):
"""Lensing potential.
:param a: float, semi-major axis, must be positive
:param b: float, semi-minor axis, must be positive
:param psi: float, orientation in radian
:param sigma_0: float, surface mass density, must be positive
:param center_x: float, center on the x axis
:param center_y: float, center on the y axis
"""
kwargs_slice = {
"center_x": center_x,
"center_y": center_y,
"a": a,
"b": b,
"psi": psi,
"sigma_0": sigma_0,
}
x_ = x - center_x
y_ = y - center_y
x_rot = x_ * np.cos(psi) + y_ * np.sin(psi)
y_rot = -x_ * np.sin(psi) + y_ * np.cos(psi)
try:
len(x_)
except:
if (x_rot**2 / a**2) + (y_rot**2 / b**2) <= 1:
return self.pot_in(x_, y_, kwargs_slice)
else:
return self.pot_ext(x_, y_, kwargs_slice)
else:
f = np.array(
[
self.pot_in(x_[i], y_[i], kwargs_slice)
if (x_rot[i] ** 2 / a**2) + (y_rot[i] ** 2 / b**2) <= 1
else self.pot_ext(x_[i], y_[i], kwargs_slice)
for i in range(len(x_))
]
)
return f
[docs]
def derivatives(self, x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0):
"""Lensing deflection angle.
:param a: float, semi-major axis, must be positive
:param b: float, semi-minor axis, must be positive
:param psi: float, orientation in radian
:param sigma_0: float, surface mass density, must be positive
:param center_x: float, center on the x axis
:param center_y: float, center on the y axis
"""
kwargs_slice = {
"center_x": center_x,
"center_y": center_y,
"a": a,
"b": b,
"psi": psi,
"sigma_0": sigma_0,
}
x_ = x - center_x
y_ = y - center_y
x_rot = x_ * np.cos(psi) + y_ * np.sin(psi)
y_rot = -x_ * np.sin(psi) + y_ * np.cos(psi)
try:
len(x_)
except:
if (x_rot**2 / a**2) + (y_rot**2 / b**2) <= 1:
return self.alpha_in(x_, y_, kwargs_slice)
else:
return self.alpha_ext(x_, y_, kwargs_slice)
else:
defl = np.array(
[
self.alpha_in(x_[i], y_[i], kwargs_slice)
if (x_rot[i] ** 2 / a**2) + (y_rot[i] ** 2 / b**2) <= 1
else self.alpha_ext(x_[i], y_[i], kwargs_slice)
for i in range(len(x_))
]
)
return defl[:, 0], defl[:, 1]
[docs]
def hessian(self, x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0):
"""Lensing second derivatives.
:param a: float, semi-major axis, must be positive
:param b: float, semi-minor axis, must be positive
:param psi: float, orientation in radian
:param sigma_0: float, surface mass density, must be positive
:param center_x: float, center on the x axis
:param center_y: float, center on the y axis
"""
diff = 0.000000001
alpha_ra, alpha_dec = self.derivatives(
x, y, a, b, psi, sigma_0, center_x, center_y
)
alpha_ra_dx, alpha_dec_dx = self.derivatives(
x + diff, y, a, b, psi, sigma_0, center_x, center_y
)
alpha_ra_dy, alpha_dec_dy = self.derivatives(
x, y + diff, a, b, psi, sigma_0, center_x, center_y
)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
[docs]
@staticmethod
def sign(z):
"""Sign function.
:param z: complex
"""
x = z.real
y = z.imag
if x > 0 or (x == 0 and y >= 0):
return 1
else:
return -1
[docs]
def alpha_in(self, x, y, kwargs_slice):
"""Deflection angle for (x,y) inside the elliptical slice.
:param kwargs_slice: dict, dictionary with the slice definition
(a,b,psi,sigma_0)
"""
z = complex(x, y)
zb = z.conjugate()
psi = kwargs_slice["psi"]
e = (kwargs_slice["a"] - kwargs_slice["b"]) / (
kwargs_slice["a"] + kwargs_slice["b"]
)
sig_0 = kwargs_slice["sigma_0"]
e2ipsi = c.exp(2j * psi)
I_in = (z - e * zb * e2ipsi) * sig_0
return I_in.real, I_in.imag
[docs]
def alpha_ext(self, x, y, kwargs_slice):
"""Deflection angle for (x,y) outside the elliptical slice.
:param kwargs_slice: dict, dictionary with the slice definition
(a,b,psi,sigma_0)
"""
z = complex(x, y)
r, phi = param_util.cart2polar(x, y)
zb = z.conjugate()
psi = kwargs_slice["psi"]
a = kwargs_slice["a"]
b = kwargs_slice["b"]
f2 = a**2 - b**2
sig_0 = kwargs_slice["sigma_0"]
median_op = False
# when (x,y) is on one of the ellipse axis, there might be an issue when calculating the square root of
# zb ** 2 * e2ipsi - f2. When the argument has an imaginary part ==0, having 0. or -0. may return different
# answers. Therefore, for points (x,y) close to one axis, we take 3 points (one is x,y ; another one is a delta
# away from this position, perpendicularly to the axis ; another one is at -delta perpendicularly away from
# x,y). We calculate the function for each point and take the median. This avoids any singularity for points
# along the axis but it slows down the function.
if (
np.abs(np.sin(phi - psi)) <= 10**-10
or np.abs(np.sin(phi - psi - np.pi / 2.0)) <= 10**-10
): # very close to one of the ellipse axis
median_op = True
e2ipsi = c.exp(2j * psi)
eipsi = c.exp(1j * psi)
if median_op is True:
eps = 10**-10
z_minus_eps = complex(r * np.cos(phi - eps), r * np.sin(phi - eps))
zb_minus_eps = z_minus_eps.conjugate()
z_plus_eps = complex(r * np.cos(phi + eps), r * np.sin(phi + eps))
zb_plus_eps = z_plus_eps.conjugate()
I_out_minus = (
2
* a
* b
/ f2
* (
zb_minus_eps * e2ipsi
- eipsi
* self.sign(zb_minus_eps * eipsi)
* c.sqrt(zb_minus_eps**2 * e2ipsi - f2)
)
* sig_0
)
I_out_plus = (
2
* a
* b
/ f2
* (
zb_plus_eps * e2ipsi
- eipsi
* self.sign(zb_plus_eps * eipsi)
* c.sqrt(zb_plus_eps**2 * e2ipsi - f2)
)
* sig_0
)
I_out_mid = (
2
* a
* b
/ f2
* (
zb * e2ipsi
- eipsi * self.sign(zb * eipsi) * c.sqrt(zb**2 * e2ipsi - f2)
)
* sig_0
)
I_out_real = np.median([I_out_minus.real, I_out_plus.real, I_out_mid.real])
I_out_imag = np.median([I_out_minus.imag, I_out_plus.imag, I_out_mid.imag])
else:
I_out = (
2
* a
* b
/ f2
* (
zb * e2ipsi
- eipsi * self.sign(zb * eipsi) * c.sqrt(zb**2 * e2ipsi - f2)
)
* sig_0
)
I_out_real = I_out.real
I_out_imag = I_out.imag
# buf = zb ** 2 * e2ipsi - f2 ##problem with 0. and -0. giving different answers
# if buf.real == 0:
# buf = complex(0, buf.imag)
# if buf.imag == 0:
# buf = complex(buf.real, 0)
return I_out_real, I_out_imag
[docs]
@staticmethod
def pot_in(x, y, kwargs_slice):
"""Lensing potential for (x,y) inside the elliptical slice.
:param kwargs_slice: dict, dictionary with the slice definition
(a,b,psi,sigma_0)
"""
psi = kwargs_slice["psi"]
a = kwargs_slice["a"]
b = kwargs_slice["b"]
sig_0 = kwargs_slice["sigma_0"]
e = (a - b) / (a + b)
rE = (a + b) / 2.0
pot_in = (
0.5
* (
(1 - e) * (x * np.cos(psi) + y * np.sin(psi)) ** 2
+ (1 + e) * (y * np.cos(psi) - x * np.sin(psi)) ** 2
)
* sig_0
)
cst = sig_0 * rE**2 * (1 - e**2) * np.log(rE)
return pot_in + cst
[docs]
def pot_ext(self, x, y, kwargs_slice):
"""Lensing potential for (x,y) outside the elliptical slice.
:param kwargs_slice: dict, dictionary with the slice definition
(a,b,psi,sigma_0)
"""
z = complex(x, y)
# zb = z.conjugate()
psi = kwargs_slice["psi"]
a = kwargs_slice["a"]
b = kwargs_slice["b"]
sig_0 = kwargs_slice["sigma_0"]
r, phi = param_util.cart2polar(x, y)
median_op = False
# when (x,y) is on one of the ellipse axis, there might be an issue when calculating the square root of
# z ** 2 * em2ipsi - f2. When the argument has an imaginary part ==0, having 0. or -0. may return different
# answers. Therefore, for points (x,y) close to one axis, we take 3 points (one is x,y ; another one is a delta
# away from this position, perpendicularly to the axis ; another one is at -delta perpendicularly away from
# x,y). We calculate the function for each point and take the median. This avoids any singularity for points
# along the axis but it slows down the function.
if (
np.abs(np.sin(phi - psi)) <= 10**-10
or np.abs(np.sin(phi - psi - np.pi / 2.0)) <= 10**-10
): # very close to one of the ellipse axis
median_op = True
e = (a - b) / (a + b)
f2 = a**2 - b**2
emipsi = c.exp(-1j * psi)
em2ipsi = c.exp(-2j * psi)
if median_op is True:
eps = 10**-10
z_minus_eps = complex(r * np.cos(phi - eps), r * np.sin(phi - eps))
z_plus_eps = complex(r * np.cos(phi + eps), r * np.sin(phi + eps))
pot_ext_minus = (
(1 - e**2)
/ (4 * e)
* (
f2
* c.log(
(
self.sign(z_minus_eps * emipsi) * z_minus_eps * emipsi
+ c.sqrt(z_minus_eps**2 * em2ipsi - f2)
)
/ 2.0
)
- self.sign(z_minus_eps * emipsi)
* z_minus_eps
* emipsi
* c.sqrt(z_minus_eps**2 * em2ipsi - f2)
+ z_minus_eps**2 * em2ipsi
)
* sig_0
)
pot_ext_plus = (
(1 - e**2)
/ (4 * e)
* (
f2
* c.log(
(
self.sign(z_plus_eps * emipsi) * z_plus_eps * emipsi
+ c.sqrt(z_plus_eps**2 * em2ipsi - f2)
)
/ 2.0
)
- self.sign(z_plus_eps * emipsi)
* z_plus_eps
* emipsi
* c.sqrt(z_plus_eps**2 * em2ipsi - f2)
+ z_plus_eps**2 * em2ipsi
)
* sig_0
)
pot_ext_mid = (
(1 - e**2)
/ (4 * e)
* (
f2
* c.log(
(
self.sign(z * emipsi) * z * emipsi
+ c.sqrt(z**2 * em2ipsi - f2)
)
/ 2.0
)
- self.sign(z * emipsi) * z * emipsi * c.sqrt(z**2 * em2ipsi - f2)
+ z**2 * em2ipsi
)
* sig_0
)
pot_ext = np.median(
[pot_ext_minus.real, pot_ext_plus.real, pot_ext_mid.real]
)
else:
pot_ext = (
(1 - e**2)
/ (4 * e)
* (
f2
* c.log(
(
self.sign(z * emipsi) * z * emipsi
+ c.sqrt(z**2 * em2ipsi - f2)
)
/ 2.0
)
- self.sign(z * emipsi) * z * emipsi * c.sqrt(z**2 * em2ipsi - f2)
+ z**2 * em2ipsi
)
* sig_0
).real
return pot_ext