__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., 'b': 0., 'psi': -90./180.*np.pi,'center_x':-100.,'center_y':-100.}
upper_limit_default = {'a': 100., 'b': 100., 'psi': 90. / 180. * np.pi, 'center_x': 100., 'center_y': 100.}
[docs] def function(self,x, y, a, b, psi, sigma_0, center_x=0., center_y=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., center_y=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., center_y=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] def sign(self, 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.)) <= 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] def pot_in(self,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.
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.)) <= 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.)
- 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.)
- 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.)
- 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.)
- self.sign(z * emipsi) * z * emipsi * c.sqrt(
z ** 2 * em2ipsi - f2) + z ** 2 * em2ipsi) * sig_0).real
return pot_ext