__author__ = "sibirrer"
import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
from lenstronomy.Util import derivative_util as calc_util
__all__ = ["CoredDensity"]
[docs]
class CoredDensity(LensProfileBase):
"""
class for a uniform cored density dropping steep in the outskirts
This profile is e.g. featured in Blum et al. 2020 https://arxiv.org/abs/2001.07182v1
..math::
\\rho_c(r) = \\frac{2}{\\pi} \\Sigma_{c} R_c^3 \\left(R_c^2 + r^2 \\right)^{-2}
with the convergence profile as
..math::
\\kappa_c(\\theta) = \\left(1 + \\frac{\\theta^2}{\\theta_c^2} \\right)^{-3/2}.
An approximate mass-sheet degeneracy can then be written as
..math::
\\kappa_{\\lambda_c}(\\theta) = \\lambda_c \\kappa(\\theta) + (1-\\lambda_c) \\kappa_c(\\theta).
"""
_s = 0.000001 # numerical limit for minimal radius
param_names = ["sigma0", "r_core", "center_x", "center_y"]
lower_limit_default = {
"sigma0": -1,
"r_core": 0,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"sigma0": 10,
"r_core": 100,
"center_x": 100,
"center_y": 100,
}
[docs]
def function(self, x, y, sigma0, r_core, center_x=0, center_y=0):
"""Potential of cored density profile.
:param x: x-coordinate in angular units
:param y: y-coordinate in angular units
:param sigma0: convergence in the core
:param r_core: core radius
:param center_x: center of the profile
:param center_y: center of the profile
:return: lensing potential at (x, y)
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
r = np.maximum(r, self._s)
return (
2
* sigma0
* r_core**2
* (2 * np.log(r) - np.log(np.sqrt(r**2 + r_core**2) - r_core))
)
[docs]
def derivatives(self, x, y, sigma0, r_core, center_x=0, center_y=0):
"""Deflection angle of cored density profile.
:param x: x-coordinate in angular units
:param y: y-coordinate in angular units
:param sigma0: convergence in the core
:param r_core: core radius
:param center_x: center of the profile
:param center_y: center of the profile
:return: alpha_x, alpha_y at (x, y)
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
r = np.maximum(r, self._s)
alpha_r = self.alpha_r(r, sigma0, r_core)
f_x = alpha_r * x_ / r
f_y = alpha_r * y_ / r
return f_x, f_y
[docs]
def hessian(self, x, y, sigma0, r_core, center_x=0, center_y=0):
"""
:param x: x-coordinate in angular units
:param y: y-coordinate in angular units
:param sigma0: convergence in the core
:param r_core: core radius
:param center_x: center of the profile
:param center_y: center of the profile
:return: Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y)
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
r = np.maximum(r, self._s)
d_alpha_dr = self.d_alpha_dr(r, sigma0, r_core)
alpha = self.alpha_r(r, sigma0, r_core)
dr_dx = calc_util.d_r_dx(x_, y_)
dr_dy = calc_util.d_r_dy(x_, y_)
f_xx = d_alpha_dr * dr_dx * x_ / r + alpha * calc_util.d_x_diffr_dx(x_, y_)
f_yy = d_alpha_dr * dr_dy * y_ / r + alpha * calc_util.d_y_diffr_dy(x_, y_)
f_xy = d_alpha_dr * dr_dy * x_ / r + alpha * calc_util.d_x_diffr_dy(x_, y_)
return f_xx, f_xy, f_xy, f_yy
[docs]
@staticmethod
def alpha_r(r, sigma0, r_core):
"""Radial deflection angle of the cored density profile.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: deflection angle
"""
return (
2 * sigma0 * r_core**2 / r * (1 - (1 + (r / r_core) ** 2) ** (-1.0 / 2))
)
[docs]
@staticmethod
def d_alpha_dr(r, sigma0, r_core):
"""Radial derivatives of the radial deflection angle.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: dalpha/dr
"""
return (
2
* sigma0
* (
((1 + (r / r_core) ** 2) ** (-3.0 / 2))
- (r_core / r) ** 2 * (1 - (1 + (r / r_core) ** 2) ** (-1.0 / 2))
)
)
[docs]
@staticmethod
def kappa_r(r, sigma0, r_core):
"""Convergence of the cored density profile. This routine is also for testing.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: convergence at r
"""
return sigma0 * (1 + (r / r_core) ** 2) ** (-3.0 / 2)
[docs]
@staticmethod
def density(r, sigma0, r_core):
"""Rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-2)
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: density at radius r
"""
return 2 / np.pi * sigma0 * r_core**3 * (r_core**2 + r**2) ** (-2)
[docs]
def density_lens(self, r, sigma0, r_core):
"""Computes the density at 3d radius r given lens model parameterization. The
integral in the LOS projection of this quantity results in the convergence
quantity.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: desnity at radius r
"""
return self.density(r, sigma0, r_core)
[docs]
def density_2d(self, x, y, sigma0, r_core, center_x=0, center_y=0):
"""Projected density at projected radius r.
:param x: x-coordinate in angular units
:param y: y-coordinate in angular units
:param sigma0: convergence in the core
:param r_core: core radius
:param center_x: center of the profile
:param center_y: center of the profile
:return: projected density
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
r = np.maximum(r, self._s)
return self.kappa_r(r, sigma0, r_core)
[docs]
def mass_2d(self, r, sigma0, r_core):
"""Mass enclosed in cylinder of radius r.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: mass enclosed in cylinder of radius r
"""
return self.alpha_r(r, sigma0, r_core) * np.pi * r
[docs]
@staticmethod
def mass_3d(r, sigma0, r_core):
"""Mass enclosed 3d radius.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: mass enclosed 3d radius
"""
return (
8
* sigma0
* r_core**3
* (np.arctan(r / r_core) / (2 * r_core) - r / (2 * (r**2 + r_core**2)))
)
[docs]
def mass_3d_lens(self, r, sigma0, r_core):
"""Mass enclosed a 3d sphere or radius r given a lens parameterization with
angular units For this profile those are identical.
:param r: radius (angular scale)
:param sigma0: convergence in the core
:param r_core: core radius
:return: mass enclosed 3d radius
"""
return self.mass_3d(r, sigma0, r_core)