__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
3d rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-2)
"""
_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./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./2)) - (r_core/r) ** 2 * (1 - (1+(r/r_core)**2) ** (-1./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./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)