__author__ = 'lucateo'
import numpy as np
import scipy.interpolate as interp
from scipy.special import exp1, erf
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
import lenstronomy.Util.constants as const
__all__ = ['CoredDensityExp']
[docs]class CoredDensityExp(LensProfileBase):
"""
this class contains functions concerning an exponential cored density profile,
namely
..math::
\\rho(r) = \\rho_0 \\exp(- (\\theta / \\theta_c)^2)
"""
_s = 0.000001 # numerical limit for minimal radius
param_names = ['kappa_0', 'theta_c', 'center_x', 'center_y']
lower_limit_default = {'kappa_0': 0, 'theta_c': 0, 'center_x': -100, 'center_y': -100 }
upper_limit_default = {'kappa_0': 10, 'theta_c': 100, 'center_x': 100, 'center_y': 100 }
[docs] def rhotilde(self, kappa_0, theta_c):
"""
Computes the central density in angular units
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:return: central density in 1/arcsec
"""
return kappa_0 / (np.sqrt(np.pi) * theta_c)
[docs] def function(self, x, y, kappa_0, theta_c, center_x=0, center_y=0):
"""
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: lensing potential (in arcsec^2)
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_** 2 + y_** 2)
r = np.maximum(r, self._s)
Integral_factor = 0.5 * exp1( (r/theta_c)**2) + np.log( (r/theta_c))
function = kappa_0 * theta_c**2 * Integral_factor
return function
[docs] @staticmethod
def alpha_radial(r, kappa_0, theta_c):
"""
returns the radial part of the deflection angle
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:return: radial deflection angle
"""
prefactor = kappa_0 * theta_c**2 / r
return prefactor * (1 - np.exp(- (r/theta_c)**2 ))
[docs] def derivatives(self, x, y, kappa_0, theta_c, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function (lensing potential), which are
the deflection angles
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: deflection angle in x, deflection angle in y
"""
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
R = np.maximum(R,0.00000001)
f_x = self.alpha_radial(R, kappa_0, theta_c ) * x_ / R
f_y = self.alpha_radial(R, kappa_0, theta_c) * y_ / R
return f_x, f_y
[docs] def hessian(self, x, y, kappa_0, theta_c, center_x=0, center_y=0):
"""
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
R = np.maximum(R,0.00000001)
prefactor = kappa_0 * theta_c**2
expFactor = np.exp( - (R/theta_c)**2)
factor1 = (1 - expFactor)/R**4
factor2 = 2/(R**2 * theta_c**2) * expFactor
f_xx = prefactor * ( factor1 * (y_**2 - x_**2) + factor2 * x_**2 )
f_yy = prefactor * ( factor1 * (x_**2 - y_**2) + factor2 * y_**2 )
f_xy = prefactor * ( - factor1 * 2 * x_ * y_ + factor2 *x_*y_ )
return f_xx, f_xy, f_xy, f_yy
[docs] def density(self, R, kappa_0, theta_c):
"""
three dimensional density profile in angular units
(rho0_physical = rho0_angular \Sigma_crit / D_lens)
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:return: rho(R) density
"""
rhotilde = self.rhotilde(kappa_0, theta_c)
return rhotilde * np.exp( -(R/theta_c)**2)
[docs] def density_lens(self, r, kappa_0, theta_c):
"""
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 x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:return: density rho(r)
"""
return self.density(r, kappa_0, theta_c)
[docs] def kappa_r(self, R, kappa_0, theta_c):
"""
convergence of the cored density profile. This routine is also for testing
:param R: radius (angular scale)
:param kappa_0: convergence in the core
:param theta_c: core radius
:return: convergence at r
"""
expFactor = np.exp( - (R/theta_c)**2)
return kappa_0 * expFactor
[docs] def density_2d(self, x, y, kappa_0, theta_c, center_x = 0, center_y = 0):
"""
projected two dimensional ULDM profile (convergence * \Sigma_crit), but given our
units convention for rho0, it is basically the convergence
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:return: Epsilon(R) projected density at radius R
"""
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
return self.kappa_r(R, kappa_0, theta_c)
[docs] @staticmethod
def mass_3d( R, kappa_0, theta_c):
"""
mass enclosed a 3d sphere or radius r
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:param R: radius in arcseconds
:return: mass of soliton in angular units
"""
integral_factor = np.sqrt(np.pi) * erf(R/theta_c)/2 - R/theta_c * np.exp(-(R/theta_c)**2)
m_3d = 2* np.sqrt(np.pi) * kappa_0 * theta_c**2 * integral_factor
return m_3d
[docs] def mass_3d_lens(self, r, kappa_0, theta_c):
"""
mass enclosed a 3d sphere or radius r
:param kappa_0: central convergence of profile
:param theta_c: core radius (in arcsec)
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: mass
"""
m_3d = self.mass_3d(r, kappa_0, theta_c)
return m_3d
[docs] def mass_2d(self, R, kappa_0, theta_c):
"""
mass enclosed a 2d sphere of radius r
returns
.. math::
M_{2D} = 2 \\pi \\int_0^r dr' r' \\int dz \\rho(\\sqrt(r'^2 + z^2))
:param kappa_0: central convergence of soliton
:param theta_c: core radius (in arcsec)
:return: M_2D (ULDM only)
"""
return self.alpha_radial(R, kappa_0, theta_c) * np.pi * R