__author__ = "lucateo"
import numpy as np
from scipy.special import exp1, erf
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__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]
@staticmethod
def rhotilde(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 r: 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 R: projected 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 r: 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]
@staticmethod
def kappa_r(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) :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