import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ['PJaffe']
[docs]class PJaffe(LensProfileBase):
"""
class to compute the DUAL PSEUDO ISOTHERMAL ELLIPTICAL MASS DISTRIBUTION
based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A
Module name: 'PJAFFE';
An alternative name is dPIED.
The 3D density distribution is
.. math::
\\rho(r) = \\frac{\\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)}
with :math:`Rs > Ra`.
The projected density is
.. math::
\\Sigma(R) = \\Sigma_0 \\frac{Ra Rs}{Rs-Ra}\\left(\\frac{1}{\\sqrt{Ra^2+R^2}} - \\frac{1}{\\sqrt{Rs^2+R^2}} \\right)
with
.. math::
\\Sigma_0 = \\pi \\rho_0 \\frac{Ra Rs}{Rs + Ra}
In the lensing parameterization,
.. math::
\\sigma_0 = \\frac{\Sigma_0}{\\Sigma_{\\rm crit}}
"""
param_names = ['sigma0', 'Ra', 'Rs', 'center_x', 'center_y']
lower_limit_default = {'sigma0': 0, 'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'sigma0': 10, 'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100}
def __init__(self):
"""
"""
LensProfileBase.__init__(self)
self._s = 0.0001
[docs] def density(self, r, rho0, Ra, Rs):
"""
computes the density
:param r: radial distance from the center (in 3D)
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: density at r
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
rho = rho0 / ((1 + (r / Ra) ** 2) * (1 + (r / Rs) ** 2))
return rho
[docs] def density_2d(self, x, y, rho0, Ra, Rs, center_x=0, center_y=0):
"""
projected density
:param x: projected coordinate on the sky
:param y: projected coordinate on the sky
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:param center_x: center of profile
:param center_y: center of profile
:return: projected density
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
sigma0 = self.rho2sigma(rho0, Ra, Rs)
sigma = sigma0 * Ra * Rs / (Rs - Ra) * (1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2))
return sigma
[docs] def mass_3d(self, r, rho0, Ra, Rs):
"""
mass enclosed a 3d sphere or radius r
:param r: radial distance from the center (in 3D)
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: M(<r)
"""
m_3d = 4 * np.pi * rho0 * Ra ** 2 * Rs ** 2 / (Rs ** 2 - Ra ** 2) * (Rs * np.arctan(r / Rs) - Ra * np.arctan(r / Ra))
return m_3d
[docs] def mass_3d_lens(self, r, sigma0, Ra, Rs):
"""
mass enclosed a 3d sphere or radius r given a lens parameterization with angular units
:param r: radial distance from the center (in 3D)
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: M(<r) in angular units (modulo critical mass density)
"""
rho0 = self.sigma2rho(sigma0, Ra, Rs)
return self.mass_3d(r, rho0, Ra, Rs)
[docs] def mass_2d(self, r, rho0, Ra, Rs):
"""
mass enclosed projected 2d sphere of radius r
:param r: radial distance from the center in projection
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: Sigma(<r)
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
sigma0 = self.rho2sigma(rho0, Ra, Rs)
m_2d = 2 * np.pi * sigma0 * Ra * Rs / (Rs - Ra) * (np.sqrt(Ra ** 2 + r ** 2) - Ra - np.sqrt(Rs ** 2 + r ** 2) + Rs)
return m_2d
[docs] def mass_tot(self, rho0, Ra, Rs):
"""
total mass within the profile
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: total mass
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
sigma0 = self.rho2sigma(rho0, Ra, Rs)
m_tot = 2 * np.pi * sigma0 * Ra * Rs
return m_tot
[docs] def grav_pot(self, r, rho0, Ra, Rs):
"""
gravitational potential (modulo 4 pi G and rho0 in appropriate units)
:param r: radial distance from the center (in 3D)
:param rho0: density normalization (see class documentation above)
:param Ra: core radius
:param Rs: transition radius from logarithmic slope -2 to -4
:return: gravitational potential (modulo 4 pi G and rho0 in appropriate units)
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
pot = 4 * np.pi * rho0 * Ra ** 2 * Rs ** 2 / (Rs ** 2 - Ra ** 2) * (Rs / r * np.arctan(r / Rs) - Ra / r * np.arctan(r / Ra)
+ 1. / 2 * np.log((Rs ** 2 + r ** 2) / (Ra ** 2 + r ** 2)))
return pot
[docs] def function(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0):
"""
lensing potential
:param x: projected coordinate on the sky
:param y: projected coordinate on the sky
:param sigma0: sigma0/sigma_crit (see class documentation above)
:param Ra: core radius (see class documentation above)
:param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above)
:param center_x: center of profile
:param center_y: center of profile
:return: lensing potential
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
f_ = -2*sigma0 * Ra * Rs / (Rs - Ra) * (np.sqrt(Rs ** 2 + r ** 2) - np.sqrt(Ra ** 2 + r ** 2) + Ra * np.log(Ra + np.sqrt(Ra ** 2 + r ** 2)) - Rs * np.log(Rs + np.sqrt(Rs ** 2 + r ** 2)))
return f_
[docs] def derivatives(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0):
"""
deflection angles
:param x: projected coordinate on the sky
:param y: projected coordinate on the sky
:param sigma0: sigma0/sigma_crit (see class documentation above)
:param Ra: core radius (see class documentation above)
:param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above)
:param center_x: center of profile
:param center_y: center of profile
:return: f_x, f_y
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
if isinstance(r, int) or isinstance(r, float):
r = max(self._s, r)
else:
r[r < self._s] = self._s
alpha_r = 2*sigma0 * Ra * Rs / (Rs - Ra) * self._f_A20(r / Ra, r / Rs)
f_x = alpha_r * x_/r
f_y = alpha_r * y_/r
return f_x, f_y
[docs] def hessian(self, x, y, sigma0, Ra, Rs, center_x=0, center_y=0):
"""
Hessian of lensing potential
:param x: projected coordinate on the sky
:param y: projected coordinate on the sky
:param sigma0: sigma0/sigma_crit (see class documentation above)
:param Ra: core radius (see class documentation above)
:param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above)
:param center_x: center of profile
:param center_y: center of profile
:return: f_xx, f_xy, f_yx, f_yy
"""
Ra, Rs = self._sort_ra_rs(Ra, Rs)
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
if isinstance(r, int) or isinstance(r, float):
r = max(self._s, r)
else:
r[r < self._s] = self._s
gamma = sigma0 * Ra * Rs / (Rs - Ra) * (2 * (1. / (Ra + np.sqrt(Ra ** 2 + r ** 2)) - 1. / (Rs + np.sqrt(Rs ** 2 + r ** 2))) -
(1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2)))
kappa = sigma0 * Ra * Rs / (Rs - Ra) * (1 / np.sqrt(Ra ** 2 + r ** 2) - 1 / np.sqrt(Rs ** 2 + r ** 2))
sin_2phi = -2*x_*y_/r**2
cos_2phi = (y_**2 - x_**2)/r**2
gamma1 = cos_2phi*gamma
gamma2 = sin_2phi*gamma
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
def _f_A20(self, r_a, r_s):
"""
equation A20 in Eliasdottir (2007)
:param r_a: r/Ra
:param r_s: r/Rs
:return: f(R/a, R/s)
"""
return r_a/(1+np.sqrt(1 + r_a**2)) - r_s/(1+np.sqrt(1 + r_s**2))
[docs] def rho2sigma(self, rho0, Ra, Rs):
"""
converts 3d density into 2d projected density parameter, Equation A4 in Eliasdottir (2007)
:param rho0: density normalization
:param Ra: core radius (see class documentation above)
:param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above)
:return: projected density normalization
"""
return np.pi * rho0 * Ra * Rs / (Rs + Ra)
[docs] def sigma2rho(self, sigma0, Ra, Rs):
"""
inverse of rho2sigma()
:param sigma0: projected density normalization
:param Ra: core radius (see class documentation above)
:param Rs: transition radius from logarithmic slope -2 to -4 (see class documentation above)
:return: 3D density normalization
"""
return (Rs + Ra) / Ra / Rs / np.pi * sigma0
@staticmethod
def _sort_ra_rs(Ra, Rs):
"""
sorts Ra and Rs to make sure Rs > Ra
:param Ra:
:param Rs:
:return: Ra, Rs in conventions used
"""
if Ra >= Rs:
Ra, Rs = Rs, Ra
if Ra < 0.0001:
Ra = 0.0001
if Rs < Ra + 0.0001:
Rs += 0.0001
return Ra, Rs