import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ['Hernquist']
[docs]class Hernquist(LensProfileBase):
"""
class to compute the Hernquist 1990 model, which is in 3d:
rho(r) = rho0 / (r/Rs * (1 + (r/Rs))**3)
in lensing terms, the normalization parameter 'sigma0' is defined such that the deflection at projected RS leads to
alpha = 2./3 * Rs * sigma0
"""
_diff = 0.00000001
_s = 0.00001
param_names = ['sigma0', 'Rs', 'center_x', 'center_y']
lower_limit_default = {'sigma0': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'sigma0': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100}
[docs] def density(self, r, rho0, Rs):
"""
computes the 3-d density
:param r: 3-d radius
:param rho0: density normalization
:param Rs: Hernquist radius
:return: density at radius r
"""
rho = rho0 / (r/Rs * (1 + (r/Rs))**3)
return rho
[docs] def density_lens(self, r, sigma0, Rs):
"""
Density as a function of 3d radius in lensing parameters
This function converts the lensing definition sigma0 into the 3d density
:param r: 3d radius
:param sigma0: rho0 * Rs (units of projected density)
:param Rs: Hernquist radius
:return: enclosed mass in 3d
"""
rho0 = self.sigma2rho(sigma0, Rs)
return self.density(r, rho0, Rs)
[docs] def density_2d(self, x, y, rho0, Rs, center_x=0, center_y=0):
"""
projected density along the line of sight at coordinate (x, y)
:param x: x-coordinate
:param y: y-coordinate
:param rho0: density normalization
:param Rs: Hernquist radius
:param center_x: x-center of the profile
:param center_y: y-center of the profile
:return: projected density
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
X = r/Rs
sigma0 = self.rho2sigma(rho0, Rs)
if isinstance(X, int) or isinstance(X, float):
if X == 1:
X = 1.000001
else:
X[X == 1] = 1.000001
sigma = sigma0 / (X**2-1)**2 * (-3 + (2+X**2)*self._F(X))
return sigma
[docs] def mass_3d(self, r, rho0, Rs):
"""
mass enclosed a 3d sphere or radius r
:param r: 3-d radius within the mass is integrated (same distance units as density definition)
:param rho0: density normalization
:param Rs: Hernquist radius
:return: enclosed mass
"""
mass_3d = 2*np.pi*Rs**3*rho0 * r**2/(r + Rs)**2
return mass_3d
[docs] def mass_3d_lens(self, r, sigma0, Rs):
"""
mass enclosed a 3d sphere or radius r for lens parameterisation
This function converts the lensing definition sigma0 into the 3d density
:param sigma0: rho0 * Rs (units of projected density)
:param Rs: Hernquist radius
:return: enclosed mass in 3d
"""
rho0 = self.sigma2rho(sigma0, Rs)
return self.mass_3d(r, rho0, Rs)
[docs] def mass_2d(self, r, rho0, Rs):
"""
mass enclosed projected 2d sphere of radius r
:param r: projected radius
:param rho0: density normalization
:param Rs: Hernquist radius
:return: mass enclosed 2d projected radius
"""
sigma0 = self.rho2sigma(rho0, Rs)
return self.mass_2d_lens(r, sigma0, Rs)
[docs] def mass_2d_lens(self, r, sigma0, Rs):
"""
mass enclosed projected 2d sphere of radius r
Same as mass_2d but with input normalization in units of projected density
:param r: projected radius
:param sigma0: rho0 * Rs (units of projected density)
:param Rs: Hernquist radius
:return: mass enclosed 2d projected radius
"""
X = r/Rs
alpha_r = 2*sigma0 * Rs * X * (1-self._F(X)) / (X**2-1)
mass_2d = alpha_r * r * np.pi
return mass_2d
[docs] def mass_tot(self, rho0, Rs):
"""
total mass within the profile
:param rho0: density normalization
:param Rs: Hernquist radius
:return: total mass within profile
"""
m_tot = 2*np.pi*rho0*Rs**3
return m_tot
[docs] def function(self, x, y, sigma0, Rs, center_x=0, center_y=0):
"""
lensing potential
:param x: x-coordinate position (units of angle)
:param y: y-coordinate position (units of angle)
:param sigma0: normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
:param Rs: Hernquist radius in units of angle
:param center_x: x-center of the profile (units of angle)
:param center_y: y-center of the profile (units of angle)
: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)
X = r / Rs
f_ = sigma0 * Rs ** 2 * (np.log(X ** 2 / 4.) + 2 * self._F(X))
return f_
[docs] def derivatives(self, x, y, sigma0, Rs, center_x=0, center_y=0):
"""
:param x: x-coordinate position (units of angle)
:param y: y-coordinate position (units of angle)
:param sigma0: normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
:param Rs: Hernquist radius in units of angle
:param center_x: x-center of the profile (units of angle)
:param center_y: y-center of the profile (units of angle)
:return: derivative of function (deflection angles in x- and y-direction)
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
r = np.maximum(r, self._s)
X = r/Rs
if isinstance(r, int) or isinstance(r, float):
#f = (1 - self._F(X)) / (X ** 2 - 1) # this expression is 1/3 for X=1
if X == 1:
f = 1./3
else:
f = (1 - self._F(X)) / (X ** 2 - 1)
else:
f = np.empty_like(X)
f[X == 1] = 1./3
X_ = X[X != 1]
f[X != 1] = (1 - self._F(X_)) / (X_ ** 2 - 1)
alpha_r = 2 * sigma0 * Rs * f * X
f_x = alpha_r * x_/r
f_y = alpha_r * y_/r
return f_x, f_y
[docs] def hessian(self, x, y, sigma0, Rs, center_x=0, center_y=0):
"""
Hessian terms of the function
:param x: x-coordinate position (units of angle)
:param y: y-coordinate position (units of angle)
:param sigma0: normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
:param Rs: Hernquist radius in units of angle
:param center_x: x-center of the profile (units of angle)
:param center_y: y-center of the profile (units of angle)
:return: df/dxdx, df/dxdy, df/dydx, df/dydy
"""
alpha_ra, alpha_dec = self.derivatives(x, y, sigma0, Rs, center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Rs, center_x, center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, Rs, center_x, center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
[docs] def rho2sigma(self, rho0, Rs):
"""
converts 3d density into 2d projected density parameter
:param rho0: 3d density normalization of Hernquist model
:param Rs: Hernquist radius
:return: sigma0 defined quantity in projected units
"""
return rho0 * Rs
[docs] def sigma2rho(self, sigma0, Rs):
"""
converts projected density parameter (in units of deflection) into 3d density parameter
:param sigma0: density defined quantity in projected units
:param Rs: Hernquist radius
:return: rho0 the 3d density normalization of Hernquist model
"""
return sigma0 / Rs
def _F(self, X):
"""
function 48 in https://arxiv.org/pdf/astro-ph/0102341.pdf
:param X: r/rs
:return: F(X)
"""
c = self._s
if isinstance(X, int) or isinstance(X, float):
X = max(X, c)
if X < 1 and X > 0:
a = 1. / np.sqrt(1 - X ** 2) * np.arctanh(np.sqrt(1 - X**2))
elif X == 1:
a = 1.
elif X > 1:
a = 1. / np.sqrt(X ** 2 - 1) * np.arctan(np.sqrt(X**2 - 1))
else: # X == 0:
a = 1. / np.sqrt(1 - c ** 2) * np.arctanh(np.sqrt((1 - c ** 2)))
else:
a = np.empty_like(X)
X[X < c] = c
x = X[X < 1]
a[X < 1] = 1 / np.sqrt(1 - x ** 2) * np.arctanh(np.sqrt((1 - x**2)))
x = X[X == 1]
a[X == 1] = 1.
x = X[X > 1]
a[X > 1] = 1 / np.sqrt(x ** 2 - 1) * np.arctan(np.sqrt(x**2 - 1))
# a[X>y] = 0
return a
[docs] def grav_pot(self, x, y, rho0, Rs, center_x=0, center_y=0):
"""
#TODO decide whether these functions are needed or not
gravitational potential (modulo 4 pi G and rho0 in appropriate units)
:param x: x-coordinate position (units of angle)
:param y: y-coordinate position (units of angle)
:param rho0: density normalization parameter of Hernquist profile
:param Rs: Hernquist radius in units of angle
:param center_x: x-center of the profile (units of angle)
:param center_y: y-center of the profile (units of angle)
:return: gravitational potential at projected radius
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
M = self.mass_tot(rho0, Rs)
pot = M / (r + Rs)
return pot