import numpy as np
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["Hernquist"]
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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
Examples for converting angular to physical mass units
------------------------------------------------------
>>> from lenstronomy.Cosmo.lens_cosmo import LensCosmo
>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05)
>>> lens_cosmo = LensCosmo(z_lens=0.5, z_source=1.5, cosmo=cosmo)
Here we compute the angular scale of Rs on the sky (in arc seconds) and the deflection the normalization sigma0 from
the total stellar mass in M_sol and Rs in [Mpc]:
>>> sigma0, rs_angle = lens_cosmo.hernquist_phys2angular(mass=10**11, rs=0.02)
And here we perform the inverse calculation given Rs_angle and alpha_Rs to return the physical halo properties.
>>> m_tot, rs = lens_cosmo.hernquist_angular2phys(sigma0=sigma0 rs_angle=rs_angle)
The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses
>>> from lenstronomy.LensModel.Profiles.hernquist import Hernquist
>>> hernquist = Hernquist()
>>> alpha_x, alpha_y = hernquist.derivatives(x=1, y=1, Rs=rs_angle, sigma0=sigma0, center_x=0, center_y=0)
"""
_diff = 0.00001
_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}
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@staticmethod
def density(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
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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)
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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
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@staticmethod
def mass_3d(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
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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 r: 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.mass_3d(r, rho0, Rs)
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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)
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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
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@staticmethod
def mass_tot(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
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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 / 4.0) + 2 * self._F(X))
return f_
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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 = 1.0 / 3
else:
f = (1 - self._F(X)) / (X**2 - 1)
else:
f = np.empty_like(X)
f[X == 1] = 1.0 / 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
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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
"""
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(
x + diff / 2, y, sigma0, Rs, center_x, center_y
)
alpha_ra_dy, alpha_dec_dy = self.derivatives(
x, y + diff / 2, sigma0, Rs, center_x, center_y
)
alpha_ra_dx_, alpha_dec_dx_ = self.derivatives(
x - diff / 2, y, sigma0, Rs, center_x, center_y
)
alpha_ra_dy_, alpha_dec_dy_ = self.derivatives(
x, y - diff / 2, sigma0, Rs, center_x, center_y
)
f_xx = (alpha_ra_dx - alpha_ra_dx_) / diff
f_xy = (alpha_ra_dy - alpha_ra_dy_) / diff
f_yx = (alpha_dec_dx - alpha_dec_dx_) / diff
f_yy = (alpha_dec_dy - alpha_dec_dy_) / diff
return f_xx, f_xy, f_yx, f_yy
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@staticmethod
def rho2sigma(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
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@staticmethod
def sigma2rho(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 0 < X < 1:
a = 1.0 / np.sqrt(1 - X**2) * np.arctanh(np.sqrt(1 - X**2))
elif X == 1:
a = 1.0
elif X > 1:
a = 1.0 / np.sqrt(X**2 - 1) * np.arctan(np.sqrt(X**2 - 1))
else: # X == 0:
a = 1.0 / 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] = 1.0
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
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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