__author__ = "sibirrer"
# this file contains a class to compute the Navaro-Frenk-White profile
import numpy as np
import scipy.interpolate as interp
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["NFW"]
[docs]
class NFW(LensProfileBase):
"""This class contains functions concerning the NFW profile.
relation are: R_200 = c * Rs
The definition of 'Rs' is in angular (arc second) units and the normalization is put
in with regard to a deflection angle at 'Rs' - 'alpha_Rs'. To convert a physical
mass and concentration definition into those lensing quantities for a specific
redshift configuration and cosmological model, you can find routines in
`lenstronomy.Cosmo.lens_cosmo.py`
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 angle at Rs (in arc seconds):
>>> Rs_angle, alpha_Rs = lens_cosmo.nfw_physical2angle(M=10**13, c=6)
And here we perform the inverse calculation given Rs_angle and alpha_Rs to return the physical halo properties.
>>> rho0, Rs, c, r200, M200 = lens_cosmo.nfw_angle2physical(Rs_angle=Rs_angle, alpha_Rs=alpha_Rs)
The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses
>>> from lenstronomy.LensModel.Profiles.nfw import NFW
>>> nfw = NFW()
>>> alpha_x, alpha_y = nfw.derivatives(x=1, y=1, Rs=Rs_angle, alpha_Rs=alpha_Rs, center_x=0, center_y=0)
"""
profile_name = "NFW"
param_names = ["Rs", "alpha_Rs", "center_x", "center_y"]
lower_limit_default = {"Rs": 0, "alpha_Rs": 0, "center_x": -100, "center_y": -100}
upper_limit_default = {"Rs": 100, "alpha_Rs": 10, "center_x": 100, "center_y": 100}
[docs]
def __init__(self, interpol=False, num_interp_X=1000, max_interp_X=10):
"""
:param interpol: bool, if True, interpolates the functions F(), g() and h()
:param num_interp_X: int (only considered if interpol=True), number of interpolation elements in units of r/r_s
:param max_interp_X: float (only considered if interpol=True), maximum r/r_s value to be interpolated
(returning zeros outside)
"""
self._interpol = interpol
self._max_interp_X = max_interp_X
self._num_interp_X = num_interp_X
super(NFW, self).__init__()
[docs]
def function(self, x, y, Rs, alpha_Rs, 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 Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: lensing potential
"""
rho0_input = self.alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
f_ = self.nfw_potential(R, Rs, rho0_input)
return f_
[docs]
def derivatives(self, x, y, Rs, alpha_Rs, center_x=0, center_y=0):
"""Returns df/dx and df/dy of the function (integral of NFW), 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 Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
: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
"""
rho0_input = self.alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
f_x, f_y = self.nfw_alpha(R, Rs, rho0_input, x_, y_)
return f_x, f_y
[docs]
def hessian(self, x, y, Rs, alpha_Rs, 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 Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
: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
"""
rho0_input = self.alpha2rho0(alpha_Rs=alpha_Rs, Rs=Rs)
if Rs < 0.0000001:
Rs = 0.0000001
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
kappa = self.density_2d(R, 0, Rs, rho0_input)
gamma1, gamma2 = self.nfw_gamma(R, Rs, rho0_input, x_, y_)
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
[docs]
@staticmethod
def density(R, Rs, rho0):
"""Three-dimensional NFW profile.
:param R: radius of interest
:type R: float/numpy array
:param Rs: scale radius
:type Rs: float
:param rho0: density normalization (characteristic density)
:type rho0: float
:return: rho(R) density
"""
return rho0 / (R / Rs * (1 + R / Rs) ** 2)
[docs]
def density_lens(self, r, Rs, alpha_Rs):
"""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: 3d radios
:param Rs: turn-over radius of NFW profile
:param alpha_Rs: deflection at Rs
:return: density rho(r)
"""
rho0 = self.alpha2rho0(alpha_Rs, Rs)
return self.density(r, Rs, rho0)
[docs]
def density_2d(self, x, y, Rs, rho0, center_x=0, center_y=0):
"""Projected two-dimensional NFW profile (kappa)
:param x: x-coordinate
:param y: y-coordinate
:param Rs: scale radius
:type Rs: float
:param rho0: density normalization (characteristic density)
:type rho0: float
:param center_x: x-centroid position
:param center_y: y-centroid position
:return: Epsilon(R) projected density at radius R
"""
x_ = x - center_x
y_ = y - center_y
R = np.sqrt(x_**2 + y_**2)
x = R / Rs
Fx = self.F_(x)
return 2 * rho0 * Rs * Fx
[docs]
def mass_3d(self, r, Rs, rho0):
"""Mass enclosed a 3d sphere or radius r.
:param r: 3d radius
:param Rs: scale radius
:param rho0: density normalization (characteristic density)
:return: M(<r)
"""
Rs = float(Rs)
m_3d = 4.0 * np.pi * rho0 * Rs**3 * (np.log((Rs + r) / Rs) - r / (Rs + r))
return m_3d
[docs]
def mass_3d_lens(self, r, Rs, alpha_Rs):
"""Mass enclosed a 3d sphere or radius r. This function takes as input the
lensing parameterization.
:param r: 3d radius
:param Rs: scale radius
:param alpha_Rs: deflection (angular units) at projected Rs
:return: M(<r)
"""
rho0 = self.alpha2rho0(alpha_Rs, Rs)
m_3d = self.mass_3d(r, Rs, rho0)
return m_3d
[docs]
def mass_2d(self, R, Rs, rho0):
"""Mass enclosed a 2d cylinder or projected radius R.
:param R: projected radius
:param Rs: scale radius
:param rho0: density normalization (characteristic density)
:return: mass in cylinder.
"""
x = R / Rs
gx = self.g_(x)
m_2d = 4 * rho0 * Rs * R**2 * gx / x**2 * np.pi
return m_2d
[docs]
def mass_2d_lens(self, R, Rs, alpha_Rs):
"""
:param R: projected radius
:param Rs: scale radius
:param alpha_Rs: deflection (angular units) at projected Rs
:return: mass enclosed 2d cylinder <R
"""
rho0 = self.alpha2rho0(alpha_Rs, Rs)
return self.mass_2d(R, Rs=Rs, rho0=rho0)
[docs]
def nfw_potential(self, R, Rs, rho0):
"""Lensing potential of NFW profile (Sigma_crit D_OL**2)
:param R: radius of interest
:type R: float/numpy array
:param Rs: scale radius
:type Rs: float
:param rho0: density normalization (characteristic density)
:type rho0: float
:return: Epsilon(R) projected density at radius R
"""
x = R / Rs
hx = self.h_(x)
return 2 * rho0 * Rs**3 * hx
[docs]
def nfw_alpha(self, R, Rs, rho0, ax_x, ax_y):
"""Deflection angle of NFW profile (times Sigma_crit D_OL) along the projection
to coordinate 'axis'.
:param R: radius of interest
:type R: float/numpy array
:param Rs: scale radius
:type Rs: float
:param rho0: density normalization (characteristic density)
:type rho0: float
:param ax_x: projection to either x- or y-axis
:type ax_x: same as R
:param ax_y: projection to either x- or y-axis
:type ax_y: same as R
:return: Epsilon(R) projected density at radius R
"""
R = np.maximum(R, 0.00000001)
x = R / Rs
gx = self.g_(x)
a = 4 * rho0 * Rs * gx / x**2
return a * ax_x, a * ax_y
[docs]
def nfw_gamma(self, R, Rs, rho0, ax_x, ax_y):
"""Shear gamma of NFW profile (times Sigma_crit) along the projection to
coordinate 'axis'.
:param R: radius of interest
:type R: float/numpy array
:param Rs: scale radius
:type Rs: float
:param rho0: density normalization (characteristic density)
:type rho0: float
:param ax_x: projection to either x- or y-axis
:type ax_x: same as R
:param ax_y: projection to either x- or y-axis
:type ax_y: same as R
:return: Epsilon(R) projected density at radius R
"""
c = 0.000001
R = np.maximum(R, c)
x = R / Rs
gx = self.g_(x)
Fx = self.F_(x)
a = (
2 * rho0 * Rs * (2 * gx / x**2 - Fx)
) # /x #2*rho0*Rs*(2*gx/x**2 - Fx)*axis/x
return a * (ax_y**2 - ax_x**2) / R**2, -a * 2 * (ax_x * ax_y) / R**2
[docs]
def F_(self, X):
"""Computes h()
:param X:
:return:
"""
if self._interpol:
if not hasattr(self, "_F_interp"):
x = np.linspace(0, self._max_interp_X, self._num_interp_X)
F_x = self._F(x)
self._F_interp = interp.interp1d(
x,
F_x,
kind="linear",
axis=-1,
copy=False,
bounds_error=False,
fill_value=0,
assume_sorted=True,
)
return self._F_interp(X)
else:
return self._F(X)
@staticmethod
def _F(X):
"""Analytic solution of the projection integral.
:param X: R/Rs
:type X: float >0
"""
if isinstance(X, int) or isinstance(X, float):
if X < 1 and X > 0:
a = (
1
/ (X**2 - 1)
* (
1
- 2
/ np.sqrt(1 - X**2)
* np.arctanh(np.sqrt((1 - X) / (1 + X)))
)
)
elif X == 1:
a = 1.0 / 3
elif X > 1:
a = (
1
/ (X**2 - 1)
* (
1
- 2
/ np.sqrt(X**2 - 1)
* np.arctan(np.sqrt((X - 1) / (1 + X)))
)
)
else: # X == 0:
c = 0.0000001
a = (
1
/ (-1)
* (1 - 2 / np.sqrt(1) * np.arctanh(np.sqrt((1 - c) / (1 + c))))
)
else:
a = np.empty_like(X)
x = X[(X < 1) & (X > 0)]
a[(X < 1) & (X > 0)] = (
1
/ (x**2 - 1)
* (1 - 2 / np.sqrt(1 - x**2) * np.arctanh(np.sqrt((1 - x) / (1 + x))))
)
a[X == 1] = 1.0 / 3.0
x = X[X > 1]
a[X > 1] = (
1
/ (x**2 - 1)
* (1 - 2 / np.sqrt(x**2 - 1) * np.arctan(np.sqrt((x - 1) / (1 + x))))
)
# a[X>y] = 0
c = 0.0000001
a[X == 0] = (
1 / (-1) * (1 - 2 / np.sqrt(1) * np.arctanh(np.sqrt((1 - c) / (1 + c))))
)
return a
[docs]
def g_(self, X):
"""Computes h()
:param X: R/Rs
:type X: float >0
:return:
"""
if self._interpol:
if not hasattr(self, "_g_interp"):
x = np.linspace(0, self._max_interp_X, self._num_interp_X)
g_x = self._g(x)
self._g_interp = interp.interp1d(
x,
g_x,
kind="linear",
axis=-1,
copy=False,
bounds_error=False,
fill_value=0,
assume_sorted=True,
)
return self._g_interp(X)
else:
return self._g(X)
@staticmethod
def _g(X):
"""Analytic solution of integral for NFW profile to compute deflection angel and
gamma.
:param X: R/Rs
:type X: float >0
"""
c = 0.000001
if isinstance(X, int) or isinstance(X, float):
if X < 1:
x = max(c, X)
a = np.log(x / 2.0) + 1 / np.sqrt(1 - x**2) * np.arccosh(1.0 / x)
elif X == 1:
a = 1 + np.log(1.0 / 2.0)
else: # X > 1:
a = np.log(X / 2) + 1 / np.sqrt(X**2 - 1) * np.arccos(1.0 / X)
else:
a = np.empty_like(X)
X[X <= c] = c
x = X[X < 1]
a[X < 1] = np.log(x / 2.0) + 1 / np.sqrt(1 - x**2) * np.arccosh(1.0 / x)
a[X == 1] = 1 + np.log(1.0 / 2.0)
x = X[X > 1]
a[X > 1] = np.log(x / 2) + 1 / np.sqrt(x**2 - 1) * np.arccos(1.0 / x)
return a
[docs]
def h_(self, X):
"""Computes h()
:param X: R/Rs
:type X: float >0
:return: h(X)
"""
if self._interpol:
if not hasattr(self, "_h_interp"):
x = np.linspace(0, self._max_interp_X, self._num_interp_X)
h_x = self._h(x)
self._h_interp = interp.interp1d(
x,
h_x,
kind="linear",
axis=-1,
copy=False,
bounds_error=False,
fill_value=0,
assume_sorted=True,
)
return self._h_interp(X)
else:
return self._h(X)
@staticmethod
def _h(X):
"""Analytic solution of integral for NFW profile to compute the potential.
:param X: R/Rs
:type X: float >0
"""
c = 0.000001
if isinstance(X, int) or isinstance(X, float):
if X < 1:
x = max(0.001, X)
a = np.log(x / 2.0) ** 2 - np.arccosh(1.0 / x) ** 2
else: # X >= 1:
a = np.log(X / 2.0) ** 2 + np.arccos(1.0 / X) ** 2
else:
a = np.empty_like(X)
X[X <= c] = 0.000001
x = X[X < 1]
a[X < 1] = np.log(x / 2.0) ** 2 - np.arccosh(1.0 / x) ** 2
x = X[X >= 1]
a[X >= 1] = np.log(x / 2.0) ** 2 + np.arccos(1.0 / x) ** 2
return a
[docs]
@staticmethod
def alpha2rho0(alpha_Rs, Rs):
"""Convert angle at Rs into rho0.
:param alpha_Rs: deflection angle at RS
:param Rs: scale radius
:return: density normalization (characteristic density)
"""
rho0 = alpha_Rs / (4.0 * Rs**2 * (1.0 + np.log(1.0 / 2.0)))
return rho0
[docs]
@staticmethod
def rho02alpha(rho0, Rs):
"""Convert rho0 to angle at Rs.
:param rho0: density normalization (characteristic density)
:param Rs: scale radius
:return: deflection angle at RS
"""
alpha_Rs = rho0 * (4 * Rs**2 * (1 + np.log(1.0 / 2.0)))
return alpha_Rs