__author__ = "sibirrer"
import numpy as np
import scipy.special as special
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["SPP"]
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class SPP(LensProfileBase):
"""Class for circular power-law mass distribution."""
param_names = ["theta_E", "gamma", "center_x", "center_y"]
lower_limit_default = {
"theta_E": 0,
"gamma": 1.5,
"center_x": -100,
"center_y": -100,
}
upper_limit_default = {
"theta_E": 100,
"gamma": 2.5,
"center_x": 100,
"center_y": 100,
}
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def function(self, x, y, theta_E, gamma, center_x=0, center_y=0):
"""
:param x: set of x-coordinates
:type x: array of size (n)
:param y: set of y-coordinates
:type y: array of size (n)
:param theta_E: Einstein radius of lens
:type theta_E: float.
:param gamma: power law slope of mass profile
:type gamma: <2 float
:returns: function
:raises: AttributeError, KeyError
"""
gamma = self._gamma_limit(gamma)
x_ = x - center_x
y_ = y - center_y
E = theta_E / ((3.0 - gamma) / 2.0) ** (1.0 / (1.0 - gamma))
# E = phi_E_spp
eta = -gamma + 3
p2 = x_**2 + y_**2
s2 = 0.0 # softening
return 2 * E**2 / eta**2 * ((p2 + s2) / E**2) ** (eta / 2)
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def derivatives(self, x, y, theta_E, gamma, center_x=0.0, center_y=0.0):
gamma = self._gamma_limit(gamma)
xt1 = x - center_x
xt2 = y - center_y
r2 = xt1 * xt1 + xt2 * xt2
a = np.maximum(r2, 0.000001)
r = np.sqrt(a)
alpha = theta_E * (r2 / theta_E**2) ** (1 - gamma / 2.0)
fac = alpha / r
f_x = fac * xt1
f_y = fac * xt2
return f_x, f_y
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def hessian(self, x, y, theta_E, gamma, center_x=0.0, center_y=0.0):
gamma = self._gamma_limit(gamma)
xt1 = x - center_x
xt2 = y - center_y
E = theta_E / ((3.0 - gamma) / 2.0) ** (1.0 / (1.0 - gamma))
# E = phi_E_spp
eta = -gamma + 3.0
P2 = xt1**2 + xt2**2
if isinstance(P2, int) or isinstance(P2, float):
a = max(0.000001, P2)
else:
a = np.empty_like(P2)
p2 = P2[P2 > 0] # in the SIS regime
a[P2 == 0] = 0.000001
a[P2 > 0] = p2
kappa = (
1.0
/ eta
* (a / E**2) ** (eta / 2 - 1)
* ((eta - 2) * (xt1**2 + xt2**2) / a + (1 + 1))
)
gamma1 = (
1.0
/ eta
* (a / E**2) ** (eta / 2 - 1)
* ((eta / 2 - 1) * (2 * xt1**2 - 2 * xt2**2) / a)
)
gamma2 = (
4 * xt1 * xt2 * (1.0 / 2 - 1 / eta) * (a / E**2) ** (eta / 2 - 2) / E**2
)
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
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@staticmethod
def rho2theta(rho0, gamma):
"""Converts 3d density into 2d projected density parameter.
:param rho0:
:param gamma:
:return:
"""
fac = (
np.sqrt(np.pi)
* special.gamma(1.0 / 2 * (-1 + gamma))
/ special.gamma(gamma / 2.0)
* 2
/ (3 - gamma)
* rho0
)
# fac = theta_E**(gamma - 1)
theta_E = fac ** (1.0 / (gamma - 1))
return theta_E
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@staticmethod
def theta2rho(theta_E, gamma):
"""Converts projected density parameter (in units of deflection) into 3d density
parameter.
:param theta_E:
:param gamma:
:return:
"""
fac1 = (
np.sqrt(np.pi)
* special.gamma(1.0 / 2 * (-1 + gamma))
/ special.gamma(gamma / 2.0)
* 2
/ (3 - gamma)
)
fac2 = theta_E ** (gamma - 1)
rho0 = fac2 / fac1
return rho0
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@staticmethod
def mass_3d(r, rho0, gamma):
"""Mass enclosed a 3d sphere or radius r.
:param r:
:param rho0:
:param gamma:
:return:
"""
mass_3d = 4 * np.pi * rho0 / (-gamma + 3) * r ** (-gamma + 3)
return mass_3d
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def mass_3d_lens(self, r, theta_E, gamma):
"""
:param r:
:param theta_E:
:param gamma:
:return:
"""
rho0 = self.theta2rho(theta_E, gamma)
return self.mass_3d(r, rho0, gamma)
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def mass_2d(self, r, rho0, gamma):
"""Mass enclosed projected 2d sphere of radius r.
:param r:
:param rho0:
:param gamma:
:return:
"""
alpha = (
np.sqrt(np.pi)
* special.gamma(1.0 / 2 * (-1 + gamma))
/ special.gamma(gamma / 2.0)
* r ** (2 - gamma)
/ (3 - gamma)
* 2
* rho0
)
mass_2d = alpha * r * np.pi
return mass_2d
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def mass_2d_lens(self, r, theta_E, gamma):
"""
:param r: projected radius
:param theta_E: Einstein radius
:param gamma: power-law slope
:return: 2d projected radius enclosed
"""
rho0 = self.theta2rho(theta_E, gamma)
return self.mass_2d(r, rho0, gamma)
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def grav_pot(self, x, y, rho0, gamma, center_x=0, center_y=0):
"""Gravitational potential (modulo 4 pi G and rho0 in appropriate units)
:param x:
:param y:
:param rho0:
:param gamma:
:param center_x:
:param center_y:
:return:
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
mass_3d = self.mass_3d(r, rho0, gamma)
pot = mass_3d / r
return pot
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@staticmethod
def density(r, rho0, gamma):
"""Computes the density.
:param r:
:param rho0:
:param gamma:
:return:
"""
rho = rho0 / r**gamma
return rho
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def density_lens(self, r, theta_E, gamma):
"""Computes the density at 3d radius r given lens model parameterization.
The integral in projected in units of angles (i.e. arc seconds) results in the
convergence quantity.
"""
rho0 = self.theta2rho(theta_E, gamma)
return self.density(r, rho0, gamma)
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@staticmethod
def density_2d(x, y, rho0, gamma, center_x=0, center_y=0):
"""Projected density.
:param x:
:param y:
:param rho0:
:param gamma:
:param center_x:
:param center_y:
:return:
"""
x_ = x - center_x
y_ = y - center_y
r = np.sqrt(x_**2 + y_**2)
sigma = (
np.sqrt(np.pi)
* special.gamma(1.0 / 2 * (-1 + gamma))
/ special.gamma(gamma / 2.0)
* r ** (1 - gamma)
* rho0
)
return sigma
@staticmethod
def _gamma_limit(gamma):
"""Limits the power-law slope to certain bounds.
:param gamma: power-law slope
:return: bounded power-law slopte
"""
return gamma