__author__ = 'sibirrer'
import numpy as np
import scipy.special as special
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ['SPP']
[docs]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}
[docs] 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 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. - gamma) / 2.) ** (1. / (1. - gamma))
# E = phi_E_spp
eta= -gamma + 3
p2 = x_**2+y_**2
s2 = 0. # softening
return 2 * E**2/eta**2 * ((p2 + s2)/E**2)**(eta/2)
[docs] def derivatives(self, x, y, theta_E, gamma, center_x=0., center_y=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.)
fac = alpha/r
f_x = fac*xt1
f_y = fac*xt2
return f_x, f_y
[docs] def hessian(self, x, y, theta_E, gamma, center_x=0., center_y=0.):
gamma = self._gamma_limit(gamma)
xt1 = x - center_x
xt2 = y - center_y
E = theta_E / ((3. - gamma) / 2.) ** (1. / (1. - gamma))
# E = phi_E_spp
eta = -gamma + 3.
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./eta*(a/E**2)**(eta/2-1)*((eta-2)*(xt1**2+xt2**2)/a+(1+1))
gamma1 = 1./eta*(a/E**2)**(eta/2-1)*((eta/2-1)*(2*xt1**2-2*xt2**2)/a)
gamma2 = 4*xt1*xt2*(1./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
[docs] @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. / 2 * (-1 + gamma)) / special.gamma(gamma / 2.) * 2 / (3 - gamma) * rho0
#fac = theta_E**(gamma - 1)
theta_E = fac**(1. / (gamma - 1))
return theta_E
[docs] @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. / 2 * (-1 + gamma)) / special.gamma(gamma / 2.) * 2 / (3 - gamma)
fac2 = theta_E**(gamma - 1)
rho0 = fac2 / fac1
return rho0
[docs] @staticmethod
def mass_3d(r, rho0, gamma):
"""
mass enclosed a 3d sphere or radius r
:param r:
:param a:
:param s:
:return:
"""
mass_3d = 4 * np.pi * rho0 /(-gamma + 3) * r ** (-gamma + 3)
return mass_3d
[docs] 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)
[docs] def mass_2d(self, r, rho0, gamma):
"""
mass enclosed projected 2d sphere of radius r
:param r:
:param rho0:
:param a:
:param s:
:return:
"""
alpha = np.sqrt(np.pi) * special.gamma(1. / 2 * (-1 + gamma)) / special.gamma(gamma / 2.) * r ** (2 - gamma)/(3 - gamma) *np.pi * 2 * rho0
mass_2d = alpha*r
return mass_2d
[docs] 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)
[docs] 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 a:
:param s:
: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
[docs] @staticmethod
def density(r, rho0, gamma):
"""
computes the density
:param x:
:param y:
:param rho0:
:param a:
:param s:
:return:
"""
rho = rho0 / r**gamma
return rho
[docs] 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)
[docs] @staticmethod
def density_2d(x, y, rho0, gamma, center_x=0, center_y=0):
"""
projected density
:param x:
:param y:
:param rho0:
:param a:
:param s:
: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./2*(-1+gamma))/special.gamma(gamma/2.) * 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