class TestMassAngleConversion(object):
"""
test angular to mass unit conversions
"""
def setup(self):
self.composite = CompositeSersicNFW()
self.sersic = Sersic()
self.nfw = NFW_ELLIPSE()
def test_convert(self):
theta_E = 1.
mass_light = 1 / 2.
Rs = 5.
n_sersic = 2.
r_eff = 0.7
theta_Rs, k_eff = self.composite.convert_mass(theta_E, mass_light, Rs,
n_sersic, r_eff)
alpha_E_sersic, _ = self.sersic.derivatives(theta_E,
0,
n_sersic,
r_eff,
k_eff=1)
alpha_E_nfw, _ = self.nfw.derivatives(theta_E,
0,
Rs,
theta_Rs=1,
q=1,
phi_G=0)
a = theta_Rs * alpha_E_nfw + (k_eff * alpha_E_sersic)
b = theta_Rs * alpha_E_nfw / (k_eff * alpha_E_sersic)
npt.assert_almost_equal(a, theta_E, decimal=10)
npt.assert_almost_equal(b, mass_light, decimal=10)
def test_function(self):
theta_E = 1.
mass_light = 1 / 2.
Rs = 5.
n_sersic = 2.
r_eff = 0.7
q, phi_G = 0.9, 0
q_s, phi_G_s = 0.7, 0.5
x, y, = 1, 1
f_ = self.composite.function(x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0)
npt.assert_almost_equal(f_, 1.1983595285200526, decimal=10)
def test_derivatives(self):
theta_E = 1.
mass_light = 1 / 2.
Rs = 5.
n_sersic = 2.
r_eff = 0.7
q, phi_G = 0.9, 0
q_s, phi_G_s = 0.7, 0.5
x, y, = 1, 1
f_x, f_y = self.composite.derivatives(x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0)
npt.assert_almost_equal(f_x, 0.54138666294863724, decimal=10)
npt.assert_almost_equal(f_y, 0.75841883763728535, decimal=10)
def test_hessian(self):
theta_E = 1.
mass_light = 1 / 2.
Rs = 5.
n_sersic = 2.
r_eff = 0.7
q, phi_G = 0.9, 0
q_s, phi_G_s = 0.7, 0.5
x, y, = 1, 1
f_xx, f_yy, f_xy = self.composite.hessian(x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0)
npt.assert_almost_equal(f_xx, 0.43275276043197586, decimal=10)
npt.assert_almost_equal(f_yy, 0.37688935994317774, decimal=10)
npt.assert_almost_equal(f_xy, -0.46895575042671389, decimal=10)
class TestNFWELLIPSE(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.nfw = NFW()
self.nfw_e = NFW_ELLIPSE()
def test_function(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
theta_Rs = 1.
q = 1.
phi_G = 0
values = self.nfw.function(x, y, Rs, theta_Rs)
values_e = self.nfw_e.function(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(values[0], values_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
q = .8
phi_G = 0
values = self.nfw_e.function(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(values[0], 0, decimal=4)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.nfw_e.function(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(values[0], 1.8827504143588476, decimal=5)
npt.assert_almost_equal(values[1], 2.6436373117941852, decimal=5)
npt.assert_almost_equal(values[2], 3.394127018818891, decimal=5)
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
theta_Rs = 1.
q = 1.
phi_G = 0
f_x, f_y = self.nfw.derivatives(x, y, Rs, theta_Rs)
f_x_e, f_y_e = self.nfw_e.derivatives(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(f_x[0], f_x_e[0], decimal=5)
npt.assert_almost_equal(f_y[0], f_y_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
theta_Rs = 0
f_x, f_y = self.nfw_e.derivatives(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(f_x[0], 0, decimal=5)
npt.assert_almost_equal(f_y[0], 0, decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
theta_Rs = 1.
q = .8
phi_G = 0
values = self.nfw_e.derivatives(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(values[0][0], 0.32458737284934414, decimal=5)
npt.assert_almost_equal(values[1][0], 0.9737621185480323, decimal=5)
npt.assert_almost_equal(values[0][1], 0.76249351329615234, decimal=5)
npt.assert_almost_equal(values[1][1], 0.38124675664807617, decimal=5)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
theta_Rs = 1.
q = 1.
phi_G = 0
f_xx, f_yy, f_xy = self.nfw.hessian(x, y, Rs, theta_Rs)
f_xx_e, f_yy_e, f_xy_e = self.nfw_e.hessian(x, y, Rs, theta_Rs, q,
phi_G)
npt.assert_almost_equal(f_xx[0], f_xx_e[0], decimal=5)
npt.assert_almost_equal(f_yy[0], f_yy_e[0], decimal=5)
npt.assert_almost_equal(f_xy[0], f_xy_e[0], decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
q = .8
phi_G = 0
values = self.nfw_e.hessian(x, y, Rs, theta_Rs, q, phi_G)
npt.assert_almost_equal(values[0][0], 0.26998576668768592, decimal=5)
npt.assert_almost_equal(values[1][0],
-0.0045328224507201753,
decimal=5)
npt.assert_almost_equal(values[2][0], -0.16380454531672584, decimal=5)
npt.assert_almost_equal(values[0][1], -0.014833136829928151, decimal=5)
npt.assert_almost_equal(values[1][1], 0.31399726446723619, decimal=5)
npt.assert_almost_equal(values[2][1], -0.13449884961325154, decimal=5)
class TestNFWELLIPSE(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.nfw = NFW()
self.nfw_e = NFW_ELLIPSE()
def test_function(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw.function(x, y, Rs, alpha_Rs)
values_e = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], values_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], 0, decimal=4)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], 1.8690403434928538, decimal=5)
npt.assert_almost_equal(values[1], 2.6186971904371217, decimal=5)
npt.assert_almost_equal(values[2], 3.360273255326431, decimal=5)
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
f_x, f_y = self.nfw.derivatives(x, y, Rs, alpha_Rs)
f_x_e, f_y_e = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_x[0], f_x_e[0], decimal=5)
npt.assert_almost_equal(f_y[0], f_y_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
alpha_Rs = 0
f_x, f_y = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_x[0], 0, decimal=5)
npt.assert_almost_equal(f_y[0], 0, decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
alpha_Rs = 1.
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0][0], 0.31473652125391116, decimal=5)
npt.assert_almost_equal(values[1][0], 0.9835516289184723, decimal=5)
npt.assert_almost_equal(values[0][1], 0.7525519008422061, decimal=5)
npt.assert_almost_equal(values[1][1], 0.39195411502198224, decimal=5)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
f_xx, f_xy, f_yx, f_yy = self.nfw.hessian(x, y, Rs, alpha_Rs)
f_xx_e, f_xy_e, f_yx_e, f_yy_e = self.nfw_e.hessian(
x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_xx[0], f_xx_e[0], decimal=5)
npt.assert_almost_equal(f_yy[0], f_yy_e[0], decimal=5)
npt.assert_almost_equal(f_xy[0], f_xy_e[0], decimal=5)
npt.assert_almost_equal(f_yx[0], f_yx_e[0], decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.hessian(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0][0], 0.26355306825820435, decimal=5)
npt.assert_almost_equal(values[3][0], -0.008064660050877137, decimal=5)
npt.assert_almost_equal(values[1][0], -0.159949276046234, decimal=5)
npt.assert_almost_equal(values[0][1], -0.01251554415659939, decimal=5)
npt.assert_almost_equal(values[3][1], 0.32051139520206107, decimal=5)
npt.assert_almost_equal(values[1][1], -0.13717027513848734, decimal=5)
def test_mass_3d_lens(self):
R = 1
Rs = 3
alpha_Rs = 1
m_3d = self.nfw_e.mass_3d_lens(R, Rs, alpha_Rs)
npt.assert_almost_equal(m_3d, 1.1573795105019022, decimal=8)
class TestNFWELLIPSE(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.nfw = NFW()
self.nfw_e = NFW_ELLIPSE()
def test_function(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw.function(x, y, Rs, alpha_Rs)
values_e = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], values_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], 0, decimal=4)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.nfw_e.function(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0], 1.8827504143588476, decimal=5)
npt.assert_almost_equal(values[1], 2.6436373117941852, decimal=5)
npt.assert_almost_equal(values[2], 3.394127018818891, decimal=5)
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
f_x, f_y = self.nfw.derivatives(x, y, Rs, alpha_Rs)
f_x_e, f_y_e = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_x[0], f_x_e[0], decimal=5)
npt.assert_almost_equal(f_y[0], f_y_e[0], decimal=5)
x = np.array([0])
y = np.array([0])
alpha_Rs = 0
f_x, f_y = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_x[0], 0, decimal=5)
npt.assert_almost_equal(f_y[0], 0, decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
alpha_Rs = 1.
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.derivatives(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0][0], 0.32458737284934414, decimal=5)
npt.assert_almost_equal(values[1][0], 0.9737621185480323, decimal=5)
npt.assert_almost_equal(values[0][1], 0.76249351329615234, decimal=5)
npt.assert_almost_equal(values[1][1], 0.38124675664807617, decimal=5)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
alpha_Rs = 1.
q = 1.
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
f_xx, f_xy, f_yx, f_yy = self.nfw.hessian(x, y, Rs, alpha_Rs)
f_xx_e, f_xy_e, f_yx_e, f_yy_e = self.nfw_e.hessian(
x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(f_xx[0], f_xx_e[0], decimal=5)
npt.assert_almost_equal(f_yy[0], f_yy_e[0], decimal=5)
npt.assert_almost_equal(f_xy[0], f_xy_e[0], decimal=5)
npt.assert_almost_equal(f_yx[0], f_yx_e[0], decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
q = .8
phi_G = 0
e1, e2 = param_util.phi_q2_ellipticity(phi_G, q)
values = self.nfw_e.hessian(x, y, Rs, alpha_Rs, e1, e2)
npt.assert_almost_equal(values[0][0], 0.26998576668768592, decimal=5)
npt.assert_almost_equal(values[3][0],
-0.0045328224507201753,
decimal=5)
npt.assert_almost_equal(values[1][0], -0.16380454531672584, decimal=5)
npt.assert_almost_equal(values[0][1], -0.014833136829928151, decimal=5)
npt.assert_almost_equal(values[3][1], 0.31399726446723619, decimal=5)
npt.assert_almost_equal(values[1][1], -0.13449884961325154, decimal=5)
def test_mass_3d_lens(self):
R = 1
Rs = 3
alpha_Rs = 1
m_3d = self.nfw_e.mass_3d_lens(R, Rs, alpha_Rs)
npt.assert_almost_equal(m_3d, 1.1573795105019022, decimal=8)
class TestNFWELLIPSE(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.tnfw = TNFW()
self.nfw_e = NFW_ELLIPSE()
self.tnfw_e = TNFW_ELLIPSE()
def test_function(self):
x = np.linspace(start=0.1, stop=10, num=10)
y = np.linspace(start=0.1, stop=10, num=10)
# test round case against TNFW
kwargs_tnfw_e_round = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 5,
'e1': 0.,
'e2': 0
}
kwargs_tnfw_round = {'Rs': 1, 'alpha_Rs': 0.1, 'r_trunc': 5}
f_e = self.tnfw_e.function(x, y, **kwargs_tnfw_e_round)
f_r = self.tnfw.function(x, y, **kwargs_tnfw_round)
npt.assert_almost_equal(f_e, f_r, decimal=5)
# test elliptical case with r_trunc -> infinity against NFW_ELLIPSE
kwargs_tnfw_e = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 500,
'e1': 0.2,
'e2': -0.01
}
kwargs_nfw_e = {'Rs': 1, 'alpha_Rs': 0.1, 'e1': 0.2, 'e2': -0.01}
f_te = self.tnfw_e.function(x, y, **kwargs_tnfw_e)
f_e = self.nfw_e.function(x, y, **kwargs_nfw_e)
npt.assert_almost_equal(f_te, f_e, decimal=3)
def test_derivatives(self):
x = np.linspace(start=0.1, stop=10, num=10)
y = np.linspace(start=0.1, stop=10, num=10)
# test round case against TNFW
kwargs_tnfw_e_round = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 5,
'e1': 0.,
'e2': 0
}
kwargs_tnfw_round = {'Rs': 1, 'alpha_Rs': 0.1, 'r_trunc': 5}
f_xe, f_ye = self.tnfw_e.derivatives(x, y, **kwargs_tnfw_e_round)
f_xr, f_yr = self.tnfw.derivatives(x, y, **kwargs_tnfw_round)
npt.assert_almost_equal(f_xe, f_xr, decimal=5)
npt.assert_almost_equal(f_ye, f_yr, decimal=5)
# test elliptical case with r_trunc -> infinity against NFW_ELLIPSE
kwargs_tnfw_e = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 500,
'e1': 0.2,
'e2': -0.01
}
kwargs_nfw_e = {'Rs': 1, 'alpha_Rs': 0.1, 'e1': 0.2, 'e2': -0.01}
out_te = self.tnfw_e.derivatives(x, y, **kwargs_tnfw_e)
out_e = self.nfw_e.derivatives(x, y, **kwargs_nfw_e)
npt.assert_almost_equal(out_te, out_e, decimal=3)
def test_hessian(self):
x = np.linspace(start=0.1, stop=10, num=10)
y = np.linspace(start=0.1, stop=10, num=10)
# test round case against TNFW
kwargs_tnfw_e_round = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 5,
'e1': 0.,
'e2': 0
}
kwargs_tnfw_round = {'Rs': 1, 'alpha_Rs': 0.1, 'r_trunc': 5}
out_e = self.tnfw_e.hessian(x, y, **kwargs_tnfw_e_round)
out_r = self.tnfw.hessian(x, y, **kwargs_tnfw_round)
npt.assert_almost_equal(out_e, out_r, decimal=4)
# test elliptical case with r_trunc -> infinity against NFW_ELLIPSE
kwargs_tnfw_e = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 500,
'e1': 0.2,
'e2': -0.01
}
kwargs_nfw_e = {'Rs': 1, 'alpha_Rs': 0.1, 'e1': 0.2, 'e2': -0.01}
out_te = self.tnfw_e.hessian(x, y, **kwargs_tnfw_e)
out_e = self.nfw_e.hessian(x, y, **kwargs_nfw_e)
npt.assert_almost_equal(out_te, out_e, decimal=3)
def test_mass_3d_lens(self):
with npt.assert_raises(ValueError):
kwargs_tnfw_e = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 5,
'e1': 0.1,
'e2': -0.02
}
self.tnfw_e.mass_3d_lens(1, **kwargs_tnfw_e)
def test_density_lens(self):
with npt.assert_raises(ValueError):
kwargs_tnfw_e = {
'Rs': 1,
'alpha_Rs': 0.1,
'r_trunc': 5,
'e1': 0.1,
'e2': -0.02
}
self.tnfw_e.density_lens(1, **kwargs_tnfw_e)
class CompositeSersicNFW(object):
"""
class for a composite model (Sersic and NFW profile combined)
with joint center and parameterization of Einstein radius
"""
def __init__(self):
self.sersic = SersicEllipse()
self.nfw = NFW_ELLIPSE()
def function(self,
x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0):
"""
:param theta_E:
:param mass_light:
:param Rs:
:param q:
:param phi_G:
:param n_sersic:
:param r_eff:
:param center_x:
:param center_y:
:return:
"""
theta_Rs, k_eff = self.convert_mass(theta_E, mass_light, Rs, n_sersic,
r_eff)
f_s = self.sersic.function(x, y, n_sersic, r_eff, k_eff, q_s, phi_G_s,
center_x, center_y)
f_nfw = self.nfw.function(x, y, Rs, theta_Rs, q, phi_G, center_x,
center_y)
return f_s + f_nfw
def derivatives(self,
x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0):
"""
:param theta_E:
:param mass_light:
:param Rs:
:param q:
:param phi_G:
:param n_sersic:
:param r_eff:
:param center_x:
:param center_y:
:return:
"""
theta_Rs, k_eff = self.convert_mass(theta_E, mass_light, Rs, n_sersic,
r_eff)
f_x_s, f_y_s = self.sersic.derivatives(x, y, n_sersic, r_eff, k_eff,
q_s, phi_G_s, center_x,
center_y)
f_x_nfw, f_y_nfw = self.nfw.derivatives(x, y, Rs, theta_Rs, q, phi_G,
center_x, center_y)
return f_x_s + f_x_nfw, f_y_s + f_y_nfw
def hessian(self,
x,
y,
theta_E,
mass_light,
Rs,
q,
phi_G,
n_sersic,
r_eff,
q_s,
phi_G_s,
center_x=0,
center_y=0):
"""
:param theta_E:
:param mass_light:
:param Rs:
:param q:
:param phi_G:
:param n_sersic:
:param r_eff:
:param center_x:
:param center_y:
:return:
"""
theta_Rs, k_eff = self.convert_mass(theta_E, mass_light, Rs, n_sersic,
r_eff)
f_xx_s, f_yy_s, f_xy_s = self.sersic.hessian(x, y, n_sersic, r_eff,
k_eff, q_s, phi_G_s,
center_x, center_y)
f_xx_nfw, f_yy_nfw, f_xy_nfw = self.nfw.hessian(
x, y, Rs, theta_Rs, q, phi_G, center_x, center_y)
return f_xx_s + f_xx_nfw, f_yy_s + f_yy_nfw, f_xy_s + f_xy_nfw
def convert_mass(self, theta_E, mass_light, Rs, n_sersic, r_eff):
"""
convert global parameters theta_E and mass_light to specific ones theta_Rs and k_eff
:param theta_E:
:param mass_light:
:param Rs:
:param n_sersic:
:param r_eff:
:return:
"""
if theta_E < 0.0000001:
return 0, 0
alpha_E_sersic, _ = self.sersic.derivatives(theta_E,
0,
n_sersic,
r_eff,
k_eff=1,
q=1,
phi_G=0)
alpha_E_nfw, _ = self.nfw.derivatives(theta_E,
0,
Rs,
theta_Rs=1,
q=1,
phi_G=0)
#f_xx_s, f_yy_s, _ = self.sersic.hessian(r_eff, 0, n_sersic, r_eff, k_eff=1)
#f_xx_n, f_yy_n, _ = self.nfw.hessian(r_eff, 0, Rs, theta_Rs=1, q=1, phi_G=0)
#kappa_eff_sersic = (f_xx_s + f_yy_s) / 2.
#kappa_eff_nfw = (f_xx_n + f_yy_n) / 2.
# equations must satisfy:
# theta_Rs * alpha_E_nfw + k_eff * alpha_E_sersic = theta_E
# theta_Rs * kappa_eff_nfw / (k_eff * kappa_eff_sersic) = mass_light
#k_eff = theta_E * kappa_eff_nfw / (alpha_E_sersic*kappa_eff_nfw+mass_light*alpha_E_nfw*kappa_eff_sersic)
#theta_Rs = (theta_E - k_eff*alpha_E_sersic)/alpha_E_nfw
theta_Rs = theta_E / alpha_E_nfw / (1 + 1. / mass_light)
k_eff = (theta_E - theta_Rs * alpha_E_nfw) / alpha_E_sersic
return theta_Rs, k_eff