class PJaffe_Ellipse(LensProfileBase):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
param_names = ['sigma0', 'Ra', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'sigma0': 0,
'Ra': 0,
'Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'sigma0': 10,
'Ra': 100,
'Rs': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = PJaffe()
self._diff = 0.000001
super(PJaffe_Ellipse, self).__init__()
def function(self, x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_ = self.spherical.function(x_, y_, sigma0, Ra, Rs)
return f_
def derivatives(self,
x,
y,
sigma0,
Ra,
Rs,
e1,
e2,
center_x=0,
center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_x_prim, f_y_prim = self.spherical.derivatives(x_,
y_,
sigma0,
Ra,
Rs,
center_x=0,
center_y=0)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
"""
alpha_ra, alpha_dec = self.derivatives(x, y, sigma0, Ra, Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Ra,
Rs, e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, Ra,
Rs, e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
#f_yx = (alpha_dec_dx - alpha_dec)/diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_yy, f_xy
def mass_3d_lens(self, r, sigma0, Ra, Rs, e1=0, e2=0):
"""
:param r:
:param sigma0:
:param Ra:
:param Rs:
:param q:
:param phi_G:
:return:
"""
return self.spherical.mass_3d_lens(r, sigma0, Ra, Rs)
class PJaffe_Ellipse(LensProfileBase):
"""
class to compute the DUAL PSEUDO ISOTHERMAL ELLIPTICAL MASS DISTRIBUTION
based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A
with the ellipticity implemented in the potential
Module name: 'PJAFFE_ELLIPSE';
An alternative name is dPIED.
The 3D density distribution is
.. math::
\\rho(r) = \\frac{\\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)}
with :math:`Rs > Ra`.
The projected density is
.. math::
\\Sigma(R) = \\Sigma_0 \\frac{Ra Rs}{Rs-Ra}\\left(\\frac{1}{\\sqrt{Ra^2+R^2}} - \\frac{1}{\\sqrt{Rs^2+R^2}} \\right)
with
.. math::
\\Sigma_0 = \\pi \\rho_0 \\frac{Ra Rs}{Rs + Ra}
In the lensing parameterization,
.. math::
\\sigma_0 = \\frac{\\Sigma_0}{\\Sigma_{\\rm crit}}
"""
param_names = ['sigma0', 'Ra', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'sigma0': 0,
'Ra': 0,
'Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'sigma0': 10,
'Ra': 100,
'Rs': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = PJaffe()
self._diff = 0.000001
super(PJaffe_Ellipse, self).__init__()
def function(self, x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
x_, y_ = param_util.transform_e1e2_square_average(
x, y, e1, e2, center_x, center_y)
f_ = self.spherical.function(x_, y_, sigma0, Ra, Rs)
return f_
def derivatives(self,
x,
y,
sigma0,
Ra,
Rs,
e1,
e2,
center_x=0,
center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_, y_ = param_util.transform_e1e2_square_average(
x, y, e1, e2, center_x, center_y)
e = param_util.q2e(q)
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
f_x_prim, f_y_prim = self.spherical.derivatives(x_,
y_,
sigma0,
Ra,
Rs,
center_x=0,
center_y=0)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
alpha_ra, alpha_dec = self.derivatives(x, y, sigma0, Ra, Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Ra,
Rs, e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, Ra,
Rs, e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
def mass_3d_lens(self, r, sigma0, Ra, Rs, e1=0, e2=0):
"""
:param r:
:param sigma0:
:param Ra:
:param Rs:
:param e1:
:param e2:
:return:
"""
return self.spherical.mass_3d_lens(r, sigma0, Ra, Rs)
class PJaffe_Ellipse(object):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
def __init__(self):
self.spherical = PJaffe()
self._diff = 0.000001
def function(self, x, y, sigma0, Ra, Rs, q, phi_G, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_ = self.spherical.function(x_, y_, sigma0, Ra, Rs)
return f_
def derivatives(self,
x,
y,
sigma0,
Ra,
Rs,
q,
phi_G,
center_x=0,
center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = min(abs(1. - q), 0.99)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_x_prim, f_y_prim = self.spherical.derivatives(x_,
y_,
sigma0,
Ra,
Rs,
center_x=0,
center_y=0)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, sigma0, Ra, Rs, q, phi_G, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
"""
alpha_ra, alpha_dec = self.derivatives(x, y, sigma0, Ra, Rs, q, phi_G,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Ra,
Rs, q, phi_G, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, Ra,
Rs, q, phi_G, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
#f_yx = (alpha_dec_dx - alpha_dec)/diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_yy, f_xy
def mass_3d_lens(self, r, sigma0, Ra, Rs, q=1, phi_G=0):
"""
:param r:
:param sigma0:
:param Ra:
:param Rs:
:param q:
:param phi_G:
:return:
"""
return self.spherical.mass_3d_lens(r, sigma0, Ra, Rs)
class TestP_JAFFW(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.profile = PJaffe()
def test_function(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.function(x, y, sigma0, Ra, Rs)
assert values[0] == 0.87301557036070054
x = np.array([0])
y = np.array([0])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.function(x, y, sigma0, Ra, Rs)
assert values[0] == 0.20267440905756931
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.profile.function(x, y, sigma0, Ra, Rs)
assert values[0] == 0.87301557036070054
assert values[1] == 1.0842781309377669
assert values[2] == 1.2588604178849985
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
f_x, f_y = self.profile.derivatives(x, y, sigma0, Ra, Rs)
assert f_x[0] == 0.11542369603751264
assert f_y[0] == 0.23084739207502528
x = np.array([0])
y = np.array([0])
f_x, f_y = self.profile.derivatives(x, y, sigma0, Ra, Rs)
assert f_x[0] == 0
assert f_y[0] == 0
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.profile.derivatives(x, y, sigma0, Ra, Rs)
assert values[0][0] == 0.11542369603751264
assert values[1][0] == 0.23084739207502528
assert values[0][1] == 0.19172866612512479
assert values[1][1] == 0.063909555375041588
def test_hessian(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
f_xx, f_yy, f_xy = self.profile.hessian(x, y, sigma0, Ra, Rs)
assert f_xx[0] == 0.077446121589827679
assert f_yy[0] == -0.036486601753227141
assert f_xy[0] == -0.075955148895369876
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.profile.hessian(x, y, sigma0, Ra, Rs)
assert values[0][0] == 0.077446121589827679
assert values[1][0] == -0.036486601753227141
assert values[2][0] == -0.075955148895369876
assert values[0][1] == -0.037260794616683197
assert values[1][1] == 0.052668405375961035
assert values[2][1] == -0.033723449997241584
def test_mass_tot(self):
rho0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.mass_tot(rho0, Ra, Rs)
npt.assert_almost_equal(values, 2.429441083345073, decimal=10)
def test_mass_3d_lens(self):
mass = self.profile.mass_3d_lens(r=1, sigma0=1, Ra=0.5, Rs=0.8)
npt.assert_almost_equal(mass, 0.87077306005349242, decimal=8)
def test_grav_pot(self):
x = 1
y = 2
rho0 = 1.
Ra, Rs = 0.5, 0.8
grav_pot = self.profile.grav_pot(x,
y,
rho0,
Ra,
Rs,
center_x=0,
center_y=0)
npt.assert_almost_equal(grav_pot, 0.89106542283974155, decimal=10)
class TestP_JAFFW(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.profile = PJaffe()
def test_function(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.function(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(values[0], 0.87301557036070054, decimal=8)
x = np.array([0])
y = np.array([0])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.function(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(values[0], 0.20267440905756931, decimal=8)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.profile.function(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(values[0], 0.87301557036070054, decimal=8)
npt.assert_almost_equal(values[1], 1.0842781309377669, decimal=8)
npt.assert_almost_equal(values[2], 1.2588604178849985, decimal=8)
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
f_x, f_y = self.profile.derivatives(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(f_x[0], 0.11542369603751264, decimal=8)
npt.assert_almost_equal(f_y[0], 0.23084739207502528, decimal=8)
x = np.array([0])
y = np.array([0])
f_x, f_y = self.profile.derivatives(x, y, sigma0, Ra, Rs)
assert f_x[0] == 0
assert f_y[0] == 0
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.profile.derivatives(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(values[0][0], 0.11542369603751264, decimal=8)
npt.assert_almost_equal(values[1][0], 0.23084739207502528, decimal=8)
npt.assert_almost_equal(values[0][1], 0.19172866612512479, decimal=8)
npt.assert_almost_equal(values[1][1], 0.063909555375041588, decimal=8)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
sigma0 = 1.
Ra, Rs = 0.5, 0.8
f_xx, f_xy, f_yx, f_yy = self.profile.hessian(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(f_xx[0], 0.077446121589827679, decimal=8)
npt.assert_almost_equal(f_yy[0], -0.036486601753227141, decimal=8)
npt.assert_almost_equal(f_xy[0], -0.075955148895369876, decimal=8)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.profile.hessian(x, y, sigma0, Ra, Rs)
npt.assert_almost_equal(values[0][0], 0.077446121589827679, decimal=8)
npt.assert_almost_equal(values[3][0], -0.036486601753227141, decimal=8)
npt.assert_almost_equal(values[1][0], values[2][0], decimal=8)
def test_mass_tot(self):
rho0 = 1.
Ra, Rs = 0.5, 0.8
values = self.profile.mass_tot(rho0, Ra, Rs)
npt.assert_almost_equal(values, 2.429441083345073, decimal=10)
def test_mass_3d_lens(self):
mass = self.profile.mass_3d_lens(r=1, sigma0=1, Ra=0.5, Rs=0.8)
npt.assert_almost_equal(mass, 0.87077306005349242, decimal=8)
def test_grav_pot(self):
x = 1
y = 2
rho0 = 1.
r = np.sqrt(x**2 + y**2)
Ra, Rs = 0.5, 0.8
grav_pot = self.profile.grav_pot(r, rho0, Ra, Rs)
npt.assert_almost_equal(grav_pot, 0.89106542283974155, decimal=10)