class TestHernquistEllipseCSE(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.hernquist = Hernquist()
self.hernquist_cse = HernquistEllipseCSE()
def test_function(self):
x = np.linspace(0.01, 2, 10)
y = np.zeros_like(x)
kwargs = {'sigma0': 2, 'Rs': 2, 'center_x': 0, 'center_y': 0}
f_nfw = self.hernquist.function(x, y, **kwargs)
f_cse = self.hernquist_cse.function(x, y, e1=0, e2=0, **kwargs)
npt.assert_almost_equal(f_cse / f_nfw, 1, decimal=5)
def test_derivatives(self):
x = np.linspace(0.01, 2, 10)
y = np.zeros_like(x)
kwargs = {'sigma0': 0.5, 'Rs': 2, 'center_x': 0, 'center_y': 0}
f_x_nfw, f_y_nfw = self.hernquist.derivatives(x, y, **kwargs)
f_x_cse, f_y_cse = self.hernquist_cse.derivatives(x, y, e1=0, e2=0, **kwargs)
npt.assert_almost_equal(f_x_cse, f_x_nfw, decimal=5)
npt.assert_almost_equal(f_y_cse, f_y_nfw, decimal=5)
def test_hessian(self):
x = np.linspace(0.01, 5, 30)
y = np.zeros_like(x)
kwargs = {'sigma0': 0.5, 'Rs': 2, 'center_x': 0, 'center_y': 0}
f_xx_nfw, f_xy_nfw, f_yx_nfw, f_yy_nfw = self.hernquist.hessian(x, y, **kwargs)
f_xx_cse, f_xy_cse, f_yx_cse, f_yy_cse = self.hernquist_cse.hessian(x, y, e1=0, e2=0, **kwargs)
npt.assert_almost_equal(f_xx_cse / f_xx_nfw, 1, decimal=2)
npt.assert_almost_equal(f_xy_cse, f_xy_nfw, decimal=5)
npt.assert_almost_equal(f_yx_cse, f_yx_nfw, decimal=5)
npt.assert_almost_equal(f_yy_cse, f_yy_nfw, decimal=5)
def test_mass_3d_lens(self):
R = 1
Rs = 3
alpha_Rs = 1
m_3d_nfw = self.hernquist.mass_3d_lens(R, Rs, alpha_Rs)
m_3d_cse = self.hernquist_cse.mass_3d_lens(R, Rs, alpha_Rs)
npt.assert_almost_equal(m_3d_nfw, m_3d_cse, decimal=8)
class TestHernquist(object):
def setup(self):
self.profile = Hernquist()
def test_function(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
sigma0 = 0.5
values = self.profile.function(x, y, sigma0, Rs)
npt.assert_almost_equal(values[0], 0.66514613455415028, decimal=8)
x = np.array([0])
y = np.array([0])
Rs = 1.
sigma0 = 0.5
values = self.profile.function(x, y, sigma0, Rs)
npt.assert_almost_equal(values[0], 0, decimal=6)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.profile.function(x, y, sigma0, Rs)
npt.assert_almost_equal(values[0], 0.66514613455415028, decimal=8)
npt.assert_almost_equal(values[1], 0.87449395673649566, decimal=8)
npt.assert_almost_equal(values[2], 1.0549139073851708, decimal=8)
def test_derivatives(self):
x = 1
y = 2
Rs = 1.
sigma0 = 0.5
f_x, f_y = self.profile.derivatives(x, y, sigma0, Rs)
npt.assert_almost_equal(f_x, 0.11160641027573866, decimal=8)
npt.assert_almost_equal(f_y, 0.22321282055147731, decimal=8)
x = np.array([0])
y = np.array([0])
f_x, f_y = self.profile.derivatives(x, y, sigma0, Rs)
npt.assert_almost_equal(f_x, 0, decimal=8)
npt.assert_almost_equal(f_y, 0, decimal=8)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
Rs = 1.
sigma0 = 0.5
f_xx, f_xy, f_yx, f_yy = self.profile.hessian(x, y, sigma0, Rs)
npt.assert_almost_equal(f_xx[0], 0.0779016004481825, decimal=6)
npt.assert_almost_equal(f_yy[0], -0.023212809452388683, decimal=6)
npt.assert_almost_equal(f_xy[0], -0.0674096084507525, decimal=6)
npt.assert_almost_equal(f_xy, f_yx, decimal=8)
def test_mass_tot(self):
rho0 = 1
Rs = 3
m_tot = self.profile.mass_tot(rho0, Rs)
npt.assert_almost_equal(m_tot, 169.64600329384882, decimal=6)
def test_grav_pot(self):
x, y = 1, 0
rho0 = 1
Rs = 3
grav_pot = self.profile.grav_pot(x,
y,
rho0,
Rs,
center_x=0,
center_y=0)
npt.assert_almost_equal(grav_pot, 42.411500823462205, decimal=8)
def test_sigma0_definition(self):
Rs = 2.
sigma0 = 0.5
f_x, f_y = self.profile.derivatives(Rs, 0, sigma0, Rs)
alpha = f_x
npt.assert_almost_equal(alpha, 2 / 3. * sigma0 * Rs, decimal=5)
class Hernquist_Ellipse(object):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
def __init__(self):
self.spherical = Hernquist()
self._diff = 0.000001
def function(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
return f_
def derivatives(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
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, 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, Rs, q, phi_G,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Rs,
q, phi_G, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, 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
class Hernquist_Ellipse(LensProfileBase):
"""
this class contains functions for the elliptical Hernquist profile. Ellipticity is defined in the potential.
"""
param_names = ['sigma0', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'sigma0': 0,
'Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'sigma0': 100,
'Rs': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = Hernquist()
self._diff = 0.0000000001
super(Hernquist_Ellipse, self).__init__()
def function(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
return f_
def derivatives(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
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, 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, Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Rs,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, 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 density(self, r, rho0, Rs, e1=0, e2=0):
"""
computes the 3-d density
:param r: 3-d radius
:param rho0: density normalization
:param Rs: Hernquist radius
:return: density at radius r
"""
return self.spherical.density(r, rho0, Rs)
def density_lens(self, r, sigma0, Rs, e1=0, e2=0):
"""
Density as a function of 3d radius in lensing parameters
This function converts the lensing definition sigma0 into the 3d density
:param r: 3d radius
:param sigma0: rho0 * Rs (units of projected density)
:param Rs: Hernquist radius
:return: enclosed mass in 3d
"""
return self.spherical.density_lens(r, sigma0, Rs)
def density_2d(self, x, y, rho0, Rs, e1=0, e2=0, center_x=0, center_y=0):
"""
projected density along the line of sight at coordinate (x, y)
:param x: x-coordinate
:param y: y-coordinate
:param rho0: density normalization
:param Rs: Hernquist radius
:param center_x: x-center of the profile
:param center_y: y-center of the profile
:return: projected density
"""
return self.spherical.density_2d(x, y, rho0, Rs, center_x, center_y)
def mass_2d_lens(self, r, sigma0, Rs, e1=0, e2=0):
"""
mass enclosed projected 2d sphere of radius r
Same as mass_2d but with input normalization in units of projected density
:param r: projected radius
:param sigma0: rho0 * Rs (units of projected density)
:param Rs: Hernquist radius
:return: mass enclosed 2d projected radius
"""
return self.spherical.mass_2d_lens(r, sigma0, Rs)
def mass_2d(self, r, rho0, Rs, e1=0, e2=0):
"""
mass enclosed projected 2d sphere of radius r
:param r: projected radius
:param rho0: density normalization
:param Rs: Hernquist radius
:return: mass enclosed 2d projected radius
"""
return self.spherical.mass_2d(r, rho0, Rs)
def mass_3d(self, r, rho0, Rs, e1=0, e2=0):
"""
mass enclosed a 3d sphere or radius r
:param r: 3-d radius within the mass is integrated (same distance units as density definition)
:param rho0: density normalization
:param Rs: Hernquist radius
:return: enclosed mass
"""
return self.spherical.mass_3d(r, rho0, Rs)
class Hernquist_Ellipse(object):
"""
this class contains functions concerning the NFW profile
relation are: R_200 = c * Rs
"""
param_names = ['sigma0', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'sigma0': 0,
'Rs': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'sigma0': 100,
'Rs': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = Hernquist()
self._diff = 0.0000000001
def function(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
return f_
def derivatives(self, x, y, sigma0, 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 = abs(1 - q)
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, Rs)
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, 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, Rs, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, sigma0, Rs,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, sigma0, 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