def alpha(x, y, b, q, t):
"""
Converts the parameters from lenstronomy definitions (as defined in PEMD) to the definitions of Tessore+(2015)
:param x: x-coordinate (angle)
:param y: y-coordinate (angle)
:param theta_E: Einstein radius (angle), pay attention to specific definition!
:param e1: eccentricity component
:param e2: eccentricity component
:param gamma: logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
:param center_x: x-position of lens center
:param center_y: y-position of lens center
:return: complex derotated coordinate, rescaled Einstein radius, powerlaw index, elliptical axis ratio and angle
"""
zz = x * q + 1j * y
R = np.abs(zz)
phi = np.angle(zz)
Omega = omega(phi, t, q)
alph = (2 * b) / (1 + q) * nan_to_num((b / R)**t * R / b) * Omega
return alph
def alpha(x, y, b, q, t, Omega=None):
"""
Calculates the complex deflection
:param x: x-coordinate (angle)
:param y: y-coordinate (angle)
:param b: Einstein radius (angle), pay attention to specific definition!
:param q: axis ratio
:param t: logarithmic power-law slope. Is t=gamma-1
:param Omega: If given, use this Omega (to avoid recalculations)
:return: complex deflection angle
"""
zz = x * q + 1j * y
R = np.abs(zz)
phi = np.angle(zz)
if Omega is None:
Omega = omega(phi, t, q)
# Omega = omega(phi, t, q)
alph = (2 * b) / (1 + q) * nan_to_num((b / R)**t * R / b) * Omega
return alph
def hessian(x, y, theta_E, gamma, e1, e2, center_x=0., center_y=0.):
"""
:param x: x-coordinate (angle)
:param y: y-coordinate (angle)
:param theta_E: Einstein radius (angle), pay attention to specific definition!
:param gamma: logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
:param e1: eccentricity component
:param e2: eccentricity component
:param center_x: x-position of lens center
:param center_y: y-position of lens center
:return: Hessian components f_xx, f_yy, f_xy
"""
z, b, t, q, ang_ell = param_transform(x, y, theta_E, gamma, e1, e2,
center_x, center_y)
ang = np.angle(z)
r = np.abs(z)
zz_ell = z.real * q + 1j * z.imag
R = np.abs(zz_ell)
phi = np.angle(zz_ell)
u = nan_to_num(
(b / R)**t
) # I remove all factors of (b/R)**t to only have to remove nans once
kappa = (2 - t) / 2
Roverr = np.sqrt(np.cos(ang)**2 * q**2 + np.sin(ang)**2)
Omega = omega(phi, t, q)
alph = (2 * b) / (1 + q) / b * Omega
gamma_shear = -np.exp(2j * (ang + ang_ell)) * kappa + (1 - t) * np.exp(
1j * (ang + 2 * ang_ell)) * alph * Roverr
f_xx = (kappa + gamma_shear.real) * u
f_yy = (kappa - gamma_shear.real) * u
f_xy = gamma_shear.imag * u
# Fix the nans if x=y=0 is filled in
return f_xx, f_xy, f_xy, f_yy