def _one_dim_lens_eq_calcs(args, phi):
"""Calculates intermediate quantities that are needed for several of the subsequent functions"""
b, t, y1, y2, q, gamma1, gamma2 = args
y = (y1 + y2 * 1j)
rhat = 1 / (1 - gamma1**2 - gamma2**2) * (
((1 + gamma1) * np.cos(phi) + gamma2 * np.sin(phi)) + 1j *
(gamma2 * np.cos(phi) + (1 - gamma1) * np.sin(phi)))
thetahat = 1 / (1 - gamma1**2 - gamma2**2) * (
((1 + gamma1) * np.sin(phi) - gamma2 * np.cos(phi)) + 1j *
(gamma2 * np.sin(phi) - (1 - gamma1) * np.cos(phi)))
frac_Roverrsh, phiell = pol_to_ell(1, phi, q)
Omega = omega(phiell, t, q)
const = 2 * b / (1 + q)
if abs(t - 1) > 1e-4:
b_over_r_pow_tm1 = -cdot(y, thetahat) / (const * cdot(Omega, thetahat))
R = b * np.abs(b_over_r_pow_tm1)**(1 /
(1 - t)) * np.sign(b_over_r_pow_tm1)
else:
Omega_ort = 1j * Omega
x = ((1 - gamma1) * np.cos(phi) -
gamma2 * np.sin(phi)) + 1j * (-gamma2 * np.cos(phi) +
(1 + gamma1) * np.sin(phi))
R = cdot(Omega_ort, y) / cdot(Omega_ort, x) * frac_Roverrsh
r, theta = ell_to_pol(R, phiell, q)
return Omega, const, phiell, q, r, rhat, t, b, thetahat, y
def _one_dim_lens_eq_unsmooth(phi, args):
"""Calculates the not-smooth 1-dimensional lens equation to to solve - to be used by a root-finder.
For some parameters and solutions, numerical issues make solving this one feasible while the other is not."""
Omega, const, phiell, q, r, rhat, t, b, thetahat, y = _one_dim_lens_eq_calcs(
args, phi)
rr, thetaa = ell_to_pol(1, phiell, q)
ip = cdot(y, rhat) * cdot(Omega, thetahat) - cdot(Omega, rhat) * cdot(
y, thetahat)
eq_notsmooth = ps(rr * b, 1 - t) * (cdot(y, thetahat) / const) * np.abs(
ip)**t + ip * np.abs(cdot(Omega, thetahat))**(+t)
return eq_notsmooth
def _one_dim_lens_eq(phi, args):
"""Calculates the smooth 1-dimensional lens equation to solve - to be used by a root-finder"""
Omega, const, phiell, q, r, rhat, t, b, thetahat, y = _one_dim_lens_eq_calcs(
args, phi)
rr, thetaa = ell_to_pol(1, phiell, q)
ip = cdot(y, rhat) * cdot(Omega, thetahat) - cdot(Omega, rhat) * cdot(
y, thetahat)
eq = (rr * b)**(2 / t - 2) * ps(
(cdot(y, thetahat) / const), 2 / t) * ip**2 + ps(ip, 2 / t) * cdot(
Omega, thetahat)**2
return eq
def _one_dim_lens_eq_both(phi, args):
"""Calculates and returns simultaneously both the smooth and the not-smooth 1-dimensional lens equation
that needs to be solved"""
Omega, const, phiell, q, r, rhat, t, b, thetahat, y = _one_dim_lens_eq_calcs(
args, phi)
rr, thetaa = ell_to_pol(1, phiell, q)
ip = cdot(y, rhat) * cdot(Omega, thetahat) - cdot(Omega, rhat) * cdot(
y, thetahat)
# The derivations are lost somewhere in my notes...
eq = (rr * b)**(2 / t - 2) * ps(
(cdot(y, thetahat) / const), 2 / t) * ip**2 + ps(ip, 2 / t) * cdot(
Omega, thetahat)**2
eq_notsmooth = ps(rr * b, 1 - t) * (cdot(y, thetahat) / const) * np.abs(
ip)**t + ip * np.abs(cdot(Omega, thetahat))**(+t)
return eq, eq_notsmooth