def mikkola_coe(k, p, ecc, inc, raan, argp, nu, tof):
a = p / (1 - ecc**2)
n = np.sqrt(k / np.abs(a)**3)
# Solve for specific geometrical case
if ecc < 1.0:
# Equation (9a)
alpha = (1 - ecc) / (4 * ecc + 1 / 2)
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
else:
alpha = (ecc - 1) / (4 * ecc + 1 / 2)
M0 = F_to_M(nu_to_F(nu, ecc), ecc)
M = M0 + n * tof
beta = M / 2 / (4 * ecc + 1 / 2)
# Equation (9b)
if beta >= 0:
z = (beta + np.sqrt(beta**2 + alpha**3))**(1 / 3)
else:
z = (beta - np.sqrt(beta**2 + alpha**3))**(1 / 3)
s = z - alpha / z
# Apply initial correction
if ecc < 1.0:
ds = -0.078 * s**5 / (1 + ecc)
else:
ds = 0.071 * s**5 / (1 + 0.45 * s**2) / (1 + 4 * s**2) / ecc
s += ds
# Solving for the true anomaly
if ecc < 1.0:
E = M + ecc * (3 * s - 4 * s**3)
f = E - ecc * np.sin(E) - M
f1 = 1.0 - ecc * np.cos(E)
f2 = ecc * np.sin(E)
f3 = ecc * np.cos(E)
f4 = -f2
f5 = -f3
else:
E = 3 * np.log(s + np.sqrt(1 + s**2))
f = -E + ecc * np.sinh(E) - M
f1 = -1.0 + ecc * np.cosh(E)
f2 = ecc * np.sinh(E)
f3 = ecc * np.cosh(E)
f4 = f2
f5 = f3
# Apply Taylor expansion
u1 = -f / f1
u2 = -f / (f1 + 0.5 * f2 * u1)
u3 = -f / (f1 + 0.5 * f2 * u2 + (1.0 / 6.0) * f3 * u2**2)
u4 = -f / (f1 + 0.5 * f2 * u3 + (1.0 / 6.0) * f3 * u3**2 +
(1.0 / 24.0) * f4 * (u3**3))
u5 = -f / (f1 + f2 * u4 / 2 + f3 * (u4 * u4) / 6.0 + f4 *
(u4 * u4 * u4) / 24.0 + f5 * (u4 * u4 * u4 * u4) / 120.0)
E += u5
if ecc < 1.0:
nu = E_to_nu(E, ecc)
else:
if ecc == 1.0:
# Parabolic
nu = D_to_nu(E)
else:
# Hyperbolic
nu = F_to_nu(E, ecc)
return nu
def danby_coe(k, p, ecc, inc, raan, argp, nu, tof, numiter=20, rtol=1e-8):
semi_axis_a = p / (1 - ecc**2)
n = np.sqrt(k / np.abs(semi_axis_a)**3)
if ecc == 0:
# Solving for circular orbit
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
M = M0 + n * tof
nu = M - 2 * np.pi * np.floor(M / 2 / np.pi)
return nu
elif ecc < 1.0:
# For elliptical orbit
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
M = M0 + n * tof
xma = M - 2 * np.pi * np.floor(M / 2 / np.pi)
E = xma + 0.85 * np.sign(np.sin(xma)) * ecc
else:
# For parabolic and hyperbolic
M0 = F_to_M(nu_to_F(nu, ecc), ecc)
M = M0 + n * tof
xma = M - 2 * np.pi * np.floor(M / 2 / np.pi)
E = np.log(2 * xma / ecc + 1.8)
# Iterations begin
n = 0
while n <= numiter:
if ecc < 1.0:
s = ecc * np.sin(E)
c = ecc * np.cos(E)
f = E - s - xma
fp = 1 - c
fpp = s
fppp = c
else:
s = ecc * np.sinh(E)
c = ecc * np.cosh(E)
f = s - E - xma
fp = c - 1
fpp = s
fppp = c
if np.abs(f) <= rtol:
if ecc < 1.0:
sta = np.sqrt(1 - ecc**2) * np.sin(E)
cta = np.cos(E) - ecc
else:
sta = np.sqrt(ecc**2 - 1) * np.sinh(E)
cta = ecc - np.cosh(E)
nu = np.arctan2(sta, cta)
break
else:
delta = -f / fp
delta_star = -f / (fp + 0.5 * delta * fpp)
deltak = -f / (fp + 0.5 * delta_star * fpp +
delta_star**2 * fppp / 6)
E = E + deltak
n += 1
else:
raise ValueError("Maximum number of iterations has been reached.")
return nu
def delta_t_from_nu(nu, ecc, k=1.0, q=1.0, delta=1e-2):
"""Time elapsed since periapsis for given true anomaly.
Parameters
----------
nu : float
True anomaly.
ecc : float
Eccentricity.
k : float
Gravitational parameter.
q : float
Periapsis distance.
delta : float
Parameter that controls the size of the near parabolic region.
Returns
-------
delta_t : float
Time elapsed since periapsis.
"""
assert -np.pi <= nu < np.pi
if ecc < 1 - delta:
# Strong elliptic
E = nu_to_E(nu, ecc) # (-pi, pi]
M = E_to_M(E, ecc) # (-pi, pi]
n = np.sqrt(k * (1 - ecc)**3 / q**3)
elif 1 - delta <= ecc < 1:
E = nu_to_E(nu, ecc) # (-pi, pi]
if delta <= 1 - ecc * np.cos(E):
# Strong elliptic
M = E_to_M(E, ecc) # (-pi, pi]
n = np.sqrt(k * (1 - ecc)**3 / q**3)
else:
# Near parabolic
D = nu_to_D(nu) # (-∞, ∞)
# If |nu| is far from pi this result is bounded
# because the near parabolic region shrinks in its vicinity,
# otherwise the eccentricity is very close to 1
# and we are really far away
M = D_to_M_near_parabolic(D, ecc)
n = np.sqrt(k / (2 * q**3))
elif ecc == 1:
# Parabolic
D = nu_to_D(nu) # (-∞, ∞)
M = D_to_M(D) # (-∞, ∞)
n = np.sqrt(k / (2 * q**3))
elif 1 + ecc * np.cos(nu) < 0:
# Unfeasible region
return np.nan
elif 1 < ecc <= 1 + delta:
# NOTE: Do we need to wrap nu here?
# For hyperbolic orbits, it should anyway be in
# (-arccos(-1 / ecc), +arccos(-1 / ecc))
F = nu_to_F(nu, ecc) # (-∞, ∞)
if delta <= ecc * np.cosh(F) - 1:
# Strong hyperbolic
M = F_to_M(F, ecc) # (-∞, ∞)
n = np.sqrt(k * (ecc - 1)**3 / q**3)
else:
# Near parabolic
D = nu_to_D(nu) # (-∞, ∞)
M = D_to_M_near_parabolic(D, ecc) # (-∞, ∞)
n = np.sqrt(k / (2 * q**3))
elif 1 + delta < ecc:
# Strong hyperbolic
F = nu_to_F(nu, ecc) # (-∞, ∞)
M = F_to_M(F, ecc) # (-∞, ∞)
n = np.sqrt(k * (ecc - 1)**3 / q**3)
else:
raise RuntimeError
return M / n