def gooding_coe(k, p, ecc, inc, raan, argp, nu, tof, numiter=150, rtol=1e-8):
# TODO: parabolic and hyperbolic not implemented cases
if ecc >= 1.0:
raise NotImplementedError(
"Parabolic/Hyperbolic cases still not implemented in gooding.")
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
semi_axis_a = p / (1 - ecc**2)
n = np.sqrt(k / np.abs(semi_axis_a)**3)
M = M0 + n * tof
# Start the computation
n = 0
c = ecc * np.cos(M)
s = ecc * np.sin(M)
psi = s / np.sqrt(1 - 2 * c + ecc**2)
f = 1.0
while f**2 >= rtol and n <= numiter:
xi = np.cos(psi)
eta = np.sin(psi)
fd = (1 - c * xi) + s * eta
fdd = c * eta + s * xi
f = psi - fdd
psi = psi - f * fd / (fd**2 - 0.5 * f * fdd)
n += 1
E = M + psi
return E_to_nu(E, ecc)
def markley_coe(k, p, ecc, inc, raan, argp, nu, tof):
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
a = p / (1 - ecc**2)
n = np.sqrt(k / a**3)
M = M0 + n * tof
# Range between -pi and pi
M = (M + np.pi) % (2 * np.pi) - np.pi
# Equation (20)
alpha = (3 * np.pi**2 + 1.6 * (np.pi - np.abs(M)) /
(1 + ecc)) / (np.pi**2 - 6)
# Equation (5)
d = 3 * (1 - ecc) + alpha * ecc
# Equation (9)
q = 2 * alpha * d * (1 - ecc) - M**2
# Equation (10)
r = 3 * alpha * d * (d - 1 + ecc) * M + M**3
# Equation (14)
w = (np.abs(r) + np.sqrt(q**3 + r**2))**(2 / 3)
# Equation (15)
E = (2 * r * w / (w**2 + w * q + q**2) + M) / d
# Equation (26)
f0 = _kepler_equation(E, M, ecc)
f1 = _kepler_equation_prime(E, M, ecc)
f2 = ecc * np.sin(E)
f3 = ecc * np.cos(E)
f4 = -f2
# Equation (22)
delta3 = -f0 / (f1 - 0.5 * f0 * f2 / f1)
delta4 = -f0 / (f1 + 0.5 * delta3 * f2 + 1 / 6 * delta3**2 * f3)
delta5 = -f0 / (f1 + 0.5 * delta4 * f2 + 1 / 6 * delta4**2 * f3 +
1 / 24 * delta4**3 * f4)
E += delta5
nu = E_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 pimienta_coe(k, p, ecc, inc, raan, argp, nu, tof):
q = p / (1 + ecc)
# TODO: Do something to increase parabolic accuracy?
n = np.sqrt(k * (1 - ecc)**3 / q**3)
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
M = M0 + n * tof
# Equation (32a), (32b), (32c) and (32d)
c3 = 5 / 2 + 560 * ecc
a = 15 * (1 - ecc) / c3
b = -M / c3
y = np.sqrt(b**2 / 4 + a**3 / 27)
# Equation (33)
x_bar = (-b / 2 + y)**(1 / 3) - (b / 2 + y)**(1 / 3)
# Coefficients from equations (34a) and (34b)
c15 = 3003 / 14336 + 16384 * ecc
c13 = 3465 / 13312 - 61440 * ecc
c11 = 945 / 2816 + 92160 * ecc
c9 = 175 / 384 - 70400 * ecc
c7 = 75 / 112 + 28800 * ecc
c5 = 9 / 8 - 6048 * ecc
# Precompute x_bar powers, equations (35a) to (35d)
x_bar2 = x_bar**2
x_bar3 = x_bar2 * x_bar
x_bar4 = x_bar3 * x_bar
x_bar5 = x_bar4 * x_bar
x_bar6 = x_bar5 * x_bar
x_bar7 = x_bar6 * x_bar
x_bar8 = x_bar7 * x_bar
x_bar9 = x_bar8 * x_bar
x_bar10 = x_bar9 * x_bar
x_bar11 = x_bar10 * x_bar
x_bar12 = x_bar11 * x_bar
x_bar13 = x_bar12 * x_bar
x_bar14 = x_bar13 * x_bar
x_bar15 = x_bar14 * x_bar
# Function f and its derivatives are given by all the (36) equation set
f = (c15 * x_bar15 + c13 * x_bar13 + c11 * x_bar11 + c9 * x_bar9 +
c7 * x_bar7 + c5 * x_bar5 + c3 * x_bar3 + 15 * (1 - ecc) * x_bar - M)
f1 = (15 * c15 * x_bar14 + 13 * c13 * x_bar12 + 11 * c11 * x_bar10 +
9 * c9 * x_bar8 + 7 * c7 * x_bar6 + 5 * c5 * x_bar4 +
3 * c3 * x_bar2 + 15 * (1 - ecc))
f2 = (210 * c15 * x_bar13 + 156 * c13 * x_bar11 + 110 * c11 * x_bar9 +
72 * c9 * x_bar7 + 42 * c7 * x_bar5 + 20 * c5 * x_bar3 +
6 * c3 * x_bar)
f3 = (2730 * c15 * x_bar12 + 1716 * c13 * x_bar10 + 990 * c11 * x_bar8 +
504 * c9 * x_bar6 + 210 * c7 * x_bar4 + 60 * c5 * x_bar2 + 6 * c3)
f4 = (32760 * c15 * x_bar11 + 17160 * c13 * x_bar9 + 7920 * c11 * x_bar7 +
3024 * c9 * x_bar5 + 840 * c7 * x_bar3 + 120 * c5 * x_bar)
f5 = (360360 * c15 * x_bar10 + 154440 * c13 * x_bar8 +
55440 * c11 * x_bar6 + 15120 * c9 * x_bar4 + 2520 * c7 * x_bar2 +
120 * c5)
f6 = (3603600 * c15 * x_bar9 + 1235520 * c13 * x_bar7 +
332640 * c11 * x_bar5 + 60480 * c9 * x_bar3 + 5040 * c7 * x_bar)
f7 = (32432400 * c15 * x_bar8 + 8648640 * c13 * x_bar6 +
1663200 * c11 * x_bar4 + 181440 * c9 * x_bar2 + 5040 * c7)
f8 = (259459200 * c15 * x_bar7 + 51891840 * c13 * x_bar5 +
6652800 * c11 * x_bar3 + 362880 * c9 * x_bar)
f9 = (1.8162144e9 * c15 * x_bar6 + 259459200 * c13 * x_bar4 +
19958400 * c11 * x_bar2 + 362880 * c9)
f10 = (1.08972864e10 * c15 * x_bar5 + 1.0378368e9 * c13 * x_bar3 +
39916800 * c11 * x_bar)
f11 = 5.4486432e10 * c15 * x_bar4 + 3.1135104e9 * c13 * x_bar2 + 39916800 * c11
f12 = 2.17945728e11 * c15 * x_bar3 + 6.2270208e9 * c13 * x_bar
f13 = 6.53837184 * c15 * x_bar2 + 6.2270208e9 * c13
f14 = 1.307674368e13 * c15 * x_bar
f15 = 1.307674368e13 * c15
# Solving g parameters defined by equations (37a), (37b), (37c) and (37d)
g1 = 1 / 2
g2 = 1 / 6
g3 = 1 / 24
g4 = 1 / 120
g5 = 1 / 720
g6 = 1 / 5040
g7 = 1 / 40320
g8 = 1 / 362880
g9 = 1 / 3628800
g10 = 1 / 39916800
g11 = 1 / 479001600
g12 = 1 / 6.2270208e9
g13 = 1 / 8.71782912e10
g14 = 1 / 1.307674368e12
# Solving for the u_{i} and h_{i} variables defined by equation (38)
u1 = -f / f1
h2 = f1 + g1 * u1 * f2
u2 = -f / h2
h3 = f1 + g1 * u2 * f2 + g2 * u2**2 * f3
u3 = -f / h3
h4 = f1 + g1 * u3 * f2 + g2 * u3**2 * f3 + g3 * u3**3 * f4
u4 = -f / h4
h5 = f1 + g1 * u4 * f2 + g2 * u4**2 * f3 + g3 * u4**3 * f4 + g4 * u4**4 * f5
u5 = -f / h5
h6 = (f1 + g1 * u5 * f2 + g2 * u5**2 * f3 + g3 * u5**3 * f4 +
g4 * u5**4 * f5 + g5 * u5**5 * f6)
u6 = -f / h6
h7 = (f1 + g1 * u6 * f2 + g2 * u6**2 * f3 + g3 * u6**3 * f4 +
g4 * u6**4 * f5 + g5 * u6**5 * f6 + g6 * u6**6 * f7)
u7 = -f / h7
h8 = (f1 + g1 * u7 * f2 + g2 * u7**2 * f3 + g3 * u7**3 * f4 +
g4 * u7**4 * f5 + g5 * u7**5 * f6 + g6 * u7**6 * f7 +
g7 * u7**7 * f8)
u8 = -f / h8
h9 = (f1 + g1 * u8 * f2 + g2 * u8**2 * f3 + g3 * u8**3 * f4 +
g4 * u8**4 * f5 + g5 * u8**5 * f6 + g6 * u8**6 * f7 +
g7 * u8**7 * f8 + g8 * u8**8 * f9)
u9 = -f / h9
h10 = (f1 + g1 * u9 * f2 + g2 * u9**2 * f3 + g3 * u9**3 * f4 +
g4 * u9**4 * f5 + g5 * u9**5 * f6 + g6 * u9**6 * f7 +
g7 * u9**7 * f8 + g8 * u9**8 * f9 + g9 * u9**9 * f10)
u10 = -f / h10
h11 = (f1 + g1 * u10 * f2 + g2 * u10**2 * f3 + g3 * u10**3 * f4 +
g4 * u10**4 * f5 + g5 * u10**5 * f6 + g6 * u10**6 * f7 +
g7 * u10**7 * f8 + g8 * u10**8 * f9 + g9 * u10**9 * f10 +
g10 * u10**10 * f11)
u11 = -f / h11
h12 = (f1 + g1 * u11 * f2 + g2 * u11**2 * f3 + g3 * u11**3 * f4 +
g4 * u11**4 * f5 + g5 * u11**5 * f6 + g6 * u11**6 * f7 +
g7 * u11**7 * f8 + g8 * u11**8 * f9 + g9 * u11**9 * f10 +
g10 * u11**10 * f11 + g11 * u11**11 * f12)
u12 = -f / h12
h13 = (f1 + g1 * u12 * f2 + g2 * u12**2 * f3 + g3 * u12**3 * f4 +
g4 * u12**4 * f5 + g5 * u12**5 * f6 + g6 * u12**6 * f7 +
g7 * u12**7 * f8 + g8 * u12**8 * f9 + g9 * u12**9 * f10 +
g10 * u12**10 * f11 + g11 * u12**11 * f12 + g12 * u12**12 * f13)
u13 = -f / h13
h14 = (f1 + g1 * u13 * f2 + g2 * u13**2 * f3 + g3 * u13**3 * f4 +
g4 * u13**4 * f5 + g5 * u13**5 * f6 + g6 * u13**6 * f7 +
g7 * u13**7 * f8 + g8 * u13**8 * f9 + g9 * u13**9 * f10 +
g10 * u13**10 * f11 + g11 * u13**11 * f12 + g12 * u13**12 * f13 +
g13 * u13**13 * f14)
u14 = -f / h14
h15 = (f1 + g1 * u14 * f2 + g2 * u14**2 * f3 + g3 * u14**3 * f4 +
g4 * u14**4 * f5 + g5 * u14**5 * f6 + g6 * u14**6 * f7 +
g7 * u14**7 * f8 + g8 * u14**8 * f9 + g9 * u14**9 * f10 +
g10 * u14**10 * f11 + g11 * u14**11 * f12 + g12 * u14**12 * f13 +
g13 * u14**13 * f14 + g14 * u14**14 * f15)
u15 = -f / h15
# Solving for x
x = x_bar + u15
w = x - 0.01171875 * x**17 / (1 + ecc)
# Solving for the true anomaly from eccentricity anomaly
E = M + ecc * (-16384 * w**15 + 61440 * w**13 - 92160 * w**11 + 70400 *
w**9 - 28800 * w**7 + 6048 * w**5 - 560 * w**3 + 15 * w)
return E_to_nu(E, ecc)
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 recseries_coe(k,
p,
ecc,
inc,
raan,
argp,
nu,
tof,
method="rtol",
order=8,
numiter=100,
rtol=1e-8):
# semi-major axis
semi_axis_a = p / (1 - ecc**2)
# mean angular motion
n = np.sqrt(k / np.abs(semi_axis_a)**3)
if ecc == 0:
# Solving for circular orbit
# compute initial mean anoamly
M0 = nu # For circular orbit (M = E = nu)
# final mean anaomaly
M = M0 + n * tof
# snapping anomaly to [0,pi] range
nu = M - 2 * np.pi * np.floor(M / 2 / np.pi)
return nu
elif ecc < 1.0:
# Solving for elliptical orbit
# compute initial mean anoamly
M0 = E_to_M(nu_to_E(nu, ecc), ecc)
# final mean anaomaly
M = M0 + n * tof
# snapping anomaly to [0,pi] range
M = M - 2 * np.pi * np.floor(M / 2 / np.pi)
# set recursion iteration
if method == "rtol":
Niter = numiter
elif method == "order":
Niter = order
else:
raise ValueError(
"Unknown recursion termination method ('rtol','order').")
# compute eccentric anomaly through recursive series
E = M + ecc # Using initial guess from vallado to improve convergence
for i in range(0, Niter):
En = M + ecc * np.sin(E)
# check for break condition
if method == "rtol" and (abs(En - E) / abs(E)) < rtol:
break
E = En
return E_to_nu(E, ecc)
else:
# Parabolic/Hyperbolic orbits are not supported
raise ValueError("Parabolic/Hyperbolic orbits not supported.")
return nu
def markley_bulk(k, tof, pp, eecc, iinc, rraan, aargp, nnu0):
""" Solves the kepler problem by a non iterative method. Relative error is
around 1e-18, only limited by machine double-precission errors.
Parameters
----------
k : float
Standar Gravitational parameter
r0 : array
Initial position vector wrt attractor center.
v0 : array
Initial velocity vector.
tof : float
Time of flight.
Returns
-------
rf: array
Final position vector
vf: array
Final velocity vector
Note
----
The following algorithm was taken from http://dx.doi.org/10.1007/BF00691917.
"""
# Solve first for eccentricity and mean anomaly
# p, ecc, inc, raan, argp, nu = rv2coe(k, r0, v0)
# check of all orbits are elipses
# if eecc.max() >= 1:
# raise ValueError("All eccentricities must be smaller than 1")
EE = nu_to_E(nnu0, eecc)
MM0 = E_to_M(EE, eecc)
aa = pp / (1 - eecc**2)
nn = np.sqrt(k / aa**3)
MM = MM0 + nn * tof
# Range between -pi and pi
MM = (MM + np.pi) % (2 * np.pi) - np.pi
# MM = MM % (2 * np.pi)
# for i, MM_elmt in enumerate(MM):
# if MM_elmt > np.pi:
# MM[i] = -(2 * np.pi - MM_elmt)
# Equation (20)
aalpha = (3 * np.pi**2 + 1.6 * (np.pi - np.abs(MM)) /
(1 + eecc)) / (np.pi**2 - 6)
# Equation (5)
dd = 3 * (1 - eecc) + aalpha * eecc
# Equation (9)
qq = 2 * aalpha * dd * (1 - eecc) - MM**2
# Equation (10)
rr = 3 * aalpha * dd * (dd - 1 + eecc) * MM + MM**3
# Equation (14)
ww = (np.abs(rr) + np.sqrt(qq**3 + rr**2))**(2 / 3)
# Equation (15)
EE = (2 * rr * ww / (ww**2 + ww * qq + qq**2) + MM) / dd
# Equation (26)
f0 = _kepler_equation(EE, MM, eecc)
f1 = _kepler_equation_prime(EE, MM, eecc)
f2 = eecc * np.sin(EE)
f3 = eecc * np.cos(EE)
f4 = -f2
# Equation (22)
delta3 = -f0 / (f1 - 0.5 * f0 * f2 / f1)
delta4 = -f0 / (f1 + 0.5 * delta3 * f2 + 1 / 6 * delta3**2 * f3)
delta5 = -f0 / (f1 + 0.5 * delta4 * f2 + 1 / 6 * delta4**2 * f3 +
1 / 24 * delta4**3 * f4)
EE += delta5
nnu = E_to_nu(EE, eecc)
return nnu
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
