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import numpy as np
from astropy import units as u
from poliastro.bodies import Earth
from poliastro.core.angles import (
E_to_nu,
_kepler_equation,
_kepler_equation_prime,
nu_to_M,
)
from poliastro.core.elements import rv2coe
from poliastro.examples import iss
from poliastro.twobody.propagation import cowell
from poliastro.twobody.orbit import Orbit
def kepler_improved(k, r0, v0, tof):
""" Solves the kepler problem by a non iteraiteve method.
Parameters
----------
k: float
Gravitational parameter
r0: vector 1x3
Initial position vector
v0: vector 1x3
Initial velocity vector
tof: float
Time of flight
Returns
-------
rf: vector 1x3
Final position vector
vf: vector 1x3
Final velocity vector
"""
# Convert to international unit system
k = k.to(u.m ** 3 / u.s ** 2).value
r0 = r0.to(u.m).value
v0 = v0.to(u.m / u.s).value
tof = tof.to(u.s).value
# Solve first for eccentricity and mean anomaly
p, ecc, inc, raan, argp, nu = rv2coe(k, r0, v0)
M0 = nu_to_M(nu, ecc)
a = p / (1 - ecc ** 2)
n = np.sqrt(k / a ** 3)
M = M0 + n * tof
# Range between -pi and pi
M = M % (2 * np.pi)
if M > np.pi:
M = -(2 * np.pi - M)
# ------ ALGORITHM STARTS -------
# 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)
E1 = (2 * r * w / (w ** 2 + w * q + q ** 2) + M) / d
# Equation (26)
f0 = _kepler_equation(E1, M, ecc)
f1 = _kepler_equation_prime(E1, M, ecc)
f2 = ecc * np.sin(E1)
f3 = ecc * np.cos(E1)
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
)
# Finally
E5 = E1 + delta5
nu = E_to_nu(E5, ecc)
return nu * u.rad
def hard_orbit():
""" Propagates the hard orbit. """
r = [8.0e3, 1.0e3, 0.0] * u.km
v = [-0.5, -0.5, 0.0] * u.km / u.s
ss = Orbit.from_vectors(Earth, r, v)
print(ss.classical())
a, ecc, inc, raan, argp, nu = ss.classical()
# Cowell propagator (integrates two-body ODE)
ss_cowell = ss.propagate(1 * u.h, method=cowell)
nu_expected = ss_cowell.nu
print("Nu nu_expected:", nu_expected)
print(ss_cowell.classical())
# Kepler_improved propagator (Defined in the first script function)
nu = kepler_improved(ss.attractor.k, ss.r, ss.v, 1 * u.h)
ss_kepler_improved = Orbit.from_classical(Earth, a, ecc, inc, raan, argp, nu)
print("Nu obtained:", nu)
print(ss_kepler_improved.classical())
if __name__ == '__main__':
hard_orbit()
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