First, using the data available in the reports, we try to compute some of the properties of orbit #2. This is not enough to completely define the trajectory, but will give us information later on in the process.
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from astropy import units as u
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T_ref = 150 * u.day
T_ref
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from poliastro.bodies import Earth, Sun, Venus
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k = Sun.k
k
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import numpy as np
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a_ref = np.cbrt(k * T_ref**2 / (4 * np.pi**2)).to(u.km)
a_ref.to(u.au)
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energy_ref = (-k / (2 * a_ref)).to(u.J / u.kg)
energy_ref
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from poliastro.twobody import Orbit
from poliastro.util import norm
from astropy.time import Time
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flyby_1_time = Time("2018-09-28", scale="tdb")
flyby_1_time
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r_mag_ref = norm(Orbit.from_body_ephem(Venus, epoch=flyby_1_time).r)
r_mag_ref.to(u.au)
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v_mag_ref = np.sqrt(2 * k / r_mag_ref - k / a_ref)
v_mag_ref.to(u.km / u.s)
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d_launch = Time("2018-08-11", scale="tdb")
d_launch
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ss0 = Orbit.from_body_ephem(Earth, d_launch)
ss1 = Orbit.from_body_ephem(Venus, epoch=flyby_1_time)
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tof = flyby_1_time - d_launch
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from poliastro import iod
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(v0, v1_pre), = iod.lambert(Sun.k, ss0.r, ss1.r, tof.to(u.s))
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v0
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v1_pre
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norm(v1_pre)
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from poliastro.threebody.flybys import compute_flyby
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V = Orbit.from_body_ephem(Venus, epoch=flyby_1_time).v
V
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h = 2548 * u.km
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d_flyby_1 = Venus.R + h
d_flyby_1.to(u.km)
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V_2_v_, delta_ = compute_flyby(v1_pre, V, Venus.k, d_flyby_1)
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norm(V_2_v_)
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def func(theta):
V_2_v, _ = compute_flyby(v1_pre, V, Venus.k, d_flyby_1, theta * u.rad)
ss_1 = Orbit.from_vectors(Sun, ss1.r, V_2_v, epoch=flyby_1_time)
return (ss_1.period - T_ref).to(u.day).value
There are two solutions:
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import matplotlib.pyplot as plt
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theta_range = np.linspace(0, 2 * np.pi)
plt.plot(theta_range, [func(theta) for theta in theta_range])
plt.axhline(0, color='k', linestyle="dashed")
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func(0)
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func(1)
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from scipy.optimize import brentq
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theta_opt_a = brentq(func, 0, 1) * u.rad
theta_opt_a.to(u.deg)
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theta_opt_b = brentq(func, 4, 5) * u.rad
theta_opt_b.to(u.deg)
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In [34]:
V_2_v_a, delta_a = compute_flyby(v1_pre, V, Venus.k, d_flyby_1, theta_opt_a)
V_2_v_b, delta_b = compute_flyby(v1_pre, V, Venus.k, d_flyby_1, theta_opt_b)
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norm(V_2_v_a)
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norm(V_2_v_b)
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ss01 = Orbit.from_vectors(Sun, ss1.r, v1_pre, epoch=flyby_1_time)
ss01
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The two solutions have different inclinations, so we still have to find out which is the good one. We can do this by computing the inclination over the ecliptic - however, as the original data was in the International Celestial Reference Frame (ICRF), whose fundamental plane is parallel to the Earth equator of a reference epoch, we have change the plane to the Earth ecliptic, which is what the original reports use.
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ss_1_a = Orbit.from_vectors(Sun, ss1.r, V_2_v_a, epoch=flyby_1_time)
ss_1_a
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ss_1_b = Orbit.from_vectors(Sun, ss1.r, V_2_v_b, epoch=flyby_1_time)
ss_1_b
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Let's define a function to do that quickly for us, using the get_frame function from poliastro.frames:
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from astropy.coordinates import CartesianRepresentation
from poliastro.frames import Planes, get_frame
def change_plane(ss_orig, plane):
"""Changes the plane of the Orbit.
"""
ss_orig_rv = ss_orig.frame.realize_frame(
ss_orig.represent_as(CartesianRepresentation)
)
dest_frame = get_frame(ss_orig.attractor, plane, obstime=ss_orig.epoch)
ss_dest_rv = ss_orig_rv.transform_to(dest_frame)
ss_dest_rv.representation_type = CartesianRepresentation
ss_dest = Orbit.from_vectors(
ss_orig.attractor,
r=ss_dest_rv.data.xyz,
v=ss_dest_rv.data.differentials['s'].d_xyz,
epoch=ss_orig.epoch,
plane=plane,
)
return ss_dest
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change_plane(ss_1_a, Planes.EARTH_ECLIPTIC)
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change_plane(ss_1_b, Planes.EARTH_ECLIPTIC)
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Therefore, the correct option is the first one.
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ss_1_a.period.to(u.day)
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ss_1_a.a
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And, finally, we plot the solution:
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from poliastro.plotting import OrbitPlotter
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frame = OrbitPlotter()
frame.plot(ss0, label=Earth)
frame.plot(ss1, label=Venus)
frame.plot(ss01, label="#0 to #1")
frame.plot(ss_1_a, label="#1 to #2");