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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Analyzing the Parker Solar Probe flybys
1. Modulus of the exit velocity, some features of Orbit #2
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.
Step1:  $$ T = 2 \pi \sqrt{\frac{a^3}{\mu}} \Rightarrow a = \sqrt[3]{\frac{\mu T^2}{4 \pi^2}}$$
Step2:  $$ \varepsilon = -\frac{\mu}{r} + \frac{v^2}{2} = -\frac{\mu}{2a} \Rightarrow v = +\sqrt{\frac{2\mu}{r} - \frac{\mu}{a}}$$
Step3:  2. Lambert arc between #0 and #1
To compute the arrival velocity to Venus at flyby #1, we have the necessary data to solve the boundary value problem.
Step4:  3. Flyby #1 around Venus
We compute a flyby using poliastro with the default value of the entry angle, just to discover that the results do not match what we expected.
Step5:  4. Optimization
Now we will try to find the value of $\theta$ that satisfies our requirements.
Step6:  There are two solutions
Step7:  5. Exit orbit
And finally, we compute orbit #2 and check that the period is the expected one.
Step8:  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.
Step10:  Let's define a function to do that quickly for us, using the get_frame function from poliastro.frames
Step11:  Therefore, the correct option is the first one.
Step12:  And, finally, we plot the solution
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Python Code:
from astropy import units as u
T_ref = 150 * u.day
T_ref
from poliastro.bodies import Earth, Sun, Venus
k = Sun.k
k
import numpy as np
Explanation: Analyzing the Parker Solar Probe flybys
1. Modulus of the exit velocity, some features of Orbit #2
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.
End of explanation
a_ref = np.cbrt(k * T_ref**2 / (4 * np.pi**2)).to(u.km)
a_ref.to(u.au)
Explanation: $$ T = 2 \pi \sqrt{\frac{a^3}{\mu}} \Rightarrow a = \sqrt[3]{\frac{\mu T^2}{4 \pi^2}}$$
End of explanation
energy_ref = (-k / (2 * a_ref)).to(u.J / u.kg)
energy_ref
from poliastro.twobody import Orbit
from poliastro.util import norm
from astropy.time import Time
flyby_1_time = Time("2018-09-28", scale="tdb")
flyby_1_time
r_mag_ref = norm(Orbit.from_body_ephem(Venus, epoch=flyby_1_time).r)
r_mag_ref.to(u.au)
v_mag_ref = np.sqrt(2 * k / r_mag_ref - k / a_ref)
v_mag_ref.to(u.km / u.s)
Explanation: $$ \varepsilon = -\frac{\mu}{r} + \frac{v^2}{2} = -\frac{\mu}{2a} \Rightarrow v = +\sqrt{\frac{2\mu}{r} - \frac{\mu}{a}}$$
End of explanation
d_launch = Time("2018-08-11", scale="tdb")
d_launch
ss0 = Orbit.from_body_ephem(Earth, d_launch)
ss1 = Orbit.from_body_ephem(Venus, epoch=flyby_1_time)
tof = flyby_1_time - d_launch
from poliastro import iod
(v0, v1_pre), = iod.lambert(Sun.k, ss0.r, ss1.r, tof.to(u.s))
v0
v1_pre
norm(v1_pre)
Explanation: 2. Lambert arc between #0 and #1
To compute the arrival velocity to Venus at flyby #1, we have the necessary data to solve the boundary value problem.
End of explanation
from poliastro.threebody.flybys import compute_flyby
V = Orbit.from_body_ephem(Venus, epoch=flyby_1_time).v
V
h = 2548 * u.km
d_flyby_1 = Venus.R + h
d_flyby_1.to(u.km)
V_2_v_, delta_ = compute_flyby(v1_pre, V, Venus.k, d_flyby_1)
norm(V_2_v_)
Explanation: 3. Flyby #1 around Venus
We compute a flyby using poliastro with the default value of the entry angle, just to discover that the results do not match what we expected.
End of explanation
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
Explanation: 4. Optimization
Now we will try to find the value of $\theta$ that satisfies our requirements.
End of explanation
import matplotlib.pyplot as plt
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")
func(0)
func(1)
from scipy.optimize import brentq
theta_opt_a = brentq(func, 0, 1) * u.rad
theta_opt_a.to(u.deg)
theta_opt_b = brentq(func, 4, 5) * u.rad
theta_opt_b.to(u.deg)
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)
norm(V_2_v_a)
norm(V_2_v_b)
Explanation: There are two solutions:
End of explanation
ss01 = Orbit.from_vectors(Sun, ss1.r, v1_pre, epoch=flyby_1_time)
ss01
Explanation: 5. Exit orbit
And finally, we compute orbit #2 and check that the period is the expected one.
End of explanation
ss_1_a = Orbit.from_vectors(Sun, ss1.r, V_2_v_a, epoch=flyby_1_time)
ss_1_a
ss_1_b = Orbit.from_vectors(Sun, ss1.r, V_2_v_b, epoch=flyby_1_time)
ss_1_b
Explanation: 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.
End of explanation
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
change_plane(ss_1_a, Planes.EARTH_ECLIPTIC)
change_plane(ss_1_b, Planes.EARTH_ECLIPTIC)
Explanation: Let's define a function to do that quickly for us, using the get_frame function from poliastro.frames:
End of explanation
ss_1_a.period.to(u.day)
ss_1_a.a
Explanation: Therefore, the correct option is the first one.
End of explanation
from poliastro.plotting import OrbitPlotter
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");
Explanation: And, finally, we plot the solution:
End of explanation
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