analyzing-the-parker-solar-probe-flybys.myst.md

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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:

from astropy import units as u

T_ref = 150 * u.day
T_ref

from hapsira.bodies import Earth, Sun, Venus

k = Sun.k
k

import numpy as np

$$ T = 2 \pi \sqrt{\frac{a^3}{\mu}} \Rightarrow a = \sqrt[3]{\frac{\mu T^2}{4 \pi^2}}$$

a_ref = np.cbrt(k * T_ref**2 / (4 * np.pi**2)).to(u.km)
a_ref.to(u.au)

$$ \varepsilon = -\frac{\mu}{r} + \frac{v^2}{2} = -\frac{\mu}{2a} \Rightarrow v = +\sqrt{\frac{2\mu}{r} - \frac{\mu}{a}}$$

energy_ref = (-k / (2 * a_ref)).to(u.J / u.kg)
energy_ref

from astropy.time import Time

from hapsira.ephem import Ephem
from hapsira.util import norm

flyby_1_time = Time("2018-09-28", scale="tdb")
flyby_1_time

r_mag_ref = norm(Ephem.from_body(Venus, flyby_1_time).rv()[0].squeeze())
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)

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:

d_launch = Time("2018-08-11", scale="tdb")
d_launch

r0, _ = Ephem.from_body(Earth, d_launch).rv()
r1, V = Ephem.from_body(Venus, flyby_1_time).rv()

r0 = r0[0]
r1 = r1[0]
V = V[0]

tof = flyby_1_time - d_launch

from hapsira import iod

v0, v1_pre = iod.lambert(Sun.k, r0, r1, tof.to(u.s))

v0

v1_pre

norm(v1_pre)

3. Flyby #1 around Venus

We compute a flyby using hapsira with the default value of the entry angle, just to discover that the results do not match what we expected:

from hapsira.threebody.flybys import compute_flyby

V.to(u.km / u.day)

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_)

4. Optimization

Now we will try to find the value of $\theta$ that satisfies our requirements:

from hapsira.twobody import Orbit

def func(theta):
    V_2_v, _ = compute_flyby(v1_pre, V, Venus.k, d_flyby_1, theta * u.rad)
    orb_1 = Orbit.from_vectors(Sun, r1, V_2_v, epoch=flyby_1_time)
    return (orb_1.period - T_ref).to(u.day).value

There are two solutions:

from matplotlib import 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[0], Venus.k, d_flyby_1, theta_opt_a)
V_2_v_b, delta_b = compute_flyby(v1_pre, V[0], Venus.k, d_flyby_1, theta_opt_b)

norm(V_2_v_a)

norm(V_2_v_b)

5. Exit orbit

And finally, we compute orbit #2 and check that the period is the expected one:

ss01 = Orbit.from_vectors(Sun, r1, v1_pre, epoch=flyby_1_time)
ss01

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 to change the plane to the Earth ecliptic, which is what the original reports use:

orb_1_a = Orbit.from_vectors(Sun, r1, V_2_v_a, epoch=flyby_1_time)
orb_1_a

orb_1_b = Orbit.from_vectors(Sun, r1, V_2_v_b, epoch=flyby_1_time)
orb_1_b

from hapsira.frames import Planes

orb_1_a.change_plane(Planes.EARTH_ECLIPTIC)

orb_1_b.change_plane(Planes.EARTH_ECLIPTIC)

Therefore, the correct option is the first one:

orb_1_a.period.to(u.day)

orb_1_a.a

And, finally, we plot the solution:

:tags: [nbsphinx-thumbnail]

from hapsira.plotting import OrbitPlotter

frame = OrbitPlotter(plane=Planes.EARTH_ECLIPTIC)

frame.plot_body_orbit(Earth, d_launch)
frame.plot_body_orbit(Venus, flyby_1_time)
frame.plot(ss01, label="#0 to #1", color="C2")
frame.plot(orb_1_a, label="#1 to #2", color="C3")