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.

In [1]:
from astropy import units as u
In [2]:
T_ref = 150 * u.day
T_ref
Out[2]:
$150 \; \mathrm{d}$
In [3]:
from poliastro.bodies import Earth, Sun, Venus
In [4]:
k = Sun.k
k
Out[4]:
$1.3271244 \times 10^{20} \; \mathrm{\frac{m^{3}}{s^{2}}}$
In [5]:
import numpy as np
\[T = 2 \pi \sqrt{\frac{a^3}{\mu}} \Rightarrow a = \sqrt[3]{\frac{\mu T^2}{4 \pi^2}}\]
In [6]:
a_ref = np.cbrt(k * T_ref**2 / (4 * np.pi**2)).to(u.km)
a_ref.to(u.au)
Out[6]:
$0.55249526 \; \mathrm{AU}$
\[\varepsilon = -\frac{\mu}{r} + \frac{v^2}{2} = -\frac{\mu}{2a} \Rightarrow v = +\sqrt{\frac{2\mu}{r} - \frac{\mu}{a}}\]
In [7]:
energy_ref = (-k / (2 * a_ref)).to(u.J / u.kg)
energy_ref
Out[7]:
$-8.0283755 \times 10^{8} \; \mathrm{\frac{J}{kg}}$
In [8]:
from poliastro.twobody import Orbit
from poliastro.util import norm

from astropy.time import Time
In [9]:
flyby_1_time = Time("2018-09-28", scale="tdb")
flyby_1_time
Out[9]:
<Time object: scale='tdb' format='iso' value=2018-09-28 00:00:00.000>
In [10]:
r_mag_ref = norm(Orbit.from_body_ephem(Venus, epoch=flyby_1_time).r)
r_mag_ref.to(u.au)
Out[10]:
$0.72573132 \; \mathrm{AU}$
In [11]:
v_mag_ref = np.sqrt(2 * k / r_mag_ref - k / a_ref)
v_mag_ref.to(u.km / u.s)
Out[11]:
$28.967364 \; \mathrm{\frac{km}{s}}$

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.

In [12]:
d_launch = Time("2018-08-11", scale="tdb")
d_launch
Out[12]:
<Time object: scale='tdb' format='iso' value=2018-08-11 00:00:00.000>
In [13]:
ss0 = Orbit.from_body_ephem(Earth, d_launch)
ss1 = Orbit.from_body_ephem(Venus, epoch=flyby_1_time)
In [14]:
tof = flyby_1_time - d_launch
In [15]:
from poliastro import iod
In [16]:
(v0, v1_pre), = iod.lambert(Sun.k, ss0.r, ss1.r, tof.to(u.s))
In [17]:
v0
Out[17]:
$[9.5993373,~11.298552,~2.9244933] \; \mathrm{\frac{km}{s}}$
In [18]:
v1_pre
Out[18]:
$[-16.980821,~23.307528,~9.1312908] \; \mathrm{\frac{km}{s}}$
In [19]:
norm(v1_pre)
Out[19]:
$30.248465 \; \mathrm{\frac{km}{s}}$

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.

In [20]:
from poliastro.threebody.flybys import compute_flyby
In [21]:
V = Orbit.from_body_ephem(Venus, epoch=flyby_1_time).v
V
Out[21]:
$[648499.74,~2695078.4,~1171563.7] \; \mathrm{\frac{km}{d}}$
In [22]:
h = 2548 * u.km
In [23]:
d_flyby_1 = Venus.R + h
d_flyby_1.to(u.km)
Out[23]:
$8599.8 \; \mathrm{km}$
In [24]:
V_2_v_, delta_ = compute_flyby(v1_pre, V, Venus.k, d_flyby_1)
In [25]:
norm(V_2_v_)
Out[25]:
$27.755339 \; \mathrm{\frac{km}{s}}$

4. Optimization

Now we will try to find the value of \(\theta\) that satisfies our requirements.

In [26]:
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:

In [27]:
import matplotlib.pyplot as plt
In [28]:
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")
Out[28]:
<matplotlib.lines.Line2D at 0x7f5f3bddd2b0>
../_images/examples_Analyzing_the_Parker_Solar_Probe_flybys_36_1.png
In [29]:
func(0)
Out[29]:
-9.142672330001195
In [30]:
func(1)
Out[30]:
7.09811543934556
In [31]:
from scipy.optimize import brentq
In [32]:
theta_opt_a = brentq(func, 0, 1) * u.rad
theta_opt_a.to(u.deg)
Out[32]:
$38.598709 \; \mathrm{{}^{\circ}}$
In [33]:
theta_opt_b = brentq(func, 4, 5) * u.rad
theta_opt_b.to(u.deg)
Out[33]:
$279.3477 \; \mathrm{{}^{\circ}}$
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)
In [35]:
norm(V_2_v_a)
Out[35]:
$28.967364 \; \mathrm{\frac{km}{s}}$
In [36]:
norm(V_2_v_b)
Out[36]:
$28.967364 \; \mathrm{\frac{km}{s}}$

5. Exit orbit

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

In [37]:
ss01 = Orbit.from_vectors(Sun, ss1.r, v1_pre, epoch=flyby_1_time)
ss01
Out[37]:
0 x 1 AU x 18.8 deg (HCRS) orbit around Sun (☉)

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.

In [38]:
ss_1_a = Orbit.from_vectors(Sun, ss1.r, V_2_v_a, epoch=flyby_1_time)
ss_1_a
Out[38]:
0 x 1 AU x 25.0 deg (HCRS) orbit around Sun (☉)
In [39]:
ss_1_b = Orbit.from_vectors(Sun, ss1.r, V_2_v_b, epoch=flyby_1_time)
ss_1_b
Out[39]:
0 x 1 AU x 13.1 deg (HCRS) orbit around Sun (☉)

Let’s define a function to do that quickly for us, using the `get_frame <https://docs.poliastro.space/en/latest/safe.html#poliastro.frames.get_frame>`__ function from poliastro.frames:

In [40]:
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
In [41]:
change_plane(ss_1_a, Planes.EARTH_ECLIPTIC)
Out[41]:
0 x 1 AU x 3.5 deg (HeliocentricEclipticJ2000) orbit around Sun (☉)
In [42]:
change_plane(ss_1_b, Planes.EARTH_ECLIPTIC)
Out[42]:
0 x 1 AU x 13.1 deg (HeliocentricEclipticJ2000) orbit around Sun (☉)

Therefore, the correct option is the first one.

In [43]:
ss_1_a.period.to(u.day)
Out[43]:
$150 \; \mathrm{d}$
In [44]:
ss_1_a.a
Out[44]:
$82652115 \; \mathrm{km}$

And, finally, we plot the solution:

In [45]:
from poliastro.plotting import OrbitPlotter
In [46]:
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");
/home/juanlu/Personal/poliastro/poliastro-library/src/poliastro/twobody/orbit.py:434: UserWarning:

Frame <class 'astropy.coordinates.builtin_frames.icrs.ICRS'> does not support 'obstime', time values were not returned

../_images/examples_Analyzing_the_Parker_Solar_Probe_flybys_59_1.png