Is the JWST halo orbit period sync'd with earth's elliptical orbit about sun?

I was looking at the JWST halo and wondering if getting the halo orbit phased with earths elliptical orbit would be a way of keeping gravitational variations regular and periodic so that station keeping burns could be more predicable, preserving fuel and thereby extending useable life of JWST. Right now JWST is near max +Y travel... when earth is just past Periapsis. Is this a coincidence? The halo could be tailored with precision burns to get max Y at Periapsis and min Y at midpoint between Periapsis and Apoapsis. At Apoapsis JWST would be at max Y again, 1 halo orbit in 1/2 year.

and so the question: Is (or will) the JWST halo orbit period sync'd with earth's elliptical orbit about sun?

EDIT: More directly about my "Sync'd orbit" question... Is there a need or benefit to have JWST at a specific place in its halo when earth is at a specific place in its orbit.

• @BradV FYI, I just estimated the period from the Horizon's data we looked at earlier. It comes out at 6.07 months. It's not quite a closed orbit so there's no exact number. I'd be surprised if the Earth's orbit was eccentric enough to cause much difference over 10 years. The Moon would have a lot more effect but it's period is not close JWST's in any simple ratio. Jan 28, 2022 at 22:06
• @BrendanLuke15 I've adjusted the wording of the title to match the sentiment of the question, basically from "is?" to "could?"
– uhoh
Jan 29, 2022 at 2:01
• I suspect (but do not know with enough certainty) that the answer to this question is that the JWST Flight Dynamics Team (FDT) already does what the OP is asking about. While there are additional complexities to the Elliptical Restricted Three Body Problem (ER3BP) compared to the Circular Restricted Three Body Problem (CR3BP), numerous papers have been written on the ER3BP. The FDT would be remiss if they didn't account for the eccentricity of the Earth's orbit. They would also be remiss if they didn't account for the Moon. This isn't a three body problem; It's a restricted four body problem. Jan 29, 2022 at 12:17
• @uhoh: Sorry, but I'd rather not have folks edit my questions and discussion without first discussing with me. Most of the time I have a specific reasoning for structuring things the way I do. Also... I've learned to 'see past' minor things like capitalizations and mis-spellings if they do not affect meaning or context. I don't mean to be snarky here, but I'm not pleased when what appears to be my questions is actually someone elses. Jan 29, 2022 at 15:07
• @uhoh: thanks for putting up with "five year old" sitting in on a PhD level discussion! I realize I'm in well over my head. I've looked at several halos and they seem to be 2:1 but I have no idea if positioning of object within halo is related to and sync'd with M2 orbit.. Jan 30, 2022 at 1:43

Is the JWST halo orbit period sync'd with earth's elliptical orbit about sun?

Wow, It's surprisingly close!

At first I figured it would be closer to 6.5 or 7 months due to past information and questions and answers based on an earlier JWST trajectory, but wow, trying to define the period of a complicated 3 or 4 body orbit (including the Moon's effect) in a simple way, I've come up with 180 days!

I took the new predicted orbit for JWST in Horizons

Revised: Jan 28, 2022
2Y_SCHEDULE_2022027000000_01U.OEM.V0.1   2022-Jan-27 00:01  2024-Jan-27 00:01


downloaded the heliocentric positions of JWST and Earth, subtracted Earth/Moon barycenter from JWST then rotated by Earth's angle to be in a pseudo-synodic frame (it rotates with EM/Bary angle, not steadily), and got the following (data starts 2022-Feb-08, ends 2024-Jan-27):

The offset in X is the average distance from Earth, close to the 1.5 million km of the classical L2 distance.

If I pick off the maxima and calculate a period for X, Y and Z motion I get 179.3, 180.0, and 180.7 days!

As this is an ad hoc method, I'll say that these periods are *currently indistinguishable from a half year so my answer is a qualified but quite surprised "yes!"

import numpy as np
import matplotlib.pyplot as plt

fnames = ('Earth orbit heliocentric horizons_results.txt',
'JWST orbit heliocentric horizons_results.txt')

JDs, datas = [], []
for fname in fnames:
n_offset = 10
with open(fname, 'r') as infile:
a = [i for (i, line) in enumerate(lines) if 'SOE' in line][0]
b = [i for (i, line) in enumerate(lines) if 'EOE' in line][0]
lines = lines[a+1+n_offset: b]
info = [line.split(',') for line in lines]
JD = np.array([float(line[0]) for line in info])
data = np.array([ [float(thing) for thing in line[2:8]] for line in info])
JDs.append(JD)
datas.append(data)

JD = JDs[0]
Earth, JWST = [thing.T.copy() for thing in datas]

dJWST = JWST - Earth

z = dJWST[2]

th = np.arctan2(Earth[1], Earth[0])

s, c = [f(-th) for f in (np.sin, np.cos)]

x = dJWST[0] * c - dJWST[1] * s
y = dJWST[1] * c + dJWST[0] * s

fig, axes = plt.subplots(3, 1)
for thing, name, ax in zip([x, y, z], 'XYZ', axes):
ax.plot((JD - JD[0]) / 365.2564, thing)
ax.set_ylabel(name + ' (km)')
plt.show()

maxima = [np.where((p[2:] < p[1:-1]) * (p[1:-1] >= p[:-2])) for p in (x, y, z)]

maxima = [thing[0] for thing in maxima]

periods = [(m[-1] - m[0]) / (len(m) - 1) for m in maxima]

print(periods)

• While within narrow confines uhoh's answer is correct in finding a 2:1 relationship... my question was about JWST being at a specific location in halo relative to a specific location on earth orbit. Feb 3, 2022 at 0:57
• @BradV Why don't you go ahead and further clarify your question's title. I'd suggest something like "Is JWST's halo orbit's 2:1 relationship with Earth's orbit phased with respect to Earth's orbit's apses? If so, was this done intentionally?" I think something like that is necessary to completely capture all that you are after, which I think is really interesting! I'm not sure I can do much better answering myself, but if not I can certainly add a bounty or to to your question to help get it attention and perhaps answered.
– uhoh
Feb 3, 2022 at 1:41