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As I feel a little less uncomfortable with "halo" orbits, with this question, I would like to explore the practical aspects, in particular those related to the design of the James Webb Space Telescope (JWST) operational orbit.

This (huge) spacecraft is scheduled to be launched this month, while its inception dated back to the end of the 90s (with an orginal target launch in 2007!). Its operational position will be in the vicinity of the Earth-Sun Lagrangian point "L2", about 1.5 millions km from Earth. For a vizualitation of the targetted JWST's trajectory in orbit around the Sun and around the Earth-Sun L2 point, see this NASA animation.

For an introduction to Lagrangian points (aka libration points), see What are Lagrangian points.

For an introduction to "halo" orbits, see Farquhar's classic "The flight of ISEE-3/ICE:Origin, mission History and a Legacy".

Note that there are other types of orbits in literature that are also connected to libration points, Lyapunov and Lissajous orbits. As far as I understood, halo orbits form a subclass of the Lissajous class. I propose that we focus here on this subclass (see sub-note below).

A previous question by @uhoh Are some Halo orbits actually stable? shed light on the existence of a subset of halo orbits that have the nice property of being (theoretically) stable. "Theoretical" means that we make simplifications to the real-word equations of forces, reducing the problem to a more tractable mathematical model. The model of interest is the so-called Circular Restricted 3-Body model (also designated as CR3BP, "P" for problem), which seems to be invariably used as a starting point for the preliminary mission analysis of halo orbits.

Intuitively, stability means that we do not need to spend any energy to keep a spacecraft from moving out of a prescribed (bounded) area. The magic of gravity suffices, in theory. In practice, it means we need to spend "very little", or at least much less than what we would have to spend had we targetted an unstable trajectory.

MY QUESTION

Was the JWST halo orbit chosen from a stable halo "template", and in "broad terms", what were the trade-offs between JWST's scientific mission requirement, orbital life-time and cost of station-keeping?

By "broad terms", I mean I would be satisfied with the results of an analysis comparing the Delta-v per year, amount of fuel embarked,... versus the "amplitude/size/excursion" (whatever this means, is up to you) of the halo orbit and its period. It would be nice if you could factor in the Deep Space Antenna Network constraints (how to maximize the visibility and link connections with Earth) and the launch window.

Note:

Related question Halo vs Lissajous, which station-keeping strategy and when?


UPDATE (6 Dec. 2021)

First, some hints on the approach they used to design the JWST orbit are given in the JWST User Documentation Home Page.

In particular (emphasis added):

The L2 orbit shape is not constrained, so torus orbits, halo orbits, or Lissajous orbits are acceptable and are determined primarily by the launch's time of day and day of year. This freedom in the L2 orbit design allows for multiple launch opportunities for most months and minimizes the velocity needed to get to orbit.

This corroborates well with the paper by Brown et. al. Seasonal variations of the JWST orbital dynamics . Thanks to @PearsonArtPhoto

This paper (published in 2015) provided (Fig. 5) an example of orbit realizations if the JWST had been launched in October 2018. The paragraph before the illustrative Figure reads:

The launch window is defined by continuous time intervals between 11:30 and 14:00 UTC each day for which all the previous orbital constraints are met, and includes margin for model uncertainty (e.g. propulsion system performance). The previous launch window analysis demonstrated that over half of the launch readiness period (October thru December 2018) provides viable launch opportunities that satisfy all orbit requirements. However, the geometry of the orbits varies significantly.

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    $\begingroup$ I'm trying to digest it, but I think the answer can be found at ntrs.nasa.gov/api/citations/20150017756/downloads/… $\endgroup$
    – PearsonArtPhoto
    Dec 3, 2021 at 15:36
  • $\begingroup$ @PearsonArtPhoto, seems to be good. I had thought that we have to search for more "vintage" documentations! $\endgroup$
    – Ng Ph
    Dec 3, 2021 at 15:59
  • $\begingroup$ @PearsonArtPhoto, skimming over the document (Brown et.al. 2015), it looks rather like an investigation on the impact of the launch date. At least, I didn't find the keywords I am looking for. Will try perusing ... $\endgroup$
    – Ng Ph
    Dec 3, 2021 at 17:48
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    $\begingroup$ Thanks for the very well composed question. Many useful links ! $\endgroup$
    – Woody
    Dec 3, 2021 at 18:21
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    $\begingroup$ "So, my question... seems to be settled by these sources." All of your new material belongs in a new answer post, not in the question. Just copy/paste the new text into an answer post, then go to the edit history and click "roll back". $\endgroup$
    – uhoh
    Dec 6, 2021 at 11:07

2 Answers 2

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Has the JWST halo orbit been chosen from a stable halo "template", and in "broad terms", what were the trade-offs between JWST's scientific mission requirement, orbital life-time and cost of station-keeping?

By "broad terms", I mean I would be satisfied with the results of an analysis comparing the Delta-v per year, amount of fuel embarked,... versus the "amplitude/size/excursion" (whatever this means, is up to you) of the halo orbit and its period. It would be nice if you could factor in the Deep Space Antenna Network constraints (how to maximize the visibility and link connections with Earth) and the launch window.

This is a big, beautiful question and I am not sure I can answer all of it, but I'll add my two cents.

I'm not aware of any major delta-v-specific trade-offs against scientific requirements, but there may be some interaction between observers and station-keepers related to rotation of the telescope around its axis (see below). Perceived1 budgetary limits always compete head to head with scientific capabilities and perceived1 schedules, but to the best of my current understanding the designers of the telescopes "bus" (navigation, attitude control, station keeping, power management, thermal management, etc.) were simply given a list of requirements and a minimum primary mission duration2 of 10 years and they delivered a really beautiful solution!

The SOHO spacecraft:

(planned for 2 years, now in it's 26th year despite almost being lost) is a great first instructive example of station-keeping for a halo orbit.

We know that real world halo orbits are always unstable and need to be station-kept regularly. They are exponentially unstable so a spacecraft's drift away from its "ideal"3 halo orbit accelerates rapidly. I recall that for SOHO its doubling time was only two weeks. The beauty of delay-doppler measurements with a spacecraft having a coherent transponder is that deviations in position and velocity can be detected on the scale of meters and millimeters per second, so one still has a few months to recover if there is sufficient fuel to do what needs to be done.

By today's standards SOHO used a relatively simple station-keeping strategy. It's a solar telescope and needs to point at the Sun to do its job. It has a Sun direction sensor and hardware that keeps its axis pointed in that direction, and since it's at Sun-Earth L1 that means the back end where it's main antenna and main thrusters are will be roughly pointed in Earth's general direction.

SOHO sits just beyond its ideal orbit so that if there is no station-keeping it would start spiraling out towards Earth on the orbit's unstable manifold. That means that station-keeping is simple; Earth measures its rate of drift away from its ideal orbit and towards Earth and then calculates how many seconds to burn its thrusters to push it back towards the Sun.

It's a simple and potentially very efficient 1D station keeping solution that has worked for 26 years!

For more on all of this see Roberts 2002 and other sources4

The present space telescope being discussed:

can benefit from both all of the amazing initial work on halo orbit-based missions and decades of thinking and research and improvement of all the bits of space technology that go into a modern spacecraft bus.

And in an interesting twist, the fact that it is an infrared telescope has provided an extra bit of fuel-saving orbital mechanical wizardry.

Like SOHO, its orbit is designed to always sit just slightly closer to Earth than its ideal halo orbit. And like SOHO it will be constantly monitored from Earth via delay-doppler and receive instructions regularly from Earth to manage its orbit.

But since it is at L2 rather than L1, unlike SOHO it will sit slightly closer to the Sun than its ideal orbit.

However, the fact that its an infrared telescope requiring cold optics and cold sensors achieved with a giant multilayer, reflective sunshield (and passive radiative cooling to space) provides a source of free propulsion in the direction away from the Sun and Earth (at L2 now) and towards its ideal halo orbit. The spacecraft can play the two effects against each other; if it is drifting away from ideal and towards Earth it can orient its sunshield more towards the Sun and increase solar photon thrust back towards its orbit. If it is moving too fast back towards its ideal orbit it can orient its sunshield somewhat obliquely to the Sun's direction, reducing thrust and letting orbital instability mechanics slow its rate of drift down.

But how can it change the direction of the Sun shield without changing the direction that the telescope is pointing?

Here is where some degree of balance between science and delta-v might be found, but it's a soft one. For many situations the telescope can simply rotate around its optical axis and still make the same observation. There are a range of such rotations where the sunshield can block the Sun's heat but vary the direction of the reflected light and thus modulate the photon thrust's magnitude along the Sun-Earth axis. For some observations, the flexibility may be less than for others, but these things will be managed in detail as part of the observation proposal and scheduling phases.

In short, some of the delta-v budget is paid for without the use of propellant, by clever maneuvering of the telescope's sunshield by rotating around the telescope's optical axis.

The rest will be done using the spacecraft's thrusters.

For more on all of this refer to:


1Of course the accuracy of perception of the limits on the budget and schedule during the design phase were inaccurate in a Mark Twain sort of way (and even that's a misquote it seems 1, 2, 3).

2If all goes well, it's likely that the telescope will continue to be able to do science well beyond this limit, Getting into that is a great topic for a new question if it's not covered by existing Q&A.

3an "ideal" halo orbit might be the one that has a minimum amount of delta-v for station keeping. There's no exact definition.

4 There is a lot of material about SOHO in and the relevant documents here are

There is a page of recovery docs or you can read about it in Aerospace America May 1999: Saving SOHO or the article by ESA's F.C. Vandenbussche SOHO’s Recovery – An Unprecedented Success Story or for more of the technical details; Roberts 2002 The SOHO Mission L1 Halo Orbit Recovery From the Attitude Control Anomalies of 1998.

with particular emphasis on Roberts 2002.

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  • $\begingroup$ Interesting! Lots of info to digest. I retain the idea that JWST may use a kind of solar-sail station-keeping (but looks complicated and risky, at first examination). $\endgroup$
    – Ng Ph
    Dec 3, 2021 at 23:29
  • $\begingroup$ @NgPh there really is no choice. The giant reflective sunshield is necessary to keep the infrared telescope cold, and so the solar photon thrust is always going to be there and have to be managed. If you don't use it, you have fight it, and that would cost a lot more propellant. Exactly how much more would be the basis of a great new question! $\endgroup$
    – uhoh
    Dec 3, 2021 at 23:33
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My question was based on my presumption that the designers of JWST chose an “ideal” orbit as a target. Furthermore, I thought that their key driver would be the required fuel for station-keeping. From this, a logical deduction would be that this ideal target, if it exists, would be an orbit that requires no station-keeping at all, when the real world perturbations are neglected in a first step. Then a station-keeping strategy would be engineered to take care of these perturbations, assumed anyway small compared to the main forces included in the ideal model used to derive the "zero station-keeping" orbit.

My question also presumed that JWST would be launched and maneuvered to follow a halo orbit. This is because a previous discussion, following @uhoh's question Are some Halo orbits actually stable?, confirmed that a family of stable halo orbits does exist at Earth-Sun L2, at least theoretically. This, class of orbits, however small, I thought, would represent the starting point for the design of the JWST orbit.

Note that, contrarily to ISEE-3 that was kept at an L1 orbit for only 4 years, the goal for JWST is up to 10 years of operation. Farquhar told us that ISEE-3 required 10 m/s/year of Delta-v for station-keeping. The station-keeping budget for JWST can not exceed 83.5 m/s of Delta-v (150 m/s overall minus 66.5 m/s for transfer to libration point orbit). The reaction wheels used for attitude control will take a part of this “in-orbit” remaining Delta-v budget.

Subsequent researches I conducted revealed the following:

  • The designers of JWST’s libration point orbit did not restrict themselves to a solution in the subclass of halo orbits.

This information is available from the orbit page of JWST User Documentation Home Page . The page has a sentence that reads:

The L2 orbit shape is not constrained, so torus orbits, halo orbits, or Lissajous orbits are acceptable and are determined primarily by the launch's time of day and day of year.

Postponing for the time being the details on what these 3 classes of orbits (“torus”, “halo” and “Lissajous”) mean exactly, we can settle already on the conclusion that the designers of JWST did not restrict themselves to the class of halo orbits, nor to any other subclass. Rather, any of the mentioned classes has a member that satisfies the mission requirements, including the 10-year station-keeping. Furthermore, we learn here that which of the 3 classes will be the actual followed by JWST is determined primarily by the launch epoch.

The paper from J. Brown, J. Petersen, B. Villac and W. Yu (2015) Seasonal variations of the JWST Orbital Dynamics provides additional details on this dependence. Here we learn that for each day there is a launch window starting at 11:30 UTC and ending at 14:00 UTC. In particular on page 6, can be read:

Lissajous orbits are common early in the launch window, slowly transforming into halo orbits around 13:00, and finally becoming quasi-halo at the end of the daily window.

(confusingly, this page from NASA states: Webb's launch trajectory sets it up into a halo orbit. Perhaps what is stated is just an example of the many possible launch trajectories and the statement was taken out of context)

Observe that, through out Brown’s paper, nowhere can we find a mention of “stable halo orbits”, nor can we find a mention of a stable Lissajous orbit. Rather, whenever the concept of stability was discussed in the paper, it was used together with the term “manifold”. For example, on page 13, can be read:

Figure 13 presents sample trajectories in the stable manifold of a halo orbit. Other LPOs have corresponding stable manifolds, including quasi-periodic orbits.

enter image description here

A stable/unstable manifold for a class of orbits with a given geometry is not synomymous of “stable/unstable orbits” in that particular geometry. My current “take away” is that, if we are looking for some form of stability, all three classes of LPO possess “stable manifolds”, and this would “do the job” according to Brown et.al. Doing the job means (thanks to @uhoh's comment) that if the spacecraft is on the stable manifold connecting to a given orbit, sooner or later it will reach that orbit, with zero fuel spent. On the other hand, if there is an unstable manifold that connects to the orbit, a spacecraft on the orbit and displaced slightly by a small perturbation force from the orbit will leave the orbit (exponentially in time). Hence the job of station-keeping is to keep the spacecraft from inadvertently "stepping into" a nearby unstable manifold.

  • So, how did they design it? (and who were they?)

A paper from David Folta, Steven Foley and Kathleen Howell (2001) Trajectory Design Strategies for the NGST L2 Libration Point Mission, shed further light on the approach used. NGST (New Generation Space Telescope) is the former name of the JWST.

First, we are reminded of the main requirements of the NGST (aka JWST) mission (Table 1).

enter image description here

Then we are told that, instead of the traditional “shooting method” approach, an improved one based on Dynamical System Theory (DST) was exploited. This is where the concept of “manifold” comes in the picture, for example quote (page 4):

An invariant manifold is defined as an n-dimensional surface such that an orbit starting on the surface remains on the surface throughout its dynamical evolution. So, an invariant manifold is a set of orbits that form a surface. Invariant manifolds, in particular stable, unstable, and center manifolds, are key components in the analysis of the phase space.

How these geometrical concepts (stable/unstable, center manifolds) are rigorously defined and how they are actually computed require quite advanced mathematical tools. In fact the paper doesn’t go into deep explanations (at least IMO) on these different types of manifolds. Nevertheless, the paper revealed some unique constraints of the JWST orbital design. In particular:

  • The size of Y-amplitude has a large impact on the Delta-v for orbital transfer (Table 3): Although a 400 km Lissajous Y-amplitude exists, it would require 123 m/s of Delta-v transfer.
  • Station-keeping thrusts are only allowed to push the JWST away from Sun. This is to avoid contaminating the optical instrument.
  • Stability and station-keeping using single-axis maneuvers only: X-axis (I don’t understand whether this is a requirement or a feasibility or a desirable solution)
  • Station-keeping taking into account the constant Sun Radiation Pressure (SRP).
  • No Eclipse-Avoidance maneuvers budgeted.

This partial answer obviously contains holes. I hope that somebody will be able to fill these.

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    $\begingroup$ "Whether a stable manifold for a class of orbits with a given geometry is synomymous of 'stable orbits' in that particular geometry, I don’t know." Searching my user ID user:12102 plus "manifold" and "unwind" finds two discussions of what the manifolds are. Basically they are "tubes" made up of all the trajectories into (stable) and out of (unstable) halo orbits, not the orbits themselves. To get into a halo orbit you want to insert into one of the paths that make up the stable manifold, then you'll just "slide in" $\endgroup$
    – uhoh
    Dec 13, 2021 at 22:35
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    $\begingroup$ @uhoh, thanks for the explanation on stable/unstable invariant manifolds. So, I understand: stable -> leads you to an orbit (halo or else); unstable-> leads you out. Hence, to go (effortlessly) to a target orbit, put yourself on its stable manifold(s) and stop maneuvering. Likewise, to escape from an orbit, choose an unstable manifold connecting to the orbit, step into it and there you go (exponentially). Enlightening! But then, for a stable halo orbit there is no unstable manifold connecting to it, correct? $\endgroup$
    – Ng Ph
    Dec 14, 2021 at 12:48
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    $\begingroup$ OK, so it launched at 12:20 UTC on Dec 25th (YAY!). So I guess that means it is in a Lissajous orbit? How/when/where can we get more precise descriptions of the orbit and implications for the rest of the questions here? Many thanks for all your insights, folks, including the cool code from @uhoh! $\endgroup$
    – nealmcb
    Dec 26, 2021 at 4:57
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    $\begingroup$ @uhoh I've explored many of the answers, but I only found ones from before the actual launch. And they suggest that launches before 13:00 actually lead to non-halo Lissajous orbits. Has anyone plugged in the actual launch time and presented a best guess for the expected orbit? $\endgroup$
    – nealmcb
    Dec 26, 2021 at 19:21
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    $\begingroup$ @nealmcb then you are way ahead of me! This sounds interesting, I'd never read that so "I'm all ears" so to speak. I suppose being "a little bit non-halo" i.e. the in/out-of plane periods slightly different might not make any difference; if it goes around ~20 times in its ten year mission and it's shape in the synodic frame looks somewhat elliptical but there are no sunlight issues nor significant rise in station keeping delta-v per year, then it doesn't really matter much. Do you think this rises to the level of a new question? $\endgroup$
    – uhoh
    Dec 26, 2021 at 19:40

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