4
$\begingroup$

The question Did DSCOVR travel “along the stable manifold of its future SE L1 Halo orbit” to get there? is specific to DSCOVR's trajectory from Earth to its primarily heliocentric orbit near Sun-Earth L1, which is a Lissajous oribit.

Here I would just like to ask the general question: Do non-halo1 Lissajous orbits have stable/unstable manifolds? For those specific orbits where the horizontal and vertical periods are rational fractions (e.g. 3/2, 5/4) would the manifold just be a "cylinder-like" extension of the closed, periodic orbit? For example, if the ratio were 2:1 would it look like a figure-eight just extruded and stretched into the 3rd dimension?


1Halo orbits are a subset of Lissajous orbits where the oscillations in the plane of the two massive bodies (e.g. Earth's orbit around the Sun) and the oscillations out of plane (perpendicular to Earth's orbit) are the same, so that in the CR3BP approximation the orbit is perfectly closed and repeating. Lissajous orbits have that constraint relaxed, so the periods could be different by a tiny amount up to tens of percent or even more.

$\endgroup$
8
  • 1
    $\begingroup$ It is a pity that this question have not received enough attention. My intuition would say that they have (they are quite similar to Halos except for the periodicity) but it is merely opinion based $\endgroup$
    – Julio
    Aug 26 '18 at 0:23
  • 3
    $\begingroup$ In short- yes they do. In fact, last Wednesday at the Astrodynamics Specialists Conference in Snowbird, Utah, Ryan Russell from the University of Texas presented "Europa Lander Trajectory Design Using Lissajous Staging Orbits" by Ricardo Restrepo, Ryan Russell (both UT) and Martin Lo (JPL), in which he discussed Lissajous manifolds. In the paper, the authors show images of the manifold tubes-- over one rev of the Liss (or quasi-halo), the orbit looks very much like a not-quite-connected halo, and the manifold tube looks very much like a halo manifold tube. $\endgroup$
    – Diane
    Aug 27 '18 at 14:39
  • 2
    $\begingroup$ The final papers from the conference will be available in a month or so. $\endgroup$
    – Diane
    Aug 27 '18 at 14:41
  • 1
    $\begingroup$ I just found it online searching on the title at scholar.google.com. $\endgroup$
    – Diane
    Apr 22 '19 at 12:48
  • 1
    $\begingroup$ Unstable, definitely yes. Stable, nope not for Earth-Sun Lagrange points. There are quasi-stable manifolds, but not truly stable ones. Here "stable" would have limits for any orbit but if the orbit decay on the order of months, I don't really consider that stable when Earth orbiting spacecraft have stability for many years or even much longer. $\endgroup$ Feb 24 '21 at 16:22
2
$\begingroup$

It seems that they do. As @Diane noted in a comment, this paper (PDF freely available) discusses Lissajous orbits and approximation of their invariant manifolds: Europa Lander Trajectory Design Using Lissajous Staging Orbits

Abstract:

Lissajous orbits and approximation of their invariant manifolds are used to generate landing trajectories to the surface of Europa. Each lissajous is discretized into individual revolutions that each resemble a periodic orbit. The unstable man- ifolds of each individual revolution propagated forward in time generate more surface coverage than manifolds of simple libration point orbits such as halo or Lyapunov orbits. The stable manifolds propagated backwards in time from the individual lissajous revolutions provide direct connections to the last phase of a moon tour. The strategy developed produces ballistic landing trajectories with a wide surface coverage, and allows for the decoupling of the landing and moon tour phase by using the lissajous as an intermediate staging orbit. The multiple revo- lutions of the lissajous, multiple departure times along each revolution, multiple quasi periodic options at each energy, and multiple energies of the lissajous family provide many degrees of freedom in the design process.

Below are two figures as examples of the workflow to give a rough idea of what's being done, but a careful reading of the full paper is really necessary.

screenshot of figure X from Europa Lander Trajectory Design Using Lissajous Staging Orbits https://trs.jpl.nasa.gov/handle/2014/48660

above: "Figure 8. The approximated unstable invariant manifold of a full lissajous orbit (Ay = 2000 km, Az = 3000 km) and potential landing trajectories. a) 200 rev lissajous, 1 rev's manifold, 100 periapsis, b) 200 rev lissajous, 100 rev's manifold, 20000 periapsis"

below: "Figure 9. Manifold trajectories of a single lissajous rev with multiple periapsis passes. a) Manifold of a single lissajous rev (Ay=1000 km, Az = 5000 km, 𝜙=171). b) latitude/longitude map of periapsis."

screenshot of figure X from Europa Lander Trajectory Design Using Lissajous Staging Orbits https://trs.jpl.nasa.gov/handle/2014/48660

$\endgroup$
4
  • $\begingroup$ (@uhoh too) - If I understand correctly, "invariant" manifolds can be either stable or unstable (but not both?). This depends on the direction of their connection to the "fix" object they are associated to (an L-point, a periodic trajectory, a quasi-periodic one, ...). Leading to = stable, Leading out="unstable". All examples given in the above answers seem to be manifolds leading out of an orbit (an invariant) towards Europa. They do not demonstrate the existence of a stable manifold leading to a Lissajous orbit. $\endgroup$
    – Ng Ph
    Jan 9 at 11:31
  • $\begingroup$ For ex., if we can prove that Ariane (+ the first 2 MCC) took the JWST to a stable manifold associated to the final JWST orbit and if this orbit turns out to be a Lissajous, then we have demonstrated that we do not need to aim at a Halo orbit to get "zero fuel" navigation to the target orbit (from the point of insertion to the said manifold) $\endgroup$
    – Ng Ph
    Jan 9 at 11:40
  • $\begingroup$ @NgPh Here's my partial, non-expert understanding; the terms "stable" and "unstable" just indicate the direction of time. Since the system is lossless (no friction) Newtonian mechanics tells us it can run in either direction in time. The two sets of manifolds occur together because of this symmetry, so all you've got to do is run the problem in the other direction to have the "stable" manifolds. They didn't do it here simply because it wasn't the point of the paper. In the CR3BP "backwards in time" just means the two massive bodies just rotate in the opposite direction. $\endgroup$
    – uhoh
    Jan 9 at 17:42
  • $\begingroup$ @NgPh that's why all those red/green plots of in/out manifolds have mirror symmetry. e.g. i.stack.imgur.com/8g57B.jpg and i.stack.imgur.com/Fqr72.png $\endgroup$
    – uhoh
    Jan 9 at 17:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.