How to modify patched conics approximation to include Lagrangian points?

[Urwumpe]

I am just looking for some really simple model to describe interplanetary trajectories, and of course, the simplest and coarsest model there is the patched conics approximation.

The problem is for the task that I plan to use, I would also need some simplified way to describe Lagrangian points and halo orbits around it, which are impossible to describe because of the chaotic nature of them. I can live with some huge errors there, because the model is supposed to be just a first-order approximation, but I just can't find a good way to include them into the model.

So far, my idea is to have different propagation models: Kepler orbits and weak stability boundary, and switch between the two models in the propagation.

For example:

Earth elliptic -> WSB (Lunar L1)

Earth hyperbolic -> WSB -> Solar elliptic orbit -> WSB -> Jupiter hyperbolic -> WSB -> Solar hyperbolic orbit -> WSB -> Saturn hyperbolic -> WSB -> Titan hyperbolic

But this just feels a bit complicated, also describing targets and transfers in the WSB, like a L1 point or transfer between L1 and L2 is still not solved that way.

I don't want to use an iterative N-body solver for the task, since this makes it harder to put the navigation into its context of the mission (Telling where you are under the assumptions of the flight plan)

 

[Miner1]

You can fairly accurately describe a halo orbit using the Poincare-Lindstedt method of successive approximations. As a reference, see

Richardson, David L. "Analytic construction of periodic orbits about the collinear points." Celestial mechanics 22.3 (1980): 241-253.

This paper also provides a constraint on the orbit amplitude within and out of the orbit plane of the primaries. The constraint is necessary to ensure a closed orbit, and if not met, the frequencies of oscillation within and out of the orbit plane will not match and you will end up with a Lissajous orbit instead.

As for getting to and from a halo orbit, Dr. K. C. Howell has used invariant manifold theory for generating low thrust trajectories to reach orbits about the Lagrange points. She has also done considerable research in methods of stationkeeping in those orbits. Here are a few references

Howell, Kathleen Connor. "Three-dimensional, periodic,‘halo’orbits." Celestial Mechanics 32.1 (1984): 53-71.


Barden, B. T., K. C. Howell, and M. W. Lo. "Application of dynamical systems theory to trajectory design for a libration point mission." Journal of the Astronautical Sciences 45.2 (1997): 161-178.

Howell, K. C., and H. J. Pernicka. "Station-keeping method for libration point trajectories." Journal of Guidance, Control, and Dynamics 16.1 (1993): 151-159.


M. T. Ozimek and K. C. Howell. "Low-Thrust Transfers in the Earth-Moon System, Including Applications to Libration Point Orbits", Journal of Guidance, Control, and Dynamics, Vol. 33, No. 2 (2010), pp. 533-549.

Hopefully that helps a little.

 

[Keithth G]

Basically, you can't modify patched conics to include Lagrange points.

The underlying principle of patched conics is the standard Keplerian two-body system of one object orbiting another. The reference frame for patched conics is inertial. And the minor planetary perturbations arising from the gravitational interaction of third bodies is modelled by a system of 'patches'.

The under principle of Lagrange points is a three-body system - with two massive bodies in a standard Keplerian orbit around each other (e.g. the Earth and the Moon; or the Sun and Jupiter; or the Earth/Moon system and the Sun) with a light third body, a spacecraft say, in motion close to one of the Lagrange points. Now, the Lagrange points only make sense in a rotating (non-inertial) reference system co-rotating with the massive bodies and where the gravitational influence of the two massive bodies mostly offset each other (and the centrifugal force arising from the shift to a rotating reference frame. Close to the Lagrange points, Keplerian motion (i.e. conics) simply isn't a useful description of the motion.

If you are close to a Lagrange point the Poincare-Linstedt procedure will give you an approximate analytical description of orbits around Lagrange points. In effect, for Lagrange points, these approximate solutions replace the Keplerian system of patched conics. The solutions also describe the stable and unstable manifolds of the orbits about the Lagrange points. The stable manifold is the surface in space-velocity space that one has to be on if one wants a 'free ride' to the Lagrange point. And the unstable manifolds describe the 'free ride' surfaces away from the Lagrange point.

Now, the stable and unstable manifolds can be thought of as surfaces that extend far out into space. And since the Lagrange points of different systems (e.g. Earth/Moon, Sun/Jupiter) will have their own stable and unstable manifolds, these manifolds will intersect at certain points. Moving between the Lagrange points of different systems, then, consists of leaving the Lagrange point of one system along its unstable manifold, and then at the point of intersection, transitioning to the stable manifold of the Lagrange point of the second system. These manoeuvres provide a system of low energy transfers between planetary systems and, for this reason, are of interest.

It goes without saying that none of the current suite of navigational aids available to Orbiter 2010 address this problem.

 

[HopDavid]

I am just looking for some really simple model to describe interplanetary trajectories, and of course, the simplest and coarsest model there is the patched conics approximation.

The problem is for the task that I plan to use, I would also need some simplified way to describe Lagrangian points and halo orbits around it, which are impossible to describe because of the chaotic nature of them. I can live with some huge errors there, because the model is supposed to be just a first-order approximation, but I just can't find a good way to include them into the model.

So far, my idea is to have different propagation models: Kepler orbits and weak stability boundary, and switch between the two models in the propagation.

For example:

Earth elliptic -> WSB (Lunar L1)

Earth hyperbolic -> WSB -> Solar elliptic orbit -> WSB -> Jupiter hyperbolic -> WSB -> Solar hyperbolic orbit -> WSB -> Saturn hyperbolic -> WSB -> Titan hyperbolic

But this just feels a bit complicated, also describing targets and transfers in the WSB, like a L1 point or transfer between L1 and L2 is still not solved that way.

I don't want to use an iterative N-body solver for the task, since this makes it harder to put the navigation into its context of the mission (Telling where you are under the assumptions of the flight plan)

There is a Farquhar route from LEO to EML2 that takes about 9 days.

It takes about 3.1 km/s to depart LEO then a .18 and a .15 km/s burn to get to EML2.

The orbit's time reversible so about .33 to drop from EML2 to a near earth perigee. The ship would be moving just under escape at perigee, about 10.8 km/s.

An elliptical orbit with a 300 km altitude perigee and a 320,000 km apogee (EML1 height) has perigee velocity of about 10.8 km/s and apogee velocity of .22 km/s. EML1 moves at the same angular velocity as the moon so it is moving only .87 km/s rather the 1.1 km you'd expect if it were a 2-body Keplerian orbit. .87 - .22 = .65. It takes about .7 km to drop from EML1 to a near earth perigee. At perigee it'd be traveling about 10.8 km/s

A 300 km altitude LEO is about 7.7 km/s. 10.8 - 7.7 is 3.1. So a trans EML1 burn from LEO would be about 3.1 km/s. Then at apogee it takes about .7 km/s to park at EML1.

Whether dropping from EML1 or EML2, a 300 km altitude perigee velocity is about 10.8 km/s in both cases. That can help in determining insertion burns into hyperbolic orbits. For example Trans Mars Injection at 300 km altitude is about 11.3 km/s. 11.3 - 10.8 is .5. So if moving 10.8 km/s at perigee, another .5 km/s suffices for Trans Mars Injection.