How to Orbit a Lagrangian Point

from Michael Khan, 28. May 2009, 09:31

I was tempted to title this post "Lagrange Orbits for Dummies" but then thought better of it ... not just because that wouldn't have been very original, but more importantly, because anyone who is interested enough to want to learn about such topics is anything but a dummy.

(Hier geht es zur deutschen Version dieses Artikels)

If you followed the news lately you'll know that we're not discussing an arcane theoretical concept here. On May 14, the European Space Agency sent two space telescopes, Herschel und Planck, on the way to the Lagrange point L2. The James Webb Space Telescope, the ESA's astrometry mission Gaia und certainly a host of other spacecraft will follow in the years to come. So what are these Lagrange points and what is so special about them?

In the eighteenth century the Italian mathematician Herschel 3D-Modell, Quelle: ESAJoseph-Louis Lagrange discovered that in a system of two massive bodies (such as the Sun and a planet, but also a planet and one of the moons orbiting this planet) there are five "equilibrium points", which were called the Lagrangian points in honour of their discoverer. Out of these five, two are of particular interest for man-made spacecraft: L1, located between Earth and Sun ad L2, located "behind" the Earth as seen from the Sun. Each is at a distance of about 1.5 million km from the Earth.

L1 is useful for solar observatories, while L2 is the perfect location for space-based telescopes. It offers an unobstructed view of the entire celestial vault in just half a year. Moon and Earth are far away and more or less in the some apparent direction as the Sun. It is easy to ensure that none of these bight bodies enter the field of view of sensitive detectors on board the spacecraft.  TYhe hill spheres and the Lagrangian points, source: Wikipedia 

In the following, when I say that an object is placed "in a Lagrangian point", don't take that literally and keep in mind that we are talking about regions that have a somewhat fuzzy boundary.

An object placed "in a Lagrangian point" is in fact in an orbit around the Sun. This is a very fundamental point; you have to keep it in mind to understand the following explanations. 

The laws of celestial mechanics dictate that the lower an orbit (= the smaller its orbital radius), the higher its velocity. An object in the L1 point is closer to the Sun than the Earth is. So it should be flying faster than the Earth does. Conversely, an object in the L2 point is slower than the Earth. In both cases, the distance from the object to the body to the Earth will increase (I won't go into the difference between vectorial and angular velocity here,  but strictly speaking, I should). In fact, for any other type of heliocentric orbit, this is what one would see, but here, we have an exception from the general rule. 

Also, an orbit around a body of a low mass has a smaller velocity than an orbit of the same size around a heavy body. 

Picture an object in the L2 point. This will be drawn "inwards" by a force: the solar gravity. But wait - there is an additional force! The Earth's gravity is pulling the same direction. Of course, the Earth has a much smaller mass than the Sun. But it also is much closer to the object, only 1.5 million km. The Sun is about 100 times farther away. If we combine both forces, it will appear to the object as if it were orbiting around a star that is slightly larger than the Sun actually is. Therefore, the object will have a slightly higher speed than actually corresponds to its distance from the Sun, and it will complete one orbital revolution in exactly 365.25 days, as the Earth does. Without the added effect of the Earth gravity, it would be a bit slower and need about 5.5 days more to circle the Sun once. 

The situation in the L1 point is the exact opposite. There, the Sun and Earth gravitational attraction are acting in opposite directions. So to the object in L1, it appears as if the Sun were a but smaller than it actually is and its orbital velocity is slightly reduced - it again requires exactly 365.25 days to revolve around the Sun once. 

Obviously, this "trick" only works if Sun, Earth and object are almost aligned such that the forces are super-imposed as described. The more an object is displaced from the ideal location, the faster it will move away. This is the case of an unstable equilibrium, similar to that of a marble balanced on the tip of a sugar cone.  

Gaia-Astrometrie-Teleskop, Quelle:ESAIn reality, it is not possible to place anything directly in the Lagrangian points, because these are not stationary locations relative to the Earth, but vary because of the eccentricity of the Earth orbit and the fact that additional forces are acting. It wouldn't be desirable to have a spacecraft directly in the Sun-Earth-line. In the L1-point, the spacecraft, seen from the Earth, would always appear against the intensely bright backdrop of the Sun, rendering radio communications difficult. Conversely, in the L2 point, the spacecraft would permanently fly in the Earth penumbra. What's more, maintaining the position exactly in these theoretical points would require a prohibitive amount of propellant. 

Luckily, it isn't necessary to remain exactly in the equilibrium points. If the heliocentric orbit of the spacecraft is slightly eccentric and a bit inclined with respect to the Earth orbit plane, its trajectory, seen from the Earth, will appear to trace an ellipse-like shape (Lissajous curve) about the respective Lagrangian point. There are different types of Lissajous curves. Without going into too much detail, I'll subdivide them into "wide loops" and "narrow loops". On these curves, the spacecraft's distance to the actual Lagrangian point can be anywhere between several hundreds of thousands up to well over a million km. You see that it is not necessary to remain too close - it's enough to ensure that the spacecraft doesn't stray from the region surrounding the actual Lagrangian point.

Grafische Darstellung einer Lissajous-Bahn samt Transfer, Quelle: ESA One revolution on such an "apparent loop" (that is really just an effect of presenting the perturbed orbit of the spacecraft around the Sun in a coordinate frame rotating with the Earth along its orbit) takes half a year. The spacecraft's speed relative to the Earth is low; we have a case of near-formation flying here. This explains why it takes two months or more from launch to reach the operational orbit: The orbital energy is only just sufficient to keep the spacecraft from falling back towards the Earth, so at large distances, the velocity is extremely low and the spacecraft will hardly advance - its relative speed will be a mere few hundreds of meters per second.  

Whether a "wide loop" or a "narrow loop" is chosen depends on the technical and scientific mission requirements. The infrared telescope Herschel will be inserted into a wide Lissajous curve, the Planck spacecraft needs a "narrow loop".

Both types are unstable. If it were possible to compute even the most minute orbit perturbations exactly and to obtain an absolute precise launch, then they would theoretically remain on their orbits forever without any corrective manoeuvres. This is of course unfeasible in practice; real-world Lagrangian point orbits need to be controlled. However, a small amount of propellant suffices to ensure that the spacecraft will not escape into deep space or fall back towards the Earth during its mission lifetime. 

The dynamics of orbits in these unstable equilibrium regions is tricky, but manageable. You will hear people say that this region is "chaotic", which means that it would be unpredictable and therefore uncontrollable. This is not true - the solar observatory SoHO has been flying on its special form of wide loop (a Halo orbit) about the L1 point  for over ten years.

Controlling the orbits of Herschel and Planck will be a challenge for sure, but one that can be met. An interesting problem for the spacecraft engineers and flight dynamicists.

Herschel's (blue) and Planck's (red) tranjectories. Source: ESANot only the operational target orbit is of interest, but also the way to get to it. The orbits about Lagrangian point fork into energetically similar shapes, so-called manifolds. These manifolds can lead to very different places in the solar system, which has a very concrete significance: It suffices to place a spacecraft onto a manifold, and months later it will end up in the associated target orbit, requiring just minimal manoeuvres with the onboard rocket motors. Difficult to envisage, but mathematically and even operationally proven - and an extremely useful feature. (Aside: The existence of such manifolds is also the basis for theories concerning the "interplanetary superhighway" where multi-body dynamics is employed to fly across the solar system with lesser propellant expenditure - but probably very long flight durations.) 

The "wide loop" that Herschel shall take forks into a manifold that approaches the Earth at an altitude of a few hundreds of kilometres, so by launching at the right time and into the right direction and final velocity, the spacecraft is placed right into the manifold and is on the right way.  

Transfer in engle-Lissajous-Bahn mitttels Mond-Swingby, Quelle: NASA-GSFC With Planck's "narrow loop" things are a bit trickier. The manifold forking away from the operational orbit does not get close to the Earth but passes at a distance several tens of thousands of kilometers. Reaching this manifold implies inserting first into a very eccentric ellipse and then, when crossing the manifold, firing the spacecraft's rocket motors, applying a significant manoeuvre to "turn" into the manifold. Alternatively, as the manifold intersects the lunar orbit, one could get the timing exactly right and achieve the "turn" into the manifold via a "lunar swingby", a close pass near the moon such that the lunar gravity re-directs the spacecraft trajectory and thus imparts the required momentum.  

Planck will use its engines, but later missions will certainly make use of the lunar swingby and thus save propellant.

Although I tried to keep it simple and make do without calculus and physics, some of the issues involved may appear complicated and even somewhat counter-intuitive. However, you should start getting used to the concept, because we'll be seeing lots of spacecraft going to L2 in the years to come. It really is an ideal place for space telescopes - and it has another advantage: Because of the unstable equilibrium conditions, Lagrangian points are inherently self-cleaning. Unlike low Earth orbit, defective or obsolete spacecraft, upper stages and other space debris cannot accumulate there.

Das James Webb Space Telescope, Quelle: ESAL2 will remain clean.

Further Information

In this presentation the mathematician and ESA mission analyst Martin Hechler, one of the leading experts on the design and calculation of orbits about the Lagrangian points, gives a comprehensive insight into the topic, delving deeply into the mathematics I tried to avoid in this post. 


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  1. Nat Well Done
    07.06.2009 | 14:15

    This is conceptually helpful, and much appreciated. Thanks for taking the time to explain this.

  2. Corneel orbit around at the sun
    01.07.2009 | 12:21

    At the risk of sounding like a complete nitwit:

    I always thought that telescopes and other observatory spacecraft were in earth orbit, just like communication satellites. So that's one thing you set me straight on. The part about Lagrangian points is a nice bonus, haha.

  3. Michael Khan @corneel
    02.07.2009 | 09:49

    Communications satellites are in many kinds of different orbits. Some are in geostationary orbits, almost 36000 km above the equiator. But others are in much lower, circular orbits, and other again are in highly eccentric orbits.

    Orbital telescopes also mostly have been in Earth orbits up to now, some in low, circular orbits, some in eccentric orbits.

    So in fact, your perception was quite right: orbital telescopes in fact are in the same kinds of orbits as (many) communications satellites.

    Now, we will see more and more of them in Lagrange orbits, but still, you were correct.

    In fact, if I may just add that one remark: Intuitively,one might think that the farther out you go, the more propellant is required. That is true to a point, but not quite: Actually, it costs less propellant to place a satellite in L2 than, for instance, in geostationary orbit, or even in a highly eccentric orbit, when the perigee is later raised so it won't graze the Earth atmosphere.

    This is certainly counter-intuitive, but nevertheless true. That means that with the same rocket, one gets more payload to L2 than into many eccentric orbits.

    Just one of the many unexpected aspects of celestial mechanics.

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