L2 orbit question.

[Fozzie]

At http://news.bbc.co.uk/1/hi/sci/tech/8040073.stm there is a diagram which shows the Herchel and Planck spacecraft orbiting the L2 lagrange point.

 

I am puzzled by this as there is no solid body with any mass in which to orbit around. Could somebody please explain what is going on to a non-scientific mind? Thanks.

[D H]

The math gets pretty hairy. This paper, http://www.aoe.vt.edu/~cdhall/papers/KIHAPaperAAS.pdf goes into the math in all its hairy detail.

 

The easiest way to get a picture of what is going on is to look at things from the perspective of a rotating reference frame with origin at the L2 point. The reference frame rotates at the Earth's orbital rate. Ignoring the eccentricity of the Earth's orbit, the Earth and Sun will have fixed positions in this frame. The Sun, Earth, and L2 point are collinear. Call this line the rotating frame's x axis, with positive x pointing away from the Sun and Earth. The rotating frame's z axis is normal to the Earth's orbital plane, with positive z pointing along the the Earth's angular momentum. Finally, the y axis is normal to x and z, choosing the direction to form a right-handed coordinate system.

 

Now consider what happens to an object located between the L2 point and the Earth (x coordinate is negative, y and z are zero) that is moving in the +y direction in the rotating frame. If the object's instantaneous velocity (in this reference frame) was zero, the object would experience an apparent force directed away from the L2 point (i.e., the -x direction). However, if the object's velocity is sufficiently large velocity, the coriolis force will counter this null velocity force, making the total apparent force be in the +x direction. The object will curve inward.

 

A tiny bit of time later, the object will have moved a tiny bit in the +y direction. It's velocity vector will have a small +x component. The gravitational force will have a small -y component. The x component of the acceleration will have changed a little, but is still pointed in the +x direction. The object again curves inward.

 

If you make that initial velocity just right, the object will follow a curved path around the L2 point and will come right back to where it started. In other words, it follows an orbit. This is called a Lyapunov orbit. The next step in complexity is to make the initial position below L2 and a bit out of plane. An initial velocity can once again be found that makes the object follow a closed path, this time called a halo orbit. The paths followed by Herschel and Planck are even more complex: They are called Lissajous orbits. These are not true orbits in the sense that they aren't a closed path. They are orbits in the sense that the state remains within some bounded region.

 

An object located exactly at L2 and moving exactly with L2 will stay there forever -- but that placement has to be exact and ignores perturbations induced by the Moon and the other planets, by non-circular nature of the Earth's orbit. The object could stay close to L2, but this would require constant application of thrust.

 

Just as L2 is unstable, so are Lyapunov orbits, halo orbits, and Lissajous orbits about it. However, the energy needed to maintain those orbits is less, a lot less, than the energy needed to stay exactly at L2. All vehicles that operate in the vicinity of one of the collinear libration points are placed in some kind of orbit about the point.

[Fozzie]

Thanks, but you lost me in the first sentence. I didn't do physics at school. What is a reference frame? Imagine you are explaining to your 90 yr old granny.

[SH3RL0CK]

How about a simpler (albeit incomplete) explaination? Consider the orbit to be around both the earth and the sun. Or maybe consider the orbit to be between the earth and sun. In other words, a satelite orbiting the L2 point is really orbiting both the earth and the sun at the same time.

[D H]

Thanks, but you lost me in the first sentence. I didn't do physics at school. What is a reference frame? Imagine you are explaining to your 90 yr old granny.

Do you understand the notion of an orbital plane? The Earth orbits the Sun in a plane (more or less). Planes in three dimensional space can be characterized in terms of a point on the plane and a normal to the plane. In the case of the Sun-Earth orbital plane, the Sun and Earth are points on the plane. Now think of what this orbit looks like from the perspective of the Sun. Hold your right hand with your thumb extended and fingers curled. Orient your hand so that the your curled fingers represent the Earth's motion about the Sun. Your thumb is pointing normal to the orbital plane. Call this direction in which your thumb is pointing "+z".

 

Now imagine you have a magical spacesuit that lets you stay at the L2 point indefinitely. Imagine orienting yourself such that the line from your feet to your head is pointing in the +z direction and the Earth and Sun are behind you. Now imagine you are rotating (with respect to the fixed stars) so that the Earth and Sun are always behind you. You are the origin of a reference frame, with +x forward, +y to your left, and +z up.

 

This is the rotating reference frame in which the diagram in the article is represented.

 

If you want to learn more about reference frames, google the term "reference frame". You really need to understand the concept if you want to have an explanation beyond "that's how it works".

[abilash81]

The Italian-French mathematician Joseph-Louis Lagrange discovered five special points in the vicinity of two orbiting masses where a third, smaller mass can orbit at a fixed distance from the larger masses. More precisely, the Lagrange Points mark positions where the gravitational pull of the two large masses precisely equals the centripetal force required to rotate with them.

 

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