| From: Chris W (Plebeian) | 4/05/99
9:12:05 |
| Subject: Lagrangian Points in Sun/Earth System | post id:
9470 |
| I was reading about the solar
observatory probe, SOHO, and how it sits around the L1 Lagrangian Point
about 1.5 million km from Earth towards the Sun. In a rough sense, I can
see that gravitaional force from the Sun and Earth balance at this point
and tend to keep objects there. There are three other points roughly: +/- 60o on the orbit of Earth, and a point outside the orbit of Earth but on a line with the Sun and Earth. Can someone enlighten me as to why these points qualify, and whether they are stable? | |
| From: Chris (Avatar) | 4/05/99
11:39:13 |
| Subject: re: Lagrangian Points in Sun/Earth System | post id:
9496 |
Imagine launching a satellite into space to orbit the sun instead of earth. Now imagine we want that satellite to stay between the earth and the sun to monitor effects of the solar wind before they reach earth (for example). Well, Huston, there is a problem! Consider two objects orbiting the sun, E and S. E is the earth and has an orbital radius R. S is the launched satellite and its orbital radius is less than R because we want it to be closer to the sun. Now according to Kepler's laws of planetary motion the object with the smaller orbital radius will have a smaller orbital period. This means it won't stay between the earth and sun, but will move ahead of the earth and away from it. That is if Kepler's laws were spot on correct! The "Lagrangian" is a method of mathematically analysing dynamic systems without using forces. There are a series of Lagrangian equations which deal with perturbations in Kepler's laws and attempt to fix them. By exploiting a perturbation, we can get an answer to our problem above! A satellite which sits directly between the earth and sun will be pulled towards the sun by the sun's gravity and towards the earth by the earth's gravity. In effect, the earth's gravity is reducing the sun's. There is a point on this line where the sun's gravity is sufficiently reduced that the solar orbit of a satellite would slow down to equal that of the earth. This is the first Lagrangian point, L1. There is a similar point behind the earth - this time earth's gravity adds to the sun's which speeds up the satellite until its orbital period equals earth's. From memory I think this is L3. L4 and L5 occur in the orbital of the moon, but splayed by 60 degrees either side. These positions are quite stable due to being equidistant from earth and moon. An object which is displaced from either L4 or L5 would tend to return to L4 or L5 - in this sense they act like minimum potentials. There has been some suggestion that perhaps one of these points would be a candidate for an artificial space civilisation, but not in the next few decades!! Hope this helps! Chris | |
| From: Chris W (Plebeian) | 4/05/99
12:02:32 |
| Subject: re: Lagrangian Points in Sun/Earth System | post id:
9503 |
| Yep, bollocks up that last
post... It certainly does, thankyou. Did I miss L2 in that discussion? I found a reasonable picture at Nine Planets: Lagrange points in case anybody is wondering what we're talking about. It seems there are similar points on the orbits of the planets (seems general to a three body system). Jupiter has collections of asteroids called the Trojans in the vicinity of two of its Langrangian points. | |
| From: Chris (Avatar) | 4/05/99
12:18:01 |
| Subject: re: Lagrangian Points in Sun/Earth System | post id:
9507 |
Sure is. That was why I used the neutral references E and S (although we used them in terms of the earth and a satellite). What you need to make it work are some sizable differentials. The sun is much bigger than the earth or Jupiter, etc, and the satellite must be small enough that its mass doesn't contribute another order of perturbations! If you want to see the Langrangian differential and partial differential equations, you could have a quick look http://scienceworld.wolfram.com/physics/LagrangesPlanetaryEquations.html (be warned - they're presented and worked, not explained!). L2 lies on the sun-earth line as well, near L1, but uses the moon as well, I think. I also forgot to mention earlier that L1 and L3 are not stable, probes in these positions need position accounting rockets. You will also find that the probe at L1 sits just off L1 for reasons of attenae positioning, and so requires additional corrections. Hope this helps! Chris | |