So these Lagrangian Points are staples of the pulpy science fiction books that I’m so fond of. I get why they would work theoretically- they’re the balance points between the two massive objects in question. What I don’t understand is how anything can get balanced so perfectly on such a fine point- it seems like you’d need a miracle for something to slide gently into this orbit and stick there, perched on the precipice of two gravity wells. But I know that it happens, because I know that Jupiter has a whole bunch of asteroids in its L4 and L5 points and I know that we’ve put satellites in our own Lagrangian points. So what gives? How do we balance these things so perfectly?
Also, we’ve been discussing the Ptolemaic universe in my Intro Astronomy class (I know, my school misspells “Science Distribution That Even Theatre Majors Can’t Fail”) and I’ve seen that Ptolemy has Venus on an epicycle centered on an imaginary line between the sun and Earth. Could you reasonably orbit Venus around the Lagrangian point between the Earth and the sun and come up with an orbit for a planet that hung out near the sun in the sky like Venus and otherwise conformed to all basic naked-eye observational activity of Venus? I understand that this would be easily falsifiable with a telescope that could see the phases of Venus, but failing that, could such an orbit exist?
Okay, one thing that you do need to understand about these ‘balance points’ - they’re not just points where gravity is balanced, they’re orbital spots where gravity and velocity make it possible for a third body (of negligible mass) to enter a relatively steady state with respect to a system containing two gravitational masses.
L1, L2, and L3 are ‘saddle shaped’ balances - the forces towards and away from the planets (or stars) in question are balanced, but it’s easy to slip out sideways, which is why they’re generally unstable. L4 and L5 are actually balanced in a very slightly self-correcting way - if you’re not quite in the correct L4 orbit or you get a foreign force pulling you very slightly away, the gravity of the two major bodies will act to bring you closer back in to the L4 spot.
This is the secret of how foreign bodies can actually get captured by a trojan point, as I understand it. You need to have something flying into almost the right orbit on its own devices, (or charted into such a course by a rocket booster,) but the ways the laws of gravity and inertia feed into such points makes it easier to land on that summit. It’s not a precipice so much as a bowl-shaped plateau, to mangle your analogy slightly.
I am not a rocket scientist. (How often do you get a chance to use that one?)
And to illustrate it, here’s a contour plot (2nd figure down) of the gravitational potential in the coordinate system co-revolving with the secondary mass. (The secondary mass would be the Earth, if the primary mass is the Sun.) The L4 and L5 points are stable, because they’re at the bottom of “valleys” in the potential function.
In the past we have put space probes at Earth’s L1 and L2 points. Although these are not stable in the way L4 and L5 are, they do require only a modest amount of fuel consumption to keep the probes in the general vicinty.
(I just noticed there’s another contour plot, possibly more helpful, at the second link.)
If I understand your question correctly . . . all of the Earth Lagrangian points are fixed with respect to the Earth. So, you couldn’t “orbit” Venus from an Earth Lagrangian point, because Venus is revolving around the Sun with a different period.
The two Stable Lagrage points (L4 and L5) are located very nearly 60 degrees ahead of the planet (or moon) and very nearly 60 degrees behind it in the orbit, so you can’t put your satellite just anywhere – these are pretty well defined locations. It’s the reason George O. Smith called his Lagrange-orbit station at the L4 or L5 point in Venus’ orbit “Venus Equilateral” in the series of stories he wrote in the 1940s (with two checking in much later).
Objects located at these points make kidney-shaped orbits, bounded by gravity and (if you go into the rotating coordinate system) “centrifugal force” and “Coriolis Force”. If you pick up too much energy you’ll break out of that delicate orbit, because it’s not a deep gravity well. Getting an object caught precisely at that point must be a real trick – like that carnival game where you try to pitch pennies into a plate, and get to keep it if the coin stays on the plate. Most of the time a random object gets to the spot, its velocity will be too high or too low, and it’ll just jump out. That there actually are “trojan” asteroids at Jupiter’s Trojan points is, I think, a tribute to the power of large numbers – throw enough debris at that spot and eventually something will have just the right combination of position and velocity to stick.
Some clarification on “hilltops”: The OP should know that, in science and engineering, there are two ways to balance an object. It can be statically balanced or dynamically balanced. In simple terms, I was taught to think of the former as a ball at rest at the bottom of a valley, and the latter as a ball at rest at a hilltop…as described above. Hope that may help complete the picture.
Keep in mind that whether a point is a “hill” or “valley” depends on your frame of reference. For example, if you just look at the two body system of the earth and the moon, the moon is obviously in a stable position, as it is not currently moving either toward earth or away from it (okay, yes, actually it is, but not so fast that it’s going to become unstable anytime soon). However, imagine that the moon was somehow removed, and a new object was placed at the moon’s former position. Whether it would orbit the earth, crash into it, or drift off into deep space, depends on the velocity of the object. Same thing with the Lagrange points. The L4 and L5 points are stable only for an object corotating with the two larger bodies.
(How often do you get a chance to use that one?)