And, just out of interest, the trajectory of an object falling off the inner colinear Lagrangian:

L1escape.jpg

It ends up confined to the inner slope of the potential surface, rather than moving locally around the Lagrangian point. In the non-rotating frame, of course, it's in an elliptical orbit with a period considerable shorter than the co-rotating frame. But it's doomed to have a close encounter with the secondary mass at some time in the future.

Grant Hutchison

d*rn straight! I'm in. I suppose the first step is to calculate something...

You can really see the instability/stability/instability aspects of this complex system from your pictures. It's obviously hard to make too many sweeping generalizations about what will happen. I also note that for L1, it only takes one orbit to get to the other side of the system (assuming each swoop is an orbit), whereas for the horseshoe orbits around L4, it takes about 10. Nevertheless, that's only an order of magnitude difference. Maybe it takes a pretty big perturbation to get into the horseshoe mode? A pretty neat simulation. You really ought to be able to buy rotating surfaces bent to the shape of that effective potential, and be able to launch beads or whatever to see these kinds of orbits in a demonstration or a tabletop toy.

You wouldn't want to use an effective potential map, as in Cornish's article, though; just a model of the gravitational potential of the two objects. Otherwise you have centrifugal force both embedded in the model and arising from the model's rotation.

Grant Hutchison

In fact, won't you need to distort your model gravity field in order to ameliorate the centrifugal force arising from its own rotation? Centrifugal force increases with the square of omega, whereas Coriolis force varies directly. (I'm assuming the reduction in modelled r will be offset by the reduction in modelled v.) If your model is to turn in less than the period of revolution of the modelled system (which is presumably desirable), it's going to need an "anticentrifugal" bias built in, isn't it?

Grant Hutchison

Oops. The hills have disappeared.

Oops, you're right, I was still thinking in terms of the magnetic field.

That might be a clever way to go, but I think more straightforward would be to simply tune omega to the "orbital period", as it were. Of course that wouldn't be Jupiter's orbital period, but you can tune it by altering the steepness of the potential hills. Choose the omega, then set the steepness to keep the bead in equilibrium at L4. But this is even better than an effective potential, as people can better picture the real potential I should think.

In fact, how cool would an amusement park ride based on this principle be? You'd have a very steep and slippery potential hill, and spin it up, and let people walk or slide around somehow. They'd find, despite their intuition about the shape of the hills, that they followed your orbits!

No, my worry was that coriolis and centrifugal seemed to follow different scaling laws in their relationship with omega, so the behaviour of a slow rotator wouldn't be reproduced by a fast rotator.
This didn't seem to make sense, however, since the stability considerations for the Trojan vicinity involve only the mass ratio and the assumption of circular orbits.
As an example, say we modelled the Jupiter-Sun Trojans by reducing the linear dimensions by a factor of 10¹¹ and increasing the angular velocity by a factor of 10⁷. This would produce something 16m across rotating once in 37s, so it would fit with your fairground ride.
Centrifugal force would scale with ω²r, so would increase by a factor of 10³. This could be compensated by reducing the modelled masses by a factor of 10¹⁹, so that gravity, scaling with M/r², would also increase by a factor of 10³.
So far so good. But Coriolis, scaling with ωv, looked like it should scale as 10⁷x10^-11 = 10^-4. Yikes. Didn't make sense, from the stability criteria.
What I was missing is that the velocity needs to scale not just with the change in linear dimensions, but also with the change in the characteristic time for the system. That gives me another factor of 10⁷ for the Coriolis force, and it settles into line with the other forces, with a thousand-fold increase.

Grant Hutchison

Let's make it on the order of a meter in radius--we can sell it to every math/science/exploritorium in the country.

Found this simulator which looks interesting. (Takes a while to load sometimes). I found it interesting to watch the L1,2 and 3 satellites' paths once they were perturbed from their initial positions.

http://www.princeton.edu/~rvdb/JAVA/...y/Galaxy1.html

Load the L1-L5 M/m=40 simulation and let it run until L3 starts wandering.

Yeah, that's cute, and yeah, there's a lot of money to made on the rotating plastic sheet!

Nice links!