DaveC426913 said:
Here's a (very) rough diagram of Jupiter's Trojans.
This diagram is of Jupiter's
present day stable orbit, after it's finished migrating...But let's take a look at it and see if we notice something of significance. As you say -
the Lagrangian points - which range from … a billion and a half miles along it's orbit.
Yet in the diagram you can see that the Lagrangian points are very well defined geometrically as being sixty degrees ahead of and behind Jupiter in it's orbit, making equilateral triangles with Jupiter and the Sun. Only we see the asteroids spread out over a long distance in the orbit. Does this mean that the region of stability extends throughout this region?
No! So then what is happening? It turns out that Jupiter, with it's Trojan asteroids, at a distance of 5 A.U. from the sun and an orbital period of ten years, pass by Saturn, at a distance of 10 A.U. and period of thirty years. Every time the leading edge of the Trojans approach Saturn, Saturn's gravity tugs on them causing them to advance a little more along their orbit, and every time the trailing edge passes by Saturn it's motion is retarded a little bit, and the cumulative effect of this is to cause the Trojans to spread out in their orbit. It's a type of tidal force called dynamical friction. By the way,
DaveC426913 said:
(As witnessed by the estimated almost quarter million bodies larger than 2 km. each).
But according to the same Wikipedia entry on Jupiter's Trojan asteroids that the diagram came from: quote - "According to the new estimates, the total number of Jupiter Trojans with a diameter larger than 2 km. is 6,300 and 3,400 in the L4 and L5 swarms, respectively."
The migration of Jupiter...is maybe a thousand miles per year.
See if you can draw Jupiter moving 1000 miles away over the course of it's year.
A Jovian year is ten Earth years, so over the course of a Jovian year, it should have moved
ten thousand miles! By choosing an arbitrarily small timescale, you're minimizing the effect that is occurring, similar to Zeno's paradox. What you're really doing amounts to selection bias. According to the migration hypothesis, Jupiter started out forming at a distance of 350 million miles from the sun, then migrated inward to 150 million miles, then, under the influence of Saturn, out to it's present distance of 500 million miles. If the Trojans migrated with Jupiter from it's former position to it's present one that would also mean that they keep the same angular relationship in order to maintain stability at the L points.
DaveC426913 said:
The Trojans didn't go to sleep and wake up to discover Jupiter migrated in their absence.
Let's not anthropomorphize; their motion is constrained by the laws of gravity, and as Jupiter moves away, two things happen - one, the gravitational attraction weakens so their trajectory is influenced less and two, the geometric relationship between them is distorted so they're no longer in a region of gravitational stability. Also, there's a third effect, and that is as their orbit separates from Jupiter's their orbital periods are no longer in sync, which will reduce the stability even more. Finally, if I haven't convinced you, let me quote from the Wikipedia entry on the formation and evolution of the solar system referencing the migration theory: "...when Jupiter and Saturn were near their 1:2 resonance, the orbits of pre-existing Jupiter Trojans became unstable. This process was also reversible allowing...numerous objects...to enter this region and be captured. These
new Trojans..." So evidently, the migration theory model finds that the original Trojan's orbits become unstable and new Trojans come in and replace them at the Jupiter's new orbit.