Why are Jupiter's trojans even remotely stable?

Question

The spacecraft Lucy is en route to explore Jupiter's trojan asteroids. Lucy is so named because the trojan asteroids are believed to be fossil remnants from the formation of the solar sytem. Presumably this means that the astroids have been occupying the Lagrange points for a very long time.

At Saturn's closest approach to Jupiter's Lagrange points, it is actually slightly closer than Jupiter (~650 million km vs. 780 million km). Saturn has only 30% the mass of Jupiter, so its gravitational effects are smaller. Nevertheless, for periods of many months, Saturn will exert on the asteroids a force of up to 43% of the gravitational force of Jupiter. So how do these asteroids manage to remain in the vicinity of the Lagrange points given such large perturbing forces?

I suppose the reality of their apparent longevity must be supported by simulations, but are there some simple arguments or observations that give some insight into why, given the presence of Saturn, the trojans are still there after five billion years?

Answer 1

I'm answering my own question here based largely on the comment from @CuteKItty_pleaseStopBArking

Saturn's influence can be separated into a radial component (near Saturn's conjunction with the Lagrange point) and a tangential component (before and after conjunction). The radial force is perpendicular to the orbital motion and does not change the orbital energy or period. However, it will induce an eccentricity into the Trojan's orbit. The tangential forces are first prograde and then retrograde, balancing to zero. Again there is no change in orbital energy or period. However, at each interaction, the Trojan will advance slightly in position along its orbit.

The effect of small perturbations around the L4/L5 points are dealt with in a paper by Neal Cornish. In a linearized system in the rotating frame, two eigenmodes can be excited. If Jupiter's mass is $m = 1/1047 << 1$, expressed in solar masses, then the two eigenfrequencies reduce to $1-27m/8 = 0.9968$ and $sqrt(27m/4) = 0.0803$. The first mode is 1.0032 times longer than Jupiter's orbital period and the second is 12.45 times longer than Jupiter's orbital period.

From simulations, the radial disturbance primarily excites the first mode resulting in a kidney-bean orbit around the Lagrange point. The tangential disturbance primarily excites the second mode causing a slow motion back and forth along the orbit (approaching and receding from Jupiter). In neither case is there simple integer relationship between these periods and the interval of 1.674 between Jupiter-Saturn conjunctions.

The conclusion is that the large disturbances caused by Saturn certainly do not accumulate linearly. Nor do they even seem to accumulate randomly (like a random walk). The disturbances would seem to be applied roughly uniformly around the trojan orbit and thus to closely cancel over a long period of time. For example, a radial disturbance results in an eccentricity in the trojan orbit, but the axis of the eccentricity will precess uniformly with respect to the excitation creating the eccentricity (which occurs at every Jupiter-Saturn conjunction).

Answer 2

A little perspective might be borne in mind: In terms of relative mass and distance of disturbing planets, Jupiter's trojans are less disturbed than their counterparts with any of the other five neighborhood-clearing planets for which at least a temporary trojan is known:

Venus/2013 ND15: Earth passes by with a slightly larger mass and about half the minimum distance versus the distance from Venus to the Lagrange point. Jupiter's much larger mass also creates a significant perturbation even over a longer distance (also for Earth and Mars, see below).

Earth/2010 TK7, 2020 XL5: Same passage as above: Venus is slightly smaller than Earth in mass, but the Earth-Lagrange point distance is greater than its counterpart in the Venus orbit.

Mars/14 asteroids: Earth again is the major inner-planet actor, with a much larger mass than Mars as well as a nearest approach distance only about one-third of the planet/Lagrange point distance.

Uranus/2011 QF99, 2014 YX49: Saturn is much more massive than Uranus and approaches the Lagrange points at a minimum distance nearly equal to Saturn's own.

Neptune/22 asteroids: Uranus approaches with a nearly equal mass and nearly equal approach distance; the maximum relative gravitational influence is about 70% as opposed to the 43% quoted for Saturn's disturbance of Jupiter. Like Jupiter and Mars, Neptune may have added trojan stability by not being between two relatively nearby large planets.