Newton’s Great... Oversight
Galileo’s Falling Bodies and Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits

6          TROJAN POINTS AND
ASTRONOMY IN THE 21^ST CENTURY

SECTIONS

6.1 Historical Digression - Why “Trojan” Points?

6.2 Trojan Point Astronomy in the 20^th Century

6.3 Trojan Point Astronomy in the 21^st Century

6.1         Historical Digression - Why “Trojan” Points?

Lagrange apparently called the astronomical bodies he predicted “Trojan Planets” because there was a hidden quality to them, reminiscent of the (Greeks in the) Trojan horse of Homer’s Illiad. This hollow wooden horse was constructed at the suggestion of the crafty Odysseus to seem to be a sacrifice to Athena. After it was constructed outside the walls of the city of Troy, the Greeks appeared to leave in their ships, “defeated”. The horse actually hid Greek warriors inside from the Trojans who took the horse into their city to crown their victory over the Greeks, even though the cursed prophetess Cassandra warned them not to. We can note with further irony that the curse on Cassandra caused people not to believe her. The Greek warriors hidden in the horse came out when the people were asleep and unlocked the city gates to let in the rest of the Greek warriors who had returned under cover of night. Together they all sacked and slew the city and, “among other things”, gave rise to Virgil’s tale of the Æneid, which lead to the founding of Rome.

The hiddeness of the Trojan points has to do with a characteristic of Lagrange’s conception of the astronomical problem. When Lagrange developed perturbation theory and found the (theoretical) Trojan points, he was mathematically looking at the Sun, at Jupiter which was very much smaller (almost “infinitesimal”, relative to the Sun), and at what would happen if a body infinitesimal with respect to Jupiter were perturbed from its position in the Trojan points leading and following Jupiter. Since any such bodies would be small, they would be invisible by the astronomical standards of his day, so he called them “Trojan points” after the (Greeks in the) Trojan horse. Or so I vaguely remember reading a long time ago.

This is more than historically interesting since it brings up an important difference between the approach given here and Lagrange’s perturbation theory (based on differential equations). Lagrange was not thinking in terms of arbitrary non-infinitesimal and even potentially equal masses at each vertex of the equilateral triangle. And astronomers since have not, either. Astronomy is extremely subject to the psychological reality that if you don’t know where to look, or don’t know what to look for or how to look, then it is very easy to remain blind to even the most obvious events.

SECTIONS

6.1 Historical Digression - Why “Trojan” Points?

6.2 Trojan Point Astronomy in the 20^th Century

6.3 Trojan Point Astronomy in the 21^st Century

6.2         Trojan Point Astronomy in the 20^th Century

It was in February 1906, more than a century after Lagrange, that the astronomer Max Wolf — credited as the first astronomer to use photography to do astronomy — finally proved that Lagrange was correct 134 years earlier by finding 588 Achilles in the leading point of Jupiter. Within a year August Kopff had found 617 Patroclus and 624 Hector. Today, some 400+ Trojan asteroids have been named/numbered and somewhat studied, but there are estimates of 2300 ± 500 Trojan asteroids with diameters greater than 15 kilometers, about 1300 in the leading point, L4, and 1000 in the following point, L5. (Apologies; I’ve lost track of where I found this estimate.) Many more are suspected to exist. In fact, Trojan asteroids are known to exist not only in the orbit of Jupiter, but in the orbits of some of Jupiter’s moons, and even in the orbit of Mars, where Eureka was discovered in 1990. These asteroids slowly orbit their respective Trojan points in “unusual” but relatively stable non-elliptical orbits called “tadpole” orbits. (See APPENDIX and Figure 5.) Jupiter’s Trojan asteroids can take hundreds of years to complete such an orbit.

After the first few were discovered it was decided to name all Trojan planets/asteroids after the heroes of the Trojan war, with the leading point bodies named after Greek heroes and the following point bodies named after Trojan heroes. All, that is, except Hector in the leading group and Patroclus in the following group. They are today considered “spies” in the other camps. (If you are trying to remember, it was the Trojan Hector, son of King Priam, who killed the Greek Patroclus, a friend of Achilles, after which Achilles started fighting again and, as the war came to a close in its tenth year, finally killed Hector. (And don’t forget Helen and... “We’ll always have Paris.”)

Although theoretically Trojan planets, moons or asteroids could exist in the orbit of any planet or moon, no exhaustive search for them seems to have been performed — at least not with modern telescopes — and this despite the fact that searching Trojan points for asteroids is an obviously easier study than searching for asteroids in general (much smaller volume of space to examine). But as late as 1990 Dr. Hannes Alfven (1908-1995), Nobel Laureate in Physics (for “contributions and fundamental discoveries in magnetohydrodynamics”), was suggesting that such asteroids do exist in the orbit of Earth, both leading and following Earth, to encourage such a search. And ironically, in that same year the American astronomers David H. Levy and Henry E. Holt reported finding the first Trojan asteroids in the orbit of Mars.

The physics of Trojan points relates to the dynamics of those fascinating “horseshoe” and “tadpole” shaped orbits of asteroid size bodies relative to the Earth and other planets. (See the Encyclopedia of the Solar System (hyperlink unfortunately no longer seems valid), pp. 815-7; Weissman, McFadden, Johnson, Eds.; Academic Press, 1999.) The new and very simple approach to Trojan point dynamics given here could conceivably yield new insights, but at the very least it can help generate popular interest in the physics of these fascinating astronomical phenomena.

SECTIONS

6.1 Historical Digression - Why “Trojan” Points?

6.2 Trojan Point Astronomy in the 20^th Century

6.3 Trojan Point Astronomy in the 21^st Century

6.3         Trojan Point Astronomy in the 21^st Century

Computers open up new worlds of possibilities, even to an important extent to amateur astronomers and physicists. They can be used to study intractable problems that have no closed form solutions, the deceptively “simple”, 3‑body gravitational problem being a classical example. Lagrange made computationally convenient, simplifying assumptions that are not realistic, such as the infinitesimality of the 3^rd body. The problem would have been intractable for him, in his day, i.e. before computers, if he had not done so. Over time such assumptions... well, computers can be used to good effect to study the stability of Trojan systems outside of Lagrange’s limits and assumptions. This might have applicability in looking for ternary star systems (see Figure 6a and Figure 6b), looking for “inhabitants” of Trojan points of seemingly “binary” systems, etc.

It is not well appreciated in current astronomy, but the detailed study of Trojan bodies — made feasible by computers — can potentially shed light on the evolution of the Solar System in a way that the study of other asteroids would not. The times at which they were trapped in orbits around the Trojan points could be important clues to the timetable of Solar System development. For example, if Trojan asteroids occur very close to their equilibrium positions with close to zero velocities at the Trojan point(s) in the orbit of Pluto, or even of Jupiter, it would mean that they had probably been equilibrating for a very, very long time (since the atmospheric viscosity of “empty space” is very slight, even if greater than zero). When we eventually study them up close using space probes, e.g. studying their composition, the combined information could yield great insights. Ironically, though, the further they are from equilibrium, the easier it might be to estimate how long they had been actually trapped into approaching that equilibrium point. E.g. their Trojan-point-orbital velocities would be more readily measurable, at least with percentage-wise much greater accuracy.

Trojan points are in unusually precise positions in the volume of Solar System space by astronomical standards, and it should take only a small fraction of the effort to search for asteroids in the tadpole orbits near them compared to doing general searches. Well, perhaps not as precise as one might wish since Trojan asteroids in the orbit of Jupiter seem to range angularly before and behind both Trojan points by ~ 15º, and to range radially over ~ 1 AU (an Astronomical Unit, the distance between the Earth and the Sun). Most of those known and plotted near L4 (the Trojan point leading Jupiter) seem to be closer to the Sun than Jupiter’s ~ 5.2 AU, and most of those around L5 (the Trojan point following Jupiter) seem to be further from the Sun. (This estimate was made from a crude visual analysis of a plot of actual positions of 132 known Trojan asteroids made in 1990; it can be found in Fraknoi, Morrison, Wolff, Voyages Through the Universe, p. 255, Saunders College Publishing, 1997; which see since plot not shown here; plot courtesy of Edward Bowell, Lowell Observatory. There is now a 2^nd edition, by Harcourt College Publishers. The link given above points to one of their web pages.) And of course those asteroids in horseshoe orbits would range almost as far and wide as the more usual sort.

It would be very important to detect as many Trojan asteroids and their trajectories as possible to begin to approximate their cumulative effects on each other over time, and in order to factor that into an extrapolated past history. Perhaps it would be practical for our modern space telescopes such as Hubble to be used to search for them, but even if the space telescopes are busier doing other things, amateur astronomers could very likely find both studying known Trojan asteroids and searching for new ones rewarding.

One other fascinating possibility is that there are micro- or mini- asteroids or other interesting space debris accumulating near the Trojan points of Earth’s Moon. This is close enough that we could send a space probe to not only photograph it, but pick up and return micro-asteroids to Earth with more sterility than picking up meteorites from Earth’s surface. And the study of their atmospheres would be interesting. The L4 and L5 points might have important differences in both debris and atmospheres, clues to their role in Solar System evolution.

So we have several possibilities for the astronomical study of Trojan points/bodies in the 21^st Century:

•  use computers — which Lagrange didn’t have — to do a more complete analysis of the more general case of 3 non-infinitesimal bodies with no restrictions on relative masses; in particular, look for stabilities that Lagrange may not have been able to find because of his computationally convenient but limiting assumptions

•  try to determine if the orbital data of Trojan asteroids can indicate the time that the asteroids were captured by the Trojan point in its “tadpole orbit”/energy well, and study what this might indicate about the evolution of the Solar System

•  relatedly, Trojan points and their associated orbits/energy wells are a good point of focus to study — both theoretically and observationally — the dynamic viscosity/drag of space (from space debris and from the tenuous but turbulent Solar System atmosphere, both of which will tend to concentrate there if their energies are low enough), so...

•  compared to other projects, it would be relatively easy to send a robotic space probe to pick up and return with micro-/mini- asteroids from the almost certainly concentrated space debris near one or both Trojan points of the Earth-Moon system, and...

•  at the same time make estimates of the amount, distribution and orbits of accumulated space debris, check for the existence of the Trojan point atmosphere(s), and study their composition, viscosity, etc. If both points can be visited, study the differences in debris and atmospheres.

The simple mathematical approach presented here can help make beginning study of the fascinating Trojan points in the 21^st Century accessible not only to:

•  high school physics students

•  amateur astronomers (who might be inspired to look for Trojan bodies close to home, e.g. in the Moon’s orbit around the Earth or the Earth’s orbit around the Sun, and to use computers to study the general Trojan body problem)

•  the popular science reading public,
  but, importantly, to

•  all professional astronomers and physicists, who (usually) do not wish to spend the extra graduate study it takes to learn the rather arcane and difficult perturbation theory to a useful extent.

And professional astronomers may benefit from this simple approach also, because:

•  It makes obvious certain properties of the dynamics of Trojan points and Trojan asteroids which are not made obvious by the differential equation/perturbation theory approach; i.e. many astronomers and physicists are under the misimpression left by Lagrange’s approach that the dynamics of Trojan points must depend on the 3 bodies having vastly different masses, e.g. a massive Sun, a relatively small 2^nd body (m₂ < ~0.04 m₁) such as Jupiter in orbit around it, and a 3^rd body of asteroid size that is “infinitesimal” in the usual sense of “negligible mass” (which is neither theoretically nor actually realistic if we consider long time periods). But, in fact, at least “unperturbed stability” pertains for 3 arbitrary masses, and it may be possible to extend this simple but more general approach to a more general determination of stability (or to inspire the search for such, with the improved understanding that can result even from “fruitless” research). The non-infinitesimality of the 3^rd body could conceivably make Lagrange’s result inapplicable, e.g. it may be that any instability of more equally distributed mass may take an astronomically significant time to show itself, and that astronomers may be able to find more variety in Trojan systems than currently assumed.

•  This has implications for e.g. binary star systems and co-orbiting galaxies; their Trojan points can be more closely examined for e.g. planets or micro-galaxies of even very large size, for globular “asteroid clusters” (similar to globular star clusters) or other accumulated space debris, condensed “nebular” or small particulate or atmospheric matter, or, especially in the case of galaxies, for condensed “dark matter”. Also, ternary star systems may be more stable than Lagrange’s theory seems to predict (see Figure 6a and Figure 6b).

Newton’s Great... Oversight, the difference in falling rates of lighter and heavier bodies, and this related and simple approach to the physics of Trojan points with their tadpole and horseshoe orbits, just may form the beginnings of a popular and fruitful — and heavenly — body of study for astronomers, and even physicists, in the 21^st Century.