TROJAN PLANETS, a group of asteroids revolving round the sun in the same period as Jupiter. Joseph Louis Lagrange, in his essay on the problem of Three Bodies (1772), noted certain simple cases in which an exact solution could be found by elemen tary methods. In some of these the bodies remained in a rotating straight line ; these cases are unstable, and of little practical im portance, though it has been suggested that the Gegenschein, or Counterglow, may be due to particles connected in this manner with the sun and earth. The most interesting case is that in which three masses (equal or unequal) S, J, T, whose centre of gravity is G, are situated at the angular points of an equilateral triangle. Then, if they are projected in the plane of the triangle, with veloc ities proportional to GS, GJ, GT, in directions making equal angles with GS, GJ, GT, in the same direction of angular motion, they will describe ellipses in the same period, exactly as though they were particles moving under accelerations directed towards G and proportional to their distances from G; further, the triangle SJT will remain equilateral throughout the motion, though it will vary in size unless the orbits round G are circular.
This case, like the others, was regarded simply as a mathemati cal curiosity till the year 1906. On Feb. 22 in that year Prof. Max Wolf, in the course of the systematic search for new planets that is carried on by photography at Konigstuhl Observatory, Heidel berg, found a faint planet, provisionally lettered TG, which sub sequently received the permanent number 588 and the name Achilles. Its orbit was calculated by Dr. Berberich at the Berlin Recheninstitut ; he found that its period was almost exactly the same as that of Jupiter (11.86 years). Thereupon Prof. Charlier, of Lund Observatory, noting that Achilles was about in front of Jupiter, as seen from the sun, suggested that this was an ex ample in the heavens of the equilateral configuration which La grange had deduced by theory. It was seen, of course, that the
ideal conditions were not exactly fulfilled ; the inclination of Achil les to the ecliptic is and its eccentricity f—much larger than the corresponding elements of Jupiter's orbit. But it was conjec tured that since the equilateral configuration is stable, these devia tions would simply produce periodic oscillations about that con figuration.
Achilles did not long remain the sole example of this type of motion; No. 617 Patroclus was found in the same year, and Hector in the following year. Three more members were subse quently found, 659 Nestor, 884 Priamus and 911 Agamemnon. Of these Achilles, Hector, Nestor, Agamemnon are about 6o° ahead of Jupiter (greater longitude), while Patroclus and Priamus are 6o° behind. It is unfortunate that the obvious plan was not followed of giving Greek names to those on one side, Trojan names to the other, but it would cause confusion to change now.
The Trojans seem to be among the larger members of the aster oid group. Their diameters, estimated from their light, are of the order of 150 miles, which is greater than that of Juno, one of the original four asteroids. This diameter, however, is only one five hundredth of that of Jupiter, so that, even if we assign to them double the density of Jupiter, the mass of each of them is only about one fifty-millionth of his. This is, of course, too small to produce any sensible effect on Jupiter, and the point G in the tri angle may be taken as the centre of gravity of the sun and Jupiter.
This is distant from the sun's centre by 1 of the side of the triangle, or 461,000 miles, which places it just outside the sun's surface. This is the point which would be the centre of revolution under the strict Lagrangian configuration. It is obvious, however, that the actual motion is far from exhibiting the Lagrangian sim plicity and uniformity ; the inclination of two of the Trojans is over 20°, and their eccentricities are also considerable.