On 28/10/12 20:30, Dr J R Stockton wrote:
[...] Do you know if Gascheau's 1843
article is accessible on the Web?

I've never pursued any historical research into these
areas, just read what I needed to for my own research and other
interests. Ie, I don't know, and your googling is likely to be
at least as well-directed as mine.

Danby, Astron J, 69, 4, pp.294-6, May 1964 (GIF) looks a good read - for
some. Its abstract indicates that the same holds for non-circular
orbits, but with a varying numerical constant. A.N.Other implied that
moderately elliptical orbits are stable for moderately lighter
secondaries, but no further.

It would be very surprising if it was not so. The usual
restricted circular result is not "fragile", but comes merely from
the shape of the [modified] equipotentials, esp near the maximum
at the Trojan points. If the Trojan asteroids are stable to small
perturbations in their own orbits, we can expect them to be stable
equally to small perturbations in Jupiter's orbit. Putting some
numbers in to that is a whole lot harder, though; not least
because co-rotating co-ordinates are no longer rotating uniformly.
But a computer simulation can't be hard, and could be Quite
Interesting, esp for highly elliptic orbits.

On looking at Danby's paper, eg at

http://articles.adsabs.harvard.edu/c...J.....69..165D

his Figure 1 [p171] is interesting; the shape of the stability
region, with a minimum at Danby's point B [where only the circular
case is stable] and a cusp at D [where some elliptical cases are
stable even when the circular case isn't] is more complicated
than might be expected.

--
Andy Walker,
Nottingham.