[sci.astro] Finding Lagrange's Libration Points

dalex@eleazar.dartmouth.edu (Dave Alexander) (01/19/89)

In article <1989Jan18.044744.18328@sq.uucp>
                                        msb@sq.com (Mark Brader) writes:
> Lagrange found that there are exactly 5 positions that the tertiary
> can occupy with respect to the primary and secondary.  The three
> bodies can be in a straight line: position L1 has the tertiary in the
> middle, L2 the secondary, and L3 the primary.  Or they can be in an
> equilateral triangle, with the tertiary either leading or trailing the
> secondary as they move around the primary; these positions are L4 and
> L5.

...

> The L in each of these positions stands for libration, as a body near
> those positions may librate or oscillate around them, and not for
> Lagrange.

...

> This is as far as I have gotten on my own.  Now, I observe that if a
> body is placed near one of the L1, L2, L3 points but slightly off the
> straight line, then there will be a sideways force on it, so it makes
> sense that, as I have read, these are unstable equilibria.

So if L1, L2, and L3 are loci of unstable equilibrium, how can we expect
an object to librate or oscillate about any of them?


                        -- Dave Alexander

--
"Experience has proved that some people indeed know everything."
                                     -- Russell Baker

henry@utzoo.uucp (Henry Spencer) (01/21/89)

In article <11854@dartvax.Dartmouth.EDU> dalex@eleazar.dartmouth.edu (Dave Alexander) writes:
>So if L1, L2, and L3 are loci of unstable equilibrium, how can we expect
>an object to librate or oscillate about any of them?

You can't, and it won't, without help.  An object *precisely* at one of
those points, with *no* perturbations, would stay there, but in the real
world, that doesn't work.

There are trajectories near those points -- so-called "halo orbits" --
which are *almost* stable, requiring only very slight corrections to
maintain.  ISEE-3 was in a halo orbit around the Earth-Sun L1 point for
some years, to study the solar wind "upwind" of Earth.
-- 
Allegedly heard aboard Mir: "A |     Henry Spencer at U of Toronto Zoology
toast to comrade Van Allen!!"  | uunet!attcan!utzoo!henry henry@zoo.toronto.edu

dalex@eleazar.dartmouth.edu (Dave Alexander) (01/22/89)

In article <1989Jan20.180839.7800@utzoo.uucp>
                                henry@utzoo.uucp (Henry Spencer) writes:
> In article <11854@dartvax.Dartmouth.EDU>
                    dalex@eleazar.dartmouth.edu (Dave Alexander) writes:
>> In article <1989Jan18.044744.18328@sq.uucp>
                                        msb@sq.com (Mark Brader) writes:
>>> The L in each of these positions stands for libration, as a body
>>> near those positions may librate or oscillate around them, and not
>>> for Lagrange.

>> So if L1, L2, and L3 are loci of unstable equilibrium, how can we
>> expect an object to librate or oscillate about any of them?

> You can't, and it won't, without help.  An object *precisely* at one
> of those points, with *no* perturbations, would stay there, but in the
> real world, that doesn't work.

I understand that.  The question that I was *really* asking was "How can
you say that the `L' in L1-5 stands for `libration,' when 60% of such
points do not exhibit that behavior?"  In light of that, I question Mr.
Brader's assertion that the `L' stands for `libration' and not
`Lagrange.'


                         -- Dave Alexander

--
"Experience has proved that some people indeed know everything."
                                     -- Russell Baker

joe@cit-vax.Caltech.Edu (Joe Beckenbach) (01/24/89)

In his article dalex@eleazar.dartmouth.edu (Dave Alexander) writes:
>The question that I was *really* asking was "How can
>you say that the `L' in L1-5 stands for `libration,' when 60% of such
>points do not exhibit that behavior?"  In light of that, I question Mr.
>Brader's assertion that the `L' stands for `libration' and not
>`Lagrange.'

	I too thought that the 'L' was for the describer of the points,
Lagrange. And I do remember a bumper sticker from several years back,
in a space-enthusiast-humorous context:
		"FREE THE LAGRANGE FIVE!!"

	Whether he had it right, respondent saith not.
		:-) :-) :-)
-- 
Joe Beckenbach	joe@csvax.caltech.edu	Caltech 256-80, Pasadena CA 91125
Users I'd like to see (:      camera%observer@commsat.mars.solar-system
	postmaster@link1.L5.edu		daemon@probe.titan.solar-system

dsmith@hplabsb.HP.COM (David Smith) (01/27/89)

In article <11854@dartvax.Dartmouth.EDU> dalex@eleazar.dartmouth.edu (Dave Alexander) writes:
>So if L1, L2, and L3 are loci of unstable equilibrium, how can we expect
>an object to librate or oscillate about any of them?

I didn't work through the mathematics, but:
In an article on ISEE, Aviation Week pointed out that L1 was at a saddle
point.  The spacecraft was attracted to the Earth-Sun line, orbiting it.
In the direction of the line, position was unstable, but quite manageable
to the fuel budget.
-- 

			David R. Smith, HP Labs
			dsmith@hplabs.hp.com
			(415) 857-7898