Re: Something I don't understand about Lagrange points


"Steve Willner" <willner@xxxxxxxxxxxxxxx> wrote in message
news:42dea9d4$1@xxxxxxxxxxxxxxxxxxxxxxxxxx
> In article <1121771337.499398.129770@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> vturiserra@xxxxxxxxx writes:
>> Hello everybody. There's something about the Lagrange points I don't
>> understand. Let's consider a body in L4 or L5 in the orbit of Jupiter
>> (for example, a Trojan or a Greek asteroid). The forces that affect
>> that body are:
>>
>> 1) gravitacional attraction to the Sun
>> 2) centripetal force
>> 3) gravitational attraction to Jupiter
>>
>> Forces 1 and 2 are the same strength and opposite direction, and are
>> balanced.
>
> No. Force 2 is the vector sum of forces 1 and 3. The interesting
> property of all five Lagrange points is that (for circular orbits)
> the centripetal force is exactly perpendicular to the orbit velocity.
> This means there is an equilibrium orbit at all five Lagrange points
> (and only at those points). _Stability_ of these equilibria is more
> complicated.
>
> --

The original post worried me, because I studied Lagrange points as part of
perturbation theory 30 years ago, and thought I understood why they worked.
The OP has asked a very simple question, and I still don't know the answer.

Lets assume Jupiter and the asteroids are in a circular orbit, and the mass
of the asteroid << mass of Jupiter << mass of Sun (as is obviously the
case).

Then the gravitational force of the Sun and the "centripedal" force on the
asteroid point in oppsoite directions - both vectors lie on the radial line
which joins the asteroid and the Sun, but point in opposite directions. The
gravitational pull of Jupiter on the asteroid is a vector which clearly
doesn't lie entirely on that radial line. It must therefore have a component
which is perpendicular to that line, which means the asteroid must have some
nett acceleration in the direction of its travel (tangentially) - presumably
falling towards Jupiter. Yet this doesn't happen. Why not?

Here's a variation of that same argument. Lets imagine an asteroid at a
Lagrange point. Poof, Jupiter dissappears. The asteroid is till in the same
stable orbit. Poof, Jupiter re-appears. Asteroid still in same stable orbit.
How come this bloody great hulking gas giant has no apparent gravitational
effect on the asteroid - the orbit is the same whether its there or not?




.

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