mcv writes:

Henry Spencer wrote:
In article ,
mcv wrote:

I've also always wanted to know why the Lagrange points work the way
they do. I have no idea.
[...]
Sloppily speaking, a circular orbit around an unaccompanied planet is a
balance between gravity and centrifugal force. But in (say) the
Earth-Moon system, there are three forces involved, one centrifugal and
two gravitational.

The "in-line" Lagrange points, along the axis joining the Earth and Moon,
are not too hard to grasp. They arise from various combinations of the
three forces adding up to zero by simple arithmetic. For example, the L1
point between Earth and Moon (caution, astronomers and space engineers
don't number the points the same way) is where centrifugal force *plus*
the Moon's gravity exactly balances Earth's gravity.

This makes sense. And as I was reading this, I realised that the L2 point
(on the other side of the moon) is where the earth's gravity plus the
moon's gravity balances the centrifugal force, which is larger because
its orbital velocity is higher.

I still have no idea how L1 and L2 can possibly be stable, however.

Simple --- they _aren't_. L1, L2, and L3 are all "saddle points," unstable
to perturbations along the Earth-Moon line. (However, active station-keeping
to hold a body near L1, L2, or L3 costs less propellant than trying to "hover"
in a powered orbit anywhere else in the Earth-Moon system...)

[...]
So an object at a Trojan point, at the same distance *from Earth's
center* as the Moon, is not quite at the same distance from the point
it actually orbits around -- it is slightly farther out than the Moon.
And from its viewpoint, the point it orbits is slightly to one side,
the Moonward side, of Earth's center. So to maintain its orbit,
it needs to be pulled inward a bit harder than Earth alone can manage,
and it needs to be pulled off to the Moonward side of Earth too.

Alright, now I think I understand that too. It's easier than I thought.
But now I don't understand why this holds specifically for the L4 and
L5 points, and not for an infinite number of other points between those
and the L3 point.

At any point other than the Lagrange points, the acceleration due to gravity
has the wrong strength and direction for it to be possible for a body to
maintain a constant distance from both the Earth and the Moon, since it will
not point toward the barycenter and/or provide the correct centripetal
acceleration for the body to move in a 29.5 day circular orbit.


-- Gordon D. Pusch

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