In article ,
mcv wrote:
Other than that, I'm pretty much like you: an interested layman looking
for a clear explanation of principles with math that I can still understand
(and I'm only a computer scientist, and not a physicist or mathematician).

I've also always wanted to know why the Lagrange points work the way they
do. I have no idea.

The finer questions, like their stability and why there aren't any more of
them, get messy. But the basics of why they exist are not too hard.

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.

The "Trojan" Lagrange points, in the Moon's orbit 60deg ahead of and
behind it, are more subtle. At first glance, it looks like they shouldn't
work -- centrifugal force balances Earth's gravity just like it does for
the Moon, but nothing balances the Moon's gravity and it ought to pull
objects away from the point.

The key thing to understand is that objects in the Earth-Moon system don't
orbit the Earth. They orbit the barycenter of the system, basically the
system's overall center of mass, and that is displaced somewhat toward the
Moon from the Earth's center.

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.

The Moon's gravity obviously pulls to the Moonward side, and since the
Moon is only 60deg away from Earth as seen from the Trojan point, it also
pulls toward Earth a little bit. Here too the forces balance, but you
need to use vector addition rather than just arithmetic, and you must
remember that the center of rotation (which defines the centrifugal force)
is the barycenter, not Earth's center.
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MOST launched 1015 EDT 30 June, separated 1046, | Henry Spencer
first ground-station pass 1651, all nominal! |