How big is L5?

In article ,
Alex Terrell wrote:
The Earth Moon L5 (or L4) position is often mentioned as a
gravitationally stable region ideal for space colonies. I assume that
means that items in the vicinity of L5 (or L4) are attracted to, or
orbit about, the central point.

They orbit around it, more or less. The underlying story is actually
quite complex, but that's the bottom line.

But can anyone tell me how big the region is? How many kilomters
towards Earth would you need to be before you drifted away.

There isn't a sharp boundary. Quite large orbits around the point are
stable in principle (although stability in practice is a much more
complicated question, given things like solar perturbations).

Also, any
indications of the gravitational gradients within L5? How far would
objects need to be apart so as not to drift into the central area and
bump into each other?

As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to drift
toward the central point and then stop.
--
MOST launched 1015 EDT 30 June, separated 1046, | Henry Spencer
first ground-station pass 1651, all nominal! |

Henry Spencer wrote:

They orbit around it, more or less. The underlying story is actually
quite complex, but that's the bottom line.
...
As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to
drift
toward the central point and then stop.

Ideally (no other objects than the primary and secondary, test particle
has zero mass), with what period do the objects orbit around the Trojan
point? Does that period depend on their distance from the point?

--
Erik Max Francis && && http://www.alcyone.com/max/
__ San Jose, CA, USA && 37 20 N 121 53 W && &tSftDotIotE
/ \ That's what I'm about / Holding out / Holding out for my baby
\__/ Sandra St. Victor

"Henry Spencer" wrote:
In article ,
Alex Terrell wrote:
Also, any
indications of the gravitational gradients within L5? How far would
objects need to be apart so as not to drift into the central area and
bump into each other?

As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to drift
toward the central point and then stop.

Which is a pretty obvious conclusion if you think about it.
We're talking gravitational potentials here, so think hills
and valleys. Except in space there's no friction, so if
you fall into a valley (and for some reason don't fall out)
you're going to roll around in it, up and down each side,
rather than fall toward the exact bottom and stick there.
There are forces directing you toward the bottom but no
forces that'll make you stick there once you arrive.

"Christopher M. Jones" wrote in message ...
"Henry Spencer" wrote:
In article ,
Alex Terrell wrote:
Also, any
indications of the gravitational gradients within L5? How far would
objects need to be apart so as not to drift into the central area and
bump into each other?

As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to drift
toward the central point and then stop.

Which is a pretty obvious conclusion if you think about it.
We're talking gravitational potentials here, so think hills
and valleys. Except in space there's no friction, so if
you fall into a valley (and for some reason don't fall out)
you're going to roll around in it, up and down each side,
rather than fall toward the exact bottom and stick there.
There are forces directing you toward the bottom but no
forces that'll make you stick there once you arrive.

I believe L1, L2 and L3 are saddles or hilltops, whilst L4 and L5 are
troughs. The question really is how steep are the sides of the
"troughs", and how far does one have to go "uphill" before one starts
going "downhill".

If I put a space station 10 km from L5, I assume it will orbit about
L5. What about 100km? What about 1,000km? I guess 10,000km towards
Earth, and the space station will leave L5 and orbit about the Earth
independently.

"Christopher M. Jones" writes:

"Henry Spencer" wrote:
In article ,
Alex Terrell wrote:
Also, any
indications of the gravitational gradients within L5? How far would
objects need to be apart so as not to drift into the central area and
bump into each other?

As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to drift
toward the central point and then stop.

Which is a pretty obvious conclusion if you think about it.
We're talking gravitational potentials here, so think hills
and valleys. Except in space there's no friction, so if
you fall into a valley (and for some reason don't fall out)
you're going to roll around in it, up and down each side,
rather than fall toward the exact bottom and stick there.
There are forces directing you toward the bottom but no
forces that'll make you stick there once you arrive.

....Except that that _isn't_ how L4 and L5 work !!!

If you examine the "effective potential" in the co-rotating frame,
you will find that L4 and L5 are both at the summits of "hills,"
not the troughs of "valleys." What "stabilizes" orbits about L4 or L5
is the "coriolis force:" As a body falls "downhill," it gains velocity,
and since the coriolis "force" is both perpendicular to and proportional
to elocity, the body is deflected until it finds itselef moving "uphill"
and slowing down again. (The dynamics are quite similar to those of a
charged particle in a Penning trap: The electrostatic forces produce
stable oscillations perpendicular to the X-Y plane and unstable motion
in the X-Y plane; the magnetic field along the Z-axis deflects the motion
of particle in the X-Y plane so that particles falling "outward" get
whipped around until they are moving "inward" again, so that they move
in a cycloidal path around the Z-axis.)


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'

In article ,
Erik Max Francis wrote:
They orbit around it, more or less. The underlying story is actually
quite complex, but that's the bottom line...

Ideally (no other objects than the primary and secondary, test particle
has zero mass), with what period do the objects orbit around the Trojan
point?

In the general case, it's actually quite a complex motion, because the
particle oscillates around the point along all three axes, but the period
can be *different* on each axis. Only in special cases does the particle
follow something approximating a simple orbit around the point.

Does that period depend on their distance from the point?

In some cases -- perhaps all, I don't remember for sure -- no, it's
independent of distance.
--
MOST launched 1015 EDT 30 June, separated 1046, | Henry Spencer
first ground-station pass 1651, all nominal! |

Hop David wrote:

I used to have that misconception. The L1, 2 & 3 are saddlepoints and
L4
& 5 are hill tops.
http://www.physics.montana.edu/facul.../lagrange.html

The stability is due to Coriolis effect, not a valley between the
walls
of gravity and centrifugal force.

You're not alone; it's an extremely common misunderstanding, simply
because of the weirdness of the stability of the Trojan points (as you
say, due to Coriolis pseudoforces, not the usual mechanism). Plenty of
informative Web sites have it wrong, as well. It "seems" obvious it's a
valley, first perceptions can be wrong.

--
Erik Max Francis && && http://www.alcyone.com/max/
__ San Jose, CA, USA && 37 20 N 121 53 W && &tSftDotIotE
/ \ You are inspiration to my life / You are the reason why I smile
\__/ India Arie

Hop David wrote in message ...
Alex Terrell wrote:
"Christopher M. Jones" wrote in message ...

"Henry Spencer" wrote:

In article ,
Alex Terrell wrote:

Also, any
indications of the gravitational gradients within L5? How far would
objects need to be apart so as not to drift into the central area and
bump into each other?

As noted above, what happens is that they oscillate around the point,
rather than drifting toward it. There is no tendency for them to drift
toward the central point and then stop.

Which is a pretty obvious conclusion if you think about it.
We're talking gravitational potentials here, so think hills
and valleys. Except in space there's no friction, so if
you fall into a valley (and for some reason don't fall out)
you're going to roll around in it, up and down each side,
rather than fall toward the exact bottom and stick there.
There are forces directing you toward the bottom but no
forces that'll make you stick there once you arrive.


I believe L1, L2 and L3 are saddles or hilltops, whilst L4 and L5 are
troughs. The question really is how steep are the sides of the
"troughs", and how far does one have to go "uphill" before one starts
going "downhill".

I used to have that misconception. The L1, 2 & 3 are saddlepoints and L4
& 5 are hill tops.
http://www.physics.montana.edu/facul.../lagrange.html

The stability is due to Coriolis effect, not a valley between the walls
of gravity and centrifugal force.

Hop
http://clowder.net/hop/index.html

One learns something new every time.

Would that mean that it's difficulty to place something exactly on the
Lagrange Point?

Or is the effect still that the combined potential and kinetic
energies of something in a Lagrange point are minimised. How much
delta V is needed to actually escape from L4/L5?

(Henry Spencer) writes:

In article ,
Erik Max Francis wrote:
They orbit around it, more or less. The underlying story is actually
quite complex, but that's the bottom line...

Ideally (no other objects than the primary and secondary, test particle
has zero mass), with what period do the objects orbit around the Trojan
point?

In the general case, it's actually quite a complex motion, because the
particle oscillates around the point along all three axes, but the period
can be *different* on each axis. Only in special cases does the particle
follow something approximating a simple orbit around the point.

Such special cases of "closed orbits" are called _resonant_ orbits, because
the periods of the orbit in all three degrees of freedom are "commensurate"
(i.e., related by simple integer multiples). However, it shoul be noted that
these closed orbits are all unstable to small perturbations, and in general
the motion of a "test particle" in the volume near the L4 or L5 points
would be "chaotic," even if the gravitational perturbations of the Sun and
the non-circularity of the Moon's orbit could be neglected (which they can't).

Does that period depend on their distance from the point?

In some cases -- perhaps all, I don't remember for sure -- no, it's
independent of distance.

This is true only of small amplitude oscillations. As the amplitude
increases and the orbit approaches the separatrix enclosing L4 / L5
(which sadly is not shown on the otherwise excellent figure at
http://www.physics.montana.edu/faculty/cornish/lagrange.html ---
thank you "Hop David" for posting this link!), then in the absence of
external perturbations (such as the Sun, which actually can't be neglected)
the periods of those special families of "closed orbits" (as viewed from
the co-rotatiing frame) all approach infinity, with the limiting case
being a particle that follows a branch of the separatrix in to L3,
taking an infinite amount of time to get there. (However in practice,
well before the amplitude reached the separatrix, the external
gravitational perturbations that convert L1, L2, and L3 from neat
"saddle points" into heteroclinic fixed points would have resulted
in the "test particle" being ejected from the Earth / Moon system.)


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'

How would these issues effect tidal pull on a very large space
station? The main benefit of L5 (I thought / think) is that your (For
eg) O'Niell cylinder can always point towards the sun, without the
Earth's tidal pull pulling it around.

(Gordon D. Pusch) wrote in message ...
(Henry Spencer) writes:

In article ,
Erik Max Francis wrote:
They orbit around it, more or less. The underlying story is actually
quite complex, but that's the bottom line...

Ideally (no other objects than the primary and secondary, test particle
has zero mass), with what period do the objects orbit around the Trojan
point?

In the general case, it's actually quite a complex motion, because the
particle oscillates around the point along all three axes, but the period
can be *different* on each axis. Only in special cases does the particle
follow something approximating a simple orbit around the point.

Such special cases of "closed orbits" are called _resonant_ orbits, because
the periods of the orbit in all three degrees of freedom are "commensurate"
(i.e., related by simple integer multiples). However, it shoul be noted that
these closed orbits are all unstable to small perturbations, and in general
the motion of a "test particle" in the volume near the L4 or L5 points
would be "chaotic," even if the gravitational perturbations of the Sun and
the non-circularity of the Moon's orbit could be neglected (which they can't).

Does that period depend on their distance from the point?

In some cases -- perhaps all, I don't remember for sure -- no, it's
independent of distance.

This is true only of small amplitude oscillations. As the amplitude
increases and the orbit approaches the separatrix enclosing L4 / L5
(which sadly is not shown on the otherwise excellent figure at
http://www.physics.montana.edu/faculty/cornish/lagrange.html ---
thank you "Hop David" for posting this link!), then in the absence of
external perturbations (such as the Sun, which actually can't be neglected)
the periods of those special families of "closed orbits" (as viewed from
the co-rotatiing frame) all approach infinity, with the limiting case
being a particle that follows a branch of the separatrix in to L3,
taking an infinite amount of time to get there. (However in practice,
well before the amplitude reached the separatrix, the external
gravitational perturbations that convert L1, L2, and L3 from neat
"saddle points" into heteroclinic fixed points would have resulted
in the "test particle" being ejected from the Earth / Moon system.)


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'