How big is L5?

(Karl Hallowell) writes:

Erik Max Francis wrote in message ...

snip

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.

I'm still not getting this. So what everybody is saying is that L4 and
L5 are normal stable points (ie, is a "valley") with respect to the
usual differential equations describe the dynamics of the orbits,

No, we are saying L4 and L5 are =NOT= in "valleys" --- they are both
sitting at the summits of "hills" !!! (Or, at least they are when
viewed from the co-rotating frame.) Hence, if it weren't for the coriolis
pseudo-force terms, the L4 and L5 points would be UNSTABLE, =NOT= stable.


but that certain force-like terms of the equations that help insure
stability of the solution aren't actual forces when you look at the
system from a non-rotating frame of reference

That is more or less correct. The coriolis force is a "fictious" force
that one must add in a non-inertial reference frame to make it _appear_
as if Newton's equations are still valid in that non-inertial frame;
however, this is a "cheat" --- Newton's equations are strictly valid
only relative to a non-rotating, non-acceleratoing frame of reference.


(hence isn't a "valley" wrt that frame of reference)?

Relative to the non-rotating frame, the L4 and L5 are =NEITHER= "valleys"
=NOR= "hilltops" --- they are points on the slopes of the combined Earth / Moon
gravity wells haveing the property that the vector sum of the gravitational
accelerations produced by the Earth and Moon points toward the Earth / Moon
barycenter, and has the correct magnitude such that a body placed at that
distance from the Earth / Moon barycenter and moving with the correct
velocity perpendicular to that acceleration in the plane of the Moon's
orbit about the Earth will be in a circular orbit about the barycenter
with the same period as the Moon's orbit (neglecting the eccentricity
of the Moon's orbit and the tidal forces exerted by the Sun.)
For sufficiently small deviations from these conditions, a test particle
will appear to execute bounded amplitude oscillation =RELATIVE= to L4 or L5
(again neglecting Lunar eccentricity and Solar tides), but if the initial
distance or velocity error are too large, the test particle will either
impact the Earth, the Moon, or be slung out of the Earth / Moon system into
a Solar orbit.


-- Gordon D. Pusch

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

(Gordon D. Pusch) wrote in message ...
(Karl Hallowell) writes:

Erik Max Francis wrote in message ...

snip

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.

I'm still not getting this. So what everybody is saying is that L4 and
L5 are normal stable points (ie, is a "valley") with respect to the
usual differential equations describe the dynamics of the orbits,

No, we are saying L4 and L5 are =NOT= in "valleys" --- they are both
sitting at the summits of "hills" !!! (Or, at least they are when
viewed from the co-rotating frame.) Hence, if it weren't for the coriolis
pseudo-force terms, the L4 and L5 points would be UNSTABLE, =NOT= stable.

Ok, I see what's going on. I think we are making this unnecessarily
complicated though. If I were piloting spacecraft in the neighborhood
of L4 or L5, I would treat the coriolis pseudo-force (which is just an
artifact of my co-rotating frame of reference) as a real force. That's
because in my frame of reference, it looks like a force. Then I am
moving in a "valley". However, if I were to build a local model of
this region and neglected the coriolis force then my model would
appear as a saddle or perhaps hill depending on what coordinate system
I was using.

In other words, people expect to find valleys with stable points and
saddles and hills with unstable points. That is a natural and
effective analogy that IMHO we should work with rather than against.
It seems better to me to point out that these stable points are in
valleys with the sides partly consisting of pseudo-forces rather than
to state in a confusing manner that the stable point is not a valley,
but that it is when you add pseudo-forces, but that you shouldn't
consider it as a valley even though it does look like a valley
locally. I just don't see gain from this mental pretzel.


Karl Hallowell

(Karl Hallowell) writes:

(Gordon D. Pusch) wrote in message ...
(Karl Hallowell) writes:

Erik Max Francis wrote in message ...

snip

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.

I'm still not getting this. So what everybody is saying is that L4 and
L5 are normal stable points (ie, is a "valley") with respect to the
usual differential equations describe the dynamics of the orbits,

No, we are saying L4 and L5 are =NOT= in "valleys" --- they are both
sitting at the summits of "hills" !!! (Or, at least they are when
viewed from the co-rotating frame.) Hence, if it weren't for the coriolis
pseudo-force terms, the L4 and L5 points would be UNSTABLE, =NOT= stable.

Ok, I see what's going on. I think we are making this unnecessarily
complicated though. If I were piloting spacecraft in the neighborhood
of L4 or L5, I would treat the coriolis pseudo-force (which is just an
artifact of my co-rotating frame of reference) as a real force. That's
because in my frame of reference, it looks like a force. Then I am
moving in a "valley".

No, I'm afraid that is =NOT= the case. Example: Suppose you bring yourself
to rest relative to L4 or L5 as viewed from the co-rotating frame. Since
your velocity vanishes (relative to the co-rotating frame), the coriolis
pseudo-force =ALSO= vanishes, since it is given by 2 M V x \Omega.
However, the combined gravitational force plus the "centrifugal"
pseudo-force does =NOT= vanish --- and it points =AWAY= from L4 or L5 !!!
Therefore, an object placed "at rest" relative to L4 or L5 will initially
"fall" _AWAY_ from L4 or L5, =NOT= toward them !!! No matter how you
try to twist it, L4 and L5 are _THE TOPS OF HILLS_ --- =NOT= the "bottoms"
of "valleys" !!!


In other words, people expect to find valleys with stable points and
saddles and hills with unstable points. That is a natural and
effective analogy that IMHO we should work with rather than against.

Sadly, REAL physical phenomena do =NOT= always conform to our naive
expectations --- which more nearly correspond to Aristotle's incorrect physics
than to Newton's correct physics. This is one example where intuition
breaks down. L4 and L5 are =NOT= "valleys," in ANY sense of the word !!!


It seems better to me to point out that these stable points are in
valleys with the sides partly consisting of pseudo-forces rather than
to state in a confusing manner that the stable point is not a valley,
but that it is when you add pseudo-forces,

I'm sorry, but stable points are =NOT= necessarily "valleys" in 3-space,
even when you "add the pseudo-force." If you begin from an incorrect premise,
you will reach incorrect conclusions.


I just don't see gain from this mental pretzel.

The "gain" is that you are using a PHYSICALLY CORRECT model, instead of a
PHYSICALLY WRONG model. If people are going to live and work in space, they
are going to have to learn to understand REAL Newtonian physics on a intuitive
level, not the quasi-Aristotelean pseudo-physics that living at the bottom
of a gravity well has incorrectly conditioned them to expect. When intuition
disagrees with reality, it is the INTUITION that must change, not the reality.
People who live and work in space are going to need to become INSTINCTIVELY
familiar with "An object in motion remains in motion, Every action has an
equal and opposite reaction, and Force equals mass times acceleration,"
and the orbital dynamics mantra, "In takes you East, East takes you Out,
Out takes you West, West takes you In. North and South bring you back"
on the same gut intuitive level that we groundhogs FALSELY expect that
an object at rest remains at rest, an object in motion comes to rest,
and velocity is proportional to force. We will have no choice but to
UN-learn our FALSE intuitions, and learn the ones that are correct for the
space environment --- because those who =DON'T= un-learn their groundhog
Aristotelean expectations WILL DIE IN SPECTACULARLY MESSY WAYS.
It's that simple.


-- Gordon D. Pusch

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

(Gordon D. Pusch) wrote in message ...
(Karl Hallowell) writes:

snip

Ok, I see what's going on. I think we are making this unnecessarily
complicated though. If I were piloting spacecraft in the neighborhood
of L4 or L5, I would treat the coriolis pseudo-force (which is just an
artifact of my co-rotating frame of reference) as a real force. That's
because in my frame of reference, it looks like a force. Then I am
moving in a "valley".

No, I'm afraid that is =NOT= the case. Example: Suppose you bring yourself
to rest relative to L4 or L5 as viewed from the co-rotating frame. Since
your velocity vanishes (relative to the co-rotating frame), the coriolis
pseudo-force =ALSO= vanishes, since it is given by 2 M V x \Omega.
However, the combined gravitational force plus the "centrifugal"
pseudo-force does =NOT= vanish --- and it points =AWAY= from L4 or L5 !!!
Therefore, an object placed "at rest" relative to L4 or L5 will initially
"fall" _AWAY_ from L4 or L5, =NOT= toward them !!! No matter how you
try to twist it, L4 and L5 are _THE TOPS OF HILLS_ --- =NOT= the "bottoms"
of "valleys" !!!

snip

Ok, I read through the discussion of the physics of the stability of
these points. One beef I have is that a number of them stop short of
proving stability. Ie, it's common to see linearization of the
equations to yield an evolution equation (ie, dx/dt = Ax, where A is a
matrix and x a vector function of time t). However, this yields
completely imaginary eigenvalues which indication rotation occurs
around the equilibrium point but imply neither indicate stability nor
instability in themselves. Ie, the stability is nonlinear. See for
example:

http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html
http://map.gsfc.nasa.gov/ContentMedia/lagrange.ps (a detailed
analysis)
http://scienceworld.wolfram.com/phys...ngePoints.html

The last one (from the World of Physics) refers to the following text,
"Fundamentals of Celestial Mechanics, 2nd ed., rev. ed.", by J. M.
Danby, where a discussion of this nonlinear stability occurs. Does
anyone have an opinion on this book?

In other words, people expect to find valleys with stable points and
saddles and hills with unstable points. That is a natural and
effective analogy that IMHO we should work with rather than against.

Sadly, REAL physical phenomena do =NOT= always conform to our naive
expectations --- which more nearly correspond to Aristotle's incorrect physics
than to Newton's correct physics. This is one example where intuition
breaks down. L4 and L5 are =NOT= "valleys," in ANY sense of the word !!!

snip

I agree with your statements now. We have complex rotating orbits
around this point. One would need to be careful when piloting through
such a region.

Also, here's a answer of sorts to the original question. It's an
arXiv.org article which estimates the size of the stability region of
the Sun-Jupiter L4 point. This is probably just the tip of the
iceberg, but what I found with some casual searching.

http://www.arxiv.org/abs/astro-ph/0012225

Abstract:

"We study the spatial circular restricted problem of three bodies in
the light of Nekhoroshev theory of stability over large time
intervals. We consider in particular the Sun-Jupiter model and the
Trojan asteroids in the neighborhood of the Lagrangian point $L_4$. We
find a region of effective stability around the point $L_4$ such that
if the initial point of an orbit is inside this region the orbit is
confined in a slightly larger neighborhood of the equilibrium (in
phase space) for a very long time interval. By combining analytical
methods and numerical approximations we are able to prove that
stability over the age of the universe is guaranteed on a realistic
region, big enough to include one real asteroid. By comparing this
result with the one obtained for the planar problem we see that the
regions of stability in the two cases are of the same magnitude."

-----


Karl Hallowell