Re: My paper on dynamical stability of an Earth ring

abdul.ahad_at_ntlworld.com
Date: 12/10/04 

Date: Fri, 10 Dec 2004 11:01:13 +0000 (UTC)

Ralph Hartley wrote:
> abdul.ahad@ntlworld.com wrote:
> > In my paper, case [1] and [2] are proved beyond all doubt when one
> > considers simple geocentric orbital motion in 3D. Only case [3] is
> > (initially) 'suspect'.
>
> That isn't enough. You need to prove (or at least show) that there
aren't
> any *other* cases. You haven't shown that there can be *no* stable
ring in
> *any* orbit.

Yes you're quite right here, I have only picked a few "example"
orientations to computationally show the instability. There are a
couple of scenarios where, unless one plugs in some numbers and does
the sums, one may be mistaken into thinking there's some room for a
stable ring. First is where the inclination i=63.5 degrees, where by
equation (2) the term (2 - 5/2 * sin^2 i) would equal zero. So for even
an elliptical ring system at that inclination, the line of apsides
would not precess at all. However, by equation (1) an inclination of
i=63.5 degrees would still precess the line of nodes, since the term in
cos(i) still has a value, and the rates of precession for an outer
versus inner ring particle (P1 and P2 as per my existing nomenclature
in the given diagrams) would be different. So ring system not stable.

I have just noticed that a perfectly polar ring system would be a
special case. So I have inserted this as an addendum to case [1]:

"By equations (1) and (2) above, there is one unique case where a ring
system *could* theoretically hold stable: where the inclination, i=90
degrees (exactly polar) and the eccentricity, e=0 (exactly circular).
That orientation would however cause the ring plane to experience
maximum solar light pressure (if oriented face on relative to the Sun)
and the stipulation here is based on only a *first order* dynamical
model that ignores perturbative influences from the Sun and the Moon."

>
> > "Orbits of uncontrolled GEO objects oscillate around the stable
> > Laplacian plane,
> > which has an inclination of 7.5 =E2=97=A6 with respect to the ..."
- resear=
> > ch
> > reference [3] "A Geosynchronous Orbit Search Strategy" - Africano
J.;
> > Schildknecht T.; Matney M.; Kervin P.; Stansbery E.; Flury W. ,
> > 2000-01-01
> >
> > Above paper on Google listed as an abstract here: -
> > http://www.google.com/search?hl=3Den&q=3Dafricano+laplacian
>
> The way I read this leaves open the possibility that your conclusion
is
> incorrect.
>
> Note the phrase "the stable Laplacian plane". The implication is that
a
> ring in *that* plane, not the planes you looked at, might be stable.
I
> don't know for sure if that is so, but it wouldn't surprise me.

The Laplacian plane is an instantaneous *average* that passes through
the "invariable" momentum plane of the Earth-Moon-Sun gravity force, so
is constantly shifting relative to the Earth's equatorial plane, as the
Moon and Sun change orientation. I think it is also affected by changes
in the Earth-Moon barycentre (center of mass) as the Moon revolves
around the Earth every 27 days.

So I don't accept the Laplacian plane to be "fixed" relative to
anything. The Earth goes around the Sun, the Moon goes around the
Earth, all the planes are inclined at various angles to each other and
are in constant motion.

If the ring system is at a "fixed" inclination relative to the Earth's
equator, it cannot be then said to be at a "fixed" inclination relative
to the Laplacian plane... the two are rotating (precessing) all the
time.

>
> It looks like a set of particles released in a single oscillating
plane
> would continue to oscillate together, producing a ring that wobbles,
but
> does not spread. (Old geosynchronous satellites aren't like that,
because
> they are all released at different times.)

Consider that the ring has finite *width* where the inner and the outer
particles of the ring are spaced apart and have differentially
precessing orbits with respect to their lines of *nodes* and *apsides*.

>
> As for your underlying question, "would it pose an unacceptable hazard to
> navigation?" I suspect that, unless it was *extremely* narrow, even a

> completely stable ring would be unacceptable.
>
> Remember that every orbit must pass through the ring plane twice in each
> revolution. To be safe, each of those crossings must be either inside or
> outside the ring. That completely rules out circular orbits with the same
> period as the ring.
>
> Also, it would be incompatible with low thrust propulsion, such as solar
> sails and ion drives. To pass the ring, any spacecraft would have to raise
> (or lower) its apogee by at least the thickness of the ring within one
> orbit. That would put an absolute lower limit on the thrust to mass ratio
> of all spacecraft passing the ring.
>
> Ralph Hartley

The overall conclusion seems to be that, with the greatest of respect
for their work, those guys who produced that Sandia National
Laboratories research article about possible climate change due to
Earth rings were likely to have been flawed in their opinion. The Earth
cannot maintain a stable ring system and all future orbital engineering
projects will need to be mindful of how they scatter unwanted debris
around the planet, 'cos they aint gonna be contained in a neat ring
plane - fact!

Abdul Ahad

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