Article posted: 9 October 2004, with subsequent revisions
Future orbital engineering projects, such as the construction of space stations and orbital colonies, may result in unwanted debris accumulating in a ring formation around the Earth. To enable safe passage of interplanetary and lunar spacecraft to and from the Earth, it would be highly desirable to contain the ring particles from such projects in a neat plane so that they do not randomly scatter around the planet over time and pose a hazard to safe spaceflight. This paper illustrates that such a ring system around the Earth is unlikely to remain stable over any length of time and the degree and pace of scatter of ring material is a function of the orbital ellipticity and inclination of such a ring system.
For a particle, P, of point mass circling about the Earth, the elements of its unperturbed orbit can be visualized thus:
Dynamical models have shown that due to secular perturbations of the first order, the line of nodes of such a particle will precess (rotate) along the equator over time according to the equation :-
[where J2 is the second zonal harmonic in Earth's gravitational potential (approx. 0.001), R is
the equatorial radius of the Earth (6378km), p is the "parameter" of the orbit
(= a*(1-e^2)), n is the mean motion of the orbit (=sqrt(mu/a^3)), and i is
the inclination. (And "a" is the semi-major axis of the orbit, equal to
its radius for a circular orbit, "e" is the eccentricity, 0 for a circular
orbit, and "mu" is is the standard gravitational parameter for the Earth and equal to 398600.4, Earth's mass times the universal
gravitational constant.) The result is in radians per second if you've
been consistent with the other units.]
Similar models have yielded that the secular rate of change of the argument of perigee for the in-plane rotation of the orbit (i.e. the line of apsides) is given by:-
The orbit precessions described by these equations are due to uneven mass (and gravity) distribution of an oblate  Earth.
Based on computations using these equations, three seprate hypothetical ring orientations are considered here to identify if an optimum solution exists for achieving dynamical stability of such a ring system over a period of time.
Case 1 - Circular ring system, inclined to equator
In figure 2 below, consider a perfectly concentric, circular ring system inclined at an angle i-degrees relative to the Earth's equatorial plane, whose boundaries are marked by two ring particles P1 and P2 orbiting at the outer and inner edges of the ring, respectively (the purple ring). At time t1, the particles are at positions P1-1 and P2-1, relative to a common longitude of ascending node omega-1 as shown. Now, by equation (1) above we expect the ascending nodes of the two particles to precess at slightly different rates, depending on the distance separating them. At some future epoch, t2, we expect the particles to be at P1-2 and P2-2 relative to their new separate longitudes of ascending nodes, omega1-2 and omega2-2. From this illustration, it can be seen that the new ring formation (blue) is no longer concentric and the two particles have scattered away from the original (purple) ring formation.
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 radiation and solar wind particle pressure (when 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.
Case 2 - Circular ring system, in equatorial plane
This scenario is illustrated below, by figures 3(a) and 3(b), where since the ring system is perfectly circular (the line of *apsides* for every constituent particle is undefined) and precisely co-planer with the Earth's equatorial plane (the line of *nodes* for every constituent particle is undefined), the differential rates of precession between individual particles within the ring system stipulated by dynamical equations (1) and (2) will not affect its overall shape and stability.
Case 3 - Elliptical ring system in equatorial plane
In this scenario, illustrated in figure 4 below, the in-plane precession component stipulated by equation (2) above will be the determinant factor that causes the non-circular ring system to scatter its constituent particles over time.
From a first order dynamical model's perspective, one would conclude that a perfectly spherical, co-planer set of rings orbiting at a decay-immune altitude of say 40,000 km above the equator ought to remain fully stable over time (as depicted in case  above).
However, studies of the behaviour of geostationary satellites, whose thrusters for station-keeping duties had exhausted, have shown  that the perturbing influences of the Sun and the Moon cause these satellites to drift within a belt of approximately +/- 15 degrees geocentric latitude either side of the equator with an oscillation period of 54 years. This is due to a combined precessional effect between the Sun's perturbing influence (running along the ecliptic plane whose inclination is 23.4 degrees relative to the Earth's equatorial plane) and the Moon's perturbing influence (running along its orbital plane which is inclined at 5.1 degrees relative to the ecliptic plane). In actual fact, the satellite orbits oscillate around the Laplacian plane of the Earth-Moon system, which runs between the ecliptic and equatorial planes and is inclined at an angle of 7.3 degrees to the Earth's equator with an intersection along the line of equinoxes.
Computational Proof & Conclusions
Numerical results from two of the more critical scenarios outlined above are summarised below. Please click on each image thumb nail to download a more fully readable version:
Thus the combined perturbing influences of the Sun and the Moon together would cause ring particles even in the apparently most stable orientation [case (2) above] to scatter over many years into a belt of nearly +/- 15 degrees around the equator (7.3 degrees relative to the Laplacian plane, which itself is inclined at 7.3 degrees and precessing relative to the Earth's equatorial plane).
In conclusion, maintaining a dynamically stable ring system around the Earth is not possible in virtually all orientations owing to the complex interactive forces at play between the Earth, the Moon and the Sun.
However.... many experts have hitherto suggested that such a ring system could in fact have existed for hundreds of thousands of years, causing climatic effects on the Earth, as postulated here  and here.
This article was featured in the following links and discussion forums:-
 From the usenet thread "Precession of polar satellites" as supplied by Henry Spencer on sci.space.tech, 2003-11-07. Originally from A.E. Roy's "Orbital Motion", 3rd ed.
 "Variations in the Earth's oblateness during the past 28 years" - JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B09402 2004-09-16
 "A Geosynchronous Orbit Search Strategy" - Africano J.; Schildknecht T.; Matney M.; Kervin P.; Stansbery E.; Flury W.
 News article "Rings around the Earth: A clue to climate change?" - Peter J. Fawcett, of the University of New Mexico, and Mark B.E. Boslough, of the U.S. Department of Energy’s Sandia National Laboratories.