THE ORBITAL CHARACTERISTICS OF DEBRIS PARTICLE RINGS AS DERIVED FROM
IDE OBSERVATIONS OF MULTIPLE ORBIT INTERSECTIONS WITH LDEF
William J. Cooke,
John P. Oliver and Charles G. Simon
Institute for Space Science and Technology
Gainesville, FL 32609
Phone: 904/371-4778, Fax: 904/372-5042
SUMMARY
During the first 346 days of the LDEF's almost 6 year stay in space, the
metal oxide silicon detectors of the Interplanetary Dust Experiment (IDE)
recorded over 15,000 impacts, most of which were separated in time by
integer multiples of the LDEF orbital period (called multiple orbit event
sequences, or MOES). Simple celestial mechanics provides ample reason to
expect that a good deal of information about the orbits of the impacting
debris particles can be extracted from these MOES, and so a procedure,
based on the work of Greenberg, has been developed and applied to one of
these events, the so-called May swarm. This technique, the
Method of Differential Precession allows for the determination of
the geometrical elements of a particle orbit from the change in the
position of the impact point with time. The application of this approach
to the May swarm gave the following orbital elements for the orbit of the
particles striking LDEF during this MOES: a = 6746.5 km; e between 0.0165
and 0.025; i = 66°.55; right ascension of ascending node = 179°.0 ± 0°.2;
argument of perigee = 178°.1 ± 0°.2.
INTRODUCTION
For 346 days after the deployment of the LDEF satellite on April 7, 1984,
the tape recorder belonging to the Interplanetary Dust Experiment (IDE)
stored information on over 15,000 impacts made by submicron and
larger-size particles on its metal oxide silicon (MOS) detectors . These
detectors were mounted on trays facing in six orthogonal directions - LDEF
ram and trailing edge, the poles of the LDEF orbit (north and south), and
radially inward (towards the Earth) and outward (towards space). The 13.1
second time resolution provided by the IDE electronics, combined with the
high sensitivity of the MOS detectors and large collecting area (~ 1 sq.
m) of the experiment, conclusively showed that the small particle
environment at the LDEF altitude of 480 km was highly time-variable, with
particle fluxes spanning over four orders of magnitude.
It is highly desirable to learn as much as possible about the orbital
characteristics of the particles which struck the IDE trays. At a
minimum, these characteristics can determine whether the particles were
interplanetary in origin or debris from a satellite or spent rocket. If
the particles can be identified as debris, then it becomes possible to
determine their parent body, which gives a clue as to which objects in
earth orbit are major contributors to the orbital debris population.
Unfortunately, the IDE data permit the unique determination of only the
position of the impacted particle (which is the same as that of LDEF at
the time of impact), whereas an unambiguous determination of the
particle’s orbit requires a knowledge of both the position and the
particle velocity. The IDE sensors were threshold detectors, triggered by
any particle with sufficient energy to damage the detector dielectric, and
so were rough indicators of particle energy, not velocity. It is
therefore impossible to use the IDE data to obtain an orbit for a single
impacting particle. This situation improves, however, for an impacting
group of particles which have the same orbit. In this case, the particles
will strike multiple IDE trays, permitting a rough determination of the
direction of the group’s velocity, which, when combined with the position
information, yields a family of possible candidate orbits for the
particles. The situation improves even more if the orbit of the group is
such that it encounters LDEF multiple times, for then the change in the
LDEF position at the encounter times can be used to produce a family of
possible orbits, which can be further constrained by the velocity
direction information.
Fortunately, most of the 15,000 impacts recorded by IDE occurred in such
groups, which we term events. These events were of two types - the
spikes, which were single, isolated events of high intensity and
the multiple orbit event sequences (MOES), which were series of
events with the events separated in time by integer multiples of the LDEF
orbital period. The spikes are discussed in another paper in these
proceedings; here we shall concentrate on the multiple orbit event
sequences, as they were produced by particles with orbital characteristics
such that the group had multiple encounters with LDEF. Even though the
spikes were generally more intense, the MOES could be quite long-lived,
some lasting for many days. As discussed in the previous paragraph, it is
these MOES which can yield the most information about the particles’
orbit.
Figure 1 is a "seismograph" plot of a typical MOES; time increases to the
right along the horizontal axis, and the intensities of the events are
roughly indicated by the extent of the vertical lines. A cursory glance
reveals two important bits of information about the particle orbits
involved in MOES:
- the particle orbits are eccentric; if they were circular, the IDE
detectors would register the group twice each orbit, as a circular orbit
would intersect LDEF’s orbit (which is essentially circular) at two
points, and
- the particles must be "smeared out" along the orbit in some ring-like
or torus structure. If the particles were concentrated in a clump,
the encounters with LDEF would not occur at integer multiples of the LDEF
orbital period, unless the period of the particle orbit was the same as
that of LDEF, a highly unlikely circumstance.
Figure 1: Typical MOES (not the May swarm). Impacts on the south
(So4h), ram (Le4h), and trailing (Tr4h) surfaces are shown. The tick
marks along the top and bottom are spaced at intervals of a LDEF orbital
period.
This information is about all that can be determined from a visual
inspection of the MOES in the IDE data set. Clearly, it is necessary to
develop a technique that will extract additional information about the
particles’ orbit. We have arrived at such a technique, the Method of
Differential Precession, which shall be summarized and applied in the
following pages.
THE METHOD OF DIFFERENTIAL PRECESSION
Overview
The goal of the Method of Differential Precession is to obtain the orbital
characteristics of the particles which struck the IDE detectors during a
MOES by an analysis of the time variation of the LDEF position over the
series of encounters. This analysis makes use of the fact that the
non-sphericity of the Earth induces the pole of an object’s orbit to
precess, resulting in a cyclic change in the position of the line of nodes
of the orbit (in the case of LDEF, the period of this precession is
approximately 53 days). The oblateness of the Earth also causes the line
of apsides of the orbit to precess, the point of perigee advancing if the
orbital inclination is low and regressing otherwise. In general, bodies
in different orbits will have different rates of these precessions, and
should two of these orbits intersect, the differences in the precession
rates will cause the point(s) of intersection to vary with time. If the
characteristics of one of the intersecting orbits are known, the migration
of the point of intersection may be used to determine the precession rates
and orientation of the unknown orbit, which then may be used to calculate
a family of candidate orbits.
This concept is illustrated more clearly in figures 2 and 3, which depict
the geometry of the situation with regard to LDEF. Following conventional
notation, 
represents the position of the
ascending node of the LDEF orbit (which is known) and 
represents the ascending node of the unknown orbit of
the impacting particles. The position of the perigee of the unknown orbit
is represented by 
; LDEF’s orbit is
essentially circular (e ~ 0.0001) and so has no perigee. The inclinations
of the two orbits are 
and 
, and 
and 
denote the arguments of latitude of the point of
intersection, measured counterclockwise along the orbits from the
respective ascending nodes. We are using the argument of latitude rather
than the more conventional true anomaly, 
, due
to the fact that one of the orbits is circular. In the case of an
elliptical orbit, the two quantities are related by 
. Figure 2 shows that, for any given time, the known
quantities and
determine the location of the unknown orbit’s node, , provided that the inclination of the unknown orbit is
specified. Similarly, figure 3 shows that , and determine if is specified. As time progresses, the orbits will precess
at different rates, resulting in the movement of the point of impact
(intersection), with a corresponding change in
and .
Figure 2: Differential precession of the lines of nodes
Figure 3: Precession of the line of apsides
Assuming that the inclination of the particle orbit is known, our
knowledge of the LDEF orbit enables us to compute and for each impact
occurring in a given MOES. The variation in with time directly yields the precession rate of the
line of nodes of the particle orbit, , which can then be used to
construct a family of possible candidate orbits by means of the well-known
relation
1)
where 
is the radius of the Earth and 
is the second gravitational harmonic. The semi-major
axis and eccentricity of the particle orbit are denoted by a and e, while
n is the mean motion of the particles. The family of candidate orbits
will have values of a and e specified by equation (1), and can be
constrained by the simple fact that any candidate orbit must intersect
that of LDEF at some point. Information about the direction of the
particle velocity obtained from the numbers of impacts on the IDE trays
during the MOES can also be employed to derive the vector intercept of the
particles, which further constrains the range of allowed orbits. It
should noted that even though this technique can completely determine the
orientation of the particle orbit (, , and ), the lack of
velocity information still prohibits a unique determination of the orbit’s
size and shape.
Unfortunately, the inclination of the particle orbit is not known, forcing
the adoption of an iterative scheme in order to achieve a solution. One
starts by assuming a reasonable value for the particle inclination, which
will enable the determination of the ’s and
’s at the times of impact, and,
consequently, . It is also necessary to obtain the rate of the perigee
advance of the particle orbit. This can be done by realizing that, at
each impact, the position of LDEF must be the same as that of the
particle, thus
2)
Equation (2) implies that 
, and we see
that the time variation of the argument of latitude of the impact point,
measured along the particle orbit, is equal to the rate of the advance of
the perigee. The ratio 
can now be
formulated and compared to the theoretical value, which is given by
3)
If the ratios are not equal, then a new is
calculated according to Newton’s method or some similar scheme, and the
process repeated until the values agree. In addition to the assumption of
no non-gravitational forces, this method also requires that all particles
striking LDEF during the MOES share the same orbit, which is perfectly
reasonable in light of the short duration of each event belonging to a
MOES.
Methodology
Based on the above discussion, one may obtain the family of possible
particle orbits by proceeding as follows:
- Obtain the arguments of latitude () of
LDEF at all impact times in the MOE. This is a simple matter, given the
LDEF orbital elements and an orbit propagation code.
- Assume an inclination () for the orbit of
the particles. Good starting values are 30º, 65º, or 82º, as these are
representative of most satellite orbits.
- The difference between the right ascension of the particle orbit
ascending node () and that of LDEF () is determined by the argument of latitude of
LDEF at the time of impact (), the inclination
of the particle's orbit, and the inclination of LDEF's orbit (). The relevant expressions are
4)
At each impact time, equations (4) may be solved for 
(and hence, ) via
Newton's method or some other scheme.
- The slope of a line fit to the 's and
their associated impact times gives the rate of regression of the nodal
line of the particle orbit,
.
- Once the 's have been determined for
the impacts, the corresponding arguments of latitude for the particle
orbit are found from
5)
- In the absence of non-gravitational forces, the semi-major axis (a)
and the eccentricity (e) of the particle orbit remain constant over time.
Therefore, equation (2) requires that
must also be constant, or 
. The slope of
a line obtained by a linear regression performed on the ’s and their associated times yields the progression or
regression of the line of apsides, 
.
- Compute the ratio and compare to the
theoretical ratio obtained from equation (3), which is a function of only
the inclination of the particle orbit. If the two are not equal to within
a specified tolerance, compute a new by means
of Newton’s method and repeat steps 3) through 6) until the values agree.
- Use the the values of and to determine a family of possible candidate orbits in
(a,e) space by means of equation (1). Constrain the range of potential
candidates by imposing the requirements that the particle orbit must
intersect that of LDEF and must not enter the atmosphere (i.e., the
perigee must be greater than 200 km).
Application to the May Swarm MOES
One of the most prominent multiple orbit event sequences observed by IDE
began on May 13, 1984, and so has become known as the "May swarm." This
MOES can be characterized as being of low intensity (~3 impacts per orbit)
and long duration, lasting for over 20 days (300 LDEF orbits), with
several hundred impacts recorded on the IDE trays facing in the LDEF ram
direction and towards the south pole of the orbit, the majority occuring
on the south-facing tray. The long duration of this MOES made it an
especially suitable choice for analysis by the differential precession
technique, the only drawback being the low intensity of the events. To
avoid contamination by the occasional "random" impact, the times chosen
were those in which the high sensitivity (0.4 micron dielectric thickness)
IDE detectors on the south tray recorded multiple impacts within the same
IDE clock "tick" (13.1 seconds). This resulted in a total of 38 points
for use in the analysis, spanning a time interval of some 18 days.
The procedure outlined in the previous section was then applied to these
data, with the inclination converging to a value of 66°.55 after only a
couple of iterations. Table 1 lists the resulting longitudes of ascending
node and arguments of latitude of the impact points for the particle
orbit, along with the times of impact (in decimal days from LDEF deploy)
and the LDEF arguments of latitude of the impact points.
Table 1: May swarm times of impact (days from LDEF deploy) and the
corresponding LDEF arguments of latitude, with the values of the
longitudes of the ascending node of the particle orbit and the particle
arguments of latitude for =
66°.55.
The two linear regressions (see figures 4 and 5), involving the impact
times, , and ,
yielded
These four quantities, along with , uniquely
specify the orientation of the particle orbit at any given time. Note
that the initial value of the argument of the perigee indicates that these
particles are striking LDEF near apogee, a somewhat surprising result.
Figure 4: Linear fit to determine nodal line properties
Figure 5: Linear fit to determine apsidal line properties
Next, the precession rate of the particle orbit line of nodes was used in
equation (1) to determine the family of possible candidate orbits. These
results are displayed in figure 6. Note that the semi-major axis varies
little with the eccentricity; in this case, the variation in a is so small
that we could confidently set a = 6746.5 km, regardless of the
eccentricity. The dual requirement that the candidate orbits have
perigees of greater than 200 km in altitude and intersect the LDEF orbit
placed strict limits on the allowed values of the eccentricity, which must
be between 0.0165 and 0.025.
Figure 6: Candidate orbits for the May swarm
One of the candidate orbits (e = 0.017) was then chosen for a series of
checks on the results of the method. The first check involved the
computation of the particle velocity of impact over the duration of the
May swarm. These velocities were then resolved into components along the
LDEF body axes in order to determine the impact speeds on the IDE trays.
For this particular orbit, only the south tray and the ram-facing tray
were struck, with the south impact speed being larger than that for the
other tray (see figure 7). This is in good agreement with the IDE
observations of the May swarm, in which these same two trays recorded
large numbers of impacts, with the south tray receiving the most hits.
The second check consisted of a comparison of the sky track of the points
of closest approach between the two orbits to the sky positions of the
individual impacts comprising the May swarm. As can be seen from figure
8, the agreement is excellent, with the sky track of close approach
passing neatly through diffuse band of impact positions.
Figure 7: Particle impact speeds along IDE tray normals for test
orbit.
Figure 8: Sky track of close approach between the test orbit and that
of LDEF. The particles are moving in a northerly direction, whereas LDEF
is moving along its orbit from left to right. At the onset of the May
swarm, the impacts are located at the position labeled "Onset", with the
impact positions gradually moving towards the lower right as time
progresses.
In summary, it would seem that the particles impacting LDEF during the May
swarm MOES have an orbit that can be characterized by the following
parameters:
CONCLUSION
It has been pointed out numerous times in the literature that the surface
area to mass ratio of a micron-sized particle (the size of many of the
LDEF impactors) is large, thus causing the particle to experience
significant perturbations from forces such as radiation pressure and
atmospheric drag. Indeed, numerical calculations indicate that radiation
pressure can cause a one micron diameter particle, initially in an orbit
similar to that of LDEF, to enter Earth’s atmosphere after only a very few
(<10) orbital revolutions. Such calculations leave us hard-pressed to
explain how a MOES like the May swarm, which we postulate to be caused by
a ring of micron and sub-micron particles, can persist for many days. The
only reasonable explanation is that the ring must be replenished by debris
from some source during the time spanned by the MOES.
To obtain the geometrical chararcteristics of this ring, the technique of
differential precession looks at the time evolution of the point of
intersection with the LDEF orbit. It does not matter that the particles
exist in the ring for only a short time; the only requirement is that the
orbit shared by the particles at the times of impact with LDEF be
similar. In general, this orbit would not be the same as that of the
source of the debris particles, for non-gravitational forces would have
rapidly acted to alter the particles’ orbit from that of the parent body.
It should be realized that if the parent body (whose orbit is presumably
stable) continually produced particles of similar properties, these
particles would have experienced the same perturbations as their
predecessors and would therefore have undergone a similar orbital
evolution. If any of the future orbits intersected that of LDEF, a MOES
would have been observed by IDE, this MOES lasting as long as the source
produced particles, or until the geometry of both orbits changed such that
there was no longer a point of contact.
The IDE data set is rich, with many MOES of varying characteristics that
await analysis by some procedure. The method of differential precession
is such a technique, one that appears to be able to extract a good deal of
information about the particle orbit involved in a MOES. We fully expect
that its application to the other MOES will not only shed some light on
possible sources of orbital debris, but will also yield quite a few
surprises.
This research was supported by NASA grant NAG-1-1218
References
- Greenberg, R., Orbital Interactions: A New Geometrical
Formalism, Astronomical Journal, 87, pp. 184-195 (1982)
- "IDE Spatio-Temporal Impact Fluxes and High Time-Resolution Studies of
Multi-Impact Events and Long-Lived Debris Clouds", J.D. Mulholland, S.F.
Singer, J.P. Oliver, J.L. Weinberg, W.J. Cooke, P.C. Kassel, J.J. Wortman,
N.L. Montague, W.H. Kinard, LDEF - 69 Months in Space: First LDEF
Post-Retrieval Symposium, (NASA CP-3134), January, 1992, pp. 517-528
- See "LDEF Interplanetary Dust
Experiment (IDE) Results", J.P. Oliver et al., this archive.
- For details on the IDE detectors, see "Long-term Particle Flux
Variability Indicated by Comparison of Interplanetary Dust Experiment
(IDE) Timed Impacts for LDEF's First Year in Orbit with Impact Data for
the Entire 5.77 Year Orbital Lifetime", C.G. Simon, J.D. Mulholland, W.J.
Cooke, J.P. Oliver, P.C. Kassel,
, (NASA CP-3194), April, 1993, pp.
693-704
- Greenberg, R., Orbital Interactions: A New Geometrical
Formalism, Astronomical Journal, 87, pp. 184-195 (1982)
This document produced by Dr. Bill Cooke; bjc@ufl.edu