From: tkelso@afit.af.mil (TS Kelso)
Subject: Two-Line Element Set Questions
Keywords: Satellite, Orbital Elements, Keplerian
Organization: Air Force Institute of Technology
Date: Thu, 9 Dec 1993 07:36:37 GMT

Why is it that separate element sets are generated for both STS-61 and the
Hubble Space Telescope (HST) when they are docked together?  Why aren't these
element sets identical?

Since these questions have come up several times during the STS-61 mission to
service the Hubble Space Telescope and arise from time to time with regard to
the Mir space station, I thought I'd take this opportunity to answer them in
some detail here.

NORAD has responsibility for tracking all man-made objects in earth orbit.  In
performing this function, NORAD maintains a catalog of each individual it
tracks, even if that object is subsequently docked to another object.  Such is
the case for STS-61 and the HST or the various modules of the Mir space
station (Mir, Kvants 1 and 2, Kristall, and various Soyuz and Progress).  And
they also maintain separate element sets for each object.

Of course, these objects are physically locked together and the observations
collected for one object could be used for updating the elements for all
objects.  But that's not necessarily how it works.  Since NORAD has to track
over 7,000 objects every day, the process is automated.  Here's how it works.

Observations are collected for an object and used to generate a two-line
element set using a process known as differential corrections.  The two-line
element set is used to predict the observations (using the SGP4 orbital model)
and the differences between actual and predicted observations are compared.  A
correction to the two-line element set is determined which moves to minimize
these differences (something like doing a multi-dimensional Newton search).
This process is repeated until convergence is reached.

The element set generated by this process will be used until subsequent
observations show its predictions to be off by more than some preset amount.
For the data I make available, that is five kilometers (90 percent confidence
interval).  Five kilometers may seem like a large distance until you realize
that most of the error will be in the in-track direction and that it amounts
to less than one second's travel in near-earth orbits.  Once the current
element set is determined to be out of tolerance, a new element set is
generated.

Now, since all observations have error, the element sets generated will remain
in tolerance for differing periods of time.  How long will depend on the site
used to collect the observations, the viewing geometry, changes in atmospheric
density or spacecraft attitude, maneuvers, and so on.  As such, updates to
separate components of a docked structure will be updated at different times.

Okay, so even if NORAD tracks each object separately and maintains separate
element sets, why are the element sets different?  To understand the answer to
this question requires a basic understanding of satellite orbits.  Only ideal
orbits describe true ellipses and have constant orbital elements.  Real orbits
experience both periodic and secular (trending) effects in their elements.
Because of this, there are various ways of describing orbital elements.

One way is to use osculating (touching) or instantaneous orbital elements,
that is, the instantaneous values of each element at the specified epoch
(time).  The other is to use mean orbital elements.  The NORAD two-line
element sets use mean orbital elements.  In *neither* case will the elements
generated be constant over time.  In the SGP4 orbital model, the mean is
extracted in a particular way to not only allow combination with the various
perturbing forces (geopotential, atmospheric drag, solar/lunar gravitational
effects, and solar radiation pressure) to generate accurate predictions but to
also permit quick calculations (it is an analytical model which means you can
plug in a time and get out a position and velocity *without* numerical
integration).

To develop a better understanding of this concept, try to describe the mean x
and y position of an object traveling along a circle.  Taken over a long
enough period of time, it is the origin of the circle -- not terribly useful.
To be more specific, you must specify a time associated with your mean --
change the time and you change the mean.  Therefore, if two or more objects
are docked together in earth orbit, their orbital elements will not match
unless the epoch of the element sets are identical.  And even that assumes the
elements were generated from identical observations with identical update
processes.  However, predictions of each object's position and velocity, using
the NORAD SGP4 orbital model, should be the same within the error tolerance.

Remember, I did not develop this process but am only explaining it.  While I
can think of ways to improve it, you must also remember that it has evolved
slowly over time and is resistant to radical changes (as is any bureaucracy).
You may be thinking that it would make more sense to generate only one element
set for a group of docked objects, realizing that the error in prediction can
be on the order of five kilometers!  You might think that a single element set
would result in no error.  However, the error is still there -- a single
element set just seems more consistent.  Actually, multiple element sets
provide the advantage of redundancy -- protecting you from the occasional bad
element set.

Before I quit, I'd like to take this opportunity to explain why it is
important to use an orbital prediction program based upon the NORAD SGP4
orbital model when using the two-line element sets.  As with fitting any kind
of data, the coefficients determined depend upon the model used.  Imagine a
set of points (with noise) that you want to fit a curve to and determine the
constant term.  If you use a linear fit, you get one value; a quadratic fit
yields another; and a cubic fit yet another.  If you take the value for the
cubic fit and apply a linear projection, you won't get very good results.

The same is true for the NORAD two-line element sets.  If you take the mean
values from these element sets (the constant terms) and put them into a
different orbital model (say, a simple two-body model), you won't get very
good predictions.  Likewise, if you take some other data (for example, the
AMSAT data) and "convert" it to the two-line format, that does NOT make it
two-line data compatible with the SGP4 model.  I *strongly* discourage people
from converting other data into the two-line format, particularly if you are
reposting it, because the two-line format implies that the data was generated
with the NORAD SGP4 orbital model and will produce compatible results.  You
will cause other users a great disservice by distributing converted two-line
element sets since they will *not* produce accurate predictions.

I have made, and will continue to make, every effort to make NORAD two-line
element sets and the associated orbital models available to the satellite
community.  I welcome you to take advantage of these materials to produce the
most accurate predictions readily available.  I hope this answer your
questions.  - TS
--
Dr TS Kelso                           Assistant Professor of Space Operations
tkelso@afit.af.mil                    Air Force Institute of Technology