[Server-sky] sanity check, please: orbit intersections

[Keith Lofstrom]

Tony, good work!

On Sat, Mar 28, 2009 at 10:36:48AM -0700, Tony Rick wrote:
> Can anybody check my approach here?
> 
> parameters:
>     omega = argument of periapsis (clockwise angle from line of ascending
> node to semi-major axis)
>     a = semi-major axis
>     e = eccentricity
>     ta = true anomaly (counter-clockwise angle from semi-major axis to
> object position on orbit)
> 
> - omega is provided in TLE (as argument of perigee)
> - a is recovered by norad lib (cOrbit instance)
> - e is provided in TLE

We will need to be careful to differentiate lower case omega (ω), argument
of perigee, and and upper case Omega (Ω) the longitude of the ascending node, 
in our writing.  You got it right there.


> Assuming that the true anomaly of interest is on the line of the ascending
> node,
>     ta = 360 - omega
> 
> Earth center is the primary focus of the orbit ellipse.
> 
> Radius from a focus to a point on the ellipse is given by
>     r = (a * (1 - e^2)) / (1 + (e * cos(ta)))

There are two true anomalies of interest, the ascending and descending nodes.
>From the trig identities,
The ascending  node cosine is cos( 360-Ω ) = cos( -Ω ) = cos( Ω )
The descending node cosine is cos( 180-Ω ) = -cos( Ω )

> Critical region is 12289  >= r >= 13289
> 
> Applying this to Keith's 'interesting' TLE data yields no critical
> intersections.

That is good for a first order approximation.  The server-sat orbits for
a fully populated constellation will have some small amounts of inclination,
peeking above and below the orbital plane by as much as ±500km , with 
inclinations of about 2.3 degrees  ( 2.3° = sin¯¹(e) ) .   So all orbits
with inclinations of less than 2.3 degrees and perigee/apogees bracketing
the region of interest are troublesome.   Other troublesome orbits with
small inclinations with ascending/descending nodes "close" to our orbits
will be troublesome.  

That gets messy, and for the purpose of constructing the "junk density"
graph versus radius, we should just consider the actual equatorial plane
crossings for now, for all 70 million TLEs, ascending and descending nodes,
and plot their density versus radius on a log/linear graph (with perhaps
50 km bins).   There should actually be two plots on the graph, one for
junk and one for satellites in use.  We don't want to hit junk ( which
creates more junk) and we especially don't want to hit expensive assets
(which creates lawsuits).

Another interesting graph, involving a heck of a lot more calculation,
would involve extrapolating the decay (first and second derivative of
mean motion) of highly eccentric orbits with apogees currently higher
than m288.  Those would eventually become problems.  

> Side note: norad lib appears to provide orbital distance from surface of
> Earth instead of
> center in some cases.  Perigees of sample TLEs are reported as in the range
> 200 - 5000 km.
> I'm adding mean Earth radius 6371km, but I'm not sure I have this right.

Question:  The TLE examples I saw seem to give mean motion in orbits per
day (sidereal I assume);  from that we can compute the semimajor axis,
and with the eccentricity we can compute perigee and apogee.  I don't
see anything in the TLE specifying apogee or perigee directly.  Those
might pop out of one of the programs, perhaps as altitude, so the 
question is what the programs are assuming.  The value used in the
SGP4 fortran program on wikipedia is  XKMPER = 6378.135 .   To think, I
used to write ugly fortran like that (now I write ugly C and ugly Perl).

Of course, the Earth is actually oblate, which causes highly inclined
orbits to precess (that is how sun synchronous orbits work ) but this only
shifts the angles of ascending and descending nodes, not their radii .
Other interesting effects are tides from the sun and moon.  For actually
computing collisions, rather than their probabilities, much more sophisticated
descriptions of orbits will be needed, and orbiting bodies will need to be
characterized with size and shape and other ballistic parameters.  The
probabilities will tell us which subset of the 70 million will need to
be closely scrutinized and real-time tracked for collision avoidance.

Keith

P.S.  You may notice some non-8-bit-ascii in the above email (or not,
depending on what you display it with).  These are from the extended
two byte ASCII set, which contain hundreds of non-Latin characters. 
Using the vim editor, these can be entered as digraphs, typically a
control K followed by two characters.  ^K D G is a degree symbol,
^K W * is capital Omega.  I have relatives in Sweden, and I often end
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- - - - Keith Löfström 

-- 
Keith Lofstrom          keithl at keithl.com         Voice (503)-520-1993
KLIC --- Keith Lofstrom Integrated Circuits --- "Your Ideas in Silicon"
Design Contracting in Bipolar and CMOS - Analog, Digital, and Scan ICs