(a) links to the first four quarterly Space Ornithology newsletters and the first three pages of
the SPACE BIRDS user's manual,
1. When the Adobe Acrobat Reader asks you if you want it to go to full screen mode, click
"Yes." You can hit the escape key at any time to go back to the Acrobat Reader frame.
2. Use the keyboard's right arrow and left arrow keys to page forward and back,
respectively, through the slide show. The space bar acts like the right arrow key.
*Did you notice the month and day, in the date given on the last line of the primer? -- There is no
such society as the "Maculate Order of Space Ornithologists" (M.O.S.O.)
The Space Ornithology Newsletter (SON) was published quarterly on the 15th of January, April, July, and
October during the years 1988 through 1991, as information for SPACE BIRDS computer program purchasers.
Each newsletter typically contained a feature article followed by "Tips for Birdbaggers" and "The Space
Ornithologist's Library."
The Space Ornithology Kit consists of a CD-ROM and printed materials. The WinZip executable on the CD will write about 8 Megabytes of content to your computer's hard disk. Suggestion: unzip to a flash drive for maximum portability and ease of use.
Warning: This program is not "user friendly"
by today's standards. It was designed back
in the days of MS-DOS. That is, it was designed
to run on the MS-DOS command line of an
IBM PC, using a text editor to prepare input
files and view output files.
SPACE BIRDS can be run as a DOS program at
the DOS prompt, or executed directly from its icon
in its Windows folder.
SPACE BIRDS runs under all DOS versions and has
been verified to run under Windows 98SE, 2000, XP,
and Vista.
During 1988-1991, SPACE BIRDS was shipped with
the PC-Write text editor, for use in preparing
input files and viewing output files. Windows
users can now better use the Windows accessory
programs Notepad and Wordpad for this same
purpose.
One PDF "slide show" document per year
(four quarterly issues per year).
However, due to changes in U.S. federal and state laws regarding the collection and payment of sales taxes by U.S. retailers, the burden of compliance is so great, relative to the expected net return on sales, that it is no longer possible for Astronomical Data Service to offer publications such as the Space Ornithology Kit for sale directly to consumers.
Postal mailing address:
ASTRONOMICAL DATA SERVICE
P.O. Box 885
Palmer Lake, CO 80133-0885
U.S.A.
Back to Top
"Predicting Iridium Flares" Presentation at DDA Conference, Boulder, Colorado, April 30, 2008 -- See
http://astroger.com/
Case Study 1: Iridium Birds and Flares. The Iridium constellation of 66 operational global mobile
telephony satellites is arguably the most exciting "flock" of space birds that space ornithologists can
study right now. This is because each Iridium satellite has two highly reflective solar panels, plus
three large Main Mission Antennas (MMAs) that can reflect sunlight back to earthbound observers in a
way that can be predicted. Also, the spacecraft body itself, which is triangular in cross-section,
reflects sunlight well.
In the past two years I have put several pieces of content online as regards Iridium satellite
observing, to include an illustration of a typical Iridium satellite and some of the mathematical
details of flare prediction. Also, Tom Bisque (of
Software Bisque, Inc.)
has captured some really amazing videos of Iridium flares using his Wright-Schmidt telescope and
Paramount ME robotic telescope mount -- see the related links and references at the end of this
case study.
What I want to do now is to:
(a) tell you how to predict Iridium flares using Software Bisque's new
TheSkyX
computer program, and then
(b) show you how the SPACE BIRDS program provides a concise summary of just about everything a
seasoned satellite orbital analyst might want to know as regards Iridium satellite visibility --
especially before observing a visible pass during which an Iridium bird is predicted to flare. (And of
course, the Iridium birds are just one class of space bird that you can study with SPACE BIRDS.)
I'm going to resort now to the format that every experienced space operations analyst uses when working
a satellite pass: an operational checklist.
CHECKLIST FOR PREDICTING IRIDIUM FLARES AND GENERATING FLARE PASS SUMMARIES
__1. Go to T.S. Kelso's Celestrak website, at
http://celestrak.com,
and download the three-line orbital elements (TLE) for the Iridium satellites.
By "download" I mean: when your browser is displaying the Iridium birds' orbital elements in the window
presented by Celestrak, (a) "select all" and "copy" the elements to the Windows clipboard, (b) then
open a Windows Notepad window, "paste" the elements there, and (c) "save" the elements to a Notepad
text file.
When I took this step on October 1, 2008, I named my own Notepad file "IRIDIUM CELESTRAK
2008-10-01.txt".
__2. Launch TheSkyX and predict the Iridium flares visible from your observing location
over the next seven days.
To do this, you need to go to TheSkyX's Input menu to set the location and date (in this case, October
1, 2008), and then click on the Satellites submenu to import the TLE from the file you saved in
Checklist Step 1.
When you have imported the Iridium TLE, click on the Iridium Satellites tab and then click on Find
Flares. When I did so, I found a magnitude -2.9 flare was predicted on Sunday, October 5, 2008. Then I
clicked on the Watch Flare button to see an animation of the flare. Below is a screen capture that I
obtained from TheSkyX's Watch Flare animation.
Figure. Iridium 65 flare visible from Colorado Springs, Colorado U.S.A. on October 5, 2008,
as predicted and depicted by Software Bisque's TheSkyX.
__3. Create Orbital Elements, Observer Location, and Run Control
Information files for SPACE BIRDS, for a day on which an Iridium bird of interest will flare.
To create the Orbital Elements file, I opened the Notepad TLE file from Checklist Step 1, searched for
the Iridium 65 TLE, then copied/pasted the TLE to a text file that I named IRIDSAT.TXT. Here, in
boldface type, is the text of that file:
IRIDIUM 65 [+]
1 25288U 98021D 08274.64871166 -.00000065 00000-0 -30155-4 0 2797
2 25288 86.3913 113.6449 0002272 82.5170 277.6282 14.34218014549001
I had already prepared the Observer Location file -- its name is COS.TXT. Here, in boldface
type, is the text of that file:
COLORADO SPRINGS, COLORADO U.S.A.
+38.833 -104.817 1.981 -06.00 COLOSPGS
LATI LONGI HEIGHT D UTC ABBREV'D
TUDE TUDE (KM) (HR) LOCATION
F7.3 F10.3 F8.3 F8.2 3X, A8
o ONLY THE FIRST TWO LINES ARE READ BY THE PROGRAM, UNLESS IN QUEUE MODE.
o FIRST LINE IS TEXT DESCRIPTOR FOR YOUR LOCATION, UP TO 72 CHARACTERS.
o INPUT LATITUDE NEGATIVE IF SOUTH; INPUT LONGITUDE NEGATIVE IF WEST.
o D UTC IN HOURS IS -5.00 FOR EST; -6.00 FOR CST; -7.00 FOR MST;
-8.00 FOR PST; -9.00 FOR AST; -10.00 FOR HST.
(ADD ONE HOUR FOR DAYLIGHT SAVING TIME.)
o ABBREVIATED LOCATION DESCRIPTOR CAN CONTAIN UP TO 8 CHARACTERS.
(Only the first two lines are read by the program.
Subsequent lines are just "Help" comments in the COS.TXT file.)
I named the Run Control Information file RUN.TXT, and it looks like this:
279- 1 3.00 YES YES 3D3D 10.00 10.00 YES 0.00 NO
START NBR PNTS WANT WANT PASS LOWEST LOWEST VISBLE TWI INVISB
DAY OF PER SCREEN DISK MODE EL MDP EL POINTS THRESH MIDP
DAYS MIN OUTPUT OUTPT CODE (DEG) (DEG) ONLY (MIN) REJECT
I5 I6 F6.2 5X,A3 4X,A3 3X,A4 F7.2 F8.2 5X,A3 F7.2 5X,A3
o ONLY FIRST TWO LINES ARE READ BY PROGRAM; SECOND LINE MUST BE BLANK UNLESS
IN QUEUE MODE. ALL DATA SHOULD BE RIGHT-ALIGNED IN FIELDS.
o PASS MODE CODE: _ALL=ALL PASSES; _DAY=ALL DAY PASSES;
NITE=ALL NIGHT PASSES; 3D3D=THREE DAWN & THREE DUSK PASSES.
o VISIBLE POINTS ONLY=YES REJECTS POINTS FOR WHICH EITHER SATELLITE IS NOT
IN SUNLIGHT OR OBSERVER IS NOT IN DARKNESS.
o SEE SPACE BIRDS MANUAL, SECTION 5B, FOR DETAILED INSTRUCTIONS.
IMPORTANT! A MINUS SIGN MUST APPEAR IN COLUMN 6, RIGHT AFTER THE START DAY,
TO TELL SPACE BIRDS THAT THE COMMON NAME APPEARS BEFORE THE TWO-LINE
ELEMENTS (TLE) IN THE ORBITAL ELEMENTS INPUT FILE, RATHER THAN AFTER THEM.
(Only the first line and the blank line below it are read by the program.
Subsequent lines are just "Help" comments in the RUN.TXT file.)
Note that the start day is counted from the beginning of the year, i.e., January 1 is day 1, and
October 1, 2008 is day 279, since 2008 is a leap year. (A table of days since the beginning of the year
is given on p. 20 of the SPACE BIRDS user's manual.)
__4. Run SPACE BIRDS, specifying the three files IRIDSAT.TXT, COS.TXT, RUN.TXT.
Output went to the SPACE BIRDS output file AVES.OUT, and it looks like this:
PROGRAM AVES - ACQUISITION OF VISIBLE EARTH SATELLITES
COPYRIGHT 2008, BY ASTRONOMICAL DATA SERVICE V. 2.12
RUN CONTROL INFORMATION:
START DAY 279
NUMBER OF DAYS TO RUN 1
POINTS PER MINUTE 3.00
SEND OUTPUT TO SCREEN YES
SEND OUTPUT TO DISK FILE YES
PASS MODE CODE 3D3D
MINIMUM ELEVATION (DEG) 10.00
MINIMUM MIDPASS ELEVATION (DEG) 10.00
VISIBLE POINTS ONLY YES
TWILIGHT THRESHOLD (MIN) .00
REJECT PASS IF MIDPASS INVISIBLE NO
SATELLITE ORBITAL INFORMATION:
SATELLITE NAME IRIDIUM 65 [+]
REVOLUTION NUMBER AT EPOCH 54900
YEAR OF EPOCH 8
EPOCH (DAY AND FRACTION OF DAY) 274.64871166
SEMIMAJOR AXIS (E.R.) 1.1221655
ORBITAL ECCENTRICITY .0002272
ARGUMENT OF PERIGEE (DEG) 82.5170
ORBITAL INCLINATION (DEG) 86.3913
R.A. OF ASCENDING NODE (DEG) 113.6449
MEAN ANOMALY (DEG) 277.6282
MEAN MOTION (REVS/DAY) 14.34218014
MEAN MOTION, 1ST DERIV. (REVS/DAY**2) -.650000D-06
MEAN MOTION, 2ND DERIV. (REVS/DAY**3) .000000D+00
R.A. ASC. NODE, 1ST DERIV. (DEG/DAY) -.41909317D+00
NODAL PERIOD (MIN) 100.4666
OBSERVER INFORMATION:
COLORADO SPRINGS, COLORADO U.S.A.
LATITUDE (DEGREES, NEGATIVE IF SOUTH) 38.833
LONGITUDE (DEGREES, NEGATIVE IF WEST) -104.817
HEIGHT ABOVE SEA LEVEL (KM) 1.981
HOURS FAST (+) OR SLOW (-) ON UTC -6.00
==============================================================================
8 10-05 279 COLOSPGS SUNRISE= 659/26 SUNSET=1834/58 DUTCH= -6.00 IRIDIUM
25288 54968 .0 NO ACQUISITION ON THIS REV
ASC. NODE TIME= 325/51 LONG= -44.2
25288 54969 .0 NO ACQUISITION ON THIS REV
ASC. NODE TIME= 506/19 LONG= -69.4
25288 54970 35.4 SETS TOO CLOSE TO SUNRISE
ASC. NODE TIME= 646/47 LONG= -94.7
25288 54977 83.5 PERCENT ILL. MOON AT MIDPASS = 37.6
ASC. NODE TIME=1830/03 LONG= 88.8
DAY TIME LATI LONGI HEIGHT ELEV AZI RANGE %ILL SEP. R.A. DEC.
NBR HHMM/SS TUDE TUDE (KM) ATION MUTH (KM) SAT MOON HHMM DGMN
279 1904/20 57.1 -105.3 793.4 10.9 359.1 2283.2 51.5 143.5 713 6203
279 1904/40 55.9 -105.2 793.0 12.8 359.3 2153.8 51.9 142.0 712 6356
279 1905/00 54.7 -105.0 792.6 14.8 359.6 2025.3 52.2 140.2 710 6558
279 1905/20 53.5 -104.9 792.2 17.0 359.8 1898.1 52.6 138.3 708 6813
279 1905/40 52.3 -104.8 791.8 19.5 .1 1772.6 53.1 136.2 705 7041
279 1906/00 51.1 -104.6 791.4 22.3 .5 1649.0 53.6 133.8 701 7326
279 1906/20 50.0 -104.5 791.0 25.4 .9 1528.0 54.2 131.0 653 7631
279 1906/40 48.8 -104.5 790.6 28.9 1.4 1410.2 54.8 127.8 640 7959
279 1907/00 47.6 -104.4 790.2 32.9 2.0 1296.8 55.5 124.2 605 8353
279 1907/20 46.4 -104.3 789.8 37.6 2.7 1188.8 56.3 119.9 303 8731
279 1907/40 45.2 -104.2 789.4 43.1 3.8 1088.1 57.2 114.8 2119 8454
279 1908/00 44.0 -104.2 788.9 49.5 5.2 996.9 58.2 108.8 2020 7844
279 1908/20 42.8 -104.1 788.5 56.9 7.5 918.0 59.3 101.8 2001 7117
279 1908/40 41.6 -104.0 788.1 65.4 11.6 854.9 60.3 93.7 1952 6236
279 1909/00 40.4 -104.0 787.7 74.7 21.2 811.4 61.1 84.6 1946 5250
279 1909/20 39.2 -104.0 787.3 82.9 58.2 790.7 61.7 74.7 1943 4218
279 1909/40 38.1 -103.9 786.9 80.5 137.7 794.7 62.0 64.8 1941 3133
279 1910/00 36.9 -103.9 786.5 71.4 159.2 822.8 61.8 55.4 1939 2111
279 1910/20 35.7 -103.9 786.1 62.3 166.0 872.9 61.3 47.1 1938 1143
279 1910/40 34.5 -103.8 785.7 54.1 169.3 941.4 60.7 40.1 1937 0322
279 1911/00 33.3 -103.8 785.4 47.0 171.2 1024.6 59.9 34.4 1936 -0350
279 1911/20 32.1 -103.8 785.0 40.9 172.5 1119.2 59.2 30.0 1935 -0959
279 1911/40 30.9 -103.8 784.6 35.7 173.4 1222.5 58.5 26.8 1935 -1513
279 1912/00 29.7 -103.7 784.3 31.2 174.1 1332.4 57.8 24.5 1934 -1943
279 1912/20 28.5 -103.7 783.9 27.4 174.7 1447.5 57.2 23.1 1934 -2337
279 1912/40 27.3 -103.7 783.6 24.0 175.1 1566.5 56.7 22.3 1934 -2701
279 1913/00 26.1 -103.7 783.3 21.0 175.5 1688.5 56.2 22.0 1934 -3001
279 1913/20 24.9 -103.7 783.0 18.3 175.8 1812.9 55.7 22.0 1933 -3241
279 1913/40 23.7 -103.7 782.7 15.9 176.0 1939.3 55.3 22.4 1933 -3506
279 1914/00 22.5 -103.7 782.4 13.8 176.3 2067.1 55.0 22.9 1933 -3717
279 1914/20 21.3 -103.7 782.1 11.8 176.5 2196.1 54.6 23.5 1933 -3916
25288 54978 9.9 MIDPASS ELEVATION TOO LOW
ASC. NODE TIME=2010/31 LONG= 63.6
25288 54979 .0 NO ACQUISITION ON THIS REV
ASC. NODE TIME=2150/59 LONG= 38.4
Please note the following:
a. I adjusted the run control information so that the visible points would be output every 20 seconds,
and in particular, at 19:10:40 MDT. Also, I set the twilight threshold to 0.0 minutes. It normally
would be about 30 minutes, but this bird rises less than 30 minutes after sunset (as you can see from
the AVES.OUT output).
b. Note that SPACE BIRDS and TheSkyX don't give exactly the same azimuth and elevation at 19:10:40 MDT.
This is because TheSkyX uses the SGP4 orbit propagation model, while SPACE BIRDS uses the much
simpler (to understand and to implement), but less accurate GP1 model. The GP1 model is
documented in my textbook,
Topics in Astrodynamics,
and in my 1987 AAS paper, as supplied with the Space Ornithology Kit.
c. SPACE BIRDS tells me that the satellite is visible, at better than 50% illumination by the sun, from
the time it rises above 10 degrees elevation, all the way up to the time it sets below 10 degrees
elevation, a total of about ten minutes of naked-eye visibility.
So if I track the satellite with the naked eye or binoculars, I should be able to see the entire pass,
and I will be treated with a flare at 7:10:40 p.m. The SPACE BIRDS summary tells me that the Iridium
satellite number is 25288, that it is on orbital revolution 54977, and that the ascending nodal
crossing time and west longitude were 18:30:03 (24-hour clock) and 88.8 degrees, respectively.
d. Iridium birds are easy to observe, and interesting to observe, even when no flare is predicted on a
given pass of interest. This is because flares are only reliably predicted for the MMAs, and possibly
even the solar panels themselves (if you have access to precise information about solar panel
orientation as a function of time), but the spacecraft body by itself can generate unpredicted flashes
of reflected sunlight.
e. Although we set up the SPACE BIRDS input files here for just one satellite, we can in fact set up
SPACE BIRDS to generate visibility data (pass data only, not flares -- you need TheSkyX for the flare
predictions) for the entire Iridium satellite constellation.
We can do this by employing SPACE BIRDS in "queue mode." In queue mode, the entire file from Checklist
Step 1 is used as the Orbital Elements input file. Suppose that there are 92 satellites (operational,
non-operational, and spare). Then we copy the two observer location lines 91 times in the Observer
Location file, and copy the run control line 91 times in the Run Control information file, followed by
one blank line to terminate the program.
In other words, queue mode means that we can "queue up" visibility prediction requests for varying
satellite, observer, and run control combinations. Admittedly, this is cumbersome by today's standards,
but SPACE BIRDS was originally designed at a time when generating a day's visibility data on 92
satellites was unthinkable -- it would have taken a prohibitive amount of machine time on a desktop
computer with the the Intel 8088/8087 chip set. Nowadays, the task takes just takes a few seconds.
Case Study Summary. Although the SPACE BIRDS program is more than two decades old now, there is
much information, concisely presented, in the SPACE BIRDS output summary to please today's orbital
analyst. And even though SPACE BIRDS is an MS-DOS program, the Windows Notepad editor makes it easy to
set up the input files and run the program from a Windows folder.
Related Links and References
1. Predicting Iridium Flares was the title of both my DDA 2008 Boulder presentation and a
subsequent article in the
May 2008 issue of PTC Express,
the monthly online newsletter for Pro/Engineer, Mathcad, and other Parametric Technology Corporation
(PTC) product users. You can download the presentation and publicity flyer directly from the following
links:
Presentation,
Publicity Flyer.
2. Tom Bisque, one of the four
Brothers Bisque
(from left to right in the photo: Matt, Tom, Dan, and Steve),
has done some truly remarkable satellite tracking with his
robotic telescope setup,
which uses TheSky for telescope control via its
Paramount ME telescope mount.
You can find out more by visiting
Tom's Corner.
3. T.S. ("TS") Kelso has produced a trilogy of webinars on satellite visibility (November 28, 2006),
Iridium flares (December 5, 2006), and satellite transits of the Sun and Moon (December 12, 2006). See
AGI Webinars.
By the way, TS provides the operational status of the Iridium birds at his Celestrak website:
[+] denotes "operational," [-] denotes "non-operational," and [S] denotes "spare."
The status is on the same line as the satellite common name, i.e., it is on the first line of the TLE.
(Did you notice the [+] on the SATELLITE NAME line of the SPACE BIRDS output summary?)
Why is operational status important to a visual observer? Because (a) operational Iridium birds flare
predictably; (b) non-operational birds are not maintained at nominal "long axis down, MMA#1 forward"
attitude, and so do not flare predictably; (c) spare Iridium birds are kept in lower orbits, but are
boosted to operational altitude before being used for the global mobile telephony mission.
Tom, TS, and I have all observed Iridium spares to flare predictably. This obviously implies that the
on-orbit attitude of each of the Iridium spares is being maintained according to the "long axis down, MMA#1
forward" control law.
Back to Top
Case Study 2: Angles-Only Orbit Determination. There are several widely-known, well-documented angles-only orbit determination methods out there for artificial Earth satellites: Gauss, Lagrange-Gauss-Gibbs, and Laplace readily come to mind.
But the method that I like best is not on this short list. The method that I like best is Herget's method.
Herget's Method is well-known to those who use Earth-based telescopes to measure the celestial positions of comets and asteroids (thanks to Tony Danby, and more recently, to Bill J. Gray and his Project Pluto). But it is not so well-known to those of us who determine artificial Earth satellite orbits. (Perhaps in the not-too-distant future it will be.)
Therefore, in this case study I will apply Herget's method to the problem of determining a preliminary orbit for a LEO (low-Earth orbit) satellite.
The basic idea of Herget's method is to take a set of topocentric RA (right ascension) and DEC (declination) measurements of an asteroid or comet, guess the topocentric distances rho1 and rho2, and then iterate on these initial distance estimates via equations that seek to minimize their residuals in the sense of "observed minus computed." Herget's method is thus a two-parameter fit of rho1 and rho2 to as many observations as are available.
Embedded within Herget's method is Gauss's two-position-vector-and-time (TPV&T) orbit determination method. Gauss's TPV&T method solves this problem: given two position vectors of the secondary in a two-body system, and given the time of flight of the secondary from the first position to the second, find the velocity of the secondary at its first position.
Upon analyzing the mathematics of Herget's method, starting with Herget's original article in the Astronomical Journal (AJ, 1965), I found that I could write Gauss's hypergeometric X-function as a quotient of c-functions, whereas Herget, in his AJ article, uses the older truncated-series representation of the X-function.
Further, my own adaptation of Herget's method uses f and g functions of Stumpff's c-functions to propagate position and velocity, and thus it does not assume that the orbit is elliptical. The orbit can be parabolic or hyperbolic, and this does not affect the mathematics or the solution vector. (The derivation of my improvements to Gauss's TPV&T method, which also therefore constitute improvements to Herget's method, can be found in Chapter 14 of my textbook,
Topics in Astrodynamics.)
An advantage of Herget's method over the other three methods cited above is that it can use all of the available observations in a given track. One does not have to judiciously pick out just three angles-only observations. Plus, one gets a two-parameter fit over all available observations.
To build a LEO test case that illustrates Herget's method, we need look no farther than the SPACE BIRDS output for the Iridium 65 satellite dealt with in Case Study 1. There are 31 angles-only "observations" in that case study. Since the SPACE BIRDS-predicted RA "measurements" are rounded to the nearest minute of time, and the DEC "measurements" are rounded to the nearest minute of arc, they have built into them a random error of +/- half a time-minute in RA, and a random error of +/- half an arc-minute in DEC. Observations comprised of these two measurements should not, therefore, be called perfect observations.
My implementation of Herget's method for artificial Earth satellites, called GH1/GHC, gives the following elements after three GHC iterations (final RMS error was 1.251 km). Epoch is at the time of the first observation.
Element GH1/GHC*** SPACE BIRDS**
MEAN MOTION, REV/DAY 14.26271254 14.33761220
PERIOD, MIN 100.9625621 100.4351338
ECENTRICITY 0.003385770 0.000227526
INCLINATION, DEG 86.39510 86.39130
R.A. OF ASC. NODE, DEG 111.36445 111.38349
ARG. OF PERIGEE, DEG 137.10095 64.90861
MEAN ANOMALY, DEG 345.87781 57.98234
MEAN ARG. OF LATITUDE, DEG* 122.97876 122.89095
*At first glance it would appear that the GH1/GHC argument of perigee and mean anomaly do not agree with those predicted by SPACE BIRDS.
However, since the eccentricity is quite small, we need to add the argument of perigee to the mean anomaly in both cases, thereby obtaining the mean argument of latitude. We see, then, that on the line labeled MEAN ARG. OF LATITUDE, we get good agreement between Herget's method and SPACE BIRDS.
**Two-line elements were propagated to the time of first observation via the GP1 model, then position and velocity at this time were transformed to classical elements.
***My implementation of Herget's method is documented on the Web at
Mathcad Worksheets by Astroger.
(To find the downloadable files, Google with quotes: "Herget's Method with Cassini's Earth Flyby" -- GH1.mcd is the geocentric Herget's method manual initiation worksheet; GHC.mcd is the manual iteration worksheet.) If you have trouble finding and downloading the two worksheets at PTC's new Mathcad website, then contact me directly by e-mail.
Conclusion. We can conclude from Case Study 2 that Herget's method, despite being only a two-parameter fit, leads to a good, single-track preliminary orbit solution for the LEO example chosen.
Back to Top
Case Study 3: Differential Correction of an Orbit. Herget's method, as noted in Case Study 2, does a two-parameter fit to all of the available observations. Differential correction (DC), as done for this case study, constitutes a six-parameter fit (position and velocity).
The initial estimate needed to start the DC is the Herget's method solution from the previous case study. Here are the results after two GDC iterations (final RMS error was 0.895 km).
Element GH1/GHC GD1/GDC*** SPACE BIRDS*
MEAN MOTION, REV/DAY 14.26271254 14.29461632 14.33761220
PERIOD, MIN 100.9625621 100.73722637 100.43513383
ECENTRICITY 0.003385770 0.001945470 0.000227526
INCLINATION, DEG 86.39510 86.40017 86.39130
R.A. OF ASC. NODE, DEG 111.36445 111.37563 111.38349
ARG. OF PERIGEE, DEG 137.10095 135.33931 64.90861
MEAN ANOMALY, DEG 345.87781 347.60332 57.98234
MEAN ARG. OF LATITUDE, DEG** 122.97876 122.94263 122.89095
X, E.R. 0.16680776 0.16705553 0.16708559
Y, E.R. -0.58910198 -0.58914326 -0.58932048
Z, E.R. 0.94069292 0.94048493 0.94007247
XDOT, E.R./MIN 0.02373656 0.02373112 0.02372859
YDOT, E.R./MIN -0.05408231 -0.05405251 -0.05401576
ZDOT, E.R./MIN -0.03814725 -0.03811022 -0.03806313
*Two-line elements were propagated to the time of first observation via the GP1 model, then position and velocity at this time were transformed to classical elements.
**At first glance it would appear that the Herget's method argument of perigee and mean anomaly agree reasonably well with the GD1/GDC argument of perigee and mean anomaly, but not with those same elements as predicted by SPACE BIRDS.
However, since the eccentricity is quite small, we need to add the argument of perigee and mean anomaly in all three cases, thereby obtaining the mean argument of latitude, and look for agreement there. We see, then, that on the line labeled MEAN ARG. OF LATITUDE, we do get good agreement for all three sets of predictions.
***My implementation of a single-track, geocentric, two-body DC is called GD1/GDC and is documented on the Web at
Mathcad Worksheets by Astroger.
(To find the downloadable files, Google with quotes: "Tracking Data Reduction for Galileo's Earth 1 Flyby" -- GD1.mcd is the geocentric DC manual initiation worksheet; GDC.mcd is the geocentric DC manual iteration worksheet.) If you have trouble finding and downloading the two worksheets at PTC's new Mathcad website, then contact me directly by e-mail.
Conclusion. We can conclude from Case Study 3 that the Herget's method solution, as obtained from Case Study 2, can be improved somewhat via a DC that employs two-body orbital modeling.
However, we should note that in the real world, the observations would be actual observations from an optical sensor, and the DC mathematics would not be limited to two-body modeling. The DC would incorporate at least SGP or SGP4 orbital modeling, and would therefore account for orbital perturbations by atmospheric drag as well as by the J2, J3, and J4 terms of the geopotential.
Back to Top
(c) 2008-2018 by Astronomical Data Service. Last updated 2018 October 17.
E-mail: astrocourse@att.net
Accesses:
|