Assorted interesting examples of orbit determination
The following are all situations in which Find_Orb has been used to handle "unusual" cases (objects other than minor planets). In each case, astrometry is either provided or referenced so you can duplicate the results. This can be educational, useful, or entertaining... sometimes all three.
Astrometry for the Cassini flyby, 18-20 August
On 19 August 1999, I received the following e-mail from Gordon Garradd:
Hi Bill, I obtained astrometry on an interesting NEO and include the
observations and a Find_orb derived orbit.
COD 422
OBS G. J. Garradd
TEL 0.45-m f/5.4 reflector + CCD
NET GSC-ACT
Cassini C1999 08 18.51899 23 28 23.37 -05 04 56.9 16.2 422
Cassini C1999 08 18.52503 23 28 21.35 -05 04 57.3 16.7 422
Cassini C1999 08 18.54081 23 28 15.69 -05 04 56.6 16.5 422
Cassini C1999 08 18.57286 23 28 03.14 -05 04 51.7 17.0 422
Cassini C1999 08 18.58419 23 27 58.51 -05 04 48.9 16.4 422
Cassini C1999 08 18.59442 23 27 54.32 -05 04 45.8 16.7 422
Cassini C1999 08 19.72619 23 29 03.21 -05 01 06.4 19.0 422
Cassini C1999 08 19.73325 23 29 02.24 -05 01 04.8 19.2 422
Cassini C1999 08 19.74565 23 29 00.47 -05 01 02.1 19.0 422
Cassini C1999 08 19.77046 23 28 57.46 -05 00 56.8 19.1 422
Cassini C1999 08 19.77853 23 28 56.63 -05 00 55.2 19.1 422
Orbital elements:
Cassini
Perihelion 1999 Jul 17.333671 TT
Epoch 1999 Aug 10.0 TT = JDT 2451400.5
q 0.8162838 (2000.0) P Q
Peri. 128.03399 0.04893115 -0.99875763
Node 144.76719 -0.92142838 -0.04878227
e 1.0167922 Incl. 0.93662 -0.38545490 -0.01017236
>From 11 observations 1999 Aug. 18-19; RMS error 0.318 arcseconds
As you may have guessed it is Cassini! you may like to use it as a real
life example on your CDs for Find_orb
I posted a composite pic
on my page from images on 990818, and the image generated so many hits I
was asked to remove it from the server in Tx where I had it stored.
Cheers, Gordon
(There is also a copy of the image at the European Space Agency science site.)
This e-mail prompted me to ask those on the Minor Planet Mailing List if anyone else had some Cassini astrometry. Rob McNaught responded, sending the following four positions:
COD 413
OBS R. H. McNaught
MEA R. H. McNaught
TEL 1.0m f/8 reflector + CCD
NET USNO-A2.0
Cassini C1999 08 20.66391 23 29 34.89 -05 00 47.0 19.1 R 413
Cassini C1999 08 20.66724 23 29 34.49 -05 00 47.2 19.4 R 413
Cassini C1999 08 20.67062 23 29 34.06 -05 00 47.0 19.4 R 413
Cassini C1999 08 20.69003 23 29 31.66 -05 00 46.5 19.4 R 413
If you save this page, either as text or as an HTML file, you can load it into the FIND_ORB orbit determination program and analyze the orbit of this probe. After loading the data, your first step is to click on "Herget step" until the orbit settles down. Then change the epoch to JD 2451409.5 = 1999 Aug 19.0, right around the middle of the observations. Of course, this object has just passed the Earth, and we can assume that perturbations by the Earth and Moon will be significant; so you should click those check-boxes in the "Perturbers" section of FIND_ORB.
Now click on "Full Step" a few times. You will get an orbit such as the following:
Orbital elements: Cassini Perihelion 1999 Jul 16.403976 TT Epoch 1999 Aug 19.0 TT = JDT 2451409.5 M 4.11450 (2000.0) P Q n 0.12246997 Peri. 131.94718 0.11658913 0.99315416 a 4.0159008 Node 144.74619 -0.91487380 0.11021386 e 0.7847029 Incl. 0.71434 -0.38653965 0.03870031 P 8.05 H 27.1 G 0.15 q 0.8646120 From 15 observations 1999 Aug. 18-20; RMS error 0.321 arcseconds
This "post-flyby" orbit shows that the aphelion of Cassini's orbit is now out at (2a-q) = 8.03-.86 = 7.17, well past Jupiter. You'll notice that the RMS error is surprisingly low, too.
The next step of interest is to move the epoch back to near the perihelion. Set it to JD 2451408.7 = 1999 Aug 18.2. This will bring Cassini back to a time when it is being very heavily perturbed (by Earth), so it's necessary to cut down the step size drastically. The three-day steps are great for objects in interplanetary space, but in this case, a step size of .01 day (14.4 minutes) is about right. Change the epoch and step size, click on "Full Step" a few times, and the solution converges to...
Orbital elements:
Cassini
Perigee 1999 Aug 18.145405 TT
Epoch 1999 Aug 18.2 TT = JDT 2451408.7
q 7554.431 km (2000.0) P Q
Peri. 248.51355 -0.31982399 -0.94718637
Node 3.13188 -0.62921508 0.19381266
e 5.8755378 Incl. 25.43489 -0.70837914 0.25548900
From 15 observations 1999 Aug. 18-20; RMS error 0.321 arcseconds
FIND_ORB has decided that this object is no longer really orbiting the Sun; it is so close to the earth that it makes more sense to speak of a geocentric orbit than a heliocentric orbit. (Though the geocentric orbit does have an eccentricity of 5.88... something you never see with natural objects in heliocentric orbits! For these, e is rarely much greater than 1.)
You'll also notice that q = 7554 km is greater than the earth's radius of 6378 km, confirming that Cassini really did miss us by about 1176 km. (JPL came up with a figure of 1171 km, but they probably had more data to work with.)
If you now change the epoch back to JD 2451407.5, a day or so before the flyby, and "Full Step" a few times, you'll get this orbit:
Orbital elements: Cassini Perihelion 1999 Jun 29.637939 TT Epoch 1999 Aug 17.0 TT = JDT 2451407.5 M 21.98564 (2000.0) P Q n 0.45460520 Peri. 102.17546 -0.39155937 0.92008890 a 1.6751220 Node 144.76676 -0.85136694 -0.35779056 e 0.5685862 Incl. 1.07747 -0.34907821 -0.15944383 P 2.17 H 27.1 G 0.15 q 0.7226708 From 15 observations 1999 Aug. 18-20; RMS error 0.321 arcseconds
...an orbit that would have brought Cassini, at aphelion, out to (2a-q) = 3.35-.72 = 2.53 AU from the Sun, in the main belt. So you can see that the flyby boosted the orbit by quite a lot, moving the aphelion from 2.53 AU out to 7.17 AU.
You can also load the post-encounter orbit into Guide, or any similar star charting/planetarium program, and get a Jovicentric ephemeris that shows that Cassini is indeed headed straight for Jupiter, passing it in early 2001. For example, here's part of a Jovicentric ephemeris from Guide, using the post-encounter orbit:
Cassini Date RA declination r delta mag Elong ---- -- ----------- - ----- --- ----- 11 Dec 2000 16h05m15.77s -19 37' 58.0" 4.890 0.151 55.1 4.2 16 Dec 2000 16h34m58.63s -21 02' 11.3" 4.916 0.126 65.8 2.5 21 Dec 2000 17h19m02.22s -22 28' 40.8" 4.942 0.103 40.3 12.4 26 Dec 2000 18h24m23.53s -23 13' 19.0" 4.968 0.085 34.2 27.0 31 Dec 2000 19h52m42.01s -21 34' 08.6" 4.994 0.076 30.9 47.0 5 Jan 2001 21h25m13.03s -16 35' 29.6" 5.020 0.078 29.2 69.0 10 Jan 2001 22h37m54.89s -10 33' 36.1" 5.045 0.091 28.7 87.2 15 Jan 2001 23h27m23.02s - 5 39' 02.5" 5.071 0.111 28.6 99.9 20 Jan 2001 0h00m18.55s - 2 10' 41.3" 5.096 0.135 28.7 108.4
Unfortunately, the above is an unperturbed ephemeris. So it shows Cassini passing within .076 AU in January 2001, which is not quite right. (It probably does not help that the observations span only two days.) The full ephemeris does show that the elongation is never much more than a few degrees for most of the trip; Cassini basically is "lifted" straight out toward Jupiter, which then catches up to it and slings it on out to Saturn.
Gordon pointed out that you can get a view of Cassini's path from Earth to Jupiter (27K). This screen shot is a view from 10 AU "above" the plane of the ecliptic, showing the motion of the Earth, Cassini, Jupiter, and Mars between 25 Aug 1999 and 6 Jan 2001, when Cassini (according to Guide) arrives at Jupiter. Tick marks show the position at 20-day intervals. Cassini doesn't really pass close to Mars, but I added the Mars trail for perspective.
This view also shows that Cassini is going very fast indeed right now, but the sun's gravity will rob it of most of that speed by the time it gets to Jupiter (the ticks are widely spaced at the start of the Cassini path, but are close together by the time it gets to Jupiter.) If I had a "post-Jupiter-flyby" orbit, I could show how it gets a significant boost after it goes by Jupiter.
Another point that should be mentioned: the post-flyby orbit is probably of quite good quality. The pre-flyby one is probably close to garbage; any small errors in the orbit would get magnified by the earth's gravity.
Astrometry and orbits for a very high-orbiting Earth satellite
Recently, Gordon Garradd found and tracked an object moving about .7 arcseconds/second (.7 degrees/hour), which he temporarily named GL010. Gordon contacted Rob McNaught, who was observing about 185 km away; communicating by e-mail, it soon became apparent from the parallax between their observations that the object was about 180,000 km away, half the distance to the moon. And running it through Find_Orb showed that it was in Earth orbit. Gareth V. Williams, at the Minor Planet Center, got a similar orbit, checked with a "local satellite expert" at MPC, and found that "the object is probably a PROGNOZ magnetospheric satellite." The Russians launched several of these, in high orbits, some with apogees out at 800,000 km, twice the distance to the moon.
Here is the astrometry Gordon (422), Rob (413), and the Bisei Spaceguard Center (BATTeRS) got:
COD 422
OBS G. J. Garradd
TEL 0.45-m f/5.4 reflector + CCD
NET GSC-ACT
GL010 C2000 02 28.71008 11 16 38.09 +02 12 08.7 15.5 B 422
GL010 C2000 02 28.71127 11 16 37.97 +02 10 56.5 15.6 B 422
GL010 C2000 02 28.71244 11 16 37.89 +02 09 46.2 15.6 B 422
GL010 C2000 02 28.71354 11 16 37.81 +02 08 40.5 15.6 B 422
GL010 C2000 02 28.71403 11 16 37.78 +02 08 10.1 15.2 B 422
GL010 C2000 02 28.72351 11 16 37.66 +01 58 40.4 15.3 B 422
GL010 C2000 02 28.72591 11 16 37.77 +01 56 17.1 15.1 B 422
GL010 C2000 02 28.72792 11 16 37.91 +01 54 16.4 15.2 B 422
GL010 C2000 02 28.73325 11 16 38.46 +01 48 57.7 15.0 B 422
GL010 C2000 02 28.73768 11 16 39.16 +01 44 33.1 15.0 B 422
COD 413
OBS R. H. McNaught
MEA R. H. McNaught
TEL 1.0m f/8 reflector + CCD
ACK GL010 updated
NET GSC-ACT
GL010 C2000 02 28.73624211 16 48.79 +01 45 56.1 15.3 V 413
GL010 C2000 02 28.73764111 16 48.95 +01 44 32.6 15.3 V 413
GL010 C2000 02 28.73966711 16 49.20 +01 42 31.5 15.4 V 413
GL010 C2000 02 28.74115311 16 49.46 +01 41 02.5 15.5 V 413
GL010 C2000 02 28.74303511 16 49.71 +01 39 12.1 15.5 V 413
GL010 C2000 02 28.74464911 16 50.01 +01 37 35.5 15.6 V 413
GL010 C2000 02 28.79338911 17 13.77 +00 49 31.3 15.3 V 413
GL010 C2000 02 28.79482111 17 14.94 +00 48 06.9 15.2 V 413
GL010 C2000 02 28.79814011 17 17.75 +00 44 52.7 15.3 V 413
GSC-1.1, USNO-A2.0
GL010 C2000 02 28.80635 11 18 30.60 -01 28 39.8 14.8 V 300
GL010 C2000 02 28.80737 11 18 31.02 -01 29 36.7 15.0 V 300
GL010 C2000 02 28.80835 11 18 31.62 -01 30 32.4 15.2 V 300
GL010 C2000 02 28.80931 11 18 32.07 -01 31 28.1 15.2 V 300
GL010 C2000 02 28.82250 11 18 39.42 -01 44 00.9 15.0 V 300
GL010 C2000 02 28.82502 11 18 41.14 -01 46 24.6 15.2 V 300
GL010 C2000 02 28.82745 11 18 42.68 -01 48 42.8 15.1 V 300
Putting all this together and feeding it to Find_Orb, you wind up with:
Orbital elements: GL010 Perigee 2000 Feb 26.855598 TT Epoch 2000 Feb 28.8 TT = JDT 2451603.3 M 169.36794 (2000.0) P Q n 87.10541604 Peri. 2.45666 0.98898510 0.02933272 a 108622.262 km Node 350.68693 -0.14643033 0.05084323 e 0.7719220 Incl. 63.70173 -0.02160188 0.99827579 P 4.13d H 29.2 G 0.15 q 24774.347 km From 26 observations 2000 Feb. 28-28; RMS error 0.992 arcseconds
Here's the same data stored in .TLE ("Two-Line Element") form, suitable for use in most planetarium/satellite software (including Guide):
GL010 1 05941U 72029A 00059.80000000 -.00000000 +00000-0 +00000-0 0 00016 2 05941 063.7017 350.6869 7719220 2.4566 169.3679 00.24195877889002
Astrometry and orbits for the three new satellites of Uranus
Last summer, discovery circumstances and some astrometic data were published in IAU Circulars for three newly-found outer satellites of Uranus. These joined the two outer Uranian satellites Caliban and Sycorax, found in 1997.
Naturally, I fed the published astrometry into Find_Orb and tried to determine the orbits for all three objects. I met with total failure; Find_Orb insisted that all three were best matched to Centaur (distant asteroid) orbits around the sun, rather than to uranocentric orbits. The main problem was that, at that time, not all the data had been published.
Brett Gladman, one of the discoverers of these objects, posted the data at this Web site. From here, you can click on the 'All known astrometry of 1999 U1, U2, and U3' to download all the astrometry for all three of these objects. There is also a link to the discovery story, both for these and the 1997 objects.
With this data, plus a copy of Find_Orb, it is quite straightforward to get the orbit of S/1999 U 1. With a little extra effort, you can get those of S/1999 U 2 and S/1999 U 3.
Your first step should be to visit the above link and download the astrometric data into a text file. As is noted at the site, you'll have to edit the data a little, because the objects were called "U99satcand1" through "U99satcand3" at first, when they were just 'candidate' satellites. But you can fix this in a text editor, converting the names to a consistent scheme so that Find_Orb will understand that there are three objects here and not six.
Next, fire up Find_Orb, click 'Open', and select the file. The three objects should appear in the object list. Double-click on the first, and a preliminary (totally absurd!) orbit will appear.
Obviously, you'll have to set the 'Uranus' check-box, so that Guide will include perturbations from that object. Do that, and then click "Herget step". And do it again, and again... Find_Orb starts out with the assumption that the object is in the main belt; it takes about a dozen iterations before it pushes the object out to Uranus and figures out that it ought to be using an uranocentric orbit.
Once it does so, and the orbit no longer changes much with an Herget step, you can do a few 'Full Step' iterations. It's probably a good idea to set the check-boxes for Jupiter, Saturn, and Neptune, too... though the observed arcs are a bit short for this to really matter right now!
Despite the fact that the starting assumption of a main-belt orbit is totally absurd for this object, Find_Orb still gets a solution for the first new satellite. But the program needs a little help with the remaining two. It turns out that both satellites are closer to Uranus, and need better starting values for R1 and R2 (distance to the target at the times of the first and last observation). I tackled this by setting R1 and R2 to be the distances to Uranus at the times of first and last observation. (If the satellite is at all close to Uranus, this is a decent first approximation.) I also tried subtracting .05 AU from both R1 and R2, then adding .05 AU to R1 and R2, the idea being that the object might be a bit closer or farther away than Uranus.
Feed Find_Orb these 'better' initial values for R1 and R2, and you can click on 'Herget step' and quickly get a good orbit; then on 'Full Step' and get a better orbit. After a few minutes of experimenting, I found that using R1=19 and R2=19.05 produced good results for S/1999 U 2 * and R1=19.1 and R2=19.7 works for S/1999 U 3.
A few 'full steps' should converge on orbits resembling those shown below.
Orbital elements for S/1999 U 1, U 2, and U 3
Orbital elements: U99_1 Periuranus 2003 Aug 14.646264 TT Epoch 1999 Oct 9.0 TT = JDT 2451460.5 M 223.00244 (2000.0) P Q n 0.09746233 Peri. 317.01012 0.24165583 -0.91062011 a 0.1645334 Node 243.63047 -0.75748612 -0.39294164 e 0.0695029 Incl. 158.02804 -0.60647938 0.12793704 P 10.11 H 10.2 G 0.15 q 0.1530979 From 20 observations 1999 Jul. 18-Oct. 14; RMS error 0.584 arcseconds Orbital elements: U99_2 Periuranus 1999 Mar 3.969800 TT Epoch 1999 Aug 30.0 TT = JDT 2451420.5 M 71.17275 (2000.0) P Q n 0.39754604 Peri. 126.66828 0.39917396 0.90595093 a 0.0644496 Node 195.19307 0.57024285 -0.12478585 e 0.2012919 Incl. 147.42162 0.71797161 -0.40457559 P 2.48 H 10.9 G 0.15 q 0.0514764 From 20 observations 1999 Jul. 18-Sep. 2; RMS error 0.441 arcseconds Orbital elements: U99_3 Periuranus 2002 May 25.508367 TT Epoch 1999 Oct 9.0 TT = JDT 2451460.5 M 204.25245 (2000.0) P Q n 0.16232016 Peri. 185.04569 -0.71859953 0.60521364 a 0.1171018 Node 320.26807 0.65531987 0.75416767 e 0.2671330 Incl. 147.59599 0.23274575 -0.25484814 P 6.07 H 10.1 G 0.15 q 0.0858201 From 17 observations 1999 Jul. 18-Oct. 15; RMS error 0.363 arcseconds
A footnote on S/1999 U 2: if instead of using R1=19 and R2=19.05, you use R1=18.9 and R2=18.95 (corresponding to "the nearer side" of Uranus), Find_Orb can determine a prograde orbit:
Orbital elements: U99_2 Periuranus 1999 Dec 27.555841 TT Epoch 1999 Aug 30.0 TT = JDT 2451420.5 M 288.21844 (2000.0) P Q n 0.60040198 Peri. 340.79738 -0.00661262 0.38072738 a 0.0490000 Node 276.83325 -0.75151843 -0.61189995 e 0.5938975 Incl. 68.63506 -0.65967895 0.69327131 P 1.64 H 11.0 G 0.15 q 0.0198990 From 20 observations 1999 Jul. 18-Sep. 2; RMS error 0.436 arcseconds
The Minor Planet Center has posted, in IAU Circular 7385, orbits for these three satellites; in this, the prograde solution is given.
I'd lean in favor of the retrograde solution. You'll notice that all four of the other satellites have retrograde orbits with inclinations of about 145 degrees, and the e=.6 value for the prograde orbit would be peculiar (though not unheard of; e=.75 for Nereid.) We won't really know until the object is recovered in late May 2000, but it seems to me that a retrograde orbit is a better match to the other four outer Uranians.
I didn't find any reasonable prograde orbits for S/1999 U 1 or S/1999 U 3.
Orbital elements for S/1997 U 1 and U 2 (Sycorax and Caliban)
For comparison, the elements for Caliban and Sycorax follow. I'm fairly certain that I'm missing a lot of observations. However, the MPC did publish some measurements made from 1984 plates, so the arcs are far longer than they are for the three new objects.
Orbital elements: U97_2 (Sycorax) Periuranus 1997 Jun 14.005812 TT Epoch 1997 Nov 28.0 TT = JDT 2450780.5 M 46.61042 (2000.0) P Q n 0.27911402 Peri. 17.96273 -0.50196916 -0.74367434 a 0.0816525 Node 255.63170 -0.83888241 0.29439449 e 0.5042946 Incl. 152.88243 -0.21048343 0.60023358 P 3.53 H 7.1 G 0.15 q 0.0404756 From 33 observations 1984 Jun. 1-2000 May. 28; RMS error 0.921 arcseconds Orbital elements: U97_1 (Caliban) Periuranus 1998 Apr 19.157631 TT Epoch 1997 Nov 28.0 TT = JDT 2450780.5 M 271.72805 (2000.0) P Q n 0.62094416 Peri. 339.06472 -0.95398163 -0.29462481 a 0.0479132 Node 175.05474 -0.08300926 0.43832689 e 0.0815479 Incl. 139.64873 -0.28814669 0.84915591 P 1.59 H 8.8 G 0.15 q 0.0440060 From 18 observations 1984 Jun. 1-2000 May. 28; RMS error 0.707 arcseconds
Comments on the results: Keep in mind that the above elements for the 1999 satellites are very rough. The real hope is that they'll be good enough to indicate where to look in Spring 2000 when Uranus emerges from behind the sun, and recovery of the three is attempted. Then we'll have fresh astrometry, better orbits, and it might even be possible to hook up to older data, as was done for Caliban and Sycorax.
That said, certain patterns are emerging. The orbits all have pretty much the same inclination: steeply retrograde, around 145 degrees. Their periods are longer than those associated with other distant satellites; omit Caliban, and the remaining four have the longest periods of any satellite in the solar system. I was mildly astonished to see the ten-year orbit for S/1999 U 1... I expect that number to drop a lot once the recovery astrometry is available, though.
Astrometry and orbits for the new satellite of Jupiter, S/1999 J 1
The Minor Planet Center has posted, in IAU Circular 7460, information about the discovery of a new outer satellite of Jupiter. This was found by the University of Arizona's Spacewatch program, and originally received the (minor planet) designation 1999 UX18. As data accumulated, it was no longer possible to fit it to a heliocentric orbit, and it was eventually announced as a satellite in the above Circular. (There's also a press release on the MPC site describing the discovery process.)
The IAU Circular also provides the raw astrometry. You can load these into Find_Orb and determine the orbit from them. The procedure is quite similar to that for the Uranian satellites described above. One small added hurdle: the astrometry is not given in the "standard" MPC format. Do a little editing with a text editor, and add in the Spacewatch MPC code of 691, and you can fix that problem.
You can then load the observations into Find_Orb, click on the "Jupiter" check-box so perturbations from that planet will be included, and take a guess for R1 and R2 (distance to the target object at the time of the first and last observation). I started by finding the distance to Jupiter on both dates (essentially 4 AU on both days), and trying R1=R2=3.9 and R1=R2=4.1 (that is, guessing that the target was "a little closer to us than Jupiter", and "a little farther away than Jupiter").
After setting initial guesses for R1 and R2, you then have to do a single Herget step (which gives you very rough initial orbits), then do a series of "full-steps" to refine the result. As often happens with orbits this preliminary, that Herget step results in weird "Jovian fly-by" orbits that look totally meaningless and have terrible residuals. But they're good enough to supply initial data for the "full step" process.
As it turns out, in this case, both use of R1=R2=3.9 and of R1=R2=4.1 converge to solutions. The first solution looks a lot better than the second (and is the only one MPC presumably felt was worth publishing). Click here for details on the secondary solution, and why I'm betting on the primary solution.