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Re: Historical Lunars : take in account 'delta-T' or ignore it ?
From: Paul Hirose
Date: 2009 Dec 13, 22:03 -0800
From: Paul Hirose
Date: 2009 Dec 13, 22:03 -0800
(I received your reply to my message, but I was already well into this one, so I'll finish it tonight. If you're in France, there will be a great waste of time if I hold this message until tomorrow. Here in California we're 8 hours behind Greenwich. Also, since I'm in a hurry to get this done, I have not worked the computations twice to verify them. So if something "does not compute", it's probably my blunder.) This compares your computation with HORIZONS for the case ∆T = 0. antoine.m.couette@club-internet.fr wrote: > -*-*- Second, I am submitting here-after a fictitious example of an upcoming Lunar which shows that - if we were to ignore to-day the effects of delta-T on Lunars - we would significantly increase our error in UT determination, and therefore our Longitude uncertainty. > > 23 Dec 2009 Height of Eye = 17 ' T = 59�F , P = 29.92 ' Estimated UT / Position : 16h58m53.0s / N3707.1W01253.7 - All observations assumed to occur at the same time - Distance SUN-MOON = 77�53'7 , SUNL = 5�50'1 , MOONL = 50�05'0 . All sextant measures are corrected for Instrument error (and only instrument error), i.e. all other corrections (dip, refraction, SD, Parallax ... ) need to be performed from the "raw data" hereabove. Dip = .0667°, therefore lower limb apparent altitude = 5.7683° (Sun) and 50.0166° (Moon). Refraction (Nautical Almanac formula) = .1451° (Sun) and .0138° (Moon). Lower limb observed altitude = 5.6232° (Sun) and 50.0029 (Moon). > *** If we compute observer's position from the observed heights : > > o With delta-T = 0.0 second : > > Lunar Distance UT = 17h01m07.4s - i.e. TT = 17h01m07.4s - hence UT error = -2m14.4s, cleared distance = 78�34.00', and > We also get the following position at time of Lunar Distance : > N3659.9W01317.1 (result of 2 body fix), and As explained in my last message, I assume JPL HORIZONS uses ∆T = 66.06 seconds. To simulate ∆T = 0, I move your longitude east by 66.06 * 15 * 1.002738 arc seconds, or .2760°. HORIZONS says Sun azimuth = 234.6730, unrefracted altitude = 5.8962, semidiameter = .2710. Altitude - semidiameter = 5.6252 = unrefracted lower limb altitude. The observed altitude was .0020° (.12') less. HORIZONS says Moon azimuth = 155.5489, unrefracted altitude = 50.2545, semidiameter = .2517. Altitude - semidiameter = 50.0028 = unrefracted lower limb altitude. The observed altitude was .0003° (.02') less. When we apply refraction to the HORIZONS altitudes, we get 6.0358 (Sun center) and 50.2682 (Moon center). The refracted center to center separation angle = 78.4162. Subtract the sum of the semidiameters (.5227) to get the predicted limb to limb angle, 77.8935. The observed angle was 77.8950, .0015° (.09') greater. The predicted altitudes and lunar distance accurately match the observations when I use your ∆T = 0 solution. Good work! Now let's return to your original ∆T = 66.70. Here is the output from my program. (The "target" body is the body other than the Moon; the Sun in this case.) Program Lunar2, by Paul S. Hirose. Initial conditions. estimated time: 2009-12-23T16:58:53.00 UT1 2009-12-23T16:59:59.68 Terrestrial Time 66.700 seconds delta T estimated position: +37.1183° - 12.8950° north lat, east lon - 13.1737° ephemeris east lon 5 meters above ellipsoid atmosphere: 15° C (59° F) at observer 1013.3 mb (29.92" Hg) altimeter setting 1012.6 mb (29.90" Hg) actual pressure Moon altitude observation: 50.0167° observed lower limb altitude 0.0133° refraction 0.2517° unrefracted semidiameter 50.2550° unrefracted altitude of center 50.0853° predicted altitude 0.1697° intercept 155.3688° predicted azimuth Sun altitude observation: 5.7683° observed lower limb altitude 0.1428° refraction 0.2710° unrefracted semidiameter 5.8965° unrefracted altitude of center 5.9377° predicted altitude - 0.0412° intercept 234.5518° predicted azimuth Moon to Sun predicted separation angle: 78.5147° center to center, unrefracted 0.1037° refraction 78.4110° center to center, refracted 0.2517° Moon near limb refracted semidiameter 0.2680° target near limb refracted semidiameter 77.8913° Moon near limb to Sun near limb 77.8950° observed angle 0.0037° observed - predicted separation angle rate of change: +0.00493° per minute (topocentric) 75% of total angular velocity -------------------- Solution, after 3 iterations. corrected time: 2009-12-23T16:59:36.04 UT1 2009-12-23T17:00:42.76 Terrestrial Time 66.700 seconds delta T corrected position: +36.9973° - 12.9033° north lat, east lon - 13.1820° ephemeris east lon 79° LOP crossing angle geocentric coordinates (true equator and equinox): 23.30103h + 0.4240° Moon RA and dec. 0.2487° apparent semidiameter 18.14559h -23.4235° Sun RA and dec. 0.2710° semidiameter geocentric separation angle and rate: 78.5632° center to center +0.00767° per minute 84% of total angular velocity illumination conditions: 234.7° 5.9° Sun unrefracted az, el 265° Moon to Sun position angle (0 = 12 o'clock) 101° Moon phase angle (0 = full, 180 = new) position angles: 265° Moon to Sun 40° Sun to Moon recommended limbs: Use Moon lower limb. Use Moon near limb. Moon altitude observation: 50.0167° observed lower limb altitude 0.0133° refraction 0.2518° unrefracted semidiameter 50.2550° unrefracted altitude of center 50.2550° predicted altitude 0.0000° intercept 155.5538° predicted azimuth Sun altitude observation: 5.7683° observed lower limb altitude 0.1428° refraction 0.2710° unrefracted semidiameter 5.8965° unrefracted altitude of center 5.8965° predicted altitude 0.0000° intercept 234.6733° predicted azimuth Moon to Sun predicted separation angle: 78.5192° center to center, unrefracted 0.1045° refraction 78.4147° center to center, refracted 0.2517° Moon near limb refracted semidiameter 0.2680° target near limb refracted semidiameter 77.8950° Moon near limb to Sun near limb 77.8950° observed angle 0.0000° observed - predicted separation angle rate of change: +0.00492° per minute (topocentric) 75% of total angular velocity Now check this with HORIZONS. The longitude is moved west by .0027° to correct for the discrepancy in ∆T (66.70 vs. 66.06). HORIZONS says Sun azimuth = 234.6733, unrefracted altitude = 5.8965, semidiameter = .2710. Altitude - semidiameter = 5.6255 = unrefracted lower limb altitude. The observed altitude was .0023° (.14') less. This is not a really bad result, but my program should be more accurate. The problem is refraction. The Sun lower limb apparent altitude (i.e., with correction for dip) is 5.7683° The Nautical Almanac formula says refraction is .1451° at this altitude. But my program uses the Astronomical Almanac low altitude formula, which gives .1428°. A 1.6% difference between refraction formulas seems reasonable at this low altitude. In fact, I would avoid shooting a lunar with one body so near the horizon. If I use my program's refraction value to correct the observation, the unrefracted lower limb altitude = 5.6255, a perfect match to the value from HORIZONS. HORIZONS says Moon azimuth = 155.5539, unrefracted altitude = 50.2550, semidiameter = .2517. Altitude - semidiameter = 50.0033 = unrefracted lower limb altitude. The observed altitude was .0004° (.02') less. When we apply refraction to the HORIZONS altitudes, we get 6.0362 (Sun center) and 50.2686 (Moon center). The refracted center to center separation angle = 78.4129. Subtract the sum of the semidiameters (.5227) to get the predicted limb to limb angle, 77.8902. The observed angle was 77.8950, .0048° (.29') greater. Again, at this low altitude the solution is sensitive to the refraction formula. If both bodies had more altitude, I'm sure the results from my program would not be so disappointing. Of course, for these tests I could simply use the same refraction formulas as my program. Then all the discrepancies would disappear. But I think the real world refraction variations will be greater than the differences between formulas, so this is a good lesson in avoiding low altitude lunars. I should have explained in my last message that the HORIZONS Web site is http://ssd.jpl.nasa.gov/?horizons . There is a Web interface for the system, but if you do many computations of the same type, I think it's less trouble to use the email interface. You can save your old command emails and simply modify the values as needed. My lunar distance program is available at my Web site: http://home.earthlink.net/~s543t-24dst/sofajplNet/LunarDist2.html But it's not a "real program" because I'm not a "real lunars man". I wrote it to demonstrate my positional astronomy DLL, and also so I could run my own computations when lunars are discussed. There is no user interface. To set the input data, you modify the values in the Basic source code, re-compile the program, and run it! This is primitive, but computers are so powerful now, I can barely stand up from my chair before the recompile and execution are complete. Since I don't use the program much, it's not worthwhile to write a proper user interface. However, anyone is welcome to use this as a basis for a real program. All the source code is free. By the way, your first email has some strange format problems. There are many • in the message and the degree symbol (°) doesn't appear. When I view the message on the Web, • appears as a bullet (•), but there's a little empty box where the degree symbols should appear. There are many interesting points in your reply, but unfortunately I don't have time to get to them before tomorrow. -- -- NavList message boards: www.fer3.com/arc Or post by email to: NavList@fer3.com To , email NavList+@fer3.com