NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Lunar distance solution by linear equations
From: Paul Hirose
Date: 2009 Feb 05, 16:34 -0800
From: Paul Hirose
Date: 2009 Feb 05, 16:34 -0800
Last summer there was a long thread with the subject "My first Lunar": (http://groups.google.com/group/NavList/browse_thread/thread/3d6cb12c5f97d5ba?hl=en#) I'm half a year late, but here are my results: time: 1999 Jan 26 22:19:32.090 UTC position: N 14.56993° W 61.66016° To test those results, I have azimuths and altitudes (from the JPL HORIZONS online calculator) for that time and place. If my solution is correct, the Moon and Jupiter altitudes and the lunar distance should match the observations. (In order to minimize roundoff error in this evaluation, I've used greater precision than is normally reasonable.) HORIZONS says the Moon azimuth is 83.2772° and refracted altitude is 65.7533°. Compare that to the altitude observation: 66.15500 Moon upper limb, sextant .12833 dip .27388 semidiameter (HORIZONS value) ------- 65.75279 observed refracted center altitude 65.7533 HORIZONS altitude -------- -.00051 intercept HORIZONS says Jupiter azimuth is 250.9957° and refracted altitude is 45.2444°. Compare that to the altitude observation: 45.37167 Jupiter, sextant .12833 dip ------- 45.24333 observed refracted center altitude 45.2444 HORIZONS altitude -------- -.00107 intercept The observed altitudes are 2 and 4 arc seconds different from the HORIZONS predictions. From the HORIZONS azimuths and altitudes I calculate 68.59563° refracted lunar distance, center to center. Compare that to the observation: 68.32333 sextant, near limb to center .27388 Moon semidiameter (HORIZONS value) -------- 68.59721 observed center to center refracted angle 68.59563 HORIZONS -------- -.00158 difference The observed lunar distance is 6 arc seconds different from the HORIZONS value. In terms of practical sextant accuracy, the solution is almost perfect. That is, when you give HORIZONS the time and position I computed, its Moon and Jupiter positions match all three observed angles within a tenth of an arc minute. Here is the full output of my program, in units and precision more appropriate to navigation: Program Lunar2, by Paul S. Hirose. Initial conditions. estimated time: 1999-01-26T22:20:00.0 UTC 1999-01-26T22:21:04.0 Terrestrial Time 63.492 seconds delta T estimated position: +20°00.0' - 70°00.0' north lat, east lon - 70°15.9' ephemeris east lon 19 meters above ellipsoid atmosphere: 15° C (59° F) at observer 1013.3 mb (29.92" Hg) altimeter setting 1011.0 mb (29.85" Hg) actual pressure Moon altitude observation: 66°01.6' observed upper limb altitude 0.4' refraction 16.4' unrefracted semidiameter 65°44.8' unrefracted altitude of center 57°57.9' predicted altitude 7°46.8' intercept 91°49.7' predicted azimuth Jupiter altitude observation: 45°14.6' observed center altitude 0.9' refraction 45°13.7' unrefracted altitude of center 50°06.9' predicted altitude - 4°53.2' intercept 239°30.1' predicted azimuth Moon to Jupiter predicted separation angle: 68°42.7' center to center, unrefracted 1.3' refraction 68°41.4' center to center, refracted 16.4' Moon near limb refracted semidiameter 68°25.0' Moon near limb to Jupiter center 68°19.4' observed angle - 0°05.6' observed - predicted separation angle rate of change: +24" per minute (topocentric) 94% of total angular velocity -------------------- Solution, after 5 iterations. corrected time: 1999-01-26T22:19:32.0 UTC 1999-01-26T22:20:36.4 Terrestrial Time 63.492 seconds delta T corrected position: +14°34.2' - 61°39.6' north lat, east lon - 61°55.5' ephemeris east lon 12° LOP crossing angle geocentric coordinates (true equator and equinox): 4h14.72m +16°00'48" Moon RA and dec. 16.2' apparent semidiameter 23h48.75m - 2°29'12" Jupiter RA and dec. 0.3' semidiameter geocentric separation angle and rate: 68°13.1' center to center +35" per minute 97% of total angular velocity illumination conditions: 251.9° -4.7° Sun unrefracted az, el 347° Moon to Sun position angle (0 = 12 o'clock) 61° Moon phase angle (0 = full, 180 = new) 181° Jupiter to Sun position angle 9° Jupiter phase angle position angles: 351° Moon to Jupiter 5° Jupiter to Moon recommended limbs: Use Moon upper limb. Use Jupiter lower limb. Use Moon near limb. Use Jupiter far limb. Moon altitude observation: 66°01.6' observed upper limb altitude 0.4' refraction 16.4' unrefracted semidiameter 65°44.7' unrefracted altitude of center 65°44.7' predicted altitude 0°00.0' intercept 83°16.6' predicted azimuth Jupiter altitude observation: 45°14.6' observed center altitude 0.9' refraction 45°13.7' unrefracted altitude of center 45°13.7' predicted altitude 0°00.0' intercept 250°59.7' predicted azimuth Moon to Jupiter predicted separation angle: 68°37.2' center to center, unrefracted 1.4' refraction 68°35.8' center to center, refracted 16.4' Moon near limb refracted semidiameter 68°19.4' Moon near limb to Jupiter center 68°19.4' observed angle 0°00.0' observed - predicted separation angle rate of change: +22" per minute (topocentric) 93% of total angular velocity With large errors in the initial estimates for time and position and the narrow angle between the LOPs, this problem was a good challenge for the computer. The program needed five iterations to reach a solution. For the comparison with HORIZONS, I requested .00001° accuracy, causing the program to iterate six times. Its algorithm belongs to today's era of cheap computing power, when iterating to the desired accuracy can be more practical than a direct solution which demands less from the machine but more from the programmer. There's no attempt to reduce the lunar to the equivalent geocentric observation. Instead, the algorithm works directly with the three topocentric observables (one separation angle and two altitudes) to deduce the three unknowns (time, latitude, and longitude). The assumptions are 1) you can predict the three observables, given a set of values assigned to the unknowns, 2) fairly good initial estimates of the unknowns are at hand, 3) a small change in any unknown (the other two remaining constant) causes a proportional change in the three predicted observables. This implies that you can describe the response of the observables with three linear equations in the three unknowns. Solving the equations yields corrections to the initial estimates of the unknowns. That's the end of the problem, if the relationship between the observables and unknowns is truly linear. In reality, it's not, so what you get are improved estimates of the unknowns. By repeating this process the lunar can be solved to arbitrary accuracy. That's the principle of my program. The source code is available my site: http://home.earthlink.net/~s543t-24dst/sofajplNet/LunarDist2.html In a followup message I'll try to explain the algorithm in more detail. -- I block messages that contain attachments or HTML. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---