# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Polynomial Almanac**

**From:**Gordon Talge

**Date:**1996 Sep 02, 15:45 EDT

This is probably my last post on polynomials approximations. Thanks to everyone who responded with feed back and tried them out. I learned alot about approximations and the Nautical Almanacs. For example, sometimes the Air Almanac and the Nautical Almanac have different values listed for the same date and time. Sept 19, 1996 00h 00m 00s the Air Almanac has for the Sun: GHA 181d 32.5min Dec. N 1d 27.4min and the Nautical Almanac has: GHA 181d 32.6min Dec. N 1d 27.5min. ------------------------------------------ The Sept poly gave GHA 181d 32.5min and Dec N 1d 27.4min The Floppy Almanac, ICE and MICA sometimes give slightly different answers, so I decided not to sweat it, if my polynomials didn't match up to exactly 0.1 min on everything. Gordon Since getting some feed back from the list, I have revised my polynomial for the Sun, and have added one for the Moon. The Moon has a longer polynomial and is valid for only about a week. Power Series for the Moon Dates : Sep. 1 to Sep. 8 1996 A = 4.0 W = 1 Moon Moon H.P. S.D. GHA Dec. Term 0 1710.0627 17.8118 0.9213 0.2510 1 1392.4493 3.5367 -0.0416 -0.0113 2 1.2898 -6.8970 0.0206 0.0062 3 0.3901 0.1067 0.0026 -0.0003 4 0.0828 0.3447 -0.0026 -0.0028 5 -0.1331 -0.0606 0.0006 0.0033 6 -0.0059 0.0075 0.0000 0.0015 7 0.0122 -0.0002 -0.0001 -0.0022 sums 3104.1479 14.8496 0.9008 0.2454 The power series for the Moon is used exactly the same way explained below for the Sun, except that their are 8 terms instead of 6 in the Moon power series. Example: Moon Sept 4 17h 32m 16s t = 4.730740741 x = ( ( t - 1 ) / 4 ) - 1 = -0.067314815 this is the argument for the power series. GHA = 1616.335961 or removing multiples of 360 we get GHA = 176.3359610 degs or ** 176 degs 20.2 mins ** Using the same argument for the Dec. H.P. and S.D. we get Dec. = 17.5425 degs or ** N 17 degs 32.5 mins ** H.P. = 0.9242 degs or ** 55.5 mins ** S.D. = 0.2518 degs or ** 15.1 mins ** ------------------------------------------------------------------ ------------------------------------------------------------------ Power Series for the Sun Dates : Sep. 1 to Oct. 1 1996 A = 16.0 W = 1 Sun Sun Sun GHA Dec. SD Term 0 5941.3640 2.2319 0.2652 1 5761.4229 -6.1847 0.0012 2 0.0046 -0.0946 0.0001 3 -0.0506 0.0653 0.0000 4 -0.0051 -0.0004 0.0000 5 -0.0024 -0.0008 0.0000 sums 11702.7334 -3.9833 0.2665 Dates: Oct 1 to Nov 1 1996 A = 16.0 W = 1 Sun Sun Sun GHA Dec. SD Term 0 5943.6509 -9.2726 0.2675 1 5760.8108 -5.8514 0.0012 2 -0.3099 0.2859 0.0000 3 -0.0427 0.0749 -0.0001 4 -0.0004 0.0002 0.0000 5 -0.0018 -0.0009 0.0000 sums 11704.0969 -14.7639 0.2686 Dates: Nov 1 to Dec 1 1996 A = 16.0 W = 1 Sun Sun Sun GHA Dec. SD Term 0 5943.7628 -18.9951 0.2696 1 5759.2133 -3.9009 0.0010 2 -0.4361 0.7241 -0.0001 3 0.0191 0.0727 0.0000 4 0.0066 -0.0042 0.0000 5 -0.0008 -0.0012 0.0000 sums 11702.9574 -22.1046 0.2705 The way to use these series is first convert UT into decimal then t = d + 24 /UT, where d is the day of the month. Next, get x where x in between -1 and +1. ie [-1,+1] using the formula x = ( (t - W ) / A ) -1, use x as the argument to evaluate the polynomial. f(x) = a0 + a1*x + a2 * x^2 + a3 * x^3 + a4 * x^4 + a5 *x^5 . This can be better evaluated as f(x) = a0 + x*(a1+x*(a2+x*(a3+x*(a4+x*a5)))). Once the GHA, Dec. or S.D. is obtained, remove any multiples of 360 degrees and convert to degrees and mins. or in the case of the S.D. mins, round off to the nearest .1 min Example: GHA for Sept 18th 7h 28m 19s UT 7h 28m 19s = 7.471944444 hours or 7.471944444 / 24 = 0.311331019 parts of a day. Since d= 18 we have for t, t= 18.311331019 days x = (( 18.311331019 - 1 )/ 16 ) -1 or x = 0.081958189 This is what we use for the argument of the polynomial. Evaluating the GHA polynomial at 0.081958189 we get GHA = 6413.559790 . Removing multiples of 360 degrees we get 293.5597896d or ** 293d 33.6 mins ** So the GHA for Sept 18th 7h 28m 19s is 293d 33.6min By the 1996 Air Almanac I get for Sept 18th 7h 20mins 291d 28.8 min The correction for 8min 19s is 2d 4.8min so the ** GHA is 293d 33.6 min ** Using the same argument for the Dec. Series I get 1.724183513 or ** N 1d 43.5 ** (Note: N is + and S is - ) The Air Almanac give N 1d 43.6 for 7h 20min and N 1d 43.4 for 7h 30min Since 28min 19sec is between 20 and 30mins and is closer to 30 but still less and the dec is going down, N 1d 43.5 seems resonable. I don't know what the Nautical Almanac says with it's d correction. I have tried to fit the polynomial to give the proper value with an error of not more then 0.1 min The series is ONLY valid for the month stated, NOT before and NOT after. The sums at the bottom are NOT used in calculations, they are used to check that you entered the coefficients correctly. What I did was use a programable calculator to evaluate the polynomial by loading in the coefficients into the storage registers and then running it through. The reference ephemeris used is JPL's DE200 which is the background basis for the Astronomial Almanac and Nautical Almanacs. -- Gordon ------------------------------------------------------------------------ This mail list is managed by the majordomo program. 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