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

**Re: how are the tables for declination generated ? equation ? etc......**

**From:**Bill Murdoch

**Date:**2005 Feb 11, 21:10 EST

In a message dated 2/11/05 6:41:30 A.M. Eastern Standard Time,
jim2@JIMTHOMPSON.NET writes:

What type of calculator are you thinking of using? Something like a Texas

Instruments programmable calculator? eg the TI-80's series:

http://education.ti.com/us/product/tech/84p/features/features.html.

Programming yourself:

The formulas for calculating the GHA, dec, and SD of the sun are short
enough to enter easily (?) by hand into a programmable calculator. I
wrote two articles for doing that. The first "A Sun Sight Calculator
for UKL 17", was published by

__Practical Boat Owner__in February 1994. A month or two later there was a correction of a typesetting error that covered part of the program with a photo. The second, "Create Your Own Sun-Sight Reduction Program", was published by__Cruising World__in March 1996. It too was bitten by the typesetting bug and has corrections in a later issue. The English article was written for the TI-67 and the American for the TI-81. In those days when hand held programmable calculators were "the thing", I wrote a "does everything" program for the TI-82 and sold it by word of mouth for several years. The real fun of these things, at least for me, was figuring out how the calculations are made and accomplishing them myself. If you want to do that, here is a little bit to get you started or at least show you what is involved. The internet will no doubt butcher the layout of the tables, but here goes. In order to reduce a sight of the sun it is
necessary to know three astronomical values and two corrections. The astro
values are the sun's Greenwich hour angle, declination, and semidiameter.
The corrections are the refraction correction and the dip of the horizon.
All can be found in the Nautical Almanac, but they can also be calculated
directly. The necessary calculations are too long to be done by hand, but
they can be easily done with a programmable calculator or computer
program. After finding these five numbers, the sight reduction formulas
can be used to reduce a sun sight to an azimuth and intercept.

The problem is broken into six parts. The
common units of time are reduced to a single unit of time. The position of
the sun is calculated in the ecliptic coordinates of celestial latitude and
longitude. The celestial coordinates are converted to right ascension and
declination. The Greenwich hour angle of Aries is calculated and used to
convert the right ascension of the sun to Greenwich hour angle. Values are found
for the semidiameter of the sun, the refraction correction, and the dip of the
horizon. Finally, the azimuth and intercept are calculated.

The common units of time are hopelessly
complicated. There are too many of them; years, months, days, hours,
minutes, and seconds. Time must be changed into a single unit. The
chosen unit is the number of centuries after noon 1 January 2000. Time has
a value of -.5 at 1200 on 1 January 1950, 0 at 1200 on 1 January 2000, and +.5
at 1200 on 1 January 2050.

Van Flandern and Pulkkinen give a short formula
for converting common UTC (or GMT) time to UTC centuries that is valid from
March 1900 to February 2100.

Tu = (367*yr-trunc(7*(yr+trunc((mo+9)/12))/4)+trunc(275*mo/9)

+day+(hr+(min+(sec/60))/60)/24-730531.5)/36525

Tu = (367*yr-trunc(7*(yr+trunc((mo+9)/12))/4)+trunc(275*mo/9)

+day+(hr+(min+(sec/60))/60)/24-730531.5)/36525

The formula fails outside those dates because 1900
and 2100 are both years divisible by 4 which are not leap years. The
function trunc is the integer part of the number within the brackets; any
fractional part is dropped. To have an accuracy of one second, the value
of time must have 11 significant digits because there are over 6 billion seconds
in two centuries.

The way the formula works is a little vague and is
best explained working backwards. At the end of the formula, 36525
converts the units from days to centuries. The 730531.5 makes the formula
have a value of 0 at noon on 1 January 2000. The term
"day+(hr+(min+(sec/60))/60)/24" converts the date of the month and time into the
number of days since the beginning of the last day of the previous month.
The remaining part of the formula handles the changing number of days in the
months and accounts for leap years. You can get an idea of how it works by
solving it for the first day of each month for four consecutive years writing
down the numbers from within each set of parentheses.

We must keep up with two kinds of time. One
is UTC and is related to the rotation of the earth. The other is ephemeris
time which is related to the speed with which the earth revolves around the
sun. These two kinds of time are slowly drifting apart because the earth's
rotational speed has been slowing down for the last several decades. Every
time a leap second is inserted into UTC the two get farther apart.

An equation for converting centuries of UTC time
to ephemeris time is

Te = Tu+((63+60*Tu)/3,200,000,000)

Te = Tu+((63+60*Tu)/3,200,000,000)

This says that the difference between the two
kinds of time is 63 seconds in January 2000 and is increasing by 60 seconds per
century. The formula has been accurate for the last few decades and will
probably be accurate for several more.

With time out of the way, the next step is to
calculate the apparent ecliptic longitude of the sun. Because the orbit of
the earth around the sun is perturbed by the nearby planets and because the
earth is pulled about by the moon, we must first make rough estimates of the
positions of the earth, moon, and planets.

The positions of venus, earth, mars, and jupiter
are each calculated as the mean anomaly which is the angle between the planet's
perihelion and position. The results are in degrees with values oftentimes
greater than 360 or less than 0. That does not matter to most
calculators, but you may find it necessary to subtract the extra revolutions
before taking the sines of the angles in later calculations.

V = 50+(58517*Te)

E = 357.52558+(35999.04974*Te)

M = 20+(19140*Te)

J = 19.9+(3034.6*Te)

V = 50+(58517*Te)

E = 357.52558+(35999.04974*Te)

M = 20+(19140*Te)

J = 19.9+(3034.6*Te)

A few more short formulas give the longitude of
the moon's ascending node and twice the sun's mean longitude.

N = 125.0-(1934.1*Te)

L = 200.9+(72001.7*Te)

N = 125.0-(1934.1*Te)

L = 200.9+(72001.7*Te)

With these intermediate values in hand, it is
possible to calculate the apparent ecliptic longitude of the sun. The
formula is long and somewhat repetitious.

EL =
E+(1018585.1+(6191.2*Te)+(1.1*Te2)

+6892.8*sin(E-0.0018)

+72.0*sin(2*E)

-17.4*Te*sin(E)

+7.2*sin(E-J-90.5)

+6.5*sin((445267.1*Te)-62.1)

-6.4*sin((20.2*Te)+71.4)

+5.5*sin((2*E)-(2*V)-58)

-4.8*sin(E-V-29)

-2.7*sin((2*E)-(2*J)-3)

-2.6*sin(J+7)

-2.5*sin((3*E)-(2*V)-46)

+2.0*sin((2*E)-(2*M)+74)

-1.9*sin((150*Te)+28)

+1.8*sin(E-(2*M)-70)

-1.6*sin(E-(2*J)+20)

-1.6*sin((4*E)-(3*V)-75)

+1.0*sin(3*E)

-1.0*sin((5*E)-(3*V)-48)

-20.5

-17.2*sin(N)

-1.3*sin(L))3600

+6892.8*sin(E-0.0018)

+72.0*sin(2*E)

-17.4*Te*sin(E)

+7.2*sin(E-J-90.5)

+6.5*sin((445267.1*Te)-62.1)

-6.4*sin((20.2*Te)+71.4)

+5.5*sin((2*E)-(2*V)-58)

-4.8*sin(E-V-29)

-2.7*sin((2*E)-(2*J)-3)

-2.6*sin(J+7)

-2.5*sin((3*E)-(2*V)-46)

+2.0*sin((2*E)-(2*M)+74)

-1.9*sin((150*Te)+28)

+1.8*sin(E-(2*M)-70)

-1.6*sin(E-(2*J)+20)

-1.6*sin((4*E)-(3*V)-75)

+1.0*sin(3*E)

-1.0*sin((5*E)-(3*V)-48)

-20.5

-17.2*sin(N)

-1.3*sin(L))3600

The result is in degrees, but the units of each of
the summed elements are seconds of arc. The first line calculates the mean
ecliptic longitude of the sun. The next eighteen lines move the sun in an
elliptical orbit and correct for the tugs and pulls of the moon and
planets. Because we see the sun where it was about 8 minutes ago, 20.5
seconds of arc must be subtracted. The last two lines correct for the
nutation of the earth's axis.

The next task is to change from ecliptic to
equatorial coordinates. If we assume that the sun's celestial latitude is
zero, the formulas for right ascension and declination are short, and we need
only know the obliquity of the ecliptic.

Ob = 23.43929-(0.01300*Te)+(0.00256*cos(N))+(0.00016*cos(L))

RA = tan-1(tan(EL)*cos(Ob))

if 90<EL<270, then RA = 180+RA

Dec = sin-1(sin(EL)*sin(Ob))

Ob = 23.43929-(0.01300*Te)+(0.00256*cos(N))+(0.00016*cos(L))

RA = tan-1(tan(EL)*cos(Ob))

if 90<EL<270, then RA = 180+RA

Dec = sin-1(sin(EL)*sin(Ob))

The GHA of Aries can be calculated from the
universal time in centuries with a relatively short formula. The answer is
in degrees and may need to be reduced to an angle between 0 and 360. This
formula requires at least 12 significant digits to keep an accuracy of 0.1'
between 1900 and 2100. The GHA of the sun is the difference between the
GHA of Aries and the right ascension of the sun.

ARIES = 360*(0.7790573+(36625.0021390*Tu)+(0.0000011*Tu2)

-(0.0000122*sin(N))-(0.0000009*sin(L)))

GHA = ARIES-RA

ARIES = 360*(0.7790573+(36625.0021390*Tu)+(0.0000011*Tu2)

-(0.0000122*sin(N))-(0.0000009*sin(L)))

GHA = ARIES-RA

The semidiameter of the sun varies with its
distance from the earth and that depends on the position of the sun. The
semidiameter of the sun can be calculated from its mean anomaly.

SD = sin-1 (0.004659/(1-0.0167*cos(E)))

SD = sin-1 (0.004659/(1-0.0167*cos(E)))

The almanac has a formula for the height of eye
correction with the height of eye in meters and the dip of the horizon in
degrees.

D = 0.0293*HE

D = 0.0293*HE

The sun's apparent altitude is the total of the
sextant altitude, the index correction, and the negative of the dip. The
apparent altitude, sextant altitude, and dip are all in degrees. The index
correction is in minutes of arc.

Ha = Hs+(IC/60)-D

Ha = Hs+(IC/60)-D

The almanac also has a formula for the refractive
correction.

The pressure is in millibar, and the temperature is in Celsius.

Rc = (0.28*Pres/(Temp+273))*0.0167/tan(Ha+(7.31/(Ha+4.4)))

The pressure is in millibar, and the temperature is in Celsius.

Rc = (0.28*Pres/(Temp+273))*0.0167/tan(Ha+(7.31/(Ha+4.4)))

The horizontal parallax of the sun is small and
can be ignored, but for completeness we can include it. Both the parallax
in altitude and apparent altitude are in degrees.

PA = 0.0024*cos(Ha)

PA = 0.0024*cos(Ha)

With all the usual Nautical Almanac data now in
hand, the well known sight reduction formulas can be used to find the calculated
altitude and azimuth. The latitude and longitude of the dead reckoning
position are in degrees with north and east positive and with south and west
negative.

LHA = GHA+Long

Hc = sin-1((cos(LHA)*cos(Lat)*cos(Dec))+(sin(Lat)*sin(Dec)))

Zc = cos-1((sin(Dec)-(sin(Lat)*sin(Hc)))/(cos(Lat)cos(Hc)))

if 0°<LHA<180°, Zn = 360-Zc

if 180°<LHA<360°, Zn = Zc

LHA = GHA+Long

Hc = sin-1((cos(LHA)*cos(Lat)*cos(Dec))+(sin(Lat)*sin(Dec)))

Zc = cos-1((sin(Dec)-(sin(Lat)*sin(Hc)))/(cos(Lat)cos(Hc)))

if 0°<LHA<180°, Zn = 360-Zc

if 180°<LHA<360°, Zn = Zc

The intercept in nautical miles is finally
calculated. The sign of the semidiameter depends on the limb
observed. The sign is + for the lower limb and - for the upper limb.

p = 60*(Ha-Rc+PA±S-HC)

p = 60*(Ha-Rc+PA±S-HC)

It is easy to incorporate the formulas in a
calculator program, a spreadsheet, or a PC program. To help in debugging
such a program, two examples are given in the table below. They were
calculated with an Excel spreadsheet. The results might be slightly
different with other calculators or programs.

The astro formulas give results that closely match
the Nautical Almanac. The accuracy seems to be 0.1' or better. In
December and January the results for the sun's GHA may appear worse, but
remember that the Nautical Almanac entries are slightly in error then to avoid a
'v' correction for the sun.

The formulas used in this article came from the
from the Nautical Almanac, from Van Flandern and Pulkkinen, "Low Precision
Formulae for Planetary Positions", The Astrophysical Supplement Series, vol 41,
p 391, (1979), and from Montenbruck and Phleger, Astronomy on the Personal
Computer, Springer-Verlag, Berlin, 1991. They were put into the form used
by B. Emerson in N.A.O. Technical Note Number 47 - Approximate Solar
Coordinates, Her Majesty's Nautical Almanac Office, November 1978.

Test Problems

yr 1972 1994

mo 6 4

day 23 8

hr 0 21

min 17 54

sec 52 9

Tu -0.275249489 -0.057319299

Te -0.275249475 -0.05731928

V -16056.77351 -3304.15233

E -9551.193949 -1705.914046

M -5248.274945 -1077.091027

J -815.3720558 -154.0410883

N 657.3600089 235.8612202

L -19617.5301 -3926.18563

EL -9268.363115 -1421.172693

Ob 23.44388492 23.43873142

RA -88.21592359 17.37102545

RA 91.78407641 17.37102545

Dec 23.43374638 7.375208356

ARIES -3628884.262 -755474.5373

ARIES 275.7376754 165.4626822

GHA 183.953599 148.0916567

SD 0.262639337 0.266624737

HE 3.4 2.2

D 0.054026531 0.043458923

Hs 50.02000000 2.53000000

IC 10.2 -5.8

Ha 50.13597347 2.38987441

Pres 1010 1030

Temp 22 40

Rc 0.013305383 0.254021969

PA 0.001538323 0.002397913

Long 172 -58

LHA 355.953599 90.0916567

Lat -16.1 13

Hc 50.2688665 1.566109477

Zc 5.813557565 82.79151131

Zn 5.813557565 277.2084887

Limb lower upper

p 7.078755034 18.33096838

A Comparison of the Calculator Almanac and the Nautical Almanac

mo 6 4

day 23 8

hr 0 21

min 17 54

sec 52 9

Tu -0.275249489 -0.057319299

Te -0.275249475 -0.05731928

V -16056.77351 -3304.15233

E -9551.193949 -1705.914046

M -5248.274945 -1077.091027

J -815.3720558 -154.0410883

N 657.3600089 235.8612202

L -19617.5301 -3926.18563

EL -9268.363115 -1421.172693

Ob 23.44388492 23.43873142

RA -88.21592359 17.37102545

RA 91.78407641 17.37102545

Dec 23.43374638 7.375208356

ARIES -3628884.262 -755474.5373

ARIES 275.7376754 165.4626822

GHA 183.953599 148.0916567

SD 0.262639337 0.266624737

HE 3.4 2.2

D 0.054026531 0.043458923

Hs 50.02000000 2.53000000

IC 10.2 -5.8

Ha 50.13597347 2.38987441

Pres 1010 1030

Temp 22 40

Rc 0.013305383 0.254021969

PA 0.001538323 0.002397913

Long 172 -58

LHA 355.953599 90.0916567

Lat -16.1 13

Hc 50.2688665 1.566109477

Zc 5.813557565 82.79151131

Zn 5.813557565 277.2084887

Limb lower upper

p 7.078755034 18.33096838

A Comparison of the Calculator Almanac and the Nautical Almanac

Sun
Aries
Date Time
G.H.A.
Declination
Semidiameter
G.H.A.

N.A. Calc. Error N.A. Calc. Error N.A. Calc. Error N.A. Calc. Error

° ' ° ' ° ' ° ' ' ' ' ° ' ° ' '

1 Jan 95 0000 179 12.0 179 12.2 +.2 S23 03.2 S23 03.3 +.1 16.3 16.3 0 100 10.7 100 10.7 0

2 Jun 94 0100 195 32.5 195 32.6 +.1 N22 07.9 N22 07.9 0 15.8 15.8 0 265 16.6 265 16.6 0

27 Feb 93 0200 206 48.0 206 48.0 0 S 8 23.2 S 8 23.2 0 16.2 16.2 0 186 55.3 186 55.3 0

3 Sep 93 0300 225 08.4 225 08.4 0 N 7 34.8 N 7 34.8 0 15.9 15.9 0 27 15.8 27 15.9 +.1

20 Mar 92 0400 238 07.5 238 07.4 -.1 S 0 04.7 S 0 04.7 0 16.1 16.1 0 237 56.5 237 56.5 0

10 Oct 92 0500 258 15.1 258 15.0 -.1 S 6 44.1 S 6 44.1 0 16.0 16.0 0 94 03.2 94 03.2 0

23 Apl 91 0600 270 23.5 270 23.4 -.1 N12 22.5 N12 22.5 0 15.9 15.9 0 300 47.3 300 47.3 0

16 Nov 91 0700 288 49.7 288 49.7 0 S18 37.7 S18 37.7 0 16.2 16.2 0 159 51.5 159 51.5 0

8 May 90 0800 300 52.9 300 52.9 0 N17 03.0 N17 02.9 -.1 15.9 15.9 0 345 53.6 345 53.5 -.1

13 Dec 90 0900 316 29.5 316 29.5 0 S23 08.5 S23 08.5 0 16.3 16.3 0 216 47.5 216 47.4 -.1

26 May 89 1000 330 45.6 330 45.7 +.1 N21 09.5 N21 09.5 0 15.8 15.8 0 33 57.2 33 57.2 0

6 Jun 84 1100 345 20.2 345 20.3 +.1 N22 41.8 N22 41.8 0 15.8 15.8 0 60 02.3 60 02.3 0

N.A. Calc. Error N.A. Calc. Error N.A. Calc. Error N.A. Calc. Error

° ' ° ' ° ' ° ' ' ' ' ° ' ° ' '

1 Jan 95 0000 179 12.0 179 12.2 +.2 S23 03.2 S23 03.3 +.1 16.3 16.3 0 100 10.7 100 10.7 0

2 Jun 94 0100 195 32.5 195 32.6 +.1 N22 07.9 N22 07.9 0 15.8 15.8 0 265 16.6 265 16.6 0

27 Feb 93 0200 206 48.0 206 48.0 0 S 8 23.2 S 8 23.2 0 16.2 16.2 0 186 55.3 186 55.3 0

3 Sep 93 0300 225 08.4 225 08.4 0 N 7 34.8 N 7 34.8 0 15.9 15.9 0 27 15.8 27 15.9 +.1

20 Mar 92 0400 238 07.5 238 07.4 -.1 S 0 04.7 S 0 04.7 0 16.1 16.1 0 237 56.5 237 56.5 0

10 Oct 92 0500 258 15.1 258 15.0 -.1 S 6 44.1 S 6 44.1 0 16.0 16.0 0 94 03.2 94 03.2 0

23 Apl 91 0600 270 23.5 270 23.4 -.1 N12 22.5 N12 22.5 0 15.9 15.9 0 300 47.3 300 47.3 0

16 Nov 91 0700 288 49.7 288 49.7 0 S18 37.7 S18 37.7 0 16.2 16.2 0 159 51.5 159 51.5 0

8 May 90 0800 300 52.9 300 52.9 0 N17 03.0 N17 02.9 -.1 15.9 15.9 0 345 53.6 345 53.5 -.1

13 Dec 90 0900 316 29.5 316 29.5 0 S23 08.5 S23 08.5 0 16.3 16.3 0 216 47.5 216 47.4 -.1

26 May 89 1000 330 45.6 330 45.7 +.1 N21 09.5 N21 09.5 0 15.8 15.8 0 33 57.2 33 57.2 0

6 Jun 84 1100 345 20.2 345 20.3 +.1 N22 41.8 N22 41.8 0 15.8 15.8 0 60 02.3 60 02.3 0

Bill Murdoch

(My wife and I have a Crealock 34
in Northwest Creek Marina not far from
you.)