# Position of Moon to 0.3 degrees

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## Overview

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"The sun must not catch up the moon, nor does the night outstrip the day. Each one is travelling in an orbit with its own motion"
(Al Qur'an 36:40)

"the sun and the moon (are subjected) to calculations"
(Al Qur'an 55:5)

This page provides a QBASIC program for calculating a rough position for the Moon, given your location and the date and time (UT). The program listing below gives the geocentric longitude and latitude of the Moon's centre, plus the geocentric and topocentric right ascension and declination, as well as the lunar parallax. The program is a 'test bed' for a programmable calculator program - the aim is to spend less time in front of computer screens!

The method used is based on a trigonometric series for the longitude, latitude and parallax of the Moon found on page D46 of the 1997 Astronomical Almanac. The position of the Moon in the sky can be predicted to within a third of its apparent diameter of the 'true' position as found using sophisticated methods.

I have compared the results obtained with this BASIC program (modified so as to direct output to a file) with topographic positions produced by Monzur Ahmed's Mooncalc program, and with geocentric positions taken from the NASA Twelve Year Planetary Ephemeris. The results suggest that short term errors are less than 0.015 Hours in RA and 0.12 degrees in declination. This is better than claimed in the Astronomical Almanac.

A similar series exists for the position of the Sun, and this will be added soon to allow calculations of the phase, position angle and parallactic angle of the Moon.

## BASIC listing

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Below is the QBASIC listing for the Moon position program. The program is not 'optimised' at all, and is designed to be easy to read and adapt for other languages including the limited syntax found on programmable calculators.

The user is prompted for the latitude and longitude of their location, and the year, month, day and time in UT for which the Moon's position is wanted. The longitude is given in decimal degrees, with West negative.

The output produced by a trial session is shown after the BASIC listing. Note that you will need to alter the values of `glong` and `glat` to reflect your own location.

The easiest way to transfer the listing into QBASIC would be to use `copy and paste` into a text editor such as Notepad (or SimpleText), from where the listing can be saved with a `.BAS` file extension.

```'   **********************************************************
'   Moon positions to a quarter of a degree on the sky
'   From page D76 of Astronomical Almanac
'
'   max RA error 97 (seconds of time), rms error 22 secs
'   max dec error 811 arcseconds, rms error 224 arcseconds
'	compared to Integrated Computer Ephemeris for 18.6 years
'	either side of J2000
'
'   definitions and functions
'
DEFDBL A-Z
pi = 4 * ATN(1)
twopi = 2 * pi
degs = 180 / pi
'
'   FNday only works between 1901 to 2099 - see Meeus chapter 7
'
DEF FNday (y, m, d, h) = 367 * y - 7 * (y + (m + 9) \ 12) \ 4 + 275 * m \ 9 + d - 730531.5 + h
/ 24
'
'   the atn2 function below returns an angle in the range 0 to two pi
'   depending on the signs of x and y.
'
DEF FNatn2 (y, x)
a = ATN(y / x)
IF x < 0 THEN a = a + pi
IF (y < 0) AND (x > 0) THEN a = a + twopi
FNatn2 = a
END DEF
'
'   the function below returns the true integer part,
'   even for negative numbers
'
DEF FNrange (x) = x - INT(x / twopi) * twopi
'
CLS
'
'    get the date and planet number from the user
'
INPUT "    lat : ", glat
INPUT "   long : ", glong
INPUT "  year  : ", y
INPUT "  month : ", m
INPUT "  day   : ", day
INPUT "hour UT : ", h
INPUT " minute : ", mins
h = h + mins / 60
d = FNday(y, m, day, h)
t = d / 36525
'
'   calculate the geocentric longitude
'	writing the coefficients in the series as angles in rads
'	would save 29 multiplications per position!
'
l = FNrange(rads * (218.32# + 481267.883# * t))
l = l + rads * 6.29# * SIN(FNrange((134.9# + 477198.85# * t) * rads))
l = l - rads * 1.27# * SIN(FNrange((259.2# - 413335.38# * t) * rads))
l = l + rads * .66# * SIN(FNrange((235.7# + 890534.23# * t) * rads))
l = l + rads * .21# * SIN(FNrange((269.9# + 954397.7# * t) * rads))
l = l - rads * .19# * SIN(FNrange((357.5# + 35999.05# * t) * rads))
l = l - rads * .11# * SIN(FNrange((186.6# + 966404.05# * t) * rads))
l = FNrange(l)
'
'   calculate the geocentric latitude
'
bm = rads * 5.13# * SIN(FNrange((93.3# + 483202.03# * t) * rads))
bm = bm + rads * .28# * SIN(FNrange((228.2# + 960400.87# * t) * rads))
bm = bm - rads * .28# * SIN(FNrange((318.3# + 6003.18# * t) * rads))
bm = bm - rads * .17# * SIN(FNrange((217.6# - 407332.2# * t) * rads))
'
'   get the parallax
'
gp = gp + rads * .0518# * COS(FNrange((134.9# + 477198.85# * t) * rads))
gp = gp + rads * .0095# * COS(FNrange((259.2# - 413335.38# * t) * rads))
gp = gp + rads * .0078# * COS(FNrange((235.7# + 890534.23# * t) * rads))
gp = gp + rads * .0028# * COS(FNrange((269.9# + 954397.7# * t) * rads))
'
'   from the parallax, get the semidiameter and the radius vector
'
sdia = .2725 * gp
rm = 1 / (SIN(gp))
xg = rm * COS(l) * COS(bm)
yg = rm * SIN(l) * COS(bm)
zg = rm * SIN(bm)
'
'   rotate to equatorial coords
'   obliquity of ecliptic
'
ecl = (23.4393 - 3.563E-07 * d) * rads

xe = xg
ye = yg * COS(ecl) - zg * SIN(ecl)
ze = yg * SIN(ecl) + zg * COS(ecl)
'
'   geocentric RA and Dec
'
ra = FNatn2(ye, xe)
dec = ATN(ze / SQR(xe * xe + ye * ye))
'
'   topocentric RA and DEC (spherical earth)
'   Local Siderial Time in degrees
'
lst = 100.46 + .985647352# * d + h * 15 + glong
xtop = xe - COS(glat) * COS(lst)
ytop = ye - COS(glat) * SIN(lst)
ztop = ze - SIN(glat)
rtop = SQR(xtop * xtop + ytop * ytop + ztop * ztop)
ratop = FNatn2(ytop, xtop)
dectop = ATN(ztop / SQR(xtop * xtop + ytop * ytop))
'
'   topocentric distance
'
rmtop = SQR(xtop * xtop + ytop * ytop + ztop * ztop)
'
'   print output in readable form - the 'formatting strings'
'	only work with Microsoft basics and FirstBas compiler
'	remove them for other basics.
'
PRINT
pr1\$ = "\                                      \ ###.###  ###.###"
pr2\$ = "\                                      \ #######.####"
PRINT USING pr2\$; "days from J2000 ....................... "; d
PRINT USING pr1\$; "geocentric ecliptic coords (l,b) ...... "; l * degs; bm * degs
PRINT USING pr2\$; "lunar parallax (deg) .................. "; gp * degs
PRINT USING pr2\$; "diameter (arcmin) ..................... "; sdia * degs * 60 * 2
PRINT USING pr2\$; "geocentric distance (lun dia) ......... "; rm
PRINT USING pr2\$; "equator of date (deg) ................. "; ecl * degs
PRINT USING pr1\$; "geocentric equatorial coords (ra, dec). "; ra * degs / 15; dec * degs
PRINT USING pr2\$; "local siderial time (hrs) ............. "; lst * degs / 24
PRINT USING pr1\$; "topocentric equatorial coords (ra, dec) "; ratop * degs / 15; dectop * degs
PRINT USING pr2\$; "topocentric lunar distance (lun dia) .. "; rmtop
END
'*********************************************************************************************
```

Here is the output of a sample session. Finding the position of the Moon for 11:56 UT on August 9th 1998, as seen from Birmingham UK. Note that only the first three decimal places of angle are actually meaningful! This is an approximate method!

```    lat : -1.91667
long : 52.5
year  : 1998
month : 8
day   : 9
hour UT : 11
minute : 56

days from J2000 .......................     -510.0028
geocentric ecliptic coords (l,b) ......  335.206   -0.244
lunar parallax (deg) ..................        0.9909
diameter (arcmin) .....................       32.4039
geocentric distance (lun dia) .........       57.8223
equator of date (deg) .................       23.4395
geocentric equatorial coords (ra, dec).   22.475   -9.830
local siderial time (hrs) .............        7.8865
topocentric equatorial coords (ra, dec)   22.510   -9.656
topocentric lunar distance (lun dia) ..       58.6530

```

For comparison, Mooncalc (by Monzur Ahmed) gives the following positions for the Moon.

```BIRMINGHAM 52:30N 1:55W TZ:0.0 Height:236m              Geo   Refrac OFF
Julian Day Number:  2451034.5                     Date: Sun 9 Aug  1998
JD of Conjunction:  2451018.0728                  Time: 11h  56m 00s UT
Apparent Sunrise:   4h   39m 47s UT    Apparent Sunset: 19h  45m 27s UT
---------------------------------1 of 4----------------------------------
Moon Alt:  -43.721 -43d  43m 17s     Moon Azi: 328.781  328d 46m 52s
Moon Dec:   -9.909 -9d   54m 31s     Moon RA:   22.481  22h  28m 51s
Sun Alt:    53.172  53d  10m 20s     Sun Azi:  173.100  173d 05m 59s
Sun Dec:    15.830  15d  49m 46s     Sun RA:     9.277  9h   16m 37s
Rel Alt:   -96.894 -96d  53m 37s     Rel Azi:  155.681  155d 40m 53s
Elongation 161.440  161d 26m 23s     Moon Age: 406.19h  16D  22H 11M
Phase:0.9741  Width:31.57m  Semi-Diam:0.270  Earth-Moon Dist:368652.93km
-------------------------------------------------------------------------
Moon Rise:          20h  29m 33s UT  Azimuth:           104d 09m 30s
Moon Set:           6h   14m 09s UT  Azimuth:           251d 47m 14s
Sunrise-Moonrise:   15h  49m 46s     Sunset-Moonset:   -13h  31m 18s
-------------------------------------------------------------------------
New Moon:           23 July 1998                        13h  44m 48s TD
Full Moon:          9  July 1998                        16h  01m 53s TD
-------------------------------------------------------------------------

```

On this occasion, the method is different from Mooncalc by a third of a minute of time (about 5 arcmin) in RA and about 5 arcminutes in DEC. The results can be worse than this, see the following section.

## Accuracy

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I compared geocentric positions produced by the method shown here with a three month set of daily geocentric positions taken from the NASA Twelve Year Planetary Ephemeris. I took the error in each coordinate as 'TYPE - low precision method' and then plotted the errors as a time series against day number for RA and for DEC using a spreadsheet.

The main features of the RA error graph are;

• Maximum error is +0.01 hour and -0.015 hour over the whole three month period.
• Error changes sign 12 times in the 91 day period.
• Error is multiply periodic, there are short term variations of period 10 days or so of small amplitude.
• Longer period variation of 30 days or so have a larger amplitude.
• The whole three month period shows signs of being a part of a much longer cycle.

The main features of the DEC error are;

• Maximum error is +0.12 degree and -0.15 degree over the whole three month period.
• Error changes sign 11 times in the 91 day period.
• The DEC error does not appear to have any definite phase relationship with the RA error.
• There are multiple period terms present in the DEC error, but these terms seem to be of similar amplitude.
• The whole three month period shows signs of being a part of a much longer cycle.

I then compared the topocentric positions adjusted for refraction obtained with Mooncalc and with the method shown here for each hour on April 13th 1997. By comparing the approximate method with refraction adjusted positions, I am attempting to compare the position of the Moon as observed against the stars with the position predicted by the low precision method.

The results for a complete day are shown in the table below. The topocentric RA and DEC produced by Mooncalc are listed in columns 2 and 3, and the estimates produced by the program given here are in columns 4 and 5. Column 6 shows the RA error, defined as (error=Mooncalc - estimate). Column 7 shows the RA error in degrees for comparison with the DEC error in column 8. The *'s show when the Moon is above the horizon.

```April 13th 1997

Mooncalc	D46		Error
UT	RA	Dec	RA	Dec	RA	RA degs	DEC
0	6.218	17.413	6.228	17.391	-0.01	-0.15	0.022
1	6.258	17.363	6.268	17.34	-0.01	-0.15	0.023
2	6.301	17.317	6.311	17.294	-0.01	-0.15	0.023
3	6.346	17.278	6.355	17.254	-0.009	-0.135	0.024
4	6.391	17.246	6.401	17.222	-0.01	-0.15	0.024
5	6.438	17.224	6.447	17.197	-0.009	-0.135	0.027
6	6.484	17.21	6.493	17.183	-0.009	-0.135	0.027
7	6.53	17.205	6.539	17.177	-0.009	-0.135	0.028
8	6.575	17.208	6.584	17.178	-0.009	-0.135	0.03
9	6.618	17.217	6.627	17.186	-0.009	-0.135	0.031
10*	6.658	17.231	6.667	17.199	-0.009	-0.135	0.032
11*	6.697	17.247	6.706	17.214	-0.009	-0.135	0.033
12*	6.733	17.263	6.742	17.228	-0.009	-0.135	0.035
13*	6.766	17.275	6.775	17.24	-0.009	-0.135	0.035
14*	6.797	17.281	6.806	17.246	-0.009	-0.135	0.035
15*	6.826	17.28	6.835	17.243	-0.009	-0.135	0.037
16*	6.853	17.268	6.862	17.231	-0.009	-0.135	0.037
17*	6.88	17.244	6.888	17.207	-0.008	-0.12	0.037
18*	6.905	17.209	6.914	17.171	-0.009	-0.135	0.038
19*	6.932	17.161	6.94	17.122	-0.008	-0.12	0.039
20*	6.959	17.102	6.967	17.063	-0.008	-0.12	0.039
21*	6.987	17.034	6.995	16.993	-0.008	-0.12	0.041
22*	7.017	16.957	7.026	16.916	-0.009	-0.135	0.041
23*	7.05	16.874	7.058	16.833	-0.008	-0.12	0.041
24*	7.085	16.789	7.093	16.747	-0.008	-0.12	0.042
```

As you can see, the errors are getting larger in DEC and smaller in RA throughout the day. These changes are probably part of a larger periodic error variation over time. There is not any sudden change in error value near Moonrise.

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