The Museum of HP Calculators
The Museum of HP Calculators
This program is Copyright © 2002 by Jean-Marc Baillard and is used here by permission.
This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
-Nowadays, it's easy to calculate the positions of the Sun, the Moon and
the planets with a great accuracy
by mean of Fourier series A.cos(B.T+C) and Poisson series.
Unfortunately, this requires many bytes and a more economical approach
is proposed here:
-Kepler's equation is solved first and then, a few periodic terms are added
to the heliocentric longitude,
the heliocentric latitude and the radius vector. ( exception: the Moon's
geocentric coordinates are computed directly )
-The result is an ephemeride over the interval 1000-3000 with an accuracy
of about 0.01° in the heliocentric longitudes.
( except for Pluto: its coordinates are obtained with this precision
over 1880-2110 only )
-The Sun's coordinates are more accurate. However, errors in the geocentric
coordinates are increased when the distance between the Earth and the planet
is small and in this case, for Venus and Mars, the error in the obtained
geocentric longitude can reach 1 arcminute, perhaps a little more.
-All these coordinates are referred to the mean ecliptic and equinox of
date.
-The elements of planetary orbits and the perturbations are taken from the
planetary theory VSOP87 by P. Bretagnon, JL. Simon and G. Francou
who are French astronomers at the "Bureau des Longitudes" ( cf
www.bdl.fr )
-The Moon's coordinates are based on the ELP2000-85 theory by M.
Chapront-Touzé and J. Chapront
who are French astronomers at the "Bureau des Longitudes" too.
-Pluto's position is derived from simplified formulae I constructed myself
( thanks to a numerical integration of the JPL DE200/LE200 and to my HP-48!
)
( I'm not at the Bureau des Longitudes ... )
-In order to save bytes, perturbations are written under the form
S sin µ + K cos µ instead of A.cos(B.T+C)
The phase µ is actually a linear combination of the mean mean
longitudes of the planets.
Furthermore, the R-P is quite useful: for instance, if the phase µ
is in register R10 , 2 sin µ + 4 cos µ is computed
by:
RCL 10
2
P-R
ST+ X
+
-This trick is extensively used troughout all these programs ( except for
the Moon )
-I also made several approximations which are satisfactory here, but would
be inadequate to obtain a great accuracy.
-However, the whole set of programs occupies about 3000 bytes and therefore,
some of these programs are to be saved in extended memory.
-These programs are presented as follows:
1°) The Sun ( 98 bytes )
2°) The "L" "N" "O" "K" subroutines which are called by
the main programs and must stay in on-line memory for use. ( 419 bytes )
3°) The Moon ( 557 bytes )
4°) Mercury-Venus-Mars-Pluto ( 531 bytes )
5°) Jupiter-Neptune ( 522 bytes )
6°) Saturn ( 475 bytes )
7°) Uranus ( 344 bytes )
DATA REGISTERS:
-Registers R00 thru R15 are used by every program.
R00 = T = the time expressed in thousands of years since 2000/01/01
0h ET is to be initialized before executing any program
R01 and R02 contain the Sun's rectangular coordinates. They are calculated
and stored by the "SUN" program.
R03 = geocentric longitude in decimal degrees
R04 = geocentric latitude ------------------
R05 = distance to the Earth in Astronomical Units ( except for the
Moon: R05 = the Moon's parallax in sexagesimal degrees )
R06 = Right Ascension in hh.mnss
R07 = Declination
in ° ' ''
R08 = heliocentric longitude in decimal degrees ( except
for the Sun and the Moon )
R09 = heliocentric latitude
------------------- ---------------------------------
R10 = radius vector in Astronomical
Units
---------------------------------
-Registers are also used for temporary data storage, for instance:
R05 thru R10 contain the mean elements of the planets: semi-major axis,
eccentricity, inclination,
mean longitude, longitude of perihelion, longitude of the ascending node.
R03 , R04 , R11 contain perturbations in longitude, radius vector and
latitude respectively.
-And for each program, you have:
| STACK | INPUTS | OUTPUTS |
| Z | / | distance to the earth* ( AU ) |
| Y | / | declination ( ° ' '' ) |
| X | / | right ascension ( hh.mnss ) |
*or Moon's parallax ( ° ' " )
INSTRUCTIONS:
1-Store the time T ( in thousands of years since 2000/01/01 0h ET (
not 12h ) ) into register R00.
This can be performed by one of these short routines:
LBL "T1" if
you
have
LBL "T2" if you don't have a TIME module
HR
a TIME module
or
HR
and for dates before 1582/10/15
24
24
/
/
X<>Y
X<>Y
1.012
XEQ "J0" ( cf "Phases of the Moon for the HP-41"
)
DDAYS
+
-
365250
365250
/
/
STO 00
STO
00
END
END
Then,
date ENTER^
hh.mnss XEQ "T1" or XEQ "T2"
2-XEQ "SUN" first: This is needed to calculate the geocentric coordinates of the planets ( of course, this doesn't apply to the Moon ).
3-XEQ "MO" for the Moon and "ME" "VE" "MA" "JU" "SA" "UR" "NE" PL" for Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.
NB: The HP-41 must be set in DEG mode.
REFERENCES:
-If you need a better accuracy, I strongly recommend these 3 books:
1-Planetary Programs and tables from -4000 to +2800 - Bretagnon and
Simon - Willmann-Bell ISBN 0-943396-08-5
2-Lunar Tables and Programs from 4000 B.C. to A.D. 8000 -
Chapront-Touzé and Chapron - Willmann-Bell ISBN 0-943396-33-6
3-Astronomical Algorithms - J Meeus - Willmann-Bell ISBN
0-943396-35-2
"Planetary Programs and Tables ... " also provides simple formulae for
corrections of aberration and nutation.
The Sun
01 LBL "SUN"
02 XEQ "O"
03 COS
04 RCL 08
05 RCL 05
06 -
07 SIN
08 +
09 RCL 13
10 SIN
11 +
12 ST+ X
13 RCL 12
14 SIN
15 LASTX
16 ST+ X
17 SIN
18 +
19 2
20 SQRT
21 *
22 -
23 STO 03
24 RCL 13
25 COS
26 PI
27 %
28 STO 04
29 17195
30 RCL 00
31 STO Z
32 46
33 *
34 +
35 *
36 77063
37 -
38 STO 09
39 CLX
40 42
41 +
42 *
43 CHS
44 1671
45 +
46 STO 06
47 SIGN
48 STO 05
49 CLX
50 33
51 /
52 *
53 ST+ 08
54 CLST
55 STO 01
56 STO 02
57 STO 07
58 XEQ "L"
59 RCL 03
60 RCL 05
61 P-R
62 STO 01
63 X<>Y
64 STO 02
65 RCL 05
66 RCL 07
67 RCL 06
68 END
( 98 bytes / SIZE 016 )
Example: Calculate the Sun's position on 2100 January 1st at 0h TD ( Dynamical Time or Ephemeris Time )
thus T = 0.1 0.1 STO 00 ( or 1.012100 ENTER^ 0 XEQ "T1" or 2100.0101 ENTER^ 0 XEQ "T2" )
then:
XEQ "SUN" yields
18h46m09s ( right ascension
) = R06 ( execution time = 36s
)
RDN
-23°00'10''
( declination ) = R07
RDN
0.983356 AU ( radius vector
) = R05
we have also: R01 =
0.181015 and R02 =
-0.966552 ( the Sun's rectangular coordinates )
R03 = -79.393° ( the
Sun's longitude )
R04 =
0
( the Sun's latitude is always 0 )
4 Subroutines: "L" "N" "O" "K"
001 LBL "L"
002 +
003 E3
004 ST/ 03
005 ST/ 04
006 ST/ 07
007 ST/ 09
008 ST/ 11
009 /
010 STO 10
011 E5
012 ST/ 06
013 9
014 STO 15
015 RCL 08
016 360
017 MOD
018 13.971
019 RCL 00
020 *
021 +
022 RCL 09
023 -
024 ENTER^
025 ENTER^
026 LBL 00 (
solving Kepler's equation )
027 SIN
028 RCL 06
029 R-D
030 *
031 +
032 DSE 15
033 GTO 00
034 2
035 /
036 1
037 RCL 06
038 +
039 1
040 LASTX
041 -
042 /
043 SQRT
044 P-R
045 LASTX
046 /
047 R-P
048 RDN
049 ST+ X
050 1
051 R^
052 ST+ X
053 COS
054 RCL 06
055 *
056 -
057 RCL 05
058 *
059 RCL 04
060 +
061 X<> 10
062 STO 04
063 -
064 RCL 09
065 +
066 COS
067 RCL 07
068 LASTX
069 SIN
070 P-R
071 X<>Y
072 ASIN
073 RCL 11
074 +
075 STO 09
076 X<> Z
077 R-P
078 CLX
079 RCL 04
080 +
081 RCL 03
082 +
083 STO 08
084 RCL 10
085 XEQ 01
086 RCL 02
087 ST+ Z
088 CLX
089 RCL 01
090 +
091 XEQ 02
092 STO 05
093 RDN
094 STO 03
095 X<>Y
096 STO 04
097 LBL "N" (
this line is used for the Moon only )
098 RCL 00
099 13
100 %
101 23.4393
102
-
( ecliptic / equatorial transformation )
103 RCL 04
104 RCL 03
105 RCL 05
106 XEQ 01
107 RDN
108 R-P
109 X<> Z
110 ST- Y
111 X<> Z
112 P-R
113 R^
114 XEQ 02
115 X<> Z
116 HMS
117 STO 07
118 X<>Y
119 15
120 /
121 24
122 MOD
123 HMS
124 STO 06
125 RTN
126 LBL
01 ( spherical
/ rectangular conversion )
127 X<>Y
128 RDN
129 P-R
130 R^
131 X<>Y
132 P-R
133 RTN
134 LBL 02 (
rectangular / spherical conversion )
135 R-P
136 X<>Y
137 RDN
138 R-P
139 R^
140 X<>Y
141 RTN
142 LBL "O"
143 XEQ "K"
144 CLX
145 STO 04
146 STO 11
147 4452671.115
148 RCL 00
149 6
150 /
151 -
152 RCL 00
153 *
154 68.245
155 -
156 STO 13
157 ST+ X
158 STO 14
159 585178.159
160 RCL 00
161 *
162 181.179
163 +
164 STO 07
165 359993.727
166 RCL 00
167 *
168 80.027
169 -
170 STO 08
171 -
172 STO 12
173 LASTX
174 191402.993
175 RCL 00
176 *
177 4.829
178 -
179 STO 09
180 ST+ X
181 -
182 4
183 *
184 RCL 05
185 3
186 *
187 +
188 STO 10
189 RTN
190 LBL "K"
191 30349.057
192 RCL 00
193 *
194 34.31
195 +
196 STO 05
197 12221.14
198 RCL 00
199 *
200 50.061
201 +
202 STO 06
203 -
204 STO 10
205 LASTX
206 -
207 STO 11
208 RCL 10
209 +
210 STO 12
211 RCL 06
212 ST+ X
213 -
214 STO 09
215 ST+ X
216 STO 03
217 4284.67
218 RCL 00
219 *
220 45.951
221 -
222 STO 07
223 2184.862
224 RCL 00
225 *
226 55.654
227 -
228 STO 08
229 -
230 STO 14
231 LASTX
232 -
233 STO 15
234 RCL 06
235 RCL 07
236 3
237 *
238 -
239 STO 04
240 LASTX
241 RCL 09
242 RCL 06
243 -
244 +
245 STO 13
246 RCL 09
247 5
248 END
( 419 bytes / SIZE 016 )
The Moon
001 LBL "MO"
002 XEQ "O"
003 3.22
004 RCL 00
005 STO Z
006 65
007 /
008 +
009 *
010 77.06
011 -
012 ST- 08
013 CLX
014 90
015 +
016 *
017 477198868
018 +
019 %
020 128.43
021 +
022 STO 07
023 ST+ X
024 STO 12
025 CLX
026 34
027 *
028 483202018
029 -
030 %
031 86.66
032 -
033 STO 09
034 ST+ X
035 STO 06
036 6036
037 RCL 14
038 RCL 07
039 -
040 STO 03
041 COS
042 58
043 *
044 -
045 RCL 14
046 COS
047 46
048 *
049 -
050 RCL 12
051 COS
052 9
053 *
054 -
055 RCL 14
056 RCL 12
057 -
058 STO 15
059 COS
060 RCL 07
061 COS
062 82
063 *
064 -
065 4
066 *
067 +
068 RCL 14
069 RCL 08
070 -
071 STO 11
072 COS
073 RCL 14
074 RCL 07
075 +
076 STO 10
077 COS
078 +
079 3
080 *
081 -
082 1
083 %
084 1/X
085 ASIN
086 HMS
087 STO 05
088 RCL 07
089 RCL 09
090 -
091 SIN
092 281
093 *
094 RCL 07
095 RCL 09
096 +
097 SIN
098 278
099 *
100 +
101 RCL 14
102 RCL 09
103 +
104 SIN
105 173
106 *
107 +
108 RCL 03
109 RCL 09
110 -
111 SIN
112 55
113 *
114 +
115 RCL 03
116 RCL 09
117 +
118 SIN
119 46
120 *
121 +
122 RCL 14
123 RCL 09
124 -
125 SIN
126 33
127 *
128 +
129 RCL 12
130 RCL 09
131 -
132 SIN
133 17
134 *
135 +
136 RCL 10
137 RCL 09
138 +
139 SIN
140 RCL 12
141 RCL 09
142 +
143 SIN
144 +
145 9
146 *
147 +
148 RCL 11
149 RCL 09
150 +
151 SIN
152 RCL 09
153 SIN
154 641
155 *
156 -
157 8
158 *
159 +
160 RCL 15
161 RCL 09
162 +
163 SIN
164 RCL 10
165 RCL 09
166 -
167 SIN
168 +
169 4
170 *
171 +
172 RCL 14
173 RCL 08
174 +
175 RCL 09
176 +
177 SIN
178 3
179 *
180 -
181 RCL 11
182 RCL 09
183 -
184 STO 04
185 RCL 07
186 -
187 SIN
188 LASTX
189 RCL 06
190 +
191 SIN
192 +
193 RCL 04
194 SIN
195 +
196 ST+ X
197 +
198 STO 04
199 RCL 07
200 SIN
201 6289
202 *
203 RCL 14
204 SIN
205 658
206 *
207 +
208 RCL 12
209 SIN
210 214
211 *
212 +
213 RCL 06
214 SIN
215 114
216 *
217 +
218 RCL 15
219 SIN
220 59
221 *
222 +
223 RCL 03
224 RCL 08
225 -
226 SIN
227 57
228 *
229 +
230 RCL 10
231 SIN
232 53
233 *
234 +
235 RCL 11
236 SIN
237 46
238 *
239 +
240 RCL 08
241 RCL 07
242 -
243 SIN
244 41
245 *
246 -
247 RCL 03
248 SIN
249 98
250 *
251 RCL 07
252 RCL 06
253 -
254 SIN
255 -
256 13
257 *
258 +
259 RCL 07
260 RCL 06
261 +
262 SIN
263 RCL 14
264 ST+ X
265 RCL 07
266 -
267 SIN
268 +
269 11
270 *
271 +
272 RCL 03
273 ST+ X
274 SIN
275 9
276 *
277 +
278 RCL 03
279 RCL 08
280 +
281 SIN
282 8
283 *
284 -
285 RCL 14
286 RCL 08
287 +
288 SIN
289 7
290 *
291 -
292 RCL 08
293 SIN
294 RCL 00
295 37
296 -
297 *
298 RCL 13
299 SIN
300 7
301 *
302 -
303 RCL 07
304 RCL 08
305 +
306 SIN
307 6
308 *
309 -
310 RCL 14
311 RCL 06
312 +
313 SIN
314 3
315 *
316 +
317 RCL 07
318 RCL 12
319 +
320 SIN
321 ST+ X
322 +
323 RCL 13
324 RCL 07
325 -
326 SIN
327 -
328 RCL 13
329 RCL 08
330 +
331 SIN
332 +
333 5
334 *
335 +
336 RCL 10
337 RCL 08
338 -
339 SIN
340 RCL 14
341 RCL 12
342 +
343 SIN
344 +
345 RCL 14
346 ST+ X
347 SIN
348 +
349 RCL 03
350 RCL 12
351 -
352 SIN
353 +
354 RCL 00
355 23
356 *
357 1
358 -
359 R-D
360 SIN
361 -
362 4
363 *
364 +
365 RCL 08
366 RCL 12
367 -
368 SIN
369 RCL 03
370 RCL 06
371 -
372 SIN
373 +
374 3
375 *
376 -
377 RCL 11
378 RCL 08
379 -
380 SIN
381 LASTX
382 RCL 07
383 -
384 SIN
385 +
386 RCL 11
387 RCL 12
388 -
389 SIN
390 +
391 RCL 13
392 RCL 07
393 +
394 SIN
395 -
396 RCL 12
397 RCL 08
398 +
399 SIN
400 -
401 RCL 08
402 ST+ X
403 SIN
404 -
405 RCL 00
406 ST+ X
407 133
408 -
409 RCL 00
410 *
411 4812678813
412 +
413 RCL 00
414 *
415 211729
416 +
417 E3
418 /
419 360
420 MOD
421 SIN
422 ST+ X
423 ST- 04
424 X<> L
425 STO 03
426 RCL 09
427 +
428 SIN
429 +
430 ST+ X
431 +
432 E3
433 ST/ 04
434 /
435 ST+ 03
436 XEQ "N"
437 END
( 557 bytes / SIZE 016 )
Example:
( 2100/0101 0h TD ) ( T = 0.1 )
XEQ "MO"
>>>>
10h38m03s ( right ascension
) =
R06
( execution time = 63s )
RDN
9°47'60" ( declination
)
= R07
RDN
0°59'01" ( Moon's
parallax ) = R05
and R03 =
157.404° ( geocentric longitude )
R04
=
1.092° ( geocentric
latitude )
Mercury-Venus-Mars-Pluto
001 LBL "ME" All the perturbations are
neglected in the motion of Mercury.
002
CLX
If you want to take the most important one into account,
003 STO 03 add
the following instructions after line 005:
004 STO 04
63563 RCL 00 * 87 + COS ST+ X STO
03
005 STO 11 and
delete line 003
006 .3871
007 STO 05
008 18
009 RCL 00
010 STO Z
011 2
012 *
013 -
014 *
015 7005
016 +
017 STO 07
018 CLX
019 3
020 %
021 1494726.751
022 +
023 *
024 250.205
025 +
026 STO 08
027 CLX
028 20
029 *
030 20563
031 +
032 STO 06
033 CLX
034 30
035 *
036 15564
037 +
038 *
039 77456
040 +
041 STO 09
042 CLX
043 18
044 *
045 11861
046 +
047 *
048 48331
049 GTO 14 ( synthetic )
050 LBL "VE"
051 XEQ "O"
052 RCL 12
053 SIN
054 2
055 SQRT
056 *
057 RCL 12
058 3
059 *
060 SIN
061 ST+ X
062 -
063 RCL 12
064 ST+ X
065 SIN
066 PI
067 *
068 -
069 STO 03
070 .72333
071 STO 05
072 RCL 07
073 RCL 00
074 STO Z
075 X^2
076 32
077 /
078 +
079 STO 08
080 CLX
081 48
082 -
083 *
084 677
085 +
086 STO 06
087 CLX
088 18
089 +
090 *
091 CHS
092 2337
093 +
094 *
095 6
096 *
097 131564
098 +
099 STO 09
100 CLX
101 10
102 *
103 3395
104 +
105 STO 07
106 CLX
107 41
108 *
109 9010
110 +
111 *
112 76680
113 GTO 14 ( synthetic )
114 LBL "MA"
115 XEQ "O"
116 5
117 P-R
118 3
119 *
120 -
121 RCL 10
122 SIN
123 ST+ X
124 RCL 00
125 *
126 +
127 RCL 09
128 RCL 05
129 -
130 STO 13
131 SIN
132 7
133 *
134 -
135 RCL 13
136 RCL 05
137 -
138 5
139 P-R
140 2
141 /
142 -
143 +
144 RCL 07
145 ST+ X
146 RCL 13
147 7
148 *
149 -
150 RCL 05
151 +
152 RCL 06
153 6
154 *
155 -
156 3.6
157 P-R
158 -
159 +
160 RCL 08
161 RCL 09
162 -
163 STO 14
164 LASTX
165 -
166 STO 15
167 3
168 P-R
169 -
170 -
171 RCL 13
172 ST+ X
173 SIN
174 4
175 *
176 +
177 RCL 14
178 RCL 15
179 +
180 SIN
181 RCL 14
182 SIN
183 -
184 ST+ X
185 +
186 RCL 09
187 3
188 *
189 RCL 07
190 -
191 COS
192 RCL 03
193 2
194 /
195 RCL 10
196 ST+ X
197 -
198 COS
199 +
200 RCL 10
201 RCL 09
202 -
203 COS
204 +
205 2
206 SQRT
207 *
208 -
209 STO 03
210 1.52368
211 STO 05
212 RCL 09
213 RCL 00
214 STO Z
215 X^2
216 32
217 /
218 +
219 STO 08
220 RDN
221 CHS
222 90
223 +
224 *
225 9340
226 +
227 STO 06
228 CLX
229 13
230 *
231 18410
232 +
233 *
234 23940
235 -
236 STO 09
237 CLX
238 6
239 -
240 *
241 1850
242 +
243 STO 07
244 SIGN
245 +
246 *
247 ST+ X
248 7720
249 +
250 *
251 49558
252 GTO 14 ( synthetic )
253 LBL "PL"
254 XEQ "K"
255 RCL 05
256 1451.7
257 RCL 00
258 *
259 238.925
260 +
261 STO 08
262 -
263 STO 10
264 SIN
265 7
266 *
267 RCL 08
268 COS
269 4
270 *
271 -
272 RCL 08
273 ST+ X
274 STO 09
275 SIN
276 ST+ X
277 -
278 RCL 06
279 RCL 08
280 -
281 STO 11
282 SIN
283 4
284 *
285 +
286 STO 03
287 RCL 09
288 1
289 P-R
290 RCL 08
291 SIN
292 -
293 ST+ X
294 +
295 RCL 10
296 COS
297 5
298 *
299 +
300 RCL 11
301 COS
302 ST+ X
303 +
304 STO 04
305 RCL 00
306 117
307 *
308 3
309 -
310 RCL 08
311 COS
312 5
313 *
314 -
315 STO 11
316 39.489
317 STO 05
318 24900
319 STO 06
320 17140
321 STO 07
322 13971
323 RCL 00
324 *
325 135925
326 -
327 STO 09
328 246232
329 LBL 14
330 XEQ "L"
331 END
( 531 bytes / SIZE 016 )
Example: if T = 0.1 ( 2100/01/01 0h TD )
1°) Mercury:
XEQ "ME"
>>>>
19h19m18s ( right ascension
)
= R06 ( execution time =
25s )
RDN
-24°18'31" ( declination
)
= R07
RDN
1.38603 AU ( Mercury's distance to the Earth
) = R05
and R03 =
-71.984° ( geocentric longitude
)
R08 = -54.797° (
heliocentric longitude )
R04 =
-2.113° ( geocentric
latitude
)
R09 =
-6.791°
( heliocentric latitude )
R10 = 0.43213 AU ( radius vector )
2°) Venus:
XEQ "VE"
>>>>
21h32m22s ( right ascension
)
= R06 ( execution time =
36s )
RDN
-16°32'22" ( declination
)
= R07
RDN
1.1256 AU ( Venus' distance to
the Earth ) = R05
and R03 =
-39.923° ( geocentric longitude
)
R08 =
19.733° ( heliocentric
longitude )
R04 =
-1.853° ( geocentric
latitude
)
R09 =
-2.876°
( heliocentric latitude )
R10 = 0.7252 AU ( radius vector )
3°) Mars:
XEQ "MA"
>>>>
1h48m30s ( right ascension
)
= R06 ( execution time =
43s )
RDN
12°11'29" ( declination
)
= R07
RDN
0.8699 AU ( Mars' distance to the Earth
) = R05
and R03 =
29.535° ( geocentric longitude
)
R08 =
67.577° ( heliocentric
longitude )
R04 =
0.950° ( geocentric
latitude
)
R09 =
0.548°
( heliocentric latitude )
R10 = 1.5095 AU ( radius vector )
4°) Pluto:
XEQ "PL"
>>>>
2h23m50s ( right ascension
)
= R06 ( execution time =
33s )
RDN
-3°37'43" ( declination
)
= R07
RDN
48.576 AU ( Pluto's distance to the Earth
) = R05
and R03 =
32.394° ( geocentric longitude
)
R08 =
33.511° ( heliocentric
longitude )
R04 =
-16.921° ( geocentric
latitude
)
R09 =
-16.794°
( heliocentric latitude )
R10 = 48.934 AU ( radius
vector )
Jupiter-Neptune
001 LBL "JU"
002 XEQ "K"
003 X^2
004 P-R
005 5
006 SQRT
007 /
008 +
009 RCL 09
010 RCL 00
011 ST+ X
012 P-R
013 5
014 LN
015 *
016 -
017 -
018 RCL 00
019 *
020 RCL 09
021 37
022 P-R
023 3.4
024 *
025 -
026 +
027 RCL 05
028 RCL 09
029 +
030 STO 08
031 12
032 P-R
033 6
034 /
035 +
036 -
037 RCL 11
038 6
039 SQRT
040 P-R
041 +
042 +
043 RCL 12
044 2
045 P-R
046 +
047 +
048 RCL 09
049 RCL 05
050 -
051 STO 07
052 PI
053 P-R
054 ST+ X
055 -
056 +
057 RCL 03
058 COS
059 ST+ X
060 +
061 RCL 00
062 *
063 RCL 09
064 325
065 P-R
066 9
067 D-R
068 *
069 +
070 -
071 RCL 10
072 ST+ X
073 STO 14
074 SIN
075 56
076 *
077 +
078 RCL 08
079 3
080 P-R
081 14
082 *
083 +
084 +
085 RCL 11
086 36
087 P-R
088 6
089 /
090 -
091 +
092 RCL 12
093 12
094 P-R
095 5
096 LN
097 *
098 -
099 +
100 RCL 10
101 SIN
102 22
103 *
104 -
105 RCL 07
106 15
107 P-R
108 5
109 SQRT
110 /
111 +
112 -
113 RCL 11
114 ST+ X
115 2
116 P-R
117 ST+ X
118 +
119 +
120 RCL 10
121 RCL 14
122 +
123 SIN
124 5
125 *
126 +
127 RCL 10
128 RCL 12
129 +
130 2
131 P-R
132 ST+ X
133 -
134 -
135 RCL 06
136 7
137 SQRT
138 P-R
139 +
140 +
141 RCL 05
142 RCL 14
143 +
144 SIN
145 RCL 12
146 RCL 14
147 +
148 COS
149 +
150 RCL 13
151 SIN
152 +
153 RCL 03
154 SIN
155 +
156 3
157 *
158 +
159 RCL 11
160 RCL 06
161 -
162 2
163 P-R
164 -
165 -
166 STO 03
167 RCL 08
168 SIN
169 ST+ X
170 RCL 14
171 COS
172 3
173 *
174 -
175 RCL 12
176 SIN
177 -
178 RCL 07
179 COS
180 -
181 RCL 10
182 COS
183 +
184 STO 04
185 RCL 08
186 COS
187 5
188 *
189 RCL 07
190 COS
191 PI
192 *
193 -
194 STO 11
195 1303
196 RCL 00
197 STO Z
198 55
199 *
200 -
201 STO 07
202 CLX
203 45
204 /
205 *
206 RCL 05
207 +
208 STO 08
209 CLX
210 5.203
211 STO 05
212 INT
213 CHS
214 *
215 163
216 +
217 *
218 4849
219 +
220 STO 06
221 CLX
222 4
223 *
224 CHS
225 103
226 +
227 *
228 16126
229 +
230 *
231 14331
232 +
233 STO 09
234 CLX
235 40
236 *
237 10210
238 +
239 *
240 100464
241 XEQ "L"
242 RTN
243 LBL "NE"
244 XEQ "K"
245 RCL 15
246 583
247 P-R
248 18
249 /
250 -
251 RCL 15
252 3
253 RCL 00
254 *
255 P-R
256 3
257 *
258 -
259 -
260 RCL 14
261 3
262 -
263 SIN
264 71
265 *
266 +
267 RCL 15
268 ST+ X
269 24
270 P-R
271 6
272 /
273 +
274 +
275 RCL 14
276 RCL 15
277 +
278 STO 13
279 SIN
280 22
281 *
282 +
283 RCL 05
284 RCL 08
285 -
286 STO 10
287 SIN
288 9
289 *
290 +
291 RCL 15
292 RCL 08
293 -
294 STO 12
295 4
296 P-R
297 +
298 +
299 RCL 06
300 RCL 08
301 -
302 STO 09
303 SIN
304 5
305 *
306 +
307 RCL 14
308 ST+ X
309 STO 11
310 SIN
311 RCL 13
312 RCL 15
313 +
314 SIN
315 -
316 ST+ X
317 -
318 STO 03
319 RCL 15
320 COS
321 8
322 *
323 RCL 14
324 1
325 P-R
326 17
327 *
328 +
329 -
330 RCL 10
331 COS
332 RCL 13
333 COS
334 -
335 5
336 *
337 +
338 RCL 09
339 COS
340 3
341 *
342 +
343 RCL 12
344 1
345 P-R
346 -
347 -
348 RCL 11
349 COS
350 +
351 STO 04
352 RCL 14
353 6
354 P-R
355 +
356 CHS
357 RCL 12
358 5
359 P-R
360 .7
361 /
362 -
363 -
364 STO 11
365 30.07
366 STO 05
367 899
368 RCL 00
369 STO Z
370 6
371 *
372 +
373 STO 06
374 RDN
375 CHS
376 93
377 -
378 *
379 1770
380 +
381 STO 07
382 CLX
383 32
384 /
385 *
386 ST+ 08
387 CLX
388 38
389 *
390 14263
391 +
392 *
393 48124
394 +
395 STO 09
396 CLX
397 26
398 *
399 11022
400 +
401 *
402 131784
403 XEQ "L"
404 END
( 522 bytes / SIZE 016 )
Example: T =
0.1 ( 2100/01/01 0h TD )
1°) Jupiter:
XEQ "JU"
>>>>
13h20m20s ( right ascension
)
= R06 ( execution time =
53s )
RDN
-7°05'09" ( declination
)
= R07
RDN
5.546 AU ( Jupiter's distance
to the Earth ) = R05
and R03 =
-158.788° ( geocentric longitude
)
R08 = 190.996° (
heliocentric longitude )
R04 =
1.278° ( geocentric
latitude
)
R09 =
1.300°
( heliocentric latitude )
R10 = 5.451 AU ( radius vector )
2°) Neptune:
XEQ "NE"
>>>>
11h14m44s ( right ascension
)
= R06 ( execution time =
45s )
RDN
5°54'25" ( declination
)
= R07
RDN
29.808 AU ( Neptune's distance
to the Earth ) = R05
and R03 =
167.288° ( geocentric longitude
)
R08 = 165.575° (
heliocentric longitude )
R04 =
0.964° ( geocentric
latitude
)
R09 =
0.951°
( heliocentric latitude )
R10 = 30.210 AU ( radius vector )
Saturn
001 LBL "SA"
002 XEQ "K"
003 P-R
004 LASTX
005 LN
006 *
007 -
008 RCL 00
009 *
010 RCL 09
011 61
012 P-R
013 5
014 SQRT
015 /
016 +
017 -
018 RCL 11
019 ST+ X
020 STO 08
021 COS
022 9
023 *
024 +
025 RCL 09
026 RCL 06
027 -
028 STO 14
029 4
030 P-R
031 .8
032 /
033 -
034 +
035 RCL 03
036 SIN
037 ST+ X
038 +
039 RCL 00
040 *
041 RCL 09
042 91
043 P-R
044 3.41
045 *
046 -
047 -
048 RCL 08
049 61
050 P-R
051 7
052 /
053 -
054 +
055 RCL 11
056 7
057 P-R
058 .8
059 /
060 +
061 -
062 RCL 14
063 21
064 P-R
065 5
066 SQRT
067 /
068 +
069 -
070 RCL 11
071 RCL 06
072 -
073 STO 15
074 SIN
075 RCL 13
076 COS
077 -
078 RCL 14
079 RCL 06
080 -
081 COS
082 -
083 RCL 08
084 RCL 09
085 +
086 SIN
087 -
088 ST+ X
089 +
090 RCL 12
091 2
092 P-R
093 -
094 +
095 RCL 03
096 7
097 SQRT
098 P-R
099 ST+ X
100 -
101 +
102 RCL 00
103 *
104 RCL 09
105 800
106 P-R
107 9
108 D-R
109 *
110 +
111 +
112 RCL 08
113 72
114 P-R
115 3
116 *
117 +
118 -
119 RCL 11
120 117
121 P-R
122 32
123 SQRT
124 /
125 -
126 -
127 RCL 14
128 9
129 P-R
130 5
131 *
132 -
133 -
134 RCL 15
135 6
136 P-R
137 ST+ X
138 -
139 +
140 RCL 12
141 9
142 P-R
143 PI
144 /
145 +
146 -
147 RCL 10
148 2
149 P-R
150 4
151 *
152 +
153 +
154 RCL 10
155 ST+ X
156 SIN
157 9
158 *
159 -
160 RCL 13
161 SIN
162 8
163 *
164 -
165 RCL 03
166 7
167 P-R
168 3
169 /
170 +
171 -
172 RCL 04
173 COS
174 7
175 *
176 -
177 RCL 04
178 RCL 06
179 +
180 5
181 P-R
182 .6
183 *
184 +
185 +
186 RCL 06
187 RCL 07
188 -
189 ST+ X
190 SIN
191 4
192 *
193 +
194 RCL 05
195 COS
196 RCL 14
197 RCL 06
198 -
199 SIN
200 -
201 RCL 08
202 RCL 09
203 +
204 COS
205 +
206 3
207 *
208 +
209 RCL 03
210 RCL 04
211 -
212 2
213 P-R
214 +
215 +
216 RCL 04
217 RCL 07
218 +
219 COS
220 RCL 06
221 RCL 07
222 -
223 SIN
224 -
225 RCL 05
226 RCL 12
227 +
228 SIN
229 -
230 ST+ X
231 +
232 STO 03
233 RCL 14
234 RCL 00
235 P-R
236 ST+ X
237 -
238 RCL 08
239 RCL 00
240 P-R
241 5
242 *
243 +
244 -
245 RCL 09
246 SIN
247 RCL 00
248 *
249 -
250 RCL 08
251 18
252 P-R
253 3
254 /
255 -
256 -
257 RCL 10
258 COS
259 8
260 *
261 +
262 RCL 11
263 1
264 P-R
265 5
266 *
267 +
268 +
269 RCL 14
270 4
271 P-R
272 6
273 /
274 +
275 -
276 RCL 09
277 COS
278 4
279 *
280 +
281 RCL 10
282 ST+ X
283 COS
284 +
285 RCL 15
286 SIN
287 +
288 STO 04
289 RCL 08
290 8
291 P-R
292 4
293 /
294 RCL 00
295 *
296 +
297 RCL 14
298 4
299 P-R
300 +
301 -
302 RCL 12
303 COS
304 ST+ X
305 -
306 RCL 00
307 *
308 RCL 14
309 9
310 P-R
311 .6
312 /
313 -
314 -
315 RCL 08
316 5
317 P-R
318 3.8
319 *
320 +
321 -
322 RCL 12
323 5
324 P-R
325 .7
326 *
327 -
328 -
329 RCL 09
330 SIN
331 4
332 *
333 +
334 RCL 10
335 COS
336 ST+ X
337 +
338 RCL 15
339 COS
340 3
341 *
342 -
343 STO 11
344 9.543
345 STO 05
346 84
347 RCL 00
348 STO Z
349 5
350 *
351 +
352 *
353 19638
354 +
355 *
356 93057
357 +
358 STO 09
359 CLX
360 5.2
361 %
362 *
363 RCL 06
364 +
365 STO 08
366 CLX
367 6
368 *
369 CHS
370 347
371 -
372 *
373 5553
374 +
375 STO 06
376 RDN
377 ST+ X
378 CHS
379 37
380 -
381 *
382 2489
383 +
384 STO 07
385 CLX
386 6
387 +
388 *
389 ST+ X
390 CHS
391 8771
392 +
393 *
394 113666
395 XEQ "L"
396 END
( 475 bytes / SIZE 016 )
Example: ( T = 0.1
) ( 2100/01/01 0h TD )
XEQ "SA"
>>>>
13h38m36s ( right ascension
)
= R06 ( execution time =
71s )
RDN
-7°38'38" ( declination
)
= R07
RDN
9.875 AU ( Saturn's distance to
the Earth ) = R05
and R03 =
-154.370° ( geocentric longitude
)
R08 = 199.986° (
heliocentric longitude )
R04 =
2.424° ( geocentric
latitude
)
R09 =
2.476°
( heliocentric latitude )
R10 = 9.667 AU ( radius vector
)
Note: Here are the results of 2001 tests over the time-span
1000-3000 ( thanks to my HP-48 )
| interval | maximum error in heliocentric longitude | maximum error in heliocentric latitude |
maximum error in radius vector |
| 1000-1100 | 0.0074° | 0.0061° | 0.0024 AU |
| 1100-1200 | 0.0071° | 0.0062° | 0.0021 AU |
| 1200-1300 | 0.0080° | 0.0054° | 0.0025 AU |
| 1300-1400 | 0.0091° | 0.0050° | 0.0029 AU |
| 1400-1500 | 0.0098° | 0.0053° | 0.0030 AU |
| 1500-1600 | 0.0091° | 0.0057° | 0.0031 AU |
| 1600-1700 | 0.0080° | 0.0062° | 0.0030 AU |
| 1700-1800 | 0.0080° | 0.0059° | 0.0027 AU |
| 1800-1900 | 0.0085° | 0.0052° | 0.0028 AU |
| 1900-2000 | 0.0064° | 0.0051° | 0.0024 AU |
| 2000-2100 | 0.0071° | 0.0048° | 0.0025 AU |
| 2100-2200 | 0.0045° | 0.0048° | 0.0027 AU |
| 2200-2300 | 0.0059° | 0.0047° | 0.0028 AU |
| 2300-2400 | 0.0088° | 0.0049° | 0.0037 AU |
| 2400-2500 | 0.0104° | 0.0043° | 0.0039 AU |
| 2500-2600 | 0.0085° | 0.0049° | 0.0038 AU |
| 2600-2700 | 0.0084° | 0.0045° | 0.0034 AU |
| 2700-2800 | 0.0104° | 0.0047° | 0.0036 AU |
| 2800-2900 | 0.0094° | 0.0045° | 0.0036 AU |
| 2900-3000 | 0.0087° | 0.0046° | 0.0021 AU |
| mean error | 0.0028° | 0.0017° | 0.0010 AU |
Uranus
001 LBL "UR"
002 XEQ "K"
003 RCL 15
004 4
005 P-R
006 .3
007 /
008 -
009 RCL 04
010 RCL 07
011 +
012 STO 11
013 4
014 P-R
015 .7
016 *
017 +
018 +
019 RCL 04
020 3
021 P-R
022 .7
023 /
024 +
025 -
026 RCL 13
027 COS
028 ST+ X
029 +
030 RCL 00
031 *
032 RCL 15
033 861
034 P-R
035 18
036 /
037 -
038 -
039 RCL 14
040 2
041 ST* 08
042 *
043 STO 10
044 210
045 P-R
046 E2
047 /
048 -
049 -
050 RCL 11
051 9
052 P-R
053 .23
054 /
055 +
056 -
057 RCL 08
058 39
059 P-R
060 5.5
061 /
062 +
063 -
064 RCL 04
065 5
066 P-R
067 7
068 *
069 -
070 -
071 RCL 15
072 ST+ X
073 35
074 P-R
075 6
076 /
077 +
078 -
079 RCL 13
080 SIN
081 8
082 *
083 +
084 RCL 05
085 RCL 07
086 -
087 SIN
088 RCL 10
089 RCL 15
090 +
091 STO 09
092 SIN
093 -
094 3
095 *
096 RCL 14
097 RCL 15
098 +
099 SIN
100 -
101 5
102 *
103 +
104 RCL 14
105 RCL 10
106 +
107 STO 06
108 SIN
109 4
110 *
111 +
112 RCL 10
113 RCL 07
114 +
115 STO 12
116 SIN
117 6
118 *
119 RCL 14
120 SIN
121 5
122 *
123 -
124 RCL 07
125 RCL 08
126 +
127 SIN
128 +
129 RCL 10
130 ST+ X
131 SIN
132 +
133 RCL 09
134 RCL 08
135 -
136 SIN
137 -
138 RCL 15
139 RCL 08
140 -
141 SIN
142 +
143 RCL 07
144 SIN
145 -
146 ST+ X
147 +
148 RCL 13
149 RCL 07
150 -
151 3
152 SQRT
153 P-R
154 +
155 -
156 RCL 11
157 RCL 07
158 +
159 STO 13
160 6
161 P-R
162 3
163 /
164 +
165 +
166 STO 03
167 RCL 10
168 COS
169 34
170 *
171 RCL 08
172 1
173 P-R
174 6
175 *
176 -
177 -
178 RCL 11
179 6
180 P-R
181 LASTX
182 /
183 -
184 -
185 RCL 05
186 RCL 07
187 -
188 COS
189 5
190 *
191 +
192 RCL 13
193 COS
194 PI
195 *
196 +
197 RCL 09
198 COS
199 RCL 15
200 COS
201 -
202 RCL 12
203 COS
204 -
205 ST+ X
206 +
207 RCL 04
208 SIN
209 -
210 RCL 14
211 COS
212 +
213 RCL 06
214 COS
215 -
216 STO 04
217 RCL 08
218 2
219 P-R
220 .4
221 /
222 -
223 RCL 10
224 COS
225 6
226 *
227 +
228 STO 11
229 19.192
230 STO 05
231 14864
232 RCL 00
233 STO Z
234 21
235 *
236 +
237 *
238 173005
239 +
240 STO 09
241 CLX
242 33
243 /
244 *
245 RCL 07
246 +
247 STO 08
248 CLX
249 27
250 -
251 *
252 4630
253 +
254 STO 06
255 CLX
256 4
257 *
258 8
259 +
260 *
261 773
262 +
263 STO 07
264 CLX
265 18
266 -
267 *
268 CHS
269 134
270 +
271 *
272 5211
273 +
274 *
275 74006
276 XEQ "L"
277 END
( 344 bytes / SIZE 016 )
Example: ( T = 0.1
) ( 2100/01/01 0h TD )
XEQ "UR"
>>>>
1h06m24s ( right ascension
)
= R06 ( execution time =
57s )
RDN
6°22'39" ( declination
)
= R07
RDN
19.824 AU ( Uranus' distance to the
Earth ) = R05
and R03 =
17.739° ( geocentric longitude
)
R08 =
20.540° ( heliocentric
longitude )
R04 =
-0.628° ( geocentric
latitude
)
R09 =
-0.623°
( heliocentric latitude )
R10 = 19.970 AU ( radius vector )
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