Overview
0°) 2 Subroutines: "M" & "K"
1°) Sun+Moon
2°) Planets ( Mercury, Venus, .....
, Pluto , Xena )
3°) Sidereal Time + Sun + Moon + Subroutines
-The following programs calculate the geocentric and heliocentric longitudes
and latitudes of the Sun, the Moon and the major planets,
including Pluto & Xena ( Lilah or Eris or more exactly:
Asteroid
136199 Eris (2003 UB313) )
-The subroutine "K" listed in §0°) leads to coordinates which
are referred to the mean equinox of the date.
-In paragraph 3°) "K" is improved to compute the apparent coordinates:
-The nutation, aberration, time-light and parallax are taken into account
to get the topocentric azimuthal coordinates.
-A small correction has been made to take the new precession formulas into account.
-For the Sun, Mercury & Venus, truncated series from the VSOP87/VSOP09
theories are used.
-For Jupiter, Saturn, Uranus & Neptune, the perturbations come
from the new theory TOP2010.
-They are expressed as functions of a unique variable µ with
µ = 20°599908944 T
-The perturbations of Mars are expressed with µ too.
-For Jupiter and Saturn, these series converge more rapidly than the
usual perturbations expressed with the mean mean longitudes.
-The smallest retained terms are of order of 0°002 ( sometimes
even smaller )
-However, a comparison between DE406 and DE422 also suggests another correction in longitudes of Uranus & Neptune:
dL = -0°0007 T for Uranus and dL = 0°0024T for Neptune. These corrections have been added to the mean longitude.
-The coordinates of Pluto & Xena are obtained by a least-squares
fit of DE406, thanks to my HP48.
-For the Moon, the series are those given by Jean Meeus in his "Astronomical
Algorithms".
-The results are valid at least between 1000 and 3000 for the Sun, the
Moon and the major planets.
-And over more limited time-spans: [1880-2100] for Pluto and [1900-2100]
for the "tenth planet" ( dwarf planet )
• PLUTO
L = 238°927 + 1465°671 T
i = 17°140
a = 39.489
omegabar = -135°925 + 13°971 T
e = 0.249
OMEGA = omegabar + 246°232
-After solving Kepler's equation with these orbital elements, the following corrections are added:
dL = 0°007 Cos ( 28897 T + 65 ) + 0°004 Cos ( 10769 T
+76 ) max error
~ 0°009
dB = -0°004 + 0°086 T
max error ~ 0°0066
between 1880 and 2100
dR = -0.001 + 0.021 T
max error ~ 0.011
• XENA
L = 20°862 + 655.265 T
i = 43°867
a = 68.049
omegabar = OMEGA + 151.210
e = 0.43364
OMEGA = 36.039 + 13.970 T
-After solving Kepler's equation with these orbital elements, the following corrections are added:
dL = 0°014 - 1°175 T + 3°080 T^2
max error ~ 0°0067
dB = -0°124 T + 1°358 T^2 + 26°587 T^3
max error ~ 0°0032
between 1900 and 2100
dR = 0.008 - 0.960 T - 11.605 T^2
max error ~ 0.010
-Other orbital elements ( valid over larger time-spans ) may also be
used for Pluto & Xena, cf paragraph 2°)
Data Registers
-Registers R00 thru R15 are used.
R00 = T = time since 2000/01/01 0h TT is to be initialized before
executing this program (
Unit = 1000 years = 365250 days )
R01, R02 contain the Sun's rectangular ecliptic coordinates.
So, first execute "SUN" if you want to get correct geocentric coordinates.
R03 = geocentric longitude in decimal degrees
R04 = geocentric latitude in decimal degrees
R05 = distance to the Earth in Astronomical Units (
parallax in sexagesimal degrees for the Moon )
R06 = heliocentric longitude in decimal degrees = L
R07 = heliocentric latitude in decimal degrees
= B
R08 = radius vector in Astronomical Units
= R
R09 = semi-major axis
R10 = eccentricity
R11 = inclination over the ecliptic.
R12 = mean longitude
R13 = longitude of the perihelion
R14 = longitude of the ascending node
R03 R04 & R05 are also used for corrections in the longitude,
latitude and radius vector.
R15 = temporary data storage ( actually, R15 is not used
for the Sun, Mercury, Venus, Mars , Pluto and Xena )
>>>> The HP-41 must be set in DEG mode.
0°) 2 Subroutines: "M" & "K"
-After executing "M" with T in register R00, we get:
R03 = 31 µ R06 = 1473
µ R09 = 287 µ
R12 = 208 µ R15 = µ
( R00-R01-R02 are unchanged )
R04 = 8 µ
R07 = 593 µ R10 = 1760
µ R13 = 106 µ
R05 = 4 µ
R08 = 880 µ R11 =
19 µ R14 = 102 µ
-"K" solves Kepler's equation, calculates the heliocentric ecliptic coordinates and finally the geocentric ecliptic longitude, latitude and distance to the Earth.
-Line 59 ( 12 ) is the number of iterations to solve the equation of
Kepler.
-It's only useful for the 10th planet.
-Otherwise, 12 may be replaced by 9.
-Lines 151-152 & 126-127 are not essential and may be deleted.
-Of course, you can add program lines before the END to compute the
right-ascension, declination ...
| 01 LBL "M"
02 102 03 RCL 00 04 20.599909 05 * 06 STO 15 07 * 08 STO 14 09 106 10 LASTX 11 * 12 STO 13 13 + 14 STO 12 15 19 16 RCL 15 17 * 18 STO 11 19 593 20 LASTX 21 * 22 STO 07 23 LASTX 24 880 25 * 26 STO 08 27 + 28 STO 06 29 LASTX 30 RCL 07 31 - |
32 STO 09
33 + 34 STO 10 35 4 36 RCL 15 37 * 38 STO 05 39 ST+ X 40 STO 04 41 LASTX 42 31 43 * 44 STO 03 45 RTN 46 LBL "K" 47 + 48 E3 49 ST/ 03 50 ST/ 04 51 ST/ 05 52 ST/ 11 53 ST/ 12 54 ST/ 13 55 / 56 STO 14 57 E5 58 ST/ 10 59 12 60 STO 08 61 RCL 12 62 RCL 13 |
63 -
64 360 65 MOD 66 ENTER 67 ENTER 68 LBL 10 69 SIN 70 RCL 10 71 R-D 72 * 73 + 74 DSE 08 75 GTO 10 76 2 77 / 78 1 79 RCL 10 80 + 81 1 82 LASTX 83 - 84 / 85 SQRT 86 P-R 87 LASTX 88 / 89 R-P 90 RDN 91 ST+ X 92 1 93 R^ |
94 ST+ X
95 COS 96 RCL 10 97 * 98 - 99 RCL 09 100 * 101 RCL 05 102 + 103 STO 08 104 CLX 105 RCL 14 106 - 107 RCL 13 108 + 109 COS 110 RCL 11 111 LASTX 112 SIN 113 P-R 114 X<>Y 115 ASIN 116 RCL 04 117 + 118 STO 07 119 X<> Z 120 R-P 121 CLX 122 RCL 14 123 + 124 RCL 03 |
125 +
126 360 127 MOD 128 STO 06 129 RCL 08 130 X<>Y 131 RDN 132 P-R 133 R^ 134 X<>Y 135 P-R 136 RCL 02 137 ST+ Z 138 CLX 139 RCL 01 140 + 141 R-P 142 X<>Y 143 RDN 144 R-P 145 R^ 146 X<>Y 147 STO 05 148 X<> Z 149 STO 04 150 X<>Y 151 360 152 MOD 153 STO 03 154 END |
( 206 bytes / SIZE 016 )
1°) Sun + Moon
-To get a better precision and faster results, "SUN" calls "SUN3" listed
in "Astronomical Ephemeris(I).
-Simply replace line 106 in the "SUN3" listing ( 30010 )
by 30009 to take into account the new precession formulae.
| 01 LBL "SUN"
02 XEQ "SUN3" 03 STO 05 04 X<>Y 05 STO 03 06 X<>Y 07 P-R 08 STO 01 09 X<>Y 10 STO 02 11 RCL 05 12 0 13 STO 04 14 RCL 03 15 RTN 16 LBL "MO" 17 RCL 00 18 359990.503 19 RCL 00 20 65 21 / 22 - 23 * 24 2.964 25 - 26 STO 08 27 CLX 28 .188 29 * 30 CHS 31 4452671.114 32 + 33 * 34 68.245 35 - 36 STO 13 37 ST+ X 38 STO 14 39 CLX 40 70 41 / 42 .874 43 + 44 * 45 4771988.675 46 + 47 * 48 128.431 49 + 50 STO 07 51 ST+ X 52 STO 12 53 CLX 54 .365 55 * 56 4832020.175 57 - 58 * 59 86.657 60 - 61 STO 09 62 ST+ X 63 STO 06 64 6036 65 RCL 14 66 RCL 07 67 - 68 STO 03 69 COS 70 58 71 * 72 - 73 RCL 14 74 COS 75 46 76 * 77 - 78 RCL 12 79 COS 80 9 81 * 82 - 83 RCL 14 84 RCL 12 |
85 -
86 STO 15 87 COS 88 RCL 07 89 COS 90 82 91 * 92 - 93 4 94 * 95 + 96 RCL 14 97 RCL 08 98 - 99 STO 11 100 COS 101 RCL 14 102 RCL 07 103 + 104 STO 10 105 COS 106 + 107 3 108 * 109 - 110 1 111 % 112 1/X 113 ASIN 114 HMS 115 STO 05 116 RCL 07 117 RCL 09 118 - 119 SIN 120 281 121 * 122 RCL 07 123 RCL 09 124 + 125 SIN 126 278 127 * 128 + 129 RCL 14 130 RCL 09 131 + 132 SIN 133 173 134 * 135 + 136 RCL 03 137 RCL 09 138 - 139 SIN 140 55 141 * 142 + 143 RCL 03 144 RCL 09 145 + 146 SIN 147 46 148 * 149 + 150 RCL 14 151 RCL 09 152 - 153 SIN 154 33 155 * 156 + 157 RCL 12 158 RCL 09 159 - 160 SIN 161 17 162 * 163 + 164 RCL 10 165 RCL 09 166 + 167 SIN 168 RCL 12 |
169 RCL 09
170 + 171 SIN 172 + 173 9 174 * 175 + 176 RCL 11 177 RCL 09 178 + 179 SIN 180 RCL 09 181 SIN 182 641 183 * 184 - 185 8 186 * 187 + 188 RCL 15 189 RCL 09 190 + 191 SIN 192 RCL 10 193 RCL 09 194 - 195 SIN 196 + 197 4 198 * 199 + 200 RCL 14 201 RCL 08 202 + 203 RCL 09 204 + 205 SIN 206 PI 207 * 208 - 209 RCL 11 210 RCL 09 211 - 212 STO 04 213 RCL 07 214 - 215 SIN 216 LASTX 217 RCL 06 218 + 219 SIN 220 + 221 RCL 04 222 SIN 223 + 224 RCL 09 225 RCL 07 226 - 227 STO 04 228 RCL 08 229 + 230 SIN 231 - 232 RCL 04 233 RCL 14 234 ST+ X 235 + 236 SIN 237 + 238 RCL 08 239 RCL 09 240 - 241 SIN 242 - 243 RCL 06 244 RCL 09 245 + 246 SIN 247 + 248 ST+ X 249 + 250 STO 04 251 RCL 07 252 SIN |
253 6289
254 * 255 RCL 14 256 SIN 257 658 258 * 259 + 260 RCL 12 261 SIN 262 214 263 * 264 + 265 RCL 06 266 SIN 267 114 268 * 269 + 270 RCL 15 271 SIN 272 59 273 * 274 + 275 RCL 03 276 RCL 08 277 - 278 SIN 279 57 280 * 281 + 282 RCL 10 283 SIN 284 53 285 * 286 + 287 RCL 11 288 SIN 289 46 290 * 291 + 292 RCL 08 293 RCL 07 294 - 295 SIN 296 41 297 * 298 - 299 RCL 03 300 SIN 301 98 302 * 303 RCL 07 304 RCL 06 305 - 306 SIN 307 - 308 13 309 * 310 + 311 RCL 07 312 RCL 06 313 + 314 SIN 315 RCL 14 316 ST+ X 317 RCL 07 318 - 319 SIN 320 + 321 11 322 * 323 + 324 RCL 03 325 ST+ X 326 SIN 327 9 328 * 329 + 330 RCL 03 331 RCL 08 332 + 333 SIN 334 8 335 * 336 - |
337 RCL 14
338 RCL 08 339 + 340 SIN 341 7 342 * 343 - 344 RCL 08 345 SIN 346 RCL 00 347 37 348 - 349 * 350 RCL 13 351 SIN 352 7 353 * 354 - 355 RCL 07 356 RCL 08 357 + 358 SIN 359 6 360 * 361 - 362 RCL 14 363 RCL 06 364 + 365 SIN 366 3 367 * 368 + 369 RCL 07 370 RCL 12 371 + 372 SIN 373 ST+ X 374 + 375 RCL 13 376 RCL 07 377 - 378 SIN 379 - 380 RCL 13 381 RCL 08 382 + 383 SIN 384 + 385 5 386 * 387 + 388 RCL 10 389 RCL 08 390 - 391 SIN 392 RCL 14 393 RCL 12 394 + 395 SIN 396 + 397 RCL 14 398 ST+ X 399 SIN 400 + 401 RCL 03 402 RCL 12 403 - 404 SIN 405 + 406 RCL 00 407 23 408 * 409 1 410 - 411 R-D 412 SIN 413 - 414 4 415 * 416 + 417 RCL 08 418 RCL 12 419 - 420 SIN |
421 RCL 03
422 RCL 06 423 - 424 SIN 425 + 426 3 427 * 428 - 429 RCL 11 430 RCL 08 431 - 432 SIN 433 LASTX 434 RCL 07 435 - 436 SIN 437 + 438 RCL 11 439 RCL 12 440 - 441 SIN 442 + 443 RCL 13 444 RCL 07 445 + 446 SIN 447 - 448 RCL 12 449 RCL 08 450 + 451 SIN 452 - 453 RCL 08 454 ST+ X 455 SIN 456 - 457 RCL 06 458 RCL 10 459 + 460 SIN 461 - 462 RCL 14 463 RCL 06 464 - 465 SIN 466 - 467 RCL 00 468 ST+ X 469 158 470 - 471 RCL 00 472 * 473 4812678812 474 + 475 RCL 00 476 * 477 211728 478 + 479 E3 480 / 481 360 482 MOD 483 SIN 484 ST+ X 485 ST- 04 486 X<> L 487 STO 03 488 RCL 09 489 + 490 SIN 491 + 492 ST+ X 493 + 494 E3 495 ST/ 04 496 / 497 ST+ 03 498 RCL 05 499 RCL 04 500 RCL 03 501 END |
( 663 bytes / SIZE 016 )
| STACK | INPUTS | OUTPUTS |
| Z | / | distance to the earth ( AU )* |
| Y | / | geocentric latitude ( deg ) |
| X | / | geocentric longitude ( deg ) |
* or parallax for the Moon in ° ' "
Example: T = 1.2
i-e 3200/01/10 0h TT
• XEQ "SUN" >>>>
Long = 288°786
= R03
R01 = X = 0.316992
RDN Lat = 0
= R04 and
R02 = Y = -0.931927
RDN Dist = 0.984364 AU
= R05
• XEQ "MO" or R/S >>>>
Long = 307°037
= R03
RDN Lat = -0°686
= R04
RDN parallax = 1°01'14"
= R05
-The Sun position may also be calculated without calling "SUN3":
-For example, replace lines 01 to 16 by
| 01 LBL "SUN"
02 RCL 00 03 4452671 04 * 05 68 06 - 07 STO 03 08 SIN 09 329645 10 RCL 00 11 * 12 23 13 - 14 COS 15 - 16 202 17 RCL 00 18 * |
19 18
20 - 21 COS 22 - 23 ST+ X 24 RCL 00 25 225184 26 * 27 81 28 + 29 SIN 30 LASTX 31 ST+ X 32 SIN 33 - 34 2 35 SQRT 36 * |
37 +
38 X<> 03 39 COS 40 PI 41 % 42 STO 05 43 RCL 00 44 360007697 45 RCL 00 46 30 47 * 48 + 49 * 50 80026 51 - 52 STO 12 53 CLX 54 42 |
55 +
56 * 57 CHS 58 1671 59 + 60 STO 10 61 SIGN 62 STO 09 63 CLX 64 46 65 * 66 17194 67 + 68 * 69 77063 70 - 71 STO 13 72 CLST |
73 STO 01
74 STO 02 75 STO 04 76 STO 11 77 XEQ "K" 78 RCL 03 79 RCL 05 80 P-R 81 STO 01 82 X<>Y 83 STO 02 84 RCL 05 85 RCL 04 86 RCL 03 87 RTN 88 LBL "MO" 89 . . . |
Same example: T = 1.2
i-e 3200/01/10 0h TT
• XEQ "SUN" >>>>
Long = 288°786
= R03
R01 = X = 0.316992
RDN Lat = 0
= R04 and
R02 = Y = -0.931926
RDN Dist = 0.984362 AU
= R05
Notes:
-The results are slightly less accurate and the routine runs slower,
but it saves bytes !
-Since several arguments also appear to calculate the positions of
Venus and Mars,
one can write another subroutine that would be called by "SUN"
, "VE" , "MA"
-These arguments are: 225184 T + 81 and 202 T - 18
-Another argument ( rounded to 4452671 T - 68 for the Sun ) is also
used by "MO" ...
2°) The Planets
-Of course, this huge program may be split in several parts...
| 01 LBL "ME"
02 RCL 00 03 63563 04 * 05 SIN 06 ST+ X 07 CHS 08 STO 03 09 CLX 10 STO 04 11 STO 05 12 .3871 13 STO 09 14 RCL 00 15 18 16 RCL 00 17 2 18 * 19 - 20 * 21 7005 22 + 23 STO 11 24 CLX 25 30 26 * 27 1494740721 28 + 29 * 30 250205 31 + 32 STO 12 33 CLX 34 20 35 * 36 20563 37 + 38 STO 10 39 CLX 40 30 41 * 42 15563 43 + 44 * 45 77456 46 + 47 STO 13 48 CLX 49 18 50 * 51 11860 52 + 53 * 54 48331 55 XEQ "K" 56 RTN 57 LBL "VE" 58 RCL 00 59 225184 60 * 61 81 62 + 63 STO 03 64 3 65 * 66 SIN 67 ST+ X 68 RCL 03 69 ST+ X 70 SIN 71 PI 72 * 73 - 74 RCL 03 75 SIN 76 2 77 SQRT 78 * 79 - 80 STO 03 81 CLX 82 STO 04 83 STO 05 84 .72333 85 STO 09 86 RCL 00 87 585192129 88 RCL 00 89 31 90 * 91 + 92 * 93 181178 94 + 95 STO 12 96 CLX 97 48 98 - 99 * 100 677 101 + 102 STO 10 103 CLX 104 10 105 * 106 3395 107 + 108 STO 11 109 CLX 110 18 111 + 112 * 113 CHS 114 6 115 * 116 14021 117 + 118 * 119 131564 120 + 121 STO 13 122 CLX 123 41 124 * 125 9010 126 + 127 * 128 76680 129 XEQ "K" 130 RTN 131 LBL "MA" 132 XEQ "M" 133 RCL 15 134 2215 135 * 136 71 137 - 138 COS 139 RCL 15 140 9301 141 * 142 STO 05 143 COS 144 + 145 RCL 05 146 RCL 11 147 - 148 37 149 + 150 COS 151 + 152 2 153 SQRT 154 * 155 RCL 11 156 28 157 * 158 49 159 + 160 COS 161 RCL 03 162 264 163 * 164 STO 05 165 13 166 + 167 COS 168 + 169 RCL 15 170 7077 171 * 172 ST- 05 173 SIN 174 - 175 RCL 15 176 10 177 * 178 STO 04 179 SIN 180 RCL 00 181 * 182 + 183 ST+ X 184 + 185 RCL 09 186 22 187 * 188 RCL 03 189 + 190 ST+ 06 191 12 192 - 193 COS 194 6 195 * 196 - 197 RCL 05 198 21 199 + 200 COS 201 RCL 06 202 ST+ X 203 11 204 + 205 COS 206 - 207 4 208 * 209 + 210 RCL 06 211 48 212 + 213 COS 214 7 215 * 216 + 217 RCL 04 218 18 219 - 220 COS 221 15 222 * 223 + 224 STO 03 225 CLX 226 STO 04 227 STO 05 228 1.5237 229 STO 09 230 RCL 00 231 191416964 232 RCL 00 233 31 234 * 235 + 236 * 237 4830 238 - 239 STO 12 240 RDN 241 CHS 242 90 243 + 244 * 245 9340 246 + |
247 STO 10
248 CLX 249 6 250 - 251 * 252 1850 253 + 254 STO 11 255 CLX 256 13 257 * 258 18409 259 + 260 * 261 23940 262 - 263 STO 13 264 CLX 265 SIGN 266 + 267 * 268 ST+ X 269 7719 270 + 271 * 272 49558 273 XEQ "K" 274 RTN 275 LBL "JU" 276 XEQ "M" 277 RCL 11 278 49 279 - 280 SIN 281 PI 282 * 283 RCL 00 284 * 285 RCL 09 286 COS 287 ST+ X 288 - 289 RCL 06 290 RCL 11 291 - 292 STO 15 293 23 294 - 295 COS 296 5 297 * 298 + 299 RCL 11 300 32 301 - 302 COS 303 41 304 * 305 + 306 RCL 00 307 * 308 RCL 06 309 39 310 - 311 COS 312 2 313 SQRT 314 * 315 + 316 RCL 07 317 ST+ X 318 67 319 + 320 COS 321 ST+ X 322 - 323 RCL 04 324 RCL 05 325 + 326 SIN 327 RCL 14 328 3 329 * 330 76 331 + 332 COS 333 + 334 RCL 11 335 ST+ X 336 75 337 + 338 COS 339 + 340 RCL 06 341 RCL 15 342 + 343 30 344 - 345 SIN 346 + 347 RCL 06 348 RCL 10 349 + 350 11 351 - 352 SIN 353 + 354 3 355 * 356 + 357 RCL 07 358 COS 359 RCL 09 360 RCL 10 361 + 362 69 363 - 364 COS 365 + 366 4 367 * 368 + 369 RCL 09 370 ST+ X 371 65 372 - 373 SIN 374 RCL 08 375 RCL 10 376 + 377 62 378 - 379 SIN 380 + 381 5 382 * 383 + 384 RCL 06 385 RCL 11 386 + 387 STO 04 388 12 389 + 390 SIN 391 16 392 * 393 - 394 RCL 08 395 73 396 + 397 COS 398 22 399 * 400 + 401 RCL 10 402 RCL 07 403 - 404 41 405 + 406 SIN 407 23 408 * 409 - 410 RCL 09 411 75 412 - 413 SIN 414 37 415 * 416 + 417 RCL 15 418 61 419 - 420 SIN 421 42 422 * 423 + 424 RCL 10 425 33 426 - 427 SIN 428 56 429 * 430 + 431 RCL 11 432 7.3 433 - 434 SIN 435 329 436 * 437 - 438 STO 03 439 RCL 04 440 18 441 + 442 COS 443 4 444 * 445 RCL 15 446 63 447 - 448 SIN 449 5 450 * 451 + 452 STO 04 453 RCL 15 454 29 455 + 456 SIN 457 ST+ X 458 RCL 10 459 57 460 + 461 SIN 462 3 463 * 464 + 465 CHS 466 STO 05 467 RCL 00 468 1303 469 RCL 00 470 55 471 * 472 - 473 STO 11 474 CLX 475 22 476 * 477 30363027 478 + 479 * 480 34310 481 + 482 STO 12 483 CLX 484 5.203 485 STO 09 486 INT 487 CHS 488 * 489 163 490 + 491 * 492 4849 |
493 +
494 STO 10 495 CLX 496 4 497 * 498 CHS 499 103 500 + 501 * 502 16126 503 + 504 * 505 14331 506 + 507 STO 13 508 CLX 509 40 510 * 511 10209 512 + 513 * 514 100464 515 XEQ "K" 516 RTN 517 LBL "SA" 518 XEQ "M" 519 RCL 03 520 48 521 - 522 SIN 523 RCL 11 524 ST+ X 525 STO 13 526 39 527 - 528 COS 529 - 530 RCL 10 531 RCL 07 532 - 533 STO 05 534 30 535 - 536 COS 537 - 538 ST+ X 539 RCL 09 540 COS 541 6 542 * 543 + 544 RCL 11 545 41 546 + 547 COS 548 RCL 00 549 * 550 RCL 07 551 RCL 11 552 + 553 4 554 - 555 SIN 556 - 557 8 558 * 559 + 560 RCL 09 561 ST+ X 562 27 563 - 564 COS 565 21 566 * 567 - 568 RCL 11 569 32 570 - 571 COS 572 101 573 * 574 - 575 RCL 00 576 * 577 RCL 09 578 RCL 11 579 - 580 56 581 + 582 COS 583 RCL 08 584 RCL 10 585 + 586 44 587 + 588 COS 589 + 590 RCL 07 591 RCL 12 592 - 593 STO 14 594 ST+ 14 595 COS 596 - 597 RCL 14 598 RCL 15 599 + 600 78 601 - 602 COS 603 + 604 RCL 12 605 RCL 03 606 - 607 39 608 + 609 SIN 610 - 611 ST+ X 612 + 613 RCL 07 614 RCL 13 615 - 616 36 617 + 618 COS 619 RCL 05 620 RCL 13 621 + 622 STO 04 623 COS 624 - 625 RCL 15 626 7 627 * 628 28 629 - 630 SIN 631 + 632 RCL 07 633 59 634 + 635 COS 636 - 637 RCL 06 638 14 639 + 640 COS 641 + 642 PI 643 * 644 + 645 RCL 14 646 11 647 + 648 SIN 649 RCL 07 650 RCL 03 651 - 652 COS 653 + 654 6 655 * 656 - 657 RCL 03 658 21 659 - 660 COS 661 RCL 13 662 15 663 - 664 SIN 665 + 666 RCL 03 667 RCL 11 668 - 669 SIN 670 - 671 RCL 08 672 30 673 - 674 COS 675 + 676 8 677 * 678 + 679 RCL 10 680 54 681 + 682 COS 683 RCL 05 684 25 685 + 686 COS 687 + 688 9 689 * 690 + 691 RCL 09 692 RCL 11 693 + 694 STO 06 695 2 696 - 697 SIN 698 14 699 * 700 + 701 RCL 07 702 RCL 11 703 + 704 STO 14 705 50 706 - 707 SIN 708 45 709 * 710 + 711 RCL 09 712 14 713 + 714 COS 715 118 716 * 717 + 718 RCL 07 719 RCL 11 720 - 721 30 722 + 723 COS 724 228 725 * 726 + 727 RCL 11 728 7.3 729 - 730 SIN 731 810 732 * 733 + 734 STO 03 735 RCL 06 736 76 737 - 738 COS |
739 RCL 08
740 48 741 - 742 COS 743 + 744 3 745 * 746 RCL 04 747 67 748 + 749 SIN 750 ST+ X 751 - 752 RCL 11 753 SIN 754 RCL 09 755 ST+ X 756 STO 13 757 46 758 + 759 SIN 760 RCL 00 761 * 762 - 763 4 764 * 765 + 766 RCL 05 767 28 768 - 769 COS 770 6 771 * 772 + 773 RCL 14 774 20 775 + 776 COS 777 18 778 * 779 - 780 RCL 13 781 33 782 + 783 COS 784 20 785 * 786 + 787 STO 04 788 RCL 09 789 73 790 - 791 COS 792 6 793 * 794 RCL 11 795 COS 796 RCL 14 797 50 798 - 799 COS 800 + 801 4 802 * 803 - 804 RCL 08 805 20 806 - 807 COS 808 8 809 * 810 + 811 RCL 13 812 60 813 - 814 COS 815 19 816 * 817 + 818 12 819 - 820 STO 05 821 9.555 822 STO 09 823 RCL 00 824 347 825 RCL 00 826 6 827 * 828 + 829 * 830 CHS 831 5551 832 + 833 STO 10 834 CLX 835 52 836 * 837 12235109 838 + 839 * 840 50061 841 + 842 STO 12 843 RDN 844 ST+ X 845 CHS 846 37 847 - 848 * 849 2489 850 + 851 STO 11 852 CLX 853 5 854 * 855 84 856 + 857 * 858 19637 859 + 860 * 861 93057 862 + 863 STO 13 864 CLX 865 6 866 + 867 * 868 ST+ X 869 CHS 870 8770 871 + 872 * 873 113665 874 XEQ "K" 875 RTN 876 LBL "UR" 877 XEQ "M" 878 RCL 04 879 41 880 - 881 SIN 882 RCL 04 883 RCL 05 884 + 885 54 886 + 887 SIN 888 - 889 ST+ X 890 RCL 12 891 RCL 03 892 - 893 65 894 + 895 COS 896 RCL 03 897 44 898 + 899 COS 900 + 901 8 902 * 903 + 904 RCL 12 905 RCL 05 906 - 907 21 908 + 909 COS 910 10 911 * 912 + 913 RCL 05 914 52 915 - 916 COS 917 27 918 * 919 + 920 RCL 00 921 * 922 RCL 03 923 RCL 05 924 - 925 SIN 926 RCL 03 927 RCL 12 928 + 929 59 930 - 931 COS 932 + 933 RCL 04 934 RCL 12 935 + 936 SIN 937 + 938 RCL 14 939 ST+ X 940 STO 11 941 ST+ X 942 STO 15 943 46 944 - 945 COS 946 + 947 RCL 15 948 RCL 04 949 - 950 50 951 - 952 COS 953 - 954 RCL 12 955 65 956 + 957 COS 958 + 959 ST+ X 960 + 961 RCL 11 962 RCL 04 963 - 964 63 965 - 966 SIN 967 PI 968 * 969 + 970 RCL 11 971 RCL 14 972 + 973 60 974 - 975 COS 976 4 977 * 978 + 979 RCL 14 980 RCL 05 981 - 982 78 983 + 984 SIN |
985 5
986 * 987 - 988 RCL 09 989 RCL 14 990 + 991 RCL 05 992 - 993 STO 10 994 28 995 + 996 COS 997 6 998 * 999 + 1000 RCL 04 1001 RCL 05 1002 + 1003 80 1004 - 1005 COS 1006 7 1007 * 1008 + 1009 RCL 14 1010 10 1011 + 1012 SIN 1013 10 1014 * 1015 - 1016 RCL 11 1017 RCL 12 1018 + 1019 20 1020 - 1021 SIN 1022 12 1023 * 1024 + 1025 RCL 06 1026 RCL 12 1027 - 1028 10 1029 - 1030 COS 1031 RCL 12 1032 RCL 04 1033 - 1034 COS 1035 - 1036 15 1037 * 1038 + 1039 RCL 03 1040 73 1041 + 1042 SIN 1043 RCL 04 1044 39 1045 + 1046 SIN 1047 + 1048 35 1049 * 1050 - 1051 RCL 05 1052 RCL 12 1053 + 1054 STO 07 1055 11 1056 - 1057 COS 1058 39 1059 * 1060 + 1061 RCL 12 1062 RCL 03 1063 - 1064 STO 15 1065 51 1066 - 1067 COS 1068 40 1069 * 1070 + 1071 RCL 11 1072 19 1073 + 1074 SIN 1075 209 1076 * 1077 - 1078 RCL 05 1079 62.2 1080 - 1081 SIN 1082 858 1083 * 1084 + 1085 STO 03 1086 RCL 11 1087 33 1088 + 1089 COS 1090 RCL 07 1091 SIN 1092 - 1093 6 1094 * 1095 RCL 05 1096 39 1097 + 1098 SIN 1099 ST+ X 1100 - 1101 STO 04 1102 RCL 06 1103 RCL 12 1104 - 1105 80 1106 + 1107 COS 1108 5 1109 * 1110 RCL 10 1111 SIN 1112 PI 1113 * 1114 - 1115 RCL 11 1116 RCL 05 1117 - 1118 SIN 1119 ST+ X 1120 - 1121 RCL 07 1122 11 1123 - 1124 SIN 1125 RCL 15 1126 49 1127 - 1128 SIN 1129 + 1130 6 1131 * 1132 + 1133 RCL 11 1134 19 1135 + 1136 COS 1137 34 1138 * 1139 + 1140 26 1141 - 1142 STO 05 1143 19.218 1144 STO 09 1145 RCL 00 1146 8 1147 RCL 00 1148 4 1149 * 1150 + 1151 * 1152 773 1153 + 1154 STO 11 1155 CLX 1156 27 1157 - 1158 * 1159 4630 1160 + 1161 STO 10 1162 CLX 1163 30 1164 * 1165 4298642 1166 + 1167 * 1168 45955 1169 - 1170 STO 12 1171 CLX 1172 21 1173 * 1174 14862 1175 + 1176 * 1177 172993 1178 + 1179 STO 13 1180 RDN 1181 CHS 1182 18 1183 + 1184 * 1185 134 1186 + 1187 * 1188 5210 1189 + 1190 * 1191 74005 1192 XEQ "K" 1193 RTN 1194 LBL "NE" 1195 XEQ "M" 1196 RCL 14 1197 RCL 04 1198 - 1199 61 1200 + 1201 COS 1202 RCL 14 1203 ST+ X 1204 24 1205 + 1206 SIN 1207 - 1208 RCL 14 1209 COS 1210 RCL 05 1211 38 1212 + 1213 SIN 1214 9 1215 * 1216 + 1217 RCL 00 1218 * 1219 - 1220 ST+ X 1221 RCL 07 1222 RCL 13 1223 - 1224 STO 10 1225 16 1226 + 1227 COS 1228 RCL 05 1229 RCL 13 1230 + |
1231 77
1232 - 1233 COS 1234 + 1235 5 1236 * 1237 + 1238 RCL 06 1239 RCL 13 1240 - 1241 STO 08 1242 COS 1243 9 1244 * 1245 + 1246 RCL 14 1247 RCL 05 1248 - 1249 STO 09 1250 11 1251 - 1252 COS 1253 22 1254 * 1255 + 1256 RCL 04 1257 50 1258 - 1259 COS 1260 24 1261 * 1262 + 1263 RCL 14 1264 7 1265 + 1266 SIN 1267 71 1268 * 1269 + 1270 RCL 05 1271 27.8 1272 + 1273 COS 1274 584 1275 * 1276 + 1277 STO 03 1278 RCL 04 1279 RCL 14 1280 + 1281 69 1282 - 1283 SIN 1284 RCL 14 1285 54 1286 + 1287 SIN 1288 - 1289 9 1290 * 1291 STO 04 1292 RCL 09 1293 11 1294 - 1295 SIN 1296 RCL 08 1297 SIN 1298 - 1299 5 1300 * 1301 RCL 10 1302 16 1303 + 1304 SIN 1305 3 1306 * 1307 - 1308 RCL 05 1309 63 1310 - 1311 COS 1312 8 1313 * 1314 + 1315 RCL 14 1316 6 1317 + 1318 COS 1319 17 1320 * 1321 - 1322 41 1323 - 1324 STO 05 1325 30.11 1326 STO 09 1327 RCL 00 1328 899 1329 RCL 00 1330 6 1331 * 1332 + 1333 STO 10 1334 CLX 1335 93 1336 + 1337 * 1338 CHS 1339 1770 1340 + 1341 STO 11 1342 CLX 1343 31 1344 * 1345 2198828 1346 + 1347 * 1348 55654 1349 - 1350 STO 12 1351 CLX 1352 38 1353 * 1354 14260 1355 + 1356 * 1357 48124 1358 + 1359 STO 13 1360 CLX 1361 26 1362 * 1363 11021 1364 + 1365 * 1366 131784 1367 XEQ "K" 1368 RTN 1369 LBL "PL" 1370 RCL 00 1371 28897 1372 * 1373 65 1374 + 1375 COS 1376 7 1377 * 1378 RCL 00 1379 10769 1380 * 1381 76 1382 + 1383 COS 1384 4 1385 * 1386 + 1387 STO 03 1388 RCL 00 1389 86 1390 * 1391 4 1392 - 1393 STO 04 1394 RCL 00 1395 21 1396 * 1397 STO 05 1398 39.489 1399 STO 09 1400 24900 1401 STO 10 1402 17140 1403 STO 11 1404 1465671 1405 RCL 00 1406 * 1407 238927 1408 + 1409 STO 12 1410 RCL 00 1411 13971 1412 * 1413 135925 1414 - 1415 STO 13 1416 246232 1417 XEQ "K" 1418 RTN 1419 LBL "XE" 1420 RCL 00 1421 1358 1422 RCL 00 1423 26587 1424 * 1425 + 1426 * 1427 124 1428 - 1429 * 1430 STO 04 1431 CLX 1432 3080 1433 * 1434 1175 1435 - 1436 * 1437 14 1438 + 1439 STO 03 1440 CLX 1441 11605 1442 * 1443 960 1444 + 1445 * 1446 CHS 1447 8 1448 + 1449 STO 05 1450 CLX 1451 655265 1452 * 1453 20862 1454 + 1455 STO 12 1456 68.049 1457 STO 09 1458 43364 1459 STO 10 1460 503 1461 + 1462 STO 11 1463 RCL 00 1464 13970 1465 * 1466 187249 1467 + 1468 STO 13 1469 208790 1470 XEQ "K" 1471 END |
( 2168 bytes / SIZE 016 )
| STACK | INPUTS | OUTPUTS |
| Z | / | distance to the earth ( AU ) |
| Y | / | geocentric latitude ( deg ) |
| X | / | geocentric longitude ( deg ) |
Example1: If you are using the subroutine
"K" listed in paragraph 0°)
T = 1.2 = R00
i-e 3200/01/10 0h TT
• Mercury
XEQ "ME" >>>> Long =
308°394
= R03
R06 = 41°700 = L
RDN Lat = -0°917
= R04 and
R07 = -2°516 = B
RDN Dist = 0.908313 AU
= R05
R08 = 0.331214 AU = R
• Venus
R/S or XEQ "VE" >>>> Long
= 334°440
= R03
R06 = 51°123
RDN Lat = -1°714
= R04 and
R07 = -2°025
RDN Dist = 0.85506 AU
= R05
R08 = 0.72386 AU
• Mars
R/S or XEQ "MA" >>>> Long
= 343°729
= R03
R06 = 19°233
RDN Lat = -0°963
= R04 and
R07 = -1°176
RDN Dist = 1.6951 AU
= R05
R08 = 1.3878 AU
• Jupiter R/S
or XEQ "JU" >>>> Long =
116°734
= R03
R06 = 115°221
RDN Lat = 0°066
= R04 and
R07 = 0°054
RDN Dist = 4.177 AU
= R05
R08 = 5.154 AU
• Saturn R/S
or XEQ "SA" >>>> Long =
326°093
= R03
R06 = 329°536
RDN Lat = -0°975
= R04 and
R07 = -1°050
RDN Dist = 10.702 AU
= R05
R08 = 9.937 AU
• Uranus R/S or
XEQ "UR" >>>> Long =
66°500
= R03
R06 = 68°428
RDN Lat = -0°172
= R04 and
R07 = -0°165
RDN Dist = 18.940 AU
= R05
R08 = 19.679 AU
• Neptune R/S or XEQ
"NE" >>>> Long = 61°159
= R03
R06 = 62°558
RDN Lat = -1°680
= R04 and
R07 = -1°642
RDN Dist = 29.113 AU
= R05
R08 = 29.785 AU
-For the 2 following planets, store T = 0.1 in R00
( 2100/01/01 0h TT ) and execute "SUN" first
• Pluto R/S
or XEQ "PL" >>>>
Long = 32°402
= R03
R06 = 33°518
RDN Lat = -16°921
= R04 and
R07 = -16°793
RDN Dist = 48.583 AU
= R05
R08 = 48.941 AU
• Xena R/S
or XEQ "XE" >>>>
Long = 43°192
= R03
R06 = 43°761
RDN Lat = 6°180
= R04 and
R07 = 6°141
RDN Dist = 83.419 AU
= R05
R08 = 83.950 AU
Example2: If you are using the subroutine
"K" listed in paragraph 3°) below, see the example at the bottom of
this paragraph.
Notes:
-The following elements may also be used for Pluto & Xena:
• PLUTO
L = 238°927 + 1465°744 T - 0°957 T^2 + 3°457
T^3
i = 17°141 -
0°056 T
a = 39.487 + T/22
omegabar = -135°925 + 13°889 T + 0°030 T^2
e = 0.24885 + 0.00060 T
OMEGA = omegabar + 246°232 -0°383 T
-After solving Kepler's equation with these orbital elements, add the following corrections:
dL = 0°007 Cos ( 28881 T + 64 ) + 0°007 Cos ( 1452 T
- 79 )
max error ~ 0°015
dB = 0
max error ~ 0°007 between
1700 and 2300
dR = 0
max error ~ 0.018
• XENA
L = 21°030 + 659°569 T + 2°408 T^2 - 24°460 T^3
- 55°036 T^4
i = 43°993 +
0°099 T
a = 67.842
omegabar = 187°200 + 13°429 T
e = 0.43846 + 0.00169 T
OMEGA = 35°978 + 13°878 T
-After solving Kepler's equation with these orbital elements, add the following corrections:
dL = 0
max error ~ 0°010
dB = 0
max error ~ 0°006 between
1793 and 2177
dR = 0.025 + 0.036 Cos ( 966 T +177 )
max error ~ 0.010
-The accuracy is lesser but the domain of validity is larger.
>>>> These formulas are used in the listing hereunder, which can
replace lines 1369 to 1471 in the listing above.
| 01 LBL "PL"
02 RCL 00 03 28881 04 * 05 64 06 + 07 COS 08 RCL 00 09 1452 10 * 11 79 12 - 13 COS 14 + 15 7 16 * 17 STO 03 18 CLX 19 STO 04 20 STO 05 21 RCL 00 22 39.487 23 RCL 00 24 22 25 / 26 + |
27 STO 09
28 CLX 29 60 30 * 31 24885 32 + 33 STO 10 34 CLX 35 56 36 * 37 17141 38 X<>Y 39 - 40 STO 11 41 CLX 42 3457 43 * 44 957 45 - 46 * 47 1465744 48 + 49 * 50 238927 51 + 52 STO 12 |
53 CLX
54 30 55 * 56 13889 57 + 58 * 59 135925 60 - 61 STO 13 62 X<>Y 63 383 64 * 65 - 66 246232 67 XEQ "K" 68 RTN 69 LBL "XE" 70 67.842 71 STO 09 72 CLX 73 STO 03 74 STO 04 75 25 76 RCL 00 77 966 78 * |
79 PI
80 - 81 COS 82 36 83 * 84 - 85 STO 05 86 RCL 00 87 24460 88 RCL 00 89 55036 90 * 91 + 92 * 93 CHS 94 2408 95 + 96 * 97 659569 98 + 99 * 100 21030 101 + 102 STO 12 103 CLX 104 169 |
105 *
106 43846 107 + 108 STO 10 109 CLX 110 99 111 * 112 43993 113 + 114 STO 11 115 CLX 116 13429 117 * 118 187200 119 + 120 STO 13 121 CLX 122 13878 123 * 124 35978 125 XEQ "K" 126 END |
( 257 bytes )
Example:
-With T = 0.176 ( i-e 2176/01/02 0h TT ) this variant gives the following ecliptic heliocentric coordinates:
• Pluto L = 109°531
B = -0°969
R = 41.791 AU
• Xena L = 76°411
B = 30°737
R = 61.237 AU
3°) Sidereal Time + Sun + Moon + Subroutines
-The nutation, aberration & time-light and parallax are taken into
account to get the topocentric azimuthal coordinates.
-The geocentric apparent coordinates are referred to the true equinox
of the date.
-For the nutation, however, only the largest term is used.
Data Registers: ( Registers R16 thru R18 are to be initialized before executing these programs )
R00 = T = time since 2000/01/01 0h TT is initialized by
executing "ST"
( Unit = 1000 years = 365250 days )
R01, R02 contain the Sun's rectangular ecliptic coordinates.
R03 = geocentric longitude in decimal degrees
R04 = geocentric latitude in decimal degrees
R05 = distance to the Earth in Astronomical Units ( for
the Moon too )
R06 = heliocentric longitude in decimal degrees = L
R07 = heliocentric latitude in decimal degrees
= B
R08 = radius vector in Astronomical Units
= R
R09 = right-ascension ( hh.mnss )
R10 = declination ( ° ' " )
R11 = Azimuth ( ° ' " ) reckoned clockwise from South.
R12 = height ( ° ' " )
R13 = height ( ° ' " ) corrected for refraction
R14 = elongation from the Sun ( deg )
R15: temp
• R16 = Longitude ( ° ' " )
• R17 = Latitude ( ° ' " )
• R18 = Delta T = TT - UT ( in seconds )
R19 = local true sidereal time ( hh.mnss )
R20 = true obliquity of the ecliptic ( deg )
R21 = nutation in longitude ( deg )
R22 = apparent longitude of the Sun ( deg )
R23 = distance Sun-Earth ( AU )
Flags: F01 F02 F03
SF 01 = the programs return the heliocentric coordinates
L , B , R in registers X , Y , Z respectively
CF 01 & SF 02 = the true geocentric coordinates l
, b , r are in X , Y , Z and the elongation from the Sun is in T-register.
CF 01 & CF 02 = the topocentric coordinates Az , h
, h0 , and the elongation from the Sun are returned in
X , Y , Z , T respectively
CF 03 = the aberration & light-time are taken into
account
SF 03 = no aberration & no light-time correction
Subroutines:
"SUN3" ( cf "Astronomical Ephemeris (I)" ) Replace
line 106 in the "SUN3" listing ( 30010 ) by 30009 to
take into account the new precession formulae.
"J0" or "J1" or "J2" ( cf "Julian & Gregorian Calendars"
)
| 01 LBL "ST"
02 DEG 03 HR 04 STO Z 05 X<>Y 06 XEQ "J0" 07 24 08 * 09 + 10 8766 E3 11 / 12 STO 00 13 .259 14 % 15 24000.51337 16 + 17 * 18 6.66452 19 + 20 + 21 RCL 18 22 315576 E5 23 / 24 RCL 00 25 + 26 STO 00 27 X^2 28 6 29 * 30 1301 31 - 32 RCL 00 33 * 34 E4 35 / 36 23.4393 37 + 38 19341 39 RCL 00 40 * 41 125 42 - 43 STO 20 44 COS 45 391 46 / 47 + 48 COS 49 LASTX 50 X<> 20 51 SIN 52 209 53 / 54 STO 21 55 * 56 RCL 16 57 HR 58 + 59 15 60 / 61 + 62 24 63 MOD 64 HMS 65 STO 19 66 RTN 67 LBL "SUN" 68 XEQ "SUN3" 69 STO 05 70 STO 23 71 X<>Y 72 STO 03 73 X<>Y 74 P-R 75 STO 01 76 X<>Y 77 STO 02 78 CLX 79 STO 04 80 RCL 03 81 176 82 1/X 83 - 84 RCL 21 85 + 86 STO 03 87 STO 22 88 XEQ 08 89 RTN 90 LBL "MO" 91 RCL 00 92 359990.503 93 RCL 00 94 65 95 / 96 - 97 * 98 2.964 99 - 100 STO 08 101 CLX 102 .188 103 * 104 CHS 105 4452671.114 106 + 107 * 108 68.245 109 - 110 STO 13 111 ST+ X 112 STO 14 113 CLX 114 70 115 / 116 .874 117 + 118 * 119 4771988.675 120 + 121 * 122 128.431 123 + 124 STO 07 125 ST+ X 126 STO 12 127 CLX 128 .365 129 * 130 4832020.175 131 - 132 * 133 86.657 134 - 135 STO 09 136 ST+ X 137 STO 06 138 2574 139 RCL 07 140 COS 141 28 142 * 143 RCL 14 144 RCL 07 145 - 146 STO 03 147 COS 148 5 149 * 150 + 151 RCL 14 152 COS 153 4 154 * 155 + |
156 5
157 * 158 - 159 RCL 12 160 COS 161 4 162 * 163 - 164 RCL 14 165 RCL 12 166 - 167 STO 15 168 COS 169 RCL 14 170 RCL 08 171 - 172 STO 11 173 COS 174 - 175 2 176 SQRT 177 * 178 + 179 RCL 07 180 RCL 14 181 + 182 STO 10 183 COS 184 - 185 RCL 03 186 RCL 08 187 - 188 COS 189 - 190 E6 191 / 192 STO 05 193 RCL 07 194 RCL 09 195 - 196 SIN 197 281 198 * 199 RCL 07 200 RCL 09 201 + 202 SIN 203 278 204 * 205 + 206 RCL 14 207 RCL 09 208 + 209 SIN 210 173 211 * 212 + 213 RCL 03 214 RCL 09 215 - 216 SIN 217 55 218 * 219 + 220 RCL 03 221 RCL 09 222 + 223 SIN 224 46 225 * 226 + 227 RCL 14 228 RCL 09 229 - 230 SIN 231 33 232 * 233 + 234 RCL 12 235 RCL 09 236 - 237 SIN 238 17 239 * 240 + 241 RCL 10 242 RCL 09 243 + 244 SIN 245 RCL 12 246 RCL 09 247 + 248 SIN 249 + 250 9 251 * 252 + 253 RCL 11 254 RCL 09 255 + 256 SIN 257 RCL 09 258 SIN 259 641 260 * 261 - 262 8 263 * 264 + 265 RCL 15 266 RCL 09 267 + 268 SIN 269 RCL 10 270 RCL 09 271 - 272 SIN 273 + 274 4 275 * 276 + 277 RCL 14 278 RCL 08 279 + 280 RCL 09 281 + 282 SIN 283 PI 284 * 285 - 286 RCL 11 287 RCL 09 288 - 289 STO 04 290 RCL 07 291 - 292 SIN 293 LASTX 294 RCL 06 295 + 296 SIN 297 + 298 RCL 04 299 SIN 300 + 301 RCL 09 302 RCL 07 303 - 304 STO 04 305 RCL 08 306 + 307 SIN 308 - 309 RCL 04 310 RCL 14 |
311 ST+ X
312 + 313 SIN 314 + 315 RCL 08 316 RCL 09 317 - 318 SIN 319 - 320 RCL 06 321 RCL 09 322 + 323 SIN 324 + 325 ST+ X 326 + 327 STO 04 328 RCL 07 329 SIN 330 6289 331 * 332 RCL 14 333 SIN 334 658 335 * 336 + 337 RCL 12 338 SIN 339 214 340 * 341 + 342 RCL 06 343 SIN 344 114 345 * 346 + 347 RCL 15 348 SIN 349 59 350 * 351 + 352 RCL 03 353 RCL 08 354 - 355 SIN 356 57 357 * 358 + 359 RCL 10 360 SIN 361 53 362 * 363 + 364 RCL 11 365 SIN 366 46 367 * 368 + 369 RCL 08 370 RCL 07 371 - 372 SIN 373 41 374 * 375 - 376 RCL 03 377 SIN 378 98 379 * 380 RCL 07 381 RCL 06 382 - 383 SIN 384 - 385 13 386 * 387 + 388 RCL 07 389 RCL 06 390 + 391 SIN 392 RCL 14 393 ST+ X 394 RCL 07 395 - 396 SIN 397 + 398 11 399 * 400 + 401 RCL 03 402 ST+ X 403 SIN 404 9 405 * 406 + 407 RCL 03 408 RCL 08 409 + 410 SIN 411 8 412 * 413 - 414 RCL 14 415 RCL 08 416 + 417 SIN 418 7 419 * 420 - 421 RCL 08 422 SIN 423 RCL 00 424 37 425 - 426 * 427 RCL 13 428 SIN 429 7 430 * 431 - 432 RCL 07 433 RCL 08 434 + 435 SIN 436 6 437 * 438 - 439 RCL 14 440 RCL 06 441 + 442 SIN 443 3 444 * 445 + 446 RCL 07 447 RCL 12 448 + 449 SIN 450 ST+ X 451 + 452 RCL 13 453 RCL 07 454 - 455 SIN 456 - 457 RCL 13 458 RCL 08 459 + 460 SIN 461 + 462 5 463 * 464 + 465 RCL 10 |
466 RCL 08
467 - 468 SIN 469 RCL 14 470 RCL 12 471 + 472 SIN 473 + 474 RCL 14 475 ST+ X 476 SIN 477 + 478 RCL 03 479 RCL 12 480 - 481 SIN 482 + 483 RCL 00 484 23 485 * 486 1 487 - 488 R-D 489 SIN 490 - 491 4 492 * 493 + 494 RCL 08 495 RCL 12 496 - 497 SIN 498 RCL 03 499 RCL 06 500 - 501 SIN 502 + 503 3 504 * 505 - 506 RCL 11 507 RCL 08 508 - 509 SIN 510 LASTX 511 RCL 07 512 - 513 SIN 514 + 515 RCL 11 516 RCL 12 517 - 518 SIN 519 + 520 RCL 13 521 RCL 07 522 + 523 SIN 524 - 525 RCL 12 526 RCL 08 527 + 528 SIN 529 - 530 RCL 08 531 ST+ X 532 SIN 533 - 534 RCL 06 535 RCL 10 536 + 537 SIN 538 - 539 RCL 14 540 RCL 06 541 - 542 SIN 543 - 544 RCL 00 545 ST+ X 546 158 547 - 548 RCL 00 549 * 550 4812678812 551 + 552 RCL 00 553 * 554 211728 555 + 556 E3 557 / 558 360 559 MOD 560 SIN 561 ST+ X 562 ST- 04 563 X<> L 564 STO 03 565 RCL 09 566 + 567 SIN 568 + 569 ST+ X 570 + 571 E3 572 ST/ 04 573 / 574 RCL 21 575 + 576 ST+ 03 577 XEQ 08 578 RTN 579 GTO "ME" 580 LBL "M" 581 102 582 RCL 00 583 20.599909 584 * 585 STO 15 586 * 587 STO 14 588 106 589 LASTX 590 * 591 STO 13 592 + 593 STO 12 594 19 595 RCL 15 596 * 597 STO 11 598 593 599 LASTX 600 * 601 STO 07 602 LASTX 603 880 604 * 605 STO 08 606 + 607 STO 06 608 LASTX 609 RCL 07 610 - 611 STO 09 612 + 613 STO 10 614 4 615 RCL 15 616 * 617 STO 05 618 ST+ X 619 STO 04 620 LASTX |
621 31
622 * 623 STO 03 624 RTN 625 LBL "K" 626 + 627 E3 628 ST/ 03 629 ST/ 04 630 ST/ 05 631 ST/ 11 632 ST/ 12 633 ST/ 13 634 / 635 STO 14 636 E5 637 ST/ 10 638 12 639 STO 08 640 RCL 12 641 RCL 13 642 - 643 360 644 MOD 645 ENTER 646 ENTER 647 LBL 10 648 SIN 649 RCL 10 650 R-D 651 * 652 + 653 DSE 08 654 GTO 10 655 2 656 / 657 1 658 RCL 10 659 + 660 1 661 LASTX 662 - 663 / 664 SQRT 665 P-R 666 LASTX 667 / 668 R-P 669 RDN 670 ST+ X 671 1 672 R^ 673 ST+ X 674 COS 675 RCL 10 676 * 677 - 678 RCL 09 679 * 680 RCL 05 681 + 682 STO 08 683 CLX 684 RCL 14 685 - 686 RCL 13 687 + 688 COS 689 RCL 11 690 LASTX 691 SIN 692 P-R 693 X<>Y 694 ASIN 695 RCL 04 696 + 697 STO 07 698 X<> Z 699 R-P 700 CLX 701 RCL 14 702 + 703 RCL 03 704 + 705 360 706 MOD 707 STO 06 708 RCL 08 709 RCL 07 710 RCL 06 711 FS? 01 712 RTN 713 RCL 08 714 XEQ 07 715 RCL 02 716 ST+ Z 717 CLX 718 RCL 01 719 + 720 XEQ 06 721 STO 05 722 CLX 723 RCL 21 724 + 725 STO 03 726 X<>Y 727 STO 04 728 FS? 03 729 GTO 08 730 RCL 05 731 X^2 732 RCL 23 733 X^2 734 STO 11 735 - 736 RCL 08 737 X^2 738 + 739 RCL 05 740 ST+ X 741 / 742 RCL 09 743 1 744 RCL 10 745 X^2 746 - 747 * 748 SQRT 749 RCL 08 750 X^2 751 / 752 RCL 11 753 1/X 754 - 755 * 756 RCL 05 757 RCL 11 758 / 759 + 760 176 761 / 762 ST- 03 763 LBL 08 764 RCL 03 765 RCL 22 766 - 767 COS 768 RCL 04 769 COS 770 * 771 ACOS 772 STO 14 773 RCL 05 774 RCL 04 775 RCL 03 |
776 FS? 02
777 RTN 778 RCL 20 779 CHS 780 RCL 04 781 RCL 03 782 XEQ 05 783 X<>Y 784 HMS 785 X<>Y 786 15 787 / 788 24 789 MOD 790 HMS 791 STO 09 792 STO 11 793 X<>Y 794 STO 10 795 STO 12 796 RCL 19 797 HR 798 RCL 11 799 HR 800 - 801 15 802 * 803 RCL 17 804 HR 805 90 806 - 807 RCL 12 808 HR 809 90 810 R^ 811 - 812 XEQ 05 813 90 814 X<>Y 815 - 816 360 817 MOD 818 STO 11 819 X<>Y 820 1 821 P-R 822 STO 13 823 CLX 824 23455 825 RCL 05 826 * 827 1/X 828 RCL 17 829 HR 830 STO 12 831 SIN 832 X^2 833 298.257 834 STO 15 835 / 836 CHS 837 1 838 + 839 * 840 ST- Y 841 RCL 12 842 ST+ X 843 SIN 844 RCL 15 845 / 846 * 847 STO 15 848 RCL 11 849 COS 850 RCL 13 851 * 852 ST+ X 853 - 854 RCL 15 855 * 856 RCL 13 857 X^2 858 + 859 SQRT 860 R-P 861 X<>Y 862 HMS 863 STO 12 864 RCL 11 865 RCL 13 866 P-R 867 RCL 15 868 - 869 R-P 870 RDN 871 HMS 872 STO 11 873 X<>Y 874 HR 875 19.272 876 RCL Y 877 6.942 878 + 879 / 880 + 881 5.459 882 X<>Y 883 / 884 + 885 TAN 886 1/X 887 62.6 888 / 889 X<0? 890 CLX 891 + 892 HMS 893 STO 13 894 RCL 14 895 X<>Y 896 RCL 12 897 RCL 11 898 RTN 899 LBL 05 900 1 901 XEQ 07 902 RDN 903 R-P 904 X<> Z 905 ST- Y 906 X<> Z 907 P-R 908 R^ 909 XEQ 06 910 RDN 911 RTN 912 LBL 06 913 R-P 914 X<>Y 915 RDN 916 R-P 917 R^ 918 X<>Y 919 RTN 920 LBL 07 921 X<>Y 922 RDN 923 P-R 924 R^ 925 X<>Y 926 P-R 927 END |
( 1286 bytes / SIZE 024 )
| STACK | INPUTS | OUTPUTS |
| T | / | Elg = elongation from the Sun ( deg ) |
| Z | / | h0 = apparent altitude ( ° ' '' ) |
| Y | / | h = true altitude ( ° ' '' ) |
| X | / | Az = Azimuth ( ° ' '' ) |
Example: Suppose we are at the US Naval Observatory
at Washington ( D.C. ) on 2100/01/01 7h41m UT , assuming TT-UT
= 203 seconds
• Longitude = 77°03'56"
W = -77°03'56" STO 16
• Latitude =
38°55'17" N = +38°55'17" STO 17
•
DELTA T = TT - UT = 203s
STO 18
CF 01 CF 02 CF 03
• True local sidereal time
2100.0101 ENTER^
7.41
XEQ "ST" >>>> TLST = 9h16m57s3 = R19
-We also have
true obliquity of the ecliptic = 23°4288 = R20
nutation in longitude
= 0°0008 = R21
• The Sun
R/S or XEQ "SUN" >>>>
Az = -105°36'34" = R11
RDN h = -54°33'15" = R12
RDN h0 = -54°33'15" = R13
( when h0 < 0 , h0 is meaningless )
RDN Elg = 0
R03 = l = 280°932
R09 = R.A. = 18h47m33s
R04 = b = 0
R10 = Decl = 22°58'44"
R05 = r = 0.983349 AU
the rectangular coordinates of the Sun: R01 = X = 0.186563
& R02 = Y = -0.965489
apparent longitude of Sun = R22 = 280°932
distance Earth-Sun = R23 = 0.983349 AU
• The Moon:
R/S or XEQ "MO"
>>>> Az = -41°28'13" = R11
RDN h = 51°16'54" = R12
RDN h0 = 51°17'40" = R13
RDN Elg = 119°01 = R14
R03 = l = 161°921
R09 = R.A. = 10h54m23s
R04 = b = 0°699
R10 = Decl = 7°44'04"
R05 = r = 0.002483 AU
• Mercury:
R/S or XEQ "ME"
>>>> Az = -112°29'21" = R11
RDN h = -61°33'32" = R12
RDN h0 = -61°33'32" = R13
RDN Elg = 7°88 = R14
R03 = l = -71°472
R06 = L = 306°251
R09 = R.A. = 19h21m32s
R04 = b = -2°120
R07 = B = -6°822
R10 = Decl = -24°14'53"
R05 = r = 1.38310 AU
R08 = R = 0.43077 AU
• Venus:
R/S or XEQ "VE"
>>>> Az = 169°29'37" = R11
RDN h = -67°11'25" = R12
RDN h0 = -67°11'25" = R13
RDN Elg = 39°56 = R14
R03 = l = -39°542
R06 = L = 20°246
R09 = R.A. = 21h33m52s
R04 = b = -1°845
R07 = B = -2°860
R10 = Decl = -16°24'46"
R05 = r = 1.1236 AU
R08 = R = 0.7252 AU
• Mars:
R/S or XEQ "MA"
>>>> Az = 113°32'57" = R11
RDN h = -8°43'04" = R12
RDN h0 = -8°43'04" = R13
RDN Elg = 108°73 = R14
R03 = l = 29°662
R06 = L = 67°745
R09 = R.A. = 1h48m59s
R04 = b = 0°957
R07 = B = 0°553
R10 = Decl = 12°14'35"
R05 = r = 0.8729 AU
R08 = R = 1.5100 AU
• Jupiter:
R/S or XEQ "JU"
>>>> Az = -65°15'06" = R11
RDN h = 17°20'45" = R12
RDN h0 = 17°23'46" = R13
RDN Elg = 79°69 = R14
R03 = l = -158°757
R06 = L = 191°020
R09 = R.A. = 13h20m27s
R04 = b = 1°277
R07 = B = 1°298
R10 = Decl = -7°05'55"
R05 = r = 5.540 AU
R08 = R = 5.451 AU
• Saturn:
R/S or XEQ "SA"
>>>> Az = -68°05'24" = R11
RDN h = 13°42'40" = R12
RDN h0 = 13°46'29" = R13
RDN Elg = 75°30 = R14
R03 = l = -154°353
R06 = L = 199°997
R09 = R.A. = 13h38m40s
R04 = b = 2°424
R07 = B = 2°475
R10 = Decl = -7°39'01"
R05 = r = 9.868 AU
R08 = R = 9.665 AU
• Uranus:
R/S or XEQ "UR"
>>>> Az = 116°49'08" = R11
RDN h = -20°18'51" = R12
RDN h0 = -20°18'51" = R13
RDN Elg = 96°81 = R14
R03 = l = 17°745
R06 = L = 20°547
R09 = R.A. = 1h06m25s
R04 = b = -0°631
R07 = B = -0°626
R10 = Decl = 6°22'39"
R05 = r = 19.834 AU
R08 = R = 19.974 AU
• Neptune:
R/S or XEQ "NE"
>>>> Az = -46°29'42" = R11
RDN h = 47°36'39" =
R12
RDN h0 = 47°37'32"
= R13
RDN Elg = 113°64 = R14
R03 = l = 167°288
R06 = L = 165°577
R09 = R.A. = 11h14m44s
R04 = b = 0°966
R07 = B = 0°953
R10 = Decl = 5°54'31"
R05 = r = 29.801 AU
R08 = R = 30.209 AU
• Pluto:
R/S or XEQ "PL"
>>>> Az = 95°34'30" = R11
RDN h = -12°35'27" = R12
RDN h0 = -12°35'27" = R13
RDN Elg = 110°50 = R14
R03 = l = 32°402
R06 = L = 33°519
R09 = R.A. = 2h23m51s
R04 = b = -16°919
R07 = B = -16°793
R10 = Decl = -3°37'21"
R05 = r = 48.588 AU
R08 = R = 48.941 AU
• Xena:
R/S or XEQ "XE"
>>>> Az = 113°18'36" = R11
RDN h = 5°45'32" =
R12
RDN h0 = 5°53'56"
= R13
RDN Elg = 122°05 = R14
R03 = l = 43°193
R06 = L = 43°761
R09 = R.A. = 2h34m58s
R04 = b = 6°180
R07 = B = 6°141
R10 = Decl = 21°40'31"
R05 = r = 83.424 AU
R08 = R = 83.950 AU
Notes:
-For Pluto & Xena, the 1st formulas at the top of this page have been used.
-The formulae for the aberration and light-time are only approximate
( lines 730 to 762 ).
-A better method is to compute the true positions of the Sun and the
planet at the instant ( t minus light-time ), but it would be much slower.
-Lines 820 to 873 calculate the correction for parallax ( cf "Parallax
for the HP-41" )
-Lines 874 to 892 may be replaced by XEQ "H-H0" where
"H-H0" is listed in "Astronomical Refraction"
-The LBL 05 LBL 06 LBL 07 ( lines 899 to 927
) are actually the routines "EE" "R-S" "S-R"
listed in "Transformation of Coordinates for the HP-41"
References:
[1] Jean Meeus - "Astronomical Algorithms" - Willmann-Bell
- ISBN 0-943396-61-1
[2] VSOP87D Series ftp://ftp.imcce.fr/pub/ephem/planets/vsop87/
[3] VSOP09 & TOP2010 Series http://www.imcce.fr/~jlsimon
[4] A fantastic software by Aldo Vitagliano: "SOLEX" which may
be downloaded freely from
http://chemistry.unina.it/~alvitagl/solex/