1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
|
#include "erfa.h"
void eraNut00b(double date1, double date2, double *dpsi, double *deps)
/*
** - - - - - - - - - -
** e r a N u t 0 0 b
** - - - - - - - - - -
**
** Nutation, IAU 2000B model.
**
** Given:
** date1,date2 double TT as a 2-part Julian Date (Note 1)
**
** Returned:
** dpsi,deps double nutation, luni-solar + planetary (Note 2)
**
** Notes:
**
** 1) The TT date date1+date2 is a Julian Date, apportioned in any
** convenient way between the two arguments. For example,
** JD(TT)=2450123.7 could be expressed in any of these ways,
** among others:
**
** date1 date2
**
** 2450123.7 0.0 (JD method)
** 2451545.0 -1421.3 (J2000 method)
** 2400000.5 50123.2 (MJD method)
** 2450123.5 0.2 (date & time method)
**
** The JD method is the most natural and convenient to use in
** cases where the loss of several decimal digits of resolution
** is acceptable. The J2000 method is best matched to the way
** the argument is handled internally and will deliver the
** optimum resolution. The MJD method and the date & time methods
** are both good compromises between resolution and convenience.
**
** 2) The nutation components in longitude and obliquity are in radians
** and with respect to the equinox and ecliptic of date. The
** obliquity at J2000.0 is assumed to be the Lieske et al. (1977)
** value of 84381.448 arcsec. (The errors that result from using
** this function with the IAU 2006 value of 84381.406 arcsec can be
** neglected.)
**
** The nutation model consists only of luni-solar terms, but
** includes also a fixed offset which compensates for certain long-
** period planetary terms (Note 7).
**
** 3) This function is an implementation of the IAU 2000B abridged
** nutation model formally adopted by the IAU General Assembly in
** 2000. The function computes the MHB_2000_SHORT luni-solar
** nutation series (Luzum 2001), but without the associated
** corrections for the precession rate adjustments and the offset
** between the GCRS and J2000.0 mean poles.
**
** 4) The full IAU 2000A (MHB2000) nutation model contains nearly 1400
** terms. The IAU 2000B model (McCarthy & Luzum 2003) contains only
** 77 terms, plus additional simplifications, yet still delivers
** results of 1 mas accuracy at present epochs. This combination of
** accuracy and size makes the IAU 2000B abridged nutation model
** suitable for most practical applications.
**
** The function delivers a pole accurate to 1 mas from 1900 to 2100
** (usually better than 1 mas, very occasionally just outside
** 1 mas). The full IAU 2000A model, which is implemented in the
** function eraNut00a (q.v.), delivers considerably greater accuracy
** at current dates; however, to realize this improved accuracy,
** corrections for the essentially unpredictable free-core-nutation
** (FCN) must also be included.
**
** 5) The present function provides classical nutation. The
** MHB_2000_SHORT algorithm, from which it is adapted, deals also
** with (i) the offsets between the GCRS and mean poles and (ii) the
** adjustments in longitude and obliquity due to the changed
** precession rates. These additional functions, namely frame bias
** and precession adjustments, are supported by the ERFA functions
** eraBi00 and eraPr00.
**
** 6) The MHB_2000_SHORT algorithm also provides "total" nutations,
** comprising the arithmetic sum of the frame bias, precession
** adjustments, and nutation (luni-solar + planetary). These total
** nutations can be used in combination with an existing IAU 1976
** precession implementation, such as eraPmat76, to deliver GCRS-
** to-true predictions of mas accuracy at current epochs. However,
** for symmetry with the eraNut00a function (q.v. for the reasons),
** the ERFA functions do not generate the "total nutations"
** directly. Should they be required, they could of course easily
** be generated by calling eraBi00, eraPr00 and the present function
** and adding the results.
**
** 7) The IAU 2000B model includes "planetary bias" terms that are
** fixed in size but compensate for long-period nutations. The
** amplitudes quoted in McCarthy & Luzum (2003), namely
** Dpsi = -1.5835 mas and Depsilon = +1.6339 mas, are optimized for
** the "total nutations" method described in Note 6. The Luzum
** (2001) values used in this ERFA implementation, namely -0.135 mas
** and +0.388 mas, are optimized for the "rigorous" method, where
** frame bias, precession and nutation are applied separately and in
** that order. During the interval 1995-2050, the ERFA
** implementation delivers a maximum error of 1.001 mas (not
** including FCN).
**
** References:
**
** Lieske, J.H., Lederle, T., Fricke, W., Morando, B., "Expressions
** for the precession quantities based upon the IAU /1976/ system of
** astronomical constants", Astron.Astrophys. 58, 1-2, 1-16. (1977)
**
** Luzum, B., private communication, 2001 (Fortran code
** MHB_2000_SHORT)
**
** McCarthy, D.D. & Luzum, B.J., "An abridged model of the
** precession-nutation of the celestial pole", Cel.Mech.Dyn.Astron.
** 85, 37-49 (2003)
**
** Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M.,
** Francou, G., Laskar, J., Astron.Astrophys. 282, 663-683 (1994)
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double t, el, elp, f, d, om, arg, dp, de, sarg, carg,
dpsils, depsls, dpsipl, depspl;
int i;
/* Units of 0.1 microarcsecond to radians */
static const double U2R = ERFA_DAS2R / 1e7;
/* ---------------------------------------- */
/* Fixed offsets in lieu of planetary terms */
/* ---------------------------------------- */
static const double DPPLAN = -0.135 * ERFA_DMAS2R;
static const double DEPLAN = 0.388 * ERFA_DMAS2R;
/* --------------------------------------------------- */
/* Luni-solar nutation: argument and term coefficients */
/* --------------------------------------------------- */
/* The units for the sine and cosine coefficients are */
/* 0.1 microarcsec and the same per Julian century */
static const struct {
int nl,nlp,nf,nd,nom; /* coefficients of l,l',F,D,Om */
double ps,pst,pc; /* longitude sin, t*sin, cos coefficients */
double ec,ect,es; /* obliquity cos, t*cos, sin coefficients */
} x[] = {
/* 1-10 */
{ 0, 0, 0, 0,1,
-172064161.0, -174666.0, 33386.0, 92052331.0, 9086.0, 15377.0},
{ 0, 0, 2,-2,2,
-13170906.0, -1675.0, -13696.0, 5730336.0, -3015.0, -4587.0},
{ 0, 0, 2, 0,2,-2276413.0,-234.0, 2796.0, 978459.0,-485.0,1374.0},
{ 0, 0, 0, 0,2,2074554.0, 207.0, -698.0,-897492.0, 470.0,-291.0},
{ 0, 1, 0, 0,0,1475877.0,-3633.0,11817.0, 73871.0,-184.0,-1924.0},
{ 0, 1, 2,-2,2,-516821.0, 1226.0, -524.0, 224386.0,-677.0,-174.0},
{ 1, 0, 0, 0,0, 711159.0, 73.0, -872.0, -6750.0, 0.0, 358.0},
{ 0, 0, 2, 0,1,-387298.0, -367.0, 380.0, 200728.0, 18.0, 318.0},
{ 1, 0, 2, 0,2,-301461.0, -36.0, 816.0, 129025.0, -63.0, 367.0},
{ 0,-1, 2,-2,2, 215829.0, -494.0, 111.0, -95929.0, 299.0, 132.0},
/* 11-20 */
{ 0, 0, 2,-2,1, 128227.0, 137.0, 181.0, -68982.0, -9.0, 39.0},
{-1, 0, 2, 0,2, 123457.0, 11.0, 19.0, -53311.0, 32.0, -4.0},
{-1, 0, 0, 2,0, 156994.0, 10.0, -168.0, -1235.0, 0.0, 82.0},
{ 1, 0, 0, 0,1, 63110.0, 63.0, 27.0, -33228.0, 0.0, -9.0},
{-1, 0, 0, 0,1, -57976.0, -63.0, -189.0, 31429.0, 0.0, -75.0},
{-1, 0, 2, 2,2, -59641.0, -11.0, 149.0, 25543.0, -11.0, 66.0},
{ 1, 0, 2, 0,1, -51613.0, -42.0, 129.0, 26366.0, 0.0, 78.0},
{-2, 0, 2, 0,1, 45893.0, 50.0, 31.0, -24236.0, -10.0, 20.0},
{ 0, 0, 0, 2,0, 63384.0, 11.0, -150.0, -1220.0, 0.0, 29.0},
{ 0, 0, 2, 2,2, -38571.0, -1.0, 158.0, 16452.0, -11.0, 68.0},
/* 21-30 */
{ 0,-2, 2,-2,2, 32481.0, 0.0, 0.0, -13870.0, 0.0, 0.0},
{-2, 0, 0, 2,0, -47722.0, 0.0, -18.0, 477.0, 0.0, -25.0},
{ 2, 0, 2, 0,2, -31046.0, -1.0, 131.0, 13238.0, -11.0, 59.0},
{ 1, 0, 2,-2,2, 28593.0, 0.0, -1.0, -12338.0, 10.0, -3.0},
{-1, 0, 2, 0,1, 20441.0, 21.0, 10.0, -10758.0, 0.0, -3.0},
{ 2, 0, 0, 0,0, 29243.0, 0.0, -74.0, -609.0, 0.0, 13.0},
{ 0, 0, 2, 0,0, 25887.0, 0.0, -66.0, -550.0, 0.0, 11.0},
{ 0, 1, 0, 0,1, -14053.0, -25.0, 79.0, 8551.0, -2.0, -45.0},
{-1, 0, 0, 2,1, 15164.0, 10.0, 11.0, -8001.0, 0.0, -1.0},
{ 0, 2, 2,-2,2, -15794.0, 72.0, -16.0, 6850.0, -42.0, -5.0},
/* 31-40 */
{ 0, 0,-2, 2,0, 21783.0, 0.0, 13.0, -167.0, 0.0, 13.0},
{ 1, 0, 0,-2,1, -12873.0, -10.0, -37.0, 6953.0, 0.0, -14.0},
{ 0,-1, 0, 0,1, -12654.0, 11.0, 63.0, 6415.0, 0.0, 26.0},
{-1, 0, 2, 2,1, -10204.0, 0.0, 25.0, 5222.0, 0.0, 15.0},
{ 0, 2, 0, 0,0, 16707.0, -85.0, -10.0, 168.0, -1.0, 10.0},
{ 1, 0, 2, 2,2, -7691.0, 0.0, 44.0, 3268.0, 0.0, 19.0},
{-2, 0, 2, 0,0, -11024.0, 0.0, -14.0, 104.0, 0.0, 2.0},
{ 0, 1, 2, 0,2, 7566.0, -21.0, -11.0, -3250.0, 0.0, -5.0},
{ 0, 0, 2, 2,1, -6637.0, -11.0, 25.0, 3353.0, 0.0, 14.0},
{ 0,-1, 2, 0,2, -7141.0, 21.0, 8.0, 3070.0, 0.0, 4.0},
/* 41-50 */
{ 0, 0, 0, 2,1, -6302.0, -11.0, 2.0, 3272.0, 0.0, 4.0},
{ 1, 0, 2,-2,1, 5800.0, 10.0, 2.0, -3045.0, 0.0, -1.0},
{ 2, 0, 2,-2,2, 6443.0, 0.0, -7.0, -2768.0, 0.0, -4.0},
{-2, 0, 0, 2,1, -5774.0, -11.0, -15.0, 3041.0, 0.0, -5.0},
{ 2, 0, 2, 0,1, -5350.0, 0.0, 21.0, 2695.0, 0.0, 12.0},
{ 0,-1, 2,-2,1, -4752.0, -11.0, -3.0, 2719.0, 0.0, -3.0},
{ 0, 0, 0,-2,1, -4940.0, -11.0, -21.0, 2720.0, 0.0, -9.0},
{-1,-1, 0, 2,0, 7350.0, 0.0, -8.0, -51.0, 0.0, 4.0},
{ 2, 0, 0,-2,1, 4065.0, 0.0, 6.0, -2206.0, 0.0, 1.0},
{ 1, 0, 0, 2,0, 6579.0, 0.0, -24.0, -199.0, 0.0, 2.0},
/* 51-60 */
{ 0, 1, 2,-2,1, 3579.0, 0.0, 5.0, -1900.0, 0.0, 1.0},
{ 1,-1, 0, 0,0, 4725.0, 0.0, -6.0, -41.0, 0.0, 3.0},
{-2, 0, 2, 0,2, -3075.0, 0.0, -2.0, 1313.0, 0.0, -1.0},
{ 3, 0, 2, 0,2, -2904.0, 0.0, 15.0, 1233.0, 0.0, 7.0},
{ 0,-1, 0, 2,0, 4348.0, 0.0, -10.0, -81.0, 0.0, 2.0},
{ 1,-1, 2, 0,2, -2878.0, 0.0, 8.0, 1232.0, 0.0, 4.0},
{ 0, 0, 0, 1,0, -4230.0, 0.0, 5.0, -20.0, 0.0, -2.0},
{-1,-1, 2, 2,2, -2819.0, 0.0, 7.0, 1207.0, 0.0, 3.0},
{-1, 0, 2, 0,0, -4056.0, 0.0, 5.0, 40.0, 0.0, -2.0},
{ 0,-1, 2, 2,2, -2647.0, 0.0, 11.0, 1129.0, 0.0, 5.0},
/* 61-70 */
{-2, 0, 0, 0,1, -2294.0, 0.0, -10.0, 1266.0, 0.0, -4.0},
{ 1, 1, 2, 0,2, 2481.0, 0.0, -7.0, -1062.0, 0.0, -3.0},
{ 2, 0, 0, 0,1, 2179.0, 0.0, -2.0, -1129.0, 0.0, -2.0},
{-1, 1, 0, 1,0, 3276.0, 0.0, 1.0, -9.0, 0.0, 0.0},
{ 1, 1, 0, 0,0, -3389.0, 0.0, 5.0, 35.0, 0.0, -2.0},
{ 1, 0, 2, 0,0, 3339.0, 0.0, -13.0, -107.0, 0.0, 1.0},
{-1, 0, 2,-2,1, -1987.0, 0.0, -6.0, 1073.0, 0.0, -2.0},
{ 1, 0, 0, 0,2, -1981.0, 0.0, 0.0, 854.0, 0.0, 0.0},
{-1, 0, 0, 1,0, 4026.0, 0.0, -353.0, -553.0, 0.0,-139.0},
{ 0, 0, 2, 1,2, 1660.0, 0.0, -5.0, -710.0, 0.0, -2.0},
/* 71-77 */
{-1, 0, 2, 4,2, -1521.0, 0.0, 9.0, 647.0, 0.0, 4.0},
{-1, 1, 0, 1,1, 1314.0, 0.0, 0.0, -700.0, 0.0, 0.0},
{ 0,-2, 2,-2,1, -1283.0, 0.0, 0.0, 672.0, 0.0, 0.0},
{ 1, 0, 2, 2,1, -1331.0, 0.0, 8.0, 663.0, 0.0, 4.0},
{-2, 0, 2, 2,2, 1383.0, 0.0, -2.0, -594.0, 0.0, -2.0},
{-1, 0, 0, 0,2, 1405.0, 0.0, 4.0, -610.0, 0.0, 2.0},
{ 1, 1, 2,-2,2, 1290.0, 0.0, 0.0, -556.0, 0.0, 0.0}
};
/* Number of terms in the series */
const int NLS = (int) (sizeof x / sizeof x[0]);
/*--------------------------------------------------------------------*/
/* Interval between fundamental epoch J2000.0 and given date (JC). */
t = ((date1 - ERFA_DJ00) + date2) / ERFA_DJC;
/* --------------------*/
/* LUNI-SOLAR NUTATION */
/* --------------------*/
/* Fundamental (Delaunay) arguments from Simon et al. (1994) */
/* Mean anomaly of the Moon. */
el = fmod(485868.249036 + (1717915923.2178) * t, ERFA_TURNAS) * ERFA_DAS2R;
/* Mean anomaly of the Sun. */
elp = fmod(1287104.79305 + (129596581.0481) * t, ERFA_TURNAS) * ERFA_DAS2R;
/* Mean argument of the latitude of the Moon. */
f = fmod(335779.526232 + (1739527262.8478) * t, ERFA_TURNAS) * ERFA_DAS2R;
/* Mean elongation of the Moon from the Sun. */
d = fmod(1072260.70369 + (1602961601.2090) * t, ERFA_TURNAS) * ERFA_DAS2R;
/* Mean longitude of the ascending node of the Moon. */
om = fmod(450160.398036 + (-6962890.5431) * t, ERFA_TURNAS) * ERFA_DAS2R;
/* Initialize the nutation values. */
dp = 0.0;
de = 0.0;
/* Summation of luni-solar nutation series (smallest terms first). */
for (i = NLS-1; i >= 0; i--) {
/* Argument and functions. */
arg = fmod( (double)x[i].nl * el +
(double)x[i].nlp * elp +
(double)x[i].nf * f +
(double)x[i].nd * d +
(double)x[i].nom * om, ERFA_D2PI );
sarg = sin(arg);
carg = cos(arg);
/* Term. */
dp += (x[i].ps + x[i].pst * t) * sarg + x[i].pc * carg;
de += (x[i].ec + x[i].ect * t) * carg + x[i].es * sarg;
}
/* Convert from 0.1 microarcsec units to radians. */
dpsils = dp * U2R;
depsls = de * U2R;
/* ------------------------------*/
/* IN LIEU OF PLANETARY NUTATION */
/* ------------------------------*/
/* Fixed offset to correct for missing terms in truncated series. */
dpsipl = DPPLAN;
depspl = DEPLAN;
/* --------*/
/* RESULTS */
/* --------*/
/* Add luni-solar and planetary components. */
*dpsi = dpsils + dpsipl;
*deps = depsls + depspl;
return;
}
/*----------------------------------------------------------------------
**
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** All rights reserved.
**
** This library is derived, with permission, from the International
** Astronomical Union's "Standards of Fundamental Astronomy" library,
** available from http://www.iausofa.org.
**
** The ERFA version is intended to retain identical functionality to
** the SOFA library, but made distinct through different function and
** file names, as set out in the SOFA license conditions. The SOFA
** original has a role as a reference standard for the IAU and IERS,
** and consequently redistribution is permitted only in its unaltered
** state. The ERFA version is not subject to this restriction and
** therefore can be included in distributions which do not support the
** concept of "read only" software.
**
** Although the intent is to replicate the SOFA API (other than
** replacement of prefix names) and results (with the exception of
** bugs; any that are discovered will be fixed), SOFA is not
** responsible for any errors found in this version of the library.
**
** If you wish to acknowledge the SOFA heritage, please acknowledge
** that you are using a library derived from SOFA, rather than SOFA
** itself.
**
**
** TERMS AND CONDITIONS
**
** Redistribution and use in source and binary forms, with or without
** modification, are permitted provided that the following conditions
** are met:
**
** 1 Redistributions of source code must retain the above copyright
** notice, this list of conditions and the following disclaimer.
**
** 2 Redistributions in binary form must reproduce the above copyright
** notice, this list of conditions and the following disclaimer in
** the documentation and/or other materials provided with the
** distribution.
**
** 3 Neither the name of the Standards Of Fundamental Astronomy Board,
** the International Astronomical Union nor the names of its
** contributors may be used to endorse or promote products derived
** from this software without specific prior written permission.
**
** THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
** "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
** LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
** FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
** COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
** INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
** BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
** LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
** CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
** LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
** ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
** POSSIBILITY OF SUCH DAMAGE.
**
*/
|
