starfield 0.12.5

Astronomical data reduction toolkit with star catalogs, coordinate systems, and star finding algorithms (inspired by skyfield)
Documentation 
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
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406

//! Nutation computations based on the IAU 2000A model
//!
//! Implements nutation in longitude (delta-psi) and obliquity (delta-epsilon),
//! mean obliquity of the ecliptic, the nutation rotation matrix, and the
//! complementary terms of the equation of equinoxes.
//!
//! Full IAU 2000A model: 678 lunisolar terms + 687 planetary terms.

#[cfg(all(test, feature = "python-tests"))]
mod python_tests;

mod iau2000a_data;

use crate::constants::{ASEC2RAD, J2000, TAU};
use nalgebra::Matrix3;

/// Conversion factor from 0.1 microarcsecond to radians
const TENTH_USEC_2_RAD: f64 = ASEC2RAD / 1e7;

/// Planetary fundamental argument coefficients [constant, rate] in radians
/// From Souchay et al. (1999) and IERS Conventions (2003)
const PLANETARY_FA: [[f64; 2]; 14] = [
    // Moon: l
    [2.35555598, 8328.6914269554],
    // Sun: l'
    [6.24006013, 628.301955],
    // Moon: F
    [1.627905234, 8433.466158131],
    // Moon-Sun: D
    [5.198466741, 7771.3771468121],
    // Moon: Omega
    [2.18243920, -33.757045],
    // Mercury
    [4.402608842, 2608.7903141574],
    // Venus
    [3.176146697, 1021.3285546211],
    // Earth
    [1.753470314, 628.3075849991],
    // Mars
    [6.203480913, 334.0612426700],
    // Jupiter
    [0.599546497, 52.9690962641],
    // Saturn
    [0.874016757, 21.3299104960],
    // Uranus
    [5.481293871, 7.4781598567],
    // Neptune
    [5.321159000, 3.8127774000],
    // General precession in longitude
    [0.02438175, 0.00000538691],
];

/// Fundamental argument polynomial coefficients
///
/// Each row: [constant, t^1, t^2, t^3, t^4] in arcseconds
/// From IERS Conventions (2003), Chapter 5
#[rustfmt::skip]
const FA_COEFFS: [[f64; 5]; 5] = [
    // Mean Anomaly of the Moon (l)
    [485868.249036, 1717915923.2178, 31.8792, 0.051635, -0.00024470],
    // Mean Anomaly of the Sun (l')
    [1287104.79305, 129596581.0481, -0.5532, 0.000136, -0.00001149],
    // Mean Longitude of Moon - Ascending Node (F)
    [335779.526232, 1739527262.8478, -12.7512, -0.001037, 0.00000417],
    // Mean Elongation of Moon from Sun (D)
    [1072260.70369, 1602961601.2090, -6.3706, 0.006593, -0.00003169],
    // Mean Longitude of Ascending Node of Moon (Omega)
    [450160.398036, -6962890.5431, 7.4722, 0.007702, -0.00005939],
];

/// Compute the five fundamental arguments (Delaunay variables) in radians
///
/// # Arguments
/// * `t` - TDB/TT time in Julian centuries since J2000.0
pub fn fundamental_arguments(t: f64) -> [f64; 5] {
    let mut fa = [0.0f64; 5];
    for (i, coeffs) in FA_COEFFS.iter().enumerate() {
        let val = coeffs[0] + (coeffs[1] + (coeffs[2] + (coeffs[3] + coeffs[4] * t) * t) * t) * t;
        fa[i] = val * ASEC2RAD;
    }
    fa
}

/// Compute the 14 fundamental arguments for planetary nutation in radians
///
/// Arguments 0-4: Delaunay variables (l, l', F, D, Omega)
/// Arguments 5-12: Planetary mean longitudes (Mercury through Neptune)
/// Argument 13: General precession in longitude (multiplied by t)
fn planetary_fundamental_arguments(t: f64) -> [f64; 14] {
    let mut fa = [0.0f64; 14];
    for (i, coeffs) in PLANETARY_FA.iter().enumerate() {
        fa[i] = coeffs[0] + coeffs[1] * t;
    }
    // General precession in longitude gets an extra factor of t
    fa[13] *= t;
    fa
}

/// Compute nutation angles (delta-psi, delta-epsilon) in radians
///
/// Full IAU 2000A model: 678 lunisolar + 687 planetary nutation terms.
///
/// # Arguments
/// * `tt_jd` - TT Julian date
///
/// # Returns
/// * `(d_psi, d_eps)` - nutation in longitude and obliquity, in radians
pub fn iau2000a_nutation(tt_jd: f64) -> (f64, f64) {
    let t = (tt_jd - J2000) / 36525.0;
    let fa = fundamental_arguments(t);

    let mut d_psi = 0.0;
    let mut d_eps = 0.0;

    // Lunisolar nutation (678 terms)
    for i in 0..iau2000a_data::NALS_T.len() {
        let mut arg = 0.0;
        for (j, &fa_val) in fa.iter().enumerate() {
            arg += iau2000a_data::NALS_T[i][j] as f64 * fa_val;
        }

        let sin_arg = arg.sin();
        let cos_arg = arg.cos();

        d_psi += (iau2000a_data::LUNISOLAR_LONGITUDE[i][0]
            + iau2000a_data::LUNISOLAR_LONGITUDE[i][1] * t)
            * sin_arg
            + iau2000a_data::LUNISOLAR_LONGITUDE[i][2] * cos_arg;
        d_eps += (iau2000a_data::LUNISOLAR_OBLIQUITY[i][0]
            + iau2000a_data::LUNISOLAR_OBLIQUITY[i][1] * t)
            * cos_arg
            + iau2000a_data::LUNISOLAR_OBLIQUITY[i][2] * sin_arg;
    }

    // Planetary nutation (687 terms)
    let fa14 = planetary_fundamental_arguments(t);
    for i in 0..iau2000a_data::NAPL_T.len() {
        let mut arg = 0.0;
        for (j, &fa_val) in fa14.iter().enumerate() {
            arg += iau2000a_data::NAPL_T[i][j] as f64 * fa_val;
        }

        let sin_arg = arg.sin();
        let cos_arg = arg.cos();

        d_psi += iau2000a_data::PLANETARY_LONGITUDE[i][0] * sin_arg
            + iau2000a_data::PLANETARY_LONGITUDE[i][1] * cos_arg;
        d_eps += iau2000a_data::PLANETARY_OBLIQUITY[i][0] * sin_arg
            + iau2000a_data::PLANETARY_OBLIQUITY[i][1] * cos_arg;
    }

    (d_psi * TENTH_USEC_2_RAD, d_eps * TENTH_USEC_2_RAD)
}

/// Compute the mean obliquity of the ecliptic in radians
///
/// From Capitaine et al. (2003), Astronomy and Astrophysics 412, 567-586.
///
/// # Arguments
/// * `jd_tdb` - TDB Julian date
pub fn mean_obliquity(jd_tdb: f64) -> f64 {
    let t = (jd_tdb - J2000) / 36525.0;

    let epsilon = ((((-0.0000000434 * t - 0.000000576) * t + 0.00200340) * t - 0.0001831) * t
        - 46.836769)
        * t
        + 84381.406;

    epsilon * ASEC2RAD
}

/// Build the nutation rotation matrix
///
/// Matches Skyfield's construction using mean obliquity, true obliquity,
/// and nutation in longitude (d_psi), all as positive angles.
///
/// # Arguments
/// * `mean_obliquity_rad` - mean obliquity of ecliptic in radians
/// * `d_psi` - nutation in longitude in radians
/// * `d_eps` - nutation in obliquity in radians
pub fn build_nutation_matrix(mean_obliquity_rad: f64, d_psi: f64, d_eps: f64) -> Matrix3<f64> {
    let eps_mean = mean_obliquity_rad;
    let eps_true = eps_mean + d_eps;

    let (sobm, cobm) = eps_mean.sin_cos();
    let (sobt, cobt) = eps_true.sin_cos();
    let (spsi, cpsi) = d_psi.sin_cos();

    #[rustfmt::skip]
    let n = Matrix3::new(
        cpsi,          -spsi * cobm,                      -spsi * sobm,
        spsi * cobt,    cpsi * cobm * cobt + sobm * sobt,  cpsi * sobm * cobt - cobm * sobt,
        spsi * sobt,    cpsi * cobm * sobt - sobm * cobt,  cpsi * sobm * sobt + cobm * cobt,
    );

    n
}

/// Number of complementary terms for equation of equinoxes
const N_EQEQ_TERMS: usize = 33;

/// Complementary terms argument multipliers (14 fundamental arguments each)
#[rustfmt::skip]
const KE0: [[i32; 14]; N_EQEQ_TERMS] = [
    [  0,   0,   0,   0,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   0,   0,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,  -2,   3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,  -2,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,  -2,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,   0,   3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,   0,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   0,   0,   3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,   0,   0,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,   0,   0,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   0,   0,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   0,   0,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,   2,  -2,   3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,   2,  -2,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   4,  -4,   4,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   1,  -1,   1,   0,  -8,  12,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,   0,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   2,   0,   3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   2,   0,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,  -2,   0,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,  -2,   2,  -3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,  -2,   2,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   0,   0,   0,   0,   8, -13,   0,   0,   0,   0,   0,  -1],
    [  0,   0,   0,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  2,   0,  -2,   0,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   0,  -2,   1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   1,   2,  -2,   2,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,   0,  -2,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   4,  -2,   4,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  0,   0,   2,  -2,   4,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,  -2,   0,  -3,   0,   0,   0,   0,   0,   0,   0,   0,   0],
    [  1,   0,  -2,   0,  -1,   0,   0,   0,   0,   0,   0,   0,   0,   0],
];

/// Complementary terms sine coefficients (arcseconds)
#[rustfmt::skip]
const SE0_SIN: [f64; N_EQEQ_TERMS] = [
    2.640960e-03, 6.352000e-05, 1.175000e-05, 1.121000e-05, -4.550000e-06,
    2.020000e-06, 1.980000e-06, -1.720000e-06, -1.410000e-06, -1.260000e-06,
    -6.300000e-07, -6.300000e-07, 4.600000e-07, 4.500000e-07, 3.600000e-07,
    -2.400000e-07, 3.200000e-07, 2.800000e-07, 2.700000e-07, 2.600000e-07,
    -2.100000e-07, 1.900000e-07, 1.800000e-07, -1.000000e-07, 1.500000e-07,
    -1.400000e-07, 1.400000e-07, -1.400000e-07, 1.400000e-07, 1.300000e-07,
    -1.100000e-07, 1.100000e-07, 1.100000e-07,
];

/// Complementary terms cosine coefficients (arcseconds)
#[rustfmt::skip]
const SE0_COS: [f64; N_EQEQ_TERMS] = [
    -3.900000e-07, -2.000000e-08, 1.000000e-08, 1.000000e-08, 0.0,
    0.0, 0.0, 0.0, -1.000000e-08, -1.000000e-08,
    0.0, 0.0, 0.0, 0.0, 0.0,
    -1.200000e-07, 0.0, 0.0, 0.0, 0.0,
    0.0, 0.0, 0.0, 5.000000e-08, 0.0,
    0.0, 0.0, 0.0, 0.0, 0.0,
    0.0, 0.0, 0.0,
];

/// Single time-dependent complementary term coefficients
const SE1_SIN: f64 = -0.87e-6;
const SE1_COS: f64 = 0.0;

/// Single time-dependent complementary term argument [l, l', F, D, Omega, ...]
const KE1: [i32; 14] = [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0];

/// Compute the complementary terms of the equation of equinoxes in radians
///
/// From IERS Conventions (2010), Chapter 5, Table 5.2e.
///
/// # Arguments
/// * `tt_jd` - TT Julian date
pub fn equation_of_the_equinoxes_complementary_terms(tt_jd: f64) -> f64 {
    let t = (tt_jd - J2000) / 36525.0;

    // Compute the 14 fundamental arguments
    let mut fa = [0.0f64; 14];

    // Moon and Sun arguments (Delaunay variables)
    fa[0] = (485868.249036
        + (715923.2178 + (31.8792 + (0.051635 + (-0.00024470) * t) * t) * t) * t)
        * ASEC2RAD
        + (1325.0 * t).rem_euclid(1.0) * TAU;

    fa[1] = (1287104.793048
        + (1292581.0481 + (-0.5532 + (0.000136 + (-0.00001149) * t) * t) * t) * t)
        * ASEC2RAD
        + (99.0 * t).rem_euclid(1.0) * TAU;

    fa[2] = (335779.526232
        + (295262.8478 + (-12.7512 + (-0.001037 + (0.00000417) * t) * t) * t) * t)
        * ASEC2RAD
        + (1342.0 * t).rem_euclid(1.0) * TAU;

    fa[3] = (1072260.703692
        + (1105601.2090 + (-6.3706 + (0.006593 + (-0.00003169) * t) * t) * t) * t)
        * ASEC2RAD
        + (1236.0 * t).rem_euclid(1.0) * TAU;

    fa[4] = (450160.398036
        + (-482890.5431 + (7.4722 + (0.007702 + (-0.00005939) * t) * t) * t) * t)
        * ASEC2RAD
        + (-5.0 * t).rem_euclid(1.0) * TAU;

    // Planetary longitudes (Mercury through Neptune)
    fa[5] = 4.402608842 + 2608.7903141574 * t;
    fa[6] = 3.176146697 + 1021.3285546211 * t;
    fa[7] = 1.753470314 + 628.3075849991 * t;
    fa[8] = 6.203480913 + 334.0612426700 * t;
    fa[9] = 0.599546497 + 52.9690962641 * t;
    fa[10] = 0.874016757 + 21.3299104960 * t;
    fa[11] = 5.481293872 + 7.4781598567 * t;
    fa[12] = 5.311886287 + 3.8133035638 * t;

    // General accumulated precession in longitude
    fa[13] = (0.024381750 + 0.00000538691 * t) * t;

    // Reduce to [0, 2π)
    for val in &mut fa {
        *val = val.rem_euclid(TAU);
    }

    // Evaluate the single time-dependent term
    let mut a = 0.0;
    for j in 0..14 {
        a += KE1[j] as f64 * fa[j];
    }
    let mut c_terms = SE1_SIN * a.sin() + SE1_COS * a.cos();
    c_terms *= t;

    // Evaluate the 33 constant terms
    for i in 0..N_EQEQ_TERMS {
        let mut arg = 0.0;
        for j in 0..14 {
            arg += KE0[i][j] as f64 * fa[j];
        }
        c_terms += SE0_SIN[i] * arg.sin() + SE0_COS[i] * arg.cos();
    }

    c_terms * ASEC2RAD
}

#[cfg(test)]
mod tests {
    use super::*;
    use approx::assert_relative_eq;

    #[test]
    fn test_fundamental_arguments_at_j2000() {
        let fa = fundamental_arguments(0.0);
        assert_relative_eq!(fa[0], FA_COEFFS[0][0] * ASEC2RAD, epsilon = 1e-10);
        assert_relative_eq!(fa[4], FA_COEFFS[4][0] * ASEC2RAD, epsilon = 1e-10);
    }

    #[test]
    fn test_nutation_at_j2000() {
        let (d_psi, d_eps) = iau2000a_nutation(J2000);
        let d_psi_asec = d_psi / ASEC2RAD;
        assert!(d_psi_asec.abs() < 20.0, "d_psi = {d_psi_asec} arcseconds");
        let d_eps_asec = d_eps / ASEC2RAD;
        assert!(d_eps_asec.abs() < 15.0, "d_eps = {d_eps_asec} arcseconds");
    }

    #[test]
    fn test_mean_obliquity_at_j2000() {
        let eps = mean_obliquity(J2000);
        let eps_deg = eps.to_degrees();
        assert_relative_eq!(eps_deg, 23.4393, epsilon = 0.001);
    }

    #[test]
    fn test_nutation_matrix_orthogonality() {
        let eps = mean_obliquity(J2000);
        let (d_psi, d_eps) = iau2000a_nutation(J2000);
        let n = build_nutation_matrix(eps, d_psi, d_eps);
        let product = n.transpose() * n;
        for i in 0..3 {
            for j in 0..3 {
                let expected = if i == j { 1.0 } else { 0.0 };
                assert_relative_eq!(product[(i, j)], expected, epsilon = 1e-14);
            }
        }
    }

    #[test]
    fn test_nutation_matrix_determinant() {
        let eps = mean_obliquity(J2000);
        let (d_psi, d_eps) = iau2000a_nutation(J2000);
        let n = build_nutation_matrix(eps, d_psi, d_eps);
        assert_relative_eq!(n.determinant(), 1.0, epsilon = 1e-14);
    }

    #[test]
    fn test_complementary_terms_small() {
        let c = equation_of_the_equinoxes_complementary_terms(J2000);
        let c_asec = c / ASEC2RAD;
        assert!(
            c_asec.abs() < 0.01,
            "complementary terms = {c_asec} arcseconds"
        );
    }
}