double nutation_in_obliquity(date d)
{
double t; // time measured in Julian centuries between given date and the beggingig of the epoch J2000
double da[TOTAL_DELAUNAY_ARGUMENTS]; // array holding Delaunay arguments
double lan; // longitude of lunar ascending node (☊) referred to the mean equinox of date
int i; // loop index variable
double a; // argument of each term of the serie
double n; // result nutation in obliquity value
// computing time since the beginning of the epoch J2000 to the given measured in Julian centuries
t = (julian_date(d) - J2000) / DAYS_IN_JULIAN_CENTURY;
// computing Delaunay arguments as given by semi-analytic lunar theory ELP
compute_delaunay_arguments(t, FULL_SERIES_TOTAL_TERMS, da);
// computing longitude of the lunar ascending node (☊)
// Source: P.K. Seidelman. 1980 IAU Theory of Nutation: The Final Report of the IAU Working Group on Nutation,
// Celestial Mechanics, vol. 27, May 1982, p. 20.
lan = 450160.28 - 6962890.539 * t + 7.455 * t * t + 0.008 * t * t * t;
// computing nutation serie (see top comment for more information)
for (i = 0, n = 0.0; i < TOTAL_NUTATION_TERMS; i++) {
// computing argument of the term of the serie
a = nutation_multipliers[i][0] * da[L];
a += nutation_multipliers[i][1] * da[LP];
a += nutation_multipliers[i][2] * da[F];
a += nutation_multipliers[i][3] * da[D];
a += nutation_multipliers[i][4] * lan;
// converting argument from arcseconds to radians (π = 648000")
a *= M_PI / 648000.0;
// accumulation nutation values
n += (nutation_coefficients[i][3] + nutation_coefficients[i][4] * t) * cos(a);
}
// coefficients are given in 10⁻⁵", converting to degrees (1° = 36000000×10⁻⁵")
n /= 36000000.0;
// converting nutation values from arcseconds to radians (π = 180°)
n *= M_PI / 180.0;
return n;
}
spherical_point geocentric_moon_position(double t)
{
double delaunay_arguments[TOTAL_DELAUNAY_ARGUMENTS]; // Delaunay arguments
double elp2000_arguments[TOTAL_ELP2000_ARGUMENTS]; // ELP2000 arguments
double planetary_arguments[TOTAL_PLANETARY_ARGUMENTS]; // planetary arguments
double zeta; // argument of the precession precession (ζ)
spherical_point sp; // result position of the Moon
// each coordinate (longitude, latitude and radial distance) is computed by adding together results of each serie:
// Main Porblem and all perturbations; then, Moon's mean mean longitude (W₁) must be added to the value of the
// longitude to find the actual position
// computing Delaunay arguments (non reduced)
compute_delaunay_arguments(t, FULL_SERIES_TOTAL_TERMS, delaunay_arguments);
// computing Main Problem
sp.longitude = compute_serie_a_sin(delaunay_arguments, main_problem_longitude_multipliers,
main_problem_longitude_coefficients, TOTAL_MAIN_PROBLEM_LONGITUDE_TERMS);
sp.latitude = compute_serie_a_sin(delaunay_arguments, main_problem_latitude_multipliers,
main_problem_latitude_coefficients, TOTAL_MAIN_PROBLEM_LATITUDE_TERMS);
sp.distance = compute_serie_a_cos(delaunay_arguments, main_problem_distance_multipliers,
main_problem_distance_coefficients, TOTAL_MAIN_PROBLEM_DISTANCE_TERMS);
// recomputing reduced Delaunay arguments
compute_delaunay_arguments(t, LINEAR_SERIES_TOTAL_TERMS, delaunay_arguments);
// computing ELP2000 arguments
compute_elp2000_arguments(t, LINEAR_SERIES_TOTAL_TERMS, elp2000_arguments);
// computing precession argument
zeta = compute_precession_argument(t);
// computing Earth figure perturbations (constant)
sp.longitude += compute_serie_b(zeta, delaunay_arguments, earth_figure_longitude_0_multipliers,
earth_figure_longitude_0_coefficients, TOTAL_EARTH_FIGURE_LONGITUDE_0_TERMS);
sp.latitude += compute_serie_b(zeta, delaunay_arguments, earth_figure_latitude_0_multipliers,
earth_figure_latitude_0_coefficients, TOTAL_EARTH_FIGURE_LATITUDE_0_TERMS);
sp.distance += compute_serie_b(zeta, delaunay_arguments, earth_figure_distance_0_multipliers,
earth_figure_distance_0_coefficients, TOTAL_EARTH_FIGURE_DISTANCE_0_TERMS);
// computing Earth figure perturbations (linear)
sp.longitude += compute_serie_b(zeta, delaunay_arguments, earth_figure_longitude_0_multipliers,
earth_figure_longitude_0_coefficients, TOTAL_EARTH_FIGURE_LONGITUDE_0_TERMS) * t;
sp.latitude += compute_serie_b(zeta, delaunay_arguments, earth_figure_latitude_0_multipliers,
earth_figure_latitude_0_coefficients, TOTAL_EARTH_FIGURE_LATITUDE_0_TERMS) * t;
sp.distance += compute_serie_b(zeta, delaunay_arguments, earth_figure_distance_0_multipliers,
earth_figure_distance_0_coefficients,TOTAL_EARTH_FIGURE_DISTANCE_0_TERMS) * t;
// computing planetary arguments
compute_planetary_arguments(t, planetary_arguments);
// computing planetary 1 perturbations (constant)
sp.longitude += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_longitude_0_multipliers,
planetary1_longitude_0_coefficients, TOTAL_PLANETARY1_LONGITUDE_0_TERMS);
sp.latitude += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_latitude_0_multipliers,
planetary1_latitude_0_coefficients, TOTAL_PLANETARY1_LATITUDE_0_TERMS);
sp.distance += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_distance_0_multipliers,
planetary1_distance_0_coefficients, TOTAL_PLANETARY1_DISTANCE_0_TERMS);
// computing planetary 1 perturbations (linear)
sp.longitude += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_longitude_1_multipliers,
planetary1_longitude_1_coefficients, TOTAL_PLANETARY1_LONGITUDE_1_TERMS) * t;
sp.latitude += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_latitude_1_multipliers,
planetary1_latitude_1_coefficients, TOTAL_PLANETARY1_LATITUDE_1_TERMS) * t;
sp.distance += compute_serie_c(planetary_arguments, delaunay_arguments, planetary1_distance_1_multipliers,
planetary1_distance_1_coefficients, TOTAL_PLANETARY1_DISTANCE_1_TERMS) * t;
// computing planetary 2 perturbations (constant)
sp.longitude += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_longitude_0_multipliers,
planetary2_longitude_0_coefficients, TOTAL_PLANETARY2_LONGITUDE_0_TERMS);
sp.latitude += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_latitude_0_multipliers,
planetary2_latitude_0_coefficients, TOTAL_PLANETARY2_LATITUDE_0_TERMS);
sp.distance += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_distance_0_multipliers,
planetary2_distance_0_coefficients, TOTAL_PLANETARY2_DISTANCE_0_TERMS);
// computing planetary 2 perturbations (linear)
sp.longitude += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_longitude_1_multipliers,
planetary2_longitude_1_coefficients, TOTAL_PLANETARY2_LONGITUDE_1_TERMS) * t;
sp.latitude += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_latitude_1_multipliers,
planetary2_latitude_1_coefficients, TOTAL_PLANETARY2_LATITUDE_1_TERMS) * t;
sp.distance += compute_serie_d(planetary_arguments, delaunay_arguments, planetary2_distance_1_multipliers,
planetary2_distance_1_coefficients, TOTAL_PLANETARY2_DISTANCE_1_TERMS) * t;
// computing tidal effects (constant)
sp.longitude += compute_serie_b(zeta, delaunay_arguments, tidal_longitude_0_multipliers,
tidal_longitude_0_coefficients, TOTAL_TIDAL_LONGITUDE_0_TERMS);
sp.latitude += compute_serie_b(zeta, delaunay_arguments, tidal_latitude_0_multipliers,
tidal_latitude_0_coefficients, TOTAL_TIDAL_LATITUDE_0_TERMS);
sp.distance += compute_serie_b(zeta, delaunay_arguments, tidal_distance_0_multipliers,
tidal_distance_0_coefficients, TOTAL_TIDAL_DISTANCE_0_TERMS);
// computing tidal effects (linear)
sp.longitude += compute_serie_b(zeta, delaunay_arguments, tidal_longitude_1_multipliers,
tidal_longitude_1_coefficients, TOTAL_TIDAL_LONGITUDE_1_TERMS) * t;
sp.latitude += compute_serie_b(zeta, delaunay_arguments, tidal_latitude_1_multipliers,
tidal_latitude_1_coefficients, TOTAL_TIDAL_LATITUDE_1_TERMS) * t;
sp.distance += compute_serie_b(zeta, delaunay_arguments, tidal_distance_1_multipliers,
tidal_distance_1_coefficients, TOTAL_TIDAL_DISTANCE_1_TERMS) * t;
// computing Moon figure perturbations
sp.longitude += compute_serie_b(zeta, delaunay_arguments, moon_figure_longitude_multipliers,
moon_figure_longitude_coefficients, TOTAL_MOON_FIGURE_LONGITUDE_TERMS);
sp.latitude += compute_serie_b(zeta, delaunay_arguments, moon_figure_latitude_multipliers,
moon_figure_latitude_coefficients, TOTAL_MOON_FIGURE_LATITUDE_TERMS);
sp.distance += compute_serie_b(zeta, delaunay_arguments, moon_figure_distance_multipliers,
moon_figure_distance_coefficients, TOTAL_MOON_FIGURE_DISTANCE_TERMS);
// computing relativistic perturbations
sp.longitude += compute_serie_b(zeta, delaunay_arguments, relativistic_longitude_multipliers,
relativistic_longitude_coefficients, TOTAL_RELATIVISTIC_LONGITUDE_TERMS);
sp.latitude += compute_serie_b(zeta, delaunay_arguments, relativistic_latitude_multipliers,
relativistic_latitude_coefficients, TOTAL_RELATIVISTIC_LATITUDE_TERMS);
sp.distance += compute_serie_b(zeta, delaunay_arguments, relativistic_distance_multipliers,
relativistic_distance_coefficients, TOTAL_RELATIVISTIC_DISTANCE_TERMS);
// computing planetary perturbations (solar eccentricity) (quadratic)
sp.longitude += compute_serie_b(zeta, delaunay_arguments, planetary_longitude_2_multipliers,
planetary_longitude_2_coefficients, TOTAL_PLANETARY_LONGITUDE_2_TERMS) * t * t;
sp.latitude += compute_serie_b(zeta, delaunay_arguments, planetary_latitude_2_multipliers,
planetary_latitude_2_coefficients, TOTAL_PLANETARY_LATITUDE_2_TERMS) * t * t;
sp.distance += compute_serie_b(zeta, delaunay_arguments, planetary_distance_2_multipliers,
planetary_distance_2_coefficients, TOTAL_PLANETARY_DISTANCE_2_TERMS) * t * t;
// recomputing full ELP2000 arguments
compute_elp2000_arguments(t, FULL_SERIES_TOTAL_TERMS, elp2000_arguments);
// adding mean mean longitude of the Moon (W₁)
sp.longitude += elp2000_arguments[W1];
return sp;
}
