/// <summary>
/// Gets precessional elements to convert equatorial coordinates of a point from one epoch to another.
/// Equatorial coordinates of a point must be reffered to system FK5.
/// </summary>
/// <param name="jd0">Initial epoch, in Julian Days.</param>
/// <param name="jd">Target (final) epoch, in Julian Days.</param>
/// <returns>
/// <see cref="PrecessionalElements"/> to convert equatorial coordinates of a point from
/// one epoch (<paramref name="jd0"/>) to another (<paramref name="jd"/>).
/// </returns>
/// <remarks>
/// This method is taken from AA(I), chapter 20 ("Precession", topic "Rigorous method").
/// </remarks>
public static PrecessionalElements ElementsFK5(double jd0, double jd)
{
PrecessionalElements p = new PrecessionalElements();
p.InitialEpoch = jd0;
p.TargetEpoch = jd;
double T = (jd0 - 2451545.0) / 36525.0;
double t = (jd - jd0) / 36525.0;
double T2 = T * T;
double t2 = t * t;
double t3 = t2 * t;
// all values in seconds of arc
p.zeta = (2306.2181 + 1.39656 * T - 0.000139 * T2) * t
+ (0.30188 - 0.000344 * T) * t2 + 0.017998 * t3;
p.z = (2306.2181 + 1.39656 * T - 0.000139 * T2) * t
+ (1.09468 + 0.000066 * T) * t2 + 0.018203 * t3;
p.theta = (2004.3109 - 0.85330 * T - 0.000217 * T2) * t
- (0.42665 + 0.000217 * T) * t2 - 0.041833 * t3;
// convert to degress
p.zeta /= 3600;
p.z /= 3600;
p.theta /= 3600;
return(p);
}
/// <summary>
/// Performs reduction of equatorial coordinates from one epoch to another
/// with using of precessional elements.
/// </summary>
/// <param name="eq0">Equatorial coordinates for initial epoch.</param>
/// <param name="p">Precessional elements for reduction from initial epoch to target (final) epoch.</param>
/// <returns>Equatorial coordinates for target (final) epoch.</returns>
/// <remarks>
/// This method is taken from AA(I), formula 20.4.
/// </remarks>
public static CrdsEquatorial GetEquatorialCoordinates(CrdsEquatorial eq0, PrecessionalElements p)
{
CrdsEquatorial eq = new CrdsEquatorial();
double sinDelta0 = Math.Sin(Angle.ToRadians(eq0.Delta));
double cosDelta0 = Math.Cos(Angle.ToRadians(eq0.Delta));
double sinTheta = Math.Sin(Angle.ToRadians(p.theta));
double cosTheta = Math.Cos(Angle.ToRadians(p.theta));
double sinAlpha0Zeta = Math.Sin(Angle.ToRadians(eq0.Alpha + p.zeta));
double cosAlpha0Zeta = Math.Cos(Angle.ToRadians(eq0.Alpha + p.zeta));
double A = cosDelta0 * sinAlpha0Zeta;
double B = cosTheta * cosDelta0 * cosAlpha0Zeta - sinTheta * sinDelta0;
double C = sinTheta * cosDelta0 * cosAlpha0Zeta + cosTheta * sinDelta0;
eq.Alpha = Angle.ToDegrees(Math.Atan2(A, B)) + p.z;
eq.Alpha = Angle.To360(eq.Alpha);
if (Math.Abs(C) == 1)
{
eq.Delta = Angle.ToDegrees(Math.Acos(A * A + B * B));
}
else
{
eq.Delta = Angle.ToDegrees(Math.Asin(C));
}
return(eq);
}
/// <summary>
/// Calculates ecliptical coordinates of Triton, largest moon of Neptune.
/// </summary>
/// <param name="jd">Julian Day of calculation</param>
/// <param name="neptune">Ecliptical coordinates of Neptune for the Julian Day specified.</param>
/// <returns>Ecliptical coordinates of Triton for specified date.</returns>
/// <remarks>
///
/// The method is based on following works:
///
/// 1. Harris, A.W. (1984), "Physical Properties of Neptune and Triton Inferred from the Orbit of Triton" NASA CP-2330, pages 357-373:
/// http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1984NASCP2330..357H&defaultprint=YES&filetype=.pdf
///
/// 2. Seidelmann, P. K.: Explanatory Supplement to The Astronomical Almanac,
/// University Science Book, Mill Valley (California), 1992,
/// Chapter 6 "Orbital Ephemerides and Rings of Satellites", page 373, 6.61-1 Triton
/// https://archive.org/download/131123ExplanatorySupplementAstronomicalAlmanac/131123-explanatory-supplement-astronomical-almanac.pdf
///
/// </remarks>
private static CrdsEcliptical TritonPosition(double jd, CrdsEcliptical neptune)
{
NutationElements ne = Nutation.NutationElements(jd);
double epsilon = Date.TrueObliquity(jd, ne.deltaEpsilon);
// convert current coordinates to J1950 epoch, as algorithm requires
CrdsEquatorial eq = neptune.ToEquatorial(epsilon);
PrecessionalElements pe1950 = Precession.ElementsFK5(jd, Date.EPOCH_J1950);
CrdsEquatorial eqNeptune1950 = Precession.GetEquatorialCoordinates(eq, pe1950);
const double t0 = 2433282.5; // 1.0 Jan 1950
const double a = 0.0023683; // semimajor axis of Triton, in a.u.
const double n = 61.2588532; // nodal mean motion, degrees per day
const double lambda0 = 200.913; // longitude from ascending node through the invariable plane at epoch
const double i = 158.996; // inclination of orbit to the invariable plane
const double Omega0 = 151.401; // angle from the intersection of invariable plane with the earth's
// equatorial plane of 1950.0 to the ascending node
// of the orbit through the invariable plane
const double OmegaDot = 0.57806; // nodal precision rate, degrees per year
// Calculate J2000.0 RA and Declination of the pole of the invariable plane
// These formulae are taken from the book:
// Seidelmann, P. K.: Explanatory Supplement to The Astronomical Almanac,
// University Science Book, Mill Valley (California), 1992,
// Chapter 6 "Orbital Ephemerides and Rings of Satellites", page 373, 6.61-1 Triton
double T = (jd - 2451545.0) / 36525.0;
double N = ToRadians(359.28 + 54.308 * T);
double ap = 298.72 + 2.58 * Sin(N) - 0.04 * Sin(2 * N);
double dp = 42.63 - 1.90 * Cos(N) + 0.01 * Cos(2 * N);
// Convert pole coordinates to J1950
CrdsEquatorial eqPole1950 = Precession.GetEquatorialCoordinates(new CrdsEquatorial(ap, dp), pe1950);
ap = eqPole1950.Alpha;
dp = eqPole1950.Delta;
// take light-time effect into account
double tau = PlanetPositions.LightTimeEffect(neptune.Distance);
double lambda = To360(lambda0 + n * (jd - t0 - tau));
double omega = Omega0 + OmegaDot * (jd - t0 - tau) / 365.25;
// cartesian state vector of Triton
var r =
Matrix.R3(ToRadians(-ap - 90)) *
Matrix.R1(ToRadians(dp - 90)) *
Matrix.R3(ToRadians(-omega)) *
Matrix.R1(ToRadians(-i)) *
new Matrix(new[, ] {
{ a *Cos(ToRadians(lambda)) }, { a *Sin(ToRadians(lambda)) }, { 0 }
});
// normalize by distance to Neptune
r.Values[0, 0] /= neptune.Distance;
r.Values[1, 0] /= neptune.Distance;
r.Values[2, 0] /= neptune.Distance;
// offsets vector
var d =
Matrix.R2(ToRadians(-eqNeptune1950.Delta)) *
Matrix.R3(ToRadians(eqNeptune1950.Alpha)) *
r;
// radial component, positive away from observer
// converted to degrees
double x = ToDegrees(d.Values[0, 0]);
// semimajor axis, expressed in degrees, as visible from Earth
double theta = ToDegrees(Atan(a / neptune.Distance));
// offsets values in degrees
double dAlphaCosDelta = ToDegrees(d.Values[1, 0]);
double dDelta = ToDegrees(d.Values[2, 0]);
double delta = eqNeptune1950.Delta + dDelta;
double dAlpha = dAlphaCosDelta / Cos(ToRadians(eqNeptune1950.Delta));
double alpha = eqNeptune1950.Alpha + dAlpha;
CrdsEquatorial eqTriton1950 = new CrdsEquatorial(alpha, delta);
// convert J1950 equatorial coordinates to current epoch
// and to ecliptical
PrecessionalElements pe = Precession.ElementsFK5(Date.EPOCH_J1950, jd);
CrdsEquatorial eqTriton = Precession.GetEquatorialCoordinates(eqTriton1950, pe);
CrdsEcliptical eclTriton = eqTriton.ToEcliptical(epsilon);
// calculate distance to Earth
eclTriton.Distance = neptune.Distance + x / theta * a;
return(eclTriton);
}
/// <summary>
/// Calculates ecliptical coordinates of Nereid, the third-largest moon of Neptune.
/// </summary>
/// <param name="jd">Julian Day of calculation</param>
/// <param name="neptune">Ecliptical coordinates of Neptune for the Julian Day specified.</param>
/// <returns>Ecliptical coordinates of Nereid for specified date.</returns>
/// <remarks>
///
/// The method is based on work of F. Mignard (1981), "The Mean Elements of Nereid",
/// The Astronomical Journal, Vol 86, Number 11, pages 1728-1729
/// The work can be found by link: http://adsabs.harvard.edu/full/1981AJ.....86.1728M
///
/// There are some changes from the original algorithm were made,
/// to be compliant with ephemeris provided by Nasa JPL Horizons system (https://ssd.jpl.nasa.gov/?ephemerides):
///
/// 1. Other value of mean motion (n) is used:
/// - original work : n = 0.999552
/// - implementation: n = 360.0 / 360.1362 (where 360.1362 is an orbital period)
///
/// 2. Rotation around Z axis by angle OmegaN should by taken with NEGATIVE sign,
/// insted of POSITIVE sign in original work (possible typo?),
/// note the NEGATIVE sign for "Ne" angle (same meaning as "OmegaN" in original work) in the book:
/// Seidelmann, P. K.: Explanatory Supplement to The Astronomical Almanac,
/// University Science Book, Mill Valley (California), 1992,
/// Chapter 6 "Orbital Ephemerides and Rings of Satellites", page 376, formula 6.62-3
///
/// </remarks>
private static CrdsEcliptical NereidPosition(double jd, CrdsEcliptical neptune)
{
NutationElements ne = Nutation.NutationElements(jd);
double epsilon = Date.TrueObliquity(jd, ne.deltaEpsilon);
// convert current coordinates to J1950 epoch, as algorithm requires
CrdsEquatorial eq = neptune.ToEquatorial(epsilon);
PrecessionalElements pe1950 = Precession.ElementsFK5(jd, Date.EPOCH_J1950);
CrdsEquatorial eqNeptune1950 = Precession.GetEquatorialCoordinates(eq, pe1950);
const double jd0 = 2433680.5; // Initial Epoch: 3.0 Feb 1951
const double a = 0.036868; // Semi-major axis, in a.u.
const double e0 = 0.74515; // Orbit eccentricity for jd0 epoch
const double i0 = 10.041; // Inclination of the orbit for jd0 epoch, in degrees
const double Omega0 = 329.3; // Longitude of the node of the orbit for jd0 epoch, in degrees
const double M0 = 358.91; // Mean anomaly for jd0 epoch, in degrees
const double n = 360.0 / 360.1362; // Mean motion, in degrees per day
const double OmegaN = 3.552; // Longitude of ascending node of the orbit of Neptune, for J1950.0 epoch, in degrees
const double gamma = 22.313; // Inclination of the orbit of Neptune, for J1950.0 epoch, in degrees
// take light-time effect into account
double tau = PlanetPositions.LightTimeEffect(neptune.Distance);
double t = jd - tau - jd0; // in days
double T = t / 36525.0; // in Julian centuries
double psi = ToRadians(To360(282.9 + 2.68 * T));
double twoTheta = ToRadians(To360(107.4 + 0.01196 * t));
// Equation to found omega, argument of pericenter
Func <double, double> omegaEquation = (om) => To360(282.9 + 2.68 * T - 19.25 * Sin(2 * psi) + 3.23 * Sin(4 * psi) - 0.725 * Sin(6 * psi) - 0.351 * Sin(twoTheta) - 0.7 * Sin(ToRadians(2 * om) - twoTheta)) - om;
// Solve equation (find root: omega value)
double omega = ToRadians(FindRoots(omegaEquation, 0, 360, 1e-8));
// Find longitude of the node
double Omega = Omega0 - 2.4 * T + 19.7 * Sin(2 * psi) - 3.3 * Sin(4 * psi) + 0.7 * Sin(6 * psi) + 0.357 * Sin(twoTheta) + 0.276 * Sin(2 * omega - twoTheta);
// Find orbit eccentricity
double e = e0 - 0.006 * Cos(2 * psi) + 0.0056 * Cos(2 * omega - twoTheta);
// Find mean anomaly
double M = To360(M0 + n * t - 0.38 * Sin(2 * psi) + 1.0 * Sin(2 * omega - twoTheta));
// Find inclination
double cosi = Cos(ToRadians(i0)) - 9.4e-3 * Cos(2 * psi);
double i = Acos(cosi);
// Find eccentric anomaly by solving Kepler equation
double E = SolveKepler(M, e);
double X = a * (Cos(E) - e);
double Y = a * Sqrt(1 - e * e) * Sin(E);
Matrix d =
Matrix.R2(ToRadians(-eqNeptune1950.Delta)) *
Matrix.R3(ToRadians(eqNeptune1950.Alpha)) *
Matrix.R3(ToRadians(-OmegaN)) *
Matrix.R1(ToRadians(-gamma)) *
Matrix.R3(ToRadians(-Omega)) *
Matrix.R1(-i) *
Matrix.R3(-omega) *
new Matrix(new double[, ] {
{ X / neptune.Distance }, { Y / neptune.Distance }, { 0 }
});
// radial component, positive away from observer
// converted to degrees
double x = ToDegrees(d.Values[0, 0]);
// offsets values in degrees
double dAlphaCosDelta = ToDegrees(d.Values[1, 0]);
double dDelta = ToDegrees(d.Values[2, 0]);
double delta = eqNeptune1950.Delta + dDelta;
double dAlpha = dAlphaCosDelta / Cos(ToRadians(eqNeptune1950.Delta));
double alpha = eqNeptune1950.Alpha + dAlpha;
CrdsEquatorial eqNereid1950 = new CrdsEquatorial(alpha, delta);
// convert J1950 equatorial coordinates to current epoch
// and to ecliptical
PrecessionalElements pe = Precession.ElementsFK5(Date.EPOCH_J1950, jd);
CrdsEquatorial eqNereid = Precession.GetEquatorialCoordinates(eqNereid1950, pe);
CrdsEcliptical eclNereid = eqNereid.ToEcliptical(epsilon);
// semimajor axis, expressed in degrees, as visible from Earth
double theta = ToDegrees(Atan(a / neptune.Distance));
// calculate distance to Earth
eclNereid.Distance = neptune.Distance + x / theta * a;
return(eclNereid);
}
public static CrdsEcliptical Position(double jd, GenericSatelliteOrbit orbit, CrdsEcliptical planet)
{
NutationElements ne = Nutation.NutationElements(jd);
double epsilon = Date.TrueObliquity(jd, ne.deltaEpsilon);
// convert current coordinates to epoch, as algorithm requires
CrdsEquatorial eq = planet.ToEquatorial(epsilon);
PrecessionalElements peEpoch = Precession.ElementsFK5(jd, Date.EPOCH_J2000);
CrdsEquatorial eqPlanetEpoch = Precession.GetEquatorialCoordinates(eq, peEpoch);
// ecliptical pole
CrdsEquatorial pole = new CrdsEcliptical(0, 90).ToEquatorial(epsilon);
double distance0;
double distance = planet.Distance;
CrdsEcliptical eclSatellite;
do
{
distance0 = distance;
// take light-time effect into account
double tau = PlanetPositions.LightTimeEffect(distance);
double t = jd - tau - orbit.jd;
double M = To360(orbit.M + orbit.n * t);
double omega = To360(orbit.w + t * 360.0 / (orbit.Pw * 365.25));
double node = To360(orbit.Om + t * 360.0 / (orbit.POm * 365.25));
// Find eccentric anomaly by solving Kepler equation
double E = SolveKepler(M, orbit.e);
double X = orbit.a * (Cos(E) - orbit.e);
double Y = orbit.a * Sqrt(1 - orbit.e * orbit.e) * Sin(E);
// cartesian state vector of satellite
var d =
Matrix.R2(ToRadians(-eqPlanetEpoch.Delta)) *
Matrix.R3(ToRadians(eqPlanetEpoch.Alpha)) *
Matrix.R3(ToRadians(-pole.Alpha - 90)) *
Matrix.R1(ToRadians(pole.Delta - 90)) *
Matrix.R3(ToRadians(-node)) *
Matrix.R1(ToRadians(-orbit.i)) *
Matrix.R3(ToRadians(-omega)) *
new Matrix(new double[, ] {
{ X / distance }, { Y / distance }, { 0 }
});
// radial component, positive away from observer
// converted to degrees
double x = ToDegrees(d.Values[0, 0]);
// semimajor axis, expressed in degrees, as visible from Earth
double theta = ToDegrees(Atan(orbit.a / distance));
// offsets values in degrees
double dAlphaCosDelta = ToDegrees(d.Values[1, 0]);
double dDelta = ToDegrees(d.Values[2, 0]);
double delta = eqPlanetEpoch.Delta + dDelta;
double dAlpha = dAlphaCosDelta / Cos(ToRadians(delta));
double alpha = eqPlanetEpoch.Alpha + dAlpha;
CrdsEquatorial eqSatelliteEpoch = new CrdsEquatorial(alpha, delta);
// convert jd0 equatorial coordinates to current epoch
// and to ecliptical
PrecessionalElements pe = Precession.ElementsFK5(Date.EPOCH_J2000, jd);
CrdsEquatorial eqSatellite = Precession.GetEquatorialCoordinates(eqSatelliteEpoch, pe);
eclSatellite = eqSatellite.ToEcliptical(epsilon);
// calculate distance to Earth
distance = planet.Distance + x / theta * orbit.a;
}while (Abs(distance - distance0) > 1e-6);
eclSatellite.Distance = distance;
return(eclSatellite);
}
