public static AASSaturnRingDetails Calculate(double JD)
{
//What will be the return value
AASSaturnRingDetails details = new AASSaturnRingDetails();
double T = (JD - 2451545) / 36525;
double T2 = T * T;
//Step 1. Calculate the inclination of the plane of the ring and the longitude of the ascending node referred to the ecliptic and mean equinox of the date
double i = 28.075216 - 0.012998 * T + 0.000004 * T2;
double irad = AASCoordinateTransformation.DegreesToRadians(i);
double omega = 169.508470 + 1.394681 * T + 0.000412 * T2;
double omegarad = AASCoordinateTransformation.DegreesToRadians(omega);
//Step 2. Calculate the heliocentric longitude, latitude and radius vector of the Earth in the FK5 system
double l0 = AASEarth.EclipticLongitude(JD);
double b0 = AASEarth.EclipticLatitude(JD);
l0 += AASFK5.CorrectionInLongitude(l0, b0, JD);
double l0rad = AASCoordinateTransformation.DegreesToRadians(l0);
b0 += AASFK5.CorrectionInLatitude(l0, JD);
double b0rad = AASCoordinateTransformation.DegreesToRadians(b0);
double R = AASEarth.RadiusVector(JD);
//Step 3. Calculate the corresponding coordinates l,b,r for Saturn but for the instance t-lightraveltime
double DELTA = 9;
double PreviousEarthLightTravelTime = 0;
double EarthLightTravelTime = AASElliptical.DistanceToLightTime(DELTA);
double JD1 = JD - EarthLightTravelTime;
bool bIterate = true;
double x = 0;
double y = 0;
double z = 0;
double l = 0;
double b = 0;
double r = 0;
while (bIterate)
{
//Calculate the position of Saturn
l = AASSaturn.EclipticLongitude(JD1);
b = AASSaturn.EclipticLatitude(JD1);
l += AASFK5.CorrectionInLongitude(l, b, JD1);
b += AASFK5.CorrectionInLatitude(l, JD1);
double lrad = AASCoordinateTransformation.DegreesToRadians(l);
double brad = AASCoordinateTransformation.DegreesToRadians(b);
r = AASSaturn.RadiusVector(JD1);
//Step 4
x = r * Math.Cos(brad) * Math.Cos(lrad) - R * Math.Cos(l0rad);
y = r * Math.Cos(brad) * Math.Sin(lrad) - R * Math.Sin(l0rad);
z = r * Math.Sin(brad) - R * Math.Sin(b0rad);
DELTA = Math.Sqrt(x * x + y * y + z * z);
EarthLightTravelTime = AASElliptical.DistanceToLightTime(DELTA);
//Prepare for the next loop around
bIterate = (Math.Abs(EarthLightTravelTime - PreviousEarthLightTravelTime) > 2E-6); //2E-6 corresponds to 0.17 of a second
if (bIterate)
{
JD1 = JD - EarthLightTravelTime;
PreviousEarthLightTravelTime = EarthLightTravelTime;
}
}
//Step 5. Calculate Saturn's geocentric Longitude and Latitude
double lambda = Math.Atan2(y, x);
double beta = Math.Atan2(z, Math.Sqrt(x * x + y * y));
//Step 6. Calculate B, a and b
details.B = Math.Asin(Math.Sin(irad) * Math.Cos(beta) * Math.Sin(lambda - omegarad) - Math.Cos(irad) * Math.Sin(beta));
details.a = 375.35 / DELTA;
details.b = details.a * Math.Sin(Math.Abs(details.B));
details.B = AASCoordinateTransformation.RadiansToDegrees(details.B);
//Step 7. Calculate the longitude of the ascending node of Saturn's orbit
double N = 113.6655 + 0.8771 * T;
double Nrad = AASCoordinateTransformation.DegreesToRadians(N);
double ldash = l - 0.01759 / r;
double ldashrad = AASCoordinateTransformation.DegreesToRadians(ldash);
double bdash = b - 0.000764 * Math.Cos(ldashrad - Nrad) / r;
double bdashrad = AASCoordinateTransformation.DegreesToRadians(bdash);
//Step 8. Calculate Bdash
details.Bdash = AASCoordinateTransformation.RadiansToDegrees(Math.Asin(Math.Sin(irad) * Math.Cos(bdashrad) * Math.Sin(ldashrad - omegarad) - Math.Cos(irad) * Math.Sin(bdashrad)));
//Step 9. Calculate DeltaU
double U1 = Math.Atan2(Math.Sin(irad) * Math.Sin(bdashrad) + Math.Cos(irad) * Math.Cos(bdashrad) * Math.Sin(ldashrad - omegarad), Math.Cos(bdashrad) * Math.Cos(ldashrad - omegarad));
double U2 = Math.Atan2(Math.Sin(irad) * Math.Sin(beta) + Math.Cos(irad) * Math.Cos(beta) * Math.Sin(lambda - omegarad), Math.Cos(beta) * Math.Cos(lambda - omegarad));
details.DeltaU = AASCoordinateTransformation.RadiansToDegrees(Math.Abs(U1 - U2));
//Step 10. Calculate the Nutations
double Obliquity = AASNutation.TrueObliquityOfEcliptic(JD);
double NutationInLongitude = AASNutation.NutationInLongitude(JD);
//Step 11. Calculate the Ecliptical longitude and latitude of the northern pole of the ring plane
double lambda0 = omega - 90;
double beta0 = 90 - i;
//Step 12. Correct lambda and beta for the aberration of Saturn
lambda += AASCoordinateTransformation.DegreesToRadians(0.005693 * Math.Cos(l0rad - lambda) / Math.Cos(beta));
beta += AASCoordinateTransformation.DegreesToRadians(0.005693 * Math.Sin(l0rad - lambda) * Math.Sin(beta));
//Step 13. Add nutation in longitude to lambda0 and lambda
//double NLrad = AASCoordinateTransformation.DegreesToRadians(NutationInLongitude/3600);
lambda = AASCoordinateTransformation.RadiansToDegrees(lambda);
lambda += NutationInLongitude / 3600;
lambda = AASCoordinateTransformation.MapTo0To360Range(lambda);
lambda0 += NutationInLongitude / 3600;
lambda0 = AASCoordinateTransformation.MapTo0To360Range(lambda0);
//Step 14. Convert to equatorial coordinates
beta = AASCoordinateTransformation.RadiansToDegrees(beta);
AAS2DCoordinate GeocentricEclipticSaturn = AASCoordinateTransformation.Ecliptic2Equatorial(lambda, beta, Obliquity);
double alpha = AASCoordinateTransformation.HoursToRadians(GeocentricEclipticSaturn.X);
double delta = AASCoordinateTransformation.DegreesToRadians(GeocentricEclipticSaturn.Y);
AAS2DCoordinate GeocentricEclipticNorthPole = AASCoordinateTransformation.Ecliptic2Equatorial(lambda0, beta0, Obliquity);
double alpha0 = AASCoordinateTransformation.HoursToRadians(GeocentricEclipticNorthPole.X);
double delta0 = AASCoordinateTransformation.DegreesToRadians(GeocentricEclipticNorthPole.Y);
//Step 15. Calculate the Position angle
details.P = AASCoordinateTransformation.RadiansToDegrees(Math.Atan2(Math.Cos(delta0) * Math.Sin(alpha0 - alpha), Math.Sin(delta0) * Math.Cos(delta) - Math.Cos(delta0) * Math.Sin(delta) * Math.Cos(alpha0 - alpha)));
return details;
}
public static AASSaturnRingDetails Calculate(double JD, bool bHighPrecision)
{
//What will be the return value
AASSaturnRingDetails details = new AASSaturnRingDetails();
double T = (JD - 2451545) / 36525;
double T2 = T * T;
//Step 1. Calculate the inclination of the plane of the ring and the longitude of the ascending node referred to the ecliptic and mean equinox of the date
double i = 28.075216 - 0.012998 * T + 0.000004 * T2;
double irad = AASCoordinateTransformation.DegreesToRadians(i);
double omega = 169.508470 + 1.394681 * T + 0.000412 * T2;
double omegarad = AASCoordinateTransformation.DegreesToRadians(omega);
//Step 2. Calculate the heliocentric longitude, latitude and radius vector of the Earth in the FK5 system
double l0 = AASEarth.EclipticLongitude(JD, bHighPrecision);
double b0 = AASEarth.EclipticLatitude(JD, bHighPrecision);
l0 += AASFK5.CorrectionInLongitude(l0, b0, JD);
double l0rad = AASCoordinateTransformation.DegreesToRadians(l0);
b0 += AASFK5.CorrectionInLatitude(l0, JD);
double b0rad = AASCoordinateTransformation.DegreesToRadians(b0);
double R = AASEarth.RadiusVector(JD, bHighPrecision);
//Step 3. Calculate the corresponding coordinates l,b,r for Saturn but for the instance t-lightraveltime
double DELTA = 9;
double PreviousEarthLightTravelTime = 0;
double EarthLightTravelTime = AASElliptical.DistanceToLightTime(DELTA);
double JD1 = JD - EarthLightTravelTime;
bool bIterate = true;
double x = 0;
double y = 0;
double z = 0;
double l = 0;
double b = 0;
double r = 0;
while (bIterate)
{
//Calculate the position of Saturn
l = AASSaturn.EclipticLongitude(JD1, bHighPrecision);
b = AASSaturn.EclipticLatitude(JD1, bHighPrecision);
l += AASFK5.CorrectionInLongitude(l, b, JD1);
b += AASFK5.CorrectionInLatitude(l, JD1);
double lrad = AASCoordinateTransformation.DegreesToRadians(l);
double brad = AASCoordinateTransformation.DegreesToRadians(b);
r = AASSaturn.RadiusVector(JD1, bHighPrecision);
//Step 4
x = r * Math.Cos(brad) * Math.Cos(lrad) - R * Math.Cos(l0rad);
y = r * Math.Cos(brad) * Math.Sin(lrad) - R * Math.Sin(l0rad);
z = r * Math.Sin(brad) - R * Math.Sin(b0rad);
DELTA = Math.Sqrt(x * x + y * y + z * z);
EarthLightTravelTime = AASElliptical.DistanceToLightTime(DELTA);
//Prepare for the next loop around
bIterate = (Math.Abs(EarthLightTravelTime - PreviousEarthLightTravelTime) > 2e-6); //2e-6 corresponds to 0.17 of a second
if (bIterate)
{
JD1 = JD - EarthLightTravelTime;
PreviousEarthLightTravelTime = EarthLightTravelTime;
}
}
//Step 5. Calculate Saturn's geocentric Longitude and Latitude
double lambda = Math.Atan2(y, x);
double beta = Math.Atan2(z, Math.Sqrt(x * x + y * y));
//Step 6. Calculate B, a and b
details.B = Math.Asin(Math.Sin(irad) * Math.Cos(beta) * Math.Sin(lambda - omegarad) - Math.Cos(irad) * Math.Sin(beta));
details.a = 375.35 / DELTA;
details.b = details.a * Math.Sin(Math.Abs(details.B));
details.B = AASCoordinateTransformation.RadiansToDegrees(details.B);
//Step 7. Calculate the longitude of the ascending node of Saturn's orbit
double N = 113.6655 + 0.8771 * T;
double Nrad = AASCoordinateTransformation.DegreesToRadians(N);
double ldash = l - 0.01759 / r;
double ldashrad = AASCoordinateTransformation.DegreesToRadians(ldash);
double bdash = b - 0.000764 * Math.Cos(ldashrad - Nrad) / r;
double bdashrad = AASCoordinateTransformation.DegreesToRadians(bdash);
//Step 8. Calculate Bdash
details.Bdash = AASCoordinateTransformation.RadiansToDegrees(Math.Asin(Math.Sin(irad) * Math.Cos(bdashrad) * Math.Sin(ldashrad - omegarad) - Math.Cos(irad) * Math.Sin(bdashrad)));
//Step 9. Calculate DeltaU
details.U1 = AASCoordinateTransformation.MapTo0To360Range(AASCoordinateTransformation.RadiansToDegrees(Math.Atan2(Math.Sin(irad) * Math.Sin(bdashrad) + Math.Cos(irad) * Math.Cos(bdashrad) * Math.Sin(ldashrad - omegarad), Math.Cos(bdashrad) * Math.Cos(ldashrad - omegarad))));
details.U2 = AASCoordinateTransformation.MapTo0To360Range(AASCoordinateTransformation.RadiansToDegrees(Math.Atan2(Math.Sin(irad) * Math.Sin(beta) + Math.Cos(irad) * Math.Cos(beta) * Math.Sin(lambda - omegarad), Math.Cos(beta) * Math.Cos(lambda - omegarad))));
details.DeltaU = Math.Abs(details.U1 - details.U2);
if (details.DeltaU > 180)
{
details.DeltaU = 360 - details.DeltaU;
}
//Step 10. Calculate the Nutations
double Obliquity = AASNutation.TrueObliquityOfEcliptic(JD);
double NutationInLongitude = AASNutation.NutationInLongitude(JD);
//Step 11. Calculate the Ecliptical longitude and latitude of the northern pole of the ring plane
double lambda0 = omega - 90;
double beta0 = 90 - i;
//Step 12. Correct lambda and beta for the aberration of Saturn
lambda += AASCoordinateTransformation.DegreesToRadians(0.005693 * Math.Cos(l0rad - lambda) / Math.Cos(beta));
beta += AASCoordinateTransformation.DegreesToRadians(0.005693 * Math.Sin(l0rad - lambda) * Math.Sin(beta));
//Step 13. Add nutation in longitude to lambda0 and lambda
lambda = AASCoordinateTransformation.RadiansToDegrees(lambda);
lambda += NutationInLongitude / 3600;
lambda = AASCoordinateTransformation.MapTo0To360Range(lambda);
lambda0 += NutationInLongitude / 3600;
lambda0 = AASCoordinateTransformation.MapTo0To360Range(lambda0);
//Step 14. Convert to equatorial coordinates
beta = AASCoordinateTransformation.RadiansToDegrees(beta);
AAS2DCoordinate GeocentricEclipticSaturn = AASCoordinateTransformation.Ecliptic2Equatorial(lambda, beta, Obliquity);
double alpha = AASCoordinateTransformation.HoursToRadians(GeocentricEclipticSaturn.X);
double delta = AASCoordinateTransformation.DegreesToRadians(GeocentricEclipticSaturn.Y);
AAS2DCoordinate GeocentricEclipticNorthPole = AASCoordinateTransformation.Ecliptic2Equatorial(lambda0, beta0, Obliquity);
double alpha0 = AASCoordinateTransformation.HoursToRadians(GeocentricEclipticNorthPole.X);
double delta0 = AASCoordinateTransformation.DegreesToRadians(GeocentricEclipticNorthPole.Y);
//Step 15. Calculate the Position angle
details.P = AASCoordinateTransformation.RadiansToDegrees(Math.Atan2(Math.Cos(delta0) * Math.Sin(alpha0 - alpha), Math.Sin(delta0) * Math.Cos(delta) - Math.Cos(delta0) * Math.Sin(delta) * Math.Cos(alpha0 - alpha)));
return(details);
}