KeplerianAnomalyUtility.java
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*
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* Unless required by applicable law or agreed to in writing, software
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package org.orekit.orbits;
import org.hipparchus.exception.MathIllegalStateException;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.MathUtils;
import org.hipparchus.util.SinCos;
import org.orekit.errors.OrekitMessages;
/**
* Utility methods for converting between different Keplerian anomalies.
* @author Luc Maisonobe
* @author Andrew Goetz
*/
public final class KeplerianAnomalyUtility {
/** First coefficient to compute elliptic Kepler equation solver starter. */
private static final double A;
/** Second coefficient to compute elliptic Kepler equation solver starter. */
private static final double B;
static {
final double k1 = 3 * FastMath.PI + 2;
final double k2 = FastMath.PI - 1;
final double k3 = 6 * FastMath.PI - 1;
A = 3 * k2 * k2 / k1;
B = k3 * k3 / (6 * k1);
}
/** Private constructor for utility class. */
private KeplerianAnomalyUtility() {
// Nothing to do
}
/**
* Computes the elliptic true anomaly from the elliptic mean anomaly.
* @param e eccentricity such that 0 ≤ e < 1
* @param M elliptic mean anomaly (rad)
* @return elliptic true anomaly (rad)
*/
public static double ellipticMeanToTrue(final double e, final double M) {
final double E = ellipticMeanToEccentric(e, M);
final double v = ellipticEccentricToTrue(e, E);
return v;
}
/**
* Computes the elliptic mean anomaly from the elliptic true anomaly.
* @param e eccentricity such that 0 ≤ e < 1
* @param v elliptic true anomaly (rad)
* @return elliptic mean anomaly (rad)
*/
public static double ellipticTrueToMean(final double e, final double v) {
final double E = ellipticTrueToEccentric(e, v);
final double M = ellipticEccentricToMean(e, E);
return M;
}
/**
* Computes the elliptic true anomaly from the elliptic eccentric anomaly.
* @param e eccentricity such that 0 ≤ e < 1
* @param E elliptic eccentric anomaly (rad)
* @return elliptic true anomaly (rad)
*/
public static double ellipticEccentricToTrue(final double e, final double E) {
final double beta = e / (1 + FastMath.sqrt((1 - e) * (1 + e)));
final SinCos scE = FastMath.sinCos(E);
return E + 2 * FastMath.atan(beta * scE.sin() / (1 - beta * scE.cos()));
}
/**
* Computes the elliptic eccentric anomaly from the elliptic true anomaly.
* @param e eccentricity such that 0 ≤ e < 1
* @param v elliptic true anomaly (rad)
* @return elliptic eccentric anomaly (rad)
*/
public static double ellipticTrueToEccentric(final double e, final double v) {
final double beta = e / (1 + FastMath.sqrt(1 - e * e));
final SinCos scv = FastMath.sinCos(v);
return v - 2 * FastMath.atan(beta * scv.sin() / (1 + beta * scv.cos()));
}
/**
* Computes the elliptic eccentric anomaly from the elliptic mean anomaly.
* <p>
* The algorithm used here for solving hyperbolic Kepler equation is from Odell,
* A.W., Gooding, R.H. "Procedures for solving Kepler's equation." Celestial
* Mechanics 38, 307–334 (1986). https://doi.org/10.1007/BF01238923
* </p>
* @param e eccentricity such that 0 ≤ e < 1
* @param M elliptic mean anomaly (rad)
* @return elliptic eccentric anomaly (rad)
*/
public static double ellipticMeanToEccentric(final double e, final double M) {
// reduce M to [-PI PI) interval
final double reducedM = MathUtils.normalizeAngle(M, 0.0);
// compute start value according to A. W. Odell and R. H. Gooding S12 starter
double E;
if (FastMath.abs(reducedM) < 1.0 / 6.0) {
E = reducedM + e * (FastMath.cbrt(6 * reducedM) - reducedM);
} else {
if (reducedM < 0) {
final double w = FastMath.PI + reducedM;
E = reducedM + e * (A * w / (B - w) - FastMath.PI - reducedM);
} else {
final double w = FastMath.PI - reducedM;
E = reducedM + e * (FastMath.PI - A * w / (B - w) - reducedM);
}
}
final double e1 = 1 - e;
final boolean noCancellationRisk = (e1 + E * E / 6) >= 0.1;
// perform two iterations, each consisting of one Halley step and one
// Newton-Raphson step
for (int j = 0; j < 2; ++j) {
final double f;
double fd;
final SinCos sc = FastMath.sinCos(E);
final double fdd = e * sc.sin();
final double fddd = e * sc.cos();
if (noCancellationRisk) {
f = (E - fdd) - reducedM;
fd = 1 - fddd;
} else {
f = eMeSinE(e, E) - reducedM;
final double s = FastMath.sin(0.5 * E);
fd = e1 + 2 * e * s * s;
}
final double dee = f * fd / (0.5 * f * fdd - fd * fd);
// update eccentric anomaly, using expressions that limit underflow problems
final double w = fd + 0.5 * dee * (fdd + dee * fddd / 3);
fd += dee * (fdd + 0.5 * dee * fddd);
E -= (f - dee * (fd - w)) / fd;
}
// expand the result back to original range
E += M - reducedM;
return E;
}
/**
* Accurate computation of E - e sin(E).
* <p>
* This method is used when E is close to 0 and e close to 1, i.e. near the
* perigee of almost parabolic orbits
* </p>
* @param e eccentricity
* @param E eccentric anomaly (rad)
* @return E - e sin(E)
*/
private static double eMeSinE(final double e, final double E) {
double x = (1 - e) * FastMath.sin(E);
final double mE2 = -E * E;
double term = E;
double d = 0;
// the inequality test below IS intentional and should NOT be replaced by a
// check with a small tolerance
for (double x0 = Double.NaN; !Double.valueOf(x).equals(Double.valueOf(x0));) {
d += 2;
term *= mE2 / (d * (d + 1));
x0 = x;
x = x - term;
}
return x;
}
/**
* Computes the elliptic mean anomaly from the elliptic eccentric anomaly.
* @param e eccentricity such that 0 ≤ e < 1
* @param E elliptic eccentric anomaly (rad)
* @return elliptic mean anomaly (rad)
*/
public static double ellipticEccentricToMean(final double e, final double E) {
return E - e * FastMath.sin(E);
}
/**
* Computes the hyperbolic true anomaly from the hyperbolic mean anomaly.
* @param e eccentricity > 1
* @param M hyperbolic mean anomaly
* @return hyperbolic true anomaly (rad)
*/
public static double hyperbolicMeanToTrue(final double e, final double M) {
final double H = hyperbolicMeanToEccentric(e, M);
final double v = hyperbolicEccentricToTrue(e, H);
return v;
}
/**
* Computes the hyperbolic mean anomaly from the hyperbolic true anomaly.
* @param e eccentricity > 1
* @param v hyperbolic true anomaly (rad)
* @return hyperbolic mean anomaly
*/
public static double hyperbolicTrueToMean(final double e, final double v) {
final double H = hyperbolicTrueToEccentric(e, v);
final double M = hyperbolicEccentricToMean(e, H);
return M;
}
/**
* Computes the hyperbolic true anomaly from the hyperbolic eccentric anomaly.
* @param e eccentricity > 1
* @param H hyperbolic eccentric anomaly
* @return hyperbolic true anomaly (rad)
*/
public static double hyperbolicEccentricToTrue(final double e, final double H) {
return 2 * FastMath.atan(FastMath.sqrt((e + 1) / (e - 1)) * FastMath.tanh(0.5 * H));
}
/**
* Computes the hyperbolic eccentric anomaly from the hyperbolic true anomaly.
* @param e eccentricity > 1
* @param v hyperbolic true anomaly (rad)
* @return hyperbolic eccentric anomaly
*/
public static double hyperbolicTrueToEccentric(final double e, final double v) {
final SinCos scv = FastMath.sinCos(v);
final double sinhH = FastMath.sqrt(e * e - 1) * scv.sin() / (1 + e * scv.cos());
return FastMath.asinh(sinhH);
}
/**
* Computes the hyperbolic eccentric anomaly from the hyperbolic mean anomaly.
* <p>
* The algorithm used here for solving hyperbolic Kepler equation is from
* Gooding, R.H., Odell, A.W. "The hyperbolic Kepler equation (and the elliptic
* equation revisited)." Celestial Mechanics 44, 267–282 (1988).
* https://doi.org/10.1007/BF01235540
* </p>
* @param e eccentricity > 1
* @param M hyperbolic mean anomaly
* @return hyperbolic eccentric anomaly
*/
public static double hyperbolicMeanToEccentric(final double e, final double M) {
// Solve L = S - g * asinh(S) for S = sinh(H).
final double L = M / e;
final double g = 1.0 / e;
final double g1 = 1.0 - g;
// Starter based on Lagrange's theorem.
double S = L;
if (L == 0.0) {
return 0.0;
}
final double cl = FastMath.sqrt(1.0 + L * L);
final double al = FastMath.asinh(L);
final double w = g * g * al / (cl * cl * cl);
S = 1.0 - g / cl;
S = L + g * al / FastMath.cbrt(S * S * S + w * L * (1.5 - (4.0 / 3.0) * g));
// Two iterations (at most) of Halley-then-Newton process.
for (int i = 0; i < 2; ++i) {
final double s0 = S * S;
final double s1 = s0 + 1.0;
final double s2 = FastMath.sqrt(s1);
final double s3 = s1 * s2;
final double fdd = g * S / s3;
final double fddd = g * (1.0 - 2.0 * s0) / (s1 * s3);
double f;
double fd;
if (s0 / 6.0 + g1 >= 0.5) {
f = (S - g * FastMath.asinh(S)) - L;
fd = 1.0 - g / s2;
} else {
// Accurate computation of S - (1 - g1) * asinh(S)
// when (g1, S) is close to (0, 0).
final double t = S / (1.0 + FastMath.sqrt(1.0 + S * S));
final double tsq = t * t;
double x = S * (g1 + g * tsq);
double term = 2.0 * g * t;
double twoI1 = 1.0;
double x0;
int j = 0;
do {
if (++j == 1000000) {
// This isn't expected to happen, but it protects against an infinite loop.
throw new MathIllegalStateException(
OrekitMessages.UNABLE_TO_COMPUTE_HYPERBOLIC_ECCENTRIC_ANOMALY, j);
}
twoI1 += 2.0;
term *= tsq;
x0 = x;
x -= term / twoI1;
} while (x != x0);
f = x - L;
fd = (s0 / (s2 + 1.0) + g1) / s2;
}
final double ds = f * fd / (0.5 * f * fdd - fd * fd);
final double stemp = S + ds;
if (S == stemp) {
break;
}
f += ds * (fd + 0.5 * ds * (fdd + ds / 3.0 * fddd));
fd += ds * (fdd + 0.5 * ds * fddd);
S = stemp - f / fd;
}
final double H = FastMath.asinh(S);
return H;
}
/**
* Computes the hyperbolic mean anomaly from the hyperbolic eccentric anomaly.
* @param e eccentricity > 1
* @param H hyperbolic eccentric anomaly
* @return hyperbolic mean anomaly
*/
public static double hyperbolicEccentricToMean(final double e, final double H) {
return e * FastMath.sinh(H) - H;
}
}