CartesianOrbit.java

  1. /* Copyright 2002-2022 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.orbits;

  19. import java.io.Serializable;
  20. import java.util.stream.Stream;

  22. import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  23. import org.hipparchus.exception.LocalizedCoreFormats;
  24. import org.hipparchus.exception.MathIllegalStateException;
  25. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  26. import org.hipparchus.geometry.euclidean.threed.Rotation;
  27. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  28. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  29. import org.hipparchus.util.FastMath;
  30. import org.hipparchus.util.SinCos;
  31. import org.orekit.annotation.DefaultDataContext;
  32. import org.orekit.data.DataContext;
  33. import org.orekit.errors.OrekitMessages;
  34. import org.orekit.frames.Frame;
  35. import org.orekit.time.AbsoluteDate;
  36. import org.orekit.utils.CartesianDerivativesFilter;
  37. import org.orekit.utils.FieldPVCoordinates;
  38. import org.orekit.utils.PVCoordinates;
  39. import org.orekit.utils.TimeStampedPVCoordinates;


  42. /** This class holds Cartesian orbital parameters.

  44.  * <p>
  45.  * The parameters used internally are the Cartesian coordinates:
  46.  *   <ul>
  47.  *     <li>x</li>
  48.  *     <li>y</li>
  49.  *     <li>z</li>
  50.  *     <li>xDot</li>
  51.  *     <li>yDot</li>
  52.  *     <li>zDot</li>
  53.  *   </ul>
  54.  * contained in {@link PVCoordinates}.
  55.  *

  57.  * <p>
  58.  * Note that the implementation of this class delegates all non-Cartesian related
  59.  * computations ({@link #getA()}, {@link #getEquinoctialEx()}, ...) to an underlying
  60.  * instance of the {@link EquinoctialOrbit} class. This implies that using this class
  61.  * only for analytical computations which are always based on non-Cartesian
  62.  * parameters is perfectly possible but somewhat sub-optimal.
  63.  * </p>
  64.  * <p>
  65.  * The instance <code>CartesianOrbit</code> is guaranteed to be immutable.
  66.  * </p>
  67.  * @see    Orbit
  68.  * @see    KeplerianOrbit
  69.  * @see    CircularOrbit
  70.  * @see    EquinoctialOrbit
  71.  * @author Luc Maisonobe
  72.  * @author Guylaine Prat
  73.  * @author Fabien Maussion
  74.  * @author V&eacute;ronique Pommier-Maurussane
  75.  */
  76. public class CartesianOrbit extends Orbit {

  78.     /** Serializable UID. */
  79.     private static final long serialVersionUID = 20170414L;

  81.     /** Indicator for non-Keplerian derivatives. */
  82.     private final transient boolean hasNonKeplerianAcceleration;

  84.     /** Underlying equinoctial orbit to which high-level methods are delegated. */
  85.     private transient EquinoctialOrbit equinoctial;

  87.     /** Constructor from Cartesian parameters.
  88.      *
  89.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  90.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  91.      * use {@code mu} and the position to compute the acceleration, including
  92.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  93.      *
  94.      * @param pvaCoordinates the position, velocity and acceleration of the satellite.
  95.      * @param frame the frame in which the {@link PVCoordinates} are defined
  96.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  97.      * @param mu central attraction coefficient (m³/s²)
  98.      * @exception IllegalArgumentException if frame is not a {@link
  99.      * Frame#isPseudoInertial pseudo-inertial frame}
  100.      */
  101.     public CartesianOrbit(final TimeStampedPVCoordinates pvaCoordinates,
  102.                           final Frame frame, final double mu)
  103.         throws IllegalArgumentException {
  104.         super(pvaCoordinates, frame, mu);
  105.         hasNonKeplerianAcceleration = hasNonKeplerianAcceleration(pvaCoordinates, mu);
  106.         equinoctial = null;
  107.     }

  109.     /** Constructor from Cartesian parameters.
  110.      *
  111.      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
  112.      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
  113.      * use {@code mu} and the position to compute the acceleration, including
  114.      * {@link #shiftedBy(double)} and {@link #getPVCoordinates(AbsoluteDate, Frame)}.
  115.      *
  116.      * @param pvaCoordinates the position and velocity of the satellite.
  117.      * @param frame the frame in which the {@link PVCoordinates} are defined
  118.      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
  119.      * @param date date of the orbital parameters
  120.      * @param mu central attraction coefficient (m³/s²)
  121.      * @exception IllegalArgumentException if frame is not a {@link
  122.      * Frame#isPseudoInertial pseudo-inertial frame}
  123.      */
  124.     public CartesianOrbit(final PVCoordinates pvaCoordinates, final Frame frame,
  125.                           final AbsoluteDate date, final double mu)
  126.         throws IllegalArgumentException {
  127.         this(new TimeStampedPVCoordinates(date, pvaCoordinates), frame, mu);
  128.     }

  130.     /** Constructor from any kind of orbital parameters.
  131.      * @param op orbital parameters to copy
  132.      */
  133.     public CartesianOrbit(final Orbit op) {
  134.         super(op.getPVCoordinates(), op.getFrame(), op.getMu());
  135.         hasNonKeplerianAcceleration = op.hasDerivatives();
  136.         if (op instanceof EquinoctialOrbit) {
  137.             equinoctial = (EquinoctialOrbit) op;
  138.         } else if (op instanceof CartesianOrbit) {
  139.             equinoctial = ((CartesianOrbit) op).equinoctial;
  140.         } else {
  141.             equinoctial = null;
  142.         }
  143.     }

  145.     /** {@inheritDoc} */
  146.     public OrbitType getType() {
  147.         return OrbitType.CARTESIAN;
  148.     }

  150.     /** Lazy evaluation of equinoctial parameters. */
  151.     private void initEquinoctial() {
  152.         if (equinoctial == null) {
  153.             if (hasDerivatives()) {
  154.                 // getPVCoordinates includes accelerations that will be interpreted as derivatives
  155.                 equinoctial = new EquinoctialOrbit(getPVCoordinates(), getFrame(), getDate(), getMu());
  156.             } else {
  157.                 // get rid of Keplerian acceleration so we don't assume
  158.                 // we have derivatives when in fact we don't have them
  159.                 equinoctial = new EquinoctialOrbit(new PVCoordinates(getPVCoordinates().getPosition(),
  160.                                                                      getPVCoordinates().getVelocity()),
  161.                                                    getFrame(), getDate(), getMu());
  162.             }
  163.         }
  164.     }

  166.     /** Get the position/velocity with derivatives.
  167.      * @return position/velocity with derivatives
  168.      * @since 10.2
  169.      */
  170.     private FieldPVCoordinates<UnivariateDerivative2> getPVDerivatives() {
  171.         // PVA coordinates
  172.         final PVCoordinates pva = getPVCoordinates();
  173.         final Vector3D      p   = pva.getPosition();
  174.         final Vector3D      v   = pva.getVelocity();
  175.         final Vector3D      a   = pva.getAcceleration();
  176.         // Field coordinates
  177.         final FieldVector3D<UnivariateDerivative2> pG = new FieldVector3D<>(new UnivariateDerivative2(p.getX(), v.getX(), a.getX()),
  178.                                                                new UnivariateDerivative2(p.getY(), v.getY(), a.getY()),
  179.                                                                new UnivariateDerivative2(p.getZ(), v.getZ(), a.getZ()));
  180.         final FieldVector3D<UnivariateDerivative2> vG = new FieldVector3D<>(new UnivariateDerivative2(v.getX(), a.getX(), 0.0),
  181.                                                                new UnivariateDerivative2(v.getY(), a.getY(), 0.0),
  182.                                                                new UnivariateDerivative2(v.getZ(), a.getZ(), 0.0));
  183.         return new FieldPVCoordinates<>(pG, vG);
  184.     }

  186.     /** {@inheritDoc} */
  187.     public double getA() {
  188.         final double r  = getPVCoordinates().getPosition().getNorm();
  189.         final double V2 = getPVCoordinates().getVelocity().getNormSq();
  190.         return r / (2 - r * V2 / getMu());
  191.     }

  193.     /** {@inheritDoc} */
  194.     public double getADot() {
  195.         if (hasDerivatives()) {
  196.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  197.             final UnivariateDerivative2 r  = pv.getPosition().getNorm();
  198.             final UnivariateDerivative2 V2 = pv.getVelocity().getNormSq();
  199.             final UnivariateDerivative2 a  = r.divide(r.multiply(V2).divide(getMu()).subtract(2).negate());
  200.             return a.getDerivative(1);
  201.         } else {
  202.             return Double.NaN;
  203.         }
  204.     }

  206.     /** {@inheritDoc} */
  207.     public double getE() {
  208.         final double muA = getMu() * getA();
  209.         if (muA > 0) {
  210.             // elliptic or circular orbit
  211.             final Vector3D pvP   = getPVCoordinates().getPosition();
  212.             final Vector3D pvV   = getPVCoordinates().getVelocity();
  213.             final double rV2OnMu = pvP.getNorm() * pvV.getNormSq() / getMu();
  214.             final double eSE     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(muA);
  215.             final double eCE     = rV2OnMu - 1;
  216.             return FastMath.sqrt(eCE * eCE + eSE * eSE);
  217.         } else {
  218.             // hyperbolic orbit
  219.             final Vector3D pvM = getPVCoordinates().getMomentum();
  220.             return FastMath.sqrt(1 - pvM.getNormSq() / muA);
  221.         }
  222.     }

  224.     /** {@inheritDoc} */
  225.     public double getEDot() {
  226.         if (hasDerivatives()) {
  227.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  228.             final FieldVector3D<UnivariateDerivative2> pvP   = pv.getPosition();
  229.             final FieldVector3D<UnivariateDerivative2> pvV   = pv.getVelocity();
  230.             final UnivariateDerivative2 r       = pvP.getNorm();
  231.             final UnivariateDerivative2 V2      = pvV.getNormSq();
  232.             final UnivariateDerivative2 rV2OnMu = r.multiply(V2).divide(getMu());
  233.             final UnivariateDerivative2 a       = r.divide(rV2OnMu.negate().add(2));
  234.             final UnivariateDerivative2 eSE     = FieldVector3D.dotProduct(pvP, pvV).divide(a.multiply(getMu()).sqrt());
  235.             final UnivariateDerivative2 eCE     = rV2OnMu.subtract(1);
  236.             final UnivariateDerivative2 e       = eCE.multiply(eCE).add(eSE.multiply(eSE)).sqrt();
  237.             return e.getDerivative(1);
  238.         } else {
  239.             return Double.NaN;
  240.         }
  241.     }

  243.     /** {@inheritDoc} */
  244.     public double getI() {
  245.         return Vector3D.angle(Vector3D.PLUS_K, getPVCoordinates().getMomentum());
  246.     }

  248.     /** {@inheritDoc} */
  249.     public double getIDot() {
  250.         if (hasDerivatives()) {
  251.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  252.             final FieldVector3D<UnivariateDerivative2> momentum =
  253.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity());
  254.             final UnivariateDerivative2 i = FieldVector3D.angle(Vector3D.PLUS_K, momentum);
  255.             return i.getDerivative(1);
  256.         } else {
  257.             return Double.NaN;
  258.         }
  259.     }

  261.     /** {@inheritDoc} */
  262.     public double getEquinoctialEx() {
  263.         initEquinoctial();
  264.         return equinoctial.getEquinoctialEx();
  265.     }

  267.     /** {@inheritDoc} */
  268.     public double getEquinoctialExDot() {
  269.         initEquinoctial();
  270.         return equinoctial.getEquinoctialExDot();
  271.     }

  273.     /** {@inheritDoc} */
  274.     public double getEquinoctialEy() {
  275.         initEquinoctial();
  276.         return equinoctial.getEquinoctialEy();
  277.     }

  279.     /** {@inheritDoc} */
  280.     public double getEquinoctialEyDot() {
  281.         initEquinoctial();
  282.         return equinoctial.getEquinoctialEyDot();
  283.     }

  285.     /** {@inheritDoc} */
  286.     public double getHx() {
  287.         final Vector3D w = getPVCoordinates().getMomentum().normalize();
  288.         // Check for equatorial retrograde orbit
  289.         if ((w.getX() * w.getX() + w.getY() * w.getY()) == 0 && w.getZ() < 0) {
  290.             return Double.NaN;
  291.         }
  292.         return -w.getY() / (1 + w.getZ());
  293.     }

  295.     /** {@inheritDoc} */
  296.     public double getHxDot() {
  297.         if (hasDerivatives()) {
  298.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  299.             final FieldVector3D<UnivariateDerivative2> w =
  300.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity()).normalize();
  301.             // Check for equatorial retrograde orbit
  302.             final double x = w.getX().getValue();
  303.             final double y = w.getY().getValue();
  304.             final double z = w.getZ().getValue();
  305.             if ((x * x + y * y) == 0 && z < 0) {
  306.                 return Double.NaN;
  307.             }
  308.             final UnivariateDerivative2 hx = w.getY().negate().divide(w.getZ().add(1));
  309.             return hx.getDerivative(1);
  310.         } else {
  311.             return Double.NaN;
  312.         }
  313.     }

  315.     /** {@inheritDoc} */
  316.     public double getHy() {
  317.         final Vector3D w = getPVCoordinates().getMomentum().normalize();
  318.         // Check for equatorial retrograde orbit
  319.         if ((w.getX() * w.getX() + w.getY() * w.getY()) == 0 && w.getZ() < 0) {
  320.             return Double.NaN;
  321.         }
  322.         return  w.getX() / (1 + w.getZ());
  323.     }

  325.     /** {@inheritDoc} */
  326.     public double getHyDot() {
  327.         if (hasDerivatives()) {
  328.             final FieldPVCoordinates<UnivariateDerivative2> pv = getPVDerivatives();
  329.             final FieldVector3D<UnivariateDerivative2> w =
  330.                             FieldVector3D.crossProduct(pv.getPosition(), pv.getVelocity()).normalize();
  331.             // Check for equatorial retrograde orbit
  332.             final double x = w.getX().getValue();
  333.             final double y = w.getY().getValue();
  334.             final double z = w.getZ().getValue();
  335.             if ((x * x + y * y) == 0 && z < 0) {
  336.                 return Double.NaN;
  337.             }
  338.             final UnivariateDerivative2 hy = w.getX().divide(w.getZ().add(1));
  339.             return hy.getDerivative(1);
  340.         } else {
  341.             return Double.NaN;
  342.         }
  343.     }

  345.     /** {@inheritDoc} */
  346.     public double getLv() {
  347.         initEquinoctial();
  348.         return equinoctial.getLv();
  349.     }

  351.     /** {@inheritDoc} */
  352.     public double getLvDot() {
  353.         initEquinoctial();
  354.         return equinoctial.getLvDot();
  355.     }

  357.     /** {@inheritDoc} */
  358.     public double getLE() {
  359.         initEquinoctial();
  360.         return equinoctial.getLE();
  361.     }

  363.     /** {@inheritDoc} */
  364.     public double getLEDot() {
  365.         initEquinoctial();
  366.         return equinoctial.getLEDot();
  367.     }

  369.     /** {@inheritDoc} */
  370.     public double getLM() {
  371.         initEquinoctial();
  372.         return equinoctial.getLM();
  373.     }

  375.     /** {@inheritDoc} */
  376.     public double getLMDot() {
  377.         initEquinoctial();
  378.         return equinoctial.getLMDot();
  379.     }

  381.     /** {@inheritDoc} */
  382.     public boolean hasDerivatives() {
  383.         return hasNonKeplerianAcceleration;
  384.     }

  386.     /** {@inheritDoc} */
  387.     protected TimeStampedPVCoordinates initPVCoordinates() {
  388.         // nothing to do here, as the canonical elements are already the Cartesian ones
  389.         return getPVCoordinates();
  390.     }

  392.     /** {@inheritDoc} */
  393.     public CartesianOrbit shiftedBy(final double dt) {
  394.         final PVCoordinates shiftedPV = (getA() < 0) ? shiftPVHyperbolic(dt) : shiftPVElliptic(dt);
  395.         return new CartesianOrbit(shiftedPV, getFrame(), getDate().shiftedBy(dt), getMu());
  396.     }

  398.     /** {@inheritDoc}
  399.      * <p>
  400.      * The interpolated instance is created by polynomial Hermite interpolation
  401.      * ensuring velocity remains the exact derivative of position.
  402.      * </p>
  403.      * <p>
  404.      * As this implementation of interpolation is polynomial, it should be used only
  405.      * with small samples (about 10-20 points) in order to avoid <a
  406.      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
  407.      * and numerical problems (including NaN appearing).
  408.      * </p>
  409.      * <p>
  410.      * If orbit interpolation on large samples is needed, using the {@link
  411.      * org.orekit.propagation.analytical.Ephemeris} class is a better way than using this
  412.      * low-level interpolation. The Ephemeris class automatically handles selection of
  413.      * a neighboring sub-sample with a predefined number of point from a large global sample
  414.      * in a thread-safe way.
  415.      * </p>
  416.      */
  417.     public CartesianOrbit interpolate(final AbsoluteDate date, final Stream<Orbit> sample) {
  418.         final TimeStampedPVCoordinates interpolated =
  419.                 TimeStampedPVCoordinates.interpolate(date, CartesianDerivativesFilter.USE_PVA,
  420.                                                      sample.map(orbit -> orbit.getPVCoordinates()));
  421.         return new CartesianOrbit(interpolated, getFrame(), date, getMu());
  422.     }

  424.     /** Compute shifted position and velocity in elliptic case.
  425.      * @param dt time shift
  426.      * @return shifted position and velocity
  427.      */
  428.     private PVCoordinates shiftPVElliptic(final double dt) {

  430.         // preliminary computation
  431.         final Vector3D pvP   = getPVCoordinates().getPosition();
  432.         final Vector3D pvV   = getPVCoordinates().getVelocity();
  433.         final double r2      = pvP.getNormSq();
  434.         final double r       = FastMath.sqrt(r2);
  435.         final double rV2OnMu = r * pvV.getNormSq() / getMu();
  436.         final double a       = getA();
  437.         final double eSE     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(getMu() * a);
  438.         final double eCE     = rV2OnMu - 1;
  439.         final double e2      = eCE * eCE + eSE * eSE;

  441.         // we can use any arbitrary reference 2D frame in the orbital plane
  442.         // in order to simplify some equations below, we use the current position as the u axis
  443.         final Vector3D u     = pvP.normalize();
  444.         final Vector3D v     = Vector3D.crossProduct(getPVCoordinates().getMomentum(), u).normalize();

  446.         // the following equations rely on the specific choice of u explained above,
  447.         // some coefficients that vanish to 0 in this case have already been removed here
  448.         final double ex        = (eCE - e2) * a / r;
  449.         final double ey        = -FastMath.sqrt(1 - e2) * eSE * a / r;
  450.         final double beta      = 1 / (1 + FastMath.sqrt(1 - e2));
  451.         final double thetaE0   = FastMath.atan2(ey + eSE * beta * ex, r / a + ex - eSE * beta * ey);
  452.         final SinCos scThetaE0 = FastMath.sinCos(thetaE0);
  453.         final double thetaM0   = thetaE0 - ex * scThetaE0.sin() + ey * scThetaE0.cos();

  455.         // compute in-plane shifted eccentric argument
  456.         final double thetaM1 = thetaM0 + getKeplerianMeanMotion() * dt;
  457.         final double thetaE1 = meanToEccentric(thetaM1, ex, ey);
  458.         final SinCos scTE    = FastMath.sinCos(thetaE1);
  459.         final double cTE     = scTE.cos();
  460.         final double sTE     = scTE.sin();

  462.         // compute shifted in-plane Cartesian coordinates
  463.         final double exey   = ex * ey;
  464.         final double exCeyS = ex * cTE + ey * sTE;
  465.         final double x      = a * ((1 - beta * ey * ey) * cTE + beta * exey * sTE - ex);
  466.         final double y      = a * ((1 - beta * ex * ex) * sTE + beta * exey * cTE - ey);
  467.         final double factor = FastMath.sqrt(getMu() / a) / (1 - exCeyS);
  468.         final double xDot   = factor * (-sTE + beta * ey * exCeyS);
  469.         final double yDot   = factor * ( cTE - beta * ex * exCeyS);

  471.         final Vector3D shiftedP = new Vector3D(x, u, y, v);
  472.         final Vector3D shiftedV = new Vector3D(xDot, u, yDot, v);
  473.         if (hasNonKeplerianAcceleration) {

  475.             // extract non-Keplerian part of the initial acceleration
  476.             final Vector3D nonKeplerianAcceleration = new Vector3D(1, getPVCoordinates().getAcceleration(),
  477.                                                                    getMu() / (r2 * r), pvP);

  479.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  480.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  481.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  482.             final double   fixedR2 = fixedP.getNormSq();
  483.             final double   fixedR  = FastMath.sqrt(fixedR2);
  484.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  485.                                                   dt, nonKeplerianAcceleration);
  486.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  487.                                                   1, nonKeplerianAcceleration);

  489.             return new PVCoordinates(fixedP, fixedV, fixedA);

  491.         } else {
  492.             // don't include acceleration,
  493.             // so the shifted orbit is not considered to have derivatives
  494.             return new PVCoordinates(shiftedP, shiftedV);
  495.         }

  497.     }

  499.     /** Compute shifted position and velocity in hyperbolic case.
  500.      * @param dt time shift
  501.      * @return shifted position and velocity
  502.      */
  503.     private PVCoordinates shiftPVHyperbolic(final double dt) {

  505.         final PVCoordinates pv = getPVCoordinates();
  506.         final Vector3D pvP   = pv.getPosition();
  507.         final Vector3D pvV   = pv.getVelocity();
  508.         final Vector3D pvM   = pv.getMomentum();
  509.         final double r2      = pvP.getNormSq();
  510.         final double r       = FastMath.sqrt(r2);
  511.         final double rV2OnMu = r * pvV.getNormSq() / getMu();
  512.         final double a       = getA();
  513.         final double muA     = getMu() * a;
  514.         final double e       = FastMath.sqrt(1 - Vector3D.dotProduct(pvM, pvM) / muA);
  515.         final double sqrt    = FastMath.sqrt((e + 1) / (e - 1));

  517.         // compute mean anomaly
  518.         final double eSH     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(-muA);
  519.         final double eCH     = rV2OnMu - 1;
  520.         final double H0      = FastMath.log((eCH + eSH) / (eCH - eSH)) / 2;
  521.         final double M0      = e * FastMath.sinh(H0) - H0;

  523.         // find canonical 2D frame with p pointing to perigee
  524.         final double v0      = 2 * FastMath.atan(sqrt * FastMath.tanh(H0 / 2));
  525.         final Vector3D p     = new Rotation(pvM, v0, RotationConvention.FRAME_TRANSFORM).applyTo(pvP).normalize();
  526.         final Vector3D q     = Vector3D.crossProduct(pvM, p).normalize();

  528.         // compute shifted eccentric anomaly
  529.         final double M1      = M0 + getKeplerianMeanMotion() * dt;
  530.         final double H1      = meanToHyperbolicEccentric(M1, e);

  532.         // compute shifted in-plane Cartesian coordinates
  533.         final double cH     = FastMath.cosh(H1);
  534.         final double sH     = FastMath.sinh(H1);
  535.         final double sE2m1  = FastMath.sqrt((e - 1) * (e + 1));

  537.         // coordinates of position and velocity in the orbital plane
  538.         final double x      = a * (cH - e);
  539.         final double y      = -a * sE2m1 * sH;
  540.         final double factor = FastMath.sqrt(getMu() / -a) / (e * cH - 1);
  541.         final double xDot   = -factor * sH;
  542.         final double yDot   =  factor * sE2m1 * cH;

  544.         final Vector3D shiftedP = new Vector3D(x, p, y, q);
  545.         final Vector3D shiftedV = new Vector3D(xDot, p, yDot, q);
  546.         if (hasNonKeplerianAcceleration) {

  548.             // extract non-Keplerian part of the initial acceleration
  549.             final Vector3D nonKeplerianAcceleration = new Vector3D(1, getPVCoordinates().getAcceleration(),
  550.                                                                    getMu() / (r2 * r), pvP);

  552.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  553.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  554.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  555.             final double   fixedR2 = fixedP.getNormSq();
  556.             final double   fixedR  = FastMath.sqrt(fixedR2);
  557.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  558.                                                   dt, nonKeplerianAcceleration);
  559.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  560.                                                   1, nonKeplerianAcceleration);

  562.             return new PVCoordinates(fixedP, fixedV, fixedA);

  564.         } else {
  565.             // don't include acceleration,
  566.             // so the shifted orbit is not considered to have derivatives
  567.             return new PVCoordinates(shiftedP, shiftedV);
  568.         }

  570.     }

  572.     /** Computes the eccentric in-plane argument from the mean in-plane argument.
  573.      * @param thetaM = mean in-plane argument (rad)
  574.      * @param ex first component of eccentricity vector
  575.      * @param ey second component of eccentricity vector
  576.      * @return the eccentric in-plane argument.
  577.      */
  578.     private double meanToEccentric(final double thetaM, final double ex, final double ey) {
  579.         // Generalization of Kepler equation to in-plane parameters
  580.         // with thetaE = eta + E and
  581.         //      thetaM = eta + M = thetaE - ex.sin(thetaE) + ey.cos(thetaE)
  582.         // and eta being counted from an arbitrary reference in the orbital plane
  583.         double thetaE        = thetaM;
  584.         double thetaEMthetaM = 0.0;
  585.         int    iter          = 0;
  586.         do {
  587.             final SinCos scThetaE = FastMath.sinCos(thetaE);

  589.             final double f2 = ex * scThetaE.sin() - ey * scThetaE.cos();
  590.             final double f1 = 1.0 - ex * scThetaE.cos() - ey * scThetaE.sin();
  591.             final double f0 = thetaEMthetaM - f2;

  593.             final double f12 = 2.0 * f1;
  594.             final double shift = f0 * f12 / (f1 * f12 - f0 * f2);

  596.             thetaEMthetaM -= shift;
  597.             thetaE         = thetaM + thetaEMthetaM;

  599.             if (FastMath.abs(shift) <= 1.0e-12) {
  600.                 return thetaE;
  601.             }

  603.         } while (++iter < 50);

  605.         throw new MathIllegalStateException(LocalizedCoreFormats.CONVERGENCE_FAILED);

  607.     }

  609.     /** Computes the hyperbolic eccentric anomaly from the mean anomaly.
  610.      * <p>
  611.      * The algorithm used here for solving hyperbolic Kepler equation is
  612.      * Danby's iterative method (3rd order) with Vallado's initial guess.
  613.      * </p>
  614.      * @param M mean anomaly (rad)
  615.      * @param ecc eccentricity
  616.      * @return the hyperbolic eccentric anomaly
  617.      */
  618.     private double meanToHyperbolicEccentric(final double M, final double ecc) {

  620.         // Resolution of hyperbolic Kepler equation for Keplerian parameters

  622.         // Initial guess
  623.         double H;
  624.         if (ecc < 1.6) {
  625.             if (-FastMath.PI < M && M < 0. || M > FastMath.PI) {
  626.                 H = M - ecc;
  627.             } else {
  628.                 H = M + ecc;
  629.             }
  630.         } else {
  631.             if (ecc < 3.6 && FastMath.abs(M) > FastMath.PI) {
  632.                 H = M - FastMath.copySign(ecc, M);
  633.             } else {
  634.                 H = M / (ecc - 1.);
  635.             }
  636.         }

  638.         // Iterative computation
  639.         int iter = 0;
  640.         do {
  641.             final double f3  = ecc * FastMath.cosh(H);
  642.             final double f2  = ecc * FastMath.sinh(H);
  643.             final double f1  = f3 - 1.;
  644.             final double f0  = f2 - H - M;
  645.             final double f12 = 2. * f1;
  646.             final double d   = f0 / f12;
  647.             final double fdf = f1 - d * f2;
  648.             final double ds  = f0 / fdf;

  650.             final double shift = f0 / (fdf + ds * ds * f3 / 6.);

  652.             H -= shift;

  654.             if (FastMath.abs(shift) <= 1.0e-12) {
  655.                 return H;
  656.             }

  658.         } while (++iter < 50);

  660.         throw new MathIllegalStateException(OrekitMessages.UNABLE_TO_COMPUTE_HYPERBOLIC_ECCENTRIC_ANOMALY,
  661.                                             iter);
  662.     }

  664.     /** Create a 6x6 identity matrix.
  665.      * @return 6x6 identity matrix
  666.      */
  667.     private double[][] create6x6Identity() {
  668.         // this is the fastest way to set the 6x6 identity matrix
  669.         final double[][] identity = new double[6][6];
  670.         for (int i = 0; i < 6; i++) {
  671.             identity[i][i] = 1.0;
  672.         }
  673.         return identity;
  674.     }

  676.     @Override
  677.     protected double[][] computeJacobianMeanWrtCartesian() {
  678.         return create6x6Identity();
  679.     }

  681.     @Override
  682.     protected double[][] computeJacobianEccentricWrtCartesian() {
  683.         return create6x6Identity();
  684.     }

  686.     @Override
  687.     protected double[][] computeJacobianTrueWrtCartesian() {
  688.         return create6x6Identity();
  689.     }

  691.     /** {@inheritDoc} */
  692.     public void addKeplerContribution(final PositionAngle type, final double gm,
  693.                                       final double[] pDot) {

  695.         final PVCoordinates pv = getPVCoordinates();

  697.         // position derivative is velocity
  698.         final Vector3D velocity = pv.getVelocity();
  699.         pDot[0] += velocity.getX();
  700.         pDot[1] += velocity.getY();
  701.         pDot[2] += velocity.getZ();

  703.         // velocity derivative is Newtonian acceleration
  704.         final Vector3D position = pv.getPosition();
  705.         final double r2         = position.getNormSq();
  706.         final double coeff      = -gm / (r2 * FastMath.sqrt(r2));
  707.         pDot[3] += coeff * position.getX();
  708.         pDot[4] += coeff * position.getY();
  709.         pDot[5] += coeff * position.getZ();

  711.     }

  713.     /**  Returns a string representation of this Orbit object.
  714.      * @return a string representation of this object
  715.      */
  716.     public String toString() {
  717.         // use only the six defining elements, like the other Orbit.toString() methods
  718.         final String comma = ", ";
  719.         final PVCoordinates pv = getPVCoordinates();
  720.         final Vector3D p = pv.getPosition();
  721.         final Vector3D v = pv.getVelocity();
  722.         return "Cartesian parameters: {P(" +
  723.                 p.getX() + comma +
  724.                 p.getY() + comma +
  725.                 p.getZ() + "), V(" +
  726.                 v.getX() + comma +
  727.                 v.getY() + comma +
  728.                 v.getZ() + ")}";
  729.     }

  731.     /** Replace the instance with a data transfer object for serialization.
  732.      * <p>
  733.      * This intermediate class serializes all needed parameters,
  734.      * including position-velocity which are <em>not</em> serialized by parent
  735.      * {@link Orbit} class.
  736.      * </p>
  737.      * @return data transfer object that will be serialized
  738.      */
  739.     @DefaultDataContext
  740.     private Object writeReplace() {
  741.         return new DTO(this);
  742.     }

  744.     /** Internal class used only for serialization. */
  745.     @DefaultDataContext
  746.     private static class DTO implements Serializable {

  748.         /** Serializable UID. */
  749.         private static final long serialVersionUID = 20170414L;

  751.         /** Double values. */
  752.         private double[] d;

  754.         /** Frame in which are defined the orbital parameters. */
  755.         private final Frame frame;

  757.         /** Simple constructor.
  758.          * @param orbit instance to serialize
  759.          */
  760.         private DTO(final CartesianOrbit orbit) {

  762.             final TimeStampedPVCoordinates pv = orbit.getPVCoordinates();

  764.             // decompose date
  765.             final AbsoluteDate j2000Epoch =
  766.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  767.             final double epoch  = FastMath.floor(pv.getDate().durationFrom(j2000Epoch));
  768.             final double offset = pv.getDate().durationFrom(j2000Epoch.shiftedBy(epoch));

  770.             if (orbit.hasDerivatives()) {
  771.                 this.d = new double[] {
  772.                     epoch, offset, orbit.getMu(),
  773.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  774.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ(),
  775.                     pv.getAcceleration().getX(), pv.getAcceleration().getY(), pv.getAcceleration().getZ()
  776.                 };
  777.             } else {
  778.                 this.d = new double[] {
  779.                     epoch, offset, orbit.getMu(),
  780.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  781.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ()
  782.                 };
  783.             }

  785.             this.frame = orbit.getFrame();

  787.         }

  789.         /** Replace the deserialized data transfer object with a {@link CartesianOrbit}.
  790.          * @return replacement {@link CartesianOrbit}
  791.          */
  792.         private Object readResolve() {
  793.             final AbsoluteDate j2000Epoch =
  794.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  795.             if (d.length >= 12) {
  796.                 // we have derivatives
  797.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  798.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  799.                                                                        new Vector3D(d[6], d[ 7], d[ 8]),
  800.                                                                        new Vector3D(d[9], d[10], d[11])),
  801.                                           frame, d[2]);
  802.             } else {
  803.                 // we don't have derivatives
  804.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  805.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  806.                                                                        new Vector3D(d[6], d[ 7], d[ 8])),
  807.                                           frame, d[2]);
  808.             }
  809.         }

  811.     }

  813. }
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