CartesianOrbit.java

  1. /* Copyright 2002-2023 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;

  21. import org.hipparchus.analysis.differentiation.UnivariateDerivative2;
  22. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  23. import org.hipparchus.geometry.euclidean.threed.Rotation;
  24. import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  25. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  26. import org.hipparchus.linear.MatrixUtils;
  27. import org.hipparchus.util.FastMath;
  28. import org.hipparchus.util.SinCos;
  29. import org.orekit.annotation.DefaultDataContext;
  30. import org.orekit.data.DataContext;
  31. import org.orekit.frames.Frame;
  32. import org.orekit.time.AbsoluteDate;
  33. import org.orekit.utils.FieldPVCoordinates;
  34. import org.orekit.utils.PVCoordinates;
  35. import org.orekit.utils.TimeStampedPVCoordinates;


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

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

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

  75.     /** Serializable UID. */
  76.     private static final long serialVersionUID = 20170414L;

  78.     /** 6x6 identity matrix. */
  79.     private static final double[][] SIX_BY_SIX_IDENTITY = MatrixUtils.createRealIdentityMatrix(6).getData();

  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(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  = 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 (isElliptical()) {
  210.             // elliptic or circular orbit
  211.             final Vector3D pvP   = 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 Vector3D initPosition() {
  388.         // nothing to do here, as the canonical elements are already the Cartesian ones
  389.         return getPVCoordinates().getPosition();
  390.     }

  392.     /** {@inheritDoc} */
  393.     protected TimeStampedPVCoordinates initPVCoordinates() {
  394.         // nothing to do here, as the canonical elements are already the Cartesian ones
  395.         return getPVCoordinates();
  396.     }

  398.     /** {@inheritDoc} */
  399.     public CartesianOrbit shiftedBy(final double dt) {
  400.         final PVCoordinates shiftedPV = isElliptical() ? shiftPVElliptic(dt) : shiftPVHyperbolic(dt);
  401.         return new CartesianOrbit(shiftedPV, getFrame(), getDate().shiftedBy(dt), getMu());
  402.     }

  404.     /** Compute shifted position and velocity in elliptic case.
  405.      * @param dt time shift
  406.      * @return shifted position and velocity
  407.      */
  408.     private PVCoordinates shiftPVElliptic(final double dt) {

  410.         // preliminary computation
  411.         final PVCoordinates pv = getPVCoordinates();
  412.         final Vector3D pvP     = pv.getPosition();
  413.         final Vector3D pvV     = pv.getVelocity();
  414.         final Vector3D pvM     = pv.getMomentum();
  415.         final double r2        = pvP.getNormSq();
  416.         final double r         = FastMath.sqrt(r2);
  417.         final double rV2OnMu   = r * pvV.getNormSq() / getMu();
  418.         final double a         = r / (2 - rV2OnMu);
  419.         final double muA       = getMu() * a;

  421.         // compute mean anomaly
  422.         final double eSE    = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(muA);
  423.         final double eCE    = rV2OnMu - 1;
  424.         final double E0     = FastMath.atan2(eSE, eCE);
  425.         final double M0     = E0 - eSE;

  427.         final double e         = FastMath.sqrt(eCE * eCE + eSE * eSE);
  428.         final double sqrt      = FastMath.sqrt((1 + e) / (1 - e));

  430.         // find canonical 2D frame with p pointing to perigee
  431.         final double v0     = 2 * FastMath.atan(sqrt * FastMath.tan(E0 / 2));
  432.         final Vector3D p    = new Rotation(pvM, v0, RotationConvention.FRAME_TRANSFORM).applyTo(pvP).normalize();
  433.         final Vector3D q    = Vector3D.crossProduct(pvM, p).normalize();

  435.         // compute shifted eccentric anomaly
  436.         final double M1     = M0 + getKeplerianMeanMotion() * dt;
  437.         final double E1     = KeplerianAnomalyUtility.ellipticMeanToEccentric(e, M1);

  439.         // compute shifted in-plane Cartesian coordinates
  440.         final SinCos scE    = FastMath.sinCos(E1);
  441.         final double cE     = scE.cos();
  442.         final double sE     = scE.sin();
  443.         final double sE2m1  = FastMath.sqrt((1 - e) * (1 + e));

  445.         // coordinates of position and velocity in the orbital plane
  446.         final double x      = a * (cE - e);
  447.         final double y      = a * sE2m1 * sE;
  448.         final double factor = FastMath.sqrt(getMu() / a) / (1 - e * cE);
  449.         final double xDot   = -factor * sE;
  450.         final double yDot   =  factor * sE2m1 * cE;

  452.         final Vector3D shiftedP = new Vector3D(x, p, y, q);
  453.         final Vector3D shiftedV = new Vector3D(xDot, p, yDot, q);
  454.         if (hasNonKeplerianAcceleration) {

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

  460.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  461.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  462.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  463.             final double   fixedR2 = fixedP.getNormSq();
  464.             final double   fixedR  = FastMath.sqrt(fixedR2);
  465.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  466.                                                   dt, nonKeplerianAcceleration);
  467.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  468.                                                   1, nonKeplerianAcceleration);

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

  472.         } else {
  473.             // don't include acceleration,
  474.             // so the shifted orbit is not considered to have derivatives
  475.             return new PVCoordinates(shiftedP, shiftedV);
  476.         }

  478.     }

  480.     /** Compute shifted position and velocity in hyperbolic case.
  481.      * @param dt time shift
  482.      * @return shifted position and velocity
  483.      */
  484.     private PVCoordinates shiftPVHyperbolic(final double dt) {

  486.         final PVCoordinates pv = getPVCoordinates();
  487.         final Vector3D pvP   = pv.getPosition();
  488.         final Vector3D pvV   = pv.getVelocity();
  489.         final Vector3D pvM   = pv.getMomentum();
  490.         final double r2      = pvP.getNormSq();
  491.         final double r       = FastMath.sqrt(r2);
  492.         final double rV2OnMu = r * pvV.getNormSq() / getMu();
  493.         final double a       = getA();
  494.         final double muA     = getMu() * a;
  495.         final double e       = FastMath.sqrt(1 - Vector3D.dotProduct(pvM, pvM) / muA);
  496.         final double sqrt    = FastMath.sqrt((e + 1) / (e - 1));

  498.         // compute mean anomaly
  499.         final double eSH     = Vector3D.dotProduct(pvP, pvV) / FastMath.sqrt(-muA);
  500.         final double eCH     = rV2OnMu - 1;
  501.         final double H0      = FastMath.log((eCH + eSH) / (eCH - eSH)) / 2;
  502.         final double M0      = e * FastMath.sinh(H0) - H0;

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

  509.         // compute shifted eccentric anomaly
  510.         final double M1      = M0 + getKeplerianMeanMotion() * dt;
  511.         final double H1      = KeplerianAnomalyUtility.hyperbolicMeanToEccentric(e, M1);

  513.         // compute shifted in-plane Cartesian coordinates
  514.         final double cH     = FastMath.cosh(H1);
  515.         final double sH     = FastMath.sinh(H1);
  516.         final double sE2m1  = FastMath.sqrt((e - 1) * (e + 1));

  518.         // coordinates of position and velocity in the orbital plane
  519.         final double x      = a * (cH - e);
  520.         final double y      = -a * sE2m1 * sH;
  521.         final double factor = FastMath.sqrt(getMu() / -a) / (e * cH - 1);
  522.         final double xDot   = -factor * sH;
  523.         final double yDot   =  factor * sE2m1 * cH;

  525.         final Vector3D shiftedP = new Vector3D(x, p, y, q);
  526.         final Vector3D shiftedV = new Vector3D(xDot, p, yDot, q);
  527.         if (hasNonKeplerianAcceleration) {

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

  533.             // add the quadratic motion due to the non-Keplerian acceleration to the Keplerian motion
  534.             final Vector3D fixedP   = new Vector3D(1, shiftedP,
  535.                                                    0.5 * dt * dt, nonKeplerianAcceleration);
  536.             final double   fixedR2 = fixedP.getNormSq();
  537.             final double   fixedR  = FastMath.sqrt(fixedR2);
  538.             final Vector3D fixedV  = new Vector3D(1, shiftedV,
  539.                                                   dt, nonKeplerianAcceleration);
  540.             final Vector3D fixedA  = new Vector3D(-getMu() / (fixedR2 * fixedR), shiftedP,
  541.                                                   1, nonKeplerianAcceleration);

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

  545.         } else {
  546.             // don't include acceleration,
  547.             // so the shifted orbit is not considered to have derivatives
  548.             return new PVCoordinates(shiftedP, shiftedV);
  549.         }

  551.     }

  553.     @Override
  554.     protected double[][] computeJacobianMeanWrtCartesian() {
  555.         return SIX_BY_SIX_IDENTITY;
  556.     }

  558.     @Override
  559.     protected double[][] computeJacobianEccentricWrtCartesian() {
  560.         return SIX_BY_SIX_IDENTITY;
  561.     }

  563.     @Override
  564.     protected double[][] computeJacobianTrueWrtCartesian() {
  565.         return SIX_BY_SIX_IDENTITY;
  566.     }

  568.     /** {@inheritDoc} */
  569.     public void addKeplerContribution(final PositionAngleType type, final double gm,
  570.                                       final double[] pDot) {

  572.         final PVCoordinates pv = getPVCoordinates();

  574.         // position derivative is velocity
  575.         final Vector3D velocity = pv.getVelocity();
  576.         pDot[0] += velocity.getX();
  577.         pDot[1] += velocity.getY();
  578.         pDot[2] += velocity.getZ();

  580.         // velocity derivative is Newtonian acceleration
  581.         final Vector3D position = pv.getPosition();
  582.         final double r2         = position.getNormSq();
  583.         final double coeff      = -gm / (r2 * FastMath.sqrt(r2));
  584.         pDot[3] += coeff * position.getX();
  585.         pDot[4] += coeff * position.getY();
  586.         pDot[5] += coeff * position.getZ();

  588.     }

  590.     /**  Returns a string representation of this Orbit object.
  591.      * @return a string representation of this object
  592.      */
  593.     public String toString() {
  594.         // use only the six defining elements, like the other Orbit.toString() methods
  595.         final String comma = ", ";
  596.         final PVCoordinates pv = getPVCoordinates();
  597.         final Vector3D p = pv.getPosition();
  598.         final Vector3D v = pv.getVelocity();
  599.         return "Cartesian parameters: {P(" +
  600.                 p.getX() + comma +
  601.                 p.getY() + comma +
  602.                 p.getZ() + "), V(" +
  603.                 v.getX() + comma +
  604.                 v.getY() + comma +
  605.                 v.getZ() + ")}";
  606.     }

  608.     /** Replace the instance with a data transfer object for serialization.
  609.      * <p>
  610.      * This intermediate class serializes all needed parameters,
  611.      * including position-velocity which are <em>not</em> serialized by parent
  612.      * {@link Orbit} class.
  613.      * </p>
  614.      * @return data transfer object that will be serialized
  615.      */
  616.     @DefaultDataContext
  617.     private Object writeReplace() {
  618.         return new DTO(this);
  619.     }

  621.     /** Internal class used only for serialization. */
  622.     @DefaultDataContext
  623.     private static class DTO implements Serializable {

  625.         /** Serializable UID. */
  626.         private static final long serialVersionUID = 20170414L;

  628.         /** Double values. */
  629.         private double[] d;

  631.         /** Frame in which are defined the orbital parameters. */
  632.         private final Frame frame;

  634.         /** Simple constructor.
  635.          * @param orbit instance to serialize
  636.          */
  637.         private DTO(final CartesianOrbit orbit) {

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

  641.             // decompose date
  642.             final AbsoluteDate j2000Epoch =
  643.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  644.             final double epoch  = FastMath.floor(pv.getDate().durationFrom(j2000Epoch));
  645.             final double offset = pv.getDate().durationFrom(j2000Epoch.shiftedBy(epoch));

  647.             if (orbit.hasDerivatives()) {
  648.                 this.d = new double[] {
  649.                     epoch, offset, orbit.getMu(),
  650.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  651.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ(),
  652.                     pv.getAcceleration().getX(), pv.getAcceleration().getY(), pv.getAcceleration().getZ()
  653.                 };
  654.             } else {
  655.                 this.d = new double[] {
  656.                     epoch, offset, orbit.getMu(),
  657.                     pv.getPosition().getX(),     pv.getPosition().getY(),     pv.getPosition().getZ(),
  658.                     pv.getVelocity().getX(),     pv.getVelocity().getY(),     pv.getVelocity().getZ()
  659.                 };
  660.             }

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

  664.         }

  666.         /** Replace the deserialized data transfer object with a {@link CartesianOrbit}.
  667.          * @return replacement {@link CartesianOrbit}
  668.          */
  669.         private Object readResolve() {
  670.             final AbsoluteDate j2000Epoch =
  671.                     DataContext.getDefault().getTimeScales().getJ2000Epoch();
  672.             if (d.length >= 12) {
  673.                 // we have derivatives
  674.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  675.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  676.                                                                        new Vector3D(d[6], d[ 7], d[ 8]),
  677.                                                                        new Vector3D(d[9], d[10], d[11])),
  678.                                           frame, d[2]);
  679.             } else {
  680.                 // we don't have derivatives
  681.                 return new CartesianOrbit(new TimeStampedPVCoordinates(j2000Epoch.shiftedBy(d[0]).shiftedBy(d[1]),
  682.                                                                        new Vector3D(d[3], d[ 4], d[ 5]),
  683.                                                                        new Vector3D(d[6], d[ 7], d[ 8])),
  684.                                           frame, d[2]);
  685.             }
  686.         }

  688.     }

  690. }
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