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    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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  package org.orekit.propagation.numerical;
  
  import java.util.ArrayList;
  import java.util.Arrays;
  import java.util.Collections;
  import java.util.List;
  
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.ode.ODEIntegrator;
  import org.hipparchus.util.FastMath;
  import org.orekit.annotation.DefaultDataContext;
  import org.orekit.attitudes.Attitude;
  import org.orekit.attitudes.AttitudeProvider;
  import org.orekit.data.DataContext;
  import org.orekit.errors.OrekitException;
  import org.orekit.errors.OrekitIllegalArgumentException;
  import org.orekit.errors.OrekitInternalError;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.forces.ForceModel;
  import org.orekit.forces.gravity.NewtonianAttraction;
  import org.orekit.forces.inertia.InertialForces;
  import org.orekit.frames.Frame;
  import org.orekit.orbits.Orbit;
  import org.orekit.orbits.OrbitType;
  import org.orekit.orbits.PositionAngle;
  import org.orekit.propagation.PropagationType;
  import org.orekit.propagation.Propagator;
  import org.orekit.propagation.SpacecraftState;
  import org.orekit.propagation.events.EventDetector;
  import org.orekit.propagation.integration.AbstractIntegratedPropagator;
  import org.orekit.propagation.integration.StateMapper;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.utils.AbsolutePVCoordinates;
  import org.orekit.utils.PVCoordinates;
  import org.orekit.utils.ParameterDriver;
  import org.orekit.utils.ParameterObserver;
  import org.orekit.utils.TimeStampedPVCoordinates;
  
  /** This class propagates {@link org.orekit.orbits.Orbit orbits} using
   * numerical integration.
   * <p>Numerical propagation is much more accurate than analytical propagation
   * like for example {@link org.orekit.propagation.analytical.KeplerianPropagator
   * Keplerian} or {@link org.orekit.propagation.analytical.EcksteinHechlerPropagator
   * Eckstein-Hechler}, but requires a few more steps to set up to be used properly.
   * Whereas analytical propagators are configured only thanks to their various
   * constructors and can be used immediately after construction, numerical propagators
   * configuration involve setting several parameters between construction time
   * and propagation time.</p>
   * <p>The configuration parameters that can be set are:</p>
   * <ul>
   *   <li>the initial spacecraft state ({@link #setInitialState(SpacecraftState)})</li>
   *   <li>the central attraction coefficient ({@link #setMu(double)})</li>
   *   <li>the various force models ({@link #addForceModel(ForceModel)},
   *   {@link #removeForceModels()})</li>
   *   <li>the {@link OrbitType type} of orbital parameters to be used for propagation
   *   ({@link #setOrbitType(OrbitType)}),
   *   <li>the {@link PositionAngle type} of position angle to be used in orbital parameters
   *   to be used for propagation where it is relevant ({@link
   *   #setPositionAngleType(PositionAngle)}),
   *   <li>whether {@link org.orekit.propagation.integration.AdditionalEquations additional equations}
   *   (for example {@link PartialDerivativesEquations Jacobians}) should be propagated along with orbital state
   *   ({@link #addAdditionalEquations(org.orekit.propagation.integration.AdditionalEquations)}),
   *   <li>the discrete events that should be triggered during propagation
   *   ({@link #addEventDetector(EventDetector)},
   *   {@link #clearEventsDetectors()})</li>
   *   <li>the binding logic with the rest of the application ({@link #getMultiplexer()})</li>
   * </ul>
   * <p>From these configuration parameters, only the initial state is mandatory. The default
   * propagation settings are in {@link OrbitType#EQUINOCTIAL equinoctial} parameters with
   * {@link PositionAngle#TRUE true} longitude argument. If the central attraction coefficient
   * is not explicitly specified, the one used to define the initial orbit will be used.
   * However, specifying only the initial state and perhaps the central attraction coefficient
   * would mean the propagator would use only Keplerian forces. In this case, the simpler {@link
   * org.orekit.propagation.analytical.KeplerianPropagator KeplerianPropagator} class would
   * perhaps be more effective.</p>
   * <p>The underlying numerical integrator set up in the constructor may also have its own
   * configuration parameters. Typical configuration parameters for adaptive stepsize integrators
   * are the min, max and perhaps start step size as well as the absolute and/or relative errors
   * thresholds.</p>
   * <p>The state that is seen by the integrator is a simple seven elements double array.
   * The six first elements are either:
   * <ul>
   *   <li>the {@link org.orekit.orbits.EquinoctialOrbit equinoctial orbit parameters} (a, e<sub>x</sub>,
  *   e<sub>y</sub>, h<sub>x</sub>, h<sub>y</sub>, λ<sub>M</sub> or λ<sub>E</sub>
  *   or λ<sub>v</sub>) in meters and radians,</li>
  *   <li>the {@link org.orekit.orbits.KeplerianOrbit Keplerian orbit parameters} (a, e, i, ω, Ω,
  *   M or E or v) in meters and radians,</li>
  *   <li>the {@link org.orekit.orbits.CircularOrbit circular orbit parameters} (a, e<sub>x</sub>, e<sub>y</sub>, i,
  *   Ω, α<sub>M</sub> or α<sub>E</sub> or α<sub>v</sub>) in meters
  *   and radians,</li>
  *   <li>the {@link org.orekit.orbits.CartesianOrbit Cartesian orbit parameters} (x, y, z, v<sub>x</sub>,
  *   v<sub>y</sub>, v<sub>z</sub>) in meters and meters per seconds.
  * </ul>
  * <p> The last element is the mass in kilograms.
  *
  * <p>The following code snippet shows a typical setting for Low Earth Orbit propagation in
  * equinoctial parameters and true longitude argument:</p>
  * <pre>
  * final double dP       = 0.001;
  * final double minStep  = 0.001;
  * final double maxStep  = 500;
  * final double initStep = 60;
  * final double[][] tolerance = NumericalPropagator.tolerances(dP, orbit, OrbitType.EQUINOCTIAL);
  * AdaptiveStepsizeIntegrator integrator = new DormandPrince853Integrator(minStep, maxStep, tolerance[0], tolerance[1]);
  * integrator.setInitialStepSize(initStep);
  * propagator = new NumericalPropagator(integrator);
  * </pre>
  * <p>By default, at the end of the propagation, the propagator resets the initial state to the final state,
  * thus allowing a new propagation to be started from there without recomputing the part already performed.
  * This behaviour can be chenged by calling {@link #setResetAtEnd(boolean)}.
  * </p>
  * <p>Beware the same instance cannot be used simultaneously by different threads, the class is <em>not</em>
  * thread-safe.</p>
  *
  * @see SpacecraftState
  * @see ForceModel
  * @see org.orekit.propagation.sampling.OrekitStepHandler
  * @see org.orekit.propagation.sampling.OrekitFixedStepHandler
  * @see org.orekit.propagation.integration.IntegratedEphemeris
  * @see TimeDerivativesEquations
  *
  * @author Mathieu Rom&eacute;ro
  * @author Luc Maisonobe
  * @author Guylaine Prat
  * @author Fabien Maussion
  * @author V&eacute;ronique Pommier-Maurussane
  */
 public class NumericalPropagator extends AbstractIntegratedPropagator {
 
     /** Force models used during the extrapolation of the orbit. */
     private final List<ForceModel> forceModels;
 
     /** boolean to ignore or not the creation of a NewtonianAttraction. */
     private boolean ignoreCentralAttraction = false;
 
     /** Create a new instance of NumericalPropagator, based on orbit definition mu.
      * After creation, the instance is empty, i.e. the attitude provider is set to an
      * unspecified default law and there are no perturbing forces at all.
      * This means that if {@link #addForceModel addForceModel} is not
      * called after creation, the integrated orbit will follow a Keplerian
      * evolution only. The defaults are {@link OrbitType#EQUINOCTIAL}
      * for {@link #setOrbitType(OrbitType) propagation
      * orbit type} and {@link PositionAngle#TRUE} for {@link
      * #setPositionAngleType(PositionAngle) position angle type}.
      *
      * <p>This constructor uses the {@link DataContext#getDefault() default data context}.
      *
      * @param integrator numerical integrator to use for propagation.
      * @see #NumericalPropagator(ODEIntegrator, AttitudeProvider)
      */
     @DefaultDataContext
     public NumericalPropagator(final ODEIntegrator integrator) {
         this(integrator,
                 Propagator.getDefaultLaw(DataContext.getDefault().getFrames()));
     }
 
     /** Create a new instance of NumericalPropagator, based on orbit definition mu.
      * After creation, the instance is empty, i.e. the attitude provider is set to an
      * unspecified default law and there are no perturbing forces at all.
      * This means that if {@link #addForceModel addForceModel} is not
      * called after creation, the integrated orbit will follow a Keplerian
      * evolution only. The defaults are {@link OrbitType#EQUINOCTIAL}
      * for {@link #setOrbitType(OrbitType) propagation
      * orbit type} and {@link PositionAngle#TRUE} for {@link
      * #setPositionAngleType(PositionAngle) position angle type}.
      * @param integrator numerical integrator to use for propagation.
      * @param attitudeProvider the attitude law.
      * @since 10.1
      */
     public NumericalPropagator(final ODEIntegrator integrator,
                                final AttitudeProvider attitudeProvider) {
         super(integrator, PropagationType.MEAN);
         forceModels = new ArrayList<ForceModel>();
         initMapper();
         setAttitudeProvider(attitudeProvider);
         clearStepHandlers();
         setOrbitType(OrbitType.EQUINOCTIAL);
         setPositionAngleType(PositionAngle.TRUE);
     }
 
     /** Set the flag to ignore or not the creation of a {@link NewtonianAttraction}.
      * @param ignoreCentralAttraction if true, {@link NewtonianAttraction} is <em>not</em>
      * added automatically if missing
      */
     public void setIgnoreCentralAttraction(final boolean ignoreCentralAttraction) {
         this.ignoreCentralAttraction = ignoreCentralAttraction;
     }
 
      /** Set the central attraction coefficient μ.
       * <p>
       * Setting the central attraction coefficient is
       * equivalent to {@link #addForceModel(ForceModel) add}
       * a {@link NewtonianAttraction} force model.
       * </p>
      * @param mu central attraction coefficient (m³/s²)
      * @see #addForceModel(ForceModel)
      * @see #getAllForceModels()
      */
     public void setMu(final double mu) {
         if (ignoreCentralAttraction) {
             superSetMu(mu);
         } else {
             addForceModel(new NewtonianAttraction(mu));
         }
     }
 
     /** Set the central attraction coefficient μ only in upper class.
      * @param mu central attraction coefficient (m³/s²)
      */
     private void superSetMu(final double mu) {
         super.setMu(mu);
     }
 
     /** Check if Newtonian attraction force model is available.
      * <p>
      * Newtonian attraction is always the last force model in the list.
      * </p>
      * @return true if Newtonian attraction force model is available
      */
     private boolean hasNewtonianAttraction() {
         final int last = forceModels.size() - 1;
         return last >= 0 && forceModels.get(last) instanceof NewtonianAttraction;
     }
 
     /** Add a force model.
      * <p>If this method is not called at all, the integrated orbit will follow
      * a Keplerian evolution only.</p>
      * @param model {@link ForceModel} to add (it can be either a perturbing force
      * model or an instance of {@link NewtonianAttraction})
      * @see #removeForceModels()
      * @see #setMu(double)
      */
     public void addForceModel(final ForceModel model) {
 
         if (model instanceof NewtonianAttraction) {
             // we want to add the central attraction force model
 
             try {
                 // ensure we are notified of any mu change
                 model.getParametersDrivers().get(0).addObserver(new ParameterObserver() {
                     /** {@inheritDoc} */
                     @Override
                     public void valueChanged(final double previousValue, final ParameterDriver driver) {
                         superSetMu(driver.getValue());
                     }
                 });
             } catch (OrekitException oe) {
                 // this should never happen
                 throw new OrekitInternalError(oe);
             }
 
             if (hasNewtonianAttraction()) {
                 // there is already a central attraction model, replace it
                 forceModels.set(forceModels.size() - 1, model);
             } else {
                 // there are no central attraction model yet, add it at the end of the list
                 forceModels.add(model);
             }
         } else {
             // we want to add a perturbing force model
             if (hasNewtonianAttraction()) {
                 // insert the new force model before Newtonian attraction,
                 // which should always be the last one in the list
                 forceModels.add(forceModels.size() - 1, model);
             } else {
                 // we only have perturbing force models up to now, just append at the end of the list
                 forceModels.add(model);
             }
         }
 
     }
 
     /** Remove all force models (except central attraction).
      * <p>Once all perturbing forces have been removed (and as long as no new force
      * model is added), the integrated orbit will follow a Keplerian evolution
      * only.</p>
      * @see #addForceModel(ForceModel)
      */
     public void removeForceModels() {
         final int last = forceModels.size() - 1;
         if (hasNewtonianAttraction()) {
             // preserve the Newtonian attraction model at the end
             final ForceModel newton = forceModels.get(last);
             forceModels.clear();
             forceModels.add(newton);
         } else {
             forceModels.clear();
         }
     }
 
     /** Get all the force models, perturbing forces and Newtonian attraction included.
      * @return list of perturbing force models, with Newtonian attraction being the
      * last one
      * @see #addForceModel(ForceModel)
      * @see #setMu(double)
      */
     public List<ForceModel> getAllForceModels() {
         return Collections.unmodifiableList(forceModels);
     }
 
     /** Set propagation orbit type.
      * @param orbitType orbit type to use for propagation, null for
      * propagating using {@link org.orekit.utils.AbsolutePVCoordinates} rather than {@link Orbit}
      */
     public void setOrbitType(final OrbitType orbitType) {
         super.setOrbitType(orbitType);
     }
 
     /** Get propagation parameter type.
      * @return orbit type used for propagation, null for
      * propagating using {@link org.orekit.utils.AbsolutePVCoordinates} rather than {@link Orbit}
      */
     public OrbitType getOrbitType() {
         return super.getOrbitType();
     }
 
     /** Set position angle type.
      * <p>
      * The position parameter type is meaningful only if {@link
      * #getOrbitType() propagation orbit type}
      * support it. As an example, it is not meaningful for propagation
      * in {@link OrbitType#CARTESIAN Cartesian} parameters.
      * </p>
      * @param positionAngleType angle type to use for propagation
      */
     public void setPositionAngleType(final PositionAngle positionAngleType) {
         super.setPositionAngleType(positionAngleType);
     }
 
     /** Get propagation parameter type.
      * @return angle type to use for propagation
      */
     public PositionAngle getPositionAngleType() {
         return super.getPositionAngleType();
     }
 
     /** Set the initial state.
      * @param initialState initial state
      */
     public void setInitialState(final SpacecraftState initialState) {
         resetInitialState(initialState);
     }
 
     /** {@inheritDoc} */
     public void resetInitialState(final SpacecraftState state) {
         super.resetInitialState(state);
         if (!hasNewtonianAttraction()) {
             // use the state to define central attraction
             setMu(state.getMu());
         }
         setStartDate(state.getDate());
     }
 
     /** {@inheritDoc} */
     public TimeStampedPVCoordinates getPVCoordinates(final AbsoluteDate date, final Frame frame) {
         return propagate(date).getPVCoordinates(frame);
     }
 
     /** {@inheritDoc} */
     protected StateMapper createMapper(final AbsoluteDate referenceDate, final double mu,
                                        final OrbitType orbitType, final PositionAngle positionAngleType,
                                        final AttitudeProvider attitudeProvider, final Frame frame) {
         return new OsculatingMapper(referenceDate, mu, orbitType, positionAngleType, attitudeProvider, frame);
     }
 
     /** Internal mapper using directly osculating parameters. */
     private static class OsculatingMapper extends StateMapper {
 
         /** Simple constructor.
          * <p>
          * The position parameter type is meaningful only if {@link
          * #getOrbitType() propagation orbit type}
          * support it. As an example, it is not meaningful for propagation
          * in {@link OrbitType#CARTESIAN Cartesian} parameters.
          * </p>
          * @param referenceDate reference date
          * @param mu central attraction coefficient (m³/s²)
          * @param orbitType orbit type to use for mapping (can be null for {@link AbsolutePVCoordinates})
          * @param positionAngleType angle type to use for propagation
          * @param attitudeProvider attitude provider
          * @param frame inertial frame
          */
         OsculatingMapper(final AbsoluteDate referenceDate, final double mu,
                          final OrbitType orbitType, final PositionAngle positionAngleType,
                          final AttitudeProvider attitudeProvider, final Frame frame) {
             super(referenceDate, mu, orbitType, positionAngleType, attitudeProvider, frame);
         }
 
         /** {@inheritDoc} */
         public SpacecraftState mapArrayToState(final AbsoluteDate date, final double[] y, final double[] yDot,
                                                final PropagationType type) {
             // the parameter type is ignored for the Numerical Propagator
 
             final double mass = y[6];
             if (mass <= 0.0) {
                 throw new OrekitException(OrekitMessages.SPACECRAFT_MASS_BECOMES_NEGATIVE, mass);
             }
 
             if (getOrbitType() == null) {
                 // propagation uses absolute position-velocity-acceleration
                 final Vector3D p = new Vector3D(y[0],    y[1],    y[2]);
                 final Vector3D v = new Vector3D(y[3],    y[4],    y[5]);
                 final Vector3D a;
                 final AbsolutePVCoordinates absPva;
                 if (yDot == null) {
                     absPva = new AbsolutePVCoordinates(getFrame(), new TimeStampedPVCoordinates(date, p, v));
                 } else {
                     a = new Vector3D(yDot[3], yDot[4], yDot[5]);
                     absPva = new AbsolutePVCoordinates(getFrame(), new TimeStampedPVCoordinates(date, p, v, a));
                 }
 
                 final Attitude attitude = getAttitudeProvider().getAttitude(absPva, date, getFrame());
                 return new SpacecraftState(absPva, attitude, mass);
             } else {
                 // propagation uses regular orbits
                 final Orbit orbit       = getOrbitType().mapArrayToOrbit(y, yDot, getPositionAngleType(), date, getMu(), getFrame());
                 final Attitude attitude = getAttitudeProvider().getAttitude(orbit, date, getFrame());
 
                 return new SpacecraftState(orbit, attitude, mass);
             }
 
         }
 
         /** {@inheritDoc} */
         public void mapStateToArray(final SpacecraftState state, final double[] y, final double[] yDot) {
             if (getOrbitType() == null) {
                 // propagation uses absolute position-velocity-acceleration
                 final Vector3D p = state.getAbsPVA().getPosition();
                 final Vector3D v = state.getAbsPVA().getVelocity();
                 y[0] = p.getX();
                 y[1] = p.getY();
                 y[2] = p.getZ();
                 y[3] = v.getX();
                 y[4] = v.getY();
                 y[5] = v.getZ();
                 y[6] = state.getMass();
             }
             else {
                 getOrbitType().mapOrbitToArray(state.getOrbit(), getPositionAngleType(), y, yDot);
                 y[6] = state.getMass();
             }
         }
 
     }
 
     /** {@inheritDoc} */
     protected MainStateEquations getMainStateEquations(final ODEIntegrator integrator) {
         return new Main(integrator);
     }
 
     /** Internal class for osculating parameters integration. */
     private class Main implements MainStateEquations, TimeDerivativesEquations {
 
         /** Derivatives array. */
         private final double[] yDot;
 
         /** Current state. */
         private SpacecraftState currentState;
 
         /** Jacobian of the orbital parameters with respect to the Cartesian parameters. */
         private double[][] jacobian;
 
         /** Simple constructor.
          * @param integrator numerical integrator to use for propagation.
          */
         Main(final ODEIntegrator integrator) {
 
             this.yDot     = new double[7];
             this.jacobian = new double[6][6];
 
             for (final ForceModel forceModel : forceModels) {
                 forceModel.getEventsDetectors().forEach(detector -> setUpEventDetector(integrator, detector));
             }
 
             if (getOrbitType() == null) {
                 // propagation uses absolute position-velocity-acceleration
                 // we can set Jacobian once and for all
                 for (int i = 0; i < jacobian.length; ++i) {
                     Arrays.fill(jacobian[i], 0.0);
                     jacobian[i][i] = 1.0;
                 }
             }
 
         }
 
         /** {@inheritDoc} */
         @Override
         public void init(final SpacecraftState initialState, final AbsoluteDate target) {
             for (final ForceModel forceModel : forceModels) {
                 forceModel.init(initialState, target);
             }
         }
 
         /** {@inheritDoc} */
         @Override
         public double[] computeDerivatives(final SpacecraftState state) {
 
             currentState = state;
             Arrays.fill(yDot, 0.0);
             if (getOrbitType() != null) {
                 // propagation uses regular orbits
                 currentState.getOrbit().getJacobianWrtCartesian(getPositionAngleType(), jacobian);
             }
 
             // compute the contributions of all perturbing forces,
             // using the Kepler contribution at the end since
             // NewtonianAttraction is always the last instance in the list
             for (final ForceModel forceModel : forceModels) {
                 forceModel.addContribution(state, this);
             }
 
             if (getOrbitType() == null) {
                 // position derivative is velocity, and was not added above in the force models
                 // (it is added when orbit type is non-null because NewtonianAttraction considers it)
                 final Vector3D velocity = currentState.getPVCoordinates().getVelocity();
                 yDot[0] += velocity.getX();
                 yDot[1] += velocity.getY();
                 yDot[2] += velocity.getZ();
             }
 
 
             return yDot.clone();
 
         }
 
         /** {@inheritDoc} */
         @Override
         public void addKeplerContribution(final double mu) {
             if (getOrbitType() == null) {
 
                 // if mu is neither 0 nor NaN, we want to include Newtonian acceleration
                 if (mu > 0) {
                     // velocity derivative is Newtonian acceleration
                     final Vector3D position = currentState.getPVCoordinates().getPosition();
                     final double r2         = position.getNormSq();
                     final double coeff      = -mu / (r2 * FastMath.sqrt(r2));
                     yDot[3] += coeff * position.getX();
                     yDot[4] += coeff * position.getY();
                     yDot[5] += coeff * position.getZ();
                 }
 
             } else {
                 // propagation uses regular orbits
                 currentState.getOrbit().addKeplerContribution(getPositionAngleType(), mu, yDot);
             }
         }
 
         /** {@inheritDoc} */
         public void addNonKeplerianAcceleration(final Vector3D gamma) {
             for (int i = 0; i < 6; ++i) {
                 final double[] jRow = jacobian[i];
                 yDot[i] += jRow[3] * gamma.getX() + jRow[4] * gamma.getY() + jRow[5] * gamma.getZ();
             }
         }
 
         /** {@inheritDoc} */
         @Override
         public void addMassDerivative(final double q) {
             if (q > 0) {
                 throw new OrekitIllegalArgumentException(OrekitMessages.POSITIVE_FLOW_RATE, q);
             }
             yDot[6] += q;
         }
 
     }
 
     /** Estimate tolerance vectors for integrators when propagating in absolute position-velocity-acceleration.
      * @param dP user specified position error
      * @param absPva reference absolute position-velocity-acceleration
      * @return a two rows array, row 0 being the absolute tolerance error and row 1
      * being the relative tolerance error
      * @see NumericalPropagator#tolerances(double, Orbit, OrbitType)
      */
     public static double[][] tolerances(final double dP, final AbsolutePVCoordinates absPva) {
 
         final double relative = dP / absPva.getPosition().getNorm();
         final double dV = relative * absPva.getVelocity().getNorm();
 
         final double[] absTol = new double[7];
         final double[] relTol = new double[7];
 
         absTol[0] = dP;
         absTol[1] = dP;
         absTol[2] = dP;
         absTol[3] = dV;
         absTol[4] = dV;
         absTol[5] = dV;
 
         // we set the mass tolerance arbitrarily to 1.0e-6 kg, as mass evolves linearly
         // with trust, this often has no influence at all on propagation
         absTol[6] = 1.0e-6;
 
         Arrays.fill(relTol, relative);
 
         return new double[][] {
             absTol, relTol
         };
 
     }
 
     /** Estimate tolerance vectors for integrators when propagating in orbits.
      * <p>
      * The errors are estimated from partial derivatives properties of orbits,
      * starting from a scalar position error specified by the user.
      * Considering the energy conservation equation V = sqrt(mu (2/r - 1/a)),
      * we get at constant energy (i.e. on a Keplerian trajectory):
      * <pre>
      * V² r |dV| = mu |dr|
      * </pre>
      * <p> So we deduce a scalar velocity error consistent with the position error.
      * From here, we apply orbits Jacobians matrices to get consistent errors
      * on orbital parameters.
      *
      * <p>
      * The tolerances are only <em>orders of magnitude</em>, and integrator tolerances
      * are only local estimates, not global ones. So some care must be taken when using
      * these tolerances. Setting 1mm as a position error does NOT mean the tolerances
      * will guarantee a 1mm error position after several orbits integration.
      * </p>
      * @param dP user specified position error
      * @param orbit reference orbit
      * @param type propagation type for the meaning of the tolerance vectors elements
      * (it may be different from {@code orbit.getType()})
      * @return a two rows array, row 0 being the absolute tolerance error and row 1
      * being the relative tolerance error
      */
     public static double[][] tolerances(final double dP, final Orbit orbit, final OrbitType type) {
 
         // estimate the scalar velocity error
         final PVCoordinates pv = orbit.getPVCoordinates();
         final double r2 = pv.getPosition().getNormSq();
         final double v  = pv.getVelocity().getNorm();
         final double dV = orbit.getMu() * dP / (v * r2);
 
         return tolerances(dP, dV, orbit, type);
 
     }
 
     /** Estimate tolerance vectors for integrators when propagating in orbits.
      * <p>
      * The errors are estimated from partial derivatives properties of orbits,
      * starting from scalar position and velocity errors specified by the user.
      * <p>
      * The tolerances are only <em>orders of magnitude</em>, and integrator tolerances
      * are only local estimates, not global ones. So some care must be taken when using
      * these tolerances. Setting 1mm as a position error does NOT mean the tolerances
      * will guarantee a 1mm error position after several orbits integration.
      * </p>
      * @param dP user specified position error
      * @param dV user specified velocity error
      * @param orbit reference orbit
      * @param type propagation type for the meaning of the tolerance vectors elements
      * (it may be different from {@code orbit.getType()})
      * @return a two rows array, row 0 being the absolute tolerance error and row 1
      * being the relative tolerance error
      * @since 10.3
      */
     public static double[][] tolerances(final double dP, final double dV,
                                         final Orbit orbit, final OrbitType type) {
 
         final double[] absTol = new double[7];
         final double[] relTol = new double[7];
 
         // we set the mass tolerance arbitrarily to 1.0e-6 kg, as mass evolves linearly
         // with trust, this often has no influence at all on propagation
         absTol[6] = 1.0e-6;
 
         if (type == OrbitType.CARTESIAN) {
             absTol[0] = dP;
             absTol[1] = dP;
             absTol[2] = dP;
             absTol[3] = dV;
             absTol[4] = dV;
             absTol[5] = dV;
         } else {
 
             // convert the orbit to the desired type
             final double[][] jacobian = new double[6][6];
             final Orbit converted = type.convertType(orbit);
             converted.getJacobianWrtCartesian(PositionAngle.TRUE, jacobian);
 
             for (int i = 0; i < 6; ++i) {
                 final double[] row = jacobian[i];
                 absTol[i] = FastMath.abs(row[0]) * dP +
                             FastMath.abs(row[1]) * dP +
                             FastMath.abs(row[2]) * dP +
                             FastMath.abs(row[3]) * dV +
                             FastMath.abs(row[4]) * dV +
                             FastMath.abs(row[5]) * dV;
                 if (Double.isNaN(absTol[i])) {
                     throw new OrekitException(OrekitMessages.SINGULAR_JACOBIAN_FOR_ORBIT_TYPE, type);
                 }
             }
 
         }
 
         Arrays.fill(relTol, dP / FastMath.sqrt(orbit.getPVCoordinates().getPosition().getNormSq()));
 
         return new double[][] {
             absTol, relTol
         };
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected void beforeIntegration(final SpacecraftState initialState, final AbsoluteDate tEnd) {
 
         if (!getFrame().isPseudoInertial()) {
 
             // inspect all force models to find InertialForces
             for (ForceModel force : forceModels) {
                 if (force instanceof InertialForces) {
                     return;
                 }
             }
 
             // throw exception if no inertial forces found
             throw new OrekitException(OrekitMessages.INERTIAL_FORCE_MODEL_MISSING, getFrame().getName());
 
         }
 
     }
 
 }