/* Copyright 2002-2017 CS Systèmes d'Information
    * Licensed to CS Systèmes d'Information (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,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.orekit.propagation.semianalytical.dsst.forces;
  
  import java.io.NotSerializableException;
  import java.io.Serializable;
  import java.util.ArrayList;
  import java.util.Collections;
  import java.util.HashMap;
  import java.util.List;
  import java.util.Map;
  import java.util.Set;
  import java.util.SortedMap;
  import java.util.SortedSet;
  import java.util.TreeMap;
  
  import org.hipparchus.analysis.differentiation.DSFactory;
  import org.hipparchus.analysis.differentiation.DerivativeStructure;
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.MathUtils;
  import org.orekit.attitudes.AttitudeProvider;
  import org.orekit.errors.OrekitException;
  import org.orekit.forces.gravity.potential.UnnormalizedSphericalHarmonicsProvider;
  import org.orekit.forces.gravity.potential.UnnormalizedSphericalHarmonicsProvider.UnnormalizedSphericalHarmonics;
  import org.orekit.frames.Frame;
  import org.orekit.frames.Transform;
  import org.orekit.orbits.Orbit;
  import org.orekit.propagation.SpacecraftState;
  import org.orekit.propagation.events.EventDetector;
  import org.orekit.propagation.semianalytical.dsst.utilities.AuxiliaryElements;
  import org.orekit.propagation.semianalytical.dsst.utilities.CoefficientsFactory;
  import org.orekit.propagation.semianalytical.dsst.utilities.GHmsjPolynomials;
  import org.orekit.propagation.semianalytical.dsst.utilities.GammaMnsFunction;
  import org.orekit.propagation.semianalytical.dsst.utilities.JacobiPolynomials;
  import org.orekit.propagation.semianalytical.dsst.utilities.ShortPeriodicsInterpolatedCoefficient;
  import org.orekit.propagation.semianalytical.dsst.utilities.hansen.HansenTesseralLinear;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.utils.TimeSpanMap;
  
  /** Tesseral contribution to the central body gravitational perturbation.
   *  <p>
   *  Only resonant tesserals are considered.
   *  </p>
   *
   *  @author Romain Di Costanzo
   *  @author Pascal Parraud
   */
  public class DSSTTesseral implements DSSTForceModel {
  
      /** Minimum period for analytically averaged high-order resonant
       *  central body spherical harmonics in seconds.
       */
      private static final double MIN_PERIOD_IN_SECONDS = 864000.;
  
      /** Minimum period for analytically averaged high-order resonant
       *  central body spherical harmonics in satellite revolutions.
       */
      private static final double MIN_PERIOD_IN_SAT_REV = 10.;
  
      /** Number of points for interpolation. */
      private static final int INTERPOLATION_POINTS = 3;
  
      /** Retrograde factor I.
       *  <p>
       *  DSST model needs equinoctial orbit as internal representation.
       *  Classical equinoctial elements have discontinuities when inclination
       *  is close to zero. In this representation, I = +1. <br>
       *  To avoid this discontinuity, another representation exists and equinoctial
       *  elements can be expressed in a different way, called "retrograde" orbit.
       *  This implies I = -1. <br>
       *  As Orekit doesn't implement the retrograde orbit, I is always set to +1.
       *  But for the sake of consistency with the theory, the retrograde factor
       *  has been kept in the formulas.
       *  </p>
       */
      private static final int I = 1;
  
      /** Provider for spherical harmonics. */
      private final UnnormalizedSphericalHarmonicsProvider provider;
  
      /** Central body rotating frame. */
      private final Frame bodyFrame;
  
     /** Central body rotation rate (rad/s). */
     private final double centralBodyRotationRate;
 
     /** Central body rotation period (seconds). */
     private final double bodyPeriod;
 
     /** Maximal degree to consider for harmonics potential. */
     private final int maxDegree;
 
     /** Maximal degree to consider for short periodics tesseral harmonics potential (without m-daily). */
     private final int maxDegreeTesseralSP;
 
     /** Maximal degree to consider for short periodics m-daily tesseral harmonics potential . */
     private final int maxDegreeMdailyTesseralSP;
 
     /** Maximal order to consider for harmonics potential. */
     private final int maxOrder;
 
     /** Maximal order to consider for short periodics tesseral harmonics potential (without m-daily). */
     private final int maxOrderTesseralSP;
 
     /** Maximal order to consider for short periodics m-daily tesseral harmonics potential . */
     private final int maxOrderMdailyTesseralSP;
 
     /** List of resonant orders. */
     private final List<Integer> resOrders;
 
     /** Maximum power of the eccentricity to use in summation over s. */
     private int maxEccPow;
 
     /** Maximum power of the eccentricity to use in summation over s for
      * short periodic tesseral harmonics (without m-daily). */
     private final int maxEccPowTesseralSP;
 
     /** Maximum power of the eccentricity to use in summation over s for
      * m-daily tesseral harmonics. */
     private final int maxEccPowMdailyTesseralSP;
 
     /** Maximum value for j. */
     private final int maxFrequencyShortPeriodics;
 
     /** Maximum power of the eccentricity to use in Hansen coefficient Kernel expansion. */
     private int maxHansen;
 
     /** Keplerian period. */
     private double orbitPeriod;
 
     /** Ratio of satellite period to central body rotation period. */
     private double ratio;
 
     // Equinoctial elements (according to DSST notation)
     /** a. */
     private double a;
 
     /** ex. */
     private double k;
 
     /** ey. */
     private double h;
 
     /** hx. */
     private double q;
 
     /** hy. */
     private double p;
 
     /** lm. */
     private double lm;
 
     /** Eccentricity. */
     private double ecc;
 
     // Common factors for potential computation
     /** &Chi; = 1 / sqrt(1 - e²) = 1 / B. */
     private double chi;
 
     /** &Chi;². */
     private double chi2;
 
     // Equinoctial reference frame vectors (according to DSST notation)
     /** Equinoctial frame f vector. */
     private Vector3D f;
 
     /** Equinoctial frame g vector. */
     private Vector3D g;
 
     /** Central body rotation angle θ. */
     private double theta;
 
     /** Direction cosine α. */
     private double alpha;
 
     /** Direction cosine β. */
     private double beta;
 
     /** Direction cosine γ. */
     private double gamma;
 
     // Common factors from equinoctial coefficients
     /** 2 * a / A .*/
     private double ax2oA;
 
     /** 1 / (A * B) .*/
     private double ooAB;
 
     /** B / A .*/
     private double BoA;
 
     /** B / (A * (1 + B)) .*/
     private double BoABpo;
 
     /** C / (2 * A * B) .*/
     private double Co2AB;
 
     /** μ / a .*/
     private double moa;
 
     /** R / a .*/
     private double roa;
 
     /** ecc². */
     private double e2;
 
     /** The satellite mean motion. */
     private double meanMotion;
 
     /** List of non resonant orders with j != 0. */
     private final SortedMap<Integer, List<Integer> > nonResOrders;
 
     /** A two dimensional array that contains the objects needed to build the Hansen coefficients. <br/>
      * The indexes are s + maxDegree and j */
     private HansenTesseralLinear[][] hansenObjects;
 
     /** Fourier coefficients. */
     private FourierCjSjCoefficients cjsjFourier;
 
     /** Short period terms. */
     private TesseralShortPeriodicCoefficients shortPeriodTerms;
 
     /** Factory for the DerivativeStructure instances. */
     private final DSFactory factory;
 
     /** Simple constructor.
      * @param centralBodyFrame rotating body frame
      * @param centralBodyRotationRate central body rotation rate (rad/s)
      * @param provider provider for spherical harmonics
      * @param maxDegreeTesseralSP maximal degree to consider for short periodics tesseral harmonics potential
      *  (must be between 2 and {@code provider.getMaxDegree()})
      * @param maxOrderTesseralSP maximal order to consider for short periodics tesseral harmonics potential
      *  (must be between 0 and {@code provider.getMaxOrder()})
      * @param maxEccPowTesseralSP maximum power of the eccentricity to use in summation over s for
      * short periodic tesseral harmonics (without m-daily), should typically not exceed 4 as higher
      * values will exceed computer capacity
      * @param maxFrequencyShortPeriodics maximum frequency in mean longitude for short periodic computations
      * (typically {@code maxDegreeTesseralSP} + {@code maxEccPowTesseralSP and no more than 12})
      * @param maxDegreeMdailyTesseralSP maximal degree to consider for short periodics m-daily tesseral harmonics potential
      *  (must be between 2 and {@code provider.getMaxDegree()})
      * @param maxOrderMdailyTesseralSP maximal order to consider for short periodics m-daily tesseral harmonics potential
      *  (must be between 0 and {@code provider.getMaxOrder()})
      * @param maxEccPowMdailyTesseralSP maximum power of the eccentricity to use in summation over s for
      * m-daily tesseral harmonics, (must be between 0 and {@code maxDegreeMdailyTesseralSP - 2},
      * but should typically not exceed 4 as higher values will exceed computer capacity)
      * @exception OrekitException if degrees or powers are out of range
      * @since 7.2
      */
     public DSSTTesseral(final Frame centralBodyFrame,
                         final double centralBodyRotationRate,
                         final UnnormalizedSphericalHarmonicsProvider provider,
                         final int maxDegreeTesseralSP, final int maxOrderTesseralSP,
                         final int maxEccPowTesseralSP, final int maxFrequencyShortPeriodics,
                         final int maxDegreeMdailyTesseralSP, final int maxOrderMdailyTesseralSP,
                         final int maxEccPowMdailyTesseralSP)
         throws OrekitException {
 
         // Central body rotating frame
         this.bodyFrame = centralBodyFrame;
 
         //Save the rotation rate
         this.centralBodyRotationRate = centralBodyRotationRate;
 
         // Central body rotation period in seconds
         this.bodyPeriod = MathUtils.TWO_PI / centralBodyRotationRate;
 
         // Provider for spherical harmonics
         this.provider  = provider;
         this.maxDegree = provider.getMaxDegree();
         this.maxOrder  = provider.getMaxOrder();
 
         //set the maximum degree order for short periodics
         checkIndexRange(maxDegreeTesseralSP, 2, maxDegree);
         this.maxDegreeTesseralSP       = maxDegreeTesseralSP;
 
         checkIndexRange(maxDegreeMdailyTesseralSP, 2, maxDegree);
         this.maxDegreeMdailyTesseralSP = maxDegreeMdailyTesseralSP;
 
         checkIndexRange(maxOrderTesseralSP, 0, maxOrder);
         this.maxOrderTesseralSP        = maxOrderTesseralSP;
 
         checkIndexRange(maxOrderMdailyTesseralSP, 0, maxOrder);
         this.maxOrderMdailyTesseralSP  = maxOrderMdailyTesseralSP;
 
         // set the maximum value for eccentricity power
         this.maxEccPowTesseralSP       = maxEccPowTesseralSP;
 
         checkIndexRange(maxEccPowMdailyTesseralSP, 0, maxDegreeMdailyTesseralSP - 2);
         this.maxEccPowMdailyTesseralSP = maxEccPowMdailyTesseralSP;
 
         // set the maximum value for frequency
         this.maxFrequencyShortPeriodics = maxFrequencyShortPeriodics;
 
         // Initialize default values
         this.resOrders = new ArrayList<Integer>();
         this.nonResOrders = new TreeMap<Integer, List <Integer> >();
         this.maxEccPow = 0;
         this.maxHansen = 0;
 
         this.factory = new DSFactory(1, 1);
 
     }
 
     /** Check an index range.
      * @param index index value
      * @param min minimum value for index
      * @param max maximum value for index
      * @exception OrekitException if index is out of range
      */
     private void checkIndexRange(final int index, final int min, final int max)
         throws OrekitException {
         if (index < min || index > max) {
             throw new OrekitException(LocalizedCoreFormats.OUT_OF_RANGE_SIMPLE, index, min, max);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public List<ShortPeriodTerms> initialize(final AuxiliaryElements aux, final boolean meanOnly)
         throws OrekitException {
 
         // Keplerian period
         orbitPeriod = aux.getKeplerianPeriod();
 
         // Set the highest power of the eccentricity in the analytical power
         // series expansion for the averaged high order resonant central body
         // spherical harmonic perturbation
         final double e = aux.getEcc();
         if (e <= 0.005) {
             maxEccPow = 3;
         } else if (e <= 0.02) {
             maxEccPow = 4;
         } else if (e <= 0.1) {
             maxEccPow = 7;
         } else if (e <= 0.2) {
             maxEccPow = 10;
         } else if (e <= 0.3) {
             maxEccPow = 12;
         } else if (e <= 0.4) {
             maxEccPow = 15;
         } else {
             maxEccPow = 20;
         }
 
         // Set the maximum power of the eccentricity to use in Hansen coefficient Kernel expansion.
         maxHansen = maxEccPow / 2;
 
         // Ratio of satellite to central body periods to define resonant terms
         ratio = orbitPeriod / bodyPeriod;
 
         // Compute the resonant tesseral harmonic terms if not set by the user
         getResonantAndNonResonantTerms(meanOnly);
 
         //initialize the HansenTesseralLinear objects needed
         createHansenObjects(meanOnly);
 
         final int mMax = FastMath.max(maxOrderTesseralSP, maxOrderMdailyTesseralSP);
         cjsjFourier = new FourierCjSjCoefficients(maxFrequencyShortPeriodics, mMax);
 
         shortPeriodTerms = new TesseralShortPeriodicCoefficients(bodyFrame, maxOrderMdailyTesseralSP,
                                                                  maxDegreeTesseralSP < 0, nonResOrders,
                                                                  mMax, maxFrequencyShortPeriodics, INTERPOLATION_POINTS,
                                                                  new TimeSpanMap<Slot>(new Slot(mMax, maxFrequencyShortPeriodics,
                                                                                                 INTERPOLATION_POINTS)));
 
         final List<ShortPeriodTerms> list = new ArrayList<ShortPeriodTerms>();
         list.add(shortPeriodTerms);
         return list;
 
     }
 
     /** Create the objects needed for linear transformation.
      *
      * <p>
      * Each {@link org.orekit.propagation.semianalytical.dsst.utilities.hansenHansenTesseralLinear HansenTesseralLinear} uses
      * a fixed value for s and j. Since j varies from -maxJ to +maxJ and s varies from -maxDegree to +maxDegree,
      * a 2 * maxDegree + 1 x 2 * maxJ + 1 matrix of objects should be created. The size of this matrix can be reduced
      * by taking into account the expression (2.7.3-2). This means that is enough to create the objects for  positive
      * values of j and all values of s.
      * </p>
      *
      * @param meanOnly create only the objects required for the mean contribution
      */
     private void createHansenObjects(final boolean meanOnly) {
         //Allocate the two dimensional array
         final int rows     = 2 * maxDegree + 1;
         final int columns  = maxFrequencyShortPeriodics + 1;
         this.hansenObjects = new HansenTesseralLinear[rows][columns];
 
         if (meanOnly) {
             // loop through the resonant orders
             for (int m : resOrders) {
                 //Compute the corresponding j term
                 final int j = FastMath.max(1, (int) FastMath.round(ratio * m));
 
                 //Compute the sMin and sMax values
                 final int sMin = FastMath.min(maxEccPow - j, maxDegree);
                 final int sMax = FastMath.min(maxEccPow + j, maxDegree);
 
                 //loop through the s values
                 for (int s = 0; s <= sMax; s++) {
                     //Compute the n0 value
                     final int n0 = FastMath.max(FastMath.max(2, m), s);
 
                     //Create the object for the pair j, s
                     this.hansenObjects[s + maxDegree][j] = new HansenTesseralLinear(maxDegree, s, j, n0, maxHansen);
 
                     if (s > 0 && s <= sMin) {
                         //Also create the object for the pair j, -s
                         this.hansenObjects[maxDegree - s][j] =  new HansenTesseralLinear(maxDegree, -s, j, n0, maxHansen);
                     }
                 }
             }
         } else {
             // create all objects
             for (int j = 0; j <= maxFrequencyShortPeriodics; j++) {
                 for (int s = -maxDegree; s <= maxDegree; s++) {
                     //Compute the n0 value
                     final int n0 = FastMath.max(2, FastMath.abs(s));
 
                     this.hansenObjects[s + maxDegree][j] = new HansenTesseralLinear(maxDegree, s, j, n0, maxHansen);
                 }
             }
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public void initializeStep(final AuxiliaryElements aux) throws OrekitException {
 
         // Equinoctial elements
         a  = aux.getSma();
         k  = aux.getK();
         h  = aux.getH();
         q  = aux.getQ();
         p  = aux.getP();
         lm = aux.getLM();
 
         // Eccentricity
         ecc = aux.getEcc();
         e2 = ecc * ecc;
 
         // Equinoctial frame vectors
         f = aux.getVectorF();
         g = aux.getVectorG();
 
         // Central body rotation angle from equation 2.7.1-(3)(4).
         final Transform t = bodyFrame.getTransformTo(aux.getFrame(), aux.getDate());
         final Vector3D xB = t.transformVector(Vector3D.PLUS_I);
         final Vector3D yB = t.transformVector(Vector3D.PLUS_J);
         theta = FastMath.atan2(-f.dotProduct(yB) + I * g.dotProduct(xB),
                                 f.dotProduct(xB) + I * g.dotProduct(yB));
 
         // Direction cosines
         alpha = aux.getAlpha();
         beta  = aux.getBeta();
         gamma = aux.getGamma();
 
         // Equinoctial coefficients
         // A = sqrt(μ * a)
         final double A = aux.getA();
         // B = sqrt(1 - h² - k²)
         final double B = aux.getB();
         // C = 1 + p² + q²
         final double C = aux.getC();
         // Common factors from equinoctial coefficients
         // 2 * a / A
         ax2oA  = 2. * a / A;
         // B / A
         BoA  = B / A;
         // 1 / AB
         ooAB = 1. / (A * B);
         // C / 2AB
         Co2AB = C * ooAB / 2.;
         // B / (A * (1 + B))
         BoABpo = BoA / (1. + B);
         // &mu / a
         moa = provider.getMu() / a;
         // R / a
         roa = provider.getAe() / a;
 
         // &Chi; = 1 / B
         chi = 1. / B;
         chi2 = chi * chi;
 
         //mean motion n
         meanMotion = aux.getMeanMotion();
     }
 
     /** {@inheritDoc} */
     @Override
     public double[] getMeanElementRate(final SpacecraftState spacecraftState) throws OrekitException {
 
         // Compute potential derivatives
         final double[] dU  = computeUDerivatives(spacecraftState.getDate());
         final double dUda  = dU[0];
         final double dUdh  = dU[1];
         final double dUdk  = dU[2];
         final double dUdl  = dU[3];
         final double dUdAl = dU[4];
         final double dUdBe = dU[5];
         final double dUdGa = dU[6];
 
         // Compute the cross derivative operator :
         final double UAlphaGamma   = alpha * dUdGa - gamma * dUdAl;
         final double UAlphaBeta    = alpha * dUdBe - beta  * dUdAl;
         final double UBetaGamma    =  beta * dUdGa - gamma * dUdBe;
         final double Uhk           =     h * dUdk  -     k * dUdh;
         final double pUagmIqUbgoAB = (p * UAlphaGamma - I * q * UBetaGamma) * ooAB;
         final double UhkmUabmdUdl  =  Uhk - UAlphaBeta - dUdl;
 
         final double da =  ax2oA * dUdl;
         final double dh =    BoA * dUdk + k * pUagmIqUbgoAB - h * BoABpo * dUdl;
         final double dk =  -(BoA * dUdh + h * pUagmIqUbgoAB + k * BoABpo * dUdl);
         final double dp =  Co2AB * (p * UhkmUabmdUdl - UBetaGamma);
         final double dq =  Co2AB * (q * UhkmUabmdUdl - I * UAlphaGamma);
         final double dM = -ax2oA * dUda + BoABpo * (h * dUdh + k * dUdk) + pUagmIqUbgoAB;
 
         return new double[] {da, dk, dh, dq, dp, dM};
     }
 
     /** {@inheritDoc} */
     @Override
     public void updateShortPeriodTerms(final SpacecraftState... meanStates)
         throws OrekitException {
 
         final Slot slot = shortPeriodTerms.createSlot(meanStates);
 
         for (final SpacecraftState meanState : meanStates) {
 
             initializeStep(new AuxiliaryElements(meanState.getOrbit(), I));
 
             // Initialise the Hansen coefficients
             for (int s = -maxDegree; s <= maxDegree; s++) {
                 // coefficients with j == 0 are always needed
                 this.hansenObjects[s + maxDegree][0].computeInitValues(e2, chi, chi2);
                 if (maxDegreeTesseralSP >= 0) {
                     // initialize other objects only if required
                     for (int j = 1; j <= maxFrequencyShortPeriodics; j++) {
                         this.hansenObjects[s + maxDegree][j].computeInitValues(e2, chi, chi2);
                     }
                 }
             }
 
             // Compute coefficients
             // Compute only if there is at least one non-resonant tesseral
             if (!nonResOrders.isEmpty() || maxDegreeTesseralSP < 0) {
                 // Generate the fourrier coefficients
                 cjsjFourier.generateCoefficients(meanState.getDate());
 
                 // the coefficient 3n / 2a
                 final double tnota = 1.5 * meanMotion / a;
 
                 // build the mDaily coefficients
                 for (int m = 1; m <= maxOrderMdailyTesseralSP; m++) {
                     // build the coefficients
                     buildCoefficients(meanState.getDate(), slot, m, 0, tnota);
                 }
 
                 if (maxDegreeTesseralSP >= 0) {
                     // generate the other coefficients, if required
                     for (final Map.Entry<Integer, List<Integer>> entry : nonResOrders.entrySet()) {
 
                         for (int j : entry.getValue()) {
                             // build the coefficients
                             buildCoefficients(meanState.getDate(), slot, entry.getKey(), j, tnota);
                         }
                     }
                 }
             }
 
         }
 
     }
 
     /** Build a set of coefficients.
      *
      * @param date the current date
      * @param slot slot to which the coefficients belong
      * @param m m index
      * @param j j index
      * @param tnota 3n/2a
      */
     private void buildCoefficients(final AbsoluteDate date, final Slot slot,
                                    final int m, final int j, final double tnota) {
         // Create local arrays
         final double[] currentCijm = new double[] {0., 0., 0., 0., 0., 0.};
         final double[] currentSijm = new double[] {0., 0., 0., 0., 0., 0.};
 
         // compute the term 1 / (jn - mθ<sup>.</sup>)
         final double oojnmt = 1. / (j * meanMotion - m * centralBodyRotationRate);
 
         // initialise the coeficients
         for (int i = 0; i < 6; i++) {
             currentCijm[i] = -cjsjFourier.getSijm(i, j, m);
             currentSijm[i] = cjsjFourier.getCijm(i, j, m);
         }
         // Add the separate part for δ<sub>6i</sub>
         currentCijm[5] += tnota * oojnmt * cjsjFourier.getCijm(0, j, m);
         currentSijm[5] += tnota * oojnmt * cjsjFourier.getSijm(0, j, m);
 
         //Multiply by 1 / (jn - mθ<sup>.</sup>)
         for (int i = 0; i < 6; i++) {
             currentCijm[i] *= oojnmt;
             currentSijm[i] *= oojnmt;
         }
 
         // Add the coefficients to the interpolation grid
         slot.cijm[m][j + maxFrequencyShortPeriodics].addGridPoint(date, currentCijm);
         slot.sijm[m][j + maxFrequencyShortPeriodics].addGridPoint(date, currentSijm);
 
     }
 
     /** {@inheritDoc} */
     @Override
     public EventDetector[] getEventsDetectors() {
         return null;
     }
 
      /**
       * Get the resonant and non-resonant tesseral terms in the central body spherical harmonic field.
       *
       * @param resonantOnly extract only resonant terms
       */
     private void getResonantAndNonResonantTerms(final boolean resonantOnly) {
 
         // Compute natural resonant terms
         final double tolerance = 1. / FastMath.max(MIN_PERIOD_IN_SAT_REV,
                                                    MIN_PERIOD_IN_SECONDS / orbitPeriod);
 
         // Search the resonant orders in the tesseral harmonic field
         resOrders.clear();
         nonResOrders.clear();
         for (int m = 1; m <= maxOrder; m++) {
             final double resonance = ratio * m;
             int jRes = 0;
             final int jComputedRes = (int) FastMath.round(resonance);
             if (jComputedRes > 0 && jComputedRes <= maxFrequencyShortPeriodics && FastMath.abs(resonance - jComputedRes) <= tolerance) {
                 // Store each resonant index and order
                 resOrders.add(m);
                 jRes = jComputedRes;
             }
 
             if (!resonantOnly && maxDegreeTesseralSP >= 0 && m <= maxOrderTesseralSP) {
                 //compute non resonant orders in the tesseral harmonic field
                 final List<Integer> listJofM = new ArrayList<Integer>();
                 //for the moment we take only the pairs (j,m) with |j| <= maxDegree + maxEccPow (from |s-j| <= maxEccPow and |s| <= maxDegree)
                 for (int j = -maxFrequencyShortPeriodics; j <= maxFrequencyShortPeriodics; j++) {
                     if (j != 0 && j != jRes) {
                         listJofM.add(j);
                     }
                 }
 
                 nonResOrders.put(m, listJofM);
             }
         }
     }
 
     /** Computes the potential U derivatives.
      *  <p>The following elements are computed from expression 3.3 - (4).
      *  <pre>
      *  dU / da
      *  dU / dh
      *  dU / dk
      *  dU / dλ
      *  dU / dα
      *  dU / dβ
      *  dU / dγ
      *  </pre>
      *  </p>
      *
      *  @param date current date
      *  @return potential derivatives
      *  @throws OrekitException if an error occurs
      */
     private double[] computeUDerivatives(final AbsoluteDate date) throws OrekitException {
 
         // Potential derivatives
         double dUda  = 0.;
         double dUdh  = 0.;
         double dUdk  = 0.;
         double dUdl  = 0.;
         double dUdAl = 0.;
         double dUdBe = 0.;
         double dUdGa = 0.;
 
         // Compute only if there is at least one resonant tesseral
         if (!resOrders.isEmpty()) {
             // Gmsj and Hmsj polynomials
             final GHmsjPolynomials ghMSJ = new GHmsjPolynomials(k, h, alpha, beta, I);
 
             // GAMMAmns function
             final GammaMnsFunction gammaMNS = new GammaMnsFunction(maxDegree, gamma, I);
 
             // R / a up to power degree
             final double[] roaPow = new double[maxDegree + 1];
             roaPow[0] = 1.;
             for (int i = 1; i <= maxDegree; i++) {
                 roaPow[i] = roa * roaPow[i - 1];
             }
 
             // SUM over resonant terms {j,m}
             for (int m : resOrders) {
 
                 // Resonant index for the current resonant order
                 final int j = FastMath.max(1, (int) FastMath.round(ratio * m));
 
                 // Phase angle
                 final double jlMmt  = j * lm - m * theta;
                 final double sinPhi = FastMath.sin(jlMmt);
                 final double cosPhi = FastMath.cos(jlMmt);
 
                 // Potential derivatives components for a given resonant pair {j,m}
                 double dUdaCos  = 0.;
                 double dUdaSin  = 0.;
                 double dUdhCos  = 0.;
                 double dUdhSin  = 0.;
                 double dUdkCos  = 0.;
                 double dUdkSin  = 0.;
                 double dUdlCos  = 0.;
                 double dUdlSin  = 0.;
                 double dUdAlCos = 0.;
                 double dUdAlSin = 0.;
                 double dUdBeCos = 0.;
                 double dUdBeSin = 0.;
                 double dUdGaCos = 0.;
                 double dUdGaSin = 0.;
 
                 // s-SUM from -sMin to sMax
                 final int sMin = FastMath.min(maxEccPow - j, maxDegree);
                 final int sMax = FastMath.min(maxEccPow + j, maxDegree);
                 for (int s = 0; s <= sMax; s++) {
 
                     //Compute the initial values for Hansen coefficients using newComb operators
                     this.hansenObjects[s + maxDegree][j].computeInitValues(e2, chi, chi2);
 
                     // n-SUM for s positive
                     final double[][] nSumSpos = computeNSum(date, j, m, s, maxDegree,
                                                             roaPow, ghMSJ, gammaMNS);
                     dUdaCos  += nSumSpos[0][0];
                     dUdaSin  += nSumSpos[0][1];
                     dUdhCos  += nSumSpos[1][0];
                     dUdhSin  += nSumSpos[1][1];
                     dUdkCos  += nSumSpos[2][0];
                     dUdkSin  += nSumSpos[2][1];
                     dUdlCos  += nSumSpos[3][0];
                     dUdlSin  += nSumSpos[3][1];
                     dUdAlCos += nSumSpos[4][0];
                     dUdAlSin += nSumSpos[4][1];
                     dUdBeCos += nSumSpos[5][0];
                     dUdBeSin += nSumSpos[5][1];
                     dUdGaCos += nSumSpos[6][0];
                     dUdGaSin += nSumSpos[6][1];
 
                     // n-SUM for s negative
                     if (s > 0 && s <= sMin) {
                         //Compute the initial values for Hansen coefficients using newComb operators
                         this.hansenObjects[maxDegree - s][j].computeInitValues(e2, chi, chi2);
 
                         final double[][] nSumSneg = computeNSum(date, j, m, -s, maxDegree,
                                                                 roaPow, ghMSJ, gammaMNS);
                         dUdaCos  += nSumSneg[0][0];
                         dUdaSin  += nSumSneg[0][1];
                         dUdhCos  += nSumSneg[1][0];
                         dUdhSin  += nSumSneg[1][1];
                         dUdkCos  += nSumSneg[2][0];
                         dUdkSin  += nSumSneg[2][1];
                         dUdlCos  += nSumSneg[3][0];
                         dUdlSin  += nSumSneg[3][1];
                         dUdAlCos += nSumSneg[4][0];
                         dUdAlSin += nSumSneg[4][1];
                         dUdBeCos += nSumSneg[5][0];
                         dUdBeSin += nSumSneg[5][1];
                         dUdGaCos += nSumSneg[6][0];
                         dUdGaSin += nSumSneg[6][1];
                     }
                 }
 
                 // Assembly of potential derivatives componants
                 dUda  += cosPhi * dUdaCos  + sinPhi * dUdaSin;
                 dUdh  += cosPhi * dUdhCos  + sinPhi * dUdhSin;
                 dUdk  += cosPhi * dUdkCos  + sinPhi * dUdkSin;
                 dUdl  += cosPhi * dUdlCos  + sinPhi * dUdlSin;
                 dUdAl += cosPhi * dUdAlCos + sinPhi * dUdAlSin;
                 dUdBe += cosPhi * dUdBeCos + sinPhi * dUdBeSin;
                 dUdGa += cosPhi * dUdGaCos + sinPhi * dUdGaSin;
             }
 
             dUda  *= -moa / a;
             dUdh  *=  moa;
             dUdk  *=  moa;
             dUdl  *=  moa;
             dUdAl *=  moa;
             dUdBe *=  moa;
             dUdGa *=  moa;
         }
 
         return new double[] {dUda, dUdh, dUdk, dUdl, dUdAl, dUdBe, dUdGa};
     }
 
     /** Compute the n-SUM for potential derivatives components.
      *  @param date current date
      *  @param j resonant index <i>j</i>
      *  @param m resonant order <i>m</i>
      *  @param s d'Alembert characteristic <i>s</i>
      *  @param maxN maximum possible value for <i>n</i> index
      *  @param roaPow powers of R/a up to degree <i>n</i>
      *  @param ghMSJ G<sup>j</sup><sub>m,s</sub> and H<sup>j</sup><sub>m,s</sub> polynomials
      *  @param gammaMNS &Gamma;<sup>m</sup><sub>n,s</sub>(γ) function
      *  @return Components of U<sub>n</sub> derivatives for fixed j, m, s
      * @throws OrekitException if some error occurred
      */
     private double[][] computeNSum(final AbsoluteDate date,
                                    final int j, final int m, final int s, final int maxN, final double[] roaPow,
                                    final GHmsjPolynomials ghMSJ, final GammaMnsFunction gammaMNS)
         throws OrekitException {
 
         //spherical harmonics
         final UnnormalizedSphericalHarmonics harmonics = provider.onDate(date);
 
         // Potential derivatives components
         double dUdaCos  = 0.;
         double dUdaSin  = 0.;
         double dUdhCos  = 0.;
         double dUdhSin  = 0.;
         double dUdkCos  = 0.;
         double dUdkSin  = 0.;
         double dUdlCos  = 0.;
         double dUdlSin  = 0.;
         double dUdAlCos = 0.;
         double dUdAlSin = 0.;
         double dUdBeCos = 0.;
         double dUdBeSin = 0.;
         double dUdGaCos = 0.;
         double dUdGaSin = 0.;
 
         // I^m
         @SuppressWarnings("unused")
         final int Im = I > 0 ? 1 : (m % 2 == 0 ? 1 : -1);
 
         // jacobi v, w, indices from 2.7.1-(15)
         final int v = FastMath.abs(m - s);
         final int w = FastMath.abs(m + s);
 
         // Initialise lower degree nmin = (Max(2, m, |s|)) for summation over n
         final int nmin = FastMath.max(FastMath.max(2, m), FastMath.abs(s));
 
         //Get the corresponding Hansen object
         final int sIndex = maxDegree + (j < 0 ? -s : s);
         final int jIndex = FastMath.abs(j);
         final HansenTesseralLinear hans = this.hansenObjects[sIndex][jIndex];
 
         // n-SUM from nmin to N
         for (int n = nmin; n <= maxN; n++) {
             // If (n - s) is odd, the contribution is null because of Vmns
             if ((n - s) % 2 == 0) {
 
                 // Vmns coefficient
                 final double vMNS   = CoefficientsFactory.getVmns(m, n, s);
 
                 // Inclination function Gamma and derivative
                 final double gaMNS  = gammaMNS.getValue(m, n, s);
                 final double dGaMNS = gammaMNS.getDerivative(m, n, s);
 
                 // Hansen kernel value and derivative
                 final double kJNS   = hans.getValue(-n - 1, chi);
                 final double dkJNS  = hans.getDerivative(-n - 1, chi);
 
                 // Gjms, Hjms polynomials and derivatives
                 final double gMSJ   = ghMSJ.getGmsj(m, s, j);
                 final double hMSJ   = ghMSJ.getHmsj(m, s, j);
                 final double dGdh   = ghMSJ.getdGmsdh(m, s, j);
                 final double dGdk   = ghMSJ.getdGmsdk(m, s, j);
                 final double dGdA   = ghMSJ.getdGmsdAlpha(m, s, j);
                 final double dGdB   = ghMSJ.getdGmsdBeta(m, s, j);
                 final double dHdh   = ghMSJ.getdHmsdh(m, s, j);
                 final double dHdk   = ghMSJ.getdHmsdk(m, s, j);
                 final double dHdA   = ghMSJ.getdHmsdAlpha(m, s, j);
                 final double dHdB   = ghMSJ.getdHmsdBeta(m, s, j);
 
                 // Jacobi l-index from 2.7.1-(15)
                 final int l = FastMath.min(n - m, n - FastMath.abs(s));
                 // Jacobi polynomial and derivative
                 final DerivativeStructure jacobi =
                         JacobiPolynomials.getValue(l, v, w, factory.variable(0, gamma));
 
                 // Geopotential coefficients
                 final double cnm = harmonics.getUnnormalizedCnm(n, m);
                 final double snm = harmonics.getUnnormalizedSnm(n, m);
 
                 // Common factors from expansion of equations 3.3-4
                 final double cf_0      = roaPow[n] * Im * vMNS;
                 final double cf_1      = cf_0 * gaMNS * jacobi.getValue();
                 final double cf_2      = cf_1 * kJNS;
                 final double gcPhs     = gMSJ * cnm + hMSJ * snm;
                 final double gsMhc     = gMSJ * snm - hMSJ * cnm;
                 final double dKgcPhsx2 = 2. * dkJNS * gcPhs;
                 final double dKgsMhcx2 = 2. * dkJNS * gsMhc;
                 final double dUdaCoef  = (n + 1) * cf_2;
                 final double dUdlCoef  = j * cf_2;
                 final double dUdGaCoef = cf_0 * kJNS * (jacobi.getValue() * dGaMNS + gaMNS * jacobi.getPartialDerivative(1));
 
                 // dU / da components
                 dUdaCos  += dUdaCoef * gcPhs;
                 dUdaSin  += dUdaCoef * gsMhc;
 
                 // dU / dh components
                 dUdhCos  += cf_1 * (kJNS * (cnm * dGdh + snm * dHdh) + h * dKgcPhsx2);
                 dUdhSin  += cf_1 * (kJNS * (snm * dGdh - cnm * dHdh) + h * dKgsMhcx2);
 
                 // dU / dk components
                 dUdkCos  += cf_1 * (kJNS * (cnm * dGdk + snm * dHdk) + k * dKgcPhsx2);
                 dUdkSin  += cf_1 * (kJNS * (snm * dGdk - cnm * dHdk) + k * dKgsMhcx2);
 
                 // dU / dLambda components
                 dUdlCos  +=  dUdlCoef * gsMhc;
                 dUdlSin  += -dUdlCoef * gcPhs;
 
                 // dU / alpha components
                 dUdAlCos += cf_2 * (dGdA * cnm + dHdA * snm);
                 dUdAlSin += cf_2 * (dGdA * snm - dHdA * cnm);
 
                 // dU / dBeta components
                 dUdBeCos += cf_2 * (dGdB * cnm + dHdB * snm);
                 dUdBeSin += cf_2 * (dGdB * snm - dHdB * cnm);
 
                 // dU / dGamma components
                 dUdGaCos += dUdGaCoef * gcPhs;
                 dUdGaSin += dUdGaCoef * gsMhc;
             }
         }
 
         return new double[][] {{dUdaCos,  dUdaSin},
                                {dUdhCos,  dUdhSin},
                                {dUdkCos,  dUdkSin},
                                {dUdlCos,  dUdlSin},
                                {dUdAlCos, dUdAlSin},
                                {dUdBeCos, dUdBeSin},
                                {dUdGaCos, dUdGaSin}};
     }
 
     /** {@inheritDoc} */
     @Override
     public void registerAttitudeProvider(final AttitudeProvider attitudeProvider) {
         //nothing is done since this contribution is not sensitive to attitude
     }
 
     /** Compute the C<sup>j</sup> and the S<sup>j</sup> coefficients.
      *  <p>
      *  Those coefficients are given in Danielson paper by substituting the
      *  disturbing function (2.7.1-16) with m != 0 into (2.2-10)
      *  </p>
      */
     private class FourierCjSjCoefficients {
 
         /** Absolute limit for j ( -jMax <= j <= jMax ).  */
         private final int jMax;
 
         /** The C<sub>i</sub><sup>jm</sup> coefficients.
          * <p>
          * The index order is [m][j][i] <br/>
          * The i index corresponds to the C<sub>i</sub><sup>jm</sup> coefficients used to
          * compute the following: <br/>
          * - da/dt <br/>
          * - dk/dt <br/>
          * - dh/dt / dk <br/>
          * - dq/dt <br/>
          * - dp/dt / dα <br/>
          * - dλ/dt / dβ <br/>
          * </p>
          */
         private final double[][][] cCoef;
 
         /** The S<sub>i</sub><sup>jm</sup> coefficients.
          * <p>
          * The index order is [m][j][i] <br/>
          * The i index corresponds to the C<sub>i</sub><sup>jm</sup> coefficients used to
          * compute the following: <br/>
          * - da/dt <br/>
          * - dk/dt <br/>
          * - dh/dt / dk <br/>
          * - dq/dt <br/>
          * - dp/dt / dα <br/>
          * - dλ/dt / dβ <br/>
          * </p>
          */
         private final double[][][] sCoef;
 
         /** G<sub>ms</sub><sup>j</sup> and H<sub>ms</sub><sup>j</sup> polynomials. */
         private GHmsjPolynomials ghMSJ;
 
         /** &Gamma;<sub>ns</sub><sup>m</sup> function. */
         private GammaMnsFunction gammaMNS;
 
         /** R / a up to power degree. */
         private final double[] roaPow;
 
         /** Create a set of C<sub>i</sub><sup>jm</sup> and S<sub>i</sub><sup>jm</sup> coefficients.
          *  @param jMax absolute limit for j ( -jMax <= j <= jMax )
          *  @param mMax maximum value for m
          */
         FourierCjSjCoefficients(final int jMax, final int mMax) {
             // initialise fields
             final int rows    = mMax + 1;
             final int columns = 2 * jMax + 1;
             this.jMax         = jMax;
             this.cCoef        = new double[rows][columns][6];
             this.sCoef        = new double[rows][columns][6];
             this.roaPow       = new double[maxDegree + 1];
             roaPow[0] = 1.;
         }
 
         /**
          * Generate the coefficients.
          * @param date the current date
          * @throws OrekitException if an error occurs while generating the coefficients
          */
         public void generateCoefficients(final AbsoluteDate date) throws OrekitException {
             // Compute only if there is at least one non-resonant tesseral
             if (!nonResOrders.isEmpty() || maxDegreeTesseralSP < 0) {
                 // Gmsj and Hmsj polynomials
                 ghMSJ = new GHmsjPolynomials(k, h, alpha, beta, I);
 
                 // GAMMAmns function
                 gammaMNS = new GammaMnsFunction(maxDegree, gamma, I);
 
                 final int maxRoaPower = FastMath.max(maxDegreeTesseralSP, maxDegreeMdailyTesseralSP);
 
                 // R / a up to power degree
                 for (int i = 1; i <= maxRoaPower; i++) {
                     roaPow[i] = roa * roaPow[i - 1];
                 }
 
                 //generate the m-daily coefficients
                 for (int m = 1; m <= maxOrderMdailyTesseralSP; m++) {
                     buildFourierCoefficients(date, m, 0, maxDegreeMdailyTesseralSP);
                 }
 
                 // generate the other coefficients only if required
                 if (maxDegreeTesseralSP >= 0) {
                     for (int m: nonResOrders.keySet()) {
                         final List<Integer> listJ = nonResOrders.get(m);
 
                         for (int j: listJ) {
                             buildFourierCoefficients(date, m, j, maxDegreeTesseralSP);
                         }
                     }
                 }
             }
         }
 
         /** Build a set of fourier coefficients for a given m and j.
          *
          * @param date the date of the coefficients
          * @param m m index
          * @param j j index
          * @param maxN  maximum value for n index
          * @throws OrekitException in case of Hansen kernel generation error
          */
         private void buildFourierCoefficients(final AbsoluteDate date,
                final int m, final int j, final int maxN) throws OrekitException {
             // Potential derivatives components for a given non-resonant pair {j,m}
             double dRdaCos  = 0.;
             double dRdaSin  = 0.;
             double dRdhCos  = 0.;
             double dRdhSin  = 0.;
             double dRdkCos  = 0.;
             double dRdkSin  = 0.;
             double dRdlCos  = 0.;
             double dRdlSin  = 0.;
             double dRdAlCos = 0.;
             double dRdAlSin = 0.;
             double dRdBeCos = 0.;
             double dRdBeSin = 0.;
             double dRdGaCos = 0.;
             double dRdGaSin = 0.;
 
             // s-SUM from -sMin to sMax
             final int sMin = j == 0 ? maxEccPowMdailyTesseralSP : maxEccPowTesseralSP;
             final int sMax = j == 0 ? maxEccPowMdailyTesseralSP : maxEccPowTesseralSP;
             for (int s = 0; s <= sMax; s++) {
 
                 // n-SUM for s positive
                 final double[][] nSumSpos = computeNSum(date, j, m, s, maxN,
                                                         roaPow, ghMSJ, gammaMNS);
                 dRdaCos  += nSumSpos[0][0];
                 dRdaSin  += nSumSpos[0][1];
                 dRdhCos  += nSumSpos[1][0];
                 dRdhSin  += nSumSpos[1][1];
                 dRdkCos  += nSumSpos[2][0];
                 dRdkSin  += nSumSpos[2][1];
                 dRdlCos  += nSumSpos[3][0];
                 dRdlSin  += nSumSpos[3][1];
                 dRdAlCos += nSumSpos[4][0];
                 dRdAlSin += nSumSpos[4][1];
                 dRdBeCos += nSumSpos[5][0];
                 dRdBeSin += nSumSpos[5][1];
                 dRdGaCos += nSumSpos[6][0];
                 dRdGaSin += nSumSpos[6][1];
 
                 // n-SUM for s negative
                 if (s > 0 && s <= sMin) {
                     final double[][] nSumSneg = computeNSum(date, j, m, -s, maxN,
                                                             roaPow, ghMSJ, gammaMNS);
                     dRdaCos  += nSumSneg[0][0];
                     dRdaSin  += nSumSneg[0][1];
                     dRdhCos  += nSumSneg[1][0];
                     dRdhSin  += nSumSneg[1][1];
                     dRdkCos  += nSumSneg[2][0];
                     dRdkSin  += nSumSneg[2][1];
                     dRdlCos  += nSumSneg[3][0];
                     dRdlSin  += nSumSneg[3][1];
                     dRdAlCos += nSumSneg[4][0];
                     dRdAlSin += nSumSneg[4][1];
                     dRdBeCos += nSumSneg[5][0];
                     dRdBeSin += nSumSneg[5][1];
                     dRdGaCos += nSumSneg[6][0];
                     dRdGaSin += nSumSneg[6][1];
                 }
             }
             dRdaCos  *= -moa / a;
             dRdaSin  *= -moa / a;
             dRdhCos  *=  moa;
             dRdhSin  *=  moa;
             dRdkCos  *=  moa;
             dRdkSin *=  moa;
             dRdlCos *=  moa;
             dRdlSin *=  moa;
             dRdAlCos *=  moa;
             dRdAlSin *=  moa;
             dRdBeCos *=  moa;
             dRdBeSin *=  moa;
             dRdGaCos *=  moa;
             dRdGaSin *=  moa;
 
             // Compute the cross derivative operator :
             final double RAlphaGammaCos   = alpha * dRdGaCos - gamma * dRdAlCos;
             final double RAlphaGammaSin   = alpha * dRdGaSin - gamma * dRdAlSin;
             final double RAlphaBetaCos    = alpha * dRdBeCos - beta  * dRdAlCos;
             final double RAlphaBetaSin    = alpha * dRdBeSin - beta  * dRdAlSin;
             final double RBetaGammaCos    =  beta * dRdGaCos - gamma * dRdBeCos;
             final double RBetaGammaSin    =  beta * dRdGaSin - gamma * dRdBeSin;
             final double RhkCos           =     h * dRdkCos  -     k * dRdhCos;
             final double RhkSin           =     h * dRdkSin  -     k * dRdhSin;
             final double pRagmIqRbgoABCos = (p * RAlphaGammaCos - I * q * RBetaGammaCos) * ooAB;
             final double pRagmIqRbgoABSin = (p * RAlphaGammaSin - I * q * RBetaGammaSin) * ooAB;
             final double RhkmRabmdRdlCos  =  RhkCos - RAlphaBetaCos - dRdlCos;
             final double RhkmRabmdRdlSin  =  RhkSin - RAlphaBetaSin - dRdlSin;
 
             // da/dt
             cCoef[m][j + jMax][0] = ax2oA * dRdlCos;
             sCoef[m][j + jMax][0] = ax2oA * dRdlSin;
 
             // dk/dt
             cCoef[m][j + jMax][1] = -(BoA * dRdhCos + h * pRagmIqRbgoABCos + k * BoABpo * dRdlCos);
             sCoef[m][j + jMax][1] = -(BoA * dRdhSin + h * pRagmIqRbgoABSin + k * BoABpo * dRdlSin);
 
             // dh/dt
             cCoef[m][j + jMax][2] = BoA * dRdkCos + k * pRagmIqRbgoABCos - h * BoABpo * dRdlCos;
             sCoef[m][j + jMax][2] = BoA * dRdkSin + k * pRagmIqRbgoABSin - h * BoABpo * dRdlSin;
 
             // dq/dt
             cCoef[m][j + jMax][3] = Co2AB * (q * RhkmRabmdRdlCos - I * RAlphaGammaCos);
             sCoef[m][j + jMax][3] = Co2AB * (q * RhkmRabmdRdlSin - I * RAlphaGammaSin);
 
             // dp/dt
             cCoef[m][j + jMax][4] = Co2AB * (p * RhkmRabmdRdlCos - RBetaGammaCos);
             sCoef[m][j + jMax][4] = Co2AB * (p * RhkmRabmdRdlSin - RBetaGammaSin);
 
             // dλ/dt
             cCoef[m][j + jMax][5] = -ax2oA * dRdaCos + BoABpo * (h * dRdhCos + k * dRdkCos) + pRagmIqRbgoABCos;
             sCoef[m][j + jMax][5] = -ax2oA * dRdaSin + BoABpo * (h * dRdhSin + k * dRdkSin) + pRagmIqRbgoABSin;
         }
 
         /** Get the coefficient C<sub>i</sub><sup>jm</sup>.
          * @param i i index - corresponds to the required variation
          * @param j j index
          * @param m m index
          * @return the coefficient C<sub>i</sub><sup>jm</sup>
          */
         public double getCijm(final int i, final int j, final int m) {
             return cCoef[m][j + jMax][i];
         }
 
         /** Get the coefficient S<sub>i</sub><sup>jm</sup>.
          * @param i i index - corresponds to the required variation
          * @param j j index
          * @param m m index
          * @return the coefficient S<sub>i</sub><sup>jm</sup>
          */
         public double getSijm(final int i, final int j, final int m) {
             return sCoef[m][j + jMax][i];
         }
     }
 
     /** The C<sup>i</sup><sub>m</sub><sub>t</sub> and S<sup>i</sup><sub>m</sub><sub>t</sub> coefficients used to compute
      * the short-periodic zonal contribution.
      *   <p>
      *  Those coefficients are given by expression 2.5.4-5 from the Danielson paper.
      *   </p>
      *
      * @author Sorin Scortan
      *
      */
     private static class TesseralShortPeriodicCoefficients implements ShortPeriodTerms {
 
         /** Serializable UID. */
         private static final long serialVersionUID = 20151119L;
 
         /** Retrograde factor I.
          *  <p>
          *  DSST model needs equinoctial orbit as internal representation.
          *  Classical equinoctial elements have discontinuities when inclination
          *  is close to zero. In this representation, I = +1. <br>
          *  To avoid this discontinuity, another representation exists and equinoctial
          *  elements can be expressed in a different way, called "retrograde" orbit.
          *  This implies I = -1. <br>
          *  As Orekit doesn't implement the retrograde orbit, I is always set to +1.
          *  But for the sake of consistency with the theory, the retrograde factor
          *  has been kept in the formulas.
          *  </p>
          */
         private static final int I = 1;
 
         /** Central body rotating frame. */
         private final Frame bodyFrame;
 
         /** Maximal order to consider for short periodics m-daily tesseral harmonics potential. */
         private final int maxOrderMdailyTesseralSP;
 
         /** Flag to take into account only M-dailies harmonic tesserals for short periodic perturbations.  */
         private final boolean mDailiesOnly;
 
         /** List of non resonant orders with j != 0. */
         private final SortedMap<Integer, List<Integer> > nonResOrders;
 
         /** Maximum value for m index. */
         private final int mMax;
 
         /** Maximum value for j index. */
         private final int jMax;
 
         /** Number of points used in the interpolation process. */
         private final int interpolationPoints;
 
         /** All coefficients slots. */
         private final transient TimeSpanMap<Slot> slots;
 
         /** Constructor.
          * @param bodyFrame central body rotating frame
          * @param maxOrderMdailyTesseralSP maximal order to consider for short periodics m-daily tesseral harmonics potential
          * @param mDailiesOnly flag to take into account only M-dailies harmonic tesserals for short periodic perturbations
          * @param nonResOrders lst of non resonant orders with j != 0
          * @param mMax maximum value for m index
          * @param jMax maximum value for j index
          * @param interpolationPoints number of points used in the interpolation process
          * @param slots all coefficients slots
          */
         TesseralShortPeriodicCoefficients(final Frame bodyFrame, final int maxOrderMdailyTesseralSP,
                                           final boolean mDailiesOnly, final SortedMap<Integer, List<Integer> > nonResOrders,
                                           final int mMax, final int jMax, final int interpolationPoints,
                                           final TimeSpanMap<Slot> slots) {
             this.bodyFrame                = bodyFrame;
             this.maxOrderMdailyTesseralSP = maxOrderMdailyTesseralSP;
             this.mDailiesOnly             = mDailiesOnly;
             this.nonResOrders             = nonResOrders;
             this.mMax                     = mMax;
             this.jMax                     = jMax;
             this.interpolationPoints      = interpolationPoints;
             this.slots                    = slots;
         }
 
         /** Get the slot valid for some date.
          * @param meanStates mean states defining the slot
          * @return slot valid at the specified date
          */
         public Slot createSlot(final SpacecraftState... meanStates) {
             final Slot         slot  = new Slot(mMax, jMax, interpolationPoints);
             final AbsoluteDate first = meanStates[0].getDate();
             final AbsoluteDate last  = meanStates[meanStates.length - 1].getDate();
             if (first.compareTo(last) <= 0) {
                 slots.addValidAfter(slot, first);
             } else {
                 slots.addValidBefore(slot, first);
             }
             return slot;
         }
 
         /** {@inheritDoc} */
         @Override
         public double[] value(final Orbit meanOrbit) throws OrekitException {
 
             // select the coefficients slot
             final Slot slot = slots.get(meanOrbit.getDate());
 
             // Initialise the short periodic variations
             final double[] shortPeriodicVariation = new double[6];
 
             // Compute only if there is at least one non-resonant tesseral or
             // only the m-daily tesseral should be taken into account
             if (!nonResOrders.isEmpty() || mDailiesOnly) {
 
                 //Build an auxiliary object
                 final AuxiliaryElements aux = new AuxiliaryElements(meanOrbit, I);
 
                 // Central body rotation angle from equation 2.7.1-(3)(4).
                 final Transform t = bodyFrame.getTransformTo(aux.getFrame(), aux.getDate());
                 final Vector3D xB = t.transformVector(Vector3D.PLUS_I);
                 final Vector3D yB = t.transformVector(Vector3D.PLUS_J);
                 final Vector3D  f = aux.getVectorF();
                 final Vector3D  g = aux.getVectorG();
                 final double currentTheta = FastMath.atan2(-f.dotProduct(yB) + I * g.dotProduct(xB),
                                                             f.dotProduct(xB) + I * g.dotProduct(yB));
 
                 //Add the m-daily contribution
                 for (int m = 1; m <= maxOrderMdailyTesseralSP; m++) {
                     // Phase angle
                     final double jlMmt  = -m * currentTheta;
                     final double sinPhi = FastMath.sin(jlMmt);
                     final double cosPhi = FastMath.cos(jlMmt);
 
                     // compute contribution for each element
                     final double[] c = slot.getCijm(0, m, meanOrbit.getDate());
                     final double[] s = slot.getSijm(0, m, meanOrbit.getDate());
                     for (int i = 0; i < 6; i++) {
                         shortPeriodicVariation[i] += c[i] * cosPhi + s[i] * sinPhi;
                     }
                 }
 
                 // loop through all non-resonant (j,m) pairs
                 for (final Map.Entry<Integer, List<Integer>> entry : nonResOrders.entrySet()) {
                     final int           m     = entry.getKey();
                     final List<Integer> listJ = entry.getValue();
 
                     for (int j : listJ) {
                         // Phase angle
                         final double jlMmt  = j * meanOrbit.getLM() - m * currentTheta;
                         final double sinPhi = FastMath.sin(jlMmt);
                         final double cosPhi = FastMath.cos(jlMmt);
 
                         // compute contribution for each element
                         final double[] c = slot.getCijm(j, m, meanOrbit.getDate());
                         final double[] s = slot.getSijm(j, m, meanOrbit.getDate());
                         for (int i = 0; i < 6; i++) {
                             shortPeriodicVariation[i] += c[i] * cosPhi + s[i] * sinPhi;
                         }
 
                     }
                 }
             }
 
             return shortPeriodicVariation;
 
         }
 
         /** {@inheritDoc} */
         @Override
         public String getCoefficientsKeyPrefix() {
             return "DSST-central-body-tesseral-";
         }
 
         /** {@inheritDoc}
          * <p>
          * For tesseral terms contributions,there are maxOrderMdailyTesseralSP
          * m-daily cMm coefficients, maxOrderMdailyTesseralSP m-daily sMm
          * coefficients, nbNonResonant cjm coefficients and nbNonResonant
          * sjm coefficients, where maxOrderMdailyTesseralSP and nbNonResonant both
          * depend on the orbit. The j index is the integer multiplier for the true
          * longitude argument and the m index is the integer multiplier for m-dailies.
          * </p>
          */
         @Override
         public Map<String, double[]> getCoefficients(final AbsoluteDate date, final Set<String> selected)
             throws OrekitException {
 
             // select the coefficients slot
             final Slot slot = slots.get(date);
 
             if (!nonResOrders.isEmpty() || mDailiesOnly) {
                 final Map<String, double[]> coefficients =
                                 new HashMap<String, double[]>(12 * maxOrderMdailyTesseralSP +
                                                               12 * nonResOrders.size());
 
                 for (int m = 1; m <= maxOrderMdailyTesseralSP; m++) {
                     storeIfSelected(coefficients, selected, slot.getCijm(0, m, date), "cM", m);
                     storeIfSelected(coefficients, selected, slot.getSijm(0, m, date), "sM", m);
                 }
 
                 for (final Map.Entry<Integer, List<Integer>> entry : nonResOrders.entrySet()) {
                     final int           m     = entry.getKey();
                     final List<Integer> listJ = entry.getValue();
 
                     for (int j : listJ) {
                         for (int i = 0; i < 6; ++i) {
                             storeIfSelected(coefficients, selected, slot.getCijm(j, m, date), "c", j, m);
                             storeIfSelected(coefficients, selected, slot.getSijm(j, m, date), "s", j, m);
                         }
                     }
                 }
 
                 return coefficients;
 
             } else {
                 return Collections.emptyMap();
             }
 
         }
 
         /** Put a coefficient in a map if selected.
          * @param map map to populate
          * @param selected set of coefficients that should be put in the map
          * (empty set means all coefficients are selected)
          * @param value coefficient value
          * @param id coefficient identifier
          * @param indices list of coefficient indices
          */
         private void storeIfSelected(final Map<String, double[]> map, final Set<String> selected,
                                      final double[] value, final String id, final int... indices) {
             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
             keyBuilder.append(id);
             for (int index : indices) {
                 keyBuilder.append('[').append(index).append(']');
             }
             final String key = keyBuilder.toString();
             if (selected.isEmpty() || selected.contains(key)) {
                 map.put(key, value);
             }
         }
 
         /** Replace the instance with a data transfer object for serialization.
          * @return data transfer object that will be serialized
          * @exception NotSerializableException if an additional state provider is not serializable
          */
         private Object writeReplace() throws NotSerializableException {
 
             // slots transitions
             final SortedSet<TimeSpanMap.Transition<Slot>> transitions     = slots.getTransitions();
             final AbsoluteDate[]                          transitionDates = new AbsoluteDate[transitions.size()];
             final Slot[]                                  allSlots        = new Slot[transitions.size() + 1];
             int i = 0;
             for (final TimeSpanMap.Transition<Slot> transition : transitions) {
                 if (i == 0) {
                     // slot before the first transition
                     allSlots[i] = transition.getBefore();
                 }
                 if (i < transitionDates.length) {
                     transitionDates[i] = transition.getDate();
                     allSlots[++i]      = transition.getAfter();
                 }
             }
 
             return new DataTransferObject(bodyFrame, maxOrderMdailyTesseralSP,
                                           mDailiesOnly, nonResOrders,
                                           mMax, jMax, interpolationPoints,
                                           transitionDates, allSlots);
 
         }
 
 
         /** Internal class used only for serialization. */
         private static class DataTransferObject implements Serializable {
 
             /** Serializable UID. */
             private static final long serialVersionUID = 20160319L;
 
             /** Central body rotating frame. */
             private final Frame bodyFrame;
 
             /** Maximal order to consider for short periodics m-daily tesseral harmonics potential. */
             private final int maxOrderMdailyTesseralSP;
 
             /** Flag to take into account only M-dailies harmonic tesserals for short periodic perturbations.  */
             private final boolean mDailiesOnly;
 
             /** List of non resonant orders with j != 0. */
             private final SortedMap<Integer, List<Integer> > nonResOrders;
 
             /** Maximum value for m index. */
             private final int mMax;
 
             /** Maximum value for j index. */
             private final int jMax;
 
             /** Number of points used in the interpolation process. */
             private final int interpolationPoints;
 
             /** Transitions dates. */
             private final AbsoluteDate[] transitionDates;
 
             /** All slots. */
             private final Slot[] allSlots;
 
             /** Simple constructor.
              * @param bodyFrame central body rotating frame
              * @param maxOrderMdailyTesseralSP maximal order to consider for short periodics m-daily tesseral harmonics potential
              * @param mDailiesOnly flag to take into account only M-dailies harmonic tesserals for short periodic perturbations
              * @param nonResOrders lst of non resonant orders with j != 0
              * @param mMax maximum value for m index
              * @param jMax maximum value for j index
              * @param interpolationPoints number of points used in the interpolation process
              * @param transitionDates transitions dates
              * @param allSlots all slots
              */
             DataTransferObject(final Frame bodyFrame, final int maxOrderMdailyTesseralSP,
                                final boolean mDailiesOnly, final SortedMap<Integer, List<Integer> > nonResOrders,
                                final int mMax, final int jMax, final int interpolationPoints,
                                final AbsoluteDate[] transitionDates, final Slot[] allSlots) {
                 this.bodyFrame                = bodyFrame;
                 this.maxOrderMdailyTesseralSP = maxOrderMdailyTesseralSP;
                 this.mDailiesOnly             = mDailiesOnly;
                 this.nonResOrders             = nonResOrders;
                 this.mMax                     = mMax;
                 this.jMax                     = jMax;
                 this.interpolationPoints      = interpolationPoints;
                 this.transitionDates          = transitionDates;
                 this.allSlots                 = allSlots;
             }
 
             /** Replace the deserialized data transfer object with a {@link TesseralShortPeriodicCoefficients}.
              * @return replacement {@link TesseralShortPeriodicCoefficients}
              */
             private Object readResolve() {
 
                 final TimeSpanMap<Slot> slots = new TimeSpanMap<Slot>(allSlots[0]);
                 for (int i = 0; i < transitionDates.length; ++i) {
                     slots.addValidAfter(allSlots[i + 1], transitionDates[i]);
                 }
 
                 return new TesseralShortPeriodicCoefficients(bodyFrame, maxOrderMdailyTesseralSP, mDailiesOnly,
                                                              nonResOrders, mMax, jMax, interpolationPoints,
                                                              slots);
 
             }
 
         }
 
     }
 
     /** Coefficients valid for one time slot. */
     private static class Slot implements Serializable {
 
         /** Serializable UID. */
         private static final long serialVersionUID = 20160319L;
 
         /** The coefficients C<sub>i</sub><sup>j</sup><sup>m</sup>.
          * <p>
          * The index order is cijm[m][j][i] <br/>
          * i corresponds to the equinoctial element, as follows: <br/>
          * - i=0 for a <br/>
          * - i=1 for k <br/>
          * - i=2 for h <br/>
          * - i=3 for q <br/>
          * - i=4 for p <br/>
          * - i=5 for λ <br/>
          * </p>
          */
         private final ShortPeriodicsInterpolatedCoefficient[][] cijm;
 
         /** The coefficients S<sub>i</sub><sup>j</sup><sup>m</sup>.
          * <p>
          * The index order is sijm[m][j][i] <br/>
          * i corresponds to the equinoctial element, as follows: <br/>
          * - i=0 for a <br/>
          * - i=1 for k <br/>
          * - i=2 for h <br/>
          * - i=3 for q <br/>
          * - i=4 for p <br/>
          * - i=5 for λ <br/>
          * </p>
          */
         private final ShortPeriodicsInterpolatedCoefficient[][] sijm;
 
         /** Simple constructor.
          *  @param mMax maximum value for m index
          *  @param jMax maximum value for j index
          *  @param interpolationPoints number of points used in the interpolation process
          */
         Slot(final int mMax, final int jMax, final int interpolationPoints) {
 
             final int rows    = mMax + 1;
             final int columns = 2 * jMax + 1;
             cijm = new ShortPeriodicsInterpolatedCoefficient[rows][columns];
             sijm = new ShortPeriodicsInterpolatedCoefficient[rows][columns];
             for (int m = 1; m <= mMax; m++) {
                 for (int j = -jMax; j <= jMax; j++) {
                     cijm[m][j + jMax] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
                     sijm[m][j + jMax] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
                 }
             }
 
         }
 
         /** Get C<sub>i</sub><sup>j</sup><sup>m</sup>.
          *
          * @param j j index
          * @param m m index
          * @param date the date
          * @return C<sub>i</sub><sup>j</sup><sup>m</sup>
          */
         double[] getCijm(final int j, final int m, final AbsoluteDate date) {
             final int jMax = (cijm[m].length - 1) / 2;
             return cijm[m][j + jMax].value(date);
         }
 
         /** Get S<sub>i</sub><sup>j</sup><sup>m</sup>.
          *
          * @param j j index
          * @param m m index
          * @param date the date
          * @return S<sub>i</sub><sup>j</sup><sup>m</sup>
          */
         double[] getSijm(final int j, final int m, final AbsoluteDate date) {
             final int jMax = (cijm[m].length - 1) / 2;
             return sijm[m][j + jMax].value(date);
         }
 
     }
 
 }