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Inclinati on Functionin Satellite Dynamics,Inclination function is a kind of special function commonly-used in thesatellite dynamics. With the technical development and intensive study, theorder of inclination function to be calculated becomes higher and higher.For the high-precision computation of inclination function, this bookletintroduced the available methods, put forward new methods, gave theirFORTRAN program, and studied their stability.

This booklet can be used as a reference for scholars of astronomy and earthscience, and can be also used as a textbook of the graduates.
 

Preface
1 Introduction
1.1 Introduction of inclination function
1.2 Other definitionsofIncFun
1.3 Requirements of satellite dynamics for lncFun
1.3.1 Normalized IncFun and its derivatives
1.3.2 Kernel oflncFun
1.3.3 Calculating order and storage of lncFun
1.4 About this book
2 Expressions of lncFun and its derivative
2.1 Frequentlyused notations oflncFun
2.1.1 Normalized lncFun'
2.1.2 Quasinormalized IncFun
2.1.3 Kernel of lncFun
2.1.4 Gooding's notation
2.1.5 Emeljanov's notation
2.2 Series expressions oflncFun
2.2.1 Single summation expression
2.2.2 Dual summation expression
2.2.3 Triple summation expression
2.3 Definite integral expression oflncFun
2.4 Jacobi polynomial expression oflncFun
2.5 Hypergeometric series expression oflncFun
2.5.1 Expressions in three areas
2.5.2 Expression suited to areas A and B
2.5.3 Unified hypergeometric series expression
2.6 dfunction expression oflncFun
2.6.1 dfunction expression
2.6.2 Expression of inclination matrix element
2.6.3 Timoshkova's expression
2.6.4 Kinoshita's expression
2.6.5 Comparison of four expressions
2.7 Tisserand polynomial expression oflncFun
2.8 Calculating method of the derivatives of IncFun
2.8.1 The 1st method
2.8.2 The 2nd method
2.8.3 The 3rd method
2.9 Primary properties oflncFun
3 Recnrsion of inclination function
3.1 Classification ofrecursion
3.1.1 Classification with special function used
3.1.2 Classification with recursion index
3.1.3 Classification with recursive function
3.2 Starting values for recursion oflncFun
3.2.1 Starting values for Lplane recursion
3.2.2 Starting values for Mplane recursion
3.3 Recursion using recursion relations of Legendre polynomial: Giacaglia's formulae
3.3.1 The first set of formulae
3.3.2 The second set of formulae
3.3.3 The third set of formulae
3.4 Recursion using recursion relations of Jacobi polynomial
3.4.1 Primary recursion relations of Jacobi polynomial
3.4.2 Several practical recursions
3.4.3 Formulae of Allah's recursion
3.4.4 Formulae ofGooding's recursion
3.5 Recursion using recursion formula of hypergeometric series
3.5.1 Important property of recursion formula
3.5.2 Several practical rccursions
3.6 Recursion using recursion formula oldfunction
3.6.1 Blanco recursion
3.6.2 Risbo recursion
3.7 Stability analysis ofrecursion
3.8 Preliminary analysis for algorithms with highprecision and highstability
3.8.1 Difficulty of the Mk(l+) recursion
3.8.2 Methods ofovercomingthis difficulty
4 Computation method of inclination function
4.1 Analytical method
4.1.1 For case mk 4.1.2 For case mk>O
4.1.3 Loss of precision of analytical method and its causes
4.2 Definite integral method
4.2.1 Outline of the method
4.2.2 Computation method of normalized Lcgendre polynomials
4.2.3 Selection of numerical integral formula
4.3 KosteleckyWnuk's method
4.4 Giacaglia's method
4.5 Gooding's method
4.6 Modified Gooding's method
4.7 Emeljanov's method
4.8 Modified Emeljanov's method
4.9 Jacobi polynomial method
4.10 dfunction method
4.11 Lplane recursion method
5 Computation program
5.1 Analytical method
5.2 Definite integral method
5.3 KosteleckyWnuk's method
5.4 Giacaglia's method
5．5 SimplifiedGoodingsmethod
5．6 Modified Goodings meod
5.7 Emeljanovs method
5.8 Modified Emeljanovs method
5.9 Jacobi polynomial method
5．10 dfunctionmethod
5．11 Lplane recursion method
6 Comparison and evaluation
6．1 Computation accuracy
6.1.1 Assessment standard and accuracy index
6.1.2 Computation results ofthe accuracy index
6.1.3 Comparisonindex
6.1.4 Note about the treatment 0fs and c
6.1.5 Precision ofthe derivatives oflncFun
6．2 Computation speed
6．3 Requirement to memo~
6.4 Method selection for the computation of IncFun in satellite dynamics
6.4.1 Method fbr computation without singularity
6.4.2 Method ofdistinguishing large inclination from small inclination
6．5 A quadprecision calculation test and prospects for future research
Reftrences
AppendixA Proofof series expressions ofinclination function
A1 Single summation formula
A2 DuaI summation formula
A3 Kaulas triple summation formula
A4 Analytical expression ofloworder inclination functions
Appendix B Contiguousfunction relations ofhypergeometric series
B1 Gausss contiguous function relations
B2 CFR needed for the recursion of ioclination function
B3 CFR for m．recursion
B4 CFR fnr／recursion
B5 CFR forprecursion
Appendix C The quadprecision program for calculation of inclination function
C1 The quadprecision program of Jacobi polynomial method
C2 The quadprecision program of Lplane method
Acknowledgements

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