# Comparison between N-Body and Semi-analytic code

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## Energy dissipation in the primary

Several parameters were introduced in the tidal acceleration. First, the Love Number k2 appears as a constant in the equation. But the assumption that the tidal potential raised by a satellite is independent from the satellite’s orbit or its mass can not always hold. Instead one can assume a frequency dependant tidal response, for which k2 is a function of the main tidal frequency χ = 2| p − n|. (1.13). Also, one important feature of the tidal theory is the quality factor Q introduced in the time lag (Equation 1.11). It is a measure of energy dissipated during a tidal cycle (Goldreich and Soter, 1966).

### The disturbing function as a sum of cosines

Starting from the main perturbing acceleration (Equation 2.17), our main interest in defining the disturbing function is to express it in terms of the orbital elements of the body and the perturber. This task was subjected to centuries of research and enhancement after the basis of orbital mechanics was laid by Isaac Newton in his Principia Mathematica (Newton, 1687). A few notable works in which the authors have been working on expanding the disturbing function are Laplace, 1785 and Delaunay, 1860. In general, one has Equation 2.17 ¨ri + G(M + mi) r3 i ri = ∇riRj. for a perturbing body j acting on a body i. Now, as we have shown in chapter 2 or in Appendix C, the disturbing function is slightly different whether one studies a body perturbing an inner or an outer body. However, one can show that, using several expansion technics, such as Fourier or Taylor series, a general disturbing function R can be expanded as a trigonometric series.

#### Dependence on the Sun’s mean longitude

Equation 3.16 averages the disturbing function over the longitude of the satellite and the Sun. But one can also average on the satellite’s mean longitude only, avoiding further approximation for the Sun. The reason is that the period of the Sun is quite big compared to the satellite’s period and acts on a time scale between the fast orbital period of the satellites and the slow variations of their nodes and pericentres. After averaging only over the satellite’s mean longitude, we are left with the eight secular terms Table 3.2 plus some terms involving the mean longitude of the Sun in the form k × λ⊙. Because of the large number of terms and the fact that k × λ⊙ has a short period when k is big, we only choose terms in 1 × λ⊙. Table 3.3 gathers all of theses terms. We have R⊙,λ⊙  = n2 ⊙a2 s X35 i=1 Si cos(φi).

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Acknowledgements
Introduction
I Theoretical framework
1 Tides and tidal acceleration in gravitational systems
1.1 General overview
1.1.1 Differential acceleration, the source of tides
1.1.2 Tidal bulge
1.1.3 Energy dissipation and angular momentum exchange
1.2 Formalism
1.3 Energy dissipation in the primary
1.4 Constant lag versus frequency dependent tidal response
1.5 Summary and Conclusion
2 Equations of motion for the system of Saturn
2.1 Saturnian system as a N-Body system
2.2 Saturn’s gravity field
2.3 Inner Moons
2.4 Summary of forces on a moon
2.5 Summary of the numerical model
3 Semi-analytic modelling
3.1 General statements
3.1.1 The disturbing function as a sum of cosines
3.2 Averaging the flattening effect
3.3 Solar terms
3.3.1 Secular
3.3.2 Dependence on the Sun’s mean longitude
3.4 Titan-Iapetus interaction
3.4.1 Secular terms
3.4.2 Resonant terms
3.5 Comparison between N-Body and Semi-analytic code
3.6 Legendre Expansion
3.7 Summary
4 The Laplace Plane
II The 5:1 Mean Motion Resonance
5 Sorting out results
5.1 Preliminary assessment
5.1.1 Initial conditions
5.1.2 Chaos
5.2 Simulations
5.2.1 N-Body code Q=20, Q=50, Q=75 Q=100, Q=200, Q=400, Q=600 and Q=1500
5.2.2 Semi-analytic code
5.3 Trends
5.4 Summary and conclusion
6.1.1 First model results
6.1.2 Starting with a tilt
6.2 Axial precession of Saturn
6.3 Hyperion and Jupiter
Conclusion
A Numerical tools
A.1.1 Presentation
A.2 Algorithm
A.3.1 Derivation
A.3.2 Roots specification
A.4 Numerical implementation
A.4.1 A few changes
A.4.2 Evaluation of the perturbing acceleration
B Lagrange Planetary Equations in Non-singular coordinates
B.1 Generalities on orbital elements
B.2 Lagrange Planetary Equations
C Disturbing Function
C.1 Introduction
C.2 Elliptical expansion
C.3 Legendre Expansion
C.4 Expansion using explicit expressions
C.5 Alternative Expansion
D Laplace coefficient
D.1 Introduction
D.2 Computation
D.3 Relation between coefficients
D.4 Implementation
Bibliography

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