^{1}

^{2}

^{3}

The probability of capture of Mimas-Tethys in 2:4 resonance was found to be 0.76 by Champenois when they considered the orbit of Tethys to be elliptical (that is eccentricity of Tethys to be 0.0008) and chaos was taken into account. It means probability of capture in
i<sup>2</sup><sub style="margin-left:-6px;">1</sub> or
i<sup>2</sup><sub style="margin-left:-6px;">3</sub> resonance is 0.24 (
*i.e.* probability of non capture in i
_{1}i
_{3 }resonance). Here we have done the comparative study of the dynamics of Mimas-Tethys system at i
_{1}i
_{3}, i
_{1}i
_{3}e
_{3},
i<sup>2</sup><sub style="margin-left:-6px;">1</sub>e
_{3},
i<sup>2</sup><sub style="margin-left:-6px;">3</sub>e
_{3} and
i<sup>2</sup><sub style="margin-left:-6px;">1</sub>, i
_{1}i
_{3}e
_{3},
i<sup>2</sup><sub style="margin-left:-6px;">1</sub>e
_{3},
i<sup>2</sup><sub style="margin-left:-6px;">3</sub>e
_{3} resonances along with secular resonance of Saturn’s six inner satellites and Saturn’s oblateness. We have drawn Poincare surface of sections and Time series graphs to compare their effect.

Allan [

Vienne and Duriez [

Greenberg [

Champenois and Vienne [

Jha and Agrawal [

Thomas, P. C. et al. [

Czechowski et al. [

The physical model was taken to be Mimas and Tethys on eccentric orbits inclined on the equatorial plane of the Saturn. Saturn’s gravitational momenta are essential as they provide the main contribution to the orbital precession rates and we had to take into account the lowest degree oblateness terms

Here the notations

Here we are extending the work of Jha and Agrawal [

n (rad/yr) | i (deg) | M_{s}/M | ae (km) | E | J_{2 } | J_{4 } | J_{6} | ||
---|---|---|---|---|---|---|---|---|---|

Mimas | 6.34 × 10^{−8} | 2422.44 | 1.62 | - | - | 0.0194 | |||

Enceladus | 0.15 × 10^{−6} | ||||||||

Tethys | 1.06 × 10^{−6} | 1213.17 | 1.093 | - | - | 0.009 | |||

Dione | 1.963 × 10^{−6} | ||||||||

Rhea | 4.32 × 10^{−6} | - | - | - | - | - | |||

Titan | 236.638 × 10^{−6} | ||||||||

Saturn | - | - | - | 3498.79 | 60330 | - | 0.01298 | 0.000915 | 0.000095 |

Saturn Oblateness.

(Equations of motion is taken from [

where,

and

where

The value of

where

and

Argument | I | ||
---|---|---|---|

0 | −1.65088068 | ||

1 | 5.23786953 | ||

2 | 9.70821605 | ||

3 | 0.22188903 | ||

4 | 0.82544034 |

With

The values of

With

(We are not considering any changes in semi major axis of any satellites)

and

Now we will find the terms due to Oblateness of Saturn. Saturn’s gravitational momenta are quite important so that we have, in order to get the full variations of the mean longitudes, nodes and pericentres due to the oblateness, taken into account the lowest-degree terms with

taken constant. Values of

pair

We then get (

1 - 2 | 0.78026 | 1.2473 | 1.3674 | −5.4695 | 1.0996 |

1 - 3 | 0.63064 | 1.1306 | 0.38952 | −1.5581 | 0.55451 |

1 - 4 | 0.49258 | 1.0706 | 0.15366 | −0.61463 | 0.33609 |

1 - 5 | 0.35283 | 1.0335 | 0.059940 | −0.23976 | 0.20479 |

1 - 6 | 0.15223 | 1.0059 | 0.0090810 | −0.036324 | 0.078148 |

2 - 3 | 0.80824 | 1.2807 | 1.8535 | −7.4138 | 1.2978 |

3 - 4 | 0.78108 | 1.2482 | 1.3790 | −5.5159 | 1.1047 |

3 - 5 | 0.55948 | 1.0960 | 0.23856 | −0.95425 | 0.42538 |

3 - 6 | 0.24140 | 1.0151 | 0.024460 | −0.097840 | 0.12912 |

Our equations were integrated backwards in time. The initial conditions are taken from TASS1.6 [

Vienne and Duriez [

Mimas. Champenois and Vienne [

Here, we have analyzed that the system is more chaotic if considered to be (at presently it is) locked in

Mishra, H.K., Jha, G.K. and Jha, S. (2017) A Comparative Stu- dy of Resonances in the Dynamics of the Mi- mas-Tethys System. International Journal of Astronomy and Astrophysics, 7, 28-37. https://doi.org/10.4236/ijaa.2017.71003