IJAAInternational Journal of Astronomy and Astrophysics2161-4717Scientific Research Publishing10.4236/ijaa.2017.71003IJAA-74796ArticlesPhysics&Mathematics A Comparative Study of Resonances in the Dynamics of the Mimas-Tethys System HemantKumar Mishra1GovindKumar Jha2SaritaJha3Department of Mathematics, Vinoba Bhave University, Hazaribag, IndiaDepartment of Mathematics, K. B. Women’s College, Hazaribag, IndiaDepartment of Mathematics, Chatra College, Chatra, India2501201707012837December 13, 2016Accepted: March 18, March 21, 2017© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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 1i 3 resonance). Here we have done the comparative study of the dynamics of Mimas-Tethys system at i 1i 3, i 1i 3e 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 1i 3e 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.

Secular Resonance Three Body Problem Disturbing Function Oblateness1. Introduction

Allan [1] and Sinclair [2] investigated the dynamics of the Mimas-Tethys system under the hypothesis of circular orbits. Allan found (backward in time) the values for the satellites inclinations before capture in the present resonance to be and and Sinclair found the probability of capture to be 0.04.

Vienne and Duriez [3] discovered, using the frequency analysis method developed by Laskar [4] , a particularly interesting 200 yr period in the mean longitude of Mimas which came because oblateness of Saturn and the interaction between Mimas, Enceladus, Tethys, Dione, Rhea and Titan (See [4] [5] [6] [7] ) set the eccentricity of Tethys 0.000235, but with some uncertainties, it could amount to 0.001. This is a very small value, which certainly explains why the previous studies would assume Tethys moving on a circular orbit. However, Champenois and Vienne showed that in spite of such a small value, the third order terms with arguments

, and

brought chaos in the dynamics of Mimas-Tethys system because of near vanishing frequency.

Greenberg [8] analyzed the effect of and resonances on Mimas-Te- thys system along with resonance.

Champenois and Vienne [5] [6] numerically investigated the role of 200 year period and found that the inclination of Mimas before capture might have been higher (up to 0.7 degree) or lower (down to 0.03 degree) than that of previously considered 0.42 degree. Also Tethys’s eccentricity on capture may have been quite higher (0.0008 versus 0). This value of eccentricity was found by a method which took chaos into account. They also found that probability of capture in resonance was 0.76 when eccentricity of Tethys was 0.0008.

Jha and Agrawal [9] [10] have done the comparative study of the dynamics of Mimas-Tethys at, and resonances along with three third order resonances with and without considering the secular term of all inner satellites and oblateness of Saturn. Jha and Jha [11] [12] [13] [14] have studied the secular resonance effect of Dione, Rhea and Titan on this system.

Thomas, P. C. et al. [15] have measured the shapes and sizes of six icy satellites by Cassini imaging subsystem (ISS) data, employing limb coordinates and stereogrammetic control points. Mimas, Enceladus, Tethys, Dione and Rhea are all well described by triaxial ellipsoids; Iapetus is best represented by an oblate spheroid.

Czechowski et al. [16] found that the temperature of Mimas interior was significantly lower than that of Enceladus. Czechowski and Losiak [17] have investigated the early thermal history of Rhea and have found that liquid state convection could delay the differentiation for hundreds of millions years.

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

and (see Table 1 for its values), also the actions of the Japet in the equations whereas the Sun and the small satellites of Saturn are not taken into account because of their weak effects in the generations of the pertur- bative frequency.

Here the notations and are orbital semi-major axis, mean motion, eccentricity, inclination, sine of semi inclination, longitudes of periapse, longitude of ascending node and mean longitude of Mimas respectively. Corresponding notions with subscript 2, 3, 4, 5 and 6 refers to the Enceladus, Tethys, Dione, Rhea and Titan’s orbital elements. Small

and stands for Mimas, Enceladus, Tethys, Dione, Rhea and Titan’s masses with unit of Saturn.

Here we are extending the work of Jha and Agrawal [9] in and resonance by considering the effect of secular term of all inner satellites along with

Parameters of the three-body system Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Saturn<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x30.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x30.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x31.png" xlink:type="simple"/></inline-formula>and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x30.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x31.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x32.png" xlink:type="simple"/></inline-formula> are the masses of the considered satellite, Saturn and the Sun, respectively
n (rad/yr)i (deg)Ms/Mae (km)EJ2 J4 J6
Mimas6.34 × 10−82422.441.62--0.0194
Enceladus0.15 × 10−6
Tethys1.06 × 10−61213.171.093--0.009
Dione1.963 × 10−6
Rhea4.32 × 10−6-----
Titan236.638 × 10−6
Saturn---3498.7960330-0.012980.0009150.000095

Saturn Oblateness.

2. Equation of Motion When the System Is Locked in<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x35.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x35.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x36.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x35.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x36.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x37.png" xlink:type="simple"/></inline-formula>and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x35.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x36.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x37.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x38.png" xlink:type="simple"/></inline-formula> Resonances

(Equations of motion is taken from [9] )

where, (1)

,

where,

and

where is the frequency of at time t. is the value of at time (J2000) (see [6] [7] ). Here (for values see Table 2).

, and

The value of, and depend on the Laplace’s Coefficients and its values are given in Table 2.

3. Equation of Motion When the System Is Locked in<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x62.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x62.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x63.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x62.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x63.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x64.png" xlink:type="simple"/></inline-formula>and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x62.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x63.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x64.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x65.png" xlink:type="simple"/></inline-formula> Resonances

where

,

and

Analytical expressions of the functions <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x70.png" xlink:type="simple"/></inline-formula> for the argument
ArgumentI
0−1.65088068
15.23786953
29.70821605
30.22188903
40.82544034

With where and

and.

The values of, and depend on the Laplace’s Coefficients and given in Table 2.

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 and (See Table 1 for its values) as a factor (Vienne and Duriez [3] [5] ). The other terms are

taken constant. Values of and for every

pair involving Mimas, Enceladus, Tethys, Dione, Rhea and Titan are given in Table 3.

We then get (is the equatorial radius of Saturn).

Value of <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x105.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x105.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x106.png" xlink:type="simple"/></inline-formula> for every pair <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x105.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x106.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x107.png" xlink:type="simple"/></inline-formula> involving Mimas, Enceladus, Tethys, Dione, Rhea and Titan <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x105.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x106.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x107.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x108.png" xlink:type="simple"/></inline-formula> (CV [<xref ref-type="bibr" rid="scirp.74796-ref6">6</xref>] )
1 - 20.780261.24731.3674−5.46951.0996
1 - 30.630641.13060.38952−1.55810.55451
1 - 40.492581.07060.15366−0.614630.33609
1 - 50.352831.03350.059940−0.239760.20479
1 - 60.152231.00590.0090810−0.0363240.078148
2 - 30.808241.28071.8535−7.41381.2978
3 - 40.781081.24821.3790−5.51591.1047
3 - 50.559481.09600.23856−0.954250.42538
3 - 60.241401.01510.024460−0.0978400.12912
4. Numerical Integration and Surface of Sections

Our equations were integrated backwards in time. The initial conditions are taken from TASS1.6 [7] for J2000. and are taken to be fixed. Here we have integrated it for.

Figure 1 and Figure 2 are Poincare surface of sections with and without obla- teness of Saturn respectively at and resonances. Figure 3 and Figure 4 are the time-series graph for the same. Figure 5 and Figure 6 are the Poincare surface of sections with and without oblateness of Saturn at

and resonances and Figure 7 and Figure 8 are the time series graph for the same.

5. Discussion

Vienne and Duriez [3] discovered the 200 year period in the mean longitude of

Poincare surface of section at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x127.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x127.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x128.png" xlink:type="simple"/></inline-formula> resonances with oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Poincare surface of section at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x132.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x132.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x133.png" xlink:type="simple"/></inline-formula> resonances without oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Time-series graph at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x137.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x137.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x138.png" xlink:type="simple"/></inline-formula> resonances with oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Time-series graph at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x142.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x142.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x143.png" xlink:type="simple"/></inline-formula> resonances without oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Poincare surface of section at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x147.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x147.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x148.png" xlink:type="simple"/></inline-formula> resonances with oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Poincare surface of section at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x152.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x152.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x153.png" xlink:type="simple"/></inline-formula> resonances without oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Time-series graph at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x157.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x157.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x158.png" xlink:type="simple"/></inline-formula> resonances withoblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Time-series graph at <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x162.png" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x162.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-4500630x163.png" xlink:type="simple"/></inline-formula> resonances without oblateness of Saturn. Secular resonances of all inner satellites have been considered a

and.

Mimas. Champenois and Vienne [5] [6] have investigated the role of 200 year long period on the dynamics of Mimas-Tethys system when considered to be presently trapped in resonance with probability of capture 0.76 at 2:4 commensurability. They found that the sources of this period were both the oblateness of Saturn and the interaction between its six inner satellites. They also realized that considering an eccentric orbit of Tethys upseted the earlier vision of dynamics of the Mimas-Tethys system.

Here, we have analyzed that the system is more chaotic if considered to be (at presently it is) locked in resonance compared with resonance; we consider the effect of Saturns oblateness and the interaction of its six inner satellites, by the help of Poincare surface of section and time-series graphs which confirm the earlier results too. Figure 2 and Figure 6 are Poincare surface of sections and Figure 4 and Figure 8 are time series graphs without Oblateness of Saturn. Although we could not quantify the chaos, on the basis of above figures, we can say that oblateness of Saturn plays a significant role in the dynamics of Mimas-Te- thys system and also it partially controls the chaos.

Cite this paper

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

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