Influence of a second satellite on the rotational dynamics of an oblate moon
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Abstract
The gravitational influence of a second satellite on the rotation of an oblate moon is numerically examined. A simplified model, assuming the axis of rotation perpendicular to the (Keplerian) orbit plane, is derived. The differences between the two models, i.e. in the absence and presence of the second satellite, are investigated via bifurcation diagrams and by evolving compact sets of initial conditions in the phase space. It turns out that the presence of another satellite causes some trajectories, that were regular in its absence, to become chaotic. Moreover, the highly structured picture revealed by the bifurcation diagrams in dependence on the eccentricity of the oblate body’s orbit is destroyed when the gravitational influence is included, and the periodicities and critical curves are destroyed as well. For demonstrative purposes, focus is laid on parameters of the Saturn–Titan–Hyperion system, and on oblate satellites on loweccentric orbits, i.e. \(e\approx 0.005\).
Keywords
Chaos Planets and satellites Rotation Bifurcation diagrams1 Introduction
Saturn’s seventh moon, Hyperion (also known as Saturn VII), was discovered in the XIX century by Bond (1848) and Lassel (1848), but only due to Voyager 2 (Smith et al. 1982) and Cassini (Thomas 2010) missions it became apparent that it is the biggest known highly aspherical celestial body in the Solar System, with a highly elongated shape and dimensions \(360\,\times \,266\,\times \,205\) km. Since the rotational state of Hyperion was predicted to remain in the chaotic zone (Wisdom et al. 1984) based on the spinorbit coupling theory (Goldreich and Peale 1966), further analyses and observations, regarding Hyperion as well as other Solar System satellites, were conducted on a regular basis.
Hyperion’s longterm observations were carried out twice in the post Voyager 2 era. In 1987, Klavetter (1989a, (1989b) performed photometric R band observations over a timespan of more than 50 days, resulting in 38 highquality data points. In 1999 and 2000, Devyatkin et al. (2002) conducted C (integral), B, V and R band observations. The objective of both analyses was to determine whether Hyperion’s rotation is chaotic and to fit a solution of the equation of motion to the observations. To the best of the author’s knowledge (Melnikov, priv. comm.) there were no other longterm observations that resulted in a lightcurve allowing the determination of Hyperion’s rotational state (see also Strugnell and Taylor 1990; Dourneau 1993 for a list of earlier observations). Although, shortly after the Cassini 2005 passage a groundbased BVR photometry was conducted (Hicks et al. 2008), resulting in 6 nights of measurements (and additional 3 nights of R photometry alone) over a monthlong period. Unfortunately, this data was greatly undersampled and period fitting procedures yielded several plausible solutions.
The theoretical and numerical treatment of the rotational dynamics of an oblate satellite have been performed widely. After the seminal paper of Wisdom et al. (1984), Boyd et al. (1994) applied the method of close returns to a sparse and shortterm simulated observations of Hyperion’s lightcurve. Black et al. (1995) performed numerical experiments using the full set of Euler equations to model longterm dynamical evolution. Beletskii et al. (1996) considered a number of models, including the gravitational, magnetic and tidal moments as well as rotation in gravitational field of two centers. A model with a tidal torque was examined analytically using Melnikov’s integrals and assymptotic methods (Khan et al. 1998). The stability of resonances with application to the Solar System satellites was inferred based on a series expansion of the terms in the equation of rotational motion (Celletti and Chierchia 1998, 2000). The Lyapunov exponents and spectra were exhaustively examined for a number of satellites^{1} (Shevchenko 2002; Shevchenko and Kouprianov 2002; Kouprianov and Shevchenko 2003, 2005). A model of an oblate satellite with dissipation was used to examine the basins of attraction in case of low eccentricities, especially with application to the Moon (Celletti and Chierchia 2008). The dynamical stability was examined for all known satellites by Melnikov and Shevchenko (2010). Again the dynamical modeling using the full Euler equations was conducted by Harbison et al. (2011), who also analyzed the moments of inertia in light of the precessional period. Finally, Tarnopolski (2015a) argued that in order to extract a maximal Lyapunov exponent from the photometric lightcurve of Hyperion, at least one year of dense data is required.
Physical parameters of the Saturn–Titan–Hyperion system
Parameter  Symbol  Value  References 

Saturn’s mass  M  \(5.68\times 10^{26}\,\mathrm{kg}\)  Jacobson et al. (2006) 
Titan’s mass  \(m_1\)  \(1.35\times 10^{23}\,\mathrm{kg}\)  Jacobson et al. (2006) 
\(m_1/M\)  \(2.4\times 10^{4}\)  
Hyperion’s major semiaxis  a  1 429 600 km  
Titan’s major semiaxis  \(a_0\)  1 221 865 km  
\(a_0/a\)  0.855  
Hyperion’s oblateness  \(\omega ^2\)  0.79  Wisdom et al. (1984) 
Hyperion’s eccentricity  e  0.1  Wisdom et al. (1984) 
Hyperion’s orbital period  T  21.3 d  Thomas et al. (2007) 
This paper is organized in the following manner. In Sect. 2 the rotational models in case of the absence and presence of a second satellite’s gravitation are derived. In Sect. 3 the phase space is briefly described. Section 4 presents the methods used: the correlation dimension and its benchmark testing, and the bifurcation diagrams. The results are presented in Sect. 5, which is followed by discussion and conclusions gathered in Sect. 6. The computer algebra system mathematica ^{®} is applied throughout this paper.
2 Models
2.1 Rotational model of an oblate moon
 1.the orbit of the satellite around the planet is Keplerian with eccentricity e and major semiaxis a:where \(f_H\) is the true anomaly given by$$\begin{aligned} r=\frac{a\left( 1e^2\right) }{1+e\cos f_H}, \end{aligned}$$(1)with M the mass of the planet and the overdot denotes differentation with respect to time;$$\begin{aligned} \dot{f}_H=\frac{\sqrt{GM}}{\left[ a\left( 1e^2\right) \right] ^{3/2}}\left( 1+e\cos f_H\right) ^2, \end{aligned}$$(2)
 2.
in general, the physical model of the satellite is a triaxial ellipsoid; however, to simplify calculations, the satellite is simulated by a double dumbbell with four mass points 1–4 (see Fig. 1) with equal mass m arranged in the orbital plane. The principal moments of inertia are \(A>B>C\);
 3.
the satellite’s spin axis is fixed and perpendicular to the orbit plane; the spin axis is aligned with the shortest physical axis, i.e. the one corresponding to the largest principal moment of inertia.
In case of Hyperion, the first assumption is not precisely valid, as it is well known that due to gravitational interaction with Titan the eccentricity of Hyperion oscillates from \(\sim 0.08\) to \(\sim 0.12\) with an 18.8year period (Taylor et al. 1987). However, as the analysis herein is performed on a time span much shorter than this period (i.e., \(<1\,\mathrm{yr}\)), the effect of this interaction will be negligible and as such is omitted (Black et al. 1995; Shevchenko and Kouprianov 2002). The second assumption, while might look like an oversimplification at first, does not affect the final equation of motion, which is the same as the one obtained directly from the Euler equations (Danby 1962; see also Appendix 1 for a remark on the moments of inertia in both models). The third assumption is justified for most satellites as the angular momentum is assumed to be constant with great accuracy. However, it should be noted that Wisdom et al. (1984) showed that the chaotic state is attitude unstable, and also the analysis of Voyager 2 images showed that the axis of rotation was far from being perpendicular to the orbital plane. Therefore, the models derived herein are a first approximation that will be expected to give initial insight into the dynamics of the satellite, and the parameters corresponding of the Saturn–Titan–Hyperion system are used for demonstrative reasons.
2.2 Introducing a second satellite
3 Phase space properties
In this Section the structure of the phase space of the dynamical system given by Eqs. (3)–(5) is briefly described. This will allow an insight into how does the gravitational interaction with the second satellite influence the oblate moon’s rotation.
As is common in Hamiltonian systems, the phase space is divided into regions occupied with chaotic trajectories, and regions of regular (periodic or quasiperiodic) motion. There are also motions called sticky orbits, when the trajectory initially behaves in a regular manner and diverges into the chaotic zone after some time. (See also Melnikov 2014 for the emergence of strange attractors when dissipation is introduced.) The phase space in Fig. 2a is dominated by a large chaotic sea, formed by merging the chaotic zones surrounding spinorbit resonances from p=1:2 to p=2:1 when \(\omega ^2\) increases (Wisdom et al. 1984). Quasiperiodic motions are represented by smooth curves and by closed curves, e.g. the ones surrounding the synchronous \(p\,=\,\)1:1 resonance (see Table 1 in Black et al. 1995 for locations of the surviving resonances). When the IC is located near the boundary between regular motion and the chaotic sea, sticky motion occurs. A narrow chaotic zone is present also in Fig. 2b, c obtained for smaller values of the eccentricity and oblateness, respectively. In fact, every Solar System satellite has a chaotic period in its past (Spohn et al. 2014).
4 Methods
4.1 Correlation dimension
The algorithm and programme for computing the correlation dimension are briefly described in Sect. 4.1.1, which is followed by the description of methodology and discussion of the results of the benchmark testing in Sect. 4.1.2.
4.1.1 Algorithm
A fractal dimension (or, more precisely, the Hausdorff dimension; Hausdorff 1919; Theiler 1990; Ott 2002) is often measured with the correlation dimension, \(d_C\) (Grassberger and Procaccia 1983; Grassberger 1986; Theiler 1990; Alligood et al. 2000; Ott 2002), which takes into account the local densities of the points in the examined dataset. For usual 1D, 2D or 3D cases the \(d_C\) is equal to 1, 2 and 3, respectively. Typically, a fractional correlation dimension is obtained for fractals (Mandelbrot 1983).
4.1.2 Benchmark testing
Results of the correlation dimension benchmark testing for uniformly sampled regions I and II; \(\sigma \) denotes the standard deviation of the sample
Region  \(\langle d_C\rangle \)  \(\sigma \) 

I  1.988  0.027 
II  1.988  0.023 
Results of the correlation dimension benchmark testing when clustering is introduced
Region  \(\langle d_C\rangle \)  \(\sigma \) 

I  1.981  0.026 
II  1.974  0.022 
4.2 Bifurcation diagrams
In dynamical systems theory, a bifurcation occurs when an infinitesimal change of a (nonlinear) parameter governing the system leads to a topological change in its behaviour. Generally speaking, at a bifurcation the stability of equilibria, periodic orbits or other invariant sets is changed. The theory of bifurcation is a vast field (Crawford 1991; Lichtenberg and Lieberman 1992; Baker and Gollub 1996; Alligood et al. 2000; Ott 2002; Kuznetsov 2004; Peitgen et al. 2004). Herein focus will be laid on the pitchfork bifurcations, that are present e.g. in the logistic map (May 1976; Feigenbaum 1979) and that constitute one of the routes to chaos. Let us consider a system of differential equations in the form \(\dot{x}=f(x;\alpha )\), where \(\alpha \) is a parameter, and assume that given \(x_0\) as an IC, for \(\alpha <\alpha _1\) the orbit is 1periodic. A bifurcation at \(\alpha =\alpha _1\) is a point where the trajectory begins to be 2periodic and maintains its periodicity up to \(\alpha =\alpha _2\). Similarly, at \(\alpha =\alpha _2\) a bifurcation occurs on each of the two branches, hence the orbit becomes 4periodic. This scheme, called a perioddoubling (pitchfork) bifurcation cascade, continues until at \(\alpha =\alpha _{\infty }<\infty \) the orbit becomes chaotic. However, in the chaotic zone, \(\alpha >\alpha _{\infty }\), windows of periodic motion with arbitrary period occur. E.g., when a 3periodic trajectory emerges from the chaotic zone it also undergoes the perioddoubling, hence produces orbits that are 6periodic, 12periodic, and so on. A bifurcation diagram is a diagram illustrating this complex mechanism with respect to the nonlinear parameter \(\alpha \). Finally, bifurcations may also occur when \(\alpha \) is decreasing (periodhalving bifurcations) as well as when \(\alpha \) is decreasing or increasing.
Usually, on the bifurcation diagrams there appear to be some curves running through the plot in the chaotic region. These are the so called critical curves (Peitgen et al. 2004) defined by \(x=f^n(x_0;\alpha )\).
5 Results
5.1 Correlation dimension
 1.
IC1 – a total of \(101\times 101=10,201\) ICs distributed uniformly (with a step of \(10^{3}\)) on a \(0.1\times 0.1\) square centered on \((\pi /2,0.55)\);
 2.
IC2 – similar to IC1, but centered on \((\pi /2,1.5)\).

both models given by Eqs. (3) and (10) are Hamiltonian and hence cannot posses a strange attractor (Greiner 2010) that could be characterized by a fractional correlation dimension; assymptotically any set of ICs leading to chaotic orbits should occupy a 2dimensional subset in the phase space;

the number of points, \(N=10,201\), used to estimate the \(d_C\) is relatively small and hence might bias the outcome (Tarnopolski 2014);

local densities exceeding the average density (clustering) affect the correlation dimension such that it is lower than the dimension of the embedding space (see Sect. 4.1.2 and Tarnopolski 2015b).
5.2 Bifurcation diagrams
6 Discussion and conclusions
The aim of this paper was to investigate how does the gravitational interaction with a second satellite influence the rotational dynamics of an oblate moon. A simplified model was designed, resulting in the equation on motion given in Eq. (10), being basically a perturbation of the well known Eq. (3). The derived equation of motion introduces a third parameter, the mass ratio \(m_1/M\), additional to the oblateness \(\omega ^2\) and eccentricity e. To allow comparison, two sets of ICs distributed uniformly in a \(0.1\times 0.1\) square in the phase space were evolved, in case of the absence and presence of the additional source of gravitation. In case of the set IC1 [centered at \((\pi /2,0.55)\)] the difference between the two models is qualitative in nature: when the second satellite was absent all trajectories were quasiperiodic (first row in Fig. 4), as indicated by the \(d_C=1\) in Fig. 5a. Interestingly, when its presence was taken into account, one could observe leaking of the orbits into the chaotic sea (second row in Fig. 4). This phenomenon manifests itself also through a higher \(d_C\) attained in Fig. 5b. Hence, it turns out that an additional satellite has the ability to change quasiperiodic orbits into chaotic ones, i.e. it enlarges the chaotic domain. On the other hand, when the set IC2, located in the center of the chaotic region [an \(0.1\times 0.1\) square centered at \((\pi /2,1.5)\)], was considered, no long term (assymptotic) differences could be observed (third and fourth rows in Fig. 4), and the correlation dimension for both models reached a plateau at \(d_C\approx 1.75<2\) (Fig. 5c, d), likely due to clustering. However, this is not that surprising, given that the gravitational influence under investigation was three orders of magnitude smaller than the planet’s, and that the rotational model in absence of the second satellite is dominated by the chaotic zone, hence it would be highly unexpected for it to have the ability to change chaotic motion into a regular one.
The bifurcation diagrams, especially interesting when \(m_1/M\) and \(\omega ^2\) were fixed and e was varied, when small values of the eccentricity (i.e., \(e<0.03\)) are considered lead to a conclusion that the regular and highly structured picture (Fig. 6d) becomes much more messy, and the transition to chaos occurs for smaller eccentricities than in the case when additional gravitation source is neglected (Fig. 7d). This is consistent with the results of the other method (i.e., evolving the sets IC1 and IC2) in the sense that the second satellite changes regular motion into chaotic. The differences in case when \(\omega ^2\) was varied was not that much remarkable (Figs. 6a, 7a), and both models lead to chaotic motion when larger e are considered (Fig. 6c, 7c). The bifurcation diagram in dependence on the ratio \(m_1/M\) was mostly structureless (Fig. 7b). Eventually, the destruction of regular rotation caused by the second satellite might be ascribed to the destruction of the invariant tori (Tabor 1989; see also Celletti and Chierchia 1998 and references therein).
 1.
the additional source of gravitation can change some regular orbits into chaotic ones, and
 2.
it destroys the regularity, particularly the periodicities and critical curves, in the bifurcation diagram for small eccentricities \(e<0.03\).
Footnotes
Notes
Acknowledgments
The author acknowledges support in form of a special scholarship of Marian Smoluchowski Scientific Consortium MatterEnergyFuture from KNOW funding, grant number KNOW/48/SS/PC/2015.
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