Illuminating dot-satellite motion around the natural moons of planets using the concept of ER3BP with variable eccentricity

Abstract

In the current study, we explore stability of satellite motion around the natural moons of planets in solar system using the novel concept of ER3BP with variable eccentricity. This concept was introduced earlier when novel type of ER3BP (Sun–planet–satellite) was investigated with variable spin state of secondary planet correlated implicitly to the satellite motion (in the synodic co-rotating Cartesian coordinate system) for its trapped orbit near the secondary planet (which is involved in Kepler’s duet “Sun–planet”). However, it is of real interest to explore another kind of aforedescribed problem, ER3BP (planet–moon–satellite) with respect to investigation of satellite motion m around the natural moon m_moon of planet with variable eccentricity of the moon in its motion around the planet. Therefore, we consider here two primaries, M_planet and m_moon, the latter orbiting around their common barycenter on quasi-elliptic orbit with slow-changing, not constant eccentricity (on a large-time scale) due to tidal phenomena. Our aim is to investigate the motion of a small dot satellite around the natural moon of planet on quasi-stable elliptic orbit. Both novel theoretical and numerical findings (for various cases of trio “planet–moon–satellite”) are presented in the current research.

Appendix (mathematical procedure of derivation of Eq. (11)).

Let us present mathematical procedure of derivation of Eq. (11) as follows:

Since we consider that main contribution influencing on the orbit of moon in its motion around the host planet stems from the tides raised on the surface of planet by the moon orbiting in the 1:1 spin–orbit resonance around the planet, we can use formulae (3.2) obtained in [14], e.g., dynamical invariant which interrelates semimajor axis with respect to the eccentricity ({a₁, e₁} = {a_p (0), e (0)} = const):

$$ a_{{\text{p}}} = a_{1} \cdot \left( {\frac{e}{{e_{1}^{{}} }}} \right)^{\frac{8}{19}} \cdot \exp \left( {\frac{51}{{19}}(e^{2} - e_{1}^{2} )} \right), $$

(14)

where the term: exp ((51/19)⋅(e² − e₁²)) ≅ 1. For the reason that time t has not been involved to be presented in expressions in both parts of (14), we can change the independent variable t → f in (14) for {a_p (t), e(t)} → {a_p(f), e(f)} (and vice versa) without losing a generality for the dynamical invariant (14). Let us present Eq. (1) in other form as below

$$ \frac{{{\text{d}}f}}{{{\text{d}}t}} = \left( {\frac{{{\text{GM}}}}{{a_{{{\kern 1pt} 1}}^{{{\kern 1pt} 3}} \cdot {\kern 1pt} \left( {\frac{e}{{e_{1}^{{}} }}} \right)^{{\frac{24}{{19}}}} \cdot \exp \left( {\frac{153}{{19}}(e(f)^{2} - e_{{{\kern 1pt} 1}}^{2} )} \right) \cdot (1 - e(f)^{{{\kern 1pt} 2}} )^{3} }}} \right)^{\frac{1}{2}} \cdot (1 + e(f) \cdot \cos f)^{2} $$

(15)

which can be transformed in case of low-eccentricity orbit e ≅ 0 (by neglecting of terms of second-order smallness in (15)) as follows

$$ \begin{aligned} \frac{{{\text{d}}f}}{{{\text{d}}t}}{\kern 1pt} &= \left( {\frac{e}{{e_{1}^{\,} }}} \right)^{{ - \,\frac{12}{{19}}}} \cdot \left( {\frac{{{\text{GM}}}}{{a_{1}^{{{\kern 1pt} 3}} }}} \right)^{\frac{1}{2}} \cdot (1 + e(f) \cdot \cos f)^{2} \\ & \Rightarrow \left\{ {C = \left( {\frac{{{\text{G}}{\kern 1pt} {\text{M}}}}{{a_{1}^{3} }}} \right)^{\frac{1}{2}} ,\left( {\frac{e}{{e_{1}^{{}} }}} \right)^{{ - \frac{12}{{19}}}} \cong 1} \right\} \\ & \Rightarrow \int {\frac{{{\text{d}}f}}{(1 + 2e \cdot \cos f)} \cong Ct} \\ \end{aligned} $$

(16)

(let us remind that we have chosen e₁ = e₀, a₁ = a₀ in (11) just for simplicity of presentation of the final result). While in (16) eccentricity e is a very slowly varying function on a long-time scale period, it could be considered equal to constant in Eq. (16) for the sufficiently large period of changing of true anomaly f. Thus, in (16) we have obtained the equation solution of which approximately results to (9) as follows (see [8]):

$$ \begin{aligned} & \int {\frac{{{\text{d}}f}}{(1 + 2e \cdot \cos f)} \cong Ct \Rightarrow } \frac{2}{{\sqrt {1 - 4e^{2} } }}\arctan \left( {\frac{(1 - 2e)\tan (f/2)}{{\sqrt {1 - 4e^{{{\kern 1pt} 2}} } }}} \right) \cong Ct \\ & \Rightarrow 2\arctan \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} (1 - 2e)\tan (f/2)\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right) \cong Ct\quad {\text{or}}\quad f \cong 2\arctan \left( {\tan \left( \frac{Ct}{2} \right)(1 + 2e)} \right) \\ \end{aligned} $$