Free and forced inclinations of orbits perturbed by the central body’s oblateness and an inclined third body

Abstract

The effect of inclined third-body perturbation on the perturbed body’s inclination evolution has been investigated extensively. However, under various initial orbital elements, the exact effect of the $J_2$ perturbation and the inclined third-body perturbation on perturbed body’s inclination is still an open problem. In this paper, we use a simplified model for long-term orbital variations to analyze the free and forced inclinations for almost circular orbits with low inclinations. We analyze the influence of the $J_2$ perturbation on the inclination evolution under various initial orbital elements. The results show that the $J_2$ perturbation consistently decreases the maximum value of the inclination. Besides, the $J_2$ perturbation either decreases the amplitude of the inclination’s oscillation, or enables the perturbed body to orbit closer to the equator of the central body. These effects enable the objects with initially low inclinations to maintain their orbits near the equator of the central body for a long time. The results of the study are applied to circum-Jovian orbits and the triple asteroid system 87 Sylvia.

Proof of inequality \({I}_{\text {1max}, {{J}_{2}}=0} \ge {I}_{\text {1max}, {{J}_{2}}\ne 0}\) for \(0<\gamma <1\)

We aim to prove that the inequality ${I}_{\text {1max} ,{{J}_{2}}=0}\ge {I}_{\text {1max} ,{{J}_{2}}\ne 0}$ holds for $0<\gamma <1$.

Step 1: Start with the given inequality Eqs. (32 and 39):

$$\begin{aligned} {{I}_{20}}+\sqrt{I_{10}^{2}+I_{20}^{2}-2{{I}_{10}}{{I}_{20}}\cos \Delta \Omega }\text { }-\gamma {{I}_{20}}-\sqrt{I_{10}^{2}+{{\gamma }^{2}}I_{20}^{2}-2\gamma {{I}_{10}}{{I}_{20}}\cos \Delta \Omega }\ge 0.\nonumber \\ \end{aligned}$$

(A1)

Step 2: Add $\sqrt{I_{10}^{2}+{{\gamma }^{2}}I_{20}^{2}-2\gamma {{I}_{10}}{{I}_{20}}\cos \Delta \Omega }$ on both sides:

$$\begin{aligned} \left( 1-\gamma \right) {{I}_{20}}+\sqrt{I_{10}^{2}+I_{20}^{2}-2{{I}_{10}}{{I}_{20}}\cos \Delta \Omega }\ge \sqrt{I_{10}^{2}+{{\gamma }^{2}}I_{20}^{2}-2\gamma {{I}_{10}}{{I}_{20}}\cos \Delta \Omega }.\nonumber \\ \end{aligned}$$

(A2)

Step 3: Square both sides and simplify:

$$\begin{aligned} \begin{aligned}&{{\left( 1-\gamma \right) }^{2}}I_{20}^{2}+2\left( 1-\gamma \right) {{I}_{20}}\sqrt{I_{10}^{2}+I_{20}^{2}-2{{I}_{10}}{{I}_{20}}\cos \Delta \Omega } \\&+\left( 1-{{\gamma }^{2}} \right) I_{20}^{2}-2\left( 1-\gamma \right) {{I}_{10}}{{I}_{20}}\cos \Delta \Omega \ge 0. \\ \end{aligned} \end{aligned}$$

(A3)

Step 4: Since $1-\gamma >0$, divide both sides by $(1-\gamma )/2$ and simplify:

$$\begin{aligned} \sqrt{I_{10}^{2}+I_{20}^{2}-2{{I}_{10}}{{I}_{20}}\cos \Delta \Omega }\ge {{I}_{10}}\cos \Delta \Omega -{{I}_{20}}. \end{aligned}$$

(A4)

Step 5: Classified discussion:

If ${{I}_{10}}\cos \Delta \Omega -{{I}_{20}}<0$. Inequality (A4) holds, which implies that inequality ${I}_{\text {1max} ,{{J}_{2}}=0}\ge {I}_{\text {1max} ,{{J}_{2}}\ne 0}$ holds for $0<\gamma <1$.

If ${{I}_{10}}\cos \Delta \Omega -{{I}_{20}}\ge 0$, square both sides of inequality (A4) and simplify. The derived inequality is:

$$\begin{aligned} {{\cos }^{2}}\Delta \Omega \le 1, \end{aligned}$$

(A5)

which holds for all real numbers. This implies that inequality ${I}_{\text {1max} ,{{J}_{2}}=0}\ge {I}_{\text {1max} ,{{J}_{2}}\ne 0}$ holds for $0<\gamma <1$.

More generally, consider two particles A and B, where $\gamma =\gamma _1$ for A and $\gamma =\gamma _2$ for B, and $\gamma _1<\gamma _2$. Other initial conditions are the same for A and B. Then we have:

$$\begin{aligned} {I}_{\text {1max} ,\textrm{B}}\ge {I}_{\text {1max} ,\textrm{A}}. \end{aligned}$$

(A6)

We omit the proof for Eq. (A6) here, since this process is similar to the one described above.