A semi-analytic model for the study of 1/1 resonant dynamics of the planar elliptic restricted co-orbital problem
Bibliographic record
Abstract
Abstract Mean motion resonances (MMRs) are widespread in our Solar System. Moreover, resonant dynamics has always been an essential topic in planetary research. Recently, the research about exoplanets and the potential Planet Nine with large eccentricity has given rise to our interests in the secular dynamics inside MMRs in the elliptic model. In this paper, we study the fixed points of the averaged Hamiltonian and the long-term stable regions of the 1/1 resonance (or co-orbital motion) in the elliptic restricted three-body problem (ERTBP) systematically. Numerical integrations prove those test particles trapped in “apsidal co-rotation”, where both the resonant angle ϕ res and the secular angle ⧍ ϖ (or apsidal longitude differences) librate simultaneously, always survive the long-term simulations. Furthermore, utilizing a semianalytical method based on the adiabatic approach, three families of long-term fixed points of the averaged Hamiltonian of the planar ERTBP inside the 1/1 resonance have been found. We call them QS-points, Hpoints, and T-points here, whose values of the ( ϕ res , ⧍ ϖ ) are (0°, 180°), (180°, 0°), and (± 60°, ± 60°), respectively. All the fixed points of the averaged Hamiltonian of the co-orbital motion in the ERTBP are presented in the e - e ′ plane (′ represents the elements of the planet in this paper). We find that QS-points and T-points always exist for the arbitrary eccentricity of a planet, while H-points only exist for the cases of low e ′ and very high e . Furthermore, we measure the libration width in terms of eccentricity, ⧍ e , around these stable equilibrium points in the e -⧍ ϖ phase-space portraits. The “apsidal co-rotation” around all the stable equilibrium points is presented in the e - e ′ plane. All these results are effectively confirmed by numerical experiments. The long-term stable zones around these periodic orbits in the e - e ′ plane are significant for the research of the co-orbital motion in the ERTBP. Above all, these practical approaches that we proposed can also be used to study the secular dynamics of other MMRs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".