Bibliographic record
Abstract
Abstract The traditional approach to analyzing mean motion resonances (MMRs) is through the canonical perturbation theory. While this is a powerful method, its generality leads to complicated combinations of variables that are challenging to interpret and require looking up numerical coefficients particular to every different resonance. In this paper, we develop simpler scaling relations in the limit where orbits are closely spaced (period ratios ≲2), and interplanetary interactions can be approximated by only considering the close approaches each time the inner planet overtakes the outer at the conjunction. We develop geometric arguments for several powerful results: (i) that p : p − q MMRs of the same order q are all rescaled versions of one another, (ii) that the general case of two massive planets on closely spaced, eccentric, coplanar orbits can be approximately mapped onto the much simpler case of an eccentric test particle perturbed by a massive planet on a coplanar circular orbit, and (iii) that, while the effects of consecutive conjunctions add up coherently for first-order ( p : p − 1) MMRs, they partially cancel for p : p − q MMRs with order q > 1, providing a physical explanation for why these higher-order MMRs are weaker and can often be ignored. Finally, we provide simple expressions for the widths of MMRs and their associated oscillation frequencies that are universal to all closely spaced MMRs of a given order q , in the pendulum approximation.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".