Secular dynamics of hierarchical quadruple systems: the case of a triple system orbited by a fourth body
Bibliographic record
Abstract
We study the secular gravitational dynamics of quadruple systems consisting of a hierarchical triple system orbited by a fourth body. These systems can be decomposed into three binary systems with increasing semimajor axes, binaries A, B and C. The Hamiltonian of the system is expanded in ratios of the three binary separations, and orbit averaged. Subsequently, we numerically solve the equations of motion. We study highly hierarchical systems that are well described by the lowest order terms in the Hamiltonian. We find that the qualitative behaviour is determined by the ratio |$\mathcal {R}_0$| of the initial Kozai–Lidov (KL) time-scales of the binary pairs AB and BC. If |$\mathcal {R}_0\ll 1$|, binaries AB remain coplanar if this is initially the case, and KL eccentricity oscillations in binary B are efficiently quenched. If |$\mathcal {R}_0\gg 1$|, binaries AB become inclined, even if initially coplanar. However, there are no induced KL eccentricity oscillations in binary A. Lastly, if |$\mathcal {R}_0\sim 1$|, complex KL eccentricity oscillations can occur in binary A that are coupled with the KL eccentricity oscillations in B. Even if binaries A and B are initially coplanar, the induced inclination can result in very high eccentricity oscillations in binary A. These extreme eccentricities could have significant implications for strong interactions such as tidal interactions, gravitational wave dissipation, and collisions and mergers of stars and compact objects. As an example, we apply our results to a planet+moon system orbiting a central star, which in turn is orbited by a distant and inclined stellar companion or planet, and to observed stellar quadruples.
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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) | 0.002 | 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".