Gravitational stability of spherical self-gravitating relaxation models
Bibliographic record
Abstract
The gravitational stability of spherical, self-gravitating, hydrostatically pre-stressed planetary models remains a subject of active interest. Love (1907, 1911) was the first to show that purely elastic models can become unstable when values of rigidity and bulk modulus are insufficient to counteract self-gravitational collapse. We revisit his calculations and extend his work to show that so-called dilatational (or ‘D’) modes of a viscoelastic sphere can also become unstable to self-gravitation in a specific region of Lamé parameter space. As an example, we derive a marginal stability curve for the dilatational modes of a homogeneous planetary model at spherical harmonic degree two. We demonstrate that the stability conditions are independent of viscosity and that the instability will occur only when the homogeneous earth model is already unstable to the elastic instability described by Love (1907, 1911). Finally, we also consider a class of Rayleigh–Taylor (or ‘RT’) instabilities related to unstable density stratification in planetary models. This convective instability is explored using both a homogeneous Maxwell viscoelastic sphere (which has an unstable layering at all depths) and a suite of Maxwell earth models that adopt the elastic and density structure of the seismic model PREM (which has regions of unstable density stratification within the upper mantle). We argue that previous studies have significantly overestimated the potential importance of these modes to Earth evolution. For example, suggestions that the timescale of the RT modes is short relative to the age of the Earth face the fundamental problem that the ensuing convective instability would have long ago destroyed the unstable layering and produced an adiabatic profile. We predict that at low degrees the RT instabilities for a PREM density profile and realistic viscosity stratification have timescales comparable to the age of the Earth. It is unclear, in any event, whether the unstable density layering in the PREM upper mantle is robust.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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) | 0.001 | 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".