Inspiral time probability distribution for two black holes captured by emitting gravitational radiation
Bibliographic record
Abstract
If two initially unbound black holes of masses ${M}_{1}$ and ${M}_{2}$, total mass $M={M}_{1}+{M}_{2}$, reduced mass $\ensuremath{\mu}={M}_{1}{M}_{2}/({M}_{1}+{M}_{2})$, and initial relative velocity $v\ensuremath{\ll}c(4\ensuremath{\mu}/M)$ in otherwise empty space are captured into a bound orbit by emitting gravitational radiation, the inspiral time to coalescence increases monotonically to infinity as the impact parameter $b$ approaches from below the critical impact parameter ${b}_{c}=[340\ensuremath{\pi}{G}^{7}{M}^{6}\ensuremath{\mu}/(3{c}^{5}{v}^{9}){]}^{1/7}=[(85\ensuremath{\pi}/384)(4\ensuremath{\mu}/M){]}^{1/7}(2GM/{c}^{2})(v/c{)}^{\ensuremath{-}9/7}$ for capture. Assuming a uniform flux of impinging black holes with $b\ensuremath{\le}{b}_{c}$, the cumulative probability for impact parameters smaller than some value $b$, conditional upon the impact parameter being smaller than ${b}_{c}$, is $P=(b/{b}_{c}{)}^{2}$. Then it is shown that the inspiral time for $[M{v}^{2}/(4\ensuremath{\mu}{c}^{2}){]}^{2/7}\ensuremath{\ll}P<1$ is $T=(2\ensuremath{\pi}GM/{v}^{3}){P}^{21/4}\ensuremath{\zeta}(3/2,1\ensuremath{-}{P}^{7/2})$, and closed-form approximate expressions for the inverse function $P(T/{T}_{0})$ with ${T}_{0}=2\ensuremath{\pi}GM/{v}^{3}$ are also given.
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 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) | 0.008 | 0.001 |
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".