Bibliographic record
Abstract
We derive a spatiotemporal analytical resolution of the dark night sky, or Olbers' paradox, first showing that in an infinitely large universe the cumulative solid angle of the light that is projected upon the celestial sphere by an infinite population of directly observable stars is indeed finite. Using the GAIA DR2 data, we show that the number radial density of the stars that are directly visible has a radial distribution that is quasi‐unimodal, and that such a radial distribution is constrained by the inverse square law, which asymptotically limits the magnitude of the cumulative solid angle of the light that is projected upon the celestial sphere. Second, we show that the temporal summation parameter of the imaging device that samples the celestial sphere projection further limits the cumulative apparent brightness of the image formed on the detector surface of the imaging device. The unaided human eye is a biological imaging device, which in the mode of achromatic dim light sensing known as scotopic vision, has a short temporal summation window of approximately 650 ms, which is determined by retinal photochemistry (Holmes et al., Vision Res., 2017, 140, 33–43; Lamb & Pugh Jr, Prog. Retin. Eye Res., 2004, 23(3), 307–380). This short temporal summation window makes the eye act as a high‐pass power filter, making it incapable of rendering visible those stars that project an apparent brightness below a certain threshold. We show analytically that astronomical objects that individually have an apparent brightness mG > 8 will not contribute to the cumulative apparent brightness of the night sky as seen with the fully dark‐adapted unaided eye under ideal seeing conditions, even when the celestial sphere is tiled completely with an infinite number of such faint objects. Using numerical simulations, we also show that a large‐scale homogeneous star field and a Galactic‐scale inhomogeneous field project collectively a cumulative apparent brightness upon the human retina that is approximately 100 million times fainter than the apparent brightness of the Sun. We show that the cumulative solid angle projected by all the unaided eye visible stars is approximately 5.95×10−13 sr, and the asymptotic cumulative solid angle limiting distance is approximately 2 kpc, which is the radial distance beyond which no significant increase in the cumulative apparent brightness of the night sky can be detected with the unaided eye, even in a large‐scale homogeneous and infinitely large universe with an infinite number of stars.
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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 | 0.014 |
| 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 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".