Atmospheric extinction, sky glow and background

A path can remove source signal and add sky signal

Atmospheric extinction reduces radiation retained in a selected source beam through absorption and scattering out of that direction. Sky background adds radiation to the selected measurement. Natural emission, scattered light and artificial illumination can contribute, depending on wavelength and conditions.

Use one elementary rate model, C = T S₀ + K. Here S₀ is the expected source-electron rate that the otherwise fixed instrument would have without atmospheric loss, T is dimensionless transmission for the selected band and direction, and K is the added sky-electron rate within the extraction region. All rate terms use electrons per second.

For invented S₀ =100, T =0.8 and K =50, the total expected rate is 130 electrons/s. Subtracting the exactly known mean sky rate leaves 80 electrons/s from the transmitted source. It does not return the 20 electrons/s lost from that source-direction contribution.

Dividing the 80 by a known 0.8 can infer the 100-electron/s pre-loss source rate within this model. That is a model-based estimate, with its noise and transmission uncertainty propagated. It is not a physical recovery of the removed photons or a complete atmospheric transfer solution.

A finite sky region adds estimation uncertainty

A real background mean is often estimated from pixels outside the source region. Those pixels contain a fluctuating sample, so the inferred mean has uncertainty. Subtracting that estimate adds another uncertainty term to the source measurement.

Consider a synthetic uniform sky with expected b =90 electrons per pixel. The source extraction contains n =10 pixels and 100 expected source electrons. Estimate the mean sky level from m =40 separate independent sky pixels with the same exposure and response, containing no source light.

For independent Poisson sky pixels, the per-pixel mean estimate has variance b/m =90/40 =2.25 in electron-count variance units. Subtracting n times that estimated mean adds n²b/m =225 to the extraction variance.

The source and sky fluctuations inside the aperture already contribute 100 +10 ×90 =1000. Including the finite-estimate term gives total variance 1225, standard deviation 35 electrons, and model SNR 100/35≈2.86. Treating the sky mean as exact would omit that additional 225.

This derivation assumes a uniform background, independent disjoint samples and no source wings, covariance, read noise or dark terms. Gradients, unresolved sources and response variation can make a distant sky region an imperfect reference. More sky pixels do not by themselves cure a biased background model.

A star field can support a bounded observational exercise

A useful beginner exercise compares a preselected star field under documented conditions. Record the date, field, observer, unaided method, viewing altitude, Moon, cloud, transparency and dark-adaptation conditions. Keep uncertain detections distinct from confident ones.

In an original hypothetical comparison, the selected reference set contains bright, intermediate and faint stars in that supplied brightness order. Session A records the bright and intermediate references but not the faint one. Session B records only the bright reference.

The selected set therefore has two detections in A and one in B. That observation can motivate further comparisons of background and conditions. It does not establish an absolute sky radiance, prove that artificial light alone caused the difference or calibrate a universal faintest visible star.

The documented IAU activity provides a starting point for such comparisons. The interpretation here retains observer and condition limits; it does not adopt an uncalibrated transparency-hole magnitude scale, a universal eye threshold or an all-sky extrapolation from one small field.

Visibility, point-source flux and sky brightness are different measures

A reference star’s received flux is a point-source quantity. Sky surface brightness describes radiation per angular area. A count of visible reference stars is an observer-and-condition-dependent record. These measurements can inform one another through a model, but they are not interchangeable units.

A stronger sky-brightness claim needs suitable calibrated measurements and known response, with wavelength range and uncertainty. Repeated matched observations can test consistency while retaining variation in transparency, seeing, Moon and observer response.

For instrument data, preserve the distinction between subtracting a mean, estimating that mean and accounting for its fluctuations. For visual records, preserve the selected field and observation method. Both practices connect a conclusion to the actual evidence.

The next lesson examines separated collecting elements. Interferometric spatial sampling introduces another kind of measurement limit, alongside the source, path, image and background limits developed here.

Worked examples

Try the reasoning

1. Does subtracting the known sky mean remove the random sky-arrival fluctuations in that exposure?

   Show the answer and reasoning

   No.

   The mean contribution and the exposure’s fluctuating contribution are different parts of the model.

2. Should a background estimated from a finite independent sky sample always be treated as exact?

   Show the answer and reasoning

   No.

   Its uncertainty propagates into the corrected source estimate.

3. Do two selected detections in A and one in B prove a calibrated all-sky loss caused only by artificial light?

   Show the answer and reasoning

   No.

   The selected sample, observer response, conditions and lack of radiance calibration limit that conclusion.

4. Is a star’s total received flux the same quantity as sky brightness per angular area?

   Show the answer and reasoning

   No.

   They have different spatial meanings and require the appropriate calibration and units.

What to carry forward

• Extinction attenuates a source contribution while sky background adds another contribution.
• Background fluctuations and finite mean-estimation error remain in a corrected measurement.
• Documented visibility comparisons support bounded claims; quantitative sky brightness needs its own calibration.