The Sun Does Not Shrink

Recession can make the Sun small. It cannot make it comfortable.

1 · No. The diameter does not change.

Put a solar filter on the lens and measure. Across an afternoon the Sun’s angular diameter holds constant. (An arcminute is a sixtieth of a degree and an arcsecond a sixtieth of that; the Sun is about 32 arcminutes, or 1,920 arcseconds, across.)

The globe model does predict a change within a single day, and it is in Q8’s favour: the Sun genuinely does recede as it sets. An observer with it overhead stands one Earth radius nearer than one seeing it on the horizon — the solar parallax, 8.794 arcseconds — so the horizon Sun is 0.082 arcseconds smaller, or 0.0043 per cent, off a disc of 1,920. Q8’s own geometry demands a drop of 1,448 arcseconds between 45° and 10°. The two are apart by a factor of about eighteen thousand.

2 · What “shrinks due to perspective” has to mean, in numbers

Perspective’s entire quantitative content is one relation: for an object of fixed size S at distance r, the visual angle is θ ≈ S/r. “The Sun shrinks due to linear perspective” and “the Sun recedes” are one claim in two vocabularies, and it can be worked. Put a lamp of diameter S at height h above a flat plane. Seen at elevation α, the slant distance is exactly r = h/sin α, so θ(α) = S sin α/h and θ(α)/θ(90°) = sin α. The height cancels out of every ratio. There is no free parameter — 700 miles, 3,000 miles or a figure never stated, the required shrink between any two elevations is the same.

And the rate peaks where nobody looks for it. At noon the Sun is at its closest, its distance momentarily stationary, and the predicted change is zero; near the horizon the fractional change dies away again. The peak sits where the horizontal distance equals the Sun’s height — 45° altitude — either side of noon.

Computed for a lamp 5,000 km up with its sub-point moving at 1,670 km/h, the smallest defensible speed; the flat model’s own map makes every figure larger (Method notes). None of it touches the θ ∝ sin α ratio, in which no speed and no height appear.

So the test is mid-morning or mid-afternoon, which is also the cleanest air of the day: total refraction at 45° is one arcminute, and the differential across the disc — the part that distorts shape — is far smaller again. The model makes its loudest prediction under the conditions least able to excuse it: a sixth of the diameter within the hour, in still, clear, high air. Two filtered frames sixty minutes apart are enough to check, and the disc is the same size in both.

3 · What does change is the brightness, by a factor of tens to hundreds

Something real happens as the Sun descends, and it is not size. Two mechanisms, neither of which moves the Sun.

Extinction along the slant path. Sunlight at 45° crosses 1.41 air masses — an air mass being the path through the atmosphere relative to looking straight up; at one degree it crosses 26. Each air mass costs a fixed fraction of the beam, so the loss compounds, at a rate set by the extinction coefficient k, the loss per air mass in magnitudes. His stipulated conditions — clear, dry, high — put k near 0.12; a good clear night at sea level is nearer 0.20, and typical sea-level air about 0.28, which makes the dimming around 2,300× at half a degree. Both columns are given because the difference between them is larger than most of the argument.

Kasten–Young air mass; dimming is 10^{0.4k(X−1.41)}. Atmosphere only — occlusion by ground, sea or haze bank is §7’s subject. The last two rows are indicative rather than predictive: below about two degrees the whole slant path lies inside the aerosol layer and a single k stops being a good model. Cleaner air makes the dimming smaller and does nothing whatever to the diameter.

The cosine law. Illuminance on a horizontal surface — the ground, a cloud top, a downward-pointing camera — falls as the cosine of the zenith angle, which is the sine of the elevation, before any absorption at all. Between 45° and 10° that is a factor of 4.07 on its own; multiplied by the table’s extinction — 1.6× in his high, dry air, 2.2× in good sea-level air — it gives the sixfold-to-ninefold darkening of a landscape that Q8 notices, and the whole content of the time-lapse at p. 171, where the cloud deck goes dark while the Sun is still in full view and the caption asks “Local light?” No: the cloud top is a horizontal surface, and the Sun lighting it is unchanged. That 4.07 is the same number as the required shrink — both are sine ratios between 45° and 10° — but one is a projection effect on a surface and the other would be a distance effect on a source, and §4 tells them apart.

4 · The Sun’s size does change — annually — and that is how we know the method works

Earth’s orbit is an ellipse, so the Sun is nearer in January than in July:

An amateur photographed exactly that. Anthony Ayiomamitis shot the Sun through a TeleVue Pronto with a Canon EOS 300D and a Baader ND5 filter on 2 January and 5 July 2005 — same telescope, camera and filter — and published the pair side by side, labelled 32.53′ and 31.46′. He does not say whether those figures came off his frames or an ephemeris, so read them as labels; the evidence is two exposures at one focal length in which the 3.40 per cent is plainly there to be measured. See for yourself.

The professional record is older than the classical proof the book spends most of its energy on. A generation before Eratosthenes set up his gnomon, Aristarchus had the Sun at a 720th of the zodiac circle — 1,800 arcseconds, six per cent under the modern 1,919 — and Archimedes bracketed it between 1,620 and 1,976 arcseconds in The Sand Reckoner, having first corrected for the fact that “the eye does not see from one point but from a certain area.” Twenty-three centuries of better instruments have narrowed that band without moving it: Auwers synthesised a century of nineteenth-century heliometer, meridian-circle, eclipse-contact and micrometric work into a semi-diameter of 959.63″; SODISM II at Calern gives 959.78 ± 0.19″ from over 20,000 observations — a telescope on a mountain in France, not in orbit.

Three different things get called the size of the Sun. The annual variation is 3.40%. The diurnal recession is 0.0043%, and that is the only one this page claims. The Auwers-to-SODISM agreement is 0.016% — between values standardised to 1 AU across 130 years of instruments, a statement about measurement consistency, not about what a disc does between lunch and dinner.

5 · Dimming and shrinking are the same number — and the Sun does one without the other

This section answers p.37 on p.37’s own terms. (The book calls the attenuation an exponential decrease and points to the inverse-square law; that law is a power law, and the difference is the discriminator §6(c) turns into a measurement, because absorption is exponential and recession is not.)

Under pure recession the two effects are locked together. Flux falls as 1/r². Diameter falls as 1/r. Therefore, for a resolved source,

diameter ratio = √(flux ratio).

You cannot select one and decline the other. The flat geometry’s requirement — 4.07× smaller at 10° than at 45° — also demands the direct beam fall by 4.07² = 16.6×, to about 6% of its noon value. Clear-sky direct beam at 10° is only between about 1.9× and 2.9× down on its 45° value, depending on site altitude and aerosol load — the §3 law for measured extinction coefficients from 0.16 at 2 km altitude to 0.28 at sea level. Recession over-predicts the observed beam dimming by a factor of roughly six to nine; to reconcile it, atmospheric extinction would have to be negative.

Then the deeper theorem, which removes the escape rather than narrowing it. Specific intensity — flux per unit solid angle, which is to say surface brightness — is conserved along a ray in a medium that neither absorbs, emits nor scatters: precisely the vacuum leg between the Sun and the top of the atmosphere, where any recession would occur. Condon and Ransom’s Essential Radio Astronomy states it with the Sun as the worked example: the angular size of the Sun depends on the distance to the camera; the photons falling on the detector per unit area per unit time per unit solid angle do not.

Absorption and scattering are what reduce a disc’s brightness per unit area, and they operate inside the atmosphere, where the theorem does not apply. What the setting Sun does is become dim per unit area at full width: a 32-arcmin orange disc you can look at directly. That is extinction’s signature, and the reverse of recession’s.

6 · The measurements that were made for other reasons

The strongest evidence against a claim is usually an instrument built by someone who had never heard of it.

(a) Cassini’s meridian line, San Petronio, Bologna, 1655. Gian Domenico Cassini set a gnomon hole 27.07 m up in the fourth vault of the basilica and laid a 66.71 m brass line across the floor. His purpose, twenty-two years after Galileo’s trial, was to test whether the Sun’s slower summer motion was merely greater distance — Kepler’s second law.

The pinhole geometry is the cleanest discriminator we have. The image’s transverse width is

width = θ × L, where L = pinhole-to-floor distance = H / sin α.

For a distant Sun, θ is fixed and the width goes as 1/sin α: small in summer when the Sun is high, large in winter when it is low. For a lamp at fixed height h above a flat plane, θ = S·sin(α)/h, so the width becomes

(S·sin α / h) × (H / sin α) = S·H/h

— a constant, at every hour of every day of every year, for every possible lamp height.

The Bologna Observatory reports the floor image as 26 cm across in summer and 168 × 64 cm in winter. The recession model requires 26 cm and 26 cm. The measured transverse widths differ by a factor of 2.46. (The elongation recovers Bologna’s solstice geometry; our forward reconstruction runs 6% high on the winter figure — Method notes, §8. The argument rests on the ratio, which needs no absolute scale.) Cassini’s own accuracy on the diameter was about one arcminute, right at the 1.1-arcminute size of the real eccentricity effect, which is why it took a 67-metre instrument to see it.

(b) The 150-foot solar tower, Mount Wilson, daily since 4 January 1917. A solar image roughly 42 cm across is projected onto a drawing table; observers trace sunspots by hand onto a 25.5 × 50 cm pad, a drawing “sometimes requiring hours to complete.” Over 27,000 are archived. Eighteen cardboard Stonyhurst disks cover the year’s B-angle (the tilt of the Sun’s axis toward Earth) and, in the source’s words, “earth orbital distances from the sun” — the ±1.7% eccentricity effect, resolved in cardboard. Recession would take that 42 cm limb circle to 10.3 cm partway through a single afternoon’s drawing, with the paper clamped down; the annual variation the observers do allow for is 1.4 cm. Whether the table can be moved to rescale the image we could not establish (§8); no refocusing scheme rescues a four-fold change mid-trace.

(c) The Langley plot — the standard photometer calibration: plot the logarithm of the signal against air mass through a morning and extrapolate to zero air mass. Every AERONET reference instrument — the master photometers at Mauna Loa and Izaña, from which every field calibration in the network descends — is calibrated this way as the airmass runs from about 7 down to 2, solar elevations of roughly 7.8° to 29.9°, a run of 1.5 to 2.25 hours. It works because Beer–Lambert — the exponential loss law behind the §3 table — makes that plot straight in airmass. Under recession the beam would go as 1/r² ∝ sin²α ∝ 1/X², straight in ln X rather than X, and would fall by 92% across that run where clean-air extinction at optical depth 0.08–0.10 falls by 33–39%. Ten-plot-averaged calibrations reproduce to 0.25–0.5%, single plots to 0.7% and 0.9% — in exactly the elevation band where recession requires the disc to shrink 3.6×.

(d) The almanac. The Sun’s semi-diameter is tabulated in the Nautical Almanac by date only — six values across the year, 15.8′ to 16.3′. Refraction is the altitude-dependent correction in the sextant tables; semi-diameter is not, and never has been. USNO computes sunrise and sunset at geometric zenith distance 90.8333°: 34 arcmin of horizon refraction plus a flat 16 arcmin of semi-diameter. Two centuries of navigators have applied a semi-diameter correction that does not know what altitude the sight was taken at.

The same tables turn an absent correction into a present one. The Moon does get an altitude-dependent semi-diameter correction, called augmentation: the excess of the topocentric disc (seen from the observer’s spot on the surface) over the geocentric one (seen from Earth’s centre). An observer with the Moon overhead stands one Earth radius nearer to it than one with the Moon on the horizon — 6,378 km against 384,400, which is 1.66 per cent — so the disc really is bigger at the zenith, by about 0.27 arcminutes at mean distance and 0.3 near perigee, carried in the Moon tables as a term in sin(altitude). The Sun’s version of the identical effect is 6,378 divided by 149.6 million, which is 0.0043 per cent, or four hundredths of an arcsecond, and so the Sun gets no such term. The correction is applied to exactly the body whose distance makes it measurable, and withheld from exactly the body whose distance does not — by the ratio the globe model predicts, in tables two centuries older than this argument.

(e) Six telescopes, every minute, since 1995. The Global Oscillation Network Group runs six identical instruments at Big Bear, Learmonth, Udaipur, Teide, Cerro Tololo and Mauna Loa, imaging the full solar disc at 2.5″ per pixel once a minute at about 90 per cent duty cycle, spread in longitude so that the Sun is always high somewhere and low somewhere else. A 16.7-per-cent-per-hour dependence of radius on airmass would be the largest signal in that dataset, reversing sign at local noon at every station, and would have made helioseismology — reading the Sun’s interior from its surface oscillations, which is what GONG is for — impossible for thirty years. It is not in the data.

(f) Concentrator optics. The thermodynamic ceiling on solar concentration is C = 1/sin²θ for the Sun’s angular radius θ — about 46,200 at 4.653 mrad (milliradians), a fixed design constant throughout the concentrating-solar-power literature. A Sun whose angular radius fell fourfold through the afternoon would lift that ceiling sixteenfold, and late-afternoon operation would beat noon.

7 · Why the sunset is the wrong place to look

Everything below happens in the last degree or two of altitude, where the flat model’s own demand has fallen to under 6 per cent an hour and every optical confound is at its worst. This page does not rest on it; it is here because it is checkable and because the book puts its case there.

Q8’s second half says the Sun “disappears completely from a lack of visibility.” What it does instead: the lower limb goes first. The disc is cut by a straight edge. Its horizontal width is undiminished right down to the last sliver, and the upper limb is the brightest thing in the frame at the moment it goes. Any filtered sunset time-lapse — a common genre — shows the sequence, and the timing is checkable against any almanac (Method notes).

Three things follow that recession cannot produce.

(a) The green flash exists because atmospheric dispersion sets the red, green and blue images of the upper limb at slightly different moments behind a hard edge; a source fading uniformly with distance has no edge to set behind. It is a mirage effect, not a transparency one: on Andrew Young’s classification roughly two-thirds to three-quarters of observed flashes are inferior-mirage flashes and most of the remainder mock-mirage flashes. Clean air is necessary and nowhere near sufficient.

(b) The stand-up test. Dip of the horizon is 1.753·√(height in metres) arcmin, tabulated in every nautical almanac. Rising from lying (0.2 m, dip 0.78′) to standing (1.7 m, dip 2.29′) lifts the horizon 1.50 arcmin, which at 45°N at an equinox — altitude changing at 10.61 arcmin per minute — returns about eight seconds of Sun. A beach to a 100 m cliff buys 15.2 arcmin and about 1.4 minutes.

(c) The shape. The one real distortion of the setting Sun is anisotropic. Differential refraction lifts the lower limb about 6 arcmin more than the upper — Bennett’s formula gives 34.48′ at apparent altitude 0° against 28.42′ at 0.533°, a differential of 6.05′ — squashing the vertical diameter to roughly 26 arcmin, a flattening of up to roughly 19% — the “about 20%” Cowley quotes — for a normal temperature profile with the observer near sea level (a strong inversion can produce other distortions entirely, so this is a rule with a domain). The horizontal diameter is untouched. Recession is isotropic and cannot make a squash of either sign.

Which hands the page its cleanest prescription: measure the horizontal diameter, the axis refraction leaves alone, and the confound disappears.

The book’s other Sun: p.145 needs a stretch

At p.145 the perspective rescue is extended from Polaris to the Sun: elevation angles are said to be compressed, so the Sun is really higher than it looks. Our Celestial Globes page works out the optical field that requires — a vertical magnification of 1.57× at 45° elevation, 2.18× at 30° and 3.27× at 10° — a low Sun drawn as a vertical egg roughly 3.3 times taller than wide. Q8 requires the Sun 4.07× smaller in both axes at that same 10°. In the vertical the two claims differ by a factor of 13.3 and point in opposite directions. They sit in different chapters, but any single physical account of the sky has to choose one of them.

8 · Where this page could be wrong

• The vertical diameter genuinely does change. Refraction flattens the low Sun: about 0.8 per cent with the lower limb at 10°, 2.4 per cent at 5°, and up to roughly 19 per cent with it on the horizon. Every constancy claim on this page is about the horizontal axis.
• Our Bologna winter reconstruction is 6 per cent out. Working forward from the quoted 27.07 m pinhole height we get a winter floor image 68.3 cm across, against the 64 cm reported. Our best guess is floor geometry at the far end of the line, and it is a guess. The argument rests on the ratio, which needs no absolute scale.
• The horizon extinction figures extrapolate a law past its range. Flagged in §3; nothing in the argument depends on them.
• We have no published amateur time series of the diameter through a day. We computed what an amateur rig makes possible — 751 pixels across the disc with a 300 mm lens on APS-C, one pixel being 0.13 per cent of the diameter — but could not find a dataset that has done it. The pixel arithmetic is checkable by anyone; the dataset is a gap.
• We could not establish whether the Mount Wilson image scale is adjustable. Our argument uses only the clamped paper and the hours-long drawing, both documented.
• We take “linear perspective” to mean what it means in optics. Some writers in this tradition use “the laws of perspective” for a compression of everything toward an eye-level vanishing point, which is not θ ∝ 1/r. We answer the phrase Miller used; any other perspective has to arrive with its own equations, and the last item in §9 stands ready to work them.
• We derived θ ∝ sin α from a lamp at fixed height above a flat plane. If Miller’s model has a Sun of varying height, or a surface that is not a flat plane, the derivation does not apply as written. We would redo it.

9 · What would change our mind

Specified in advance, cheap, and falsifiable. Any one of these would move us.

A filtered photograph pair. Same camera, lens and afternoon, solar filter fitted: the Sun at 45° and at 10°. Measure the horizontal diameter in pixels. A 300 mm lens on APS-C puts the disc across 751 pixels, one pixel being 0.13% of the diameter. The recession model predicts 751 px and 185 px; we predict 751 px and 751 px, the second frame simply darker and redder. A horizontal diameter differing by more than about 1% between those elevations would mean this page needs rewriting.

The pinhole version — Cassini’s experiment at domestic scale. Punch a hole in card and project onto a screen at a fixed throw (a hole a millimetre or two across adds its own width to the image; the sizes below assume an ideal pinhole, and the variation survives either way). At 2 m the solar image is 18.6 mm across; at 5 m, 46.5 mm; at 10 m, 93.0 mm. There are two predictions, not three, because a fixed-height lamp above a flat plane and a receding Sun are the same model:

The fixed throw makes this read the opposite way from Bologna, where the throw itself scales as 1/sin α: same geometry, different mounting, inverted signature. Neither needs a camera.

A published dataset showing solar radius correlated with airmass by more than the ~20 milliarcsec SODISM II’s own error budget already allows at zenith distances under 70°. That would not vindicate recession, but it would demolish this page’s central claim.

A clear-air sunset sequence over a sea horizon, filtered, in which a full-width circular disc fades to nothing without ever being cut by the horizon line. Q8 says that is what happens; photograph it under the conditions Q8 specifies and we are wrong.

A nautical almanac, of any nation or century, tabulating the Sun’s semi-diameter as a function of apparent altitude rather than of date. Produce one and §6 collapses.

A Langley run that comes out straight in ln(airmass) with slope −2 rather than straight in airmass, from airmass 7 to 2 on a clear morning. The equipment is a photodiode and a clock.

A demonstration that the setting Sun’s surface brightness is unchanged from midday — flux per unit solid angle, not total flux. If the low Sun is as blinding per square arcminute at 5° as at noon, §5 is wrong and the mechanism is recession after all. The test is a photometer and a known angular aperture.

A statement from Miller of the Sun’s height and diameter in his model, if it implies something other than θ ∝ sin α. We would work it.

10 · The bottom line

Something does happen to the Sun on a clear afternoon, and it deserves the account Q8 asks for. It is not size: through a filter the disc holds its diameter, where the flat model’s own geometry demands it lose a sixth every hour at 45°. It is the light getting weaker, sixteenfold in his own high, dry air and nearer a hundredfold at sea level by one degree, with a cosine law costing another 4.07 on any horizontal surface. And the two cannot be swapped, because recession fixes the ratio between them and does not change surface brightness — so it cannot produce the one thing that actually happens, a full-width Sun gone soft enough to look at.

And at the very end, in the worst part of the sky: the Sun does not fade out. It is cut off, bottom first, by a hard line that can be raised again by standing up — the one thing a receding object could never permit.

Sources & further reading

• Levi Miller, *Globe Deconstruction?* (prerelease draft, 2026) — Q8 at printed p.51; light attenuation and the inverse-square law at p.37; the perspective correction extended from Polaris to the Sun at p.145; the SUN OBSERVATIONS list at p.165; the above-the-clouds time-lapse at p.171; the ‘linear perspective of the Sun’ figure at p.172. Page numbers are the printed folio; PDF page indices differ between renders of the prerelease and are not given.
• Archimedes, The Sand Reckoner, trans. T. L. Heath (1897) — https://sacred-texts.com/cla/archim/sand/sandreck.htm. Aristarchus’ ‘about a 720th part of the circle of the zodiac’ (1,800 arcsec); Archimedes’ bracket of between a 200th and a 164th part of a right angle (1,620 to 1,976 arcsec, containing the modern 1,919); “the eye does not see from one point but from a certain area.”
• Arthur Auwers (1891), adopted solar semidiameter 959.63 arcsec at 1 au — https://solareclipses.com/pages/Auwers.html. With modern figures near 959.90–959.97, the ~0.34 arcsec total spread across 130 years.
• M. Meftah et al., ‘Ground-based measurements of the solar diameter during the rising phase of solar cycle 24’, *Astronomy & Astrophysics* 569, A60 (2014) — https://www.aanda.org/articles/aa/full_html/2014/09/aa23598-14/aa23598-14.html. SODISM II / PICARD-SOL radius 959.78 ± 0.19 arcsec from over 20,000 observations; refraction under 20 milliarcsec in the error budget for zenith distances inside 70°; turbulence bias ~0.6 arcsec at r0 ≈ 3 cm.
• Bologna Astronomical Observatory, ‘La meridiana di San Petronio 1655–2005’ — http://stelle.bo.astro.it/archivio/2005-anno-cassiniano/meridian_ing.htm; corroborated by IMCCE, https://ssp.imcce.fr/meridienne/italie.php?lang=en. The 27.07 m gnomon height, the 66.71 m line, the 26 cm summer and 168 × 64 cm winter floor image, and Cassini’s ~1 arcminute accuracy.
• UCLA / Mount Wilson 150-Foot Solar Tower, daily sunspot drawing programme — http://obs.astro.ucla.edu/150_draw.html. The ~42 cm image, drawings since 4 January 1917, over 27,000 archived, the 25.5 × 50 cm clamped pad, drawings ‘sometimes requiring hours to complete’, eighteen Stonyhurst disks for B-angle and Earth–Sun distance.
• Anthony Ayiomamitis, ‘Sun at Aphelion and Perihelion’, Perseus astrophotography archive — https://www.perseus.gr/Astro-Solar-Scenes-Aph-Perihelion.htm. TeleVue Pronto, Canon EOS 300D, Baader ND5 solar filter, Losmandy G-11 mount; 2 January 2005 (12:53:52 UT+2) and 5 July 2005 (13:09:24 UT+3) at 1/2500 and 1/800 second; labelled 32.53′ and 31.46′. Whether those figures were measured from the frames or taken from an ephemeris is not stated; this page relies only on the single unchanged optical configuration.
• C. Toledano et al., ‘Assessment of Sun photometer Langley calibration at the high-elevation sites Mauna Loa and Izaña’, *Atmospheric Chemistry and Physics* 18, 14555 (2018) — https://acp.copernicus.org/articles/18/14555/2018/. The airmass 7-to-2 range, 1.5–2.25 hour runs, reproducibility 0.25–0.5% on ten-plot averages (0.7% Mauna Loa, 0.9% Izaña single plots).
• J. J. Condon & S. M. Ransom, *Essential Radio Astronomy*, §2.2 — https://www.cv.nrao.edu/~sransom/web/Ch2.html. Invariance of specific intensity along a ray in empty space, with the Sun as the worked example.
• J. L. Heilbron, The Sun in the Church: Cathedrals as Solar Observatories (Harvard University Press, 1999), and his article from it in The Sciences 39:5 (1999) — https://catholiceducation.org/en/science/faith-and-science/the-sun-in-the-church.html. Cassini’s purpose in his own words: ‘to give astronomers an accurate and simple procedure directly from observation for establishing the “orbit” of the sun’; the line made it ‘possible to measure the disk diameter accurately enough to distinguish the two theories’, the observations agreeing with Kepler. Cassini’s values: 31′08″ at apogee, 32′10″ at perigee, against the modern 31.46′ and 32.53′. The 27.07 m and 66.71 m dimensions are quoted from the Bologna observatory and given identically by Caltech Library citing Heilbron; we have not seen them on a numbered page of the book.
• A. T. Young, ‘Green Flashes at a Glance’ and observing pages at San Diego State — https://aty.sdsu.edu/papers/Zenit/glance.html and https://aty.sdsu.edu/observing/advice.html. About two-thirds to three-quarters of observed green flashes are inferior-mirage flashes, most of the rest mock-mirage; transparency is a transmission condition, not the cause.
• On the Moon’s augmentation, §6(d): USNO Astronomical Almanac glossary, “the increase in the topocentric apparent semidiameter of a celestial body compared to its apparent semidiameter when viewed from the geocenter” (https://aa.usno.navy.mil/faq/asa_glossary); the Nautical Almanac’s Moon altitude-correction tables carry it as a term proportional to sin(altitude), of order 0.016 of the semi-diameter (NavList, https://navlist.net/Nautical-Almanac-Moon-Correction-Tables-FrankReed-jul-2006-g897); standard practice: “the distance between the sun and the observer being very large there is no need to take into account such augmentation for the sun.” The 0.27′ at mean distance and 0.3′ near perigee are computed from 6,378 km over 384,400 and 356,500 km.
• US Naval Observatory, ‘Rise, Set, and Twilight Definitions’ — https://aa.usno.navy.mil/faq/RST_defs. Sunrise and sunset at geometric zenith distance 90.8333°: 34 arcmin of refraction plus a flat 16 arcmin of semi-diameter.
• NavList on the Nautical Almanac daily pages — https://navlist.net/Semidiameter-Nautical-Almanac-LaPook-dec-2009-g11176. The Sun’s semi-diameter tabulated by date only, six values from 15.8′ to 16.3′.
• F. Kasten & A. T. Young (1989), relative optical airmass X = 1/[sin α + 0.50572(α + 6.07995°)^−1.6364]. The §3 table: 1.413 at 45°, 5.586 at 10°, 37.92 at the horizon.
• Asterism.org, ‘Atmospheric Extinction and Refraction’ — https://asterism.org/resources/atmospheric-extinction-and-refraction/. Broadband visual extinction ~0.28 mag per airmass at sea level, 0.24 at 0.5 km, 0.21 at 1 km, 0.16 at 2 km — the range behind the 1.9×–2.9× beam figure in §5 — and ~40 airmasses at the horizon.
• Les Cowley, Atmospheric Optics, ‘Flattened Suns’ — https://atoptics.co.uk/blog/flattened-suns/. Vertical flattening reaches about 20% ‘for a normal atmospheric temperature profile and an observer close to sea level’.
• National Solar Observatory, Global Oscillation Network Group — https://nso.edu/telescopes/nisp/gong/. Six telescopes at Big Bear, Learmonth, Udaipur, Teide, Cerro Tololo and Mauna Loa; 2.5 arcsec per pixel every minute since 1995 at ~90% duty cycle.
• R. Pitz-Paal, ‘Concept and status of Concentrating Solar Power systems’, *EPJ Web of Conferences* 189, 00008 (2018) — https://www.epj-conferences.org/articles/epjconf/pdf/2018/24/epjconf_eps-sif2018_00008.pdf. The concentration ceiling C = 1/sin²θ ≈ 46,200 for the Sun’s 4.653 mrad half-angle.