Fun With Science  /  Globe Deconstruction  /  Q8 · page 51  /  Draft

The Sun Does Not Shrink

Q8 asks why the Sun fades instead of setting. The fading is real and large; the shrinking that would have to come with it is absent to two decimal places.

This page grants Question 8 its instincts. It does not grant its observation. The instincts are good ones: the conditions Miller specifies — clear, dry, high — are the right conditions to test this under, the reach for the inverse-square law on p. 37 is the reach a receding source deserves, and the low Sun genuinely does change in a way that reads to the eye as retreat. What it does not do is get smaller. "Shrinks due to linear perspective" is a statement about distance, because perspective diminution is nothing other than the statement that a fixed object's visual angle falls as one over its distance. Put a lamp of fixed size at any fixed height above a plane and the arithmetic closes with no adjustable parameter: its apparent diameter must be proportional to the sine of its elevation. From a 45° midday to 10° that is a required shrink of 4.07×, and it is 4.07× whether the lamp is 700 miles up, 3,000 miles up, or at a height the book never states. We measured that requirement against what the disc is observed to do — in the Bologna basilica since 1655, on a Mount Wilson drawing table since 1917, across six GONG telescopes since 1995, and in the sextant-correction tables of two centuries of navigators — and we could not find it. That check is not a modern one, and it is older than the classical proof the book spends most of its energy on: a generation before Eratosthenes set up his gnomon, Aristarchus already had the Sun at a 720th of the zodiac circle — 1,800 arcseconds, six per cent under the modern 1,919 — and Archimedes, Eratosthenes’ own correspondent, 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. And perspective diminution is a claim about relative distance — a thing shrinks only when you get meaningfully further from it — so a disc that holds its measured diameter is the cheapest disproof of a local Sun there is. It needs no curvature, no eclipse timing, no agency’s word, and no instrument the ancients did not already have. The fading, meanwhile, is entirely there, and has a size: a cosine law worth a factor of four on any horizontal surface between 45° and 10°, and atmospheric extinction worth another 1.6× in his own high, dry air or 2.2× at sea level — call it a sixfold darkening of the landscape under his conditions and closer to ninefold at the coast. Our conclusion is that the setting Sun does exactly one of the two things the claim requires. It dims. We could not establish that it shrinks, and the physics of a receding source does not permit dimming without shrinking.

Diameter constant; only the brightness changesDemands 16.7% an hour at 45° — absentThe “shrinking” is bloom collapse, not the discSame rig resolves the 3.4% annual changeObservation of dimming: grantedConditions well chosen: grantedInverse-square instinct: grantedSun really does recede: 0.0043%
Where this lands

The conditions Miller specifies — clear, dry, high — are the right ones to test this under, and half of what he reports seeing is there. The dimming is real and large: about eightfold on a horizontal surface between 45° and 10° elevation, with the Sun's own output unchanged. His instinct to reach for the inverse-square law on p.37 is the correct instinct, because that is the law a receding source obeys. (He calls it an exponential decrease; it is a power law. Not pedantry here — it is exactly the discriminator §6(c) turns into a measurement, because absorption is exponential and recession is not.) But invoking it commits the model to a second number that is not optional. For a resolved source, flux falls as 1/r² and diameter falls as 1/r, so the diameter ratio is the square root of the flux ratio, always. The flat geometry's own requirement — 4.07× smaller at 10° than at 45° — therefore also demands the direct beam fall 16.6×, to about 6% of its noon value, where clear-sky measurement puts it at roughly a half. And the disc that must have shrunk by 1,448 arcseconds has been measured, by unrelated techniques across 130 years, to within about a third of an arcsecond. The two claims are separated by a factor of roughly four thousand. Separately, and independently of any of that, the Sun at sunset does not fade away: it is cut, bottom-first, by a straight edge, at undiminished horizontal width, with its upper limb the brightest thing in the frame at the moment it goes — and it can be brought back by standing up, which is the one thing a genuinely receding object could never permit.

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, and the constancy is not marginal.

The globe model does predict a change within a single day, and it belongs first because it is real and it is in his 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. That makes the horizon Sun 0.082 arcseconds smaller, or 0.0043 per cent, off a disc of 1,920.

So the Sun really does shrink as it sets, in exactly the direction Q8 claims, and the honest thing is to say so. It shrinks by 0.082 arcseconds. 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. Diminution is the statement that angular subtense falls as one over distance. “The Sun shrinks due to linear perspective” and “the Sun recedes” are one claim in two vocabularies, and the claim can be worked.

So work it. Put a lamp of diameter S at height h above a 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 has a maximum, which is not where anyone looks for it. At noon the Sun is at its closest, so its distance is momentarily stationary and the predicted change is zero. Far towards the horizon distance grows nearly in proportion to time and the fractional change dies away again. The peak sits where the horizontal distance equals the Sun’s height — 45° altitude — symmetric either side of noon.

How the table is computed, and on whose numbers. For an observer under the track, the fractional rate is (v/h) · sin 2α / 2, which is where both the zero at noon and the peak at 45° come from. The table takes the lamp 5,000 km up with its sub-point crossing at 1,670 km/h — and that figure is the globe’s own equatorial rotation speed, chosen deliberately because it is the smallest defensible number and we would rather understate his demand than inflate it. The flat model’s own map makes it larger: on the azimuthal-equidistant projection the equinox sun-track is a circle of radius 10,019 km completed in 24 hours, so the sub-point moves at 2,623 km/h and the 45° demand becomes 26.2 per cent an hour, not 16.7. Drop the lamp to 3,000 km and it is 43.7. Every figure in the table is a floor. Two caveats on the table itself: it assumes the sub-point passes overhead, true at the equator at equinox and differing in detail but not in character elsewhere; and at 10° with a 5,000 km lamp the horizontal distance is 28,400 km, which is past the rim of the disc on most flat maps. None of that touches the θ ∝ sin α ratio argument above, in which no speed and no height appear at all.

So the test is mid-morning or mid-afternoon, which is convenient, because that 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 is therefore making its loudest prediction under the conditions least able to excuse it. It says the disc should lose a sixth of its diameter within the hour, in still, clear, high air, with no atmospheric effect anywhere near large enough to hide the loss or to fake it. Two filtered frames sixty minutes apart are enough to check, and the disc is the same size in both.

One coincidence to disarm before it causes trouble. The flux requirement of 16.6× in §5 and the 16.7 per cent hourly demand here are unrelated quantities that happen to land on the same digits. The first is 4.07². The second is v/2h for one illustrative height. Nothing connects them.

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 atmospheres of air; at one degree it crosses 26. Each air mass costs a fixed fraction of the beam, so the loss compounds. His own stipulated conditions — clear, dry, high — put the extinction coefficient k near 0.12 magnitudes per air mass; a good clear night at sea level is nearer 0.20, and typical sea-level air runs to about 0.28 — which makes the dimming larger still, around 2,300× at half a degree. Both 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)}. This is atmosphere only — nothing here is the Sun being cut off by ground, sea or haze bank, which is a separate effect and the one §7 is about. The last two rows should be read as indicative rather than predictive: below about two degrees the whole slant path lies inside the aerosol layer, so a single k stops being a good model, and the disc is by then heavily refracted and usually partly occluded anyway. Cleaner air makes the dimming smaller and does nothing whatever to the diameter.

The cosine law — cosine of the zenith angle, which is the sine of the elevation, so the two names describe one thing. Illuminance on a horizontal surface — the ground, a cloud top, a downward-pointing camera — falls as sin(elevation) before any absorption at all. Between 45° and 10° that is a factor of 4.07 on its own, with the Sun’s output untouched. Multiplied by extinction it gives the sixfold to ninefold darkening of a landscape that Q8 correctly 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. Illuminance on a horizontal cloud top falls as the sine of the elevation before a single photon is absorbed, and the Sun lighting it is unchanged.

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

It would be wrong to claim the disc is a fixed size. 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, same camera, same filter, six months apart — and published the pair side by side, labelled 32.53′ and 31.46′. He does not say whether those figures were taken off his own frames or off an ephemeris, so read them as labels rather than as an independent determination. The evidence is the images: 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 says the same thing from the other direction. Auwers synthesised roughly a century of nineteenth-century heliometer measures, meridian-circle transits, eclipse contact timings and micrometric work and obtained a semi-diameter of 959.63″; SODISM II at Calern gives 959.78 ± 0.19″ from over 20,000 observations — and that is a telescope on a mountain in France, not an instrument in orbit, which is why it is here at all. This page uses ground-based measurements throughout and cites the one space result it mentions explicitly as an illustration it does not rely on.

Three different things get called the size of the Sun; do not let them run together. The annual variation is 3.40%. The diurnal recession is 0.0043%. The Auwers-to-SODISM agreement is 0.016% — but that is between values standardised to 1 AU across 130 years of instruments, and it is 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, and it is the strongest physics on the page.

Under pure recession the two effects are locked together with no freedom at all. 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 own requirement — 4.07× smaller at 10° than at 45° — therefore also demands the direct beam fall by 4.07² = 16.6×, to about 6% of its noon value. Measured clear-sky direct beam at 10° is between about 1.9× and 2.9× down on its 45° value, depending on site altitude and aerosol load. Recession over-predicts the observed dimming by a factor of roughly six to nine. To reconcile it, the model would need atmospheric extinction to be negative.

Then the deeper theorem, which removes the escape rather than merely 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. That is 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, but the photons falling on the detector per unit area per unit time per unit solid angle do not.

Absorption and scattering along the path are the mechanisms that reduce a disc's brightness per unit area — and they operate inside the atmosphere, where the invariance theorem does not apply, which is exactly the point. 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 it is 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. Here there are several, and two of them predate the debate by centuries.

(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 declared purpose was the length of the tropical year for the calendar; his working 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 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's own description of the instrument 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.

Two honesty notes. The reported elongation, 168/64 = 2.63, independently recovers 1/sin(22.02°) = 2.67 — Bologna's winter-solstice noon elevation at latitude 44.49° and obliquity 23.49° — which confirms we are reading the figures as the geometry says we should. And a first-principles reconstruction from the quoted 27.07 m pinhole height gives 26.6 cm and 68.3 cm: the summer figure lands, the winter figure runs about 6% high, most likely from floor geometry at the far end of the line. We could not resolve that discrepancy and we are not smoothing it over. The argument rests on the ratio, which needs no absolute scale at all. Cassini's own accuracy on the solar diameter was about one arcminute, which is 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 different cardboard Stonyhurst disks are cut to cover the year's B-angle and, in the source's own words, "earth orbital distances from the sun" — that is, the ±1.7% eccentricity effect is resolved in cardboard. A check falls out for free: 42 cm at 9.301 mrad implies a focal length of 45.2 m, which is 148 feet, in a tower named for 150.

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 have to allow for is 1.4 cm.

(c) The Langley plot. Every AERONET reference instrument — the master photometers at Mauna Loa and Izaña, from which every field calibration in the network descends — is calibrated by measuring the direct beam through a morning as the airmass runs from about 7 down to 2 — solar elevations of roughly 7.8° up to 29.9°, a run of 1.5 to 2.25 hours — and extrapolating ln(signal) against airmass to zero airmass. The method works because Beer–Lambert makes that plot straight in airmass. Under recession the beam would go as 1/r² ∝ sin²α ∝ 1/X², which is straight in ln X, not in 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 at Mauna Loa and Izaña reproduce to 0.25–0.5%, and single plots to 0.7% and 0.9%. That is exactly the elevation band in which the recession model 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 as the moment the Sun's centre reaches 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. (e) Concentrator optics. The thermodynamic ceiling on solar concentration is C = 1/sin²θ for the Sun's angular radius θ; at 4.653 mrad that is about 46,200, quoted as a fixed design constant throughout the concentrating-solar-power literature. If the Sun's angular radius fell fourfold through the afternoon, that ceiling would rise 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. It is the weakest ground for both sides, which is why this page does not rest on it. It is recorded because it is checkable, and because the book puts its case there. And here is the part that turns an absent correction into a present one. The Moon does get an altitude-dependent semi-diameter correction in the same tables, and it has a name: augmentation. An observer with the Moon overhead stands one Earth radius nearer to it than an observer 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. The almanac’s Moon tables carry it 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 not missing. It 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. The network exists to watch the Sun oscillate, not to measure its diameter, and the six sites are spread in longitude precisely so that the Sun is always high somewhere and always low somewhere else. A 16.7-per-cent-per-hour dependence of measured radius on airmass would therefore not be a subtle systematic in that dataset. It would be the largest signal in it, present at every station, reversing sign at local noon, and it would have made helioseismology impossible for thirty years. It is not in the calibration literature because it is not in the data.

Q8's second half says the Sun "disappears completely from a lack of visibility." We looked at 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.

The timing is checkable against any almanac. At an equinox, for a Sun setting due west, altitude falls at 15·cos(latitude) arcmin per minute, so a 32-arcmin disc takes 2.13 minutes to cross at the equator, 3.02 minutes at 45°N, and 3.43 minutes at 51.5°N.

Three things follow that recession cannot produce.

(a) The green flash. It is an upper-limb phenomenon that exists because atmospheric dispersion sets the red, green and blue images of the limb at slightly different moments behind a hard edge. A source fading uniformly with distance has no edge to set behind. A green flash also needs transparent air to get any green through at all — but the flash itself is a mirage effect rather than 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 are mock-mirage flashes, each requiring a particular thermal structure at the horizon. 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 because no sextant sight is usable without it. 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 — where altitude changes at 10.61 arcmin per minute — returns about eight seconds of Sun. Going from 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′ of refraction 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 about 20% for a normal temperature profile with the observer near sea level. Under a strong inversion the profile can produce other distortions entirely, so this is a rule with a domain, not a law. 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.

Short section, and the one place this page interlocks directly with our Celestial Globes page.

At p.145 the perspective rescue is extended from Polaris to the Sun explicitly: elevation angles are said to be compressed, so the Sun is really higher than it looks. We worked out on that page what optical field such a compression requires — a vertical magnification of 1.57× at 45° elevation, 2.18× at 30° and 3.27× at 10° — and showed it would render the low Sun a vertical egg roughly 3.3 times taller than wide.

Q8 requires the Sun 4.07× smaller in both axes at that same 10° elevation. In the vertical, the two claims differ by a factor of 13.3 and point in opposite directions: one stretches the disc, the other shrinks it.

8 · Where this page could be wrong

The standing of this review rests on our arithmetic checking out, so here is where it is weakest.

• Three different quantities are all called “the size of the Sun”. The annual variation is 3.40 per cent; the diurnal recession is 0.0043 per cent; the Auwers-to-SODISM agreement is 0.016 per cent, and that last one is between values standardised to 1 AU across 130 years of instruments, not a statement about a single afternoon. Only the middle one is our claim. If any sentence here reads as though the disc were a fixed size, it is an error and we want to know which one.
• 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. Any that is not so qualified is wrong.
• 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 — but a reader is entitled to weigh the 6 per cent against us.
• The horizon extinction figures extrapolate a law past its range. Flagged in §3 where they appear, and repeated here because they are the softest arithmetic on the page. 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 we could not find a dataset that has actually done it, and we are not implying one exists. 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 of which are documented.
• We take “linear perspective” to mean what it means in optics. Some writers in this tradition use “the laws of perspective” for a different thing: a compression of everything toward an eye-level vanishing point, which is not θ ∝ 1/r. We answer the phrase Miller actually used. Linear perspective is by definition the projective model in which a fixed object’s visual angle falls as one over its distance, and that is what this page works. Any other perspective is no longer an appeal to established optics and has to arrive with its own equations — at which point the last item in §9 already stands ready to work them.
• We derived θ ∝ sin α from a lamp at fixed height above a plane. If Miller’s model has a Sun of varying height, or a surface that is not a plane, the derivation does not apply to it 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, same lens, same afternoon, solar filter fitted: the Sun at 45° and at 10°. Measure the horizontal diameter in pixels, because that is the axis refraction does not touch. 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, with the second frame simply darker and redder. A horizontal diameter differing by more than about 1% between those elevations would mean we have missed a real effect and this page needs rewriting. The pinhole version — Cassini's experiment at domestic scale, correctly stated. Punch a hole in card and project onto a screen at a fixed throw. Use a hole a millimetre or two across, or subtract its diameter from what you measure — the hole’s own width adds to the image, and the sizes below assume an ideal pinhole. The variation the test turns on 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 plane and a receding Sun are the same model:

Model	Image at 45°	Image at 10°
Sun at fixed distance	18.6 mm	18.6 mm
Any lamp at fixed height above a plane	18.6 mm	4.6 mm

Note that the fixed throw is what makes this test read the opposite way from Bologna. There, the screen is the floor and the throw itself scales as 1/sin α, so the fixed-height lamp predicts a constant floor image and the distant Sun predicts a growing one. Same geometry, different mounting, inverted signature. Both are decisive; 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 — no bottom-first occlusion, no straight edge, no bright upper limb at the last instant. 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. If two hundred years of navigators had needed an altitude-dependent semi-diameter, it would be in the tables. 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. That is the difference between a receding source and an absorbing atmosphere, plotted. 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 dim only because it is small, and its disc is as blinding per square arcminute at 5° as at noon, then §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 honest bottom line

The instincts behind Q8 are sound and the conditions were well chosen. Something does happen to the Sun on a clear afternoon, it is large, and it deserves the account Q8 asks for. What it is not is the thing Q8 names.

It is not size. Through a filter the disc holds its diameter, and the flat model’s own geometry — θ ∝ sin α, with the height cancelling and no parameter left to adjust — demands that it lose a sixth of that diameter every hour at 45° elevation. Five times the annual variation, in roughly a four-thousandth of the time, on equipment that has already been shown to catch the annual variation. It is not there.

What happens instead is that the light gets weaker: 1.41 air masses at 45° against 5.59 at 10° and 26 at one degree, dimming the disc by a factor of sixteen in his own high, dry air and nearer a hundred at sea level before it reaches the horizon, with a cosine law costing another 4.07 on any horizontal surface. That is a real, large, measured effect with the Sun standing still.

The diameter has been measured to within about a third of an arcsecond by unrelated techniques across 130 years, drawn at 42 cm on a Mount Wilson table every clear day since 1917, and cast on a Bologna floor since 1655 by an instrument built to test Kepler. We could not find the shrink.

And at the very end, for what it is worth 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 — which is 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. Establishes the claim under review, in the book's own words. Page numbers are the folio printed on the page, read off a per-page render in each case, and every one was checked individually. We give no PDF page indices: the prerelease circulates in renders whose leading pages differ, so an index that is right against one copy is off by one against the next. The folio is the only citation that survives that, which is why it is the only one here.
• Archimedes, The Sand Reckoner, trans. T. L. Heath (1897) — https://sacred-texts.com/cla/archim/sand/sandreck.htm. Establishes the two oldest figures cited in the opening: Archimedes' report that Aristarchus 'discovered that the sun appeared to be about a 720th part of the circle of the zodiac' (1,800 arcsec), and Archimedes' own bracket on the solar diameter of between a 200th and a 164th part of a right angle (1,620 to 1,976 arcsec, containing the modern 1,919), together with his stated correction for the fact that “the eye does not see from one point but from a certain area.”
• J.-P. Rozelot & C. Damiani, 'A brief history of the solar diameter measurements', arXiv:1609.02710 — https://arxiv.org/pdf/1609.02710. Corroborates the Aristarchus and Archimedes values in modern units and carries the record forward through Ibn al-Shatir's 29′30″–32′32″ apogee-to-perigee range to the present. Establishes that the band the ancients set has been narrowed rather than moved.
• Arthur Auwers (1891), adopted solar semidiameter 959.63 arcsec at 1 au — https://solareclipses.com/pages/Auwers.html. A synthesis of roughly a century of heliometer, meridian-circle, eclipse-contact and micrometric determinations. Establishes the historical baseline value and, 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. Establishes the SODISM II / PICARD-SOL radius of 959.78 ± 0.19 arcsec from over 20,000 observations, and — critically for this page — the published error budget placing refraction under 20 milliarcsec for zenith distances inside 70°, with turbulence bias of ~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. Establishes the 27.07 m gnomon height, the 66.71 m meridian line, and the reported floor-image dimensions of 26 cm in summer against 168 × 64 cm in winter, together with Cassini's ~1 arcminute accuracy on the annual diameter variation.
• UCLA / Mount Wilson 150-Foot Solar Tower, daily sunspot drawing programme — http://obs.astro.ucla.edu/150_draw.html. Establishes the ~42 cm projected image, drawings since 4 January 1917, over 27,000 archived, the 25.5 × 50 cm clamped pad, drawings 'sometimes requiring hours to complete', and the eighteen cardboard Stonyhurst disks cut to cover the year's B-angle and Earth–Sun distance variation.
• Anthony Ayiomamitis, 'Sun at Aphelion and Perihelion', Perseus astrophotography archive — https://www.perseus.gr/Astro-Solar-Scenes-Aph-Perihelion.htm. Establishes the amateur side-by-side cited in §4: TeleVue Pronto, Canon EOS 300D, Baader ND5 solar filter, Losmandy G-11 mount, imaged 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′. The page does not state whether those figures were measured from the frames or taken from an ephemeris, and this page does not rely on them as an independent determination — only on the imaging being at a single unchanged optical configuration, which the equipment line establishes.
• 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/. Establishes the airmass 7-to-2 calibration range, the 1.5–2.25 hour run length, and reproducibility of 0.25–0.5% on ten-plot averages (0.7% Mauna Loa, 0.9% Izaña for single plots).
• J. J. Condon & S. M. Ransom, *Essential Radio Astronomy*, §2.2 — https://www.cv.nrao.edu/~sransom/web/Ch2.html. Establishes the invariance of specific intensity along a ray in empty space, stated with the Sun as its worked example: angular size depends on distance to the camera; photons per unit area per unit time per unit solid angle do not.
• J. L. Heilbron, The Sun in the Church: Cathedrals as Solar Observatories (Harvard University Press, 1999), and the author’s own article drawn from it in The Sciences 39:5 (1999) — https://catholiceducation.org/en/science/faith-and-science/the-sun-in-the-church.html. Establishes Cassini’s purpose at San Petronio in his own account: he set out ‘to give astronomers an accurate and simple procedure directly from observation for establishing the “orbit” of the sun’, and with the meridian line ‘it became possible to measure the disk diameter accurately enough to distinguish the two theories’ — the observations agreeing with Kepler. Cassini’s own values were 31′08″ at apogee and 32′10″ at perigee, against the modern 31.46′ and 32.53′. The 27.07 m and 66.71 m dimensions are quoted here from the Bologna observatory, and are given identically by Caltech Library citing Heilbron; we have not seen them on a numbered page of the book itself.
• A. T. Young, 'Green Flashes at a Glance' and the observing pages at San Diego State — https://aty.sdsu.edu/papers/Zenit/glance.html and https://aty.sdsu.edu/observing/advice.html. Establishes the classification used in §7: about two-thirds to three-quarters of observed green flashes are inferior-mirage flashes and most of the rest mock-mirage flashes, each requiring a specific thermal structure at the horizon, with transparency a transmission condition rather than the cause.
• On the Moon’s augmentation, cited in §6(d): the USNO Astronomical Almanac glossary defines it as “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 implement 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); and standard navigational practice states plainly that “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 here from 6,378 km over 384,400 and 356,500 km respectively.
• US Naval Observatory, Astronomical Applications Department, 'Rise, Set, and Twilight Definitions' — https://aa.usno.navy.mil/faq/RST_defs. Establishes that published sunrise and sunset are computed at a geometric zenith distance of 90.8333° — 34 arcmin of horizon refraction plus a flat 16 arcmin of semi-diameter, independent of altitude.
• NavList discussion of the Nautical Almanac daily pages and altitude-correction tables — https://navlist.net/Semidiameter-Nautical-Almanac-LaPook-dec-2009-g11176. Establishes that the Sun's semi-diameter is tabulated by date only, six values across the year from 15.8′ to 16.3′, while refraction carries the altitude dependence.
• F. Kasten & A. T. Young (1989), relative optical airmass formula X = 1/[sin α + 0.50572(α + 6.07995°)^−1.6364]. The standard reference formula in solar and atmospheric work. Establishes the airmass table used in §3: 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/. Establishes typical broadband visual extinction coefficients of ~0.28 mag per airmass at sea level, 0.24 at 0.5 km, 0.21 at 1 km and 0.16 at 2 km, and corroborates ~40 airmasses at the horizon.
• Les Cowley, Atmospheric Optics, 'Flattened Suns' — https://atoptics.co.uk/blog/flattened-suns/. Establishes that vertical flattening of the setting Sun reaches about 20% 'for a normal atmospheric temperature profile and an observer close to sea level' — the qualification is the source's own, and it is why this page states the rule with its domain.
• National Solar Observatory, Global Oscillation Network Group — https://nso.edu/telescopes/nisp/gong/. Establishes six identical telescopes at Big Bear, Learmonth, Udaipur, Teide, Cerro Tololo and Mauna Loa, full-disk imaging at 2.5 arcsec per pixel every minute since 1995 at ~90% duty cycle.
• J. R. Kuhn et al., 'The Precise Solar Shape and Its Variability', *Science* 337, 1638 (2012); summarised at https://physicstoday.aip.org/news/the-suns-oblateness-appears-to-be-constant. Establishes an equatorial-minus-polar radius of 7.2 ± 0.5 milliarcsec from SDO/HMI. Cited as an illustration of the precision the quantity supports, and explicitly not relied upon, being a space measurement.
• 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. Establishes the thermodynamic concentration ceiling C = 1/sin²θ ≈ 46,200 for the Sun's 4.653 mrad half-angle, quoted as a fixed design constant.
• Kopp & Lean (2011), *Geophysical Research Letters* 38, L01706 — https://spot.colorado.edu/~koppg/TSI/. Establishes total solar irradiance at 1 au of 1360.8 W/m² at the 2008 solar minimum, the quantity a Langley extrapolation recovers from the ground.
• Companion pages on this site: docs/celestial-globes/index.html (§3 vertical-magnification table, and the structural result that a lamp at finite height above a plane reaches elevation zero only at infinite distance); docs/bottom-up-observations/index.html (angular size does not depend on eye height); docs/rampion/index.html (hull-first occlusion at the same horizon line); docs/luminaries/index.html (Q9, the preceding entry in this series).