Fun With Science / Globe Deconstruction / Q5 + Q6 · pages 48–49
Globe Deconstruction Review — Q5: why everything in the sky appears to move at the same speed across one night, and what does not.
Q5 on the diurnal sweep: correctQ5 on stars vs planets: contradictedQ5's conclusion: does not follow
Where this lands
The claim (Q5, p. 48): “Why is there no difference in visual speed between stars and planets as the Earth spins during a single night? A large distance gap requires that visual speeds will not be equal to the observer.”
The verdict. There is a difference, and it is large. Measured against the star background rather than the horizon, planets at opposition (Earth passing between them and the Sun) drift between four and fifty-four arcseconds an hour in inverse proportion to their distance, while stars drift by nothing — thirteen to one across the planets, five hundred to one once the Moon is included, ordered by distance throughout. (An arcsecond is 1/3,600 of a degree, an arcminute 1/60; the full Moon is about 30 arcminutes, or 1,800 arcseconds, across.) What the question gets right is the part it names: the diurnal sweep, the once-a-day turning of the whole sky, really is identical for everything, to one part in ten billion even for the nearest star — not a coincidence but a consequence of what rotation is, a change in where the observer points, which contains no distance term. His intuition that translation gives a rate going as one over distance is the right law, correctly remembered; it is applied to the wrong motion, and where it does apply, the effect it predicts is there.
Q6, on whether Jupiter’s moon shadows line up the way a single distant Sun requires, is answered on its own page.
The Moon covers eight of its own diameters against the background between dusk and dawn — naked eye, no instrument, one night, exactly as asked.
A line of sight changes for two independent reasons: the observer's orientation changes, or the observer's position changes. Earth's rotation does both, in wildly unequal amounts.
The orientation change is a full turn per day: 360.9856123 degrees per UT1 day (UT1 being the day as Earth’s actual rotation defines it), or 15.041067 arcseconds per second. The official expression for it, in the method notes at the end, has one property worth noticing: nothing in it refers to the object being observed. The same rotation that swings a telescope past a lamppost a kilometre away swings it past a quasar, and the swing rate is the telescope's, not the lamppost's. The only distance-dependent departure for a star is diurnal parallax, and for the nearest star, Proxima Centauri, it is a wobble of about 65.5 microarcseconds peak-to-peak against a horizon-to-horizon sweep of 180 degrees — 648,000 arcseconds. One part in ten billion. Miller is not exaggerating. He is rounding down.
The train-window rule — angular rate from an observer's translation is the perpendicular velocity divided by the distance — is real, and the error is narrow: Earth's rotation is overwhelmingly a change in where the observer points, not where the observer is. The position change is the observer being carried around Earth's axis on a lever arm of at most 6,378.1366 km. That part is distance-dependent; its name is diurnal parallax, and it is priced below.
The rate carries its own check: 86,400 / 1.00273781191135448 = 86,164.0989 seconds, or 23h 56m 04.1s, so the stars return 235.90 seconds — three minutes fifty-six — early every day. That is why mounts ship with a sidereal drive rate (sidereal: reckoned against the stars rather than the Sun) distinct from a solar one, and why the constellations shift through the seasons; the companion page on whether the Earth spins or the sky does derives it at length.
One qualification, which a hostile expert would raise. It is not true that the diurnal rate is identical for everything. What is invariant, for objects distant enough that the Earth-radius-over-distance ratio is negligible, is the angular rate about the celestial axis. A star at declination δ (its angular distance north or south of the celestial equator) sweeps a linear track at cos δ of the equatorial rate; a star near the pole barely moves at all. Atmospheric refraction compresses altitudes near the horizon, by up to roughly 35 arcminutes at the horizon itself. Solar-system bodies carry diurnal parallax on top of that, and the Moon and planets add their own orbital motion. Miller's claim is very nearly true for stars, and "very nearly" is where all the physics lives.
Whether objects at different distances move at visibly different apparent speeds depends on which coordinate you measure in. Relative to the horizon you are measuring the observer's rotation, common to everything. Relative to the star background the spread is enormous.
| Body | Drift against the stars (deg/day) | Distance recovered by Kepler III (au) |
|---|---|---|
| Moon | 13.176 | — |
| Mercury | 4.092 | 0.3871 |
| Venus | 1.602 | 0.7233 |
| Sun | 0.986 | — |
| Mars | 0.524 | 1.5237 |
| Jupiter | 0.0831 | 5.2014 |
| Saturn | 0.0335 | 9.5360 |
| Uranus | 0.0117 | 19.1828 |
| Neptune | 0.00598 | 30.0577 |
From the Moon to Neptune, that is a range of 13.176 / 0.00598 = 2,203 to 1 in apparent speed. A factor of two thousand.
The left column comes from the sidereal periods; the right column is the same numbers pushed through Kepler's third law, a = P^(2/3) with P in years and a in astronomical units. Nothing else goes in, and the recovered distances match the accepted semi-major axes (mean orbital radii) to better than 0.05 per cent — a few parts in ten thousand, the residual being mostly the planet-mass term Kepler III leaves out. The apparent-speed difference Miller reports as absent is one of the most productive measurements in the history of astronomy. It sits in a different coordinate — motion against the stars rather than against the horizon — and turning the handle on it hands you the scale of the solar system. Miller's question, pushed one level further, is the distance ladder.
The standing reply, and what it concedes. A flat-plane reader will say the planets are “wandering stars” that move by themselves, so their drift is their own motion and not a distance signal, and that the right-hand column recovers distances only because Kepler’s third law was assumed. The second half is fair as far as it goes: the au values are a model output, and the orbital elements the rates come from are ephemeris inputs (tabulated orbital data) handed to both models alike — the site’s rule on that is stated once, on the method page, not re-argued here. The first half does not survive the table. The ordering needs no model: nine drift rates read off the sky, spanning 2,203 to 1, and the one body whose distance is triangulated from Earth with no orbital theory at all — the Moon, by its 57-arcminute parallax below — is the fastest of them. Self-moving lights over a flat plane predict no ordering; any wanderer may go at any speed, and nine of them in strict sequence is a coincidence the model does nothing to explain. The globe with the train-window law predicts drift falling as one over distance, 4 to 54 arcseconds an hour across the planets at opposition, which is what the next table holds. The distances are model-dependent; the ordering that makes Q5’s premise false is not.
The table above is degrees per day; the question says during a single night. Take a planet at opposition — Earth passing directly between it and the Sun, as against quadrature, where the planet stands ninety degrees from the Sun. Its motion against the stars is then dominated by our own orbital velocity sliding past it, and to first order
where Δ is the distance from Earth. Beyond the asteroid belt vplanet is small, so the rate is essentially 29.8 km/s divided by the distance: the train-window rule with nothing else in it, the exact law Miller invokes, in the exact timeframe he specifies, inversely proportional to distance because it is a translation and not a rotation. Around opposition that drift runs westward against the stars — retrograde — as Earth overtakes the planet.
| At opposition | Distance, au | Apparent drift | Over an eight-hour night |
|---|---|---|---|
| Moon (eastward, not retrograde) | 0.0026 | 1,977″/hr | 4.39° — about eight lunar diameters |
| Mars | 0.524 | 54″/hr | 0.119° |
| Vesta | 1.361 | 38″/hr | 0.084° |
| Ceres | 1.766 | 33″/hr | 0.074° |
| Jupiter | 4.20 | 20″/hr | 0.044° |
| Saturn | 8.54 | 12″/hr | 0.026° |
| Uranus | 18.18 | 6″/hr | 0.014° |
| Neptune | 29.06 | 4″/hr | 0.009° |
| Any star | — | 0 | 0 |
Computed here from v = 29.7847 km/s / √a and Δ = a − 1 au. Cross-check: these give peak retrograde rates of 0.36°/day for Mars, 0.13° for Jupiter and 0.028° for Neptune, which are the published figures. The Moon’s entry is its mean sidereal drift rather than an opposition rate, and it runs the other way round the sky.
So within one night the sky is not moving at one speed. It is moving at a range of speeds spanning five hundred to one, ordered by distance, in the direction the inverse-distance law requires. Nothing about brightness or colour predicts that ordering; distance predicts all of it, and inverting the relation hands the distance back.
The minor planets are the cleanest case, because finding them is this measurement. A main-belt asteroid near opposition drifts 33 to 38 arcseconds an hour. Photograph the same field twice an hour apart, blink the frames, and everything stellar sits still while the asteroid has walked across it — a century of discovery on photographic plates, and routine amateur work now. Near-Earth objects are faster still: one passing at a hundredth of an astronomical unit, at the tens of kilometres per second typical of such encounters, moves at roughly 1.4 degrees per hour, and a close approach inside the Moon’s distance can cross several degrees in an hour, visible in real time at the eyepiece.
These motions are the basis of an entire observational discipline; the slowest entry in the table is nine times the measurement error of the survey that would record it. (Astrometric RMS is a survey’s typical position error on one frame; σ is the motion in multiples of it.)
| At opposition | Moves in 15 minutes | Against Pan-STARRS 1’s 0.12″ astrometric RMS† |
|---|---|---|
| Mars | 13.40″ | 112σ |
| Vesta | 9.47″ | 79σ |
| Ceres | 8.35″ | 70σ |
| Jupiter | 4.95″ | 41σ |
| Saturn | 2.92″ | 24σ |
| Uranus | 1.57″ | 13σ |
| Neptune | 1.05″ | 9σ |
| Any star | 0 | 0 |
† Motion is a difference between two frames, so it carries √2 times the single-detection RMS; on that stricter convention the significances fall by about 30 per cent, and Neptune’s entry becomes roughly 6σ rather than 9. Still a clean detection. Fifteen minutes is the spacing the surveys actually use.
The way these objects are found is the measurement Miller says is missing. Pan-STARRS images each field four times a night; its instrument paper says the four frames of a “quad” are taken with “a transient time interval (TTI) of about 15 minutes between sequential detections”, giving “a typical arc length of about 45 minutes”, after which the pipeline searches for tracklets (short chains of detections that line up as one moving object). ATLAS does the same with “four exposures (over a 1 hour interval)”. A solar-system object is defined, operationally, as the thing that moved between frames taken within the hour while the stars did not. If planets and asteroids drifted at the star field’s rate, these surveys would return nothing.
The IAU Minor Planet Center — run at the Smithsonian Astrophysical Observatory, funded by NASA — holds 547.0 million astrometric observations, 21.9 million of them submitted this year and 1.7 million this month, every one a measured position of a moving solar-system body at a recorded instant. A large share is amateur work, and the accuracy bar is public. The MPC’s guide for new observers asks that they “consistently produce observations with a consistency of <1″” and that “your pixel scale is no greater than 2″/pixel (preferably) or 3″/pixel (at worst)”; an observatory code “will typically not be assigned if your astrometry shows large post-fit residuals”. Against that one-arcsecond bar, Neptune, the slowest object in the solar system, outruns it four times over in an hour and thirty-four times over across a night; Mars, fifty-four times over in the hour. The measured RMS of the professional surveys runs from 0.12″ for Pan-STARRS 1 to 0.69″ for Catalina, all small against motions of several arcseconds per quarter-hour.
The sharpest amateur case is stellar occultation timing, in which the International Occultation Timing Association — “a large multinational network of amateur astronomers” — times an asteroid passing in front of a star. Since Gaia became the reference catalogue those results reach roughly the 10-milliarcsecond level, an improvement of about a factor of five; nobody can predict where to stand for such an event without knowing the asteroid’s position and motion to a small fraction of an arcsecond, hours ahead.
Now the small effect, which is the very thing Miller is looking for. An object at distance d shows a horizontal parallax of arcsin(R_Earth / d). With R_Earth = 6,378.1366 km:
So the correct statement is not "distance does not matter" but "the distance-dependent term is smaller than the distance-blind term by roughly the ratio of Earth's radius to the object's distance." For the nearest body it is applied daily by people with no stake in this argument. The Nautical Almanac prints the Moon's horizontal parallax hour by hour on its daily pages, ranging roughly 54 to 61.4 arcminutes, and no Moon sight reduces correctly without the parallax-in-altitude correction, p = HP × cos(altitude), which can exceed a full degree of altitude — sixty nautical miles of position error if left out.
What does it do to apparent speed, Miller's actual variable? From moonrise, where p is the full 3,422 arcseconds, to a near-zenith transit, where p is essentially zero, the Moon's apparent position shifts by 3,422 arcseconds relative to where an infinitely distant object on the same geocentric line (the line from Earth’s centre, as against the topocentric line from the observer’s own spot on the surface) would sit. Spread over six hours — 21,600 seconds — that averages 0.158 arcseconds per second, against the Moon's mean apparent diurnal rate of 14.492 arcseconds per second. That is a 1.09 percent rate difference, tied directly to distance, on the nearest object in the sky — exactly the effect Q5 predicts, corrected for in commercial navigation, and for a star unmeasurable from the ground.
One night is the narrowest window there is, and the one in which the sky looks most like a fixed pattern being turned. Widen it and the picture inverts: the sky is full of slow, ordered, predictable motion that no rotating dome accounts for.
The large velocities quoted at flat-earth audiences — and mocked, reasonably enough, as absurd things nobody feels — are outputs, not assumptions: what was measured in each case is a small angle, of the kind a careful observer can check, and the speed is that angle times a distance. The first was found by running this page’s own sorting exercise. In the 1720s James Bradley went looking for stellar parallax in gamma Draconis, found an annual swing far too large to be parallax, and ruled parallax out because the swing was ninety degrees out of phase and a star at nearly opposite right ascension showed the same amplitude rather than a distance-scaled one. It was aberration, which does not depend on the distance of the object, whereas parallax does.
Nobody proposes that you should feel 67,000 mph; the figure exists because somebody noticed a twenty-arcsecond wobble and did the arithmetic. The scepticism is aimed at the output while the input — a small, stubborn, repeatable deviation from where a thing should have been — is exactly the sort of observation this book’s own method is built to respect. And it completes the answer to Q5: the moment the window widens past one night, distance-dependent motion appears everywhere — annual parallax scaling as 1/d, proper motions separating near stars from far, a reference frame of quasars chosen because they are too distant to show either. The single night is the one exposure in which the distance signal is guaranteed to be too small to see.
The drift rates are computed, not observed by us. The opposition rates come from published orbital elements and our own arithmetic, not from frames we took; they are checkable by anyone with a camera and two nights. The MPC counters are read, not audited: the 547.0 million, 21.9 million and 1.7 million figures are live counters read on 23 August 2026, not verified against a database dump. Two figures rest on secondary sources: the Nautical Almanac’s 54 to 61.4 arcminute range is quoted from a compilation rather than from the almanac’s daily pages, which we could not fetch; and we could not fetch the ASCOM drive-rate enumeration, so we claim only that the standard defines several distinct rates, which the AAVSO material supports.
The rotation rate. The IERS Conventions fix the Earth Rotation Angle as ERA(Tu) = 2π × (0.7790572732640 + 1.00273781191135448 × Tu), which is 360 × 1.00273781191135448 = 360.9856123 degrees per UT1 day, or 15.041067 arcseconds per second. No term in it refers to the object observed.
Q6 figures; the derivation is on the Jupiter page. At the April 2026 quadrature our computation puts Io's shadow 1.15 Jupiter radii from Io, Ganymede's 2.91, Callisto's 5.12 — Callisto's lands off the disk while Callisto is still transiting, so moon-to-shadow distances measured without each moon's orbital radius give four different answers. Shadow sizes: Callisto's umbral core about 1,584 km against a moon 4,821 km wide (33 percent), penumbra some 8,058 km; Io's umbra about 3,017 km against 3,643 km (83 percent), narrow penumbra — a sharp black dot against a grey smudge. NASA's caption for the Hubble triple transit of 24 January 2015 adds, in parentheses, “The moons' orbital velocities are proportionally slower with increasing distance from the planet” — correct as a gloss on “slower-moving” and “fastest-moving”, probably what is meant, but not the cause of the offsets, which is the Sun-Jupiter-Earth angle, zero at opposition whatever the orbital velocities. A reader who noticed that the caption never states the cause was noticing something real.
Sources for the Q6 figures in the method notes are on the Jupiter page.