Fun With Science / Globe Deconstruction / Q5 + Q6 · pages 48–49 / Draft
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
Q5 asks why stars and planets show no difference in visual speed across one night. They do show one. Measured against the star background rather than the horizon, planets at opposition drift between four and fifty-four arcseconds an hour in inverse proportion to their distance, while stars drift by nothing — a range of thirteen to one across the planets alone, and five hundred to one once the Moon is included, ordered by distance throughout, and among the most heavily measured quantities in astronomy. What the question gets right is the part it names: the diurnal sweep, the turning of the whole sky, really is identical for everything, to one part in ten billion even for the nearest star. That is not a coincidence to be explained away 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.
We will start with what we grant, because on Q5 the granted part is precise and everything below depends on the reader seeing exactly where the line falls. In one sentence, in advance: the diurnal sweep is common to everything, and the drift against the stars is not.
Q5's observation is true of the motion it names, and understated rather than overstated. “As the Earth spins” points at the diurnal sweep, and about that sweep he is exactly right. Earth carries the whole sky round at 360.9856123 degrees per UT1 day. For stars, nothing in colour, brightness, or catalogued distance shows up in that rate. 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 — that is 648,000 arcseconds. One part in ten billion. Miller is not exaggerating. He is rounding down. The intuition underneath Q5 is a real law, correctly remembered. Angular rate produced by an observer's translation is the perpendicular velocity divided by the distance, and it genuinely does fall off as one over distance. That is the train-window rule, and it is right. The error is narrow and specific: Earth's rotation is overwhelmingly a change in where the observer points, not where the observer is, and a change of pointing contains no distance term at all. Q6 rests on a real appearance, not an invented one. At the April 2026 quadrature our own computation puts Io's shadow 1.15 Jupiter radii from Io, Ganymede's 2.91, and Callisto's 5.12. Callisto's shadow lands well off the disk while Callisto is still transiting. Anyone who measured moon-to-shadow distances on such an image without knowing each moon's orbital radius would get four different answers and would reasonably conclude they do not share a source. The shadows genuinely are not the same size as their moons. Callisto's umbral core is about 1,584 km across against a moon 4,821 km wide — 33 percent — wrapped in a penumbra some 8,058 km across. Io's umbra is about 3,017 km against 3,643 km, 83 percent, with a narrow penumbra. Io's shadow is a sharp black dot; Callisto's is a grey smudge. That mismatch is real data. And the standard published explanation is loose. NASA's caption for the Hubble triple transit of 24 January 2015 describes the moons and their shadows and then adds, in parentheses, “The moons' orbital velocities are proportionally slower with increasing distance from the planet.” That is the only causal-sounding sentence in it. Read as a gloss on “slower-moving” and “fastest-moving” it is perfectly correct, and that is probably what is meant. Read as an account of why the shadows sit where they do — which its position after the shadow sentence makes available — it would be wrong, because the offset is set by the Sun-Jupiter-Earth angle, which goes to zero at opposition whatever the orbital velocities are. We do not press the second reading. The point that survives either way is that the caption never states the actual cause. A reader who noticed that the standard caption does not actually explain the offset was noticing something real.So: Q5 needs an explanation, not a rebuttal. Q6 needs an arithmetic check, which either passes or fails.
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, and the official expression for it has a property worth noticing before you read it: nothing in it refers to the object being observed. The IERS Conventions fix it 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. There is no term for what you are looking at, how far away it is, or what it is made of. It is a property of the observer alone. 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 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, exactly as Miller's intuition demands. Its name is diurnal parallax. We come to it in a moment.
Spinning in place, everything sweeps together. Walking past, near things sweep and far things crawl. Earth does both — but the spin is a full turn per day and the walk is one Earth radius.
One consequence of that rate is itself a check on the whole picture. Divide a day by it: 86,400 / 1.00273781191135448 = 86,164.0989 seconds, or 23h 56m 04.1s. The stars return 235.90 seconds — three minutes fifty-six — early every day. That is why telescope mounts ship with a sidereal drive rate distinct from a solar one, and why the constellations shift through the seasons. (Our companion page on whether the Earth spins or the sky does derives this at length; we do not repeat it here.)
One correction we must make to our own side of the argument, because a hostile expert would make it for us. 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 traces a smaller circle and sweeps a linear track across the sky at cos δ of the equatorial rate; a star near the pole barely moves at all. Atmospheric refraction differentially 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 the phrase "very nearly" is where all the physics lives.
Here is the part of the answer we think Miller would find most interesting, and it is why we put it before the small effects rather than after.
Miller says objects at different distances ought to move at visibly different apparent speeds, and that they do not. The first half of that is correct and the second half depends entirely on which coordinate you measure the motion in. Measure motion relative to the horizon and you are measuring the observer's rotation, which is common to everything. Measure motion relative to the star background and 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. Not a subtle effect. A factor of two thousand.
The drift rates in the left column come 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. Those recovered distances match the accepted semi-major axes 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.
This is the turn of the whole Q5 section. The apparent-speed difference Miller reports as absent is the single most productive measurement in the history of the subject. It is not missing. It is in a different coordinate — motion against the stars rather than motion against the horizon — and when you write it down and turn the handle, it hands you the scale of the solar system. Miller's question, pushed one level further, is the distance ladder.
The same holds for the stars themselves, in a weaker form: nearby stars are measurably displaced against distant ones over years, which is exactly what a distance ordering predicts. We have deliberately not quoted a proper-motion figure here, because the value we had was taken from a compilation rather than from the Gaia archive record itself, and this page's rule is that we cite what we pulled.
The table above is degrees per day, and the question says during a single night. That distinction is worth taking seriously rather than waving away, because the answer survives it and gets sharper in the process.
Take a planet at opposition. Earth is passing between it and the Sun, so the apparent motion against the stars is dominated by our own orbital velocity sliding past it, and to first order the rate is
where Δ is the distance from Earth. For anything beyond the asteroid belt vplanet is small, so the rate is essentially 29.8 km/s divided by the distance. That is the train-window rule with nothing else in it — the exact law Miller invokes, in the exact timeframe he specifies, and it is inversely proportional to distance because it is a translation and not a rotation.
| 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 |
† 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 — Neptune’s entry becomes roughly 6σ rather than 9. Still a clean detection; we give the looser convention in the table and the stricter one here rather than quote whichever flatters the argument.
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. The Moon covers eight of its own diameters against the background between dusk and dawn — naked eye, no instrument, one night, exactly as asked.
And the minor planets are the cleanest case of all, 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 two frames, and everything stellar sits still while the asteroid has walked a visible distance across it. That is not a specialist facility technique — it is how these objects were discovered for a century on photographic plates, and it is routine amateur work now, with astrometry from back gardens feeding the Minor Planet Center. 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, fast enough to see the motion against the stars in real time at the eyepiece.
The ordering in that table is the whole answer to Q5 in one column. Nothing about brightness or colour predicts it. Distance predicts all of it, and inverting the relation hands the distance back.
A table of rates is only worth having if somebody can actually read them off the sky, so here is the instrumental side. The short answer is that these motions are not near anyone’s detection threshold. They are the basis of an entire observational discipline, and the slowest entry in the table is nine times larger than the measurement error of the survey that would record 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 |
Fifteen minutes is not an arbitrary interval. It is the spacing the surveys actually use, for exactly this reason.
The way these objects are found is the measurement Miller says is missing. Pan-STARRS images each field four times a night, and its own instrument paper states the design plainly: 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. 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 same apparent rate as the star field, these surveys would return nothing at all.
And the output is not a handful of specialist results. The IAU Minor Planet Center — run at the Smithsonian Astrophysical Observatory, funded by NASA — holds 547.0 million astrometric observations, of which 21.9 million were submitted this year and 1.7 million this month. Every one of them is a measured position of a moving solar-system body at a recorded instant, which is to say every one of them is a datum in exactly the comparison Q5 asks about.
A large share of that 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)”, and it says an observatory code “will typically not be assigned if your astrometry shows large post-fit residuals”. Set that one-arcsecond bar against the table above: 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 outruns it 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 of them small against motions of several arcseconds per quarter-hour.
The sharpest version of the amateur case is stellar occultation timing, where networks including the International Occultation Timing Association — described in the literature as “a large multinational network of amateur astronomers” — time 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. You cannot predict where to stand for one of those events without knowing the asteroid’s position and its motion to a small fraction of an arcsecond, hours ahead.
Now the small effect, which is small but not zero, and 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."
And 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 — a range we quote from a secondary compilation rather than from the almanac's daily pages, which we could not fetch. No Moon sight reduces correctly without the parallax-in-altitude correction, p = HP × cos(altitude). That correction can exceed a full degree of altitude, which is sixty nautical miles of position error if you leave it out.
What does it do to apparent speed, which is 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 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. It is exactly the effect Q5 predicts. It is real. It is corrected for in commercial navigation. It is one percent, and for a star it is unmeasurable from the ground.Four artefacts. None of them was made to win a debate.
Satellite dishes do not track — and lower ones must. A body at 35,786 km altitude in an orbit of one sidereal day — ESA's own description is "23 hours 56 minutes and 4 seconds ... the duration of a sidereal day" — has an apparent diurnal rate of effectively zero — within a dish’s beamwidth, since real satellites hold a station-keeping box and inclined-orbit birds do need tracking — which is why an ordinary fixed dish points at a constant azimuth and elevation and needs no drive. For contrast, the ISS at roughly 420 km altitude, in a 93.0-minute orbit, crosses near the zenith at something like 0.98 degrees per second, about 235 times the sidereal rate. We flag that ISS figure as an idealised zenith pass and an order-of-magnitude illustration; real passes off-zenith are slower. The point survives the imprecision. Distance and orbital motion absolutely do govern apparent sky speed — for objects near enough that the observer's rotation radius and the object's own orbital motion matter — and the effect saturates at the sidereal rate for everything beyond. A global telecommunications industry is built on knowing precisely where that transition sits. Mounts ship with different rates because the rates differ. The AAVSO puts it plainly: the solar tracking rate is 15 arcseconds per clock second, while the sidereal rate is 15 arcseconds per sidereal second. The three-minute-fifty-six-second difference between the two days is encoded as a switchable firmware setting in consumer equipment. (We could not fetch the ASCOM drive-rate enumeration this session to confirm its exact numeric definitions, so we claim only that the standard defines several distinct rates.) ICRF3. The fundamental celestial reference frame is built from 4,588 extragalactic radio sources, 303 of them defining, with a 0.03 mas noise floor. They were chosen precisely because at those distances parallax and proper motion are undetectable. The frame is a funded, operational statement that apparent motion dies with distance. VLBI. Against that frame, Earth's rotation is measured to better than 2×10⁻⁴ arcseconds, which is under 0.6 cm projected onto Earth's surface, for geodesy, plate tectonics and spacecraft navigation.One last note, offered as a compliment rather than a rebuttal. In the 1720s James Bradley went looking for stellar parallax in gamma Draconis and found an annual swing far too large to be parallax. He worked out that it could not be parallax on two grounds: it was ninety degrees out of phase with what parallax requires, and a star at nearly opposite right ascension showed the same amplitude rather than a distance-scaled one. What he had found was aberration — and the distinguishing property of aberration is precisely that it does not depend on the distance of the object, whereas parallax does. Bradley met exactly the puzzle Q5 poses, an apparent motion that stubbornly refused to scale with distance, and resolved it by sorting effects into those that can carry a distance term and those that cannot. This page is running Bradley's sorting exercise on Miller's observation. We have quoted none of Bradley's own figures here because our source for them was secondary and we did not reach a primary; the argument does not need them.
Everything above works inside the frame Q5 sets: one night, one object, one pass across the sky. That is the narrowest window available, and it is the one in which the sky looks most like a fixed pattern being turned. Widen it and the picture inverts. The sky is not static furniture with one nightly rotation applied to it; it is full of slow, ordered, predictable motion, most of which needs no instrument at all to notice and none of which a rotating dome accounts for.
Across a year, by eye. Any given star rises about three minutes fifty-six seconds earlier each night — roughly two hours a month, a full circuit in a year. That is why Orion belongs to winter and Scorpius to summer, and why every agricultural calendar in the ancient world was built on which stars were up before dawn. This is not a subtle measurement. It is the most widely observed astronomical fact in human history, and it is a second periodicity, orbital rather than diurnal, sitting on top of the nightly turn.
Across centuries, from naked-eye catalogues. In 1718 Edmond Halley compared the positions Ptolemy and Hipparchus recorded for Sirius, Arcturus and Aldebaran against the positions in his own day and found they had moved — Arcturus by more than a degree, twice the width of the full Moon. Nothing in that comparison required a telescope. It required only two catalogues eighteen centuries apart, and it established that the “fixed” stars are not fixed relative to one another. A pattern being spun overhead cannot do that; only individually moving objects can.
Within a lifetime, from a camera. Barnard’s Star crosses 10.3 arcseconds a year — a full lunar diameter in about 181 years, and a visibly different position between two photographs taken a decade apart on the same amateur rig. Blinking a pair of your own images is the modern version of Halley’s comparison, and it is well inside what a back-garden setup can do.
At the limit, from Gaia. The same quantity, measured for 1.46 billion stars at microarcsecond-per-year precision. Gaia did not discover that the stars move. It measured how, for most of the ones you can see.
The large velocities that get quoted at flat-earth audiences — and mocked, reasonably enough, as absurd things nobody feels — are not assumptions anyone started from. Every one of them is an output. What was measured in each case is a small angle, of the kind a careful observer can check, and the speed is that angle multiplied by a distance.
So the honest description of the position is close to the opposite of the one usually attacked. 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.
It also completes the answer to Q5 itself. The question asks why apparent motion does not scale with distance over a single night. It does not, and the page above says why. But the moment the window widens, distance-dependent motion appears everywhere — annual parallax that scales exactly as 1/d, proper motions that separate nearby stars from far ones, and a reference frame built from quasars chosen precisely 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.
Stated with thresholds, in advance, and capable of going against us.
We would rather flag these ourselves than have them found.
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, which is the point of quoting them.
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.
The drive-rate claim rests on a secondary source. 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.
This list is shared with the companion page on Jupiter’s shadows; not every entry is used here.