A solar radius measured past a mountain range

The Sun’s angular radius is about 959.6 arcseconds, and measuring it to better than a tenth of an arcsecond is surprisingly hard. Direct imaging is limited by the atmosphere, by the instrument’s own scattered light, and by the fact that the Sun has no edge — its brightness falls steeply over a few hundred kilometres rather than stopping.

The sharpest available method is an eclipse. At second contact the last sliver of photosphere disappears behind the Moon; at third it reappears; and the interval between them, timed from a known place on the Earth with a known lunar ephemeris, gives the difference between the two bodies’ angular sizes to a few hundredths of an arcsecond.

The difficulty is that the Moon’s edge is not a circle.

The measurement is therefore differential in an awkward way. Everything about the Sun that is wanted has to be extracted from a timing that is dominated by the Moon, and the lunar part is not a smooth correction but a rugged one that changes from eclipse to eclipse. An eclipse is a shadow rather than a phase, and the edge of that shadow is cast by a landscape.

Timing an edge that is a landscape #

What happens at second contact is not a smooth disappearance. The Moon’s limb has mountains and crater rims on it, and sunlight persists in the valleys between them after it has been cut off by the peaks. The result is a chain of bright points that shrink and go out one by one — Baily’s beads — over a few seconds.

Which valley goes out last, and when, depends on the profile at that position angle, and on the libration of the Moon that day, which decides which part of the limb is presented. So the timing of second contact is not a property of the two bodies’ sizes alone. It is a property of the sizes and of a specific piece of lunar topography.

The correction is therefore not a small refinement. Two arcseconds of limb relief against a measurement aiming at a few hundredths means the profile must be known to about one per cent of its own amplitude — a few tens of metres of lunar mountain — before the solar radius is accessible at all.

There is a second effect of the beads that is worth stating because it changes what “contact” means. With a jagged limb there is no single instant at which sunlight stops; there is a sequence over a few seconds, and different observers pick different points in it depending on their instrument’s sensitivity and their own convention. Photoelectric timing of the total light curve is objective and gives a well-defined moment; visual timing is not, and the historical record is largely visual. A systematic difference of a second between two observers’ conventions is 0.02 arcseconds in the derived radius, which is at the level of the modern measurement and well below the historical scatter.

The modern technique sidesteps some of this by not timing contacts at all. Recording the beads photometrically at many stations across the edge of the path of totality gives many independent determinations of where the umbral edge fell, each tied to a specific lunar valley whose depth the altimeter map supplies. That converts a handful of contact times into hundreds of constraints, and it is what took the precision below a tenth of an arcsecond.

Where the profile used to come from #

For most of the twentieth century the profile came from Watts’s charts: a set of limb profiles derived by Chester Watts from measurements of some seven hundred photographic plates of the Moon’s limb, published in 1963 after decades of work. They gave the limb’s radius as a function of position angle for every libration, and they were the standard for forty years.

They were a heroic piece of work and they had two known problems. The plates were measured against a reference that was itself uncertain, so the charts carried a datum error of a few hundred metres — a systematic offset in the mean radius that propagates directly into the solar radius. And the resolution was limited by the plate scale, so small-scale relief was smoothed, which biases the extremes rather than the mean and therefore biases contact timings in a particular direction.

Both errors are of the size that matters. A datum error of three hundred metres is 0.16 arcseconds at the Moon’s distance, which is larger than the precision the eclipse method claims.

It is worth being fair about why the charts were used for so long. There was nothing better, the errors were known and quoted, and for most purposes — predicting where a total eclipse would be visible, or when the beads would appear — they were entirely adequate. What they were not adequate for was the one application that pushed them past their design accuracy: extracting a solar radius to a hundredth of an arcsecond from a quantity in which the lunar term is twenty times larger. A reference is only as good as the demand placed on it, and the demand here grew after the reference was built.

What changed #

A laser altimeter in lunar orbit measured the Moon’s topography directly, to a vertical precision of about a metre and with dense global coverage. From that map a limb profile can be computed for any date and any observer, with no plates, no datum question and no smoothing.

Replacing one profile with the other in the reduction of eclipse timings moved the answer.

Two features of that plot are worth reading carefully, because together they diagnose what happened.

It is a step, not a drift. The change occurs at the date the reduction method changed, not gradually over the series. A physical variation in the Sun would have no reason to do that, and a systematic in the reduction has every reason to.

The scatter fell as well as the mean moved. A pure datum error would shift every point by the same amount and leave the scatter alone. The scatter falling says that the old profile was also wrong in a way that varied from eclipse to eclipse — which is exactly what a libration-dependent error looks like, since each eclipse presents a different part of the limb.

Two further checks were available and both were applied, because a step in a series is exactly the kind of result that has to be defended against the accusation of being an artefact of the new method rather than a correction of the old one. The first was to reduce a subset of the old eclipses against the new profile: their radii move to the new value, which shows the step is in the reduction rather than in the epoch. The second was to compare the eclipse radius against radii from completely different techniques — transits of Mercury and Venus, which time a planet’s silhouette against the solar limb and involve no lunar topography at all. Those give a value consistent with the new eclipse determinations and inconsistent with the old ones, which is the independent confirmation that the correction went the right way.

What was actually measured #

The reason any of this attracted attention is a claim that would be important if it were true.

Several analyses of historical eclipse timings, going back to the seventeenth century, reported that the solar radius had been shrinking — by a few tenths of an arcsecond per century in the most-cited version. A change of that size implies a change in the Sun’s total luminosity or in its internal structure on a timescale of centuries, which would be a substantial result about stellar physics and about the solar constant.

The analyses were careful and the effect was real in the data. What was not established was that the effect was in the Sun. Historical timings were reduced against whatever limb information was available at the time, by observers whose timing conventions differed, at eclipses whose lunar librations differed — and the sign and size of the reported trend are within the range that a drift in reduction practice can produce.

The modern reductions, using one profile of known quality across a homogeneous set of eclipses, find no significant secular variation. The upper limit is a few hundredths of an arcsecond per century, which is an order of magnitude below the claimed effect.

The episode is a good illustration of how a systematic masquerades as a discovery. Every step in the historical analyses was defensible. The timings were real, the reductions were competent, the trend was statistically significant against the quoted errors, and the physical implications were interesting enough to justify the attention. What was missing was a way to check the reference — and there was none available until an unrelated mission mapped the Moon.

The general form of the trap is worth naming: a long time series assembled from measurements reduced by different people, at different times, against different versions of a reference, will show a trend if the reference drifted. The trend has the same statistical signature as a real one. Distinguishing them requires either re-reducing everything against one reference, which is what happened here, or finding a second method with a different reference, which is what the several independent routes to the same distance do for a different quantity.

Where the picture stops #

Three, and the second is the one that limits the method now.

The Sun has no edge. What is timed is the disappearance of the photosphere, and the photosphere’s brightness falls over a scale of about four hundred kilometres. Defining “the radius” requires a convention — the inflection point of the limb-darkening profile is the usual one — and different conventions differ by tenths of an arcsecond. Comparing a radius from an eclipse with one from a heliometer or from helioseismology means first checking that the same convention was used, and historically it often was not.

The lunar ephemeris is now the second-largest term. With the limb profile solved, the accuracy of a contact timing depends on knowing where the Moon was, and laser ranging gives that to centimetres in distance but rather less well in the direction transverse to the line of sight. The residual is small and is no longer negligible.

And the observer’s position and the atmosphere still enter. A contact time depends on the observer’s altitude and on the refraction along a low line of sight if the eclipse is near the horizon. Both are modelled, and both are the reason amateur timings from many sites, which are the bulk of the data, carry larger errors than a single professional measurement.

A fourth sits alongside them, and it is about coverage rather than precision. Total solar eclipses happen somewhere on Earth about every eighteen months, and the ones with good instrumental coverage are fewer. A homogeneous modern series therefore contains perhaps a dozen points spread over two decades, which is a thin basis for any statement about variation on the timescale of a solar cycle. Improving the systematic has not removed the fundamental limitation, which is that the events are rare — the same limitation that makes the eighteen-year cycle of similar eclipses so useful for planning and so restrictive for measurement.

Why the profile is the interesting half #

The reason to file this argument with eclipses rather than with solar physics is that the Sun is the easy part.

A solar radius from an eclipse is a difference of two edges, and the whole difficulty is that one of the two is not smooth, not known and not the same at every eclipse. When the second edge was mapped properly, the measurement improved by a factor of three in precision and moved by more than its old error bar. Nothing about the Sun contributed to either change.

That is a shape worth recognising, because it recurs whenever a measurement is differential. An occultation measures a body’s shape against the edge of a shadow and inherits everything uncertain about the star being occulted. A transit measures a planet’s radius as a ratio to a stellar one, and improved stellar radii moved every planet radius at once. In each case the published quantity carries the errors of a reference object nobody was interested in.

There is a second reason the profile is the interesting half, and it is about how such an error is found. Nothing in the eclipse data itself could have revealed the problem: the timings were internally consistent, the formal errors were honest, and the scatter was attributed to observing conditions. The error was only visible once an external, independent measurement of the reference existed. That is the general condition of a systematic — it is invisible in the data it corrupts, and it is found by measuring something else. Two instruments blind in opposite directions is the same idea stated as a design principle rather than as a post-mortem.

The general lesson is that the reference deserves as much attention as the target, and usually gets less — because it is somebody else’s subject. The people measuring the Sun were not lunar topographers, and the lunar topographers were not measuring the Sun. What resolved a fifty-year question about solar variability was a laser altimeter flown for a completely unrelated purpose.

A concluding observation about who benefits. The altimeter map that resolved this was flown to characterise the Moon: to map its topography, to find its permanently shadowed craters, and to support future landings. Nothing in its objectives mentioned the Sun. That a lunar mission settled a fifty-year question in solar physics is not a coincidence so much as a consequence of the structure of the problem — when a measurement is a difference between two objects, an improvement to either one improves the answer, and the improvement usually comes from whichever community happens to be funded. The same thing happened to every planet radius when the stars were measured better, and it will happen again to some quantity currently limited by a reference nobody working on it is in a position to improve.

Be explicit about the size of the correction relative to the quantity, because the ratio is what makes the measurement so demanding. The Sun subtends about sixteen arcminutes and the disputed variation in its radius is of order a tenth of an arcsecond — one part in ten thousand. A lunar mountain a kilometre high subtends about half an arcsecond seen from Earth, which is five times the signal. So the limb profile is not a small correction to a clean timing; it is a correction several times larger than the effect being sought, and every conclusion about a varying solar diameter is a conclusion about how well a lunar landscape was known at that particular position angle on that particular date. That ordering — correction larger than signal — is the same one the microwave dipole imposes on the anisotropies, and it demands the same discipline: the correction must be known well rather than merely estimated, and its own uncertainty belongs in the error bar rather than in a footnote.

Note too that the same lunar profile is what every occultation timing needs, so the improvement paid for itself several times over in measurements that had nothing to do with the Sun.

Where the ladder goes next #

The rung above this one is the convention itself: what “the edge of the Sun” means when the brightness falls over four hundred kilometres, and how the eclipse definition relates to the one helioseismology uses, which differs by about half an arcsecond and is not a disagreement. The rung after it is the same measurement made from space, where a spacecraft passing through the Moon’s shadow times contacts with no atmosphere and a known position — and where the limb profile is still the limiting term.