The Sun is three hundred kilometres larger seen edge-on

Timing an eclipse is the most accurate way to measure the Sun’s diameter, once the mountains on the Moon’s edge are accounted for. What is timed is the moment the last sunlight vanishes behind the Moon, and that moment is a statement about where the Sun’s light ends. Measurements of that kind, from eclipses, from transits of Mercury and Venus, and from images taken by telescopes in space, give a radius near 696,000 kilometres.

The Sun’s oscillations give a different answer. The waves that ring through the Sun’s surface layers have frequencies that depend on its radius, and the radius they imply is about three hundred kilometres smaller. For a while in the late 1990s that looked like a discrepancy needing explanation in the physics of the solar interior.

It needed an explanation in the physics of a line of sight. A star has no surface; what looks like one is the level where its atmosphere becomes opaque. Opaque depends on the direction of the look. Looked at from straight above, the Sun becomes opaque at one level. Looked at edge-on, as every measurement of its edge must, it becomes opaque far higher up, and the edge an instrument records is the edge of that higher layer.

The edge, computed #

The photosphere is a layer of hot gas whose opacity — mostly from the negative hydrogen ion, an electron loosely attached to a hydrogen atom — falls steeply with height, because both the gas and its free electrons thin out upward. The optical depth looking straight down, measured from outside, rises by a factor of $e$ every sixty kilometres or so in the upper photosphere. Somewhere in that steep rise it passes one, and that level is the conventional surface: the depth from which light escapes straight up.

The brightness at each point of an image of the Sun is the light emerging along one line of sight, and it can be computed by following that line through the atmosphere, adding up the light emitted at each point and dimmed by everything between there and the observer. The figure does exactly that, through a grey model atmosphere in which the temperature at each depth follows the classical Eddington relation and the source of light at each point is the blackbody brightness at its temperature.

Near the centre of the disc the line of sight is nearly vertical, it reaches optical depth one at the conventional surface, and the brightness is set by the temperature there. Towards the limb the line of sight is slanted, it reaches optical depth one higher up where the gas is cooler, and the brightness falls — the limb darkening that every image of the Sun shows. At the very edge the line of sight is tangential, skimming the atmosphere, and the brightness collapses to nothing over a couple of hundred kilometres.

The steepest point of that collapse is the inflection point of the profile, and it is the conventional definition of the Sun’s photometric edge. In the computation it lies 335 kilometres above the level where the vertical optical depth is one.

Why the edge is three hundred kilometres up #

The height has a one-line explanation. A ray skimming the limb at a height $h$ passes through the atmosphere along a chord, and because the atmosphere’s thickness is tiny compared with the Sun’s radius, the chord’s length within a scale height of its lowest point is about $\sqrt{2\pi R H}$, where $R$ is the radius and $H$ the scale height. The vertical path through the same layer is $H$. The tangential path is longer by $\sqrt{2\pi R/H}$, which for a radius of 696,000 kilometres and a scale height of sixty is about 270.

So a tangential ray accumulates optical depth 270 times faster than a vertical one at the same height, and becomes opaque — reaches optical depth one along its own length — where the vertical optical depth is only a 270th. In an atmosphere whose optical depth falls by $e$ every sixty kilometres, that level is $60 \ln 270$, about 336 kilometres, above the vertical surface. The computed inflection point, 335 kilometres, is that number: the edge of the Sun is where the grazing view first becomes opaque.

This is the resolution of the old discrepancy. Helioseismology measures the radius through the frequencies of surface gravity waves, whose properties depend on the density and pressure where the atmosphere becomes opaque in the vertical sense — the level the waves themselves feel. Images and eclipses measure the edge where the grazing view becomes opaque. The two differ by the height just computed, and when the difference was calculated with a full model atmosphere in the late 2000s it came out at about 330 kilometres, closing the gap. In 2015 the International Astronomical Union adopted a nominal solar radius of 695,700 kilometres, explicitly as a convention, so that the radius used to scale every star by the Sun would no longer depend on which edge was meant.

The edge depends on the question #

Once the edge is understood as a convention, it becomes clear how many conventions there are.

The level of unit vertical optical depth is where the photosphere’s energy escapes and is the right radius for a star’s luminosity: it is the radius in the Stefan–Boltzmann law, where the effective temperature is defined. The inflection point is the photometric radius, 335 kilometres above it. The level where the brightness has fallen to half its value just inside the edge is almost the same, fifteen kilometres higher. The levels where it has fallen to a hundredth, a thousandth and a ten-thousandth of the disc centre are 555, 695 and 830 kilometres up, and those are the edges a detector with each sensitivity would see at an eclipse.

None of these is the true radius, because there is none. Each answers a different question — where the energy escapes, where the image is steepest, where the light can no longer be detected — and a measurement is only comparable with another made to the same definition. At the level of hundreds of kilometres, which is four tenths of an arcsecond seen from the Earth, every published solar radius has to say which edge it means.

Blur moves the edge #

An image of the Sun is blurred, by the atmosphere if taken from the ground and by the telescope’s optics in any case, and the blur moves the edge.

Blurring averages the brightness over neighbouring points. On a symmetric edge it would leave the steepest point where it was; on the Sun’s edge, which falls gently on the disc side and steeply on the dark side, it shifts the steepest point by an amount that depends on both slopes and on the width of the blur. In the computation the inflection moves by up to twenty kilometres for blurs from a fifth of an arcsecond to an arcsecond, in a direction that changes as the blur grows.

Twenty kilometres is small, but it is three hundredths of an arcsecond, and the historical record of solar radius measurements from the ground, made over centuries with different instruments in different seeing, varies by tenths of an arcsecond. Some of the claims that the Sun’s radius changes with the solar cycle, or over decades, have been traced to the blur of different instruments interpreted without a profile model — the same atmosphere that limits every ground-based image, moving the edge it was trying to measure. From space, with a stable instrument, the radius has been found constant to within a few hundredths of an arcsecond over a cycle.

A different Sun in every colour #

The edge is set by the opacity, and the opacity depends on wavelength, so the Sun is a different size at every wavelength.

The negative hydrogen ion absorbs most strongly in the visible and least near 1.6 micrometres in the infrared, where its bound–free absorption has run out and its free–free absorption has not yet grown. At that wavelength a line of sight penetrates deepest, and the edge lies lowest. In the near ultraviolet, where crowds of metal lines add their opacity, the line of sight stops higher and the Sun is larger. In the computation, with representative opacity ratios, the inflection is about fifty kilometres lower at 1.6 micrometres than at 500 nanometres and fifty higher at 350.

The opacity ratios drawn are representative rather than computed from a detailed atmosphere, and real measurements show the same direction with somewhat different sizes. They also show something the grey model cannot: within spectral lines, where the opacity is hundreds of times larger, the Sun’s edge lies a thousand kilometres or more higher, in the chromosphere, and the radius in a strong line is a different object altogether. The surface is a depth, and the depth depends on the colour of the light used to look for it.

A second of eclipse is three hundred kilometres #

An eclipse measures the edge by time rather than position, and the conversion makes the definitions concrete.

The Moon moves across the Sun at about half an arcsecond per second, and an arcsecond on the Sun is 725 kilometres, so the Moon’s limb crosses a few hundred kilometres of the Sun’s atmosphere every second. The difference between the inflection point and the edge a detector actually loses is therefore a difference of a second or two in the recorded time of contact, and it depends on the detector’s sensitivity: a hundredth of the centre’s brightness, about a second; a ten-thousandth, about two.

An eclipse timed to a tenth of a second therefore measures the Sun’s radius to a few tens of kilometres only if the limb profile is modelled, and the reduction to the inflection point is part of every published eclipse radius. The Moon’s mountains add their own corrections, measured now by a laser altimeter, and Baily’s beads — the last points of sunlight shining through lunar valleys — are the profile and the landscape together, seen one valley at a time.

Every edge in astronomy is seen edge-on #

The Sun is the only star whose edge can be seen directly, but it is not the only edge measured along a grazing ray, and the same geometry turns up wherever a radius is measured.

A planet transiting its star blocks light in proportion to the area of its disc, and the disc’s edge is where the planet’s atmosphere becomes opaque along a ray grazing it — the slant path again, longer than the vertical by the square root of $2\pi R/H$ for the planet’s radius and its atmosphere’s scale height. For a hot Jupiter that ratio is tens, and the transit radius lies several scale heights above the level where the atmosphere is vertically opaque. Because the opacity differs from one wavelength to another, so does that height, and the planet’s radius depends on the colour it is measured in by exactly the mechanism that makes the Sun smaller at 1.6 micrometres — which is what transmission spectroscopy measures.

Stars resolved by interferometry show the same thing in the other direction. What an interferometer measures is the brightness distribution across the stellar disc, and converting it to a radius requires a model of the limb profile: a limb-darkened disc, whose edge is defined by a convention like the inflection point. For giant stars, whose atmospheres are extended and whose scale heights are a sizeable fraction of their radii, the difference between definitions reaches several per cent — and every planet radius is a stellar radius multiplied by a ratio, so the convention propagates.

And every radius of every star is ultimately expressed in solar units, with the Sun’s radius as the yardstick. Until 2015 different groups used different solar radii, differing by up to a part in a thousand, and stellar radii quoted to better than that were silently inconsistent. The nominal radius was adopted precisely so that the unit would stop depending on the edge.

What was actually measured #

Photometric radii. Measurements from eclipses, from the transits of Mercury in 2003, 2006, 2016 and 2019 and of Venus in 2004 and 2012, and from space telescopes imaging the full disc, give inflection-point radii of about 959.6 to 960 arcseconds at one astronomical unit — about 696,000 kilometres — with differences between methods of a few hundredths of an arcsecond, comparable to the blur and wavelength effects computed here.

The helioseismic radius. The frequencies of the surface gravity waves, measured from space with years of continuous data, imply a radius about three hundred kilometres smaller than the photometric one, at the level where the vertical optical depth is near one.

The reconciliation. Models of the solar atmosphere computing the inflection point’s height above the vertical surface found about 330 kilometres, in agreement with the computation here and with the gap between the two measured radii. The nominal radius adopted in 2015, 695,700 kilometres, is close to the vertical surface.

Constancy. A satellite built to measure the radius at several wavelengths over a solar cycle found it constant to within a few hundredths of an arcsecond, ruling out the larger variations claimed from historical records.

Why a few hundred kilometres matter #

Three hundred kilometres is four parts in ten thousand of the Sun’s radius, and it would be reasonable to ask whether a difference that small is worth the trouble.

It is, because of what the radius calibrates. Models of the solar interior are tuned to reproduce the Sun’s luminosity, age and radius, and the frequencies helioseismology measures are compared with those of the tuned model to test its physics. A model forced to the photometric radius rather than the vertical one is three hundred kilometres too large in its outer layers, and the surface-wave frequencies it predicts differ from the measured ones by more than their errors. The discrepancy was first noticed that way, as an unexplained offset in the frequencies, and it was resolved not by changing the interior physics but by changing which radius the model was fitted to.

The effective temperature is the other place the radius enters. The Sun’s luminosity is measured directly, from the solar constant; the effective temperature is the temperature of a blackbody of the Sun’s radius that would emit it. A radius uncertain by three hundred kilometres makes the effective temperature uncertain by about a kelvin, small but larger than the precision with which the luminosity is now known. The nominal value of 5,772 kelvin was defined together with the nominal radius, for that reason.

Where the model stops #

Grey and static. The model atmosphere is grey — the opacity is the same at every wavelength apart from a scale factor — and has the Eddington temperature law, no granulation and no magnetic structure. Real limb profiles are computed with detailed atmospheres and three-dimensional convection, which change the profile’s shape near the edge by tens of kilometres.

One opacity scale height. The opacity’s scale height is taken as sixty kilometres throughout, which is right in the upper photosphere and wrong elsewhere. The 335 kilometres depends on it logarithmically, so a scale height ten per cent different moves the inflection by about thirty-five kilometres.

Representative opacity ratios. The wavelength figure scales the opacity by assumed ratios, not by a computed continuum. The direction of the effect is robust; its size is indicative.

A sharp detector threshold. The contact figure treats the detector as switching from seeing light to not at one brightness. A real timing fits the light curve of the last seconds, which uses the whole profile and is less sensitive to a threshold.

A surface that depends on how it is approached #

The general lesson is about edges defined by opacity. Wherever a boundary is the place where a medium becomes opaque — a stellar photosphere, a cloud deck, a planetary atmosphere seen in an occultation — the boundary moves with the direction of the look, because opacity accumulates along the path and the path is longer when it grazes. An edge seen edge-on is higher than the same edge seen face-on, by a scale height times the logarithm of the ratio of path lengths, and for the Sun that ratio is hundreds.

A boundary defined by opacity has no position, only a position for each line of sight, and comparing two measurements of it means comparing their lines of sight first. The three-hundred-kilometre discrepancy between the Sun’s two radii was not a failure of either measurement. It was the geometry of the grazing ray, and the same geometry sets the edge of a planet’s atmosphere in an occultation and the radius of every transiting planet, which is measured edge-on through its own atmosphere.

Still open: the edge seen from space during an eclipse #

The best eclipse measurements are made from the ground, through the atmosphere, with the Moon’s limb profile supplied by an altimeter and the Sun’s limb profile by a model. A spacecraft passing through the Moon’s shadow — or two spacecraft flying in formation, one occulting the Sun for the other, as a mission launched in 2024 does to study the corona — times contacts with no atmosphere at all and from a known position. How precisely such timings can measure the Sun’s radius, and whether they can detect the small changes in the edge’s height that the solar cycle’s magnetic activity might produce, is limited in the end by the same limb profile computed here, which no observation of the edge can avoid.