Astronomy & Celestial Mechanics
the sky, computed
Starlight

A temperature that depends on where the observer stands

A star turning near its break-up rate is half again as wide as it is tall, and its equator is thousands of degrees cooler than its poles. Neither of those is a small correction to a spectrum — the effective temperature and the luminosity such a star appears to have are partly statements about which way its axis happens to point.

Assumes Internal rotation, Hydrostatic equilibrium and Interferometry.

17 min read 8 figures Light is the only messengerMeasuring the unreachable

A star is usually treated as a sphere with one temperature, so that its colour is a thermometer and nothing else. That is a very good approximation for the Sun, whose equator moves at two kilometres a second, and it fails completely for a star whose equator moves at three hundred.

The failure is not subtle. A star rotating near the rate at which its equator would fly off is measurably not round, and because it is not round it does not have a single surface gravity, and because the emergent flux is tied to the local gravity it does not have a single temperature. What an instrument reports as such a star’s effective temperature is an average over the visible hemisphere, weighted by projection — and the visible hemisphere depends on where the observer is.

A star 1.24 times wider than it is tall, and 19 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.9257 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.245 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.768, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.329 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.320 at the theoretical exponent 0.25 and 1.232 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.19 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 1 Left, the meridional section of a star turning at 93 per cent of its critical angular velocity, computed from the Roche potential. The equator sits at 1.245 polar radii and the effective gravity there is a third of its polar value. Right, the apparent bolometric brightness against viewing angle: the same star is 19 per cent brighter seen pole-on than edge-on. The two dots marked at the pole and the equator are not decoration — the surface is coded by its own local temperature, and the gradient between them is what the rest of the essay is about.

The shape is not a fit

The first thing to establish is that the distortion is a consequence rather than a parameter.

Take a star whose mass is centrally concentrated enough that the gravity outside the bulk of it is that of a point — which for a real star is a good approximation, since the outer layers weigh almost nothing. Add rotation. The surface must be an equipotential of gravity plus the centrifugal term:

GMr+12Ω2r2sin⁡2θ=GMRpole.\frac{GM}{r} + \frac{1}{2}\Omega^2 r^2 \sin^2\theta = \frac{GM}{R_{\rm pole}}.

In units of the polar radius this is a cubic with one physical root, and it has a property worth pausing on: at the critical rotation rate, the equatorial radius is exactly 1.5 times the polar one. Not approximately, and not depending on what the star is made of. The number falls out of the potential, in the same way that the tunnel through a uniform Earth takes the same time from anywhere — it is a property of an inverse-square field rather than of a body.

That gives a clean definition of “how fast” a star is turning, and immediately a trap. There are two conventions in use, and they are not the same number.

The first is ω=Ω/Ωcrit\omega = \Omega/\Omega_{\rm crit}, the angular velocity as a fraction of the critical one. The second is veq/vcritv_{\rm eq}/v_{\rm crit}, the equatorial speed as a fraction of the critical equatorial speed. These differ by the distortion itself, since the critical star’s equator sits further out:

veqvcrit=ω Req(ω)1.5 Rpole.\frac{v_{\rm eq}}{v_{\rm crit}} = \omega\,\frac{R_{\rm eq}(\omega)}{1.5\,R_{\rm pole}}.

Regulus is quoted at 0.86 in the second convention and is at 0.97 in the first. A figure that prints one under the other’s name is wrong by an amount that looks like a rounding error and is not — and the difference between 0.86 and 0.97 is the difference between a star with some room left and a star at the edge.

Why the equator is cooler

The temperature gradient follows from a result of von Zeipel’s that is one line long once the right thing is assumed.

In a rotating star in radiative equilibrium, the emergent flux at any point on the surface is proportional to the local effective gravity:

F∝geff,soTeff∝geff1/4.F \propto g_{\rm eff}, \qquad\text{so}\qquad T_{\rm eff} \propto g_{\rm eff}^{1/4}.

The reasoning is that radiative energy transport is driven by the temperature gradient, that the temperature gradient in hydrostatic equilibrium is tied to the pressure gradient, and that the pressure gradient is the effective gravity. Where the effective gravity is small — at the equator, where the centrifugal term subtracts from the attraction and the radius is larger besides — less flux emerges, and the surface is cooler.

For the star drawn above, the equatorial gravity is 0.33 of the polar value, so von Zeipel predicts a temperature ratio of 0.33−1/4=1.320.33^{-1/4} = 1.32. On a star with a polar temperature of 8,500 kelvin that is an equator at 6,450, a difference of two thousand degrees across one object.

A star 1.35 times wider than it is tall, and 30 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.98 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.350 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.882, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.164 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.571 at the theoretical exponent 0.25 and 1.410 at the 0.19 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.30 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 2 The same construction pushed nearer the limit, at 98 per cent of critical. The distortion and the gravity contrast both accelerate as the critical rate is approached — the equatorial gravity goes to zero there, and with it the equatorial temperature — so the observable consequences are not a smooth function of how fast the star turns. A star at 0.9 is a curiosity; a star at 0.99 is a different object, with a nearly dark equator and a pole that carries almost all of the light.

What was actually measured, and what the measurement disagreed with

Both halves of this were confirmed, and one of them was confirmed to be wrong.

The shape was measured by long-baseline optical interferometry. A single telescope cannot resolve a star that subtends a few hundredths of an arcsecond; an array of telescopes measures the Fourier transform of the brightness distribution at the spatial frequency each pair’s separation corresponds to, and enough pairs at enough orientations reconstruct an image.

A radius needs a distance: 47 mas against parallax. The radius that follows from an angular diameter of 47 milliarcseconds, drawn against the parallax used to convert it. R = θd/2 and d = 1/ϖ, so the curve is an exact inverse proportionality and the radius is only ever as good as the distance: Hipparcos's 5.95 mas puts the star at 168 pc and gives 849 R☉, the revision's 4.51 mas puts the star at 222 pc and gives 1120 R☉. Those two determinations differ by 271 R☉. The shaded band above the curve is the other uncertainty — the same measured null fitted with a limb-darkened disc instead of a uniform one, 4.9% more angle and so 42 R☉ more radius at the better parallax. The distance dominates by a factor of 6.5, which is why an interferometric radius is quoted with a parallax attached and why revising the parallax revised the star.
Fig. 3 How a stellar diameter comes out of an interferometer. The visibility — the contrast of the fringes a pair of telescopes produces — falls with baseline in a way that depends on the source’s angular size, and the first null locates the diameter. A star is not a uniform disc, so the fit has to include limb darkening, and the recovered diameter depends on the assumed profile at the level of a few per cent. For a rotating star the answer also depends on the orientation of the baseline, which is how the oblateness is detected rather than assumed.

Imaging of the brightest rapid rotators returned axial ratios near 1.25 and a pole-to-equator temperature contrast — and the contrast is smaller than von Zeipel predicts. Where the gravity ratio implies an exponent of 0.25, the fitted exponent comes out near 0.19. The disagreement is not marginal and it is systematic across the objects measured.

The explanation is that von Zeipel’s derivation assumes strictly radiative energy transport and a barotropic structure, and a rapidly rotating star satisfies neither exactly. Any circulation driven by the rotation itself carries energy sideways, which is precisely what flattens the temperature contrast. The measured exponent is therefore a measurement of how much energy is being redistributed across latitude, made from outside a star nobody can enter.

A star 1.04 times wider than it is tall, and 2 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.5 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.042 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.347, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.844 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.043 at the theoretical exponent 0.25 and 1.032 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.02 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 4 The same construction at half the critical rate. The star is 1.042 polar radii across the equator and only two per cent brighter seen pole-on than equator-on — which is to say the effect is nearly invisible here, and that is worth seeing before the extreme cases. The distortion goes as the square of the rotation rate and the brightness contrast faster still, so a star at half of break-up is very nearly round and a star at nine tenths is not. Nothing between these two drawings is a different physics.

The speed, and the angle nobody knows

The other measurable is the rotation speed, and it arrives incomplete.

A rotating star’s absorption lines are broadened, because different parts of the disc are moving at different line-of-sight speeds. The broadening kernel has a hard edge — a semicircle, modified by limb darkening — set by the equatorial speed times the sine of the inclination. Only the product is measurable. This is where the two techniques earn their keep together. Interferometry delivers the axial ratio and the orientation of the projected ellipse; the line profile delivers vsin⁡iv\sin i; and together they close the system and return the star’s actual rotation rate, its inclination, and its two radii separately. Neither does it alone. Compare the three broadening mechanisms that can be tuned to look alike: a width on its own is ambiguous, and the ambiguity is only ever broken by another measurement.

What one image showed

The general argument is worth grounding in a particular object, because the first resolved images of a rapid rotator settled several things at once and raised one.

Altair is a bright A-type star twenty light years away, rotating fast enough to be visibly oblate. Imaged with a four-telescope array in the near infrared, it resolves into a disc with an axial ratio near 1.2 and a brightness distribution that is not symmetric about its centre: one end of the projected ellipse is measurably brighter than the other.

That asymmetry is the gravity darkening seen directly. The pole is hot and the equator cool, and because the star is inclined the two are not symmetrically placed on the disc — so the reconstructed image has a bright region displaced from the geometric centre, exactly as a tilted, darkened spheroid should.

Three quantities came out of the fit at once. The inclination, from where the bright region sits. The two radii, from the projected ellipse. And the darkening exponent, from how fast the brightness falls from the bright region to the dim limb.

The exponent was the surprise, and it is the one this essay’s earlier section is about: it came out well below the von Zeipel value, and the same has been found for every rapid rotator imaged since. That is a measurement of a stellar interior — of how much energy is being carried across latitude — made by resolving a disc a few milliarcseconds across.

What it raised is a calibration problem. Altair had been used for decades as a spectroscopic standard, and a star whose temperature depends on inclination is a poor standard. Its catalogued effective temperature is an average over its visible hemisphere at its particular inclination, and the same star seen from a different direction would have been catalogued differently. Several of the brightest stars in the sky are rapid rotators, and their tabulated parameters carry that.

The place on the diagram that is not a place

The consequence for the rest of the subject is that a rapid rotator’s position on the diagram that sorted the stars is partly about the observer. Seen pole-on, the star shows its hot bright pole filling the disc and its cool equator edge-on and foreshortened: it looks hotter and more luminous than it is. Seen equator-on, it looks cooler and fainter, and its projected area is larger besides. For the star in the opening figure the apparent brightness swings by nineteen per cent between the two, and the apparent effective temperature by several hundred kelvin.

That propagates. A luminosity read off the diagram feeds a mass through the relation in which mass decides everything, and an age through the isochrones.

A lever 24 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 24 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 1.2·10⁻¹⁰ of its angular momentum in the same year — a ratio of 5,260, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 8.3·10⁹ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.
Fig. 5 Where the rotation went. A magnetised wind is forced to co-rotate out to the Alfvén surface, so every gram that leaves carries the specific angular momentum of that radius rather than of the star’s own surface — and at twenty-four stellar radii that is a lever nearly six hundred times as effective per unit mass as the surface would be. A star can lose almost all its angular momentum while losing almost none of its mass, which is the reason a star of this age rotates slowly at all and therefore the reason gravity darkening is a rarity rather than the rule.

The one thing the picture cannot show

Every figure in this essay draws the star as an axisymmetric surface with a smooth temperature gradient, and the last part of that is the part least supported.

Interferometric images of the brightest rapid rotators are reconstructions from a modest number of baselines, and they are regularised — smoothed, because an unregularised reconstruction from sparse Fourier coverage is noise. The pole-to-equator gradient is robust because it is a low-order feature that the visibility amplitudes constrain directly. Anything smaller than that is model. A rapid rotator could have latitudinal structure, spots, or a circulation pattern imprinted on its surface, and the measurements would not currently distinguish it.

A star 1.10 times wider than it is tall, and 6 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.7 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.095 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.511, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.674 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.104 at the theoretical exponent 0.25 and 1.078 at the 0.19 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.06 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 6 The same effect at a more ordinary rotation rate. At seven-tenths of break-up the star is ten per cent wider at the equator than at the poles and the pole-to-equator temperature difference is a few hundred kelvin — small enough that the inferred temperature moves by less than most catalogues’ error bars, and large enough that it does not vanish. The dependence on inclination is what makes it a systematic rather than a scatter: a population of fast rotators observed at random angles has a biased mean temperature, not merely a noisy one.

The planet that maps the surface

There is a second route to the same measurement that needs no interferometer, and it works on stars far too distant to resolve.

A planet transiting a gravity-darkened star crosses a surface whose brightness varies from place to place. Where its path takes it over the hot pole it blocks bright light and the transit is deep; where it crosses the cool equator it blocks dim light and the transit is shallow. So the light curve is asymmetric in time, and the shape of the asymmetry encodes the path the planet took across a non-uniform disc.

That is a great deal of information from one light curve. The asymmetry fixes the angle between the planet’s orbital plane and the star’s rotation axis — not the projected angle a spectroscopic measurement gives, but the true three-dimensional obliquity — and it fixes it without any spectroscopy at all.

The method works only on rapid rotators, which means on stars hotter than the convective boundary, which are exactly the stars whose spectroscopic obliquity measurements are hardest because their lines are broad and shallow. So the photometric route reaches the population the spectroscopic route struggles with, and the two have been compared on the handful of systems where both apply.

The results agree, and they have produced some of the most striking obliquity measurements in the field: transiting planets on orbits nearly perpendicular to their star’s equator, detected from the shape of a dip in a light curve.

A distortion that corrupts the star’s own parameters is, for a transiting system, the measurement — and it is available for any system where the star turns fast enough, which is a large fraction of the hot ones.

There is a cost attached and it is the usual one for a photometric method: the signal is a small asymmetry in a light curve whose overall shape is set by several other things. Limb darkening produces its own curvature, the planet’s own path produces its own asymmetry if the orbit is eccentric, and the star’s oblateness changes the transit’s duration in a way that trades against the impact parameter.

So the measurement is a simultaneous fit with more parameters than a spherical star’s transit needs, and the obliquity is recovered along with the inclination, the two radii and the darkening exponent. Where the star’s rotation is slow the asymmetry vanishes and the fit becomes degenerate — which is the same statement as the method only working on rapid rotators, arrived at from the direction of the algebra rather than of the physics.

The published cases are therefore few and well characterised rather than numerous, and the sample is selected on the star being distorted rather than on anything about the planet. That is a selection nobody would choose and it happens to be harmless here: whether a star spins fast has no bearing on how its planet’s orbit was oriented, which is the quantity being measured.

One further consequence of the geometry is worth recording, because it is the only case in which gravity darkening helps rather than hinders. A planet crossing the hot pole of a rapidly rotating star blocks a disproportionate share of the light, so its transit is deeper than its size alone would give — and a planet crossing the cool equator produces a transit that is shallower. Fitting a spherical, uniform star to either returns the wrong radius ratio, by several per cent in the extreme cases.

Since the radius ratio is the one quantity a transit is supposed to deliver cleanly, that is a systematic in the planet’s radius produced entirely by the star’s rotation. It has been found in a handful of systems and it is not applied routinely, because doing so requires knowing the obliquity, which requires the fit the systematic is corrupting.

A star 1.39 times wider than it is tall, and 34 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.99 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.390 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.917, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.114 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.720 at the theoretical exponent 0.25 and 1.504 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.34 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.
Fig. 7 And a hundredth below break-up. The equator is 1.390 polar radii — approaching the 1.5 that is exact at the critical rate whatever the star is made of — and the pole is 34 per cent brighter than the equator. The ratio 1.5 is a property of the Roche potential and not of any stellar model, which is why it is the one number in this subject that can be quoted without a model behind it, and why a measured axis ratio above it would refute the whole picture rather than adjust it.

What happens at the limit

The critical rate is not merely the end of the axis; it is a place stars actually reach, and reaching it does something visible.

At ω=1\omega = 1 the effective gravity at the equator is exactly zero. Material there is held by nothing, and any further addition of angular momentum — or any expansion of the star as it evolves — leaves it behind. The star does not fly apart, because the layers below the equator are still bound; what happens instead is that mass leaks off the equator into orbit, building a thin disc in the equatorial plane.

That is the standard account of the Be stars: rapid rotators of early spectral type, seen with hydrogen emission lines that come from a disc rather than from the star. The emission appears and disappears on timescales of years, which says that the disc is being fed episodically and drains when the feeding stops. The measured rotation rates of these stars cluster near, but mostly a little below, critical — and how far below is contested, precisely because the two conventions above disagree there and because the gravity darkening biases the very vsin⁡iv\sin i measurement used to place them. The circularity is worth naming. A near-critical star is the hardest one to measure the rotation of, because the same rotation that is being measured has put material in front of the star, distorted the surface being averaged over, and made the temperature depend on the angle. The best-measured rapid rotators are therefore the ones a little short of the limit, and the population nearest the limit is the one whose rates are least certain.

There is a way out of it, and it is the one the interferometric measurements above supply. An image resolves the star’s shape directly, so the oblateness gives the rotation rate without any line profile being involved — and the oblateness is a geometric quantity that a disc in front of the star biases in a different direction from the way it biases a line width. Where both have been done on the same object they broadly agree, which is the reason the exponent measured on a handful of bright stars is taken to apply to a population that cannot be imaged at all.

A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 1000, 2500, 4570 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.35: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star.
Fig. 8 And the consequence of that braking, as a clock. Rotation period against colour for five cluster ages: the isochrones do not cross, and they are separated at every colour by the age ratio raised to a fixed power, which is what makes a measured period an age. The stars this essay is about sit far off the top-left of this diagram — too hot to have a convective envelope, so no magnetised wind, so no braking. The stars that are gravity-darkened are exactly the stars this clock cannot read, and the two facts have one cause.

Where the ladder goes

The immediate extension is population-level. Rapid rotators are common among stars hotter than the convective boundary — precisely because those stars have no magnetic brake — so the systematic distortion described here applies to a large fraction of the early-type main sequence, and the errors it introduces do not average out, because the pole-on cases are brighter and are over-represented in any brightness-limited sample.

The deeper thread runs into structure. Von Zeipel’s exponent is a statement about how a star transports energy, and the measured departure from it is a measurement of the circulation that rotation drives. That circulation also carries chemical elements: it mixes processed material from the burning core outwards, which lengthens the star’s life and changes its surface composition. The same rotation that makes the surface hard to measure is the reason the interior is not what a non-rotating model says, and the observed nitrogen enrichment of rapidly rotating massive stars is the check on it.

Where this goes next

Three profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere. The same width for three different reasons Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another. The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them. Resolution without a mirror Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those. The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius. The diagram that sorted the stars, by plotting two things against each other Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Closure phaseCritical rotationEffective gravityEffective temperatureGravity darkeningHydrostatic equilibriumInclinationInterferometryLimb darkeningOblatenessProjected rotation velocityRoche modelStellar radiusVon zeipel theorem