A rotation found where the transform goes dark

A spectral line from a star is never as narrow as the atom that made it. The absorbing gas moves — thermally, in small-scale turbulence, in large convective flows across the surface — and the whole star turns, so that one limb approaches and the other recedes. Each motion Doppler-shifts the absorption by a different amount at different places on the disc, and the line that reaches the telescope is the sum of all of them, further smeared by the spectrograph.

Three different widenings can produce the same width: thermal motion, rotation and pressure can be traded against each other to give lines of the same area, and only the shape tells them apart. For two of those broadenings — rotation and the large-scale surface flows called macroturbulence — even the shape is a weak guide. Both widen the core and spread the wings, both scale with a velocity, and a star fitted with a little more rotation and a little less turbulence looks almost the same as one fitted the other way round.

This essay is about a way to separate them that does not use the profile’s shape directly. A line profile is a convolution of broadening kernels, and a convolution in velocity is a product in Fourier space. The Fourier transform of a rotating disc’s kernel has zeros. The others do not.

A valley of fits #

The degeneracy is clearest as a fit. Take a line from a star that rotates at six kilometres a second and has three kilometres a second of macroturbulence, observed at a resolving power of sixty thousand with the usual thermal and microturbulent broadening, and fit it with a model in which the macroturbulence is fixed at some assumed value and the rotation is free.

The fit is not hopeless; the valley has a bottom, at the truth, and its walls rise. But they rise slowly. At a signal-to-noise ratio of two hundred, which is a good spectrum, the rotation is uncertain by ten per cent and the turbulence by fifty, and the two errors are perfectly correlated: an analysis that assumes the wrong turbulence gets the wrong rotation, with a fit statistic that does not object.

For many purposes ten per cent is enough. For some it is not. Slowly rotating Sun-like stars spin at two or three kilometres a second, the same order as their macroturbulence, and their rotation is what dates them through the slow spin-down of magnetic braking; a v sin i uncertain by a kilometre a second is an age uncertain by billions of years. And for those stars the valley is wider still.

The two valleys differ in kind, not only in width. For the faster star the rotation is the larger broadening and the fit is mostly sensitive to it; the turbulence is the parameter that floats. For the slow star the turbulence dominates, and the rotation floats almost freely, because a small rotation added to a large Gaussian-like broadening changes the profile’s shape by an amount that scales roughly as the square of the ratio between them. The quantity of interest is the one the profile is least able to see.

Why Fourier space #

The observed profile is the intrinsic line convolved with each broadening in turn:

$$D(v) = I(v) * T(v) * G(v) * M(v) * P(v),$$

the intrinsic line, the thermal and microturbulent Gaussian, the rotation kernel, the macroturbulence and the instrument. Each convolution smooths the profile, and the order does not matter. The Fourier transform of a convolution is the product of the transforms,

$$d(\sigma) = i(\sigma)\, t(\sigma)\, g(\sigma)\, m(\sigma)\, p(\sigma),$$

with $\sigma$ the frequency conjugate to velocity, in cycles per kilometre per second. In the profile, the broadenings are tangled together. In the transform, they are factors.

That alone would not help if all the factors looked alike. They do not. The thermal broadening, the instrument and macroturbulence are all close to Gaussian, or are sums of Gaussians, and the transform of a Gaussian is a Gaussian: positive at every frequency, smoothly decreasing, never zero. The rotation kernel is different.

The shape of a rotating disc #

A star rotating at an equatorial speed whose projection on the line of sight is $v\sin i$ has every strip of its disc parallel to the rotation axis moving at one line-of-sight velocity, from $-v\sin i$ at one limb to $+v\sin i$ at the other. The rotation kernel is the light-weighted length of those strips: a half-ellipse for a uniformly bright disc, and something between a half-ellipse and a parabola for a disc darker at its limb.

The zeros are the important property. A function confined to a finite interval with steep edges — a box, a half-ellipse — has a transform that oscillates through zero, with the first zero at a frequency inversely proportional to the interval’s width. For a uniformly bright disc the first zero is at $0.610/v\sin i$; for a solar limb darkening, $0.660/v\sin i$. A Gaussian, which has no edges, has no zeros.

So in the product $d(\sigma)$, the rotation factor goes to zero at its zeros and drags the whole product to zero with it, whatever the other factors are doing, because zero times anything is zero. The Gaussian factors only reduce the amplitude. The first zero of the observed transform therefore sits exactly where the rotation kernel’s first zero sits, and it measures $v\sin i$ without reference to how much macroturbulence there is.

A method older than the spectrographs that use it #

The idea of reading rotation from the transform of a line is nearly a century old. It was proposed in 1928 and worked out in the early 1930s, when computing a Fourier transform of a photographic line profile meant hours of arithmetic by hand, and the instruments of the time could not resolve the features it relied on. The analysis of a line as a convolution — intrinsic absorption smeared by successive broadening functions — was the natural language for it even then; what was missing was the resolving power to see the zeros and the computers to find them.

Both arrived in the 1970s, with photoelectric detectors at high resolution and the fast Fourier transform, and the method became a standard tool for measuring the rotation of moderately slow stars and the macroturbulence of their atmospheres. Its practitioners also established the empirical facts about macroturbulence that the rest of the subject now relies on: that it is larger in hotter stars and in giants, that it is comparable to the sound speed in the photosphere, and that its profile is better described by radial and tangential flows than by isotropic turbulence — the model used in the figures here.

Reading the zero #

The procedure, developed in its modern form in the 1970s and refined since, is simple to state. Take the observed profile of an unblended line, transform it, divide out the transform of the instrumental profile measured from a calibration lamp, and find the first frequency at which the result drops to the noise. That frequency, multiplied by $v\sin i$ and divided by $0.660$, gives $v\sin i$.

The macroturbulence is then measured from the shape of the transform before the zero, with the rotation already fixed: the rotation kernel’s transform is now known exactly, and dividing it out leaves the product of the Gaussian-like factors, whose fall-off with frequency gives the turbulent velocity. The degeneracy of the profile fit is broken because the two quantities are measured from different features of the transform — one from where it vanishes, the other from how it falls.

What the turbulence turns out to be #

Once the rotation is fixed by the zero, the remaining width belongs to the other broadenings, and the shape of the transform before the zero measures it. What it measures is the velocity field of the photosphere’s convection.

In a star like the Sun, the photosphere is the top of the convective envelope, and its surface is a pattern of granules: bright columns of hot gas rising at a kilometre or two a second, a thousand kilometres across, surrounded by dark lanes where cooled gas sinks. Seen as a whole, unresolved, that pattern Doppler-shifts every line by a spread of velocities with a characteristic shape — the radial–tangential pattern, in which rising and falling flows dominate at disc centre and horizontal flows towards the limb. The macroturbulence of the line analysis is that pattern, summarised as one velocity.

The Fourier measurements found it growing with temperature along the main sequence and larger in giants than in dwarfs at the same temperature, in step with the expectation that convective velocities scale with the energy the convection must carry. Three-dimensional hydrodynamic simulations of stellar surfaces now compute it from first principles, and they reproduce both the amplitude and the asymmetry of observed lines — the asymmetry being something the symmetric kernels used here cannot capture, because rising granules are brighter than sinking lanes and contribute more light. That asymmetry, which shifts and bends the line’s core, is the convective blueshift that, varying with a star’s activity, hides inside the radial velocities of planet searches.

What limb darkening does to the zero #

The zero’s position depends on one thing besides the rotation: how the star’s brightness falls towards its limb. A disc darker at the edge gives less weight to the fastest-moving strips at the limbs, so its kernel is rounder and its effective width smaller, and its zeros move to higher frequency.

Seventeen per cent across the full range of limb darkening is the systematic error of the method if the darkening is unknown. It is not unknown: the limb-darkening coefficient for a star of given temperature and gravity at a given wavelength is computed from model atmospheres, and measured directly for the Sun, for eclipsing binaries and for stars with interferometric diameters, and for ordinary stars the uncertainty is a few hundredths in the coefficient — a per cent or two in the zero.

The darkening is also wavelength-dependent, stronger in the blue than in the red, so lines at different wavelengths in the same star have their zeros in slightly different places for the same rotation. Measuring several lines across the spectrum and checking that they give one v sin i is a test of the darkening adopted.

How slow a rotation can be seen #

The method has a floor, and it is set by the factors that are not rotation. The thermal and instrumental Gaussians multiply the transform by a factor that falls off as $\exp[-(\pi\sigma_{\rm G}\,\sigma)^2]$, where $\sigma_{\rm G}$ is their combined width; at high frequency they crush the transform below the noise. A slow rotator’s first zero is at high frequency, and if the transform has already been crushed there, the zero is invisible — there is nothing left to go to zero.

The floor is why the Sun’s own rotation, two kilometres a second at the equator, is at the edge of what the method could measure from an unresolved solar spectrum. For the slowest rotators the Fourier method gives way to other techniques: the rotational modulation of starspots, which gives the period directly when the star is active enough to have spots; the shadow a transiting planet casts across a rotating line, which maps the velocity across the disc; and asteroseismology, which measures the rotation of the interior through the splitting of oscillation frequencies.

Many lines, one transform #

The noise in a single line’s transform is what sets the floor, and the obvious remedy is to use many lines. A spectrum of a cool star has thousands of lines, all broadened by the same rotation and almost the same turbulence, and their transforms differ mainly in their amplitudes and in their intrinsic shapes.

There are two ways to combine them. One is to transform each and average the transforms after dividing out each line’s intrinsic profile — which needs a model of the intrinsic profile, and reintroduces some of the dependence on the atmosphere the method was meant to avoid. The other is to combine the lines into a single mean profile first, by a least-squares deconvolution that treats the observed spectrum as a list of line positions and depths convolved with one common broadening function, and then transform that. The mean profile has a signal-to-noise ratio higher by roughly the square root of the number of lines used, and its transform carries the same zeros.

The combination works only if every line has the same broadening, which is close to true for rotation and less true for turbulence: lines formed at different heights in the atmosphere sample different convective velocities. Lines are therefore usually chosen from a narrow range of strengths, so that they form at similar depths and the part of the line not in its depth — the saturated core — does not distort the comparison.

The fast end #

At the other extreme the method is easiest. A star rotating at a hundred kilometres a second has its first zero at a frequency of 0.0066, well below where any thermal or instrumental broadening has any effect, and the second and third zeros are visible too. Their spacing is a check on the first: the zeros of a limb-darkened rotation kernel are spaced almost but not quite uniformly, and their ratios constrain the limb darkening independently.

Fast rotators raise a different problem. A star spinning near its breakup speed is flattened and darker at its equator than at its poles, and its rotation kernel is no longer the symmetric half-ellipse of a sphere with uniform limb darkening. The zeros move, and the ratio of the second to the first moves differently from the ratio for a slow rotator. That shift has been used to detect differential rotation — equators turning faster or slower than higher latitudes — in stars whose rotation is fast enough for the second zero to be measured.

Why the zero is not simply the width #

It would be natural to think the zero is just another measure of the line’s width, and that a width is a width whatever its source. The figures say otherwise. The two stars in the comparison had the same width to within a per cent, and their first zeros differed by twenty per cent — exactly the ratio of their rotations, unaffected by the fact that one had nearly twice the other’s turbulence.

The reason is that the zero responds to the edges of a broadening function and not to its spread. The rotation kernel has edges at plus and minus v sin i: no point of the disc is moving faster than that, and the kernel stops dead there. A Gaussian has no edges; its wings extend to every velocity, thinning smoothly. In the profile, where everything is added, the Gaussian’s wings blur the rotation’s edges into invisibility. In the transform, where everything is multiplied, the edges survive as zeros and the wings only as a gentle overall decline. The method converts a property of the profile that convolution hides — where the broadening stops — into one that multiplication preserves.

That is also why the method fails gracefully rather than abruptly as rotation becomes small. A slow rotation is a narrow kernel, whose edges are close together; its zeros move out to high frequencies, where the Gaussian factors have already pushed the transform down towards the noise. The zero is still there in principle. It is the noise, not the physics, that stops it being found.

Where the model stops #

One unblended line. The figures transform a single weak line, isolated, with a known continuum. A real line has neighbours whose wings overlap its own, and the continuum it is measured against was drawn by hand. Blends add their own transforms; a misplaced continuum adds a spurious component at low frequency. Both move the zero less than they move a profile fit, which is part of the method’s appeal, but neither leaves it untouched.

Kernels that are not strictly separable. The convolution model assumes every point on the disc has the same intrinsic line shape and the same macroturbulence, shifted only by its rotation velocity. In a real atmosphere the intrinsic line changes from centre to limb, and the macroturbulence, which is the projection of convective flows, depends on the viewing angle — the radial–tangential model used here builds that in, approximately. Three-dimensional simulations of convection show line profiles with asymmetries no convolution of symmetric kernels can produce, and at the highest precision the Fourier method is replaced by fitting simulated profiles directly.

A noise floor drawn as a level. The noise in a transform is not a fixed number; it depends on the number of points across the line and on how the noise in the spectrum is correlated. The figures use a representative level to show where a zero stops being findable, and the floor in the last figure is an estimate of the scale, not a calculation for any instrument.

An idealised instrument. The instrumental profile is a Gaussian with the width the resolving power implies. Real instrumental profiles have wings and asymmetries, are measured from calibration exposures and divided out, and the division amplifies the noise at high frequency, which is why the method needs a well-characterised spectrograph as much as a high resolving power.

The surprising connection #

The first zero of a rotation kernel’s transform is the same mathematics as the first null of a stellar interferometer’s visibility: the transform of a limb-darkened disc, seen as a function of frequency, falls to zero at a frequency inversely proportional to the disc’s size. In interferometry the disc is the star’s image on the sky and the frequency is the baseline over the wavelength; the null gives the angular diameter. In spectroscopy the disc is the star’s velocity field and the frequency is in reciprocal kilometres per second; the null gives the rotation speed. The limb-darkening correction to the null — 1.22 for a uniform disc in an interferometer’s language, and the shift from 0.610 to 0.660 here — is the same correction for the same reason, applied to a velocity instead of an angle.

Still open: the slowest rotators #

The Fourier floor leaves the slowest Sun-like stars — the ones whose rotation dates them most precisely — below the reach of the method at ordinary resolving powers. Spectrographs built for planet searches now reach resolving powers above a hundred thousand with signal-to-noise ratios in the hundreds, which lowers the floor towards two kilometres a second, and averaging the transforms of many lines lowers the noise. Whether the thermal width of the lines themselves, which no instrument can remove, leaves the zero of a star rotating at one or two kilometres a second permanently unreadable — so that the rotation of the oldest, quietest stars can only come from their spots, their oscillations or the planets that cross them — depends on whether lines formed in the coolest parts of the atmosphere, with the smallest thermal widths, can be found and measured cleanly enough to carry the method down.