The same width for three different reasons

An absorption line’s equivalent width is one number, and the curve of growth is the story of how little that number can be made to say — two decades of abundance passing through a stretch where the area changes by a factor of two. That is the standing result of the rung below this one, and it is about the width.

The complement is the subject here. A profile is a function and not a number, and reducing it to an area throws almost all of it away — not noise, but a rotation speed, a pressure and a temperature, in a form that survives holding the area fixed.

An area is one number and a profile is a function #

The equivalent width integrates the line away on purpose: a spectrograph of poor resolution smears a profile out, making it shallower and wider, and leaves the integral alone. That is bought by discarding the shape, and the shape is where the mechanism is. The three profiles above have the same area because they were made to; their half widths are 1.11 km s⁻¹ for the thermal one, 1.35 for the rotational and 0.68 for the collisional, a factor of 1.98 between widest and narrowest. Equal equivalent width is not equal width, and the counter-example lives inside the quantity the title is about.

Three mechanisms that share no variable #

Each of the three widths is short enough to write out completely, and the interesting thing about them is that they share no argument.

Thermal motion. Atoms of mass $m$ at temperature $T$ have a most probable line-of-sight speed $\sqrt{2kT/m}$, the shifts are distributed as a Maxwellian, and the absorption profile is therefore a Gaussian of width

$$\frac{\Delta\lambda_D}{\lambda} = \frac{1}{c}\sqrt{\frac{2kT}{m}}.$$

For iron at 6000 K that is 1.34 km s⁻¹. Note what is in it: a temperature and an atomic mass, and no property of the star. Hydrogen, 55.8 times lighter, gets a width 7.5 times larger in the same gas — which is why two species in one spectrum are a thermometer before anything else.

Rotation. Every point on a rotating disc is Doppler shifted by its own projected velocity, so the line seen from far away is the local profile convolved with the distribution of projected speeds across the disc. Integrating along strips of constant shift gives, for a limb-darkening coefficient $\varepsilon$,

$$G(x) \propto 2(1-\varepsilon)\sqrt{1-x^2} + \frac{\pi\varepsilon}{2}\,(1-x^2), \qquad x = \frac{\Delta\lambda}{\lambda\,v\sin i/c},$$

and the important feature is the domain rather than the formula. $G$ is exactly zero for $|x| > 1$, because no part of the disc moves faster than its own limb. A rotating star’s line has an edge, at $\Delta\lambda_{\max} = \lambda\,v\sin i/c$, and neither of the other two mechanisms has one.

Collisions. A radiating atom perturbed by its neighbours loses phase coherence, the transition acquires a finite width $\gamma$, and the profile is a Lorentzian, $1/(1 + (2\Delta\lambda/\gamma)^2)$. What $\gamma$ depends on is the density of perturbers, which is a pressure — so the third width is not about motion at all, but about how crowded the gas is.

Equal areas are not equal widths #

An area is a width times the shape’s own dimensionless integral, so the width giving a required area goes inversely with that integral — $\pi$ for a Lorentzian, $\sqrt{\pi}$ for a Gaussian, smaller still for the rotation kernel. Read that as a statement about where the area comes from.

A shape with wings buys most of its area far from the centre, so to reach a given total it needs a narrower core; a shape with an edge must buy all of its area inside the edge, so it has to be wide. That is the ordering the figure returns — collisional 0.68 km s⁻¹, thermal 1.11, rotational 1.35 — so a spectroscopist who measures a width and infers a velocity can be wrong by a factor of two with no arithmetical error anywhere.

Six decades apart at five half widths #

The separation between the shapes is not a matter of careful fitting. It is enormous, and it grows outward from the core.

A factor of $10^6$ in depth is not a subtle discrimination, and it is why the shape of a line stays diagnostic long after its width has stopped being so: the core of a strong line and its wings are measured for entirely different purposes, because two parts of one feature are dominated by different physics.

The figure also states a warning about itself. Eleven per cent of the Lorentzian’s equivalent width lies beyond the right-hand edge of that plot and is drawn nowhere — a power-law tail is never wholly on the page, and a quadrature truncated where the picture stops returns a number the picture is not of.

A wing width is a surface gravity, and a surface gravity is a distance #

The collisional width measures a pressure, and in an atmosphere a pressure is not a free quantity. Integrating the balance that holds a star up inward to optical depth $\tau$ gives, for opacity $\kappa$ and surface gravity $g$,

$$P \approx \frac{g\,\tau}{\kappa},$$

so the pressure where a line forms is fixed by $g$ and the opacity and by nothing about the star’s size or distance. The power of $P$ that $\gamma$ follows depends on which interaction dominates — neutral perturbers give roughly $\gamma \propto P$, and the linear Stark effect that broadens hydrogen follows a different power of the electron density — but every one of them rises with pressure. So a damping wing is a measurement of $\log g$. Here is the part that is genuinely unexpected. A luminosity class in the Morgan–Keenan system — V for a dwarf, III for a giant, Ia or Ib for a supergiant — is assigned by reading $\log g$ off the profiles, and it therefore fixes where on the diagram that sorted the stars the object sits, which fixes an absolute magnitude, which with an apparent one is a distance. So the shape of a line is a rangefinder. No parallax and no standard candle is involved: a brightness becomes a distance once something is known, and what is known here is the curvature of a wing.

What is actually measured #

None of the three quantities above is observed. What is observed is a column of counts: a cross-dispersed échelle spectrograph spreads starlight across a detector and records, for each pixel, a number of electrons. At a resolving power of 100,000 one resolution element at 500 nm is 3.00 km s⁻¹ wide, and every profile the instrument returns is the star’s own convolved with an instrumental one — measured separately, from a thorium–argon lamp or a laser frequency comb, never perfectly. Those counts become a residual intensity only after division by a continuum, and in a crowded spectrum no wavelength is free of absorption, so the continuum is a fitted envelope whose systematic error biases every width the same way. What comes out of the fit is a parameter of a forward model rather than a reading. Deriving $v\sin i$ requires a rigidly rotating spherical disc, a chosen limb-darkening coefficient — 0.6 in the figures here — a value for macroturbulence, and the assumption that every other broadening is already known. Composition read from a spectrum needs the same model atmosphere, so an error in $\log g$ propagates into the abundances, and an error in the abundances changes the opacity and so the pressure at which the wings formed. And the rotation that emerges is never a rotation: it is $v\sin i$, an equatorial speed times the sine of an inclination nobody has measured.

The rotation that sinks below the floor #

Rotational broadening has a floor, and the floor is worth more than the method.

Below 4.00 km s⁻¹ the profile’s shape belongs to the other mechanisms. The measurement is not merely imprecise there; the quantity being fitted has stopped controlling the thing fitted to, and a better spectrum improves nothing.

That number is computed rather than quoted, from a stated budget — iron’s thermal width at 6000 K, no microturbulence, 3 km s⁻¹ of macroturbulence, the instrument at $R = 100{,}000$. The value usually quoted in the literature is “about 5”, and the difference is the budget rather than the physics: restoring the Sun’s own microturbulence of roughly a kilometre a second moves the floor up, and a lower resolving power moves it up further.

The consequence is worth sitting with. The Sun’s equator turns once in about 25 days, which at its radius is 2.0 km s⁻¹, measured from sunspot drift and from the Doppler shift of opposite limbs. Seen from far enough away to be a point source, the star whose rotation is best known in the sky would have no measurable $v\sin i$.

The fourth mechanism, which splits rather than broadens #

Three mechanisms share the profile and there is a fourth that is usually absent and, when present, dominates everything.

A magnetic field lifts the degeneracy of an atomic level: states that had the same energy no longer do, and a single transition becomes several at slightly different wavelengths. The separation is proportional to the field strength and to the square of the wavelength,

$$\Delta\lambda_B = \frac{e}{4\pi m_e c^2}\,g_{\rm eff}\,\lambda^2 B,$$

with $g_{\rm eff}$ the line’s effective Landé factor — a number of order one that differs from line to line and is calculable from the atomic structure.

Two features of that expression make it usable. The $\lambda^2$ means the effect grows quadratically towards the infrared while thermal and rotational broadening grow only linearly, so a magnetic star’s lines are conspicuously wider in the infrared than a non-magnetic star’s and identical in the blue. And the $g_{\rm eff}$ means that two lines of the same element, formed at the same depth, at the same temperature, with the same thermal and rotational widths, have different magnetic widths — so comparing a magnetically sensitive line against an insensitive one isolates the field with no model of the atmosphere required at all.

For a sunspot, with a field of about 3,000 gauss, the splitting at 1.5 microns exceeds the thermal width and the components are visibly separated rather than merely broadened. For the Sun as a whole, where the field averages to a few gauss with cancelling polarity, the splitting is far below every other width and is recovered only in polarised light — the components are circularly polarised in opposite senses, so the difference of the two circular polarisations shows a signal where the total intensity shows a smooth line.

That is the general escape from a crowded budget: when four mechanisms contribute to one width, find the observable that only one of them produces. Rotation is separated in Fourier space, magnetism in polarisation, and the difficulty of separating thermal from collisional broadening is precisely that neither of them offers such an observable.

The practical consequence is a choice of instrument rather than a choice of analysis. Detecting a field of a few hundred gauss on an ordinary star means either a high-resolution infrared spectrograph, where the quadratic wavelength dependence has made the splitting comparable to the other widths, or a spectropolarimeter at any wavelength, where the field is isolated by a property no other broadening mechanism has. Both exist, both are expensive, and the choice between them is decided by whether the star’s field is likely to be organised — a cancelling field of mixed polarity is invisible in polarisation and still broadens the lines.

The Sun is the case where both were tried and both are needed, since its disc can be resolved: a spot is measured by splitting, and the quiet photosphere between the spots is measured in polarisation and turns out to carry a tangled field of a hundred gauss or so that no unresolved measurement of the Sun as a star would ever have shown.

What the picture cannot show #

An inclination. Every rotational width here is $v\sin i$, and a slow-looking star may be a fast one seen pole-on. Averaged over random orientations $\langle \sin i \rangle = \pi/4$, so a sample can be corrected and an individual star cannot. The figures draw a kernel and cannot draw the axis it turns about.

A rounded edge. The hard edge follows from rigid rotation on a spherical, uniformly limb-darkened disc, and real stars are none of the three. The Sun’s equator turns about a fifth faster than its high latitudes, and differential rotation softens the very feature the measurement leans on hardest. The sharpest thing in the picture is the part of it that is most model.

A depth of formation. The three mechanisms do not act in one place. A strong line’s core forms high where the pressure is low and its wings far deeper where it is high, so one profile is a weighted mixture of gravities and temperatures rather than a reading of either — the limitation the curve of growth has, one level further in.

A magnetic field. Zeeman splitting of an unresolved line in a field of a few kilogauss widens it symmetrically and looks much like a slightly faster rotation. The two are separable, but not from one profile at one wavelength: the splitting grows as $\lambda^2$ and rotational broadening as $\lambda$, so the discriminator is the same line in two colours, or polarisation, and neither is here.

The three mechanisms are separated above by shape at fixed area, and there is a fourth reading of the same profiles that runs the other way: hold the mechanism fixed and ask what happens to the area as more atoms are added. That is the curve of growth, and it has three regimes. While the line is optically thin the equivalent width grows in proportion to the number of absorbers, because every atom removes its own share of photons — this is the regime in which an abundance can be read straight off a measured width. Once the core saturates the growth nearly stops: the centre of the line is already black, more atoms deepen nothing, and the width creeps up only as the square root of the logarithm. Then, when the column is large enough that the damping wings rise above the continuum’s noise, growth resumes as the square root of the abundance, because a Lorentzian wing’s area goes as the square root of its strength.

The practical consequence is that the same measured equivalent width can correspond to abundances differing by orders of magnitude depending on which regime the line is in, and the regime is decided by quantities — the pressure, the microturbulence, the temperature — that this essay’s three mechanisms are exactly about. An abundance from a single line is a shape measurement wearing a number’s clothes, and the standard defence is to use many lines of a range of strengths and require them to give one answer, which is a consistency condition on all three broadening mechanisms at once.

The same three shapes, at galactic size #

The argument generalises past a stellar atmosphere, and it generalises by shape rather than by scale. A spiral galaxy’s 21-centimetre line is broadened by the rotation of its disc, and has the same hard-edged character for the same reason — no gas in it moves faster than the flat part of the rotation curve — which is why the profile is double-horned and why its width, not its area, is the measurement. That width is then a distance, by the same two-step logic as a luminosity class. An elliptical galaxy has no such edge, because it is held up by disorder rather than by rotation, and its lines are Gaussian — a rotation kernel against a Maxwellian, the distinction surviving intact from a photosphere to a galaxy.

The stellar half has an independent check from a different observable entirely. The frequency of maximum power in a star’s comb of oscillation modes scales as $g/\sqrt{T_{\rm eff}}$, so a years-long photometric series measures the same $\log g$ a damping wing measures in one exposure — and where a wing’s curvature and a comb’s spacing disagree, it is the model atmosphere being tested.

The fast rotators were found first, and that was not an accident #

William Abney proposed in 1877 that rotation would broaden a stellar line, half a century before any instrument could test it. The test came in 1929, when Grigory Shajn and Otto Struve measured rotational velocities from line profiles and found values in the hundreds of kilometres a second.

The order of discovery followed from the floor. Those first rotators were early-type stars turning at 100 to 300 km s⁻¹, tens of times clear of the limit above, while the slow rotators — every solar-type star, which is to say almost every star — stayed unmeasured for decades because they sit within a factor of a few of it.

Morgan, Keenan and Kellman’s 1943 atlas made the other half routine at almost the same moment, turning pressure-broadened profiles into a numeral before the physics behind them was worked out.

A last mechanism belongs on the list, because it is the one that has no astrophysical content at all and is present in every measurement. The spectrograph has a profile of its own — set by the slit width, the grating and the detector’s pixels — and what is recorded is the stellar profile convolved with it. At a resolving power of a hundred thousand that instrumental width is three kilometres a second, which is comparable with the macroturbulent floor and larger than the rotation of a slowly turning star. Removing it is a deconvolution, and deconvolution amplifies noise at exactly the high spatial frequencies the narrow features live at, so the practice is to convolve the model rather than to deconvolve the data. That choice is why a published rotation velocity is inseparable from the instrument it was measured on, and why the same star observed at two resolving powers can be reported twice with different answers and no error at all in either.

Where the ladder goes next #

Two rungs are owed. The first is separating rotation from macroturbulence, which cannot be done in wavelength and can be done in Fourier space: the rotation kernel’s transform has zeros at frequencies fixed by $v\sin i$ alone and a Gaussian has none, so two kernels hopelessly degenerate in a profile come apart in its transform. The second is the profile watched over time — a spot crossing the disc distorts the line at a wavelength tracking its longitude, so a series of profiles maps a surface no telescope resolves.