The valve that has to sit at the right depth

A Cepheid’s period predicts its luminosity, and that relation is the rung on which the distance ladder’s whole middle section stands. It is an empirical fact, discovered by Leavitt in 1908 by plotting periods against magnitudes for stars in the Small Magellanic Cloud, and it works.

The question this rung answers is why such a relation can exist at all — and, having answered it, why only stars in a narrow band of the Hertzsprung–Russell diagram obey one.

A period is a density #

The dimensional argument is short and does all the work.

A pulsation is a sound wave crossing the star and returning. The crossing time is $R/c_s$, and in a body held up by its own gravity the pressure is of order $G M^2/R^4$ and the density $M/R^3$, so $c_s^2 \sim P/\rho \sim GM/R$. The period is then

$$\Pi \sim \frac{R}{\sqrt{GM/R}} = \sqrt{\frac{R^3}{GM}} \sim \frac{1}{\sqrt{G\bar\rho}}.$$

Nothing about the star’s composition, temperature or internal structure appears. The period of a pulsating star is a measurement of its mean density and of nothing else, up to the dimensionless constant $Q$, which depends only on which mode is being excited and weakly on the run of density inside.

That is what makes the period–luminosity relation possible. A mean density is a mass and a radius; a radius with a temperature is a luminosity, through $L = 4\pi R^2\sigma T^4$; and if the pulsators all lie in a narrow band of temperature and follow a mass–luminosity relation, then the period and the luminosity are related by two substitutions.

The relation is therefore not a coincidence and not a calibration. It is a period–mean-density relation composed with two others, and each of the three carries its own scatter, which is why the observed relation is tight but not perfect and why it has a colour term.

The engine #

A star that pulsates is a heat engine, and a heat engine needs a working substance that does the right thing at the right moment in the cycle.

Consider a layer being compressed. Compression heats it, and a hotter layer radiates more freely; so the energy flowing through the layer increases at maximum compression and the layer loses the heat that would have driven it outwards again. That is damping, and it is what almost every layer of almost every star does.

The exception is a layer whose opacity rises when it is compressed. Then compression dams the flux, the layer absorbs energy at maximum compression, and it pushes outwards harder than it was pushed inwards. Over a cycle it does net work on the layers above it. That is the κ-mechanism — Eddington’s “valve”, proposed in 1926 and identified with a specific layer thirty years later. The mechanism therefore requires a partial ionisation zone: a layer in which the gas is neither fully neutral nor fully ionised. Compressing such a layer ionises more of it, and the energy that would have raised the temperature goes into the ionisation instead. The temperature barely rises, the density does, and the opacity — which rises with density and falls with temperature — goes up.

There are two such zones in a normal stellar envelope. Hydrogen and the first ionisation of helium overlap in a region near 11,000 to 15,000 kelvin; the second ionisation of helium sits near 40,000 kelvin, deeper down. The second is the one that drives the classical Cepheids, and the opacity model in the figure above does not resolve it — the mechanism is exactly the one drawn, at a zone this particular calculation does not contain.

Why the strip is a strip #

The mechanism is available in every star that has a partial ionisation zone, which is every star cooler than about 20,000 kelvin at the surface. Almost none of them pulsate. The reason is where the zone sits.

A star’s surface temperature fixes the depth at which any given interior temperature is reached. Make the star hotter and the 40,000-kelvin layer moves outwards, towards the surface; make it cooler and the layer sinks.

The blue edge is where the zone has moved so close to the surface that there is almost no mass in it. The driving is proportional to the mass of the driving layer, and a layer with nothing in it drives nothing; above that temperature the damping from every other layer wins, and the star is stable. That boundary is computable, and computed blue edges agree with the observed ones to a couple of hundred kelvin.

The red edge is not computable in the same way, and the reason is instructive. As the star cools, the zone sinks and gains mass, so the driving strengthens — and simultaneously the envelope becomes convective, and convection carries the flux around the valve. A layer whose radiative flux has been short-circuited by convection cannot dam anything. That is worth stating plainly because it is unusual. The blue edge is a boundary of a well-posed calculation and the red edge is a boundary of the theory: it is set by time-dependent convection, and every published red edge depends on a treatment of that which its own authors describe as provisional. One side of the instability strip is physics and the other is a modelling assumption, and the observed width of the strip is one of the few constraints available on the assumption.

The second condition, and why the zone cannot be too shallow either #

There is a second requirement on the driving zone, distinct from the opacity condition, and it is the one that makes the argument quantitative.

The layer must not only dam the flux; it must dam enough of it to matter. The work done per cycle is proportional to the mass of the driving layer times the fractional flux modulation, and the damping it has to overcome is contributed by the whole envelope. So the driving zone must contain a non-negligible fraction of the envelope’s mass.

There is a third condition too, and it is about timing. The heat has to be stored and released in phase with the compression — which means the layer’s thermal timescale must be comparable to the pulsation period. A zone so shallow that it thermalises in seconds simply follows the compression and stores nothing; a zone so deep that its thermal timescale is millennia never notices the pulsation at all. The driving region is bracketed above by mass and below by thermal inertia, and the two brackets are what confine the whole phenomenon to a strip rather than a half-plane.

What crosses the strip, and how often #

A star does not choose to be in the strip. It passes through it, and how many times and how quickly is a question about evolution rather than about pulsation.

A star of five solar masses leaves the main sequence, expands and cools, and crosses the strip within a few thousand years — far too quickly to be caught. It then settles into core helium burning, and during that long phase it executes a blue loop: the surface temperature rises, the star crosses the strip a second time going blueward, and crosses it a third time coming back. The second and third crossings take of order a hundred thousand years each, which is a thousandth of the star’s life and is why Cepheids exist at all. The rate of change of the period during the crossing is measurable, and it is one of the more direct tests available on stellar evolution. A Cepheid crossing blueward is contracting, so its mean density is rising and its period is falling, at a few seconds per century. Polaris’s period has increased by about eight seconds in a century — it is on a redward crossing — and δ Cephei’s has decreased slowly. A period change is a direct measurement of a star’s radius changing, on a timescale of decades, and there are very few of those.

What the strip contains #

The band is not populated by one kind of star. What it contains at a given point is decided by the star’s mass and its evolutionary state, and the pulsation constant separates them.

Classical Cepheids sit at high luminosity — masses of 4 to 12 solar, periods of 1 to 100 days, $Q$ near 0.04 for the fundamental mode. RR Lyrae stars sit lower, at 0.6 to 0.8 solar masses on the horizontal branch, with periods of 0.2 to 1 day and $Q$ near 0.035. δ Scuti stars sit at the strip’s intersection with the main sequence, at 1.5 to 2.5 solar masses, with periods of hours.

All three are the same mechanism at the same zone in stars of very different masses, and the pulsation constant is what says so. A star at a given position in the strip has a definite radius and temperature; its period then depends only on its mass, through the density. Two stars at the same place in the diagram with different periods have different masses, and the period is the mass measurement.

The relation’s slope and its zero point are two different quantities and they fail in two different ways, which is worth drawing.

Two periods at once, and the mass they give #

A star pulsating in one mode gives one period, and one period is one equation. Some pulsate in two at the same time, and two periods are two equations — which is enough to solve for something that is otherwise unobtainable.

The modes in question are the fundamental and the first overtone: the whole star breathing, and the whole star breathing with one stationary surface partway out. Their period ratio depends on how the density is distributed through the star, and therefore on the star’s mass and radius, in a way a single period cannot separate.

For the double-mode Cepheids the ratio sits near 0.70, and reading a mass off it was one of the standard uses of the theory. It produced, for two decades, an embarrassment: the mass discrepancy. Masses derived from pulsation came out systematically lower than masses derived from evolutionary tracks for the same stars — by twenty to forty per cent, far outside either method’s stated errors.

The resolution was not in either calculation. It was in the opacity tables both of them used. New computations of atomic opacities in the early 1990s found substantially more absorption around two hundred thousand kelvin than the older tables had, from the many bound–bound transitions of partially ionised iron — an element present at a thousandth of the abundance of hydrogen and dominating the opacity there because it has so many available transitions.

With the revised opacities both masses moved and the discrepancy largely closed.

That episode is worth carrying because of what was wrong. Neither the pulsation theory nor the stellar evolution theory was at fault; a table of input physics was, and it had been used by everybody, so the two methods disagreed while sharing the error that caused it. Agreement between two calculations is only evidence when they do not depend on the same tables.

The other things that pulsate #

The instability strip in this essay is defined by one mechanism operating in one ionisation zone, and it is worth naming the neighbours, because the diagram has several pulsating populations and they do not share a cause.

δ Scuti stars sit at the strip’s low-luminosity end, driven by the same helium valve at shorter periods — hours rather than days — and often in many modes at once.

β Cephei stars are hot and massive and lie nowhere near the strip. Their driving is the iron opacity bump of the previous section rather than helium ionisation: the same kind of valve, in a different material, at a different depth and temperature.

Solar-like oscillators, including the Sun, are not valves at all. Their modes are damped, and they are excited stochastically by the convection beneath them — a broadband noise source ringing every resonance it can reach, which is why the amplitudes are tiny and the frequencies are many.

The distinction matters for what each population is good for. A valve-driven star pulsates in one or a few modes at large amplitude, which makes it easy to find and gives a clean period; a stochastically excited star rings in hundreds of modes at amplitudes of centimetres a second, which makes it hard to detect and gives, once detected, a far richer record of its interior.

The instability strip is therefore not the pulsating part of the diagram, only the part where one particular mechanism runs away — and the diagram’s other pulsators were each discovered separately and explained separately before it was noticed how much of the machinery they share.

There is a practical corollary for anybody reading a light curve. A star pulsating in one large-amplitude mode has a light curve that repeats and is described by a period and a shape; a star ringing in hundreds of modes has one that never repeats and is described by a power spectrum. Those are different analyses on different instruments, and a survey designed to find one is nearly blind to the other — which is why the two populations were catalogued a century apart despite being on the same diagram.

The same split runs through what each population is used for: the large-amplitude pulsators are distance indicators, found by the thousand in other galaxies, and the stochastic ones are interior probes, measured one at a time on stars bright enough to be worth the telescope time.

Both uses are downstream of the same question this essay asks, which is where in a star a layer can act as a valve — and the answer decides not only whether the star pulsates but what the pulsation is good for.

Where the model stops #

Four limits.

The pulsation constant is not quite constant, and the figure shows it: $Q$ drifts from 0.026 at the shortest period to 0.054 at the longest, a factor of two across a factor of five hundred. Some of that is real — $Q$ depends weakly on the density concentration, which changes along the sequence — and some of it is the masses, which for the Cepheids come from evolutionary models rather than from any dynamical measurement.

The pulsation is treated as linear, and the observed light curves are not. A Cepheid’s light curve is sawtoothed, with a rapid rise and a slow decline, and around a period of ten days it develops a secondary bump whose phase moves systematically — the Hertzsprung progression, which is a resonance between the fundamental mode and the second overtone and which a linear theory cannot contain.

The strip’s red edge depends on the convection theory, as above, and different treatments move it by several hundred kelvin.

And the whole picture is for radial pulsation, in which the star remains spherical. Most pulsating stars, including the Sun, pulsate non-radially in thousands of modes simultaneously, and for those the period is not a mean density but a probe of the interior structure at a particular depth. Two more readings separate what the relation asserts from what the sample happens to contain.

Where this ladder goes next #

This rung establishes why a period is a density, why only a narrow band of stars has one, and which of the strip’s two edges is a calculation.

Above it lies non-linear pulsation: the amplitude, which linear theory cannot predict at all, is set by where the driving is balanced by dissipation, and computing it requires following the shock that forms in the outer envelope every cycle.

Beside it lies the use of the strip as a test. The instability boundaries depend on the opacity, and the discovery in the early 1990s that the OPAL opacities were substantially larger than their predecessors near 200,000 kelvin — the iron-group bump — resolved a long-standing failure to explain the β Cephei and slowly pulsating B stars, whose driving zone is that bump. A class of pulsators that theory could not produce turned out to be evidence about atomic physics, which is an unusually direct payment from one subject to another.