Cepheid variable stars: the perfect natural oscillators

What is a Cepheid variable star?

Cepheid variable stars as any astronomers here know are God’s gift to cosmology – almost a design argument by themselves since they’re so incredibly useful to human star-gazers. Why? – because they tell you exactly how far away they are. Cepheids pulsate in brightness, and the frequency of their pulsation (in the range of days to months) is nicely correlated to their size. So if you see a Cepheid anywhere in the sky, measure its pulsation rate and you have its size (actual luminosity). Combine this with its apparent brightness and bingo – you know its distance from us. Priceless information for mapping the size and layout (and expansion) of the visible universe.

Cepheid variable stars were discovered by  Henrietta Swan Leavitt, publishing a major paper on the subject in 1908, where she deciphered the relationship between their oscillation rate and size and their resulting enormous utility for cosmic mapping. In fact, Cepheids helped to reveal the vast scale of the universe beyond our own local galaxy, the Milky Way – a step forward in understanding the nature and magnitude of the universe on a level with the discovery of Copernicus and Galileo that the earth was not the center of everything. This made our knowledge vastly greater, and ourselves so much smaller.


Cepheid variables are one of the handful of key discoveries that led to the understanding of the scale and layout of the universe – along with the red shift, the microwave background and one or two others.

It struck me that Cepheid variable stars were a nice example of a natural system that oscillates from internal dynamics, not requiring any external forcing. What external forcing could a star possibly experience? Unless it happened to be next door to a pulsar or another Cepheid – which would be rare. (But then – how would that object be pulsating …?) So the strong and regular oscillations in diameter and brightness of Cepheids must be a pure example of oscillation from internal nonlinear dynamics.

Why is a Cepheid variable variable?

The mechanism of Cepheid pulsation is nicely explained at the wiki page:


It’s about the different opacity of the different ions of helium. Helium at solar temperatures can lose either one (He+) or both (He2+) of its electrons. The electron-bald He2+ ion needs higher temperatures to form than He+, and is also more opaque to radiation. These two facts together make helium ionisation in a Cepheid Star take on the nonlinear pattern characteristics of a Turing reaction – one that becomes oscillatory in time. Here’s the oscillation sequence:

1. Radiative heating forms He2+ in the star’s exterior

2. He2+ ions have higher opacity, so trap more radiation, heating itself up further

3. The increasing temperature forces expansion of the star

4. Expansion diminishes radiation intensity, so cooling occurs and some of the He2+ reverts to the more transparent He+.

5. Greater transparency lets more radiation escape, further increasing the cooling

6. The cooling star contracts

7. Contraction once again increases radiation intensity to the star’s peripheral helium, heating it back up.

Return to 1 …

The positive feedback by which He2+ ions drive heating of the Cepheid star’s exterior, is self-terminating, leading to cyclic oscillation. The operation of the opacity driven heating feedback sets in motion processes – most notably expansion of the star – which counteracted the heating by diminishing radiation intensity in the cloud of helium. So it did not lead to endless warming but instead it was self-cancelling and led to a regular oscillation, a series of short runs of positive feedback each one self-terminating and reversing to a starting position.

Excitability and friction

This is an important general insight into the role and effect of positive feedbacks in complex nonlinear systems. It is well known to chemists who study nonlinear oscillatory systems such as the Belousov-Zhabotinsky thin film reaction [ref. 1] and the surface catalysed oxidation of CO on a platinum substrate (what happens in a car exhaust’s catalytic converter) [ref. 2]. Positive feedbacks in such complex-chaotic systems are always intermittent and set in motion their immediate reversal, cyclically, leading to regular monotonic oscillation. Negative feedbacks by contrast dampen regular monotonic oscillations (they are referred to as friction or damping) and cause emergence of more complex chaotic pattern. The combination of positive and negative feedbacks in a complex-chaotic system result in the elusive apparent oscillations that feint at certain patterns or frequencies but are nested in chaos. These patterns don’t necessarily come from external forcing. They can arise from the Feigenbaum numbers of pure chaos mathematics [ref. 3]

The Cepheid variable star is a perfect example of positive feedback leading to monotonic oscillation in a dissipative out-of-equilibrium nonlinear system. It never reaches equilibrium and its predominantly positive feedback landscape makes its oscillatory pattern monotonic rather than chaotic-quasi periodic. This is great for astronomers – the Cepheids’ oscillation period is related to their size, which combined with their apparent brightness, establishes their distance and allows mapping of distant star systems and galaxies.

From stars to oceans

In the terminology of chaos dynamics, the positive feedbacks in a complex system that energise oscillations are said to make the system an “excitable medium”. The ocean is an excitable medium. The earth’s ocean-driven climate system contains well-known examples of oscillatory systems driven intermittently by positive feedbacks. One is ENSO, where the excitatory positive feedback is the Peruvian upwelling-trade wind mutual reinforcement. This operates on an annual to decadal time scale – exceptionally fast for the ocean. The (intermittent) positive feedback loop at the core of ENSO that makes the equatorial east Pacific an excitable medium, is:

La Niña:
1. Peruvian upwelling strengthens leading to east Pacific surface cooling;

2. Surface temperature gradient east-west (cooler east warmer west) strengthens the trade winds since cold sea surface water increases air pressure;

3. East to west blowing trades impel more upwelling at the Pacific eastern margin (Peru);

Back to 1

El Niño:
1. Peruvian upwelling falters

2. East equatorial Pacific surface water warms

3. East-west surface temperature gradient is weakened, diminishing or in extreme cases (e.g. 1998) reversing the trade winds;

4. Weakened trades add to weakening of Peruvian upwelling;

Back to 1

The timescale of ENSO is so short since although ocean circulation and vertical mixing are a component, it also involves atmospheric circulation with its much shorter and more evanescent process times.


The El Niño oceanic oscillation was discovered and named by Peruvian anchovy fishermen, whose livelihoods are closely bound up with the ocean’s vertical mixing and its strong effect on the abundance of the anchovy.

Another more purely oceanic nonlinear oscillation with a much longer century to millennial timescale is the AMOC (Atlantic Meridional Overturning Circulation). This gives rise to what colloquially is called the “Gulf Stream”. While this is running in its strong phase, it enhances transport of ocean heat from the tropical Atlantic to the Arctic, causing both Arctic and northern hemisphere (NH) warming. Its weakening causes both to cool. At the heart of AMOC is another intermittent positive feedback involving salinity and downwelling. It works like this:

1. The Gulf Stream brings high salinity water from the Caribbean to the North Atlantic;

2. When it cools between Norway and Greenland its higher salinity makes it exceptionally dense so that it downwells all the way down to the ocean floor;

3. This deep cold dense water flows south, completing the loop of the AMOC;

4. South flowing bottom water in turn propels the northward Gulf Stream up on the surface, reinforcing the whole circuit with positive feedback;

Back to 1

Now if this salinity-downwelling AMOC positive feedback is so effective, why is it intermittent? Why does it stop? Why should it not endlessly reinforce itself and strengthen, until the whole North Atlantic comes to resemble a swirling maelstrom like the scene at the end of the movie “Pirates of the Caribbean – at World’s End”? The reason is that the enhanced transport of warm equatorial water poleward and resultant warming of the North Atlantic and Arctic sets in motion a process that will oppose and weaken the AMOC. What is this opposing feedback (or damping, or friction)? Greenland ice melt. Warming water around Greenland increases ice melt resulting eventually in a surface floating raft of fresh meltwater that interferes with the process of downwelling of cold high-saline super dense water to the ocean floor, and thus choking off the deep water formation.

So in this case Greenland ice melt is the negative feedback that reacts to choke off runs of the positive AMOC feedback. The result is oscillatory behaviour of the AMOC. For instance, a moderate alternation between stronger and weaker phases of the Gulf Stream that we refer to as the Atlantic Multidecadal Oscillation or AMO. We are currently just past the peak of the AMO and beginning the downside of the wave to a weakening Gulf Stream.

This self-reversing behaviour of the AMOC is most starkly visible if we look back to ice age times. The Dansgaard-Oesger dramatic episodes of northern hemisphere warming followed by cooling over a few centuries, that happen periodically during glacial periods, are an example of such a self-terminating excursions of the AMOC.

(The last of the D-O events in the last glacial period has been called the Bolling-Alerod, and the 1000-year interval between it and the subsequent inception of the Holocene, has been called the Younger Dryas.)


The essential insight provided by Cepheid variable stars is that, in a complex-chaotic system, feedback never leads to a simple linear unidirectional change. That’s why they’re called nonlinear. Going back to the case of CO2, one could argue “yes, but here there is a steady linear increase in a system parameter – CO2 concentration. This should lead to a corresponding steady linear change in the system as a whole, with nonlinear oscillations just adding noise and wiggle around the long term unidirectional trend.” But this response is a superficial dismissal of nonlinear dynamics, and a failure to grasp at how deep a level they operate. One must think of a complex nonlinear dynamic system as a multi-dimensional phase space. Every system variable is a dimension, including CO2. By moving a parameter like CO2, one moves the whole system in its multi-dimensional phase space. What will the result of this be? To claim to know this is somewhat arrogant, it’s fundamentally hard to predict the effect of perturbing such a complex system. However, from anecdotal experience of how such systems behave, we can say that the least likely outcome is a unidirectional change of the system as a whole. Such systems tend to gravitate toward a fixed point in the multi-dimensional phase space – the attractor. These attractors often prove robust, and movement from the attractor triggers feedbacks that chaperone the system back to the attractor.

(With CO2, an important question is of course “how significant a player is it? How strong is it’s radiative warming effect?” But that Pandora’s box is not the point of this article!)

Every point in the phase space however has a different dynamic, so moving a chaotic-nonlinear system can have unexpected results. To quote Murray Walker, the popular former TV commentator for Formula One car racing, “anything can happen – and it usually does”.

But an important generalisation from the systems looked at above – Cepheid variables, ENSO and AMOC oceanic oscillations, is that they are just that – oscillations. Such complex systems are robustly drawn to an attractor and will tend to oscillate around it. Moving the system away from the attractor sets in motion system changes and feedbacks that return the system back to the attractor. That’s why it’s called an attractor.

Scientists such as Ferenc Miskolczi and Richard Lindzen have proposed something similar. Namely, that changes to atmospheric radiative dynamics caused by CO2 will move the atmosphere in its phase space, and in so doing set in motion processes, such as changes to the emission height, or interactions with the much stronger greenhouse gas water vapour, that will oppose and eventually reverse any CO2 induced changes. This is exactly the behaviour that we would expect of a complex-chaotic nonlinear system.

I’m not saying that the attractor-seeking Lyapunov-stability of the climate system means that it can never move in response to forcing. It clearly does, most obviously in the Milankovitch forcing of the glacial-interglacial cycle over the current Pleistocene epoch. Here – as Javier has shown – the dominant forcing is the obliquity cycle (lagged by 6500 years by the oceans’ “thermal inertia”).

But is the climate’s response to obliquity forcing immediate and passive? That whatever obliquity does, global temperature and glaciation will obediently do also? No – the Pleistocene climate record is not smoothly sinusoidal, mirroring obliquity for instance, but is ragged and chaotic, showing sharp alternation between two attractors, glacial and interglacial. There is a rhythm there – but it is elusive. Is it following eccentricity? Or obliquity, or precession? Or some combination of all three? Again this is exactly what we expect from a complex-chaotic nonlinear system in which there is a constant tension between excitability and friction; that is, between positive and negative feedbacks. The system feints intermittently at certain rhythms and frequencies, giving hope to those who would explain all climate’s ups and downs as a consequence of astrophysical inputs. But the carrot of apparent wiggle-correlation, once dangled, is snatched away, and chaos returns.

There are natural nonlinear systems which do oscillate with a pure monotonic frequency. The Cepheid variable stars are a nice example, where feedbacks are dominated by a single strong positive feedback between He2+ ion formation and the star’s thermal expansion-contraction. But in earth’s climate there are a host of feedbacks of which a significant number are negative, providing frictional damping which cause the emergence of complex pattern.

This makes the analysis of the apparent rhythmicity of the climate frustratingly elusive. It will also disappoint expectations that the climate will change in a steady unidirectional way in response to a steady unidirectional forcing. It won’t.


1. Lin AL, Bertram M, Martinez K, Swinney HL, Ardelea A, Carey GF. Resonant phase patterns in a reaction-diffusion system. Physical Review Letters. 2000 May 1;84(18):4240.

Click to access 1063306441.pdf

2. Kim M, Bertram M, Pollmann M, von Oertzen A, Mikhailov AS, Rotermund HH, Ertl G. Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic CO oxidation on Pt (110). Science. 2001 May 18;292(5520):1357-60.


3. Testa J, Pérez J, Jeffries C. Evidence for universal chaotic behavior of a driven nonlinear oscillator. Physical Review Letters. 1982 Mar 15;48(11):714.

Click to access qt18m504mf.pdf

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