Week 4: stellar pulsation and variable stars

Variable stars

Some stars are found to vary more or less sinusoidally with fairly consistent periods, but they don't have light curves like eclipsing binaries, so it must be a variation in the star itself. Some rotational modulation is possible, but variable stars tend to vary by a large degree, and with a consistent light curve that is indicative of pulsation rather than rotation. One particularly important type of pulsation is radial pulsation, where the entire star pulsates in phase, and the pulsation period is related to the free-fall time, which is also related to the sound crossing time. So we have a deviation of the force balance from the static solution and we need to include a dynamical term that is essentially Newton's "m A".

Dynamical vs. pulsational instability

A dynamical instability is when small perturbations cause the force balance to get farther and farther out of whack, so that the dynamical behavior "runs away", and you get a calamitous rapid change. This rarely happens in stars, which tend to have extremely short force equilibration timescales. When the force equilibration timescale is much less than the energy transport timescale, we have adiabatic contraction, where the release of gravitational energy during contraction easily increases the internal kinetic energy of non-relativistic gas enough to reverse the contraction. Even when a degenerate core exceeds the Chandrasekhar limit, and contracts under its own gravity, causing a supernova, or when a post-main-sequence isothermal core of helium ash exceeds about 10% of the stellar mass, exceeding the Schonberg-Chandrasekhar limit and also causing the core to go out of pressure balance and contract on the free-fall time, but causing only a "subgiant" expansion of the envelope, not a supernova. These are examples of unstable equilibria in the global force balance, so a star that is momentarily virialized that cannot stably maintain its configuration.

However, a much more subtle type of instability is also possible, called a pulsational instability, wherein you have a configuration that is in stable force balance, so tiny perturbations cause oscillation not runaway, but the oscillations receive a driving that exceeds their damping. Most stars are both in a stable force balance, and experience damping in their radial oscillations that exceeds the driving, or else are not driven at all. Such stars are both dynamically and pulsationally stable. However, in a few narrow strips in the H-R diagram, called "instability strips", we find stars that are dynamically stable but pulsationally unstable. These stars have radial modes of oscillation that are driving more strongly than they are damped, so tiny pertubations that become oscillations are eventually magnified until the whole star is beating like a heart. At this point, the oscillation saturates, because the driving can no longer exceed the damping. These are the variable stars like Cepheids, Miras, delta Scutis, and beta Cepheids.

How pulsational instability works

In order to feed energy into a pulsation mode, the key is to get the full cycle of the oscillation to do PdV work on its surroundings. A cycle that does work on its surroundings will channel energy into the mode, causing the mode to increase in amplitude, because the work will convert thermal energy (presumably drawn from the stellar luminosity) into bulk motion, and the bulk motion is the oscillation itself. It turns out that a cycle will do work if it receives heat when it is in its compressed portion of the cycle, and gives off heat when it is in its expanded portion.

There are two ways you can picture why that is the necessary ingredient to do work. One is to consider a Carnot engine, which is a cycle involving compression and expansion of a gas, whereby the gas takes heat from a hot reservoir when it is compressed, and gives heat off to a cool reservoir when it is expanded. This allows it to cycle in such a way that does net work. The reason it does net work is that you want the cycle to happen spontaneously, so it must increase the entopy of the universe. But if you take the same heat from a hot reservoir, and give it to a cooler reservoir, that implies an excess increase in entropy (corrct?)-- you don't need that much entropy increase to allow it to happen. So you don't need to drop all the heat off in the cool reservoir-- you can siphon some of that energy off as work done, and still have the total entropy increase. The efficiency of the Carnot engine is set by the need for the total entropy to not decrease, and a perfectly efficient engine pulls off as much work as will still allow enough heat to be cropped into the cool reservoir to balance the entropy lost from the hot one. So we see that to get work done by a cycle of expansion and contraction, we need to add heat in the hotter contracted phase, and expel heat in the cooler expanded phase, of the cycle.

An equivalent way to see that you need to add heat when contracted is to consider the PdV work done in a full cycle, as in the diagram in Ch. 14. When the P is higher, that's the hot contracted phase, and that's when you want it to be even hotter-- you want to add heat to raise P in that portion of the PdV integral. Then once you reach the low P expanded phase, you want to remove heat, so the return path in P-V space follows a path of lower P than the path taken during expansion. That will induce hysteresis in the P-V path, and allow a positive PdV work to be done by the cycle, just as happens in a Carnot cycle.

So, to have pulsational instability, you first need dynamical stability, so you get an oscillation instead of a runaway, but then you also need to add heat to the oscillation when compressed, and take heat out when expanded. That will drive the oscillation to higher amplitude, so even the tiniest initial perturbation will be ammplified until the whole star is beating like a heart. At that point, the oscillation amplitude saturates, because the rate that work is being done is balanced by the rate of dissipation. If no balance like that is ever struck, the star shakes apart like the Tacoma Narrows bridge (see the YouTube on that one), but this does not typically happen-- instead, the oscillation saturates at various different degrees, and we get the various flavors of variable stars like Cepheids (and the beta and delta flavors of Cepheids), delta Scutis, and Miras. There's even a pulsational type of white dwarf! The question then becomes, what mechanisms can add heat to the compressed phase, and why do they only happen in narrow strips in the H-R diagram?

The epsilon mechanism

One way to add heat to the compressed part of the cycle is to compress the core and increase the nuclear burning rate. However, if that were the answer, it would be a ubiquitous aspect of main sequence stars, whereas the only main sequence stars that pulsate are the delta Scuti stars (which can also be giants, like delta Scuti itself). What's more, radial oscillations do not extend into the central region, just because of the spherical symmetry and high density in the core, so there's very little compression of the core and little extra fusion. The epsilon mechanism is not generally the explanion for variable stars.

The kappa mechanism

Another way to get extra heating during a compressed phase of the star is if the opacity rises upon compression. That would tend to trap the stellar luminosity, and drive the oscillation. However, the opacity due to bound electrons in metals is normally governed by "Kramers opacity", which means it is proportional to density, and inversely proportional to temperature to the 3.5 power. Physically, the Kramers form comes from the fact that high density and low temperature both tend to cause metal recombination, increasing the availability of bound electrons to generate opacity in the metals (the opacity is generally not from hydrogen or helium, despite their abundance, because they lack prominent lower-energy transitions, and they tend to get stripped of their electrons before they are important for blocking the stellar continuum). The strong temperature sensitivity (from that 3.5 power) of this form means that when the gas is hot and compressed, its opacity is less, not more, because the temperature dependence wins out over the density dependence, and the metals lose bound electrons, so this will not drive oscillations. Again, that is why most stars do not show large radial pulsations.

However, sometimes this trend can be reversed, and the opacity can rise when the gas is compressed and heated. This is when much of the compressed gas is in a partially ionized zone of hydrogen or helium, which are both abundant enough to require a lot of energy to ionize. When partially ionized gas is compressed, the need to increase the ionization of the hydrogen or helium has a thermostatic effect, much as the need to melt ice will keep ice water from raising its temperature when you heat it. If the increasing ionization in hydrogen or helium keeps the temperature from changing too much, the increased density will actually cause the metals to recombine, even as the hydrogen and helium is ionizing. That increases the opacity and traps radiation, adding heat to the compressed gas in just the way needed to drive pulsations. This is called the kappa mechanism simply because kappa is the common symbol for cross section per gram.

Note we can now understand why the instability strips are so narrow-- the depth of the partially ionized zones depend on the surface T of the star, and if the zones are too high up, they fall in the convective regions of stars hotter than late F type, and the stellar luminosity is not carried by radiation in those regions so the kappa mechanism is not effective (for really hot stars, there is no surface convection, but hydrogen and helium are always highly ionized). For cooler stars than F type, the partially ionized zones fall too deep in the star, where the pulsations only weakly penetrate, and the driving is too weak to amplify the pulsation.

So it is pretty much F type main-sequence stars (or F and G giants) that have ionization zones at the right depths to drive pulsations, although there are some hotter (beta Cepheid) and cooler (Mira) stars that can also pulsate, because there can also be other reasons why the opacity increases upon compression. For example, in the beta Cepheid stars (which are B stars, either main-sequence or giants), the oscillation is driven by the "iron opacity bump", which is an increase in iron opacity when the temperature is between 100,000 K and 200,000 K, breaking from the overall behavior of Kramers opacity and allowing the opacity to rise when the T rises under the appropriate circumstances (suitable to B stars). This type of star is also given to nonradial pulsations, which are even more complicated and somewhat rare.

The gamma mechanism

Although the kappa mechanism goes a long way toward explaining variable stars, it receives help from a subtle effect whereby an ionization zone that is undergoing a thermostatic maintenance of its temperature will, during the compressive phase, find itself surrounded by hotter gas that did not contain partially ionized gas so had no thermostat. That hotter environment for the partially ionized zone causes conduction (by radiation) into the partically ionized zone, and since the partially ionized zone already has to be situation in the sweet spot of the pulsation, this adds additional heat to the oscillating gas during times of compression. So that further helps to drive the pulsational instability, and is called the gamma mechanism because gamma is the symbol that determines how much the temperature varies when you compress the gas adiabatically.

The significance of variable stars

The significance of variable stars is that their pulsational timescale is closely related to their sound crossing time, which is also similar to their free-fall timescale. Simple Newtonian physics indicates that the free-fall time depends only on the density, like density to the minus 1/2 power. So by looking at the pulsation rate, we can tell the density of the stellar envelope, and determine if the star is a dense dwarf or a huge giant or supergiant. When properly callibrated, this leads to a "period-luminosity relationship," and if we know the star's intrinsic luminosity, we can infer its distance by observing its apparent brightness. So variable stars are distance indicators, and indeed Hubble used Cepheids to determine that Andromeda was so far it had to be its own galaxy, and to determine the Hubble law.


The Sun is not given to pulsational instability because it is cool enough that its last helium ionization zone is too deep to drive it. But the Sun contains resonant cavities in its envelope where sound waves can get reflected and trapped, leading to global oscillation modes like a musical instrument. Trapped waves set up standing oscillations that are of low amplitude because they are not driven, yet are still observable by careful photometry because sound waves are compressive and create brightenings at the photosphere. Close analysis of these brightenings are a bit like using a seismometer to study earthquakes, and tell us about the propagation speed of sound waves inside the Sun. That in turn gives depth-dependent temperature information, because of the dependence of sound speed on temperature. It also gives differential rotation information, owing to the Doppler shifts of sound waves propagating with and against the rotation at each depth, which slightly split the frequencies of those global modes. Helioseismology has helped us model the interior of the Sun, and with accurate photometry of other stars offered by Kepler, promises to be extended to many other stars as well.

In addition to the pressure waves ("p" modes), there is another type of oscillation that is important for learning about the Sun, and that is the "g" modes. These are internal gravity waves whose restoring force is buoyancy, and unlike the p modes, depend on the local gradients rather than just the local sound speed. Sound waves have a common propagation speed, but gravity waves have a common frequency, called the "Brunt-Vaisala frequency", which is kind of a gravitational analog to the plasma frequency in plasma oscillations in the sense that it is a kind of "spring" that is built right into the gas and produces sloshing at that frequency. The explicit dependence on gradients appears in the form of a dependence on the "polytrope exponent" (given by the logarithmic derivative d log pressure / d log density, where this refers to gradients in the plasma, not variations due to a local perturbation or the passage of a wave). When this exponent is smaller than the adiabatic index (which is d log pressure / d log density referring to local adiabatic perturbations in pressure and density), there is a restoring force on any gas that is adiabatically displaced, and this restoring force sets up oscillations at the Brunt-Vaisala frequency, allowing for global gravity modes. The B-V frequency is given by the ratio of the local gravity to the local sound speed, times the square root of the quantity, the ratio of the local polytrope exponent to the adiabatic index, minus 1, end square root. In addition to being capable of diagnosing the local polytrope exponent, g modes penetrate deeper into the star, and hence give complementary information to the p modes, which largely report on the surface layers and convection zone of the Sun. However, g modes are more difficult to observe.