3.4 Stellar Pulsations and Variable Stars

Pulsating Variable Stars

Some stars aren't static. They pulsate, changing in brightness over regular or semi-regular timescales. These variable stars serve as both probes of stellar interiors and critical rungs on the cosmic distance ladder. Understanding why and how stars pulsate connects directly to their evolutionary state, internal structure, and the physics of opacity and energy transport.

Cepheid Variables and RR Lyrae Stars

Cepheid variables undergo periodic expansion and contraction of their outer layers, producing regular brightness changes. Their pulsation periods range from about 1 to 100 days, and crucially, period correlates with intrinsic luminosity. This makes them powerful standard candles for extragalactic distance measurements.

RR Lyrae stars pulsate on shorter timescales, typically 0.2 to 1 day. Compared to Cepheids, they are:

• Lower mass (roughly 0.6–0.8 $M_\odot$)
• Older, metal-poor Population II stars
• Commonly found in globular clusters and the galactic halo
• More uniform in absolute magnitude ($M_V \approx +0.6$), which makes them useful distance indicators even without a tight period-luminosity slope

Both classes occupy the instability strip on the HR diagram, but at different luminosities. Cepheids are supergiants crossing the strip during post-main-sequence evolution, while RR Lyrae stars are horizontal branch stars burning helium in their cores.

Delta Scuti and Beta Cephei Stars

Delta Scuti stars sit on or just above the main sequence, with masses of roughly 1.5–2.5 $M_\odot$. They pulsate with short periods (0.03–0.3 days) and typically exhibit multiple simultaneous pulsation modes, both radial and non-radial. Their location at the intersection of the instability strip and the main sequence makes them accessible asteroseismic targets.

Beta Cephei stars are hotter and more massive (8–20 $M_\odot$), with pulsation periods of about 0.1–0.3 days. Their pulsations are driven by the kappa mechanism operating on the iron opacity bump near $T \sim 2 \times 10^5$ K. This is a different driving zone than the helium ionization layers responsible for Cepheid and RR Lyrae pulsations.

Both classes provide valuable constraints on stellar interiors through asteroseismology, but they probe very different mass and evolutionary regimes.

Slowly Pulsating B Stars

Slowly Pulsating B stars (SPB stars) are B-type stars with masses of 3–8 $M_\odot$. They show non-radial gravity-mode (g-mode) pulsations with longer periods of 0.5–5 days. Like Beta Cephei stars, their pulsations are driven by the kappa mechanism acting on the iron opacity bump.

The key distinction: Beta Cephei stars predominantly exhibit pressure modes (p-modes), while SPB stars exhibit gravity modes. G-modes penetrate deeper into the stellar interior, making SPB stars particularly valuable for probing near-core regions, including convective core boundaries and internal mixing processes. Many SPB stars show multiple pulsation modes simultaneously, which strengthens the asteroseismic constraints you can extract.

Pulsation Mechanisms

Radial and Non-radial Pulsations

Radial pulsations involve the entire star expanding and contracting spherically. The star breathes in and out while maintaining its spherical shape. These are the simplest modes and are characterized by the number of radial nodes (shells within the star where the displacement is zero). The fundamental mode has no nodes; the first overtone has one, and so on.

Non-radial pulsations are more complex. Different parts of the stellar surface move in different directions simultaneously. These modes are described by spherical harmonics with quantum numbers $\ell$ (angular degree) and $m$ (azimuthal order):

• p-modes (pressure modes): restoring force is pressure. These have higher frequencies and are most sensitive to conditions in the outer layers.
• g-modes (gravity modes): restoring force is buoyancy. These have lower frequencies and probe deeper into the stellar interior.
• Mixed modes can occur in evolved stars, behaving like g-modes in the core and p-modes in the envelope. These are especially powerful diagnostics of core conditions.

Many variable stars exhibit both radial and non-radial pulsations simultaneously, producing complex light curves that require Fourier decomposition to interpret.

The Instability Strip and Driving Mechanisms

The instability strip is a nearly vertical region on the HR diagram, running from the main sequence up through the supergiant branch at effective temperatures of roughly 6,000–8,000 K. Stars crossing this region become pulsationally unstable.

The primary driving mechanism for classical pulsators (Cepheids, RR Lyrae, Delta Scuti) is the kappa mechanism ($\kappa$-mechanism). Here's how it works:

1. A star in the instability strip has partial ionization zones (primarily He II $\rightarrow$ He III) at a critical depth in its envelope.
2. When the star contracts, these layers are compressed and heated. Instead of becoming more transparent (as most gas does), the partially ionized material becomes more opaque because the increased temperature drives further ionization, increasing the number of absorbing particles.
3. The increased opacity traps radiative energy, building up pressure.
4. This excess pressure drives the layers outward, causing the star to expand.
5. As the star expands and cools, the opacity drops, the trapped energy escapes, and the star falls back inward.
6. The cycle repeats, sustaining the pulsation.

The gamma mechanism ($\gamma$-mechanism) is a related effect: in partial ionization zones, absorbed heat goes into ionization rather than raising the temperature, which changes the local adiabatic exponent ($\gamma$). This makes the gas more compressible and contributes to the driving.

For Beta Cephei and SPB stars, the same kappa mechanism operates, but in the iron opacity bump at $T \sim 2 \times 10^5$ K, where a forest of iron-group spectral lines produces a local opacity peak.

Period-Luminosity Relation

The period-luminosity (P-L) relation is what makes pulsating variables so important for cosmology. Henrietta Leavitt established this relation in 1912 by studying Cepheids in the Small Magellanic Cloud, where all stars are at approximately the same distance.

The relation takes the form:

$M_V = a \log P + b$

where $M_V$ is the absolute visual magnitude, $P$ is the pulsation period in days, and $a$ and $b$ are empirically calibrated constants. For classical Cepheids, $a \approx -2.8$.

The physical basis: more luminous Cepheids have larger radii and lower mean densities. Since the pulsation period scales roughly as $P \propto \rho^{-1/2}$ (the period-mean-density relation), more luminous stars pulsate more slowly.

Different classes of pulsating variables follow distinct P-L relations. This means you need to correctly classify a variable star before using it as a distance indicator. The Cepheid P-L relation is a foundational calibrator for the cosmic distance ladder, linking geometric distance methods (parallax) to extragalactic scales.

Stellar Oscillations and Structure

Asteroseismology: Principles and Techniques

Asteroseismology uses observed oscillation frequencies to infer the internal structure of stars. The core idea is that oscillation modes propagate through the stellar interior, and their frequencies depend on the local sound speed, density, and composition along their paths. Different modes sample different depths, so a rich set of detected frequencies lets you build a profile of conditions throughout the star.

Observationally, stellar oscillations appear as tiny periodic variations in brightness (photometry) or surface velocity (spectroscopy). Amplitudes can be extremely small: solar-like oscillations produce brightness variations of only a few parts per million.

The analysis pipeline typically involves:

1. Obtaining long, continuous time-series photometry or radial velocity measurements (space missions like Kepler and TESS have been transformative here).
2. Applying Fourier analysis to extract individual oscillation frequencies from the data.
3. Identifying the modes by their quantum numbers ($n$, $\ell$, $m$).
4. Comparing observed frequencies against theoretical stellar models to constrain internal properties.

Key frequency patterns include the large separation ($\Delta \nu$), which is sensitive to mean stellar density, and the small separation ($\delta \nu$), which probes core composition and evolutionary state.

Applications of Asteroseismology

Asteroseismology determines fundamental stellar parameters with remarkable precision. Masses and radii can be constrained to a few percent, and ages to roughly 10–15%, far better than most other methods.

Specific applications include:

• Probing stellar cores: Mixed modes in red giants reveal whether a star is burning hydrogen in a shell (RGB) or helium in its core (red clump), a distinction that's nearly impossible from surface observations alone.
• Internal rotation: Rotational splitting of oscillation frequencies maps how fast different layers of a star rotate, constraining angular momentum transport mechanisms.
• Convective overshooting: The frequencies of g-modes and mixed modes are sensitive to the extent of mixing beyond convective core boundaries, a major uncertainty in massive star evolution.
• Galactic archaeology: Precise asteroseismic ages for large samples of red giants (from Kepler and TESS) enable mapping the age structure and chemical evolution of the Milky Way.

Helioseismology and Stellar Structure

Helioseismology is asteroseismology applied to the Sun. Because the Sun is so close, millions of individual oscillation modes can be resolved across its disk, providing an extraordinarily detailed picture of its interior.

Major results from helioseismology:

• The base of the solar convection zone lies at $0.713 \pm 0.001 \, R_\odot$, pinned down with remarkable precision.
• The Sun's internal rotation profile shows differential rotation in the convection zone (equator rotates faster than poles) transitioning to nearly solid-body rotation in the radiative interior, with a sharp transition layer called the tachocline.
• Helioseismic sound-speed profiles initially disagreed with solar models using revised (lower) heavy-element abundances. This "solar abundance problem" remains an active area of research and has implications for opacity calculations and stellar modeling broadly.

Helioseismology serves as the benchmark for all of stellar physics. Every asteroseismic technique and stellar structure model must first pass the test of reproducing what we observe in the Sun before being applied to other stars.