6.4 Pulsating variables and cataclysmic variables

Pulsating Variables

Mechanisms of Stellar Pulsations

Stellar pulsations occur when a star's internal structure becomes unstable, causing it to rhythmically expand and contract. These oscillations change the star's brightness, temperature, and radial velocity over time. Understanding the driving mechanisms tells you why certain stars pulsate while others don't.

The κ-mechanism is the most important driver of pulsations in many variable stars. It works through opacity changes in partial ionization zones deep inside the star. Here's the cycle:

1. As the star contracts, gas in the ionization zone gets compressed and heats up.
2. Instead of becoming more transparent (which would let energy escape), the gas partially ionizes and its opacity increases.
3. The trapped radiation builds up pressure, pushing the outer layers outward.
4. As the star expands and cools, the opacity drops, radiation escapes, and the star contracts again.
5. The cycle repeats, sustaining the pulsation.

This mechanism only works when an ionization zone sits at the right depth in the star, which is why pulsating variables cluster in specific regions of the HR diagram (the instability strip).

Convective driving is a separate mechanism where energy transported by convective motions couples to the pulsation. This becomes important in cooler stars where convection dominates energy transport, such as Mira variables and pulsating red giants.

Pulsation modes come in two broad categories:

• Radial pulsations are spherically symmetric: the entire star expands and contracts together. Cepheids and RR Lyrae stars pulsate this way.
• Non-radial pulsations involve asymmetric oscillations where some parts of the surface move inward while others move outward. These are described by spherical harmonics and are common in Delta Scuti stars and pulsating white dwarfs.

Within non-radial modes, two types propagate through different stellar regions:

• Pressure modes (p-modes) are driven by pressure as the restoring force and propagate through the outer layers.
• Gravity modes (g-modes) use buoyancy as the restoring force and probe the deep interior.

A fundamental relationship connects pulsation period to mean density:

$P \propto \rho^{-1/2}$

This period-density relation means denser stars pulsate faster. A compact star like a pulsating white dwarf has periods of minutes, while a bloated red giant pulsates over months.

Classification of Pulsating Variables

Pulsating variables span nearly the entire HR diagram. Each class occupies a different region and has distinct physical properties.

Cepheid variables are luminous supergiants that obey a tight period-luminosity (P-L) relationship: brighter Cepheids pulsate with longer periods. This makes them invaluable as standard candles for measuring distances to other galaxies. Two subtypes exist:

• Classical (Type I) Cepheids are young, metal-rich, Population I stars found in the disk. Delta Cephei is the prototype.
• Type II Cepheids are older, metal-poor, Population II stars. They follow a different P-L relation and are about 1.5 magnitudes fainter at the same period, so confusing the two types leads to serious distance errors.

RR Lyrae stars are low-mass, horizontal-branch stars with short periods (roughly 0.2 to 1 day). They have a nearly constant absolute magnitude ($M_V \approx +0.6$), making them useful distance indicators for globular clusters and for mapping the structure of the Milky Way's halo.

Mira variables are cool, evolved stars on the asymptotic giant branch (AGB) with very large amplitude pulsations, sometimes brightening by several magnitudes. Their periods range from about 100 to 1000 days. Mira (Omicron Ceti) is the prototype.

Delta Scuti stars are A- to F-type stars on or near the main sequence, sitting at the lower end of the instability strip. They pulsate with short periods (roughly 0.02 to 0.25 days) in both radial and non-radial modes. Altair shows Delta Scuti-type oscillations.

Beta Cephei stars are hot, massive B-type stars with short-period pulsations (3 to 8 hours), driven by the κ-mechanism operating in iron-group opacity bumps rather than hydrogen or helium ionization zones.

Slowly Pulsating B (SPB) stars are also B-type but exhibit longer-period g-mode oscillations (roughly 0.5 to 5 days), probing their deep interiors.

RV Tauri variables are post-AGB supergiants with a distinctive light curve pattern: alternating deep and shallow minima. R Scuti is a well-known example.

Pulsating white dwarfs come in three spectral classes:

• DAV (ZZ Ceti) stars have hydrogen atmospheres and pulsate with periods of minutes.
• DBV stars have helium atmospheres.
• DOV (GW Vir) stars are the hottest, with carbon/oxygen atmospheres.

All three types pulsate in non-radial g-modes and are used for asteroseismology, probing white dwarf internal structure.

Cataclysmic Variables

Characteristics of Cataclysmic Variables

Cataclysmic variables (CVs) are close binary systems where a white dwarf (the primary) accretes matter from a low-mass companion (the secondary), usually a main-sequence or slightly evolved star. The dramatic brightness changes in these systems come from the accretion process itself and from explosive nuclear burning on the white dwarf surface.

Mass transfer begins when the secondary star fills its Roche lobe, the teardrop-shaped gravitational boundary around it. Material streams through the inner Lagrangian point (L1) toward the white dwarf. Because the infalling gas carries angular momentum, it doesn't fall straight onto the white dwarf but instead spirals inward, forming an accretion disk.

The main classes of CVs are:

• Classical novae erupt when enough hydrogen accumulates on the white dwarf surface to trigger a thermonuclear runaway. The explosion ejects the accreted shell at high velocity but does not destroy the white dwarf. Some novae recur on human timescales; RS Ophiuchi, for example, has erupted roughly every 15 to 20 years.
• Dwarf novae show repeated outbursts (brightening by 2 to 5 magnitudes) driven not by nuclear burning but by a disk instability mechanism. SS Cygni is the prototype, with outbursts every few weeks.
• Magnetic CVs have white dwarfs with strong magnetic fields that disrupt or prevent disk formation. Polars (AM Herculis systems) have fields strong enough ($B \sim 10\text{–}200 \text{ MG}$) to lock the white dwarf's rotation to the orbital period and channel accretion directly along field lines. Intermediate polars have weaker fields; a partial disk forms, but the inner region is magnetically truncated.

Orbital period distribution reveals important evolutionary physics. CVs are found at periods from roughly 80 minutes up to about 10 hours, but there's a conspicuous period gap between about 2 and 3 hours where very few systems exist. The standard explanation involves a change in the mechanism driving angular momentum loss: above the gap, magnetic braking of the secondary dominates; within the gap, the secondary becomes fully convective, magnetic braking weakens, and mass transfer nearly stops. The system continues to shrink through gravitational radiation alone until contact resumes below the gap. At the shortest periods (near 80 minutes), the secondary becomes degenerate, and the period begins to increase again, creating a period minimum.

Outbursts in Cataclysmic Variables

The energy source for CV luminosity is ultimately gravitational. As material spirals inward through the accretion disk, gravitational potential energy converts to thermal energy and then radiation. The accretion luminosity is:

$L_{\text{acc}} \approx \frac{G M_{\text{WD}} \dot{M}}{R_{\text{WD}}}$

where $M_{\text{WD}}$ is the white dwarf mass, $\dot{M}$ is the mass accretion rate, and $R_{\text{WD}}$ is the white dwarf radius. Because white dwarfs are compact ($R_{\text{WD}} \sim 10^{-2} R_\odot$), even modest accretion rates produce significant luminosity.

Nova eruptions follow a specific sequence:

1. Hydrogen-rich material accumulates on the white dwarf surface over time.
2. The base of the accreted layer is compressed and heated by the white dwarf's gravity.
3. When the temperature reaches roughly $10^7$ K, hydrogen fusion ignites under degenerate conditions.
4. Because degenerate gas doesn't expand when heated, a thermonuclear runaway develops: fusion heats the gas further, accelerating the reaction.
5. The energy release eventually lifts the degeneracy, and the accreted envelope is violently ejected.
6. The light curve shows a rapid rise (hours to days) followed by a slow decline (weeks to months).

Recurrence timescales depend on the white dwarf mass and the accretion rate. More massive white dwarfs need less accreted material to reach ignition conditions, so they recur faster. Recurrent novae like RS Ophiuchi have massive white dwarfs (near the Chandrasekhar limit) and high accretion rates, giving recurrence times of decades. Classical novae with lower-mass white dwarfs may take thousands to millions of years between eruptions.

Dwarf nova outbursts involve a completely different mechanism. The accretion disk itself cycles between two states:

• In quiescence, the disk is cool and matter accumulates.
• When enough mass builds up, a thermal instability triggers a transition to a hot, high-viscosity state. The disk drains rapidly onto the white dwarf, releasing a burst of energy.
• Once the disk is depleted, it cools and returns to quiescence.

This disk instability model explains the quasi-periodic outburst cycles seen in systems like SS Cygni (outbursts every ~50 days) and other dwarf novae.

In magnetic CVs, accretion doesn't proceed through a disk (or only through a partial disk). Instead, material follows magnetic field lines and crashes onto the white dwarf's magnetic poles at near free-fall velocities. The resulting accretion shock heats the gas to temperatures of $\sim 10^8$ K, producing hard X-ray emission. This is why magnetic CVs are prominent X-ray sources, and their X-ray light curves often show modulation at the white dwarf's spin period.