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Degenerate Objects

Evolution of Cataclysmic Variables

While massive stars end their lives with a bang, low-mass stars end theirs with a whimper. Thermonuclear fusion in a low-mass star does not end with the total exhaustion of the star's thermonuclear fuel and the subsequent collapse of the star's core, as it does in a large star, but with the formation of a stable core that supports itself against gravity through the pressure exerted by degenerate electrons. The gentle transition from a hot core supported by thermal pressure to a cold core supported by degeneracy pressure preserves the mass of the star.

This evolution of the low-mass star makes the evolution of a compact binary system containing two low-mass stars a rather simple process. The transition from a system containing two fusion-powered stars to a system containing one fusion-powered star and one degenerate dwarf star does not disrupt the binary system, because the transition does not cause the system to lose mass. Even earlier in the evolution, where the more massive star losses a small quantity of gas when it transitions from hydrogen fusion to helium fusion, the amount of mass loss is small compared to the mass of the star, so the binary system is not disrupted. This makes the evolution of a binary system containing two low-mass star fundamentally different from a system containing at least one high-mass star; in those systems, the mass lost by the high-mass star during a supernova is sufficient to disrupt the binary system.

With two low-mass stars, the star with more mass ends it life of thermonuclear fusion first, becoming a degenerate dwarf. Its companion star remains on the main-sequence, since its life of thermonuclear fusion is much longer. If the system is born with the stars close enough together, the orbit can decay enough to cause the main-sequence star to overflow its Roche lobe, dumping gas onto the degenerate dwarf. This transfer of mass from one star to the other is stable, because the mass flow is from the lower-mass star to the higher-mass star. We see these systems as cataclysmic variables.

Cataclysmic variables are driven by the loss of angular momentum and energy from their orbit. The loss is necessary, because the natural reaction of the binary system to mass transfer from the lower mass star to the higher mass star is to widen the separation between the stars, while the natural reaction of the mass donor is to shrink in radius. The loss of orbital angular momentum and energy causes the binary system to shrink rather than expand as mass transfer occurs, so that the main sequence star continues to overflow its Roche lobe. The interesting point about Roche Lobe overflow is that it locks the separation of the stars and the period of the binary orbit to the mass of the donor star in a predictable way. If the main-sequence star changes its character at a particular mass, it also changes its character at a particular orbital period.

Two mechanisms drive the loss of orbital energy and angular momentum. The first is gravitational radiation. As described elsewhere, this mechanism is important when the two stars are very close together. Farther apart, however, another mechanism is at work that drives the stars together in a reasonable time (less than the remaining lifetime of fusion-powered star). This mechanism is thought to be a stellar wind driven from the main-sequence star. We know that many stars have winds, including our own Sun. If the wind carries a magnetic field, it can extract a considerable amount of angular momentum from a binary system, because the torque exerted by the gas flowing away from the star is exerted back onto the star by the magnetic field. This lost angular momentum and energy comes from the binary system as a whole, because the degenerate dwarf exerts a torque on the main sequence star's Roche lobe that causes the main-sequence star to rotate synchronously with its orbit around the degenerate dwarf.

The loss of mass by the main sequence star is rapid enough to prevent the star from maintaining the stable configuration it would have away from a binary system. The reason is that the star loses mass faster than energy within the star can diffuse through the star, so the temperature gradient near the surface becomes steeper than it would be in an isolated main-sequence star. The effect of this is to cause the donor star to puff-up larger than it would be if isolated.

Cataclysmic variables begin their lives with long orbital periods, and evolve to shorter periods. The systems we see have periods ranging from as high as 15 hours to as low as 80 minutes. We don't see systems orbiting with periods shorter than 80 minutes. More unusual, we see very few cataclysmic variables with periods between 2 and 3 hours.

This period gap is thought to be caused by the shutting-down of the stellar wind, which dramatically slows the loss of orbital energy and angular momentum from the system. The decay of the binary orbit therefore slows, and the donor star suddenly has enough time to come into thermal equilibrium. This causes the star to shrink back to its normal main-sequence radius, ending mass transfer to the companion. The binary system remains invisible to us until the orbit shrinks enough to case the star to again overflow its Roche lobe. In this picture, the stellar wind ceases when the binary period is 3 hours, and Roche lobe overflow recommences when the binary period is 2 hours. Because the period of the system is tied by the Roche lobe overflow to the mass of the donor star, the wind ceases when the donor falls to a certain mass.

Below the period gap, the donor star continues to shrink as it loses mass until the electrons at its core become degenerate. When this occurs, the core of the star supports itself against gravity through degeneracy pressure rather than through thermal pressure. This changes how the star responds as it loses mass. In this state, the star expands rather than contracts as it loses mass. The binary system's reaction to mass transfer is naturally to expand, so the donor star's expansion can be accommodated; the binary system, however, reacts to the mass transfer by increasing the separation between the donor star and the degenerate dwarf, which increases the orbital period. This mechanism gives the cataclysmic variables its minimum orbital period, which is observed at around 80 minutes.

Cataclysmic variables have two possible ends, one violent, the other quiet. The transfer of mass from the main-sequence star to the degenerate dwarf can push the degenerate dwarf over the Chandrasekhar limit, creating a type 1a supernova. This end can occur any time during the binary system's evolution. The other end is to simply disappear from sight, which occurs if there is insufficient mass in the main-sequence star to implode the degenerate dwarf. As the mass of the main-squence star drops, the center of mass of the system moves towards the more-massive degenerate dwarf, making gravitational radiation less efficient at removing orbital energy and angular momentum from the system. This causes mass transfer to slow to the point that the system becomes invisible. Eventually mass transfer stops as the donor star grows cold and crystallizes.

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