Two-Body Problem: Double Stars

The solution to the gravitational two-body problem in Newtonian dynamics was provided by Newton himself: the relative motion of the two bodies forms a conic section, a circle, ellipse, parabola, or hyperbola. However, this solution only holds for point masses, as well as for bodies that are completely spherically symmetric. For bodies with more complicated shapes, the relative motion is more complex. Especially when the shapes of the bodies are influenced by their relative motion, as is the case with tidal interactions between stars or planets or moons, the resultant perturbed two-body problem has to be solved either numerically, or analytically through a perturbation analysis. Another complication arises when one or both of the bodies loose mass, through explosions or more gradual outflows.

Tidal Stability

Intuitively, it is clear that a double star configuration forms a stable equilibrium when the two stars are co-planar, synchronized (where their spin axes align with the orbital angular momentum, and the spin periods equal the orbital period), and they are in a circular orbit. What is not clear, however, is whether this is the only equilibirum configuration, and under which conditions the equilibrium configuration(s) are stable. This question was addressed for the first time in the full three dimensional configuration in the paper Stability of Tidal Equilibrium, by Hut, P., 1980, Astron. Astrophys. 92, 167-170.

Tidal Evolution

Charles Darwin

How does a double star evolve, under the influence of tidal perturbations, caused by the tides that each one raises on the other? One of the simplest models for for the height of the tidal bulges assumes that their shape has a constant time lag with respect to an instantaneous hydrodynamic response. This model had been analyzed first by George Darwin, the son of Charles Darwin (an interest in evolution seems to run in the family). Since he was interested mostly in planetary motions, where the orbits are nearly circular, he applied a perturbation treatment, in terms of powers in eccentricity. In the case of double stars, arbitrary high eccentricities can occur, which makes it useful to apply more general techniques.

One hundred and two years after George Darwin's publication, a treatment for general eccentricity was provided in Tidal Evolution in Close Binary Systems, by Hut, P., 1981, Astron. Astrophys. 99, 126-140 (when I got the page proofs, I had to change the reference to Darwin back to 1879; the editor thought it was a typo, and had changed it to 1979).

Pseudo-synchronization illustrated with Mercury's spin-orbit resonance

In this paper I also introduced the new concept pseudo-synchronization, a generalization of what has happened in the solar system with Mercury, where its spin and orbit are approximately in sync during pericenter. This concept has proved useful in the analysis of, for example, eccentric X-ray binaries.

A follow-up paper treated the case of extreme eccentricity, in which the orbits are close to parabolic. This case has applications to tidal capture, in which two unbound stars pass close enough to each other in order to let their tides dissipate enough energy to form a bound orbit: Tidal Evolution in Close Binary Systems for High Eccentricity, by Hut, P., 1982, Astron. Astrophys. 110, 37-42 [note the erratum in Astron. Astrophys. 116, 351].

While the constant time-lag model was attractive in its simplicity, no clear physical basis had been provided. This situation was improved by a more general treatment in The Equilibrium-Tide Model for Tidal Friction , by Eggleton, P.P., Kiseleva, L.G. & Hut, P. 1998, Astrophys. J. 499, 853-870. We showed that the simple equation for tidal friction, based on the picture of the tidal bulge lagging the line of centers by some small constant time, follows quite directly from a more physical model in which the dissipation is related to a positive-semidefinite function of the rate of change of the tidal deformation (as measured by the quadrupole tensor) in the frame that rotates with the star. Our analysis gave the effective lag time as a function of the dissipation rate and the quadrupole moment.

In many close binaries, the rate at which the two stars spiral in toward each other significantly exceeds the rate given by gravitational radiation losses. One mechanism that is often invoked is magnetic braking, where the stellar wind of one of the stars is forced to co-rotate with the magnetic field anchored in that star. If this co-rotation holds at distances far larger than the radius of the star, the slingshot effect of the magnetic field lines cause a large amount of angular momentum loss for a relatively small amount of mass loss. In many earlier treatments, the subsequent energy dissipation in the star was neglected, even though it could be comparable to the energy generated by nuclear burning. We analyzed the conditions under which this assumption is correct in the paper Magnetic Braking and Tidal Energy Dissipation in Close Binaries, by Verbunt, F. & Hut, P., 1983, Astron. Astrophys. 127, 161-163.

Explosive Mass Loss

When a star in a double star slowly loses mass, on a time scale much larger than the orbital time, the system will not be disrupted, no matter how much mass is lost. But when the mass loss happens almost instantaneously, the double star will be disrupted if more than half of the total mass of the system is lost. As a practical application, a type II supernova typical sheds far more mass than is retained in the black hole or neutronstar left behind. If a companion star is light enough, it will no longer remain bound to the supernova remnant. But what happens in the intermediate case, in which the mass loss time scale and orbital time scale are comparable? Using a mathematical technique known as asymptotic expansions in a two time-scale method, we give a quantitative answer in the paper Explosive Mass Loss in Binary Stars: the Two Time-Scale Method, by Hut, P. & Verhulst, F., 1981, Astron. Astrophys. 101, 134-137.