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Two-Body Problem
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
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
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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.
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