Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational radiation — where it appears

The energy and angular momentum a binary loses to gravitational waves, dominated by whatever happens near periastron. It removes the two at rates that do not keep step, which is why an isolated pair arrives at merger circular.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Every pair arrives circular. Eccentricity against gravitational-wave frequency for four binaries of 30 and 30 solar masses, each starting at 0.2 astronomical units with an eccentricity of 0.3, 0.7, 0.9, 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. The curves are Peters' closed solution a(e), and they are checked against Peters' differential equation at three eccentricities on each track rather than against the integral they came from. The ordering is preserved — a pair that starts rounder stays rounder — but by the time the orbit is radiating at 10 hertz, where a ground-based detector begins to hear it, the four eccentricities are 8.8·10⁻⁸, 5.6·10⁻⁷, 3.6·10⁻⁶, 1.5·10⁻⁴. All four are far below anything a detector could measure. That is the figure's whole content and it is a strong statement: radiation reaction removes angular momentum faster, relative to energy, than a circular orbit would need, so eccentricity is destroyed on the way in. A binary observed to be eccentric in band therefore cannot have spent long shrinking quietly, and must have been put on that orbit recently — by a third body, or in the crowded centre of a cluster. The figure assumes the two bodies are points and nothing else acts on them, which is exactly the assumption an eccentric detection would refute.

Every pair arrives circular

Gravitational radiation drains a binary's energy and its angular momentum at rates that do not keep step, and the mismatch destroys eccentricity far faster than it shrinks the orbit. A pair that starts at 0.99 and spirals in from a fifth of an astronomical unit is round to better than a part in a hundred million by the time a detector can hear it.

gravitation · Gravitational waves The last parsec, priced in stars. The two timescales that shrink a pair of 10⁸-solar-mass black holes at the centre of a merged galaxy, against their separation in parsecs, both axes logarithmic, for a stellar velocity dispersion of 200 km/s and a central density of 500 solar masses per cubic parsec. Ejecting stars hardens the binary at a rate proportional to the separation, so that timescale grows as the orbit shrinks; gravitational radiation goes as the fourth power of the separation, so its timescale collapses. Neither alone finishes the job and the crossing of the two is where the answer is set. With the loss cone kept full, the crossing is at 2.9e-2 parsecs and the total is 4.2·10⁸ years, inside a Hubble time. With the supply of low-angular-momentum orbits emptied by a factor of 100 — which is what happens in a smooth spherical nucleus, because the stars that could interact have already been thrown out and two-body relaxation refills the orbits far too slowly — the crossing moves outward to 7.3e-2 parsecs and the total becomes 1.7·10¹⁰ years. That is the final-parsec problem, and it is not a problem about gravity: it is a problem about supply. Real nuclei are not spherical, and the figure cannot show what a triaxial potential does, which is to keep feeding the binary orbits it has not already used.

The last parsec, and the stars that are not there

Two galaxies merge and their central black holes sink towards each other, and then stop. From about a parsec apart, friction no longer works and gravitational radiation is not yet strong enough — and the only mechanism in between throws away the stars it depends on faster than they can be replaced.

galaxies · Galactic nuclei A cycle and a radiation, racing for the same pericentre. Two integrations of the same compact triple, in units of the cycle's own timescale at the starting separation. The dashed curves are the semi-major axis and the solid ones the eccentricity. Radiation alone shrinks the orbit slowly and smoothly, reaching a tenth of its starting size after 1136 cycle times — which the closed form for a circular inspiral predicts to within a few per cent, and which is the check that the integrator is right. With the cycle running, the distant companion drives the eccentricity to 0.9104, the pericentre falls by two orders of magnitude, and the radiation — whose rate goes as the inverse seventh power of one minus the square of the eccentricity — empties the orbit in 18. That is a factor of 62, obtained without changing a single parameter of either mechanism. The two do not merely add: the cycle manufactures, at its own peak, exactly the configuration in which the radiation is most efficient, and the radiation then shrinks the orbit until the relativistic precession switches the cycle off.

A cycle that builds what the wave needs

Gravitational radiation empties an orbit at a rate that goes as the inverse seventh power of one minus the square of the eccentricity. A distant companion drives exactly that eccentricity, over and over. The two are not independent mechanisms that happen to act on the same binary — the first manufactures, at its own peak, the configuration the second is most efficient in.

orbits · Kozai–Lidov

Named alongside it

The objects these essays reach for when they reach for this one.

Angular momentumKozai lidov mechanismPeters equationsApsidal precessionBinary black holeChirp massCircularisationCompact binaryCore scouringDynamical formationDynamical frictionFinal parsec problem

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