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Newtonian Mechanics in our Universe means simple systems in orbital motion last forever, although dynamical chaos can make a system of three or more masses evolve increasingly unpredictably as time passes. Large systems of masses behave more like a thin gas than a bunch of stars, dissipating over the aeons – Galaxies evaporate.
In Einstein’s Universe, where General Relativity applies, even the simplest orbital systems of binary stars loses energy and angular momentum as gravitational waves, contracting their orbital separations over trillions of trillions of years.
Last time, I reviewed the underlying dynamical equations. Chief is this equation:
For a pair of binary stars of masses \(M_1\) & \(M_2\) separated by rotation radius \(r\), the Inspiral time is:
$$(t_f – t_o) = \left[\frac{5}{256}\left(\frac{c^5}{G^3}\right)\frac{r^4}{(M_1 M_2)(M_1+M_2)}\right]_{r_f}^{r_o}$$
We’ll also use the Schwarzschild Radius, which is the radius of a black hole’s event horizon, defined for any mass \(M\) as:
$$r_s = \frac{2GM}{c^2}$$
To get a feel for the orders of magnitude involve, if we substitute in the Mass of the Sun \(M_\odot\) and for the rotation radius \(r\) use Astronomical Units (149,597,870,700 metres) the Inspiral Time becomes:
$$(t_f – t_o) = \left[\left(\tau\right)\frac{r^4}{(M_1 M_2)(M_1+M_2)}\right]_{r_f}^{r_o}$$
Where \(\tau\) is a time constant equal to 1.013445269E+25 seconds, which is about ~3.2E+17 (320 quadrillion) years.
Two solar mass stars will spiral together in half that time if 1 AU apart.
An Earth would spiral into a Sun in about 332,950 times longer, some 1023 years.
Swap out the Sun for Sagittarius A*, which is our Galactic core’s Super-Massive Black Hole that masses 4.2 million solar masses, and Earth would In-Spiral within 6 billion years ~ (1023/(4.2 x 106)2).
Two Sag A* mass Black Holes 1 AU apart would be 38 hours from their mutual Innermost Stable Circular Orbits, after which they’d “plunge” together in mere hours.
Since the mass-ratio of the objects is key to their mutual orbit lifetimes, it means heavier orbiting objects circling a much larger mass would sweep through lower mass objects as they in-spiral.
Imagine an SMBH that acts as the anchor mass for a vast collection of planet-rings like those described by Sean Raymond in his PlanetPlanet blog post: The Ultimate Engineered Solar System. In the deepest Far Future such locations will be a way of supplying energy via Dark Matter decay, if at least a fraction of Dark Mass is neutrinos. Even in Milgrom’s MOND there’s some percentage of non-baryonic mass required so it’s a fairly generic feature of both Dark Matter and Modified Gravity theories. Such an SMBH eco-sphere would need to be carefully arranged to last the hoped for trillion trillion years the neutrino energy is estimated to last. If the neutrino halo masses 1015 solar masses, the anchor hole will mass ~ 1010 solar masses, with an Event Horizon 200 AU in radius. For a 5 Earth-mass Super-Earth the minimum radius to in-spiral in 1024 years is a bit over 4 light-years, about 262,000 AU. Rings of thousands of planets could be positioned outwards from there, as they need to be 14 Hill Radii apart. At 262,000 AU that’s about 29 AU. About 9,000 planets per ring, then rings 29 AU apart out from there.