Dozens of Habitable Worlds in One System?

Question

I want to know if this is physically possible. So at first I thought I was nuts. Then I started reading about Hill Radii, and googled left and right. Turns out a Hill Sphere is the region around an astronomical body where it is dominant in regards to attracting satellites. The Sun has a Hill "Sphere" of sorts, but close enough to Earth, the Earth's Hill Sphere is dominant.

$r_H = a(1-e)\sqrt[\leftroot{-2}\uproot{2}3]{\frac{m}{3M}}$

which simplifies to

$a\sqrt[\leftroot{-2}\uproot{2}3]{\frac{m}{3M}}$

for circular orbits, where e is eccentricity, a is the semimajor axis of the smaller object's orbit around the larger body, and m and M are the masses of the smaller and the larger object, respectively.

It occurs to me that as M grows larger and larger, the area where the gravitation of the mass-m orbiter dominates over the larger M shrinks more and more, and the degree to which planets perturb their neighbors shrinks and shrinks.

So here's the idea:

• One supermassive black hole (think > 100,000 $M_{\odot}$)
• Orbiting that, place One (1) coorbital-ring system of 7+ sun-like stars, a safe distance from the black hole, in a stupidly stable orbit.
• Farther out, a whole coterie of dozens (hundreds?) of worlds orbiting within say 0.1 AU of each other, their Hill Radii safely shrunk to the point where they barely interfere with each other.

Potential downsides:

• Planets may get tidally locked, and may get a bit, uh, unspherical.
• Despite doing the math, I still have a hard time believing that the planets would not perturb each other.

Potential upsides:

• At really ridiculously low Hill Radii, you could (have not done the math) have coorbiting planets, and have them almost touching. A Niven ring made out of planets.
• If the perturbations really bother you, just add more suns to the Sun-ring, and push the Goldilocks zone out to multiple AU.

But this could never just happen naturally! Well, duh.

Answer 1

Only dozens of habitable planets in one solar system, eh? Well, you're thinking small. The work for a solar system with one million Earths has already been done by astrophysicist Sean Raymond. His model, like the one postulated in his question, is centred on a supermassive black hole.

Let’s start with a supermassive black hole, like in the Black Hole Ultimate Solar System. It’s 1 million times the Sun’s mass. A behemoth!

Now let’s see what rings of planets we can make. As we saw in the Ultimate Engineered Solar System, a ring of 42 Earths orbiting a Sun-like star is stable.

Sean Raymond's One Million Earths model solar system also considers the role of the Hills radius.

The requirements for a stable rings of planets are simple (technical details here and here):

The planets on a given ring must all have the same mass, There must be at least 7 of them, and They must be evenly spaced along a circular orbit and separated by at least 12 Hill radii. (The Hill radius is the distance inside which a planet’s gravity dominates over its star’s.)

Around a supermassive black hole (of 1 million Sun masses) the Hill radius shrinks to 1/100th of its value around the Sun. That means that 100 times more planets can fit on the same ring around a black hole!

Persons seeking more information about the One Million Earth solar system should refer to his website.

However, his basic model can be detailed as follows: --

Let’s start simple. In the Black Hole Solar System I replaced the Sun with a Sun+black hole system. Let’s do the same thing again. Of course, in the Black Hole Solar System I used a Solar-mass black hole and here we will use a million Solar-mass black hole. But it doesn’t make all that much difference.

A supermassive black hole is not very big. Its event horizon (or Schwartzschild radius) — the distance inside which light can’t escape — is only 2% of the Earth-Sun distance (that is, 0.02 AU, or 3 million km). That’s about 4 times bigger than the Sun. The innermost stable circular orbit around the black hole is three times that distance, about 0.06 AU. To avoid anything too crazy, I’m going to put a single Sun three times farther out, at 0.2 AU.

In this system the central black hole+star weigh 1 million Suns (well, 1,000,001 to be annoyingly precise). But they produce the same amount of energy as one Sun. That means that the habitable zone is at the same place as for the Sun.

Let’s break out the planets. If we mega-pack rings, with alternating rings on retrograde orbits, we can fit up to 689 rings in the habitable zone. Each ring can hold up to 4154 planets. That makes for a maximum of 2.86 million planets! I’m going to be a cautious (I know, that is not my usual m.o.) and space things out more. Let’s use 400 rings with 2500 planets each. That’s a million Earths in the habitable zone!

In conclusion, solar systems with dozens and all the way to around one million habitable planets does seem to be physical plausible. The probability this could happen in nature is presumably remarkably low. This doesn't necessarily prevent a sufficiently advanced technological civilization building its own solar system with a suitable plenitude of habitable planets. This is much more achievable if the primary barycentre is a supermassibe black hole.

Answer 2

While I clearly am not going to make nearly as useful a post as that above, here is something to consider as a heuristic for most sci fi authors in general. If you are writing anything not hard sci fi, then why not play around with questionable possibility? Strictly possible is likely a moving target (ask Burgess) but some of the better sci fi pushes those boundaries and no one fault Clark because Jupiter can become a star on its own. If something is inspiring, why not ride that wave? Especially if you've yet to publish, just getting words on paper is a good thing.

Just saying