Welcome to what might very well be the culmination of the Building the Ultimate Solar System series. Teaser: the system built in this post could also be called the Ultimate Engineered Black Hole Eyeball Ringworld Solar System (if you’re really not into the whole brevity thing).
Our Solar System has one habitable planet. A few dozen known exoplanet systems (like Kepler-186) host a candidate habitable planet. The Trappist-1 system has three Earth-sized planets in the habitable zone (not too shabby — although we don’t know whether they have life, of course).
How many potentially habitable worlds could one system have? That question is at the heart of the Ultimate Solar System project I’ve been working on for the past several years. I first built Ultimate Solar Systems with 24 and 36 habitable planets. I upgraded in a big way with the Ultimate Engineered Solar System and packed 416 habitable planets in one star’s habitable zone. In the Black Hole Ultimate Solar System I used a supermassive black hole to cram planets 100 times closer together than I could around the Sun, and got up to 550 Earths orbiting in a single habitable zone.
Today I will build what I believe to be the Ultimate Ultimate System of this kind. I don’t think it’s possible to out-Ultimate this one.
Here we go.
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.
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! It looks like this:
See how the planets are so close the symbols are overlapping in the plot on the left? Well, that plot is still only showing one planet out of ever 50! (And their sizes are hugely exaggerated so you can see them). The zoom on the right shows just how close together the planets actually are (sizes still exaggerated, but by much less).
There is another big deal consequence of the shrunk-down Hill radius. Around the Sun 6 Earth-mass planets can fit on stable orbits within the habitable zone. Pack them any tighter and they’re unstable (more detail in this post).
Well, around a supermassive black hole a lot more planets can fit within any given region. Too many to easily visualize. Let’s just take a look at a small slice of the habitable zone:
The black lines are maximally-packed orbits of Earth-mass planets around the Sun. The green lines are around a supermassive black hole. In this little slice, two Earths can fit around the Sun. Or 145 can fit around the black hole!
Now we can use two tricks that we know and love (details here and here). First, each orbit can hold a whole ring of up to 4000+ planets. We’ll get to the numbers later, but you can already tell this is going to add up to a *lot* of planets (hence the title of this post). Second, if each neighboring ring of planets orbits in the opposite direction — such that rings 1, 3, 5, … orbit clockwise and rings 2, 4, 6, … orbit counterclockwise — then the rings can be more tightly-packed while remaining stable.
What should we do about illumination? We need sunlight to keep these planets habitable. I can’t choose, so I’m going to build three separate systems.
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!
UPDATE: It was pointed out to me by Phil Armitage, astrophysicist (and photographer) extraordinaire, that the Sun would tidally disrupt anywhere within about half an AU of the black hole. (I should have realized this since he and I had recently worked together on tidal disruption of ‘Oumuamua.) Million-Earth system 1 would not survive with the Sun on a 0.2 AU-wide orbit. Instead, the Sun would be torn to shreds and spiral onto the black hole in an accretion disk. Let’s transform the Sun into that disk, and for now regulate its rate of infall to make it give off the same amount of light as the normal Sun. Now it looks like this:
In Million Solar System 2 we’ll just change things up a tiny bit. As in the Black Hole Ultimate Solar System, we’ll add a stable ring with 9 Sun-like stars evenly spaced along the same orbit at 0.5 AU. Since the central “star” is 9 times brighter than the Sun, the habitable zone is three times farther away than the Sun’s. Apart from that we have the same setup of planets as before. It looks like this:
Let’s switch up the illumination again. This time I’ll put a ring of stars on an exterior orbit. I’ll put 36 Suns on a 6 Astronomical Unit-wide orbit. That means that a planet on a 1 AU-wide orbit will still receive the same amount of total illumination as Earth. Now it looks like this:
With an outer ring of Suns, the source of illumination is separate from the central object around which planets are orbiting. Each planet is bathed in sunlight from all sides. The planets have no night side! It’s like Asimov’s permanent-daytime planet Kalgash!
The most magnificent view in this system would come from the North (or South) pole. All 36 stars would be at the horizon in permanent sunrise/sunset. You could only see the stars from the poles if you could block out all the stars.
You would never feel alone in these systems. The other planets would loom huge in the sky! Neighboring planets are about ten times closer than the Moon (on an Earth-like orbit in systems 1 and 3). Earth is about 4 times larger than the Moon, so neighboring planets would be about 40 times larger in the sky than the full Moon. That’s 20 degrees in the sky. That’s about the size of a laptop computer held at arm’s reach, only up in the sky!
Meanwhile, planets on adjacent rings of orbits would zip across the sky, growing larger than the full Moon before shrinking and fading in a matter of minutes, only to make room for the next planet. I explained what this would look like for the Black Hole Ultimate Solar System — now it would be amplified by rings of planets.
There are some big differences between these million-Earth systems and the Solar System.
Everything moves ridiculously fast. Instead of taking 365 days to orbit the Sun, a planet on Earth’s orbit would take just 9 hours to complete an orbit around the supermassive black hole! The planet is orbiting at about 10% the speed of light. At these super-high speeds, relativity starts to matter. Starlight would also be stretched by the black hole’s gravity. Stars closer to the black hole would appear redder and those farther from the black hole would appear bluer. Time would move more slowly for planets on different rings around the black hole. Two babies born at the same instant on different rings would age at slightly different rates. The baby on the inner ring would age slightly more slowly. This effect is small in this system, but it is huge in the super-extreme environment of Miller’s planet in the (awesome) movie Interstellar.
There would also be a gravitational lensing effect every time a star passed behind the black hole. As seen from the planets, stars are distorted into arcs and then into rings when the star is aligned just behind the black hole. Here is an animation of what this would look like specifically for the 9-star ring setup of the Black Hole Ultimate Solar System (done by @GregroxMun using Space Engine):
In all these systems, the planets are tidally-locked to the central black hole.
In Million-Earth systems 1 and 2, all planets may be Eyeball planets. The illumination comes from the same direction as the black hole, so one side of each planet is in permanent daytime while the opposite side is in permanent darkness. A perfect setup for Eyeball worlds! (Of course, whether or not they become Eyeball-like depends on details like their atmospheric thickness and total water contents).
In Million-Earth system 3 the planets will not be Eyeball worlds because their source of illumination is separate from the strong gravity they feel. Since the ring of Suns is external these planets are in permanent daylight. There is no hot/cold side of the planet.
The planets in all three systems would not be spheres. The black hole’s gravity would squish them. The side closer to the star would be pulled on more strongly than the side opposite the black hole. This is the tidal force, and it would stretch the planets out. (It’s the same kind of force that makes planets always show the black hole the same face).
Time for a twist.
Imagine living in million-Earth Solar System 1 or 3. The size of your planet’s orbit is the same as Earth’s around the Sun. There are a ton of other planets in the sky. But the relative speeds of different planets is enormous. You cannot travel to another planet because the speeds involved are beyond the capabilities of any current (or even yet envisioned) technology.
But thousands of planets share your orbital ring. The closest ones huge in the sky, 40 times as big as the full Moon. Given that all planets are locked to the black hole, these giant neighboring worlds never move in the sky.
This presents an opportunity. The sides of each planet could be joined. Despite the huge orbital speeds relative to the black hole and to neighboring rings of planets, the relative speeds between neighbors is almost zero. If they could be joined you could travel between the planets. A space elevator could connect planets at the bull’s eye points, where the worlds are closest to each other. It’s kind of like the setup in the book Hothouse by Brian Aldiss, in which the Moon and the (tidally-locked) Earth are connected, although in that case it it by a planet’s spider-web like branches.
Imagine that each pair of neighboring planets along a given ring was connected. It would almost make a Ringworld:
The difference between this setup and the Ringworld from Larry Niven‘s book is that in this case there is no livable surface area in between the planets. The space elevators only serve to connect the different worlds.
People living on all the planets on a given ring can intermix. But the huge relative speeds between rings means that all other rings of planets are off-limits. It would be prohibitively expensive (in terms of energy) to land on a planet from a separate ring. Then again, it wouldn’t be hard to communicate with them (or, for that matter, to launch pretty devastating bombs at them).
This could make a great setting for science fiction stories, a blend between the Ultimate Engineered Solar System, Hothouse and Ringworld (and featuring Eyeball planets!). That’s why I’m also putting this post in the Real-life Sci-Fi worlds series.
A final big question: where would such a system come from?
In the Ultimate Engineered Solar System, I argued that systems with rings of planets are not likely to form naturally. There are way too “just-so”. Instead, they must be engineered by super-advanced civilizations that can make their own planets and systems (like Slartibartfast from the Hitchhiker’s Guide to the Galaxy).
Of course, you might ask: if super-intelligent aliens can make their own Solar Systems, couldn’t they just make any planet they want habitable? To which I reply once again (very loudly): sheesh, stop raining on my parade!
I can imagine super-advanced aliens creating a system like the million-Earth Solar System as a cosmic work of art. Kind of like the art of skyscrapers or painted icebergs. A way to say “look how fancy we are” on the grandest scale.
Or maybe aliens would create this kind of system as a zoo. They could have a gradient in climates from the hottest to coldest, and stock the planets with all sorts of creatures they collect across the Universe. Of course, they’d have to be careful not to put the wrong combinations of space-creatures on the same ring of planets, because that wouldn’t end well!
There you have it: the million-Earth Solar System(s). BOOM!
Questions? Comments? Words of wisdom?
Technical issues. I don’t want to hide anything under the rug, so I’ll mention that there are a couple of small potential issues with this system. The first is related to stability of compact planetary systems in the face of strong tides. The planets are simultaneously stretched by the black hole’s gravity (which is super strong) as well as their closest neighboring planets (which are super close-by). This can make the planets want to point their bulges in two places at once, and repeated tidal flexing can cause energy losses that translate into orbital shifts that on long timescales can sometimes destabilize planetary systems (details here). In such an extreme environment I’m not convinced this is a real problem, as it’s possible that the planets will actually get stretched not into cigar-shapes but instead into ninja star-shapes, with longer parts pointing both toward their neighbors toward and away from the black hole.
Of course, there is an easy way to circumvent this issue. Tides are strongest for planets super close to their stars (or the largest mass, here the black hole). As in million-Earth system 2, we could just put a ring containing as many bright stars as possible to move the habitable zone outward. For example, with a ring of 64 stars, each with twice the Sun’s mass (and 16 times its brightness), the habitable zone would be moved out past Neptune’s orbit. A million Earths would still fit in the habitable zone, they would just be somewhat farther apart. For instance, neighbors on the same ring would now be a little farther than the Earth and Moon.
It’s also worth mentioning that the concept of stable rings of planets was developed in Newtonian gravity, without accounting for the extra effects that arise from general relativity. While those should not in principle have a strong effect on circular orbits, this environment is so extreme that it’s possible that additional effects could make a difference.
- Building the Ultimate Solar System page
- Real-life Sci-Fi Worlds page: testing ideas from science fiction (and coming up with new ones)
- The Ultimate Engineered Solar System: introduction to stable rings of planets
- The Black Hole Solar System: an introduction to black holes
- The Black Hole Ultimate Solar System: first crazy system with a supermassive black hole
- Real-life Sci-Fi World 2: the Hot Eyeball planet
- The movie Interstellar (with supermassive black hole Gargantua)
- Hothouse by Brian Aldiss
- Ringworld by Larry Niven
- Kalgash by Isaac Asimov. Related posts here and here.
- The Hitchhiker’s Guide to the Galaxy by Douglas Adams.
- Thanks again to @GregroxMun for making that sweet animation of the stars passing behind the black hole (and also to Space Engine)
46 thoughts on “The Million Earth Solar System”
For an ultimater system you could add more rings of suns to make more habitable zones.
You are totally right and I’ve started working on this — you can have a whole constellation with rings of stars with HZ planets. Just need a_HZ < R_Hill for this to work.
“a_HZ < R_Hill"
With the planets orbiting the stars? Another cool idea.
I was thinking of ring of stars, habitual zone, ring of stars, habitable zone, etc.
This is awesome. As you can tell, my idea was a constellation of zillions of rings of stars, each of which has its own HZ planets. You could fit a ton of planets, each around its own star, within some finite system size (say, 1000 AU).
But your idea is like a giant “millefeuille” (or mega-sandwich) system! I love it. You would have a zillion habitable zones squashed in between rings of stars. Very cool contrast!
I suppose these two contrasting mega-systems might be worth making into a blog post at some point (when I’m less burned out — all the posts over the last week took a lot of time!). When I start to write it up I’ll be in touch to see you if you have any other ideas.
Or to make it scarier put some of the ring in retrograde orbits.
There are probably issues with relativity with that idea though.
I don’t have a good enough feel to know whether rings of retrograde orbits would make a difference with GR. Do you?
I was just thinking that black holes form from rotating objects and was speculating that that might affect the retrograde objects differently than the prograde. I don’t really know where the idea came from though.
Given the 4D nature and effects such as frame dragging I don’t even know how to conceptualize that… 0_o I don’t even know if it is possible to do a retrograde orbit as by opposing the motion of the system well space would be warped strangely.
Hmm looking it up there is a bias regarding direction for pro and retrograde so they probably are incompatible. From one paper I found that mentioned this there should be a effective “space time wind” that is going to be “blowing” in the direction of the blackholes rotation repelling or pushing retrograde objects outwards and presumably pulling prograde objects inwards. But since these inwards and outwards are spacetime itself I don’t ave any grasp of what that actually means….
Here’s a simulation I made in Space Engine of a retrograde planet flying by its prograde neighbor, seen from the retrograde planet. (single planet per orbit, in an orbital ring set up this would be happening all the time!). This isn’t even with a doubling of the orbits which you could do with the retrograde system, it’s just the normal distance.
I hope this gets across just how incredibly fast these orbits are. The relative velocity between a prograde and retrograde planet here is about 0.115c (34392 km/s). A spent rocket stage carelessly left in orbit between the planets, say, an Apollo S-IVB Third Stage, would release a kinetic energy of 6.603e18 Joules if it hit a planet going the wrong way. That’s 31 Tsar Bombas! Even the energy of a loose screw would be as energetic as 143 tons of TNT. The Chelyabinsk meteor would have been a mass extinction event. The dinosaur killer would destroy the entire planet.
Let’s not do retrograde planets this time.
(Hmm, a habitable planet orbiting a starless black hole that gets its heat from relativistic meteors? A little far fetched…)
What about the 60 Tons Of Cosmic Dust Fall To Earth Every Day? That’s 10 Gigatons of TNT equivalent, 10x the world nuclear arsenal every 24 hours.
Though that’s only 0.1% the energy the Earth gets from the sun each day, so it wouldn’t mess with the habitable zone…
What affect would ‘Oumuamua have passing through? With 1 million targets it would probably hit someone and hitting someone going 10% the speed of light sounds like it would make something uninhabitable. At 0.1c, even bumping into cosmic dust sounds like a bad time.
With so many planets providing “moonlight” would they shift their own habitable zone? Moonlight * 1 million seems like enough to provide a significant heating effect.
Also that’s going to be a lot of eclipses. Would Million Earth Solar System 3 have a nighttime just by having so many eclipses?
What’s the lifespan on the lifespan on this system? The rings of suns is going to cap it in the billions, but if you’re using rings of stars can’t you put much longer lived stars in super dense orbital rings and get the same illumination?
This system could provide a model of an Ultimate Deep Future Habitat. There’s a possible energy source that’ll last long after the stars all sputter out – neutrino annihilation. To tap it, a planet needs a big iron core, not unlike a Super-Mercury would have. Good for at least a trillion trillion years. To concentrate the neutrinos sufficiently for life, the planets would be better anchored to an SMBH at the core of a Galaxy Cluster, thus the Ultimate system would have thousands (millions?) of rings.
Now I see several possible challenges to overcome.
The first is how would your highly advanced aliens art project handle the mass differentiation where more massive objects get drawn closer to the galactic center over long timescales (billions of years and greater)?
Over time the supermassive blackhole ought to build up a collection of highly dense objects namely stellar remnants or stellar mass blackholes neutron stars and white dwarfs. All of those would be catastrophic for the potential habitability of this system on truly long timescales.
The second has to due with the blackhole “feeding” which is necessary if you want a source of illumination. This will eventually be depleted meaning you probably will not be able to keep it a constant source.
Blackholes are known to be very messy eaters and they tend to self quench their feeding by emitting high energy radiation that cuts off the flow of material until it sufficiently cools enough to begin accreting again. This should give the accretion disk a relativly periodic luminosity.
Now regarding that light emission in particular I remember running fits for Dr. Fukumura in XSPEC when I was in undergrad. Part of that involved a blackbody radiation profile which to fit the observational data from real supermassive blackholes is so hot that the curve actually emits a sizable chunk of kev (or kilo-electron volt) range X-rays.
Thus your planets will need some way to protect themselves from that X-ray “disklight”. (As it’s not really sunlight anymore, 😉 right? )
And this hasn’t even addressed those high energy relativistic particles emitted by the accreting black hole which while they would predominately be ejected outwards away from the orbital plane along the magnetic field lines I couldn’t rule out some of them causing problems depending on the amount of atmosphere the planets would need to safely block them form reaching the surface. So while not a deal breaker it is safe to say at the very least the planets would likely have crazy weather in terms of thunderstorms.
Are there no free supermassive blackholes that aren’t surrounded by galaxies? Setting up around a host blackhole out in the intergalactic void seems like a way of avoiding annoying neighbours.
That is a good question but one we don’t really have an answer for as it is pretty much impossible to see a isolated Black Hole. But we have at the very least seen supermassive blackholes in the process of being ejected at highly relativistic speeds due to interactions and or mergers with other supermassive blackholes during galaxy mergers so at the very least they should exist.
We have also detected some dwarf galaxies with supermassive blackholes including one with comparable mass to Sagittarius A* so they likely exist it would just be a matter of finding one. So on that note the answer would probably be yes you would just need to add yet another relativistic component 😉
Wow, thanks for the info!
Thanks for the comment — all good points worth thinking about.
If you moved the rings farther out so the hill radius was bigger you could use double planets instead of the eyeball planets to get a larger habitable area.
Another good point!
Some fun with math:
If population increases 1% a year a civilization could double to fill a second planet in 70 years, grows a two thousand fold to fill all of those in one ring in 770, and grow a million fold to fill all of the planets in 1400 years.
You can have day and night in such a system. Use the design with the outer ring of stars–but half the ring is dead stars.
Sure, there are plenty of variations that would change the illumination patterns.
This would make a great sci-fi book, although there are definitely a lot of nuances that need to be ironed out and overall I still think that this would not be a livable environment,
For one, this is assuming that the blackhole is absolutely free of any accretion disk or falling in debris from the outside because even a tiny asteroid or a comet falling into it will result in X-ray radiation stripping all of those planets barren over time
Also the time dilation wouldn’t be as great as some people might imagine
the difference of orbital speed between inner and outer rings would be 29000 to 24000 km/s so that’s about from 10% of speed of light to about 8% which calculated to about 2.4 months of time dilation for every 100 years or 0.21% time dilation difference. Important in science and calculations, not really that noticeable in daily life
Lastly, this would create a potential energy difference between the prograde and retrograde side of the planet as the prograde side will receive more energy (slightly more, but enough to matter overtime) due to blue shifting and the opposite side will receive less due to red shifting, so hypothetically, this would create a lot of other chaos that we can’t predict right now
I’m sure there are a lot of other things here that would make it very difficult to be a livable ,functional “habitable” zone but these were the first that came to mind.
All good points! I really like the time dilation idea but the setup needs to be way way more extreme to make much of a difference, like on Miller’s planet in Interstellar. And your point about the redshift/blueshift dichotomy across the planet is also interesting. Lots to think about.
On a separate note, I just saw your video of the Ultimate Engineered Solar System (https://youtu.be/-BvvouoPJW4). Very cool!
Here’s a thought. Rather than just one central blackhole, couldn’t you also have a bunch of “planetary” black holes surrounded by habitable moons to pack the solar system even more densely?
On a side note, since Sean mentioned it, Miller’s Planet is *sadly* uninhabitable – not due to radiations from Gargantua, but due to the blue-shifted CMB! Details here… https://arxiv.org/abs/1601.02897
Are these planets actually habitable??
1) They must be separated by at least 12 Hill radii.
2) Neighboring planets are about ten times closer than the Moon
Ok, the planets are 25,000 miles apart, which is 12 Hill radii, thus the Hill radius must be just over 2000 miles.
Thus you have planets whose Hill radius is far below their surface. I’m not at all sure a planet survives that but lets figure it works. The atmosphere is another matter, I can’t imagine how atmosphere outside the planet’s Hill radius doesn’t get lost quickly.
Am I misunderstanding something badly, or is there a math error here?
Your concerns are valid plus a whole lot more. In fact, very basic astrophysics blows this article’s concepts out of the water, at least for intelligent life, and probably most primitive life. Of the 3000+ exoplanets found so far, we can’t find ONE good candidate that isn’t like super, Saturn, or a monster Rocky planet, or water covered 10-100 miles deep. A black hole could never host a habitable solar type system.
Well, there are indeed many exoplanets found so far that are ~Earth-sized and likely to be rocky (e.g., https://en.wikipedia.org/wiki/List_of_nearest_terrestrial_exoplanet_candidates). I’ve written a long series of blog posts explaining the ideas that culminate with the idea of a system of rocky planets orbiting a black hole (see https://planetplanet.net/the-ultimate-solar-system/). Naturally it’s a stretch to expect to find anything like this million-planet system in nature, as I discuss in the post, but all the ingredients are scientifically valid.
Granted on the rocky planets. And, fairly near in size, though they virtually all are 1.4x or larger — but that is close. Catch there, is that (just read Japanese researchers found that) the magnetic field of rocky planets increases up to near 1.4x of earth, but then rapidly drops off. Ya gotta have a magnetic field for life to exist, at least any advanced life. My frustration with looking for life is that it takes SO many conditions to be met. Even a rocky exoplanet, which is so far, found as either solid rock (no real water) or covered with super deep ocean of say, 50+ miles deep, won’t work. It has been determined that our 79 water/21 land ratio of earth is the best ratio possible. I probably can get you the research citation if desired. But, you have to have both land and water. This hasn’t been found, yet. Plus, besides having the water and UV habitable zone, there is also the photosynthesis habitable zone, plus many others.
… I meant Jupiter, not super. And, a black hole could never host a habitable solar type system.
Yes, this couldn’t happen in nature, and that’s beside the point.
Since they do not dominate their orbits doesn’t that mean that none of these are planets by definition? What would that make this system? An ultimate fantasy asteroid ring?
To have any chance at habitability, you (the aliens) would have to move the stars and planetary disk much farther out for several reasons.
1) A habitable planet must have a magnetic field, which is generated by the planet’s rotation. Therefore the planets must be far enough out to not be tidally locked with the black hole.
2) In systems 1 & 3, and to a lesser extent in system 2, a significant portion of the energy from the stars would be blocked by eclipses. Once you get more than a few rings into the disk, eclipses by planets closer to the suns would block so much heat that nearly all of the rings would be frozen and thus uninhabitable. The disk must be far enough out that planetary spacing is sufficient to minimize the heat blocking effects of eclipses. Either that or the orbit of the stars must be at a 90 degree angle to that of the planetary disk, but I can’t imagine that such a system could be stable.
3) To maintain a stable axis, which is essential to avoiding frequent (in a cosmological sense) mass extinctions that would prevent the evolution of intelligent life, a planet must have a large moon. Therefore the Hill radii need to be large enough to allow a moon to orbit each planet. Not sure if a binary planet would also satisfy both 1) and 3).
4) I’m pretty sure that the stars must be far enough out that their entire atmospheres fit within their Hill radii. Otherwise, the black hole would strip off the coronas and gradually eat the stars.
As you move the stars farther out, you must account for the varying distances to those stars in calculating the habitable zone. For example, if the star ring is at 12 AU, a planet at 11 AU would receive 529X more heat from the nearest star (1 AU away) than from the farthest one (23 AUs away). In your scenarios, you implicitly averaged the distance to the stars to find the heating level, but that doesn’t work because a star’s apparent luminosity and heat are not linear; they are a function of the square of the distance to the star. This would also cause much greater seasonality than we have on Earth as a planet in conjunction with the nearest star would receive far more energy than when it is equidistant from the two closest stars, assuming that the distance between stars is not significantly less than the distance between the planet and the nearest star, though perhaps the short duration of such seasons would mitigate this effect.
Thanks for your thoughts. I can see that you are coming from the point of view of “Rare Earth” — a book by Peter Ward and Don Brownlee — that argues that a long list of specific criteria needs to be met for habitability. Based on N=1, it’s really uncertain whether these things are needed. For instance, Earth’s magnetic field is sometimes argued to be key for life (your first point), but a large number of researchers think that’s not really true. And exactly what starts a magnetic field is not even 100% clear. And it is unlikely that the stability of Earth’s axial tilt really impacts the planet’s potential for life (e.g., climate simulations find plenty of liquid water with basically any axial tilt you want). For your second point, you’re right that some light is lost due to eclipses but I thought I had calculated that for another comment somewhere — I didn’t think it was a big enough amount to matter terribly. As for your point 4, you are definitely right! If the stars are too close to the BH they will be torn apart, or at least slowly drain onto the BH. Did I not choose them to be far enough away? If not, the whole system can simply be scaled outward… But I see that you’ve gone into this and we must be careful not to mess things up!