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I was wondering if there could theoretically exist systems in which an Earth-like planet was in the habitable zone of one (or two) Sun-like star(s), while actually not orbiting around any star. To be more precise, I started asking myself whether a planet like earth could be placed in the inner orbit of a black hole, while one or a pair of Suns would be revolving around the black hole too, on an outer orbit.

My question is not whether such a system could actually exist somewhere, but only if it could theoretically exist, even if it is probabilistically not likely to exist.

So I found this sets of issues:

  1. Planet must be far enough from the black hole ( > Roche Limit )
  2. Harmful radiations from black hole's accretion disk
  3. Planet-Star relative distance must not vary too much...which I guess in turn means that the planet-blackhole distance must be small compared to the average blackhole-star distance

Given these limitations, how would you choose the stars and blackhole masses and their relative distances to the planet, in order to have it suitable for complex, perhaps intelligent forms of life? Do you see other issues to this purpose?

Below I add what I've been able to collect so far (more like a collection of unsolved questions..)

So, about the issue #1, I thought about a rather lightweight blackhole, with M < 10 Solar masses. I once read in a thread that when considering a central body = 10 solar masses, and a satellite with a density of 5000 kg/m^3 (Earth's average density), then the Roche Limit would be as little as 0.8% of an AU. That would be conveniently small, though I guessed that using Earth's average density may not guarantee that the whole planet is kept together (it may lose its atmosphere...). So, just to be sure, would you rather increase this distance to 1/20 of an AU, perhaps less, perhaps more? I'm asking here because I'm not able to conduct these calculation myself.

Then again, I'm not an expert and I don't know if staying out of the Roche Limit is a sufficient condition for orbiting around a blackhole of this small size without things becoming nasty. I know of issue #2 though,and about that I read someone suggesting that If the planet were subjected to tidal lock, then at least the face looking away from the blackhole would be spared from the harming X rays. But how close a planet should get to a blackhole in order to experience a tidal lock within a reasonable amount of time? Would it likely be tidally locked if it stayed at 0.05 AU away from a 10 solar masses black hole? I guess the answer to be yes, but I wanted some validations from the experts here.

Then about issue #3 again, I thought that If there were two Suns (making another binary system of their own), then there would be also a larger insulating flux, perhaps allowing us to keep the Suns more far away (by a quick calculation I think I found that two Suns at 1.4 AU would give the same flux as one Sun at 1 AU). Another possibility is to have a brighter Star class, at expense of its life-time. So for simplicity let's say we can go up to 2 AU (maybe even more) without the Planet getting too cold, or the Stars too short-lived.

Summarizing, I now have a 10 Solar Masses black hole and a (likely) tidally locked earth-like planet orbiting at 0.05 AU from it. Tidal forces should still be small enough not to suck the planet's atmosphere. Perhaps having a planet slightly bigger than the Earth would also help in this account. Then we have a couple of Stars orbiting together around the black hole, at a distance of 2 AU. Relative distance from the twin Stars to the Planet is 2 +/- 0.05 AU. I guess that would be within the limits of the habitable zone, right?

One last curiosity: assuming the black hole has an "average" spin, would it be realistic if the two Stars had a tilted orbit with respect to the black hole spinning axis?

I'm sorry for the many questions, by any means please don't assume I'm expecting a single person to answer all of them.

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  • $\begingroup$ One problem is that the formation of the black hole would blow any planets away, or certainly burn them to a crisp. A black hole with a habitable planet around it would need to be captured and that's probably a rare event. Another problem is 10 solar masses and 0.05 AU is inside the roche limit. The planet would be torn apart. You should move it 10 or 15 times further away. For a star, twice the mass of our sun would be 16 times as bright, so it could be 4 AU away in a kind of Jupiter orbit and you'd get similar sunlight that Earth gets. Slightly bluer but otherwise similar. $\endgroup$ – userLTK Sep 9 '16 at 10:20
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The Roche limit for a solar-mass (or 10-solar-mass) black hole is much larger than the Schwarzschild radius, or even the last stable orbit (which is a few times the Schwarzschild radius). So the Roche radius is a good limit (and, yes, you're right about the limit for an Earth-like planet and a 10-solar-mass BH.)

You needn't assume the BH has an accretion disk; I suspect most of the ones wandering around in our galaxy don't at the moment, just because accretion disks accrete (go into the black hole) over time, and so you only see them when there's some relatively recent source of gas to create the disk. (Most of the Milky Way black holes we know about are in close binaries, where the companion star is close to the Roche limit and is a regular source of gas.) Now, in principle the stellar wind(s) from the star(s) in your system could contribute to a very weak, tenuous accretion disk, but in practice it's unlikely to produce enough radiation to be of any concern, especially as you're assuming a planet with an atmosphere.

So there's no need to worry about whether or not the planet is tidally locked to the black hole, unless you want to specify the length of the day-night period. Of course, if the planet is only a few tenths of an AU away from the BH, it probably will be tidally locked before very long.

The variation in distance from the star(s) you suggest should be fine in "habitable zone" terms. In fact, given that the planet orbits the BH quite rapidly (about 1.3 days for a 0.05 AU orbit around a 10 solar-mass BH), it won't spend much time at either extreme, so you could probably have it go inside and outside of the formal habitable zone and still be habitable, as long as the average insolation is in the habitable range. This would create some interesting day/night/seasonal effects, particularly in the tidally locked case: e.g., at a certain time of "year", the "eastern" hemisphere would experience noon when the stars were only 1.95 AU away, while for the "western" hemisphere noon would happen when the stars were 2.05 AU away. Half a year later, the situation would be reversed.

One last curiosity: assuming the black hole has an "average" spin, would it be realistic if the two Stars had a tilted orbit with respect to the black hole spinning axis?

That really depends on how the system formed in the first place. If the initial pre-BH star and the other star(s) formed from the same collapsing cloud, they would probably have similar spin and orbital angular momenta; if the spin of the BH is entirely inherited from its progenitor (which goes supernova to form the BH), then the orbital plane might be similar to the BH's spin. But if the system got put together later (maybe an encounter between a binary consisting of the BH and another star and the current binary pair, in which the BH's original companion star gets ejected), then the two stars could have a tilted orbit.

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  • $\begingroup$ Perfect answer, I think you put everything in its place. Thank you! $\endgroup$ – Damjam Kajim Sep 9 '16 at 13:56
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To start with, a (non rotating) black hole and a star of the same mass creates the same gravitational potential when you are far away enough. Since we observe planets at $0.05\mathrm{AU}$ around stars of about the same mass as the sun, I think you can expect planets in this region around a similar mass black hole, according to stability criterion. Moreover, the smallest black holes you can physically create have masses of about $3-4 M_\odot$.

There are multiple issues here though. First is how do you physically bring the planet that close to the black hole? I see two possibilities. Either the planet formed directly in the accretion disk of the black hole, and staid there, or the planet formed further away (say around the binary star system) and was captured by the black hole. If you assume that at $0.05\mathrm{AU}$, the accretion disk of the black hole is not that different than the accretion disk around a protostar at early ages, then you can maybe create a planet there (I can think of a few scenario that would prevent it, but it would require a more important study). If the planet formed further away in the disk or in the binary system, it is possible to make it migrate inward. This is actually called ex-situ formation in planetary systems.

If you imagine that the planet is tidally locked (you can try fiddling around with the formula $t_{\text{lock}} \approx \frac{\omega a^6 I Q}{3 G m_p^2 k_2 R^5}$ from wikipedia that gives the timescale for the planet to become tidally locked to get an estimate of how long it would take) and that there is no accretion disk anymore, I guess you could manage to make the system stable.

The final question is then to get the whole system stable on a large. Even though it may extremely unlikely, I think there has to be some stable orbits in the system using 3 body resonance, but this is not my field of expertise. having one or two stars is not making a huge difference, between the flux received from them and the gravitational pull are both in $1/r^2$, so that two stars at 1.4AU give the same flux and the same mean gravitational pull than one star at 1AU. The only difference is in the stability of the system.

To conclude, I guess that the system is very unlikely to exist. Yet if you manage to find a planet there, you'd require to have life:

  1. the accretion disk of the black hole has to be cleared
  2. the planet should be tidally locked to the black hole to be protected from the radiations
  3. the planet should be trapped in a very unlikely stable orbit so that it doesn't travel around in the system AND it has time to get tidally locked to the black hole
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  • $\begingroup$ I'm sorry if I may have not described the system clearly enough.. but the two stars are supposed to be twins very close to each other, orbiting around their own barycentre, while this barycentre orbits the black hole (or rather the centre of mass of the whole system). I think it should be stable this way, but the black hole must have a relatively large mass in order to keep the system centre in its surroundings. I was aware of the planet formation issues though I didn't consider them since they make the system just more rare, but still not unrealistic. Thank you anyway $\endgroup$ – Damjam Kajim Sep 9 '16 at 11:47
  • $\begingroup$ There will probably be lots of stable orbits in such a system. There are, for example, multiple binary systems (two tight binaries each forming one part of a large-scale binary), and that in a sense is what the question is positing (with the black hole--planet pair being an extreme-mass-ratio binary). $\endgroup$ – Peter Erwin Sep 9 '16 at 11:56
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    $\begingroup$ What's the reason for wanting the planet to be tidally locked to the black hole? What "radiations" is that going to protect the planet from? [Edit: OK, the "harmful radiation" is from the accretion disk in the original question. The solution is to say there's no accretion disk.] $\endgroup$ – Peter Erwin Sep 9 '16 at 11:58

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