Ramifications of black hole stellar system

Recently, I got around to seeing the movie Interstellar. In it, the characters of the movie visit a stellar system that appears to be built around a black hole instead of a star. On top of this, their mission is to find a habitable planet in this system. Question: I was wondering how stable the system, as a whole, would be in terms of operation and would any of the planets be habitable at all due to the obvious differences between a star and a black hole.

In the movie the black hole is supposed to be a 'supermassive rotating black hole'. As for being a binary star system, it's never stated as such nor depicted as such. It's depicted visually as the black hole alone being the center. However one of the characters does mention a "Neutron Star" as part of the system so it could possibly be a binary star system. The ambient lighting for the planets is generated by the accretion disk of the black hole. The size and rotation speed of the event horizon are not defined in the movie. At least not that I can remember.

As for the planets, their proximity varies, with the first planet depicted as being so close to the event horizon that it is effected by time dilation. The distances of the other two are not specified directly, but traveling to the second planet out seemingly takes days, while traveling to the third planet out is stated to take months, if I remember correctly. As for their surface gravity, the first planet is depicted as having higher surface gravity than Earth but not so much higher that movement is impossible, just strained. The second planet is depicted as being '80% Earth's gravity', if memory serves me correctly.

To better help define certain variables relevant to the question I found an info-graphic related to the movie that illustrates the size of the black hole and it's rotational speed: http://tinyurl.com/pqph8wl

• @TildalWave Oof, no spoilers are going to be tough. The system - from what I got out of the movie - wasn't a binary system per se, in the same way that Cygnus X-1 is a binary system. The black hole was the only object that could be considered the primary, and I'm not sure if any other planets were mentioned. – HDE 226868 Jan 6 '15 at 23:13
• @HDE226868 Then either the movie is terribly dark, or it simply doesn't make much sense? I mean, if it's not the former, then where does the light come from? Torchlights? I don't know much about the movie's plot but I got the impression (also from this question) that astronauts spend quite some time in that system? Or it's just some FTL jump to the place, "Right, nothing to see here!" (pun ... optional), and they're off to another system? – TildalWave Jan 6 '15 at 23:34
• @TildalWave I got the impression that the black hole's accretion disk was apparently emitting so much light that the planet was illuminated. . . Not sure how plausible that was. For the last bit of your comment - they use a wormhole opening near Jupiter to travel to a variety of worlds. (Cringe). Matthew McConaughey also travels through dimensions, so aside from the stuff Kip Thorne was consulted on, there's some scientific inaccuracy - "artistic license". They end up spending - for outside observers - quite some years on the planet, though apparently the dilation makes it hours for them. – HDE 226868 Jan 6 '15 at 23:39
• @HDE226868 I see! OK the accretion glow seems plausible but I'm not so sure about huge time dilation. Not one with the other at least, because accretion does imply a fast spinner, if for nothing else then due to angular momentum of the in-falling matter and you wouldn't be able to survive so close next to a fast spinning black hole at all, just the frame drag alone would cause all kinds of colorful effects on you. And black holes also tend to be rather messy eaters, if it eats matter, then lots of it is also all over the place, at near light speed relative velocities. I.e. not habitable LOL – TildalWave Jan 6 '15 at 23:56
• @TildalWave Yeah, I just did the time dilation calculations for a Schwarzschild black hole - screw rotation for now - and that planet is in trouble. Even without taking into account those effects. I suppose you could avoid the "messy eating" by having the planet orbit above the disk's plane, but that would be tough, and unlikely. – HDE 226868 Jan 6 '15 at 23:59

For those who haven't seen it:

Some human explorers land on a planet orbiting a black hole. The black hole is surrounded by a large accretion disk. The planet orbits at a distance such that going any closer to the black hole will mean that your odds of getting out are slim; it's also composed of water. Finally, time dilation from the black hole means that even though the characters spend about two hours on the planet, a decade or so passes for their colleague on board.

The basic answer is that a planet can orbit a black hole. There are stable circum-black-hole orbits, just as there are stable orbits around just about any celestial body. There's a problem: A black hole typically forms as a result of a supernova. This will eject most nearby planets out of the stellar system. Alternatively, it's unlikely that a planet could be captured by a black hole and be in a stable orbit, so the whole premise - while possible - is highly unlikely. Then again, it's improbable that a planet will be made out of water, a wormhole will open up near Jupiter, or Matthew McConaughey will star in a decent sci-fi movie, so why should anything else in the story be normal?

However, you can't just put a planet anywhere near a black hole, give it a strong enough push, and hope it orbits. The innermost orbit is at the boundary of the photon sphere. On this sphere, only photons can orbit. Inside it, nothing can orbit. However, the only stable orbit is twice as far away, at $$2r_p$$.

The radius of the sphere is $$r_p=\frac{3GM}{c^2}$$

We'll assume that the object is not rotating (I don't remember exactly if it is or isn't rotating, but it's simpler in this demo to say it isn't.). The formula for gravitational time dilation is $$t_0=t_f \sqrt{1-\frac{3r_0}{2r_f}}$$ where $$r_0=\frac{2GM}{c^2}$$ Assuming that $$t_0$$ (the time for the observer inside the field) is two hours (7200 seconds) and $$t_f$$ is ten years (315360000 seconds), $$\frac{t_0}{t_f}=\frac{1}{43800}=1-\frac{(3)2GM}{(2)r_fc^2}$$ Simplifying, and saying that $$\frac{2GM}{c^2}=\frac{2}{3}r_p$$, $$\frac{r_p}{r_f}=\frac{43799}{43800}$$ $$r_f \approx 1.0000228315715 r_p$$ which is outside the photon sphere, but just barely. However, it's well inside $$2r_p$$, and so most likely instable. The planet is, in short, not going to survive for long. And so giant waves - mini-spoiler - are the least of Matthew McConaughey's problems.

Post-question-edit modifications:

It couldn't have been a supermassive black hole; these form at the center of galaxies. It could have been a stellar-mass black hole, though an intermediate-mass black hole is also likely - if not likelier, if the massiveness is emphasized.

The existence of the neutron star is interesting. If the black hole were intermediate-mass, I would expect that it would have gobbled up the neutron star by now - and the planets, too. So I'd bet the black hole is a slightly-more-massive-than-average stellar-mass black hole. I highly doubt that multiple planets could orbit a black hole - for the reason I gave above; the supernova would have destroyed them or flung them out of the system.

Would any of the planets be habitable? I doubt it. The accretion disk could heat up enough to provide some light, but there probably wouldn't be a lot. I'll write up the calculations either later today or possibly tomorrow, as I'm a bit LaTeXed-out after writing a math-heavy answer on Worldbuilding to find the luminosity, but I suspect it'll be negligible - as will Hawking radiation, in case any smart-aleck was planning on bringing that up.

• +1 for saving me the bother to watch the movie :) Also, wait... giant waves? As in, not stationary w.r.t. the surface tidal bulges but proper, moving waves? How tall did they stretch the poor Matthew McConaughey as the tides came? – TildalWave Jan 7 '15 at 0:21
• The black hole is almost certainly rotating, but for the nonrotating case, the innermost stable circular orbit is at $r = 6M$, not the $r = 3M$ photon sphere. The photon sphere is the innermost possible circular orbit, but it is always unstable. – Stan Liou Jan 8 '15 at 0:10
• The calculations in this answer are done for a non-rotating black hole. For a rotating BH, there are stable orbits much closer to it. Kip Thorne has said that the BH in the movie is the rotating kind. So the orbit of the water planet can be stable. – Florin Andrei Jan 8 '15 at 2:21
• What Florin said--rotating black holes can have orbits much closer to the event horizon, see the equation for the "innermost stable circular orbit" on this page. And Kip Thorne's book "The Science of Interstellar" indicates that the black hole is a supermassive one, with a mass of about 100 million Suns, and that it's supposed to be spinning by only 1 part in 100 trillion less than the rotation rate of an "extremal" rotating black hole (beyond that rate it becomes a naked singularity). – Hypnosifl Jan 8 '15 at 20:12
• If anyone's interested in the detailed calculations showing that the math makes sense for the planet to be in a stable orbit close enough to the event horizon to have such a large time dilation factor, see this thread, especially post #8 where I worked out the numbers given some equations that had been posted by stevebd1 in post #2. – Hypnosifl Jan 8 '15 at 20:14

One consequence of being so close to a black hole as Miller's planet in the movie is that the cosmic background radiation would be blueshifted and contribute significantly to keeping the planet warm. In their paper "Life under a black sun", Tomáš Opartrný, Lukáš Richterek and Pavel Bakala estimate that Miller's planet would be uncomfortably hot:

The bad news for the visiting astronauts is that it is too much energy: the incoming flux density (power per unit area perpendicular to the incoming radiation) is $$\Phi \approx 420\ \rm kW/m^2$$, i.e., about 300 times bigger than the solar constant. This value can be used to find the equilibrium temperature of a planet radiating its energy as a black body, $$T = \sqrt[4]{\Phi/(4\sigma)} \approx 890 \rm ^{\circ}C$$. Thus, the tidal waves observed on the planet might be, e.g., of melted aluminum. Moreover, the astronauts would be grilled by extreme-UV radiation.

A follow-up paper, "Habitable zones around almost extremely spinning black holes (black sun revisited)" by Pavel Bakala, Jan Docekal and Zuzana Turonova further explores the scenario and determine that black holes with masses exceeding 1.63×108 solar masses would have their habitable zones (solely due to blueshifted background radiation) beyond the tidal disruption radius for an Earth-like planet. They note that on such a planet, the bulk of the radiation would be in the ultraviolet part of the spectrum, but there would still be some flux in the visible and infrared. They also estimate the timescale for orbital decay of the planet due to gravitational radiation, estimating that the timescale to cross the habitable zone is ~1010 years.

The scenario including an accretion disc is explored in "Life on Miller's Planet: The Habitable Zone Around Supermassive Black Holes" by Jeremy D. Schnittman. Unsurprisingly, adding extra radiation from the accretion disc makes things even worse for conditions on Miller's planet.

The temperature [of the accretion disk] scales with black hole mass and accretion rate like $$T_{\rm peak} \approx 2 \times 10^5 \left( \frac{M}{10^8\ M_\odot} \right)^{-1/2} \left( \frac{\dot{M}}{0.1 \dot{M}_{\rm Edd}} \right)^{1/4} \rm K$$ So if we want the accretion disk to look more like a main sequence star (indeed, the visualization of Gargantua’s accretion disk does appear a very similar color to our own Sun), we need to scale down the accretion rate by a factor of a million. Yet even after doing this, Miller’s planet, orbiting just outside the horizon, will be completely surrounded by a 6000-degree blackbody radiation field: hardly hospitable to life!

The scenario depicted in the movie, where the planet is located outside the accretion disc does not seem too likely.

Schnittman also raises the issue that a supermassive black hole in a galactic core would be surrounded by stars, and these would also be blueshifted. The apparent temperatures of sun-like stars would be extremely high, which means the bulk of the energy received would be in the form of damaging short-wavelength radiation:

For a planet in our own galactic center, the night sky would actually be 100,000× brighter than that of Earth! In Figure 12, we plot the mean flux on the planet’s surface as a function of distance from the BH horizon for $$a/M = 1$$. The habitable zone for a tidally locked planet is marked in red. Note that the HZ for planets irradiated by blueshifted starlight is at a relatively large radius of $$1.1r_g$$, where the time dilation is a measly factor of 23, but the blackbody temperature of a sun-like star would still be a whopping 600,000K. Again, technically habitable from an energy balance point of view, but challenging from a photochemical perspective.

It's definitely an interesting scenario for thought experiments and science fiction stories, though I suspect it is not a scenario that is going to be frequently realised in real life!

If you are orbiting a 1-sun-mass black hole at 1 AU, gravity-wise it would be just like it is for us orbiting the sun. Gravity is gravity and mass is mass. The planets would have to have been captured after formation of the black hole or moved in from distant orbits. However, the accretion disk would be likely to be emitting lots of X-rays and not anywhere near enough visible light, so life on that planet would be pretty grim.