# How close could a planet get to a super massive black hole before being destroyed

I wanted to create a scene where two god-like characters are talking or fighting on a planet or fragment of planet to which they've teleported that's about to be obliterated by a black hole.

The characters are effectively gods so heat and gravity won't affect them too much. I wanted the black hole and stars being engulfed to fill the sky.

At what distance could a planet still have some surface to stand on as it orbits the black hole before turning completely into dust?

• Depends on how big the black hole is. en.wikipedia.org/wiki/Roche_limit – John Dvorak Feb 5 at 16:42
• I was thinking super massive centre of galaxy size, I will update the question – rag haran Feb 5 at 16:43
• As this scenario seems similar to Interstellar (2014) it may be worth reading The Science of Interstellar by Kip Thorne – Morrison Chang Feb 5 at 17:38
• It's worth noting that if stars are being drawn in by the black hole, the resultant emissions would probably tear apart anything else in the vicinity without worrying about the Roche limit at all. – jdunlop Feb 5 at 18:27
• "I wanted the black hole and stars being engulfed to fill the sky" Exactly what that would look like is probably another question you should ask. Light does weird things near that much mass. – Ray Feb 5 at 18:33

For this, we need to know two things: how close you can get to the center of mass of the black hole before the tidal forces tear you apart, and the radius of the event horizon.

The first is determined by the Roche limit: $$d = R_m (2 M_M/M_m)^{1/3}$$, where $$R_m$$ is the radius of the smaller object, $$M_M$$ is the mass of the larger object, and $$M_m$$ is the mass of the smaller object. This assumes certain simplified conditions, so in practice, you may want to add a margin of error.

The second is defined by the Schwarzschild radius: $$r_s = 2GM/c^2$$, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.

The Schwarzschild radius grows faster than the Roche Limit does, so as the black hole changes in mass, the distance between them changes as well.

Let's assume the planet is roughly comparable to Earth, and look at how close you can get to three different black holes: Cygnus X-1, the supermassive Sagittarius A*, and the ultramassive black hole at the center of NGC 1277.

# Cygnus X-1

This was the first black hole discovered, and is a nice normal-sized example. Its mass is approximately 15 times that of the sun.

$$d \approx 1.29 \times 10^9 m$$.

$$r_s \approx 73 \times 10^3 m$$.

The black hole fits into an area smaller than Earth, but its Roche Limit is a 17 thousand farther away than the event horizon. By the time you're anywhere near the event horizon, the effects of the tidal forces are significant.

# Sagittarius A*

This is the supermassive black hole at the center of the Milky Way. It masses about 4 million times more than the sun.

$$d \approx 8.32 \times 10^{10} m$$

$$r_s \approx 1.27 \times 10^{10} m$$

The two are the same order of magnitude at this size. The black hole is about the size of the Sun, and the Roche Limit is a bit more than the orbit of Mercury at its widest.

# NGC 1277's black hole

The largest black hole ever found is the one in NGC 1277. It has a mass of 17 billion solar masses and makes up 14% of the mass of its host galaxy.

$$d \approx 1.35 \times 10^{12} m$$

$$r_s \approx 5.05 \times 10^{13} m$$

$$d < r_s$$.

Which means that for a large enough black hole, so as long as you aren't actually inside the black hole, the planet isn't going to break apart from the tidal forces.

Orbits near a black hole aren't stable, though. You'll need an orbital radius of at least $$3 r_s$$ if you want it to last. This is only a relevant factor in the largest of the three cases.

Of course, your orbital velocity at the minimum "safe" orbit around a supermassive black hole is around 0.3-0.4c, and you're constantly accelerating in the direction of the black hole, so you'll probably get some weird relativistic effects (I don't know enough general relativity to tell you what they'd be, though). And there's probably no way the planet will be able to keep an atmosphere. And if the black hole is still actively feeding, there will be probably be horrible levels of gamma radiation; I'd recommend an inactive black hole if you want to keep the temperature somewhat reasonable. But for a large enough black hole, as long the planet is outside the event horizon, it'll be an intact uninhabitable, superaccelerated, and possibly molten ball of iron.

Edit: Fixed a major error in calculations that was in the original version (Roche Limit was calculated as $$(R_m * 2 M_M/M_m)^{1/3}$$ instead of $$R_m (2 M_M/M_m)^{1/3}$$, which would have placed the Roche Limit inside the event horizon).

Edit 2: There exists a black hole large enough that the original (and much more awesome) conclusion applies: there does exist a black hole where the Roche Limit is inside the event horizon!

• I'd even say $d \ll r_s$, but that's probably mostly semantics in this case... – a CVn Feb 5 at 20:15
• thanks @Ray, great answer. A molten ball of iron speeding towards an in active black hole, using gravitational lensing to suggest it's position isn't what I was originally thinking but it would make a great scene. – rag haran Feb 5 at 20:38
• Wow! I knew a person could survive the tides near a supermassive black hole, I didn't realize a planet would be basically unscathed. I see no reason the atmosphere would be lost, the only real threat is if the black hole is active. – Loren Pechtel Feb 6 at 3:37
• @ragharan For a normal black hole, the math works out differently. Note that the Schwarzschild radius scales with mass, and the Roche limit with the cube root of mass. For a small enough black hole, the Roche limit of a given body is outside the event horizon, but as you look at bigger black holes, the event horizon expands faster. The article on spaghettification Ray linked has more detail. – Cadence Feb 6 at 21:54
• @LorenPechtel I'd heard that as well...which is why I didn't double check my arithmetic. The Roche limit for Sagittarius A* is outside the event horizon. But fortunately, there does exist at least one black hole large enough that you can survive near the event horizon (somewhat less fortunately for anyone who was in the 14% of its galaxy that it's eaten). But for any size, it's worse for a human than it is for a planet. The Roche limit increases linearly with the radius of the infalling object and inversely only with respect to the cube root of the mass. – Ray Feb 6 at 23:41

You want to know how close your planet would get to a supermassive black hole.

# The Planet wouldn't get near the black hole itself. Not even near the Roche Limit.

And I'll prove it.

• The vicinity of a super massive black hole as the one at the centre of our galaxy is an extremley hot, turbulent, and magneticaly charged zone, it would also compass jets of charged particles, gamma radiation and (very hot) dust.

• The Black hole itself is thought to be in the order of 100,000 solar masses, the theoretical maximum limit being thought to be in the region of 50,000,000,000 (50 billion solar masses) for an ultramassive black hole.

The Accretion disk:

• This is proportional to the size of black hole, some are speculated to be thin and comparativley cool, just like a planetary disk. The one at the centre of the galaxy is wide, thick and hot.

• Velikhov-Chandrasekhar instability (or Balbus-Hawley instability) means that differential magnetic field densities in the disk make the material towards the centre of the disk move faster than that on the outside - more than would be accounted for by different orbital velocities at these distances. This signifies that there is huge friction surrounding huge vortices of superheated turbulent material constantly swirling in a dance around the centre.

• The plasma of the disk, highly electricaly conductive, carries currents of inconcievable magnitude, sporadically discharging to nearby regions of different charge in colossal lightning bolts as the maelstrom whirls about it's centre, ejecting a jet of energy from the poles of the black hole.

How much energy is released in an accretion disk?

Accretion process can convert about 10 percent to over 40 percent of the mass of an object into energy as compared to around 0.7 percent for nuclear fusion processes.

• That is, it (mass for mass) converts nearly 60 times more of the matter going into it than the sun converts it's own mass into free energy - heat, light, gamma rays, the energy of the jet being ejected from the poles.

The Polar Jets:

These radiate energy in a concentrated beam on the axis of rotation of the disk. In extreme cases, the total energy radiated by the disk and by the polar jets can equal thousands of times the total radiant light from all the stars in the rest of the galaxy combined. They can be seen shining brightly from across the farthest reaches of the universe that can be seen.

Relativistic beaming of the jet emission results in strong and rapid variability of the [jet's] brightness.

# Conclusion:

On approaching the accretion disk:

• The planet would approach the disk boundry and melt, quickly being pulled apart by the magnetic fields and dissolve in the swirling motion of the disk in a blaze of gas plasma.

On approaching the polar jet:

• The planet entering the jet would be almost instantly vapourised and be carried away with the jet's (near light-speed) motion.

QED.

The Planet wouldn't get near the black hole itself.

References:

• brilliant answer @FaySuggers thank you, great description of the awaiting destruction – rag haran Feb 5 at 20:45
• @ragharan Thank you, I didn't write anything specific about the gods surfing the lightning or being confounded by the sudden explosive stormwinds as they are taken by the eddies, as that is the business of the author, not of the answerer here. Please feel free to upvote my answer, and to award accepted answer as you see fit after the 24 hour grace period to whom you chose. – Chickens are not cows Feb 5 at 22:22