Discover the Black Hole Growth Rate: Time Frame for Consuming Planetary Mass

[Ryan_m_b]

If a black hole was placed at the centre of an object (like a planet) how long will it takes for the mass of that planet to be consumed?

I’ve tried having a look around the internet for an answer for this (kind of silly) question but can’t find one. I thought maybe of working out the flow of mass in based on the size but drew a blank on how to figure out the rate of in falling matter. Any help would be appreciated :)

Obviously some mass will end up as an accretion disk but really I’m looking for if there is a touchy way of working out the time frame.

 

[Vanadium 50]

Wouldn't it be the same as the time it takes for something at the surface to fall to the center if unconstrained? For the Earth I think it's like 20 or 30 minutes.

 

[Ryan_m_b]

I thought of that but it if the matter is falling to a narrow point presumably it doesn’t all fall at the same rate. Some particles will be pushed ahead and others behind as the area shrinks.

 

[Ibix]

It would depend on the mass of the black hole, surely? Which would have to be fairly small so that you could transport it to the core without disrupting your initial conditions somewhat. You aren't going to get a stellar mass hole in there without ripping the planet apart long before it gets there.

So assuming it's a micro black hole, I don't quite agree V50's answer. We've got a kind of reverse slinky drop going on here - when the hole starts swallowing the core (I'm assuming we've moved the hole in very very quickly) the crust won't be affected until mechanical waves propagate to the surface. That's going to take twenty minutes at least - then it starts to fall. I also don't think the matter will free-fall, because there's too much matter in the outer regions to fit in the inner ones. So a mechanical model of dense matter falling into a black hole is needed, I think.

There's also the thorny question of what you mean by "when the Earth has been swallowed". Naive interpretation of that can be answered with "never", since defining "at the same time as something crosses the event horizon" is a hairy question and at least one possible answer is "nothing ever does cross according to external observers". You could do something like consider the time for all matter to pass the photon sphere, which is outside the event horizon but no inwardly free falling object can escape it (rockets could).

Reviewing that, I'm afraid I don't really have anything more than further questions for you. But perhaps that explains why you can't find an answer easily!

 

[Vanadium 50]

I am assuming an Earth mass black hole. I am further assuming it got in the exact center, and did so by magic. Sorcery, I think, although it might be voodoo.

 

[berkeman]

I am assuming an Earth mass black hole. I am further assuming it got in the exact center, and did so by magic.

It would be an interesting simulation to see what the Earth-mass BH (diameter = 9mm according to Google) collision would look like. The closing velocity could be varied to see what the impact crater effect looks like, with the subsequent infalling Earth matter. I wonder how much matter would escape such a collision...

 

[PeterDonis]

Wouldn't it be the same as the time it takes for something at the surface to fall to the center if unconstrained?

No, because the fall is not unconstrained. Also, not all the matter that falls towards the center will actually hit the hole--that is, not if we assume the hole starts out with a mass much, much smaller than the mass of the Earth, so the initial condition will not severely disrupt the matter in the Earth. Such a hole will be tiny--probably smaller than a single atom--and therefore will be a very hard target to hit. Instead, there will be a lot of matter getting stirred up in the center and forming something like an accretion disk or accretion sphere (note that the Earth is rotating, which means the infalling matter has angular momentum, which makes it even harder for it to hit the hole), which could take a very long time to actually swirl down into the hole.

 

[PeterDonis]

There's also the thorny question of what you mean by "when the Earth has been swallowed".

This isn't as hard as you're making it out to be. Consider the last little piece of the Earth that crosses the hole's horizon. The event where that piece of matter crosses the horizon--call it event L--has a past light cone. Say you are safely out in space watching all this. Once your worldline exits the past light cone of event L, the Earth has been swallowed as far as you're concerned.

The question of what you will see of the swallowing process is a different question, and most of what you say is really about that question. But that question is not the same as the question of what you mean by "when the Earth has been swallowed". You can easily gather enough data from your observations to calculate, with reasonable precision, at what time, by your clock, the condition I describe above will be satisfied, and you can quite justifiably say that, for you, the Earth had been completely swallowed by that time.

 

[jedishrfu]

There was a sci-fi short story by Larry Niven about this notion. Basically some alien communication device had suspended a microscopic black hole in a containment field. The black hole would vibrate to create gravitational waves for message communication.

The device was discovered by some explorers and the black hole was inadvertently freed and it falls to the center of the planet now where it begins to swallow up the planet.

https://en.wikipedia.org/wiki/The_Hole_Man

and here's some discussion on finding these critters:

https://phys.org/news/2010-03-mini-black-holes.html

I couldn't find anything on their consumption rate as it appears they might evaporate faster than they can consume.

 

[PeterDonis]

I couldn't find anything on their consumption rate as it appears they might evaporate faster than they can consume.

Yes, this is a point that I didn't take into account in my previous post. Hawking radiation creates an outward radiation pressure that pushes matter away from the hole; if the hole's mass is small enough, so the radiation intensity is large enough (it gets larger as the hole's mass gets smaller), this pressure is sufficient to prevent any matter from falling in. So a sufficiently small hole will just evaporate away in the center without accreting any matter.

 

[Ibix]

This isn't as hard as you're making it out to be. Consider the last little piece of the Earth that crosses the hole's horizon. The event where that piece of matter crosses the horizon--call it event L--has a past light cone. Say you are safely out in space watching all this. Once your worldline exits the past light cone of event L, the Earth has been swallowed as far as you're concerned.

That's a fairly aggressive simultaneity condition, isn't it? And has the rather odd property that if you are closer to the hole than I am and I send you a radio message saying "the last bit fell in right now" you will receive the message at the same time the last bit fell in. I agree it's a defensible claim and there probably isn't any other, but it doesn't really feel like "it's not in my causal future anymore" is quite the same thing as "it's happened".

 

[PeterDonis]

That's a fairly aggressive simultaneity condition, isn't it?

Viewing it as a "simultaneity condition" might be part of the problem. Why do you care about "simultaneity"?

it doesn't really feel like "it's not in my causal future anymore" is quite the same thing as "it's happened"

If you accept the lesson of relativity that only invariants matter, then "it's not in my causal future any more" is the same thing as "it's happened, as far as I am concerned". Note that I put that last qualifier in there (as I did in my previous post), whereas you did not. It makes a difference.

if you are closer to the hole than I am and I send you a radio message saying "the last bit fell in right now" you will receive the message at the same time the last bit fell in

I will receive the message at the same time by my clock as the time I calculated that I would be just passing out of the past light cone of event L, i.e., the time that the last bit would have fallen in, as far as I was concerned. Which is exactly what I should expect: I am closer to the hole than you are, so you can't possibly tell me about anything that happens at the hole according to you, before it has happened according to me.

Note, again, that I put the qualifier "as far as I am concerned" in, whereas you did not; you just said "the same time the last bit fell in". Adding the qualifier makes a difference.

 

[BillTre]

Hawking radiation creates an outward radiation pressure that pushes matter away from the hole

Would not radiation released as matter falls in also produce an outward pressure?
I am thinking of the quasar blaster beams (don't know what they are really called) coming out of the poles of black holes that are actively eating stuff.

 

[Ibix]

If you accept the lesson of relativity that only invariants matter, then "it's not in my causal future any more" is the same thing as "it's happened, as far as I am concerned".

Is it? "It's happened as far as I'm concerned" is true in an invariant sense once the event is in my past light cone (which it never is in this case, of course). "It hasn't happened yet as far as I'm concerned" is true in an invariant sense until the event leaves my future light cone. I'm uncomfortable saying that "not in my future light cone" means "has happened for me" because one can make the same argument in reverse from the past light cone - "not in my past light cone" means "hasn't happened yet for me". And that leaves all events spacelike separated from me as both has and hasn't happened for me, depending on my choice of starting point.

 

[PeterDonis]

Would not radiation released as matter falls in also produce an outward pressure?

Yes. This is an important reason why accretion disks and other concentrations of matter form around black holes, and remain stable for long periods of time.

 

[jedishrfu]

Would not radiation released as matter falls in also produce an outward pressure?
I am thinking of the quasar blaster beams (don't know what they are really called) coming out of the poles of black holes that are actively eating stuff.

I think they're called jets:

https://en.wikipedia.org/wiki/Astrophysical_jet

 

[PeterDonis]

"It's happened as far as I'm concerned" is true in an invariant sense once the event is in my past light cone (which it never is in this case, of course).

I would say that "I know that it has happened" is true in an invariant sense once the event is in my past light cone. Knowing that something has happened is not the same thing as that something happening.

If "it has happened as far as I'm concerned" bothers you, try "I can no longer do anything about it". We're just talking about ordinary language anyway, not physics. You can't causally affect an event once you are out of its past light cone, and you can't know an event happened until you are in its future light cone; that's the physics. There is no invariant fact of the matter in the physics about "when something happens" beyond that.

that leaves all events spacelike separated from me as both has and hasn't happened for me, depending on my choice of starting point.

Or it means that there is no physical fact of the matter at all about "when something happens"; the only physical facts of the matter are what events you can or can't do something about, and what events you can or can't know about.

 

[mfb]

We had this discussion a while ago. The matter that falls into the black hole heats up before and emits radiation, creating a pressure that slows further mass. How much should depend a lot on the spin of the black hole. The matter that falls in come with a very low angular velocity.

 

[Ibix]

Or it means that there is no physical fact of the matter at all about "when something happens"; the only physical facts of the matter are what events you can or can't do something about, and what events you can or can't know about.

Fair enough. But that simply seems to be an even stronger restatement of what I said in #4 that you objected to - rather than "when does it fall in" being a thorny question, you're first narrowing it to "...for me" and then saying the question's meaningless even restated that way and we can only answer "when's the last moment I could (in principle) stop the last piece falling in".

Of course, since the answer seems to be "a long time because an accretion disc forms" this may be a purely academic issue.

 

[anorlunda]

There was a sci-fi short story by Larry Niven about this notion.

I remember that story, or one like it from Analog SF. The story was fun. The BH fell in from space. It went into a highly elliptical Earth orbit that penetrated the Earth. It didn't collide with any mass because it sucked in material creating a vacuum around its trajectory. Needless to say, every time it entered the Earth, and every time it left, it caused local havoc. And it became more massive with each orbit.

I don't remember how the story ended.

 

[jedishrfu]

I remember that story, or one like it from Analog SF. The story was fun. The BH fell in from space. It went into a highly elliptical Earth orbit that penetrated the Earth. ...

I think that's a different story Larry Niven's story is on Mars and is a kind of murder mystery where two explorers are arguing over what's in the containment field and one explorer turns it off causing it to kill the other one.

Here's a list of stories with Black Hole plots:

https://en.wikipedia.org/wiki/Black_holes_in_fiction

Maybe you can find it there.

 

[Klystron]

I remember that story, or one like it from Analog SF. The story was fun. The BH fell in from space. It went into a highly elliptical Earth orbit that penetrated the Earth. It didn't collide with any mass because it sucked in material creating a vacuum around its trajectory. Needless to say, every time it entered the Earth, and every time it left, it caused local havoc. And it became more massive with each orbit.

I don't remember how the story ended.

I read that Analog story or one like it a million (subjective) years ago.

In one version the Earth people capture the hole in a giant EM field and install it in a power plant. Earth children visit and throw tiny rocks at it analogous to feeding herring scraps to California sea lions at an aquatic park. I forget why radiation is not an issue.

In another story by SF author John Varley, the BH is conscious, intelligent and, if memory serves, female or at least anthropomorphic . She approaches the solar system, messes with some outlying space stations, reveals herself to the crews then takes off to explore the Milky Way galaxy. Story title "The Black Hole Passes" anthologized in "The Persistence of Vision".

I think the story was titled "The Black Hole Approaches", possibly part of Varley's "Ophiuci Hotline" series .

 

[stefan r]

Material from the equator won't fall into the black hole until it is de-spun.

The material that does fall into the hole releases a considerable amount of energy. the Penrose process is supposed release up to 20% of matter's rest mass. The energy released could be much lower. Our Sun fuses 6.2 x 10¹¹ kg of hydrogen per second which releases 4.23 x 10⁹ kg of mass per second. If the black hole is gaining around 10¹² kg/s then the hole should have much higher luminosity than the Sun.

The presence of an Earth mass black hole increases the gravitational binding energy. But it is currently only 2.2 x 10³² J. If matter was converted to energy at 100% then you only need 2.5 x 10¹⁵ kg to explode Earth or 10¹⁶ since the mass is doubled and binding energy is 4X. Even at only 1% matter to energy conversion 10¹⁸ kg explodes Earth. The Earth has 6 x 10²⁴ kg mass. So the first estimate should be that much less than 1 part per million actually falls into the black hole. Most of the rest will blow out into space. The jets allow energy to escape without lifting other material so significant quantities material could collect in an accretion disc. The disc will still only be a fraction of the original mass.

 

[PeterDonis]

that simply seems to be an even stronger restatement of what I said in #4 that you objected to

I view it as showing that the question you said in #4 was "thorny" is not actually a meaningful question at all, and that the meaningful questions involved are easier to resolve. I can't stop people from poking themselves in the eye with a pointed stick, but I can at least point out ways in which they could avoid doing so.

 

[OCR]

Larry Niven's story is on Mars...

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Lol... just a narration, but interesting. .

The Hole Man - Larry Niven - YouTube

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