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### Author Topic: How long would it take for a black hole on Earth to swallow the Sun?  (Read 8865 times)

#### nietzsche

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##### How long would it take for a black hole on Earth to swallow the Sun?
« on: 21/05/2010 12:30:02 »
Brent Kee  asked the Naked Scientists:

Hi,

Theoretically, if a black hole were to form on earth right now, about how long would it take for the sun to be sucked in, and even the whole solar system, maybe out to the Kuiper belt....??

Thanks,

Brent

What do you think?
« Last Edit: 21/05/2010 12:30:02 by _system »

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #1 on: 21/05/2010 13:00:34 »
This is not an easy question to answer. A micro-black hole would evaporate so cause no damage. We don't know how to make a bigger (more dangerous) one should anyone want to. When formed naturally it takes something rather bigger than our sun to collapse. If it were posible to create a stable BH by some means, I guess it would suck in the surroundings fairly quickly and I guess would break up the earth. I suspect that that the whole lot would survive a long time as a swirling mass of debris and that this mass, being no heavier than the earth from which it was created, would continue to circle the sun for a very long time (maybe billions of years).

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #2 on: 21/05/2010 16:42:19 »
If a Black Hole were to be formed on Earth then it could not exceed the mass of the Earth, and so it would make no difference to the Sun, or any of the other planets and moons.

#### imatfaal

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #3 on: 21/05/2010 18:26:46 »
Lee - you are clearly correct;  why then is the common (and incorrect) perception of black holes sucking up everything around them? Surely anything that gets sucked up by a black hole would have crashed into the star if the star hadn't undergone gravitational collapse - is this fallacy just a part of science fiction colouring our minds or is there a more rational explanation?

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #4 on: 21/05/2010 18:51:01 »
Its residual angular momentum that stops things getting swallowed up. I already said it would probably take billions of years for the debris associated with the earth to conjoin with the sun. I sometimes think my posts must be so boring that nobody reads them!

However, the poster asked what would happen if a BH was formed on the earth. This would have to be done artificially as there is no natural way for it to happen (as far as we know). I would guess it would result in considerable damage to the earth.

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #5 on: 22/05/2010 18:18:31 »
graham.d, I don't think your posts are so boring that nobody reads them; far from it, in fact.  But that's not to say that I necessarily agree with every word

According to Hawkin's theory on BH evaporation, the smaller a BH is, the hotter it is, and as a consequence, the quicker it would evaporate.  However, this doesn't mean that a microscopic sized BH would evaporate immediately; it would depend upon just how small it was when created.  If it were to be able to start accreting mass before it evaporated then it would grow, which would slow down the rate of evaporation.  If the rate of accretion is greater than the rate of evaporation then the BH will continue to grow until there is no more local mass to accrete.

Now one of the things in your first post that I'm not sure I'd agree with is the idea that if a BH was created on Earth and started accreting the Earth's mass it would result in a swirling cloud of debris.

While I'm pretty sure that the in-falling matter would create a degree of radiation back-pressure, I don't think it would be strong enough to liberate enough of the Earth's matter to produce a debris cloud of any significant size.

I also think that...

Quote
I would guess it would result in considerable damage to the earth

...is a bit of an understatement.

Re the perception that BH sucks up everything around them, well this is true, at least to the extent that every gravity well will attract matter towards it.  The difference between BHs and ordinary stars is that the mass is concentrated into a much smaller volume, which means that the gravitational gradient can become much greater than with a 'normal' stellar body while still being 'outside' it.

For example, if the Earth were to be turned into a BH it would have a diameter of just under 9 mm, instead of 12742 km.  Now, if you were located 6371 km from a 9 mm Earth-mass BH you'd still only experience a 1 G force, which is what we experience on the Earth's surface.  However, because the BH is only 9 mm in diameter, we can get much closer than 6371 km from it.  In fact we could get to just under 4.5 mm away and as a consequence we would experience much higher gravitational forces, from which it would be correspondingly harder to escape.

It is because matter can come much closer to a BH then, where it experiences much higher gravitational forces, that a BH can 'suck' everything in, but matter that doesn't come any closer to the BH than it would have done before the BH formed will just experience the same gravitational force as it did before.

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #6 on: 22/05/2010 20:10:26 »
I'm glad somebody reads them Lee :-)

My thought was that a black hole formed somewhere on the earth's surface would start to accrete and move towards the centre of the earth, however Coriolis forces would divert it. It is too difficult to imagine, and a challenging simulation for a potential PhD student, but I suspect there would be much slicing up of the earth but that the BH, and much accreted material would end up somewhere near the gravitational centre. I think you are right that much of the earth would be drawn in to an accretion disk. My thinking was that there would be sufficient angular momentum to hold a significant mass at some distance (the moon, for example, would reamain intact I think).

#### imatfaal

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #7 on: 24/05/2010 15:39:30 »
The same was said by Andrew Pontzen (naked astronomy expert) on a similar post that looked at the sun being magically changed into a black hole.  there is no reason for anything in orbit to change its position - its a strange thought; all those artificial satellites, the moon, the iss, and the space garbage orbiting a black-hole the size of an aniseed ball.  And I did read your post - and my question wasn't meant as a slant.

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #8 on: 24/05/2010 19:26:55 »
graham.d - I don't think, and didn't say that much of the Earth would end up in an accretion disk (an accretion disk is a spinning disk of material outside the Event Horizon of a rotating Black Hole).  In fact, I don't think an accretion disk of any significance would form at all.

If the scenario has the micro BH forming on the surface of the Earth then what I believe would happen is that the micro BH would start accelerating towards the Earth's center, initially at 9.81 m/s2, but with its rate of acceleration slowing as it approached the center.  Note that although the rate of acceleration would slow as the BH approached the center of the Earth this wouldn't mean that the BH would slow down; it would still be moving at its fastest as it passed through the center.

Neither would the micro BH be slowed by the material of the Earth as it passed through it because the BH wouldn't be pushing aside the material in its way but absorbing it.  After it has passed the center of the Earth it would then start to slow down but because it wouldn't have been slowed by its passage through the Earth it would re-emerge near the antipode of its entry point, come to a stop, then sink back through the Earth again, starting a cycle of oscillation back and forth until the Earth is entirely absorbed.

I don't think that Coriolis forces would come into play very much but because the Earth would continue to rotate, and assuming that the BH would have had some initial angular moment with respect to the Earth, it wouldn't follow the same path through the Earth on each transit.  As such, it wouldn't 'slice' up the Earth, but would leave a microscopically small tunnel behind it, at least near the surface of the Earth; deeper down the pressure would collapse the tunnel left by the BH's passing.

Because the BH would initially be so small it would only accrete material correspondingly slowly at first.  However, once that process has started there's no way of stopping it and as the process continues the rate at which further material is accreted will accelerate.  As the BH grows, the tunnel through the Earth that it leaves behind it will get bigger and bigger and, as it then collapses under the gravity induced pressure from the material above it, the Earth will start to shrink.

Eventually, nearly all of the Earth will be accreted by the BH, with just an insignificant proportion of it being forced away from the BH by the back-pressure of radiation released by the in-falling matter.

I think it's even likely that many of the artificial satellites in orbit around the Earth would survive and, as the Earth's atmosphere eventually gets 'hoovered' up by the BH, some of those in relatively low decaying orbits may actually have their orbital decay halted.  That would depend upon how quickly the Earth's atmosphere was accreted though, and I've just got no idea how quickly that would happen.

#### syhprum

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #9 on: 24/05/2010 20:03:55 »
Go to it LHC it looks like we are in for an interesting time.

#### Bored chemist

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #10 on: 24/05/2010 21:41:38 »
Just a thought. If by some bizarre glitch, the Earth turned into a black hole (of the same mass as the Earth)and we then waited for the Sun to go red giant, what would happen?

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #11 on: 24/05/2010 23:13:16 »
Lee, any BH that is formed anywhere other than at the poles would not fall in a straight line to the centre because of the Coriolis force. If formed on the equator, for example, it has a significant tangential velocity. Also remember there is no mechanism for the rapid loss of the earth's angular momentum which would be a very rapid rotation of the earth's mass if only a few millimeters diameter. I am not sure whether all the earth's mass would be accreted or whether some would remain in orbit. I don't know how you can be so sure without a simulation or elaborate calculation. Anyway it's all speculation (I hope).

#### Geezer

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #12 on: 25/05/2010 03:23:45 »
Also remember there is no mechanism for the rapid loss of the earth's angular momentum which would be a very rapid rotation of the earth's mass if only a few millimeters diameter.

Oooo! That's a great point. I realize much of this discussion in conflict with physics, but if the the earth was compressed to something the size of a ping-pong ball for example, it would have to rotate rather quickly, so the "day" would be quite a bit shorter. I'll have to "crack the books" to try to figure out how short.

#### syhprum

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #13 on: 25/05/2010 06:08:04 »
My back of an envolope calculation says it would be rotating at 10^15 rpm with a surface speed of 1.7c.

#### Geezer

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #14 on: 25/05/2010 06:30:44 »
My back of an envolope calculation says it would be rotating at 10^15 rpm with a surface speed of 1.7c.

Isn't that a bit quick?

(I don't mean the frequency. I was referring to the speed of your computation.)

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #15 on: 25/05/2010 14:25:25 »
Just a thought. If by some bizarre glitch, the Earth turned into a black hole (of the same mass as the Earth)and we then waited for the Sun to go red giant, what would happen?

I think the BH Earth would win.

In the same way that the microscopic BH wouldn't have been slowed by passing through the Earth, it wouldn't be slowed by orbiting through the outer layers of a red giant.  This isn't to say that the BH Earth's orbit would be unchanged but I don't think it would spiral in; instead, you'd get mass transfer from RG Sol to BH Earth, shifting their common barycenter away from RG Sol and towards BH Earth.

At least, I think that's what would happen.

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #16 on: 25/05/2010 14:59:31 »
Lee, any BH that is formed anywhere other than at the poles would not fall in a straight line to the centre because of the Coriolis force. If formed on the equator, for example, it has a significant tangential velocity. Also remember there is no mechanism for the rapid loss of the earth's angular momentum which would be a very rapid rotation of the earth's mass if only a few millimeters diameter. I am not sure whether all the earth's mass would be accreted or whether some would remain in orbit. I don't know how you can be so sure without a simulation or elaborate calculation. Anyway it's all speculation (I hope).

Well, the Coriolis Force isn't really a force at all; it just describes the apparent motion of something that's traveling in a straight line from a rotating frame of reference, so that the object that's moving relative to the rotating observer appears to move along a curved non-straight path.

I do agree though, that if the BH was formed anywhere off-axis then it wouldn't fall straight down because, as I mentioned, it would have acquired some angular velocity from the Earth's rotation.

That's a good point about the conservation of angular momentum, and syhprum's calculations are very interesting.  With that > 'c' result, they suggest that there's something else that's not yet occurred to us that still needs to be factored in.

In the end you're right: it is all speculation, so perhaps we'll just have to agree to disagree about how much of the Earth avoids direct accretion.  I'm still going to hold out for near total accretion, at least for the time being, because the only mechanisms I can think of that could disperse any of the matter of the Earth away from the BH are either the radiation back pressure from the in-falling matter, or by some of the matter being catapulted away from the BH in near-miss gravity assist slingshot events (which would mean that matter accelerated in this way would have to avoid colliding with any other matter, which could only happen at or near the surface of the Earth before it totally collapsed), neither of which I think would liberate a significant proportion of the Earth.  I haven't tried to do the maths with real numbers though, so these two mechanisms might possibly have a greater effect.  Of course, if someone can come up with other mechanisms I might change my mind

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #17 on: 25/05/2010 15:00:40 »
Lee, the centre of gravity of the whole system will not change. The centre of rotation will be the same so it will tend to look like the BH will spiral in. This is no normal BH but one that has a tiny mass by comparison with "normal" BHs.

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #18 on: 25/05/2010 15:11:10 »
Lee, the centre of gravity of the whole system will not change. The centre of rotation will be the same so it will tend to look like the BH will spiral in. This is no normal BH but one that has a tiny mass by comparison with "normal" BHs.

Do you mean in BC's Red Giant Sol scenario?

At first, I agree that there would be little change, but as the BH Earth would be inside the outer layers of the RG Sol mass transfer would occur, so there would be a shift in the barycenter away from Sol and towards Earth.  BH Earth might start off as being tiny compared with Sol, or even other 'normal' BHs, but while it can accrete mass from RG Sol it's not going to stay that way and I think the outcome, that of BH Earth absorbing RG Sol, is inevitable and just a question of time.

#### syhprum

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #19 on: 25/05/2010 15:33:44 »
I rather hoped that the Solar systems salvation would be that the Earth mass BH would have a short life but the Hawking equation predicts 5.668638e+50 years.
For it to start to acreate mass presumably it would have to begin life with a surface temperature of less than 300°K, this would correspond to an initial mass of 4.090677e+17tons.
« Last Edit: 25/05/2010 15:43:45 by syhprum »

#### graham.d

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #20 on: 25/05/2010 15:50:11 »
"[...] so there would be a shift in the barycenter away from Sol and towards Earth". No there wouldn't, Lee. The Centre of Gravity of the total system will stay the same as would the total angular momentum. These can only change by outside influence.

#### syhprum

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #21 on: 25/05/2010 16:10:29 »
My figure for the rotational speed may be somewhat in error as I took the the radius of an Earth mass BH as 0.5 cm whereas Hawking quotes 0.8864564 cm, perhaps someone would care to repeat the calculation with greater precision.

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #22 on: 25/05/2010 16:11:47 »
I rather hoped that the Solar systems salvation would be that the Earth mass BH would have a short life but the Hawking equation predicts 5.668638e+50 years.
For it to start to acreate mass presumably it would have to begin life with a surface temperature of less than 300°K, this would correspond to an initial mass of 4.090677e+17tons.

Yes, even though a BH Earth would only be ~9 mm in diameter it would be far too big to evaporate quickly.  I'm not sure where you've got that 300K figure from though.  It seems far too low and I'd expect a microscopic BH to be able to accrete matter at much higher temperatures, but I'm sure you've used that figure for good reason.

#### imatfaal

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #23 on: 25/05/2010 16:54:17 »
I hope I have got this right - please be gentle with corrections its been a long time since i have tried this

and putting in figures

Hopefully no one read the first attempt with the wrong second radius

Matthew

couldnt cope with ascii equations.

« Last Edit: 25/05/2010 16:58:08 by imatfaal »

#### LeeE

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #24 on: 25/05/2010 17:31:14 »
"[...] so there would be a shift in the barycenter away from Sol and towards Earth". No there wouldn't, Lee. The Centre of Gravity of the total system will stay the same as would the total angular momentum. These can only change by outside influence.

It doesn't require outside influence to change the CoG of the system.  Consider a pair of scales, with with two open jars of water, of equal mass in each pan, so that the scales are balanced.  Now imagine that the the chains supporting the two pans are of unequal length; this won't affect the balance of the scales because the vectors of the two masses are still identical i.e. downwards, but it does allow us to put a siphon tube between the two so that the water flows from the higher vessel to the lower one.  As the water flows the scales will become unbalanced, even though the total mass of the system hasn't changed and no external forces have acted upon the system.

From: http://en.wikipedia.org/wiki/Center_of_mass#Barycenter_in_astronomy

Quote
The center of mass of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The center of mass is closer to the more massive object

...so while the relative masses of Sol and BH Earth remain unchanged their barycenter will stay where it is, apart from the usual perturbations from the other planets.

However, once Sol becomes a red giant, extending out to, and probably slightly beyond the Earth's orbit, then BH Earth will start to accrete mass from RG Sol.  As BH Earth acquires mass and RG Sol loses it, their barycenter must change too.

For the barycenter to stay in the same place it would require BH Earth to move inwards, equivalent to shortening the arm of the scales supporting the lower vessel, while RG Sol must start to move outwards, equivalent to lengthening the arm supporting the higher vessel.  However, while Sol doesn't have appreciable proper motion, being close to the initial barycenter, the Earth does.  Simply accreting more mass from RG Sol won't slow BH Earth down, but it will increase BH Earth's momentum with the result that BH Earth will start to move outwards from Sol and not towards it, as it would need to do if the barycenter was to remain unchanged.

In fact, I think it will eventually end up, once most of RG Sol has been accreted, with BH Earth trundling away from where our solar system is currently located, possibly accompanied by a few of the outer planets.  Going back to the scales analogy, this proper motion is represented by the movement of the arms as the scale becomes unbalanced.

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##### How long would it take for a black hole on Earth to swallow the Sun?
« Reply #24 on: 25/05/2010 17:31:14 »