B Can nuclear bomb(s) change trajectory of small black hole?

[eggchess]

TL;DR Summary

A 4 ft diameter black hole (1.5 to 2 times the mass of Saturn) is headed toward Earth. Will Earth be able to defend itself using missiles with nuclear bombs? How to calculate such a problem?

A 4 ft diameter black hole (1.5 to 2 times the mass of Saturn) is headed toward Earth. Will Earth be able to defend itself using missiles with nuclear bombs? How to calculate such a problem?

 

[jedishrfu]

It would depend on when they were applied and how they were applied.

If you apply the nuclear explosion at right angle to its trajectory toward Earth then you might nudge a bit off course. Thats a very slight nudge and to be effective it would have to be done very far from Earth.

Classical physics could be used to work out an answer. If you consider the bomb as a kind of rocket fuel expelled in one direction pushing the black hole in the opposite direction. The problem is though that the black hole doesn't have a "surface" to push off from but I'm thinking that may not matter since it would absorb the blast material and perhaps its momentum via conservation of momentum although now it gets trickier.

Where did you get this problem from? Is this homework related?

 

[mathman]

Doubtful. Mass is too big.

 

[phinds]

The bomb would either (1) be so far away from the BH that it the arc subtended by the BH would mean that the portion of the force of the explosion that actually affected the BH would be trivial in comparison to its mass or (2) The bomb would be close enough that the BH would tear it apart before it got close enough to cause any change in the BH's trajectory.

So, no.

Also, I doubt your calculations. Can you show how you got your diameter based on the mass you stipulate?

 

[eggchess]

It would depend on when they were applied and how they were applied.

If you apply the nuclear explosion at right angle to its trajectory toward Earth then you might nudge a bit off course. Thats a very slight nudge and to be effective it would have to be done very far from Earth.

Classical physics could be used to work out an answer. If you consider the bomb as a kind of rocket fuel expelled in one direction pushing the black hole in the opposite direction. The problem is though that the black hole doesn't have a "surface" to push off from but I'm thinking that may not matter since it would absorb the blast material and perhaps its momentum via conservation of momentum although now it gets trickier.

Where did you get this problem from? Is this homework related?

It's a problem I made up myself that I'm working on. We've all seen the movie Armageddon, an asteroid coming at the Earth. What about a small black hole?

 

[phinds]

What about a small black hole?

Same problem as before AND you didn't answer my question.

On PF, when someone asks you a question it is expected that you will answer it.

 

[eggchess]

Same problem as before AND you didn't answer my question.

On PF, when someone asks you a question it is expected that you will answer it.

Sorry to be slow. I didn't do the calculation myself. I asked an astrophysicist at Univ Arizona. He sent this:
"Assuming the diameter is the Schwarzschild radius, then
rS = 2 GM/c^2 => M = rS c^2 / 2G ~ 1e30 g.
This is about 2 times of the mass of Saturn".
I'll be the first to admit that my math skills aren't this high. I'm trying jto gather information.

 

[Ibix]

Well, a 4ft diameter black hole would correspond to a Schwarzschild radius of 2ft, no? That comes out at $4×10^{26}\mathrm{kg}$, which is about 2/3 the mass of Saturn. I'd say your prof is off by a factor of 2.

For a black hole this small you'd have to detonate very close in for it to absorb any momentum from the detonation. And for a black hole this small the tidal effects close in would probably be enough to shred the bomb. Also, your target has the mass of a planet. A nuke going off would be less than a bacterium hitting at 18-wheeler, and the result would be similar. If you spotted the thing coming early enough (and the problem with black holes is that they're black against a black background) a tiny nudge would work, but it would need to be a long time in advance.

You might be better to use your nuke as an Orion drive and ram something solid into the hole. That largely solves your delivery problem, but not the enormous mass problem.

 

[phinds]

For a black hole this small you'd have to detonate very close in for it to absorb any momentum from the detonation. And for a black hole this small the tidal effects close in would probably be enough to shred the bomb.

Gee. Why didn't I think of that?

 

[Ibix]

Gee. Why didn't I think of that?

I thought I had acknowledged that I was agreeing with you, but apparently I edited that out. Oops...

 

[jbriggs444]

Rather than attempting to deflect a black hole with the mass of Saturn and a negligible size, would it not be more effective to deflect the Earth? Much less massive, much more surface area to push on and much easier to get to. Let the inhabitants worry about the reduction in real estate values.

Or deflect the moon and let lunar gravity pull the Earth to one side.

 

[mfb]

It doesn't work even if we ignore all bomb design questions.

Let's give the bomb a mass of 10 tonnes, all of it undergoes fusion (there is no structural mass, fission trigger, or anything else really) and all the energy is perfectly split among two halves flying in opposite directions, one of these hits the black hole with all its mass. That's 5 tonnes hitting at ~5% the speed of light. It will change the velocity of a Saturn mass black hole by 5000 kg*0.05c/(mass of Saturn) = 1.3*10^-16 m/s. If the impact happens one year in advance we change its trajectory by 4 nanometers.

Even with a billion bombs and 100 years we only achieve a deflection by 400 meters, which is essentially nothing compared to the diameter of Earth.

 

[Delta2]

I think the nuke missile will be dismantled to inactive particles by the intense gravitational field before it hits the black hole.

 

[jedishrfu]

Looks like its better to move the Earth (thank you @jbriggs444) until the threat passes and then move it back like nothing happened. That would be an awesome scifi story, with potential for nuclear contamination, extreme weather effects, earthquakes, volcanic activity just like the movie 2012.

 

[PeroK]

Looks like its better to move the Earth (thank you @jbriggs444) until the threat passes and then move it back like nothing happened. That would be an awesome scifi story, with potential for nuclear contamination, extreme weather effects, earthquakes, volcanic activity just like the movie 2012.

If you could organise simultaneous mass intercourse for the entire adult population of the Earth, then perhaps the Earth would move. That would be alternative awesome story.

 

[jbriggs444]

If you could organise simultaneous mass intercourse for the entire adult population of the Earth

Sounds like a rational response to a difficult situation.

 

[mfb]

Changing Earth's position by 10000 km in 100 years can be achieved with a close fly-by of an asteroid with a mass of 2*10¹⁵ tonnes, which is an object with a diameter of ~100 km. Chiron at ~200 km is large enough and in a somewhat unstable orbit, but we would have to move it by something like 100 million km to make a close pass at a planet (which would then send it towards Earth). If we assign another 100 years for that maneuver we need 10²⁰ kg m/s initial momentum change. That's two orders of magnitude lower than the momentum change Earth will get. As Chiron is larger than the asteroid in the first calculation it will shift Earth's position faster - something like 10 years might be enough.

10 MT TNT equivalent is enough energy to accelerate 8*10¹⁰ kg to 1 km/s, providing ~10¹⁴ kg m/s momentum change under ideal conditions. We only need a million of these weapons.

We can do better if we bombard Chiron with smaller asteroids. Plan 100 impacts of 2*10¹¹ tonne asteroids, that's probably more than needed. Moving each asteroid by a billion kilometers over 100 years would only need ~100 bombs each, or 10,000 total. Better: Start with bigger impacts, then use smaller impacts to fine-tune the orbit of Chiron. Even course corrections could need thousands of bombs so we don't want to get that wrong.

tl;dr: It's not completely impossible to move Earth's location in space by 10000 km over something like 100 years with near-future technology, but it would be a project far larger than everything we have ever done.

Chiron making such a close pass would be an amazing sight. We'll need to remove it - or its fragments - from that orbit afterwards (more asteroids impacts?), otherwise it will impact Earth at some point and end all life here.

 

[256bits]

Changing Earth's position by 10000 km

Is that enough?
The resulting tidal forces from an object comparable to Saturn that close to Earth would wreck havoc.
1 M miles might be a better guess, certainly outside the moons orbit to a fair extent, even then...

 

[256bits]

That's 5 tonnes hitting at ~5% the speed of light. It will change the velocity of a Saturn mass black hole by 5000 kg*0.05c/(mass of Saturn) = 1.3*10-16 m/s. If the impact happens one year in advance we change its trajectory by 4 nanometers.

Don't you mean, " 5 tonnes ejected from the Saturn mass + bomb".
The momentum change would occur from the half of the bomb traveling away from the gravitationally bound system of bomb+ Saturn mass.

I am not convinced that an explosion itself does anything to change the trajectory Saturn mass object, since the potential energy released is already part and parcel of that gravitational attraction between the two objects. Momentum ( in these cases ) changes to normal matter occur due to electromagnetic repulsion as explosion fragments hit and stick the surface, or deflect from it. A black hole has none of that.

A change in trajectory would have come about as the bomb approaches the 'hole'. If detonated on approach, the explosion would push it back the other way, negating some, if not all depending on whether all the matter is still gravitationally bound, as noted above.
Better to put the bomb in orbit, and detonate on the 'far side' so both effects - the movement from gravitational attraction + the momentum change from the ejecta of the explosion on the system - complement each other.

 

[mfb]

Is that enough?
The resulting tidal forces from an object comparable to Saturn that close to Earth would wreck havoc.
1 M miles might be a better guess, certainly outside the moons orbit to a fair extent, even then...

It depends on the relative velocity and how much damage we are willing to accept. It's easy to adjust the values as needed.

Don't you mean, " 5 tonnes ejected from the Saturn mass + bomb".

Doesn't matter which system you look at. You can see it as half the bomb hitting the black hole in an inelastic collision, or the bomb+black hole system emitting matter, same result.
You won't make a bomb directional enough to achieve that (or sturdy enough to get close), but I only looked at the theoretical maximum there.

Entering an orbit doesn't change anything besides requiring an absurd amount of propulsion. The approach speed is essentially zero compared to the speed of the ejected matter.