Relativistic Speeds Question

[kaisersean]

I just finished reading The Forge of God by Greg Bear (scifi novel, not very good in my opinion) and I got to thinking about something.

What kind of effects could you expect to see from a small mass (say 10 kg) moving at relativistic speeds (say .75c) through the atmosphere? Would it be like an object hitting water at a really high speed (i.e. explode upon striking the air) or would it go straight through? And if it would go through, what sort of effects could we expect to see?

 

[geni]

I just finished reading The Forge of God by Greg Bear (scifi novel, not very good in my opinion) and I got to thinking about something.

What kind of effects could you expect to see from a small mass (say 10 kg) moving at relativistic speeds (say .75c) through the atmosphere? Would it be like an object hitting water at a really high speed (i.e. explode upon striking the air) or would it go straight through? And if it would go through, what sort of effects could we expect to see?

It would vaporise. I expect you would get a reasonable explosion.

 

[kaisersean]

Hmm. I rather expected as much.

 

[Zombified]

This would be on the order of 10¹⁷ joules of kinetic energy, the mass equivalent of around 5kg of matter converted to energy. That's bigger than an atom bomb. And at about 450 MeV/nucleon, it's probably going to be producing particles by nuclear collisions when it hits the atmosphere, probably covering hundreds or even thousands of square miles with radiation...

 

[Atlas]

I don't know how to calculate the energy nor do I believe I can appreciate just how big a blast this or a nuke would be. Is "Scary" a magnitude?

The question to me is would the explosion happen in less than a tenth of a second....

Let's say that the earth was not solid but had an empty cylinder from pole to pole such that this speeding object could go through the atmosphere, through the earth, back through the atmosphere and out again into deep space... if it doesn't explode, that is.

The ionosphere goes up to 800 miles but at 60 the sky turns black because the atmosphere is so thin. 60 is how hi Spaceship1 traveled recently.

If we say the atmosphere is 500 miles deep and the Earth is 8000 miles in diameter, the object would fly through 9000 miles before reaching outer space again.

An object traveling at .75c travels about 14,000 miles in a tenth of a second.

Now a fast moving asteroid, let's say 72,000 mph or 20 miles per second would rip through 500 miles of atmosphere in 25 seconds.

Things don't get interesting until the last 10 miles or so of Troposphere. The Tugunska asteroid blew up in the air at about 6 miles high I think. It wasn't moving quite so fast either.

I'm just thinking that a tenth of a second (or a twentieth) is a pretty short time period and that the mass might even survive. Sure, hitting the atmosphere might be like hitting a brick wall but it's possible that it might even survive that.

 

[Zombified]

According to Wikipedia a megaton is on the order of magnitude of 10¹⁵J. So this object is about as powerful as the largest nuclear weapons ever built, but not really more so. It's a lot more powerful that the 15 kiloton Hiroshima bomb.

It's hard to say how far the object gets before it disintegrated completely, but I'm guessing not very far. It's got enough energy to generate radioactive showers as soon as it hits any atoms at all. 450 MeV/nucleon is the lower end of cosmic ray energy and is more than enough to produce pi mesons by nuclear collisions.

A comet with 10¹³ kg of mass hitting the atmosphere at 10⁴ m/sec (see here) would have about 10²¹ joules (order of magnitude) - a lot more energy, but no radiation, since individual nuclear particles don't have much energy.

(Edit to fix superscript formatting)

Edit to add: 1/10 of a second is all the time in the world for nuclear interactions...

 

[Rob Lister]

I'm trying to imagine how this object survived long enough to even get to earth or how it could achieve such a high relative velocity.

Imagine is all I can do 'cause I can't do the math...

So y'all help

But, assuming it was made of iron (7.2g/cm3 according to two agreeing web sources),
and given the stated mass of 10kg
then the area is 1388 cm3.
If spherical then the diameter is 6.91 cm so

Help!

Now, assume deep space density is about one hydrogen atom per cm3 (two web sources that agreed.)

How long would it survive?

 

[Zombified]

A diameter of 7 cm gives you a cross-section of about 40cm². There's no Lorentz contraction orthogonal to the direction of travel, and as always, I'm rounding a lot here. Each meter, the sphere will sweep through 4000cm³, striking that many protons. So in one second, that's going to be 4000 atom/meter * 3E8 meters/second or on the order of 10¹³ atoms/second.

Now the big question is how many iron atoms will a relativistic proton ablate from a solid target? I honestly don't know. Let's start by just comparing the numbers of atoms.

10kg of iron, which has an atomic weight of 55.8, is about 180 moles, or about 10²⁶ atoms. Assuming a 1:1 ablation ratio, the object lasts around 10¹³ seconds. There's 3E7 seconds in a year, so we're looking at 1 million years, order of magnitude.

That's long enough to cross the galaxy. The probability of encountering a region of denser gas or even dust is quite high, so the real lifetime is probably a lot lower. But 1:1 is probably ridiculously low.

It might also be interesting to look at how quickly the object heats up. Vacuum is a very good insulator; you lose heat only by radiating it. 10¹³ nucleons * 500 MeV/nucleon works out to about 10²¹ eV or 100 joules (again, rounding to near powers of 10...).

Order-of 100 joules is a macroscopic quantity of energy - it will boil a gram of water. It takes around 10⁶J to vaporize a mole of iron if the iron is already at the vaporization temperature, so that would take about 10⁴ seconds, or about 10⁶ seconds to boil the whole thing. Now we're down to about 4 months. But that assumes that the object absorbs all the energy, and is already at the boiling temperature of iron, and it ignores cooling from evaporation and radiation.

That the object will vaporize seems certain purely on thermodynamic grounds without even considering nuclear reactions: a flux of 500MeV particles has a temperature of trillions of degrees, so the object is not going to reach thermal equilibrium with its environment without vaporizing.

 

[Rob Lister]

A diameter of 7 cm gives you a cross-section of about 40cm². There's no Lorentz contraction orthogonal to the direction of travel, and as always, I'm rounding a lot here. Each meter, the sphere will sweep through 4000cm³, striking that many protons. So in one second, that's going to be 4000 atom/meter * 3E8 meters/second or on the order of 10¹³ atoms/second.

Now the big question is how many iron atoms will a relativistic proton ablate from a solid target? I honestly don't know. Let's start by just comparing the numbers of atoms.

10kg of iron, which has an atomic weight of 55.8, is about 180 moles, or about 10²⁶ atoms. Assuming a 1:1 ablation ratio, the object lasts around 10¹³ seconds. There's 3E7 seconds in a year, so we're looking at 1 million years, order of magnitude.

That's long enough to cross the galaxy. The probability of encountering a region of denser gas or even dust is quite high, so the real lifetime is probably a lot lower. But 1:1 is probably ridiculously low.

It might also be interesting to look at how quickly the object heats up. Vacuum is a very good insulator; you lose heat only by radiating it. 10¹³ nucleons * 500 MeV/nucleon works out to about 10²¹ eV or 100 joules (again, rounding to near powers of 10...).

Order-of 100 joules is a macroscopic quantity of energy - it will boil a gram of water. It takes around 10⁶J to vaporize a mole of iron if the iron is already at the vaporization temperature, so that would take about 10⁴ seconds, or about 10⁶ seconds to boil the whole thing. Now we're down to about 4 months. But that assumes that the object absorbs all the energy, and is already at the boiling temperature of iron, and it ignores cooling from evaporation and radiation.

That the object will vaporize seems certain purely on thermodynamic grounds without even considering nuclear reactions: a flux of 500MeV particles has a temperature of trillions of degrees, so the object is not going to reach thermal equilibrium with its environment without vaporizing.

err...

Wow! Thanks.

 

[kaisersean]

Sweet. Thanks a bunch, ya'll.