How fast do I have to go to get away from an explosion in space

[rben13]

I'm a science fiction writer, as yet unpublished. I really like to try to get as much of the science right as I can.

I'm working on a story in which an asteroid hits a kilometer long spacecraft that is loaded up with 130 30m long 10m diameter tanks. They hold hydrogen and oxygen (in separate tanks, of course). The ship is hit by a nickel-iron asteroid that's also about a kilometer across and quite hot. (It went through the upper part of Earth's atmosphere, leaving a nice trail of fire, before exiting back out into space.)

If I have a smaller spacecraft nearby, how much lead time does it need, and how fast does it need to go, to avoid being blown up?

This all happens about 36,000km from Earth. How much lead time would I need for my smaller ship to dive towards Earth and get around back behind it before debris could reach it?

Finally, any idea how much velocity a kilometer-wide asteroid would lose if it went through the atmosphere as I've described? Currently, I'm assuming the asteroid hits the atmosphere with a velocity of about 60 meters per second, gets to within 8k meters of the surface, and exits with a velocity of about 18 meters per second.

Any help with these questions and how to figure the answers would be greatly appreciated.

 

[bombadil]

Hello rben13, that’s a fun question. Calculating something like that requires you to assume lots of things though. Here’s a list of (some of) my assumptions:

1) All of the oxygen and hydrogen are used up in the chemical reaction explode.
2) The part of the blast wave moving towards the ship is not “used up” to explode the asteroid or much of the ship. Instead it will just be carrying the water formed from the combination of oxygen and hydrogen
3) The density contained in the cylindrical tanks is that of water (1 gram per cc).
4) All the energy of the explosion is converted into the kinetic energy of the blast wave

Here’s a sketch of my calculations for the chemistry and physics people.

Chemistry stuff:
According to my calculations (I love *saying* that), there is a total of $3 \times 10^8$ kg of oxygen and hydrogen (with 2 hydrogen atoms for every 1 oxygen atom). This gives $2 \times 10^{34}$ atoms of hydrogen. If 118 kcal are released for every 4 moles of hydrogen, our explosion has an energy of about $4 \times 10^{15}$ J.

Physics stuff:
Here you need http://en.wikipedia.org/wiki/Blast_wave" to tell you the velocity of the ejecta. This sort of analysis is routinely used to analyze supernova remnants and atomic bomb blasts. What happens is the ejecta expands freely (at constant velocity) until the mass the blast wave has swept up equals the original mass from the original explosion. For simplicity’s sake I’m going to forgo the taylor-sedov stuff and just assume the ejecta is moving along at constant velocity. Here’s the energy equation:

$$E=\frac{1}{2}m v^2$$

E is the energy of the explosion, m is the mass of the stuff (in our case mostly hydrogen and oxygen) and v is the speed of the ejecta (that the spaceship has to escape from). So…

$$v=\sqrt{\frac{2 E}{m}}= 1-5km/s$$

I’v done these calculations really fast, so feel free to correct the numbers or the physics. Spacecraft shouldn’t have any trouble reaching 5 km/s, but I’m not sure how long they need to attain a few kilometers per second from a standstill. Guestimating, I would assume the craft would need about 15-30 seconds to react and accelerate to top speed (I’m limiting acceleration to human constraints, not engineering ones). This means the escape craft has to be about 100 km away from the explosion to escape the blast wave.

Of course, what happens if the craft is engulfed? At some point the blast wave will be so wimpy the other craft will be able to withstand the explosion. The amount of energy the craft will absorb follows the http://en.wikipedia.org/wiki/Inverse_square_law" . If we say the spacecraft can withstand about 5000 Joules per square meter (that’s kinetic energy of a 1 kg ball thrown at 100 m/s hitting a 1 square meter of spacecraft ) then once it is about 200-300 km away it will be safe (it would take about a minute for the craft to reach this distance).

Summary:
Speed of blast wave: 5 km/s

Necessary distance between explosion and escaping spacecraft : 100km-ish (the spacecraft should be able to attain a speed of 5 km/s easily so it just needs enough time to reach the speed of 5 km/s and then it doesn’t need to hide behind Earth)

(If this explosion isn't exciting enough for you, you can always make it more powerful by saying it's a nuclear explosion)

 

[rogermunns]

Isn't the assumption about asteroid velocity very low? I thought a speed more like 50km a second was likely. Or 1000 times as large as the OP's figure.

 

[ideasrule]

Isn't the assumption about asteroid velocity very low? I thought a speed more like 50km a second was likely. Or 1000 times as large as the OP's figure.

50 km/s is on the high end, but 60 m/s is certainly much too low. A typical speed for an impactor is 10 km/s. In geosynchronous orbit, the spacecraft 's orbital speed is only
460 m/s, so the asteroid is likely to zip right through the cargo ship before it has time to heat up the ship's contents. That's OK, though; the heat and sparks created by such a violent impact should be enough to set off an explosion. So you don't really need the asteroid to pass through Earth's atmosphere first.