Let's assume that the entire planet of Krypton is made of Kryptonite. With the wide variety of kryptonite this doesn't even feel like such a stretch of the imagination.

So the planet explodes causing the mass that was the planet to shoot outwards and expand as a shell of debre. If we can figure out the fraction of that shell to hit the earth than we know the amount of kryptonite to hit earth (if we know the original mass of the planet krypton. Let's go with the 10 times greater than earth that Mohitparikh guessed instead of my 25 times (although for estimating it doesn't really matter!)).

We need to know the area of the shell. That depends on how far it's expanded. So we have to ask ourselves: How far is Krypton from Earth?

Well, how long did it take Kal-El to get to Earth? The fastest his rocket (it's usually a rocket) could possibly travel is the speed of light. You don't have to know the speed of light to know that a light-year is a unit of distance. It is the distance that light can travel in 1 year. So if Kal-El was put in the rocket (that goes the speed of light) as a new born and he arrived on earth as a yearling then Krypton is 1 light-year away. If he arrives as a 10 year old then Krypton is 10 light-years away. Here's a fact: No solar system is as close as 1 light-year away from us. Usually Kal-El is not 10 years old when the Kents find him so let's take the geometric mean and say that he was a toddler of 3 when he crashed on Earth and therefore Krypton is 3 light-years away.

Some more facts: Proxima Centauri is 4.2 light-years from us. Sirius A and B are 8.6 light-years. Vega is 25 light-years. So by guessing 3 light-years we are saying that Krypton was one of Earth's closest neighbors in a really big neighborhood.

Let's find out what 3 light-years are in useful units. Everyone rememembers the speed of light? It's 3 * 10^8 m/s. Here's the coolest trick I learnt from the Guesstimation book: You can either to all the slow multiplying of 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, 365 days in a year --- OR --- you can memorize that there are pi * 10^7 s in every year. It's a really good estimate. It's easy to remember that pi is about 3.14 and there are 3.15 * 10^7 sec / year. Cool, eh?

Anyway, how many meters to Krypton? 3 light-years times 3 * 10^8 m/s * pi * 10^7 s/year equals 27 * 10^{15} m (cuz pi=3, remember?). Which we round to 3 * 10^{16} m.

So what is the area of the shell of the exploded Krypton when it reaches earth? Well, we just found the radius of the shell (3 * 10^{16} m) and the total area of a shell is A=4 * pi * R^2. Which means the area is 4 * 3 * (3 * 10^{16})^2 = 4 * 3 * 9 * 10^{32} = 100 * 10^{32} = 10^{34} m^2.

We only need one more piece of information: How much of that hits earth?

All the Kryptonite is kind of evenly spread over the sphere so the density is d = M / A = 6* 10^{25}kg / 10^{34} m^2 = 6 * 10^{-9} kg/m^2. Now we know how many kg pre square meter and all we need is the area of the Earth that this hits. It's really the silhouette area or the cross sectional area A = pi * r^2. To do this you need to know or guess the radius of the Earth. I happen to know that it is r = 6000 km = 6 * 10^6 m.

That means that the cross section of earth is A = 3 * (6 * 10^6 m)^2 = 3 * 6 * 6 * 10^{12} = 100 * 10^{12} = 10^{14} m^2. When we multiply the area by the density we will have the mass of Kryptonite that hit the Earth. 6 * 10^{-9} kg/m^2 * 10^{14} m^2 = 6 * 10^5 kg. Holy Rao, Superman. That's 600 tons! But if you think about spread over the entire earth most of it would be dust and really we never made any guess about what fraction of the mass on Krypton was Kryptonite.

Notice, the Wikipedia page on Krypton says that Superman's home planet is 50 light-years away. If we ignore what this would mean about how fast Kal-El's rocket would travel (not to mention the Kryptonite rocks themselves) and redo the calculation then there's only 2 tons of Kryptonite on Earth. BUT even if it moved at 10% of the speed of light, it would take the Kryptonite 500 years to get here.

## 2009-06-28

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