Evaluating an exponential of a really large negative number

[flatmaster]

As a kid, I remember my father saying "there's a small chance that an electron in your body is on the moon" Well, today I decided to calculate the odds. Among the assumptions I made to make math easier.

*Ground state wave function of Hydrogen
*the moon is a cube of sides 2r. where r is the radius of the moon.
*Ignore gravity

Setting the origin at earth, you simply integrate over the volume of the moon in spherical coordinates.

I don't have the result on me now, so I'm guessing at what I got. I think I eliminated some pi's and constants because it's an order of magnitude kind of situation.

{ (rm^2)e^(-ro/ao) } / (ro^2)

rm - radius of the moon
ro - radius of the moon's orbit
ao - bohr radius

My question, the term e^(-ro/ao) is an exponential of a huge negative number. The grapher on this mac makes things look pretty, but it can't crunch numbers. How would you evaluate this?

 

[mathman]

For numerical evaluation of large (negative or positive) powers of e, convert first to a power of 10, by multiplying the exponent by log₁₀e. The rest is obvious.

 

[flatmaster]

So I want to take both sides of the equation to the log10e power?

 

[diazona]

Well... uh, sort of. You'll want to use the identity
$$e^{-r_0/a_0} = 10^{-(r_0/a_0)\log e}$$
So just figure out what $(-(r_0/a_0)\log e)$ is, and then if it's, say, -1000000000 you'll have an answer like "10 to the -1000000000 power" or whatever. Since it's just an order-of-magnitude thing, that's good enough.

 

[flatmaster]

Ok. I atleast see that that equation is balanced. Good enough for me.

What a crazy chance diazona, My calculations end up with...

p = 10^-(10^8) = 10^-10,000,000

 

[flatmaster]

I'd like to see a cubic moon actually.