Not So Simply Climate: The Love Song of J. Alfred "Proof"rock

I have always felt that there are a couple of jumping off points for people and mathematics.  A point where the teacher says “just do this” and someone says “…What?” and that is it for them.  The first is most like when the teacher says “then we just solve for ‘x’”.  The second is when they explain proofs, “We’ll be proving this is true;” “Prove there are 180^o in a triangle.” There is no why or how, just do.  These Not so Simply Climate posts will hopefully show you that it really isn’t that bad and can even be fun in the right context.

Proofs are like building blocks in math and science; for your house it is plywood and studs, for a bank it may be cinderblock, for science it is proofs.  If we want to build up large, powerful ideas we need much smaller ones to build from. If you are going to build predictions or solutions from these ideas, you want to make sure your foundation is strong.  You can’t build a skyscraper from poorly made cement.  In our last blog post we said that as the earth gets closer to the sun it gets warmer.  Can you prove that?  How much warmer?  Even simple ideas such as these need to be proven.

Finally remember Math and Science are like a game, or puzzle.  You are encouraged to be clever and thoughtful while finding your way through.  Sometimes you get stuck, but they will never lie to you.

-

.

That was what I would call a rigorous proof.  We had a well-defined problem statement and cleanly proved its validity   Just knowing that this information is true doesn’t really serve us and purpose however, now we must use it to further our understanding of the natural world!  We will use our previous proof to help support the last blog post, when the planet gets closer or further from the sun it sees much more or less heat than the distance closer. 

Inverse Square Law

This proof isn’t formal or super rigorous but it will validate what we expect and show the process of scientific thinking.  Say I told you that "the further you get away from a heat source the less hot you get" you would say "Of course you goof".  Unfortunately "Of course" or "Because I believe it" is not enough to base our reality on.  

The surface area of a sphere is 4πR² (this also requires a proof which is linked at the end)

4π is simply a number ~12.  R is the radius of the sphere.  The radius is the distance from the very center of the sphere to its surface.  If we could actually dig through the center of the earth, the Radius of the earth would be half the distance you would dig.  

When you are ordering tile for a floor you measure two sides of the floor, multiply them together and order that many "square feet".  If you were going to tile a sphere however you would measure across it and divide that number by two and then plug that number into 4πR²  and order that many square feet. This is what we mean by “Surface Area”.

Now imagine you had a sphere of heat where the heat was the same at all points on the surface, and only on the surface, like a balloon of heat.  We will say the total heat at the surface is some number H^o and the sphere has some radius R.  Since we said the heat at any point on the surface is the same, the heat at any point on the surface is the total heat divided by its surface area or 

H_point = H^o/4πR²

No let’s pretend we have the exact same situation except the radius of this sphere is R+200, it is larger than the old sphere. Now:

This is like inflating the balloon.  Same amount of heat, but spread out more.

Since the radius is always positive we know R²+ 400*R+40000 > R² .  Now we use our proof from earlier!

The larger and larger our radius, the less heat at any point on the surface sees.  Also the smaller the radius the more heat any point sees.  We also know that this difference goes with square of the distance (R²) so if you move 10ft away, you reduce the heat by 100 times etc…

Sure this proof isn’t perfect.  Sure we took some liberties; obviously the sun doesn’t have uniform heat across its surface, all of its heat isn’t only located on its surface.  The last little part about H_point-new < H_point requires some more algebra to really prove, however all of this is close enough to reality to prove what we needed to prove, i.e. the basis of natural occurring global warming.