As an engineering undergrad I had to take a course in probability and statistics. This course made me crazy because I could seldom get the right answer to the assigned problems. These were inevitably questions about counting certain groups of events and I would come up with a counting scheme that seemed logical but was often wrong. I could look at the provided answer which counted in some other way and it made sense BUT I could not see why my answer was wrong.
I was reminded of this when I showed N-body to a friend and his first question was “Why don’t the orbits spiral in?”. I kept trying to repeat the answers from my physics education but we kept coming back to “I’m sure your answer is correct, but why is mine wrong?”. To deal with that it finally occurred to me to ask him why he though they were spirals – instead of continuing to blather on about ellipses.
“If I throw a ball (even really far) – it hits the ground. This would happen in a vacuum too. Is this not a spiral path?” was the response. This is actually a good observation – which when linked with other observations about decaying orbits of satellites could easily leave you with the idea that orbits are spirals.
The explanation I came up with was that if the earth was replaced by a single point of the same mass and you threw the ball then the path would be an ellipse and it would come back to where it was. When the real earth is in the way, the ball hits the earth. To a person on the earth it does appear that the object is getting closer to the centre – it is. If you looked really carefully you would discover the path you thought was a spiral was actually a segment of an ellipse.
Why an ellipse? That’s something I used to be able to derive. It turns out to be not as simple a process as you might hope. After all it took someone with the stature of Newton to figure it out. The way we now do that calculation using differential equations is quite different from the geometric approach Newton used. Ironically it draws far more on the Calculus notation of Leibnitz (who independently developed Calculus, something Newton never acknowledged and worked hard to undermine).
As part of developing N-body I tried to dust off that part of my brain and sat down with a pad of paper to see how far I could get. Embarrassingly, I was a little rusty since my PhD is from the mid-nineties and since then I have not done much math. Nevertheless, I peeked only as much as needed and did finally get to the answer. I’ll post about that as a separate topic.
The details do really matter. That’s what science IS. One of my favourite illustrations of this viewpoint is in the movie Insignificancewhich has the characters of Marilyn Monroe and Albert Einstein. Marylyn explains special relativity in essentially correct lay-person terms and says:
Marilyn: I understand the results and the premise. I guess that’s the main thing, huh?
Einstein: That’s nothing.