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Just a little more Lagrange

September 1, 2016 at 10:05 PM by Dr. Drang

I don’t want to turn this into a blog exclusively about Lagrange points,1 but I got what I believe is the correct answer to the question I posed a few days ago:

Why, in A Fall of Moondust, did Arthur C. Clarke get the numbering the first two Lagrange points of the Earth-Moon system backward?

While it’s certainly true that the numbering of the points is arbitrary, the convention I’ve always seen has been to put L1 between the Earth and the Moon and L2 beyond the Moon. Clarke, no neophyte when it came to orbital mechanics, turned those numbers around.

Today, the likely answer arrived in my Twitter feed. Here’s Peter Demarest with a bit of scholarship:

@drdrang Szebehely’s “Theory of Orbits” used that numbering for the Langrange points, but it fell out of fashion long ago.
— Peter Demarest (@pdemarest) Sep 1 2016 9:47 AM

It didn’t take me long to find Victor Szebehely and his wonderfully focused book, Theory of Orbits: The Restricted Problem of Three Bodies, published in 1967 by Academic Press. On page 133, we see this:

Szebehely page 133

As in the analysis we went through a couple of weeks ago, Szebehely sets his origin at the barycenter. But as Peter tweeted, Szebehely defines L1 as the colinear point beyond the smaller mass and L2 as the colinear point between the masses. Clarke may have used Szebehely directly or he may have referred to works that used the same nomenclature. Either way, it wasn’t a mistake.

Theory of Orbits is a delightful book, very much in keeping with the era in which it was written. I have many engineering texts from this period, and it’s common for them to treat numerical methods as something exotic. Here’s a passage from page 135 of Szebehely, where he gives advice on the most effective ways to solve the fifth-order algebraic equation that leads to the positions of the colinear Lagrange points:

Szebehely page 135

Take a look at the paragraph after Equation 20. He’s describing a direct iteration technique and actually concerns himself with the time savings that can be achieved by eliminating a single iteration. How quaint!

Thanks to Peter for solving the mystery and introducing me to this wonderful book.

  1. On the other hand, success in blogging is often said to be the result of a tight focus on a single topic. Maybe an all-Lagrange blog would be the big break I’ve been waiting for. ↩

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