Nerds in Bed
Usually Girl dives right into the ocean of sleep and it is Boy whose mind wanders in the dark from island to island.
One time near the beginning, Girl was the one who couldn’t sleep. She tossed and turned in their bed, which was so narrow that she had no room to toss and turn. Boy decided to help her sleep by telling her a story. The story was about Fourier Transforms. In a low, even voice, he said to her, “Many times it is convenient to solve an equation with sine waves. But what if a function is not a sine wave? Well, it turns out that any function can be represented as a sum of sine waves. All the multiples of a certain frequency are used. For each frequency, you just have to figure out the amplitude and phase of each sine wave. Now, how do you suppose you find those amplitudes?”
Boy had to repeat the question. Gently, he again said, “How do you suppose you find those amplitudes?”
Girl settled deeper in Boy’s arms and muttered, “You use a Fourier Transform…”
And Boy hardly had to say more about how to calculate an integral transform before he could hear the rhythmic, mono-frequency of Girl breathing in her sleep.
Later, Girl took a graduate course in image processing and knew all about Fourier transforms herself, so it was good that she did not need any help sleeping. Many years passed. Every now and then she would say, “I can’t sleep,” and he would say, “Do you want to hear about Fourier Transforms?” and he would hear her snoring before he could say another word.
But one night a few months ago, she was rolling and thrashing like a ship running headlong into a train of deep-water surface gravity waves, and he knew something needed to be done. “Don’t worry,” he said, “I’ll get you to sleep.”
“You mean… Fourier series?” she asked.
“I haven’t been thinking much about Fourier series lately. I’ve got something that I think will work equally well. Do you know about wet bulb temperature?”
She had not. This allowed Boy to tell her about ambient temperature, and evaporative cooling, and putting socks on thermometers, until she was fast asleep.
However, Girl was becoming increasingly restless. Just a month or two later, she needed to be bored calmed again. This time Boy was tired of wet bulb temperature and his mind had drifted to orbital mechanics, so he started talking about that.
“Once upon a time, a French guy named Joseph-Louis Lagrange was thinking about the three-body problem, which is what happens when three objects are floating around in space tugging on each other with gravity. Actually, Lagrange was an Italian guy named Giuseppe Luigi Lagrangia, but he moved to France just in time for the French Revolution. He did better in the Revolution than his friend Antoine Lavoisier, who was beheaded by the revolutionaries, and he helped invent the Metric System.
“Anyway, the problem with the three-body problem is that it is impossible to solve, but Lagrange figured out equilibrium states for the case when two bodies are big, like the Sun and the Earth, and one body is small, like an asteroid or a spaceship. If the Earth is orbiting the Sun, then there are five ‘Lagrange Points’ where the asteroid can orbit the Sun with the same period as the Earth. For instance, there is a spot in between the two where the tug of the Earth and Sun cancels out and an asteroid can stay in that spot as it orbits. And there is another point, in the same orbit as the Earth, but on the opposite side of the Sun from the Earth…”
Girl abruptly turned around and faced Boy. “Wait, how can something keep orbiting the sun on the opposite side of the Earth?”
“Oh, that one makes sense. My problem is with the two other points that don’t line up with the Earth and the Sun, L4 and L5.”
“I don’t know, the opposite side of the Sun one doesn’t seem right.” Girl got out of bed. “Where are you going?” asked Boy. “My chair,” Girl answered, referring to the recliner in the living room. “Instead of putting me to sleep, now you got me wondering about Lagrange Points! I need to stay up for a little while.” Boy muttered an apology for being too interesting, put his head back down on the pillow and was soon asleep.
The next day they were having dinner at Siena’s Restaurant in Rockville. When their youngest son was young, pizza was one of the few things he ate, and so they went to Siena’s every Tuesday for most of his life. Even after he grew up, started eating other food, and went off to college, Boy and Girl continued to come back to Siena’s.
While waiting for the pizza to arrive at their table, Boy started drawing on a napkin. “Look, here’s L5. Gravity from the Sun is pointing towards the Sun. Centrifugal force from L5 orbiting the Sun is pointing away from the Sun. Those can balance. But gravity from the Earth is pointing this way, so it can’t be cancelled by the Sun or by centrifugal force. There’s no other force. It doesn’t make sense.”
“Hey, there’s a guy with a NASA t-shirt,” Girl said, looking across the room at a young man sitting at another table. “I wonder if he is from Goddard or it’s just a tourist shirt,” answered Boy. NASA’s Goddard Space Flight Center was about 20 miles away from Siena’s. “Plate o’ shrimp! It’s strange that the first time we see a guy in a NASA shirt here its when we’re talking about L5.”
Later the man walked by their table on the way out of the restaurant. Boy waved to him and said “Hey, are you a NASA person?”
“Yeah, I work at Goddard Space Flight Center.”
“Do you understand why L5 exists?”
For a few seconds he looked into space – the space inside Siena’s and the space inside the solar system – and said, “It’s just the balance between gravity and centrifugal force.” He turned around and walked out of the restaurant.
“Yeah, just like I was saying!” said Boy after he left, “But I can’t make it work.”
Back at home, Boy read through some more explanations about Lagrange Points. Looking over the mathematical derivation of the locations of L4 and L5, he realized which piece of information he was forgetting. It was a seemingly small part of how orbits worked, but it made the possibility of L4 and L5 clear. He wrote a short article describing what he found.
And he thought, next time Girl was having trouble sleeping, maybe he should just talk about Fourier Transforms again.
Image from https://phys.org/news/2012-10-wise-unknowns-jupiter-asteroids.html.