Mid-afternoon coffee-and-scone time. As I step into his coffee shop Al’s quizzing Cathleen about something in one of his Astronomy magazines. “This Lucy space mission they just sent up, how come it looks like they’re shooting at either side of Jupiter instead of hitting it straight-on? And it’s got this crazy butterfly orbit that crosses the whole Solar System a couple of times. What sense does that make?”

“It shoots to either side because there’s interesting stuff out there. We think the Solar System started as a whirling disk of dust that gradually clumped together. The gravity from Jupiter’s clump scarfed up the lion’s share of the leftovers after the Sun coalesced. The good news is, not all of Jupiter’s hoard wound up in the planet. Some pieces made it to Jupiter’s orbit but then collected in the Trojan regions ahead and behind it. Looking at that material may teach us about the early Solar System.”

“Way out there? Why not just fall into Jupiter like everything else did?”

I do Physics, I can’t help but cut in. “It’s the many‑body problem in its simplest case, just the Sun, Jupiter and an asteroid in a three‑body interaction—”

Cathleen gives me a look. “Inappropriate physicsplaining, Sy, we’re talking Astronomy here. Al’s magazine is about locating and identifying objects in space. These asteroids happen to cluster in special locations roughly sixty degrees away from Jupiter.”

“But Al’s question was, ‘Why?‘ You told him why we’re sending Lucy to the Trojans, but Physics is why they exist and why that mission map looks so weird.”

“Good point, go ahead. OK with you, Al?”

“Sure.”

I unholster Old Reliable, my tricked‑out tablet, and start sketching on its screen. “OK, orange dot’s Jupiter, yellow dot’s the Sun. Calculating their motion is a two-body problem. Gravity pulls them together but centrifugal force pulls them apart. The forces balance when the two bodies orbit in ellipses around their common center of gravity. Jupiter’s ellipse is nearly a circle but it wobbles because the Sun orbits their center of gravity. Naturally, once Newton solved that problem people turned to the next harder one.”

“That’s where Lucy comes in?”

“Not yet, Al, we’ve still got those Trojan asteroids to account for. Suppose the Jupiter‑Sun system’s gravity captures an asteroid flying in from somewhere. Where will it settle down? Most places, one body dominates the gravitational field so the asteroid orbits that one. But suppose the asteroid finds a point where the two fields are equal.”

“Oh, like halfway between, right?”

“Between, Al, but not halfway.”

“Right, Cathleen. The Sun/Jupiter mass ratio and Newton’s inverse‑square law put the equal‑pull point a lot closer to Jupiter than to the Sun. If the asteroid found that point it would hang around forever or until it got nudged away. That’s Lagrange’s L₁ point. There are two other balance points along the Sun‑Jupiter line. L₂ is beyond Jupiter where the Sun’s gravity is even weaker. L₃ is way on the other side of the Sun, a bit inside Jupiter’s orbit.”

“Hey, so those 60° points on the orbit, those are two more balances because they’re each the same distance from Jupiter and the Sun, right?”

“There you go, Al. L₄ leads Jupiter and L₅ runs behind. Lagrange published his 5‑point solution to the three‑body problem in 1762, just 250 years ago. The asteroids found Jupiter’s Trojan regions billions of years earlier.”

“We astronomers call the L₄ cluster the Trojan camp and the L₅ cluster the Greek camp, but that’s always bothered me. It’d be OK if we called the planet Zeus, but Jupiter’s a Roman god. Roman times were a millennium after classical Greece’s Trojan War so the names are just wrong.”

“I hadn’t thought about that, Cathleen, but you’re right. Anyway, back to Al’s diagram of Lucy’s journey. <activating Old Reliable’s ‘Animate’ function> Sorry, Al, but you’ve been misled. The magazine’s butterfly chart has Jupiter standing still. Here’s a stars-eye view. It’s more like the Trojans will come to Lucy than the reverse.”

~~ Rich Olcott