Hohmann transfer orbit – Hard Science Ain't Hard

“Vinnie, those nifty-looking transfer orbits that Hohmann invented but didn’t get to patent — you left something out.”

“What’s that, Sy?”

“The geometry looks lovely — a rocket takes off tangent to its orbit around one planet or something and inserts along a tangent to an orbit around something else. Very smooth and I can see how that routing avoids having to spend fuel to turn corners. But that ignores speeds.”

“What difference does that make?”

“It makes a difference whether or not you can get into the orbit you’re aiming for. Any orbit is a trade-off between gravity’s pull and the orbiter’s kinetic energy. Assuming you’re going for a circular orbit, there’s a strict relationship between your final height and your approach speed when you’re finally flying on the horizontal. You don’t want to come in too fast or too slow.”

“First thing I learned in pilot school. But that relationship’s an equation, ain’t it?”

“A couple, actually, but they’re simple. Let’s back into the problem. Say your mission is to put a communications relay satellite into lunastationary orbit around the Moon—”

“Lunastationary?”

“Like geostationary, but with the Moon. The satellite’s supposed to hover permanently above one spot on the Moon’s equator, so its orbital period has to equal the Moon’s ‘day,’ <pulling out Old Reliable, tapping> which is 27.322 days. Your satellite must loop around the Moon in exactly that much time. Either it’s scooting at low altitude or it’s ambling along further up. If we knew the speed we could find the radius, and if we knew the radius we could find the speed. We need some math.”

“I knew it. You’re gonna throw calculus at me.”

“Relax, Vinnie, it’s only algebra and we’re only going to combine two formulas and you already know one of them. The one you don’t know connects the speed, which I’m calling v, with the radius, R. They’re tied together by the Moon’s mass, M and Newton’s gravitational constant G. The formula is v²=2G×M/R. You can handle that, right?”

“Lessee … that says if I either double the mass or cut the distance by two, the speed has to be four times larger. Makes sense ’cause that’s about being in a deeper gravity well or getting closer in. Am I on track?”

“Absolutely. Next formula is the one you know, the circumference of a circle or in this case, the distance around that orbit.”

“That’s easy, 2πR.”

“And that’s also speed times the time, T so I’ll set those equal. <tapping on Old Reliable> Okay, the first formula says v² so I square the circumference equation and solve that for v² . You still with me?”

“You’re gonna set those two v-squareds equal, I suppose.”

“You’re still on track. Yup and then I gather the Rs on one side and everything else on the other. That gives me something in R³ but that’s okay. Plug in all the numbers, take the cube root and we get that you need to position that satellite 111 megameters out from the Moon’s center, flying at 296 meters per second. Think you can manage that?”

“Given the right equipment, sure. Seventy thousand miles out from the Moon … pretty far.”

“It’s about ¾ of the way to the Moon from Earth.”

“Cool. Does that R³ formula work for the planets?”

“Sure. Works for the Sun, too, but that’s so massive and spins so fast the sol‑stationary orbit’s half way to Mercury. An orbiter would have to fly 205 000 miles an hour to keep up with an equatorial sunspot. Flying something‑stationary over other planets offers problems beyond targeting the orbit, though.”

“Besides how long the trip would be?”

“Well, that, yes, but here’s another one. Suppose you’re going to Mars, aiming at an ares‑stationary orbit. It’ll be 20 megameters, 12500 miles from the center. You need to make your tangential injection at a Mars‑relative speed of 1439 meters per second. Problem is, you left Earth from a geostationary orbit at 3075 m/s relative to Earth. At the classic Hohmann positions, Earth’s going 5710 m/s relative to Mars, Somehow you’re going to have to shed 7346 m/s per second of excess speed.”

~~ Rich Olcott

“C’mon, Vinnie, you’re definitely doing Sy stuff. I ask you a question about how come rockets can get to the Moon easier than partway and you go round the barn with ballistics and cruisers. Stop dodging.”

“Now, Al, Vinnie’s just giving you background, right. Vinnie?”

“Right, Sy, though I gotta admit a lot of our talks have gone that way. So what’s your answer?”

“Nice try, Vinnie. You’re doing fine, so keep at it.”

“Okay. <deep breath> It has to do with vectors, Al, combination of amount and direction, like if you’re going 3 miles north that’s a vector. You good with that?”

“If you say so.”

“I do. Then you can combine vectors, like if you’re going 3 miles north and at the same time 4 miles east you’ve gone 5 miles northeast.”

“That’s a 3-4-5 right triangle, even I know that one. But that 5 miles northeast is a vector, too, right?”

“You got the idea. Now think about fueling a rocket going up to meet the ISS.”

“Sy said it’s 250 miles up, so we need enough fuel to punch that far against Earth’s gravity.”

“Not even close. If the rocket just went straight up, it’d come straight down again. You need some sideways momentum, enough so when you fall you miss the Earth.”

“Miss the Earth? Get outta here!”

“No, really. Hey, Sy, you tell him.”

“Vinnie’s right, Al. That insight goes back to Newton. He proposed a thought experiment about building a powerful cannon to fire horizontally from a very tall mountain. <sketching on paper napkin> A ball shot with a normal load of powder might hit the adjacent valley. Shoot with more and more powder, balls would fly farther and farther before hitting the Earth. Eventually you fire with a charge so powerful the ball flies far enough that its fall continues all around the planet. Unless the cannon blows up or the ball shatters.”

“That’s my point, Al. See, Newton’s cannon balls started out going flat, not up. To get up and into orbit you need up and sideways velocity, like on the diagonal. You gotta calculate fuel to do both at the same time.”

“So what’s that got to do with easier to get to the Moon than into orbit?”

“‘S got everything to do with that. Not easier, though, just if you aim right the vectors make it simpler and cheaper to carry cargo to the Moon than into Earth orbit.”

“So you just head straight for the Moon without going into orbit!”

“Not quite that simple, but you got the general idea. Remember when I brought that kid’s top in here and me and Sy talked about centrifugal force?”

“Do I? And I made you clean up all those spit-wads.”

“Yeah, well, suppose that cannon’s at the Equator <adding dotted lines to Sy’s diagram> and aimed with the Earth’s spin and suppose we load in enough powder for the ball to go straight horizontal, which is what it’d do with just centrifugal force.”

“If I’m standing by the cannon it’d look like the ball’s going sideways.”

“Yup. Basically, you get the up‑ness for free. We’re not talking about escape velocity here, that’s different. We’re talking about the start of a Hohmann orbit.”

“Who’s Hohmann?”

“German engineer. When he was a kid he read sci‑fi like the rest of us and that got him into the amateur rocketry scene. Got to be a leader in the German amateur rocket club, published a couple of leading‑edge rocket science books in the 1920s but dropped out of the field when the Nazis started rolling and he figured they’d build rocket weapons. Anyhow, he invented this orbit that starts off tangent to a circle around one planet or something, follows an ellipse to end tangent to a circle around something else. Smooth transitions at both ends, cheapest way you can get from here to there. Kinks in the routing cost you fuel and cargo capacity to turn. Guy shoulda patented it.”

“Wait, an orbit’s a mathematical abstraction, not a thing.”

“Patent Office says it’s a business method, Sy. Check out PAS-22, for example.”

“Incredible.”

~ Rich Olcott

Thanks to Ken, who asked another question.

Hard Science Ain't Hard

Tag: Hohmann transfer orbit

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