The Cold Equation

Afternoon break time. I’m enjoying one of Al’s strawberry scones when he plops one of his astronomy magazines on Vinnie’s table. “Vinnie, you bein’ a pilot and all, could you ‘splain some numbers which I don’t understand? It’s this statistics table for super‑heavy lifter rockets. I think it says that some of them can carry more cargo to the Moon than if they only go partway there. That’s nuts, right?”

Vehicle Payload to LEO GSO Payload TLI Payload
Energia 100 20 32
Falcon Heavy 64 27 28
NASA’s SLS 1b 105 42
SpaceX Starship 100
Yenisei 103 26 28
Yenisei Don 140 30 33
LEO=Low Earth Orbit, GSO = Geosynchronous Orbit, TLI=Trans Lunar Injection
Payloads in metric tons (megagrams)

“Lemme think … LEO is anywhere up to about 2000 kilometers. GSO is about 36000 kilometers out, so it makes sense that with the same amount of fuel and stuff you can’t lift as much out there. TLI … that’s not to the Moon, that’s to a point where you can switch from orbiting the Earth to orbiting the Moon so, yeah, that’s gonna be way farther out, like a couple hundred thousand kilometers or more depending.”

“Depending on what?”

“Oh, lots of things — fuel, orbit, design philosophy—”

“Now wait, you been taking Sy lessons. Philosophy?”

“No, really. There’s two basic ways to do space travel, either you’re ballistic or you’re cruising. All the spacecraft blast‑offs you’ve seen are ballistic. Use up most of your fuel to get a good running start and then basically coast the rest of the way to your target. Ballistic means you gotta aim careful from the get‑go. That’s the difference between ballistic and cruise missiles. Cruisers keep burning fuel and accelerating. That lets ’em change directions whenever.”

“Cruisers are better, right, so you can point at different asteroids? I read about that weird orbit they had to send the Lucy mission on.”

“Actually, Lucy used the ballistic‑and‑coast model. NASA spent a bucketful of computer time calculating exactly where to point and when to lift off so Lucy could visit all those asteroids.”

“Why not just use a cruise strategy and skip around?”

“Cruisers are just fine once you’re up between planets. NASA’s Dawn mission to the Vesta and Ceres asteroids used a cruise drive — but only after the craft rode a boostered Delta‑II ballistic up to low Earth orbit. Nine boosters worth of ballistic. The problem is you’re caught in a double bind. You need to burn fuel to get the payload off the planet, but you need to burn fuel to get the fuel off, too. ‘S called diminishing returns. Hey, Sy, what’s that guy’s name?”

“Which guy?”

“The rocket equation guy, the Russian.”

“Ah. Tsiolkovsky. Lived in a log cabin but wrote a lot about space travel. Everything from rocket theory to airlocks and space stations. What about him?”

“I’m tellin’ Al about rockets. Tsiol… That guy’s equation says if you know how much you need to change velocity and you know your payload mass, you can figure how much fuel you need to burn to do that.”

“With some conditions, Vinnie. There’s a multiplier in there you have to calibrate for fuel, engine design. even whether you’re traveling through water or vacuum or different atmospheres. Then, the equation doesn’t figure in gravity. Oh, and it only works with straight‑line velocity change. If you want to change direction you need to use calculus to figure the—”

“Hey, I just realized why they use boosters!”

“Why’s that, Al?”

“The gravity thing. Gravity’s strongest near the Earth, right? Once the beast gets high enough, you’re not fighting as much gravity. You don’t need the extra power.”

“True, but that’s not the whole picture. The ISS orbit’s about 250 miles up, which puts it about 4250 miles from the planet’s center. Newton’s Law of Gravity says the field all the way up there is still about 88% of what’s at the surface. The real reason is that a booster’s basically a fuel tank. Once you’ve burned the fuel you don’t need the tank and that’s a lot of weight to carry for nothing.”

“Right, tank and engine don’t count as payload so dump ’em.”

“Seems cold‑hearted, though.”

~~ Rich Olcott

Sail On, Silver Bird

Big excitement in Al’s coffee shop. “What’s the fuss, Al?”

Lightsail 2, Sy. The Planetary Society’s Sun-powered spacecraft. Ten years of work and some luck and it’s up there, way above Hubble and the ISS, boosting itself higher every day and using no fuel to do it. Is that cool or what?”

“Sun-powered? Like with a huge set of solar panels and an electric engine?”

“No, that’s the thing. It’s got a couple of little panels to power its electronics and all, but propulsion is all direct from the Sun and that doesn’t stop. Steady as she goes, Skipper, Earth to Mars in weeks, not months. Woo-hoo!”

Image by Josh Spradling / The Planetary Society

Never the rah-rah type, Big Vinnie throws shade from his usual table by the door. “It didn’t get there by itself, Al. SpaceX’s Falcon Heavy rocket did the hard work, getting Lightsail 2 and about 20 other thingies up to orbit. Takes a lot of thrust to get out of Earth’s gravity well. Chemical rockets can do that, puny little ion drives and lightsails can’t.”

“Yeah, Vinnie, but those ‘puny’ guys could lead us to a totally different travel strategy.” A voice from the crowd, astrophysicist-in-training Newt Barnes. “Your big brawny rocket has to burn a lot of delta-v just to boost its own fuel. That’s a problem.”

Al looks puzzled. “Delta-v?”

“It’s how you figure rocket propellant, Al. With a car you think about miles per gallon because if you take your foot off the gas you eventually stop. In space you just keep going with whatever momentum you’ve got. What’s important is how much you can change momentum — speed up, slow down, change direction — and that depends on the propellant you’re using and the engine you’re putting it through. All you’ve got is what’s in the tanks.”

Al still looks puzzled. I fill in the connection. “Delta means difference, Al, and v is velocity which covers both speed and direction so delta-v means — “

“Got it, Sy. So Vinnie likes big hardware but bigger makes for harder to get off the ground and Newt’s suggesting there’s a limit somewhere.”

“Yup, it’s gotten to the point that the SpaceX people chase an extra few percent performance by chilling their propellants so they can cram more into the size tanks they use. I don’t know what the limit is but we may be getting close.”

Newt’s back in. “Which is where strategy comes in, Vinnie. Up to now we’re mostly using a ballistic strategy to get to off-Earth destinations, treating the vehicle like a projectile that gets all its momentum at the beginning of the trip. But there’s really three phases to the trip, right? You climb out of a gravity well, you travel to your target, and maybe you make a controlled landing you hope. With the ballistic strategy you burn your fuel in phase one while you’re getting yourself into a transfer orbit. Then you coast on momentum through phase two.”

“You got a better strategy?”

“In some ways, yeah. How about applying continuous acceleration throughout phase two instead of just coasting? The Dawn spacecraft, for example, was rocket-launched out of Earth’s gravity well but used a xenon-ion engine in continuous-burn mode to get to Mars and then on to Vesta and Ceres. Worked just fine.”

“But they’re such low-thrust –“

“Hey, Vinnie, taking a long time to build up speed’s no problem when you’re on a long trip anyway. Dawn‘s motor averaged 1.8 kilometer per second of delta-v — that works out to … about 4,000 miles per hour of increased speed for every hour you keep the motor running. Adds up.”

“OK, I’ll give you the ion motor’s more efficient than a chemical system, but still, you need that xenon reaction mass to get your delta-v. You still gotta boost it up out of the well. All you’re doing with that strategy is extend the limit.”

Al dives back in. “That’s the beauty of Lightsail, guys. No delta-v at all. Just put it up there and light-pressure from the Sun provides the energy. Look, I got this slick video that shows how it works.”

Video courtesy of The Planetary Society.

~~ Rich Olcott