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

Log-rhythmic gymnastics

I recognized the knock.  “Come on in, Jeremy, the door’s open.”

“Hi, Mr Moire.  Can you believe this weather?  Did Miss Anne like her gelato?  What’s this funny ruler thing that my Dad sent me?  He said they used it to send men to the moon.”

log rhythm

“No, yes, it’s called a slide rule, and he’s right — back in the 1960s engineers used slip-sticks like that when they couldn’t get to a four-function mechanical calculator.  Now, though, they’re about as useful as a cast-iron bath towel.  Kind of a shame, because the slide rule is based on mathematical principles that are fundamental to just about all of mathematical physics.”

“Like what?”

“The use of exponents, for one.  Add exponents to multiply, subtract to divide.  Quick — what’s 100×100×100?”

“Uhh…  Ten million?”

“Nup.  But if I recast that as 102×102×102=102+2+2?”

“106.  Oh, that’s a million.”

“See how easy?  We’ve known that kind of arithmetic since Archimedes.  The big advance was in the early 1600s when John Napier realized that the exponents didn’t have to be integers.  Take square roots, for example.  What’s the square root of 100?”


“Sure — √100=√(102)=102/2=101=10.  Now write √10 with exponents.”

“Would it be 101/2?”

“Let’s see.  Do you have a spreadsheet app on that tablet you carry?”


“OK, bring it up.  Poke =10^(0.5) into cell A1, and =A1^2 into A2.  What do you get?”

“Gimme a sec … the first cell says 3.162278 and the second says … exactly 10.”

“Or as exact as that software is set up for.  So what we’ve got is that 0.5 is a perfectly good power of ten, and exponent arithmetic works the same with it and all the other rational numbers that it does with integers.  Too big a leap, or are you OK with that?”

“OK, I suppose, but what does that have to do with this gadget getting people to the Moon?”

“Take a good look at at the C scale, the lowest one on the middle ruler that slides back and forth.  Are the numbers evenly spaced out?”

“No, they’re stretched out at the low end, scrunched together at the high end.”Slide rule 3“Look for 3.16 on there.  You read it like a ruler — the number before the decimal point shows as a digit, then you locate the fractional part with the high and low vertical lines.”

“Got it.  About halfway across.”

“It’s exactly on center if that’s a good slide rule.  A number’s distance along the scale should be proportional to the exponent of 10 (we call it the logarithm) that gives you that number.  The C scale’s left end is 1.0, its right end is 10.0, and 3.162 is halfway.”

“Ah, I see how it works.  Adding distances is like adding exponents.  So if I want to multiply 2 by 3 I slide the middle ruler until its 1 is against 2 on the D scale, then I look for 3 on the C scale and, yes! it’s right next to 6 on the D scale!  Oh and the A and B scales wrap twice in the same distance so they must be logarithms for squares?  Hah, there’s 10 on B right above where I found 3.16 on CK wraps three times so it must be cubes, but why did they call it K?”

“Blame the Germans, who spell ‘cube‘ with a ‘k‘.  What do you suppose CI does?”

“Hmm, it runs backwards.  Adding with CI would be like subtracting distances which would be like dividing, so … I’ll bet it’s ‘C-Inverse‘!”

“You win the mink-lined frying pan.  So you see how even a simple 5- or 6-scale device can do a lot of calculation.  The really fancy ones had as many as a dozen scales on each side, ready for doing trigonometry, compound interest, all kinds of things.  That’s the quick compute power the rocket engineers used back in the 50s.”

“Logarithms did all that, eh?”

“Yup, that and the inverse operation, exponentiation.  Of course, you don’t have to build your log and exponent system around 10.  If you’re into information theory you might use powers of 2.  If you’re doing physics or pure math you’re probably going to use a different base, Euler’s number e=2.71828.  Looks weird, but it’s really useful because calculus.”

“So logarithms do calculating.  You said something about physical principles?”

“Calculating growth, for instance…”

~~ Rich Olcott