Hiding Among The Hill Spheres

Bright Spring sunlight wakes me earlier than I’d like. I get to the office before I need to, but there’s Jeremy waiting at the door. “Morning, Jeremy. What gets you here so soon after dawn?”

“Good morning, Mr Moire. I didn’t sleep well last night, still thinking about that micro black hole. Okay, I know now that terrorists or military or corporate types couldn’t bring it near Earth, but maybe it comes by itself. What if it’s one of those asteroids with a weird orbit that intersects Earth’s orbit? Could we even see it coming? Aren’t we still in danger of all those tides and quakes and maybe it’d hollow out the Earth? How would the planetary defense people handle it?”

“For so early in the day you’re in fine form, Jeremy. Let’s take your barrage one topic at a time, starting with the bad news. We know this particular object would radiate very weakly and in the far infrared, which is already a challenge to detect. It’s only two micrometers wide. If it were to cross the Moon’s orbit, its image then would be about a nanoarcsecond across. Our astrometers are proud to resolve two white‑light images a few milliarcseconds apart using a 30‑meter telescope. Resolution in the far‑IR would be about 200 times worse. So, we couldn’t see it at a useful distance. But the bad news gets worse.”

“How could it get worse?”

“Suppose we could detect the beast. What would we do about it? Planetary defense people have proposed lots of strategies against a marauding asteroid — catch it in a big net, pilot it away with rocket engines mounted on the surface, even blast it with A‑bombs or H‑bombs. Black holes aren’t solid so none of those would work. The DART mission tried using kinetic energy, whacking an asteroid’s moonlet to divert the moonlet‑asteroid system. It worked better than anyone expected it to, but only because the moonlet was a rubble pile that broke up easily. The material it threw away acted as reaction mass for a poorly controlled rubble rocket. Black holes don’t break up.”

“You’re not making getting to sleep any easier for me.”

“Understood. Here’s the good news — the odds of us encountering anything like that are gazillions‑to‑one against. Consider the probabilities. If your beast exists I don’t think it would be an asteroid or even from the Kuiper Belt. Something as exotic as a primordial black hole or a mostly‑evaporated stellar black hole couldn’t have been part of the Solar System’s initial dust cloud, therefore it wouldn’t have been gathered into the Solar System’s ecliptic plane. It could have been part of the Oort cloud debris or maybe even flown in on a hyperbolic orbit from far, far away like ‘Oumuamua did. Its orbit could be along any of an infinite number of orientations away from Earth’s orbit. But it gets better.”

“I’ll take all the improvement you can give me.”

“Its orbital period is probably thousands of years long or never.”

“What difference does that make?”

“You’ve got to be in the right place at the right time to collide. Earth is 4.5 billion years old. Something with a 100‑year orbit would have had millions of chances to pass through a spot we happen to occupy. An outsider like ‘Oumuamua would have only one. We can even figure odds on that. It’s like a horseshoe game where close enough is good enough. The object doesn’t have to hit Earth right off, it only has to pierce our Hill Sphere.”

“Hill Sphere?”

“A Hill Sphere is a mathematical abstract like an Event Horizon. Inside a planet’s Sphere any nearby object feels a greater attraction to the planet than to its star. Velocities permitting, a collision may ensue. The Sphere’s radius depends only on the average planet–star distance and the planet and star masses. Earth’s Hill Sphere radius is 1.5 million kilometers. Visualize Hill Spheres crowded all along Earth’s orbit. If the interloper traverses any Sphere other than the one we’re in, we survive. It has 1 chance out of 471 . Multiply 471 by 100 spheres sunward and an infinity outward. We’ve got a guaranteed win.”

“I’ll sleep better tonight.”

~~ Rich Olcott

A Tug at The Ol’ Gravity Strings

“Why, Jeremy, you’ve got such a stunned look on your face. What happened? Is there anything I can do to help?”

“Sorry, Mr Moire. I guess I’ve been thinking too much about this science fiction story I just read. Which gelato can I scoop for you?”

“Two dips of mint, in a cup. Eddie went heavy with the garlic on my pizza this evening. What got to you in the story?”

“The central plot device. Here’s your gelato. In the story, someone locates a rogue black hole hiding in the asteroid belt. Tiny, maybe a few thousandths of a millimeter across, but awful heavy. A military‑industrial combine uses a space tug to tow it to Earth orbit for some kind of energy source, but their magnetic grapple slips and the thing falls to Earth. Except it doesn’t just fall to Earth, it’s so small it falls into Earth and now it’s orbiting inside, eating away the core until everything crumbles in. I can’t stop thinking about that.”

“Sounds pretty bad, but it might help if we run the numbers.” <drawing Old Reliable from its holster> “First thing — Everything about a black hole depends on its mass, so just how massive is this one?” <tapping on Old Reliable’s screen with gelato spoon> “For round numbers let’s say its diameter is 0.002 millimeter. The Schwartzschild ‘radius’ r is half that. Solve Schwartschild’s r=2GM/c² equation for the mass … plug in that r‑value … mass is 6.7×1020 kilograms. That’s about 1% of the Moon’s mass. Heavy indeed. How did they find this object?”

“The story didn’t say. Probably some asteroid miner stumbled on it.”

“Darn lucky stumble, something only a few microns across. Not likely to transit the Sun or block light from any stars unless you’re right on top of it. Radiation from its accretion disk? Depends on the history — there’s a lot of open space in the asteroid belt but just maybe the beast encountered enough dust to form one. Probably not, though. Wait, how about Hawking radiation?”

“Oh, right, Stephen Hawking’s quantum magic trick that lets a black hole radiate light from just outside its Event Horizon. Does Old Reliable have the formulas for that?”

“Sure. From Hawking’s work we know the object’s temperature and that gives us its blackbody spectrum, then we’ve got the Bekenstein‑Hawking equation for the power it radiates. Mind you, the spectrum will be red‑shifted to some extent because those photons have to crawl out of a gravity well, but this’ll give us a first cut.” <more tapping> “Chilly. 170 kelvins, that’s 100⁰C below room temperature. Most of its sub‑nanowatt emission will be at far infrared wavelengths. A terrible beacon. But suppose someone did find this thing. I wonder what’ll it take to move it here.”

“Can you calculate that?”

“Roughly. Suppose your space tug follows the cheapest possible flight path from somewhere near Ceres. Assuming the tug itself has negligible mass … ” <more tapping> “Whoa! That is literally an astronomical amount of delta-V. Not anything a rocket could do. Never mind. But where were they planning to put the object? What level orbit?”

“Well, it’s intended to beam power down to Earth. Ions in the Van Allen Belts would soak up a lot of the energy unless they station it below the Belts. Say 250 miles up along with the ISS.”

“Hoo boy! A thousand times closer than the Moon. Force is inverse to distance squared, remember. Wait, that’s distance to the center and Earth’s radius is about 4000 miles so the 250 miles is on top of that. 250,000 divided by 4250 … quotient squared … is a distance factor of almost 3500. Put 1% of the Moon that close to the Earth and you’ve got ocean tides 36 times stronger than lunar tides. Land does tides, too, so there’d be earthquakes. Um. The ISS is on a 90‑minute orbit so you’d have those quakes and ocean tides sixteen times a day. I wouldn’t worry about the black hole hollowing out the Earth, the tidal effect alone would do a great job of messing us up.”

“The whole project is such a bad idea that no-one would or could do it. I feel better now.”

~~ Rich Olcott

The Sky’s The Limit

Another meeting of the Acme Pizza and Science Society, at our usual big round table in Pizza Eddie’s place on the Acme Building’s second floor. (The table’s also used for after‑hours practical studies of applied statistics, “only don’t tell nobody, okay?“) It’s Eddie’s turn to announce the topic for the evening. “This one’s from my nephew, guys. How high up is the sky on Mars?”

General silence ensues, then Al throws in a chip. “Well, how high up is the sky on Earth?”

Being a pilot, Vinnie’s our aviation expert. “Depends on who’s defining ‘sky‘ and why they did that. I’m thinking ‘the sky’s the limit‘ and for me that’s the highest altitude I can get up to legal‑like. Private prop planes generally stay below 10,000 feet, commercial jets aren’t certified above 43,000 feet, private jets aren’t supposed to go above 51,000 feet.”

Eddie counters. “How about the Concorde? And those military high-flyers?”

“They’re special. The SST has, um, had unique engineering to let it go up to 60,000 feet ’cause they didn’t want sonic boom complaints from ground level. But it don’t fly no more anyhow. I’ve heard that the Air Force’s SR-71 could hit 85,000 feet but it got retired, too.”

Al’s not impressed. “All that’s legal stuff. There’s a helicopter flying on Mars but the FAA don’t make the rules there. What else we got?”

Geologist Kareem swallows his last bite of cheese melt. “How about the top of the troposphere? That’s the lowest layer of our atmosphere, the one where most of our weather and sunset colors happen. If you look at clouds in the sky, they’re inside the troposphere.”

“How high is that?”

“It expands with heating, so the top depends where you’re measuring. At the Equator it can be as high as 18½ kilometers; near a pole in local winter the top squeezes down to 6 kilometers or so. And to your next question — above the troposphere we’ve got the stratosphere that goes up to 50 kilometers. What’s that in feet, Sy?”

<drawing Old Reliable and screen-tapping…> “Says about 31.2 miles or 165,000 feet. Let’s keep things in kilometers from here on, okay?”

“Then you’ve got the mesosphere and the exosphere but the light scattering that gives us a blue sky happens below them so I’d say the sky stops at 50 kilometers.”

Al’s been rummaging through his astronomy magazines. “I read somewhere here that you’re not an astronaut unless you’ve gone past either 80 or 100 kilometers, which is weird with two cut‑offs. Who came up with those?”

Vinnie’s back in. “Who came up with the idea was a guy named von Kármán. One of the many Hungarians who came to the US in the 30s to get away from the Nazis. He did a bunch of advanced aircraft design work, helped found Aerojet and JPL. Anyway, he said the boundary between aeronautics and astronautics is how high you are when the atmosphere gets too thin for wings to keep you up with aerodynamic lift. Beyond that you need rockets or you’re in orbit or you fall down. He had equations and everything. For the Bell X‑2 he figured the threshold was around 52 miles up. What’s that in kilometers, Sy?”

“About 84.”

“So that’s where the 80 comes from. NASA liked that number for their astronauts but the Europeans rounded it up to 100. Politics, I suppose. Do von Kármán’s equations apply to Mars as well as Earth?”

“Now we’re getting somewhere, Vinnie. They do, sort of. It’s complicated, because there’s a four‑way tug‑of‑war going on. Your aircraft has gravity pulling you down, lift and centrifugal force pulling you up. Lift depends on the atmosphere’s density and your vehicle’s configuration. The fourth player is the kicker — frictional heat ruining the craft. Lift, centrifugal force and heating all get stronger with speed. Von Kármán based his calculations on the Bell X‑2’s configuration and heat‑management capabilities. Problem is, we’re not sending an X‑2 to Mars.”

“Can you re‑calibrate his equation to put a virtual X‑2 up there?”

“Hey, guys, I think someone did that. This magazine says the Karman line on Mars is 88 kilometers up.”

“Go tell your nephew, Eddie.”

~~ Rich Olcott

The Frame Game

A familiar footstep outside my office, “C’mon in, Vinnie, the door’s open.”

“Hi, Sy, how ya doin’?”

“Can’t complain. Yourself?”

“Fine, fine. Hey, I been thinking about something you said while Al and us were talking about rockets and orbits and such. You remember that?”

“We’ve done that in quantity. What statement in particular?”

“It was about when you’re in the ISS, you still see like 88% of Earth’s gravity. But I seen video of those astronauts just floating around in the station. Seems to me those two don’t add up.”

“Hah! We’re talking physics of motion here. What’s the magic word?”

“You’re saying it’s frames? I thought black holes did that.”

“Black holes are an extreme example, but frame‑thinking is an essential tool in analyzing any kind of relative motion. Einstein’s famous ‘happy thought‘ about a man in a free‑falling elevator—”

“Whoa, why is that a happy thought? I been nervous about elevators ever since that time we got stuck in one.”

“At least it wasn’t falling, right? Point is, the elevator and whoever’s in it agree that Newton’s First Law of Motion is valid for everything they see in there.”

“Wait, which Law is that?”

“‘Things either don’t move or else they move at a steady pace along a straight line.’ Suppose you’re that guy—”

“I’d rather not.”

“… and the elevator is in a zero‑gravity field. You take something out of your pocket, put it the air in front of you and it stays there. You give it a tap and it floats away in a straight line. Any different behavior means that your entire frame — you, the elevator and anything else in there — is being accelerated by some force. Let’s take two possibilities. Case one, you and the elevator are resting on terra firma, tightly held by the force of gravity.”

“I like that one.”

“Case two, you and the elevator are way out in space, zero‑gravity again, but you’re in a rocket under 1-g acceleration. Einstein got happy because he realized that you’d feel the same either way. You’d have no mechanical way to distinguish between the two cases.”

“What’s that mean, mechanical?”

“It excludes sneaky ways of outside influence by magnetic fields and such. Anyhow, Einstein’s insight was key to extending Newton’s First Law to figuring acceleration for an entire frame. Like, for instance, an orbiting ISS.”

“Ah, you’re saying that floating astronauts in an 88% Earth-gravity field is fine because the ISS and the guys share the frame feeling that 88% but the guys are floating relative to that frame. But down here if we could look in there we’d see how both kinds of motion literally add up.”

“Exactly. It’s just much easier to think about only one kind at a time.”

“Wait. You said the ISS is being accelerated. I thought it’s going a steady 17500 miles an hour which it’s got to do to stay 250 miles up.”

“Is it going in a straight line?”

“Well, no, it’s going in a circle, mostly, except when it has to dodge some space junk.”

“So the First Law doesn’t apply. Acceleration is change in momentum, and the ISS momentum is constantly changing.”

“But it’s moving steady.”

“But not in a straight line. Momentum is a vector that points in a specific direction. Change the direction, you change the momentum. Newton’s Second Law links momentum change with force and acceleration. Any orbiting object undergoes angular acceleration.”

“Angular acceleration, that’s a new one. It’s degrees per second per second?”

“Yup, or radians. There’s two kinds, though — orbiting and spinning. The ISS doesn’t spin because it has to keep its solar panels facing the Sun.”

“But I’ve seen sci-fi movies set in something that spins to create artificial gravity. Like that 2001 Space Odyssey where the guy does his running exercise inside the ship.”

“Sure, and people have designed space stations that spin for the same reason. You’d have a cascade of frames — the station orbiting some planet, the station spinning, maybe even a ballerina inside doing pirouettes.”

“How do you calculate all that?”

“You don’t. You work with whichever frame is useful for what you’re trying to accomplish.”

“Makes my head spin.”

~~ Rich Olcott

Climbing Out Of A Well

Al can’t contain himself. “Wait, it’s gravity!”

Vinnie and I are puzzled. “Come again?”

“Sy, you were going on about how much speed a rocket has to shed on the way to some special orbit around Mars, like that’s a big challenge. But it’s not. The rocket’s fighting the Sun’s gravity all the way. That’s where the speed goes. The Earth’s gravity, too, a little bit early on, but mostly the Sun’s, right?”

“Good point, Al. Sun gravity’s what bends the rocket onto a curve instead of a straight line. Okay, Sy, you got a magic equation that accounts for the shed speed? Something’s gotta, ’cause we got satellites going around Mars.”

“Good point, Vinnie, and you’re right, there is an equation. It’s not magic, you’ve already seen it and it ties kinetic energy to gravitational potential energy.”

“Wait, if I remember right, kinetic energy goes like mass times velocity squared. How can you calculate that without knowing how big the rocket is?”

“Good question. We get around that by thinking things through for a unit mass, one kilogram in SI units. We can multiply by the rocket’s mass when we’re done, if we need to. The kinetic energy per unit mass, we call that specific kinetic energy, is just ½v². Look familiar?”

“That’s one side of your v²=2GM/R equation except you’ve got the 2 on the other side.”

“Good eye, Al. The right-hand side, except for the 2, is specific gravitational potential energy, again for unit mass. But we can’t use the equation unless we know the kinetic energy and gravitational potential are indeed equal. That’s true if you’re in orbit but we’re talking about traveling between orbits where you’re trading kinetic for potential or vice versa. One gains what the other loses so Al’s right on the money. Traveling out of a gravity well is all about losing speed.”

Al’s catching up. “So how fast you’re going determines how high you are, and how high you are says how fast you have to be going.”

Vinnie frowns a little. “I’m thinking back to in‑flight refueling ops where I’m coming up to the tanker from below and behind while the boom operator directs me in. That doesn’t sound like it’d work for joining up to a satellite.”

“Absolutely. If you’re above and behind you could speed up to meet the beast falling, or from below and ahead you could slow down to rise. Away from that diagonal you’d be out of luck. Weird, huh?”

“Yeah. Which reminds me, now we’re talking about this ‘deeper means faster‘ stuff. How does the deep‑dive maneuver work? You know, where they dive a spacecraft close to a planet or something and it shoots off with more speed than it started with. Seems to me whatever speed it gains it oughta give up on the way out of the well.”

“It’s a surprise play, alright, but it’s actually two different tricks. The slingshot trick is to dive close enough to capture a bit of the planet’s orbital momentum before you fly back out of the well. If you’re going in the planet’s direction you come out going faster than you went in.”

“Or you could dive in the other direction to slow yourself down, right?”

“Of course, Al. NASA used both options for the Voyager and Messenger missions. Vinnie, I know what you’re thinking and yes, theoretically stealing a planet’s orbital momentum could affect its motion but really, planets are huge and spacecraft are teeny. DART hit the Dimorphos moonlet head-on and slowed it down by 5%, but you’d need 66 trillion copies of Dimorphos to equal the mass of dinky little Mercury.”

“What’s the other trick?”

“Dive in like with the slingshot, but fire your rocket engine when you’re going fastest, just as the craft approaches its closest point to the planet. Another German rocketeer, Hermann Oberth, was the first to apply serious math to space navigation. This trick’s sometimes called the Oberth effect, though he didn’t call it that. He showed that rocket exhaust gets more effective the faster you’re going. The planet’s gravity helps you along on that, for free.”

“Free help is good.”

~~ Rich Olcott

Not Too Fast, Not Too Slow

“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—”


“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

Carefully Considered Indirection

“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.”


~ Rich Olcott

  • Thanks to Ken, who asked another question.

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

DARTing to A Conclusion

The park’s trees are in brilliant Fall colors, the geese in the lake dabble about as I walk past but then, “Hey Moire, I gotta question!”

“Good morning, Mr Feder. What can I do for you?”

“NASA’s DART mission to crash into Diddy’s mos’ asteroid—”

“The asteroid’s name is Didymos, Mr Feder, and DART was programmed to crash into its moon Dimorphos, not into the asteroid itself.”

“Whatever. How’d they know it was gonna hit the sunny side so we could see it? If it hits in the dark, nobody knows what happened. They sent that rocket up nearly a year ago, right? How’d they time that launch just right? Besides, I thought we had Newton’s Laws of Motion and Gravity to figure orbits and forces. Why this big‑dollar experiment to see if a rocket shot would move the thing? Will it hit us?”

“You’re in good form today, Mr Feder.” <unholstering Old Reliable> “Let me pull some facts for you. Ah, Didymos’ distance from the Sun ranges between 1.01 and 2.27 astronomical units. Earth’s at 1.00 AU or 93 million miles, which means that the asteroid’s orbit is 930 000 miles farther out than ours, four times our distance to the Moon. That’s just orbits; Earth is practically always somewhere else than directly under Didymos’ point of closest approach. Mm… also, DART flew outward from Earth’s orbit so if the impact has any effect on the Didymos‑Dimorphos system it’ll be to push things even farther away from the Sun and us. No, I’m not scared, are you?”

“Who me? I’m from Jersey; scare is normal so we just shrug it off. So why the experiment? Newton’s not good enough?”

“Newton’s just fine, but collisions are more complicated than people think. Well, people who’ve never played pool.”

“That’s our national sport in Jersey.”

“Oh, right, so you already know about one variable we can’t be sure of. When the incoming vector doesn’t go through the target’s center of mass it exerts torque on the target.”

“We call that ‘puttin’ English on it.'”

“Same thing. If the collision is off‑center some of the incoming projectile’s linear momentum becomes angular momentum in the target object. On a pool table a simple Newtonian model can’t account for frictional torque between spinning balls and the table. The balls don’t go where the model predicts. There’s negligible friction in space, you know, but spin from an off‑center impact would still waste linear momentum and reduce the effect of DART’s impact. But there’s another, bigger variable that we didn’t think much about before we actually touched down on a couple of asteroids.”

“And that is…?”

“Texture. We’re used to thinking of an asteroid as just a solid lump of rock. It was a surprise when Ryugu and Bennu turned out to be loose collections of rocks, pebbles and dust all held together by stickiness and not much gravity. You hit that and surface things just scatter. There’s little effect on the rest of the mass. Until we do the experiment on a particular object we just don’t know whether we’d be able to steer it away from an Earth‑bound orbit.”

“Okay, but what about the sunny‑side thing?”

“Time for more facts.” <tapping on Old Reliable> “Basically, you’re asking what are the odds the moonlet is in eclipse when DART arrives on the scene. Suppose its orbit is in the plane of the ecliptic. Says here Dimorphos’ orbital radius is 1190 meters, which means its orbit is basically a circle 3740 meters long. The thing is approximately a cylinder 200 meters long and 150 meters in diameter. Say the cylinder is pointed along the direction of travel. It occupies (200m/3740m)=5% of its orbit, so there’s a 5% chance it’s dark, 95% chance it’s sunlit.”

“Not a bad bet.”

“The real odds are even better. The asteroid casts a shadow about 800 meters across. Says here the orbital plane is inclined 169° to the ecliptic so the moonlet cycles up and down. At that tilt and 1190 meters from Didymos, 200‑meter Dimorphos dodges the shadow almost completely. No eclipses. DART’s mission ends in sunlight.”

~~ Rich Olcott

  • Thanks to my brother Ken, who asked the question but more nicely.