Category: Physics: Newton and Gravity

The Coriolis Mandala

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment

“Cathleen, this Coriolis Effect‘s rule is so easy. The force always twists you to the right no matter which way you travel. Going north, you swerve eastward; going south you swerve westward, always to the right. But what about the in‑betweens? What if I’m going east or west? Does the force just go away?”

“Of course not, Cal, but it mutates. Mind you, the rules you quoted only apply in Earth’s northern hemisphere. If you’re south of the Equator they go the other direction. Anyway, the northbound and southbound cases everybody writes about are the simplest to describe. You might be surprised how it varies across all the compass points separating them.” <pulling another image up on her tablet> “Here’s a chart I did for my class.”

“Whoa, that’s complicated.”

“Let me break it down for you. The circles around the outside represent different travel orientations — north’s at the top, then northeast to its left, directly east at the three-o’clock position, and so on. The brown arrows point to the corresponding direction. They’re all the same length to imply equal travel speed because why not.”

“Southbound’s at the bottom, just like on a map, right?”

“Right. Now, the blue arrow in a circle shows the relative Coriolis force experienced by an object traveling in that circle’s direction. See how the leftward force falls off from full‑strength if you’re northbound, to 70% if you’re going northeast, so zero for eastbound motion.”

“I’ll take your word for the 70%, but yeah, that’s what the picture shows. Wait, right after you turn the corner from east to south that blue arrow flips like the rule says.”

“It’s not like there’s a big toggle switch somewhere. If you’re going a little bit north of due east, there’s just a little bit of east‑pushing Coriolis; drive just a little bit south of due east there’s just a little bit of west‑pushing Coriolis. Build a Coriolis swerve‑meter. Swivel it all around the compass. It’ll show peak force when pointed north or south. The force smoothly fades away through zero, grows to a peak in the other direction, fades away, then re‑grows as you come back full circle.”

“So that’s what the pink and green disks are about, huh? Strong to fade‑away east both sides in the top half, strong to fade‑away west in the south?”

“You’ve got it.”

“What’s the red and blue stuff about then?”

“Re‑think the picture to 3‑D. Visualize each of those circles as a sphere. The thin black arrows are x‑y‑z pointers coming out from the sphere’s center. Can you see that?”

“The arrows coming out an angle, that’s like coming out of the screen?”

“Mm‑hm. Suppose you’re driving straight east in a vehicle that doesn’t care about gravity. What would happen?”

“Heh‑heh‑heh, lift‑off!”

“Exactly. It’s not centrifugal force, technically, but the Coriolis Effect’s equation establishes a trade‑off between Coriolis acting horizontally and a centrifuge‑like sibling acting vertically. Both sibs act to increase or decrease an object’s effective inertia depending on the object’s direction of motion relative to Earth’s rotation. The vertical component strengthens as the horizontal component weakens and vice‑versa.”

“The vertical one is the tubes coming out?”

“Yes, that’s what those are about. Long tubes in the diagram indicate high relative strength, short tubes signify weaker intensity, which is why the northbound and southbound spheres don’t have any tubes at all.” <grins> “Sy would appreciate my color choices.”

“I don’t get the joke.”

“Over there on the left side, the vertical component is upward, towards us. If a star’s coming towards its spectrum is shifted to shorter wavelengths, bluer than it it were standing still. No surprise, we call that a blue shift. Being an astronomer, I made those tubes blue‑ish.”

“I suppose the right side has red tubes because the vertical component pushes down when you’re westbound?”

“You broke the code, Cal. Red‑shifted stars are flying away from us. I’m being careful here — the red tubes don’t represent a true centripetal force. There’s no pushing and the effect doesn’t depend on distance from the rotation axis. But it makes a westbound truck act a bit like the road’s coming up to meet it.”

~ Rich Olcott

Belts And Zones

On By rolcottIn Astronomy: Planetary Science, Chemistry, Physics: Newton and Gravity, UncategorizedLeave a comment

“This cold-brew latte concoction — good choice, Cal. Thanks.”

“You’re welcome, Cathleen. Sounded like it’d fit your mood. Hey, Sy’s latitude racetracks got me wondering. Do Jupiter’s stripes follow the same pattern?”

“The ‘racetracks’ exist, sort of, but they’re a very small part of a very complicated picture. Jupiter’s belts and zones are each wider than Earth. Maybe half an Earth‑width deep, too. That’s just a small fraction of Jupiter’s fluid interior. A grand arena for a host of dueling pseudo‑forces.”

“Lotsa ways for winds to work, then.”

“Indeed. Ferocious drafts everywhere, up, down and sideways. Water in all its forms is a major weather driver here on Earth. Jupiter has water, too, in multiple forms acting at multiple atmospheric levels. It also has ammonia, sulfides, phosphines, tholins, hydrocarbons, a potpourri of chemicals flavoring its hydrogen and helium. Ammonia has gas/liquid and liquid/solid transitions like water does but they happen at their own temperatures and pressures. Jupiter’s white zones are mostly ammonia ice floating a couple hundred kilometers above its brown belts of ammonium sulfides and tholins. We’re still learning about how the planet’s complicated ammonia‑sulfides system works. Compared to Jupiter, Earth’s atmosphere is a kid’s toy.”

Earth’s landmasses in natural colors.
Image by Daniel R Strebe, licensed under CC A-SA 3.0

“Earth’s a lot smaller, for sure. Do we have belts and zones?”

“Oh, yes. You’ve seen evidence for them on a world map, but maybe you haven’t noticed it.” <pulls up an image on her tablet> “All those deserts stretching across low latitudes north and south of the forested Equator and below the boreal forests. Pretty distinctive pattern, right?”

“How about that? My astronomy magazines carry photos from Chile’s waterless Atacama observatories all the time. I’ve never connected those with the Australian outback or Namibia’s desert. All at the same level, aren’t they? And the Sahara’s tan blotch goes all the way east to the Gobi and matches northern Mexico and USA’s high plains on the other side. The green areas must get all the water that the deserts don’t. How does that square with your vapor‑ice water pump theory?”

“Sideways, actually. I don’t claim that molecules evaporating near the Equator make it all the way to an ice sheet in a single pass. The heat energy does, eventually, but the molecules get waylaid by the Coriolis force. That’s where the racetracks come in. I can’t do better than this graphic. Each of those flattened blue ovals is a slice through an air‑mass donut that dominates its latitude range.”

Earth’s atmospheric circulation patterns
After an image by Kaidor, lic. under CC A-SA 3.0

“That third‑down donut pretty much covers the Sahara. Is that why it’s dry?”

“It’s a big part of the reason, but you’re way ahead of me. Look at the colored arrows inside that donut’s slice on the right. Why do they point where they do?”

“Well, we’re hottest at the Equator. Hot air rises which is why the red arrow goes up. We said rising air over the ocean carries evaporated water with it, right, which the grey arrow will drop close by as the Earth spins eastward. That’s why the Equator’s got the forest. How’m I doin’?”

“Just fine. How about the yellow and orange arrows?”

“Um, the yellow’s colliding with the next donut north so it’s gotta go down?”

“There’s more to it than that. The air up high gets chilled which makes it more dense. That’s the major reason it descends. When it gets down to ground level, though, how much water does it hold?”

“Not much, ’cause it rained out over the Equator’s forests. The orange arrow’s gonna be thirsty so it’ll pick up more water sweeping over the ocean.”

“But what if it sweeps over land?”

“Ah‑hah. It’ll suck the land dry and THAT’s why the Sahara is where it is.”

“Right. Now, those black arrows over the same cell…?”

“Northbound twisting to the right — that’s Coriolis in action. The gray arrow up top must skew eastward. By the rules, the southbound orange arrow at sea level skews west. Hey, that’s those white‑arrow trade winds. Cool.”

“Those blue donuts could be Earth’s version of Jupiter’s brown belts.”

~ Rich Olcott

Tropical beach with palm trees next to icy polar region with glaciers.

The Big Water Pump

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment

“Springtime! Could you make me a lilac latte, Cal?”

“Maybe if I left out the coffee, Cathleen. Lilac’s too delicate to stand up to coffee’s punch. How about a cold brew of light roast? I just made a batch. Plenty of caffeine in there, not too much intensity and you can imagine the flowery part.”

“I’ll have that, and a lemon scone, please.”

“Here you go, fresh from the filter. Hey, you sure lit a fire under Sy. He’s done a whole string of posts about Coriolois Effects.”

“Tsk, Cal, the scientist’s name was Coriolis. I’m not surprised there’s been multiple posts — the same pseudo‑force shows up in many ways.”

“Pseudo‑force?”

<looks around> “Good, Sy’s not here. He’d talk our ears off about inertial frames. My quick answer from a planet scientist perspective is that real forces are the ones that make things happen in systems where everything’s moving in straight lines at a steady pace.”

“Like on a pool table?”

“Mm-hm. You can generally make good predictions on systems like that, which is how pool sharks make their money. But if part of the system is accelerating in some way, maybe it’s rotating, you’ve got two choices for predicting how the system will behave. The hard way is to calculate each individual component’s motion in a single coordinate system using just the real forces. The easier way is to group components that have a common acceleration. Pick a convenient group to serve as your base subsystem. Define another subsystem for the components that all have the same acceleration relative to the first subsystem and so on. Then you pretend a pseudo‑force drives the interactions between your subsystems.”

“Like Earth and our Moon make a subsystem ’cause they orbit the Sun together and you said rotation’s a kind of acceleration. The pseudo‑force is centrifugal, fighting against the Sun’s gravity to keep Earth’s subsystem in orbit!”

“I love it when that kind of connection‑making happens in my classroom. Thank you, Cal.”

“You’re welcome. So your subsystems are what Sy calls frames?”

“Pretty much. Skipping some technical caveats, that’s the idea. When I think about atmosphere dynamics, I could try to calculate the planet’s whole atmosphere as an incredibly messy collection of atoms. I prefer to think of the Earth as a subsystem hosting some number of air mass subsystems, all embedded in the Universe system. The Universe enforces straight‑line inertia and the Earth adds rotational acceleration but the air masses are constrained to the planet’s spherical geometry. The Coriolis pseudo‑force summarizes all three effects. The calculation’s still messy, but it’s a lot more manageable. And then there’s water.”

“Water?”

“The piston that drives the climate. Water molecules are small so they move easily through the atmosphere. The important thing is, they’re good at transporting heat energy.”

“How’s that? They’d be the same temperature as everything else.”

“Temperature doesn’t always measure energy. Water molecules like to hold onto other water molecules. It takes energy to get them apart. When they get back together, the energy’s released so it’s like the freed‑up molecules store heat energy. In solid water, every molecule is locked into position. Melting a given mass of water amounts to breaking those locks. The liquid mass at freezing temperature contains more energy than the ice did. When liquid water evaporates, the gas contains even more energy, because the molecules can roam even more freely. Visualize a bucket of water someplace warm.”

“A Hawai’ian beach.”

“That bucketful absorbs heat energy as it evaporates, cooling the Pacific Ocean. Winds sweep up the gas and carry it north to the Arctic where it freezes. In the process it warms the ice cap by giving up its liquid‑to‑gas heat and also its solid‑to‑liquid heat. Water’s two active phase transitions make it a far more efficient heat transporter than dry air alone.”

“One bucket’s teeny in the ocean, though.”

“Multiply that by gazillions. We have gigatons of surface water. The evaporate/freeze/melt process cycles as the icecaps degrade, continuously acting to moderate Earth’s temperature differences. If Earth were dry, the gradient would be far steeper. Thermal gradients drive air movement. A dry Earth’s extreme temperature discrepancies would generate permanent gale‑force winds towards the poles.”

~ Rich Olcott

Atmospheric Jiu-jitsu

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment

A gorgeous early Spring day for a walk by the lake — blue sky, air just the right side of crisp, trees showing their young green leaves, geese goosily paddling around. As I pass the park bench I hear a familiar voice. “Hello, Mr Moire.”

“Morning, Walt. It’s been a while. What do your people want to know about now?”

“We’ve been reading your series of posts about the Coriolis Effect. You have masses of air pushing each other around, you have pendulums twisting about, and you have objects flying weird orbits around latitudes instead of the planet’s center. Which is the real Effect?”

“All three.”

“They’re so different. How can all three be right?”

“Knowing something of your interests, I can think of another Coriolis application will help clarify the connection. How do you steer an old‑fashioned artillery shell?”

“You don’t, you aim it.” <His eyes are looking inward.> “Those old howitzers, you traverse to the target’s coordinates, set the elevation for the distance and your munitions and whatever your barrel’s still good for, load ‘er up, let ‘er rip and wait for the forward observer to tell you how to adjust.”

“No correction for the Earth turning west‑to‑east beneath the shell’s trajectory?”

“No need. Inside a max 10‑mile range, the artillery, target and shell all share the same initial eastward vector. Windage and temperature inversions are more of a problem than Coriolis forces. That’s where judgement, feedback and reload speed come in.”

“Now stand that up against a cruise missile.”

“Very different situation. With cannons, all the propulsion happens at the start. That’s why they call it ballistic. Cruise missiles have an extended boost phase, maybe more than one, so they can do in‑flight steering. On the other hand, range is hundreds of miles or more so you do need to figure in relative easting.”

“And the easting correction takes power, right?”

“Of course.”

“From the missile’s point of view, that power goes to counteract the Coriolis force pushing it off‑course. You don’t see it from the ground but the missile does. Clearer now?”

“Give me a minute.” <sketches on his notepad> “Okay, counteracting attempt to deflect course — got it. Hmm, a pendulum’s even simpler, because it’s not trying to keep in sync with the Earth’s rotation. No forces in play crosswise to the swing plane so it can maintain orientation relative to the Universe. To museum visitors it looks like something’s twisting, but it’s us doing the moving. The air masses, though … forces are in play with that one.”

“It’s always important to keep track of who’s doing what to whom. That system has four distinct frames of reference: the Earth, a moving air mass, the air mass it collides with, and the Universe.”

“The Universe?”

“Sets the stage for Newton’s First Law, about conservation of linear momentum. Say there’s an air mass hovering over Dallas, latitude 30° north. Relative to the Earth it’s stationary, but relative to the Sun and the rest of the Universe it has an eastward vector clocking 1450 km/hour. Now suppose that mass moves north relative to the Earth.”

“But there’s already an airmass taking up space there, say in Manitoba. There’ll be a collision, northbound momentum against Manitoban inertia.”

“Here’s where Coriolis gets into the game. Manitoba may have zero motion relative to Earth, but Manitoba and its air mass are also moving eastward relative to the Universe. Manitoba’s speed is slower than Dallas’ but it’s not zero. Manitoba’s momentum deflects the Dallas mass into an even more easterly vector.”

“You’re saying that Coriolis plays jiu‑jitsu with the atmosphere.”

“I wouldn’t have come up with that interpretation, but it’s reasonable.”

“What about the weird orbits?”

“Not really orbits, more like equilibrium bands. The concept comes from the theoretical notion that every latitude along a meridian has a natural equilibrium speed where air pressure balances other forces. The bands would be parallel circles around the globe except for geography and transient disturbances. Dallas’ 1450‑km/h number was an example. If you exceed your local natural speed, centrifugal force moves you towards the Equator; if you’re a slowpoke, you’re shoved towards the nearest pole. Real weather’s more complicated.”

“Everything’s always more complicated.”

~ Rich Olcott

The Polar Expression

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, Uncategorized2 Comments

“Good afternoon, Mr … Moire, yes?”

“The same. Can I help you?”

“Yes. I am Tomas Frashko. I am new to this University. I could not help overhearing—”

“The whole neighborhood couldn’t help overhearing.”

“Mmm, yes. My sympathy. But I have some questions, if you have a moment.”

“My coffee mug’s not empty yet. Please sit down. I’ll help if I can.”

“Thank you. I have often seen the Coriolis Effect explained as an atmospheric effect — northbound air with high‑speed low‑latitude momentum deflected eastward by slower‑moving air already at higher latitudes. The last part of your recent post goes to some trouble to avoid that explanation. Why is that?”

“Because the Effect doesn’t only play with the atmosphere. It drives gyre currents in the oceans and probably the magma flows deep inside Earth’s mantle.”

“So fluids, not just air. But it is still a matter of fluid with a velocity in one direction being diverted by fluid with a different velocity. Also, these cases are planet‑scale effects operating over large distances. Surely systems at small scale do not experience a measurable amount of Coriolis force.”

“But they do. Museum Foucault pendulums swing on a scale measured in meters. There’s dozens of them on display all over the world, they act just as Coriolis’ ideas predicted, and the host institutions go to a great deal of trouble to ensure the steady swinging isn’t disturbed by rushing air.”

“Ah, yes. I have seen the pendulum exhibit in our museum in the city where I grew up. A hypnotic thing, swinging back and forth on its wire, each swing a little closer to knocking down a pin … finally! Then slowly turning direction to knock down another one. The museum docent said the plane of the pendulum’s swing pivots to demonstrate Earth’s rotation, but then she mentioned that the full circle takes more than a day to complete. She couldn’t explain why.”

“If it were swinging from a point above the North or South Pole it would be a one-day completion, 15 arcseconds per second.. Scientists tried mounting one at the South Pole and that’s exactly what they determined. The poles are the only points on Earth’s surface where the the pendulum’s inertial frame matches Earth’s so it looks like the Earth is simply turning beneath the pendulum. On the other hand, along the Equator the Coriolis force doesn’t affect a pendulum’s motion at all.”

“Not at all?”

“Nope. Centrifugal force, a little bit, but not Coriolis force.”

“Does the one become the other?”

“Oh no, they’re quite different. Centrifugal force represents competition between dissimilarly rotating frames; Coriolis force represents their coupling. If you’re riding on a merry‑go‑round—”

“A what?”

“Mm, you’d probably call it a carousel.”

“Ah. Yes, go on.”

“If you’re riding on a carousel, your straight‑line inertia in the fairgrounds frame tries to drive you forward. To stay in position on the rotating carousel, you fight that outward inertial impetus by holding onto something. In the ride’s rotating frame, that looks like you’re exerting centripetal force to counterbalance a centrifugal force that the fairgrounds frame doesn’t see.”

“Yes, yes, but how does that differ from Coriolis force?”

Cycle time = 24 hours / Sin(latitude)

“Centrifugal force depends on an object’s distance from the center of rotation. Coriolis force doesn’t care about that. It rises with the sine of the angle between the object’s vector and the axis of rotation. On a sphere the relevant angle is the latitude. A northbound object, could be a pendulum bob, arrives at the North Pole traveling perpendicular to the Earth’s axis. Perpendicular angles have the maximum sine, 1.0. The Coriolis coupling is strongest there and that’s why a pendulum’s reference frame is locked to the Earth’s 24‑hour period. At the equator a northbound object moves parallel to the polar axis. Parallel lines have zero angle with zero sine so the Coriolis coupling’s zero. A pendulum’s plane of motion at the equator stays where it started, infinite precession completion time.”

“And in‑between?”

“In between. A pendulum’s cycle would run 27.7 hours in Helsinki, more than 60 hours at the Tropic of Cancer.”

“Small coupling, not much swerving.”

~ Rich Olcott

  • Thanks to Ric Werme for his thoughtful comments and suggestions.

Directional Reset

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, Uncategorized1 Comment

Professor of Astronomy Cathleen O’Meara barges into Cal’s Coffee Shop. “There you are, Sy Moire! You numbskull! You addlepate! You … nincompoop! “

We’ve known each other since we were kids but I’ve rarely seen her this angry. “What have I done this time, Cathleen? I apologize, but what for?”

“That last post you put up. One of the hardest things to get across to planet science students is the Coriolis Effect. You got it exactly backwards, you lummox! Confused the be-jeepers out of half my students and it’s going to take a whole class period to unwind it.”

All those exclamation points sting when they strike home. “It did feel funny. All the sources I checked said Coriolis skews travel to the right in the northern hemisphere but I worked hard for hours on that video and it clearly shows ‘left‘.”

<sniff> “Stupid waste of time, chump! That video doesn’t show Coriolis.” <she grabs one of Cal’s graph-paper napkins and starts sketching> “Your balloon or whatever isn’t traveling north along Earth’s surface. It’s going out into space. That dark line tracks the thing’s shadow, or it would if you had the Sun behind it instead of off to the side. It has nothing at all to do with the cloud stream at the top of the hurricane and by the way those winds in the picture are outward, not inward as you’d’ve known if you’d’ve thought about for even a moment, blockhead! Here, look at a sideways view.”

“You’re saying my balloon’s not following the surface, it’s vectored away from the surface parallel to the north‑south axis. Also that the shadow points that I plotted on Earth trend westward only because the Earth turns west‑to‑east underneath the balloon. … Okay, I can see that. Goes so high up I guess it can’t be a balloon, huh?”

“Don’t try to deflect the conversation, nitwit. Figure out what you got wrong and put up a correction post that gives a proper account of Coriolis. Sorry, Cal, I’ll need my coffee in a sippy‑cup. Gotta go revise my lesson plan, again.”

She grabs her caffeine to‑go, flings me a final “Dolt! ” and storms out the door trailing a cloud of grumbles.

Vinnie’s open-mouthed. “Geez, Sy, she does have a temper.”

“You know it, Vinnie. Fortunately she saves it up for deserving occasions but don’t ever get her started on politics. So let’s see, what part of what I posted did I get right?”

“Well, there’s the part about Helsinki’s rotation around the Earth runs fewer kilometers per hour than Quito’s. That’s just fact, can’t argue with it.”

“Yeah, Mr Moire, and there’s Conservation of Momentum.”

“Right, Jeremy.” Synapses connect in my head. “Got it! Vinnie, what’s the rule between speed and orbit size?”

“The closer the faster. The Moon’s a quarter‑million miles away, takes a month to go round the Earth; the ESS is 250 miles up, circles us every 90 minutes. If you’re in some orbit and wanna go lower, you gotta speed up. Took me an hour to convince Larry that’s the way it works. He was all about centrifugal force forcing you outward, but if you want to get deeper in the gravity well you need the extra speed to balance the extra gravity.”

“That’s the rule for space orbits, alright, but things work exactly the opposite for travel on the surface of a rotating sphere. Gravity pulls centerward with the same strength everywhere so gravity’s not what balances the centrifugal force.”

“What does?”

“Geometry. In space orbits, velocity and kinetic energy increase toward the core. On a sphere’s surface, the highest velocity is farthest away from the rotational axis, at the equator. Velocity falls off to zero at both poles. Every latitude has its characteristic velocity and kinetic energy. Suppose you’re loose on Earth’s northern hemisphere and moving east too fast for your latitude. You’ll drift southward, away from the axis, until you hit a latitude that matches your speed. Meanwhile, because you’re moving east the landscape will flow westward beneath you. The blend is the Coriolis Effect.”

“So if I’m slower than my latitude I drift north and Coriolis sends me east?”

“Cathleen would agree, Jeremy.”

~ Rich Olcott

When It’s Not The Same Frame – Never Mind

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment
  • Author‘s note — Please ignore everything below the separator line. It’s bogus. No excuses, it’s just wrong. I intend to embarrass Vinnie and Sy just as soon as I get my head straight. My apologies to every reader, especially teachers, that I’ve confused.

“Hey, Sy, I couldn’t help overhearing—”

<chuckle> “Of course not, Cal. Overhearing what?”

“When you said Quito goes round the world twice as fast as Helsinki. That can’t be true! Things would collide and we’d get all kinds of earthquakes and stuff.”

“Well, sure, Cal, if those two airports moved relative to each other. But they don’t, they’re stuck 10750 kilometers apart just like they’ve always been. I hated flying that route. Mountains to dodge at both ends, in between there’s bad weather a lot of the time and no place good to set down if something goes wrong. … Wait — different speeds — it’s frames again, ain’t it, Sy?”

“Exactly, Vinnie, even though it’s not black holes for a change. Relative to an inertial frame on the Earth’s surface, the Earth itself doesn’t move and neither does either city. Relative to a Sun‑centered frame, though, the Earth spins on its axis once every 24 hours. In the Sun’s frame, Quito on Earth’s 40‑thousand kilometer Equator does 1666 kilometers per hour. Helsinki’s at 60° North. Its circle around the spin axis is only 20 thousand kilometers so its linear speed is 833 kilometers per hour even though it does the same 15 degrees per hour that Quito does.”

“Hi, Mr Moire. Welcome back. I couldn’t help overhearing—”

<chuckle> “Of course not, Jeremy. Overhearing what?”

“You talking about places on Earth moving different speeds. We just studied about that in Dr O’Meara’s planet science class but it’s still loose in my head. It has to do with why storms go counterclockwise, right?”

“It has everything to do with that, except the counterclockwise storms are only in the northern hemisphere. Southern hemisphere storms rotate the other way.”

“I got this, Sy. Bring up that movie you got on Old Reliable, the one that shows the northern hemisphere. Yeah, that one. Jeremy, some guy in a balloon is the dark line on his way from Kansas to the North Pole to meet Santa. In his frame the earth is moving left‑to‑right relative to his northbound course. See how the red star’s moving?”

“Yeah, it’s moving towards sunrise so his movie’s got the rotation right. Why Kansas?”

“‘Cause he’s got a good long shot over flatlands before any mountains or big lakes get in the way, okay? So, the other section of Sy’s movie is like it was shot from a satellite in geostationary orbit. In its frame the Earth is standing still, but the balloon guy’s swerving to his left which is west. Also counterclockwise.”

“Mmm, okay. So you’re saying that in our earthbound frame we see northerly winds getting twisted to their left which is west but it’s really the Earth turning under the atmosphere and that’s why hurricanes turn the way they do.”

“There are other ways to analyze it, guys.”

“Like what, Sy?”

“Let’s get back to Quito and Helsinki. In the northern hemisphere the latitude lines make shorter circles as you go north so your distance traveled per day gets smaller.”

“Makes sense, yeah.”

“Right. Your balloon guy’s at rest somewhere in the Earth’s frame before he starts his trip so the satellite sees him traveling eastward at say 1200 kilometers per hour. The atmosphere around him is doing about the same. Suppose he suddenly moves a few hundred kilometers north where the atmosphere’s moving significantly slower but he still has his original eastward momentum. What happens?”

“He gets slowed down.”

“Why?”

“Drag from the slower air. He dumps some of his momentum to the air molecules.”

“Conservation of Momentum does apply, Vinnie. That’s an explanation I see a lot in the pop‑sci press, but I’m not happy with it. An astronaut in a shuttlecraft going point‑to‑point across the airless Moon would see the same between‑frames contrast.”

“Oh! Newton’s First Law says you can’t change momentum unless an external force acts on you. So that’s the Coriolis Force, Mr Moire?”

“It’s related, Jeremy. Gravity restricts planet‑bound travelers to surface motion. Geometry and the force of gravity give that westward push in the planet’s frame to northbound objects in the northern hemisphere. The balloon guy and the astronaut don’t observe the Coriolis Effect unless they look out the window.”

~ Rich Olcott

When It’s Not The Same Frame

On By rolcottIn Astronomy: Planetary Science, Physics: Newton and Gravity, UncategorizedLeave a comment
  • Author‘s note — Please ignore everything below the separator line. It’s bogus. No excuses, it’s just wrong. I intend to embarrass Vinnie and Sy just as soon as I get my head straight. My apologies to every reader, especially teachers, that I’ve confused.

“Hey, Sy, I couldn’t help overhearing—”

<chuckle> “Of course not, Cal. Overhearing what?”

“When you said Quito goes round the world twice as fast as Helsinki. That can’t be true! Things would collide and we’d get all kinds of earthquakes and stuff.”

“Well, sure, Cal, if those two airports moved relative to each other. But they don’t, they’re stuck 10750 kilometers apart just like they’ve always been. I hated flying that route. Mountains to dodge at both ends, in between there’s bad weather a lot of the time and no place good to set down if something goes wrong. … Wait — different speeds — it’s frames again, ain’t it, Sy?”

“Exactly, Vinnie, even though it’s not black holes for a change. Relative to an inertial frame on the Earth’s surface, the Earth itself doesn’t move and neither does either city. Relative to a Sun‑centered frame, though, the Earth spins on its axis once every 24 hours. In the Sun’s frame, Quito on Earth’s 40‑thousand kilometer Equator does 1666 kilometers per hour. Helsinki’s at 60° North. Its circle around the spin axis is only 20 thousand kilometers so its linear speed is 833 kilometers per hour even though it does the same 15 degrees per hour that Quito does.”

“Hi, Mr Moire. Welcome back. I couldn’t help overhearing—”

<chuckle> “Of course not, Jeremy. Overhearing what?”

“You talking about places on Earth moving different speeds. We just studied about that in Dr O’Meara’s planet science class but it’s still loose in my head. It has to do with why storms go counterclockwise, right?”

“It has everything to do with that, except the counterclockwise storms are only in the northern hemisphere. Southern hemisphere storms rotate the other way.”

“I got this, Sy. Bring up that movie you got on Old Reliable, the one that shows the northern hemisphere. Yeah, that one. Jeremy, some guy in a balloon is the dark line on his way from Kansas to the North Pole to meet Santa. In his frame the earth is moving left‑to‑right relative to his northbound course. See how the red star’s moving?”

“Yeah, it’s moving towards sunrise so his movie’s got the rotation right. Why Kansas?”

“‘Cause he’s got a good long shot over flatlands before any mountains or big lakes get in the way, okay? So, the other section of Sy’s movie is like it was shot from a satellite in geostationary orbit. In its frame the Earth is standing still, but the balloon guy’s swerving to his left which is west. Also counterclockwise.”

“Mmm, okay. So you’re saying that in our earthbound frame we see northerly winds getting twisted to their left which is west but it’s really the Earth turning under the atmosphere and that’s why hurricanes turn the way they do.”

“There are other ways to analyze it, guys.”

“Like what, Sy?”

“Let’s get back to Quito and Helsinki. In the northern hemisphere the latitude lines make shorter circles as you go north so your distance traveled per day gets smaller.”

“Makes sense, yeah.”

“Right. Your balloon guy’s at rest somewhere in the Earth’s frame before he starts his trip so the satellite sees him traveling eastward at say 1200 kilometers per hour. The atmosphere around him is doing about the same. Suppose he suddenly moves a few hundred kilometers north where the atmosphere’s moving significantly slower but he still has his original eastward momentum. What happens?”

“He gets slowed down.”

“Why?”

“Drag from the slower air. He dumps some of his momentum to the air molecules.”

“Conservation of Momentum does apply, Vinnie. That’s an explanation I see a lot in the pop‑sci press, but I’m not happy with it. An astronaut in a shuttlecraft going point‑to‑point across the airless Moon would see the same between‑frames contrast.”

“Oh! Newton’s First Law says you can’t change momentum unless an external force acts on you. So that’s the Coriolis Force, Mr Moire?”

“It’s related, Jeremy. Gravity restricts planet‑bound travelers to surface motion. Geometry and the force of gravity give that westward push in the planet’s frame to northbound objects in the northern hemisphere. The balloon guy and the astronaut don’t observe the Coriolis Effect unless they look out the window.”

~ Rich Olcott

Not Even A Sneeze in A Hurricane

On By rolcottIn General: Fun Stuff, Physics: Newton and Gravity, Space Travel, UncategorizedLeave a comment

Quite a commotion at the lakeshore this morning. I walk over to see what’s going on. Not surprised at who’s involved. “Come away from there, Mr Feder, you’re too close to their goslings.” Doesn’t work, of course, so I resort to stronger measures. “Hey, Mr Feder, any questions for me?”

That did the trick. “Hey, yeah, Moire, I got one. There’s this big problem with atomic power ’cause there’s leftovers when the fuel’s all used up and nobody wants it buried their back yard and I unnerstand that. How about we just load all that stuff into one of Musk’s Starships and send it off to burn up in the Sun? Or would that make the Sun blow up?”

“Second part first. Do you sneeze?”

“What kinda question is that? Of course I sneeze. Everyone sneezes.”

“Ever been in a hurricane?”

“Oohyeah. Sandy, back in 2012. Did a number on my place in Fort Lee. Took out my back fence, part of the roof, branches down all over the place—”

“Did you sneeze during the storm?”

“Who remembers that sort of thing?”

“If you had, would it have made any difference to how the winds blew?”

“Nah, penny‑ante compared to what else was going on. Besides, the storm eye went a couple hundred miles west of us.”

“Well, there you go. The Sun’s surface is covered by about a million granules, each about the size of Texas, and each releasing about 400 exawatts—”.

“Exawha?”

“Exawatt. One watt is one joule of energy per second. Exa– means 1018. So just one of those granules releases 400×1018 joules of energy per second. By my numbers that’s about 2300 times the total energy that Earth gets from the Sun. There’s a million more granules like that. Still think one of our rockets would make much difference with all that going on?”

“No difference anybody’d notice. But that just proves it’d be safe to send our nuclear trash straight to the Sun.”

“Safe, yes, but not practical.”

“When someone says ‘practical’ they’re about to do numbers, right?”

“Indeed. How much nuclear waste do you propose to ship to the Sun?”

“I dunno. How much we got?”

“I saw a 2022 estimate from the International Atomic Energy Agency that our world‑wide accumulation so far is over 265 000 tonnes, mostly spent fuel. Our heaviest heavy‑lift vehicle is the SpaceX Starship. Maximum announced payload to low‑Earth orbit is 400 tonnes for a one‑way trip. You ready to finance 662 launches?”

“Not right now, I’m a little short ’til next payday. How about we just launch the really dangerous stuff, like plutonium?”

“Much easier rocket‑wise, much harder economics‑wise.”

“Why do you say that?”

“Because most of the world’s nuclear power plants depend on MOX fuel, a mixture of plutonium and uranium oxides. Take away all the plutonium, you mess up a significant chunk of our carbon‑free‑mostly electricity production. But I haven’t gotten to the really bad news yet.”

“I’m always good for bad news. Give.”

“Even with the best of intentions, it’s an expensive challenge to shoot a rocket straight from Earth into the Sun.”

“Huh? It’d go down the gravity well just like dropping a ball.”

“Nope, not like dropping a ball. More like flinging it off to the side with a badly‑aimed trebuchet. Guess how fast the Earth moves around the Sun.”

“Dunno. I heard it’s a thousand miles an hour at the Equator.”

“That’s the planet’s rotation on its own axis. My question was how fast we go taking a year to do an orbit around the Sun. I’ll spare you the arithmetic — the planet speeds eastward at 30 kilometers per second. Any rocket taking off from Earth starts with that vector, and it’s at right angles to the Earth‑Sun line. You can’t hit the Sun without shedding all that lateral momentum. If you keep it, the rules of orbital mechanics force the ship to go faster and faster sideways as it drops down the well — you flat‑out miss the Sun. By the way, LEO delta‑v for SpaceX’s most advanced Starship is about 7 km/s, less than a fifth of the minimum necessary for an Earth‑to‑Sun lift.”

~ Rich Olcott

Two’s Company, Three Is Perturbing

On By rolcottIn Mathematics, Physics: Newton and Gravity, Space Travel, UncategorizedLeave a comment

Vinnie does this thing when he’s near the end of his meal. He mashes his pizza crumbs and mozzarella dribbles into marbles he rolls around on his plate. Mostly on his plate. Eddie hates it when one escapes onto his floor. “Vinnie, you lose one more of those, you’ll be paying extra.”

“Aw, c’mon, Eddie, I’m your best customer.”

“Maybe, but there’ll be a surcharge for havin’ to mop extra around your table.”

Always the compromiser, I break in. “How about you put on less sauce, Eddie?”

Both give me looks you wouldn’t want.
  ”Lower the quality of my product??!?”
    ”Adjust perfection??!?”

“Looks like we’ve got a three‑body problem here.” Blank looks all around. “You two were just about to go at it until I put in my piece and suddenly you’re on the same side. Two‑way interaction predictable results, three‑way interaction hard to figure. Like when Newton calculated celestial orbits to confirm his Laws of Gravity and Motion. They worked fine for the Earth going around the Sun, not so good for the Moon going around the Earth. The Sun pulls on the Moon just enough to play hob with his two‑body Earth‑Moon predictions.”

“Newton again. So how did he solve it?”

“He didn’t, not exactly anyway.”

“Not smart enough?”

“No, Eddie, plenty smart. Later mathematicians have proven that the three‑body problem simply doesn’t have a general exact solution.”

“Ah-hah, Sy, I heard weaseling — general?”

“Alright, Vinnie, there are some stable special cases. Three bodies at relative rest in an equilateral triangle; certain straight‑line configurations; two biggies circling each other and a third, smaller one in a distant orbit around the other two’s center of gravity. There are other specials but none stable in the sense that they wouldn’t be disrupted by a wobbly gravity field from a nearby star or the host galaxy.”

A special-case solution
to the three-body problem
by MaxwellMolecule,
licensed under CC ASA 4.0

“So if NASA’s mission planners are looking at a four‑body Sun‑Jupiter‑Europa‑Juno situation, what’re they gonna do? ‘Give up’ ain’t an option.”

“Sure not. There’s a grand strategy with variations. The oldest variation goes back to before the Egyptian builders and everybody still uses it. Vinnie, when you fly a client to Tokyo, do you target a specific landing runway?”

“Naw, I aim for Japan, contact ATC Narita when I get close and they vector me in to wherever they want me to land.”

“How about you, Eddie? How do you get that exquisite balance in your flavoring?”

“Ain’t easy, Sy. Every batch of each herb is different — when it was picked, how it was stored, even the weather while it was growing. I start with an average mix which is usually close, then add a pinch of this and a little of that until it’s right.”

“For both of you, the critical word there was ‘close’. Call it in‑flight course adjustments, call it pinch‑and‑taste, everybody uses the ‘tweaking’ strategy. It’s a matter of skill and intuition, usually hard to generalize and even harder to teach in a systematic fashion. Engineers do it a lot, theoretical physicists work hard to avoid it.”

“What’ve they got that’s better?”

” ‘Better’ depends on your criteria. The method’s called ‘perturbation theory’ and strictly speaking, you can only use it for certain kinds of problems. Newton’s, for instance.”

“Good ol’ Newton.”

“Of course. Newton’s calculations almost matched Kepler’s planetary observations, but finagling the ‘not quite’ gave Newton headaches. More than 150 years passed before Laplace and others figured out how to treat a distant object as a perturbation of an ideal two‑body situation. It starts with calculating the system’s total energy, which wasn’t properly defined in Newton’s day. A perturbation factor p controls the third body’s contribution. The energy expression lets you calculate the orbits, but they’re the sum of terms containing powers of p. If p=0.1, p2=0.01, p3=0.001 and so on. If p isn’t zero but is still small enough, the p3 term and maybe even the p2 term are too small to bother with.”

“I’ll stick with pinch‑and‑taste.”

“Me and NASA’ll keep course‑correcting.”

~ Rich Olcott