Astrometers Are Wobble-Watchers

letter A Hi, Sy, what’s going on in Cathleen’s seminar?

You were right, Al.
It’s about exoplanets and how to find them.
Jeremy’s pitching astrometry.
That’s about measuring star locations in the sky.
I’ll fill you in later.

“So that’s my cultural colonialism rant, thanks for listening. On to the real presentation. Maria showed us how to look for exoplanets when they wobble along our line of sight. But what if they wobble perpendicular to that? Careful measurement should show that, right? The ancients thought that holy forces had permanently set the positions of all the stars except for the planets so they didn’t measure that close. Tycho Brahe took meticulous measurements with room‑sized instruments—”

<voice from the back> “Room‑sized? What difference does that make?”

“What if I told you that two stars are 3 millimeters apart in the sky?”

<another voice> “How far out’s your ruler? Sky stuff, you need to talk angles because that’s all you got.”

“Well there you go. That’s why Tycho went for maximum angle‑measuring accuracy. He built a sextant with a 5‑foot radius. He used an entire north‑south wall as a quadrant. His primary instrument was an armillary sphere three yards across.”

<first voice again> “Wait, a sphere, like a big bubble? Why north‑south? What’s a quadrant?”

  • I give him a nudge. “He’s just a kid, Mr Feder. Be nice. One question at a time.”
  • “But I got so many!”

“Think about Tycho’s goal. Like astrometers before him, he wanted to build an accurate map of the heavens. Native Americans a thousand years or more ago carved free‑hand star maps on cave ceilings and turtle shells. Tycho followed the Arabic and Chinese quantitative mapping traditions. There’s two ways to do that. One is to measure and map the visual angles between many pairs of stars. That strategy fails quickly because errors accumulate. Four or five steps along the way you’re plotting the same star in two different locations.”

<Feder’s voice again> “There’s a better way?”

“Yessir. Measure and map each star relative to a standard coordinate system. If your system’s a good one, errors tend to average out. The latitude‑longitude system works well for locating places on Earth. Two thousand years ago the Babylonians used something similar for places in the crystal sphere they thought supported the stars above us. Where the equinoctial Sun rose on the horizon was a special direction. Their buildings celebrated it. Starting from that direction the horizontal angle to a star was its longitude. The star’s latitude was its angle up from the horizon towards the zenith straight above. But those map coordinates don’t work for another part of the world. Astrometers needed something better.”

<Feder again> “So what did they do already?”

“They may or may not have believed the Earth itself is round, but they recognized the Pole Star’s steady position that the rest of the sky revolved around. They also noticed that as each month went by the constellations played ring‑a‑rosie in a plane perpendicular to the north‑south axis. Call that the Plane of The Ecliptic. Pick a star, measure its angle away from the Ecliptic and you’ve got an ecliptic latitude. Measure its angle around the Ecliptic away from a reference star and you’ve got a ecliptic longitude. Tycho’s instruments were designed to measure star coordinates. His quadrant was a 90° bronze arc he embedded in that north‑south wall, let him measure a star’s latitude as it crossed his meridian. His ‘Sphere’ was simply a pair of calibrated metal rings on a gimbal mounting so he could point to target and reference stars and measure the angle between them. If his calibration used degree markings they’d be about 25 millimeters apart. His work was the best of his time but the limit of his accuracy was a few dozen arcseconds.”

“Is that bad?”

“It is if you’re looking for exoplanets by watching for stellar wobble. Maria’s Jupiter example showed the Sun wobbling by 1½ million kilometers. I worked this example with a bigger wobble and a star that would be mid‑range for most of our constellations. Best case, we’d see its image jiggling by about 90 microarcseconds. Tycho’s instruments weren’t good enough for wobbles.”

~~ Rich Olcott

The Stars from A Different Viewpoint

“Thank you, Maria, nice job showing us why the Doppler method had such a hard time finding exoplanets. Next up, Jeremy. You’re not going to talk about black holes, are you?”

“No, ma’am, my subject today is astrometry, but that’s useful for both exoplanets and black holes. I have to be careful when I say the word because it sounds so much like astronomy but they’re different things. It helped when I looked the words up. Turns out that ‘astronomy‘ means ‘naming stars‘ but ‘astrometry‘ means ‘measuring‘ them. Not weighing one or any of that, just measuring accurately where that star is in the sky at a certain moment. Everyone on Earth has the sky above. In the days before city life and city lights brought their eyes down, cultures all over the world were doing astronomy and astrometry. Professional astronomers generally use Greek and Arabic names, but that’s Eurocentrism and it got silly.”

<voice from the back> “Like how?”

“The Greeks couldn’t name constellations in the southern hemisphere’s skies because they never saw those stars. Polynesian navigators and Indigenous Australians saw them. Those cultures had their own perfectly good constellations. Did official Astronomy ask any of those people? Of course not, so we’ve got contrived designations like The Microscope and The Air Pump. Some of you know that I’m doing a research project with Professor Begaye to correlate constellations from different cultures. I’ve found some surprises.”

<voice from the back> “Like what?”

“Practically everyone in the northern hemisphere has a special image for the Pole Star and the stars close to it. Europeans picked out Ursa Minor, the Little Bear. For us Navajos the same stars make up The Northern Fire in the sky’s dome like the fire in our traditional domed hogan homes. Staying close to the Northern Fire we see two human figures, a woman and a man. One surprise for me was that the woman’s most prominent stars are the same ones the Greeks chose for Cassiopeia, also a female. The man’s image includes many of the same stars that Europeans call Ursa Major, the Big Bear. Did you know that the word ‘Artic‘ comes from the Greek word ‘arktos‘ which means ‘bear‘? Anyway, further out there’s a winter constellation containing three bright stars in a straight line plus a few more that could be shoulders and knees.”

<voice from the back> “Orion!”

“Mm-hm. We have almost exactly the same constellation. It’s also a hunter, except that the Greeks picture the three stars as his belt and we say it’s the quiver for his arrows. Right in front of the hunter are—”

<voice from the back> “The Pleaides!”

“But for us they’re Dilyehe, the Planting Stars. When they go below the horizon it’s time to plant corn. Which gets me to astrometry. The stars and constellations have always been clocks and calendars for the world’s cultures. Typically they compare the position of the Sun or certain stars with special structures.”

<voice from the back> “Like Stonehenge and the Pyramids!”

“There’s claims and doubts about both of those. People have searched out apparent special locations, like ‘This doorway and that window were placed to show a certain star rising on Midsummers Eve,’ but without explicit markings there’s no way to be sure it wasn’t just accidental. Besides, both structures were built with huge stone blocks, a real challenge to place accurately enough to pick out just one star on one day. We Navajos don’t build structures to track special times. We use mountains.”

<voice from the back> “What, you move mountains around?”

“No, we honor and respect the natural landscape for its beauty. What we do is find the special places that help the mountains and other landmarks tell us what time of year it is. My favorite example is the Double Sunset.”

<voice from the back> “Can’t have two!”

“Yes, you can, if the mountains are sharp and stand close to one another. On the right day of the year, the Sun sets behind one mountain, then peeks for just a minute through the cleft between the two. You just have to know where to stand to see that.”

~ Rich Olcott

Useful Eccentricity

“Hi, Al. What’s the hubbub in the back room?”

“Cathleen’s doing another astronomy class group seminar. This one’s about exoplanets. I’d like to listen in but I’ve got to tend the cash register here. Take notes, okay?”

“Sure, no problem.”

Professor Cathleen’s at the podium. “Okay, class, settle down. I hope everyone’s ready with their presentations. Maria, you’ve got a good topic to start us off.”

“Thank you. Everyone here knows I’ve been interested in spectroscopy since I was a student intern at Arecibo. It is such a powerful thing to know that a particular kind of atom, anywhere in the Universe, absorbs or gives off exactly the same pattern of light frequencies. Suppose you are looking at the spectrum of a star or a galaxy and you recognize a pattern, like sodium’s yellow doublet or hydrogen’s Lyman series. The pattern won’t be at its normal frequencies because of the Doppler effect. That’s good because the amount of blue‑shift or red‑shift tells us how quick the object is moving toward or away from us. That was how Dr Hubble proved that most other galaxies are flying away.”

<casts a slide to Al’s video screen> “I’ll begin with a review of some class material. The spectroscopy we see in the sky is light that was emitted at some peak wavelength lambda. Lambda with the little ‘o‘ is what we see for the same emission or absorption process in the laboratory. The wavelength difference between sky and laboratory is the absolute shift. Divide that by the laboratory wavelength to get the relative shift, the z‑scale. All the light from one object should have the same z value. It is important that z also gives us the object’s velocity if we multiply by the speed of light.”

<voice from the rear> “What’s the ‘fe ka‘ stuff about?”

“I was getting to that. Those two lines describe a doublet, a pair of peaks that always appear together. This is in the X‑ray spectrum of iron which is Fe for the chemists. K-alpha is a certain process inside the iron atom. Astronomers like to use that doublet because it’s easy to identify. Yes, profesora?”

“Two additional reasons, Maria. Iron’s normally the heaviest element in a star because stellar nuclear fusion processes don’t have enough energy to make anything heavier than that. Furthermore, although every element heavier than neon generates a K-alpha doublet, the peak‑to‑peak split increases with atomic mass. Iron’s doublet is the widest we see from a normal star.”

“Thank you. So, the arithmetic on the rest of the slide shows how Dr Hubble might have calculated the speed of a galaxy. But that’s steady motion. Exoplanets orbiting a star appear to speed ahead then fall behind the star, yes? We need to think about how a planet affects its star. This next slide talks about that. My example uses numbers for the Sun and Jupiter. We say Jupiter goes around the Sun, but really, they both go around their common center of gravity, their barycenter. You see how it’s calculated here — MP is the planet’s mass, MS is the star’s mass, dSP is the star-to-planet distance and dB is the distance from the star’s center to the barycenter. I’ve plugged in the numbers. The barycenter is actually ten thousand kilometers outside the Sun!”

“So you could say that our Sun counterbalances Jupiter by going in a tight circle around that point.”

“Exactly! For my third slide I worked out whether a distant astronomer could use Doppler logic to detect Sun‑Jupiter motion. The first few lines calculate the size of the Sun’s circle and than how fast the Sun flies around it. Each Jupiter year’s blue shift to red shift totals only 79 parts per billion. The Sun’s iron K‑alpha1 wavelength varies only between 193.9980015 and 193.9979985 picometers. This is far too small a change to measure, yes?”

<dramatic pause> “I summarize. To make a good Doppler signal, a star must have a massive exoplanet that’s close enough to push its star fast around the barycenter but far enough away to pull the barycenter outside of the star.”

“Thank you, Maria.”

“X” marks the barycenter

~~ Rich Olcott

A Nightcap And Secrets

“A coffee nightcap, Sy? It’s decaf so Teena can have some.”

“Sounds good, Sis.”

“Why didn’t Mr Einstein like entanglement, Uncle Sy? Thanks, Mom. A little more cream in it, please.”

“I’ll bet it has to do with the instant-effect aspect, right, Sy?”

“Thanks, Sis, and you’re right as usual. All of Relativity theory rests on the claim that nothing, not light or gravity or causality itself, can travel faster than light in a vacuum. There’s good strong arguments and evidence to support that, but Einstein himself proved that entanglement effects aren’t constrained to lightspeed. Annoyed him no end.”

“Well, your coin story‘s very nice, but it’s just a story. Is there evidence for entanglement?”

“Oh, yes, though it was fifty years after Einstein’s entanglement paper before our technology got good enough to do the experiments. Since then a whole industry of academics and entrepreneurs has grown up to build and apply devices that generate entangled systems.”

“Systems?”

“Mm-hm. Most of the work has been done with pairs of photons, but people have entangled pairs of everything from swarms of ultra‑cold atoms to electrons trapped in small imperfect diamonds. It’s always a matter of linking the pair members through some shared binary property.”

“Binary! I know what that is. Brian has a computer toy he lets me play with. You tell it where to drive this little car and it asks for decisions like left‑right or go‑stop and they’re all yes or no and the screen shows your answer as ‘0’or ‘1’ and that’s binary, right?”

“Absolutely, Teena. The entangled thingies are always created in pairs, remember? Everything about each twin is identical except for that one property, like the two coin metals, so it’s yes, no, or some mixture. Cars can’t do mixtures because they’re too big for quantum.”

“What kinds of properties are we talking about? It’s not really gold and silver, is it?”

“No, you’re right about that, Sis. Transmutation takes way too much power. Entangled quantum states have little or no energy separation which is one reason the experiments are so hard. Photons are the easiest to work with so that’s where most of the entanglement work has been done. Typically the process splits a laser beam into two rays that have contrasting polarizations, say vertical and horizontal. Or the researchers might work with particles like electrons that you can split into right‑ and left‑handed spin. Whatever, call ’em ones and zeroes, you’ve got a bridge between quantum and computing.”

“Brian says binary can do secret codes.”

“He’s right about that. Codes are about hiding information. Entanglement is real good at hiding quantum information behind some strict rules. Rule one is, if you inspect an entangled particle, you break the entanglement.”

“Sounds reasonable. When you measure it you make it part of a big system and it’s not quantum any more.”

“Right, Sis. Rule two, an entanglement only links pairs. No triples or broadcasts. Rule three is for photons — you can have two independent ways to inspect a property, but you need to use the same way for both photons or you’ve got a 50% chance of getting a mismatch.”

“Oho! I see where the hiding comes in. Hmm… From what I’ve read, encryption’s big problem is guarding the key. I think those three rules make a good way to do that. Suppose Rocky and Bullwinkle want to protect their coded messages from Boris Badinoff. They share a series of entangled photon pairs. and they agree to a measurement protocol based on the published daily prices for a series of stocks — for each photon in a series, measure it with Method 1 if the corresponding price is an odd number, Method 2 if it’s even. Rocky measures his photon. If he measures a ‘1’ then Bullwinkle sees a ‘0’ for that photon and he knows Rocky saw a ‘1.’ Rocky encrypts his message using his measured bit string. Bullwinkle flips his bit string and decrypts.”

“Brilliant. Even if Boris knows the proper sequence of measurements, if he peeks at an entangled photon that breaks the entanglement. When Bullwinkle decodes gibberish Rocky has to build another key. Your Mom’s a very smart person, Teena.”

~ Rich Olcott

Tiramisu And Gemstones

“Sis, you say there’s dessert?”

“Of course there is, Sy. Teena, please bring in the tray from the fridge.”

“Tiramisu! You did indeed go above and beyond. Thank you, Teena. Your Mom’s question must be a doozey.”

“I’ll let you enjoy a few spoonfulls before I hit you with it.” <minutes with spoon noises and yumming> “Okay. tell me about entanglement.”

“Whoa! What brought that on?”

“I’ve seen the word bandied about in the popular science press—”

“And pseudoscience—”

“Well, yes. I’m writing something where the notion might come in handy if it makes sense.”

“How can you tell what’s pseudoscience?”

“Good question, Teena. I look for gee-whiz sentences, especially ones that include weasely words like ‘might‘ and ‘could.’ Most important, does the article make or quote big claims that can’t be disproven? I’d want to see pointers to evidence strong enough to match the claims. A respectable piece would include comments from other people working in the same field. Things like that.”

“What your Mom said, and also has the author used a technical term like ‘energy‘ or ‘quantum‘ but stretched it far away from its home base? Usually when they do that and you have even an elementary idea what the term really means, it’s pretty clear that the author doesn’t understand what they’re writing about. That goes double for a lot of what you’ll see on YouTube and social media in general. It’s just so easy to put gibberish up there because there’s no‑one to contradict a claim, or if there is, it’s too late because the junk has already spread. ‘Entanglement‘ is just the latest buzzword to join the junk‑science game.”

“So what can you tell us about entanglement that’s non‑junky?”

“First thing is, it’s strictly a microscopic phenomenon, molecule‑tiny and smaller. Anything you read about people or gemstones being entangled, you can stop reading right there unless it’s for fun.”

“Weren’t Rapunzel and the prince entangled?

“They and all the movie’s other characters were tangled up in the story, yes, but that’s not the kind of entanglement your Mom’s asking about. This kind seems to involve something that Einstein called ‘spooky action at a distance‘. He didn’t like it.”

“‘Seems to‘?”

“Caught me, Sis, but it’s an important point. You make a system do something by acting on it, right? We’re used to actions where force is transmitted by direct contact, like hitting a ball with a bat. We’ve known how direct contact works with solids and fluids since Newton. We’ve extended the theory to indirect contact via electric and other fields thanks to Maxwell and Einstein and a host of other physicists. ‘Action at a distance‘ is about making something happen without either direct or indirect contact and that’s weird.”

“Can you give us an example?”

“How about an entanglement story? Suppose there’s a machine that makes coins, nicely packaged up in gift boxes. They’re for sweethearts so it always makes the coins in pairs, one gold and one silver. These are microscopic coins so quantum rules apply — every coin is half gold and half silver until its box is opened, at which point it becomes all one pure metal.”

“Like Schrödinger’s asleep‑awake kitty‑cat!”

“Exactly, Teena. So Bob buys a pair of boxes, keeps one and gives the other to Alice before he flies off in his rocket to the Moon. Quantum says both coins are both metals. When he lands, he opens his box and finds a silver coin. What kind of coin does Alice have?”

“Gold, of course.”

“For sure. Bob’s coin‑checking instantly affected Alice’s coin a quarter‑million miles away. Spooky, huh?”

“But wait a minute. Alice’s coin doesn’t move. It’s not like Bob pushed on it or anything. The only thing that changed was its composition.”

“Sis, you’ve nailed it. That’s why I said ‘seems to‘. Entanglement’s not really action at a distance. No energy or force is exerted, it’s simply an information thing about quantum properties. Which, come to think of it, is why there’s no entanglement of people or gemstones. Even a bacterium has billions and billions of quantum‑level properties. Entanglement‑tweaking one or two or even a thousand atoms won’t affect the object as a whole.”

~~ Rich Olcott

Metrological Extremes

Al’s coffee shop smells festive. “Hiya, Sy. Can I interest you in a peppermint latte this morning?”

Adapted from a YouTube video contributed by NPL(UK)

“You know me better than that, Al. My usual black mud, please. Hmm… What flavor’s hiding under the chocolate frosting on the scone rack?”

“Strawberry.”

“In that case I’ll take two. Your latest artwork behind the cash register is more a scroll than a poster.”

“You noticed. Yeah, it’s very cool but I don’t understand a couple things.”

“Oh? Like what?”

“Like what’s NPL, for starters, but mostly what the poster’s even about. I get that it’s science-y and my Physics and Astronomy customers chuckle at it, but…”

“Well, for starters, NPL is the United Kingdom’s National Physical Laboratory. In USA terms they’re a little bit like a mixture of NIST and what used to be Bell Labs with a side order of DARPA. They were early supporters of high‑precision instrumentation, computer and network tech, lots of cutting‑edge stuff until they were privatized and the company that mostly bought them lost a whole lot of money. Now they’re back to a government plus academy structure but they’re still a going concern, one of the major drivers behind the SI conventions.”

“You wrote about that a while ago, din’tcha?”

“Did a whole series that started with revising the official mass standard and wound up at the full set of Système International basic and derived units. Pretty boring until you realize that precise measurement has been crucial to practically all manufacturing since the introduction of mass production. And it’s important to use a consistent set of units. One of NASA’s worst black eyes was the Mars Climate Orbiter failure when one team used Imperial feet‑and‑pounds units and everyone else was on the metric system.”

“I gotta use both sets. Most of my baking supplies come in pounds, but the coffee beans and some of the flavorings come in kilograms. I gotta use my computer to resize a recipe.”

“That’s the thing with the metric system. It’s all about powers of ten. No dividing by 12 or is it 16 or even 5280 to get to a different size range — just move the decimal and you’re done. I don’t know why people have so much trouble with it.”

“It’s something new, Sy.”

“Yeah, but it’s not been new since the 1800s. It’s a long time since doctors prescribed by the scruple or minim. All there’s been for generations is milligrams and microliters. Gas prices being what they are these days I’m surprised the oil companies haven’t been pushing to sell by the liter — price per unit volume would drop by nearly a quarter.”

“I see ‘milli’ and ‘micro’ ornaments on one of those Christmas trees. Is that what they’re about?”

“That’s the ‘divide by a thousand’ tree. You already know ‘milli’ as the first cut‑down from grams or whatever the unit is. Divide by another thousand, you’ve got ‘micro’, which is one millionth or 10‑6. You’ve seen the ‘nano’ prefix by now — it’s 10‑9 and I like the nano‑nine connection. The ornaments on that tree display the prefixes for smaller and smaller subdivisions. The gold ones near the bottom are new this year. ‘Quecto’ is 10‑30, which would take you 30 digits if you wrote the number out.”

“So I guess the other tree is ‘multiply by a thousand.‘ Yup, there’s the ‘kilo’ for a thousand grams. Someone once told me I get about ten thousand beans in a kilogram bag.”

“Ten beans to a gram, then. That makes each bean a tenth of a gram or 100 milligrams. See how easy? Try figuring that in ounces.”

“Nice. Hey, I recognize ‘mega’ next to … a million. Counting’s hard without the commas in there.”

“Some people use spaces. You probably remember ‘giga’ and ‘tera’ from gigabytes and terabytes, you being a computer user.”

“Gigabucks, too. I read the news, you know. Politicians and CEOs play in the billions. But who needs numbers as big as ‘quetta’? That’s what, 1030?”

“Scientists and computer storage managers, mostly. Jupiter’s just shy of two quettagrams, and civilization’s on the path to generating a ronnabyte of data.”

~~ Rich Olcott

Dinner Rolls And Star Dust

“MAH-ahm! Uncle Sy’s here! Hi, Uncle Sy, dinner’s almost ready. I’ve saved up some questions for you”

“Hi, Teena, let’s have—”

“Now Teena, we said we’d hold the questions until after the meal. Hi, Sy.”

“Hi, Sis. Smells wonderful. One of Mom’s recipes?”

“Nope, I’m experimenting. Mom’s pasta sauce, though. You toss the salad and we’ll dig in.”

<later> “Wow. Sis, that lasagna was amazing. Five different meats, I think, and four different cheeses? Every mouthful was a new experience. A meal that Mom would’ve been proud of.”

“Six meats, you missed one. Full credit — Teena did the dinner rolls, from scratch, and she composed the salad.”

“Well, young lady, I think your grandma would be proud of you, too. You’ve earned questions. I may stay awake long enough to answer them.”

“Yay.”

“First the dishes, guys, then to the living room.”

“Sure, Sis. And you get a question, too.”

“As a matter of fact…”

<later> “Okay, Teena, question number one.”

“Alright. Umm. Brian tries to annoy me by saying over and over that the Sun’s gonna supernova into a black hole. That’s not true, is it?”

“You can tell Brian that the Sun’s way too small to make either a supernova or a black hole. Yes, the Sun will collapse in something like five billion years, but when that happens it’ll only be a garden‑variety nova. When things calm down there’ll be a white dwarf in the middle of our Solar System, not a black hole. Supernovas come from really big stars and they leave neutron stars behind or sometimes just emptiness. To get a black hole you need a star at least half again bigger than ours. D’ya think that’ll shut Brian down?”

“No-o, because there’s other things he says to annoy me.”

“Like what?”

“That our galaxy’s gonna collide with another one and we’ll all burn up in the explosion.”

“He’s got a thing for disasters, doesn’t he? Well, he’s partially right but mostly wrong. Yes, galaxy Andromeda is on a collision course with the Milky Way. But that collision won’t be anything like what he’s talking about. Remember those bird flocks we talked about?”

“Oh that was so long ago. What was the word? Mur, mur .. something?”

“Murmuration. That was your favorite word back then.”

“Oh, yes. It still is, now that I remember it.” <Sis and I give each other a look.> “What do birds have to do with galaxies?”

“Imagine two flocks colliding. Think there’ll be feathers all over the place?”

“No, the flocks would pass right through each other, except maybe some birds from one flock might fly off with the other one.”

“That’s pretty much what will happen with us and Andromeda. Stars in each galaxy are lightyears apart, hundreds of star‑widths apart, like cars miles apart on a highway. Star‑star collisions during a galaxy collision will be very rare. The galaxy’s own shapes will be distorted and gravity will pull stars from one galaxy to the other, but that’s about the extent of it. Anyway, that’s also about five billion years into the future. So Brian’s off on that prediction, too. Anything else?”

“Actually, yes. He says we’re made of stardust. I thought we’re made of atoms.”

“Indeed we are, but the atoms come from stars. Quick story about how stars work. The oldest and most common kind of atom is hydrogen. Back at the beginning of the Universe that’s all there was. If you shove hydrogen atoms together with enough heat and pressure, like inside stars, they combine to form heavier atoms like carbon and oxygen. You’re made of hydrogen and carbon and oxygen and such, but all your atoms except hydrogen were cooked up inside stars.”

“But how did they get inside me?”

“Remember those novas and supernovas? Doesn’t matter which kind of star collapses, half or more of its atoms spray into the Universe. They become star dust adrift in the winds of space, waiting to become part of another solar system and whatever’s in it. Brian’s right on this one, you are made of star dust.”

“Whooo, that’s awesome!”

“My question’s after dessert, Sy.”

~~ Rich Olcott

  • Thanks to the young Museum visitors who asked these questions.

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—”

“Lunastationary?”

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

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

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

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

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

“That’s easy, 2πR.”

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

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

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

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

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

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

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

“Besides how long the trip would be?”

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

~~ Rich Olcott