It’s in The Book

On By rolcottIn Chemistry, Physics: Entropy and Thermodynamics, Physics: Quantum Mechanics, UncategorizedLeave a comment

A young man’s knock, eager yet a bit hesitant. “Door’s open, Jeremy, c’mon in.”

“Hi, Mr Moire, I’ve got something to show you. It’s from my acheii, my grandfather. He said he didn’t need it any more now he’s retired so he gave it to me. What do you think?”

“Wow, the CRC Handbook of Chemistry And Physics, in the old format, not the 8½×11″ monster. An achievement award, too — my congratulations to your grandfather. Let’s see … over 3000 pages, and that real thin paper you can read through. It’s still got the math tables in front — they moved those to an Appendix by the time I bought my copy. Oooh yeah, lots of data in here, probably represents millions of grad student lab hours. Tech staff, too. And then their bosses spent time checking the work before publishing.”

“Acheii said I’d have to learn a lot before I could use it properly. I see lots of words in there I don’t recognize.” <opens book to a random page> “See, five- and six‑figure values for, what’re Specific Heat and Enthalpy?”

“Your grandfather’s absolutely correct. Much of the data’s extremely specialized. Most techs, including me, have a few personal‑favorite sections they use a lot, never touch the rest of the book. These particular pages, for instance, would be gold for a someone who designs or operates steam‑driven equipment.”

“But what do these numbers mean?”

“Specific Heat is the amount of heat energy you need to put into a certain mass of something in order to raise its temperature by a certain amount. In the early days the Brits, the Scots really, defined the British Thermal Unit as the amount of energy it took to raise the temperature of one pound of liquid water by one degree Fahrenheit. You’d calculate a fuel purchase according to how many BTUs you’d need. Science work these days is metric so these pages tabulate Specific Heat for a substance in joules per gram per °C. Tech in the field moves slow so BTUs are still popular inside the USA and outside the lab.”

“But these tables show different numbers for different temperatures and they’re all for water. Why water? Why isn’t the Specific Heat the same number for every temperature?”

“Water’s important because most power systems use steam or liquid water as the working fluid or coolant. Explaining why heat capacity varies with temperature was one of the triumphs of 19th‑century science. Turns out it’s all about how atomic motion but atoms were a controversial topic at the time. Ostwald, for instance—”

“Who?”

“Wilhelm Ostwald, one of science’s Big Names in the late 1800s. Chemistry back then was mostly about natural product analysis and seeing what reacted with what. Ostwald put his resources into studying chemical processes themselves, things like crystallization and catalysis. He’s regarded as the founder of Physical Chemistry. Even though he invented the mole he steadfastly maintained that atoms and molecules were nothing more than diffraction‑generated illusions. He liked a different theory but that one didn’t work out.”

“Too bad for him.”

“Oh, he won the first Nobel Prize in Chemistry so no problem. Anyway, back to Specific Heat. In terms of its molecules, how do you raise something’s temperature?”

“Um, temperature’s average kinetic energy, so I’d just make the molecules move faster.”

“Well said, except in the quantum world there’s another option. The molecules can’t just waggle any which way. There are rules. Different molecules do different waggles. Some kinds of motion take more energy to excite than others do. Rule 1 is that the high‑energy waggles don’t get to play until the low‑energy ones are engaged. Raising the temperature is a matter of activating more of the high‑energy waggles. Make sense?”

“Like electron shells in an atom, right? Filling the lowest‑energy shells first unless a photon supplies more energy?”

“Exactly, except we’re talking atoms moving within a molecule. Smaller energies, by a factor of 100 or more. My point is, the heat capacity of a substance depends on which waggles activate as the temperature rises. We didn’t understand heat capacity until we applied quantum thinking to the waggles.”

“What about ‘Enthalpy’ then?”

Crystal photo by JJ Harrison

~ Rich Olcott

Behold, a square?

On By rolcottIn General: Fun Stuff, Uncategorized1 Comment

It’s been a while since I heard that footstep in the hall outside my office. “Door’s open, Vinnie, c’mon in.”

“Hi, Sy. Brought you a thing.” <lays a card on my desk> “So the question is, how is this a square?”

“Is this another puzzle you got from Larry?”

“Yeah. He said you could ‘splain it.”

“Well, the idea’s clear — four right angles, four equal sides, sounds square-ish to me.”

“Yeah, but is the picture lying to us the way that other one did?”

“Fair question. Let’s see whether we can construct it with some real numbers. Both of those arcs seem to be parts of concentric circles so I’ll assume that.” <drawing on card> “The one that’s most of a circle has a radius I’ll call r.”

“You’re gonna do equations, ain’t you? You know I hate equations.”

“You asked the question. Bear with me, this won’t take long. Those two straight lines seem to run radially out from the almost‑circle’s center. I’ll call the angle between them a. By the way, if the lines are indeed radial then we’re guaranteed that all four of those ‘right angle’ markers are truthful. Any radius meets its circumference in a right angle, right?”

“Learned that in Geometry class.”

“I certainly hope so. Okay, the radius of the outer arc is 1 plus the radius of the inner arc so the length of the outer arc is the angle times that or a(1+r) —”

“Wait, where did that come from? You can’t just multiply the angle and radius together like that.”

“Sure you can. What’s the formula for a circle’s circumference?”

“2πr.”

“Which is an angle, 2π, times the radius.”

“How is 2π an angle? Should be 360°.”

“It’s like feet and meters ‑ same value, different units. Physicists like radians. 180° is π radians and the length of a semicircle is πr. Other arcs work the same way. It’s perfectly legal to multiply angle and radius if you express the angle in radians. So that outer arc length is a(1+r) and that’s 1 according to the diagram. Are you with me?”

“I suppose.”

“Now for the almost‑circle. Its angle is 2π minus that bit that got stretched out. Are we agreed that the arc length is (2π-a)r?”

“And that’s also 1.”

“Right. So we have two unknowns a and r, and two equations to settle them with: a(1+r)=1 and (2π-a)r=1. Simple high school algebra but I’ll spare you the pain and just ask Old Reliable for the result.”

“Thank you.”

“So there’s your answer. Yes, the keyhole figure can be truthful if the angle is 48.4° and the sticky‑out part is about 5½ times longer than the almost‑circle’s radius. Any other angle or radius and the diagram’s wrong. Happy?”

“Yeah.” <quiet moment> “Hey, I just figured out a different way. The latitude lines and longitude lines always cross at right angles, right?”

“Right.”

“So you could do a keyhole ‘square’ on the Earth, right? Circle the North Pole at some latitude, except take a detour straight south, then straight west for a while, then straight back north just in time to meet your part‑circle’s starting point. I’ve flown crazy routes a little like that but that’s always been point‑to‑point. How do you from‑scratch figure something like that so that all the sides are the same length?”

“Whoa, that’s a much harder problem. You’re flying over Earth’s surface so r is constant but now you’ve got two angular variables, latitude and longitude. The north‑south tracks are pretty straight‑forward — you’re good if one starts at the same latitude the other stops at. The tough part is how to split the 360° of longitude between the two east‑west tracks so that the southern arc is the same length as the northern one and they both match the north‑south distance which depends on the start‑stop latitudes. That’s not quadratic equations any more, we’re looking at transcendental equations involving trig functions. There may not be a closed‑form solution. To get those angles we’d need a load of computer time doing successive approximations toward a numerical solution. Surely keyhole‑square routes exist but they’re well‑hidden.”

“Regular squares’re much easier. Colorado or Wyoming’d be no problem.”

~~ Rich Olcott

Not Enough Monkeys

On By rolcottIn General: Fun Stuff, UncategorizedLeave a comment

“Morning, Sy. You see the news about the Infinite Monkey thing?”

“No, Cal, with everything else going on I seem to have missed that.”

“Understandable. I only heard about it from a ‘lighter side of the news’ piece on the radio. Something about disproving what everybody used to believe. You wrote about it a while ago, didn’t you?”

“Mm-hm. Did a lot of arithmetic for that one. The idea is that if you somehow managed to get an infinite number of monkeys banging away on typewriters, sooner or later one of them would produce the complete works of Shakespeare. The piece I did, gee, years ago, used Terry Pratchett’s idea of a library that contains all the books that have been written, all those that will be written, and all those that would have been written but the author thought better of it. I asked, how big is that library?”

“That’s gotta be a lot of books. Here’s your coffee.”

“Thanks. I guessed maybe a billion, maximum. The Library of Congress has only 30‑some million, last I looked, and that’s real books. Anyhow, I decided to compare that to the number of possible books, printed up using some configuration of 500 characters.”

“500? What else besides ‘a, b, c‘?”

“Upper case, lower case, blanks, punctuation, math symbols, alphabets from other languages, whatever. No pictographic systems like Japanese kanji and Chinese but you can’t have everything. I defined ‘possible book’ as 500 pages, 4000 characters per page so two million per book.”

“All my books are shorter than that and they don’t scramble alphabets from different languages.”

“Short books you could pad to 500 characters with blanks at the end. Some of the experimental fanfic I’ve seen is pretty creative. At any rate, I calculated 5002,000,000 = 105,397,940 different possible books. Limit the library to 250 pages and 100 characters in, say, Spanish with no math that’d be 1001,000,000 = 102,000,000 different possible books, which is still huge, right?”

“My calculator doesn’t do numbers up in the air like that. I’ll believe you, it’s a big number. So where are you going with this?”

“So even a billion‑book library would be swamped by the other 105,397,931 books in an all‑possible‑books library. My point in that old post was that the monkeys could indeed type up Shakespeare but you wouldn’t be able to find it in the welter of absolute nonsense books.”

He’s an Ape, not a monkey

“Looks good to me, so what’d these guys prove?”

“Dunno, haven’t seen their paper yet. Give me a minute with Old Reliable … Ah, here it is, ‘A numerical evaluation of the Finite Monkeys Theorem‘ by Woodcock and Falletta. Aand it’s not paywalled!” <reading> “Wait, finite — that’s different.”

“How’s it different? Arithmetic’s arithmetic, right?”

“Until you get into infinities. True infinity operates differently than ‘large beyond anything we can measure’. I highlighted the difference in a tech note I wrote a few years ago. How would you bet if someone suggested there’s an exact duplicate Earth existing somewhere else in the Universe?”

“That’s what that goofy ‘Everything Everywhere’ movie was all about, right? Multiverses?”

“Mmm, no, the bet’s about only in our Universe.”

“Knowing you, I’d stay out of the betting.”

“Wise choice. The right answer is ‘It depends’. I calculated that there could be 1.54×10154 possible Earths with exactly the same atom count that we have, just arranged differently, maybe swap one nickel atom with one iron atom inside a hematite rock. So 1.54×10154 chances for an identical copy of you. If the Universe is infinite, then you’re guaranteed to have not just one, but an infinite number of identical copies, each of whom thinks they’re the only you.”

“That’s comforting, somehow.”

“On the other hand, if the Universe is finite, then the planet creation process would have to run through something like 10150 creations before it had a good shot at re‑making you. Vanishingly small odds.”

“So what’s this got to do with finite monkeys?”

“Woodcock and Falletta maintain that there’s only a limited number of monkeys and they’re time‑constrained. Under those conditions, there’s vanishingly small odds for Shakespeare or even the word ‘bananas’.”

~ Rich Olcott

Caged But Free

On By rolcottIn Astronomy, UncategorizedLeave a comment

Afternoon coffee time. Cal waves a handful of astronomy magazines at us as Cathleen and I enter his shop. “Hey, guys, there’s a ton of black hole stuff in the news all of a sudden.”

Cathleen plucks a scone from the rack. “Not surprised, Cal. James Webb Space Telescope looks harder and deeper than we ever could before and my colleagues have been feasting on the data. Black holes are highly energetic so the most extreme ones show up well. The Hubble and JWST folks find new extremes every week.”

Cal would be disappointed if I didn’t ask. “So what’s the new stuff in there?”

<flipping through the magazines> “This seems to be quasar jet month. We’ve got a new champion jet and this article says M87’s quasar makes novas.”

“Remind me, Cathleen, what’s a quasar?”

“A quasi‑stellar object, Sy, except we now know it’s a galaxy with a supermassive black hole—”

“I thought they all had super‑massives.”

“Most do, but these guys are special. For reasons researchers are still arguing about, they emit enormous amounts of energy, as much as a trillion average stars. Quasar luminosity is more‑or‑less flat all across the spectrum from X-rays down as low as we can measure. Which isn’t easy, because the things are so far away that Universe expansion has stretched their waves by z‑factors of 6 or 8 or more. We see their X‑ray emissions in the infrared range, which is why JWST’s optimized for infrared.”

“What does ‘flat’ tell you?”

“Sy’d give a better answer than I would. Sy?”

“Fun fact, Cal. Neither atoms nor the Sun have flat spectra and for the same reason: confinement. Electromagnetic waves come from jiggling charges, right? In an atom the electron charge clouds are confined to specific patterns centered on the nucleus. Each pattern holds a certain amount of energy. The atom can only move to a different charge pattern by emitting or absorbing a wave whose energy matches the difference between the pattern it’s in and some alternate pattern. Atomic and molecular spectra show peaks at the energies where those transitions happen.”

“But the Sun doesn’t have those patterns.”

“Not in the stepped energy‑difference sense. The Sun’s made of plasma, free electrons and nuclei all bouncing off each other, moving wherever but confined to the Sun’s spherical shape by gravity. Any particle that’s much more or less energetic than the local average eventually gets closer to average by exchanging energy with its neighbors. Free charged particles radiate over a continuous, not stepwise, spectrum of energies. The free‑particle combined spectrum has a single peak that depends on the average temperature. You only get flat spectra from systems that aren’t confined either way.”

“What I get from all that is a jet’s flat spectrum says that its electrons or whatever aren’t confined. But they must be — the things are thin as a pencil for thousands of lightyears. Something’s gotta be holding them together but why no peaks?”

“Excellent question, Cal. By the way, jets can be even longer than you said. I’ve read about your champion jet. It extends 23 million lightyears, more than a hundred times the width of the Milky Way galaxy. Straight as a string, no kinks or wiggles during a billion years of growth. I think what’s going on is that the charged particles are confined side‑to‑side somehow but they’re free to roam along the jet’s axis. If that’s the case, the flat‑spectrum light ought to be polarized. I’m sure someone is working on that test now. Your thoughts, Sy?”

“As a physicist I’m interested in the ‘somehow.’ We only know of four forces. The distances are too big for weak and strong nuclear forces. Gravity’s out, too, because it acts equally in all directions, not just crosswise to the axis. That leaves electromagnetic fields in some super‑strong self‑reinforcing configuration. The particles must be spiraling like mad about that central axis. I’ll bet that explains Cal’s quasar galaxy concentrating novae close to its SMBH jet axis. A field that strong could generate enough interference to wreak havoc on an unstable star’s plasma.”

Hubble’s view of the M87 galaxy and jet
Credit NASA and the Hubble Heritage Team (STScI/AURA)

~ Rich Olcott

The Importance of Saving Data

On By rolcottIn General: Serious Stuff, Uncategorized1 Comment
  • A repost from 8 years ago, but it’s become timely again. Eight years ago my concern was data related to Public Health and animal welfare. Subsequent event proved that concern was well‑founded. This time around the climate and Public Health issues are still with us but the likelihood of ideological meddling spreads much more broadly, to research related to psychoactives, guns, citizenship status and more. Forewarned is forearmed.

Sorry, but I’ve got to break into my normal Monday-morning stream to spread this around.  It’s a ProPublica document (click on the link to pull down a copy) detailing safe ways to leak information.

When I first heard about the data-stashing “parties” I thought it was something of an over-reaction.  Climate scientists and students organizing a massive effort to copy important data out of government files in case the new Administration decided to cover it all up somehow.

I’ve changed my mind.

What changed it was USDA’s suddenly blocking access to their animal welfare database, the one that keeps inspection records on research labs, companies, zoos, circuses, and animal transporters and how well they adhere to the Animal Welfare Act.

The agency said in a statement that it revoked public access to the reports “based on our commitment to being transparent …”  Being transparent by blocking information — there’s a certain Orwellian flavor to that, but it gets better.

I followed this article‘s link to see the original statement.  Well, I tried to follow it.  FireFox flat-out refused to show me the page because “Your connection is not secure. The owner of acis.aphis.educ.usda.gov has configured their website improperly.”  The error code was “SEC_ERROR_UNKNOWN_ISSUER.” Funny that an official .gov site mucked up its security certificate.

Then I tried Microsoft’s  edge browser, which has less alert security than my beefed-up FireFox.  edge showed me an imposing and somewhat threatening USDA e-Login page including the statements that “Unauthorized or improper use of this system may result in disciplinary action, as well as civil and criminal penalties…. You have no reasonable expectation of privacy regarding any communications or data transiting or stored on this information system…. Your consent is final and irrevocable…”

disappearing-lorem-ipsum

All this before Mr Sonny Perdue III is confirmed as the new Secretary of Agriculture. That name rings a bell, right?  Yeah, Perdue Farms, the country’s #3 poultry farmer. It’s hard not to connect dots to the Department suddenly wanting to hide farm inspection records.

So, it’s now pretty clear that we can expect other government-funded databases to disappear without warning, especially databases even remotely related to climate change, drug safety, water supply degradation, … you know, the things that there are regulations about that get in the way when your object is to maximize profits.

So — if you’re in science and you have possession of or access to data (databases, files, whatever) that might be in jeopardy

  1. Get it to an offsite and secure backup ASAP
  2. When/if it becomes clear that your or the public’s access to that data is about to be restricted, take one or more of the actions laid out in the ProPublica document.

Sometimes it’s rational to be paranoid.

~~ Rich Olcott

The Spaghettification Zone

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

Vinnie’s still wincing. “That neutron star pulling all the guy’s joints apart — yuckhh! So that’s spaghettification? I thought that was a black hole thing.”

“Yes and no, in that order. Spaghettification’s a tidal phenomenon associated with lopsided gravity fields, black holes or otherwise. You know what causes the tides, of course.”

“Sure, Sy. The Sun pulls up on the water underneath it.”

“That’s not quite it. The Sun’s direct‑line pull on a water molecule is less than a part per million of the Earth’s. What really happens is that the Sun broadly attracts water molecules north‑south east‑west all across the Sun‑side hemisphere. There’s a general movement towards the center of attraction where molecules pile up. The pile‑up’s what we call the tide.”

“What explains the high tide on the other side of the Earth? You can’t claim the Sun pushes it over there.”

“Of course not. It goes back to our lopsided taste of the Sun’s gravitational field. If it weren’t for the Sun’s pull, sea level would be a nice round circle where centrifugal force balances Earth’s gravity. The Sun’s gravity puts its thumb on the scale for the near side, like I said. It’s weaker on the other side, though — balance over there tilts toward the centrifugal force, makes for a far‑side bulge and midnight tides. We get lopsided forces from the moon’s gravity, too. That generates lunar tides. The solar and lunar cycles combine to produce the pattern of tides we experience. But tides can get much stronger. Ever hear of the Roche effect?”

“Can’t say as I have.”

“Imagine the Earth getting closer to the Sun but ignore the heat. What happens?”

Cygnus X-1 Illustration
Credit: NASA, CXC, Melissa Weiss (CXC)

“Sun‑side tides get higher and higher until … the Sun pulls the water away altogether!”

“That’s the idea. In the mid‑1800s Édouard Roche noticed the infinity buried in Newton’s F=GMm/r² equation. He realized that the forces get immense when the center‑to‑center distance, r, gets tiny. ‘Something’s got to give!’ he thought so he worked out the limits. The center‑to‑center force isn’t the critical one. The culprit is the tidal force which arises from the difference in the gravitational strength on either side of an object. When the force difference exceeds the forces holding the object together, it breaks up.”

“Only thing holding the ocean to Earth is gravity.”

“Exactly. Roche’s math applies strictly to objects where gravity’s the major force in play. Things like rubble‑pile asteroids like Bennu and Dimorphos or a black hole sipping the atmosphere off a neighboring blue supergiant star. We relate spaghettification to rubble piles but it can also compete with interatomic electronic forces which are a lot stronger.”

“You’re gonna get quantitative, right?”

“Of course, that’s how I operate.” <tapping on Old Reliable’s screen> “Okay, suppose Niven’s guy Shaffer is approaching some object from far away. I’ve set up tidal force calculations for some interesting cases. Turns out if you know or can estimate an object’s mass and size, you can calculate its density which is key to Roche’s distance where a rubble pile flies apart. You don’t need density for the other thresholds. Spagettification sets in when tidal force is enough to bend a molecule. That’s about 500 newtons per meter, give or take a factor of ten. I estimated the rip‑apart tidal force to be near the tensile strength of the ligaments that hold your bones together. Sound fair?”

“Fair but yucky.”

“Mm‑hm. So here’s the results.”

“What’s with the red numbers?”

“I knew you’d ask that first. Those locations are inside the central object so they make no sense physically. Funny how Niven picked the only object class where stretch and tear effects actually show up.”

“How come there’s blanks under whatever ‘Sgr A*’ is?”

“Astronomer‑ese for ‘Sagittarius A-star,’ the Milky Way’s super‑massive black hole. Can’t properly calculate its density because the volume’s ill‑defined even though we know the Event Horizon’s diameter. Anyhow, look at the huge difference between the Roche radii and the two thresholds that affect chemical bonds.”

“Hey, Niven’s story had Shaffer going down to like 13 miles, about 20 kilometers. He’d’ve been torn apart before he got there.”

“Roughly.”

~~ Rich Olcott

Stretch

On By rolcottIn Physics: Newton and Gravity, Space Travel, Uncategorized2 Comments

It’s a chilly day as I take my favorite elevator up to my office on the Acme Building’s 12th floor. Vinnie’s on my sofa, reading an old paperback. “Morning, Sy. Whaddaya think of Larry Niven?”

“One of the grand old men of hard science fiction. I gather you’re reading something of his there?”

“Yup, been bingeing on his Known Space series. His Neutron Star short story here won a Hugo back in 1967. It’s got so many numbers I wonder how good they are.”

“Probably pretty good. He and Heinlein both enjoyed showing off their celestial mechanics chops. What numbers stick out to you? Wait, what’s the story line again?”

“Story line? Most of Niven’s shorts were puzzles. When he had a good one he’d wrap some hokey story around it. This one, there’s a magical space ship that’s supposed to be invulnerable. Says here nothing can get through the hull, ‘no kind of electromagnetic energy except visible light. No kind of matter, from the smallest subatomic particle to the fastest meteor’ except something reached in and squashed two people to death in the nose of their ship. Our hero Mr Shaeffer’s in a ship just like theirs and has to figure out what the something was before it gets him, too.”

“Ah. What numbers did Niven give us?”

“Shaeffer’s ship was heading towards a neutron star. Lessee… ah, says the star’s mass is 1.3 times the Sun’s, diameter’s about 12 miles, and the ship’s on a fast in‑and‑out orbit, closest approach just a mile above the surface. Oh, and early on he drifts forward like something’s pulling on him but not on the ship. What does that tell you?”

“Enough to solve the puzzle, not enough to check his numbers. Anything about speed?”

“Mmm, he says the ship popped into the system a million miles out and it’d take 12 hours to reach the close‑approach point. The average speed’s just arithmetic, right?”

“Not really. A simple average doesn’t take account of acceleration changes or relativity effects. It’s easier and more accurate to apply conservation of energy. Okay with you if I assume the ship ‘pops into the system’ with zero velocity relative to the star and then free‑falls towards it?”

“That fits with the story, mostly.”

“Good. So right after the pop‑in” <tapping on Old Reliable’s screen> “the ship’s gravitational potential energy is ‑1.08×105 joules/kilogram—”

“Negative?”

“It’s defined as the potential energy Shaeffer’d gave up en route from infinitely far away. At 13 miles from the star’s center, that’s zoomed to ‑8.3×109 J/kg. The potential energy’s converted to kinetic energy ½mv² except we’re talking per kilogram so m is 1.0 and the velocity is —whoa!— 129 thousand kilometers/second. That’s 43% of lightspeed!”

“Well, Shaeffer did see the background stars shift blue even before he got deep into the gravity well. So, how about Niven’s 12‑hour, million‑mile claim?”

“That distance in that time works out to 37 miles per second, way less than lightspeed’s 186 000. Shaeffer was dawdling. You need calculus to figure the actual travel time — integrate 1/v between here and there. Ugly problem to solve manually but Old Reliable’s up to it. Given the appropriate orbit equation and the numbers we’ve worked out so far, Old Reliable says the trip should have taken him about 17 seconds.”

“HAW! I knew something seemed off. Wait, you said you’d solved the puzzle. What’s your answer?”

“Tides. That’s what moved him forward relative to the ship.”

“Yeah, that’s what Niven wrote, but I don’t see why what Shaeffer did saved him.”

“What did Shaeffer do?”

“Spread-eagled himself across a gangway at the ship’s center of gravity.”

“Brilliant — minimized his thickness along the star‑to‑ship line. Gravity’s pull on his sternum wasn’t much different from the pull on his spine. If he’d oriented himself perpendicular to that, his feet would feel a stronger pull than his head would have. Every transverse joint from neck to ankles would crackle or even tear. Talk about chiropractic.”

Vinne winces. “Why does thickness matter?”

“Tidal force reflects how center‑to‑center force changes with distance. Center‑to‑center force rises with 1/r². Tidal force goes up as 1/r³. Cube grows faster than square. Small r, big tides.”

~ Rich Olcott

Competing Curves

On By rolcottIn Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

It’s still October but there’s a distinct taste of oncoming November in the air — grey, gusty with a moist chill as I step into Cal’s coffee shop. “You’re looking a bit grumpy, Cal.”

“Sure am, Sy. Some lady come in here, wanted pumpkin spice. The nerve! I sell good honest high‑quality coffee, special beans and everything, no goofy flavors. You want peppermint or apple brown betty, go down to the mermaid place. Here’s your mugfull, double‑dark as always. By the way, fair warning — Richard Feder’s in town and looking for you. He’s at that corner table.”

“Thanks, Cal.” <sound of footsteps> “Morning, Mr Feder. How’d things go in Fort Lee?

“Nicely, nicely… I got a question, Moire.”

“Of course you do.”

“I been reading your stuff, you had a graph in one post looks just like the graph in a different post. Here, I printed ’em out. What’s up with that?”

“But they plot entirely different things, brightness against distance in one, atom loss against time in the other, completely different equations.”

“Yeah, yeah, but the shapes are the same I don’t care you say they got different equations. Look, they even both go through the same points at x=2 and 4. What’re you trying to pull here?”

“Not pulling anything. Those two curves are similar, yes, but they’re not identical.” <quickly building charts on Old Reliable> “Here, I’ve laid them both on the same axis. For good measure I’ve extended the x‑axis into a second panel with a stretched‑out y‑axis. What do you see?”

“Well, the orange one goes up and stops but it looks like the blue one’s headed for the sky.”

“It is. But where on the x-axis do those things happen?”

“Zero and one. Okay so the blue line squoze in a little.”

“How about out there at the x=8 end? Looks like they’re close, I’ll grant you, but check the y‑values at at the left of the second panel.”

“Uhh… Looks like blue’s four times higher than orange. Then the orange line flattens out but the blue line not so much.”

“Mm‑hm. So they behave differently at that end, too.”

“Yeah, but what about in the middle here” <jabs finger at Old Reliable’s screen> “where they’re real close and even cross over each other a couple times and you could just draw a straight line?”

“You’ve put your finger on something that challenges every theoretician and research experimentalist who works in a quantitative field. How do you connect the dots? Sure, you can eyeball a straight line through observed points sometimes, there are even statistical techniques for locating the best possible straight line, but is a straight line even appropriate? Sometimes it is, sometimes it’s not, and often we don’t know.”

“How can you not know? Everything starts with a straight line, shortest distance between two points, right?”

“Only if they’re the right points. Real observations are always uncertain. Lenses are never perfect, adjustment screws have a little bit of play, detector pixels are larger than a perfect point would be, whatever. Good experimentalists put enormous amounts of time and care into eliminating or at least controlling for every imaginable error source, but perfect measurements just don’t happen.”

“So it’ll be a fuzzy straight line.”

“For some range of ‘fuzzy’, mm‑hm. Now we get into the theory issues. We’ve already seen the simplest one — range of validity. Your straight‑line approximation might be good enough for some purposes in the x‑range between 2 and 4, but things get out of hand outside of that range.”

“Okay, in graphs. But these two curves both look good. Why choose one over the other?”

“That’s where theory and data collude. Sometimes theories tell us what data to look for, sometimes the data challenges us to develop an explanatory theory, sometimes we just try curve after curve until we find one that works across the full range that experiment can reach but we don’t know why. What’s exciting is when we get to use the data to determine which of several competing theories is the correct one. Or least incorrect.”

“I got other ways to get excited.”

“Of course you do.”

~~ Rich Olcott

A Quick Email Note

On By rolcottIn UncategorizedLeave a comment

To those who subscribe to this blog via email —

I’ve come to realize a couple of things about how WordPress (the software this blog sits on top of) handles what you receive in your Inbox.

First, every post I build includes some theme‑related artwork as a “Featured Image” up top. The image appears above the title when you look at the post online, but it’s not included in the emails. At all. Worse (from the image’s point of view), how much of the FI you see depends on whether you’re using your phone, a tablet, or full‑screen or partial window on a deskside computer. When I figured that out I started sticking a second, smaller copy of the image onto the bottom of the post. That way everyone sees some version of the artwork, like this one that should have gone on the bottom of the 14 Oct 2024 post (The Oldest Clock Ticks Slowest):

Second, WordPress recently enhanced its offering with an Expand On Click option. I’ve been selecting that for charts, calculations and such so they don’t take up a lot of screen space but readers can get a bigger view by clicking on the small version. Except that operation doesn’t work on the email distributions. I hope they’ll extend that functionality Real Soon Now.

In the meantime, if the emailed post you see has a chart or side‑bar that’s too small to read, please click on the post’s title. That’s a link that should transfer activity to your browser, then take you to the corresponding online post where you can click on the too‑small item to see the larger version.

And thanks for your interest in Hard Science.

~ Rich Olcott

The Oldest Clock Ticks Slowest

On By rolcottIn Astronomy: Techniques, Chemistry, UncategorizedLeave a comment

<Cliff‑hanger Cathleen strikes again> “How can you even measure a 2‑million year half‑life?”

Kareem’s right back at her. “What’s a half‑life?”

“Start a clock, weigh a sample, wait around for a while and then weigh it again to see how much is still there. When half of it’s gone, stop the clock and you’ve measured a half‑life. Simple.”

“Simple but not that simple or maybe a bit simpler. For one thing, you don’t have to wait for a full half‑life. For spontaneous radioactivity, all you have to know is the interval and whatever fraction disappeared. There’s a nice equation that ties those two to the half‑life.”

“Spontaneous? Like there’s another kind?”

“Stimulated radioactivity. That’s what nuclear reactors do — spew neutrons at uranium‑235 atoms, for instance, transmuting them to uranium‑236 so they’ll split into krypton and barium atoms and release energy and more neutrons. How often that happens depends on neutron concentration. Without that provoking push, the uranium nuclei would just split when they felt like it and that’s the natural half‑life.”

“Wait. I know that curve, it’s an exponential. Why isn’t e in the equation?”

“It could be. Would you prefer e-0.69315*t/half‑life? Works just as well but it’s clumsier. Base‑2 makes more sense when you’re talking halves. Usually when you say ‘exponential’ people visualize an increase. Here we’re looking at a decrease by a constant percentage rate but yeah, that’s an exponential, too. You get a falling curve like that from a Geiger counter and you’re watching counts per minute from someone’s thyroid that’s been treated with iodine‑131. Its 8‑day half‑life is slow enough to track that way. Really short half‑lives I don’t know much about; I care about the slow disintegraters that are either primordial or generated by some process.”

“Primordial — that means ‘back to the beginning’, which in your specialty would mean the beginning of the Solar System. We’re pretty sure the Sun’s pre‑planetary disk was built from dust broadcast by stars that went nova. Isotopes in the dust must be the primordial isotopes, right? Which ones are the other kind?”

“Mmm, aluminum‑26 is a good example. The half‑life equation still applies even for million‑year intervals. Half‑life of aluminum‑26 is about 0.7 million years and it decays to magnesium‑26. Whatever amount got here from the stars would have burnt down to a trillionth of that within the first 30 million years or so after arrival. Any aluminum‑26 we find today couldn’t be primordial. On the other hand, cosmic rays can smack a proton and neutron out of a silicon‑28 nucleus and voila! a new aluminum‑26. There’s a steady rain of cosmic rays out in space so there’s a steady production of aluminum‑26 out there. Not here on Earth, though, because our atmosphere blocks out most of the rays. Very nice for us geologists who can compare measured aluminum‑26 to excess magnesium‑26 to determine when a meteorite fell.”

“Excess?”

“Background magnesium is about 10% magnesium‑26 so we have subtract that to get the increment which came from aluminum‑26. A lot of the arguments in our field hinge on how much of which isotope is background or was background when a given rock formed. That’s one reason you see so much press about tiny but rugged zircons. They’re key to uranium‑lead dating. Crystallizing zirconium silicate doesn’t allow lead ions into its structure but it happily incorporates uranium ions. Uranium‑235 and uranium‑238 both decay to crystal‑trapped lead, but each isotope goes to a different lead isotope and with a different half‑life. The arithmetic’s simpler and the results are more definitive when you know that the initial lead content was zero.”

“So that aluminum‑magnesium trick’s not your only tool?”

“Hardly. The nuclear chemists have given us a long list of isotope chains, what decays to what with what half‑life and how much energy the radiating particle gets. Nuclei flit between quantum energy levels just like atoms and molecules do, except a spectrum of alpha or beta particles is a different game from the light‑wave spectrum. Tell me a radiated particle’s energy and I can probably tell you which isotope spat it out and disappeared.”

“Your ladder rungs are Cheshire Cat grins.”

~~ Rich Olcott