Tag: Mathematics: Spherical Harmonics

A Spherical Bandstand

On By rolcottIn Astronomy: Planetary Science, Mathematics, Mathematics: Spherical Harmonics, UncategorizedLeave a comment
Radial harmonics Rn = Ln e-nr

“Whoa, Sy, something’s not right. Your zonal harmonics — I can see how latitudes go from pole to pole and that’s all there are. Your sectorial harmonic longitudes start over when they get to 360°, fine. But this chart you showed us says that the radius basically disappears crazy close to zero. The radius should keep going forever, just like x, y and z do.”

“Ah, I see the confusion, Susan. The coordinate system and the harmonic systems and the waves are three different things, um, groups of things. You can think of a coordinate system as a multilevel stage where chords of harmonic musicians can interact to play a composition of wave signals. The spherical system has latitude and longitude levels for the brass and woodwind players, plus one in back for the linear percussion section. Whichever direction the brass and woodwinds point, that’s where the signals go out, but it’s the percussion that determines how far they get. Sure, radius lines extend to infinity but except for R0 radial harmonics damp out pretty quickly.”

“Signals… Like Kaski’s team interpreted Juno‘s orbital twitches as a signal about Jupiter’s gravitational unevenness. Good thing Juno got close enough to be inside the active range for those radial harmonics. How’d they figure that?”

“They probably didn’t, Cathleen, because radial harmonics don’t fit easily into real situations. First problem is scale — what units do you measure r in? There’s an easy answer if the system you’re working with is a solid ball, not so easy if it’s blurry like a protein blob or galaxy cluster.”

“What makes a ball easy?”

“Its rigid surface that doesn’t move so it’s always a node. Useful radial harmonics must have a node there, another node at zero and an integer number of nodes between. Better yet, with the ball’s radius as a natural length unit the r coordinate runs linearly between zero at the center and 1.0 at the surface. Simplifies computation and analysis. In contrast, blurries usually don’t have convenient natural radial units so we scrabble around for derived metrics like optical depth or mixing length. If we’re forced into doing that, though, we probably have worse challenges.”

“Like what?”

“Most real-world spherical systems aren’t the same all the way through. Jupiter, for instance, has separate layers of stratosphere, troposphere, several chemically distinct cloud‑phases, down to helium raining on layers of hydrogen in liquid, maybe slushy or even solid form. Each layer has its own suite of physical properties that put kinks into a radial harmonic’s smooth curve. Same problem with the Sun.”

“How about my atoms? The whole Periodic Table is based on atoms having a shell structure. What about the energy level diagrams for atomic spectra? They show shells.”

“Well, they do and they don’t, Susan. Around the turn of the last Century, Lyman, Balmer, Paschen, Brackett and Pfund—”

“Sounds like a law firm.”

“<ironically> Ha, ha. No, they were experimental physicists who gave the theoreticians an important puzzle. Over a 40‑year period first Balmer and then the others, one series at a time, measured the wavelengths of dozens of lines in hydrogen’s spectrum. ’Okay, smarties, explain those!‘ So the theoreticians invented quantum mechanics. The first shot did a pretty good job for hydrogen. It explained the lines as transitions between discrete states with different energy levels. It then explained the energy levels in terms of charge being concentrated at different distances from the nucleus. That’s where the shell idea came from. Unfortunately, the theory ran into problems for atoms with more than one electron.”

“Give us a second… Ah, I get why. If one electron avoids a node, another one dives in there and that radius isn’t a node any more.”

“Got it in one, Cathleen. Although I prefer to think of electrons as charge clouds rather than particles. Anyhow, when an atom has multiple charge concentrations their behavior is correlated. That opens the door to a flood of transitions between states that simply aren’t options for a single‑electron system. That’s why the visible spectrum of helium, with just one additional electron, has three times more lines than hydrogen does.”

“So do we walk away from spherical harmonics for atoms?”

“Oh, no, Susan, your familiar latitude and longitude harmonics fit well into the quantum framework. These days, though, we mostly use combinations of radial fade‑aways like my Sn00 example.”

~~ Rich Olcott

Jupiter And The Atoms

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, UncategorizedLeave a comment

“Okay, Sy, what’s your third solution?”

“Solution to what, Susan?”

“These harmonic thingies. They’re about angles so it makes sense to chart them in polar or spherical coordinates, but when they take on negative values the radius goes the wrong way. You said one solution was to chart the negatives in a different color. That’s confusing, though. Another solution is to square all the values to get everything into positive territory. That’s okay for chemists like me because the peaks and nodes we care about stay in the same places. What’s the third option?”

“One that gets to why these ‘harmonic thingies’ are interesting at all. When Juno‘s orbiting Jupiter, does it feel each of Kaspi’s Jn shapes individually?”

“No, of course not, she just reacts to how they all add togethherrr … Oh! So you’re saying we can handle negative values from one harmonic by adding it to another one that’s more positive and plotting the combination.”

<pointing to paper napkin> “Bingo! Remember this linear plot of J2 where I colored its negative section pink?” <pointing to display on Old Reliable> “When you multiply J2 by C0 you get S220. I added that to four helpings of Sn00 to get this combination.”

“Ah, that negative region in S220‘s middle shaves back the equator on Sn00‘s sphere while the positive part adds bumps top and bottom.” <Susan gives me the side‑eye> “Why’d you pick that 4‑to‑1 ratio, and what’s with those n subscripts instead of numbers?”

“Getting a little ahead of myself. For the moment let’s concentrate on Juno‘s experience with Jupiter’s gravity. One reason I chose that ratio was that it’s pretty easy to see in the picture. In real cases the physical system determines the ratios. Kaspi’s team derived their ratios experimentally. They used math to fit a model to Juno‘s very slightly wobbly orbit. Their model of Jupiter’s gravity field started from the spherical J0 shape. They tweaked that by adding different ratios of J2 through J40, adjusting the ratios until the model’s total gravity field predicted an orbit that matched the real‑world one. J2‘s share was about 15 parts per thousand but most of the rest contributed less than a part per million. Jupiter probably uses multiple mass blobs to make the J2 shape. The point is, the planet’s really a mess but we can analyze the mess in terms of the harmonics.”

“So that’s how you drew what Cal called your wiggle-waggles — you followed Kaski’s Jn recipe and then added some constant to push the polar plot out far enough that the negatives didn’t poke out the wrong side. That constant — what value did you use and why that one?”

“That’s exactly what I did do, Cathleen. Frankly, I don’t even remember what constant I added, just something that was big enough to make the negatives behave nicely, not so large that the peaks vanished by comparison. Calibrating accurately to Jupiter’s J0 would shrink the peaks down to parts‑per‑thousand invisibility. After all, I was more concerned with peak position than peak size.”

“Now we’re back to your 4‑to‑1 ratio. Was that arbitrary, too?”

“No, it wasn’t, Susan. Would it have been closer to Chemistry if I’d labeled that figure as 1s22s22p1?”

“Two electrons in the 1s‑shell, two in the 2s‑shell plus one 2p electron … that’s a boron atom? But you’re showing only one radial shell, not two separate ones.”

“True, but that’s to make another point. There isn’t an electron in the 1s shell, or even a pair of them nicely staying on opposite sides. The atom’s charge, all five electrons‑worth of it, is smeared out as a wave pattern across the entire structure. The Sn00 pattern captures everything that’s spherical. The S220 pattern gets what’s left.”

“But what about the radial nodes? Isn’t that the difference between 1s and 2s, that 2s has a node?”

“Oh there are nodes, alright, but they don’t have much effect. Each radial harmonic is the product of two factors — a polynomial and an exponential. The exponential part squeezes the polynomial so hard that adjacent peaks and valleys are barely bumps and dents.”

“So Jupiter and atoms use the same math, huh?”

“So does the Sun.”

~~ Rich Olcott

Shapes And Numbers

On By rolcottIn Mathematics: Spherical Harmonics, Physics: Quantum Mechanics, UncategorizedLeave a comment

I’m nursing my usual mug of eye‑opener in Cal’s Coffee Shop when astronomer Cathleen and chemist Susan chatter in and head for my table. Susan fires the first volley.

“Sy, those spherical harmonics you’ve posted don’t look anything like the atomic orbitals in my Chem text. Shouldn’t they?”
 ”How do you add and multiply shapes together?”
  ”What does the result even mean?”
   ”And what was that about solar seismology?”

The angular portion of the lowest‑energy spherical harmonics.
Adapted from work by Inigo.quilez, under CCA SA 3.0 license

“Whoa, have you guys taken interrogation lessons from Mr Feder? One at a time, please. Let’s start with the basics.” <sketching on a paper napkin> “For example, the J2 zonal harmonic depends only on the latitude, not on the longitude or the distance from the center, so whatever it does encircles the axis. Starting at the north pole and swinging down to the south pole, the blue line shows how J2 varies from 1.0 down to some negative decimal. At any latitude, whatever else is going on will be multiplied by the local value of J2.”

“The maximum is 1.0, huh? Something multiplied by a number less than 1 becomes even smaller. But what happens where J2 is zero? Or goes negative?”

“Wherever Jn‘s zero you’re multiplying by zero which makes that location a node. Furthermore, the zero extends along its latitude all around the sphere so the node’s a ring. J2‘s negative value range does just what you’d expect it to — multiply by the magnitude but flip the product’s sign. No real problem with that, but can you see the problem in drawing a polar graph of it?”

“Sure. The radius in a polar graph starts from zero at the center. A negative radius wouldn’t make sense mixed in with positives in the opposite direction.”

“Well it can, Cathleen, but you need to label it properly, make the negative region a different color or something. There are other ways to handle the problem. The most common is to square everything.” <another paper napkin> “That makes all the values positive.”

“But squaring a magnitude less than 1 makes it an even smaller multiplier.”

“That does distort the shapes a bit but it has absolutely no effect on where the nodes are. ’Nothin’ times nothin’ is nothin’,’ like the song says. Many of the Chem text orbital illustrations I’ve seen emphasize the peaks and nodes. That’s exactly what you’d get from a square‑everything approach. Makes sense in a quantum context, because the squared functions model electron charge distributions.”

“Thanks for the nod, Sy. We chemists care about charge peaks and nodes around atoms because they control molecular structures. Chemical bonds and reactions tend to localize near those places.”

“I aim for fairness, Susan. There is another way to handle a negative radius but it needs more context to look reasonable. Meanwhile, we’ve established that at any given latitude each Jn is just a number so let’s look at longitudes.” <a third paper napkin> “Here’s the first two sectorial harmonics plotted out in linear coordinates.”

“Looks familiar.”

“Mm‑hm. Similar principles, except that we’re looking at a full circle and the value at 360° must match the value at 0°. That’s why Cm always has an even node count — with an odd number you’d have -1 facing +1 and that’s not stable. In polar coordinates,” <the fourth paper napkin> “it’s like you’re looking down at the north pole. C0 says ‘no directional dependence,’ but C1 plays favorites. By the way, see how C1‘s negative radii in the 90°‑270° range flip direction to cover up the positives?”

“Ah, I see where you’re going, Sy. Each of these harmonics has a numeric value at each angle around the center. You’re going to tell us that we can multiply the shapes by multiplying their values point by point, one for each latitude for a J and each longitude for a C.”

“You’re way ahead of me as usual, Cathleen. You with us, Susan?”

“Oh, yes. In my head I multiplied your J2 by C0 and got a pz orbital.”

“I’m impressed.”
 ”Me, too.”

“Oh, I didn’t do it numerically. I just followed the nodes. J2 has two latitude nodes, C0 has no longitude nodes. There it is, easy‑peasy.”

~~ Rich Olcott

A Loose-end Lagniappe

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, UncategorizedLeave a comment

<chirp, chirp> “Moire here.”

“We have some loose ends to tie up. Too early for pizza. Coffee at Cal’s?”

“Hello, ‘Walt‘. Fifteen minutes?”

“Confirmed.”

He’s at a back table, facing the door, of course. He points to the steaming mug and strawberry scone beside it on the table. I nod to acknowledge. ”So, Walt, what are these loose ends?”

“My people say that Juno‘s not on a 53‑day orbit any more. NASA’s jiggled it down to 33 days. What’s that do to the numbers you gave me?”

<sliding a folded paper scrap across the table> “I had a hunch you’d want more so I worked up estimates. Juno started with a 53‑day orbit but a Ganymede flyby dropped it to 43 days. A Europa flyby took Juno to a 38‑day orbit. Now it’s swerved by Io and we’re at 33 days. I threw in the 23‑day line for grins, no extra charge.”

“Half the orbit size but no significant change in the close‑in specs. That’s surprising.”

“Not really. It’s like a dog’s butt wagging its tail. At close approach, we call it perijove, Juno is only 76 500 kilometers out from Jupiter’s center. Its orbit thereabouts is pretty much nailed down by the big guy’s central field. But there’s no second attractor to constrain the orbit’s other extreme millions of kilometers out. Do an Oberth burn near perijove or arrange for a gravity tweak from a convenient moon, you get a big difference at the far end.”

“That wraps that.” <reaches for his cane, then settles back to do a Columbo> “Just one more thing, Moire. I came in with a question about the Sun’s effect on Juno. You took care of that pretty quick but spent a load of my time and consultancy budget on these spherical harmonics. How come?”

“As I recall, you and your people kept coming back for more detail. Also, the 225 000‑kilometer radius I got from R2‘s structure was essential in calculating these close‑in numbers. You’re getting your money’s worth. I’ll even throw in a lagniappe.”

“A free gift? I never trust them.”

“Such a mean world you live in, Walt.” <displaying an image on Old Reliable> “Here it is, take it or leave it.”

Top: F000 plus a time-varying contribution from F660
Bottom: C0 plus a time-varying contribution from C4

“What is it?”

“It’s a bridge between the physics of light and sound, and the physics of atoms and stars. When I say ‘coordinates,’ what words spring to your mind?”

“Traverse and elevation.”

“Interesting choice. Any other systems?”

“Mm, latitude, longitude and altitude. And x‑y‑z if you’re in a classroom.”

“Way beyond the classroom. You use spreadsheets, right?”

“Doesn’t everyone?”

“Rows‑columns‑sheets is x‑y‑z. On digital screens, pixels‑lines‑luminosity is x‑y‑z. Descarte’s rectilinear invention is so deeply embedded in our thinking we don’t even notice it. Perpendicular straight‑line coordinates fit things that are flat or nearly so, not so good for spheres and central‑force problems. Movement there is mostly about rotation, which is why your first two picks were angular instead of linear.”

“Okay, but our choice of coordinates is our choice. What have x‑y‑z or your Fnnm to do with natural things?”

“Overtones and resonance. Look at that black line in the movie. It could be a guitar string or a violin string, doesn’t matter. One end’s fixed to the instrument’s bridge, the other end’s under somebody’s finger. All other points on the string are free to move, subject to tension along the string. Then someone adds energy to the string by plucking or bowing it.”

“At one of those peaks or valleys, right?”

“Nope, anywhere, which goes to my point. The energy potentially could contort the string to any shape. Doesn’t happen. The only stable shapes are combinations of sine waves with an integer number of nodes, like C4‘s quartet. Adding even more energy gives you overtones, waves that add in‑between nodes to lower‑energy waves. C0‘s no‑nodes black line could run along x, y or z in any flat system.”

“So you’re going to tell me that your C‘s, J‘s and R‘s support wave structures for spheres.”

“Indeed. All four giant planets have stripes along their J arcs. Solar seismologists have uncovered C, R and maybe J wave structures inside the Sun.”

“Bye.”

“Don’t mention it.”

~ Rich Olcott

Sectorial Setbacks

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, UncategorizedLeave a comment

<chirp, chirp> “Moire here.”

“Moire, you were holding out on me. Eddie’s, fifteen minutes.”

“Not so fast, Walt. That wasn’t me holding out, that was you leaving too soon. From now on you’re paying quite a bit more. And it’ll be thirty minutes.”

“So we’re negotiating, hmm?”

“That’s about the size of it. You still interested?”

“My people are, they sent me back here. Oh well. Thirty minutes.”

Thirty-three minutes later I walk into Eddie’s. Walt’s already gotten a table. He beckons, points to the freshly‑served pizza, raises an eyebrow.

“Apology accepted. What made your people unhappy?”

“You told me flat‑out that the Sun’s gravity couldn’t affect those zonal harmonics. Do you have anything to back that up?”

“Symmetry. Zonal harmonics and latitude are about north‑south. Each Jn is a pole‑to‑pole variation pattern. The only way solar gravity can tilt Jupiter’s north‑south axis is to exert torque along the zonal harmonics. Jupiter’s equator is within 3° of edge‑on to the Sun.” <showing an image on Old Reliable’s screen> “Here’s what the Sun sees looking at J10, for instance. Solar pull on any northern zone segment, say, would be counteracted by an equal pull on the corresponding southern segment of the same zone. No net torque, no tilt. J0‘s the only exception. It’s simply a sphere that doesn’t vary across the whole planet. The Sun’s pull along J0‘s arc can’t tilt Jupiter.”

“Okay, so the zonal picture’s too simple. Just one set of waves, running up and down the planet—”

“No, not running. One way to characterize a wave is by how its components change with time. You’re thinking like ocean waves that move from place to place as time goes by. There’s also standing waves like on a guitar string, where individual points move but the peaks and valleys don’t. There’s time‑only waves like how the day length here changes through the year. And there’s static waves where time’s not even in the equation. Jupiter’s stripes don’t move, they’re peaks and valleys in a static wave pattern. By definition, the zonal harmonic system is static like that. But you’re right, it’s only part of the picture.”

“Give me the part the Sun’s gravitational field does play with.”

“That’d be two parts — sectorial and radial harmonics. Sectorial is zonal’s perpendicular twin. Zonal wave patterns show variation along the polar axis; sectorial wave patterns Cm vary around it. I’m keeping it non‑technical for you but Cm‘s actually cos(m*x) where x is the longitude.”

“Just don’t let it go any farther.”

“I’ll try not to. My point is that each sector pattern can be labeled with a positive integer just like we did with the zones.”

“If the Jn arcs aren’t affected by solar gravity, why would I care about these Cms?”

“You wouldn’t, except for the fact that mass distribution across Jupiter’s sectors is probably lumpy. We know the Great Red Spot holds its position in the southern hemisphere and the planet’s magnetic field points way off to the side. Maybe those features mark off‑center mass deficits and concentrations. Suppose a particular sectorial wave’s peak sits directly over a mass lump or hole. Everything under that harmonic’s influence is tugged back and forth by solar gravity each time the wave traverses the day side. Juno in its N‑S path just isn’t an efficient sensor for those tugs. Good sectorial sensing would require an orbiter on an E‑W path, preferably right over the equator.  Any orbital wobbles we’d see could be fed into a sectorial gravity map. Cross that with the zonal map and we’d be able to locate underlying mass variations by latitude and longitude.”

“Not a good idea. Gravity’s not the only field in play. You’ve just mentioned Jupiter’s magnetic field. I’ve read it’s stronger than any other planet’s. If your E‑W orbiter’s built with even a small amount of iron, you’d have a hard time deciding which field was responsible for any observed irregularities.”

“Good point. The idea’s even worse than you think, though. Jupiter’s sulfur‑coated moon—”

“Io. Yes, your induction‑heating idea might even be real. What about it?”

“I haven’t written yet about the high‑voltage Io‑to‑Jupiter bridge made of sulfur, oxygen and hydrogen ions. Jupiter’s magnetism plays a complicated game with them but the result is a chaotic sheet of radiating plasma around the planet’s equator. An E‑W orbiter in there would be tossed about like a paper boat on the ocean.”

~~ Rich Olcott

Screaming Out Of Space

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, UncategorizedLeave a comment

Cal (formerly known as Al) comes over to our table in his coffee shop. “Lessee if I got this right. Cathleen is smug twice. First time because the new results from Juno‘s data say her hunch is right that Jupiter’s atmosphere moves like cylinders inside each other. Nearly cylinders, anyhow. Second smug because Sy used the Juno data to draw a math picture he says shows the Great Red Spot but I’m lookin’ at it and I don’t see how your wiggle‑waggles show a Spot. That’s a weird map, so why’re you smug about it, Cathleen?”

“The map’s weird because it’s abstract and way different from the maps you’re used to. It’s also weird because of how the data was collected. Sy, you tell him about the arcs.”

“Okay. Umm… Cal, the maps you’re familiar with are two‑dimensional. City maps show you north‑south and east‑west, that’s one dimension for each direction pair. Maps for bigger‑scale territories use latitude for north‑south and longitude for east‑west but the principle’s the same. The Kaspi group’s calculations from Juno‘s orbit data give us a recipe for only a one‑dimensional map. They show how Jupiter’s gravity varies by latitude, nothing about longitude. We could plot that as a rectangle, latitude along the x‑axis, relative strength along the y‑axis. I thought I’d learn more by wrapping the x‑axis around the planet so we could look for correlations with Jupiter’s geography. I found something and that’s why Cathleen’s smug. Me, too.”

“Why latitude but nothing about longitude?”

“Because of the way Juno‘s orbit works. The spacecraft’s not hovering over the planet or even circling it like the ISS circles Earth. NASA wanted to minimize Juno‘s exposure to Jupiter’s intense magnetic and radiation fields. The craft spends most of its 53‑day orbit at extreme distance, up to millions of kilometers out. When it approaches, it screams in at about 41 kilometers per second, that’s 91 700 mph, on a mostly north‑to‑south vector so it sees all latitudes from a few thousand kilometers above the cloud‑tops. Close approach lasts only about three hours, for the whole planet, and then the thing is on its way out again. During that three hours, the planet rotates about 120° underneath Juno so we don’t have a straight vertical N‑S pass down the planet’s face. Gathering useful longitude data’s going to take a lot more orbits.”

“So you’re sayin’ Juno felt gravity glitches at all different angles going pole to pole, but only some of the angles going round and round.”

“Exactly.”

“So now explain the wiggle‑waggles.”

“They represent parts‑per‑million variations in the field pulling Juno towards Jupiter at each latitude. Where the craft is over a more massive region it’s pulled a bit inwards and Sy’s map shows that as a green bump. Over a lighter region Juno‘s free to move outward a little and the map shows a pink dip. Kaspi and company interpret the heaviness just north of the equator to be a dense inward flow of gas all around the planet. Maybe it is. Sy and I think the pink droplet south of the equator could reflect the Great Red Spot lowering the average mass at its latitude. Maybe it is. As always, we need more data, okay? Now I’ve got questions for you, Sy.”

“Shoot.”

“You built your map by multiplying each Jn‑shape by its Kaspi gravitational intensity then adding the multiplied shapes together. But you only used Jn‑shapes with integer names. Is there a J½?”

“Some mathematicians play with fractional J‑thingies but I’ve not followed that topic.”

“Understandable. Next question — the J‘s look so much like sine waves. Why not just use sine‑shapes?”

“I used Jn‑shapes because that’s how Kaspi’s group stated their results. They had no choice in the matter. Jn‑shapes naturally appear in spherical system math. The nice thing about Jn‑shapes is that n provides a sort of wavelength scale. For instance, J35 divides Jupiter’s pole‑to‑pole arc into 36 segments each as wide as Earth’s diameter. Here’s a plot of intensity against n.”

Adapted from Kaspi, Figure 2a

“Left to right, red light to blue.”

“Exactly.”

~ Rich Olcott

Zoning Out over Jupiter

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, Physics: Waves, UncategorizedLeave a comment

I’m nursing my usual mug of eye‑opener in Cal’s Coffee Shop when astronomer Cathleen and chemist Susan chatter in. “Morning, ladies. Cathleen, prepare to be even more smug.”

“Ooo, what should I be smug about?”

“Your Jupiter suggestion. Grab some coffee and a couple of chairs.” <screen‑tapping on Old Reliable> “Ready? First step — purple and violet. You’ll never see violet or purple light coming from a standard video screen.”

“He’s going spectrum‑y on us, right, Cathleen?”

“More like anti‑spectrum‑y, Susan. Purple light doesn’t exist in the spectrum. We only perceive that color when we see red mixed with blue like that second band on Sy’s display. Violet light is a thing in nature, we can see it in flowers and dyes and rainbows beyond blue. Standard screens can’t show violet because their LEDs just emit red, blue and green wavelengths. Old Reliable uses mixtures of those three to fake all its colors. Where are you going with this, Sy?””

“Deeper into Physics. Cast your eyes upon the squiggles to the right. The one in the middle represents the lightwave coming from purple‑in‑the‑middle. The waveform’s jaggedy, but if you compare peaks and troughs you can see its shape is the sum of the red and blue shapes. I scaled the graphs up from 700 nanometers for red and 450 for blue.”

“Straightforward spectroscopy, Sy, Fourier analysis of a complicated linear waveform. Some astronomers make their living using that principle. So do audio engineers and lots of other people.”

“Patience, Cathleen, I’m going beyond linear. Fourier’s work applies to variation along a line. Legendre and Poisson extended the analysis to—”

“Aah, spherical harmonics! I remember them from Physical Chemistry class. They’re what gives shapes to atoms. They’ve got electron shells arranged around the nucleus. Electron charge stays as close to the nucleus as quantum will let it. Atoms absorb light energy by moving charge away from there. If the atom’s in a magnetic field or near other atoms that gives it a z-axis direction then the shells split into wavey lumps going to the poles and different directions and that’s your p-, d– and f-orbitals. Bigger shells have more room and they make weird forms but only the transition metals care about that.”

The angular portion of the lowest-energy spherical harmonics
Credit: Inigo.quilez, under CCA SA 3.0 license
Adapted from Wieczorek and Meschede under CC BY-NC-ND 4.0

“Considering you left out all the math, Susan, that’s a reasonable summary. I prefer to think of spherical harmonics as combinations of wave shapes at right angles. Imagine a spherical blob of water floating in space. If you tap it on top, waves ripple down to the bottom and back up again and maybe back down again. Those are zonal waves. A zonal harmonic averages over all E‑W longitudes at each N‑S latitude. Or you could stroke the blob on the side and set up a sectorial wave pattern that averages latitudes.”

“How about center‑out radial waves?”

“Susan’s shells do that job. My point was going to be that what sine waves do for characterizing linear things like sound and light, spherical harmonics do for central‑force systems. We describe charge in atoms, yes, but also sound coming from an explosion, heat circulating in a star, gravity shaping a planet. Specifically, Jupiter. Kaspi’s paper you gave me, Cathleen, I read it all the way to the Results table at the tail end. That was the rabbit‑hole.”

“Oh? What’s in the table?”

“Jupiter’s zonal harmonics — J‑names in the first column, J‑intensities in the second. Jn‘s shape resembles a sine wave and has n zeroes. Jupiter’s never‑zero central field is J0. Jn increases or decreases J0‘s strength wherever it’s non‑zero. For Jupiter that’s mostly by parts per million. What’s cool is the pattern you see when you total the dominating Jeven contributions.”

Data from Kaspi, et al.

Cathleen’s squinting in thought. “Hmm… green zone A would be excess gravity from Jupiter’s equatorial bulge. B‘s excess is right where Kaspi proposed the heavy downflow. Ah‑HAH! C‘s pink deficit zone’s right on top of the Great Red Spot’s buoyant updraft. Perfect! Okay, I’m smug.”

~ Rich Olcott

The Shapes of Fuzziness

On By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

Egg murmuration 1“That was a most excellent meat loaf, Sis.  Flavor balance was perfect.”

“Glad you liked it, Sy.  Mom’s recipe, of course, with the onion soup mix.”

“Yeah, but there was an extra tang in there.”

“Hah, you caught that!  I threw in some sweet pickle relish to brighten it some.”

“Mommy, Uncle Sy told me about quantum thingies and how they hide behind barriers and shoot rainbows at us.”

Sis gives me that What now? look so I must defend myself.  “Whoa, Teena, that’s not even close to what I said.”

“I know, Uncle Sy, but it’s more fun this way.  Little thingies going, ‘Pew! Pew! Pew!’”

“Hey, get me out of trouble with your Mom, here.  What did I say really?”

<sigh> “Everything’s made of these teeny-weeny quantum thingies, smaller even than a water-bear egg — so small — and they have to obey quantum rules.  One of the rules is, um, if a lot of them get together to make a big thing, the big thing has to follow big-thing rules even though the little things follow quantum rules.”

“Nicely put, Sweetie.”

“And sometimes the quantum thingies act like waves and sometimes they act like real things and no-one knows how they do that.  And, uh, something about barriers making forbidden places that colors come out of and I’m mixed up about that.”

“Excellent summary, young lady.  That deserves an extra —” <sharp look from Sis who has a firm ‘No rewarding with food!‘ policy> “— chase around the block the next time we go scootering.”

“Yay!  But can you unconfuse me about the forbidden areas and colors?”

“Well, I can try.  Tell you what, bring your toy box over by the stairway, OK?  We’ll pick it all up when we’re done, Sis, I promise.  Ready, Teena?”

“Ready!”

“OK, put your biggest marble on the bottom step. Yes, it is pretty.  Now put a tennis ball and that dumbbell-shaped thing on the second step.  Oh, it’s a yo-yo?  Cool.  And that ring-toss ring, put it on the second step, too.  Now for the third step.  Put the softball there and … umm … take some of those Legos and make a little ring inside a big ring.  Thanks, Sis, just half a cup.  Ready, Teena?”

“Just a sec… ready!”

“Perfect.  Oh, Teena, you forgot to tell Mommy about the murmuration.”

“Oh, she’s seen them.  You know, Mommie, thousands of birds flying in a big flock and they have rules so they keep together but not too close and they make big pictures in the sky.”

“Yes, I have, sweetheart, but what does that have to do with quantum, Sy?”

“How would you describe their shapes?”

“Oh, they make spirals, and swirls… I’ve seen balls and cones and doughnuts and wide flowing sheets, and other shapes we simply don’t have names for.”

“These shapes on the stairs are the first few letters in science’s alphabet for describing complex shapes like atoms.  It’s like spelling a word.  That ball on the first step is solid.  The tennis ball is a hollow shell.  Pretend the softball is hollow, too, with a hollow ping-pong ball at its center.  If you pretend that each of these is a murmuration, Teena, does that make you think of anything?”

“Mmm..  There aren’t any birds flying outside of the marble, or outside or inside of the tennis ball.  And I guess there aren’t any flying between the layers in the ping-softball.  Are those forbidden areas?”

“C’mere for a high-five!  That’s exactly where I’m going with this.  The marble has one forbidden region infinitely far away.  The tennis ball has that one plus a second one at its middle.  The softball-ping-pong combo has three and so on.  We can describe any spherical fuzziness by mixing together shapes like that.”Combining shapes

“So what about the rings and that dumbbell yo-yo?”

“That’s the start of our alphabet for fuzziness that isn’t perfectly round.  Math has given us a toolkit of spheres, dumbbells, rings and fancier figures that can describe any atom.  Plain and fancy dumbbells stretch the shape out, rings bulge its equator, and so on.  Quantum scientists use the shapes to describe atoms and molecules.”

“Why the stairsteps?”

“What about my colors?”

~ Rich Olcott

To Bond Or Not To Bond, That Is The Question

On By rolcottIn Chemistry, Physics: Quantum Mechanics, Physics: Waves, UncategorizedLeave a comment

Vinnie’s pushing pizza crumbs around his plate, watching them clump together.  “These molecular orbitals gotta be pretty complicated.  How do you even write them down?”

“Combinations.  There’s a bunch of different strategies, but they all go back to Laplace’s spherical harmonics.  Remember, he showed that every possible distribution around a central attractor could be described as a combination of his patterns.  Turn on a field, like from another atom, and you just change what combination is active.  Here’s a sketch of the simplest case, two hydrogen atoms — see how the charge on each one bulges toward the other?  The bulge is a combination of a spherical orbital and a dumbbell one.  The molecular orbitals are combinations of orbitals from both atoms, describing how the charges overlap, or not.”Hydrogen molecule

“What’s that blue in the other direction?”

“Another possible combination.  You can combine atomic orbitals with pluses or minuses.  The difference is that the minus combination will always have an additional node in between.  Extra nodes mean higher energy, harder to activate. When the molecule’s in the lowest energy state, charge will be between the atoms where that extra node isn’t.”

“So the overlapped charge here is negative, right, and it pulls the two positive nucleusses —”

“Nuclei”

“Whatever, it pulls ’em together.  Why don’t they just merge?”

“Positive-positive repulsion counts, too.  At the equilibrium bond distance, the nuclei repel each other exactly as much as the shared charge pulls them together.”

Eddie’s still hovering by our table.  “You said that there’s this huge number of possible atomic orbitals.  Wouldn’t there be an even huger number of molecular orbitals?”

“Sure.  The trick is in figuring out which of them are lowest-energy and activated and how that relates to the molecule’s configuration.  Keep track of your model’s total energy as you move the atoms about, for instance, and you can predict the equilibrium distance where the energy is a minimum.  In principle you can calculate configuration changes as two molecules approach each other and react.”

“Looks like a lot of work.”

“For sure, Eddie.  Even a handful of atoms has lots of atomic orbitals to keep track of.  That can burn up acres of compute time.”

Vinnie pushes three crumbs into a triangle.  “You got three distances, you can figure their angles.  So you got the whole shape of the thing.”

“Right, but like Eddie said, that’s a lot of computer work.  Chemists had to come up with shortcuts.  As a matter of fact, they had the shortcuts way before the computers came along.”

“They used, like, abacuses?”

“Funny, Vinnie.  No, no math at all.  And it’s why they still show school-kids those Bohr diagrams.”

“Crazy Eights.”

“Eddie, you got games on the brain.  But yeah, eights.  Or better, quartets of pairs.  One thing I’ve not mentioned yet is that even though they’ve got the same charge, electrons are willing to pair up.”

“How come?”

“That’s the thing of it, Vinnie.  There’s a story about Richard Feynman, probably the foremost physicist of the mid-20th Century.  Someone asked him to explain the pairing-up without using math.  Feynman went into his office for a week, came back out and said he couldn’t do it.  The math demands pairing-up, but outside of the math all we can say is experiments show that’s how it works.”

“HAH, that’s the reason for the ‘two charge units per orbital’ rule!”

“Exactly, Eddie.  It’s how charge can collect in that bonding molecular orbital in the first place.  It’s also the reason that helium doesn’t form molecules at all.  Imagine two helium atoms, each with two units of charge.  Suppose they come close to each other like those hydrogens did.  Where would the charge go?”

“OK, you got two units going into that in-between space, ahh, and the other two activating that blue orbital and pulling the two atoms apart.  So that adds up to zero?”

“Uh-huh.  They just bounce off and away.”

“Cool.”

“Hey, I got a question.  Your sketch has a ball orbital combining with a dumbbell.  But they’ve got different node counts, one and two.  Can you mix things from different shells?”

“Sure, Vinnie, if there’s enough energy.  The electron pair-up can release that much.”

“Cool.”

~~ Rich Olcott

  • A friend pointed out that I’m doing my best to avoid saying the word “electron.” He’s absolutely right.  At least in this series I’m taking Bohr’s side in his debate with Einstein — electrons in atoms don’t act like little billiard balls, they act like statistical averages, smeared-out ones at that.  It’s closer to reality to talk about where the charge is so that’s how I’m writing it.

The Shell You Say

On By rolcottIn Physics: Quantum Mechanics, Physics: Waves, UncategorizedLeave a comment

Everyone figures Eddie started his pizza place because he likes to eavesdrop.  No surprise, he wanders over to our table.  “I heard you guys talking about atoms and stuff and how Sy here don’t like Bohr’s model of electrons in atoms even though Bohr’s model and the shell model both account for hydrogen’s spectrum.  Why’s the shell model better?”

Vinnie comes back quick.  “Because it’s not physically impossible, for one thing.”

I’m on it.  “Because the shell model extends smoothly to atoms and ions in an electric or magnetic field.  Better yet, shell methods can be applied to molecules.”

“What do fields have to do with it?”

“It’ll help to know that some of those electron patterns come in sets.  The 2-node shell has three dumbbell shapes, for instance — one each along the x, y and z axes. Think about an atom all alone in space with no fields around.  How does it know which way z goes?””

“It don’t.  Everything’s gotta be in all directions, like spherical.”

Vinnie’s back in.  “I’m seeing an atom in an electric field, say up-to-down, it’s going to pull charge in one direction, say down.  So now the atom don’t look like no ball no more, right?”

orbital in a field
Vertical field on the right

“Right.  Once the atom’s got a special direction, those three dumbbells stop being equivalent.  We say that the field mixes together the spherical pattern (in atoms we’d call it an s-orbital) with that direction’s dumbbell (we’d call it a p-orbital) to make two combination orbitals.  One combination has a lump of charge stretched downwards and the other combination has a bowl of diminished charge stretched upwards.  The stronger the field, the wider the energy split between those two.”

“What about the other two dumbbells?”

“They’re still equivalent, Eddie.  If there’s charge in them it’s spread evenly around the equator like a doughnut.  Energy-wise they’re in between the two s±p combinations.”

“IF there’s charge, like maybe there ain’t?”

“Ever suspicious, eh, Vinnie?  You’re right, and that’s a good point.  Orbitals are only a way to describe the chaos inside the atom, like notes are a way to describe music.  There are 3-node orbitals and 47-node orbitals, all the way up, but most of the time they’re not charge-activated just like a piano’s top note hardly ever gets played.”

“How do we know whether an orbital’s activated?”

“We’ve got rules for that, Eddie.  Maximum of two units of charge per orbital, lowest energy first.  Unless some light wave has deactivated a deeper orbital and activated a higher one.”

“You’re being careful again, not saying an electron’s here or an electron’s there.”

“Darn right, Vinnie.  It’s that chaos thing — charge is smeared all over the atom like air molecules jiggle all over the place to carry a sound wave.  Chemists and physicists may talk about ‘the electron in the 2s-orbital’ but that’s shorthand.  They know it’s really not like that.”

“I’m doing arithmetic over here.  So there’s two electrons, OK, call it two units of charge for that 1-node ball orbital, plus two units for the 2-node ball, plus two units each for the three dumbbells, uses up five orbitals.  That’s the same 2+8 stable mix that Bohr came up with.”

“Yeah, Eddie, but that field Sy talked about could be any strength.  Run the energy  equations backwards and the astronomers get a way to check a star’s fields.”

“Exactly, Vinnie.  Transitions involving combination orbitals have slightly different energy jumps than the ones we see in isolated atoms.  Electric and magnetic fields split each line in an element’s spectrum into multiplets.  Measure their splittings and you can work back to the field strengths that caused them.  The shell theory offers more predictions and more scientific insights than Bohr’s model ever dreamed of.”

“You said shell theory can handle molecules, too.  How’s that work?”

“Same as that electric field, but a lot messier.  Every nucleus exerts a field, mostly electric, on the rest of the molecule.  So does all the electron charge, but it’s more diffuse and includes more magnetism.  Molecular orbitals span the whole thing.  Works like atoms but much harder to calculate.”

“Figuring tips is easier,” hints Eddie.

~~ Rich Olcott