Tag: Inertial frame

Sharpening The Image

On By rolcottIn Astronomy: Stellar Science, Astronomy: Techniques, UncategorizedLeave a comment

“One coffee, one latte and two scones, Cal. Next time is Cathleen’s turn. Hey, you’ve got a new poster behind the cash register. What are we looking at?”

“You like it, Sy? Built the file myself from pics in my astronomy magazines, used the Library’s large‑format printer for the frameable copy. Came out pretty well, didn’t it, Cathleen?”

“Mm‑hm. Sy, you should recognize the pebbly-looking one. It’s granules at the bottom of the Sun’s atmosphere. The image came from the Inouye Solar Telescope at Haleakala Observatory on Maui, probably Earth’s best ground‑based facility for studying the Sun. I showed the image to your niece in that phone call. For scale, those granules of super‑heated rising gas are each about the size of Texas.”

“My magazine article didn’t mention Texas but it said there’s about ten million granules. What it was mostly about was the IST and its resolution. Those edges in the picture are as narrow as 18 miles across. It’s that good ’cause the beast has a 4‑meter mirror, which used to be amazing, but they made it even better with active and adaptive optics.”

“Hmm. It’s obvious that the bigger the mirror, the better it is for catching photons. If someone’s going to build a big mirror they’re going to put it behind a big aperture, which is important for resolving points that are close together. But what are ‘active and adaptive optics’ and why did you say that like they’re two different things?”

” ‘Cause they are two different things, Sy. Different jobs, different time‑scales. Gravity here on Earth can make a big mirror sag, and the sag changes depending on where the machine is pointed and maybe part of it gets the wrong temperature. Active optics is about keeping the whole mirror in the right shape to focus the photons where they’re supposed to go. There’s a bunch of actuators rigged up to give adjustable support at different points behind the mirror. The astronomer tells the system to watch a certain guide point and there’s a computer that directs each actuator’s pushing to sharpen the point’s image.”

“And adaptive optics?”

“That’s about solving a different problem. Stars twinkle, right, and the reason they twinkle is because of the atmosphere. One part refracts light one way, another part maybe warmer or with different humidity sends the light another way. Everything moves second to second. By the time a light‑wave gets down to us it’s been jiggled a lot. Adaptive optics is a small mirror, also with a lot of actuators, placed up in the light path after the primary mirror. Again with a guide point and a computer, the little mirror’s job is to cancel the jiggles so the scope’s sensors see a smooth wave. Adaptive works a lot faster than active, which sounds backwards, but I guess active came first.”

“The granules must be in the Sun’s disk somewhere. The other two images look like they’re on the edge.”

“That’s right, Sy. The bottom one is from the Solar Dynamic Observatory satellite a few years ago. That’s not visible light, it’s EUV—”

“EUV?”

“Extreme UltraViolet, light‑waves too short even for hydrogen so it’s mostly from iron atoms heated to millions of degrees. SDO had to be a satellite to catch that part of the spectrum because the atmosphere absorbs it. Of course, up there there’s no need for active or adaptive optics but imaging EUV has its own problems.”

“How tall is that photogenic tree?”

“It’s a prominence. The article said it’s about twenty times Earth’s diameter.”

“What about the pink one?”

“That’s new, Cathleen, from another Maui telescope. Adaptive optics were in play but there’s a problem. If you’re probing inside the corona there’s no fixed guide point. The team focused their adjustment system on corona features where they were a few seconds ago. The article said the process was ‘tricky,’ but look at the results. The loop is about the size of Earth, and those fine lines are about the width of Vancouver Island. They discovered details no‑one’s ever seen before.”

Top left: Schmidt et al./NJIT/NSO/AURA/NSF;
Top right: NSO/AURA/NSF under CC A4.0 Intl license;
Bottom: NASA/SDO

~ Rich Olcott

Deep Dive

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

“Sy, I’m trying to get my head wrapped around how the potential‑kinetic energy thing connects with your enthalpy thing.”

“Alright, Vinnie, what’s your cut so far?”

“It has to do with scale. Big things, like us and planets, we can see things moving and so we know they got kinetic energy. If they’re not moving steady in a straight line we know they’re swapping kinetic energy, give and take, with some kind of potential energy, probably gravity or electromagnetic. Gravity pulls things into a circle unless angular momentum gets in the way. How’m I doing so far?”

“I’d tweak that a little, but nothing to argue with. Keep at it.”

“Yeah, I know the moving is relative to whether we’re in the same reference frame and all that. Beside the point, gimme a break. So anyway, down to the quantum level. Here you say heat makes the molecules waggle so that’s kinetic energy. What’s potential energy like down there?”

<grabs another paper napkin> “Here’s a quick sketch of the major patterns.”

“Hmm. You give up potential energy when you fall and gravity’s graph goes down from zero to more negative forever, I guess, so gravity’s always attracting.”

“Pretty much, but at this level we don’t have to bother with gravity at all. It’s about a factor of 1038 weaker than electric interactions. Molecular motions are dominated by electromagnetic fields. Some are from a molecule’s other internal components, some from whatever’s around that brandishes a charge. We’ve got two basic patterns. One of them, I’m labeling it ‘Waggle,’ works like a pendulum, sweeping up and down that U‑shape around some minimum position, high kinetic energy where the potential energy’s lowest and vice‑versa. You know how water’s H‑O‑H molecules have that the V‑shape?”

“Yeah, me you and Eddie talked about that once.”

“Mm‑hm. Well, the V‑shape gives that molecule three different ways to waggle. One’s like breathing, both sides out then both sides in. If the hydrogens move too far from the oxygen, that stretches their chemical bonds and increases their potential energy so they turn around and go back. If they get too close, same thing. Bond strength is about the depth of the U. The poor hydrogens just stretch in and out eternally, swinging up and down that symmetric curve.”

“Awww.”

“That’s a chemist’s picture. The physics picture is cloudier. In the quantum version, over here’s a trio of fuzzy quarks whirling around each other to make a proton. Over there’s a slightly different fuzzy trio pirouetting as a neutron. Sixteen of those roiling about make up the oxygen nucleus plus two more for the hydrogens plus all their electrons — imagine a swarm of gnats. On the average the oxygen cloud and the two hydrogen clouds configure near the minimum of that U‑shaped potential curve but there’s a lot of drifting that looks like symmetrical breathing.”

“What about the other two waggles?”

“I knew you’d ask. One’s like the two sides of a teeterboard, oscillating in and out asymmetrically. The other’s a twist, one side coming toward you and then the other side. Each waggle has its own distinct set of resistance forces that define its own version of waggle curve. Each kind interacts with different wavelengths of infrared light which is how we even know about them. Waggle’s official name is ‘harmonic oscillator.’ More complicated molecules have lots of them.”

“What’s that ‘bounce’ curve about?”

“Officially that’s a Lennard-Jones potential, the simplest version of a whole family of curves for modeling how molecules bounce off each other. Little or no interaction at large distances, serious repulson if two clouds get too close, and a little stickiness at some sweet-spot distance. If it weren’t for the stickiness, the Ideal Gas Law would work even better than it does. So has your head wrapped better?”

“Sorta. From what I’ve seen, enthalpy’s PV part doesn’t apply in quantum. The heat capacity part comes from your waggles which is kinetic energy even if it’s clouds moving. Coming the other way, quantum potential energy becomes enthalpy’s chemical part with breaking and making chemical bonds. Did I bridge the gap?”

“Mostly, if you insist on avoiding equations.”

~ Rich Olcott

EROs Atop A Ladder

On By rolcottIn Astronomy: Techniques, UncategorizedLeave a comment

“‘That’s where the argument started‘? That’s right up there with ‘Then the murders began.’ Cathleen Cliff‑hanger strikes again.”

<giggling> “Gotcha, Sy, just like always. Sorry, Kareem, we’ve had this thing since we were kids.”

“Don’t mind me, but do tell him what’s awry with the top of your galactic distance ladder.”

“I need to fill you in first about the ladder’s framework. We know the distances to special ‘standard candles’ scattered across the Universe, but there’s oodles of other objects that aren’t special that way. We can’t know their distances unless we can tie them to the candles somehow. Distance was Edwin Hubble’s big thing. Twenty years after Henrietta Swan Leavitt identified one kind of candle, Hubble studied the light from them. The farthest spectra were stretched more than the closest ones. Better yet, there was a strict relationship between the amount of stretch, we call it the z factor, and the candle’s distance. Turns out that everything at the intergalactic scale is getting farther from everything else. He didn’t call that expansion the Hubble Flow but we do. It comes to about 7% per billion lightyears distance. z connects candle spectrum, object spectrum and object distance. That lets us calibrate successive overlapping steps on the distance ladder, one candle type to the next one.”

“A constant growth rate — that’s exponential, by definition. Like compound interest. The higher it gets, it gets even higher faster.”

“Right, Kareem, except that in the past quarter-century we’ve realized that Hubble was an optimist. The latest data suggests the expansion he discovered is accelerating. We don’t know why but dark energy might have something to do with it. But that’s another story.”

“Cathleen, you said the distance ladder’s top rung had something to do with surface brightness. Surface of what?”

“Galaxies. Stars come at all levels of brightness. You can confirm that visually, at least if you’re in a good dark‑sky area. But a galaxy has billions of stars. When we assess brightness for a galaxy as a whole, the brightest stars make up for the dimmest ones. On the average it’ll look like a bunch of average stars. The idea is that the apparent brightness of some galaxy tells you roughly how many average stars it holds. In turn, that gives you a rough estimate of the galaxy’s mass — our final step up the mass ladder. Well, except for gravitational lensing, but that’s another story.”

“So what’s wrong with that candle?”

“We didn’t think anything was wrong until recently. Do you remember that spate of popular science news stories a year ago about giant galaxies near the beginning of time when they had no business to exist yet?”

“Yeah, there was a lot of noise about we’ll have to revise our theories about how the Universe evolved from the Big Bang, but the articles I saw didn’t have much detail. From what you’ve said so far, let me guess. These were new galaxy sightings, so probably from James Webb Space Telescope data. JWST is good at infra‑red so they must have been looking at severely stretched starlight—”

“z-factor near 8″

“— so near 13 billion lightyears old, but the ‘surface brightness’ standard candle led the researchers to claim their galaxies held some ridiculous number of stars for that era, at least according to current theory. How’d I do?”

“Good guess, Sy. That’s where things stood for almost a year until scientists did what scientists do. A different research group looking at even more data as part of a larger project came up with a simpler explanation. Using additional data from JWST and several other sources, the group concentrated on the most massive galaxies, starting with low‑z recent ones and working back to z=9. Along the way they found some EROs — Extremely Red Objects where a blast of infra‑red boosts their normal starlight brightness. The researchers attribute the blast to hot dust associated with a super‑massive black hole at each ERO’s center. The blast makes an ERO appear more massive than it really is. Guess what? The first report’s ‘ridiculously massive’ early galaxies were EROs. Can’t have them in that top rung.”

“Kareem, how about the rungs on your ladder?”

~~ Rich Olcott

One Step After Another

On By rolcottIn Astronomy: Techniques, UncategorizedLeave a comment

Mid-afternoon, time for a coffee break. As I enter Cal’s shop, I see Cathleen and Kareem chuckling together behind a jumble of Cal’s distinctive graph‑lined paper napkins. “What’s the topic of conversation, guys?”

“Hi, Sy. Kareem and I are comparing ladders.”

I look around, don’t see anything that looks like construction equipment.

“Not that kind, Sy. What’s your definition of a ladder?”

“Getting down to definitions, eh, Kareem? Okay, it’s a framework with steps you can climb up towards something you can’t reach.”

“Well, there you go.”

“Not much help, Cathleen. What are you really bantering about?”

“Each of our fields of study has a framework with steps that let us measure something that’d be way out of reach without it.”

“You’ll appreciate this, Sy — our ladders even use different math. The steps on Cathleen’s ladder are mostly linear, mine are mostly exponential.”

“And they’re both finicky — you have to be really careful when using them.”

“And they’ve both recently had adjustments at the top end.”

“I can see the fun, I think. How about some specifics?”

They exchange a look, Kareem gestures ‘after you‘ and Cathleen opens. “Mine’s in astrometry, Sy, the precise recording of relative positions. Tycho Brahe’s numbers were good to a few dozen arcseconds—”

“Arcsecond?”

“1/60 of an arcminute which is 1/60 of a degree which is 1/360 of a full circle around the sky. Good enough in Newton’s day for him to explain planetary orbits, but we’ve come <ahem> a long way since then. The Gaia telescope mission can resolve certain objects down to a few microarcseconds but that’s only half the problem.”

“Let me guess — you have angles but you don’t have distances.”

“Bingo. Distance is astrometry’s biggest challenge.”

“Wait, Newton’s Law of Gravity includes r as the distance between objects. For that matter, Kepler’s Laws use r² and r³. Couldn’t you juggle them around to evaluate r?”

“Nope. Kepler did ratios, not absolute values. Newton’s Law has r² but you can rewrite it as F ² = GMm/r² = G(M/r)(m/r), G times the product of two mass‑to‑distance ratios. Newton’s G is our least‑accurate physical constant and we don’t have good handles on either of those numerators. Before space flight we just had mass ratios like M/m. We only discovered the Moon’s absolute mass when we orbited it with spacecraft of known mass. That’s the lowest rung on our mass ladder. Inside the Solar System we go step by step with orbit ratios. Outside the system everything’s measured relative to Solar mass.”

“I’m getting the ladder idea. So how do you distances?”

“Lowest rung is parallax, like binocular vision. You look at something from two different points a known distance apart. Measure the angle between the sight‑lines. Figure the triangles to get the something’s distance. The earliest example I know of was in the mid‑1700s when astrometers thousands of miles apart on Earth watched Venus cross the Sun’s disk. Each recorded the precise time they saw Venus touch the Sun’s disk. Given the time shift and the on‑Earth distance, some trigonometry gave them the Earth‑Venus distance. That put a scale to Newtonian orbital diagrams. Parallax across the width of Earth’s orbit yielded stellar distances out to thousands of lightyears with Hubble. We expect ten times better from Gaia.”

“That gets you maybe across the Milky Way. What about farther out?”

“Several ingenious variations on the parallax idea, but mostly standard candles.”

“Candles?”

“Suppose you measure the brightness of a candle that’s a known distance away and there’s an equally luminous candle some unknown distance away. Measured brightness falls as the square of the distance, so if the second candle appears half as bright it’s four times the distance and so on. Climbing the cosmic distance ladder is going from one kind of uniformly‑luminous candle to another kind farther away.”

“Such as?”

“We know how brightness relates to bright‑dim‑bright cycle time for several types of variable stars. That gets us out to 30 million lightyears or so. Type I‑a supernovas act as useful candles out to a billion lightyears. Beyond that we can use galaxy surface brightness. That’s where the recent argument started.”

~ Rich Olcott

  • Thanks to Ken Burke for mentioning tellurium‑128’s septillion‑year half‑life.

A Non-political Polarizing Topic

On By rolcottIn Astronomy: Techniques, Physics: Black Holes and Relativity, Physics: Waves, UncategorizedLeave a comment

Vinnie gets the deck next, but first thing he does is plop a sheet of paper onto the table. “Topic is black holes, of course. Everybody’s seen this, right?”

“Sure, it’s the new view of the Milky Way’s super-massive black hole with the extra lines. So deal already.”

“Hold your horses, Cal.” <Vinnie starts dealing.> “I’m looking for explanations. Where’d those lines come from? They swirl across the accretion disk like so much rope, right? Why aren’t they just going straight in orderly‑like? The whole thing just don’t make sense to me.”

Susan bets a few chips. “I saw a similar pop‑sci article, Vinnie. It said the lines trace out polarization in the light waves the Event Horizon Telescope captured. Okay, radio waves — same thing just longer wavelength. Polarized radio waves. I’ve measured concentrations of sugar and amino acid solutions by how much the liquid rotates polarized light, but the light first went through a polarizing filter. How does a black hole make polarized waves?”

Kareem matches Susan’s bet. “Mm‑hm. We use polarized light passing through thin sections of the rocks we sample to characterize the minerals in them. But like Susan says, we don’t make polarized light, we use a filter to subtract out the polarization we don’t want. You’re the physicist, Sy, how does the black hole do the filtering?”

Plane‑polarized electromagnetic wave
 Electric (E) field is red
 Magnetic (B) field is blue
(Image by Loo Kang Wee and Fu-Kwun Hwang from Wikimedia Commons)

My hand’s good so I match the current ante. “It doesn’t. There’s no filtering, the light just comes out that way. I’d better start with the fundamentals.” <displaying Old Reliable> “Does this look familiar, Vinnie?”

“Yeah, Sy, you’ve used it a lot. That blue dot in the back’s an electron, call it Alice, bobbing straight up and down. That’s the polarization it’s puttin’ on the waves. The red lines are the force that another electron, call it Bob, feels at whatever distance away. Negative‑negative is repelling that so Bob goes down where the red line goes up but you get the basic idea.”

“The blue lines are important here.”

“I’m still hazy on those. They twist things, right?”

“That’s one way to put it. Hendrik Lorentz put it better when he wrote that Bob in this situation experiences one force with two components. There’s the red‑line charge‑dependent component, plus the blue‑line component that depends on the charge and Bob’s motion relative to Alice. If the two are moving in parallel—”

“The same frame, then. I knew frames would get into this somehow.”

“It’s hard to avoid frames when motion’s the subject. Anyway, if the two electrons are moving in parallel, the blue‑line component has zero effect. If the two are moving in different directions, the blue‑line component rotates Bob’s motion perpendicular to Alice’s red‑line polarization plane. How much rotation depends on the angle between the two headings — it’s a maximum when Bob’s moving perpendicular to Alice’s motion.”

“Wait, if this is about relative motion, then Bob thinks Alice is twisting, too. If she thinks he’s being rotated down, then he thinks she’s being rotated up, right? Action‑reaction?”

“Absolutely, Vinnie. Now let’s add Carl to the cast.”

“Carl?”

Alice and Bob’s electromagnetic interaction
begets motion that generates new polarized light.

“Distant observer at right angles to Alice’s polarization plane. From Carl’s point of view both electrons are just tracking vertically. Charges in motion generate lightwaves so Carl sees light polarized in that plane.”

Cathleen’s getting impatient, makes her bet with a rattle of chips. “What’s all this got to do with the lines in the EHT image?”

“The hole’s magnetic field herds charged particles into rotating circular columns. Faraday would say each column centers on a line of force. Alice and a lot of other charged particles race around some column. Bob and a lot of other particles vibrate along the column and emit polarized light which shows up as bright lines in the EHT image.”

“But why are the columns twisted?”

“Orbit speed in the accretion disk increases toward its center. I’d bet that’s what distorts the columns. Also, I’ve got four kings.”

“That takes this pot, Sy.”

~~ Rich Olcott

A Million Times Weaker Than Moonlight

On By rolcottIn Cosmology, Gravitational astronomy, Physics: Waves, UncategorizedLeave a comment

Big Vinnie’s getting downright antsy, which is something to behold. “C’mon, Sy. We get it that sonication ain’t sonification and molecules bumping into each other can carry a sound wave across space if the frequency’s low enough and that can maybe account for galaxies having spiral arms, but you said the Cosmic Hum is a sound, too. That’s a gravity thing, not molecules, right?”

“Not quite what I said, Vinnie. The Hum’s sound‑related, but it’s not ‘sound’ even by our extended definition.”

“Then what’s the connection?”

“Waves.”

“Not frames like always?”

“Not frames, for a change.”

“So it’s waves, but they go though empty space. Can’t happen like sound waves from molecules bumping into each other ’cause molecules are too small to have enough gravity do that when they’re so far apart. What’s carrying the waves?”

“Good question. Einstein figured out one answer. A whole cohort of mid‑20th‑century theoreticians came to a slightly different conclusion.”

“Okay, I’ll bite. What was Einstein’s answer?”

“Relativity, of course. Gravity’s the effect we see from mass deforming nearby space. Moving a mass drives corresponding changes in the shapes of space where it was and where it has moved to. The shape‑changes generate follow‑on gravitational effects that propagate outward over time. Einstein even showed that the gravitational propagation speed is equal to lightspeed.”

“Gimme a sec … Okay, that black hole collision signal LIGO picked up back in 2015, the holes lost a chunk of their combined mass all of a sudden. Quick drop in the gravity thereabouts. You’re saying it took time for the missing gravity strength to get noticed where we’re at. If I remember right, the LIGO people said the event was something like a billion lightyears away so that tells me it happened about a billion years ago and what the LIGO gadget picked up was space waves, right?”

“Right, but it wasn’t just the mass loss, it was the rapid and intense waggles in the gravitational field as those two enormous bodies, each 30 times as massive as the Sun, whirled around each other multiple times per second. The ever‑faster whirling shook the field with a frequency that swept upward to the ‘POP‘ when your mass‑loss happened. LIGO eventually picked up that signal. Einstein would say there’s no ‘action at a distance‘ in the collision‑LIGO interaction, because the objects acted on the gravitational field which acted on the LIGO system.”

“Like using a towel to pop someone in the locker room. The towel’s just transmi– ulp.”

“An admission of guilt if I ever heard one. Yes, like that, except a towel pop carries all the initial energy to its final destination. Gravitational waves spread their energy across the surface of an expanding sphere. The energy per unit area goes down as the square of the distance.” <keying a calculation on Old Reliable> “Suppose the collision releases 10 solar masses worth of energy, we’re a billion lightyears away, and the ‘POP‘ signal is delivered in a tenth of a second. We’d see a signal power … about a millionth as strong as moonlight.”

“Not much there.”

“Right, which is why LIGO and its kin have been such pernickety instruments to build and run. LIGO’s sensors are mirrors roughly a meter across. You get a million times more power sensitivity if your detector’s diameter is a mile across. That was part of the NANOGrav team’s strategy, but they went much bigger.”

“Yeah, that’s the multi-telescope thing, so NANOGrav faked a receiver the size of the Earth, like the Event Horizon Telescope.”

“Much bigger. Their receiver is the entire Milky Way. Instead of LIGO’s mirrors, NANOGrav’s signal generators are neutron stars a dozen or more miles wide scattered across the galaxy.”

“Gotcha, Sy. Two ways. Neutron stars are billions heavier than a LIGO mirror so they’d be less power‑sensitive, not more. Then, power is about moving stuff closer or farther but if I got you right these space waves don’t really do that anyway, right?”

“Right and right, Vinnie, but not relevant. What NANOGrav’s been watching for is pulsar beams being twitched by a gravitational wave. A waltzing black hole pair should generate years‑long or decades‑long wavelengths. NANOGrav may have found one.”

~~ Rich Olcott

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

Completing The Triad

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

Walt’s mustache bristles as he gives me the eye. ”You claim three harmonics control how the Sun’s gravity could affect spacecraft orbits around a target planet like Jupiter. You said we don’t have to care about Jupiter’s gravitational zones and isolating the sectors probably isn’t doable. What’s the third?”

Time to twist the screws. ”Three harmonic systems, Walt, all working together and you’ve got their names wrong. They control nothing, they’re a framework for analysis. And Jupiter’s special. Solar gravity doesn’t affect its zonal harmonic arcs but that’s only because Jupiter’s polar axis is nearly perpendicular to its orbital plane. Zonal‑effect N‑S twisting at Jupiter is pennies on a C‑note. Any mission we send to Mars, Saturn or Uranus we’ll care a lot about their zonal harmonics because their axes have more tilt. An 82° tilt for Uranus, can’t get much more tilted than that. Sectorial harmonics may still help us navigate there because Uranus probably has a lot less magnetism than Jupiter.”

That rocks him but he comes back strong. ”The third kind of harmonic?!! C’mon, give!”

“Radial, the center‑out dimension. The gravitational force between bodies depends on center‑to‑center distances so yeah, your people would be interested.”

“I presume radial harmonics have numbers like Jn and Cm do?”

“They do. Sorry, this’ll get technical again but I’ll go as light as I can. Each radial harmonic is the product of two factors. You know about factors, right?”

“Sure, force multipliers.”

“You would know that kind. More generally, factors are things that get multiplied together. I’ll call the general radial harmonic Rn. It’s the product of two factors. The first is a sum of terms that begin with rn, where r is the distance. For instance, R3‘s first factor would look like a*r³+b*r²+c*r+d, where the a,b,c,d are just some numbers. Different radial harmonics have different exponents in their lead terms. You still with me?”

“Polynomials from high school algebra. Tell me something new.”

“The second factor decreases exponentially with n*r. No matter how large rn gets, when you multiply an rn polynomial by something that decreases exponentially, the (polynomial)×(exponential) product eventually gets really small.”

“Give me a second. … So what you’re saying is, at a big enough distance these radial harmonics just die away.”

“That’s where I was going.”

“How far is ‘enough’?”

“Depends on n. Higher values of n shut down faster.”

“So these Cms and Jns and Rns just add together?” <pauses, squints at me suspiciously> “Is there some reason you used n for both Jn and Rn?”

“No but yes, and yes. You combine a C, a J and an R using multiplication to get a full harmonic F, except there are rules. The J and R must belong to the same n. The m can’t be larger than n. From far away we’d model Jupiter’s gravity as F000=R0×J0×C0, which is an infinite sphere — R0 never dies away and J0×C0 says ‘no angular dependence.’ The Sun’s gravity acts along R0 and that’s what keeps Jupiter in orbit. If the problem demands combining full harmonics, you use addition.” <rousing a display on Old Reliable> “Here’s how a particular pair of harmonics combine to increase or decrease spherical gravity in specific directions.”

“But Juno doesn’t see those gravity lumps until it gets close‑in. How close?”

“R2‘s down to less than a part per thousand at three planetary radii, call it 225 000 kilometers away from the planet’s center.”

“How much time is it closer than that distance?”

“Complicated question. A precise answer requires some calculus — is your smart phone set up for elliptic integrals?”

“Of course not. A good estimate will do.”

“Okay, here’s the plan. What we’d like is total time spent while Juno travels along the ellipsoidal arc between points A and D where the orbit crosses the 225 000‑km circle. Unfortunately, Juno speeds up approaching point P, slows down going away — calculating the A‑D time is tricky. I’ll assume Juno travels straight lines AB and CD at the A-speed. I’ll also approximate the orbit’s close pass as a semicircle at P‑speed.” <tapping> “I get a 3.6-hour duration, less than 0.3% of the full 53-day orbit. Will that satisfy your people?”

“You’ll know if it doesn’t.”

~~ Rich Olcott

Not Silly-Season Stuff, Maybe

On By rolcottIn Chemistry, Physics: Electromagnetism, UncategorizedLeave a comment

“Keep up the pace, Mr Feder, air conditioning is just up ahead.”

“Gotta stop to breathe, Moire, but I got just one more question.”

“A brief pause, then. What’s your question?”

“What’s all this about LK99 being a superconductor? Except it ain’t? Except maybe it is? What is LK99, anyway, and how do superconductors work? <puffing>”

“So many question marks for just one question. Are you done?”

“And why do news editors care?”

“There’s lots of ways we’d put superconductivity to work if it didn’t need liquid‑helium temperatures. Efficient electric power transmission, portable MRI machines, maglev trains, all kinds of advances, maybe even Star Trek tricorders.”

“Okay, I get how zero‑resistance superconductive wires would be great for power transmission, but how do all those other things have anything to do with it?”

“They depend on superconductivity’s conjoined twin, diamagnetism.”

“Dia—?”

“Means ‘against.’ It’s sort of an application of Newton’s Third Law.”

“That’s the one says, ‘If you push on the Universe it pushes back,’ right?”

“Very good, Mr Feder. In electromagnetism that’s called Lenz’ Law. Suppose you bring a magnet towards some active conductor, say a moving sheet of copper. Or maybe it’s already carrying an electric current. Either way, the magnet’s field makes charge carriers in the sheet move perpendicular to the field and to the prevailing motion. That’s an eddy current.”

“How come?”

“Because quantum and I’m not about to get into that in this heat. Emil Lenz didn’t propose a mechanism when he discovered his Law in 1834 but it works. What’s interesting is what happens next. The eddy current generates its own magnetic field that opposes your magnet’s field. There’s your push‑back and it’s called diamagnetism.”

“I see where you’re going, Moire. With a superconductor there’s zero resistance and those eddy currents get big, right?”

“In theory they could be infinite. In practice they’re exactly strong enough to cancel out any external magnetic field, up to a limit that depends on the material. A maglev train’s superconducting pads would float above its superconducting track until someone loads it too heavily.”

“What about portable MRI you said? It’s not like someone’s gonna stand on one.”

“A portable MRI would require a really strong magnet that doesn’t need plugging in. Take that superconducting sheet and bend it into a doughnut. Run your magnet through the hole a few times to start a current. That current will run forever and so will the magnetic field it generates, no additional power required. You can make the field as strong as you like, again within a limit that depends on the material.”

“Speaking of materials, what’s the limit for that LK99 stuff?”

“Ah, just in time! Ahoy, Susan! Out for a walk yourself, I see. We’re on our way to Al’s for coffee and air conditioning. Mr Feder’s got a question that’s more up your Chemistry alley than my Physics.”

“LK99, right? It’s so newsy.”

“Yeah. What is it? Does it superconduct or not?”

“Those answers have been changing by the week. Chemically, it’s basically lead phosphate but with copper ions replacing some of the lead ions.”

“They can do that?”

“Oh yes, but not as neatly as we’d like. Structurally, LK99’s an oxide framework in the apatite class — a lattice of oxygens with phosphorus ions sitting in most of the holes in the lattice, lead ions in some of the others. Natural apatite minerals also have a sprinkling of hydroxides, fluorides or chlorides, but the reported synthesis doesn’t include a source for any of those.”

“Synthesis — so the stuff is hand‑made?”

“Mm‑hm, from a series of sold‑state reactions. Those can be tricky — you grind each of your reactants to a fine powder, mix the powders, seal them in a tube and bake at high temperature for hours. The heat scrambles the lattices. The atoms can settle wherever they want, mostly. I think that’s part of the problem.”

“Like maybe they don’t?”

“Maybe. There are uncontrollable variables — grinding precision, grain size distribution, mixing details, reaction tube material, undetected but critical impurities — so many. That’s probably why other labs haven’t been able to duplicate the results. Superconductivity might be so structure‑sensitive that you have to prepare your sample j‑u‑s‑t right.”

~~ Rich Olcott

Not A Hum, A Rumble

On By rolcottIn Cosmology, Gravitational astronomy, UncategorizedLeave a comment

Vinnie taps on the magazine. “Sy, you’ve done it again. We ask you one question, you spend a lot of time talking about something else entire. They got this article here” <tap> “says the NANOGrav team captured the hum of the Universe. Al and me, we ask you about that and you get us discussing pulsars. Seems to me,” <tap tap> “that if you got a pulsar and the pulses got only a 3% duty cycle they’d sound more like clicks and,” <taptaptaptap> “if it’s a 10 millisecond pulsar that’s a hundred per second and they’d be more like a low‑pitched buzz, nothing like a hum.”

“One more short detour, Vinnie, sorry. Remember when we discussed the VLA, the Very Large Array of radio telescopes in northern New Mexico?”

“Sorta. I do remember visiting the place, out in the desert miles away from anywhere. They’ve got a couple dozen dish antennas each as wide as a four‑lane road, all spread out along railroad tracks. Big dishes for catching weak signals I understand, but I forget why there’s lots of dishes instead of one huge one or how that even works.”

“One reason is simple mechanics. A huge dish would try to sail away in the desert wind. VLA admins even have to safe‑mount those 25‑meter ones when things get gusty. But the real reason goes to how the array works as one big instrument. Here’s a hint — the dishes can be miles apart and lightspeed isn’t infinite.”

“Ah, that joggled my memory. It’s about when a signal comes in from some nova or something, each dish registers it with a slightly different arrival time and then the computers play match‑up games with all the time differences to figure exactly what angle the signal came from, right?”

“Roughly. The VLA’s multi‑dish design is about being able to resolve signal sources that are close together in the sky so yeah, slightly different angles. The Event Horizon Telescope team used the same strategy and a collection of radio dishes all over the world to produce those orange‑ring images of supermassive black holes. NANOGrav and the other Pulsar Timing Arrays sort of the flip the strategy.”

“At last we get to NANOGrav. Wait, they use lots of antennas to send signals to a star?”

“Nothing like that, Al. No, they use just a few antennas but they track the timing of many pulsars. About 70 at last count.”

“But we know what the timing is, to nanoseconds you said.”

“One word, Vinnie. ‘Frames‘.”

“Aw geez, Sy. Again?”

“Mm-hm. In the pulsar’s frame, it’s majestically rotating at a steady pace, tens or hundreds of times per second relative to its neighbors. Its beam proudly announces its presence on an absolutely regular schedule save for a small but steady slow‑down. In our frame, though, things can happen to a pulse as it heads our way.”

“Like what?”

“It might pass through a molecular cloud. We know those exist. Photons in the right wavelength ranges could interact with cloud components. That’d delay them, stretch the pulse, might even create interference between successive pulses. On the theory side, some cosmologists think the Universe may hold objects like cosmic strings or curvature‑induced domain walls that could delay, deflect or otherwise mess up a pulse. The best possibility, though, is that a gravitational wave could cross the path of a pulse en route to us.”

“Why is that a good thing?”

“Because they’d interact to alter that pulse’s timing. Gravitational waves stretch and squeeze time as they squeeze and stretch space. If a wave crosses a traveling pulse, the pulse will get here either early or late depending. Better yet, if we track enough pulsars scattered across the sky we might even see a parade of offset timings as the wave encounters different pulse beams. Hasn’t happened yet, though. The NANOGrav reports so far are about the background variation as waves from everywhere traverse the paths we’re watching.”

“The article says a hum.”

“Hum sounds come in waves per second. The gravitational background happens in waves per decade, such a low frequency even elephants couldn’t hear it.”

“OK, it’s rumble, not a hum. But why either one?”

~~ Rich Olcott