Tag: spectrum

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

The Venetian Blind Problem

On By rolcottIn Astronomy: Planetary Science, Chemistry, Physics: Light and Relativity, UncategorizedLeave a comment

Susan Kim gives me the side‑eye. “Sy, I get real suspicious when someone shows me a graph with no axis markings. I’ve seen that ploy used too often by people pushing a bias — you don’t know what happens offstage either side and you don’t know whether an effect was large or small. Your animated chart was very impressive, how that big methane infrared absorption peak just happens to fill in the space between CO2 and H2O peaks. But how wide is the chart compared to the whole spectrum? Did you cherry‑pick a region that just happens to make your point?”

“Susan, how could you accuse me of such underhanded tactics? But I confess — you’re right, sort of. <more tapping on Old Reliable’s keyboard> The animation only covered the near‑IR wavelengths from 1.0 to 5.0 micrometers. Here’s the whole strip from 0.2 micrometers in the near UV, out to 70 micrometers in the far IR. Among other things, it explains the James Webb Space Telescope, right, Al?”

Spectrum of Earth’s atmosphere. Adapted
under the Creative Commons 3.0 license
from Robert Wohde’s work
with the HITRAN2004 spectroscopic database,

“I know the Webb’s set up for IR astronomy from space, Sy. Wait, does this graph say there’s too much water vapor blocking the galaxy’s IR and that’s why they’re putting the scope like millions of miles away out there?”

“Not quite. The mission designers’ problem was the Sun’s heat, not Earth’s water vapor. The solution was to use Earth itself to shield the device from the Sun’s IR emissions. The plan is to orbit the Webb around the Earth‑Sun L2 point, about a million miles further out along the Sun‑Earth line. Earth’s atmosphere being only 60 miles thick, most of it, the Webb will be quite safe from our water molecules. No, our steamy atmosphere’s only a problem for Earth‑based observatories that have to peer through a Venetian blind with a few missing slats at very specific wavelengths.”

“Don’t forget, guys, the water spectrum is a barrier in both directions. Wavelengths the astronomers want to look at can’t get in, but also Earth’s heat radiation at those wavelengths can’t get out. Our heat balance depends on the right amount of IR energy making it out through where those missing slats are. That’s where Sy’s chart comes in — it identifies the wavelengths under threat by trace gases that aren’t so trace any more.”

“And we’re back to your point, Susan. We have to look at the whole spectrum. I heard one pitch by a fossil fuel defender who based his whole argument on the 2.8‑micrometer CO2 peak. ‘It’s totally buried by water’s absorption,‘ he claimed. ‘Can’t possibly do us any further damage.’ True, so far as it goes, but he carefully ignored CO2‘s other absorption wavelengths. Pseudoscience charlatan, ought to be ashamed of himself. Methane’s not as strong an absorber as CO2, but its peaks are mostly in the right places to do us wrong. Worse, both gas concentrations are going up — CO2 is 1½ times what it was in Newton’s day, and methane is 2½ times higher.”

Data from EPA Climate Indicators Explorer

“Funny how they both go up together. I thought the CO2 thing was about humanity burning fossil fuels but you said methane operations came late to that game.”

“Right on both counts, Al. Researchers are still debating why methane’s risen so bad but I think they’re zeroing in on cow gas — belches and farts. By and large, industry has made the world’s population richer over the past two centuries. People who used to subsist on a grain diet can now afford to buy meat so we’ve expanded our herds. Better off is good, but there’s an environmental cost.”

Al gets a far-away look. “Both those gases have carbon in them, right? How about we burn methane without the carbon in, just straight hydrogen?”

Susan gets excited. “Several groups in our lab are working on exactly that possibility, Al. The 2H2+O2→2H2O reaction yields 30% more energy per oxygen atom than burning methane. We just need to figure out how to use hydrogen economically.”

~~ Rich Olcott

Dark Glasses

On By rolcottIn Physics: Light and Relativity, Uncategorized2 Comments

My office door THUMPs as Richard Feder barrels in. Vinnie’s half out of his chair with his fists balled up but he settles back down when he sees who it is. “Moire, I gotta question.”

“Afternoon, Mr Feder. What brings you to the 12th floor of the Acme Building?”

“My dentist’s up here. They gave me these really dark glasses for when they aimed a bright light in my mouth to harden something in there so I wondered why’re they so dark an’ what about those glasses that can’t make up their minds?”

“Well, Mr Feder, as usual you’ve asked a jumbled question. Let’s see. The answers all boil down to what light is made of and what the glasses are made of.”

“I thought it was photon particles, Sy. The light, I mean.”

“It is, Vinnie, but photons only act like particles when they’re emitted and when they’re absorbed. In between, they act like waves. Dark glasses are all about photons as waves. The simplest case is the plain dark glasses.”

“Yeah, Moire, simple’s good.”

“They’re black because they’ve been doped with black chemicals. If your glasses are actually made of glass, the manufacturer probably dumped iron and sulfur into the melt. When heated those elements combine to form black iron sulfide particles spread throughout the mass. If the glasses are plastic, the manufacture mixed black dye into the formula. Either way, the more dopant added, the blacker the product and the fewer waves make it through the lens.”

“Great, Sy, but how come the black? I remember that Sun-spectrum poster that Al had up in his shop once. Lotsa sharp dark lines that Cathleen said were from different elements absorbing little slices of that rainbow background. But there were plenty of colors left over to make white.”

“Impressive memory, Vinnie. That was what, three years ago? Anyhow, those absorption lines come from separated atoms floating in the hot gas of the solar atmosphere. Quantum mechanics says that an isolated atom has a characteristic set of electron configurations, each with its own energy level. Say an incoming photon meets a gas atom. If the photon’s energy just matches the difference between the atom’s current configuration and some other configuration, suddenly the atom’s in the new configuration and no more photon. It has to match just right or no absorption. Those sharp lines come from that selectivity, OK?”

“So how do you get total black from selective atoms?”

“You don’t. You get black from less‑selective molecules and larger structures. Atoms right next to each other bring entanglement into the action — which electron is where on which atom? Many more configurations, many more differences between energy level pairs, many more lines that can overlap to make broad absorption bands. Suppose you’ve got some glass or plastic doped to have a single band sucking up everything between orange and green. Shine white light into it. Only red light and blue light come through. We see that as purple, a color that’s not even in the spectrum. Make that band even broader like it is with metals and rocks and iron sulfide; nothing gets through.”

“Then how do they do those glasses that get dark or light depending? The factory can’t put chemicals in but take ’em out temporary‑like when you walk inside.”

“Good point. In fact, the glass composition stays the same, sort of. The factory puts in chemicals that change their structure depending on the light level. If you dope optical glass with silver chloride crystallites, for instance, UV light can energize a chloride’s electron up to where it can leave the chloride and be captured by a silver ion. Do that with enough silver ions in the crystallite and you have a tiny piece of silver metal. Enough pieces and the glass looks gray, at least until heat energy joggles things back to the silver chloride ground state. For plastic lenses they use a subtler strategy — large‑ish molecules with spread‑out electron structures. UV light energizes an electron to another level and the molecule twitches to an opaque alternate form that relaxes when heat shakes it down.”

“Heat, huh? No wonder mine don’t work so good on the beach.”

~~ Rich Olcott

Trio for Rubber Ruler

On By rolcottIn Astronomy: Stellar Science, Astronomy: Techniques, Cosmology, Physics: Fundamentals, Physics: Quantum Mechanics, UncategorizedLeave a comment

“It’s all about how lightwaves get generated and then what happens.”

“Sy and me talked about that, Cathleen.  Lightwaves come from jiggling electrons, right?”

“Any kind of charged particles, Vinnie, but there’s different ways that can happen.  Each leads to its own kind of spectrum.”

“Different kinds of spectrum?  Do you mean like visible versus infrared and ultraviolet, Cathleen?”

“No, I don’t, Sy.  I’m referring to the thing’s overall appearance in every band.  A hundred and fifty years ago Kirchoff pointed out that light from a source can have lines of color, lines without color, or a smooth display without lines.”

“Like that poster that Al put up between the physicist and astronomer corners?”  (We’re still chatting at a table in Al’s coffee shop.  I’m on my fourth scone.)

Astroruler with solar spectrum
Based on N.A.Sharp, NOAO/NSO/Kitt Peak FTS/AURA/NSF

“Kind of.  That’s based on a famous image created at Kitt Peak Observatory.  In the background there you see a representation of what Kirchoff called a continuous or black-body spectrum, where all the colors fade smoothly into each other in classic rainbow order.  You’re supposed to ignore the horizontal dark lines.”

“And the vertical lines?”

“They form what Kirchoff called an absorption spectrum.  Each dark vertical represents an isolated color that we don’t get from the Sun.”

“You’re saying we get all the other colors but them, right?”

“Exactly, Vinnie.  The Sun’s chromosphere layer filters those specific wavelengths before they get from the deeper photosphere out into space.”

“Complicated filter.”

“Of course.  The Sun contains most of the elements lighter than nickel.  Each kind of atom absorbs its own collection of frequencies.”

“Ah, that’s the quantum thing that Sy and me talked about, right, Sy?”

“Mm-hm.  We only did the hydrogen atom, but the same principles apply.  An electromagnetic wave tickles an atom.  If the wave delivers exactly the right amount of energy, the atom’s chaotic storm of electrons resonates with the energy and goes a different-shaped storm.  But each kind of atom has a limited set of shapes.  If the energy doesn’t match the energy difference between a pair of levels, there’s no absorption and the wave just passes by.”

“But I’ll bet the atom can’t hold that extra energy forever.”

“Good bet, Vinnie.  The flip side of absorption is emission.  I expect that Cathleen has an emission spectrum somewhere on her laptop there.”Emission spectrum“You’re right, Sy.  It’s not a particularly pretty picture, but it shows that nice strong sodium doublet in the yellow and the broad iron and hydrogen lines down in the green and blue.  I’ll admit it, Vinnie, this is a faked image I made to show my students what the solar atmosphere would look like if you could turn off the photosphere’s continuous blast of light.  The point is that the atoms emit exactly the same sets of colors that they absorb.”

“You do what you gotta do, Cathleen.  But tell me, if each kind of atom does only certain colors, where’s that continuous rainbow come from?  Why aren’t we only getting hydrogen colors?”

“Kirchoff didn’t have a clue on that, Vinnie.  It took 50 years and Einstein to solve it.  Not just where the light comes from but also its energy-wavelength profile.”

“So where does the light come from?”

“Pure heat.  You can get a continuous spectrum from a hot wire, molten lava, a hole through the wall of a hot oven, even the primordial chaos of the Big Bang.  It doesn’t matter what kind of matter you’re looking at, the profile just depends on the temperature.  You know that temperature measures the kinetic energy stored in particle random motion, Vinnie?”

“Well, I wouldn’t have put it that way, but yeah.”

“Well, think about the Sun, just a big ball of really hot atoms and electrons and nuclei, all bouncing off each other in frantic motion.  Every time one of those changes direction it affects the electromagnetic field, jiggles it as you say.  The result of all that jiggling is the continuous spectrum.  Absorption and emission lines come from electrons that are confined to an atom, but heat motion is unconfined.”

“How about hot metal?”

“The atoms are locked in their lattice, but heat jiggles the whole lattice.”

~~ Rich Olcott

aLIGO and eLISA: Tuning The Instrument

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

Oh, it’s good to see Big News in hard science get big attention in Big Media.  The LIGO story and Columbia’s Dr Brian Greene even made it to the Stephen Colbert Late Show.  Everyone chuckled at the final “boowee-POP” audio recording (simulation at 7:30 into this clip; get for-real traces and audio from this one).

There’s some serious science in those chirps, not to mention serious trouble for any alien civilization that happened to be too close to the astronomical event giving rise to them.

LIGO trace 3
Adapted from the announcement paper by Abbot et al

The peaks and valleys in the top LIGO traces represent successive spatial compression cycles generated by two massive bodies orbiting each other.  There’s one trace for each of the two LIGO installations.  The spectrograms beneath show relative intensity at each frequency.  Peaks arrived more rapidly in the last 100 milliseconds and the simulated sound rose in pitch because the orbits grew smaller and faster.  The audio’s final POP is what you get from a brief but big disturbance, like the one you hear when you plug a speaker into a live sound system.  This POP announced two black holes merging into one, converting the mass-energy of three suns into a gravitational jolt to the Universe.

Scientists have mentioned in interviews that LIGO has given us “an ear to the Universe.”  That’s true in several different <ahem> senses.  First, we’ve seen in earlier posts that gravitational physics is completely different from the electromagnetism that illuminates every kind of telescope that astronomers have ever used.  Second, black hole collisions generate signals in frequencies that are within our auditory range.  Finally, LIGO was purposely constructed to have peak sensitivity in just that frequency range.

Virtually every kind of phenomenon that physicists study has a characteristic size range and a characteristic frequency/duration range.  Sound waves, for instance, are in the audiophile’s beloved “20 to 20,000” cycles per second (Hz).  Put another way, one cycle of a sound wave will last something between 1/20 and 1/20,000 second (0.05-0.000 05 second).  The speed of sound is roughly 340 meters per second which puts sound’s characteristic wavelength range between 17 meters and 17 millimeters.

No physicist would be surprised to learn that humans evolved to be sensitive to sound-making things in that size range.  We can locate an oncoming predator by its roar or by the snapping twig it stepped on but we have to look around to spot a pesky but tiny mosquito.

So the greenish box in the chart below is all about sound waves.  The yellowish box gathers together the classes of phenomena scientists study using the electromagnetic spectrum.  For instance, we use infra-red light (characteristic time range 10-15-10-12 second) to look at (or cause) molecular vibrations.

RegimesWe can investigate things that take longer than an instrument’s characteristic time by making repeated measurements, but we can’t use the instrument to resolve successive events that happen more quickly than that.  We also can’t resolve events that take place much closer together than the instrument’s characteristic length.

The electromagnetic spectrum serves us well, but it has its limitations.  The most important is that there are classes of objects out there that neither emit nor absorb light in any of its forms.  Black holes, for one.  They’re potentially crucial to the birth and development of galaxies.  However, we have little hard data on them against which to test the plethora of ideas the theoreticians have come up with.

Dark matter is another.  We know it’s subject to gravity, but to our knowledge the only way it interacts with light is by gravitational lensing.  Most scientists working on dark matter wield Occam’s Razor to conclude it’s pretty simple stuff.  Harvard cosmologist Dr Lisa Randall has suggested that there may be two kinds, one of which collects in disks that clothe themselves in galaxies.

That’s where LIGO and its successors in the gray box will help.  Their sensitivity to gravitational effects will be crucial to our understanding of dark objects.  Characteristic times in tens and thousands of seconds are no problem nor are event sizes measured in kilometers, because astronomical bodies are big.

GrWave Detectors
Gravitational instrumentation, from Christopher Berry’s blog and Web page

This is only the beginning, folks, we ain’t seen nothin’ yet.

~~ Rich Olcott