# Terzetto for Rubber Ruler

“So you’re telling me, Cathleen, that you can tell how hot a star is by looking at its color?”

“That’s right, Vinnie.  For most stars their continuous spectrum is pretty close to the blackbody equation tying peak wavelength to temperature.”

“But you can’t do that with far-away stars, right, because the further they are, the more stretched-out their lightwaves get.  Won’t that mess up the peak wavelength?”

“The key is Kirchhoff’s other kinds of spectrum.”

“You’re talking the bright-line and dark-line kinds.”

“Exactly.  Each kind of spectrum comes from a different process — each is affected differently by the object in question and the environment it’s embedded in.  A continuous spectrum is all about charged particles moving randomly in response to the heat energy they’re surrounded by.  It doesn’t matter what kind of particles they are or even whether they’re positive or negative.  Whenever a particle changes direction, it twitches the electromagnetic field and gives off a wave.”

“Right — the higher the temperature the less time between twitches; the wave can’t move as far before things change so the wavelength’s shorter; any speed’s possible so you can turn that dial wherever; I got all that.  So what’s different with the bright-line and dark-line spectrums?”

Cathleen and I both blurt out, “Spectra!” at the same time and give each other a look.  We’re grown-ups now.  We don’t say, “Jinx!” to each other any more.

“Alright, spectra.  But how’re they different?”

I pick up the story.  “Like Cathleen said, continuous spectra from same–temperature stuff look identical no matter what kind of stuff’s involved because heat is motion and each particle moves as a unit  The other kinds of spectrum are about transitions within particles so they’re all about which kind of stuff.  A given kind of atom can only absorb certain wavelengths of light and it can only relax by giving off exactly the same wavelengths.  There’s no in-betweens.”

She cuts in.  “Sodium, for instance.  It has two strong lines in the yellow, at 588.995 and 589.592 nanometers.  Whether in a star or a meteor or fireworks, sodium gives off exactly those colors.  Conversely, in an interstellar cloud or in a star’s outermost layers sodium absorbs exactly those colors from any continuous-spectrum light passing through.”

I’m back in.  “And there’s the key to your unmixing question, Vinnie.  We’ve talked about frames, remember?  Your far-away star’s light-generating layers emit a continuous spectrum that describes its temperature.  If we were right next to it, that’s the spectrum we’d see.  But as you say, we’re a long way away and in our frame the light’s been stretched.  It still looks like the black-body curve but it’s red-shifted because of our relative motion.”

Cathleen’s turn.  “But if there are sodium atoms in the star’s upper layers, their absorptions will cut a pair of notches in that emitted spectrum.  It won’t be a smooth curve, there’ll be two sharp dips in it, close together, with the blue-side one twice as strong as the other one.  Easy to recognize and measure the redshift.  The blackbody peak is redshifted by exactly the same amount so with some arithmetic you’ve got the peak’s original wavelength and the star’s temperature.”

Mine.  “See, because we know what the sodium wavelengths were in the star’s frame, we can divide the dip wavelengths we measure by the rest-frame numbers we know about.  The ratios give us the star’s redshift.”

Cathleen turns to her laptop and starts tapping keys.  “Let’s do an example.  Suppose we’re looking at a star’s broadband spectrogram.  The blackbody curve peaks at 720 picometers.  There’s an absorption doublet with just the right relative intensity profile in the near infra-red at 1,060,190 and 1,061,265 picometers.  They’re 1,075 picometers apart.  In the lab, the sodium doublet’s split by 597 picometers.  If the star’s absorption peaks are indeed the sodium doublet then the spectrum has been stretched by a factor of 1075/597=1.80.  Working backward, in the star’s frame its blackbody peak must be at 720/1.80=400 picometers, which corresponds to a temperature of about 6,500 K.”

“Old Reliable calculates from that stretch factor and the Hubble Constant the star’s about ten billion lightyears away and fleeing at 240,000 km/s.”

“All that from three peaks.  Spectroscopy’s pretty powerful, huh?”

Cathleen and me: “For sure!    Jinx!”

~~ Rich Olcott

# Zarzuela for Rubber Ruler

“Hey, Cathleen, if the expansion of the Universe stretches light’s wavelengths, how do you know when you see a color in a star what you’re looking at?”

“Excuse me, Professor, but your office-mate said you’d be here at the coffee shop and I have a homework question.”

“Good heavens, look at the time!  It’s my office hours, I should be over there.  Oh well, you’re here, Maria, what’s the question?”

“You showed us this chart and asked us to write an essay on it.  I don’t know where to begin.”

“Ah.  Hang on, Vinnie, this bears on your question, too.  OK, Maria, what can you tell me about the chart?”

“Well, there are five peaked curves, labeled with different temperatures.  Can I assume the green curve peaks, too, not continuing straight up?”

“Yes.  What else?”

“The horizontal axis, sorry I don’t know the word —”

“abscissa”

“Oh, we have almost the same word in Spanish!  Anyhow, the abscisa says it shows wavelengths.  It goes from a tenth of a nanometer to maybe 10 micrometers.  The chart must have to do with light, because sound waves can’t get that short.  The … ordinada…?”

“Ordinate”

“Thank you.  The ordinate says ‘Intensity’ so the chart must show light spectra at different temperatures.  But there’s only one peak at each temperature.”

“Is that Kirchhoff’s ‘continuous spectrum,’ Cathleen?”

“Right, Vinnie, a smoothly-varying cascade of every wavelength, photons arising from heat-generated motion of charged particles.”

Ah, ya lo veo — this is blackbody spectra given off by hot objects.  You showed us one in class and here we have several.”

“Good, Maria.  Now —”

“But all the peaks look exactly the same, Cathleen.  The hot objects ought to be brighter.  A really hot flame, you can’t even look at it.  Something’s phony.”

“Good eye, Vinnie.  I divided each curve in the graph by its peak height to put them all on an even footing.  That’s why the axis is labeled ‘Intensity profile‘ instead of ‘Intensity.'”

“I’ve got a different issue, Cathleen.  Hot objects have more energy to play with.  Shouldn’t the hotter peaks spread over a wider wavelength range?  These are all the same width.”

“I think I know the answer to that one, Mr Moire.  In class la profesora showed us how the blackbody curve’s equation has two factors, like B=W*X.  The W factor depends only on wavelength and grows bigger as the wavelength gets smaller.  That’s the ‘ultraviolet catastrophe,’ right, ma’am?”

“Mm-hm.  Go on, Maria.”

“But the X factor gets small real fast as the wavelength gets small.  In fact, it gets small so fast that it overpowers W‘s growth — the W*X product gets small, too.  Do you have that movie you showed us on your laptop there, ma’am?”

“Sure.  Here it is…”

“OK, the blue line is that W factor.  Oh, by the way, the ordinate scale here is logarithmic, so the value at the left end of the blue line is 1027/106 or about 1021 times bigger than it is at the right end even though it looks like a straight line.  The green line is that temperature-dependent factor.  See how it pulls down the orange lines’ values for cold objects, but practically goes away for very hot objects?”

“Yeah, that shows it real good, right, Sy?  That orange peak moves to the left just like Cathleen’s picture shows.  It answers your question, too.”

“It does, Vinnie?  How so?”

“‘Cause the peaks get broader as they get higher.  It’s like the intensity at the, umm, microwave end hardly changes at all and the whole rest of the curve swings up and out from there.”

“Keep in mind, guys, that we’re talking really large numbers here.  Vinnie’s ‘hardly changes at all’ is actually a factor of 40,000 or so.  Those pretty peaks in my homework chart are only pretty because the spread-out tails are so small relative to the peaks.”

“Alright, Cathleen, but how does Maria’s question tie in with mine?”

“They both hinge on wavelength.  The blackbody equation lets us measure a star’s temperature by looking at its color.  Do you have enough to start on that essay, Maria?”

“Yes, ma’am.  Gracias.”

De nada.  Now run along and get to work on it.”

~~ Rich Olcott