Rhythm Method

A warm Summer day.  I’m under a shady tree by the lake, watching the geese and doing some math on Old Reliable.  Suddenly a text-message window opens up on its screen.  The header bar says 710-555-1701.  Old Reliable has never held a messaging app, that’s not what I use it for.  The whole thing doesn’t add up.  I type in, Hello?

Hello, Mr Moire.  Remember me?

Suddenly I do.  That sultry knowing stare, those pointed ears.  It’s been a yearHello, Ms Baird.  What can I do for you?

Another tip for you, Mr Moire.  One of my favorite star systems — the view as you approach it at near-lightspeed is so ... meaningful.  Your astronomers call it PSR J0337+1715.

So of course I head over to Al’s coffee shop after erasing everything but that astronomical designation.  As I hoped, Cathleen and a few of her astronomy students are on their mid-morning break.  Cathleen winces a little when she sees me coming.  “Now what, Sy?  You’re going to ask about blazars and neutrinos?”

I show her Old Reliable’s screen.  “Afraid not, Cathleen, I’ll have to save that for later.  I just got a message about this star system.  Recognize it?”

“Why, Sy, is that a clue or something?  And why is the lettering in orange?”

“Long story.  But what can you tell me about this star system?”

“Well, it’s probably one of the most compact multi-component systems we’re ever going to run across.  You know what compact objects are?”

“Sure.  When a star the size of our Sun exhausts most of its hydrogen fuel, gravity wins its battle against heat.  The star collapses down to a white dwarf, a Sun-full of mass packed into a planet-size body.  If the star’s a bit bigger it collapses even further, down to a neutron star just a few miles across.  The next step would be a black hole, but that’s not really a star, is it?”

“No, it’s not.  Jim, why not?”

“Because by definition a black hole doesn’t emit light.  A black hole’s accretion disk or polar jets might, but not the object itself.”

“Mm-hm.  Sy, your ‘object’ is actually three compact objects orbiting  around each other.  There’s a neutron star with a white dwarf going around it, and another white dwarf swinging around the pair of them.  Vivian, does that sound familiar?”

“That’s a three-body system, like the Moon going around the Earth and both going around the Sun.  Mmm, except really both white dwarfs would go around the neutron star because it’s heaviest and we can calculate the motion like we do the Solar System.”

“Not quite.  We can treat the Sun as motionless because it has 99% of the mass.  J0337+1715’s neutron star doesn’t dominate its system as much as the Sun does ours.  That outermost dwarf has 20% of its system’s mass.  Phil, what does that suggest to you?”

“It’d be like Pluto and Charon.  Charon’s got 10% of their combined mass and so Pluto and Charon both orbit a point 10% of the way out from Pluto.  From Earth we see Pluto wobbling side to side around that point.  So the neutron star must wobble around the point 20% outward towards the heavy dwarf.  Hey, star-wobble is how we find exoplanets.  Is that what this is about, Mr Moire?  Did someone measure its red-shift behavior?”PSR J0337+1715Cathleen saves me from answering.  “Not quite.  The study Sy’s chasing is actually a cute variation on red-shift measurements.  That ‘PSR‘ designation means the neutron star is a pulsar.  Those things emit electromagnetic radiation pulses with astounding precision, generally regular within a few dozen nanoseconds.  If we receive slowed-down pulses then the object’s going away; sped-up and it’s approaching, just like with red-shifting.  The researchers  derived orbital parameters for all three bodies from the between-pulse durations.  The heavy dwarf is 200 times further out than the light one, for instance.  Not an easy experiment, but it yielded an important result.”

My ears perk up.  “Which was…?”

“The gravitational force between the pulsar and each dwarf was within six parts per million of what Newton’s Laws prescribe.  That observation rules out whole classes of theories that tried to explain galaxies and galaxy clusters without invoking dark matter.”

Cool, huh?

Uh-huh.

~~ Rich Olcott

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The Biggest Telescope in The Universe

Vinnie rocks back in his chair.  “These gravitational lenses, Cathleen.  How do you figure their apertures and f-numbers, space being infinite and all?”

She takes a breath to answer, but I cut in.  “Whoa, I never got past a snapshot camera.  How about you explain Vinnie’s question before you answer it?”Bird and lenses

“You’re right, Sy, most people these days just use their cellphone camera and have no clue about what it does inside.  Apertures and f-numbers are all just simple geometry.  Everything scales with the lens’ focal length.”

“That’s how far away something is that you’re taking a picture of?”

“No, it’s a characteristic of the lens itself.  It’s the distance between the midpoint of the lens and its focal plane, which is where you’d want to put the sensor chip or film in a camera.  The aperture is the diameter of the light beam entering the lens.  The optimal aperture, the image size, even the weight of the lens, all scale to the lens focal length.”

“I can see image size thing — the further back the focal plane, the bigger the image by the the time it gets there.  It’s like a lever.”

“Sort of, Vinnie, but you’ve got the idea.”

“The aperture scales to focal length?  I’d think you could make a lens with any diameter you like.”

“Sure you could, Sy, but remember you’d be using a recording medium of some sort and it’s got an optimum input level.  Too much light and you over-expose, too little and you under-expose.  To get the right amount of light when you take the shot the aperture has to be right compared to the focal length.”

“Hey, so that’s the reason for the old ‘Sunny 16‘ rule.  Didn’t matter if I had a 35mm Olympus or a big ol’ Rollei, if it was a sunny day I got good pictures with an f/16 aperture.  ‘Course I had to balance the exposure time with the film’s speed rating but that was easy.”

“Exactly, Vinnie.  If I remember right, the Rollei’s images were about triple the size of the little guy’s.  Tripled focal length meant tripled lens size.  You could use the same speed-rated film in both cameras and use the same range of f-stops.  The rule still works with digital cameras but you need to know your sensor’s ISO rating.”

“Ya got this, Sy?  Can we move on to Cathleen’s gravity lenses?”

“Sure, go ahead.”

“Well, they’re completely different from … I’ll call them classical lenses. That kind has a focal plane and a focal length and an aperture and only operates along one axis.  Gravitational lenses have none of that, but they have an infinite number of focal lines and rings.”

Gravitational lens and galaxy“Infinite?”

“At least in principle.  Any observation point in the Universe has a focal line running to a massive object’s center of gravity.  At any point along the line, you could look toward an object and potentially see all or part of a ring composed of light from some bright object behind it.  Einstein showed that a completed ring’s  visual angle depends on the deflector’s mass and the three distances between the observer, the deflector and the bright object.”

“The way you said that, there could be a bunch of rings.”

“Sure, one for each bright object shining onto the lens.  For that matter, the deflector itself could be complex — the gravity of a whole cluster of galaxies rather than the single black hole we’ve been assuming as an example.”

“That diagram reminds me of Galileo’s telescope, just a three-foot tube with an objective lens at the far end and an eyepiece lens to look through.  But it was enough to show him the rings of Saturn and the moons of Jupiter.”

“Right, Sy.  His objective lens was maybe a couple of inches across.  If its focal point was halfway down the tube, his scope’s light-gathering power would match an f/9 camera lens.  Gravitational lenses don’t have apertures so not an issue.”

“So here we are like Galileo, with a brand new kind of telescope.”

“Poetic, Vinnie, and so right.  It’s already shown us maybe the youngest galaxy, born 13 billion years ago.  We’re just getting started.”

~~ Rich Olcott

The Fellowship of A Ring

Einstein ring 2018

Hubble photo from NASA’s Web site

Cathleen and I are at a table in Al’s coffee shop, discussing not much, when Vinnie comes barreling in.  “Hey, guys.  Glad I found you together.  I just saw this ‘Einstein ring’ photo.  They say it’s some kind of lensing phenomenon and I’m thinking that a lens floating out in space to do that has to be yuuuge.  What’s it made of, and d’ya think aliens put it there to send us a message?”

Astronomer Cathleen rises to the bait.  I sit back to watch the fun.  “No, Vinnie, I don’t.  We’re not that special, the rings aren’t signals, and the lenses aren’t things, at least not in the way you’re thinking.”

“There’s more than one?”

“Hundreds we know of so far and it’s early days because the technology’s still improving.”

“How come so many?”

“It’s because of what makes the phenomenon happen.  What do you know about gravity and light rays?”

Me and Sy talked about that a while ago.  Light rays think they travel in straight lines past a heavy object, but if you’re watching the beam from somewhere else you think it bends there.”

I chip in.  “Nice summary, good to know you’re storing this stuff away.”Gravitational lens 1

“Hey, Sy, it’s why I ask questions is to catch up.  So go on, Cathleen.”

She swings her laptop around to show us a graphic.  “So think about a star far, far away.  It’s sending out light rays in every direction.  We’re here in Earth and catch only the rays emitted in our direction.  But suppose there’s a black hole exactly in the way of the direct beam.”

“We couldn’t see the star, I get that.”

“Well, actually we could see some of its light, thanks to the massive black hole’s ray-bending trick. Rays that would have missed us are bent inward towards our telescope.  The net effect is similar to having a big magnifying lens out there, focusing the star’s light on us.”

“You said, ‘similar.’  How’s it different?”Refraction lens

“In the pattern of light deflection.  Your standard Sherlock magnifying lens bends light most strongly at the edges so all the light is directed towards a point.  Gravitational lenses bend light most strongly near the center.  Their light pattern is hollow.  If we’re exactly in a straight line with the star and the black hole, we see the image ‘focused’ to a ring.”

“That’d be the Einstein ring, right?”

“Yes, he gets credit because he was the one who first set out the equation for how the rays would converge.  We don’t see the star, but we do see the ring.  His equation says that the angular size of the ring grows as the square root of the deflecting object’s mass.  That’s the basis of a widely-used technique for measuring the masses not only of black holes but of galaxies and even larger structures.”

“The magnification makes the star look brighter?”

“Brighter only in the sense that we’re gathering photons from a wider field then if we had only the direct beam.  The lens doesn’t make additional photons, probably.”

Suddenly I’m interested.  “Probably?”

“Yes, Sy, theoreticians have suggested a couple of possible effects, but to my knowledge there’s no good evidence yet for either of them.  You both know about Hawking radiation?”

“Sure.”

“Yup.”

“Well, there’s the possibility that starlight falling on a black hole’s event horizon could enhance virtual particle production.  That would generate more photons than one would expect from first principles.  On the other hand, we don’t really have a good handle on first principles for black holes.”

“And the other effect?”

“There’s a stack of IFs under this one.  IF dark matter exists and if the lens is a concentration of dark matter, then maybe photons passing through dark matter might have some subtle interaction with it that could generate more photons.  Like I said, no evidence.”

“Hundreds, you say.”

“Pardon?”

“We’ve found hundreds of these lenses.”

“All it takes is for one object to be more-or-less behind some other object that’s heavy enough to bend light towards us.”

“Seein’ the forest by using the trees, I guess.”

“That’s a good way to put, it, Vinnie.”

~~ Rich Olcott

RIP, Dr Hawking

Today I depart from my normal schedule and the current story line and science line.  A giant has left us and I want to pay proper tribute.

Dr Stephen Hawking enjoyed telling people of his fortunate birth date, exactly 300 years after Galileo Galilei passed away.  He liked a good joke, and I think he’d be tickled with this additional connection to the man whose work made Hawking’s work possible:
RIP Hawking

The equation in the center of this cut is Hawking’s favorite result, which he wanted to be carved on his gravestone.  It links a black hole’s entropy (S) to its surface area (A).  The other letters denote a collection of constants that have been central to the development of theoretical Physics over the past century and a half:

  • k is Boltzmann’s constant, which links temperature with kinetic energy
  • c is the speed of light, the invariance of which led Einstein to Relativity
  • G is Newton’s universal gravitational constant
  • h is Planck’s constant, the “quantum of action”

Hawking spent much of his career thinking deeply about the implications of Einstein’s concepts.  Newton’s equations support excellent descriptions of everyday physical motions, from the fall of raindrops to the orbits of solar systems.  Einstein’s equations led to insights about conditions at the most extreme — velocities near lightspeed, masses millions of times the Sun’s but packed into a volume only a few dozen miles wide.

But Hawking also pondered extremes of the ultimately large and the ultimately small — the edge of the Universe and distances far smaller than atomic nuclei.  Because his physical condition prohibited speech or quick jottings, he was forced to develop extraordinary powers of concentration and visualization that enabled him to encapsulate in a few phrases insights that would take others books to develop and communicate.

Hawking wrote books, too, of course, of a quality and clarity that turned his name and Science into watchwords for the general public as well as the physics community.  By his life and how he lived it he was an inspiration to many, abled and otherwise.  Science needs its popularizers, though some in the field deprecate them as hangers-on.  Hawking managed to bridge that gap with ease and grace, a giant with standing on either side.

Requiescat in pace, Dr Hawking.  Thank you.

~~ Rich Olcott

Gravity from Another Perspective

“OK, we’re looking at that robot next to the black hole and he looks smaller to us because of space compression down there.  I get that.  But when the robot looks back at us do we look bigger?”

We’re walking off a couple of Eddie’s large pizzas.  “Sorry, Mr Feder, it’s not that simple.  Multiple effects are in play but only two are magnifiers.”

“What isn’t?”

“Perspective for one.  That works the same in both directions — the image of an object shrinks in direct proportion to how far away it is.  Relativity has nothing to do with that principle.”

“That makes sense, but we’re talking black holes.  What does relativity do?”

“Several things, but it’s complicated.”

“Of course it is.”

“OK, you know the difference between General and Special Relativity?”

“Yeah, right, we learned that in kindergarten.  C’mon.”

“Well, the short story is that General Relativity effects depend on where you are and Special Relativity effects depend on how fast you’re going.  GR says that the scale of space is compressed near a massive object.  That’s the effect that makes our survey robot appear to shrink as it approaches a black hole.  GR leaves the scale of our space larger than the robot’s.  Robot looks back at us, factors out the effect of perspective, and reports that we appear to have grown.  But there’s the color thing, too.”

“Color thing?”

“Think about two photons, say 700-nanometer red light, emitted by some star on the other side of our black hole.  One photon slides past it.  We detect that one as red light.  The other photon hits our robot’s photosensor down in the gravity well.  What color does the robot see?”

“It’s not red, ’cause otherwise you wouldn’t’ve asked me the question.”

“Check.”

“Robot’s down there where space is compressed…  Does the lightwave get compressed, too?”

“Yup.  It’s called gravitational blue shift.  Like anything else, a photon heading towards a massive object loses gravitational potential energy.  Rocks and such make up for that loss by speeding up and gaining kinetic energy.  Light’s already at the speed limit so to keep the accounts balanced the photon’s own energy increases — its wavelength gets shorter and the color shifts blue-ward.  Depending on where the robot is, that once-red photon could look green or blue or even X-ray-colored.”

“So the robot sees us bigger and blue-ish like.”Robots and perspective and relativity 2“But GR’s not the only player.  Special Relativity’s in there, too.”

“Maybe our robot’s standing still.”

“Can’t, once it gets close enough.  Inside about 1½ diameters there’s no stable orbit around the black hole, and of course inside the event horizon anything not disintegrated will be irresistibly drawn inward at ever-increasing velocity.  Sooner or later, our poor robot is going to be moving at near lightspeed.”

“Which is when Special Relativity gets into the game?”

“Mm-hm.  Suppose we’ve sent in a whole parade of robots and somehow they maintain position in an arc so that they’re all in view of the lead robot.  The leader, we’ll call it RP-73, is deepest in the gravity well and falling just shy of lightspeed.  Gravity’s weaker further out — trailing followers fall slower.  When RP-73 looks back, what will it see?”

“Leaving aside the perspective and GR effects?  I dunno, you tell me.”

“Well, we’ve got another flavor of red-shift/blue-shift.  Speedy RP-73 records a stretched-out version of lightwaves coming from its slower-falling followers, so so it sees their colors shifted towards the red, just the opposite of the GR effect.  Then there’s dimming — the robots in the back are sending out n photons per second but because of the speed difference, their arrival rate at RP-73 is lower.  But the most interesting effect is relativistic aberration.”

“OK, I’ll bite.”

“Start off by having RP-73 look forward.  Going super-fast, it intercepts more oncoming photons than it would standing still.”

“Bet they look blue to it, and really bright.”

“Right on.  In fact, its whole field of view contracts towards its line of flight.  The angular distortion continues all the way around.  Rearward objects appear to swell.”

“So yeah, we’d look bigger.”

“And redder.  If RP-73 is falling fast enough.”

~~ Rich Olcott

  • Thanks to Timothy Heyer for the question that inspired this post.

A Perspective on Gravity

“I got another question, Moire.”

“Of course you do, Mr Feder.”

“When someone’s far away they look smaller, right, and when someone’s standing near a black hole they look smaller, too.  How’s the black hole any different?”

“The short answer is, perspective depends on the distance between the object and you, but space compression depends on the distance between the object and the space-distorting mass.  The long answer’s more interesting.”

“And you’re gonna tell it to me, right?”

“Of course.  I never let a teachable moment pass by.  Remember the August eclipse?”

“Do I?  I was stuck in that traffic for hours.”

“How’s it work then?”

“The eclipse?  The Moon gets in front of the Sun and puts us in its shadow. ‘S weird how they’re both the same size so we can see the Sun’s corundum and protuberances.”

“Corona and prominences.  Is the Moon really the same size as the Sun?”

“Naw, I know better than that.  Like they said on TV, the Moon’s about ¼ the Earth’s width and the Sun’s about 100 times bigger than us.  It’s just they look the same size when they meet up.”

“So the diameter ratio is about 400-to-1.  Off the top of your head, do you know their distances from us?”

“Millions of miles, right?”

“Not so much, at least for the Moon.  It’s a bit less than ¼ of a million miles away.  The Sun’s a bit less than 100 million miles away.”

“I see where you’re going here — the distances are the same 400-to-1 ratio.”

“Bingo.  The Moon’s actual size is 400 times smaller than the Sun’s, but perspective reduces the Sun’s visual size by the same ratio and we can enjoy eclipses.  Let’s try another one.  To keep the arithmetic simple I’m going to call that almost-100-million-mile distance an Astronomical Unit.  OK?”

“No problemo.”

“Jupiter’s diameter is about 10% of the Sun’s, and Jupiter is about 5 AUs away from the Sun.  How far behind Jupiter would we have to stand to get a nice eclipse?”

“Oh, you’re making me work, too, huh?  OK, I gotta shrink the Sun by a factor of 10 to match the size of Jupiter so we gotta pull back from Jupiter by the same factor of 10 times its distance from the Sun … fifty of those AUs.”

“You got it.  And by the way, that 55 AU total is just outside the farthest point of Pluto’s orbit.  It took the New Horizons spacecraft nine years to get there.  Anyhow, perspective’s all about simple ratios and proportions, straight lines all the way.  So … on to space compression, which isn’t.”

“We’re not going to do calculus, are we?”

“Nope, just some algebra.  And I’m going to simplify things just a little by saying that our black hole doesn’t spin and has no charge, and the object we’re watching, say a survey robot, is small relative to the black hole’s diameter.  Of course, it’s also completely outside the event horizon or else we couldn’t see it.  With me?”

“I suppose.”

“OK, given all that, suppose the robot’s as-built height is h and it’s a distance r away from the geometric center of an event horizon’s sphere.  The radius of the sphere is rs.  Looking down from our spaceship we’d see the robot’s height h’ as something smaller than h by a factor that depends on r.  There’s a couple of different ways to write the factor.  The formula I like best is h’=h√[(r-rs)/r].”

“Hey, (r-rs) inside the brackets is the robot’s distance to the event horizon.”

“Well-spotted, Mr Feder.  We’re dividing that length by the distance from the event horizon’s geometric center.  If the robot’s far away so that r>>rs, then (r-rs)/r is essentially 1.0 and h’=h.  We and the robot would agree on its height.  But as the robot closes in, that ratio really gets small.  In our frame the robot’s shrinking even though in its frame its height doesn’t change.”

“We’d see it getting smaller because of perspective, too, right?”

“Sure, but toward the end relativity shrinks the robot even faster than perspective does.”

“Poor robot.”

~~ Rich Olcott

  • Thanks to Carol, who inspired this post by asking Mr Feder’s question but in more precise form.

Shopping The Old Curiosity

“Still got questions, Moire.”

“This’ll be your last shot this year, Mr Feder.  What’s the question?”

“They say a black hole absorbs all the light that falls on it. But the theory of blackbody radiation says a perfect absorber is also a perfect radiator. Emission should be an exact opposite flow to the incoming flow in every direction. Wouldn’t a black hole be shiny like a ball bearing?”Black hole as ball bearing 1
“A perfectly good question, but with crucial imperfections. Let’s start with the definition of a perfect absorber — it’s an object that doesn’t transmit or reflect any light. Super-black, in other words. So by definition it can’t be a mirror.”

“OK, maybe not a mirror, but the black hole has to send out some kind of exact opposite light to balance the arriving light.”

“Yes, but not in the way you think. Blackbody theory does include the assumption that the object is in equilibrium, your ‘exact opposite flow.’ The object must indeed send out as much energy as it receives, otherwise it’d heat up or cool down. But the outbound light doesn’t necessarily have to be at the same frequencies as the inbound light had. In fact, it almost never will.”

“How come not?”

“Because absorption and emission are two different processes and they play by different rules. If we’re including black holes in the discussion there are four different processes. No, five.  Maybe six.”

“I’m listening.”

“Good. Blackbody first. When a photon is absorbed by regular matter, it affects the behavior of some electron in there. Maybe it starts spending more time in a different part of the molecule, maybe it moves faster — one way or another, the electron configuration changes and that pulls the atomic nuclei away from where they were and the object’s atoms wobble differently. So the photon raises the object’s internal kinetic energy, which means raising its temperature, and we’ve got energy absorption, OK?”

“Yeah, and…?”

“At some later time, to keep things in equilibrium that additional energy has to be gotten rid of. But you can’t just paint one bit of energy red, say it’s special and follow it until it’s emitted. The whole molecule or crystal or whatever has excess energy as the result of all the incoming photons. When the total gets high enough, something has to give.  The object emits some photons to get rid of some of the excess. The only thing you can say about the outbound photons is that they generally have a lower energy than the incoming ones.”

“Why’s that?”

“Think of a bucket that’s brim-full and you’re dumping in cupfuls of water. Unless you’re pouring slowly and carefully, the dribbles escaping over the bucket’s rim will generally be many small amounts sloshing out more often than those cupfuls come in.  For light that’s fluorescence.”

“I suppose. What about the black hole?”

“The problem with a black hole is the mystery of what’s inside its event horizon. It probably doesn’t contain matter in the form of electrons and nuclei but we don’t know. There are fundamental reasons why information about what’s inside can’t leak out to us. All we can say is that when a light wave encounters a black hole, it’s trapped by the intense gravity field and its energy increments the black hole’s mass.  The mechanism … who knows?”

“Like I said, it gets absorbed. And gets emitted as Hawking radiation.”

“Sorry, that’s exactly what doesn’t happen. Hawking radiation arises from a different pair of processes. Process 1 generates pairs of virtual particles, which could be photons, electrons or something heavier. That happens at a chaotic but steady rate throughout the Universe.  Usually the particle pairs get back together and annihilate.  However, right next to the black hole’s event horizon there’s Process 2, in which one member of a virtual pair flies inward and the other member flies outward as a piece of Hawking radiation. Neither process even notices incoming photons. That’s not mirroring or even fluorescence.”

“Phooey, it was a neat idea.”

“That it was, but facts.”

~~ Rich Olcott

  • Thanks to lifeisthermal for inspiring this post.
  • Thus endeth a full year of Sy Moire stories.  I hope you enjoyed them.  Here’s to a new year and new ideas for all.