Would the CIA want a LIGO?

On February 22, 2016November 29, 2025 By rolcottIn Astronomy: Techniques, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Electromagnetism, Physics: Fundamentals, Physics: Light and Relativity, UncategorizedLeave a comment

So I was telling a friend about the LIGO announcement, going on about how this new “device” will lead to a whole new kind of astronomy. He suddenly got a far-away look in his eyes and said, “I wonder how many of these the CIA has.”

The CIA has a forest of antennas, but none of them can do what LIGO does. That’s because of the physics of how it works, and what it can and cannot detect. (If you’re new to this topic, please read last week’s post so you’ll be up to speed on what follows. Oh, and then come back here.)

There are remarkable parallels between electromagnetism and gravity. The ancients knew about electrostatics — amber rubbed by a piece of cat fur will attract shreds of dry grass. They certainly knew about gravity, too. But it wasn’t until 100 years after Newton wrote his Principia that Priestly and then Coulomb found that the electrostatic force law, F = k_e·q₁·q₂ / r², has the same form as Newton’s Law of Gravity, F = G·m₁·m₂ / r². (F is the force between two bodies whose centers are distance r apart, the q‘s are their charges and the m‘s are their masses.)

Almost a century later, James Clerk Maxwell (the bearded fellow at left) wrote down his electromagnetism equations that explain how light works. Half a century later, Einstein did the same for gravity.

But interesting as the parallels may be, there are some fundamental differences between the two forces — fundamental enough that not even Einstein was able to tie the two together.

One difference is in their magnitudes. Consider, for instance, two protons. Running the numbers, I found that the gravitational force pulling them together is a factor of 10³⁶ smaller than the electrostatic force pushing them apart. If a physicist wanted to add up all the forces affecting a particular proton, he’d have to get everything else (nuclear strong force, nuclear weak force, electromagnetic, etc.) nailed down to better than one part in 10³⁶ before he could even detect gravity.

But it’s worse — electromagnetism and gravity don’t even have the same shape.

Electromagneticwave3D — Electric (red) and magnetic (blue) fields in a linearly polarized light wave
(graphic from WikiMedia Commons, posted by Lookang and Fu-Kwun Hwang)

A word first about words. Electrostatics is about pure straight-line-between-centers (longitudinal) attraction and repulsion — that’s Coulomb’s Law. Electrodynamics is about the cross-wise (transverse) forces exerted by one moving charged particle on the motion of another one. Those forces are summarized by combining Maxwell’s Equations with the Lorenz Force Law. A moving charge gives rise to two distinct forces, electric and magnetic, that operate at right angles to each other. The combined effect is called electromagnetism.

The effect of the electric force is to vibrate a charge along one direction transverse to the wave. The magnetic force only affects moving charges; it acts to twist their transverse motion to be perpendicular to the wave. An EM antenna system works by sensing charge flow as electrons move back and forth under the influence of the electric field.

Gravitostatics uses Newton’s Law to calculate longitudinal gravitational interaction between masses. That works despite gravity’s relative weakness because all the astronomical bodies we know of appear to be electrically neutral — no electrostatic forces get in the way. A gravimeter senses the strength of the local gravitostatic field.

Gravitodynamics is completely unlike electrodynamics. Gravity’s transverse “force” doesn’t act to move a whole mass up and down like Maxwell’s picture at left. Instead, as shown by Einstein’s picture, gravitational waves stretch and compress while leaving the center of mass in place. I put “force” in quotes because what’s being stretched and compressed is space itself. See this video for a helpful visualization of a gravitational wave.

LIGO is neither a telescope nor an electromagnetic antenna. It operates by detecting sudden drastic changes in the disposition of matter within a “small” region. In LIGO’s Sept 14 observation, 10³¹ kilograms of black hole suddenly ceased to exist, converted to gravitational waves that spread throughout the Universe. By comparison, the Hiroshima explosion released the energy of 10^-6 kilograms.

Seismometers do a fine job of detecting nuclear explosions. Hey, CIA, they’re a lot cheaper than LIGO.

~~ Rich Olcott

LIGO, a new kind of astronomy

On February 15, 2016November 29, 2025 By rolcottIn Astronomy: Techniques, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Fundamentals, UncategorizedLeave a comment

Like thousands of physics geeks around the world, I was glued to the tube Thursday morning for the big LIGO (Laser Interferometer Gravitational-Wave Observatory) announcement. As I watched the for-the-public videos (this is a good one), I was puzzled by one aspect of the LIGO setup. The de-puzzling explanation spotlit just how different gravitational astronomy will be from what we’re used to.

There are two LIGO installations, 2500 miles apart, one near New Orleans and the other near Seattle. Each one looks like a big L with steel-pipe arms 4 kilometers long. By the way, both arms are evacuated to eliminate some sources of interference and a modest theoretical consideration.

The experiment consists of shooting laser beams out along both arms, then comparing the returned beams.

Some background: Einstein conquered an apparent relativity paradox. If Ethel on vehicle A is speeding (like, just shy of light-speed speeding) past Fred on vehicle B, Fred sees that Ethel’s yardstick appears to be shorter than his own yardstick. Meanwhile, Ethel is quite sure that Fred’s yardstick is the shorter one.

Einstein explained that both observations are valid. Fred and Ethel can agree with each other but only after each takes proper account of their relative motion. “Proper account” is a calculation called the Lorenz transformation. What Fred (for instance) should do is divide what he thinks is the length of Ethel’s yardstick by √[1-(v/c)²] to get her “proper” length. (Her relative velocity is v, and c is the speed of light.)

Suppose Fred’s standing in the lab and Ethel’s riding a laser beam. Here’s the puzzle: wouldn’t the same Fred/Ethel logic apply to LIGO? Wouldn’t the same yardstick distortion affect both the interferometer apparatus and the laser beams?

Well, no, for two reasons. First, the Lorenz effect doesn’t even apply, because the back-and-forth reflected laser beams are standing waves. That means nothing is actually traveling. Put another way, if Ethel rode that light wave she’d be standing as still as Fred.

The other reason is that the experiment is less about distance traveled and more about time of flight.

Suppose you’re one of a pair of photons (no, entanglement doesn’t enter into the game) that simultaneously traverse the interferometer’s beam-splitter mirror. Your buddy goes down one arm, strikes the far-end mirror and comes back to the detector. You take the same trip, but use the other arm.

The beam lengths are carefully adjusted so that under normal circumstances, when the two of you reach the detector you’re out of step. You peak when your buddy troughs and vice-versa. The waves cancel and the detector sees no light.

Now a gravitational wave passes by (red arcs in the diagram). In general, the wave will affect the two arms differently. In the optimal case, the wave front hits one arm broadside but cuts across the perpendicular one. Suppose the wave is in a space-compression phase when it hits. The broadside arm, beam AND apparatus, is shortened relative to the other one which barely sees the wave at all.

The local speed of light (miles per second) in a vacuum is constant. Where space is compressed, the miles per second don’t change but the miles get smaller. The light wave slows down relative to the uncompressed laboratory reference frame. As a result, your buddy in the compressed arm takes just a leetle longer than you do to complete his trip to the detector. Now the two of you are in-step. The detector sees light, there is great rejoicing and Kip Thorne gets his Nobel Prize.

But the other wonderful thing is, LIGO and neutrino astronomy are humanity’s first fundamentally new ways to investigate our off-planet Universe. Ever since Galileo trained his crude telescope on Jupiter the astronomers have been using electromagnetic radiation for that purpose – first visible light, then infra-red and radio waves. In 1964 we added microwave astronomy to the list. Later on we put up satellites that gave us the UV and gamma-ray skies.

The astronomers have been incredibly ingenious in wringing information out of every photon, but when you look back it’s all photons. Gravitational astronomy offers a whole new path to new phenomena. Who knows what we’ll see.

~~ Rich Olcott

Smoke and a mirror

On February 8, 2016November 28, 2025 By rolcottIn General: Fun Stuff, Physics: Fundamentals, Uncategorized1 Comment

Grammie always grimaced when Grampie lit up one of his cigars inside the house. We kids grinned though because he’d soon be blowing smoke rings for us. Great fun to try poking a finger into the center, but we quickly learned that the ring itself vanished if we touched it.

My grandfather can’t take credit for the smoke ring on the left — it was “blown” by Mt Etna. Looks very like the jellyfish on the right, doesn’t it? When I see two such similar structures, I always wonder if the resemblance comes from the same physics phenomenon.

This one does — the physics area is Fluid Dynamics, and the phenomenon is a vortex ring. We need to get a little technical and abstract here: to a physicist a fluid is anything that’s composed of particles that don’t have a fixed spatial relationship to each other. Liquid water is a fluid, of course (its molecules can slide past each other) and so is air. The sun’s ionized protons and electrons comprise a fluid, and so can a mob of people and so can vehicle traffic (if it’s moving at all). You can use Fluid Dynamics to analyze motion when the individual particles are numerous and small relative to the volume in question.

Ring x-section — Adapted from a NOAA page

You get a vortex whenever you have two distinct fluids in contact but moving at sufficiently different velocities. (Remember that “velocity” includes both speed and direction.) When Grampie let out that little puff of air (with some smoke in it), his fast-moving breath collided with the still air around him. When the still air didn’t get out of the way, his breath curled back toward him. The smoke collected in the dark gray areas in this diagram.

That curl is the essence of vorticity and turbulence. The general underlying rule is “faster curls toward slower,” just like that skater video in my previous post. Suppose fluid is flowing through a pipe. Layers next to the outside surface move slowly whereas the bulk material near the center moves quickly. If the bulk is going fast enough, the speed difference will generate many little whorls against the circumference, converting pump energy to turbulence and heat. The plant operator might complain about “back pressure” because the fluid isn’t flowing as rapidly as expected from the applied pressure.

But Grampie didn’t puff into a pipe (he’s a cigar man, right?), he puffed into the open air. Those curls weren’t just at the top and bottom of his breath, they formed a complete circle all around his mouth. If his puff didn’t come out perfectly straight, the smoke had a twist to it and circulated along that circle, the way Etna’s ring seems to be doing (note the words In and Out buried in the diagram’s gray blobs). When a vortex closes its loop like that, you’ve got a vortex ring.

A vortex ring is a peculiar beast because it seems to have a life of its own, independent of the surrounding medium. Grampie’s little puff of vortical air usually retained its integrity and carried its smoke particles for several feet before energy loss or little fingers broke up the circulation.

To show just how special vortex rings are, consider the jellyfish. Until I ran across this article, I’d thought that jellyfish used jet propulsion like octopuses and squids do — squirt water out one way to move the other way. Not the case. Jellyfish do something much more sophisticated, something that makes them possibly “the most energy-efficient animals in the world.”

Thanks to a very nice piece of biophysics detective work (read the paper, it’s cool, no equations), we now know that a jellyfish doesn’t just squirt. Rather, it relaxes its single ring of muscle tissue to open wide. That motion pulls in a pre-existing vortex ring that pushes against the bell. On the power stroke, the jellyfish contracts its bell to push water out (OK, that’s a squirt) and create another vortex ring rolling in the opposite direction. In effect, the jellyfish continually builds and climbs a ladder of vortex rings.

Vortex rings are encapsulated angular momentum, potentially in play at any size in any medium.

~~ Rich Olcott

The Force(s) of Geometry

On February 1, 2016November 28, 2025 By rolcottIn Mathematics, Physics: Fundamentals, UncategorizedLeave a comment

There’s a lot more to Geometry than congruent triangles. Geometry can generate hurricanes and slam you to the floor.

It all starts (of course) with Newton. His three laws boil down to

Effect is to Cause as Change of Motion is to Force.

They successfully account for the physical movement of pretty much everything bigger than an atom. But sometimes the forces are a bit weird and it takes Geometry to understand them.

For instance, suppose Fred and Ethel collaborate on a narwhale research project. Fred is based in San Diego CA and Ethel works out of Norfolk VA. They fly to meet their research vessel at the North Pole. Fred’s plane follows the green track, Ethel’s plane follows the yellow one. At the start of the trip, they’re on parallel paths going straight north (the dotted lines). After a few hours, though, Ethel notices the two planes pulling closer together.

Ethel calls on her Newton knowledge to explain the phenomenon. “It can’t be Earth’s gravity moving us together, because that force points down to Earth’s center and this is a sideways motion. Our planes each weigh about 2000 kilograms and we’re still 2,000 kilometers apart. By Newton’s F = G m₁m₂/r² equation, the gravitational force between us should be (6.7×10^-11 N m²/kg²) x (2000 kg) x (2000 kg) / (2,000 m)² = 6.7×10^-11 newtons, way too small to account for our speed of approach. Both planes were electrically grounded when we fueled up, so we’re both carrying a neutral electric charge and it can’t be an electrostatic force. If it were magnetic my compass would be going nuts and it’s not. Woo-hoo, I’ve discovered a new kind of force!”

See what I did there? Fred and Ethel would have stayed a constant distance apart if Earth were a cylinder. Parallel lines running up a cylinder never meet. But Earth is a sphere, not a cylinder. Any pair of lines on a sphere must meet, sooner or later. Ethel’s “sideways force” is a product of Geometry.

Hurricanes, too. This video shows a day in the life of Hurricane Sandy. Weather geeks will find several interesting details there, but for now just notice the centers of counter-clockwise rotation (the one off the Florida coast is Sandy). Storm centers in the Northern Hemisphere virtually always spin counterclockwise. Funny thing is, in the Southern Hemisphere those centers go clockwise instead.

The difference has to do with angular momentum. We could get all formal vector math here, but the easy way is to consider how fast the air is moving in different parts of the world.

We’ve all seen at least one ice show act where skaters form a spinning line. The last skater to join up (usually it’s a short girl) has to push like mad to catch the end of that moving line and everyone applauds her success. Meanwhile the tall girl at the center of the line is barely moving except to fend off dizziness.

The line rotates as a unit — every skater completes a 360^o rotation in the same time. Similarly, everywhere on Earth a day lasts for exactly 24 hours.

Skaters at the end of the line must skate faster than those further in because they have to cover a greater distance in the same amount of time. The same geometry applies to Earth’s atmosphere. The Earth is 25,000 miles around at the equator but only 12,500 miles around near the latitude of Whitehorse, Canada. By and large, a blob of air at the equator must move twice as fast as a blob at 60^o north.

Now suppose our speedy skater hits a slushy patch of ice. Her end of the line is slowed down, so what happens to the rest of the line? It deforms — there’s a new center of rotation that forces the entire line to curl around towards the slow spot. Similarly, that blob near the Equator in the split-Earth diagram curls in the direction of the slower-moving air to its north, which is counter-clockwise.

In the Southern hemisphere, “slower” is southward and clockwise.

If not for Geometry (those differing circle sizes), we wouldn’t have hurricanes. Or gravity — but that’s another story.

~~ Rich Olcott

Smack-dab in the middle

On January 25, 2016November 28, 2025 By rolcottIn Cosmology, General: Fun Stuff, Mathematics: Dimensions, UncategorizedLeave a comment

See that little guy on the bridge, suspended halfway between all the way down and all the way up? That’s us on the cosmic size scale.

I suspect there’s a lesson there on how to think about electrons and quantum mechanics.

Let’s start at the big end. The physicists tell us that light travels at 300,000 km/s, and the astronomers tell us that the Universe is about 13.7 billion years old. Allowing for leap years, the oldest photons must have taken about 4.3×10¹⁷ seconds to reach us, during which time they must have covered 1.3×10²⁶ meters. Double that to get the diameter of the visible Universe, 2.6×10²⁶ meters. The Universe probably is even bigger than that, but far as I can see that’s as far as we can see.

At the small end there’s the Planck length, which takes a little explaining. Back in 1899, Max Planck published his epochal paper showing that light happens piecewise (we now call them photons). In that paper, he combined several “universal constants” to derive a convenient (for him) universal unit of length: 1.6×10^-35 meters. It’s certainly an inconvenient number for day-to-day measurements (“Gracious, Junior, how you’ve grown! You’re now 8×10³⁴ Planck-lengths tall.”). However, theoretical physicists have saved barrels of ink and hours of keyboarding by using Planck-lengths and other such “natural units” in their work instead of explicitly writing down all the constants.

Furthermore, there are theoretical reasons to believe that the smallest possible events in the Universe occur at the scale of Planck lengths. For instance, some theories suggest that it’s impossible to measure the distance between two points that are closer than a Planck-length apart. In a sense, then, the resolution limit of the Universe, the ultimate pixel size, is a Planck length.

So that’s the size range of the Universe, from 1.6×10^-35 up to 2.6×10²⁶ meters. What’s a reasonable way to fix a half-way mark between them?

It makes no sense to just add the two numbers together and divide by two the way we’d do for an arithmetic average. That’d be like adding together the dime I owe my grandson and the US national debt — I could owe him 10¢ or $10, but either number just disappears into the trillions.

The best way is to take the geometrical average — multiply the two numbers and take the square root. I did that. It’s the X in the sizeline, at 6.5×10^-5 meters, or about the diameter of a fairly large bacterium. (In the diagram, VSC is the Vega Super Cluster, AG is the Andromeda Galaxy, and the numbers are those exponents of 10.)

That’s worth marveling at. Sixty orders of magnitude between the size of the Universe and the size of the ultimate pixel. Yet from blue whales to bacteria, Earth’s life just happens to occupy the half-dozen orders right in the middle of the range. We think that’s it.

Could this be another case of the geocentric fallacy? Humans were so certain that Earth was the center of the Universe, before Brahe and Galileo and Newton proved otherwise. Is there life out there at scales much larger or much smaller than we imagine?

Who knows? But here’s an intriguing physics/quantum angle I’d like to promote. We know a lot about structures bigger than us — solar systems and binary stars and galaxy clusters on up. We know a few sizes and structures a bit smaller — viruses and molecules and atoms. We’re aware of quarks and gluons that reside inside protons and atomic nuclei, but we don’t know their size or structure.

Even a proton is huge on the Planck-length scale. At 1.8×10^-15 meters the proton measures some 10²⁰ Planck-lengths. There’s as much scale-space between the Planck-length and the proton as there is between the Earth (1.3×10⁷ meters) and the Universe.

It’s hard to believe that Terra infravita’s area has no structure whereas Terra supravita is so … busy. The Standard Model’s “ultimate particles,” the electrons and photons and neutrinos and quarks and gluons, all operate down there somewhere. It’s reasonable to suppose that they reflect a deeper architecture somewhere on the way down to the Planck-length foam.

Newton wrote (in Latin), “I do not make hypotheses.” But golly, it’s tempting.

~~ Rich Olcott

There’s a lot of not much in Space

On January 18, 2016March 18, 2020 By rolcottIn Cosmology, General: Fun Stuff, Mathematics: Dimensions, Uncategorized2 Comments

A while ago I drove from Denver to Fort Worth, and I was impressed. See, there’s a lot of not much in eastern Colorado. It’s pretty much the same in western Oklahoma except there’s less not much because there’s less of Oklahoma – but Texas has way more not much than anybody.

That gives Texas not much to brag about, but they do the best they can, bless their hearts.

What got me started on this rant was a a pair of astronomical factoids Katherine Kornei wrote in the Nov 2014 Discover magazine.

“If galaxies were shrunk to the size of apples, neighboring galaxies would be only a few meters apart….”
“If the stars within galaxies were shrunk to the size of oranges, they would be separated by 4,800 kilometers (3,000 miles).”

So there’s a lot of not much between galaxies, but a whole lot more not much, relatively speaking, within them. I just measured an apple and an orange in my kitchen. They’re both about the same size, 3 inches in diameter, so I have no idea why she chose different fruits – perhaps she wanted to avoid comparing apples and oranges.

Anyway, if you felt like doing the galaxy visualization you could put two apple galaxies on the floor about 12 feet apart and then line up about 50 apples between them. A fair amount of space for more galaxies.

To see inside a galaxy you could put one orange star in Miami FL, and its on-the-average nearest orange neighbor in Seattle WA. Then you could set out a long skinny row of just about 63 million oranges in between. Oh, and on this scale the nearest galaxy would be about 2 billion miles (or 43 quadrillion oranges) away. Way more not much inside a galaxy than between two neighboring ones.

So if we squeeze all those apples and oranges together we’d get rid of all the empty space, right?

Not by a long shot. Nearly all those stars are balls of very hot gas, which means they’re made up of atoms crossing empty space inside the star to collide with other atoms. Relative to the size of the atoms, how much empty space is there inside the star?

For example, every chemistry student learns that 6×10²³ molecules of any gas take up a volume of 22.4 liters at normal Earth temperature and pressure. For a single-atom gas like helium that works out to about 22 atom-widths between atoms.

Now think about emptiness inside the Sun. If it’s a typical star (which it is) and if all of its atoms are hydrogen (which they mostly are) and if the average density of the Sun (1408 kg/m³) applied all the way down to the center of the Sun (which it doesn’t), and if we believe NASA’s numbers for the Sun (hey, why not?), then the average density works out to about 0.7 atom-widths between neighbors.

So no empty space to squeeze out of the Sun, eh? Well, actually there is quite a lot, because those atoms are mostly empty space, too.

OK, I cheated up there about the Sun, because virtually all of the Sun’s atoms have been dissociated into separated electrons and nuclei. The nucleus is much smaller than than its atom – by a factor of 60,000 or so. Think of a grape seed in the middle of a football field.

To sum it upward, we’ve got a set of Russian matryoshka dolls, one inside the next. At the center is a collection of grape seeds, billions and billions of them, each in their own football field. The football fields are all balled into a stellar orange (or maybe an apple), but there are billions of those crammed into a galactic apple (or maybe an orange) that’s about ten feet away from the nearest other piece of fruit.

As Douglas Adams wrote in Hitchhiker’s Guide to The Galaxy,

“Space … is big. Really big. You just won’t believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space…”

The thing to realize is that the function of all that space is to keep everything from being in the same place. That’s important.

~~ Rich Olcott

Sir Isaac, The Atom And The Whirlpool

On January 11, 2016November 28, 2025 By rolcottIn Physics: Fundamentals, Physics: Newton and GravityLeave a comment

Newton definitely didn’t see that one coming. He has an excuse, though. No-one in in the 17th Century even realized that electricity is a thing, much less that the electrostatic force follows the same inverse-square law that gravity does. So there’s no way poor Isaac would have come up with quantum mechanics.

Lemme ‘splain. Suppose you have a mathematical model that’s good at predicting some things, like exactly where Jupiter will be next week. But if the model predicts an infinite value under some circumstances, that tells you it’s time to look for a new model for those particular circumstances.

For example, Newton’s Law of Gravity says that the force between two objects is proportional to 1/r², where r is the distance between their centers of mass. The Law does a marvelous job with stars and satellites but does the infinity thing when r approaches zero. In prior posts I’ve described some physics models that supercede Newton’s gravity law at close distances.

Electrical forces are same song second verse with a coda. They follow the 1/r² law, so they also have those infinity singularities. According to the force law, an electron (the ultimate “particle” of negative charge) that approaches another electron would feel a repulsion that rises to infinity. The coda is that as an electron approaches a positive atomic nucleus it would feel an attraction that rises to infinity. Nature abhors infinities, so something else, some new physics, must come into play.

I put that word “particle” in quotes because common as the electron-is-a-particle notion is, it leads us astray. We tend to think of the electron as this teeny little billiard-ballish thing, but it’s not like that at all. It’s also not a wave, although it sometimes acts like one. “Wavicle” is just a weasel-word. It’s far better to think of the electron as just a little traveling parcel of energy. Photons, too, and all those other denizens of the sub-atomic zoo.

An electron can’t crumble or leak mass or deform to merge the way that sizable objects can. What it does is smear. Quantum mechanics is all about the smear. Much more about that in later posts.

If Newton loved anything (and that question has been discussed at length), he loved an argument. His battle with Leibniz is legendary. He even fought with Descartes, who was a decade dead when Newton entered Cambridge.

Descartes had grabbed “Nature abhors a vacuum” from Aristotle and never let it go. He insisted that the Universe must be filled with some sort of water-like fluid. He know the planets went round the Sun despite the fluid getting in the way, so he reasoned they moved as they did because of the fluid.

Surely you once played with toy boats in the bathtub. You may have noticed that when you pulled your arm quickly through the water little whirlpools followed your arm. If a whirlpool encountered a very small boat, the boat might get caught in it and move in the same direction. Descartes held that the Solar System worked like that, with the Sun as your arm and the planets caught in Sun-stirred vortices within that watery fluid.

Newton knew that couldn’t be right. The planets don’t run behind the Sun, they share the same plane. Furthermore, comets orbit in from all directions. Crucially, Descartes’ theory conflicted with his own and that settled the matter for Newton. Much of Principia‘s “Book II” is about motions of and through fluid media. He laid out there what a trajectory would look like under a variety of conditions. As you’d expect, none of the paths do what planets, moons and comets do.

From Newton’s point of view, the only use for Book II was to demolish Descartes. For us in later generations, though, he’d invented the science of hydrodynamics.

Which was a good thing so long as you don’t go too far upstream towards the center of the whirlpool. As you might expect (or I wouldn’t even be writing this section), Book II is littered with 1/rⁿ formulas that go BLOOIE when the distances get short. What happens near the center? That’s where the new physics of turbulence kicks in.

~~ Rich Olcott

Squeezing past Newton’s infinity

On January 4, 2016March 17, 2020 By rolcottIn Astronomy: Planetary Science, Physics: Fundamentals, Physics: Newton and GravityLeave a comment

One of the most powerful moments in musical theater — Philip Quast in his Les Miz role of Inspector Javert, praising the stars for the steadfastness and reverence for law that they signify for him. The performance is well worth a listen.

Javert’s certitude came from Newton’s sublimely reliable mechanics — the notion that every star’s and planet’s motion is controlled by a single law, F~(1/r²). The law says that the attractive force between any pair of bodies is inversely proportional to the square of the distance between their centers. But as Javert’s steel-clad resolve hid a fatal spark of mercy towards Jean Valjean, so Newton’s clockworks hold catastrophe at their axles.

Newton’s gravity law has a problem. As the distance approaches zero, the predicted force approaches infinity. The law demands that nearby objects accelerate relentlessly at each other to collide with infinite force, after which their combined mass attracts other objects. In time, everything must collapse in a reverse of The Big Bang.

Victor Hugo wrote Les Misérables about 180 years after Newton published his Principia. A decade before Hugo’s book, Professeur Édouard Roche (pronounced rōsh) solved at least part of Newton’s problem.

Roche realized that Newton had made an important but crucial simplification. Early in the Principia, he’d proven that for many purposes you can treat an entire object as though all of its mass were concentrated at a single point (the “center of mass”). But in real gravity problems every particle of one object exerts an attraction for every particle of the other.

That distinction makes no difference when the two objects are far apart. However, when they’re close together there are actually two opposing forces in play:

gravity, which preferentially affects the closest particles, and
tension, which maintains the integrity of each structure.

contact_binary_1 — Binary star pair demonstrating Roche lobes, image courtesy of Cronodon.com

Roche noted that the gravity fields of any pair of objects must overlap. There will always be a point on the line between them where a particle will be tugged equally in either direction. If two bodies are close and one or both are fluid (gases and plasmas are fluid in this sense), the tension force is a weak competitor. The partner with the less intense gravity field will lose material across that bridge to the other partner. Binary star systems often evolve by draining rather than collision.

Now suppose both bodies are solid. Tension’s game is much stronger. Nonetheless, as they approach each other gravity will eventually start ripping chunks off of one or both objects. The only question is the size of the chunks — friable materials like ices will probably yield small flakes, as opposed to larger lumps made from silicates and other rocky materials. Roche described the final stage of the process, where the less-massive body shatters completely. The famous rings of Saturn and the less famous rings of Neptune, Uranus and Jupiter all appear to have been formed by this mechanism.

Roche was even able to calculate how close the bodies need to be for that final stage to occur. The threshold, now called the Roche Limit, depends on the size and mass of each body. You can get more detail here.

And then there’s spaghettification. That’s a non-relativistic tidal phenomenon that occurs near an extremely dense body like a neutron star or a black hole. Because these objects pack an enormous amount of mass into a very small volume, the force of gravity at a close-in point is significantly greater than the force just a little bit further out. Any object, say a Klingon Warbird that ignored peril markings on a space map (Klingons view warnings as personal challenges), would find itself stretched like a noodle between high gravity on the side near the black hole and lower gravity on the opposite side. (In this cartoon, notice how the stretching doesn’t care which way the pin-wheeling ship is pointed.)

Nature abhors singularities. Where a mathematical model like Newton’s gravity law predicts an infinity, Nature generally says, “You forgot something.” Newton assumed that objects collide as coherent units. Real bodies drain, crumble, or deform to slide together. Look to the apparent singularities to find new physics.

~~ Rich Olcott

Prime years and such

On December 28, 2015November 28, 2025 By rolcottIn General: Fun Stuff, Uncategorized3 Comments

I’ve liked 4s and 6s ever since when, but lately 3s and 7s have been cropping up. A lot. And they have a really weird connection with 2016.

For me New Year has always been an opportunity to inspect the upcoming year’s number for interesting properties.

Maybe the easiest way for a number to be interesting is to be prime, that is, not divisible by anything other than itself and one. My Uncle Harold once proved to me that all odd numbers are prime.

“One’s a prime, and so are three and five. How about seven? Seven’s prime. Nine? Not a prime but we can throw that one out as experimental error. Eleven? Prime. Thirteen? Prime. Case closed.”

They use that logic a lot in politics nowadays.

There are a few prime-ity tests that just need a quick glance. Take 2015 for example. Ends with a “5” so it’s got to be divisible by five. Not a prime. A number ending with a “0” is like ending with twice five so it’s not prime either.

Take 2016. Ends in an even digit so it’s divisible by two. Not a prime. Moreover, it fails the “nines test” — add up all the digits (2+0+1+6=9). If the total is nine or divisible by nine then the number itself is divisible by nine (and by three) so it’s non-prime. 2016 is also divisible by seven but that’s not as easy to diagnose.

That’s about it for quickies. Beyond those tests you have to slog through dividing the target by every prime number from three up to the target’s square root. Why stop there? Because any factor bigger than the square root will have a partner smaller than the square root.

Remember Party Like It’s 1999 (prime)? Very popular when the Artist Then Known As Prince produced it in 1982 (not a prime). Unfortunately, we who were working on Y2K projects were too busy to party that year so we couldn’t celebrate 1999 being prime until it was all over.

Y2K itself, 2000, definitely wasn’t prime. If you know that 1999 is prime you know 2000 can’t be because after you get past 1-2-3, no two adjacent numbers can be prime — one of them would have to be even. Next-but-one can work, though: both 1997 and 1999 are prime. Primes separated by two like that are twin primes.

If 2016 won’t be a prime year, is there another way it can be special? Hmmm… 2016 isn’t a perfect square, nor is it the sum of two squares. Neither its square nor its cube are particularly noteworthy, but the square PLUS the cube is kinda cute: their sum is 8,197,604,352 which contains every digit just once.

According to The On-Line Encyclopedia of Integer Sequences, 2016 is a hexagonal number. Start with a dot. Make that dot one corner of a hexagon of dots. Then add a hexagon around that, one more dot per side, keeping the original dot as a corner (like the plan for a starter motte-and-bailey castle)… Keep going until the outermost hexagon has 32 dots along each edge. All the hexagons together will have exactly 2016 dots.

The OEIS says that 2016 is a participant in at least 925 more special sequences, so I guess it’s a pretty cool number after all.

Those 3s and 7s? Here they come….

My nominee for Puzzle King of The World is my good friend Jimmy. I challenged him once to find the connection between

the British Army’s WWII section number (2701) for Alan Turing’s super-secret cryptography unit at Bletchley Park, and
Jean Valjean’s prisoner number (24601) in Les Misérables

Turns out it’s all about the primes. 2701 is the product of two primes: 73×37. 24601 is also the product of two primes 73×337. Better yet, both of the product expressions are palindromes in their digits (7337, 73337). To put whipped cream on top, I first noticed the connection during my 73rd year.

So then of course I went looking for other 3…7 and 7…3 primes. There aren’t a lot of them. Going all the way out to 10³⁷ I found:

37	73
337	733
(3,337 is 47×71, not a prime)	7,333
333,337	733,333

Pretty good symmetry there.

OK, back to number 2016. I asked Mathematica®, “How many different pairs of primes, like 1999 and 17, sum to 2016?”

What do you suppose the answer was? Yup, “73.”

Oh, and the next prime year is 2017. It’ll be great.

~~ Rich Olcott

The direction Newton avoided facing

On December 21, 2015July 26, 2025 By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

Reading Newton’s Philosophiæ Naturalis Principia Mathematica is less challenging than listening to Vogon poetry. You just have to get your head working like a 17th Century genius who had just invented Calculus and who would have deep-fried his right arm in rancid skunk oil before he’d admit to using any of his rival Leibniz’ math notations or techniques.

Newton was essentially a geometer. These illustrations (from Book 1 of the Principia) will give you an idea of his style. He’d set himself a problem then solve it by constructing sometimes elaborate diagrams by which he could prove that certain components were equal or in strict proportion.

For instance, in the first diagram (Proposition II, Theorem II), we see an initial glimpse of his technique of successive approximation. He defines a sequence of triangles which as they proliferate get closer and closer to the curve he wants to characterize.

The lines and trig functions escalate in the second diagram (Prop XII, Problem VII), where he calculates the force on a body traveling along a hyperbola.

The third diagram is particularly relevant to the point I’ll finally get to when I get around to it. In Prop XLIV, Theorem XIV he demonstrates something weird. Suppose two objects A and B are orbiting around attractive center C, but B is moving twice as fast as A. If C exerts an additional force on B that is inversely dependent on the cube of the B-C distance, then A‘s orbit will be a perfect circle (yawn) but B‘s will be an ellipse that rotates around C, even though no external force pushes it laterally.

In modern-day math we’d write the additional force as F∼(1/r_BC³), but Newton verbalized it as “in a triplicate ratio of their common altitudes inversely.” See what I mean about Vogon poetry?

Now, about that point I was going to get to. It’s C, in the center of that circle. If the force is proportional to 1/r³, what happens when r approaches zero? BLOOIE, the force becomes infinite.

In the previous post we used geometry to understand the optical singularity at the center of the Christmas ball. I said there that my modeling project showed me a deeper reason for a BLOOIE. That reason showed up partway through the calculation for the angle between the axis and the ring of reflected light. A certain ratio came out to be (1-x)/2x, where x is proportional to the distance between the LED and the ball’s center. Same problem: as the LED approaches the center, x approaches zero and BLOOIE. (No problem when x is one, because the ratio is 0/2 which is zero which is OK.)

Singularities happen when the formula for something goes to infinity.

Now, Newton recognized that his central-force (1/rⁿ)-type equations covered gravity and magnetism and even the inward force on the rim of a rotating wheel. It’s surprising that he didn’t seem too worried about BLOOIE.

I think he had two excuses. First, he was limited by his graphical methodology. In most of his constructions, when a certain distance goes to zero there’s a general catastrophe — rectangles and triangles collapse to lines or even points, radii whirl aimlessly without a vertex to aim at… His lovely derivations devolve into meaninglessness. Further advances would depend on the algebraic approach to Calculus taken by the detested Leibniz.

Second (here’s the hook for this post’s title), Newton was looking outward, not inward. He was considering the orbits of planets and other sizable objects. r is always the distance between object centers. For sizable objects you don’t have to worry about r=0 because “center-to-center equals zero” never occurs. If the Moon (radius 1080 miles) were to drop down to touch the Earth (radius 3960 miles), their centers would still be 5000 miles apart. No BLOOIE.

Actually, there would be CRUMBLE instead of BLOOIE because a different physical model would apply — but that’s a tale for another post.

The moral of the story is this. Mathematical models don’t care about infinities, but Nature does. Any conditions where the math predicts an infinite value (for instance, where a denominator can become zero) are prime territory for new models that make better predictions.

~~ Rich Olcott

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