Category: Physics: Newton and Gravity

LIGO: Gravity Waves Ain’t Gravitational Waves

On By rolcottIn Astronomy: Techniques, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Foundations of Measurement, Physics: Newton and Gravity, UncategorizedLeave a comment

Sometimes the media get sloppy.  OK, a lot of times, especially when the reporters don’t know what they’re writing about.  Despite many headlines that “LIGO detected gravity waves,” that’s just not so.  In fact, the LIGO team went to a great deal of trouble to ensure that gravity waves didn’t muck up their search for gravitational waves.

Spring2A wave happens in a system when a driving force and a restoring force take turns overshooting an equilibrium point AND the away-from-equilibrium-ness gets communicated around the system.  The system could be a bunch of springs tied together in a squeaky old bedframe, or labor and capital in an economic system, or the network of water molecules forming the ocean surface, or the fibers in the fabric of space (whatever those turn out to be).

If you  were to build a mathematical model of some wavery system you’d have to include those two forces plus quantitative descriptions of the thingies that do the moving and communicating.  If you don’t add anything else, the model will predict motion that cycles forever.  In reality, of course, there’s always something else that lets the system relax into equilibrium.

The something else could be a third force, maybe someone sitting on the bed, or government regulation in an economy, or reactant depletion for a chemical process.  But usually it’s friction of one sort or another — friction drains away energy of motion and converts it to heat.  Inside a spring, for instance, adjacent crystallites of metal rub against each other.  There appears to be very little friction in space — we can see starlight waves that have traveled for billions of years.

Physicists pay attention to waves because there are some general properties that apply to all of them.  For instance, in 1743 Jean-Baptiste le Rond d’Alembert proved there’s a strict relationship between a wave’s peakiness and its time behavior.  Furthermore, Jean-Baptiste Joseph Fourier (pre-Revolutionary France must have been hip-deep in physicist-mathematicians) showed that a wide variety of more-or-less periodic phenomena could be modeled as the sum of waves of differing frequency and amplitude.

Monsieur Fourier’s insight has had an immeasurable impact on our daily lives.  You can thank him any time you hear the word “frequency.”  From broadcast radio and digitally recorded music to time-series-based business forecasting to the mode-locked lasers in a LIGO device — none would exist without Fourier’s reasoning.

Gravity waves happen when a fluid is disturbed and the restoring force is gravity.  We’re talking physicist fluid here, which could be sea water or the atmosphere or solar plasma, anything where the constituent particles aren’t locked in place. Winds or mountain slopes or nuclear explosions push the fluid upwards, gravity pulls it back, and things wobble until friction dissipates that energy.

Gravitational waves are wobbles in gravity itself, or rather, wobbles in the shape of space.  According to General Relativity, mass exerts a tension-like force that squeezes together the spacetime immediately around it.  The more mass, the greater the tension.

Binary BH with AENAn isolated black hole is surrounded by an intense gravitational field and a corresponding compression of spacetime.  A pair of black holes orbiting each other sends out an alternating series of tensions, first high, then extremely high, then high…

Along any given direction from the pair you’d feel a pulsing gravitational field that varied above and below the average force attracting you to the pair.  From a distance and looking down at the orbital plane, if you could see the shape of space you’d see it was distorted by four interlocking spirals of high and low compression, all steadily expanding at the speed of light.

The LIGO team was very aware that the signal of a gravitational wave could be covered up by interfering signals from gravity waves — ocean tides, Earth tides, atmospheric disturbances, janitorial footsteps, you name it.  The design team arrayed each LIGO site with hundreds of “seismometers, accelerometers, microphones, magnetometers, radio receivers, power monitors and a cosmic ray detector.”  As the team processed the LIGO trace they accounted for artifacts that could have come from those sources.

So no, the LIGO team didn’t discover gravity waves, we’ve known about them for a century.  But they did detect the really interesting other kind.

~~ Rich Olcott

Sir Isaac, The Atom And The Whirlpool

On By rolcottIn Physics: Fundamentals, Physics: Newton and GravityLeave a comment

Newton and atomNewton 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.

Newton in whirlpoolIf 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 By rolcottIn Astronomy: Planetary Science, Physics: Fundamentals, Physics: Newton and GravityLeave a comment

One of the most powerful moments in musical theater — Philip Quast Quastin 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.

Klingon3And 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

The direction Newton avoided facing

On 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 II-II ellipseNewton 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.

Newton XII-VII hyperbolaFor 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.

Newton XLIV-XIV precessionThe 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/rBC3), 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