# Gentle pressure in the dark

“C’mon in, the door’s open.”

Vinnie clomps in and he opens the conversation with, “I don’t believe that stuff you wrote about LIGO.  It can’t possibly work the way they say.”

“Well, sir, would you mind telling me why you have a problem with those posts?”  I’m being real polite, because Vinnie’s a smart guy and reads books.  Besides, he’s Vinnie.

“I’m good with your story about how Michelson’s interferometer worked and why there’s no æther.  Makes sense, how the waves mess up when they’re outta step.  Like my platoon had to walk funny when we crossed a bridge.  But the gravity wave thing makes no sense.  When a wave goes by maybe it fiddles space but it can’t change where the LIGO mirrors are.”

“Gravitational wave,” I murmur, but speak up with, “What makes you think that space can move but not the mirrors?”

“I seen how dark energy spreads galaxies apart but they don’t get any bigger.  Same thing must happen in the LIGO machine.”

“Not the same, Vinnie.  I’ll show you the numbers.”

“Ah, geez, don’t do calculus at me.”

“No, just arithmetic we can do on a spreadsheet.” I fire up the laptop and start poking in  astronomical (both senses) numbers.  “Suppose we compare what happens when two galaxies face each other in intergalactic space, with what happens when two stars face each other inside a galaxy.  The Milky Way’s my favorite galaxy and the Sun’s my favorite star.  Can we work with those?”

“Yeah, why not?”

“OK, we’ll need a couple of mass numbers.  The Sun’s mass is… (sound of keys clicking as I query Wikipedia) … 2×1030 kilograms, and the Milky Way has (more key clicks) about 1012 stars.  Let’s pretend they’re all the Sun’s size so the galaxy’s mass is (2×1030)×1012 = 2×1042 kg. Cute how that works, multiplying numbers by adding exponents, eh?”

“Cute, yeah, cute.”  He’s getting a little impatient.

“Next step is the sizes.  The Milky Way’s radius is 10×104 lightyears, give or take..  At 1016 meters per lightyear, we can say it’s got a radius of 5×1020 meters.  You remember the formula for the area of a circle?”

“Sure, it’s πr2.” I told you Vinnie’s smart.

“Right, so the Milky Way’s area is 25π×1040 m2.  Meanwhile, the Sun’s radius is 1.4×109 m and its cross-sectional area must be 2π×1018 m2.  Are you with me?”

“Yeah, but what’re we doing playing with areas?  Newton’s gravity equations just talk about distances between centers.”  I told you Vinnie’s smart.

“OK, we’ll do gravity first.  Suppose we’ve got our Milky Way facing another Milky Way an average inter-galactic distance away.  That’s about 60 galaxy radii,  about 300×1020 meters.  The average distance between stars in the Milky Way is about 4 lightyears or 4×1016 meters.  (I can see he’s hooked so I take a risk)  You’re so smart, what’s that Newton equation?”

“Alright, I’m impressed.  Let’s go for force.”

“Force equals Newton’s G times the product of the masses divided by the square of the distance.”

“Full credit, Vinnie.  G is about 7×10-11 newton-meter²/kilogram², so we’ve got a gravity force of (typing rapidly) (7×10-11)×(2×1042)×(2×1042)/(300×1020)² = 3.1×1029 N for the galaxies, and (7×10-11)×(2×1030)×(2×1030)/(4×1016)² = 1.75×1017 N for the stars.  Capeesh?”

“Yeah, yeah.  Get on with it.”

“Now for dark energy.  We don’t know what it is, but theory says it somehow exerts a steady pressure that pushes everything away from everything.  That outward pressure’s exerted here in the office, out in space, everywhere.  Pressure is force per unit area, which is why we calculated areas.

“But the pressure’s really, really weak.  Last I saw, the estimate’s on the order of 10-9 N/m².  So our Milky Way is pushed away from that other one by a force of (10-9)×(25π×1040) ≈ 1031 N, and our Sun is pushed away from that other star by a force of (10-9)×(2π×1018) ≈ 1010 N with rounding.  Here, look at the spreadsheet summary…”

Force, newtons Between Galaxies Between stars
Gravity 3.1×1029 1.75×1017
Dark energy 1031 1010
Ratio 3.1×10-2 17.5×106

“So gravity’s force pulling stars together is 18 million times stronger than dark energy’s pressure pushing them apart.  That’s why the galaxies aren’t expanding.”

“Gotta go.”

(sound of door-slam )

“Don’t mention it.”

~~ Rich Olcott

# Titan’s Atmosphere Is A Gas

One year ago I kicked off these weekly posts with some speculations about how Life might exist on Saturn’s moon Titan. My surmises were based on reports from NASA’s Cassini-Huygens mission, plus some Physical Chemistry expectations for Titan’s frigid non-polar mix of liquid ethane and methane. Titan offers way more fun than that.

The environment on Titan is different from everything we’re used to on Earth. For instance, the atmosphere’s weird.Titan’s atmosphere is heavy-duty compared with Earth’s — 6 times deeper and about 1½ times the surface pressure. When I read those numbers I thought, “Huh? But Titan’s diameter is only 40% as big as Earth’s and its surface gravity is only 10% of ours. How come it’s got such a heavy atmosphere?”

Wait, what’s gravity got to do with air pressure? (I’m gonna use “air pressure” instead of “surface atmospheric pressure” because typing.) Earth-standard sea level air pressure is 14.7 pounds of force per square inch. That 14.7 pounds is the total weight of the air molecules above each square inch of surface, all the way out to space.

(Fortunately, air’s a hydraulic fluid so its pressure acts on sides as well as tops. Otherwise, a football’s shape would be even stranger than it is.)

Newton showed us that weight (force) is mass times the the acceleration of gravity. Gravity on Titan is 1/10 as strong as Earth’s, so an Earth-height column of air on Titan should weigh about 1½ pounds.

But Titan’s atmosphere (measured to the top of each stratosphere) goes out 6 times further than Earth’s. If we built out that square-inch column 6 times taller, it’d weigh only 9 pounds on Titan, well shy of the 22 pounds the Huygens lander measured. Where does the extra weight come from?

My first guess was, heavy molecules. If gas A has molecules that are twice as heavy as gas B’s, then a given volume of A would weigh twice as much as the same volume of B. An atmosphere composed of A will press down on a planet’s surface twice as hard as an atmosphere composed of B.

Good guess, but doesn’t apply. Earth’s atmosphere is 78% N2 (molecular weight 28) and 21% O2 (molecular weight 32) plus a teeny bit of a few other things. Their average molecular weight is about 29. Titan’s atmosphere is 98% N2 so its average molecular weight (28) is virtually equal to Earth’s. So no, those tarry brown molecules that block our view of Titan’s surface aren’t numerous enough to account for the high pressure.

My second guess is closer to the mark, I think. I remembered the Ideal Gas Law, the one that says, “pressure times volume equals the number of molecules times a constant times the absolute temperature.” In symbols, P·V=n·R·T.

Visualize one gas molecule, Fred, bouncing around in a cube sized to match the average volume per molecule, V/n=R·T/P. If Fred goes outside his cube in any direction he’s likely to bang into an adjacent molecule. If Fred has too much contact with his neighbors they’ll all stick together and become a liquid or solid.

The equation tells us that if the pressure doesn’t change, the size of Fred’s cube rises with the temperature. Just for grins I calculated the cube’s size for standard Earth conditions: (22.4 liters/mole)×(1 cubic meter/1000 liters)×(1 mole/6.02×1023 molecules)=37.2×10-27 cubic meter/molecule. The cube root of that is the length of the cube’s edge — 3.3 nanometers, about 8.3 times Fred’s 0.40-nanometer diameter.

Earth-standard surface temperature is about 300°K (absolute temperatures are measured in Kelvins). Titan’s surface temperature is only 94°K. On Titan that cube-edge would be 8.3*(94/300)=2.6 times Fred’s diameter — if air pressure were Earth-standard.

But really Titan’s air pressure is 1.5 times higher because its column is so tall and contains so much gas. The additional pressure squeezes Fred’s cube-edge down to 2.6*(1/1.5)=1.8 times his diameter. Still room enough for Fred to feel well-separated from his neighbors and continue acting like a proper gas.

The primary reason Titan’s atmosphere is so dense is that it’s chilly up there. Also, there’s a lot of Freds.

~~ Rich Olcott

– For the technorati… The cube-root of the Van der Waals volume for N2. And yeah, I know I’m almost writing about Mean Free Path but I think the development’s simpler this way.

# The question Newton couldn’t answer

250 years ago, when people were getting used to the idea that the planets circle the Sun and not the other way around, they wondered how that worked.  Isaac Newton said, “I can explain it with my Laws of Motion and my Law of Gravity.”

The first Law of Motion is that an object will move in a straight line unless acted upon by a force.  If you’re holding a ball by a string and swing the ball in a circle, the reason the ball doesn’t fly away is that the string is exerting a force on the ball.  Using Newton’s Laws, if you know the mass of the ball and the length of the string, you can calculate how fast the ball moves along that circle.

Newton said that the Solar System works the same way.  Between the Sun and each planet there’s an attractive force which he called gravity.  If you can determine three points in a planet’s orbit, you can use the Laws of Motion and the Law of Gravity to calculate the planet’s speed at any time, how close it gets to the Sun, even how much the planet weighs.

Astronomers said, “This is wonderful!  We can calculate the whole Solar System this way, but… we don’t see any strings.  How does gravity work?”

Newton was an honest man.  His response was, “I don’t know how gravity works.  But I can calculate it and that should be good enough.”

And that was good enough for 250 years until Albert Einstein produced his Theories of Relativity.  This graphic shows one model of Einstein’s model of “the fabric of space.”  According to the theory, light (the yellow threads) travels at 186,000 miles per second everywhere in the Universe.

As we’ve seen, the theory also says that space is curved and compressed near a massive object.  Accordingly, the model’s threads are drawn together near the dark circle, which could represent a planet or a star or a black hole.  If you were standing next to a black hole (but not too close). you’d feel fine because all your atoms and the air you breathe would shrink to the same scale.  You’d just notice through your telescope that planetary orbits and other things in the Universe appear larger than you expect.

This video shows how a massive object’s space compression affects a passing light wave.  The brown dot and the blue dot both travel at 186,000 miles per second, but “miles are shorter near a black hole.”  The wave’s forward motion is deflected around the object because the blue dot’s miles are longer than the miles traveled by the brown dot.

When Einstein presented his General Theory of Relativity in 1916, his calculations led him to predict that this effect would cause a star’s apparent position to be altered by the Sun’s gravitational field.

An observer at the bottom of this diagram can pinpoint the position of star #1 by following its light ray back to the star’s location.  Star #2, however, is so situated that its light ray is bent by our massive object.  To the observer, star #2’s apparent position is shifted away from its true position.

In 1919, English physicist-astronomer Arthur Eddington led an expedition to the South Atlantic to test Einstein’s prediction.  Why the South Atlantic?  To observe the total eclipse of the sun that would occur there.  With the Sun’s light blocked by the Moon, Eddington would be able to photograph the constellation Taurus behind the Sun.

Sure enough, in Eddington’s photographs the stars closest to the Sun were shifted in their apparent position relative to those further way.  Furthermore, the sizes of the shifts were almost embarrassingly close to Einstein’s predicted values.

Eddington presented his photographs to a scientific conference in Cambridge and thus produced the first public confirmation of Einstein’s theory of gravity.

Wait, how does an object bending a light ray connect with that object’s pull on another mass?  Another piece of Einstein’s theory says that if a light ray and a freely falling mass both start from the same point in spacetime, both will follow the same path through space.  American physicist John Archibald Wheeler said, “Mass bends space, and bent space tells mass how to move.”

~~ Rich Olcott

# Gravity and other fictitious forces

In this post I wrote, “gravitational force is how we we perceive spatial curvature.”
Here’s another claim — “Gravity is like centrifugal force, because they’re both fictitious.”   Outrageous, right?  I mean, I can feel gravity pulling down on me now.  How can it be fictional?

“Fictitious,” not “fictional,” and there’s a difference.  “Fictional” doesn’t exist, but a fictitious force is one that, to put it non-technically, depends on how you look at it.

Newton started it, of course.  From our 21st Century perspective, it’s hard to recognize the ground-breaking impact of his equation F=a.  Actually, it’s less a discovery than a set of definitions.  Its only term that can be measured directly is a, the acceleration, which Newton defined as any change from rest or constant-speed straight-line motion.  For instance, car buffs know that if a vehicle covers a one-mile half-mile (see comments) track in 60 seconds from a standing start, then its final speed is 60 mph (“zero to sixty in sixty”).  Furthermore, we can calculate that it achieved a sustained acceleration of 1.47 ft/sec2.

Both F and m, force and mass, were essentially invented by Newton and they’re defined in terms of each other.  Short of counting atoms (which Newton didn’t know about), the only routes to measuring a mass boil down to

• compare it to another mass (for instance, in a two-pan balance), or
• quantify how its motion is influenced by a known amount of force.

Conversely, we evaluate a force by comparing it to a known force or by measuring its effect on a known mass.

Once the F=a. equation was on the table, whenever a physicist noticed an acceleration they were duty-bound to look for the corresponding force.  An arrow leaps from the bow?  Force stored as tension in the bowstring.  A lodestone deflects a compass needle?  Magnetic force.  Objects accelerate as they fall?  Newton identified that force, called it “gravity,” and showed how to calculate it and how to apply it to planets as well as apples.  It was Newton who pointed out that weight is a measure of gravity’s force on a given mass.

Incidentally, to this day the least accurately known physical constant is Newton’s G, the Universal Gravitational Constant in his equation F=G·m1·m2/r2.  We can “weigh” planets with respect to each other and to the Sun, but without an independently-determined accurate mass for some body in the Solar System we can only estimate G.  We’ll have a better value when we can see how much rocket fuel it takes to push an asteroid around.

But there are other accelerations that aren’t so easily accounted for.  Ever ride in a car going around a curve and find yourself almost flung out of your seat?  This little guy wasn’t wearing his seat belt and look what happened.  The car accelerated because changing direction is an acceleration due to a lateral force.  But the guy followed Newton’s First Law and just kept going in a straight line.  Did he accelerate?

This is one of those “depends on how you look at it” cases.  From a frame of reference locked to the car (arrows), he was accelerated outwards by a centrifugal force that wasn’t countered by centripetal force from his seat belt.  However, from an earthbound frame of reference he flew in a straight line and experienced no force at all.

Suppose you’re investigating an object’s motion that appears to arise from a new force you’d like to dub “heterofugal.”  If you can find a different frame of reference (one not attached to the object) or otherwise explain the motion without invoking the “new force,” then heterofugalism is a fictitious force.

Centrifugal and centripetal forces are fictitious.  The  “force” “accelerating” one plane towards another as they both fly to the North Pole in this tale is actually geometrical and thus also fictitious   So is gravity.

In this post you’ll find a demonstration of gravity’s effect on the space around it.  Just as a sphere’s meridians give the effect of a fictitious lateral force as they draw together near its poles, the compressive curvature of space near a mass gives the effect of a force drawing other masses inward.

~~ Rich Olcott

# Gargh, His Heirs, and the AAAD Problem

Gargh, proto-humanity’s foremost physicist 2.5 million years ago, opened a practical investigation into how motion works.  “I throw rock, hit food beast, beast fall down yes.  Beast stay down no.  Need better rock.”  For the next couple million years, we put quite a lot of effort into making better rocks and better ways to throw them.  Less effort went into understanding throwing.

There seemed to be two kinds of motion.  The easier kind to understand was direct contact — “I push rock, rock move yes.  Rock stop move when rock hit thing that move no.”  The harder kind was when there wasn’t direct contact — “I throw rock up, rock hit thing no but come back down.  Why that?

Gargh was the first but hardly the last physicist to puzzle over the Action-At-A-Distance problem (a.k.a. “AAAD”).  Intuition tells us that between pusher and pushee there must be a concrete linkage to convey the push-force.  To some extent, the history of physics can be read as a succession of solutions to the question, “What linkage induces this apparent case of AAAD?”

Most of humanity was perfectly content with AAAD in the form of magic of various sorts.  To make something happen you had to wish really hard and/or depend on the good will of some (generally capricious) elemental being.

Aristotle wasn’t satisfied with anything so unsystematic.  He was just full of theories, many of which got in each other’s way.  One theory was that things want to go where they’re comfortable  because of what they’re made of — stones, for instance, are made of earth so naturally they try to get back home and that’s why we see them fall downwards (no concrete linkage, so it’s still AAAD).

Unfortunately, that theory didn’t account for why a thrown rock doesn’t just fall straight down but instead goes mostly in the direction it’s thrown.  Aristotle (or one of his followers) tied that back to one of his other theories, “Nature hates a vacuum.”  As the rock flies along, it pushes the air aside (direct contact) and leaves a vacuum behind it. More air rushes in to fill the vacuum and pushes the rock ahead (more direct contact).

We got a better (though still AAAD) explanation in the 17th Century when physicists invented the notions of gravity and inertia.

Newton made a ground-breaking claim in his Principia.  He proposed that the Solar System is held together by a mysterious AAAD force he called gravity.  When critics asked how gravity worked he shrugged, “I do not form hypotheses” (though he did form hypotheses for light and other phenomena).

Inertia is also AAAD.  Those 17th Century savants showed that inertial forces push mass towards the Equator of a rotating object.  An object that’s completely independent of the rest of the Universe has no way to “know” that it’s rotating so it ought to be a perfect sphere.  In fact, the Sun and each of its planets are wider at the equator than you’d expect from their polar diameters.  That non-sphere-ness says they must have some AAAD interaction with the rest of the Universe.  A similar argument applies to linear motion; the general case is called Mach’s Principle.

The ancients knew of the mysterious AAAD agents electricity and its fraternal twin, magnetism.  However, in the 19th Century James Clerk Maxwell devised a work-around.  Just as Newton “invented” gravity, Maxwell “invented” the electromagnetic field.  This invisible field isn’t a material object.  However, waves in the field transmit electromagnetic forces everywhere in the Universe.  Not AAAD, sort of.

It wasn’t long before someone said, “Hey, we can calculate gravity that way, too.”  That’s why we now speak of a planet’s gravitational field and gravitational waves.

But the fields still felt like AAAD because they’re not concrete.  Some modern physicists stand that objection on its head.  Concrete objects, they say, are made of atoms which themselves are nothing more than persistent fluctuations in the electromagnetic and gravitational fields.  By that logic, the fields are what’s fundamental — all motion is by direct contact.

Einstein moved resolutely in both directions.  He negated gravity’s AAAD-ness by identifying mass-contorted space as the missing linkage.  On the other hand, he “invented” quantum entanglement, the ultimate spooky AAAD.

~~ Rich Olcott

# LIGO: Gravity Waves Ain’t Gravitational Waves

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.

A 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.

An 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