Tag: Newton

Gravity’s Real Rainbow

On By rolcottIn Astronomy: Multi-messenger, Astronomy: Stellar Science, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Waves, UncategorizedLeave a comment

Some people are born to scones, some have scones thrust upon them.  As I stepped into his coffee shop this morning, Al was loading a fresh batch onto the rack.  “Hey, Sy, try one of these.”

“Uhh … not really my taste.  You got any cinnamon ones ready?”

“Not much for cheddar-habañero, huh?  I’m doing them for the hipster trade,” waving towards all the fedoras on the room.  “Here ya go.  Oh, Vinnie’s waiting for you.”

I navigated to the table bearing a pile of crumpled yellow paper, pulled up a chair.  “Morning, Vinnie, how’s the yellow writing tablet working out for you?”

“Better’n the paper napkins, but it’s nearly used up.”

“What problem are you working on now?”

“OK, I’m still on LIGO and still on that energy question I posed way back — how do I figure the energy of a photon when a gravitational wave hits it in a LIGO?  You had me flying that space shuttle to explain frames and such, but kept putting off photons.”

“Can’t argue with that, Vinnie, but there’s a reason.  Photons are different from atoms and such because they’ve got zero mass.  Not just nearly massless like neutrinos, but exactly zero.  So — do you remember Newton’s formula for momentum?”

“Yeah, momentum is mass times the velocity.”

“Right, so what’s the momentum of a photon?”

“Uhh, zero times speed-of-light.  But that’s still zero.”

“Yup.  But there’s lots of experimental data to show that photons do carry non-zero momentum.  Among other things, light shining on an an electrode in a vacuum tube knocks electrons out of it and lets an electric current flow through the tube.  Compton got his Nobel prize for that 1923 demonstration of the photoelectric effect, and Einstein got his for explaining it.”

“So then where’s the momentum come from and how do you figure it?”

“Where it comes from is a long heavy-math story, but calculating it is simple.  Remember those Greek letters for calculating waves?”

(starts a fresh sheet of note paper) “Uhh… this (writes λ) is lambda is wavelength and this (writes ν) is nu is cycles per second.”

“Vinnie, you never cease to impress.  OK, a photon’s momentum is proportional to its frequency.  Here’s the formula: p=h·ν/c.  If we plug in the E=h·ν equation we played with last week we get another equation for momentum, this one with no Greek in it:  p=E/c.  Would you suppose that E represents total energy, kinetic energy or potential energy?”

“Momentum’s all about movement, right, so I vote for kinetic energy.”

“Bingo.  How about gravity?”

“That’s potential energy ’cause it depends on where you’re comparing it to.”

light-in-a-gravity-well“OK, back when we started this whole conversation you began by telling me how you trade off gravitational potential energy for increased kinetic energy when you dive your airplane.  Walk us through how that’d work for a photon, OK?  Start with the photon’s inertial frame.”

“That’s easy.  The photon’s feeling no forces, not even gravitational, ’cause it’s just following the curves in space, right, so there’s no change in momentum so its kinetic energy is constant.  Your equation there says that it won’t see a change in frequency.  Wavelength, either, from the λ=c/ν equation ’cause in its frame there’s no space compression so the speed of light’s always the same.”

“Bravo!  Now, for our Earth-bound inertial frame…?”

“Lessee… OK, we see the photon dropping into a gravity well so it’s got to be losing gravitational potential energy.  That means its kinetic energy has to increase ’cause it’s not giving up energy to anything else.  Only way it can do that is to increase its momentum.  Your equation there says that means its frequency will increase.  Umm, or the local speed of light gets squinched which means the wavelength gets shorter.  Or both.  Anyway, that means we see the light get bluer?”

“Vinnie, we’ll make a physicist of you yet.  You’re absolutely right — looking from the outside at that beam of photons encountering a more intense gravity field we’d see a gravitational blue-shift.  When they leave the field, it’s a red-shift.”

“Keeping track of frames does make a difference.”

Al yelled over, “Like using tablet paper instead of paper napkins.”

~~ Rich Olcott

A Shift in The Flight

On By rolcottIn Gravitational astronomy, Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

I heard a familiar squeak from the floorboard outside my office.

“C’mon in, Vinnie, the door’s open.  What can I do for you?”

“I still got problems with LIGO.  I get that dark energy and cosmic expansion got nothin’ to do with it.  But you mentioned inertial frame and what’s that about?”

earth-moon“Does the Moon go around the Earth or does the Earth go around the Moon?”

“Huh?  Depends on where you are, I guess.”

“Well, there you are.”

“Waitaminnit!  That can’t be all there is to it!”

“You’re right, there’s more.  It all goes back to Newton’s First Law.”  (showing him my laptop screen)  “Here’s how Wikipedia puts it in modern terms…”

In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

“That’s really a definition rather than a Law.  If you’re looking at an object and it doesn’t move relative to you or else it’s moving at constant speed in a straight line, then you and the object share the same inertial frame.  If it changes speed or direction relative to you, then it’s in a different inertial frame from yours and Newton’s Laws say that there must be some force that accounts for the difference.”

“So another guy’s plane flying straight and level with me has a piece of my inertial frame?”

“Yep, even if you’re on different vectors.  You only lose that linkage if either airplane accelerates or curves off.”

“So how’s that apply to LIGO’s laser beams?  I thought light always traveled in straight lines.”

“It does, but what’s a straight line?”

“Shortest distance between two points — I been to flight school, Sy.”

“Fine.  So if you fly from London to Mexico City on this globe here you’d drill through the Earth?”mex-atl-jfk-lgw

“Of course not, I’d take the Great Circle route that goes through those two cities.  It’s the shortest flight path.  Hey, how ’bout that, the circle goes through NYC and Atlanta, too.”

“Cool observation, but that line looks like a curve from where I sit.”

“Yeah, but you’re not sittin’ close to the globe’s surface.  I gotta fly in the flight space I got.”

“So does light.  Photons always take the shortest available path, though sometimes that path looks like a curve unless you’re on it, too.  Einstein predicted that starlight passing through the Sun’s gravitational field would be bent into a curve.  Three years later, Eddington confirmed that prediction.”

“Light doesn’t travel in a straight line?”

“It certainly does — light’s path defines what is a straight line in the space the light is traveling through.  Same as your plane’s flight path defines that Great Circle route.  A gravitational field distorts the space surrounding it and light obeys the distortion.”

“You’re getting to that ‘inertial frames’ stuff, aren’t you?”

“Yeah, I think we’re ready for it.  You and that other pilot are flying steady-speed paths along two navigation beams, OK?”

“Navigation beams are radio-frequency.”

“Sure they are, but radio’s just low-frequency light.  Stay with me.  So the two of you are zinging along in the same inertial frame but suddenly a strong gravitational field cuts across just your beam and bends it.  You keep on your beam, right?”

“I suppose so.”

“And now you’re on a different course than the other plane.  What happened to your inertial frame?”

“It also broke away from the other guy’s.”

“Because you suddenly got selfish?”

“No, ’cause my beam curved ’cause the gravity field bent it.”

“Do the radio photons think they’re traveling a bent path?”

“Uh, no, they’re traveling in a straight line in a bent space.”

“Does that space look bent to you?”

“Well, I certainly changed course away from the other pilot’s.”

“Ah, but that’s referring to his inertial frame or the Earth’s, not yours.  Your inertial frame is determined by how those photons fly, right?  In terms of your frame, did you peel away or stay on-beam?”

“OK, so I’m on-beam, following a straight path in a space that looks bent to someone using a different inertial frame.  Is that it?”

“You got it.”

(sounds of departing footsteps and closing door)

“Don’t mention it.”

~~ Rich Olcott

Breathing Space

On By rolcottIn Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Fundamentals, Physics: Newton and Gravity, UncategorizedLeave a comment

It was December, it was cold, no surprise.  I unlocked my office door, stepped in and there was Vinnie, standing at the window.  He turned to me, shrugged a little and said, “Morning, Sy.”  That’s Vinnie for you.

“Morning, Vinnie.  What got you onto the streets this early?”

“I ain’t on the streets, I’m up here where it’s warm and you can answer my LIGO question.”

“And what’s that?”

“I read your post about gravitational waves, how they stretch and compress space.  What the heck does that even mean?”

gravwave
An array of coordinate systems
floating in a zero-gravity environment,
each depicting a local x, y, and z axis

“Funny thing, I just saw a paper by Professor Saulson at Syracuse that does a nice job on that.  Imagine a boxful of something real light but sparkly, like shiny dust grains.  If there’s no gravitational field nearby you can arrange rows of those grains in a nice, neat cubical array out there in empty space.  Put ’em, oh, exactly a mile apart in the x, y, and z directions.  They’re going to serve as markers for the coordinate system, OK?”

“I suppose.”

“Now it’s important that these grains are in free-fall, not connected to each other and too light to attract each other but all in the same inertial frame.  The whole array may be standing still in the Universe, whatever that means, or it could be heading somewhere at a steady speed, but it’s not accelerating in whole or in part.  If you shine a ray of light along any row, you’ll see every grain in that row and they’ll all look like they’re standing still, right?”

“I suppose.”

“OK, now a gravitational wave passes by.  You remember how they operate?”

“Yeah, but remind me.”

(sigh)  “Gravity can act in two ways.  The gravitational attraction that Newton identified acts along the line connecting the two objects acting on each other.  That longitudinal force doesn’t vary with time unless the object masses change or their distance changes.  We good so far?”
long-and-transverse-grav
“Sure.”

“Gravity can also act transverse to that line under certain circumstances.  Suppose we here on Earth observe two black holes orbiting each other.  The line I’m talking about is the one that runs from us to the center of their orbit.  As each black hole circles that center, its gravitational field moves along with it.  The net effect is that the combined gravitational field varies perpendicular to our line of sight.  Make sense?”

“Gimme a sec…  OK, I can see that.  So now what?”

“So now that variation also gets transmitted to us in the gravitational wave.  We can ignore longitudinal compression and stretching along our sight line.  The black holes are so far away from us that if we plug the distance variation into Newton’s F=m1m2/r² equation the force variation is way too small to measure with current technology.

“The good news is that we can measure the off-axis variation because of the shape of the wave’s off-axis component.  It doesn’t move space up-and-down.  Instead, it compresses in one direction while it stretches perpendicular to that, and then the actions reverse.  For instance, if the wave is traveling along the z-axis, we’d see stretching follow compression along the x-axis at the same time as we’d see compression following stretching along the y-axis.”

gravwave-2“Squeeze in two sides, pop out the other two, eh?”

“Exactly.  You can see how that affects our grain array in this video I just happen to have cued up.  See how the up-down and left-right coordinates close in and spread out separately as the wave passes by?”

“Does this have anything to do with that ‘expansion of the Universe’ thing?”

“Well, the gravitational waves don’t, so far as we know, but the notion of expanding the distance between coordinate markers is exactly what we think is going on with that phenomenon.  It’s not like putting more frosting on the outside of a cake, it’s squirting more filling between the layers.  That cosmological pressure we discussed puts more distance between the markers we call galaxies.”

“Um-hmm.  Stay warm.”

(sound of departing footsteps and door closing)

“Don’t mention it.”

~~ Rich Olcott

Gentle pressure in the dark

On By rolcottIn Cosmology, Gravitational astronomy, Physics: Black Holes and Relativity, Physics: Newton and Gravity, UncategorizedLeave a comment

“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.”de-vs-gravity

“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?”

“Force or potential energy?”

“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

Plutonic Goofyness

On By rolcottIn Astronomy: Planetary Science, UncategorizedLeave a comment
goofy-and-dp-pluto
Is Pluto wearing a space helmet?
No, that helmet is Pluto.
(Based on a cartoon by Andy Diehl)

Andy Diehl brings up a question worth considering over a tasty beverage.  How come Pluto’s a dog and Goofy’s a dog but Pluto gets the collar end of the leash?  Hardly seems fair.

Which brings us to that other controversial Pluto, the one that NASA’s New Horizon spacecraft visited last July.  (News flash — on 28 October, NASA announced that they’d received the very last of the data NH accumulated during that 2½-hour visit.)  Official Astronomy has reclassified Pluto from “planet” to “dwarf planet,” but NH honcho Alan Stern and much of the rest of the world say, “No way!”

The traditionalist position is, “But we’ve always called Pluto the ninth planet.”  Well, “always” only goes back to when the preternaturally persistent Clyde Tombaugh discovered the object in 1930.  At the time he found it Pluto was indeed the ninth “planet” out from the Sun.  However, it spends about 10% of each orbit* closer to the Sun than the eighth planet, Neptune.  So should we call it the “seven-and-a-fraction-th” planet?

No, because (1) that contravenes Official Astronomy’s rules, and (2) it’d be silly.

So what are the rules for what’s a planet?

  1. The object must be in orbit around its star.
  2. The object must be massive enough to be rounded by its own gravity.
  3. It must have cleared the neighborhood around its orbit.

“Rounded” is a bit tricky.  It doesn’t mean “spherical” because if you spin a  sphere, centrifugal forces move mass towards its equator.  Earth’s equator is 13.3 miles further away from its center than its poles are.  Miller’s Planet in the Interstellar movie is also a spheroid, even further deformed by elongation towards the black hole it orbits, yet it still rates as “rounded by its gravity” and qualifies as a planet.

“Clearing the neighborhood” means “my gravity dominates the motion of everything in my orbit.”  Earth and Jupiter, both acknowledged planets, each have retinues of asteroids in the Trojan positions, at the same distance from the Sun as the host planet but in regions 60º ahead of or behind it.  Even so, both planets often suffer messy encounters (remember Chicxulub and Chelyabinsk?) with asteroids and such that hadn’t gotten the memo.

Neptune meets all three criteria.  Its gravity dominates Pluto’s motion even though Pluto’s in a separate orbit.  For every three of Neptune’s trips around the Sun, Pluto makes exactly two.  The gravitational converse doesn’t hold, though.  Pluto’s mass is 0.1% of Neptune’s so the big guy doesn’t care.

pluto-orbits-1This video, from an Orbits Table display at the Denver Museum of Nature and Science, shows a different Plutonian weirdness.  We’re circling the Solar System at about 50 times Earth’s distance from the Sun (50 AU).  Reading inward, the white lines represent the orbits of Neptune, Uranus, Saturn and Jupiter.  The Asteroid Belt is the small greenish ring close to the Sun.  The four terrestrial planets are even further in.  The Kuiper Belt is the greenish ring that encloses the lot.

The  yellow-orange line is Pluto’s orbit.  Most of the Solar System lies within a thin pancake, the Plane of The Ecliptic. Pluto’s orbit is inclined 17º out of the Plane.  That’s odd.

Theory says that the System evolved from an eddy in a primordial cloud of dust and gas.  Gravity shrank that blob of stuff to form a disk at the eddy’s equator as it drew 99.9% of the system’s mass to form the Sun at the disk’s center.

Newton’s First Law is all about Conservation of Momentum.  When applied to circular motion, it says that if you’re whirling in a certain plane, you’ll continue whirling in that plane unless something knocks you out of that plane.  Hence, the Plane of The Ecliptic.

Pluto’s path is a puzzling challenge to the theory.  It was only a minor puzzle until the 1990’s when astronomers discovered a plethora of Pluto-type objects outside of Neptune’s orbit.  Most run way out of the Plane.  Worst is Eris, at inclination 44º .  Clearly, Pluto’s not special.  It belongs to a large tribe that Astrophysicists must explain if they’re to claim to  understand the Solar System.

~~ Rich Olcott

* – During its current 248-year orbit, Pluto was inside Neptune’s orbit between 1979 and 1999.

The Titanic Winds of Titan (And Venus)

On By rolcottIn Astronomy: Planetary Science, Uncategorized2 Comments

Last week we saw that the atmosphere of Saturn’s moon Titan wasn’t quite as weird as we thought.  But there another way it’s really weird, completely unlike Earth but yet very much like Venus.  Titan’s a superrotater, a world whose atmosphere circles the planet much faster than its surface does.

Let’s start with a relatively simple Earthside phenomenon, a hurricane.  Warm air rises, right?  When the warmth comes from bathtub-temperature sea-water, it’s wet warm air.  As the air rises it cools and releases the moisture as rain.  But the air can’t just keep rising forever or we’d squirt out all our atmosphere. So where does it go?

From a physicist’s perspective, that’s the key question.   If we can track/predict the path of a small parcel of air molecules through a weather system, then we’ve got at least a rough understanding of how that system works.

For the past half-century, atmosphere physicists have been engaged on a project grandly entitled the General Circulation Model (GCM), a software mash-up of the Ideal Gas Law, Newton’s Laws of motion, thermodynamic data for solid/liquid/gas transformations, the notoriously difficult Navier-Stokes equations for viscous fluids, and careful data management for input streams from thousands of disparate sources.  Oh, and it’s important that the Earth is a rotating spheroid rather than a flat plane.

hurricane-en-svg
How a tropical cyclone works
Illustration by Kevin Song, from Wikimedia Commons

Kevin Song’s diagram summarizes much of what we know about hurricanes.  An air packet rises until it hits the tropopause (the top edge of the troposphere), then expands horizontally.  While the packet’s spreading out, the planet’s rotation generates Coriolis “forces” that bend straight-line radial paths into the spirals we’ve seen so often in satellite photos.

A hurricane may look big on your weathercaster’s screen, but it’s less than 0.1% of Earth’s surface area.  Nonetheless, many of the same principles that drive a hurricane underlie global weather patterns.

wind-cells-and-jets-2Air warmed by the equatorial Sun rises, only to sink as it heads poleward.  Our packet loops between the Equator and about 30ºN (see the diagram).

Actually that loop is a slice through a big doughnut that stretches all the way around the Earth.  Another doughnut lies southward just below the Equator.  Two more pairs of doughnuts reside polewards of those as indicated by the other arrows in the diagram.  The doughnuts act like a set of interlocking gears, each reinforcing and moderating the motion of its neighbors.

Thanks to the same geometric phenomenon that spins a hurricane, air packets in these doughnuts don’t loop back to the points they started from.  The Earth turns under the packets as they journey, so each packet takes a spiraling tour around the planet.

Because of all those doughnuts, on average Earth wears a set of cloud-top necklaces.  Regions within 15º of the Equator are rain-forested, as are the Canadian and Siberian forested belts near 60ºN.  The world’s most prominent deserts cluster beneath the dry downdrafts near the 30º latitudes.  Jupiter, “the Easter egg planet,” gets its pink and blue bands from similar doughnuts except that Jupiter has room for many more of them.

Those green circles in the diagram are important, too.  They also represent Earth-circling doughnuts, but ones whose winds flow parallel to Earth’s surface rather than perpendicular to it.  The ones close to the surface give rise to the trade winds.  The high-altitude ones are the jet streams that steer storm systems and give the weathercasters something to talk about, especially in the wintertime.

Jet streams flow briskly — 60 to 200 mph, on a par with a middling hurricane.  Here’s a benchmark: Earth’s equatorial circumference is 25,000 miles, so Ecuadorian palm trees circle the planet at (25000 miles/24 hours)=1041 mph.  Our jet streams go about 15% of that.  Theory and GCM agree that the jets are powered by the Coriolis effect — spiraling air packets in the primary donuts cooperate to push jet stream air packets like oars on a galley ship.  That adds up.

Titan and Venus can’t possibly work that way.  Both of them rotate much more slowly than Earth (Titan about 30 mph, Venus only 4), so Coriolis forces are negligible.  But Titan’s jet streams do 75 mph and Venus’ race at 185.  What powers them?  The physicists are still arguing.

~~ Rich Olcott

Titan’s Atmosphere Is A Gas

On By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

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.earth-vx-titanTitan’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.∗

titan-boxes
Fred and neighbors

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.

Superluminal Superman

On By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Comic book and movie plotlines often make Superman accelerate up to lightspeed and travel backward in time.  Unfortunately, well-known fundamental Physics principles forbid that.  But suppose Green Lantern or Dr Strange could somehow magic him past the Lightspeed Barrier.  Would that let him do his downtimey thing?

Light_s hourglass
Light’s Hourglass

A quick review of Light’s Hourglass.  According to Einstein we live in 4D spacetime.  At any moment you’re at a specific time t relative to some origin time t=0 and a specific 3D location (x,y,z) relative to a spatial origin (0,0,0).  Your spacetime address is (ct,x,y,z) where c is the speed of light.  This diagram shows time running vertically into the future, plus two spatial coordinates x and y.  Sorry, I can’t get z into the diagram so pretend it’s zero.

The two cones depict all the addresses which can communicate with the origin using a flash of light.  Any point on either cone is at just the right distance d=√(x²+y²+z²) to match the distance that light can travel in time t.  The bottom cone is in the past, which is why we can see the light from old stars.  The top cone is in the future, which is why we can’t see light from stars that aren’t born yet.

If he obeys the Laws of Physics as we know them, Superman can travel anywhere he wants to inside the top cone.  He goes upward into the future at the rate of one second per second, just like anybody.  On the way, he can travel in space as far from (ct,0,0,0) as he likes so long as it’s not farther than the distance that light can travel the same route at his current t.

From our perspective, the Hourglass is a stack of circles (spheres in 3D space) centered on (ct,0,0,0).  From Supey’s perspective at time t he’s surrounded by a figure with radius ct that Physics won’t let him break through.  That’s his Lightspeed Barrier, like the Sonic Barrier but 900,000 times faster.

Suppose Green Lantern has magicked Supey up to twice lightspeed along the x-axis.  At moment t, he’s at (ct,2ct,0,0), twice as far as light can get.  In the diagram he’s outside the top cone but above the central disk.

Now GL pours on the power to accelerate Superman.  Each increment gets the Man of Steel closer to that disk.  He’s always “above” it, though, because he’s still moving into the future.  Only if he were to get to infinite speed could he reach the disk.

However, at infinite speed he’d go anywhere/everywhere instantaneously which would be confusing to even his Kryptonian intellect.  On the way he might run into things (stars, black holes,…) with literally zero time to react.

But the plotlines have Tall-Dark-and-Muscular flying into the past, breaching that disk and traveling downwards into the bottom cone.  Can GL make that happen?

Enter the Lorentz correction.  If you have rest mass m0 and you’re traveling at speed v, your effective mass is m=m0/√[1-(v/c)²]. That raises a couple of issues when you exceed lightspeed.

Suppose GL decelerates Superluminal Supey down towards lightspeed.  The closer he approaches c from higher speeds, the smaller that square root gets and the greater the effective mass.  It’s the same problem Superman faced when accelerating up to lightspeed.  That last mile per second down to c requires an infinite amount of braking energy — the Lightspeed Barrier is impermeable in both directions.

The other problem is that if v>c there’s a negative number inside that square root.    Above lightspeed, your effective mass becomes Bombelli-imaginary.  Remember Newton’s famous F=m·a?  Re-arrange it to a=F/m.  A real force applied to an object with imaginary mass produces an imaginary acceleration.  “Imaginary” in Physics generally means “perpendicular in some sense” and remember we’re in 4D here with time perpendicular to space.

GL might be able to shove Superman downtime, but he’d have to

  1. squeeze inward at hiper-lightspeed with exactly the same force along all three spatial dimensions, to make sure that “perpendicular” is only along the time axis
  2. start Operation Squish at some time in his own future to push towards the past.

Nice trick.  Would Superman buy in?

~~ Rich Olcott

The question Newton couldn’t answer

On By rolcottIn Physics: Black Holes and Relativity, Physics: Newton and Gravity, UncategorizedLeave a comment

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.

Fabric of Space 4a

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.FoS wave

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. Fabric of Space 4b

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

On By rolcottIn Physics: Fundamentals, Physics: Newton and Gravity, Uncategorized3 Comments

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 triangle
A fictitious triangle

“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=m·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=m·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.

CoasterBut 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.

Side forceSuppose 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