Ya got potential, kid, but how much?

Dusk at the end of January, not my favorite time of day or year.  I was just closing up the office when I heard a familiar footstep behind me.  “Hi, Vinnie.  What’s up?”

“Energy, Sy.”

“Energy?”

“Energy and LIGO.  Back in flight school we learned all about trading off kinetic energy and potential energy.  When I climb I use up the fuel’s chemical energy to gain gravitational potential energy.  When I dive I convert gravitational potential energy into  kinetic energy ’cause I speed up.  Simple.”

“So how do you think that ties in with LIGO?”

“OK, back when we pretended we was in those two space shuttles (which you sneaky-like used to represent photons in a LIGO) and I got caught in that high-gravity area where space is compressed, we said that in my inertial frame I’m still flying at the same speed but in your inertial frame I’ve slowed down.”

“Yeah, that’s what we worked out.”

“Well, if I’m flying into higher gravity, that’s like diving, right, ’cause I’m going where gravity is stronger like closer to the Earth, so I’m losing gravitational potential energy.  But if I’m slowing down I’ve gotta be losing kinetic energy, too, right?  So how can they both happen?  And how’s it work with photons?”

“Interesting questions, Vinnie, but I’m hungry.  How about some dinner?”shuttle-escape-1

We took the elevator down to Eddie’s pizza joint on the second floor.  I felt heavier already.  We ordered, ate and got down to business.

“OK, Vinnie.  Energy with photons is different than with objects that have mass, so let’s start with the flying-objects case.  How do you calculate gravitational potential energy?”

“Like they taught us in high school, Sy, ‘little g’ times mass times the height, and ‘little g’ is some number I forget.”

“Not a problem, we’ll just suppose that ‘little g’ times your plane’s mass is some convenient number, like 1,000.  So your gravitational potential energy is 1000×height, where the height’s in feet and the unit of energy is … call it a fidget.  OK?”

“Saves having to look up that number.”

sfo-to-den
Vinnie’s route, courtesy of Google Earth

“Fine.  Let’s suppose you’re flying over San Francisco Bay and your radar altimeter reads 20,000 feet.  What’s your gravitational potential energy?”

“Uhh… twenty million fidgets.”

“Great.  You maintain level flight to Denver.  As you pass over the Rockies you notice your altimeter now reads 6,000 feet because of that 14,000-foot mountain you’re flying over.  What’s your gravitational potential energy?”

“Six million fidgets.  Or is it still twenty?”

“Well, if God forbid you were to drop out of the sky, would you hit the ground harder in California or Colorado?”

“California, of course.  I’d fall more than three times as far.”

“So what you really care about isn’t some absolute amount of potential energy, it’s the relative amount of smash you experience if you fall down this far or that far.  ‘Height’ in the formula isn’t some absolute height, it’s height above wherever your floor is.  Make sense?”

“Mm-hm.”

“That’s an essential characteristic of potential energy — electric, gravitational, chemical, you name it.   It’s only potential.  You can’t assign a value without stating the specific transition you’re interested in.  You don’t know voltages in a circuit until you put a resistance between two specific points and meter the current through it.  You don’t know gravitational potential energy until you decide what location you want to compare it with.”

“And I suppose a uranium atom’s nuclear energy is only potential until a nuke or something sets it off.”

“You got the idea.  So, when you flew into that high-gravity compressed-space sector, what happened to your gravitational potential energy?”

“Like I said, it’s like I’m in a dive so I got less, right?”

“Depends on what you’re going to fall onto, doesn’t it?”

“No, wait, it’s definitely less ’cause I gotta use energy to fly back out to flat space.”

“OK, you’re comparing here to far away.  That’s legit.  But where’s that energy go?”

“Ahh, you’re finally getting to the kinetic energy side of my question –”

“Whoa, look at the time!  Got a plane to catch.  We’ll pick this up next week.  Bye.”

“Hey, Sy, your tab! …  Phooey, stuck for it again.”

~~ Rich Olcott

Life and energy on Titan, maybe

Say you’re an astrobiologist tasked with designing a world that would be able to support life we’d be able to recognize as such.  What absolute essentials would you need to include?

Abundant liquid water?  Biologists have found algae thriving inside desert rocks, moistened only by dew seeping in through microscopic pores.  A comfortable temperature?  We’ve found bacteria living in environments as cold as 5ºF and as warm as 250ºF.  A solid surface to grow on?  Arthur C Clarke (A Meeting with Medusa) wrote about complex life-forms floating in the 3,000-mile-deep atmosphere of Jupiter.  OK, that’s science fiction, but Clarke’s the guy who invented geostationary satellites for telecommunications and GPS.gibbs-energies

Many scientists would say that the obvious essential is a source of chemical energy.  I’d add, “and an efficient mechanism to convert the source energy to a form that can be transported within an organism.”  To my knowledge, all life-forms now on Earth have met the second prerequisite by using the ATP molecule for intra-cellular energy transport.  But life has been amazingly creative in finding ways to build those ATPs.  The tall diagram lists some biologic energy sources in decreasing order of how much energy is released.

All the Biology textbooks tell us that Earth’s energy cycle starts with the Sun.  Solar photons energize plant photosynthesis which creates loads of ATP molecules.  Some of them power a multistep process which combines CO2 and H2O to release O2 and create carbohydrates (CH2O)x.  (Glucose, for instance, is (CH2O)6.  Guess where the term “carbohydrate” came from.)  Earth’s biologic carbon cycle completes when other life “burns” carbohydrates to exploit the energy stored therein.  On this chart, “burn” means “combine with O2” and usually doesn’t involve fire.

Notice that “Make (CH2O)x” is at the bottom of the chart — that process absorbs a lot of energy per carbon atom.  Conversely, “Burn (CH2O)x” releases energy which is why we like sugar too much.

In the past couple of decades we’ve learned that’s not the only way, or maybe even the dominant way, that Earth-life makes its ATPs.  Microbes have evolved a surprising number of “front ends” to the energy machinery.  Here in Colorado we’ve got problems in old mines where microbes build ATPs by oxidizing iron pyrite (FeS) to sludgy rust (Fe2O3) and sulfuric acid (H2SO4).  Works great for them, not so good for downstream organisms.

Iron compounds are such a good energy source that many scientists believe (it’s still controversial) that Earth’s hematite and magnetite deposits were laid down half-a-billion years ago by archaea, microorganisms that preceded bacteria.

Way down on the energy-source scale are the methanogens, archaea that use molecular hydrogen to convert CO2 to methane (CH4).  They only live in zero-oxygen environments — peat bogs, ocean-bottom hydrothermal vents and subsurface veins that are perilous to mine.

Earthly biology participates in many cyclic processes.  The bi-level diagram below highlights two — oxygen cycling between O2 and oxygen compounds, and carbon cycling between CO2 and living tissues (which contain carbohydrates).

If it weren’t for light-driven photosynthesis ( ~~ is a photon), pretty soon all our O2 would be locked up in the ground where it came from.  In a sense, Earth uses life and carbon to get oxygen back up into the atmosphere.  Astronomers look for O2 in a planetary atmosphere as a sign of life.titan-cycles-2

Maybe Titan does something similar.  Titan’s atmosphere contains methane (CH4) and H2 but the quantities aren’t right.  The purple “Lyman α” and blue “Balmer α” lines on the energy chart denote particularly strong solar photons that can break up C-H bonds and generate H2 in Titan’s upper atmosphere.  We understand the relevant processes pretty well and can calculate how much methane, acetylene (C2H2) and H2 should be up there.

The calculated quantities pretty much match what astronomers found in Titan’s upper atmosphere.  But they’re not what Cassini-Huygens found on the ground.  Acetylene just isn’t there, and a (somewhat precarious) computer simulation indicates that there’s much less ground-side H2 than you’d expect from simple diffusion.  Dr Chris McKay has put those clues and the energy stack together to suggest that something on Titan inhales acetylene and hydrogen and exhales methane.

Something alive, maybe?

~~ Rich Olcott

And now for some completely different dimensions

Terry Pratchett wrote that Knowledge = Power = Energy = Matter = Mass.  Physicists don’t agree because the units don’t match up.

Physicists check equations with a powerful technique called “Dimensional Analysis,” but it’s only theoretically related to the “travel in space and time” kinds of dimension we discussed earlier.

Place setting LMTIt all started with Newton’s mechanics, his study of how objects affect the motion of other objects.  His vocabulary list included words like force, momentum, velocity, acceleration, mass, …, all concepts that seem familiar to us but which Newton either originated or fundamentally re-defined. As time went on, other thinkers added more terms like power, energy and action.

They’re all linked mathematically by various equations, but also by three fundamental dimensions: length (L), time (T) and mass (M). (There are a few others, like electric charge and temperature, that apply to problems outside of mechanics proper.)

Velocity, for example.  (Strictly speaking, velocity is speed in a particular direction but here we’re just concerned with its magnitude.)   You can measure it in miles per hour or millimeters per second or parsecs per millennium — in each case it’s length per time.  Velocity’s dimension expression is L/T no matter what units you use.

Momentum is the product of mass and velocity.  A 6,000-lb Escalade SUV doing 60 miles an hour has twice the momentum of a 3,000-lb compact car traveling at the same speed.  (Insurance companies are well aware of that fact and charge accordingly.)  In terms of dimensions, momentum is M*(L/T) = ML/T.

Acceleration is how rapidly velocity changes — a car clocked at “zero to 60 in 6 seconds” accelerated an average of 10 miles per hour per second.  Time’s in the denominator twice (who cares what the units are?), so the dimensional expression for acceleration is L/T2.

Physicists and chemists and engineers pay attention to these dimensional expressions because they have to match up across an equal sign.  Everyone knows Einstein’s equation, E = mc2. The c is the velocity of light.  As a velocity its dimension expression is L/T.  Therefore, the expression for energy must be M*(L/T)2 = ML2/T2.  See how easy?

Now things get more interesting.  Newton’s original Second Law calculated force on an object by how rapidly its momentum changed: (ML/T)/T.  Later on (possibly influenced by his feud with Liebniz about who invented calculus), he changed that to mass times acceleration M*(L/T2).  Conceptually they’re different but dimensionally they’re identical — both expressions for force work out to ML/T2.

Something seductively similar seems to apply to Heisenberg’s Area.  As we’ve seen, it’s the product of uncertainties in position (L) and momentum (ML/T) so the Area’s dimension expression works out to L*(ML/T) = ML2/T.

SeductiveThere is another way to get the same dimension expression but things aren’t not as nice there as they look at first glance.  Action is given by the amount of energy expended in a given time interval, times the length of that interval.  If you take the product of energy and time the dimensions work out as (ML2/T2)*T = ML2/T, just like Heisenberg’s Area.

It’s so tempting to think that energy and time negotiate precision like position and momentum do.  But they don’t.  In quantum mechanics, time is a driver, not a result.  If you tell me when an event happens (the t-coordinate), I can maybe calculate its energy and such.  But if you tell me the energy, I can’t give you a time when it’ll happen.  The situation reminds me of geologists trying to predict an earthquake.  They’ve got lots of statistics on tremor size distribution and can even give you average time between tremors of a certain size, but when will the next one hit?  Lord only knows.

File the detailed reasoning under “Arcane” — in technicalese, there are operators for position, momentum and energy but there’s no operator for time.  If you’re curious, John Baez’s paper has all the details.  Be warned, it contains equations!

Trust me — if you’ve spent a couple of days going through a long derivation, totting up the dimensions on either side of equations along the way is a great technique for reassuring yourself that you probably didn’t do something stupid back at hour 14.  Or maybe to detect that you did.

~~ Rich Olcott