Life and energy on Titan, maybe

On October 24, 2016March 17, 2020 By rolcottIn Astronomy: Planetary Science, Chemistry, Uncategorized2 Comments

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.

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.

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

A Brief History of Atmospheres

On October 17, 2016March 17, 2020 By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

Long ago in a far-away career, I taught a short-course about then-current theories on the origin of life. The lab portion of the course centered on the 1952 Miller-Urey experiment the first demonstration that amino acids could be produced abiotically.

Imagine my surprise when I learned that Miller’s original lab apparatus is on exhibit at the Denver Museum of Nature and Science, where I volunteer now.

The diagram’s notes describe the basic experiment. Load up the system with whatever gases you think might have been in the primeval atmosphere. Start cooling water running through the condenser (double-wall tubing below the upper sphere) and gently heat the water sample you’ve put in the bottom sphere. Vapors travel up the tube into the top sphere where there’s a spark arcing between two electrodes (the black lines at 45º). Water vapor passing by sweeps any gas-spark reaction products back down to the bottom sphere.

Let the whole thing stew for a while (Miller ran his for a week, we let ours go for two), then draw off and analyze a sample of the solution in the bottom sphere. In Stanley’s day his analytical techniques found 5 amino acids. In 1971 (I think) we found (I think) 8 or 9. More recent work-ups of Miller’s sealed original samples found 25, including all 20 considered essential to life. So yeah, if you supply enough energy to a methane-ammonia-water system (Miller added hydrogen to that; we didn’t, for safety reasons) you can make the building blocks for proteins.

The experiment has been repeated probably thousands of times by different researchers in the last half-century. Some replications were duplicates of Miller’s, some started with recipes derived from other theories about what Earth’s early atmosphere looked like.

And there’s the problem. In Miller’s day we thought that Earth’s atmosphere was basically comet-tail concentrate. That’d be mostly water vapor along with a couple of volatiles like methane and ammonia. Later on we realized that much of our atmosphere is volcano belch — a hodgepodge of carbon dioxide, carbon monoxide, methane, hydrogen, sulfur gases, nitrogen, argon, helium, various acids, multiple kinds of rock dust…. Some of that is left-overs from Earth’s initial stages; some has been generated by subsequent geological processes like serpentinization, which can generate both methane and hydrogen.

If you’re running a Miller experiment you’re free to load the apparatus with whatever mixture you think other scientists might think is reasonable for an Earth on the verge of Biology. No oxygen, though — the chemistry of ancient rocks rules out significant atmospheric O₂ before about 3½ billion years ago.

Now the researchers are playing variations on the theme, asking whether conditions on (or within) Titan could also generate complex compounds that given a billion years could self-organize into anything we’d recognize as life. So what did Titan’s atmosphere look like a few billion years ago?

That’s a toughie, because we don’t have on-the-ground (or out-of-the-ground) data like we have for Earth. We’ll have to make do with theory, which starts with this chart.

solar_system_escape_velocity_vs_surface_temperature-svg — Molecular escape velocities

At any given temperature you can calculate the average energy per gas molecule (any kind of gas). Combine that with the known mass of a specific kind of molecule and you can compute its speed.

On any given world you can calculate the minimum speed (the escape velocity) that an object (rocket, rock or gas molecule) needs to have in order to overcome the world’s gravitational pull.

The chart combines both calculations for some important molecules for worlds in the Solar System that have atmospheres. For instance, Earth’s average temperature (give or take a few dozen degrees) is 300ºK=27ºC≈80ºF. From the chart, hydrogen and helium should be able to (and do) leave our atmosphere quickly. However, Earth’s gravity is sufficient to hold onto its original dowry of the heavier species. By contrast, the four massive planets would have to warm up by hundreds of degrees before they lose even the light gases.

Sure enough, Titan’s atmosphere is mostly nitrogen. The astronomers measure its methane and hydrogen content in parts per thousand but those concentrations aren’t the same going from top to bottom of Titan’s atmosphere. Therein lies an intriguing tale, but it’ll have to wait for the next post.

~~ Rich Olcott

Fifty Shades of Ice

On October 10, 2016November 28, 2025 By rolcottIn Astronomy: Planetary Science, Chemistry, UncategorizedLeave a comment

This week we dip a little deeper into Titan’s weirdness trove. Check the diagram. Two kinds of ice??!? What’s that about?

diamond-and-ice — Carbon allotropes
and water polymorphs

As everyone knows, diamonds and pencil lead (graphite, and I loved learning that graphite is an actual dug-from-the-ground mineral whose name came from the Greek verb “to write”) are both pure carbon, mostly. Same atoms, just arranged differently.

Graphite‘s carbon atoms are laid out in sheets of hexagons. Adjacent sheets are bonded together but not as strongly as are the atoms within each sheet. Sheets can slide past each other, which is why we use graphite as a lubricant and why pencils can write and erasers can unwrite.

Diamond‘s atoms are also laid out in sheets of (rumpled) hexagons, but now the bonds between sheets are identical to the bonds within a sheet. Turn your head sideways and you’ll see that sheets run vertical, too. In fact, each carbon atom participates in four sheets, three vertical and one horizontal. All that symmetrical bonding makes diamond one of the hardest substances we know of.

Neighboring carbon atoms form bonds by sharing electrons between their positive nuclei. Neighboring water molecules (H₂O) don’t share electrons but they do tend to line up with their somewhat positive hydrogen atoms pointing towards nearby somewhat negative oxygens. That’s a loose rule in liquid water but it dominates when the molecules freeze into ice.

Most of the ice on Earth has an Ice-I_h structure, where the oxygen atoms are arranged in the same pattern as the carbons are in graphite. Water’s hexagonal sheets aren’t quite flat, but the 6-fold symmetry gives us snowflakes. There’s a hydrogen atom between each pair of oxygens, but it’s not half-way between. Instead, each oxygen tightly holds its own two hydrogens while it pulls at further-away hydrogens owned by two neighboring oxygens.

But water molecules have other ways to arrange themselves. A small fraction of Earth’s ice has a diamond-like Ice-I_c structure with each oxygen participating in four hexagon sheets. Again, hydrogens are on the lines between them.

Water’s such a versatile molecule that it doesn’t stop with two polymorphs. Ice scientists recognize seventeen distinct crystalline varieties, plus three where the molecules don’t line up neatly. (None of them is Vonnegut’s “Ice-nine.”) Each polymorph exists in a unique temperature and pressure range; each has its own set of properties. As you might expect, ices formed at high pressure are denser than liquid water. Fortunately, Ice-I_h is lighter than water and so ice cubes and icebergs float.

As cold as Titan is and as high as the pressure must be under 180 miles of Ice-I_h and watery ammonia sea (even at 10% of Earth’s gravity), it’s quite likely* that there’s a thick layer of Ice-something around Titan’s rocky core.

The primary reason we think Titan is so wet is that Titan’s density is about halfway between rock and water. We know there are other light molecules on Titan — ammonia, methane, etc. We don’t know how much of each. Those compounds don’t have water’s complex phase behavior but many can dissolve in it. That’s why that hypothetical “Ammonia sea” is in the top diagram.

But wait, there’s more. Both graphite and water ice are known to form complex polymorphs, clathrates, that host other molecules. This diagram gives a hint of how that can happen. Frozen water under pressure forms a large number of more-or-less ordered cage clathrate structures that can host Titan’s molecular multitude.

At Titan’s temperatures ice rocks would be about as hard as granite. Undoubtedly they’ll have surprising chemistry and interesting histories. We can expect clathrate geology on Titan to be as complex as silicate geology is on Earth.

Geochemist heaven, except for the space suits.

~~ Rich Olcott

* – A caveat: we know a great deal about Earth’s structure because we live here and have been studying it scientifically for centuries. On the other hand, most of what we think we know about Titan’s interior comes from mathematical models based on gravitational observations from the Cassini mission, plus 350 photos relayed back from the Huygens lander, plus experiments in Earth-bound chemistry labs. Expect revisions on some of this stuff as we learn more.

The Titanic Winds of Titan (And Venus)

On October 3, 2016November 28, 2025 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.

Air 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 September 26, 2016November 28, 2025 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.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.

Superluminal Superman

On September 19, 2016November 28, 2025 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?

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

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
start Operation Squish at some time in his own future to push towards the past.

Nice trick. Would Superman buy in?

~~ Rich Olcott

Superman flying at lightspeed? Umm… no

On September 12, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Back when I was in high school I did a term paper for some class (can’t remember which) ripping the heck out of Superman physics. Yeah, I was that kind of kid. If I recall correctly, I spent much of it slamming his supposed vision capabilities — they were fairly ludicrous even to a HS student and that was many refreshes of the DC universe ago.But for this post let’s consider a trope that’s been taken off the shelf again and again since those days, even in the movies. This rendition should get the idea across — Our Hero, in a desperate effort to fix a narrative hole the writers had dug themselves into, is forced to fly around the Earth at faster-than-light speeds, thereby reversing time so he can patch things up.

So many problems… Just for starters, the Earth is 8000 miles wide, Supey’s what, 6’6″?, so on this scale he shouldn’t fill even a thousandth of a pixel. OK, artistic license. Fine.

Second problem, only one image of the guy. If he’s really passing us headed into the past we should see two images, one coming in feet-first from the future and the other headed forward in both space and time. Oh, and because of the Doppler effect the feet-first image should be blue-shifted and the other one red-shifted.

Of course both of those images would be the wrong shape. The FitzGerald-Lorentz Contraction makes moving objects appear shorter in the direction of motion. In other words, if the Man of Steel were flying just shy of the speed of light then 6’6″ tall would look to us more like a disk with a short cape.

Tall-Dark-And-Muscular has other problems to solve on his way to the past. How does he get up there in the first place? Back in the day, DC explained that he “leaped tall buildings in a single bound.” That pretty much says ballistic high-jump, where all the energy comes from the initial impulse. OK, but consider the rebound effects on the neighborhood if he were to jump with as much energy as it would take to orbit a 250-lb man. People would complain.

Remember Einstein’s famous E=mc²? That mass m isn’t quite what most people think it is. Rather, it’s an object’s rest mass m₀ modified by a Lorentz correction to account for the object’s kinetic energy. In our hum-drum daily life that correction factor is basically 1.00000… When you get into the lightspeed ballpark it gets bigger.

Here’s the formula with the Lorentz correction in red: m=m₀/√[1-(v/c)²]. The square root nears zero as Superman’s velocity v approaches lightspeed c. When the divisor gets very small the corrected mass gets very large. If he got to the Lightspeed Barrier (where v=c) he’d be infinitely massive.

So you’ve got an infinite mass circling the Earth about 7 times a second — ocean tides probably couldn’t keep up, but the planet would be shaking enough to fracture the rock layers that keep volcanoes quiet. People would complain.

Of course, if he had that much mass, Earth and the entire Solar System would be orbiting him.

On the E side of Einstein’s equation, Superman must attain that massive mass by getting energy from somewhere. Gaining that last mile/second on the way to infinity is gonna take a lot of energy.

But it’s worse. Even at less than lightspeed, the Kryptonian isn’t flying in a straight line. He’s circling the Earth in an orbit. The usual visuals show him about as far out as an Earth-orbiting spacecraft. A GPS satellite’s stable 24-hour orbit has a 26,000 mile radius so it’s going about 1.9 miles/sec. Superman ‘s traveling about 98,000 times faster than that. Physics demands that he use a powered orbit, continuously expending serious energy on centripetal acceleration just to avoid flying off to Vega again.

The comic books have never been real clear on the energy source for Superman’s feats. Does he suck it from the Sun? I sure hope not — that’d destabilize the Sun and generate massive solar flares and all sorts of trouble.

Not even the DC writers would want Superman to wipe out his adopted planet just to fix up a plot point.

~~ Rich Olcott

Light’s hourglass

On September 5, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

Terry Pratchett’s anthropomorphic character Death (who always speaks in UPPER CASE with a voice that sounds like tombstones falling) has a thing about hourglasses. So do physicists, but theirs don’t have sand in them. And they don’t so much represent Eternity as describe it. Maybe.

The prior post was all about spacetime events (an event is the combination of a specific (x,y,z) spatial location with a specific time t) and how the Minkowski diagram divides the Universe into mutually exclusive pieces:

“look but don’t touch” — the past, all the spacetime events which could have caused something to happen where/when we are
“touch but don’t look” — the future, the events where/when we can cause something to happen
“no look, no touch” — the spacelike part that’s so far away that light can’t reach us and we can’t reach it without breaching Einstein’s speed-of-light constraint
“here and now” — the tiny point in spacetime with address (ct,x,y,z)=(0,0,0,0)

Last week’s Minkowski diagram was two-dimensional. It showed time running along the vertical axis and Pythagorean distance d=√(x²+y²+z²) along the horizontal one. That was OK in the days before computer graphics, but it loaded many different events onto the same point on the chart. For instance, (0,1,0,0), (0,-1,0,0), (0,0,1,0) and (0,0,0,1) (and more) are all at d=1.

This chart is one dimension closer to what the physicists really think about. Here we have x and y along distinct axes. The z axis is perpendicular to all three, and if you can visualize that you’re better at it than I am. The xy plane (and the xyz cube if you’re good at it) is perpendicular to t.

That orange line was in last week’s diagram and it means the same thing in this one. It contains events that can use light-speed somehow to communicate with the here-and-now event. But now we see that the line into the future is just part of a cone (or a hypercone if you’re good at it).

If we ignite a flash of light at time t=0, at any positive time t that lightwave will have expanded to a circle (or bubble) with radius d=c·t. The circles form the “future” cone.

Another cone extends into the past. It’s made up of all the events from which a flash of light at time at some negative t would reach the here-and-now event.

The diagram raises four hotly debated questions:

Is the pastward cone actually pear-shaped? It’s supposed to go back to The Very Beginning. That’s The Big Bang when the Universe was infinitesimally small. Back then d for even the furthest event from (ct,0,0,0) should have been much smaller than the nanometers-to-lightyears range of sizes we’re familiar with today. But spacetime was smaller, too, so maybe everything just expanded in sync once we got past Cosmic Inflation. We may never know the answer.
What’s outside the cones? You think what you see around you is right now? Sorry. If the screen you’re reading this on is a typical 30 inches or so distant, the light you’re seeing left the screen 2½ nanoseconds ago. Things might have changed since then. We can see no further into the Universe than 14 billion lightyears, and even that only tells us what happened 14 billion years ago. Are there even now other Earth-ish civilizations just 15 billion lightyears away from us? We may never know the answer.
How big is “here-and-now”? We think of it as a size=zero mathematical point, but there are technical grounds to think that the smallest possible distance is the Planck length, 1.62×10^-35 meters. Do incidents that might affect us occur at a smaller scale than that? Is time quantized? We may never know the answers.
Do the contents of the futureward cone “already” exist in some sense, or do we truly have free will? Einstein thought we live in a block universe, with events in future time as fixed as those in past time. Other thinkers hold that neither past not future are real. I like the growing block alternative, in which the past is real and fixed but the future exists as maybes. We may never know the answer.

~~ Rich Olcott

You can’t get there from here

On August 29, 2016November 28, 2025 By rolcottIn Mathematics: Dimensions, Physics: Light and Relativity, UncategorizedLeave a comment

In this series of posts I’ve tried to get across several ideas:

By Einstein’s theories, in our Universe every possible combination of place and time is an event that can be identified with an “address” like (ct,x,y,z) where t is time, c is the speed of light, and x, y and z are spatial coordinates
The Pythagorean distance between two events is d=√[(x₁-x₂)²+(y₁-y₂)²+(z₁-z₂)²]
The Minkowski interval between two events is √[(ct₁-ct₂)²– d²]
When i=√(-1) shows up somewhere, whatever it’s with is in some way perpendicular to the stuff that doesn’t involve i

That minus sign in the third bullet has some interesting implications. If the time term is bigger then the spatial term, then the interval (that square root) is a real number. On the other hand, if the time term is smaller, the interval is an imaginary number and therefore is in some sense perpendicular to the real intervals.

We’ll see what that means in a bit. But first, suppose the two terms are exactly equal, which would make the interval zero. Can that happen?

Sure, if you’re a light wave. The interval can only be zero if d=(ct₁-ct₂). In other words, if the distance between two events is exactly the distance light would travel in the elapsed time between the same two events.

In this Minkowski diagram, we’ve got two number lines. The real numbers (time) run vertically (sorry, I know we had the imaginary line running upward in a previous post, but this is the way that Minkowski drew it). Perpendicular to that, the line of imaginary numbers (distance) runs horizontally, which is why that starscape runs off to the right.
The smiley face is us at the origin (0,0,0,0). If we look upwards toward positive time, that’s the future at our present location. Looking downward to negative time is looking into the past. Unless we move (change x and/or y and/or z), all the events in our past and future have addresses like (ct,0,0,0).

What about all those points (events) that aren’t on the time axis? Pick one and use its address to figure the interval between it and us. In general, the interval will be a complex number, part real and part imaginary, because the event you picked isn’t on either axis.

The two orange lines are special. Each of them is drawn through all the events for which the distance is equal to ct. All the intervals between those events and us are zero. Those are the events that could be connected to us by a light beam.

The orange connections go one way only — someone in our past (t less than zero) can shine a light at us that we’ll see when it reaches us. However, if someone in our future tried to shine a light our way, well, the passage of time wouldn’t allow it (except maybe in the movies). Conversely, we can shine a light at some spatial point (x,y,z) at distance d=√(x²+y²+z²) from us, but those photons won’t arrive until the event in our future at (d,x,y,z).

The rest of the Minkowski diagram could do for a Venn diagram. We at (0,0,0,0) can do something that will cause something to happen at (ct,x,y,z) to the left of the top orange line. However, we won’t be able to see that effect until we time-travel forward to its t. That region is “reachable but not seeable.”

Similarly, events to the left of the bottom orange line can affect us (we can see stars, for instance) but they’re in our past and we can’t cause anything to happen then/there. The region is “seeable but not reachable.”

Then there’s the overlap, the segment between the two orange lines. Events there are so far away in spacetime that the intervals between them and us are imaginary (in the mathematical sense). To put it another way, light can’t get here from there. Neither can cause and effect.

Physicists call that third region space-like, as opposed to the two time-like regions. Without a warp drive or some other way around Einstein’s universal speed limit, the edge of “space-like” will always be The Final Frontier.

~~ Rich Olcott

Does a photon experience time?

On August 22, 2016November 28, 2025 By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Fundamentals, Physics: Light and Relativity, Uncategorized2 Comments

My brother Ken asked me, “Is it true that a photon doesn’t experience time?” Good question. As I was thinking about it I wondered if the answer could have implications for Einstein’s bubble.

When Einstein was a grad student in Göttingen, he skipped out on most of the classes given by his math professor Hermann Minkowski. Then in 1905 Einstein’s Special Relativity paper scooped some work that Minkowski was doing. In response, Minkowski wrote his own paper that supported and expanded on Einstein’s. In fact, Minkowski’s contribution changed Einstein’s whole approach to the subject, from algebraic to geometrical.

But not just any geometry, four-dimensional geometry — 3D space AND time. But not just any space-AND-time geometry — space-MINUS-time geometry. Wait, what?

Early geometer Pythagoras showed us how to calculate the hypotenuse of a right triangle from the lengths of the other two sides. His a²+b²= c² formula works for the diagonal of the enclosing rectangle, too.

Extending the idea, the body diagonal of an x×y×z cube is √(x²+y²+z²) and the hyperdiagonal of a an ct×x×y×z tesseract is √(c²t²+x²+y²+z²) where t is time. Why the “c“? All terms in a sum have to be in the same units. x, y, and z are lengths so we need to turn t into a length. With c as the speed of light, ct is the distance (length) that light travels in time t.

But Minkowski and the other physicists weren’t happy with Pythagorean hyperdiagonals. Here’s the problem they wanted to solve. Suppose you’re watching your spacecraft’s first flight. You built it, you know its tip-to-tail length, but your telescope says it’s shorter than that. George FitzGerald and Hendrik Lorentz explained that in 1892 with their length contraction analysis.

What if there are two observers, Fred and Ethel, each of whom is also moving? They’d better be able to come up with the same at-rest (intrinsic) size for the object.

Minkowski’s solution was to treat the ct term differently from the others. Think of each 4D address (ct,x,y,z) as a distinct event. Whether or not something happens then/there, this event’s distinct from all other spatial locations at moment t, and all other moments at location (x,y,z).

To simplify things, let’s compare events to the origin (0,0,0,0). Pythagoras would say that the “distance” between the origin event and an event I’ll call Lucy at (ct,x,y,z) is √(c²t²+x²+y²+z²).

Minkowski proposed a different kind of “distance,” which he called the interval. It’s the difference between the time term and the space terms: √[c²t² + (-1)*(x²+y²+z²)].

If Lucy’s time is t=0 [her event address (0,x,y,z)], then the origin-to-Lucy interval is √[0²+(-1)*(x²+y²+z²)]=i√(x²+y²+z²). Except for the i=√(-1) factor, that matches the familiar origin-to-Lucy spatial distance.

Now for the moment let’s convert the sum from lengths to times by dividing by c². The expression becomes √[t²-(x/c)²-(y/c)²-(z/c)²]. If Lucy is at (ct,0,0,0) then the origin-to-Lucy interval is simply √(t²)=t, exactly the time difference we’d expect.

Finally, suppose that Lucy departed the origin at time zero and traveled along x at the speed of light. At any time t, her address is (ct,ct,0,0) and the interval for her trip is √[(ct)²-(ct)²-0²-0²] = √0 = 0. Both Fred’s and Ethel’s clocks show time passing as Lucy speeds along, but the interval is always zero no matter where they stand and when they make their measurements.

One more step and we can answer Ken’s question. A moving object’s proper time is defined to be the time measured by a clock affixed to that object. The proper time interval between two events encountered by an object is exactly Minkowski’s spacetime interval. Lucy’s clock never moves from zero.

So yeah, Ken, a photon moving at the speed of light experiences no change in proper time although externally we see it traveling.

Now on to Einstein’s bubble, a lightwave’s spherical shell that vanishes instantly when its photon is absorbed by an electron somewhere. We see that the photon experiences zero proper time while traversing the yellow line in this Feynman diagram. But viewed from any other frame of reference the journey takes longer. Einstein’s objection to instantaneous wave collapse still stands.

~~ Rich Olcott

Hard Science Ain't Hard

Author: rolcott

Life and energy on Titan, maybe

A Brief History of Atmospheres

Fifty Shades of Ice

The Titanic Winds of Titan (And Venus)

Titan’s Atmosphere Is A Gas

Superluminal Superman

Superman flying at lightspeed? Umm… no

Light’s hourglass

You can’t get there from here

Does a photon experience time?