physics, thought experiment

The Highest Mountain

One of my favorite things about writing this blog is that it gives me a chance to answer the questions my irritating eight-year-old self wouldn’t quit asking. It’s like an exorcism: I’m putting to rest the ghost of my incredibly nerdy childhood.

Some of those questions have proven more stubborn than others. For instance: How fast can a planet spin? I talked about that in a previous post, but I’m still not satisfied, so trust me, the centrifuge planet will return.

Another of those questions has actually gotten *harder* to answer the more I’ve learned. It was sparked by a poster in one of my classrooms. The class must have been excruciatingly boring, because although I couldn’t tell you what subject it was or who the teacher was, I can picture that poster very clearly in my head.

It must have been a geometry poster, because it had all sorts of different geometric objects on it: A cube, a hyperboloid, a cone, a flat plane, and a sphere. But what caught my eye was that, to demonstrate the shapes of the surfaces, they put maps of the Earth on them. I spent what felt like hours staring at that poster (probably scaring the hell out of the teacher) while I wondered what it would be like to stand on an Earth shaped like a giant Pringle. Recently, the memory of the poster was reignited by the fantastic, infuriating, and pleasantly inexpensive game HyperRogue, which takes place on a hyperbolic plane that’s a bit like the Pringle-shaped surface.

But until I can hunt down a copy of that poster on e-Bay (and worry people by staring at it for many more hours), my questions about life on a hyperbolic planet will have to go unanswered. Instead, I want to put to rest another stubborn question:


Seriously! How high can a mountain get? I’m not sure why I’m so angry about this, but I think it’s because my own attempts to figure it out have been repeatedly foiled, either by circumstance (read: laziness), or by the fact that geology is more complicated than you might think. I’ve tried to write this very post over half a dozen times, and failed every time. Now, though, I think I can finish what I started.

At first, I assumed the only real limit to a mountain’s height would be the strength of the rock it was made of. As we’ll see by the end, that *does* turn out to be the most important limit, but there are a lot of other important constraints.

The phrase “the tip of the iceberg” is one of those cliches that’s been run over so many times it’s almost invisible on the page. It’s like that one pair of underwear that’s so worn out a harem-girl could use it as a really bizarre veil. But you know what? The phrase is still valid. It’s hard to find proper images of what the rest of an iceberg looks like, but we know it’s far bigger than the tip. We know that because of buoyancy.

For an object to float, it has to displace its own weight’s worth of the medium it’s floating in. That means the floating object has to be less dense (averaged over its whole volume) than the medium. This is why you can float a cannonball in a bucket of mercury. It’s also why the ridiculous-looking Civil War ironclads like this


The CSS Virginia or Merrimac

The CSS Virginia or “Merrimac” (Source.)

could float. The mercury is almost one and a half times as dense as the iron cannonball. The ironclad is made of thick metal plates, but those plates are wrapped around a large volume of air and humans, making the overall density low enough that it floats. (Apparently, though, its density was only just barely low enough. When the ship was fully-loaded, its upper deck would be covered in a few inches of water, making it the world’s worst submarine.)

You might be wondering why the hell I’m talking about icebergs all of a sudden. I’ll explain in a moment. For now, though, some calculations. Let’s assume a cubic iceberg. The density of sea ice varies depending on the age of the ice (which determines how much dense salt water is trapped in its pores), but let’s say 1.000 g/cc. The density of average seawater is about 1.027 g/cc.

It turns out that, in order to balance its own weight, a cubic iceberg must have over 97% of its edge length submerged. An iceberg that would only just fit inside a stadium (a 100-meter edge length) would stick up a little higher than the average adult man. 97 meters of it would be underwater. “The tip of the iceberg” is a cliche, yeah, but that doesn’t make it wrong.

I imagine many of you are swearing at me right now. I don’t blame you: I went from talking about a weird poster in my elementary school math class, then promised to talk about mountains, then got sidetracked talking about icebergs. Believe it or not, I’m actually getting at something, because, as it turns out, mountains float like icebergs.

“Mountains float like icebergs.” That sounds like it belongs in a really awful poem. But in a geological sense, it’s kinda true. It’s called isostasy. On geologic timescales, solid rock flows like an incredibly viscous fluid. (Technically it’s a rheid or a viscoelastic solid.) The mantle is much dense than the crust. The crust’s density averages 2.75 g/cc, while the mantle-like transition zone supporting it is around 3.40 g/cc. And luckily, it turns out that, in order to find out how deep the roots of a mountain have to go (I stole that phrase from Tolkein, I’ll admit), we can use the same math we used for icebergs. For the densities I gave above, the math says that a mountain’s root (or foundation, if you prefer) has to go 5.231 times deeper than the mountain’s height.

(Before anyone says anything, I *am* aware that there’ll be a contribution from the mountain’s buoyancy in air, but I’ve already digressed too much as it is.)

Does that mean we can have a mountain as high as we want as long as the roots go deep enough? No, unfortunately. The rock that makes up continental crust is fairly strong and solid down to a depth of about 200 kilometers, where the crust gives way to the asthenosphere. Asthenosphere comes from the Greek for “weak sphere.” It’s a good name: in the asthenosphere, rock gets, well, weak. It also gets mushy, a little like clay. It’s not actually *molten* (in fact, most of the mantle isn’t molten, which I’m still really annoyed my geology teachers never told me), but it’s soft enough that it’s likely to flatten out under pressure.

That means that the pressure exerted by the mountain’s weight and the buoyancy force will squeeze the foundation sideways until the foundation’s shallower than the asthenosphere boundary. On the surface, that means the mountain will sink until it’s about one-fifth as high as the asthenosphere is deep. So no mountain on Earth can reasonably reach higher than 38.234 kilometers.

That’s still damned impressive. Even weather balloons would have to steer around a mountain that high, nevermind airliners. You wanna talk about the death zone? The tip of this mountain would be way above the Armstrong limit. Up there, not only is there too little oxygen to support human life, but the pressure’s so low that *water boils at body temperature*, which is bad news to exposed, wet organs like, say, lungs.

Actually, the mountain might be able to go a little higher than this. Our math so far assumes that the mountain is supported only by its foundation’s buoyancy. It’s like a gigantic peg set in a hole in a continent. In reality, the mountain will be anchored to the crust, by friction if nothing else. That crust is just as reluctant to sink as the crust that makes up the mountain, and plus, it’s got a bit of bending strength. After all, the crust is a mass of rock somewhere between 20 and 100 kilometers thick. This so-called “flexural isostasy” (which doesn’t even look like real words, but is). Imagine a yardstick (or meter-stick, if you swing that way) sticking off the edge of a table. It bends some, but holds itself up. That’s flexural isostasy. Its effect is much harder to calculate, but we know it would help support the mountain, so it would make a positive contribution to a mountain’s maximum height.

But like I said at the beginning, isostasy isn’t the real limiting factor. The limiting factor is much simpler: rocks, like the piers of a bridge, can only support so much weight before they crack. No matter how wide your pillar, if you over-load it, the pressure will squeeze its base outward until its outer layer breaks off, putting all the weight on an even smaller cross-section, which increases the pressure and makes the next layer break off, and so on.

Mountains aren’t columns, though. They’re better approximated as cones. Time to dust off my calculus and figure out the pressure on the base of a conical mountain.

It turns out that the pressure doesn’t depend at all on the steepness of the cone’s sides. This makes sense, if you think about it: A narrow pointy mountain has less mass and therefore less weight, but that weight is concentrated on a much smaller area. A broader mountain is much heavier, but it also has a much bigger cross-section. The formula for the pressure at the base of a mountain is dead-simple:

(1/3) * (density of the rock) * (height of the mountain) * (acceleration due to gravity)

If the pressure exerted by the mountain is larger than the compressive strength of the rock (I’m assuming granite for this calculation), then the rock at the bottom will be very likely to shatter and peel off, and over less-than-geologic time, that’ll trigger rockslides that eat away the mountain’s sides until the mountain is shorter and surrounded by a pile of broken rock. And before you try to say we could just build the whole mountain out of broken rock, that won’t work: gravel flows under pressure, so we’d have the same problem.

Granite’s compressive strength is around 130 megapascals, according to the Engineering Toolbox. Plug that into the formula, and we get a maximum height of 14.461 kilometers. That would be an impressive mountain. If it sat where London is, you’d be able to see the summit from Paris. Still, airliners would only barely have to detour around it, and it’s not even twice the height of Mount Everest. Nothing like the insane mountain in that one H.P. Lovecraft story (the physics of which I’ll forgive because Lovecraft is awesome).

But what if we exploited that flexural isostasy by saying the crust around the mountain is just really absurdly thick and strong? No dice. The transition from crust to asthenosphere happens above a particular pressure and temperature, which means a particular depth below the surface. Say we have an Australia-sized plateau that sits twenty kilometers higher than the surrounding crust. That just means that, beneath the plateau, the asthenosphere will start twenty kilometers above the asthenosphere of the normal crust, which means the whole damn plateau will undergo plastic deformation and sink like a deflating marshmallow until it’s close to the same height as the crust around it. That’s hydrostatic equilibrium at work. It’s like trying to build a tower out of mud: for any reasonable height, it’s just going to flatten itself out.

But if you’re like me and you *demand* a really heigh mountain, you could try varying the parameters in the equation. You could build your mountain from something less dense, like ice, which decreases the weight, which seems like it should decrease the maximum height. Unfortunately, not only is ice much weaker under compression than granite, but ice is a rheid, and flows like a liquid under pressure, over shorter-than-geologic timescales. That’s one of the reasons glaciers look like rivers: they almost *are*.

The only other parameter we can control is the acceleration due to gravity. That means that, if you want a really enormous mountain, you have to go to another planet.

Mars proves my point with Olympus Mons, an ancient shield volcano the size of France. It reaches 22 kilometers high. Earth’s highest mountain (measured relative to sea level) is Mount Everest, at 8.848 kilometers.

Mars has two advantages when it comes to building huge mountains: it’s got a lower surface gravity, which means the rock weighs less; and its interior is a lot cooler, meaning the asthenosphere (if it even exists) starts much deeper.

For a long time, Olympus Mons was calle dthe highest mountain in the solar system. That, though, is kinda hard to say with a straight face. Olympus Mons is the clear winner if its altitude is 26 kilometers, which is its altitude as measured from the low-lying plains nearby. Olympus Mons, though, is such a wide mountain that those plains are a thousand kilometers from the summit. At that distance, we’re no longer dealing with “pointy mountain rising above a plain” as much as “lump of putty squished against a globe.” I presume the 22-kilometer height is measured against Mars’s effective sea level or its geoid or something like that, which in my mind is a more sensible way to measure height.

But even though Olympus Mons might still be the tallest, it’s probably going to have to share the title. It almost loses out to the bizarre ridge that runs around [Iapetus’s] equator. Iapetus has always kind of creeped me out. It looks *wrong*. It looks like the ancient seed of a universe-devouring demon-vine. Maybe that’s just me. But either way, the equatorial ridge rises 20 kilometers above the surface, almost dethroning Olympus, and making a visible bump in Iapetus’s silhouette


Iapetus. (Source.)

But Olympus’s best competitor isn’t found on a planet. It’s found on the asteroid slash dwarf planet Vesta: Rheasilvia Mons, rising 22 kilometers above the local topography.


Rheasilvia region, with Rheasilvia Mons just above center. (Source.)

This, though, is as much of a cheat as saying Olympus Mons is 26 kilometers high, because Vesta is a very lumpy object, and the local topography of Rheasilvia is (probably) a gigantic planet-shattering impact basin the size of Sri Lanka. It covers most of Vesta’s southern hemisphere, and it’s so big and so deep that it squishes Vesta way out of sphericality, which makes deciding what standard to use when measuring Rheasilvia’s height a little tricky.

Still, Olympus Mons, Iapetus’s creepy Winter Wall, and the apocalyptic mountain on Vesta prove two things: even the most sterile airless bodies are amazing, and that I was right: mountain height is mostly constrained by gravitational acceleration, which determines how high a mountain gets before it exceeds its rock’s compressional strength.

That’s done. Time to go find some more childhood demons to exorcise.


Return of the submarine-car!

About a month ago, I tried to figure out how fast my 2007 Toyota Yaris would be able to drive underwater. I came up with 15 MPH (23 km/h, 6.52 m/s), which is respectable for a submerged economy car, but not that impressive for an underwater vehicle. Plus, as commenter Azure James correctly pointed out, propelling a car underwater isn’t as simple as having the horsepower to overcome drag: you have to make sure your drivetrain, from engine to gearbox to wheels, can generate enough torque to overcome the drag force.

As usual, I took this as an opportunity to imagine something ridiculous and draw a stupid picture. I present to you, the Hobo Sullivan Cruiser, the world’s most impractical underwater vehicle!

Underwater Car

It would be about the size of a large van or a small delivery truck, and that’s mostly because it has to accommodate a spherical pressure hull. Because if I’m going to make an underwater car, I’m for damn sure going to make it able to drive into the Challenger Deep! A lot of the other stuff is self-explanatory. The outer hull has a half-torpedo shape (Or if you prefer, half-cigar shape. Or, if you’re really picky, half a Sears-Haack body.) That’s to minimize drag. I chose to power it with a fuel cell and electric motors because those allow a greater range of rotation speeds (which, as my commenter correctly pointed out, would limit my speed on Mars, since a normal gearbox can’t get the wheel speeds up high enough to propel an ordinary car at 300+ MPH). Plus, with electric motors, exhaust isn’t as serious a problem.

The power for the electric motors is provided by a methanol-oxygen fuel cell. I chose methanol for one reason: space. Hydrogen, even compressed and cooled to cryogenic temperatures, takes up a lot of room. If you look at the hydrogen-burning stages of the Saturn V, the tanks are mostly hydrogen with a little spheroid at the end for the oxygen. Methanol fuel cells don’t provide as much power, but methanol’s a hell of a lot easier to transport. I’d still have to do something about the exhaust from the fuel cell, but for now, let’s assume I run the fuel cell hot enough that the pressure pushes it out a one-way valve.

You know, the more I look at it, the more “screw wheels” makes it look like I’ve got some weird grudge against wheels. In reality, those funny-looking objects are screw-drive wheels, which are so good at all-terrain they make tanks drive home to their hangars and weep transmission fluid into their berths.

That glorious contraption is the 1929 Fordson snow machine. It’s got huge pontoon-shaped wheels that turn on an axis parallel to the direction of travel, rather than perpendicular, with the force provided by many small segments of the helical screw pushing against the snow as the wheels turn. Because the pontoon wheels have such a large footprint, they don’t sink even in deep snow. And there is, of course, the cool factor, which is almost off the scale here. And now, let me send it careening completely off the scale by showing you the Russian incarnation of the screw-drive all-terrain vehicle:

When they say this is an all-terrain vehicle, they mean all terrain. Including the surface of a fucking lake. It’s hard not to be impressed.

Okay. So screw-wheels are cool. Is that reason enough to complicate the design of my submarine car to throw four of them on there? Well, yes. The seafloor is unpredictable. Without vegetation to anchor it or ordinary algae and fungi to bind it, the seafloor can have a wide variety of textures, and almost all of them would be a nightmare for a wheeled vehicle: silt six feet deep, loose sand, gravel rocks, weird clay, mud, sulfide sludge spewed from seafloor volcanoes… A wheel is very likely to sink into a morass like that. I could make the wheels larger, but a pneumatic wheel will deflate at seafloor pressures. I could make the wheels non-pneumatic, but they’re still big and bulky and taking up a lot of room, and they’re going to increase drag, which I’m already spending all my energy fighting. Tracks might work, but they contain an awful lot of joints and moving parts, and would probably get silted up.

Of course, screws as small as the one in the illustration probably wouldn’t fare too well in soft silt, either, so I’d have to make them larger, but nonetheless, I’d say screw wheels are the optimal solution. And besides that, they’re fucking awesome!

Of course, all this is really moot. There’s a very good physics reason we don’t turn all cars into planes (And plenty of other reasons, if you see how people drive on the ground.): it’s hard to generate lift in air. For a long time, many believed heavier-than-air flying machines were impossible, and they only became really successful a century ago. Air isn’t all that dense. An average 80-kilogram (172-pound) human could comfortably stand in a box 2 meters on an edge. That box would be seriously oversized, but the air in it would still only weigh 10 kilograms.

A box of water 2 meters on an edge, though, would weigh 8,000 kilograms, or about the same amount as an African elephant. Water is dense. Denser fluids resist being pushed around, and therefore more readily generate force. In my earlier article, I figured out that my car would theoretically be able to go 135 miles an hour in air, but only 15 miles an hour in water. But when it comes to flying, that resistive force comes in handy, because you don’t need very big wings to turn your submarine car into a submarine…submarine. Here’s a picture to illustrate how much better wings work in water than they do in air:



That is a boat. Being lifted almost completely out of the water by wings that look to me to be smaller than the wings on a Cessna.

And there’s another thing that makes an underwater vehicle confined to a surface kind of silly: buoyancy. Like I said, air’s not very dense, and for a bag full of gas to lift anything, it has to be less dense than the surrounding medium. There aren’t many gases less dense than air. There are hydrogen and helium, of course, but both are made of such small atoms that they leak out of containers no matter what you do, and hydrogen is flammable. There’s hot air, but that’s a lot denser than hydrogen or helium, and after a certain altitude, it gets hard to heat it up enough to provide useful lift.

Now here are some things that are less dense than water: Myself. Wood. Warmer water. Air. Compressed air. Methanol. Ethanol. Propanol. Butanol. Low-density polyethylene. In other words, around half the matter (by volume) in a submarine is less dense than water. Submarines actually have to take on extra water to sink. I had to stick depleted uranium ballast weights in my ocean car just to make sure it wouldn’t float.

But really, why shouldn’t it float? It’s not like I couldn’t stick landing skids or water-filled pontoons on a regular submarine and then have a submarine that can go to the bottom. And with a little creativity and plenty of power (provided by a wide variety of sources), you can adjust your buoyancy on-the-fly.

So forget the submarine car. I just want a submarine.

Actually, that’s a lie. I want a submarine car with screw-wheels. But I’m going to use it on land. Because, holy shit, look at those motherfuckers in the videos up above!