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Sundiving, Part 2

(NOTE: After re-reading this post in 2021, I’m starting to doubt the validity of the math and physics here. I’m keeping the post up for posterity, but I’m warning you: read this with a very critical eye.)

In the previous post, I figured out how to get a spacecraft to an altitude of one solar radius (meaning one solar radius above the Sun’s surface, and two solar radii from its center). That’s nice and all, but unless we figure out how to get it the rest of the way down intact, then we’ve essentially done the same thing as a flight engineer who sends an astronaut into orbit in a fully-functional space capsule, but forgets to put a parachute on it. (Not that I know anything about that. Cough cough Kerbal Space Program cough…)

The sun is vicious. Anyone who’s ever had a good peeling sunburn knows this, and cringes at the thought. And anyone who, in spite of their parents’ warnings, has looked directly at the sun, also knows this. But I’ve got a better demonstration. I have a big 8.5 x 11-inch Fresnel lens made to magnify small print. I also have a lovely blowtorch that burns MAP-Pro, a gas that’s mostly propylene, which is as close as a clumsy idiot like me should ever come to acetylene. Propylene burns hot. About 2,200 Kelvin. I turned it on a piece of gravel and a piece of terra cotta. It got them both orange-hot, but that was the best it could do. The Fresnel lens, a cheap-ass plastic thing I bought at a drugstore, melted both in seconds (albeit in very small patches), using nothing more than half a square foot of sunlight.

Actually, the area of that magnifier is handy to have around. It’s 0.0603 square meters. On the surface of the Earth, we get (very roughly) 1,300 watts per square meter of sunlight (that’s called the solar constant). To melt terra cotta, I have to get the spot down to about a centimeter across. The lens intercepts about 80 Watts. If those 80 Watts are focused on a circle a centimeter across, then the target is getting irradiated with 770 solar constants, which, if it was a perfect absorber, would raise its temperature to 2,000 Kelvin. If I can get the spot is half a centimeter across, then we’re talking 3,070 solar constants and temperatures approaching 3,000 Kelvin.

And while I was playing around with my giant magnifier, I made a stupid mistake. Holding the lens with one hand, I reached down to re-position my next target. The light spot, about the size of a credit card, fell on the back of my hand. I said words I usually reserve for when I’ve hit my finger with a hammer. This is why you should always be careful with magnifying lenses. Even small ones can burn you and start fires.

The area of a standard credit card is about 13 times smaller than the area of my lens, so my hand was getting 13 solar constants. And even a measly 13 solar constants was more than enough to sting my skin like I was being attacked by a thousand wasps. Even at the limit of my crappy Fresnel lens, somewhere between 770 and 3,070 solar constants, we’re already in stone-melting territory.

At an altitude of 1 solar radius, our Sundiver will be getting to 11,537 solar constants. Enough to raise a perfect absorber to 4,000 Kelvin, which can melt every material we can make in bulk. Our poor Sundiver hasn’t even reached the surface and already it’s a ball of white-hot slag.

Except that I’ve conveniently neglected one thing: reflectivity. If the Sundiver was blacker than asphalt, sure, it would reach 4,000 Kelvin and melt. But why on Earth would we paint an object black if we’re planning to send it to the place where all that heat-producing sunlight comes from? That’s even sillier than those guys you see wearing black hoodies in high summer.

My first choice for a reflective coating would be silver. But there’s a massive problem with silver. Here are two graphs to explain that problem:

(Source.)

(Source is obvious.)

The top spectrum shows the reflectivities of aluminum (Al), silver (Ag, because Latin), and gold (Au) at wavelengths between 200 nanometers (ultraviolet light; UV-C, to be specific: the kind produced by germicidal lamps) and 5,000 nanometers (mid-infrared, the wavelength heat-seeking missiles use).

The bottom spectrum is the blackbody spectrum for an object at a temperature of 5,778 Kelvin, which is a very good approximation for the solar spectrum. See silver’s massive dip in reflectivity around 350 nanometers? See how it happens, rather inconveniently, right around the peak of the solar emission spectrum? Sure, a silver shield would be good at reflecting most of the infra-red light, but what the hell good is that if it’s still soaking up all that violet and UV?

Gold does a little better (and you can see from that spectrum why they use gold in infrared mirrors), but it still bottoms out right where we don’t want it to. (Interesting note: see how gold is fairly reflective between 500 nanometers and 1,000 nanometers, but not nearly as reflective between 350 nanometers and 500 nanometers? And see how silver stays above 80% reflectivity between 350 and 1,000? That’s the reason gold is gold-colored and silver is silver-colored. Gold absorbs more green, blue, indigo, and violet than it does red, orange, and yellow. Silver is almost-but-not quite constant across this range, which covers the visible spectrum, so it reflects all visible light pretty much equally. Spectra are awesome.)

Much to my surprise, our best bet for a one-material reflector is aluminum. My personal experiences with aluminum are almost all foil-related. My blowtorch will melt aluminum, so it might seem like a bad choice, but in space, there’s so little gas that almost all heat transfer is by radiation, so it might still work. And besides, if you electropolish it, aluminum is ridiculously shiny.

(Image from the Finish Line Materials & Processes, Ltd. website.)

That’s shiny. And it’s not just smooth to the human eye–it’s smooth on scales so small you’d need an electron microscope to see them. They electro-polish things like medical implants, to get rid of the microscopic jagged bits that would otherwise really annoy the immune system. So get those images of crinkly foil out of your head. We’re talking a mirror better than you’ve ever seen.

Still, aluminum’s not perfect. Notice how its reflectivity spectrum has an annoying dip at about 800 nanometers. The sun’s pretty bright at that wavelength. Still, it manages 90% or better across almost all of the spectrum we’re concerned about. (Take note, though: in the far ultraviolet, somewhere around 150 nanometers, even aluminum bottoms out, and the sun is still pretty bright even at these short wavelengths. We’ll have to deal with that some other way.)

So our aluminum Sun-shield is reflecting 90% of the 15.7 million watts falling on every square meter. That means it’s absorbing the other 10%, or 1.57 million watts per square meter.

Bad news: even at an altitude of 1 solar radius, and even with a 90% reflective electropolished aluminum shield, the bastard’s still going to melt. It’s going to reach over 2,000 Kelvin, and aluminum melts at 933.

We might be able to improve the situation by using a dielectric mirror. Metal mirrors reflect incoming photons because metal atoms’ outer electrons wander freely from one atom to another, forming a conductive “sea”. Those electrons are easy to set oscillating, and that oscillation releases a photon of similar wavelength, releasing almost all the energy the first photon deposited. Dielectric mirrors, on the other hand, consist of a stack of very thin (tens of nanometers) layers with different refractive indices. For reference, water has a refractive index of 1.333. Those cool, shiny bulletproof Lexan windows that protect bank tellers have a refractive index of about 1.5. High-grade crystal glassware is about the same. Diamonds are so pretty and shiny and sparkly because their refractive index is 2.42, which makes for a lot of refraction and internal reflection.

These kind of reflections are what make dielectric mirrors work. The refractive index measures how fast light travels through a particular medium. It travels at 299,792 km/s through vacuum. It travels at about 225,000 km/s through water and about 124,000 km/s through diamond. This means, effectively, that light has farther to go through the high-index stuff, and if you arrange the layers right, you can set it up so that a photon that makes it through, say, two layers of the stack will have effectively traveled exactly three times the distance, which means the waves will add up rather than canceling out, which means they’re leaving and taking their energy with them, rather than canceling and leaving their energy in your mirror.

This, of course, only works for a wavelength that matches up with the thickness of your layers. Still, close to the target frequency, a dielectric mirror can do better than 99.9% reflectivity. And if you use some scary algorithms to optimize the thicknesses of the different layers, you can set it up so that it reflects over a much broader spectrum, by making the upper layers very thin to reflect short-wavelength light (UV, et cetera) and the deeper layers reflect red and infra-red. The result is a “chirped mirror,” which is yet another scientific name that pleases me in ways I don’t understand. Here’s the reflection spectrum of a good-quality chirped mirror:

(Source.)

Was I inserting that spectrum just an excuse to say “chirped mirror” again? Possibly. Chirped mirror.

Point is, the chirped mirror does better than aluminum for light between 300 and 900 nanometers (which covers most or all of the visible spectrum). But it drops below 90% for long enough that it’s probably going to overheat and melt. And there’s another problem: even at an altitude of 1 solar radius, the Sundiver’s going to be going upwards of 400 kilometers per second. If the Sundiver crosses paths with the smallest of asteroids (thumbnail-sized or smaller), or even a particularly bulky dust grain, there’s going to be trouble. To explain why, here’s a video of a peanut-sized aluminum cylinder hitting a metal gas canister at 7 kilometers per second, 57 times slower than the Sundiver will be moving:

We have a really, really hard time accelerating objects anywhere near this speed. We can’t do too much better than 10 to 20 km/s on the ground, and in space, we can at best double or triple that, and only if we use gravity assists and clever trajectories. On the ground, there are hypersonic dust accelerators, which can accelerate bacterium-sized particles to around 100 km/s, which is a little better.

But no matter the velocity, the news is not good. A 5-micron solid particle will penetrate at least 5 microns into the sunshade (according to Newton’s impact depth approximation). Not only will that rip straight through dozens of layers of our carefully-constructed chirped mirror, but it’s also going to deposit almost all of its kinetic energy inside the shield. A particle that size only masses 21.5 picograms, so its kinetic energy (according to Wolfram Alpha) is about the same required to depress a computer key. Not much, but when you consider that this is a bacterium-sized mote pressing a computer key, that’s a lot of power. It’s also over 17,000 times as much kinetic energy as you’d get from 21.5 picograms of TNT.

As for a rock visible to the naked eye (100 microns in diameter, as thick as a hair), the news just gets worse. A particle that size delivers 110.3 Joules, twenty times as much as a regular camera’s flash, and one-tenth as much as one of those blinding studio flashbulbs. All concentrated on a volume too small to squeeze a dust mite into.

And if the Sundiver should collide with a decent-sized rock (1 centimeter diameter, about the size of a thumbnail), well, you might as well just go ahead and press the self-destruct button yourself, because that pebble would deliver as much energy as 26 kilos (over 50 pounds) of TNT. We’re talking a bomb bigger than a softball. You know that delicately-layered dielectric mirror we built, with its precisely-tuned structure chemically deposited to sub-nanometer precision? Yeah. So much for that. It’s now a trillion interestingly-structured fragments falling to their death in the Sun.

My point is that a dielectric mirror, although it’s much more reflective than a metal one, won’t cut it. Not where we’re going. We have to figure out another way to get rid of that extra heat. And here’s how we’re going to do it: heat pipes.

The temperature of the shield will only reach 2,000 Kelvin if its only pathway for getting rid of absorbed heat is re-radiating it. And it just so happens that our ideal shield material, aluminum, is a wimp and can’t even handle 1,000 Kelvin. But aluminum is a good conductor of heat, so we can just thread the sunshade with copper pipes, sweep the heat away with a coolant, and transfer it to a radiator.

But how much heat are we going to have to move? And has anybody invented a way to move it without me having to do a ridiculous handwave? To find that out, we’re going to need to know the area of our sunshade. Here’s a diagram of that sunshade.

I wanted to make a puerile joke about that, but the more I look at it, the less I think “sex toy” and the more I think “lava lamp.” In this diagram, the sunshade is the long cone. The weird eggplant-shaped dotted line is the hermetically-sealed module containing the payload. That payload will more than likely be scientific instruments, and not a nuclear bomb with the mass of Manhattan island, because that was probably the most ridiculous thing about Sunshine. Although (spoiler alert), Captain Pinbacker was pretty out there, too.

The shield is cone-shaped for many reasons. One is that, for any given cross-sectional radius, you’re going to be absorbing the same amount of heat no matter the shield’s area, but the amount you can radiate depends on total, not cross-sectional, area. Let’s say the cone is 5 meters long and 2 meters in diameter at the base. If it’s made of 90% reflective electropolished aluminum, it’s going to absorb 4.93 megawatts of solar radiation at an altitude of 1 solar radius. Its cross-section is 3.142 square meters, but its total surface area is 16.02 square meters. That means that, to lose all its heat by radiation alone, the shield would have to reach a blackbody temperature of 1,500 Kelvin. Still almost twice aluminum’s melting point, but already a lot more bearable. If we weren’t going to get any closer than an altitude of 1 solar radius, we could swap the aluminum mirror out for aluminum-coated graphite and we could just let the shield cool itself. I imagine this is why the original solar probe designs used conical or angled bowl-shaped shields: small cross-sectional area, but a large area to radiate heat. But where we’re going, I suspect passive cooling is going to be insufficient sooner or later, so we might as well install our active cooling system now.

Heat pipes are awesome things. You can find them in most laptops. They’re the bewildering little copper tubes that don’t seem to serve any purpose. But they do serve a purpose. They’re hollow. Inside them is a working fluid (which, at laptop temperatures, is usually water or ammonia). The tube is evacuated to a fairly low pressure, so that, even near its freezing point, water will start to boil. The inner walls of the heat pipe are covered with either a metallic sponge or with a series of thin inward-pointing fins. These let the coolant wick to the hot end, where it evaporates. Evaporation is excellent for removing heat. It deposits that heat at the cold end, where something (a passive or active radiator or, in the case of a laptop, a fan and heat sink) disposes of the heat.

Many spacecraft use heat pipes for two reasons. 1) The absence of an atmosphere means the only way to get rid of heat is to radiate it, either from the spacecraft itself, or, more often, by moving the heat to a radiator and letting it radiate from there; heat pipes do this kind of job beautifully; 2) most heat pipes contain no moving parts whatsoever, and will happily go on doing their jobs forever as long as there’s a temperature difference between the ends, and as long as they don’t spring a leak or get clogged.

On top of this, some heat pipes can conduct heat even better than solid copper. Copper’s thermal conductivity is 400 Watts per meter per Kelvin difference, which is surpassed only by diamond (and graphene, which we can’t yet produce in bulk). But heat pipes can do better than one-piece bulk materials: Wikipedia says 100,000 Watts per meter per Kelvin difference, which my research leads me to believe is entirely reasonable. (Fun fact: high-temperature heat pipes have been used to transport heat from experimental nuclear reactor cores to machinery that can turn that heat into electricity. These heat pipes use molten frickin’ metal and metal vapor as their working fluids.)

The temperature difference is going to be the difference between the temperature of the shield (in this case, around 1,500 Kelvin at the beginning) and outer space (which is full of cosmic background radiation at an effective temperature of 2.3 Kelvin, but let’s say 50 Kelvin to account for things like reflected light off zodiacal dust, light from the solar corona, and because it’s always better to over-build a spacecraft than to under-build it).

When you do the math, at an altitude of 1 solar radius, we need to transport 4.93 megawatts of heat over a distance of 5 meters across a temperature differential of 1,450 Kelvin. That comes out to 680 Watts per meter per Kelvin difference. Solid copper can’t quite manage it, but a suitable heat pipe could do it with no trouble.

But we still have to get rid of the heat. For reasons that will become clear when Sundiver gets closer to the Sun, the back of the spacecraft has to be very close to a flat disk. So we’ve got 3.142 square meters in which to fit our radiator. Let’s say 3 square meters, since we’re probably going to want to mount things like thruster ports and antennae on the protected back side. Since we’re dumping 4.93 megawatts through a radiator with an area of 3 square meters, that radiator’s going to have to be able to handle a temperature of at least 2,320 Kelvin. Luckily, that’s more than manageable. Tungsten would work, but graphite is probably our best choice, because it’s fairly tough, it’s unreactive, and it’s a hell of a lot lighter than tungsten, which is so dense they use it in eco-friendly bullets as a replacement for lead (yes, there’s such a thing as eco-friendly bullets). Let’s go with graphite for now, and see if it’s still a good choice closer to the Sun. (After graphite, our second-best choice would be niobium, which is only about as dense as iron, with a melting point of 2,750 Kelvin. I’m sticking with graphite, because things are going to get hot pretty fast, and the niobium probably won’t cut it. (Plus, “graphite radiator” has a nicer ring to it than “niboium radiator.”)

Our radiator’s going to be glowing orange-hot. We’ll need a lot of insulation to minimize thermal contact between the shield-and-radiator structure and the payload, but we can do that with more mirrors, more heat pipes, and insulating cladding made from stuff like like calcium silicate or thermal tiles filled with silica aerogel.

Of course, all the computations so far have been done for an altitude of 1 solar radius. And I didn’t ask for a ship that could survive a trip to 1 solar radius. I want to reach the freakin surface! Life is already hard for our space probe, and it’s going to get worse very rapidly. So let’s re-set our clock, with T=0 seconds being the moment the Sundiver passes an altitude of 1 solar radius.

Altitude: 0.5 solar radii

T+50 minutes, 46 seconds

Speed: 504 km/s

Solar irradiance: 28 megawatts per square meter (20,600 solar constants)

Temperature of a perfect absorber: 4,700 Kelvin (hot enough to boil titanium and melt niobium)

Total heat flux: 8.79 megawatts

Temperature of a 90% reflective flat shield: 2,700 Kelvin (almost hot enough to boil aluminum)

Temperature of Sundiver’s conical shield (radiation only): 1,764 Kelvin (still too hot for aluminum)

Radiator temperature: 2,600 Kelvin (manageable)

Required heat conductivity: 1,000 Watts per meter per Kelvin difference (manageable)

Altitude: 0.25 solar radii

T+1 hour, 0 minutes, 40 seconds

Speed: 553 km/s

Solar irradiance: 40.3 megawatts per square meter (29,600)

Temperature of a perfect absorber: 5,200 Kelvin (hot enough to boil almost all metals. Not tungsten, though. Niobium boils.)

Total heat flux: 12.66 megawatts

Temperature of a 90% reflective flat shield: 2,900 Kelvin (more than hot enough to boil aluminum)

Temperature of Sundiver’s conical shield (radiation only): 1,900 Kelvin (way too hot for aluminum)

Radiator temperature: 2,900 Kelvin (more than manageable, but the radiant heat would probably hurt your eyes)

Required heat conductivity: 1,400 Watts per meter per kelvin difference (manageable)

Altitude: 0.1 solar radii

T+1 hour, 5 minutes, 25 seconds

Speed: 589 km/s

Solar irradiance: 52 megawatts per square meter

Temperature of a perfect absorber: 5,500 Kelvin (tungsten melts, but still doesn’t boil; tungsten’s tough stuff; niobium is boiling)

Total heat flux: 16.34 megawatts

Temperature of a flat shield: 3,000 Kelvin (tungsten doesn’t melt, but it’s probably uncomfortable)

Temperature of our conical shield: 2,000 Kelvin (getting uncomfortably close to aluminum’s boiling point)

Radiator temperature: 3,100 Kelvin (tungsten and carbon are both giving each other worried looks; the shield can cause fatal radiant burns from several meters)

Required heat conductivity: 1,600 watts per meter per kelvin difference (still manageable, much to my surprise)

Altitude: 0.01 solar radii (1% of a solar radius)

T+1 hour, 8 minutes, 11 seconds

Speed: 615 km/s

Irradiance: 61 megawatts per square meter

Temperature of a perfect absorber: 5,700 Kelvin (graphite evaporates, but tungsten is just barely hanging on)

Total heat flux: 19.38 megawatts

Temperature of a flat shield: 3,200 Kelvin (most materials have melted; tungsten and graphite are still holding on)

Temperature of our conical shield: 2,100 Kelvin (titanium melts)

Radiator temperature: 3,200 Kelvin (tungsten and graphite are still stable, but at this point, the radiator itself is almost as much of a hazard as the Sun)

Required heat conductivity: 1,900 Watts per meter per Kelvin difference (we’re still okay, although we’re running into trouble)

The Sundiver finally strikes the Sun’s surface traveling at 618 kilometers per second. Except “strike” is a little melodramatic. The Sundiver’s no more striking the Sun than I strike the air when I jump off a diving board. The Sun’s surface is (somewhat) arbitrarily defined as the depth the sun’s plasma gets thin enough to transmit over half the light that hits it. At an altitude of 0 solar radii, the Sun’s density is a tenth of a microgram per cubic centimeter. For comparison, the Earth’s atmosphere doesn’t get that thin until you get 60 kilometers (about 30 miles) up, which is higher than even the best high-altitude balloons can go. Even a good laboratory vacuum is denser than this.

But even this thin plasma is a problem. The problem isn’t necessarily that the Sundiver is crashing into too much matter, it’s that it’s that the matter it is hitting is depositing a lot of kinetic energy. Falling at 618 kilometers per second, it encounters solar wind protons traveling the opposite direction at upwards of 700 kilometers per second, for a total velocity of 1,300 kilometers per second. Even at photosphere densities, when the gas is hitting you at 1,300 kilometers per second, it transfers a lot of energy. We’re talking 17 gigawatts per square centimeter, enough to heat the shield to a quarter of a million Kelvin.

This spells the end for the Sundiver. It might survive a few seconds of this torture, but its heat shield is going to be evaporating very rapidly. It won’t get more than a few thousand kilometers into the photosphere before the whole spacecraft vaporizes.

In fact, even at much lower densities (a million hydrogen atoms per cubic centimeter), the energy flux due to the impacts of protons alone is greater than one solar constant. (XKCD’s What-If, the inspiration for this whole damn blog, pointed this out when talking about dropping tungsten countertops into the sun.) At 1.0011 solar radii, the proton flux is more than enough to heat the shield up hotter than a lightning bolt. As a matter of fact, when the solar wind density exceeds 0.001 picograms per cubic centimeter (1e-15 g/cc), the energy flux from protons alone is going to overheat the shield. It’s hard to work out at what altitude this will happen, since we still don’t know very much about the environment and the solar wind close to the sun (one of the questions Solar Probe+ will hopefully answer when (if) it makes its more pedestrian and sensible trip to 8 solar radii.) But we know for certain the shield will overheat by the time we hit zero altitude. The whole Sundiver will turn into a wisp of purplish-white vapor that’ll twist and whirl away on the Sun’s magnetic field.

But even if heating from the solar wind wasn’t a problem, the probe was never going to get much deeper than zero altitude. Here’s a list of all the problems that would kill it, even if the heat from the solar wind didn’t:

1) This close to the Sun, the sun’s disk fills half the sky, meaning anything that’s not inside the sunshade is going to be in direct sunlight and get burned off. That’s why I said earlier that the back of the Sundiver had to be very close to flat.

2) The radiator will reach its melting point. Besides, we would probably need high-power heat pumps rather than heat pipes to keep heat flowing from the 2,000 Kelvin shield to the 3,000-Kelvin radiator. And even that might not be enough.

3) Even if we ignore the energy added by the proton flux, those protons are going to erode the shield mechanically. According to SRIM, the conical part of the shield (which has a half-angle of 11 degrees) is going to lose one atom of aluminum for every three proton impacts. At this rate, the shield’s going to be losing 18.3 milligrams of aluminum per second to impacts alone. While that’s not enough to wear through the shield, even if it’s only a millimeter thick, my hunch is that all that sputtering is going to play hell with the aluminum’s structure, and probably make it a lot less reflective.

4) Moving at 618 kilometers per second through a magnetic field is a bad idea. Unless the field is perfectly uniform (the Sun’s is the exact opposite of uniform: it looks like what happens if you give a kitten amphetamines and set it loose on a ball of yarn), you’re going to be dealing with some major eddy currents induced by the field, and that means even more heating. And we can’t afford any extra heating.

5) This is related to 1): even if the Sun had a perfectly well-defined surface (it doesn’t), the moment Sundiver passed through that surface, its radiator would be less than useless. In practical terms, the vital temperature differential between the radiator and empty space would vanish, since even in the upper reaches of the photosphere, the temperature exceeds 4,000 Kelvin. There simply wouldn’t be anywhere for the heat to go. So if we handwaved away all the other problems, Sundiver would still burn up.

6) Ram pressure. Ram pressure is what you get when the fluid you’re moving through is too thin for proper fluid dynamics to come into play. The photosphere might be, as astronomers say, a red-hot vacuum, but the Sundiver is moving through it at six hundred times the speed of a rifle bullet, and ram pressure is proportional to gas density and the square of velocity. Sundiver is going to get blown to bits by the rushing gas, and even if it doesn’t, by the time it reaches altitude zero, it’s going to be experiencing the force of nine Space Shuttle solid rocket boosters across its tiny 3.142-square-meter shield. For a 1,000-kilogram spacecraft, that’s a deceleration of 1,200 gees and a pressure higher than the pressure at the bottom of the Mariana Trench. But at the bottom of the trench, at least that pressure would be coming equally from all directions. In this case, the pressure at the front of the shield would be a thousand atmospheres and the pressure at the back would be very close to zero. Atoms of spacecraft vapor and swept-up hydrogen are going to fly from front to back faster than the jet from a pressure washer, and they’re going to play hell with whatever’s left of the spacecraft.

Here’s the closest I could come to a pretty picture of what would happen to Sundiver. Why do my thought experiments never have happy endings?

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The weather in hell.

Neutron stars are horrifying things. They’re born in supernova explosions, which can shine with the light of 10 billion suns for weeks on end. But even after the fireworks, they’re still scary as hell. A neutron star compacts over 1.44 times the sun’s mass into a sphere about 20 kilometers across (about the size of a city). Apart from black holes, they’re the most extreme objects we know about (There may actually be more extreme variants, but none have been conclusively observed, and thank goodness: what we’re dealing with is scary enough.).

Everything about a neutron star is horrifying. Their surfaces broil at temperatures of over 100,000 Kelvin, which is twice as hot as the hottest stars. The young ones can get hotter than 1,000,000 Kelvin. Red-hot metal emits red light because most hot objects emit a so-called blackbody spectrum, with the most intense wavelength (color) of light depending on temperature. Iron near its melting point (1811 Kelvin) emits strongest at a wavelength of 1.6 microns, which is in the near-infrared (which is not the kind of infra-red the Predator used to hunt Arnold Scwharzenegger). To our eyes, the iron would look bright red-orange. The sun, with a blackbody temperature of about 5800 Kelvin, emits most strongly in the blue-green part of the spectrum (it looks yellow on Earth partly because of atmospheric scattering and partly because the human eye is a lot less sensitive to indigo and violet than it is to blue, yellow, orange, and red). At 100,000 to 1,000,000 Kelvin (and above), neutron stars would have about the same purplish-blue color as lightning bolts or the most powerful electric arcs, and they would emit most of their light in the deadly far-ultraviolet and really deadly X-ray. If you replaced the Sun with a 1,000,000-Kelvin neutron star, the Earth would only receive five times less energy than it does from the Sun. That’s ridiculous, considering all that energy is coming from an object the size of downtown Tokyo. (Speaking of which, that gives me an idea for a new Godzilla movie…) Of course, almost all of that energy would be in the x-ray and ultraviolet portions of the spectrum, and therefore would blow off the ozone layer and kill us all. But what would this blog be if I didn’t kill humanity every article?

But let’s handwave the radiation away (in true Star Trek fashion) and say you were able to get close to the neutron star.

Sorry. You’re still dead. Say you replaced Earth with a 2-solar-mass neutron star. If you were in the International Space Station, you’d probably survive the tidal forces, but they would be enough that if you oriented yourself feet-down, you’d feel a noticeable and unpleasant stretch: your head and feet would experience a difference in acceleration of about 0.4 gees. If you were orbiting at 4,000 kilometers, you’d experience a very painful 1.5-gee stretch (and, incidentally, you’d be completing one orbit every 3 seconds, which is ridiculous). By the time you got within 1,000 kilometers of the neutron star, your feet and your head would experience a difference in acceleration of 100 gees, more than enough to pull half your blood to your feet and the other half to your brain, stopping your heart and giving you lethal cerebral hemorrhages at the same time.

But as bad as things are in the neutron star’s neighborhood, they’re even worse on its surface. This is where the “Hell” part comes from. I’ve read through a few papers on neutron star structure (and skimmed many more). It’s hard to get a straight answer on what neutron stars are like on the inside (partly because it’s really hard to simulate the imponderable conditions near the core), but here’s neutron star structure as I understand it:

At the top is the atmosphere, crushed to a thickness of between 10 centimeters and 10 meters (definite numbers are hard to find) by a gravity of 200 billion gees. Most neutron star atmospheres are pure hydrogen or pure helium, but sometimes, because of the insane pressures and temperatures, the atoms undergo fusion, leaving behind a carbon atmosphere. Below that is a crust that, thanks to the way nuclei get smashed together in supernovae, is mostly iron ions, their electrons wandering around with little regard for the nuclei.

The uppermost layers of the crust are made of almost-normal mater, albeit crushed to insane densities. But as you get deeper, the weight of the crust above puts extreme pressure on the nuclei. Suddenly, nuclei that have half-lives measured in seconds on Earth become as stable as ordinary gold or lead, because the pressure keeps decay products from escaping. The higher the pressure, the easier it is for extra neutrons to slip into nuclei. The nuclei get heavier and heavier and closer and closer together as you go down. Then, a few hundred meters below the surface, you encounter the “neutron drip,” where neutrons start leaking out of nuclei and roaming free. (On Earth, loose neutrons decay with a half-life of about 10 minutes. In the neutron star, once again, the pressure makes them stable.) This region gives us one of the coolest scientific terms ever devised: “nuclear pasta.”

Say it. Nuclear pasta. That sounds like something from an awesomely shitty ’70’s superhero movie. But, according to our current physics, it’s a real thing.

Because like charges repel, a nucleus wants to fly apart, since it’s full of chargeless neutrons (which don’t attract or repel anything (electrostatically)) and positive protons (which fiercely repel each other (electrostatically)). The strong nuclear force (or Strong Force, which sounds like the name of a badly-translated kung fu movie) provides the glue that holds nuclei together. At small distances, it’s far stronger than the electromagnetic force that causes like charges to repel (thus the name). But it decays very quickly with distance: beyond about a millionth of a nanometer, it gets vanishingly weak. Nuclei that are too large for the strong force to span their whole diameter tend to be unstable.

For this reason, nuclei also repel each other when brought close together. But deep in a neutron star’s crust, the pressure begins to overwhelm the repulsion. The protons still repel each other, but now there are places where they’re forced so close together that the strong force starts winning out. The nuclei grow oblong, and then they fuse into long tubes of protons and neutrons, like subatomic sausage (Subatomic Sausage. Another good band name.). This is the eponymous “pasta phase.” As you go down and the pressure goes up, these tubes get closer together and adjacent ones merge into two-dimensional sheets. This is called the “lasagna phase.” If you go on Google Scholar and type in “nuclear pasta”, you can find an actual peer-reviewed university-supported scientific paper that uses the phrase “lasagna phase” with a straight face. That makes me smile.

Deeper down, parts of the sheets come into contact, and you end up with a weird latticework of nuclear-matter tubes called the “gyroid phase.” Below that is a phase where you have cylindrical holes surrounded by nucleus-stuff, a sort of negative of the pasta phase. Then you get spherical holes. Then, the electromagnetic repulsion just gives up entirely, and the neutron-star matter reaches the density of an atomic nucleus, which is so huge (2 x 10^17 kilograms per cubic meter) that a piece the size of a grain of sand (a cube 500 microns on an edge) would weigh as much as a small cargo ship (25,000 metric tons). Of course, without the pressure provided by a whole neutron star to compress it, that grain would expand rapidly, and you would be spread over a large area. So don’t go removing pieces of neutron stars and carrying them around. Ain’t safe.

This is the outer part of the core. So far, the electrons have pretty much been minding their own business, ignoring the pained cries of the protons and neutrons as they were squeezed unnaturally close (the bastards). Deeper down, the pressures get so high that the electrons and protons combine to form neutrons, releasing a neutrino. Below a certain depth, it’s just a soup of neutrons crammed shoulder-to-shoulder. This soup has some weird properties, which we’ll get to later.

Below the neutron fluid, physicists aren’t quite sure what happens. Funnily enough, we here on Earth don’t have a lot of experience with soups of pure atomic-nucleus fluid. It’s possible that, in the depths, quarks could leak out of the individual neutrons, or even weirder stuff could happen, but we just don’t know. Most diagrams of neutron star structures either just put a question mark at the center or list a bunch of exotic particle names and then put a question mark. I won’t even attempt to guess what’s going on down there. I’ve got hellish weather to talk about.

But would there even be weather on a neutron star? That’s hard to say. Neutron stars have powerful magnetic fields, which could very well hold the plasma in the atmosphere in place, or at least make it awfully hard for it to move around. But, if you think about it, the temperature difference between the bottom and the top of a neutron star atmosphere is between 900,000 and 2,500,000 Kelvin, which is several thousand times the 350-Kelvin temperature difference that drive’s Earth’s weather. I’ll be working under the assumption that the atmosphere of a neutron star is able to circulate. If anybody has better information than me, feel free to set me straight.

In the last article, I talked about “scale height,” which is a nifty number that tells you how high you have to go for the atmosphere on a planet to be e times (about 2.7 times) less dense. Because their gravity is so monstrous, even the superheated plasma in a neutron star’s atmosphere is crushed tight against the surface. The Earth’s scale height is about 8,500 meters. The scale height in a neutron star’s atmosphere (depending on whether it’s made of hydrogen, helium, carbon, or something else, and depending on temperature) can range from a fraction of a millimeter to a few centimeters. On Earth, if you go up two scale heights, you’re higher than most airliners ever fly. If you go up 12 scale heights, you’re in space. I couldn’t find any really reliable numbers on the surface density of neutron-star atmospheres, but let’s assume it’s the same as the density of the sun’s core: 150,000 kilograms per cubic meter. The gravity is so strong that even the deepest neutron star atmospheres reach outer-space densities within half a meter. If I could stand on a neutron star (which, I will remind you, is a bad idea), the atmosphere would just about reach my knees. Right before I evaporated and then collapsed into a one-atom-thin layer of plasma.

Lucky for us, since the density profile of an atmosphere is exponential, many of its features will scale nicely from more familiar examples, like the atmospheres of stars and planets. Here, I’m pretty much making shit up, but the shit I’m making up is informed by some real-world knowledge. It’s also based on a few assumptions. I’m assuming, for one thing, that our neutron star is a pulsar, a neutron star spun up like a top by the infall of matter from a binary companion. I’m further assuming that this neutron star has about the same rotation rate as PSR J1748-2446ad, the fastest-spinning pulsar yet discovered (as of June 2014). It spins 716 times a second. Let’s imagine there was a single spot on the pulsar that emitted radio waves (in reality, there’s one at each magnetic pole, but one spot is usually looks brighter to us on Earth). If you could hear the signal it emitted, it would sound about like this (TURN DOWN YOUR SPEAKERS! It’s not a pretty sound.):

716-Hertz Pulsar

Even for an object like a pulsar, which is small in astronomical terms, spinning 716 times per second is ridiculous. It means that the matter at the equator is moving at 15% of the speed of light. It also means the Coriolis effect is gonna come kick some ass.

For those of you who find the Coriolis effect as confusing as I once did, here’s a brief explanation. Imagine you roll a ball inward from the edge of a frictionless spinning carousel. In the reference frame of the ground, the ball just travels straight to the center, across the other side, and off the edge. But if you were rotating with the carousel, the ball wouldn’t appear to travel in a straight line. This is because, according to you, the ball has both the inward velocity the outside observer sees and a radial velocity tangent to the circle’s circumference, since it’s not moving and the carousel is. This radial velocity is at its highest at the carousel’s edge, so as the ball approaches the center, it’s moving faster than the carousel’s surface, since the inner parts have lower radial velocities. Therefore, it appears to curve across the carousel as though acted upon by a force, pass through the center, and curve back out, describing a (roughly) semicircular trajectory. I know that’s the dunderhead layman’s explanation, but since I’m a dunderheaded layman, what do you expect?

The Coriolis effect is why low-pressure systems swirl anti-clockwise over Earth’s Northern hemisphere (and clockwise over the Southern hemisphere): the Earth is a sphere, so as you move towards the poles, you’re closer to the Earth’s axis of rotation. The faster the rotation of the body in question, the stronger the Coriolis effect and the tighter the circulation. Since our pulsar is spinning so damn fast, the circulation will be very tight, and since the bottom of the atmosphere is so much hotter than the top, the motion will be quite violent. Here’s my guess at what the pulsar’s atmosphere will look like:

Here, I’m calling upon the concept of inertial circles. The radius of an inertial circle is given by:

(speed of the moving fluid) / (2 * angular velocity of planet * sin(latitude, with the poles being plus or minus 90 degrees and the equator 0))

An inertial circle is the path a body would take on a planet’s surface under the influence of the Coriolis effect alone. On Earth, if you assume a wind speed of 100 MPH (about 44 meters per second), then the inertial circle at a latitude of 45 degrees has a radius of about 480 kilometers, which is about right for a hurricane. I’ll make the very naive assumption that the winds on a neutron star will scale up in proportion to the increase in the temperature difference (from about 320 Kelvin (and 44 meters per second) on Earth to as much as 2,500,000 Kelvin on a neutron star). Neutron star winds will therefore have speeds on the order of 1,700 kilometers per second (Dorothy’s not going to make it to Oz in this tornado. Forget an F-5. We’re looking at an F-68000.) At a latitude of 45 degrees, a neutron star hurricane will have a radius of about 250 meters:

Imagine standing up to your knees in glowing gas. Spreading out around you is a brilliant electric-purple hurricane the size of a football stadium. At its center, it has an eye a few meters across, which feeds down into a needle-thin funnel with gas swirling at 0.1% of the speed of light.

I would guess that our pulsar wouldn’t have features like jet streams. For one thing, pretty much any movement in the atmosphere is going to be twisted into a circle by the Coriolis effect and turn into a cyclone. For another thing, the magnetic field would probably put the brakes on big swathes of moving fluid. A neutron star, having such a violent, hot atmosphere, would have hellish weather, and its surface would be paved with hurricanes. I imagine it would look something like this:

(Yes, I mixed up North and South and scribbled them out. Yes, I do know that I’m an idiot.)

The magnetic poles of neutron stars are often not aligned with the geographic poles (also true of Earth!), so there would probably be spots where the emerging magnetic field, all bunched up and concentrated, would stop the gas from moving much at all, even if the field was weak enough to let it move around everywhere else. These are also the spots where material tends to fall onto neutron stars, so they would have perpetual hot high-pressure systems (much like Louisiana or North Carolina). I took the liberty of adding magnetically-dampened high-pressure cyclones around these poles, and putting hurricanes everywhere else.

But this isn’t the only possible weather on a neutron star. Wherever you have fluid, gravity, and a density and/or temperature gradient, you can have weather-like phenomena. They happen in Earth’s atmosphere, they happen in Earth’s oceans, they happen on Venus, and they happen on the Sun. I’ve just spent quite a while talking about the weather on the surface of a neutron star, but there could also be weather in the interior. Beneath the crust, where the protons and electrons combine and it’s almost all neutrons, the material stops being solid and becomes a neutron liquid. But it’s not just any liquid. It’s a neutron superfluid. Superfluids are weird shit. They behave more or less like liquids, except that they have zero viscosity. Bizarre, but true. Water has a non-zero viscosity: as your pipe gets smaller, it gets harder and harder for water to move at a given velocity. Viscosity is pretty much the internal friction within the moving fluid. Viscosity determines the minimum size a stable vortex can have. Water has a viscosity of about 8.9 x 10^-4 pascal-seconds. But superfluids like liquid helium-4 have a viscosity of zero. The viscosity isn’t just very small, it’s actually zero. Superfluids can slip through any hole (that sounds dirty), and because of capillary action (which allows a wet spot on a paper towel to spread out), and because there’s no viscous friction to oppose it, they can crawl up and out of containers.

That’s all awesome, but, for my money, one of the coolest things about superfluids is what happens when they start swirling. Normally, when you rotate a container of fluid, the fluid starts to rotate with the container. Essentially, the whole container of fluid becomes one giant vortex.

This doesn’t happen in superfluids. No matter how large or small the rotating container, the superfluid forms lots and lots of extremely tiny vortices, their number depending only on the spin rate. If spin a glass of water at 1 revolution per minute, you’ll get one big, slow vortex. If you fill the same glass with superfluid helium-4 and rotate it at 1 RPM, you’ll get thousands of them. Don’t know what the hell I’m talking about? Here’s a truly beautiful video of swirling superfluid helium in action:

And here’s a terrible picture I drew of the same phenomenon:

This is basically quantum mechanics acting on a large scale, which often happens when things get cold enough. Since superfluid helium has zero viscosity, when it has to form vortices, the vortices are infinitely small, or rather, as small as the fact that the helium is made of atoms will allow them to be. This is called “quantization of vortices,” and is extremely weird, and most likely also happens in the superfluid interiors of neutron stars. These tiny vortices will be oriented along the axis of rotation, so they’ll be parallel to the crust near the equator and perpendicular near the poles (with additional changes depending on the magnetic field and whether the rotation is faster in some regions than in others, which is usually how it works out). So if you look at it from the bottom up, you get knee-deep plasma tornadoes the size of football stadiums. And if you look at it from the top down, you get a field of weird nuclear pasta crawling with trillions and trillions of microscopic tornadoes piercing a neutron sea. I’ve said it before and you know I’ll say it again: astronomy is awesome.

Uncategorized

Endless sky.

I’ve been a nerd pretty much my whole life, so as a kid, I wondered nerdy things. I wondered, for instance, if it was possible to have an endless sky: clouds below you and clouds above. As a slightly older kid, I realized that skies don’t work that way. A sky is just an atmosphere, just a layer of gas wrapped around a solid (or liquid) planet.

Now, though, with a fair bit of obscure physics knowledge under my belt, I can finally decide not only whether or not an all-sky planet is possible, but if it is, what that planet would be like. I knew there had to be some good things about being a grownup.

Because I want my planet to be all sky, I’m going to build it from dry air and water vapor (the water vapor is so we can have clouds). Can you build a planet out of nothing but air? Well-informed intuition suggests you can: after all, Jupiter and the Sun are mostly hydrogen, and hydrogen is less dense and therefore harder to squeeze together than air. But as it turns out, we can get more precise. We can ask how large a cloud of air would have to be to collapse into a planet. This number is called the “Jeans length” or “Jeans radius” (and thanks to XKCD for making me aware of this formula). For air at Earth surface density (1.2 kilograms per cubic meter) and room temperature (68 Fahrenheit, 20 Celsius, or 293.15 Kelvin), the Jeans length is 35,400 kilometers, which is just over half the equatorial radius of Jupiter. You might think that would mean that the silly planet known as Endless Sky would be quite massive. In fact, it’s only got three times the mass of the moon.

Unfortunately, this manageably-small gas planet wouldn’t collapse very much under its own gravity. Its gravity would be weak, and as it collapsed, compression would heat it up and probably evaporate most or all of it. It’s important to note that the Jeans equation was intended to be used on nebulae, not inexplicable pockets of air floating in space for no reason. I guess Sir James Jeans just wasn’t thinking ahead. (Incidentally, Sir James Jeans would be a really pretentious name for a line of denim pants).

But no matter the details, what would Endless Sky actually be like? This is where the fun begins.

As anybody who’s ever looked into hydrodynamics knows, fluids are an enormous pain in the ass to deal with. They’re always swirling around and compressing and carrying pressure waves and expanding and contracting. You can simulate fluid behavior using the Navier-Stokes equations, which are frightening:

(From the Wikipedia article.)

We’ve got partial derivatives and dot products and divergence operators all over the fucking place. And to actually turn these equations into a computer program, you need a whole set of conservation equations, as well as initial conditions and boundary values. These equations are complicated because things like air and water are swirly and their mass moves around all the time. So you’d think the equation for the pressure in an atmosphere would be horrifying. Actually, it’s not. It’s

(pressure at the surface) * exp(-1 * (altitude) / (scale height))

The scale height is the altitude at which the pressure is e times smaller than it is at the surface (where e is about 2.71). This means that pressure corresponds quite closely to altitude. The scale height on Earth is (roughly) 8500 meters, so at an altitude of 8.5 kilometers, the pressure is 1/e atmospheres (0.37 atmospheres or 37 kilopascals). Because this dependency is pretty stable, you can also say that, if you’re measuring a pressure of 37 kilopascals, you’re at an altitude of 8.5 kilometers.

We can use this nifty formula to figure out what our Endless Sky will be like. Before I get to the amusing pictures, here’s a caveat: I’m assuming constant Earth surface gravity throughout the atmosphere, which is inaccurate. That’s why this is a thought experiment, and why I majored in English instead of physics.

But here’s roughly, what the atmosphere of Endless Sky would look like:

You can learn two things from this picture right away: First, how thin tropospheres are, which tend to be where human beings and similar organisms hang out. Second, don’t buy cheap graph-paper notebooks, because the paper can apparently detect when you’re trying to tear it out gently and violently self-destructs.

Because I decided to make the parameters of my endless sky pretty much the same as Earth’s atmosphere, it all looks pretty familiar from the 1 atmosphere level to the 0.00001 atmosphere level (where we pass the Kármán line and go into space). Because Endless Sky has an oxygen atmosphere and (I’ve just decided) a sun, it’ll have an ozone layer much like Earth’s. I always wondered why thunderstorms grow to a certain height and then stop and spread out. It turns out to be because, below a certain level (called the tropopause), the higher you go, the colder the air gets. But above this level, and in fact throughout the stratosphere, the air actually gets warmer as you go higher. This is partly because of the ozone layer. Thunderstorms happen when hot, moist air rises into the lower-pressure air above, expands, and its water condenses. But this can only happen if it’s warmer than the surrounding air, and that pretty much becomes impossible at the tropopause, since the air above is already warmer.

Basically, if you were flying around Endless Sky in a hot-air balloon, and you sat so that the gondola obscured the horizon, it would be easy to think you were flying around Earth: bright blue skies, fluffy white clouds (no happy trees, though, unfortunately).

But if you looked down, you would see something horrifying. Namely, you would see no ground. Depending on atmospheric conditions below you, you might get different kinds of exotic clouds, but these would be smaller than the ones you see on Earth, because the higher pressure wouldn’t allow them to expand as much.

Looking into the sky below you would probably be quite a lot like looking into a deep, clear ocean: it would grow a deeper and darker blue. The blue is due to Rayleigh scattering, which (combined with the spectrum of light the Sun puts out) is why the sky above is blue. The darkness is because less of the sunlight would make it through all that air.

You might think that the pressure would just keep going up and up, and the air would just get denser and denser as you got deeper. On most planets, solid, liquid, and gaseous alike, temperatures tend to go up as you go down. In the case of rocky planets, this is partly because of radioactive decay. But no matter what kind of planet, there’s internal heat left over from its formation. Therefore, it gets pretty hot down there. And when the temperature and pressure rise above the so-called critical point of the gases involved, something magical happens: the gases don’t quite liquefy, but they don’t remain gaseous, either. They sort of forget what they are, and become supercritical fluids. Supercritical fluids are amazing. They can be as dense as water (or denser), but they compress and expand like a gas and fill their containers. Here’s a video from awesome YouTuber Ben Krasnow showing you what supercritical carbon dioxide looks like:

Nitrogen’s critical point is lower than oxygen’s, so at a depth of about 128 kilometers (below the 1-atmosphere level), you would encounter a broiling opalescent sea of semi-liquid nitrogen containing a lot of dissolved oxygen. Looking down, you would see city-sized Bénard cells, which look approximately like this:

(Image courtesy of NOAA, the US government atmospheric science people, who are pretty neat.)

They probably wouldn’t be quite that orderly, though. But there would be nothing but broiling opalescent clouds rising and falling as far as the eye could see, twisted into peculiar shapes or into alien jet streams by Coriolis forces.

Being denser, the supercritical ocean would attenuate light much faster than the gaseous part of the atmosphere. You would see deep, dark blue down to the opalescent layer, and then nothing. But, farther down still (about 200 kilometers deep), the sky would no longer be endless: the pressure would pass 88,000 atmospheres, which is the pressure at which oxygen solidifies into pretty blue crystals. This means there would be another layer of weather on Endless Sky: down there, the crust of oxygen and nitrogen “ice” would evaporate into the supercritical fluid above, rise, and cool, driving powerful convection currents and stirring the deep, and where it fell and cooled, the supercritical fluid would condense into oxygen and nitrogen snow. Perhaps it would form enormous dune fields like the ones we see on Earth and, amazingly, on Saturn’s moon Titan:

(Earth dunes on top. Titan dunes (probably made of water ice) on the bottom. Image courtesy of NASA, via Wikipedia.)

I hope my eight-year-old self is vindicated. He’s probably dancing around like a lunatic, the hyperactive little freak…

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Tag Archives: atmosphere

Sundiving, Part 2

The weather in hell.

Endless sky.