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A Certain Scientific Analysis

Happy New Year everyone! I had meant to write this sooner, but holiday festivities interfered. No more shall they stand in my way, however!

Anyway, I bring to your attention the show To Aru Kagaku no Railgun (“A Certain Scientific Railgun”).

This is the main character:

Misaka, the main character

… and this is her railgun.

railgun

Essentially, the main character, Mikoto Misaka, has electromagnetic manipulation as a superpower. Her “signature” move is to take a coin, and to flick it forward while using her powers to accelerate it towards the target; as a result, she has earned the nickname “The Railgun”.

And now, J will apply his MIT education to something totally frivolous. Allow me to present: A Certain Scientific Analysis.

Warning: Those of you with an aversion to science and/or math may wish to turn back now, but it’d be nice if you stay. I’ll keep it simple, I promise!

Disclaimer: some of what follows was exhaustively researched via Wikipedia. Subs courtesy of Mazui.

This is an actual railgun:

400px-Railgun_usnavy_2008

And here’s how it works:

Essentially, when you run a current through a loop of wire, it produces a magnetic field (B) in the loop. This field interacts with the current going through the loop to produce a force that pushes outwards on the loop (relative to the center of the loop). It doesn’t matter which way the current is going.

If you run a lot of current through the loop, the loop rips apart due to the forces! Well, it would, except usually what happens is that all the current going through the wires instead causes the loop to catch fire and/or explode. That’s no good.

Let’s replace the loop with two long “rails” of metal parallel to each other bolted firmly to the ground, with a thing that can slide down the rails connecting them. When we turn the current on, each section of the loop tries to go outwards; the rails are bolted down, but the sliding thing travels down the rails just fine due to the forces. Hooray! If you make the rails really long, you can use this system to go places, but it’ll use a lot of power.

If you make the rails a bit too short, the sliding thing goes flying off the end. This is a railgun.

“But why bother!” you ask. “Can’t I do tons of damage by say, shooting an explosive artillery shell out of a cannon?” You could. The thing is, normal guns have an upper limit to how fast they can shoot the bullet because they’re powered by exploding gases (basically). A railgun doesn’t have this limitation, so it can shoot solid shells really, really fast and destroy targets with kinetic energy alone. BOOM!

The railgun shown above has a muzzle velocity of 2520 m/s (about 5,600 miles/hr, or about 7.5 times the speed of sound), with a muzzle energy of 10.64 megajoules (source). This is roughly the amount of energy in an average car traveling at 250mph.

For comparison, the AK-47 has a muzzle velocity of 715m/s (~1600mph, about twice the speed of sound), with an energy of 2010 joules.

What about Misaka?

According to the anime and manga, Misaka has a “muzzle velocity” of 1030 m/s:

railgun_velocity

This isn’t particularly impressive as far as railguns go; real railguns are capable of achieving speeds of around 5000m/s, and possibly more. Let’s leave that aside for now. (Aside: Misaka’s firing rate is given as 8 shots / minute, which actually is about how fast normal railguns can fire.)

Misaka is throwing these:

coin2

The coin is about the size of a U.S. quarter, so it seems safe to assume that it has similar mass, about 5 grams.

This means that Misaka’s “muzzle energy” is about

E = ½(mass)(velocity)² = ½(0.005kg)(1030m/s)² = 2652.25 joules

Not particularly impressive when compared with a real railgun, but still nothing to scoff at when it’s coming from a middle-schooler. Assuming that all the power to shoot the “railgun” is coming from Misaka (it’s probably not; who can say with superpowers?), and that the conversion from biological energy to electromagnetic energy is 100% efficient, Misaka is burning (2652.25 joules) * (1 Calorie / 4184 joules) = 0.633 Calories per shot. If she shoots a hojillion coins she’ll lose weight!

Also consider the recoil. According to Newton’s 3rd Law, in the absence of friction Misaka should be scooting backwards at (bullet mass) * (bullet velocity) / (Misaka’s mass) meters per second. Assuming Misaka weighs ~45 kg (I’d use a more precise figure, but I couldn’t find one), The recoil has her sliding backwards at (0.005kg) * (1030m/s) / (45kg) = 0.1144m/s = 0.25 mph. But since it’s the electrical powers and not the thumb-flick that accelerates the coin, it’s doubtful that she’d experience this recoil directly; more likely, the recoil will blow lots of charged particles backwards, creating a dramatic wind and flipping up Misaka’s skirt to reveal her panties shorts.

For comparison, the M-16 rifle fires 4 gram bullets at 975 m/s, for an expected energy of 1901.25 joules. So Misaka should be doing a bit more damage than a good assault rifle.

Instead, she does this much damage:

railgun_damage2

Does auto insurance cover damage from electromagnetic weapons?

Obviously, there’s something wrong here.

The first thing, of course, is that Misaka’s ability doesn’t look anything like a railgun. Did you see any rails? I didn’t think so.

There are other ways of shooting projectiles using electricity, of course. Moving electric charges (currents) produce magnetic fields, so producing magnetic fields should be child’s play for Misaka. If she makes a giant magnetic field in the air, and makes a smaller magnetic field in the coin, the coin will go zipping through the air … along with every other magnetic object in the area.

Another possiblity is that Misaka is effectively creating rails out of thin air. By this I mean not that she’s summoning I-beams or something out of somewhere, but that she is making the air conduct electricity and thus serve as rails. Normally, air doesn’t conduct electricty, which is why you don’t get zapped until you stick your finger in the electrical socket. At high enough voltages, though, the air becomes conducting, and electricity can arc through the air. This is why, although current is what’s actually deadly, all of the warning signs near power stations read “DANGER: HIGH VOLTAGE”.

Well, arcs of electricty (like lightning bolts) are crooked, so this approach probably wouldn’t make very good rails. In fact, the manga specifically says that she uses the “create an electromagnet” approach. If the magnetic field were sufficiently focused, she could accelerate only the coin and possibly a few nearby objects (No metal buttons or jewelry, and keep the spare ammo far away from the firing hand!).

The “ionize the air” trick lets us do another interesting calculation though: how much voltage can Misaka generate?

zappity zap zap.

zappity zap zap.

The dielectric strength of air, the electric field strength at which air stops being an insulator and starts conducting electricity, is about 300,000 volts/meter. Misaka’s fingers look to be about 10 cm apart in the above image, so the voltage difference between her fingers is (300000V/m) * (0.1m) = 30,000 volts. This gives a lower bound on the maximum voltage she can produce.

The manga says that Misaka is capable of manipulating 100 million volts. Theoretically, then she should be able to electrocute people from (100 000 000 v) / (300 000 V/m) = 333.3 meters = 1,093 feet away! Of course, the attack is easily deflected by the presence of any electrically grounded objects in the area, so attempting this trick would instead cause lightning to strike at her feet instead. No wonder she grabs people before shocking them!

Back to the issue of Misaka’s “railgun” doing more damage than it “should”. Suppose the given speed of 1030m/s is incorrect? Let’s try to calculate how fast the coins are actually going.

Approach 1: Frame-Watching

Consider the first episode, when Misaka fires her “railgun” at an oncoming car. We see the coin fly down this street:

railgun_distance1

Assuming (generously) that Kuroko is 150cm tall, then those posts are about 3 meters apart.

railgun_distance2

railgun_distance3

Between these two consecutive frames, the orange streak appears to travel about 10 posts down the street, or about 30 meters. Assuming that the anime is shot at a conventional 24 frames / second, this yields a speed of about (30m) / (1/24 s) = 720 m/s, even slower! Possibly, the coin is moving even faster than that and the orange line, like the groove in the ground, serves only to indicate where it has been. Let’s try another approach.

Approach 2: Impact Energy

Misaka’s power is said to be so great that measurements have to be performed using the school’s pool to absorb the energy of the shot. Here is that pool:

A commentary on the apparent power of Misaka's railgun?

A commentary on the apparent power of Misaka's railgun?

The pool appears to be about 14 Kurokos long, or 21 meters (a bit on the small side), and about 11 Kurokos wide, or 16.5 meters, juding by the lines on the floor. It also appears to be about 1 Kuroko deep:

pool_depth

This is what happens when Misaka shoots the pool:

Shabam!

Shabam!

We could attempt to integrate over the volume of the wall of water, assume it’s half air, find it’s weight … or we could be lazy.

Let’s assume that 1/1000th of the pool’s water is shot up into the air each time Misaka fires. The pool contains (1.5m)*(21m)*(16.5m) = 519.75 m³ of water. The “center of mass” of the wall of water appears to be near the top of the building, which going by the tall stories and the height of the doors appears to be perhaps 10m tall. The energy it takes to lift an object is (mass of object) * (acceleration due to gravity) * (height you’re lifting it to). In this case, then, if we assume that all the energy of the shot goes into lifting the water (ignoring the energy spent warming the water up, creating a loud explosion, casuing dramatic winds to swirl about…), then we have

Energy = (1/1000) * (519.75 m³) * (1000 kg/m³ water) * (9.8 m/s²) * (10m) = 50935.5 joules

Plugging this backwards into the kinetic energy equation, Misaka’s “muzzle velocity” now becomes:

(50935.5 J) * 2  / (0.005kg) =4514 m/s !

This value happens to be around what “modern” railguns are capable of. Granted, at these sorts of speeds, depending on the composition of the coin, it may already have turned into molten slag due to friction heating from the air.

Regardless, we must respect the canon value of 1030m/s given in the anime. Perhaps the damage caused by Misaka’s ability is due to electromagnetic effects caused by the coin flying through the air?

I’ll do another one of these again sometime, if someone has suggestions for another good series to analyze. Once again, happy new year! I look forward (with trepidation) to your responses…

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11 comments to A Certain Scientific Analysis

  • Jennifu Jennifu

    STOP RUINING MY ANIME WITH YOUR NITPICKING, NERD.

    No, just kidding, it was actually really interesting. Cool to see how you estimate this kind of thing (and how MIT can actually be applied to the useful, important things in life like anime).

    Dangit, I want railgun powers…

    As for another series to analyze, the only one I can think of immediately is 20th Century Boys, which claims to have a depiction of how a giant robot would actually look and function in real life. It’s pretty long though, and the robot isn’t featured that much, so I don’t know how well that’d work.

  • yomikoma yomikoma

    So a Kuroko is 0.88 Smoots?

  • Mimi Mimi

    Coincidentally, I was listening to this when reading the article.
    http://www.youtube.com/watch?v=ta58G3ixdIs

    I must check out the anime now! :D :D

  • Lulu Lulu

    Woah! An awesome analysis!

    A possible explanation about the higher muzzle velocity when Misaka fires in the pool may be because she was firing in the pool. Water is already a much better conductor than air, and a chlorine-infested swimming pool would probably be a much better medium for Misaka to manipulate, given the amount of strong ionic atoms (lots of free electrons) in the waters (*coughchlorinecough*). So her 1030 cap might be just due to being mortal – if the authors didn’t want to make Misaka completely like a real railgun.

    Which brings to the question (since I haven’t seen the anime), when and how was Misaka’s ‘muzzle velocity’ calculated? If the pool was used to calculate it, the calculations could either be attributed to human error, or the animation artists took a considerable amount of creative license in drawing that huge wall of pool water.

    Also! I went digging around Wikipedia and found a method of ionization that Misaka could be triggering, called “Avalanche breakdown“. The process basically occurs when a really strong electric field (brought about by really high voltages) sends free electrons flying, transferring high amounts of energy that make new ions. Rolling this snowball (‘electron avalanche’ that occurs in picoseconds) down would result in huge currents, and thus a huge magnetic field.

    I’m definitely adding this anime to my list of to-be-watched! :D

    • J J

      She wasn’t actually firing in the pool; rather, she was firing into the pool, with the water to stop the bullet and absorb its energy. While firing in a chlorinated pool would give her more ions to move around, I surmise that her ability is based on moving electrons. Furthermore, firing using the electromagnet in the pool would result in a jet of ionized pool water in addition to the flying coin.

      Misaka used Water Gun! It's not very effective...

      I’m guessing they measured her speed with a high-speed camera (in an Faraday cage); another possibility is a pair of detectors that check when the bullet passes through them and then divides by the distance between them, but I’ve heard stories of that kind getting confused by muzzle flash from a railgun.

      I don’t think avalanche breakdown is controllable, so it’s probably not how Misaka goes about making the fields.

  • Ivan Ivan

    In the manga, it said that she shoots the coin up to 3 times the speed of sound which is about the same as the 1030 m/s figure above. That means that it’s pretty much a sure thing that this wasn’t a typographical mistake. Either Misaka is using a very heavy coin made up of really dense Phlebotinum to get that massive amount of kinetic energy or the author didn’t do his research. I’m betting on the latter.

    Anyway it’s just fiction. I’m willing to suspend my disbelief to enjoy a good show. If I wanted to see a hard sci-fi anime, I would be watching Planetes right now.

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