Madoka Magica and Science

This page is dedicated to the scientific references that was made throughout the series.

Thermodynamics and Entropy

This BBC special is an excellent refresher course on Entropy.

Assume Incubators are trying to sustain the entire universe using the misery of little girls. According to data gathered by the Wilkinson Microwave Anisotrophy Probe (WMAP), energy density of the universe comes down to roughly 9.9 x 10^-30 g/cm3, or 0,889767627 joules/km^3. The expansion rate of the universe, again as determined by WMAP observations is roughly 71.9 (km/s)/Mpc, ignoring other studies. The minimum radius of the universe, as determined by cosmic microwave background radiation data, is 12 gigaparsecs. Consequently, the universe gains at least 862800 kilometers of radius per second, which comes down to a volume gain of 1.48656647 × 10^54 kilometers cubed per second.

Thus, in order to maintain the energy density of the universe (admittedly, though, there are better ways to preserve life) 1.32269872 × 10^54 joules is required every second. Let us use high estimates and say that there is one magical girl for every 1,000 people. As of now, there are roughly 6,9 billion living people, so there are 6,900,000 magical girls across the world. Assume that every magical girl defeats a witch every two days, and hands over her Grief Seed to an Incubator for its energy to be collected. This comes down to roughly 40 witches being defeated every second. Therefore, if the energy density of the universe is to be maintained, every Grief Seed needs to have an energy output of at least 3.31249766 × 10^52 joules. What can you do with this sort of energy?

A. Supply the Earth’s energy needs for roughly 8,6 octillion years, a timespan 6,2802117 × 10^17 times the current age of the universe.

B. Create 3,68565071 × 10^35 kilograms of any matter (except cheese.) This comes down to about 184027 sun-sized stars, 61 billion Earth-sized planets or a single supermassive black hole with a radius of 547298,036 kilometers, roughly 1,4 times the distance between Earth and the Moon.

C. In a single year, enough energy is collected to create roughly 10 galaxies the size of our Milky Way.

D. Create a single explosion a few hundred thousand times stronger than any gamma ray burst we’ve observed so far. This would end most, if not all life (at least carbon-based life as we know it) pretty much everywhere from Earth to Andromeda galaxy.”

Personally, the only reason I can muster for this bizzare turn is that the writers want to change the setting. In the previous episodes, QB has become a bit too obviously antagonistic, making him a villain of sorts. But this development changes things, making QB not a villain but an act of god. Something outside human power. So the objective here becomes not “winning,” but rather “surviving”

Biology and the Food Chain

Kyouko: "You know about the food chain, right? I'm sure you learned about it in school. The witches eat the weak humans. And we eat those witches. That's the way stuff works, right?"

Both the Magic Girl and Witch populations draws their numbers from the general population. For this population model we assume that both are in a vast minority comparing to the overall human population, and the loses that result from magical activities contribute insignificant changes.

Population Modeling

The Magical Girl population [M(t)] increases by the number of new girls QB contracted [C], and decreases by the rate of them dying [D] or becoming [B] witches. The Witch population [W(t)] increase by the number [N] of familiars maturing per witch as well as the number of Magic Girls that became witch, but decreases when Magic Girls hunt and kill [K] them. These can be summarized as following:

• ΔM(t)= C - D * M(t) - B *M(t) = C - (D + B) * M(t)
• ΔW(t)= B * M(t) + N * W(t) - K * M(t) = N * W(t) + (B - K) * M(t)

At Equilibrium

If the populations are at equilibrium (that is, both ΔM(t) & ΔW(t) are zero), it follows that the number of Magic Girls contracted need to balance the number that died or become witch in the same period [C = (D + B) M(t)], and the number of witches matured from minions balances out numbers contributed by Magic Girls [N * W(t) = (K - B) M(t)]. Observations:

• Every dead Magic Girl is one that did not become a witch. For Kyuubey, whose goal is the unlocked energy from the Magic Girl-to-Witch transformation, every death meant time wasted contracting the girl. It makes sense then to design a system where most Witches are weaker than Magic Girls, to avoid casualty rate. [B > D]
• If Magic Girls are stronger than Witches, it follows that the typical Magic Girl will kill multiple Witches before becoming one, thus K - B is positive. Because of this, there's a need for familiars maturing to supplement the Witch population. This may explain why even though Kyuubey only harvest energy from Magic Girl-Witches transformation, such a secondary system for Witch generation exist.
• At equilibrium, the Magic Girl population is equal to C /(D + B).
• At equilibrium, the Witch population is (B - K) M(t). If we assume that the Magic Girl population is also in equilibrium, then the Witch population is C*(B - K)/(D + B).

General Solution

The general solution of the Witch and Magic Girl populations are:

• 
• 

for some constant α and β. If we assume Witches and Magic Girls are introduced by QB's civilization, thus their initial numbers were zero [M(0)=0 W(0)=0], W(t) and M(t) becomes:

Observations:

Astronomy

Anonymous observes that almost all the Witches and Familiars have names corresponding to celestial objects within the Solar System:

Name	Celestial Object	Description
Gertrud	710 Gertrud	minor planet orbiting the sun
Anthony	272 Antonia	asteroid
Adelbert	719 Albert	Amor asteroid
Suleika	563 Suleika	minor planet orbiting the sun
Ulla	375 Ursula	large main-belt asteroid
Charlotte	Charles	a lunar crater
Pyotr	2720 Pyotr Pervyj	main belt asteroid
Elly	2650 Elinor	an asteroid
Daniyyel	Daniell	a lunar crater
Albertine	1290 Albertine	asteroid
Anja	265 Anna	asteroid
Elsa Maria	182 Elsa 170 Maria	both asteroids
Sebastians	1482 Sebastiana	asteroid
Gisela	352 Gisela	asteroid
Dora	668 Dora	main belt asteroid
Uhrmann
Bartels	17823 Bartels	a main belt asteroid
Oktavia	598 Octavia	an asteroid
Holger	9266 Holger	main-belt asteroid
Walpurgis	256 Walpurga	asteroid
Isadel	Possibly 210 Isabella	a main belt asteroid
Patricia	436 Patricia	a main belt asteroid
Roberta	335 Roberta	a large main belt asteroid
Kriemhild Gretchen	242 Kriemhild	a main belt asteroid.