Madoka Magica and Science

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

File:Madoka Starry Sky.jpg

Madoka amongst the stars

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	a minor planet orbiting the sun
Anthony	272 Antonia	an asteroid
Adelbert	719 Albert	An asteroid
Suleika	563 Suleika	a minor planet orbiting the sun
Ulla	375 Ursula	a large main-belt asteroid
Charlotte	Charles	a lunar crater
Pyotr	2720 Pyotr Pervyj	a main-belt asteroid
Elly	2650 Elinor	an asteroid
Daniyyel	Daniell	a lunar crater
Jennifer	6249 Jennifer	an asteroid
Albertine	1290 Albertine	an asteroid
Anja	265 Anna	an asteroid
Gisela	352 Gisela	an asteroid
Dora	668 Dora	a main-belt asteroid
Elsa Maria	182 Elsa 170 Maria	both asteroids
Sebastian	1482 Sebastiana	an asteroid
Uhrmann	possibly Hermann or 10239 Hermann	a lunar crater/a main belt asteroid.
Bartels	17823 Bartels	a main-belt asteroid
Oktavia	598 Octavia	an asteroid
Holger	9266 Holger	a main-belt asteroid
Walpurgis	256 Walpurga	an asteroid
Isadel	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.

Biology and the Food Chain

File:Population model.png

Graphical representation of population model used.

"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?"
-Kyouko in Episode 5

Both the magical girl and witch populations draw their numbers from the general human population. For this population model we assume that both are in a vast minority in comparison to the overall human population, and the loses that result from magical activities contribute insignificant changes to it.

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 of matured familiars [F] per witch as well as the rate of magic girls that became witch, but decreases when magic girls hunt and kill [K] them. These premises can be summarized as following:

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

Where D, B, F, K are values between [0,1], while C, M(t), and W(t) are whole numbers.

Population at Equilibrium

If both populations are in equilibrium, then there is no changes to both the witch and magical girl population
[ΔM(t) = 0 & ΔW(t) = 0]. 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 to harvest energy from the magic girl-to-witch transformation, every death meant time wasted contracting the girl. It makes sense for him to design a system where most Witches are weaker than Magic Girls in order to reduce magic girl 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 [K - B > 0]. Because of this, there's a need for an alternative source for creating witches. This explains why while Kyuubey cares only of the energy from Magic Girl-Witches transformation, the supplementary system of witch growth from familiars exist.
• At equilibrium, the magic girl population is equal to C / (D + B).
• At equilibrium, the witch population is (B - K) M(t). If the Magic Girl population is also in equilibrium, then the witch population is C*(B - K)/(D + B).

General Solution

The general solution for 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 initial population of both at the system's introduction were zero [M(0)=0 W(0)=0]. W(t) and M(t) becomes:

• 
• 

Observations:

• The magic girl population will always reach equilibrium at C / (D + B), regardless of initial number of magical girls or actual values of C, D, or B. This restrain by Kyuubey's ability to contract new girls may explain why there seems to be so few magical girls around.
• Even when a significant number of magic girl is introduced initially, their numbers will eventually whithered down to sustainable population level of C/ (D + B). This is a natural consequence of limited linear growth rate and independent of environmental or crowding stresses.
• In contrast, the witch population never reach equilibrium. The number of witches will fluctuate minutely at the initial period, but inevitably the minion growth [e^Nt] will outpace all other variables, and the population will explode into exponential growth. The growth may be capped due to human depopulation from witch activities, but at this point the initial premises of the model breaks down.
• Due to the above, any planet with this magic girl and witch system in place is inevitably doomed to extinction. It is not possible to save the human population, with exception of exterminating all witches and magic girls. This may explain why Kyuubey simply abandoned Earth rather than attempt sustainable farming of the planet.

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”