How Strong is Celestia? · 11:50pm Apr 30th, 2014
Once again, it seems I have decided to dedicate considerable thought and mathematical analysis to a show about pastel-colored, magical, cartoon horses. My target this time is that loftiest of targets, subject of infinite speculation and praise: Princess Celestia herself.
"I predate all of the equations you're going to use. All of them."
I'm going to primarily base my analysis on her stated task of raising the Sun (and, for 1,000 years, the Moon as well). To do that, we first need to establish what the Equestrian solar system actually looks like. This is a subject I've speculated on before, but this time I have a much better idea of what I'm doing.
Any good analysis starts with a statement of assumptions, so here are the first few of mine:
Assumption 1: The physics in Equestria are broadly equivalent to the physics in the real world. Any exceptions will be noted. This includes things like fundamental constants and physical laws.
Assumption 2: Equestria’s Sun and Moon orbit the planet, rather than the other way around. This is, as will soon become apparent, to keep the numbers from exploding even further into the realms of implausibility. Besides, if the Equestrian solar system behaved in the same way ours does, there would be no need for two Princesses to keep things in check.
It’s not the easiest to see, but this image clearly shows both the Sun and Moon in the sky at the same time. They’re moving at the same angular speed, but in opposite directions, suggesting their orbits are counter to each other. They also appear to be the same size, which will be important later.
Further evidence for counter-orbits comes in this image, from “Princess Twilight Sparkle - Part 1”, where the Moon is rising above the same horizon in which the pink glow of sunset can be seen.
In the entire series so far, the only times we’ve seen the Sun and Moon in the sky together have involved Discord, Nightmare Moon, or the Summer Sun Celebration. Each time, it’s been an event of some significance, suggesting that the rotation of the planet and the orbits of the Sun and Moon are such that only one can normally appear over Equestria at a time.
We need a system where both the Sun and the Moon can orbit in opposite directions with the same orbital period, while the two points where their orbits “cross” have to maintain fixed angles to the longitude of Canterlot. There are a few ways to handle this, but most of them are needlessly complicated and don’t provide any tangible numbers. For the sake of simplicity, I’ll make another assumption:
Assumption 3: Equestria’s Sun and Moon are very nearly the same size, and they have very nearly the same orbital features.
The Moon is closer than the Sun, as proven when Luna causes and eclipse in the flashback in “Princess Twilight Sparkle - Part 1”, so it must have a slightly shorter orbital period. To counter this difference and keep the Sun and Moon from showing up out-of-sync, Equestria’s planet must rotate slowly in the direction of the Moon’s motion, lengthening its apparent path while commensurately shortening the Sun’s. This rotation won’t significantly affect our analysis, so I’m going to ignore it.
Now that we have a model to work with, let’s introduce some numbers. I’m going to focus on the Moon first, because the numbers are slightly simpler. The period of a gravitational orbit is given by:
T = 2pi(a^3/GM)^(½)
Where T is the period, a is the semi-major axis (half of the distance across the long direction of an ellipse), G is the gravitational constant, and M is the mass of the larger body. To solve this for the Moon’s distance from the planet, we need another assumption:
Assumption 4: Equestria’s planet has the same physical characteristics as Earth.
For a period of one day, or 86,400 seconds, a mass of 5.97219*10^24 kg, and a gravitational constant of 6.67*10^-11, we obtain a semi-major axis of 77,930 km. Our own Moon has a semi-major axis of 384,400 km, nearly five times greater.
Equestria’s Moon appears the same size in the sky as our Moon does to us, so they must have the same ratio of radius to semi-major axis. Our Moon has a radius of 1,737.1 km, so applying the properties of similar triangles gives Equestria’s Moon a radius of 352 km and a volume of 1.83*10^17 m^3. If both Moons have the same density of 3346.4 kg/m^3, then Equestria’s Moon has a mass of 6.12*10^20 kg. For comparison, our Moon has a mass of 7.35*10^22 kg, more than one hundred times greater.
For completeness, if I’d allowed for the planet’s rotation, the Moon’s orbit would be slightly smaller, shrinking the Moon itself accordingly.
”Your Princess is suspicious of these new-fangled orbital equations.”
At a height of 77,930 km and an orbital period of 86,400 seconds, the Moon is thus traveling over the surface of the planet at 25.61 meters per second, or roundabout 57 miles per hour. Its kinetic energy relative to the planet is given by:
KE = (m*v^2)/2
For a value of 2.0077*10^23 Joules.
This is all well and good, but how does it relate to our Princess of the Sun? Well, in “Princess Twilight Sparkle - Part 1” (I’m referencing this episode quite a bit, I realize), both the Sun and Moon stop moving after Princess Celestia and Princess Luna disappear. When Celestia disappears, the night has just begun, and from Twilight’s and Spike’s reactions upon waking, morning should still be a ways away. Further, as the Summer Sun Celebration, the longest day (and thus shortest night) is just over the horizon, the night in which the Sun and Moon stopped could have been as short as eight hours. Let’s make another assumption:
Assumption 5: Celestia and Luna have to actively prevent the Sun and Moon from coming to a halt. Without their influence, drag, friction, or some other force will bring these bodies to a stop in about eight hours.
It’s conceivable that Discord (or his plunder seeds) could have hastened the stoppage, but it’s also possible that the Sun and Moon stopped moving in the middle of the night while Twilight was still asleep. Eight hours, or 28,800 seconds, will serve as a reasonable estimate.
To keep an object moving against the resistive force of friction takes as much power as that object would lose if allowed to come to a halt. Power is the rate at which energy is developed. When the Moon comes to a stop, it has lost all 2.0077*10^23 Joules of kinetic energy it possessed, and it did so in 28,800 seconds, meaning that it requires 6.97*10^18 Watts to keep moving at a constant speed.
”Power! Unlimited POWER!”
This is a number so large as to be entirely incomprehensible, so let’s put it into perspective. To keep the Moon in its orbit, Luna is putting out the equivalent of the entire yearly electricity consumption of South Korea five times every second.
What about the Sun? Well, from our assumptions above, we already know its semi-major axis and its orbital velocity, but what about its mass? It’s tempting to use the average density of our Sun here (1,408 kg/m^3, in case you were wondering), but stars aren’t that straightforward. No main-sequence star could sustain itself at a paltry radius of 352 km; its gravitational attraction wouldn’t be able to compress its hydrogen enough to create fusion, and it would either collapse into a degenerate dwarf or, more likely, dissipate into a loose cloud of hydrogen and trace amounts of helium.
It seems we must look to more exotic objects to fit our facts. White dwarfs, the remnants of long-dead stars that lacked the mass to collapse into black holes or neutron stars, can be very small indeed. The smallest may even be smaller than our Earth; it’s not too much of a stretch to imagine one just a bit smaller than that, even if it would be tricky to explain how it could have formed. As an added bonus, white dwarfs still emit light, though the energy comes from latent heat rather than fusion. This is probably just as well; just look at Mercury for an example of the dangers of being too close to a full-size star.
White dwarfs are dense. They can pack the mass of the Sun into a volume the size of the Earth. A white dwarf the size we’re looking for would weigh in at a colossal 1.8*10^24 kg, fully one-third the mass of the Earth. It would be next to impossible for our Moon to hold a steady orbit this close to such a significant object, but that isn’t even the worst part. If the same rules about energy apply here, then Celestia is putting out 2.08*10^22 Watts.
Celestia sneezed once. There were no survivors.
Again, this is an utterly meaningless figure, so let’s put it in terms of our favorite librarian.
ALWAYS RELEVANT
As I’ve explained before, Twilight’s peak power output lies around 570 kW. Let’s suppose that every one of the 7 billion people on Earth can match this power output, and can do it continuously, day and night. To match the amount of power Celestia could be putting out here, we would need a fully-populated Earth for every person in Norway. If Equestria’s Sun is a white dwarf, Celestia is delivering the energy of the Chicxulub impact (the one that wiped out the dinosaurs) every twenty-five seconds.
There is another, saner option. Equestria’s Sun could actually be another moon, one with a surface made of ice instead of regolith. Ice is highly reflective; if our Moon had an icy surface, the light it would reflect from the Sun would make night brighter than day. Perhaps our entire Equestria-Sun-Moon system orbits a distant star, the Sun reflecting its light to brighten the world. If the Sun’s surface were more reflective, say by being metallic, the actual star could be even further away, making it appear to be just a particularly bright star in the sky.
This seems much more reasonable than the previous, “living matter-antimatter reaction” scenario. Even so, Celestia and Luna can each take control of both Sun and Moon, should they need to, so they can both put out at least 1.4*10^19 Watts, putting them on the order of 20 trillion times more powerful than anything else we’ve seen on the show.
Basically, don’t mess with a Princess.
As an addendum, this means that Shining Armor loves Cadence, literally, with the heat of a hundred Suns. The Earth receives 1.7*10^17 Joules of energy from the Sun every second; multiply that by one hundred, and you match Celestia's power output. That's how much power Chrysalis would have needed to overcome Celestia at full power, which means that she must have absorbed that much power from Shining Armor's love for Cadence. Forget Celestia; Cadence is the Princess to watch, if only because her husband will utterly destroy you.
You forgot to add the amount of power that Celestia used to move both sun and moon. Or I missed it? Anyway, that's must be a huge amount of power.
I also have assumed that both Sun and Moon orbit the planet. I thinks it's the only way possible to explain the need of Celestia and Luna to move them.
2065389 Actually, I did account for both celestial (heh) bodies. Each one takes 6.97*10^18 W, so moving both requires 1.4*10^19 W. If I go for the dramatically-higher value for the Sun, then the work to move the Moon is insignificant in comparison.
Actually, I think the most impressive feat Celestia has performed in the show to date is when she lifts her teacup in "A Bird in the Hoof". The level of control required compared to her strength would be a bit like trying to light a match by blowing up Krakatoa without burying Ketimbang in burning ash.
2065505
Oh, right. I didn't notice it before.
I wonder how much destruction was when she was learning to control her power? Maybe Discord wasn't the cause of all the chaos back when he ruled Equestria?
... Has anyone ever told you that you put a bit too much thought into this show?
If you ask me, she is only as strong as the plot demands her to be.
Need her to be and overly powerful sue? Gotcha.
Need her to be crippled at the slightest danger? Why not?
Need I say more?
Whilst I love the amount of effort you put into these calculations, I feel like you should explore some other reasonable alternatives. It's possible that instead of controlling the satellites every motion, that they 'give them a push', and thus 'raise' the sun and moon, and they take over the rest themselves.
This would mean perhaps it has a much smaller amount of friction, and takes longer to slow down; thus in the princess twilight episode, the celestial bodies had not been given a 'nudge' since the night/day before, meaning the time taken to grind to a halt would be much larger than a mere 8 hours, reducing the amount of power required to give them a 'tap' every so often?
Theorycrafting here, not mathhammering, I'm useless at the latter
I would also argue that in your theory, Celestia and Luna would be outputting such massive amounts of energy constantly that it would actually leave them potentially incredibly weak a majority of the time, perhaps a reason for how Chrysalis could defeat Celestia, without also having an insane amount of power (this is presuming that love isn't so efficiently harvested in such vast quantities by Chrysalis/Shining Armour that it could beat something with the power to move the sun).
This would also help explain why Celestia and Luna do so little, and send the Elements so frequently; to deal with a threat, they would not only have to relinquish their hold on the sun/moon for a period of time, but would then have to attempt to focus that power against a single target -without- annihilating half of Equestria, or more.
2066321 You know, I had considered that exact scenario, with them only exerting themselves at sunrise and sunset. It arguably makes more sense from the context of what we know. My post was already running a bit long, however, so I left it out.
You'll find that it makes little difference. If the Moon and Sun take longer to slow down, then the power requirements are lower when averaged across the entire day. However, the time frame to deliver that power is shorter, so their peak power output would be almost the same no matter what.
2065877 I wouldn't say it's "too much thought", as I really enjoy doing maths like this. For me, it's just another way to enjoy the show.
@title: Strong enough to hoof-wrestle Big Mac to a standstill without breaking a sweat, according to the comics. Wow!
More relevantly, this interpretation still doesn't account for some of the things we see. I mean, forget the tidal forces of an orbiting white dwarf or the lack thereof from mini-moons, we still have to explain that shot whre both sun and moon are up and half of Equestria is dark. Meaning the moon emits darkness somehow, if you go with a traditional satellite explanation.
Maybe they don't actually move anything and instead just bend the light coming in from either body such that it looks like they're moving?