How Strong is Twilight? Redux · 10:51pm Nov 12th, 2013

I’ve talked before about how strong Twilight Sparkle is. Based on her lifting of an Ursa Minor in S1E06 “Boast Busters”, I put the value of one Sparkle at about 86 kW. I also said that I could look at her rage-splosion in “Feeling Pinkie Keen” to get another estimate, but that I wasn’t ready to take on the maths quite yet. Well, I’m ready now.

Again, there aren’t any concrete numbers to work with here, so I’m going to be making a few guesses. I’m also, as before, going to assume that Equestrian physics are basically the same as our own, and that things like surface gravity and air composition are similar. I must also preface this analysis with the disclaimer that I am not a physicist; my experience in this field consists basically of a few college-level courses and a few hours’ research on Wikipedia.

With that out of the way, let’s talk about light bulbs. The typical incandescent light bulb consists of a thin, tungsten filament surrounded by inert gas and encased in a bulb of glass or quartz. An electrical current flowing through the filament causes it to heat up, whereupon it produces light through incandescence. In this analysis, I will basically treat Twilight as though she were a very large, very oddly-shaped light bulb filament.

Pictured: filament.

Hot objects emit light through thermal radiation. A perfect emitter, or black body, emits radiation in direct relation to its temperature. Actual objects will emit somewhat less than a true black body would, based on things like reflectivity and chemical composition. Fortunately, skin and hair are fairly close to ideal, so we can proceed as though they were.

The formula for power emitted by a black body, in units of power per unit area, is j=σ(T^4), where σ is the Stefan-Boltzmann constant, approximately 5.67*(10^-8) W/(m^2 * K^4) and T is the absolute temperature in Kelvins. If we multiply by the surface area, we find how much power a given black body will radiate.

So, what is Twilight’s surface area? The average surface area of an adult human is about 2 m^2. This is a good starting point; the proportions and the difference in height should about balance out, and Twilight doesn’t yet have her wings at this point in the series.

What about her temperature? Well, when it comes to incandescence, the color is the biggest clue. Material has a bit of an impact, but not as much as you might think. The glow we see on Pinkie’s and Spike’s faces is a sort of red-orange, meaning the temperature is somewhere between 1000K and 2000K. Let’s go for the middle, say 1500K, and see where that gets us.

Bam. Red-orange.

Plugging these numbers into our formula tells us that Twilight is putting out about 570 kW, or 6.6 Sparkles. Subtracting the radiation she’s absorbing from the air doesn’t change it substantially, so this is a good number to work with. That was surprisingly easy, but we’re not done yet. What does that sort of power output look like?

Again, I’m treating Twilight as a very odd light bulb. A typical light bulb gives off light at an efficacy of about 15 lumens per Watt. This depends on temperature, and is almost certainly an overestimate in our case, but it’s enough to get an idea. Twilight, then, emits about 8,600,000 lumens of visible light. The lumen is a bit of an odd unit from a pure mathematics viewpoint, but it works here because it already accounts for the way our eyes perceive visible light. It is not, however, a measure of “brightness”; for that, we need lumens per square meter, or lux.
Pinkie and Spike are, by my estimate, about 2 meters away from Twilight when she flares up. Light expands spherically, and the amount of light an object receives depends on its surface area. Thus, brightness decreases with the square of the distance from the light source. At a radius of two meters, the light emitted by Twilight will be spread out to about 170,000 lumens per square meter (170 kilolux). If Pinkie’s front has an area of 0.5 square meters, she’s illuminated by 86,000 lumens (this is more than half of 170,000, but that’s because of rounding; I’m working with more significant figures than I’m showing). Coincidentally, this means that Pinkie’s illumination (let’s call it one Rapidash) divided by one Sparkle is equal to one lumen-second per Newton-meter. Units are weird.

So, Pinkie sees Twilight’s brightness as 170,000 lux. That, by itself, is an almost meaningless number, so let’s compare it to some things we encounter every day. The darkest time most of us are likely to see is a moonless, overcast night, with an illumination of about 0.0001 lux. The full moon on a clear night can be up to 1.0 lux. Typical office lighting goes up to 500 lux, and a bright day can be as high as 25,000 lux. Direct sunlight is 130,000 lux, tops. This means that, to Pinkie, Twilight is outshining the sun. That’s not even the whole story; visible light comprises about 45% of sunlight, with almost all of the rest consisting of infrared light. Twilight, however, isn’t hot enough to put out that much visible light, so most of her output is going to be lower-frequency infrared. In fact, only about 2% of her power is going towards the visible spectrum (incandescent light bulbs are terribly inefficient light sources).

Infrared radiation is what we feel as heat; along with convection in the surrounding air, it’s what makes fire such a useful heat source. The 98% of Twilight’s power that isn’t visible will come out as infrared light. This means that Pinkie is being hit with about 5,600 W of radiant heat, which is much more than the 125 W of light she can see. For reference, an electric oven uses about 4000 W for cooking. To Pinkie, it would feel as though she had stuck her head into a burning house, albeit only for a few seconds.

Twilight was on fire for about four seconds. During that time, she emitted 2.3 MJ of energy. This is roughly equivalent to the kinetic energy of two two-ton trucks moving at 70 miles per hour, plus a police escort keeping pace. One kilowatt-hour, the unit by which electricity is commonly sold, is equal to 3.6 MJ. Had Twilight kept burning for a full hour, she would have produced about 575 kilowatt-hours of energy (2 GJ), which is more than the energy contained in some lightning bolts. It’s also enough energy to launch a small child (~33 kg) from Earth’s surface at escape velocity.

It'd look like this, minus the suffocation. Image is "Blue Marble".

The obvious question, then, is where Twilight gets this energy from. The first law of thermodynamics states that energy is always conserved in a closed system, so it has to come from somewhere. Animals get their energy from the foods they eat, so Twilight must have to eat constantly to fuel her magic, right? Actually, I was surprised to find that it’s not nearly as impossible as I’d thought. 2.3 MJ is about equivalent to 560 kcal (food calories are measured in kilocalories, so a 100-calorie snack actually contains 100,000 calories of energy); this is roughly the amount of energy in a Big Mac. So, worst case, Twilight is a bit hungry after her rage-splosion.

Not this Big Mac. He contains considerably more calories, plus he's holding a cupcake.

IN SUMMARY:
Power: 570 kW (6.6 Sparkles)
Brightness at 2m distant: 170,000 lx (2 Rapidashes per square meter)
Energy: 2.3 MJ (1.02 Big Macs)

Now, if you’ll excuse me, I have to go and prepare my offering to appease the Purple One. I know she’s pretty reasonable, but with that kind of power, it never hurts to be careful.

Note: A more reasonable value for Twilight’s luminous efficacy is around 3 lu/W, rather than 15. Her power output is unchanged, but her brightness from Pinkie’s perspective is now about 35,000 lx. She’s still brighter than daylight, but she probably wouldn’t hurt your eyes as much.

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