Prefix
In order to uncover how powerful a My Little Pony: Friendship is Magic unicorn is, we must first reveal the astronomical and cosmetically involved aspects of the providence Equestria. It is prospectively smaller than Earth, perhaps the size of Mars. The lower gravity would explain how pegasi can fly with such small wings. It would also allow for more fantastic architecture (the type we see demonstrated in Canterlot). Due to its small size, the planet would have quickly run out of geothermal heat. It would rely almost entirely on the sun for warmth.
Being a geologically dead planet, Equestria's tectonic plates would have 'frozen' in place, forever fused together. The solidifying of the planet's core would make it less susceptible to tidal forces. Smaller tides would mean that global weather patterns would be dampened, vindicating why the pegasi need to manage the weather. Therefore, with the core of the planet dead, it would be lacking a magnetic field. This means that Equestria could not have a sun like Earth's, as the solar wind would slowly strip away all of Equestria's atmospheric water. A significantly smaller star would not have as much coronal mass ejections and would be safer for Equestria. However, this would mean the star would also need to be closer to make up for the lower output.
The sun would likely be below the threshold for self-sustained fusion. If forced to do such a thing, it might sputter and burn for a few hundred years, but never ignite into a true celestial fusion furnace. Thus, magic is needed to sustain the sun. Given this understanding, if Celestia died, this would not mark the sun's destruction, but rather its regression to a black dwarf. It would be possible to reignite the corpse of Celestia's 'fake' sun.
Facts
-The apparent brightness of Earth’s sun is the same as that as the sun controlled by Celestia.
-The apparent size of Earth’s sun is the same as the sun controlled by Celestia.
-The blackbody curve of Earth’s sun is the same as the sun controlled by Celestia.
-Therefore, Equestria’s need for pegasi to maintain weather is due to the planet being geologically dead. This has caused it
to have a mostly stagnate climate (with places like the Everfree forest being the exception because of various factors (mountain ranges, bodies of water, etc...).
-Celestia’s sun was not a true star. Assuming Equestria is geologically dead would mean that it has no protection from coronal mass ejections (unless the moon has a magnetic field).
-Celestia’s sun requires a constant flux of energy to maintain. It is not simply ‘running’ of a preexisting well of energy. This rules out class F white dwarfs. This means Celestia needs to constantly feed the sun energy. This also means her sun needs to be relatively small (so that gravitational pressure is not causing fusion).
-The fact that Celestia’s cutie mark is a sun leads me to believe maintaining the sun is her special talent. I will assume her efficiency is 100%, i.e., all of the energy she pours into her sun is converted to luminous output.
-Celestia would put her sun in a position as to minimize the burden it would place upon her.
Because the geothermal energy fluxes of Earth, it is several orders of magnitude lower than the solar energy flux, which for the purpose of energy calculations we do not need to worry about if Equestria is geologically active. Any discrepancies that exist could be accounted for by higher levels of greenhouse gas (perhaps also helping to explain the lift generation of pegasi).
Physics Application
For simplicity, ESun will mean Earth’s sun. CSun will mean Celestia’s sun.
The brightness multiplied by 4pi times the distance squared will give us the luminosity (or energy output). We will assume that the brightness of Esun and Csun are equal so we can find the luminosity as a function of solar distance. To reduce the required energy to run the sun, we will be assuming that Celestia keeps it close to the planet. I am assuming a Low Earth Orbit (or in this case, Low Equestria Orbit) of 2000km.
This states that the power output (Luminosity) of the CSun needs to be 210 Petawatts (thereabouts). I will put this in perspective a bit, but we can now use the Stefan-Boltzmann law to find the size of her sun.
The Stefan–Boltzmann law, states that the total energy radiated per unit surface area of a black body per unit time (also known as the black-body irradiation, or emissive power), is directly proportional to the fourth power of the black body's thermodynamic temperature. Thus at 100 K the energy flux density is 5.67 W/m2, at 1000 K 56,700 W/m2. Convert it, and the distance is 9.4km.
This is only about twice the size of Manhattan (quite small), only about 0.5% the size of our moon. This puts it on par with Neutron Stars in terms of size. I should also mention that at Low Equestrian Orbit it is significantly closer than the moon. We now have a good understanding of what the sun might be like. I would assume its mass is only large enough to hold it together against the outwards energy flux (I haven’t done those calculations yet, and it may well turn out it will need to be massless).
To put the energy output into perspective:
121 Petawatts is how many joules of energy per second the sun is emitting. Not all of it reaches Equestria; some just goes out into space.
Humanity is used on average 16 Terawatts of power (or 16 Terajoules) of energy a second during 2010. This is 1300 times less than our value of 210 Petawatts.
Things Celestia could accomplish by redirecting the energy she sends to the sun for one second:
Power a hurricane for 16 minutes.
Cause a 60 megaton explosion (the largest hydrogen bomb ever detonated was 50 megatons, most are much smaller than that. She can do this once every second).
Launch three billion kilograms to the moon (about six billion pounds).
Since her special talent is the sun, not sending things to the moon, causing hurricanes, or blowing things up, she is probably less efficient at those things. However, even assuming 1% efficiency at non-solar activities, it is still very impressive.
Twilight Sparkle in Season 1:
Based on the Ursa Minor’s size, and assuming its density is the same as a pony, the Ursa Minor weighs around 100,000 kilos. Add this weight to the water tower she was also levitating, and you get 2,185,000 kilos. Assuming she was near her limit, this puts Twilight`s peak output at around 3 megawatts (assuming that her apparent struggle is not an issue of finesse, or control). I suspect the average power output of a skilled unicorn (like Trixie) is closer to fifty kilowatts, and your average unicorn is capable of 1/2 that.
Chapter Summary:
-Princess Celestia was capable of creating a 60 megaton explosion every second (a 60 megaton explosion can send the Hawaiian Islands into fallout). A 1 megaton explosion is 80 times as powerful as the atomic bomb dropped on Hiroshima.
-In episode 6, Twilight Sparkle could create a 0.0000108 megaton explosion.
Author`s Note-
Please comment or PM me about any errors you see, grammar or scientific. Expect Chapter 2- The Fundamental Nature of Magic- tomorrow!
The Fundamental Nature of Magic
Facts
Magic is a force that acts at a distance; it does not require contact.
A unicorn's horn is somehow involved in the ability to use magic.
The 'color' of magic can vary from pony to pony.
In some cases, using magic to lift things seems to apply a reactive response force (Twilight Sparkle needed to dig her hooves into the ground to lift and Ursa Minor).
In other scenarios, there does not seem to be a response force (The unicorn filly recently born to the Cake family was able to levitate itself. It's possible that this is some type of anti-gravity or mass-manipulation, but that seems unlikely). This seems, at first, to be a violation of one of the basic laws of motion.
I will make three assumptions:
We can, in fact, explain all aspects of magic within a logical and self-consistent mathematical framework.
We can accomplish the above while remaining consistent with our current understanding of physics.
In order for assumption 1 and 2 to be true, we will need to formulate our theory of magic with physics as our inspiration.
To begin with, let us take one of the most fundamental spells that all unicorns learn while still just a foal:
Telekinesis
Telekinesis is the ability to move objects without touching them through the power of one's mind by utilizing magic. Ordinarily, the object being moved is enveloped in a glow of magic the same color as the unicorn’s horn. Telekinesis is able to move an object in all three spatial directions and affect rotation.
What makes telekinesis so odd and difficult to understand is that every other physical force that acts at a distance is either an attractive or repulsive force. Telekinesis seems to have much more freedom in its directionality.
We will now be relying on assumption 3. All forces that are capable of acting at a distance are actually forces between fundamental 'charges.' In the case of electromagnetism, this is either an attractive or repulsive between positive and negative electric charges. In the case of gravity, it is an attractive force between positive 'charges' of mass. Negative mass matter is another theory, due to its pure conjecture, but we will be ignoring that for the sake of this discussion.
Obviously, telekinesis is not just repulsive or attractive. It can also move objects perpendicular to the vector between the two objects.
Vectors
We also have the fact that most objects do not seem to be constantly affecting each other magically. In the case of telekinesis, it is only when a unicorn actively uses the magic that anything happens. A unicorn may be able to levitate two boxes, but the boxes will not levitate one another.
Therefore, I posit that there must be two types of magical charge:
Inactive magical charge. This will in no way interact with other inactive magical charges, but will respond to magical fields. However, it will not create a magical field of its own. The magnitude of inactive charge in an object will affect how easy the object is to affect with magic. Therefore we can also refer to inactive charge as magical conductivity.
Active magical charge. Active magical charge will exert a force on inactive magical charges. Active magical charge is the source of magical fields.
And now onto fact 2: the importance of a unicorn’s horn. There are two possible explanations at this point of the role the horn plays in magic.
The horn is like an active magical charge battery. The exterior surface of the horn acts as type of reverse faraday cage. A faraday cage is a construction which can protect whatever is inside from external electric fields. When the unicorn wishes to perform magic, some component of the pony's biology 'breaks' the cage allowing the internal magical field to escape from the confines of the horn. This would allow for the magical field to be directed in a specific direction. Further exploration of this possibility will need to wait for later.
Accelerating inactive magical charge results in a magical field. Spirals in the horn conduct this charge, resulting in the generation in a field. This is the same basic way in which an electromagnet works. Once again, further investigation of this will wait for later.
While we will need to hold off on any definitive decision regarding the importance of a unicorn's horn, we nonetheless have established two reasonable possibilities.
We still have not explained the directionality of telekinesis, only its selectivity.
Because telekinesis is beyond a simple attractive and repulsive force, active magical charge, at least, must be more complicated than we have so far presumed.
Let us assume that active magical charge is made of three components, an X, Y, and Z component. These components compose an 'active magical charge vector.' However, these components are dependent on the directionality of the main magic vector. The direction the telekinetic force applies will then be the cross product between the active magical charge vector and the unit vector along the direction between the unicorn and object it is levitating.
To get the magnitude of the force we must also multiply it by the magical conductivity (keeping in mind that we are being inspired by the electromagnetic force, ordinarily you multiply two charges together, but here we are multiplying two different types of charges together).
The effect of the distance, at worst, results in the magical force decreasing as the inverse of the distance squared. However, this assumes the magical field is beaming off in all directions and that the field lines are decreasing in density. A capable unicorn can 'tighten' his or her magical field and reduce the exponent on the distance to a theoretical zero at maximum tightness. Of course, this ignores the magical interference caused by intervening air, which we will discuss later.
In practice, because objects are not of infinitesimal size, the unicorn will need to allow some widening of the magical field in order to encompass an object. Ironically, because an object's angular size decreases with distance, a unicorn will be able to use his or her magic more efficiently at greater distances.
Chapter Summary
F = S*(((X*Xhat+Y*Yhat+Z*Zhat) of object 1 x Rhat) * C of object 2) + ((X*Xhat+Y*Yhat+Z*Zhat) of object 2 x Rhat) * C of object 1))/(R^A)
Key:
F- The force between the objects (a vector)
S- The sparkles constant, a number that I have yet to measure X, Y, Z; letters that represent the X, Y, and Z components of an active magical charge
Xhat, Yhat, Zhat- Directions in the x, y, and z direction
Rhat- the direction between the two objects
C- The magical conductivity (inactive magical charge) of an object
R- The distance between the objects
A- A parameter between 2 and 0 based on the 'tightness' and focus of the magical field.
Note that this formulation obeys Newton's law that all forces must have an equal and opposite force.
