The Earth, of mass $m$, being split into two 'earth-chunks', each of mass $\frac{m}{2}$
Firstly, let us consider the change in energy between the two systems. I don't know anything about general relativity yet, so I'm going to stick completely to Newton's Gravitational Law, hopefully it'll be a good enough approximation. The minimum energy required to shift from the left side to the right side will just be the change in gravitational potential energy between the two states. (as the total kinetic energy doesn't change).
The left hand side's potential energy is:
$$E_1 = -\frac{GM_{sun}M_{earth}}{R_{SE}} + E_{SI} + E_{E1}$$
and the right hand side's potential energy is:
$$E_2 = -\frac{GM_{sun}M_{earth}}{R_{SE}} -\frac{GM_{earth}^2}{2R_{SE}} + E_{SI} + 2E_{E2}$$
Where $M_{sun}$ is the mass of the sun, $M_{earth}$ is the mass of the earth, $R_{SE}$ is the radius of the earth's orbit (which I will assume to be roughly circular), $E_{SI}$ is the internal gravitational potential energy of the sun (the potential energy held between parts of the sun itself), $E_{E1}$ is the internal gravitational potential energy of the Earth before it gets split, and $E_{E2}$ is the gravitational potential energy of one of the two 'earth-chunks' after it gets split. Taking the difference, we find that:
$$\Delta E = -\frac{GM_{earth}^2}{2R_{SE}} + 2E_{E2} - E_{E1}$$
The first term is easy enough to compute, just a matter of plugging numbers into the formula. The first term ends up being roughly $-7.956 * 10^{27} J$. To compute the other two terms, however, certain assumptions have to be made. At first, I considered assuming the earth was of uniform density, but that seemed too ridiculous an assumption to yield any meaningful result, so instead I found a graph of the density of the earth versus the radius:
-courtesy of AllenMcC from Wikipedia-
Instead of assuming constant density, I instead assumed the Earth was made from separate layers (i.e. inner core, outer core, mantle, and crust), and that each of these layers had a uniform density (which is not too harsh of an approximation). Using this assumption, and the numbers from hyperphysics, I calculated the mass of the earth (from my approximation) to be $5.973 * 10^{24} kg$, not too far from the real value of $5.972*10^{24} kg$, though that COULD just be a coincidence I guess. I also assumed, that when splitting into the two 'earth-chunks', each 'earth-chunk' would receive half of each layer (i.e. would contain 50% of the total mantle by volume, 50% of the total core by volume, and 50% of the total crust by volume).
Anyhow, with this new information, these are the values I calculated (using integration):
$$E_1 = -2.447 * 10^{32} J$$
$$2*E_2 = -1.541 * 10^{32} J$$
This is mind boggling. When I started the problem I thought with conviction that the contribution from the internal energy would be smaller than the contribution from the separation of the two Earth planets. Instead it's about $10^4$ greater. (In other words, splitting the Earth into two 'earth-chunks' spends a lot more energy than sending one 'earth-chunk' to the other side of the sun). Finding the total change in energy:
$$\Delta E = 9.06*10^{31} J$$
Exactly how much energy is this? The International Energy Agency estimated the world energy consumption of 2012 to be about $5.598*10^{20}J$. The amount of energy required to split the earth requires on the order of $10^{11}$ years. That's on the order of 100 billion years. In comparison, the 'age' of the universe is estimated to be only 13.8 billion years.
But that's OK, you say, because we have nuclear bombs! Here is a graphic depicting the strength of nuclear bombs in kilotons of TNT:
1 Kiloton of TNT gives on the order of $10^{12}J$
Let's be generous, and say that ALL the bombs currently being held by nations worldwide are on the top of the scale ($10^5$ kilotons worth of TNT). That means each nuclear bomb gives $10^{17}J$ of energy. According to Ploughshares, there is a total of $15,695$ nuclear weapons stockpiled. Let's be extra generous and say that there are $100,000$ in total (many are just unreported and hidden, say underground). That means our entire nuclear stockpile gives a total of $10^{22}J$ of energy. That is extremely pitiful. We would need a BILLION times as many nuclear weapons before we could reach the amount of energy required.
The total mass of all humans currently can be estimated to be roughly $10^{11}kg$. ($7$ billion people, average weight is roughly $50-70kg$ for adults). The amount of energy required would also be equivalent to converting 10,000 times the world population's mass into pure energy (by $E = mc^2$). That's a lot.
There are however many factors I have ignored, and many things I have assumed to be true that might not be true. For instance, I haven't considered the Earth's rotational energy and what happens to it when the Earth splits. Using my 'layered-earth' assumption, I calculated the Earth's rotational energy to be on the order of $10^{29}$. That's certainly enough to make a significant impact on the process. We already use the Earth's rotational energy in launching rockets (when we launch them near the equator to utilize the Earth's rotational speed). (-related link).
If there's one thing I've taken from this experience, it's that writing a blog post isn't rocket science, but trying to see how much energy it would require to split the earth IS rocket science.
-Note to self: when doing future stuff, REMEMBER TO CONVERT TO SI UNITS YOU NOOB gah caused me so much grief when I forgot to convert the radii of the layers of the earth from km to metres in like the first step, and I kept getting strange nonsensical answers-
