End of the Earth VI: Nanobot destruction

Let's destroy the earth with technology.

A while ago, I read the novel Postsingular by Rudy Rucker, and in the first chapter the Earth gets destroyed, and then undestroyed, and then the novel unfolds and the Earth's likelihood is threatened again, and it looks like the Earth will be destroyed, but it isn't.

How does all of this craziness happen you might ask: nanobots! The story revolves around little self-replicating robots. The story explores what it would be like to live in a world where every surface on Earth was coated in little computers, all of which were networked together. It's certainly a neat idea, but whenever you have self-replicating things, you need to worry a bit about what might happen if they get out of control.

Earth Day - Earth Units

In honor of Earth day, I thought I would take a look at what it would mean to do physics in 'Earth' units. What do I mean by that? Well lets be anti-Copernican here, in fact lets assume the opposite of the
Copernican principle, and state that the Earth is privileged in the universe and define all of our units around the Earth.

So, I will put a little subscript earth on all of the 'earth' units. They are to be read as 'earth meters' or 'earth amps', etc. We will take as our starting point the mass, radius and day of the earth, normalizing all of our standards to that. This gives us our initial conversion factors
\[ 1 g_{\oplus} = M_{\oplus} = 5.9742 \times 10^{34} \text{ kg} \]
\[ 1 m_{\oplus} = R_{\oplus} = 6378.1 \text{ km} \]
\[ 1 s_{\oplus} = T_{\oplus} = 86,400 \text{ s} \]

From this, we can figure out what all of the other 'earth' units would be.