Units

Some rules for natural units. The type that we use are fairly simplistic, the Planck natural units.

$$c = 1$$

$$\hbar = 1$$

$$G = 1$$

So what consequences does this have? As for \(c = 1\), time and distance are the same units, [E] = [m]. As for \(\hbar = 1\), this means that Energy are inversely related to distance or time. Some important numbers:

$$m_{proton} \sim 1 \textrm{Gev}$$

$$\alpha = \frac{1}{137}$$

$$m_{electron} \sim 10^{-3} \textrm{GeV}$$

$$\alpha_{G} = G_{N} m^{2}_{proton} \sim 10^{-39}$$

Such as,

$$E \sim - \frac{Z \alpha}{r} + \frac{1}{m_{electron} r^{2}}$$

$$p \times r \sim 1$$

$$r_{atom} \sim \frac{1}{Z \alpha m_{electron}}$$

$$r_{nucleus} \sim Z^{1/3} r_{proton} \sim \frac{Z^{1/3}}{m_{proton}}$$

$$\rho_{solid} \sim \frac{Zm_{proton}}{r^{3}_{atom}}$$

$$p_{e} \sim \frac{1}{r_{atom}} \sim m_{electron}$$

$$E_{atom} \sim \frac{Z \alpha}{r_{atom}} \sim Z^{2} \alpha^{2} m_{electron}$$

For Planetary stuff:

$$E_{gravity} \sim \frac{G_{N} M^{2}_{Planet}}{R_{Planet}}$$

$$P_{gravity} \sim \frac{G_{N} M^{2}_{Planet}}{R^{4}_{Planet}}$$

$$M_{Planet} \sim \rho_{solid} \times R^{3}_{Planet} \sim \frac{Zm_{proton} R^{3}_{Planet}}{r^{3}_{atom}}$$

Remember that any P_solid = E/V = F/A, and rho_solid = M/V