Higher powers in physics formulae A huge number of formulae involve second powers (squares).  For example, the very famous:
$$e = m c^2$$
Third powers (cubes) are also common e.g. volumes:
$$V = \frac{4}{3} \pi r^3$$
Fourth powers occur occasionally.  E.g. Stefan-Boltzmann:
$$P = \sigma T^4$$
For a long time, that was the only example of a fourth power that I knew.  I thought that I saw another example recently in this group but I cannot find it now.
Today, I saw a reference to a 6th power in a comment to a question in Astronomy.
Do planets lose energy while rotating?
"Be aware that time to tidal lock scales, to a first approximation, with the sixth (6th) power of the orbital radius (SMA)."
Any examples of even higher powers?  Any more examples of fourth powers and higher?
 A: Another famous fourth power is found in Rayleigh scattering, where the cross section goes up with the frequency to the fourth: $\nu^4$
For sixth and higher powers you got the famous Lennard-Jones molecular potential
$$V_\text{LJ}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right]$$
Usually in molecular potentials you can find high power of the distance because of multipolar expansions.
I cannot think of higher examples, but I think you can find more things like that $1/r^6$ in tidal locking by looking at the physics of precision physics using general relativity or quantum field theory, as any phenomena involving these theories should be vanishing in regular conditions (for not-so-small distances, masses and velocities).
A: In higher dimensions you get arbitrarily high powers, because of the geometry. For example, Coulomb's law in dimension $d$ scales as
$$\vec{F} =\frac{k q_1 q_2}{r^{d-1}}\hat{r}$$
An example of arbitrarily high powers without more spacial dimensions is the field of an electric multipole:

*

*A monopole is created by the electric field of a single particle, and scales as $\propto \frac{1}{r^2}$

*An electric dipole has two opposing charges, and its electric field scales as $\propto \frac{1}{r^3}$

*An electric quadropole consists of two positive and two negative charges,  its field scales as $\frac{1}{r^4}$
In general the electric field drops as $log_2(n) + 2$, where $n$ is the number of particles in the multipole. Out of curiosity, if you take $n$ to be of order Avogadro's number $n=10^{23}$, you get
$$|\vec{E}| \propto \frac{1}{r^{78}}$$
...which is not all that bad for so many particles!
A: The Toda potential is exponential in the coordinates so contains all powers of the coordinates.
