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The equation with the highest exponent I could find was the coefficient of energy loss of light scattered in an optical fiber: $$ {\displaystyle \alpha _{\text{scat}}={\frac {8\pi ^{3}}{3\lambda ^{4}}}n^{8}p^{2}kT_{\text{f}}\beta } $$ which has a power of $8$ on $n$, the index of refraction.

My question is why is this an outlier? Most physical laws I can think of only scale at most to the power $\pm 3$, so why would seeing an exponent of say, $50$ be regarded as possibly unphysical? I have seen the answers to this question, which are close, but don't answer my question.

Stated another way, is there any reason why the magnitude of the highest order exponent in a formula is inversely correlated to how commonly such an exponent appears in any given formula?

Nine Thousand
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  • Some other examples of high powers are stellar nucleosynthesis rates, where for example for triple alpha process has a rate that goes like $T^{40}$. – jacob1729 Jun 17 '20 at 22:22
  • Don't forget that there are many physical laws which contain exponential functions, and an exponential grows or decays faster than any power. In one sense, something which changes very fast with a high power, but not exponentially, is a weird special case, since fundamental forces that act over "large" distances (e.g. gravitation or electrostatics) follow inverse square laws. – alephzero Jun 17 '20 at 23:29
  • The Lennard-Jones Potential has a $12th$ power – mmesser314 Jun 18 '20 at 00:21

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