# Role of power laws in astronomy?

I often see astronomers fitting data to power laws. What about power laws makes them so useful in astronomy? Why are so many astronomical observations well-fit by power laws? I know it's a relation between two quantities, but why is a power law the go-to model?

### Scale invariance and self-similarity

Power laws basically mean that there is no preferred scale, i.e. that a physical property is scale invariant. Any deviation from a power law means that the Universe somehow thinks that the scale where it breaks down has some special significance. In other words, a power law describes self-similarity$$^\dagger$$.

You can see this mathematically by considering a scaling of the independent variable $$x$$ in a power law $$y(x)=x^n$$ by a constant factor $$\lambda$$; the result is just a scaling of the dependent variable by another factor $$y(\lambda x) = (\lambda x)^n = \lambda^n x^n = \lambda^n y.$$

Of course, nothing is infinitely self-similar, but many processes are to certain degree. And if you have a process that depends on many sub-processes that are themselves described by power laws within some domain, the result will evolve toward a power law as well.

### Power laws in astronomy…

In astronomy, you often see that power laws are a good description of some relation out to a certain scale, after which you see an "exponential cut-off", i.e. where the relation transitions to the steeper exponential.

A notable example of this is the halo mass function (HMF) of structures in the Universe, where the distributions of (dark matter halo) masses are well described by power law from the lowest masses out to a "characteristic" mass, after which the number density of object decreases rapidly with mass (Press & Schechter 1974). In fact, many relations in astronomy can be traced back to this hierarchical formalism which has its origin in the assumption that overdensities are Gaussian distributed. Scaling relations between masses, sizes, star formation rates, luminosities, feedback mechanisms, etc. then result in new power laws.

Assuming a constant mass-to-light ratio $$M/L$$ of galaxies, the luminosity function (LF) of galaxies should follow closely the MF. However, observationally we find that the LF is suppressed at both higher and lower mass scales than the characteristic mass. The reason is thought to be feedback preventing star formation by heating the gas and/or pushing it out of the galaxy; at the high-mass end, you have powerful active galactic nuclei (AGN), while at the low-mass end, where the gravitational potential is smaller, stellar activity can do the job.

Predicted LF assuming a Jenkins et al (2001) with a constant $$M/L$$ (cyan), and observed $$K$$ band LF (yellow). Figure from Benson et al. (2003) with my own annotations.

The Milky Way, it turns out, is rather close to being such a characteristic mass and luminosity, which is why you sometimes hear that the Milky Way is a "typical" galaxy.

The LF is typically described by a Schechter function. In recent years though we are seeing evidence that the exponential cut-off is a bit too drastic, and that the high-mass end may be better described by another power law, albeit with a steep index, i.e. a so-called broken power law (see e.g. Oesch et al. 2018).

### …and in nature in general

Power laws are common not only in astronomy, but in all of nature, and even in "artificial" settings, such as linguistics (the occurrence of words in a language), economics (e.g. income distributions), and sociology (e.g. city sizes).

In some cases, the reason can be attributed to the fact that even non-power laws often behave like power laws in limited intervals. Other functions that occur in nature, such as $$\exp(x)$$ and $$\sin(x)$$, can be Taylor-expanded, and near "critical points" the first term dominates.

I'm actually not sure if the reason for the ubiquity of power laws in nature is entirely understood. I think you can explain it in many cases, but not in general. One example that can be explained, mathematically, is that if stochastic processes with exponential growth in expectation are "killed" (􏰒or observed􏰀) randomly, the distribution of the killed or observed state exhibits power-law behavior in one or both tails (Reed & Hughes 2002).

### But beware!

That said, I think there sometimes is a tendency to assume a power law relation between variables where none is necessarily expected. A power law is a linear relation if you take the logarithm of both $$x$$ and $$y$$, and probably the fact that deviations from some relation are suppressed in a log-log plot is sometimes a bit too appealing…

$$\dagger$$To be fair, scale invariance and self-similarity is not exactly the same, in that the former describes a continuous scaling, while the latter includes discrete scaling such as fractals which are identical to themselves only in certain steps and certain regions.

I have to admit that power-laws (in general) used to be my shtick so I am happy to shed some light on their general importance in physics which obviously also hold for astronomy.

The main idea of a power law is nicely written in Wikipedia, but the essential part is the I highlighted in the following quote:

[A Powerlaw is] a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities

The (mathematically) nice part is exactly this, that over a really large range of the x-values, the y-values follow the same dependency. Usually, "really large range" means values spreading over 3 to even 10 powers of ten.

When fitting power laws in practice, there are effects on the sides of the scale, meaning for small and large $$x$$-values, where usually attributed to "finite size effects".

$$P=K\rho ^{{(n+1)/n}}$$
where $$P$$ is pressure, $$\rho$$ is density and $$K$$ is a constant of proportionality. The constant $$n$$ is known as the polytropic index; note however that the polytropic index has an alternative definition as with $$n$$ as the exponent.