Equations: Formulas for obfuscation

Graham Doskoch
Jul 24, 2017 · 5 min read

I realized recently that I do have a favorite answer of mine on Stack Exchange. It’s one of two answers I’ve written to a question I asked almost two years ago on Worldbuilding Stack Exchange, entitled Can stars that are not powered by nuclear fusion exist? The question boils down to that title: Is it possible to have an object that looks and behaves like a normal star, but does not generate its energy from nuclear fusion?

I wasn’t particularly happy with the answers I received (although the question did get quite a bit of attention), so I did plenty of thinking on my own over time. The first solution I came up with involved a massive body oscillating in and out of a plane containing a disk of gas. However, the answer had several issues:

  1. It was incredibly contrived and unlikely to form in nature.
  2. I wasn’t able to get great quantitative results.
  3. The whole setup seemed unstable.

About two months ago, I came up with a much better solution involving an object called a quasi-star — a hypothetical ball of gas from the early universe surrounding a black hole. Models of quasi-stars have been created in some detail, though no observations have yet been made. My idea was that a very low-mass quasi-star could resemble a massive giant or supergiant star. I did some calculations and simulations, found that the idea was probably feasible (and not too contrived!), and wrote an answer explaining it.

I’m very happy with that solution. I haven’t been able to get anyone else to check the calculations — if you know someone who can, help would be obliged! — but I’ve gone through them enough that I’m reasonably confident that it will work. Assuming I haven’t made a grievous mistake anywhere, I’ve proved that the object meets all the requirements set forth in my question. It works! I like to think that I’m conscious of my own fallibility, so I assume that there’s a mistake somewhere in there, but my results seem to be correct.

That said, I don’t like the answer, and the reason is simple: It’s terribly written. There are too many numbers, too many equations, and too many other things that make most people’s eyes glaze over. I did ask people for advice, and I did some restructuring to make things, clearer, but it’s still not as good as I’d like. That’s arguably a great failure on the part of me, the writer.

The answer is long, one of the longer ones I’ve written over the past couple of years. But that’s not the problem — after all, good science writers can make long articles quite appealing and interesting. The problem, I think, is with the equations.

A famous cartoon by Sidney Harris

Equations are not always clear the first time you see them. There are two approaches you can take when trying to understand them: looking at the physical meaning of each term and variable (which is time-consuming), or take the “Shut up and calculate!” approach, which showcases the beauty of the equations but loses their connections to the physical world. In either case, it can be rather difficult to understand what an equation means right off the bat.

When writing for a lay audience, the optimal solution is to simply use words. Words have elegance — which equations do, too — simplicity, and accessibility. They can make even the most complicated subjects seem appealing.

But words do have some downsides. For instance, words aren’t quite as handy when they’re used to explain a proof, or the results of calculations. My answer requires mathematical results, and I’m not a huge fan of simply adding in the end of my calculations at the end. Anyone who’s gotten points docked on a test for not showing work can sympathize — you need to prove that you’re not just pulling the numbers out of the ether.

Let me take a look at Equation 1 in my answer. It’s a sort of equation of state for a polytrope. Polytropes are idealized fluids with a simple relationship between the density and the pressure inside the fluid:

That K is a constant of some sort, and the n is called the polytropic index, which determines a lot about the structure of the object — just through this simple equation! Well, once you take some other things into account.

The equation looks simple, but is actually more confusing once you dive into it. How do you calculate K? What happens to the units when the density is raised to a power than isn’t an integer? And what in the name of sanity is the thing you get when you substitute the above into more complex equations?

You get the Lane-Emden equation, actually.

My answer simply gets worse from there, using phrases like “analytical solutions”, “Runge-Kutta method” and “coupled differential equations”. All of these are needed to make the equations turn into something remotely sensible, but unless you really dig in, they, too, can make you want to gouge your eyes (or mine) out. Unfortunately, my answer needs them to prove its conclusions.

Here, I think, is what I’m trying to say: My newer answer seems to be much more physically sound than the previous one. I’ve checked the calculations quite a few times, and it seems to pass all the sanity checks I put it through. I think it’s the best answer I’ve ever written on Stack Exchange — certainly my favorite — because it solves a problem I worked on for a long time.

It is, however, not the best piece of writing I’ve ever posted on Stack Exchange, especially not for a lay audience. It needs revision — and while the equations and numbers will have to stay, the accompanying words need fixing. Eventually, it doesn’t matter if it’s right or not, if nobody ever reads it.

Graham Doskoch

Written by

Astrophysics undergrad, currently looking at the skies through gamma-rays and x-rays. I write about deep sky objects and the future of observational astronomy.

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