Thursday, January 12, 2017

When You Think Upon a Star

Among the sciences, astronomy benefits from widespread public appeal. Hilariously large numbers and gorgeous images make it an attractive source for science news. The result is that some difficult notions from astronomy have managed to penetrate successfully into public awareness. For example, this meme, which I've run across several times:

I got this image here (which, incidentally, is a blog post doing exactly what I'm about to do), but I've seen this meme in other forms elsewhere and have no idea what its original source is.
I'd like to say that I feel conflicted about this meme—that I'm happy the joke relies on knowledge of astronomy (the immense size of the universe versus the finite speed of light), despite the specific fact it calls upon being incorrect (visible stars are almost certainly still alive)—but that would be a lie, because I'm an enormous pedant.

However, in this post I'm going to steer my pedantry in what I hope is a slightly more interesting (and less annoying) direction, toward mathematical reasoning. That is, while I think it's great that the public has been able to learn certain specific facts about astronomy (and other sciences), I think it would be far more valuable if the public learned how to apply mathematical reasoning to claims they encounter.

Here's why: as a recent graduate with an official degree in astronomy and all that jazz, I happen to simply know the fact that, in general, the stars we can see with the naked eye are close enough, and live long enough, to still be alive by the time their light reaches us.

But even if I didn't know that fact, I could arrive at it by constructing an argument from some more readily available facts. And this argument, although mathematical in nature, doesn't involve anything more complicated than a bit of algebra, such that anyone who gets out of high school should be able to reach the same conclusion.

Now, the joke's humor relies on some common facts from astronomy: stars are far away, light is slow compared to the size of the universe, stars eventually die. But before we get into the mathematical meat of evaluating this claim, let's think about another common fact: our sun is 4.5 billion years old (give or take), and it's roughly halfway through its life, so it's got another several billion years to go.

In order for the sun to be dead by the time an alien civilization see its light, that civilization would have to be farther away in light years than the sun's remaining age in years. That is, the alien civilization would have to be many billions of light years away. So if we take our sun as typical, then the above meme is only true if we can see, by the naked eye, stars that are billions of light years away. We can't, and as I'll show in a bit, we don't even have to assume our sun is typical for this argument to work (which it isn't, really). But this is the structure of the mathematical argument: compare the lifetimes of stars we can see with the naked eye to their distances from us.

A few more astronomical facts are necessary to work this out, some of which can be gotten by a bit of googling, and one which, I admit, most people probably aren't aware of. This fact, which makes evaluating the claim very easy, is that the more luminous a (main sequence) star is, the shorter it lives. This means the most luminous stars (which are the most likely to be visible by the naked eye at great light travel times) are the best candidates for stars that are dead by the time their light reaches us. If the claim fails for these stars, it fails for all stars.

The most luminous stars live about a million years and are about a million times brighter than the sun. Now, it's always possible that a star we're seeing just happens to be at the end of its life, but all else being equal, if we pick stars at random out of the sky, then on average they will be halfway through their lives, just like (coincidentally) our sun (not strictly true, because there is some selection bias to the stars we can see).

To be visible by the naked eye, a star needs to have an apparent magnitude of 6 or lower.

For the sun to be magnitude 6 (it's currently an obscenely bright -27), it would have to be about 60 light years away. (There's some math involving logarithms here, but there are tools online that could get you this answer.)

How bright a star appears to us is proportional to its intrinsic brightness and inversely proportional to the square of its distance from us. That is, if star A and star B are identical but star B is twice as far away, it looks 1/4 as bright as star A.

And that's everything we need to evaluate the claim. Now here's how we construct the argument. A star is dead by the time its light reaches us if its remaining lifetime in years is less than its distance in light years. It is visible with the naked eye if its intrinsic brightness relative to the Sun is greater than the square root of its distance relative to the Sun's distance at magnitude 6.

Let me unpack that second statement a bit. Say a star is intrinsically four times as bright as the sun. If it's also magnitude 6 (just visible), then it needs to be farther away than the sun. Specifically, a star four times as bright as the Sun will be just visible at twice the distance (square root of 4) of the magnitude 6 sun: 120 light years. If it is farther away, it is too dim for us to wish upon it.

The brightest stars are 1,000,000 times more luminous than the sun, which means they are the same apparent brightness as the Sun when they are 1,000 times farther away. If the sun is just visible at 60 light years, then the brightest stars are just visible at 60,000 light years. Is 60,000 light years greater than the (on average) half a million years the star will have left to live? No. At that distance, the star could only be dead by the time we see it if it were already 95% of the way through its life. For less luminous stars which live longer, that percentage gets even higher, which makes it much less likely that we ever see such a star.

When we learned algebra via word problems, we were supposed to be learning how to solve problems like these. And while most of us probably managed to get through those word problems successfully, it's been my observation that most of us don't apply this kind of analysis outside of school, to things like evaluating claims that have mathematical content. While it's not vital to the health of a democracy that we be pedantic about random Facebook memes, it might be useful for us to be able to think carefully about scientific claims, at least when the facts and math involved don't require a PhD.

Outside of learning a bunch of astronomical facts, one of the most valuable (academic) lessons I acquired while getting my degree was learning how to bring mathematical tools to bear on a problem. I'm sure this blog post doesn't really have what it takes to impart that same lesson on others, but I hope it reveals a bit of the process. If I could wish upon a star (and I were feeling altruistic), I might wish for an educational system that did a better job of that.