I needed a book

One of my students asked me this, and I told him that I thought stars
would win...but I figured you have probably actually thought about this...

-p-

On Tue, 28 Sep 2004, Michael Milligan wrote:

> Hey folks!
>
> Being as it's expenseive and hard to find, and since you (hopefully)
> only take grad e+m once and always do the homework with other people
> anyway, I never bothered to acquire a copy of Jackson.
>
> Yet, as fate and my big mouth would have it, I've landed a line of
> research that would be made easier by having one handy for the next few
> months.  I.e., while I'm in Israel.
>
> Anyone feel like loaning me their copy for a while?  It'll be about as
> likely as I am to come back in good condition.
>
> Thanks,
> ...Milligan

Ah, a challenge. The best way to win a book. Or, as I've found, a good way to annoy students while teaching them about scientific notation. Naturally, if contrary to the usual wisdom, sand throws down.

Pity he had to add the "on the Earth" bit, because I was all ready to throw in the snarky point that there are vastly more grains of sand than stars. Figure an Earthlike, water-eroded planet contains around <N> grains. Well, it's got to be true that more than 1/<N> stars will harbor such a planet. Especially when we consider just how big that is.

The question Paul really asked, though, is how compares to what we'll call N_*, the number of stars contained in our local Hubble volume. NASA thinks this number is around 10²¹, although 2dF survey data suggest it's closer to 10²³; personally, I'm a little leery of the effect of cosmic variance on the 2dF sample, though -- that's rather a lot larger than previous estimates.

Giving the stars the benefit of the doubt, let's say a typical grain of sand occupies a cubic millimeter (they're generally a little smaller). That puts 10¹⁸ grains in a cubic kilometer.

Let's say a typical beach extends 100 m inland. I don't know how deep the sand generally is, but I'd imagine 10 m is a reasonable estimate. That gives a beach a cross-sectional area of about .001 km². A book and documentary by PBS suggests there are 1.6 million km of coastline on Earth; we'll call it 1 million since not all of that is sandy beach, for a total of 1000 cubic kilometers of beach sand, or about 10²¹ grains.

Already we're up to the lower value of N_*; to push <N> over the top, consider that continental shelves are also sandy, if submerged, and extend 1-10 km from the shore. So chalk up at least another two orders of magnitude for the ground-up rock.

Now, just to be mean, we'll throw in the deserts, where the real sand is hiding. There are at least four really big ones: the Sahara, the Gobi, the Austalian Outback, and our very own Chihuahan-Permian desert. Each is on the order of a thousand miles on a side (some more, some less), and in deep desert, I wouldn't be surprised if the sand goes down rather more than 10 m, perhaps to 100 m or so. A few times a million square kilometers times a reasonable fraction of 100 m comes out to 100,000 cubic kilometers of sand, as much or more as in all the planet's continental shelves, and at least as many as the most generous estimate of N_*.