Date: Wed, 29 Sep 2004 07:40:20 -0500 (CDT) From: "P.E. Robinson"To: Michael Milligan Subject: Re: being cheap You can borrow my copy if... you can give me a quantitative answer to this: are there more stars in the universe than grains of sand on the Earth?

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 _{*}, the number of stars contained in our local Hubble volume. NASA thinks this number is around 10^{21}, although 2dF survey data suggest it's closer to 10^{23}; personally, I'm a little leery of the effect of cosmic variance on the 2dF sample, though -- that's rather a

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 10^{18} 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 km^{2}. 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 10^{21} 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

Now, just to be mean, we'll throw in the deserts, where the _{*}.

That's similar to the reasoning that I came up with

for such a calculation, though I never actually went through with it. You are a big nerd.

-p-

AH HA! A flaw in your assumptions my dear sir! Paul asked about the number of stars in the _UNIVERSE_ which is different from our Hubble Volume. Hence, you'd have to find out if Paul favors an open or closed universe (and which geometry) before answering this question. The answer should be: You have provided insufficent data to answer this question!

This is why I started with the obvious point that there are more sand grains in the universe than stars, no matter how large it is.

Since it's almost certain that the universe is open, if not infinite in some less trivial way, there's clearly going to be more stars, in toto, than any finite number.

Unless, of course, you stick to the strict relativistic notion of time, in which case there are only as many stars as we can see, since in our temporal frame of reference recombination is only just now taking place ~14 GLy away. In which case our local Hubble volume -- or more strictly, the visible universe -- actually is the whole thing just at the moment.

Universes are like parachutes. Open ones are the only way to go. :)

May I please say how incredibly pleased I am that you're geeking up the internet? Thanks. =)

As if the internet needed any more geeking up! But at least he's jumping on the blog bandwagon.

-p-