Stars and Sand Grains

Are there more actually more stars in the sky, than there are
grains of sand on all the world's beaches?
I think most of us have heard that perennial estimate of the number of stars in the Universe being greater than all of the grains of sand in all of Earth’s beaches.

Sitting on Limantour Beach at Point Reyes awhile back, watching the waves slosh in and out, listening to gulls and feeling very lazy, I found myself looking about me at all that sand, and wondering how it could possibly be true. Reaching out, scooping up a mere handful of grains and letting--what?--a few hundred thousand of the would-be star proxies fall through my fingers, the notion seemed even more absurd.

Raising my eyes from the bit of the cosmos cupped in my hand and taking in the comparatively vast reaches of sand about me--a hundred or so feet between me and the waves, at least a mile or two of beach visible to the north, another stretch to the south, and who knows how many feet of depth beneath the surface? I simply couldn’t believe it. So, I pulled out my journal and started to write down some figures, working out the problem rationally.

So, is it true? Well, here's what I came up with:

Stars: Astronomers have estimated that there are about 200 billion stars in the Milky Way Galaxy. Galaxies come in many sizes, both much larger and considerably smaller than our home galaxy. I don't know what the average number of stars in each galaxy is, but for the sake of this calculation I chose a conservative 10 billion stars per galaxy. Astronomers have also estimated that there are between 50 billion and 100 billion galaxies in the Universe, based on observations made by the Hubble Space Telescope. Again being conservative, I chose the lower figure of 50 billion. So, with those numbers, I calculate a number of stars in the Universe at 10 billion times 50 billion, or 500 billion billion---or in exponential notation, 5 X 10²⁰.

So how does the number of sand grains in the entire world's beaches stack up against that?

To get to that number, I had to do some estimation. First, pulling some numbers out of the air, I decided that an average sandy beach is 30 meters wide (about 100 feet), and 10 meters deep (about 33 feet). Some beaches are wider, some much less so. I don't imagine that the sand on the average beach is as deep as 10 meters---but I've never taken a shovel and found out, either.

Next, I assume that the average sand grain is a millimeter across, giving it a volume of about a cubic millimeter. With that number, I figure the sand grain density to be 1000³, or one billion, sand grains per cubic meter of beach.

The final piece of the equation--after density, width, and depth--is length: the total length of beach shorelines in the entire world. Here's where I made some serious assumptions. Starting with the total length of shorelines of all continents and islands in the world, I got a figure of 356,000 kilometers from the CIA World Factbook. That's 356 million meters.

Now here's where my estimate becomes truly conservative. In my final calculation, I assumed that all 356 million meters of world coastline consisted of sandy beaches-- which is not the case, of course; there are plenty of coastlines that are rocky, pebbly, gravely, ice-covered, or sheer cliffs, all without much, if any, sand.

So what were my results? Well, doing the math, 1 billion grains per cubic meter times a 30 meter beach width times a 10 meter beach depth times a 356 million meter beach length and assuming 100% of the coastlines consist of my hypothetical average beach, I get:

1 billion x 30 x 10 x 356 million x 100% = 1.068 x 10²⁰ grains of sand

Compared to the estimate of stars in the Universe, that's about 5 times as many stars in the Universe as grains of sand in all the beaches in the world! I guess the old adage was not only right, but somewhat of an understatement...

But it's all a thing of scale. I also calculated that there are about 3000 times as many water molecules in a glass of water than there are stars in the Universe...

37.8148 -122.178