grain-of-sand:earth

“The total number of stars in the Universe is larger than all the grains of sand on all the beaches of the planet Earth.”
~ Carl Sagan

When I was young I would imagine that if the earth were the size of a baseball then perhaps the sun would be the size of a beach-ball and that they’d be about 20-ft (6 meters) apart at that scale. This seems about right, but it turns out to be very VERY wrong. If the earth was the size of a baseball, the sun would be almost 27-ft (8 meters) in diameter, already more than the distance I imagined separated the two. The distance between the baseball-size earth and the 27-ft diameter sun is about a half-mile.

In other words, imagine a ball that’s nearly three stories high, and you’d have to walk a half-mile to find a baseball-sized earth. The baseball:earth scale isn’t practical to construct a model.

But do astronomical models ever meaningful scale? That is, can we construct a scale model of the solar system and some nearby stars? Let’s see what happens.

Scale the earth down to a single grain of sand on the beach. Imagine a normal sandy beach like the one pictured above, and use an an average sand particle of about 1 mm, nothing exceptional.

At this grain-of-sand:earth scale, we would have a softball-size sun about 38-ft (12 meters) away from the grain of sand earth. This model would fit within most actual beaches, except on this beach, there’d only be 31 grains of sand that have been discovered so far (and four pieces of gravel, and a bunch of silt, but we’ll get to that).

Our moon on this scale would be about an inch away from the grain-of-sand earth, it would be an even smaller particle of sand. The furthest human beings have set foot is only one-inch on this beach.

Mars is another grain of sand, and its moons are so small they wouldn’t be visible, too small even to be called silt.

Jupiter would be 161-ft (49 meters) away from the grain-of-sand earth, but Jupiter would be too large to be considered sand, it would appropriately be called gravel. Jupiter would be the size of a small marble. About 50 meters away from the grain-of-sand earth is a marble-size Jupiter. This beach has four marbles revolving around a softball-size sun. Interestingly, there is more sand revolving around the marbles than there is sand revolving around the softball-size sun. Most of the objects on this beach are tiny bits of silt, i.e., particles too small to be considered grains of sand.

The furthest man-made object, the Voyager spacecrafts, would be far too microscopic to be visible on this beach, but these microscopic spacecraft would be almost a mile away from the softball-size sun.

The grain-of-sand-scale model so far is a pretty lonely beach. This beach would go about a mile inland, a vast and open beach with 31 grains of sand, four marbles of gravel, countless silt particles thrown about (most of it would be invisible to the naked eye). The English language has precise words for silt, sand, and gravel; unfortunately, for solar system objects the English language isn’t as discriminating. Astronomically, we lump gravel together with sand and if they happen to be spherical and revolve around a star we call them “planets”. Some grains of sand are not planets only because they revolve around gravel. A bit silly, and if you’ve ever wondered why Pluto isn’t considered a planet, remember that it’s smaller than Earth’s moon, and would barely be visible as a grain of sand on this scale. Debates about Pluto completely miss the point: our knowledge of the solar system is far deeper than “there are 9 planets, no wait, 8 planets”.

Looking at the grain-of-sand earth, this is about the smallest reasonable scale that we can model, and so far this model fills a one-mile radius. We can count 31 grains of sand, four marbles, and bands of silt revolving within a mile-radius around the softball-size sun.

And this is just our solar system, we’re not into the universe, not yet. Let’s venture out to the closest star.

On this grain-of-sand-scale, the nearest star, Alpha Centauri, would be a bit larger than a softball (about 5.3 inches in diameter). And if our lonely beach with a tiny handful of sand, silt, and gravel were in Los Angeles, then you’d have to walk all the way to Tennessee (somewhere between Memphis and Nashville) to get to Alpha Centauri.

Walking from Los Angeles to Tennessee is far but not unreasonable with basic provisions. Unfortunately, at this scale, the speed of light would also be scaled down. We tend to think that the speed of light is fast, but at this grain-of-sand-scale, the speed of light is slower than a sloth. It’s about 0.05 miles-per-hour, about 84 meters-per-hour (277 feet-per-hour). How long would it take a sloth to get from Los Angeles to Nashville? It doesn’t matter, because at this scale the sloth would be faster than light.

84 meters per hour, that’s the speed that light would travel at this tiny scale, and hence it would take over 16 hours to get across the beach (from the sun to the edge of the solar system).

Those Voyager spacecraft, on this grain-of-sand-scale, are traveling less than half-a-centimeter every hour. That is slower than bamboo grows. When you imagine the solar system, realize that these objects are so far apart that both light and gravity are moving at a snails-pace relative to the distances; and that these scaled down objects would move slower than a plant grows. How long would it take a plant to grow from Los Angeles to Nashville?

Let’s look at the night sky, what about the north star, Polaris?

On this grain-of-sand-scale, Polaris is much bigger than the softball-sized sun, it’s about 16-ft (almost 5 meters) in diameter, and it’d be about 321,000 km away … so even at this grain-of-sand-scale, even though we have to go cross-country to get to the nearest softball-size star, for other stars we’d leave the earth. In the case of Polaris we’d almost be to the moon, and we wouldn’t find a softball, we’d find a 16-foot diameter bolder representing Polaris.

And what about the larger objects in our galactic neighborhood, for example, the star Betelgeuse would be over 220-ft (67 meters) in diameter, 474,000 km away from the grain-of-sand earth. On a clear night you may be able to see the Andromeda galaxy, on our grain-of-sand-scale this Andromeda model would be so large as to fill our actual inner solar system, but it’d be 1.8 billion kilometers away. The brightest quasar viewable from earth, 3C 273, on a grain-of-sand scale would be 1.8 trillion kilometers away.

Even at this tiny scale, a model of the solar system fits within a mile-wide beach, but to model our neighboring stars we’d leave earth and our model becomes as large as the thing we’re trying to model.

The problem with scale models of astronomy is that we try to model the physical stuff and forget that the largest and most interesting thing to model is the empty space itself. The size of a solar system, the size of a galaxy; like our lonely beach with 31 particles of sand; it’s mostly empty space.

This is why astronomical models aren’t to scale, the range of size within human intuition is simply too narrow, we have to continually abstract and abstract and can lose our bearings on just how big and how far away these objects are. Fortunately, it is within the poetry of mathematics that we can artfully express these abstractions. Mathematics becomes the language to convey these otherwise non-intuitive concepts, opening the universe to intelligence beyond scaled models.

References

For this scaling, we’re using the following size descriptors:

  • Silt: 0.002 mm to 0.0625 mm
  • Sand: 0.0625 mm to 2 mm
  • Gravel: 2 mm to 64 mm

To simplify the scaling, imagine the earth is a 1mm grain of sand, this puts the earth:grain-of-sand scale at 12756200000 : 1

Using that scale,

  • A silt particle models any object 25 km to 797 km in diameter
  • A sand particle models any object 797 km to 25,512 km in diameter
  • A piece of gravel models objects up to 816,397 km in diameter

This gives us the following models,

  • Gravel: Jupiter, Saturn, Uranus, Neptune
  • Sand: Mercury, Venus, Earth, Luna, Mars, Ganymede, Titan,
  • Silt: Dysnomia, Chaos, Enceladus, Proteus, Hale-Bopp,

At this scale, the model of the Sun (which is 1.391 million km) is too large for gravel; and scaled down to 4.3 inches is about the size of a softball, but to keep with the metaphor we would call this a Cobble stone.