Which is Bigger, the Sun or the Earth?  Check it Yourself!

Once you’ve convinced yourself the Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon.  It just takes a couple of triangles, and a bit of Moon-gazing.

Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two.  That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.

It’s Just a Phase

The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again.  The phases arise because there are two simple ways of dividing the Moon in half:

  • At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
  • At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half.  The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)  

The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night.   See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle. 

Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.

When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky. 

At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.

When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit.  Why not “First Half?”  Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?

This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!

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How to Figure Out the Distance to the Moon Yourself

Last time I described an easy way for you to determine the size of the Moon — easier than the famous techniques used by the classical Greeks. (We don’t need to know the Earth’s circumference, as they did, if we’re ok with a moderately precise estimate.) Once you’ve done that, there’s an simple method, well known since classical times, for figuring out how far away the Earth’s companion is. That’s what I’ll describe in this short post.

(What’s not so easy is to determine the distance and size of the Sun. The classical Greeks failed in their efforts. We’ll need a more modern approach… but that’s for next week.)

Size Versus Distance

Even the early classical Greeks knew something about the Sun, just from the fact that the Moon and Sun appear roughly the same size to our eyes — that is, they occupy about the same amount of sky. If the Sun is twice as far away as the Moon, its diameter must be twice as big, in order that it appear the same size. That’s illustrated in the figure below. If it is ten times as far away, its diameter must be ten times as big. If it’s four hundred times as far away, its diameter is four hundred times as big. (Spoiler: that last one’s the truth; but we’ll get to it later.)

If the Moon is a distance L away from you, and another object twice as far away appears to be the same size in the sky, then that object’s diameter must be twice the Moon’s diameter D. This logic applies more generally to objects further and nearer than the Moon.

You can run this logic in the other direction; if something perfectly blocks the Moon, then if it’s ten times closer than the Moon its diameter must be ten times smaller. If it’s a billion times closer than the Moon, it must be a billion times smaller.

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How to Figure Out the Size of the Moon Yourself


Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?

If the Moon were close to Earth, then at any one time it would only be visible over a small part of the Earth, as indicated in light blue. But in fact (except at new moon) about half the Earth can see it at a time.

Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.

The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)

The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.

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The Earth’s Shape and Size? You Can Measure it Yourself — Part 2

In my last post, I showed, using only simple arithmetic, that the observed atmospheric effects from the January 15th volcanic explosion in the Kingdom on Tonga are consistent with a round Earth. From the timing of the observed spikes in pressure, seen around the world, one can work out how long the pressure wave took … Read more

Geometry From Within: Evidence for a Round Earth

It’s a lot easier to map the Earth than it used to be.  Before satellites, you had to do many careful measurements of distances and directions, at many different locations around the world, and combine them all to build a picture of a world you couldn’t see.  That’s part of why maps and globes made … Read more

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