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. 

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!

“First Quarter” Versus “Half Lit”

The reason for this nomenclature is that the “Quarter” refers to how far the Moon has traveled in its cycle.  If we divide the cycle of 29¹/₂ Earth days between one New Moon and the next into four quarters, each 7³/₈ days long, then after the first quarter of the cycle the Moon is roughly half lit, after one half the cycle comes the fully lit Full Moon, and at the third quarter the Moon is half lit again. To say this another way, at New Moon the angle between Sun and Moon is nearly zero. After the First Quarter, the angle between the Sun and Moon has increased to about 90 degrees; at Full Moon the angle is about 180 degrees, and by Third Quarter it’s decreased again to about 90 degrees.

These statements would be exactly true if the Moon’s orbit were a perfect circle and lay in the plane of the diagram above. But neither is the case. The orbit is slightly squashed into an ellipse. Consequently the Moon’s speed changes, moving faster when it’s closer to Earth and slower when it is further, and the full Moon seems sometimes a little larger and sometimes a little smaller in the sky.  [I’m not referring here to the difference between the Moon’s appearance at the horizon vs overhead, which is some combination of psychology and atmospheric effects, but rather to its apparent size when directly overhead, which does vary as the Earth-Moon distance varies.]  The orbit’s orientation relative to the line between Earth and Sun changes as the months go by, so each monthly cycle’s a little different. And the small tilt between the plane of the Moon’s orbit and the Sun-Earth line means that the angles at New and Full Moon aren’t quite 0 or 180 degrees, differing instead by up to 5 degrees depending on the month.

So the statements that the First Quarter is 1/4 of a cycle, and happens when the Moon and Sun are 90 degrees apart from Earth’s perspective, aren’t exactly right. The time from New Moon to the time when the Moon and Sun are at a 90 degree angle varies between 6⁵/₈ days and 8¹/₈ days. That’s about a 10% variation, small but not small enough to ignore completely.

Nevertheless, to get a rough estimate, let’s assume the simple statements for now. Once the geometry is clear, we’ll see how to do better.

The appearance of the Sun doesn’t tell us its size and distance. Though its angular size in the sky is about the same as the Moon’s, that doesn’t teach us whether it’s twice as far and twice as large as the Moon (and thus smaller than Earth, Figure 2), or two thousand times as far and two thousand times as large as the Moon. The fact that the apparent size of an object doesn’t change if you increase its size and its distance by the same factor helped us determine the distance to the Moon using a penny). 

Figure 3 figure indicates the Moon’s motion around the Earth (assuming the Moon’s orbit is a circle, and ignoring the relative motion of the Earth and Sun, which would make the figure more complicated to draw but wouldn’t change the conceptual points).  Compare the figure’s two triangles.

• The Moon is half lit — the near-side is equally divided into lit and unlit — when the Moon, Earth and Sun  form a right-angle triangle, with the right angle at the location of the Moon.
• The Moon reaches first quarter — it has traveled 1/4 of its cycle, and lies 90 degrees from the Earth-Sun line — when the Moon, Earth and Sun form a a different right-angle triangle, this time with the right angle at the location of the Earth.

These triangles are not the same! Half-Lit is not First Quarter! As the figure shows, if the Sun were only twice as far as the Moon, the half-lit Moon would occur well before the first-quarter Moon, by a couple of days.  And at first quarter, a little more than seven days after New Moon, the Moon would be considerably more than half lit. 

A subtle point: shouldn’t I be drawing my triangles to the edge of the Earth, not to its center? That’s where you and I would be standing, after all. Here the fact that I’m not drawing distances to scale is important. Remember the Moon is 30 times further away than the diameter of the Earth! In Figure 4 below, I have in fact shown the sizes and distances almost to scale. As a result it’s impossible to even see the Moon, much less see its phases. Earth’s so small, barely visible, that whether I draw the triangles to its center or to its edge makes no difference.

Figure 4 shows, for three different distances to (and sizes for) the Sun, the triangles that indicate where the Moon would be located when half lit. You can see clearly that the further the Sun is from Earth, the closer the half-lit Moon occurs to the first quarter position — the 90 degree position — in its cycle. For a far, large Sun, half lit does equal first quarter.

But you can also see that the difference between half lit and first quarter is already very small once we put the Sun several times further than the Moon. This method won’t easily be able to tell us whether the Sun is ten times, a hundred times, or a million times further away than the Moon; the angles don’t change enough. That’s what kept the classical Greeks from measuring the Sun.

Just to be sure the geometry is clear, I’ve redrawn in Figure 5 the middle triangle of Figure 4, at the misleading scale of Figure 3, with the phase of the Moon clearly indicated. You can see that the half lit position puts the Moon closer to its first quarter position than in Figure 3.

How Big and Far is the Sun? A First Try.

Now you can check for yourself, or just look at a lunar calendar like the one at right, that First Quarter as measured by timing (¹/₄ of a cycle, or about 7³/₈ days) occurs at nearly the same time as the Moon is half-lit. If half-lit occurred a day earlier than first quarter (and a day later than third quarter), that would prove the Sun is close by.  But the calendar shows that this is not the case. The fact that first quarter and half-lit agree to within about a day tells you the Sun can’t be close and small. (You can check this yourself later this week: New Moon was at 1638 UT [1238 New York time] on March 2nd; 7³/₈ days later is March 10 at 138 UT [March 9 at 2138 (9:38 p.m.) New York time.])

How far, and how large, then, might the Sun be?  You might guess that since half-lit seems to agree with first quarter to within about a day, and that’s one out of seven days between New Moon and First Quarter Moon, that the Sun must be at least 7 times further away from Earth than the Moon is.  That’s a good guess, but it turns out there’s a factor of π/2 which brings the real answer down a bit below 5. 

Still, that’s enough.  That means the Sun’s at least five times further than the Moon, and at least five times larger, too… which makes it bigger than the Earth, and at least a million miles (1.6 million km) away.

How Big and Far is the Sun? A Second Try.

But actually, you can do better than this, if instead of counting the days you’re willing to watch the sky and use very simple geometry.  The figure below shows you that we don’t really care if the orbit’s an ellipse, or what ellipse it is, or whether and how it’s tilted relative to the plane of the diagram.  We can just ask either one of two related questions:

• What is the angle that the Moon appears to make with the Sun when it is half lit?

or

• What fraction of the Moon is lit when its position in the sky is 90 degrees away from the Sun’s position?

If the answer to the first question were “less than 90 degrees”, and the answer to the second question were “more than half lit”, that would tell us the Sun can’t be much further than the Moon.  But you can check yourself that the answer to the first question is “very close to 90 degrees” and the answer to the second question is “very close to one half.” 

What’s good about this method is that the details of the Moon’s orbit don’t matter so much, so it gives a more reliable measure of how far the Sun is.  For instance, to see if the Moon’s 90 degrees from the Sun, you can take an L-shaped tool (often called a carpenter’s square), point one end of the L at the Sun, and try to point the other at the Moon.  If you succeed, then they’re 90 degrees apart.  If it doesn’t quite work, then the Moon’s not in the right place in its orbit; either you should wait a few hours, or you missed your chance and will have to wait two weeks for the next opportunity, half a a cycle away. Anyway, once you succeed, look at the Moon: it will be half lit, the edge between its lit and unlit portions appearing as a straight line down the middle.

By the way, doing this measurement while the Sun is up should be no problem. A half lit Moon can be easily seen in full sunlight, if you know roughly where to look.  Around First Quarter, the Moon follows the Sun by six hours, and it precedes it by six hours near Third Quarter. (For instance, if it’s 3 pm on a day near first quarter Moon, then the Moon will be near where the Sun was at 9 am; the reverse is true near third quarter Moon.)  The next first quarter Moon is on March 9th or 10th (depending on your time zone), and the next third quarter Moon is around the 24th. Beyond that, you get two chances a month. So even with the vagaries of weather and timing, you shouldn’t have to wait long to give this a try.

In this way you should be able to convince yourself that the Moon’s within just a few percent of half lit when it’s 90 degrees from the Sun.  And from that, I think you can conclude the Sun’s at least ten times further than the Moon — at least a couple of million miles away.  Since the Moon’s diameter is ¹/₄ of the Earth’s diameter, that makes the Sun’s diameter at least ¹⁰/₄ = 2.5 times as big as the Earth’s.

We’re still nowhere near figuring out how big and far the Sun really is; that’s harder, and for another time. But look how much we can learn with nothing more than eyeballs, grade school math, and a little thought!