A Fact Often Eclipsed

Eclipses fascinate us because they play with our very definition of “reality.”  We define “day” as when the Sun is above the horizon.  So, even when there is heavy cloud cover and rain, we still see light during the day.  Except, of course, when the Moon inserts itself directly between the Sun and Earth.  Then, for a little while, the Moon’s shadow passes over Earth.  And if that shadow passes over us, the sky will become dark despite the Sun’s position above the horizon.

For a few moments, we experience nighttime during daytime.

It’s that contradiction that plays with our reptile brain.  And it’s the reason why our ancestors created elaborate mythologies around the phenomenon.  For, unless you understand that the Moon goes around the Earth and the Earth goes around the Sun – something most of our ancestors did not – it’s pretty hard to explain what’s going on during an eclipse.

But there is something even more magical than the nighttime-for-daytime situation during solar eclipses that rarely gets mentioned.  If you look at pictures of total solar eclipses, you’ll see that the Moon covers up the Sun exactly.  In other words, from Earth, the Moon covers up exactly as much of the sky as the Sun does.

There is nothing in physics that says this has to be the case.

For example, when Venus gets between the Sun and the Earth, it barely covers a small percentage of the Sun’s disk and looks like a small dot to us on Earth.  So how does the Moon cover up so much of the Sun while Venus, a much larger body about the size of the Earth, does not?

That’s a bit like asking who will appear larger:  the 4 foot 5 inch Peter Dinklage standing 1 yard from you or the 6 foot 6 inch Michael Jordan standing 100 yards away?  It’s pretty obvious that the nearby Dinklage will fill more of your field of view than the distant Jordan and therefore will appear larger, despite the obvious size differences.  Similarly, this is how the tiny, but close, Moon can appear larger than Venus to us on Earth.  Or how the Moon can appear as large as the far more massive, but also far-away, Sun.  So large, in fact, that through a heavenly coincidence they appear to be the same size.

We can use this observation to estimate the size of the Sun.  First we make a diagram to get a sense of our geometry.  This diagram does not need to be drawn to scale.  It only needs to indicate how we (on Earth) will see the diameter of the Moon to be the same as the diameter of the Sun. Do you see the two triangles formed?  One is created by using the Sun’s diameter as the base and us, on Earth, as the apex point.  The other is created using the Moon’s diameter as the base; this second triangle is indicated in blue.  The drawing reveals how the Moon’s edges can line up with those of the Sun because it is closer to us.

The drawing also gives us the intuitive feel that the two triangles are proportional to one another.  We could prove that fact rigorously using 2,300-year-old Euclidean geometry.  Let’s stay focused, however, and instead assume our intuition is correct (and it is).

Now, because the triangles are proportional, their dimensions must be in identical ratios – that’s the very definition of proportional!  In particular, the ratio of the triangle heights (the distance to Earth) is the same as the ratio of their bases (the diameters):Now let’s put numbers into this equation.  You may remember from science class that the Sun is 149,600,000 km (roughly 93,000,000 miles) from us and the Moon is 384,400 km (roughly 240,000 miles) from us.  And we also learn from NASA that the Moon’s diameter is 3,475 km (about 2,160 miles).  That means, to maintain the proper ratio, we estimate the Sun’s diameter to be 1,352,000 km (840,000 miles).  This number is within 3% of the Sun’s actual diameter 1,391,000 km (864,000 miles).

Not bad for a simple estimate!

But we can do better.  In our distance estimates, we assumed we were standing at the center of the Earth.  I don’t know about you, but I usually stand on the Earth’s surface, not its center.  That means we are closer to both the Sun and the Moon by the radius of the Earth, a distance of 6,371 km (3,960 miles).  This number is a minuscule amount compared to the distance to the Sun but it’s nearly 2% of the distance to the Moon.  So if we re-calculate the distance from the Moon to the Earth’s surface as 378,000 km, our revised ratio estimate of the Sun’s diameter is 1,375,000 km (854,000 miles).  This new estimation is accurate to the actual number by 1%!

When you consider that we modeled the elliptical (oval) orbits of the Earth around the Sun and the Moon around the Earth as perfectly circular (with a single radius that doesn’t change during the orbit), the agreement between our simple estimate and the observed measurement is even more impressive.

Why is it that the Sun is both 400 times wider than the Moon and also 400 times farther from the Earth?  It’s a cosmic coincidence.  Literally.  Because of it, the curvatures of the two disks, Sun and Moon, match precisely.  As a result, scientists have been able to cleanly observe and study the Sun’s atmosphere (the corona) and the Moon’s mountains and valleys during the few precious minutes of totality.

So, whether you watch a solar eclipse on a news feed or travel to the shadow itself, remember this marvelous celestial dance springs from the rigorous certainty of classical physics with a sprinkling of random correlation.  The vastness and diversity of the Universe allows for many possibilities.  We are fortunate one of those possibilities, a stellar synchronicity, is so near.  It provides us something to enjoy.  And to ponder.