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.)
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.
Take a U.S. penny, for instance. Its diameter is about 19 millimeters (1.9 centimeters, 0.75 inches). As you can confirm using the methods in my last post, the Moon’s diameter is about 2200 miles (3500 kilometers), which is larger than a penny by about 180,000,000 times. Now if you hold a penny at arm’s length on a night when the Moon is full enough that you can tell how big it is, you’ll see that the penny appears to be between two and three times wider than the Moon, depending on the length of your arm. So if it’s two or three arm’s lengths away — say, about two meters, or about 100 times further away than its own size — then it will appear the same size as the Moon against the sky.
In turn that means the distance to the moon is about 180,000,000 times larger than 2 meters, which comes out to 360,000,000 meters, or 360,000 kilometers (225,000 miles). How close did we get?
In fact the Moon’s orbit isn’t circular, but mildly elliptical. At its closest, the Moon is about 360,000 km (225,000 miles) away, and at its furthest it’s about 400,000 km (250,000 miles) away from the Earth’s center (and depending on the time of night you may be a few thousand km (miles) closer to the Moon.) The average is about 385,000 km (240,000 miles).
So our estimate is well within 10% of the average, even while being pretty careless. I didn’t try to measure the “2 meter” distance from my eye to the penny at all precisely. You could certainly try to do better, if you wanted, and perhaps even confirm, by seeing the Moon grow and shrink slightly, that the Moon’s orbit isn’t circular. But sometimes it doesn’t pay to be too exact.
We’ve now checked that the Moon’s about 1/4 the size of the Earth, and its distance from Earth is about 30 Earth diameters (60 Earth radii.) Just think about those numbers for a moment. That’s like you and your toddler being two hundred feet (sixty meters) apart. There’s a lot of empty space out there. You can get a sense of that in this beautiful, haunting image of the Earth and Moon, taken in 2017 by the OSIRIS-REx spacecraft on its way to visit an asteroid. This craft will return, bringing a souvenir back to Earth, just 18 months from now.