Eclipses and the Size of the Sun
 
Henry Hamburger

The ring of fire at full solar eclipse is the last of
five comprehensible steps to figuring out the size of the sun.
Click here for a more detailed account by a physics prof!

  1. Size of the earth. Ancient Greeks realized that the earth might be essentially spherical, since its shadow on the moon is round during eclipses in all seasons, with sun from various angles. Even right here on earth, they noticed that departing ships descend into the horizon. More impressively they were aware that change in latitude results in a change of angle of elevation of the north star, aka Polaris.

    Eratosthenes used a similar idea, the varying peak angle of elevation of the sun, to get the size of the earth. He had heard of a place where sunlight could be seen to strike the bottom of a well - was directly overhead - just once a year. That well would have been located on what we now call the tropic of Cancer, and that day we call the summer solstice. On that day, but not in that place, he measured the maximum elevation angle, a, of the sun in radians. He did this at a point that was a substantial and carefully measured distance x due north of the well. Assuming a spherical earth, its radius is x/a.

    Let's hear it for Eratosthenes! This year is the 2200th anniversary of his death.

  2. Size of the moon. First assume that the earth, in its own vicinity - where the moon is - casts a roughly cylindrical shadow in the direction opposite the sun, with no appreciable penumbra. This is not strictly correct. Neither the full shadow nor the region of partial shadow is cylindrical, and they differ from each other. Moreover it is less accurate than it would be if only the sun were even further away and/or smaller. But then it would be colder, so let's not complain.

    During a maximum-duration total eclipse of the moon, record the time-interval from the first touch of darkness to the beginning of total eclipse. During this time, the point that will be the last to darken has moved to the earth's shadow from a moon-diameter away, so this time interval is proportional to the moon's size. (Velocity is also involved; we'll get back to that).

    Also measure the time interval from the first touch of darkness until the moon starts to emerge from shadow. In other words, get the time it takes for the leading point of the moon to move through the earth's shadow. Since we chose a maximum eclipse, it's going through the center of the shadow, and therefore takes a time proportional to the earth's diameter. Both of these time intervals are also affected by the moon's speed but in exactly the same way (inversely proportional to it). So the ratio of the time intervals equals the ratio of the diameters of moon and earth. We had one of them from #1 above, so now we know both.

  3. Distance to the moon. This one is easy, just a matter of similar triangles (having the same angles and proportional sides). Hold up a finger to the moon. Don't be crude. Straighten your arm until the finger just barely, but completely, blocks the moon from your vision in one eye. You better close the other one. If your fingers are too fat for this, use a pencil or get a longer arm or a friend. The ratio of the width of the finger to its distance from your eye is the same as that of the moon's diameter to its distance.

  4. Distance to the sun. During an exact half moon, the sun must be directly off to the lit-up side. In other words, there is a right triangle with a right angle at the moon between the lines from there to the sun and to the observer on earth. At the observer's vertex of that same triangle, the moon and sun form an angle slightly under a right angle. That angle's cosine is the ratio of distances of moon and sun, one of which is known from #3, thus yielding the other. This one requires great precision, because small errors in either the half-moon determination or the angle measurement can yield wildly inaccurate results.

  5. Size of the Sun. The photo of a solar eclipse dramatizes the fact - which can actually be demonstated under ordinary circumstances - that the diameters of the sun and moon have roughly the same ratio as their distances from earth. That plus the above results and some arithmetic yield the size of the sun. To get the necessary facts without waiting for a total eclipse, use the method in #3, which would provide enough information even if the two ratios were not roughly equal.