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Last night, using the methods I described as part of my check-it-yourself astronomy series, I estimated the distance to the planet Jupiter using nothing more than my eyes, a protractor, and a simple calculator. It took about 30 seconds of measuring something before and after sunset, and about 15 more seconds using my cell phone’s calculator. You can do it too, if you have clear skies over the weekend.

The day on which you should make the measurement is the one when Earth, Jupiter and the Sun make a right triangle as in Figure 1. You can estimate this date quite easily. Once every 13 months or so, Jupiter is at “opposition”, meaning that there is a straight line connecting the Sun, Earth and Jupiter, as at the far left of Figure 1. At opposition, Jupiter is directly overhead at midnight. Let’s call the time between oppositions the length of Jupiter’s cycle. Then, about 1/4 or 3/4 of the way through the cycle, the three heavenly bodies form a right-angle triangle, drawn in Figure 1, and the angle between Jupiter and the Sun as seen from Earth is enough to determine the ratios of the triangle’s sides.

Looking at Figure 1, you can see that if RJS were just a little larger than RES, the angle A would be close to 45 degrees. If RJS were enormously larger than RES, then the angle would be very close to 90 degrees. But wandering into your backyard just after sunset, and looking up, reveals the angle to be much larger than 45 degrees, yet clearly less than 90.

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