Sky & Telescope - August 2021 - 27

within a few days of being exact recurrences. As you may
have guessed, fi nding an exact recurrence period for Mars
isn't possible for the same reason that an irrational number
cannot be reduced to a simple fraction. But it is possible to
fi nd recurrence periods that are as close as we desire to any
chosen margin of error.
Using only the length of the year and the Martian synodic
period (779.94 days) as inputs, the program found the same
Goal-Year periods as the Babylonian astronomers: 22 cycles
in 47 years, and 37 cycles in 79 years. But as we'll see, there's
an error lurking here that we haven't yet quantifi ed. In other
words, the hands of our imaginary two-planet clock line
up, but not exactly. And it's the " not exactly " part that is of
particular interest.
Using Errors to Good Advantage
The Babylonians knew their Goal-Year periods weren't exact,
but did they know by how much they were off? Scholars suspect
they reckoned their errors to refi ne their predictions far
into the future. Let's see how this might have worked.
My program found that the mean deviations, or errors in
the Goal-Year periods for Mars, are -8.07 days (early) over 22
cycles in 47 years, and +3.03 days (late) over 37 cycles spanning
79 years. This means that Mars shows up at the same
place in the zodiac more than a week early after 47 years,
and a little more than 3 days late after 79 years. So, though
neither estimate is dead-on, the 79-year return is the more
accurate of the two.
Now here's the fun part. Scholars have surmised that the
Babylonians combined several Goal-Year cycles to cancel out
these errors and gain predictive accuracy that spans centuries.
I tested this with the error values churned out by the program.
For example, if the Babylonians knew about the errors
mentioned above, they could have mapped out a 133-cycle
return (284 years) this way: 22 + 37 + 37 + 37 = 133 synodic
cycles. Totting up the numbers we see the cumulative error
for 133 synodic cycles amounts to about 1 day across almost
three centuries: -8 + 3 + 3 + 3 = +1 day.
The error from the fi rst 22-cycle run becomes the starting
line for the 37-cycle run, and so on. The plus and minus
errors aggregate, and can amplify or cancel, depending on

http://www.skyandtelescope.org

Sky & Telescope - August 2021

Table of Contents for the Digital Edition of Sky & Telescope - August 2021

Contents
Sky & Telescope - August 2021 - Cover1
Sky & Telescope - August 2021 - Cover2
Sky & Telescope - August 2021 - 1
Sky & Telescope - August 2021 - Contents
Sky & Telescope - August 2021 - 3
Sky & Telescope - August 2021 - 4
Sky & Telescope - August 2021 - 5
Sky & Telescope - August 2021 - 6
Sky & Telescope - August 2021 - 7
Sky & Telescope - August 2021 - 8
Sky & Telescope - August 2021 - 9
Sky & Telescope - August 2021 - 10
Sky & Telescope - August 2021 - 11
Sky & Telescope - August 2021 - 12
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Sky & Telescope - August 2021 - 14
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Sky & Telescope - August 2021 - 27
Sky & Telescope - August 2021 - 28
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Sky & Telescope - August 2021 - 31
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Sky & Telescope - August 2021 - Cover3
Sky & Telescope - August 2021 - Cover4
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