Sky & Telescope - August 2021 - 28

Computational Astronomy
view, the result is completely expected given the nature of
periodic motion.
Refi ning our search to errors of within about two days of
exact recurrence, we identify some other close instances in
the long haul that illustrate the method:
Synodic Cycle Variances
Number of cycles
96
133
229
362
According to Otto Neugebauer, the pioneering 20thInterval
(years)
205
284
489
773
Variance (days)
-2.01
+1.02
-0.99
+0.03
Glancing at the table above, you doubtless noticed intriguing
possibilities for combining cycles. Take one 96-cycle and
add it to one 133-cycle and you get a 229-cycle period accurate
to about 1 day. Add another 133 and you arrive at 362
with negligible error over 773 years.
Plotting the Closest Synodic Recurrences
Curious about the patterns of these periodic phenomena,
I went ahead about 500 years in the computer program -
beyond the known range of the extrapolated Babylonian
Goal-Year periods - and charted the sequences that fall
within the range of 10 days ahead or behind schedule of
Mars's same-place opposition point. The chart on
page 27 shows the day variances along the
y-axis, and the synodic cycles are plotted
on the x-axis. The numbered dots closest
to zero are the best passages of
the planet, that is, those closest to
the same place in the sky where
Mars was at the start of any
chosen observation series. Begin
exploring the chart by looking
for the 22- and 37-cycle returns
and identifying their variances.
You can spot the main 133-cycle
return and use some mental
math to estimate how other
combinations above and below the
horizontal axis might cancel out.
I found (not surprisingly)
that longer-term results are the
most sensitive to slight changes
in the initial assumptions. We
don't know whether the Babylonians
were aware of this, but they
appear to have settled on the 133cycle
(284-year) recurrence period
as good enough for their larger
ambitions.
28 AUGUST 2021 * SKY & TELESCOPE
century translator and scholar of ancient mathematics, the
Mars-related tablets explicitly show the 47- and 79-year
Goal-Year periods, revealing a precise period for Mars as a
linear combination of one 47-year set (22 synodic cycles)
combined with three 79-year periods (each containing 37
synodic cycles) for a total of 284 years. Similarly, the Babylonians
found exact periods for Jupiter and for Saturn. These
results appear to have functioned as baseline parameters for
the creation of long-term ephemerides.
By combining sets of Goal Years to cancel errors, the Babylonians
could thus leapfrog far into the future beyond actual
observation to derive long-term synodic recurrence periods.
For Mars, they could confi dently predict that 133 synodic
arcs will fi t in 151 Martian orbits over the course of 284
Earth years. There's a satisfying symmetry here: 133 complete
synodic arcs comprise 6,480°, which happens to be 18 whole
revolutions around the 360° ecliptic.
The fi rst full-fl edged Babylonian astronomical ephemeris
appeared probably in the 3rd century BC - a milestone
in their evolving planetary theory. Planetary motion along
elliptical orbits is nonuniform, so creating an ephemeris is
a bigger challenge than forecasting synodic returns. Yet the
Babylonians exhibited remarkable cleverness here, too. They
used exact periods as mean parameters for the planets, which
they could then augment or diminish with a small correction
according to the planet's place (and thus, speed) in its orbit.
They created speed zones. Of the outer planets, sluggish
Jupiter and Saturn each required only two speed
zones; Mars, the perennial problem child
whose orbit is most elliptical, needed six.
The speed adjustments for Mars were
in simple ratios for the six 60° zones
of the zodiac, a sequence that
Dutch astronomer and Babylonian
expert Teije de Jong called
" a miracle of numerical elegance
that continues to baffl e me. "
KEPLER'S MARS This is a modern rendering of Johannes
Kepler's epic, hand-calculated drawing of the path of Mars
from his 1609 Astronomia Nova, " on the assumption that the
earth stands still, as Ptolemy and Brahe would have it. " Kepler
incorporated 16 years of Tycho Brahe's observational data
(from 1580 to 1596) to make this diagram. The number-oriented
Babylonians were observing and calculating Martian paths
2,000 years before Kepler.
Mathematical Time
Travel
To look at the path of Mars
in a modern context, I loaded
into my long-suffering laptop
80 years of orbital data from
1940 to 2020, drawn from the
NASA/JPL Horizons online
database. The result is a complex
image resembling the top view
of a woven Japanese basket.
Johannes Kepler's epochal graph
of Mars's path in his 1609
Astronomia Nova, using 16 years
of Tycho Brahe's data, inspired
the diagram. Kepler imagined
COURTESY WILLIAM DONAHUE AND GREEN LION PRESS
Sky & Telescope - August 2021

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