John Pazmino
 NYSkies Astronomy Inc
 2001 August 1
   At the Observing Group meeting of 15 July 2000 Ken Brown and Susan 
Andreoli asked about the interval between conjunctions of Jupiter and 
Saturn. Chair Bruce Kamiat explained that the interval is based on the 
orbital periods of the two planets such that conjunctions occur about 
every twenty years. 
    Altho the question was in connexion with the invisible event on 31 
May 2000, when the conjunction occurred right next to the Sun, the 
behavior of the two planets takes on special meaning right now. For 
they are in fall 2000 dancing together in the sky at nightfall in 
downtown Taurus. They are featured targets at our starviewing 
sessions, including the new series at the Hayden Planetarium 
Synodic periods 
    We have Jupiter's period of 11.86y and Saturn's of 29.42y. Bruce 
used rounded numbers, 12 and 30, to figure out the interval in his 
head but the method is the same. 
    It's the textbook exercise of doing 
    (Jup-Sat per) = ((Jup per) * (Sat per)) / ((Jup per) - (Sat per)) 
                  = ((11.86y) * (29.42y)) / ((11.86y) - (29.42y)) 
                  = -19.87y 
                 -> 19.87 years
    If we take the subtraction in reverse order we merely get the 
opposite signum. Only the absolute value matters in this situation, so 
the signum is tossed. This is called the synodic period of the Jupiter 
relative to Saturn, or of Saturn relative to Jupiter. 
    The interval in arc, the synodic arc, along the zodiac between 
conjunctions comes directly from noting that Jupiter rounds the zodiac 
once between conjunctions and thereafter it's the excess motion that 
displaces the event among the stars. So 
    excess = (19.87y) - (11.86y) 
           = 8.01 year
    arc = (8.01y) / (11.87y) 
        = 0.675 circle
       -> 16h 12m
    We could have banked off of Saturn, noting that Saturn doesn't 
complete a whole round before the next conjunction occurs. 
    excess = (19.87y) - (29.72y) 
           = -9.55 year
    arc = (-9.55y) / (29.42y) 
        = -0.325 circle
       -> -7h 48m
    The displacement is upstream from Saturn, which is exactly the 
same as the downstream displacement from Jupiter. That is, you get to 
the same point in the zodiac but approaching from opposite directions. 
Babylonian planetary motion
    However, after ragchewing a bit it turns out that this interval 
was known long before we understood the solar system. In fact, the 
Babylonian culture had no concept of a system of planets at all. The 
planets were mere meandering points on the celestial sphere. 
    How could they have known this 19.87 year synodic period for 
Jupiter-Saturn conjunctions? 
    A touch of history. Astronomy in Babylonia started in about 2,500 
BC as an effort to document the planets (which included the Moon and 
Sun). The people were concerned with the occurrence of certain 
alignments of the planets, like elongations and conjunctions. They 
noted when by date and where by star background these happened. 
    This chronicling of the heavens was motivated by the divination of 
the country's fate and fortune. Accompanying the observations were 
interpretations, often pretty ludicrous, for the events. 
    After many many centuries they built up a database of 
observations. Note well that this set was far from ideal and perfect. 
The observers, altho trained under the ruler, made mistakes. They 
mistook stars or misread calendars. The weather in Mesopotamia was 
fickle. The horizon was shmutzed with haze and mists. 
Modern reconstructions
    In the bad old days the calculations of planet positions to 
correspond with the description in an ancient tablet was a hideous 
chore. Today it is a trivial matter by any of the better planetarium 
computer programs. 
    For example, from a book on Babylonian astronomy I see a [modern 
typeset version of] a tablet noting that Venus was just west of Mars 
in Aries, the Moon was in Scorpius, and Antares was within a halo 
around the Moon. This supposedly happened on the 19th day of the 12th 
month of -651. The year is 'astronomical', counting a year 0; it is 
numericly one LESS than the 'BC' year. 
    On what day, on the modern calendar, did this occur -- if at all! -
- so we can date the inscription and other events of the people 
associated with it. From a planetarium (any is valid so long as it 
employs a correct longterm algorithm for the planets) I find that such 
an event actually did occur on -650 February 28 in Babylon. 
    What looks like a wrong year, -650, is really a rollover from -651 
within the one year. In Babylonia the year began in what we now call 
March. The last, 12th, month was February. Think of 'September', 
'October' and so on, yet an other legacy handed down to us from those 
Confidence in celestial events
    This innocent diary entry brings out a very profound feature of 
Earthly civilization. A peach-fuzzed boy from Norway can in a couple 
minutes determine with perfect certainty the truth of the planet 
positions. But no boy, no man, no army of men, no world of armies can 
ever reconstruct the weather over Babylon 26 centuries ago and see if 
a lunar halo did shine. 
    Storms, drought, insect invasion, animal migration, crop blight --
all events vital and crucial for the wellbeing of early humankind --
dismally failed of certain and dependible foretelling. 
    Only the stars and the planets obeyed definite rules. And so 
astronomy rose above the other attempts of doping out the ways and 
wiles of nature to become humankind's first true science. It and it 
alone worked. 
Jupiter-Saturn conjunctions 
    I do not have at hand ample historical accounts for conjunctions 
between Jupiter and Saturn. I fudged a set of records by hopping among 
Dance of the Planets, Earth Centered Universe, and SkyGlobe. I come up 
with the list here for the span of 700 BC to 300 BC. 
  Universal Time     Sepn  2000 RA   Universal Time     Sepn  2000 RA 
  -----------------  ----  ------- | -----------------  ----  -------
  01 Aug -741  *:*   ~220  02h 23m | 16 Dec -522 20:04    65  14h 04m  
  22 Feb -740 00:13    73  02h 29m | 14 Mar -521 11:40    65  13h 58m
                                   | 10 Jul -521 14:24    71  13h 46m 
  24 Jan -720 18:49    41  19h 28m | 
  10 Aug -720  *:*   ~160  19h 27m | 25 May -502 22:40    64  05h 18m
  21 Dec -701  *:*   ~230  12h 12m | 06 Mar -482 01:11     7  21h 26m                                
  30 Jul -700 15:18    49  12h 20m | 07 Sep -482  *:*    ~90  21h 18m
  31 May -681 00:44    70  03h 22m | 09 Oct -462 16:04    70  14h 23m                                
  12 Dec -681  *:*   ~160  03h 20m |
                                   | 23 Sep -443  *:*   ~110  05h 55m                                
  25 Nov -661 08:57    36  19h 47m | 04 Apr -442 12:56    61  05h 49m
  01 Nov -641 01:23    55  13h 04m | 28 Dec -423 10:59     2  21h 42m                                
  03 May -640  *:*    ~60  12h 52m |
                                   | 09 Feb -402  *:*    ~70  14h 58m                                
  11 Apr -621 09:06    72  03h 53m | 09 Aug -402 16:37    77  14h 41m
  16 Feb -601 10:38    24  20h 30m | 06 Jul -383 03:13    49  06h 42m                                
  24 Aug -601  *:*   ~110  20h 23m | 04 Feb -382  *:*   ~210  06h 52m
  10 Sep -581 05:00    63  13h 25m | 19 Mar -363 15:17    12  22h 19m                                
                                   | 21 Sep -363  *:*   ~100  22h 12m
  27 Jul -562 00:07    68  04h 47m |                                                                 
  01 Nov -562 05:39    75  04h 37m | 30 Oct -343 21:49    73  15h 18m
  10 Feb -561 04:14    76  04h 25m |
                                   | 16 May -323 19:01    42  07h 10m                                
  13 Dec -542 22:04    17  20h 46m |
                                   | 10 Jan -303 14:02    20  22h 34m 
    The date and hour are in UT, separation is in arcminutes, location 
is the right ascension of Jupiter. The actual hour is not critical in 
as much as the timescale of planet motion is a full day. 
    I chose this span because it begins near the ascension of king
Nabonassar, starting a regime of political and social stability in 
Babylonia. Under him the skywatching mission was systematized and 
records were more carefully made and kept. 
    Some works on Babylonian history relate that in Nabonassar's first 
year of rule there was a Moon-multiplanet convention that officiated 
the power. Modern planetarium programs do not show anything out of the 
ordinary during this timeframe, so the story may be an allegorical 
appeal to celestial validation 
    Babylonia went thru many disasters and conquests since Nabonassar, 
yet the skywatching mission was kept up by his successors, even by 
those from the foreign powers. 
Babylonian planetary theory
    In the latter part of this span pure skywatching shifted to 
analysis of the records and the invention of mathematical methods to 
foretell in advance the various phaenomena. Among the artifacts from 
the 400s BC forward we find tablets of calculations, instructions, and 
ephemerides. Tony Hoffman's 'What's Up' column in NYSkies is a direct 
descendant from these. 
    In yet an other sobering feature of human existence on Earth, the 
Babyloniam calculations for the planet positions is humankind's first 
use of mathematics to successfully describe events in nature. It would 
be fully 19 centuries later, in the early 1600s, before maths were 
applied to formulate the first true laws of physics; this was 
performed by Galileo. 
    After about 300 BC, Babylonian culture was subsumed under Greek 
culture. It is thru the tradition of Greece from this time that we owe 
so very much a debt to Mesopotamian astronomy. 
Conjunctions and close appulses
    In my gathering of conjunctions, I found an account of a grand 
spectacle in the predawn sky of November -701, or 702 BC. Jupiter, 
Mars, and Saturn assembled on the Leo-Virgo frontier! Such a sight 
would have been an omen of some great importance. 
    When I searched for this conjunction I found the meeting of Mars 
with Jupiter and with Saturn. Nothing turned up for Jupiter meeting 
with Saturn! A closer inspection of this event showed that Jupiter and 
Saturn did approach closely but never passed each other. 
    Today we define 'conjunction' as the achievement of equal ecliptic 
longitude (sometimes equal right ascension). This is what programs 
search for when asked to list conjunctions. What happened in 702 BC 
was that Jupiter creeped up from the west near to Saturn, rounded its 
station point, and backed away on its retrograde loop! 
    The closest approach came on 21 December 702 BC with no true 
conjunction. Right off of the bat I had a mess on my hands. I had to 
find all the other close appulses of Jupiter and Saturn because these 
were noted in the tablets even tho they were not what we would call a 
conjunction. Thus, the entries with no time, '*.*', are these 
appulses, with the separation being approximate. 
Cyclical pattern of conjunctions 
    OK, from this database of 'observations' of Jupiter against Saturn 
how could the Babylonian astronomer suss out a pattern in the absence 
of any notion of a planetary system? It actually ends up being very 
simple. First, a few statistics. 
    The interval between the first conjunction in -740 (taking only 
true conjunctions) and the last is quite 436.88 years. The number of 
conjunctions, counting a group of passes as one event, since -740 is 
22. The number of circuits of the zodiac made by these conjunctions is 
7.16 and these laps were made in reverse order of the zodiac. 
    This third stat is a bit important, so listen up. The first 
conjunction occurs near 2-1/2h RA. The next, near 19-1/2h; the third, 
near 12h; the fourth, near 3-1/2h. That's one lap backwards in the 
zodiac. Stepping thru the list we come up with 7+. 
    Then we have 
    (synodic period) = (total years)/(count of conj) 
                     = (436.88y) / (22) 
                     = 19.86 year
    (synodic arc) = (count of circ)/(count of conj) 
                  = (-7.16) / (22) 
                  = -0.325 circle 
                 -> -7h 49m 
The minus sign means the arc is stepped westward, backward, upstream 
along the zodiac. 
No orbital concepts 
    Note very well that there is nothing of a theory of orbit around 
the Sun, Earth, Alcyone nor any use of cycles or revolutions. In fact, 
to the Babyklonian astronomer the overall behavior of Jupiter and 
Saturn, not pertaining to their mutual meetings, is not defined. It 
was the Greeks who first conceived of a continuous motion for the 
planets and devised models for simulating it. 
    Hence, Jupiter-Saturn conjunctions occur every 19.86 years with 
each being 7h 49m westward in the zodiac. And this is essentially what 
we get by the modern method involving orbital motion.