John Pazmino
 NYSkies Astronomy Inc
 2010 December 18 initial
 2011 December 25 current 
    The NYSkies Astronomy Seminar on 17 December 2010 previewed the 
celestial activity for the coming year 2011. The motion of the planets 
was part of the discussion, with demonstration of the synodic and 
sidereal cycles. The dialog was protracted to bring along the 
newcomers of our profession. In this tuition there was nothing really 
innovative or original in the treatment. 
    Over the adjacent weekend I found that a good summary of the 
synodic behavior of the planets appears in pieces thruout websites, 
mainly associated with a special event with a this or that planet. One 
example is the triple conjunction of Jupiter with Uranus in progress 
in fall 2010 thru January 2011. 
    Most modern astronomy books skim thru the planet motions, given 
that for the most part solar system astronomy centers on studies on 
and around the very planets. Older works thru the mid 20th century 
devoted a chapter or more to the gyrations of the planets. 
Planetary models
    Today all of the movement and placement of the planets is derived 
from the dynamical model of the solar system. The Sun is near the 
center of the planetary orbits, from his humongous gravity. The center 
of gravity of the solar system always sits within the solar globe. 
This allows us to simplify matters by putting the Sun at the geometric 
center of the solar system. 
    The planet antics were recognized for millennia before the 
heliocentric model was adopted. In fact, there was no need for a 
kinetic, let alone dynamic, model at all to monitor the planets and 
foretell their behavior. The raw plots or log of planet positions was 
sufficient to build a very adequate theory of the planets 
    The Mesopotamian astronomers were among the first people to carry 
out deliberate careful observations and analysis of the planets. 
Earlier cultures are known to be aware of the planets but so far we 
have limited understanding of them. 
    Meospotamia had nothing of a physical model of the planets, like 
orbits, distance, linear speed. It worked only with the placement and 
displacement of the planets on the celestial sphere. 
    The Greeks were the first to construct a kinetic, if not physical, 
model of the planets. They built the deferents and epicycles, in 
various elaborations, culminating in the Ptolemaeus theory. This had 
the Earth at the center of the cosmos. 
    That scheme endured thru the Dark and Middle Ages until Copernicus 
proposed the Sun-centered model. Once the Newton gravity concept was 
accepted, and it did take a few decades for that, only the dynamical 
solar system is used to work out the planet behavior. Never the less, 
for home astronomers, the Mesopotamian methods are quite handy for 
mental or back-of-envelope calculations. 
The planets
    All of the planets, with Earth, circulate around the Sun in nearly 
one plane. The orbits are titled at most by a couple degrees from 
Earth's. This makes the line of sight from Earth to any planet aim 
into a band around the sky of only 16 degree width. This width covers 
the greatest wandering of a planet off of the Earth's orbit plane. 
    This band is the zodiac and it passes thru, by tradition, twelve 
constellations. Today with the delineated frontiers there are thirteen 
constellations within the zodiac. The odd one, Ophiuchus, is never 
counted as a zodiac constellation in any traditional treatment. 
    The centerline of the zodiac is the trace of the Earth's orbit. In 
the sky this is reflected by the apparent path of the Sun around the 
Earth. In deed, in spite of the centuries of experience with a Sun-
centered solar system, our vocabulary still holds to an Earth-centered 
    This centerline is the ecliptic, dimensioned in degrees along it 
from 0 thru 360. The zero point is at the place the Sun occupies on 
the first day of spring, the vernal equinox. The downrange distance 
along the ecliptic is longitude. 
    Displacement north or south of the ecliptic is also measured in 
degrees as latitude. The Sun latitude is always zero because he stays 
on the ecliptic, never deviating from it. All other planets can wander 
north or south and have a nonzero latitude. 
    The planets travel in the same sense, anticlockwise in their 
orbits as seen from north of the solar system plane, with south in the 
background. This trend is prograde or direct motion. 
    In the sky the planets run from west to east, right to left when 
looking south from northern latitudes. As a zeroth approximation, a 
planet is farther downrange, into higher longitude, as time passes. 
This motion is also called prograde or direct. 
    The terms are easily mixed up. In the solar system all planets 
carry out pure direct motion. They do not ever retrace or reverse 
their pace in their orbits. In the sky, they are viewed from a moving 
Earth. A planet does seem to slow, stop, retreat to lower longitude on 
certain occasions. After a few weeks the planet resumes its downrange 
motion. This reversal in the sky is retrograde motion. 
    Because the planets circulate around the heavens along a narrow 
band, it was natural to build a system of measurement on this band. It 
seemed in the very early eras easy to specify the place of a planet 
within the zodiac constellation. Because the constellations were not 
formal constructs, only rough regions honoring various aspects of a 
culture, and because the constellations occupied various reaches of 
the zodiac, the recorded locations were not easy to work with. 
    To simply the measurements and calculations, the constellations 
were normalized into twelve zones, each of 30 degrees length. When the 
zones were standardized in about 150 BC, they lined up with the 
constellations. The zones are the signs of the zodiac. Degrees within 
each sign are numbered 0 thru 29. Some authors say 1 thru 30 but this 
louses up any maths performed on the angles. 
    In the very early era of astronomy the vernal equinox sat in 
constellation Taurus, so the signs were listed starting from there.  
As precession pulled the vernal equinox out of Taurus into Aries, its 
location was duly cited as in such-&-such degree of Aries. The signs 
were attached to the stars, not the equinox. 
    A couple millennia later, near the beginning of the current era, 
the equinox slided to the western end of sign Aries. At this time the 
signs were tied to the vernal equinox at Aries 0. By today, an other 
two thousand years later, the equinox -- dragging with it sign Aries -
- is in constellation Pisces. 
    The shift is about one full sign. The sign is one step AHEAD, 
EAST, of the constellation it now overlies. The constellation is one 
step BEHIND, WEST, of the sign sitting over it.  
    Rather than jump all over about the astronomy-astrology argument, 
it's just as well to consider the signs as divisions of the zodiac 
like the months are divisions of the year. Recall that October is no 
longer the 8th month, not any lass so than Libra sign no longer sits 
on Libra constellation. If you argue against zodiac signs in 
astrology, you may want to campaign for renaming the months. 
    The correspondence of signs and longitude is: 
        sign-deg      ecl lon | sign degree        ecl lon 
        -----------   ------- | ----------------   -------
        Aries 0-29  =   0- 29 | Libra 0-29       = 180-209 
        Taurus 0-29 =  30- 59 | Scorpius 0-29    = 210-239 
        Gemini 0-29 =  60- 89 | Sagittarius 0-29 = 240-259 
        Cancer 0-29 =  90-119 | Capricornus 0-29 = 270-299 
        Leo 0-29    = 120-149 | Aquarius 0-29    = 300-329 
        Virgo 0-29  - 150-179 | Pisces 0-29      = 330-359 
    The zero point of the ecliptic lat-lon dimension is not fixed in 
the stars. It drifts steadily thru the zodiac, completing one circuit 
in 25,800 years. The precise value differs by a hundred or so years 
among authors because of various theories of the Earth-Moon system. 
    For this article, precession is neglected. It is only a small 
change of longitude each year, quite 50 arcseconds. This accumulates 
to 1d 23m per century. While this is a substantial amount for many 
astronomy functions, for casual stargazing it doesn't show. You can, 
as example, enjoy a star-finding book from the 19th century. 
    Precession increases the longitudes of stars because the zero 
point is dragged uprange, putting greater distance between it and a 
given star. Eventually, the longitude wraps around 0-360 degrees to 
begin a new cycle of drift. A star at the west side of the vernal 
equinox, of longitude 359+, or 360-, will flip to 0+ degree when the 
equinox passes over it. 
    Latitudes are not affected. The latitude of a star remains the 
same for indefinite time, altered by the star's own spatial motion in 
the Galaxy and the slight change in the Earth axis tilt. Both factors 
for here are ignored. 
    With precession neglected, the longitude along the ecliptic can be 
printed on starcharts. Stating the longitude of a planet gives its 
location within the zodiac constellations. 
    The lat-lon of many zodiac stars within +/-8 degree latitude is 
listed here. These are specificly for the year 2000. Precession will 
increase each of the longitudes one degree by year 2072. 
        star     sign    lon  lat  magn  name 
        -------  ------  ---  ---  ----  ----
        bet Ari  Tau  4   34   +8  +2.6  Sheratan 
        M45 Tau  Gem  0   60   +4  +1.2  Pleiades 
        eta Tau  Gem  0   60   +5  +2.9  Alcyone 
        alp Tau  Gem 10   70   -5  +0.9  Aldebaran 
        bet Tau  Gem 23   83   +5  +1.7  Alnath 
        zet Tau  Gem 25   85   -2  +3.0  Al Hecka 
        mu  Gem  Cnc  5   95   -1  +2.9  Tejat Posterior 
        gam Gem  Cnc  9   99   -7  +2.0  Alhena 
        eps Gem  Cnc 10  100   +2  +3.0  Mebsuta 
        bet Gem  Cnc 23  113   +7  +1.1  Pollux 
        M44 Cnc  Leo  4  124   +2  +3.7  Praesepe 
        alp Leo  Vir  0  150    0  +1.4  Regulus 
        alp Vir  Lib 24  204   -2  +1.0  Spica 
        alp Lib  Sco 15  225    0  +2.8  Zuben Elgenubi 
        bet Lib  Sco 19  229   +8  +2.6  Zuben Elschemali 
        del Sco  Sgr  3  243   -2  +2.4  Dschubba 
        pi  Sco  Sgr  3  243   -5  +2.9  ---
        bet Sco  Sgr  3  243   +1  +2.6  Graffias 
        sig Sco  Sgr  8  248   -4  +2.9  Alniyat 
        alp Sco  Sgr 10  250   -5  +1.0  Antares 
        tau Sco  Sgr 12  252   -6  +2.8  ---
        eta Oph  Sgr 18  258   +7  +2.4  Sabik 
        M8  Sgr  Cap  1  271   -1  +6.0  Spiculum 
        gam Sgr  Cap  1  271   -7  +3.0  Nash 
        del Sgr  Cap  5  275   -6  +2.7  Kaus Meridionalis 
        lam Sgr  Cap  6  276   -2  +2.8  Kaus Borealis 
        M22 Sgr  Cap  8  278   -1  +5.1  Facies 
        sig Sgr  Cap 12  282   -3  +2.0  Nunki 
        zet Sgr  Cap 14  284   -7  +2.6  Ascella 
        pi  Sgr  Cap 16  286   +1  +2.9  Albadah 
        del Cap  Aqr 24  324   -3  +2.9  Deneb Algedi 
        bet Aqr  Aqr 24  324   +9  +2.9  Sadalsuud        
Stability of orbits 
    Now here is one crucial factor often missed from other discussion 
of planet behavior. The orbits of the planets are stable for long 
spans of time, many millennia at least. Except for really delicate 
work, we are comfortable knowing that the action of Jupiter today is 
that several centuries from now or many centuries ago. 
    Early cultures had no proof, as we do thru astrodynamics, and, of 
course, had no inkling of what a 'planet' really was. Some pundits 
claim that today we still don't know. 
    After sussing out how Jupiter does his thing for a few centuries, 
the early astronomers exercised a leap of faith. They presumed that 
his actions will endure into the future and the behavior figured out 
from previous records is still valid in their own day. 
    With the discovery of some strange orbital action at certain 
planetary stars, it is one of the incredible blessings of humanity 
that we live in a stable solar system. An idea of what could have been 
is gleaned from our experience with comets. We simply never figured 
out what's with comets until Newton applied his gravity model to them 
in the late 1600s. 
Inner and outer planet 
    The planets are disposed in two main groups, inner and outer, 
inferior and superior. The groups were originally based on a 
traditional ranking of the planets outward from Earth. The planetary 
turfs were stacked in rings or shells upward in the order of Moon, 
Mercury, Venus, Sun, Mars, Jupiter, Saturn. 
    This was based on the angular speed thru the zodiac. The turfs 
were placed adjacently with no 'wasted' room between them, only that 
they were broad enough to prevent 'collisions' when one planet passed 
an other in the sky. 
    Mercury and Venus were 'below' the Sun in this cosmography. Mars, 
Jupiter, Saturn were 'above' him. We keep these terms in the sense of 
closer and farther orbits around the Sun compared to Earth. The closer 
planets do have orbits, viewed from Earth that pass 'under' the Sun 
while those with farther orbits are always 'above' the Sun. 
    So far there remain only two inferior planets, there being no new 
ones found within Earth's radius from the Sun. Many asteroids are 
found in this inner region but I can't recall them ever being called 
inferior asteroids. All newer planets, Uranus, Neptune, Pluto, are 
superior planets. 
    At any moment the planets and Sun are disposed along the zodiac, 
each at its proper longitude. The difference of longitude, planet 
minus Sun, is the planet's elongation from the Sun. Elongation is 
usually stated east or west from the Sun rather than round the entire 
360 degrees. An east elongation is positive; west, negative. The 
latter is best for computations but the final answer is converted to 
an east-west elongation. 
    Occasionally elongation is the diagonal or great-circle distance 
between planet and Sun. For elongations greater than 30 degrees the 
discrepancy is minor. The diagram here shows the distinction with the 
triangle planet-ecliptic-Sun. 
                    |<-------- elongation ---------| Sun 
    - -ecliptic - - +- - - - - - - - - - - - - - - -O - - - - - 
                    |-latitude            / / / / / 
                   \|/            / / / / /  
                    * / / / / / / separation 
    For just about all home astronomy playing with just the longitudes 
will put the planet in the correct place in the zodiac within a degree 
or so. As a matter of history, the Babylonian astronomers neglected 
   A planet as it circulates around the zodiac must at some time pass 
by the Sun. The moment of bypass, when the planet and Sun longitudes 
are equal or the elongation is zero, is conjunction. 
    An outer planet has one conjunction in its round of the zodiac. 
This occurs when the planet in its orbit is farther, above, the Sun. 
This bypass is the superior conjunction. Sometimes, because there is 
only one conjunction for an outer planet, 'superior' is omitted. 
'Jupiter's conjunction with the Sun is on June 18th.' 
    An inner planet hs two conjunctions during its cycle of the 
zodiac. One is when the planet is between us and the Sun, below the 
Sun, for an inferior conjunction. The other is when the planet is 
beyond, above, the Sun for superior conjunction. 'Superior' and 
'inferior' are needed here to tell the two events apart. 
    At and near conjunction the planet is out of sight in daylight or 
strong twilight. The planet also rises and sets together with the Sun. 
We treat this interval near conjunction as the Sun gap, when the 
planet finished a previous tour of visibility, apparition, and will 
soon begin a new apparition. 
    In early astronomy the apparition began when the planet was first 
spotted in dawn twilight, for a superior planet as an example. For a 
few days after geometric conjunction the planet is hidden in strong 
twilight. On a particular morning the planet is removed just far 
enough from the Sun to rise in a sky just dark enough to let it shine 
thru and be seen. A minute later the oncoming dawn veils it again. 
This event of first sighting is the heliacal rising of the planet. 
    The end of the apparition was marked by the heliacal setting. The 
planet is seen to set in twilight until one day, as conjunction 
approaches, the twilight is too bright. The planet sets just before 
the sky darkens enough to otherwise let it shine thru. 
    Nowayears we no longer record or calculate the heliacal events. 
The are driven by factors such as the eyesight of the observer and the 
shmutz along the horizon. We apply a nominal or recognition interval 
before and after conjunction when the planet is reasonably smothered 
in twilight. This interval separates the two adjacent apparitions. One 
month is a common value. 
Visibility or elongation chart
    A very handy chart is one that plots solar elongation by date 
within a year. A skeleton chart is shown here in one common layout. 
       east elongation - evening sky   | west elongation - morning sky 
       |.. 19  18  17  16  15  14  13  12  11  10  09  08  07  06 ..  
       |  105  90  75  60  45  30  15  0   15  30  45  60  75  90     
 Jan  0|        .          \     | . \ |-Sun          . 
     10|          .         \    |   . |                . 
     20|            .   Mars-\     \   | \                . 
     30|              .       \     \  | .|-Mer             . 
 Feb  9|                .      \     \ | / .           star C-. 
     19|                  .     \     \|     .                  . 
 Mar  2|                    .    \    /|\      .-star B           . 
     12|               star A-.   \  / | \-Ven   .                  . 
     22|                        .  \|  |  \        . 
 Apr  1|                          . \ \|   |         . 
       ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~~  ~  ~  ~  
    In this chart the Sun is placed on the vertical centerline at 0 
degree elongation and 12 hours of time. Hours of the day march from 
24 (midnight) at the left thru 0 at the previous midnight on the 
right. Elongations march from 180 degrees west at the right thru 0 at 
the Sun thru 180 degrees east at the left. In this skeleton the hours 
and degrees are cropped at left and right ends. 
    The hour indicates when its vertical column crosses the south 
meridian, aligns straight south. The corresponding elongation is also 
on the meridian. I must mention that just about all such charts are 
for northern observers. There are 'up-side-down' visibility charts for 
southern observers, but they are rarely seen in northern lands. 
    Many curved lines run down the chart. Two corkscrew around the 
Sun. These are Mercury (tighter screw) and Venus (looser screw). You 
may read directly against the horizontal row for any date the 
elongation of these planets from the Sun. You may also read out the 
hour (approximate) when these planets cross your meridian. 
    An outer planet's curve is a lazy wavy line trending from left to 
right down the chart. When it crosses the E180d/24h or W180d/0h column 
it opposes the Sun. Note that an outer planet can touch all 
elongations from 0 thru E180 and W180, while Mercury and Venus are 
confined to a narrow range of elongation centered on the Sun. 
    Since on the left half of the chart the hours are after the Sun's 
transit, a planet there is behind, later than, the Sun and 1s in the 
evening sky after local sunset. When a planet is on the right side, it 
is ahead, earlier than, the Sun and is in the morning sky before 
    When a planet's curve crosses the 0d/12h centerline it makes 
conjunction with the Sun. Some authors will break the curve at the Sun 
to indicate a superior conjunction and leave it continuous for an 
inferior conjunction. 
    Extra curves can be plotted like the three for stars A, B, C. 
Since stars do not move against the celestial sphere, their plots are 
straight lines drifting left to right at the seasonal rate of quite 1 
degree/day or about 30 degrees/month. Besides stars, these lines can 
represent cardinal points of the ecliptic, frontiers of zodiacal 
constellations, deepsky objects. Only a few of these extras should be 
plotted to avoid excess clutter on the chart. 
    Omitted in this skeleton but common on elongation charts is the 
curve for the Moon. The Moon behaves quite differently from planets by 
rounding the ecliptic in 29-1/2 days relative to the Sun. This is her 
synodic period or cycle of phases or lunation. Her curves are gentle 
upward sloping lines from left to right, 12-1/3 laps per year. 
    Lunar phases are taken from the elongation or transit times: 
0d/12h for new Moon; E90d/18h, first quarter; 180d/24h (either sense), 
full Moon; W90d/06h, 3rd quarter. 
    Where any two curves intersect, there is a conjunction with their 
associated objects. The curve for Mercury crosses that for star B in 
early February to mark a conjunction between the two. Because latitude 
is ignored, the two in the sky may be almost touching or spaced with 
many degrees of latitude between them. In the extreme case, the Moon 
may actually cross in front of a planet or star to hide it from view. 
    This visibility chart shows at a glance when planets are visible 
by hour of the day or proximity to the Sun. In general a planet within 
15 degrees elongation from the Sun is not easily visible. 
    Some astronomers have trouble seeing the circular properties of 
this chart. One way to make things clear is to trim the printed chart 
and roll it around to match up the left and right edges. Make sure to 
line up the dates! Taping this cylinder allows you to turn it and see 
how the planet wraps around the zodiac. If you're ambitious attach to 
the top and bottom the charts for previous and future years to build a 
pipe of celestial action for several years. 
Other targets 
    In addition to conjunctions and, in general, elongation against 
the Sun, a planet may conjunct with an other planet or a star. It may 
also conjunct with the Moon but she is a ways beyond this article to 
handle. We note such conjunctions in almanacs for curiosity and to aid 
in identifying a planet. 
    On the other hand we almost never routinely publish or compute 
elongations of a planet from an other or a star. If you need such an 
other elongation you must subtract their published solar elongations 
or actually measure it from a computer planetarium program. 
    There is no convention for the sense of the elongation between 
planets and stars. You must be careful about stating the 'from' and 
'to' body. 'Mars is 15 degrees east of Spica.' 'From Jupiter to Venus 
is 6 degrees.' 
    Astrologers care very much about these other elongations. Certain 
values, not just 0 (conjunction) and 180 (opposition), are called 
aspects. Astrologers developed computer programs to tabulate or graph 
elongations among planets and stars. 
    Some programs can look for desired elongations among specific 
targets. Such a program can query 'When is Jupiter 15 degrees west of 
Mars?' 'When does Venus oppose the Beehive cluster with a western 
elongation from the Sun'? 
    For this reason, it can be kosher for an astronomer to play with 
an astrology program if he needs elongations other than only against 
the Sun. One situation is to plan an astrophotography opportunity. 
Motion near conjunction
    When an outer planet rounds superior conjunction or an inferior 
planet rounds inferior conjunction the planet is moving along the 
zodiac slower than the Sun. It lags behind the Sun. The planet comes 
into view a while after conjunction in the eastern sky at dawn before 
sunrise. It drifts into a darker region of twilight after it rises 
before the Sun, allowing it to be seen for the first time after 
    When an inner planet rounds superior conjunction it is running 
faster than the Sun and leads him. It first comes into view in the 
western sky at dusk after sunset. It advances far enough ahead of the 
Sun, setting later after him, in a darker part of twilight. 
    The jargon can be confusing for the newcomer. The rounding of 
superior conjunction is in the prograde direction. The planet runs 
east, downrange, thru the zodiac at a pace slower, less degrees per 
day, than the Sun. For this article I set the Sun's speed at a uniform 
one degree per day thruout the year. 
    For the inferior conjunction the planet, an inner one, is sliding 
in retrograde motion. It's moving east to west, uprange in the zodiac. 
What makes the concept of motion so difficult at first is the frame of 
reference, the Sun or the stars. This is discussed below for sidereal 
and synodic motion. 
    The diagrams show what's going on with the planet moving relative 
to the Sun  During this action the stars behind the Sun flow east to 
west at 1 degree/day. 
          sup conj             sup conj       inf conj 
        - - -+- - - >      < - - -+- - -    - - -+- - - > 
             O                    O              O 
        outer planet        inner planet    inner planet
Observing conjunctions
    If really isn't possible to observe a planet at or very near solar 
conjunction. An exception is Venus for her extreme brilliance and thin 
crescent shape. These make her more easily found in the daytime. When 
looking for a planet near the Sun it is ABSOLUTELY ESSENTIAL to 
physicly prevent ever looking into the very Sun.
    The usual method is to hide the Sun behind a fixed barrier, a roof 
or wall, so that the Sun moves, by diurnal motion, FARTHER BEHIND the 
barrier. NEVER place the Sun to move OUT OF the barrier into open sky. 
Conjunctions with SOHO 
    Since 1996 home astronomers have a far simpler, safer, and more 
fun way to follow planets near conjunction. In that year a solar 
monitoring satellite, SOHO, began service as a joint NASA-ESA project. 
Among its instruments is a camera, LASCO, that centers on the Sun to 
photograph the middle and outer corona. 
    The corona extends many degrees from the Sun so the camera images 
a field of 7 degree radius. The pictures are good enough to record 
background stars. SOHO has several cameras. The one you want is the 
LASCO C3 camera, with blue sky background in its pictures. 
    The image quality is pretty awful but adequate to reveal stars to 
about 7th magnitude. When a planet conjuncts the Sun it shows up as a 
moving dot in the pictures day by day. 
    The images are posted in the SOHO website for anyone to examine. 
In fact, some home astronomers study them to find comets that are 
impossible to see in daylight. In this way two thousand new comets of 
the Kreutz sungrazing class were found, adding valuable knowledge 
about these peculiar objects. 
    This trick works because SOHO is orbiting the Sun at the Sun-Earth 
L1 Lagrange point. It has the same line of sight on the Sun as Earth 
does, unlike an other solar observatory STEREO. That one has two 
satellites, one each at the L4 and L5 points, way off the line of 
sight from Earth to Sun. 
     Here's a sketch of ae SOHO LASCO C3 iamge 
    |    *               *      *  | 
    |            #     @  #        | 
    |     *         #    #  #  *   | 
    |  #   #         #  #  #  *    | 
    |        # * #  # / \ # #      | 
    |               #( O )   # #   | 
    |           *  # #\ /\  #     | 
    |           #    #  # \\   *   | 
    |               #   #   \\     | 
    |date & hour             a\\ a | 
    The Sun is covered by a paddle, a-a, protruding in from the lower 
right corner. This blocks the Sun's overwhelming brilliance from the 
camera. On the paddle is a circle, O, marking the size and location of 
the Sun's disc behind it. 
    The #s are the corona, in agitation as seen in successive 
pictures. To keep close watch on the corona SOHO takes pictures every 
few hours. The date and hour, in Universal Time, are posted in the 
lower left corner of the frame.  
    The stars, *, are scattered around the Sun according as his 
location in the zodiac. North is up and the ecliptic is the horizontal 
centerline thru the Sun. 
    @ is the planet on a picture taken at conjunction. You recognize 
it by being the odd 'star' in the field. A common trick is to play a 
planetarium program next to the SOHO webpage. Turn off the daytime sky 
and zoom in to about the same scale as the SOHO picture. Now flip 
between the two scenes to identify the stars and planet. 
    You can verify that the behavior of planets near conjunction 
follows the discussion above by fetching SOHO pictures for recent 
conjunctions. Track the planet for a few days in each case. 
    A superior planet's orbit encloses Earth's to allow the planet to 
circulate completely thru all 360 degrees of elongation from the Sun. 
At some point in its round the planet stands 180 degrees, opposite on 
the ecliptic, from the Sun. This is opposition. 
    The planet is in a night sky and is observable all night long, It 
rises near sunset, stands in the south near midnight, sets at sunrise. 
In the solar system, the planet and Earth are adjacent in their orbits 
on the same side of the Sun. From a telescope observer's concern the 
planet is closest to Earth, largest in angular diameter, and brightest. 
All these factors favor easier inspection in small telescopes. 
Motion near opposition 
    A superior planet's normal run thru the zodiac is direct or 
prograde, advancing steadily downrange toward increasing longitude. 
Near opposition it executes a weird reversal of direction. For a while 
it moves uprange, against the signs. This is the retrograde motion. 
    A trace of the planet on the sky is a swing to-&-fro, an 'S' 
curve, or a loop. The shape comes from the action of latitude during 
the opposition period. 
    The planet begins retrograde motion a few weeks before the moment 
of opposition by slowing down its prograde pace. It eventually comes 
to a complete halt on a certain date to attain the western station. 
    On the next day it starts its reverse movement, speeding up thru 
the opposition point. It runs fastest at opposition, then slows down 
again. It comes to a second halt at eastern station a few weeks later. 
    The planet after a day's dwell at eastern station picks up its 
prograde movement again. Then after it glides thru the zodiac toward 
its next conjunction with the Sun. This process is illustrated below. 
                *             / - - - -<- - - - - -<- - A 
                        *   /      *                   * 
            B - -<- - - - / - -<- - - - - - -<- \   * 
            west station-|     *                  \ 
                           \              * C      |-east station 
                        *    \                     / 
                               \ - ->- - -+- ->- -    * 
                             *            | 
    Early astronomers had no explanation for this behavior but they 
thoroly appreciated it and took full consideration of it in their 
planetary work. The action is a direct consequence of the heliocentric 
system, but it is NOT, as some authors claim, a proof of it. The 
retrograde motion can be replicated in a geocentric scheme with a more 
complicated mechanism. The Ptolemaeus model really worked well enough 
to impede acceptance of the Copernicus model for two centuries after 
it was proposed. 
    The planet comes along in prorade motion at A several weeks before 
opposition. It passes star C in a local conjunction along the upper 
path and continues eastward thru the zodiac. It slows down, in this 
instance by turning south, to stop its forward advance at the western 
station. Why the 'west' station is at the east end of the loop is 
explained a little later. 
    The planet rounds the west station and enters the loop. It is now 
running west thru the zodiac against wayside stars like C. It passes C 
again within the loop. In this made-up case this happens at 
opposition. At this moment its elongation from the Sun is 180 degrees. 
    The planet during the retrograde stage has the fastest pace of its 
whole circuit around the zodiac, several degrees per day for Mars, as 
example. This speed is soon tempered as the planet nears its eastern 
station, where it slows, stops, and leaves the loop. 
    After leaving the loop the planet resumes its normal eastward, 
direct, prograde movement, passing star C for the third time. At point 
B it is running in the normal downrange path. 
    The length of the loop, its shape (here stylized as a real loop), 
abd its duration between the station points are functions of the 
planet and the part of the orbit the loop occurs in. 
Station points 
    The existence of stations on a planet's path was a genuine mystery 
for early astronomers, who had no spatial picture of the cosmos. Their 
simplistic nesting of planet spheres or rings was of no help to explain 
the stations. The cause of stations is demonstrated trivially with the 
heliocentric model. 
    During a station passage the planet is heading recta mente to or 
from Earth along our line of sight. It for the moment has no cross 
motion. That's why it seems to stop on the celestial sphere while it 
is really moving at planetary speed, many kilometers per second, in 
its own orbit around the Sun. 
    The reversal of direction comes from the relative speed of Earth 
and planet in their orbits. Earth is in a smaller faster orbit to run 
by the planet and leave it behind in the sky. The ago-old analogy for 
home astronomers in New York City is the interplay of a local and 
express subway train. Every kid experiences the apparent sliding 
backward of the other train against his own even tho both are speeding 
forward on adjacent tracks. This explanation makes no sense for any 
other town's astronomers, who lack subways or have only simple ones. 
    A more universal analog may be a carnival ride with seats on 
concentric rings around a central hub. The rings have their own motors 
to adjust their speed, like to load and unload riders. You in an inner 
seat must turn your head to follow a specific outer seat. 
    Sometimes you may look forward to see the other seat, then must 
twist round to look back. A film of the other seat taken from yours 
does given the impression it is swinging forward and backward relative 
to you. The outer seat goes thru its retrograde loop. 
East is west, west is east
    One of the most perplexing features of planet alignments is the 
use of 'east' and 'west'. Consider the case of a planet coming away 
from solar conjunction in the morning dawn. You are looking to the 
east but the planet has a west elongation. An other example is the 
planet when it leaves the retrograde loop. It's then at the west end 
of the loop, at its eastern station. 
    The mixup comes from two factors. First is that diagrams of the 
planets are drawn on flat surfaces, seemingly of indefinite extent. The 
celestial sphere is, uh, spherical. Displacement in one direction 
eventually cycles back to the starting place. Such is the case of 
planets circulating thru the zodiac. They repeatedly pass the vernal 
equinox or a given star. 
    East and west have no absolute location, only a direction. They 
are like 'uptown' and 'downtown' on Manhattan. Even on Earth, in 
geography, things get mixed up. The West Indies are in the east part 
of America. The East Indies are far west of America. 
    The other factor is the dance-of-three of the horizon, Sun on the 
ecliptic, background stars. For many sky scenes the Sun is down, out 
of sight and out of mind. In others the stars are veiled by twilight 
or daylight. Applying a directional concept with the wrong reference 
WILL royally disorient you. 
    In a planetarium program turn off the horizon and daylight. Now 
zoom way out to show the whole ecliptic. Move a superior planet to the 
start of its retrograde loop. The planet's solar elongation is lass 
than 180 degrees west and greater than 180 degrees east. The adit 
station is the western one by its elongation from the Sun. 
    Now move the planet thru the loop to its exit station point. The 
elongation is less than 180 degree east and more than 180 degrees 
west. The exit station is the eastern station. 
Inferior planet loop
    An inferior planet also has retrograde loops but they are usually 
not observed due to twilight or daylight. Near inferior conjunction 
the planet races uprange to get from the east to the west side of the 
Sun. If the stars were visible, as they are when a planetarium program 
has its daylight turned off, the planet is seen to rush westward. 
    There are stations, too, where the inner planet enters and leaves 
the retrograde loop. These can be observed if they occur far enough 
from the Sun in dark sky, away from bright twilight, so background 
stars are visible. 
    There are two other points of interest for an inner planet, the 
greatest elongation east and west. The inner planet in constrained in 
the sky to a zone centered on the Sun, as delimited by the diameter of 
its orbit. The planet rounds superior conjunction, arcs away to the 
east of the Sun, then attains a greatest eastern elongation. 
    It then starts reducing its elongation as it heads toward inferior 
conjunction. All this happens with the planet in the evening sky. 
    In the morning sky the planet rounds inferior conjunction and 
swings away to a farthest western elongation. After then it swings 
back, decreasing its elongation, to reach superior conjunction. 
    A common mistake is to consider the greatest elongation, east or 
west, as the stations. The problem comes from missing the reference of 
the stars, which can be obscured by twilight. Lo here the diagram. 
                     greatest east elon             * 
                        *    3               *   2           sup vonj 
                            -+- - - - - - -<- - -+ - - - -<- - -+- - 
                   *     /     *         O     O     O    O    O 
           east station-|4 *         *   5     4     3 *  2    1 
                      *  \       
                        *   - - ->- - -  + - - - - - ->- - 
                                *    inf conj         * 
    The Sun, the O's, is running along the ecliptic at one degree per 
day. Venus crosses superior conjunction at solar location 1. She has 
the speed (made up here) of 1-2/3 degree per day. She gains on the Sun 
2/3 degree/day, pulling ahead of him in the zodiac and increasing the 
elongation between the two. 
    When the Sun advances to position 2, farther along the ecliptic, 
Venus accumulated many degrees of elongation and is east of the Sun. 
It starts to slow its forward motion to 1-1/3 degree/day and reduce 
her lead on the Sun to only 1/3 degree/day. 
    At solar location 3 Venus is slowed to one degree per day, pacing 
the Sun and no longer advancing ahead of him. At this moment she 
attains her greatest eastern elongation. 
    Note very well that against the stars, a few spotted around her in 
the illustration, Venus is still advancing downrange at 1 degree/day, 
the same as the Sun. She is NOT at a standstill among the stars! 
Because greatest elongation can occur in a dark sky for Venus, you may 
confirm this on the next occasion by examining Venus and her nearby 
stars thru binoculars. 
    When the Sun is at location 4 Venus slowed to zero degree per day 
in approach to her retrograde loop. Now, a ways after the greatest 
elongation, she is stationary in the stars. The Sun now takes the gain 
on her with his steady 1 degree/day motion. 
    After the station Venus runs retrograde with a negative speed of 
1-1/2 degree/day. She closes the elongation from the Sun by 2-1/2 
degree/day, her own negative 1-1/2 minus the Sun's 1. She collapses 
the elongation to zero at solar location 5. rounding her inferior 
conjunction. She continues her retrograde movement for a while longer 
until she reaches her western station, not shown here. 
    The size, shape, duration of the retrograde loop has the widest 
variation for Mercury from his highly elliptical orbit. This feature 
of Mercury upset most models in early times, on top of his overall 
difficulty of observation. Mercury is hardly ever in a dark sky to 
bank his location against stars.
Planet cycles 
    There are really two ways to track a planet's motion and location. 
One is against the stars; the other, the Sun. In as much as the Sun 
moves along the ecliptic, the two yield different motion and location. 
    The cycle against the Sun is successive returns of the planet to 
the same elongation. This is the synodic cycle. Other explanations 
insist that the interval be taken between conjunctions or oppositions. 
This is too limiting. The cycle applies to ANY specific elongation. 
    The sidereal period is the time for the planet to complete a lap 
of its orbit against the background stars. This is the orbital period 
or length of year in tables of planet facts & figures. 
    Off hand it seems that we can not directly observe a sidereal 
period from Earth. When a planet returns to the initial point in its 
orbit, we see it from a different angle, against a new set of stars. 
There is a way to capture a true sidereal period thru Earth-bound 
observation. I explain this later. 
    Early astronomers, having no concept of solar system, found the 
sidereal period by a remarkably simple method. But they never linked 
this number to a cycle of the planet around Sun, Earth, other center. 
It was just a parameter that factored into the motion and location of 
the planet in the sky. 
    The cycles are not exactly equal from a one to the next because 
the planets have slightly elliptical orbits and a varying speed in 
these orbits. A longer term factor is real alteration of the orbits by 
gravity tugs from other planets. 
    Mercury's orbit is too excentric for a simple treatment, a 
situation that bedeviled early astronomers. Their method for tracking 
Mercury was always the most convoluted of the planets. Not that we in 
modern times were completely accurate. It took Einstein physics to 
close the last remaining errors in Mercury's motion. 
    What saves home astronomers is that Mercury is usually too close 
to the Sun for easy viewing. We skip it in skywatching unless there is 
some extra feature about him to look for. Mercury is not used in 
classical celestial navigation for this reason. 
Orbit irregularities
    To illustrate how orbit parameters vary over time, I list the orbit 
radius, in AU, for the planets during the years 2010 to 2012. The 
fluctuations are tiny, yet they can corrupt longterm projection of 
planet movement. 
  date        | Mercury  | Venus    | Mars     | Jupiter  | Saturn 
  2010 Jan  4 | 0.387098 | 0.723330 | 1.523691 | 5.202776 | 9.511343 
  2010 Jun 28 | 0.387098 | 0.723329 | 1.523732 | 5.202798 | 9.509762 
  2011 Feb  8 | 0.387098 | 0.723328 | 1.523602 | 5.202752 | 9.509322 
  2011 Aug 27 | 0.387099 | 0.723329 | 1.523645 | 5.202858 | 9.510149 
  2012 Feb  2 | 0.387098 | 0.723329 | 1.523616 | 5.202880 | 9.512027 
  2012 Aug 20 | 0.387099 | 0.723326 | 1.523714 | 5.202770 | 9.515472 
    Just from this sample you see why it's impossible to use a fixed 
set of parameters for more than a few decades. They are good for 
casual skywatching but not to replicate a scene many centuries away. 
    You must employ a dynamical model of the solar system for precise 
and faithful simulation in the far past or future. Until the 1990s 
home astronomers had no such tools. Neither did historians and 
archaeologists. This lack leaded to some erroneous deductions about 
ancient historical events linked to celestial activity. 
Remarkable formula 
    In the Copernicus model the sidereal and synodic cycles are easily 
demonstrated. The ancient astronomers knew nothing of this model and 
there was a time when the geocentric model was still in the future. 
Yet the sidereal and synodic cycles were worked out to a stunning 
degree of accuracy. 
    As a direct consequence of orbital motion around the Sun there is 
a binding between the cycles of a planet banked off of the stars and 
the Sun. It is not casually possible to observe the sidereal cycle 
because when a planet completes one revolution around the Sun it is 
seen from a very different direction from Earth and stands in a very 
other place in the zodiac. Returns of a planet to conjunction with a 
given star are NOT one sidereal period apart. 
    The synodic cycle was easy to observe directly. It's the interval 
between returns of a planet to the SAME elongation from the Sun. It 
doesn't have to be only conjunction or opposition as is commonly 
described. After a few rounds of the planet in synodic cycles, the 
irregularities due to the elliptical orbits net out to obtain a mean 
synodic motion. 
    It works out that the relation between the two cycles is amazingly 
    (Earth sid cycles) = (plan sid cycles) + (plan syn cycles) 
For an inferior planet the synodic cycles are negative; superior, 
positive. The cycles are the COUNT OF LAPS, not the duration of each 
lap. This formula is valid for any interval of time, so long as the 
counts are correct. They don't have to be only integers. 
    A sidereal cycle of Earth is ONE YEAR long by the way we define 
our calendar, One year IS one lap of Earth around the Sun. Remember 
that we ignore precession, so the year of equinox-to-equinox is the 
same as that from star to star. The former, tropical year, is about 20 
minutes shorter than the latter, sidereal year. 
    As an example Mars in the two years 2010-2011 went thru 0.9365 
synodic cycles and 1.0599 sidereal cycles. Plugging these into the 
syn-sid formula above, we have 
    (Earth sid cycles) = (Mars sid cycles) + (Mars syn cycles) 
                       = (1.0634( + (+0.9365) 
                       = (1.9999) 
which is the two Earth years, off by rounding. 
    A second example with Venus for the same two years gives 
    (Earth sid cycles) = (3.2510) + (-1.2510) 
                       = (2.0000) 
again the two cycles, two years, for Earth. 
    This is how an astronomer knowing nothing what so ever about 
orbital motion may obtain the planet's sidereal cycles. Observe the 
SYNODIC cycles of the planet for a given number of Earth SIDEREAL 
cycles, Earth years. Plug the numbers into the syn-sid formula and 
solve for the planet SIDEREAL cycle. 
    Imagine we did not know the sidereal cycle for Venus. In the stead 
we record when Venus crosses the same elongation from the Sun during, 
say, 15 years. I take many years to smear out the orbit irregularities, 
which ancient astronomers may have treated as observational errors. 
    Doing this for 1996 thru 2011. I find that Venus hit inferior 
conjunction on 1996 June 10 and 2010 October 28, completing 9 synodic 
laps. The number of Earth sidereal cycles is the number of years 
between these two dates, 14.3819 years. 
    The syn-sid formula becomes 
    (14.3819) = (Venus sid cycles) + (-9) 
    (Venus sid cycles) = (14.3819) - (-9) 
                       = (23.3819) 
    These cycles were accomplished in Earth's 14.3819 years. The 
duration of a Venus sidereal cycle is (14.3819 yr)/(23.3819 sid cyc) = 
(0.6151 yr/sid cyc). The synodic period is similarly calculated as 
(14.3819 yr)/(-9 syn cyc) = (-1.5980 yr/syn cyc) 
School formula
    You likely know the 'one-over-one-over' formula relating sidereal 
and synodic periods: 
    (plan syn per) = (1)/((1/Earth sid per)-(1/plan sid per)) 
I call this the 'school formula' because it seems intended to favor 
maths mistakes on school homework. There is no need for such a jinxed 
formula. By algebra this formula reduces to: 
    (plan syn per) = ((plan sid per) * (Earth sid per))
                    / ((plan sid per) - (Earth sid per))
which is easier to work with and has far less chance of mistakes. When 
you realize that Earth's sidereal period is one year, it reduces to 
the even simpler form: 
    (plan syn per) = (plan sid per [yr2]) / ((plan sid per) - (1))
    Note carefully the units! The numerator is [year2] because it's a 
multiply of [year]*[1 year]. The denominator is [year]. The division 
is [year2]/[year] = [year]. 
    This is an application of dimensional analysis to check the 
validity of an equation. If it fails the dimension test, it IS wrong. 
The converse is not always true because the operations may be wrong, 
yet the dimensions could still come out correctly. 
    Consider Mars with sidereal period of 1.8807 yr. 
    (Mar syn per) = (1.8807 yr2) / ((1.8807 yr) - (1 yr)) 
                  = (2.1355 yr) 
The procedure is the same for both inferior and superior planet with 
no problem with algebraic signum. 
    To find the synodic period for tow other planets, replace Earth's 
period with that of the other planet. The synodic period of Venus as 
seen from Mars is 
    (Ven/Mar syn per) = ((0.6152 yr) * (1.8807 yr)) 
                       / ((0.6152 yr) - (1.8807 yr)) 
                      = (-0.9143 yr) 
The negative result means Venus is an inner planet. The period is in 
EARTH years, NOT Mars or Venus years. 
Synodic arc 
    Altho the Venus synodic cycle in the example above was an integer, 
the sidereal cycle was not. Venus's place in the stars migrated around 
the zodiac for each cycle, not returning to the same place again in 
this particular 14 year span. 
    Each cycle was finished downrange by an angular displacement of 
    (Ven syn displace) = ((Ven syn per) - (whole years))
                        * (360 deg/yr) 
                       = ((-1.5980 yr) - (-1 yr)) * (360 deg/cyc) 
                       = (-215.2800 deg/cyc) 
    The 360 deg/yr is the mean motion of the Sun, who 'pushes' the 
synodic event forward in the zodiac. It is proportioed to the excess 
part of the next year beyond whole cycles. 
    The negative amount means the displacement is uprange, but by 
knocking out complete rounds of 360 degrees, we get a more normal 
    (-215.2800 deg/cyc) + (360 deg/cyc) = (144.7200 deg/cyc) 
where 360 is one round of the zodiac, leaving the incomplete second 
lap. Each inferior conjunction occurs 144.7200 degrees downrange from 
the previous one. This is the synodic arc and it applies to all other 
elongations of Venus. 
    The synodic arcs for the planets, based on mean motions, is given 
    planet   | sid year | syn year | syn arc 
    Mercury  |  0.2409  | -0.3173  | 245.7542 
    Venus    |  0.6152  | -1.5988  | 144.4491 
    Mars     |  1.8807  | +2.1355  |  48.7658 
    Jupiter  | 11.8624  | +1.0921  |  33.1418 
    Saturn   | 29.3090  | +1.0353  |  12.7160 
    If you see Jupiter in his western station next to Regulus, 
longitude 150 deg, in one year, the next western station is near 
longitude 183 deg. This is 3 deg east of the autumnal equinox.. 
    The Venus figures in this table differ from the example above. The 
example used a particular cycle of Venus while the table is based mean 
motion of Venus. 
Observable sidereal period 
    The heliocentric model offered a way to directly observe a 
planet's sidereal period. The orbit of a planet is slightly inclined 
to Earth's orbit. The planet crosses the plane of Earth's orbit, once 
heading south to north and then north to south. It does both crossings 
each per revolution or sidereal period. 
    In the sky the planet crosses the ecliptic, the edgeon view of 
Earth's orbit. The crossing from south to north is the ascending node; 
north to south, descending node. By watching when the planet returns 
to the same node we obtain directly its sidereal period. Kepler was 
the first to do this in working out his rules for planet motion. 
    For the Venus example in 2010-2011, the dates of ascending node 
crossing are 2010 April 13, 2010 November 23, and 2011 July 6. The 
interval between the first and last is 1.2320 years. Since this spans 
two cycles, we have 
    (Ven sid per) = (Earth years) / (sid cyc) 
                  = (1.2293 yr) / (2 sid cyc) 
                  = (0.6146 yr/sid cyc) 
    This is quite close, given that I took only two cycles and noted 
only the date, but not the hour. Looking at a longer time interval, 
like a few decades would yield a better value which is how ancient 
astronomers dealt with such situations. 
Integer cycles? 
    Can there be INTEGER synodic cycles plus INTEGER sidereal cycles 
that equal INTEGER years? If so, then the synodic events take place 
again in the same places in the zodiac and on the same calendar dates. 
    In truth the answer is flat out 'no' because the periods are fully 
irrational. There is no pair of integer cycles that exactly make an 
integer number of years. 
    If we relax the precision of fit, like by a tolerance of 
observational error, then we can find such pairs. This was done by 
Babylonians by a method we go thru here. It did NOT demand continuous 
direct observation of the planets for thousands of years, as is 
sometimes claimed. Records of only a few centuries were enough. 
    Consider Jupiter. After one year his synodic cycle isn't complete 
yet, falling short by 0.0921 cycle. The numbers can get tricky, so we 
build a table of trials:. 
         yrs | combination | syn cyc  | sid cyc | sid exc 
           1 | 1           |   0.9079 |  0.0843 | +0.0843 
          12 | 1 * 12      |  10.9884 |  1.0116 | +0.0116 
          71 | 6 * 12 - 1  |  65.0147 |  5.9853 | -0.0147 
          83 | 7 * 12 - 1  |  76.0031 |  6.9969 | -0.0031 
          95 | 8 * 12 - 1  |  86.9915 |  8.0085 | +0.0085 
         166 | 14 * 12 - 2 | 152.0062 | 13.9938 | -0.0062 
         261 | 22 * 12 - 3 | 238.9977 | 22.0023 | +0.0023 
         344 | 29 + 12 - 4 | 315.0008 | 28.9992 | -0.0008 
         427 | 36 * 12 - 5 | 391.0039 | 35.9961 | -0.0039 
    We're mixing and matching pairs of sidereal and synodic cycles in 
hopes of finding a set as close to integers as possible. As an index 
of fit, the excess (over or under) sidereal cycles is noted. This 
ideally must be zero, 
    After a few trials, we arrive at some promising pairs. Each entry 
is a combination trying to net out the plus and minus excesses. We 
come to an excellent pair of 315 synodic cycles and 29 sidereal cycles 
over a 344 year span. 
    In the 345th year the events of year 1 repeat almost exactly in 
the same elongation, longitude, date. A master table of events can be 
compiled for each of the 344 years. An event of the instant has its 
proper place in the 344-year grand cycle. From there any other event 
can be found by counting off the years from the instant event to the 
other event. 
    Such a master table would look like this skeleton: 
    synodic event | date  | long | elon 
    year 1 of 344
    west station  |  xxx  | yyy  | zzz
    opposition    |  aaa  | bbb  | ccc
    east station  |  ddd  | eee  | fff 
    conjunction   |  ggg  | hhh  | iii
    ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~
    ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~  ~
    year 344 of 344
    west station  |  lll  | mmm  | nnn
    opposition    |  ooo  | ppp  | qqq
    east station  |  rrr  | sss  | ttt 
    conjunction   |  uuu  | vvv  | www
    year 345 of 344 = 1 of 2nd set of 344 
    west station  |  xxx  | yyy  | zzz
    opposition    |  aaa  | bbb  | ccc
    east station  |  ddd  | eee  | fff 
    conjunction   |  ggg  | hhh  | iii
    The repetition is not perfect. The discrepancy, by displacement in 
the zodiac, is only 0.288 degree. This is within tolerance for just 
about all naked-eye observing. 
    The dance of the planets is one of the happiest features of the 
sky for home astronomers. By watching the planets by eye or binoculars 
you carry on a tradition that reaches back to the dawn of astronomy. 
    By working thru the concepts and examples here you duplicate the 
method employed by ancient astronomers in their effort to understand 
the planets. You also realize that the periods and cycles and events 
do not imply a knowledge of a Sun-centered solar system. 
    You may actually try building a master synodic table for a planet. 
It will be a bit tedious but a calculette or computer language code 
will do the maths and generate the entries for the table. You may 
include the synodic events of interest, not only the traditional ones 
around opposition and conjunction. 
    Test this table by homing it on a known event in the instant year. 
Then see if an earlier event in the table actually occurred on the 
tabular date, elongation, longitude. This is easiest done with a 
planetarium program. Also see if a future event is predicted