SIDEREAL AND SYNODIC CYCLES
-------------------------
John Pazmino
NYSkies Astronomy Inc
www.nyskies.org
nyskies@nyskies.org
2010 December 18 initial
2011 December 25 current
Introduction
----------
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
viewpoint.
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.
Zodiac
----
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
---------------------------------------------------
Precession
--------
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.
Elongation
--------
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
planet
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
latitude.
Conjunction
---------
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.
Visibility
-------
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
sunrise.
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
conjunction.
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.
Opposition
--------
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
* \ /
\ - ->- - -+- ->- - *
* |
opposition
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
simple.
(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
displacement:
(-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
below.
----------------------------------------
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.
Conclusion
--------
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
correctly.