SIDEREAL AND SYNODIC CYCLES ------------------------- John Pazmino NYSkies Astronomy Inc www.nyskies.org firstname.lastname@example.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.