FROM TIME to time a planet, as it is observed from the earth, moves into alignment with the sun or with another planet. These alignments are termed conjunctions or oppositions, according to the relative positions of the earth and the sun. They can be predicted by a graphical technique that has been devised by B. E. Johnson (2233 81 Avenue SE, Mercer Island, Wash. 98040).

Conjunctions and oppositions have attracted interest throughout recorded history. To astrologers they were portents, usually dire. Today the forces that arise from such groupings are taken into account by digital computers to help determine the course of space probes.

To amateur observers conjunctions and oppositions form interesting patterns in the night sky. Moreover, a conjunction marks the onset of spectacular motions that appear to carry the planets involved along a looping path against the background of the fixed stars. The looping motion can be followed by making nightly observations for a few weeks.

The mathematical prediction of planetary alignments can be tedious, even with the aid of a pocket calculator. No graphical technique can yield an exact prediction. Moreover, planetary motions are not exactly proportional to the passage of time. Even so, Johnson's method generates results of sufficient accuracy to satisfy most amateur needs. Johnson, who is an electrical engineer, describes his scheme as follows.

"Planetary alignments are of four kinds, depending on the distance of the planets from the sun. The two 'inferior' planets, Mercury and Venus, are closer to the sun than the earth is. Mercury completes one orbit around the sun in 87.969 earth-days; Venus, in 224.7 earth-days. These intervals are the sidereal periods of the planets. (The sidereal period of the earth is 365.2563 days. The interval between two successive vernal equinoxes of the earth is the tropical year. It spans 365.2422 days.)

"The sidereal period of each planet increases with the planet's distance from the sun. The inclinations of the orbital planes of the planets differ from one another by only a few degrees, with the exception of the orbital plane of Pluto. The orbit of Pluto is inclined about 17 degrees to the ecliptic, the plane in which the earth travels.

"An apparent alignment of the inferior planets occurs when Mercury or Venus moves exactly between the earth and the sun. (Sometimes both of them move.) These events are called inferior conjunctions. Continued rotation carries each planet at its characteristic velocity to the opposite side of its orbit to points where the sun lies in the line between the planets and the earth. These positions are termed superior conjunctions.

"The 'superior' planets (Mars, Jupiter, Saturn, Uranus, Neptune and Pluto) are, in that order, increasingly distant from the sun and have correspondingly longer sidereal periods. A superior planet reaches conjunction when it moves to the point where the sun lies between the planet and the observer's meridian at noon. At that position the planet is lost from view in the sun's glare.

"Continued rotation carries each superior planet to the point where the earth is eventually aligned between the planet and the sun. The planet is then said to be in opposition. The position appears on the observer's meridian at midnight. Periodically several superior planets reach opposition simultaneously. Inferior planets can never reach opposition, and superior planets cannot reach inferior conjunction [see illustration at left].

"The interval of time between successive conjunctions or successive oppositions of a planet is its synodic period. This period can be calculated easily by either of two methods. For a superior planet the reciprocal of the synodic period is equal to the difference between the reciprocal of the earth's sidereal period and the reciprocal of the planet's sidereal period. With Mars as an example the reciprocal of the synodic period is 1/365.25 - 1/686.98 = 1/ 779.9. The interval between successive conjunctions or successive oppositions of Mars is 779.9 earth-days.

"The synodic period of the inferior planets is given by the difference between the reciprocal of the planet's sidereal period and the period of the earth. For Venus the reciprocal of the synodic period is 1/224.7 - 1/365.25 = 1/ 583.94. The interval between successive superior or inferior conjunctions of Venus is 583.94 earth-days.

"A simple graph can also display the synodic period of a planet [see illustration at right]. Draw a pair of rectangular coordinates. Calibrate the abscissa in intervals of one year for, say, 236 years and the ordinate in units of 360 degrees for at least 720 degrees. The fact that the earth completes an orbit around the sun (traverses 360 degrees) in one year can be represented by a graph in the form of a diagonal line drawn from zero on the abscissa to intercept 360 degrees at one year. The slope of the graph is equal to the quotient of 360 degrees divided by the number of days in a year, or .9856 degree per day.

"The graph of any planet can be similarly plotted on the same coordinates. For example, the motion of Mars can be represented by a straight line that begins at zero on the abscissa and intercepts the line representing 360 degrees at the sidereal period of Mars (686.99 earth-days). The slope of the Martian graph is equal to the quotient of 360 degrees divided by the number of earth-days in one sidereal year of Mars, or .5240 degree per day. The difference in the rate at which the two planets travel around the sun is indicated by the difference in the slopes of the graphs.

"Periodically the difference between the graphs as measured on the ordinate amounts to a multiple of 360 degrees. The multiples occur at time intervals equal to the synodic period of the planet and are equal to 360 degrees divided by the difference between the slopes of the graphs:'360/ (.9856 - .5240) = 779.9, which is the synodic period of Mars. Graphically the synodic period can be measured along the abscissa. It amounts to 2.135 sidereal earth-years.

"A simple modification can adapt this graphical technique for displaying both past and future conjunctions and oppositions Limit the ordinate to one interval of 360 degrees. Divide the abscissa into any number of yearly intervals. The accuracy of the predictions increases with the size of the graph.

"The graph of each planet originates on the abscissa at the beginning of the sidereal period and terminates at the point where the end of the period intercepts 360 degrees as measured by the ordinate. Continued motion of the planet is depicted by initiating a new graph at zero degrees and at the first day of the next sidereal interval as suggested by the broken line 'Earth" and the solid line 'Mars" in the illustration. The result is a sawtooth pattern on the graph of each planet.

"The slope of the 'teeth' and their number vary with the sidereal period of each planet. The accompanying graph [illustration at left] depicts motions of the earth and Mars through the decade ending in 1979. The ordinate of the graph is divided into 12 intervals of 30 degrees. Each interval represents the partial orbit of the planet around the sun in roughly one month. Each interval of 30 degrees is labeled to indicate approximately the constellation in which the sun appears at the time and also the corresponding months during which the constellations appear on the meridian.

"The abscissa of the graph is divided into 10 intervals that represent the years 1969 through 1978. To avoid cluttering the graph with fine detail the yearly intervals have not been subdivided into intervals representing months. A black line that represents the motion of the earth for l969 is drawn on the graph beginning on January 1, 1969, at zero degrees and sloping diagonally to December 31 at 360 degrees.

"A colored line begins at approximately 75 degrees on the ordinate and January 1 on the time scale. This graph depicts the motion of Mars. It intercepts the time scale approximately 548 days later (about July 2, 1970). On that date the new graph of Mars begins at zero degrees, as measured along the ordinate In contrast, graphs of the earth's motion begin and end with each calendar year.

"Note the periodic intersections of the two lines every 779.9 earth-days. The intersections mark the dates when Mars is in opposition, as it will be on December 15, 1975. Conjunctions can be depicted by displacing the graph of the earth's motion by an interval of six months with respect to the abscissa, as indicated by the broken line. Dates when Mars has been or will be in conjunction are indicated by the intersections of the broken and the solid lines.

"As a convenience I rule a pattern of horizontal and vertical lines 9-3/4 by 4-1/2 inches in size on standard 11 by 8-3/4inch paper and label the 12 horizontal lines with the months in the second column from the left [see illustration at right]. At the right margin the same 12 lines are labeled in multiples of 90 degrees from zero through 360 degrees. Duplicates of this form are made on an office copying machine. The accompanying drawing of the form depicts the graphs of the nine planets. It illustrates how the display tends to become crowded with planets within 200 million miles of the sun as well as those that are on the order of a billion or more miles away.

"The slope of any graph of this km can be plotted most accurately by locating the origin at zero degrees and zero years and terminating it either at the intersection of the planet's sidereal period and 360 degrees or at the point of the orbit (expressed in degrees) that the planet would reach within the maximum number of years displayed by the abscissa. In the accompanying example the slopes of the graphs of Mercury, Venus, the earth and Mars were determined by drawing straight lines between the point of origin and the 360-degree points that corresponded to the respective sidereal periods of the four planets within 200 million miles of the sun.

"In this example the abscissa spans 10 years, or 3,652.5 days. To what point along the right margin of the chart should a graph be drawn to represent the slope of Pluto? The sidereal period of Pluto is 90,465 earth-days. In one day Pluto would rotate 360/ 90,465 = .0039794 degree. In 10 years it would orbit through 3,652.5 days multiplied by .0039794 degree per day, or 14.53 degrees. A straight line drawn from this point on the right margin of the coordinates to the point of origin at the left displays the slope of Pluto's graph.

"Because the synodic period is a function of the sidereal period, I calibrate the coordinates of the graphs in terms of the sidereal year of 365.2563 days and initiate the orbit at zero degrees on January 1 instead of at the vernal equinox, as would be done in the case of the tropical year of 365.2422 days. With the slopes determined, the graphs of the planets must then be displaced in time from the point of origin as required to locate the intersections of the planet and earth graphs at the exact dates of known conjunctions and oppositions. Almanacs list the dates of these events for each calendar year. A reference that is remarkably complete and easy to use is The American Ephemeris and Nautical Almanac. The volume is updated and reissued annually. It can be obtained for $6.20 postpaid from the Superintendent of Documents (U.S. Government Printing Office, Washington, D.C. 20402). The British edition, The Astronomical Ephemeris, is published by Her Majesty's Stationery Office and is available from 49 High Holborn, London W.C. 1.

"Having plotted graphs of the earth's motion for as many yearly intervals as are provided by the coordinates, I put one or more points on the graphs to indicate oppositions of the planet to be plotted. The graph of the planet is drawn through these points at the slope previously determined for the planet. A line that has been drawn lightly from the zero point to determine the slope of a planet can be conveniently transferred to intercept the graph of the earth at points of known conjunctions or oppositions by the instrument known to navigators as the parallel ruler.

"Planets can be in conjunction with each other as well as with the sun. These events appear as intersections between the graphs of the planets at points removed from the graph of the earth. For example, the accompanying graph [right] indicates that Mars and Jupiter were in conjunction about July 2. Because their intersection appears above the graph of the earth they were seen as morning stars. Evening events appear below the graph of the earth's position."

THE history of technology is full of instances in which more than one person makes the same invention at about the same time. A recent example is the Princeton "sail wing," an efficient airfoil of light and flexible construction that has found application in hang gliders and sailboats [see "The Amateur Scientist," SCIENTIFIC AMERICAN, December, 1974]. The effectiveness of the sail wing in powering small boats is indicated by the following letter from a yacht designer, Peter Hodgins (248 Roger Road, Ottawa, Ontario, Canada, K1H 5C6).

"'Blow me down!' to coin a phrase, if the top drawing on page 141 of your December 1974 issue does not show a sail wing configuration almost identical with the mainsail I designed and rigged on a dinghy. Based on its performance, I have named the dinghy Dreamboat. Naturally thought the sail wing was my own invention. It is too bad for me that I was not the first with the principle. On the other hand, it was great to see in your magazine the wind-tunnel test results that I suspected but could never afford to make.

"My sail differs in a number of details from the wing of the hang glider you described in 1974. Essentially the details involve the way in which the basic principle is exploited. For example, because of the competitiveness of the sailboat market I was obliged to simplify the cut of the sails, yet the rigging details had to be arranged in such a way that the sail could be hoisted and lowered easily by a halyard while the boat was under way

"As yachtsmen know, the gradient of wind speeds and the direction of the apparent wind differ along a sail's height. To compensate for these differences the head of the sail should be allowed to sag off to leeward with respect to the position of the boom at the foot. In other words, it is generally agreed that the sail should have some twist. For this reason the control mechanism used in Dreamboat's mainsail luff, or leading edge, is set slightly loose to allow for the seeming difference in angle of attack. The design maximizes forward thrust. Drag to leeward is minimized.

"Another aerodynamic refinement that helps to account for the dinghy's remarkable performance, which is rare in such a craft, is a masthead jib with ample vision below. A light wood fitting at the peak of the mast functions as an end plate that minimizes the vortex normally shed by the sail. The end plate thus minimizes induced drag that would aggravate the tendency of the boat to heel, or list to leeward. Somewhat similar panels, which perform much the same function at the foot of the mast, are sewn into both the mainsail and the jib.

"Three halyards (for mainsail, jib and spinnaker) are placed between the two surfaces of the sail wing to minimize drag. The halyards could have been run inside the aluminum mast. Instead the mast was sealed to provide buoyancy in the event the boat capsized. The wood end plate at the head of the mast contributes still more buoyancy.

"In spite of these attractive features, sales of the boat have not been large compared with sales of traditional craft in the same price range. Yachtsmen tend to be bound by tradition. The craft somehow looks 'different.' Therefore I am designing a new model with an improved hull that will combine the sail wing with a more traditional appearance."

Bibliography

SAILING THEORY AND PRACTICE. C. A. Marchaj. Dodd, Mead & Co., 1964.

INTRODUCTION TO PLANETARY PHYSICS. William M. Kaula. John Wiley & Sons, Inc., 1968.

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