Fig. 8.—Photographing the circumpolar stars.—Barnard.Fig. 8.—Photographing the circumpolar stars.—Barnard.
13.Photographing the stars.—Exercise 7.—For any student who uses a camera. Upon some clear and moonless night point the camera, properly focused, at Polaris, and expose a plate for three or four hours. Upon developing the plate you should find a series of circular trails such as are shown inFig. 8, only longer. Each one of these is producedby a star moving slowly over the plate, in consequence of its changing position in the sky. The center indicated by these curved trails is called the pole of the heavens. It is that part of the sky toward which is pointed the axis about which the earth rotates, and the motion of the stars around the center is only an apparent motion due to the rotation of the earth which daily carries the observer and his camera around this axis while the stars stand still, just as trees and fences and telegraph poles stand still, although to the passenger upon a railway train they appear to be in rapid motion. So far as simple observations are concerned, there is no method by which the pupil can tell for himself that the motion of the stars is an apparent rather than a real one, and, following the custom of astronomers, we shall habitually speak as if it were a real movement of the stars. How long was the plate exposed in photographingFig. 8?
14.Finding the stars.—OnPlate I, oppositepage 124, the pole of the heavens is at the center of the map, near Polaris, and the heavy trail near the center ofFig. 8is made by Polaris. See if you can identify from the map any of the stars whose trails show in the photograph. The brighter the star the bolder and heavier its trail.
Find from the map and locate in the sky the two bright stars Capella and Vega, which are on opposite sides of Polaris and nearly equidistant from it. Do these stars share in the motion around the pole? Are they visible on every clear night, and all night?
Observe other bright stars farther from Polaris than are Vega and Capella and note their movement. Do they move like the sun and moon? Do they rise and set?
In what part of the sky do the stars move most rapidly, near the pole or far from it?
How long does it take the fastest moving stars to make the circuit of the sky and come back to the same place? How long does it take the slow stars?
15.Rising and setting of the stars.—A study of the sky along the lines indicated in these questions will show that there is a considerable part of it surrounding the pole whose stars are visible on every clear night. The same star is sometimes high in the sky, sometimes low, sometimes to the east of the pole and at other times west of it, but is always above the horizon. Such stars are said to be circumpolar. A little farther from the pole each star, when at the lowest point of its circular path, dips for a time below the horizon and is lost to view, and the farther it is away from the pole the longer does it remain invisible, until, in the case of stars 90° away from the pole, we find them hidden below the horizon for twelve hours out of every twenty-four (seeFig. 9). The sun is such a star, and in its rising and setting acts precisely as does every other star at a similar distance from the pole—only, as we shall find later, each star keeps always at (nearly) the samedistance from the pole, while the sun in the course of a year changes its distance from the pole very greatly, and thus changes the amount of time it spends above and below the horizon, producing in this way the long days of summer and the short ones of winter.
Fig. 9.—Diurnal motion of the northern constellations.Fig. 9.—Diurnal motion of the northern constellations.
How much time do stars which are more than 90° from the pole spend above the horizon?
We say in common speech that the sun rises in the east, but this is strictly true only at the time when it is 90° distant from the pole—i. e., in March and September. At other seasons it rises north or south of east according as its distance from the pole is less or greater than 90°, and the same is true for the stars.
16.The geography of the sky.—Find from a map the latitude and longitude of your schoolhouse. Find on the map the place whose latitude is 39° and longitude 77° west of the meridian of Greenwich. Is there any other place in the world which has the same latitude and longitude as your schoolhouse?
The places of the stars in the sky are located in exactly the manner which is illustrated by these geographical questions, only different names are used. Instead of latitude the astronomer saysdeclination, in place of longitude he saysright ascension, in place of meridian he sayshour circle, but he means by these new names the same ideas that the geographer expresses by the old ones.
Imagine the earth swollen up until it fills the whole sky; the earth's equator would meet the sky along a line (a great circle) everywhere 90° distant from the pole, and this line is called thecelestial equator. Trace its position along the middle of the map oppositepage 190and notice near what stars it runs. Every meridian of the swollen earth would touch the sky along an hour circle—i. e., a great circle passing through the pole and therefore perpendicular to the equator. Note that in the map one of these hour circles is marked 0. It plays the same part in measuring right ascensions as does the meridian of Greenwich in measuring longitudes; it is the beginning, from which they are reckoned. Note also, at the extreme left end of the map, the four bright stars in the form of a square, one side of which is parallel and close to the hour circle, which is marked 0. This is familiarly called the Great Square in Pegasus, and may be found high up in the southern sky whenever the Big Dipper lies below the pole. Why can it not be seen when Ursa Major is above the pole?
Astronomers use the right ascensions of the stars not only to tell in what part of the sky the star is placed, but also in time reckonings, to regulate their sidereal clocks, andwith regard to this use they find it convenient to express right ascension not in degrees but in hours, 24 of which fill up the circuit of the sky and each of which is equal to 15° of arc, 24 × 15 = 360. The right ascension of Capella is 5h. 9m. = 77.2°, but the student should accustom himself to using it in hours and minutes as given and not to change it into degrees. He should also note that some stars lie on the side of the celestial equator toward Polaris, and others are on the opposite side, so that the astronomer has to distinguish between north declinations and south declinations, just as the geographer distinguishes between north latitudes and south latitudes. This is done by the use of the + and - signs, a + denoting that the star lies north of the celestial equator, i. e., toward Polaris.
Fig. 10.—From a photograph of the Pleiades.Fig. 10.—From a photograph of the Pleiades.
Find onPlate II, oppositepage 190, the Pleiades(Plēadēs), R. A. = 3h. 42m., Dec. = +23.8°. Why do they not show onPlate I, oppositepage 124? In what direction are they from Polaris? This is one of the finest star clusters in the sky, but it needs a telescope to bring out its richness. See how many stars you can count in it with the naked eye, and afterward examine it with an opera glass. Compare what you see withFig. 10. Find Antares, R. A. = 16h. 23m. Dec. = -26.2°. How far is it, in degrees, from the pole? Is it visible in your sky? If so, what is its color?
Find the R. A. and Dec. of α Ursæ Majoris; of β Ursæ Majoris; of Polaris. Find the Northern Crown,Corona Borealis, R. A. = 15h. 30m., Dec. = +27.0°; the Beehive,Præsepe, R. A. = 8h. 33m., Dec. = +20.4°.
These should be looked up, not only on the map, but also in the sky.
17.Reference lines and circles.—As the stars move across the sky in their diurnal motion, they carry the framework of hour circles and equator with them, so that the right ascension and declination of each star remain unchanged by this motion, just as longitudes and latitudes remain unchanged by the earth's rotation. They are the same when a star is rising and when it is setting; when it is above the pole and when it is below it. During each day the hour circle of every star in the heavens passes overhead, and at the moment when any particular hour circle is exactly overhead all the stars which lie upon it are said to be "on the meridian"—i. e., at that particular moment they stand directly over the observer's geographical meridian and upon the corresponding celestial meridian.
An eye placed at the center of the earth and capable of looking through its solid substance would see your geographical meridian against the background of the sky exactly covering your celestial meridian and passing from one pole through your zenith to the other pole. InFig. 11the inner circle represents the terrestrial meridian of a certain place,O, as seen from the center of the earth,C, and the outer circle represents the celestial meridian ofOas seen fromC, only we must imagine, what can not be shown on the figure, that the outer circle is so large that the inner one shrinks to a mere point in comparison with it. IfC Prepresents the direction in which the earth's axis passes through the center, thenC Eat right angles to it must be the direction of the equator which we suppose to be turned edgewise toward us; and ifC Ois the direction of some particular point on the earth's surface, thenZdirectly overhead is called thezenithof that point, upon the celestial sphere. The lineC Hrepresents a direction parallel to the horizon plane atO, andH C Pis the angle which the axis of the earth makes with this horizon plane. The arcO Emeasures the latitude ofO, and the arcZ Emeasures the declination ofZ, and since by elementary geometry each of these arcs contains the same number of degrees as the angleE C Z, we have the
Fig. 11.—Reference lines and circles.Fig. 11.—Reference lines and circles.
Theorem.—The latitude of any place is equal to the declination of its zenith.
Corollary.—Any star whose declination is equal to your latitude will once in each day pass through your zenith.
18.Latitude.—From the construction of the figure
∠E C Z+ ∠Z C P=90°∠H C P+ ∠Z C P=90°
from which we find by subtraction and transposition
∠E C Z= ∠H C P
and this gives the further
Theorem.—The latitude of any place is equal to the elevation of the pole above its horizon plane.
An observer who travels north or south over the earth changes his latitude, and therefore changes the angle between his horizon plane and the axis of the earth. What effect will this have upon the position of stars in his sky? If you were to go to the earth's equator, in what part of the sky would you look for Polaris? Can Polaris be seen from Australia? From South America? If you were to go from Minnesota to Texas, in what respect would the appearance of stars in the northern sky be changed? How would the appearance of stars in the southern sky be changed?
Fig. 12.—Diurnal path of Polaris.Fig. 12.—Diurnal path of Polaris.
Exercise 8.—Determine your latitude by taking the altitude of Polaris when it is at some one of the four points of its diurnal path, shown inFig. 12. When it is at1it is said to be at upper culmination, and the star ζ Ursæ Majoris in the handle of the Big Dipper will be directly below it. When at2it is at western elongation, and the star Castor is near the meridian. When it is at3it is at lower culmination, and the star Spica is on the meridian. When it is at4it is at eastern elongation, and Altair is near the meridian. All of these stars are conspicuous ones, which the student should find upon the map and learn to recognize in the sky. The altitude observed at either2or4may be considered equal to the latitude of the place, but the altitude observed when Polaris is at the positions marked1and3must be corrected for the star's distance from the pole, which may be assumed equal to 1.3°.
The plumb-line apparatus described atpage 12is shown inFig. 6slightly modified, so as to adapt it to measuring the altitudes of stars. Note that the board with the screweye at one end has been transferred from the box to the vertical standard, and has a screw eye at each end. When the apparatus has been properly leveled, so that the plumb line hangs at the middle of the hole in the box cover, the board is to be pointed at the star by sighting through the centers of the two screw eyes, and a pencil line is to be ruled along its edge upon the face of the vertical standard. After this has been done turn the apparatus halfway around so that what was the north side now points south, level it again and revolve the board about the screw which holds it to the vertical standard, until the screw eyes again point to the star. Rule another line along the same edge of the board as before and with a protractor measure the angle between these lines. Use a bicycle lamp if you need artificial light for your work. The student who has studied plane geometry should be able to prove that one half of the angle between these lines is equal to the altitude of the star.
After you have determined your latitude from Polaris, compare the result with your position as shown upon the best map available. With a little practice and considerable care the latitude may be thus determined within one tenth of a degree, which is equivalent to about 7 miles. If you go 10 miles north or south from your first station you should find the pole higher up or lower down in the sky by an amount which can be measured with your apparatus.
19.The meridian line.—To establish a true north and south line upon the ground, use the apparatus as described atpage 13, and when Polaris is at upper or lower culmination drive into the ground two stakes in line with the star and the plumb line. Such a meridian line is of great convenience in observing the stars and should be laid out and permanently marked in some convenient open space from which, if possible, all parts of the sky are visible. June and November are convenient months for this exercise, since Polaris then comes to culmination early in the evening.
20.Time.—What isthe timeat which school begins in the morning? What do you mean by "the time"?
The sidereal time at any moment is the right ascension of the hour circle which at that moment coincides with the meridian. When the hour circle passing through Sirius coincides with the meridian, the sidereal time is 6h. 40m., since that is the right ascension of Sirius, and in astronomical language Sirius is "on the meridian" at 6h. 40m. sidereal time. As may be seen from the map, this 6h. 40m. is the right ascension of Sirius, and if a clock be set to indicate 6h. 40m. when Sirius crosses the meridian, it will show sidereal time. If the clock is properly regulated, every other star in the heavens will come to the meridian at the moment when the time shown by the clock is equal to the right ascension of the star. A clock properly regulated for this purpose will gain about four minutes per day in comparison with ordinary clocks, and when so regulated it is called a sidereal clock. The student should be provided with such a clock for his future work, but one such clock will serve for several persons, and a nutmeg clock or a watch of the cheapest kind is quite sufficient.
THE HARVARD COLLEGE OBSERVATORY, CAMBRIDGE, MASS.THE HARVARD COLLEGE OBSERVATORY, CAMBRIDGE, MASS.
Exercise 9.—Set such a clock to sidereal time by means of the transit of a star over your meridian. For this experiment it is presupposed that a meridian line has been marked out on the ground as in§ 19, and the simplest mode of performing the experiment required is for the observer, having chosen a suitable star in the southern part of the sky, to place his eye accurately over the northern end of the meridian line and to estimate as nearly as possible the beginning and end of the period during which the star appears to stand exactly above the southern end of the line. The middle of this period may be taken as the time at which the star crossed the meridian and at this moment the sidereal time is equal to the right ascension of the star. The difference between this right ascension and the observedmiddle instant is the error of the clock or the amount by which its hands must be set back or forward in order to indicate true sidereal time.
A more accurate mode of performing the experiment consists in using the plumb-line apparatus carefully adjusted, as inFig. 7, so that the line joining the wire to the center of the screw eye shall be parallel to the meridian line. Observe the time by the clock at which the star disappears behind the wire as seen through the center of the screw eye. If the star is too high up in the sky for convenient observation, place a mirror, face up, just north of the screw eye and observe star, wire and screw eye by reflection in it.
The numerical right ascension of the observed star is needed for this experiment, and it may be measured from the star map, but it will usually be best to observe one of the stars of the table at the end of the book, and to obtain its right ascension as follows: The table gives the right ascension and declination of each star as they were at the beginning of the year 1900, but on account of the precession (seeChapter V), these numbers all change slowly with the lapse of time, and on the average the right ascension of each star of the table must be increased by one twentieth of a minute for each year after 1900—i. e., in 1910 the right ascension of the first star of the table will be 0h. 38.6m. + (10/20)m. = 0h. 39.1m. The declinations also change slightly, but as they are only intended to help in finding the star on the star maps, their change may be ignored.
Having set the clock approximately to sidereal time, observe one or two more stars in the same way as above. The difference between the observed time and the right ascension, if any is found, is the "correction" of the clock. This correction ought not to exceed a minute if due care has been taken in the several operations prescribed. The relation of the clock to the right ascension of the starsis expressed in the following equation, with which the student should become thoroughly familiar:
A=T±U
Tstands for the time by the clock at which the star crossed the meridian.Ais the right ascension of the star, andUis the correction of the clock. Use the + sign in the equation whenever the clock is too slow, and the - sign when it is too fast.Umay be found from this equation whenAandTare given, orAmay be found whenTandUare given. It is in this way that astronomers measure the right ascensions of the stars and planets.
DetermineUfrom each star you have observed, and note how the several results agree one with another.
21.Definitions.—To define a thing or an idea is to give a description sufficient to identify it and distinguish it from every other possible thing or idea. If a definition does not come up to this standard it is insufficient. Anything beyond this requirement is certainly useless and probably mischievous.
Let the student define the following geographical terms, and let him also criticise the definitions offered by his fellow-students: Equator, poles, meridian, latitude, longitude, north, south, east, west.
Compare the following astronomical definitions with your geographical definitions, and criticise them in the same way. If you are not able to improve upon them, commit them to memory:
The Polesof the heavens are those points in the sky toward which the earth's axis points. How many are there? The one near Polaris is called the north pole.
The Celestial Equatoris a great circle of the sky distant 90° from the poles.
The Zenithis that point of the sky, overhead, toward which a plumb line points. Why is the word overhead placed in the definition? Is there more than one zenith?
The Horizonis a great circle of the sky 90° distant from the zenith.
An Hour Circleis any great circle of the sky which passes through the poles. Every star has its own hour circle.
The Meridianis that hour circle which passes through the zenith.
A Vertical Circleis any great circle that passes through the zenith. Is the meridian a vertical circle?
The Declinationof a star is its angular distance north or south of the celestial equator.
The Right Ascensionof a star is the angle included between its hour circle and the hour circle of a certain point on the equator which is called theVernal Equinox. From spherical geometry we learn that this angle is to be measured either at the pole where the two hour circles intersect, as is done in the star map oppositepage 124, or along the equator, as is done in the map opposite page 190. Right ascension is always measured from the vernal equinox in the direction opposite to that in which the stars appear to travel in their diurnal motion—i. e., from west toward east.
The Altitudeof a star is its angular distance above the horizon.
The Azimuthof a star is the angle between the meridian and the vertical circle passing through the star. A star due south has an azimuth of 0°. Due west, 90°. Due north, 180°. Due east, 270°.
What is the azimuth of Polaris in degrees?
What is the azimuth of the sun at sunrise? At sunset? At noon? Are these azimuths the same on different days?
The Hour Angleof a star is the angle between its hour circle and the meridian. It is measured from the meridian in the direction in which the stars appear to travel in their diurnal motion—i. e., from east toward west.
What is the hour angle of the sun at noon? What isthe hour angle of Polaris when it is at the lowest point in its daily motion?
22.Exercises.—The student must not be satisfied with merely learning these definitions. He must learn to see these points and lines in his mind as if they were visibly painted upon the sky. To this end it will help him to note that the poles, the zenith, the meridian, the horizon, and the equator seem to stand still in the sky, always in the same place with respect to the observer, while the hour circles and the vernal equinox move with the stars and keep the same place among them. Does the apparent motion of a star change its declination or right ascension? What is the hour angle of the sun when it has the greatest altitude? Will your answer to the preceding question be true for a star? What is the altitude of the sun after sunset? In what direction is the north pole from the zenith? From the vernal equinox? Where are the points in which the meridian and equator respectively intersect the horizon?
23.Star maps.—Select from the map some conspicuous constellation that will be conveniently placed for observation in the evening, and make on a large scale a copy of all the stars of the constellation that are shown upon the map. At night compare this copy with the sky, and mark in upon your paper all the stars of the constellation which are not already there. Both the original drawing and the additions made to it by night should be carefully done, and for the latter purpose what is called the method of allineations may be used with advantage—i. e., the new star is in line with two already on the drawing and is midway between them, or it makes an equilateral triangle with two others, or a square with three others, etc.
A series of maps of the more prominent constellations, such as Ursa Major, Cassiopea, Pegasus, Taurus, Orion, Gemini, Canis Major, Leo, Corvus, Bootes, Virgo, Hercules, Lyra, Aquila, Scorpius, should be constructed in this manner upon a uniform scale and preserved as a part of the student's work. Let the magnitude of the stars be represented on the maps as accurately as may be, and note the peculiarity of color which some stars present. For the most part their color is a very pale yellow, but occasionally one may be found of a decidedly ruddy hue—e. g., Aldebaran or Antares. Such a star map, not quite complete, is shown inFig. 13.
So, too, a sharp eye may detect that some stars do not remain always of the same magnitude, but change theirbrightness from night to night, and this not on account of cloud or mist in the atmosphere, but from something in the star itself. Algol is one of the most conspicuous of thesevariable stars, as they are called.
Fig. 13.—Star map of the region about Orion.Fig. 13.—Star map of the region about Orion.
24.The moon's motion among the stars.—Whenever the moon is visible note its position among the stars by allineations, and plot it on the key map oppositepage 190. Keep a record of the day and hour corresponding to each such observation. You will find, if the work is correctly done, that the positions of the moon all fall near the curved line shown on the map. This line is called the ecliptic.
After several such observations have been made and plotted, find by measurement from the map how many degrees per day the moon moves. How long would it require to make the circuit of the heavens and come back to the starting point?
On each night when you observe the moon, make on a separate piece of paper a drawing of it about 10 centimeters in diameter and show in the drawing every feature of the moon's face which you can see—e. g., the shape of the illuminated surface (phase); the direction among the stars of the line joining the horns; any spots which you can see upon the moon's face, etc. An opera glass will prove of great assistance in this work.
Use your drawings and the positions of the moon plotted upon the map to answer the following questions: Does the direction of the line joining the horns have any special relation to the ecliptic? Does the amount of illuminated surface of the moon have any relation to the moon's angular distance from the sun? Does it have any relation to the time at which the moon sets? Do the spots on the moon when visible remain always in the same place? Do they come and go? Do they change their position with relation to each other? Can you determine from these spots that the moon rotates about an axis, as the earth does? In what direction does its axis point? How long does it take to make one revolution about the axis? Is there any day and night upon the moon?
Each of these questions can be correctly answered from the student's own observations without recourse to any book.
25.The sun and its motion.—Examine the face of the sun through a smoked glass to see if there is anything there that you can sketch.
By day as well as by night the sky is studded with stars, only they can not be seen by day on account of the overwhelming glare of sunlight, but the position of the sunamong the stars may be found quite as accurately as was that of the moon, by observing from day to day its right ascension and declination, and this should be practiced at noon on clear days by different members of the class.
Exercise 10.—The right ascension of the sun may be found by observing with the sidereal clock the time of its transit over the meridian. Use the equation in§ 20, and substitute in place ofUthe value of the clock correction found from observations of stars on a preceding or following night. If the clock gains or loseswith respect to sidereal time, take this into account in the value ofU.
Exercise 11.—To determine the sun's declination, measure its altitude at the time it crosses the meridian. Use either the method ofExercise 4, or that used with Polaris inExercise 8. The student should be able to show fromFig. 11that the declination is equal to the sum of the altitude and the latitude of the place diminished by 90°, or in an equation
Declination = Altitude + Latitude - 90°.
If the declination as found from this equation is a negative number it indicates that the sun is on the south side of the equator.
The right ascension and declination of the sun as observed on each day should be plotted on the map and the date, written opposite it. If the work has been correctly done, the plotted points should fall upon the curved line (ecliptic) which runs lengthwise of the map. This line, in fact, represents the sun's path among the stars.
Note that the hours of right ascension increase from 0 up to 24, while the numbers on the clock dial go only from 0 to 12, and then repeat 0 to 12 again during the same day. When the sidereal time is 13 hours, 14 hours, etc., the clock will indicate 1 hour, 2 hours, etc., and 12 hours must then be added to the time shown on the dial.
If observations of the sun's right ascension and declinationare made in the latter part of either March or September the student will find that the sun crosses the equator at these times, and he should determine from his observations, as accurately as possible, the date and hour of this crossing and the point on the equator at which the sun crosses it. These points are called the equinoxes, Vernal Equinox and Autumnal Equinox for the spring and autumn crossings respectively, and the student will recall that the vernal equinox is the point from which right ascensions are measured. Its position among the stars is found by astronomers from observations like those above described, only made with much more elaborate apparatus.
Similar observations made in June and December show that the sun's midday altitude is about 47° greater in summer than in winter. They show also that the sun is as far north of the equator in June as he is south of it in December, from which it is easily inferred that his path, the ecliptic, is inclined to the equator at an angle of 23°.5, one half of 47°. This angle is called the obliquity of the ecliptic. The student may recall that in the geographies the torrid zone is said to extend 23°.5 on either side of the earth's equator. Is there any connection between these limits and the obliquity of the ecliptic? Would it be correct to define the torrid zone as that part of the earth's surface within which the sun may at some season of the year pass through the zenith?
Exercise 12.—After a half dozen observations of the sun have been plotted upon the map, find by measurement the rate, in degrees per day, at which the sun moves along the ecliptic. How many days will be required for it to move completely around the ecliptic from vernal equinox back to vernal equinox again? Accurate observations with the elaborate apparatus used by professional astronomers show that this period, which is called atropical year, is 365 days 5 hours 48 minutes 46 seconds. Is this the same as the ordinary year of our calendars?
26.The planets.—Any one who has watched the sky and who has made the drawings prescribed in this chapter can hardly fail to have found in the course of his observations some bright stars not set down on the printed star maps, and to have found also that these stars do not remain fixed in position among their fellows, but wander about from one constellation to another. Observe the motion of one of these planets from night to night and plot its positions on the star map, precisely as was done for the moon. What kind of path does it follow?
Both the ancient Greeks and the modern Germans have called these bodies wandering stars, and in English we name them planets, which is simply the Greek word for wanderer, bent to our use. Besides the sun and moon there are in the heavens five planets easily visible to the naked eye and, as we shall see later, a great number of smaller ones visible only in the telescope. More than 2,000 years ago astronomers began observing the motion of sun, moon, and planets among the stars, and endeavored to account for these motions by the theory that each wandering star moved in an orbit about the earth. Classical and mediæval literature are permeated with this idea, which was displaced only after a long struggle begun by Copernicus (1543A. D.), who taught that the moon alone of these bodies revolves about the earth, while the earth and the other planets revolve around the sun. The ecliptic is the intersection of the plane of the earth's orbit with the sky, and the sun appears to move along the ecliptic because, as the earth moves around its orbit, the sun is always seen projected against the opposite side of it. The moon and planets all appear to move near the ecliptic because the planes of their orbits nearly coincide with the plane of the earth's orbit, and a narrow strip on either side of the ecliptic, following its course completely around the sky, is called thezodiac, a word which may be regarded as the name of a narrow street (16° wide) within which all the wanderings of the visibleplanets are confined and outside of which they never venture. Indeed, Mars is the only planet which ever approaches the edge of the street, the others traveling near the middle of the road.
Fig. 14.—The apparent motion of a planet.Fig. 14.—The apparent motion of a planet.
27.A typical case of planetary motion.—The Copernican theory, enormously extended and developed through the Newtonian law of gravitation (seeChapter IV), has completely supplanted the older Ptolemaic doctrine, and an illustration of the simple manner in which it accounts for the apparently complicated motions of a planet among the stars is found in Figs.14and15, the first of which represents the apparent motion of the planet Mars through the constellations Aries and Pisces during the latter part of theyear 1894, while the second shows the true motions of Mars and the earth in their orbits about the sun during the same period. The straight line inFig. 14, with cross ruling upon it, is a part of the ecliptic, and the numbers placed opposite it represent the distance, in degrees, from the vernal equinox. InFig. 15the straight line represents the direction from the sun toward the vernal equinox, and the angle which this line makes with the line joining earth and sun is called the earth's longitude. The imaginary line joining the earth and sun is called the earth's radius vector, and the pupil should note that the longitude and length of the radius vector taken together show the direction and distance of the earth from the sun—i. e., they fix the relative positions of the two bodies. The same is nearly true for Mars and would be wholly true if the orbit of Mars lay in the same plane with that of the earth. How doesFig. 14show that the orbit of Mars does not lie exactly in the same plane with the orbit of the earth?
Exercise 13.—Find fromFig. 15what ought to have been the apparent course of Mars among the stars during the period shown in the two figures, and compare what you find withFig. 14. The apparent position of Mars among the stars is merely its direction from the earth, and this direction is represented inFig. 14by the distance of the planet from the ecliptic and by its longitude.
Fig. 15.—The real motion of a planet.Fig. 15.—The real motion of a planet.
The longitude of Mars for each date can be found fromFig. 15by measuring the angle between the straight lineS Vand the line drawn from the earth to Mars. Thus for October 12th we may find with the protractor that the angle between the lineS Vand the line joining the earth to Mars is a little more than 30°, and inFig. 14the position of Mars for this date is shown nearly opposite the cross line corresponding to 30° on the ecliptic. Just how far below the ecliptic this position of Mars should fall can not be told fromFig. 15, which from necessity is constructed as if the orbits of Mars and the earth lay in the same plane, andMars in this case would always appear to stand exactly on the ecliptic and to oscillate back and forth as shown inFig. 14, but without the up-and-down motion there shown. In this way plot inFig. 14the longitudes of Mars as seen from the earth for other dates and observe how the forward motion of the two planets in their orbits accounts for the apparently capricious motion of Mars to and fro among the stars.