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 in Fig.13.

So, too, a sharp eye may detect that some stars do not remain always of the same magnitude, but change their brightness 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 these variable stars, as they are called.

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 opposite page190. 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 sun among 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 of U the value of the clock correction found from observations of stars on a preceding or following night. If the clock gains or loses with respect to sidereal time, take this into account in the value of U.Exercise 11.—To determine the sun's declination, measure its altitude at the time it crosses the meridian. Use either the method of Exercise4, or that used with Polaris in Exercise8. The student should be able to show from Fig.11 that 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 declination are 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 a tropical 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 (1543 A.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 the zodiac, a word which may be regarded as the name of a narrow street (16° wide) within which all the wanderings of the visible planets 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.

27. A typical case of planetary motion.—The Copernican theory, enormously extended and developed through the Newtonian law of gravitation (see ChapterIV), 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.14 and15, the first of which represents the apparent motion of the planet Mars through the constellations Aries and Pisces during the latter part of the year 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 in Fig.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. In Fig.15 the 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 does Fig.14 show that the orbit of Mars does not lie exactly in the same plane with the orbit of the earth?Exercise 13.—Find from Fig.15 what 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 with Fig.14. The apparent position of Mars among the stars is merely its direction from the earth, and this direction is represented in Fig.14 by the distance of the planet from the ecliptic and by its longitude.

The longitude of Mars for each date can be found from Fig.15 by measuring the angle between the straight line SV and the line drawn from the earth to Mars. Thus for October 12th we may find with the protractor that the angle between the line SV and the line joining the earth to Mars is a little more than 30°, and in Fig.14 the 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 from Fig.15, which from necessity is constructed as if the orbits of Mars and the earth lay in the same plane, and Mars in this case would always appear to stand exactly on the ecliptic and to oscillate back and forth as shown in Fig.14, but without the up-and-down motion there shown. In this way plot in Fig.14 the 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.

28. The orbits of the planets.—Each planet, great or small, moves in its own appropriate orbit about the sun, and the exact determination of these orbits, their sizes, shapes, positions, etc., has been one of the great problems of astronomy for more than 2,000 years, in which successive generations of astronomers have striven to push to a still higher degree of accuracy the knowledge attained by their predecessors. Without attempting to enter into the details of this problem we may say, generally, that every planet moves in a plane passing through the sun, and for the six planets visible to the naked eye these planes nearly coincide, so that the six orbits may all be shown without much error as lying in the flat surface of one map. It is, however, more convenient to use two maps, such as Figs.16 and17, one of which shows the group of planets, Mercury, Venus, the earth, and Mars, which are near the sun, and on this account are sometimes called the inner planets, while the other shows the more distant planets, Jupiter and Saturn, together with the earth, whose orbit is thus made to serve as a connecting link between the two diagrams. These diagrams are accurately drawn to scale, and are intended to be used by the student for accurate measurement in connection with the exercises and problems which follow.

In addition to the six planets shown in the figures the solar system contains two large planets and several hundred small ones, for the most part invisible to the naked eye, which are omitted in order to avoid confusing the diagrams.29. Jupiter and Saturn.—In Fig.16 the sun at the center is encircled by the orbits of the three planets, and inclosing all of these is a circular border showing the directions from the sun of the constellations which lie along the zodiac. The student must note carefully that it is only the directions of these constellations that are correctly shown, and that in order to show them at all they have been placed very much too close to the sun. The cross lines extending from the orbit of the earth toward the sun with Roman numerals opposite them show the positions of the earth in its orbit on the first day of January (I), first day of February (II), etc., and the similar lines attached to the orbits of Jupiter and Saturn with Arabic numerals show the positions of those planets on the first day of January of each year indicated, so that the figure serves to show not only the orbits of the planets, but their actual positions in their orbits for something more than the first decade of the twentieth century.

The line drawn from the sun toward the right of the figure shows the direction to the vernal equinox. It forms one side of the angle which measures a planet's longitude.

Exercise 14.—Measure with your protractor the longitude of the earth on January 1st. Is this longitude the same in all years? Measure the longitude of Jupiter on January1, 1900; on July 1, 1900; on September 25, 1906.

Draw neatly on the map a pencil line connecting the position of the earth for January1, 1900, with the position of Jupiter for the same date, and produce the line beyond Jupiter until it meets the circle of the constellations. This line represents the direction of Jupiter from the earth, and points toward the constellation in which the planet appears at that date. But this representation of the place of Jupiter in the sky is not a very accurate one, since on the scale of the diagram the stars are in fact more than 100,000 times as far off as they are shown in the figure, and the pencil mark does not meet the line of constellations at the same intersection it would have if this line were pushed back to its true position. To remedy this defect we must draw another line from the sun parallel to the one first drawn, and its intersection with the constellations will give very approximately the true position of Jupiter in the sky.Exercise 15.—Find the present positions of Jupiter and Saturn, and look them up in the sky by means of your star maps. The planets will appear in the indicated constellations as very bright stars not shown on the map.

Which of the planets, Jupiter and Saturn, changes its direction from the sun more rapidly? Which travels the greater number of miles per day? When will Jupiter and Saturn be in the same constellation? Does the earth move faster or slower than Jupiter?

The distance of Jupiter or Saturn from the earth at any time may be readily obtained from the figure. Thus, by direct measurement with the millimeter scale we find for January 1, 1900, the distance of Jupiter from the earth is 6.1 times the distance of the sun from the earth, and this may be turned into miles by multiplying it by 93,000,000, which is approximately the distance of the sun from the earth. For most purposes it is quite as well to dispense with this multiplication and call the distance 6.1 astronomical units, remembering that the astronomical unit is the distance of the sun from the earth.Exercise 16.—What is Jupiter's distance from the earth at its nearest approach? What is the greatest distance it ever attains? Is Jupiter's least distance from the earth greater or less than its least distance from Saturn?

On what day in the year 1906 will the earth be on line between Jupiter and the sun? On this day Jupiter is said to be in opposition—i.e., the planet and the sun are on opposite sides of the earth, and Jupiter then comes to the meridian of any and every place at midnight. When the sun is between the earth and Jupiter (at what date in 1906?) the planet is said to be in conjunction with the sun, and of course passes the meridian with the sun at noon. Can you determine from the figure the time at which Jupiter comes to the meridian at other dates than opposition and conjunction? Can you determine when it is visible in the evening hours? Tell from the figure what constellation is on the meridian at midnight on January 1st. Will it be the same constellation in every year?30. Mercury, Venus, and Mars.—Fig.17, which represents the orbits of the inner planets, differs from Fig.16 only in the method of fixing the positions of the planets in their orbits at any given date. The motion of these planets is so rapid, on account of their proximity to the sun, that it would not do to mark their positions as was done for Jupiter and Saturn, and with the exception of the earth they do not always return to the same place on the same day in each year. It is therefore necessary to adopt a slightly different method, as follows: The straight line extending from the sun toward the vernal equinox, V, is called the prime radius, and we know from past observations that the earth in its motion around the sun crosses this line on September 23d in each year, and to fix the earth's position for September 23d in the diagram we have only to take the point at which the prime radius intersects the earth's orbit. A month later, on October 23d, the earth will no longer be at this point, but will have moved on along its orbit to the point marked 30 (thirty days after September 23d). Sixty days after September 23d it will be at the point marked 60, etc., and for any date we have only to find the number of days intervening between it and the preceding September 23d, and this number will show at once the position of the earth in its orbit. Thus for the date July 4, 1900, we find

1900, July 4 - 1899, September 23 = 284 days,

and the little circle marked upon the earth's orbit between the numbers 270 and 300 shows the position of the earth on that date.

In what constellation was the sun on July 4, 1900? What zodiacal constellation came to the meridian at midnight on that date? What other constellations came to the meridian at the same time?

The positions of the other planets in their orbits are found in the same manner, save that they do not cross the prime radius on the same date in each year, and the times at which they do cross it must be taken from the following table:

Table of Epochs

The first line of figures in this table shows the number of days that each of these planets requires to make a complete revolution about the sun, and it appears from these numbers that Mercury makes about four revolutions in its orbit per year, and therefore crosses the prime radius four times in each year, while the other planets are decidedly slower in their movements. The following lines of the table show for each year the date at which each planet first crossed the prime radius in that year; the dates of subsequent crossings in any year can be found by adding once, twice, or three times the period to the given date, and the table may be extended to later years, if need be, by continuously adding multiples of the period. In the case of Mars it appears that there is only about one year out of two in which this planet crosses the prime radius.

After the date at which the planet crosses the prime radius has been determined its position for any required date is found exactly as in the case of the earth, and the constellation in which the planet will appear from the earth is found as explained above in connection with Jupiter and Saturn.

The broken lines in the figure represent the construction for finding the places in the sky occupied by Mercury, Venus, and Mars on July 4, 1900. Let the student make a similar construction and find the positions of these planets at the present time. Look them up in the sky and see if they are where your work puts them.31. Exercises.—The "evening star" is a term loosely applied to any planet which is visible in the western sky soon after sunset. It is easy to see that such a planet must be farther toward the east in the sky than is the sun, and in either Fig.16 or Fig.17 any planet which viewed from the position of the earth lies to the left of the sun and not more than 50° away from it will be an evening star. If to the right of the sun it is a morning star, and may be seen in the eastern sky shortly before sunrise.

What planet is the evening star now? Is there more than one evening star at a time? What is the morning star now?

Do Mercury, Venus, or Mars ever appear in opposition? What is the maximum angular distance from the sun at which Venus can ever be seen? Why is Mercury a more difficult planet to see than Venus? In what month of the year does Mars come nearest to the earth? Will it always be brighter in this month than in any other? Which of all the planets comes nearest to the earth?

The earth always comes to the same longitude on the same day of each year. Why is not this true of the other planets?

The student should remember that in one respect Figs.16 and17 are not altogether correct representations, since they show the orbits as all lying in the same plane. If this were strictly true, every planet would move, like the sun, always along the ecliptic; but in fact all of the orbits are tilted a little out of the plane of the ecliptic and every planet in its motion deviates a little from the ecliptic, first to one side then to the other; but not even Mars, which is the most erratic in this respect, ever gets more than eight degrees away from the ecliptic, and for the most part all of them are much closer to the ecliptic than this limit.