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 gla.s.s 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 cla.s.s.

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 subst.i.tute 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 alt.i.tude at the time it crosses the meridian. Use either the method of Exercise 4, or that used with Polaris in Exercise 8. The student should be able to show from Fig. 11 that the declination is equal to the sum of the alt.i.tude and the lat.i.tude of the place diminished by 90, or in an equation

Declination = Alt.i.tude + Lat.i.tude - 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 alt.i.tude 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 pa.s.s 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. Cla.s.sical and mediaeval 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...o...b..t with the sky, and the sun appears to move along the ecliptic because, as the earth moves around its...o...b..t, 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...o...b..t, 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.

[Ill.u.s.tration: 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 (see Chapter IV), has completely supplanted the older Ptolemaic doctrine, and an ill.u.s.tration of the simple manner in which it accounts for the apparently complicated motions of a planet among the stars is found in Figs. 14 and 15, 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.

[Ill.u.s.tration: FIG. 15.--The real motion of a planet.]

The longitude of Mars for each date can be found from Fig. 15 by measuring the angle between the straight line _S V_ 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 _S V_ 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.

[Ill.u.s.tration: FIG. 16.--The orbits of Jupiter and Saturn.]

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 pa.s.sing 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 and 17, 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...o...b..t 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.

[Ill.u.s.tration: FIG. 17.--The orbits of the inner planets.]

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 January 1, 1900; on July 1, 1900; on September 25, 1906.

Draw neatly on the map a pencil line connecting the position of the earth for January 1, 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 pa.s.ses 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...o...b..t. A month later, on October 23d, the earth will no longer be at this point, but will have moved on along its...o...b..t 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...o...b..t. 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...o...b..t 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

----------------------------------------------------------- A. D.

Mercury.

Venus.

Earth.

Mars.

--------+------------+-----------+------------+------------ Period

88.0 days.

224.7 days.

365.25 days.

687.1 days.

1900

Feb. 18th.

Jan. 11th.

Sept. 23d.

April 28th.

1901

Feb. 5th.

April 5th.

Sept. 23d.

...

1902

Jan. 23d.

June 29th.

Sept. 23d.

March 16th.

1903

April 8th.

Feb. 8th.

Sept. 23d.

...

1904

March 25th.

May 3d.

Sept. 23d.

Feb. 1st.

1905

March 12th.

July 26th.

Sept. 23d.

Dec. 19th.

1906

Feb. 27th.

March 8th.

Sept. 23d.

...

1907

Feb. 14th.

May 31st.

Sept. 23d.

Nov. 6th.

1908

Feb. 1st.

Jan. 11th.

Sept. 23d.

...

1909

Jan. 18th.

April 4th.

Sept. 23d.

Sept. 23d.

1910

Jan. 5th.

June 28th.

Sept. 23d.

...

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...o...b..t 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.