Part 3

Since celestial longitude and right ascension are both measured from the first point of Aries, the longitude and right ascension of the stars are slowly changing from year to year. It will be seen, from Figs. 38 and 39, that the declination is also slowly changing.

30. _Nutation._--The gyratory motion of the earth's axis is not perfectly regular and uniform. The earth's axis has a slight tremulous motion, oscillating to and fro through a short distance once in about nineteen years. This tremulous motion of the axis causes the pole of the heavens to describe an undulating curve, as shown in Fig. 40, and gives a slight unevenness to the motion of the equinoxes along the ecliptic. This nodding motion of the axis is called _nutation_.

31. _Refraction._--When a ray of light from one of the heavenly bodies enters the earth's atmosphere obliquely, it will be bent towards a perpendicular to the surface of the atmosphere, since it will be entering a denser medium. As the ray traverses the atmosphere, it will be continually passing into denser and denser layers, and will therefore be bent more and more towards the perpendicular. This bending of the ray is shown in Fig. 41. A ray which started from _A_ would enter the eye at _C_, as if it came from _I_: hence a star at _A_ would appear to be at _I_.

Atmospheric refraction displaces all the heavenly bodies from the horizon towards the zenith. This is evident from Fig. 42. _OD_ is the horizon, and _Z_ the zenith, of an observer at _O_. Refraction would make a star at _Q_ appear at _P_: in other words, it would displace it towards the zenith. A star in the zenith is not displaced by refraction, since the rays which reach the eye from it traverse the atmosphere vertically. The farther a star is from the zenith, the more it is displaced by refraction, since the greater is the obliquity with which the rays from it enter the atmosphere.

At the horizon the displacement by refraction is about half a degree; but it varies considerably with the state of the atmosphere. Refraction causes a heavenly body to appear above the horizon longer than it really is above it, since it makes it appear to be on the horizon when it is really half a degree below it.

The increase of refraction towards the horizon often makes the sun, when near the horizon, appear distorted, the lower limb of the sun being raised more than the upper limb. This distortion is shown in Fig. 43. The vertical diameter of the sun appears to be considerably less than the horizontal diameter.

32. _Parallax._--_Parallax_ is the displacement of an object caused by a change in the point of view from which it is seen. Thus in Fig. 44, the top of the tower _S_ would be seen projected against the sky at _a_ by an observer at _A_, and at _b_ by an observer at _B_. In passing from _A_ to _B_, the top of the tower is displaced from _a_ to _b_, or by the angle _aSb_. This angle is called the parallax of _S_, as seen from _B_ instead of _A_.

The _geocentric parallax_ of a heavenly body is its displacement caused by its being seen from the surface of the earth, instead of from the centre of the earth. In Fig. 45, _R_ is the centre of the earth, and _O_ the point of observation on the surface of the earth. Stars at _S_, _S'_, and _S''_, would, from the centre of the earth, appear at _Q_, _Q'_, and _Q''_; while from the point _O_ on the surface of the earth, these same stars would appear at _P_, _P'_ and _P''_, being displaced from their position, as seen from the centre of the earth, by the angles _QSP_, _Q'S'P'_, and _Q''S''P''_. It will be seen that parallax displaces a body from the zenith towards the horizon, and that the parallax of a body is greatest when it is on the horizon. The parallax of a heavenly body when on the horizon is called its _horizontal parallax_. A body in the zenith is not displaced by parallax, since it would be seen in the same direction from both the centre and the surface of the earth.

The parallax of a body at _S'''_ is _Q'''S'''P_, which is seen to be greater than _QSP_; that is to say, the parallax of a heavenly body increases with its nearness to the earth. The distance and parallax of a body are so related, that, either being known, the other may be computed.

33. _Aberration._--_Aberration_ is a slight displacement of a star, owing to an apparent change in the direction of the rays of light which proceed from it, caused by the motion of the earth in its orbit. If we walk rapidly in any direction in the rain, when the drops are falling vertically, they will appear to come into our faces from the direction in which we are walking. Our own motion has apparently changed the direction in which the drops are falling.

In Fig. 46 let _A_ be a gun of a battery, from which a shot is fired at a ship, _DE_, that is passing. Let _ABC_ be the course of the shot. The shot enters the ship's side at _B_, and passes out at the other side at _C_; but in the mean time the ship has moved from _E_ to _e_, and the part _B_, where the shot entered, has been carried to _b_. If a person on board the ship could see the ball as it crossed the ship, he would see it cross in the diagonal line _bC_; and he would at once say that the cannon was in the direction of _Cb_. If the ship were moving in the opposite direction, he would say that the cannon was just as far the other side of its true position.

Now, we see a star in the direction in which the light coming from the star appears to be moving. When we examine a star with a telescope, we are in the same condition as the person who on shipboard saw the cannon-ball cross the ship. The telescope is carried along by the earth at the rate of eighteen miles a second: hence the light will appear to pass through the tube in a slightly different direction from that in which it is really moving: just as the cannon-ball appears to pass through the ship in a different direction from that in which it is really moving. Thus in Fig. 47, a ray of light coming in the direction _SOT_ would appear to traverse the tube _OT_ of a telescope, moving in the direction of the arrow, as if it were coming in the direction _S'O_.

As light moves with enormous velocity, it passes through the tube so quickly, that it is apparently changed from its true direction only by a very slight angle: but it is sufficient to displace the star. This apparent change in the direction of light caused by the motion of the earth is called _aberration of light_.

34. _The Planets._--On watching the stars attentively night after night, it will be found, that while the majority of them appear _fixed_ on the surface of the celestial sphere, so as to maintain their relative positions, there are a few that _wander_ about among the stars, alternately advancing towards the east, halting, and retrograding towards the west. These wandering stars are called _planets_.

Their motions appear quite irregular; but, on the whole, their eastward motion is in excess of their westward, and in a longer or shorter time they all complete the circuit of the heavens. In almost every instance, their paths are found to lie wholly in the belt of the zodiac.

Fig. 48 shows a portion of the apparent path of one of the planets.

II. THE SOLAR SYSTEM.

I. THEORY OF THE SOLAR SYSTEM.

35. _Members of the Solar System._--The solar system is composed of the _sun_, _planets_, _moons_, _comets_, and _meteors_. Five planets, besides the earth, are readily distinguished by the naked eye, and were known to the ancients: these are _Mercury_, _Venus_, _Mars_, _Jupiter_, and _Saturn_. These, with the _sun_ and _moon_, made up the _seven planets_ of the ancients, from which the seven days of the week were named.

The Ptolemaic System.

36. _The Crystalline Spheres._--We have seen that all the heavenly bodies appear to be situated on the surface of the celestial sphere. The ancients assumed that the stars were really fixed on the surface of a crystalline sphere, and that they were carried around the earth daily by the rotation of this sphere. They had, however, learned to distinguish the planets from the stars, and they had come to the conclusion that some of the planets were nearer the earth than others, and that all of them were nearer the earth than the stars are. This led them to imagine that the heavens were composed of a number of crystalline spheres, one above another, each carrying one of the planets, and all revolving around the earth from east to west, but at different rates. This structure of the heavens is shown in section in Fig. 49.

37. _Cycles and Epicycles._--The ancients had also noticed that, while all the planets move around the heavens from west to east, their motion is not one of uniform advancement. Mercury and Venus appear to oscillate to and fro across the sun, while Jupiter and Saturn appear to oscillate to and fro across a centre which is moving around the earth, so as to describe a series of loops, as shown in Fig. 50.

The ancients assumed that the planets moved in exact circles, and, in fact, that all motion in the heavens was circular, the circle being the simplest and most perfect curve. To account for the loops described by the planets, they imagined that each planet revolved in a circle around a centre, which, in turn, revolved in a circle around the earth. The circle described by this centre around the earth they called the _cycle_, and the circle described by the planet around this centre they called the _epicycle_.

38. _The Eccentric._--The ancients assumed that the planets moved at a uniform rate in describing the epicycle, and also the centre in describing the cycle. They had, however, discovered that the planets advance eastward more rapidly in some parts of their orbits than in others. To account for this they assumed that the cycles described by the centre, around which the planets revolved, were _eccentric_; that is to say, that the earth was not at the centre of the cycle, but some distance away from it, as shown in Fig. 51. _E_ is the position of the earth, and _C_ is the centre of the cycle. The lines from _E_ are drawn so as to intercept equal arcs of the cycle. It will be seen at once that the angle between any pair of lines is greatest at _P_, and least at _A_; so that, were a planet moving at the same rate at _P_ and _A_, it would seem to be moving much faster at _P_. The point _P_ of the planet's cycle was called its _perigee_, and the point A its _apogee_.

As the apparent motion of the planets became more accurately known, it was found necessary to make the system of cycles, epicycles, and eccentrics exceedingly complicated to represent that motion.

The Copernican System.

39. _Copernicus._--Copernicus simplified the Ptolemaic system greatly by assuming that the earth and all the planets revolved about the sun as a centre. He, however, still maintained that all motion in the heavens was circular, and hence he could not rid his system entirely of cycles and epicycles.

Tycho Brahe's System.

40. _Tycho Brahe._--Tycho Brahe was the greatest of the early astronomical observers. He, however, rejected the system of Copernicus, and adopted one of his own, which was much more complicated. He held that all the planets but the earth revolved around the sun, while the sun and moon revolved around the earth. This system is shown in Fig. 52.

Kepler's System.

41. _Kepler._--While Tycho Brahe devoted his life to the observation of the planets. Kepler gave his to the study of Tycho's observations, for the purpose of discovering the true laws of planetary motion. He banished the complicated system of cycles, epicycles, and eccentrics forever from the heavens, and discovered the three laws of planetary motion which have rendered his name immortal.

42. _The Ellipse._--An _ellipse_ is a closed curve which has two points within it, the sum of whose distances from every point on the curve is the same. These two points are called the _foci_ of the ellipse.

One method of describing an ellipse is shown in Fig. 53. Two tacks, _F_ and _F'_, are stuck into a piece of paper, and to these are fastened the two ends of a string which is longer than the distance between the tacks. A pencil is then placed against the string, and carried around, as shown in the figure. The curve described by the pencil is an ellipse. The two points _F_ and _F'_ are the foci of the ellipse: the sum of the distances of these two points from every point on the curve is equal to the length of the string. When half of the ellipse has been described, the pencil must be held against the other side of the string in the same way, and carried around as before.

The point _O_, half way between _F_ and _F'_, is called the _centre_ of the ellipse; _AA'_ is the _major axis_ of the ellipse, and _CD_ is the _minor axis_.

43. _The Eccentricity of the Ellipse._--The ratio of the distance between the two foci to the major axis of the ellipse is called the _eccentricity_ of the ellipse. The greater the distance between the two foci, compared with the major axis of the ellipse, the greater is the eccentricity of the ellipse; and the less the distance between the foci, compared with the length of the major axis, the less the eccentricity of the ellipse. The ellipse of Fig. 54 has an eccentricity of 1/8. This ellipse scarcely differs in appearance from a circle. The ellipse of Fig. 55 has an eccentricity of 1/2, and that of Fig. 56 an eccentricity of 7/8.

44. _Kepler's First Law._--Kepler first discovered that _all the planets move from west to east in ellipses which have the sun as a common focus_. This law of planetary motion is known as _Kepler's First Law_. The planets appear to describe loops, because we view them from a moving point.

The ellipses described by the planets differ in eccentricity; and, though they all have one focus at the sun, their major axes have different directions. The eccentricity of the planetary orbits is comparatively small. The ellipse of Fig. 54 has seven times the eccentricity of the earth's orbit, and twice that of the orbit of any of the larger planets except Mercury; and its eccentricity is more than half of that of the orbit of Mercury. Owing to their small eccentricity, the orbits of the planets are usually represented by circles in astronomical diagrams.

45. _Kepler's Second Law._--Kepler next discovered that a planet's rate of motion in the various parts of its orbit is such that _a line drawn from the planet to the sun would always sweep over equal areas in equal times_. Thus, in Fig. 57, suppose the planet would move from _P_ to _P^1_ in the same time that it would move from _P^2_ to _P^3_, or from _P^4_ to _P^5_; then the dark spaces, which would be swept over by a line joining the sun and the planet, in these equal times, would all be equal.

A line drawn from the sun to a planet is called the _radius vector_ of the planet. The radius vector of a planet is shortest when the planet is nearest the sun, or at _perihelion_, and longest when the planet is farthest from the sun, or at _aphelion_: hence, in order to have the areas equal, it follows that a planet must move fastest when at perihelion, and slowest at aphelion.

_Kepler's Second Law_ of planetary motion is usually stated as follows: _The radius vector of a planet describes equal areas in equal times in every part of the planet's orbit_.

46. _Kepler's Third Law._--Kepler finally discovered that the periodic times of the planets bear the following relation to the distances of the planets from the sun: _The squares of the periodic times of the planets are to each other as the cubes of their mean distances from the sun_. This is known as _Kepler's Third Law_ of planetary motion. By _periodic time_ is meant the time it takes a planet to revolve around the sun.

These three laws of Kepler's are the foundation of modern physical astronomy.

The Newtonian System.

47. _Newton's Discovery._--Newton followed Kepler, and by means of his three laws of planetary motion made his own immortal discovery of the _law of gravitation_. This law is as follows: _Every portion of matter in the universe attracts every other portion with a force varying directly as the product of the masses acted upon, and inversely as the square of the distances between them._

48. _The Conic Sections._--The _conic sections_ are the figures formed by the various plane sections of a right cone. There are four classes of figures formed by these sections, according to the angle which the plane of the section makes with the axis of the cone.

_OPQ_, Fig. 58, is a right cone, and _ON_ is its axis. Any section, _AB_, of this cone, whose plane is perpendicular to the axis of the cone, is a _circle_.

Any section, _CD_, of this cone, whose plane is oblique to the axis, but forms with it an angle greater than _NOP_, is an _ellipse_. The less the angle which the plane of the section makes with the axis, the more elongated is the ellipse.

Any section, _EF_, of this cone, whose plane makes with the axis an angle equal to _NOP_, is a _parabola_. It will be seen, that, by changing the obliquity of the plane _CD_ to the axis _NO_, we may pass uninterruptedly from the circle through ellipses of greater and greater elongation to the parabola.

Any section, _GH_, of this cone, whose plane makes with the axis _ON_ an angle less than _NOP_, is a _hyperbola_.

It will be seen from Fig. 59, in which comparative views of the four conic sections are given, that the circle and the ellipse are _closed_ curves, or curves which return into themselves. The parabola and the hyperbola are, on the contrary, _open_ curves, or curves which do not return into themselves.

49. _A Revolving Body is continually Falling towards its Centre of Revolution._--In Fig. 60 let _M_ represent the moon, and _E_ the earth around which the moon is revolving in the direction _MN_. It will be seen that the moon, in moving from M to N, falls towards the earth a distance equal to _mN_. It is kept from falling into the earth by its orbital motion.

The fact that a body might be projected forward fast enough to keep it from falling into the earth is evident from Fig. 61. _AB_ represents the level surface of the ocean, _C_ a mountain from the summit of which a cannon-ball is supposed to be fired in the direction _CE_. _AD_ is a line parallel with _CE_; _DB_ is a line equal to the distance between the two parallel lines _AD_ and _CE_. This distance is equal to that over which gravity would pull a ball towards the centre of the earth in a minute. No matter, then, with what velocity the ball _C_ is fired, at the end of a minute it will be somewhere on the line _AD_. Suppose it were fired fast enough to reach the point _D_ in a minute: it would be on the line _AD_ at the end of the minute, but still just as far from the surface of the water as when it started. It will be seen, that, although it has all the while been falling towards the earth, it has all the while kept at exactly the same distance from the surface. The same thing would of course be true during each succeeding minute, till the ball came round to _C_ again, and the ball would continue to revolve in a circle around the earth.

50. _The Form of a Body's Orbit depends upon the Rate of its Forward Motion._--If the ball _C_ were fired fast enough to reach the line _AD_ beyond the point _D_, it would be farther from the surface at the end of the second than when it started. Its orbit would no longer be circular, but _elliptical_. If the speed of projection were gradually augmented, the orbit would become a more and more elongated ellipse. At a certain rate of projection, the orbit would become a _parabola_; at a still greater rate, a _hyperbola_.

51. _The Moon held in her Orbit by Gravity._--Newton compared the distance _mN_ that the moon is drawn to the earth in a given time, with the distance a body near the surface of the earth would be pulled toward the earth in the same time; and he found that these distances are to each other inversely as the squares of the distances of the two bodies from the centre of the earth. He therefore concluded that _the moon is drawn to the earth by gravity_, and that the _intensity of gravity decreases as the square of the distance increases_.

52. _Any Body whose Orbit is a Conic Section, and which moves according to Kepler's Second Law, is acted upon by a Force varying inversely as the Square of the Distance._--Newton compared the distance which any body, moving in an ellipse, according to Kepler's Second Law, is drawn towards the sun in the same time in different parts of its orbit. He found these distances in all cases to vary inversely as the square of the distance of the planet from the sun. Thus, in Fig. 62, suppose a planet would move from _K_ to _B_ in the same time that it would move from _k_ to _b_ in another part of its orbit. In the first instance the planet would be drawn towards the sun the distance _AB_, and in the second instance the distance _ab_. Newton found that _AB : ab = (SK)^2 : (Sk)^2_. He also found that the same would be true when the body moved in a parabola or a hyperbola: hence he concluded that _every body that moves around the sun in an ellipse, a parabola, or a hyperbola, is moving under the influence of gravity_.

[Transcriber's Note: In Newton's equation above, (SK)^2 means to group S and K together and square their product. In the original book, instead of using parentheses, there was a vinculum, a horizontal bar, drawn over the S and the K to express the same grouping.]

53. _The Force that draws the Different Planets to the Sun Varies inversely as the Squares of the Distances of the Planets from the Sun._--Newton compared the distances _jK_ and _eF_, over which two planets are drawn towards the sun in the same time, and found these distances to vary inversely as the squares of the distances of the planets from the sun: hence he concluded that _all the planets are held in their orbits by gravity_. He also showed that this would be true of any two bodies that were revolving around the sun's centre, according to Kepler's Third Law.

54. _The Copernican System._--The theory of the solar system which originated with Copernicus, and which was developed and completed by Kepler and Newton, is commonly known as the _Copernican System_. This system is shown in Fig. 64.

II. THE SUN AND PLANETS.

I. THE EARTH.

Form and Size.

55. _Form of the Earth._--In ordinary language the term _horizon_ denotes the line that bounds the portion of the earth's surface that is visible at any point.

(1) It is well known that the horizon of a plain presents the form of a circle surrounding the observer. If the latter moves, the circle moves also; but its form remains the same, and is modified only when mountains or other obstacles limit the view. Out at sea, the circular form of the horizon is still more decided, and changes only near the coasts, the outline of which breaks the regularity.