The geometry of motion: How the tilt between the equator and the ecliptic governs earth's cycles

“ We have already seen (chapter I., § 10) that, in addition to the daily motion (from east to west) which it shares with the rest of the celestial bodies, and of which we need here take no further account, the sun has also an annual motion on the celestial sphere in the reverse direction (from west to east) in a path oblique to the equator, which was early recognised as a great circle, called the ecliptic. It must be remembered further that the celestial sphere, on which the sun appears to lie, is a mere geometrical fiction introduced for convenience ”

“ The faint star just below Alnitak is composed of an exquisite group of delicate stars of various colors.
If the north pole of our earth was shot out like a dart, it would hit the dome of our sky somewhere in the vicinity of the North Star. Likewise, if the earth's equator could be conceived of as swelling and swelling and swelling, it would hit Orion just about at the top star of his belt. The circular line where this imaginary terrestrial hoop would fit against the heavens is called the celestial equator and it, of course, is in a position equally distant between the two heavenly poles. ”

“ Because the earth is tilted on its axis twenty-three and a half degrees from the perpendicular, the ecliptic runs through the heavens in an oblique circle, crossing the line of the equator at two points called the vernal and autumnal equinoxes. The equator in the heavens is the great circle extending around the celestial sphere half-way between the north and south poles. It is always practically ninety degrees from the north star, and the points at which the ecliptic intersects it are called the equinoxes. ”

“ In other words, if we were situated on the sun instead of on the earth, we would see the earth travelling round the sky in the circle of the ecliptic. We must keep this fact, that the ecliptic indicates the plane of the earth's orbit, firmly in mind, in order to understand what follows.
44The ecliptic is not coincident with the celestial equator, for the following reason: The axis of the earth's daily rotation is not parallel to, or does not point in the same direction as, the axis of its yearly revolution round the sun. ”

“ We have next to point out the relation of the co-ordinates we have described to the annual motion of the earth around the sun. In consequence of this motion the sun appears to us to describe annually a great circle, called the ecliptic, round the celestial sphere, among the stars, with a nearly uniform motion, of somewhat less than 1° in a day. Were the stars visible in the daytime in the immediate neighbourhood of the sun, this motion could be traced from day to day. The ecliptic intersects the celestial equator at two opposite points, the equinoxes, at an angle of 23° 27′. ”

“ The ecliptic is inclined to the celestial equator approximately 231⁄2°, an angle which obviously measures the inclination of the plane of the earth’s equator to the plane of its orbit, since the celestial equator and the ecliptic are great circles on the celestial sphere formed by extending the planes of the earth’s equator and its orbit to an infinite distance. ”

“ The ecliptic is a great circle, in which the earth performs its annual revolutions around the sun. It passes through the centre of the earth and the centre of the sun. It is found, by observation, that the earth does not lie with its axis at right angles to the plane of the ecliptic, so as to make the equator coincide with it, but that it is turned about twenty-three and a half degrees out of a perpendicular direction, making an angle with the plane itself of sixty-six and a half degrees. ”

“ The axis of the earth is inclined to the ecliptic nearly twenty-three and one-half degrees, so that we receive the sun’s rays with varying degrees of obliquity during our yearly journeying around it, which is the cause of our agreeable change of seasons. Venus travels with her axis so slightly inclined to her orbit (a little more than three degrees) that each particular parallel of latitude receives practically the same amount of sunlight every day in the year, though at different parallels the sun’s rays strike with varying degrees of obliquity. ”

“ The turning of the earth on its own axis makes the differences of day and night; it being day in those parts of the earth which are turned towards the sun; and night in those parts which are in the shade, or turned from the sun.
The annual revolution of the earth in the ecliptic is the cause of the different seasons, and of the several lengths of days and nights, in every part of the world, in the course of the year.
If the diameter of the sun be to the diameter of the earth as 48 to 1, (as by some it is computed) , the disk of the sun is above 2000 times bigger than the disk of the earth ”

“ The Obliquity of the Ecliptic
The effect of the obliquity of the earth's equator to its path round the sun, upon which depend our varying seasons and the inequality of day and night throughout all the temperate zones, is very generally known. But it is not usually considered that this obliquity is of any great importance as regards the suitability of the earth for the development and maintenance of life; and it seems to have been passed over as an accident hardly worth notice, because almost any other obliquity or none at all would have been equally advantageous. ”

“ If, therefore, the equator had coincided with the ecliptic, as would have been the case, had the earth's axis been perpendicular to the plane of its orbit, the diurnal motion of the sun would always have been in the equator, and the days and nights would have been equal all over the globe. To the inhabitants of the equatorial parts of the earth, the sun would always have appeared to move in the prime vertical, rising directly in the east, passing through the zenith at noon, and setting in the west. In the polar regions, the sun would always have appeared to revolve in the horizon ”

“ To understand the equatorial mounting it is necessary to remember that the rising and setting motions of the heavenly bodies are apparent ones only, and due in reality to the turning of the earth on its own axis. As the earth goes around, it carries observer, telescope, and observatory past the stars fixed upon the distant sky. Consequently, to keep a telescope pointed continuously at a given star, it is merely necessary to rotate it steadily backward upon a suitable axis just fast enough to neutralize exactly the turning of our earth. ”

“ The general position of the equator in the sky is always readily found according to the latitude you are in. If you are in 40° N. latitude, the celestial equator would intersect the celestial sphere at a point 40° South of you. Knowing this, the angular distance of a star North or South of the equator (which is its declination) should be easily found. Remember, however, that the equator in the sky is a curved line and hence a star in the East or West which looks to be slightly North of you may actually be South of you. ”

“ Therefore every celestial body which rises or sets to the north of the equator will cut the horizon between the east or west point and the north point; those bodies which do not set will, of course, not cut the horizon at all.
The sun, and stars near the equator, in such a latitude as that of Thebes, will appear to rise or set at no very considerable angle to the vertical; but when we deal with stars rising or setting near to the north or south points of the horizon they will seem to skim along the horizon instead of rising or setting vertically. ”

“ The paths of all the planets, save one, lie always within three degrees of the ecliptic.
Having once learned the constellations of the zodiac and, approximately, the line of the ecliptic, it is not necessary for the ordinary observer to keep in mind the exact location of the vernal equinox. It is, however, an important point for the student of mathematical astronomy. ”

“ A good deal of careful observation of the kind we have been describing must, however, have been necessary before it was ascertained that the sun's annual path on the celestial sphere (see fig. 4) is a great circle (that is, a circle having its centre at the centre of the sphere) . This great circle is now called the ecliptic (because eclipses take place only when the moon is in or near it) , and the angle at which it cuts the equator is called the obliquity of the ecliptic. ”

“ In the epicyclic theory, uneven motion is represented by motion round a circle whose centre travels round another circle, the latter is called the deferent. Ecliptic: The plane of the earth’s orbital motion about the sun, which cuts the heavens in a great circle. It is so called because obviously eclipses can only occur when the moon is also approximately in this plane, besides being in conjunction or opposition with the sun. Epicycle: A point moving on the circumference of a circle whose centre describes another circle, traces an epicycle with reference to the centre of the second circle. ”

“ Revolution.—The Earth revolves about the Sun in an elliptical orbit having the Sun at one focus. The plane of the orbit is termed the ecliptic; it is inclined to the Earth’s equator at an angle termed the obliquity, and the points of intersection of the equator and ecliptic are termed the equinoctial points. The major axis of the ellipse is the line of apsides; when the Earth is nearest the Sun it is said to be in perihelion, when farthest it is in aphelion. The mean distance of the Earth from the Sun is a most important astronomical constant, since it is the unit of linear measurement ”

“ Moreover, a line drawn from this intersection, or terrestrial vernal equinox, through the center of the sun and extended to the celestial sphere would strike the corresponding intersection of the ecliptic and the equinoctial or celestial equator—the celestial vernal equinox. The arrival of the earth at this position is the signal of spring for the northern hemisphere, likewise it announces the advent of autumn to our southern neighbors below the “Line.” The sun this day rises in the east (approximately) and passing through the zenith, sets in the west for those living on the equator. ”

“ The equality of days and nights when the sun is on the celestial equator is also evident from the following considerations: one-half of the earth is in sunshine all of the time; when the sun is on the celestial equator, it is directly over the equator of the earth, and the illumination extends from pole to pole, as is evident from Figs. 70 and 71, in the former of which the sun is represented as on the eastern horizon at a place along the central line of the figure, and in the latter as on the meridian along the same line. ”

“ It will, I think, appear likely, when we come to study Plates XIX. and XX., that as early as 4600 B.C. constellations were imagined to honour and mark the equinoctial as well as the solstitial seasons.
Perhaps then, at that date the constellation Sagittarius was extended to its present dimensions; and it may be that some centuries later, when the colure of the winter solstice had passed into the constellation Aquarius, some astronomers desired—like Gudea of Lagash and Tchuen-Hio in China—to honour that [223] season, and to make it the beginning of the year. ”

“ Precession.—The axis of the earth does not preserve an invariable direction in space, but in a certain time it describes a cone, in much the same manner as the axis of a top spinning out of the vertical. The equator, which preserves approximately the same inclination to the ecliptic (there is a slight variation in the obliquity which we shall mention later) , must move so that its intersections with the ecliptic, or equinoctial points, pass in a retrograde direction, i.e. opposite to that of the Earth. ”

“ The ordinary approximate statement concerning the earth's axis is that it remains parallel to itself, i.e. has a fixed direction as the earth moves round the sun. But if, instead of being thus fixed, it be supposed to have a slow movement of revolution, so that it traces out a cone in the course of about 26,000 years, then, since the equator of course goes with it, the motion of its intersection with the fixed ecliptic is so far accounted for. That is to say, the precession of the equinoxes is seen to be dependent on, and caused by, a slow conical movement of the earth's axis. ”

“ Now, the sun’s attraction upon such a spheroidal body is not quite the same as it would be upon a perfect sphere. When the sun is at either equinox—that is, just over the equator—the attraction exercised upon our earth is the same as if that body were spherical; but when the sun is at or near the upper tropic, its action upon the terrestrial matter which bulges at the equator has a tendency to pull that matter towards the ecliptic, and to make the axis of the earth approach to a vertical to the ecliptic. ”

“ If we consider, that the Earth is not a perfect sphere, but an oblate spheroid, having its Axis shorter than its Equatoreal diameter; and that the Sun and Moon are constantly acting obliquely upon the greater quantity of matter about the Equator, pulling it, as it were, towards a nearer and nearer co-incidence with the Ecliptic ”

“ The immediate effect of the gravitation is so modified by the rapid rotation of the top, that, in obedience to a profound dynamical principle, the axis of the top revolves in a cone rather than fall down, as it would do were the top not spinning. In a similar manner the immediate effect of the sun's attraction on the protuberance at the equator would be to bring the pole of the earth's axis towards the pole of the ecliptic, but the rapid rotation of the earth modifies this into the conical movement of precession. ”

“ There is another equally important and instructive result of this precession of the equinox. For the earth's axis is always perpendicular to the plane of the equator, and if the latter moves, the former [Pg 98] must too, and change its position with respect to the axis of the ecliptic, which remains immovable. And the ends of these axes, or the points they occupy among the stars, called their poles, will change in the same way; the pole of the equator, round which the heavens appear to move, describing a curve about the pole of the ecliptic ”

“ The effect of the attraction of the sun and moon upon the equatorial protuberance of the earth is to draw it round from west to east by a very slow motion, and make the ecliptic cross the equator each year about one minute of arc to the east of where it crossed it the year before. So, then, the sidereal year, or interval between the times at which the sun is in a certain position amongst the stars, is longer than the solar year, or interval between the times at which the sun crosses the equator at the vernal equinox. ”

“ The effect of other planets, called planetary precession, tends to produce a much smaller motion eastward along the ecliptic. The resultant motion, called general precession, is westward along the ecliptic at the rate of about 50.3" per year. The component of general precession along the celestial equator, called precession in right ascension, is about 46.1" per year and the component along a celestial meridian, called precession in declination, is about 20.0" per year. ”

“ The moment of force thus exerted tends to tilt the earth, which, like a top, “precesses” instead of falling over. As this disturbing force on the earth is very small relatively to its mass, this turning of its axis about the pole of the ecliptic takes place at about the rate of 50.1′′ per annum, and it therefore requires 25,868 years for the equinoxes to describe a complete circle on the ecliptic. ”

“ If the direction of the earth's axis and with it that of the celestial sphere, did not change at all, then the celestial poles and the poles of the ecliptic would always retain the same relative positions in the sky; but, in fact, an exterior force, acting upon the earth, causes a gradual change in the direction of its axis, and in consequence of this change the celestial poles, whose position depends upon that of the earth's poles, have a slow motion of revolution about the poles of the ecliptic, in a circle of 231⁄2° radius. ”

“ As the earth travels in its annual path around the sun, the axis of the earth is directed to a point in the heavens which is 23-1⁄2° from the pole of the ecliptic. The precessional effect of the sun is to cause this point—the pole of the earth—to revolve, always preserving the same angular distance from the pole of the ecliptic; and thus we have a motion of the type represented in the diagram. As the ecliptic occupies a position which for our present purpose we may regard as fixed in space, it follows that the pole of the ecliptic is a fixed point on the surface of the heavens ”

“ All Stars that have their distance from the Ecliptick Northward not more then 23 degrees and an half (which is the greatest distance of the Ecliptick from the Equator) may in progression of time have declination Southward, and move beyond the Equator: but if any Star hath just this distance of 23 and an half (as hath Capella on the back of Ericthonius) it may hereafter move under the Equinoctial; and the same will happen respectively unto Stars which have declination Southward. And therefore many Stars may be visible in our Hemisphere, which are not so at present ”

“ We must keep in mind, that the celestial equator and celestial ecliptic are here understood, and we may imagine them to be two great circles delineated on the face of the sky. On comparing observations made at different periods, for more than two thousand years, it is found, that the obliquity of the ecliptic is not constant, but that it undergoes a slight diminution, from age to age, amounting to fifty-two seconds in a century, or about half a second annually. ”

“ The plane of the ecliptic itself, though assumed to be fixed at a given epoch for the convenience of astronomical computation, is subject to a minute secular variation of 52".109, occasioned by the reciprocal action of the planets; but as this is also periodical, the terrestrial equator, which is inclined to it at an angle of about 23° 28’, will never coincide with the plane of the ecliptic; so there never can be perpetual spring.
The rotation of the earth is uniform ”

“ On the other hand, if our own system be at rest, and any of the Stars in real motion, this must vary their positions; and the more so, the nearer they are to us, or the swifter their motions are; or the more proper the direction of their motion is, for our perception.
The Ecliptic less oblique now to the Equator than formerly. 405. The obliquity of the Ecliptic to the Equinoctial is found at present to be above a third part of a degree less than Ptolemy found it. And most of the observers after him found it to decrease gradually down to Tycho’s time. ”

“ The action of the planets on one another and on the sun, occasions a very slow variation in the position of the plane of the ecliptic, which affects its inclination on the plane of the equator, and gives the equinoctial points a slow but direct motion on the ecliptic of 0"·312 annually, which is entirely independent of the figure of the earth, and would be the same if it were a sphere. Thus the sun and moon, by moving the plane of the equator, cause the equinoctial points to retrograde on the ecliptic ”

“ Nutation.—In treating precession we have stated that the axis of the Earth traces a cone, and it follows that the pole describes a circle (approximately) on the celestial sphere, about the pole of the ecliptic. This is not quite true. Irregularities in the attracting forces which occasion precession also cause a slight oscillation backwards and forwards over the mean precessional path of the pole, the pole tracing a wavy line or nodding. Both the Sun and Moon contribute to this effect. Solar nutation depends upon the position of the Sun on the ecliptic ”

“ The laws of motion of the ecliptic and equator are stated in the article Precession of the Equinoxes.
Attempts have been made by Laplace and his successors to fix certain limits within which the obliquity of the ecliptic shall always be confined. The results thus derived are, however, based on imperfect formulae. When the problem is considered in a rigorous form, it is found that no absolute limits can be set. It can, however, be shown that the obliquity cannot vary more than two or three degrees within a million of years of our epoch. ”

“ It is so combined with the latter, however, that though quite extensive in itself, amounting to more than 4°, it can never change the tilt of the ecliptic to the equator by more than 1°, one third on either side, so that our seasons can never differ greatly from what we know them now. They can neither be materially accentuated nor proportionately reduced.
Though the obliquity of the ecliptic is oscillatory, the motion of the ecliptic pole keeps persistently, though unsteadily, on always in the same sense. Its wanderings trace out an elliptic spiral which never returns into itself. ”