CHAPTER IV.

THE EARTH’S ORBIT.

EXACT SHAPE OF THE ORBIT.—It will be clear that if we made our annual journey in a circle we should always be at the same distance from the sun, and the apparent size of that luminary would never vary. This, however, is not the case. Exact measurements, which are best made by means of the transit instrument, indicate variations which, though not perceptible to the unassisted eye, establish a want of circularity. The observations bearing on this point consist of a measurement of the time required for the sun to cross the meridian—the larger its apparent diameter, the longer it will obviously be in passing the meridian. An observation of the sidereal time at which the centre of the sun passes the meridian determines the right ascension, and from this one can calculate the sun’s longitude.

[Illustration:

FIG. 13.—_Elliptic Form of Earth’s Orbit._ ]

If such observations be made at intervals during a year, we can utilise them for determining the shape of the earths orbit independently of a knowledge of the actual size. In Fig. 13 let us suppose the sun to be situated at the point S; from S we draw a line, S A, representing the line joining the earth and sun at the vernal equinox when the sun’s longitude is zero. If our observations include a measure of the sun’s diameter on that day, let S A be drawn on some convenient scale. To plot the observations for other days, we must draw S F, S E, etc., at angles A S F, A S E, etc., equal to the sun’s longitude, and make the lengths inversely proportional to the apparent diameters, on the same scale as S A. The other observations can be plotted in the same way, and the earths orbit is then found to be an ellipse with the sun in one of its foci. Actually, the earth’s orbit is much more nearly circular than is shown in Fig. 13, and in illustration of this the following numerical data may be given:—

1896. Jan. 1 Greatest apparent diameter of sun = 32′ 35″·2 in long. 281° July 3 Least „ „ „ = 31′ 30″·6 „ 102° March 29 Mean „ „ „ = 32′ 4″ „ 9° Oct. 5 „ „ „ „ = 32′ 4″ „ 193°

It thus appears that in 1896 we were nearest to the sun on January 1, as on that day the sun’s apparent diameter was greatest, while we were furthest removed on July 3.

[Illustration:

FIG. 14.—_The Ellipse._ ]

The ellipse is a curve of such importance in astronomy that an understanding of some of its properties is essential for further progress. This beautiful closed curve lies in one plane, and its figure is such that the sum of the distances of any point upon it from two fixed points within the curve is constant. These two fixed points, F F′ (Fig. 14), are called the foci of the ellipse, and we have, for example, the sum of the lengths P F and P F′, equal to the sum of P′ F and P′ F′. The line A B passing through the foci is the greatest distance across the ellipse, and is called the major axis; at right angles to this is the minor axis C D.

Following our definition of the ellipse, we see that as B is a point upon its circumference, B F + B F′ must be equal to the sum of the distances of any point P from the foci. But since B F is of the same length as A F′, the sum of the distances of the point B from the foci, and therefore of all other points, is equal to the major axis. Hence the average or mean distance of the focus F from all points on the ellipse is half the length of the major axis. It follows also that C F is equal to the semi-major axis O B.

At the point O, where the axes intercept each other, we have the centre of the ellipse, and the ratio between the distance from the centre to either of the foci and the semi-major axis, _i.e._, (O F)/(O B) is called the eccentricity of the ellipse. Thus, in an ellipse of eccentricity 0·5, the foci would lie midway between the centre of the ellipse and the extremities of the major axis. The eccentricity is always less than unity; if it become unity, the two foci merge together, and the curve becomes a circle.

[Illustration:

FIG. 15.—_How to draw an Ellipse._ ]

To draw an ellipse, two pins may be stuck into a piece of paper at the points intended as foci. A loop of thread is then made and thrown over the pins. A pencil placed inside the loop, so as to stretch it, and traced completely round, will outline an ellipse. The size and shape of the ellipse may be varied by changing the length of the thread and the distance between the pins. Such, then, is the curve in which our earth performs its annual journey round the sun, the sun being relatively fixed in one of the foci.

APHELION AND PERIHELION.—When the earth is in that part of its orbit where it makes its nearest approach to the sun, it is said to be in _perihelion_; when at the point furthest removed from the sun it is in _aphelion_. The line joining these two points is obviously the major axis of the earth’s orbit, and when this is imagined to be prolonged indefinitely into space it is called the _line of apsides_, or _apse line_. When the earth is in perihelion, the sun’s apparent diameter will be the greatest possible, and when in aphelion it will be at a minimum. A knowledge of these limiting values of the apparent solar diameter enables us to determine the eccentricity of the orbit of the earth. The sun’s apparent diameter when the earth is in perihelion amounts to 32′ 35″·2, and to 31′ 30″·6, when the earth is in aphelion, from which it results that the value of e is 0·0167.

UNEQUAL SPEED OF THE EARTH.—The observations by which we are enabled to determine the true form of the earth’s orbit are not quite exhausted of their usefulness; we can utilise them still further for studying the varying rate of the earth’s motion. If the earth moved through equal angles every day, the apparent movement of the sun would always be uniform, and in that case the sun’s daily increase of longitude would be constant.

The following figures, however, prove that this uniformity does not exist:—

1896. Sun’s daily motion in longitude. Jan. 1 1° 1′ 8″·5 Mar. 29 1° 0′ 6″·7 July 3 0° 57′ 12″·1

Facts such as these led Kepler in 1609 to the discovery of his famous second law of planetary motion, namely, that the radius vector (the line joining the sun and earth in the case of the earth’s orbit) describes equal areas in equal times. For the sake of clearness, imagine the earth’s orbit to be represented by the elongated ellipse in Fig. 16, with the sun in the focus _S_. When the earth is near perihelion, it will move over a certain distance, _a b_, in a given time; some time afterwards it will be in another part of the orbit, and in the same interval as before it will traverse the distance _c d_; again, in another equal interval of time, it will move from the point _e_ to the point _f_. The law tells that the areas _S a b_, _S c d_, and _S e f_, are equal so long as equal times are in question; in different parts of its path, then, the earth’s rate of motion must vary, _c d_, for example, being smaller than _a b_. It will be seen that the motion is most rapid when the earth is in perihelion, and least rapid when in aphelion.

[Illustration:

FIG. 16.—_Illustrating Kepler’s Second Law._ ]

CHANGES IN THE EARTH’S ORBIT.—Owing to disturbances caused by the proximity of other bodies, the earth’s orbit is not always of the same shape. The eccentricity is steadily diminishing, and in about 24,000 years the orbit will be very nearly a circle; it will afterwards become more elliptical again, until in another 40,000 years or so the eccentricity will be about 0·02. So far as our knowledge goes, the eccentricity will never exceed 0·07.

The direction of the major axis of the earth’s orbit, that is, the line of apsides, moves forward at the rate of about 11″ per annum, so that at this speed a whole revolution will be made in a period of 108,000 years.

On account of precession, the equinox moves backwards along the orbit at the rate of 50″·2 per annum, so that the movement of the apse line with regard to the equinox is 61′ in a year; or, in other words, the perihelion point of the earth’s orbit makes a complete revolution with respect to the equinoctial point in a little over 20,000 years. The earth at present passes through perihelion in our northern winter, but owing to this motion of the apse line it will in 10,000 years time be at aphelion in winter. Northern winters will then be somewhat colder than at present. The plane of the orbit itself is subject to changes, with the result that the obliquity of the ecliptic is variable in amount. In the course of ages the obliquity may oscillate between the limits 24° 35′ 58″ and 21° 58′ 36″. The mean value during 1896 was 23° 27′ 9″·9.

THE EARTH’S REAL PATH.—In this and preceding chapters, we have had occasion to consider various features of the earth’s orbit, but it must now be pointed out that what we call the orbit of the earth is not quite the same thing as the earth’s actual path in space. The earth, as we know, is accompanied by the moon, and these two bodies are bound together in such a way that it is really the centre of gravity of the earth and moon which describes an elliptic orbit round the sun; the moon is so small in relation to the earth that the centre of gravity of the two companions lies within the earth’s surface, but, nevertheless, an oscillatory displacement of the earth’s centre in space is produced by the moon’s monthly circuit round the earth. We judge of the earth’s movement by the apparent movement of the sun, and we actually find a monthly inequality in the sun’s apparent motion. A very good illustration of this may be found in the varying celestial latitude of the sun. It will be clear that if the earth always moved in the plane of the ecliptic, the sun’s latitude would always be zero. If, on the other hand, the earth has a motion round the common centre of gravity, it will be above the ecliptic when the moon is below, and _vice versâ_; the sun will, therefore, not always appear to be in the ecliptic, and its latitude will depend upon that of the moon. The following figures from the “Nautical Almanac” will illustrate this point:

Sun’s apparent latitude. Moon’s latitude. 1896, April 1 0″·70 S. 5° 9′ S. „ 10 0″·01 N. 1° 41′ N. „ 16 0″·39 N. 5° 6′ N. „ 22 0″·07 S. 0° 48′ N. „ 29 0″·74 S. 5° 1′ S.

The displacement in right ascension amounts to a little over 6″, and is, therefore, large enough to be directly measurable.

On account of this association with her satellite, the earth’s centre moves some hundreds of miles above and below the plane of the ecliptic.

The so-called “perturbations,” or disturbing effects of the other planets, also cause the earth to depart more or less from the plane of the ecliptic and from a geometrical elliptic path. Nevertheless, these disturbances can be calculated and allowed for, so that when we speak of the earth’s orbit we really mean the path which the centre of gravity of the earth and moon would traverse if subject only to the influence of the sun.