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.