THE ABERRATION OF LIGHT.
The Real and Apparent Movements of the Stars--How they can be
Discriminated--Aberration produces Effects dependent on the
Position of the Stars--The Pole of the Ecliptic--Aberration makes
Stars seem to Move in a Circle, an Ellipse, or a Straight Line
according to Position--All the Ellipses have Equal Major Axes--How
is this Movement to be Explained?--How to be Distinguished from
Annual Parallax--The Apex of the Earth's Way--How this is to be
Explained by the Velocity of Light--How the Scale of the Solar
System can be Measured by the Aberration of Light.
We have in this chapter to narrate a discovery of a recondite character,
which illustrates in a forcible manner some of the fundamental truths of
Astronomy. Our discussion of it will naturally be divided into two
parts. In the first part we must describe the nature of the phenomenon,
and then we must give the extremely elegant explanation afforded by the
properties of light. The telescopic discovery of aberration, as well as
its explanation, are both due to the illustrious Bradley.
The expression _fixed_ star, so often used in astronomy, is to be
received in a very qualified sense. The stars are, no doubt, well fixed
in their places, so far as coarse observation is concerned. The
lineaments of the constellations remain unchanged for centuries, and, in
contrast with the ceaseless movements of the planets, the stars are not
inappropriately called fixed. We have, however, had more than one
occasion to show throughout the course of this work that the expression
"fixed star" is not an accurate one when minute quantities are held in
estimation. With the exact measures of modern instruments, many of these
quantities are so perceptible that they have to be always reckoned with
in astronomical enquiry. We can divide the movements of the stars into
two great classes: the real movements and the apparent movements. The
proper motion of the stars and the movements of revolution of the binary
stars constitute the real movements of these bodies. These movements are
special to each star, so that two stars, although close together in the
heavens, may differ in the widest degree as to the real movements which
they possess. It may, indeed, sometimes happen that stars in a certain
region are animated with a common movement. In this phenomenon we have
traces of a real movement shared by a number of stars in a certain
group. With this exception, however, the real movements of the stars
seem to be governed by no systematic law, and the rapidly moving stars
are scattered here and there indiscriminately over the heavens.
The apparent movements of the stars have a different character, inasmuch
as we find the movement of each star determined by the place which it
occupies in the heavens. It is by this means that we discriminate the
real movements of the star from its apparent movements, and examine the
character of both.
In the present chapter we are concerned with the apparent movements
only, and of these there are three, due respectively to precession, to
nutation, and to aberration. Each of these apparent movements obeys laws
peculiar to itself, and thus it becomes possible to analyse the total
apparent motion, and to discriminate the proportions in which the
precession, the nutation, and the aberration have severally contributed.
We are thus enabled to isolate the effect of aberration as completely as
if it were the sole agent of apparent displacement, so that, by an
alliance between mathematical calculation and astronomical observation,
we can study the effects of aberration as clearly as if the stars were
affected by no other motions.
Concentrating our attention solely on the phenomena of aberration we
shall describe its particular effect upon stars in different regions of
the sky, and thus ascertain the laws according to which the effects of
aberration are exhibited. When this step has been taken, we shall be in
a position to give the beautiful explanation of those laws dependent
upon the velocity of light.
At one particular region of the heavens the effect of aberration has a
degree of simplicity which is not manifested anywhere else. This region
lies in the constellation Draco, at the pole of the ecliptic. At this
pole, or in its immediate neighbourhood, each star, in virtue of
aberration, describes a circle in the heavens. This circle is very
minute; it would take something like 2,000 of these circles together to
form an area equal to the area of the moon. Expressed in the usual
astronomical language, we should say that the diameter of this small
circle is about 40.9 seconds of arc. This is a quantity which, though
small to the unaided eye, is really of great relative magnitude in the
present state of telescopic research. It is not only large enough to be
perceived, but it can be measured, with an accuracy which actually does
not admit of a doubt, to the hundredth part of the whole. It is also
observed that each star describes its little circle in precisely the
same period of time; and that period is one year, or, in other words,
the time of the revolution of the earth around the sun. It is found that
for all stars in this region, be they large stars or small, single or
double, white or coloured, the circles appropriate to each have all the
same size, and are all described in the same time. Even from this alone
it would be manifest that the cause of the phenomenon cannot lie in the
star itself. This unanimity in stars of every magnitude and distance
requires some simpler explanation.
Further examination of stars in different regions sheds new light on the
subject. As we proceed from the pole of the ecliptic, we still find that
each star exhibits an annual movement of the same character as the stars
just considered. In one respect, however, there is a difference. The
apparent path of the star is no longer a circle; it has become an
ellipse. It is, however, soon perceived that the shape and the position
of this ellipse are governed by the simple law that the further the star
is from the pole of the ecliptic the greater is the eccentricity of the
ellipse. The apparent path of the stars at the same distance from the
pole have equal eccentricity, and of the axes of the ellipse the shorter
is always directed to the pole, the longer being, of course,
perpendicular to it. It is, however, found that no matter how great the
eccentricity may become, the major axis always retains its original
length. It is always equal to about 40.9 seconds--that is, to the
diameter of the circle of aberration at the pole itself. As we proceed
further and further from the pole of the ecliptic, we find that each
star describes a path more and more eccentric, until at length, when we
examine a star on the ecliptic, the ellipse has become so attenuated
that it has flattened into a line. Each star which happens to lie on the
ecliptic oscillates to and fro along the ecliptic through an amplitude
of 40.9 seconds. Half a year accomplishes the journey one way, and the
other half of the year restores the star to its original position. When
we pass to stars on the southern side of the ecliptic, we see the same
series of changes proceed in an inverse order. The ellipse, from being
actually linear, gradually grows in width, though still preserving the
same length of major axis, until at length the stars near the southern
pole of the ecliptic are each found to describe a circle equal to the
paths pursued by the stars at the north pole of the ecliptic.
The circumstance that the major axes of all those ellipses are of equal
length suggests a still further simplification. Let us suppose that
every star, either at the pole of the ecliptic or elsewhere, pursues an
absolutely circular path, and that all these circles agree not only in
magnitude, but also in being all parallel to the plane of the ecliptic:
it is easy to see that this simple supposition will account for the
observed facts. The stars at the pole of the ecliptic will, of course,
show their circles turned fairly towards us, and we shall see that they
pursue circular paths. The circular paths of the stars remote from the
pole of the ecliptic will, however, be only seen somewhat edgewise, and
thus the apparent paths will be elliptical, as we actually find them. We
can even calculate the degree of ellipticity which this surmise would
require, and we find that it coincides with the observed ellipticity.
Finally, when we observe stars actually moving in the ecliptic, the
circles they follow would be seen edgewise, and thus the stars would
have merely the linear movement which they are seen to possess. All the
observed phenomena are thus found to be completely consistent with the
supposition that every star of all the millions in the heavens describes
once each year a circular path; and that, whether the star be far or
near, this circle has always the same apparent diameter, and lies in a
plane always parallel to the plane of the ecliptic.
We have now wrought the facts of observation into a form which enables
us to examine into the cause of a movement so systematic. Why is it that
each star should seem to describe a small circular path? Why should that
path be parallel to the ecliptic? Why should it be completed exactly in
a twelvemonth? We are at once referred to the motion of the earth around
the sun. That movement takes place in the ecliptic. It is completed in a
year. The coincidences are so obvious that we feel almost necessarily
compelled to connect in some way this apparent movement of the stars
with the annual movement of the earth around the sun. If there were no
such connection, it would be in the highest degree improbable that the
planes of the circles should be all parallel to the ecliptic, or that
the time of revolution of each star in its circle should equal that of
the revolution of the earth around the sun. As both these conditions are
fulfilled, the probability of the connection rises to a value almost
infinite.
The important question has then arisen as to why the movement of the
earth around the sun should be associated in so remarkable a manner with
this universal star movement. There is here one obvious point to be
noticed and to be dismissed. We have in a previous chapter discussed the
important question of the annual parallax of stars, and we have shown
how, in virtue of annual parallax, each star describes an ellipse. It
can further be demonstrated that these ellipses are really circles
parallel to the ecliptic; so that we might hastily assume that annual
parallax was the cause of the phenomenon discovered by Bradley. A single
circumstance will, however, dispose of this suggestion. The circle
described by a star in virtue of annual parallax has a magnitude
dependent on the distance of the star, so that the circles described by
various stars are of various dimensions, corresponding to the varied
distances of different stars. The phenomena of aberration, however,
distinctly assert that the circular path of each star is of the same
size, quite independently of what its distance may be, and hence annual
parallax will not afford an adequate explanation. It should also be
noticed that the movements of a star produced by annual parallax are
much smaller than those due to aberration. There is not any known star
whose circular path due to annual parallax has a diameter one-twentieth
part of that of the circle due to aberration; indeed, in the great
majority of cases the parallax of the star is an absolutely insensible
quantity.
There is, however, a still graver and quite insuperable distinction
between the parallactic path and the aberrational path. Let us, for
simplicity, think of a star situated near the pole of the ecliptic, and
thus appearing to revolve annually in a circle, whether we regard either
the phenomenon of parallax or of aberration. As the earth revolves, so
does the star appear to revolve; and thus to each place of the earth in
its orbit corresponds a certain place of the star in its circle. If the
movement arise from annual parallax, it is easy to see where the place
of the star will be for any position of the earth. It is, however, found
that in the movement discovered by Bradley the star never has the
position which parallax assigns to it, but is, in fact, a quarter of the
circumference of its little circle distant therefrom.
A simple rule will find the position of the star due to aberration. Draw
from the centre of the ellipse a radius parallel to the direction in
which the earth is moving at the moment in question, then the extremity
of this radius gives the point on its ellipse where the star is to be
found. Tested at all seasons, and with all stars, this law is found to
be always verified, and by its means we are conducted to the true
explanation of the phenomenon.
We can enunciate the effects of aberration in a somewhat different
manner, which will show even more forcibly how the phenomenon is
connected with the motion of the earth in its orbit. As the earth
pursues its annual course around the sun, its movement at any moment may
be regarded as directed towards a certain point of the ecliptic. From
day to day, and even from hour to hour, the point gradually moves along
the ecliptic, so as to complete the circuit in a year. At each moment,
however, there is always a certain point in the heavens towards which
the earth's motion is directed. It is, in fact, the point on the
celestial sphere towards which the earth would travel continuously if,
at the moment, the attraction of the sun could be annihilated. It is
found that this point is intimately connected with the phenomenon of
aberration. In fact, the aberration is really equivalent to drawing each
star from its mean place towards the Apex of the Earth's Way, as the
point is sometimes termed. It can also be shown by observation that the
amount of aberration depends upon the distance from the apex. A star
which happened to lie on the ecliptic will not be at all deranged by
aberration from its mean place when it happens that the apex coincides
with the star. All the stars 10 deg. from the apex will be displaced each by
the same amount, and all directly in towards the apex. A star 20 deg. from
the apex will undergo a larger degree of displacement, though still in
the same direction, exactly towards the apex; and all stars at the same
distance will be displaced by the same amount. Proceeding thus from the
apex, we come to stars at a distance of 90 deg. therefrom. Here the amount
of displacement will be a maximum. Each one will be about twenty seconds
from its average place; but in every case the imperative law will be
obeyed, that the displacement of the star from its mean place lies
towards the apex of the earth's way. We have thus given two distinct
descriptions of the phenomenon of aberration. In the first we find it
convenient to speak of a star as describing a minute circular path; in
the other we have regarded aberration as merely amounting to a
derangement of the star from its mean place in accordance with specified
laws. These descriptions are not inconsistent: they are, in fact,
geometrically equivalent; but the latter is rather the more perfect,
inasmuch as it assigns completely the direction and extent of the
derangement caused by aberration in any particular star at any
particular moment.
The question has now been narrowed to a very definite form. What is it
which makes each star seem to close in towards the point towards which
the earth is travelling? The answer will be found when we make a minute
enquiry into the circumstances in which we view a star in the telescope.
The beam of rays from a star falls on the object-glass of a telescope;
those rays are parallel, and after they pass through the object-glass
they converge to a focus near the eye end of the instrument. Let us
first suppose that the telescope is at rest; then if the telescope be
pointed directly towards the star, the rays will converge to a point at
the centre of the field of view where a pair of cross wires are placed,
whose intersection defines the axis of the telescope. The case will,
however, be altered if the telescope be moved after the light has passed
through the objective; the rays of light in the interior of the tube
will pursue a direct path, as before, and will proceed to a focus at the
same precise point as before. As, however, the telescope has moved, it
will, of course, have carried with it the pair of cross wires; they will
no longer be at the same point as at first, and consequently the image
of the star will not now coincide with their intersection.
The movement of the telescope arises from its connection with the earth:
for as the earth hurries along at a speed of eighteen miles a second,
the telescope is necessarily displaced with this velocity. It might at
first be thought, that in the incredibly small fraction of time
necessary for light to pass from the object-glass to the eye-piece, the
change in the position of the telescope must be too minute to be
appreciable. Let us suppose, for instance, that the star is situated
near the pole of the ecliptic, then the telescope will be conveyed by
the earth's motion in a direction perpendicular to its length. If the
tube of the instrument be about twenty feet long, it can be readily
demonstrated that during the time the light travels down the tube the
movement of the earth will convey the telescope through a distance of
about one-fortieth of an inch.[42] This is a quantity very distinctly
measurable with the magnifying power of the eye-piece, and hence this
derangement of the star's place is very appreciable. It therefore
follows that if we wish the star to be shown at the centre of the
instrument, the telescope is not to be pointed directly at the star, as
it would have to be were the earth at rest, but the telescope must be
pointed a little in advance of the star's true position; and as we
determine the apparent place of the star by the direction in which the
telescope is pointed, it follows that the apparent place of the star is
altered by the motion of the earth.
Every circumstance of the change in the star's place admits of complete
explanation in this manner. Take, for instance, the small circular path
which each star appears to describe. We shall, for simplicity, refer
only to a star at the pole of the ecliptic. Suppose that the telescope
is pointed truly to the place of the star, then, as we have shown, the
image of the star will be at a distance of one-fortieth of an inch from
the cross wires. This distance will remain constant, but each night the
direction of the star from the cross wires will change, so that in the
course of the year it completes a circle, and returns to its original
position. We shall not pursue the calculations relative to other stars;
suffice it here to say that the movement of the earth has been found
adequate to account for the phenomena, and thus the doctrine of the
aberration of light is demonstrated.
It remains to allude to one point of the utmost interest and importance.
We have seen that the magnitude of the aberration can be measured by
astronomical observation. The amount of this aberration depends upon the
velocity of light, and on the velocity with which the earth's motion is
performed. We can measure the velocity of light by independent
measurements, in the manner already explained in Chapter XII. We are
thus enabled to calculate what the velocity of the earth must be, for
there is only one particular velocity for the earth which, when combined
with the measured velocity of light, will give the measured value of
aberration. The velocity of the earth being thus ascertained, and the
length of the year being known, it is easy to find the circumference of
the earth's path, and therefore its radius; that is, the distance from
the earth to the sun.
Here is indeed a singular result, and one which shows how profoundly the
various phenomena of science are interwoven. We make experiments in our
laboratory, and find the velocity of light. We observe the fixed stars,
and measure the aberration. We combine these results, and deduce
therefrom the distance from the earth to the sun! Although this method
of finding the sun's distance is one of very great elegance, and admits
of a certain amount of precision, yet it cannot be relied upon as a
perfectly unimpeachable method of deducing the great constant. A perfect
method must be based on the operations of mere surveying, and ought not
to involve recondite physical considerations. We cannot, however, fail
to regard the discovery of aberration by Bradley as a most pleasing and
beautiful achievement, for it not only greatly improves the calculations
of practical astronomy, but links together several physical phenomena of
the greatest interest.