eye-piece, which are pierced by the light; an astronomer, applying his
eye to the tube and looking for the origin of the disturbance, sees it
apparently, but not in its real position--its apparent direction is
displaced in the direction of the telescope's motion; by an amount
depending on the ratio of the velocity of the earth to the velocity of
light, and on the angle between those two directions.
[Illustration: FIG. 78.--Aberration diagram. The light-ray L penetrates
the object-glass of the moving telescope at O, but does not reach the
eye-piece until the telescope has travelled to the second position.
Consequently a moving telescope does not point out the true direction of
the light, but aims at a point a little in advance.]
But how minute is the displacement! The greatest effect is obtained when
the two motions are at right angles to each other, _i.e._ when the star
seen is at right angles to the direction of the earth's motion, but even
then it is only 20", or 1/180th part of a degree; one-ninetieth of the
moon's apparent diameter. It could not be detected without a cross-wire
in the telescope, and would only appear as a slight displacement from
the centre of the field, supposing the telescope accurately pointed to
the true direction.
But if this explanation be true, it at once gives a method of
determining the velocity of light. The maximum angle of deviation,
represented as a ratio of arc / radius, amounts to
1 1
------------ - .0001 = ------
180 x 57-1/3 10,000
(a gradient of 1 foot in two miles). In other words, the velocity of
light must be 10,000 times as great as the velocity of the earth in its
orbit. This amounts to a speed of 190,000 miles a second--not so very
different from what Roemer had reckoned it in order to explain the
anomalies of Jupiter's first satellite.
Stars in the direction in which the earth was moving would not be thus
affected; there would be nothing in mere approach or recession to alter
direction or to make itself in any way visible. Stars at right angles to
the earth's line of motion would be most affected, and these would be
all displaced by the full amount of 20 seconds of arc. Stars in
intermediate directions would be displaced by intermediate amounts.
But the line of the earth's motion is approximately a circle round the
sun, hence the direction of its advance is constantly though slowly
changing, and in one year it goes through all the points of the compass.
The stars, being displaced always in the line of advance, must similarly
appear to describe little closed curves, always a quadrant in advance of
the earth, completing their orbits once a year. Those near the pole of
the ecliptic will describe circles, being always at right angles to the
motion. Those in the plane of the ecliptic (near the zodiac) will be
sometimes at right angles to the motion, but at other times will be
approached or receded from; hence these will oscillate like pendulums
once a year; and intermediate stars will have intermediate motions--that
is to say, will describe ellipses of varying excentricity, but all
completed in a year, and all with the major axis 20". This agreed very
closely with what was observed.
The main details were thus clearly and simply explained by the
hypothesis of a finite velocity for light, "the successive propagation
of light in time." This time there was no room for hesitation, and
astronomers hailed the discovery with enthusiasm.
Not yet, however, did Bradley rest. The finite velocity of light
explained the major part of the irregularities he had observed, but not
the whole. The more carefully he measured the amount of the deviation,
the less completely accurate became its explanation.
There clearly was a small outstanding error or discrepancy; the stars
were still subject to an unexplained displacement--not, indeed, a
displacement that repeated itself every year, but one that went through
a cycle of changes in a longer period.
The displacement was only about half that of aberration, and having a
longer period was rather more difficult to detect securely. But the
major difficulty was the fact that the two sorts of disturbances were
co-existent, and the skill of disentangling them, and exhibiting the
true and complete cause of each inequality, was very brilliant.
For nineteen years did Bradley observe this minor displacement, and in
that time he saw it go through a complete cycle. Its cause was now clear
to him; the nineteen-year period suggested the explanation. It is the
period in which the moon goes through all her changes--a period known to
the ancients as the lunar cycle, or Metonic cycle, and used by them to
predict eclipses. It is still used for the first rough approximation to
the prediction of eclipses, and to calculate Easter. The "Golden Number"
of the Prayer-book is the number of the year in this cycle.
The cause of the second inequality, or apparent periodic motion of the
stars, Bradley made out to be a nodding motion of the earth's axis.
The axis of the earth describes its precessional orbit or conical
motion every 26,000 years, as had long been known; but superposed upon
this great movement have now been detected minute nods, each with a
period of nineteen years.
The cause of the nodding is completely accounted for by the theory of
gravitation, just as the precession of the equinoxes was. Both
disturbances result from the attraction of the moon on the non-spherical
earth--on its protuberant equator.
"Nutation" is, in fact, a small perturbation of precession. The motion
may be observed in a non-sleeping top. The slow conical motion of the
top's slanting axis represents the course of precession. Sometimes this
path is loopy, and its little nods correspond to nutation.
The probable existence of some such perturbation had not escaped the
sagacity of Newton, and he mentions something about it in the
_Principia_, but thinks it too small to be detected by observation. He
was thinking, however, of a solar disturbance rather than a lunar one,
and this is certainly very small, though it, too, has now been observed.
Newton was dead before Bradley made these great discoveries, else he
would have been greatly pleased to hear of them.
These discoveries of aberration and nutation, says Delambre, the great
French historian of science, secure to their author a distinguished
place after Hipparchus and Kepler among the astronomers of all ages and
all countries.
NOTES TO LECTURE XI
_Lagrange_ and _Laplace_, both tremendous mathematicians, worked very
much in alliance, and completed Newton's work. The _Mecanique Celeste_
contains the higher intricacies of astronomy mathematically worked out
according to the theory of gravitation. They proved the solar system to
be stable; all its inequalities being periodic, not cumulative. And
Laplace suggested the "nebular hypothesis" concerning the origin of sun
and planets: a hypothesis previously suggested, and to some extent,
elaborated, by Kant.
A list of some of the principal astronomical researches of Lagrange and
Laplace:--Libration of the moon. Long inequality of Jupiter and Saturn.
Perturbations of Jupiter's satellites. Perturbations of comets.
Acceleration of the moon's mean motion. Improved lunar theory.
Improvements in the theory of the tides. Periodic changes in the form
and obliquity of the earth's orbit. Stability of the solar system
considered as an assemblage of rigid bodies subject to gravity.
The two equations which establish the stability of the solar system
are:--
_Sum (me^2[square root]d) = constant,_
and
_Sum (m tan^2[theta][square root]d) = constant;_
where _m_ is the mass of each planet, _d_ its mean distance from the
sun, _e_ the excentricity of its orbit, and [theta] the inclination
of its plane. However the expressions above formulated may change for
individual planets, the sum of them for all the planets remains
invariable.
The period of the variations in excentricity of the earth's orbit is
86,000 years; the period of conical revolution of the earth's axis is
25,800 years. About 18,000 years ago the excentricity was at a maximum.
LECTURE XI
LAGRANGE AND LAPLACE--THE STABILITY OF THE SOLAR SYSTEM, AND THE NEBULAR
HYPOTHESIS
Laplace was the son of a small farmer or peasant of Normandy. His
extraordinary ability was noticed by some wealthy neighbours, and by
them he was sent to a good school. From that time his career was one
brilliant success, until in the later years of his life his prominence
brought him tangibly into contact with the deteriorating influence of
politics. Perhaps one ought rather to say trying than deteriorating; for
they seem trying to a strong character, deteriorating to a weak one--and
unfortunately, Laplace must be classed in this latter category.
It has always been the custom in France for its high scientific men to
be conspicuous also in politics. It seems to be now becoming the fashion
in this country also, I regret to say.
The _life_ of Laplace is not specially interesting, and I shall not go
into it. His brilliant mathematical genius is unquestionable, and almost
unrivalled. He is, in fact, generally considered to come in this respect
next after Newton. His talents were of a more popular order than those
of Lagrange, and accordingly he acquired fame and rank, and rose to the
highest dignities. Nevertheless, as a man and a politician he hardly
commands our respect, and in time-serving adjustability he is comparable
to the redoubtable Vicar of Bray. His scientific insight and genius
were however unquestionably of the very highest order, and his work has
been invaluable to astronomy.
I will give a short sketch of some of his investigations, so far as they
can be made intelligible without overmuch labour. He worked very much in
conjunction with Lagrange, a more solid though a less brilliant man, and
it is both impossible and unnecessary for us to attempt to apportion
respective shares of credit between these two scientific giants, the
greatest scientific men that France ever produced.
First comes a research into the libration of the moon. This was
discovered by Galileo in his old age at Arcetri, just before his
blindness. The moon, as every one knows, keeps the same face to the
earth as it revolves round it. In other words, it does not rotate with
reference to the earth, though it does rotate with respect to outside
bodies. Its libration consists in a sort of oscillation, whereby it
shows us now a little more on one side, now a little more on the other,
so that altogether we are cognizant of more than one-half of its
surface--in fact, altogether of about three-fifths. It is a simple and
unimportant matter, easily explained.
The motion of the moon may be analyzed into a rotation about its
own axis combined with a revolution about the earth. The speed of
the rotation is quite uniform, the speed of the revolution is not
quite uniform, because the orbit is not circular but elliptical,
and the moon has to travel faster in perigee than in apogee (in
accordance with Kepler's second law). The consequence of this is
that we see a little too far round the body of the moon, first on
one side, then on the other. Hence it _appears_ to oscillate
slightly, like a lop-sided fly-wheel whose revolutions have been
allowed to die away so that they end in oscillations of small
amplitude.[23] Its axis of rotation, too, is not precisely
perpendicular to its plane of revolution, and therefore we
sometimes see a few hundred miles beyond its north pole, sometimes
a similar amount beyond its south. Lastly, there is a sort of
parallax effect, owing to the fact that we see the rising moon from
one point of view, and the setting moon from a point 8,000 miles
distant; and this base-line of the earth's diameter gives us again
some extra glimpses. This diurnal or parallactic libration is
really more effective than the other two in extending our vision
into the space-facing hemisphere of the moon.
These simple matters may as well be understood, but there is
nothing in them to dwell upon. The far side of the moon is probably
but little worth seeing. Its features are likely to be more blurred
with accumulations of meteoric dust than are those of our side, but
otherwise they are likely to be of the same general character.
The thing of real interest is the fact that the moon does turn the same
face towards us; _i.e._ has ceased to rotate with respect to the earth
(if ever it did so). The stability of this state of things was shown by
Lagrange to depend on the shape of the moon. It must be slightly
egg-shape, or prolate--extended in the direction of the earth; its
earth-pointing diameter being a few hundred feet longer than its visible
diameter; a cause slight enough, but nevertheless sufficient to maintain
stability, except under the action of a distinct disturbing cause. The
prolate or lemon-like shape is caused by the gravitative pull of the
earth, balanced by the centrifugal whirl. The two forces balance each
other as regards motion, but between them they have strained the moon a
trifle out of shape. The moon has yielded as if it were perfectly
plastic; in all probability it once was so.
It may be interesting to note for a moment the correlative effect of
this aspect of the moon, if we transfer ourselves to its surface in
imagination, and look at the earth (cf. Fig. 41). The earth would be
like a gigantic moon of four times our moon's diameter, and would go
through its phases in regular order. But it would not rise or set: it
would be fixed in the sky, and subject only to a minute oscillation to
and fro once a month, by reason of the "libration" we have been speaking
of. Its aspect, as seen by markings on its surface, would rapidly
change, going through a cycle in twenty-four hours; but its permanent
features would be usually masked by lawless accumulations of cloud,
mainly aggregated in rude belts parallel to the equator. And these
cloudy patches would be the most luminous, the whitest portions; for of
course it would be their silver lining that we would then be looking
on.[24]
Next among the investigations of Lagrange and Laplace we will mention
the long inequality of Jupiter and Saturn. Halley had found that Jupiter
was continually lagging behind its true place as given by the theory of
gravitation; and, on the other hand, that Saturn was being accelerated.
The lag on the part of Jupiter amounted to about 34-1/2 minutes in a
century. Overhauling ancient observations, however, Halley found signs
of the opposite state of things, for when he got far enough back Jupiter
was accelerated and Saturn was being retarded.
Here was evidently a case of planetary perturbation, and Laplace and
Lagrange undertook the working of it out. They attacked it as a case of
the problem of three bodies, viz. the sun, Jupiter, and Saturn; which
are so enormously the biggest of the known bodies in the system that
insignificant masses like the Earth, Mars, and the rest, may be wholly
neglected. They succeeded brilliantly, after a long and complex
investigation: succeeded, not in solving the problem of the three
bodies, but, by considering their mutual action as perturbations
superposed on each other, in explaining the most conspicuous of the
observed anomalies of their motion, and in laying the foundation of a
general planetary theory.
[Illustration: FIG. 79.--Shewing the three conjunction places in the
orbits of Jupiter and Saturn. The two planets are represented as leaving
one of the conjunctions where Jupiter was being pulled back and Saturn
being pulled forward by their mutual attraction.]
One of the facts that plays a large part in the result was known to
the old astrologers, viz. that Jupiter and Saturn come into
conjunction with a certain triangular symmetry; the whole scheme
being called a trigon, and being mentioned several times by Kepler.
It happens that five of Jupiter's years very nearly equal two of
Saturn's,[25] so that they get very nearly into conjunction three
times in every five Jupiter years, but not exactly. The result of
this close approach is that periodically one pulls the other on and
is itself pulled back; but since the three points progress, it is
not always the same planet which gets pulled back. The complete
theory shows that in the year 1560 there was no marked
perturbation: before that it was in one direction, while afterwards
it was in the other direction, and the period of the whole cycle of
disturbances is 929 of our years. The solution of this long
outstanding puzzle by the theory of gravitation was hailed with the
greatest enthusiasm by astronomers, and it established the fame of
the two French mathematicians.
Next they attacked the complicated problem of the motions of Jupiter's
satellites. They succeeded in obtaining a theory of their motions which
represented fact very nearly indeed, and they detected the following
curious relationship between the satellites:--The speed of the first
satellite + twice the speed of the second is equal to the speed of the
third.
They found this, not empirically, after the manner of Kepler, but as a
deduction from the law of gravitation; for they go on to show that even
if the satellites had not started with this relation they would sooner
or later, by mutual perturbation, get themselves into it. One singular
consequence of this, and of another quite similar connection between
their positions, is that all three satellites can never be eclipsed at
once.
The motion of the fourth satellite is less tractable; it does not so
readily form an easy system with the others.
After these great successes the two astronomers naturally proceeded to
study the mutual perturbations of all other bodies in the solar system.
And one very remarkable discovery they made concerning the earth and
moon, an account of which will be interesting, though the details and
processes of calculation are quite beyond us in a course like this.
Astronomical theory had become so nearly perfect by this time, and
observations so accurate, that it was possible to calculate many
astronomical events forwards or backwards, over even a thousand years or
more, with admirable precision.
Now, Halley had studied some records of ancient eclipses, and had
calculated back by means of the lunar theory to see whether the
calculation of the time they ought to occur would agree with the record
of the time they did occur. To his surprise he found a discrepancy, not
a large one, but still one quite noticeable. To state it as we know it
now:--An eclipse a century ago happened twelve seconds later than it
ought to have happened by theory; two centuries back the error amounted
to forty-eight seconds, in three centuries it would be 108 seconds, and
so on; the lag depending on the square of the time. By research, and
help from scholars, he succeeded in obtaining the records of some very
ancient eclipses indeed. One in Egypt towards the end of the tenth
century A.D.; another in 201 A.D.; another a little before Christ; and
one, the oldest of all of which any authentic record has been preserved,
observed by the Chaldaean astronomers in Babylon in the reign of
Hezekiah.
Calculating back to this splendid old record of a solar eclipse, over
the intervening 2,400 years, the calculated and the observed times were
found to disagree by nearly two hours. Pondering over an explanation of
the discrepancy, Halley guessed that it must be because the moon's
motion was not uniform, it must be going quicker and quicker, gaining
twelve seconds each century on its previous gain--a discovery announced
by him as "the acceleration of the moon's mean motion." The month was
constantly getting shorter.
What was the physical cause of this acceleration according to the theory
of gravitation? Many attacked the question, but all failed. This was the
problem Laplace set himself to work out. A singular and beautiful result
rewarded his efforts.
You know that the earth describes an elliptic orbit round the sun: and
that an ellipse is a circle with a certain amount of flattening or
"excentricity."[26] Well, Laplace found that the excentricity of the
earth's orbit must be changing, getting slightly less; and that this
change of excentricity would have an effect upon the length of the
month. It would make the moon go quicker.
One can almost see how it comes about. A decrease in excentricity means
an increase in mean distance of the earth from the sun. This means to
the moon a less solar perturbation. Now one effect of the solar
perturbation is to keep the moon's orbit extra large: if the size of its
orbit diminishes, its velocity must increase, according to Kepler's
third law.
Laplace calculated the amount of acceleration so resulting, and found it
ten seconds a century; very nearly what observation required; for,
though I have quoted observation as demanding twelve seconds per
century, the facts were not then so distinctly and definitely
ascertained.
This calculation for a long time seemed thoroughly satisfactory, but it
is not the last word on the subject. Quite lately an error has been
found in the working, which diminishes the theoretical
gravitation-acceleration to six seconds a century instead of ten, thus
making it insufficient to agree exactly with fact. The theory of
gravitation leaves an outstanding error. (The point is now almost
thoroughly understood, and we shall return to it in Lecture XVIII).
But another question arises out of this discussion. I have spoken of the
excentricity of the earth's orbit as decreasing. Was it always
decreasing? and if so, how far back was it so excentric that at
perihelion the earth passed quite near the sun? If it ever did thus pass
near the sun, the inference is manifest--the earth must at one time have
been thrown off, or been separated off, from the sun.
If a projectile could be fired so fast that it described an orbit round
the earth--and the speed of fire to attain this lies between five and
seven miles a second (not less than the one, nor more than the
other)--it would ever afterwards pass through its point of projection
as one point of its elliptic orbit; and its periodic return through that
point would be the sign of its origin. Similarly, if a satellite does
_not_ come near its central orb, and can be shown never to have been
near it, the natural inference is that it has _not_ been born from it,
but has originated in some other way.
The question which presented itself in connexion with the variable
ellipticity of the earth's orbit was the following:--Had it always been
decreasing, so that once it was excentric enough just to graze the sun
at perihelion as a projected body would do?
Into the problem thus presented Lagrange threw himself, and he succeeded
in showing that no such explanation of the origin of the earth is
possible. The excentricity of the orbit, though now decreasing, was not
always decreasing; ages ago it was increasing: it passes through
periodic changes. Eighteen thousand years ago its excentricity was a
maximum; since then it has been diminishing, and will continue to
diminish for 25,000 years more, when it will be an almost perfect
circle; it will then begin to increase again, and so on. The obliquity
of the ecliptic is also changing periodically, but not greatly: the
change is less than three degrees.
This research has, or ought to have, the most transcendent interest for
geologists and geographers. You know that geologists find traces of
extraordinary variations of temperature on the surface of the earth.
England was at one time tropical, at another time glacial. Far away
north, in Spitzbergen, evidence of the luxuriant vegetation of past ages
has been found; and the explanation of these great climatic changes has
long been a puzzle. Does not the secular variation in excentricity of
the earth's orbit, combined with the precession of the equinoxes, afford
a key? And if a key at all, it will be an accurate key, and enable us to
calculate back with some precision to the date of the glacial epoch;
and again to the time when a tropical flora flourished in what is now
northern Europe, _i.e._ to the date of the Carboniferous era.