approximately double the distance of the inner adjacent planet from
the sun, but that the rate of increase is distinctly slower than
this for the inner ones; consequently a better approximation will
be obtained by adding a constant to each term of an appropriate
geometrical progression. Thus, form a doubling series like this,
1-1/2, 3, 6, 12, 24, &c. doubling each time; then add 4 to each,
and you get a series which expresses very fairly the relative
distances of the successive planets from the sun, except that the
number for Mercury is rather erroneous, and we now know that at the
other extreme the number for Neptune is erroneous too.
I have stated it in the notes above in a form calculated to give
the law every chance, and a form that was probably fashionable
after the discovery of Uranus; but to call the first term of the
doubling series 0 is evidently not quite fair, though it puts
Mercury's distance right. Neptune's distance, however, turns out to
be more nearly 30 times the earth's distance than 38.8. The others
are very nearly right: compare column D of the table preceding
Lecture III. on p. 57, with the numbers in the notes on p. 294.
The discovery of Uranus a few years afterwards, in 1781, at 19.2 times
the earth's distance from the sun, lent great _eclat_ to the law, and
seemed to establish its right to be regarded as at least a close
approximation to the truth.
The gap between Mars and Jupiter, which had often been noticed, and
which Kepler filled with a hypothetical planet too small to see, comes
into great prominence by this law of Bode. So much so, that towards the
end of last century an enthusiastic German, von Zach, after some search
himself for the expected planet, arranged a committee of observing
astronomers, or, as he termed it, a body of astronomical detective
police, to begin a systematic search for this missing subject of the
sun.
[Illustration: FIG. 90.--Planetary orbits to scale; showing the
Asteroidal region between Jupiter and Mars. (The orbits of satellites
are exaggerated.)]
In 1800 the preliminaries were settled: the heavens near the zodiac
were divided into twenty-four regions, each of which was intrusted to
one observer to be swept. Meanwhile, however, quite independently of
these arrangements in Germany, and entirely unknown to this committee, a
quiet astronomer in Sicily, Piazzi, was engaged in making a catalogue of
the stars. His attention was directed to a certain region in Taurus by
an error in a previous catalogue, which contained a star really
non-existent.
In the course of his scrutiny, on the 1st of January, 1801, he noticed a
small star which next evening appeared to have shifted. He watched it
anxiously for successive evenings, and by the 24th of January he was
quite sure he had got hold of some moving body, not a star: probably, he
thought, a comet. It was very small, only of the eighth magnitude; and
he wrote to two astronomers (one of them Bode himself) saying what he
had observed. He continued to observe till the 11th of February, when he
was attacked by illness and compelled to cease.
His letters did not reach their destination till the end of March.
Directly Bode opened his letter he jumped to the conclusion that this
must be the missing planet. But unfortunately he was unable to verify
the guess, for the object, whatever it was, had now got too near the sun
to be seen. It would not be likely to be out again before September, and
by that time it would be hopelessly lost again, and have just as much to
be rediscovered as if it had never been seen.
Mathematical astronomers tried to calculate a possible orbit for the
body from the observations of Piazzi, but the observed places were so
desperately few and close together. It was like having to determine a
curve from three points close together. Three observations ought to
serve,[27] but if they are taken with insufficient interval between
them it is extremely difficult to construct the whole circumstances of
the orbit from them. All the calculations gave different results, and
none were of the slightest use.
The difficulty as it turned out was most fortunate. It resulted in the
discovery of one of the greatest mathematicians, perhaps the greatest,
that Germany has ever produced--Gauss. He was then a young man of
twenty-five, eking out a living by tuition. He had invented but not
published several powerful mathematical methods (one of them now known
as "the method of least squares"), and he applied them to Piazzi's
observations. He was thus able to calculate an orbit, and to predict a
place where, by the end of the year, the planet should be visible. On
the 31st of December of that same year, very near the place predicted by
Gauss, von Zach rediscovered it, and Olbers discovered it also the next
evening. Piazzi called it Ceres, after the tutelary goddess of Sicily.
Its distance from the sun as determined by Gauss was 2.767 times the
earth's distance. Bode's law made it 2.8. It was undoubtedly the missing
planet. But it was only one hundred and fifty or two hundred miles in
diameter--the smallest heavenly body known at the time of its discovery.
It revolves the same way as other planets, but the plane of its orbit is
tilted 10 deg. to the plane of the ecliptic, which was an exceptionally
large amount.
Very soon, a more surprising discovery followed. Olbers, while searching
for Ceres, had carefully mapped the part of the heavens where it was
expected; and in March, 1802, he saw in this place a star he had not
previously noticed. In two hours he detected its motion, and in a month
he sent his observations to Gauss, who returned as answer the calculated
orbit. It was distant 2.67, like Ceres, and was a little smaller, but it
had a very excentric orbit: its plane being tilted 34-1/2 deg., an
extraordinary inclination. This was called Pallas.
Olbers at once surmised that these two planets were fragments of a
larger one, and kept an eager look out for other fragments.
In two years another was seen, in the course of charting the region of
the heavens traversed by Ceres and Pallas. It was smaller than either,
and was called Juno.
In 1807 the persevering search of Olbers resulted in the discovery of
another, with a very oblique orbit, which Gauss named Vesta. Vesta is
bigger than any of the others, being five hundred miles in diameter, and
shines like a star of the sixth magnitude. Gauss by this time had become
so practised in the difficult computations that he worked out the
complete orbit of Vesta within ten hours of receiving the observational
data from Olbers.
For many weary years Olbers kept up a patient and unremitting search for
more of these small bodies, or fragments of the large planet as he
thought them; but his patience went unrewarded, and he died in 1840
without seeing or knowing of any more. In 1845 another was found,
however, in Germany, and a few weeks later two others by Mr. Hind in
England. Since then there seems no end to them; numbers have been
discovered in America, where Professors Peters and Watson have made a
specialty of them, and have themselves found something like a hundred.
Vesta is the largest--its area being about the same as that of Central
Europe, without Russia or Spain--and the smallest known is about twenty
miles in diameter, or with a surface about the size of Kent. The whole
of them together do not nearly equal the earth in bulk.
The main interest of these bodies to us lies in the question, What is
their history? Can they have been once a single planet broken up? or are
they rather an abortive attempt at a planet never yet formed into one?
The question is not _entirely_ settled, but I can tell you which way
opinion strongly tends at the present time.
Imagine a shell travelling in an elliptic orbit round the earth to
suddenly explode: the centre of gravity of all its fragments would
continue moving along precisely the same path as had been traversed by
the centre of the shell before explosion, and would complete its orbit
quite undisturbed. Each fragment would describe an orbit of its own,
because it would be affected by a different initial velocity; but every
orbit would be a simple ellipse, and consequently every piece would in
time return through its starting-point--viz. the place at which the
explosion occurred. If the zone of asteroids had a common point through
which they all successively passed, they could be unhesitatingly
asserted to be the remains of an exploded planet. But they have nothing
of the kind; their orbits are scattered within a certain broad zone--a
zone everywhere as broad as the earth's distance from the sun,
92,000,000 miles--with no sort of law indicating an origin of this kind.
It must be admitted, however, that the fragments of our supposed shell
might in the course of ages, if left to themselves, mutually perturb
each other into a different arrangement of orbits from that with which
they began. But their perturbations would be very minute, and moreover,
on Laplace's theory, would only result in periodic changes, provided
each mass were rigid. It is probable that the asteroids were at one time
not rigid, and hence it is difficult to say what may have happened to
them; but there is not the least reason to believe that their present
arrangement is derivable in any way from an explosion, and it is certain
that an enormous time must have elapsed since such an event if it ever
occurred.
It is far more probable that they never constituted one body at all, but
are the remains of a cloudy ring thrown off by the solar system in
shrinking past that point: a small ring after the immense effort which
produced Jupiter and his satellites: a ring which has aggregated into a
multitude of little lumps instead of a few big ones. Such an event is
not unique in the solar system; there is a similar ring round Saturn.
At first sight, and to ordinary careful inspection, this differs from
the zone of asteroids in being a solid lump of matter, like a quoit. But
it is easy to show from the theory of gravitation, that a solid ring
could not possibly be stable, but would before long get precipitated
excentrically upon the body of the planet. Devices have been invented,
such as artfully distributed irregularities calculated to act as
satellites and maintain stability; but none of these things really work.
Nor will it do to imagine the rings fluid; they too would destroy each
other. The mechanical behaviour of a system of rings, on different
hypotheses as to their constitution, has been worked out with consummate
skill by Clerk Maxwell; who finds that the only possible constitution
for Saturn's assemblage of rings is a multitude of discrete particles
each pursuing its independent orbit. Saturn's ring is, in fact, a very
concentrated zone of minor asteroids, and there is every reason to
conclude that the origin of the solar asteroids cannot be very unlike
the origin of the Saturnian ones. The nebular hypothesis lends itself
readily to both.
The interlockings and motions of the particles in Saturn's rings are
most beautiful, and have been worked out and stated by Maxwell with
marvellous completeness. His paper constituted what is called "The Adams
Prize Essay" for 1856. Sir George Airy, one of the adjudicators
(recently Astronomer-Royal), characterized it as "one of the most
remarkable applications of mathematics to physics that I have ever
seen."
There are several distinct constituent rings in the entire Saturnian
zone, and each perturbs the other, with the result that they ripple and
pulse in concord. The waves thus formed absorb the effect of the mutual
perturbations, and prevent an accumulation which would be dangerous to
the persistence of the whole.
The only effect of gravitational perturbation and of collisions is
gradually to broaden out the whole ring, enlarging its outer and
diminishing its inner diameter. But if there were any frictional
resistance in the medium through which the rings spin, then other
effects would slowly occur, which ought to be looked for with interest.
So complete and intimate is the way Maxwell works out and describes the
whole circumstances of the motion of such an assemblage of particles,
and so cogent his argument as to the necessity that they must move
precisely so, and no otherwise, else the rings would not be stable, that
it was a Cambridge joke concerning him that he paid a visit to Saturn
one evening, and made his observations on the spot.
NOTES TO LECTURE XIV
The total number of stars in the heavens visible to a good eye is about
5,000. The total number at present seen by telescope is about
50,000,000. The number able to impress a photographic plate has not yet
been estimated; but it is enormously greater still. Of those which we
can see in these latitudes, about 14 are of the first magnitude, 48 of
the second, 152 of the third, 313 of the fourth, 854 of the fifth, and
2,010 of the sixth; total, 3,391.
The quickest-moving stars known are a double star of the sixth
magnitude, called 61 Cygni, and one of the seventh magnitude, called
Groombridge 1830. The velocity of the latter is 200 miles a second. The
nearest known stars are 61 Cygni and [alpha] Centauri. The distance
of these from us is about 400,000 times the distance of the sun. Their
parallax is accordingly half a second of arc. Sirius is more than a
million times further from us than our sun is, and twenty times as big;
many of the brightest stars are at more than double this distance. The
distance of Arcturus is too great to measure even now. Stellar parallax
was first securely detected in 1838, by Bessel, for 61 Cygni. Bessel was
born in 1784, and died in 1846, shortly before the discovery of Neptune.
The stars are suns, and are most likely surrounded by planets. One
planet belonging to Sirius has been discovered. It was predicted by
Bessel, its position calculated by Peters, and seen by Alvan Clark in
1862. Another predicted one, belonging to Procyon, has not yet been
seen.
A velocity of 5 miles a second could carry a projectile right round the
earth. A velocity of 7 miles a second would carry it away from the
earth, and round the sun. A velocity of 27 miles a second would carry a
projectile right out of the solar system never to return.
LECTURE XIV
BESSEL--THE DISTANCES OF THE STARS, AND THE DISCOVERY OF STELLAR PLANETS
We will now leave the solar system for a time, and hastily sketch the
history of stellar astronomy from the time of Sir William Herschel.
You remember how greatly Herschel had changed the aspect of the heavens
for man,--how he had found that none of the stars were really fixed, but
were moving in all manner of ways: some of this motion only apparent,
much of it real. Nevertheless, so enormously distant are they, that if
we could be transported back to the days of the old Chaldaean
astronomers, or to the days of Noah, we should still see the heavens
with precisely the same aspect as they wear now. Only by refined
apparatus could any change be discoverable in all those centuries. For
all practical purposes, therefore, the stars may still be well called
fixed.
Another thing one may notice, as showing their enormous distances, is
that from every planet of the solar system the aspect of the heavens
will be precisely the same. Inhabitants of Mars, or Jupiter, or Saturn,
or Uranus, will see exactly the same constellations as we do. The whole
dimensions of the solar system shrink up into a speck when so
contemplated. And from the stars none of the planetary orbs of our
system are visible at all; nothing but the sun is visible, and that
merely as a twinkling star, brighter than some, but fainter than many
others.
The sun and the stars are one. Try to realize this distinctly, and keep
it in mind. I find it often difficult to drive this idea home. After
some talk on the subject a friendly auditor will report, "the lecturer
then described the stars, including that greatest and most magnificent
of all stars, the sun." It would be difficult more completely to
misapprehend the entire statement. When I say the sun is one of the
stars, I mean one among the others; we are a long way from them, they
are a long way from each other. They need be no more closely packed
among each other than we are closely packed among them; except that some
of them are double or multiple, and we are not double.
It is highly desirable to acquire an intimate knowledge of the
constellations and a nodding acquaintance with their principal
stars. A description of their peculiarities is dull and
uninteresting unless they are at least familiar by name. A little
_viva voce_ help to begin with, supplemented by patient night
scrutiny with a celestial globe or star maps under a tent or shed,
is perhaps the easiest way: a very convenient instrument for the
purpose of learning the constellations is the form of map called a
"planisphere," because it can be made to show all the
constellations visible at a given time at a given date, and no
others. The Greek alphabet also is a thing that should be learnt by
everybody. The increased difficulty in teaching science owing to
the modern ignorance of even a smattering of Greek is becoming
grotesque. The stars are named from their ancient grouping into
constellations, and by the prefix of a Greek letter to the larger
ones, and of numerals to the smaller ones. The biggest of all have
special Arabic names as well. The brightest stars are called of
"the first magnitude," the next are of "the second magnitude," and
so on. But this arrangement into magnitudes has become technical
and precise, and intermediate or fractional magnitudes are
inserted. Those brighter than the ordinary first magnitude are
therefore now spoken of as of magnitude 1/2, for instance, or .6,
which is rather confusing. Small telescopic stars are often only
named by their numbers in some specified catalogue--a dull but
sufficient method.
Here is a list of the stars visible from these latitudes, which are
popularly considered as of the first magnitude. All of them should
be familiarly recognized in the heavens, whenever seen.
Star. Constellation.
Sirius Canis major
Procyon Canis minor
Rigel Orion
Betelgeux Orion
Castor Gemini
Pollux Gemini
Aldebaran Taurus
Arcturus Booetes
Vega Lyra
Capella Auriga
Regulus Leo
Altair Aquila
Fomalhaut Southern Fish
Spica Virgo
[alpha] Cygni is a little below the first magnitude. So,
perhaps, is Castor. In the southern heavens, Canopus and [alpha]
Centauri rank next after Sirius in brightness.
[Illustration: FIG. 91.--Diagram illustrating Parallax.]
The distances of the fixed stars had, we know, been a perennial problem,
and many had been the attempts to solve it. All the methods of any
precision have depended on the Copernican fact that the earth in June
was 184 million miles away from its position in December, and that
accordingly the grouping and aspect of the heavens should be somewhat
different when seen from so different a point of view. An apparent
change of this sort is called generally parallax; _the_ parallax of a
star being technically defined as the angle subtended at the star by the
radius of the earth's orbit: that is to say, the angle E[sigma]S;
where E is the earth, S the sun, and [sigma] a star (Fig. 91).
Plainly, the further off [sigma] is, the more nearly parallel will
the two lines to it become. And the difficulty of determining the
parallax was just this, that the more accurately the observations were
made, the more nearly parallel did those lines become. The angle was, in
fact, just as likely to turn out negative as positive--an absurd result,
of course, to be attributed to unavoidable very minute inaccuracies.
For a long time absolute methods of determining parallax were attempted;
for instance, by observing the position of the star with respect to the
zenith at different seasons of the year. And many of these
determinations appeared to result in success. Hooke fancied he had
measured a parallax for Vega in this way, amounting to 30" of arc.
Flamsteed obtained 40" for [gamma] Draconis. Roemer made a serious
attempt by comparing observations of Vega and Sirius, stars almost the
antipodes of each other in the celestial vault; hoping to detect some
effect due to the size of the earth's orbit, which should apparently
displace them with the season of the year. All these fancied results
however, were shown to be spurious, and their real cause assigned, by
the great discovery of the aberration of light by Bradley.
After this discovery it was possible to watch for still outstanding very
minute discrepancies; and so the problem of stellar parallax was
attacked with fresh vigour by Piazzi, by Brinkley, and by Struve. But
when results were obtained, they were traced after long discussion to
age and gradual wear of the instrument, or to some other minute
inaccuracy. The more carefully the observation was made, the more nearly
zero became the parallax--the more nearly infinite the distance of the
stars. The brightest stars were the ones commonly chosen for the
investigation, and Vega was a favourite, because, going near the zenith,
it was far removed from the fluctuating and tiresome disturbances of
atmospheric refraction. The reason bright stars were chosen was because
they were presumably nearer than the others; and indeed a rough guess at
their probable distance was made by supposing them to be of the same
size as the sun, and estimating their light in comparison with sunlight.
By this confessedly unsatisfactory method it had been estimated that
Sirius must be 140,000 times further away than the sun is, if he be
equally big. We now know that Sirius is much further off than this; and
accordingly that he is much brighter, perhaps sixty times as bright,
though not necessarily sixty times as big, as our sun. But even
supposing him of the same light-giving power as the sun, his parallax
was estimated as 1".8, a quantity very difficult to be sure of in any
absolute determination.
Relative methods were, however, also employed, and the advantages of one
of these (which seems to have been suggested by Galileo) so impressed
themselves upon William Herschel that he made a serious attempt to
compass the problem by its means. The method was to take two stars in
the same telescopic field and carefully to estimate their apparent
angular distance from each other at different seasons of the year. All
such disturbances as precession, aberration, nutation, refraction, and
the like, would affect them both equally, and could thus be eliminated.
If they were at the same distance from the solar system, relative
parallax would, indeed, also be eliminated; but if, as was probable,
they were at different distances, then they would apparently shift
relatively to one another, and the amount of shift, if it could be
observed, would measure, not indeed the distance of either from the
earth, but their distance from each other. And this at any rate would be
a step. It might be completed by similarly treating other stars in the
same field, taking them in pairs together. A bright and a faint star
would naturally be suitable, because their distances were likely to be
unequal; and so Herschel fixed upon a number of doublets which he knew
of, containing one bright and one faint component. For up to that time
it had been supposed that such grouping in occasional pairs or triplets
was chance coincidence, the two being optically foreshortened together,
but having no real connection or proximity. Herschel failed in what he
was looking for, but instead of that he discovered the real connection
of a number of these doublets, for he found that they were slowly
revolving round each other. There are a certain number of merely optical
or accidental doublets, but the majority of them are real pairs of suns
revolving round each other.
This relative method of mapping micrometrically a field of neighbouring
stars, and comparing their configuration now and six months hence, was,
however, the method ultimately destined to succeed; and it is, I
believe, the only method which has succeeded down to the present day.
Certainly it is the method regularly employed, at Dunsink, at the Cape
of Good Hope, and everywhere else where stellar parallax is part of the
work.