THE EFFECT OF COMMENSURATE PERIODS
The Asteroids and Saturn’s Rings
Ever inquiring, ever fertile, his mind turned to seek the explanation of
divers astronomical phenomena. In 1912, for example, under the title
“Precession and the Pyramids,” we find him discussing in the _Popular
Science Monthly_ the pyramid of Cheops as an astronomical observatory,
with its relation to the position of the star then nearest to the North
Pole, its lines of light and shadow, in a great gallery constructed with
the object of recording the exact changes in the seasons.
But leaving aside these lesser interests, and the unbroken systematic
observation of the planets, his attention in the later years of his life
was chiefly occupied by two subjects, not unconnected, but which may be
described separately. They are, first, the influence over each other’s
position and orbits of two bodies, both revolving about a far larger
one; and, second, the search for an outer planet beyond the path of
Neptune. Each of these studies involved the use of mathematics with
expanding series of equations which no one had better attempt to follow
unless he is fresh and fluent in such forms of expression. For accurate
and quantitative results they are absolutely essential, but an
impression of what he was striving to do may be given without them.
Two bodies revolving about a common centre at different distances, and
therefore different rates of revolution, will sometimes be on the same
side of the central body, and thus nearer together; sometimes on
opposite sides, when they will be much farther apart. Now it is clear
that the attraction of gravity, being inversely as the square of the
distance, will be greatest when they are nearest together; and if this
happens at the same point in their orbits every time they approach each
other the effect will be cumulative, and in the aggregate much larger
than if they approach at different parts of their orbits and hence pull
each other sometimes in one direction and sometimes in another. To use a
homely, and not altogether apt, illustration: If a man, starting from
his front door, walk every day across his front lawn in the same track
he will soon make a beaten path and wear the grass away. If, instead, he
walk by this path only every other day and on the alternate days by
another, he will make two paths, neither of which will be so much worn.
If he walk by three tracks in succession the paths will be still less
worn; and if he never walk twice in the same place the effect on the
grass will be imperceptible.
Now, if the period taken by the outer body to complete its orbit be just
twice as long as that taken by the inner, they will not come close
together again until the outer one has gone round once to the inner
one’s twice, and they will always approach at the same point in their
orbits. Hence the effects on each other will be greatest. If the outer
one take just two turns while the inner takes three they will approach
again only at the same point, but less frequently; so that the pull will
be always the same, but repeated less often. This will be clearly true
whenever the rates of the revolution differ by unity: _e.g._, 1 to 2, 2
to 3, 3 to 4, 4 to 5, etc.
Take another case where the periods differ by two; for example, where
the inner body revolves about the central one three times while the
outer one does so once; in that case the inner one will catch up with
the outer when the latter has completed half a revolution and the inner
one and a half; and again when the outer has completed one whole
revolution and the inner three. In this case there will be two strong
pulls on opposite sides of the orbits, and, as these pulls are not the
same, the total effect will be less than if there were only one pull in
one direction. This is true whenever the periods of revolution differ by
two, _e.g._, 1 to 3, 3 to 5, 5 to 7. If the periods differ by three the
two bodies will approach three times,—once at the starting point, then
one third way round, and again two thirds way round, before they reach
the starting point; three different pulls clearly less effective.
In cases like these, where the two bodies approach in only a limited
number of places in their orbits the two periods of revolution are
called commensurate, because their ratio is expressed by a simple
fraction. The effect is greater as the number of such places in the
orbit is less, and as the number of revolutions before they approach is
less. But it is clearly greater than when the two bodies approach always
at different places in their orbits, never again where they have done so
before. This is when the two periods are incommensurate, so that their
ratio cannot be expressed by any vulgar fraction. One other point must
be noticed. The commensurate orbit, and hence the distance from the Sun,
and the period of revolution, of the smaller and therefore most affected
body, may not be far from a distance where the orbits would be
incommensurate. To take the most completely incommensurate ratio known
to science, that of the diameter of a circle to the circumference, which
has been carried out to seven hundred decimal places without repetition
of the figures. This is expressed by the decimal fraction .314159 etc.
and yet this differs from the simple commensurate 1/3 or .333333 etc. by
only about five per cent.; so that a smaller body may have to be pulled
by the larger, only a very short way before it reaches a point where it
will be seriously affected no more.
The idea that commensurateness affects the mutual attraction of bodies,
and hence the perturbations in their orbits, especially of the smaller
one, was not new; but Percival carried it farther, and to a greater
degree of accuracy, by observation, by mathematics and in its
applications. The most obvious example of its effects lay in the
influence of Jupiter upon the distribution of the asteroids, that almost
innumerable collection of small bodies revolving about the Sun between
the orbits of Jupiter and Mars, of which some six hundred had been
discovered. These are so small, compared with Jupiter, that, not only
individually but in the aggregate, their influence upon it may be
disregarded, and only its effect upon them be considered. In its
immediate neighborhood the commensurate periods, Percival points out,
come so close together (100 to 101, 99 to 100, etc.) that although
occasions of approach would be infrequent they would be enough in time
to disturb any bodies so near, until the planet had cleared out
everything in its vicinity that did not, by revolving around it, become
its own satellite.
Farther off Jupiter’s commensurate zones are less frequent, but where
they occur the fragments revolving about the Sun would be so perturbed
by the attraction of the planet as to be displaced, mainly, as Percival
points out, to the sunward side. This has made gaps bare of such
fragments, and between them incommensurate spaces where they could move
freely in their solar orbits. Here they might have gathered in a nucleus
and, collecting other fragments to it, form a small planet, were it not
that the gaps were frequent enough to prevent nuclei of sufficient size
arising anywhere. Thus the asteroids remained a host of little bodies
revolving about the Sun, with gaps in their ranks—as he puts it “embryos
of planets destined never to be born.”
The upper diagram in the plate opposite page 166 shows the distribution
and relative densities of the asteroids, with the gaps at the
commensurate points. The plate is taken from his “Memoir on Saturn’s
Rings,”[35] and brings us to another study of commensurate periods with
quite a different set of bodies obeying the same law. Indeed, among the
planets observed at Flagstaff not the least interesting was Saturn, and
its greatest peculiarity was its rings.
In Bulletin No. 32 of the Observatory (Nov. 24, 1907) Percival had
written: “Laplace first showed that the rings could not be, as they
appear, wide solid rings inasmuch as the strains due to the differing
attraction of Saturn for the several parts must disrupt them. Peirce
then proved that even a series of very narrow solid rings could not
subsist and that the rings must be fluid. Finally Clerk-Maxwell showed
that even this was not enough and that the rings to be stable must be
made up of discrete particles, a swarm of meteorites in fact. But, if my
memory serves me right, Clerk-Maxwell himself pointed out that even such
a system could not eternally endure but was bound eventually to be
forced both out and in, a part falling upon the surface of the planet, a
part going to form a satellite farther away.
“Even before this Edward Roche in 1848 had shown that the rings must be
composed of discrete particles, mere dust and ashes. He drew this
conclusion from his investigations on the minimum distance at which a
fluid satellite could revolve around its primary without being disrupted
by tidal strains.
“The dissolution which Clerk-Maxwell foresaw can easily be proved to be
inevitable if the particles composing the swarm are not at considerable
distances from one another, which is certainly not the case with the
rings as witnessed by the light they send us even allowing for their
comminuted form. For a swarm of particles thus revolving round a primary
are in stable equilibrium _only in the absence of collisions_. Now in a
crowded company collisions due either to the mutual pulls of the
particles or to the perturbations of the satellites must occur. At each
collision although the moment of momentum remains the same, energy is
lost unless the bodies be perfectly elastic, a condition not found in
nature, the lost energy being converted into heat. In consequence some
particles will be forced in toward the planet while others are driven
out and eventually the ring system disappears.
“Now the interest of the observations at Flagstaff consists in their
showing us this disintegration in process of taking place and
furthermore in a way that brings before us an interesting case of
celestial mechanics.”
He examines the rings mathematically, as the result of perturbations
caused by the two nearest of the planet’s satellites, Mimas and
Enceladus.
The effect is the same that occurs in the case of Jupiter and the
asteroids, Saturn taking the place of the Sun, his satellites that of
Jupiter, and the rings that of the asteroids. In spite of repetition it
may be well to state in his own words the principle of commensurate
periods and its application to the rings:[36]
“The same thing can be seen geometrically by considering that the two
bodies have their greatest perturbing effect on one another when in
conjunction and that if the periods of the two be commensurate they will
come to conjunction over and over in these same points of the orbit and
thus the disturbance produced by one on the other be cumulative. If the
periods are not commensurate the conjunctions will take place in ever
shifting positions and a certain compensation be effected in the
outstanding results. In proportion as the ratio of periods is simple
will the perturbation be potent. Thus with the ratio 1:2 the two bodies
will approach closest only at one spot and always there until the
perturbations induced themselves destroy the commensurability of period.
With 1:3 they will approach at two different spots recurrently; with 1:4
at three, and so on....
“We see, then, that perturbations, which in this case will result in
collisions, must be greatest on those particles which have periods
commensurate with those of the satellites. But inasmuch as there are
many particles in any cross-section of the ring there must be a
component of motion in any collision tending to throw the colliding
particles out of the plane of the ring, either above or below it.
“Considering, now, those points where commensurability exists between
the periods of particle and satellite we find these in the order of
their potency:
With Mimas, 1:2
1:3
1:4
With Enceladus, 1:3
2:3 of Mimas and 1:2; 2:3 of Enceladus falling outside the ring system.
1:2 of Mimas and 1:3 of Enceladus fall in Cassini’s division, which
separates ring A from ring B.... 1:3 of Mimas’ period falls at the
boundary of ring B and ring C at 1:50 radii of Saturn from the centre.”
In the following years this supposition was reinforced by the discovery
of six new divisions in the rings. Three of them were in ring A and
three in ring B, two of them in each case seen by Percival for the first
time. This led to very careful measurements of Saturn’s ball and rings
in 1913-14 and again in 1915; recorded in Bulletins 66 and 68 of the
Observatory. Careful allowance was made for irradiation, and the results
checked by having two sets of measurements, one made by Percival, the
other by Mr. E. C. Slipher. The observations were, of course, made when
the rings were so tilted to the Earth as to show very widely, the tilt
on March 21, 1915, showing them at their widest for fifteen years.
But unfortunately, as it seemed, the divisions in the rings did not come
quite where the commensurate ratios with the two nearest satellites
should place them. They came in the right order and nearly where they
ought to be, but always a little farther from Saturn. It occurred to
Percival that this might be due to an error in the calculation of the
motion of the rings, that if the attraction of Saturn were slightly more
than had been supposed the revolutions of all parts of the rings would
be slightly faster, and the places in them where the periods would be
commensurate with the satellites would be slightly farther out, that is
where the divisions actually occur. Everyone knows that the earth is not
a perfect sphere but slightly elliptical, or oblate, contracted from
pole to pole and enlarged at the equator; and the same is even more true
of Saturn on account of its greater velocity of rotation. Now its
attraction on bodies as near it as the rings, and to a less extent on
its satellites, is a little greater than it would be if it were a
perfect uniform sphere; and it would be greater still if it were not
uniform throughout, but composed of layers increasing in density, in
rapidity of rotation, and hence in oblateness, toward the centre.
Percival made, therefore, a highly intricate calculation on what the
attraction of such a body would be (“Observatory Memoir on Saturn’s
Rings,” Sept. 7, 1915), and found that it accounted almost exactly for
the discrepancy between the points of computed commensurateness and the
observed divisions in the rings. Such a constitution of Saturn is by no
means improbable in view of its still fluid condition and the process of
contraction that it is undergoing. He found it noteworthy that a study
of the perturbations of the rings by the satellites should bring to
light the invisible constitution of the planet itself:
“Small discrepancies are often big with meaning. Just as the more
accurate determination of the nitrogen content of the air led Sir
William Ramsay to the discovery of argon; so these residuals between the
computed and the observed features of _Saturn’s_ rings seem to lead to a
new conception of _Saturn’s_ internal constitution. That the mere
position of his rings should reveal something within him which we cannot
see may well appear as singular as it is significant.” (p. 5); and he
concludes: (pp. 20-22).
“All this indicates that _Saturn_ has not yet settled down to a uniform
rotation. Not only in the spots we see is the rate different for
different spots but from this investigation it would appear that the
speed of its spin increases as one sinks from surface to centre.[37]
“The subject of this memoir is of course two-fold: first, the observed
discrepancy, and second, the theory to account for it. The former
demands explanation and the latter seems the only way to satisfy it.
From the positions of the divisions in its rings we are thus led to
believe that _Saturn_ is actually rotating in layers with different
velocities, the inside ones turning the faster. If these layers were two
only, or substantially two, this would result in _Saturn’s_ being
composed of a very oblate kernel surrounded by a less oblate husk of
cloud.”
[Illustration: ASTEROIDS and SATURN’S RINGS]
MAJOR AXES
DISTANCES AT WHICH PERIODS ARE COMMENSURATE WITH THAT OF JUPITER
1/4 1/3 3/8 2/5 3/7 4/9 5/11 1/2 3/5 2/3
DISTANCES AT WHICH PERIODS ARE COMMENSURATE WITH THOSE OF MIMAS AND
ENCELADUS
1/4 1/3 1/4E. 3/8 2/5 3/7 4/9 5/11 1/2 2/5E. 3/5 3/7E.
The divisions so made in Saturn’s rings by its satellites may be seen in
the lower of the two diagrams opposite; the three fractions followed by
an E indicating the divisions caused by Enceladus, the rest those caused
by Mimas. The upper diagram represents, as already remarked, the similar
effects by Jupiter on the asteroids. A slight inspection shows their
coincidence.