_Pallas_ was the existence of two planets at nearly the same distance
from the sun, and apparently crossing the ecliptic in the same part of
the heavens, or having the same node. On account of this singularity,
Dr. Olbers was led to conjecture that Ceres and Pallas are only
fragments of a larger planet, which had formerly circulated at the same
distance, and been shattered by some internal convulsion. The hypothesis
suggested the probability that there might be other fragments, whose
orbits might be expected to cross the ecliptic at a common point, or to
have the same node. Dr. Olbers, therefore, proposed to examine
carefully, every month, the two opposite parts of the heavens in which
the orbits of Ceres and Pallas intersect one another, with a view to the
discovery of other planets, which might be sought for in those parts
with a greater chance of success, than in a wider zone, embracing the
entire limits of these orbits. Accordingly, in 1804, near one of the
nodes of Ceres and Pallas, a third planet was discovered. This was
called _Juno_, and the character ([Planet: Juno]) was adopted for its
symbol, representing the starry sceptre of the Queen of Olympus.
Pursuing the same researches, in 1807 a fourth planet was discovered, to
which was given the name of _Vesta_, and for its symbol the character
([Planet: Vesta]) was chosen,--an altar surmounted with a censer holding
the sacred fire.
The _average distance_ of these bodies from the sun is two hundred and
sixty-one millions of miles; and it is remarkable that their orbits are
very near together. Taking the distance of the earth from the sun for
unity, their respective distances are 2.77, 2.77, 2.67, 2.37. Their
_times_ of revolution around the sun are nearly equal, averaging about
four and a half years.
In respect to the _inclination of their orbits_ to the ecliptic, there
is also considerable diversity. The orbit of Vesta is inclined only
about seven degrees, while that of Pallas is more than thirty-four
degrees. They all, therefore, have a higher inclination than the orbits
of the old planets, and of course make excursions from the ecliptic
beyond the limits of the zodiac. Hence they have been called the
_ultra-zodiacal planets_. When first discovered, before their place in
the system was fully ascertained it was also proposed to call them
_asteroids_, a name implying that they were planets under the form of
stars. Their title, however, to take their rank among the primary
planets, is now generally conceded.
The _eccentricity of their orbits_ is also, in general, greater than
that of the old planets. You will recollect that this language denotes
that their orbits are more elliptical, or depart further from the
circular form. The eccentricities of the orbits of Pallas and Juno
exceed that of the orbit of Mercury. The asteroids differ so much,
however, in eccentricity, that their orbits may cross each other. The
orbits of the old planets are so nearly circular, and at such a great
distance apart, that there is no danger of their interfering with each
other. The earth, for example, when at its nearest distance from the
sun, will never come so near it as Venus is when at its greatest
distance, and therefore can never cross the orbit of Venus. But since
the average distance of Ceres and Pallas from the sun is about the same,
while the eccentricity of the orbit of Pallas is much greater than that
of Ceres, consequently, Pallas may come so near to the sun at its
perihelion, as to cross the orbit of Ceres.
The _small size_ of the asteroids constitutes one of their most
remarkable peculiarities. The difficulty of estimating the apparent
diameter of bodies at once so very small and so far off, would lead us
to expect different results in the actual estimates. Accordingly, while
Dr. Herschel estimates the diameter of Pallas at only eighty miles,
Schroeter places it as high as two thousand miles, or about the diameter
of the moon. The volume of Vesta is estimated at only one fifteen
thousandth part of the earth's, and her surface is only about equal to
that of the kingdom of Spain.
These little bodies are surrounded by _atmospheres_ of great extent,
some of which are uncommonly luminous, and others appear to consist of
nebulous matter, like that of comets. These planets shine with a more
vivid light than might be expected, from their great distance and
diminutive size; but a good telescope is essential for obtaining a
distinct view of their phenomena.
Although the great chasm which occurs between Mars and Jupiter,--a chasm
of more than three hundred millions of miles,--suggested long ago the
idea of other planetary bodies occupying that part of the solar system,
yet the discovery of the asteroids does not entirely satisfy expectation
since they are bodies so dissimilar to the other members of the series
in size, in appearance, and in the form and inclinations of their
orbits. Hence, Dr. Olbers, the discoverer of three of these bodies, held
that they were fragments of a single large planet, which once occupied
that place in the system, and which exploded in different directions by
some internal violence. Of the fragments thus projected into space, some
would be propelled in such directions and with such velocities, as,
under the force of projection and that of the solar attraction would
make them revolve in regular orbits around the sun. Others would be so
projected among the other bodies in the system, as to be thrown in very
irregular orbits, apparently wandering lawless through the skies. The
larger fragments would receive the least impetus from the explosive
force, and would therefore circulate in an orbit deviating less than any
other of the fragments from the original path of the large planet; while
the lesser fragments, being thrown off with greater velocity, would
revolve in orbits more eccentric, and more inclined to the ecliptic.
Dr. Brewster, editor of the 'Edinburgh Encyclopedia,' and the well-known
author of various philosophical works, espoused this hypothesis with
much zeal; and, after summing up the evidence in favor of it, he remarks
as follows: "These singular resemblances in the motions of the greater
fragments, and in those of the lesser fragments, and the striking
coincidences between theory and observation in the eccentricity of their
orbits, in their inclination to the ecliptic, in the position of their
nodes, and in the places of their perihelia, are phenomena which could
not possibly result from chance, and which concur to prove, with an
evidence amounting almost to demonstration, that the four new planets
have diverged from one common node, and have therefore composed a single
planet."
The same distinguished writer supposes that some of the smallest
fragments might even have come within reach of the earth's attraction,
and by the combined effects of their projectile forces and the
attraction of the earth, be made to revolve around this body as the
larger fragments are carried around the sun; and that these are in fact
the bodies which afford those _meteoric stones_ which are
occasionally precipitated to the earth. It is now a well-ascertained
fact, a fact which has been many times verified in our own country, that
large meteors, in the shape of fire-balls, traversing the atmosphere,
sometimes project to the earth masses of stony or ferruginous matter.
Such were the meteoric stones which fell at Weston, in Connecticut, in
1807, of which a full and interesting account was published, after a
minute examination of the facts, by Professors Silliman and Kingsley, of
Yale College. Various accounts of similar occurrences may be found in
different volumes of the American Journal of Science. It is for the
production of these wonderful phenomena that Dr. Brewster supposes the
explosion of the planet, which, according to Dr. Olbers, produced the
asteroids, accounts. Others, however, as Sir John Herschel, have been
disposed to ascribe very little weight to the doctrine of Olbers.
FOOTNOTES:
[13] Altissimum planetam tergeminum observavi. Or, as transposed,
Smaismrmilme poeta leumi bvne nugttaviras.
[14] In imitation of Galileo, Huyghens announced his discovery in this
form: a a a a a a a c c c c c d e e e e e g h i i i i i i i l l l l m m
n n n n n n n n n o o o o p p q r r s t t t t t u u u u u; which he
afterwards recomposed into this sentence: _Annulo cingitur, tenui,
plano, nusquam cohærente, ad eclipticam inclinato._
[15] Dick's 'Celestial Scenery.'
[16] Dick's 'Celestial Scenery.'
LETTER XXIV.
THE PLANETARY MOTIONS.----KEPLER'S LAWS.----KEPLER.
"God of the rolling orbs above!
Thy name is written clearly bright
In the warm day's unvarying blaze,
Or evening's golden shower of light;
For every fire that fronts the sun,
And every spark that walks alone
Around the utmost verge of heaven,
Was kindled at thy burning throne."--_Peabody._
IF we could stand upon the sun and view the planetary motions, they
would appear to us as simple as the motions of equestrians riding with
different degrees of speed around a large ring, of which we occupied the
centre. We should see all the planets coursing each other from west to
east, through the same great highway, (the Zodiac,) no one of them, with
the exception of the asteroids, deviating more than seven degrees from
the path pursued by the earth. Most of them, indeed, would always be
seen moving much nearer than that to the ecliptic. We should see the
planets moving on their way with various degrees of speed. Mercury would
make the entire circuit in about three months, hurrying on his course
with a speed about one third as great as that by which the moon revolves
around the earth. The most distant planets, on the other hand, move at
so slow a pace, that we should see Mercury, Venus, the Earth, and Mars,
severally overtaking them a great many times, before they had completed
their revolutions. But though the movements of some were comparatively
rapid, and of others extremely slow, yet they would not seem to differ
materially, in other respects: each would be making a steady and nearly
uniform march along the celestial vault.
Such would be the simple and harmonious motions of the planets, as they
would be seen from the sun, the centre of their motions; and such they
are, in fact. But two circumstances conspire to make them appear
exceedingly different from these, and vastly more complicated; one is,
that we view them out of the centre of their motions; the other, that we
are ourselves in motion. I have already explained to you the effect
which these two causes produce on the apparent motions of the inferior
planets, Mercury and Venus. Let us now see how they effect those of the
superior planets, Mars, Jupiter, Saturn, and Uranus.
Orreries, or machines intended to exhibit a model of the solar system,
are sometimes employed to give a popular view of the planetary motions;
but they oftener mislead than give correct ideas. They may assist
reflection, but they can never supply its place. The impossibility of
representing things in their just proportions will be evident, when we
reflect that, to do this, if in an orrery we make Mercury as large as a
cherry, we should have to represent the sun six feet in diameter. If we
preserve the same proportions, in regard to distance, we must place
Mercury two hundred and fifty feet, and Uranus twelve thousand five
hundred feet, or more than two miles from the sun. The mind of the
student of astronomy must, therefore, raise itself from such imperfect
representations of celestial phenomena, as are afforded by artificial
mechanism, and, transferring his contemplations to the celestial regions
themselves, he must conceive of the sun and planets as bodies that bear
an insignificant ratio to the immense spaces in which they circulate,
resembling more a few little birds flying in the open sky, than they do
the crowded machinery of an orrery.
The _real_ motions of the planets, indeed, or such as orreries usually
exhibit, are very easily conceived of, as before explained; but the
_apparent_ motions are, for the most part, entirely different from
these. The apparent motions of the inferior planets have been already
explained. You will recollect that Mercury and Venus move backwards and
forwards across the sun, the former never being seen further than
twenty-nine degrees, and the latter never more than about forty-seven
degrees, from that luminary; that, while passing from the greatest
elongation on one side, to the greatest elongation on the other side,
through the superior conjunction, the apparent motions of these planets
are accelerated by the motion of the earth; but that, while moving
through the inferior conjunction, at which time their motions are
retrograde, they are apparently retarded by the earth's motion. Let us
now see what are the apparent motions of the superior planets.
Let A, B, C, Fig. 62, page 294, represent the earth in different
positions in its orbit, M, a superior planet, as Mars, and N R, an arc
of the concave sphere of the heavens. First, suppose the planet to
remain at rest in M, and let us see what apparent motions it will
receive from the real motions of the earth. When the earth is at B, it
will see the planet in the heavens at N; and as the earth moves
successively through C, D, E, F, the planet will appear to move through
O, P, Q, R. B and F are the two points of greatest elongation of the
earth from the sun, as seen from the planet; hence, between these two
points, while passing through its orbit most remote from the planet,
(when the planet is seen in superior conjunction,) the earth, by its own
motion, gives an apparent motion to the planet in the order of the
signs; that is, the _apparent_ motion given by the _real_ motion of the
earth is _direct_. But in passing from F to B through A, when the planet
is seen in opposition, the apparent motion given to the planet by the
earth's motion is from R to N, and is therefore _retrograde_. As the arc
described by the earth, when the motion is direct, is much greater than
when the motion is retrograde, while the apparent arc of the heavens
described by the planet from N to R, in the one case, and from R to N,
in the other, is the same in both cases, the retrograde motion is much
swifter than the direct, being performed in much less time.
[Illustration Fig. 62.]
But the superior planets are not in fact at rest, as we have supposed,
but are all the while moving eastward, though with a slower motion than
the earth. Indeed, with respect to the remotest planets, as Saturn and
Uranus, the forward motion is so exceedingly slow, that the above
representation is nearly true for a single year. Still, the effect of
the real motions of all the superior planets, eastward, is to increase
the direct apparent motion communicated by the earth, and to diminish
the retrograde motion. This will be evident from inspecting the figure;
for if the planet _actually_ moves eastward while it is _apparently_
carried eastward by the earth's motion, the whole motion eastward will
be equal to the sum of the two; and if, while it is really moving
eastward, it is apparently carried westward still more by the earth's
motion, the retrograde movement will equal the difference of the two.
If Mars stood still while the earth went round the sun, then a second
opposition, as at A, would occur at the end of one year from the first;
but, while the earth is performing this circuit, Mars is also moving the
same way, more than half as fast; so that, when the earth returns to A,
the planet has already performed more than half the same circuit, and
will have completed its whole revolution before the earth comes up with
it. Indeed Mars, after having been seen once in opposition, does not
come into opposition again until after two years and fifty days. And
since the planet is then comparatively near to us, as at M, while the
earth is at A, and appears very large and bright, rising unexpectedly
about the time the sun sets, he surprises the world as though it were
some new celestial body. But on account of the slow progress of Saturn
and Uranus, we find, after having performed one circuit around the sun,
that they are but little advanced beyond where we left them at the last
opposition. The time between one opposition of Saturn and another is
only a year and thirteen days.
It appears, therefore, that the superior planets steadily pursue their
course around the sun, but that their apparent retrograde motion, when
in opposition, is occasioned by our passing by them with a swifter
motion, of which we are unconscious, like the apparent backward motion
of a vessel, when we overtake it and pass by it rapidly in a steam-boat.
Such are the real and the apparent motions of the planets. Let us now
turn our attention to the _laws of the planetary orbits_.
There are three great principles, according to which the motions of the
earth and all the planets around the sun are regulated, called KEPLER'S
LAWS, having been first discovered by the astronomer whose name they
bear. They may appear to you, at first, dry and obscure; yet they will
be easily understood from the explanations which follow; and so
important have they proved in astronomical inquiries, that they have
acquired for their renowned discoverer the appellation of the
'_Legislator of the Skies_.' We will consider each of these laws
separately; and, for the sake of rendering the explanation clear and
intelligible, I shall perhaps repeat some things that have been briefly
mentioned before.
[Illustration Fig. 63.]
FIRST LAW.--_The orbits of the earth and all the planets are ellipses,
having the sun in the common focus._ In a circle, all the diameters are
equal to one another; but if we take a metallic wire or hoop, and draw
it out on opposite sides, we elongate it into an ellipse, of which the
different diameters are very unequal. That which connects the points
most distant from each other is called the _transverse_, and that which
is at right angles to this is called the _conjugate_, axis. Thus, A B,
Fig. 63, is the transverse axis, and C D, the conjugate of the ellipse A
B C. By such a process of elongating the circle into an ellipse, the
centre of the circle may be conceived of as drawn opposite ways to E and
F, each of which becomes a _focus_, and both together are called the
_foci_ of the ellipse. The distance G E, or G F, of the focus from the
centre is called the _eccentricity_ of the ellipse; and the ellipse is
said to be more or less eccentric, as the distance of the focus from the
centre is greater or less. Figure 64 represents such a collection of
ellipses around the common focus F, the innermost, A G D, having a small
eccentricity, or varying little from a circle, while the outermost, A C
B, is an eccentric ellipse. The orbits of all the bodies that revolve
about the sun, both planets and comets, have, in like manner, a common
focus, in which the sun is situated, but they differ in eccentricity.
Most of the planets have orbits of very little eccentricity, differing
little from circles, but comets move in very eccentric ellipses. The
earth's path around the sun varies so little from a circle, that a
diagram representing it truly would scarcely be distinguished from a
perfect circle; yet, when the comparative distances of the sun from the
earth are taken at different seasons of the year, we find that the
difference between their greatest and least distances is no less than
three millions of miles.
[Illustration Fig. 64.]
SECOND LAW.--_The radius vector of the earth, or of any planet,
describes equal areas in equal times._ You will recollect that the
radius vector is a line drawn from the centre of the sun to a planet
revolving about the sun. This definition I have somewhere given you
before, and perhaps it may appear to you like needless repetition to
state it again. In a book designed for systematic instruction, where all
the articles are distinctly numbered, it is commonly sufficient to make
a reference back to the article where the point in question is
explained; but I think, in Letters like these, you will bear with a
little repetition, rather than be at the trouble of turning to the Index
and hunting up a definition long since given.
[Illustration Fig. 65. ]
In Figure 65, _E a_, _E b_, _E c_, &c., are successive representations
of the radius vector. Now, if a planet sets out from _a_, and travels
round the sun in the direction of _a b c_, it will move faster when
nearer the sun, as at _a_, than when more remote from it, as at _m_;
yet, if _a b_ and _m n_ be arcs described in equal times, then,
according to the foregoing law, the space _E a b_ will be equal to the
space _E m n_; and the same is true of all the other spaces described in
equal times. Although the figure _E a b_ is much shorter than _E m n_,
yet its greater breadth exactly counterbalances the greater length of
those figures which are described by the radius vector where it is
longer.
THIRD LAW.--_The squares of the periodical times are as the cubes of the
mean distances from the sun._ The periodical time of a body is the time
it takes to complete its orbit, in its revolution about the sun. Thus
the earth's periodic time is one year, and that of the planet Jupiter
about twelve years. As Jupiter takes so much longer time to travel round
the sun than the earth does, we might suspect that his orbit is larger
than that of the earth, and of course, that he is at a greater distance
from the sun; and our first thought might be, that he is probably twelve
times as far off; but Kepler discovered that the distance does not
increase as fast as the times increase, but that the planets which are
more distant from the sun actually move slower than those which are
nearer. After trying a great many proportions, he at length found that,
if we take the squares of the periodic times of two planets, the greater
square contains the less, just as often as the cube of the distance of
the greater contains that of the less. This fact is expressed by saying,
that the squares of the periodic times are to one another as the cubes
of the distances.
This law is of great use in determining the distance of the planets from
the sun. Suppose, for example, that we wish to find the distance of
Jupiter. We can easily determine, from observation, what is Jupiter's
periodical time, for we can actually see how long it takes for Jupiter,
after leaving a certain part of the heavens to come round to the same
part again. Let this period be twelve years. The earth's period is of
course one year; and the distance of the earth, as determined from the
sun's horizontal parallax, as already explained, is about ninety-five
millions of miles. Now, we have here three terms of a proportion to find
the fourth, and therefore the solution is merely a simple case of the
rule of three. Thus:--the square of 1 year : square of 12 years :: cube
of 95,000,000 : cube of Jupiter's distance. The three first terms being
known, we have only to multiply together the second and third and divide
by the first, to obtain the fourth term, which will give us the cube of
Jupiter's distance from the sun; and by extracting the cube root of this
sum, we obtain the distance itself. In the same manner we may obtain the
respective distances of all the other planets.
So truly is this a law of the solar system, that it holds good in
respect to the new planets, which have been discovered since Kepler's
time, as well as in the case of the old planets. It also holds good in
respect to comets, and to all bodies belonging to the solar system,
which revolve around the sun as their centre of motion. Hence, it is
justly regarded as one of the most interesting and important principles
yet developed in astronomy.
But who was this Kepler, that gained such an extraordinary insight into
the laws of the planetary system, as to be called the 'Legislator of the
Skies?' John Kepler was one of the most remarkable of the human race,
and I think I cannot gratify or instruct you more, than by occupying the
remainder of this Letter with some particulars of his history.
Kepler was a native of Germany. He was born in the Duchy of Wurtemberg,
in 1571. As Copernicus, Tycho Brahe, Galileo, Kepler, and Newton, are
names that are much associated in the history of astronomy, let us see
how they stood related to each other in point of time. Copernicus was
born in 1473; Tycho, in 1546; Galileo, in 1564; Kepler, in 1571; and
Newton, in 1642. Hence, Copernicus was seventy-three years before
Tycho, and Tycho ninety-six years before Newton. They all lived to an
advanced age, so that Tycho, Galileo, and Kepler, were contemporary for
many years; and Newton, as I mentioned in the sketch I gave you of his
life, was born the year that Galileo died.