Masses of Planets that have no Satellites determined from their
Perturbations—Masses of the others obtained from the Motions of their
Satellites—Masses of the Sun, the Earth, of Jupiter and of the Jovial
System—Mass of the Moon—Real Diameters of Planets, how obtained—Size
of Sun, Densities of the Heavenly Bodies—Formation of Astronomical
Tables—Requisite Data and Means of obtaining them.
THE masses of such planets as have no satellites are known by comparing
the inequalities they produce in the motions of the earth and of each
other, determined theoretically, with the same inequalities given by
observation; for the disturbing cause must necessarily be proportional
to the effect it produces. The masses of the satellites themselves may
also be compared with that of the sun by their perturbations. Thus, it
is found, from the comparison of a vast number of observations with La
Place’s theory of Jupiter’s satellites, that the mass of the sun is no
less than 65,000,000 times greater than the least of these moons. But,
as the quantities of matter in any two primary planets are directly as
the cubes of the mean distances at which their satellites revolve, and
inversely as the squares of their periodic times (N. 135), the mass of
the sun and of any planets which have satellites may be compared with
the mass of the earth. In this manner it is computed that the mass of
the sun is 354,936 times that of the earth; whence the great
perturbations of the moon, and the rapid motion of the perigee and nodes
of her orbit (N. 136). Even Jupiter, the largest of the planets, has
been found by Professor Airy to be 1047·871 times less than the sun;
and, indeed, the mass of the whole Jovial system is not more than the
1054·4th part of that of the sun. So that the mass of the satellites
bears a very small proportion to that of their primary. The mass of the
moon is determined from several sources—from her action on the
terrestrial equator, which occasions the nutation in the axis of
rotation; from her horizontal parallax; from an inequality she produces
in the sun’s longitude; and from her action on the tides. The three
first quantities, computed from theory and compared with their observed
values, give her mass respectively equal to the 1/71, 1/74·2, and
1/69·2, part of that of the earth, which do not differ much from each
other. Dr. Brinkley has found it to be 1/80 from the constant of lunar
nutation: but, from the moon’s action in raising the tides, her mass
appears to be about the 1/75 part of that of the earth—a value that
cannot differ much from the truth.
The apparent diameters of the sun, moon, and planets are determined by
measurement; therefore their real diameters may be compared with that of
the earth; for the real diameter of a planet is to the real diameter of
the earth, or 7926 miles, as the apparent diameter of the planet to the
apparent diameter of the earth as seen from the planet, that is, to
twice the parallax of the planet. According to Bessel, the mean apparent
diameter of the sun is 1923ʺ·64, and with the solar parallax 8ʺ·5776, it
will be found that the diameter of the sun is about 886,877 miles.
Therefore, if the centre of the sun were to coincide with the centre of
the earth, his volume would not only include the orbit of the moon, but
would extend nearly as far again; for the moon’s mean distance from the
earth is about sixty times the earth’s equatorial radius, or 238,793
miles: so that twice the distance of the moon is 477,586 miles, which
differs but little from the solar radius; his equatorial radius is
probably not much less than the major axis of the lunar orbit. The
diameter of the moon is only 2160 miles; and Jupiter’s diameter of
88,200 miles is very much less than that of the sun; the diameter of
Pallas does not much exceed 79 miles, so that an inhabitant of that
planet, in one of our steam carriages, might go round his world in a few
hours. The diameters of Lutetia and Atalanta are only 8 and 4 miles
respectively; but the whole of the 55 telescopic planets are so small,
that their united mass is probably not more than the fifth or sixth part
of that of the moon.
The densities of bodies are proportional to their masses, divided by
their volumes. Hence, if the sun and planets be assumed to be spheres,
their volumes will be as the cubes of their diameters. Now, the apparent
diameters of the sun and earth, at their mean distance, are 1923ʺ·6 and
17ʺ·1552, and the mass of the earth is the 354,936th part of that of the
sun taken as the unit. It follows, therefore, that the earth is four
times as dense as the sun. But the sun is so large that his attractive
force would cause bodies to fall through about 334·65 feet in a second.
Consequently, if he were habitable by human beings, they would be unable
to move, since their weight would be thirty times as great as it is
here. A man of moderate size would weigh about two tons at the surface
of the sun; whereas at the surface of some of the new planets he would
be so light that it would be impossible to stand steady, since he would
only weigh a few pounds. The mean density of the earth has been
determined by the following method. Since a comparison of the action of
two planets upon a third gives the ratio of the masses of these two
planets, it is clear that, if we can compare the effect of the whole
earth with the effect of any part of it, a comparison may be instituted
between the mass of the whole earth and the mass of that part of it. Now
a leaden ball was weighed against the earth by comparing the effects of
each upon a pendulum; the nearness of the smaller mass making it produce
a sensible effect as compared with that of the larger: for by the laws
of attraction the whole earth must be considered as collected in its
centre. By this method it has been found that the mean density of the
earth is 5·660 times greater than that of water at the temperature of
62° of Fahrenheit’s thermometer. The late Mr. Baily, whose accuracy as
an experimental philosopher is acknowledged, was unremittingly occupied
nearly four years in accomplishing this very important object. In order
to ascertain the mean density of the earth still more perfectly, Mr.
Airy made a series of experiments to compare the simultaneous
oscillations of two pendulums, one at the bottom of the Harton coal-pit,
1260 feet deep, in Northumberland, and the other on the surface of the
earth immediately above it. The oscillations of the pendulums were
compared with an astronomical clock at each station, and the time was
instantaneously transmitted from one to the other by a telegraphic wire.
The oscillations were observed for more than 100 hours continuously,
when it was found that the lower pendulum made 2-1/2 oscillations more
in 24 hours than the upper one. The experiment was repeated for the same
length of time with the same result; but on this occasion the upper
pendulum was taken to the bottom of the mine and the lower brought to
the surface. From the difference between the oscillations at the two
stations it appears that gravitation at the bottom of the mine exceeds
that at the surface by the 1/19190 part, and that the mean density of
the earth is 6·565, which is greater than that obtained by Mr. Baily by
·89. While employed on the trigonometrical survey of Scotland, Colonel
James determined the mean density of the earth to be 5·316, from a
deviation of the plumb-line amounting to 2ʺ, caused by the attraction of
Arthur’s Seat and the heights east of Edinburgh: it agrees more nearly
with the density found by Mr. Baily than with that deduced from Mr.
Airy’s experiments. All the planets and satellites appear to be of less
density than the earth. The motions of Jupiter’s satellites show that
his density increases towards his centre. Were his mass homogeneous, his
equatorial and polar axes would be in the ratio of 41 to 36, whereas
they are observed to be only as 41 to 38. The singular irregularities in
the form of Saturn, and the great compression of Mars, prove the
internal structure of these two planets to be very far from uniform.
Before entering on the theory of rotation, it may not be foreign to the
subject to give some idea of the methods of computing the places of the
planets, and of forming astronomical tables. Astronomy is now divided
into the three distinct departments of theory, observation, and
computation. Since the problem of the three bodies can only be solved by
approximation, the analytical astronomer determines the position of a
planet in space by a series of corrections. Its place in its circular
orbit is first found, then the addition or subtraction of the equation
of the centre (N. 48) to or from its mean place gives its position in
the ellipse. This again is corrected by the application of the principal
periodic inequalities. But, as these are determined for some particular
position of the three bodies, they require to be corrected to suit other
relative positions. This process is continued till the corrections
become less than the errors of observation, when it is obviously
unnecessary to carry the approximation further. The true latitude and
distance of the planet from the sun are obtained by methods similar to
those employed for the longitude.
As the earth revolves equably about its axis in 24 hours, at the rate of
15° in an hour, time becomes a measure of angular motion, and the
principal element in astronomy, where the object is to determine the
exact state of the heavens and the successive changes it undergoes in
all ages, past, present, and to come. Now, the longitude, latitude, and
distance of a planet from the sun are given in terms of the time, by
general analytical formulæ. These formulæ will consequently give the
exact place of the body in the heavens, for any time assumed at
pleasure, provided they can be reduced to numbers. But before the
calculator begins his task the observer must furnish the necessary data,
which are, obviously, the forms of the orbits, and their positions with
regard to the plane of the ecliptic (N. 57). It is therefore necessary
to determine by observation, for each planet, the length of the major
axis of its orbit, the excentricity, the inclination of the orbit to the
plane of the ecliptic, the longitudes of its perihelion and ascending
node at a given time, the periodic time of the planet, and its longitude
at any instant arbitrarily assumed, as an origin from whence all its
subsequent and antecedent longitudes are estimated. Each of these
quantities is determined from that position of the planet on which it
has most influence. For example, the sum of the greatest and least
distances of the planet from the sun is equal to the major axis of the
orbit, and their difference is equal to twice the excentricity. The
longitude of the planet, when at its least distance from the sun, is the
same with the longitude of the perihelion; the greatest latitude of the
planet is equal to the inclination of the orbit: the longitude of the
planet, when in the plane of the ecliptic in passing towards the north,
is the longitude of the ascending node, and the periodic time is the
interval between two consecutive passages of the planet through the same
node, a small correction being made for the precession of the node
during the revolution of the planet (N. 137). Notwithstanding the
excellence of instruments and the accuracy of modern observers,
unavoidable errors of observation can only be compensated by finding the
value of each element from the mean of a thousand, or even many
thousands of observations. For as it is probable that the errors are not
all in one direction, but that some are in excess and others in defect,
they will compensate each other when combined.
However, the values of the elements determined separately can only be
regarded as approximate, because they are so connected that the
estimation of any one independently will induce errors in the others.
The excentricity depends upon the longitude of the perihelion, the mean
motion depends upon the major axis, the longitude of the node upon the
inclination of the orbit, and _vice versâ_. Consequently, the place of a
planet computed with the approximate data will differ from its observed
place. Then the difficulty is to ascertain what elements are most in
fault, since the difference in question is the error of all; that is
obviated by finding the errors of some thousands of observations, and
combining them, so as to correct the elements simultaneously, and to
make the sum of the squares of the errors a minimum with regard to each
element (N. 138). The method of accomplishing this depends upon the
Theory of Probabilities; a subject fertile in most important results in
the various departments of science and of civil life, and quite
indispensable in the determination of astronomical data. A series of
observations continued for some years will give approximate values of
the secular and periodic inequalities, which must be corrected from time
to time, till theory and observation agree. And these again will give
values of the masses of the bodies forming the solar system, which are
important data in computing their motions. The periodic inequalities
derived from a great number of observations are employed for the
determination of the values of the masses till such time as the secular
inequalities shall be perfectly known, which will then give them with
all the necessary precision. When all these quantities are determined in
numbers, the longitude, latitude, and distance of the planet from the
sun are computed for stated intervals, and formed into tables, arranged
according to the time estimated from a given epoch, so that the place of
the body may be determined from them by inspection alone, at any instant
for perhaps a thousand years before and after that epoch. By this
tedious process, tables have been computed for all the great planets,
and several of the small, besides the moon and the satellites of
Jupiter. In the present state of astronomy the masses and elements of
the orbits are pretty well known, so that the tables only require to be
corrected from time to time as observations become more accurate. Those
containing the motions of Jupiter, Saturn, and Uranus have already been
twice constructed within the last thirty years, and the tables of
Jupiter and Saturn agree almost perfectly with modern observation. The
following prediction will be found in the sixth edition of this book,
published in the year 1842: “Those of Uranus, however, are already
defective, probably because the discovery of that planet in 1781 is too
recent to admit of much precision in the determination of its motions,
or that possibly it may be subject to disturbances from some unseen
planet revolving about the sun beyond the present boundaries of our
system. If, after a lapse of years, the tables formed from a combination
of numerous observations should be still inadequate to represent the
motions of Uranus, the discrepancies may reveal the existence, nay, even
the mass and orbit, of a body placed for ever beyond the sphere of
vision.”[5]
That prediction has been fulfilled since the seventh edition of this
book was published. Not only the existence of Neptune, revolving at the
distance of three thousand millions of miles from the sun, has been
discovered from his disturbing action on Uranus, but his mass, the form
and position of his orbit in space, and his periodic time had been
determined before the planet had been seen, and the planet itself was
discovered in the very point of the heavens which had been assigned to
it. It had been noticed for years that the perturbation of Uranus had
increased in an unaccountable manner (N. 139). After the disturbing
action of all the known planets had been determined, it was found that,
between the years 1833 and 1837, the observed and computed distance of
Uranus from the sun differed by 240,000 miles, which is about the mean
distance of the moon from the earth, while, in 1841, the error in the
geocentric longitude of the planet amounted to 96ʺ. These discrepancies
were therefore attributed to the attraction of some unseen and unknown
planet, consequently they gave rise to a case altogether unprecedented
in the history of astronomy. Heretofore it was required to determine the
disturbing action of one known planet upon another. Whereas the inverse
problem had now to be solved, in which it was required to find the place
of an unknown body in the heavens, at a given time, together with its
mass, and the form and position of its orbit, from the disturbance it
produced on the motions of another. The difficulty was extreme, because
all the elements of the orbit of Uranus were erroneous from the action
of Neptune, and those of Neptune’s orbit were unknown. In this dilemma
it was necessary to form some hypothesis with regard to the unknown
planet; it was therefore assumed, according to Bode’s empirical law on
the mean distances of the planets, that it was revolving at twice the
distance of Uranus from the sun. In fact, the periodic time of Uranus is
about 84 years, and, as the discrepancies in his motions increased
slowly and regularly, it was evident that it would require a planet with
a much longer periodic time to produce them—moreover, it was clear that
the new planet must be exterior to Uranus, otherwise it would have
disturbed the motions of Saturn.
Another circumstance tended to lessen the difficulty; the latitude of
Uranus was not much affected, therefore it was concluded that the
inclination of the orbit of the unknown body must be very small, and, as
that of the orbit of Uranus is only 46ʹ 28ʺ·4, both planets were assumed
to be moving in the plane of the ecliptic, and thus the elements of the
orbit of the unknown planet were reduced from six to four. Having thus
assumed that the unknown body was revolving in a circle in the plane of
the ecliptic, the analytical expression of its action on the motion of
Uranus, when in numerous points of its orbit, was compared with the
observed longitude of Uranus, through a regular series of years, by
means of which the faulty elements of the orbit of Uranus were
eliminated, or got rid of, and there only remained a relation between
the mass of the new planet and three of the elements of its orbit; and
it then was necessary to assume such a value for two of them as would
suit the rest. That was accomplished so dexterously, that the
perturbations of Uranus were perfectly conformable to the motions of
Neptune, moving in the orbit thus found, and the place of the new planet
exactly agreed with observation. Subsequently its orbit and motions have
been determined more accurately.
The honour of this admirable effort of genius is shared by Mr. Adams and
M. Le Verrier, who, independently of each other, arrived at these
wonderful results. Mr. Adams had determined the mass and apparent
diameter of Neptune, with all the circumstances of its motion, eight
months before M. Le Verrier had terminated his results, and had also
pointed out the exact spot where the planet would be found; but the
English observers neglected to look for it till M. Le Verrier made known
his researches, and communicated its position to Dr. Galle, at Berlin,
who found it the very first night he looked for it, and then it was
evident that it would have been seen in the place Mr. Adams had assigned
to it eight months before had it been looked for. So closely did the
results of these two great mathematicians agree.
Neptune has a diameter of 39,793 miles, consequently he is nearly 200
times larger than the earth, and may be seen with a telescope of
moderate power. His motion is retrograde at present, and six times
slower than that of the earth. At so great a distance from the sun it
can only have the 1/1300th part of the light and heat the earth
receives; but having a satellite, the deficiency of light may in some
measure be supplied.
The prediction may now be transferred from Uranus to Neptune, whose
perturbations may reveal the existence of a planet still further
removed, which may for ever remain beyond the reach of telescopic
vision—yet its mass, the form and position of its orbit, and all the
circumstances of its motion may become known, and the limits of the
solar system may still be extended hundreds of millions of miles.
The mean distance of Neptune from the sun has subsequently proved to be
only 2893 millions of miles, and the period of his revolution 166 years,
so that Baron Bode’s law, of the interval between the orbits of any two
planets being twice as great as the inferior interval and half of the
superior, fails in the case of Neptune, though it was useful on the
first approximation to his motions; and since Bode’s time it has led to
the discovery of fifty-five telescopic planets revolving between the
orbits of Mars and Jupiter, some by chance, others by a systematic
search on the faith that these minute planets are fragments of a larger
body that has exploded, because their distances from the sun are nearly
the same; the lines of the nodes of some of their orbits terminate in
the same points of the heavens, and the inclinations of their orbits are
such as might have taken place from their mutual disturbances at the
time of the explosion, and while yet they were near enough for their
forms to affect their motions. The orbits of the more recently
discovered asteroids show that this hypothesis is untenable.
The tables of Mars, Venus, and even those of the sun, have been greatly
improved, and still engage the attention of our Astronomer Royal, Mr.
Airy, and other eminent astronomers. We are chiefly indebted to the
German astronomers for tables of the four older telescopic planets,
Vesta, Juno, Ceres, and Pallas; the others have only been discovered
since the year 1845.
The determination of the path of a planet when disturbed by all the
others, a problem which has employed the talents of the greatest
astronomers, from Newton to the present day, is only successfully
accomplished with regard to the older planets, which revolve in nearly
circular orbits, but little inclined to the plane of the ecliptic. When
the excentricity and inclination of the orbits are great, their analysis
fails, because the series expressing the co-ordinates of the bodies
become extremely complicated, and do not converge when applied to comets
and the telescopic planets. This difficulty has been overcome by Sir
John Lubbock, and other mathematicians, who have the honour of having
completed the theory of planetary motion, which becomes every day of
more importance, from the new planets that have been discovered, and
also with regard to comets, many of which return to the sun at regular
intervals, and from whose perturbations the masses of the planets will
be more accurately determined, and the retarding influence of the
ethereal medium better known.