dx/dt = f'1(a, b, c, d, t) = x'
dy/dt = f'2(a, b, c, d, t) = y' (3)
The symbols x' and y' are used for brevity to mean the velocities
expressed by the differential coefficients. The arbitrary constants,
a, b, c and d, are the elements of the orbit, or any quantities from
which these elements can be obtained. We note that, in the actual
process of integration, no geometric construction need enter.
[Illustration: fig. 2.]
Let us next consider the problem in another form. Conceive that
instead of the orbit of the planet, there is given a position P (fig.
2), through which the planet passed at an assigned moment, with a
given velocity, and in a given direction, represented by the
arrowhead. Logically these data completely determine the orbit in
which the planet shall move, because there is only one such orbit
passing through P, a planet moving in which would have the given
speed. It follows that the elements of the orbit admit of
determination when the co-ordinates of the planet at an assigned
moment and their derivatives as to time are given. Analytically the
elements are determined from these data by solving the four equations
just given, regarding a, b, c and d as unknown quantities, and x, y,
x', y' and t as given quantities. The solution of these equations
would lead to expressions of the form
a = [phi]1(x, y, x', y', t)
b = [phi]2(x, y, x', y', t) (4)
&c. &c.
one for each of the elements.
The general equations expressing the motion of a planet considered as
a material particle round a centre of attraction lead to theorems the
more interesting of which will now be enunciated.
(1) The motion of such a planet may take place not only in an ellipse
but in any curve of the second order; an ellipse, hyperbola, or
parabola, the latter being the bounding curve between the other two. A
body moving in a parabola or hyperbola would recede indefinitely from
its centre of motion and never return to it. The ellipse is therefore
the only closed orbit.
(2) The motion takes place in accord with Kepler's laws, enunciated
elsewhere.
(3) _Whewell's theorem_: if a point R be taken at a distance from the
sun equal to the major axis of the orbit of a planet and, therefore,
at double the mean distance of the planet, the speed of the latter at
any point is equal to the speed which a body would acquire by falling
from the point R to the actual position of the planet. The speed of
the latter may, therefore, be expressed as a function of its radius
vector at the moment and of the major axis of its orbit without
introducing any other elements into the expression. Another corollary
is that in the case of a body moving in a parabolic orbit the velocity
at any moment is that which would be acquired by the body in falling
from an infinite distance to the place it occupies at the moment.
(4) If a number of bodies are projected from any point in space with
the same velocity, but in various directions, and subjected only to
the attraction of the sun, they will all return to the point of
projection at the same moment, although the orbits in which they move
may be ever so different.
(5) At each distance from the sun there is a certain velocity which a
body would have if it moved in a circular orbit at that distance. If
projected with this velocity in any direction the point of projection
will be at the end of the minor axis of the orbit, because this is the
only point of an ellipse of which the distance from the focus is equal
to the semi-major axis of the curve, and therefore the only point at
which the distance of the body from the sun is equal to its mean
distance.
(6) The relation between the periodic time of a planet and its mean
distance, approximately expressed by Kepler's third law, follows very
simply from the laws of centrifugal force. It is an elementary
principle of mechanics that this force varies directly as the product
of the distance of the moving body from the centre of motion into the
square of its angular velocity. When bodies revolve at different
distances around a centre, their velocities must be such that the
centrifugal force of each shall be balanced by the attraction of the
central mass, and therefore vary inversely as the square of the
distance. If M is the central mass, n the angular velocity, and a the
distance, the balance of the two forces is expressed by the equation
an^2 = M/a^2,
whence a^3n^2 = M, a constant.
The periodic time varying inversely as n, this equation expresses
Kepler's third law. This reasoning tacitly supposes the orbit to be a
circle of radius a, and the mass of the planet to be negligible. The
rigorous relation is expressed by a slight modification of the law.
Putting M and m for the respective masses of the sun and planet, a for
the semi-major axis of the orbit, and n for the mean angular motion in
unit of time, the relation then is
a^3n^2 = M + m.
What is noteworthy in this theorem is that this relation depends only
on the sum of the masses. It follows, therefore, that were any portion
of the mass of the sun taken from it, and added to the planet, the
relation would be unchanged. Kepler's third law therefore expresses
the fact that the mass of the sun is the same for all the planets, and
deviates from the truth only to the extent that the masses of the
latter differ from each other by quantities which are only a small
fraction of the mass of the sun.
_Problem of Three Bodies._--As soon as the general law of gravitation
was fully apprehended, it became evident that, owing to the attraction
of each planet upon all the others, the actual motion of the planets
must deviate from their motion in an ellipse according to Kepler's
laws. In the _Principia_ Newton made several investigations to
determine the effects of these actions; but the geometrical method
which he employed could lead only to rude approximations. When the
subject was taken up by the continental mathematicians, using the
analytical method, the question naturally arose whether the motions of
three bodies under their mutual attraction could not be determined
with a degree of rigour approximating to that with which Newton had
solved the problem of two bodies. Thus arose the celebrated "problem
of three bodies." Investigation soon showed that certain integrals
expressing relations between the motions not only of three but of any
number of bodies could be found. These were:--
First, the law of the conservation of the centre of gravity. This
expresses the general fact that whatever be the number of the bodies
which act upon each other, their motions are so related that the
centre of gravity of the entire system moves in a straight line with a
constant velocity. This is expressed in three equations, one for each
of the three rectangular co-ordinates.
Secondly, the law of conservation of areas. This is an extension of
Kepler's second law. Taking as the radius vector of each body the line
from the body to the common centre of gravity of all, the sum of the
products formed by multiplying each area described, by the mass of the
body, remains a constant. In the language of theoretical mechanics,
the moment of momentum of the entire system is a constant quantity.
This law is also expressed in three equations, one for each of the
three planes on which the areas are projected.
Thirdly, the entire _vis viva_ of the system or, as it is now called,
the energy, which is obtained by multiplying the mass of each body
into half the square of its velocity, is equal to the sum of the
quotients formed by dividing the product of every pair of the masses,
taken two and two, by their distance apart, with the addition of a
constant depending on the original conditions of the system. In the
language of algebra putting m1, m2, m3, &c. for the masses of the
bodies, r_1.2, r_1.3, r_2.3, &c. for their mutual distances apart;
v1, v2, v3, &c., for the velocities with which they are moving at any
moment; these quantities will continually satisfy the equation
m1m2 m1m3 m2m3
1/2(m1[v1]^2 + m2[v2]^2 + ...) = ----- + ----- + ----- + ... + a constant.
r_1.2 r_1.3 r_2.3
The theorems of motion just cited are expressed by seven integrals, or
equations expressing a law that certain functions of the variables and
of the time remain constant. It is remarkable that although the seven
integrals were found almost from the beginning of the investigation,
no others have since been added; and indeed it has recently been shown
that no others exist that can be expressed in an algebraic form. In
the case of three bodies these do not suffice completely to define the
motion. In this case, the problem can be attacked only by methods of
approximation, devised so as to meet the special conditions of each
case. The special conditions which obtain in the solar system are such
as to make the necessary approximation theoretically possible however
complex the process may be. These conditions are:--(1) The smallness
of the masses of the planets in comparison with that of the sun, in
consequence of which the orbit of each planet deviates but slightly
from an ellipse during any one revolution; (2) the fact that the
orbits of the planets are nearly circular, and the planes of their
orbits but slightly inclined to each other. The result of these
conditions is that all the quantities required admit of development in
series proceeding according to the powers of the eccentricities and
inclinations of the orbits, and the ratio of the masses of the several
planets to the mass of the sun.
_Perturbations of the Planets._--Kepler's laws do not completely
express the motion of a planet around a central body, except when no
force but the mutual attraction of the two bodies comes into play.
When one or more other bodies form a part of the system, their action
produces deviations from the elliptic motion, which are called
_perturbations_. The problem of determining the perturbations of the
heavenly bodies is perhaps the most complicated with which the
mathematical astronomer has to grapple; and the forms under which it
has to be studied are so numerous that they cannot be easily arranged
under any one head. But there is one conception of perturbations of
such generality and elegance that it forms the common base of all
those methods of determining these deviations which have high
scientific interest. This conception is embodied in the method of
"variation of elements," originally due to J.L. Lagrange. The simplest
method of presenting it starts with the second view of the elliptic
motion already set forth.
We have shown that, when the position of a planet and the direction
and speed of its motion at a certain instant are given, the elements
of the orbit can be determined. We have supposed this to be done at a
certain point P of the orbit, the direction and speed being expressed
by the variables x, y, x' and y'. Now, consider the values of these
same variables expressing the position of the planet at a second point
Q, and the speed with which it passes that point. With this position
and speed the elements of the orbit can again be determined. Since the
orbit is unchanged so long as no disturbing force acts, it follows
that the elements determined by means of the two sets of values of the
variables are in this case the same. In a word, although the position
and speed of the planet and the direction of its motion are constantly
changing, the values of the elements determined from these variables
remain constant. This fact is fully expressed by the equations (4)
where we have constants on one side of the equation equal to functions
of the variables on the other. Functions of the variables possessing
this property of remaining constant are termed _integrals_.
Now let the planet be subjected to any force additional to that of the
sun's attraction,--say to the attraction of another planet. To fix the
ideas let us suppose that the additional attraction is only an impulse
received at the moment of passing the point P. The first effect will
evidently be to change either the velocity or the direction in which
the planet is moving at the moment, or both. If, with the changed
velocity we again compute the elements they will be different from the
former elements. But, if the impulse is not repeated, these new
elements will again remain invariable. If repeated, the second impulse
will again change the elements, and so on indefinitely. It follows
that, if we go on computing the elements a, b, c, d from the actual
values of x, y, x' and y', at each moment when the planet is subject
to the attraction of another body, they will no longer be invariable,
but will slowly vary from day to day and year to year. These ever
varying elements represent an ever varying elliptic orbit,--not an
orbit which the planet actually describes through its whole course,
but an ideal one in which it is moving at each instant, and which
continually adjusts itself to the actual motion of the planet at the
instant. This is called the _osculating_ orbit.
The essential principle of Lagrange's elegant method consists in
determining the variations of this osculating ellipse, the
co-ordinates and velocities of the planet being ignored in the
determination. This may be done because, since the elements and
co-ordinates completely determine each other, we may concentrate our
attention on either, ignoring the other. The reason for taking the
elements as the variables is that they vary very slowly, a property
which facilitates their determination, since the variations may be
treated as small quantities, of which the squares and products may be
neglected in a first solution. In a second solution the squares and
products may be taken account of, and so on as far as necessary.
If the problem is viewed from a synthetic point of view, the stages of
its solution are as follows. We first conceive of the planets as
moving in invariable elliptic orbits, and thus obtain approximate
expressions for their positions at any moment. With these expressions
we express their mutual action, or their pull upon each other at any
and every moment. This pull determines the variations of the ideal
elements. Knowing these variations it becomes possible to represent by
integration the value of the elements as algebraic expressions
containing the time, and the elements with which we started. But the
variations thus determined will not be rigorously exact, because the
pull from which they arise has been determined on the supposition that
the planets are moving in unvarying orbits, whereas the actual pull
depends on the actual position of the planets. Another approximation
is, therefore, to be made, when necessary, by correcting the
expression of the pull through taking account of the variations of the
elements already determined, which will give a yet nearer
approximation to the truth. In theory these successive approximations
may be carried as far as we please, but in practice the labour of
executing each approximation is so great that we are obliged to stop
when the solution is so near the truth that the outstanding error is
less than that of the best observations. Even this degree of precision
may be impracticable in the more complex cases.
The results which are required to compare with observations are not
merely the elements, but the co-ordinates. When the varying elements
are known these are computed by the equations (2) because, from the
nature of the algebraic relations, the slowly varying elements are
continuously determined by the equations (4), which express the same
relations between the elements and the variables as do the equations
(2) and (3). This method is, therefore, in form at least, completely
rigorous. There are some cases in which it may be applied unchanged.
But commonly it proves to be extremely long and cumbrous, and
modifications have to be resorted to. Of these modifications the most
valuable is one conceived by P.A. Hansen. A certain mean elliptic
orbit, as near as possible to the actual varying orbit of the planet,
is taken. In this orbit a certain fictitious planet is supposed to
move according to the law of elliptic motion. Comparing the longitudes
of the actual and the fictitious planet the former will sometimes be
ahead of the latter and sometimes behind it. But in every case, if at
a certain time t, the actual planet has a certain longitude, it is
certain that at a very short interval dt before or after t, the
fictitious planet will have this same longitude. What Hansen's method
does is to determine a correction dt such that, being applied to the
actual time t, the longitude of the fictitious planet computed for the
time t + dt, will give the longitude of the true planet at the time t.
By a number of ingenious devices Hansen developed methods by which dt
could be determined. The computations are, as a general rule, simpler,
and the algebraic expressions less complex, than when the computations
of the longitude itself are calculated. Although the longitude of the
fictitious planet at the fictitious time is then equal to that of the
true planet at the true time, their radii vectores will not be
strictly equal. Hansen, therefore, shows how the radius vector is
corrected so as to give that of the true planet.
In all that precedes we have considered only two variables as
determining the position of the planet, the latter being supposed to
move in a plane. Although this is true when there are any number of
bodies moving in the same plane, the fact is that the planets move in
slightly different planes. Hence the position of the plane of the
orbit of each planet is continually changing in consequence of their
mutual action. The problem of determining the changes is, however,
simpler than others in perturbations. The method is again that of the
variation of elements. The position and velocity being given in all
three co-ordinates, a certain osculating plane is determined for each
instant in which the planet is moving at that instant. This plane
remains invariable so long as no third body acts; when it does act the
position of the plane changes very slowly, continually rotating round
the radius vector of the planet as an instantaneous axis of rotation.
_Secular and Periodic Variations._--When, following the preceding
method, the variations of the elements are expressed in terms of the
time, they are found to be of two classes, _periodic_ and _secular_.
The first depend on the mean longitudes of the planets, and always
tend back to their original values when the planets return to their
original positions in their orbits. The others are, at least through
long periods of time, continually progressive.
A luminous idea of the nature of these two classes of variation may be
gained by conceiving of the motion of a ship, floating on an ocean
affected by a long ground swell. In consequence of the swell, the ship
is continually pitching in a somewhat irregular way, the oscillations
up and down being sometimes great and sometimes small. An observer on
board of her would notice no motion except this. But, suppose the tide
to be rising. Then, by continued observation, extended over an hour or
more, it will be found that, in the general average, the ship is
gradually rising, so that two different kinds of motion are
superimposed on each other. The effect of the rising tide is in the
nature of a secular variation, while the pitching is periodic.
But the analogy does not end here. If the progressive rise of the ship
be watched for six hours or more, it will be found gradually to cease
and reverse its direction. That is to say, making abstraction of the
pitching, the ship is slowly rising and falling in a total period of
nearly twelve hours, while superimposed upon this slow motion is a
more rapid motion due to the waves. It is thus with the motions of the
planets going through their revolutions. Each orbit continually
changes its form and position, sometimes in one direction and
sometimes in another. But when these changes are averaged through
years and centuries it is found that the average orbit has a secular
variation which, for a number of centuries, may appear as a very slow
progressive change in one direction only. But when this change is more
fully investigated, it is found to be really periodic, so that after
thousands, tens of thousands, or hundreds of thousands of years, its
direction will be reversed and so on continually, like the rising and
falling tide. The orbits thus present themselves to us in the words of
a distinguished writer as "Great clocks of eternity which beat ages as
ours beat seconds."
The periodic variations can be represented algebraically as the
resultant of a series of harmonic motions in the following way: Let L
be an angle which is increasing uniformly with the time, and let n be
its rate of increase. We put L0 for its value at the moment from which
the time is reckoned. The general expression for the angle will then
be