Latitudinal Co-ordinate; Latitude or Ecliptic Polar Distance.
Longitudinal " Longitude.
_Relation of the Diurnal Motion to Spherical Co-ordinates._--The
vertical line at any place being the fundamental axis of the apparent
system of co-ordinates, this system rotates with the earth, and so
seems to us as fixed. The other two systems, including the vernal
equinox, are fixed on the celestial sphere, and so seem to us to
perform a diurnal revolution from east towards west. Regarding the
period of the revolution as 24 hours, the apparent motion goes on at
the rate of 15 deg. per hour. Here we have to make a distinction of
fundamental importance between the diurnal motions of the sun and of
the stars. Owing to the unceasing apparent motion of the sun toward
the east, the interval between two passages of the same star over the
meridian is nearly four minutes less than the interval between
consecutive passages of the sun. The latter is the measure of the day
as used in civil life. In astronomical practice is introduced a day,
termed "sidereal," determined, not by the diurnal revolution of the
sun, but of the stars. The year, which comprises 365.25 solar days,
contains 366.25 sidereal days. The latter are divided into sidereal
hours, minutes and seconds as the solar day is. The conception of a
revolution through 360 deg. in 24 hours is applicable to each case.
The sun apparently moves at the rate of 15 deg. in a solar hour; the
stars at the rate of 15 deg. in a sidereal hour. The latter motion
leads to the use, in astronomical practice, of time instead of angle,
as the unit in which the right ascensions are to be expressed.
Considering the position of the vernal equinox, and also of a star on
the celestial sphere, it will be seen that the interval between the
transits of these two points across the meridian may be used to
measure the right ascension of a star, since the latter amounts to 15
deg. for every sidereal hour of this interval. For example, if the
right ascension of a star is exactly 15 deg., it will pass the
meridian one sidereal hour after the vernal equinox. For the relations
thus arising, and their practical applications, see TIME, MEASUREMENT
OF.
_Theoretical Astronomy._
Theoretical Astronomy is that branch of the science which, making use of
the results of astronomical observations as they are supplied by the
practical astronomer, investigates the motions of the heavenly bodies.
In its most important features it is an offshoot of celestial mechanics,
between which and theoretical astronomy no sharp dividing line can be
drawn. While it is true that the one is concerned altogether with
general theories, it is also true that these theories require
developments and modifications to apply them to the numberless problems
of astronomy, which we may place in either class.
Among the problems of theoretical astronomy we may assign the first
place to the determination of orbits (q.v.), which is auxiliary to the
prediction of the apparent motions of a planet, satellite or star. The
computations involved in the process, while simple in some cases, are
extremely complex in others. The orbit of a newly-discovered planet or
comet may be computed from three complete observations by well-known
methods in a single day. From the resulting elements of the orbit the
positions of the body from day to day may be computed and tabulated in
an ephemeris for the use of observers. But when definitive results as
to the orbits are required, it is necessary to compute the
perturbations produced by such of the major planets as have affected
the motions of the body. With this complicated process is associated
that of combining numerous observations with a view of obtaining the
best definitive result. Speaking in a general way, we may say that
computations pertaining to the orbital revolutions of double stars, as
well as the bodies of our solar system, are to a greater or less
extent of the classes we have described. The principal modification is
that, up to the present time, stellar astronomy has not advanced so
far that a computation of the perturbations in each case of a system
of stars is either necessary or possible, except in exceptional cases.
_Celestial Mechanics_.
Celestial Mechanics is, strictly speaking, that branch of applied
mathematics which, by deductive processes, derives the laws of motion of
the heavenly bodies from their gravitation towards each other, or from
the mutual action of the parts which form them. The science had its
origin in the demonstration by Sir Isaac Newton that Kepler's three laws
of planetary motion, and the law of gravitation, in the case of two
bodies, could be mutually derived from each other. A body can move round
the sun in an elliptic orbit having the sun in its focus, and describing
equal areas in equal times, only under the influence of a force directed
towards the sun, and varying inversely as the square of the distance
from it. Conversely, assuming this law of attraction, it can be shown
that the planets will move according to Kepler's laws.
Thus celestial mechanics may be said to have begun with Newton's
_Principia_. The development of the science by the successors of Newton,
especially Laplace and Lagrange, may be classed among the most striking
achievements of the human intellect. The precision with which the path
of an eclipse is laid down years in advance cannot but imbue the minds
of men with a high sense of the perfection reached by astronomical
theories; and the discovery, by purely mathematical processes, of the
changes which the orbits and motions of the planets are to undergo
through future ages is more impressive the more fully one apprehends the
nature of the problem. The purpose of the present article is to convey a
general idea of the methods by which the results of celestial mechanics
are reached, without entering into those technical details which can be
followed only by a trained mathematician. It must be admitted that any
intelligent comprehension of the subject requires at least a grasp of
the fundamental conceptions of analytical geometry and the infinitesimal
calculus, such as only one with some training in these subjects can be
expected to have. This being assumed, the hope of the writer is that the
exposition will afford the student an insight into the theory which may
facilitate his orientation, and convey to the general reader with a
certain amount of mathematical training a clear idea of the methods by
which conclusions relating to it are drawn. The non-mathematical reader
may possibly be able to gain some general idea, though vague, of the
significance of the subject.
The fundamental hypothesis of the science assumes a system of bodies
in motion, of which the sun and planets may be taken as examples, and
of which each separate body is attracted toward all the others
according to the law of Newton. The motion of each body is then
expressed in the first place by Newton's three laws of motion (see
MOTION, LAWS OF, and MECHANICS). The first step in the process shows
in a striking way the perfection of the analytic method. The
conception of force is, so to speak, eliminated from the conditions of
the problem, which is reduced to one of pure kinematics. At the
outset, the position of each body, considered as a material particle,
is defined by reference to a system of co-ordinate axes, and not by
any verbal description. Differential equations which express the
changes of the co-ordinates are then constructed. The process of
discovering the laws of motion of the particle then consists in the
integration of these equations. Such equations can be formed for a
system of any number of bodies, but the process of integration in a
rigorous form is possible only to a limited extent or in special
cases.
The problems to be treated are of two classes. In one, the bodies are
regarded as material particles, no account being taken of their
dimensions. The earth, for example, may be regarded as a particle
attracted by another more massive particle, the sun. In the other
class of problems, the relative motion of the different parts of the
separate bodies is considered; for example, the rotation of the earth
on its axis, and the consequences of the fact that those parts of a
body which are nearer to another body are more strongly attracted by
it. Beginning with the first branch of the subject, the fundamental
ideas which it is our purpose to convey are embodied in the simple
case of only two bodies, which we may call the sun and a planet. In
this case the two bodies really revolve round their common centre of
gravity; but a very slight modification of the equations of motion
reduces them to the relative motion of the planet round the sun,
regarding the moving centre of the latter as the origin of
co-ordinates. The motion of this centre, which arises from the
attraction of the planet on the sun, need not be considered.
In the actual problems of celestial mechanics three co-ordinates
necessarily enter, leading to three differential equations and six
equations of solution. But the general principles of the problem are
completely exemplified with only two bodies, in which case the motion
takes place in a fixed plane. By taking this plane, which is that of
the orbit in which the planet performs its revolution, as the plane of
xy, we have only two co-ordinates to consider. Let us use the
following notation:
x, y, the co-ordinates of the planet relative to the sun as the origin.
M, m, the masses of the attracting bodies, sun and planet.
r, the distance apart of the two bodies, or the radius vector of m
relative to M. This last quantity is analytically defined by the
equation--
r^2 = x^2 + y^2
t, the time, reckoned from any epoch we choose.
The differential equations which completely determine the changes in
the co-ordinates x and y, or the motion of m relative to M, are:--
d^2x (M + m)x
---- = - --------
dt^2 r^3
d^2y (M + m)y
---- = - --------
dt^2 r^3
These formulae are worthy of special attention. They are the
expression in the language of mathematics of Newton's first two laws
of motion. Their statement in this language may be regarded as
perfect, because it completely and unambiguously expresses the naked
phenomena of the motion. The equations do this without expressing any
conception, such as that of force, not associated with the actual
phenomena. Moreover, as a third advantage, these expressions are
entirely free from those difficulties and ambiguities which are met
with in every attempt to express the laws of motion in ordinary
language. They afford yet another great advantage in that the
derivation of the results requires only the analytic operations of the
infinitesimal calculus.
The power and spirit of the analytic method will be appreciated by
showing how it expresses the relations of motion as they were
conceived geometrically by Newton and Kepler. It is quite evident that
Kepler's laws do not in themselves enable us to determine the actual
motion of the planets. We must have, in addition, in the case of each
special planet, certain specific facts, viz. the axes and eccentricity
of the ellipse, and the position of the plane in which it lies.
Besides these, we must have given the position of the planet in the
orbit at some specified moment. Having these data, the position of the
planet at any other time may be geometrically constructed by Kepler's
laws. The third law enables us to compute the time taken by the radius
vector to sweep over the entire area of the orbit, which is identical
with the time of revolution. The problem of constructing successive
radii vectores, the angles of which are measured off from the radius
vector of the body at the original given position, is then a geometric
one, known as Kepler's problem.
In the analytic process these specific data, called elements of the
orbit, appear as arbitrary constants, introduced by the process of
integration. In a case like the present one, where there are two
differential equations of the second order, there will be four such
constants. The result of the integration is that the co-ordinates x
and y and their derivatives as to the time, which express the
position, direction of motion and speed of the planet at any moment,
are found as functions of the four constants and of the time. Putting
a, b, c, d,
for the constants, the general form of the solution will be
x = f1(a, b, c, d, t)
y = f2(a, b, c, d, t) (2)