(8)

CHAPTER II.

MOTION RELATIVE TO THE EARTH.

9. When a number of particles are in motion under their mutual attractions or other forces, and the motion relatively to one of them is required, we must bring that one to rest, and then keep it at rest without altering the relative motions of the others with respect to it.

Now, firstly, the chosen particle will be brought to rest by giving it at any instant a velocity equal and opposite to that which it has at that instant; secondly, it will be kept at rest by applying to it accelerating forces equal and opposite to those which act upon it.

Therefore give the same velocity and apply the same accelerating forces to all the bodies of the system, and the absolute motions about the chosen body, which is now at rest, will be the same as their relative motions previously.

Problem of Two Bodies.

10. As the sun disturbs the moon's motion with respect to the earth, it is important to know what that motion would have been if no disturbance had existed, or generally :—

Two bodies attracting one another with forces varying directly as the mass and inversely as the square of the distance, to determine the orbit of one relatively to the other.

Let M, M' be the masses of the bodies, r the distance between them at any time t, M' being the body whose motion relatively to M is required.

M

The accelerating force of M on M' equals acting towards

TWO BODIES.

M'

9

M, while that of M' on M equals in the opposite direction. Therefore, by the principle above stated, we must apply to both M and M' accelerating forces equal and opposite to this latter force, and M' will move about M fixed, the accelerating force

e and a being constants to be determined by the circumstances of the motion at any given time.

This is the equation to a conic section referred to its focus,

the eccentricity being e, the semi-latus rectum

made by the apse line with the prime radius a.

h2

μ

and the angle

In the relative motion of the moon, or in that of the sun about the earth, the orbit would, as observation informs us, be an ellipse with small eccentricity, that of the moon being about and that of the sun.

-

11. The angle a between the radius vector and the apse line is called the true anomaly.

If n is the angular velocity of a radius vector which moving uniformly would accomplish its revolution in the same time as the true one, both passing through the apse at the same instant, then nt + ε a is called the mean anomaly, a being a constant depending on the instant when the body is at the apse, its value being also equal to the angle between the prime radius and the uniformly revolving one when t

= 0.

Thus, if MY (fig. 3) be the fixed line or prime radius,

A the apse,

M' the moving body at time t,

Mu the uniformly revolving radius at same time, the

direction of motion being represented by the arrow.

Let MD be the position of Mμ when t = 0,

then

TMD = ε and is called the epoch,*
DMμnt,

ΥΜΑ =α= longitude of the apse;

therefore, mean anomaly = AMμ = nt + ε

true anomaly = AMM' = YMM' — YMA = 0 — a.

12. To express the mean anomaly in terms of the true in a series ascending according to the powers of e, as far as e*.

h

n

=

h

1

{1 + e cos(0 − a)}”

(1 − e")* {1 + e cos(0 − a)} ̃o

—1⁄2 (1 − fe3) {1 − 2e cos(0 − a) + 3ea cos”(0 − a)}

n

1

— — (1 − 2e cos(0 − a) + §e2 cos2 (0 − a)} ;

=3

n

-

therefore nt = 0 2e sin(0-a) + že2 sin2 (0-a),

or (nt+e-a) = (0 − a) — 2e sin (0 − a) + že2 sin2 (0 — a),

13. To express the true anomaly in terms of the mean to the same order of approximation.

0 - a = nt + e − a + 2e sin (0 — a) — že sin 2 (0 - a) ... (1);

.. 0 a = nt + - a first approximation.

*The introduction of the epoch is avoided in the Lunar theory by a particular assumption (Art. 34); but in the Planetary it forms one of the elements of the orbit.

THREE BODIES.

Substituting this in the first small terms of (1), we get

11

0 — a = nt + e − a + 2e sin (nt + e − a)...a second approximation.

Substitute the second approximation in that small term of (1) which is multiplied by e, and the first approximation in that multiplied by e2; the result will be correct to that term, and gives

Ꮎ - a = nt + ε - a + 2e sin {nt + e − a + 2e sin (nt + e − a)}

=

= nt + ɛ − a + 2e sin (nt + e − a)

e sin 2(nt +ε-a)

+4e*cos(nt + e − a) sin(nt + e − a) — şe2 sin 2 (nt + e − a)

= nt + e − a + 2e sin (nt + e − a) + že2 sin2 (nt + e − a),

the required relation.

The development could be carried on by the same process to any power of e, but in what follows we shall not require it further than e2.

Problem of Three Bodies.

14. In order to fix the position of the moon with respect to the centre of the earth, which, by means of the process described in Art. (9), is reduced to and kept at rest, we must have some determinate invariable plane passing through the earth's centre to which the motion may be referred.

The plane which passes through the earth's centre and the direction of the sun's motion at any instant is called the true ecliptic; and as a first rough result of calculation, obtained on supposition of the sun and earth being the only bodies in the universe, this plane, in which, according to the last section, an elliptic orbit would be described, is a fixed plane: but this is no longer the case when we take into account the disturbances produced by the moon and planets, and it becomes necessary to substitute some other plane of reference unaffected by these disturbances.

Theory teaches us that such a plane exists, but as its determination can only be the work of time, the following theorem will supply us with a plane whose motion is extremely slow, and it may for a long period and to a degree of approximation far beyond that to which we shall carry our investigations, be considered as fixed and coinciding with its position at present.

15. The centre of gravity of the earth and moon describes relatively to the sun an orbit very nearly in one plane and elliptic; the square of the ratio of the distances of the moon and sun from the earth being neglected.†

S

M

E

G

Let S, E, M be the centres of the sun, earth, and moon, G the centre of gravity of the two last. Now the motion of G is the same as if the whole mass E+ M were collected there and acted on by forces equal and parallel to the moving forces which act on E and M. The whole force on G is therefore in the plane SEM; join SG.

* See Poinsot, “Théorie et détermination de l'équateur du système solaire,” where he proves that an invariable plane exists for the solar system, that is, a plane whose position relatively to the fixed stars will always be the same whatever changes the orbits of the planets may experience; but as its position depends on the moments of inertia of the sun, planets, and satellites, and therefore on their internal conformation, it cannot be determined à priori, and ages must elapse before observation can furnish sufficient data for doing so à posteriori.

This result Poinsot obtains on the supposition that the solar system is a free system; but it is possible, as he furthermore remarks, nay probable, that the stars exert some action upon it, it follows that this invariable plane may itself be variable, though the change must, according to our ideas of time and space, be indefinitely slow and small.

This ratio is about and, as we shall see Art. (21), such a quantity we shall consider as of the 2nd order of small quantities, and its square therefore of the 4th order. Our investigations are carried to the 2nd order only.