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= μ

r2 = Sp2 + pm2, and if MCP = u,
SpCp CS= a cos u — e',

Spa (cos u - e)2.


the result is

or making

Again, but hence

r2 = a2 (1 − e2) (1 — cos2 u) + a2 (cos u — e)3,



r = a {1 e cos u}.

This value of r and its differential being substituted in the value of

dt it becomes

e= —


pm2 = b2. sin2 u = b2 (1 - cos2 u);
bae a2 (1 − e2);


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the integral of which is t + k =



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du (1

k being an arbitrary constant quantity.

This equation gives u and consequently r in terms of t, and as x, y, z are given in functions of r, the values of these co-ordinates

are known at


any instant.

1 the values of dt and he become


the conic sections. Let AmP, fig. 75, be an ellipse whose major axis is 2a, its minor axis 26, the eccentricity CS =e, and the radius vector Sm = r.

Let the circle PMA be described on the major axis, draw the perpendicular Mp through m, and

a &

and when substituted in dv.

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- e cos u)

{u - e sin u};

a (1 − e2)


dr. Ja(1-e2)

and a (1-e),

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a(1 − e2 ) 1



which is the general equation to the conic sections, when the origin of r the radius vector is in the focus; a is half the greater axis, and

cos (v

a) = cos (Sm

SP), fig. 77.

1-e cos (v-)'

Elements of the Orbit.

378. Thus the finite values of the equations of elliptical motion are completely determined,

Six arbitrary constant quantities have been introduced, namely,

2a, the greater axis of the orbit.'

e, the ratio of the eccentricity to half the greater axis.

,, the projection of the longitude of the perihelion.

e, the longitude of the ascending node.

, the inclination of the orbit on the plane of the ecliptic, and

e, the longitude of the epoch.

The two first determine the nature of the orbit, the three following its position in space, and the last is relative to the position of the body at a given epoch; or, which is the same thing, it depends on the instant of its passage at the perihelion.

Equations of Elliptical Motion.

379. It now becomes necessary to determine three equations which will give values of the longitude and latitude Sm, mSp, and the distance Sm, fig. 72, in terms of the time from whence tables of the elliptical motions of the planets and satellites may be computed.


380. The motion of a body in an ellipse is not uniform, its velocity is greatest at the perihelion, and least at the aphelion, varying with the angle PSm, which is the true angular motion of the planet; but if the circle PBAD, fig. 75, be described from the centre of the ellipse with the semigreater axis CP, or mean distance from S as radius, the motion of the planet in this circle would be uniform. This is called the mean motion of a body.

381. Were the motion of a planet uniform, the angle PSm described by the planet in any interval of time after leaving perihelion might be found by simple proportion from knowing the periodic time, or time in which it describes 360°; but in order to preserve the equable description of areas, the true place of the planet will be before the mean place in going from perihelion to aphelion; and from aphelion to perihelion the true place will be behind the mean place. These angles are estimated from west to east, the direction in which the bodies of the system move, beginning at the perihelion. If, however, they are estimated from the aphelion, it is only necessary to add 180° to each.

382. The angular distance PCB between the perihelion and the mean place, is the mean anomaly, PSm the angular distance between the true place and the perihelion is the true anomaly; and mSB the angle at the sun, contained between the true and the mean place is called the equation of the centre. If then the mean anomaly be increased or diminished by the equation of the centre, the result will be the true place of the planet in its orbit. The equation of the centre is zero, both at the perihelion and aphelion, for at these points the true and mean places of the planets coincide; it is greatest when the planet is in quadratures, and at its maximum it is equal to an angle measured by twice the eccentricity of the orbit.

383. The mean place of a planet at any given time may be found by simple proportion from its periodic time. The true place of the planet in its orbit, and its distance from the sun, may be found in terms of its mean place by help of the angle PCM, called the eccentric anomaly.

If the time be estimated from the perihelion, 70, which reduces equation (95) to


(u - e. sin u), or nt = u — e sin u, if n =

whence COS v =

Where nt

If the angles u and v be estimated from the perihelion, a comparison of the values of r in article 377, gives

1 - e2


1+e cos v


v = PSm


u = PCM









e cos u=

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- e cos u


tan v=

sin v =

1+ e



tan v


384. The motions of the celestial bodies in elliptical orbits are therefore obtained from the three equations

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sin u.


e cos u)

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- e

mean anomaly, fig. 75,

true anomaly,

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e cos u





eccentric anomaly,

radius vector,

mean distance, and

the ratio of the eccentricity to the mean dis


385. It appears from these expressions that when u becomes u+360°, r remains the same; and as v is then augmented by 360°, the planet returns to the same point of its orbit, having moved

through four right angles, and the time becomes T=

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√ μ

that the time of a complete revolution is independent of the eccentricity, and only depends on 2a, the major axis of the orbit; it is consequently the same as if the planet described a circle at its mean distance from the sun; for in this case e= 0, ra, unt, v=u, consequently vnt; the arcs described are therefore proportional to the time, and the planet moves uniformly in the circle whose radius is a. Generally nt represents the arc that a body would

describe in the time t, if it set out from the perihelion at the same instant with a planet m, and moved with a uniform velocity represented by n in a circle described on the major axis of the orbit as diameter. This body would pass the perihelion and aphelion at the same instant with the planet m, but in one half of its revolution the planet would precede the body, and in the other half it would fall behind it. If a=1, μ=1, then n=1, and vt, the time will therefore be expressed by the arcs described by the planet in the circle whose radius is unity.

Astronomers generally compare the motions of the solar system with those of the earth; they take the mean distance of the sun from the earth as the unit of distance, the sum of the masses of the sun and earth as the unit of mass; and supposing the time to be estimated in mean solar days, the unit of time will be represented by the arc that the earth describes round the sun in one day with its mean motion.

Determination of the Eccentric Anomaly in functions of the Mean Anomaly.

386. If a value of u could be found in terms of nt from the first of these equations, both r and v, and consequently the place of the planet in its orbit at any instant, would be known from the two last.

Now an arc and its sine are incommensurate quantities, so that the one can only be obtained in functions of the other by an infinite series. Therefore a value of u in terms of nt must be found by an infinite series from the first of the preceding equations; but unless the terms of the series decrease rapidly in value u cannot be obtained, for a few of the first terms being computed, the value of the remaining part of the series must be so small that it may be neglected without sensible error. The small eccentricities of the orbits of the planets and satellites afford the means of approximation, for e the ratio of the eccentricity to half the greater axis is still smaller, consequently the powers of such quantities decrease rapidly, and therefore the second part of the equation unt + e sin u may be expanded into a series in functions of the time, and according to the powers of e, which will be sufficiently convergent. This may be

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