Part 18 (2/2)
3. The fact that one attracting body acts on all the planets with an inverse square law, causes the cubes of their mean distances to be proportional to the squares of their periodic times.
Not only these but a mult.i.tude of other deductions follow rigorously from the simple datum that every particle of matter attracts every other particle with a force directly proportional to the ma.s.s of each and to the inverse square of their mutual distance. Those dealt with in the _Principia_ are summarized above, and it will be convenient to run over them in order, with the object of giving some idea of the general meaning of each, without attempting anything too intricate to be readily intelligible.
[Ill.u.s.tration: FIG. 70.]
No. 1. Kepler's second law (equable description of areas) proves that each planet is acted on by a force directed towards the sun as a centre of force.
The equable description of areas about a centre of force has already been fully, though briefly, established. (p. 175.) It is undoubtedly of fundamental importance, and is the earliest instance of the serious discussion of central forces, _i.e._ of forces directed always to a fixed centre.
We may put it afresh thus:--OA has been the motion of a particle in a unit of time; at A it receives a knock towards C, whereby in the next unit it travels along AD instead of AB. Now the area of the triangle CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being the perpendicular height of the triangle from the base AC. (Fig. 70.) Now the blow at A, being along the base, has no effect upon _h_; and consequently the area remains just what it would have been without the blow. A blow directed to any point other than C would at once alter the area of the triangle.
One interesting deduction may at once be drawn. If gravity were a radiant force emitted from the sun with a velocity like that of light, the moving planet would encounter it at a certain apparent angle (aberration), and the force experienced would come from a point a little in advance of the sun. The rate of description of areas would thus tend to increase; whereas in reality it is constant. Hence the force of gravity, if it travel at all, does so with a speed far greater than that of light. It appears to be practically instantaneous. (Cf. ”Modern Views of Electricity,” -- 126, end of chap. xii.) Again, anything like a r.e.t.a.r.ding effect of the medium through which the planets move would const.i.tute a tangential force, entirely un-directed towards the sun.
Hence no such frictional or r.e.t.a.r.ding force can appreciably exist. It is, however, conceivable that both these effects might occur and just neutralize each other. The neutralization is unlikely to be exact for all the planets; and the fact is, that no trace of either effect has as yet been discovered. (See also p. 176.)
The planets are, however, subject to forces not directed towards the sun, viz. their attractions for each other; and these perturbing forces do produce a slight discrepancy from Kepler's second law, but a discrepancy which is completely subject to calculation.
No. 2. Kepler's first law proves that this central force diminishes in the same proportion as the square of the distance increases.
To prove the connection between the inverse-square law of distance, and the travelling in a conic section with the centre of force in one focus (the other focus being empty), is not so simple. It obviously involves some geometry, and must therefore be left to properly armed students.
But it may be useful to state that the inverse-square law of distance, although the simplest possible law for force emanating from a point or sphere, is not to be regarded as self-evident or as needing no demonstration. The force of a magnetic pole on a magnetized steel sc.r.a.p, for instance, varies as the inverse cube of the distance; and the curve described by such a particle would be quite different from a conic section--it would be a definite cla.s.s of spiral (called Cotes's spiral).
Again, on an iron filing the force of a single pole might vary more nearly as the inverse fifth power; and so on. Even when the thing concerned is radiant in straight lines, like light, the law of inverse squares is not universally true. Its truth a.s.sumes, first, that the source is a point or sphere; next, that there is no reflection or refraction of any kind; and lastly, that the medium is perfectly transparent. The law of inverse squares by no means holds from a prairie fire for instance, or from a lighthouse, or from a street lamp in a fog.
Mutual perturbations, especially the pull of Jupiter, prevent the path of a planet from being really and truly an ellipse, or indeed from being any simple re-entrant curve. Moreover, when a planet possesses a satellite, it is not the centre of the planet which ever attempts to describe the Keplerian ellipse, but it is the common centre of gravity of the two bodies. Thus, in the case of the earth and moon, the point which really does describe a close attempt at an ellipse is a point displaced about 3000 miles from the centre of the earth towards the moon, and is therefore only 1000 miles beneath the surface.
No. 3. Kepler's third law proves that all the planets are acted on by the same kind of force; of an intensity depending on the ma.s.s of the sun.
The third law of Kepler, although it requires geometry to state and establish it for elliptic motion (for which it holds just as well as it does for circular motion), is very easy to establish for circular motion, by any one who knows about centrifugal force. If _m_ is the ma.s.s of a planet, _v_ its velocity, _r_ the radius of its...o...b..t, and _T_ the time of describing it; 2[pi]_r_ = _vT_, and the centripetal force needed to hold it in its...o...b..t is
mv^2 4[pi]^2_mr_ -------- or ----------- _r_ T^2
Now the force of gravitative attraction between the planet and the sun is
_VmS_ -----, r^2
where _v_ is a fixed quant.i.ty called the gravitation-constant, to be determined if possible by experiment once for all. Now, expressing the fact that the force of gravitation _is_ the force holding the planet in, we write,
4[pi]^2_mr_ _VmS_ ----------- = ---------, T^2 r^2
whence, by the simplest algebra,
r^3 _VS_ ------ = ---------.
T^2 4[pi]^2
The ma.s.s of the planet has been cancelled out; the ma.s.s of the sun remains, multiplied by the gravitation-constant, and is seen to be proportional to the cube of the distance divided by the square of the periodic time: a ratio, which is therefore the same for all planets controlled by the sun. Hence, knowing _r_ and _T_ for any single planet, the value of _VS_ is known.
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