| r3mr | = | VS | . |
| T2 | 4π2 |
The mass of the planet has been cancelled out; the mass 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.
No. 4. So by knowing the length of year and distance of any planet from the sun, the sun's mass can be calculated, in terms of that of the earth.
No. 5. For the satellites, the force acting depends on the mass of their central body, a planet. Hence the mass of any planet possessing a satellite becomes known.
The same argument holds for any other system controlled by a central body—for instance, for the satellites of Jupiter; only instead of S it will be natural to write J, as meaning the mass of Jupiter. Hence, knowing r and T for any one satellite of Jupiter, the value of VJ is known.
Apply the argument also to the case of moon and earth. Knowing the distance and time of revolution of our moon, the value of VE is at once determined; E being the mass of the earth. Hence, S and J, and in fact the mass of any central body possessing a visible satellite, are now known in terms of E, the mass of the earth (or, what is practically the same thing, in terms of V, the gravitation-constant). Observe that so far none of these quantities are known absolutely. Their relative values are known, and are tabulated at the end of the Notes above, but the finding of their absolute values is another matter, which we must defer.
But, it may be asked, if Kepler's third law only gives us the mass of a central body, how is the mass of a satellite to be known? Well, it is not easy; the mass of no satellite is known with much accuracy. Their mutual perturbations give us some data in the case of the satellites of Jupiter; but to our own moon this method is of course inapplicable. Our moon perturbs at first sight nothing, and accordingly its mass is not even yet known with exactness. The mass of comets, again, is quite unknown. All that we can be sure of is that they are smaller than a certain limit, else they would perturb the planets they pass near. Nothing of this sort has ever been detected. They are themselves perturbed plentifully, but they perturb nothing; hence we learn that their mass is small. The mass of a comet may, indeed, be a few million or even billion tons; but that is quite small in astronomy.
But now it may be asked, surely the moon perturbs the earth, swinging it round their common centre of gravity, and really describing its own orbit about this point instead of about the earth's centre? Yes, that is so; and a more precise consideration of Kepler's third law enables us to make a fair approximation to the position of this common centre of gravity, and thus practically to "weigh the moon," i.e. to compare its mass with that of the earth; for their masses will be inversely as their respective distances from the common centre of gravity or balancing point—on the simple steel-yard principle.
Hitherto we have not troubled ourselves about the precise point about which the revolution occurs, but Kepler's third law is not precisely accurate unless it is attended to. The bigger the revolving body the greater is the discrepancy: and we see in the table preceding Lecture III., [on page 57], that Jupiter exhibits an error which, though very slight, is greater than that of any of the other planets, when the sun is considered the fixed centre.
Let the common centre of gravity of earth and moon be displaced a distance x from the centre of the earth, then the moon's distance from the real centre of revolution is not r, but r-x; and the equation of centrifugal force to gravitative-attraction is strictly