To close this chapter, we remark on a different view that could be taken of the dimensional analysis we have just done. We organized it in such a way as to show how, starting from the observations of Galilleo and Keplar, and completely constraining our dimensional relations with some more modern evidence from astronauts on the moon, it is possible to estimate many parameters of the solar system. If we had been in a position to make use of some of Newton's reasoning that we will see in later chapters, we could even have dispensed with the modern evidence, and reached the same conclusions.
Alternatively, we could organize the arguments so as to draw the maximum power from the simplicity of the mere dimensional content of Newton's constant . is not necessarily an easy number to remember, but its dimensions are. And the radius of the earth, and Galilleo's are easy numbers to reconstruct from common experience, if we ever forget them. But those can be used in turn to recover the explicit value of , from only its dimensions. Simply knowing the dimensions of then forces Keplar's laws upon us (in a sort of inverse to the way Keplar's laws told Newton those dimensions for the first time). If we then estimate the mass of the earth from its radius, and take Keplar's laws from the dimensions of , we can recover the radius of the moon, knowing that it orbits in a month.
Then, even if we didn't have the particular orbital value of the planetary constant from Keplar's observation, we could recover that, and the mass of the sun, from what we have here. This is because the mass of the sun is related to two things we know. From its opening angle in the sky and its density, the mass of the sun relates to its radius and thus to its distance from us. But entirely independently of that, it relates, through and the earth's orbital period of a year, to that same distance. This closed loop makes it possible to find the only value of that distance which is consistent with both the earth's orbital period and the sun's opening angle. All we need to assume is an approximate value for the sun's density, which amounts to an assumption about whether or not it is reasonably typical solid matter.
In other words, we can compute the mass of the sun sitting in an armchair, recalling common everyday occurrences and maybe timing the fall of a penny, if we remember the dimensions of Newton's . That is an example of the power of Dimensional Analysis.