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.

Thu Aug 31 12:01:42 CDT 1995