Finally, having illustrated the principles and the limitations of Dimensional Analysis, and the methods of thinking it makes possible and powerful, we close this chapter with a few additional examples. They happen to be of great historical importance, because they were some of the first systems that, in spite of their remoteness and therefore seeming inaccessibility, provided the compensation that they were so simple that it was possible to recognize the features of their behavior and in that way make inroads to progress. In addition to their historical interest, they yield some of the simplest and most useful laws of physics, and provide us a jumping-off place from the Earth, where we realize that using the observations we make here, we can begin to explore the cosmos by reason.
The problem of interest is understanding the motions of the planets, moons and other orbiting things. Historically, Galilleo had made the initial discoveries along two lines of inquiry that will enable us to solve this problem dimensionally, though at the time he had no way to anticipate this. First, as we have mentioned, he recognized the pattern in the motions of fallers near earth as one of a consistent acceleration. Second, and of interest to us now, he discovered that Jupiter has moons.
It had already been understood by Copernicus that the really useful
way to describe planetary motions was as nearly circular orbits around
a single body, namely the sun. It was interesting to compare
the motions to one another because they had something in common,
specifically that they shared this orbital center. It had also been
recognized that the moon orbits the earth in much the same way as any
planet orbits the sun, and that it has yet another orbital
period
(the time to complete an orbit,
much like the period of a cycle). However, in spite of the fact that
the moon is closest to the earth, and therefore easiest to observe, it
was difficult to say anything useful about its motion, because there
was nothing else large enough to see, that was also orbiting the
earth, against whose motion it could be compared. Further, since the
moon orbits the earth, rather than the sun directly, it is difficult
to predict what relations those motions might be expected to have.
Unlike Earth, however, Jupiter has many moons and not just one. Thus, it is like a small copy of the sun, but is different from the actual sun. We can look for relations among the motions of the moons of Jupiter in the same was as we can look for relations among the planetary motions. Then if we find them, we can compare the motions of the moons of Jupiter to those of the planets, and see if those have anything in common. Precisely such a series of measurements was made by Johannes Keplar over the course of a lifetime, and was available to Newton as a set of exceedingly high-quality experimental evidence at the beginning of his work.
Several thing were available to measure. Orbital motions, like cyclic motions, have an associated notion of length. For circular orbits, it is simply the distance of the orbiting thing from the center about which it moves. Because the moons of Jupiter are all within a fairly small range of Jupiter itself, compared to the distance of all of them from us, it is easy to measure relations among these orbital radii simply by looking through a telescope at how far the moons get from their center when they look most distant (i.e., when we see them most sideways). We are simply observing a revolving motion from the side. Also like cyclic motions, because they repeat themselves, orbital motions have a definite notion of period. This can be measured most cleanly by noting when a given moon passes just in front of the planet in each orbit. Keplar was able to determine that, for Jupiter, there seemed to be a certain constant of the system for all of the moons. The particular combination that was always the same depended on only the orbital period and the orbital radius, and took the form
Much more remarkably than that, he was able to perform a similar class
of observations for the motions of the planets around the sun (which
require much more ingenuity to do accurately, because the earth
changes position more quickly than do the outer planets, and their
radii are not much smaller than their distance from us). The result
of the latter was that these also possessed a constant of the system,
of the same dimensional form , but it was a different
constant than the one obeyed by Jupiter's moons. Thus the scaling behavior was the same, but the particular scale factor
seemed to depend on the thing at the center. The existence of such a
scaling relation between the periods and radii of orbiting bodies is,
not surprisingly, known as Keplar's Law.
There were now two patterns (Galilleo's and Keplar's) recognized about
the motions of fallers, though at the time no-one understood that
planets even were fallers.
From this body of accumulated data, Newton made his great leap of
intellect. He proposed that, not only were orbital motions the
motions of free fall, but that the constants that described them were
related to the same gravitation as the constant Galilleo had
discovered for fallers on Earth. If that were the case, even though
there is nothing else orbital to compare it against, the moon
should also obey a rule that its equals some special
constant. Further, that specail constant should be related to
Galilleo's constant
. We can recognize that the only thing special
about the situations Galilleo measured is, that they were all right
here on the surface of the earth, so they were of necessity at a
fixed distance from its center, namely the radius of the earth. The
moon, being in free fall in empty space, had no such constraint, so
its orbital radius took whatever value circumstances had given it.
Then, recalling the dimensions of
, there is a unique guess
for what should be the
constant of the moon,
Few things about astronomy have been known longer than the time it
takes the moon to complete a cycle. Now with this law, we have a
prediction, from nothing more than the behavior of free fallers near
earth and the length of a month, of how far away the moon is from
the earth. This is because we have a prediction for the
constant and we know the
. The prediction is correct, too. It is
breathtaking that seemingly so little directly measured experience,
together with a few simple laws, can make it possible to predict a
such a number correctly. We note, in passing, that the particular
value we get for this constant is different, both from the one for
Jupiter's moons, and from the one for the planets around the sun, so
again it seems relevant that the moon orbits the earth, and not one of
these other bodies.
We now pursue this kind of reasoning through several more steps,
because with the work of Keplar, much more than the distance to the
moon can be understood. The fact that the is shared by
planets that orbit a common body, but different for each body that
they orbit, says that this constant tells something about the body at
the center. The fact that the pattern is the same, though,
while the sun is a very differentlt-composed object from the earth (or
as we now know, from Jupiter as well), makes it tempting to speculate
that the value of this constant depends only on something very simple,
like the mass of what is at the center.
It is hard to measure the mass of a planet directly, though. You
can't go there easily, and there is certainly no ``bathroom scale'' in
which to set it. So we make use of some of what we have already seen
to constrain this dimensionally. We know, from our previous
prediction, how far away the moon is. And just by looking, we can see
how big an angle it makes in the sky (called its ``opening angle'').
Thus we can easily estimate its radius. The moon seems to be a solid
object, so it is reasonable to estimate its density as one typical of
solid objects (this is intrinsically a Fermi problem; the densities of
rock and water are similar enough that it doesn't matter which we use,
for purposes of estimation). Thus, from the radius, which we can
infer by eye, and its density, which we can infer by assuming its
composition, we can estimate its mass as
. The radius of the earth is
much easier to estimate directly, as we have already seen, and we know
that the earth seems to be made mostly of rock, so we can estimate its
mass by the same means, and somewhat more confidently.
Now we would like to know if there is a relation between the masses
and the orbital constants. In a previous section we made use of
something Newton did not have, which was a direct TV picture that
allowed us to infer the value of on the moon (after all, Newton
was the one who got us there). Thus we know, in addition to the
masses, the values of their respective
s. The two, in this case,
turn out to be proportional. In other words, for objects falling
toward some massive body, their fall is characterized by a
. Since
also tells us,
independently of mass, about Keplar's orbital constant, we expect that
the orbital constant also is proportional to the mass, and by the same
scale factor. (We will see, in the chapter on Newton's work itself,
that there is another reason to expect this linear proportionality.
This other reason was what enabled Newton to propose such a
relationship, without the benefit of pictures sent back from the moon.
For now, since we have the observation, we will simply use it here,
and regard the linear scaling relation as an experimental observation,
to see where it leads.)
Knowing the orbital constant for the planets around the sun, or of its
moons around Jupiter, and the scale factor inferred from our own
and the radius of and mass of the earth, we can predict the
masses of Jupiter and of the sun. In terms of direct
inaccessibility, this is even a more stunning accmplishment than
predicting the distance to the moon. We have something more
remarkable, though, even than these predictions. (One remark is in
order here. While, with sufficient cleverness and patience, it was
possible to infer the distances of other planets from the sun by
their projections against the backdrop of the much more ``fixed''
stars, the distance of the earth from the sun is not easily
measureable in any similar way. Knowing the earth's orbital period (a
year), this distance was one of the other numbers Keplar's law enabled
us to predict, making use of the radii of the other planets to
determine the constant.)
In discovering the linear relation between the orbital constant and
the mass of the thing at the center, we seem now to have accounted for
all of the observable properties that matter in controlling
orbital motions. The pattern we are left with seems to be a constant of all these systems, irrespective of the particular
motions, and even of which objects we choose to combine. We will see
this constant again in the chapter on Newton's laws, where we will
start to recognize its importance. For now, since it seems to tell us
something about Gravitational interactions, we will simply give it a
name, and note its dimensions. For all orbital motions of all bodies
about some central body of mass , the measureable combination
is always the same.
Finally, we make one last note, as a consistency check and to tie things together. Solar eclipses happen. Furthermore, they are interesting. This has been known for long enough that it is easy to forget what a remarkable accident it is. And, viewed on a scale of cosmic times, it is only true for a while longer that they can happen. This is because, for the sun to be eclipsed, the moon must be big enough to cover the disk of the sun. For the eclipses to still be interesting, it must be small enough that it just covers the main disk of the sun, leaving the more diffuse gases at the fringe visible. That requires that the opening angles of the moon and the sun, to a remarkably good accuracy be the same. (Eclipses won't be possible forever because the same action that causes the tides we see is also transferring some of the earth's rotation, ever so gradually, to the moon, and pushing it further away, thus making it seem smaller. This process will only stop when the moon goes around the earth at the same rate that the earth turns. Thus, every day will be a month long, though a month will also be longer. Furthermore, eclipses haven't always been as interesting as they are now. When the earth was young, and the moon substantially closer, its disk was large enough to obscure the sun and go a long way beyond as well.) (Note to myself: check Mr. Sagan's arithmetic here at some point, to see if the scales are what he claims.)
Thus, the same process we used to estimate the mass of the moon from its distance can be used to estimate the radius, of the sun from its distance. Whereas direct observations of the moon make it fairly a reliable guess that the moon is made of the same rock as the earth, the sun is a much different object. Thus we could use the mass we infer for the sun, as extrapolated from Keplar's law and the earth's orbital period, and the volume, as inferred from that distance and its opening angle, to check whether the density of the sun is comparable to the densities we know for solid objects, or something wildly different. The answer is, in fact, that despite its great difference in properties, the density of the sun is consistent, to the accuracy of these approximations, with the densities of other liquids and solids that we know. This check, in some sense, closes a loop of inferences, and establishes that the many relations we have assumed, and the wide variety of distances and times that we have extracted from them, are mutually consistent.