At the end of the excerpt Dr. Peter Shelus says that the measurements show that the shape of the moon's orbit is not quite as expected by "Newton's" theories, which I interpreted to mean that the orbit is non-Keplerian, but possibly he meant something else.
That's a bad assumption. He undoubtably meant something else. Even Kepler knew the Moon's orbit about the Earth was not Keplerian. Newton tried to explain this non-Keplerian behavior in terms of perturbations from the Sun. He did manage to explain some of the key lunar anomalies, but the math of his time was not quite up to the challenge. The full Newtonian explanation would have to wait a couple of hundred years, culminating in the work of Ernest William Brown.
To develop his (classical) lunar theory, Brown had to account for gravitational perturbations from a number of different objects: The Sun, Venus, Jupiter (and to a lesser extent, the other planets), and also the non-spherical nature of the Earth's and the Moon's gravity fields. Even with that, Brown had to introduce a fudge factor to account for a 10 arc second fluctuation in the Moon's angular position. It turned out that the need for this fudge was the transfer of angular momentum from the Earth's rotation to the Earth's orbit. This slows down the Earth's rotation rate (making time based on the the Earth's rotation a less than stellar idea) and makes the Moon recede from the Earth. Brown eventually added these concepts to his lunar theory.
Brown's student Wallace J. Eckert later extended Brown's lunar theory. Eckert was the first person to apply digital computers (as opposed to human computers) to the problem of predicting the Moon's orbit. He was also the person who developed the lunar ephemerides used by the Apollo program. Eckert's work was still classical as opposed to relativistic.
The retroreflectors placed on the Moon by the U.S. and the Soviet Union during the 1960s and 1970s finally necessitated the use of general relativity to explain the Moon's orbit. The effects are extremely subtle, but they are there. In particular, those retroreflectors serve as one of the highest precision tests of a key precept of general relativity, the strong equivalence principle. The equivalence principle comes in a number of forms:
- The Galilean (or Newtonian) equivalence principle, which addresses the equivalence of inertial and gravitational mass, and hence the universality of free fall.
- The weak equivalence principle, which says the motion of test bodies with negligible self-gravity is independent of their properties (this essentially restates the Galilean equivalence principle in relativistic terms),
- The Einstein equivalence principle, which says that a state of rest in homogeneous gravitational field is physcially equivalent to a state of uniform acceleration in gravity-free space,
- The strong equivalence principle, which says that at any point of spacetime, physics is locally described by special relativity and is not affected by the presence of a gravitational field.
Given sufficiently accurate measurements, the Moon's weird composition forms an excellent basis for testing both the various forms of the equivalence principle. The Moon's far side has a much thicker crust than does its near side. This makes for a two kilometer offset between the Moon's center of mass and center of figure. The thick crust on the far side (mostly silicon, oxygen, and aluminum) the offset core on the near side (mostly iron and nickel) would subtly affect the Moon's orbit if weak form of the equivalence principle didn't apply. In fact, the Moon's orbit as measured by lunar laser ranging experiments provides an excellent basis for testing even the strong equivalence principle.
Note that Newtonian mechanics disagrees with all but the Galilean equivalence principle. Newtonian mechanics also fails to match the observed orbit of Mercury, and to a lesser extent, the observed orbit of the Moon. Viable alternatives to general relativity agree with the weak form of the equivalence principle, and most agree with the Einstein form of the equivalence principle. General relativity is consistent with the strong equivalence principle (as are a very small number of alternatives to general relativity).