Lecture 3 extras.
Uranus, Neptune, Pluto, and Charo.

Below some historical facts to illustrate how incredibly accurate
the calculations by astronomers were (and are). Note that much of
this is pre-computer analytical work.  We will see elementary
versions [full technique, but applied to simple examples]  of the
methods later on.

Neptune was first observed in 1846 (Berlin), guided by earlier (1845)
  predictions by Urbain Le Verrier in Paris and John Couch Adams in
  Cambridge. Note that, in fact, the planet had been observed earlier
  but had not been recognized as a planet.

Pluto was first observed in 1930 (Flagstaff),  guided by earlier
  calculations by Percival Lowell and W.H. Pickering. Lowell had made
  his calculations nearly 10 years earlier, and searched for Pluto,
  but was unable to find it. The reason being that Pluto is very small
  and very difficult to observe [a new technique was needed].
  Turned out that Pluto's mass was not enough to explain the observed
  perturbations to Uranus and Neptune's orbits. But on 1978 Charon,
  Pluto's moon, was discovered. The two together then had enough mass.
EOF