  We study the evolution of long-period comets by numerical integration of
their orbits, following comets from their origin in the Oort cloud until their
final escape or destruction, in a model solar system consisting of the Sun, the
four giant planets and the Galactic tide. We also examine the effects of
non-gravitational forces and the gravitational forces from a hypothetical solar
companion or circumsolar disk. We confirm the conclusion of Oort and other
investigators that the observed distribution of long-period comet orbits does
not match the expected steady-state distribution unless there is fading or some
similar process that depletes the population of older comets. We investigate
several simple fading laws. We can match the observed orbit distribution if the
fraction of comets remaining observable after m apparitions is proportional to
m to the power -0.6 +/- 0.1 (close to the fading law originally proposed by
Whipple 1962); or if approximately 95% of comets live for only a few (~6)
returns and the remainder last indefinitely. Our results also yield statistics
such as the expected perihelion distribution, distribution of aphelion
directions, frequency of encounters with the giant planets and the rate of
production of Halley-type comets.
