  Our concern here is the nature of secondary resonances--commensurabilities
between apsidal and libration periods lying within first-order mean motion
resonances [mmr] in the solar system. At the 4/3 and 3/2 mmr in the asteroid
belt, we find in general that it is possible to identify the positions of their
considerably large number and to determine the degree of chaos that they
develop. The severity of the latter corresponds to the absence of observed
asteroids at low eccentricity, e, in both of these mmr and even more so at 2/1.
Chaos at higher e is present but weaker so that real bodies can remain over the
age of the solar system. This reduced chaos, we suggest, arises because the
denser though weaker secondaries have widths that allow them to overlap and so
provide a continuous quite constant level of chaos over a broad eccentricity
range. We also consider the similar first-order 1/2 mmr with Neptune in the
Kuiper belt at 47.8 AU and find considerable chaos, probably once again induced
by secondary resonances, among the many orbits with e < 0.10. The instability
so generated seems very likely to correspond in realistic models to the slow
escape from 1/2 of a fair number of its once captured bodies over the solar
system's age.
