  We investigate the gravitational instability (GI) of dust-ring structures and
the formation of planetesimals by their gravitational collapse. The normalized
dispersion relation of a self-gravitating ring structure includes two
parameters that are related to its width and line mass (the mass per unit
length). We survey these parameters and calculate the growth rate and
wavenumber. Additionally, we investigate the planetesimal formation by growth
of the GI of the ring that is formed by the growth of the secular GI of the
protoplanetary disk. We adopt a massive, dust rich disk as a disk model. We
find the range of radii for the fragmentation by the ring GI as a function of
the width of the ring. The inner-most radius for the ring GI is smaller for the
smaller ring width. We also determine the range of the initial planetesimal
mass resulting from the fragmentation of the ring GI. Our results indicate that
the planetesimal mass can be as large as 10^28 g at its birth after the
fragmentation. It can be as low as about 10^25 g if the ring width is 0.1% of
the ring radius and the lower limit increases with the ring width. Furthermore,
we obtain approximate formulas for the upper and lower limits of the
planetesimal mass. We predict that the planetesimals formed by the ring GI have
prograde rotations because of the Coriolis force acting on the contracting
dust. This is consistent with the fact that many trans-Neptunian binaries
exhibit prograde rotation.
