  We carry out two searches for periodic gravitational waves using the most
sensitive few hours of data from the second LIGO science run. The first search
is targeted at isolated, previously unknown neutron stars and covers the entire
sky in the frequency band 160-728.8 Hz. The second search targets the accreting
neutron star in the low-mass X-ray binary Scorpius X-1, covers the frequency
bands 464-484 Hz and 604-624 Hz, and two binary orbit parameters. Both searches
look for coincidences between the Livingston and Hanford 4-km interferometers.
  For isolated neutron stars our 95% confidence upper limits on the
gravitational wave strain amplitude range from 6.6E-23 to 1E-21 across the
frequency band; For Scorpius X-1 they range from 1.7E-22 to 1.3E-21 across the
two 20-Hz frequency bands. The upper limits presented in this paper are the
first broad-band wide parameter space upper limits on periodic gravitational
waves using coherent search techniques. The methods developed here lay the
foundations for upcoming hierarchical searches of more sensitive data which may
detect astrophysical signals.
