  Recent theoretical predictions hint at an implementation of a superradiant
laser based on narrow optical clock transitions by using a filtered thermal
beam at high density. Corresponding numerical studies give encouraging results
but the required very high densities are sensitive to beam collimation errors
and inhomogeneous shifts. Here we present extensive numerical studies of
threshold conditions and the predicted output power of such a superradiant
laser involving realistic particle numbers and velocities along the cavity
axis. Detailed studies target the threshold scaling as a function of
temperature as well as the influence of eliminating the hottest part of the
atomic distribution via velocity filtering and the benefits of additional
atomic beam guiding. Using a cumulant expansion approach allows us to quantify
the significance of atom-atom and atom-field correlations in such
configurations. We predict necessary conditions to achieve a certain threshold
photon number depending on the atomic temperature and density. In particular,
we show that the temperature threshold can be significantly increased by using
more atoms. Interestingly, a velocity filter removing very fast atoms has only
almost negligible influence despite their phase perturbing properties. On the
positive side an additional conservative optical guiding towards cavity mode
antinodes leads to significantly lower threshold and higher average photon
number. Interestingly we see that higher order atom-field and direct atom-atom
quantum correlations play only a minor role in the laser dynamics, which is a
bit surprising in the superradiant regime.
