We examine flame balls that are not optically thin so that radiation f
rom the hot burned gas is not lost but is reabsorbed by the surroundin
g cool gas. The steady solution is shown to be multivalued in general
and defines an intrinsic (apparatus independent) flammability limit fo
r a sufficiently small ignition source. Beyond the flammability limit
a steady solution exists for arbitrarily weak mixtures, but is unreali
zable since it is unstable to one-dimensional perturbations. An unstea
dy propagating flame solution is also possible for arbitrarily weak mi
xtures, but can only be generated by a large ignition source. Observat
ions of flame-strings in SF6-diluted mixtures are reported and suggest
that the peculiar nature of the radiative properties of SF6 is respon
sible for peculiar dynamics.