  Recent angle-resolved photoemission experiments have observed a
proximity-induced superconducting gap in the helical surface states of a thin
film of the 3D topological insulator Bi$_2$Se$_3$ grown on a superconducting
NbSe$_2$ substrate. The superconducting coherence peaks in the electronic
density of states are strongly suppressed when the topological insulator is
doped with magnetic Mn impurities, which was interpreted as the complete
destruction of helical superconductivity in the topological surface states.
Motivated by these experiments, we explore a different possibility: gapless
helical superconductivity, where a gapless electronic density of states
coexists with a nonzero helical superconducting order parameter. We study a
model of superconducting Dirac fermions coupled to random magnetic impurities
within the Abrikosov-Gor'kov framework, and find finite regions of gapless
helical superconductivity in the phase diagram of the system for both
proximity-induced and intrinsic superconductivity. For the latter, we derive
universal rates of supression of the superconducting transition temperature due
to magnetic scattering and, for a Fermi level at the Dirac point, a universal
rate of increase of the quantum critical attraction strength.
