  The Habitable Exoplanet Imaging Mission (HabEx) concept requires an optical
coronagraph that provides deep starlight suppression over a broad spectral
bandwidth, high throughput for point sources at small angular separation, and
insensitivity to temporally-varying, low-order aberrations. Vortex coronagraphs
are a promising solution that perform optimally on off-axis, monolithic
telescopes and may also be designed for segmented telescopes with minor losses
in performance. We describe the key advantages of vortex coronagraphs on
off-axis telescopes: 1) Unwanted diffraction due to aberrations is passively
rejected in several low-order Zernike modes relaxing the wavefront stability
requirements for imaging Earth-like planets from <10 to >100 pm rms. 2) Stars
with angular diameters >0.1 $\lambda/D$ may be sufficiently suppressed. 3) The
absolute planet throughput is >10%, even for unfavorable telescope
architectures. 4) Broadband solutions ($\Delta\lambda/\lambda>0.1$) are readily
available for both monolithic and segmented apertures. The latter make use of
grayscale apodizers in an upstream pupil plane to provide suppression of
diffracted light from amplitude discontinuities in the telescope pupil without
inducing additional stroke on the deformable mirrors. We set wavefront
stability requirements on the telescope, based on a stellar irradiance
threshold set at an angular separation of 3$\pm$0.5 $\lambda/D$ from the star,
and discuss how some requirements may be relaxed by trading robustness to
aberrations for planet throughput.
