| Abstract | In order to establish the Lyα absorption cross section of present-day galaxies, we have identified 38 galaxies with z = 0-0.08 that lie within 40-500 h^-1^ kpc of the line of sight to a QSO observed with the Faint Object Spectrograph aboard the Hubble Space Telescope (HST). Including three galaxies in the field of 3C 273 investigated by previous authors, we find that nine of 41 galaxies have associated Lyα absorption. If the identified Lyα absorption systems are genuinely associated with the galaxies, then the covering factor of gas around galaxies remains roughly constant at ~40% between 100 and 300 h^-1^ kpc. Beyond 300 h^-1^ kpc, the incidence of absorption drops sharply. We conclude that (1) nearby galaxies do not possess Lyα-absorbing halos beyond 300 h^-1^ kpc in radius and (2) the covering factor of present-day galaxies between 50 and 300 h^-1^ kpc is 44% at an equivalent width limit of W >= 0.3 A. For the nine galaxies with associated Lyα absorption lines, differences in the galaxies systemic velocities and the velocity of the absorption line, {DELTA}v, range over +/- 300 km s^-1^, consistent with the distribution found at redshifts > 0.1 by Lanzetta et al. and Le Brun, Bergeron, & Boisse. Values of {DELTA}v spanning several hundred km s^-1^ are probably real for some of the QSO-galaxy pairs, however, and do not simply arise from errors in measuring cz_gal_ or cz_abs_. This suggests that the absorbing clouds are kinematically tied to the galaxy disks and that the distribution of {DELTA}v may arise because of the effects of galaxy inclination. We find no evidence for a correlation between Lyα equivalent width and galaxy line-of-sight separation, which weakens the argument that the identified galaxies are directly associated with the Lyα lines. Also, we find that absorption does not arise only from bright galaxies, since there are several examples in which low-luminosity galaxies apparently cause absorption. Yet we show that the absorbing halos around galaxies cannot be independent of galaxy luminosity because if all low- redshift galaxies were surrounded by extended halos, the number of Lyα absorption systems found in HST spectra would be much larger than has recently been determined. This result leads us to question whether the galaxies are actually responsible for the Lyα absorption lines or whether the association in redshift is fortuitous. Our results support the picture suggested by others that Lyα lines arise in filaments or sheets that connect and contain the overdense regions that galaxies inhabit. This conclusion, however, remains at odds with the results of Lanzetta et al., who find that Lyα systems are more intimately linked with galaxies. It may be, therefore, that we are seeing an evolution of the way in which Ly& lines associate with galaxies over the last few billion years. We suggest that at least some of the Lyα clouds that are not associated with galaxies at z ~ 0.5 have been slowly merging with galaxies over time, creating larger but less uniform gas distributions around the galaxies we see today. |