Sizes of Lyman-Alpha-Emitting Galaxies and Their Rest-Frame Ultraviolet
  Components at z=3.1
  We present a rest-frame ultraviolet analysis of ~120 z~3.1 Lyman Alpha
Emitters (LAEs) in the Extended Chandra Deep Field South (ECDF-S). Using Hubble
Space Telescope (HST) images taken as part of the Galaxy Evolution From
Morphology and SEDS (GEMS) survey, Great Observatories Origins Deep Survey
(GOODS), and Hubble Ultradeep Field (HUDF) surveys, we analyze the sizes of
LAEs, as well as the spatial distribution of their components, which are
defined as distinct clumps of UV-continuum emission. We set an upper limit of
~1 kpc (~0.1") on the rms offset between the centroids of the continuum and
Lyman-alpha emission. The star formation rates of LAE components inferred from
the rest-frame ultraviolet continuum range from ~0.1 M_{sun}/yr to ~5$
M_{sun}/yr. A subsample of LAEs with coverage in multiple surveys (at different
imaging depths) suggests that one needs a signal-to-noise ratio, S/N>~30, in
order to make a robust estimate of the half-light radius of an LAE system. The
majority of LAEs have observed half-light radii <~2 kpc, and LAE components
typically have observed half-light radii <~1.5 kpc (<~ 0.2"). Although only
~50% of the detected LAE components are resolved at GOODS depth, the brightest
(V<~26.3) are all resolved in both GOODS and GEMS. Since we find little
evidence for a correlation between the rest-UV sizes and magnitudes of LAEs,
the majority should be resolved in a deeper survey at the ~0.05" angular
resolution of HST. Most of the multi-component LAEs identified in shallow
frames become connected in deeper images, suggesting that the majority of the
rest-UV "clumps" are individual star-forming regions within a single system.
