Searching for the connection between ionizing-photon escape and the
  surface density of star formation at z~3
  The connection between the escape fraction of ionizing photons ($f_{\rm
esc}$) and star-formation rate surface density ($\Sigma_{\rm SFR}$) is a key
input for reionization models, but remains untested at high redshift. We
analyse 35 z~3 galaxies from the Keck Lyman Continuum Survey (KLCS) covered by
deep, rest far-UV spectra of the Lyman continuum (LyC) and high-resolution HST
V$_{606}$ imaging. The dataset enables estimates of both ionizing escape
fractions and rest-UV sizes. Using S\'ersic profile fits to HST images and
spectral-energy distribution fits to multi-band photometry, we measure
effective sizes and star-formation rates for the galaxies in our sample, and
separate the sample into two bins of $\Sigma_{\rm SFR}$. Based on composite
spectra, we estimate <$f_{\rm esc}$> for both $\Sigma_{\rm SFR}$ subsamples,
and find no significant difference in <$f_{\rm esc}$> between the two. To test
the representativeness of the KLCS HST sample and the robustness of this
result, we attempt to recover the well-established correlation between $f_{\rm
esc}$ and Ly$\alpha$ equivalent width. This correlation is not significant
within the KLCS HST sample, indicating that the sample is not sufficient for
correlating $f_{\rm esc}$ and galaxy properties such as $\Sigma_{\rm SFR}$. We
perform stacking simulations using the KLCS parent sample to determine the
optimal sample size for robust probes of the $f_{\rm esc}$-$\Sigma_{\rm SFR}$
connection to inform future HST LyC observing programs. For a program with a
selection that is independent of ionizing properties, >= 90 objects are
required; for a program preferentially observing strongly-leaking LyC sources,
>= 58 objects would be required. More generally, measuring the connection
between $f_{\rm esc}$ and $\Sigma_{\rm SFR}$ requires a larger, representative
sample spanning a wide dynamic range in galaxies properties such as
$\Sigma_{\rm SFR}$.
