Surface Brightness Profile of Lyman-$\alpha$ Halos out to 320 kpc in
  HETDEX
  We present the median-stacked Lyman-$\alpha$ surface brightness profile of
968 spectroscopically selected Lyman-$\alpha$ emitting galaxies (LAEs) at
redshifts $1.9<z<3.5$ in the early data of the Hobby-Eberly Telescope Dark
Energy Experiment (HETDEX). The selected LAEs are high-confidence
Lyman-$\alpha$ detections with large signal-to-noise ratios observed with good
seeing conditions (point-spread-function full-width-at-half-maximum $<1.4"$),
excluding active galactic nuclei (AGN). The Lyman-$\alpha$ luminosities of the
LAEs are $10^{42.4}-10^{43}\, \mathrm{erg}\, \mathrm{s}^{-1}$. We detect faint
emission in the median-stacked radial profiles at the level of $(3.6\pm
1.3)\times
10^{-20}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}\,\mathrm{arcsec}^{-2}$
from the surrounding Lyman-$\alpha$ halos out to $r\simeq 160$ kpc (physical).
The shape of the median-stacked radial profile is consistent at
$r<80\,\mathrm{kpc}$ with that of much fainter LAEs at $3<z<4$ observed with
the Multi Unit Spectroscopic Explorer (MUSE), indicating that the
median-stacked Lyman-$\alpha$ profiles have similar shapes at redshifts $2<z<4$
and across a factor of $10$ in Lyman-$\alpha$ luminosity. While we agree with
the results from the MUSE sample at $r<80\,\mathrm{kpc}$, we extend the profile
over a factor of two in radius. At $r>80\,\mathrm{kpc}$, our profile is flatter
than the MUSE model. The measured profile agrees at most radii with that of
galaxies in the Byrohl et al. (2021) cosmological radiative transfer simulation
at $z=3$. This suggests that the surface brightness of a Lyman-$\alpha$ halo at
$r\lesssim 100$ kpc is dominated by resonant scattering of Lyman-$\alpha$
photons from star-forming regions in the central galaxy, whereas at $r > 100$
kpc it is dominated by photons from galaxies in surrounding dark matter halos.
