Mapping the Polarization of the Radio-Loud Ly$\alpha$ Nebula B3
  J2330+3927
  Lya nebulae, or "Lya blobs", are extended (up to ~100 kpc), bright (L[Lya] >
10^43 erg/s) clouds of Lya emitting gas that tend to lie in overdense regions
at z ~ 2--5. The origin of the Lya emission remains unknown, but recent
theoretical work suggests that measuring the polarization might discriminate
among powering mechanisms. Here we present the first narrowband, imaging
polarimetry of a radio-loud Lya nebula, B3 J2330+3927 at z=3.09, with an
embedded active galactic nucleus (AGN). The AGN lies near the blob's Lya
emission peak and its radio lobes align roughly with the blob's major axis.
With the SPOL polarimeter on the 6.5m MMT telescope, we map the total (Lya +
continuum) polarization in a grid of circular apertures of radius 0.6"
(4.4kpc), detecting a significant (>2sigma) polarization fraction P in nine
apertures and achieving strong upper-limits (as low as 2%) elsewhere. P
increases from <2% at ~5kpc from the blob center to ~17% at ~15-25kpc. The
detections are distributed asymmetrically, roughly along the nebula's major
axis. The polarization angles theta are mostly perpendicular to this axis.
Comparing the Lya flux to that of the continuum, and conservatively assuming
that the continuum is highly polarized (20-100%) and aligned with the total
polarization, we place lower limits on the polarization of the Lya emission
P(Lya) ranging from no significant polarization at ~5 kpc from the blob center
to ~ 3--17% at 10--25kpc. Like the total polarization, the Lya polarization
detections occur more often along the blob's major axis.
