ALMA maps the Star-Forming Regions in a Dense Gas Disk at z~3
  We exploit long-baseline ALMA sub-mm observations of the lensed star-forming
galaxy SDP 81 at z=3.042 to investigate the properties of inter-stellar medium
on scales of 50-100pc. The kinematics of the CO gas within this system are well
described by a rotationally-supported disk with an inclination-corrected
rotation speed, v=320+/-20km/s and a dynamical mass of M=(3.5+/-1.0)x10^10Mo
within a radius of 1.5 kpc. The disk is gas rich and unstable, with a Toomre
parameter, Q=0.30+/-0.10 and so should collapse in to star-forming regions with
Jeans length L_J~130pc. We identify five star-forming regions within the ISM on
these scales and show that their scaling relations between luminosity,
line-widths and sizes are significantly offset from those typical of molecular
clouds in local Galaxies (Larson's relations). These offsets are likely to be
caused by the high external hydrostatic pressure for the interstellar medium
(ISM), P/kB=(40+/-20)x10^7K/cm3, which is ~10,000x higher than the typical ISM
pressure in the Milky Way. The physical conditions of the star-forming ISM and
giant molecular clouds appears to be similar to the those found in the densest
environments in the local Universe, such as those in the Galactic center.
