Where infall meets outflows: turbulent dissipation probed by CH$^+$ and
  Ly$\alpha$ in the starburst/AGN galaxy group SMM J02399$-$0136 at z$\sim$2.8
  We present a comparative analysis of the $\rm CH^+$(1-0) and $\rm Ly \alpha$
lines, observed with the Atacama Large Millimeter Array (ALMA) and Keck
telescope respectively, in the field of the submillimetre-selected galaxy (SMG)
SMM\,J02399$-$0136 at $z\sim2.8$, which comprises a heavily obscured starburst
galaxy and a broad absorption line quasar, immersed in a large $\rm Ly \alpha$
nebula. This comparison highlights the critical role played by turbulence in
channeling the energy across gas phases and scales, splitting the energy trail
between hot/thermal and cool/turbulent phases in the circum-galactic medium
(CGM). The unique chemical and spectroscopic properties of $\rm CH^+$ are used
to infer the existence of a massive ($\sim 3.5 \times 10^{10}$ ${\rm
M}_\odot$), highly turbulent reservoir of diffuse molecular gas of radius $\sim
20\,$kpc coinciding with the core of the $\rm Ly \alpha$ nebula. The whole cool
and cold CGM is shown to be inflowing towards the galaxies at a velocity $\sim$
400 km$\,s^{-1}$. Several kpc-scale shocks are detected tentatively in $\rm
CH^+$ emission. Their specific location in space and velocity with respect to
the high-velocity $\rm Ly \alpha$ emission suggests that they lie at the
interface of the inflowing CGM and the high-velocity $\rm Ly \alpha$ emission,
and signpost the feeding of CGM turbulence by AGN- and stellar-driven outflows.
The mass and energy budgets of the CGM require net mass accretion at a rate
commensurate with the star formation rate (SFR). From this similarity, we infer
that the merger-driven burst of star formation and black-hole growth are
ultimately fuelled by large-scale gas accretion.
