A quantitative explanation of the radio - X-ray correlation in black-hole X-ray binaries

Accretion of matter onto compact objects is tightly connected with ejection of matter from them.  After all, it is the accretion flow that feeds the outflow.  Despite many efforts, this connection has not become clear.  In particular, the observed correlation between the radio and X-ray fluxes in the hard state of black-hole X-ray binaries (BHXRBs) has been around for more than two decades now. It is currently accepted that the hard X-rays in BHXRBs come from Comptonization in the corona and the radio emission from the jet. The jet and the corona, however, are separate entities with hardly any communication between them, apart from the fact that the jet is fed from the corona.  It is also widely accepted that the accretion flow around black holes in BHXRBs consists of an outer thin disk and an inner hot flow. From this hot inner flow, an outflow must emanate in the hard and hard-intermediate states of the source. By considering Compton up-scattering of soft disk photons in the outflow (i.e., in an outflowing "corona") as the mechanism that produces the hard X-ray spectrum, we have been able to explain quantitatively a number of observed correlations. Here, we investigate whether this outflowing "corona" can explain the observed radio - X-ray correlation also.  
We consider a parabolic outflow and compute the radio emission at 8.6 GHz coming from it, as well as the power-law photon-number spectral index Γ of the Comptonized hard X-rays produced in it. Thus, we have a correlation between the computed radio flux F_R at 8.6 GHz and the computed spectral index Γ of the hard X-ray spectrum. This correlation is actually a theoretical prediction, since both F_R and Γ are computed from the model and, to our knowledge, no such correlation has been constructed from observations for the hard and the hard-intermediate states. This prediction can be confirmed or rejected in future outbursts of GX 339-4. From observations of GX 339-4, we also produce a correlation between the observed X-ray flux F_X and the observed index Γ. Thus, for each value of Γ, observed/computed, we have the corresponding values of the observed F_X and the computed F_R, which we plot one against the other.  We find that, in the hard state of GX 339-4, our idea reproduces the observed correlation of F_R ∝ (F_X)^0.6 . In addition, in the hard-intermediate state of GX 339-4, we predict that, in future outbursts of the source, the F_R will exhibit first a sudden increase and then a sharp drop within a very narrow range of values of F_X . Also, since in a parabolic outflow the density is largest at its bottom, the transverse optical depth at the bottom of the outflow is very large (typically 10 to 100) and so the soft input photons, that are Comptonized there, see something like a "slab" above their emission. This may explain naturally the observed X-ray polarization from BHXRBs.