A quantitative explanation of the radio – X-ray correlation in black-hole X-ray binaries
Authors: Kylafis Nikolaos D. , Reig Pablo , Tsouros Alexandros 
Published: September 2023

Context. The observed correlation between the radio and X-ray fluxes in the hard state of black-hole X-ray binaries (BHXRBs) has been in existence for over two decades. It is currently accepted that the hard X-rays in BHXRBs come from Comptonization in the corona and the radio emission from the relativistic jet (Lorentz γ ≫ 1), which is a narrow structure of a few Rg = GM/c2 at its base. The jet and the corona, however, are separate entities with hardly any communication between them, apart from the fact that both are fed from the accreting matter.
Aims. It is also widely accepted that the accretion flow around the black holes in BHXRBs consists of a thin outer disk and a hot inner flow. From this hot inner flow, which has a positive Bernoulli integral, 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 the outflowing “corona”, which is a wider structure of tens to hundreds of Rg at its base) as the mechanism that produces the hard X-ray spectrum, we have been able to quantitatively explain a number of observed correlations. Here, we investigate whether this outflowing corona can also explain the observed radio – X-ray correlation. We remark that the outflowing corona (wide, with a low Lorentz γ) is completely separate from the relativistic jet (narrow, with a high Lorentz γ). The two may coexist, with the jet at the rotation axis and the corona around it.
Methods. We considered parabolic outflow models, which we have successfully used in the explanation of other correlations regarding GX 339-4 in the hard and hard-intermediate states, and computed the radio emission at 8.6 GHz coming from them, as well as the power-law photon-number spectral index Γ of the Comptonized hard X-rays produced in them. Thus, we have a correlation between the computed radio flux FR at 8.6 GHz and the computed spectral index Γ of the hard X-ray spectrum. This correlation is a theoretical prediction, since both FR and Γ are computed from the model and, to our knowledge, no such correlation has been constructed from observations. This prediction can be confirmed or proven wrong in future outbursts of GX 339-4. Based upon observations, we also produced a correlation between the observed hard X-ray flux FX and the observed index Γ. Thus, for each value of Γ, observed or computed, we have the corresponding values of the observed FX and the computed FR, which we plotted against each other.
Results. For GX 339-4, we found that our model calculations for FR and Γ, with Γ as the link between the observed FX and the computed FR, successfully reproduce the observed correlation of FR ∝ FX0.6 in the hard state. In addition, in the hard-intermediate state of GX 339-4, this correlation breaks down and we predict that, in future outbursts of the source, the FR will exhibit first a sudden increase and then a sharp drop within a very narrow range of values of FX. Such a sharp drop of the FR has been observed in other sources.
Conclusions. In our picture both the radio and the hard X-ray emission come from the same region, namely the outflow, and it is therefore not surprising that they are correlated. Since in a parabolic outflow with constant outflow speed the density is largest at its bottom, the soft photons, coming from below, see something appearing similar to a “slab”, with a moderate optical depth (up to ten in the hard state) along the outflow and an order of magnitude larger in the perpendicular direction. We remark that it is a slab geometry that is invoked to explain the observed X-ray polarization from BHXRBs. Because of this, we predict that the X-ray polarization of GX 339-4 will be parallel to the outflow in the hard state and perpendicular to it in the hard-intermediate state.