The delayed radio emission in the black hole X-ray binary MAXI J1348$-$630

As shown in Fig. 2, during the rising phase, the onsets of both the radio emission and X-ray Comptonization were detected around MJD 58510. After that, the X-ray Comptonization first peaks around MJD 58516, and then fades towards the hard-to-soft state transition, showing a typical X-ray Compton variation in the rising hard state (You et al., 2021). Surprisingly, after the X-ray Compton peaking, the radio emission continuously increases with time, and appears to peak around MJD 58520. The ICCF analysis reveals that the radio emission lags the X-ray Comptonization emission by about 3 days. Considering the scenario that the radio delay is caused by jet traveling, we estimate the time lag to be the order of tens of minutes (You et al., 2023). This is significantly shorter than the observed radio lag of $\sim$ 3 days. Thus, the daily optical/radio delays in this work cannot be attributed to the jet traveling, and we rule out the possibility that X-rays are emitted by the jet in MAXI J348-630. The discovered radio delay suggests that the Compton emission originates from the ADAF rather than the jet during the rising phase in this source, and the Compton luminosity light curve indicates the evolution (the building up and fading away) of the ADAF.

During the state transition, the ADAF is accreting at a rate very close to the critical rate, and the radial energy advection is negligible (Narayan & Yi, 1995). Thus, almost all dissipated gravitational power in the ADAF can be radiated out. The flux of the inner ADAF is $F_{\rm ADAF}\propto\dot{m}(1/r_{\rm in}-1/r_{\rm tr})$, where $r_{\rm tr}$ is the truncated radius between inner ADAF and the outer thin disk. In the hard-to-soft state transition, the mass accretion rate increases with decreasing truncated radius (Esin et al., 1997). This implies that the ADAF radiates at its peak flux $F_{\rm ADAF}^{\rm peak}$ while the increase of the accretion rate is balanced by the decline of the truncated radius $r_{\rm tr}$. We note that the radio emission from the jets in this source is rather strong while it radiates at the peak flux in the hard X-ray band. However, the radio flux continuously rises till its peak at about 3 days after the X-ray peak. We conjecture that the large-scale magnetic field has already been sufficiently strong to arrest the accretion flow while the inner ADAF is radiating at its peak flux (Igumenshchev et al., 2003; Narayan et al., 2003; Begelman et al., 2022). The magnetic field of the MAD can be estimated as $B_{\rm MAD}\propto\dot{m}^{1/2}$ (see Narayan et al. 2003, and Equation S8 in You et al., 2023). The accretion rate keeps on increasing after the hard X-ray peak, and then the field in the MAD increases accordingly. The radio flux increases with the field strength of the MAD if the jets are assumed to be launched via the Blandford-Znajek mechanism (Blandford & Znajek, 1977). The truncated radius is approaching the ISCO with an increasing accretion rate, which leads to less amplification of the field in the shrinking ADAF (Cao, 2011). Ultimately, the field in the inner edge of the ADAF is too weak to meet the MAD criterion, and therefore, the radio emission from the jets declines afterward.

The production of transient ejections has been observed in a number of sources during short excursions from the soft to the intermediate state, and back to the soft state, when crossing the ‘jet line’ (e.g. Brocksopp et al., 2013). During the hard state, the compact jets are rarely resolved in radio, with their extent $\lesssim 10^{15}\,\rm cm$. However, during the hard-to-soft state transition, the transient jets could propagate far away from the core, reaching the distances of $\sim 10^{18}\,\rm cm$, which are at least three orders of magnitudes further away than the compact jet (see Zdziarski & Heinz, 2024, and the references therein). The power of the compact jet in the hard state, if assuming electron-positron pair dominance, is less than the MAD limit (Zdziarski & Heinz, 2024), which is consistent with the argument of a low magnetic field in a recent study of X-ray polarization (Barnier & Done, 2024). This implies that the compact jet in the bright hard state is launched from the standard and normal evolution (SANE) accretion flow (Narayan et al., 2012), which agrees with the argument of Fragile et al. (2023). It was then proposed in Zdziarski & Heinz (2024) that the difference in the propagation length between the compact jet during the bright hard state and the transient jet during the state transient could be attributed to both the jet composition and the power. Note that, in the case of the faint hard state, e.g., when a BHXRB leaves the soft state, the accretion mass was low compared to the bright hard state, and it was discovered to decay exponentially with time. In this case, the ADAF was observationally inferred to expand. In this case, the criteria for reaching the MAD state decreases with time (Narayan et al., 2003), whereas the magnetic field amplification becomes significant (You et al., 2023). It was proposed that the inner region of the ADAF would become magnetically dominated, forming the MAD in the decaying hard state.

As for MAXI J1348-630 studied in this work, the jet power during the hard-to-soft state transition was estimated to be at the MAD limit (Zdziarski et al., 2023), in which case the ejection was observed up to almost a parsec (Carotenuto et al., 2021b), similar to MAXI J1820+070 (Zdziarski & Heinz, 2024). In contrast, the compact jets in this source were not resolved and certainly did not propagate to large distances. Similar to MAXI J1820+070 discussed above, such different propagation lengths of the compact and the transient jets, may imply that the compact jet in the bright hard state is launched from the SANE while the transient jet during the state transient is associated with the MAD.

We also note that, as the radio emission decreases during the soft state, the optically thin radio emission from the core location was again detected between MJD 58573 and 58582, with the flux density increasing by at least two orders of magnitude (Carotenuto et al., 2021b). Interestingly, during such a short radio flare, the X-ray flux changed by less than about 30%. The physics mechanism behind these radio and X-ray activities in the soft state is interesting but unclear, which should be paid attention to in future studies of jet physics.

It is well known that the jet behavior correlates with the X-ray spectral states (Fender et al., 2004). The jet quenching is usually associated with the hard-to-soft state transition, while the hard X-ray from the ADAF/corona decreases and the soft X-ray from the disk still increases (Fender et al., 1999). Current observations show that the relation between the jet quenching and the decreasing of hard X-rays is complex. The nearly simultaneous hard X-ray/radio emission during the outburst rise of MAXI J1820+070 indicates that the jet quenching was almost simultaneous with the weakening of the ADAF/corona (You et al., 2023), which was also observed in the 1998 outburst of GX 339-4 (Fender et al., 1999). However, the radio-X-ray time lag in MAXI J1348-630 indicates that the jet started to be quenched at about 3 days after the state transition, which is similar to the 1999 outburst of XTE J1859+226 (Brocksopp et al., 2002). Yan & Yu (2012) even discovered that the jet quenching started 10 days before the state transition. Russell et al. (2020) reported an onset of jet quenching in the infrared band in MAXI J1535-571, suggesting that the jet quenching may start earlier than predicted by radio detection. To better understand the coupling between the jet and hot corona/ADAF, more high-cadence multi-wavelength observations across the state transition are required in the future.

The ICCF analysis shows that the X-ray Comptonization and radio synchrotron emission appear to be nearly simultaneous. The simultaneous hard X-ray/radio emissions could be attributed to the possibility that the X-ray emission originates from the Comptonization within the radio jet (Markoff et al., 2005). Alternatively, the X-ray emission may come from the ADAF, rather than the radio jet (Dai et al., 2023; You et al., 2023). In this case, the accretion flow and the magnetic field threading it in the outer thin disk are efficiently dragged to pass through the ADAF with large radial velocity, towards the BH (Cao, 2011).

During the outburst decay of MAXI J1820+070, the radio luminosity started to decrease after the magnetic field reached the MAD criterion, since both the accretion rate and the relevant magnetic field fed into the ADAF decreased with time (You et al., 2023), which led to the 8 days delay between the radio and Comptonization luminosity. The different cross-correlations between radio and Comptonization in MAXI J1348-630 and MAXI J1820+070 implies that the coupling between the ADAF and jet are complex, which may depend on some properties of the different BHXRB systems.

Correlations between the radio and X-ray luminosity (R-X correlation, $L_{\mathrm{R}}\propto L_{\mathrm{X}}^{\beta}$) in the BHXRBs were also explored to study the accretion-jet coupling. The X-ray luminosity in a narrow-band 1–10 keV (or 3–9 keV) was usually used in the studies of R-X correlations (Corbel et al., 2003; Russell et al., 2006). Carotenuto et al. (2021a) studied the R-X correlations using the X-ray luminosity in 1–10 keV obtained from Swift/XRT. They obtained a slope $\beta=0.95$ for $L_{\mathrm{X}}>6.3\times 10^{35}\rm erg\,s^{-1}$. Here, we revisit the R-X correlations, using the observations of Insight-HXMT. We convert the radio and X-ray comptonization fluxes to the radio and comptonization luminosities, using a distance of 2.2 kpc (Chauhan et al., 2021) ¹¹1Note that the distance to the source was also estimated to be $D\approx 3.4\pm 0.3$ kpc, based on X-ray detections of a dust-scattering halo (Lamer et al., 2021)., which is consistent with Carotenuto et al. (2021a). It turns out that we derived the same slope of $\beta=0.95$, if also using the X-ray luminosity in 1–10 keV. However, this limited-band X-ray luminosity inevitably contains the contributions from the Comptonization by the ADAF/jet, as well as, the reflection by the accretion disk which is illuminated by the Comptonized photons out of the ADAF/jet (You et al., 2021).

During the rising phase, the X-ray energy spectrum gradually softens over time, with the photon index $\Gamma$ increasing from 1.53 to 2.22. Meanwhile, the reflection component strengthens, as the reflection fraction (defined as the ratio of the coronal intensity illuminating the disk to the coronal intensity reaching the observer) increases from 0.3 to 2.0. As a result, the portion of the Comptonization in the total luminosity decreases over time. The ratio of the 1–10 keV comptonization luminosity to the 1–10 keV total luminosity decreases from 40% to 5%. Thus, the limited-band X-ray luminosity doesn’t directly probe the radiation output from the Comptonization source, if the reflection component is comparable to the Comptonzation component. Moreover, the majority of the overall Comptonization in the hard state is emitted around the hard X-ray band (Zdziarski et al., 2004; Islam & Zdziarski, 2018), which is thus underestimated if only considering the narrow-band 1–10 keV (or 3–9 keV) flux. In You et al. (2023), the Comptonization luminosity $L_{\mathrm{C}}$ in the broadband 0.1-100 keV is estimated by the spectral fits, which is utilized in the studies of the radio/X-ray correlation $L_{\mathrm{R}}\propto L_{\mathrm{C}}^{\beta}$ for MAXI J1820+070. They found that the correlation during the soft-to-hard state transition apparently deviates from the one in the hard state.

In this work, we use the estimated Comptonization luminosity $L_{\mathrm{C}}$ in 0.1-100 keV and the radio luminosity $L_{\mathrm{R}}$ from the core location to investigate the R-X correlation in the outburst of MAXI J1348+630. We find that:

• •

  $L_{\mathrm{R}}\propto L_{\mathrm{C}}^{3.04\pm 0.93}$,   for the rising phase;

• •

  $L_{\mathrm{R}}\propto L_{\mathrm{C}}^{1.11\pm 0.15}$,   for the mini-outburst.

The corresponding correlations are plotted in Fig. 4. Note that, for the rising phase, the time-delay of $\sim$ 3 days between the radio and X-ray is taken into account in the radio-Compton correlation analysis, which results in a strong correlation with Spearman rank correlation coefficients of $\sim 0.8$ and null hypothesis probability of $\sim 0.3\%$. In comparison, if not considering this time delay of $\sim$ 3 days, the correlation becomes worse with Spearman rank correlation coefficients of $\sim 0.6$ and null hypothesis probability (of no correlation) of $\sim 4\%$. For MAXI J1820+070, You et al. (2023) found that $L_{\mathrm{C}}$ and $L_{\mathrm{R}}$ follow a correlation $L_{\mathrm{R}}\propto L_{\mathrm{C}}^{0.53\pm 0.06}$ during the rising hard state during which there is no daily delay. However, during the soft-to-hard state transition, the radio emission was discovered to lag the X-ray Comptonization by $\sim$ 8 days. Moreover, if the radio lag of $\sim$ 8 days is not taken into account, the luminosity $L_{\mathrm{C}}$ and $L_{\mathrm{R}}$ apparently deviate from the above correlation during the rising hard state. In this work, we revisit the radio/X-ray correlation during the soft-to-hard state transition and the decaying hard state of MAXI J1820+070, taking the radio lag of $\sim$ 8 days into account. It turns out that the slope is $\beta=0.80$, with Spearman rank correlation coefficients of $\sim 0.98$ and null hypothesis probability less than $\sim 0.1\%$. This is steeper than the derived slope $\beta=0.53$ during the rising hard state (You et al., 2023). This highlights the necessity of considering the time delay in the R-X correlation analysis if the time delay is detected. As for the decaying phase, the radio-Compton relation does not follow a single power law (see Figure 4). This has also been found in Carotenuto et al. (2021a) that R-X relation is broken at the 1–10 keV luminosity $L\sim 6.3\times 10^{35}\,\rm erg\,s^{-1}$. Therefore, we cannot derive a monotonic correlation during the decaying phase.

As for the steep R-X correlation with $\beta\sim 3$ derived above, one of the possible scenarios is the jet Doppler boosting (e.g. Russell et al., 2014), which requires a low inclination of the source. Up to date, the inclination angle of MAXI J1348-630 is not well constrained, but it was estimated to be low in previous studies, e.g., $\lesssim 46^{\circ}$ (Carotenuto et al., 2021b), and $\sim 29^{\circ}$ (Carotenuto et al., 2022). The alternative cause of such a steep correlation is attributed to the outer thin disk in the truncation scenario being radiation pressure dominated (Y.H. Jiang et al. 2024, in preparation). In this case, the field generated in the thin disk will depend on the accretion rate as $B\sim\dot{m}$. In comparison, if the thin disk is gas press dominated, the field generated will depend on the accretion rate as $B\sim\dot{m}^{3/5}$. Given that the field in the outer disk is dragged through ADAF towards BH, this makes the radio luminosity (jet power) vary more sensitively with the accretion rate, and finally leads to a steeper radio/X-ray correlation.

As for the mini-outburst, the R-X correlation slope of $\sim 1.1$ is much flatter than the one for the rising phase of the main outburst. Furthermore, there is most likely no delay between the radio and X-ray Compotonization. These indicate an intrinsic difference in the accretion-jet coupling between the main outburst and the mini-outburst, which will be studied in future work.