The MAVERIC Survey: Variable Jet-accretion Coupling in Luminous Accreting Neutron Stars in Galactic Globular Clusters

We obtained VLA observations of NGC 6712 over seven epochs, each of duration 1 or 2 hr, in 2014 April and May shown in Table 1. X1850–087 is strongly variable in the radio over this time span, with a mixture of tight upper limits and clear detections.

First we discuss our quasi-simultaneous VLA and Swift radio and X-ray observations, with the VLA data obtained on 2014 April 5 and the X-ray data on 2014 April 6. The source is not detected with the VLA at either 5.0 or 7.4 GHz during these observations, with 3σ upper limits of <12.9 μJy and <12.3 μJy, respectively. Averaging the subbands gives a 3σ upper limit of <9.0 μJy at an average frequency of 6.2 GHz.

The quasi-simultaneous Swift/XRT X-ray observations are well fit by an absorbed power law with Γ = 2.30 ± 0.13, giving an unabsorbed flux of (1.6 ± 0.1) × 10⁻¹⁰ erg s⁻¹ cm⁻². This flux corresponds to an X-ray luminosity of 1.2 × 10³⁶ erg s⁻¹. We note that a photon index of Γ ∼ 2.3 is atypically soft at this X-ray luminosity (Wijnands et al. 2015), perhaps suggesting that the true spectrum is instead a composite that has contributions from both a power-law and a thermal disk spectrum. Unfortunately, the quality of the Swift data is not high enough to distinguish between this hypothesis and the simpler single power law. If we try such a two-component fit with the photon index fixed to Γ = 1.7, both the fit quality and inferred flux are essentially identical, indicating that the presence of a two-component X-ray spectrum is plausible, but unproven. If indeed X1850–087 is in the hard state, then Figure 1 shows that this source has the most stringent quasi-simultaneous 3σ radio continuum upper limit (L_R ∼ 3 × 10²⁸ erg s⁻¹) for a hard state neutron star LMXB at L_X ≳ 10³⁶ erg s⁻¹.

While X1850–087 is only marginally detected in daily MAXI (4–10 keV) and Swift/BAT (15–50 keV) observations in this time frame, binned light curves do show detections (Figure 4). These light curves show that there is no meaningful variability in the flux or hardness of X1850–087 in the data surrounding the 2014 April 5–6 radio and X-ray observations, suggesting the quasi-simultaneous Swift/XRT data are representative.

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We also delve into the 2014 April 5 radio nondetection by imaging the data in individual 10 minute scans, averaging together both basebands in the uv plane. The source was undetected in all of the individual scans, with a typical per-scan rms of about 9 μJy (and hence a per-scan 3σ limit of ≲27 μJy). We conclude that the limit in the entire 2 hr block is representative and that the binary is indeed not detected in these radio continuum data at this time.

In contrast to the strong radio upper limit in 2014 April, X1850–087 is clearly detected in several other radio observations in 2014 May at >10σ significance (Table 2). Of the six VLA epochs obtained during 2014 May, in five of these X1850–087 has a flux density >100 μJy. In one epoch (2014 May 13), X1850–087 is again not detected (with a formal 3σ upper limit of <9.5 μJy at a combined average frequency of 6.2 GHz), but is well detected in bracketing observations on May 9 and May 18. This demonstrates that the radio flux density of X1850–087 is changing by at least a factor of ∼20 in just a few days.

While there is no Swift/XRT X-ray coverage of these 2014 May epochs, the binned MAXI and Swift/BAT light curves indicate that an X-ray brightening and a transition to a softer state occurred around the time of these 2014 May VLA observations (Figure 4). Hence, one interpretation of the overall higher radio flux density from 2014 May 5–9 and May 18–20 is that it is associated with this transition, perhaps due to the launching of discrete ejecta. A challenge to this simple interpretation is the nondetection on May 13, which would require a separate transition to the baseline fainter harder state (or perhaps an intermediate hard state) between May 9 and 13 and a reflaring to a softer higher state by May 18. As we lack daily Swift/XRT data during this period we cannot definitively rule out this possibility, but it is not supported by the binned MAXI and Swift/BAT light curves, and would also represent phenomenology not previously observed in the Rossi X-ray Timing Explorer (RXTE) light curve for the source (see the next subsection).

Absent this fast flaring behavior, we know of no straightforward explanation for such dramatic variability on these timescales in an ultracompact system, which are much longer than the 0.014 days (20.6 minutes) orbital period but much shorter than, for example, the ≳100 days superorbital periods observed in some other ultracompact systems such as 4U 1820–30 (Section 3.1).

In the May 13 radio data for X1850–087, as in the April 5 data, the source is not detected in 10 minute individual scans during the 2 hr observation block. This is consistent with the idea that any orbital variations—which ought to be averaged out over individual blocks that represent ∼6 orbital periods—are not the primary cause of the variability of X1850–087.

We also imaged the individual scans for the observing blocks adjacent to the May 13 block in which the source was not detected. Each of these is a 1 hr block with five 8 minute scans on source. On May 9, there is no evidence for significant variation among the scans. On May 18, X1850–087 is brighter in the last two scans than in the first three at a formal level of about 3.5σ (5.0 GHz flux density of 235 ± 17 μJy in the last two, compared to 168 ± 9 μJy in the first three). Given its marginal significance, we do not attach too much import to the change, but if it were a real increase in the flux density, it could be short-term variability due to a recent jet ejection event.

We report spectral indices in Table 2 for our 2014 May observations of X1850–087. The spectral index ranged from inverted (brighter at higher frequencies) to flat, varying on short timescales: between May 5 and May 8 the source changed from an inverted spectral index (α = +0.50 ± 0.15) to a flatter value (α = −0.13 ± 0.11), then quickly back to inverted (α = +0.46 ± 0.13) on May 9, just one day later. The detection on May 18 was likewise flat, with α = +0.06 ± 0.13. We also calculated the per-scan spectral indices on May 18, finding a mean value of α = −0.04 ± 0.20 for the three early scans and α = +0.30 ± 0.16 for the two final scans. This could be consistent with a transition to more inverted emission during the putative flux density increase at the end of the May 18 observing block, though the evidence for a spectral index change is not formally significant. We discuss possible interpretations of the spectral index variations between different epochs in Section 4.2.3.

To improve the time baseline in the radio, we also make use of archival VLA data for X1850–087, reducing all of the C band data taken in the high-resolution (A or B) VLA configurations. This resulted in three additional epochs, all at 4.9 GHz, spanning 1989–1998 (Table 2). In 1989 February and 1991 December, X1850–087 was clearly detected, with flux densities consistent with our 2014 detections. In 1998 September, the source was not detected, but the 3σ upper limit (<135 μJy) is not particularly constraining as to whether order-of-magnitude variability was present in these older data.

Some previous papers on the 1989 February radio detection have claimed that X1850–087 is spatially resolved in these data (Lehto et al. 1990; Machin et al. 1990), which if true would have important implications for its interpretation. In our re-reduction of these data, given the moderate signal-to-noise ratio of the source, we find that either a point source or extended morphology offer reasonable fits. In our substantially deeper 2014 detections while the VLA was in its highest resolution A configuration, we find that in all epochs the morphology of X1850–087 is consistent with a point source.

There is at least partial X-ray data at these earlier epochs to contextualize these radio detections and limits. A Ginga/ASM (1–20 keV) light curve is available from 1987 March to 1991 October, and while it does not have spectral information, this X-ray light curve shows only modest variability (Figure 3). Two epochs of Ginga/LAC data in this time interval, from 1989 April and 1990 September, suggest an unabsorbed 1–10 keV luminosity of L_X ∼ 1.8 × 10³⁶ erg s⁻¹ (Kitamoto et al. 1992), consistent with the Swift/XRT data from 2014.

RXTE/ASM (1–10 keV) offers a well-sampled light curve of X1850–087 from 1995 to 2012, with evidence for a number of distinct episodes of brightening on typical timescales of weeks to months, as well as lower-level variability. In the vicinity of the 1998 September radio upper limit, the source has an X-ray flux typical of the long-term value (Cartwright et al.2013).

The most notable feature of the long-term X-ray light curve of X1850–087 is that it spends a relatively short amount of time in obvious bright states. As a rough estimate, we use the RXTE/ASM light curve, which has the best sampling and decent sensitivity of the long-term light curves. If we make a somewhat arbitrary designation that the source is in a bright state if the X-ray luminosity is ≳3 × 10³⁶ erg s⁻¹, then it spends only ∼6% in such bright states (see also Cartwright et al. 2013). This percentage increases to ∼15% if a more conservative luminosity of ≳2 × 10³⁶ is used, though Figure 3 shows that this value would likely also encompass periods of typical persistent activity outside of a bright state. Hence the long-term X-ray light curve shows that it is unlikely that two random radio continuum observations in 1989 and 1991 would have been taken during this uncommon flaring/bright state purely by chance.

Overall, we conclude that a straightforward flaring/state transition model for the dramatic radio variability of X1850–087 cannot be firmly ruled out. However, such a scenario also does not readily explain the data.

GRS 1747–312 is a transient LMXB first observed in 1990 by Granat (Pavlinsky et al. 1994). The compact object is a neutron star that shows Type I X-ray bursts (in't Zand et al. 2003a). Observations of X-ray eclipses have allowed the determination of a precise orbital period of 12.360 hr (in't Zand et al. 2003b). GRS 1747–312 has unusually frequent outbursts, clustering with a recurrence time around 4.5 months, though some outbursts occur much more quickly or slowly than this typical value. While the optical companion is unknown—due primarily to the high optical extinction toward Terzan 6—the relatively long orbital period and fast recurrence time suggest a somewhat high mass transfer rate from a subgiant donor (in't Zand et al. 2003b).

Here, we report three quasi-simultaneous (within ∼1 d) epochs of radio and X-ray observations of GRS 1747–312. The ephemerides of the X-ray eclipses are known with sufficient precision (in't Zand et al. 2003b) such that we can determine with certainty that none of the X-ray data discussed below occur during eclipses. For one of the five VLA epochs (2018 June 3), and for the ATCA observations, there is a partial overlap with the predicted eclipse times. The radio emission at these frequencies (5/7 GHz for VLA; 5.5/9 GHz for ATCA) is unlikely to be eclipsed even during X-ray eclipses (Maccarone et al. 2020), and in any case only a small amount of data—about 13% of the 2018 June 3 VLA epoch and ≲4% of the ATCA data—were taken during a predicted eclipse, so do not affect our results or conclusions.

In ATCA observations that began 2015 April 17 (labeled as 1 in Figure 1), the binary is well detected with 5.5 and 9.0 GHz flux densities of 213.3 ± 5.1 μJy and 172.4 ± 5.2 μJy, respectively, implying a radio spectral index α = −0.43 ± 0.08. We use this well-measured spectral slope to infer a 5.0 GHz flux density of 221.1 ± 6.4 μJy. The data quality of all three Swift/XRT observations discussed in this section (including the quasi-simultaneous one obtained on 2015 April 16) is relatively low, and fitting an absorbed power law to these X-ray data gives a strong degeneracy between Γ and N_H . The presence of variable dips for this source means that a constant N_H cannot necessarily be assumed. Therefore we have taken the approach of fitting two models to each of the Swift/XRT observations: one with fixed Γ and free N_H , and the other with both Γ and N_H fixed, the latter to the estimated foreground of 1.4 × 10²² cm⁻². Motivated by the typical photon indices for low-mass X-ray binaries at these X-ray luminosities and the spectral fitting results of Vats et al. (2018) for the source in quiescence, for the models with Γ fixed, we assume Γ = 1.7.

For the 2015 April 16 Swift/XRT data, the free N_H fit has ${N}_{H}={7.8}_{-4.0}^{+6.4}\times {10}^{22}$ cm⁻² and an unabsorbed X-ray flux of ${5.3}_{-2.4}^{+4.0}\times {10}^{-12}$ erg s⁻¹ cm⁻², corresponding to ${L}_{{\rm{X}}}={2.9}_{-1.3}^{+2.2}\times {10}^{34}$ erg s⁻¹ at the assumed distance of 6.8 kpc. For the fit with both N_H and Γ fixed, we find an unabsorbed flux of ${2.1}_{-0.8}^{+1.0}\times {10}^{-12}$ erg s⁻¹ cm⁻² (${L}_{{\rm{X}}}={1.2}_{-0.4}^{+0.5}\times {10}^{34}$ erg s⁻¹). These two flux estimates are consistent within the uncertainties; we plot the former (free N_H ) fit in Figure 1, but none of our conclusions depend on this choice.

At this first quasi-simultaneous epoch, GRS 1747–312 is the most radio-luminous neutron star LMXB ever observed at X-ray luminosities <10³⁵ erg s⁻¹, with a 5.0 GHz radio luminosity of (6.1 ± 0.2) × 10²⁸ erg s⁻¹. It sits near the upper envelope of the well-defined black hole radio/X-ray correlation (Figure 1).

Two VLA epochs in 2018 were obtained quasi-simultaneously with additional Swift/XRT data. On 2018 March 25 (labeled as 2 in Figure 1), GRS 1747–312 is not detected with the VLA at either 5.0 or 7.1 GHz, with 3σ upper limits of < 12.9 μJy and <11.7 μJy respectively. We fit the Swift/XRT observations on 2018 March 26 using the two models discussed above: Γ = 1.7 and N_H free, or Γ = 1.7 and N_H = 1.4 × 10²² cm⁻². For the former model we find ${N}_{H}={3.6}_{-2.1}^{+4.2}\times {10}^{23}$ cm⁻² and an unabsorbed X-ray flux of ${8.0}_{-4.8}^{+13.6}\times {10}^{-12}$ erg s⁻¹ cm⁻², equivalent to ${L}_{{\rm{X}}}={4.4}_{-2.6}^{+7.5}\times {10}^{34}$ erg⁻¹. For the fixed N_H model, the unabsorbed flux is ${1.0}_{-0.4}^{+0.6}\times {10}^{-12}$ erg s⁻¹ (${L}_{{\rm{X}}}={5.5}_{-2.4}^{+3.3}\times {10}^{33}$ erg⁻¹). For this observation, the X-ray fluxes from the two models are not quite consistent within the uncertainties. We conservatively plot the free N_H fit in Figure 1, but again none of our conclusions depend on the precise X-ray luminosity plotted for this radio upper limit.

GRS 1747–312 is again detected in the radio on 2018 April 30 (labeled as 3 in Figure 1) with 5.0 and 7.1 GHz flux densities of 26.9 ± 4.3 μJy and 32.1 ± 4.1 μJy, respectively, and a measured spectral index of α = +0.51 ± 0.57. In quasi-simultaneous Swift/XRT data from 2018 May 1, it is marginally detected, with an unabsorbed X-ray flux from the free N_H model of ${0.9}_{-0.6}^{+1.1}\times {10}^{-12}$ erg s⁻¹ cm⁻² (${N}_{H}={0.5}_{-0.5}^{+1.8}\times {10}^{22}$ cm⁻²) and a fixed N_H flux of $({1.3}_{-0.8}^{+1.2})\times {10}^{-12}$ erg s⁻¹ cm⁻². Since the former fit yields an N_H below the expected foreground, for this epoch we use the fixed N_H result in Figure 1, again emphasizing that this choice does not affect any of our conclusions. This gives ${L}_{{\rm{X}}}={7.1}_{-4.4}^{+6.4}\times {10}^{33}$ erg s⁻¹.

The VLA nondetection in March 2018 is more typical of the upper limits or detections for neutron star LXMBs at an L_X of ∼5 × 10³³ to 5 × 10³⁴ erg s⁻¹. However, the VLA detection about a month later (2018 April 30), at a fainter X-ray and radio luminosity than the bright 2015 detection, again approaches the regime more typically occupied by black hole LMXBs. In subsequent VLA epochs, taken from 2018 May 21 to June 3, the source is well detected at a comparable radio luminosity to the first VLA detection (Table 3), also showing modest (less than a factor of ∼2) variability. We have no simultaneous X-ray data at these latter radio epochs. We stack the four detected VLA epochs in the uv plane to get our best estimate of the VLA source position (Table 3). The spectral index in the stack is α = −0.51 ± 0.28, entirely consistent with the ATCA value. This suggests a similar origin for the radio continuum emission in both data sets, despite a factor of ∼6 drop in radio luminosity.

As we discuss below for AC 211 in M15 (Section 3.6.1), given the high orbital inclination of GRS 1747–312, it is possible that its unabsorbed X-ray luminosity is being underestimated, which might make GRS 1747–312 somewhat less of an outlier in the radio/X-ray correlation. Mitigating this possibility is that for GRS 1747–312 the central accretor is visible (unlike the case for AC 211), implying less "extra" obscuration for GRS 1747–312, if any.

It is worth considering whether we are sure that the radio and X-ray fluxes indeed reflect those of GRS 1747–312. The source was not in X-ray outburst during any of our observations, and the X-ray data were taken outside of eclipse.

While no other luminous X-ray sources are definitively known in Terzan 6, there are some puzzling observations that could be consistent with the existence of sources other than GRS 1747–312. In a 2009 Suzaku observation, Saji et al. (2016) observe a single source with L_X ∼ 6 × 10³⁴ erg s⁻¹, but see no evidence for an eclipse at the predicted time for GRS 1747–312, leading them to argue that this X-ray emission might arise from a different source.

Only a single X-ray source is evident in our Swift/XRT images of Terzan 6, and its position is consistent with the Chandra/HRC position of GRS 1747–312 (in't Zand et al. 2003a). But the Swift position is at best only constrained at the level of ∼2''–3'', so this association is not definitive in the dense environment of a globular cluster.

The position of the counterpart radio source in our ATCA and VLA observations is much better constrained. Since this source shows radio variability, has a radio spectral slope consistent with an LMXB, and already has a relatively high ratio of radio to X-ray luminosity, it seems fair to assert that this radio emission indeed arises from the same source as in the quasi-simultaneous Swift/XRT observations. The ATCA and VLA positions of the source can be found in Table 3, where we have conservatively assumed the 1σ uncertainties to be 10% of the beam. The ATCA and VLA positions are entirely consistent with each other, and the VLA position is about 07 from the best-fit Chandra/HRC position ¹¹ . This X-ray position has a formal 95% uncertainty of 04 (in't Zand et al. 2003a) based on an astrometric solution from three X-ray sources in the field. Hence, formally our radio continuum position is inconsistent with the Chandra one. Since absolute astrometry with Chandra is challenging, it seems plausible that the uncertainty is slightly underestimated, which could reconcile the positions.

A piece of evidence in favor of this argument is that neither the Chandra/HRC position of GRS 1747–312 nor our radio position are right at the center of the cluster: rather, they are northwest of the center in an HST/WFC3 F110W image (corrected to ICRS using Gaia DR2 stars), perhaps ∼14–2'' from the center and approaching the edge of the core. While it is possible that there are two unrelated X-ray binaries only 07 from each other but offset from the cluster center, an unrelated binary could plausibly be located anywhere in the core.

Finally, an XMM observation of Terzan 6 taken when GRS 1747–312 was not in outburst showed evidence for a single X-ray source with L_X ∼ 10³³ erg s⁻¹ (assuming our distance of 6.8 kpc). A spectral fit to this source required a two-component model with both a neutron star atmosphere and a hard power law, consistent with ongoing low-level accretion onto a neutron star (Vats et al. 2018). This X-ray luminosity is well below that of our Swift/XRT observations, which range from L_X ∼ 7 × 10³³ to 4 × 10³⁴ erg s⁻¹. Hence, to explain our Swift/XRT data, a non-GRS 1747–312 source would need to have a typical luminosity (in both 2015 and 2018) around L_X ∼ 10³⁴ erg s⁻¹. Being in neither quiescence nor full outburst, such luminosities are quite unusual for LMXBs, but are not unexpected for GRS 1747–312 given its very frequent outbursts. A hypothetical non-GRS 1747–312 source would also need to be X-ray and radio variable, as well as being very close to the position of GRS 1747–312. This verges on implausibility, and the most parsimonious explanation—pending, of course, additional X-ray and radio data—is that the only X-ray binary clearly detected thus far in Terzan 6 is indeed GRS 1747–312.

AC 211 (X2127+119) is an LMXB with a 17.1 hr period (Ilovaisky et al. 1993). It is one of the most famous accretion disk corona sources, where an edge-on inclination leads to the outer disk blocking the view of the central accretion flow, and only scattered emission is observed. The compact object is generally assumed to be a neutron star but due to the obscuration this has not been confirmed, since the normal identification tools (e.g., Type I X-ray bursts) cannot be used. AC 211 is a variable source that shows eclipses and flares (e.g., Hannikainen et al. 2005), with milliarcsecond-scale radio outflows or ejection events observed during outbursts (Kirsten et al. 2014).

AC 211 is detected in all five epochs of radio continuum imaging in 2011 (see Table 4) and is strongly variable, with its 5.0 GHz flux density ranging from 271 to 794 μJy on timescales as short as a few days. Radio eclipses, while a priori unlikely at these frequencies (Maccarone et al. 2020), are one possible explanation for this variability. Unfortunately, the existing ephemerides (Ioannou et al. 2003) extrapolated to the epoch of observation are not of sufficient precision, with a 1σ uncertainty of 0.23 days. While the absolute phase cannot be ascertained, the better-determined relative phases of the three low flux density epochs (<300 μJy) also do not line up, with a spread of ϕ = 0.35 in the mid-exposure times. Hence, if radio eclipses are the main explanation for the variability, they would need to last for longer than a third of the orbit, which is quite unlikely. Improved ephemerides would allow a test of this scenario.

In the single quasi-simultaneous epoch, we find an unabsorbed Chandra/HRC X-ray flux of (4.5 ± 0.1) × 10⁻¹⁰ erg s⁻¹ cm⁻² (using the parameters of the power-law model of White & Angelini 2001), equivalent to (5.7 ± 0.1) × 10³⁶ erg s⁻¹, and a 5.0 GHz radio flux density of 292.0 ± 6.3 μJy. This places it among the more radio-luminous neutron star LMXBs. At the higher, nonsimultaneous value of 794 μJy, the radio luminosity is a factor of 2.7 higher and hence more typical of stellar-mass black holes than neutron stars, though this higher radio flux density could be associated with a flare rather than steady emission.

It is worth noting that the X-ray luminosity measurement is almost certainly an underestimate, given the obscuration that prevents us from seeing the central source; the true L_X could be at least an order of magnitude higher than we estimate. We indicate this in Figure 1 by plotting the L_X measurement as a lower limit. Nonetheless, even at a much higher value of L_X, AC 211 would still be the among the most radio-luminous accreting neutron stars observed, especially in the latter nonsimultaneous epoch.

One possible explanation is that AC 211 has an intrinsic L_X that is around two orders of magnitude higher than observed and is a "Z source" accreting near the Eddington luminosity. The nearest and best-studied member of this class, Sco X-1, shows rapid variations in radio flux density and spectral index that are somewhat similar to those that we observe in AC 211 (Hjellming et al. 1990; Fomalont et al. 2001).

Another, speculative possibility is that AC 211 contains a black hole rather than a neutron star, which could potentially explain its radio-loud nature. This is statistically unlikely, and evidence from optical spectroscopy tends to favor a neutron star primary (van Zyl et al. 2004a, 2004b), but this prospect cannot be definitively ruled out.

White & Angelini (2001) used Chandra data to solve the mystery of why the obscured source AC 211 occasionally showed X-ray bursts that ought not to have been visible: these bursts are associated with a very nearby (separated by <3'') luminous LMXB, M15 X-2. White & Angelini (2001) found an ultraviolet-bright counterpart to M15 X-2 with a 23 minute periodicity identified with the orbital period (Dieball et al. 2005), showing that M15 X-2 is an ultracompact X-ray binary—one of only four confirmed persistent ultracompact systems known among Galactic globular clusters.

M15 X-2 had bright flaring events in both 2011 and 2013 (Sivakoff et al. 2011; Pooley & Homan 2013), with X-ray bursts detected during both events (Negoro et al. 2016). The 2011 flare is the one covered by the same five epochs of VLA data as for AC 211. M15 X-2 was strongly variable during this event (see Table 4), dropping from a 5.0 GHz flux density ∼150 to 64 μJy from 2011 May 22 to May 26, then back up to 205.0 ± 6.3 μJy on 2011 May 30. When re-visited in late August 2011, only upper limits were found, equivalent to a 3σ upper limit of <12.7 μJy at 5.0 GHz if both nondetection epochs are combined.

This radio time series clearly shows that the data from 2011 May 30 reflect an elevated level of radio emission during the brightening event. These data share the same quasi-simultaneous Chandra/HRC data set used for AC 211. At this epoch the unabsorbed Chandra flux was (9.46 ± 0.03) × 10⁻¹⁰ erg s⁻¹ cm⁻² (using the power-law model of White & Angelini 2001), equivalent to L_X = 1.2 × 10³⁷ erg s⁻¹. The radio/X-ray luminosity ratio on 2011 May 30 for M15 X-2 is high, close to the upper envelope of other accreting neutron star binaries observed at this X-ray luminosity.

We note that since these Chandra data were obtained with HRC, no spectral information was available to assess the state of M15 X-2 at this epoch. Instead, we analyze a Swift/XRT observation from 2011 May 30 that is strictly simultaneous with this VLA epoch. The Swift data have a lower spatial resolution than those from Chandra and hence also include AC 211. Since the count rate of AC 211 in the Chandra/HRC data is about 10% of that of M15 X-2, it makes a small but nonnegligible contribution to the Swift/XRT spectrum. Using this relative count rate and the power-law spectral model of White & Angelini (2001) for AC 211, we fit a combination of AC 211 and M15 X-2 to the Swift spectrum. We find that M15 X-2 is well fit by a composite disk (diskbb) + power-law model (Γ = 1.6 ± 0.1) with an unabsorbed X-ray flux of (1.04 ± 0.04) × 10⁻⁹ erg s⁻¹ (L_X = 1.3 × 10³⁷ erg s⁻¹). About three quarters of the flux in this band is contributed by the power law, suggesting that M15 X-2 is closer to the hard state than the soft state at the time of the radio observations that are plotted in Figure 1. The radio spectral index on 2011 May 30 was α = −0.02 ± 0.12, consistent with the presence of a compact jet at this epoch (see additional discussion in Section 4.2.1).

For additional context, Figure 6 shows the 2011–2014 MAXI and Swift/BAT X-ray light curves for M15. The 2011 and 2013 M15 X-2 flaring events are clearly visible in the MAXI X-ray light curve. The BAT data are noisy even when heavily binned, and AC 211 likely makes a substantial or even dominant contribution to the hard X-ray flux at all epochs, making it difficult to quantitatively interpret the plotted hardness ratio. Given the results of the Swift/XRT spectral analysis above, and that the 2011 May 30 radio data were obtained on the rise to peak, a reasonable interpretation is that these data were taken during a transition from a hard state to a softer one, with the jet not yet quenched.

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M15 X-3 is a so-called "very faint X-ray transient", peaking at ∼10³⁴ erg s⁻¹ (Arnason et al. 2015) rather than the value ≳10³⁶ erg s⁻¹ typical of LMXB transients. Even though it belongs to a different class than the other sources discussed in this paper, we include it for completeness given our use of deep radio continuum data for M15.

Arnason et al. (2015) show that the secondary of M15 X-3 is likely a low-mass main-sequence star but generally find no obvious explanation for the unusually faint X-ray outbursts, settling on propeller-mode accretion onto a rapidly rotating neutron star as perhaps the most likely explanation. They also consider the possibility that M15 X-3 could belong to the unusual class of "transitional millisecond pulsars," which show flat-spectrum radio emission in their subluminous accretion disk states. The detection of flat-spectrum radio emission would be suggestive evidence that M15 X-3 could belong to this subclass of millisecond pulsars.

We report a 3σ 5.0 GHz radio upper limit of <16.1 μJy on 2011 May 30, somewhat deeper than previously published radio continuum upper limits (Heinke et al. 2009). Quasi-simultaneous Chandra X-ray observations on 2011 May 31 yield an unabsorbed X-ray flux of $({3.5}_{-0.8}^{+1.4})\times {10}^{-13}$ erg s⁻¹ cm⁻² (Arnason et al. 2015), equivalent to ∼4 × 10³³ erg s⁻¹ at the distance of M15.

Our quasi-simultaneous radio upper limit of <16.1 μJy is ≲10²⁸ erg s⁻¹ at 5.0 GHz, which is not very constraining in the context of the known transitional millisecond pulsars or candidates, being comparable only to the detected radio emission in 3FGL J0427.9–6704 (Li et al. 2020). If we assume the system stayed in a similar state for the longer period of 2011 May 22 to 2011 August 22, we can use the deeper, nonsimultaneous radio data from Strader et al. (2012b) to set a 3σ upper limit of <6.3 μJy (≲4 × 10²⁷ erg s⁻¹ at 5.0 GHz). This limit is closer to the radio flux density observed for the transitional millisecond pulsar XSS J12270–4859 (Hill et al. 2011) but a factor of a few higher than that seen for the original transitional system PSR J1023 + 0038 (Deller et al. 2015).

We conclude the existing radio data cannot strongly constrain whether M15 X-3 exhibits transitional millisecond pulsar like radio properties, but that somewhat deeper data—perhaps challenging to obtain before the ngVLA era—would be interesting and useful.

For the 2015 ATCA observation of 4U 1820–30 we find α = −0.26 ± 0.05 in the hard state at L_X ∼ 10³⁷ erg s⁻¹. This well-measured spectral index has a slightly steep value, which could suggest it is partially optically thin at these frequencies. In addition, the 5.0 GHz radio flux density is essentially identical to that observed with ATCA in 2014 when 4U1820–30 was in a brighter (L_X ∼ 8 × 10³⁷ erg s⁻¹) soft state (Díaz Trigo et al. 2017). We note that when the present paper was close to submission, a new paper appeared with a comprehensive analysis of new and archival radio continuum and X-ray observations of 4U 1820–30 (Russell et al. 2021). Their results are consistent with, but more extensive than, the spectral index analysis in our paper, and suggest a transition from flat-spectrum steady compact jet emission in the low (island) state to steeper, possibly transient emission from jet ejecta in the high (banana) state.

The radio spectral index measurements for M15 X-2 all come from its X-ray brightening event in 2011 May. While we do not have X-ray spectral information for the first radio epoch, the radio continuum measurements are consistent with the likely initial presence of a discrete optically thick synchrotron blob (α = +0.90 ± 0.17) that then fades before recovering in radio luminosity to a flat spectrum (α = −0.02 ± 0.12) consistent with a compact jet at L_X ∼ 10³⁷ erg s⁻¹. As discussed in Section 3.6.2, at this latter epoch the system is consistent either being in a hard state or in a transition from a hard state to a soft state on the rise to the peak of the brightening event. We cannot definitively decide between these possibilities with these data: there is no evidence for the jet being quenched in the May 30 VLA observation. Overall, in 2011 May M15 X-2 shows "classic" radio behavior for LMXB in the initial stages of an X-ray flare/outburst.

The spectral index data for AC 211 are from the same data set as M15 X-2. However, the AC 211 data are not well explained by a standard hard/island state accreting neutron star model. In the first two observations (2011 May 22 and 26) the source shows a flat spectral index, consistent with a jet, while its radio luminosity increased by a factor of ∼3 between the first and second epochs. On 2011 May 30 (four days later), a steeper spectrum is observed (α = −0.55 ± 0.10) despite little change in the 5.0 GHz flux density. The next radio data available are three months later, on 2011 August 21, with both the spectrum and flux matching the previous epoch. But only one day later (2011 August 22) the 5.0 GHz flux doubles while the spectrum remains steep. To summarize, AC 211 shows substantial variation in both its radio luminosity and spectral index, but with no clear correlation between these.

Previously published observations of AC 211 also show evidence for variability. A pre-upgrade VLA image of M15 found a 1.4 GHz source with a flux density of ∼1.8 mJy at a position consistent with AC 211 (Kulkarni et al. 1990); this measurement is roughly consistent with the brightest 5.0 GHz flux density we see, assuming an α = −0.7 spectral index. However, very long baseline imaging finds a typical flux density of around 200 μJy at 1.6 GHz (Kirsten et al. 2014), which would be more consistent with a flat spectral index and a fainter overall flux level. An interpretation of AC 211 as a Z source accreting at close to the Eddington limit would give a straightforward explanation for the variability in the radio flux density and spectral index as due to discrete cases of jet ejection events (e.g., Hjellming et al. 1990; Fomalont et al. 2001). This interpretation appears to offer a more plausible explanation for the high radio luminosity of AC 211 than the speculative idea that the primary is a black hole.

For X1850–087 the interpretation of its spectral indices is mixed together with our interpretation of its fast radio variability, which is somewhat confusing (Section 3.2). The strongly inverted to flat spectral indices observed from 2014 May 5–9 appear to coincide with an X-ray flare, so can be explained as transient optically thick synchrotron emission. However, the source is undetected in the radio on 2014 May 13 before becoming bright again, with a flat to inverted spectral index, on May 18–20. Since the system was not detected in the radio in an apparently normal hard state (L_X ∼ 10³⁶ erg s⁻¹) on 2014 April 5, one possibility is that this system has no jet in the hard state at this L_X, and that it transitioned to the hard (or intermediate hard) state by May 13, before undergoing a fast reflare by May 18 associated with partially optically thick synchrotron emission. This scenario is not generally consistent with previous observations of X1850–087 in the X-ray and radio (Section 3.2) but cannot be ruled out either. The level of X-ray variability observed in X1850–087 and other persistent ultracompact LMXBs is already challenging to explain (e.g., in't Zand et al. 2007; Maccarone et al. 2010; Cartwright et al. 2013), and these new radio results are intriguing, highlighting the need for more coordinated radio and X-ray observations of this source.

The final source with spectral index measurements is GRS 1747–312. It was well detected in many radio epochs, despite being relatively faint with L_X ∼ 7 × 10³³ to 4 × 10³⁴ erg s⁻¹. At the first (and radio-brightest) epoch in 2015, sitting at the upper edge of the black hole radio/X-ray correlation, it had α = −0.43 ± 0.08 in ATCA data, more consistent with optically thin synchrotron than the flat/inverted emission associated with a steady jet. In an average of the 2018 VLA detections, it was fainter, with a mean α = −0.51 ± 0.29, consistent with the previous ATCA epoch. However, the range of values measured in the individual epochs was enormous (−1.77 ± 0.56 to +0.53 ± 0.50) and consistent with an intrinsic spread in the spectral indices, though this cannot be proven with these data due to the large uncertainties on the individual measurements.

The radio spectral indices and extreme radio variability are not consistent with expectations for a steady compact jet. One possibility is that the radio emission is associated with individual luminous discrete ejecta events; in principle it might be possible to image the brightest events with very long baseline interferometry. Other explanations may also be feasible. For example, the transitional millisecond pulsar PSR J1023 + 0038 is radio loud in quiescence with a variable spectral index that changes on short timescales, though it is not as radio loud as GRS 1747–312, and typically has a flatter/inverted spectral index (Deller et al. 2015). For PSR J1023 + 0038, the radio emission likely comes from nonsteady synchrotron bubbles created near the interface between the inner disk and the neutron star (Bogdanov et al. 2018). PSR J1023 + 0038 is not alone: other confirmed and candidate transitional systems show luminous radio emission in their subluminous disk states (Hill et al. 2011; Jaodand 2019; Li et al. 2020). It is not clear whether the same physical mechanism powers the radio emission in all of these systems, and a more "standard" propeller mechanism, producing a radio outflow, could instead be at play in a subset of them—or in GRS 1747–312.

In Figure 7, we show the spectral index of our sources as a function of their 5.0 GHz radio luminosity. More sources are plotted here than in Figure 1, since we do not require a simultaneous X-ray measurement to plot them here. Epochs with only radio upper limits are not plotted, since these have no spectral index measurements or constraints.

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As discussed for the individual sources above, there is no clear, strong relationship between radio luminosity and spectral index for individual sources or for the sample as a whole. There may be weak evidence for a change at the highest radio luminosities: those with L_R > 10²⁹ erg s⁻¹ have a mean α = −0.39 ± 0.02, while those below this radio luminosity have a mean α = +0.01 ± 0.04. However, this result is not very robust as it is dominated by the unusual source AC 211 at high luminosities. We also do not see evidence that the relationship between radio luminosity and spectral index varies substantially as a function of broad bins of X-ray luminosity. However, this is also not straightforward to interpret, since we do not have simultaneous X-ray data (and hence a proper spectral classification) for each measurement.