LAE galaxies are defined by a high equivalent width (EW > 20 Å) Lyα line and are believed to be composed of extremely large regions of active star formation. Many efforts have been made to detect and characterize LAE galaxies (e.g., Conselice et al. 2003; Conselice 2004; Ravindranath et al. 2006; Shimasaku et al. 2006; Bournaud et al. 2007; Ouchi et al. 2008, 2017; Elmegreen et al. 2009a, 2009b; Tacconi et al. 2010; Gronwall et al. 2011; Kashikawa et al. 2011; Mandelker et al. 2014; Moody et al. 2014; Guo et al. 2015). In general, these galaxies appear as clusters of bright clumps, sometimes with a background of continuum emission. Evidence suggests that these clumps are larger and brighter than most star-forming regions in nearby low-redshift galaxies (Elmegreen et al. 2009a). Efforts have been made in quantifying mass, star formation rates, gas composition, and kinematics, as well as other LAE properties (e.g., Nilsson et al. 2009; Ono et al. 2010a, 2010b; Swinbank et al. 2010; Tacconi et al. 2010; Shibuya et al. 2014; Livermore et al. 2015; Nakajima et al. 2016; Hashimoto et al. 2017). These have revealed a wealth of information about the early universe, but they are ultimately limited by LAE surface brightnesses. Most studies rely upon stacks of galaxies and can draw only limited inferences about individual LAEs. Other studies show that LAE dust content, particularly clumpy dust, in the interstellar medium (ISM) can have an impact on most LAE observables (Kobayashi et al. 2007, 2010; Verhamme et al. 2008; Duval et al. 2014). Finkelstein et al. (2009) showed that clumpy dust models can provide a good fit to a set of z ∼ 4.5 LAEs, although they invoked a multiphase ISM that may be unlikely to form in nature (Laursen et al. 2013). Nevertheless, dust in LAE galaxy ISM could cause some of the irregularity in LAE surface-brightness profiles (Buck et al. 2017). With limited resolution, however, it is difficult to make this distinction. A further challenge to morphological studies is that the clump sizes are near the resolution limit of instrumental point spread functions (PSFs) and often cannot be distinguished from point sources (Guo et al. 2015). As a result, direct imaging studies cannot decisively determine whether the clumps are different in nature from star-forming regions in our local universe or if the larger apparent size is merely an artifact of insufficient resolution (Shibuya et al. 2014; Kobayashi et al. 2016; Tamburello et al. 2017; Fisher et al. 2017).


As described in Sect. 1, the galaxy protocluster associated with 7C 1756+6520 is characterized by a high fraction of AGN protocluster members: seven AGN, including the central radio galaxy, have been spectroscopically confirmed in close proximity both spatially and in redshift space of the protocluster. This high AGN fraction detected so far, ~23%, makes the overdensity around 7C 1756+6520 similar to the interesting and well-studied cluster around the radio galaxy PKS 1138−262 from this point of view as well (Pentericci et al. 2002; Croft et al. 2005), in addition to the extension (see Sect. 5.2). The source PKS 1138−262 is a massive forming radio galaxy at z ~ 2.16 (Pentericci et al. 1998) that is surrounded by overdensities of Lyα emitters (Pentericci et al. 2000), extremely red objects (Kurk et al. 2004a; Koyama et al. 2013), Hα emitters (Kurk et al. 2004b), X-ray emitters (Pentericci et al. 2002), and an overdensity of dusty starbursts (Dannerbauer et al. 2014; Rigby et al. 2014), several of which are spectroscopically confirmed to be close to the radio galaxy redshift. Five of the 18 X-ray sources (~28%) detected by Pentericci et al. (2002) are AGN, including the central radio galaxy. From the soft X-ray luminosity function of AGN, Pentericci et al. (2002) estimated how many sources are expected in a given region of the cluster PKS 1138−262, finding that it contains about twice the number of expected AGN. More recently, Pentericci et al. (2013) also found high AGN fractions by studying eight galaxy groups from z ~ 0.5 to z ~ 1.1. They found that the fraction of AGN with Log LH > 42 erg s−1 in galaxies with MR  −20 varies from less than 5% to 22%, with an average value of 6.3%, which is more than double the fraction for massive cluster at similar high redshifts (e.g., Overzier et al. 2005). Martini et al. (2013) estimated that the cluster AGN fraction in a sample of 13 clusters of galaxies at 1  z  1.5 is ~3% for AGN with rest-frame, hard X-ray luminosity greater than LX, H ≥ 1044 erg s−1. Based on these findings, the galaxy protocluster around 7C 1756+6520 seems to be a particularly interesting object.


Various surveys of SNRs in our Galaxy and nearby galaxies have been carried out at radio, X-ray, Infrared (IR), and optical wavelengths. The first extragalactic SNR candidates were identified in the LMC by Mathewson & Healey (1964) and later confirmed with a combination of radio and optical techniques by Westerlund & Mathewson (1966). To date, a total of 60 SNRs have been confirmed in the LMC with an additional 14 suggested candidates (Maggi et al. 2016; Bozzetto et al. 2017; Maitra et al. 2019). However, sensitivity and resolution limitations severely reduce the effectiveness of the past and present generations of radio and X-ray searches for SNRs in galaxies beyond the Small and Large Magellanic Clouds (MCs) (Goss et al. 1980; Long et al. 1981; Cowan & Branch 1985; Matonick et al. 1997; Matonick & Fesen 1997; Millar, White & Filipovic 2012; Galvin & Filipovic 2014; Sasaki et al. 2018; Lin et al. 2020; Sasaki 2020). As a result, optical studies have produced the largest number (∼1200) of new extra-Magellanic SNR candidates. Optical extragalactic searches for SNRs are mainly done by using an emission line ratio criterion of the form [S ii]/H α > 0.4–0.5 (Mathewson & Clarke 1973; Dodorico, Dopita & Benvenuti 1980; Fesen 1984; Blair & Long 1997; Matonick & Fesen 1997; Dopita et al. 2010b; Lee & Lee 2014; Vučetić et al. 2019b, a, 2018; Lin et al. 2020). This criterion separates shock-ionization from photoionization in SNRs from H ii regions and Planetary Nebulae (PNe) (Frew & Parker 2010). SNR radiative shocks collisionally excite sulphur ions in the extended recombination region resulting in S+, hence the larger contribution of [S ii] accounting for an increase of the [S ii] to H α ratio. In typical H ii regions, sulphur exists predominantly in the form of S++, yielding low [S ii] to H α emission ratios. Ratios from narrow-band imaging are usually verified spectroscopically, since [N ii] lines at 6548 and 6584 Å can contaminate the H α images at an unknown and variable level. Spectroscopic observations of such emission nebulae also can provide other evidence of shock heating, such as strong [O i] λ6300 emission, elevated [N ii] to H α with respect to H ii regions, or high [O iii] electron temperatures, verifying the candidate as being an SNR (Blair, Kirshner & Chevalier 1981, 1982; Long et al. 1990; Smith et al. 1993; Blair & Long 1997). Although somewhat biased as an isolated criterion, this method is proven and a good way of identifying ordinary radiatively cooling SNRs in nearby galaxies. We note that young, Balmer-dominated SNRs (Chevalier, Kirshner & Raymond 1980) would be missed by this criterion.


We assumed a screen geometry and used PYNEB3 (Luridiana et al. 2015) to correct line fluxes for dust extinction via the Hα/Hβ ratio, adopting the O’Donnell (1994) reddening law with RV = 3.1 and a theoretical Hα/Hβ = 2.86. The extinction-corrected emission line luminosities of the H II regions were then computed using the distances reported in Table 1. For every H II region in our catalog, we also estimated the gas-phase metallicity and the gas ionization parameter by using extinction-corrected emission line fluxes. The gas-phase metallicity O/H was calculated using the Pilyugin & Grebel (2016) S calibration (Scal hereafter). This calibration relies on three diagnostic line ratios (i.e., [N II]/Hβ, [S II]/Hβ, and [O III]/Hβ) and provides an empirical calibration against H II regions that have direct constraints on their nebular temperatures and hence their abundances. This calibration is relatively insensitive to changes in gas pressure or ionization parameter, and we adopted it as fiducial approach in this paper (see Kreckel et al. 2019, for a discussion). In addition, for each galaxy we fit the radial metallicity gradient by using an unweighted least-square linear fitting of the trend between 12 + log(O/H) and the deprojected galactocentric radius (see Fig. A.19). The gas ionization parameter represents the ratio between the ionizing photon flux density and the gas hydrogen density. In this paper, we express the ionization parameter as q = U × c = Q(H0)/4πR2n, where c is the speed of light, U is the dimensionless ionization parameter, Q(H0) is the number of hydrogen ionizing photons (E >  13.6 eV) emitted per second, R is the empty (wind-blown) radius of the H II region, and n its hydrogen density. The ionization parameter is ultimately defined by the structure of an H II region (e.g., size, gas density, filling factor) and the properties of its ionizing source. Photoionization models show that it can be extracted via different diagnostic lines (Kewley & Dopita 2002; Dors et al. 2011). In this paper we use the calibration proposed by Diaz et al. (1991) based on the [S III](9069+9532)/[S II](6717+6713) line ratio. As the [S III]λ9532 Å line falls outside the wavelength range covered by MUSE, we assumed that [S III]λ9532 Å = 2.47[S III]λ9069 Å according to default atomic data in PYNEB (Luridiana et al. 2015). It should be noted that for about 3000 H II regions we are not able to estimate the ionization parameter due to lack of detection of the [S III]λ9532 Å emission line.


One has to be sure to rule out cases where inorganic chemistry can mimic the presence of life (“false positives”). Potential abiotic ozone production on Venus- and Mars-like planets has been discussed by Schindler & Kasting (2000, and references therein). While this is based on photolysis of e.g., CO2 and H2O and is thus limited in extent, a sustainable production of abiotic O3 which could build up to a detectable level has been suggested by Domagal-Goldman & Meadows (2010) for a planet within the habitable zone of AD Leonis with a specific atmospheric composition. Indeed, other studies confirm that abiotic buildup of ozone is possible (e.g., Hu et al. 2012; Tian et al. 2014); however, detectable levels are unlikely if liquid water is abundant, as e.g. rainout of oxidized species would keep atmospheric O2 and O3 low (Segura et al. 2007), unless the CO2 concentration is high and both H2 and CH4 emissions are low (Hu et al. 2012). False-positive detection of molecules such as CH4 and O3 is discussed by von Paris et al. (2011). Seager et al. (2013) present a biosignature gas classification. Since abiotic processes cannot be ruled out for individual molecules (e.g. for O3), searches for biosignature molecules should search for multiple biosignature species simultaneously. It has been suggested that the simultaneous presence of O2 and CH4 can be used as an indication for life (Sagan et al. 1993, and references therein). Similarly, Selsis et al. (2002) suggest a so-called “triple signature”, where the combined detection of O3, CO2 and H2O would indicate biological activity. Domagal-Goldman & Meadows (2010) suggest to simultaneously search for the signature of O2, CH4, and C2H6. Of course, care has to be taken to avoid combinations of biosignature molecules which can be generated abiotically together (see e.g. Tian et al. 2014). The detectability of biosignature molecules is discussed, e.g. by von Paris et al. (2011) and Hedelt et al. (2013). In particular, the simulation of the instrumental response to simulated spectra for currently planned or proposed exoplanet characterization missions has shown that the amount of information the retrieval process can provide on the atmospheric composition may not be sufficient (von Paris et al. 2013).


Characterizing the differential rotation (DR) realized at the base and in the convective envelope of solar-type stars is central to the understanding of their magnetic field generation, activity level, and rotation, as it is directly linked to the Ω effect (e.g., stretching of the poloidal magnetic field lines by large-scale shear). Hence, the role of DR in driving the star’s magnetic activity level and field properties should be clarified (Donahue et al. 1996). Doppler imaging (Donati & Collier Cameron 1997; Barnes et al. 2005), asteroseismology (Gizon & Solanki 2004; Reinhold et al. 2013; García et al. 2014), classical spot models (Lanza et al. 2014), and short-term Fourier transform (Vida et al. 2014) are methods to infer DR. The combination of all these observations on stellar rotation and magnetism helps constrain the trends linking rotation with stellar DR and magnetic activity. Various analyses of stellar DR revealed different dependencies between DR and star’s rotation (ΔΩ ∝ Ω
n
), with n varying between 0.15 and 0.7 (Barnes et al. 2005; Reiners 2006; Reinhold et al. 2013). There is no clear consensus in the community for now; some authors are even advocating that such laws should be derived per spectral stellar classes and that the confusion comes from mixing together F and K stars (Balona & Abedigamba 2016). Saar (2011), Brandenburg & Giampapa (2018) also propose that the dependency of the DR with the rotation rate may not be monotonic, with a break near Rossby equals unity. By contrast, a more systematic and stronger dependency is observed with the star’s temperature (

ΔΩ∝Teff8.92

, Barnes et al. 2005, Reinhold et al. 2013; and

ΔΩ∝Teff8.6

, Collier Cameron 2007). Hence, we expect large-scale shear to vary both in amplitude and profile (as a function of latitude and depth) as the global stellar parameters change. Some recent studies have confirmed this is happening in solar-type stars by inverting seismically their profile (Benomar et al. 2018), pointing to a possible antisolar DR state (e.g., slow equator/fast poles), which was possibly already guessed in F stars (Reiners 2007) and advocated to exist in numerical simulations (Matt et al. 2011; Gastine et al. 2014; Brun et al. 2015, see below).

short-term Fourier transform
are methods to infer DR.


ALMA is capable of resolving internal structures of molecular clouds even in external galaxies. In particular, the Large Magellanic Cloud (LMC) is an ideal laboratory to investigate high-mass star formation thanks to its nearly face-on view (Balbinot et al. 2015) and the close distance, ∼50 kpc (Schaefer 2008; de Grijs et al. 2014). It is also a great advantage to directly compare the distributions of molecular gas observed by ALMA and positions of massive YSOs identified by Spitzer and Herschel (e.g., Gruendl & Chu 2009; Chen et al. 2010; Seale et al. 2014) without any serious contamination in the line of sight. Earlier studies using the H i gas observations by Fukui et al. (2017) found that there are supergiant shells (Kim et al. 1999, 2003) and kiloparsec-scale gas flows caused by the last tidal interaction between the LMC and the Small Magellanic Cloud (SMC). Therefore, we may be able to examine the relation between such large-scale gas kinematics and the local star formation activities. Our present target in this paper is the N159W-South clump, which was discovered by our previous ALMA Cycle 1 observations (Fukui et al. 2015, hereafter Paper I) with an angular resolution of ∼1″ (∼0.24 pc) toward a GMC in the N159W region (e.g., Johansson et al. 1998; Minamidani et al. 2008, 2011). Paper I revealed that the GMC is composed of many filamentary molecular clouds and discovered the first example of protostellar outflows in the external galaxies. Paper I also found that the protostellar source with a stellar mass of ∼37 M⊙ in the N159W-South clump is located toward an intersection of two filaments and suggested that the filament–filament collision triggered the protostar formation. Although the ALMA observations significantly improved our understanding of molecular cloud structures and star formation in this object, much higher-angular-resolution studies are needed to further resolve the filamentary structures down to a width of ≲0.1 pc (see, Arzoumanian et al. 2011, 2019) and investigate the star formation activities therein.


A spatial coincidence of the radio jet morphology and velocity dispersion of the ionised gas has already been reported for spatially-resolved spectroscopy of more luminous radio-quiet AGN (e.g. Husemann et al. 2013; Villar-Martín et al. 2017) and powerful compact radio sources (e.g. Roche et al. 2016), but it has been correctly proposed that the fast moving plasma itself can lead to radio emission that mimics jet activity (Zakamska & Greene 2014; Hwang et al. 2018). In the case of HE 1353−1917 we can rule out that the ionised plasma is creating the radio emission because the high-velocity ionised gas traced by [O III] is significantly displaced compared to the observed jet-like radio emission. Hence, we think that the radio jet is transferring its energy and momentum to the ambient medium through an extended shock front, which creates turbulence in a dense clumpy ISM. Such a great impact of the radio jet has been observationally shown in many cases (e.g. Villar-Martín et al. 1999, 2014; O’Dea et al. 2002; Nesvadba et al. 2006; Holt et al. 2008; Guillard et al. 2012; Harrison et al. 2015; Santoro et al. 2018; Tremblay et al. 2018; Jarvis et al. 2019) and theoretically supported through detailed hydrodynamic simulations (e.g. Krause & Alexander 2007; Sutherland & Bicknell 2007; Wagner & Bicknell 2011; Wagner et al. 2012; Cielo et al. 2018; Mukherjee et al. 2018). As we discussed in Sect. 3.6, the jet power alone is sufficient to energetically drive the outflow because only a small fraction of the AGN luminosity would impact the thin disc of the galaxy implying conversion efficiencies of more than 10% of Lbol. Hopkins & Elvis (2010) proposed a two-stage process for efficient radiation-driven outflows. They describe a scenario in which an initial weak wind in the hot gas phase, possibly initiated by an accretion disc wind or a radio jet, creates additional turbulence in the surrounding medium so that massive gas clouds will subsequently expand and disperse. This expansion of gas clouds would significantly increase their apparent cross-section with respect to incident radiation field of the AGN. Such a two-stage process may increase the coupling efficiency by an order of magnitude. While we cannot directly confirm this process with our observations, the close alignment of the jet axis and the ionisation cone greatly suggest that the outflow is driven jointly by both mechanical and radiative energy with an unknown ratio of the two. The open question is whether the same powerful outflow could have developed without the fast radio jet impacting the cold gas directly given its unique orientation.


In addition to secular evolutionary processes, a disc galaxy like ours is expected to have experienced several accretion events in its recent and early past (Bullock & Johnston 2005; De Lucia & Helmi 2008; Stewart et al. 2008; Cooper et al. 2010; Font et al. 2011; Brook et al. 2012; Martig et al. 2012; Pillepich et al. 2015; Deason et al. 2016; Rodriguez-Gomez et al. 2016). While some of these accretions are currently being caught in the act, like for the Sagittarius dwarf galaxy (Ibata et al. 1994) and the Magellanic Clouds (Mathewson et al. 1974; Nidever et al. 2010; D’Onghia & Fox 2016), we need to find the vestiges of ancient accretion events to understand the evolution of our Galaxy and how its mass growth has proceeded over time. Events that took place in the far past are expected to have induced a thickening of the early Galactic disc, first by increasing the in-plane and vertical velocity dispersion of stars (Quinn et al. 1993; Walker et al. 1996; Villalobos & Helmi 2008, 2009; Zolotov et al. 2009; Purcell et al. 2010; Di Matteo et al. 2011; Qu et al. 2011; Font et al. 2011; McCarthy et al. 2012; Cooper et al. 2015; Welker et al. 2017), and second by agitating the gaseous disc from which new stars are born, generating early stellar populations with higher initial velocity dispersions than those currently being formed (Brook et al. 2004, 2007; Forbes et al. 2012; Bird et al. 2013; Stinson et al. 2013). These complementary modes of formation of the Galactic disc can be imprinted on kinematics-age and kinematics-abundance relations (Strömberg 1946; Spitzer & Schwarzschild 1951; Nordström et al. 2004; Seabroke & Gilmore 2007; Holmberg et al. 2007, 2009; Bovy et al. 2012a, 2016; Haywood et al. 2013; Sharma et al. 2014; Martig et al. 2016; Ness et al. 2016; Mackereth et al. 2017; Robin et al. 2017), and distinguishing between them requires full 3D kinematic information for several million stars, in order to be able to separate the contribution of accreted from in-situ populations, and to constrain impulsive signatures that are typical of accretions (Minchev et al. 2014) versus a more quiescent cooling of the Galactic disc over time. Accretion events that took place in the more recent past of our Galaxy can also generate ripples and rings in a galactic disc (Gómez et al. 2012b), as well as in the inner stellar halo (Jean-Baptiste et al. 2017). Such vertical perturbations of the disc are further complicated by the effect of spiral arms (D’Onghia et al. 2016; Monari et al. 2016b), which together with the effect of accretion events might explain vertical wave modes as observed in SEGUE andRAVE (Widrow et al. 2012; Williams et al. 2013; Carrillo et al. 2018), as well as in-plane velocity anisotropy (Siebert et al. 2012; Monari et al. 2016b). Mapping the kinematics out to several kiloparsec from the Sun is crucial for understanding whether signs of these recent and ongoing accretion events are visible in the Galactic disc, to ultimately understand to what extent the Galaxy can be represented as a system in dynamical equilibrium (Häfner et al. 2000; Dehnen & Binney 1998), at least in its inner regions, or to recover the nature of the perturber and the time of its accretion instead from the characteristics and strength of these ringing modes (Gómez et al. 2012b).


Lenticular galaxies seems to have a wider range of properties compared to ellipticals that resemble more the old definition of ETGs. However, even in ellipticals, large differences prevail. Recent observations of elliptical galaxies with Spitzer and Herschel (Temi et al. 2005, 2007a, 2007b, 2009; Smith et al. 2012; Agius et al. 2013; Mathews et al. 2013) have revealed that the far-infrared (FIR) luminosity LFIR from these galaxies can vary by ∼100 among galaxies with similar optical luminosity. The 70 μm band luminosities (from Temi et al. 2007a, 2009), is a good example of such a huge scatter in the FIR luminosity of elliptical galaxies. Some of the high L70 galaxies are members of a small subset of ellipticals having radio detections of neutral and molecular gas. A few others may be S0 galaxies which, because of their rotationally supported disks, often contain large masses of cold gas and dust. Ellipticals containing large excess masses of dust and cold gas probably result from significant galaxy mergers in the past. However a fraction of elliptical galaxies appear to be completely normal but L70 in these galaxies still ranges over a factor of ∼30, far larger than can be explained by uncertainties in the estimate of the FIR spectral energy distribution (SED) due to local stellar mass loss. While a significant fraction of the cold gas mass in low- to intermediate-mass ETGs is thought to have an external, merger-related origin (e.g., Davis et al. 2011), in the most massive ETGs the cold gas phases are presumably generated internally (Davis et al. 2011; David et al. 2014; Werner et al. 2014). Mergers with gas- and dust-rich galaxies have often been suggested for the origin of dust in all elliptical galaxies (e.g., Forbes 1991). Although the merger explanation is almost certainly correct in some cases, mergers cannot explain most of the observed scatter in L70. A crucial element in our understanding of the evolution of galaxies toward ETGs is the mutual role played by the major merging of galaxies and the secular star formation quenching. Neither of these scenarios yet accounts for all the observational evidence, and one could assume both contributing to some extent.


We derived the photospheric stellar parameters using three different techniques: the curve-of-growth approach, spectral synthesis match, and empirical calibration. The first method minimizes the trend of iron abundances (obtained from the equivalent width, EW, of each line) with respect to excitation potential and reduced EW respectively, to obtain the effective temperature and the microturbulent velocity, ξt. The gravity log g is obtained by imposing the same average abundance from neutral and ionized iron lines. We obtained the EW measurements using ARESv2
 5 (Sousa et al. 2015). We used the local thermodynamic equilibrium (LTE) code MOOG
 6 (Sneden 1973) for the line analysis, together with the ATLAS9 grid of stellar model atmosphere from Castelli & Kurucz (2003). The whole procedure is described in more detail in Sousa (2014). We performed the analysis on a co-added spectrum (SNR > 600), and after applying the gravity correction from Mortier et al. (2014) and adding systematic errors in quadrature (Sousa et al. 2011), we obtained Teff = 5346 ± 69 K, log g =4.60 ± 0.12, [Fe/H] =−0.40 ± 0.05, and ξt =0.78 ± 0.08 km s−1. The spectral synthesis match was performed using the Stellar Parameters Classification tool (SPC, Buchhave et al. 2012, 2014). It determines effective temperature, surface gravity, metallicity, and line broadening by performing a cross-correlation of the observed spectra with a library of synthetic spectra, and interpolating the correlation peaks to determine the best-matching parameters. For technical reasons, we ran the SPC on the 62 GTO spectra only7: the SNR is so high that the spectra are anyway dominated by systematic errors, and including the A40TAC_23 spectra would not change the results. We averaged the values measured for each exposure, and we obtained Teff =5389 ± 50 K, log g =4.49 ± 0.10, [M/H] = −0.36 ± 0.08, and v sin i  2 km s−1. We finally used CCFpams,8 a method based on the empirical calibration of temperature, metallicity, and gravity on several CCFs obtained with subsets of stellar lines with different sensitivity to temperature (Malavolta et al. 2017b). We obtained Teff = 5293 ± 70 K, log g =4.50 ± 0.15, and [Fe/H] = −0.40 ± 0.05, after applying the same gravity and systematic corrections as for the EW analysis. We list the final spectroscopic adopted values, i. e. the weighted averages of the three methods, in Table 3. From the co-added HARPS-N spectrum, we also derived the chemical abundances for several refractory elements (Na, Mg, Si, Ca, Ti, Cr, Ni). We used the ARES + MOOG method assuming LTE, as described earlier. The reference for solar values was taken from Asplund et al. (2009), and all values in Table 3 are given relative to the Sun. Details on the method and line lists are described in Adibekyan et al. (2012) and Mortier et al. (2013). This analysis shows that this iron-poor star is alpha-enhanced. Using the average abundances of magnesium, silicon, and titanium to represent the alpha-elements and the iron abundance from the ARES + MOOG method (for consistency), we find that [α/Fe] = 0.23.


Along with the time-domain detections, we identified J173438.35-504550.4 as a potential host galaxy for FRB 20201123A using robust statistical treatment given the relatively small localization error region. At face value, the low redshift of J173438.35-504550.4 appears at odds with the large dispersion measure for FRB 20201123A (${\rm DM}_{\rm FRB}\approx 434 \, {\rm pc \, cm^{-3}}$). Our Galaxy, however, contributes ${\rm DM}_{\rm ISM}\approx 200 \, {\rm pc \, cm^{-3}}$ (NE2001 gives 229 ${\rm pc \, cm^{-3}}$ and YMW16 gives 162 ${\rm pc \, cm^{-3}}$) from its interstellar medium and a presumed ${\rm DM}_{\rm Halo}\sim 50 \, {\rm pc \, cm^{-3}}$ from its halo (Prochaska & Zheng 2019). This leaves ${\approx}180 \, {\rm pc \, cm^{-3}}$ for the cosmos (DMcosmic) and the host (DMhost). At z = 0.05, the average cosmic contribution is $\langle {\rm DM}_{\rm cosmic}\rangle \sim 42\, {\rm pc \, cm^{-3}}$ (Macquart et al. 2020) but the intrinsic scatter in this quantity is predicted to be large. Adopting the best-fitting model to the Macquart relation by Macquart et al. (2020), the 95 per cent confidence interval is ${\rm DM}_{\rm cosmic}= [15,125] \, {\rm pc \, cm^{-3}}$. Allowing for the maximum value of this interval (which would imply a significant foreground galaxy halo), we recover a minimum host contribution of ${\rm DM}_{\rm host, min} \approx 60~\rm pc~cm^{-3}$. This is consistent with estimates for host galaxy contributions from theoretical and empirical treatments (Prochaska & Zheng 2019; James et al. 2022). For a true DMcosmic value of this sightline closer to (or below) the mean, the host contribution would exceed $100 \, {\rm pc \, cm^{-3}}$. Such values are inferred for other FRB hosts (e.g. FRB 20121102A; Tendulkar et al. 2017). In conclusion, we find no strong evidence to rule out the association with J173438.35-504550.4 based on its redshift and DMFRB. The significant host contribution to the DM, combined with the scattering in FRB 20201123A possibly originating in the host, shows that it shares similarities with other highly active, repeating FRBs like FRB 20121102A and FRB 20190520A and potentially resides in a turbulent and dense environment within the host.


The observation of massive neutron stars Demorest et al. (2010), Antoniadis et al. (2013) indicates that the EoS of nuclear matter must be very stiff in the regime of high densities and low temperatures. The degree of stiffness in the nuclear matter EoS is directly related to the repulsive interaction among particles at high densities, as well as to the particle content in the core of the stars. In particular, it has been extensively discussed in the literature whether exotic degrees of freedom might populate the core of neutron stars. On the one hand, it is more energetically favorable for the system to populate new degrees of freedom, such as hyperons (Dexheimer & Schramm 2008; Ishizuka et al. 2008; Bednarek et al. 2012; Fukukawa et al. 2015; Gomes et al. 2015; Maslov et al. 2015; Oertel et al. 2015; Lonardoni et al. 2015, 2016); Biswal et al. 2016; Burgio & Zappalà 2016; Chatterjee & Vidana 2016; Mishra et al. 2016; Vidaña 2016; Yamamoto et al. 2016; Tolos et al. 2017); Torres et al. 2017), delta isobars (Fong et al. 2010;Schurhoff et al. 2010; Drago et al. 2014; 2016; Cai et al. 2015; Zhu et al. 2016), and meson condensates (Ellis et al. 1995; Menezes et al. 2005; Takahashi 2007; Ohnishi et al. 2009; Alford et al. 2010; Fernandez et al. 2010; Mesquita et al. 2010; Mishra et al. 2010; Lim et al. 2014; Muto et al. 2015), in order to lower its Fermi energy (starting at about two times the saturation density). On the other hand, the EoS softening due to the appearance of exotica might turn some nuclear models incompatible with observational data, in particular with the recently measured massive neutron stars. One possible way to overcome this puzzle is the introduction of an extra repulsion in the YY interaction Schaffner & Mishustin (1996), Bombaci (2016), allowing models with hyperons to be able to reproduce massive stars (Dexheimer & Schramm 2008; Bednarek et al. 2012; Weissenborn et al. 2012; Banik et al. 2014; Bhowmick et al. 2014; Gusakov et al. 2014; Lopes & Menezes 2014; van Dalen et al. 2014; Yamamoto et al. 2014; Gomes et al. 2015). Another possible solution is the introduction of a deconfinement phase transition at high densities Bombaci (2016), with a stiff EoS for quark matter, usually associated with quark vector interactions (see Klähn et al. 2013 and references therein).


We have rigorously developed the embedded gravitational lensing theory for point mass lenses in a series of recent papers (Chen et al. 2010, 2011, 2015; Kantowski et al. 2010, 2012, 2013) including the embedded lens equation, time delays, lensing magnifications, shears, etc. We successfully extended the lowest-order embedded point mass lens theory to arbitrary spherically symmetric distributed lenses in Kantowski et al. (2013). The gravitational correctness of the theory follows from its origin in Einstein’s gravity. The embedded lens theory is based on the Swiss cheese cosmologies (Einstein & Straus 1945; Schücking 1954; Kantowski 1969). The idea of embedding (or Swiss cheese) is to remove a co-moving sphere of homogeneous dust from the background Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology and replace it with the gravity field of a spherical inhomogeneity, maintaining the Einstein equations. In a Swiss cheese cosmology the total mass of the inhomogeneity (up to a small curvature factor) is the same as that of the removed homogeneous dust sphere. For a galaxy cluster, embedding requires the overdense cluster to be surrounded by large underdense regions often modeled as vacuum. For a cosmic void, embedding requires the underdense interior to be “compensated” by an overdense bounding ridge, i.e., a compensated void (Sato & Maeda 1983; Bertschinger 1985; Thompson & Vishniac 1987; Martínez-González et al. 1990; Amendola et al. 1999; Lavaux & Wandelt 2012). A low-density region without a compensating overdense boundary, or with an overdense boundary not containing enough mass to compensate the interior mass deficit, has a negative net mass (with respect to the homogeneous background) and is known as an “uncompensated” or “undercompensated” void (Fillmore & Goldreich 1984; Bertschinger 1985; Sheth & van de Weygaert 2004; Das & Spergel 2009).3

3
This dichotomy between compensated and uncompensated voids is slightly different from one based on the classification of the small initial perturbations from which voids are thought to be formed. The initial perturbation can be compensated or uncompensated, which leads to different void growth scenarios (Bertschinger 1985), but if the evolved void formed from either perturbation is surrounded by an overdense shell that “largely” compensates the underdense region (i.e., the majority of the void mass is swept into the boundary shell in the snowplowing fashion when the void is growing), we still call it compensated because the small mass deficit originating in the initial perturbation is unimportant for gravitational lensing.
 Similarly, an overcompensated void has positive net mass with respect to the homogeneous FLRW background. Numerical or theoretical models of over-or undercompensated voids do commonly exist (e.g., Sheth & van de Weygaert 2004; Cai et al. 2010, 2014; Ceccarelli et al. 2013; Hamaus et al. 2014). We focus on compensated void models in this paper, given that uncompensated void models do not satisfy Einstein’s equations. The critical difference between an embedded lens and a traditional lens lies in the fact that embedding effectively reduces the gravitational potential’s range, i.e., partially shields the lensing potential because the lens mass is made a contributor to the mean mass density of the universe and not simply superimposed upon it. At lowest order, this implies that the repulsive bending caused by the removed homogeneous dust sphere must be accounted for when computing the bending angle caused by the lens mass inhomogeneity and legitimizes the prior practice of treating negative density perturbations as repulsive and positive perturbations as attractive. In this paper we investigate the gravitational lensing of cosmic voids using the lowest-order embedded lens theory (Kantowski et al. 2013). We introduce the embedded lens theory in Section 2, build the simplest possible lens model for a void in Section 3, and study the lensing of the CMB by individual cosmic voids in Section 4. Steps we outline can be followed for many void models of current interest.


On the other hand, the distribution of the orbital inclination of the third companion with respect to the inner binary in the triple i3 (bottom panel) is found to peak at ∼100°, but with non-negligible tails. For comparison, isolated triples that merge due to the KL mechanism show a very pronounced peak at ∼90°, with only a few mergers happening in low-inclination systems (Antonini et al. 2017). There are two possible caveats to this. First, possible resonances between nodal precession and KL oscillations can arise. As shown in Hamers & Lai (2017), this could make even low-inclination systems merge. Secondly, there are three different KL mechanisms competing, thus the eccentricity of the inner binary of the CO triple can be pumped up by the torque either of the SMBH or the third companion in the CO triple, for which the KL time-scale is shorter (e.g. Hamers et al. 2015). We define the following ratios
(18)
\begin{eqnarray*}
\mathcal {R}_{12M}=\frac{T_{KL}^{123}}{T_{KL}^{12M}}
\end{eqnarray*}
(19)
\begin{eqnarray*}
\mathcal {R}_{123M}=\frac{T_{KL}^{123}}{T_{KL}^{123M}}
\end{eqnarray*}
(20)
\begin{eqnarray*}
\mathcal {R}_{M}=\frac{T_{KL}^{123M}}{T_{KL}^{12M}}\ ,
\end{eqnarray*}
which we plot in Fig. 4 for the BH–BH binaries that merge in our simulations in Models MW, α = 2, a3, max = 50 au and different values of β. Clearly, the shape of the three distributions does not depend on the assumed value of β, and we find that $\mathcal {R}_{12M}$, $\mathcal {R}_{123M}$, and $\mathcal {R}_{M}$ peak at ∼10−4, ∼5 × 10−3, and ∼10−1, respectively. We note that, however, KL cycles in a given CO triple may be inactive if the relative inclinations are not in the KL window. As a consequence, the initial dynamical evolution of the CO triple can ultimately be dictated by a KL cycle that takes place on a long time-scale. The latter can in turn excite the relative inclination of one of the other orbits, which could activate KL cycles that take place on a shorter time-scale. Thus, the CO triple may experience rather different and rich dynamical paths, which tend to lead to chaotic behaviour (Grishin et al. 2018).


NGC 2110. NGC 2110 is another nearby (z = 0.00779, Gallimore et al. 1999), X-ray bright Seyfert galaxy. Diniz et al. (2015) report a black hole mass of

2
.

7

−
2.1

+
3.5

×

10
8

M
⊙

$ 2.7^{+ 3.5}_{- 2.1} \times 10^{8}\,{{M}_{\odot}} $

, from the relation with the stellar velocity dispersion. From BeppoSAX data, Malaguti et al. (1999) found the X-ray spectrum to be affected by complex absorption. This has been later confirmed by Evans et al. (2007), who find the Chandra+XMM–Newton data to be well fitted with a neutral, three-zone, partial-covering absorber. Rivers et al. (2014) find the Suzaku data to be well fitted with a stable full-covering absorber plus a variable partial-covering absorber. A soft excess below 1.5 keV is also present (Evans et al. 2007), and possibly due to extended circumnuclear emission seen with Chandra (Evans et al. 2006). No Compton reflection hump has been detected with Suzaku (Rivers et al. 2014) or NuSTAR (Marinucci et al. 2015), despite the presence of a complex Fe Kα line. According to the multi-epoch analysis of Marinucci et al. (2015), the Fe Kα line is likely the sum of a constant component (from distant, Compton-thick material) and a variable one (from Compton-thin material). Concerning the high-energy cut-off, ambiguous results have been reported in literature (see Table 1). Ricci et al. (2017) report a value of

448

−
55

+
63

$ 448^{+63}_{-55} $

 keV, while Lubiński et al. (2016) report a coronal temperature of

230

−
57

+
51

$ 230^{+51}_{-57} $

 keV and an optical depth of

0
.

52

−
0.13

+
0.14

$ 0.52^{+ 0.14}_{- 0.13} $

. From 2008–2009 INTEGRAL data, Beckmann & Do Cao (2010) report a cut-off of ∼80 keV with a hard photon index, but these results are not confirmed by NuSTAR (Marinucci et al. 2015). Indeed, only lower limits to the high-energy cut-off have been found with NuSTAR (210 keV: Marinucci et al. 2015), Suzaku (250 keV: Rivers et al. 2014) and BeppoSAX (143 keV: Risaliti 2002). No hard X-ray spectral variability has been detected by Caballero-Garcia et al. (2012) and Soldi et al. (2014) from BAT data, despite the significant flux variability.


Figure 13 shows the temporal evolution of the density, ρ, plasma flow velocity, vx, position of the wave crest, PosA, phase speed, vw, and magnetic field component in the z-direction, Bz, for the primary waves in every different case of initial amplitude, ρIA. In Figure 13(a), we observe that the amplitude of the density remains approximately constant at their initial values until the time when the shock is formed and the density amplitude of the primary wave starts decreasing (see Vršnak & Lulić 2000), i.e., at t ≈ 0.03 (blue), t ≈ 0.04 (red), and t ≈ 0.055 (green). For the case of ρIA = 1.3 (magenta), a decrease of the amplitude of the primary can hardly be observed, as expected for low-amplitude wave (Warmuth 2015). One can see that the larger the initial amplitude, ρIA, the stronger the decrease of the primary wave’s amplitude, which is consistent with observations (Warmuth & Mann 2011; Muhr et al. 2014; Warmuth 2015). The amplitudes decrease to values of ρ ≈ 1.6 (blue), ρ ≈ 1.5 (red), and ρ ≈ 1.4 (green) until the primary wave starts entering the CH. Due to the fact that the waves with larger initial amplitude enter the CH earlier than those with small initial amplitude, we can see in Figure 13(a) that the tracking of the parameters of the faster waves stops at an earlier time than the one for the slower waves. A similar behavior to the one of the density, ρ, can be observed for the plasma flow velocity, vx, in Figure 13(b) and the magnetic field component, Bz, in Figure 13(e). Here, the amplitudes decrease from vx = 0.75, Bz = 1.9 (for ρIA = 1.9, blue), vx = 0.6, Bz = 1.7 (for ρIA = 1.7, red), vx = 0.45, Bz = 1.5 (for ρIA = 1.5, green), and vx = 0.27, Bz = 1.3 (for ρIA = 1.3, magenta) to vx = 0.55, Bz = 1.6 (for ρIA = 1.9, blue), vx = 0.46, Bz = 1.5 (for ρIA = 1.7, red), vx = 0.36, Bz = 1.4 (for ρIA = 1.5, green), and vx = 0.25, Bz = 1.25 (for ρIA = 1.3, magenta). Figure 13(c) shows how the primary waves propagate in the positive x-direction. In all four cases of different initial amplitude, ρIA, the phase speed decreases slighty (consistent with observations; see Warmuth et al. 2004 and Warmuth 2015) until the waves enter the CH at different times, i.e., the values for the phase speed start at vw ≈ 2.2 (for ρIA = 1.9, blue), vw ≈ 1.9 (for ρIA = 1.7, red), vw ≈ 1.7 (for ρIA = 1.5, green), and vw ≈ 1.4 (for ρIA = 1.3, magenta) and decrease to vw ≈ 1.5 (for ρIA = 1.9, blue), vw ≈ 1.39 (for ρIA = 1.7, red), vw ≈ 1.2 (for ρIA = 1.5, green), and vw ≈ 1.13 (for ρIA = 1.3, magenta).


Synchrotron radiation emitted from one relativistic electron population with density Nrel gyrating along a magnetic field B produces a power-law radio spectrum (S ∝ να) with spectral slope αthin 0 above a critical frequency νbreak that depends on B and Nrel. Below this critical frequency self-absorption effects become important and the emission becomes optically thick with a spectral slope αthick = 2.5 (e.g. Kaiser 2006). For instance, the spectrum of the radio outburst in the pulsar system PSR B1259−63 has a spectral index α = −0.7, which is consistent with optically thin synchrotron emission (Fig. 3 in Connors et al. 2002). In contrast, the jets of microquasars often show flat/inverted radio spectra as demonstrated by Fender (2001). Plasma and magnetic field variations along the jet in fact may create regions with different B and Nrel values resulting in spectral components with different turn-over frequencies νbreak. The overlap of the optically thin part of one spectral component with the self-absorbed part of the adjacent one will result in a flat spectrum when observed with a spatial resolution insufficient to resolve the jet (Kaiser 2006). In the microquasar Cygnus X-1, radio emission has been measured with a flat spectrum, i.e. α = −0.06 ± 0.05 and α = 0.07 ± 0.04 (Fender et al. 2000). In Cygnus X-1 the black hole is powered by accretion of the stellar wind of its supergiant companion star. Since the companion is close to filling its Roche lobe, the wind is not symmetric but strongly focused towards the black hole (Miškovičová et al. 2011, and references therein). In contrast, as discussed above in the context of an accretion scenario, accretion in LS I +61°303  occurs in two particular orbital phases due to the eccentric orbit around the Be star. Is the accretion phase and subsequent ejection in LS I +61°303  able to develop a microquasar jet with a flat spectrum? As shown by Corbel et al. (2013), in the early phase of jet formation the low density particles in the jet produce an optically thin synchrotron power-law spectrum at GHz frequencies. Only when the density of the jet plasma increases, a transition to higher optical depth occurs resulting in flat/inverted radio spectra (α ≥ 0). In LS I +61°303 strong evidence for a flat radio spectrum does exist. Early VLA observations at four epochs revealed a flat spectrum between 1.5 and 22 GHz (Gregory et al. 1979). Furthermore, Strickman et al. (1998) carried out multi-frequency VLA observations sparsely covering one orbit and found deviations in the radio spectrum from a simple power law during the outburst. Finally, Massi & Kaufman Bernadó (2009) measured even inverted, optically thick spectra between 2.2 GHz and 8.3 GHz. The primary aim of the current work is to extend previous radio observations of LS I +61°303 and systematically study – for the first time – broad-band cm/mm (2.6–32 GHz) radio spectra and their evolution during a complete outburst. The densely sampled observations of LS I +61°303 were performed over a period of about three weeks (a total of 24 days) using the Effelsberg 100 m telescope at a total of seven frequency bands. The paper is structured as follows. In Sect. 2 we briefly present the observations and data reduction procedures. In Sect. 3 we present the results. Section 4 provides a short discussion and our conclusions.


The catalog reported by Lada & Lada (2003) for embedded clusters is consistent with the mass-size relation R ~ M1/3 − M1/2 for low-mass clusters as pointed out by Murray (2009), where R and M are the gas radius and mass, respectively. While the stellar mass of an embedded cluster is not directly observable and is obtained by assuming an underlying universal initial mass function (IMF), Adams et al. (2006) compiled data from Lada & Lada (2003) and Carpenter (2000) and showed a number-size relation \hbox{$R_* \sim N_*^{1/2}$}R∗~N∗1/2 between the radius of the cluster and the number of objects it contains. The results of Gutermuth et al. (2009) are also compatible with this relation. The number-size relation is reasonably equivalent to the mass-size relation if we adopt a universal IMF and thus similar averaged mass among clusters. Despite the slight uncertainty in the power-law exponent, varying from 1/3 to 1/2, and the scatter of the data, it is clear that embedded clusters follow some mass-size relation. Larger data sets of star-forming clumps, identified as gaseous protoclusters, also exhibit a mass-size relation. Fall et al. (2010) compiled the observations of Shirley et al. (2003), Faúndez et al. (2004), and Fontani et al. (2005) in their Fig. 1, and found via least-squares regression the relation R ∝ M0.38. A regression fit on the ATLASGAL survey (Urquhart et al. 2014) data gives a dependence of R ∝ M0.50. In their work, they fitted log (M) to log (R) and found M ∝ R1.67, of which the power-law exponent is not the inverse of what we inferred. This indicates that the clump properties are differently dispersed in size and mass, and thus there exist some uncertainties in the power-law dependence. Meanwhile, both studies exhibit mass scatter of about 1 dex and are compatible with constant gas surface density, that is, R ∝ M0.5. These observations are performed with molecular lines and dust continuum of the star-forming gas, suggesting that the mass-size relation and probably some other properties of the stellar cluster are established as early as the gas-dominated phase. Pfalzner et al. (2016) pointed out that the mass-size relation for embedded clusters and gaseous protoclusters follow the same trend with different absolute value, which could be explained with star formation efficiency or cluster expansion. One interesting question to ask would be what physical processes actually determine this mass-size relation. Its existence suggests that this primary phase of cluster formation, the gaseous protocluster, is very likely in some equilibrium state governed by the molecular cloud environment in which it resides, and is crucial for understanding the nature of the cluster and, more generally, star formation. An analytical study by Hennebelle (2012) yielded, by linking the gaseous protocluster to properties of the parent cloud, R ~ M1/3 or R ~ M1/2 for protoclusters with different accretion schemes. We stress that we are emphasizing a global equilibrium, which does not imply that the structure is not locally collapsing. More precisely, we propose that the large scales are supported by a combination of rotation and turbulence that sets the M-R relation.


We have presented the analysis of the current X-ray observations of the disc wind in MCG-03-58-007. Here, multiple and variable wind components with velocities ranging from $\sim \! -0.08\, c$ to $\sim \! - 0.2\, c$ (and potentially up to $0.35\, c$) are seen at different times. Multi-epoch observations of disc winds, like the one presented here, are crucial for revealing all the possible phases of the disc wind. For example, over a decade worth of observations of PDS 456 revealed that the wind is most likely clumpy and/or stratified with the ionization ranging from log ($\xi /\rm {erg\, cm \, s^{-1})}\sim 2$ erg cm s−1 up to log ($\xi /\rm {erg\, cm \, s^{-1})}=6$ erg cm s−1 and velocities ranging from $\sim \! -0.2\, c$ up to $\sim \! -0.46\, c$ (Reeves et al. 2016, 2018a, 2020). It is possible, as suggested in other examples of ultra fast disc winds, that we are looking at a stratified wind, where multiple components are launched at different disc radii, but not all of them are always detected. This adds MCG-03-58-007 to the small but growing list of multiphase fast X-ray winds. Other examples of AGN with at least two variable phases of the X-ray winds are PG 1211+143 (Pounds et al. 2016; Reeves et al. 2018b), IRAS 13224-3809 (Parker et al. 2017; Chartas & Canas 2018; Pinto et al. 2018), 1H 0707-495 (Kosec et al. 2018), IRAS 17020+4544 (Longinotti et al. 2015), and PG 1114+445 (Serafinelli et al. 2019). In those cases, multiple phases with a common or different outflowing velocities are detected in the X-ray band. In contrast to most of the cases reported so far, neither of the two phases seen in MCG-03-58-007 requires a different ionization (aside from slice B) suggesting that we are seeing different streamlines of the same highly ionized flow. The only exception could be the eclipsing event seen in 2015, where a solution is found with a lower ionization for the Fe K intervening absorber. However, what we most likely see during this occultation event is a higher density and lower ionization clump of the wind, which could be faster because its higher opacity makes it easier to accelerate (Waters et al. 2017). Note that this does not imply that the soft X-ray wind components, like the ones seen for example in PDS 456 or PG 1211+143, are not present; in contrast to the other examples, MCG-03-58-007 is seen through a relatively high column density (NH ∼ 2 × 1023 cm−2, see Table 2) neutral absorber, therefore these phases may be hidden behind it. MCG-03-58-007 is not the only example where multiple Fe-K zones with the same ionization and outflowing with different velocities had been detected in a single observation. For instance, two simultaneous Fe-K phases were detected at least twice in PDS456 (Reeves et al. 2018a, 2020) and possibly in PG 1211+143 (Pounds et al. 2016) and IRAS 13349+2438 (Parker et al. 2020).


Accretion physics has been extensively studied for decades, particularly in terms of the theoretical aspects including semi-analytic investigations as well as global numerical simulations, in an effort to further understand its physical nature and observational consequences. Many of the works on BH accretion have, in general, revealed an important generic feature of accretion, i.e., the formation of shocks as an accreting plasma is subject to outward forces via a number of decelerating mechanisms (e.g., Abramowicz & Prasanna 1990) and develops a shock front at r = rsh within the radius of the inner edge of a magnetized accretion disk7

7
Armitage et al. (2001) found an ISCO-like edge in their pseudo-Newtonian MHD accretion simulations.
, perhaps equivalent to an innermost stable circular orbit (ISCO) for a pure HD Keplerian disk, before crossing an event horizon at r = rH. Previous studies include hydrodynamic shocks (e.g., Nobuta & Hanawa 1994; Lu et al. 1997; Chakrabarti 1990; Fukumura & Tsuruta 2004) and magnetohydrodynamic (MHD) shocks (e.g., Koide et al. 1998, 2000; Das & Chakrabarti 2007; Takahashi et al. 2002, 2006, hereafter T02, T06, respectively; Fukumura & Kazanas 2007b; Fukumura et al. 2007, hereafter F07; Takahashi & Takahashi 2010). In particular, extensive theoretical studies of various types of shocks have been conducted to date in an attempt to understand their dynamical behavior; e.g., shock oscillation in the context of quasi-periodic oscillations and its spectroscopic signatures (e.g., Chakrabarti & Titarchuk 1995; Molteni et al. 1996, 1999; Acharya et al. 2002; Okuda et al. 2004, 2007; Nagakura & Yamada 2008) that may be relevant for X-ray Binaries (XRBs), for example. Independent general relativistic (GR) MHD (GRMHD) simulations of the tilted accretion disk clearly show that the compression of the plunging plasma in the inner region(

) leads to the formation of standing shocks (e.g., Fragile et al. 2007; Fragile & Blaes 2008; Generozov et al. 2014) depending on the characteristics of the disk geometry and the BH spin (e.g., Morales et al. 2014). The expected highly magnetized shocked region may perhaps correspond to the magnetically arrested plasma seen in other large-scale simulations (e.g., Tchekhovskoy et al. 2011).

Previous studies include
and magnetohydrodynamic (MHD) shocks


Large-scale magnetic fields with strength of the order of 1–10 μG have been observed in disk galaxies (e.g., Beck et al. 1996; Fletcher 2010; Beck 2012; Beck & Wielebinski 2013; Van Eck et al. 2015). The origin of these fields can be explained through mean-field dynamo theory (Ruzmaikin et al. 1988; Beck et al. 1996; Brandenburg & Subramanian 2005a; Kulsrud & Zweibel 2008). The conservation of magnetic helicity is one of the key constraints in these models, and also leads to the suppression of the α-effect. The operation of the mean-field dynamo automatically leads to the growth of magnetic helicity of opposite signs between the large-scale and small-scale magnetic fields (Pouquet et al. 1976; Gruzinov & Diamond 1994; Blackman & Field 2002). To avoid catastrophic suppression of the dynamo action (α-quenching), the magnetic helicity due to the small-scale magnetic field should be removed from the system (Blackman & Field 2000, 2001; Kleeorin et al. 2000). Mechanisms suggested to produce these small-scale magnetic helicity fluxes are: advection of magnetic fields by an outflow from the disk through the galactic fountain or wind (Shukurov et al. 2006; Sur et al. 2007; Chamandy et al. 2014), magnetic helicity flux from anisotropy of the turbulence produced by differential rotation (Vishniac & Cho 2001; Subramanian & Brandenburg 2004, 2006; Sur et al. 2007; Vishniac & Shapovalov 2014), and through diffusive flux (Kleeorin et al. 2000, 2002; Brandenburg et al. 2009; Mitra et al. 2010; Chamandy et al. 2014). The outflow of magnetic helicity from the disk through dynamo operation leads to the formation of a corona (Blackman & Field 2000). According to Taylor's hypothesis, an infinitely conducting corona would resistively relax to force-free field configurations under the constraint of global magnetic helicity conservation (Woltjer 1960; Taylor 1974; Finn & Antonsen 1983; Berger & Field 1984; Mangalam & Krishan 2000). In this paper, we include advective and diffusive fluxes in a simple semi-analytic model of a galactic dynamo that transfers magnetic helicity outside the disk and consequently builds up a force-free corona in course of time. We first solve the time-dependent dynamo equations by expressing them as separable in variables r and z. The radial part of the dynamo equation is solved using an eigenvector expansion constructed using the steady-state solutions of the dynamo equation. The eigenvalues of the z part of the solution are obtained by solving a fourth-order algebraic equation, which primarily depends upon the turbulence parameters and the magnetic helicity fluxes. Once the dynamo solutions are written out as parametric functions of these parameters, the evolution of the mean magnetic field is computed numerically by simultaneously solving the dynamical equations for α-quenching and the growth of large-scale coronal magnetic helicity. Since the large-scale magnetic field lines cross the boundary between the galactic disk and the corona, the magnetic helicity of the large-scale magnetic field in the disk volume is not well defined. Hence we use the concept of gauge-invariant relative helicity (Finn & Antonsen 1983; Berger & Field 1984; Berger 1985) to estimate the large-scale magnetic helicity in the disk and the corona. Here the gauge-invariant relative helicity for the cylindrical geometry is calculated using the prescription given in Low (2006, 2011). We then investigate the dependence of the saturated mean magnetic field strength and its geometry on the magnetic helicity fluxes within the disk and the corresponding evolution of the force-free field in the corona.


Regardless of the cluster orientation, the spectral index observed for the halo at all available frequencies suggests that it is a USSRH. Despite the number of detected USSRH is still low, radio halos with steep indices are being discovered more and more frequently in the last years thanks to the improved observational capabilities of low-frequency instruments such as GMRT, MWA (Murchison Widefield Array) and LOFAR (Shimwell et al. 2016; Wilber et al. 2018; Bruno et al. 2021; Di Gennaro et al. 2021; Duchesne et al. 2022). An in-depth analysis of all radio halos hosted in Planck clusters and observed in LoTSS, including A1550, has recently been presented in Botteon et al. (2022). USSRH are a prediction of turbulent re-acceleration models (Cassano et al. 2006; Brunetti et al. 2008), in which particles are re-accelerated by turbulence (Brunetti et al. 2001, 2017; Petrosian 2001; Brunetti & Lazarian 2011). On the other hand, the detection of such steep indices is not expected from hadronic (or secondary) models, in which the emission of halos comes from the production of secondary electrons from hadronic collisions between thermal and CR protons (Blasi & Colafrancesco 1999; Dolag & Enßlin 2000; Pfrommer et al. 2008). Given that the integrated spectral index observed for the USSRH with LOFAR is

α

54

MHz

144

MHz

∼
−
1.6

$ \alpha_{54\,\rm MHz}^{144\,\rm MHz} \sim -1.6 $

, we expect an index for the spectral energy distribution8δ = 2α − 1 = −4.2. If there is no break in the spectrum, the energy budget for these particles would be untenable (Brunetti et al. 2008). Therefore, a break at low energies (∼GeV) should exist, suggesting a possible interplay between radiative losses and turbulent re-acceleration during the lifetime of emitting electrons (Brunetti & Jones 2014). Moreover, re-acceleration models predict that a large fraction of halos associated with clusters of masses between 4 and 7 × 1014 M⊙ should exhibit steep spectra (Cassano et al. 2010, 2012; Brunetti & Jones 2014; Cuciti et al. 2021). The mass of A1550 of ∼6 × 1014 M⊙ estimated from Planck Collaboration XXVII (2016) falls in this range9.


We carry out analysis of the orbital properties of our clumps only using the sample as detected by the DDS, as this method is sensitive to most clump masses and semimajor axes. The total semimajor axis evolution of all clumps is shown in the left-hand panel of Fig. 3, which we have already discussed, and refer the reader back to. Circles mark surviving clumps (including clumps that subsume another clump), squares mark destroyed clumps and triangles mark merged clumps. Larger markers correspond to more massive clumps. For destroyed clumps, we take the last measured mass. Roughly half of our most massive clumps migrate radially inwards, which is consistent with migration in locally isothermal discs, as objects exchange angular momentum with the surrounding gas and move inwards. However, about half of our most massive clumps migrate radially outwards. This is known to be possible in radiative discs (Kley & Nelson 2012), but requires either large torques or steep surface density gradients (D'Angelo & Lubow 2010). Large torques can have many sources, but in massive, self-gravitating discs they are likely to be in the form of global spiral arms. We carried out a Fourier analysis on the density structure of our discs, to determine the Fourier amplitude of each m mode (where m is the number of spiral arms). The amplitude, Am, of each mode, m, is calculated by
(4)
\begin{equation}
A_{\mathrm{m}} = \Bigg | \sum _{i=1}^{N_{\mathrm{region}}} \frac{{\rm e}^{-im\phi _i}}{N_{\mathrm{region}}} \Bigg |,
\end{equation}
where Nregion is the number of particles in the region we are considering (for our case, R = 20 to R = 100 au), and ϕi is the azimuthal angle of the ith particle. Some example amplitudes are shown in Fig. 9, which shows the first 10 Fourier components of the density structure of two discs in their initial state (i.e. when they have just begun to fragment), marked in red, and the same two discs in their final state, marked in black. The discs are from simulation 1 and simulation 5, and their final state can be seen in their column density plots, shown in Fig. 1. These discs were selected because they ran for the same length of time, and they have contrasting final m-modes states, simulation 1 ultimately peaks in the m = 2 mode and simulation 5 ultimately peaks in the m = 6 mode.


The above qualitative theoretical reasoning raises the question about why would Swift J1834.9−0846 be the only magnetar so far powering a wind nebula, given that previous searches around individual magnetars have returned no sign of extended emission attributable to wind nebulae (e.g., Viganò et al. 2014). With only one observed so far, it is difficult to draw any firm conclusions. Nevertheless, Swift J1834.9−0846 has some interesting characteristics that are not shared with the entire magnetar population. First, the environment of Swift J1834.9−0846 is extremely crowded, with a Fermi GeV source, an H.E.S.S. TeV source, an SNR, a GMC, and an OH maser in its vicinity (Frail et al. 2013; H.E.S.S. Collaboration et al. 2015). The relationship between all these sources is unclear. However, it is tempting to speculate that environmental effects from such a rich field could be playing a role in the production of this wind nebula (e.g., triggering of pair cascade by external gamma rays from a nearby source; Shukre & Radhakrishnan 1982; Istomin and Sob’yanin 2011). Second, the Swift J1834.9−0846 X-ray luminosity in quiescence is

 erg s−1. Only five other magnetars (SGR 0418+5729, SGR 1745−2900, XTE J1810−197, Swift J1822.3−1606, 3XMM J185246.6+003317)20

20

http://www.physics.mcgill.ca/ pulsar/magnetar/main.html

 have luminosities ≲1032 erg s−1. Among these five, three have the smallest surface B fields measured (SGR 0418+5729, Swift J1822.3−1606, 3XMM J185246.6+003317;

 G), and only one source, SGR 1745−2900, has a rotational energy loss rate

 similar to Swift J1834.9−0846, while the rest have

 at least an order of magnitude lower. Hence, from an observational point of view, it seems that the combination of very weak X-ray luminosity, a magnetar-like B-field strength, and a somewhat large

 (properties that are only shared by the Galactic center magnetar SGR 1745−2900) may favor wind nebula production. Another possibility is that the Swift J1834.9−0846 magnetar/nebula system is an older analog to the Kes 75 system, where the central pulsar evolves into a magnetar while preserving its originial PWN.


Nuclear star clusters (NSCs) are dense and massive clusters observed with high frequency ($\gt 80{{\ \rm per\ cent}}$) at the centre of galaxies with stellar masses 109–1011 M⊙ (e.g. Böker et al. 2002, 2004; Côté et al. 2006; Turner et al. 2012; Baldassare et al. 2014; den Brok et al. 2014; Georgiev & Böker 2014; Sánchez-Janssen et al. 2019; Pechetti et al. 2020). These extreme environments often harbour a central supermassive black hole (as in the case of our Galaxy), and represent the most dense stellar systems in the Universe. We refer to the recent and complete review by Neumayer, Seth & Böker (2020) for more details about NSCs. There are two main formation channels that are thought to compete in NSC formation: in situ formation via fragmentation of gaseous clouds in the galactic centre (e.g. Loose, Kruegel & Tutukov 1982), or via orbital segregation and merger of massive star clusters that migrate towards the galactic centre via dynamical friction (Tremaine, Ostriker & Spitzer 1975; Capuzzo-Dolcetta 1993). The latter formation channel, named dry-merger scenario, has been widely explored theoretically and numerically (Capuzzo-Dolcetta & Miocchi 2008a, b). For instance, high-resolution N-body models suggested that the Milky Way (MW) NSC might have formed through this mechanism (Antonini et al. 2012; Tsatsi et al. 2017; Arca-Sedda et al. 2020), which might explain both the structure and kinematics of the Galactic NSC. The dry-merger scenario also provides a successful explanation for the potential formation of NSCs in young galaxies and for the seemingly absence of nucleated regions in small dwarf and massive ellipticals (Arca-Sedda & Capuzzo-Dolcetta 2014, 2017). In fact, several semi-analytical models have shown that the dry-merger scenario leads to correlations between the NSC and the host galaxy properties pretty similar to the observed one (Antonini 2013; Arca-Sedda & Capuzzo-Dolcetta 2014; Gnedin, Ostriker & Tremaine 2014; Capuzzo-Dolcetta & Tosta e Melo 2017). None the less, several features of NSCs seem hard to explain as the result of star cluster merging events. For instance, NSCs exhibit a complex star formation history that seems to suggest the occurrence of several episodic star formation events over the entire course of their lifetime (Neumayer et al. 2020). Such a feature can also be easily explained with in situ formation (Böker et al. 2004), thus suggesting that the formation and evolution of NSCs is likely the result of both scenarios operating in concert. Here, we present our numerical approach to test this scenario in one of the clearest examples of two massive star clusters caught in the process of merging in the nucleus of the nearby MW-like spiral galaxy NGC 4654 shown in Fig. 1 and a zoom in on its nucleus in Fig. 2. Their projected separations, photometric mass, and assumption for the local velocity field of an MW-like galaxy suggest that they should be on a short, few tens of Myr, collision course before they completely merge (Georgiev & Böker 2014). However, this pure analytical expectation needs to be tested in order to gain a deeper knowledge on the following: (1) what will be the merging time-scales given the cluster current observational properties; (2) how the physical and observational properties of the final product depend on the merger dynamics and how such properties compare to those of current NSCs in galaxies of similar mass and type as NGC 4654; and (3) in the hypothesis that the two clusters contain different stellar populations, what are the expected distributions and fractions of the stars coming from the two progenitors in the merger product.


The age of a neutron star is difficult to measure, as for many other astronomical sources. The most robust way to do it is by identifying the birth supernova of the neutron star. However, this can be done precisely only for a very small number of objects, as 5–10 supernovae have historically been observed in our galaxy (Stephenson & Green 2005), and neutron stars are faint sources–practically undetectable at distances beyond the Magellanic Clouds. The explosions, however, leave imprints in the interstellar medium that can remain visible at radio wavelengths for 10 − 100 kyr (Sarbadhicary et al. 2017), thereby allowing the association of pulsars with supernova remnants (SNRs). However, pulsars are rarely found at the centre of SNRs (Frail et al. 1994), as most are expelled like bullets during the explosions possibly due to asymmetries in the process (e.g. Socrates et al. 2005). The transverse velocities of pulsars (based on proper motion and distance estimates) are particularly large, with a mean close to 310 km s−1 (Hobbs et al. 2005), which is at least ten times larger than the average velocities for stars in the solar neighbourhood (e.g. Gaia Collaboration 2018). Moreover, some measured velocities range as high as 1000 km s−1 (Chatterjee et al. 2005; Deller et al. 2019). Thus, associations between SNRs and pulsars are not always straightforward to make (e.g. see the chapter on young pulsars in Lyne & Graham-Smith 2012). The farther the pulsar is from the explosion site, the higher the possibility that the pulsar and SNR are unrelated. In order to confirm an association, it could be necessary to account for up to 100 kyr of evolution of the SNR (that we assume as the maximum possible age of a SNR), and movement across the Galaxy of the pulsar (e.g. Suzuki et al. 2021). In some situations, proper motion measurements for the pulsars can shed light on the matter. For an association to be secure, the pulsar must be moving away from where the explosion took place (usually adopted as the centre of the SNR), and the time necessary to move the pulsar to its current position must match the age of the system. If such time coincided with an independent age measurement of the SNR or the pulsar, or both, then the association would be concretely confirmed. However, this is rarely possible as SNR and pulsar ages are hard to obtain.


While the status of the S8 discrepancy is perhaps somewhat less clear than that of the H0 tension, it is beyond question that there overall is some disagreement between high- and low-redshift probes of the amplitude of matter fluctuations (see, for instance, Di Valentino et al. 2020c, for a concise review of the problem). It is thus worthwhile to investigate whether new physics might solve or at least alleviate the S8 discrepancy, a possibility that has been investigated in several works. Models that have been contemplated in this sense include for example active and sterile neutrinos (Battye & Moss 2014; MacCrann et al. 2015; Feng, Zhang & Zhang 2017; Vagnozzi et al. 2017; Mccarthy et al. 2018), ultra-light axions (Hlozek et al. 2015), decaying dark matter (DM; Enqvist et al. 2015; Chudaykin, Gorbunov & Tkachev 2018; Di Valentino et al. 2018; Abellán et al. 2020; Chen et al. 2021; Pandey, Karwal & Das 2020; Xiao et al. 2020; Abellán, Murgia & Poulin2021), extended or exotic DM and/or dark energy (DE) models and interactions (Kunz, Nesseris & Sawicki 2015; Kumar & Nunes 2016; Pourtsidou & Tram 2016; Gariazzo et al. 2017; Benetti, Graef & Alcaniz 2018; Buen-Abad et al. 2018; Kumar, Nunes & Yadav 2018, 2020a, b; Poulin et al. 2018; Archidiacono et al. 2019; Di Valentino et al. 2019b, 2020b; Lambiase et al. 2019; Vagnozzi et al. 2019; Chamings et al. 2020; Dutta et al. 2020; Heimersheim et al. 2020; Jiménez et al. 2020; Choi, Yanagida & Yokozaki 2021) including unified dark sector models (Camera, Martinelli & Bertacca 2019), modified gravity models (Dossett et al. 2015; De Felice & Mukohyama 2017; Nesseris, Pantazis & Perivolaropoulos 2017; Kazantzidis & Perivolaropoulos 2018, 2019; Barros et al. 2020; De Felice, Nakamura & Tsujikawa 2020; Skara & Perivolaropoulos 2020; Zumalacarregui 2020; Marra & Perivolaropoulos 2021), and more generally extended parameter spaces (Di Valentino & Bridle 2018; Di Valentino, Melchiorri & Silk 2020), among the others. It is also worth noting that most of the models invoked to address the S8 discrepancy do so at the expense of worsening the H0 tension, and vice versa (see e.g. Vagnozzi et al. 2018; Poulin et al. 2018; Kumar et al. 2019a; Hill et al. 2020; Alestas & Perivolaropoulos 2021), highlighting the importance of a conjoined analysis of the two tensions (Di Valentino et al. 2020b; Di Valentino, Linder & Melchiorri 2020a).


With the increase in precision of CRSF measurements, the dependence of CRSFs on luminosity and spin phase has become obvious and is one of the important areas of current X-ray pulsar research, as it potentially allows probing of emission region geometry and properties (Staubert et al. 2019). Of particular interest is the luminosity dependence of the line properties, as it might be used to probe the mechanism of plasma deceleration and the transition between so-called sub- and supercritical accretion regimes associated with the onset of an accretion column first suggested by Basko & Sunyaev (1976). As a result, two main accretion regimes, super- and subcritical accretion, are expected depending on whether the source luminosity is higher or lower than a “critical luminosity” Lcrit, which strongly depends on the magnetic field of the NS (Basko & Sunyaev 1976; Becker et al. 2012; Mushtukov et al. 2015a). In the supercritical regime, for L > Lcrit, an accretion column forms, and the falling plasma is decelerated by a radiation shock that forms at a certain distance from the NS surface. In this case, the emission height increases with increasing luminosity. In the subcritical regime, for L Lcrit, the infalling matter is presumably decelerated by Coulomb interaction, forming a region whose height decreases with increasing luminosity. At even lower luminosity, the description of the deceleration process of falling material is not very conclusive. A reasonable scenario is that there exists a transition for the regime from Coulomb-stopping to gas-shock dominance beyond which only a small accretion mound forms on the NS surface. Based on this picture, and considering that the local magnetic field strength decreases with height, the correlation between the CRSF line energy and luminosity can be used to trace the accretion regimes. For instance, a transition between the sub- and supercritical regimes was recently reported by Doroshenko et al. (2017) and Vybornov et al. (2018) for V0332+53 during a giant outburst.


Moreover, there could also be some contribution to the detected temperature asymmetry from high-latitude gas clouds in our Galaxy along the line of sight toward M 81. In this respect we note that M 81 is at about 40.9° north of the Galactic disk, where contamination from the Milky Way is expected to be low. However, interpretation of astronomical observations is often hampered by the lack of direct distance information. Indeed, it is often not easy to judge whether objects on the same line of sight are physically related or not. Since the discovery of the Arp’s Loop (Arp 1965) the nature of the interstellar clouds in this region has been debated; in particular whether they are related to the tidal arms around the galaxy triplet (Sun et al. 2005; de Mello et al. 2008) or to Galactic foreground cirrus (Sollima et al. 2010; Davies et al. 2010). Already Sandage et al. (1976) presented evidence showing that we are observing the M 81 triplet through widespread Galactic foreground cirrus clouds and de Vries et al. 1987 built large-scale HI, CO, and dust maps that showed Galactic cirrus emission toward the M 81 region with NH ≃ 1−2  ×  1020 cm-2. The technique used to distinguish between the emission from extragalactic or Galactic gas and dust relies on spectral measurements and on the identification of the line of sight velocities, which are expected to be different in each case. Unfortunately, in the case of the M 81 Group, this technique appears hardly applicable since the radial velocities of extragalactic and Galactic clouds share a similar LSR (local standard of rest) velocity range (Heithausen 2012). Several small-area molecular clouds (SAMS), that is, tiny molecular clouds in a region where the shielding of the interstellar radiation field is too low (so that these clouds cannot survive for a long time), have been detected by Heithausen (2002) toward the M 81 Group. More recently, data from the Spectral and Photometric Imaging Receiver (SPIRE) instrument onboard Herschel ESA space observatory and Multiband Imaging Photometer for Spitzer (MIPS) onboard Spitzer allowed the identification of several dust clouds north of the M 81 galaxy with a total hydrogen column density in the range 1.5–5 × 1020 cm-2 and dust temperatures between 13 and 17 K (Heithausen 2012). However, since there is no obvious difference among the individual clouds, there was no way to distinguish between Galactic or extragalactic origin although it is likely that some of the IR emission both toward M 81 and NGC 3077 is of Galactic origin. Temperature asymmetry studies in Planck data may be indicative of the bulk dynamics in the observed region provided that other Local (Galactic) contamination in the data is identified and subtracted. This is not always possible, as in the case of the M 81 Group, and therefore it would be important to identify and study other examples of dust clouds where their origin, either Galactic or extragalactic, is not clear. One such example might be provided by the interacting system toward NGC 4435/4438 (Cortese et al. 2010) where the SAMS found appear more consistent with Galactic cirrus clouds than with extragalactic molecular complexes. Incidentally, the region A1 within R0.50 has been studied by Barker et al. (2009), who found evidence for the presence of an extended structural component beyond the M 81 optical disk, with a much flatter surface brightness profile, which might contain ≃10–15% of the M 81 total V-band luminosity. However, the lack of both a similar analysis in the other quadrants (and at larger distances from the M 81 center) and the study of the gas and dust component associated to this evolved stellar population, hamper our understanding of whether this component may explain the observed temperature asymmetry toward the M 81 halo.


Table 1 depicts the heliolatitudes of the solar offset points on different experiment days and the corresponding heliocentric distances. It is to be noted that owing to the trajectory constraints of the spacecraft along the ecliptic plane, a pure latitudinal dependence (at a constant heliocentric distance) of the spectral index cannot be assessed here. However, variations in the spectral index values with respect to the complementary heliolatitudes of the proximate points of the ray path between 5° and 39° N can be observed, where the higher-heliolatitude spectra are flatter (lower αf) than the lower-heliolatitude spectra (see Table 1). Low-heliolatitude coronal regions have streamer structures where the slow solar winds, which have a highly turbulent structure, are generated (Woo & Martin 1997). The fast solar winds emanate from coronal holes at higher heliolatitudes (Zirker 1977). The coronal holes, which generally have a higher magnetic field, lead to a lower sound wave to Alfvénic wave ratio and a relatively lower turbulence cascading energy. Consequently, the turbulence requires a longer time (and hence larger heliocentric distances) to achieve a fully developed turbulence spectrum. Thus, at a given heliocentric distance (but varying heliolatitude) of a proximate point of the satellite radio ray path, the fast (high-latitude) solar wind has flatter spectra compared with slow (low-latitude) solar wind spectra, as reported in earlier studies (Pätzold et al. 1996; Efimov et al. 2008, 2017). Efimov et al. (2008) and Chashei et al. (2007) observed that this change of turbulence regime is prominent during periods of minimum solar activity, as coronal holes (the source of the high-latitude fast solar wind) exist in abundance, and there is relative lack of the active regions that generate slow solar winds. It is, however, intriguing that we notice this aspect in our study, which was conducted during a period of relatively low solar activity.


In this work, the difference between G and NG clusters plays a major role. In a first step, we focus our attention on characterizing the structure and distribution of galaxy member properties in each class. In Fig. 1, we show the distributions of: (a) logarithmic virial mass ($\rm log(\mathit{ M}_{200}/M_{\odot })$); (b) r-band absolute magnitude; (c) a proxy for the concentration of stellar mass in each cluster, defined as $R_{80}/R_{20}$, where $R_{x}$ is the projected radius within which the stellar mass represents x per cent of the total stellar mass within $R_{200}$; (d) velocity dispersion along the LOS; and (e) the distribution of cluster mean stellar mass of bright galaxies within $R_{200}$ ($\langle M_{\mathrm{ stellar}}^{C} \rangle$).3 We compare the distributions using two different statistical tests: AD and Wilcoxon Rank Test (Wlx; see Engmann & Cousineau 2011 and Gehan 1965 for a review of both).4 The results are shown in each panel. Aside from the histograms, we add a kernel smoothed curve, shown as shaded area, which is derived directly from the data set using an Epanechnikov Kernel Density estimator (Silverman 1986) with a bandwidth equal to 1.5 times the bin size. We find that G and NG clusters have statistically different distributions. In panel (a), we note that NG clusters tend to have higher values of $M_{200}$ in comparison to G clusters. Namely, we find that 32.4 per cent (11/34) of NG clusters have $\mathrm{ log}(M_{200}/\mathrm{ M}_{\odot })\gt 14.75$, while this fraction decreases to 6.3 per cent (9/143) in G clusters. The majority of G systems ($\sim 71.3{{\ \rm per\ cent}}$) have $\mathrm{ log}(M_{200}/\mathrm{ M}_{\odot })\lt 14.5$. Panel (b) shows an excess of fainter galaxies5 in NG clusters in comparison to G systems. We find that 30.5 per cent of NG cluster members have $M_{r} \ge -19.5$, while the equivalent cut in absolute magnitude yields 14.1 per cent of galaxies in G systems. Panel (c) shows the distribution of concentration in G and NG clusters, revealing that NG cluster galaxies are less concentrated than their G cluster counterparts. Only one NG cluster reaches C > 4.7. In panel (d), we observe an excess of NG clusters with higher velocity dispersion in comparison to G clusters. NG clusters are presumed to be found in a non-virialized state, so that the expected velocity dispersions are higher, and, possibly, the estimated mass may also be an overestimate of the real cluster mass. In any case, this is one more piece of evidence regarding the different state of NG clusters with respect to G systems. Finally, we note in panel (f) an excess of NG clusters with mean stellar mass of $\rm 3 \le \langle \mathit{ M}_{stellar}^{\mathit{ C}} \rangle \lt 4 \times 10^{11}\, M_{\odot }$. In other words, we find that NG clusters have higher virial mass and radii, an excess of fainter galaxies, are less concentrated, contain more massive B galaxies and have higher velocity dispersion in comparison to G systems.


To identify the fibers seen in the PPV space, we apply the agglomerative clustering implementation in the scikit-learn package
7

7

https://scikit-learn.org/stable/
 (Pedregosa et al. 2011) to the PPV points for each transition. The agglomerative clustering algorithm groups points in N-dimensional space via recursively merging points or clusters (i.e., groups of points) into higher-order clusters such that the pair of points or clusters to be merged minimally increases the linkage distance (Ward 1963). To make the clustering procedure adjustable, we introduce two parameters t

x
 and t

v
 in defining the linkage distance of two PPV points:
7

slink,ij=[tx(xi−xj)]2+yi−yj2+vi−vjtv2,

where x and y are the spatial coordinates in physical units, and v is the LoS velocity. For each transition, we adjust the parameters (t

x
, t

v
) and run the algorithm with the modified coordinates of the PPV points as input. By practice we found that the best values of (t

x
, t

v
) for identifying the fiber structures in the H13CO+, N2H+, and NH2D data are (2.8 pc, 1.0 km s−1pc−1), (0.5 pc, 0.8 km s−1pc−1), and (1.0 pc, 2.0 km s−1pc−1), respectively, with which the DR21(OH) ridge is decomposed into 3, 3, and 2 fibers in the PPV space. Figure 4 shows the identified fibers projected on the PoS and their PV plots, with the color coding representing different velocities. In the H13CO+ data, three fibers with roughly north–south orientations are clearly seen; these fibers have distinct velocities, forming a trident with the junction approximately coincident with the DR21(OH) core. We name the three fibers as f1, f2, and f3 from east to west. Among the fibers, f1 is clearly tracing the central-densest part of the dust ridge well known from previous dust-continuum observations (Motte et al. 2007; Hennemann et al. 2012); f2 and f3 are relatively new, not seen in the dust emission, but previous single-dish H13CO+ (1 − 0) observations showed that the ridge slight moves from east to west with the velocity increasing from −5 km s−1 to 0 km s−1 (Schneider et al. 2010), in a manner consistent with the positions and velocities of the three fibers identified here. In particular, f3 is also discernible in the H13CO+(1 − 0) map in Schneider et al. (2010; see their Figure A3), though at a much lower resolution. Given the LoS velocities of the fibers and the existing observations suggesting that the DR21(OH) ridge is in a global collapse (Schneider et al. 2010), one may expect that f1 and f3 are on the far side and the near side along LoS, respectively, and that f2 is probably in the middle. The three fibers in the N2H+ line are less prominent yet are still clearly seen with positions and velocities consistent with those in the H13CO+ line, indicating that they are tracing the same physical entities. While the whole f1 and the southern part of f3 are detected, f2 is fragmented into several parts, probably due to the regional variations of the abundances of the two molecular tracers. On the other hand, in the NH2D line, the whole f1 (though fragmented into parts) and the northern part of f2 are detected, and f3 is completely absent. The appearance of the fibers are also different from that in the H13CO+ and N2H+ lines by their more compact morphologies and smaller widths. In addition, in contrast to what is seen in the H13CO+ and N2H+ lines, the southern end of f1 in the NH2D line is shifted to the west and does not coincide with the DR21(OH) core, indicating that active high-mass star formation has destroyed most of NH2D. The detection information of the fibers in the three transitions is summarized in Table 1.


On the other hand, at separations ≲ 0.01 pc, smaller than those characteristics of spectroscopic binary candidates, many theoretical studies have predicted a significant variability in the observed nuclear light curve due to different physical processes. For example, in studies of the evolution of MBHBs in circumnuclear discs, a modulated gas inflow from the outer gas distribution periodically fuels the accretion discs within the Hill radii of the individual MBHs (that, being smaller than the surrounding circumbinary disc, are commonly referred to as ‘mini-discs’ in the literature, Artymowicz & Lubow 1994; Ivanov, Papaloizou & Polnarev 1999; Hayasaki, Mineshige & Ho 2008; Cuadra et al. 2009; Roedig et al. 2011, 2012; D’Orazio, Haiman & MacFadyen 2013; Farris et al. 2015; D’Orazio et al. 2016; Miranda, Muñoz & Lai 2017; Tang, MacFadyen & Haiman 2017; Bowen et al. 2018; d’Ascoli et al. 2018) as a consequence of the non-axisymmetric and time-dependent potential of the binary. Such modulated inflow could result in a similarly variable luminosity, depending on the properties of the inflowing gaseous streams and of the preexisting mini-discs (see the discussion in Sesana et al. 2012). An alternative cause of observed variability could be the plunging of a very eccentric secondary MBH on to the primary disc, as proposed by Valtonen et al. (2008) for the observed variability of OJ287. Finally, even in the absence of periodic inflows or very eccentric binaries (as expected in the case of a low-mass secondary; D’Orazio et al. 2016; Duffell et al. 2020), variability can be caused by the relativistic Doppler boost of the emitted spectrum during the orbit of the MBHB, resulting in a variable flux observed in fixed observational bands, as proposed for PG 1302-102 in D’Orazio, Haiman & Schiminovich (2015). This last model has the peculiarity of predicting different variability amplitudes at different wavelengths, as demonstrated for the UV versus optical light curves of PG 1302-102 (Xin et al. 2019).


It may be appropriate to single out at this point the recent and interesting works by Lin & Ishak (2017a,b) in which the authors run a so-called (dis)cordance test based on using a proposed index of inconsistency (IOI) tailored at finding possible inconsistencies/tensions between two or more data sets in a systematic and efficient way. For instance, it is well known that there is a persistent discrepancy between the Planck CMB measurements of H0 and the local measurements based on distance ladder (Riess et al. 2016, 2018b). At the same time, if one compares what is inferred from Planck 2015 best-fitting values, the LSS/RSD measurements generally assign smaller power to the LSS data parametrized in terms of the weighted linear growth rate f(z)σ8(z). This feature is of course nothing but the σ8-tension we have been addressing in this paper. It is therefore natural to run the IOI test for the different kinds of H0 measurements and also to study the consistency between the H0 and the growth data. For example, upon comparing the constraints on H0 from different methods, Lin & Ishak (2017b) observe a decrease of the IOI when the local H0 measurement is removed. From this fact they conclude that the local measurement of H0 is an outlier compared to the others, what would favour a systematics-based explanation. This situation is compatible with the observed improvement in the statistical quality of the fitting analysis by Solà, Gómez-Valent & de Cruz Pérez (2017b,c) when the local H0 measurement is removed from the overall fit of the data using the RVM and the ΛCDM. In this respect, let us mention that a recent model-independent analysis of data on cosmic chronometers and an updated compilation of SNIa seem to favour the lower range of H0 (Gómez-Valent & Amendola 2018), what would be more along the line of the results found here, which favour a theoretical interpretation of the observed σ8 and H0 tensions in terms of vacuum dynamics and in general of DDE (cf. Fig. 10).


Using the RADEX3 non local thermodynamic equilibrium (LTE) modeling program (van der Tak et al. 2007) with collisional rate coefficients from Wiesenfeld & Faure (2013), we modeled the relation between the gas kinetic temperature and the measured average of para-H2CO 0.5 × [(322–221 + 321–220)/303–202] ratios, adopting a 2.73 K background temperature, an average observational linewidth of 4.0 km s−1, and column densities N(para-H2CO) = 2.7 × 1012 and 3.7 × 1012 cm−2 for N113 and N159W, respectively. The results are shown in Fig. 5. The values of the para-H2CO column density were obtained with APEX data (beam size ~30″; Tang et al. 2017b), which cover similar regions. Different column densities of para-H2CO only weakly affect derived kinetic temperatures (see Fig. 3 in Tang et al. 2017b or Fig. 4 in Tang et al. 2018a; this was also shown in Fig. 13 and discussed in Sect. 4.3.1 of Mangum & Wootten 1993) as long as all lines are optically thin. Considering that the relation between the gas temperature and the para-H2CO line ratio may vary at different spatial densities (see Fig. 2 in Tang et al. 2017b), we modeled it at spatial densities 104, 105, and 106 cm−3 in Fig. 5. It appears that Tkin at n(H2) = 105 cm−3 is consistently lower than values at 104 and 106 cm−3 by ≲23% and ≲34%, respectively, for Tkin ≲ 100 K. Local thermodynamic equilibrium (LTE) is a good approximation for the H2CO level populations under optically thin and high-density conditions (Mangum & Wootten 1993; Tang et al. 2017a,b, 2018b). Following the method applied by Tang et al. (2017b) in their Eq. (2), we plot the relation between the LTE kinetic temperature, TLTE, and the para-H2CO (3–2) line ratio in Fig. 5. Apparently, TLTE agrees well with Tnon-LTE at volume densities n(H2) ~ 105 cm−3 as long as Tkin ≲ 100 K. Previous observations show that para-H2CO (3–2) is sensitive to gas temperature at density 105 cm−3 (Ginsburg et al. 2016; Immer et al. 2016; Tang et al. 2017b). The spatial density measured with para-H2CO (303–202) and C18O (2–1) in N113 and N159W is n(H2) ~ 105 cm−3 on a size of ~30″ (Tang et al. 2017b). Therefore, here we adopt 105 cm−3 as an averaged spatial gas density in the N113 and N159W regions.


To distinguish the emission from AGNs and star formation we use the fact that star-forming galaxies (SFGs) are known to exhibit radio–IR correlation across a wide range of luminosities and redshifts (see Condon 1992; Appleton et al. 2004; Basu et al. 2015). The correlation between 1.4 GHz luminosity and IR luminosity (monochromatic IR luminosity as well as bolometric IR luminosity between 8.0 and 1000 $\mu$m) in SFGs is attributed to the fact that both radio and IR emission are closely related to star formation (Ivison et al. 2010). AGNs with predominant radio emission from jet deviate from radio–IR correlation by showing radio-excess (Morić et al. 2010; Del Moro et al. 2013). Therefore, we examine if our NLS1s show radio-excess in the radio–IR correlation. We note that radio–IR correlation can be represented as the ratio of IR flux to 1.4 GHz radio flux density i.e. q=log(SIR/$S_{\rm 1.4 \, GHz}$) (see Appleton et al. 2004). For our radio-detected NLS1s we estimate q$_{\rm 22\, {\mu }m}$=log($S_{\rm 22 {\mu }m}$/$S_{\rm 1.4 \, GHz}$), where IR flux at 22 $\mu$m is taken from Wide-field Infrared Survey Explorer (WISE),2 and 1.4 GHz flux density is taken from FIRST whenever available, otherwise from NVSS. WISE is an all sky survey carried out at four MIR photometric bands namely W1 [3.6 $\mu$m], W2 [4.6 $\mu$m], W3 [12 $\mu$m], and W4 [22 $\mu$m], with 5σ sensitivity of 0.08, 0.11, 1.0, and 6.0 mJy, and angular resolution of 6.1, 6.4, 6.5, and 12.0, respectively (Wright et al. 2010). Using the most recent version of WISE source catalogue (AllWISE3 data release) we obtain MIR counterparts for 481, 480, 440, and 354 of our 498 radio-detected NLS1s with SNR ≥ 5.0 in W1, W2, W3, and W4 bands, respectively. The WISE counterparts of our radio-detected NLS1s are searched within a circle of 2.0 arcsec radius centred at SDSS optical positions. q$_{\rm 22 {\mu }m}$ is estimated using k-corrected fluxes, where IR spectral index is derived using W3 and W4 band fluxes, and radio spectral index is derived using 1.4 GHz and 150 MHz flux densities, whenever available, otherwise an average radio spectral index of −0.7 is considered.


The equilibrium between the MFR and the two magnetic charges can lead to an infinite twist on the surface of the MFR because no toroidal field exists outside the MFR. The magnetic field of the line current I0 is included to decrease this to reasonable values, typically less than 4π, suggested by many observations (Liu & Alexander 2009; Wang et al. 2015; Guo et al. 2012). Combining contributions from line current I0 and the poloidal current inside the MFR, the expression for the toroidal field is written as (7)\begin{eqnarray} \label{eq:bt} B_{t}&\approx& \frac{\mu_0 I_0}{2\pi}\sqrt{\frac{2 \alpha I^2}{a^2 I_0^2}\left(1-\frac{r_{\rm a}^2}{a^2}\right)+\frac{1}{R^2}},~r_{\rm a}\leq a,\nonumber \\ &=& \frac{\mu_0 I_0}{2\pi r_{\perp}},~r_{\rm a}>a. \end{eqnarray}Bt≈μ0I02π2αI2a2I021−ra2a2+1R2,ra≤a,=μ0I02πr⊥,ra>a.For an analytical expression of the poloidal magnetic field, we follow the practice of Titov & Démoulin (1999), in which poloidal field is given in terms of the vector potential as in (8)\begin{equation} B_{p}=-\frac{\partial A_{t}}{\partial x}\frac{\vec{r}_{\perp}}{r_{\perp}}+\left(\frac{\partial A_t}{\partial r_{\perp}}+\frac{A_t}{r_{\perp}}\right)\vec{x}. \end{equation}Bp=−∂At∂xr⊥r⊥+∂At∂r⊥+Atr⊥x.The toroidal At outside the MFR (ρ ≥ a) is (9)\begin{equation} A_{t}(x,r_{\perp}) \approx \frac{\mu_{0}I}{2\pi}\sqrt{\frac{R}{r_{\perp}}}\mathcal{A}(k), \end{equation}At(x,r⊥)≈μ0I2πRr⊥𝒜(k),and inside (ρa) is (10)\begin{equation} A_{t}(x,r_{\perp}) \approx \frac{\mu_{0}I}{2\pi}\sqrt{\frac{R}{r_{\perp}}} (\mathcal{A}(k_{\rm a})+\mathcal{A}'(k_{\rm a})(k-k_{\rm a})). \end{equation}At(x,r⊥)≈μ0I2πRr⊥(𝒜(ka)+𝒜′(ka)(k−ka)).Here, \begin{eqnarray} &&\mathcal{A}(k)=k^{-1}[(2-k^2)K(k)-2E(k)], \\ &&k=2\sqrt{\frac{r_{\perp}R}{(r_{\perp}+R)^2+x^2}}, \\ &&k_{\rm a}=2\sqrt{\frac{r_{\perp}R}{4r_{\perp}R+a^2}}, \end{eqnarray}𝒜(k)=k-1[(2−k2)K(k)−2E(k)],k=2r⊥R(r⊥+R)2+x2,ka=2r⊥R4r⊥R+a2,and its derivative (14)\begin{equation} \mathcal{A}'(k)=\frac{(2-k^2)E(k)-2(1-k^2)K(k)}{k^2(1-k^2)}\cdot \end{equation}𝒜′(k)=(2−k2)E(k)−2(1−k2)K(k)k2(1−k2)·These expressions contain the complete elliptic integrals of the first and the second kinds, K(k) and E(k).