The ARGO-YBJ detector, hosted in a building at the YangBaJing Cosmic Ray Observatory (Tibet, China, 90°31′50″E, 30°06′38″N), 4300 m above sea level, has been designed for very high-energy (VHE) gamma-ray astronomy and cosmic-ray observations. It is made up of a single layer of resistive plate chambers (RPCs) operated in streamer mode, 2.850 m × 1.225 m each, organized in a modular configuration to cover a surface of about 5600 m2 with an active area of about 93%. The RPCs detect the charged particles in air showers with an efficiency ≥98%. To improve the shower reconstruction, other chambers are deployed around the central carpet for a total instrumented area of 100 m × 110 m. A highly segmented readout is performed by means of 55.6 cm × 61.8 cm external electrodes, called “pads,” whose fast signals are used for triggering and timing purposes. These pads provide the digital readout of the detector up to 22 particles m−2, allowing the count of the air shower charged particles without any significant saturation up to primary cosmic-ray energies of about 200 TeV (Bartoli et al. 2012a). In order to extend the dynamical range to PeV energies each RPC is also equipped with two large size pads (139 cm × 123 cm), allowing the collection of the total charge developed by the particles hitting the detector (Aielli et al. 2012). The digital output of each pad is splitted in two signals sent to the logic chain that builds the trigger and to the 18,360 multi-hit time-to-digital converters, which are routinely calibrated with 0.4 ns accuracy by means of an off-line method using cosmic-ray showers (He et al. 2007; Aielli et al. 2009a). More details about the detector and the RPC performance can be found in Aielli et al. (2006, 2009b). The detector is connected to two independent acquisition systems corresponding to two different operation modes, referred to as the shower mode and the scaler mode (Aielli et al. 2008). The data used in this paper were recorded by the digital readout in shower mode. This mode is implemented by means of an inclusive trigger based on the time correlation between the pad signals, depending on their relative distance. In this way the data acquisition is triggered when at least 20 pads in the central carpet are fired in a time window of 420 ns. By means of this trigger the energy threshold for gamma-induced showers can go down to 300 GeV with an effective area depending on the zenith angle (see Figure 1 in Bartoli et al. 2013).


The GMC associated with G305 is one of the most massive and luminous clouds in the Milky Way (Fig. 1). It is located in the Galactic plane at l ~ 305°, b ~ 0° and at a kinematic distance of 4 kpc (derived from a combinationof radio and Hα observationsby Clark & Porter (2004); Davies et al. (2012) measured its spectrophotometric distance to be 3.8 ± 0.6 kpc and most recently Borissova et al. (2019) measured the Gaia DR2 average distance to be 3.7 ± 1.2 kpc); this places it in the Scutum-Crux spiral arm. Given this distance, the complex has a diameter of ~ 30 pc (Clark & Porter 2004) and a molecular mass of ~6 × 105 M⊙ (Hindson et al. 2010). The G305 complex consists of a large central cavity that has been cleared by the winds from massive stars belonging to two visible central clusters (Danks 1 and 2) and the Wolf-Rayet star (WR48a; Clark & Porter 2004; Davies et al. 2012). The cavity is surrounded by a thick layer of molecular gas (traced by CO and NH3 emission; Hindson et al. 2010, 2013). Radio continuum observations by Hindson et al. (2012) have revealed that the cavity is filled with ionized gas and identified six ultra-compact HII (UC HII) regions and also one bright rimmed cloud (BRC) at the periphery of the cavity, indicating molecular gas irradiated by UV radiation (Sugitani & Ogura 1994; Thompson et al. 2004), which may cause implosion (Bertoldi 1989) or evaporation. A number of studies havereported star formation tracers (water and methanol masers, HII regions and massive young stellar objects, MYSOs; Clark & Porter 2004; Lumsden et al. 2013; Urquhart et al. 2014; Green et al. 2009, 2012). Furthermore, Hindson et al. (2010) found the concentration of star formation tracers to be enhanced inside a clump of NH3 bearing molecular gas that faces the ionizing sources, which is consistent with the hypothesis that the star formation has been triggered. Analysis of the stellar clusters in the complex reveals them to have ages of 1.5 Myr for Danks 1 and 3 Myr for Danks 2,with the former possibly being triggered by the latter (Davies et al. 2012). Additionally, a diffuse population of evolved massive stars was also found to exist within the confines of the G305 complex that had formed around the same time as the two clusters (Leistra et al. 2005; Shara et al. 2009; Mauerhan et al. 2011; Davies et al. 2012; Faimali et al. 2012; Borissova et al. 2019).


Cometary activity is driven by the gas pressure P(s), which does not depend on the ice abundance, but only on the gas temperature: also minor species can drive cometary activity, provided that heat transfer inside a pebble is faster than ice depletion. This condition fixes an upper limit for the refractory-to-ice mass ratio δi (Fulle 2021)
(8)$$\begin{eqnarray*}
\delta _i \lt {\lambda _s \over {3 ~Q ~R ~c_p}} - 1
,
\end{eqnarray*}$$where cp ≈ 103 J kg−1 K−1 is the heat capacity of the pebbles (Blum et al. 2017). In order to fulfil equation (2) during the whole inbound orbit, cometary activity is driven by at least the five ices listed in Table 2, reporting the nucleus and gas coma parameters computed by means of the activity model (Fulle et al. 2020b). Thus, the less abundant the ice, the shorter the rh-range where it can drive cometary activity (e.g. ethane in Table 2). The upper limit of each water-to-ice mass ratio is δi/δw, and in the protoplanetary disc δw ≈ 5 (Cambianica et al. 2020). The probable activity due to further ices not listed in Table 2 and the decrease of cp with the temperature (Takahashi & Westrum 1970; Shulman 2004; Bouziani & Jewitt 2022) significantly increase the upper limits of δi. However, e.g. formaldehyde has unknown thermodynamical parameters (Fray & Schmitt 2009), and e.g. ethylene and nitric oxide were not detected in 67P (Rubin et al. 2020), thus preventing a realistic computation of e.g. evolving cp values. According to the activity model, the distribution of water-ice is very inhomogeneous in the nucleus (Ciarniello et al. 2022), with water-rich pebbles, depleted of supervolatiles, embedded in a matrix of water-poor pebbles, which are rich of the ices listed in Table 2. Therefore, at rh > 3.8 au, dust is ejected from water-poor pebbles only, characterized by δw ≈ 50 (Fulle 2021), so that equation (2) is surely verified. In water-poor pebbles the water ice is less abundant than all other ices (excluded ethane, Table 2), so that these ices cannot be trapped inside a less abundant water ice. The observed activity of C/2017K2 at rh = 23.7 au (Jewitt et al. 2017) excludes also that ices may be trapped inside CO2-ice, because the CO2-driven activity onsets at rh = 13 au (Table 2).


TYC 9486-927-1 was observed by Torres et al. (2006) as part of the Search for Associations Containing Young stars (SACY) programme (Torres et al. 2008). They assigned a spectral type of M1 and measured a radial velocity of vrad = 8.7 ± 4.6 km s−1 from 10 observations. The large uncertainty is likely due to the star's high rotational velocity (vsin i = 43.5 ± 1.2 km s−1); suggesting it is either a single rapid rotator or a spectroscopic binary with blended lines. TYC 9486-927-1 also shows signs of activity in X-ray (Thomas et al. 1998), H α emission (Torres et al. 2006) and the UV (using GALEX data from Martin et al. 2005 we find log FFUV/FJ = −2.49, log FNUV/FJ = −2.11). 2MASS J2126−8140 is an L3 first identified by Reid et al. (2008, although referencing Cruz, Kirkpatrick & Burgasser 2009 as the discovery paper). Subsequently Faherty et al. (2013) classified it as a low gravity L3γ (using the gravity classification system of Cruz et al. 2009). Recent VLT/ISAAC observations by Manjavacas et al. (2014) find it is a good match to the young, L3 companion CD-35 2722B (Wahhaj et al. 2011). These authors also used the spectral indices of Allers & Liu (2013) to confirm that the 2MASS J2126−8140 is an L3 and shows low gravity spectral features. Manjavacas et al. (2014) also used the BT-Settl-2013 atmospheric models (Allard, Homeier & Freytag 2012) to derive Teff = 1800 ± 100 K, log g = 4.0 ± 0.5 dex, albeit with better fits to supersolar metallicity models. Filippazzo et al. (2015) use photometry, a trigonometric parallax of 31.3 ± 2.6 mas (referenced to Faherty et al., in preparation) and evolutionary models to derive an effective temperature of 1663 ± 35 K. They also derived a mass of 23.80 ±1 5.19 MJ assuming a broad young age range of 10–150 Myr. Gagné et al. (2014) listed 2MASS J2126−8140 as a high probability candidate member of Tucana–Horologium association (TucHor) but noted that its photometric distance would be in better agreement with its TucHor kinematic distance if it were an equal mass binary.


Douchin & Haensel (2001; DH) formulated a unified EoS for NS on the basis of the SLy4 Skyrme nuclear effective force (Chabanat et al. 1998), where some parameters of the Skyrme interaction were adjusted to reproduce the Wiringa et al. calculation of neutron matter (Wiringa et al. 1988) above saturation density. Hence, the DH EoS contains certain microscopic input. In the DH model the inner crust was treated in the CLDM approach. More recently, unified EoSs for NS have been derived by the Brussels-Montreal group (Chamel et al. 2011; Pearson et al. 2012; Fantina et al. 2013; Potekhin et al. 2013). They are based on the BSk family of Skyrme nuclear effective forces (Goriely et al. 2010). Each force is fitted to the known masses of nuclei and adjusted among other constraints to reproduce a different microscopic EoS of neutron matter with different stiffness at high density. The inner crust is treated in the extended Thomas-Fermi approach with trial nucleon density profiles including perturbatively shell corrections for protons via the Strutinsky integral method. Analytical fits of these neutron-star EoSs have been constructed in order to facilitate their inclusion in astrophysical simulations (Potekhin et al. 2013). Quantal Hartree calculations for the NS crust have been systematically performed by (Shen et al. 2011b,a). This approach uses a virial expansion at low density and a RMF effective interaction at intermediate and high densities, and the EoS of the whole NS has been tabulated for different RMF parameter sets. Also recently, a complete EoS for supernova matter has been developed within the statistical model (Hempel & Schaffner-Bielich 2010). We shall adopt here the EoS of the BSk21 model (Chamel et al. 2011; Pearson et al. 2012; Fantina et al. 2013; Potekhin et al. 2013; Goriely et al. 2010) as a representative example of contemporary EoS for the complete NS structure, and a comparison with the other EoSs of the BSk family (Chamel et al. 2011; Pearson et al. 2012; Fantina et al. 2013; Potekhin et al. 2013) and the RMF family (Shen et al. 2011b,a) shall be left for future study.


Despite the differences of the dust analogues retained in previous studies (Mennella et al. 1998; Boudet et al. 2005; Coupeaud et al. 2011) and in the present one, the spectroscopic properties and temperature dependent behavior of all these analogues remain qualitatively similar: the MAC value is correlated with the dust temperature and has a complex spectral shape differing from a simple asymptotic behavior in λ-2. At the same time, because of these differences between the samples in terms of internal structure, composition, porosity, homogeneity, the shape of the MAC in the FIR varies. Consequently, the MAC value at a given wavelength of all samples from this study and from previous studies spans a large range of values (see Table 1 for the MAC value of the samples studied here and Table 1 from Demyk et al. 2013, for previous studies at 1 mm). It must be noted that the MAC in Boudet et al. (2005) has been corrected for grain shape effect whereas in Coupeaud et al. (2011) and Mennella et al. (1998) and in the present study this correction was not applied (see Sect. 2.3). This partly explains why the MAC value are smaller in Boudet’s study, together with the effect of differences in the samples in terms of composition and structure. At 100 μm, the MAC values of all the samples from this study and from previous work are similar at 300 and 10 K (50–300 cm2 g-1 at 300 K and 40–260 cm2 g-1 at 10 K). At 500 μm, the MAC is greater at 300 K than at 10 K (2–20 cm2 g-1 at 300 K and 0.7–13 cm2 g-1 at 10 K). This is also the case at 1 mm where the MAC is in the range 0.1–11 cm2 g-1 at 300 K and 0.12–7 cm2 g-1 at 10 K. We note that for each sample the MAC at 300 K is always greater than at 10 K. Furthermore, the fact that the range of values of the MAC at 300 K and 10 K overlaps results from the dispersion of the measured spectra. This diversity of the MAC value and spectral shape that vary as a function of the materials studied can be problematic for astrophysical applications since one has to decide what analog is the most relevant and should be used in the modeling. This is considered in more detail in Sect. 5.


Surveys of radio bursts in decimetric wavelengths is presented in papers by Isliker & Benz (1994) and Jirička et al. (2001), within 1–3 GHz and 0.8–2.0 GHz frequency ranges, respectively. Some of these bursts are still not well understood. This is a case of the slowly positively drifting bursts (SPDBs). They appear in groups or as single bursts, with a duration of an individual burst from 1 to several seconds and their frequency drift is lower than about 100 MHz s−1 (Jirička et al. 2001). The SPDBs seem to be similar to the reverse type III bursts (Aschwanden 2002) but their frequency drift is much smaller. The majority of observed SPDBs are connected to solar flares (Jirička et al. 2001), and they appear many times at the very beginning of the flares (Benz & Simnett 1986; Kotrč et al. 1999; Kaltman et al. 2000; Karlický et al. 2018). Kaltman et al. (2000) reported on several SPDBs observed during three solar flares in the 0.8–2 GHz frequency range. They found frequency drifts of the observed SPDBs to be within the 20–180 MHz s−1 range. Kotrč et al. (1999) studied one of those flares. By combining the radio and spectral plus imaging Hα observations, they explained the observed SPDBs as radio emission generated by downwards propagating shock waves. Based on numerical simulations of the formation of thermal fronts in solar flares, Karlický (2015) proposed that SPDBs observed in the 1–2 GHz range could be a signature of a thermal front. Furthermore, Karlický et al. (2018) reported the observation of an SPDB (1.3–2.0 GHz) observed during the impulsive phase of an eruptive flare. They found time coincidence between the SPDB occurrence, an appearance of an ultraviolet (UV)/EUV multithermal plasma blob moving down along the dark Hα loop at approximately 280 km s−1, and the observed change of Hα profile at the footpoint of that dark loop. Combining these observations they concluded that observed SPDB was likely generated by the thermal front formed in front of the falling EUV blob.


The QBOs are also found in the local wavelet power spectra of the Bx, By, and Bz that are displayed in the left panel of Figure 2. As this figure shows, the significant regions of above 95% confidence level in the range of 256–512 days (peak at 1.01 yr) for Bx intermittently appear during 1975–2000, and those in the range of 256–600 days (peak at 1.02 yr) for By are also intermittently found during 1975–2000. The QBOs for Bz only appear in solar cycle 22, which corresponds to the significant period of 0.90 yr, and also peak at 1.58 and 2.77 yr. The possible period of 3.45 yr for Bx does not show the corresponding significant region in the local wavelet power spectrum. The significant region that corresponds to the possible period of 2.37 yr for By appears near 1992. Early study proved that the solar QBOs can be transmitted into interplanetary space by the open magnetic flux (Lockwood 2001; Bazilevskaya et al. 2014); thus, the QBOs found in the Bx, By, and Bz components should be related to the solar QBOs. On the other hand, the annual variation in the IMF polarity in the band of 256–512 days, which is referred to the Rosenberg–Coleman effect (Rosenberg & Coleman 1969), only appears in the ascending phase of solar cycles by studying the IMF polarity date for the years 1927–2002 through the wavelet analysis method (Echer & Svalgaard 2004). The authors indicated that such a result should be attributed to a more stable and flat heliospheric current sheet that only appears in the ascending phase of solar cycles and strong disturbance of the heliospheric current sheet that is present in the declining phase and minimum time of solar cycles, and the Rosenberg–Coleman effect had been confirmed by the IMF Bx component during 1964–2002. The wavelet power spectra of the IMF Bx and By components in Figure 2 show that the annual variation in two time series can also be found in some, but not all, ascending phases of solar cycles, which is partly similar to the wavelet map of the IMF polarity in Echer & Svalgaard (2004). Thus, the annual variation in the IMF Bx and By components is also modulated by the stability or strong disturbance of the heliospheric current sheet. However, the wavelet power spectra of the IMF Bz show a different result, which indicates that the annual variation in this time series should not be related to the heliospheric current sheet.


Magnetohydrodynamic (MHD) shallow water equations are an alternative to complete system of MHD equations for plasma. In this approximation, a thin layer of plasma with a free boundary in a gravity field is studied (Gilman 2000). Shallow water flows with rotation are usually considered in case of large-scale flows in plasma astrophysics (Gilman 1967) and play an important role in understanding of various astrophysical objects. Such a model is used to study large-scale flows of the solar tachocline (a thin layer inside the Sun, located above the convective zone) (Gilman 2000; Dikpati & Gilman 2001; Miesch & Gilman 2004; Hughes, Rosner & Weiss 2007), flows of accreting matter in neutron stars (Inogamov & Sunyaev 1999, 2010; Spitkovsky, Levin & Ushomirsky 2002), dynamics of the atmospheres of neutron stars (Heng & Spitkovsky 2009), and magnetoactive tidally locked atmospheres of exoplanets (Cho 2008; Heng & Showman 2015; Batygin, Stanley & Stevenson 2017; Pierrehumbert & Hammond 2019). MHD shallow water equations are obtained by depth-averaging three-dimensional MHD equations of plasma layer with respect to pressure being hydrostatic and the height of the layer being much smaller than characteristic horizontal scale of motion. However, the depth-averaging procedure does not always lead to shallow water equations. Although the equations obtained by integrating over the height of the studied layer of the liquid will have a form similar to the equations of shallow water (Klimachkov & Petrosyan 2017b).1 Equations obtained in this paper are similar to MHD shallow water equations, but they have additional terms and equations referred to vertical magnetic field (Klimachkov & Petrosyan 2017a; Fedotova, Klimachkov & Petrosyan 2020). These new equations transform into common MHD shallow water equations (Gilman 2000) in case of zero vertical magnetic field. Due to this, hereinafter we name our approximation as quasi-two-dimensional MHD approximation and equations are named as quasi-two-dimensional equations for rotating astrophysical plasma with vertical magnetic field.2 Detailed review of wave processes in magnetohydrodynamics of astrophysical plasma with vertical magnetic field is presented in Petrosyan et al. (2020). Extensions of magnetohydrodynamics in plasma astrophysics for vertically stratified flows with vertical magnetic field can be found in Fedotova, Klimachkov & Petrosyan (2021).


In this work, the parameter ranges are for 68% credibility interval unless specifically mentioned. With Equation (1), the amount of the post-merger gravitational radiation can be reasonably/qualitatively evaluated as long as

 (i.e., the EoS) is known. However, various EoS models have been proposed in the literature and it is not possible to be uniquely determined even in the foreseeable future. Fortunately, under the reasonable assumption that all NSs follow the same EoS, their properties can be jointly/reliably constrained with the nuclear data, the GW data, the measured masses, and the estimated radii of some NSs (e.g., Lattimer & Prakash 2016; Abbott et al. 2017a; Tews et al. 2017; Most et al. 2018; Landry & Essick 2019). The masses of NSs in some binary systems have been accurately measured and there is a robust lower limit on MTOV ≥ 2 M⊙ (Cromartie et al. 2020; Kandel & Romani 2020). The radii of NSs, however, usually are just evaluated indirectly and suffer from large systematical uncertainties. Thanks to the successful performance of the Neutron Star Interior Composition Explorer, the situation has changed and very recently the first-ever accurate measurement of mass and radius together for PSR J0030+0451, a nearby isolated quickly rotating NS, has been achieved (Miller et al. 2019; Riley et al. 2019), which favor a stiffer EoS than the data of GW170817. Hence, GW170817, PSR J0030+0451, some nuclear data as well as the lower limit on MTOV can be combined to reliably constrain the EoS as well as the bulk properties of NSs. This can be done either in the EoS parameterizing methods (Raaijmakers et al. 2019; Jiang et al. 2020) or the nonparametric approach (Essick et al. 2020; Landry et al. 2020), and the results are well consistent with each other. Here we directly adopt posterior samples of {MA, MB, ΛA, ΛB} obtained in Jiang et al. (2020) to calculate the

 for GW170817. Note that the region of

 represents the prompt black hole formation, which is irrelevant to GW170817 because of the delayed collapse of the remnant (Metzger 2019). So we neglect the posterior samples that give

 for GW170817. At 90% credible level, for the piecewise polytropic expansion method we have

, while for the spectral decomposition method we have

. We also adopt the method described in Kumar & Landry (2019) to evaluate the nonparametric posterior of PSRs+GWs+x-ray/Riley case in Landry et al. (2020) and get

 for GW1708017. The incorporation of the strong phase transition possibility by Landry et al. (2020) favors lower k2T than ours (note that the

 region is excluded), but has an overall agreement with the piecewise result and the spectral result (as shown in upper panel of Figure 1). For this reason we combine an equal sample of k2T calculated from these three different parameterization methods to perform all the calculations in this work unless specially specified. Clearly, the inferred

 is well within the region that predicts the very prominent post-merger GW radiation, which is rather encouraging. The post-merger energy is then estimated to be

 (90% confidence level) when the fitting error is considered for the combined posterior (see the upper panel of Figure 2). For B1534+12, B2127+11C, J1757-1854, and J0453+1559, we expect that they would emit almost the same amount of energy as GW170817 in the post-merger phase. This is understandable considering the comparable total mass of these systems. For lighter BNS systems like J0514-4002A, the expected post-merger energy is relatively small because of the large

. While for heavier BNS systems like GW190425, we predict a prompt collapse scenario and thus emit a small amount of energy in the post-merger phase (see the lower panels of Figures 1 and 2). To our knowledge, ours is the first study to combine the intriguing numerical finding of Zappa et al. (2018) with the EoS constrained with the multimessenger information of NSs and then to demonstrate that GW170817 is likely the most efficient post-merger GW emitter among the observed BNS GW events.


In order to determine the unknown rate constant that will be used in the network, we used a methodology developed in previous articles (Loison et al. 2014a,b, 2015) and summarized in Appendix A. This methodology includes an extensive literature review, various Density Functional Theory (DFT) and ab initio calculations for critical gas phase reactions, namely H + l-C3H2, H + t-C3H2, O + c-C3H2, O + l-C3H2, N + C3, N + c-C3H2, N + l-C3H2, H + C3O, O + C3O, O + c-C3H3+, OH + C3, OH + c-C3H2, H2 + l-C3H and H2 + c-C3H, to determine the presence, or not, of a barrier. When there is no barrier in the entrance valley and exothermic bimolecular exit channels, we chose to use the capture rate constant or sometimes a fraction of the capture rate constant by comparison with similar reactions (the capture rate is the upper limit of the rate constants for barrierless reactions). The dissociative recombination (DR) of c,l-C3H2+ and c,l-C3H3+ is an important source of c,l-C3H and is the main source c,l,t-C3H2 in our network. The first step of c,l-C3H2+ and c,l-C3H3+ DR is the formation of highly excited C3H2** and C3H3**, which leads to bond fragmentation. Angelova et al. (2004) have shown that the DR of c-C3H2+ leads to 87.5 per cent of C3Hx and 12.5 per cent of C2Hy + CHz, and the DR of c-C3H3+ leads to 90.7 per cent of C3Hx and 9.3 per cent of C2Hy + CHz. Moreover, in DR processes, the H ejection is in general favoured more than H2 ejection (Plessis et al. 2010, 2012; Janev & Reiter 2004). Considering the exothermicity for the ejection of two hydrogen atoms (endothermic for c-C3H3+ DR and only slightly exothermic for l-C3H2+, c-C3H2+ and l-C3H3+ DR, see Appendix B), this process will have a low branching ratio. Then, the dissociation of C3H2** and C3H3** will produce mainly C3H + H and C3H2 + H, both C3H and C3H2 species being also excited considering the exothermicity of the DR and the fact that the hydrogen atom will carry only a limited part of the available energy through kinetic energy. Part of the excited C3H* and C3H2* molecules will lead to dissociation when they are populated above the dissociation limit, but most of them will relax through radiative emission of an infrared photon. As noted by Herbst et al. (2000), the typical time-scale for isomeric conversion is much shorter than that for relaxation by one infrared photon. Thus, because radiative relaxation occurs slowly, isomeric conversion leads to equilibrated isomeric (c-C3H $\leftrightarrows $ l-C3H, c-C3H2$\leftrightarrows $ l-C3H2) abundances at each internal energy. The final balance is determined at or near the effective barrier to isomerization, which corresponds to the energy of the transition state. The ratio between the isomeric forms are then approximated by the ratio of the rovibrational densities of states of the isomers at the barrier to isomerization calculated using the MESMER program (Glowacki et al. 2012). Fig. 2 shows the isomerization pathway calculated at the DFT level. The calculated geometries of the stationary point can be found in Appendix A. The t-C3H2 has a triplet ground state and its production in the excited singlet ground state is neglected here. The production of t-C3H2 from the DR of c,l-C3H3+ is supposed to come from c,l-C3H3+ + e− → c,l-C3H3 → t-C3H2 + H, which is supposed to be a minor channel in comparison to c,l-C3H3+ + e− → c,l-C3H3 → c,l-C3H2 + H (t-C3H2 has a ground triplet state, in contrast to c,l-C3H2, which has a singlet ground state).


The formation and evolution of S0 galaxies are very important for understanding the formation and evolution of galaxies, but they are still an open question (e.g., the recent review by D’Onofrio et al. 2015). Currently, there are two possible scenarios on the origin of S0 galaxies. One is that S0 galaxies are transformed from spiral galaxies, where spirals lose their gas and star formation is rapidly quenched. The other is that S0 galaxies are intrinsically different from spiral galaxies since their formation (Kormendy & Kennicutt 2004; Barway et al. 2009; van den Bergh 2009). The transformation origin may be associated with intra-cluster medium and neighboring galaxies, via minor mergers, slow encounters, galaxy harassments (Moore et al. 1996), or tidal effects in the dense environment (Gunn & Gott 1972; Larson et al. 1980; Dressler & Sandage 1983; Mihos & Hernquist 1994; Moore et al. 1998, 1999; Neistein et al. 1999; Shioya et al. 2002). A lot of studies have discussed the environmental dependence of galaxy evolution. Generally speaking, early-type galaxies tend to be in dense environments and have low star-formation rates (SFRs; Dressler 1980; Balogh et al. 1997, 1998, 2000; Poggianti et al. 1999; Treu et al. 2003). The fraction of S0 galaxies in the field is only about 15%, while spirals are the majority (Naim et al. 1995). Within the group environment, spirals and S0s are both about 40%–45% (Postman & Geller 1984). Furthermore, S0s become dominant in dense environments, the fraction grows up to 60% in clusters (Dressler 1980; Postman & Geller 1984). Dressler et al. (1997), Fasano et al. (2000), and Desai et al. (2007) also found that the galaxy morphological distributions change abruptly in clusters at

, about 50 ∼ 70% of spirals at high redshift (z > 0.4) are transformed into S0s, while a fraction of ellipticals, about 25%, remains nearly constant between z = 0.8 and z = 0.0. However, Wilman et al. (2009) showed that the fraction of S0s in groups is the same as in clusters but it is much higher than in the field at z = 0.4, which might suggest that S0s are formed in groups or subgroups.


Extended radio emission in galaxies is associated with both radio jets and lobes and with outflows, seen often as aligned radio sources in the opposite directions with respect to the central compact radio core. Giant radio galaxies (GRG) are extreme cases of this phenomenology with jets and lobes extending on ∼ Mpc scales suggesting that they are either very powerful or very old site for electron acceleration. In this respect, GRGs have a crucial role in the acceleration of cosmic rays over large cosmic scales (e.g. Kronberg et al. 2004), in the feedback mechanism of AGNs into the intergalactic and intracluster medium (e.g. Subrahmanyan et al. 2008) and in the seeding of large-scale magnetic fields in the universe (e.g. Kronberg et al. 2004) and they are excellent sites to determine the total jet/lobe energetics in AGN-dominated structures (see e.g. Colafrancesco 2008, Colafrancesco & Marchegiani 2011). To date our knowledge of GRGs (see e.g. Ishwara-Chandra & Saikia 1999, 2002; Lara et al. 2001; Machalski, Jamrozy & Zola 2001; Schoenmakers et al. 2001; Kronberg et al. 2004; Saripalli et al. 2005; Malarecki et al. 2013; Butenko et al. 2014) is limited by their sparse numbers and by the difficulty of detecting them over large areas of the sky. Low-frequency radio observations have an enhanced capacity to detect the extended old electron population in these objects (see e.g. the recent Low Frequency Array – LOFAR – observation of the GRG UGC095551), but high-frequency radio observations are less efficient in this task due to the steep-spectra of giant radio lobes. In this context these sources will be ideal targets for the next coming deep, wide-field surveys like, e.g. the ATLAS survey of the Australia Telescope Network Facility (ATNF; see Norris et al. 2009) or the Square Kilometre Array (SKA) deep surveys that will have the potential to study their population evolution up to high redshifts and thus clarifying their role on the feedback for the evolution of non-thermal processes in large-scale structures.


The Virgo Cluster radio galaxy M87 (NGC 4486), located at a distance of

 Mpc (Mei et al. 2007) and believed to harbour a BH of mass

, was the first extragalactic source detected at VHE energies (Aharonian et al. 2003). Given its proximity, M87 has been a prime target to probe scenarios for the formation of relativistic jets with high-resolution radio observations exploring scales down to some tens of rg, and much effort has recently been dedicated in this direction (e.g., Acciari et al. 2009; Doeleman et al. 2012; Hada et al. 2014, 2016; Akiyama et al. 2015, 2017; Kino et al. 2015). At VHE energies, M87 has revealed at least three active γ-ray episodes, during which day-scale flux variability (i.e.,

) has been observed (Aharonian et al. 2006; Albert et al. 2008; Acciari et al. 2009; Abramowski et al. 2012; Aliu et al. 2012). The VHE spectrum is compatible with a relatively hard power law (photon index ∼2.2) extending from 300 GeV to beyond 10 TeV, while the corresponding TeV output is relatively moderate, with an isotropic equivalent luminosity of

 erg s−1. The inner, parsec-scale jet in M87 is considered to be misaligned by

–25°, resulting in modest Doppler boosting of its jet emission and creating challenges for conventional jet models to account for the observed VHE characteristics (see, e.g., Rieger & Aharonian 2012 for review and references). Gap-type emission models offer a promising alternative and different realizations have been proposed in the literature (e.g., Neronov & Aharonian 2007; Levinson & Rieger 2011; Broderick & Tchekhovskoy 2015; Vincent 2015; Ptitsyna & Neronov 2016). M87 is overall highly underluminous with characteristic estimates for its total nuclear (disk and jet) bolometric luminosity not exceeding

 erg s−1 by much (e.g., Owen et al. 2000; Whysong & Antonucci 2004; Prieto et al. 2016), suggesting that accretion onto its BH indeed occurs in a non-standard, advective-dominated (ADAF) mode characterized by an intrinsically low radiative efficiency (e.g., Di Matteo et al. 2003; Nemmen et al. 2014), with inferred accretion rates possibly ranging up to

 (e.g., Levinson & Rieger 2011) and a BH spin parameter close to its maximum one (e.g., Feng & Wu 2017). For these values of the accretion rate, the soft photon field (see Equations (2) and (3)) is sufficiently sparse that the maximum Lorentz factor

 of the magnetospheric particles is essentially determined by the curvature mechanism. The observed VHE variability is in principle compatible with

, so that the different dependence of the gap power on β, Equation (20), does not necessarily (in the absence of other, intrinsic considerations of gap closure) imply a strong difference in the extractable gap powers. Figure 2 shows a representative point for M87 (taking

). The observed VHE luminosity of M87 is some orders of magnitudes lower than the maximum possible gap power (given by the dotted line) and within the bound imposed by ADAF considerations (vertical line). The observed VHE flaring events thus appear consistent with a magnetospheric origin. VLBI observations of (delayed) radio core flux enhancements indeed provide support for the proposal that the variable VHE emission in M87 originates at the jet base very near to the BH (e.g., Acciari et al. 2009; Beilicke 2012; Hada et al. 2012, 2014).


In the tidal torque model used in this work, the alignment of spiral galaxies is purely due to their orientation, which in turn is related to the angular momentum correlation of neighbouring galaxies relative to the line of sight (Croft & Metzler 2000; Crittenden et al. 2001). Angular momentum correlations are mainly build up at early times during structure formation and are thus due to initial correlations (Catelan & Theuns 1996, 1997; Theuns & Catelan 1997). The correlated angular momenta result into to correlated inclination angles of neighbouring galaxies and thus ultimately into correlated ellipticities (Catelan, Kamionkowski & Blandford 2001). Assuming that the symmetry axis of the galactic disc coincides with the direction of the angular momentum $\hat{L} = \boldsymbol{L}/L$, the ellipticity can be written as the alignment of spiral galaxies is purely due to the orientation of their circular discs, which in turn is related to the angular momentum correlation of neighbouring galaxies relative to the line of sight (Croft & Metzler 2000; Crittenden et al. 2001). Angular momentum correlations are mainly build up at early times during structure formation and are thus due to initial correlations (Catelan & Theuns 1996, 1997; Theuns & Catelan 1997). The correlated angular momenta result into correlated inclination angles of neighbouring galaxies and thus ultimately into correlated ellipticities (Catelan et al. 2001). Assuming that the symmetry axis of the galactic disc coincides with the direction of the angular momentum $\hat{L} = \boldsymbol{L}/L$, the ellipticity can be written as
(17)$$\begin{eqnarray*}
\epsilon = \frac{\hat{L}^2_x - \hat{L}^2_y}{1+ \hat{L}^2_z} + 2\mathrm{i}\frac{\hat{L}_x\hat{L}_y}{1+\hat{L}^2_z}\,\, .
\end{eqnarray*}$$Angular momentum is generated by a torque exerted by the ambient large-scale structure on to the protogalactic halo, a mechanism called tidal torquing (White 1984; Barnes & Efstathiou 1987; Schaefer 2009; Stewart et al. 2013). For Gaussian random fields, the autocorrelation of angular momenta is given by (Lee & Pen 2001)
(18)$$\begin{eqnarray*}
\left\langle \hat{L}_\alpha \hat{L}_\beta \right\rangle = \frac{1}{3}\left(\frac{1+ A}{3}\delta _{\alpha \beta } - A \hat{\Phi }_{\alpha \mu }\hat{\Phi }_{\mu \beta } \right)\,\, .
\end{eqnarray*}$$The free parameter A determines the strength of the coupling between alignment and tidal torque. Since the correlation is determined by the traceless part of the shear tensor $\hat{\Phi }_{\alpha \beta }$ the resulting effect is clearly due to orientation effects only. For a Gaussian distribution $p(\hat{L}|\hat{\Phi }_{\alpha \beta })\mathrm{d}\hat{L}$ and the use of equation (17), one can express the ellipticity in terms of the tidal field
(19)$$\begin{eqnarray*}
\epsilon (\hat{\Phi }) = \frac{A}{2} \left(\hat{\Phi }_{x\alpha }\hat{\Phi }_{\alpha x} - \hat{\Phi }_{y\alpha }\hat{\Phi }_{\alpha y} -2\mathrm{i}\hat{\Phi }_{x\alpha }\hat{\Phi }_{\alpha y}\right).
\end{eqnarray*}$$Correlations in the ellipticities can thus be traced back to the four-point function of the shear field, which is given in equation (11). For keeping a correct relative normalization of the shape correlations, we scale the resulting angular ellipticity spectra $C^{\mathrm{s},II}_{ij}(\ell)$ with the squared number of spiral galaxies $n_s^2$. It is remarkable that the shapes of spiral galaxies in the quadratic alignment model are in fact sensitive to tidal shear components parallel to the line of sight; in fact, those components determine the magnitude of the alignment effect, in contrast to the alignment of elliptical galaxies in the linear alignment model or to gravitational lensing, which reflect purely the tidal shear components perpendicular to the line of sight.


Galaxy colour is known to be sensitive to local density. Generally, sheets are denser than fields and filaments are denser than sheets. So the dependence of red and blue fractions on the geometry of large-scale environment shown in Fig. 5 may partly arise due to dependence of galaxy colour on local density. We need to decorrelate the effect of local density in order to test the role of large-scale structures on galaxy colours. We address this issue by calculating local number density of galaxies using kth nearest neighbour method (Casertano & Hut 1985). We compute the local number density using equation (2) with k = 5. We plot the local density against the local dimension of galaxies in three different volume-limited samples in Fig. 7. The top left, middle left, and bottom left panels of Fig. 7 show the relations between local dimension and local density for the three volume-limited samples when local dimensions are computed using $R_2=10 {\, h^{-1}\, {\rm Mpc}}$. The results in these panels show that the environments with larger local dimension indeed tend to have a lower local density. However, these relationships show very large scatters. The environments with local dimension up to D = 2.5 can have a wide range of local densities and it is difficult to assign a specific density range to the environments with different local dimensions. The three panels in the middle column and three panels in the right column of Fig. 7 show the relations between local density and local dimension in these samples for $R_2=40 {\, h^{-1}\, {\rm Mpc}}$ and $R_2=70 {\, h^{-1}\, {\rm Mpc}}$. They show a similar trend as seen in the three panels in the left column of the same figure. It may be noted that galaxies with smaller local dimension are progressively absent when the geometry of environments are characterized on larger length-scales. This points out to the emergence of a homogeneous network of galaxies on larger length-scales as mentioned earlier.


The 3 month γ-ray light curve of 4FGL J1510.1+5702 reveals that it is at a high flux state in an epoch of several tens of days in 2018; meanwhile, optical flux densities of GB 1508+5714 in two bands rise in the same epoch. To further investigate the relationship between these two domains of emissions, a 3 day time bin γ-ray light curve is presented, together with the zoomed-in ZTF light curves; see Figure 4. In spite of the limited statistics and no Fermi-LAT observation toward the target at the exact time of the optical flares, the time bin (i.e., centered at MJD 58288.5) with the largest TS value in the 3 day γ-ray light curve is very close to the peaking time of the optical flares. Since optical variations are likely from the jet because of the large variability amplitude, based on the simultaneous γ-ray and optical brightening, we conclude that 4FGL J1510.1+5702 is the γ-ray counterpart of GB 1508+5714. There are three other time bins in the 3 month γ-ray light curve with TS values ≥10, centered at MJD 54909, 57265, and 56540, respectively. The first two time intervals do not fall into the operation time range of iPTF/ZTF. Moreover, no iPTF/ZTF data of GB 1508+5714 around MJD 56,540 are available; see Figure 3. Theoretically, in the leptonic radiation scenario, the optical and GeV γ-ray emissions of low synchrotron peaked blazars, including FSRQs, are proposed to be from the same population of emitting electrons. It is supported by correlated optical/γ-ray flares in FSRQs (e.g., Abdo et al. 2010; Bonning et al. 2012). Meanwhile, the redder-when-brighter spectral variability behavior has been detected in the optical wavelengths of γ-ray FSRQs, which is explained by the influence of the blue and slowly varying accretion disk emission (e.g., Bonning et al. 2012; Fan et al. 2018). For GB 1508+5714, a similar trend is shown. The optical spectral color, rmag–imag, is 0.66 mag in MJD 58286, while it is 0.43 mag in MJD 58272, despite the relatively large photometric uncertainties. The optical spectral color scaled by rmag corresponding to observations in different epochs are plotted, as shown in Figure 4. Its i-band variability amplitude is larger than that in the r band. All of these facts suggest that the contribution of the jet emission becomes significant at the optical wavelengths of GB 1508+5714 when the jet activity is intense with rising γ-ray emission. Moreover, the significant γ-ray emission together with the rapid optical variation in the r band provide information of the emitting jet blob. The radius of the emitting blob is constrained by the variability timescale,

, where

 days for the current event. Meanwhile, assuming that the optical and γ-ray photons of GB 1508+5714 are from the same region, to avoid serious absorption on γ-rays from soft photons via the

 process, the corresponding optical depth should not be high,
1

where

 is the scattering Thomson cross section,

 is the differential comoving number density of the target photon per energy,

 is the energy of the target photon in dimensionless units, and

 is the absorption length (Dondi & Ghisellini 1995; Begelman et al. 2008). The soft photons from the jet itself could be responsible for the absorption. Since the highest energy of the detected γ-ray photons of GB 1508+5714 is ∼8 GeV, the corresponding soft photons are those detected at a few keV (3

  erg s−1; Marcotulli et al. 2020), and the absorption length can be set the same as the radius of the emitting blob. A constraint of the Doppler factor of the jet blob,

, is given.


Finally, our models assume the absence of planetary magnetic fields. The early paradigm considering the evolution of terrestrial planets has implied that the planetary magnetic field is necessary to protect planetary atmospheres and reduce the atmospheric mass loss (see e.g. Dehant et al. 2007, and references therein). The later studies, however, show that this point of view is ambiguous. Thus, the effect of the magnetic field on the atmospheric escape can be considered as a result of the two concurring processes: reducing the escape by capturing the ionised atmospheric species within the closed magnetic field lines, and enhancing the escape of the atmospheric ions through the regions of the open magnetic lines (polar cusps, in the case of a dipole field) and the reconnection on the night-side (see, e.g. Khodachenko et al. 2015; Sakai et al. 2018; Carolan et al. 2021). Thus, for planets in the Solar System, it was shown both in the observations (Gunell et al. 2018; Ramstad & Barabash 2021) and by modelling (Sakai et al. 2018; Egan et al. 2019) that the presence of a weak magnetic field can intensify atmospheric escape. These results, however, should be taken with caution for young planets, and in particular those in the sub-Neptune range, because of the different atmospheric structures and the non-thermal mechanisms dominating the atmospheric mass loss in the Solar System, which are contrary to the planets considered in this study (see, e.g. Scherf & Lammer 2021, for the discussion). For hot Jupiters, Khodachenko et al. (2015) predict a significant suppression of escape for intrinsic magnetic fields larger than 0.3 G. The model with the closest setup to this study by Carolan et al. (2021) predicts, however, for the 0.7Mjup planet experiencing XUV (thermally) driven atmospheric escape, a small increase in the atmospheric mass-loss rate with increasing dipole field strength (about twice between 0 and 5 G). We therefore expect that the possible effect from the planetary intrinsic magnetic field depends largely on the strength and configuration of the planetary and stellar magnetic fields, but, according to the numbers reported in the literature, might not affect our results dramatically. The lack of studies for close-in sub-Neptune-like planets, however, holds us from making final conclusions.


Numerous investigations of tidal activity on extrasolar planets have been conducted, with a range of topics from the behavior of gas giants (e.g., Běhounková et al. 2010, 2011; Remus et al. 2012a, 2012b; Storch & Lai 2014), to tidal alterations of system dynamics (e.g., Lecoanet et al. 2009; Matsumura et al. 2010; Cébron et al. 2011; Bolmont et al. 2015; Turbet et al. 2017), to tidal alterations of habitability (Barnes et al. 2008, 2013; Jackson et al. 2008a, 2008b; Heller & Armstrong 2014; Kopparapu et al. 2014), issues of spin dynamics (Correia et al. 2008; Ferraz-Mello et al. 2008; Efroimsky 2012b; Cunha et al. 2015), and the role of tides on exomoons (Namouni 2010; Heller & Barnes 2013). Many such studies naturally begin with frequency-independent internal models, but an increasing number consider viscoelastic models (Henning et al. 2009; Běhounková et al. 2010, 2011; Remus et al. 2012a, 2012b; Auclair-Desrotour et al. 2014; Correia et al. 2014; Henning & Hurford 2014; Makarov & Efroimsky 2014; Shoji & Kurita 2014; Driscoll & Barnes 2015; Makarov 2015). Countless more studies rely upon reasonable selections of tidal dissipation terms in order to inform simulations of system dynamics. For solid planetary objects, a detailed study is eventually needed to constrain which rheological models are best under the stress, pressure, and compositional conditions that are applicable to exoplanets and exomoons. Indeed, studies of the Earth tell us that multiple rheological models may be needed as one goes deeper into an exoplanet’s interior. Higher pressures will surely change the microphysical mechanisms that govern the rheological response (Karato & Spetzler 1990). We currently must rely mainly on analytical and numerical modeling when exploring the interiors of extrasolar planets, particularly worlds in the super-Earth category not represented in our solar system (Valencia et al. 2007). It is not yet known how well laboratory results on the viscosity of peridotite can extend to high-pressure phases such as post-perovskite (Murakami 2004), which may play a large role in super-Earths.


Cygnus A, at z = 0.0562, is 10 times closer than the next radio galaxy of similar radio luminosity.4

4
Radio luminosity >1045 erg s−1.
 The nuclear regions in Cygnus A have been observed extensively at radio through X-ray wavelengths (Carilli & Barthel 1996). The inner few arcseconds is a complex mix of optically obscuring dust clouds (Vestergaard & Barthel 1993; Whysong & Antonucci 2004; Lopez-Rodriguez et al. 2014; Merlo et al. 2014), atomic gas seen in narrow line emission (Stockton et al. 1994; Taylor et al. 2003), H i 21 cm absorption toward the inner radio jets, with a neutral atomic column density >1023 cm−2, depending on H i excitation temperature (Struve & Conway 2010), polarized, broad optical emission lines due to scattering by dust (Antonnuci et al. 1994; Ogle et al. 1997), and a highly absorbed hard X-ray spectrum with a total gas column density of ∼3 × 1023 cm−2 (Ueno et al. 1994; Reynolds et al. 2015). VLBI radio observations at 0.05 mas resolution reveal highly collimated jets originating on scales ∼200 times the Schwarzschild radius (Boccardi et al. 2016). Tadhunter et al. (2003), derive a black hole mass of 2.5 ± 0.7 × 109 M⊙ from HST and Keck spectroscopy of Pa-α and [O iii], and conclude that Cygnus A contains an AGN with a bolometric luminosity of order 1046 erg s−1, comparable to high redshift quasars (Runnoe et al. 2012). This AGN is highly obscured in the optical due to dust along our line of sight, with Av > 50 magnitudes, based on near-IR spectroscopy (Imanishi & Ueno 2000). Studies of the mid- to far-IR spectral and polarization properties have led to a model of a clumpy, dusty torus obscuring the AGN in Cygnus A, with a radius of at least 130 pc, although these conclusions are based on spatially integrated properties; these observations did not have the spatial resolution to resolve the torus, and hence are partially contaminated by emission from the radio core-jet (Privon et al. 2012; Lopez-Rodriguez et al. 2018).


A significant limitation of the model is that it ignores the well-established hydrodynamic evolution of the loop during the cooling process, involving the substantial transfer of mass between the chromosphere and the corona. For large downward heat fluxes, the transition region is unable to radiate the supplied energy, resulting in the deposition of thermal energy in the dense chromosphere. The resulting two to three orders-of-magnitude temperature enhancements create a large pressure gradient that drives an upward enthalpy flux of “evaporating” plasma. However, as the loop cools, the decreased heat flux becomes insufficient to sustain the radiation emitted in the now-dense transition region and hence an inverse process of downward enthalpy flux starts to occur. It has been suggested (Klimchuk et al. 2008) that the enthalpy fluxes associated with both evaporating and condensing plasma are at all times in approximate balance with the excess or deficit of the heat flux relative to the transition region radiation loss rate. This basic idea has allowed the development of global “Enthalpy-Based Thermal Evolution of Loops” (EBTEL) models that describe the evolution of the average temperature and density in the coronal part of the loops; these models are generally in good agreement with one-dimensional hydrodynamic simulations (Klimchuk et al. 2008; Cargill et al. 2012a, 2012b). It is, in principle, possible to include the effects of a turbulence-controlled heat flux in EBTEL (or 1D hydrodynamic) models. If this heat flux is reduced sufficiently relative to its collisional value, then, for the reasons explained above, there will be a significant impact on the thermal evolution of the loop. Doing so, however, would still require a numerical treatment, which is beyond the scope of the present work (but which it is our intention to carry out in a future work). Instead, we adopt a simpler approach that allows a systematic and fairly transparent quantitative analysis of the impact of turbulence on the thermodynamics of post-flare loops.


The first step in searching for dynamically correlated minor bodies, particularly those resulting from break-ups, is to get a clear characterization of what the expectations may be. The outcome of cometary disruption is well documented through two well-studied examples, those of the comets 73P/Schwassmann-Wachmann 3 and D/1993 F2 (Shoemaker-Levy 9). For reasons that still remain unclear, comet 73P started to break apart in 1995 and dozens of fragments were observed in 2006 and 2007 (see e.g. Crovisier et al. 1996; Weaver et al. 2006; Reach et al. 2009; Hadamcik & Levasseur-Regourd 2016). Some of these fragments have been recovered in 2010–2011 (Harker et al. 2011, 2017; Sitko et al. 2011) and 2016–2017 (e.g. Kadota et al. 2017;1 Williams 2017); it may consist of hundreds of pieces now (68 of them have orbit determinations). This fragmentation process can be described as gentle and progressive. In striking contrast, comet Shoemaker-Levy 9 experienced a sudden, violent fragmentation event triggered by strong tidal forces during a close encounter with Jupiter in 1992 July (see e.g. Sekanina, Chodas & Yeomans 1994, 1998; Asphaug & Benz 1996; Sekanina 1997). Most fragments collided with Jupiter over a period of a week (1994 July 16–22); 21 of them have orbit determinations. Quite different may have been the collisional event that led to the formation of the Haumea family (Brown et al. 2007; Schlichting & Sari 2009; Leinhardt, Marcus & Stewart 2010; Lykawka et al. 2012; Ortiz et al. 2012) perhaps more than 1 Gyr ago (Ragozzine & Brown 2007; Volk & Malhotra 2012). Fragments of recently disrupted minor bodies must have very similar values of their semimajor axis, a, eccentricity, e, inclination, i, longitude of the ascending node, $\mathit {\Omega }$, argument of perihelion, ω, and time of perihelion passage, τq, but $\mathit {\Omega }$, ω and τq tend to become increasingly randomized over time. In contrast, recently unbound pairs resulting from binary dissociation events might have relatively different values of a and e, but very similar values of i, $\mathit {\Omega }$ and ω, the difference in τq may initially range from weeks to centuries, but grows rapidly over time (see e.g. de León, de la Fuente Marcos & de la Fuente Marcos 2017; de la Fuente Marcos, de la Fuente Marcos & Aarseth 2017).


Braukmuller et al. (2018) proposed that all elements fall into one of four categories based on their condensation temperature: refractory elements (50% condensation temperature, Tc,50 > 1400 K), which exhibit approximately uniform enrichments in their Si-normalized concentrations in CC chondrites compared to CI chondrites by a factor of ∼1–1.4; main component elements (1300 K  Tc,50  1400 K), which have approximately the same Si-normalized elemental abundances in CC chondrites as CI chondrites (differ by a factor of ∼0.8–1.1); slope-volatile elements (800 K  Tc,50  1300 K), which exhibit monotonically decreasing Si-normalized concentrations with decreasing Tc,50 compared to CI chondrites; and plateau volatile elements (Tc,50  800 K), which display uniform depletions in Si-normalized concentrations compared to CI chondrites by a factor of ∼0.1–0.7 that are characteristic of each CC chondrite group. Given their uniform nature with Tc,50 and comparatively well-constrained isotopic and elemental compositions, we chose to focus on the concentrations of refractory, main component, and plateau volatile elements in this study. For the refractory and main component elements in CC chondrites, we examine the elemental and isotopic compositions of Ti and Cr, respectively, because these are lithophile elements whose isotopic compositions have been measured precisely for a large number of chondrites and their components (Trinquier et al. 2007, 2009; Qin et al. 2010; Olsen et al. 2016; Van Kooten et al. 2016; Gerber et al. 2017; Davis et al. 2018; Zhu et al. 2019; Schneider et al. 2020; Williams et al. 2020). For CC iron meteorites, we examine the isotopic compositions of Mo and Ni, respectively, because these are siderophile elements (so are therefore present in appreciable concentrations in iron meteorites, unlike Ti and Cr) whose compositions have also been relatively well studied in a number of iron meteorites as well as chondrites and their components (Burkhardt et al. 2011; Budde et al. 2016; Kruijer et al. 2017; Bermingham et al. 2018; Nanne et al. 2019; Budde et al. 2019; Worsham et al. 2019; Brennecka et al. 2020; Spitzer et al. 2020). For the plateau volatile elements, we examine the elemental compositions of six elements (Bi, Ag, Pb, Zn, Te, and Sn) that exhibit a number of desirable properties: their concentrations have been relatively well constrained in CC chondrites; they show a range of lithophile, siderophile, and chalcophile behaviors; their concentrations do not appear to be strongly dependent on redox state; they show minimal variability among NC chondrite groups. Our reasoning for not considering the isotopic compositions of these elements is discussed in Section 2.3. The adopted isotopic and chemical composition of each element used in this study in CC chondrites, CC iron meteorites, CAIs, CI chondrites, and NC chondrites are included in Table 1. Uncertainties on elemental concentrations have not been routinely reported throughout the literature, although these values are typically ±5 wt% (e.g., Lodders 2003; Palme et al. 2014). CAIs can be categorized into six groups based on their compositions (Stracke et al. 2012). For the purposes of this study, we adopt the composition of type I CAIs as the representative value of refractory objects because they are seemingly the most abundant type and lack the characteristic elemental depletions of other CAI groups (e.g., Stracke et al. 2012; Brennecka et al. 2020). We also focus largely on ordinary chondrites (OC) as representative NC meteorites rather than enstatite chondrites (EC) or Rumuruti chondrites (RC). This is because EC chondrites formed under more reducing conditions than OC and RC chondrites, which introduced a compositional signature for some elements to EC chondrites that is not present in OC, RC, or CC chondrites (presumably due to their formation in more oxidizing environments) so is not representative of large-scale mixing in the disk (Alexander 2019b). Additionally, the isotopic compositions of RC chondrites are sparsely measured compared to OC and EC chondrites. NC meteorites could have experienced a number of processes (e.g., mixing, chondrule formation, volatile loss, the addition of refractory materials, etc.) that gave these meteorites their specific chemical and isotopic signatures (Alexander 2019b). We do not explore these processes in this study and simply adopt the measured elemental and isotopic compositions of NC chondrites as potential end-members for the compositions of CC meteorites.


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.


The best-fit parameters are shown in Table 3. We find that the uncertainties in M* and

 are considerably large due to a degeneracy between the two parameters when all parameters are variable. Plotted in Figure 1 is the faint-end slope α as a function of redshift. Our results indicate that the best-fit values of α are about −2 at

 to 10, which are steeper than those at lower redshift (e.g.,

 at

 in Bouwens et al. 2015). We show the fitting results at

, 8, 9, and 10 in Figures 2, 3, 4, and 5, respectively. The top and bottom panels present the observed number densities and the best-fit luminosity functions in the image plane and the source plane, respectively. We also plot the results of previous blank-field surveys (Ouchi et al. 2009; Bradley et al. 2012; McLure et al. 2013; Oesch et al. 2013; Schenker et al. 2013; Bowler et al. 2014; Bouwens et al. 2015; Finkelstein et al. 2015; Calvi et al. 2016) and recent HFF results in other studies (Atek et al. 2015a; Laporte et al. 2016; McLeod et al. 2016). The best-fit parameters are consistent with those in previous studies. In the top panel of Figure 2, there may be an excess in the observed surface number density at

. The reason for this excess is not clear, although using a size–luminosity relation that gives smaller sizes at faint magnitudes may reduce this excess. At

, the observed number densities at the bright end are slightly larger than the number densities in the simulation. This is probably due to the existence of an overdense region of

 dropouts in the Abell 2744 cluster field. We discussed the properties of the overdensity in Ishigaki et al. (2016; see also Zheng et al. 2014 and Atek et al. 2015b). At

, although we detect no galaxies, we can place a constraint on the luminosity function from the non-detection. Based on the best-fit parameters where only

 is variable, ∼1.4 galaxies are expected to be detected in the HFF fields. The middle panels of Figures 2–5 show histograms of the number of the dropouts. It is seen that our samples push the magnitude limits of the luminosity functions significantly by up to ∼3 magnitude. The correlations between M* and α at z ∼6–7 and 8 are presented in Figure 6.


The Gibbs free energy of the system, and thus the composition of the solids formed depends on the pressure and temperature at which condensation occurs. In order to consider reasonable pressures and temperatures for the inner regions of the PPD, and in order to convert these temperatures and pressures into radial locations within the disc and formation times for the solid condensates, we consider the simplest possible PPD model. We use the theoretical model derived in Chambers (2009), which models the viscous accretion of gas heated by the star. This model has been previously used for the modelling of planetesimal formation in PPDs (Moriarty et al. 2014; Harrison et al. 2018) and super-Earths (Alessi, Pudritz & Cridland 2017). This model ignores any vertical or radial mixing, and as will be discussed further later, any radial drift. All of these processes may be of critical importance in a realistic PPD. The Chambers model is a disc model with an alpha parameterization that divides the disc into three sections; an inner viscous evaporating region, an intermediate viscous region, and an outer irradiated region. For the calculations in this work, we have assumed disc parameters of $s_{0} = 33\, \mathrm{au}$, $\kappa _{0} = 0.3\, \mathrm{m^{2}\, kg}^{-1}$, α = 0.01, γ = 1.7, μ = 2.4, and $M_{*}=0.78\, \mathrm{M}_{\odot }$ following Chambers (2009) and Motalebi et al. (2015). We also assume that the mass of the PPD is directly proportional to the mass of the host star according to M0 = 0.1M* (Chambers 2009; Andrews et al. 2013). The temperature and radius of the star in the PPD phase are assumed to be functions of the stellar mass in the form derived in Siess, Dufour & Forestini (2000). The relations used in this work to calculate the PPD mass, the initial stellar radius, and the initial stellar temperature as a function of stellar mass are consistent with the values given in Stepinski (1998) and Chambers (2009) for a solar mass star. The analytical expressions for the pressure and temperature of the mid-plane of the PPD as a function of radial location (a) and time (tdisc) are presented in Appendix C. The temperature–radial location curves for the model disc around a star similar to HD 219134 are plotted as a function of time in Fig. 2. The pressure–temperature space mapped out by the model disc for the case of a star similar to HD 219134 is displayed in the Appendix in Fig. C1. The pressure–temperature space for the model disc of a solar mass star shows negligible differences compared to the HD 219134 case.


This conclusion can be quantified by the number of catastrophic collisions per comet (Rickman et al. 2015)
(6)
\begin{equation}
N_{{\rm coll}} = N_{\rm p} A_{\rm p} u T / V,
\end{equation}
where Np is the number of comets in the disc, Ap and u are the collision cross-section and speed, respectively, T ≈ 0.4 Gyr is the time spent from accretion to scattering beyond Neptune and V is the disc volume. A cumulative size distribution of comets with a power index of −2.0 and u ≈ 0.6 km s−1 provides Ncoll ≈ 3, assuming that ≈5 Earth masses are distributed in comets of diameters from 1 to 100 km (Rickman et al. 2015). The size distribution is probably shallower (Johansen et al. 2015), as confirmed by the shallowing size distribution of craters of diameter 10 km observed on Pluto and Charon (Singer et al. 2016; Robbins et al. 2017). We simulate the impacts on Pluto and Charon by means of Holsapple’s model (http://keith.aa.washington.edu/craterdata/scaling/index.htm) with the following target parameters: cold ice as geological type, lunar gravity, zero atmospheric pressure; and with the following projectile parameters: bulk density of 500 kg m−3, diameter of 0.1 km and impact speed of 1 km s−1. Tests performed assuming numerical values around those just listed provide craters always a factor 2 larger than the projectiles, so that the number of projectiles of 0.1 km radius is a factor from (0.1/2.5)2.4 − 3 = 7 to (0.1/2.5)1.6 − 3 = 90 below previous estimates (Morbidelli & Rickman 2015; Jutzi & Benz 2017; Jutzi et al. 2017), according to Charon’s and Pluto’s crater size distributions, respectively (Robbins et al. 2017). The same computations performed by Rickman et al. (2015), changing only the cumulative power index from −2.0 to −1.4, i.e. Charon’s crater size distribution (Robbins et al. 2017), provide Ncoll ≈ 0.5, and Ncoll ≈ 0.05 with a cumulative power index of −0.6, i.e. Pluto’s crater size distribution (Robbins et al. 2017), which make equation (6) consistent with the observed flux of 67P fractal particles. This conclusion is confirmed by the craters observed on Nix (Robbins et al. 2017), with a cumulative power index of −0.8, close to Pluto’s one, a measured cumulative crater density of 0.01 km−2 at diameters >3 km (smaller projectiles would not destroy a parent comet, according to equation (5)) and an extrapolated cumulative crater density 0.1 km−2 at diameters >0.1 km, matching that extrapolated for Charon with a steeper cumulative index (Robbins et al. 2017). Nix and Charon are less affected than Pluto by a possible significant resurfacing and their crater density suggests a negligible number of catastrophic impacts during the Solar system lifetime on bodies of cross-section 10 km2.


We show in Fig. 6 the dependence of some Ly α characteristics on absolute FUV magnitude. Our galaxies with detected LyC emission are shown by red filled circles and those with upper limits of LyC emission are represented by red open circles. All our low-mass galaxies are characterized by moderate Ly α luminosities in a narrow range between 1042.19 and 1042.74 erg s−1 (Fig. 6a), that are slightly below the values for confirmed LyC leakers (blue filled circles) by Izotov et al. (2016a,b, 2018a,b) and high-redshift galaxies (grey open circles) by Ouchi et al. (2008), Hashimoto et al. (2017), Jiang et al. (2018), Matthee et al. (2017), Matthee et al. (2018), Sobral et al. (2018), but are ∼1 order of magnitude higher than those for GPs (black asterisks; Jaskot & Oey 2014; Henry et al. 2015; Jaskot et al. 2017; Yang et al. 2017a; McKinney et al. 2019). Our low-mass galaxies have high EW(Ly α) ∼ 65–220 Å (Table 7, red symbols in Fig. 6b), similar to those in other LyC leakers (blue symbols in Fig. 6b). They are at the high end of the EW(Ly α) values for high-z LAEs by Ouchi et al. (2008), Hashimoto et al. (2017), Jiang et al. (2018), Harikane et al. (2018), Caruana et al. (2018), Pentericci et al. (2018), Matthee et al. (2017, 2018), Sobral et al. (2018) (grey open circles) and GPs (black asterisks). However, contrary to expectations for galaxies with lower stellar masses and, likely, lower masses of the neutral gas, the separation between the Ly α peaks is on average similar to that in higher mass LyC leakers (Fig. 6c), and it is higher in galaxies with upper limits of LyC emission (>400 km s−1, red open circles). Ly α escape fractions fesc(Ly α) in low-mass galaxies are also similar to those in higher mass LyC leakers, and they are lower in galaxies with upper limits of LyC emission (red open circles in Fig. 6d). We also note that the average ratio of blue and red peak fluxes of ∼25 per cent (Table 7) is somewhat lower than the value of ∼30 per cent quoted by Hayes et al. (2021) for $z\, \sim$ 0 galaxies.


An alternative approach is to map the CGM through direct imaging of the Ly α line. Theoretical models suggest that three main mechanisms should be able to generate circumgalactic Ly α emission: cooling radiation of gravitationally heated gas (e.g. Haiman, Spaans & Quataert 2000; Yang et al. 2006; Dijkstra & Loeb 2009), ultraviolet (UV) photons produced through shock mechanisms (Taniguchi & Shioya 2000; Mori, Umemura & Ferrara 2004), and recombination radiation following photoionization (often referred as fluorescence) powered by UV sources (Cantalupo et al. 2005; Geach et al. 2009; Kollmeier et al. 2010). While the fluorescent signal powered by the diffuse metagalactic UV background (Hogan & Weymann 1987; Binette et al. 1993; Gould & Weinberg 1996; Haardt & Madau 1996), with an expected surface brightness (SB) of $\rm SB_{Ly\alpha }\sim 10^{-20} \,erg \,s^{-1} \,cm^{-2} \,arcsec^{-2}$ (Cantalupo et al. 2005; Rauch et al. 2008), is still out of reach for current optical instrumentation (but see Gallego et al. 2018), Ly α fluorescence is predicted to be boosted up into the detectable regime in the vicinity of bright ionizing sources, as luminous quasars (Rees 1988; Haiman & Rees 2001; Alam & Miralda-Escudé 2002; Cantalupo et al. 2005). This theoretical prediction has been confirmed by a number of surveys targeting the fluorescent Ly α emission around luminous and radio-quiet quasars, using narrow-band (NB) filters on 8-metre class optical telescopes (e.g. Cantalupo, Lilly & Haehnelt 2012; Cantalupo et al. 2014; Martin et al. 2014; Arrigoni Battaia et al. 2016) and spectroscopic observations (e.g. Christensen et al. 2006; North et al. 2012; Herenz et al. 2015). However, these surveys have revealed giant Ly α nebulae, spanning distances from the quasar larger than 100 physical kpc (pkpc), only in less than 10 per cent of the targets (e.g. Cantalupo et al. 2014; Hennawi et al. 2015; maximum projected linear sizes >300 pkpc), and emission on smaller scales (R ≲ 50–60 pkpc) have been detected only in about 50 per cent of the cases. This 50 per cent detection rate is likely due to a combination of limits of the observational techniques, as for instance NB filter losses, spectroscopic slit losses, point spread function (PSF) losses and, most importantly, dilution of the signal of the Ly α line into the continuum flux (both background and from the host galaxy) encompassed by the width of the filter (see also Borisova et al. 2016 for a discussion).

Background Background Background Background


These trends make physical sense and agree with the limited previous measurements. Physically, elevated SFR/M
⋆ may trace more intense radiation fields and stronger heating of the gas, suggesting higher temperatures. The anticorrelation with M
⋆ may reflect the impact of dust shielding. Based on the existence of the mass–metallicity relation (e.g., Tremonti et al. 2004; Kewley & Ellison 2008), we expect the low-mass members of our sample to also have lower dust-to-gas ratios (e.g., Leroy et al. 2011; Rémy-Ruyer et al. 2014; Casasola et al. 2020) and more intense radiation fields. In literature studies, CO line ratios do appear enhanced in low-metallicity regions or galaxies (e.g., Lequeux et al. 1994; Bolatto et al. 2003; Druard et al. 2014; Kepley et al. 2016; Cicone et al. 2017, among many others). Higher SFR/L
CO may indicate poorly shielded, low-metallicity gas in which the CO persists only in the core of a molecular cloud (e.g., see discussion in Glover & Clark 2012; Schruba et al. 2012; Bolatto et al. 2013b; Rubio et al. 2015). Alternatively, higher SFR/L
CO can indicate more efficiently star-forming gas, which will often be denser gas with more nearby heating sources. These are both factors that can lead to higher line ratios, especially R
32 and R
31 (see Section 2). Given that our sample skews toward relatively massive and thus nearly solar metallicity targets, we expect that these density and heating effects likely represent the main drivers of the observed correlations. The correlations that we see agree with the results of Lamperti et al. (2020), who showed a correlation between SFR/L
CO,low and R
31, and Yajima et al. (2021), who used a subset of the data we consider here and showed a correlation between R
21 and SFR/L
CO,low. Qualitatively, Figure 5 echoes other results at low and high redshift that show a correlation between normalized star formation activity and excitation (e.g., Weiß et al. 2005; Bolatto et al. 2013b; Liu et al. 2021).


The fifth force, propagated by the scalar degree of freedom, affects the Poisson equations associated to the Newtonian potential Φ, as well as the relativistic one, Ψ, according to (Kobayashi, Watanabe & Yamauchi 2015; Crisostomi & Koyama 2018; Dima & Vernizzi 2018),
(1)$$\begin{eqnarray*}
\frac{\text{d} \Phi (r)}{\text{d}r} = \frac{G M(r)}{r^2} \left[1+\frac{3}{4}Y_1\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\left(2+\frac{\text{d}\ln \rho }{\text{d}\ln r}\right)\right],
\end{eqnarray*}
$$(2)$$\begin{eqnarray*}
\frac{\text{d} \Psi (r)}{\text{d}r} =\frac{G M(r)}{r^2}\left[1-\frac{15}{4}Y_2\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\right].
\end{eqnarray*}
$$In the above equations, we have assumed spherical symmetry. $\bar{\rho }(r)$ is the (spatially) average density at radius r from the centre of the galaxy cluster, and Y1, Y2 correspond to the dimensionless fifth-force couplings. Finally, G is the Newton’s constant. Although the dynamics of member galaxies in the cluster is governed by the potential Φ, lensing is sourced by the combination
(3)$$\begin{eqnarray*}
\frac{\mathrm{ d}}{\mathrm{ d}r} \Phi _{\rm {lens}} = \frac{1}{2}\frac{\mathrm{ d}}{\mathrm{ d}r}(\Phi + \Psi).
\end{eqnarray*}
$$Therefore, kinematical observations allow for contraints on Y1, while lensing constrains both Y1 and Y2. The right-hand side of above equation can be expressed in terms of the density profile ρ(r) according to the relevant equations for Φ and Ψ above. The dominant source of pressureless matter density in the cluster comes from dark matter, which density we choose to model with a Navarro-Frenk-White (NFW) of Navarro, Frenk & White (1997) profile as
(4)$$\begin{eqnarray*}
\rho (r)=\frac{\rho _\text{s}}{r/r_\text{s}(1+r/r_\text{s})^2},
\end{eqnarray*}
$$with ρs is a characteristic density and rs the radius at which the logarithmic derivative of the density profile takes the value −2. The NFW profile has been shown to provide an overall good agreement with observations and simulations over a broad range of scales in GR (e.g. Biviano et al. 2013; Umetsu et al. 2016; Peirani et al. 2017) and in MG (e.g. Lombriser et al. 2012a; Wilcox et al. 2016). Moreover, the GR analyses with lensing and internal kinematics of both clusters indicate that the total mass profile is well fitted by the NFW model (Biviano et al. 2013; Umetsu et al. 2016; Caminha et al. 2017; Sartoris et al. 2020). Under the assumption of a NFW profile, we can re-write the equation for the potential Φ in an effective way as
(5)$$\begin{eqnarray*}
\frac{\text{d}\Phi }{\text{d}r} \equiv \frac{G M_{\text{dyn}}}{r^2}=\frac{G}{r^2}\left[ M_{\rm {NFW}}(r)+M_1(r)\right],
\end{eqnarray*}
$$which serves as a definition of the dynamical mass Mdyn. Notice that, G here is still Newton’s constant as measure in the Solar system. The fifth-force contribution M1 is defined in terms of the NFW parameters as
(6)$$\begin{eqnarray*}
M_1(r)= M_{200}\frac{Y_1}{4}\frac{r^2(r_\text{s}-r)}{(r_\text{s}+r)^3}\times [\ln (1+c)- c/(1+c)]^{-1}.
\end{eqnarray*}
$$where c = r/rs is the concentration and M200 is the mass of a sphere of radius r200 enclosing an average density 200 times the critical density of the universe at that redshift. In a similar fashion, the relevant expression for the lensing mass can be found by computing
(7)$$\begin{eqnarray*}
M_{\text{lens}}(r) =\frac{r^2}{2G}\left[\frac{\text{d}\Psi }{\text{d}r}+\frac{\text{d}\Phi }{\text{d}r}\right].
\end{eqnarray*}
$$$$\begin{eqnarray*}
M_{\text{lens}}=M_{\text{NFW}}+\frac{r^2M_{200}\left[Y_1(r_\text{s}-r)-5Y_2(r_\text{s}+r)\right]}{4[\log (1+c_{200})-c_{200}/(1+c_{200})]}\frac{1}{(r_\text{s}+r)^{3}},
\end{eqnarray*}
$$which can be effectively re-expressed in terms of the dynamical mass as
(8)$$\begin{eqnarray*}
M_{\text{lens}} \equiv M_{\text{dyn}}+M_2,
\end{eqnarray*}
$$with M2 the contribution from the fifth force defined through
(9)$$\begin{eqnarray*}
M_2=\frac{r^2M_{200}}{8(r_\text{s}+r)^{3}}\frac{Y_1(r-r_\text{s})-5Y_2(r_\text{s}+r)}{[\ln (1+c)-c/(1+c)]}.
\end{eqnarray*}
$$In view of the above equations, it is important to emphasize again that, although the fifth force effect enters the dynamical mass only through the coupling Y1, the lensing mass is affected by both Y1 and Y2. This is expected, since lensing is sourced by the combination of the two potentials Φ and Ψ, equation (3). Note also that, with gravitational lensing observations, one reconstructs the projected surface mass density profile Σ(R), where R is the projected radius from the cluster centre. We refer to e.g. Umetsu (2020) for an explicit discussion of the physics and mathematical framework.


The last decade and a half has seen a revolution in the study of overdensities in the early Universe. While the study and careful characterization of large associations of galaxies in the local Universe has been possible for nearly a century, and in the intermediate redshift Universe for a significant fraction of that time (e.g., Shapley & Ames 1926; Shapley 1930; Zwicky 1937; Abell 1958; Zwicky et al. 1961), the study of their progenitors presented several practical problems which have prevented their study until relatively recently. The primary problem, inherent to the study of nearly all galaxy populations in the early Universe, is the extreme apparent faintness of galaxy populations at these distances. While some phenomena exist in the early Universe, such as quasars or radio galaxies, which are so powerful and intrinsically bright that they have been able to serve as beacons to early searches near the epoch of H I reionization (z ~ 5.5–10, Becker et al. 2001, 2015; Planck Collaboration XIII 2016), the bulk of the galaxy population residing in the early Universe does not contain such phenomena (Miley & De Breuck 2008; Ouchi et al. 2008; Lemaux et al. 2014b; Ueda et al. 2014; Talia et al. 2017). As such, the first searches for these more typical primeval galaxies were largely doomed to failure (Davis & Wilkinson 1974; Partridge 1974; Pritchet & Hartwick 1987; Parkes et al. 1994). It was not until the advent of the 10m-class ground-based telescopes largely used in conjunction with the Hubble Space Telescope (HST) that the prospect of detecting and characterizing moderate samples of such galaxies became even remotely feasible (e.g., Steidel et al. 1999; Shapley et al. 2003; Giavalisco et al. 2004; Stanway et al. 2004; Malhotra & Rhoads 2004; Malhotra et al. 2005; Vanzella et al. 2005, Le Fèvre et al. 2005). With this, the prospect of finding and characterizing analogs of the progenitors of the massive clusters and superclusters of galaxies scattered throughout the local Universe began to come within the realm of possibility.


The size of ENLRs have been defined in different ways in the literature. Bennert et al. (2002) and Schmitt et al. (2003b) used the Hubble Space Telescope (HST) to obtain narrow band images of $\rm [O\, III]$, and adopted the maximum 3σ detected radius as the radius of the ENLR. This method is subject to the instrumental sensitivity limit that could be very different in different observations. Studies with long-slit spectroscopic observations define the radius based on isophote (Greene et al. 2011; Hainline et al. 2013, 2014), or the distance at which the ionization state changes from AGN to star-forming activities (Bennert et al. 2006a,b). The long-slit based observations also have drawbacks: the morphology of ENLR is sometimes irregular so that the derived size depends on the orientation of slits (Greene et al. 2011; Husemann et al. 2013). We have compared the measured size based on the IFU and the mock long-slit observation in Fig. 4 following the method discussed below. In most cases, long-slit observations tend to underestimate the true size of ENLR. IFU spectroscopic data allow us to use 2D maps to define the sizes of ENLRs. Common definitions include the radius of a specified $\rm [O\, III]$ surface brightness isophote (Liu et al. 2013, 2014), or the $\rm [O\, III]$ flux weighted radius (Husemann et al. 2013, 2014; Bae et al. 2017). We followed the same method as Liu et al. (2013) but chose a different threshold. The isophote threshold of 10−15$\rm erg\, s^{-1}cm^{-2}arcsec^{-2}$ was used for quasars related studies. This is suitable for such bright objects but are not as useful for fainter Syferts in our sample as it will leave a large number of AGN undetected. The typical 3σ depth of the MaNGA observation in $\rm [O\, III]$ surface brightness can reach 10−17$\rm erg\, s^{-1}cm^{-2}arcsec^{-2}$. For our AGN sample, the majority of AGN spaxels have surface brightnesses above 10−16$\rm erg\, s^{-1}cm^{-2}arcsec^{-2}$ which is thus adopted in this work as the threshold to define the sizes of the ENLRs (hereafter R16). If all spaxels are above this threshold, we extrapolated the fitted $\rm [O\, III]$ surface brightness profile to determine R16 (see Section 3.4 for more detail). It should be noted that the surface brightness can be affected by cosmological dimming, which has a scale factor of (1 + z)4 (Liu et al. 2013; Hainline et al. 2014). That is important for works trying to compare sample with different redshift, especially for high redshift quasars.


We derived stellar parameters, [C/M], and [N/M] using SLAM. To avoid bad fits at the edges of the parameter space, we exclude stars with spectral S/N in the g band less than 50, and metallicity less than −1.4. The derived C and N abundances are shown in Figure 8. Clearly, in the top panel, the CH-strong, CH-normal, and metal-poor field stars are separated, and their relative distribution in the N–C parameter space is similar to the case of APOGEE abundances (left panel of Figure 7): (1) metal-poor field stars form a sequence in the lower left of the top panel. As evolved stars ascend the RGB, C and N abundances may be changed by first dredge-up (Iben 1964, 1967) and extra mixing (Gratton et al. 2000; Charbonnel & Lagarde 2010). Given that for a typical halo/thick-disk star of 1 M⊙, the first dredge-up occurs around Teff = 5200 K (Boothroyd & Sackmann 1999), and most of our sample stars have

 K and log g  2.5, we infer that most stars have already undergone first dredge-up. On the other hand, the C and N abundances of these stars could be altered by extra mixing. Stars with brighter K-band absolute magnitudes tend to have higher [N/Fe] and lower [C/Fe] (middle and bottom panels of Figure 8), which is consistent with extra-mixing theory and observation (Gratton et al. 2000; Charbonnel & Lagarde 2010); (2) CH-normal stars show an enhanced median N abundance and slightly depleted median C abundance. Clearly, the median N abundance of CH-normal stars is enhanced compared to that of normal metal-poor field stars with similar C abundances. In other words, the enhanced N abundances in CH-normal stars cannot be explained by the extra-mixing effect alone. We notice that a few CH-normal stars may have low N abundances, probably due to large uncertainties when spectra of a particular type are scarce in the training set, i.e., high-N metal-poor stars. The statistical similarity between APOGEE C and N abundances and LAMOST-derived C and N abundances further strengthens our statement above.


The giant planet metallicity correlation supports the core-accretion scenario for the formation of planets (Pollack et al. 1996; Ida & Lin 2004; Mordasini et al. 2009), in which it is assumed that planetesimals are formed by the condensation of heavy elements. The discovery of this correlation has led to an increased interest on the abundances of other elements in planet hosts(e.g. Sadakane et al. 2002; Bodaghee et al. 2003; Beirão et al. 2005; Gilli et al. 2006; Ecuvillon et al. 2006; Bond et al. 2006, 2008; Robinson et al. 2006; Gonzalez & Laws 2007; Takeda et al. 2007; Delgado Mena et al. 2010; González Hernández et al. 2010, 2013; Kang et al. 2011; Brugamyer et al. 2011; Adibekyan et al. 2015; da Silva et al. 2015; Mishenina et al. 2016; Suárez-Andrés et al. 2018). Using the same sample of this study, Adibekyan et al. (2012b) found that the [X/Fe] ratios of Mg, Al, Si, Sc, and Ti both for giant and low-mass planet hosts are systematically higher than those of stars without detected planets at low metallicities ([Fe/H] ≲ from –0.2 to 0.1 dex depending on the element). Furthermore, this work confirmed the previous suggestion by Haywood (2009) that planets form preferentially in the thick disk rather than in the thin disk at lower metallicities. A plausible explanation for this behaviour is that if the amount of iron is low, it needs to be compensated with other elements that are important for planet formation, such as Mg and Si, and these elements are more abundant in the thick disk Adibekyan et al. (2012b,a). In a subsequent work, Delgado Mena et al. (2018) focused on heavier elements among the same sample of stars with and without planets, we found that planet hosts present higher abundances of Zn for [Fe/H]  –0.1 dex, as a consequence of thick-disk stars having enhanced [Zn/Fe] ratios (Delgado Mena et al. 2017). Moreover, Delgado Mena et al. (2018) also found a statistically significant underabundance of Ba for low-mass planet hosts that had been previously suggested by Mishenina et al. (2016).


However, the transition between the two extremes of modern (metal-rich) and primordial (metal-poor) star formation, and in particular the role of dust coupling and stellar radiation feedback at low metallicity, has thus far received limited exploration. Krumholz (2011) present analytical models for radiation feedback and predict a weak scaling of IMF peak mass with metallicity, while Myers et al. (2011) and Bate (2014, 2019) carry out radiation-hydrodynamic simulations of star formation over a metallicity range from $0.01{-}3\, Z_{\rm {\odot }}$ and find negligible effects on gas fragmentation. However, these studies do not explore lower metallicities, despite available evidence for the existence of a low-metallicity ISM in the past through the discovery of stars with metallicities as low as $10^{-4}\, \rm {Z_{\odot }}$ (Caffau et al. 2011; Starkenburg et al. 2018), as well as several others with $\rm {[Fe/H]} \lt -5$ (Christlieb et al. 2004; Keller et al. 2014; Frebel et al. 2015; Aguado et al. 2017, 2018; Ezzeddine et al. 2019; Nordlander et al. 2019). Coming from the opposite direction, Bromm et al. (2001), Omukai et al. (2005), and Omukai, Hosokawa & Yoshida (2010) consider the thermodynamics of gas of increasing metallicity, and find that dust and metal line cooling permits fragmentation to reach masses ≲1 M⊙ only once the metallicity exceeds ∼10−3.5 Z⊙. Dust is a more efficient coolant than metal lines, and permits fragmentation to lower masses at lower metallicity (e.g. Meece, Smith & O’Shea 2014; Chiaki & Yoshida 2020; Shima & Hosokawa 2021), but exactly by how much depends on the poorly known distribution of dust grain sizes in the early Universe (Schneider et al. 2006, 2012; Omukai et al. 2010; Schneider & Omukai 2010; Chiaki et al. 2015). However, the early Universe star formation models are fundamentally misanalogous to the modern ones that consider decreasing metallicity, in that the early Universe models consider dust solely as a coolant that enables fragmentation, whereas the modern ones assign it a more nuanced role, as both a source of cooling and later, once stellar feedback begins, a source of heating – a changeover that seems crucial to explaining why the IMF in the present-day Universe peaks at ${\sim}0.2\, \rm {M_{\odot }}$ rather than ${\sim}10^{-2}\, \rm {M_{\odot }}$ (Kroupa 2001; Chabrier 2003, 2005).


Blazars are a subclass of active galactic nuclei (AGNs) with relativistic jets of high-energy particles pointing near our line of sight (e.g., Urry & Padovani 1995). Their nonthermal emission is generally detected across the entire electromagnetic spectrum from radio to γ-ray bands. Blazars are subclassified into flat-spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs), according to the equivalent width of the emission lines in their optical spectrum (Stickel et al. 1991; Stocke et al. 1991; Marcha et al. 1996). These two subclasses of blazars are thought to be intrinsically different, perhaps based on their accretion mode (Dermer & Giebels 2016). FSRQs have high luminosity and a thin and radiatively efficient black hole accretion disk (Malkan & Moore 1986), while BL Lacs are powered by an advection-dominated, low radiative efficiency accretion flow (Dermer & Giebels 2016; Blandford et al. 2019). The jet emission is relativistically beamed (Ghisellini 2019), with a Doppler boosting factor corresponding to a bulk Lorentz factor of several to greater than 10 (Pushkarev et al. 2009). In both cases, the broadband spectra consist of two broad humps, one peaking in the IR-to-X-ray regime and the other peaking in the γ-ray regime. The low-energy peak is believed to be due to synchrotron emission, while the high-energy peak is likely due to inverse Compton scattering of low-energy photons of either the same synchrotron photons (for BL Lacs) or external photons from the disk/BLR (for FSRQs) (e.g., Dutka et al. 2017). However, some blazars might not necessarily be detected in γ-rays (e.g., Paliya et al. 2017). Indeed, a recent study showed that blazars undetected in γ-rays are likely to have relatively smaller Doppler factors and more disk dominance (Paliya et al. 2017). In the case of strong Compton scattering, the beaming of γ-rays could be larger than, e.g., that seen in the radio (Dermer 1995), leading to the possible nondetection (or reduced detection efficiency) of γ-rays from sources not seen exactly pole-on.


Without knowledge of nucleus shape and spin parameters (pole orientation and shape), a thermophysical model is pointless, and we instead adopted a NEATM (Near Earth Asteroid Thermal Model) model, used extensively for asteroids (Harris 1998) and TNOs (Müller et al. 2020, and references therein). NEATM is based on the asteroid standard thermal model (STM; Lebofsky et al. 1989) but accounts for phase angle effects; additionally, the temperature distribution is modified by an adjustable η−1/4 factor, which represents the combined and opposed effects of roughness (η  1) and thermal inertia (η > 1). For fixed surface (thermal inertia, roughness) and spin properties, η is also a function of the subsolar temperature, and, therefore, of the heliocentric distance (e.g., Spencer et al. 1989; Lellouch et al. 2013). Given the rh = 20 au distance of our measurements (and the expected large size of 2014 UN271), we adopted a beaming factor η = 1.175 ± 0.42, based on measurements of 85 Centaurs and TNOs (Lellouch et al. 2013, 2017). We also specified a bolometric emissivity ϵb = 0.90 ± 0.06 and a relative radio emissivity ϵr = ϵmm/ϵb = 0.70 ± 0.13, as inferred from combined Spitzer/Herschel/ALMA measurements of nine objects (Brown & Butler 2017; Lellouch et al. 2017). The lower-than-unity relative radio emissivity is interpreted as resulting from (i) the sounding of a colder dayside subsurface and (ii) the loss of outgoing thermal radiation due to volume scattering in the subsurface and/or Fresnel reflection at the surface. The few available radio observations of cometary nuclei also generally indicate radio emissivities lower than 1, for example ∼0.5 for Hale-Bopp (Fernández 2002) and  0.8 for 8P/Tuttle (Boissier et al. 2011). Comets are also found to have low thermal inertias (e.g.,  10,  30, and  45 MKS for 8P/Tuttle, 22P/Kopff, and 9P/Tempel 1, respectively; Boissier et al. 2011; Groussin et al. 2009, 2013), consistent with a beaming factor, η, of order unity. Based on NEATM analysis of a large sample of comet nuclei observed with Spitzer at rh = 3.5–6 au, Fernández et al. (2013) find a mean η of 1.03 ± 0.11. The large 29P/Schwassmann-Wachmann nucleus (D = 65 km) has η = 1.1 ± 0.2 (Schambeau et al. 2021). These numbers are fully consistent with our choice of η. Given the values of rh, η, and ϵb, NEATM calculations indicate that the object’s spectral index over 224–242 GHz is 1.93, slightly lower than the Rayleigh-Jeans limit of 2.


A strong property of the isothermal gas that defines the interstellar medium (ISM) is the absence of a characteristic mass scale owing to the scale-free nature of gravity. From the collapse condition of an isothermal sphere, one can derive the Jeans mass as
(2)$$\begin{eqnarray*}
M_{\mathrm{J}} = \frac{4 \pi}{3} \rho \left(c_{\mathrm{s}} t_{\rm ff}\right)^3
\end{eqnarray*}$$with cs being the sound speed, given by
(3)$$\begin{eqnarray*}
c_{\mathrm{s}} = \sqrt{\frac{k_{\mathrm{B}} T}{\mu \, m_{\mathrm{H}}}}
\end{eqnarray*}$$and the free-fall time tff is defined here as
(4)$$\begin{eqnarray*}
t_{\mathrm{ff}} = \sqrt{\frac{3\pi }{32G \rho }}.
\end{eqnarray*}$$One sees immediately that for an isothermally collapsing gas, we cannot choose a unique characteristic density that would result in a unique characteristic mass. The larger the density of the gas during collapse and fragmentation, the smaller the Jeans mass. At very large densities, though, owing to the increased dust absorption coefficient, the gas becomes opaque to its own radiation. This is sometimes referred to as the opacity limit (Low & Lynden-Bell 1976; Rees 1976). At this point, the fragmentation is halted, setting a scale below which no fragments form. It is possible to estimate the characteristic density at which this process occurs by requiring the optical depth of one Jeans radius to be unity
(5)$$\begin{eqnarray*}
\tau = \kappa _{\rm dust} \rho R_{\rm J} = 1\,\,\,{\rm with}\,\,\,R_{\rm J} = c_{\mathrm{s}} t_{\rm ff}.
\end{eqnarray*}$$For typical molecular clouds conditions in the Milky Way with T = 10 K, μ = 2.2, and κdust = 0.1 cm2 g−1, we find the critical density that defines the opacity limit
(6)$$\begin{eqnarray*}
\rho _{\rm crit} = \frac{32 G}{3 \pi \kappa _{\rm dust}^2 c_\mathrm{ s}^2 } \simeq 5 \times 10^{-14} \,{\rm g}\,{\rm cm}^{-3} .
\end{eqnarray*}$$At this critical density, we obtain a unique Jeans mass of the order of
(7)$$\begin{eqnarray*}
M_{\mathrm{J}} = 0.670 \frac{c_{\mathrm{s}}^3}{\sqrt{G^3 \rho _{\mathrm{crit}}}} \approx 6 \times 10^{-4} \, \mathrm{M_\odot }.
\end{eqnarray*}$$Note that the critical density can also be derived using slightly more complicated arguments, leading to a very similar value for typical Milky Way conditions (Krumholz 2017). This mass corresponds to the smallest gas clumps that can overcome the pressure gradients and collapse. At this characteristic density, the gas evolution transitions from isothermal to adiabatic, which prevents the collapse of smaller fragments, even at higher densities. The Jeans mass at the opacity limit is obviously much smaller than the observed characteristic mass of the IMF. It does not even correspond to the minimal mass of a star, as the smallest of these fragments will not be able to collapse enough to reach stellar densities in their centres.