The first direct proof of the existence of magnetic fields in large-scale extragalactic environments, i.e., galaxy clusters, dates back to the 1970s with the discovery of extended, diffuse, central synchrotron sources called radio halos (see, e.g., Feretti et al. 2012 for a review). Later, indirect evidence of the existence of intracluster magnetic fields has been given by several statistical studies on the effect of the Faraday rotation on the radio signal from background galaxies or galaxies embedded in galaxy clusters (Lawler & Dennison 1982; Vallée et al. 1986; Clarke et al. 2001; Johnston-Hollitt 2003; Clarke 2004; Johnston-Hollitt & Ekers 2004). On scales up to a few Mpc from the nearest galaxy cluster, possibly along filaments, only a few diffuse synchrotron sources have been reported (Harris et al. 1993; Bagchi et al. 2002; Kronberg et al. 2007; Giovannini et al. 2013, 2015). Magnetic fields with strengths on the order of 10-15 G in voids might be indicated by γ-ray observations (see Neronov & Vovk 2010; Tavecchio et al. 2010; Takahashi et al. 2012, 2013; but see Broderick et al. 2014a,b for alternative possibilities). Nevertheless, up to now, a robust confirmed detection of magnetic fields on scales that are much larger than clusters is not available. Stasyszyn et al. (2010) and Akahori et al. (2014a) investigated the possibility of statistically measuring Faraday rotation from intergalactic magnetic fields with present observations, showing that only the Square Kilometre Array (SKA) and its pathfinders are likely to succeed in this respect. By comparing the observations with single-scale magnetic field simulations, Pshirkov et al. (2015) infer an upper limit of 1.2 nG for extragalactic large-scale magnetic fields, while the Planck Collaboration XIX (2016) derived a more stringent upper limit for primordial large-scale magnetic fields of B 0.67 nG from the analysis of the Cosmic Microwave Background (CMB) power spectra and the effect on the ionization history (but see also Takahashi et al. 2005; Ichiki et al. 2006).


Magnetic field fluctuations in the solar wind are highly turbulent. The measured power spectral density (PSD) of the fluctuating magnetic field always exhibits power laws k−α, where k is the wavenumber, and α is the spectral index. A single spacecraft measures the PSD as a function of f−α in the frequency domain, which can be converted to the spatial domain under the Taylor hypothesis. According to the physical processes at different scales, the PSD in the solar wind can be divided into several segments, which can be fitted with different α. The inertial range, which is dominated by magnetohydrodynamic (MHD) turbulence, follows the cascade models with spectral indices αi from around 3/2 to 5/3 (Bruno & Carbone 2013; Chen et al. 2020). The PSDs become steepened below the ion scales (ion thermal gyroradius ρi or ion inertial length di), where kinetic mechanisms should be taken into account. Sometimes a sharp transition range is observed with αt ∼ 4 (Sahraoui et al. 2010; Bowen et al. 2020a). This transition range may be caused by imbalanced turbulence (Voitenko & Keyser 2016; Meyrand et al. 2021), energy dissipation of kinetic waves (Howes et al. 2008), ion-scale coherent structures (Lion et al. 2016), or a reconnection dominated range (Mallet et al. 2017). At smaller scales, a flatter sub-ion kinetic range forms with the spectral index αk ∼ 7/3, which can be explained as the MHD Alfvénic turbulence developing into a type of kinetic wave turbulence, e.g., kinetic Alfvén waves (KAWs; Schekochihin et al. 2009) or whistler waves (Cho & Lazarian 2004). Intermittency in the kinetic range could lead to an −8/3 spectrum (Boldyrev & Perez 2012; Zhao et al. 2016). Ion-cyclotron-wave (ICW) turbulence could lead to a steeper −11/3 spectrum (Krishan & Mahajan 2004; Galtier & Buchlin 2007; Meyrand & Galtier 2012; Schekochihin et al. 2019).The kinetic range always behaves as the KAW turbulence with the slope of −2.8 in the near-Earth space (Bale et al. 2005; Chen et al. 2013; Chen 2016). The spectral indices increase again beyond the electron kinetic scales in observations, indicating the conversion of turbulence energy to electrons (Sahraoui et al. 2009; Alexandrova et al. 2012; Chen et al. 2019) or transitions to a further cascade (Schekochihin et al. 2009; Chen & Boldyrev 2017). In simulations, Meyrand & Galtier (2013) obtained a −8/3 spectrum at electron scales under the 3D electron–MHD model.


In this Letter, we directly test the relationship between proton kinetic instabilities and plasma turbulence in the solar wind using a hybrid expanding box model that allows us to study self-consistently physical processes at ion scales. In the hybrid expanding box model, a constant solar wind radial velocity vsw is assumed. The radial distance R is then

, where R0 is the initial position and

 is the initial value of the characteristic expansion time

 Transverse scales (with respect to the radial direction) of a small portion of plasma, comoving with the solar wind velocity, increase ∝ R. The expanding box uses these comoving coordinates, approximating the spherical coordinates by the Cartesian ones (Liewer et al. 2001; Hellinger & Trávníček 2005). The model uses the hybrid approximation where electrons are considered as a massless, charge-neutralizing fluid and ions are described by a particle-in-cell model (Matthews 1994). Here, we use the two-dimensional (2D) version of the code, fields and moments are defined on a 2D x–y grid 2048 × 2048, and periodic boundary conditions are assumed. The spatial resolution is Δx = Δy = 0.25dp0, where

 is the initial proton inertial length (vA0: the initial Alfvén velocity, Ωp0: the initial proton gyrofrequency). There are 1024 macroparticles per cell for protons that are advanced with a time step

, while the magnetic field is advanced with a smaller time step

 The initial ambient magnetic field is directed along the radial z-direction, perpendicular to the simulation plane

, and we impose a continuous expansion in the x- and y-directions. Due to the expansion, the ambient density and the magnitude of the ambient magnetic field decrease as

 (the proton inertial length dp increases ∝ R; the ratio between the transverse sizes and dp remains constant; the proton gyrofrequency Ωp decreases as ∝R−2). A small resistivity η is used to avoid accumulation of cascading energy at grid scales; initially, we set

 (μ0 being the magnetic permittivity of vacuum) and η is assumed to be

 The simulation is initialized with an isotropic 2D spectrum of modes with random phases, linear Alfvén polarization (

), and vanishing correlation between magnetic and velocity fluctuation. These modes are in the range 0.02 ≤ kdp ≤ 0.2 and have a flat one-dimensional (1D) power spectrum with rms fluctuations = 0.24 B0. For noninteracting zero-frequency Alfvén waves, the linear approximation predicts

 (Dong et al. 2014). Protons initially have the parallel proton beta

 and the parallel temperature anisotropy

 as typical proton parameters in the solar wind in the vicinity of 1 AU (Hellinger et al. 2006; Marsch et al. 2006). Electrons are assumed to be isotropic and isothermal with βe = 0.5 at t = 0.


We can use the equivalent width of Hα (EWHα, see Table 4) to put additional constraints on the recent star formation history of the AGNs. To do so, we first need to estimate the largest fraction of the Hα luminosity that could arise from star formation, and for this we use the group of AGNs with the highest EWHα. There are a number of reasons why AGNs could have lower EWHα, but the maximum values will only be found when both AGN and star formation contribution are maximal. The group of six AGNs with the highest EWHα consists of ESO 137-G034, NGC 3081, NGC 2110, NGC 7582, NGC 2992, and NGC 5728, and their median EWHα is 70 Å. These also tend to have the highest log [OIII]/Hβ, with a median ratio of 1.07. Similarly, their median log [NII]/Hα is 0.01. To assess the star formation contribution to the Hα flux, we consider lines from the most extreme AGN photoionisation models of Groves et al. (2004), to the location of solar metallicity star-forming galaxies (see also Davies et al. 2014b). Since the AGNs are very close to the highest [OIII]/Hβ ratios that the models can produce, we find that at most 10% of their Hα flux could be due to star formation. Applying this correction reduces the EWHα that is associated with star formation to  7 Å. We also need to make a correction for the old stellar population, since EWHα in models compares the line flux to the continuum due to stars associated with the line emission. We find that 5% of the continuum is associated with young stars. We therefore estimate that the maximum EWHα associated with the most recent star-forming episode is  7 Å/5%, which is  140 Å. Comparing this to Starburst99 models (specifically Figs. 83 and 84 in Leitherer et al. 1999) rules out continuous star formation since it would imply a timescale exceeding several Gyr, inconsistent with the stellar population synthesis results. Instead, it means that the recent star formation must have ceased. Based on the rate at which the EWHα falls after star formation stops, we estimate that the end of star formation must have typically happened at least 6 Myr prior.


Even before studying the relationship between the IMF and the CMF, it is important to realize that how the IMF originates from the observed CMF depends directly on the definition of the cores, assumed to be the gas mass reservoir used for the formation of each star or binary system. As shown by Louvet et al. (2021), defining this mass reservoir may seem obvious in the observed map of a cloud, but core characteristics (size, mass) depend heavily on the spatial scales probed by the observations. In addition, the theoretical definition of cores also depends on whether the star-formation scenario is quasi-static or dynamic. In the former scenario, cores are gas condensations sufficiently dense to be on the verge of gravitational collapse, and they convert the core gas into stars (Shu et al. 1987; Chabrier 2003; McKee & Ostriker 2007; André et al. 2014). After a quasi-static phase of concentration of the cloud gas into cores, cores become distinct from their surrounding cloud and start to collapse, and their future stellar content becomes independent of the properties of the parental cloud. In the latter scenario, dynamics play a major role during all phases of the star-formation process (e.g., Ballesteros-Paredes et al. 2007; Hennebelle & Falgarone 2012; Padoan et al. 2014). In particular, global infall of filament networks and gas inflow toward cores are expected to be important drivers of star formation (e.g., Smith et al. 2009; Vázquez-Semadeni et al. 2019; Padoan et al. 2020). In this framework, filaments, cores, and stellar embryos simultaneously accrete gas, and the gas reservoir associated with star formation largely exceeds the extent of the observed cores. This so-called clump-fed scenario was proposed in various recent papers and described in detail in the review by Motte et al. (2018a, see references therein). One of the main objectives of the ALMA-IMF Large Program is to discriminate between the quasi-static and dynamic scenarios by quantifying the role of cloud kinematics in defining core mass and in possibly changing it over time.


Analytical and semianalytical approaches have been developed to describe this process and estimate the width of the overshooting layer (e.g., Schmitt et al. 1984; Zahn 1991; Rempel 2004). With the improvement of computational methods and resources, an increasing number of studies have been devoted to numerical simulations of convection and overshooting using realistic stellar conditions (geometry, luminosity, thermal diffusivity, equation of state, opacities, etc.). A commonly used tactic to increase the efficiency and improve the stability of these simulations is to artificially increase the luminosity (or nuclear energy for convective cores or burning shells) and to modify the thermal diffusivity of the reference stellar model. This tactic is common and has been used, for example, in Rogers et al. (2006, 2013), Meakin & Arnett (2007), Tian et al. (2009), Brun et al. (2011, 2017), Hotta (2017), Cristini et al. (2017), Edelmann et al. (2019), Horst et al. (2020). This approach is used to increase the Mach number of the convective flow, reducing the disparity between advective and acoustic timescales, improving the efficiency of time-explicit codes limited by the Courant–Friedrich–Levy constraint. It is also used to provide numerical stability or to accelerate the thermal relaxation. But no examination of its potentially far-reaching impact has been conducted. Rempel (2004) pointed out that enhanced energy flux in numerical simulations could lead to unrealistically vigourous convection, which could impact the properties of the overshooting layer and could explain some of the discrepancies between analytical models and numerical simulations. Numerical simulations also suggest an increase in the overshooting depth with increasing flux (Hotta 2017; Käpylä 2019). Determining scaling laws of the overshooting depth as a function of the energy input could thus allow for an extrapolation of the results to more realistic stellar conditions and help to estimate the overshooting depth in real stars, as suggested by Hotta (2017), for example, for the Sun. But Käpylä (2019) also shows that an artificial modification of the heat conductivity in the radiative and overshooting regions could impact the overshooting process. In some computational studies, both the luminosity and the thermal diffusivity are enhanced by the same factor to ensure that the thermal structure is unchanged compared to the reference stellar structure and with the expectation that the larger thermal diffusivity counterbalances the larger energy flux. This procedure has been proposed as a way to provide a good representation of the true dynamics of the system (e.g., Rogers et al. 2006, 2013; Tian et al. 2009). But this expectation has never been demonstrated. Another expectation concerns internal waves, excited by convective motions and by flows penetrating the convective boundary. Simulations with artificially modified luminosity and thermal diffusivity are also used to perform the analysis of internal waves, either for convective envelopes (e.g., Rogers et al. 2006; Brun et al. 2011; Alvan et al. 2014) or for convective cores (e.g., Rogers & McElwaine 2017; Edelmann et al. 2019; Horst et al. 2020). None of these works have examined whether the wave spectrum of a realistic star is accurately predicted by such simulations.


The RADYN code (Carlsson & Stein 1992, 1995, 1997; Allred et al. 2015) is a one-dimensional radiative hydrodynamic code that can be used to study the interaction of particle beams with the solar atmosphere. It uses the Fokker–Planck formalism (McTiernan & Petrosian 1990), which takes into account the beam energy losses due to Coulomb collisions and pitch-angle diffusion when incorporating relativistic effects. RADYN includes a six-level hydrogen atom, a nine-level helium atom, and a six-level calcium atom. A return current has also been included in the simulations. We generated a set of models that simulates the conditions in the solar atmosphere during weak to intense WLFs. The beam fluxes used had values of 3 × 109, 1 × 1010, and 3 × 1010 erg cm−2 s−1, while the low-energy cutoff EC covered the parameter space where flare values are usually found (20–120 keV; Warmuth & Mann 2016). The spectral index δ was equal to 3 for all models. The beams were applied continuously and the outputs were analyzed at t = 20 s. This value is consistent with the X-ray analysis of the best observed type II WLF to date (Procházka et al. 2018). The initial atmosphere used in this work has the transition region placed at a height of 1200 km above the photospheric floor and has a coronal temperature of 3 MK at 10 Mm (QS.SL.HT loop described in Allred et al. 2015). The beams were injected at the top of a half-loop with a Gaussian distribution with an HWHM of 235. Line synthesis was carried out using the RH code (Uitenbroek 2001) incorporating partial redistribution that is particularly important for resonance line profiles. We used the 20-level hydrogen atom and the 6-level calcium atom to produce synthetic spectra for the Lyman and Balmer line and continuum diagnostics. A VOIGT profile was used for Ca ii K line, Lyman γ the higher-order Lyman lines. The Ca ii H and Lyα and β profiles were modeled in PRD. The Balmer lines had profiles of type VOIGT_VCS_STARK, which incorporates the unified Stark effect theory (Kowalski et al. 2017). The spectra were synthesized by setting a minimal spectral resolution of 0.05 nm.


In the present survey of GCSs in BCGs, we use the color index

 (from here on we drop the accents on the SDSS indices). In the following discussion it will be useful to have a calibration of this index versus cluster metallicity [Fe/H]. To do this, we would ideally need to have GC photometry of the same clusters in both the Kron-Cousins and SDSS systems, in addition to spectroscopically based metallicity measurements. At present, there are no ideal solutions to that problem. Galaxies satisfying all three of these criteria are rare; in principle the Milky Way GC databases could be used, but cluster-to-cluster foreground reddenings differ strongly, the published SDSS indices (Vanderbeke et al. 2014) show considerable scatter versus metallicity, and the variety of studies from which the UBVRI indices were derived are completely different from the SDSS survey, so that aperture-size mismatches are significant. Similar problems affect the M31 GC sample. The best option at the present time for developing a

 transformation is likely to be from the nearby early-type giant galaxy NGC 5128: here, UBVRI photometry is available from Peng et al. (2004), griz photometry from Sinnott et al. (2010), and [Fe/H] values derived through

 from Woodley et al. (2010); these [Fe/H] values are in turn well correlated with the Sloan-system spectroscopic index [MgFe]’ (see Woodley et al.). We have extracted the GCs in common from these three catalogs, with the results shown in Figure 1. The great majority of these GCs lie well outside the central few kiloparsecs of NGC 5128 and thus are unaffected by the well-known dust lane. We have therefore applied only the foreground reddening of the Galaxy, for which we adopt

 (Cardelli et al. 1989) to obtain the intrinsic colors. We note, however, that the UBVRI measurements were done on

 aperture diameters corrected to

 through median curves of growth (Peng et al. 2004), while the griz measures were done through

 apertures (Sinnott et al. 2010), which means that a small aperture mismatch may exist here as well that affects the zero-point of

.


Alfvén waves of arbitrary amplitude with constant total pressure are known to provide an exact solution to the compressible magnetohydrodynamic (MHD) system in a homogeneous plasma, in that nonlinearities are turned off and there no couplings with compressible modes. However, such a dynamical system is linearly unstable to parametric instabilities and large-amplitude Alfvén waves are known to decay into compressible and secondary Alfvénic modes through three- or four-wave resonances that lead to a variety of parametric instabilities, depending on the plasma beta and dispersive effects, such is the case of parametric decay (Galeev & Oraevskii 1973; Derby 1978), modulational, and beat instabilities (Sakai & Sonnerup 1983; Wong & Goldstein 1986; Jayanti & Hollweg 1993; Nariyuki & Hada 2007). Parametric instabilities of Alfvén waves (or of a spectrum of Alfvén waves) have been widely studied over the years through theoretical approaches (Goldstein 1978; Jayanti & Hollweg 1993; Malara & Velli 1996), and numerical simulations adopting both MHD (Ghosh et al. 1994; Malara et al. 2000; Del Zanna et al. 2001; Tenerani et al. 2017) and kinetic models (Terasawa et al. 1986; Matteini et al. 2010; Nariyuki et al. 2012; Verscharen et al. 2012), although most often in one-dimensional setups. In particular, the traditional parametric decay instability has attracted much attention over the years in the context of both turbulence and plasma heating. This type of decay is most efficient at low values of the plasma beta and it essentially involves the decay of a pump Alfvén wave into a lower-frequency reflected Alfvén wave and a forward sound wave. For this reason, parametric decay remains an appealing process because it provides a natural mechanism for the production of reflected modes, which is essential for the triggering of a turbulent cascade. Indeed, recently it has been proposed as a viable mechanism to initiate the turbulent cascade in the solar wind acceleration region (Chandran 2018; Réville et al. 2018), while global MHD simulations of the solar wind have also shown that the parametric decay instability can contribute substantially to solar wind heating and acceleration, thanks to the generation of compressible modes that, in the absence of kinetic effects, naturally steepen into shocks (see, e.g., Shoda et al. 2019). The traditional parametric decay has been also invoked as a possible source for the generation of inward modes and solar wind turbulence in the inner heliosphere, where an increasing content of reflected waves (cross-helicity) and an evolving turbulent spectrum are observed with increasing heliocentric distance (Bavassano et al. 2000). However, expansion effects are known to inhibit its development, essentially because the parametric decay process is strongly suppressed as the plasma beta increases at larger heliocentric distances (Tenerani & Velli 2013, 2020; Del Zanna et al. 2015). Temperature anisotropies can destabilize the parametric decay at values of the plasma beta approaching unity and above, but the anisotropy in the solar wind is not large enough to affect the instability significantly (Tenerani et al. 2017).


Water ice is ubiquitous in the cold regions of the Universe, owing to the fact that hydrogen and oxygen are the two most abundant elements to form a solid such as icy dust particles and comets. It is, therefore, commonly accepted that the essential component of dust particles and planetesimals in protoplanetary discs is water ice beyond the so-called snow line, at which the temperature of gas is low enough for water vapour to condense into ices (e.g. Cyr, Sears & Lunine 1998). Reactive accretion of water ice from hydrogen and oxygen atoms on the surface of dust particles takes place in the dense core of molecular clouds where the growth of dust particles has been observed by scattering of stellar radiation (Steinacker et al. 2010). It is worthwhile noting that laboratory experiments on the coagulation growth of water-ice particles have a long history outside astronomy and planetary science, since coagulation is observed in daily life and is a plausible route to the formation of snowflakes (e.g. Faraday 1860; Hosler, Jensen & Goldshlak 1957). Recent works on laboratory measurements of cohesion between crystalline water-ice particles at vacuum conditions provided encouraging results that dust particles composed of water ice might be much more cohesive than previously believed (Gundlach et al. 2011; Gundlach & Blum 2015; Jongmanns et al. 2017). Form a theoretical point of view, Chokshi, Tielens & Hollenbach (1993) demonstrated that the JKR theory of elastic contact formulated by Johnson, Kendall & Roberts (1971) is a powerful tool for better understanding of dust coagulation. Numerical simulations incorporating the JKR theory have shown that dust aggregates consisting of submicrometre-sized water-ice particles proceed with coagulation growth even at a collision velocity of 50 m s−1 (Wada et al. 2009, 2013). As a result, the majority of recent studies on dust coagulation and planetesimal formation assume that silicate aggregates are disrupted by mutual collision at a velocity of vdisrupt ∼ 1 m s−1, but icy aggregates at vdisrupt ∼ 10 m s−1 (e.g. Birnstiel, Dullemond & Brauer 2010; Vericel & Gonzalez 2019). Such a trendy assumption led Drążkowska & Alibert (2017) to propose planetesimal formation by the ‘traffic jam’ effect at the snow line, provided that sticky water-ice particles grow faster and thus drift toward the central star faster than less-sticky bare silicate particles, implying that aggregates of the former catch up the latter at the snow line, which results in a traffic jam. However, we argue that the importance of water ice to dust coagulation is still open to debate, since water ice is not necessarily stickier than other materials such as silicates and complex organic matter (Kimura et al. 2015, 2020a; Musiolik & Wurm 2019).


Since the perturbation in equation (22) is not conservative, we shall focus on the extension of Noether's theorem to non-conservative systems by Djukic & Vujanovic (1975). It must be $F_i-q^\prime _if\ne 0$ for the conservation law to hold (Vujanovic, Strauss & Jones 1986). For the case of non-conservative systems the generators Fi, f, and the gauge Ψ need to satisfy the following relation:

(29)

\begin{eqnarray}
&&{\sum _i\left\lbrace \left( \frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {q_i}} \right) F_i + \left(\frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {q^\prime _i}} \right)( {F}^\prime _i - q^\prime _i{f}^\prime ) + Q_i(F_i-q^\prime _if)\right\rbrace} \nonumber\\
&&{\quad+ f^\prime \scr {L} + f\frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {\tau }} = {\Psi }^\prime .}
\end{eqnarray}

This equation and the condition $F_i-q^\prime _if\ne 0$ furnish the generalized NBH equations (Trautman 1967; Djukic & Vujanovic 1975; Vujanovic et al. 1986). The NBH equations involve the full derivative of the gauge function and the generators with respect to τ, meaning that equation (29) depends on the partial derivatives of Ψ, Fi, and f with respect to time, the coordinates, and the velocities. By expanding the convective terms the NBH equations decompose in the system of Killing equations:

(30)

\begin{eqnarray}
&&{ \scr {L} \frac{\mathrm{\partial} {f}}{\mathrm{\partial} {q_j^\prime }} +\sum _i \frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {q^\prime _i}}\left( \frac{\mathrm{\partial} {F_i}}{\mathrm{\partial} {q_j^\prime }} - q^\prime _i\frac{\mathrm{\partial} {f}}{\mathrm{\partial} {q_j^\prime }} \right) = \frac{\mathrm{\partial} {\Psi }}{\mathrm{\partial} {q_j^\prime }},} \nonumber \\
&&{ \frac{\mathrm{\partial} {}}{\mathrm{\partial} {\tau }}(f\scr {L} - \Psi ) + \sum _i \Bigg\lbrace \frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {q_i}}F_i + \scr {L}\frac{\mathrm{\partial} {f}}{\mathrm{\partial} {q_i}}q_i^\prime + Q_i(F_i-q_i^\prime f)+ \frac{\mathrm{\partial} {\scr {L}}}{\mathrm{\partial} {q_i^\prime }}} \nonumber \\
&&{\times\, \left[ \frac{\mathrm{\partial} {F_i}}{\mathrm{\partial} {\tau }} - q_i^\prime \frac{\mathrm{\partial} {f}}{\mathrm{\partial} {\tau }} + \sum _j\left( \frac{\mathrm{\partial} {F_i}}{\mathrm{\partial} {q_j}}q_j^\prime - q_i^\prime q_j^\prime \frac{\mathrm{\partial} {f}}{\mathrm{\partial} {q_j}} \right) \right] - \frac{\mathrm{\partial} {\Psi }}{\mathrm{\partial} {q_i}}q_i^\prime \Bigg \rbrace = 0.}\nonumber\\
\end{eqnarray}
The system (30) decomposes in three equations that can be solved for the generators Fρ, Fθ, and f given a certain gauge. If the transformation defined in equation (26) satisfies the NBH equations, then the system admits the integral of motion (28).


A well-studied way to look at antenna spectral requirements is from the perspective of foreground avoidance in power spectrum space. In the avoidance scheme, smooth-spectrum foregrounds should, in the ideal case, occupy a wedge–shaped region of the two–dimensional power spectrum space (where the wavenumber k can be decomposed into k⊥ and $k_\Vert$ that are the transverse and line-of-sight wave numbers, respectively) whereas the remaining area – the so-called EoR window – is dominated by the 21 cm emission (Datta, Bowman & Carilli 2010; Morales et al. 2012; Trott, Wayth & Tingay 2012; Vedantham, Udaya Shankar & Subrahmanyan 2012; Thyagarajan et al. 2013; Liu, Parsons & Trott 2014a,b). Sources have most of their emission at low k∥ values although, due to the inherent chromatic interferometric response, this area increases with baseline length to resemble a characteristic wedge-like shape (Parsons et al. 2012; Trott et al. 2012; Vedantham et al. 2012; Liu et al. 2014a,b; Thyagarajan et al. 2015). In the most optimistic scenario, the EoR power spectrum can be directly measured in the EoR window, whose boundaries are set by the so-called horizon limit, i.e. the maximum delay that an astrophysical signal can experience, given by the separation of the two receiving elements. In practice, the boundaries of the EoR window can be narrowed by a number of mechanisms that spread power from the foreground dominated region into the EoR window, in particular calibration errors (Barry et al. 2016; Patil et al. 2016), leakage of foreground polarization (Bernardi et al. 2010; Jelić et al. 2010; Moore et al. 2013; Asad et al. 2015, 2016) and intrinsic chromaticity of the instrumental response. Recent attention has indeed been given to simulate and characterize the element response, particularly for the HERA (DeBoer et al. 2017), realizing that it may be one of the critical items responsible for spilling power from the wedge into the EoR window.


When energy flows from the interior of the Sun outward into the solar atmosphere, why is the Sun’s outer atmosphere, the corona, much hotter than the inner atmosphere, the underlying chromosphere and photosphere? This is the long-standing problem of the coronal heating, which is one of the eight key mysteries in modern astronomy (Kerr 2012). For about 80 yr since the discovery of the extremely hot corona around the late 1930s (Grotian 1939; Edlen 1945), people have worked hard on addressing this issue, and great advances have been made in observation and theoretical studies (Parnell & De Moortel 2012; Amari et al. 2015; Arregui 2015; Cargill et al. 2015; De Moortel & Browning 2015; Jess et al. 2015; Klimchuk 2015; Longcope & Tarr 2015; Peter 2015; Schmelz & Winebarger 2015; Velli et al. 2015; Wilmot-Smith 2015). Especially during recent decades, high-resolution observations of solar super-fine structures indicate that small spicules, minor hot jets along small-scale magnetic channels from the low atmosphere upwards to the corona, petty tornados and cyclones, and small explosive phenomena such as mini-filament eruptions and micro- and nano-flares—all of these small-scale magnetic activities contribute greatly to coronal heating (De Pontieu et al. 2011; 2018; Zhang & Liu 2011; Parnell & De Moortel 2012; Klimchuk 2015; Peter 2015; Schmelz & Winebarger 2015; Henriques et al. 2016; Li et al. 2018a). Additionally, contributions of MHD waves to heating the corona have been observationally illustrated (van Ballegooijen et al. 2011; Jess et al. 2015; Kubo et al. 2016; Morton et al. 2016; Soler et al. 2017; Morgan & Hutton 2018). Meanwhile, with the progress of observational studies, two groups of theoretical models, magnetic reconnection models and magnetohydrodynamic wave models, have traditionally attempted to explain coronal heating, but so far no models can address the key mystery perfectly (van Ballegooijen et al. 2011; Arregui 2015; Cargill et al. 2015; Peter 2015; Velli et al. 2015; Wilmot-Smith 2015). Maybe we do not need to intentionally take to heart such the classical dichotomy, because waves and reconnections may interact with each other (De Moortel & Browning 2015; Velli et al. 2015). Additionally, statistical studies may look at coronal heating from a comprehensive perspective. Li et al. (2018b) found that the long-term variation of the heated corona, which is represented by coronal spectral irradiances, and that of small-scale magnetic activity are in lockstep, indicating that the corona should statistically be effectively heated by small-scale magnetic activity. Observational and theoretical model studies through heating channels and modes, and statistical studies by means of heating effect (performance of the heated corona), both suggest that coronal heating originates from small-scale magnetic activity.


The H α studies of Umeda et al. (2004) and Stroe et al. (2014a, 2015) are tracing instantaneous (averaged over 10 Myr) SF and little is known about SF on longer time-scales and the reservoir of gas that would enable future SF. An excellent test case for studying the gas content of galaxies within merging clusters with shocks is CIZA J2242.8+5301 (Kocevski et al. 2007). For this particular cluster unfortunately, its location in the Galactic plane, prohibits studies of the rest-frame UV or FIR tracing SF on longer time-scales, as the emission is dominated by Milky Way dust. However, the rich multiwavelength data available for the cluster give us an unprecedented detailed view on the interaction of their shock systems with the member galaxies. CIZA J2242.8+5301 is an extremely massive (M200 ∼ 2 × 1015 M⊙; Dawson et al. 2015; Jee et al. 2015) and X-ray disturbed cluster (Akamatsu & Kawahara 2013; Ogrean et al. 2013, 2014) which most likely resulted from a head-on collision of two, equal-mass systems (van Weeren et al. 2011; Dawson et al. 2015). The cluster merger induced relatively strong shocks, which travelled through the ICM, accelerated particles to produce relics towards the north and south of the cluster (van Weeren et al. 2010; Stroe et al. 2013). There is evidence for a few additional smaller shock fronts throughout the cluster volume (Stroe et al. 2013; Ogrean et al. 2014). Of particular interest is the northern relic, which earned the cluster the nickname ‘Sausage’. The relic, tracing a shock of Mach number M ∼ 3 (Stroe et al. 2014c), is detected over a spatial extent of ∼1.5 Mpc in length and up to ∼150 kpc in width and over a wide radio frequency range (150 MHz–16 GHz; Stroe et al. 2013, 2014b). There is evidence that the merger and the shocks shape the evolution of cluster galaxies. The radio jets are bent into a head–tail morphology aligned with the merger axis of the cluster. This is probably ram pressure caused by the relative motion of galaxies with respect to the ICM (Stroe et al. 2013). The cluster was also found to host a high fraction of H α emitting galaxies (Stroe et al. 2014a, 2015). The cluster galaxies not only exhibit increased SF and AGN activity compared to their field counterparts, but are also more massive, more metal rich and show evidence for outflows likely driven by SNe (Sobral et al. 2015). Stroe et al. (2015) and Sobral et al. (2015) suggest that these relative massive galaxies (stellar masses of up to ∼1010.0–10.7 M⊙) retained the metal-rich gas, which was triggered to collapse into dense star-forming clouds by the passage of the shocks, travelling at speeds up to ∼2500 km s−1 (Stroe et al. 2014c), in line with simulations by Roediger et al. (2014).


Major nearby galaxy cold gas mapping surveys (Regan et al. 2001; Wilson et al. 2009; Rahman et al. 2011; Leroy et al. 2009; Donovan Meyer et al. 2013; Bolatto et al. 2017; Sorai et al. 2019; Sun et al. 2018) have focused on observations of the molecular gas (through CO lines). Despite a few notable exceptions (e.g. Alatalo et al. 2013; Saintonge et al. 2017), these surveys observed mainly spiral or infrared-bright galaxies (i.e. galaxies with significant star formation) and have furthered our understanding of how star formation happens, rather than how it stops. This boils down to quantifying the relation between molecular gas and star formation rate (SFR), which appears nearly linear in nearby discs (Kennicutt 1998; Bigiel et al. 2008; Leroy et al. 2013; Lin et al. 2019). This relationship is often parametrised via the ratio between the SFR and the molecular gas mass (Mmol), which is called the molecular star formation efficiency (SFE = SFR/Mmol = 1∕τdep), where the inverse of the SFE is the depletion time, τdep. The depletion time indicates how much time is necessary to convert all the available molecular gas into stars at the current star formation rate. On kpc scales and in the discs of nearby star-forming galaxies, τdep is approximately constant around 1–2 Gyr (Bigiel et al. 2011; Rahman et al. 2012; Leroy et al. 2013; Utomo et al. 2017), and it appears to weakly correlate with many galactic properties such as stellar mass surface density or environmental hydrostatic pressure (Leroy et al. 2008; Rahman et al. 2012). Nevertheless, small but important deviations for a constant SFE have been noticed, which can be the first hints of star formation quenching. In some galaxies, the depletion time in the centres appear shorter (Leroy et al. 2013; Utomo et al. 2017) or longer (Utomo et al. 2017) with respectto their discs. These differences may correlate with the presence of a bar or with galaxy mergers (Utomo et al. 2017; see also Muraoka et al. 2019) and do not seem to be related to unaccounted variation in the CO-to-H2 conversion factor (Leroy et al. 2013; Utomo et al. 2017). Spiral arm streaming motions have also been observed to lengthen depletion times (Meidt et al. 2013; Leroy et al. 2015).


There are some existing studies applying machine learning to transient studies. For example, the spectral types of the SNe can be classified based on their light-curve data (Möller et al. 2016; Muthukrishna et al. 2019a; Takahashi et al. 2020; Villar et al. 2020), and transients can be identified from the astronomical survey images (Goldstein et al. 2015; Mahabal et al. 2019; Gómez et al. 2020). The light curves of SNe Ia can be well modeled by functional principal component analysis (FPCA; He et al. 2018), where it was shown remarkably that a set of FPCA eigenvectors that are independent of the photometric filters can be derived from the observed light curves of SNe Ia. There are a few studies of the application of deep learning neural networks to the spectral data of SNe. For example, Muthukrishna et al. (2019b) used a convolutional neural network (CNN) for automated SN type classification based on SN spectra. Several other works (Chen et al. 2020; Vogl et al. 2020; Kerzendorf et al. 2021) applied a Gaussian process, principal component analysis (PCA), and deep learning neural networks to radiative transfer models of SNe. Sasdelli et al. (2016) used unsupervised learning algorithms to investigate the subtypes of SNe Ia. Stahl et al. (2020) developed neural networks to predict the photometric properties of SNe Ia (phase and Δm
15) based on spectroscopic data. Saunders et al. (2018) used PCA to find low dimensional representations of the spectral sequences of 140 well-observed SNe Ia. Chen et al. (2020), in particular, built an artificial intelligence assisted inversion (AIAI) of radiative transfer models and used that to link the observed SN spectra with theoretical models. The AIAI is able to retrieve the elemental abundances and density and temperature profiles from observed SN spectra. The AIAI approach has the potential for quantitatively coupling complex theoretical models with the ever-increasing amount of high-quality observational data.


In the past, as well as in the more recent literature, there have been many attempts to explain the different chemical evolutionary paths of different MW components, particular those of the thin and thick discs. The outcome of these studies is that the observed different chemical evolutionary paths are related to differences in the main physical processes that drive galaxy evolution, the most significant of which are the gas accretion time-scale and the star formation efficiency and, possibly, radial migration (Larson 1972; Lynden-Bell 1975; Pagel & Edmunds 1981; Matteucci & Greggio 1986; Matteucci & Brocato 1990; Ferrini et al. 1994; Prantzos & Aubert 1995; Chiappini et al. 1997, 2001; Portinari & Chiosi 1999; Bekki & Tsujimoto 2011; Micali et al. 2013; Sahijpal 2014; Snaith et al. 2014; Grisoni et al. 2017; Grand et al. 2018; Spitoni et al. 2021). A good agreement between observations and theoretical predictions for the Galaxy is obtained by models that are based on the assumption that the disc formed via the infalling of gas (Chiosi 1980; Matteucci & Francois 1989; Chiappini et al. 1997). The formation of the different components is associated with distinct sequential main episodes of gas accretion (infall phases) that, at first, rapidly accumulates in the central regions and then, more slowly, in the more external ones, according to the so-called ‘inside-out scenario’ (Chiappini et al. 2001). In particular, the three-infall model, devised by Micali et al. (2013), is capable of reproducing the abundance patterns of the MW halo, thick and thin disc at once. In this model, the halo forms in a first gas infall episode of short timescale (0.2 Gyr) and mild star formation efficiency, ν = 2 Gyr−1, lasting for about 0.4 Gyr. It is immediately followed by the thick disc formation, characterised by a somewhat longer infall timescale (1.2 Gyr), a longer duration (about 2 Gyr) and a higher star formation efficiency (ν = 10 Gyr−1). Finally, star formation continues in the thin disc with a longer infall timescale (6 Gyr in the solar vicinity) and is still continuing to this day, with a star-formation efficiency of ν = 1 Gyr−1. The [O/Fe] vs. [Fe/H] path is thus continuous across the regions populated by halo, thick, and thin disc stars. While Micali et al. (2013) described the chemical enrichment as continuous across the three different infall stages, Grisoni et al. (2017) used also an alternative scheme where the thin and thick disc components evolve separately, in a parallel approach (see also Chiappini 2009). In such a parallel approach, the disc populations are assumed to form in parallel but to proceed at different rates. The gas infall exponentially decreases with a timescale that is 0.1 Gyr and 7 Gyr, for the thick and thin disc, respectively. This alternative approach better reproduces the presence of the metal-rich α-enhanced stars in the [Mg/Fe] vs. [Fe/H] diagram obtained with the recent AMBRE data (Mikolaitis et al. 2017).


Exploration of the stationary points of the [H, P, S, O] molecular system was initially carried out using Møller–Plesset second order Perturbation Theory (MP2) (Møller & Plesset 1934; Frisch et al. 1990a,b; Head-Gordon & Head-Gordon 1994; Head-Gordon et al. 1988; Sæbø & Almlöf 1989) with the 6-311++G(d, p) basis set (McLean & Chandler 1980; Krishnan et al. 1980). Following the initial survey, transition state structures connecting various isomers of [H, P, S, O] were located using MP2/6-311++G(d, p) and confirmed via an intrinsic reaction coordinate (IRC) calculation. To achieve a quantitative picture of the ground state potential energy surface, coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)] (Knowles et al. 1993, 2000) single point calculations employing the aug-cc-pV(Q+d)Z basis set were performed at the MP2 equilibrium geometries. Entrance channel pathways were explored via relaxed scans and individual searches for transition states. Following this, higher level calculations on the electronic structure, rotational constants, and harmonic vibrational frequencies of each isomer were performed using CCSD(T) together with the Dunning basis sets aug-cc-pV(X+d)Z (X = T, Q, 5) (Dunning 1989; Woon & Dunning 1993; Kendall et al. 1992), including additional tight d functions on the sulfur and phosphorous atoms (Dunning et al. 2001). CCSD(T) including all electrons was then performed using the second order Douglas-Kroll–Hess Hamiltonian in conjunction with the contracted relativistic Douglas–Kroll aug-cc-pwCVTZ-DK basis set to correct for scalar relativistic and core-correlation effects (Jorge et al. 2009). The aforementioned combinationof level of theory and basis set will be referred to as CCSD(T)-AE/TZ-DK. Finally, explicit correlation of the electrons was included using the CCSD(T)-F12b method (Werner et al. 2011) with the Dunning basis sets aug-cc-pVXZ (X = T, Q, 5). Single-reference character of the wavefunction was confirmed via single-state complete active spaceself-consistent field theory in addition to the coupled cluster T1 diagnostic. Energies and geometry parameters were then extrapolated to the complete basis set limit using the two-point (Q, 5) extrapolation scheme E(x)= ECBS + AX−3, where E(x) is the value calculated using the basis set of cardinal number X, ECBS is the extrapolated energy, and A is a parameter fit in the least-squares fitting procedure. Mulliken population analysis was performed on cis-HOPS using MOLPRO. Corrections to the rotational constants, spectroscopic data, and anharmonic vibrational data were calculated for the four lowest energy isomers at the CCSD(T)/aug-cc-pV(T+d)Z level of theory using CFOUR (Matthews et al. 2020; Stanton et al. 2019). Permanent dipole moments for each isomer were calculated using the finite field method (field strengths of 0, 0.005, and −0.005 au) as implemented in MOLPRO.


The discrete change in spin-down rate between the on and off states suggests a switching behaviour in the pulsar magnetosphere. While explanation for the phenomenon remains inconclusive, conventional treatments typically involve referring each emission state to different condition in the magnetosphere in such a way that a plasma-filled magnetosphere is related to the emission-on state and a sudden depletion of charge particles is responsible for the decrease in the spin-down rate during the emission-off state (Contopoulos 2005; Kramer et al. 2006; Beskin & Nokhrina 2007; Li et al. 2012). Kramer et al. (2006) and Beskin & Nokhrina (2007) suggest that energy loss of PSR B1931 + 24 is due only to the rotating dipole during the emission-off state where there is no current flowing, whereas current loss is responsible for the energy loss during the emission-on state. In this scenario, pulses are undetectable due to decrease in the plasma density in the emitting region to a magnetospheric state so that the pulse intensity of the pulsar decreases when it jumps to that state and resumes to its normal intensity when it returns to its initial state. While the approach has revealed unprecedented information about the pulsars, it implicitly treats the two emission states as independent entities with presumed vacuum condition for the emission-off state. Our investigation is based on a magnetospheric model that synthesizes the corotating magnetosphere model and the vacuum model (Melrose & Yuen 2014, 2016), in which multiple quasi-stable magnetospheric states are described by the parameter y such that the charge filling fraction of a state corresponds to a y value and a change in the charge filling fraction corresponds to a change of state which is signified by a change in y. The presence of plasma in the corotating magnetosphere model implies a global current flow that closes at the stellar surface within the polar cap region. By considering a simple framework for the spin-down torque as induced by the component of the global current that crosses the magnetic fields at the stellar surface, we show that discrete changes in the spin-down rates may be attributed to changes in the pulsar torque caused by variations in the global current flow when jumping takes place between different magnetospheric states in the pulsar magnetosphere. Using this idealized model, characteristics of intermittent pulsars are studied by jointly considering the two emission states based on the parameters y and α, the obliquity angle between the magnetic and rotation axes, through the ratio of the respective spin-down rate whose changes are described by the ratio of the corresponding spin-down torque in the two states. The yet unidentified links between intermittent and ordinary pulsars imply that, in addition to the parameters already under detailed examination in the literature, there may exist unknown factors that are distinctly related to intermittency and likely to be pulsar-specific. An obvious example is the large differences in the emission-on and -off cycles which range from days (Kramer et al. 2006) to years (Camilo et al. 2012). Since the correlation between these parameters and intermittency is still unclear, before including such factors, it is useful to explore the implications of the idealized model.


There are three possible explanations for the origin of these systems: (1) they formed from the fragmentation of the same molecular cloud (Elmegreen & Elmegreen 1983), (2) they were generated in distinct molecular clouds and then became bound systems after a close encounter leading to a tidal capture (Vallenari, Bettoni & Chiosi 1998; Leon, Bergond & Vallenari 1999), or (3) they are the result of division of a single star-forming region (Goodwin & Whitworth 2004; Arnold et al. 2017). Their subsequent evolution may also have different outcomes. Dynamical models and N-body simulations (see, e.g., Barnes & Hut 1986; de Oliveira, Dottori & Bica 1998, and references therein) have shown that, depending on the initial conditions, a bound pair of clusters may either become unbound, because of significant mass-loss in the early phases of stellar evolution, or merge into a single and more massive cluster on a short time-scale (≈60 Myr) due to loss of angular momentum from escaping stars (see Portegies Zwart & Rusli 2007). The final product of a merger may be characterized by a variable degree of kinematic and morphologic complexity, mostly depending on the values of the impact parameter of the pre-merger binary system (de Oliveira, Bica & Dottori 2000; Priyatikanto et al. 2016). In some cases, the stellar system resulting from the merger event may show significant internal rotation (in fact, for many years this has been the preferred dynamical route to form rotating star clusters; see Sugimoto & Makino 1989; Makino, Akiyama & Sugimoto 1991; Okumura, Ebisuzaki & Makino 1991; de Oliveira, Dottori & Bica 1998). Merger of cluster pairs has been sometimes invoked to interpret the properties of particularly massive and dynamically complex clusters (e.g. see the study of ω Centauri by Lee et al. 1999, G1 by Baumgardt et al. 2003 and NGC  2419 by Brüns & Kroupa 2011), and, more in general, as an avenue to form clusters with multiple populations with different chemical abundances both in terms of iron and light elements (e.g. van den Bergh 1996; Catelan 1997; Amaro-Seoane et al. 2013; Gavagnin, Mapelli & Lake 2016; Hong et al. 2017).


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


The most unique and characteristic observational feature of blazars’ highly variable broadband emission is the broad bimodal SED extending from the lowest accessible EM band, i.e., the radio, to the highest accessible, i.e., GeV-TeV γ-rays. The broadband SED of all blazars can be categorized into three different spectral subclasses: low-energy-peaked (LBL/LSP), intermediate-energy-peaked (IBL/ISP), and high-energy-peaked (HBL/HSP; Fossati et al. 1998; Abdo et al. 2010), based on the location of the low-energy hump. A remarkable property of each spectral subclass is the stability of the location of the two peaks despite huge variations in flux and often spectral shape. Only in a few rare instances has an appreciable shift in the location of the peaks been observed, e.g., the 1997 outburst of Mrk 501 (Pian et al. 1998; Ahnen et al. 2018) and the activity of OJ 287 from the end of 2015 to the middle of 2017 (Kushwaha et al. 2018a, 2018b). Even these two cases are remarkably different. In the case of Mrk 501, the locations of both the peaks shifted to higher energies. On the contrary, in OJ 287, a shift in the location of only the high-energy peak was observed during the 2015–2016 activity (Kushwaha et al. 2018a, 2019), while in 2016–2017 a new broadband emission component overwhelmed the overall emission, appearing as an overall shift in both the peaks as revealed in the detailed study by Kushwaha et al. (2018b). With the SED being the prime observable for exploration of the yet-debated high-energy emission mechanisms, such changes offer invaluable insights about the emission processes. For example, in Mrk 501 the shift in both peaks strongly implies the same particle distribution for the overall emission, while for OJ 287 the shift of only the high-energy peak can be reproduced by either inverse Compton scattering of the broad-line region photon field (Kushwaha et al. 2018a) or emission of hadronic origin (Oikonomou et al. 2019; Rodríguez-Ramírez et al. 2020).


Utilizing the full range of peak absolute magnitudes observed in LBV stars [M ≃  (−13)–(−9) mag; e.g. Smith et al. 2011b] provides a range of peak apparent magnitudes of m ≃  16.5–20.5 mag, for a distance of D ≃ 8 Mpc (Table 1). Hence, if the transient source (m ≈ 21.0 mag; Section 2.1.2) is an LBV star, it must have been observed a short time after its peak (Fig. 2; left-hand column; rows 4–5); quiescent LBV stars can have absolute magnitudes as low as M ≃ −6 mag (e.g. Smith et al. 2011b), or m ≃ 23.5 mag (for a distance of D ≃ 8 Mpc; Table 1). An LBV star provides an adequate explanation for the transient time-scale, the isolation (e.g. Smith & Tombleson 2015; Smith 2016), the lack of a host H ii region (Fig. 2 and 3; see below, however), and the transient/main source offset (d ≈ 0.3 kpc, for D ≃ 8 Mpc; Table 1; see below, however). In addition, (net) fading of the LBV peak event by Δm ≲ 3 mag (over a period of approximately 40 yr) may possibly produce an apparent brightness centroid shift/morphology variation in the main source (Sections 2.1.2 and 2.1.3). However, this scenario has its caveats. First, it is difficult to interpret the flux variability of the main source (Sections 2.1.4 and 2.3.1) within such a context. Secondly, although a fraction of LBV stars are isolated (e.g. Smith & Tombleson 2015; Smith 2016), in the only XMP with a documented LBV star, the LBV star appears embedded within an H ii region (DDO 68; Pustilnik et al. 2017); no clear H ii region at the location of the transient source is discernible in the HST images (Fig. 3; top), although it should be detectable given the typical lifetime of an H ii region (few Myr; e.g. Alvarez, Bromm & Shapiro 2006). Thirdly, a (net) magnitude variation of Δm ≲3 mag of the transient over a period of approximately 60 yr would result in a quiescent LBV source (m ≲ 23.5 mag) that should have been detectable in the HST images (Fig. 3; top). Lastly, as LBV stars brighten, they become redder (e.g. Sterken 2003), which appears to contradict the POSS data (Table 2; Fig. 2; rows 2–5). Consequently, as the LBV scenario explains some of the observables challenged by other scenarios (e.g. transient timeline), it remains a contender for the observed phenomenon.


The EWs were measured with the methods described in Vásquez et al. (2015). As in Paper I, we used the sum of the EWs of the two strongest CaT lines (λ8542, λ8662) as a metallicity estimator, following the Ca II triplet method of Armandroff & Da Costa (1991). Different functions have been tested in the literature to measure the EWs of the CaT lines, depending on the metallicity regime. For metal-poor stars ([Fe/H] ≲ −0.7 dex) a Gaussian function was used with excellent results (Armandroff & Da Costa 1991), while a more general function (such as a Moffat function or the sum of a Gaussian and Lorentzian, G + L) is needed to fit the strong wings of the CaT lines observed in metal-rich stars (Rutledge et al. 1997; Cole et al. 2004). Following our previous work (Gullieuszik et al. 2009; Saviane et al. 2012) we have adopted here a G+L profile fit. To measure the EWs, each spectrum was normalised with a low-order polynomial using the IRAF continuum task, and set to the rest frame by correcting for the observed radial velocity. The two strongest CaT lines were fitted by a G+L profile using a non-linear least squares fitting routine part of the R programming language. Five clusters from the sample of Saviane et al. (2012) covering a wide metallicity range were re-reduced and analysed with our code to ensure that our EWs measurements are on the same scale as the template clusters used to define the metallicity calibration. Figure 5 shows the comparison between our EWs measurements and the line strengths measured by Saviane et al. (2012) (in both cases the sum of the two strongest lines) for the five calibration clusters. The observed scatter is consistent with the internal errors of the EW measurements, computed as in Vásquez et al. (2015). The measurements show a small deviation from the unity relation, which is more evident at low metallicity. A linear fit to this trend gives the relation: ΣEW(S12) = 0.97 ΣEW(this work) + 0.21, with an rms about the fit of 0.13 Å. This fit is shown in Fig. 5 as a dashed black line. For internal consistency, all EWs in this work have been adjusted to the measurement scale of Saviane et al. (2012) by using this relation. In Table 3 we provide the coordinates, radial velocities, and the sum of the equivalent widths for the cluster member stars, both measured (“m”) and corrected (“c”) to the system of Saviane et al. 2012.


The shock is then revived by adding an energy injection following the parameterized method of Fryer et al. (2018). In this model, roughly

 was deposited into the inner

 in the first

. Some of this energy is lost through neutrino emission and the total explosion energy at late times for this model is

. This explosion is then mapped into our three-dimensional calculations, using one million SPH particles. The mapping took place when the supernova shock had moved out of the iron core and propagated into the Si–S rich shell at

. We note that our 1D methods employed for modeling the collapse, core bounce, and initial explosion do not capture the full physics of the central engine (for a discussion, see Fryer et al. 2018), and this is a source of uncertainty in our yield calculations. The details of the engine change the shock trajectories, and neutrino chemistry can change Ye values (Saez et al. 2018; Fujimoto & Nagakura 2019). The nature of the shock affects mostly the yields after the shock falls below NSE (before it falls out of NSE, the yields are set by the equilibrium values, not the time-dependent evolution). Our model captures one instance of the range of asymmetric trajectories, and it should be noted that no model at this time is sufficiently accurate to dictate exactly the properties of the asymmetries (Janka et al. 2016). In addition, any model that does not include convection-driven asymmetries from the progenitor star cannot properly capture the asymmetries (Arnett et al. 2015). The 3D explosion model used here also displays stochastic asymmetries, implying that any manner of convective asymmetry could generate similar results. If this behavior is universal, it could have important implications. These points taken together indicate that nucleosynthetic patterns arising from convection-like behavior are robust, regardless of the driver. As discussed below, this increases the utility of NSE nucleosynthesis, particularly of

 and

, as diagnostics of the conditions in the progenitor star.


Figure 2 shows the measured spectral indices of stars as functions of K magnitude, obtained from the 2MASS catalog. The CN, HK′, and CH indices increase with decreasing magnitude because the brighter RGB stars have lower temperatures and the strengths of these molecular bands generally increase with decreasing temperature. Therefore, the chemical abundances of stars are compared on the δ-index versus magnitude diagrams. It is important to note that the observed stars show a large spread in δ-index that is at least several times larger than the measurement error. The standard deviations for all sample stars are 0.23 for CN, 0.07 for HK′, and 0.07 for CH. In particular, the CN index distribution shows the largest spread. Note that a bimodality or a large spread in CN distribution is generally observed in most GCs (Norris et al. 1981; Norris 1987; Briley et al. 1992; Harbeck et al. 2003; Kayser et al. 2008; Martell et al. 2008).2

2
Although the evolutionary mixing effect can also contribute to the large spread in CN index distribution among bright RGB stars (Sweigart & Mengel 1979), this effect alone cannot explain a discrete distribution and a wide spread in the unevolved stars (see, e.g., Kayser et al. 2008).
 Therefore, we have divided subpopulations of RGB stars in NGC 5286 on the histogram of the δCN index (see Figure 3). It is clear from this histogram that RGB stars are divided into three subpopulations: CN-weak (δCN  −0.2; blue circles), CN-intermediate (−0.2 ≤ δCN  0.1; green circles), and CN-strong (0.1 ≤ δCN; red circles). The distribution of CN index into three or more subpopulations is similar to that reported in NGC 1851 (Campbell et al. 2012; Lim et al. 2015; Simpson et al. 2017). This is also consistent with the recent results from population models and spectroscopic observations that show that most GCs host three or more subpopulations (see, e.g., Jang et al. 2014; Carretta 2015). The presence of multiple populations is also observed from recent photometry using UV filters, which are mainly sensitive to N abundance (Milone et al. 2015; Piotto et al. 2015). In this regard, further observations are required to check that the trimodal CN distribution, observed in NGC 1851 and NGC 5286, is a ubiquitous feature in other GCs as well.


We attempted to find a correlation between the mean magnetic field and the outflow and minor axis of the cloud CB 17. Relative orientations between various quantities of CB 17 are presented in the first row of Table 6, along with a comparative study of the same for some dark clouds. The first column gives the cloud ID and columns 2–6 give the position angles of the mean magnetic field at the envelope ($\lt \theta ^{\rm env}_B\gt $), mean magnetic field at the core ($\lt \theta ^{\rm core}_B\gt $), outflow axis (θout), minor axis (θmin) of the core of the cloud and Galactic plane (θGP), respectively. $\lt \theta ^{\rm env}_B\gt $ of CB 17 is found to be almost aligned along the Galactic plane over that region of the sky (column 7), which indicates the dominance of the Galactic magnetic field over the envelope magnetic field of the cloud and thus we cannot infer much about the magnetic field structure from the optical study. A similar feature has also been observed in the cases of CB 34 (Das et al. 2016), L328, L673-7 (Soam et al. 2015), CB 26 (Halder et al. 2019), CB 3 and CB 246 (Ward-Thompson et al. 2009). However, $\lt \theta ^{\rm core}_B\gt $ of CB 17 (obtained by submm polarimetry) turned out to be perpendicular to $\lt \theta ^{\rm env}_B\gt $ (column 8); a similar phenomenon has been observed in the case of CB 34-C1 (Das et al. 2016), IRAM 04191 (Soam et al. 2015) and CB 54 (Wolf et al. 2003). Since, in the case of CB 17, $\lt \theta ^{\rm env}_B\gt $ is along the Galactic plane orientation, this implies that only $\lt \theta ^{\rm core}_B\gt $ (denser region) is linked with the ongoing physical phenomena in the cloud. $\lt \theta ^{\rm core}_B\gt $ is oriented perpendicular to θGP as well (column 9) and a similar orientation has been observed in the case of CB 34-C1 (Das et al. 2016), IRAM 04191 (Soam et al. 2015), CB 230 and CB 244 (Wolf et al. 2003) as well. Moreover, $\lt \theta ^{\rm core}_B\gt $ is found to be almost aligned along the minor axis of the core of the cloud; the angular offset is nearly 5.9° (column 10). The alignment of $\lt \theta ^{\rm core}_B\gt $ with the minor axis of the cloud fits the magnetically regulated star formation model, in which the magnetic field should lie along the minor axis of the cloud (Mouschovias & Morton 1991; Li 1998), and the same feature has also been observed for the clouds CB 34-C1 (Das et al. 2016) and IRAM 04191 (Soam et al. 2015). The angular offset between $\lt \theta ^{\rm core}_B\gt $ and the outflow axis is found to be 80.9° (column 11), that is, the core-scale magnetic field is oriented almost perpendicular to the outflow direction and a similar phenomenon has also been observed in the case of CB 34 (Das et al. 2016), CB 68 (Bertrang et al. 2014), B335, CB 230, CB 244 (Wolf et al. 2003) and CB 3 (Ward-Thompson et al. 2009). The angular offset between θout and θmin is found to be 75° and the same feature has been observed for CB 34-C1 (Das et al. 2016).


According to our interpretation, the observed high velocity dispersion region in zone ‘F’ associated with the zone totally empty of gas between the two opposite velocity clouds (that the 3Dbarolo model would instead predict to be filled) resembles an expanding superbubble. Superbubbles are known to be associated with very massive stars (OB-associations): strong stellar winds and subsequent SN explosions from those stars inject energy and mass into the ambient ISM, creating shock fronts that sweep-up the ISM (e.g. Castor, McCray & Weaver 1975). As the SN explosions start occurring within the cavity formed by the stellar wind bubbles, super-bubbles are created (e.g. McCray & Kafatos 1987), which may eventually reach kiloparsec sizes. These explosions never form a visible SNR, but instead expend their energy in the hot interior as sound waves. Both stellar winds and stellar explosions thus power the expansion of the superbubble in the ISM. The interstellar gas swept up by superbubbles generally cools, forming a dense shell around the cavity (observed in Hi and Hα). The bubble interior contains hot (>106 K), rarefied material, usually associated with extended diffuse X-ray emission (thus appearing as an empty cavity in cold and molecular gas). In this interpretation, the two blobs of gas above and below the observed empty cavity in region ‘F’ (at +1.5 arcsec offset, with velocities of +70 and −70 km s−1, corresponding to the two peaks in Fig. 6), can be easily explained as gas surrounding the ‘superbubble’ being thrown away in different directions by the expanding spherical shell (see Kamphuis, Sancisi & van der Hulst 1991; Boomsma et al. 2008, for detected velocities up to 100 km s−1 associated with ‘holes’ in Hi). In Fig. 13, we show different CO(2–1) velocity channels, where the ‘superbubble’, surrounded by material moving in opposite directions (e.g. peaks at −70 and 70 km s−1, not present at the systemic velocities), shows up as an empty area (encircled in the plot), with a clear shock front at its left-hand side, peaking around the systemic velocities (≃0 km s−1).


Gaia 1 is a star cluster that was recently discovered by Koposov et al. (2017) in the first Gaia data release (Gaia Collaboration 2016), alongside with another system of lower mass. Its observation and previous detections were seriously hampered by the nearby bright star Sirius, which emphasized the impressive discovery power of the Gaia mission. This object was first characterized as an intermediate-age (6.3 Gyr) and moderately metal-rich (−0.7 dex) system, based on isochrone fits to a comprehensive combination of Gaia, 2MASS (Cutri et al. 2003), WISE (Wright et al. 2010), and Pan-STARRS1 (Chambers et al. 2016) photometry. Hence, this object was characterized by Koposov et al. (2017) as a star cluster, most likely of the globular confession. Further investigation of Gaia 1 found a metallicity higher by more than 0.5 dex, which challenged the previous age measurement and rather characterized it as a young (3 Gyr), metal-rich (−0.1 dex) object, possibly of extragalactic origin given its orbit that leads it up to ~1.7 kpc above the disk (Simpson et al. 2017). Subsequently, Mucciarelli et al. (2017) measured chemical abundances of six stars in Gaia 1, suggesting an equally high metallicity, but based on their abundance study, the suggestion of an extragalactic origin was revoked. While a more metal-rich nature found by the latter authors conformed with the results by Simpson et al. (2017), the evolutionary diagrams of both studies are very dissimilar and could not be explained by one simple isochrone fit. In particular, it was noted that “the Simpson et al. (2017) stars do not define a red giant branch in the theoretical plane, suggesting that their parameters are not correct” (Fig. 1 of Mucciarelli et al. 2017). Such an inconsistency clearly emphasizes that a clear-cut chemical abundance scale is inevitable for fully characterising Gaia 1, and to further allow for tailored age determinations, even more so in the light of the seemingly well-determined orbital characteristics, Thus, this work focuses on a detailed chemical abundance analysis of four red giant members of Gaia 1, based on high-resolution spectroscopy, which we complement with an investigation of the orbital properties of this transition object. Combined with the red clump sample of Mucciarelli et al. (2017) and reaching down to the subgiant level (Simpson et al. 2017), stars in different evolutionary states in Gaia 1 are progressively being sampled.


CCSNe are also of interest for GW astronomy as targets in their own right. As the sensitivity of GW detectors increases, they will begin to detect not only binary mergers but also other lower amplitude sources of GWs such as CCSNe. Accurate knowledge of the GW emission from CCSNe will be essential for detection and parameter estimation. The GW signal from rotational core bounce has already been well covered in the literature (e.g. Dimmelmeier et al. 2008; Abdikamalov et al. 2014; Fuller et al. 2015; Richers et al. 2017). In the non-rotating case, the GW emission from the post-bounce phase has been studied using self-consistent 3D simulations by many groups (Kuroda, Kotake & Takiwaki 2016; Andresen et al. 2017, 2019; Kuroda et al. 2017, 2018; Powell & Müller 2019, 2020; Radice et al. 2019; Andresen, Glas & Janka 2020; Mezzacappa et al. 2020; Pan et al. 2020). The structure of the GW emission has shown common features in different simulations from recent years. The dominant emission feature in the GW emission is due to the quadrupolar surface f/g mode 1 of the proto-neutron star (PNS), which produces GW frequencies rising in time from a few hundred Hz up to a few kHz (Müller, Janka & Wongwathanarat 2012; Sotani et al. 2017; Kuroda et al. 2018; Morozova et al. 2018; Torres-Forné et al. 2018, 2019). In addition, some models (Kuroda et al. 2016, 2017; Andresen et al. 2017; Mezzacappa et al. 2020; Powell & Müller 2020) exhibit low-frequency GW emission due to the standing accretion shock instability (SASI; Blondin, Mezzacappa & DeMarino 2003; Blondin & Mezzacappa 2006; Foglizzo et al. 2007). In rapidly rotating models, very strong GW emission can also occur during the post-bounce phase due to a corotation instability (Takiwaki & Kotake 2018). The emerging understanding of the GW emission features has led to the formulation of universal relations for the GW emission (Torres-Forné et al. 2019) and paved the way for phenomenological modelling for CCSN signals (Astone et al. 2018). Further work is still needed, however, to extend these models to fully explore CCSN GW signals from across the progenitor parameter space. The majority of 3D simulations that include GW emission are for progenitor stars below $30\, \mathrm{M}_{\odot }$. In this paper, we perform simulations of high-mass Population III (Pop-III) stars in the pulsational pair instability regime to expand the parameter space coverage of 3D simulations and to provide further insights into the massive and very massive star remnant BH population.


Observations show that the core temperatures of powerful AGN jets are estimated to be quite high (Moellenbrock et al. 1996). So the jets are hot to start with in this paper too. The advective disc model, as in most disc models, comes with a variety of inner disc temperatures. Simulations of advective discs for high viscosity parameter produced T ≳ 1012K in the PSD (Lee et al. 2016). Moreover, in the presence of viscous dissipation in curved space–time, the Bernoulli parameter (−hut) may increase by more than 20 per cent of its value at large distance and produce very high temperatures in the PSD (Chattopadhyay & Kumar 2016). For highly rotating BHs too, the temperatures of the inner disc easily approach 1012 K. It must also be remembered that inner regions of the accretion disc can be heated by the Ohmic dissipation, reconnection, turbulence heating or MHD wave dissipation may heat up the inner disc or the base of the jet (Pudritz 2003). High temperatures in the accretion disc can induce exothermic nucleosynthesis too (Chakrabarti, Jin & Arnett 1987; Hu & Peng 2008). All these processes taken together in an advective disc will produce very hot jet base. We do not specify the exact processes that will produce very hot jet base, but would like to emphasize that it is quite possible to achieve so. One may also wonder that if the jets are indeed launched from the disc, and how justified is it to consider non-rotating jets. Phenomenologically speaking, if jets have a lot of rotation then it would not flow around the axis of symmetry and, therefore, either it has to be launched with less angular momentum or it has to lose most of the angular momentum with which it is launched. It has been shown that viscous transport removes significant angular momentum of the collimated outflow close to the axis (Lee et al. 2016). Since the jet is launched with low angular momentum and it is further removed by viscosity or by the presence of magnetic field; therefore, the assumption of non-rotating hot jet is quite feasible. Incidentally, similar to this study, there are many theoretical studies of jets that have been undertaken under similar assumptions of non-rotating, hot jets at the base (Fukue 1987b; Falcke 1996; Memola et al. 2002).


Magneto-Rossby waves arise due to the inhomogeneity of the Coriolis force depending on latitude on a sphere in rotating astrophysical plasma by analogy with a neutral fluid (Petviashvili & Pokhotelov 1992; Onishchenko et al. 2004; Vallis 2006; Onishchenko, Pokhotelov & Astafieva 2008; Zeitlin 2018). It should be noted that owing to the presence of Lorentz force full vorticity is no longer conserved and in the case of a strong magnetic field magneto-Rossby waves tend to Alfvén wave solutions almost on all wavenumber range (Zaqarashvili et al. 2021). Magneto-Rossby waves are basic mechanism in variability of various objects in plasma astrophysics. Magneto-Rossby waves determine the large-scale dynamics of the Sun and stars (Hughes et al. 2007; Zaqarashvili et al. 2007, 2011; Dikpati et al. 2020; Mandal & Hanasoge 2020; Raphaldini et al. 2020), dynamics of magnetoactive atmospheres of exoplanets captured by tides from the host star (Cho 2008), flows in accretion discs of neutron stars (Inogamov & Sunyaev 2010). Despite the difficulty of observing Rossby waves in astrophysical plasma, they have recently been detected on the Sun (McIntosh et al. 2017; Zaqarashvili & Gurgenashvili 2018; Loeptien et al. 2018; Liang et al. 2019). We also note a number of important studies on the effect of Magneto-Rossby waves on solar seasons (Lou 2000; Dikpati et al. 2017, 2018) and space weather (Dikpati & McIntosh 2020; Dikpati, McIntosh & Wing 2021). In addition, magneto-Rossby waves play a key role in the occurrence of zonal flows in two-dimensional magnetohydrodynamic turbulence (Tobias, Diamond & Hughes 2007; Zinyakov & Petrosyan 2018, 2020). The weakly non-linear theory of magneto-Rossby waves in the quasi-two-dimensional MHD single-layer model has been developed in Klimachkov & Petrosyan (2017a) and in the presence of large-scale compressibility in Klimachkov & Petrosyan (2018). In Fedotova et al. (2020), MHD shallow water approximation has been extended to the case of external vertical magnetic field and stratified flows. Linear and weakly non-linear theory of magneto-Rossby waves in a rotating stratified plasma in the quasi-two-dimensional MHD approximation in two-layer model has been developed. In Klimachkov & Petrosyan (2017a, 2018) and Fedotova et al. (2020), the phase matching conditions have been investigated and non-linear interactions of three magneto-Rossby waves have been found. In Raphaldini & Raupp (2015), Raphaldini et al. (2019, 2020) influence of non-linear dynamics of magneto-Rossby waves on cyclic nature of solar magnetic activity is studied. Non-linear interactions of magneto-Rossby waves are associated with the duration of the solar cycle. In Raphaldini et al. (2020) is shown that irregular transitions in wave amplitudes resemble the observed time series of solar activity. In Klimachkov & Petrosyan (2017a, 2018), Fedotova et al. (2020) magneto-Rossby waves have been investigated in the β-plane approximation for the Coriolis force. The β-plane approximation describes rotating spherical plasma flows in a local Cartesian coordinate system. In this case, the Coriolis parameter changes little with small changes in latitude and expands in a series up to the first order in latitude.


Another important consistency check regards the angular size distribution of the sources. Figure 6 shows the cumulative size distributions of the final catalogs combined together, in four flux density bins (yellow solid lines). Such distributions can be considered reliable only down to a flux-dependent minimum intrinsic size (see vertical gray lines), below which most of the sources cannot be reliably deconvolved and they are conventionally assigned Θ = 0. The observed distributions are compared with various realizations of the cumulative distribution function described by Eq. (6), obtained by varying either the function exponent q (left and right columns respectively) or the assumed median size – flux relations (see various black lines).The original function proposed by Windhorst et al. (1990) (Eq. (6) with q = 0.62, see left column) does provide a good approximation of the observed distributions, when assuming the original Θmed − S relation described by Eq. (7), only at flux densities S150 MHz≳10 mJy (see long-dashed lines). This is perhaps not surprising considering that this relation was calibrated at 1.4 GHz down to a few mJy fluxes. At the lowest flux densities (S150 MHz≲1 mJy) we need to assume a steepening of the parameter m (see Eq. (8)), to get a good match with observations (dotted line in the top left panel). This is consistent with what proposed for higher frequency deep surveys (as discussed earlier in this section). At intermediate fluxes (S150 MHz ~ 1−10) mJy, on the other hand, none of the discussed median size – flux relations can reproduce the observed size distribution (see second-row panel on the left). It is interesting to note, however, that if we assume a steeper exponent for the distribution function described by Eq. (7) (i.e., q = 0.80), we get a very good match with observations at all fluxes, when assuming a flux-dependent scaling factor (k = k(S); see Eq. (9)) for the Windhorst et al. (1990) median size – flux relation (black solid lines on the right). The median sizes derived from the T-RECS simulated catalogs (Bonaldi et al. 2019) also provide good results for q = 0.80 (dot-dashed lines on the right), except again at intermediate fluxes (S150 MHz ~ 1−10), where they show strong discrepancies with observations also in Fig. 5. This seems to indicate that the number density of extended radio galaxies in this flux density range is over-estimated in the T-RECS simulated catalogs.


In this work, we present well-calibrated optical photometric (−0.2 to +413 d), polarimetric (−2 to +31 d) and optical (−5 to +391 d), NIR (−5 to +22 d) spectroscopic studies of SN 2012au, based on data obtained using many observational facilities around the globe. Analysis based on our photometric observations suggests that SN 2012au appears to be one of the most luminous SNe Ib (MB, peak = −18.06 ± 0.12 mag), though fainter than the threshold limit of SLSNe I (M$_\mathrm{ g}\, \lt -$19.8 mag; Quimby et al. 2018). The MR, peak (∼ –18.67 ± 0.11 mag) of SN 2012au is brighter than the average values of SNe Ib and Ic, but closer to those reported for SNe Ic-BL (Drout et al. 2011). Similarly, the peak bolometric luminosity of SN 2012au (∼ [6.56 ± 0.70] × 1042 erg s−1) is higher than the mean peak luminosities of SNe Ib and Ic, but still lower than those of SNe Ic-BL (Lyman et al. 2016). Using the early bolometric light curve of SN 2012au, the estimated values of Mej, Ek, MNi, and T0 are ∼5.1 ± 0.7 M⊙, ∼ (4.8 ± 0.6) × 1051 erg, ∼0.27−0.30 M⊙, and ∼66.0 ± 9.4 d, respectively. These physical parameters of SN 2012au are close to those inferred for SN 2009jf (a bright SN Ib: Sahu et al. 2011) and – on average – larger than for classical SNe Ib/c but smaller than for some SNe Ic-BL. SN 2012au manifests larger Mej and MNi in comparison with most of the SNe IIb, Ib, and Ic, which may be the prime reason behind the luminous peak of SN 2012au, as seen in the case of SLSNe I (Nicholl et al. 2015). On the other hand, light-curve decline rates of SN 2012au (at phases ≥+40 d) in all the optical bands are shallower than typically observed in the case of SNe Ib and slow-decaying SLSNe I, and theoretically predicated for 56Co $\rightarrow \, ^{56}$fe decay. As SN 2012au exhibits comparatively larger Mej, a larger optical depth resulting in a larger diffusion time-scale (for the trapped energy to cross the outer envelope) could broaden the light curve. Therefore, high trapping of gamma-rays at late phases or higher opacity of massive ejecta are among the plausible interpretations for the modest luminosity decline rate of SN 2012au in comparison with other SNe Ib (Clocchiatti & Wheeler 1997). However, smoothly distributed circumstellar media up to a larger radius could be another possibility behind the late-time shallower decay rate for SN 2012au, but an absence of the CSMI in the late-time spectra ruled out this scenario (Milisavljevic et al. 2018). The late-time bolometric light curve of SN 2012au is better constrained by $L~\varpropto\, t^{-2}$, a conventional magnetic dipole equation. Hence for SN 2012au the shallower decay of the late-time light curve might be a potential indicator of a central engine powering source that is accelerating the inner ejecta.


Despite significant advances, the solar wind remains challenging to model (and importantly forecast) due to the large range of scales that need to be incorporated. It is well known that energy is injected into the corona at all scales, from magnetohydrodynamic (MHD) waves (Nutto et al. 2012; Van Doorsselaere et al. 2020) to flares and coronal mass ejections with global extent (Aschwanden et al. 2017; Green et al. 2018; Wyper et al. 2018). In addition to the range of spatial scales, heating events take place on a variety of timescales (Hollweg 1973; Viall & Klimchuk 2017). Putting aside large-scale eruptions so as to focus on the quasi-steady input of energy into the corona, it is possible to distinguish three broad magnetic configurations of the solar atmosphere for modelling purposes. These are the quiet Sun (Danilovic et al. 2010; Rempel 2014), active or enhanced regions (Carlsson et al. 2016; Chen et al. 2021), and coronal holes (Wójcik et al. 2019). In each of these configurations, energy is channelled from the convection into the low corona. Coronal holes are the dominant source of the solar wind in the heliosphere (Cranmer et al. 2017; Stansby et al. 2021), typically producing the fast solar wind (McComas et al. 2008; Ebert et al. 2009; Macneil et al. 2020a; Wang 2020). The magnetic field configuration of a coronal hole is relatively simple, compared with the quiet Sun and active regions, given that the field is principally open to the solar wind (Lowder et al. 2017; Hofmeister et al. 2019). However, there are still a range of dynamic processes taking place, such as the braiding of magnetic field lines (Wedemeyer-Böhm et al. 2012; Wedemeyer et al. 2013; Huang et al. 2018) and the emergence of new magnetic flux (Murray et al. 2009). These can trigger the formation of jets (Shen et al. 2017; Yang et al. 2017) and other phenomena, which are then observed as spicules (Martínez-Sykora et al. 2017; Bose et al. 2021) or fibrils (Hansteen et al. 2006; Leenaarts et al. 2015).


While the Loison et al. (2015) CH3C3N model corroborates the upper atmospheric abundance of C4H3N inferred by Vuitton et al. (2007) from the T5 INMS measurements (a factor of 2 higher than those derived from T40 in Vuitton et al. 2019), a large disparity between the photochemical models (and within the ensemble of models produced by Loison et al. 2015) arises in the lower atmosphere due to the poorly constrained C4H3N branching ratios and reaction rate coefficients at temperatures appropriate for Titan. Aside from electron dissociative recombination of C4H3NH+ (Vuitton et al. 2007), neutral production of CH3C3N can occur in a few ways, as found through crossed beam experiments and theoretical and photochemical modeling studies (Huang et al. 1999; Balucani et al. 2000; Zhu et al. 2003; Wang et al. 2006; Loison et al. 2015). First, through the reactions of larger hydrocarbons with CN radicals,
1

2

Similarly, with CCN radicals following their formation through H + HCCN (Takayanagi et al. 1998; Osamura & Petrie 2004) and subsequent reactions with ethylene,
3

or through the chain beginning with acetylene,
4

While both reactions (3) and (4) are found to be equally likely by Loison et al. (2015), the production of CCN via H + HCCN is not well constrained, and the synthesis of CH3C3N through CN radicals (Equations (1) and (2)) are not included in their photochemical model. Additionally, cyanoallene may be produced through reactions (1)–(4) instead of (or in addition to) methylcyanoacetylene. CH3C3N itself may form the protonated species, C4H3NH+, through reactions with the HCNH+ and C2H5+ ions producing HCN and C2H4, respectively (Vuitton et al. 2007). The other mechanism for forming C4H3NH+ is through the combination of HCN and l-C3H3+, though the reaction rate coefficient for this reaction and the abundance of l-C3H3+ are unknown (Vuitton et al. 2007). As such, the production and loss pathways for both C4H3NH+ and CH3C3N require further investigation.


Solar wind categorization schemes rely on different solar wind properties to identify the solar wind type and adopt mainly one of the following two approaches: (1) Composition-based schemes exploit the oxygen and carbon charge-state composition of the solar wind (Zhao & Fisk 2010; von Steiger et al. 2000). Based on the assumption that the charge states observed in the solar wind are determined in the solar corona and are not significantly changed thereafter, lower (higher) charge states are associated with source regions of the respective solar wind stream with comparatively low (high) electron temperatures. Because the charge state is not expected to change during the travel time of the solar wind, composition-based criteria are well suited to identify different solar source regions. Transport effects due to, for instance, compression regions in stream interaction regions and wave-particle interaction are not directly reflected in the charge-state composition. Stream interaction regions tend to be characterized by a (gradual or abrupt) transition in the oxygen charge-state compositions. From the charge-state information alone, stream interaction regions cannot be uniquely identified, and their solar source regions cannot be unambiguously determined without additional or context information. (2) Proton plasma properties provide an alternative to determine the solar wind type (Xu & Borovsky 2015; Camporeale et al. 2017). A clear advantage of this approach is that the required observables are available from more spacecraft. Unlike the charge-state composition, the proton speed, proton density, proton temperature, and magnetic field strength (and derived quantities such as the specific entropy and the Alfvén speed) are all susceptible to transport effects. In particular, these quantities show radial gradients throughout the heliosphere (Marsch et al. 1982; Bale et al. 2019; Kasper et al. 2019). Thus, a solar wind categorization based on threshold values for these quantities can be expected to depend on position. In particular, the solar wind proton temperature is not a tracer of the coronal (electron) temperature. The solar wind proton temperatures show the opposite effect (von Steiger et al. 2000): high solar wind proton temperatures are observed for coronal hole wind (which originates from comparatively cool coronal regions), while low solar wind proton temperatures appear in the slow solar wind (which likely originates in hot coronal regions). The solar wind proton temperature is probably strongly influenced by transport effects, in particular, by wave-particle interactions. In addition, proton temperature, proton density, and magnetic field strength all show characteristic variations in stream interaction regions. Solar wind categorization schemes based on proton plasma properties are therefore well suited to assess the effect of solar wind evolution during its travel time. However, these transport effects can blur the tracers of the solar origin of the solar wind. For approaches based on charge-state composition and on proton plasma, the respective threshold values are usually determined heuristically and vary in the literature. Mainly as a result of their availability, solar wind electron data (e.g., Lin et al. 1995; Wilson et al. 2018) are typically not considered for solar wind classifications, although their properties, for example, the electron temperature and the electron-proton collisional age, can be expected to be informative in this context. Future improvements on solar wind classification would most likely benefit considerably from including electron data. The collisional age (or Coulomb number) has been proposed as an ordering parameter for the solar wind in Kasper et al. (2008), Tracy et al. (2016), and Maruca et al. (2013). The collisional age can be interpreted as counting the number of 90°-equivalent collisions during the travel time from the Sun to the observing spacecraft. This notion of the collisional age relies on the simplifying assumption that the solar wind parameters are constant during the solar wind travel time. Maruca et al. (2013) introduced an improvement in the computation of the collisional age that takes into account that the underlying quantities are not constant during the travel time of the solar wind.


To investigate the particle precipitation or transport sites during the flare, we used HXR 25–50 keV and NoRH 17/34 GHz images. We chose the Pixon algorithm (Metcalf et al. 1996) for the RHESSI image reconstruction. The Pixon method is considered to be the most accurate algorithm (Hurford et al. 2002). The integration time for each image was 20 s. We utilized NoRH 5 s cadence intensity images (R+L) at 17 and 34 GHz. Figure 4 displays the HXR 25–50 keV (blue) and NoRH 17 GHz (red) contours overlaid on the AIA 1600 Å images at ∼23:51 UT. These images are used at the peak time of each of the bursts observed in the HXR 25–50 keV and microwave channels. Figure 4(a) shows a small filament rising at the flare site. We can easily identify the two legs of the filament (marked by N and S in the figure). Generally, the filaments are observed in the chromospheric images (e.g., Hα and AIA 304 Å). However, if the kink-unstable filament is heated during magnetic reconnection, it is often observed in the AIA 1600 Å channels (e.g., Kumar et al. 2012 and Kumar & Cho 2014). The locations of both the 25–50 keV sources and the 17 GHz source are almost at the quasi-circular ribbon. However, their centroids constructed at the 90% level of the peak intensity are not cospatial. We note that the 25–50 keV sources (centroid) are located close to the legs of the filament, whereas the 17 GHz source is located at the northern leg of the filament. It seems that these sources are at the footpoints of an underlying flare loop. To identify the location of the flare loop, we selected the AIA 94 Å hot channel image at 23:53 UT. In this image (panel b), we show the NoRH 34 GHz contours (yellow) overlaid on AIA 94 Å images at ∼23:53 UT. This is done to show the coronal loops associated with the eruption of the small filament. The overlying 34 GHz source is cospatial with the small loop located above the quasi-circular ribbon. This 34 GHz emission may be the evidence of trapped nonthermal electrons in the loop. The rising filament is heated during reconnection with the ambient fields and then decays into an untwisting jet, as seen in Figures 4(b)–(e). Furthermore, during ∼23:53–23:59 UT, we see the footpoint HXR sources. The NoRH 17 GHz sources cover the quasi-circular ribbon (Figures 4(c)–(d)) and could be the emission from the trapped electrons in the loop. This suggests that the bursts are most likely caused by the same population of nonthermal electrons with different energies.


The contribution to the submm background from LBGs is still poorly constrained; however, our data can finally address this question, since we now have a robust detection of the average 850 μm flux density of LBGs at three epochs, at least those with ultraviolet luminosities of L1700 ≈ 1029 erg s−1 Hz−1, characteristic of galaxies in our sample (Table 2). Using our average stacked 850 μm flux densities, we estimate surface brightness densities of 1700, 600, and 100 mJy deg−2 of LBGs at z ∼ 3, 4, and 5, respectively. By comparison, the total background at 850 μm inferred from COBE-FIRAS is 3.1–4.4 × 104 mJy deg−2 (Puget et al. 1996; Fixsen et al. 1998; Hauser & Dwek 2001; Lagache, Puget & Dole 2005). Summing these separate surface brightness densities together, we find that the LBGs with L1700 > 1029 erg s−1 Hz−1 in our z ∼ 3, 4, and 5 samples comprise around 6–8 per cent of the submm background at 850 μm (where the range of values simply reflects the uncertainty in the COBE-FIRAS result). However, the true contribution from LBG-like galaxies to the submm background will come from a wider range in redshift, not just from the rather narrow redshift slices we have sampled (see Fig. 1), and from sources falling out of our samples due to incompleteness. To determine the total (corrected) contribution from LBGs over 3  z  5, we assume that LBGs have a constant comoving number density (which is a reasonable assumption since the bright end of the luminosity function for LBGs shows little evolution over 3  z  5; Bouwens et al. 2007; McLure et al. 2009; Reddy & Steidel 2009). Starting with the z ∼ 3 sample (which is our most complete subsample of LBGs), we can integrate over the comoving volume element for this redshift range, scaling the LBG submm background contribution accordingly. We find that the total contribution to the submm background from LBGs over the redshift range 3  z  5 is likely to be closer to 14–20 per cent. This result is consistent with Webb et al. (2003), who estimated an upper limit to the contribution to the submm background from 1  z  5 of less than 20 per cent.