Once we have the completeness corrections, we can estimate the relative luminosity function of SNe Ia, which we show in Fig. 3. In the left panel we show the observed distribution of absolute magnitudes. The black histogram shows the full sample, and the red histogram shows the volume-limited sample; the dot--dashed blue histogram shows the distribution of the volume-limited LOSS sample (Li et al. 2011b). The volume-limited samples have a higher fraction of low-luminosity SNe Ia than the full magnitude-limited sample. We convert the ASAS-SN distributions into estimates of the true luminosity function using the V/Vmax method (Schmidt 1968; Huchra & Sargent 1973; Felten 1976). For each SN, we compute the maximum volume (Vmax) in which the SN could be recovered by a survey with a limiting magnitude mV = 16.8. This is an empirical limiting magnitude; this value produces a median value of V/Vmax close to 0.5, which is to be expected if sources uniformly populate the survey volume. We compute the relative luminosity function for each bin in absolute magnitude centred on M as
(2)
\begin{eqnarray*}
\Phi (M) = \sum _{i=1}^{N} \frac{1}{V_{M,i}} \times w_i \times (1+z_i),
\end{eqnarray*}
where the sum is over all the SNe within the bin. The weights $w$i correct for the incompleteness given the apparent peak magnitude of each SN, and the factor of (1 + $z$) accounts for time dilation. The results are shown in the right panel of Fig. 3. The black circles show the relative luminosity function computed from the full sample, the red squares show the results for the volume-limited sample, and the blue crosses show the control-time weighted counts from Li et al. (2011b). The luminosity functions are normalized to the bin at MV = −19. In this paper we are not aiming for an absolute rate calibration. The shape of the relative luminosity function is consistent with the volume-limited luminosity function presented in Li et al. (2011b). We fit a Schechter (1976) function
(3)
\begin{eqnarray*}
\phi (L) \propto \left(\frac{L}{L_*}\right)^{\alpha } \exp \left(-\frac{L}{L_*}\right)
\end{eqnarray*}
to the relative luminosity function of both the full and volume-limited samples, where α is the faint-end slope, and L* (alternatively M* in magnitude space) determines the ‘knee’ of luminosity function. Our fits are shown in Fig. 3 as dashed lines. We find (α, M*) corresponding to (1.3 ± 0.4, −18.1 ± 0.1) and (2.1 ± 0.3, −17.8 ± 0.1) for the full sample and the volume-limited sample, respectively.


The downsizing scenario is evident in several cases of galaxy evolution (Fontanot et al. 2009). In the case of ETGs at z ~ 0, one of the first pieces of observational evidence can be referred to the studies of Dressler et al. (1987), Faber et al. (1992) and Worthey et al. (1992), who found more massive elliptical galaxies to be more enriched in α-elements than less massive ones. These works suggested selective mass-losses, different initial mass functions (IMF) and/or different star-formation timescales as possible explanations for the high level of [α/Fe]. Subsequent studies found the same trend of [α/Fe] with mass (Carollo et al. e.g 1993; Davies et al. 1993; Bender et al. 1993; Thomas et al. 2005; 2010; McDermid et al. 2015), leading to the dominant interpretation that in more massive ETGs, the duration of star formation was substantially shorter than in less massive ones, with timescales short enough (e.g. 0.5 Gyr) to avoid the dilution of the α element abundance (produced by Type II supernovae) by the onset of Fe production by Type Ia supernovae. This is considered to be one of the main pieces of evidence for the shortening of star formation. The age of the ETG stellar populations at z ~ 0 also show evidence of downsizing, with more massive objects being older than less massive ones. These results have been derived both in clusters (Thomas et al. 2005, 2010; Nelan et al. 2005) and in the field (Heavens et al. 2004; Jimenez et al. 2007; Panter et al. 2007; see also Renzini 2006). Most of these studies are based on fitting individual spectral features with the Lick/IDS index approach (Burstein et al. 1984; Worthey et al. 1994) which allows us to mitigate the problem of the age-metallicity degeneracy (Graves & Schiavon 2008; Thomas et al. 2005, 2010; Johansson et al. 2012a; Worthey et al. 2013). However, more recently, other approaches based on the full-spectrum fitting have been developed (e.g. STARLIGHT, Cid Fernandes et al. 2005; VESPA, Tojeiro et al. 2009, 2013, and FIREFLY, Wilkinson et al. 2015), and applied to samples of ETGs at z ~ 0  (Jimenez et al. 2007; Conroy et al. 2014; McDermid et al. 2015). The results based on Lick indices are in general quite consistent with those of full spectral fitting within 10−30% (e.g. Conroy et al. 2014) and support the downsizing evolutionary pattern.


In stars, there are two components to describe the tidal interaction, equilibrium tides and dynamical tides. Equilibrium tides correspond to a large-scale hydrostatic adjustment of a body and the resulting flow due to the gravitational field of a given companion (Zahn 1966; Remus et al. 2012). It is usually employed in the framework of the constant time lag model (see Mignard 1979; Hut 1981; Eggleton et al. 1998; Bolmont et al. 2011, 2012), which allows a fast computation of the orbital evolution of the planet and works for all eccentricities (Hut 1981; Leconte et al. 2010). In this model, the dissipation of the kinetic energy of the equilibrium tide inside the star is often taken to be constant throughout the system evolution and calibrated on observations (Hansen 2010, 2012). While considering such a constant equilibrium tide dissipation is a sensible assumption, several studies have shown that this quantity might vary during the different phases of stellar evolution. For example, Zahn & Bouchet (1989) showed that the dissipation of the equilibrium tide by the turbulent friction in the convective envelope of late-type stars is strongest during their PMS. Using this theoretical framework, Villaver & Livio (2009, see also Verbunt & Phinney 1995 recalled that the variation of the semi-major axis of a planet induced by such friction can be expressed as a function of the ratio of the mass of the convective envelope to the total mass of the star, the ratio between the radius of the star and the orbital semi-major axis (to the power 8), and finally of a power of the ratio between the tidal period and the convective turnover timescale. This allows the loss of efficiency of tidal friction to be modelled for rapid tides (e.g. Zahn 1966; Goldreich & Keeley 1977). Because of the variations of these quantities during post-MS phases (e.g. Charbonnel et al. 2017), this could lead to a more efficient dissipation than during the MS. Finally, Mathis et al. (2016) demonstrated that the action of rotation on convection deeply modifies the turbulent friction it applies on the equilibrium tide. In the regime of fast rotation, which corresponds to the end of PMS and early MS phase, the friction is several orders of magnitude lower than in a model ignoring rotation. This may lead to a loss of efficiency of the dissipation of the equilibrium tide. This shows that care should be taken when assuming a calibrated constant dissipation of the equilibrium tide during the evolution of stars.


Using these quasar survey data, Clowes et al. (2013), Nadathur (2013), Einasto et al. (2014), and Park et al. (2015) found very large quasar groups and discussed the cosmological implications of the existence and properties of these extreme objects. Nadathur (2013), Einasto et al. (2014), and Park et al. (2015) pointed out that the cosmological interpretation by Clowes et al. (2013) of large quasar groups, which questions the validity of the cosmological assumption of homogeneity and isotropy, is misleading. They stressed the importance of a statistically precise analysis in order to draw conclusions on cosmological implications from the observation of one or a few extreme objects. Park et al. (2015) also emphasized that statistical comparison with cosmological simulations must be employed as well. The quasar survey data are also used for studies to constrain cosmological parameters (Han & Park 2015; Risaliti & Lusso 2015). Besides the studies directly related to cosmology, there are more studies exploring the clustering properties of quasars. Einasto et al. (2014) made a catalog of quasar groups with different linking lengths and examined their properties. They found that the characteristics of quasar groups such as number density, size, and richness, identified with linking lengths varying from 20 to 40 h−1 Mpc, are well correlated with those of galaxy superclusters. Therefore such quasar groups can be markers of galaxy superclusters. As a classical way to study clustering properties, correlation functions have been measured for quasars (AGNs in general) by many different groups (Shen et al. 2007b, 2009, 2013; Ross et al. 2009; Krumpe et al. 2010, 2012, 2015; Miyaji et al. 2011; Cappelluti et al. 2012; Richardson et al. 2012; Allevato et al. 2014; Eftekharzadeh et al. 2015). They have measured two-point cross-correlation functions (2PCCFs) between quasars and galaxies, and have found the typical mass of quasar-hosting dark matter halos (DMHs) and the dependence of the mass on quasar luminosity.


We refer to models for which the pressure P depends on the radial distance r from the central continuum source,
(2)
\begin{eqnarray*}
P(r)\propto r^{-s} \, ,
\end{eqnarray*}
as pressure-law models. In this work, we examine two limiting cases, s = 0 and s = 2, representing constant density and constant ionization parameter models, respectively. We here adopt a spherically symmetric BLR geometry spanning more than two decades in radial extent. This model is chosen for (i) its simplicity, and because (ii) we can compare our radial pressure-law models with the Local Optimally emitting Cloud model for this source presented by Korista & Goad (2001), which also adopts spherical symmetry. Here, we summarize the radial dependencies of various physical quantities for spherically symmetric pressure-law models; the derivations in this section follow Rees et al. (1989) and Goad et al. (1993). We make the simplifying assumption that the cloud temperature does not vary with radius; for solar composition, photoionization equilibrium is achieved at temperatures T ∼ 104 K across a wide range of ionization parameter (equation 5), and the gas temperature will therefore vary weakly with radius (e.g. Netzer 1990; Ilić, Kovačević & Popović 2009). For constant cloud temperatures, the cloud hydrogen gas density nH is proportional to the pressure, P, and so
(3)
\begin{eqnarray*}
n_\mathrm{H}(r)\propto r^{-s} \, .
\end{eqnarray*}
Thus, s = 0 corresponds to a constant nH throughout the BLR. The ionization parameter U is defined as
(4)
\begin{eqnarray*}
U(r)=\frac{Q_\mathrm{H}}{n_\mathrm{H}(r)4\pi r^2c} \, ,
\end{eqnarray*}
where QH is the number of hydrogen-ionizing photons emitted by the central continuum source per second. Thus, s = 2 corresponds to a constant ionization parameter model, since:
(5)
\begin{eqnarray*}
U(r)\propto r^{s-2} \, .
\end{eqnarray*}
The surface area per cloud, Ac, is proportional to $R_{\rm c}^2$, where Rc denotes the radius of a cloud. In general, Rc depends on the pressure P, and is therefore constant for s = 0. If we demand that the mass of each cloud is conserved as the clouds move radially outwards (i.e. clouds do not break up or coalesce within our region of interest), mass conservation implies that $R_c^3n_H={\rm constant}$. Thus, we obtain the relation
(6)
\begin{eqnarray*}
A_{\rm c}(r) \propto R_{\rm c}^2(r) \propto r^{2s/3} \, .
\end{eqnarray*}
The column density of each cloud, Ncol, depends on the gas density and cloud radius:
(7)
\begin{eqnarray*}
N_\mathrm{col}(r)\propto R_cn_\mathrm{H}\propto r^{-2s/3} \, .
\end{eqnarray*}
The above relations determine the local physical conditions, as parametrized by Ncol, nH and incident ionizing photon flux ΦH, at any radius in a spherically symmetric pressure-law BLR model. These conditions determine the local surface emissivity ε(r) of a BLR cloud at radius r (as determined via cloudy modeling; Section 3.1). The total luminosity for an emission line is then found by integrating over the distribution in cloud properties such that
(8)
\begin{eqnarray*}
L_{\mathrm{line}}=4\pi \int _{r_{\mathrm{in}}}^{r_{\mathrm{out}}}\epsilon (r)A_{\rm c}(r)n_{\rm c}(r)r^2{\rm d}r \, ,
\end{eqnarray*}
where rin  and rout  are the inner and outer BLR radii, respectively, Ac is the surface area of a single cloud, and nc is the local number density of clouds. As dust grains strongly absorb UV photons, routis chosen to approximately coincide with the distance at which dust grains can form and survive (≈140 light-days for NGC 5548).


Virtually all formulas that are currently used to estimate the merger time, τmer, are based on the idealised Chandraseckhar (1943) description of the deceleration caused by dynamical friction on a point mass (representing the satellite) travelling through an infinite, uniform, and collisionless medium (representing the host). They are mostly prescriptions inferred from the analytic or semi-analytic modelling of mergers (e.g. Lacey & Cole 1993; van den Bosch et al. 1999; Taffoni et al. 2003; Gan et al. 2010; Petts et al. 2015; Silva et al. 2016) or parametric equations tuned by direct comparison with the outcome of numerical simulations of collisions of live galaxy pairs (e.g. Boylan-Kolchin et al. 2008; Jiang et al. 2008; Just et al. 2011; McCavana et al. 2012; Villalobos et al. 2013), or both (e.g. Colpi et al. 1999). However, although several decades of studies of galactic mergers have allowed reaching a general consensus on what probably are the most relevant parameters in any process of this type, the extent of their impact is still a matter of debate. Factors such as continued mass losses due to tidal interactions, the drag force exerted by tidal debris, the re-accretion of some of this material onto the colliding galaxies, or the mutual tidal distortion of their internal structures are elements that introduce nontrivial uncertainties in any attempt to calculate τmer using analytical expressions. To these difficulties, which are caused by physical complexity, one must add the lack of a 100% standard methodology regarding the way mergers are tracked. The starting point of this work is precisely the definition of a new metric for calculating τmer, which is an obligatory first step to compare the outcomes of different experiments under equal conditions and derive universally valid formulas, in a form that is suitable for major mergers. With this aim, we have run a suite of nearly 600 high-resolution N-body simulations of isolated major mergers, with which we further explored the dependence of the merger time on a range of orbital parameters and on the mass-ratios, spins, and morphologies of the progenitors that are representative of such systems.


The numerical schemes in this work are essentially the same as those in Kuroda et al. (2016). Regarding the metric evolution, we evolve the standard BSSN variables (Shibata & Nakamura 1995; Baumgarte & Shapiro 1999; Marronetti et al. 2008) with a finite-difference scheme in space and with a Runge–Kutta method in time, both in fourth-order accuracy. The gauge is specified by the ‘1+log’ lapse and by the Gamma-driver-shift condition. Regarding the radiation-hydrodynamic evolution, the conservation equation $\Delta_\alpha T^{\alpha \beta }_{\rm (total)}=0$ is solved using the piecewise parabolic method (Colella & Woodward 1984; Hawke, Löffler & Nerozzi 2005). $T^{\alpha \beta }_{\rm (total)}$ is the total stress-energy tensor,
(1)\begin{equation*}T_{\rm (total)}^{\alpha \beta } = T_{\rm (fluid)}^{\alpha \beta } +\int \mathrm{ d}\varepsilon \sum _{\nu \in \nu _e, \bar{\nu }_e, \nu _x}T_{(\nu , \varepsilon )}^{\alpha \beta }, \end{equation*}
where $T_{\rm (fluid)}^{\alpha \beta }$ and $T_{(\nu , \varepsilon )}^{\alpha \beta }$ are the stress-energy tensor of the fluid and the neutrino radiation field, respectively. We consider three-flavour of neutrinos ($\nu \in \nu _e, \bar{\nu }_e, \nu _x$) with νx denoting heavy-lepton neutrinos (i.e. νμ, ντ, and their antiparticles). ϵ represents the neutrino energy measured in the comoving frame which logarithmically covers from 1 to 300 MeV with 12 energy bins. Employing an M1 analytical closure scheme (Shibata et al. 2011), we solve spectral neutrino transport of the radiation energy and momentum, based on the truncated moment formalism (e.g. Kuroda et al. 2016; Roberts et al. 2016; Ott et al. 2018). We include the gravitational red- and Doppler-shift terms to follow the neutrino radiation field in highly curved space–time around BH. Regarding neutrino opacities, the standard weak interaction set in Bruenn (1985) plus nucleon–nucleon bremsstrahlung (Hannestad & Raffelt 1998) is taken into account.


One of the first and simplest proposed Friedmann–Robertson–Walker (FRW) cosmological model is the Λ cold dark matter (ΛCDM) universe, which involves Einstein's cosmological constant Λ. This standard model of cosmology, which is also referred to as the concordance model, assumes that the total energy density ρ of the universe is made up of three components, namely matter ρm (baryonic and dark matter), radiation ρr, and dark energy or vacuum energy ρΛ, which produces the necessary gravitational repulsion. In this model, dark energy which has an equation of state (EOS) ωΛ = pΛ/ρΛ = −1, is a property of the space itself and its density ρΛ = −pΛ = Λc4/8πG is constant, such that as the universe expands the constant vacuum energy density will eventually exceed the matter density of the universe which is ever decreasing. The spatially flat ΛCDM model dominated by vacuum energy with ΩΛ ∼ 0.70, with the rest of the energy density being in the form of non-relativistic cold dark matter with Ωm ∼ 0.25 and non-relativistic baryonic matter with Ωb ∼ 0.05, fits observational data reasonably well (Riess et al. 1998; Permutter et al. 1999; Knop et al. 2003; Riess et al. 2004). However, the main problem in this model is the huge difference of about 10120 orders of magnitude between the observed value of the cosmological constant and the one predicted from quantum field theory; known as the cosmological constant problem (Weinberg 1989). Another issue is the so called coincidence problem which expresses the fact that although in this model the matter and dark energy components scale differently with redshift during the evolution of the universe, both components today have comparable energy densities, and it is unclear why we happen to live in this narrow window of time. Besides these main issues, there are other inherent problems faced by the ΛCDM, some of which arose as a result of recent observations that are in disagreement with the model's predictions. For example, in order to account for the general isotropy of the cosmic microwave background (CMB), the standard model invokes an early period of inflationary expansion (Kazanas 1980; Guth 1981; Linde 1982). However, the latest observations by Planck (Planck Collaboration XXIII 2003) indicate that there may be some problems with such an inflationary scenario (Ijjas, Steinhardt & Loeb 2013; Guth, Kaiser & Nomura 2014). It was partly due to these issues of the standard ΛCDM, that during the last decade several alternative dark energy models have been proposed and tested with observations. In these models the dark energy density component ρde is not constant and in most cases ωde = pde/ρde depends on time, redshift, or scale factor. For example in some of these so called dynamical dark energy models, late time inflation is achieved using a variable cosmological term Λ(t) (Ray et al. 2011; Basilakos 2015) sometimes taken in conjunction with a time dependent gravitational constant G(t) (Ray, Mukhopadhyay & Dutta Choudhury 2007; Ibotombi Singh, Bembem Devi & Surendra Singh 2013). Other sources of dark energy include scalar fields such as quintessence (Peebles & Ratra 2003), K-essence (Armendariz-Picon et al. 2001) and phantom fields (Singh, Sami & Dadhich 2003). An alternative approach to the dark energy problem relies on the modification of Einstein's theory itself such that in these alternative theories of gravity, cosmic acceleration is not provided solely by the matter side Tμν of the field equations, but also by the geometry of spacetime. These theories include the scalar-tensor theory with non-minimally coupled scalar fields (Barrow & Parsons 1997; Bertolami & Martins 2000), f(R) theory (Tsujikawa 2008), conformal Weyl gravity (Mannheim 2000) and higher dimensional theories such as the Randall–Sundrum (RS) braneworld model (Randall & Sundrum 1999), and the braneworld model of Dvali–Gabadadze–Porrati (DGP) (Dvali, Gabadadze & Porrati 2000). Over the last few years considerable interest has been shown in the simple FRW linearly expanding (coasting) model in Einstein's theory with a(t) ∝ t, H(z) = H0(1 + z). Like the ΛCDM the total energy density and pressure in this model are expressed in terms of matter, radiation and dark energy components, such that p = ωρ with ρ = ρm + ρr + ρde and p = pr + pde (since pm ≈ 0), but it includes the added assumption ω = −1/3, i.e. the cosmic fluid acting as the source has zero active gravitational mass. So this would definitely exclude a cosmological constant as the source of the dark energy component in this case. The model was first discussed by Kolb (1989) who referred to this zero active mass cosmic fluid as ‘K-matter’. Interest in this model has been revived recently after it was noted (Melia 2003) that in the standard model the radius of the gravitational horizon Rh(t0) (also known as the Hubble radius) is equal to the distance ct0 that light has travelled since the big bang, with t0 being the current age of the universe. In the ΛCDM this equality is a peculiar coincidence because it just happens at the present time t0. It was then proposed (Melia 2007, 2009; Melia & Shevchuk 2012) that this equality may not be a coincidence at all, and should be satisfied at all cosmic time t. This was done by an application of Birkhoff's theorem and its corollary, which for a flat universe allows the identification of the Hubble radius Rh with the gravitational radius Rh = 2GM/c2, given in terms of the Misner–Sharp mass $M = (4\pi /3)R_{{\rm h}}^{3}(\rho /c^2)$ (Misner & Sharp 1964). The added assumption of a zero active gravitational mass ρ + 3p = 0 implies (Melia & Shevchuk 2012) that Rh = ct or H = 1/t for any cosmic time t. This linear model became known as the Rh = ct universe. Unlike the ΛCDM/ωCDM which contains at least the three parameters H0, Ωm and ωde, the Rh = ct model depends only on the sole parameter H0, so that for example the luminosity distance used to fit Type Ia supernova data (Melia 2009) is given by the simple expression dL = (1 + z)Rh(t0)ln (1 + z). Also while the ΛCDM would need inflation to circumvent the well-known horizon problem, the Rh = ct universe does not require inflation. One should also point out that the condition Rh = ct is also satisfied by other linear models such as the Milne universe (Milne 1933), which however has been refuted by observations. Unlike the Rh = ct model discussed here, the Milne universe is empty (ρ = 0) and with a negative spatial curvature (k = −1). As a result of these properties its luminosity distance is given by $d_{L}^{\rm {Milne}} = R_{{\rm h}}(t_0)(1 + z)\sinh [\ln (1+z)]$, and it was shown that this is not consistent with observational data (Melia & Shevchuk 2012). In the last few years the Rh = ct universe received a lot of attention when it was shown (Melia & Maier 2013; Wei, Wu & Melia 2013, 2014a, 2014b, 2015; Melia, Wei & Wu 2015) that it is actually favoured over the standard ΛCDM (and its variant ωCDM with ω ≠ −1) by most observational data. This claim has been contested by Bilicki & Seikel (2012) and Shafer (2015) who argued that measurement of H(z) as a function of redshift and the analysis of Type Ia supernovae favoured the ΛCDM over the Rh = ct universe. However, this was later contested by Melia & McClintock (2015) who showed that the Rh = ct was still favoured when using model-independent measurements that are not biased towards a specific model. Others (see for example van Oirschot, Kwan & Lewis 2010; Lewis & van Oirschot 2012; Mitra 2014) have also criticized the model itself, particularly the validity of the EOS ω = −1/3 (Lewis 2013). These and other criticisms have been addressed by Bikwa, Melia & Shevchuk (2012); Melia (2012) (see also Melia 2015 and references therein.) As pointed out above the Rh = ct model would still require a dark energy component ρde, albeit not in the form of a cosmological constant. So the obvious question at this point would be: what are the possible sources for this component that together with the matter and radiation components will give the required total EOS, ω = −1/3? The purpose of this paper is to answer this question by discussing the various possible sources of dark energy that are consistent with this EOS. Since the radiation component ρr at the present time t0 is insignificant (at least for the ΛCDM with which this model has been compared) we assume that the total energy density ρ = ρde + ρm and the total pressure p = pde (pm ≈ 0), as is normally done in the other alternative dynamical dark energy models found in the literature. So in the next three sections we examine three possibilities for the source of dark energy in the Rh = ct model, namely a variable cosmological term Λ(t), a non-minimally coupled scalar field in Brans–Dicke theory which is equivalent to a variable gravitational constant G(t), and finally quintessence represented by a minimally coupled scalar field ϕ. We show that although the first two sources are consistent with the model, they are both unphysical, which leaves the third source of quintessence as the viable source of dark energy in the Rh = ct universe. Results are then discussed in the Conclusion. Unless otherwise noted we use units such that G = c = 1.


The terrestrial magnetosheath, downstream of the bow shock generated by the interaction between the supersonic solar wind and the Earth’s magnetosphere, is representative of turbulence downstream of collisionless plasma shock in the universe. Usually, the downstream sheath regions have larger density, temperature, mean magnetic intensity, and compressibility, and higher plasma β, as compared to the upstream solar wind (Sahraoui et al. 2006; Alexandrova 2008; Hadid et al. 2015; Huang et al. 2016, 2017). Abundant unstable kinetic wave modes and multiscale coherent structures exist in the magnetosheath. For instance, mirror mode, ion cyclotron wave, kinetic Alfvén wave, and so forth are routinely observed (Sahraoui et al. 2006; Lucek et al. 2005; Alexandrova 2008; Zhao et al. 2018; Zhang et al. 2018). Hence, the turbulent energy in the kinetic range might not only consist of energy cascading from larger scales, but also comprise direct injection of energy due to local instability at these scales. This raises an interesting question: what are the differences and similarities of turbulence intermittency between solar wind and magnetosheath? This is one goal of our research. When going down to the ion scales, PSD(δB) steepens, while PSD(δE) is relatively flattened (Chen & Boldyrev 2017; Matteini et al. 2017). The coexistence of steep PSD(B) and flat PSD(E) is caused by the ion demagnetization, which gives rise to the Hall term and/or quasi-electrostatic field (Cramer 2001; Schekochihin et al. 2009). If monofractal scaling is a common phenomenon for magnetic fluctuations at kinetic scales both in the solar wind and in the magnetosheath, what would happen to the electric field intermittent fluctuations? is the electric field intermittency monofractal as well, or is it totally different from the magnetic one? Studying the above questions would help us better understand the characteristics and nature of turbulence at kinetic scales, which is more electrostatic than that at magnetohydrodynamic (MHD) scales.


The large differences noted in Figure 9 are surely related to the different approximations underlying both the complex hydrodynamical simulations of ILLUSTRIS-1 and our simplified approach. As recently discussed by Chua et al. (2017), the distribution of noncollisional matter (i.e., stars and DM) can effectively cool down in the cores of galaxies formed in the ancient epochs, at variance with those born more recently. This provides them a much better chance of survival, or at least stronger resilience against potentially disruptive interactions that they would surely undergo throughout the cluster environment. A key role is also played by the host cluster itself, because the mass profile of this latter becomes steeper and steeper in the core due to dynamical relaxation. Consequently, the environment gets denser with time; this trend is found to be quantifiable in terms of the number of dynamical times elapsed since the cluster formation (e.g., Jiang & van den Bosch 2016). Therefore, each galaxy has to attain a sufficiently cold core in order to first survive in the cluster environment and then grow by accretion of matter from the ICM and mergers with other substructures. Given these preliminary considerations, which explain the evaporation of galaxies while falling into more massive halos, the inclusion of the BM physics in this context is crucial, and its effects are still not fully explored. Based on a systematic comparison of the full-physics and DM-only ILLUSTRIS suites, Chua et al. (2017) claimed that galactic winds and photoionization from UV radiation may effectively inhibit mass aggregation, at least at the low-mass end and in certain regimes of redshifts (see also Despali & Vegetti 2017 for similar reasoning for DM subhalos of intermediate masses inside ILLUSTRIS and EAGLE simulations; Schaye et al. 2015). In particular, photometric data inside ILLUSTRIS are bound to whether stellar particles, i.e., stellar populations, are found inside a bound subhalo. Recalling that in ILLUSTRIS-1, the average gas cell mass is 8.86 × 105 M⊙, from which integrated stellar populations might stem, and that a prescription for subgrid physics is always needed to follow up the integrated properties of all kinds of particles, these could be a substantial limit on state-of-the-art hydrodynamical simulations in describing photometry, especially for galaxies inside more recently formed DM halos of lower masses. On the other side, our numerical approach, at the basis of the buildup of samples inside clusters, provides a population of galaxies that should reflect the observational evidence of the mass–radius relation at z ∼ 0 (Chiosi et al. 2012). If observed galaxies are merely those that survived all disrupting interactions they underwent, then at least a minimum fraction of the sample (not as close to zero as the one of ILLUSTRIS-1) should have formed in an evidently large range of look-back time. However, acknowledging that our approach might not be the most refined one, we feel that the best interpretation might be a middle ground between the two models.


While previous studies mainly focused on the optical-UV properties of a MAD in RLAGNs, for the first time we try to investigate their X-ray properties in this work. The origin of X-ray emission in RLAGNs is still under debate, which may come from a corona, jet, or both. In observations, there is a big difference between the X-ray properties of radio-quiet AGNs (RQAGNs) and RLAGNs. Firstly, the average X-ray flux in RLAGNs is found to be 2–3 times higher than that in RQAGNs (e.g., Zamorani et al. 1981; Wilkes & Elvis 1987; Li & Gu 2021). Secondly, Laor et al. (1997) reported that RLAGNs have harder 2–10 kev X-ray spectra than RQAGNs by compiling a sample of 23 quasars observed with ROSAT, which was subsequently confirmed by Shang et al. (2011) with a larger sample. Comparing the X-ray spectrum of 3CRR quasars and that of radio-quiet quasars, Zhou & Gu (2020) also gave a similar result. In addition, the X-ray reflection features of RLAGNs are weaker than those of RQAGNs (Wozniak et al. 1998). All of these results seem to indicate that the contribution of a jet to X-ray spectra cannot be neglected. However, several recent works suggested a totally different result. First, the slope of LUV − LX is found to be consistent for RLAGNs and RQAGNs (Zhu et al. 2020, 2021; Li & Gu 2021). Second, Gupta et al. (2018, 2020) discovered that the distributions of X-ray photon spectral indices between RLAGNs and their radio-quiet counterpart are very similar (see Zhu et al. 2021 either). This opposite conclusion may be due to the effect of sample selection. The sample of Gupta et al. (2018, 2020) was X-ray selected (and optically selected for the sample of Zhu et al. 2021), which may lead to the radio jet power being very feeble compared to the bolometric luminosity in most of the RLAGNs. These weakly jetted RLAGNs can therefore have different X-ray photon indices compared to the strong jetted RLAGNs, such as the 3CRR quasars of Zhou & Gu (2020). P15 also indicated that the weakly jetted RLAGNs have similar αEUV as RQAGNs. However, interestingly, Markoff et al. (2005) demonstrated that both the corona model and the jet model can fit the X-ray data of some Galactic X-ray binaries well and that the jet base may be subsumed to corona in some ways. The 3CRR quasars are low frequency radio selected and have a strong jet on a large scale. However, it is still unclear whether all the objects with a strong jet harbor a MAD, or just containing MAD when jet is firstly launching millions of years ago. We focus on the RLAGNs with an EUV deficit in this work, which should possess a MAD in the inner disk region as suggested by P15. The presence of a MAD surrounding the black hole may bring a remarkable difference to the X-ray emissions since the structure of disk-corona greatly changes in the case of MAD (e.g., Tchekhovskoy et al. 2011; McKinney et al. 2012; White et al. 2019). In theory, it has been suggested that X-ray emission increases when an advection-dominated accretion flow (ADAF) becomes a MAD in its inner region (Xie & Zdziarski 2019). Nevertheless, how MAD affects the disk-corona corresponding to the X-ray emission of quasars is still an open issue. This work can constrain a future theoretical model for MAD in RLAGNs.


Here we report the X-ray spectral and timing analysis of the joint XMM–Newton and NuSTAR observations of an IRAS 04416+1215, a nearby (z = 0.0889; Boller et al. 1992) hyper-Eddington AGN. The source is part of a XMM–Newton/NuSTAR campaign that aims to constrain the broad-band X-ray properties of eight super-Eddington AGN from the best sample of bona fide super-Eddington sources available, i.e. super-Eddington accreting massive black holes (SEAMBHs; Du et al. 2014, 2015; Wang et al. 2014) that contains exclusively objects with black hole masses estimated from reverberation mapping. In this campaign we are carrying out to study the broad-band X-ray properties of super-Eddington AGN, all the sources have new NuSTAR observations performed simultaneously with XMM–Newton or Swift-X-ray Telescope (XRT). IRAS 04416+1215 has bolometric luminosity $\log (L_{\rm bol}/\rm erg\, s^{-1})=47.55$, according to Castelló-Mor, Netzer & Kaspi (2016), and $\log (L_{\rm bol}/\rm erg\, s^{-1})=45.52$, according to Liu et al. (2021). The former estimate is computed using, for the SED fitting procedure, the Slone & Netzer (2012) code, including the comparison of the observed SED with various combinations of disc SEDs covering the range of mass, accretion rate, spins, and taking into account the correction for intrinsic reddening and host galaxy contribution. In the latter estimate, the SED fitting is done using the more semplicistic templates from Krawczyk et al. (2013). The dimensionless accretion rate (Du et al. 2014) and black hole mass of the source are $\log (\dot{\mathscr {M}})$ = $2.63^{+0.16}_{-0.67}$ and log (MBH/M⊙) = $6.78^{+0.31}_{-0.06}$ with the reverberation mapping technique (Du et al. 2015), respectively, where $\dot{\mathscr {M}}\equiv \dot{M}_{\bullet }c^2/L_{\rm Edd}$, $\dot{M}_{\bullet }$ is mass accretion rates, c is speed of light, and LEdd is the Eddington luminosity. The dimensionless accretion rate is estimated by $\dot{\mathscr {M}}=20.1\, \ell _{44}^{3/2}M_7^{-2}$ from the Shakura–Sunyaev disc model (Du et al. 2015), where ℓ44 is the 5100 Å luminosity in units of $10^{44}\, {\rm erg\, s^{-1}}$ and $M_7=M_{\bullet }/10^7\, \mathrm{M}_{\odot }$. This approximation is valid for $\dot{\mathscr {M}}\lesssim 10^3$. To compute the Eddington ratio we assumed the bolometric luminosity value from Castelló-Mor et al. (2016), which is a better and more trustable estimate of the bolometric luminosity of the source, obtaining λEdd ∼ 472. This value is in perfect agreement with the dimensionless accretion rate from Du et al. (2014). However even assuming the luminosity from Liu et al. (2021), with which the value of the accretion rate would be λEdd ∼ 4.40, the source would remain a super-Eddington accreting AGN. IRAS 04416+1215 turned out to be the most peculiar of our sample, it is classified as NLS1 galaxy, showing narrow Hβ line [full width at half-maximum (FWHM) = $1670 \, \rm km \, \rm s^{-1}$; Moran, Halpern & Helfand 1996] and very broad [O iii] (FWHM = $1150 \, \rm km \, \rm s^{-1}$; Véron-Cetty, Véron & Gonçalves 2001) lines, which is typically found in sources accreting at such high Eddington accretion rates (Greene & Ho 2005; Ho 2009). The source shows a photon index in the Roentgen Satellite (ROSAT) (0.1–2.4 keV) energy band, of Γ = 2.96 ± 0.50 (Boller et al. 1992) and of $\Gamma =2.46^{+0.27}_{-0.26}$ for the rest-frame >2 keV spectrum, according to Liu et al. (2021).


Here we take the gravitational waveform of the nonrotating model with an observer viewed from the equatorial direction as an example for our HHT analysis. The gravitational waveforms of other models with different parameters are shown in the appendix (Figures B10–B20). Figure 6 (a) shows the simulated strains, followed by the wavelet spectra (b)–(c) and the stacked HHT spectra (d)–(e). The waveform starts from the core collapse and reaches the core bounce at t = 0 s. As this model is nonrotating and the proto-neutron star convection happens after the core bounce, the gravitational strain shows a loud bounce signal at time zero, and then it is followed by a low-frequency peak PNS oscillation, which could be caused by the g-mode oscillation although the nature remains controversial (Müller et al. 2013; Pan et al. 2021). Multiple low-frequency oscillation signals dominate the first ∼0.15 s, and they can be well resolved with the HHT. In comparison, the limitation of the wavelet spectrum prevents us to identify them. As the frequency of the oscillation increases with time, another high-frequency mode oscillation signal appears with similar strength as the peak PNS mode. The low-frequency PNS oscillation vanishes at t ∼ 0.5 s. Then, interestingly, it followed by the quadruple radiation of the standing accretion shock instability (SASI Blondin et al. 2003) signal, which was suggested to be seen in the GW data (Kuroda et al. 2016; Pan et al. 2021). The detailed waveform of the SASI signal is model dependent, but its time-frequency property has been shown in a few analysis either with the spectrogram or the HHT (see, e.g., Andresen et al. 2017; Mezzacappa et al. 2020; Takeda et al. 2021). In our analysis, the SASI signal can be seen on both the wavelet and the HHT spectra. The wavelet and the HHT results for other simulated CCSN events are shown in the Appendix B. The Hilbert spectra of all the data sets show much better resolution than the wavelet spectra.


With the availability of such a wide range of photo-z codes and methods, comparisons of various implementations have been performed (Hildebrandt et al. 2010; Abdalla et al. 2011; Sanchez et al. 2014). No obvious best photo-z code was named since each code displays different strengths depending on the metrics used. The focus of recent photo-z analyses has turned to improving error estimation (Oyaizu et al. 2008; Hoyle et al. 2015; Wittman, Bhaskar & Tobin 2016), the use of new statistical techniques (Lima et al. 2008; Zitlau et al. 2016), improving existing algorithms (Cavuoti et al. 2015; Sadeh et al. 2016), and the addition of extra input information to get more precise and accurate photo-z’s. With regards to the inclusion of extra information, a recent example in template methods includes using surface brightness as a prior in spectral energy distribution templates (Kurtz et al. 2007; Stabenau, Connolly & Jain 2008), motivated by the knowledge of surface brightness dimming (1 + z)4. In empirical methods this application is more straightforward, since algorithms are constructed such that it is not difficult to add extra input parameters. For example, Collister & Lahav (2004) and Wadadekar (2005) demonstrated that by including the 50 and 90 per cent Petrosian flux radii (RP50, RP90), the photometric redshift root-mean-square errors improve by 3 and 15 per cent respectively for the SDSS main galaxy sample. Tagliaferri et al. (2003) used Petrosian fluxes and radii in their work on galaxies from the SDSS early data release and calculated robust errors decreasing as much as 24 per cent. Meanwhile, Vince & Csabai (2007) included the concentration of galaxy light profiles in their study and reported that the root-mean-square error of photo-z’s on SDSS galaxies improved by 3 per cent. Wray & Gunn (2008) included surface brightness and the Sérsic index, and found improvements in variance when compared to other template fitting methods applied to the SDSS main galaxy sample previously.


Comparison of the DAXSS two temperature (2T) model fits to DEM estimates from other instruments could help validate the DAXSS simple modeling approach. Moore et al. (2018) shows that the X123 response is primarily over the temperature range of 1 MK to 10 MK, so this comparison needs to be done over a similar range, such as is accessible with using DEMs derived from extreme ultraviolet (EUV) emissions. One of the comparisons is with the DEM derived with SDO (Pesnell et al. 2012) EVE (Hock et al. 2012; Woods et al. 2012) solar EUV spectral irradiance data in the 6–37 nm range. The derivation of EVE-based DEM estimates is being developed for improving the X-ray ultraviolet Photometer System (XPS) data processing for the Solar Radiation and Climate Experiment (SORCE) (Woods et al. 2005a) and Thermosphere, Ionosphere, Mesosphere, Energetics, and Dynamics (Woods et al. 2005b) missions. The primary DEMs for this analysis is the quiet Sun (QS) and active region (AR) DEMs as needed for estimating the daily variations of the solar XUV spectral irradiance for the XPS Level 4 product (Woods et al. 2008). The Fe viii to Fe xvi lines in the SDO EVE spectra were initially used to estimate the DEMs for the reference QS and AR spectra derived with EVE data between 2010 and 2013 using the technique described by Schonfeld et al. (2017). It was then found that fitting with DEM Gaussian profiles with logarithmic temperature (K, log(T)) peaks every 0.2 and with Gaussian width of 0.42 in log(T) (FWHM of 1.0) provided more robust solutions for the DEM (a similar technique was described by Warren et al. 2013). Furthermore, fitting just specific Fe lines was providing low irradiance estimates in the 6–15 nm range. Better spectral model values for this range was found when fitting the EVE spectra over the ranges of 10–14 nm and 26–30 nm. The DEM estimate using just the rocket EVE spectral data flown with DAXSS on 2018 June 18 and the DEM based on combining the QS DEM and the AR DEM with an AR scaling factor of 0.00806 are shown in Figure 12. This scaling factor for the AR EM was determined as the best fit for the DAXSS spectral irradiance. The conversion of the EVE-based DEM (

) to EM (cm−3) for comparison to DAXSS 2T model solution is the multiplication by the solar hemisphere area (

) and by the temperature bin size (0.23 * temperature in K).


The intrinsic jet opening angles can be calculated as tan (αint/2) = tan (αapp/2) sin θ, where θ is the viewing angle to the jet axis. The latter, as well as the bulk Lorentz factor Γ, can be derived from apparent jet speed and Doppler factor using the following relations:

\begin{eqnarray*}
\theta = \arctan \frac{2\beta _{\rm app}}{\beta ^2_{\rm app}+\delta _{\rm var}^2-1}\,, \quad \Gamma = \frac{\beta _{\rm app}^2+\delta _{\rm var}^2+1}{2\delta _{\rm var}}\,.
\end{eqnarray*}
For βapp and δvar we used the fastest measured radial, non-accelerating apparent jet speed from the MOJAVE kinematic analysis (Lister et al. 2016) and the variability Doppler factor from the Metsähovi AGN monitoring programme (Hovatta et al. 2009), respectively. The corresponding overlap of the programmes comprises 55 sources, which are all members of the MOJAVE-1 sample. Variability Doppler-factors for 10 more MOJAVE-1 sources were measured within the F-GAMMA programme (Liodakis et al. 2017). The intrinsic opening angles calculated for the 65 sources range from 0$_{.}^{\circ}$1 to 9$_{.}^{\circ}$4, with a median of 1$_{.}^{\circ}$3, reflecting a very high degree of jet collimation. The intrinsic opening angles show an inverse dependence on Lorentz factor (Fig. 13), as predicted by hydrodynamical (Blandford & Königl 1979) and magnetic acceleration models (Komissarov et al. 2007) of relativistic jets. The median value of the product3 ρ = αintΓ is 0.35 rad, close to earlier estimates derived both from observations (Jorstad et al. 2005; Pushkarev et al. 2009) and from a statistical model approach (Clausen-Brown et al. 2013). The variability Doppler factors derived from variability can be underestimated due to a limited cadence of the observations. In this case, the intrinsic opening angle estimates would be smaller, while Lorentz factors would be higher if $\delta >(\beta _{\rm app}^2+1)^{1/2}$ and smaller otherwise, implying that if the variability Doppler factors are essentially underestimated, the majority of points in Fig. 13 would move downward and to the right.


There have been rather strong claims of AGN QPOs in different bands of the electromagnetic spectrum, ranging from minutes through days through months and years (e.g. Gierliński et al. 2008; Lachowicz et al. 2009; Gupta, Srivastava & Wiita 2009; Gupta et al. 2018, 2019; King et al. 2013; Gupta 2014, 2018; Ackermann et al. 2015; Pan et al. 2016; Zhou et al. 2018; Bhatta 2019; and references therein). However, many of the claimed QPOs, particularly those made earlier, were marginal detections (Gupta 2014), lasting only a few cycles, and the originally quoted statistical significances are probably overestimates (Gupta 2014; Covino, Sandrinelli & Treves 2019). Among the better recent claims of QPOs in the gamma-ray band are of ∼34.5 d in the blazar PKS 2247–131 (Zhou et al. 2018) and of ∼71 d in the blazar B2 1520+31 (Gupta et al. 2019) found as part of a continuing analysis of blazar Fermi–LAT observations. A recent claim of a ∼44 d optical band QPO in the narrow-line Seyfert 1 galaxy KIC 9650712 from densely sampled Kepler data has been made by Smith et al. (2018); it was supported by an independent analysis of the same data, indicating a QPO contribution at 52 ± 2 d (Phillipson et al. 2020). Some possibly related QPOs of a few hundred days in two widely separated bands have been reported (Sandrinelli et al. 2016a; Sandrinelli, Covino & Treves 2016b; Sandrinelli et al. 2017). However, an analysis of the Fermi–LAT and aperture photometry light curves by Covino et al. (2019) argued that some multiwaveband QPOs, along with many earlier claims of gamma-ray QPOs, are not significant. Among the gamma-ray QPOs with month-like periods, none showed simultaneous oscillations in a different wavebands. Evidence for related QPOs in multiple wavebands was observed in PG 1553+113, where a QPO was detected in the 0.1–300 GeV and the optical waveband (Ackermann et al. 2015). The observed QPO had a dominant period of ∼754 d and the source showed strong inter-waveband cross-correlations.


In order to recover the intrinsic SED, we employ the values for interstellar extinction E(B − V) from Schlafly & Finkbeiner (2011), collected on NASA's Infrared Science Archive (IRSA) web pages, to deredden the spectra. We use the relations derived by Cardelli, Clayton & Mathis (1989) as implemented in iraf, and a standard value for the ratio of the total to the selective extinction R(V) = A(V)/E(B − V) = 3.1. The corresponding values are available as the average extinction within a 2 × 2 deg2 field. This represents a severe limitation as it does not take into account potential small-scale variations in the dust structure, nor the actual amount of absorbing material in the line of sight to the target. Since CVs are intrinsically rather faint (typically, MV > 4 mag; e.g. Patterson 1984), in general the correction for the interstellar extinction will represent an upper limit. An additional uncertainty regarding the intrinsic SED is introduced by obtaining the flux calibration on the basis of a single standard star that was observed on a different night than the targets. Furthermore, the observations were not conducted at a parallactic angle and thus might be affected by chromatic slit loss due to differential refraction, since Gemini-South is not equipped with an atmospheric dispersion corrector. As a measure for the SED, we derive the exponent α of a power law F ∝ λ−α that was fitted to the spectra, restricting the continuum to wavelengths 5000–7000 Å and masking strong emission and absorption lines. During that analysis we found that most of the post-novae required a two-component fit while previously this represented an exception (Paper I, Paper IV). Furthermore, in every case the two slopes differed in the same direction, with the red part of the spectrum requiring a steeper slope than the blue part, and the points that separated the two slopes were found to be within 5850–5900 Å which marks the region that is affected by the gap between two CCDs (see Section 2). All this suggests a systematic effect in the definition of the instrumental response function, and potentially a difference in the spectral efficiency of these two CCDs. We have thus further divided the fitting range into a blue (5000  λ  5870 Å) and a red (5870  λ  7000 Å) part that are fitted individually. The consequences for the interpretation of the SEDs are further discussed in Section 4.


When the primary X-ray continuum is scattered by distant matter at several thousands of gravitational radii, the line is narrow (FWHM  10,000 km s−1) and is unlikely to carry any information about the conditions in the accretion disk and the strong-gravity regime near the SMBH. Narrow lines have been detected in the great majority of AGNs with luminosities L
X,2.0−10.0 keV  1045 erg s−1. The mean FWHM reported from Chandra High Energy Transmission Grating (HETG) spectra is ∼2000 km s−1 (Yaqoob & Padmanabhan 2004; Shu et al. 2010, 2011; see also Nandra 2006). Low equivalent width (EW) values from Suzaku observations (tens of eV; Fukazawa et al. 2011a) have also been interpreted as evidence of narrow lines originating in distant material (Ricci et al. 2014). In contrast, when the primary X-ray continuum is scattered close to the SMBH, Doppler and general relativistic effects combined may give rise to a significantly broader line (FWHM tens of thousands km s−1), reported in at least ∼36% of AGNs (de la Calle Pérez et al. 2010; see also, e.g., Porquet et al. 2004; Jiménez-Bailón et al. 2005; Guainazzi et al. 2006; Nandra et al.2007; Brenneman & Reynolds 2009; Patrick et al. 2012; Liu & Li 2015; Mantovani et al. 2016; Baronchelli et al. 2018). In this case, contributions to the line’s width become increasingly stronger as the primary continuum is scattered closer, and up to, the innermost stable circular orbit (ISCO), where the accretion disk’s inner edge is located. Since the ISCO location depends directly on black hole spin, the latter leaves an imprint on the broad line’s profile and can in principle be measured by means of relativistic modeling of the line via the X-ray reflection method of spin determination. As a result, there is a significant number of SMBH spin measurements (e.g., Brenneman 2013, and references therein). SMBH spin constraints have very significant implications for understanding both SMBHs and the way they affect their environment. Apart from potentially constituting a Kerr-metric-based test of general relativity in the strong-field regime, spin measurements can inform on jet-driving mechanisms, e.g., via extraction of SMBH rotational energy (Blandford & Znajek 1977), which in turn critically affect galactic environments and evolution.


Here, we present the GMC properties within the molecular disc of the closest giant elliptical galaxy, NGC 5128, which is the host of the radio-source Centaurus A (hereafter Cen A). Cen A is at a distance of only D ≃ 3.8 Mpc (Harris, Rejkuba & Harris 2010, 1 arcsec = 18 pc) and it is therefore by far the most adequate target in the class of giant elliptical galaxies as well as powerful radio galaxies for studies of their molecular gas with high resolution. Indeed, Cen A is a peculiar case of an elliptical galaxy whose gaseous component has been supplied a few 0.1 Gyr ago by the accretion of a HI rich galaxy (e.g. Struve et al. 2010). Along the dust lane of the elliptical galaxy, there is a molecular gas component of mass ∼109 M⊙ as probed by various molecular lines (e.g. Phillips et al. 1987; Eckart et al. 1990; Rydbeck et al. 1993; Liszt 2001; Espada et al. 2009; Espada 2013; McCoy et al. 2017), partially seen in the form of kpc scale spiral features (Espada et al. 2012). The dust lane is along the minor axis, different to other ellipticals where discs are usually along the major axis (Young 2002). The molecular gas is associated with other components of the ISM, such as ionized gas traced by the H α line (e.g. Nicholson, Bland-Hawthorn & Taylor 1992), near-infrared continuum (Quillen, Graham & Frogel 1993), submillimetre continuum (e.g. Hawarden et al. 1993; Leeuw et al. 2002), and mid-IR continuum emission (e.g. Mirabel et al. 1999; Quillen et al. 2006). In the inner hundreds of parsecs there is a circumnuclear disc (CND) of 400 pc total extent (∼24 arcsec) and a P.A. = 155°, perpendicular to the inner jet, at least as seen in projection (Espada et al. 2009). The total gas mass in this component has been estimated to be 9 × 10$^7\, \mathrm{M}_{\odot }$ (Israel et al. 2014, 2017). More detailed studies of the CND with higher resolutions of ∼5 pc in CO(3–2) and CO(6–5) have revealed the complexity of the molecular gas distribution and kinematics in that region, with multiple internal filaments and shocks (Espada et al. 2017).


During the second phase, protons are heated. For negligible heat fluxes, collisions, and fluctuations, one expects the double adiabatic behavior or CGL (Chew et al. 1956; Matteini et al. 2012): the parallel and perpendicular temperatures (with respect to the magnetic field) are expected to follow

 and

 respectively. Tp⊥ decreases slower than

 during the whole simulation; protons are heated in the perpendicular direction while in the parallel direction the heating lasts till about t ∼ 0.25te0, whereas afterward protons are cooled. The parallel and perpendicular heating rates could be estimated as (see Verscharen et al. 2015)
1

A more detailed analysis indicates that between t = 0.1te0 and t = 0.7te0 the parallel heating rate

 smoothly varies from about 0.2 Qe and

 whereas

 is about constant at ∼0.2 Qe; here,

 In total, protons are heated until t ∼ 0.7te0, and the heating reappears near the end of the simulation t ≳ 0.95te0. Note that the perpendicular heating rate is a nonnegligible fraction of that observed in the solar wind, where

 (Hellinger et al. 2013); however, the proton heating in 2D hybrid simulations is typically quite sensitive to the used electron equation of state (Parashar et al. 2014) and also to the resistivity and the number of particles per cell used (Franci et al. 2015a). The turbulent heating is, however, not sufficient to overcome the expansion-driven perpendicular cooling as in the solar wind (Matteini et al. 2007). During the third phase, t ≳ 0.7te0; there is an enhancement of the parallel cooling and perpendicular heating that cannot be ascribed to the effect of the turbulent activity. For a large parallel proton temperature anisotropy, a firehose instability is expected. The presence of such an instability is supported by the fact that the fluctuating magnetic field increases (with respect to the linear prediction), suggesting a generation of fluctuating magnetic energy at the expense of protons. To analyze the role of different processes in the system, we estimate their characteristic times (Matthaeus et al. 2014). The bottom panel of Figure 2 compares the turbulent nonlinear eddy turnover time

 at kdp = 1 (see Matthaeus et al. 2014; the expansion time te, and the linear time tl of the oblique firehose (Hellinger & Matsumoto 2000, 2001) estimated as

 where γm is the maximum growth rate calculated from the average plasma properties in the box assuming bi-Maxwellian proton velocity distribution functions (Hellinger et al. 2006). The expansion time te is much longer than tnl at kdp = 1 (as well as at the injection scales). The expanding system becomes theoretically unstable with respect to the oblique firehose around t ∼ 0.47te0 but clear signatures of a fast proton isotropization and of a generation of enhanced magnetic fluctuations appear later t ≳ 0.7te0. This is about the time when the linear time becomes comparable to the nonlinear time at ion scales. After that,

 slightly increases as a result of a saturation of the firehose instability, whereas

 at kdp = 1 is about constant (note that Ωp decreases as R−2). This may indicate that the instability has to be fast enough to compete with turbulence; however, the 2D system has strong geometrical constraints. Also the stability is governed by the local plasma properties. Figure 3 shows the evolution of the system in the plane

 During the evolution, a large spread of local values in the 2D space

 develops. Between t ≃ 0.1te0 and t ≃ 0.65te0, the average quantities evolve in time following

 This anticorrelation is qualitatively similar to in situ Helios observations between 0.3 and 1 AU (Matteini et al. 2007). During the third stage, when the strong parallel temperature anisotropy is reduced, both local and average values of

 and Ap appear to be bounded by the linear marginal stability conditions of the oblique firehose (Hellinger & Trávníček 2008), although relatively large theoretical growth rates

 are expected.


In order to investigate the signal-to-noise ratio (SNR), ρ of NS–WD binaries for LISA-type space GW detectors, we calculate the averaged square SNR $\overline{\rho ^{2}}$ over the sky location, inclination, and polarization as
(30)$$\begin{eqnarray*}
\overline{\rho ^{2}} = \int _{f_{1}}^{f_{2}}\frac{4\cdot \frac{4}{5}fA^{2}(f)}{(P_{\rm n}(f)/R(f))} \rm d (\ln \it f),
\end{eqnarray*}$$(Moore, Cole & Berry 2015; Robson, Cornish & Liu 2019), where f1 and f2 are the lower and upper limits of the integral, respectively. The factor 4 in the numerator of the integrand comes from the addition of strain noise in the detector arms and the two-way noise in each arm (Larson, Hiscock & Hellings 2000). We calculate the GW amplitudes A(f) of NS–WD binaries using the phenomenological (PhenomA) waveform model in the Fourier domain (Ajith et al. 2007; Robson et al. 2019). A(f) is expressed as
(31)$$\begin{eqnarray*}
A(f) = \sqrt{\frac{5}{24}}\frac{G^{5/6}\mathcal {M}^{5/6}}{\pi ^{2/3}c^{3/2}R_{\rm b}}f^{-7/6}\, {\rm Hz}^{-1},\,\,\,\it f\lt f_{\rm m},
\end{eqnarray*}$$
 (32)$$\begin{eqnarray*}
f_{\rm m} = \frac{0.2974\zeta ^{2}+0.04481\zeta +0.09556}{\pi (GM/c^{3})}\, {\rm Hz},
\end{eqnarray*}$$
 (33)$$\begin{eqnarray*}
\zeta = m_{1}m_{2}/M^{2},
\end{eqnarray*}$$where $\mathcal {M} \equiv m_1^{3/5} m_2^{3/5}/(m_1+m_2)^{1/5}$, and fm is the GW frequency at the point of merging. If f > fm, the index of the power-law relation between A(f) and f changes (Ajith et al. 2007) and is beyond the scope of this study. The power spectral density of total detector noise $P_{\rm n}=\frac{1}{L^{2}}\left[P_{\rm o}+2(1+\cos ^{2}(f/f_{\ast }))\frac{P_{\rm a}}{(2\pi f)^{4}}\right]$, where f* = c/(2πL), L = 2.5 × 109 m is the armlength of the detector, $P_{\rm o}=2.25\times 10^{-22} \,\rm m^{2}\left(1+(\frac{2\,mHz}{\it f})^{4}\right) \,\, \rm Hz^{-1}$ is the single-link optical metrology noise, and $P_{\rm a}=9.0\times 10^{-30} \,\rm (m\,s^{-2})^{2}\left(1+(\frac{0.4\,mHz}{\it f})^{2}\right)\left(1+(\frac{\it f}{8\,\rm mHz})^{4}\right) \,\,Hz^{-1}$ is the single test mass acceleration noise (LISA Science Study Team 2018; Robson et al. 2019). R(f) is the transfer function numerically calculated from Larson et al. (2000). The effective noise power spectral density can be defined as Sn(f) = Pn(f)/R(f). For Taiji and Tianqin, we use the sensitivity curve data in Ruan et al. (2020) and Wang et al. (2019), respectively.


To understand the distribution of Li-rich giants among different stellar components of the Galaxy, we have computed membership probability, based on the recipe given in Reddy, Lambert & Allende Prieto (2006) and references therein, for each of the stars in the selected RGB sample for belonging to one of the three main components of the Galaxy, namely; thin disc, thick disc, and halo. For this, the heliocentric velocities (U, V, W) for each of the sample giant is calculated using the Gaia astrometry (positions, parallax, and proper motions) and radial velocities (RV) from the GALAH DR2. The heliocentric velocities are corrected for the solar motion using Uo = 10, Vo = 5.3, Wo = 7.2 (km s−1) from Dehnen & Binney (1998) to get velocities with respect to the local standard of rest (ULSR, VLSR, WLSR). Entire sample is shown in Toomre Diagram of rotational velocity (VLSR) and sum of quadrature of radial and vertical velocities ([$\rm {\it U}_{\rm LSR}^2 + {\it W}_{\rm LSR}^2]^{1/2}$) (Sandage & Fouts 1987). Li-rich giants are identified by coloured symbols as shown in Fig. 4. We used kinematic boundaries for thin disc ($|V_{\rm Total}|$  = $[U_{\rm LSR}^2 + V_{\rm LSR}^2 + W_{\rm LSR}^2]^{1/2}$ ≤ 80 km s−1), thick disc (80 |VTotal| ≤ 200 km s−1) and halo (|VTotal| > 200 km s−1) which are in accordance with the results in Reddy et al. (2006). Further to quantify the number of giants belonging to different components of the Galaxy, we used probability of 70${{\ \rm per\ cent}}$ or more for any star being considered as a member of particular component. We found 223 Li-rich thin disc stars, which is about 0.8${{\ \rm per\ cent}}$ of total thin disc RGB stars with $P_{\rm thin} \ge 70 {{\ \rm per\,cent}}$ and 69 Li-rich thick disc stars consisting of 0.5${{\ \rm per\ cent}}$ of total thick disc RGB stars with $P_{\rm thick} \ge 70{{\ \rm per\,cent}}$. We have found just three Li-rich giants among 1442 halo giants with $P_{\rm halo} \ge 70{{\ \rm per\,cent}}$. This shows that Li-rich giants are rare ($\lt 1{{\ \rm per\,cent}}$) across stellar components but are relatively more prevalent among metal-rich thin disc component compared to thick disc and very metal poor halo.


In the last few decades, the dynamic properties of the quiet Sun have been thoroughly investigated using a range of substantially different techniques, allowing us to elaborate a consistent picture of the photospheric dynamics by approaching the problem from different points of view. Particularly interesting and promising are the studies involving the tracking of small-scale magnetic fields in the quiet photosphere. Such investigations reveal features that still cannot be captured by theoretical models and/or simulations because of the complexity of the system and the simultaneous coupling of a wide range of spatial and temporal scales. These studies are based on the hypothesis that magnetic fields are passively transported by the plasma flow and provide a characterisation of advection and diffusion processes in the quiet photosphere from granular to supergranular scales (see, e.g. Wang 1988; Berger et al. 1998; Cadavid et al. 1998, 1999; Hagenaar et al. 1999; Lawrence et al. 2001; Sánchez Almeida et al. 2010; Abramenko et al. 2011; Manso Sainz et al. 2011; Lepreti et al. 2012; Giannattasio et al. 2013, 2014a,b; Giannattasio et al. 2019; Keys et al. 2014; Caroli et al. 2015; Del Moro et al. 2015; Yang et al. 2015a,b; Roudier et al. 2016; Abramenko 2017; Kutsenko et al. 2018; Agrawal et al. 2018; Giannattasio & Consolini 2021). These studies reveal an anomalous scaling of magnetic field transport with a superdiffusive character consistent with a Levy walk inside supergranules and a more Brownian-like motion in their boundary. This has provided constraints on the magnetic flux emergence and evolution that models have to consider in order to fully explain the dynamics governing these environments. At the same time, the scaling laws affecting magnetic fields in the quiet Sun are crucial to understanding how such fields vary with scale size, despite the small scales at which dissipation occurs still being inaccessible with the currently available observations (see, e.g. Lawrence et al. 1994; Stenflo 2012, and references therein). For example, in the milestone work by Stenflo (2012), the spectrum of magnetic flux density in the quiet Sun was found to be consistent with a Kolmogorov power-law scaling. The scale at which scale invariance is broken lies below the current resolution limit. This latter author argued that the collapse of magnetic fields in Kilogauss flux tubes injects energy that is expected to cascade down because of the flux decay occurring via interchange instability, and to fragment into weaker ‘hidden’ fields at smaller scales (down to ∼10 km). As far as we know, no other studies have focused on the scaling properties characterising the magnetic fields in a quiet Sun region within a range of spatial and temporal scales from (sub)granular to supergranular in the time domain. In this work, for the first time we apply the structure function analysis typical of complex systems (Frisch 1995) to fill this gap. This approach complements the studies based on feature tracking mentioned above. The main difference is that, while in those works statistical properties of the photospheric plasma flows are investigated via the transport of small-scale magnetic fields in a frozen-in condition, here we directly study the magnetic field variations emerging from magnetogram time-series. The paper is organised as follows. Section 2 describes the data set used and the analysis techniques applied. Section 3 describes the obtained results, while Sect. 4 is devoted to their discussion in the light of current literature. Finally, in Sect. 5, we present our conclusions and present future perspectives.


To this we can add that Zimbardo and coworkers reported in a series of papers that spacecraft observations of SEP intensity-time profiles ahead of interplanetary shocks yielded power laws instead of the exponential profiles predicted by standard DSA. They interpreted this as a sign of superdiffusive spatial SEP transport in the shock vicinity resulting in superdiffusive shock acceleration. They predicted the superdiffusive shock acceleration to be more efficient, producing a harder accelerated power-law spectrum for SEPs, than standard DSA (Zimbardo et al. 2006; Perri & Zimbardo 2007, 2008, 2009; Zimbardo & Perri 2013; Zimbardo et al. 2017, 2020; Zimbardo & Perri 2020). Zimbardo & Perri (2020) explained such superdiffusive transport as a product of subdiffusion in pitch-angle space arising from the observation that Alfvén-wave pitch-angle scattering times in the solar wind have a power-law distribution instead of a single value. Another possibility is that there is a strong presence of SMFRs in the vicinity of heliospheric shocks so that particle trapping effects could result in anomalous diffusive transport and shock acceleration. Evidence is increasing that, in a space plasma with a significant guide/background magnetic field like the solar wind, SMFRs naturally form as part of a self-generated, quasi-two-dimensional (quasi-2D) MHD turbulence component that might dominate other MHD wave turbulence modes. This is supported by observations in the slow solar wind near 1 au (e.g., Matthaeus et al. 1990; Bieber et al. 1996; Greco et al. 2009; Zheng & Hu 2018), MHD simulations (e.g., Shebalin et al. 1983; Dmitruk et al. 2004), and nearly incompressible MHD turbulence transport theory (e.g., Zank & Matthaeus 1992, 1993; Zank et al. 2017, 2018, 2020). It is plausible that SMFRs can have a strong presence ahead of interplanetary shocks. Furthermore, there is evidence that the occurrence of SMFRs peaks at large-scale current sheets in the solar wind where magnetic reconnection produces additional SMFRs that can trap energetic particles (Khabarova et al. 2015, 2016; Hu et al. 2018). Many of these current sheets occur behind traveling shocks (Khabarova & Zank 2017). Recently, a theoretical investigation was launched to determine the transmission of quasi-2D SMFRs through a perpendicular shock (Zank et al. 2021). The results show a strong enhancement in the magnetic energy density of the SMFR magnetic island component behind the shock, thus confirming a strong presence of SMFRs downstream of the shock as well.


The hot emission line of Fe xxi 1354.09 Å and the cool emission line of Si iv 1402.77 Å have been used in many spectroscopic studies to investigate chromospheric evaporation (e.g., Tian et al. 2014, 2015; Li et al. 2015b, 2017a, 2017b; Brosius et al. 2016; Zhang et al. 2016a, 2016b). It is widely accepted that the forbidden line of Fe xxi 1354.09 Å is a hot (log T ∼ 7.05) and broad emission line during solar flares (Doschek et al. 1975; Cheng et al. 1979; Mason et al. 1986; Innes et al. 2003a, 2003b). Meanwhile, IRIS spectroscopic observations show that Fe xxi 1354.09 Å is always blended with a number of cool and narrow emission lines, which are from neutral or singly ionized species. Those blended emission lines can be easily detected at the position of the flare ribbon, including known and unknown emission lines, such as the C i line at 1354.29 Å, the Fe ii lines at 1353.02 Å, 1354.01 Å, and 1354.75 Å, the Si ii lines at 1352.64 Å and 1353.72 Å, and the unidentified lines at 1353.32 Å and 1353.39 Å (e.g., Li et al. 2015a, 2016a; Polito et al. 2015, 2016; Tian et al. 2015, 2016; Young et al. 2015; Tian 2017). In order to extract the hot line of Fe xxi 1354.09 Å and the cool line of C i 1354.29 Å (log T ∼ 4.0; Huang et al. 2014), we apply a multi-Gaussian function superimposed on a linear background to fit the IRIS spectrum at the “O i” window (e.g., Li et al. 2015a, 2016a), which has been pre-processed (i.e., IRIS spectral image deformation, bad pixel despiking and wavelength calibration) with the standard routines in Solar Soft Ware (SSW; Freeland et al. 2000). In short, the line positions and widths of these blended emission lines are fixed or constrained, and their peak intensities are tied to isolated emission lines from similar species. More details can be found in our previous papers (Li et al. 2015a, 2016a). On the other hand, the cool line of Si iv 1402.77 Å (log T ∼ 4.8) at the “Si iv” window is relatively isolated, and it can be well fitted with a single-Gaussian function superimposed on a linear background (Li et al. 2014, 2017a). Using the relatively strong neutral lines (i.e., “O i” 1355.60 Å and “S i” 1401.51 Å), we also perform an absolute wavelength calibration for the spectra at the “O i” and “Si iv” windows, respectively (Tian et al. 2015; Tian 2017). Finally, the Doppler velocities of Fe xxi 1354.09 Å, C i 1354.29 Å, and Si iv 1402.77 Å are determined by fitting line centers removed from their rest wavelengths (Cheng & Ding 2016b; Guo et al. 2017; Li et al. 2017a). As the hot Fe xxi line is absent in the non-flaring spectrum, the rest wavelength for the Fe xxi line (i.e., 1354.09 Å) is determined by averaging the line centers of the Fe XXI profiles which were used in the previous IRIS observations (Brosius & Daw 2015; Polito et al. 2015, 2016; Sadykov et al. 2015; Tian et al. 2015; Young et al. 2015; Brosius et al. 2016; Lee et al. 2017), while the rest wavelengths for the C i and Si iv lines, i.e., 1354.29 Å and 1402.77 Å, respectively, are determined from their quiet-Sun spectra (Li et al. 2014, 2015a).


One of the first models of MRI-driven accretion in protoplanetary discs was constructed by Gammie (1996), who proposed that Ohmic resistivity (due to weak coupling of charged species to the magnetic field) quenches the MRI in a ‘dead’ mid-plane region surrounded by ‘active layers’, which are ionized by cosmic rays. This layered-accretion scenario has been refined as additional non-thermal ionization sources (X-rays, UV radiation) and non-ideal magnetohydrodynamic (MHD) effects (ambipolar diffusion and the Hall effect) have been included (e.g. Igea & Glassgold 1999; Sano et al. 2000; Fromang, Terquem & Balbus 2002; Salmeron & Wardle 2003, 2005, 2008; Ilgner & Nelson 2006; Bai & Goodman 2009; Bai & Stone 2011; Wardle & Salmeron 2012). Several theoretical studies, spanning both analytical (e.g. Blaes & Balbus 1994; Desch 2004; Kunz & Balbus 2004) and numerical work (e.g. Hawley & Stone 1998; Bai & Stone 2011, 2013b; Simon et al. 2013a,b), have shown that ambipolar diffusion plays a major role in determining the level of turbulence in protoplanetary discs. In the outer regions (radii R ≳ 30 au), ambipolar diffusion acts to reduce the strength of MRI-driven turbulence close to the disc mid-plane; turbulence is stronger above this ‘ambipolar damping zone’ in a thin layer of gas strongly ionized by FUV photons (Perez-Becker & Chiang 2011; Simon et al. 2013a,b). To produce accretion rates consistent with observations, a large-scale magnetic field threading the disc perpendicular to the disc plane is required (Simon et al. 2013a,b). This net field enhances turbulence, despite the damping effect of ambipolar diffusion, and allows additional angular-momentum loss via a magnetic wind (akin to the Blandford & Payne 1982 mechanism). In the inner disc, both local (Bai & Stone 2013b) and global (Gressel et al. 2015) simulations have shown that ambipolar diffusion can quench the MRI in the purported active layers of Gammie's (1996) model. In those cases, a large-scale magnetic field is again necessary to drive accretion at the observationally inferred rates, this time solely via a magnetic wind in a manner akin to previous models of wind-driven accretion in protoplanetary discs (e.g. Wardle & Koenigl 1993; Shu et al. 1994; Salmeron, Königl & Wardle 2007).


PNs are expanding shells of the luminous gas expelled by dying stars of low and intermediate masses (LIMS). They stem from objects that have lifetimes up to gigayears. The ionized gas surrounding the central star shows emission lines of highly ionized species from which the abundances can be derived. The neon abundance can be compared and related to the results from stellar abundance analysis, as neon (and oxygen) originates from primary nucleosynthesis in massive stars (≥10 M⊙) and is therefore nearly independent of the evolution of LIMS, the progenitor stars of PNs (Henry 1989; Henry et al. 2004). For comparison, we collected several studies, where the neon abundances were obtained in PNs. We would not say that the abundances from PNs are very accurate, because the differences between abundances from collisionally excited lines and optical recombination lines can be much higher in PNs compared to those of H ii regions. For example, according to Wang & Liu (2007), in four Galactic disk PNs (He 2–118, H 1-35, NGC 6567, and M1-61), the mean neon abundance calculated from collisionally excited lines is 7.78 ± 0.23, while it is 8.72 ± 0.60 from optical recombination lines. Figure 9 presents neon abundances from different sources, including five studies focused on PNs. In those studies, where it was mentioned, we adopted neon abundances obtained from collisionally excited lines, e.g., Tsamis et al. (2003) and Wang & Liu (2007). The mean value of log ϵNe = 7.84 ± 0.24 was calculated on the basis of neon abundances in six Galactic disk PNs (Wang & Liu 2007). The mean value of the neon abundance log ϵNe = 7.99 ± 0.22 dex was calculated from six PNs (Hu 1-2, IC 418, NGC 40, NGC 2440, NGC 6543, NGC 7662) with Galactocentric distances from 7.9 kpc to 8.9 kpc from the summary of Pottasch & Bernard-Salas (2006). The value of log ϵNe = 7.76 ± 0.24 dex was derived from 16 Galactic PNs located at Galactocentric distances from 8.0 to 8.9 kpc in Stanghellini et al. (2006). The log ϵNe = 8.02 ± 0.25 dex was obtained from Marigo et al. (2003) from three PNs (NGC 6543, NGC 7027, NGC 7662), which are located at distances no more than 1.0 kpc. We adopted the mean value log ϵNe = 8.15 ± 0.21 dex, which was calculated from 16 Galactic PNs from Tsamis et al. (2003).


In order to consider both the photoionization and radiation-pressure effects from the accreting protostar, we solve the frequency-dependent radiative transfer for stellar irradiation by the following method. We inject photons from the sink cell to the computational domain at the rate
(10)$$\begin{eqnarray*}
L_{\rm *}^{\rm tot} = L_* + L_{\rm acc},
\end{eqnarray*}$$where L* is the stellar luminosity and $L_{\rm acc} \equiv G M_* \dot{M}_* / R_*$ is the accretion luminosity. The radiation spectrum is assumed to be thermal blackbody $L_{*, \nu }^{\rm tot} \propto B_\nu (T_{\rm eff})$, where Teff is the effective temperature defined as
(11)$$\begin{eqnarray*}
T_{\rm eff} = \left(\frac{L_*^{\rm tot}}{4 \pi \sigma R_*^2} \right)^{1/4} .
\end{eqnarray*}$$Assumption of the thermal blackbody radiation could overestimate the emissivity of ionizing photons at Z ∼ Z⊙ owing to the line-blanketing effect, especially for stars with M* ∼ 10 M⊙ (e.g. Diaz-Miller, Franco & Shore 1998). This approximation should be valid in our cases since we mainly consider the feedback from more massive stars exceeding 100 M⊙. As discussed in Section 2.2.3, the effective temperature is higher for lower metallicity at a given mass: at M* = 100 M⊙, for instance, Teff ≃ 105 K for Z = 0, Teff ≃ 7 × 104 K for Z = 10−2 Z⊙, and Teff ≃ 5 × 104 K for Z = Z⊙. We use a hybrid method where the direct light emitted from the central star and diffuse light re-emitted from the accretion envelope are separately solved (e.g. Kuiper et al. 2010a, b). We solve the transfer of the stellar direct component by means of the ray-tracing method,
(12)$$\begin{eqnarray*}
F_{\nu }(r) = \frac{L_{*,\nu }^{\rm tot}}{4 \pi r^2} \mathrm{ e}^{-\tau _{\nu }},
\end{eqnarray*}$$where Fν(r) is the flux at the radial position r and τν is the optical depth
(13)$$\begin{eqnarray*}
\tau _{\nu } = \int _{r_{\rm in}}^{r} [ n_{\rm H\,{\small I}} \sigma _{\rm H\,{\small I}}(\nu) + \rho \kappa _{\rm \mathrm{ d}} (\nu)] \mathrm{ d}r,
\end{eqnarray*}$$where σH i(ν) is the H i photoionization cross-section (e.g. Osterbrock 1989), κd(ν) the dust opacity given by Laor & Draine (1993), and rin the sink radius. We assume that the dust-to-gas mass ratio decreases with decreasing metallicities, linearly scaling with Z. The dust opacity is determined by the scaled dust-to-gas mass ratio. We also assume that photons freely travel without attenuation for R* r rin. We set 200 logarithmically spaced wavelength bins between 0.03 cm and 1.5 nm, achieving the higher resolution for the shorter wavelengths. The hydrogen photoionization rate is calculated by integrating the contributions by photons below the Lyman limit, i.e. 91.2 nm (Appendix A2.3). We do not include the diffuse ionizing photons, as their contribution to the disc photoevaporation is negligible compared with the stellar direct component in the massive star formation (e.g. Hosokawa et al. 2011; Tanaka, Nakamoto & Omukai 2013). As in Hosokawa et al. (2016), we do not incorporate helium ionization for simplicity. We assume that helium is in the atomic state everywhere including H ii regions. We consider the transfer of He ionizing photons above 54.4 eV emitted from the protostar, but they are consumed to photoionize hydrogen in our simulations. Regarding the photodissociation of hydrogen molecules, we do not use Fν(r) because our wavelength grids are too sparse to resolve the Lyman–Werner bands. We instead consider another monochromatic direct component representing FUV (11.2 eV ≤ hν ≤ 13.6 eV) photons
(14)$$\begin{eqnarray*}
F_{\rm FUV} (r) = \frac{L_{\rm FUV}}{4 \pi r^2} \exp (- \tau _{\rm d, FUV}),
\end{eqnarray*}$$where LFUV is the luminosity in the FUV wavelength range, and τd,FUV is the dust opacity represented by the value at λ = 100 nm. The photodissociation rate of hydrogen molecules is estimated with the flux FFUV, including the $\rm H_2$ self-shielding. The details of the calculation are shown in Appendix A2.3


Both SHINE and GPIES surveys recently discovered three new exoplanets (Macintosh et al. 2015; Chauvin et al. 2017b; Keppler et al. 2018) and a few additional higher-mass brown dwarfs (Konopacky et al. 2016; Cheetham et al. 2018). Smaller surveys using SPHERE and GPI also discovered several substellar companions (Milli et al. 2017a; Wagner et al. 2020; Bohn et al. 2020). These surveys offer unprecedented detection, astrometric, and spectrophotometric capabilities that allow us to characterize fainter and closer giant planets, such as the recent discovery of 51 Eri b (2 MJup at 14 au, T5-type, of an age of 20 Myr; Macintosh et al. 2015; Samland et al. 2017), HIP 65426 b, a young, warm, and dusty L5-L7 massive Jovian planet located at about 92 au from its host star (Chauvin et al. 2017b), and the young solar analogue PDS 70, which is now known to actually host two planets PDS 70 b discovered during the SHINE campaign (Keppler et al. 2018; Müller et al. 2018) and PDS 70 c by MUSE (Haffert et al. 2020). Such surveys also provide key spectral and orbital characterisation data for known exoplanets (e.g., De Rosa et al. 2016; Samland et al. 2017; Chauvin et al. 2018; Wang et al. 2018; Cheetham et al. 2019; Lagrange et al. 2019; Maire et al. 2019). Despite these new discoveries, SHINE and GPIES have yielded significantly fewer exoplanet detections than predicted, with their use of extrapolations of radial velocity planet populations to larger semi-major axes (e.g., Cumming et al. 2008). This results in our setting strong statistical constraints on the distribution of giant exoplanets at separations of >10 au from their stars, as well as sub-stellar companions to young stars (Nielsen et al. 2019; Vigan et al. 2021). As these systems are young (100 Myr), and thus closer to their epoch of formation than, for instance, radial-velocity planets (typically 1–10 Gyr), statistical analyses of large direct-imaging surveys can provide hints for the potential formation mechanism responsible for producing giant exoplanets on wide orbits.


Both Nel (2016) and Zhao et al. (2018) report that these parallel mean free paths are smaller during periods of high solar activity and larger during solar minima, with the opposite being true for the resulting perpendicular mean free paths. Zhao et al. (2018) also show that such solar cycle dependences would not be fully accounted for if only the solar cycle changes in the HMF were taken into consideration. These mean free paths also display complicated spatial dependences if turbulence quantities beyond 1 au, yielded by various turbulence transport models (see, e.g., Oughton et al. 2011; Usmanov et al. 2016; Wiengarten et al. 2016; Zank et al. 2017), taking into account observed latitudinal variations of turbulence quantities in the heliosphere (Forsyth et al. 1996; Bavassono et al. 2000a, 2000b; Erdös & Balogh 2005), were used as inputs for the turbulence quantities (e.g., Engelbrecht & Burger 2013a, 2015b; Chhiber et al. 2017; Moloto et al. 2018). Furthermore, mean free paths such as those discussed have, when used in conjunction with turbulence-reduced drift coefficients (see, e.g., Minnie et al. 2007b; Engelbrecht et al. 2017) in 3D stochastic modulation codes, led to computed galactic intensities in reasonable agreement with spacecraft observations at Earth (see, e.g., Engelbrecht & Burger 2013a, 2013b; Qin & Shen 2017) and even for several different solar minima (Moloto et al. 2018). The necessity of taking cosmic-ray drift effects into account, combined with the complicated spatial dependences and the fact that none of the diffusion coefficients described above display a P1 rigidity dependence, implies that a convection–diffusion or force-field approach would not be ideal to describe long-term modulation, and that the assumptions implicit to the effective diffusion coefficient used in these formulations are unrealistic. This latter point would call into question any conclusions drawn as to historic solar parameters from quantities such as the modulation potential.


Studying the dynamical properties of rotating neutron stars is a domain which brings out various interesting features when one assumes a perfect fluid. It is known that the centrifugal force of a rotating star counters gravitational force and hence one can expect massive stars to be fast rotors, at least in the initial stages of the stellar evolution. As a result of rotation a star may experience damping due to unstable oscillations such as the r-modes. The r-modes are one of many pulsating modes that exist in neutron stars and are characterized by the Coriolis force acting as the restoring force (Andersson 1998). The r-modes are unstable to emission of gravitational radiation (GR) by the Chandrashekhar-Friedman-Schutz (CFS) mechanism (Chandrasekhar 1970; Friedman & Schutz 1978). It was shown in Andersson (1998) that the r-modes are unstable for all rotating perfect fluid stars irrespective of their frequency. Dissipative effects such as shear and bulk viscosities work towards suppressing GR driven instabilities and has been studied by various authors over the past few years (Lindblom, Owen & Morsink 1998; Jones 2001; Lindblom & Owen 2002; van Dalen & Dieperink 2004; Drago, Lavagno & Pagliara 2005; Nayyar & Owen 2006; Jaikumar, Rupak & Steiner 2008; Jha, Mishra & Sreekanth 2010; Ofengeim et al. 2019) under various considerations. If the GR time-scale is shorter than the damping time-scale due to such dissipative processes, then the r-mode will be unstable and a rapidly rotating neutron star could lose a significant fraction of its rotational energy through GR. At higher temperatures (T > 109 K), the dominant dissipation is due to bulk viscosity, which arises due to density and pressure perturbations, a consequence of the star being driven out of equilibrium by oscillations. The system tries to restore equilibrium through various internal processes. In the case of r-modes, since the typical frequencies are of the order of the rotational frequencies of the stars, the reactions that dominate are the weak processes. Within these weak processes, although the modified Urca processes involving leptons are important, it has been shown that non-leptonic processes involving hyperons contribute more significantly towards bulk viscosity at temperatures lower than a few times 109 K (Lindblom & Owen 2002). Our goal here is to investigate the same using a chiral model calibrated to reproduce the desired nuclear matter properties, in particular the density content of the nuclear symmetry energy at both low and high densities.


Additional information can be derived from the abundance ratios measured in the ICM. For example, the Mn/Fe and Ni/Fe ratios are both sensitive to the electron capture rates during SNIa explosions, and are therefore crucial indicators of their progenitor channels (Seitenzahl et al. 2013a; Mernier et al. 2016a; Hitomi Collaboration 2017), while ratios of lighter elements can in principle provide constraints on the IMF and the initial metallicity of the SNcc progenitors (e.g. de Plaa et al. 2007; Mernier et al. 2016a). By pushing current observatories to their limit, recent studies derived constraints on the relative fraction of SN events that effectively contribute to the ICM enrichment. These measurements showed that SNIa and SNcc contribute relatively equally to the overall chemical enrichment in the ICM (Werner et al. 2006; de Plaa et al. 2006, 2017; Bulbul et al. 2012; Mernier et al. 2016a). The comprehensive study of Simionescu et al. (2019) compiled the most accurate abundance measurements of the Perseus cluster (taken with the XMM-Newton RGS and Hitomi SXS instruments) and compared them to state-of-the-art SNIa and SNcc yield models. Their surprising conclusion is that no current set of models is able to reproduce all the observed abundance ratios at once. In particular, the measured Si/Ar ratio tends to be systematically overpredicted by models, even when taking calibration and atomic uncertainties into account. Whereas further improvement of stellar nucleosynthesis models is expected, the non-negligible systematic errors associated to these observations and the lack of highly sensitive, spatially resolved high-resolution spectroscopy prevents us from steering any considerable change in the paradigm (see also de Grandi & Molendi 2009). In fact, measurements from currently flying missions are performed with moderate collective area, either over the whole X-ray band (0.4–10 keV) with modest spectral resolution (> 100 eV), or with higher resolution dispersive spectroscopy but over the low E band (0.3–2 keV) and without any spatial resolution, which considerably limits interpretations.


Although this penalty is small, we can still provide a quantitative estimate based on a few assumptions. The observations of M15 that measured the parallax to VLA J2130+12 (K14) employed an interferometric technique where the position of a weaker potential in-beam calibrator source (M15 S1 and VLA J2130+12) can be measured against a brighter primary calibration source, with the potential for using the in-beam calibrator to transfer more accurate calibration solutions to other in-beam targets. Based on a literature review for papers where VLBI parallax measurements were determined using an in-beam calibrator, we find 24 similar measurements, the majority of which have been used to measure parallaxes to pulsars (Fomalont et al. 1999; Brisken et al. 2003; Chatterjee et al. 2005, 2009; Ng et al. 2007; Middelberg et al. 2011; Deller et al. 2012, 2013; Ransom et al. 2014; Liu et al. 2016). As these measurements represent additional opportunities to detect the parallax of a VLA J2130+12-like object, they could be considered as potential trials. However, all of the reported in-beam calibrators were significantly brighter (4–86 mJy) than VLA J2130+12 (∼0.1–0.5 mJy). Given our conservative assumption of a uniform volume density of VLA J2130+12-like objects in the Galaxy, we should down-weight the number of trials from brighter sources by (fν,in-beam/fν,VLA J2130+12)−1.5. In that case, the number of trials penalty is only an additional ∼0.2 trials. In addition, we note that the PSRπ parallax project (a large VLBA program) has reported15

15

https://safe.nrao.edu/vlba/psrpi/

 111 additional in-beam calibrator sources that they used to measure parallaxes. Although they do not provide the flux densities of individual sources, they note that their median in-beam calibrator source is 9.2 mJy. We have measured the flux density function of secure 1–20 mJy FIRST sources (Helfand et al. 2015;

) to estimate the expected distribution of the flux densities of in-beam calibrators. We found that the minimum flux density is likely ∼3.2 mJy, and using the same down-weighting we estimate an additional penalty of ∼0.9 trials.


In recent years, several large-scale pulsar surveys have been undertaken to search for new pulsars (Cordes et al. 2006; Keith et al. 2010; Barr et al. 2013; Boyles et al. 2013; Deneva et al. 2013; Coenen et al. 2014; Stovall et al. 2014). One of the drivers for such surveys is the discovery of millisecond pulsars (MSPs). MSPs are formed through accretion from a companion during an X-ray binary phase (Alpar et al. 1982; Bhattacharya & van den Heuvel 1991) in which the pulsar is “recycled.” This accretion phase spins the pulsar up to very fast rotational rates (spin periods

 ms). Such pulsars are useful for a variety of physical applications. Examples include tests of theories of gravity using MSP–white dwarf systems such as PSR J1738+0333 and PSR J0348+0432 (Freire et al. 2012; Antoniadis et al. 2013) and triple systems like PSR J0337+1715 (Ransom et al. 2014); tests of general relativity using double neutron star systems, such as J0737−3039 (Kramer et al. 2006) and PSR B1913+16 (Weisberg et al. 2010); the study of binary systems such as eccentric MSPs like PSRs J1903+0327 (Champion et al. 2008) and J1950+2414 (Knispel et al. 2015) which are interesting due to their peculiar binary evolution; and constraining the equation of state of dense matter using measurements of neutron star masses (Demorest et al. 2010; Antoniadis et al. 2013). Another major driver for the discovery of new MSPs is the effort to detect gravitational wave emission using an array of pulsars (NANOGrav Collaboration et al. 2015; Lentati et al. 2015; Reardon et al. 2016). The large-scale pulsar surveys mentioned above, combined with targeted searches of unidentified gamma-ray sources from the Fermi Gamma-Ray Space Telescope (e.g., Hessels et al. 2011; Keith et al. 2011; Ransom et al. 2011; Kerr et al. 2012), have resulted in the discovery of about 90 new MSPs in the past 5 years, an increase of 40% in the known Galactic MSP population. A subset of the newly discovered sources are eclipsing systems that appear to fall into two categories (e.g., Freire 2005; Roberts 2012). The first category, known as black widow systems, has very low mass, degenerate companions (

) believed to be the result of ablation by the pulsar. The second, known as redback systems, has low to moderate mass, non-degenerate companions (

).


To confirm the importance of VO, water and an inversion layer (Evans et al. 2018; Mikal-Evans et al. 2019, 2020) obtained repeated HST observations of the transmission spectrum and the secondary eclipse using the STIS and WFC 3 instruments. The optical transmission spectrum displays rich variation, with multiple features consistent with VO absorption that Evans et al. (2018) could reproduce by assuming an isothermal T-P profile at 1500 K and a metallicity equivalent to 10× to 30× solar. Absorption bands of TiO appeared to be muted in the transmission spectrum, which was explained by Evans et al. (2018) as evidence of condensation of Ti-bearing species, which commences at higher temperatures than condensation of V-bearing species, producing for example, calcium titanates (Lodders 2002) while VO remains in the gas phase. Mikal-Evans et al. (2019) observed the day side emission spectrum with the G102 grism of WFC3 (0.8–1.1 μm), augmenting their earlier observations with the G141 grism. The G102 spectrum does not show the VO bands expected to be present there, and this led Mikal-Evans et al. (2019) to question the interpretation that the 1.2 μm feature is caused by VO emission. The secondary eclipse was observed at 2 μm (Kovács & Kovács 2019) and at optical wavelengths with the TESS instrument. These were analysed together with the preceding Hubble, Spitzer, and ground-based observations to yield tighter constraints on the atmospheric structure, composition and overall system parameters (Bourrier et al. 2020a; Daylan et al. 2019). These studies found that the hottest point on the day side exceeds a temperature of 3000 K, that the atmosphere is inverted on the day side, and a metallicity that is consistent with solar (Bourrier et al. 2020a) or slightly elevated (Daylan et al. 2019). Although the chemical retrievals follow different strategies (equilibriumversus free-chemistry), both indicate that a depletion of TiO relative to VO is needed to explain the observed emission spectrum, supporting the earlier findings by Mikal-Evans et al. (2019). Recently, Mikal-Evans et al. (2020) obtained new secondary-eclipse observations using the G141 grism of WFC3. Although confirming the presence of emission by H2O, a joint analysis with their previous WFC3 observations did not reproduce the emission feature at 1.2 μm, prompting the authors to entirely discard their previous interpretation of emission caused by VO.


On balance, a large abundance of baryon-dominated, dark matter cored galaxies at z ∼ 2, most strongly correlated with baryonic surface density, angular momentum, and central bulge mass, may be most naturally accounted for by the interaction of baryons and dark matter during the formation epoch of massive halos. Massive halos (log(Mhalo/M⊙) > 12) formed for the first time in large abundances in the redshift range z ∼ 1–3 (Press & Schechter 1974; Sheth & Tormen 1999; Mo & White 2002; Springel et al. 2005). At the same time, gas accretion rates were maximal (Tacconi et al. 2020). This resulted in high merger rates (Genel et al. 2008, 2009; Fakhouri & Ma 2009), very efficient baryonic angular momentum transport (Dekel et al. 2009; Zolotov et al. 2015), formation of globally unstable disks, and radial gas transport by dynamical friction (Noguchi 1999; Immeli et al. 2004; Genzel et al. 2008; Bournaud & Elmegreen 2009; Bournaud et al. 2014; Dekel & Burkert 2014). These processes enabled galaxy mass doubling on a timescale  0.4 Gyr at z ∼ 2–3, and massive bulge formation by disk instabilities and compaction events on 1 Gyr timescales. However, central baryonic concentrations would naturally also increase central dark matter densities through adiabatic contraction (Barnes & White 1984; Blumenthal et al. 1986; Jesseit et al. 2002). For adiabatic contraction to be ineffective requires the combination of kinetic heating of the central dark matter cusp by dynamical friction from in-streaming baryonic clumps (El-Zant et al. 2001; Goerdt et al. 2010; Cole et al. 2011), with feedback from winds, supernovae, and AGNs driving baryons and dark matter out again (Dekel & Silk 1986; Pontzen & Governato 2012, 2014; Martizzi et al. 2013; Freundlich et al. 2020; K. Dolag et al. 2020, in preparation). Using idealized Monte Carlo simulations, El-Zant et al. (2001) demonstrated that dynamical friction acting on in-spiraling gas clumps can provide enough energy to heat up the central dark matter component and create a finite dark matter core (see also A. Burkert et al. 2020, in preparation). They argue that dark matter core formation in massive galaxies would require that clumps be compact, such that they avoid tidal and ram-pressure disruption, and have masses of >108 M⊙. Other idealized simulations (e.g., Tonini et al. 2006) confirm these results.