The mechanism that produces large spin-up glitches such as those seen in the Vela pulsar (Dodson, Lewis & McCulloch 2007; Palfreyman et al. 2018) and PSR J0537−6910 is thought to be a sudden transfer of angular momentum from a rapidly rotating superfluid that permeates the neutron star inner crust to the rest of the star, which is slowing down from electromagnetic radiation braking (Anderson & Itoh 1975; Haskell & Melatos 2015; Graber, Andersson & Hogg 2017). Link et al. (1999) provide strong support for this theory by first showing that the fractional moment of inertia (Isf/I) of the superfluid angular momentum reservoir is related to a pulsar’s glitch activity via Isf/I ≥ 2τcAg, where I is total moment of inertia and $\tau _{\rm c}=-\nu /2\dot{\nu }$ is pulsar characteristic age. They then show that the glitch activity of pulsars like Vela gives Isf/I ≳ 0.01, and this approximately matches the theoretical fractional moment of inertia of a neutron star’s crust Icrust/I. For PSR J0537−6910, we find 2τcAg = 0.00874, which agrees with that found in previous works (Middleditch et al. 2006; Antonopoulou et al. 2018; Ferdman et al. 2018). More recently, calculations by Andersson et al. (2012) and Chamel (2013) indicate that the above relation is underestimated after accounting for superfluid entrainment and should instead be $I_{\rm sf}/I\ge 2\tau _{\rm c}A_{\rm g}\langle m_{\rm n}^\ast \rangle /m_{\rm n}$, where $\langle m_{\rm n}^\ast \rangle $ and mn are averaged effective neutron mass and neutron mass, respectively (and $\langle m_{\rm n}^\ast \rangle /m_{\rm n}\approx 4.2$; Chamel 2012), and as a result, the angular momentum required by a glitch is more than the crust can provide. One solution proposed by Andersson et al. (2012) is that the superfluid component in the crust extends into the core. Subsequently, Ho et al. (2015, 2017) show that, by combining a pulsar’s glitch activity with a measurement of true age or surface temperature, one can obtain valuable constraints on properties of the superfluid and even measure the mass of the pulsar; for the Ag and estimated age of PSR J0537−6910, the pulsar’s mass turns out to be much higher than the canonical 1.4M⊙. Very recently, Sauls, Chamel & Alpar (2020) suggest superfluid entrainment may not be as strong as found by Chamel (2012, 2017). More work is needed to resolve the issue.


Both our BOSS×BOSS and FIRAS×BOSS models require the dark matter angular power spectrum of the overdensity field, $C_{\ell }^{\delta }(z,z^{\prime })$. We calculate this angular power spectrum with the Boltzmann code CLASS (Di Dio et al. 2013, 2014), using cosmological parameters inferred from the Planck 2015 (Ade et al. 2016) temperature and low-ℓ polarization maps (TT + LowP). The Halofit routine (Smith et al. 2003) provides nonlinear corrections to the power spectrum. However, on the several-degree scales of this analysis, the fluctuations are well-described by linear perturbation theory ($k_{\rm max} \sim \frac{\ell _{\rm max}}{\chi (z)_{\rm min}} \sim \frac{50}{880}$ h/Mpc ∼0.06 h/Mpc), and nonlinear corrections are small. CLASS computes the angular power spectrum from the 3D power spectrum, P(k), according to the equation
(11)$$\begin{eqnarray*}
C_{\ell }^{A\times B}(z,z^{\prime }) = \frac{2}{\pi } \int k^2 P^{\delta }(k,z{=}0) W_A^{\rm tot}(k,z)W_B^{\rm tot}(k,z^{\prime }) \mathrm{d}k,
\end{eqnarray*}$$where Pδ(k, z = 0) is the dark matter power spectrum at the current epoch, and, if there are no redshift space distortions (RSDs),
(12)$$\begin{eqnarray*}
W_A(k, z) = b_A \int \phi _z(z^{\prime \prime }) G(z^{\prime \prime },k) j_{\ell }[k \chi (z^{\prime \prime })] \mathrm{d}z^{\prime \prime },
\end{eqnarray*}$$where bA is the bias for dark matter tracer A, ϕz(z″) is a tophat redshift selection function that is non-zero only over the range of the redshift slice centered at redshift z and normalized to integrate to 1, jℓ is a spherical Bessel function of the first kind with parameter ℓ, G(z″, k) is the growth factor, and χ(z″) is the radial comoving distance to the shell at redshift z″. Linear RSDs can be included (Fisher, Scharf & Lahav 1994; Padmanabhan et al. 2007) by replacing WA(k, z) with $W_A^{\rm tot}(k,z)$, where
(13)$$\begin{eqnarray*}
W_A^{\rm tot}(k, z) &= &W_A(k, z) + W_A^{\rm RSD}(k,z)\nonumber \\
W_A^{\rm RSD}(k,z) &=&b_A \int \beta _A(z^{\prime \prime }) \phi _z(z^{\prime \prime }) \times \nonumber \\
&& \biggl \lbrace \frac{2\ell ^2+2\ell -1}{(2\ell +3)(2\ell -1)}j_{\ell }[k \chi (z^{\prime \prime })]\nonumber \\
& &- \frac{\ell (\ell -1)}{(2\ell -1)(2\ell +1)}j_{\ell -2}[k \chi (z^{\prime \prime })]\nonumber \\
& &- \frac{(\ell +1)(\ell +2)}{(2\ell +1)(2\ell +3)}j_{\ell +2 }[k \chi (z^{\prime \prime })] \biggr \rbrace \mathrm{d}z^{\prime \prime },
\end{eqnarray*}$$where βA(z) = f(z)/bA, with f(z) being the logarithmic growth rate of linear perturbations in the matter power spectrum. The $C_{\ell }^{A\times B}(z,z^{\prime })$ that results from using $W_A^{\rm tot}(k, z)$ and $W_B^{\rm tot}(k, z)$ contains cosmological terms proportional to bAbB, proportional to (bA + bB)/2, and independent of both bA and bB. We label these terms $C_{\ell }^{(2)}(z,z^{\prime })$, $C_{\ell }^{(1)}(z,z^{\prime })$, and $C_{\ell }^{(0)}(z,z^{\prime })$ respectively. They are calculated from CLASS via linear combinations of Cℓ(z, z′) computations without RSD and bias 1, with RSD and bias 1, and with RSD and bias 0. With this formalism, the cross-power spectrum of two biased matter tracers is given by
(14)$$\begin{eqnarray*}
C_{\ell }^{A\times B}(z,z^{\prime }) = b_A b_B C_{\ell }^{(2)}(z,z^{\prime }) + \frac{b_A {+} b_B}{2}C_{\ell }^{(1)}(z,z^{\prime }) + C_{\ell }^{(0)}(z,z^{\prime }). \nonumber \\
\end{eqnarray*}$$The bias dependence of these terms is reminiscent of the Kaiser correction in power spectrum space. Indeed, these equations can be derived by including the Kaiser enhancement term in a plane-wave expansion of the power spectrum and integrating along the line-of-sight (Padmanabhan et al. 2007).


We can even further extend this analysis and constrain cosmic-ray acceleration efficiency. Using accretion shock models like Pavlidou & Fields (2006), we can estimate particle acceleration energy efficiency by comparing the kinetic power of accreted gas that follows from the model, with the power that goes into SFCR particles that is constrained by the detected neutrino and gamma-ray fluxes. For example, if we assume that SFCRs are dominantly accelerated in shocks that produce gamma-ray and neutrino spectra with index α = 2.2 (corresponding to Mach number ∼5; Bell 1978; Schlickeiser 2010), then from using the latest observed neutrino flux (with index αν = 2.46) as the upper limit, it follows that the energy efficiency of accelerating cosmic rays at these structures in the energy range 1 GeV–1 TeV is ∼40%, that is, we find that ∼40% of energy that goes into shocks gets converted into accelerated particles. This is a slightly higher efficiency compared to results from numerical models of nonlinear diffusive shock acceleration of, for example, Kang & Ryu (2013), who found that shocks of similar strength result in ≳10% acceleration energy efficiency. If, on the other hand, we consider spectral index α = 2 to represent the SFCR spectra (corresponding to Mach number ∼10) and use that to find the upper limit of SFCR flux as maximally allowed by the observed EGRB and neutrino flux with index αν = 2, we find that in this case cosmic-ray acceleration energy efficiency is  1%. This efficiency is about an order of magnitude lower compared to results from the numerical model of Kang & Ryu (2013), who found that shocks of similar strength result in an efficiency of ∼20%. Such extreme changes in efficiency that follow from our analysis are, of course, due to the fact that our constraint comes from the high-energy end that is fixed by observed neutrino fluxes, while most energy in cosmic rays comes from the low-energy end due to the power-law spectra.


In this work, we analyse a set of high-resolution stellar spectra that were acquired by the High Efficiency and Resolution Multi-Element Spectrograph (HERMES; Barden et al. 2010; Sheinis et al. 2015), a multifibre spectrograph mounted on the 3.9-metre Anglo-Australian Telescope (AAT). The spectrograph has a resolving power of R ${\sim } 28\, 000$ and covers four wavelength ranges (4713–4903, 5648–5873, 6478–6737, and 7585–7887 Å), frequently also referred to as ‘spectral arms’. Observations used in this study have been taken from multiple observing programmes that make use of the HERMES spectrograph: the main GALAH survey (De Silva et al. 2015), the K2-HERMES survey (Wittenmyer et al. 2018), and the TESS-HERMES survey (Sharma et al. 2018). Those observing programmes mostly observe stars at higher Galactic latitudes (|b| > 10°) and employ different selection functions, but share the same observing procedures, reduction, and analysis pipeline (internal version 5.3; Kos et al. 2017). All three programmes are magnitude-limited, with no colour cuts (except the K2-HERMES survey), and observations predominantly fall in the V magnitude range between 12 and 14. This leads to an unbiased sample of mostly southern stars that can be used for different population studies, such as multiple stellar systems in our case. Additionally, stellar atmospheric parameters and individual abundances derived from spectra acquired during different observing programmes are analysed with the same The Cannon procedure (internal version 182112 that uses parallax information to infer log g of a star; Ness et al. 2015; Buder et al. 2018), so they are intercomparable. The Cannon is a data-driven interpolation approach trained on a set of stellar spectra that span the majority of the stellar parameter space (for details, see Buder et al. 2018). Whenever we refer to valid or unflagged stellar parameters in the text, only stars with the quality flag flag_cannon equal to 0 were selected.


Since 2018 the Parker Solar Probe (PSP) mission is collecting solar wind plasma and magnetic field data through the inner heliosphere, reaching the closest distance to the Sun ever reached by any previous mission (Fox et al. 2016; Kasper et al. 2021). Thanks to the PSP journey around the Sun (it has completed 11 orbits) a different picture has been drawn for the near-Sun solar wind with respect to the near-Earth one (Bale et al. 2019; Kasper et al. 2019; Chhiber et al. 2020; Malaspina et al. 2020; Bandyopadhyay et al. 2022; Zank et al. 2022). Different near-Sun phenomena have been frequently encountered, with the emergence of magnetic field flips, i.e., the so-called switchbacks (Dudok de Wit et al. 2020; Zank et al. 2020), kinetic-scale current sheets (Lotekar et al. 2022), and a scale-invariant population of current sheets between ion and electron inertial scales (Chhiber et al. 2021). Going away from the Sun (from 0.17 to 0.8 au), evidence of radial evolution of different properties of solar wind turbulence (Chen et al. 2020) as the spectral slope of the inertial range (from −3/2 close to the Sun to −5/3, at distances larger than 0.4 au), an increase of the outer scale of turbulence, a decrease of the Alfvénic flux, and a decrease of the imbalance between outward (z
+) and inward (z
−) propagating components (Chen et al. 2020) has been provided. Although the near-Sun solar wind shares different properties with the near-Earth one (Allen et al. 2020; Cuesta et al. 2022), significant differences have been also found in the variance of magnetic fluctuations (about 2 orders of magnitude) and in the compressive component of inertial range turbulence. In a similar way, Alberti et al. (2020) first reported a breakdown of the scaling properties of the energy transfer rate, likely related to the breaking of the phase-coherence of inertial range fluctuations. These findings, also highlighted by Telloni et al. (2021) and Alberti et al. (2022) analyzing a radial alignment between PSP and Solar Orbiter, and PSP and BepiColombo, respectively, have been interpreted as an increase in the efficiency of the nonlinear energy cascade mechanism when moving away from the Sun. More recently, by investigating the helical content of turbulence Alberti et al. (2022) highlighted a damping of magnetic helicity over the inertial range between 0.17 and 0.6 au suggesting that the solar wind develops into turbulence by a concurrent effect of large-scale convection of helicity and creation/annihilation of helical wave structures. All these features shed new light onto the radial evolution of solar wind turbulence that urges to be considered in expanding models of the solar wind (Verdini et al. 2019; Grappin et al. 2021), and also to reproduce and investigate the role of proton heating and anisotropy of magnetic field fluctuations (Hellinger et al. 2015).


As we have shown that the Ep–Eγ relation does not significantly evolve with redshift, we can use it to calibrate GRBs. To avoid the circularity problem, the Padé method proposed by Liu & Wei (2014) is applied. The main calibrating procedures are as follows: First, derive the distance-redshift relation of SNe Ia [here we use the Union2.1 (Suzuki et al. 2012) data set] using the Padé approximation of order (3,2), i.e.

(17)

\begin{equation}
\mu (z)=\frac{\alpha _0+\alpha _1z+\alpha _2z^2+\alpha _3z^3}{1+\beta _1z+\beta _2z^2},
\end{equation}

where the coefficients (α0, α1, α2, α3, β1, β2) and the corresponding covariance matrix are derived by fitting equation (17) to the Union2.1 data set (see Liu & Wei 2014 for details). Assuming that the low-z GRBs trace the same Hubble diagram to SNe Ia, we can calculate the distance moduli of low-z GRBs directly from equation (17). The uncertainty of μ propagates from the uncertainties of the coefficients (αi, βi). Then the luminosity distance of low-z GRBs can be obtained using the relation

(18)

\begin{equation}
\mu (z)=5\log \frac{d_L(z)}{\rm {Mpc}}+25.
\end{equation}

As dL is known, the collimation-corrected energy can be further calculated from equation (11). Note that there are only 12 low-z GRBs and 12 high-z GRBs available since the others have no measurement of jet opening angle. Then we fit the Ep–Eγ relation (i.e. equation (5)) to the 12 low-z GRBs, which gives the best-fitting parameters

(19)

\begin{eqnarray}
\sigma _{\rm int}=0.161\pm 0.059,\ \ a=50.632\pm 0.062,\ \ b=1.537\pm 0.145.\nonumber\\
\end{eqnarray}

By directly extrapolating the Ep–Eγ relation to high-z GRBs, we can inversely obtain the collimation-corrected energy for 12 high-z GRBs from equation (5). Finally, calculate the luminosity distance of high-z GRBs from equation (11), and then the distance moduli from equation (18). The uncertainty of distance moduli propagates from the uncertainties of Eγ, Sbolo and Fbeam, i.e. (Schaefer 2007),

(20)

\begin{equation}
\sigma _{\mu }^2=\left(\frac{5}{2\ln 10}\right)^2\left[(\ln 10)^2\sigma _{\log E_{\gamma }}^2 + \frac{\sigma _{S_{\rm bolo}}^2}{S_{\rm bolo}^2} + \frac{\sigma _{F_{\rm beam}}^2}{F_{\rm beam}^2}\right],
\end{equation}

where

(21)

\begin{eqnarray}
\sigma _{\log E_{\gamma }}^2=\sigma _a^2 + \left(\sigma _b\log \frac{E_{p,i}}{300\ {\rm keV}}\right)^2 + \left(\frac{b}{\ln 10}\frac{\sigma _{E_{p,i}}}{E_{p,i}}\right)^2 + \sigma _{\rm int}^2.\nonumber\\
\end{eqnarray}


Several outcomes from the IllustrisTNG simulations have validated the model against observational constraints, making them suitable for the tasks at hand. Among them we highlight: the shape and width of the red sequence and the blue cloud of z = 0 galaxies (Nelson et al. 2018a), the existence and locus of the star formation main sequence (MS) at low redshifts (Donnari et al. 2019), the distribution of stellar mass across galaxy populations at z ≲ 4 (Pillepich et al. 2018), the galaxy size–mass relation for star-forming and quiescent galaxies at 0 ≤ z ≤ 2 (Genel et al. 2018), the evolution of the galaxy mass–metallicity relation (Torrey et al. 2018), and quantitatively consistent optical morphologies in comparison to Pan-STARRS data (Rodriguez-Gomez et al. 2019) as far as galaxy properties are concerned. These come in addition to observationally-consistent results in relation to the properties of massive hosts and their intra-halo gas: e.g. the X-ray signals of the hot gaseous atmospheres (Davies et al. 2020; Truong et al. 2020), the amount and distribution of highly ionized Oxygen around galaxies (Nelson et al. 2018b), and the distributions of metals in the ICM at low redshifts (Vogelsberger et al. 2018). Importantly, the new physical mechanisms included in the TNG model have been shown to return galaxy populations whose star formation activity is in better agreement with observations than previous calculations like Illustris (Genel et al. 2014; Vogelsberger et al. 2014; Sijacki et al. 2015), in galaxy colours (Weinberger et al. 2018; Nelson et al. 2018a), atomic and molecular gas content of satellite galaxies (Stevens et al. 2019), and quenched fractions of central and satellite galaxies taken together with no distinction (z ≲ 2; Donnari et al. 2019). In a companion paper (Donnari et al. 2020), we discuss in detail the level of agreement between TNG results and observations when centrals and group and cluster satellites are considered separately and show that the quenched fractions of TNG galaxies are overall consistent with observational constraints and hence the results of this paper trustworthy.


As discussed in Section 1, to identify SMGs in the simulation, it is ideal to compute individual simulated galaxies’ submm flux densities via dust radiative transfer. The mass and spatial resolution of Illustris and IllustrisTNG are insufficient to resolve the subkpc structure of the ISM, and owing to both the subgrid equation-of-state treatment of stellar feedback and resolution, such simulations tend to feature overly puffy discs and thus underestimate the attenuation (Trayford et al. 2017). Consequently, we do not perform radiative transfer directly. Instead, we use the following relation based on the results of performing radiative transfer on higher resolution hydrodynamical simulations of idealized isolated discs and mergers (Hayward et al. 2011, 2013a):
(1)$$\begin{eqnarray*}
\frac{S_{850}}{\left(\text{mJy}\right)} = 0.81 \left(\frac{\text{SFR}}{100 \, \left(\text{M}_\odot \, \text{yr}^{-1}\right)} \right)^{0.43} \left(\frac{M_{\rm dust}}{10^8 \, \left(\text{M}_\odot \right)} \right)^{0.54},
\end{eqnarray*}$$where SFR is the ‘instantaneous’ SFR associated with the star-forming gas cells (for consistency with Hayward et al. 2011) and Mdust is the dust mass. When applying the relation, following Hayward et al. (2013b), we incorporate a scatter of 0.13 dex (independent of SFR and dust mass) assuming a Gaussian distribution (Hayward et al. 2011). The submm flux densities predicted using this relation also agree well with the results of performing dust radiative transfer directly on cosmological zoom simulations (Liang et al. 2018; Cochrane et al. 2019) and a semi-analytical model ( Benson, private communication). Moreover, by combining the above with the observed SFR–M⋆ and Mdust(M⋆, z) relations, one can derive an S850–M⋆ relation (Hayward 2012; Hayward et al. 2013a) that agrees reasonably well with that observed for SMGs (Davies et al. 2013). Although given sufficient resolution and unlimited computing time, it would be preferred to perform radiative transfer directly, our approach has the advantage of trivial computational expense, and it may actually be more accurate than direct radiative transfer given the resolution of state-of-the-art large-volume simulations such as Illustris and IllustrisTNG .


Consider a rotating tornado with a straight magnetic field (Ω0 ≠ 0,J0 = 0). In this case, the dispersion relation (14) reduces to (15)\begin{eqnarray} \lefteqn{(\omega^{2}-C_{\mathrm {A}}^{2}k^{2})(\omega^{2}-C_{\rm T}^{2}k^{2})}\nonumber\\ & &+\frac{a^2}{4(C_{\mathrm {A}}^{2}+C_{\rm s}^{2})}\left[4\Omega_{0}^{2}\omega^{4}-2\Omega_0^2k^2C_{\rm s}^2\omega^{2} -2\Omega_0^2k^{4}C_{\rm s}^{2}C_{\mathrm {A}}^{2})\right]\nonumber\\ & &-\frac{a^2}{4(C_{\mathrm {A}}^{2}+C_{\rm s}^{2})}\left[(\omega^{2}-C_{\mathrm {A}}^{2}k^{2})^{2}(\omega^{2}-C_{\rm s}^{2}k^{2})\right]=\left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\ & & \times \frac{(\omega^2-k^2C_\mathrm{A}^2)(\omega^2-k^2C_{\rm s}^2)(\omega^2-k^2C_{\mathrm{Ae}}^2)a}{(C_\mathrm{A}^2+C_{\rm s}^2)m_{\rm e}}\frac{K_0(m_{\rm e}a)}{K_1(m_{\rm e}a)}\cdot \label{rotationext} \end{eqnarray}(ω2−CA2k2)(ω2−CT2k2)+a24(CA2+Cs2)[4Ω02ω4−2Ω02k2Cs2ω2−2Ω02k4Cs2CA2)]−a24(CA2+Cs2)[(ω2−CA2k2)2(ω2−Cs2k2)]=ρ0e2ρ0The second term on the left-hand side of Eq. (15) shows the effect of the internal rotation where the third term indicates the confinement of the tube. The term on the right-hand side of Eq. (15) shows the effect of the perturbations of the external medium. If we write the dispersion relation in terms of the fast and slow speeds we would have (16)\begin{eqnarray} &&\left(C_{\mathrm {A}}^{2}+C_{\rm s}^{2}+2{\cal R}C_{\mathrm {A}}^{2}\right)\left(\omega^{2}-C_{+}^{2}k^{2}\right)\left(\omega^{2}- C_{-}^{2}k^{2}\right)\nonumber\\ &&\quad-\frac{a^2}{4}\left(\omega^{2}-C_{\mathrm {A}}^{2}k^{2}\right)^{2}\left(\omega^{2}-C_{\rm s}^{2}k^{2}\right)= \left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\&&\qquad\times \frac{\left(\omega^2-k^2C_\mathrm{A}^2\right)\left(\omega^2-k^2C_{\rm s}^2\right)\left(\omega^2-k^2C_{\mathrm{Ae}}^2\right)a}{m_{\rm e}}\frac{K_0(m_{\rm e}a)}{K_1(m_{\rm e}a)}, \end{eqnarray}(CA2+Cs2+2ℛCA2)(ω2−C+2k2)(ω2−C−2k2) −a24(ω2−CA2k2)2(ω2−Cs2k2)=ρ0e2ρ0  ×(ω2−k2CA2)(ω2−k2Cs2)(ω2−k2CAe2)ameK0(mea)K1(mea),where (17)\begin{eqnarray} \label{PhaseRotation} C_{\pm}^{2}&=&C_{\mathrm {A}}^{2}\frac{(C_{\mathrm {A}}^{2}+2C_{\mathrm {s}}^{2})+{\cal R}C_{\mathrm {s}}^{2}\pm\sqrt{{\cal P}}}{2(C_{\mathrm {A}}^{2}+C_{\mathrm {s}}^{2})+4C_{\mathrm {A}}^{2}{\cal R}},\nonumber\\ {\cal P}&=&C_{\mathrm {A}}^{4}-2{\cal R}C_{\mathrm {s}}^{2}(C_{\mathrm {A}}^{2}-4C_{\mathrm {s}}^2)+{\cal R}^{2}C_{\mathrm {s}}^{2}(8C_{\mathrm {A}}^{2}+C_{\mathrm {s}}^{2}), \end{eqnarray}C±2=CA2(CA2+2Cs2)+ℛCs2±𝒫2(CA2+Cs2)+4CA2ℛ,𝒫=CA4−2ℛCs2(CA2−4Cs2)+ℛ2Cs2(8CA2+Cs2),and the dimensionless parameter, ℛ, representing the rotation is (18)\begin{equation} {\cal R}=\frac{a^2\Omega^{2}_{0}}{2\,C_{\mathrm {A}}^{2}}\cdot \end{equation}ℛ=a2Ω022 CA2·Equation (17) indicates the modification of the Alfvén speed and the tube speed by the equilibrium rotation to the fast (C+) and slow (C−) magnetosonic speeds, respectively (see also Zhugzhda & Nakariakov 1999; Vasheghani Farahani et al. 2010). As such the torsional wave would propagate with the speed C+, and the longitudinal wave would propagate with the speed C−. If we consider the dispersion to be weak (k2A0 ≪ 1), and take \hbox{$\omega^{2}\approx\,C_{\pm}^{2}k^{2}$}ω2≈ C±2k2, an explicit expression for the dispersion relation could be obtained (19)\begin{eqnarray} \label{PhaseR} \omega^{2}&\approx&\,C_{\pm}^{2}k^{2}\pm\,\frac{a^2}{4}\frac{(C_{\pm}^{2}-C_{\mathrm {A}}^{2})^{2}(C_{\pm}^{2}-C_{\mathrm {s}}^{2})}{C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}k^{4}+ \left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\&&\quad\times \frac{(C_{\pm}^2-C_\mathrm{A}^2)(C_{\pm}^2-C_{\rm s}^2)(C_{\pm}^2-C_{\mathrm{Ae}}^2)a}{GC_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}\nonumber\\&&\quad\times\frac{K_0(Ga)}{K_1(Ga)}k^4, \end{eqnarray}ω2≈ C±2k2± a24(C±2−CA2)2(C±2−Cs2)CA2 𝒫k4+ρ0e2ρ0 ×(C±2−CA2)(C±2−Cs2)(C±2−CAe2)aGCA2 𝒫 ×K0(Ga)K1(Ga)k4,where (20)\begin{eqnarray} G^2=\frac{\left(k^2C^2_{\mathrm{Ae}}-C_{\pm}^2\right)\left(k^2C^2_{\rm se}-C_{\pm}^2\right)} {\left(C^2_{\mathrm{Ae}}+C^2_{\rm se}\right)\left(k^2C^2_{\rm Te}-C_{\pm}^2\right)}, \,\mbox{ \ } C_{\rm Te}^{2}=\frac{C_{\mathrm {Ae}}^{2}C_{\rm se}^{2}}{C_{\mathrm {A}e}^{2}+C_{\rm se}^{2}}\cdot \label{me} \end{eqnarray}G2=(k2CAe2−C±2)(k2Cse2−C±2)(CAe2+Cse2)(k2CTe2−C±2), CTe2=CAe2Cse2CAe2+Cse2·The second term on the right-hand side of Eq. (19) is the dispersive correction term, while the third term on the right-hand side represents the effects of the perturbation of the external medium. Equation (19) clearly shows the interplay of the equilibrium rotation and the density contrast of the internal and external media. The modification of the speeds at k = 0 is clearly exhibited by both the rotation and the density contrast. Hence, the modification of the tube and Alfvén speeds to the slow and fast magnetoacoustic speeds is clearly observed, see also Zhugzhda & Nakariakov (1999). In the long wavelength limit where the tube radius is much smaller than the wavelength, the dispersion would be week, therefore the arguments of the modified Bessel function of the second kind K would be small. In such a regime we are able to use the first terms of the expansions of the modified Bessel function of the second kind for small arguments (Abramowitz et al. 1988) which are (21)\begin{eqnarray} K_{0}(Ga)=-\mathrm{ln}\left(\frac{1}{2}Ga\right),\,\,\,\,\,\,\,\,\, K_{1}(Ga)=\frac{1}{Ga}\cdot \label{K} \end{eqnarray}K0(Ga)=−ln12Ga,         K1(Ga)=1Ga·By substituting the terms of Eq. (21) in Eq. (19) we obtain an explicit expression that is easier to read: (22)\begin{eqnarray} \omega^{2}\approx\,C_{\pm}^{2}k^{2}\pm\,\frac{a^2}{4}\frac{(C_{\pm}^{2}-C_{\mathrm {A}}^{2})^{2}(C_{\pm}^{2}-C_{\mathrm {s}}^{2})}{C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}k^{4}- \left(\frac{\rho_{0e}}{\rho_0}\right)\nonumber\\\times \frac{a^2}{2}\frac{(C_{\pm}^2-C_\mathrm{A}^2)(C_{\pm}^2-C_{\rm s}^2)(C_{\pm}^2-C_{\mathrm{Ae}}^2)}{ C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}\mathrm{ln}\left(\frac{1}{2}Ga\right)k^4. \label{PhaseR2} \end{eqnarray}ω2≈ C±2k2± a24(C±2−CA2)2(C±2−Cs2)CA2 𝒫k4−ρ0eρ0×a22(C±2−CA2)(C±2−Cs2)(C±2−CAe2)CA2 𝒫ln12Gak4.Equation (22) represents the dispersion relation for magnetoacoustic waves propagating in a steadily rotating magnetic flux tube (e.g. a solar tornado), which is more convenient for observation purposes and coronal seismology.


Evolution of the SMD (top) and SFRD (bottom) along the cosmic history (see top axis for the corresponding redshift). For these plots, we assumed all three BBG candidates without ALMA detection to be real passive galaxies at z ∼ 6. In the top panel, the SMD of our BBG sample at z ∼ 6 (red circle) is shown in conjunction with those of star-forming (cyan symbols) and passive (magenta symbols) galaxies at lower redshifts from the literature (M13: Muzzin et al. 2013; D14: Duncan et al. 2014; S14: Straatman et al. 2014; G15: Grazian et al. 2015; S16: Song et al. 2016; D17: Davidzon et al. 2017; B19: Bhatawdekar et al. 2019; K19: Kikuchihara et al. 2019). The vertical error bar associated with our BBG data corresponds to a 1σ uncertainty propagated from the Poisson error (Gehrels 1986) for the BBG number and the SED fitting uncertainty for the stellar mass. The horizontal error bar shows the redshift range expected from our BBG color selection. In the bottom panel, the red shaded region corresponds to the SFRD expected from the progenitors of the z ∼ 6 BBGs at a 99.7% confidence level (3σ). The SFRD measurements at

 are collected from the literature (MD14: Madau & Dickinson 2014; O13: Oesch et al. 2013; O14: Oesch et al. 2014; F15: Finkelstein et al. 2015a; M16: McLeod et al. 2016; B16: Bouwens et al. 2016; I18: Ishigaki et al. 2018; O18: Oesch et al. 2018; B19: Bhatawdekar et al. 2019). All of them at

 are estimated by integrating the UVLFs down to MUV = −17 mag. The SFRD estimated at z ∼ 17 from an observed global 21 cm absorption trough (M18: Madau 2018; Bowman et al. 2018) is also shown in yellow. The functional fit to the MD14 data, which is proportional to (1 + z)−2.9 at high-z (Madau & Dickinson 2014), is superposed by the solid line. Two other power-law functions supporting an accelerated evolution at z ≳ 8 (

 Oesch et al. 2014) and a smooth evolution from lower redshift (

 Finkelstein et al. 2015a) are shown by dotted–dashed and dotted lines, respectively. The SFRD derived assuming a universal relation among the halo mass, SFR, and dark matter accretion rate (Harikane et al. 2018) is also superposed by the gray shade in its 1σ uncertainty. All of the SMD and SFRD measurements from the literature are corrected for the stellar IMF and the cosmological model to match those in this work.


I denote by λi and ϖi the mean longitude and longitude of periastron of planet i (in astrocentric coordinates), respectively. The actions canonically conjugated to the angles λi and −ϖi are the circular angular momentum Λi and the angular momentum deficit (AMD, see Laskar 2000) Di, respectively. These actions are defined as follows \begin{eqnarray} \Lambda_i &=& \beta_i\sqrt{\mu_i a_i},\\  D_i &=& \Lambda_i-G_i = \Lambda_i \left(1-\sqrt{1-e_i^2}\right), \end{eqnarray}Λi=βiμiai,Di=Λi−Gi=Λi1−1−ei2,where \hbox{$G_i=\Lambda_i\sqrt{1-e_i^2}$}Gi=Λi1−ei2 is the angular momentum of planet i, βi = mim0/ (m0 + mi), \hbox{$\mu_i = \G (m_0+m_i)$}μi = 𝒢(m0 + mi). At low eccentricities the deficit of angular momentum Di is proportional to \hbox{$e_i^2$}ei2. The Hamiltonian (Eq. (3)) can be expressed using these action-angle coordinates (6)\begin{equation} \label{eq:hamLaD}  \H = - \sum_{i=1}^n \frac{\mu_i^2\beta_i^3}{2\Lambda_i^2}  + \sum_{1\leq i<j\leq n} \H_{i,j}(\Lambda_i,\Lambda_j,D_i,D_j,\lambda_i,\lambda_j,\varpi_i,\varpi_j), \end{equation}ℋ=−∑i=1nμi2βi32Λi2+∑1≤ij≤nℋi,j(Λi,Λj,Di,Dj,λi,λj,ϖi,ϖj),where the first sum is the Keplerian part, which depends only on Λi (or equivalently ai), and ℋi,j is the perturbation between the planets i and j, which depends on the eight coordinates associated to i and j. I follow the method described in Laskar & Robutel (1995) to compute Hi,j as a power series of the eccentricities (or equivalently of \hbox{$\sqrt{D_i}$}Di and \hbox{$\sqrt{D_j}$}Dj), and a Fourrier series of the angles, where the coefficients are functions of Λi and Λj (i.e., of the semi-major axes). For a system that is close to the resonance or resonant, the semi-major axes remain close to the nominal resonant values (Kepler’s third law) (7)\begin{equation} \label{eq:nominalsma}  \frac{a_i}{a_j} \approx \frac{a_{i,0}}{a_{j,0}} = \left(\frac{k_{i,j}}{k_{j,i}}\right)^{2/3} \left(\frac{\mu_i}{\mu_j}\right)^{1/3}\cdot \end{equation}aiaj≈ai,0aj,0=ki,jkj,i2/3μiμj1/3·I introduce (8)\begin{equation} \Delta\Lambda_i = \Lambda_i - \Lambda_{i,0}, \end{equation}ΔΛi=Λi−Λi,0,where (9)\begin{equation} \Lambda_{i,0} = \beta_i\sqrt{\mu_i a_{i,0}}, \end{equation}Λi,0=βiμiai,0,and expand the Keplerian part at degree 2, and the perturbative part at degree 0 in ΔΛi(10)\begin{eqnarray} \label{eq:hamdLaD}  \H &=& \sum_{i=1}^n n_{i,0} \Delta\Lambda_i  - \frac{3}{2}\frac{n_{i,0}}{\Lambda_{i,0}}\Delta\Lambda_i^2\nonumber\\  &&\quad+ \sum_{1\leq i<j\leq n} \H_{i,j}(D_i,D_j,\lambda_i,\lambda_j,\varpi_i,\varpi_j), \end{eqnarray}ℋ=∑i=1nni,0ΔΛi−32ni,0Λi,0ΔΛi2 +∑1≤ij≤nℋi,j(Di,Dj,λi,λj,ϖi,ϖj),where ni,0 is the nominal mean motion of planet i, such that (11)\begin{equation} \frac{n_{i,0}}{n_{j,0}} = \frac{k_{j,i}}{k_{i,j}}\cdot \end{equation}ni,0nj,0=kj,iki,j·The perturbative part does not depend on Λi anymore, but is simply evaluated at Λi,0.


We made use of this catalogue, extracting three main physical properties. The first is the luminosity-weighted age (𝒜⋆,L) of the underlying stellar population, which corresponds to the first moment of the age distribution function (ADF) in light, directly provided by the stellar decomposition performed by PIPE3D. This parameter is directly connected with how sharp the SFH is in galaxies, with older stellar populations having SFHs that peak at earlier epochs (e.g. García-Benito et al. 2017). The second is the luminosity-weighted metallicity normalized to the solar value ([Z/H] or 𝒵⋆,L). This is the first moment of the metallicity distribution function (MDF) in light at the same band as the ADF (e.g. Mejía-Narváez et al. 2020). Due to the nature of the adopted SSPs, this parameter is a direct proxy of the luminosity-weighted iron abundance, [Fe/H], for a solar abundance of 12 + log(Fe/H) = 7.50 (Asplund et al. 2009). It is worth noting that we derive [Z/H] using SSP models with solar composition (i.e. [α/Fe] = 0). This may create a bias in [Fe/H] towards higher values, which can be as large as ∼30%. However, this bias affects neither our qualitative results nor the reported trends, although it may produce relative offsets in the absolute reported abundances. Hereafter we refer to this parameter as [Fe/H]. Finally, the third is the gas-phase oxygen abundance, 12 + log(O/H), derived using the calibrator presented by Ho (2019). This calibrator uses a set of emission line ratios and associates them with oxygen abundances measured using the direct method, applying a neural-network procedure. We should note that the use of this or another of the calibrators applied to derive the oxygen abundance for the considered catalogue does not qualitatively alter the results presented here (Espinosa-Ponce et al., in prep.). We transformed this value to [O/H] by adopting a solar oxygen abundance of 12 + log(O/H)⊙ = 8.69 (Asplund et al. 2009).


In more detail, the goal of this paper is to test whether preferential alignment of the CNM structures with BPOS together with fluctuations in the GMF orientation along the LOS can account fully for a number of observed statistical properties of the dust polarization over the southern low column density sky. These statistical properties include the scatter of the polarization fraction p, the dispersion of the polarization angle ψ around the large-scale GMF, and the observed dust power spectra \hbox{$\dlee$}𝒟ℓEE, \hbox{${{\cal D}_{\ell}^{BB}}$}𝒟ℓBB, and \hbox{${{\cal D}_{\ell}^{TE}}$}𝒟ℓTE at 353GHz. In doing so we extend the analysis of Planck Collaboration Int. XXXVIII (2016) and Planck Collaboration Int. XLIV (2016) to sky areas in which the filaments have very little contrast with respect to the diffuse background emission. Within the southern Galactic cap (defined here to be b ≤ −30°), in which dust and gas are well correlated (Boulanger et al. 1996; Planck Collaboration XI 2014) and the GASS HI emission is a good tracer of the dust intensity structures (Planck Collaboration Int. XVII 2014),we focus our study on the low column density portion. We choose to model the dust polarization sky at 1° resolution, which corresponds to retaining multipoles ℓ ≤ 160 where the Galactic dust dominates over the CMB B-mode signal. Also, most of the ground-based CMB experiments are focusing at a degree scale for the detection of the recombination bump of the CMB B-mode signal. Our dust polarization model should be a useful product for the CMB community. It covers 34% of the southern Galactic cap, a 3500 deg2 region containing the cleanest sky accessible to ongoing ground and balloon-based CMB experiments: ABS (Essinger-Hileman et al. 2010), Advanced ACTPol (Henderson et al. 2016), BICEP/ Keck (Ogburn et al. 2012), CLASS (Essinger-Hileman et al. 2014), EBEX (Reichborn-Kjennerud et al. 2010), Polarbear (Kermish et al. 2012), Simons array (Suzuki et al. 2016), SPIDER (Crill et al. 2008), and SPT3G (Benson et al. 2014).


Ring systems are a ubiquitous feature in planetary systems – all the gas giants in the Solar System have ring systems around them of varying optical depths (see, e.g., Tiscareno 2013; Charnoz et al. 2018), and ring systems have been detected around minor planets (e.g., Chariklo; Braga-Ribas et al. 2014), so it is reasonable that exoplanets and substellar objects host ring systems as well. Long-period eclipsing binary star systems, where one star is surrounded by an extended dark disk-like structure that periodically eclipses the other component, have already been observed, such as EE Cep (Mikolajewski & Graczyk 1999), ϵ Aurigae (Guinan & Dewarf 2002), and TYC 2505-672-1, with a companion period of 69 yr (Lipunov et al. 2016; Rodriguez et al. 2016). A large ring-like structure around a substellar companion was proposed to explain observations from 2007 from the J1407 system (Mamajek et al. 2012). 1SWASP J140747.93-394 542.6 (V1400 Cen; hereafter called “J1407”) is a young, pre-main-sequence star in the Sco-Cen OB association (Mamajek et al. 2012) with spectral type K5 IV(e) Li and is similar in size and mass to the Sun. In 2007, it displayed a complex symmetric dimming pattern of up to ~ 3 magnitudes during a 56 day eclipse. This has been attributed to the transit of a substellar companion (called “J1407 b”) with a mass of 60–100 MJup (Rieder & Kenworthy 2016) surrounded by an exoring system consisting of at least 37 rings and extending out to 0.6 au in radius (Kenworthy & Mamajek 2015). For these rings to show detectable transit signatures, they must be significantly misaligned with respect to the orbital plane of J1407 b (Zanazzi & Lai 2017). This potential ring system would be considerably larger than the ring system of Saturn, which is located within the planet’s tidal disruption radius. The proposed rings around J1407 b would even cover a significant fraction of the companion’s Hill sphere and would not be expected to be stable over gigayear timescales. If the candidate ringed companion is in a bound orbit around the star, this orbit must be moderately eccentric in order for no othereclipses to have been detected to date (Kenworthy et al. 2015), raising the possibility that there might be a second as yet undetected companion in the system that causes the implied orbital eccentricity for J1407 b. Radial velocity measurements are overwhelmed by the chromospheric noise of the star and do not place strong constraints on other substellar companions (Kenworthy et al. 2015). The transit of J1407 suggests that its orbital plane has a high inclination to our line of sight – if there are other planets inside the orbit of J1407 b, their orbits may well be coplanar with J1407 b and there is a high chance that these companions may transit J1407.


During synchrotron self-Compton emission, X-ray photons are produced by the
              scattering of the synchrotron radiation from radio to NIR on their own electron
              population. If we compare the fluxes produced by the synchrotron and SSC emissions,
              the variation of the X-ray/NIR ratio constrains the size evolution of the flaring
              source. Let us consider a spherical source of radius R with a power law
              energy distribution of relativistic electrons. Following Van der Laan (1966), the radiative transfer for the synchrotron
              radiation can be computed as (2)\begin{equation} S_\mathrm{SYN}=\int_{0}^{R}{\frac{\epsilon_\nu}{\kappa_\nu}\,\left(1-{\rm e}^{-\tau_\nu(r)}\right)\,2\pi r\,{\rm d}r} , \end{equation}

S
SYN

=

∫
0
R

ϵ
ν

κ
ν

(
1
−

e

−

τ
ν

(
r
)

)

2
πr

d
r

,

with κν ∝
                  B(p +
                    2)/2ν−(p +
                4)/2 the absorption coefficient, ϵν ∝
                  B(p +
                    1)/2ν−(p −
                1)/2 the emission coefficient, B the magnetic field
                (Lang 1999) and τν(r)
              the optical depth, which can be computed at each distance r from the sphere
              center as: (3)\begin{equation} \tau_\nu(r)=\int_{0}^{2\sqrt{R^2-r^2}}{\kappa_\nu {\rm d}l} . \end{equation}

τ
ν

(
r
)
=

∫
0

2

R

2

−

r

2

κ
ν

d
l

.

Assuming that we are in the optically
              thin regime (i.e., τν(r) ≪
                1), we utilize formula 3 of Marrone
                et al. (2008): SSYN ∝ B(p +
                  1)/2ν−(p −
                  1)/2R3. For synchrotron
              radiation, we have \hbox{$B \propto R^4\,\nu_\mathrm{m}^{5}\,S_\mathrm{m}^{-2}$}
B
∝

R
4

ν
m
5

S
m
-2

 with Sm the
              maximum flux density of the spectral energy distribution occurring at frequency
                νm (Marscher 1983). Finally, the synchrotron radiation can be expressed using
                p = 2α
                + 1 as (4)\begin{equation} S_\mathrm{SYN} \propto R^{4\alpha+7}\,\nu_\mathrm{m}^{5(\alpha+1)}\,S_\mathrm{m}^{-2(\alpha+1)}\,\nu^{-\alpha}\, . \end{equation}

S
SYN

∝

R

4
α
+
7

ν
m

5
(
α
+
1
)

S
m

−
2
(
α
+
1
)

ν

−
α

.

The SSC radiation of X-ray photons is
              (formula 4 of Marscher 1983):
                  (5)\begin{equation} S_\mathrm{SSC} \propto R^{-2(2\alpha+3)}\,\nu_\mathrm{m}^{-(3\alpha+5)}\,S_\mathrm{m}^{2(\alpha+2)}\,\ln \left(\frac{\nu_2}{\nu_\mathrm{m}}\right)\,\nu^{-\alpha} . \end{equation}

S
SSC

∝

R

−
2
(
2
α
+
3
)

ν
m

−
(
3
α
+
5
)

S
m

2
(
α
+
2
)

ln

ν
2

ν
m

ν

−
α

.

The natural logarithm in this equation
              could be approximated by c1
                  (ν2/νm)c2
              with c1 =
                1.8 and c2 = 0.201 (Eckart et al. 2012b). The synchrotron-to-SSC flux ratio is
                  (6)\begin{equation} \frac{S_\mathrm{SSC}}{S_\mathrm{SYN}} \propto R^{-(8\alpha+13)}\,\nu_\mathrm{m}^{-(8\alpha+10+c_2)}\,S_\mathrm{m}^{4\alpha+6}\, . \end{equation}

S
SSC

S
SYN

∝

R

−
(
8
α
+
13
)

ν
m

−
(
8
α
+
10
+

c
2

)

S
m

4
α
+
6

.

We therefore have three parameters that
              may vary during the flare to explain the increased ratio of X-ray and NIR flux (Fig.
                15). Considering the plasmon model, for which
              a spherical source of relativistic electrons expands and cools adiabatically, we have
                (Van der Laan 1966): νm ∝
                    R−(8α + 10)/(2α +
                  5) and Sm ∝ R−(14α +
                    10)/(2α + 5). Thus, SSSC/SSYN ∝
                  R−β with
                β ≡
                  (8α2 + (30−8
                  c2)α + 25−10
                  c2)/(2α + 5). We
              first consider the adiabatic expansion. For our observation, the ratio between the
              X-ray and the NIR flux increases during the 2014 Mar. 10 flare, implying that
                R−β must increase
              as the radius R increases. This condition is satisfied if the
              exponent β is negative and thus if the α value is lower than
                −2.5 or is between
                −2.3 and −1.25, which is inconsistent because
                α must
              be positive. The expansion case is thus likely to be rejected under the hypothesis of
              an optically thin plasmon that expands adiabatically. We can also consider the case
              where the plasmon is compressed during its motion through a bottle-neck configuration
              of the magnetic field. We can still use the equations of Van der Laan (1966), since the conservation of the magnetic flux is
              explicitly taken into account. The compression case is thus preferred, because it
              allows positive values of α for β> 5.4. Thus, for
              the SYN-SSC process, the plasmon must be adiabatically compressed with at least
                SSSC/SSYN ∝
                  R-5.4. Therefore, the observed
              increase of the X-ray-to-NIR flux ratio by a factor of 10 in 1.2 h implies a decrease
              of the radius by a factor of about 0.6. The average compression velocity is estimated
              as Vcomp =
                  ΔR/ Δt, leading to
                | Vcomp |
                  /c 0.0034
                R/Rs with
                Rs the Schwarzschild radius
                (Rs = 1.2 ×
                  1012 cm for Sgr A*, which corresponds to 0.08 au). For
              comparison, the expansion velocities computed with this model in the literature range
              between 0.0028 and 0.15c (Yusef-Zadeh et
                al. 2006a, 2009; Eckart et al. 2008), which is of the same order as the compression
              velocity computed here. Thus, the model of an adiabatic compression of a plasmon is
              the likely hypothesis to explain the variation of the ratio between X-ray and NIR
              flux, in the context of the SYN-SSC process.


The sky distribution of gamma-ray bursts (GRBs; Piran 2004; Mészáros 2006; Vedrenne & Atteia 2009; Gomboc 2012; Kouveliotou, Wijers & Woosley 2012; Kumar & Zhang 2015; Dai, Daigne & Mészáros 2017; Willingale & Mészáros 2017) has been tested intensively and early works claimed that GRBs were distributed isotropically (Meegan et al. 1992; Briggs et al. 1996; Tegmark et al. 1996). As the amount of observational data increased and various methods were applied, it was claimed that the group of short GRBs (T90  2 s; Balázs et al. 2003), which originate in mergers of compact objects such as neutron stars (Paczynski 1986; Eichler et al. 1989; Berger 2014; Abbott et al. 2017a,b,c), was distributed anisotropically (Balázs, Mészáros & Horváth 1998; Balázs et al. 1999; Magliocchetti, Ghirlanda & Celotti 2003; Vavrek et al. 2008; Tarnopolski 2017). Here, T90 is the duration during which 90  per cent of the detected counts from a GRB are accumulated (Kouveliotou et al. 1993). However, several other works claimed different results (Mészáros et al. 2000a; Mészáros, Bagoly & Vavrek 2000b; Litvin et al. 2001; Bernui, Ferreira & Wuensche 2008; Veres et al. 2010; Ukwatta & Woźniak 2016). The works that analysed GRBs with a duration of 2 ≲ T90 ≲ 10 s also found that these bursts were distributed anisotropically (Mészáros et al. 2000a,b; Litvin et al. 2001; Vavrek et al. 2008; Veres et al. 2010). However, Mészáros & Štoček (2003) and Ukwatta & Woźniak (2016) came to different conclusions. Anisotropical distribution on the sky was also proclaimed for very short GRBs (T90 ≤ 100 ms; Cline et al. 2005; Ukwatta & Woźniak 2016). In contrast, the group of long GRBs (T90 > 2 s), which are associated with the collapses of massive stars (Fruchter et al. 2006; Woosley & Bloom 2006), were found to be distributed isotropically (Balázs et al. 1998, 1999; Mészáros et al. 2000a,b; Magliocchetti et al. 2003; Vavrek et al. 2008; Ukwatta & Woźniak 2016; Tarnopolski 2017), although Mészáros & Štoček (2003) again came to a different conclusion. Mészáros et al. (2009a,b) and Mészáros (2017, 2018) have summarized these efforts.


The applicability of mid-IR and far-IR FSL as diagnostic tools of the ISM has received a boost thanks to the publication of samples of nearby galaxies observed with the infrared space observatory (ISO) and Herschel (e.g. Brauher et al. 2008; Farrah et al. 2013; Sargsyan et al. 2014; Kamenetzky et al. 2014; Cormier et al. 2015; Cigan et al. 2016; Herrera-Camus et al. 2016; Fernández-Ontiveros et al. 2016; Zhao et al. 2016; Díaz-Santos et al. 2017; Zhang et al. 2018). At high redshift (z ≳ 1), the far-IR FSL conveniently shift into the (sub)millimetre atmospheric windows. The most popular line is clearly [CII] 158 μm, followed by the [CI] 370, 609 μm lines (e.g. Walter et al. 2011; Bothwell et al. 2017). Both single dish submillimetre telescopes and interferometers have also detected the [NII] 122 μm and 205 μm lines at high redshift (Ferkinhoff et al. 2011, 2015; Combes et al. 2012; Nagao et al. 2012; Decarli et al. 2012, 2014; Béthermin et al. 2016; Pavesi et al. 2016; Lu et al. 2017; Tadaki et al. 2019; Novak et al. 2019). After a slow start, the [OIII] 88 μm line is quickly becoming a popular line to confirm redshifts of galaxies in the epoch of reionization (z ≳ 6), where it shifts into the submillimetre atmospheric windows below 500 GHz (Ferkinhoff et al. 2010; Inoue et al. 2016; Carniani et al. 2017; Marrone et al. 2018; Vishwas et al. 2018; Hashimoto et al. 2018, 2019; Walter et al. 2018; Tamura et al. 2019; Tadaki et al. 2019; Novak et al. 2019). Deep Herschel/SPIRE spectroscopy has also revealed a number of FSL detections, either in individual objects (Valtchanov et al. 2011; Coppin et al. 2012; George et al. 2013; Uzgil et al. 2016; Rigopoulou et al. 2018; Zhang et al. 2018), or in stacked spectra (Wardlow et al. 2017; Wilson et al. 2017; Zhang et al. 2018). These include the only detections of the [OI] 63 μm line at high redshift reported thus far. This [OI]3P2−3P1 line is arguably the best tracer for the star-forming gas, as it traces the very dense (

n
crit
H

$ n^{\mathrm{H}}_{\mathrm{crit}} $

 = 5×105 cm−3) neutral gas (one caveat being the frequeny presence of self-absorption observed in local ultra-luminous IR galaxies, Rosenberg et al. 2015). Like the [CII] line, the [OI] 63 μm line also shows a “deficit” in the most luminous far-IR sources, though with a higher scatter (Graciá-Carpio et al. 2011; Cormier et al. 2015; Díaz-Santos et al. 2017). Surprisingly, this bright FSL has not been frequently observed with ALMA, probably because it is only observable in the highest frequency bands. At least as surprising is that the fainter, but more accessible [OI]3P1−3P0 transition at λrest = 145 μm (

n
crit
H

$ n^{H}_{\mathrm{crit}} $

 = 9.5×104 cm−3) has only recently been detected at high redshifts (Novak et al. 2019 report a tentative detection in a z = 7.5 quasar). Also in nearby galaxies, this [OI] 145 μm line has not been observed very frequently as in most cases, it is fainter than the nearby [CII] 158 μm line. After initial detections with ISO (Malhotra et al. 2001; Brauher et al. 2008), Herschel has now detected [OI] 145 μm in significant samples of nearby galaxies (Spinoglio et al. 2015; Cormier et al. 2015; Fernández-Ontiveros et al. 2016; Herrera-Camus et al. 2018), and recently in a z = 6.5 lensed quasar (Yang et al. 2019).


It may be possible for the accretion disks to persist in systems that have ended active mass transfer. Occultations induced by persistent disks have been detected in a handful of long-period binaries. Photometric occultations by disks are often distinct from other events, as they exhibit long-duration, deep eclipse-like signals that can be identified in photometric surveys.  Aurigae is the classical example of such a system, with a post-AGB F0 supergiant with a B-star companion embedded in a dusty disk with an orbital period of 27 yr (e.g., Kuiper et al. 1937; Huang 1965; Kopal 1971). The occultation is observed in photometry (e.g., Gyldenkerne 1970; Kemp et al. 1986; Carroll et al. 1991), interferometry (Kloppenborg et al. 2010, 2015), and spectroscopy (e.g., Lambert & Sawyer 1986; Chadima et al. 2011; Leadbeater et al. 2012; Griffin & Stencel 2013; Muthumariappan et al. 2014; Strassmeier et al. 2014), and the system’s orbit and masses are constrained by the radial velocities (Stefanik et al. 2010). The 2 yr long eclipse is inferred to be from a ∼4 au diameter disk consisting of both dust and gas (e.g., Lissauer et al. 1996). The dusty component is required to explain the strong IR excess in the system (Hoard et al. 2010), while the gaseous component in the vertically extended disk shell induces a series of spectroscopic features that map out the Keplerian disk during occultation (e.g., Chadima et al. 2011; Strassmeier et al. 2014). A central brightening is seen during eclipse, leading to a flared disk geometry interpretation (e.g., Budaj 2011). Similar long-period, long-duration disk occultations in other potential mass-transfer systems have also been identified: the 96-day-period V383 Sco (Zola et al. 1994), the 5.6 yr period EE Cep (Mikolajewski & Graczyk 1999), the 468-day periodiocally recurring eclipses of OGLE-LMC-ECL-11893 (Dong et al. 2014; Scott et al. 2014), and the 1277-day periodic eclipses of OGLE-BLG182.1.162852 (Rattenbury et al. 2015). The recent advent of wide-field photometric surveys is now also enabling numerous new discoveries, including the 69 yr eclipsing disk system TYC 2505-672-1 (Rodriguez et al. 2016).


Lissauer et al. 1996
Strassmeier et al. 2014


High-precision photometric measurements have shown that the present treatment of limb darkening is not sufficiently accurate and leads to systematic errors in the derived parameters of the exoplanets (e.g., Espinoza & Jordán 2016; Morello et al. 2017a; Maxted 2018). Typically, limb darkening is represented by rather simple laws, such as a linear law (Schwarzschild 1906), a quadratic law (Kopal 1950), a square-root law (Diaz-Cordoves & Gimenez 1992), a power-2 law (Hestroffer 1997), or a four-coefficients law (Claret 2000), so that when modeling the transit light curves, limb darkening is parameterized by some set of coefficients. Ideally, these coefficients should be constrained by the stellar modeling. Consequently, many libraries of limb-darkening coefficients covering a wide range of effective temperatures (Teff), surface gravity (log g), and metal-licities (M/H) have been produced (e.g., Claret 2000; Sing 2010; Claret & Bloemen 2011; Magic et al. 2015) using various radiative transfer codes, such as ATLAS (Kurucz 1993), NextGen (Hauschildt et al. 1999), PHOENIX (Husser et al. 2013), MARCS (Gustafsson et al. 2008), and STAGGER (Magic et al. 2013). However, the limb-darkening parameters diverge between different libraries and often lead to an inadequate quality of fits to the observed transit profiles (Csizmadia et al. 2013). First, this can be due to errors in limb-darkening coefficients introduced by interpolation from the grid of stellar parameters used in these libraries to the actual stellar fundamental parameters. Second, available stellar calculations may not treat mechanisms that affect limb darkening with sufficient accuracy, for instance, convection (Pereira et al. 2013; Chiavassa et al. 2017) or magnetic activity (Csizmadia et al. 2013). Therefore, limb-darkening coefficients are often left as free parameters in a least-squares fit to observed light curves (e.g., Southworth 2008; Claret 2009; Cabrera et al. 2010; Gillon et al. 2010; Csizmadia et al. 2013; Maxted 2018). While this method usually leads to a good quality fit to observed transit profiles, it introduces additional free parameters, resulting in possible biases and degeneracies in the returned planetary radii (Espinoza & Jordán 2015; Morello et al. 2017b). The way to reduce these biases and reliably determine planetary radii is to improve theoretical computations of stellar limb darkening.


Not only are electron populations with two different temperatures and densities frequently observed in experimental plasmas (Morales & Lee 1974; Jones et al. 1975), they are also an essential feature of space plasmas (Delory, Ergun & McFadden 1998; Ergun et al. 1998; Pottelette, Berthomier & Malingre 1998). Observations of electron velocity distribution functions (VDFs) in the solar wind and magnetosphere show a clear picture of a non-Maxwellian distribution function, that is, of a dense thermal population ‘core’ superimposed on a hot superthermal population ‘halo’ (Lin 1998; Makimovic et al. 2001; Pierrard, Maksimovic & Lemaire 2001; Pagel et al. 2005; Marsch 2006; Gaelzer et al. 2008). Two-component electron populations are a common feature of interplanetary shocks in the solar wind with relative drifts parallel to the magnetic field (Feldman et al. 1975, 1983b). They are also found around the Earth's bow shock, with a comparatively dense hot component and a tenuous cold beam (Feldman et al. 1982, 1983a). Reiff & Reasoner (1975) presented a comprehensive picture of electron VDFs from in the magnetosheath at lunar distances using Apollo 14. They found two distinct populations of electron VDFs, with the hot population density higher near the bow shock, and the colder one mid-way of the magnetosheath. They found that the two components are distinct but are not completely independent, and that the high- and low-energy components grow at the expense of each other. They argued that the high-energy portion of electron VDFs originates and is accelerated at the bow shock owing to the potential difference across this shock. Marsch (2006) also showed the interdependence of the core and halo components of electron VDFs in the solar wind. Such types of electron VDFs cannot be modelled by a single electron Maxwellian population or by core and halo both modelled by a bi-Maxwellian distribution. However, solar wind electron velocity distributions comprising a bulk core and a tenuous halo population are well fitted by employing a model distribution containing a bulk Maxwellian superimposed on a kappa distribution (Nieves-Chinchilla & Vinas 2008). Schippers et al. (2008) fitted the two-population electron VDFs observed in Saturn's magnetosphere with a combination of the bi-kappa distribution function that was used by Balaku et al. (2011) to study the damping characteristics of electron acoustic waves in Saturn's magnetosphere. Saeed et al. (2017, 2017a) studied the characteristics of electron heat flux and electromagnetic electron cyclotron instabilities by considering the relative drift between the core and halo electron components in typical solar wind conditions.


To interpret precisely the finding that high-inclination Centaurs had nearly polar orbits at the end of planet formation, it is useful to recall the current understanding of early Solar system structure (Pfalzner et al. 2015). The Solar system started forming 4.6 Gyr in the past. Sometime after a few Myr, giant planets formed and radially migrated while interacting with the planetesimal disc. The migration relaxation time is thought to have been about 1 to a few 10 Myr whereas the full migration stage is thought to have lasted about a 100–500 Myr at the end of which the planets reached their final orbits. The planets’ orbital evolution in this stage is largely uncertain owing to the multitude of complex processes that operate in the first 100 Myr of planet formation (Ribeiro et al. 2020). What is certain, however, is first, that the planetesimal disc before the migration stage as well as at the end of this stage had a small inclination dispersion in order to explain the very low inclination TNOs currently observed in the Kuiper belt (known as the cold population) (Pfalzner et al. 2015). Secondly, the early planetesimal disc could not have extended beyond 30–40 au (Gomes et al. 2004) to ensure that Neptune does not end up with a much larger final semimajor axis than the current one. Thus the early scattered disc and Oort cloud were devoid of Solar system material. Our orbit determination method was run for 4.5 Gyr in the past. The fact that the planets migrated towards the end of that period is not included in our calculation because the planets’ evolution in the migration stage is largely uncertain but more importantly, because it does not affect the Centaur orbits end states in any significant way. The reason lies in the fact that the bulk of stable Centaur clones that end up on nearly polar orbits do so in the first 1 Gyr of the simulation. Therefore near 4 Gyr in the past, the polar stable clones are already located far away in the scattered disc and the inner Oort cloud. Changes in the planets’ semimajor axes over a few 10 Myr do not affect such distant polar orbits significantly, as in reality, the planets’ effect had long started to decay in the last billion years before migration even occurred.


From the list of major SEP events with a metric type II burst, we selected all those events for which adequate GOES data of the associated soft X-ray flare was available. We used GOES 1–8 Å  X-ray profiles and the daily NOAA Solar Event Reports (Edited Events) (http://www.swpc.noaa.gov/ftpmenu/warehouse.html) to determine the magnitude and the onset and peak times of the associated GOES X-ray flare. Flare locations are taken from NOAA reports or determined from images of the EIT (Delaboudinière et al. 1995) on board the Solar and Heliospheric Observatory (SOHO). The associated CMEs were identified using the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) CME catalog (http://cdaw.gsfc.nasa.gov/CME_list/index.html; Yashiro et al. 2004; Gopalswamy et al. 2009b). Our final data set consists of 59 SEP events listed in Table 1. The columns in Table 1 are: the type II date (1) and time (2), the first observation date (3) and time (4) of the associated CME, the CME space speed (5) and width (6), the location of the associated flare (7), the flare-onset date (8) and time (9), the flare peak date (10) and time (11), the magnitude of the flare (12), the calculated constant acceleration of the CME (13), and the height of the CME (14) at the type II onset calculated using the flare-onset method. The last column (15) of Table 1 gives the measured height of the CME at the type II onset obtained by 3D-fitting a spherical shock model to CME images. The onset times of the metric type II bursts listed in Table 1 are the earliest reported times compiled from the NOAA Solar Event Reports except for the onset times of the 2000 June 6 and November 8 events that are from Gopalswamy (2003). Often different observatories report different onset times for the same metric type II burst; therefore, by selecting the earliest onset times we guarantee that our estimation is the lowest CME height consistent with the radio observations. Some additional details of the entries in Table 1  are given in Sections 3 and 5 discussing data analysis methods.


The trends between the LF slope α and the aforementioned parameters, with the addition of the morphological T type, are shown in Fig. 4 together with their Spearman correlation coefficient (ρ) and their p value, indicating the probability that the two sets of data are uncorrelated. We summarize the properties for which we look for a correlation in Table 4. We define a correlation to be negligible when |ρ|=[0 − 0.2], weak when |ρ|=[0.2 − 0.4], moderate when |ρ|=[0.4 − 0.6], strong when |ρ|=[0.6 − 0.8], and very strong when |ρ|=[0.8 − 1]; using the p value to evaluate the probability that, despite showing a correlation, two variables may be uncorrelated. It should be noted that only a handful of studies so far have looked at the correlation between α and global galaxy properties: Kennicutt et al. (1989), Elmegreen & Salzer (1999), Youngblood & Hunter (1999), van Zee (2000), and Thilker et al. (2002) investigated nebular LFs as in this paper, while Liu et al. (2013) identified H II regions via Paα, and Cook et al. (2016) studied the GALEX far-ultraviolet (FUV) LFs of H II regions. While the sample of Cook et al. (2016) includes a few hundred galaxies, the other studies are based on samples ranging from 10 to 35 galaxies, similar to our study. In this section and in Sect. 6.1, we compare our results to those studies that, as in our case, applied a uniform analysis methodology on galaxy samples. It should be noted that using different tracers means probing different source ages and, as reported by Oey & Clarke (1998), older H II regions tend to have steeper LF slopes, mainly due to the short main-sequence lifetimes of the more massive stars constituting the brighter H II regions. This is the reason why, for example, FUV observations, probing H II regions with ages less than 100 Myr, are expected to deliver a steeper LF compared to Hα observations, typically probing H II regions younger than 10 Myr, and our comparison remains qualitative.


The plethora of cosmological observations has turned cosmology into a quantitative science. From combining several probes that observe the Universe at different epochs and have different systematics and statistics, emerged the ‘concordance model’ of cosmology, a six parameter model, most of them measured to the accuracy of a per cent. Among these probes, Type Ia supernovae (SNe) are a powerful cosmological tool to directly measure the expansion history of the Universe (see e.g. a recent review of Weinberg et al. 2013). The type Ia SNe are known to be standard candles and one can measure the luminosity distances to them accurately. Several SNe surveys have set strong constraints on cosmological models from the distance–redshift relation (e.g. Riess et al. 1998; Perlmutter et al. 1999; Riess et al. 2007; Sullivan et al. 2011; Campbell et al. 2013). However the measured apparent magnitudes, have a residual scatter arising from its intrinsic scatter and effects due to line-of-sight (LOS) structures. The intrinsic scatter (∼0.4 mag) can be significantly reduced to ∼0.1 mag by empirically calibrating the luminosity curves. The scatter due to photon deflection along the LOS is composed of many different physical effects. The dominant effects are peculiar velocities at z  0.1 and gravitational lensing at z ≳ 0.3. Within the realm of cosmological perturbation theory and the stochastic nature of the LOS structures, all effects are expressed as an integral over the power spectrum with appropriate kinematical factors (Ben-Dayan et al. 2012). The lensing effects create residuals from the best-fitting curve in the magnitude–redshift relation. The lensing magnifications of SNe can be extracted by correlating the residuals with the surface densities of nearby foreground galaxies. The lensing dispersion is roughly proportional to the SNe redshift z, (e.g. Holz & Linder 2005; Ben-Dayan et al. 2013), and specifically for the concordance model, the predicted value is ∼0.06z mag.


There are two possibilities resulting in the different magnetic field strengths inferred from the polarimetric and molecular-line observations: (1) the rotational-to-gravitational energy βrot is overestimated, and (2) there are additional contributions in the polarized intensity from other mechanisms, such as dust scattering. In our MHD simulations, βrot is adopted to be 0.4% based on the observational estimates of the core mass of ∼1 M and the angular speed of the core rotation of 4 × 10−14 s−1. The angular speed was estimated based on the global velocity gradient along the major axis of the dense core observed with single-dish telescopes (Saito et al. 1999; Yen et al. 2011; Kurono et al. 2013). Numerical simulations of dense cores including synthetic observations show that the specific angular momentum derived from the synthetic images of the dense cores can be a factor of 8–10 higher than their actual specific angular momentum computed by the sum of the angular momenta contributed by the individual gas parcels in the dense cores (Dib et al. 2010). In addition, if there are filamentary structures in the dense core in B335, which could not be resolved with the single-dish observations, infalling motions along the filamentary structures could also contribute to the observed velocity gradient, leading to an overestimated angular speed of the core rotation (Tobin et al. 2012). We have also performed our simulations with a lower βrot, and we find that the rotational velocity on a 100 au scale in the simulations decreases with decreasing βrot. Thus, the discrepancy in the magnetic field strengths inferred from the field structures and the gas kinematics can be reconciled, if the core rotation in B335 is overestimated by a factor of a few in the observations, and these results would suggest a weak magnetic field of initial λ of 9.6 in B335. Further observations combining single dishes and interferometers to have a high spatial dynamical range and to map the velocity structures of the entire dense core in B335 at a high angular resolution are needed to study coherent velocity features and provide a better estimate of the core rotation.


H2O ice on Pluto has long escaped spectroscopic detection, and based on initial New Horizons data appears to be exposed only in a number of specific locations, usually associated with red color, suggestive of water ice/tholin mix (Grundy et al. 2015; Cook et al. 2015). Nonetheless, water ice is likely to be ubiquitous in Pluto’s near subsurface, given its cosmogonical abundance, Pluto’s density, and its presence on Charon’s surface6. Absorption coefficients for pure water ice (kH2O) at sub-mm-to-cm wavelengths are discussed extensively by Mätzler (1998), who also provides several analytic formulations to estimate them as a function of frequency and temperature along with illustrative plots. We use the Mishima et al. (1983) formulation (see Appendix of Mätzler 1998). Its applicability is normally restricted to temperatures above 100 K, but Fig. 2 of Mätzler (1998) indicates the trend with temperature. Absorption coefficients extrapolated to 50 K (estimated as half the values at 100 K) are shown in Fig. 5. At 500 μm, our best estimate is kH2O = 0.25 cm-1, comparable to the above values for CH4 and N2 ices. The corresponding penetration length is therefore comparable to the diurnal skin depth but remains negligible compared to the seasonal skin depth, even for seasonal Γ = 25 MKS. According to these calculations, the seasonal layer would be probed only at a wavelength of ~4 mm and beyond. We also remark that the expression from Mishima et al. (1983) would give a penetration depth of 56 m at 2.2 cm, which is an order of magnitude larger than indicated by the laboratory measurements of Paillou et al. (2008). In addition, small concentrations of impurities can dramatically reduce the microwave transparency of water ice (e.g., Chyba et al. 1998 and references therein). Therefore, the above calculations likely indicate upper limits to the actual penetration depth of radiation in a H2O ice layer, from which we conclude that the seasonal layer is not reached at the Herschel wavelengths.


Furthermore, well-defined large samples of white dwarfs are an extremely useful starting point for identifying rare white dwarf types like magnetic white dwarfs (Gänsicke, Euchner & Jordan 2002; Schmidt et al. 2003; Külebi et al. 2009; Kepler et al. 2013), pulsating white dwarfs (Castanheira et al. 2004; Greiss et al. 2014), high/low-mass white dwarfs (Vennes & Kawka 2008; Brown et al. 2010; Hermes et al. 2014), white dwarfs with unresolved low-mass companions (Farihi, Becklin & Zuckerman 2005; Girven et al. 2011; Steele et al. 2013), white dwarfs with rare atmospheric composition (Schmidt et al. 1999; Dufour et al. 2010; Gänsicke et al. 2010), close white dwarf binaries (Marsh, Nelemans & Steeghs 2004; Parsons et al. 2011), metal polluted white dwarfs (Sion, Leckenby & Szkody 1990; Zuckerman & Reid 1998; Dufour et al. 2007; Koester, Gänsicke & Farihi 2014) or white dwarfs with dusty or gaseous planetary debris discs (Gänsicke et al. 2006; Farihi, Jura & Zuckerman 2009; Debes et al. 2011; Wilson et al. 2014). Because of their intrinsic low luminosities identifying a large, complete and well-defined sample of white dwarfs still remains a challenge. Much progress has been made in recent years thanks to large area surveys, first and foremost the Sloan Digital Sky Survey (SDSS; York et al. 2000; Harris et al. 2003; Eisenstein et al. 2006; Kleinman et al. 2013). The largest published catalogue of white dwarfs to date (Kleinman et al. 2013) fully exploited the spectroscopic data available at the time of SDSS Data Release 7 (DR7) and contains over 20 000 white dwarfs (of which 7424 with g ≤ 19). However not only is DR7 now outdated, but SDSS spectroscopy is only available for less than 0.01 per cent of all SDSS photometric sources. Furthermore most of SDSS's white dwarfs are only serendipitous spectroscopic targets. The true potential of SDSS's vast multiband photometric coverage still remains to be fully mined for white dwarf research, but this requires a reliable method able to select white dwarfs candidates without recourse to spectroscopy. Proper motion has been traditionally used to distinguish white dwarfs from other stellar populations. In particular many studies that contributed to the census of white dwarfs in the solar neighbourhood specifically targeted high proper motion objects (Holberg et al. 2002; Sayres et al. 2012; Limoges, Lépine & Bergeron 2013). In this paper we present a novel method which makes use of photometric data and proper motions to calculate a probability of being a white dwarf (PWD) for any photometric source within a broad region in colour space. Unlike any previous similar work, our method does not use a specific cut in colour or proper motion to generate a list of white dwarf candidates; instead it provides a catalogue of sources with an associated PWD. These PWD can then be used to create samples of white dwarf candidates best suited for different specific uses. By applying our method to the full photometric footprint of SDSS Data Release 10 (DR10), we created a catalogue which includes ∼23 000 bright (g ≤ 19) high-fidelity white dwarfs candidates (Table 1). Using this catalogue, we assess the spectroscopic completeness of the SDSS white dwarf sample.


Benzene (C6H6), the simplest aromatic hydrocarbon, is a molecule that has raised great interest in the astrophysical community for almost four decades. This is mainly because C6H6 is one of the main precursors of polycyclic aromatic hydrocarbons (PAHs) reported to be present in interstellar dust particles (Leger & Puget 1984; Allamandola et al. 1989; Tielens 2013 and references therein), carbonaceous chondrites (Pering & Ponnamperuma 1971; Hayatsu et al. 1977; Hahn et al. 1988), and other astrophysical environments, such as carbon-rich, high-temperature environments (circumstellar and carbon-rich protoplanetary nebulae; Buss et al. 1993; Clemett et al. 1994). Benzene rings easily produce more complex, polycyclic structures by the one-ring build-up mechanism (Simoneit & Fetzer 1996). In space, an analogous process to carbon soot formation occurring on Earth can be initiated through the completion of that first aromatic ring and may also lead to the synthesis of PAHs (Tielens & Charnley 1997). Mechanisms involving the addition of hydrocarbons, such as acetylene onto aromatic rings as well as the attachment of other aromatic rings, or hydrocarbon pyrolysis, have been proposed to characterize the growth process of PAHs (Bittner & Howard 1981; Frenklach & Feigelson 1989; Wang & Frenklach 1997; Cherchneff 2011 and references therein). PAH synthesis from shocked benzene has also been reported (Mimura 1995). PAHs are crucial materials involved in a variety of cosmochemical processes. For example, amino acids could be synthesized by aqueous alteration of precursor PAHs in carbonaceous chondrites (Shock & Schulte 1990). PAHs are also produced in laboratory-simulated planetary atmospheres of Titan and Jupiter (Sagan et al. 1993; Khare et al. 2002; Trainer et al. 2004), and results from these studies indicate that the formation of aromatic rings and polyaromatics may be, among other sources, a possible chemical pathway for the production of the atmospheric solid particles (Lebonnois et al. 2002; Wilson et al. 2003; Trainer et al. 2004). The formation and evolution of benzene in planetary environments or other solar system objects thus represents a fundamental primary stage of the PAH production and other subsequent relevant chemical and prebiotic processes (like soot formation). In this context, several works related to benzene have been devoted to better understand the physico-chemical processes of irradiated C6H6, in its gaseous and solid phases, and the derived products, by acquiring high-resolution astronomical spectra, carrying out detailed laboratory studies or developing theoretical modeling (Allamandola et al. 1989  and references therein; Callahan et al. 2013; Materese et al. 2015; Mouzay et al. 2021). Laboratory astrophysical investigations have mostly focused on performing vibrational spectroscopy of ion, electron, or UV irradiated C6H6 gas and C6H6 ice. Such investigations aim to provide data on the spectral properties of the irradiated C6H6 materials, compare them with spectra obtained from astronomical observations (e.g., observations of the interstellar medium), or to study photoprocessed benzene ices to understand the fate of benzene ices in Titan’s stratosphere and help understanding the formation of aerosol analogs observed in Saturn’s moon’s stratosphere (Mouzay et al. 2021).


So far, only one ancient and massive intruder has been identified unambiguously, with its body now nothing but an enormous cloud of tidal wreckage scattered throughout the inner Milky Way. The existence of a vast debris structure left behind by this merger event, known today as the Gaia-Sausage/Enceladus (GS/E), was suggested before Gaia (see Evans 2020, for the historical development). In particular, Deason et al. (2013) argued that the rapid transition in the Galactic stellar halo structural properties at break radius of 20–30 kpc is likely associated with the apocentric pileup of a relatively early (8–10 Gyr ago) and single massive accretion event. It is hypothesized that the progenitor dwarf galaxy was massive enough for its orbit to shrink and radialize quickly as a result of a complex interplay between dynamical deceleration, host recoiling, and self-friction (Vasiliev et al. 2022). Sinking deep in the heart of the Milky Way, the dwarf sprayed the bulk of its stars in a region enclosed by its last apocenter, i.e., some ∼30 kpc (see Deason et al. 2018). As a result, the region of the Galactic halo within this so-called break radius (see Deason et al. 2011) is inundated with the GS/E stars, consistently showing up as the most striking substructure even in relatively small volumes around the Sun (see examples of pre-Gaia hints in, e.g., Chiba & Yoshii 1998; Brook et al. 2003; Meza et al. 2005; Nissen & Schuster 2010; Hawkins et al. 2015). Subsequently, the Gaia data made crystal clear the unusually strong radial anisotropy of the relatively metal-rich GS/E debris and helped to reveal its dominance in the solar neighborhood (see Belokurov et al. 2018; Haywood et al. 2018b; Myeong et al. 2018b). The genesis of the GS/E debris cloud was also unambiguously confirmed by its unique chemical fingerprint (see, e.g., Helmi et al. 2018; Mackereth et al. 2019) and the large group of globular clusters (GCs) associated with it (see, e.g., Myeong et al. 2018c, 2019; Massari et al. 2019). Close to the Sun, the GS/E structure has been thoroughly scrutinized both kinematically and chemically (see Evans et al. 2019; Necib et al. 2019; Sahlholdt et al. 2019; Das et al. 2020; Feuillet et al. 2020, 2021; Kordopatis et al. 2020; Molaro et al. 2020; Monty et al. 2020; Aguado et al. 2021a; Buder et al. 2021a; Bonifacio et al. 2021; Carollo & Chiba 2021; Matsuno et al. 2021). Outside of the solar neighborhood, fewer studies exist; nonetheless, the global structure of the GS/E cloud is starting to come into focus (see, e.g., Iorio & Belokurov 2019; Lancaster et al. 2019; Simion et al. 2019; Naidu et al. 2020; Balbinot & Helmi 2021; Bird et al. 2021; Iorio & Belokurov 2021; Wu et al. 2022).


In general, the Hall time-scale for magnetic field evolution depends on the strength of the magnetic field, as seen in equation (4). For young NSs with fields below 1014 G, this time-scale may be longer than the observed SNR age. Therefore, magnetic field growth does not have a dramatic effect on these young NSs. However, many of these systems are observed to have a braking index n  3 (Espinoza 2012; Archibald et al. 2016). A possible explanation for these low braking indices may be through the emission of a relativistic particle wind (Thompson & Blaes 1998; Harding, Contopoulos & Kazanas 1999; Tong et al. 2013). However, the conclusive detection of wind nebulae around magnetars in particular is challenging due to the presence of dust-scattering haloes that accompany these X-ray bright, heavily absorbed objects (Esposito et al. 2013; Safi-Harb 2013). Only a handful of such nebulae have been proposed to be associated with highly magnetized NSs. For example, a wind nebula has been proposed to surround the magnetar Swift J1834.9–0846 in W41 (Younes et al. 2016), and the luminosity of a particle wind was estimated for SGR 1806–20 based on the X-ray and radio observations of the wind-powered nebula G10.0–0.3 (Thompson & Duncan 1996; Marsden, Rothschild & Lingenfelter 1999; Gaensler et al. 2005). In the pulsar wind model, relativistic particles load the magnetosphere with charge and distort the dipole field at large scales outside of the light cylinder. Besides affecting the NS spin-down, the emission of a relativistic wind can also offer an explanation for the significant timing noise that generally affects magnetar observations (Tsang & Konstantinos 2013). The HBPs J1119–6127 and J1846–0258 are clearly associated with pulsar wind nebulae (Gavriil et al. 2008; Kumar & Safi-Harb 2008; Ng et al. 2008; Safi-Harb & Kumar 2008; Safi-Harb 2013), suggesting that particle wind emission should play an important role in their evolution. We also expect that AXP and SGR evolution may be affected by wind emission due to candidate wind nebulae, but do not consider these models for the CCOs that do not show any evidence of PWN. However, we note that braking exclusively due to a steady particle wind produces a torque with a braking index n = 1, too low for the NSs with secure SNR associations in Table 1.


The de-projected distance that corresponds to the core-position changes of Markarian 421 is approximately 3.5 ∼ 8.7 pc (equivalent to (1.0 − 2.6) × 105 Schwarzschild radii (Rs)) for a jet-viewing angle of 2°–5° (Lico et al. 2012). One of the leading models for explaining the blazar phenomenon is the so-called internal-shock model, in which discrete ejecta with higher speeds catch up with the preceding slower ejecta, and the collision leads to a shock wave in the colliding ejecta (Spada et al. 2001; Guetta et al. 2004). Based on this model, core wandering can be naturally explained because the model predicts some scattering of the shock locations caused by the randomness of the ejecta speed. Therefore, we assume that the distances of the shock locations from the black hole immediately after the flare,

, are proportional to the product of the square of the bulk Lorentz factor (

) of the ejecta and the separation length (Δ) of the colliding ejecta. Assuming that a typical separation length is a scale of about 10 times the innermost stable circular orbit for non-rotating black holes (

), in light of numerical simulations of relativistic outflows around black holes (McKinney et al. 2013), very fast ejecta with a Lorentz factor of approximately

 are required to explain the observed magnitude of the core wandering. The

 is comparable to previously reported maximum values (Γ of 40 ∼ 50; Lister et al. 2009, 2013) based on statistical studies by a long-term monitor using VLBI. Moreover, the radio core returned to the reference stationary position after 2012 March 12 (the 10th epoch). This can be explained if the emission from the reference position is identical to the radio core that is in a quiescent state. The radio-core emission in a quiescent state is expected to be persistent. As one of the possibilities for reproducing the observed core wandering, this persistent radio core was obscured by the flare-associated radio core, which therefore must be optically thick against SSA and short lived (a lifetime of approximately 2 weeks), for several epochs. Such a flare-associated radio core must be located at least

 downstream from the persistent radio core. In addition, Koyama et al. (2015) recently conducted multi-epoch astrometric observations of Markarian 501, which is also one of the most nearby TeV blazars by using VERA. However, they detected no significant positional changes of the radio core during its quiescent state. This fact, therefore, seems to support the radio core wandering seen after the large X-ray flare.


Running penumbral waves in velocity and intensity observations were first reported by Giovanelli (1972) and Zirin & Stein (1972). Later, they were found in the photosphere as well (Musman et al. 1976), but there they appear to be more intermittent and to have higher radial phase velocity (40–90 km s−1) than the waves in Hα. Whereas the velocity amplitudes are less in the photosphere than in the chromosphere, the density is very low there and most of the wave energy lies in the photosphere and subphotosphere. Larger amplitudes on the disk-side penumbra demonstrate an alignment of the oscillations along the magnetic field. Running waves are also detected in the umbra, but the waves were believed to be unrelated to those in the penumbra (Kobanov & Makarchik 2004). In the chromosphere, the frequency of travelling waves decreases as they propagate from the umbra into the outer penumbra (e.g., Lites 1988). A similar effect is also found in measurements of the propagation velocity of travelling waves (Brisken & Zirin 1997; Sigwarth & Mattig 1997; Alissandrakis et al. 1998; Kobanov & Makarchik 2004; Tziotziou et al. 2006, 2007). Generally, the waves decelerate from 40 km s−1 near the inner part of the penumbra to 10 km s−1 or less near the outer edge of the penumbra. More recently, from a multi-wavelength study including the coronal channels of the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO), Jess et al. (2015) revealed the presence of a wide range of frequencies, with longer periodicities preferentially occurring at increasing distance from the umbra. The phase speeds also tend to decrease with increasing periodicity as the waves propagate away from the umbral barycenter. These observations also suggest that these slow waves are driven by a regular coherent source. The physical nature of running penumbral waves has been controversial. Some researchers have regarded them as trans-sunspot waves originating from umbral oscillations since they detected waves starting from the umbra and propagating through the penumbra (e.g., Alissandrakis et al. 1992; Tsiropoula et al. 1996, 2000). However, others suggest that the trans-sunspot (i.e., outward) motion is apparent to a given line of sight, and that these oscillations actually represent the upward propagation of field-guided magnetoacoustic waves from the photosphere (e.g., Christopoulou et al. 2000, 2001; Georgakilas et al. 2000; Rouppe van der Voort et al. 2003; Bogdan & Judge 2006; Kobanov et al. 2006; Bloomfield et al. 2007; Jess et al. 2013, 2015). The gradual change in the inclination of the penumbral field lines is responsible for changes in the oscillation periods and phase speeds.


Quasi-periodic pulsations (QPPs) often refer to the quasi-periodic intensity variations during solar/stellar flares (see Zimovets et al. 2021, for a recent review). In many observations, the flare QPPs were found to show a nonstationary property in the time series integrated over the whole Sun/star or over the oscillation region, for instance, each pulsation has an anharmonic and symmetric triangular profile shape (e.g., Kolotkov et al. 2015; Nakariakov et al. 2019). The signature of flare QPPs can be detected in flare light curves across a broad band of the electromagnetic spectrum, i.e., radio/microwave emissions (Ning et al. 2005; Reznikova & Shibasaki 2011; Nakariakov et al. 2018; Yu & Chen 2019), UV/EUV wavelengths (Shen et al. 2018; Hayes et al. 2019; Reeves et al. 2020; Miao et al. 2021), SXR/HXR and γ-ray channels (Nakariakov et al. 2010; Ning 2017; Hayes et al. 2020; Li et al. 2020c), and the Hα (Srivastava et al. 2008; Kashapova et al. 2020; Li et al. 2020b) or Lyα (Van Doorsselaere et al. 2011; Milligan et al. 2017; Li 2021) emissions. The quasi-periods of these QPPs were reported from subseconds to tens of minutes (e.g., Tan et al. 2010; Shen et al. 2013, 2019; Kolotkov et al. 2018; Karlický & Rybák 2020; Clarke et al. 2021). It should be stated that the observed periods are generally related to the specific channels or flare phases (Tian et al. 2016; Dennis et al. 2017; Pugh et al. 2019), suggesting that the various classes of QPPs could be produced by different generation mechanisms (e.g., Kupriyanova et al. 2020). In the literature, the flare-related QPPs were most often explained by magnetohydrodynamic (MHD) waves, more specifically sausage waves, kink waves, and slow waves (Li et al. 2020a; Nakariakov & Kolotkov 2020; Wang et al. 2021), or by a repetitive regime of magnetic reconnection that could be spontaneous (i.e., self-oscillatory process) or triggered owing to external MHD oscillations (Thurgood et al. 2017; Yuan et al. 2019; Clarke et al. 2021). They can also be interpreted in terms of the LRC-circuit oscillation in current-carrying loops (Tan et al. 2016; Li et al. 2020b) or caused by the interaction between supra-arcade downflows and flare loops (Xue et al. 2020; Samanta et al. 2021).


C-statistic (e.g. Jang & Miller 1997) is the most commonly used and the one-way analysis of variance (ANOVA; de Diego 2010) the most powerful test for verifying the presence of variability in a DLC. However, we did not employ either of these tests because, de Diego (2010) has questioned the validity of the C-test by arguing that the C-statistics does not have a Gaussian distribution and the commonly used critical value of 2.567 is too conservative. On the other hand, the ANOVA test requires a rather large number of data points in the DLC, so as to have several points within each sub-group used for the analysis. This is not feasible for our DLCs which typically have no more than about 30–45 data points. Therefore, we have instead used the F-test which is based on the ratio of variances, F = variance(observed)/variance(expected) (de Diego 2010; Villforth, Koekemoer & Grogin 2010), with its two versions : (i) the standard F-test (hereafter Fη-test, Goyal et al. 2012) and (ii) scaled F-test (hereafter Fκ-test, Joshi et al. 2011). The Fκ-test is preferred when the magnitude difference between the object and comparison stars is large (Joshi et al. 2011). Onward Paper II, we have only been using the Fη-test because our objects are generally quite comparable in brightness to their available comparison stars. An additional gain from the use of the Fη-test is that we can directly compare our INOV results with those deduced for other major AGN classes (Goyal et al. 2013). An important point to keep in mind while applying the statistical tests is that the photometric errors on individual data points in a given DLC, as returned by the algorithms in the iraf and daophot softwares are normally underestimated by the factor η which ranges between 1.3 and 1.75, as estimated in independent studies (e.g. Gopal-Krishna, Sagar & Wiita 1995; Garcia et al. 1999; Sagar et al. 2004; Stalin et al. 2004a; Bachev, Strigachev & Semkov 2005). Recently, using a large sample, Goyal et al. (2013) estimated the best-fitting value of η to be 1.5, which is adopted here. Thus, the Fη statistics can be expressed as

\begin{equation*}
F_{1}^{\eta } = \frac{\sigma ^{2}_{({\rm q-s1})}}{ \eta ^2 \langle \sigma _{{\rm q-s1}}^2 \rangle }, \hspace{5.69046pt} F_{2}^{\eta } = \frac{\sigma ^{2}_{({\rm q-s2})}}{ \eta ^2 \langle \sigma _{{\rm q-s2}}^2 \rangle }, \hspace{5.69046pt} F_{{\rm s1-s2}}^{\eta } = \frac{\sigma ^{2}_{({\rm s1-s2})}}{ \eta ^2 \langle \sigma _{{\rm s1-s2}}^2 \rangle },\end{equation*}

 where $\sigma ^{2}_{({\rm q-s1})}$, $\sigma ^{2}_{({\rm q-s2})}$ and $\sigma ^{2}_{({\rm s1-s2})}$ are the variances of the ‘quasar–star1’, ‘quasar–star2’ and ‘star1–star2’ DLCs and $\langle \sigma _{{\rm q-s1}}^2 \rangle =\sum _{\boldsymbol {i}=0}^{N}\sigma ^2_{i,{\rm err}}({\rm q-s1})/N$, $\langle \sigma _{{\rm q-s2}}^2 \rangle$ and $\langle \sigma _{{\rm s1-s2}}^2 \rangle$ are the mean square (formal) rms errors of the individual data points in the ‘quasar–star1’, ‘quasar–star2’ and ‘star1–star2’ DLCs, respectively. η is the scaling factor (= 1.5) as mentioned above.


In local galaxies, the ISM conditions are often described by some physical quantities such as ionization parameter (q), gaseous metallicity (Z) and electron density (ne). At high redshift, the ionization parameter is raised by a large flux of ionizing photons in ISM originated from hot O, B stars due to intensive star formation in relatively small galaxies. Previous studies suggest a high ionization parameter of SF galaxies at z > 2 compared to that of local galaxies (Erb et al. 2010; Nakajima et al. 2013; Masters et al. 2014; Nakajima & Ouchi 2014). Secondly, the chemical abundance of SF galaxies at z ∼ 2 is lower by 0.1–0.3 dex for a given stellar mass compared to those at low-z (Erb et al. 2006a; Sanders et al. 2014; Steidel et al. 2014). This leads to more compact and hotter O, B stars due to lower opacity (Ezer & Cameron 1971; Maeder 1987), and thus UV radiation becomes harder and produces more ionizing photons. Thirdly, the strength of collisionally excited emission lines (e.g. [O iii], [N ii]) strongly depends on the electron density. It is closely related to the number of electrons to collide since the excitation potential of this line is ∼1 eV, which is nearly the same as the energy of electrons at the virial temperature (∼104 K) (Dyson & Williams 1980). Due to low excitation potential, the transition of collisionally excited line is reliant on electron density compared to gaseous metallicity. Recent observations have suggested a high electron density (ne > 100 cm−3) in SF galaxies at z ∼ 2 (Newman et al. 2012; Masters et al. 2014; Shirazi, Brinchmann & Rahmati 2014; Wuyts et al. 2014). This value is larger than that of normal SF galaxies at low-z by an order of magnitude, and close to that of interacting galaxies which are seen as (ultra)luminous infrared (IR) galaxies in the present-day Universe (Krabbe et al. 2014). Such a large electron density contributes to the offset of galaxy distributions on the BPT diagram together with other physical parameters (Brinchmann, Pettini & Charlot 2008a). In this way, the cosmic dependence of the BPT diagram can be attributed to such physical parameters which determine ISM conditions.


The kinematic analysis was performed by using the selected sample of stars in the MUSE catalogue (see Section 3) and by following the maximum-likelihood approach described by Pryor & Meylan (1993). The method is based on the assumption that the probability of finding a star with a velocity of vi ± ϵi at a projected distance from the cluster centre Ri can be approximated as
(1)$$\begin{eqnarray*}
p(v_i,\epsilon _i,R_i) = \frac{1}{2\pi \sqrt{\sigma ^2 + \epsilon _i^2}}exp{\frac{(v_i-v_0)^2}{-2\left(\sigma ^2 + \epsilon _i^2\right)}}
\end{eqnarray*}$$where v0 and σ are the systemic radial velocity and the intrinsic dispersion profile of the cluster, respectively. Rotation was included in the analysis by adding the following angular dependence (e.g. Copin et al. 2001; Krajnović et al. 2006; Kamann et al. 2018a) to the mean velocity of equation (1).
(2)$$\begin{eqnarray*}
v_0 = v_0 + v_{rot}(R_i)sin(\theta _i - \theta _0(R_i))
\end{eqnarray*}$$where vrot and θ0 represent the projected rotation velocity and the rotation axis angle respectively as a function of the projected distance R to the cluster centre. The axis angle as well as the position angle θi of a star are measured from north through east. We restricted the prior for θ0 to a 180° wide interval and we allowed the rotation velocity to assume both positive and negative values, in order to avoid a skewed rotation velocity probability distribution in case of very small or no rotation (see also Kamann et al. 2020). In particular, we used an iterative procedure in which we adopted an angular interval [α; α + 180°) centred on the most probable rotation axis angle. This is useful to avoid skewness in the probability distribution of θ0 when its value is close to 0° or 180°. We split the sample in five concentric annuli centred on the cluster centre and with width varying in such a way that each bin contains the same number of stars (80). Only stars with r  32 arcsec were considered in the analysis as they guarantee an almost complete coverage within each annulus (Fig. 1).


For LLAGNs, there are two competing components for the emission at optical/UV wavebands. One component is the thermal emission from the outer truncated SSD, which peaks in optical or UV bands, depending on the location of the truncated radius. The other component is the Compton scattering of synchrotron emission from the inner hot accretion flow (Manmoto et al. 1997; Yuan & Narayan 2014). Since our sample is limited to sources whose λ  10−3, we argue that the emission at 2500 Å mainly comes from the inner hot accretion flow.3 Interestingly, for hot accretion flows around supermassive BHs (the LLAGN case), the optical/UV emission usually is the first Compton up-scattered bump (Manmoto et al. 1997; Yuan & Narayan 2014). The emission from such a process has a positive correlation with $\dot{m}$, as $\dot{m}$ will mainly determine the optical depth (surface density), a quantity that controls the probability of Compton scattering (Sunyaev & Titarchuk 1980; Dermer, Liang & Canfield 1991). On the other hand, the first Compton bump is even more sensitive to the energy of electrons (a.k.a. the electron temperature), as it determines the location (in wavebands) of the peak. For a given $\dot{m}$, larger α means lower density, or equivalently lower radiative cooling to the electrons. Consequently, the electrons will be more energetic at higher temperatures (Xie & Yuan 2012). The dependence of emission at optical/UV band on the parameter α relies on detailed numerical calculations, where Manmoto et al. (1997) reported that the emission at 2500 Å has a positive correlation with α. For the above reasons, we simply write LUV as
(6)
\begin{equation}
L_{\rm UV}\sim (\dot{m} \alpha )^b,
\end{equation}
where b ∼ 2. The estimation of b comes from the fact that the first-order Compton scattering (responsible for the optical/UV emission) depends on the product of the seed photon flux (synchrotron emission, $\propto \dot{m}^{0.5-1}$; cf. Mahadevan 1997; Yuan et al. 2015; note that electron temperature also depends on $\dot{m}$ for the expression in Mahadevan 1997) and optical depth in a vertical direction ($\propto \dot{m}^{1-2}$).