Under the Applegate model, the change in orbital period is directly related to the change in the companion star’s gravitational quadrupole moment Q (Applegate & Patterson 1987),
(8)$$\begin{eqnarray*}
\frac{\Delta P_{\rm orb}}{P_{\rm orb}} = -9\frac{\Delta Q}{M_{\rm c} A^2},
\end{eqnarray*}$$where A = x(1 + q)/sin i is the orbital separation. For comparison, the total quadrupole moment induced by the spin of the companion star and the tidal distortion in the pulsar’s gravitational field is (Voisin, Breton & Summers 2020a)
(9)$$\begin{eqnarray*}
\frac{Q}{M_{\rm c} A^2} = -\frac{2}{9} k_2 \left(\frac{R_{\rm c}}{A}\right)^5 \left(4 q + 1\right),
\end{eqnarray*}$$where Rc is the radius of the companion star and k2 is the apsidal motion constant, a parameter describing the deformability of the companion star (Sterne 1939). For solar-type stars k2 ∼ 0.035 (Ogilvie 2014), while if we assume that redback companions are akin to the companions in CV systems whose outer envelopes have also been stripped through accretion, then we may expect a smaller value k2 ∼ 10−3 (Cisneros-Parra 1970). For J2039, the hyperparameter $h = 3.9^{+2.2}_{-1.2}$ s corresponds to the typical fractional amplitude for the variations in orbital phase. Taking the simpler squared exponential covariance function of equation (4) corresponding to n → ∞, then the deviations in orbital period have covariance function,
(10)$$\begin{eqnarray*}
K_{\Delta P_{\rm orb}/P_{\rm orb}}(t_1,t_2) &=& \frac{\partial ^2 K}{\partial t_1 \partial t_2} \nonumber\\
&=& \frac{h^2}{l^2} \exp \left(\!-\frac{(t_1 - t_2)^2}{2\ell ^2}\!\right) \left(\!1 - \frac{(t_1 - t_2)^2}{\ell ^4}\!\right).
\end{eqnarray*}$$The typical (fractional) amplitude of the orbital period variations is therefore ΔPorb/Porb ∼ h/ℓ = (3 ± 1) × 10−7, corresponding to $\Delta Q / Q \sim 3\times 10^{-5} k_2^{-1}$. The time-varying component to the gravitational quadrupole moment is therefore required to be of order a few per cent of the total expected quadrupole moment at most to explain the observed orbital period variations. From this, it seems plausible that the observed period variations can be powered by quadrupole moment changes, without requiring that a large fraction of the star be involved in the process. The required fractional quadrupole moment changes are very similar to those recently calculated for the companion to the black widow PSR J2051–0827 by Voisin et al. (2020b), despite the large difference in their masses.


The complexity of non-adiabatic pulsations and their coupling to the convection has posed many problems since the field’s inception and still does. The main problem lies in our, so far, limited understanding of the interaction between convection and pulsations. However, several important steps forward have already been taken, and several recent reviews on the topic exist (see for example Houdek & Dupret 2015; Samadi, Belkacem & Sonoi 2015). The case of solar pulsational stability has been studied in detail both theoretically (Balmforth 1992) and observationally (Chaplin et al. 1997; Komm, Howe & Hill 2000), while the space missions CoRoT (Baglin et al. 2006) and Kepler (Borucki et al. 2010; Borucki 2016) have provided high-quality seismic data for stars of different flavours against which we can test models and further our understanding of stellar pulsations. Appourchaux et al. (2014) analysed oscillation mode linewidths for a number of Kepler main-sequence solar-like stars and found interesting relationships between linewidths, frequencies, and effective temperatures. Using CoRoT observations Samadi et al. (2012) showed that non-adiabatic effects are present and non-negligible in red giant stars. Houdek & Gough (2002) modelled damping rates and velocity amplitudes of the red giant ξ Hydrae, while Dupret et al. (2009) computed theoretical amplitudes, lifetimes, and heights in the frequency power spectrum of oscillation modes at different stages of red giant evolution, including the phase of core helium burning, and Grosjean et al. (2014) computed synthetic power spectra for mixed modes in red giants. Belkacem et al. (2012) were able to reproduce observed Γ versus Teff across the HR-diagram including both main-sequence and red giant stars. However, calculations of frequency-dependent damping rates for red giant stars have so far not been able to survive comparisons with observations. Handberg et al. (2017), hereafter referred to as H17, obtained precise frequencies and linewidths for a sample of red giants in NGC 6819 by means of extensive, careful peakbagging. Here, we compute frequency-dependent damping rates for a selection of red giant branch (RGB) stars in the H17 sample. This is done via a non-adiabatic stability calculation (Houdek et al. 1999) for which we obtain the convective fluxes from a non-local, time-dependent convection model (Gough 1977a,b) partly calibrated through 3D convection simulations (Trampedach et al. 2013).

Background Background Background Background


As scattered light imaging is sensitive to the stellar irradiation, it allows one to search for misalignments between various disk regions. While studying the morphology of the innermost disk region is challenging due to its very small radial extent, often marginally resolvable by optical interferometry (Lazareff et al. 2017), scattered light imaging of the outer disk can indirectly reveal the presence of a misaligned inner disk. In this scenario, depending on the misalignment angle, the outer disk image will show narrow shadow lanes (e.g., Pinilla et al. 2015; Stolker et al. 2016; Benisty et al. 2017; Casassus et al. 2018), broad extended shadows (Benisty et al. 2018) or low-amplitude azimuthal variations (Debes et al. 2017; Poteet et al. 2018). In some cases, studies of the CO line kinematics support a misalignment between inner and outer disk regions (Loomis et al. 2017; Pérez et al. 2018). The exact origin of such a misalignment is still unclear. In the case of T Tauri stars, if the stellar magnetic field is inclined, it can warp the innermost edge of the disk, which would then rotate at the stellar period (AA Tau; Bouvier et al. 2007). Alternatively, inner and outer disk regions can have different orientations if the primordial envelope had a different angular momentum vector orientation at the time of the inner/outer disk formation (Bate 2018). Other scenarios involve the presence of a massive companion/planet that is inclined with respect to the disk. If the companion is massive enough, the disk can break into two separate inner and outer disk regions, that can then precess differently and result in a significant misalignment between each other (e.g., Nixon et al. 2012; Facchini et al. 2013; Nealon et al. 2018; Zhu 2019). A clear example of such a scenario is the disk around HD 142527, in which an M-star companion was detected (Biller et al. 2012), likely on an inclined and eccentric orbit (Lacour et al. 2016; Claudi et al. 2019). Dedicated hydrodynamical simulations successfully reproduce most of the observed features in this disk (eccentric cavity, spiral arms, misaligned inner disk and shadows; Price et al. 2018).


Many Be-binary systems were observed during their outbursts, which offered interesting results (Bildsten et al. 1997; Reig 2011). Bright X-ray outbursts are observed in Be binaries, most likely during the periastron passage of its neutron star through the circumstellar disk of its companion. Depending on the amount of matter released from its companion and the geometry of the binary system, rare as well as regular outbursts are observed during its binary orbit (Okazaki & Negueruela 2001; Okazaki et al. 2002; Okazaki 2016). The pulse characteristics of some of these, such as EXO 2030+375 (Parmar et al. 1989), Cepheus X-4 (Mukerjee et al. 2000), and XTE J1946+274 (Paul et al. 2001), were studied in detail during their outburst activities. Studies on pulse characteristics offer information on the pulsar geometry and the mechanism underlying its emitted pulse profile. The shape of the emitted pulse depends on modes of accretion inflows, source luminosity, geometry of accretion column, and the configuration of its magnetic field with respect to an observer’s line of sight (Parmar et al. 1989). Therefore, such studies offer understanding of pulsars in binary system and disk–magnetosphere interaction during the process of mass accretion, which affects its emitted radiation. Quasi-periodic oscillations (QPOs) have been detected from many Be binaries, such as A0535+262 (Finger et al. 1996; Finger 1998), EXO 2030+375 (Angelini et al. 1989), 4U 0115+63 (Soong & Swank 1989; Heindl et al. 1999; Dugair et al. 2013), and V0032+53 (Qu et al. 2005). Studies of QPOs offer rich information about the accretion torque onto the neutron star, thermodynamic properties of the inner accretion disk, and electrodynamics of the disk–magnetosphere interaction of the neutron star. Details on sources with observed QPOs, their frequencies, and other features along with pulsar spin frequencies, etc. have been given in tabular form by Devasia et al. (2011), Ghosh (1998), and Mukerjee et al. (2001). Some of these transient Be-binary pulsars such as A0535+26 (50 mHz, 9.7 mHz), 1A 1118–61 (92 mHz, 2.5 mHz), XTE J1858+034 (110 mHz, 4.53 mHz), EXO 2030+375 (200 mHz, 24 mHz), SWIFT J1626.6–5156 (1000 mHz, 65 mHz), XTE J0111.2–7317 (1270 mHz, 32 mHz), as well as a persistent Be binary, X Per (54 mHz, 1.2 mHz), and an OB-type binary, 4U 1907+09 (69 mHz, 2.27 mHz), showed higher QPO frequency compared to their respective spin frequency as mentioned in order inside parentheses (Devasia et al. 2011). These cover an interestingly wide range of QPO frequencies between 50 and 1270 mHz for these pulsars. Studies of cyclotron absorption features, if present in the spectrum, enable us to determine the strength of the surface magnetic field of the neutron star and offer insight into the line-producing region and the structure of the accretion column and its geometry (Staubert et al. 2019). Cyclotron absorption features thus have provided an important diagnostic probe for detailed studies of neutron star binaries since their discovery in the spectrum of Her X-1 (Truemper et al. 1978). There are several reports on the detection of cyclotron absorption features in the spectrum of many Be binaries, starting at a lower energy, from ∼10 keV (Jun et al. 2012; DeCesar et al. 2013), to a higher energy, at ∼100 keV (La Barbera et al. 2001). A detailed compilation of such sources and studies is given in Staubert et al. (2019) and Maitra (2016). It has been observed in detailed studies that some sources show a wide variation in their cyclotron line energy with respect to their pulse phase and source luminosity, and time, such as Vela X-1, Cen X-3, and Her X-1 (Staubert et al. 2019). These interesting properties help in understanding the nature of these sources and also offer insight into their underlying physical properties governing such changes.


Only a few GCM studies have previously considered isolated examples of higher-order spin–orbit resonance effects on climate (Wordsworth et al. 2010; Yang et al. 2013, 2020; Turbet et al. 2016; Boutle et al. 2017; Del Genio et al. 2019b). To our knowledge, no previous work has incorporated geothermal heating into a 3D GCM in the context of evaluating IHZ limits. Haqq-Misra & Kopparapu (2014) did report the impact of a 2 W m−2 surface heating in a highly idealized GCM for a synchronous rotation planet, while Haqq-Misra & Heller (2018) conducted idealized GCM simulations of tidally heated exomoons in synchronous rotation with the host planet. Yang et al. (2013) showed that at 2:1 and 6:1 resonances with a static/slab ocean (see Section 2.2.2 of Way et al. 2017), Bond albedo is lower than it is for synchronous rotation and decreases rather than increases with incident stellar flux, thus destabilizing the climate as the planet approaches the IHZ. Turbet et al. (2016) considered a 3:2 resonance state and static ocean for Proxima Centauri b, assuming zero eccentricity. Boutle et al. (2017) simulated the same planet in 3:2 resonance and 0.3 eccentricity; that study uses a thin static ocean surface, which produced a longitudinal double-eyeball pattern of surface liquid water roughly coincident with the maxima in stellar heating. Wang et al. (2014) found zonally symmetric temperatures for 3:2 and 5:2 resonances with a static ocean, but Dobrovolskis (2015) showed that this was the result of an incorrect spatial pattern of instellation. Del Genio et al. (2019b) performed the first dynamic ocean simulation of a planet in a higher-order resonance, showing that despite the double maximum in instellation at 3:2 resonance, the resulting climate has a tropical liquid waterbelt spanning the planet because of the ocean thermal inertia and heat transport. Yang et al. (2020) used a dynamic ocean and focused on the outer edge of the habitable zone by considering the effect of sea ice drift on snowball transitions for nine exoplanets, including a sampling of the 3:2 resonance, also with zero eccentricity.


The spatial distribution of OB stars and associations, young long-period Cepheids and open clusters, star-forming regions, H ii regions, interstellar dust, and giant molecular and neutral gas clouds in the solar vicinity that have been in existence generally τ ≲ 108 yr is known to correlate with the location of the inner Sagittarius, the closest Orion, and outer Perseus spiral arm segments. (The distances for the vast majority of these spiral tracers have been determined in the literature with trigonometric or photometric methods.) The Sun is situated at the inner edge of the Orion arm (Levine et al. 2006; Hou & Han 2014; Nakanishi & Sofue 2016; Xu et al. 2018, 2021; Lallement et al. 2019; Reid et al. 2019; Skowron et al. 2019; Cantat-Gaudin et al. 2020; Fig. 2 above).3 These three spatial features nearby to the Sun appear to form part of the global spiral structure in the Galaxy. Contrary, the objects of older population with larger random velocities, for instance, main-sequence A–K stars or the oldest Cepheids and open clusters, do not currently follow the exact location of those arms (e.g. Cantat-Gaudin et al. 2020, fig. 8 therein; Griv et al. 2020, fig. 7 therein). The latter can be explained by the difference in rotation velocity between the spiral density waves and the objects. Investigating the velocity field of Xu et al.’s (2018) O and early B-type stars in the framework of the Lin–Shu density-wave proposal, we also found that the Sun lies within the Orion arm, at the inner edge of this spiral. The radial distance from the Sun to the centre of the Orion arm is ≈0.2 kpc in the direction of the Galactic anticentre, the centre of the Sagittarius arm is ≈1.8 kpc from the Sun in the direction of the GC, and the width of the arms is ≈0.5 kpc. The radial distance between the centres of the Orion and Sagittarius arms near the Sun is λrad ≈ 2 kpc (cf. Hou & Han 2014; Wu et al. 2014; Bovy et al. 2015). As for us, the nearest Orion spiral arm forms part of the dominant density-wave structure of the system.


The modulation of an extra-solar signal can, if working in terms of signal phase, be expressed as a time modulation, e.g. for a phase evolution given by
(1)
\begin{eqnarray}
\phi (t) = \phi _0 + 2\pi f_0\left( t - t_0 + \Delta \tau (t) \right),
\end{eqnarray}
where t is the time of arrival of the signal at the observer, and ϕ0 and f0 are an initial phase and frequency at the epoch t0 in a reference frame at rest with respect to the source, the time modulation term is Δτ(t). Assuming, for now, that the source is at rest with respect to the SSB, the time modulation can be expressed as a combination of terms
(2)
\begin{eqnarray}
\Delta \tau = \Delta _{\rm R} + \Delta _{\rm E} - \Delta _{\rm S},
\end{eqnarray}
where ΔR (the Roemer delay) is a geometric retardation term, ΔE (the Einstein delay) is a relativistic frame transformation term taking into account relativistic time dilation, and ΔS (the Shapiro delay) is the delay due to passing through curved space–time. These terms are discussed in, for example, chapter 5 of Lyne & Graham-Smith (1998), while Edwards, Hobbs & Manchester (2006) provide more detailed discussion of time delays accounting for more effects with particular relevance to pulsar observations. Here, for each of the terms we use the sign conventions given in the source code for the pulsar timing software tempo21 (Hobbs, Edwards & Manchester 2006) and in the LALSuite gravitational wave software library functions (LIGO Scientific Collaboration 2017), rather than those used in the equation of Edwards et al. (2006).2 The Roemer delay is given by
(3)
\begin{eqnarray}
\Delta _{\rm R} = \frac{\boldsymbol {r}\cdot \hat{\boldsymbol {s}}}{c},
\end{eqnarray}
where $\boldsymbol {r}$ is a vector giving the position of the observer with respect to the SSB, and $\hat{\boldsymbol {s}}$ is a unit vector pointing from the observer to the source. The Einstein delay (see e.g. equations 9 and 10 of Edwards et al. 2006) converts to a new time coordinate frame, and depends on the choice of frame you want, i.e. Barycentric Coordinate Time (TCB), in which the effect of the presence of the Sun's gravitational potential is removed. The Shapiro delay (for which we will only consider the contribution from the Sun) is to first order given by
(4)
\begin{eqnarray}
\Delta _{\rm S} \equiv \Delta _{\rm S_{\odot }} = -\frac{2G {\rm M}_{\odot }}{c^3}\ln {\left({\boldsymbol {r}}_{\rm se}\cdot \hat{\boldsymbol {s}} + |\boldsymbol {r}_{\rm se}| \right)}
\end{eqnarray}
for waves passing around the Sun, where $\boldsymbol {r}_{\rm se} = \boldsymbol {r}_{{\oplus}} - \boldsymbol {r}_{\odot }$ is the vector from the centre of the Sun to the geocentre.3 Unlike electromagnetic waves, gravitational waves will pass through matter, and therefore a different term is required for a wave passing through the Sun, i.e. when $\left|\boldsymbol {r}_{\rm se}\right|^2 - \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2 < R_{\odot }^2$ and $\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}} < 0$, giving
(5)
\begin{eqnarray}
\Delta _{\rm S_{\odot }} &=& -\frac{2G {\rm M}_{\odot }}{c^3}\Bigg [\ln {\left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}} + \sqrt{R_{\odot }^2 + \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2} \right)} \nonumber \\
&&- \,2\left(1 - \frac{\sqrt{\left|\boldsymbol {r}_{\rm se}\right|^2 - \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2}}{R_{\odot }}\right)\Bigg]\!,
\end{eqnarray}
where R⊙ is the radius of the Sun.


Direct measures of cascade rates in turbulent systems often employ theoretical formulations related to Kolmogorov’s “4/5” law (Kolmogorov 1941b; Frisch 1995) and its variants, in which the inertial range cascade rate is related to a signed third-order structure function. This so-called exact law is derived from the fluid equations without appeal to dimensional analysis, assumptions about scaling behavior, or any ansatz concerning timescales; however, this law does require time stationarity, spatial homogeneity, the existence of an inertial range, and a finite dissipation rate. The original formulation for isotropic incompressible hydrodynamics has been extended to magnetohydrodynamics (MHD; Politano & Pouquet 1998a,1998b) and related models. The MHD version is frequently applied to in situ observations of plasma turbulence in the solar wind (Sorriso-Valvo et al. 2007; MacBride et al. 2008; Bandyopadhyay et al. 2020) to obtain cascade rates that inform theories of heating and acceleration of the solar wind (Osman et al. 2011), providing ground truth for related approximations in space physics (Vasquez et al. 2007). Frequently a major issue in these applications is the use of formulations derived assuming isotropy in turbulence that is actually anisotropic (Verdini et al. 2015), this being the typical case for solar wind and magnetosheath turbulence. Usually this potential inconsistency is disregarded in favor of extensive averaging, whenever possible. Another more practical limitation is the challenging requirement of a sufficient volume of data (Podesta et al. 2009), a kinematic and statistical issue further complicated by potential sensitivity to the tails of the probability distribution of the fluctuations (Dudok de Wit 2004). Taking these challenges into account, we note that the ability to extract cascade rates from observational data is of increasing importance due to the centrality of fundamental questions relating to heating and dissipation in space and astrophysical plasmas (e.g., Kiyani et al. 2015). Therefore, in the present study we revisit several related issues that are pertinent to the evaluation of third-order laws using single-point or multi-point measurements. We reexamine the issue of averaging by focusing on conditions for obtaining accurate results in both isotropic and anisotropic turbulence. The strategies we examine are implemented using data from three-dimensional (3D) MHD turbulence simulations. A motivation for this approach is that for such cases we have an unambiguous determination of the underlying turbulence symmetry as well as a straightforward method to quantify the absolute dissipation rate.


A sample of 17 starless clump candidates has been selected from the Traficante et al. (2015a) as objects with Σ ≥ 0.05 g cm−2, mass ${{{M}}}\ge 300$ M⊙, bolometric luminosity over envelope mass ratio L/M ≤ 0.3 and very low dust temperature (T  15 K), indicative of very young stage of evolution (see Section 4), no (or faint) emission at 70 μm after visual inspection of each source and no counterparts in the MSX and WISE catalogues in correspondence of the Herschel dust column density peak. In addition, we checked for different masers emission associated with these clumps, as they are an indication of on-going star formation activity. We searched in the methanol multibeam survey (MMB; Green et al. 2009) and found no Class II CH3OH masers in the sources of our sample (Breen et al. 2015); from the MMB survey we also searched for hydroxyl (OH) masers at 6035 MHz (Avison et al. 2016), a transition often associated with high-mass star-forming regions, and also found no associations. We searched for CH3OH and OH masers using also the Arecibo surveys of Olmi et al. (2014), which is more sensitive than the MMB survey and it is targeted to identify weak masers associated with Hi-GAL high-mass objects. We found no CH3OH masers at distances less than 100 arcsec from the source centroids. We found one source, 34.131 + 0.075, with a weak OH maser (peak emission of 20 mJy, ≃3σ above the rms of the observations made with the Arecibo telescope, Olmi et al. 2014). Finally, we checked several surveys of water masers in the first quadrant (Merello et al., in preparation, and references therein) and found that only one source, 23.271 − 0.263, has a H2O maser association (at ≃3 arcsec from the source centroid), identified in the survey of Svoboda et al. (2016). Note that the source 18.787−0.286 is classified as starless in Traficante et al. (2015a) catalogue and has no maser associations, although it shows a 70 μm counterpart. The 70 μm source however is faint, with a peak emission of ≃60 mJy pixel−1 compared to a background of ≃130 mJy pixel−1. The clump follows all the other selection criteria and has very low dust temperature (T = 10.6 K, see Section 4), so we include it in the analysis. The clump embedded in the cloud SDC19.281−0.387, which follows the same criteria but it is not in the Traficante et al. (2015a) catalogue, was also included in our selection. The final sample of 18 clumps presented here contains some of the most massive 70 μm quiet clumps observed in the Galaxy.


Active galactic nuclei (AGN) have been discovered a century ago and still remain a hot topic of research. The theoretical predictions and the observational results suggest that the AGN have three main components namely, a central supermassive black hole (SMBH), an accretion disc around the SMBH, bipolar high relativistic jets of particles. The launching of the jets perpendicular to the accretion disc plane remains a long-standing puzzle to the astronomers. The GRMHD simulations have made some progress in understanding the launching of the jets and it is suggested to be the interplay between the accretion flow, magnetic field lines, and the SMBH (Mościbrodzka & Falcke 2013; Mościbrodzka et al. 2014). Simulations and theory together suggest that the magnetic field plays an important role in collimating the particles and giving them a jet like shape (Blandford & Königl 1979; Blandford & Payne 1982; Koide, Shibata & Kudoh 1998; Mościbrodzka, Falcke & Shiokawa 2016). The jets are believed to be highly relativistic and the process of particle acceleration inside the jets are still unclear. The AGNs are classified in various types under the AGN unification scheme developed by Urry & Padovani (1995). According to this scheme, the observer’s viewing angle with respect to jet axis divides the sources in various class such as quasars, seyfert galaxies, blazars, etc. In the case of blazars, the jet axis is within few degrees to the observer’s line of sight. Due to the direct view of the jet and because of the superluminal motion, blazars are among the brightest sources in the Universe. Blazars are classified in two main types namely flat-spectrum radio quasars (FSRQs) and the BL Lacertae (BL Lac) objects depending upon the presence or absence of optical emission lines in their spectra (Stickel et al. 1991; Weymann et al. 1991). Based on the location of the synchrotron peak, the BL Lac objects are further divided into three parts known as low BL Lac (LBL), intermediate BL Lac (IBL), and high BL Lac (HBL) by Padovani & Giommi (1995).


An outlier is defined to be an anomalous event or observation that arises from a process that differs from the majority of the data generation. Outlier detection has always been an indispensable part of data analysis; contaminated data sets without proper outlier treatment can increase modelling uncertainties and produce misleading results. A common mitigation strategy is to model the data as a Gaussian mixture of inlier and outlier distributions with differing variance (Hogg, Bovy & Lang 2010; Vallisneri & van Haasteren 2017). For example, assume a model with an inlier data model, yi = μi + ϵi, where yi is the i-th observation, μi is the mean, and ϵi is Gaussian random noise with variance $\sigma _i^2$. We can then account for possible outlier contamination in the data collection by modelling such fluctuations with $\epsilon _\mathrm{out}\sim \mathcal {N}(0, \sigma _\mathrm{out}^{2})$, where our full data model now includes a latent outlier indicator zi for each observation, where zi = 1 for an outlier and 0 otherwise. This indicator is modelled with a certain prior outlier probability, θ, e.g. zi ∼ Bernoulli(θ). This data model is a mixture of two Gaussians with different mean and different variances, which can be expressed in the form
(1)$$\begin{eqnarray}
y_i = (1-z_i) \mu _i +(1-z_i) \epsilon _i + z_i \epsilon _\mathrm{out}.
\end{eqnarray}$$Incorporating the outlier indication parameter zi allows us to assess the possibility of an individual measurement being an outlier and to formulate the likelihood as a mixture of two Gaussians that respectively model the inlier and outlier distributions:
(2)$$\begin{eqnarray}
p \left(y_{i} \mid \mu _{i}, z_{i}, \sigma _{i}, \sigma _{\mathrm{out }}\right)=\left\lbrace \begin{array}{l{@}{\quad}l}\mathrm{e}^{-\left(y_{i}- \mu _{i}\right)^{2} /\big(2 \sigma _{i}^{2}\big)} / \sqrt{2 \mathrm{\pi} \sigma _{i}^{2}}, & \text{for } z_{i}=0 \\
\mathrm{e}^{-y_{i}^{2} /\big(2 \sigma _{\mathrm{out }}^{2}\big)} / \sqrt{2 \mathrm{\pi} \sigma _{\mathrm{out }}^{2}}, & \text{for } z_{i}=1 .\end{array}\right. \\
\end{eqnarray}$$We use this model for the toy sine wave example in Section 2.3.


To overcome these limitations, in our previous paper (Chatzistergos et al. 2018b, Paper I, hereafter) we introduced a novel approach to process the historical and modern Ca II K observations, to perform their photometric calibration, to compensate for the intensity centre-to-limb variation (CLV, hereafter), and to account for various artefacts. By using synthetic data, we also showed that our method can perform the photometric calibration and account for image artefacts with higher accuracy than other methods presented in the literature. More importantly, we showed that, as long as the archives are consistent with each other, for example, they are centred at the same wavelength and employing the same bandwidth for the observations, the method can be used to derive accurate information on the evolution of plage areas without the need of any adjustments in the processing of the various archives (Chatzistergos et al. 2019b, Paper II, hereafter). In Paper II, we applied our method to 85 972 images from 9 Ca II K archives to derive plage areas and produce the first composite of plage areas over the entire 20th century. In particular, we analysed the Ca II K archives from the Arcetri, Kodaikanal (8-bit digitisation), McMath-Hulbert, Meudon, Mitaka, Mt Wilson, Rome/PSPT, Schauinsland, and Wendelstein sites. Five out of the nine analysed archives were amongst the most studied and prominent ones, while the remaining archives were from less studied data sources. There are, however, many other Ca II K archives that are available and still remain largely unexplored. These archives harbour the potential to fill gaps in the available plage series as well as to address inconsistencies among the various archives and within individual archives (e.g. change in data quality, or in the measuring instrument with time). Moreover, since the work presented in Paper II, more data from various historical and modern archives became available in digital form. In particular, historical data that have been made available in the meantime include those from the latest 16-bit digitisation of the Kodaikanal archive, Catania, Coimbra, Kenwood, Kharkiv, Kyoto, Manila, Rome, Sacramento Peak, and Yerkes observatories, as well as additional data from the Meudon and Mt Wilson archives. In this light, here we present results from the most comprehensive analysis to date of historical and modern Ca II K observations taken between 1892 and 2019 from 43 different datasets for the purposes of producing a composite plage area series.


As expected, the final luminosity (LV) of our model dwarfs strongly correlates with the shape of their formation histories. We divide our models into three categories dependent on their LV range. In the following, we will refer to them as sustained, extended and quenched. A few representative cases of each of these three categories are shown in Fig. 9. The strength of the UV-background heating is indicated by the dotted black curve. It represents the hydrogen photo-heating rate due to the UV-background photons following the model of Haardt & Madau (2012).

(a)LV > 108 L⊙, sustained: the star formation rate of those massive and luminous dwarfs increases over 1 to 2 Gyr. This period is followed by a rather constant SFR plateau. These systems are massive enough to resist the UV-background heating and, once formed, to form stars continuously. This sustained star formation activity that lasts up to z = 0 results from the self-regulation between stellar feedback and gas cooling (Revaz et al. 2009; Revaz & Jablonka 2012).
(b)106 L⊙ LV  108 L⊙, extended: in this luminosity range, the star formation is clearly affected by the UV-background. After a rapid increase, the star formation activity is damped owing to the increase of the strength of the UV-heating. However, at the exception of the h070 halo which is definitively quenched after 6.5 Gyr, the potential well of those dwarfs is sufficiently deep to avoid a complete quenching. The star formation activity extends to z = 0, however, at a much lower rate than the original one.
(c)LV  106 L⊙, quenched: the potential well of those galaxies is so shallow that the gas heated by the UV - photons escape the systems. Star formation is generally rapidly quenched after 2 or 3 Gyr. Only halo h064 shows signs of activity up to 4 Gyr. Those galaxies may be considered as true fossils of the re-ionization in the nomenclature of Ricotti & Gnedin (2005). They are all faint objects with only old stellar populations.


Figure 12 shows the results of our tidal evolution calculations. The left panel of Fig. 12 shows the planetary rotational evolution of GJ 1148 b due to star–planet tides. After ~850 Myr, GJ 1148 b reaches a rotation period that is 2∕3 of the orbital period, and remains there with Prot = 27.5 d. During the integration the planetary semi-major axis and eccentricity are mostly unaffected. An asymptotic rotation period that is shorter than synchronous and 2/3 of the orbital period is expected for eb ≳0.24 in the constant Q tidal model (Goldreich & Peale 1966; Cheng et al. 2014). The time for GJ 1148 b to reach asymptotic rotation is inversely proportional to the initial Prot, as long as the initial Prot is much less than 27.5 d, and it depends on the other parameters of GJ 1148 b according to Eq. (3) of Barnes (2017) and Eq. (15) of Cheng et al. (2014). The rotational period of GJ 1148 b is thus very likely much longer than the orbital periods of the hypothetical exomoons, which could be dynamically stable only with orbital periods between 0.7 and 2 d. The right panel of Fig. 12 shows that the longer rotational period of GJ 1148 b (Prot = 27.5 d) leadsto strong orbital decay of the stable exomoon orbits due to tidal interactions with the planet. An exomoon eventually reaches the Roche limit where it is tidally disrupted by the gas giant. Not even one hypothetical “stable” exomoon in the context of Sect. 5.3.1 had survived this test. The maximum time a Mars-like exomoon could survive is ~55 M yr, while for Titan-like moons the maximum survival time is longer, ~255 M yr. The latter is longer by roughly the mass ratio of Mars to Titan, which can be understood from Eq. (2) of Barnes (2017) and Eq. (16) of Cheng et al. (2014). These timescales are optimistic since the orbital decay would start before the planet reaches the asymptotic spin state. In both cases the survival times are much shorter than the age of the system. Therefore, given the relatively fast orbital decay in the small stable region around the planet, we conclude that exomoons around GJ 1148 b are unlikely to exist.


In order for simulations to play a role in improving our understanding of the formation and dynamics of the CGM, particularly given the complex, multiphase picture emerging from the latest observations (Tumlinson et al. 2011; Werk et al. 2014), the level of detail and sophistication in stellar feedback models must improve. In this work, we analyse the outflowing (and infalling) gas seen in the galaxies and CGM of the Feedback in Realistic Environments (FIRE) simulations,1 first presented in Hopkins et al. (2014). Unlike the subgrid recipes which involve kinetically prescribed decoupled winds and cooling-suppressed blastwaves, the FIRE simulations solve the ‘overcooling’ problem by explicitly modelling the radiation pressure, stellar winds, and ionizing feedback from young stars as taken directly from the population synthesis code starburst99 (Leitherer et al. 1999). These ‘early feedback’ mechanisms act before SNe, heating and stirring the surrounding ISM which is necessary to match conditions in star-forming regions such as Carina (Harper-Clark & Murray 2009) and 30 Dor (Lopez et al. 2011; Pellegrini, Baldwin & Ferland 2011). SNe are implemented by taking into account their energy and momentum input. When the cooling radius of SNe is resolved, SN energy injected is free to expand and generate momentum in the ISM before too much energy is radiated away. When this scale is poorly resolved, momentum accumulation from SN remnant evolution below the resolution scale is added to the surrounding gas. This model is physically realistic when it is applied on the scale of giant molecular clouds, meaning that a resolution of several to tens of parsecs is required. The physical feedback implementation in FIRE successfully regulates mass accumulation in galaxies and provides a physical explanation for the inefficiency of star formation in galactic discs (Kennicutt 1983, 1998; Genzel et al. 2010). We stress that we allow hydrodynamical interactions and cooling of all gas at all times, unlike in typical subgrid models. This is critical to make meaningful predictions for the phase structure of circumgalactic gas.


The SFG sample was constructed using the Great Observatories All-Sky LIRG Survey (GOALS sample, Armus et al. 2009), from which we extracted 158 galaxies, with data from Inami et al. (2013), who report the fine-structure lines at high resolution in the 10 − 36 μm interval, and Stierwalt et al. (2014), who include the detections of the H2 molecular lines and the PAH features at low spectral resolution. For those galaxies in the GOALS sample that have a single IRAS counterpart, but more than one source detected in the emission lines, we have added together the line or feature fluxes of all components, to consistently associate the correct line or feature emission to the total IR luminosity computed from the IRAS fluxes. To also cover lower luminosity galaxies, as the GOALS sample only includes luminous IR galaxies (LIRGs) and ultra-luminous IR galaxies (ULIRGs), we included 38 galaxies from Bernard-Salas et al. (2009) and Goulding & Alexander (2009), to reach the total sample of 196 galaxies with IR line fluxes in the 5.5 − 35 μm interval in which an AGN component is not detected. For the Bernard-Salas et al. (2009), Goulding & Alexander (2009), and the GOALS samples, we excluded all the composite starburst-AGN objects identified as those with a detection of [NeV] either at 14.3 or 24.3 μm. It is worth noting that the original samples from Goulding & Alexander (2009) and Bernard-Salas et al. (2009) have spectra solely covering the central region of the galaxies. To estimate the global SFR, we corrected the published line fluxes of the Spitzer spectra by multiplying them by the ratio of the continuum reported in the IRAS point source catalogue to the continuum measured on the Spitzer spectra extracted from the CASSIS database (Lebouteiller et al. 2015). We assumed here that the line emission scales (at first order) with the IR brightness distribution. In particular, we considered the continuum at 12 μm for the [NeII]12.8 μm and [NeIII]15.6 μm lines, and the continuum at 25 μm for the [OIV]25.9 μm, [FeII]26 μm, [SIII]33.5 μm, and [SiII]34.8 μm lines. This correction was not needed for the AGN sample and the GOALS sample because of the greater average redshift of the galaxies in the 12MGS and GOALS samples. In particular, the 12MGS active galaxy sample has a mean redshift of 0.028 (Rush et al. 1993), while the GOALS sample has a mean redshift of 0.026. The galaxies presented by Bernard-Salas et al. (2009) have instead an average redshift of 0.0074, while the sample by Goulding & Alexander (2009) has an average redshift of 0.0044. For the other lines in the 10 − 36 μm interval, Goulding & Alexander (2009) did not report a detection, and we used the data presented in Bernard-Salas et al. (2009) for a total of 15 objects. Both Bernard-Salas et al. (2009) and Goulding & Alexander (2009) reported data from the high-resolution Spitzer-IRS spectra. Data in the 50 − 205 μm interval were taken from Díaz-Santos et al. (2017). For the GOALS sample, 20 starburst galaxies were taken from Fernández-Ontiveros et al. (2016), and 23 objects were taken from the ISO-LWS observations of Negishi et al. (2001). As a result, we obtained a total sample of 193 objects. Lastly, the PAH features’ fluxes were measured from the low-resolution Spitzer-IRS spectra by Brandl et al. (2006), including 12 objects from the sample of Bernard-Salas et al. (2009) and 179 objects from Stierwalt et al. (2014).


To check for any spectral variation in the optical/IR bands, we looked for variation in the V − J band colour against the V-band brightness. This colour variation was analysed for the epochs A, B, D, and E. During epochs A and B, the source showed a ‘redder when brighter’ (RWB) behaviour. During epoch E, a bluer when brighter behaviour was observed. During epoch D, we observed a complex behaviour. Upto a V-band brightness of around 15 mag, the source showed a ‘bluer when brighter’ behaviour, but for optical brightness fainter than 15.0 mag, a ‘redder when brighter’ behaviour was observed. The colour–magnitude diagrams for all the four epochs are shown in Fig. 10. The spectral variations shown by the source are thus complex. From studies on the optical–IR colour–magnitude diagram, it is known that FSRQs in general show an RWB trend, which is attributed to them having a luminous accretion disc (Gu et al. 2006; Bonning et al. 2012). The observed optical emission is a combination of thermal blue emission from the accretion disc and non-thermal red emission from the jet. As the source gets brighter, the non-thermal emission has a more dominant contribution to the total flux, giving rise to the RWB behaviour (Bonning et al. 2012). During epochs A and B, there is a trend of the object to become RWB, irrespective of its optical brightness. The optical flares dominated by synchrotron emission processes during A and B have corresponding γ-ray flares that are produced by EC processes. However, during epochs D and E, the colour variations were found to depend on the optical brightness. During the epochs when this complex spectral behaviour was noticed, the source showed an optical/IR flare with no or a weak corresponding flare in the γ-ray band. The source showed a much larger amplitude of variability in the optical/IR bands, while in the γ-ray band it was either faint or below the detection limit of Fermi. This definitely points to some complex physical changes and could be due to a combination of changes in the bulk Lorentz factor, electron energy density, and magnetic field as seen from our SED modelling of the multiband data.


Once He core burning finishes the core contracts and hydrogen burning starts, the radius of the hot subdwarf expands beyond its Roche radius, and mass transfer starts at an orbital period close to the observed orbital period. Mass transfer will continue for ≈1 Myr at a rate exceeding 10−9 M⊙ yr−1 until hydrogen shell burning is finished and the star contracts to become a carbon–oxygen WD with a thick helium layer and a small residual layer of hydrogen. The high accretion rate will heat the accreting WD significantly (Townsley & Bildsten 2003). Models predict a Teff ≈ 50,000 K for accretion rates of 10−9 M⊙ yr−1 (Burdge et al. 2019) consistent with the high blackbody temperature of the accretor observed in ZTF J2055+4651. Accretion onto the WD companion at this rate will cause unstable hydrogen ignition after ≈10−4 M⊙ accumulates, leading to classical novae eruptions (Nomoto 1982; Nomoto et al. 2007; Wolf et al. 2013). This accretion rate predicts a recurrence time of order 105 yr for a total of approximately 10 novae. Our binary model suggests that this system is within the first ≈10% of the 1 Myr accretion phase. At the current state the orbit will shrink with

 s s−1, which will be detectable after a few years of monitoring. The right panel of Figure 5 shows the evolution of the donor through this mass transfer phase. After accretion has ceased, the orbit of the system will continue to shrink due to the radiation of gravitational waves and the system will merge in ≈30 Myr. Our models predict that there is a substantial He layer of ≈0.05 M⊙ left in the former hot subdwarf and the total mass of the system is relatively high (Mtotal ≈ 1.1 M⊙). Recent models predict that such a system explodes as a subluminous thermonuclear supernova (Perets et al. 2019; Zenati et al. 2019). If the system avoids a thermonuclear supernova it will merge and could evolve into a rapidly rotating single high-mass carbon–oxygen WD (Saio 2008; Clayton 2012; Schwab 2019).


It was found that the complex dielectric function from Pecharromán et al. (1999) for the sample obtained by heating bayerite at 1273 K, assuming a spheroid with depolarization parameters of (0.35, 0.003), produced an opacity with 11, 20, 28, and 32 μm features, so this component was included in the models. However, with only this component, the observed 20 μm features in the residual spectra were found to be wider than those in the models. By adding the opacity of the sample obtained by heating boehmite at 1173 K, the width of the 20 μm feature could be matched. This was done using the complex dielectric function for the sample obtained by heating boehmite at 1173 K from Pecharromán et al. (1999), assuming a spheroid with depolarization parameters of (0.35, 0.035). The complex dielectric functions of the samples obtained by heating bayerite and boehmite to various temperatures (Pecharromán et al. 1999) were derived by modeling the reflectance spectra of pellets obtained by pressing powders of these materials under great pressure. This method required Pecharromán et al. (1999) to assume an effective medium theory, such that a pellet is a mixture of one of their samples with a matrix of air. Pecharromán et al. (1999) noted that heating bayerite at 500°C eliminates the XRD pattern of bayerite, and they note that at 700°C, the infrared reflectance spectrum of the boehmite sample no longer shows OH− stretching bands. This must mean that the samples obtained from heating bayerite at 1273 K and from heating boehmite at 1173 K are no longer bayerite or boehmite, respectively. XRD performed by Pecharromán et al. (1999) of the sample of bayerite prepared at 1273 K suggests only θ-alumina was present, and their infrared and NMR spectroscopy confirms this. XRD of their sample obtained from heating boehmite to 1173 K (Pecharromán et al. 1999) suggests δ-alumina to be present, though some amounts of θ-alumina and α-alumina are present, as they deduce from XRD and infrared and NMR spectroscopy.


The lifetime of molecular clouds (MCs) remains an active research topic in the study of the interstellar medium and star formation, and most recent studies, both observational and theoretical, place this lifetime at a few times 107 yr for clouds in the 105–106M⊙ mass range (e.g. Blitz & Shu 1980; Kawamura et al. 2009; Zamora-Avilés, Vázquez-Semadeni & Colín 2012; Zamora-Avilés & Vázquez-Semadeni 2014; Lee, Miville-Deschênes & Murray 2016). In addition, several observational studies have suggested that the star formation rate (SFR) of the clouds appears to increase over their lifetimes. For example, studies of young clusters embedded in moderate-mass MCs (∼104M⊙) (e.g. Palla & Stahler 1999, 2000; Da Rio et al. 2010) have shown that their age histograms contain a large majority of young (1–2 Myr) objects, but also a tail of older (up to several Myr) ones suggesting an accelerating star formation activity, sometimes followed by a subsequent decline (see also Povich et al. 2016; Schneider et al. 2018). In addition, Kawamura et al. (2009) reported a clear evolutionary process over the lifetime of giant molecular clouds (GMCs; of masses ∼105–106M⊙) in the Large Magellanic Cloud, evidenced by the increasing number of massive stars across the sequence of GMC ‘classes’ proposed by those authors. Finally, on the basis of the large scatter in the observed star formation efficiency (SFE) in Milky Way GMCs, Lee et al. (2016) have concluded that the SFR in those clouds must also be time variable. Numerical simulations of MC formation and evolution also exhibit time varying, increasing SFRs during their early stages (e.g. Vázquez-Semadeni et al. 2007; Hartmann, Ballesteros-Paredes & Heitsch 2012). Also, in the presence of stellar feedback, at late times the SFRs reach a maximum and begin to decrease again (e.g. Vázquez-Semadeni et al. 2010; Colín, Vázquez-Semadeni & Gómez 2013). Vázquez-Semadeni, González-Samaniego & Colín (2017) have recently shown that the simulations of Colín et al. (2013) in fact produce stellar age histograms highly resemblant of the observed ones (Palla & Stahler 1999, 2000; Da Rio et al. 2010), and reproduce observed radial age gradients in clusters (Getman et al. 2014) as well as bottom-heavy stellar initial mass functions (IMFs) in scattered regions of massive star formation (Povich et al. 2016).


We have also considered early radio emission from gamma-ray bursts (GRBs), which can have higher brightness temperatures at early times than blazars, owing to their ultrarelativistic velocities. They can therefore be brighter and easier to measure while still at small angular sizes, and are consequently observed to show interstellar scintillation in their first days at ${\sim } 5\, \mbox{GHz}$ (Granot & van der Horst 2014). Before deceleration to Lorentz factor Γ  1/θj (before the ‘jet break’ for a jet of opening half-angle θj), the projected source angular size θ at (earth) time T after explosion of a GRB at redshift $z$ is θ ∼ 2cT Γ/DM($z$), where DM($z$) = DA($z$)(1 + $z$) is the proper motion distance, and DA($z$) the angular diameter distance. The Blandford–McKee blast wave of the ultrarelativistic shock moving into a medium of uniform external density ρ0 has radius R ≃ 2cT Γ2/(1 + $z$) and explosion energy per unit solid angle E/Ω ≃ ρ0R3Γ2c2, which gives Γ ≃ 9(Eiso, 53/n0)1/8(T/[(1 + $z$)day])−3/8, where $E=10^{53}\, \mbox{erg}(\Omega /4\pi)E_{iso,53}$ and n0 is the external density in cm−3 (Granot et al 2002, cf.). At DM($z$ = 1) = 3.3 Gpc, $\theta = [0.2,\, 1,\, 4]\, \mu \mbox{as}$ at $T=[0.1,\, 1,\, 10]\, \mbox{d}$. Thus at $\lambda \lt 4\, \mbox{cm}$ (the transition wavelength below which Milky Way scintillation becomes unimportant), the GRB will be smaller than our fiducial scattering angle $\theta =20\,\mu \mbox{as}(\lambda /30\, \mbox{cm})^{11/5}\lt 0.25\,\mu \mbox{as}$ for less than 0.1 d. During this time, the scintillation time-scale will be set by the rapidly expanding source, expanding across the refractive screen at a projected speed of ∼ΓcDl/Ds. This is many times c for our cosmological lenses with Dl ∼ 0.5Ds (but less than $1\, \mbox{km s}^{-1}$ for Milky Way interstellar plasma at $D_l\sim 100\, \mbox{pc}$, so Milky Way scintillation time-scales are dominated by gas motions in the Milky Way, not the apparent source expansion). The refractive scintillation time-scale is thus the same as the time-scale for the source to expand to a size larger than the refractive scale – i.e. the source will have only about 1 speckle before becoming too large to display refractive scintillation. This would be difficult to convincingly detect in a GRB.


We calculate the corresponding γ-ray spectrum constructed from the photons arriving at the observer within the observation time of MAGIC and plot it in the bottom panel of Fig. 5. Although it is possible to explain the very fast variability of the emission even with a moderate Lorentz factor of the blob, strong constraints are put by the level of the observed flux. If the emission region has a radius of Rb = 3 × 1014 cm, and is moving with a Lorentz factor of γb = 100, we require an energy density of ρE = 20 erg cm−3 (measured in the blob's frame of reference) to reproduce the flux observed by MAGIC and Fermi-LAT (see the bottom panel of Fig. 5). Note that such large values of the Lorentz factor of the emission region in the jet have been already postulated in terms of other models in order to explain extremely short flares observed in this source (Ackermann et al. 2014) or in the other sources, e.g. PKS 2155−304 (Aharonian et al. 2007). Such large Lorentz factors of the blob find also some observational support from the observations of the superluminal motion in PKS 1510−089 which represents similar type of blazar (Jorstad et al. 2005). We can estimate the power of the blob in the observer's reference frame on $L_{\rm blob} = \pi R_{\rm b}^2c\rho _E \gamma _{\rm b}^2\approx 1.7\times 10^{45}$ erg s−1. On the other hand, the Eddington luminosity of the black hole in PKS 1222+21, with the mass 6–8 × 108 M⊙, is LEdd = 1.3 × 1047M9 ≈ 8–10 × 1046 erg s−1. Therefore, the blob has to contain about ∼2 per cent of the Eddington power. This is quite demanding but seems not to be excluded, especially if Rb ≈ R⊥. Lower values of γb require a much higher energy density in the blob (e.g. ρE = 340 erg cm−3 for γb = 50 and Rb = 3 × 1014 cm). The strong dependence on the γb is a combined result of the transformation of the energy density to the reference frame of the observer and the beaming of the emission in a narrower cone. Note that a larger radius of the blob will lower the energy density constraint, e.g. for Rb = 1015 cm we obtain ρE = 4.9 erg cm−3 for γb = 100 and ρE = 80 erg cm−3 for γb = 50, at the assumption that there is no competing energy loss process of the electrons at such a large distance from the star.


While the angular extent of IP shocks can be directly investigated using multi-point in situ measurements, the size of coronal shocks can only be indirectly inferred via remote-sensing observations of the electromagnetic emissions associated with them. According to Nelson & Robinson (1975), the average angle subtended at the solar surface by fundamental metric type II radio emission sources is 43Â°. Aurass et al. (1994) found particular cases with larger, double type II source structures covering a separation angle beyond 90Â°. Type II radio sources often show non-radial propagation trajectories (see Mann et al. 2003, and references therein). Wave-like large-scale disturbances propagating over the solar disk in extreme ultraviolet observations (usually referred to as âEIT wavesâ or âEUV wavesâ) are in close empirical correlation with type II radio bursts (Klassen et al. 2000). Most EIT waves are accompanied by CMEs, and observations and MHD modeling suggest that they are driven by the lateral expansion of CMEs, while the ultimate nature of the phenomenon remains under debate and could consist of true waves, pseudo waves (e.g., reconnection fronts), or a combination of both (Patsourakos & Vourlidas 2012; Nitta et al. 2013b, and references therein). According to Patsourakos & Vourlidas (2012), EIT waves can reach distances up to 1.3 R (850 Mm) from the source. Single-case studies reported some EIT waves covering a whole solar hemisphere (Klassen et al. 2000; Kienreich et al. 2009; Thompson & Myers 2009). Connections between EIT waves and SEP events have been often suggested (e.g., Bothmer et al. 1997; Krucker et al. 1999), and recently Rouillard et al. (2012) hypothesized that the EIT wave can be used to track the expansion of a coronal shock responsible for particle acceleration. Other authors question the EIT wave acceleration scenario for SEPs, with many EIT waves being observed at well-connected positions having no associated SEP increase (Miteva et al. 2014).


Thermal motions of atoms produce a Doppler broadening of spectral lines with a Gaussian profile. Other unresolved velocities of a random nature are usually described as a turbulence broadening with a Gaussian or Lorentzian profile (see Rutten 2003, for a detailed description). The instrumental broadening encompasses the broadening caused by the finite spectral resolution of the instrument and is commonly approximated by a Gaussian. When other line broadening mechanisms are negligible, the total line broadening is the convolution of the line broadening profiles for the thermal and turbulence motions as well as the instrumental profile (Sect. 10.5 in  Böhm-Vitense 1989). In other words, the velocity equivalent of the observed line width, Wobs = c × Δλ/λ, at 1/e of the peak intensity results from instrumental, Doppler, and turbulence (non-thermal) broadenings by (1)\begin{equation} W_{\rm{obs}}^2 = \left(c\times \frac{\Delta\lambda}{\lambda}\right)^2 = W_{\rm{instrumental}}^2 + W_{\rm{Doppler}}^2 + W_{\rm{turbulence}}^2, \end{equation}Wobs2=c×Δλλ2=Winstrumental2+WDoppler2+Wturbulence2,assuming that all three line broadening components have a Gaussian profile (Eq. (4.17) of  Stix 2002). The POLIS instrumental width is less than 2 km s-1 (Beck 2006; Rezaei 2008; Beck et al. 2013a), while that of the Echelle spectrograph is of the same order. Assuming a generic chromospheric temperature of 104 K, we estimate a turbulence velocity using Eq. (2), (2)\begin{equation} \label{eq:one}  \Delta\lambda=\frac{\lambda}{c}\sqrt{2\,k_{\rm B}\,T/m+ W_{\rm{turbulence}}^2\,\,+\,\,W_{\rm{instrumental}}^2 }, \end{equation}Δλ=λc2 kB T/m+Wturbulence2  +  Winstrumental2,where Δλ is the observed line width, m is mass of the calcium atom, c the speed of light, and kB is the Boltzmann constant (Tandberg-Hanssen 1960). Using the measured width of Ca ii H & IR lines of 1.0 and 0.7 Å, respectively (Sect. 4), we estimate a turbulence velocity of about 45 km s-1 for Ca ii H and 24 km s-1 for Ca ii IR lines (we also note that the H1 minima of the EB profile is about 1 Å   wider than in the quiet Sun profile). The width of the emission peaks on either side of the Hα line (1 Å) amounts to a turbulence velocity of 15 km s-1. Attributing the line width to the temperature and the instrumental profiles (omitting the turbulence broadening), we get a temperature of about 5 ×  105 K, which is too hot for chromospheric heights. An increased turbulence velocity as a function of height in the chromosphere is part of standard models, either in the quiet Sun or sunspots (Kneer & Mattig 1978; Vernazza et al. 1981; Lites & Skumanich 1982). Our measured values, however, are larger than the turbulence velocity in an average atmosphere. The estimated turbulence velocity changes from one profile to another, but the general result remains the same: the observed width of the emission peaks is far in excess of any instrumental or thermal Doppler profile and to the first order has to have a turbulent nature. The Doppler width of a calcium line at 1–2 × 104 K is only about 2–3 km s-1, far from the observed widths of >20  km s-1.


As mentioned above, molecular oxygen was used to calibrate the magnetic field applied. A total of 155 Zeeman components (for the three transitions considered) was measured, with the magnetic field varied from B = 2.3 G (Itot = 0.2 Amp) to B = 113.5 G (Itot = 10 Amp). Figure 5 shows the Zeeman spectrum for the N,J = 3, 2 ← 1, 2 transition when a magnetic field of 5.7 G is applied. The fit was carried out with the program described in the previous section, with the spectroscopic parameters and g factors fixed at the values of Yu et al. (2012) and Christensen & Veseth (1978), Evenson & Mizushima (1972), respectively; the only free parameter was the correction factor to be applied to the theoretical magnetic field (see Eq. (1)). The fit reproduces in a satisfactory manner the measured Zeeman components, with a standard deviation of 73 kHz (the uncertainty for the measured frequencies was in most cases set to 70 kHz). Moving to SO, for the seven transitions considered, a total of 353 Zeeman components were measured, with the magnetic field varied from B = 5.7 Gauss (Itot = 0.5 Amp) to B = 124.8 Gauss (Itot = 11 Amp), and fitted as described above. In the fitting procedure the spectroscopic parameters (i.e., the rotational, centrifugal distortion, and fine structure constants) were kept fixed at the values derived by Lattanzi et al. (2015). The g factors resulting from the fit are given in Table 4 together with the best-estimated values discussed above, while the complete set of the measured Zeeman components is available in the Supplementary Material. Alternative fits were carried out: In the first fit, the three g factors, gs, gl, and gr, were fitted. In a second step, three different fits were carried out by fixing one of the three g factors and fitting the other two. In the last fit, only gs was determined. We note that in the first fit, based on the comparison with theory, the gs value is overestimated and the gl value is underestimated; the two terms therefore seem to be correlated. For this reason, we performed the additional fits described above. We note that in all cases, gr is determined with a limited accuracy, that is, with a relative uncertainty of ~20%. We also note that, if we fix gs at the best computed value, a gl value in good agreement with theory is obtained. However, by fixing gl at the best estimate, the resulting gs is still slightly overestimated. Simulations using values of gs in the 2.0020−2.0030 range show that the Zeeman splittings only change by a few tens of kHz, that is, in most cases within the typical uncertainty affecting the frequency measurements. The last comment concerns the standard deviation of the fits that, in all cases, is about 50 kHz.


Table 3 shows the best-fit model parameters for the baseline fit for the three observations. We note that we did not find any statistically significant neutral or ionized absorption intrinsic to the source. On the addition of an ionized absorption model generated using the latest atomic data with the photoionization modeling code CLOUDY (Ferland et al. 2013; Laha et al. 2016c, 2016b, 2017, 2019b), we did not detect any improvement in the fit, as also found in previous works on this source (Rivers et al. 2011). Two blackbody components were necessary to describe the soft excess. We detected a narrow FeKα emission line in the Suzaku observation at an energy E = 6.41 ± 0.01  keV, while a marginally broad line (σ = 0.35 ± 0.08  keV) was required in the XMM-Newton and NuSTAR observations. We detected a neutral reflection hump at energy E > 10  keV with Suzaku that was modeled with pexrav, with an improvement  Δχ2 = 32 for one additional degree of freedom. The pexrav model assumes an exponentially cutoff power-law spectrum reflected from neutral material (Magdziarz & Zdziarski 1995). The output spectrum is the sum of the cutoff power law and the reflection component. However, the reflection component alone can be obtained by setting the relative reflection value negative. The power-law cutoff energy for pexrav is assumed to be 300  keV, the abundance of the reprocessor set to solar value, and we allowed the inclination angle of this model to vary between 0° and 45° (being a Seyfert 1 galaxy). We have tied the power-law photon index normalization of pexrav to that of the primary power-law component. The pexrav reflection fraction is 0.17 ± 0.03. Note that throughout the text, we refer to the positive value of the reflection fraction. We did not detect any significant neutral reflection component in the XMM-Newton+NuSTAR spectra. Table 2 shows the fluxes of the soft excess, power law, FeKα emission line, and hard X-ray excess. In XSPEC notation, the best-fit baseline model is written as constant × tbabs × (powerlaw+bbody+pexrav+Gaussian(s)). We note that we required an emission line at 0.56  keV for XMM-Newton observation, which corresponds to O vii emission and is found in several bright AGNs, and we did not need that model for Suzaku and NuSTAR, as they do not cover that energy range.


The remaining six parameters are less constrained. They are ϵe, ϵB, n0, L (where L = ηL0), κ and Ek. We apply constraints to the range of allowed values for these parameters. ϵB has been found to be as low as 10−8 (Barniol Duran 2014; Santana, Barniol Duran & Kumar 2014) and as a fraction can be as high as 1. In practice, ϵe tends towards higher values than ϵB. We set an upper limit of 1, noting that ϵe actually refers to the electron population that is emitting synchrotron radiation, rather than the electron population as a whole, and set a lower limit of 10−3 (Kumar 2000). n0 is limited between 10−5 and 100 cm−3, in line with what has been found in these sources (Cenko et al. 2011). The upper limit of L is set by the argument in equation (5), and values of this parameter below ∼1047 erg s−1 are never energetic enough to match the data, so we set the lower limit as 1047 erg s−1. Within these limits for L, we find that EE ceases to have any influence on the light curve if κ ≲ 10−2. If EE is isotropic, and the observed luminosity is only 1 per cent of the true energy (i.e. the conversion efficiency of kinetic to potential energy in the internal shocks is 1 per cent), then the energy delivered to the synchrotron shock front could be up to 100 times higher than observed in the light curve. In practice, however, the emission is (a) unlikely to be fully isotropic, (b) likely to shock more efficiently than 1 per cent, and (c) certain to be less than 100 per cent efficient at delivering its energy to the synchrotron shock front. For these reasons, we set the upper limit of κ at a still fairly generous factor of 10. Finally, we limit the energy in the shock from prompt emission to 1048 erg  Ek  1052 erg. The arguments for these limits are identical to those used for κ, except that the prompt emission is known to be beamed (Sari, Piran & Halpern 1999; Frail et al. 2001) so the upper limit is lower, and because the injected energy at early times is negligible, Ek dominates the early light curve so the lower limit can be much less energetic before its influence vanishes. These limits are summarized in Table 3.


The data produced by the HMI instrument are of high quality; however, there are known uncertainties, limitations and systematic errors present that affect the measurement of magnetic flux. These include a sinusoidal variation in the total magnetic flux with a periodicity of 12 and 24 hr, due to a Doppler shift present in the Fe spectral line (Hoeksema et al. 2014). The main contribution to this shift is the geosynchronous orbit of SDO. The daily variation of ±3 km s−1 in spacecraft orbital velocity causes a sinusoidal variation in the total flux measured. This affects weak and strong magnetic fields in the LoS magnetograms differently, with the daily variation remaining less than 30 G for field strengths below 1000 G and less than 75 G for field strengths below 2250 G. On average during a day this is roughly ±35 G (Couvidat et al. 2015). Nevertheless, the strength of these instrumental and observational effects does not account for the strong flux cancellation we observe in AR 11226 prior to eruption. The leading edge of the positive polarity of AR 11226 reaches ∼60° from central meridian on June 6. Due to the spatially dependent sensitivity of HMI the noise level increases as a function of the center-to-limb angle and the spacecraft’s orbital velocity. This increases the value of pixels in low and moderate fields (between 250 and 750 G) by a few tens of percent and manifests itself as broad peaks that are centered at ∼±60° in the magnetic flux. This could be responsible for the increase in flux seen in Figure 5 at this time. However, the increase also coincides with the emergence of the two anti-Hale bipoles and so it is difficult to disentangle these effects. The proximity of AR 11226 to the limb on June 6 means that the magnetic flux was not measured right up until the time of the eruption but only until approximately a day beforehand. Therefore, these results provide a lower limit to the total flux canceled in the lead-up to the eruption. Furthermore, due to the fact that cancellation persists for several days, we can extrapolate that, if the cancellation process continued at the same rate (2.0 × 1019  Mx hr−1) as we observed in the period from June 1 09:00 UT to June 5 00:00 UT up until eruption, then an extra 1.1 × 1021 Mx could have been canceled. This would therefore result in a total amount of 2.8 × 1021 Mx flux canceled between two consecutive CMEs.


Analyzing optical emission lines, emanating from within the northern lobe, Tylenda et al. (2019) found a reddening with EB − V ≈ 0.9 mag or AV ≈ 2.8 mag, which we assume is mainly circumstellar in origin. Hajduk et al. (2013) observed two stars shining through the southern lobe and found AV = 3.3 − 4.4 mag with unknown contribution from the interstellar component. We assume here that those observations quantify the amount of circumstellar dust that is the main actor in shielding molecules from the central source. We recalculated the lifetimes of molecules assuming AV = 3 mag, and with (1) standard dust properties (i.e. with composition and size distribution as of interstellar dust) or (2) with larger and less opaque grains, at the gas-to-dust mass ratio of 124 (see Heays et al. 2017, for more details on the assumed dust properties). We used shielding functions from Heays et al., which include effects in lines. Results are shown in Cols. (3) and (4) of Table 3. The presence of ISM grains makes it possible for the observed molecules to survive for a very long time, longer than 350 yr. The lifetimes in the presence of the large grains considered by Heays et al. are typically a few times shorter than the age of the remnant. It is uncertain what kind of grains populate the dusty remnant of CK Vul, but given its anomalous elemental and molecular compositions and eruptive history, dust may have a peculiar chemical composition and size distribution. In such a case, the total to selective extinction law would also be different and the assumed AV may not be adequate. Nevertheless, if the molecules formed 350 yr ago and are shielded by big grains, with the calculated lifetimes a considerable fraction of molecular species would survive, except perhaps for a few most fragile ones which indeed are almost absent in the lobes. We conclude that the lifetimes in Table 3 that were calculated with an attenuated ISRF are consistent with the molecule formation 350 yr ago or more recently.


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


As regards the modeling of BGK modes, there are two main theoretical approaches: the integral solution or BGK methodology and the differential (or Schamel) technique. In the former method (BGK), one assumes that the initial particle distribution function and the electrostatic potential profiles are known, so these are substituted into the Poisson equation and the integral equation is solved to obtain the trapped particle distribution function (Bernstein et al. 1957; Aravindakshan et al. 2018a, 2018b, and the references therein). In Schamel’s approach, the form of the trapped particle distribution function and of the passing (i.e., free, nontrapped) particle distribution function is assumed and substituted in Poisson’s equation, leading to a differential equation that is then solved to obtain the form of the potential (Schamel 1986; Luque & Schamel 2005, and the references therein). A distinguishing factor in the former (BGK) approach is that it involves a condition in the form of an inequality to be satisfied by the potential parameters (width and amplitude) in order for a BGK mode to be sustained. The BGK approach will be adopted in this work. The above models tacitly assume a collisionless electron-ion plasma. These assumptions are acceptable in the Earth’s magnetosphere. However, as we move farther from near-Earth plasma environments, the presence of charged dust in the plasma cannot be neglected. In the case of Saturn, there are observations of streaming ions by the Cassini spacecraft (Badman et al. 2012a, 2012b). We know that these streaming ion flows can lead to the generation of ion holes. Electrostatic solitary waves have been observed in Saturn’s magnetosphere (Williams et al. 2006) and in the dusty environment near its moon Enceladus (Pickett et al. 2015). Williams et al. (2006) reported observations of solitary structures in the vicinity of Saturn’s magnetosphere. They detected a series of bipolar pulses and speculated that these could be either electron holes or ion holes (Williams et al. 2006). Later on, Pickett et al. (2015) observed solitary wave pulses within 10 Rs (Rs is the Saturn radius) and near Enceladus. Near the Enceladus plume, they discussed how dust impacts affected the observed solitary waves. In fact, Pickett et al. (2015) pointed out that some of the bipolar electric field pulses associated with the solitary waves observed had an inverse polarity (i.e., a positive pulse first, followed by a negative pulse in a short time period) and suggested that this might be due to either an inverse direction of propagation or to a true inverse potential pulse polarity (sign). Moreover, Farrell et al. (2017) examined the conditions that allow low-energy ions, such as those produced in the Enceladus plume, to be attracted and trapped within the sheath of negatively charged dust grains. Using particle-in-cell simulations, they showed that with dust in the system, the large electric field from the grain charge disrupts pickup and leads to ion trapping. Their simulation results also reveal that the bipolar pulses reported in the Enceladus plume by Williams et al. (2006) and Pickett et al. (2015) could most probably be ion holes. In the light of the above information, we may suggest that the formation of ion holes is highly likely in the dusty plasma of environments such as the one found in Saturn. Importantly, the instrument on board Cassini, the Radio Plasma Wave Science instrument, does not have the ability to determine the polarity of the electric fields associated with the ESWs observed with 100% certainty as it lacks the ability to perform interferometry (Williams et al. 2006).


The values rinf, ρinf, and σinf of equations (9) and (10) were computed assuming a bulge mass profile. Unlike other works that use isothermal sphere or Dehnen profiles (see e.g Volonteri, Haardt & Madau 2003; Sesana 2010; Sesana & Khan 2015; Bonetti et al. 2018b; Volonteri et al. 2020), here we decided to use a Sérsic model. This choice is motivated by observational studies that found it to be a good approximation for fitting the bulge light distribution of different galaxies (Drory & Fisher 2007; Drory & Alvarez 2008; Gadotti 2009). The analytical expressions for the Sérsic model are taken from Prugniel & Simien (1997) (see also Terzić & Graham 2005):
(11)$$\begin{eqnarray*}
\rm \rho (\mathit {r}) = \rho _0 \left(\frac{\mathit {r}}{{\it R}_e}\right)^{-{\it p}} \mathit {\rm e}^{-{\it b}\left({\mathit {r}}/{{\it R}_e}\right)^{1/{\it n}}},
\end{eqnarray*}$$(12)$$\begin{eqnarray*}
\rm \sigma ^2(\mathit {r}) &=& \frac{4 \pi G \rho _0^2 \mathrm{{\it R}_{\rm e}}^2 n^2 b^{2\mathit {n}(p-1)}}{\rho (\mathit {r})} \nonumber\\&&\times \, \int _Z^{\infty } \mathcal {Z}^{-\mathit {n}(p+1)-1} {\rm e}^{-\mathcal {Z}} \gamma (\mathit {n}(3-p),\mathcal {Z}) {\rm d}\mathcal {Z},
\end{eqnarray*}$$where $\rm {\it R}_e$ is the bulge effective radius,8 ρ0 is the central bulge density, and n is its Sérsic index. This index correlates with the central concentration of the bulge, being the bulges with smaller n, the ones less centrally concentrated. Finally, the variable γ represents the incomplete gamma function, whereas Z, p, and b are three different quantities that depend on the bulge properties: $\rm {\it Z} = {\it b}(\mathit {r}/{\it R}_e)^{1/{\it n}}$, $p = 1 - 0.6097 n^{-1} + 0.05563 n^{-2}$, and $\mathrm{\it b} = 2n - 0.33 + 0.009\,876 n^{-1}$. This Sérsic model causes that smaller MBHs spend more time in the hardening phase than the most massive ones. To guide the reader, for an MBHB system with total mass $\rm {\it M}_{bin} = 10^9\, M_{\odot }$, $q = 1$, and $e_{\rm BH} = 0.3$, the hardening time-scale is $\rm {\sim }0.2\, Gyr$. For the same system but with $\rm {\it M}_{bin} = 10^6\, M_{\odot }$, the time increases up to $\rm 10\, Gyr$. For further details, we refer to Biava et al. (2019), where a detailed study of hardening time-scales in different bulge profiles was performed.


Steady and smooth decline: Since falling from the peak of the 2012b event, SN 2009ip has only continued to fade and is now the faintest it has ever been in the optical. Immediately after the 2012 event (around day 200), it was declining somewhat slower than the rate of 56Co decay for a 56Ni mass of 0.04 $\, {\rm M}_{\odot }$ (shown by the dashed grey line in Fig. 2). During that time, its light curve and spectral properties were consistent with those of late-time SNe IIn, and it had an underlying broad-line spectrum similar to that of SN 1987A (Graham et al. 2014; Smith et al. 2014). Up to about day 1000 it continued to fade smoothly at a slower rate around 0.003 mag d−1, and spectroscopically it continued to resemble late-time interaction in SNe IIn (Fox et al. 2015; Smith et al. 2016b; Graham et al. 2017). Our new HST photometry shows that from about day 1000 to day 3000, SN 2009ip has continued to fade smoothly and steadily in the optical, at an even slower rate. While the HST cadence is obviously sparse at these late times, the filters with more than two observations (F555W and F814W) show no significant deviation from a steady decline; there is no evidence of any rebrightening or irregularity in the fading rate. From 2015 to 2021, the decline rates in the various optical filters are 0.00051 ± 0.00001 mag d−1 in F555W, 0.00068 ± 0.00002 mag d−1 in F606W, 0.00092 ± 0.00001 mag d−1 in F657N, and 0.00050 ± 0.00001 mag d−1 in F814W. (The UV is an exception, as discussed below.) While this decline is much slower than radioactive decay, such slow decline rates are not at all unusual for SNe IIn with late-time CSM interaction. SNe IIn span a wide diversity of late-time decay rates, ranging from some SNe IIn that have essentially flat light curves for many years, like SN 2005ip (Smith et al. 2009, 2017; Stritzinger et al. 2012; Fox et al. 2020), down to those that have only weak CSM interaction and faster decline rates that are difficult to distinguish from radioactive decay or light echoes. Some SNe IIn even rebrighten at late times, like SN 2006jd (Stritzinger et al. 2012). In any case, it seems as if SN 2009ip is levelling off and approaching a constant V absolute magnitude of −7.5 or so. In all three broad continuum filters, the object is now fainter than the progenitor, and the light curve has not exhibited any additional eruptive variability since the 2012b event.


Gravitational-wave astronomy has recently become a reality with the first detection of gravitational waves (GW hereafter) by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo ground-based interferemeters. To date, eleven events have been reported by the LIGO and Virgo detectors (Abbott 2018), and this number will quickly increase to tens of events in the coming years. Some of these events may correspond to gravitationally lensed events with magnification factors ranging from a few tens to a few hundreds (Dai et al. 2017; Ng et al. 2018; Li et al. 2018a; Smith et al. 2018a,b; Broadhurst et al. 2019). Recent works have studied lensing effects in the existing LIGO/Virgo O1 and O2 events (Hannuksela et al. 2019; Broadhurst et al. 2019), while Smith et al. (2018c) searched for candidate galaxy cluster lenses for the GW170814 event. The most likely lenses for such events would be massive galaxies or galaxy clusters (Ng et al. 2018; Dai et al. 2017; Smith et al. 2018b; Broadhurst et al. 2019). On the other extreme of the lens mass regime, compact objects with masses of a few hundreds to a few tens M⊙ can also act as lenses (Lai et al. 2018). In this case, the geometric optics limit is not valid since the Schwarszchild radius of the lens is comparable to the wavelength of the wave. For these relatively low masses, the lensing effect has a modest impact on the average magnification, but it can introduce a frequency dependence on the magnification (see for instance, Jung & Shin 2019; Lai et al. 2018). An even smaller mass regime was considered in Christian et al. (2018) where the authors find that lenses with a mass as low as 30 M⊙ could be detected with current experiments. They also consider future, higher-sensitivity experiments and show how they can push the limit to even smaller masses of order 1 M⊙. These conclusions are, however, obtained assuming isolated microlenses and without accounting for the effect of the macromodel, or other nearby microlenses. In the small mass regime, microlenses such as neutron stars have been also considered as scattering sources of GWs, and it is found that a GW can be focussed at a focal point near the neutron star surface (Halder et al. 2019; Stratton & Dolan 2019).


Observationally, it is a big challenge to conclude whether AGN’s feedback could regulate star formation or not, and how it regulates star formation. On one hand, if strong outflows emerge, they could clear out the star-forming gas to suppress star formation (Alexander & Hickox 2012; García-Burillo et al. 2014; Alatalo et al. 2015; Hopkins et al. 2016; Wylezalek & Zakamska 2016). The heating by jets propagating through the galaxies could prevent gas from cooling and cut-off the gas supply for further star formation (Karouzos et al. 2014; Choi et al. 2015). On the other hand, the outflows and jets interact with the gas in host galaxies and compress it to trigger new star formation (Silk 2013; Zubovas et al. 2013; Zubovas & Bourne 2017). In fact observations show either no or positive relationships between star formation rates and SMBH accretion rates but no negative trends are seen (Shi et al. 2007, 2009; Baum et al. 2010; Xu et al. 2015; Zhang et al. 2016; Mallmann et al. 2018). Whether AGN’s feedback plays the role may also depends on the spatial scale that observations could resolve and time-scale that the observed tracers could probe (Harrison et al. 2012; Cresci et al. 2015; Feruglio et al. 2015). For example, radiation from AGNs nearly instantaneously impact the surrounding ISM while the attenuation from ISM probably limits their impact to the nuclear regions (Roos et al. 2015). Outflows or jets travel slowly and may be decelerated after interactions with ambient gas, which delays their effects on star formation at large distances from the nuclei (Harrison 2017; Harrison et al. 2018). Feedback by jets or outflows on ISM also depends on their orientation relative to the dusty torus, making their effects on star formation to be anisotropic. The short duty cycle of AGNs could also make feedback by radiation from AGNs temporally variable in strength. Case studies of individual AGNs find evidence of coexistence of positive and negative feedback on star formation (Zinn et al. 2013), suggesting the complicated nature of AGN feedback.


Fermi–LAT (Atwood et al. 2009) is a pair conversion telescope, with a field of view (FoV) of above 2 sr, operating in the energy range from 20 MeV to 300 GeV. It is the most sensitive instrument available in this energy range (Ackermann et al. 2012). A few months after the launch in 2008 June, Fermi–LAT started to operate in all sky survey mode. The telescope scans the whole sky in 3 h (Atwood et al. 2009). For this paper we have analysed the Pass81 data from 2010 February 13 to 17. We analysed the data for Mrk 421 using the latest Fermi Science Tool2 software package version v10r0p5. In order to determine the flux and the spectrum of the source, maximum likelihood optimization has been used (Abdo et al. 2009). The data selection and quality checks have been made using the gtselect tool.3 Since the telescope is sensitive to γ-rays from the interactions of cosmic rays with the ambient matter, we set our maximum zenith angle at 105 deg to remove the background γ-ray events from Earth’s limb. The analysis included all photons from a circular region of 10 deg around Mrk 421, which we call the region of interest (ROI). Only photons of energy above 100 MeV were considered for further analysis. The latest LAT instrument response function ‘P8R2_SOURCE_V6’ has been used. The third Fermi–LAT catalogue (3FGL catalogue: Acero et al. 2015) has been used to include the contributions of sources inside the ROI. The spectral model of the source has been considered as a simple power law of the form $\mathrm{ d}N(E)/\mathrm{ d}E = N_0 (E/E_0)^{-\Gamma _{\mathrm{ ph}}}$, where N0 is called the prefactor, Γph is the index, and E0 is the scale in energy. The spectral parameters of the sources including Mrk 421 inside the ROI are kept free, whereas the spectral parameters of the sources beyond 10 deg from Mrk 421 are kept fixed to the values according to the 3FGL catalogue. The unbinned likelihood4 method has been used in order to estimate the detection significance of the sources. The test statistics parameter determined from the aforementioned method is given by TS = 2Δlog(L), where L denotes the likelihood function between the model with the source and without the source. According to the definition TS = 9 corresponds to a detection significance of ∼3σ (Mattox et al. 1996). All the sources with TS  9 are excluded from the likelihood analysis. In order to model the spectrum of sources, we used the latest Galactic diffuse emission model ‘gll_iem_v06’ and the isotropic background model ‘iso_P8R2_SOURCE_V6_v06’. We estimated the corresponding butterfly plots for the source using the methods mentioned in the Fermi–LAT webpage.5 VERITAS observed the high-flux state of Mrk 421 for ∼6 h. In order to obtain quasi-simultaneous data with VERITAS, we also analysed the Fermi–LAT data for this state by taking 12 h around the VERITAS observation period. In Fig. 2(d), we show the butterfly plots for both 24 and 12 h by closed butterfly (grey) and open butterfly with cross at the edges (blue), respectively, for the high-flux state.


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

2

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

or through the chain beginning with acetylene,
4

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


To solve equations (9)−(11), the pulsar rotation rate Ω0, the long-term spin-down rate $\dot{\Omega }_\infty$, the initial lag δΩ0, the mutual-friction coefficients $\mathcal {B}_{\operatorname{\text{f}}}$ and $\mathcal {B}_{\operatorname{\text{pin}}}$, and the ratios $I_{\operatorname{\text{n}}}^{\operatorname{\text{f}}}/I$ and $I_{\operatorname{\text{n}}}^{\operatorname{\text{pin}}}/I$ need to be specified. In what follows, Ω0 and $\dot{\Omega }_\infty$ are directly taken from pulsar timing. The coefficient $\mathcal {B}_{\operatorname{\text{f}}}$ in the non-pinned region is given by $\mathcal {B}_0$, and the corresponding drag-to-lift ratio by equation (8). In the pinned region, the coefficient $\mathcal {B}_{\operatorname{\text{pin}}}$ is given by equation (6), with the prescription (7) and suitable parameters. Typical values for the underlying parameters are: $\varepsilon _{\operatorname{\text{p}}}^{\operatorname{\text{pin}}}\simeq 0.05-0.2$, $\varepsilon _{\operatorname{\text{p}}}^{\operatorname{\text{f}}}\simeq 0.1-0.5$ (see e.g. Chamel & Haensel 2006; Sourie, Oertel & Novak 2016), $x_{\operatorname{\text{p}}}^{\operatorname{\text{pin}}} \simeq 0.05 - 0.1$, $x_{\operatorname{\text{p}}}^{\operatorname{\text{f}}} \simeq 0.05 - 0.4$, $\rho _{\operatorname{\text{pin}}} \simeq (0.5-2) \rho _0$, and $\rho _{\operatorname{\text{f}}} \simeq (2-6) \rho _0$, ρ0 ≃ 2.7 × 1014 g cm−3 being the nuclear saturation density (see e.g. Pearson et al. 2018). The ratios $I_{\operatorname{\text{n}}}^{\operatorname{\text{pin}}}/I$ and $I_{\operatorname{\text{n}}}^{\operatorname{\text{f}}}/I$ are computed using the relations $I_{\operatorname{\text{n}}}^{_X} /I^{_X} = 1-x_{\operatorname{\text{p}}}^{_X}$ (assuming uniform densities in each region), where $I^{_X}$ is the total moment of inertia of region X, and $I^{\operatorname{\text{f}}} = I-I^{\operatorname{\, \text{cr}}}-I^{\operatorname{\text{pin}}}$, $I^{\operatorname{\, \text{cr}}}$ denoting the crustal moment of inertia of the star. Typical values are: $I^{\operatorname{\, \text{cr}}}/I\simeq 0.01-0.05$ (Delsate et al. 2016) and $I^{\operatorname{\text{pin}}}/I\sim 0.05$ (Gügercinoğlu & Alpar 2014). Unlike the previous quantities, both the initial lag δΩ0 and number $N_{\operatorname{\text{p}}}$ of pinned fluxoids are essentially unknown. As shown in the next section, the large range of possible values for $N_{\operatorname{\text{p}}}$ could account for the very different spin-up behaviours in the Crab and Vela pulsars.