show a decrease in the distance between conjugate footpoints (Figure 10 ). The converging motion of flare ribbons is confirmed by some recent observations (Ji et al. 2006 , 2007 ; Liu et al. 2008 ; Joshi et al. 2009 ) and, thus, it has been identified as an important phenomenon occurring during the rise phase of flares. In particular, Liu et al. ( 2008 ) noted that during SOL2002-04-30, the two conjugate HXR footpoints first move toward and then away from each other, mainly parallel and perpendicular to the magnetic inversion line, respectively. Furthermore, the transition between these two phases of footpoint motions coincides with the reversal in direction of the motion of the looptop source. The explanation of ribbon motions of this kind is still beyond the scope of the standard flare model. After a prolonged rise phase, the flare exhibited the features of a “standard” large eruptive flare as described by the standard flare model. Here the main observational characteristic of the standard flare is recognized in the form of an increase in the separation of parallel flare ribbons while they move perpendicular to the PIL (Figure 11 ). Following the eruption of the flux rope, we observed the development of an arcade of coronal loops over the source region that was previously occupied by the coronal sigmoid. This phenomenon, called sigmoid-to-arcade transformation, is well established and theoretically studied in relation to the eruptive dynamics of CMEs (Gibson et al. 2002 , 2004 ). In the radio spectrum, the violent eruption of the flux ropes is followed by the occurrence of a broad, intense type II radio burst, implying the propagation of a shock wave associated with the passage of the CME (see, e.g., Cho et al. 2005 ). We, therefore, conclude that the standard flare model is well applicable to the late phase of this flare. We believe that, during this phase, the main driver of the energy release process is the large-scale magnetic reconnection driven by the erupting FR. 5. CONCLUSIONS Sigmoid-to-arcade development is considered to be an important aspect of solar eruptive phenomena. Although the association of sigmoids with eruptions is well recognized, our knowledge about the formation of these structures and their subsequent activation is still limited. In this paper, we provide a comprehensive multi-wavelength investigation of EUV, UV, H α , X-ray, radio, and magnetic measurements to probe the evolution of a transient coronal sigmoid in active region NOAA 11719, which activates and erupts, leading to a fast halo CME. In the following, we summarize the important results of this study: 1. The observations of the active region in the hot EUV channel of AIA (94 Å) reveal the formation of a large S-shaped FR through a sequence of multiple, localized brightenings involving two large pre-existing coronal loop systems. Considering the fact that the bright emission in 94 Å images corresponds to hot plasma ( T ∼ 6 MK), we attribute the multiple

O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Identifier O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength B-Wavelength B-Wavelength I-Wavelength O B-Wavelength B-Wavelength O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Telescope B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O


panel ). North is on the left-hand side of each plot. The core has been bypassed by both slices. Table 1. Flux densities of the components of 3C 328. Based on the outcome of the two-frequency VLA observations, the inverted-spectrum core must be self-absorbed, and hence very compact. Its VLBA images resulting from the observations mentioned in Sect. 2 are shown in Fig. 2 . At 5 GHz, the core of 3C 328 is point-like, while at 1.5 GHz it has an additional feature elongated southwards. As the same phase reference was used throughout those observations (J1606+3124), the absolute positions of the source at both frequencies were preserved. The core visible at 5 GHz is co-incident with the brightest part of the source in the 1.5 GHz image, and thus this image depicts a core-jet structure where the jet points to the southern lobe. This means that, apart from the lobe asymmetry, there is another similarity that 3C 328 shares with J1211+743, 3C 249.1, and 3C 334: All these AGNs have jets that point towards the lobe that is devoid of a hotspot. This raises the question of whether the light-travel-time-based model elaborated on in Paper II for J1211+743 and then used to interpret 3C 249.1 and 3C 334 in Paper III could also be applied to 3C 328. To find that out, one has to perform a quantitative test using that model since it imposes tight constraints on both the lower and upper limits of the length of the period of quiescence, t q . Fig. 2. VLBA images of 3C 328 at 1.5 GHz ( left-hand panel ) and at 5 GHz ( right-hand panel ). In both images, contours are at 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, and 12.8 mJy beam −1 . The projected angular far-side and near-side arm lengths of 3C 328 ( and ) respectively translate to 212 kpc and 232 kpc of the projected linear size, whereas the projected angular size of the jet () translates to 209 pc of the projected linear size. For these linear sizes, the lower () and upper () limits of t q were calculated using Eqs. (2) and (4) of Paper II for β adv = 0.35, three values of β jet (0.8, 0.9, and 1), and five values of the angle between the line of sight and the AGN axis θ . Since 3C 328 is a galaxy and not a QSO, only the values θ ≥ 45° are applicable to it ( Barthel 1989 ). The results are shown in Table 2 . Due to the small size of the jet in 3C 328, the calculated limits are almost independent from the adopted value of β adv . The definitions of β jet and β adv are provided in Paper II. Table 2. Lower and upper limits to the length of the quiescent period. 4. Conclusion Taking into account that for at least two restarted AGNs – 3C 293 ( Machalski et al. 2016 ) and J1835+6204 ( Konar et al. 2012 ) – the values of t q also stay within the limits shown in Table 2 , it is plausible that the above limits for t q may be valid for 3C 328, and as such the light-travel-time-based scenario can be used to explain its nature. It follows that the activity of 3C 328 has restarted and, given the minute size of its jet, that

O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O B-Telescope O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O B-CelestialObject I-CelestialObject O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O B-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-CelestialObject O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O B-Citation I-Citation O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O B-CelestialObject I-CelestialObject O B-Wavelength I-Wavelength O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O B-TextGarbage O O B-TextGarbage O O O O O O O O O O O O B-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O


shows the maximum value of the swirling strength, λ max , as a function of z for the three simulations. From it, we notice that in the convection zone the stronger the initial magnetic field, the weaker the swirls. However, the overall behavior is similar and the local peak is reached in all three cases around z ≈ −500 km. Thus, relatively strong vortices can develop in the convection zone, independently of the initial magnetic configuration of the simulation. In the photosphere, a local minimum in swirling strength is reached. It is a result of the slowdown of vortical motions in the rapidly expanding convective overshoots as a consequence of angular momentum conservation ( Nordlund et al. 1997 ). For the two MHD simulations, the minimum is reached in the lower half of the photosphere, while the Hydro simulation reaches its minimum at z ≈ 500 km. This difference can be attributed to strong arches of swirling strength in the hydrodynamical simulation, which permeate the photosphere and extend precisely up to z ≈ 500 km, as it can be seen in Fig. A.1 . These arches are less developed in the MHD simulations, probably as a consequence of the magnetic canopy. Fig. A.1. Volume rendering of the swirling strength seen from the side for three different simulations at arbitrary time instants: ( top ) MHD simulation with an initial vertical, homogeneous magnetic field of 200 G; ( middle ) MHD simulation with an initial field of 50 G; ( bottom ) purely hydrodynamical simulation. The red surface indicates the τ 500 = 1 surface. Swirls with a period larger than 10 min ( λ 2.09 × 10 −2 Hz) have been neglected and values above 1 Hz are saturated. Finally, looking at the photosphere and chromosphere, we observe that λ max increases exponentially for the MHD 50 G simulation, while this growth is only linear for the MHD 200 G simulation. In the Hydro case instead, there is a growth only in the low chromosphere up to approximately z ≈ 1200 km. Above this height, the swirls become weaker again. Clearly, magnetic fields favor the generation of swirls. However, it seems that too strong magnetic fields can also suppress the formation of strong vortices. We surmise that there exists an optimal value of the initial magnetic field for which the strength of the vortices is maximal. Fig. A.2. Distribution of the inclination cos( θ ) of the swirling strength vector as a function of altitude z for three different simulations: ( a ) MHD simulation with an initial vertical, homogeneous magnetic field of 200 G, ( b ) MHD simulation with an initial field of 50 G, and ( c ) purely hydrodynamical simulation. The inclination θ refers to the angle between the swirl axis and the vertical direction. Horizontally oriented swirls are binned close to cos( θ ) = 0, while vertically oriented swirls are close to cos( θ ) = 1. The maximum value of swirling strength λ max at each height z , for each simulation, and with a minimum of 10 occurrences per bin, is shown in panel d . The

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Model I-Model O O O O O O O O O O O O O B-Model I-Model O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O B-Model I-Model I-Model O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O B-Model I-Model I-Model I-Model O O B-Model I-Model O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


/ L ≈ 0.02. Similar results were found later by Goddard et al. ( 2016 ) with a large array of SDO data. They performed a statistical study of decaying kink oscillations in 120 coronal loops. The best-fitting results were given. In particular, with the mean period P ≈ 350 s and damping time τ ≈ 535 s, the ratio was found τ/ P ≈ 1.53 or γ/ω ≈ 0.10. The average length of analysed loops is L ≈ 240 Mm. We use the above data and the mean density contrast in oscillating coronal loops χ ≈ 0.3, given by Aschwanden et al. ( 2003 ). The aspect ratio, given above, indicates that the fundamental kink oscillations occur at dimensionless axial wavenumber k n a = π a / L ≈ 0.06. Substituting the mean density contrast in equation ( 32 ) we find that Equation ( 50 ) implies that the cut-off wavenumber exceeds the actual observable axial wavenumber of principal kink mode if the source distance d ≤ 70 a , or in terms of loop length if d ≤ 1.4 L . This is a realistic restriction, as the observable distance between the nearby flare sight or erupted filament and the impacted coronal loops is comparable with the loop length. In this case, the externally excited principal kink mode of coronal loops exists in the leaky regime and it is subjected to damping due to the wave leakage effect. With known oscillation parameters ω, γ/ω, χ and loop length L and semiwidth a , we can consider equations ( 43 ) and ( 45 ) as a system for unknown values V Ae and d . We then obtain the following estimates: V Ae ≈ 1250 km s −1 and d ≈ 0.6 L . 5 CONCLUSIONS We have studied the dispersion properties of fast oscillations in a coronal plasma slab with anchored ends. It is assumed that these oscillations are excited by a finitely distant external source. The coronal slab is considered as an enhancement in plasma density with a discontinuous boundary, embedded within uniform environment. A magnetic field is assumed to be everywhere uniform. Using the Fourier transform, the inhomogeneous boundary value problem was solved to obtain the solutions to wave variables in all the regions of the considered model. Corresponding solutions with known coefficients inside the slab contain the so-called dispersion function D (ω) in their denominators and this function completely governs the properties of the slab eigen-oscillations, excited by the external driver. The results of the study are summarized as follows. Both the kink and sausage modes are excited. There are two sets of zeros of the dispersion function. Each of these systems of roots prescribes both axially asymmetric and axially symmetric oscillations. With the appropriate condition that the distance between the source and the slab axis d is much more than the slab semiwidth a (i.e. d ≫ a ), we have shown that the first set of solutions corresponds to kink oscillations of the slab with finite amplitudes and sausage oscillations with negligible amplitudes, proportional to ∝ δ ∝ exp [− 2 m e ( d − a )] ≪ 1, whereas another set prescribes the

B-Formula I-Formula I-Formula I-Formula O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O


were likely formed in-situ at the Galactic halo (in black). Interestingly, it seems here that the stars from Gaia Enceladus are the ones closer to the extension of the linear relation, while those that deviate the most are in-situ stars. Fig. 5. Beryllium abundances as a function of metallicity. The points are the same as in the left panel of Fig. 2 , except that we colour the four extremely metal-poor stars found to likely be in-situ halo stars in Fig. 4 in black. In particular, we should remark here the position of stars BPS BS 16968 and CD-33 1173, for which we attempted to derive Be abundances for the first time. These stars are the two open blue triangles plotted over each other at about [Fe/H] ∼ −3.0 and log(Be/H) ∼ −13.8 in Fig. 5 . They are the most important points that demonstrate how stars from Gaia -Enceladus remain closer to the linear relation in a wide range of metallicities. We believe any extended discussion about Be abundances in stellar sub-components of this regime is for now precluded by our small sample. Nevertheless, two main results can be obtained from our analysis: (i) the deviation from the linear relation at [Fe/H] ∼−3.0 is real, although not universal; and (ii) the scatter we observed at these metallicities seems related to the inhomogeneous nature of the early halo. 4.3. Inhomogeneous star formation in the early Galaxy Having established our observational results, we now searched for a possible explanation to the increased scatter of points in the log(Be/H) versus [Fe/H] at the extremely metal-poor regime. To do that, we summarised results from an extensive body of investigations that have looked at the details of the first phases of Galactic chemical enrichment. The interpretation we would like to advance is that the main drivers of the observed scatter are the abundances of Fe and not those of Be. Or, in other words, that at the time of the formation of the early halo, Fe abundances (in a Galactic context) were more inhomogeneous than the Be abundances. 4.3.1. Early enrichment in beryllium The idea that abundances of Be (and of other cosmic-ray spallation products) can have smaller scatter when compared to abundances of elements produced by stellar nucleosynthesis was first suggested by the results of Suzuki et al. (1999) and Suzuki Yoshii (2001) . These authors computed chemical evolution models of an inhomogeneous early Galactic halo. Based on these models, Be abundances were suggested to be a good cosmochronometer for the early stages of the Galaxy ( Beers et al. 2000 ; Suzuki Yoshii 2001 ). This suggestion was tested by Pasquini et al. (2004 , 2007) who derived Be abundances in turn-off stars of the globular clusters NGC 6397 and NGC 6752. These measurements were very challenging, but ages obtained comparing the Be abundances with predictions of chemical evolution models (from Valle et al. 2002 ) were shown to be in excellent agreement with ages derived using isochrones. Suzuki Yoshii (2001) seem to have

O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O


and quasi-bolometric light curves of the photospheric phase and 56 Ni tail. The sample is heterogeneous, ranging over a factor of 30 in plateau luminosity and 56 Ni mass, and over a range of 2.5 mag/100 days in decline rate. Namely, the sample includes luminous, intermediate, and low-luminosity SNe II as well as SNe that were classified as II-P and as II-L. We analyze this sample, calculating the new observables that we derive here, and find the following results. 1. 56 Ni contribution is significant in most of the sample. In many SNe, it is the source of about 30% of the time-weighted integrated luminosity during the photospheric phase, but there is a single SN (2009ib) in the sample where 56 Ni contributes 72%, and several SNe where it powers about 10%. 2. 56 Ni contribution can possibly flatten and/or extend the photospheric light curve. We find that if flattening were the only effect, then SNe that are observed to have a flat plateau would have shown a decline rate of about 1 mag/100 days in the absence of 56 Ni. If, on the other hand, the only effect of 56 Ni were to extend the plateau, then without 56 Ni, the photospheric phase would have been typically shorter by a factor of about 1.15–1.35 and in the most extreme cases by almost a factor of two. 3. 56 Ni contribution cannot explain the entire range of observed decline rates, which is mostly an intrinsic property of the cooling envelope. Several independent lines of evidence suggest that the dominant effect of 56 Ni is flattening of the light curve. This includes a significant anti-correlation between the level of 56 Ni contribution and the decline rate (i.e., 56 Ni is more important in SNe with a flatter plateau). This is expected if 56 Ni is mixed throughout a significant fraction of the envelope (as suggested by Bersten et al. 2011 for SN 1999em). With this interpretation, the picture that arises from our results is that flat plateaus are not a generic feature of cooling envelope emission. Instead, many observed plateaus include a significant contribution of 56 Ni and without it, many of the SNe II-P would have shown a decline rate of up to 1 mag/100 days. Note that while not being expected a priori, there is no need for a “conspiracy,” so the combination of cooling envelope and 56 Ni emission would result in plateaus. First, SNe II exhibit a range of decline rates, thus in some cases the combined emission produces a plateau and in others it does not. Second, it seems that even without 56 Ni, the cooling envelope emission often shows a moderate decline rate where the luminosity drops by less than a factor of two over 100 days. In these cases the contribution of 56 Ni is enough to make the light curve even flatter. The coincidence in SNe II seems to be the fact that the contribution of the cooling envelope and 56 Ni-powered emission is comparable over a wide range of light curve luminosities. This is a result of the well known tight correlation between the luminosity and 56 Ni

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


because the gas is ionized. We searched for a warm ionized gas associated with GS040 using the Wisconsin H α Mapper Northern Sky Survey (WHAM-NSS) data (Haffner et al. 2003 ). The survey has an angular resolution of 1°, and, around the GS040 area, it provides H α spectra covering from −85 to +100 km s −1 at spectral resolutions of 12 km s −1 . We have examined the H α intensity map integrated over the velocity range of GS040 (−85 to −66 km s −1 ), but could not detect an associated emission ( 0.05 R where ). Note that this area is bright in the total H α intensity map with a mean intensity of about 3 R , so that we do not expect to see the faint emission associated with either GS040 or CHVC040 in the all-sky H α maps (Dennison et al. 1998 ; Finkbeiner 2003 ). We also searched for an associated hot ionized gas using the 0.1–2.4 keV image of the ROSAT All-Sky X-ray Survey (1 pixel scale = 44′′; Voges et al. 1999 ) but could not detect any emission. CHVC040 belongs to the “Galactic Center Negative” (GCN) HVC complex, which is a collection of small discrete HVCs sparsely distributed over a area within and to 10° (Wakker van Woerden 1991 ; Winkel et al. 2011 ). A kinematic model has been proposed where GCN is a smooth gas flow starting at at a heliocentric distance of 35 kpc and crossing the Galactic plane obliquely at 15 kpc (Jin 2010 ). For comparison, GS040 is probably located near the Scutum-Centaurus (Sct-Cen) arm at a distance of ∼20 kpc (Dame Thaddeus 2011 ) because the disrupted interstellar medium (ISM) is seen only at velocities below that of the Sct-Cen arm ( km s −1 ), not at higher velocities (see Figures 2 and 3 ). We examined lists of known stellar objects, H ii regions, OB stars, and SN remnants, but no known sources are likely to be associated with GS040. There is one H ii region (G039.864+00.645) in the WISE catalog of Galactic H ii regions (Anderson et al. 2014 ) that is located at west of the hub. But this H ii region has a systematic velocity of −40.9 km s −1 (Anderson et al. 2011 ) and is enclosed by a small (68′′) dust shell, so it cannot be responsible for the H i shell. Instead, the distance of ∼20 kpc is not unreasonable for CHVC040, because GCN does not have a smooth extended envelope like other HVC complexes, and it appears to be composed of several subpopulations that do not share a common origin (Winkel et al. 2011 ). Note that the geometrical center of GS040 is well below the Galactic plane ( pc where ), i.e., at while the midplane there is at because the Galactic plane is warped in the outer Galaxy (Levine et al. 2006 ). It is difficult to imagine the SN origin for a supershell at such height, and CHVC040 is most likely the energy/momentum source for GS040. The total energy deposited ( E E ) in the Galactic disk by CHVC040 can be inferred from the parameters of the GS040 supershell. The radius of GS040 is pc, while its mass at km s −1 is including the cosmic abundance of helium. If we account for the mass

O O O O O O O O O O O O O O B-CelestialObject O O B-Survey I-Survey I-Survey I-Survey I-Survey I-Survey I-Survey B-Survey O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-CelestialObject O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O B-Wavelength I-Wavelength O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O B-Survey I-Survey I-Survey I-Survey B-Formula I-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O B-Citation I-Citation O O O B-CelestialObject O O O O O B-CelestialObjectRegion B-CelestialObjectRegion B-CelestialObjectRegion O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-CelestialObject O O B-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O B-CelestialObjectRegion I-CelestialObjectRegion B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-CelestialObject O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O


the posteriors estimates of the TNM model is shown by the red line. Right-hand panel: V φ distribution for mono-age stellar particles. Colour code as in Table 2 . Table 4. TNM model parameters: w i is the mixture weight of the i -th component (i.e. young disc, old disc/thick disc, halo), while 〈 V φ 〉 and are the corresponding mean and the standard deviation of the normal distribution. Note that 0 ≤ w i ≤ 1 and ∑ i w i = 1. We also tested alternative models with a different number of components. The results for a double Gaussian distribution such as halo+disc or disc+disc are not statistically significant and do not provide good curve fitting of the data. A four-component model, aimed to represent additional disc populations, produced non-significant improvements to the model. We note that the weights of the two disc components listed in Table 4 correspond to the total mass of the young disc ( Gyr) and of the old disc/thick disc (), respectively (cf. Table 2 considering the different vertical cut applied). The comparison with the mono-age distributions shown in Fig. 7 (right-hand panel) confirms that these two disc components represent the superposition of the stellar generations formed in the age intervals 0–4 and 4–10 Gyr. Note that the old thin disc and the thick disc populations are difficult to distinguish because of their very similar velocity distributions. Moreover, the difference between the mean rotation velocity of the AqC4 discs is This value is smaller than in the MW, where (Han et al. 2020 ). Actually, this difference depends on the fact that in AqC4 the two main disc components represent the young and the old/thick discs instead of the whole thin disc and the thick disc, as in the MW. Moreover, the high rotation velocity of the AqC4 disc is consistent with its more compact structure with respect to the MW disc (Section 3.1 ). Finally, the halo component shows a small prograde rotation as observed in the inner halo of the MW, while its velocity dispersion km s −1 is about 70–100 per cent higher than in the solar neighbourhood (Re Fiorentin et al. 2015 ; Bland-Hawthorn Gerhard 2016 , and references therein). In summary, here we robustly support what is discussed in Section 3 : in the SSR, we can distinguish at least two kinematic disc components which rotate faster than what is measured in the solar neighbourhood (as discussed globally in Section 4.1 ), and with a smaller discrepancy between the mean values. These results extend the spatial characterization of AqC4 stellar disc beyond the determination of the scale lengths and scale heights, allowing us to properly investigate the complete phase-space of mono-age populations, as usually done in Galactic surveys. 4.3 Kinematics of mono-age disc populations Fig. 8 shows the three median velocity components , , and the dispersions (, , ) as a function of R , computed in radial annular bins of Δ R = 0.25 kpc, for the mono-age stellar disc populations younger than 10 Gyr and with | Z |

O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Model I-Model I-Model O O B-Model O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O B-CelestialObject O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O B-CelestialObjectRegion O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O


has a small impact on the vertical and radial density distributions of the stellar discs. These two last parameters will be constrained with observations and not by the dynamical self-consistency. Thus, these two parameters and are currently fixed before they can be more tightly constrained by kinematical observations. Then we assume that and we fixed for each stellar disc the velocity dispersion ratio σ U / σ W at the solar position, respectively 2.1 for the thin discs ( Gomez et al. 1997 ; Robin et al. 2017 ), 1.47 for the young thick disc, and 1.38 for the old thick disc ( Soubiran et al. 2003 ; Robin et al. 2017 ). Figure 1 (left) shows the vertical density distribution of the old thin disc (scale height around 230 pc) at three Galactic radii 4, 8, and 12 kpc, and Fig. 2 (left) its radial distribution at four z = 0, 200, 400, and 600 pc. The adjustment is satisfying. This is expected at R = 8 kpc since the BGM2014 version is built to be dynamically consistent up to z = 1 kpc. The fact that the fit is also satisfying at other galactic radii confirms that the choice of Einasto profiles in the BGM2014 version for the disc density was valuable with respect to the dynamics. For the other discs, the agreement between the BGM2014 and the new version is correct and is better for the thinner disc components with smaller vertical velocity dispersions. For the old thick disc, the vertical and radial density distributions (corresponding to a 795 pc scale height in the case of a s e c h 2 density law) are shown in Figs. 1 and 2 (right). The agreement between both models is still correct. We note that at R = 8 kpc and below z = 1 kpc the fit is excellent, illustrating the exactness of the dynamical consistency of the BGM2014 version. However, at larger height above the Galactic plane, both models differ, and at two scale-heights, or 1.6 kpc height, the BGM2014 density for the old thick disc is 17 % too small. Concerning the density scale length, the agreement between the two models is nearly perfect for the old thin disc. For the old thick disc, the agreement is obtained only for z lower than 1 kpc. We note that the flare present in the outskirts of the Galaxy could be naturally modelled by adjusting the vertical velocity dispersions at large Galactic radii. Table 1 gives the relative difference in percentage between the new and the old models at different heights z and at R = 8 kpc, for the old thin and old thick discs. The differences increase with z ; they become significant at three scale heights for the old thin disc and beyond 1 kpc for the old thick disc. At high z , the differences become of the order of the expected density of the stellar halo, and this illustrates the necessity of using dynamically self consistent distribution functions to properly identify and separate the Galactic stellar components. However, only observations will tell us whether our choice of nearly isothermal modelling of stellar discs is the correct one since small

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-CelestialObject O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Model O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Model O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Model O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O


The PM diagrams of Fig. 1 present spike features for high velocity stars. 2 These spikes reflect low-quality astrometric measurements, specifically due to a dishomogeneous scanning pattern of Gaia , and can be identified through the astrometric quality parameter variability_periods_used (see e.g. Lindegren et al. 2018 ). The details of the effects on kinematics of these features are described in a follow up paper (Bianchini et al. in preparation). In this work, we verify that the rotation detections reported in the next sections do not depend on these features: kinematics subsamples that include a cut in the variability_periods_used give results fully consistent with the one obtained in the following sections. 3.2 Global rotation measurements To measure the rotation on the plane of the sky, we convert the stellar positions into cartesian coordinates centred on each cluster (centres from the Harris Catalogue; Harris 1996 ), with the positive x -axis pointing towards West and the positive y -axis pointing towards North, using equation ( 1 ) from van de Ven et al. ( 2006 ). The PMs in the RA and Dec. components are converted accordingly into a cartesian coordinate system using the orthographic projection reported in equation ( 2 ) of Gaia Collaboration et al. ( 2018a ), with positive μ x in the direction of West. 3 Finally, we transform the PMs in radial and tangential components (μ r , μ t ) in polar coordinates, with positive μ r indicating velocities away from the cluster centre and positive μ t indicating counterclockwise movements. In the rest of this paper we will focus on the analysis of the tangential component of the PMs and analyse the radial component in Section 4.2 . We compute the global rotation along the tangential component within 3 R h using equation ( 2 ), as described in Section 2 . The results are reported in Table 1 and plotted in Fig. 2 . Out of the 51 GCs analysed, 11 display an average value of μ t significantly different than zero at a confidence level >3σ. These clusters are indicated as red symbols in Fig. 2 . We confirm the detection of rotation in NGC 104, NGC 5139, NGC 5272, NGC 5904, NGC 6656, NGC 6752, NGC 6809, and NGC 7078 reported by Gaia Collaboration et al. ( 2018a ), and additionally we detect rotation in NGC 4372, NGC 6273, and NGC 7089. Figure 2. Mean PM along the tangential component measured in a three half-light radii field of view around the cluster centres. The absolute value of μ t ( 3 R h ) is plotted as a function of the distance of the clusters. Red stars indicate clusters for which rotation is detected at least with a 3σ confidence level. Dotted lines indicate PM values in km s −1 for a given distance (, with μ measured in mas yr −1 , v measured in km s −1 , and d in kpc). Gaia Collaboration et al. ( 2018a ) and Lindegren et al. ( 2018 ) reported that systematics errors are present in Gaia DR2, in particular a ∼1° small-scale component, related to the Gaia scan pattern. Following the procedure

O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Dataset I-Dataset O O O O O O O O O B-Telescope O O O O O


transverse MHD wave derived from 3D MHD numerical simulations and forward modeling. The main factors affecting the intensity modulation in the host loops are the KHI and resonant absorption. Importantly, the KHI vortices (TWIKH rolls) carry over characteristics of the resonant absorption and phase mixing mechanisms, allowing these to be detected with current instrumentation. We have found that the KHI produces significant intensity changes and that emission lines with different temperature sensitivity to either the loop core or the external medium provide insights into different wave modes. At a high spatial resolution and especially in the hot line, the fine strand-like structure generated by the TWIKH rolls can be observed, as described in Antolin et al. ( 2014 ). At a lower spatial resolution corresponding to AIA, we have shown that the detected damping and period can vary depending on the emission line, which can lead to an out-of-phase behavior between the core and boundary lines. These results can be understood from the fact that the boundary line is more sensitive to the KHI dynamics, to resonant absorption and phase mixing. A decay-less oscillation is obtained in the boundary line for any LOS angle and this is the result of periodic brightenings and the coherent motion of the vortices. The vortices, in turn, result from unstable azimuthal Alfvén waves. Therefore, this decay-less oscillation is more reflective of the Alfvén waves coupled resonantly to the kink mode than the global kink mode itself. Resonant absorption transfers the energy of the kink mode to the Alfvén waves, whose dampings are expected to be much lower than that of the kink mode, since they rely on phase mixing and the turbulence resulting from the instability. The Alfvén waves therefore act as an energy reservoir for the TWIKH rolls, which persist over time, leading to the decay-less oscillation. A shorter period is found for the boundary line and is due to phase mixing. Indeed, the azimuthal Alfvén waves in the boundary layer have an increasingly higher phase speed the farther they are from the loop core. Since it is these waves that are enhanced due to resonant absorption and that become K-H unstable, their signal becomes dominant in the boundary line. The fit to the density from the numerical model ends up with a shorter period than that obtained from the core line. This is because the density profile is influenced by the KHI dynamics, which broaden the flux tube, even though the small-scale structure produced by the instability is unresolved. Accordingly, when fitting only the central section of the flux tube along the oscillation axis, which is minimally influenced by resonant absorption and by the KHI, we obtain a period of 256 s, closer to that obtained from the core line. The out-of-phase behavior between the core and boundary lines is ultimately due to the same effect that leads to the out-of-phase behavior between the LOS velocity and the POS motion seen in

O B-Model O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O B-Model O O O O O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


model the angular mask. Different assumptions on the angular selection used to construct the random catalogue might impact the measurement at small scales ( s ≲ 10 h −1 Mpc), although the effect is within the current measurement uncertainties (see Appendix A.4 ). To minimise the impact of shot noise error due to the finite number of random objects, we construct our random catalogue to be 100 times larger than the XXL cluster sample. 3.4 Covariance matrix The covariance matrix, C i , j , which measures the variance and correlation between 2PCF bins, is defined as where the subscripts i and j run over the 2PCF bins, k refers to the 2PCF of the k th of N R catalogue realisations, and is the mean 2PCF of the N R samples. The normalisation factor, , which takes into account the fact that the N R realisations might not be independent, is , , and for the subsample, jackknife, and bootstrap methods, respectively ( Norberg et al. 2009 ). We assess the XXL 2PCF covariance matrix with the bootstrap method, using 1000 realisations obtained by resampling galaxy clusters from the original catalogue, with replacement. The impact of this choice is discussed in Appendix A.3 . 4 Cosmological analysis We perform a Bayesian statistical Markov chain Monte Carlo (MCMC) analysis of the 2PCF by sampling the posterior distribution of Ω M , the only free parameter in the assumed flat ΛCDM model considered. As we verified, the current clustering uncertainties do not allow us to consider more general cosmological scenarios, for example models with free dark energy equation of state parameters, by exploiting only the XXL cluster clustering. The joint cosmological analysis of XXL cluster number counts and clustering will be presented in a forthcoming paper. We consider the commonly used likelihood function, , defined as where is the inverse of the covariance matrix estimated from the data with the bootstrap method (Eq. 5 ), N is the number of comoving separation bins at which the 2PCF ( ξ ) is estimated, and the superscripts d and m stand for data and model. The likelihood is estimated at the mean pair separations in each bin (see Sect. 3.2 ). The 2PCF model in redshift space, ξ m ( s ), is computedas where ξ DM ( r , z ) is the real-space dark matter 2PCF, which we estimated by Fourier transforming the power spectrum, P DM ( k , z ), computed with the software CAMB ( Lewis Bridle 2002 ). Since the present cluster clustering analysis focuses at sufficiently large scales, the dark matter power spectrum can be safely computed in linear theory, , with marginal effects on our results. The model depends on two free quantities, fσ 8 and b eff σ 8 (since ), and on the reference background cosmology used both to convert angles and redshifts into distances and to estimate the real-space dark matter 2PCF (see e.g. Marulli et al. 2017 ). The geometric distortions caused by an incorrect assumption of the background cosmology are modelled by the α parameter, i.e. the ratio between the

O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O B-Model O O O O O B-Model O O O O O O O O O O O O O B-Model O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Survey B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model B-Model O O O B-Model O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O B-Software O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O


relative astrometry between the initial image and each of the later sets using Astroalign . To uncover or place limits on transient emission on these timescales, we perform digital image subtraction using the HOTPANTS software package (Becker 2015 ). If an optical or NIR source is detected within the GRB localization regions (Evans et al. 2009 ; Lien et al. 2016 ), we perform aperture photometry on the residual images to determine the source brightness using the IRAF/ phot package. We determine zero-points by using point sources in common with the SDSS Data Release 12 (Alam et al. 2015 ), Pan-STARRS DR1 and DR2 (Chambers et al. 2016 ), and 2MASS (Skrutskie et al. 2006 ) or USNO-B1 (Monet et al. 2003 ) catalogs. If there are no significant residuals at or near the localization region, we obtain an upper limit from sources detected at 3 σ significance in the field. As we obtain the limit from the initial image, our values do not include additional noise from the subtraction. In total, we present 52 unpublished upper limits and three unpublished detections of 23 bursts, 10 with known redshifts (Table A1 ). 2.3. Observations from GCNs We search the GCNs of all short GRBs detected between 2013 and 2020 for constraining upper limits to ensure that our catalog covers all Swift short GRBs to date (2005–2013 is covered by the literature sample in Section 2.4 ). Considering that the range of peak kilonova luminosities based on models is ν L ν ≈ (1−5) × 10 42 erg s −1 (∼10 times the peak luminosity of AT 2017gfo; see Figure 1 ), we only include data that could feasibly fall into the range of kilonova luminosities, or fainter than a given depth, corresponding to griz ≳ 22.5 mag and JHK ≳ 21.0 mag (AB system) at z ≈ 0.25. We choose z ≈ 0.25, as 80% of known redshifts in our sample fall above this cut. Observations significantly brighter than this are not included, as they are likely dominated by afterglow emission. In total, we collect deep optical and NIR limits of 18 bursts, seven with confirmed redshifts, from the GCNs (Table A1 ). Figure 1. Light curves of AT 2017gfo and four diverse kilonova models considered in this work in the griJ bands. Light curves are shown in ν L ν (left y -axis) and absolute magnitude (right y -axis) units, and rest-frame time in days. In each panel, we show the interpolated, smoothed AT 2017gfo data set (solid dark-blue line; Villar et al. 2017 ), an NSBH kilonova model (blue dotted line; Kawaguchi et al. 2020b ), an infinite-lifetime NS remnant kilonova model (red dashed line; Kasen et al. 2015 ), an extreme optimistic magnetar-boosted kilonova model (turquoise solid line; Fong et al. 2021 ), and a model for an optimistic scenario motivated by GW190425 (magenta dashed–dotted lines; Barbieri et al. 2020 ). Black and open triangles denote upper limits of short GRBs with known and unknown redshifts, respectively, in the respective rest-frame bands at δt rest = 0.1–10 days. The most constraining limits of bursts with known

O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software B-Citation I-Citation O O O B-Wavelength O B-Wavelength O O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Software B-Software I-Software O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O B-Dataset I-Dataset O B-Dataset B-Citation I-Citation I-Citation I-Citation O O B-Survey B-Citation I-Citation I-Citation I-Citation O O B-Survey B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Wavelength O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O


for Different Dark Matter Density Profiles in NSCs Figure 9 shows the final masses of IMBHs as a function of the initial ones. For those IMBHs that could deplete NSs during one Hubble time, the final masses include the contribution from all the stellar BHs and NSs in their host NSCs. Figure 9. Same as Figure 6 but for the IMBHs in NSCs. The circles, at about 1500 M ⊙ when α = 1.5 and 7800 when α = 2.0 and 7/3, mark the masses with which an IMBH could deplete all the stellar BH in the same cluster. The solid dots are the masses corresponding to NS depletion, which is about 1500 M ⊙ when α = 2.0 and 3200 M ⊙ when α = 7/3. 4.2. Merger Rate in Nuclear Star Clusters Per Mpc3 The calculation of the merger rate per unit Mpc 3 for NSCs is almost the same as the previous calculation for GCs, but with two crucial modifications. (1) For the mass function of IMBHs, we consider two models. The first is suggested by Babak et al. ( 2017 ), which isAnother model, suggested by Gair et al. ( 2017 ), is more conservative,In this latter one, the exponent is positive, which significantly reduces the number of IMBHs. (2) When α ≥ 2.0, to calculate the M min in the integration of the mass function is determined by the solid dots in Figure 9 , because smaller IMBHs in NSCs would have depleted all the NSs. The corresponding merger rates for BHs and NSs per Mpc 3 are presented in Table 6 . Again, we find an enhancement of BH and NS merger rates in the presence of DM minispikes. Table 6 The Merger Rate per Mpc 3 of IMRIs or EMRIs with BHs R BH and Neutron Stars R NS Formed by IMBH for Different Dark Matter Density Profiles From the calculations we can also find that the results are sensitive to the BH and NS supply. With DM minispikes, the duty cycle of BHs and NSs may be shorter than the Hubble time, as the dramatically increased merger rate can lead to a high efficiency of consumption of compact objects. As a result, the appearance nowadays strongly depends on the total BH and NS supply. 5. Could Stellar Dynamics Deplete the Minispikes? We have shown in Figures 1 – 3 that dynamical hardening, by ejecting intruding stars, is important for the initial evolution of IMRI/EMRIs, when pc. It is worth discussing here whether the same process could eject DM particles and hence deplete the DM minispikes. The difference between hardening against DM and hardening against stars is that the interloper stars that we considered are gravitationally unbound to the IMBHs but the DM particles are deep in the potential well of the IMBH. The DM particles are more difficult to deplete. To show this more clearly, we calculate the hardening timescale associated with DM using the ejection timescale derived in Sesana et al. ( 2008 ) for the stars gravitationally bound to binary MBHs. They showed that it is , where P 0 is the orbital period of the binary. For example, the pink solid line in Figure 10 illustrates the dependence of this timescale on the IMBH mass. Figure 10. Same as

O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula O O O O O O O O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O B-EntityOfFutureInterest O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


one. The choice of S depends on the sampling strategy, and we recommend a fine scanning of the complete velocity spectrum which corresponds to S = 1 and would be covered by 1 + N − R bins. Then, we calculate the median for the spatial coordinates as well as for the radial velocity for each data bin. Median, as a central value estimator, is more robust than mean against the presence of outliers, and we expect to find some points away from the central value in most cases. A robust estimator of the standard deviation is given by: (see Gott et al. 2001 , for a detailed discussion of median statistics), where MAD is the median of the absolute deviations from the data median. The error of this central value can then be approached as . The b parameter is dependent on the distribution function underlying the data. In the following we will take b = 1.4826 as default. The next step is to obtain the MST graph for the R objects in each bin. The median of the edge length distribution is the first indicator of the group compactness (equation 2 ). As discussed by Maschberger Clarke ( 2011 ), median is the best estimator of the central value of the edge length when we could be facing a clumpy distribution for a given velocity range. The same is applied to a set of R randomly extracted objects from the complete sample. We repeat this step 500 times and calculate the mean and the standard deviation of the 500 medians. We chose 500 repetitions to get a reliable average of the edge length median for the R sized random sample, as recommended by Allison et al. ( 2009 ). This way we estimate the edge length median of the MST for a reference set of R objects, representing the whole sample. Then we estimate the kinematic segregation index, , as where j is the bin identification and i is the order of the first element in the bin, both being connected through the relation i = 1 + ( j − 1) × S . Supra-index 500 indicates that the average is calculated over 500 stochastic realizations. The total number of bins is given by the integer part of 1 + (( N − R )/ S ). A plot of versus the median of the radial velocity in the RV j interval for all j , shows what we name the spectrum of kinematic groupings which provides an interesting and useful tool for the exploratory analysis of the stellar system phase space. Those that verify the inequality mark the radial velocity channels, RV j , where a kinematic segregation has been detected. Here, we decided to multiply by two the standard deviation term as a conservative criterion in the detection of groupings (confidence level greater than 95 per cent). Note that is not the median of the 500 standard deviations but the standard deviation of the 500 medians. Figure 1. Schematic representation of the six test cases. From top to bottom and from left to right they are: (1) two spatially separated clusters with the same central velocity of 1 km s −1 ; (2) two clusters separated both in space and in velocity, where the mean velocities

O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


z sample imply the presence of a puffed-up torus in the low- N H NLRGs, suggesting that L / L Edd extends to lower values than those in the high- z sample. 7.2. <italic>N</italic>H Distribution The distributions of the intrinsic N H in the medium- and high- z 3CRR samples are presented in Figure 17 . The high- z sample (on the right) shows a bimodal distribution, where quasars have N H 10 22.5 cm −2 , consistent with low obscuration at face-on inclination angles, while the NLRGs show N H > 10 22.5 cm −2 , implying higher obscuration at higher inclination angles, consistent with unification schemes. There are two quasars with moderate column densities (10 22.5 N H /cm −2 10 23 ) and hard HRs (0 HR 0.5) in this sample. In the medium- z sample, the distributions of quasars and NLRGs overlap. Although the quasars show N H 10 22.5 cm −2 , similar to quasars at high redshifts, the NLRGs have a much broader range of column densities that extend to lower, quasar-like values in the low- N H NLRGs. These NLRGs possibly have low L / L Edd , which allows clouds with low column density to form in the vicinity of the central engine (Section 6.3 ). Such low L / L Edd NLRGs are missing from the high- z sample. Figure 17. Histograms of the X-ray equivalent intrinsic hydrogen column density N H for the medium- z 3CRR sample (left) and the high- z 3CRR sample (Wilkes et al. 2013 ; right). Quasars are shown in blue and NLRGs in red. Upper limits, mostly for quasars with no evidence for intrinsic absorption, are indicated by arrows. CT NLRGs in the medium- z sample with no measurement of N H due to low S/N are indicated by an underlined 3C identification. Note the low- N H (≤10 22 cm −2 ) NLRGs in the medium- z sample that are not present in the high- z sample. Although a simple unification model was sufficient to explain the X-ray data and the bimodal N H distribution in the high- z sample, this is not the case in the medium- z sample. An additional parameter, a range of L / L Edd , is required to explain the large range of N H in NLRGs seen at intermediate inclination angles, skimming the edge of the torus or accretion disk atmosphere/wind. As a result, the broad range of L / L Edd smears the N H distribution for NLRGs, removing the bimodality that was found in the high- z sample. Turning this argument around, because the unification model was sufficient for the high- z 3CRR sample, producing a bimodal and narrow N H distribution, the L / L Edd ratio must have a narrower range and higher values compared to the medium- z sample, allowing orientation effects to dominate the properties of the high- z sample. To test this hypothesis, we compiled spectra of the high- z 3CRR quasars (from the SDSS archive; Barthel et al. 1990 ; M. Vestergaard D. Stern 2021, private communication) and measured the black hole masses from the widths of the C iv and Mg ii emission lines. The masses (measured in 12 out of 20 high- z quasars) are in the range of M BH = 10 7.7 to

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O B-Tag O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-Dataset O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O B-Wavelength O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O O B-Archive I-Archive B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O


to M 0 , the total mass at t 0. The drop of the total mass over time is due to accretion of gas by sink particles. The vertical dashed line marks the onset of star formation. Interestingly, the evolution of the atomic hydrogen mass fraction is almost identical in runs FSN and FWSN, with a value around 40percent. Simulation FRWSN, however, has a mass fraction of atomic hydrogen around 70percent. This is due to the effect of photodissociation, which converts molecular into atomic gas. The mass fraction of ionized hydrogen starts to grow monotonically 10Myr after the onset of star formation from 1percent to 7percent in run FSN. This is the result of the supernova explosions, which produce an increasing amount of ionized gas. Run FWSN has a roughly constant ionized hydrogen mass fraction of 2percent. Due to the reduced SFR in this simulation, many fewer supernovae explode compared to run FSN, and the additional collisional ionization by stellar wind feedback cannot compensate for this effect. The ionized hydrogen mass fraction of run FRWSN oscillates between 3percent and 10percent as a function of time. These oscillations are very closely correlated with the variability in the radiative output from the star clusters (compare Fig. 11 ). This clearly demonstrates that photoionization from stellar radiation is the primary source of ionized gas in this simulation, with supernovae and winds contributing only a negligible amount. The molecular hydrogen mass fraction of run FSN drops from 60percent to 40percent from 40Myr onwards. The delay of 10Myr between the onset of star formation and the reduction of the molecular hydrogen mass fraction suggests that it is due to supernova feedback, since the evolution of the atomic and ionized hydrogen mass fractions show a similar behaviour. In contrast, if this reduction was primarily due to accretion of molecular gas on to the sink particles, one would not expect such a delay. For run FWSN, the molecular hydrogen mass fraction stays at around 60percent. In run FRWSN, the molecular hydrogen mass fraction oscillates around a value of 20percent. These oscillations are again indicative of photoevaporation processes caused by the stellar irradiation of the molecular clouds. 8 DEPLETION TIME Since we dynamically form H 2 in the simulations, it is interesting to check whether the relation between the molecular gas surface density H 2 and the SFR surface density SFR obtained in the simulations is in agreement with observations. The ratio of these two quantities is the depletion time . Using SFR from Fig. 7 and the instantaneous H 2 measured in face-on projections of the disc, we can plot t dep as a function of time for the three simulations. The result is shown in Fig. 14 . Figure 14. Top: depletion time for the three simulations as a function of time t . Middle: plot of SFR versus H 2 for the three simulations for all snaphots after t 40Myr. The dashed lines show constant depletion times of 0.1, 1 and 10Gyr (from

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula O O O O O O O O O O O O O


from observed frequencies and ratios r 01 and r 10 by using Equations ( 3 ) and ( 6 ). Appourchaux et al. ( 2012 ) extracted the frequencies of low- l p -modes of KIC 9812850. Using the ratio r 10 computed from the frequencies and the observed frequencies ν n ,1 , the values of A , ν 0 , and B 0 in Equation ( 3 ) are estimated to be 264 ± 62 π , 2309 ± 141 μ Hz, and 0.043 ± 0.005, respectively, for KIC 9812850. Thus, the radius r c of KIC 9812850 is predicted to be about 0.14 by Equation ( 6 ). 3. THE RADIUS OF THE CONVECTIVE CORE OF THE BEST MODEL OF KIC 9812850 3.1. Evolutionary Models In order to compare the value of r c estimated by Equations ( 3 ) and ( 6 ) for KIC 9812850 with that of the evolutionary model of KIC 9812850, we searched for the best model of KIC 9812850 that matches both nonseismic constraints and seismic characteristics in a grid of evolutionary models. We used the Yale Rotation Evolution Code (YREC; Pinsonneault et al. 1989 ; Yang Bi 2007a ; Yang et al. 2015 ) to construct the models. For the microphysics, the OPAL equation-of-state table EOS2005 (Rogers Nayfonov 2002 ) and OPAL opacity table GN93 (Iglesias Rogers 1996 ) were adopted, supplemented by the Alexander Ferguson ( 1994 ) opacity tables at low temperature. The models with a mass less than 1.30 take into account the diffusion and settling of both helium and heavy elements by using the diffusion coefficients of Thoul et al. ( 1994 ). The standard mixing-length theory is adopted to treat convection. The mixing-length parameter α is a free parameter in this work. For the Sun, the value of the α for the YREC is 1.74. The distance of the overshooting of the convective core is defined as H p , where is a free parameter and H p is the local pressure scale height. The full mixing of material is assumed in the overshooting region. The initial helium mass fraction is fixed at the standard big bang nucleosynthesis value 0.248 (Spergel et al. 2007 ) and 0.295. All models are evolved from zero-age MS to the end of MS. The values of the input parameters, mass, α , , and Z i , for the calculations are summarized in Table 1 . Table 1 Range of the Input Parameters for the Evolutionary Tracks The adiabatic oscillation frequencies ν n , l of models were computed by using the pulsation code jig7 of Guenther ( 1994 ). The effects of the near-surface effects of a model on the frequencies were calculated by using the method of Kjeldsen et al. ( 2008 ). 3.2. Observational Constraints on Models KIC 9812850 is an F8 star (Wright et al. 2003 ). The value of [Fe/H] given by Ammons et al. ( 2006 ) is , but that given by Bruntt et al. ( 2012 ) is −0.16 ± 0.06 for KIC 9812850. Combining the value of 0.023 (Grevesse Sauval 1998 ) of ( Z / X ) ⊙ of the Sun, the value of ( Z / X ) s of KIC 9812850 is estimated to be between 0.016 and 0.033 for the [Fe/H] of Ammons et al. ( 2006 ), or in the range of 0.014–0.018 for the [Fe/H] of Bruntt et al. ( 2012 ). The effective temperature of KIC

O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O B-Software I-Software I-Software I-Software B-Software B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Model I-Model I-Model I-Model B-Citation I-Citation I-Citation O O B-Model I-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-CelestialObject I-CelestialObject O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O O O O B-CelestialObject


al. ( 2013 ), blue circles of Fig. 2 and Magnelli et al. ( 2011 ), red circles of Fig. 2 . The parameters of the analytic expression (dark green dot–dashed line) we obtain from dust correcting the Schechter ( 1976 ) fit given by Cucciati et al. ( 2012 ) are φ ⋆ = 0.0033 Mpc −3 , SFR ⋆ = 16.7 M ⊙ yr −1 and α = −1.07. 3.7 The star formation rate function at <italic>z</italic> ∼ 1.15 To obtain an estimate of the SFRF at redshift z ∼ 1.15, we use the IR LFs from Gruppioni et al. ( 2013 , z ∼ 1.0–1.2) and Magnelli et al. ( 2013 , z ∼ 1.0–1.3) and the FUV LF of Cucciati et al. ( 2012 , z ∼ 1.0–1.2). Even at these intermediate redshifts, the IR surveys are unable to probe low star-forming objects. The red dashed line is the analytical SFRF obtained from the double power-law fit of the LF of Magnelli et al. ( 2013 ). The results from the FUV LF of Cucciati et al. ( 2012 ) imply a much shallower SFRF with a much lower characteristic SFR. Once again we see that the SFRFs from the extrapolations of IR studies typically have higher characteristic SFRs and are steeper for faint objects than those found by UV data. Our results for the SFRF at redshift z ∼ 1.15 are in the bottom left panel of Fig. 2 and Table 8 . Table 8. Stepwise SFR functions at z ∼ 1.15 using the data from Gruppioni et al. ( 2013 , red diamonds of Fig. 2 ) and Magnelli et al. ( 2013 , red circles in the bottom left panel of Fig. 2 ). The parameters of the analytic expression (dark green dot–dashed line) we obtain from dust correcting the Schechter ( 1976 ) fit given by Cucciati et al. ( 2012 ) are φ ⋆ = 0.006 Mpc −3 , SFR ⋆ = 8.3 M ⊙ yr −1 and α = −0.93. 3.8 The star formation rate function at <italic>z</italic> ∼ 0.9 To retrieve the SFRF at redshift z ∼ 0.9, we use the LFs of Ly et al. ( 2011 , z ∼ 0.84), Gruppioni et al. ( 2013 , ∼0.8–1.0), Cucciati et al. ( 2012 , ∼0.8–1.0), Magnelli et al. ( 2013 , z ∼ 0.7–1.0) and Sobral et al. ( 2013 , z ∼ 0.84). As discussed in Section 2.2 , the LF of Ly et al. ( 2011 ) is an Hα LF that is corrected for incompleteness and dust attenuation effects with the Hopkins et al. ( 2001 ) dust correction law. This dust correction law implies a luminosity/SFR dependent correction with the observed (non-intrinsic) luminosity/SFR. To obtain the SFRF from this Hα LF we use equation ( 2 ). The magenta filled diamonds of Fig. 2 are the stepwise determination of the SFRF for z ∼ 0.84 from the Hα LF of Ly et al. ( 2011 ). From Fig. 2 , we see that the bright end of the SFRF from Ly et al. ( 2011 ) is in excellent agreement with the results from the IR samples and this suggests that the SFR dependent dust corrections suggested by Hopkins et al. ( 2001 ) are robust. For the low star-forming objects, we see that the SFRFs derived from Ly et al. ( 2011 ) and Sobral et al. ( 2013 ) are in very good agreement. However, the SFRFs that rely on the results of Sobral et al. ( 2013 ) are highly uncertain for luminous star-forming objects and is not consistent with the SFRs of

B-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O B-Tag O O O O O O O O O O O B-Formula I-Formula I-Formula O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O B-Tag O O O O O O O O B-Formula I-Formula I-Formula O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O


closer, due to an attractive Lorentz force, in the same way as in CsTsB above. The loops then merge and seem to cross each other, because the positive toroidal field of the loop located at lower latitudes ends up appearing at higher latitudes at t = 4 days and later. From the temporal evolution of the energies that were shown in Figure 6 , it is clear that this case is the one producing the largest release of magnetic energy—because of the multiple reconnections—and thus the largest increase in kinetic energy. However, the loops do not seem to fully reconnect; the two main initial structures are still visible after eight days of evolution. However, it is still clear that the loops underwent a strong loss of magnetic intensity. Indeed, after four days, only 30% of the initial magnetic intensity remains in the loops, and only 20% remains after 5.4 days. As we shall see in the next section, even if the magnetic structures still seem to be well-confined in latitude and radius, the morphology of the emerging radial field is very different from the other cases. Figure 14. Same as Figure 7 , but for CoToBw. In this case, a strong interaction occurs between the structures: the loop of positive B φ passes through the loop of negative B φ after three to four days of evolution. 4. Consequences for Emerging Regions One of the main motivations to study interacting Ω-loops was to investigate the consequences of such interactions on emerging magnetic regions at the solar surface. We note here that, in our calculations, the top of the domain is located at r = 0.96 R ⊙ , i.e., still quite far from the solar surface—but we can get some insight on the possible implication for solar active regions by looking at the radial magnetic field close to the top of our domain in each case. In this section, we begin by focusing on the top panels of Figures 15 – 19 ; we will discuss the bottom panels (mainly the radial current maps) in the next section. Figure 15. Case CoTsBw. Top panel: zoom on the emerging radial magnetic field at r = 0.93 R ⊙ at three different times. Bottom panel: sketch of the initial configuration, with an emerging radial field at r = 0.93 R ⊙ shown on the whole Northern hemisphere and a zoomed map of the radial current J r at the last time t = 10 days. On this last panel, we superimpose contours of the radial magnetic field at 80% (thicker) and 2% (thinner) of its min and max values. The PIL, indicated by the magenta arrows, is then visible when the contours at 2% of opposite sign meet. Two emerging regions of different handedness are clearly visible here. Figure 16. Same as Figure 15 , but for case CsTsB. The black arrows on the bottom mid-panel indicate the locations of strong magnetic “necklaces.” In this merging case, the emerging radial field mostly consists of one large bipolar structure. Figure 17. Same as Figure 15 , but for case CsTsBw. In this weaker field case, the magnetic structure has been more affected by convective motions, and the

O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O