view of Event 1 observed simultaneously with AstroSat SXT, Chandra LETG/HRC-S, and HST binned to 100s each. The vertical lines in brown and yellow represents the UV and X-ray flare peaks, respectively. The amplitudes of the HST light curves are scaled to be consistent with X-ray light curves. A detailed analysis of the HST data will be presented in Schneider et al. (in preparation). However, a visual inspection of Fig. 4 indicates that the HST FUV light curve precedes the X-ray light curves by about a few minutes. This may indicate that the presence of Neupert effect (Neupert 1968 ) during the rising phase of Event 2. Using the Z-transformed discrete correlation function (ZDCF) (Alexander 1997 ), the cross-correlation of the HST and the X-ray light curves produces a time-lag between the peaks to be 300–400 s, indicating that due to electron impingement the UV light curve precedes the thermal radiation. Figure 4. Event 2 simultaneously observed with the AstroSat SXT in 0.5–4.0 keV (top panel), Chandra LETG/HRC-S zeroth order in 0.1–12.0 keV (middle panel), and HST FUV in 120–170 nm (bottom panel). 3.4 Comparison The quiescent emission measured with Einstein , EXOSAT, and ROSAT in 0.3–4.0 keV energy ranges are 7.2 × 10 −12 , 4–16 × 10 −12 , and 2.6–7.2 × 10 −12 erg s −1 cm −2 , respectively. The quiescent X-ray flux observed by Einstein and EXOSAT was reported by Haisch Linsky ( 1980 ) and Haisch et al. ( 1990 ), respectively. We used the ROSAT observations listed in the ROSAT Positive Sensitive Proportional Counter source catalogue (Voges et al. 1996 , 2000 ). Assuming a coronal temperature of 1 keV, sub-solar abundance of 0.5 and using WebPIMMS, we computed energy conversion factor ECF PSPC = erg cm −2 counts −1 to convert the observed ROSAT counts to flux in canonical 0.1–2.4 keV ROSAT energy band. Furthermore, we use the X-ray flux reported by Fuhrmeister et al. ( 2011 ) based on XMM–Newton observations. In Fig. 5 , we plot quiescent flux measured between 1979 and 2017 as a function of time. Furthermore, we assume similar temperature and abundance as before and used WebPIMMS to estimate the flux in the 0.3–4.0 keV energy range. The flux obtained from AstroSat is consistent with the previous estimates of quiescent flux for Proxima Centauri. Figure 5. Quiescent emission flux as a function of time in 0.3–4.0 keV energy range observed by several X-ray missions between 1979 and 2017. An arbitrary sine curve with a period of 7 yr is represented as the blue dashed curve. While a clear cyclic behaviour with a period of ∼7.0 yr is observed for Proxima Centauri in ASAS observations, a similar variation in the X-ray data is not apparent immediately. However, Wargelin et al. ( 2017 ) reported the evidence for a 7 yr stellar cycle using observations from several X-ray missions, principally Swift . The temporal coverage of X-ray observation, the energy resolution and uncertainties due cross-calibration restrict a detailed analysis of the cyclic

O O O O O O O B-Observatory B-Telescope B-Telescope B-Instrument O B-Telescope 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 B-Telescope O O O O O O O O B-Wavelength O O O O O O O B-Telescope 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 B-Telescope B-Wavelength O O O O B-Wavelength O O O O O O O O O O O O O O B-Model I-Model B-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O B-Telescope 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 B-Observatory B-Telescope O B-Wavelength I-Wavelength O O B-Telescope B-Instrument O O O B-Wavelength I-Wavelength O O O B-Telescope B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O O B-Telescope O B-Telescope O B-Telescope 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 B-Wavelength O O O B-Telescope O B-Telescope O O O B-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 B-Telescope O O 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 B-Citation O O O O O O O O O O O O O O B-Software 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-Telescope O O O O O B-Wavelength I-Wavelength B-Telescope O O O O O O B-Wavelength O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Software O O O O O O B-Wavelength I-Wavelength O O O O O O B-Observatory 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 I-Wavelength 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 O O O O B-CelestialObject I-CelestialObject O B-Telescope O O O O O O B-Wavelength 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 B-Wavelength O O B-Observatory O O O O O B-Wavelength O O O O O O O O O O O O O O O


no selection criteria are employed – dashed line and open squares – and the luminosity function obtained when the selection criteria previously described are used – solid line and filled squares. It is worthwhile mentioning here that the theoretical luminosity functions have been normalized to the bin of bolometric magnitude , which corresponds to the magnitude bin for which the observed white dwarf luminosity function has the smallest error bars. Thus, since this luminosity bin is very close to the maximum of the luminosity function, the normalization criterion is practically equivalent to a number density normalization. As clearly seen in this figure, the theoretical results match very well the observed data, except for a quite apparent excess of hot white dwarfs, which is discussed in detail below. We note as well that the selection criteria employed by Limoges et al. (2015) basically affect the low luminosity tail of the white dwarf luminosity function, but not the location of the observed drop-off in the white dwarf number counts, nor that of the maximum of the luminosity function. This can be more easily seen by looking at the bottom panel of Fig. 2 , where the completeness of the simulated restricted sample is shown. We found that the completeness of the entire sample is . However, the restricted sample is nearly complete at intermediate bolometric magnitudes – between and 15 – but decreases very rapidly for magnitudes larger than , a clear effect of the selection procedure employed by Limoges et al. (2015) . Nevertheless, this low-luminosity tail is populated preferentially by helium-atmosphere stars and by very massive oxygen-neon white dwarfs. The prevalence of helium-atmosphere white dwarfs at low luminosities is due to the fact that stars with hydrogen-deficient atmospheres have lower luminosities than their hydrogen-rich counterparts of the same mass and age, because in their atmospheres collision-induced absorption does not play a significant role and cool to a very good approximation as black bodies. Also, the presence of massive oxygen-neon white dwarfs is a consequence of their enhanced cooling rate, owing to their smaller heat capacity. Fig. 3 Top panel : probability test as a function of the age obtained by fitting the three faintest bins defining the cut-off of the white dwarf luminosity function. Bottom panel : white dwarf luminosity function for the best-fit age. Fig. 4 Top panel : probability test as a function of the age of the burst of star formation obtained by fitting the nine brightest bins of the white dwarf luminosity function. Bottom panel : synthetic white dwarf luminosity function for a disk age of Gyr and a recent burst of star formation (black line), compared with the observed white dwarf luminosity function of (red lines). 4.2. Fitting the age Now we estimate the age of the disk using the standard method of fitting the position of the cut-off of the white dwarf luminosity function. We did this by comparing

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


lighthouse effect. Comparing the neutrino modulation frequency with the GW frequency, we found that the neutrino peak frequency is m /2 times larger than the GW peak frequency, where m represents the deformation mode of the PNS (around the rotation axis) induced by the low- T /| W | instability. In the non-rotating model, the neutrino modulation frequency, as previously identified, is almost the same as the SASI frequency, and it is half of the GW frequency. We discussed the detectability of these neutrino modulations by IceCube and HK at a source distance of 10 kpc. If we observe our rotating models along the equatorial direction, the characteristic peak in the power spectra for the event rates at IceCube would be above the noise level. The characteristic peak due to the SASI in our non-rotating model observed along either polar or equatorial directions is also above the noise level. This analysis indicates that the joint observation of the GW and neutrino signals would be useful to extract information on growing multiple modes of non-axisymmetric instabilities such as the low- T /| W | instability and the spiral SASI activity, which is otherwise inaccessible by electromagnetic-wave observations. Finally, we shall discuss several major limitations of our work. Even when the pre-collapse core is a slow rotator (Heger et al. 2005 ) but with a sufficiently weak initial magnetic fields, the magnetic fields may be amplified to a dynamically relevant strength by the magnetorotational instability (MRI, e.g. Akiyama et al. 2003 ; Obergaulinger et al. 2009 ; Masada, Takiwaki Kotake 2015 ; Rembiasz et al. 2016 ). Although the role of the MRI on the post-bounce dynamics remains still unclear, the magnetic fields should play a key role in the angular momentum transport especially in the vicinity of the (differentially rotating) PNS surface. Possible impacts on the growth of the low- T /| W | instability with magnetic fields remain to be clarified. If progenitors have both strong initial magnetic fields ( G) and rapid rotation, the magnetic fields are amplified due to the field wrapping to dynamically relevant strength shortly after bounce, leading to the formation of magnetohydrodynamic (MHD) jets. This would also make the post-bounce dynamics such as the shock evolution and the post-shock flow very different comparing to those of the non-magnetized models. Such difference should also affect the angular momentum transport, not to mention the onset criteria of the low- T /| W | instability. Although there are a few recent 3D MHD core-collapse simulations, the development of the low- T /| W | instability has been only reported in Scheidegger et al. ( 2010 ) and suggested in Kuroda et al. ( 2020 ) (see, however, Mösta et al. 2014 ). Also, the convective dynamo may amplify the magnetic fields and modify the dynamics in the late phase of the core collapse (e.g. Raynaud et al. 2020 ). To answer these questions, one apparently need to perform 3D GR-MHD

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Instrument O B-Instrument O O O O O 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-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Model I-Model B-Model O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-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 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 B-Model 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 B-Model I-Model 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-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Model I-Model


1. Hipparcos AGB stars. This list is biased toward nearer stars, mostly within 1 kpc. McDonald et al. ( 2012 ) obtained effective temperatures and luminosities of 107,619 Hipparcos stars (ESA Publications Division 1997 ) based on photometric data covering from the UV to the far-IR. Our search was limited to Hipparcos stars restricted to T eff 3500 K and 1000 ( L / L ⊙ ) 54,000, which eventually reduced the number of candidates to 801 objects. The maximum temperature and minimum luminosity (Salaris Cassisi 1997 ) are criteria intended to reduce the contamination of the sample by RGB stars, while the upper limit of luminosity corresponds to the tip of the AGB ( M bol = − 7.1; Wood et al. 1983 ) that separates massive, evolved AGB stars from supergiants in the HR diagram. Eventually, the spectral classifications of the selected stars were examined in the SIMBAD database to exclude symbiotics, CVs, and other stars that had their temperature and/or luminosity miscalculated by McDonald et al. ( 2012 ). 2. Mass-losing AGB stars. In order to include stars with a circumstellar dust shell ( yr −1 ), a sample of 563 mass-losing AGB stars detected with the Japanese experiment IRTS (Murakami et al. 1996 ) was selected (Le Bertre et al. 2003 ). 3. S-type AGB stars. This class of objects represents the link between O- and C-rich AGB stars and includes two distinct subcategories: (i) the intrinsic S stars are genuine AGB that have their atmosphere enriched by s-process elements brought from the bottom of the convective shell by the third dredge-up process; (ii) the extrinsic S stars have accreted mass from a former AGB companion (now a WD) and generally do not show overabundances of s-process elements. We used the 2nd. edition of the General Catalog of S Stars (Stephenson 1984 ), that contains 1347 stars exhibiting ZrO or LaO bands in their spectra, obtained after various sky surveys. 4. C-rich AGB stars. C-rich stars are the result of the third dredge-up during the thermal-pulse AGB phase, which causes the abundance of carbon to exceed that of oxygen. We adopted a corrected, machine-readable version of the original General Catalog of Galactic Carbon Stars by Stephenson et al. ( 2001 ) available on the Vizier platform, containing 6891 objects. 3. AGB Counterparts in the 4XMM Catalog We cross-correlated the sample of AGB stars described above with the 4XMM-DR9 catalog 3 3 http://xmm-catalog.irap.omp.eu and assumed an X-ray source in this catalog to be a candidate counterpart of an AGB star if their angular distance is within three times the individual positional uncertainty (i.e., 3 × σ pos ). After discarding symbiotic systems, other X-ray emitters, and misclassified AGB stars, we were left with a sample of 19 AGB stars with candidate X-ray counterparts. We then proceeded to download all available XMM-Newton EPIC data of these sources from the XMM-Newton Science Archive (XSA). 4 4 http://nxsa.esac.esa.int/nxsa-web/#home A preliminary inspection revealed that

O B-Dataset I-Dataset 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 B-Telescope O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Wavelength O O B-Wavelength 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 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 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 O O O O B-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O B-Database 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 O O O O O O B-Location O B-Instrument B-Citation I-Citation I-Citation I-Citation 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 B-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-Dataset I-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 O O O O O O O O O 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 I-Dataset I-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Database O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O B-Dataset O O O B-URL 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 B-Wavelength 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 B-Telescope B-Instrument O O O O O O B-Archive I-Archive I-Archive B-Archive O O B-URL O O O O O


the mass, metallicity, effective temperature and individual frequencies in the same manner as before and computed masses and radii using the scaling relations iteratively with the reference functions described in equations ( 3 ) and ( 4 ). The results are shown in Figs 3 and A3 . These figures show that the reference functions that we present here indeed remove trends that are present in the mass and radius determinations as a function of max and mitigate the biases to a large extent. However, the standard deviations from using either equation ( 3 ) or ( 4 ) are very similar and significantly larger than in the ideal case, showing the impact of uncertainties in the observations. Figure 3. Same as Fig. 2 but now with the initial values of T eff , max and [Fe/H] perturbed with typical uncertainties (see the text for more detail). 4.1 Hare-and-hound validation We have thus far demonstrated that the new functions improve mass and radius estimations for the data they fit. In order to prove the general efficacy of our new corrections, we appraise the performance of the functions on data to which they have not been exposed. We perform a series of hare-and-hound validations using stellar models calculated with two additional evolution codes. We used garstec (Weiss Schlattl 2008 ) and mesa (Paxton etal. 2011 ) to compute grids of stellar tracks from which we draw testing data. Over 1000 models from each grid were extracted to serve as hares with the requirement that each selected model falls within the verified mass, metallicity and max ranges of the correction functions. The choice of microphysics and modelling assumptions differ between these two codes and they also differ from the code used to calculate the training data in yrec . Thus, mass and radius predictions of the hares will indicate the degree of stellar evolution code dependence in our function. Both the garstec and mesa grids were computed using OPAL opacities (Iglesias Rogers 1996 ) in conjunction with Ferguson etal. ( 2005 ) tables at low temperatures. We utilized the updated 2005 tables for the OPAL equation of state (Rogers Nayfonov 2002 ) along with the assumption of an Eddington plane-parallel atmosphere. We have taken the recommended nuclear reaction rates from the NACRE collaboration in combination with those from Caughlan Fowler ( 1988 ). The treatment of convection follows the mixing length theory as described by Kippenhahn, Weigert Weiss ( 2012 ) and models were computed with the respective solar-calibrated mixing lengths. Radial mode frequencies were calculated with the adipls pulsation package (Christensen-Dalsgaard 2008 ) for the garstec models and with the gyre pulsation package (Townsend Teitler 2013 ) for the mesa models. The large frequency separation was determined using a weighted median around the scaled max as per Bellinger etal. ( 2016 ). The grids differ in that the garstec models assume Asplund etal. ( 2009 ) solar abundances and were computed without

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row: particles reach a quasi-steady-state scale height in which downward settling is balanced by upward turbulent diffusion: . Across the columns, we vary the nondimensional turbulent viscosity: . Simulations with radiative damping: we have simulated ZVI with the initial Run_Brunt_Step profile with vertical domains from z = −3 H to z = +3 H for 2000 orbits with nonuniform vertical cooling. The simulations were local at one radius—that is, the radial extent of the computational domain was . Because of additional damping near the vertical boundaries (for stability), the effective domain is . Thick, blue vertical dashed lines indicate simulations that did not develop ZVI, whereas thick, red vertical solid lines indicate that ZVI was triggered and developed into full zombie turbulence. The vertical extent of these lines indicates the effective size of the computational domain. Note that when no part of the computational domain or only a small part of the computational domain extends into the gray region, ZVI is not triggered, while simulations in which the stratified regions of the computational domain have significant overlap with the gray regions do develop ZVI and turbulence. 4.4. Simulations of ZVI with Radiative Damping If the photon mean free path is longer than the largest scale of interest within a flow, then all scales are optically thin and radiative transfer can be modeled with a size-scale independent Newton cooling (i.e., radiated power proportional to temperature difference with respect to some equilibrium value). On the other hand, if the photon mean free path is shorter than the smallest scale of interest within a flow, then all scales are optically thick and radiative transfer can be treated as simple diffusion. A significant challenge is the case where, at the same location in space and time in the flow, small fluctuations are optically thin and large fluctuations are optically thick; using Newton cooling will overdamp large fluctuations, while using a diffusion operator will overdamp small fluctuations. Monte Carlo techniques for radiative transfer are ideal for cases that do not fit safely within the optically thin or optically thick regimes, but these methods involve a substantial investment in additional coding and computational resources. Our goal here is more modest: we want to determine how much radiative damping baroclinic critical layers can withstand and still support the development of ZVI. In Appendix A , we report on simulations with uniform stratification that confirm that ZVI requires Newton cooling times longer than a few orbital periods. In real PPDs, gravity and stratification are not uniform. Not only that, the radiative relaxation time can vary in position within the disk and is especially sensitive to how much the dust has settled. The baroclinic critical layers are most sensitive to radiative cooling; their radial extent is much less than a gas scale height and is expected to be optically thin. Thus we

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


is small compared with the speed of the flow, and the inertial term in Equation ( 2 ) can be ignored. This eliminates the variable, v θ from the induction equation, so it can be rewritten as: Therefore, expansion of the plasma () corresponds to attenuation while compression () corresponds to amplification of b θ . From the preceding discussion it follows that the latter situation is more likely to occur in downdrafts. Equation ( 4 ) looks remarkably similar to the vorticity equation for an incompressible fluid:where ω denotes vorticity in the z direction. The right-hand side of Equation ( 5 ) represents the well-known vortex stretching effect: the angular velocity of a vortex tube increases when it is stretched and decreases when it is compressed (Spurk Aksel 2008 , p. 100). In contrast, Equation ( 4 ) shows that the θ component of the magnetic field increases when the flow decelerates and decreases when it accelerates. The fact that the twist amplification stems from the induction equation emphasizes the magnetic nature of the process. Twist amplification may be thought of as a magnetic analogue of vortex stretching. In a more general situation, when the magnetic field is not weak, the twist evolution requires a more detailed treatment. In the following sections we study the twist amplification analytically as an eigenvalue problem and numerically, and analytically. We also discuss the energy source, and the physics of the amplification process. 3. UNSTABLE TWISTS IN AN EXPONENTIALLY DECAYING DOWNDRAFT The set of governing Equations ( 1 ) and ( 2 ) can be reduced to a single second order PDE for the magnetic field perturbation, b θ (Taroyan 2008 ):where , and . In deriving the above equation, we have used the equation of mass conservation. In a static medium with , the well-known wave equation can be derived from Equations ( 1 ) and ( 2 ): Equation ( 7 ) was first analyzed by Ferraro ( 1954 ) in the context of Alfvén wave propagation in the solar atmosphere. Solutions in terms of the Bessel functions were constructed for an isothermal atmosphere with an exponential density profile:where z 0 is the scale height. The corresponding Alfvén speed is given by:Note that in our notations z denotes depth. Equation ( 7 ) with the profile ( 8 ) was subsequently analyzed by different authors. They found standing wave solutions in terms of the Bessel functions (Ferraro 1954 ; An et al. 1989 ) or propagating wave solutions in terms of the Hankel functions (Hollweg 1978 ; Cally 2012 ) depending on the imposed boundary conditions. An advantage of the smooth exponential profile is the absence of any model dependent artificial reflections that may arise due to discontinuities in the Alfvén speed or its derivatives (Cally 2012 ). One of the drawbacks is the finite Alfvén travel time to (An et al. 1989 ). In studies of the convection zone, it is common to adopt a polytrope with a linear temperature. Here, we adopt the exponential profiles ( 8 ) and ( 9 ) with

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


(see Section 2.4 ). For solar metallicity μ converges to ∼2.4 in the CNM, to ∼1.2 in the WIM, and to ∼0.6 in the CGM. As we deal with only low-density environments in the simulations the assumption of an ideal gas is legitimated. Hence, γ = 5/3 in all simulations. 2.2 Self-gravity The effect of self-gravity for an arbitrary matter distribution is described by solving the well-known Poisson’s Equation for the gravitational potential ( G denotes Newton’s gravitational constant). Each mass element of contributes to . If the total mass becomes a significant fraction of the (temperature-dependent) Jeans mass M J for a particular , self-gravity cannot be neglected as it relevantly affects the dynamics of all mass elements. However, it has been shown in Sander Hensler ( 2019 ) that self-gravity has a measurable effect on the evolution of even clouds with masses well below their Jeans mass. If self-gravity is stronger than the acceleration of gas in the opposite direction (e.g. due to evaporation or advection by a streaming CGM), the net acceleration at the cloud surface points in the same direction like the density gradient. Hence, the condition for Rayleigh–Taylor instability (RTI) is not given. So, it is expected that self-gravity significantly suppresses RTI at the cloud surface (see Murray et al. 1993 ) and the typical finger-like structures do not form. According to Murray et al. ( 1993 ), the minimum mass of self-gravitating clouds above which they are stable against Kelvin–Helmholtz instability (KHI) when moving with a certain speed v rel through an ambient medium with density ρ CGM reads with being the average mass density in the cloud. To ensure that self-gravitating clouds, which are embedded in a medium with pressure P CGM , are gravitationally stable initially, their total mass must be below their Bonnor–Ebert mass (Ebert 1955 ; Bonnor 1956 ) Here, and are the average temperature and mean molecular weight in the cloud, respectively. 2.3 Metallicity Observations indicate a common metallicity for CHVCs of 0.1 to (Wakker 2001 ). In order to study extreme cases, we consider two different initial cloud metallicities of and . We note that the CGM has constant metallicity in all runs (see Section 4.3 ). The ratio of the abundances of hydrogen and helium is assumed solar and so are the ratios of metal abundances. 2.4 Dissociation and ionization of elements The simulations contain cool clouds and hot plasma, so a temperature range from a few hundreds up to several 10 6 K is covered. For T 900 K molecules exist, but they successively dissociate and finally become totally ionized with rising temperature. In our simulations, we semi-analytically calculate both dissociation and ionization for hydrogen in thermodynamic equilibrium. The fraction of H ii is then related to He and metals. We note that we do not trace single species but rather have one fluid for which the composition is calculated based upon a given metallicity. The fractions of

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 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 B-EntityOfFutureInterest I-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 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 B-Citation I-Citation I-Citation I-Citation 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 O O O O O O O O O O O O O O O O O O O O O O O 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-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 O O O O O O O O O O O O O O O O O O O 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


becomes more highly extinct and fainter. Fig. 4 LUCI J -band image of T Tauri from November 2016 displayed on a logarithmic scale. The dotted arc indicates the northern arm of the reflection nebulosity discussed by Stapelfeldt et al. (1998) . The yellow arrows indicate the tadpoles. The red arrows mark bow-shaped features. Extended features around an apparent cavity are enclosed within the black dotted lines. The green, orange, and red dots identify the locations of T Tauri N, T Tauri Sa, and T Tauri Sb, respectively. The inset shows the tadpoles and the cavity with a different gray-scale cut. Fig. 5 Zoom of Fig. 4 on the area southwest of T Tauri N showing the coiling structure and the reflection nebulosity features R2 and R3 (as shown in Fig. 2 of Kasper et al. 2016 ) in the vicinity of T Tauri S. The dotted line connects the inflection points of the coil. The red arrow indicates T Tauri Sa. Fig. 6 T Tauri H 2 line emission map obtained with LBT-LUCI in October 2016. Only features which cannot be explained by image processing artifacts are indicated and discussed in the text. The positions of the stars are indicated by colored dots (T Tauri N, Sa,and Sb are in green, orange, and red, respectively). Fig. 7 IRDIS J -band image of the coiling structure taken in December 2018. The white dotted line indicates the coiling structure’s position in December 2014 (see Kasper et al. 2016 ). The relative distance to the T Tauri stars can be determined from the orbital solution and radial velocity (RV) measurements. According to the best-fit T Tauri N-S orbit ( Schaefer et al. 2020 ), T Tauri S went through the line of nodes in 1936 with an RV difference to T Tauri N of about ± 6.5 km s −1 . The sign of the RV difference cannot be determined from an astrometric orbit solution. Duchêne et al. (2005) found that the heliocentric RVs of T Tauri Sa and Sb appear to be larger than the one of T Tauri N ( Hartmann et al. 1986 ) by 2–3 km s −1 . These values are compatible with the orbit prediction within the error bars if T Tauri S previously went through the ascending node. Since then, T Tauri S has been moving behind T Tauri N for about 80 yr and would now be located roughly 90 AU behind T Tauri N. However, the N-S orbit is still only loosely constrained and orbital solutions may exist for which T Tauri S is located in front of T Tauri N, as suggested by Beck et al. (2020) . Fig. 8 IRDIS J -band image of the R4 structure taken in December 2018. The white dotted line indicates R4’s position, and the “+” symbols indicate the positions of T Tauri Sa and Sb in December 2014 (see Kasper et al. 2016 ). Fig. 9 IRDIS J -band image of the R3 structure taken in December 2018. The white dotted line indicates R3’s position in December 2014 (see Kasper et al. 2016 ). Figure 1 also shows a bow shock at ~1′′.6 west of T Tauri. This bow shock is a signpost of the east–west outflow ( Böhm Solf 1994 ; Herbst et al. 1997 ) with the western part approaching (blue-shifted) at about

O O O O O O O O B-Instrument B-Wavelength I-Wavelength 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 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 B-CelestialObject I-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O B-CelestialObjectRegion O B-CelestialObjectRegion O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject 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 O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject B-CelestialObject B-CelestialObject O O O O O O O O O B-Instrument 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-Citation I-Citation I-Citation I-Citation 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 B-Model I-Model I-Model I-Model I-Model O B-Citation I-Citation I-Citation I-Citation O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O B-CelestialObject I-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 B-Citation I-Citation I-Citation I-Citation O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject O O O O O O O O B-CelestialObject I-CelestialObject I-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 B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O B-Citation I-Citation I-Citation I-Citation O O O B-Instrument B-Wavelength I-Wavelength O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Instrument B-Wavelength I-Wavelength O O O B-CelestialObjectRegion O O O O O O O O O O B-CelestialObjectRegion 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 B-CelestialObject I-CelestialObject O O O O O O O O O O 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


fully convective (e.g., Spada et al. 2017 ). A radius of ≈0.4 R ⊙ leads to cycle times of about 4–5 years, provided that η T remains unchanged. If η T of cooler stars is reduced with respect to the solar value, as implicitly suggested by the stellar models of Spada et al. (2017) , then the cycle time from Eq. ( 1 ) basically exceeds this limit. Mixing-length estimates of 0.33 u rms l corr for M dwarfs provide about 30% of the solar value, so that the dynamo cycle times of M dwarfs should slightly exceed the duration of the solar cycle. Indeed, the cycle time of one star in Fig. 1 slightly exceeds the 11 yr (≈4000 d) limit but this turned out to be a K2 dwarf star with a rotation period of 22.8 d ( Johnson et al. 2016 ). Figure 1 shows the observed cycle lengths for M dwarfs to be generally shorter than ≈3000 d. GJ270 (spectral type M2) shows the longest known cycle period of any M star with 2700 d (7.4 yr), while the shortest cycle is seen in GJ476 (M4) with 1060 d (2.9 yr) ( Robertson et al. 2013a ). GJ328 as an M0 dwarf with 0.69 M ⊙ shows an activity cycle of ≈2000 d, while a Jupiter-mass planet orbits it with a 4100 d period ( Robertson et al. 2013b ). We note that in the solar system the orbital period of Jupiter coincides with the length of the solar activity cycle, accidental or not. GJ581 possesses an entire planetary system with four or possibly six planets and shows an activity cycle with a length of 1600 d ( Gomes da Silva et al. 2012 ; Robertson et al. 2013a ). The rotation period of the latter star is surprisingly long with 130 ± 2 d ( Robertson et al. 2014 ). Fig. 1. Cycle times vs. rotation period (both in days) for early M stars (rhombic symbols). Among the group of rapid rotators with rotation periods less than a day ( Vida et al. 2014 ) are also the early M dwarfs EY Dra and V405 And ( Vida et al. 2013 ). The remaining data are from Suárez Mascareño et al. (2016) ; Wargelin et al. (2017) , and Distefano et al. (2017) . For GJ476 the cycle time is known but not the rotation period (blue bar). The Schwabe cycle of the Sun is indicated as an asterisk denoted SUN. Gomes da Silva et al. (2012) give further examples of M dwarfs with cycle lengths between three and five years. It is of particular importance to know the relation of the cycle time with the rotation time for a homogeneous sample of stars. Savanov (2012) used light curves of cool M dwarfs to find long-term cycle variations up to timescales of years. His sample is mixed by M stars with outer convection zones and also fully convective stars. There seems to be no systematic dependence of the cycle time on the rotation time even for subsamples with slow and fast rotations. An independent study about the coupling of rotation and cycle times for fast-rotating cool stars has been published by Vida et al. (2014) . Among the analyzed 39 Kepler light curves of late-type stars with rotation periods shorter than one day, nine examples showed cyclic variations with periods between

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 B-CelestialObject 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 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 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-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-CelestialObject O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-CelestialObject 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 B-Citation I-Citation I-Citation I-Citation O O O O O O B-CelestialObject I-CelestialObject O O O O B-CelestialObject O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest 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 B-Citation I-Citation I-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 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-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O B-Citation I-Citation I-Citation I-Citation O O B-CelestialObject O O O O O O O O O O O O O B-Model I-Model O O B-CelestialObject O O O O O O B-CelestialObject 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Telescope O O O O O O O O O O O O O O O O O O O O


by comparing the observed spectral energy distribution (SED) to the sum of the SEDs of both models. The reddening is modeled following Cardelli et al. ( 1989 ). From the total luminosity, the fixed light ratio and the other determined parameters, we then derive the individual luminosities. The final spectral fit is shown in Figure 4 . The inferred parameters are given in Table 4 . The spectroscopic mass for the O-type secondary is derived from , where G is the gravitational constant. The spectroscopic mass of the WR primary stems from mass–luminosity relations calculated for chemically homogeneous WR stars by Gräfener et al. ( 2011 ), which for a H-free star depends only on L . The error on the spectroscopic mass of the WR star is propagated from the error in L , and is possibly underestimated in its lower limit given that the WR primary may be chemically inhomogeneous, which would lower M for a given L . Figure 4. Upper panel: comparison between the observed SED (FUSE, IUE, and UBVJHK photometry, blue lines and squares) and the synthetic PoWR SED modeled for WR 133 (red dotted line). The synthetic SED is the sum of a PoWR model for the WR primary (black solid line) and O-type secondary (green dashed line). Lower panels: comparison between the observed and synthetic normalized spectra. See the text for details. Table 4 Inferred Physical Parameters of WR 133 from the PoWR Spectral Analysis Note. a The transformed radius should not be confused with the physical radius of the star; see the text for details. 5. Discussion Sana et al. ( 2012 ) found that about 75% of massive OB stars should have their evolution impacted by a companion. The general assumption used in the analysis of binary populations is that two stars will interact if the system is born with a semimajor axis of less than ∼10 au and the star evolves into a red supergiant (e.g., Sana 2017 ). During a binary interaction, the system initially becomes more compact until the mass ratio is unity and then increases until mass transfer eventually ceases. This has been seen in an active phase for binaries that could become similar to WR 133 with systems such as the LBV-like binaries MWC 314 (Richardson et al. 2016b ) and HDE 326823 (Richardson et al. 2011 ). The supergiant companion to the WR star in WR 133 could have a larger rotation rate than an average field star. The inferred value of of 96 km s −1 is similar to the average value found for O supergiants by Simón-Díaz Herrero ( 2014 ). However, if there were indeed past binary interactions, the O9I star should be rotating in the orbital plane, and thus would likely be a rapid rotator. In fact, if we use our derived value for and the inclination of the system imply a modestly fast rotation rate of 280 km s −1 , which is a reasonably fast rotation for a supergiant O star. However, if the O star had a magnetic field from the interaction, then the rotation rate could be drastically less than critical in a short time (Potter et al. 2012 ).

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0.04 and 0.03 mag for and distance modulus, with metallicities from Rey et al. , and 0.06 and 0.05 mag for the and values derived with the metallicities from Sollima et al. At the same time, results from previous studies (e.g., Cacciari Clementini 2003 ; Del Principe et al. 2006 ; Feast et al. 2008 ; Gran et al. 2015 ) suggest that distances derived on the basis of such relations can probably not yet be considered accurate to better than about mag. Thus, we adopt the latter value as a conservative estimate of the systematic error affecting the distance moduli provided in our study. Fig. 8 Fundamental (in purple circles) and first-overtone (in yellow triangles) SX Phe as classified by Olech et al. (2005) , based on their frequency spectrum. The dotted lines are the PL relations derived previously. As can be seen, there are some claimed first-overtone pulsators that also follow the relations defined by fundamental-mode pulsators. Table 4 Adopted distance moduli based on T2Cs and RRL stars. 4.2. Pulsational distance modulus from T2Cs and RRL stars Table 4 lists the weighted-average distance modulus values from T2Cs and RRL stars and sigma as the dispersion around the mean. The weights for each individual distance modulus values were defined as the inverse of the squared of the total error, including statistical and systematic errors (summed up in quadrature). As expected, the error is lower for the PL relations compared to , since the former have lower intrinsic dispersion, and is based on a more extensive dataset. Comparing both variability types, T2Cs lead to a less accurate distance modulus, probably because they are few in number, which increases the statistical error, and they have higher systematic errors since the effect of metallicity and pulsation mode separation in the derivation of the PL relations is not completely established. Considering the distance modulus values found using T2Cs stars as well as RRL stars, both in and , the weighted-average distance modulus to Cen was determined, leading to a final value of mag, where the error bar corresponds to the standard deviation of the mean. Given the discussion of systematic errors in the previous section, our final adopted distance modulus is mag, where the error bars correspond to the adopted statistic and systematic errors, respectively. Adding the errors in quadrature (as recommended, for instance, by Barlow et al. 2002 ), this corresponds to a heliocentric distance of kpc. Table 5 Distance modulus for Cen from the literature. Notes. (a) Most of the error estimates do not consider the contribution of systematic errors, which in the case of T2Cs and RRL stars can reach values of order 0.10 mag. (b) The listed error was determined considering the internal uncertainties of the mean distance modulus value, 0.02 mag, and the observational error, 0.1 mag, as described by McNamara (2000) . (c) Based on the -[Fe/H] relation from Bono et al. (2003) and mean magnitudes from Kaluzny et al.

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


environment theoretically depends on the Mach number or wave amplitude (e.g., Priest 2014 ; Vršnak et al. 2016 ). As the shock wave propagates, its amplitude decreases due to the geometric expansion of its front and the fact that its leading edge is propagating faster than its trailing edge. The amplitude decrease results in the deceleration of the shock wave. Additionally, a power-law fit is a better representation of the distance–time curves of freely propagating shock waves (Warmuth Mann 2011 ) according the the Sedov solution (Sedov 1959 ). On the other hand, we found that the speed of the reflected wave (243.1 ± 30.8 km s −1 ) is much smaller than the speed of the incident wave (which approximately equals that of the unaffected wave at around 05:13 UT: 591.0 ± 25.9 km s −1 ). In other words, the reflected wave was slower than the primary wave. There may be several possibilities for this feature. The primary wave is a nonlinear wave or shock, which is propagating with a Mach number larger than unity, while the reflected wave had changed to be a linear one traveling at the characteristic speed (Liu Ofman 2014 ; Warmuth 2015 ). The reflected wave has to propagate in the rest frame of the flow field associated with the primary wave, which may result in an apparently lower speed (Kienreich et al. 2013 ). Additionally, it may be related to the properties of the medium (magnetic separatrix) causing the shock wave to reflect. The compressible magnetic separatrix may cause the decrease in the reflected wave speed, when the shock wave reacted with the magnetic separatrix. Moreover, it is interesting to find that the refracted and diffracted waves decelerated with a bigger nonconstant deceleration rate. We find that their deceleration rate was decreasing with time, as in a damping motion. These phenomena would help us to further understand the nature of the interaction between magnetohydrodynamic shock waves and the local magnetic field. Appendix Error Estimations In this study, we determine the dynamic characteristics of EUV waves by using the method in Liu et al. ( 2010 ). The track technique ignores the heights of EUV waves by assuming that all emission originates from the spherical solar surface, and this distance measurement was corrected by the sphericity of the solar photosphere. This method would cause an overestimation in the measurement of some parameters (e.g., distance, velocity). We will supply some error estimations in subsequent sections. As Figure A1 (a) shows, a coronal structure B at a height of Δ h above the solar surface is projected onto the solar surface A observed by an observer on Earth (E). We set a new “north pole” at the disk center ( D ) and make AC = d , AC ⊥ OE, OA = OD = R ⊙ , OB = OF = R ⊙ + Δ h , and DE = 1 au. Defining θ = ∠AEO, we can get We make OG ⊥ EG and get Then, we getAccording to the law of sines, in △BOE, we can get Based on the above equations, we can get whereFirst, if the EUV wave originally

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


between 3D and 1.5D calculations might be quite influenced by ingredients of the models that require improvement. We show the relevance of these ideas commenting on two of the few papers existing on this topic. The key is that the influence of the 3D radiation field on the Hanle polarization has been posed since the beginning in terms of inhomogeneities, and not in terms of velocity gradients. The initial conclusions in this regard were obtained by Manso Sainz Trujillo Bueno ( 2011 ) with an analytical/numerical study approaching the problem in the static regime. In more recent numerical studies the action of 3D velocities in the polarization is included, but the contributions of vertical and horizontal velocity gradients are not separated. As the effects of horizontal velocity gradients and inhomogeneities are in turn blended, it is also unknown how large their relative contributions are to the differences between 1.5D and 3D calculations. Consider, for instance, Figure 3 of Štěpán Trujillo Bueno ( 2016 ), which points out that the Hanle effect at disk center tends to depolarize when it is accompanied by other symmetry breaking effect, as happens in the well-known case of the solar limb. The figure also confirms that macroscopic motions are yet an efficient polarizing source for the spectral line considered (as known from 1D simulations of the Ca ii IR triplet; Carlin et al. 2013 ). However, it is unknown whether the polarization introduced by velocity in that figure is due to the horizontal or to the vertical velocity gradients already considered in 1D calculations. This matters not only for comparing with 1D calculations but also because horizontal velocity gradients can potentially compensate and exceed the polarization created by horizontal inhomogeneities (as explained in previous paragraphs). Thus, the dominant symmetry breaking and polarizing effect at disk center could not be the inhomogeneities, as always affirmed, but the horizontal velocity gradients. Hence, we think that is key quantifying the sensitivity of such results to the model horizontal distribution of velocity gradients , which also can help detach the conclusions from the eventual lack of realism in them. Finally, it should also be noted that when those radiative transfer simulations do not add microturbulent velocity, they use the temperature distribution of MHD models that are known to represent a chromosphere cooler than the solar one. Thus, as temperature influences significantly the effect of the velocities in the LP (see Section 4.2 ), the relative strength of the polarizing effects discussed here might change accordingly. In summary, horizontal velocity gradients compete with other physical mechanisms, such as inhomogeneities and the Hanle effect, for polarizing and depolarizing the light; hence, we need to explicitly quantify whether they are relevant for inferring magnetic fields with the Hanle effect. We also need to find theoretical and observational

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O B-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


ratio moving in circular orbits, the filling of the Roche lobe is followed by violent mass transfer by the primary to the secondary and the formation of a common envelope. On the other hand, an equal-mass binary may evolve into a fully contact system of two stars, both of which fill their Roche lobes simultaneously. In the limiting case, violent mass transfer may occur on the hydrodynamic time-scale, with an amount of released energy of the order of the thermal energy of the components or the common envelope: GM 2 / a 510 48 ( M /M ) 2 ( a /R ), where M is the mass of the component or of the common envelope and a is the radius of the common envelope. For 3M common envelope mass and energy from Table 1 , the envelope radius is about 10R . The high energy of the studied explosion is a consequence of compactness and the relatively high mass of the parent system. From this point of view, it is very important to obtain observational evidence for such processes in binary systems and to determine the corresponding physical properties: the released energy, ejected mass and angular momentum. Ultraluminous red novae may be observational manifestations of violent mass transfer in the binary (Blagorodnova etal. 2017 ). We already pointed out that observation of the progenitor of V1309 Sco in the form of a contact binary with a period of 1.7d (Tylenda etal. 2011 ) provides strong evidence for the merger model of LRNe. This conclusion is also corroborated by the approximate occurrence rate of such events one event in 2040yr per galaxy like the Milky Way or M31. To show this, let us use the distribution of binary semimajor axes a (Masevich Tutukov 1981 ). This distribution has the form d N 0.2dlog( a /R ). Systems with 10 a /R 1000 can pass through the common-envelope stage during the second mass exchange stage. According to the initial distribution of binaries over semimajor axes of Masevich Tutukov ( 1981 ), N 0.2dlog( a /R ), the frequency of formation of common envelopes with radii 610R is about 0.04 per year. Systems with larger component separations are less energetic. 5 CONCLUSIONS In this article, we described the discovery of the nova M31LRN and long-term observations of its light curve with the MASTER network of robotic telescopes. It is an important point that the entire observational part of the study was performed on identical telescopes equipped with identical photometers. The resulting light curve agrees fairly well with the independent light curve published by Kurtenkov etal. ( 2015b ). An analysis of the photometric observations and velocity data for the nova M31LRN made using radiative hydrodynamic modelling showed that the rather long observable plateau (50d) could be achieved only with a higher merged stellar mass M 3M and common envelope radius R 10R . The corresponding explosion energy should be lower, 310 50 erg, whereas the total kinetic energy of the ejected envelope is lower by three orders of magnitude. The proposed

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


the literature value of 5.2 kpc, but consistent within the uncertainty, and so this star is still marginally consistent with being an LBV on the S Dor instability strip. The higher precision expected in future Gaia data releases is needed to say more. WS 1: WS 1 was classified as a bona-fide LBV by Kniazev, Gvaramadze Berdnikov ( 2015 ) based on observations of significant photometric and spectroscopic variability. The Gaia DR2 distance of 2.48 kpc is significantly lower than the baseline literature value of ∼11 kpc, although this object has a negative parallax and like the previous two, is dominated by the prior. However, the literature luminosity is also highly uncertain because it is not based on a measurement – it was derived by assuming that the star fell on the S Dor instability strip Gvaramadze et al. ( 2012 ), so perhaps a large revision is not so surprising. Gaia data suggests that the star is significantly lower in luminosity, although again DR3 data are needed for a confident result. MN48: MN48 was classified as a bona fide LBV by Kniazev et al. ( 2016 ) who identified spectroscopic and photometric variability typical of LBVs. There is large excess noise in the Gaia DR2 data for this star, and the resulting 68 per cent HDI spans 1.44–5.18 kpc. The upper end of this range is marginally consistent with the previous literature value of ∼5 kpc, although again, the literature value was highly uncertain as well. 3.2 Candidate LBVs HD 80077: This blue hypergiant was included as an LBV candidate by van Genderen ( 2001 ) based mainly on its high luminosity and spectrum. Its new Gaia DR2 distance is significantly closer than the baseline literature value, reducing its nominal luminosity by about 40 per cent, although its luminosity is still above the S Dor strip. The Gaia DR2 distance and the original cluster-fitting distance of Steemers van Genderen ( 1986 ) (2.8 ± 0.4) agree within their quoted errors. SBW1: As in the cases of Sher 25 and HD 168625, this blue supergiant is an LBV candidate because of its circumstellar nebula that bears a remarkable resemblance to the ring nebula around SN 1987A. In fact, it has been argued that SBW1 is in some respects the best Galactic analogue to the progenitor of SN 1987A, in terms of both its nebula and the properties of the weak-winded central star (Smith et al. 2013 , 2017 ). Based on various considerations, Smith et al. ( 2017 ) favoured a distance of about 7 (±1) kpc for SBW1. The Gaia DR2 parallax indicates a distance of 5.2 kpc (Table 3 ). This distance is closer, but the range of 4.0–6.8 kpc overlaps with the uncertainty of 7 (±1) kpc for the previous distance. This new distance corresponds to a somewhat lower effective single-star initial mass of 13–15 M ⊙ , as compared to ∼18 M ⊙ for the progenitor of SN 1987A (Arnett 1989 ). This confirms the notion that SBW1 is an analogue of the progenitor of SN 1987A and its nebula (Smith et al. 2013 ), although at somewhat lower initial mass.

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


by Binggeli et al. ( 1985 ) in the Virgo Cluster Catalog (VCC), although no redshift has been published. Indeed, Binggeli et al. ( 1985 , p. 1756) classified VCC 1287 as a dwarf irregular galaxy “ .. of very large size and low surface brightness. ” GCs were identified exploiting all five MegaCam bands. A catalog of sources was extracted using SExtractor (Bertin Arnouts 1996 ). Only objects detected in all five bands were considered, selecting point sources by imposing the i -band class star parameter of SExtractor to be >0.5. This removes most extended objects from the catalog, owing to the excellent (06) seeing of the images. We selected as GC candidates all point sources within the following color ranges: 0.6 ( g − z ) 1.2, 1.1 ( u − g ) 1.8, and 0.7 ( g − i ) 1.0. We selected a fairly narrow color range in order to minimize contamination from background sources (in particular, red background galaxies). However, these color ranges are consistent with the distributions seen in other studies of Virgo dwarf galaxies (e.g., Beasley et al. 2006 ; Peng et al. 2006 ). In order to exclude ultra-compact galaxies, we also imposed a magnitude cut, i > 19.5 mag, equivalent to the magnitude of the Milky Way GC ω Cen at the distance of Virgo. To determine the total number of GCs in VCC 1287, we constructed the surface density profile of the GC system and found it to extend to ∼175′′ (∼13 kpc), beyond which is a constant background with 0.2 objects per arcmin 2 . Next, all point sources within 175′′ of the galaxy center and that satisfy our selection criteria were flagged as GCs—returning 18 candidates. Star and galaxy contamination were calculated by the random placement of 1000 circles with 175′′ radii in the 1 deg 2 field surrounding VCC 1287. We found a mean of 6 objects expected within such a circle, with a standard deviation of 2.5 counts using our selection criteria, with no sky apertures with 18 objects. The identification of 18 GC candidates represents an overdensity with respect to the background level at ∼5 σ significance. Subtracting this contamination rate left 11 GC candidates down to the GC turnover magnitude ( i = 23.1, using I -band values of Kundu Whitmore 2001 and an i -band transformation from Faifer et al. 2011 ). Assuming that the GC luminosity function is bell-shaped (Harris et al. 2000 ), we doubled their number to obtain a total GC population of 22 ± 8 GCs, where the uncertainties come from the quadrature sum of the Poisson uncertainties and the background contribution. This may not sound remarkable, but when normalized to the host galaxy magnitude using the “specific frequency” ( S N ; Harris van den Bergh 1981 ), we find VCC 1287 has S N = 80 ± 29. This is a significant overabundance of GCs for VCC 1287's stellar luminosity (Figure 4 ). 2.2. GC Spectroscopy GC candidates were selected for multi-object spectroscopy with the OSIRIS instrument on the Gran Telescopio de Canarias (GTC) in La Palma. We observed this spectroscopic

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on the velocity–luminosity relation for RSGs derived by Mauron Josselin ( 2011 ), we obtain a wind velocity of ∼35 km s −1 during the RSG state of ρ Cas. Considering that the enhanced mass-loss within the RSG phase took place about 10 000 yr ago, the material would have travelled over a length of roughly 0.36 pc, which corresponds to an angular distance of ∼24 arcsec at the distance of 3.1 kpc for ρ Cas (Lobel et al. 2003 ). This angular distance is too small to be seen on, e.g. WISE images, on which ρ Cas is saturated. This distance is also much smaller than the size of a regular wind-blown bubble resulting from the previous blue supergiant phase, so that we can exclude that the material is located within the astrosphere or shock region with the interstellar medium. On the other hand, this distance is too large to be seen on the HST images presented by Schuster et al. ( 2006 ), and too large to guarantee that the material is dense and hot enough to produce measurable amounts of CO band emission. Therefore, we believe that the post-RSG wind, which is faster but much less massive, interacts with the material released during the RSG phase. This interaction can lead to a compression and heating of the old material (with the appropriate carbon isotope ratio) and hence give rise to the observed, static CO band emission. The temperature of the CO gas is much higher than the dust sublimation temperature. In the region of CO band emission, the physical parameters within the environment might hence prevent the efficient formation of dust, which could be another reason for the lack of a detectable dusty nebula around ρ Cas. For its spectral type, and considering that ρ Cas is presumably a very slow rotator, it displays extremely broad absorption lines. These were suggested to be broadened by macroturbulence. This type of line broadening is typically found in early-type (OB) supergiants in which pulsations play a significant role together with stellar rotation (see e.g. Ryans et al. 2002 ; Aerts et al. 2009 ; Simón-Díaz et al. 2010 ; Kraus et al. 2015 ), and is one of the criteria to discriminate YHGs from YSGs (de Jager 1998 ). While ρ Cas is known to pulsate, not much is known about its real rotation velocity, although Lobel et al. ( 1998 ) derived a value of v sin i ≃ 25 km s −1 from line-profile fitting. To investigate the possible contribution of rotation to the line profile broadening of ρ Cas, we analysed photospheric lines using the method of Fourier transformation. This method only works properly for symmetric profiles. However, symmetric line profile shapes are rather rare in our data of ρ Cas and basically only seen in two nights. Also, given the spectral resolution of our data, the Fourier transformation can recover only rotation velocities projected to the line of sight higher than v sin i ∼ 15 km s −1 with high confidence. The results from our analysis are shown in Fig. 8 for different elements within the same night (top panel) and for the

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stars (FRMS) has been already discussed by several authors (e.g. Decressin et al. 2007 ). This FRMS self-enrichment scenario has no problem associate with later SNe that can severely suppress secondary star formation. However, it is not clear in this FRMS scenario how SG star formation can be completed well before SN explosion can brow away the cold gas of the molecular cloud (MC). We here do not discuss how to avoid this potentially serious problem, because it is beyond the scope of this paper to investigate the FRMS scenario in detail. 4.3 Metallicity spreads in GCs Although Lee et al. ( 2009 ) investigated the colour magnitude diagrams ( V versus b − y or hk ) of the Galactic GCs and found evidence of Ca abundance spreads in the 7 GCs, Carretta et al. ( 2010 ) showed that Ca abundance spreads among FG and SG stars in 17 Galactic GCs are less than 0.02–0.03 dex. The most massive Galactic GC ω Cen and 8 GCs (e.g. M22) have been observed to show [Fe/H] spreads among GC stars so far (e.g. Freeman Rodgers 1975 ; Marino et al. 2015 ). The apparent lack of internal [Fe/H] spreads among GC stars in most GCs implies that self-enrichment by SNe did not proceed efficiently at GC formation for some physical reasons. Since these GCs show anticorrelations between light elements, which could be due to self-enrichment by AGB stars, the physical mechanisms that suppress self-enrichment by SNe in GC formation need to be understood clearly. Nakasato, Mori Nomoto ( 2000 ) investigated SFHs of proto-GC clouds with the masses of 10 6 M ⊙ , sizes of 150–300 pc, and initial temperature of 10 4 K using their original SPH simulations with feedback effects chemical enrichment by SNe. They found that although star formation in shell-like gaseous structures formed through compression of gas through SNe feedback effects is possible, self-enrichment is not seen to occur in all of their models. This study have demonstrated that although new star formation from gas contaminated by SNe is possible, the mass fraction of such stars is quite small. Furthermore, most of such stars can be formed mostly in the shocked gas that are distant from the main GC-forming regions, and therefore they cannot be finally within the central regions of GCs. As a result of this, the mass fraction of such (SG) stars with [Fe/H] by more than 0.05 dex larger than (FG) stars formed from original cold gas of GC-forming GCs is quite small within the central 10pc of GCs. This implies that the apparent lack of [Fe/H] spreads in typical GCs is due largely to SN feedback effects in GC-forming MCs. Baumgardt, Kroupa Parmentier ( 2008 ) show that the stellar masses of GCs required for self-enrichment by SNe is more than 10 7 M ⊙ , which means that original GC-hosting MC should be very massive (∼10 8 M ⊙ ) for a reasonable SFE (∼0.1). Therefore, typical GCs are unlikely to have [Fe/H] spreads. The most massive Galactic GC ω Cen has been suggested to originate from a nucleated dwarf galaxy (e.g. Freeman 1993

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timing of the first transit. As such, the mean anomaly of 159c was different for each simulation but was not a free parameter. This leaves eight unknown parameters, plus seven for the presumed third planet. This is a 15-dimensional parameter space to search. To further reduce computational requirements, we reduced this to 11 parameters. This reduction was accomplished by presuming 159b was in a circular orbit, using its orbital plane as the reference for the system, and using a constant estimate of its mass. Thus, 159b’s eccentricity, argument of periastron, longitude of the ascending node, and mass are removed from the priors. This is discussed in more detail in Section 3.4.2 . 3.2 TTVFast and MultiNest We used TTVFast (Deck et al. 2014 ), a symplectic numerical integrator, to compute the transit times for each hypothesized set of orbital parameters. These times were converted to TTVs, then compared to the observed TTV data from Holczer et al. ( 2016 ). Another set of parameters would then be chosen by MultiNest, and the values compared again. To search the parameter space efficiently, we used the Bayesian Inference algorithm MultiNest (Feroz et al. 2009 , 2013 ). This iterative process can find solutions far more quickly than searching the entire parameter space. We used the Python interface for MultiNest, PyMultiNest by Buchner et al. ( 2014 ) for this project. Due to the size and complexity of the parameter space, finding the best fits required an iterative process and dozens of sets of runs. Each ’run’ produces a single result from MultiNest, estimated from millions of samples of the parameter space and as many simulations from TTVFast. Repeated runs, even with the same priors, would often return different best-fits. This is indicative of multiple maxima in our parameter space, particularly associated with different resonances between 159d and 159c. It is known that resonances produce strong TTVs and that there can be degeneracy between resonances in this regard (Boué et al. 2012 ). To examine this effect, we performed additional runs with periods of 159d constrained to be near the resonances (periods within 10 days of resonance) of 159c. This narrowing of ranges allowed us to determine the best-fits associated with each resonance independent of the others. We used the default parameters of PyMultiNest with the exception of the evidence tolerance, which we set to 0.15 (from default of 0.5). This reduced tolerance value allows for a more fine-tuned search across the peaks of this rapidly varying landscape. 3.3 Likelihood The Likelihood, L we used is based on the usual χ 2 statistic and is given by: where x i is the calculated TTV, μ i is the observed TTV, and σ i is the error in the observed TTV for point i , respectively (Ivezic et al. 2014 ). The log-likelihood is the value we aim to maximize in the simulation In this case, summation includes transit data from both 159b and 159c. 3.4 Priors The priors used for the search were constrained

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individual column density slices along the filaments (shown in the first three panels of Fig. E.1 ). Although a Plummer may reasonably well describe slice 50 in the north-west filament (and perhaps even the centre-most portion of slice 54 in the north-east filament) the extended emission in the majority of slices renders fitting them with Gaussian or Plummer profiles inappropriate. We have evaluated further the impact of constraining the Gaussian fit parameters on the goodness of the fit itself. While setting the peak position of the Gaussian profiles to lie on the spine (i.e. x = 0) does not improve the produced fits, restricting the fitting to the inner portion of the filaments does result in better Gaussian fits of the central portion of the filament. However, filament properties (width and line mass) derived from such fits are very much dependent on the exact fitting range used. Such properties calculated from Trapezoidal integration of the asymmetrical profiles (as shown in Table 3 and Fig. 14 ) are less affected by such restrictions, and therefore more robust. Characterising filaments is fraught with difficulty, and caution must be observed whilst deriving global properties from their mean profiles (e.g. Panopoulou et al. 2017 ). Fig. E.1 Results of fitting Gaussian (red line) and Plummer (blue dashed line) profiles to individual radial column density slices (orange circles in the first , second and third panels of each row ) and the mean radial column density profile (black circles in the fourth panel of each row) of (a) the north-east filament ( top row ) and (b) the north-west filament ( bottom row ). The grey points in the background of each panel plots all of the radial column density slices simultaneously. Plummer profiles may on occasion well describe the inner regions of the filaments only, whilst Gaussian profiles fail to fit any of the distributions. Gaussian profiles are only found to reasonably fit some of the data once the fitting range is restricted tothe inner portion of the filament. Appendix F Derivation of the analytical expression of the virial ratio Consider a uniform density filament fragmenting at the length scale equal to the separation of a core, λ core . In an idealised scenario, energy is conserved during collapse whilst fragmentation is happening. However, there is a fraction of energy converted from gravitational potential energy ( E g ) into kinetic energy ( E k ) at the time the filament fragmentation starts. This conversion may be expressed in the virial equation as an efficiency ε , where if no energy is converted, ε = 0. If energy conversion does occur, this can be expressed as a fractional increase in E k , expressed as E k = E k ,0 + Δ E k , where E k ,0 is the initial kinetic energy of the fragmenting core before collapse. This fractional increase in kinetic energy can be expressed as , where E g ,0 is the initial gravitational potential energy. Substitution into the standard virial ratio of α vir = 2

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