reduced mass of the collision system. The collisional coefficients for N 2 H + are taken from Lique et al. (2015) , who computed them for the N 2 H + /p-H 2 system. The rates for the DCO + /p-H 2 system were computed by Pagani et al. (2012) , who take into account the hyperfine structure given by the deuterium nucleus. Finally, the collisions of HCO + with p-H 2 were published by Flower (1999) , from which we scaled the coefficients for HC 18 O + . In all cases, the o-H 2 is neglected,which is a reasonable assumption given that the ortho-to-para ratio (OPR) in L1544 is found to be as low as OPR = 10 −3 ( Kong et al. 2015 ). Fig. 4 Profiles of dust temperature (green), gas temperature (red), H 2 volume density (black, in logarithmic scale), and velocity (blue, in units of 0.1 km s −1 ) for the L1544 model developed in Keto et al. (2015) . The velocity in the model is negative (it represents contraction motions), but it is shown here as positive for improved readability. We tested all the 15 models described in Sect. 3.1 . The abundance profiles produced by the chemical model often systematically overestimate or underestimate the observed fluxes, and need to be multiplied by a numerical factor to find a good agreement with the observations. However, MOLLIE requires too much computational time to allow a full sampling of the parameter space. For each molecule, we therefore proceeded as follows: first, we test all the original models, varying the evolutionarystage and the thickness of the surrounding cloud, and we select the one that best reproduces the trend of the different transitions (e.g. the ratio between the peak temperatures of the brightest component of each line). We then multiply the whole profile by a numerical factor, testing approximately five values until the observed fluxes are matched. This is considered the best fit to the observations. We want to highlight that our goal is to obtain a simultaneous fit for all molecules that is roughly consistent with the observations, and a full parameter-space search for the best possible fit within the model uncertainties is beyond the scope of the present paper. In the following text, we focus on the abundance profiles that provide the best-fit solution for each molecule. As an example, wepresent in Sect. 3.2.2 the resulting spectra from several of the 15 abundance profiles for N 2 D + . 3.2.1 Best-fit solution for the observed molecules at the dust peak The best-fit solutions for each molecule are shown in red in Figs. 1 and 2 overlaying the observations. We also include here the analysis of a single-pointing observation of HCO + (1–0) performed at the dust peak (see Sect. 2 for details). Despite the fact that a map for this transition is not available, it is important to check the behaviour of the chemical model for HCO + , the main isotopologue of its family. The observed spectrum of HCO + , together with the best fit from MOLLIE, is shown in the bottom-right panel of Fig. 2 .


between the spectral and temporal index in various regimes (cooling, density regimes, or an injection from the central engine). These relations present tests without delving into details of the models (e.g., Zhang Mészáros 2004 ; Gao et al. 2013 ). Using the conventional notation, F ν ∝ t − α ν − β , we obtained α X , β X for GRB 190829A afterglow in the X-ray bands. In Table 3 , we present the indices of the flux and flux density (@ 10 keV) for the segments of the light curve. We particularly analyze segments III (flare) and IV (break) regions in detail, starting with phase IV. Table 3 The Best-fit Values of Spectral and Temporal Indices for Episodes Marked in Figure 4 Notes. We use a power-law or a smooth-broken-power-law a function to model the XRT light curves for different segments using the python module emcee (Foreman-Mackey et al. 2013 ). All quoted errors on spectral parameters correspond to the 16th and 84th percentiles. a s : smoothness parameter, α 1 : index before break t b , α 2 : index after break (Wang et al. 2018 ), log( L ): log-likelihood, n : number of data points, and k : number of the free parameters in the fit. b Excluding 3 × 10 5 –2 × 10 6 s. The segment in the flux light curve shows a shallow change (0.3 ± 0.3) in the temporal decay index. Contrary to this, in the flux density (@10 keV), α X changes by 1.1 ± 0.2. Naively, one may tend to recognize the break with a jet break, however, upon closer examination the photon index softens during ∼3 × 10 5 –2 × 10 6 s (vertical dashed lines in Figure 4 (b)). This is also reflected in the softening of the hardness ration (vertical dashed lines in Figure 4 (c)). Since after this period, the photon index settles down to its previous value before the spectral change, it suggests a rebrightening within the low-energy Swift-XRT band. The passage of some spectral break frequency is also less likely due to the same reason, because any frequency crossover will cause an irreversible change in the spectral index. Phases IV and V (excluding the region between the black dashed lines) is consistent with a typical decay with α X ∼ 1.22 ± 0.03. The UVOT data in this phase is possibly dominated by the contribution of the host. The observation in the i -band shows the rising part of a supernova (Perley Cockeram 2019c ) contemporaneous with the break t b . Consider adiabatic cooling without energy injection from the central engine. The inferred value of α X (1.31 ± 0.15) from observed β X (1.21 ± 0.1) matches with the observed value (1.22 ± 0.03) within error bars for the spectral regime ν > ν c and the interstellar or Wind mediums (Table 3 ). We estimate the value of the electron distribution index p = 2.42 ± 0.2 from p = 2 β . We calculate the kinetic energy in the jet E K,iso = 1.7 × 10 52 erg according to Equation (17) of Wang et al. ( 2015 ), which is valid for the case ν X > ( ν m , ν c ). We took a prebreak segment from phase IV (0.27–1.9 days) with a mean photon arrival time ∼0.7 day.


); and r is the distance from one specific location in the silicate's interior to its surface. 3. Results 3.1. Boundary Conditions In the standard case, we assumed spherical orthopyroxene minerals with radii of 50 μ m (Figures 2 (a)–(c)) and 500 μ m (Figures 2 (d)–(f)) condense in a region with a heliocentric distance of 1 au from the proto-Sun. The minerals are implanted by the protons in the solar nebula for 1 day, 1 month (30 days), 1 year (365 days), and 10 years (3650 days). For both silicates, the simulations were run with retention coefficients ( N R ) of 0.01, 0.1, and 0.5. Other critical boundary conditions applied for our standard case are listed in Table 1 . Figure 2. Simulated concentration profiles of water in the interior of the silicates. Spherical silicates with radii of 50 μ m (a)–(c) and 500 μ m (d)–(f) are assumed to be implanted with nebular plasma hydrogen for 1 day, 1 month (30 days), 1 year (365 days), and 10 years (3650 days). Three retention coefficients ( N R ), i.e., 0.01, 0.1, and 0.5, are applied for the implantation. With the same retention coefficient (e.g., N R = 0.01), it takes a longer time for 500 μ m sized silicate (d) to evenly distribute the projectile ions than the 50 μ m sized silicate (a). The higher retention coefficient leads to the larger production of water. For instance, implantation with a retention coefficient of 0.5 can produce more than 1000 ppm water in one year in the 50 μ m sized silicate (c), whereas when the retention coefficient is 0.01, the production is less than 200 ppm (b). The gray shaded area indicates the range of previously reported water contents (200–1000 ppm) of the ordinary chondritic minerals, which can be well produced by the implantation–diffusion processes with 0.1 ((b), (c), (e), (f)). Boundary conditions for the simulations are listed in Table 1 . Table 1 Boundary Conditions Applied for the Standard Case References. (1) Wu Judge ( 1979 ), (2) Lewis ( 1972 ), (3) Fromang et al. ( 2002 ), (4) Öberg et al. ( 2011 ), (5) Stalder Skogby ( 2003 ), (6) Dürr Höfer ( 2006 ). We first adopt a high nebular gas pressure ( P = 10 −3 bar) for the chondrule formation (Galy et al. 2000 ). The ambient temperature when chondrule minerals formed was characterized by a range of temperatures, from 600 K to >900 K (Rubin 2000 ). The presence of a primary trolite in type 3 ordinary chondrites indicates an ambient nebular temperature of 650 K (Rubin et al. 1999 ). Accordingly, we select this value as the background temperature for the implantation and diffusion. In previous studies, the ionization fraction ( f ion ) of protoplanetary disks have been reported to be with orders between 10 −12 and 10 −10 (Fromang et al. 2002 ; Öberg et al. 2011 ) and we adopt a median value of 10 −11 . Due to the lack of the parameters applicable to the low-temperature (650 ± 100 K) diffusion of hydrogen in nominally anhydrous silicate minerals, here we use the diffusion coefficient ( D 0 ) and activation energy ()


K are highlighted. The inset summarizes the distribution of all shown planets over temperature. Overlaid are two lines of constant temperature. The 1000 K line corresponds to a change in aggregation observed in our work (see below). 1500 K is approximately the upper limit of the condensation region for typical silicates. The database is still one with low statistics, and detailed dependencies on temperature might be discussed. There is, e.g., a maximum at about 500 K, but the distribution might just as well be considered flat in a wide range. None of these statements are statistically significant at the moment. Nevertheless, there is a clear gap in the occurrence of planets between about 1000 K and 1500 K. For a planet forming at a larger distance from the star, the inward drift would not need to stop at the 1000 K line. Also, from a more general perspective of late planet formation phases there is no obvious reason why planets should not form, e.g., at 1100 K as likely as at 900 K. That leaves the question whether this temperature gap is inherited from the early phases of planet formation by aggregation at elevated temperatures. More often than not, planet formation and its limits are considered in light of radial distance. It might be argued that the lack of planets at higher temperatures might be due to a closer distance and a related lack of material for planet formation. Certainly, temperature and distance are related, but we note that they are not the same. Small changes in stellar mass already lead to large luminosity and significant temperature variations. However, the local material available will not significantly change. Also, it is currently often argued that inner planetary systems are formed from material drifting inward (Boley Ford 2013 ; Wurm et al. 2013 ; Hu et al. 2014 ). Inward drift is a classical mechanism for redistribution of matter (Weidenschilling 1977 ). As a special aspect, e.g., Ebel Alexander ( 2011 ) suggested that Mercury might have formed from dusty matter (chondritic porous interplanetary particles) drifting inward that were reprocessed in a region of higher temperatures with a composition that is then very different from solar composition. Considering such work, one motivation for our experiments is a radial inward drift of a mix of materials (see below). The mass of material available for planet formation does not necessarily depend only on the radial distance to the star. We also note that current planet temperatures and temperatures at the time of dust aggregation might not be the same. They might be comparable if the relevant aggregation occurred in a passive disk. They might have been higher in a still actively accreting disk. However, accretion of dust at these active times might have removed these aggregates again. In any case, we do not claim to have the only answer to planet formation in the inner disk at high temperature, but we think it is curious that there is a temperature limit in the exoplanet


to have natural transport from one planet to the next one in the sequence of bicircular problems, and a particle could drift away from one planet to reach a neighbourhood of the following planet. After that, for transport between neighbourhoods of the libration points of the same Sun–planet problem, it is enough to consider an RTBP (Sun+planet+infinitesimal particle), in which case the existence of heteroclinic connections between libration point orbits around L 1 and L 2 is well known. These connections would allow a particle to continue its journey towards the innermost Solar system. See Fig. 5 . Figure 5. Scheme of the chain of connections between two consecutive restricted bicircular Sun–Jupiter–planet (BR4BP ip ) problems, and two consecutive restricted Sun–planet ( P ip ) problems (RTBP ip ). For the computation of the connections, we proceed as follows. Consider two consecutive bicircular problems BR4BP ip and BR4BP ip + 1 corresponding to the planets ip and ip + 1. We are interested in transits from the outer to the inner Solar system. In all the bicircular problems, OL 1 is an inner orbit than the orbit of the planet with respect to the Sun, and OL 2 is an outer orbit (see Fig. 4 ). Then, the suitable connections are those involving the invariant manifolds of and . It is well known that the invariant manifolds associated with the equilibrium points L 1 and L 2 in the RTBP have two branches: one goes inwards, while the other one goes outwards, at least for times not too big. The invariant manifolds of the dynamical substitutes have the same behaviour. According to this, we denote by the branch of the invariant manifold that goes outwards, and by the branch of the invariant manifold that goes inwards. See Fig. 6 . Figure 6. Projection in configuration space (rotating coordinates) of branches of the invariant manifold (outer corona) and (inner corona). The dynamical substitute is the periodic orbit between the two branches. See the text for further details and also Fig. 8 . Therefore, in order to find connections, in the BR4BP ip , we compute the (linear approximation of the) parametrization of the stable manifold (see equation 7 ) and we follow the branch , and in the BR4BP ip + 1 , we compute the (linear approximation of the) parametrization of the unstable manifold (again see equation 7 ) and we follow the branch . To study if there exist connections, we also fix a section Σ R = {( x , y ); x 2 + y 2 = R 2 }, where R is an intermediate value between the radii of the orbits of the planets of the two bicircular problems, a ip and a ip + 1 , that is a ip R a ip + 1 (see Fig. 7 ). Figure 7. Projection in configuration space (rotating coordinates) of the dynamical substitutes , , one orbit on the invariant manifolds and , and the section Σ R for . The main objectives are: first, to determine if both manifolds reach the section Σ, and, secondly, to study which is the minimum distance between the sets and . That is, we want to see if both


small orbital eccentricities () of the terrestrial planets. Here, we propose that the observed wide gaps in some debris disks may have been cleared by the secular resonances of a known planetary candidate (in Fomalhaut) or yet to be detected (around Vega and other systems) isolated or multiple gas giants or super Neptunes, as the secular resonances sweep across this region during the depletion of the gas disk. Analogous to the asteroids, excited eccentricities of the residual planetesimals in the region between the inner disk and outer ring are mainly damped by the hydrodynamic drag of the diminishing disk gas. In addition to the extended gap, this scenario offers a potentially exciting prospect of finding signatures of ongoing planetary assemblage in debris disks around stars with an age comparable to the formation epoch of terrestrial planets in the solar system (Su et al. 2009 ). Since not a single planet with a mass in excess of (an observationally inferred upper limit) has been found around Vega and Fomalhaut, we explore here the possibility that the extensive gaps are produced by a sweeping secular resonance of a lone-planet with a modest mass and eccentricity in the debris disks around these and other similar systems. Although systems of multiple Neptune-mass planets may also lead to a similar effect, we opt the lone-planet scenario as an idealized simplest possibility. This paper is organized as follows. In Section 2 , we briefly introduce the lone-planet possibility in the Vega system, including detailed physical models to mimic the dust-free gap opening process by the lone-planet scenario in the Vega system, develop a rough method to constrain planet candidates based on observed features, and provide predictions for future planet detection. In Section 3 , we further extend our model to Vega-like system, Fomalhaut, inferring the properties of the prospective planet and compare these with a planet candidate, Fomalhaut b. A summary of our results and an extensive discussion on the other two-belt debris disks are presented in Section 4 . 2. A Lone-planet Scenario for the Vega System 2.1. Dynamical Causes for Wide Dust-free Gaps in Debris Disks Infrared observations indicate that Vega’s debris disk contains a large gap that spans from ∼15 au to ∼110 au (Su et al. 2013 ). The relatively large size of the gap in Vega’s debris disk can be used to constrain the orbital configuration of Vega’s (hypothetical) planetary system. In our solar system, the four gas giants cleared most of the debris between the outer asteroid belt ∼3 au and the inner Kuiper Belt ∼35 au. Similarly, in the young HR 8799 system, (at least) four giant planets have been detected (Marois et al. 2008 , 2010 ) and they are believed to be the chief culprits for the large separation between the warm (∼6–15 au) and cold (∼90–300 au) debris disks (Su et al. 2009 ). Therefore, a multiple-planet configuration is a preferred scenario in the Vega and Vega-like systems, which has a


by larger quiet regions in which B varied relatively slowly, and contained small amplitude waves and turbulence. Thus, the basic structure of the VLISM observed between 2012/DOY 238 and 2020/DOY 182 consists of four jump–ramp structures associated with the four maxima in B between 2012 and 2020, separated by four quiet intervals. The timescales of the jumps that moved past V1 were ∼5.4 days and 3.3 days for the shocks sh1 and sh2 observed on ∼2012.9746 (DOY 335) and 2014.6 (DOY 236), respectively. The corresponding timescales for the jumps associated with pressure fronts were ∼35 days and ∼8 days for the pressure fronts pf1 and pf2 observed on ∼2016/346 and ∼2020/147, respectively. The corresponding timescale for the ramp associated with sh1 was ∼162 days, during which time B decreased exponentially from ∼2012.9746/335 to ∼2013.35759/131. The timescale of the ramp associated with sh2 was 276 days, when B decreased linearly (with notable fluctuations related to fine-scale features) from 2014.68212 to 2015.37669. In the ramp following the pressure front pf1, B declined nearly monotonically until 2018.41885, during an interval of 402 days. We do not yet have V1 observations during the ramp associated with pf2. We have shown that jump–ramp features of B in the VLISM and the larger quiet intervals between them are the fundamental features of the VLISM. These features were actually specially configured waves that were generated by the interaction of heliosheath with the heliopause and then propagated through the VLISM, presumably interacting with one another to produce the results observed by Voyager 1. Smaller scale waves and turbulence exist within these features. Mathematical techniques such as Hilbert transforms and wavelet transforms (Zhao et al. 2020 ) and Fourier methods (Fraternale et al. 2019 ) are very valuable tools for studying the VLISM over a wide range of scales. However, it is also essential to understand the basic structure and dynamics of the large-scale flows on scales of hours to a decade presented in this paper. This can best be done by improving the MHD models and the corresponding input observations. A number of models have been discussed in the literature, including those by Fermo et al. ( 2015 ), Zirnstein et al. ( 2018 ), and Kim et al. ( 2017 ). However, none of these models reproduce the kind of pattern of shocks, pressure pulses, and quiet regions shown in Figure 2 . Appendix New Method for Deriving Plasma Densities The Voyager PWS instrument (Scarf Gurnett 1977 ) includes a wideband waveform receiver that captures digitized waveforms in the range of 50 Hz to above 10 kHz. Waveform captures include 1600 4 bit samples obtained at a rate of 28,800 samples per second. On board, 800 of these waveform captures are obtained at a rate of one every 60 ms covering 48 seconds (one frame). Due to decreasing telemetry performance at the large distances of Voyager, beginning in 1992 November, only one of every five waveform


are computed by integrating the hydrostatic equations with the equation of state given by where p is the pressure, ρ is the mass density, G is the gravitational constant, p b is the pressure at the base of the radiative envelope, and and a is the isothermal sound velocity, and R J is the radius of Jupiter. In the convective core where p ≫ p b , we obtain corresponding to a polytrope of the index n = 1, and hence for Γ 1 ≡ (∂ln p /∂ln ρ) S ≈ 2 the square of the Brunt–Väisälä frequency where On the other hand, in the radiative envelope where p ≪ p b , we have and and c denotes the adiabatic sound speed. Since p /ρ ≈ a 2 ∝ T for an ideal gas, the temperature T is constant for a constant a , suggesting an isothermal atmosphere. In this paper, following Auclair-Desrotour Leconte ( 2018 ), for the Jovian planet model of mass M = 0.7 M J with M J being the Jovian mass, we use p b = 100 bar = 10 8 dyn cm −2 and set the outer boundary R e at p = 0.01dyn cm −2 and define the planet’s radius R = R e /1.01 = 9.31 × 10 9 cm. We also assume Γ 1 = 2 in the interior. Fig. 1 is the propagation diagram where N 2 and with l = 2 are plotted as a function of r / R where L l is the Lamb frequency. Figure 1. Squared characteristic frequencies N 2 (solid lines) and (dashed lines) plotted as a function of r / R (left-hand panel) and of log p (right-hand panel), where N and L l are, respectively, the Brunt–Väisälä frequency and Lamb frequency and δ A = 10 −4 is assumed for N in the convective core. Note that the squared frequencies are normalized by GM / R 3 where M and R are the mass and radius of the planet, G is the gravitational constant and l = 2 is used for L l . The dotted line plots log 10 | N 2 | in the convective core where N 2 0. 2.2 Perturbed basic equations The perturbed basic equations for a uniformly rotating planet may be given by where ρ, p, s, T , and c p are, respectively, the mass density, pressure, specific entropy, temperature, and specific heat at constant pressure, is the energy flux vector, is the displacement vector, α T = −(∂ln ρ/∂ln T ) P , and (′) and δ indicate, respectively, the Eulerian and Lagrangian perturbation of quantities. Note that we have applied the Cowling approximation neglecting the Euler perturbation of the gravitational potential due to self-gravity, and that we have ignored the effects of rotational deformation. The vector denotes the angular velocity vector of rotation and is assumed to be constant for uniformly rotating planets. Here, we have assumed that the perturbations are proportional to the factor e iω t + i m φ , where ω = σ + m Ω with is the oscillation frequency in the co-rotating frame of the planet and σ is the frequency observed in the inertial frame, and m denotes the azimuthal wavenumber around the axis of rotation. Since evolutionary models of irradiated Jovian planets computed with appropriate equations of state and opacities are not available to us, we employ an approximate treatment to estimate the


be directly detected. In this case, T may be adopted as a small value . Our observations require that the maximum γ -ray luminosity at the relevant age T is smaller than the upper limits, i.e.,where , and is the eight-year limit presented in the last row in Table 1 . Given a set of magnetar parameters P 0 and B p , the spin-down luminosity depends on the age of the magnetar, T . Since the earliest burst, FRB 121102, was detected in late 2012, the magnetar needs to be at least four years old until now. We consider two possibilities. First, if the pulsar was born between 2009 January 1 and 2012 November 2, one should use the data during that period of time to constrain magnetar parameters assuming different transparent times by setting in Equation ( 1 ). The transparent time depends on the amount of ejecta and opacity, which may range from minutes (for NS–NS mergers) to ∼ years (for supernovae). In our calculation, we use 10 2 s, 10 4 s, 0.1 years, 0.5 years, and 1 year, respectively. Since the upper limits are slightly different in different time bins, we use the average upper limits to derive the constraints as presented in Figure 2 . Figure 2. Constraints of the pulsar parameter by the LAT upper limit. Solid lines: cases of dotted lines: cases of dashed lines: cases of . The horizontal line denotes the break-up limit P 0 = 0.6 ms. The parameters are constrained to the regions that are above the curves/lines. The second possibility is that the magnetar was born before 2009 January 1. For these cases, the time T should be adopted as the age of the magnetar when the first observation started (in 2009 January 1). When the birth time is much earlier than 2009 January 1, one can use the tighter upper limit over a much longer time (e.g., four-year or eight-year) limit to perform the constraint. For this case, we consider T = 10 years as an example. The results for , 10 4 s, 0.1 years, 0.5 years, 1.0 year, and 10 years are plotted in the plane ( and assumed) as shown in Figure 2 . For each T , we consider , , and . The region above each curve is the parameter space allowed for the given T and η value. Since the minimum spin period of a millisecond pulsar would be the break-up limit ms (e.g., Vink Kuiper 2006 ), our results suggest that there is essentially no constraint from the upper limits if the pulsar age in the case of . This means that the current limit cannot constrain the magnetar parameter if it was born before Fermi observations started. On the other hand, for and if the magnetar age is 4–8 years and the transparent time shorter than 0.1 years, the upper limits in the first four years could place some constraints on B p of the pulsar. In particular, if the transparent time is short enough, the non-detections exclude high magnetic field strengths. As a result, if FRB 121102 progenitor was indeed born after 2009 January 1 and has a magnetar origin, it must have been born in a dirty environment with a delayed transparent time (say years) in


magnetic fields across the solar surface (Mackay Yeates 2012 ). Although synoptic maps of the stellar vector field exist for a couple of decades and have been derived for hundreds of systems (e.g. Donati Collier Cameron 1997 ; Donati et al. 1999 , 2006b ; Donati Landstreet 2009 ), there are significantly fewer synoptic maps of the vector field of the Sun. Earlier synoptic maps of the vector solar field were created on the basis of the rotational modulation of LOS magnetograms (Pevtsov Latushko 2000 ; Wang Zhang 2010 , also called ‘pseudo-vector reconstruction method’). With this method, these authors assumed that the field on large spatial scales does not change over several days. Because of this assumption, these early synoptic maps reconstructed only large-scale magnetic fields of the Sun. More recently, Gosain et al. ( 2013 ) used daily full-disc vector magnetograms from the SOLIS/Vector Spectromagnetograph (VSM) spectrograph to reconstruct the vector solar field of a series of 23 Carrington Rotations. This is a significant advance compared to the large majority of synoptic charts of the solar magnetic field, which are restricted to the LOS component of the solar magnetic field, used to derive B r . An important conclusion reached by Gosain et al. ( 2013 ) is that the solar magnetic field is not normal (i.e. purely radial) to the surface at the photospheric level. This is somewhat disturbing, as the assumption that the solar magnetic fields are purely radial are used in several studies. With the new existing capabilities in producing vector synoptic fields, we can now start to investigate the behaviour of the meridional B θ and azimuthal B φ components of the solar magnetic field and investigate how the three magnetic field components in the Sun compare to the ones derived in stellar observations (e.g. Fig. 1 ). However, direct comparison between stellar and solar magnetic field topologies are hampered by their dramatic differences in resolutions. To provide a direct comparison between solar and stellar magnetic field synoptic maps, it is first required to ‘filter out’ the small-scale field (i.e. the finely resolved magnetic features) of the solar observations, since this component is currently not accessible to ZDI studies. After this, we are left with only the solar large-scale field, whose spatial scale is comparable to those achieved by stellar observations. One way to filter out the small-scale structure of the solar field is to express the solar magnetic field as spherical harmonics of different l degrees, up to a maximum degree l max . The smallest l degrees represent the largest-scale components, e.g. l = 1 for the dipole, l = 2 for the quadrupole, l = 3 for the octupole and so on. By retaining only the components of low l degrees in the spherical harmonics decomposition, we are then able to calculate the large-scale component of the magnetic map. This approach has been extensively applied to the LOS and radial components of the


forest algorithm (TPC), while the probability estimates of our ConvNet model appear to be better calibrated. When the same network architecture is applied to the SDSS data set, the network fails to outperform TPC, but the probabilities are still slightly better calibrated. The major advantage of ConvNets is that useful features are learned automatically from images, while traditional machine learning algorithms require feature engineering as a separate process to produce accurate classifications. ConvNets have recently achieved record-breaking results in many image classification tasks (LeCun et al. 2015 ) and have been quickly and widely adopted by the computer vision community. One of the main reasons for the success is that ConvNets are general-purpose algorithms that are applicable to a variety of problems without the need for designing a feature extractor. The lack of requirement for feature extraction is a huge advantage, e.g. when the task is to classify 1000 classes in the ImageNet data set (Russakovsky et al. 2015 ), as a good feature extractor for identifying images of cats would be of little use for classifying sailboats, and it is impractical to design a separate feature extractor for each class. However, when there already exists a good feature extractor for the problem at hand, e.g. the concentration parameter, the weight-averaged spread_model parameter from DES (Desai et al. 2012 ; Crocce et al. 2016 ), or even the SE xtractor software, conventional machine learning algorithms that have been shown to be effective, such as TPC (Carrasco Kind Brunner 2013 ; Kim et al. 2015 ), remain a viable option. As the ‘no free lunch’ theorem (Wolpert 1996 ) states, there is no model that works for every problem. For the CFHTLenS data set, our ConvNet model outperforms TPC. Since the SDSS catalogue provides the concentration parameter that is highly optimized for star–galaxy classification, TPC works better for SDSS. Although we used various techniques to combat overfitting, it is possible that our ConvNet model has overfit the data. Overfitting could explain why our ConvNet model with maximal information fails to significantly outperform a standard machine learning algorithm that uses the reduced summary information from catalogues. The most effective way to prevent overfitting would be to simply collect more training images with spectroscopic follow-up, as the performance of ConvNets generally improves with more training data. However, spectroscopic observations are expensive and time-consuming, and it is unclear if sufficient training data will be available in future photometric surveys. If enough training data become available in DES or LSST, ConvNets become an attractive option because it can be applied immediately on reduced, calibrated images to produce well-calibrated posterior probabilities. We also note that using more training images will require the use of multi-GPU systems, which was beyond the scope of this work. Deep learning


model are included here for completeness. We employ a predictor–corrector integration method based on the description in Stone Gardiner ( 2009 ), along with a second-order spatial reconstruction technique with limiting applied in the characteristic variables (LeVeque 2002 ) and the HLLC Riemann solver (Toro et al. 1994 ; Batten et al. 2006 ). Radiative cooling is included via operator splitting, with the thermal energy losses accounted for after the hydrodynamic update. The cooling rates are determined using a forward Euler integration, and subcycling of the radiative cooling time step is employed to ensure a numerically robust solution. The cooling function is a piecewise parabolic fit to a solar-metallicity, collisional ionization equilibrium curve calculated using Cloudy (Ferland et al. 2013 ). Cooling is only included for gas above 10 4 K; below this cutoff a temperature floor is assumed for all the gas in the simulation. The cutoff is primarily employed to prevent collapse of the isothermal gas disk, but we note that the rarefied gas in the outflow will be subject to photoionization heating from the cosmic UV background and therefore would not be expected to cool much below 10 4 K (see Schneider Robertson 2017 ). Photoionization heating is not explicitly included in these simulations. All of the gas in the simulations is assumed to be ionized, with an adiabatic index γ = 5/3 and a mean molecular weight μ = 0.6 used in all calculations and analysis. 3. Results We begin our results by presenting a phenomenological description of the three high-resolution simulations: the central feedback model without cooling (Model A), the central feedback model with cooling (Model B), and the clustered feedback model with cooling (Model C). As explained in Section 2 , the only difference between Models A and B is the inclusion of radiative cooling in the simulation. Model A therefore represents a 3D numerical test of the original CC85 analytic model with the inclusion of a galactic disk. Similarly, Model B can be directly related to the Thompson et al. ( 2016 ) outflow model that accounts for radiative energy losses. A comparison between Figures 1 and 2 shows the dramatic effect that radiative cooling can have on the overall character of a mass-loaded outflow. In these figures, we plot density, radial velocity, and temperature slices in the centered x − z plane. Some features are common to both simulations—namely, the 10 4 K disk and the spherical gain region at the center where mass and energy are injected. In Model A, a biconical free-wind region is clearly visible in both the density and temperature slices, surrounded by a region of more turbulent, higher-temperature gas where the outflow is interacting with the disk gas. In Model B, turbulence at the disk–outflow interface is highly suppressed, leading to a free-wind region with opening angle nearly ΔΩ = 4 π . Although energy is injected into the hot wind at the same rate in Models A and B, the


the TRES spectra as described by Buchhave et al. ( 2010 ). We derived the RVs by cross correlating each spectrum order by order against the observed spectrum with the highest S/N in the wavelength range of 4520–6280 Å. EPIC 202900527 11 TRES RVs are listed in Table 1 , and the phase-folded RV curve is shown in Figure 2 . The reference (or template) spectrum is at BJD = 2457854.95084 and its RV is listed as 0.0 km s −1 in Table 1 . To allow putting the TRES RVs on an absolute scale, we determined the template spectrum absolute RV by cross correlating it with a synthetic spectrum to be −57.87 ± 0.10 km s −1 . This RV offset is not added to the RVs in Table 1 to avoid inflating their uncertainties. 3. Data Analysis and Results 3.1. Stellar Spectroscopic Parameters We derived the spectroscopic stellar parameters using the SpecMatch package (Petigura 2015 ; Petigura et al. 2017 ) and the iodine-free HIRES spectra of each star. Those include the effective temperature , surface gravity , metallicity [Fe/H], and stellar rotation projected on the line of sight , where V is the equatorial rotation and I is the stellar rotation inclination angle. We averaged the parameters extracted from all of the individual observations. The observation-to-observation parameter variance was smaller than the quoted uncertainties in all cases. The SpecMatch results are listed in Table 2 . Table 2 Fitted and Derived Parameters Notes. a Derived using SpecMatch analysis of the spectra. b Model fit parameters, fitted to the K2 light curve, RVs, and stellar isochrones. Gaussian priors are applied to [Fe/H] using the values derived from the SpecMatch spectroscopic analysis. See Section 3 for more information. c For EPIC 206155547 and EPIC 206432863 γ is the binary system’s center of mass RV since the HIRES RVs are on an absolute scale. For EPIC 202900527 the RV of the template spectrum (−57.87 ± 0.10 km s −1 ; see Section 2.3 ) needs to be added to γ to get the center of mass RV. d Parameter fixed during the model fitting process. e upper limit given when no eclipse was detected. f The transit, or primary eclipse, is at phase zero. 3.2. Global Model Fitting We performed a global modeling of the available photometric and RV measurements, along with the spectroscopic atmospheric properties of the primary star, to derive the parameters for each system. For stellar binaries, the transit-derived orbital semimajor axis normalized by the primary stellar radius is dependent on the sum of the two components’ masses and the volume of the primary star, as per Sozzetti et al. ( 2007 ):where P is the orbital period and G the gravitational constant. To take advantage of this relation, we fit directly for the masses of the two stars, the primary radius, and the secondary-to-primary radii ratio , as well as the standard transit and RV orbital parameters including the orbital period P , mid-transit time T 0 , line-of-sight orbital inclination i , orbital eccentricity parameters and


observed Galactic clouds the full spectrum of PRS shapes found in our simulations is recovered: PRS curves which show (i) local minima, (ii) an increase towards the highest N , (iii) and random or perpendicular orientation over the entire column density range. In the context of the results presented here, the clouds in these observations might still have a preferentially perpendicular orientation of the magnetic field and density structures in 3D, though not observed in 2D due to projection effects. This could in particular explain observations of Soler et al. ( 2017 ) and Soler ( 2019 ), which report significant variations in the PRS when considering different subregions of the same molecular cloud complex. Finally, we note that radiative transfer effects and imperfect dust alignment are unlikely to contribute significantly to these projection effects: As we have shown in Seifried et al. ( 2019 ), the observed magnetic field traces well the mass-weighted, LOS-integrated magnetic field. To further support this, we also calculated the PRS using this mass-weighted, LOS-integrated magnetic field instead of that inferred from the polarization maps. We find that these PRS do not differ significantly from those shown in Fig. 7 . 5.2 A critical magnetic field strength Overall, our findings of a change from a parallel to a perpendicular orientation of magnetic fields with respect to dense structures with increasing field strength is in good agreement with previous theoretical works (e.g. Heitsch et al. 2001a ; Ostriker et al. 2001 ; Li Nakamura 2004 ; Nakamura Li 2008 ; Collins et al. 2011 ; Hennebelle 2013 ; Soler et al. 2013 ; Chen Ostriker 2015 ; Li et al. 2015 ; Chen et al. 2016 ; Zamora-Avilés et al. 2017 ; Mocz Burkhart 2018 ; Chen et al. 2020 ). As pointed out by Soler Hennebelle ( 2017 ), this change of relative orientation is an indicator of compressive motions, i.e. ∇ v 0, coupled with a dynamically important magnetic field. The compressive motions could be created either by converging flows or gravitational collapse. As indicated in the Figs 1 and 3 , a stronger magnetic field results in (i) more guided motions towards the central collision interface, (ii) suppressing turbulent motions perpendicular to it, (iii) a stronger gravitation collapse (see also Heitsch et al. 2009 ; Zamora-Avilés et al. 2018 ; Iwasaki et al. 2019 ), and (iv) consequently a more pronounced perpendicular relative orientation between magnetic fields and the density structures, in good agreement with other theoretical works (Soler et al. 2013 ; Chen et al. 2016 ; Soler Hennebelle 2017 ; and Girichidis et al. (submitted)). For the CF runs, we find a critical field strength of ∼ 5 μ G above which we observe a flip in field orientation. The SILCC-Zoom simulations have an initial magnetic field strength of 3 μ G, which is close to this critical value. As demonstrated in Section 4.2 , the observed trend of ζ in the SILCC-Zoom runs supports the idea of a critical field


only increases the number count by ∼0.01. There is a dependence of number count on the cadence, and it can be used to set the actual cadence. Although the cadence of the main survey is already fixed at 27d on average, number counts in the ADS increase until the cadence reaches a few times 10 d and are more or less flat thereafter. Consequently, number counts in the ADS are not sensitive to cadence above 1–2 months. Also, we note that with just two pointings one year apart in DD fields, our number counts could be contaminated by false positives due to AGNs. However, these objects can be disentangled from high- z SNe to some degree because their luminosities are typically higher than those of SNe and fluctuate stochastically by about an order of magnitude over the time-scales that SN LCs evolve. In contrast, SNe brighten and then fade by a larger factor. 3.4.3 Scattering/absorption We treat attenuation of flux due to Ly α scattering with the model of Madau ( 1995 ) at z 6 and with Gunn Peterson ( 1965 ) for z > 6. The errors in Madau ( 1995 ) can be large but λ Ly α is only redshifted through the i band and of the z band at z 6. The large uncertainties in Madau ( 1995 ) therefore have little impact on our results because of the low numbers of counts in the i band and because they only affect part of the z band. High-redshift SNe might be obscured by dust at lower galactic altitudes. However, this will likely have little effect on the number of detections since this region of the sky is not covered by the WFD. There may also be extinction in the host galaxies. This extinction depends on the metallicity. Pop III galaxies are believed to be free of dust, so they would not obscure explosions (Rydberg et al. 2017 ). Dust in the host galaxies of Pop II SNe may dim them but the extinction is expected to be weak because they are being observed in rest-frame UV. At z ∼ 5–7 the host galaxies also likely have low metallicities and less dust. 3.4.4 Lensing model For the systemic errors due to the lensing models, we use the factor two, as mentioned in Section 2.4 , as an to denote the possible errors in the predicted numbers of SNe. A source of potentially large uncertainty in our lensing models is the fraction of the sky with 1 μ 2 (Section 2.4 ). The interpolation we use yields a rather severe lower limit because at μ = 1 it returns a lensed area that is just a few per cent of the sky. As a simple test of how robust our counts are with respect to this interpolation we have done it with a second-degree polynomial in the log 10 –log 10 plane as well. As constraints we use f (μ = 1) = 1 and f (μ = 2) = f 2 , where f(μ) is the fraction of the sky magnified by at least μ. The slope of the interpolation between μ = 2 and μ = 3 is set equal to the derivative of the polynomial. None of the counts in the WFD change because detections there always require magnifications greater than two, as seen in Table 1 . In the ADS the use of this interpolation adds on average


2012 ). Numerical simulations suggest that this sample is prevalently composed of relaxed clusters (∼70%) and largely free of orientation bias (Meneghetti et al. 2014 ). An important effect of the CLASH selection function is to significantly reduce the scatter in concentration because of their X-ray regularity (Section 6.2 ). For a lensing-unbiased subsample of 16 CLASH X-ray-selected clusters, we have examined the mean c – M relation and its intrinsic scatter using Bayesian regression methods (Section 6 ; Figures 6 and 7 ). Our analysis takes into account the correlation between the errors on mass and concentration and the effects of nonuniformity of the intrinsic mass probability distribution. Our model yields a mean concentration of at and an intrinsic scatter of , or . The normalization, slope, and scatter of the observed c – M relation are all consistent with ΛCDM predictions (Meneghetti et al. 2014 ) when the projection effects and the CLASH selection function based on X-ray morphological regularity are taken into account. Our regression results are in agreement with the SaWLens analysis of Merten et al. ( 2015 ) and the stacked shear-only analysis of Umetsu et al. ( 2014 ). This multi-probe approach for cluster lensing is one of the key advantages of the CLASH survey, providing consistency checks between different lensing methods and different data sets. We have derived an ensemble-averaged surface mass density profile at an average redshift of for the X-ray-selected subsample of 16 clusters by stacking their individual profiles (Section 5 ; Figure 3 ). The stacked lensing signal is detected at 33 σ significance over the entire radial range, R ≤ 4000 kpc h −1 , accounting for the effects of intrinsic profile variations and uncorrelated LSS along the line of sight (Figure 1 ). Our CLASH lensing determination of the cluster mass distribution provides a firm basis for a detailed comparison with theoretical models. We show that the profile is well described by a family of density profiles predicted for DM-dominated halos in gravitational equilibrium (Table 4 ; Figure 4 ), namely, the NFW, Einasto, and DARKexp models. Of these, the first two are phenomenological models and the last is a theoretically derived model (Hjorth Williams 2010 ; Williams Hjorth 2010 ). The single power-law, cored isothermal and Burkert density profiles are disfavored by the observed mass profile having a pronounced radial curvature. We find that cuspy halo models that include the large-scale two-halo contribution using the b h – M relation of Tinker et al. ( 2010 ) provide improved agreement with the data. Independent of the chosen halo density profile, we find b h ( M 200c ) ∼ 9.3 (). For the NFW halo model (NFW+LSS (i)), we measure a mean concentration of at , demonstrating consistency between complementary analysis methods (Figure 7 ). This model yields an Einstein radius of arcsec at z s = 2, which agrees within 2 σ with the observed median Einstein radius of


(electron + proton) fluid, we find that the post-shock proton temperature in the direction perpendicular to the field, which we take as the initial condition in our periodic boxes, will beThe parallel proton and electron temperatures are the same as in the pre-shock region; in particular, With regard to the initialization of electrons, we assume that the electron anisotropy is rapidly erased due to the rapid development of electron-scale instabilities, as it indeed occurs in the shock ramp. 8 8 Despite the fact that we now assume isotropic electrons, the ansatz of a Γ = 2 gas that we employed to calculate the jump conditions will still hold because the post-shock pressure is mostly contributed by protons (see Section 5 ). If this happens without any energy exchange with the protons, the isotropic electron temperature isFinally, the initial electron plasma beta can be computed from flux freezing as The physical and numerical parameters used for the periodic box experiments of this section are summarized in Table 2 (the run names have the suffix “nc” to indicate that the boxes are not compressing). We indicate the Mach number M s and plasma beta β p 0 of the corresponding shock simulations, the initialization values of , , , and employed in the periodic boxes, the number of computational particles per cell (), and the box length L y along the y -direction of the background large-scale field, in units of the proton skin depth . We employ a reduced mass ratio . From Table 2 , it is apparent that the Mach number M s has a pronounced effect on the initial proton anisotropy. In fact, in the limit of , we have and , so that from Equations ( 12 ) and ( 13 ). Table 2 Parameters for the Anisotropic Protons Box Experiments Described in Section 4 Note. We employ a fixed mass ratio and resolve the electron skin depth with seven cells, which is sufficient to capture the electron debye length. 4.2. Dependence on Ms In this subsection, we compare the periodic box simulations , , , , that correspond to shocks with fixed but different M s , ranging from 2 to 5. Our results are shown in Figure 5 . The case corresponding to M s = 2 is plotted with a dashed line to emphasize that its time axis has been reduced by a factor of two, to facilitate comparison with the other cases. Figure 5. Dependence on M s of various quantities in periodic box experiments with anisotropic protons (runs , , , ), at fixed . As a function of time in units of , we plot: (a) energy in magnetic field fluctuations, normalized to the energy of the initial field in the box (b) proton temperature anisotropy; (c) magnetic pressure in electron-scale fluctuations, normalized to the electron parallel pressure; (d) electron temperature anisotropy (solid lines for the box-averaged values, shaded regions for the 50–90 percentiles) and threshold of the electron whistler instability (dotted lines with the same color coding as the solid lines); (e) rate of violation of the electron


the observed CMD of core He-burning stars. Figure 7. CMDs of Hodge 6 and NGC 1978, the two youngest clusters in our sample. Observations are denoted with grey open triangles, synthetic HB models are overplotted in red. From left to right: Δ Y = 0 models, models with Δ Y > 0 (η R fixed to the best-fitting value used in the left-hand panel), and with Δη R > 0 ( Y fixed to the He-normal value of the left-hand panel). The arrows describe the direction along which variations in Y and mass-loss work, the amplitude being arbitrary here. If we make the assumption that η R must be constant, in agreement with the results for the other clusters in our sample, we would obtain Δ Y ∼0.06 for Hodge 6, and Δ Y ∼0.04 for NGC 1978. But without this assumption, the CMD analysis does not discriminate between a spread in η R or in Y for these two clusters. However, we also note that Hodge 6 has the largest photometric errors of any of the clusters in our sample, adding further uncertainty for this cluster. 5 DISCUSSION We have determined the total initial He abundance spread Δ Y in a sample of intermediate-age, massive LMC and SMC clusters – and the old cluster NGC 121 – by reproducing the shape and colour extension of their HB/RC stars with synthetic HB models. Our derived Δ Y values are shown in Table 2 . The typical error on these estimates of Δ Y is below 0.01. Table 2. Initial helium abundance spread derived from the HB fitting. The values in parenthesis are determined in the assumption that the RGB mass-loss does not vary amongst clusters’ stars, because of a degeneracy between the effects of mass-loss spread and He spread in these clusters (see the text for details). We can compare our results with the estimates by Lagioia et al. ( 2019 ). These authors found spreads of initial He abundances equal to 0.009 ± 0.006, 0.007 ± 0.004, 0.010 ± 0.003, and 0.000 ± 0.004 for NGC 121, NGC 339, NGC 416, and Lindsay 1, respectively. These values are clearly smaller than our results in Table 2 . But as mentioned already in the Introduction section, the method employed by Lagioia et al. ( 2019 ) most likely determines mean abundance spreads amongst the cluster subpopulations, whereas our modelling tends to determine the maximum abundance spread, irrespective of the exact distribution of initial He abundances. This is quite clear by looking at the HB of NGC 121 in Fig. 4 . The bulk of the HB population has ( m F555 W − m F814 W ) > 0.8, consistent with a negligible Δ Y with just a plume of stars extending towards bluer colours and brighter magnitudes, that is, with significantly different initial Y . The values in Table 2 are also plotted in Fig. 8 as a function of the mass of the host cluster. In the same figure, we display also the maximum initial Y spread determined for a sample of Galactic GCs by Milone et al. ( 2018 ). Milone et al. ( 2018 ) found a trend between Δ Y and the mass of the host cluster, that is clearly visible in Fig. 8 and the results for our clusters


field lines and I B / I T = O (10 −6 ) for the closed field line. Field line integrals reveal the internal structure of magnetic flux tubes. By inspecting the order of magnitude of the ratio I B / I T within the flux tube, it is possible to (i) explore the internal structure of the flux tube and (ii) verify that the field-line-averaged magnetic tension inside the tube is small. Under condition (ii), the angular velocity of the fluid enclosed in the flux tube is determined by the boundary conditions on the surface of the flux tube, which are set by the surrounding fluid. The matching to the flux tube surface occurs via viscous forces on a time-scale longer than t A . 7.3 Non-axisymmetry Non-axisymmetry arises naturally even in unmagnetized spherical Couette flow (e.g. see the phase plane in fig. 1 in Nakabayashi et al. 2002a , among many other results), and in superfluid spherical Couette flow (Melatos Peralta 2007 ). In our paper, we restrict the simulations to relatively low Reynolds number ( Re 10 3 ) and to parallel rotation and magnetic axes. We do this partly for simplicity in response to computational constraints and partly because our main application (neutron stars) is arguably a low-Reynolds-number problem with 10 −10 ≤ ΔΩ/Ω ≤ 10 −5 from measurements of rotational glitches (Espinoza et al. 2011 ). However, the low- Re assumption is certainly debatable. One can also argue that the problem is high-Re because the poorly known effective viscosity of the neutron superfluid may be lower than expected, and glitches may only relax a small fraction of the underlying shear. In that case, the flow may be fully turbulent (Melatos Peralta 2007 ), and a different style of study would be needed. Assuming low Re and parallel axes, the solution is axisymmetric, and the problem is effectively bidimensional. To test this assertion, we have performed three-dimensional runs, and verified that the solution is independent of the azimuthal coordinate φ . We have performed test runs with 180 points in the radial direction, 150 points in the θ direction, and 20 points in the φ direction for the configuration ΔΩ = 0.1 and B 0 = 1.5 × 10 −2 . Both the radial velocity and the azimuthal velocity (not plotted here) are independent of the φ variable at r = 1.04 and θ = 0.61, 0.81, and 1.21 rad. So are all the other dependent variables, e.g. magnetic field components. The system starts axisymmetric and remains so as it evolves. There are two astrophysically realistic ways to drive non-axisymmetry in the flow pattern and magnetic geometry. One way would be to tilt the magnetic axis with respect to the rotational axis, as implied by the radio polarization swings observed in pulsars. The numerical complexity of the problem depends strongly on B 0 , Re , the tilt angle α , and the grid resolution in θ and φ required to resolve turbulent meridional flows and non-axisymmetries. We perform some three-dimensional test runs with resolution 202 × 176 × 50 (in r, θ , and φ ,


of the former ones at higher redshifts. Wang et al. ( 2019 ) reported, for their sample of H-dropout-selected dusty galaxies, a space density (∼2 × 10 −5 Mpc −3 at z > 3) similar to that of our high-redshift UV-dark galaxies, in the same redshift range. However, comparing the two populations, we find that our radio-selected galaxies produce stars at a rate that is on average three times higher. In fact, the median L IR of our galaxies at z > 3 is (6.3 ± 0.5) × 10 12 , which is about three times higher than the median infrared luminosity of (2.2 ± 0.4) × 10 12 reported by Wang et al. for their sample. This means that their contribution to the cosmic SFRD is also higher. In particular, it reaches 7.0 × 10 −3 M ⊙ yr −1 Mpc −3 in the redshift range 3.0 z 4.5, and 5.4 × 10 −3 M ⊙ yr −1 Mpc −3 at 4.5 z 7.7, with uncertainties of the order of 20% (Figure 4 and Table 3 ), therefore doubling the contribution of the population of H-dropouts previously cited at the same cosmic epoch (Wang et al. 2019 ). A similar result was reported for a sample of HST-dark galaxies identified at submillimeter wavelengths in the ALPINE fields Gruppioni et al. ( 2020 ). We compared our own selection criteria also with those by Wang et al. ( 2019 ), again to estimate the possible overlap between the two samples (see the Appendix ), and we concluded that at least ∼82% of the UV-dark radio-selected galaxies are not consistent with the H-dropout selection by Wang et al. ( 2019 ) and therefore constitute a different galaxy population. It is worth noting that we find a similar percentage (83%) when focusing only on the common redshift range between the two samples (z > 3). We compared the trend toward high redshift of the cosmic SFRD of our radio-selected UV-dark galaxies with the estimate based on galaxies identified in the rest-frame UV regime (Madau Dickinson 2014 ; Bouwens et al. 2020 ). 15 15 Bouwens et al. ( 2020 ) include the SFRD contribution from ULIRG-type galaxies in the ASPECS volume, derived from various submillimeter surveys. At 3.0 z 4.5, the contribution of UV-dark radio galaxies to the SFRD corresponds to ∼10-25% of the SFRD of UV-bright galaxies. This fraction rises to ∼25%–40% at z > 4.5, where we identified 22 very-high- z candidates, whose SEDs are illustrated in Figure 5 along with their redshift likelihood distributions. We stress that the actual contribution of radio-selected UV-dark galaxies to the cosmic SFRD could be even higher, because our estimates are derived without any extrapolation to lower fluxes (hence SFRs) than our selection, which would have been extremely uncertain. In fact, the correction for incompleteness, which we apply only takes into account how representative the primary sample is with respect to our total sample of radio-selected UV-dark galaxies. Figure 5. Observed (points as in Figure 1 ) and best-fit (black line) SEDs of the highest-redshift galaxies from our primary subsample (z ≥ 4.5). Error bars are plotted for each data


the Ω 0 increases, the propensity for convection (colored red) within the postshock region clearly decreases. This inhibited convection results in weakened turbulent mass motion within the gain region, thereby reducing the GW amplitude at later times. Figure 6. Spherically averaged anisotropic velocity of the postshock region for the 12 M ⊙ progenitor. Brighter colors correspond to increased convection in the postshock region according to Equation ( 5 ). As rotational velocity increases, convective activity is inhibited. Traced in red is the radius of the PNS. Figure 7. Slices along the equator of the 12 M ⊙ progenitor at each rotational velocity. Colors correspond to the Solberg–Høiland stability criterion, R SH , from Equation ( 6 ). As rotational velocity increases, not only does the convectively stable band in the core grow (seen in blue), but the amount of convection within the postshock region (seen in red) decreases as well. The differences in shock radius evolution between Figure 6 and this figure arise because Figure 6 uses an angular average over the domain, whereas this figure uses equatorial slices. Furthermore, we recast our analysis by focusing on regions within the CCSN that emit GWs. The lower panel of Figure 8 displays the inhibited convective signal with increasing rotation, as the GW signal in the gain region becomes increasingly muted. The typical convective signals in the early postbounce regime are then quickly washed out by the postbounce ring-down of the PNS, as rotation increases. Figure 8. Time domain waveforms for the 20 M ⊙ progenitor. Each panel corresponds to the region from which the GWs are emitted. The large contribution in the top panel indicates the main source of GWs during the accretion phase is from the vibrating PNS. The lower panel displays the inhibited convective signal ∼50–100 ms postbounce that is characteristic of this quiescent phase. Under nonrotating conditions, the shock can grow unstable due to nonradial deformations exciting a vortical-acoustic cycle that leads to the growth of large-scale shock asymmetries, that is, the SASI (Blondin et al. 2003 ; Blondin Mezzacappa 2006 ; Scheck et al. 2008 ; Marek Janka 2009 ). In 2D simulations, the SASI excites large, oscillatory flows along both poles that drive changes in entropy capable of causing postshock convection. It is worth noting in 3D simulations that the SASI can excite “spiral” modes that correspond to nonzero m values (Blondin Shaw 2007 ; Kuroda et al. 2016 ). The high degree of nonlinearity among the hydrodynamic flows, neutrino interactions, and gravitational effects can yield matter flow that is quadrupolar, thereby resulting in GW emission. However, when the shock becomes restricted in the polar direction, due to centrifugal effects, SASI development is inhibited. To quantify the role of SASI, we decompose the shock front into coefficients based on the spherical harmonics, Y l m , according to Equation ( 9 ). Figure 9 illustrates the


1997, would not be significantly contaminated by magnetic activity during a minimum. While active region inflows were not discovered at this time, the authors were still aware that magnetic activity would affect the results. Nevertheless, during this observation window, cycle 23 was ascending and the monthly sunspot number was growing from 10 to 45 (see Fig. 1 ). We investigated a similar period in cycle 24 (2009, one year after Fig. 7 ) and found inflows into active regions that persisted for a number of months. These inflows are at a weaker level than in Fig. 6 and, after three rotations, have a spatial correlation coefficient of 0.25. The EW component of these inflows is what is observed in the correlation functions of Beck et al. (1998) . We note that there is a very small window (one year centered at the minimum) in which inflows do not dominate the low-latitude flow field. Interestingly, the studies of Labonte et al. (1981) and Snodgrass Howard (1984) did not report on these long-time correlations, despite the data covering cycles 20 and 21 and the supposed sensitivity of ∼1 m s −1 per wavenumber. The reported observations from Beck et al. (1998) are consistent with ours, with the exception that they incorrectly identified the large velocity cells as convective in origin. Specifically, the EW correlations have an e-folding lifetime of approximately four months, and the extent of the flows is greater in the longitude direction. Beck et al. (1998) reported EW flows elongated in longitude with an aspect ratio greater than four. We also find EW flows elongated in longitudes, however with an aspect ratio around two (Fig. 6 ). We attribute this difference to the averaging scheme. We were careful to avoid selecting large converging flow features multiple times to avoid smearing, thus our aspect ratio is smaller. The elongation of the spatial correlation of EW flows in the longitudinal direction is not surprising given the geometry of active regions. In an argument against the convective nature of these cells, Ulrich (2001) suggested that the long-lived signals in the EW flows might be due to Rossby or inertial waves. The solar equatorial Rossby waves, discovered and subsequently characterized, are not seen in the EW flow maps at low latitudes and thus cannot cause the signals originally observed by Beck et al. (1998) and Ulrich (2001) . We find that the active region inflows generate a flow pattern with azimuthal modes up to m = 9, which is consistent with the spectral analysis of Ulrich (2001) . Ulrich (2001) found that while the rotational frequencies of these modes were consistent with magnetic features, a couple modes were slower. However, in our study, we find that, at each latitude, all modes have frequencies consistent with the rotational frequency of magnetic features and that they are absent during the solar minimum, thus indicating a magnetic association for these signals. Aware of the potential affect in Doppler images, Ulrich


correlations between the measurements when using a common analysis aperture, the intrinsic scatter between X-ray and lensing masses, and the cosmology dependence of both. For a fixed cosmology and a sample of 12 clusters, we measure a lensing-to-X-ray mass ratio of r = 0.96 within the Chandra -determined r 2500 aperture with a statistical precision of 9 per cent and additional systematic uncertainty from lensing of 9 per cent, for a combined precision of ≈12 per cent. These results are for the specific Chandra calibrations tested, and cannot be easily extrapolated to other versions of the Chandra calibration. We find that our results are robust to perturbations in the sample selection and lensing analysis. We interpret the measured lensing-to-X-ray mass ratio as ruling out large departures from hydrostatic equilibrium, on the order of tens of percent, at the measurement radii in our relaxed target clusters. Since the effects of biases in the X-ray effective area calibration are degenerate with biases due to non-thermal pressure support, more robust constraints on the level of non-thermal support from lensing and X-ray data will not be possible without an independent constraint on the X-ray calibration, e.g. from precisely measured emission line ratios. By measuring the lensing-to-X-ray mass ratio within the cosmological framework of Paper II , we are able to render the results of that work independent of overall shifts in the Chandra temperature calibration. We are also able to naturally capture the cosmology dependence of the lensing-to-X-ray mass ratio by simultaneously modelling the lensing signal for the 12 clusters in this sample with the rest of the f gas data set. The improvements to cosmological constraints that the lensing enables are significant, as discussed in detail in Paper II . There are clear opportunities for improvement of the lensing-based calibration for the f gas experiment (as well as cluster counts; see discussion in Mantz et al. 2015a ) in the next few years. Efforts are under way to improve key lensing systematics, most notably with measuring shear (Kitching et al. 2011 ) and with mass modelling of clusters (LSST Dark Energy Science Collaboration 2012 ). With these improvements coming into place, the remaining limitation to an improved measurement of the lensing to X-ray mass ratio is the number of clusters with sufficient optical filter coverage for P ( z ) measurements, which eliminate a key systematic in the lensing masses used in this work (see Applegate et al. 2014 ). Expanding the sample of P ( z ) cluster masses from the current six clusters to ∼24 clusters would already improve the mass ratio precision, and thus constraints on Ω m , by ≈30 per cent. APPENDIX A: RATIO MODEL WITH INTRINSIC SCATTER AND COMMON APERTURES There are multiple, mathematically equivalent ways to implement a ratio model that includes intrinsic scatter and accounts for correlations induced from a shared measurement aperture. For


heating term varies with radius, the equilibrium is different at different radii in the disc. This variation changes the minimum density required to maintain a two-phase structure in the ISM. 2.2 Star formation and feedback The star formation recipe used here is as described in Stinson et al. ( 2006 ) with some modifications. In the recipe, stars form following a Schmidt law: where ρ * is the density of new stars formed, ρ g is the density of eligible gas, is the dynamical time and c * is the chosen efficiency (see Section 3 ). Gas is considered eligible for star formation if it lies above a set density threshold, below a maximum temperature and belongs to a converging flow. The original recipe required that gas particles be below a set temperature threshold to form stars. We now compare the effective temperature associated with the total energy (including thermal and non-cooling, as discussed below) to the threshold value. This ensures that feedback will always locally limit star formation. In this work, the temperature threshold value is taken at 100 K. The key star formation model parameters that apply here are the minimum density threshold for gas to form stars ( n th ), the star formation efficiency per dynamical time ( c * ) and the feedback efficiency (ε fb ) relative to the standard amount from SNe, as described below. We employ the SN feedback method of Agertz et al. ( 2013 ). This model differs from the older blastwave model of Stinson et al. ( 2006 ) and the newer superbubble model of Keller et al. ( 2014 ) in that it is a simpler way to deposit the energy from SNe that still produces similar results and robust feedback regulation. Specifically, rather than being a complex function of local properties, the conversion time for injected energy is a set parameter. This gives us direct knowledge of how long feedback energy remains available as effective pressure and removes the associated uncertainty in interpreting our results. In more detailed models, feedback energy might take the form of pockets of hot gas, cosmic rays, radiation pressure, winds and small-scale turbulence, all injected on initially very small scales (∼pc; see e.g. Hopkins et al. 2012 ). Many of these processes are difficult to model, convert energy rapidly from one form to another and suffer from numerical effects including excess dissipation. The associated effective pressures become important for driving dynamics when they act on scales comparable to the scaleheight. A simple model ensures a well-characterized coupling of feedback energy to the dynamics on these scales. It also allows us to sidestep the unresolved issue of the relative importance of different types of feedback energy and how efficiently it converts into different forms. We normalize our energy input relative to that of SNe. A single SN deposits 10 51 erg of energy into the surrounding ISM. Based on the stellar initial mass function of Chabrier ( 2003 ), approximately one in 100 stars


by a factor of 10. b The two photon indexes shown in bold, i.e., and , are for cases with and 10 −3 , respectively, where they are estimated from the relationship reported in Yang et al. ( 2015 ; see their Equation (5)), with (Ho 2008 ). Besides this statistical investigation of for LLAGNs, we have also calculated the Compton temperature of several individual LLAGNs. Such an investigation is a little too ambiguous, since most of these sources lack the hard X-ray and soft γ -ray observations to constrain the value of E c . So we use the theoretical SED results for such information, which is obtained by the accretion flow modeling to the sources (see Yuan Narayan 2014 for details of the accretion model of LLAGNs). We select several representative sources from Yuan et al. ( 2009 ). The results are shown in Table 3 . Again, we have tried three different electron temperatures. The Compton temperature is systematically higher than the “normal case” shown in Table 2 , but more consistent with the results of the “reduced-IR” case. The reason is that the IR flux calculated from the theoretical model is significantly weaker than that shown in Figure 1 , since we only consider the radiation from accretion flow and the jet. In other words, the Compton temperature derived here is applicable to the regions close to the black hole, as we argue above. Table 3 Compton Temperature of Individual LLAGNs Note. Observational data and theoretical modeling are taken from Yuan et al. ( 2009 ). We caution that the hard X-rays and soft γ -rays in these sources are actually from SED modeling because there are no direct observations at these energy bands. 4. The Total Radiative Heating and Cooling in LLAGNs In addition to Compton scattering, there are several additional radiative processes that can heat or cool the gas. In this section, we compare their relative magnitude. One process is the bremsstrahlung radiation ( and ), which always plays a cooling role. The other processes relate to the atomic energy-level transition, i.e., the photoionzation/recombination and line emission ( and ). The total radiative heating/cooling rate per unit volume can be expressed as (e.g., Proga et al. 2000 ; Sazonov et al. 2005 ) Here and n e are the number density of hydrogen atomics and electrons, respectively. For hot gas (i.e., temperature K) with solar abundance, . Consequently, the radiative heating/cooling rate can be expressed as The value of depends on both the metallicity (fixed to solar abundance in this work) and the ionization parameter ξ . The definition of ξ isHere R is the distance to the central black hole, is the ionization luminosity, and is the ionization luminosity factor. As shown in Figure 2 , we derive the ionization luminosity factors, i.e., , respectively, for the composite SEDs shown in Figure 1 with and . Detailed formulae of is provided in the Appendix (see Equation ( 17 )). It is a heating (cooling) term when ξ is large (small). Figure 2. Ionization


aspects of the influence of rotation on the convective flows. Therefore, the impact of the Coriolis force on the stochastic excitation of GIWs can be evaluated. In this context, recent work has established an estimate of the GIW energy flux (André et al. 2017 ). It can be used to estimate the rotational scaling of the amplitude of the wave energy flux arising from the modified properties of the convective driving. From Equation (61) of André et al. ( 2017 ), the vertical GIW energy flux can be computed from the horizontal average of the product of the vertical velocity and pressure perturbation that, given the linearization of the Boussinesq equations for monochromatic waves propagating in a selected horizontal direction, can be evaluated to bewhere v w is the magnitude of the vertical velocity of the wave. Moreover, the solution for the vertical velocity implies that the dispersion relationship iswhere f and f ss are defined above in Equation ( 20 ). Note that a reference table is given to help identify the many parameters in this section (Table 2 ). Table 2 Frequently Used Symbols in the Models of Interfacially Excited and Reynolds-stress-induced Wave Excitation Note. The abbreviations used in the table are V for vertical, H for horizontal, and PH for projected horizontal. Following P81 , further assumptions are necessary to complete the estimate of the wave energy flux. The convection is turbulent. So the fluctuating part of the velocity field is of the same order of magnitude as the convective eddy turnover velocity , which implies that convective pressure perturbations are approximately . Assuming that the pressure is continuous across the interface between the convectively stable and unstable regions, the horizontally averaged pressure perturbations of the propagating waves excited at the interface must then be equal to the turbulent pressure on the convective side of the interface. Those pressure perturbations follow from the solution for the vertical velocity and the nondiffusive Boussinesq equations (André et al. 2017 ). For plane wave solutions, the magnitude of those perturbations can then be written asNote however, the pressure matching condition fails for these modes at the pole for sub-inertial waves (). The reason is that the propagation domain of sub-inertial GIWs excludes the pole and it becomes increasingly concentrated toward the equator for faster rotation rates (e.g., Dintrans Rieutord 2000 ; Prat et al. 2016 ). Using the dispersion relationship, and equating the two pressures, yields the following equation for the vertical wave velocity:Therefore, the wave energy flux density becomes Flows in a gravitationally stratified convective medium tend to have an extent in the direction of gravity that is much larger than their extent in the transverse directions. Therefore, the horizontal wavenumber of the convective flows is much greater than the vertical wavenumber. This implies that . For efficient wave excitation, the


event (90% and 99% Poisson errors shown in blue). For this figure we assumed that the population in total produces 10 neutrino events (within a given time). While the selection bias is strong for sources detected with a single event, it quickly vanishes as soon as three or more events are detected from the same source. The effective densities of several source classes are indicated (compare Table 1 ). For each source class two different luminosity functions were adopted. The numbers in the upper line are for equally luminous sources, while a lognormal distribution with a width of one order of magnitude was assumed for the second line to show the impact of large luminosity fluctuations between individual sources. This variation in luminosity has the same impact on our results as using the observed distribution of gamma-ray luminosities for Fermi /LAT blazars which stretches over five orders of magnitudes (see Fig. 2 in Ajello et al. 2014 ) and also reproduces fluctuations observed between individual gamma-ray burst by the Swift Burst Alert Telescope ( Wanderman Piran 2010 ). This simplistic treatment does not account for correlations between the source redshift and luminosity, which have been observed for example in galaxy clusters ( Gruppioni et al. 2013 ). We find that the effect of the different redshift distributions and luminosity functions can be taken into account by using an effective density. The effective density is the density of a population of equally luminous sources without cosmic evolution which reproduces an equally large bias, i.e., the same median value. A similar calculation is performed in Murase Waxman (2016) , where the effective density quantifies the effect of the luminosity function, but not of the source evolution. The last column of Table 1 shows that the rather large luminosity fluctuations between individual sources reduce the effective source density by about one order of magnitude since a large fraction of the total neutrino flux is produced by a smaller number of sources. A detector that is not equally sensitive to all parts of the sky can be considered in the same way as the luminosity fluctuations since sources in the favored direction appear to be brighter. For example, if the detector is only sensitive in half of the sky, the effective density is reduced by a factor of two and the bias is correspondingly smaller. Figure 2 shows the size of the bias for different source densities assuming equally luminous sources and no cosmic evolution. The red bands show the expected neutrino flux of the sources where the dotted line indicates the median flux and the shaded area contains 90% and 99% of the probability distribution. The effective densities of the populations listed in Table 1 are indicated in Fig. 2 ; the width of the inner red band approximately reproduces the size of the 90% regions that were obtained for the different redshift distributions (see Table 1 ). Especially for sources detected with just a


the top panel of Fig. 2 , we show the case in which the apparent periodical variability of the spin period of LMC X-4 is tentatively described with the sinusoidal function: Here, is the mean value of the source spin period over the whole observational period (in seconds), P 3 is the characteristic time-scale of the variability, T 0 is the time of phase zero, and A the amplitude of the variability ( t , T 0 , and P 3 are all in units of days). Figure 2. Tentative description of the pulse period variations in LMC X-4 with two mathematical functions: the sinusoidal one described by equa-tion ( 1 ) (top panel) and the linear one described by equation ( 2 ) (the bottom panel). See the text for details. The equation given above describes the data reasonably well, and we obtained P 3 ≃ 5090, A = 5.3 × 10 −4 s, and s. Only the first two points obtained from the SAS observatory deviate from the qualitative periodical function. It is interesting to note that only these measurements together with the one based on the HEAO-2 data were obtained during episodes of the source flaring activity. All other spin period values were measured during quieter periods in the high superorbital phase. It is thus possible that some mechanism is at work during the flares to produce changes in the observed source pulsation period. As the available observational data (Fig. 1 ) are rather sparse, we cannot exclude that the different spin-up and spin-down episodes alternate in a more complicated fashion than a simple sinusoidal trend. Therefore, we show in the bottom panel of Fig. 2 an another example in which data are described by a sequence of power laws with identical absolute values of the slope but alternated signs: In the equation above, is the period derivative, T i is the time interval preceding when the pulse-period derivative changes sign, and P i is the pulse period in the time interval T i (in the equation all time intervals are expressed in days and all spin period values in seconds). According to this model, the characteristic rate of the pulse period change would be s −1 . 4 DISCUSSION The results presented in the previous section show that LMC X-4 undergoes cyclical transitions between spin-up and spin-down phases, with torque reversals occurring roughly every ∼6.8 yr. The spin period variability could be reasonably well described by using either a sinusoidal function, which would suggest the presence of smoothly periodical variations between two values of maximum and minimum spin periods, or a slightly more complicated function, relaxing the needs of assuming a strictly periodical phenomenon. At present, given the irregular and large spacing among all data available, it is not possible to firmly distinguish among these two possibilities. Nevertheless, below we discuss briefly several possible mechanisms that could give rise to quasi-periodical torque reversals in systems like LMC X-4. Third body. Concerning the strictly periodical case (i.e. equa-tion 1 ),


11 shows the relation between the position of a meteor in the image and the estimated errors in radiant. Since the pointing of the camera and the radiant of the simulated shower was set to the local zenith, the angular separation between the radiant and position of a meteor in the image corresponds to the elevation angle of the meteor. The median for the estimated errors in the speed for these 161 meteors was 0.12 km s −1 . The median of the estimated errors for 192 un-shuttered synthetic meteors, for which the residuals in magnitude Δmag were 0.5 m, was +0.01 m with a median deviation of 0.02 m. Fig. 11. Plot showing the relation between the angular separation and the estimated uncertainties in right ascension (filled circles) and declination (open circles). 5.2. Real meteor data We applied our software to three hours of double-station SPOSH image data acquired during an observing campaign held in Greece (i.e. at ∼37 ∘ N latitude ) on 12 August 2015 from 23 to 02 UT. The cameras were pointing to the zenith with their x -axis orientated to the north. The baseline between the two sites was 51.5 km. We reduced 177 meteor image pairs and determined their trajectories, velocities and heliocentric orbits. We focus on a 20 × 20 ∘ area centred at RA = 46 ∘ , Dec = 58 ∘ , close to the nominal radiant position of the Perseids (Fig. 12 ). To distinguish between Perseid and non-Perseid meteors, we performed a classification based on radiant position and speed as follows: 132 meteors were found to radiate from within this area. We determine the radiant of the Perseid shower as the median value of these radiants: RA = 45.96 ∘ and Dec = 57.77 ∘ . We assume that most of the meteors are Perseids and we find the 1 σ uncertainty in RA and Dec to be 3.29 ∘ and 2.27 ∘ , respectively. The geocentric speeds V G have a median value at 58.97 km s −1 and a median propagated error of 1.21 km s −1 (Fig. 13 ). We classify all meteors with speeds closer to this median speed than four times the median propagated error, as Perseids. In this way, we identified 71 meteors belonging to the Perseids meteor shower (Fig. 12 ). Their median speed V G was found to be 59.58 km s −1 with a median deviation of the observed uncertainties of 0.48 km s −1 . Statistical properties are given in Table 2 . As the aim of our example is to demonstrate the successful usage of our software using real data, we neglected the effect of radiant drift. Fig. 12. Geocentric radiants of 132 meteors originating close to the Perseid radiant. The ellipse defines the area occupied by Perseids. Meteors classified as Perseids according to their speed are shown as filled circles (•). All other meteors not meeting the requirements to be classified as Perseids are shown as open circles (∘). Fig. 13. Speed distribution of 132 meteors originating from an area close to the Perseid radiant. The dashed line shows computed median speed of 59.58 km s −1 . The bars in dark grey show the distribution of speeds for


over several campaigns (see Raiteri et al. 2013 , and references therein). The data used in this paper come from the WEBT archive. Fig. 2 Nightly averaged -band light curves (Aug. 2008–Jan. 2016) for 0716+714 and MRK 421 from Tuorla Observatory, and for BL Lac obtained combining nightly averaged data from Tuorla (black) and the WEBT observations (violet). The light curves are not host galaxy subtracted. The reported light curves match with the monitoring. Error bars are, in most cases, smaller than symbol size. The relevant nightly averaged light curves matching with the observation span are reported in Fig. 2 . The variability of MRK 421 and BL Lac is up to 5, and for 0716+714 up to a factor 10. The shapes of the light curves are grossly similar to the -ray ones, with fractional variability , 0.30, and 0.43 for 0716+714, MRK 421, and BL Lac, respectively. 3. Search for quasi-periodicities 3.1. Methods Identifying features in periodograms is a delicate task and requires several steps, i.e. a careful modeling of the noise and an analysis of the significance with respect to the noise model. Essentially, we followed the procedure described in Vaughan (2005 , 2010) and Guidorzi et al. (2016) . Power density spectra (PDS) were derived by means of generalized Lomb-Scargle periodograms (e.g. Scargle 1982 ) and, for a cross-check, by means of a completely independent technique ( Huijse et al. 2012 , 2014 ; Protopapas et al. 2015 ) based on a metric that combines the correntropy (generalized correlation) with a periodic kernel to measure similarity among samples separated by a given period. The results obtained with the two techniques are in excellent agreement. We restricted our study to long periods, i.e. in the days interval, where the limits are driven by the length of the considered light-curves, and to avoid the very noisy high-frequency part of the PDS. All the computations are carried out by procedures coded in python 4 (version 3.5.2) language. Statistical tests and functions, as well as optimization tools, are provided by the scipy 5 (V. 0.18.1) library. Lomb-Scargle periodograms are computed by means of the astropy 6 (version 1.2.1) implementation. The so-called correntropy periodograms are computed by the P4J 7 (version 0.13) python library. Leahy normalisation was adopted for the analyses of the PDS ( Leahy et al. 1983 ) while, in the figures, we normalized the resulting PDS to be within the range ( Zechmeister Kürster 2009 ). Fig. 3 From left to right , PDS of 0716+714, MRK 421, and BL Lac from the 100 MeV–300 GeV Fermi light curves, from Aug. 2008 to Jan. 2016. Lomb-Scargle spectrum of the input time-series data is given in blue and the best fit noise spectrum in red. Single frequency 95.0% and 99.7% false alarm levels are reported by purple and yellow lines. Solid and dashed lines refer to PL and AR1 models, respectively. Fig. 4 As above from the -band light curves. The noise was modeled as a simple power law (PL) or as an


to the Jet Power for Multiple-Zone Modeling of the Multiwavelength Data of Quiescent-State Emission from AP Librae The contributions of disk, broadline region (BLR), and torus radiation to the IC emission by the blob and pc-scale jet of AP Librae are considered in earlier papers. The energy of the BLR photons is blueshifted in the rest frame of the blob. The reprocessed BLR emission has been parameterized in Equation (12) of Hervet et al. ( 2015 ). The parameter values presented in Table 3 of Hervet et al. ( 2015 ) have been chosen such that at HE, the EC emission of blob-BLR origin is significant. A similar discussion is given in Zacharias Wagner ( 2016 ). Direct emissions from the disk, BLR, and torus are low as they are not detected, however, EC emission from the blob due to the reprocessed emission could be significant (depending on the parameter values chosen and the distances of the blob and jet from the black hole) after Doppler boosting. Moreover, it is discussed that, due to the outward relativistic motion of the blob and its distance from the accretion disk, the EC emission of blob electrons due to disk radiation is negligible. From Equation (A.6) of Zacharias Wagner ( 2016 ), the energy density of reprocessed emission from the dusty torus in the reference frame of the blob is . Assuming the radius of the region within which the emission is confined to be r DT = 1 pc, luminosity L DT = 10 42 erg s −1 , the Lorentz factor of the blob Γ = 7, the efficiency of reprocessing τ DT = 0.001, and the energy density in radiation u DT = 2 × 10 −8 erg cm −3 , which is much lower than the synchrotron photon energy density inside the blob, 1.28 × 10 −5 erg cm −3 . By adjusting the parameter values suitably, the energy density in reprocessed external radiation could be made even lower. Moreover, the pc jet is assumed to be located beyond the BLR and torus region, which makes EC emission insignificant. 3. Results and Discussions As shown in Figures 2 – 4 , the observed quiescent-state SED of AP Librae has been fitted by the three-zone model adopted in the present work. The optical–UV frequency data in the SED are accounted for partially by the synchrotron emission of electrons from the blob and partially by the blackbody radiation from the accretion disk. The thermal emission from the accretion disk is reproduced by assuming a Shakura–Sunyaev type disk (Shakura Sunyaev 1973 ) with a luminosity of L D = 1.3 × 10 44 erg s −1 . Incidentally, Zacharias Wagner ( 2016 ) used the same estimate for disk luminosity,The jet power requirements from each region are calculated from the expression of P jet presented in Equation ( 11 ). For the parameters used in our model, we find that the required jet power in Zone-1 is 5.7 × 10 47 erg s −1 and in Zone-2 is 3.75 × 10 47 erg s −1 . The jet power requirement in Zone-3 is 4.86 × 10 48 erg s −1 . Assuming a simple power-law proton spectrum of spectral index 2 in the energy range of 10 15 –10 18 eV, we have


field for the IC emission, the pionic γ-ray emission in our model is not able to account for both the observed γ-ray luminosities and the spectral shapes of galaxies with low SFRs such as the SMC alone. As diffusive losses become more relevant in these low-density galaxies, this leads to a steepening of the CR proton spectra (depending on the exact energy dependence incorporated in the diffusion time-scale). This necessarily results in steeper pion-decay γ-ray spectra, which would be in conflict with the observations if this were the only relevant emission channel. Hence, a leptonic contribution to the γ-ray emission from IC scattering is indispensable in our models. One of the arguments made in the literature in order to conclude a subdominant leptonic contribution to the total γ-ray emission depends on several underlying assumptions that are only valid in the nuclear regions of starbursts, i.e. for high gas densities, magnetic field strengths, and wind velocities. The magnetic field strength is either deduced from an energy equipartition argument, or from fitting several free parameters in one-zone models (see e.g. Torres 2004 ; Lacki et al. 2010 , 2011 ; Yoast-Hull et al. 2013 ) or axisymmetric two-dimensional models (Buckman et al. 2020 ) in order to simultaneously match the observations. The former approach assumes that the kinetic energy density is equal to the magnetic energy density, . Assuming turbulent velocities of and gas densities of yields in this approach magnetic field strengths of the order of . The fast cooling of CR electrons due to synchrotron emission in these strong magnetic fields has been proposed to be one of the reasons for suppressing the possible leptonic contribution to the γ-ray emission (Lacki et al. 2011 ). In contrast, γ-ray observations of the Galactic Centre suggest that CRs might not be able to penetrate into the densest regions and thus may be far from calorimetry (Crocker et al. 2011a , b ). However, those extreme properties are at most reached in the very central region of our simulated galaxies. Additionally, the energy density of the interstellar radiation field (or its equivalent magnetic field strength, see Fig. 1 , upper panels) persists throughout the whole galaxy. The cooling of the electrons is hence not fully dominated by synchrotron cooling, enabling IC losses to be non-negligible, except in the very central regions of starburst galaxies. In particular, in the low-density ISM of dwarfs, we find that the IC emission might be able to contribute up to 40 per cent of the total γ-ray luminosity, as can be seen in Fig. 4 (middle panels). Still, our models of starburst galaxies M82 and NGC 253 (that are chosen to match the observed SFRs and L γ ) also match their observed radio luminosities at 1.4 GHz. We will address the radio emission of our simulated galaxies in Paper III in more detail. Our findings suggest that SF galaxies are not reaching the calorimetric limit, and depart from this limit even


worth noting. Among them, there was a sporadic 10mag fireball, visible in the evening on 2005 October 28 or a similar event on November 30, with an orbit equivalent to the 00241 OUI October Ursae Minorids stream (Molau Rendtel 2009 ). Figure 1. Fireballs from the Southern Taurid shower registered on 2005 October 30 (left) and on 2005 October 31 (right) by the PFN05 Pozna station using the PAVO5 camera. Figure 2. Fireball of 15mag registered on 2005 November 4 at 20:19:42 ut with the Pegasus constellation in the background. Photograph taken by D. Dorosz in PFN09 abikw station (Praktica L2, Vivitar 28mm f /2.5, Konica VX200). Figure 3. Selection of fireballs registered during the peak of activity of the Taurids in 2015. 2.2 Observations in 2015 As in 2005, our research included solar longitudes ranging from 206 to 228. This time, the Moons phases were not as perfect as in 2005 with full Moon on October 27 and new Moon on November 11. From October 20 to 22, our stations did not provide any data because of heavy clouds. Between October 24 and November 4, good weather allowed the PFN to work at its fullest; the nights of November 57 were partially or completely cloudy but the nights of November 8 and 9 were clear in most of the country. Compared to 2005, the number of stations and the quality of our equipment increased significantly. We were able to gather data from 25 stations covering a period of time from October 20 to November 10. We used diverse observation equipment, the sets of the highest speed based on the Mintrons 12V6, which allowed us to observe meteors as faint as 34mag. For the first time, we used the digital DMK23GX236 megapixel camera and the Tamron 2.4-6mm f /1.2 lens. Such a set allows us to observe a field of 1283802 with a limiting magnitude of about 1 to 2 and a resolution of 19201200pixels with 25 frames per second. Table 2 summarizes a list of our stations active in 2015. Table 2. Basic data on the PFN stations that observed the Taurid shower maximum in 2015. In a period from 2015 October 20 to November 10, all PFN cameras registered 6970 meteors out of which we determined 719 orbits. They corresponded to different meteor showers and sporadic centres. Out of that number, 215 orbits belonged to the Southern Taurids and 39 to the Northern Taurids. Fig. 3 shows four bright Southern Taurids captured during 2015 maximum. The first phenomena of high brightness were observed on October 27 (similarly to 2005). Three bright fireballs were also seen on October 30. The night of October 31 brought two other incredibly bright phenomena. At 18:05 ut , a fireball from the Southern Taurid shower appeared over north-western Poland and its absolute magnitude amounted to 16mag. The fireball was visible across Central Europe and was registered by the PFNs network cameras, car cameras and people coincidentally photographing the close fly-by of the asteroid 2015 TB145 (see Olech etal. 2016 for a detailed description). The phenomenon lasted 5.6s