The overall diffuse intensity is comparable within both models. Figure 6. Fraction of successful jets in the (, ζ SN ) plane for the simple GRB model. Only a small fraction of all CCSNe can harbor a successful jet. 4. Neutrino Diffuse Emission in the Advanced GRB Model In this section, we take a somewhat more realistic model of the physics within a jet and allow Γ to vary across the jet angle from some maximum value Γ max at the center of the jet ( θ = 0) out to Γ = 1: the advanced GRB model. In this model, we do not rely on θ j = 1/Γ anymore. Because we anticipate that multiple shocks will be accelerating the protons, the resulting Γ distribution is a von-Mises–Fisher distribution,where the concentration κ ≈ 1/ σ 2 for σ small, with σ the usual standard deviation. We take motivated by random walks of the accelerated particles within the jet. In this model, we define the volume of the jet (Equation ( 3 )) with where is defined by Γ( θ max ) = 1, orWe note that for κ = Γ max , θ max has a maximum at Γ max = e . For a representative jet with Γ max = 300 this corresponds to θ max = 11°. The component of the GRB rate R ( z , Γ) introduced in Section 3 depending on Γ is assumed to follow a distribution similarly defined as in the previous section but this time this is a function of Γ max : . The constraint in Equation ( 30 ) to reproduce the observed HL-GRB rate becomes,The maximum value of is 1 and the minimum value of (the maximum value of θ ) is when Γ = 200, which is .Similarly to Equation ( 29 ), for the advanced GRB model we havewhere the beaming angle factor on each side has canceled. The diffuse neutrino intensity isThe integral goes over θ ∈ [0, θ max ], and we again require that the jet can successfully accelerate protons. The fraction of successful jets for this model is shown in the top panel of Figure 7 . One can see that the fraction of CCSNe harboring successful jets is larger in this model with respect to the simple GRB model, given the different scaling laws intrinsic to the models. Figure 7. Top panel: fraction of successful jets in the (, ζ SN ) plane for the advanced GRB model. Bottom panel: fraction of jets that are choked for the advanced GRB model. Below erg, 50% of the jets are choked; below erg, 90% of the jets are choked. The total diffuse neutrino intensity for our canonical GRB and ζ SN = 0.1 for all successful jets is plotted in the bottom panel of Figure 5 for the advanced GRB model to be compared with the one from the simple GRB model. Both models produce comparable intensities. We now turn to the issue of electromagnetically choked jets. Depending on the jet properties, we expect that there will be some successful jets that are choked or invisible. In the simple model, all of the jets are bright if they successfully form. On the other hand, among all successful jets, the fraction of choked jets for the advanced model is shown in the bottom panel of Figure 7 . Below erg, we find that 50% of the jets are choked;

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


The other possibility to explain the location and the age of the massive O3.5If* star RFS8 being nearly 69 pc from the nearest massive cluster is that the star formed in a small cluster, which has not yet been detected. To investigate if there is a cluster around RFS8, we utilize Two Micron All Sky Survey (2MASS) near-infrared JHK s imaging ( Cutri et al. 2003 ). Currently, this is the deepest photometry (allowing for extinction) available in the vicinity of RFS8. We obtain photometry of a region of 5′ surrounding RFS8. The depth of the photometry reaches roughly 16.5 mag in K s. Allowing for an extinction similar to RFS8 (of A V = 6.7), this reaches down to a spectral type of approximately mid-late A. Following the sequential sampling prescription of the stellar initial mass function (IMF) from Kroupa (2001) , we should have been able to detect a few tens of A-type stars, which might have allowed for the formation of such a massive star. This should be sufficient to identify any young cluster (3 Myr following the upper age limit of RFS8). We note that in the case of stochastic sampling of the IMF, there are no restrictions on the mass of the second most massive star in the cluster, that is, there is still a possibility of finding a cluster, but with few BA spectral type stars. From the colour–colour and colour–magnitude diagrams of stars within 5′ of RFS8, with high-quality infrared photometry, we find no evidence for a cluster sequence, and the upper locus of stars in the colour–magnitude diagram have no clear spatial correlation. In addition, we plot the stellar density (Fig. 4 ) of all point sources. From this, we find no evidence for any significant clustering around RFS8. Fig. 4. Stellar density as a function of distance from RFS8 of all infrared 2MASS sources within 5′. The lower axis shows the radius in projected distance. 4. Discussion We have shown that RFS7, a young (∼1 Myr), massive (∼100 M ⊙ ) O2.5If*/WN6 star is located ∼53 pc from the centre of the nearest massive cluster, NGC 3603 YC. Its mass is comparable to the most massive stars found within the cluster ( Roman-Lopes et al. 2016 ). From Gaia astrometry, we find that RFS7 displays a proper motion outside the locus of cluster stars of NGC 3603 YC, with a 2D projected velocity from NGC 3603 YC of 27.1 ± 1.5 km s −1 . It has a high tangential velocity (∼33 km s −1 ), and a possible bow shock located 1 pc from the star. The motion suggests the star was ejected from the cluster 1–2 Myr ago, consistent with the age of the cluster. All this indicates that RFS7 could likely be a runaway from NGC 3603 YC, making it the most massive known runaway in our Galaxy (and comparable to the behemoths found around the R 136 cluster in the Large Magellanic Cloud). A key future study in this respect would be to decipher whether the runaway was dynamically ejected, which might also help us to better understand the total mass and stellar content of the central cluster (to eject a ∼100 M ⊙

O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-Survey I-Survey I-Survey I-Survey I-Survey B-Survey B-Wavelength B-Wavelength B-Wavelength O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-CelestialObject O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O B-CelestialObject O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Model O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObject O O B-Wavelength B-Survey O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Telescope O O O O B-CelestialObject O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O B-CelestialObject O O O O O O O O B-CelestialObject I-CelestialObject O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


GHz observations, respectively. 5.3. Computational aspects Each resolve run needs ≈500 000 evaluations of the response and ≈400 000 evaluations of its adjoint. That makes the response part of the imaging algorithm a factor of ≈50 000 more expensive compared to CLEAN approaches. The good news is that the implementation of the radio response Eq. ( 7 ) in the package ducc scales well with the number of data points and that the response calls can be parallelized over the sum in Eq. ( 26 ). The resolve runs have been performed on a single node with five MPI tasks, each of which needs ≈2.2 GB main memory. Each MPI task uses four threads for the parallelization of the radio response and the Fast Fourier Transforms. The wall time for each resolve run is between 80 and 90 h. Single-scale CLEAN takes below 30 min for imaging each channel on a modern laptop. Thus, resolve is approximately 180 times slower that single-scale CLEAN here. This comparison does not include that the resolve had five times the number of CPUs available. Multi-scale CLEAN takes about 2 hours during the final round of imaging on the 13 GHz data set. This number does not account for the time taken during the initial rounds of imaging used to tune the hyper parameters and construct the mask which can be a time-consuming process. However, it should be kept in mind that CLEAN scales much better when the dimensionality of the image is much smaller than that of the data, which is not the case here. This is because CLEAN only requires about 10–30 applications of the full measurement operator and its adjoint, even including all preprocessing steps. Taking 90 min for the average multi-scale CLEAN run, resolve is 60 times slower than multi-scale CLEAN . 6. Conclusions This paper compares the output of two algorithms traditionally applied in the radio interferometry community (single-scale CLEAN and multi-scale CLEAN ) with a Bayesian approach to imaging called resolve . We demonstrate that resolve overcomes a variety of problems present in traditional imaging: The sky brightness distribution is a strictly positive quantity, the algorithm quantifies the uncertainty on the sky brightness distribution, and the weighting scheme is determined non-parametrically. Additionally, resolve provides varying resolution depending on the position on the sky into account, which enables super-resolution. We find that single-scale CLEAN and multi-scale CLEAN give similar results. In contrast, resolve produces images with higher resolution: The 4.8 GHz map has comparable resolution to the 13.4 GHz CLEAN maps. These advantages are at the cost of additional computational time, in our cases ≈90 h wall time on a single node. Future work may extend resolve to multi-frequency reconstructions where the correlation structure in frequency axis is taken into account as well in order to increase resolution. Additionally, direction-independent and antenna-based calibration may be integrated into resolve . Finally, the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Wavelength O O O O O O O B-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O B-Model O O O O O O O O O O O O O O O B-Wavelength O O B-Model I-Model O B-Model I-Model O O O O O O O O B-Model O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Model I-Model O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O B-Wavelength I-Wavelength B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


individual star. Tables 5 and 6 list the top 20 stars sorted by flare frequency f * and the number of flares N *flares , respectively. TIC43472154 is the most active star with an impressive flare frequency. If it is not coincidentally observed at an extraordinary active period, TIC43472154 can generate 233.17 flares per year. Besides, this star is not accompanied by any fainter stars or M dwarfs, according to the cross-match results from Gaia and Hipparcos data. This occurrence rate is much higher than that of KIC10422252, which is 41.6 flares per year (Shibayama et al. 2013 ). Figure 7 plots the light curve of TIC43472154. Superflares are marked with downward arrows. Figure 7. Light curve of TIC43472154. Small arrows mark those superflares selected from automatic software, and checked through a pixel-level pattern. Table 5 Stellar Properties of Flare Stars Sorted by Star Activity f * Note. Top 20 flare stars sorted by flare frequency ( f * ). The headers of this table are the same as in Table 1 . Table 6 Stellar Properties of Flare Stars Sorted by Number of Flares Note. Top 20 flare stars sorted by the number of flares on the corresponding star. The headers of this table are the same as in Table 5 . Another target, TIC364588501, was observed by all sectors of TESS and exhibits the largest number of flares (63). Additionally, like TIC43472154, this star is also a single star, according to the cross-match results from Gaia and Hipparcos data. TIC364588501 became more active than usual in the last 15 days of sector 5, and generated 15 superflares (Figure 8 ). Figure 8. Same as Figure 7 , but for TIC364588501. The most energetic superflare comes from TIC93277807, releasing 1.77 × 10 37 erg in around 1.5 hr. This value consists with the inference that energy releasing through a stellar flare is saturated at ∼2 × 10 37 erg (Wu et al. 2015 ). 3.4. Planets of Flare Stars Hot Jupiters are considered as one of the essential factors that can produce superflares on host stars (e.g., Rubenstein Schaefer 2000 ; Ip et al. 2004 ). According to these studies, it is not possible for the Sun to generate superflares. However, a lot of statistical works based on Kepler data did not detect hot Jupiters around flare stars (Maehara et al. 2012 ; Shibayama et al. 2013 ; Maehara et al. 2015 ). We made a cross-check between our 400 flare stars with the Exoplanet Follow-up Observing Program for TESS (ExoFOP– TESS 3 3 https://exofop.ipac.caltech.edu/tess/ ). Cross-matching results are listed in Table 7 . There are four stars (TIC25078924, 44797824, 257605131, and 373844472) that have planet candidates. These four stars are unlikely to affect our statistical results. We thus did not exclude them in the analysis. Besides, there is only one star (TIC410214986) that possesses a confirmed hot planet (Newton et al. 2019 ). This is an interesting planet, and we list its parameters in Table 7 , even if its hosting star (TIC410214986) is flagged as a binary system from the

O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O B-Telescope O O O O O O O O O O B-CelestialObject O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-CelestialObject O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-Telescope O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O B-Telescope O B-Telescope O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O B-Telescope O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Survey I-Survey I-Survey I-Survey I-Survey I-Survey B-Survey I-Survey O O B-URL O O O O O O O O O O O O O B-Citation B-Citation B-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O


a zoom-in of the convolution kernel. Redder wavelengths have a larger kernel width due to . Positive redshifts move the position of the diagonal convolution to the right. The final model spectrum is shown compared to a low-resolution () spectrum of a solar-like star with S/N per pixel. The additional nuisance parameters f , p b , v b only contribute during the calculation of the likelihood , and do not affect the data generated by the model. Figure 2. A faux observed spectrum (black) of and S/N of ∼20 per pixel, which was used in the toy model test. The recovered maximum a posteriori model spectrum is shown in red. The construction of constitutes a non-negligible component to the total computational budget for each probability evaluation. For this reason a few (optional) approximations have been implemented to minimize this cost. A least-recently used cacher is employed by default to minimize the number of matrix constructions. This uses a small portion of random access memory to retain the N most common sets of . If is required and is sufficiently small (i.e., below some prescribed tolerance) such that , then future calls of within the range will return the previously calculated matrix instead of reconstructing it. The tolerances are configurable, and default to the sub-km s −1 level for and z , and . Alternatively, the construction of can be completely avoided by approximating the convolution with a single kernel width σ at all , where an interpolation routine is used to calculate the expected intensity at . The extent of approximation that can be afforded will vary on the scientific objectives, but the default behavior in sick is balanced to be computationally efficient, and suitable for most scientific applications. I have produced the expected intensities for some arbitrary that can fairly represent any rotational broadening of the source, the object’s redshift, as well as the resolving power of the instrument and the location of the CCD pixels. However, these intensities are not representative of real world data. The height (i.e., photon counts) and shape of an observed spectrum is a function of the source magnitude, exposure time, instrument sensitivities, atmospheric conditions, interstellar extinction, and a host of unaddressed effects. For these reasons a function is required to normalize 3 3 The normalization process is frequently abused by stellar spectroscopists in the literature. Wherever possible, data should not be transformed. One should seek to fit a model to the data, not the other way around. the model to the data. Although the function incorporates a number of effects (e.g., source blackbody temperature, dust, instrument sensitivities), they are phenomena that usually cannot be separated without additional information, and here I only care about their combined effect. The continuum is modeled as a polynomial that enters multiplicatively,where the maximum polynomial degree is specified by the user. For data spanning

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


halos and therefore have different IGM densities, velocities, and temperature structures. These 100 independent model realizations are necessary to properly characterize the stochasticity of the IGM absorption. We forward model observational effects like finite spectral resolution, signal-to-noise ratio, and continuum-fitting errors, to make the model outputs directly comparable to observed spectra. Each spectrum extends for ±65 cMpc around the foreground quasars position, and we bin the spectra in chunks of 1 cMpc length. For each chunk, we obtain a kernel density estimate (KDE) of the probability distribution (PDF) of the transmission values in the chunk, based on the 100 independent realizations of that model. This results in smooth functions, , that describe the probability for measuring an IGM transmission F n , m in bin n , m , given the model parameter . Here, the index n identifies the different background sightlines within the sightline pattern while m denotes a certain chunk along the sightline. As long as we only focus on the quasar age, the parameter vector has only one component . However, we still denote it as a vector, because this approach allows for easy and straightforward generalization of the inference to additional parameters like quasar obscuration or orientation. Following this, each model of the H i proximity effect is fully described by a set of transmission probability functions . 5.2. Likelihood Computation Due to the high dimensionality of the observable, , i.e., several thousand individual transmission measurements, determining the likelihood poses a very challenging task. The usual approach of approximating the likelihood as a multivariate Gaussian is inadequate for our problem, because the individual transmission PDFs are not described well by Gaussians. In addition, determining the significant correlations between all the elements of { F n , m } would likely require an excessive number of model realizations. Therefore, we follow a likelihood-free approach for which we never have to actually write down an analytic form for the likelihood function. In Schmidt et al. ( 2018 ), we solved a similar problem. There, however, the dimensionality of the problem was low (at most three transmission measurements and two model parameters), which made it feasible to simply map the full likelihood space by brute-force sampling. An approach like this would be completely impossible for our current case. A fully Bayesian treatment is only achievable if the dimensionality of the problem can be drastically reduced. We do this by first computing a so-called pseudo-likelihood , which acts in many ways like a proper likelihood—except that it ignores correlations between the { F n , m }. In a second step, we then map this pseudo-likelihood to determine a proper posterior probability distribution. Our approach is inspired in many ways by Alsing et al. ( 2018 ) and Davies et al. ( 2018b ), but is customized to the problem at hand and is

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O


of HO and CO bands. This closure-phase signal is interpreted as a signature of large-scale inhomogeneities or clumps at the photospheric layers and within the extended molecular layers. These inhomogeneities or clumps contribute a few percent of the total flux. Observations taken at two epochs separated by about two months indicate a variability of the detailed structure of the inhomogeneities, while a variability of the overall radius could not be detected within our calibration uncertainties. We compared our data with the latest 1D dynamic model atmospheres based on self-excited pulsation models (CODEX models) and for the first time also with 3D radiation hydrodynamic (RHD) simulations of the convective interiors and the atmospheres of AGB stars (CO5BOLD code). For the latter we chose to use azimuthally averaged intensity profiles because our main interest in this current investigation concerns the overall extension of the atmospheres and because we aim at comparing the 3D models to the results by 1D models. In addition, we have previously shown that visibilities above a flux level of about 10% are consistent with spherically symmetric models so that the detailed 3D structure mainly concerns only relatively low flux levels. Both model attempts provide satisfactory fits to our data and, in particular reproduce the wavelength dependence of the visibility data. This means that both models include a spatially extended molecular atmosphere that is consistent with our observations. A comparison between the intensity profiles of the best-fit 1D and 3D models show that indeed both model attempts include an extended atmosphere up to roughly two to three stellar radii and that the wavelength-dependent flux contributions of the extension are consistent with observations. This agreement of both the different 1D and 3D modeling attempts with our data supports the theoretical result by Freytag Höfner (2008) that shocks induced by convection and pulsation in 3D models are roughly spherically expanding and of similar nature to those of 1D self-excited pulsations. Moreover, our results show that indeed both modeling attempts, 1D pulsation models and 3D convection simulations, can levitate the atmosphere of Mira stars to at least two to three stellar radii, which is consistent with our observations. We note that, on the contrary the same modeling attempts failed to extend the atmospheres of red supergiant stars to similarly observed extensions ( Arroyo-Torres et al. 2015 ). In addition, observed and model-predicted broad-band and are consistent for both the 1D CODEX and 3D CO5BOLD models, while the color can only be re-produced by the 1D CODEX, which includes a simple dust description, and not by the dust-free 3D CO5BOLD models, indicating that dust starts to contribute to broad-band colors from the -band onward. The detailed intensity profile across the stellar disk could not be constrained by our current snapshot observations. This would require future

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


et al. 2016 ) and can potentially suppress clumping instabilities ( Vishniac 1983 ; Ntormousi et al. 2017 ). This may well set the actual size scales of the resulting clumps in our simulation that includes the cloud. We would also expect that the actual star formation sites are filamentary, like the dense regions in the simulations of Inoue Inutsuka (2012) , and like those seen in recent observations of Pokhrel et al. (2018) . We focus here, however, on the connection to the larger scale, and refer to these other studies for details of the evolution onsmaller scales. While magnetic fields have in principle the potential to suppress the clumping instability in supershells, this is not required to explain the HI observations of the USco supershell, where the filamentary appearance of the shell suggests that the Vishniac instability is acting. Magnetic fields might, however, explain the absence of molecular gas in the HI shell around USco. We stress that the Vishniac instability is not crucial for our scenario. Suppression of instabilities at the shell surface of our global simulation would reduce the extent of the mixing zone, which is responsible for most of the radiative cooling in the bubble. This would therefore increase the size of the simulated bubble, such that a smaller number of massive stars would be required to reach the observed extent. The blow-out of the HI supershell around USco suggests that Sco–Cen formed from a large sheet of dense gas, similar to the dense sheets in large-scale simulations of the ISM (e.g. de Avillez Breitschwerdt 2004 , 2005 ; Seifried et al. 2017 ). In these simulations, these dense sheets are “walls” of superbubbles several hundred parsecs in size. Seifried et al. (2017) show that molecular gas builds up in such sheets over millions of years. Dawson et al. (2015) may have observed this process at the interface of two HI supershells in the Carina arm of the Milky Way. Giant molecular clouds formed in this way are less violently formed than in typical colliding flow simulations of molecular cloud formation (e.g. Inoue Inutsuka 2008 ; Körtgen Banerjee 2015 ), because the pressure in the surrounding superbubbles is modulated by the actual energy production of the massive stars. This might make the clouds comparatively tenuous and delay the formation of stars ( Dawson et al. 2015 ). A more gradual buildup of molecular clouds is also seen in more sophisticated, multi-phase colliding flow simulations ( Inoue Inutsuka 2012 ). Inutsuka et al. (2015) detail a scenario of magnetised molecular cloud and star formation along these lines. In general, molecular cloud formation at the interface of HI supershells is frequent ( Dawson et al. 2013 ). Near the solar circle, molecular cloud formation is only weakly coupled to spiral arm passages (e.g. Koda et al. 2016 ). How could the actively star-forming Lupus I cloud have formed in this scenario? Our analysis of multi-wavelength observations of the Lupus I molecular

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


that auroral emission excited by energetic particles is likely to dominate over airglow excited by EUV and X-rays. Nevertheless, if high-energy photons are incident at ∼3 × 10 30 erg s −1 (10 34 erg over 1 hr), the incident energy is ∼135 TW for a planet of Earth’s radius at 0.05 au. The fraction converted to airglow depends completely on the photochemistry of the upper atmosphere. If there is significant ozone present, essentially all the light shortward of 320 nm will be absorbed in the ozone Hartley band, and will potentially be available for atmospheric chemistry and excitation processes, such as excitation of the rotational–vibrational Meinel series of hydroxyl (OH) emission lines in the near infrared. Leinert et al. ( 1998 ) tabulated the strongest terrestrial airglow lines, and the OH line series is by far the strongest, at 4.5 MR (mega-Rayleighs, where R is the Rayleigh unit of surface brightness), compared to, e.g., the green oxygen line at 558 nm of 250 R, or 10 kR auroral 558 nm emission for typical zenith airglow brightnesses. With the high-energy photon flux, and the timescales for the stellar wind and CMEs being comparable, high-resolution spectroscopic searches for optical/NIR line emission are likely to benefit from filtering according to this timescale—the optimal times to seek an atmospheric response are during and for the few hours following strong flares. An instrument such as the CRyogenic high-resolution InfraRed Echelle Spectrograph (CRIRES) on the European Southern Observatory’s Very Large Telescope (VLT) would seem well suited to a search of this nature, or the High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph analyzed in detail by Luger et al. ( 2017 ). Lovis et al. ( 2017 ) discuss coupling the high-resolution spectrograph ESPRESSO to the adaptive optics instrument SPHERE at the VLT in order to measure both a spatial offset and a Doppler shift for Proxima Cen b, and conclude that such a measurement is feasible. If additional filtering on the flares were introduced, the signal-to-noise ratio could potentially be improved further. 5.4. Complexities and Opportunities The geometric distribution of flares on the surface of the star is unknown. Hunt-Walker et al. ( 2012 ) found no correlation with rotational phase for AD Leo, similar to Hawley et al. ( 2014 ), indicating a distribution of many flare sources at different longitudes and a relatively uniform flare distribution, stochastic in both outburst frequency and location. A non-uniform flare distribution could influence the illumination pattern of the space local to the star, and could work either in favor if flares are concentrated to the equatorial regions, or against if to the poles (presuming the planets to share the same equatorial plane). The planet may be illuminated by flares that we do not see, because they are both “around the back” of the star, and conversely, we may see flares that the planets do not. The degree to which flares are

O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument I-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O B-Observatory I-Observatory I-Observatory B-Telescope I-Telescope I-Telescope B-Telescope O O O O O O O O O O O O B-Instrument I-Instrument I-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Instrument O O O O O B-Instrument O O B-Telescope O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject I-CelestialObject O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


low-mass particles for the v200R1 model. The upper, middle, and bottom panels are the projected position of the each particles for φ = 30°, 45°, and 60°, respectively. The blue and red arrows show the radial velocity direction, moving toward or away from us, respectively. The size of the arrows indicates the tangential velocity. However, the number of particles approaching or moving away from us is dependent on the projected angle, but the explosion is not at all isotropic in the x -axis, which is the axis of movement of the massive particle. But the morphology of the explosion is similar to the one observed in Orion BN/KL. It is important to note that the massive particle in the Orion BN/KL explosion should be a runaway massive star with a velocity of at least 150 km s −1 , but the dynamical interaction of the massive star with clumps with a total mass comparable to the star’s mass could substantially decrease the velocity of the star at the end of the interaction. However, it is not the goal of this work to study the dynamic interaction of low-mass particles. 5. Conclusion We presented analytical and numerical solutions of the dispersion of particles due to dynamical interaction with a single massive particle. We have considered that the particles gather into a cluster and that they have a negligible mass with respect to the massive particle. The dynamical interaction with the massive particle produces a quasi-isotropic ejection of the particles. Then, we carried out a set of numerical simulations of spherical distribution of massless particles ( N -body simulations) for verifying our analytical solutions and obtaining an observational result. We have obtained a very good agreement between the numerical and analytical results. The main conclusions are that: 1. The gravitational ejection mechanism is able to accelerate small clumps or cloudlets (i.e., low-mass gas fragments) to jetlike velocities and, therefore, it should be thoroughly explored in future work. 2. The terminal velocity v ∞ of each particle is function of its own impact parameter. 3. The maximum terminal velocity is given by the limit when the particle has an impact parameter equal to zero and it is twice the velocity of the massive particle (. The minimal terminal velocity is related to the cluster radius. 4. The ejection angle of each of the particles is linearly related to the terminal velocity, and therefore is related to the impact parameter. Compact distributions of particles that are dynamically disturbed by a massive particle, which passes through the center of the distribution, produce more collimated ejections than in the case of more scattered clusters. 5. The resulting dispersion has an explosive signature, such as: (a) a small-scale common origin, (b) an isotropic distribution, and (c) velocities proportional to the distance from the common origin. This is the result of a short time interaction, which could be the mechanism that produced the explosive outflows.

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


3 (c)) accompanied by large fluctuations in the proton density and temperature (Figures 1 (d)–(e)). The coexistence of kinetic Alfvén waves and Alfvén ion cyclotron waves was demonstrated in this slow solar wind during 2018 November 6 (Huang et al. 2020a ), similarly to the observations in the fast solar wind at 1 au (e.g., He et al. 2011 , 2012 , 2015 : Podesta 2013 ; Bruno Telloni 2015 ). The plasma β varies from 0.1 to ∼10, but mostly less than 1 (Figure 1 (f)). Figure 1. Overview observations from 2018 November 4 to 6. (a) The magnitude of magnetic field, (b) the three components of magnetic field (c) the three components of the proton velocity in RTN coordinates, (d) the proton number density, (e) the proton temperature, (f) the proton β , and (g) the distance away from the Sun. To be able to study the magnetic energy spectra at the sub-ion scales for 15 minute windows, the Fourier transform was performed for the merged FGM and SCM measurements (Bowen et al. 2020a ) to obtain the magnetic energy spectra for each 15 minutes. The 15 minute duration ensures having at least one correlation time of the turbulent fluctuations (Parashar et al. 2020 ). Furthermore, to avoid possible “contamination” of the background turbulent spectra from intermittent but intense waves (e.g., ion cyclotron waves, whistlers), we excluded the magnetic spectra that showed clear spectral peaks from MHD to electron scales. One should point out that the electron measurements are not available during this time interval. But we can estimate the Doppler-shifted frequency of the electron inertial length f de ∼ 180 Hz and the Doppler-shifted frequency of the electron gyroradius f ρe ∼ 320 Hz based on the assumptions of N e = N i and T i / T e = 5. This indicates that the electron characteristic frequencies are larger than the Nyquist frequency of the magnetic field data provided by PSP, thus preventing us from investigating electron-scale turbulence. Figure 2 (a) presents an example of the resulting magnetic spectra. Three distinct frequency bands can be clearly evidenced: (i) the first one is an inertial range with the slope of −1.71 below 1 Hz, (ii) the second one is a steep ion transition range ( f −3.63 ) at the scales between the proton cyclotron frequency and the Doppler-shifted frequencies of the proton inertial length d i and gyroradius ρ i (i.e., f ∼[2, 9] Hz), (iii) the third one has a scaling f −2.68 between the ion and electron scales (∼[10, 60] Hz). These three distinct spectral features are also confirmed by all 219 selected spectra shown in Figure 2 (b). However, the spectral indices have large variations in the ion transition range. Figure 2. Power spectral densities of the magnetic field. (a) One example of magnetic spectrum during the time interval from 11:13:11 UT to 11:28:11 UT on 2018 November 6; (b) all selected spectra. The three vertical arrows from left to right correspond to the proton cyclotron frequency f ci , the Doppler-shifted frequency of the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Mission O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


the stable nature of the inclination oscillations, the incompleteness of detecting high-eccentricity and high-inclination objects, and the greater likelihood of low-inclination objects impacting the Earth (as mentioned in Section 1 ) suggest that extensive observation of high-inclination locations of vertically unstable quasi-satellite orbits may be helpful in detecting populations of potentially hazardous asteroids in advance before they reduce inclination. The long-term stable inclination oscillations associated with quasi-satellite orbits obtained in the simplified model in this study may appear to be contradictory to the shorter lifetime of , of only tens of thousands years, on its quasi-satellite motion. We do not expect long lifetimes of such objects in the real world experiencing close approaches to planets at low-inclination states (Wajer Królikowska 2012 ), but the stable nature of the inclination oscillations in the simplified model implies an ability to capture objects even in the real world. Certainly, was trapped in the quasi-satellite motion exhibiting inclination oscillations from a chaotic orbit (see fig. 5 in Kinoshita Nakai 2007 ). The stable nature is significant in our conjecture, not because of the longevity after experiencing low-inclination states, but because of the ability to capture objects in the inclination oscillations at high inclinations before reducing their inclinations. This paper focuses on the fundamental dynamics of vertical instability and inclination excitations of co-orbital orbits in the circular restricted three-body problem, but limitations exist due to the simplified assumptions of the model and additions of various perturbations, such as the eccentricity of Jupiter, and the gravitational effects of other planets could be promising directions. In such non-autonomous systems, computations of invariant manifolds associated with resonant periodic orbits (Voyatzis Antoniadou 2018 ) and quasi-periodic invariant tori (Olikara Scheeres 2012 ) would be natural extensions of the present work. 5 CONCLUSIONS This paper explored the effects of vertical instability on inclination excitations of co-orbital orbits in the Sun--Jupiter circular restricted three-body problem. Computations of invariant manifolds associated with vertical instability of the planar co-orbital periodic orbits enabled quantitative analyses of inclination excitations. Quasi-satellite orbits exhibited long-term stable, highly eccentric, substantial inclination oscillations between near zero and tens of degrees without escaping from quasi-satellite states. We pointed out the possibility that populations of undetected potentially hazardous asteroids exist at high-inclination locations of Jupiter’s vertically unstable quasi-satellite orbits. Analyses of bifurcation and phase-space structures revealed that stable, three-dimensional families of periodic quasi-satellite orbits host inclination oscillations in a stable manner. However, planar

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


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 .

O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O


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.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-CelestialObject I-CelestialObject O O O B-Wavelength O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O B-Model O O O O O B-Telescope O O O O O O O B-Software O B-Software B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Telescope O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O B-Mission O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O


); 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 ()

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


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

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


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

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


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

O O O B-TextGarbage O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-CelestialObject O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest B-CelestialObject O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O B-Wavelength O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O B-CelestialObject O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O B-CelestialObject I-CelestialObject O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O B-CelestialObjectRegion O O B-CelestialObjectRegion I-CelestialObjectRegion B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-CelestialObject O B-EntityOfFutureInterest O O O O


O B-Survey O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-URL O O O B-URL I-URL I-URL O O B-URL O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Survey O O O O O O B-Dataset I-Dataset O O O O O O O O B-Organization O O B-URL B-Organization O O B-URL O B-Observatory O O B-URL O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O B-URL I-URL B-Organization B-Survey O B-Organization O O O O O O O O O O B-Dataset I-Dataset O O O B-Organization B-Software O B-URL O O O O O O O O O O O O O B-Software O O O B-Database I-Database O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software O O O B-Software I-Software I-Software O O O O O O O O O O O B-Database I-Database I-Database O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O B-Software O B-Software B-Citation I-Citation I-Citation I-Citation O O B-Software B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


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

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


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

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


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

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