Phinney (1989) was the first to recognize that

 is not exactly the maximum orbital pericenter for a complete tidal disruption, a distance we would like to name the “physical tidal radius” (we assign it the symbol

). To remedy the neglect of internal stellar structure, he suggested that

 could be estimated by applying to

 a correction factor based on the star’s apsidal motion constant and its dimensionless binding energy. For this reason,

 is sometimes reinterpreted to be

, but without evaluating how it might differ from

 (Stone et al. 2013). Several groups have tried to include stellar structure in the calculation of

, but employing purely Newtonian dynamics on polytropic stars (e.g., Luminet & Carter 1986; Khokhlov et al. 1993; Guillochon & Ramirez-Ruiz 2013; Mainetti et al. 2017). Recently, there have been efforts beginning from genuine main-sequence stellar structures, but still restricted to Newtonian dynamics, and examining a limited range of stellar masses (only 1

 in Goicovic et al. 2019,

 and

 at several ages in Law-Smith et al. 2019,

,

, and

 at three different ages in Golightly et al. 2019). Others have explored the dependence on black hole mass induced by relativistic effects, but without any reference to internal stellar structure or the hydrodynamics of disruption (e.g., Ivanov & Chernyakova 2006; Kesden 2012; Servin & Kesden 2017). Earlier works employed a post-Newtonain approximation (e.g., Ayal et al. 2000) or explored the use of relativistic hydrodynamics simulations for strong encounters of polytropic stars, but without stellar self-gravity (e.g., Laguna et al. 1993). In some cases, relativistic effects were approximated by a “generalized Newtonian potential” (e.g., Gafton et al. 2015; Gafton & Rosswog 2019) or in terms of genuine relativistic dynamics (e.g., Frolov et al. 1994), but assuming a polytropic structure for the star and computing stellar self-gravity in an entirely Newtonian fashion (in the last case, fixing it to its initial stellar surface value). Many of these explorations of

 also computed the energy distribution dM/dE and explored the relation between the remnant mass and orbital pericenter in partial disruptions; in one case (Manukian et al. 2013), they also examined the remnants’ orbital properties. However, all this work was subject to the limitations already enumerated.


Figure 1 presents the Stokes I image from W43-MM1. The continuum emission shows a fragmented filament extending from the north to the south–west. Two bright sources at the center, A and B1, completely dominate the energy budget in W43-MM1 (with integrated fluxes of ∼2.0 and 0.5 Jy, which correspond to

 of the total flux recovered), over a number of additional fragments extending to the south–west. Also, additional sources to the east and west have been detected. Comparing with the SMA results from Sridharan et al. (2014) and the PdBI 1 mm results from Louvet et al. (2014), the ALMA observations reproduce quite well the overall morphology of W43-MM1, but with better resolution. The noise in the ALMA map is

 with a peak of 503 mJy

 obtained from Gaussian fitting. We used the getsources algorithm (Men'shchikov et al. 2012) to successfully extract 14 sources from our ALMA data (see Table 1 and Figure 1). The extraction was later compared to other methods such as clumpfind (Williams et al. 1994), FellWalker, and Reinhold (Berry et al. 2007) obtaining a good agreement with the selection produced by getsources. We used the source positions and sizes derived using getsources as initial guess for a Gaussian 2D fitting (using CASA imfit algorithm) in order to derive accurate fluxes from the extracted sources.14

14
We found that getsources tends to underestimate the recovered fluxes from our data.
 Also, we kept the same source nomenclature used by Sridharan et al. (2014), but adding numbers where higher multiplicity was discovered with respect to the SMA map. Using the standard procedure to calculate masses from dust emission (Hildebrand 1983), we computed masses for all 15 sources in our catalog assuming a dust opacity of

 cm2 g−1 (Ossenkopf & Henning 1994), a gas to dust ratio of 1:100, and a dust temperature of

 K (Bally et al. 2010), with the exception of the hot core, source A, where we used a range between

 K. Although the SMA detected CH3CN emission toward B1 and C, it was unresolved and not sufficient to derived temperatures; hence, we used

 K for these sources. Herpin et al. (2012) modeled an spectral energy distribution and derived a temperature profile for source A using all the publicly available data on W43-MM1 to date. Their model suggests a temperature of ∼150 K at 2500 au distance (05 radial;

 size) and ∼70 K at about 8000 au distance (07 radial; 14 size), which are the length scales sampled by ALMA. However, if larger spatial scales than our source size are considered, the temperature might be lower and on the order of 30 K (as suggested by Bally et al. 2010). These scales (

) are consistent with Herschel primary beam at 160 μm and, of course, larger than our deconvolved source sizes and synthesized beam. Therefore, and using this temperature range, our derived mass for source A is between 312 and 146 M

(with the exception of the 30 K temperature derived mass of 728 M

). These estimates put source A below other massive clumps such as the SDC335-MM1 mass estimate of 545 M

(Peretto et al. 2013) and G31.41+0.31 with a mass estimate of 577 M

 (Girart et al. 2009).15

15
Note that in this massive core a magnetic field strength of 10 mG was derived using n(H2) =

 cm−3.
 However, these mass estimates were derived from observations sampling larger length scales than our ALMA W43 observations. The deconvolved size of SDC335-MM1 is about 0.054, which is two times the size of source A or four times the area. A simple estimate assuming an

 density profile, will give about 272 M

 per source A size for SDC335-MM1 and about 205 M

 per source A size for G31.41+0.31.


The ALMA data were obtained in Cycle 3 through proposal 2015.1.00540.S (PI: R. A. A. Bowler). We observed the six brightest LBGs at 6.5  $z$  7.5 presented in Bowler et al. (2012, 2014). The galaxies are all within the COSMOS field (Scoville et al. 2007a),1 and were initially selected in the near-IR YJHKs data based on the UltraVISTA2 survey data release 2 (McCracken et al. 2012). The coordinates of the six galaxies, which defined each ALMA pointing, and their photometric redshifts are shown in Table 1. The galaxies were selected via SED fitting and are secure $z$ > 6.5 objects as demonstrated by the sharp spectral break observed in the photometric data at ${\simeq } 1\, {\mu} {\rm m}$ (Fig. A1). The strength of the break ($z - Y \gt 2\, {\rm mag}$) observed in the sample cannot be reproduced by the common low-redshift or brown-dwarf contaminants (e.g. Ouchi et al. 2010; Bouwens et al. 2011; Bowler et al. 2012). Each target was observed for 10 min in Band 6 (centred at $233\, {\rm GHz}$ or $1.29\, {\rm mm}$; corresponding to ∼170 ${\mu}$m in the rest frame). The integration time was chosen to provide limits on the ratio of FIR to UV luminosity sufficient to distinguish between the previous results at $z$ > 5 from Watson et al. (2015) and Capak et al. (2015). The data were taken in a compact configuration C36-2, with a maximum baseline of 330 m, on 2016 January 25. In order to maximize continuum sensitivity, the correlator was set up in time division mode with 2 GHz bandwidth across each of four spectral windows (centred at 224, 226, 240, 242 GHz) in two polarizations. The data were reduced with the ALMA pipeline of casa version 4.5.1. Calibration was performed by ALMA personnel using J1058+0133, J0948+0022, and Ganymede for bandpass, phase, and flux calibration, respectively. We then processed the data through the imaging pipeline, collapsing the full 8 GHz bandwidth and both polarizations into a single-continuum image of each field, initially using natural weighting of baselines to obtain the optimal root-mean-square (rms) sensitivity for point sources. Imaging was achieved using the tclean task, and the data were cleaned by defining a mask for any source in the field with a peak signal-to-noise ratio (SNR) > 3 that also corresponds to a Hubble Space Telescope(HST)-detected galaxy (whether the target or serendipitous), with a clean threshold of 60 ${\mu}$Jy (≃ 2 × rms). The natural weighted images have a beam size of 1.1 × 1.4 arcsec full width at half-maximum (FWHM) and reach an average rms sensitivity of $27\, {\mu} {\rm Jy}\,{\rm beam}^{-1}$, with small deviations of $2 \, {\mu} {\rm Jy}\,{\rm beam}^{-1}$ between the different pointings. To increase the sensitivity to extended flux, we also tried imaging the targets with a 1 arcsec taper, which results in a beam size of 1.4 × 1.7 arcsec and rms sensitivity of $31\, {\mu} {\rm Jy}\,{\rm beam}^{-1}$. As the natural weighted images have a smaller beam and better rms sensitivity, we chose to present this weighting throughout the paper, however comment on the tapered results where relevant. We measured fluxes using two complementary methods. First, fluxes were measured from the peak flux within an aperture of radius 1 arcsec from the HST/WFC3 centroid. The peak flux assumes that the source is unresolved in the ALMA data. For the stacks and the detected individual source, we also measured the total flux using a Gaussian fit. In the case of a non-detection at the 3σ level for the individual LBGs, we conservatively provide 2σ upper limits calculated as $f_{\rm peak} + 2\, \times \, {\rm rms}$.


The strength of the break ($z - Y \gt 2\, {\rm mag}$) observed in the sample cannot be reproduced by the common low-redshift or brown-dwarf contaminants (e.g.


A non-vanishing α1 adds to the polarization of the orbit in the same way as a non-vanishing α3, and analogous to equation (13) one finds for the change of the orbital eccentricity vector
(17)
\begin{eqnarray*}
\dot{\boldsymbol {e}} = \dot{\boldsymbol {e}}_{\hat{\alpha }_1} + \dot{\boldsymbol {e}}_{\hat{\alpha }_3} + \dot{\omega }_{\rm PN}\, \hat{\boldsymbol {k}} \times {\boldsymbol {e}} \, ,
\end{eqnarray*}
where
(18)
\begin{eqnarray*}
\dot{\boldsymbol {e}}_{\hat{\alpha }_1} \simeq \frac{\hat{\alpha }_1}{4c^2} \, \frac{M_{\rm p}- M_{\rm c}}{M_{\rm p}+ M_{\rm c}} \, n_{\rm b} \mathcal {V}_O \, {\boldsymbol w}_\perp
\end{eqnarray*}
(Damour & Esposito-Farèse 1992b) and
(19)
\begin{eqnarray*}
\dot{\boldsymbol {e}}_{\hat{\alpha }_3} \simeq \frac{3}{2\mathcal {V}_O}\, {\boldsymbol {a}}_{\hat{\alpha }_3} \times \hat{\boldsymbol {k}} = -\hat{\alpha }_3 \pi \, \frac{s_{\rm p}\nu }{\mathcal {V}_O} \, {\boldsymbol w}_\perp
\end{eqnarray*}
(Bell & Damour 1996). The velocity ${\boldsymbol w}$⊥ is the projection of the systemic CMB frame velocity into the orbital plane of the binary pulsar. To independently constrain $\hat{\alpha }_3$ from PSR J1713+0747 timing, we will include the $\hat{\alpha }_1$ limits obtained from PSR J1738+0333 by Shao & Wex (2012) in our analysis. Conversely, owing to its small orbital period, PSR J1738+0333’s possible $\hat{\alpha }_3$ effects would be much smaller than its $\hat{\alpha }_1$ effects and would have insignificant impact on the $\hat{\alpha }_1$ limits derived from this system. There is one assumption, however, that we have to make here. Since PSR J1713+0747 and PSR J1738+0333 have different masses, and therefore different sensitivities sp, we cannot assume that they would lead to identical $\hat{\alpha }_1$. For this reason, our analysis only applies to deviations from GR which exhibit only a moderate mass dependence of the strong-field parameter $\hat{\alpha }_1$, at least for neutron stars in the range of 1.3–1.5 M⊙.


In our spectroscopic analysis, we also quantify the reflection strength for each source. It is therefore interesting to compare our results to the typical assumptions made in CXB population-synthesis models. Indeed, they generally implement relatively similar assumptions. The reflection is always assumed to have a constant value

 within each population (e.g., Ballantyne et al. 2006; Treister et al. 2009; Akylas et al. 2012; Ueda et al. 2014) or possibly a function of the degree of obscuration (e.g., G07; Ueda et al. 2014; Esposito & Walter 2016) with no dispersion around a mean value. On average, we find median reflection values that are: (1) significantly lower (

) than those assumed (

) by CXB models and (2) exhibit a rather broad distribution with a median value relative to the whole sample of ∼0.4 (Table 6). Furthermore, we measure a significant anti-correlation with unabsorbed and intrinsic luminosities (the latter being more pronounced; see Figure 8 (lower panels) and Table 6). This trend is further confirmed by the findings of our companion paper on stacked NuSTAR spectra (DM17). In this context, we find sources with lower unabsorbed luminosities to have a median reflection (

) a factor of two stronger than more luminous ones (

). The broad R distribution reaches 50% percentile values a factor of two larger. When using intrinsic coronal luminosities, the differences exacerbates further by a factor of about two. A similar trend has been largely ignored by models; the one exception being the G07 model, for which QSOs have been assumed to have no reflection. G07 also assumes higher R for Type-1 sources (R = 1.3) compared to Type-2s (R = 0.88) in order to mimic a orientation-dependent disk reflection. Ueda et al. (2014) instead assumes a flat R = 0.5 from the disk and a torus-based contribution in the context of a luminosity- and redshift-dependent unified scenario in order to reproduce a total R = 1 for Seyfert galaxies.


Over the past decades, our understanding of the EoR has deepened from advances in the observational frontier of galaxies in the early universe. Dedicated surveys of high-redshift galaxies using the Hubble Space Telescope (HST) have measured a large sample of galaxies out to redshift as high as z ∼ 8 (Bouwens et al. 2015b; Finkelstein et al. 2015), which with the help of gravitational lensing has allowed the rest-frame ultraviolet (UV) galaxy luminosity function to be accurately constrained to a limiting magnitude of

M

UV

AB

≳
−
15

 (Atek et al. 2015; Bouwens et al. 2017; Yue et al. 2018). It is expected that, by the advent of the James Webb Space Telescope (JWST), not only the currently limited sample size of 9 ≲ z ≲ 12 galaxies and candidates (Ellis et al. 2013; Oesch et al. 2014, 2016, 2018) but also the constraints on the faint-end slope evolution of the UV luminosity function will be considerably enhanced (Mason et al. 2015; Yung et al. 2019). Combined with the Thomson scattering optical depth τes = 0.055 ± 0.009 inferred from the cosmic microwave background (CMB) temperature and polarization power spectra by Planck Collaboration et al. (2016a), the SFH based on a plausible faint-end extrapolation of the luminosity function suggests that the global reionization history could be explained by the “known” high-z galaxy population. If the average escape fraction of their ionizing photons into the intergalactic medium (IGM) is in the range of 10%–20% (e.g., Mason et al. 2015; Robertson et al. 2015; Bouwens et al. 2015a; Sun & Furlanetto 2016; Madau 2017; Naidu et al. 2020), there will be no need to invoke additional ionizing sources such as Population III stars and quasars. Nevertheless, the uncertainty associated with such an extrapolation indicates a fundamental limitation of surveys of individual objects—sources too faint compared with the instrument sensitivity, such as dwarf galaxies, are entirely missed by galaxy surveys, even though a significant fraction, if not the majority, of the ionizing photons are contributed by them (Wise et al. 2014; Trebitsch et al. 2018; but see also Naidu et al. 2020).


Millimeter-to-submillimeter emission tells us much about these hidden galactic nuclei, for physical scales of several parsecs to several 100 pc in this work. First, these wavelengths contain various emission lines tracing the dominant (by mass) interstellar medium in these systems, that is, cold-to-warm molecular gas (∼10 to a few 100 K). The lines tell us the physical conditions and kinematics of the gas and thereby tell us about the fueling and feedback of the nuclear activities as well as the mass distribution in the nuclei. Second, the short millimeter and submillimeter continuum is usually dominated by thermal emission from dust. It is this dust continuum that carries the vast majority of the bolometric luminosity of these nuclei. Because of that, high-resolution information available at sub/millimeter wavelengths is very useful to characterize the nuclei, complementing spatially unresolved observations at the far-IR where the continuum spectrum peaks. Third, the dust in these nuclei is moderately opaque (the opacity is around the order of unity) at sub/millimeter wavelengths to produce bright emission, and yet it also makes the nuclei penetrable except at the very center in extreme cases. As a guide, τ
1 mm = 1 corresponds to

NH2∼1025.5

 cm−2 for an environment similar to the Galactic disk (Planck Collaboration et al. 2014; Galliano et al. 2018). The structure of the nuclei can thus be probed more easily in the sub/millimeter than at far-IR and shorter wavelengths. At the same time, the continuum emission carries information about the dust opacity and hence the obscuring column density. Fourth, rotational lines from vibrationally excited HCN at sub/millimeter wavelengths can be an indicator and probe of the NGC 4418/Arp 220-type nuclei with significant obscuration (Sakamoto et al. 2009; Aalto et al. 2015, 2019; Imanishi et al. 2019). These nuclei are opaque enough to trap infrared photons to increase their inner temperature (i.e., the greenhouse effect), and the enhanced infrared radiation vibrationally excites molecules such as HCN (González-Alfonso & Sakamoto 2019). Their rotational lines in the sub/millimeter have a chance to leak out of the dusty nuclei and indicate their hot interior since dust opacity is lower at longer wavelengths. And finally, the Atacama Large Millimeter/submillimeter Array (ALMA) has dramatically enhanced our capability for high-resolution and high-sensitivity observations at sub/millimeter wavelengths. A characteristic ALMA resolution of 0.″1 is on the order of 10 pc at nearby hidden galactic nuclei at tens of megaparsecs. While the resolution does not match the accretion disks around supermassive black holes (r ≪ 1 pc), it is only an order of magnitude larger than that of young massive star clusters (Longmore et al. 2014) and corresponds to the outer radii of molecular tori around AGN (Combes et al. 2019). Therefore, ALMA observations are good at probing the obscuring interstellar medium around AGN (if any), circumnuclear star formation, and their effects on the circumnuclear material.


We should examine whether our picture is consistent with previous studies at a higher radio luminosity regime,

logL1.4GHz>27

. It is natural to assume that the unification between radio galaxies (HzRGs) and RLQSOs works at this regime. Actually, observational evidence for the unification has been found in this luminosity regime (Wylezalek et al. 2013). Hyperluminous radio galaxies/RLQSOs with

logL1.4GHz>27

 are expected to reside in the galaxy denser regions according to the L
jet − Σ relation,
16

16
The L
jet − Σ relation would break down in the extremely high-density regions, as a very large velocity dispersion of galaxies in the massive structures reduces the merger frequency; thus, SMBH spin and mass growth would be slow. as shown by arrow

C

 in Figure 7. There is definitive evidence that hyperluminous radio galaxies exist in the rich galaxy overdense regions (Venemans et al. 2007; Falder et al. 2010; Wylezalek et al. 2013; Hatch et al. 2014; Husband et al. 2016). For instance, Venemans et al. (2007) examined the environments of eight radio galaxies with log L
1.4 GHz ≥ 27.3 at z ∼ 2−5 and found six of them in overdense regions. These radio galaxies with available optical spectra are classified as either HERGs, RLQSOs, or dust-obscured RLQSOs (Eales & Rawlings 1993; Venemans et al. 2007; Drouart et al. 2016; Nesvadba et al. 2017). Hatch et al. (2014) also found that radio-loud AGNs with log L
1.4 GHz > 27.2 are significantly more present in high-density regions than radio-quiet AGNs at z ∼ 1−3. Falder et al. (2010) found that there is a positive correlation between the radio luminosities and the ambient environments of HzRGs and RLQSOs with log L
1.4 GHz = 26.65–28.65 at z ∼ 1. A similar trend is also found for seven luminous radio galaxies at z ∼ 2.2 (Husband et al. 2016). The faintest radio galaxy with log L
1.4 GHz = 26.6 in Husband et al. (2016) resides in the lowest-density region, which is comparable to that of blank fields (Figure 3), and the others with log L
1.4 GHz > 27 reside in the denser regions. On the other hand, Zeballos et al. (2018) found that some of the HzRGs with log L
1.4 GHz > 27 do not reside in the overdense regions of SMGs. However, Zeballos et al. (2018) could not deny the possibility that the overdense regions around those radio galaxies are buried due to the sample variance. In future, we could assess hyperluminous radio galaxy environments at z ∼ 4; as the survey area of the HSC-SSP will reach ∼1000 deg2, about 10 hyperluminous radio galaxies are expected to be found, assuming the radio galaxy luminosity function of Dunlop & Peacock (1990).


examined the environments of eight radio galaxies with log L
1.4 GHz ≥ 27.3 at z ∼ 2−5 and found six of them in overdense regions.


These radio galaxies with available optical spectra are classified as either HERGs, RLQSOs, or dust-obscured RLQSOs


The SFR of a gaseous clump with a density gradient is higher than if that same clump was made of uniform-density gas (Tan et al. 2006; Girichidis et al. 2011; Elmegreen 2011; Parmentier 2014, 2019). Centrally concentrated clumps actually process their gas into stars at a pace faster than expected based on their mean freefall time, because the density of their central regions is higher than the clump gas mean density. Parmentier (2019) refers to the ratio between the SFR of a clump with a density gradient, SFRclump, and the SFR of its top-hat equivalent, SFRTH, as the magnification factor ζ (see Equation (8) in Parmentier 2019):
1

The SFR of a clump can thus be written as (see Section 2 in Parmentier 2019, for a discussion)
2

In this equation, mgas is the gas mass hosted by the clump,1

1
The clump total mass mclump consists of the gas mass mgas and the mass mstars of the stars it has formed.
 and

 is the mean freefall time of the clump gas:
3

The latter is defined based on the clump gas mean density
4

with rclump the radius containing the clump gas mass mgas. ζ is the magnification factor, which quantifies by how much the density gradient of a clump enhances its SFR compared with what is expected based on its mean freefall time

. Equation (2) therefore allows one to disentangle the contribution of the density gradient, embodied by the ζ factor, from that of the star formation efficiency per freefall time itself. Parmentier (2019) coins

 the intrinsic star formation efficiency per freefall time since it is independent of the clump density gradient.2

2
This is as opposed to the (globally) measured star formation efficiency per freefall time, defined as

 (Parmentier 2019).
 For a top-hat profile, ζ = 1 and the SFR obeys
5

Note that the gas density of the top-hat equivalent is equal to the gas mean density

 of the centrally concentrated clump (same gas mass mgas enclosed within the same clump radius rclump).


refers to the ratio between the SFR of a clump with a density gradient, SFRclump, and the SFR of its top-hat equivalent, SFRTH, as the magnification factor ζ


The SFR of a clump can thus be written as (see Section 2 in


coins

 the intrinsic star formation efficiency per freefall time since it is independent of the clump density gradient.


This is as opposed to the (globally) measured star formation efficiency per freefall time, defined as


Various models of the gas and dust distribution in the GC exist: a simple bar model (Morris & Serabyn 1996); variations of a spiral arm model in which the apparent ring is formed by the inner part of two spiral arms (Sofue 1995; Sawada et al. 2004; Rodríguez-Fernández & Combes 2008; Rodríguez-Fernández 2011; Ridley et al. 2017); a closed, twisted elliptical ring detected in dust emission (Molinari et al. 2011), and a sequence of open-ended gas streams (Kruijssen et al. 2015, hereafter K15). K15 highlight the impossibility of closed orbits in extended gravitational potentials and provide a better fit to single-dish ammonia emission in position–position–velocity (PPV) space, as was confirmed by Henshaw et al. (2016b) for three molecular species in single-dish observations. Thus, we focus on this model and call it the “stream model” in contrast to the “ring model” and the “spiral arm” model. According to the stream model, the stream of GC molecular clouds oscillates radially and vertically, and can be traced for ∼1.5 revolutions around the GC. The radial oscillation periodically brings dense molecular gas closer (r ∼ 60 pc at pericenter) to the gravitational center (traced by Sgr A*) and deeper into the potential, whereas the apocenter lies at r ∼ 120 pc. As the CMZ clouds slowly evolve toward low virial ratios as the turbulent energy dissipates (Krumholz & Kruijssen 2015; Walker et al. 2015; Henshaw et al. 2016a), it is statistically more likely that cloud collapse occurs at the pericenter where the compressive tidal forces are strongest along the orbit and thus a cloud receives the final nudge for the transition to self-gravitation. Subsequent SF stages will then occur downstream from the pericenter passages and could potentially be observed as an SF sequence if the orbit is sufficiently sampled with molecular clouds that start to collapse at a similar point of their orbits. This model of triggered star formation was first proposed by Longmore et al. (2013b) based on the observation of different evolutionary SF stages along the dust ridge. The two young stellar clusters in the GC, Arches and Quintuplet, may also support this model as their orbits and ages are consistent with formation at a common point after collapse of their respective parent clouds at a pericenter passage (Figure 1 of Stolte et al. 2014; Kruijssen et al. 2015). In addition to stars and star formation tracers, cloud properties are also expected to show evolutionary behavior along the gas streams; this has not been thoroughly tested yet. Weak hints toward rising gas temperatures in the dust ridge are suggested by Ginsburg et al. (2016), while a recent paper by Kauffmann et al. (2017b) based on N2H+ data from the Galactic Center Molecular Cloud Survey can neither confirm nor exclude the possibility of triggered evolution when examining mass-size relation and SF suppression. Both analyses were based on a low number of measurements (∼35 measurements in dust ridge clouds in Ginsburg et al. 2016 and six clouds in Kauffmann et al. 2017b) and lack the statistical power to detect potential evolution in the presence of scatter.


Various models of the gas and dust distribution in the GC exist:

a closed, twisted elliptical ring detected in dust emission


based on the observation of different evolutionary SF stages along the dust ridge.

This model of triggered star formation was first proposed by


The two young stellar clusters in the GC, Arches and Quintuplet, may also support this model as their orbits and ages are consistent with formation at a common point after collapse of their respective parent clouds at a pericenter passage (Figure 1 of


based on N2H+ data from the Galactic Center Molecular Cloud Survey can neither confirm nor exclude the possibility of triggered evolution when examining mass-size relation and SF suppression.


Both analyses were based on a low number of measurements (∼35 measurements in dust ridge clouds in Ginsburg et al. 2016 and six clouds in

and lack the statistical power to detect potential evolution in the presence of scatter.


With only

 as a free parameter, different studies suggest that

 = 10%–20% at z > 6 is necessary for galaxies to fully ionize the universe (Bolton & Haehnelt 2007b; Finkelstein et al. 2012; Kuhlen & Faucher-Giguère 2012; Bouwens et al. 2015a, 2015c; Mitra et al. 2015; Robertson et al. 2015; Khaire et al. 2016; Price et al. 2016). Simulations do not agree on

 at high redshifts and find either very high (e.g., Sharma et al. 2016) or very low values (e.g., Gnedin et al. 2008; Ma et al. 2015). Furthermore, they predict a strong dependence on the mass of dark matter halos and star formation (e.g., Wise & Cen 2009; Razoumov & Sommer-Larsen 2010). Direct observational constraints on

 in the EoR are not possible because of the increasing opacity of the IGM to LyC photons at z > 4 (e.g., Madau 1995; Inoue et al. 2014). Except for one strong LyC emitter at z = 3.2 with

 > 50% (de Barros et al. 2016; Vanzella et al. 2016b), the handful of confirmed LyC emitters at z  3 all show consistently

 ≲ 8% (Steidel et al. 2001; Leitet et al. 2013; Borthakur et al. 2014; Cooke et al. 2014; Siana et al. 2015; Izotov et al. 2016a, 2016b; Leitherer et al. 2016; Smith et al. 2016). The numerous non-detections listed in the literature show upper limits of

 ∼ 2%–5% over large ranges of redshift (Vanzella et al. 2010; Sandberg et al. 2015; Grazian et al. 2016; Guaita et al. 2016; Rutkowski et al. 2016; Vasei et al. 2016). If galaxies are responsible for ionizing the universe at z > 6, then clearly their population-averaged LyC escape fraction needs to increase substantially with redshift by at least a factor of two (see also Inoue et al. 2006). What methods can we use to access

 observationally in the EoR? Radiative transfer models suggest a correlation between the ratio

/

 and

 in density-bound H ii regions (e.g., Nakajima & Ouchi 2014), and a handful of recent observational studies verify this positive correlation (de Barros et al. 2016; Izotov et al. 2016a, 2016b; Vanzella et al. 2016a, 2016b). The increased

/Hβ ratios found in z > 5 galaxies (e.g., Stanway et al. 2014; Roberts-Borsani et al. 2015; Faisst et al. 2016a) hint toward an increasing

/

 ratio for the global population of galaxies at high redshifts and therefore could be the smoking gun for a strong evolution in

(z). Currently, the

 line cannot be measured spectroscopically at z > 4, and the use of broad-band photometry to determine

 line strengths is degenerate with the 4000 Å Balmer break, which is a strong function of age and other galaxy parameters. However, local analogs of high-redshift galaxies can be used to probe the physical properties of these galaxies.


Radiative transfer models suggest a correlation between the ratio

/

 and

 in density-bound H ii regions

and a handful of recent observational studies verify this positive correlation


To detect supernova neutrinos with the KamLAND data, a cluster of DC events is required: two DC events within a 10 s window. The very low DC rate enables us to use the minimum cluster condition (two DC events). An accidental DC event cluster is a background for supernova neutrinos. Based on the DC event studies, we evaluate the number of accidental cluster events (

≡nclusteraccidental

) via Monte-Carlo (MC) simulation. Accidental cluster rates and accumulation of

nclusteraccidental

 are shown in Figure 1. Before distillation campaigns, the DC event is mainly caused by reactor -

ν¯e

, and 13C(α, n)16O interaction has a secondary contribution (Abe et al. 2008). After distillation campaigns, the DC event by 13C(α, n)16O interaction decreases but reactor -

ν¯e

 is still a dominant component (Gando et al. 2011). Japanese reactors were shut down after the Great East Japan Earthquake in 2011. In the reactor-off phase (Gando et al. 2013), the rate of reactor -

ν¯e

 events is decreased. These trends of DC event rate are also shown in Figure 1. In an energy region above 10 MeV, fast neutrons and atmospheric neutrinos have dominant contribution on the number of DC events (Abe et al. 2022b). Typically, the DC candidate rate is ∼1 day−1 in the reactor-on phase and 0.1 day−1 in the reactor-off phase. Usually, we call a KamLAND data set a “run,” which consists of typically 24 hr data. The number of expected DC events is estimated in run by run and the number of accidental clusters is calculated from that. Taking into account that time differences between each run are more than 10 s, there should be no accidental clusters that occur across two runs. As a result, the accumulated number of accidental clusters is

nclusteraccidental=0.32−0.04+0.02

 clusters, and this rate is 0.023 cluster yr−1 on average. The error of

nclusteraccidental

 includes systematic uncertainties of the number of expected DC events by reactor -

ν¯e

, atmospheric neutrino, and fast neutron.


Before distillation campaigns, the DC event is mainly caused by reactor -

ν¯e

, and 13C(α, n)16O interaction has a secondary contribution


After distillation campaigns, the DC event by 13C(α, n)16O interaction decreases but reactor -

ν¯e

 is still a dominant component


One factor that we have not taken into account so far is that the strength of an AGN may affect how strong the corresponding emission from the NLR is. Thus, it may be the case that lower-luminosity or lower-accretion-rate AGNs are more easily overwhelmed by star formation. Therefore, we next explore whether correctly and incorrectly classified X-ray AGNs differ in their intrinsic AGN properties. One of the indicators of the accretion strength of an AGN in the optical regime is the luminosity of the forbidden [O iii] line,

, which scales with the bolometric luminosity of the AGN (Mulchaey et al. 1994; Heckman et al. 2005; LaMassa et al. 2010; Azadi et al. 2017; Glikman et al. 2018). A related quantity,

 divided by the fourth power of the stellar velocity dispersion, σ4, can provide an estimate of the Eddington ratio of the AGN (Kewley et al. 2006). Trump et al. (2015) suggested that higher sSFR galaxies must have more efficient AGNs in order to be classified as BPT–AGNs because of the relative contributions from star formation. We show

 and

 versus stellar mass in Figures 8 and 9, respectively, with the BPT star-forming galaxies from GSWLC-M1 composing the backgrounds in Figures 8(a) and 9(a) and the BPT–AGNs from GSWLC-M1 making up the backgrounds in Figures 8(b) and 9(b). In Figures 8 and 9, there appear to be two distinct sequences for the two BPT types when mass is included, where in Figure 8(a), the background BPT star-forming galaxies form one sequence where

 appears to correlate with stellar mass, including galaxies with masses up to

, while the star-forming galaxies in Figure 9(a) do not show a strong dependence of proxy Eddington ratio with stellar mass, which is expected because

 from star-forming galaxies primarily comes from star formation. The BPT–AGNs form sequences in Figures 8(b) and 9(b) that have no apparent dependence on mass with masses ranging roughly from

. The existence of these two sequences is apparently the case because of the different sources of [O iii] emission that are dominant in BPT star-forming galaxies versus BPT–AGNs. The majority of the BPT–H ii are more consistent with the background star-forming galaxies in Figures 8(a) and 9(a), whereas the BPT–AGNs all appear consistent with the background BPT–AGNs shown in Figures 8(b) and 9(b), which is to be expected if the NLR dominates [O iii] emission. Six BPT–H ii are more consistent with the background BPT–AGNs in Figures 8(b) and 9(b), and if the NLR is indeed the source of their [O iii] emission, their Eddington ratios are much lower than the average BPT–AGNs, suggesting that they have relatively inefficient accretion. We will return to these outliers in Section 3.2.3.


The disk-wind outflows that accompany highly super-Eddington accretion carry a total mass-loss rate

Ṁw

, which nearly equals the entire mass-transfer rate, i.e.,

Ṁw∼Ṁ

 (e.g., Blandford & Begelman 1999; Hashizume et al. 2015), with only a small fraction making its way down to the BH/NS surface. Depending on the radial scale of their launching point in the disk or binary, the outflows can possess a range of speeds, from values similar to the binary orbital velocity, v
orb ≲ 100 km s−1, to the transrelativistic speeds v
j ≳ 0.3c that characterize jet-like outflows from the innermost radii of the disk (and similar to those observed in SS 433 and ULX; e.g., Jeffrey et al. 2016 and references therein). In ADIOS models for the radial disk structure (Blandford & Begelman 1999; Margalit & Metzger 2016), the mass accretion rate decreases as a power law with radius

Ṁ∝rp

 for

r∈[rin,rout≲Rtr]

, where r
out ≫ r
in. Assuming the outflows local to each annulus in the disk reach an asymptotic velocity equal to the local disk escape velocity, the kinetic energy-averaged wind velocity is given by (e.g., Metzger 2012)
6

vw≈p1−pGM•rinrinroutp≈0.05cM•10M⊙0.3rin6rg−0.2routR⊙−0.3,

where in the second equality we have normalized r
in to gravitational radius r
g = GM
•/c
2 and take p = 0.6 (as supported by hydrodynamical simulations of radiatively inefficient accretion flows; e.g., Yuan & Narayan 2014; Hu et al. 2022). In what follows, we take v
w ∼ 0.03c as a fiducial value, consistent with that expected for a main-sequence or moderately evolved donor star feeding a relatively compact accretion flow, r
out ∼ R
⊙. However, lower value v
w ≲ 0.01 c may be appropriate for a giant donor star (r
out ≳ 102
R
⊙) or for larger values of p. The kinetic luminosity of the wind is then
7

Lw≈12Ṁwvw2≈1042ergs−1Ṁw105ṀEdd×M•10M⊙vw,92ergs−1,

where v
w,9 = v
w/(109 cm s−1) and hereafter we adopt the short-hand notation Y
x ≡ Y/10x for quantities given in cgs units. In analytic estimates hereafter, we shall typically fix the value of M
• = 10M
⊙ and use L
w and

Ṁw

 interchangeably.


In hydrodynamic and magnetohydrodynamic systems, different fluctuations interact with each other nonlinearly, generating turbulence (Matthaeus et al. 2015). One of the most important characteristics of turbulence is the existence of intermittency among various scales. Spatial intermittency manifests as coherent structures with large gradient at small scales. In ordinary fluids, coherent structures include a tangle of vortex filaments where vorticity is highly concentrated (Frisch 1995). In the turbulent plasma environments, examples of intermittent structures include current sheets, discontinuities, shock waves, and Alfvénic vortices (Veltri & Mangeney 1999; Sundkvist et al. 2007; Lion et al. 2016; Wang et al. 2019). The intermittency will influence the measured statistical properties of the fluctuating quantities. For example, the growth of flatness with decreasing scales indicates the existence of intermittency. Plenty of simulation works (Servidio et al. 2011; Karimabadi et al. 2013; Tenbarge & Howes 2013; Wan et al. 2015, 2016; Zhang et al. 2015) have indicated that dissipation, acceleration, and thermalization of turbulent plasmas mainly take place near intermittent structures on kinetic scale, while the actual physical mechanisms behind dissipation remain unclear. There are a variety of diagnostic approaches to measure intermittency. (1) Probability density functions (PDFs) of scale-dependent field increments develop heavy tails because of intermittency, and the tail is more enhanced with increasing intermittency (Marsch & Tu 1994). (2) The scale dependency of the normalized fourth-order moment, known as flatness,

, where δv = v(t + τ)−v(t), is an alternative representation. This quantity increases as the intermittency becomes more significant along with decreasing scales. (3) The p-th order structure function,

, will be larger than that for Gaussian PDF without intermittency and the scaling exponent, ζ(p), has a nonlinear form when intermittency presents. Based on such diagnostic approaches, the properties of intermittency in plasmas have been widely investigated in the solar atmosphere, magnetosheath, solar wind, termination shock, etc. (Burlaga 1991a, 1991b; Marsch & Liu 1993; Horbury et al. 1995; Macek et al. 2011, 2017; Chasapis et al. 2018). These observational analyses have revealed that intermittency is widely existing in the heliosphere and immensely influencing plasma dynamics.


In ordinary fluids, coherent structures include a tangle of vortex filaments where vorticity is highly concentrated


In hydrodynamic and magnetohydrodynamic systems, different fluctuations interact with each other nonlinearly, generating turbulence


have indicated that dissipation, acceleration, and thermalization of turbulent plasmas mainly take place near intermittent structures on kinetic scale, while the actual physical mechanisms behind dissipation remain unclear.


Probability density functions (PDFs) of scale-dependent field increments develop heavy tails because of intermittency, and the tail is more enhanced with increasing intermittency


Based on such diagnostic approaches, the properties of intermittency in plasmas have been widely investigated in the solar atmosphere, magnetosheath, solar wind, termination shock, etc.


J0815+4729 is a main-sequence star (T

eff

=
6215
±
82

 K,

log
g

 = 4.7±0.5) with a metallicity of [Fe/H] ≤ −5.8 dex. Finding unevolved stars at this extremely low metallicity is very important since their stellar surface composition is not expected to be significantly modified by any internal mixing processes as in giant stars (Spite et al. 2005). J0815+4729 is similar to HE 1327–2326 in regard to its carbon enhancement, effective temperature, and metallicity. HE 1327–2326 is considered a turn-off/subgiant star, while J0815+4729 appears to be a dwarf. The ISIS spectrum of HE 1327–2326 indicates a metallicity of [Fe/H] ∼ −4.9 since the stellar Ca line is blended in that spectrum with the ISM features (Aguado et al. 2017b). However, the authors proposed a simple analysis taking into account the ISM absorption based on the UVES spectrum of HE 1327–2326. For J0815+4729, we require a high-resolution spectrum to clearly isolate the stellar Ca feature from possible additional ISM lines, and thus together with the detection of Fe lines, to establish the metallicity of this star. There are two other confirmed dwarf stars in this metallicity regime: one without any detectable carbon, J1029+1729 (Caffau et al. 2011), and another carbon-enhanced unevolved star, J1035+0641 (Bonifacio et al. 2015). The majority of extremely metal-poor stars shows overabundances of carbon, [C/Fe] > 0.7, and it appears that carbon-enhanced metal-poor (CEMP) stars split into two groups, with dramatically different carbon abundances (see, e.g., Beers & Christlieb 2005; Allende Prieto et al. 2015; Bonifacio et al. 2015 and references therein). The two carbon bands (high and low) studied have different origins. On the one hand, CEMP stars in the high-carbon band (A(C) ∼ 8.2) are probably produced by mass transfer from a binary companion, most likely an AGB star (Starkenburg et al. 2014). On the other hand, objects lying in the low-carbon band (A(C) ∼ 6.8) are thought to show the original carbon abundance inherited by the star from the ISM (Stancliffe 2009; Bonifacio et al. 2015; Abate et al. 2016). J0815+4729 has an abundance ratio of

[

C

/

Fe

]

≥
+
5.0

 dex  corresponding to A(C) ∼ 7.7 dex (adopting [Fe/H] ≤ −5.8). In Figure 4 (bottom panel), we show the carbon abundance ratio [C/Fe] for all stars at [Fe/H]  −4.5. All stars in this metallicity regime are considered to belong to the low-carbon band (Bonifacio et al. 2015), except for J0815+4729, which appears to be in between the low- and high-carbon bands. Both metallicity and carbon abundance are considered upper and lower limits, respectively. High-resolution spectra would be very useful to measure other elemental abundances and investigate the properties of the first supernovae. In particular, the barium abundance, or that of any other s-element, is not measurable from ISIS or OSIRIS spectra, and this is required to determine whether J0815+4729 is a CEMP-s, CEMP-r, or i-process star (Hampel et al. 2016). If we establish the abundance pattern, we will learn about the progenitor properties. Finally, the radial velocity accuracy from medium-resolution data is not enough to discard variations among different exposures, which would be indicative of binarity.


The majority of extremely metal-poor stars shows overabundances of carbon, [C/Fe] > 0.7, and it appears that carbon-enhanced metal-poor (CEMP) stars split into two groups, with dramatically different carbon abundances (see, e.g.,


On the other hand, objects lying in the low-carbon band (A(C) ∼ 6.8) are thought to show the original carbon abundance inherited by the star from the ISM


In Figure 4 (bottom panel), we show the carbon abundance ratio [C/Fe] for all stars at [Fe/H]  −4.5. All stars in this metallicity regime are considered to belong to the low-carbon band

except for J0815+4729, which appears to be in between the low- and high-carbon bands.


The distributions of separations at apoapsis and periapsis from our LSMC analysis shown in the left panel of Fig. 11 indicate a range at 68%∼18–57 au. If the companion is coplanar with the outer disk, a ∼0.25 M⊙ companion with a 50 au apoapsis and e ∼ 0.5 − 0.7 would in principle create a region of orbital instability extending out to ∼100 au (Holman & Wiegert 1999), which is the innermost possible location for the outer disk (e.g., Fukagawa et al. 2006; Casassus et al. 2012; Rameau et al. 2012; Rodigas et al. 2014; Avenhaus et al. 2017). Nevertheless, the inclination and longitude of node of HD 142527B disagree with those of the outer circumstellar disk (i = 28°, Ω = 160°, Verhoeff et al. 2011; Perez et al. 2015), as has been reported by Lacour et al. (2016). This result would at first sight rule out that HD 142527B is responsible for the outer disk truncation. However, recent hydrodynamical simulations have shown that for an eccentric companion with an almost polar inclination to the outer disk that approaches its periapsis passage, the interactions of companion and disk can reproduce several of the main observed disk features, such as its large cavity, and with the correct position angles, its spiral features and shadows (Price et al. 2018). This orbital configuration is broadly consistent with the results from our orbital analysis. On the other hand, the orbital plane of the companion is close to the plane of the inner circumstellar disk, as previously suggested by Lacour et al. (2016). The inner disk has a position angle of 110 ± 10° from CO(6-5) kinematics measured with ALMA (Casassus et al. 2015). Its mean radius is estimated to be about 10 au from near-infrared interferometric observations (Anthonioz et al., in prep.). From MIR imaging and SED modeling, Verhoeff et al. (2011) derived a maximum radial extension for the inner disk up to 30 au. However, Avenhaus et al. (2014) imaged the inner circumstellar environment down to ∼0.1″ (15 au) in polarized scattered light with NaCo, but did not detect traces of an inner disk. More recently, Avenhaus et al. (2017) imaged the disk in visible polarized light with SPHERE down to 25 mas (∼4 au) and found evidence for dust scattering close to the star, although it remains unclear if this scattering is related to the inner disk because of geometrical discrepancies with predictions from a modeling of the shadows projected onto the outer disk (Marino et al. 2015). Using our orbital analysis of the companion, we derived the distribution of its separation at periapsis (Fig. 11, right panel) and find a 68% confidence interval of ∼10 − 12 au. This results indicates that the shape of the inner disk is strongly affected by HD 142527B.


If the companion is coplanar with the outer disk, a ∼0.25 M⊙ companion with a 50 au apoapsis and e ∼ 0.5 − 0.7 would in principle create a region of orbital instability extending out to ∼100 au


which is the innermost possible location for the outer disk (e.g.,


However, recent hydrodynamical simulations have shown that for an eccentric companion with an almost polar inclination to the outer disk that approaches its periapsis passage, the interactions of companion and disk can reproduce several of the main observed disk features, such as its large cavity, and with the correct position angles, its spiral features and shadows


imaged the inner circumstellar environment down to ∼0.1″ (15 au) in polarized scattered light with NaCo, but did not detect traces of an inner disk.


We first assess the mass–size relation regulated by collision-induced turbulence, i.e. the mass–size relation of clumps in the molecular ring. In our clump–clump collision scenario, the turbulent energy injected by the collisions cascades down to small scales and counterbalances self-gravity to produce a rough virial equilibrium (i.e. the clumps are able to maintain virialization; see Section 5.1). These virialized clumps should always have their energy dissipation rates (per unit mass) $\dot{\epsilon }_{\rm diss,clump}=\frac{1}{4}\frac{\sigma _{\rm obs,los}^3}{R_{\rm clump}}$ (equation 14) match their virial energy dissipation rates (per unit mass)
(33)$$\begin{eqnarray}
\dot{\epsilon }_{\rm diss,vir} =\frac{1}{4}\left(\alpha _{\rm vir,crit}\, GM_{\rm clump}/5\right)^{3/2}R_{\rm clump}^{-5/2}
\end{eqnarray}$$(Li 2017b), i.e. $\dot{\epsilon }_{\rm diss,clump}\approx \dot{\epsilon }_{\rm diss,vir}$, where as before αvir, crit ≈ 2 is the boundary between gravitationally bound and unbound objects (Kauffmann et al. 2013, 2017). The virial energy dissipation rate (per unit mass) $\dot{\epsilon }_{\rm diss,vir}$ is the energy dissipation rate (per unit mass) a molecular gas structure would have if it were virialized. We therefore expect
(34)$$\begin{eqnarray}
\dot{\epsilon }_{\rm diss,vir}\approx \dot{\epsilon }_{\rm diss,clump}\approx \dot{\epsilon }_{\rm inject,coll}
\end{eqnarray}$$for the clumps in the molecular ring of NGC 404, where we have used $\dot{\epsilon }_{\rm diss,clump}\approx \dot{\epsilon }_{\rm inject,coll}$, i.e. the energy dissipation rates (per unit mass) of clumps are approximately equal to the energy dissipation rates (per unit mass) from clump–clump collisions in the molecular ring (see Section 4.4). We indeed find very good agreements between the estimated $\dot{\epsilon }_{\rm diss,vir}$, the measured $\dot{\epsilon }_{\rm diss,clump}$, and our predicted $\dot{\epsilon }_{\rm inject,coll}$ in the molecular ring (see panel c of Fig. 9), providing more evidence that the turbulence induced by clump–clump collisions can support and maintain molecular ring clumps in virial equilibrium.


As we see, in the magnetospheres of Crab-like pulsars the Cherenkov drift resonance might lead to the hard X-rays. In the previous section, we have seen that this emission inevitably leads to the non-vanishing pitch angles (see equation 14), generating the synchrotron emission. In Fig. 2 we show the behaviour of energy of synchrotron photons, $\epsilon _{_{{\rm GeV}}}$ versus γb in the interval [0.5; 1] × 107 for different values of Lorentz factors of the plasma component: γp = 1 (solid line), γp = 10 (dashed line), γp = 50 (dash–dotted line). We have taken into account the dipolar character of the magnetic field, $B\approx 3.2\times 10^{19}\times \sqrt{P\dot{P}}\times R^3/R_{lc}^3$ G (Goldreich & Julian 1969), where R = 106 cm is the neutron star's radius. The rest of the parameters is: $P = P_{_{{\rm Cr}}}$, $\dot{P}=\dot{P}_{_{{\rm Cr}}}$, $n_b = n_{_{GJ}}$, where $n_{_{GJ}} = B/Pec$ is the Goldreich–Julian number density (Goldreich & Julian 1969). As it is evident from the plots by increasing γp, the resulting emission energy increases as well, which directly follows from the analytical behaviour, $\epsilon _{_{{\rm GeV}}}\propto \gamma _p^{1/8}$ (this can be seen by combining equations 8, 12–14). We see that the QLD by means of the feedback of ChDI guarantees the synchrotron emission over 100 GeV. We can roughly estimate the possible VHE luminosity by multiplying the number of particles involved in the process with the single emission power, $P_{{\rm syn}}\approx 2e^4\gamma _b^2B^2\langle \psi \rangle ^2/3m^2c^3$. It is worth noting that the thickness of the layer, d, where the electrons accelerate to the highest energies, is of the order of Rlc/2γb. Then, by taking into account the corresponding number of particles, $4\pi R_{lc}^2n_{_{GJ}}d$, one can show that in the emission energy band 150 GeV the highest energy electrons lead to the bulk luminosity of the order of 1033 erg s−1 which is in a good agreement with observations (Lessard et al. 2000; Albert et al. 2008).


The rest of the parameters is: $P = P_{_{{\rm Cr}}}$, $\dot{P}=\dot{P}_{_{{\rm Cr}}}$, $n_b = n_{_{GJ}}$, where $n_{_{GJ}} = B/Pec$ is the Goldreich–Julian number density


Then, by taking into account the corresponding number of particles, $4\pi R_{lc}^2n_{_{GJ}}d$, one can show that in the emission energy band 150 GeV the highest energy electrons lead to the bulk luminosity of the order of 1033 erg s−1 which is in a good agreement with observations


The Barnard 5 (B5) dark cloud is located at the North-East of the Perseus complex (235 pc; Hirota et al. 2008). It contains four known protostars of which the most prominent is the Class I protostar IRS1. SCUBA continuum emission maps at 850 μm carried out with the JCMT only revealed dust emission towards known protostars and at about 2–4 arcmin north from IRS 1 (Hatchell et al. 2005) whilst molecular maps of CO, NH3, and other species show several chemically differentiated clumps (S. B. Charnley, in prep.). Methanol emission in B5 displays a particularly interesting distribution since it shows a bright so-called methanol hotspot at about 250 arcsec north-west from IRS 1, a region showing no infrared sources and no detected sub-mm continuum emission (see Fig. 1 in Hatchell et al. 2005; Wirstrom et al. 2014). The subsequent detection of abundant water, with absolute abundances of about 10-8, with the Herschel Space Observatory by Wirstrom et al. (2014) suggests that efficient non-thermal processes triggering the evaporation of icy methanol and water are at work. The methanol hotspot therefore represents an ideal target to detect cold COMs in dark clouds since they are supposed to be formed from methanol, either at the surface of interstellar grains, or directly in the gas phase after the evaporation of methanol. The Perseus molecular cloud was recently mapped with Herschel as part as the Gould Belt Survey key program (André et al. 2010) using the photometers PACS and SPIRE in five bands between 70 μm and 500 μm.Figure 1 presents the Barnard 5 molecular cloud as seen by Herschel/SPIRE at 250 μm. The Herschel data has been retrieved from the Herschel Science Archive. The data were calibrated and processed with the Herschel pipeline and are considered as a level-3 data product. Dust emission is compared with the integrated intensity map of the A+-CH3OH 30 − 20 transition as observed with the IRAM 30 m telescope (this work).The methanol hotspot is located between the two dense cores East 189 and East 286 (Sadavoy, priv. comm.) revealed by Herschel for the first time, and the methanol emission peaks at the edge of core East 189. East 189 and East 286 have the following properties derived through a fitting of their observed spectral energy distribution: M = 0.5M⊙, Td = 12 K, R = 3.6 × 10-2 pc, nH = 3 × 104 cm-3, and M = 0.7M⊙, Td = 9.9 K, R = 2.7 × 10-2 pc, nH = 1.2 × 105 cm-3 (Sadavoy  2013).


The subsequent detection of abundant water, with absolute abundances of about 10-8, with the Herschel Space Observatory by

suggests that efficient non-thermal processes triggering the evaporation of icy methanol and water are at work.

The methanol hotspot therefore represents an ideal target to detect cold COMs in dark clouds since they are supposed to be formed from methanol, either at the surface of interstellar grains, or directly in the gas phase after the evaporation of methanol.


The Barnard 5 (B5) dark cloud is located at the North-East of the Perseus complex (235 pc;


East 189 and East 286 have the following properties derived through a fitting of their observed spectral energy distribution: M = 0.5M⊙, Td = 12 K, R = 3.6 × 10-2 pc, nH = 3 × 104 cm-3, and M = 0.7M⊙, Td = 9.9 K, R = 2.7 × 10-2 pc, nH = 1.2 × 105 cm-3


Next, we fit the emission lines using the code described in detail in Dong et al. (2005). The optical spectra show strong, narrow emission lines such as [O ii] λ3727, [O i]

,

, [N ii]

, and [S ii]

 [O iii]

 is instead weak. As we verified above, broad

 is evidently presented (see also Greene & Ho 2007; Dong et al. 2012). The SDSS data are used to fit the blue part such as in the

+[O iii] region (as not covered by the MMT spectrum), and the MMT data for the red part such as [O i] lines, the

+[N ii] region, and [S ii] lines. We fit the MMT spectrum first. Every narrow or broad component of the emission lines is modeled with Gaussians, starting with one Gaussian and adding in more if the fit can be improved significantly according to the F-test. The [S ii]

 doublet lines are assumed to have the same profile and fixed in separation by their laboratory wavelength; the same is applied to [N ii]

 doublet lines, and to [O i]

 and [O i]

 emission lines. The flux ratio of the [N ii] doublet

 is fixed to the theoretical value of 2.96 (e.g., Acker et al. 1989; Storey & Zeippen 2000). In the best-fit model, it turns out that one Gaussian is used for the broad component of

, while two Gaussians for the narrow component of

, each line of the [N ii], and [S ii] doublet and the two [O i] lines, respectively. The reduced

 is 1.21 for the

+[N ii] region. Note that the [N ii]

 and [S ii] doublet lines show asymmetric line profiles clearly in the MMT spectrum, which are actually common in high-resolution spectra (Xiao et al. 2011); thus these narrow lines, and narrow

 also, need two Gaussians to model. Then we fit the

+[O iii] region based on the SDSS spectrum. As the spectral signal-to-noise (S/N) of this region is not high and the

 emission line is much weaker than

, we assume the broad and narrow components of

 have the same profiles as the respective components of

. The [O iii]

 doublet lines are assumed to have the same profile, and the flux ratio of

 is fixed to the theoretical value of 2.98 (e.g., Storey & Zeippen 2000; Dimitrijević et al. 2007); the best-fit model turns out to adopt two Gaussians for each [O iii] doublet line. The reduced

 is 0.95 for the

+[O iii] region. [O ii] λ3727 is fit independently, and one Gaussian is sufficient to model. We also fit the

 lines of the N and S branches of the extended emission. One Gaussian is statistically sufficient for either branch, with reduced

 of 0.95 (N) and 0.97 (S), respectively. The best-fit line parameters, centroid, FWHM (corrected for instrumental broadening), and flux of every branch agree well with those directly integrated/measured from the spectra.


The [O iii]

 doublet lines are assumed to have the same profile, and the flux ratio of

 is fixed to the theoretical value of 2.98 (e.g.,


The de-projected distance that corresponds to the core-position changes of Markarian 421 is approximately 3.5 ∼ 8.7 pc (equivalent to (1.0 − 2.6) × 105 Schwarzschild radii (Rs)) for a jet-viewing angle of 2°–5° (Lico et al. 2012). One of the leading models for explaining the blazar phenomenon is the so-called internal-shock model, in which discrete ejecta with higher speeds catch up with the preceding slower ejecta, and the collision leads to a shock wave in the colliding ejecta (Spada et al. 2001; Guetta et al. 2004). Based on this model, core wandering can be naturally explained because the model predicts some scattering of the shock locations caused by the randomness of the ejecta speed. Therefore, we assume that the distances of the shock locations from the black hole immediately after the flare,

, are proportional to the product of the square of the bulk Lorentz factor (

) of the ejecta and the separation length (Δ) of the colliding ejecta. Assuming that a typical separation length is a scale of about 10 times the innermost stable circular orbit for non-rotating black holes (

), in light of numerical simulations of relativistic outflows around black holes (McKinney et al. 2013), very fast ejecta with a Lorentz factor of approximately

 are required to explain the observed magnitude of the core wandering. The

 is comparable to previously reported maximum values (Γ of 40 ∼ 50; Lister et al. 2009, 2013) based on statistical studies by a long-term monitor using VLBI. Moreover, the radio core returned to the reference stationary position after 2012 March 12 (the 10th epoch). This can be explained if the emission from the reference position is identical to the radio core that is in a quiescent state. The radio-core emission in a quiescent state is expected to be persistent. As one of the possibilities for reproducing the observed core wandering, this persistent radio core was obscured by the flare-associated radio core, which therefore must be optically thick against SSA and short lived (a lifetime of approximately 2 weeks), for several epochs. Such a flare-associated radio core must be located at least

 downstream from the persistent radio core. In addition, Koyama et al. (2015) recently conducted multi-epoch astrometric observations of Markarian 501, which is also one of the most nearby TeV blazars by using VERA. However, they detected no significant positional changes of the radio core during its quiescent state. This fact, therefore, seems to support the radio core wandering seen after the large X-ray flare.


In addition to the processes that release SiO into the gas phase, it is also important to
        discuss the properties of the responsible shock. Shocks are divided into three types: jump
        shocks (J-type, Hollenbach & McKee 1979),
        continuous shocks (C-type, Draine 1980), and a
        mixture of both, which is a C-type shock with an introduced discontinuity (CJ-type, Chieze et al. 1998). The type of shock depends mainly on
        the presence of a magnetic field, since a magnetic field is necessary for the shock
        condition to become continuous, the ionization of the preshock gas, since only charged
        particles couple to the magnetic field, and on the velocity of the shock, since J-type
        shocks can have higher velocities than C-type shocks (Flower
          & Pineau des Forêts 2003; Tielens 2005;
          Anderl et al. 2013). A main difference between the
        types is that J shocks lead to partial or complete dissociation of molecules, such as
        molecular hydrogen. Because we observed a lot of H2 emission (see Fig. 3), pure J shocks are fairly improbable in this case, since they would dissociate
        the molecular hydrogen. Studies by Gusdorf et al.
          (2008b) have shown that even CJ-type shocks at a velocity of 25 km s-1 are energetic enough to
        dissociate the ambient hydrogen. However, in a more recent paper, Gusdorf et al. (2015) have found that CJ-type shocks reproduce
          H2 observations
        even better. They found that C-type shocks are not able to reproduce their H2 excitation temperatures. In
        conclusion, the strong observed H2 emission suggests that we are not observing J-type shocks;
        however, we cannot differentiate further between C-type or CJ-type shocks with the present
        data. Another important parameter of the shocks is the velocity. However, it is very
        difficult to estimate the shock velocity in the current case. The SiO signal in the line
        wings is present until 15 km s-1 apart from the rest velocity. This is most likely not the
        maximum shock velocity, as the strong SiO signal at ambient velocity indicates jets or
        outflows with a large velocity component in the plane of the sky. In conclusion, this means
        that SiO emission in this case can be the effect of shocks with comparably high velocities
          (vshock>
          25 km s-1), because the expected C-type or CJ-type shocks can reach
        very high velocities, depending on the preshock density (Le
          Bourlot et al. 2002). These shocks would most likely be a result of the ongoing
        star formation. However, low-velocity shocks are also able to release the SiO if it is
        present in the core mantles.


X-ray radiation observed from clusters of galaxies is emitted by hot, low-density and metal-enriched plasmas filling the space between galaxies (for a review, see Sarazin 1986). We focus attention on relaxed galaxy clusters that are not disturbed by ongoing mergers with other groups (or clusters) of galaxies. The measured X-ray surface brightness and derived plasma density profiles of such clusters are sharply centrally peaked and decrease with radius, while the derived plasma temperature profiles have minima at the centres (for a review, see e.g. Peterson  Fabian 2006, and references therein). These central low-temperature dense regions are coined as cool-cores (Molendi  Pizzolato 2001, formerly known as cooling flows) and galaxy clusters having cool cores are called cool-core clusters. If no external energy sources are present in cool cores, the radiative losses in clusters centres would lead to gas cooling well below X-ray temperatures in a fraction of the Hubble time. The absence of the low temperature emission lines in the spectra of cool-core clusters (e.g. Tamura etal. 2001) and the lack of intense star formation suggest that external energy sources for plasma heating do exist in the cool cores. A likely explanation is that the energy produced by the central active galactic nuclei (AGNs) balances radiative losses of the X-ray emitting plasma (e.g. Churazov etal. 2000; McNamara etal. 2000). The activity of the central AGNs manifests itself via the presence of X-ray cavities (or bubbles) inflated by relativistic outflows from the AGNs that are present in majority of cool-core clusters (for a review, see Fabian 2012; Bykov etal. 2015). These bubbles are typically bright in the radio band (e.g. Boehringer etal. 1993), implying that relativistic electrons (and, perhaps, positrons) are present inside them. The bubbles rise buoyantly in cluster atmospheres and pure mechanical interaction can provide efficient energy transfer from the bubble enthalpy to the gas (e.g. Churazov etal. 2000, 2001, 2002). These buoyancy arguments suggest that the energy supplied by the AGNs matches approximately the gas cooling losses. This conclusion has been confirmed by the analysis of many systems that differ in X-ray luminosity by several orders of magnitude (e.g. Brzan etal. 2004; Hlavacek-Larrondo etal. 2012).


Similar to Balachandran & Flanagan (2007), who also studied the detectability of mode resonances (see also Flanagan & Racine 2007), we assume that the frequency-domain GW signal has the form
(30)
\begin{equation}
\tilde{h}(f) = \mathcal {A} f^{-7/6} \text{e}^{\text{i}\Psi (f)},
\end{equation}
where the amplitude (Maggiore 2007)
(31)
\begin{equation}
\mathcal {A}=\left(\frac{5}{24\pi ^{4/3}}\right)^{1/2}\left(\frac{G\mathcal {M}}{c^3}\right)^{-5/6}\frac{\mathcal {K}}{d}.
\end{equation}
Here $\mathcal {M}=[(M M^{\prime })^{3}/(M+M^{\prime })]^{1/5}$ is the chirp mass, d is the distance to the source and $\mathcal {K}$ is the antenna response (for an optimally oriented source $\mathcal {K}=1$). Our model of the phase evolution Ψ(f) accounts for the zeroth-order post-Newtonian point-particle contribution and for the resonant tidal excitation of individual g-modes. We ignore higher order post-Newtonian terms, spin, the equilibrium tide and non-linear tidal effects (such as those considered in Essick, Vitale & Weinberg 2016), and assume that the signal shuts off at the GW frequency corresponding to the innermost stable circular orbit f = 2fISCO ≃ 1.6 × 103 Hz [2.8 M⊙/(M + M΄)] (Cutler & Flanagan 1994; Poisson & Will 1995). In Appendix B, we show that under these assumptions,
(32)
\begin{equation}
\Psi (f) = \Psi _{\rm pp}(f) - \sum _a\left(1-\frac{f}{f_a}\right) \delta \phi _a \Theta \left(f-f_a\right),
\end{equation}
where, by the stationary-phase approximation, the point-particle phase is (Cutler & Flanagan 1994)
(33)
\begin{equation}
\Psi _{\rm pp}(f) = 2\pi f t_{\rm c} -\phi _{\rm c}-\frac{\pi }{4} +\frac{3}{4}\left(\frac{8\pi G\mathcal {M}f}{c^3}\right)^{-5/3},
\end{equation}
δϕa is the phase shift due to the tidal resonance with a mode a with eigenfrequency fa, and Θ(f − fa) is the Heaviside step function. Here tc and ϕc are constants of integration that set a reference time and phase. The duration of each resonance is, in general, much shorter than the orbital decay time-scale due to radiation reaction (their ratio is ${\simeq } 0.1\times [(\mathcal {M}/1.2\,\mathrm{M}_{\odot })(f/500 \text{ Hz})]^{5/6}$; see Lai 1994; Balachandran & Flanagan 2007; Flanagan & Racine 2007). We therefore model the resonance as an instantaneous process, which should be a good approximation since nearly all the g-modes we consider have $f_a^{\rm (GR)} < 500\,{\rm Hz}$ (the one exception is the na = 1 hyperonic g-mode of the 1.6 M⊙ HS model which has $f_a^{\rm (GR)}=740\,{\rm Hz}$).


In Appendix B, we show that under these assumptions,
(32)
\begin{equation}
\Psi (f) = \Psi _{\rm pp}(f) - \sum _a\left(1-\frac{f}{f_a}\right) \delta \phi _a \Theta \left(f-f_a\right),
\end{equation}
where, by the stationary-phase approximation, the point-particle phase is

(33)
\begin{equation}
\Psi _{\rm pp}(f) = 2\pi f t_{\rm c} -\phi _{\rm c}-\frac{\pi }{4} +\frac{3}{4}\left(\frac{8\pi G\mathcal {M}f}{c^3}\right)^{-5/3},
\end{equation}
δϕa is the phase shift due to the tidal resonance with a mode a with eigenfrequency fa, and Θ(f − fa) is the Heaviside step function.


who also studied the detectability of mode resonances

we assume that the frequency-domain GW signal has the form
(30)
\begin{equation}
\tilde{h}(f) = \mathcal {A} f^{-7/6} \text{e}^{\text{i}\Psi (f)},
\end{equation}


The duration of each resonance is, in general, much shorter than the orbital decay time-scale due to radiation reaction (their ratio is ${\simeq } 0.1\times [(\mathcal {M}/1.2\,\mathrm{M}_{\odot })(f/500 \text{ Hz})]^{5/6}$; see


An LyC candidate at z = 3.212 in the GOODS-Southern field, named Ion2 (GDS-ID 033203.24-274518.8), was first identified by Vanzella et al. (2015). This candidate is a

 galaxy with SFR

 (sSFR

), EW(Lyα)

 Å, and EW(

)

 Å (de Barros et al. 2016). The compact star-forming region, showing strong [O iii]λλ4959,5007 emission lines (with a rest-frame equivalent width of 1500 Å) and a large Oxygen ratio

/

 (

 at 1σ, de Barros et al. 2016) measured with Keck/MOSFIRE, makes Ion2 the highest redshift “Green Pea” galaxy the currently known and, accordingly to the photoionization models (Jaskot & Oey 2013; Nakajima & Ouchi 2014), an ideal candidate LyC emitter. A plausible spectroscopic LyC detection was subsequently discussed in de Barros et al. (2016). However, the presence of a close companion not resolved with ground-based spectroscopy and imaging (02, see Figure 1) cast some doubts on the association of the observed flux with Ion2,and thus on the reliability of the LyC leakage. Thanks to the Lyβ and Lyγ detection in the Very Large Telescope (VLT)/VIMOS Ion2 spectrum (see Figure 6 panel (A), or de Barros et al. 2016, Figure 2), the spectroscopic redshift of component A is unambiguously z = 3.2. The 4 hr exposure is not enough to detect such absorption features with a signal-to-noise ratio (S/N) of

 (as observed here) for component B, which at these wavelengths is mag = 27.2 in the continuum.11

11
From ESO/Exposure Time Calculator, an S/N

 is expected at this magnitude, wavelength, and integration time. This would imply an S/N

 if Lyβ and Lyγ absorption lines arise from component B.
 Therefore these features arise from component A at z = 3.2. The [O iii] emission is also mainly associated with component A because the K-band magnitude is dominated by the strong [O iii] lines and the K-band spatial emission is more prominent from A. This is shown in Figure 2 where the VLT/HAWKI K-band image has been extracted from the HUGS survey (Fontana et al. 2014).


The observation of FRB 200428 suggests the FRBs are accompanied by XRBs. The energy ratio between radio burst and XRB is about

. If this value is valid for FRB 121102, the X-ray energy release in one period for FRB 121102 is about
17

The typical active timescale of magnetars is about 100 yr (Beloborodov & Li 2016). The total energy release of XRBs in the active phase is
18

where τ is the active timescale of the magnetar. For FRB 121102,

 is assumed. Compared to the rotational energy, the magnetic energy is the main reservoir responsible for powering FRBs. The magnetic energy of magnetar is
19

where

 G is the interior magnetic field strength, and R = 12 km is the magnetar radius. This value is smaller than the required XRB energy. The XRB energy is shared by the ultrarelativistic ejecta and the baryon shell with a subrelativistic velocity vw. It has been found that the kinetic energy of the baryon shell is comparable to the flare energy (Metzger et al. 2019; Margalit et al. 2020b). We assume that the baryon shell is subrelativistic with a typical velocity

. Based on the above assumptions, the baryonic mass ejected by magnetar in the active time can be derived as
20

This value is much larger than the typical mass (

) of a magnetar outer crust (Gudmundsson et al. 1983; Glendenning & Weber 1992). According to the structure of magnetars, the core is mainly composed of superfluid. The ejected baryon matter of the magnetar is mainly provided by crust. The baryonic mass estimated in this model is larger than the typical mass of a magnetar outer crust, indicating that the outer crust of the magnetar cannot eject enough baryonic matter required by the model. In Figure 2, we show the constraints on MB using Equation (20). If the radiation efficiency

 from FRB 200428 is used, the required baryonic mass MB is always larger than

 for the whole parameter ranges as shown in the left panel of Figure 2. If τ = 100 yr is fixed, the same result is shown in the right panel of Figure 2. The outer crust of magnetars is not sufficient to provide such a large baryonic mass, which challenges the synchrotron maser shock model. Interestingly, if the inner crust is included, the total mass of crust is about 0.01 M⊙ (Chamel & Haensel 2008), which is larger than the required baryonic mass. Theoretically, the physical mechanism and the rate of the baryonic mass ejection are both uncertain. More investigations are required. We also use the bursts of FRB 121102 observed by FAST to estimate the required baryonic mass (Li et al. 2019). The observation of FAST provides the information on the lowest-energy bursts of FRB 121102 so far. In a 56.5 hr observation, 1121 bursts were observed. The total energy of these bursts is 3.14

 erg.3

3
D. Li & P. Wang 2020, private communications.
 Therefore, the average energy release in one period is about

 erg. Using the same formulae as above, the baryon mass can be approximated as
21

which is similar to that derived from GBT observation.


The intrinsic (edge-on) ellipticity (ϵint) has been estimated using the observed V/σ and ϵ based on the theoretical prediction of intrinsic shape for varying inclination under the assumption of rotating oblate system (Binney 2005). Cappellari et al. (2007) has shown that intrinsic ellipticity correlates with the anisotropy parameter using early-type galaxies from the SAURON IFS survey (de Zeeuw et al. 2002). Cappellari et al. (2016) presented the relationship between the intrinsic and observed ellipticities:
(14)$$\begin{eqnarray*}
\epsilon = 1-\sqrt{1+\epsilon _{\rm int} (\epsilon _{\rm int}-2) \sin ^2 i}.
\end{eqnarray*}$$The observed V/σ with an inclination i can be corrected to the intrinsic (V/σ)int for an edge-on (i = 90°) view (Binney & Tremaine 1987):
(15)$$\begin{eqnarray*}
\left(\frac{V}{\sigma }\right) = \left(\frac{V}{\sigma }\right)_{\rm int} \frac{\sin i}{\sqrt{1-\delta \cos ^2 i}},
\end{eqnarray*}$$where δ is an approximation to the anisotropy parameter β such as δ ≈ β = 0.7ϵint. For each component, we generated a grid with varying ϵint and i. Then, we calculated corresponding ϵ and V/σ for each ϵint and i using equations (14) and (15). The ϵint has been derived from the grid which has the best matched ϵ and V/σ to the observation. Note that V/σ for the bulge and disk components are measured using Re, PA, and ϵ of each component in this section, whereas all kinematic measurements in the other sections except for this section and Fig. 13 used the structural parameters from galaxies. Measuring V/σ at the effective radius of each component gives more galaxies with kinematics for both components: 664 and 633 are available for measuring bulge and disk components (respectively), and 468 galaxies have kinematics available for both components. On the other hand, the above method is not valid for slowly rotating bulges (e.g. below the green line in Fig. 13) because they are not oblate rotators. In that instance, we adopt ϵint = ϵ as our best estimate of the intrinsic ellipticity (although it is strictly a lower limit).


This section discusses the results of planet–planet scattering in systems with two giant planets in four simulation sets (Table 1). The statistics use only simulations with one surviving planet after the other planet is ejected via planet–planet scattering (23.26% of all simulations; see Table 1), which is around 5500 simulations in each set. In two-planet systems, the instability boundary is very narrow. If both planets survive, its very likely they have none or very little interactions, so it is not very interesting to study them for our purpose. Simulations with planet–star (2.71% of all simulations) or planet–planet collisions (73.61% of all simulations; see Table 1) are excluded from the statistics, as obliquity evolution for the collision of fluid bodies is beyond the scope of this work (see Li & Lai 2020; Li et al. 2020). Simulations with energy error

∣

δ
E

E

∣

 greater than

(

10

−
4

)

 are also excluded (0.36% of all simulations), which is an adequate threshold for multiplanet systems (Barnes & Quinn 2004; Raymond et al. 2010). A negligible fraction of systems (0.03% of all simulations) that do not experience planet destruction in 108 yr are also excluded. Figures 5 and 6 show the distribution of the surviving planet’s final spin orientation and obliquity for simulation sets 1–3. The spin inclination is measured relative to the z-axis, which is aligned with the inner planet’s initial spin and orbit normal. The final spin and obliquity distribution is quite broad compared with the orbital inclination distribution (Figure 7). Overall, the final spin and obliquity distributions (Table 3) are broader than the inclination distributions, with 19.51% planets’ obliquity beyond 40°, and 8.77% of the surviving planets are on retrograde obliquity (i.e., the rotation is in reversed direction relative to the orbit). The inclination distribution only has 0.12% planets beyond 40°. The rate of highly oblique planets (>40°) are 17.86%, 31.98%, and 10.43% in sets 1, 2, and 3. There are 5.07% surviving planets on retrograde obliquity in the base set, 21.46% in set 2, and 3.21% in set 3. The distribution of obliquity for set 3 is more clustered at lower regions compared to the other sets. From the rate of high and retrograde obliquity in set 2 (the inner planet ends up at 0.19 au after the outer planet is ejected), we may expect a high rate of highly oblique planets close to the central star, and some oblique planets farther out (Figures 5 and 6). A moderate peak in Figure 6 for close-in planets in sets 1 and 2 between obliquities 170°–180° implies that Venus-like obliquity is possible. The simulation results imply that planets closer to the star receive more torque from it and become more evolved in obliquity.


Recently, Esteban et al. (2022) examined the abundances in the Galaxy as measured from a number of sources, including H ii  regions from Arellano-Córdova et al. (2020, 2021), with the goal of assessing the large abundance dispersion that De Cia et al. (2021) measure in Galactic neutral clouds. The direct abundances in the H ii  regions are calculated using T

e
[N ii] in the low-ionization zone and T

e
[O iii] in the high-ionization zone, only applying the Esteban et al. (2009) T

e
–T

e
 relation when one of these temperatures is missing. From the 42 Galactic regions, the dispersion about the O/H gradient is ∼0.07 dex, which is similar to the dispersion in metallicity measured from B-type stars, classical Cepheids, and young clusters in the Galaxy (Esteban et al. 2022), and significantly less than the dispersion measured in the neutral clouds. Méndez-Delgado et al. (2022a) corroborate the low dispersion in the Arellano-Córdova et al. (2020, 2021) H ii  regions when considering nonzero temperature fluctuations and updated H ii  region positions. In the spiral galaxy M101, Kennicutt et al. (2003) determine the abundance gradient from 20 H ii  regions, the majority of which have direct T

e
[O iii] and T

e
[S iii] measurements, and nine of which have direct T

e
[N ii]. Citing the tight correlation they observe between T

e
[O iii] and T

e
[S iii], Kennicutt et al. (2003) use an average of the direct T

e
[O iii] and the inferred T

e
[O iii] from T

e
[S iii] to obtain the high-ionization zone temperature, and they use the direct T

e
[N ii] in the low-ionization zone, when available. Although this sample consists of only 20 regions, there is a clear abundance gradient in M101 with very little dispersion about the gradient. The dispersion reported by CHAOS (0.097 dex; Rogers et al. 2021) is slightly larger than the gradient from Kennicutt et al. (2003) would indicate, but this could be due to the inclusion of H ii  regions with a single direct T

e
.


From the 42 Galactic regions, the dispersion about the O/H gradient is ∼0.07 dex, which is similar to the dispersion in metallicity measured from B-type stars, classical Cepheids, and young clusters in the Galaxy

and significantly less than the dispersion measured in the neutral clouds.


The newly discovered arcs and new spectroscopic redshifts have been incorporated into updated HFF+ versions of the Abell 2744 and MACSJ0416.1-2403 lensing models (Table 5); many of these have

 (see Figure 8 for a comparison of the arc redshift distributions adopted by the pre-HFF and new HFF+ lensing models; Cypriano et al. 2004; Okabe & Umetsu 2008; Zitrin et al. 2009; 2013; Okabe et al. 2010a, 2010b; Merten et al. 2011; Christensen et al. 2012; Mann & Ebeling 2012; Jauzac et al. 2014; Lam et al. 2014; Richard et al. 2014; Balestra et al. 2015; Diego et al. 2015; Grillo et al. 2015; Jauzac et al. 2015; Rodney et al. 2015; Wang et al. 2015; Kawamata et al. 2016). The incorporation of these new multiple image systems often results in a reduction in the statistical uncertainty in the galaxy magnifications for a given model. All of the public HFF lensing models provide a range of possible realizations from which the statistical uncertainty of a given model set may be calculated (typically 100 but no fewer than 30). We plot the cumulative distribution of the galaxy magnification uncertainties σ(model)/

, for the galaxies and photometric redshifts provided by the ASTRODEEP catalogs (Merlin et al. 2016; Castellano et al. 2016a) for Abell 2744 (Figure 9) and MACSJ0416.1-2403 (Figure 10). Generally, the statistical uncertainties are reduced for the models computed with the new HFF data sets, with more dramatic reductions for the methods that rely strongly upon the strong-lensing constraints. The parametric methods (CATS, Sharon, Zitrin, GLAFIC) report median statistical magnification errors of 0.2%–5%, while the non-parametric methods (Bradac Williams, Diego) report median statistical magnification errors of 2%–11% for the post-HFF calculations (green curves), versus 2%–22% and 2%–17% respectively for pre-HFF models (blue curves). (We note that the statistical errors for the MACSJ0416.1-2403 Bradac post-HFF models (Hoag et al. 2016) included additional uncertainties due to the photometric redshift uncertainties of the multiple images. These were not included in the pre-HFF Bradac model, and thus may explain why the post-HFF statistical errors are larger for this model.)


As an illustrative example we apply the general results obtained in the last section to the IGM shortly after the reionization onset at a redshift of about z = 10. To this end we calculate the effective growth rate of a Maxwell–tensor distributed electron–proton plasma with a kinetic temperature of

 for both particle species and an electron density of

 being traversed by a highly relativistic beam with a Lorentz factor of

. The corresponding fluctuations of the background plasma are dominated by a recently discovered damped, aperiodic mode

 with

 (Felten et al. 2013). Schlickeiser & Felten (2013) found that its quantitative description can be simplified by introducing the dimensionless wavenumber κ and the dimensionless frequency y defined by
79

80

where kc is the inverse of the electron skin depth
81

modified in such a way that it absorbs all remaining prefactors that occur in the dispersion relation:
82

Here,

 denotes the thermal velocity of the electrons. In terms of these variables the dispersion relation takes a simple form and is given by (Schlickeiser & Felten 2013)
83

with
84

For our present purposes it is more convenient to consider the frequency y as the independent variable rather than the wavenumber κ. This approach can easily be reversed by the approximation for the inverse dispersion relation derived by Schlickeiser & Felten (2013),
85

(This procedure was also carried out during the creation of Figures 1 and 3 shown below.) Our task is to compute the effective growth rate (78), which for aperiodic modes reads
86

Making use of definitions (79)–(80) and the dispersion relation (83) we can rewrite this as
87

where we introduced the abbreviations
88

89

90

Next, we will turn to the transversal dispersion function of the unperturbed thermal background plasma (Schlickeiser & Felten 2013)
91

Making use of (79)–(80) and (83) once more we can write the derivative as a function of y alone:
92

Plugging this expression into Equation (87) we arrive at the effective growth rate
93

where
94

Note that to zeroth order in ε this result reproduces the dispersion relation (83) of the unperturbed growth rate, and that it contains an additional first order correction term that accounts for the presence of the beam.


In terms of these variables the dispersion relation takes a simple form and is given by


This approach can easily be reversed by the approximation for the inverse dispersion relation derived by

(This procedure was also carried out during the creation of Figures 1 and 3 shown below.)


Next, we will turn to the transversal dispersion function of the unperturbed thermal background plasma


Except for WR 134, the detected frequencies are high, from 3 d−1 up to 14 d−1. This cannot be easily reconciled with an orbital period (the companion would travel inside the WR star), nor a rotation rate (it would be above break-up velocity). Therefore, the most probable culprits are pulsations. WR 66 very probably possesses a close neighbour (see Section 2.1), which could cast doubt on the identification of the pulsating star, but that is not the case for the other stars (including WR 123: see Lefèvre et al. 2005). Pulsations here are thus considered to originate from the WR star itself, and such a possibility has actually been considered before. Pushed by the observational considerations of Vreux (1985), the first pulsation models for WRs were elaborated more than 30 years ago, with predicted periods of about one hour (Maeder 1985; Scuflaire & Noels 1986). Subsequent work by Glatzel et al. (1999) focused on strange modes in small and hot helium stars (often taken as behaving similarly to WRs) and the predicted pulsations had periods of several minutes with amplitudes of several mmag. After the discovery of a 2.45 d−1 signal in WR 123 (Lefèvre et al. 2005), i.e. at a much lower frequency than expected, the models were revisited. Strange modes were extended to stars with larger radii (Dorfi, Gautschy & Saio 2006; Glatzel 2008), while predictions of g-modes excited by the κ mechanism were made for WRs by Townsend & MacDonald (2006). In both cases, the predicted periods could be made compatible with the 10-h signal observed in WR 123. The signals we observe have frequencies in the same range, even for the 2-min cadence data, which allows us to probe very high frequencies, but no coherent signal is detected above 14 d−1: WRs thus seem to pulsate only at moderately high frequencies. Furthermore, Townsend & MacDonald (2006) expected pulsations with frequencies that are higher in early WN (2–8 d−1) than in late WN (1–2 d−1). For our pulsators, only WR 7 has an early WN type and its frequency values are not particularly higher: its main signals are in the same frequency range as those of the late-type WR 66. Moreover, the highest frequencies are all found in late WN stars. However, Townsend & MacDonald (2006) models were made for ‘general/typical’ stars, not specific ones. In addition, since the WR light mostly comes from inside the wind, the filtering effect of the wind on a signal from the underlying hydrostatic surface needs to be assessed in detail. New, dedicated models will be needed for a more in-depth comparison between observations and predictions.


To obtain reaction rates in a wide temperature range

T

9

=
0.001

–10 (where T9 is in units of 109 K) for BBN calculation inputs, continuous excitation functions over

10

−
8

 to a few MeV are needed for numerical integration. Therefore we performed an R-matrix analysis to compile both the present and the previous data of these three reaction channels by using AZURE2 (Azuma et al. 2010) including all the relevant partial widths

Γ

n

,

Γ

p
0

,

Γ

p
1

, and

Γ

α

. The 7Be

(
n
,

p

0

)

 data set used for the R-matrix analysis is based on the same selection as used in Damone et al. (2018), Damone (2018; part of Dam18 + Sek76 as displayed in Figure 2), and the present THM data, Bor63, and Pop76 are added. For the 7Be

(
n
,

p

1

)

 and 7Be

(
n
,
α
)

 channels, all the labeled data shown in Figure 2 (except for p1 Tom19 and

γ
α

 Bar16) were used without any data cutoff. Appendix B shows the adopted data sets more explicitly. We included nine known levels below the 8Be excitation energy of

E

x

=
24

MeV

 (Tilley et al. 2004) and an additional nonresonant background pole, with a common channel radius of 5 fm for each channel in the same manner as Adahchour & Descouvemont (2003). The most significant channel 7Be

(
n
,

p

0

)

7

Li is dominated by four of these levels as described in detail by a former R-matrix study (Adahchour & Descouvemont 2003; Descouvemont et al. 2004); the 2− resonance near the neutron threshold (labeled as level I) principally dominates the cross section up to the BBN energies, the two 3+ states at

E

x

=
19.24

MeV

 (level II) and

E

x

=
21.5

MeV

 well characterize the corresponding single peaks, and a 2+ background pole accounts for the enhancement at high energies. The most important resonances to expand analysis to the 7Be

(
n
,

p

1

)

7

Li* and 7Be

(
n
,
α
)

4

He channels are the 1− state at

E

x

=
19.4

MeV

 (level III) and the 2+ state at

E

x

=
20.1

MeV

 (level IV), respectively; the former expresses the 7Be

(
n
,

p

1

)

 resonance behavior peaked at

E

c
.
m
.

∼
0.5

MeV

 tailing down to the thermal neutron energy by the

1

/

v

 law, and the latter forms the first peak in the 7Be

(
n
,
α
)

 spectrum around

E

c
.
m
.

∼
1

MeV

. Despite its importance in the

(
n
,
α
)

 channel, level IV is much less significant in the total cross section especially at lower energies due to its p-wave nature. Therefore we imposed some restrictions on level IV, which made the analysis much simpler; fixing

Γ

p
0

,

Γ

p
1

, and

Γ

α

 at a known ratio

Γ

α

/

Γ

p

∼
4.5

 (Tilley et al. 2004), and freeing

Γ

n

 and resonance energy (refitted to be 19.87 MeV) not to significantly exceed the known total width

Γ
=
880

keV

 (Tilley et al. 2004). The other four higher-lying levels (0+ at

E

x

=
20.2

MeV

, 2+ at

E

x

=
22.24

MeV

, 1− at

E

x

=
22

 MeV, and 2− at

E

x

=
24

 MeV) play rather supplementary roles mainly for the higher-energy behavior. The 4+ and 4− higher-spin states (at

E

x

=
19.86

MeV

 and 20.9 MeV, respectively) were not included due to their limited influences. We do not introduce the γ-emission channels to fit the Bar16 plots (representing the 7Be

n
,
γ
α

4

He reaction channel Barbagallo et al. 2016), which appears significant only below the BBN energies. See Appendix C for more details on the present R-matrix analysis.


Therefore we imposed some restrictions on level IV, which made the analysis much simpler; fixing

Γ

p
0

,

Γ

p
1

, and

Γ

α

 at a known ratio

Γ

α

/

Γ

p

∼
4.5


We included nine known levels below the 8Be excitation energy of

E

x

=
24

MeV


We first consider the case in which the star is surrounded by a gaseous atmosphere. The star surface is divided in six angular patches in magnetic colatitude, centred at the values θ = {0°, 10°, 30°, 50°, 70°, 89°}. By using the magnetic and temperature profiles previously described we compute, for each θ, the local magnetic field strength, B, the angle θB between the magnetic field and the normal to the surface, and hence the temperature, T. We then compute a set of atmospheric models corresponding to the six θ angles. Since the models are computed using different integration grids in the photon phase space (because the choice of the photon trajectories along which the radiative transfer is solved needs to be optimized to ensure fast convergence at the different values of magnetic field strength and inclination, see Lloyd 2003), we reinterpolate all model outputs over a common grid. This results in three 4D arrays for the emergent intensity $I^i_\nu (\boldsymbol k,\theta ) \equiv I^i(E,\mu _k,\phi _k,\theta )$ (i = 1, …, 3) which contain the total intensity and the intensities for the ordinary and extraordinary modes, respectively. In order to take into account the emission from the southern magnetic hemisphere of the star, we use the previous 4D arrays with the substitutions θB → π − θB and ϕk → π − ϕk, which is justified by the symmetry properties of the opacities. By using the ray tracing method described in Section 2, we can then compute light curves, phase resolved spectra and polarization fractions for each choice of the angles χ and ξ. As an example, Fig. 3 shows the X-ray light curve (0.12–0.39 keV) and the phase-dependent polarization degree in the X-ray and optical7 (B-filter) bands, for χ = 90° and ξ = 15°. For this particular viewing geometry, the X-ray pulsed fraction is 1 per cent, in agreement with the observed data and, as illustrated in the figure, the polarization degree is expected to be substantial and constant in phase.


However, this approximation holds only for luminosities â² 0.3LEdd. At higher accretion rates, the hardening factor Îº increases with luminosity and can be as high as â2.5â3 for some Galactic BHs (e.g., GRO J1655â40 at outburst peak, and GRS 1915+105) and for ULXs in the slim-disk state (Watarai & Mineshige 2003; Kawaguchi 2003; Shrader & Titarchuk 2003; Isobe et al. 2012). This is why at near-Eddington luminosities, the fitted color temperature Tin increases faster than the standard  high/soft state relation (âanomalous regimeâ; Kubota & Makishima 2004; Abe et al. 2005). Given the luminosities of M83 ULX-1 in the three XMM-Newton epochs, and the fact that a disk model with p â 0.6 provides the best fit to the data, we argue that the source was likely to be in the anomalous regime (slightly below Eddington) or in the slim disk regime (slightly above Eddington), and, therefore, we take a hardening factor Îº â 3 for BH mass estimates. Following Vierdayanti et al. (2008), we also take Î¾ â 0.353, which takes into account the transonic flow in the pseudo-Newtonian potential. Taking an average normalization constant Ndisk = (2.5 Â± 0.5) Ã 10â3 from our diskpbb fit (Table 2), we obtain a âtrueâ inner disk radius Rin â (73 Â± 15)(cosâÎ¸)â1/2âkm. This corresponds to an âapparentâ BH mass
2Finally, we need to take into account that the inner radius of a slim disk extends slightly inside the innermost stable circular orbit, so that the true mass is MBH â 1.2MX (Vierdayanti et al. 2008). This gives our final best estimate of the BH mass as
3In principle, the estimated mass can be as high as â60âM in the (implausible) extreme Kerr scenario; however, the fact that the system appears to be in the anomalous regime (upper end of the high/soft state) or in the slim disk state for a moderate bolometric luminosity â2 Ã 1039âergâsâ1 (and possibly even lower, considering the 2014 January spectrum) suggests that the BH mass is closer to â¼10â20âM. The same argument holds if we interpret the X-ray spectra as Comptonized emission from a warm, optically thick corona, which is typical of sources at LX â 1â3LEdd (Gladstone et al. 2009).