A substantial fraction of metal-poor stars that have recently evolved off the MS, e.g., giants and planetary nebulae (PNe), have been influenced by binary interactions. The IMF is significantly weighted toward low-mass stars (Bastian et al. 2010; Kroupa et al. 2013), and the MW star formation rate was ≈3 times larger ≈10 Gyr ago than it is now (Governato et al. 2007; De Lucia et al. 2014). Based on the measured IMF and modeled galactic star formation history, we estimate that ≈55% of MW giants and PNe have old, solar-type progenitors (τ* > 7 Gyr, M ≈ 0.8–1.2

). Such old, low-mass giants tend to be metal-poor (Ratnatunga & Yoss 1991; Carollo et al. 2010; Mackereth et al. 2017). The metallicity trend therefore dramatically affects the properties of low-mass evolved stars. For example, the enhanced close binary fraction of metal-poor solar-type stars substantially strengthens the conclusion that the shaping of PN morphologies is the result of binary interactions (Moe & De Marco 2006; De Marco 2009; Jones & Boffin 2017). Providing further corroboration, Badenes et al. (2015) measured the delay-time distribution of bright PNe in the LMC and discovered two distinct populations of PN progenitors: an old channel (τ* = 5–8 Gyr) deriving from solar-type stars (M ≈ 1.0–1.2

) and a young channel (35–800 Myr) evolving from late-B/early-A stars (≈2–8

). According to the measured age–metallicity relation of the LMC (Olszewski et al. 1991; Pagel & Tautvaisiene 1998; Cole et al. 2005; Carrera et al. 2011; Piatti & Geisler 2013), the old, solar-type progenitors are metal-poor ([Fe/H] ≲ −1.0) and hence have a large close binary fraction of Fclose = 40%–50%. The young progenitors have a higher metallicity of [Fe/H] ≈ −0.4 but are sufficiently massive so that they also have a large close binary fraction of Fclose = 40%–60%. Meanwhile, evolved stars with intermediate masses (M ≈ 1.2–2.0

) in the LMC have intermediate metallicities and therefore a smaller close binary fraction of Fclose ≈ 30%. If PNe derive from interactions in close binaries, then the variations in Fclose with respect to mass and metallicity can explain the observed bimodal mass/age distribution of PN progenitors in the LMC.


The recent detection of gravitational-wave (GW) emission from a merging neutron star binary (Abbott et al. 2017d) and merging black hole binaries (BHBs; Abbott et al. 2016a, 2016b, 2017a, 2017b, 2017c; The LIGO Scientific Collaboration & The Virgo Collaboration 2018) by the Laser Interferometer Gravitational-Wave Observatory (LIGO)/Virgo have ushered in an exciting new era of GW astrophysics. The astrophysical origin of the detected mergers is currently under debate, with numerous explanations proposed. These explanations can be very roughly divided into two main categories: mergers due to isolated binary evolution (e.g., Belczynski et al. 2016; de Mink & Mandel 2016; Mandel & de Mink 2016; Marchant et al. 2016), and mergers due to dynamical interactions (e.g., Portegies Zwart & McMillan 2000; Wen 2003; O’Leary et al. 2006, 2009, 2016; Antonini & Perets 2012; Kocsis & Levin 2012; Antonini et al. 2014; Antonini & Rasio 2016; Rodriguez et al. 2016; VanLandingham et al. 2016; Askar et al. 2017; Arca-Sedda & Gualandris 2018; Fragione & Kocsis 2018; Hoang et al. 2018; Randall & Xianyu 2018; Arca-Sedda & Capuzzo-Dolcetta 2019). Orbital eccentricity has been explored as a way to distinguish between these merger channels in both the LIGO/Virgo and Laser Interferometer Space Antenna (LISA) frequency bands. In contrast to mergers from isolated binary evolution, merging binaries from dynamical channels have been shown to have measurable eccentricities when they enter the LISA and/or LIGO/Virgo band, and can potentially be used as a way to distinguish between channels (e.g., O’Leary et al. 2009; Cholis et al. 2016; Gondán et al. 2018; Lower et al. 2018; Randall & Xianyu 2018; Rodriguez et al. 2018; Samsing 2018; Zevin et al. 2019). Unlike LIGO/Virgo, which can only detect merging BHBs in the final inspiral phase before merger, LISA will be able to detect eccentric stellar-mass BHBs for long timescales before they merge in the LIGO/Virgo band (e.g., O’Leary et al. 2006; Breivik et al. 2016; Nishizawa et al. 2016; Chen & Amaro-Seoane 2017; Nishizawa et al. 2017; D’Orazio & Samsing 2018; Kremer et al. 2019; Samsing & D’Orazio 2018). This provides us with invaluable insight into the dynamical evolution of eccentric binaries leading up to the merger, which has important implications about the astrophysical context in which merging binaries evolve.


After each step of calculations, the network should optimize the model based on its current and previous states to improve the subsequent mapping. Our model utilizes a computationally memory efficient optimization due to its dependence to only the first-order gradients, namely the ‘adaptive moment estimation’ (or Adam). For more details, we refer the readers to Kingma & Ba (2014). Adam optimization, compared to other gradient-based optimization, is very suitable for noisy and sparse gradients, and for simulated data that show very large scatter with respect to a given quantity of parameter (Kingma & Ba 2014). With this optimizer, we have to decide few parameters in advance. The learning step α and the parameters controlling the moving averages of the first- and second-order moments, namely β1 and β2 (both ∈[0,1)), respectively. For this purpose, we chose to minimize the MSE between the target and the prediction from the model: in what follows, we will alternatively call the MSE the ‘objective function’ f($\bf x$): with ${\bf x}$ the parameters of the model to be updated, such as weights and biases. At a given time t ≤ T, where T is the maximal learning time-step, we can update the parameters of the model as shown in the following:
(11)
\begin{eqnarray*}
g_t &=\nabla _\mathrm{ \text{$x$}} f(\mathrm{\text{$x$}}_{t-1}),
\end{eqnarray*}
(12)
\begin{eqnarray*}
\mu _{1,t} &=\beta _1 \times \mu _{1,t-1} + (1-\beta _1)\times g_t,
\end{eqnarray*}
(13)
\begin{eqnarray*}
\bar{\mu }_{1,t} &=\mu _{1,t}/(1-\beta _1^t),
\end{eqnarray*}
(14)
\begin{eqnarray*}
\mu _{2,t} &=\beta _2 \times \mu _{2,t-1} + (1-\beta _2)\times g_t^2,
\end{eqnarray*}
(15)
\begin{eqnarray*}
\bar{\mu }_{2,t} &=\mu _{2,t}/(1-\beta _2^t),
\end{eqnarray*}
(16)
\begin{eqnarray*}
\mathrm{\text{$x$}}_t &=\mathrm{\text{$x$}}_{t-1} - \alpha _t \times \bar{\mu }_{1,t}/ (\sqrt{\bar{\mu }_{2,t}} + \epsilon),
\end{eqnarray*}
where $\alpha _t=\alpha \sqrt{1-\beta _2^t}/(1-\beta _1^t)$ is the time-step at t. Equation (11) shows the gradients of the objective function at t with respect to the model parameters. Equations (12) and (14) update the estimations of the first and second moments. Our moments are biased towards the initial values; thus, we require equations (13) and (15) to account for the corrections. Finally, we update the model parameters with equation (16).


In order to provide constraints on the gas masses in these galaxies independently from the CO measurements, we can use the Rayleigh–Jeans dust continuum emission. This will provide the first constraints to the αCO conversion factor in “normal” galaxies at z > 3 in the following. The Rayleigh–Jeans dust continuum emission has been used to estimate dust and gas masses, assuming an average emissivity and dust temperature for the dominant cold dust component and a constant dust-to-gas ratio (Hildebrand 1983; Eales et al. 2012; Bourne et al. 2013; Scoville 2013; Scoville et al. 2013, 2016, 2017; Groves et al. 2015). The dependence on cold dust temperature and dust-to-gas ratio may make the Rayleigh–Jeans method less reliable than at lower redshifts (e.g., Pavesi et al. 2018a). On the other hand, the opposing effects of increasing dust temperatures and decreasing dust-to-gas ratios that may occur in “normal” galaxies at high redshift may partially compensate for each other, as also found in recent simulations that are consistent with this approach to gas mass measurement (e.g., Liang et al. 2018; Privon et al. 2018). We here adopt Equations (10) and (13) of Scoville et al. (2016) to derive gas mass estimates based on our continuum flux measurements through the same assumptions that were used in those lower-redshift samples (Scoville et al. 2016, 2017). The 34 GHz upper limits imply 3σ gas mass limits of 2.8 × 1011 M⊙ for HZ10 and 1.6 × 1011 M⊙ for LBG-1, adopting the relation derived by Scoville et al. (2016, 2017). We also use the ∼230 GHz continuum fluxes to derive approximate estimates, although these measurements may not lie on the Rayleigh–Jeans tail and therefore may not accurately trace the cold dust component. These continuum measurements would imply gas masses of ∼1.3 × 1010 M⊙ for HZ4, ∼2.5 × 1010 M⊙ for LBG-1, ∼4.4 × 1010 M⊙ for HZ9, and ∼1.1 × 1011 M⊙ for HZ10, with dominant systematic uncertainties due to the extrapolation of the method to very high redshift.


All responses calculated in this paper only account for the direct response to a perturbing potential. In general, though, the response also has an indirect component that arises from the fact that neighboring regions in the disk interact with each other gravitationally. This self-gravity of the response, which we have ignored, triggers long-lived normal-mode oscillations of the slab that are not accounted for in our treatment. Several simulation-based studies have argued that including self-gravity is important for a realistic treatment of phase spirals (e.g., Darling & Widrow 2019a; Bennett & Bovy 2021). Using the Kalnajs matrix method (Kalnajs 1977; Binney & Tremaine 2008), we have made some initial attempts to include the self-gravity of the response in our perturbative analysis, along the lines of Weinberg (1991). Our preliminary analysis shows that the self-gravitating response is a linear superposition of two terms: (i) a continuum of modes given in Equation (12), dressed by self-gravity, that undergo phase mixing and give rise to the phase spiral; and (ii) a discrete set of modes called point modes or normal modes (see Mathur 1990; Weinberg 1991) that follow a dispersion relation. The continuum response can be amplified by self-gravity when the continuum frequencies, nΩ
z
 + kv

x
, are close to the point-mode frequencies, ν. Depending on the value of k, the normal modes can be either stable or unstable. Araki (1985) finds that in an isothermal slab the bending normal mode undergoes fire hose instability below a certain critical wavelength if σ

z
/σ ≲ 0.3, while the breathing normal mode becomes unstable above the Jeans scale. In the regime of stability, the normal modes are undamped oscillation modes in absence of lateral streaming (Mathur 1990) but are Landau damped otherwise (Weinberg 1991). For an isothermal slab with typical MW-like parameter values, the point modes are strongly damped since their damping timescale (inverse of the imaginary part of ν) is of order their oscillation period (inverse of the real part of ν), which turns out to be of order the vertical dynamical time, h

z
/σ

z
. Moreover, the normal-mode oscillations are coherent oscillations of the entire system, independent of the vertical actions of the stars, and are decoupled from the phase spiral in linear theory since the full response is a linear superposition of the two. Based on the above arguments, we conclude that self-gravity has little impact on the evolution of phase spirals in the isothermal slab, at least in the linear regime. We emphasize that Darling & Widrow (2019a), who found their phase spirals to be significantly affected by the inclusion of self-gravity, assumed a perturber-induced velocity impulse with magnitude comparable to the local velocity dispersion in the solar neighborhood; hence, their results are likely to have been impacted by nonlinear effects. Moreover, the self-gravitating response of an inhomogeneous disk embedded in a DM halo, as in the simulations of Darling & Widrow (2019a), can be substantially different from that of the isothermal slab. We intend to include a formal treatment of self-gravity along the lines of Weinberg (1991) in future work.


GIADA characterizes individual dust particles by means of two independent sensors. At the instrument entrance the particle crosses a laser curtain, and is detected by photoelectric sensors (GDS, grain detection system) registering a signal (proportional to the particle cross section times the albedo) and the time at which the laser curtain is crossed. Then the particle hits the impact sensor (IS, with the same GDS cross section, A = 10−2 m2), which registers the individual particle impact momentum and its travel time from GDS to IS. The combination of GDS and IS measurements (GDS+IS particles) provides the particle mass and velocity, and constrains the particle bulk density by means of calibration curves (Della Corte et al. 2016) derived on the ground using cometary analogues (Ferrari et al. 2014). If the particle is too small to be detected by the GDS system, it may be detected by the IS sensor only (IS particles): in this case the particle momentum is converted to the mass assuming the mean value of the velocities of the GDS+IS particles in the same momentum bin, or assuming the velocities predicted by tail models (Fulle et al. 2010) if Ngds+is = 0 in that mass bin. The spacecraft velocities listed in Table 1 are always much lower than the dust velocities measured by GIADA. In this condition, in the Sun-facing coma (assumed to have uniform and R-dependent space density ρ), the dust flux from the nucleus surface corresponds to the dust flux at nadir-pointing GIADA scaled by the factor 2πR2/A. The dust number loss rate at the nucleus surface per GIADA detection is Qn = 2πR2(AΔt)−1, where Δt is the total dust collection time (Tables 5–8). In the same Tables, we show the mass loss rates Qm and the mean dust velocities already integrated in each mass bin, corresponding to the four GIADA collection periods considered in this paper: from 2015 February 19 to 28 (Table 5), from 2015 March 13 to 17 (Table 6), on 2015 March 28 (Table 7), and from 2015 August 23 to September 3 (Table 8). In Table 9 we show the data obtained during the first post-perihelion excursion at low phase angles (60  α  64°, 125  R  290 km). The R-values are too small to use the NAC DUST-MON sequences. The uncertainty affecting Afρ and the loss rates measured by GIADA and OSIRIS depends on the number of detections in each mass bin: an estimate of the relative error is given by

N

p

−
1

/

2

 and by

(

N

gds
+
is

+

N

is

)

−
1

/

2

. The dispersion of the dust velocities in Tables 2–4 provides the error affecting the dust velocities measured by OSIRIS, close to 30%. The relative error of the dust velocities provided by each GDS+IS detection is below 10%.


A widely used statistical property of the sky brightness distribution is its power spectrum (Lazarian 1995; Bharadwaj & Sethi 2001, and others). As the redshifted 21-cm signal is expected to be faint and hard to detect with imaging, estimating its power spectrum or equivalently intensity mapping gives a possible probe of the evolution of the baryonic matter distribution over cosmic time. Bharadwaj & Sethi (2001) show that visibility correlation directly measures the power spectrum. This method and its variants (Datta, Choudhury & Bharadwaj 2007; Choudhuri et al. 2014; Choudhuri et al. 2016; Bharadwaj et al. 2019; Choudhuri et al.2019, and others) have been used to estimate the angular power spectrum of the diffused galactic foreground (Ghosh et al. 2012; Choudhuri et al. 2017b; Chakraborty et al. 2019a; Choudhuri et al. 2020) as well as the power spectrum of H i distribution in nearby galaxies (Dutta et al. 2009; Dutta & Bharadwaj 2013; Nandakumar & Dutta 2020). These works propagate the uncertainties in each visibility estimate and combine that with the sample variance error in measuring the power spectrum to quote uncertainties in the power spectrum estimates. In this work, we use the estimator discussed in Choudhuri et al. (2014), where visibilities are gridded before estimating the power spectrum. Given an angular field of view of θ0 to which the telescope is sensitive, it has been shown (Bharadwaj & Sethi 2001; Bharadwaj & Ali 2005; Choudhuri et al. 2014) that the visibilities in the nearby baselines remain correlated to a baseline separation of $\Delta U \lt \frac{1}{\pi \theta _0}$. The size of the uv-grids is chosen such that they are large enough to include a sufficient number of baselines in a given uv-grid and small enough to have all visibilities in the uv-grid correlated. In each uv-grid, they estimate the power spectrum by correlating visibilities only in nearby baselines, omitting the visibility autocorrelations. This drastically reduces the noise bias in estimates of the power spectrum in uv-grids. The contribution from each uv-grid within a given annulus in $U = \mid \vec{U} \mid$ is then combined, and the real part of it is used to quote the value of the isotropic power spectrum for the baseline separation U. We may schematically write it as
(4)$$\begin{eqnarray}
\mathcal {E} \lbrace P(U)\rbrace = \mathcal {R} [\langle \tilde{V}(\vec{U})^{*} \tilde{V}(\vec{U}+\Delta \vec{U}) \rangle].
\end{eqnarray}$$Here, the average is taken over the uv-grid first and then within the annulus, as explained above. Note that the power spectrum estimator here assumes that a perfect calibration is done and the gains are all unity. In such a case, the power spectrum estimate has no bias arising from instrumental noise, and its uncertainties can be written as (Ali et al. 2008; Dutta 2011)
(5)$$\begin{eqnarray}
\sigma _P^2 = \frac{P^2(U)}{N_\mathrm{ G}} + 2\frac{P(U)\sigma _N^2}{N_\mathrm{ B}} + 2\frac{\sigma _N^4}{N_\mathrm{ B}},
\end{eqnarray}$$where NG is the number of independent estimates of the power spectrum in a given annulus bin at U, NB is the total number of visibility pairs in the bin.

Background Background Background Background


From the theoretical point of view, we normalize the disk size to a fixed black hole mass and luminosity (thus to a fixed accretion rate; see Equations (14) and 17); therefore, we expect that the disk size τ
0 is independent of AGN properties. However, there are two possibilities that may cause the observed dependence in Figure 12. First, we adopt a constant virial factor f
BLR when estimating the black hole mass using Equation (1). Dynamical modeling of broad-line regions (BLRs) generally showed that f
BLR might depend on AGN properties and change from object to object (e.g., Pancoast et al. 2011; Li et al. 2013; Pancoast et al. 2014; Grier et al. 2017; Li et al. 2018; Williams et al. 2018). Observational calibrations of the virial factor also tend to support this conclusion (e.g., Ho & Kim 2014; Mejía-Restrepo et al. 2018; Yu et al. 2019). If there are systematic correlations between f
BLR and luminosity L
5100, black hole mass M
•, or accretion rate

Ṁ

, using a constant f
BLR will lead to the apparent dependence of τ
0 on AGN properties. Such a bias can be eliminated once we have a solid understanding of the virial factor in the future. Second, it is possible that the disk’s temperature profile does change with the accretion rate. For example, the standard and slim-disk models predict different temperature profiles
12

12
However, we note that the temperature profiles of slim disks (T ∝ R
−1/2) become different from those of standard disks (T ∝ R
−3/4) only within the photon-trapping radius (e.g., Wang & Zhou 1999).  (Shakura & Sunyaev 1973; Abramowicz et al. 1988). There are also other physical processes that may contribute to the correlation between β and

Ṁ

 (see, e.g., Li et al. 2021; Kammoun et al. 2021). Figure 7 implies that τ
0 and β are highly anticorrelated. As a result, a shallower temperature profile (larger β) will lead to a smaller disk size τ
0 and vice versa. In our sample, we find that high-luminosity and high-mass black holes generally have low accretion rates, giving rise to the dependence of τ
0 on luminosity and black hole mass.


The multimessenger observations of NGC 1068 dictate the need for multiple emission zones. In this work, we will capture its inner microparsecs referring to a spherically symmetric structure for the corona of the AGN as well as an outer starburst ring with a radius of ∼1 kpc (see Figure 1). Note, that in between these two emission sites NGC 1068 shows strong indications of a jet structure on scales of up to about 1 kpc (e.g., Wilson & Ulvestad 1982; Gallimore et al. 2004, 2006), which however is not included in this work. Due to mathematical convenience we treat both spatial regions as homogeneous. For particle acceleration processes that take place on considerably shorter timescales than the energy loss in these zones, we can disentangle these processes and only describe the steady-state transport of nonthermal, accelerated electrons and protons. Hereby, we suppose that in both zones some acceleration mechanism yields a differential source rate q(T) of relativistic protons and primary electrons that can be described by a power-law distribution in momentum space up to a certain maximal kinetic energy

Tˆ

, which depends on the competing energy-loss timescales in these zones. In the case of the starburst zone, we suppose that a certain fraction

fSN

 of the total energy of the SN that releases about 1051 erg and occurs with an approximate rate
5

5
Supposing a SN rate

νSN≃0.02[SFR/(1M⊙/yr)]yr−1

 (note that Condon1992 suggested a value of 0.04 instead of 0.02 for normal galaxies) where the star formation rate SFR ≃ 17[L
IR/(1011
L
⊙)] M
⊙yr−1. (Veilleux et al. 2005)

νSN≃0.34[LIR/(1011L⊙)]yr−1

 dependent on the IR luminosity L
IR gets accelerated into CRs according to diffusive shock acceleration (DSA; e.g., Drury 1983; Protheroe 1999) by individual supernova remnants (SNR; e.g., Bell 2014, and references therein). In general, many starburst galaxies—NGC 253 is a prominent example—show a galactic superwind (e.g., Veilleux et al. 2005, and references therein) as a result of a large number of core-collapse SNe. These winds introduce another source of acceleration
6

6
Note that in these phenomena also stochastic diffuse acceleration may become relevant due to the presence of a turbulent plasma within the wind bubbles. (e.g., Anchordoqui et al. 1999; Romero et al. 2018), however, we are not aware of any observational indications of such a superwind in the starburst ring of NGC 1068. For the AGN corona, we suppose that a fraction f
inj ≪ 1 of the mass accretion rate

Ṁ=Lbol/(ηradc2)

, with a radiation efficiency of η
rad = 0.1 (Kato et al. 2008), goes into relativistic protons via stochastic diffuse acceleration (SDA; e.g., Lemoine & Malkov 2020, and references therein). For both zones, the nonthermal primary electrons are normalized by the nonthermal proton rates due to the requested quasi-neutral total charge number of the injection spectra of primary CRs above a characteristic kinetic energy of

Tˇ≃10keV

 (Schlickeiser 2002; Eichmann & Becker Tjus2016; Merten et al. 2017). Note that this corresponds to a quasi-neutral acceleration site; however, CR transport can subsequently remove CR electrons and protons in different amounts from the nonthermal energy regime. Nevertheless their charge stays conserved. Transforming the source rates from momentum space into kinetic energy T, we obtain
1

qp(T)≡dNdVdTdt=qp,0Tˇ2+2TˇEp,0×T+Ep,0T2+2TEp,0T2+2TEp,0Tˇ2+2TˇEp,0−s/2exp[−T/Tˆ],

for the injected nonthermal protons, and
2

qe±(T)=qe−,0Tˇ2+2TˇEe,0T+Ee,0T2+2TEe,0×T2+2TEe,0Tˇ2+2TˇEe,0−s/2exp[−T/Tˆ]+qe±2nd(T),

for the nonthermal electrons (e
−) and positrons (e
+). Here, the latter term

qe±2nd(T)

 introduces the source rate of secondary electrons and positrons that are generated by hadronic interaction processes, as discussed in the following. Thus, the steady-state behavior of the differential nonthermal electron and proton density n(T) in the AGN corona and the starburst zone, respectively, can be approximated by
3

−∂∂TTn(T)τcool(T)=q(T)−n(T)τesc(T).

Here, τ

cool
 refers to the total continuous energy-loss timescale, which in the case of the relativistic electrons is given by the inverse of the sum of the synchrotron (syn), inverse Compton (IC), nonthermal Bremsstrahlung (brems), and Coulomb (C) loss rates, according to
4

τcool(e)=[τsyn(e)−1+τic−1+τbrems−1+τC(e)−1]−1,

and in the case of the relativistic protons we use
5

τcool(p)=[τsyn(p)−1+τC(p)−1+τpγπ−1+τBH−1+τpp−1]−1,

including the photopion (π), Bethe–Heitler pairs (BH), and hadronic pion (pp) production loss rates. Proton synchrotron losses—as well as the associated radiation—are negligible for the considered environments. Note that these processes require additional information on the associated interaction medium, which is one of the following targets:(i)A magnetic field, which is assumed to be uniform on small scales (with respect to the particles’ gyro radius) and randomly oriented on significantly larger scales (due to isotropic Alfvénic turbulence).(ii)A photon target, which is in the case of the starburst zone dominated by the thermal IR emission due to the rescattered starlight by dust grains with a temperature θ
dust and can be described by an isotropic, diluted modified blackbody radiation field
6

nIR(E)=Cdilπ2ℏc3E2exp(E/(kBθdust))−1EE0,

where the dust clouds become optically thick above a critical energy E
0 = 8.2 meV (Yun & Carilli 2002). The constant dilution factor C

dil
 is determined from the observed IR luminosity L
IR according to the relation

LIR/(πRstr2c)=∫dEEnIR(E)

.
7

7
A more accurate approach to the IR photon spectrum has been proposed by Casey (2012), where a coupled modified greybody plus a mid-infrared power law has been used, but these modifications have no impact on our results. In case of the coronal region we used a parametrized model (Ho 2008) above 1 eV that accounts for the optical and UV emission by the disk as well as the Comptonized X-ray emission by hot thermal electrons in the corona. Hereby, the parameterization depends on the Eddington ratio (L
bol/L
Edd), i.e., the ratio of the bolometric over the Eddington luminosity, and we adopt the relation of Hopkins et al. (2007) to determine L
bol based on the intrinsic X-ray luminosity L
X between (2 and 10)keV.(iii)A thermal gas target with a given temperature θ, which is due to mathematical convenience assumed to be homogeneously distributed in both regions. For the starburst ring, Spinoglio et al. (2012) determine θ = 127 K, a gas density of n(H2) = 102.9 cm−3, and a molecular hydrogen mass of M(H2) ∼ 3.5 × 108
M
⊙.More details on the individual energy-loss timescales can be found in Appendix A.


We have performed the first detailed dissection of the stellar populations of the Milky Way disc in age, [Fe/H] and $\mathrm{[ \alpha \mathrm{/Fe]}}$ space, bridging the gap between the detailed observational understanding of MAPs (e.g. Bovy et al. 2012b, 2016b) and the plethora of studies of co-eval stellar populations in simulated galaxies (e.g Bird et al. 2013; Stinson et al. 2013; Martig et al. 2014a). We have placed novel constraints on models for the formation of the Milky Way disc by combining detailed density models fit to the mono-age, mono-[Fe/H] populations of the low and high $\mathrm{[ \alpha \mathrm{/Fe]}}$ disc, with surface mass density contributions calculated on the basis of these density fits and stellar evolution models. We summarize our key results as follows:
Radial and vertical profiles: The mono-age, mono-[Fe/H] populations of the $\mathrm{[ \alpha \mathrm{/Fe]}}$ poor disc are well fitted by a radially broken exponential, with a peak radius, Rpeak, that varies as a function of age and [Fe/H]. We find that the distance between Rpeak's of the low and high [Fe/H] populations increases with age, which we interpret as evidence for a decreasing [Fe/H] gradient with time (e.g. Anders et al. 2017). The radial variation of the stellar surface density of the high $\mathrm{[ \alpha \mathrm{/Fe]}}$ mono-age populations is found to have insignificant breaks, and they are better fit by a single exponential in this disc region. As these populations are the oldest, this may be a sign of the disc evolution washing out the density peak over time, or may point to a different formation scenario for high $\mathrm{[ \alpha \mathrm{/Fe]}}$ stars, where no density peak ever existed. These findings are in good agreement with earlier studies of MAPs (Bovy et al. 2016b). We measure an average high $\mathrm{[ \alpha \mathrm{/Fe]}}$ population scalelength of hR, in = 1.9 ± 0.1 kpc, and find scaleheights between 600 and 1000 pc, in good agreement with current measures of the $\mathrm{[ \alpha \mathrm{/Fe]}}$ rich disc scalelength and scaleheight (e.g. those outlined in Bland-Hawthorn & Gerhard 2016).Profile broadening: We show that the radial surface density profile of the low $\mathrm{[ \alpha \mathrm{/Fe]}}$ populations broadens with age in a given [Fe/H] bin, which we interpret as evidence of the gradual dispersal of mono-[Fe/H] populations, presumably due to radial migration and radial heating. The variation in shape of the broken exponential profile changes differently depending on the population [Fe/H], with low [Fe/H] populations inner profiles flattening faster, whereas the high [Fe/H] outer profiles flatten faster. We interpret this effect as tentative evidence for [Fe/H] dependent radial migration arising from pre-existing [Fe/H] gradients in the star-forming disc. We showed that our results qualitatively reproduce those of Hayden et al. (2015), finding a skewed MDF that varies as a function of R.Flaring: We find that flaring seems to be present in almost all mono-age populations, at differing levels. We have shown that the inverse flaring scalelength Rflare− 1 increases with age, meaning that the youngest populations flare most strongly. This finding appears inconsistent with that above, under the assumption that flaring is the result of radial migration. However, these results may be reconciled by invoking a more active accretion history in the early life of the disc, which could have suppressed flaring (e.g. Minchev et al. 2014b).The surface-mass density at R0: We have measured the surface mass density at the solar radius for each mono-age, mono-[Fe/H] population, finding a total surface mass density of $\Sigma _{R_0, {\rm tot}} = 20.0_{-2.9}^{+2.4}\mathrm{(stat.)}_{-2.4}^{+5.0}\mathrm{(syst.)}\ \mathrm{M_{{\odot }} \ pc^{-2}}$. Before allowing for systematics, this value is less than current estimates (e.g. Flynn et al. 2006; Bovy et al. 2012a; McKee et al. 2015), however, the systematic uncertainties are large, mainly due to a mismatch between the log g scales in APOGEE and the PARSEC models, and as such, we find our value to be consistent within the uncertainties. The relative contribution of high to low $\mathrm{[ \alpha \mathrm{/Fe]}}$ populations, $f_\Sigma$, is 18 per cent ± 5 per cent, which is consistent with existing measurements (e.g. Bland-Hawthorn & Gerhard 2016).The hZ distribution at R0: The shape of the mass-weighted hZ distribution found by this study is in good agreement with that of Bovy et al. (2012a), calling into question the existence of a vertical structural discontinuity in the Milky Way disc. The reconciliation of this finding with the discontinuity in chemical space (e.g. the bimodality in $\mathrm{[ \alpha \mathrm{/Fe]}}$ at fixed [Fe/H]: Nidever et al. 2014; Hayden et al. 2015) may shed new light on our understanding of the formation of the Galactic disc.The surface-mass density profile of the Milky Way: We have found the combined (from mono-age, mono-[Fe/H] populations at low and high $\mathrm{[ \alpha \mathrm{/Fe]}}$) surface-mass density-weighted profiles of the Milky Way disc as a function of $\mathrm{[ \alpha \mathrm{/Fe]}}$, age and [Fe/H], and found that the total surface density is also described by a broken exponential. We find that our results fail to determine the sign of the inner exponential to high significance out to ∼10 kpc, but detect a turnover to a declining exponential, at high significance, thereafter. We find evidence of a radial mean age and [Fe/H] gradient driven by the changing dominant population as a function of radius. A detailed comparison of these findings with numerical simulations is necessary for a proper interpretation. Our finding of a decline in stellar density may be consistent with that found in other studies (e.g. Reylé et al. 2009; Sale et al. 2010), albeit at shorter radii.


It is well known that star formation is inefficient on galactic scales. The observed linear correlation between molecular gas surface density and star formation rate (SFR) surface density in normal star-forming galaxies suggests a nearly constant gas depletion time-scale around ∼2 Gyr, much longer than the dynamical time-scale of galactic discs (Kennicutt 1989; Bigiel et al. 2008; Saintonge et al. 2011; Leroy et al. 2013; Genzel et al. 2015; Tacconi et al. 2018). In contrast, the SFE on GMC scales shows a large variation ranging from less than a few per cent to nearly unity (Zuckerman & Evans 1974; Krumholz & Tan 2007; Wu et al. 2010; Evans, Heiderman & Vutisalchavakul 2014; Heyer et al. 2016; Lee, Miville-Deschênes & Murray 2016; Vutisalchavakul, Evans & Heyer 2016). The origin of this large scatter is usually explained as a combination of the time variability of the SFR during the course of cloud evolution and intrinsic scatter of SFEs due to the diversity of GMC properties (Feldmann & Gnedin 2011; Kruijssen & Longmore 2014; Lee et al. 2016; Grudić et al. 2018a; Kruijssen et al. 2018). For example, recent theoretical models and high-resolution magneto-hydrodynamics simulations suggest that the SFE depends on the local virial parameters of the cloud controlled by large-scale turbulence (e.g. Krumholz & McKee 2005; Padoan, Haugbølle & Nordlund 2012). However, it has recently been recognized that large-scale turbulence can only account for an ∼0.3 dex scatter, which is not sufficient to explain the observed SFE variations (Lee et al. 2016). Another source of variation comes from different stellar feedback channels that alter the dynamical states of the GMCs (Fall, Krumholz & Matzner 2010; Murray, Quataert & Thompson 2010; Dale et al. 2014; Myers et al. 2014; Raskutti, Ostriker & Skinner 2016; Kim et al. 2017; Grudić et al. 2018b). Previous studies found that GMC simulations adopting different stellar feedback mechanisms (stellar winds, ionizing radiation, or supernovae) lead to dramatically different final SFEs. The problem has recently been recognized to be more subtle than previously thought, since even small differences in numerical treatments, such as different radiative transfer schemes, massive star sampling, and momentum and energy deposition algorithms, can lead to drastic changes for the final SFE (Dale et al. 2005; Roškar et al. 2014; Raskutti et al. 2016; Grudić et al. 2018b; Kim, Kim & Ostriker 2018). Therefore, how the SFE depends on GMC properties and the strength of stellar feedback remains an open question.


Snow lines of abundant volatile species are one of the possible origins of the observed substructures in dust continuum emission (Zhang et al. 2015; Okuzumi et al. 2016; Pinilla et al. 2017). Recent disk surveys have shown that the locations of the observed gap/ring structures do not seem to be related to the radial locations of the snow lines (Huang et al. 2018; Long et al. 2018; van der Marel et al. 2019). To estimate the positions of the snow lines in observed disks, the temperature profile is often assumed to be a simple power law (Huang et al. 2018; Long et al. 2018), which is broken if the disk has shadows on the disk surface (Dullemond et al. 2001; Baillié & Charnoz 2014; Ueda et al. 2019). Figure 5 shows the radial positions of snow lines of H2O, NH3, CO2, H2S, C2H6, CH4, and CO at

t

=
0

 and 11. For simplicity, we define the snow lines as the radial location where the midplane temperature reaches 150, 70, 55, 50, 40, 25, and 20 K for H2O, NH3, CO2, H2S, C2H6, CH4, and CO, respectively. We clearly see that the radial positions of snow lines move with time and multiple snow lines emerge even for a single species. This means that it is necessary to determine the disk temperature precisely when we evaluate the snow line locations. These snow lines would induce additional ring and gap structures and some of them would overlap with the TWI-induced substructures. It should be noted that the oscillation timescale of the TWI is much shorter than the dust radial drift timescale. Therefore, sintering-induced substructures would not coincide with the locations of the snow lines if the TWI is present. However, the dust-size variation can be induced by sublimation and recondensation, which potentially produces millimeter substructures. Since the radial locations of the snow lines are important not only for the substructure formation but also for the chemical composition of forming planets (e.g., Sato et al. 2016; Öberg & Wordsworth 2019), we should investigate how the TWI evolves in planet forming disks.


The last column in Table 1 reports the number of OB stars minus the “diffuse” population estimated from their density in the Reference field (22.5 stars per square degree): as is immediately seen, the M-star statistics is much larger than the OB star statistics. This can hardly be considered surprising, if an ordinary IMF (e.g., that from Weidner et al. 2010) is assumed for the star-formation region. In Fig. 19 we show the density ratio between M and OB stars, which provides a consistency test between our results and a plausible IMF: this ratio varies however by a large factor, close to 20, among our subregions. This might reflect differences in the respective IMFs, but also differences in completeness among the stellar samples considered for the various regions. We first note that the ratio between M and OB stars in NGC 6231 is dramatically lower than anywhere else in Sco OB1. We can indeed expect that M stars are detected less efficiently in the inner parts of NGC 6231, where the density of bright stars is very large, and their diffuse glare raises the limiting magnitude locally. As already discussed above in Sect. 4.1, this causes our sample of M stars in NGC 6231 to be highly incomplete. Moreover, we determined above that NGC 6231 is significantly more extincted, by almost half a magnitude in V, than Tr 24, and this implies a higher minimum detectable mass among NGC 6231 M stars compared to Tr 24 (see the MDA diagrams in Fig. 5); this effect reduces the completeness of the M-star sample in NGC 6231 more than in Tr 24. If Tr 24 is also slightly younger than NGC 6231, as we argue below, our M-stars in Tr 24 will reach down to lower masses than in NGC 6231, with a steep increase in the detected M-star population: adopting the IMF from Weidner et al. (2010), the predicted number of cluster M stars doubles considering the mass interval 0.25–0.5 M⊙ rather than 0.35–0.5 M⊙. If Tr 24 is younger than NGC 6231, moreover, its stars in the mass range 2.5–3 M⊙ might not have yet reached their ZAMS position as B stars, and therefore would not be counted among OB stars; this would further raisethe M/OB star ratio there by up to 30%. Therefore, the proportions of both M and OB stars that are detected in a young cluster will depend on their age and extinction, in accordance with the MDA diagrams, even for a fixed, spatially uniform photometric sensitivity. We estimated using the Weidner et al. (2010) IMF the expected range for the observed M/OB number ratio. Siess et al. (2000) predict that the latest-type B stars have a mass of ~ 3.5 M⊙ at 2 Myr, and ~ 2.2 M⊙ at 10 Myr, that is, in the range of ages expected for Sco OB1 clusters. The MDA diagrams of Fig. 5 predict that the lowest-mass stars we are able to detect using the available Sco-OB1 data have ~ 0.2 M⊙, even assuming the most favorable (and unlikely) circumstances of an age less than 2 Myr and negligible reddening. The extreme values found for the M/OB ratio are then ~ 3.8 for a minimum M-star mass as high as 0.35 M⊙ and an old age of 10 Myr, and ~ 20 for a minimum M-star massas low as 0.2 M⊙ and age of 2 Myr. These extremes are also shown as horizontal lines in Fig. 19. We note that the M/OB ratio in NGC 6231 falls well within this range; however, both Tr 24 regions are significantly richer of M stars than expected, by more than a factor of two and well above (statistical) errors. If true, then paradoxically this part of the OB association would form preferentially lower-mass stars. Of course, more detailed studies are needed to confirm this result. In the G345.45+1.50 region the M/OB ratio is highest, and far above predictions from the IMF: we may tentatively explain this since this region is very young, and some of its most massive members, like IRAS 16562-3959, are still in formation, thus decreasing the number of optically revealed OB stars. The lowest M/OB ratio in NGC 6231 is unlikely to be real, since as discussed above our M-star sample in this densest subregion is likely incomplete.


To further reduce the number of fitting parameters, we assume that no source is unique in terms of the acceleration spectral shape, i.e. all the source components have the same H spectral index α, He spectral index, maximum rigidity Rm, and He-to-H Galactic injection flux ratio χ. The spectrum of He seems harder than that of H as indicated by many spectral studies, implying, e.g. time-dependent particle acceleration (Zhang et al. 2017), or more efficient He injection in the shock acceleration process (Malkov, Diamond & Sagdeev 2012). It is thus assumed the He spectral index to be α − 0.077 according to the sub-TeV data fit (Aguilar et al. 2015b). To be consistent with the conventional scenario, we also consider that the nearby SNR can inject a total CR energy of 1050 erg into the ISM, i.e.
(15)$$\begin{eqnarray}
\int _{\text{GeV}}^{\infty }{\left[ \frac{{\rm d}R_{\text{H}}}{{\rm d}E}+\chi \left(E \right) \frac{{\rm d}R_{\text{He}}}{{\rm d}E} \right] N\left(E \right) E{\rm d}E}=10^{50}\text{ erg},
\end{eqnarray}$$where E is the particle kinetic energy, N refers to the injection spectrum of H with $N\left( \text{GV}\right) \approx 4.7\times 10^{52}\text{ GV}^{-1}$ (for the fitted parameters in Table 1). For brevity, we hereafter treat all of N and Q as quantities for H components. With these respects, in the naive approach, the spectral fitting parameters are $\chi \left( \text{GV}\right)$, α, Rm, ϕ, θ, MA, $\left| z/h \right|$, $\mu Q_{\text{d}}\left( \text{GV}\right)$, $Q_{\text{c}}\left( \text{GV}\right)$, tc, where $\theta =\arcsin \left(\rho /r_{\text{n}} \right)$ is the angle between the regular magnetic field line and the line-of-sight vector towards the nearby source. In the quasi-local approach equation (11), the alignment of the field line must uniquely be determined, leading to an additional angular parameter, which can be chosen as the angle φ around the line-of-sight vector (towards the nearby source with φ = 0° shown in Fig. 2d). The fitting results of the CR spectrum for models with the naive and quasi-local approach are shown with Figs 1(a) and 2(a), respectively.


Another way of obtaining information about the nucleosynthesis processes involved in producing carbon is to compare it with other elements that are characterised by a well-known source of production, as in the case of oxygen. In Fig. 5, we show the variation of [C/O] as a function of [Fe/H], which serves as a first-order approximation to the evolution with time. To calculate the [C/O] ratios, two oxygen abundance indicators are used independently. At subsolar metallicities, the abundance ratios with both oxygen indicators are mostly negative and show an increasing trend towards higher metallicity. This is explained by the fact that oxygen is entirely produced by SNe Type II from massive progenitors, which started to release theiryields at earlier ages in the Galaxy and, hence, at lower metallicities (e.g. Woosley & Weaver 1995). The massive stars producing carbon at low metallicities might be less massive than those producing oxygen (i.e. having a longer life), explaining a delayed contribution of carbon, hence, the negative [C/O] ratios. Alternatively, this could be explained by increasing O/C yields for more massive progenitors of SNeII. Once metallicity starts to increase, low- and intermediate-mass stars release carbon and massive stars start to eject more carbon than oxygen (Carigi et al. 2005). The [C/O] ratio seems to have a constant rise towards higher metallicities when using the forbidden oxygen line. However, in the case when the O I 6158 Å line is employed, we do observe that the maximum in [C/O] takes places close to solar metallicity to then become flat or decrease. This suggests that low-mass stars mostly contribute to carbon around solar metallicity, whereas at super-solar metallicities, massive stars produce carbon together with oxygen, thereby flattening or even decreasing the [C/O] ratio. This trend is in agreement with the metallicity dependent yields from Carigi et al. (2005), which provide higher carbon as [Fe/H] increases from massive stars (i.e. also increasing the O production) but lower carbon from low and intermediate mass stars as [Fe/H] increases (i.e. less production of C). The turning point of increased relative production of carbon from massive stars takes place at A(O) ~ 8.7 dex (see Fig. 2 of Carigi et al. 2005) which equals to [O/H] ~ 0.0 dex. This observed behaviour of [C/O] is in contrast to the steady increase of [C/O] up to [Fe/H] ~ 0.3 dex found, for example, by Franchini et al. (2021). Nevertheless, the general trend we find when using the [O I ] 6300 Å line is similar to the reported by Franchini et al. (2021), who use also that oxygen indicator. All thick-disk stars present negative [C/O] ratios and when using the oxygen line at 6158 Å thin-disk stars with [Fe/H] ≲ –0.2 have [C/O]  0 as well. Thick-disk stars and low-metallicity thin-disk stars at the same metallicity have similar [C/O] ratios, meaning that the balance between different production sites for oxygen and carbon is the same among both populations, despite [C/Fe] and [O/Fe] being systematically higher for thick-disk stars at a given metallicity.


In order to make a linear analysis, first we have to prepare the PNS models as a background. The PNS properties depend on not only the density and pressure profiles but also the distributions of temperature (or entropy per baryon) and electron fraction inside the PNS, while such profiles can be determined only via the numerical simulation of the core-collapse supernova explosion. In this study, as in Sotani & Sumiyoshi (2019), we particularly adopt the profiles obtained via the numerical simulations performed by solving the general relativistic neutrino-radiation hydrodynamics under the spherical symmetry. In the simulations, hydrodynamics and neutrino transfer in general relativity are solved simultaneously (Yamada 1997; Yamada, Janka & Suzuki 1999; Sumiyoshi et al. 2005). To describe the neutrino transfer, the Boltzmann equation is directly solved with the multi-angle and multi-energy neutrino distributions for four species, νe, $\bar{\nu }_\mathrm{ e}$, νμ/τ, and $\bar{\nu }_{\mu /\tau }$, i.e. we implement six species of neutrinos by assuming μ-type and τ-type (anti-)neutrinos have identical distributions. For the collision term associated with neutrino emission, absorption, and scattering with leptons, nucleons, and nuclei handles, the basic neutrino reactions are adopted (Bruenn 1985; Sumiyoshi et al. 2005). The metric adopted in the numerical code is given by
(1)$$\begin{eqnarray*}
\mathrm{ d}s^2 = -\mathrm{ e}^{2\Phi (t,m_\mathrm{ b})}\mathrm{ d}t^2 + \mathrm{ e}^{2\Lambda (t,m_\mathrm{ b})}\mathrm{ d}m_\mathrm{ b}^2 + r^2(t,m_\mathrm{ b})(\mathrm{ d}\theta ^2 + \sin ^2\theta \mathrm{ d}\mathrm{ }\phi ^2), \nonumber\\
\end{eqnarray*}$$where t and mb denote the coordinate time and the baryon mass coordinate, respectively (Misner & Sharp 1964). In addition, mb is related to the circumference radius (r) via the baryon mass conservation, while the metric functions, Φ(t, mb) and Λ(t, mb), are evolved together with hydrodynamical variables in the numerical simulations (Yamada 1997). The numerical simulations for core-collapse supernovae have been done with 255 grid points in the radial mass coordinate, 6 grid points in the neutrino angle, and 14 grid points in the neutrino energy. The rezoning of radial mesh is made during the simulations to resolve the accreting matter. We remark that the radial grids of mass coordinate are non-uniformly arranged to cover not only the dense region inside the central object but also the region for accreting matter.


On larger scales, the clusters in which BCGs reside can generally be divided into two categories: cool core clusters, which exhibit very peaked surface brightness distributions at X-ray wavelengths, and non cool core clusters, with similar overall X-ray luminosities but with smoother, less peaked X-ray surface brightness distributions. Some authors (e.g. Hudson et al. 2010; Santos et al. 2010) define an intermediate category called moderate or weak cool core clusters. Since cool core clusters have short radiative cooling time-scales on the order of 108 yr in their centres (e.g. Voigt & Fabian 2004; McNamara & Nulsen 2007, 2012; Hlavacek-Larrondo et al. 2012), starbursts are expected to be common at the centre of such clusters. Indeed, the central cool gas in these clusters should condense onto the BCG, forming stars at rates of hundreds of solar masses per year (e.g. Fabian 1994). However, most BCGs are relatively quiescent and those that do show evidence of star formation generally tend to have star formation rates 1 order of magnitude smaller, on the order of $1-150 \, \mathrm{M_{\odot }\, {yr}^{-1}}$ (e.g. Donahue et al. 2007; Bildfell et al. 2008; O’Dea et al. 2008, 2010; Rawle et al. 2012). This mismatch between expected and observed star-forming rates, known as the cooling flow problem, is thought to be caused by active galactic nuclei (AGNs) feedback processes from the BCG. AGNs can release copious amounts of energy into the intracluster medium (ICM) through many ways, including: jetted outflows that inflate cavities, weak shocks, sound waves, or turbulence in the ICM (e.g. Markevitch & Vikhlinin 2007; McNamara & Nulsen 2007, 2012; Zhuravleva et al. 2014; Fabian et al. 2017). Alone, the energy released by jetted outflows appears to be on the same order as the energy needed to offset cooling (e.g. Rafferty et al. 2006; McNamara & Nulsen 2007; Hlavacek-Larrondo et al. 2012), therefore suggesting that AGN feedback is a good candidate for solving the cooling flow problem.


We note in passing that recent studies address improved satellite modellimg that ameliorates many of these issues, including the core–cusp issue via non-sphericity of the stellar velocity distribution (Hayashi, Chiba & Ishiyama 2020) and the detectability of MWG satellites (Nadler et al. 2020). Other proposed solutions include those invoking baryonic physics, ranging from inclusion of baryon-contraction-induced diversity (Lazar et al. 2020), through dynamical friction-mediated coupling with baryonic clumps (El-Zant, Shlosman & Hoffman 2001; El-Zant et al. 2004; Tonini, Lapi & Salucci 2006; Romano-Díaz et al. 2008; Goerdt et al. 2010; Cole, Dehnen & Wilkinson 2011; Del Popolo et al. 2014; Nipoti & Binney 2015), or through dynamical feedback driven by starbursts or active galactic nuclei (AGNs; Read & Gilmore 2005; Mashchenko, Couchman & Wadsley 2006; Mashchenko, Wadsley & Couchman 2008; Peirani, Kay & Silk 2008; Governato et al. 2012; Pontzen & Governato 2012; Zolotov et al. 2012; Martizzi, Teyssier & Moore 2013; Teyssier et al. 2013; Madau, Shen & Governato 2014; Ogiya & Mori 2014; Pontzen & Governato 2014; El-Zant, Freundlich & Combes 2016; Silk 2017; Freundlich et al. 2020). Alternatively, modifications to the particle physics model of the dark matter have been proposed. Such proposals include ‘pre-heated’ warm dark matter (e.g. Colín, Avila-Reese & Valenzuela 2000; Bode, Ostriker & Turok 2001; Macciò et al. 2012; Schneider et al. 2012; Shao et al. 2013; Lovell et al. 2014; El-Zant, Khalil & Sil 2015) and self-interacting dark matter, whereby energy flows into the central cores of haloes through conduction (e.g. Burkert 2000; Kochanek & White 2000; Spergel & Steinhardt 2000; Miralda-Escudé 2002; Peter et al. 2013; Zavala, Vogelsberger & Walker 2013; Elbert et al. 2015). Ultralight axions, with boson mass ∼10−22 eV, have also been considered as dark matter candidates in connection with these same small (sub)galactic scale problems (e.g. Peebles 2000; Hu, Barkana & Gruzinov 2000; Peebles 2000; Marsh & Silk 2014; Schive et al. 2014b; Hui et al. 2017; Mocz et al. 2019; Nori et al. 2019; see Niemeyer 2019 for recent review). Here the zero-point momentum associated with a long de Broglie wavelength corresponding to the small mass comes along with ‘fuzziness’ in particle positions. This in turn leads to a hotter halo core with non-diverging central density and a cut-off in halo mass. Such axion fields can also be relevant for inflationary scenarios or late dark energy models. The non-thermal production implies that the axions are present with the required abundance for dark matter; they behave as cold dark matter on larger scales despite the tiny masses (Marsh 2016, 2017).


In addition to some fraction of BNS mergers masquerading as long GRBs, our sample used to constrain the DTD may suffer from other issues of incompleteness. As we rely on the modeling of host galaxies when constraining the DTD, we do not consider short GRBs that do not have a confident host association. Though the properties of short GRB hosts do not seem to deviate strongly as a function of the host-association confidence (see, e.g., Figure 4 of Nugent et al. 2022), neglecting these events may have a potential impact on both the low and high ends of our inferred DTD. For example, GRBs that are highly offset from their hosts may have afterglows with much lower luminosities, making precise localization (and therefore host identification) difficult (Perna et al. 2022). Such systems may have migrated over longer timescales to reach the highly offset locations of the burst and therefore may have longer delay times than the general population. Furthermore, the P
cc method for host identification may incorrectly associate a GRB with a faint underlying host rather than a bright host at a larger offset, though Fong et al. (2022) predicted this to be an effect only at the ≲7% level. On the other hand, if such poorly associated GRBs are instead truly associated with faint galaxies that are below detection limits, we may be excluding additional systems with short delay times as these faint, low-mass galaxies are typically star-forming. Furthermore, though Swift can detect GRBs out to z ∼ 3, there is likely some fraction of short GRBs that occur beyond this horizon, when the universe was ≲2 Gyr old. Short GRBs that occur at these early stages in the history of the universe must have short delay times, and this selection effect may bias the general population in our analysis to longer delay times. This would lead to a larger inferred

tmin

, and, due to the correlation between

tmin

 and α, more negative values of α. However, this population of high-redshift short GRBs is likely small; assuming the SFH from Madau & Fragos (2017), 10% of stars are born beyond z = 3, and the fraction of compact object binary mergers beyond this redshift will be even smaller due to the delay time between formation and merger.


In general, hybrid-kinetic models have proved to be capable of satisfactorily catching the main kinetic physics at play for a large number of problems, ranging from fluid and kinetic instabilities (Hellinger & Matsumoto 2000; Matteini et al. 2006; Califano et al. 2008; Henri et al. 2013; Kunz et al. 2014) to collisionless shocks (Lembège et al. 2009; Caprioli & Spitkovsky 2013; Weidl et al. 2016), dynamo effects (Rincon et al. 2016; St-Onge & Kunz 2018), MR (Le et al. 2016; Palmroth et al. 2017; Cerri & Califano 2017; Franci et al. 2017; Wang et al. 2019; Califano et al. 2020), and kinetic-scale turbulence (Servidio et al. 2015; Grošelj et al. 2017; Cerri et al. 2018, 2019; Hellinger et al. 2019; Wang et al. 2019). The main goal of our project is to investigate the possibility of improving the electron description in hybrid-kinetic models. The starting point is the “hybrid Vlasov–Maxwell” (HVM) code (Mangeney et al. 2002; Valentini et al. 2007), which is already equipped with a fluid model in which electrons are described as an isotropic, isothermal fluid with finite inertia. The HVM code has recently been upgraded in order to implement a more sophisticated model for the electron fluid. This includes evolution equations for the anisotropic (gyrotropic) electron pressures, p∥, e and p⊥, e (where ∥ and ⊥ refer to the local magnetic-field direction, b = B/|B|), and a Landau-fluid (LF) closure for the parallel transport of the gyrotropic electron thermal energy along field lines (i.e., parallel heat fluxes, q∥, e and q⊥, e). Hereafter, we refer to this new model as “hybrid Vlasov–Landau-fluid” (HVLF). The idea is to include within a hybrid description the relevant electron pressure-anisotropy effects and a fluid model for the electron-kinetic response that still holds in a nonlinear regime. Therefore, the LF model implemented in the HVLF code goes beyond the early attempts to include these effects in simplified settings (e.g., Hammett & Perkins 1990; Snyder et al. 1997; Passot & Sulem 2007) and is based on the approach presented by Sulem & Passot (2015).


In this paper, we report on spectroscopic CH3CN, CH3OH (methanol), and dust continuum observations with the Atacama Large Millimeter/submillimeter Array (ALMA) at 349 GHz with an angular resolution of 0′′.1. We exploit the CH3CN (19K–18K) K-ladder, with excitation energies ranging from 168 K (for K = 0) to 881 K (for K = 10), to probe, at different radii, the physical conditions in the accretion disk of an early-type young star. We targeted the star-forming region G023.01−00.41, at a trigonometric distance of 4.59

$^{+0.38}_{-0.33}$

−0.33

+0.38

 kpc from the Sun (Brunthaler et al. 2009), where we recently revealed the accretion disk around a young star of 104.6 L⊙, corresponding to a ZAMS star of 20 M⊙ (Sanna et al. 2019, their Fig. 1); the disk was imaged by means of spectroscopic ALMA observations of both CH3CN and CH3OH lines at 0′′.2 resolution inthe 230 GHz band. The disk extends up to radii of 3000 au from the central star where it warps above the midplane; here, we resolve the outer disk regions in two apparent spirals projected onto the plane of the sky. We showed that molecular gas is falling in and slowly rotating with sub-Keplerian velocities down to radii of 500 au from the central star, where we measured a mass infall rate of 6 × 10−4 M⊙ yr−1 (Sanna et al. 2019, their Fig. 5). The disk and star system drives a radio continuum jet and a molecular outflow aligned along a position angle of 57°, measured east of north (Sanna et al. 2016, their Fig. 2); their projected axis is oriented perpendicular to the disk midplane whose inclination with respect to the line-of-sight was estimated to be less than 30° (namely, the disk is seen approximately edge-on; Sanna et al. 2014, 2019). Previously, we also measured the average gas conditions over the same extent of the whole disk, by means of Submillimeter Array (SMA) observations of the CH3CN (12K–11K) emission, and we estimated a kinetic temperature of 195 K and CH3CN column density of 5.1 × 1016 cm−2 (Sanna et al. 2014, their Fig. 2 and Table 4).


However, the intrinsic effects caused by the unknown emission and acceleration mechanisms in the source could mitigate or enhance the LIV-induced time delay, which would impact the accuracy of the resulting constraints on LIV. A key challenge is then to distinguish an intrinsic time lag at the source from a delay induced by LIV. Long GRBs usually have significantly positive or negative intrinsic spectral lags and should not be used for LIV searches until reasonable progress is made on the modeling of the emission and acceleration mechanisms (Chen et al. 2005; Ukwatta et al. 2012; Bernardini et al. 2015), while short GRBs are consistent with null or negligible intrinsic spectral lag and are therefore an ideal tool to measure the LIV effect (Norris & Bonnell 2006; Bernardini et al. 2015, 2017; Xiao et al. 2022). Currently, in addition to short GRBs, active galactic nucleus (AGN) flares and gamma-ray pulsars are two other classes of astrophysical sources that have no significant intrinsic lag in general and are often used for LIV tests (Biller et al. 1999; Kaaret 1999; Aharonian et al. 2008; MAGIC Collaboration et al. 2017). It should be noted, however, that there is also evidence of intrinsic lags in some cases of AGN flares and pulsars. For example, MAGIC Collaboration during an observational campaign regarding Mkn 501 blazar found an indication of about 4 minutes time delay between the peaks at E  0.25 TeV and E > 1.2 TeV (MAGIC Collaboration et al. 2008), which may indicate a progressive acceleration of electrons in the emitting plasma blob. A robust method to study the correlations between arrival times and energy, based on a likelihood function built from the physical picture assumed for the emission, propagation, and detection of the photons was proposed by Martínez & Errando (2009). In the case of pulsars, there are some lags if the energy range is extended too much toward low energies (e.g., radio versus TeV). No real progress on the topic of intrinsic effects will be made without accurate models for production and acceleration mechanisms for each type of source. Perennes et al. (2020) first attempted to gain knowledge on source-intrinsic spectral lags of flaring AGNs at high and very high energies and on short timescales relevant for LIV searches, using leptonic AGN flare modeling. Concerning GRBs, some ways have been proposed to reduce the impact of intrinsic effects, e.g., fitting the observed spectral lags of statistical samples of GRBs at a range of different redshifts (Ellis et al. 2006; Bernardini et al. 2017; Xiao et al. 2022), or using only a limited observer-frame energy bands range corresponding to the fixed source-frame energy bands (Wei & Wu 2017). Anyway, there is no reason to think that the low and high-energy photons should be emitted simultaneously at the source, and while detecting distinct signals at different energy channels, we have no idea which one was sent first. Previous studies usually assumed that the intrinsic time delays are either an unknown constant for all GRBs considered or scale with the photon energy E according to some power-law function (Ellis et al. 2006; Biesiada & Piórkowska 2009a; Zhang & Ma 2015; Wei et al. 2017a; Acciari et al. 2020; Pan et al. 2020; Du et al. 2021).


The gravitational potential of an eccentric companion at the quadrupole order can be decomposed as a sum over circular orbits (e.g. Storch & Lai 2013; Vick, Lai & Fuller 2017):
(5)$$\begin{eqnarray*}
U\left(\boldsymbol{\mathbf {r}}, t\right) = \sum \limits _{m=-2}^2 U_{2m} \left(\boldsymbol{\mathbf {r}}, t\right) ,
\end{eqnarray*}$$(6)$$\begin{eqnarray*}
U_{2m}\left(\boldsymbol{\mathbf {r}}, t\right) &=& -\frac{GM_2 W_{2m} r^2}{D(t)^3} Y_{2m}(\theta , \phi) e^{-imf\!\!\!\:(t)}, \\
&=& -\frac{GM_2W_{2m} r^2}{a^3}Y_{2m}\left(\theta , \phi \right) \sum \limits _{N = -\infty }^\infty \!\!F_{Nm}e^{-iN\Omega t} .
\end{eqnarray*}$$Here, the coordinate system is centered on the MS star, (r, θ, ϕ) are the radial, polar, and azimuthal coordinates of $\boldsymbol{\mathbf {r}}$ respectively, $W_{2 \pm 2} = \sqrt{3\pi /10}$, W2 ± 1 = 0, $W_{20} = -\sqrt{\pi / 5}$, D(t) is the instantaneous distance to the companion, f is the true anomaly, and Ylm denote the spherical harmonics. FNm denote the Hansen coefficients for l = 2 (also denoted $X^N_{2m}$ in Murray & Dermott 1999), which are the Fourier coefficients of the perturbing function, i.e.
(7)$$\begin{eqnarray*}
\frac{a^3}{D(t)^3} e^{-imf\!\!\!\:(t)} = \sum \limits _{N = -\infty }^\infty \!\!F_{Nm} e^{-iN\Omega t}.
\end{eqnarray*}$$The FNm can be written explicitly as an integral over the eccentric anomaly (Murray & Dermott 1999; Storch & Lai 2013):
(8)$$\begin{eqnarray*}
F_{Nm} = \frac{1}{\pi }\int \limits _{0}^{\pi } \frac{\cos \left[N\left(E - e\sin E\right) - mf(E)\right]}{\left(1 - e\cos E\right)^2}\,\,\mathrm{d}E.
\end{eqnarray*}$$By considering the effect of each summand in equation (5), the total torque on the star, energy transfer in the inertial frame, and energy transfer in the star’s corotating frame (which is also the tidal heating rate) can be obtained (Storch & Lai 2013; Vick et al. 2017):
(9)$$\begin{eqnarray*}
T = \sum \limits _{N = -\infty }^\infty F_{N2}^2 T_{\rm circ}\left(N\Omega - 2\Omega _{\rm s}\right),
\end{eqnarray*}$$(10)$$\begin{eqnarray*}
\dot{E}_{\rm in} &=& \frac{1}{2}\sum \limits _{N = -\infty }^\infty \Bigg [ \left(\frac{W_{20}}{W_{22}}\right)^2 N\Omega F_{N0}^2 T_{\rm circ}\left(N\Omega \right) \\
&&+\, N\Omega F_{N2}^2 T_{\rm circ}\left(N\Omega - 2\Omega _{\rm s}\right) \Bigg ] ,
\end{eqnarray*}$$(11)$$\begin{eqnarray*}
\dot{E}_{\rm rot} = \dot{E}_{\rm in} - \Omega _{\rm s} T .
\end{eqnarray*}$$Here, dots indicate time derivatives.


In the last few years, several experiments have reported upper limits on the power spectrum of 21-cm fluctuations during reionization (Parsons et al. 2014; Patil et al. 2017; Barry et al. 2019; Mertens et al. 2020; The HERA Collaboration 2021b) and the earlier cosmic-dawn era (Eastwood et al. 2019; Gehlot et al. 2019, 2020; Garsden et al. 2021; Yoshiura et al. 2021). Scenarios in which the bulk IGM is still colder than the cosmic microwave background (CMB) during reionization give rise to the strongest fluctuations and so will be the first models to be tested as upper limits continue to improve (e.g. Parsons et al. 2014; Pober et al. 2015; Greig, Mesinger & Pober 2016). Similarly, stronger-than-expected 21-cm signals can arise if the cosmic radio background has contributions other than the CMB (Feng & Holder 2018), e.g. synchrotron emission from accreting black holes (Ewall-Wice et al. 2018), star-forming galaxies (Mirocha & Furlanetto 2019), or from decaying particles (Fraser et al. 2018; Pospelov et al. 2018). Indeed, constraints from MWA, HERA, and LoFAR disfavour models with negligible X-ray heating at z ∼ 8–9 or very strong radio backgrounds (Ghara et al. 2020, 2021; Mondal et al. 2020; Greig et al. 2021a, b; The HERA Collaboration 2021a). Of course, the recent report of an absorption signal in the sky-averaged spectrum at z ∼ 17 from EDGES (Bowman et al. 2018) requires an even colder IGM (Barkana 2018; Boddy et al. 2018; Fialkov, Barkana & Cohen 2018; Kovetz et al. 2018; Muñoz & Loeb 2018) or a brighter background (Ewall-Wice et al. 2018; Feng & Holder 2018; Fialkov & Barkana 2019; Mirocha & Furlanetto 2019) than models in ΛCDM cosmologies generally predict. However, the most stringent power spectrum upper limits from The HERA Collaboration (2021b) are derived at sufficiently low redshifts relative to EDGES (z ≲ 10 versus z ≃ 18) that they cannot yet directly address the EDGES controversy (Hills et al. 2018; Bradley et al. 2019; Singh & Subrahmanyan 2019; Sims & Pober 2020; Tauscher, Rapetti & Burns 2020; Singh et al. 2021).


Recently, our knowledge of the MW formation and evolution has been revolutionized by the massive amount of data products from the Gaia mission and large spectroscopic surveys. Several dwarf galaxies are suggested to have been accreted by the MW since its formation (e.g., Helmi et al. 2018; Belokurov et al. 2018; Myeong et al. 2019). It is expected that GCs formed in these dwarf galaxies mix up with GCs formed in situ, i.e., in the main MW progenitor, to form the current GC system. This formation history can be revealed by precise kinematics and dynamical modeling. Massari et al. (2019) placed Pal 13 into a group of GCs related to a dwarf galaxy, named Sequoia, which was likely accreted 9 Gyr ago (Myeong et al. 2019). The existence of MPs in Pal 13 supports the statement that this phenomenon is not unique in Galactic GCs formed in situ (e.g., Li & de Grijs 2019; Milone et al. 2020). Besides its proposed accreted origin, Pal 13 is suggested to be experiencing tidal stripping (Yepez et al. 2019; Piatti & Fernández-Trincado 2020). Hamren et al. (2013) found that Pal 13 has lost a considerable amount of mass, which is related to its low present-day cluster mass. Recently, the discovery of tidal tails in Pal 13 was reported by Shipp et al. (2020). Using the RR Lyrae stars, these authors estimated the initial luminosity to be

 L⊙, which is significantly larger than the current luminosity estimated by B11,

 L⊙. Due to its large distance, several detailed features of Pal 13 have been discovered just recently (e.g., Piatti & Fernández-Trincado 2020; Shipp et al. 2020) and more are still waiting for further investigation, the estimation of total mass loss or initial mass of Pal 13 should be illuminative for future studies. Besides Pal 13, several remote, low mass GCs, including Whiting 1 and Eridanus GC, also show evidence of tidal tails or extra-tidal structures(Carballo-Bello et al. 2014; Myeong et al. 2017). These features agree with the predictions of dynamical evolution of low mass GCs: it is difficult to keep stars in a shallow potential well. The relatively large stellar mass lost from low mass GCs also complicates the discussion of MPs and especially of the lower mass limit needed to maintain MPs. On the other hand, the N-rich stars located in these GCs are lost to the field, contributing to the rare N-rich field stars (e.g., Martell & Grebel 2010; Fernández-Trincado et al. 2017; Tang et al. 2019; Fernández-Trincado et al. 2019; Tang et al. 2020).


SBs are complex environments. The hot rarefied plasma is spanned by primary shocks, which decay into turbulent motions and MHD waves. Collective effects such as re-acceleration processes contribute to the acceleration of particles. SBs are also delimited by a forward shock which expands into the ISM. While the size of this shock can be very large (R ≳ 100 pc, Weaver et al. 1977), it is too slow (u ∼ 30 km s−1) to accelerate PeV protons, even assuming large magnetic fields in the ISM. More generally, in the case of strong shocks, the velocity u is identified as the velocity of the shock, as it is precisely the velocity jump at the shock discontinuity which drives the acceleration of the particles via the DSA mechanism (e.g. Drury 1983). Inside SBs, there are two types of strong primary shocks: the time-dependent SNR shocks which expand after a supernova (SN) explosion, and the wind termination shocks (WTS) which surround the individual massive stars, or the entire stellar cluster if it is compact enough. SNRs and WTSs have a typical velocity of several 1000 km s−1. In the low-density SB interior, SNRs expand to a typical radius of a few tens of pc before reaching the Sedov–Taylor phase, which is generally larger than the radius of the WTS, even in the case of a WTS powered by a very massive compact cluster. The size of the latter shock depends on the mechanical power of the stellar cluster and is typically of the order of 10 pc (Weaver et al. 1977). It is believed that, due to in situ CR acceleration, efficient magnetic field amplification takes place upstream of SNR shocks, leading to a magnetic field of up to several tens of μG. On the other hand, it is less clear if the magnetic field can be as efficiently amplified in stellar winds. In both cases, the maximum energy is generally inferred to be only ∼1 PeV (e.g. Gupta et al. 2020; Morlino et al. 2021; Vieu et al. 2022b). While this is slightly larger than the maximum energy achieved at isolated SNR, atypical conditions would be required for protons to be accelerated well beyond PeV by primary shocks embedded in SBs. A promising situation combining the advantages of both WTS and SNRs might be that of an SNR expanding in a wind profile close to a compact cluster. A powerful cluster may indeed convert a substantial amount of its mechanical energy into turbulence, amplifying magnetic fields up to hundreds of μG in its vicinity, such that particles could be accelerated up to 10 PeV by a powerful SNR propagating in the wind, even in the absence of additional magnetic-field amplification. Considering a similar scenario of SNR shock – cluster wind interaction with efficient turbulence generation, Bykov et al. (2015) found that proton energies up to 40 PeV could be reached in the case of fast shocks (U = 104 km s−1).


All coreshine observations were obtained with the IRAC instrument on board the Spitzer observatory and are gathered in Paladini (2014) and Lefèvre et al. (2014), from which we selected 21 starless cores in the Taurus-Perseus, Chamaeleon, Cepheus, and L183/L134 regions (see their Fig. 9 and Table 1). Lefèvre et al. (2014) summarised their results in two figures that we reproduce here: the 4.5 to 3.6 μm ratio, that they name “coreshine ratio”, as a function of the 3.6 μm intensity and the 2.2 to 3.6 μm ratio, that they name the “near-IR to mid-IR ratio”, as a function of the coreshine ratio. For the model cloud, we use the parameters of the control cloud defined in Sect. 3. After convolving our models with a 10″ FWHM Gaussian kernel to simulate the data analysis presented in Lefèvre et al. (2014) and following the Lehtinen & Mattila (1996) prescription to take into account the part of the ISRF+CM light that can be transmitted through the cloud3, we compute the synthetic photometry for each pixel along a radial cut through our model clouds. The results are shown in Figs. 10a and b, which present the coreshine ratio as a function of the 3.6 μm intensity for the 21 starless cores of Lefèvre et al. (2014) and in Fig. 10c and d, which displays the near- to mid-IR ratio as a function of the coreshine ratio for the Taurus-Perseus and L183/L134 regions. The CMM model has to be ruled out to explain coreshine since it only marginally fits the coreshine ratio and fails to reproduce the near- to mid-IR ratio (green areas in Fig. 10). On the contrary, the CMM+AMM and CMM+AMMI models can explain the coreshine observations (magenta and blue areas in Fig. 10, respectively). Based on the results presented in Sect. 3 and Fig. 7, the cloud parameters are as important as the dust model for explaining the dispersion in the observations. Varying the cloud external radius from Rout = 0.3 to 0.1 pc (Figs. 10a and c) and the central density from ρC = 104 to 5 × 105 H/cm3 (Figs. 10b and d) seem enough to explain most of the observed scatter.


Recently, Fraija et al. (2021a) presented the afterglow light curves generated by the deceleration of sub-relativistic masses ejected from the merger of BCOs and the death of massive stars. The authors assumed that a PL velocity distribution describes the isotropic-equivalent kinetic energy of these masses and that the sub-relativistic ejected masses were decelerated, in turn, by a stratified-density environment. As a particular case, to explain the multiwavelength observations of the gravitational event GW170817/GRB 170817A at ∼900 days, they constrained the parameter space of the synchrotron light curves of a sub-relativistic mass ejected during the merger of two NSs and decelerated in a constant-density environment. The synchrotron radiation of the sub-relativistic material was consistent with the faster blue KN afterglow. Inspired by the new observations of this GW event at 3.3 yr after the initial merger (Hajela et al. 2022), in this paper, we extend the synchrotron model presented in Fraija et al. (2021a), including the continuous energy injection from the central engine (either a spinning magnetized NS or BH remnant) into the blast wave through a numerical approach and analytic arguments. In addition, we apply the current model to potential candidates of sGRB events with evidence of a KN. The paper is organized as follows: Section 2 presents the dynamical evolution of the afterglow when the central engine continuously injects energy into the blast wave. We show an analytical solution and numerical approach. In Section 3, we show a synchrotron model with energy injection from a spinning magnetized NS and BH remnants. Section 4 shows the analysis of the multiwavelength light curves using typical values of the GRB afterglow. In Section 5, we apply our model to several potential candidates, including GW170817/GRB 170817A, and finally, in Section 6, we summarize. We consider the convention

Qx=Q10x

 in cgs units and assume for the cosmological constants a spatially flat universe Λ cold dark matter model with H
0 = 69.6 km s−1 Mpc−1, ΩM = 0.286, and ΩΛ = 0.714 (Planck Collaboration et al. 2016).


In order to establish the completeness and purity of the RR Lyrae stars confirmed by the SOS Cep&RRL pipeline and to estimate the number of new discoveries by Gaia, we performed a deep and careful comparison with the literature. As a first step, the catalogue of 140 784 confirmed sources was cross-matched against all major catalogues of known RR Lyrae stars that are available. We primarily used the OGLE catalogues for RR Lyrae stars (version IV of the survey, Soszyński et al. 2014, 2016), but we also used RR Lyrae stars by CTRS (Drake et al. 2013a,b, 2014, 2017; Torrealba et al. 2015), ASAS (Pojmanski 1997; Richards et al. 2012), ASAS-SN (Jayasinghe et al. 2018), ATLAS (Tonry et al. 2018), IOMC (Alfonso-Garzón et al. 2012), LINEAR (Palaversa et al. 2013), NSVS (Kinemuchi et al. 2006), Pann-Stars (PS1 Sesar et al. 2017), and from the works based on Kepler/K2 (Debosscher et al. 2011; Nemec et al. 2011; Molnár et al. 2015a,b, 2016) and on the Simbad database (Wenger et al. 2000). These cross-matches returned a list of 88 578 known RR Lyrae stars in our sample of 140 784 stars. The SOS Cep&RRL confirmed RR Lyrae stars were also cross-matched against catalogues of candidate RR Lyrae stars discovered by the VVV survey (Gran et al. 2016; Minniti et al. 2017; D. Minniti, priv. comm.) in the MW disc and bulge. This returned 319 VVV cross-identified sources in the MW disc and 222 in the MW bulge. We thus confirm these VVV candidates. For known RR Lyrae stars in GCs, the main reference was the catalogue of Clement et al. (2001), which was updated to the latest literature as described in Garofalo et al. (in prep.). For variables in dSphs, we used the following references: Kaluzny et al. (1995), Clementini et al. (2005), Kinemuchi et al. (2008), Dall’Ora et al. (2012) and Garofalo et al. (2013). These latter cross-matches returned a list of 1986 further known RR Lyrae stars. At the end of this cross-match procedure, of the 140 784 RR Lyrae stars that are confirmed by the SOS Cep&RRL pipeline, 90 564 were shown to be known previously, and 50 220 are new discoveries by Gaia.


Atmospheric superrotation is characterized by eastward wind at the equator, which means the atmosphere there has a higher angular momentum than the solid surface. Atmospheric superrotation is a common phenomenon across the universe. In the solar system, superrotation exists in the atmospheres of Venus, Titan, Saturn, and Jupiter, as well as the stratospheric atmosphere of Earth during the westerly phase of the quasi-biennial oscillation (e.g., Kraucunas & Hartmann 2005; Schneider & Liu 2009; Lutsko 2018; Read & Lebonnois 2018). In order to maintain atmospheric superrotation, there must be momentum transports from higher latitudes to the equator against friction or other processes, according to angular momentum conservation (Hide 1969; Held 1999; Showman et al. 2013). This up-gradient transport into the jet can result from Rossby waves, coupled Rossby–Kelvin waves, mixed Rossby–gravity waves, wave–jet resonance, barotropic instability, or baroclinic instability (Suarez & Duffy 1992; Del Genio & Zhou 1996; Joshi et al. 1997; Lee 1999; Williams 2003; Kraucunas & Hartmann 2005; Schneider & Liu 2009; Caballero & Huber 2010; Mitchell & Vallis 2010; Showman & Polvani 2010, 2011; Showman et al. 2010; Liu & Schneider 2011; Arnold et al. 2012; Pinto & Mitchell 2014; Tsai et al. 2014; Wang & Mitchell 2014; Laraia & Schneider 2015; Lutsko 2018; Read & Lebonnois 2018; Pierrehumbert & Hammond 2019). For example, Kraucunas & Hartmann (2005) suggested that in an Earth-like atmosphere, equatorial superrotation can be generated by equatorward stationary eddy momentum convergence, which is associated with zonal variations in the diabatic heating at low latitudes. Mitchell & Vallis (2010) studied the transition from current Earth-like atmospheric circulation to an equatorial superrotation state. They found that during the spin-up period, superrotation is generated by equatorward momentum convergence associated with both barotropic and baroclinic instabilities.


A critical property of the phase mask is that it needs to be able to image subapertures in off-axis interferograms. The off-axis interferograms are rather large, with size scaling with λ/Dsub, where λ is the wavelength and Dsub is the diameter of the subaperture. Therefore, imaging multiple interferograms onto separate locations (so as to avoid overlap) on the detector requires large phase tilts. This makes it difficult to manufacture classical phase implementations of a HAM phase mask for transmissive pupil planes. A solution is offered by liquid-crystal diffractive phase masks as they have an unbounded continuous phase (Escuti et al. 2016). This property enables the creation of steep phase ramps that efficiently diffract light into a single order without scattering (Oh & Escuti 2008). In Fig. 1, no noticeable second-order diffraction is seen for any off-axis interferogram. Another advantage of liquid-crystal masks is that it is possible to manufacture almost any phase pattern (Kim et al. 2015), meaning there is more design freedom. We exploited this by combining phase ramps into a single phase pattern that images a single subaperture onto multiple locations in the focal plane. This was done through multiplexing the phase ramps, and the mathematical description of multiplexing can be found in Doelman et al. (2018). An example of multiplexed subapertures is seen in the third column of Fig. 1, where multiple baselines connected to a single aperture are imaged onto different interferograms. In addition, liquid-crystal masks are diffractive because they apply a different kind of phase delay to incoming light that is independent of wavelength. These phase delays are called “geometric phase delays” and are discussed in greater detail in Sect. 3. Due to this diffraction, the location of an imaged subaperture changes with wavelength. The advantage of the diffractive nature is that, together with the right subaperture combination and fringe orientation, the wavelength smearing enables low-resolution spectroscopy. However, each interferogram can then only consist of 1D combinations of subapertures. This limits the design freedom significantly and also greatly increases the number of off-axis interferograms. As shown in Fig. 1, the fringe direction of all off-axis interferograms is orthogonal to the smearing direction for 1D combinations of subapertures. Lastly, a specific property of liquid-crystal diffractive phase masks is that they produce two off-axis interferograms for a single phase ramp with opposite location in the focal plane. This can be seen in Fig. 1, where all interferograms have an identical counterpart. The aforementioned properties have a large impact on the design of the HAM mask, which we discuss next.


The problem of the balance between driving and decay for magnetized turbulence is most acute in molecular clouds. Since these have linewidths indicating the presence of supersonic flow, the fast dissipation of turbulence found by these simulations necessitates a mechanism to reinject the energy equally quickly. A number of candidates have been proposed, including internal feedback from H ii regions (Matzner 2002; Krumholz, Matzner & McKee 2006; Goldbaum et al. 2011) or protostellar outflows (Li & Nakamura 2006; Nakamura & Li 2007; Wang et al. 2010; Federrath et al. 2014a), driving of turbulence by ongoing accretion (Klessen & Hennebelle 2010; Goldbaum et al. 2011; Lee & Hennebelle 2016) or gravitational contraction on small scales (Federrath et al. 2011b; Sur et al. 2012), thermal instability driving (Koyama & Inutsuka 2002; Hennebelle & Inutsuka 2006) and injection of energy from external supernova shocks (Mac Low & Klessen 2004; Padoan et al. 2016a,b; Pan et al. 2016). Alternately, it is possible that the linewidths do not reflect turbulent motion at all, and instead indicate global gravitational collapse (Ballesteros-Paredes et al. 2011; Zamora-Avilés & Vázquez-Semadeni 2014). Each of these proposals, however, faces challenges – internal feedback must maintain large linewidths without destroying the clouds in which they occur, driving by accretion faces the problem of what happens when the accretion eventually ends, thermal instability seems unlikely to be a viable mechanism in molecule-dominated galaxies that lack a significant warm phase, and external driving requires efficient coupling between the low-density external medium and the dense clouds. The view that clouds are in global collapse is hard to reconcile with the observed very low rates of star formation found even in gas at densities ≳ 105 cm−3 (Krumholz & Tan 2007; Federrath & Klessen 2012; Krumholz, Dekel & McKee 2012; Evans, Heiderman & Vutisalchavakul 2014; Padoan et al. 2014; Salim, Federrath & Kewley 2015; Usero et al. 2015; Heyer et al. 2016; Vutisalchavakul, Evans & Heyer 2016).


Multiwavelength observations of the GC indicate that the number of pulsars in the central few parsecs should be high (Wharton et al. 2012) and conditions are highly favourable for relativistic binaries (Faucher-Giguère & Loeb 2011). The dense nuclear star cluster surrounding Sgr A* (see e.g. Genzel, Eisenhauer & Gillessen 2010, for a review) contains a majority of older late-type stars, but contrary to expectations, massive young main-sequence stars (Ghez et al. 2003) and possible neutron star progenitors such as Wolf–Rayet stars (Paumard et al. 2001). The presence of neutron stars is further indicated by large numbers of X-ray binaries, possible pulsar wind nebulae, X-ray features such as the ‘cannonball’ and compact radio variables (Muno et al. 2005; Wang, Lu & Gotthelf 2006; Zhao, Morris & Goss 2013, 2020). Despite this only six radio pulsars have been discovered within half a degree of Sgr A* (Johnston et al. 2006; Deneva, Cordes & Lazio 2009; Eatough et al. 2013c; Shannon & Johnston 2013) even after many dedicated searches at multiple wavelengths (Kramer et al. 1996a, 2000; Klein et al. 2004; Klein 2005; Deneva 2010; Macquart et al. 2010; Eatough et al. 2013a; Siemion et al. 2013). Hyperstrong scattering of radio waves in the GC has been the principal explanation for the scarcity of detected pulsars (Cordes & Lazio 1997, 2002; Lazio & Cordes 1998a,b), however, scatter broadening measurements of PSR J1745−2900 in Spitler et al. (2014) and Bower et al. (2014) appear to contest this.1 Other authors have noted that the lack of GC pulsars is expected under a certain set of conditions and considering the sensitivity limits of existing pulsar surveys (Chennamangalam & Lorimer 2014; Liu & Eatough 2017; Rajwade, Lorimer & Anderson 2017). Alternatively, the scarcity of detected pulsars might be caused by a more complex scattering structure towards the GC (Cordes & Lazio 1997; Lazio & Cordes 1998a, b; Johnston et al. 2006; Schnitzeler et al. 2016; Dexter et al. 2017).


We first determined the orbital parameters for RS Ser, V449 Per, and V1095 Her. Further, we updated the parameters for V593 Cen and MR Del. Using the formula f  = (Ωin – Ω1)/(Ωin – Ωout), we calculated the contact factors f for RS Ser, V593 Cen, and V1095 Her as 6.5 per cent, 40 per cent, and 53 per cent, respectively. RS Ser is a contact binary with a small temperature difference of 131 K and a low contact factor. For V593 Cen, we updated the orbital parameters using more complete light curves. The orbital inclination of 83°.18 is similar to the result (82°.6) obtained by Zhao et al. (2019). The temperature of the secondary component (15 284 K) is higher than the previous result of 15 099 K. However, the mass ratio of 0.6 is lower than the previous result of 1.05 (Zhao et al. 2019). More spectroscopic observations are required to confirm the mass ratio. We confirmed that V593 Cen is an early-type contact binary with a deep contact factor as well as a black hole candidate. For MR Del, we revised the absolute parameters using its full light curve and the published radial velocities, which are similar to those published previously (Zhao et al. 2019; Pribulla et al. 2009; Djurašević et al. 2011). V1095 Her is also a contact binary with a deep contact factor of 40 per cent and a temperature difference of about 172 K. Looking over our four complete light curves, we found no evident starspot activity, and estimated variations exist over a long-term time-scale of years. V449 Per is an interesting target for detecting extra-solar and brown dwarfs using the minima timing variability of a low-mass eclipsing binary (Pribulla et al. 2012). Additional minima with higher precision are required to study its periodic variation further. Our physical parameters for RS Ser, V593 Cen, and V1095 Her are based on a light curve with a q-search-determined mass ratio. The nature of these parameters is speculative and preliminary. Radial velocities may eventually come to rescue them and provide a more definitive determination.


In Figure 2, the reduced AKARI IRC spectra of all protostars are presented; the absorption features of the H2O, CO2, and CO ices are clearly detected. All of our targets show deep and broad absorption features of H2O ice in the wavelength range 2.7–3.4 μm. In the case of AFGL 7009S, strong extinction toward the source saturates the absorption features throughout the wavelengths from 2.7 to 3.6 μm, including H2O ice. Other ice features, such as CH4 (Lacy et al. 1991; Boogert et al. 2004) and CH3OH (Grim et al. 1991; Brooke et al. 1996), were observed at 3.3−3.5 μm, but it is difficult to extract their absorption profiles from the blended features due to the low spectral resolution of AKARI IRC. An absorption feature of the CO2 ice around 4.27 μm was clearly detected toward all targets. At the wavelength around 4.6 μm for Perseus 1 and 3, RNO 91, and AFGL 7009S, there is a hint for another ice component overlapping with the CO absorption feature at 4.67 μm. Many near-infrared observations have revealed the same feature on the blue wing part of the CO absorption feature (Tegler et al. 1995; Chiar et al. 1998; Whittet et al. 2001; van Broekhuizen et al. 2005; Aikawa et al. 2012), which was suggested as an absorption feature of XCN ice. Lacy et al. (1984) and Pendleton et al. (1999) reported that the 4.62 μm absorption feature of XCN ice consists of a nitrile group and an unknown component “X”. Many laboratory studies have suggested that UV photolysis or cosmic ray irradiation of ice mantle could make the solid state OCN− on grain surfaces (Lacy et al. 1984; Grim & Greenberg 1987; Bernstein et al. 2000; Palumbo et al. 2000; Hudson et al. 2001; van Broekhuizen et al. 2004). In addition to these ice components, there are some absorption features around 4.8 and 4.9 μm. For Perseus 3 and the background star, the absorption features with a peak position around 4.78 μm are likely associated with 13CO ice (Boogert et al. 2002; Pontoppidan et al. 2003). We also detected another absorption feature at 4.83 μm toward the low-luminosity targets. However, we could not find any corresponding ice features from previous studies. The 4.9 μm absorption feature detected toward all targets was identified as solid carbonyl sulfide (OCS) ice. OCS ice can be produced when the interstellar ices containing CO and CO2 are exposed to UV photons or cosmic rays (Palumbo et al. 1997).


In order to achieve high photometric accuracy and be sensitive to low amplitude undulations, we adopted techniques from the exoplanet community, with the purpose of eliminating the systematic errors. When performing differential photometry (Sect. 3), accurate bias-subtraction and flat-fielding are of major importance. According to Irwin et al. (2007), the Poisson noise is 200 e− for a typical detector with a gain of a few e− ADU−1 and the flat illumination level is of 20 000 ADU pixel−1 = 40 000 e− pixel−1. Thus, a typical photometric aperture with a radius of 3 pixels contributes ∼1 mmag photon noise. For this reason, we obtained a considerable amount of biases (150−300 frames) and twilight flat-fields (25−100 frames) each night to reduce the Poisson noise to less than 0.2 mmag (Irwin et al. 2007). The bias frames were averaged together using the minmax in the reject option of the zerocombine task in IRAF with a view of keeping radiation events out of the master bias frame. The master flat frame was the result of combining all the frames using a median mode. The median value is an excellent way of removing the effects of hot pixels and cosmic rays, so these extreme values do not affect the calculation, as they would, if they would averaged. The reject option was set to avsigclip, in which case the “typical” sigma would have been determined from the data itself rather than an a priori knowledge of the noise characteristics of the CCD. Other related issues that can limit the photometric precision are: (i) the positioning of the telescope, (ii) fringing issues, and (iii) the differential variations on the quantum efficiency of the pixels. With the aim of minimizing the contribution of these effects, we repositioned each star almost on the same pixel of the detector using the autoguiding system of each telescope. The read-out-noise of the detectors are insignificant, as it can be as low as a few e− ( 10 e− for RISE2 and Andor Zyla cameras).


The CGM (see Tumlinson et al. 2017, for a detailed review) is now understood to be a key component in disentangling the feedback processes in active galaxies. It links the smaller-scale interstellar medium (ISM) of the galaxy to the larger-scale intergalactic medium (IGM), not only in a geometrical way but also by acting as the reservoir fueling star formation and the central black hole, where the feedback interacts with the galactic environment and where the gas recycling during galaxy evolution is controlled. This complex environment is multiphase and has been observed in numerous surveys (e.g., Tumlinson et al. 2013; Bordoloi et al. 2014; Peek et al. 2015; Borthakur et al. 2015) at low redshift. A prominent feature of the CGM around active galaxies is the Lyα (Lyman-α) emission line, which is also ubiquitously observed at high redshift (e.g., Haiman & Rees 2001; Reuland et al. 2003; van Breugel et al. 2006; Villar-Martín 2007; Humphrey et al. 2013; Cantalupo et al. 2014; Wisotzki et al. 2016, 2018; Arrigoni Battaia et al. 2018, 2019; Nielsen et al. 2020). Lyα is the transition of the hydrogen electron from the 2p orbit to its ground state. It can happen primarily through collisional excitation and recombination (see Dijkstra 2014, 2017, for a detailed review of Lyα emission mechanisms and radiative transfer). In extragalactic studies, the recombination production of Lyα emission can be generated by photoionization by young stars and/or AGN (fluorescence). This fluorescence emission on larger scales (CGM and IGM) can also be due to UV background radiation. Additionally, collisional excitation can play an important role in the emission seen in outflows and infalling gas (Ouchi et al. 2020). The bright Lyα emission line, along with other UV lines excited by the central or background sources, provides a useful tool for studying the galactic environments in the early Universe. Additionally, H I and metal absorption features observed in the CGM are powerful tracers of feedback signatures as well as tracers of infalling pristine gas (e.g., low metallicity absorption in a z ∼ 2.7 submillimeter galaxy; Fu et al. 2021). The sensitive integral field spectrographs on the largest ground-based telescopes, such as MUSE (Multi-Unit Spectroscopic Explorer; Bacon et al. 2010, 2014) and KCWI (Keck Cosmic Web Imager; Morrissey et al. 2012; see Cai et al. 2019 for observation of Lyα halos with KCWI), are perfectly suited for mapping these UV features as they move into the optical band for high-redshift sources.


The data presented here comprise quasi-simultaneous observations during secondary eclipse of WASP-33 b around the V and Y bands. The predicted planet-star flux ratio in the V-band is 0.2 ppt, four times lower than the accuracy of our measurements. Therefore, we can neglect the planet imprint and use this band to measure the stellar pulsations, and most specifically to tune their current phases (see phase shifts in von Essen et al. 2014). Particularly, our model for the light contribution of the stellar pulsations consists of eight sinusoidal pulsation frequencies with corresponding amplitudes and phases. Hence, to reduce the number of 24 free parameters and the values they can take, we use prior knowledge about the pulsation spectrum of the star that was acquired during von Essen et al. (2014). As the frequency resolution is 1/ΔT (Kurtz 1983), 3.5 h of data are not sufficient to determine the pulsations frequencies. Therefore, during our fitting procedure we use the frequencies determined in von Essen et al. (2014) as starting values plus their derived errors as Gaussian priors. As pointed out in von Essen et al. (2014), we found clear evidences of pulsation phase variability with a maximum change of 2 × 10-3 c/d. In other words, as an example after one year time a phase-constant model would appear to have the correct shape with respect to the pulsation pattern of the star, but shifted several minutes in time. To account for this, the eight phases were considered as fitting parameters. The von Essen et al. (2014) photometric follow-up started in August, 2010, and ended in October, 2012, coinciding with these LBT data. We then used the phases determined in von Essen et al. (2014) during our last observing season as starting values, and we restricted them to the limiting cases derived in Sect. 3.5 of von Essen et al. (2014), rather than allowing them to take arbitrary values. The pulsation amplitudes in δ Scuti stars are expected to be wavelength-dependent (see e.g. Daszyńska-Daszkiewicz 2008). Our follow-up campaign and these data were acquired in the blue wavelength range. Therefore the amplitudes estimated in von Essen et al. (2014), listed in Table 1, are used in this work as fixed values. This approach would be incorrect if the pulsation amplitudes would be significantly variable (see e.g., Breger et al. 2005). Nonetheless, the short time span of LBT data, and the achieved photometric precision compared to the intrinsically low values of WASP-33’s amplitudes, make the detection of any amplitude variability impossible.