The velocity distribution of the plasma motion is shown in Fig. 12, where the magnitude is scaled with respect to the Alfvén velocity, vA, which is measured based on the magnetic field strength, B0 = 2 G, and the mass density of the equilibrium current sheet, ρc = 2.81 × 10−15 g cm−3. The Alfvén timescale is measured by

t
A

=

L

¯

/

v
A

$ t_A = \bar{L}/\mathit{v}_A $

, where,

L

¯

$ \bar{L} $

 is the unit length of 109 cm. It is evident from Figs. 12a, b and c that the velocity, vx remains localized in the vicinity of the current sheet (y = 0). We note that in line with our initial single-island magnetic field perturbation, we see a pronounced rightward motion in the right half of the domain, and a leftward motion in the left half. We later see typical Petschek-like signatures in the flow fields between islands, especially about the middle x = 0, with super-Alfvénic outflow speeds bounded by slow shocks. Figure 13a represents the evolutionary nature of a current sheet in adiabatic and non-adiabatic conditions. It is clear from the figure that the instantaneous maximum velocity growth for the non-adiabatic case is more rapid than for the adiabatic conditions. The evolutionary behavior of the current sheet configuration due to thermal and tearing instabilities is shown by the black curve in Fig. 13a for plasma-β = 0.2, and a given resistivity value, η = 0.001, while the evolution for different η values are shown in Fig. 13b. As a diagnostic measurement of the instability, we determine the evolution of the instantaneous maximum absolute velocity, |vx|max. Figure 13a shows that this evolution exhibits three distinct phases: (i) the early phase (between t = 0 and 250 s), where the velocity growth occurs exponentially (linearly on the logarithmic-linear scale), which is called the linear growth regime; (ii) the middle phase between t = 250 and 665 s, where the growth rate is slower compared to the linear phase, which is called the Rutherford regime (Rutherford 1903), and (iii) the final phase, which starts at t = 665 s, where the instability suddenly develops in an explosive way, and finally saturates at a later time, which we call the post-Rutherford regime. To infer the evolution rates quantitatively for all the different phases, we calculate the growth rates by scaling them with respect to the Alfvén timescale, tA. We define the growth rate as γ = d(ln(|vx|max))/dt. To estimate the linear growth rate, γlin, we calculate the growth rate in the linear regime by taking the mean value of the slope, which gives

γ

l
i
n

=
3.76
×

10

−
1

t
A

−
1

$ \gamma_{\rm lin}=3.76 \times 10^{-1} t_A^{-1} $

. This value is larger by an order of magnitude compared to the studies of the double current sheet problem (Otto & Birk 1992; Zhang & Ma 2011; Akramov & Baty 2017; Paul & Vaidya 2021), where the radiative cooling effect (or other non-adiabatic effects, e.g., thermal conduction) is not incorporated. This implies that the higher linear growth rate can be ascribed to the non-adiabatic effects of the radiative cooling and background heating. This is also in agreement with our own study for a single current layer model reflected in Fig. 13a, where the average growth rate for the adiabatic medium is lower than the non-adiabatic case. Similarly, we estimate the average growth rates for the Rutherford regime (γRuth) and the post-Rutherford regime (γPR) for different resistivity values within the range of η = 0.0001 to 0.005. The velocity evolution for some selected resistivity values are shown in Fig. 13b. This shows that the explosive phase of the evolution starts at later times for higher resistivity values, and converges at the final stage. For a Sweet–Parker type current sheet (where the inverse aspect ratio of the current sheet follows the scaling relation

l
s

/
L
∼

S

L

−
1
/
2

$ l_s/L \sim S_{\mathrm{L}}^{-1/2} $

), the thickness of the current sheet increases with the resistivity (Loureiro et al. 2007), which reduces the growth rate of the tearing mode when it is normalized with respect to the Alfvén crossing time along the length of the current sheet (in the x-direction in our case). Hence, the explosive phase of the evolution in our simulation starts at later times for higher resistivity values. We estimated the absolute current density, |Jz| (normalized to unity) before the fragmentation stage of the current sheet (t = 214 s) by taking a vertical cut along the y-direction at x = 0 for two different resistivities, η = 0.0001 and 0.001, to confirm that the thickness of the current sheet increases with resistivity (see Fig. 14). The resistivity dependence for the different evolution phases is shown in Fig. 15. Figure 15a shows that γRuth follows a power-law dependence with the resistivity, γRuth ≈ η−0.1, with a correlation coefficient (CC) of −64.1%. The resistivity scaling relation for the post-Rutherford and the entire nonlinear regimes are shown in Figs. 15b and c respectively. We estimate the growth rate scaling relations for the post-Rutherford regime, γPR ≈ η0.03 (with CC = 59.9%), and the entire non-linear regime, γavg ≈ η0.017 (with CC = 66.7%). Previous studies by Zhang & Ma (2011), Akramov & Baty (2017), and Guo et al. (2017 and references therein) have reported the resistivity scaling relation of the non-linear growth rates for the DTM setup in the adiabatic environment, which have larger power-law indices compared to our estimation. Hence, our study infers that the resistivity dependence on the nonlinear growth rates is weaker when the thermal instability reinforces the tearing mode.


In all our simulations independent of the inflow Mach number, the radial velocity dispersion at the filament boundary amounts to about 0.85 times the total equilibrium velocity dispersion of the non-self-gravitational case for which a functional form can be found in the appendix. Thus, we can calculate the theoretical radius and central density of the filament at every line-mass and therefore we can make predictions on the fragmentation length and time-scales of cores forming in an accreting filament using the gravitational fragmentation model. This model was successfully applied to explain several observed core distances (Jackson et al. 2010; Miettinen 2012; Busquet et al. 2013; Beuther et al. 2015; Contreras et al. 2016; Heigl et al. 2016; Kainulainen et al. 2016) however it is not able to explain all observations (André et al. 2010; Kainulainen et al. 2013; Takahashi et al. 2013; Lu et al. 2014; Wang et al. 2014; Henshaw et al. 2016; Teixeira et al. 2016; Kainulainen et al. 2017; Lu et al. 2018; Palau et al. 2018; Williams et al. 2018; Zhou et al. 2019). It predicts that small density perturbations in the linear regime along the filament axis of the form:
(32)$$\begin{eqnarray*}
\rho (r, x, t) = \rho _0(r) \left(1 + \epsilon \exp (ikx -i\omega t)\right)
\end{eqnarray*}$$will grow for values of k where the dispersion relation ω2(k) is negative. Here ρ0 is the unperturbed initial density, k = 2π/λ is the wave vector with λ being the perturbation length, x is the filament axis, ω = 1/τ is the growth rate with τ being the growth time-scale, t the time variable, and ϵ the perturbation strength. The fastest growing, or dominant, fragmentation length scale λdom as well as the growth time-scale of the dominant mode τdom depend on the current line-mass as well as the current central density of the filament and are given by the pre-calculated (Nagasawa 1987) and interpolated values in Fischera & Martin (2012), shown by their Table E.1. We use these values to determine the length scale of the fastest growing mode at every line-mass for the same mass accretion rate but for different inflow Mach numbers as shown in Fig. 10. As one can see, the dominant fragmentation length changes over the evolution of the line-mass. At the boundary values it vanishes to zero and it has a maximum at about fcyl = 0.4. The figure is self-similar for different mass accretion rates, with a lower rate leading to a larger dominant fragmentation length. For a constant accretion rate, the fragmentation length does not vary much for different inflow Mach numbers. Only for large and for very low inflow Mach numbers, the fragmentation length is slightly larger. As the dominant fragmentation length constantly changes as fcyl grows, it is hard to make predictions of what will be the final distance between forming cores. But the curves have a maximum which allows us to make a prediction about the minimum number of cores that will form. For instance, a filament with an inflow Mach number of 4.0 and a length of 0.2 pc will form at least one core. As soon as the first core forms, the further evolution of the filament is also influenced by the gravitational attraction of the core. This makes the formation of additional cores even more unpredictable.


The lack of data on inelastic processes due to collisions with neutral hydrogen atoms has been a major limitation on modelling of F-, G- and K-star spectra in statistical equilibrium, and thus to reliably proceeding beyond the assumption of local thermodynamic equilibrium (LTE) in analysis of stellar spectra and the determination of elemental abundances. This problem has been well documented, e.g. see Lambert (1993); Barklem (2016a) and references therein. Significant progress has been made in recent times through detailed full-quantum scattering calculations, based on quantum chemical data, for the cases of simple atoms such as Li, Na, Mg and Ca (Belyaev & Barklem 2003; Barklem, Belyaev & Asplund 2003; Belyaev et al. 2010; Barklem et al. 2010; Belyaev et al. 2012; Barklem et al. 2012; Mitrushchenkov et al. 2017). These calculations have demonstrated the importance of the ionic-covalent curve crossing mechanism leading naturally to charge transfer processes (mutual neutralization and ion-pair production), in addition to excitation and de-excitation processes. The importance of this mechanism has allowed various simplified model approaches to be developed, which may be used in cases where suitable quantum chemistry data are not been available. In particular a semi-empirical model has been employed for Al, Si, Be and Ca (Belyaev 2013a,b; Belyaev, Yakovleva & Barklem 2014b; Yakovleva, Voronov & Belyaev 2016; Belyaev et al. 2016), and a theoretical model based on a two-electron asymptotic linear combinations of atomic orbitals (LCAO) approach, has also been employed for Ca (Barklem 2016b, 2017). Comparisons of the two methods show quite good agreement and reasonable agreement with the full quantum calculations is found, particularly for the most important processes with the largest rates (Barklem 2016b, 2017; Mashonkina, Sitnova & Belyaev 2017; Mitrushchenkov et al. 2017). Thus, the model approaches provide a useful route for obtaining estimates of the rates for these processes for many elements of astrophysical interest.


Several mechanisms have been proposed to explain the periodicity of methanol masers. van der Walt, Goedhart & Gaylard (2009) and van der Walt (2011) proposed a colliding wind binary (CWB) system for the periodic methanol masers in G9.62+0.20E and other periodic methanol maser sources with similar light curves. Using a more realistic model van den Heever et al. (2019) also showed that the CWB scenario can explain the light curves of the periodic methanol masers in G9.62+0.20E, G22.357+0.066, G37.55+0.20, and G45.473+0.134. Other mechanisms proposed are the pulsational instabilities of very young accreting high-mass stars (Inayoshi et al. 2013), spiral shocks associated with very young binary systems orbiting within a circumbinary disc (Parfenov & Sobolev 2014), periodic accretion in a very young binary system (Araya et al. 2010), and outflows in a binary system (Singh & Deshpande 2012). Maswanganye et al. (2015) evoked an eclipsing binary system involving a high-mass (primary) star and a bloated low-mass (companion) star in explaining the observed periodicity of the methanol masers in G358.460-0.391. While each of the suggested mechanisms may be sufficient for explaining periodicity in individual star-forming regions, the complexity in the nature of massive star-forming environments and the uniqueness of each region, make it difficult to evoke any one of the mechanisms for all cases. In fact, different mechanisms may be responsible for the different observed flare profiles (see van der Walt et al. 2016, for a discussion). However, the fact that it is possible to identify at least two groups of periodic sources (as explained above), each for which the light curves are very similar, suggests that there are some sources for which the underlying mechanism is the same. It is therefore reasonable to argue that there might be other similarities in star-forming regions that host periodic methanol masers with same light curves. In other words, the similarity in light curves might also manifest in other properties of the star-forming regions in addition to the maser emission.


Ultra-fast outflows (UFOs) of highly ionised gas observed at sub-parsec scales (Reeves et al. 2003; Tombesi et al. 2012) have been proposed as the likely origin of galaxy-wide outflows, interpreted as the result of the impact of UFOs on the ISM (King & Pounds 2015, and references therein). Furthermore, both models and observations of kiloparsec-scale outflows seem to indicate a UFO-ISM interaction in an energy-conserving regime, whereby the swept-up gas expands adiabatically. So far, the co-existence of a massive molecular outflow with a nuclear UFO has been confirmed in a handful of AGNs with LBol  ∼  1044 − 1046 erg s−1 (Tombesi et al. 2015; Feruglio et al. 2015; Longinotti et al. 2015) and in APM 08279+5255 (Feruglio et al. 2017), which is a gravitationally lensed QSO at z  ∼  4 with an estimated intrinsic LBol of a few times 1047 erg s−1 (Saturni et al. 2018). In all these sources the momentum boost (i.e. the momentum flux of the wind normalised to the AGN radiative momentum output, LBol/c) of the UFO is ∼1, while the momentum rate of the molecular outflow is usually ≫1, in qualitative agreement with the theoretical predictions for an energy-conserving expansion (Faucher-Giguère & Quataert 2012; Costa et al. 2014). However, these results are still limited to a very small sample and suffer from large observational uncertainties, mostly due to the relatively low signal-to-noise ratio of the UFO- or outflow-related features confirmed in spectra, or to the limited spatial resolution of sub-millimetre observations. Recent studies increasing the statistics of sources with detection of molecular outflows have widened the range of measured energetics (e.g. García-Burillo et al. 2014; Veilleux et al. 2017; Feruglio et al. 2017; Brusa et al. 2018; Barcos-Muñoz et al. 2018; Fluetsch et al. 2019). These outflows are consistent with driving mechanisms alternative to the energy-conserving expansion, such as direct radiation pressure onto the host-galaxy ISM (e.g. Ishibashi & Fabian 2014; Ishibashi et al. 2018; Costa et al. 2018).


While the Loison et al. (2015) CH3C3N model corroborates the upper atmospheric abundance of C4H3N inferred by Vuitton et al. (2007) from the T5 INMS measurements (a factor of 2 higher than those derived from T40 in Vuitton et al. 2019), a large disparity between the photochemical models (and within the ensemble of models produced by Loison et al. 2015) arises in the lower atmosphere due to the poorly constrained C4H3N branching ratios and reaction rate coefficients at temperatures appropriate for Titan. Aside from electron dissociative recombination of C4H3NH+ (Vuitton et al. 2007), neutral production of CH3C3N can occur in a few ways, as found through crossed beam experiments and theoretical and photochemical modeling studies (Huang et al. 1999; Balucani et al. 2000; Zhu et al. 2003; Wang et al. 2006; Loison et al. 2015). First, through the reactions of larger hydrocarbons with CN radicals,
1

2

Similarly, with CCN radicals following their formation through H + HCCN (Takayanagi et al. 1998; Osamura & Petrie 2004) and subsequent reactions with ethylene,
3

or through the chain beginning with acetylene,
4

While both reactions (3) and (4) are found to be equally likely by Loison et al. (2015), the production of CCN via H + HCCN is not well constrained, and the synthesis of CH3C3N through CN radicals (Equations (1) and (2)) are not included in their photochemical model. Additionally, cyanoallene may be produced through reactions (1)–(4) instead of (or in addition to) methylcyanoacetylene. CH3C3N itself may form the protonated species, C4H3NH+, through reactions with the HCNH+ and C2H5+ ions producing HCN and C2H4, respectively (Vuitton et al. 2007). The other mechanism for forming C4H3NH+ is through the combination of HCN and l-C3H3+, though the reaction rate coefficient for this reaction and the abundance of l-C3H3+ are unknown (Vuitton et al. 2007). As such, the production and loss pathways for both C4H3NH+ and CH3C3N require further investigation.


Asymptotic-giant-branch (AGB) stars have a significantly larger size (∼1 au) than their main-sequence (MS) counterparts. They have pulsating atmospheres (Vlemmings et al. 2017; Khouri et al. 2019) and may exhibit variability with long periods ranging from 200 to 1000 days (Mowlavi et al. 2018; Karambelkar et al. 2019). AGB stars are one of the major sites in galaxies that produce metals. Metals can be carried away from the AGB stars by radiation-driven AGB winds when dust forms. The speed of the AGB wind varies from 4 to 20 km s−1 (Höfner & Olofsson 2018), and a companion star may capture the wind with its gravity. In the case that there is an MS star close to an AGB star, a substantial fraction of the mass loss may be accreted onto the MS companion (Chen et al. 2017; Saladino et al. 2018, 2019). As a result, the metallicity of the companion may change. Such early stage low-mass stars become chemically peculiar, and their future evolution will be strongly affected. Carbon-enhanced-metal-poor (CEMP) stars (Beers & Christlieb 2005; Abate et al. 2013, 2015), Barium stars (Bidelman & Keenan 1951; Escorza et al. 2019), CH stars (Keenan 1942; McClure & Woodsworth 1990), and dwarf carbon stars (Dahn et al. 1977; Roulston et al. 2019) are common examples of the chemically peculiar stars. Their existence could be evidence of mass transfer during the previous AGB binary phase. The binarity of CH stars and CEMP stars has been studied (McClure & Woodsworth 1990; Starkenburg et al. 2014; Jorissen et al. 2016), confirming that many of them have companions. A number of recent studies show that the eccentricity of some of the aforementioned chemically peculiar stars may be large (Hansen et al. 2016; Jorissen et al. 2016, 2019; Van der Swaelmen 2017; Oomen et al. 2018), and their orbital periods range from hundreds to thousands of days. The nonzero eccentricity in these close binary stars indicates that some intense interactions that can pump the eccentricity may happen during their AGB binary phases. A strong correlation between a circumstellar disk and binarity has also been established in Galactic RV Tauri stars (Manick et al. 2017). Furthermore, many RV Tauri stars show a lack of refractory elements, which is called “depletion” (Giridhar et al. 1994; Van Winckel et al. 1998). Some researches suggest that the reaccretion of gas from a circumstellar disk around the post-AGB star (Gezer et al. 2019; Oomen et al. 2019) may induce the “depletion.” Besides the “smoking gun” evidence, observations also reveal that dusty circumbinary disks exist in binary systems with evolved stars (Kervella et al. 2015; Hillen et al. 2016; Homan et al. 2017; Ertel et al. 2019). The UV excess of some AGB stars also suggests that there could be accreting MS companions near them (Sahai et al. 2008; Ortiz & Guerrero 2016).


In recent years, numerous low-mass and ELM WDs have been detected in the context of relevant surveys, such as the SDSS, ELM, SPY and WASP (see, e.g., Koester et al. 2009; Brown et al. 2010, 2016, 2020; Kilic et al. 2011, 2012; Gianninas et al. 2015; Kosakowski et al. 2020). The discovery of their probable precursors, namely, the so-called low-mass pre-WDs, has triggered an interest in these types of objects because of the possibility of studying the evolution of the progenitors that lead to the WD phase. Even more interestingly, the detection of multi-periodic brightness variations in low-mass WDs (Hermes et al. 2012, 2013a,b; Kilic et al. 2015, 2018; Bell et al. 2017, 2018; Pelisoli et al. 2018), and low-mass pre-WDs (Maxted et al. 2013, 2014; Gianninas et al. 2016; Wang et al. 2020) has brought about new classes of pulsating stars known as ELMVs and pre-ELMVs, respectively (ELM and pre-ELM variables, respectively). It has allowed for the study of their stellar interiors using the tools of asteroseismology, similarly to the case of other pulsating WDs such as ZZ Ceti stars or DAVs –pulsating WDs with H-rich atmospheres – and V777 Her or DBVs – pulsating WDs with He-rich atmospheres (Winget & Kepler 2008; Fontaine & Brassard 2008; Althaus et al. 2010; Córsico et al. 2019). The pulsations observed in ELMVs are compatible with global gravity (g)-mode pulsations. In the case of pulsating ELM WDs, the pulsations have large amplitudes mainly at the core regions (Steinfadt et al. 2010; Córsico et al. 2012; Córsico & Althaus 2014a), allowing for the study of their core chemical structure. According to nonadiabatic computations (Córsico et al. 2012; Van Grootel et al. 2013; Córsico & Althaus 2016), these modes are probably excited by the κ − γ (Unno et al. 1989) mechanism acting at the H-ionization zone. In the case of pre-ELMVs, the nonadiabatic stability computations for radial (Jeffery & Saio 2013) and nonradial p- and g-mode pulsations (Córsico et al. 2016; Gianninas et al. 2016; Istrate et al. 2016b) revealed that the excitation is probably due to the κ − γ mechanism, acting mainly in the zone of the second partial ionization of He, with a weaker contribution from the region of the first partial ionization of He and the partial ionization of H. The presence of He in the driving zone is crucial to having the modes destabilized by the κ − γ mechanism (Córsico & Althaus 2016; Istrate et al. 2016b).


Because of its large mass, the discovery of GW190425 suggests that gravitational-wave analyses can access densities several times above nuclear saturation (see, e.g., Figure 4 in Douchin & Haensel 2001) and probe possible phase transitions inside the core of a neutron star (NS) (Oertel et al. 2017; Essick et al. 2019; Tews et al. 2019). However, binaries comprised of more massive stars are described, for a fixed EoS, by smaller values of the leading-order tidal contribution to the gravitational-wave phasing

 (Flanagan & Hinderer 2008). These are intrinsically more difficult to measure. For GW190425, this is exacerbated by the fairly low S/N of the event compared to GW170817. Overall, we find that constraints on tides, radius, possible p–g instabilities (Venumadhav et al. 2013; Weinberg et al. 2013; Weinberg 2016; Zhou & Zhang 2017), and the EoS from GW190425 are consistent with those obtained from GW170817 (Abbott et al. 2017b, 2019e). However, GW190425 is less constraining of NS properties, limiting the radius to only below 15 km,

 to below 1100 and only ruling out phenomenological p–g amplitudes above 1.3 times the 90% upper limit obtained from GW170817 at the same confidence level. The p–g constraints were obtained with a different high-spin prior than the rest of the results (see Appendix F.5) but the difference does not significantly change our conclusions. Spin priors can affect the inference of tidal and EoS parameters, and we note that the low-spin results are generally more constrained. Following Agathos et al. (2020), we estimate the probability of the binary promptly collapsing into a black hole (BH) after merger to be 96%, with the low-spin prior, or 97% with the high-spin prior. Repeating the analyses of Chatziioannou et al. (2017) and Abbott et al. (2019d), we find no evidence of a postmerger signal in the 1 s of data surrounding the time of coalescence. We obtain 90% credible upper limits on the strain amplitude spectral density and the energy spectral density of

 and

, respectively, for a frequency of 2.5 kHz. Similar to GW170817, this upper limit is higher than any expected post-merger emission from the binary (Abbott et al. 2019d). More details on all calculations and additional analyses are provided in Appendix F.7.


We repeated the plasma pressure calculation presented by Schwadron et al. (2011) and Fuselier et al. (2012) for the new ENA energy spectrum. The results for the downwind hemisphere and for the Voyager 1 region are summarized in Table 3. The measured intensity

j

ENA

 of neutralized hydrogen at a given energy translates into a pressure of the parent ion population in the heliosheath times the integration length along the line of sight, ΔP × l, in the following way:
3

Δ
P
×
l
=

4
π

3

n

H

m

H

v

j

ENA

(
E
)

σ
(
E
)

Δ
E

c

f

4

c

f

=

(
v
+

u

R

)

2

v

4

(

v

2

+
4

u

R

2

+
2

u

R

v
)
.

In Equation (3), ΔE denotes the width of the respective energy bin; for the typical radial velocity of solar wind in the flanks and the downwind hemisphere of the inner heliosheath, we assumed uR = 140 km s−1 as measured by Voyager 2, whereas uR = 40 km s−1 for the heliosheath in the Voyager 1 direction (Schwadron et al. 2011; Gloeckler & Fisk 2015). For the density of neutral hydrogen in the inner heliosheath a constant nH = 0.1 cm−3 is assumed (Schwadron et al. 2011; Gloeckler & Fisk 2015). The charge-exchange cross section between protons and neutral hydrogen decreases from (4 to 2) × 10−15 cm−2 for 0.015 to 1.821 keV (Lindsay & Stebbings 2005). The integration length l for ENA production in the plasma is approximately the thickness of the inner heliosheath. The part of Equation (3) without the velocity factor cf can be interpreted as stationary pressure. The total pressure or dynamic pressure is the stationary pressure times this factor. Integrating over all energy bins in Table 3, we obtain the total plasma pressure times heliosheath thickness as P × l = 304 pdyn cm−2 au for the downwind hemisphere and 66 pdyn cm−2 au for the Voyager 1 region (1 pdyn cm−2 au = 0.015 N m−1). If we want to put these numbers into the context of other studies, we face two problems. First, the uncertainty of the total pressure is large given the upper limits in the two lowest energy bins. Second, heliosheath plasma more energetic than 2 keV will produce ENAs that cannot be detected with IBEX-Lo. We therefore used the observed median j = 0 cm−2 sr−1 s−1 keV−1 for heliospheric ENAs in the two lowest energy bins of IBEX-Lo and relied on the study by Livadiotis et al. (2013). They compared the expected plasma pressure from a kappa distribution of protons with the plasma pressure derived from IBEX-Hi energy spectra: the energy range between 0.03 and 2 keV, roughly corresponding to the IBEX-Lo range, covered more than half of the total plasma pressure predicted from a kappa distribution. The authors found a total plasma pressure of P = 2.1 pdyn cm−2 for all sky directions except for the ENA Ribbon. Gloeckler & Fisk (2015) presented a multi-component plasma model for the heliosheath to explain Voyager and IBEX observations. At low energies they assumed the ENA energy spectra provided by Fuselier et al. (2012). They derived a total pressure of 2.5 pdyn cm−2 in all three plasma regions in the nose of the heliotail (Gloeckler & Fisk 2015). Pressure contributions from the slowed solar wind, magnetic pressure, and the pressure exerted from pickup ions and anomalous cosmic rays all had to be taken into account to obtain this total pressure.


The surface brightness profile for the mosaicked image of A2199 was then extracted in concentric annuli centred at the cluster centre (RA, Dec.) = (16:28:38.21,  +39:33:02.31). This corresponds to the location of the peak X-ray flux in the cluster. The radial profile of the surface brightness is shown in the upper panel of Fig. 2. In the lower panel of this figure, we present the radial profile of the azimuthally averaged electron density of A2199. The profile is recovered from the deprojection of the median surface brightness profile using the onion peeling technique (Ettori et al. 2010), and assuming that the ICM plasma is spherical symmetry. To convert from surface brightness to density, we use the following widely used approach (Eckert et al. 2012; Tchernin et al. 2016; Ghirardini et al. 2018; Ghirardini et al. 2019; Walker et al. 2020). Using xspec and the response files for XMM–Newton, the conversion factor between APEC normalization and X-ray count rate in the 0.7–1.2 keV band is found. In the 0.7–1.2 keV band, this is largely independent of the temperature of the gas, so this allows a direct conversion from the deprojected X-ray surface brightness profile to a deprojected profile of APEC normalization. The APEC normalization is related to the gas density by the equation:
(1)$$\begin{eqnarray*}
{\rm Norm} = \frac{10^{-14}}{4 \pi [d_\mathrm{A}(1+z)]^2 }\int n_\mathrm{ e} n_\mathrm{H} \mathrm{d}V,
\end{eqnarray*}$$where the electron and ion number densities are ne and nH, respectively (for a fully ionized plasma these are related by ne = 1.17nH, Asplund et al. 2009), the angular diameter distance to the cluster is dA, and z is the cluster redshift. The deprojected APEC normalizations are then converted to deprojected density in the usual fashion, assuming spherical symmetry and a constant density in each shell, by calculating the projected volumes, V, of each shell in the 2D annuli. When performing this conversion, we used a column density of 0.08 × 1022 cm−2 (from Kalberla et al. 2005), and using the abundance tables of Asplund et al. (2009).


The age distribution of the inner Galaxy is less well known. Traditionally, from photometry, the bulge has been viewed as an old structure (e.g. Zoccali et al. 2003) but this was thrown into question by spectroscopic ages of microlensed dwarfs (Bensby et al. 2013), many of which are young. Recent work by Bernard et al. (2018) constrained the age distribution of the bulge ($-5\, \mathrm{deg}\lesssim b\lt -2\, \mathrm{deg}$) from Hubble Space Telescope photometry of the main sequence turn-off stars, concluding that, although the bulge is predominantly old, approximately $10\, \mathrm{per\, cent}$ of stars are younger than $5\, \mathrm{Gyr}$. This fraction increases to $\sim 20\, \mathrm{per\, cent}$ for more metal-rich ($[\mathrm{Fe}/\mathrm{H}]\gtrsim 0.2\, \mathrm{dex}$) stars, consistent with the Bensby et al. (2013) work. Further evidence for a predominantly old ($\gtrsim 8\, \mathrm{Gyr}$) bulge comes (indirectly) from [C/N] measurements of giant stars (Bovy et al. 2019; Hasselquist et al. 2020), although, as highlighted by Hasselquist et al. (2020), age appears to correlate with both metallicity and Galactic height of the populations. Nogueras-Lara et al. (2020a) have used the luminosity of red clump stars to conclude the majority ($\sim 95\, \mathrm{per\, cent}$) of the nuclear stellar disc formed more than $8\, \mathrm{Gyr}$ ago with some evidence of a more recent ($\lt 1\, \mathrm{Gyr}$ ago) star formation burst (Matsunaga et al. 2011). This is consistent with ongoing/recent star formation within the central molecular zone (Morris & Serabyn 1996) and is broadly consistent with the conclusions of Bernard et al. (2018) on the wider bar/bulge but possibly suggesting the nuclear stellar disc is on average older than the surrounding bulge. We have fitted by-eye a very simple star formation history to the ‘cleanest’ combined fit from Bernard et al. (2018) of the form $\mathrm{sech}^2((13.5\, \mathrm{Gyr}-\tau)/4.7\, \mathrm{Gyr})$ with a truncation at $14\, \mathrm{Gyr}$. Combining the metallicity distributions and star formation histories with the PARSEC isochrones and adopting a Kroupa (2001) initial mass function, we have computed the luminosity function of the giant branch stars in the inner bulge region. We show the results in the lower panel of Fig. 4 along with a simple double Gaussian plus quadratic fit to represent the red clump stars, the red giant branch bump stars and the red giant branch stars respectively. We find that the lowest latitude bin has a red clump magnitude of $M_{K_s,\mathrm{RC}}=-1.61\, \mathrm{mag}$. This agrees well with the mean solar neighbourhood result from Chan & Bovy (2020) of $M_{K_s,\mathrm{RC}}=-1.622\, \mathrm{mag}$ and more specifically using their relations adopting the mean (J − Ks) = 0.647 (see Appendix B1) and mean metallicity $-0.18\, \mathrm{dex}$ gives $M_{K_s,\mathrm{RC}}=-1.595\, \mathrm{mag}$. The metallicity gradient with latitude produces a red clump magnitude gradient of $0.032\, \mathrm{mag}\, \mathrm{deg}^{-1}$ whilst using the change in mean metallicity in combination with the results of Chan & Bovy (2020) we would expect $0.024\, \mathrm{mag}\, \mathrm{deg}^{-1}$. At all latitudes the red clump distribution is well reproduced by a Gaussian with standard deviation $\sim 0.11\, \mathrm{mag}$. Chan & Bovy (2020) measured the solar neighbourhood red clump to have an intrinsic standard deviation of $0.097\, \mathrm{mag}$ which combined in quadrature with that arising from the metallicity variance predicts a standard deviation of $\sim 0.13\, \mathrm{mag}$, similar to the PARSEC models. The red clump peaks from the PARSEC isochrones have a slight bimodal structure arising from the bimodal metallicity distributions such that the mode typically peaks $\sim 0.03\, \mathrm{mag}$ fainter than the Gaussian mean.


Another interesting type of transient to compare DES17X1boj and DES16E2bjy with are the SN impostors. As shown in Fig. 3, SN2009ip has a short phase of re-brightening around the same phase as the secondary peak of the DES-SN transients, and its peak brightness (MV = −17.7; see e.g. Fraser et al. 2013) is similar to DES16E2bjy. However, several other features distinguish our double-peaked DES-SN transients from the SN impostors. While SN2009ip does show re-brightening, its light-curve evolution is clearly different from the DES transients. Additionally, other impostor candidates such as SN2015bh (see e.g. Elias-Rosa et al. 2016) and SN2016bdu (Pastorello et al. 2018) have very similar light curves with SN2009ip, but do not exhibit rebrightening. Furthermore, our photometric data also constrain the long-term variability of DES17X1boj to a level below what was seen in SN2009ip (Pastorello et al. 2013) and SN2016bdu (Pastorello et al. 2018) in the years before the brightest event (MV in range −13 to −14). For the more distant event DES16E2bjy, such outbursts would have been below our detection threshold. Regarding the spectroscopic data, the impostors exhibit strong, narrow H and He lines around peak brightness (see e.g. Fraser et al. 2013; Mauerhan et al. 2013; Pastorello et al. 2013). No such features are seen in either of the DES transients (see Figs 6 and 9), but it is possible that the lines are hidden in the noise.To investigate this, we estimated the limiting equivalent width (EW) for a Gaussian-shaped narrow H α line with $v$FWHM = 500  km s−1 in our spectra. H lines with similar widths are often seen in both SN impostors (Smith et al. 2011) and in SNe IIn (Taddia et al. 2013) where $v$FWHM ∼ 100−1000  km s−1 are typically measured. For the given configuration, we found limits of EW ≲ 5 Å for DES17X1boj and EW ≲ 14 Å for DES16E2bjy. In the case of SNe IIn, the line strengths are typically measured in several tens to hundreds of Ångstroms (EW≳ 40 Å; Smith, Mauerhan & Prieto 2014), and thus it is unlikely that narrow H α lines are hiding in the spectra. Due to both photometric and spectroscopic differences, it is unlikely that DES17X1boj and DES16E2bjy are events similar to SN impostors.


We use the N-body simulation package—MERCURY (Chambers 1999)—to numerically investigate the effects of photo-evaporation on the dynamical evolution of planet–satellite systems. We choose the Bulirsch–Stoer integration algorithm, which can handle close encounter accurately. It is important in the simulations, as we will see below, that many close encounters among moons and the planet are expected to happen. Collisions among moons, the planet, and the central star are also considered in simulations and treated simply as inelastic collisions without fragmentations. Each simulation consists of a central star, a planet, and some moons orbiting around the planet. The photo-evaporation is simply modeled as a slow (adiabatic) and isotropic mass-loss process of the planet. In reality, the photo-evaporation is a very slow process on a timescale of the order of 107–108 year (Owen & Wu 2013). However, it is impractical and unnecessary to perform a simulation on such a long timescale. Instead, we model the mass-loss process on a timescale of τevap, and each simulation typically lasts for several τevap. As long as the adiabatic requirement is met, i.e., the mass-loss timescale is much longer than the dynamical timescale of the system (τevap ≫ Pp, where Pp is the orbital period of the planet), one could study the dynamical effects of the mass-loss process equivalently. As we discussed in Section 3.3, the results converge if τevap > 102–103 Pp, indicating the adiabatic condition is met. Therefore, in all other simulations, we set τevap = 104 Pp. Other parameters are set to represent the typical values of Kepler planets. In particular, we consider a planet–satellite system orbiting a star of solar mass (M⋆ = M⊙) in a circular orbit (ep = 0.0) with semimajor axis of ap = 0.1 au. The orbit has a period of ∼10 days (typical value of Kepler planets), and it is sufficiently close to the central star to be subject to significant photo-evaporation effect (Owen & Wu 2013), which removes massive hydrogen envelopes of the planet. The planet has an initial mass of Mpi and a final mass of Mpf after photo-evaporation. In this paper, we adopt Mpi = 20 M⊕ and Mpf = 10 M⊕ nominally (close to the standard model adopted in Owen & Wu 2013). The mean density of the planet is set to the same as that of Neptune (1.66 g cm−3). The effect of changing the planetary density is discussed in Section 3.3. We performed a number of sets of simulations by considering different planet–satellite configurations. Similar to the definition in MERCURY, hereafter, we define “small moons” as test particles (TPs) whose mutual gravity and corresponding effects on the planet and the star are ignored, while “big moons” are gravitationally important enough that their gravitational effects are fully considered. Table 1 lists the initial setups and parameters of various simulations, whose results are presented in the following subsections.


The presumed WD cooling age is ≳5 Gyr. If it has a low mass, this age limit can be larger since after the Roche lobe detachment a proto-WD goes through the contraction phase until it reaches its cooling track (Istrate et al. 2014, 2016). The duration of this phase increases as the mass of the proto-WD decreases, and may last as long as ∼2 Gyr. The J0740 characteristic age of 3.75 Gyr (Table 1) is smaller than the WD age estimate. However, the observed pulsar period derivative and consequently its characteristic age can be biased by kinematic effects, i.e. the effects of the pulsar proper motion (Shklovskii effect; Shklovskii 1970), the acceleration towards the Galactic plane and the acceleration due to differential Galactic rotation (Nice & Taylor 1995). Using the J0740 proper motion value from Table 1, the Sun’s Galactocentric velocity and the distance (240 km s−1 and 8.34 kpc, respectively; Reid et al. 2014), we calculated these corrections to the pulsar period derivative: $\dot{P}_{\rm S}=3.0\times 10^{-21}$, $\dot{P}_{\rm G,\perp }=-1.6\times 10^{-22}$, $\dot{P}_{\rm G,p}=3.8\times 10^{-23}$ for the minimum and $\dot{P}_{\rm S}=6.9\times 10^{-21}$, $\dot{P}_{\rm G,\perp }=-2.2\times 10^{-22}$, $\dot{P}_{\rm G,p}=8.9\times 10^{-23}$ for the maximum pulsar distance estimates. The corresponding intrinsic characteristic ages are τi ∼ 5 and ∼8.5 Gyr, which are compatible with the cooling age.8 Thus, the considered binary system indeed can be very old and the presumed WD can be ultracool. This is not a unique situation. There are other examples of the objects with similar characteristics. One of them is the WD companion of PSR J0751+1807 (Bassa et al. 2006). Its colours (see Figs 2 and 3) indicate that the WD has a pure helium or mixed H/He atmosphere with a temperature T ∼ 3500–4300 K. Other examples are isolated ultracool white dwarfs WD J1102 (Hall et al. 2008; Kilic et al. 2012) and WD 0346+246 (Oppenheimer et al. 2001). These WDs have temperatures of about 3650 and 3300 K, respectively, and are best explained by the mixed atmosphere models (Gianninas et al. 2015).


Our simulations were run using the AREPO (Springel 2010; Pakmor, Bauer & Springel 2011; Pakmor et al. 2016; Weinberger, Springel & Pakmor 2020) moving-mesh magnetohydrodynamics (MHD) code. The code solves for gravity coupled with MHD. The gravity solver uses the PM-tree method (Barnes & Hut 1986) and the MHD solver uses a non-static unstructured grid formed by performing a Voronoi tesselation of the domain. AREPO has been used to produce simulations of the Universe at a wide range of scales. At the largest scales, we have uniform volume cosmological simulations such as the Illustris (Genel et al. 2014; Vogelsberger et al. 2014a; Nelson et al. 2015; Sijacki et al. 2015) and Illustris-TNG  (Marinacci et al. 2018; Naiman et al. 2018; Nelson et al. 2018; Pillepich et al. 2018b; Springel et al. 2018; Nelson et al. 2019a,b; Pillepich et al. 2019) suites. These simulations have box sizes ranging from ∼50 to ∼300 Mpc and baryonic mass resolutions ranging from ∼105 to 107 M⊙. They have been largely successful in producing galaxy and SMBH populations consistent with observations, in Illustris (Vogelsberger et al. 2014b; Sales et al. 2015; Sijacki et al. 2015) and in TNG (Genel et al. 2018; Weinberger et al. 2018; Pillepich et al. 2018b; Donnari et al. 2019; Rodriguez-Gomez et al. 2019; Torrey et al. 2019; Habouzit et al. 2021; Übler et al. 2021). At the smallest scales, we have cosmological zoom simulation suites such as AURIGA (Grand et al. 2017) for individual milky-type galaxies, and HESTIA (High-resolutions Environmental Simulations of The Immediate Area) (Libeskind et al. 2020) for the Local Group. These simulations have been successful in reproducing observational results for the internal structures of galaxies  (Blázquez-Calero et al. 2020; Cautun et al. 2020; Grand et al. 2020). All these developments make AREPO an ideal tool for the development of black hole models, which require a reliable modelling of the necessary physics over a large dynamic range.


In addition, broad-band polarization observations provide essential information about the pulsar radio emission mechanism, beam geometry, and the Galactic magneto-ionic ISM. Pulsars are among the most highly polarized radio sources known (e.g. Lyne & Smith 1968; Gould & Lyne 1998), and the polarization varies with observing frequency (e.g. Manchester, Taylor & Huguenin 1973; Johnston et al. 2008; Dai et al. 2015), providing insight into the magnetospheric emission and propagation mechanisms. In addition, the linear polarization P.A.s across pulse phase can constrain the beam size and inclination angles, with respect to the pulsar’s rotation axis and our line of sight (LoS). For example, the rotating vector model (RVM) predicts a smooth ‘S’-shape, due to the projected vectors of the magnetic field lines as they sweep across our LoS (e.g. Radhakrishnan & Cooke 1969; Lyne & Manchester 1988; Johnston et al. 2007). Many pulsars show more complex P.A. curves, particularly discontinuities with rapid jumps of ≈90°, which suggests the presence of two orthogonal polarization modes (OPM; e.g. Manchester 1975; Backer, Rankin & Campbell 1976). Furthermore, circular polarization across the pulse is observed to either remain in the same hand or change sense (e.g. Radhakrishnan & Rankin 1990), which may be intrinsic to the emission mechanism or due to propagation effects (e.g. Han et al. 1998; Kennett & Melrose 1998). Additional diagnostics of magnetospheric effects have also been investigated, including variations in Faraday rotation measure (RM) and circular polarization across the pulse (e.g. Ramachandran et al. 2004; Karastergiou 2009; Noutsos et al. 2009; Ilie, Johnston & Weltevrede 2019). Although pulsars were discovered over 50 yr ago (Hewish et al. 1968), it is clear that current understanding and models of the emission mechanism are far from replicating this wide range of observed behaviour, as well as additional emission phenomena such as nulling and mode changing.


The left-hand panel of Fig. 3 shows the scatter plot of b1 and mass, coloured by $1+\delta _{5\,h^{-1}\,{\rm Mpc}}$. There is an obvious correlation visible, with a largely vertical trend in which b1 increases monotonically with $\delta _{5\,h^{-1}\,{\rm Mpc}}$. The symbols with errors show the median bias as a function of mass, in four bins of $\delta _{5\,h^{-1}\,{\rm Mpc}}$ and averaged over 10 realizations of the default box. It is clear that, at fixed $\delta _{5\,h^{-1}\,{\rm Mpc}}$, the trend of bias with halo mass is weak. This trend is consistent with previous results in the literature, which have shown that large-scale bias is more strongly correlated with halo-centric overdensity than it is with halo mass (see e.g. Abbas & Sheth 2007; Shi & Sheth 2018). The right-hand panel of the figure explores this further, showing the Spearman rank correlation coefficient between b1 and δR as a function of halo mass, for R = 2, 3, 5 h−1 Mpc. We see that the strength of the correlation is only a weak function of mass for each smoothing scale, but monotically increases with R. This increase with R is not surprising, since our estimator for b1 itself is ultimately measuring a large-scale halo-centric overdensity, so that b1 and δR are measuring essentially the same quantity for large R. To appreciate this point better, Fig. 4 shows a visualization of the haloes in a subvolume of our high-resolution box, with haloes shown as circles whose radii scale with R200b and whose colour scales with halo bias b1 as indicated by the colour bar. The panels focus on massive (top) and low mass haloes (bottom). We discuss some connections between halo-by-halo bias and gravitational redshift measurements (see e.g. Wojtak, Hansen & Hjorth 2011; Croft 2013; Alam et al. 2017) in Appendix D. In Appendix B1, we present analytical arguments that explain the size of the scatter in b1 at fixed mass and also qualitatively reproduce the trends seen in Fig. 3.


Like normal elliptical galaxies, the analysis of the stellar populations in the inner regions of BCGs indicates that the bulk of their stars was formed rapidly in a very intense starburst at redshift z > 2 (Renzini 2006). However, despite sharing similar morphologies with normal massive ellipticals, in addition to red colours, old and metal-rich stellar populations and alpha-enhancements (Brough et al. 2008; Loubser et al. 2009; Donahue et al. 2010; Loubser & Sánchez-Blázquez 2012; Barbosa et al. 2016; Edwards et al. 2020), BCGs constitute a special category of objects with peculiar star formation histories (SFHs) seen from both observations (Tran et al. 2008; Barbosa et al. 2016) and models (Dubinski 1998; De Lucia & Blaizot 2007). The evolution of BCGs is significantly affected by their surrounding environments. Due to their central positions in the gravitational potential well of their host clusters, central cluster galaxies accrete stars and gas from satellite galaxies that orbit around them and fall in, developing extended light profiles. The more representative SFHs of their dominant stellar population include components from in situ star formation, and from the interaction with other galaxies and with the intracluster medium, where the outer regions in BCGs are continually assembling mass through minor mergers (Cooke et al. 2019). The stellar populations of a large sample of observed BCGs have been recently studied in Edwards et al. (2020), from the galaxy core into the intracluster light (ICL) out to 4 effective radii, finding old stellar populations of ∼13 Gyr and high metallicities [Fe/H] ∼0.3 in the galaxy cores, whereas the average age in the ICL is estimated to be slightly younger, ∼9.2 Gyr with lower metallicities −0.4  [Fe/H]  0.2 at 40 kpc. This broadly supports the idea of two-phase galaxy formation, with the BCG cores and inner regions formed faster and earlier than the outer regions that have formed more recently, or have accreted mass afterwards through galaxy mergers and thereby also increasing in size (Oser et al. 2010; Kubo et al. 2017; Cooke et al. 2019).


Studies of SMGs over the past few tens of years have provided valuable insights into their properties. These include the redshift distribution (e.g. Chapman et al. 2005; Aretxaga et al. 2007; Wardlow et al. 2011; Yun et al. 2012; Smolčić et al. 2012; Simpson et al. 2014, 2017; Zavala et al. 2014; Miettinen et al. 2015a; Chen et al. 2016a; Strandet et al. 2016; Danielson et al. 2017; Brisbin et al. 2017), spatial clustering and environment (e.g. Ivison et al. 2000; Blain et al. 2004; Aravena et al. 2010; Hickox et al. 2012; Miller et al. 2015; Chen et al. 2016b; Wilkinson et al. 2017; Smolčić et al. 2017), merger incidence (e.g. Conselice et al. 2003), and circumgalactic medium (Fu et al. 2016). Regarding the intrinsic physical characteristics of SMGs, the properties studied so far include the sizes and morphologies (e.g. Swinbank et al. 2010; Menéndez-Delmestre et al. 2013; Aguirre et al. 2013; Targett et al. 2013; Chen et al. 2015; Simpson et al. 2015; Ikarashi et al. 2015; Miettinen et al. 2015b, 2017b,c; Hodge et al. 2016), panchromatic spectral energy distributions (SEDs; e.g. Michałowski et al. 2010; Magnelli et al. 2012; Swinbank et al. 2014; da Cunha et al. 2015; Miettinen et al. 2017a), stellar masses (e.g. Dye et al. 2008; Hainline et al. 2011; Michałowski et al. 2012; Targett et al. 2013), gas masses (e.g. Greve et al. 2005; Tacconi et al. 2006, 2008; Engel et al. 2010; Ivison et al. 2011; Riechers et al. 2011; Bothwell et al. 2013; Huynh et al. 2017), gas kinematics (e.g. Alaghband-Zadeh et al. 2012; Hodge et al. 2012; Carilli & Walter 2013; Olivares et al. 2016), and active galactic nucleus (AGN) incidence (Alexander et al. 2003, 2005; Laird et al. 2010; Johnson et al. 2013; Wang et al. 2013). The role played by SMGs in a broader context of galaxy formation and evolution has also been investigated through models (e.g. Baugh et al. 2005; Fontanot et al. 2007; Davé et al. 2010; González et al. 2011; Hayward et al. 2013) and observational approach (e.g. Swinbank et al. 2006; Toft et al. 2014; Simpson et al. 2014).


Since the errors of the LIGO-estimated rates are dominated by Poisson statistics (Abbott et al. 2020a,b), we approximate the PDF for the expected number of detections $\mathcal {N}=\mathcal {R}VT$ (from the surveyed space–time volume VT) by $\mathrm{d}P/\mathrm{d}\mathcal {N}\propto \mathcal {N}^{k-1/2}\mathrm{e}^{-\mathcal {N}}/k!$, where k = 1 for each of the relevant cases ($\mathcal {R}_{190814}$, $\mathcal {R}_{170817}$, and $\mathcal {R}_{190425}$), and the factor of $\mathcal {N}^{-1/2}$ is from Jeffrey’s prior (Abbott et al. 2020a). From the median values of $\bar{\mathcal {R}}_{190814}=7\rm \, Gpc^{-3}\, yr^{-1}$ (Abbott et al. 2020a), $\bar{\mathcal {R}}_{\rm 170817}=760\rm \, Gpc^{-3}\, yr^{-1}$, and $\bar{\mathcal {R}}_{\rm 190425}=460\rm \, Gpc^{-3}\, yr^{-1}$ (Abbott et al. 2020b), we obtain the effective surveyed space–time volumes $VT=1.2/\bar{\mathcal {R}}$ for each of these three events (‘1.2’ is the median of $\mathrm{d}P/\mathrm{d}\mathcal {N}$). We consider both GW170817 and GW190425 as bNS mergers, because the component masses of GW190425 are not far from those of GW170817 and the nature of the merging objects makes little practical difference in our model. Thus, the PDF of the total bNS merger rate from the sum of the two is given by a convolution of the two individual PDFs
(1)$$\begin{eqnarray*}
{\mathrm{d}P\over \mathrm{d}\mathcal {R}_{\rm bns}} = \int _0^{\mathcal {R}_{\rm bns}} \mathrm{d}\mathcal {R}_1 {\mathrm{d}P\over \mathrm{d}\mathcal {R}_1} \left.{\mathrm{d}P\over \mathrm{d}\mathcal {R}_2}\right|_{\mathcal {R}_{\rm bns}-\mathcal {R}_1},
\end{eqnarray*}$$where we have written $\mathcal {R}_{1} = \mathcal {R}_{170817}$, $\mathcal {R}_{2} = \mathcal {R}_{190425}$ for brevity. We then calculate the PDF for the inverse of the total bNS merger rate $\mathrm{d}P/\mathrm{d}\mathcal {R}_{\rm bns}^{-1}=\mathcal {R}_{\rm bns}^2\mathrm{d}P/\mathrm{d}\mathcal {R}_{\rm bns}$. Finally, the PDF of the rate ratio $\beta =\mathcal {R}_{190814}/\mathcal {R}_{\rm bns}$ is given by
(2)$$\begin{eqnarray*}
{\mathrm{d}P\over \mathrm{d}\beta } = \int _0^\infty {\mathrm{d}\mathcal {R}_{3}\over \mathcal {R}_3} {\mathrm{d}P\over \mathrm{d}\mathcal {R}_{3}} \left.{\mathrm{d}P\over \mathrm{d}\mathcal {R}_{\rm bns}^{-1}}\right|_{\beta /\mathcal {R}_3},
\end{eqnarray*}$$where we have written $\mathcal {R}_{3} = \mathcal {R}_{190814}$ for brevity. We find the 90 per cent confidence interval for the rate ratio to be in the range $0.064\, \rm {per\, cent}\lt \beta \lt 2.8\, \rm {per\, cent}$.


For the magnetic field configurations considered in this work, the polarization angle can only change exactly by Δϕ = 90○ and a gradual change of the PA is not possible. There are tantalizing hints of a 90○ change in the PA in some of the GRBs, as discussed above, but the results are not yet conclusive. The result presented by Sharma et al. (2019) where the PA changes by 90○ twice over the emission is again very exciting as such a change over a single pulse can only occur for the Btor field configuration. The only difficulty, according to the modelling done here, is that both 90○ changes occur in the decaying tail of the pulse when high latitude emission dominates the flux. In the measurement presented by Sharma et al. (2019), the PA shows a change close to the peak of the emission. Another scenario in which a 90○ PA change can be obtained includes contribution from multiple pulses and when the LOS is close to the edge of the jet, such that θobs ≈ θj, along with a change in bulk Γ between the pulses which would change ξj = (Γθj)2. Alternatively, such a change in the PA can be obtained due to magnetic reconnection, e.g. in the ICMART model (Zhang & Yan 2011), where the local magnetic field orientation, which is orthogonal to the wave vector of the emitted photon, itself changes by 90○ as the field lines are destroyed and reconnected in the emission region (Deng et al. 2016). To obtain a change in the PA other than Δϕ = 90○ or to get a gradually changing PA the condition for axisymmetry must be relaxed and the magnetic field configuration or orientation in the emission region must change. One possibility is that if the different pulses that contribute to the emission arise in a ‘mini-jet’ within the outflow (e.g. Shaviv & Dar 1995; Lyutikov & Blandford 2003; Kumar & Narayan 2009; Lazar, Nakar & Piran 2009; Narayan & Kumar 2009; Zhang & Yan 2011). In this case, the different directions of the mini-jets or bright patches w.r.t. the LOS (e.g. Granot & Königl 2003; Nakar & Oren 2004) would cause the PA to also be different between the pulses even for a field that is locally symmetric w.r.t the local radial direction (e.g. B⊥ or B∥) as well as for fields that are axisymmetric w.r.t to the centre of each mini-jet (e.g. a local Btor for each mini-jet). Finally, broadly similar result would follow from an ordered field within each mini-jet (Bord) which are incoherent between different mini-jets. Time-resolved measurement in such a case would naturally yield a time-varying PA. Alternatively, as shown by Granot & Königl (2003) for GRB afterglow polarization, a combination of an ordered field component (e.g. Bord) and a random field, like B⊥, can give rise to a time-varying PA between different pulses that, e.g. arise from internal shocks. The ordered field component here would be that advected from the central engine and the random field component can be argued to be shock-generated. Notice that the ordered field component should not be axisymmetric in order for the PA to smoothly vary.


A262 (RA = 01:52:46.299, Dec. = +36:09:11.80) is a bright, nearby poor cluster at z = 0.0162 (Struble & Rood 1999) with mean ICM temperature ≈2 keV( see e.g. Vikhlinin et al. 2005, 2006; Sato, Matsushita & Gastaldello 2009; Sanders et al. 2010). Due to its low mass and temperature, it may be considered as an intermediate between clusters and groups. A262 was observed for 110.7 ks in ACIS-S and a blank-sky observation of 450 ks was used for the background fitting. We used dmcopy tool in CIAO: Chandra's data analysis system (Fruscione et al. 2006) to restrict the energy range to 0.7−7 keV for both imaging and spectral analysis in all of the four data product files: event file, blank-sky observation, RMF and ARF. This reduced the number of PI channels to 433. Also, since the output of CIAO is in fits format, we used ftools-fv6 to export the event and background files as ASCII files. We then performed a 2 arcsec binning to both the event and the background files to generate the 3D data cubes for the spectral analysis without having to re-group the energy and PI columns. We used our in-house binning software tool for binning the data; this software is also available in the online package. This reduced the size of the data to a manageable level without adversely affecting the subsequent inference of the cluster. The output is a photon counts in a grid of 256 × 256 × 433 to be read in by bayes-x. Our Bayesian framework also allows us to analyse the data from one CCD. The X-ray images (Fig. 2) were then generated by summing up the counts at each pixel. To illustrate the large-scale features in the image we also binned the events with a cell size of 16 arcsec (right-hand panel of Fig. 2). It should be noted that the images are for illustration purposes only. We applied the rmfimg tool in CIAO to convert and expand RMF and ARF files into 2D images (matrices) for the spectral analysis. Similarly, we used ftools-fv to export the 2D RMF and ARF as ASCII files to be read in by bayes-x.


As we have seen, the asymptotic reduction of the original CR propagation problem, given by Equation (9), to its isotropic part cannot proceed to higher orders of approximation using a simple asymptotic series in Equation (10) and requires a multi-time asymptotic expansion. In the Chapman–Enskog method, the operator

 is expanded instead. Its purpose is to avoid unwanted higher time derivatives to appear in higher orders of approximation. This is very similar to, e.g., a secular growth in perturbed oscillations of dynamical systems. To eliminate the secular terms, one seeks to alter (also expand in a small parameter) the frequency of the zero-order motion, which is similar to the

 expansion. One example of such an approach may be found in a derivation of hydrodynamic equations for strongly collisional, but magnetized plasmas, starting from the Boltzmann equation (Mikhailovsky 1967). The classical monograph by Chapman & Cowling (1991; Ch.VIII) gives another example of a subdivision of the

 operator for solving the transport problem in a non-uniform gas-mixture. Expanding

 operators eliminates secular terms, such as the telegraph term. Perhaps more customary today, and equivalently, is to introduce a hierarchy of formally independent time variables (e.g., Nayfeh 1981)

, so that
14

Instead of Equation (13), from Equation (9), we have
15

where the conditions

 are implied. The solution of this equation should be sought in the following form
16

where

 and

 are chosen to satisfy, respectively, the following two equations.
17

and
18

The solution for

 is as follows
19

and it can be evaluated for arbitrary n by expanding both sides of Equation (17) in a series of eigenfunctions of the diffusion operator on its lhs:

For D = 1, for example,

 are the Legendre polynomials with

,

. The time dependent coefficients

 are determined by the initial values of

 (the anisotropic part of the initial CR distribution) and the rhs of Equation (17), that depends on

, obtained at the preceding step. It is seen, however, that

 exponentially decay in time for

 and we may ignore them3

3
In fact, we must do so because our asymptotic method has a power accuracy in

, but not the exponential accuracy.
 because we are primarily interested in evolving the system over time

 and even longer. Starting from n = 0 and using Equation (15), for the slowly varying part of f, we have
20

The solubility condition for

 (obtained by integrating both sides of Equation (15) in μ) also gives a trivial result
21

so the last two conditions are consistent with the suggested decomposition in Equation (16), since from Equation (18) with n = 1, we have
22

and, thus both

 and

 are, indeed, independent of

 and

. We have introduced the function

 here by the following two relations
23

The solubility condition for

 yields the nontrivial and well-known (e.g., Jokipii 1966) result, which is actually the leading term of the

 expansion in

24

where

The solubility conditions for

,

 will generate the higher order terms of our expansion, which, after some algebra, can be manipulated into the following expressions for the third and fourth orders of approximation
25

26

We have denoted

and

. The pitch-angle diffusion coefficient

 and magnetic focusing σ are considered z-independent for simplicity, a limitation that can be easily relaxed by rearranging the operators containing

 in Equation (26). We can proceed to higher orders of approximation ad infinitum since terms containing

 can be expressed through

. According to Equations (20)–(21), of interest is the evolution of

 on the timescales

 or

; thus, as we already mentioned, the contributions of

 to all of the solubility conditions, similar to those given by Equations (24)–(26), have to be dropped (because they become exponentially small) and only

-contributions should be retained. Using Equations (20)–(21) and (24)–(26) to form the combinations

 and summing up both sides, on the lhs of the resulting equation, we simply obtain

 (see Equation (14)). Therefore, the evolution of

 up to the fourth order in ϵ takes the following form.
27

where

,

, and
28


We performed a preliminary, zeroth-order analysis of the images by constructing a syndyne–synchrone map for each observing date. From those maps, we inferred that the activation time of the asteroid should be close in time to the discovery date, owing to the absence of dust features that could have shown up at the corresponding locations of synchrones approximately two months before discovery or earlier. In particular, no neck-line or trail features appear in the image from 2016 January 7 (PlAng ∼ 0°), which could have indicated past activity. In addition, there are no dust condensations along the direction of isolated synchrones, which could have indicated short bursts of activity (e.g., the case of P/2012 F5 (Gibbs), Moreno et al. 2012a), or several separated short bursts, as in the case of P/2013 P5 (Jewitt et al. 2013; Moreno et al. 2014). According to this, it is reasonable to start the search for a minimum in the function χ defined above by placing the activation time (t0) between a few days before the discovery date (102.5 days before perihelion) and about 60 days before. Regarding the duration of the activity, the smooth variation in absolute magnitudes (from Hv = 17.88 to Hv = 18.16, see Equation (1)) over the ∼40 days period of observation and the aforementioned lack of single-synchrone dust features would suggest a long-lasting process and not an impulsive, short-duration event, such as a collision with another body. In any case, we considered both long- and short-duration events by varying HWHM in a wide range between a few days and several months in the starting simplex of the search of the five-dimensional parameter space. For the peak dust mass loss, we imposed a wide range between a minimum of 0.1 kg s−1 and 100 kg s−1, while for the velocities we set broad limits for the parameters v0, v1, and v2, so that the velocities ranged from 0 to 5 × 103 m s−1 (the mean velocity in the asteroid belt), and the parameter γ from 0.5 to 0, i.e., from typical gas drag to a nearly flat distribution of velocities.


Alternatively, if a magnetar is the central engine powering GRBs, we might expect to see periodic features in the emission. Known magnetars have clear periodic signals in their emission caused by their rotation periods (e.g. Mazets et al. 1979; Kouveliotou et al. 1998). The X-ray pulsations typically contribute to 30 per cent of the signal, with a range of 10–80 per cent (Israel et al. 1999; Kargaltsev et al. 2012; Kaspi & Beloborodov 2017). There is an energy dependence on the pulsed fraction of the signal, where low energies tend to have smaller pulsed fractions (Vogel et al. 2014). Detection of a periodic signal during the plateau phase in the X-ray light curve would provide excellent supporting evidence for the magnetar central engine model. There have been searches for a periodic signal in the prompt emission of GRBs with a number of instruments with no success, for example: Burst And Transient Source Experiment (BATSE) GRBs ( Deng & Schaefer 1997), INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) GRBs (Ryde et al. 2003), GRB 051103 (an extragalactic Soft Gamma-ray Repeater giant flare candidate detected by the Inter Planetary Network; Hurley et al. 2010) and Burst Alert Telescope (BAT) GRBs (Cenko et al. 2010; de Luca et al. 2010; Guidorzi et al. 2012). Dichiara et al. (2013) searched the prompt emission of a number of short GRBs for evidence of a precessing jet (predicted by Stone, Loeb & Berger 2013). However, these searches typically target the prompt emission and have not probed the regime where we might expect periodic signals from a magnetar central engine (i.e. during the plateau phase). Only two GRBs have been searched for periodic emission during the X-ray observations when the magnetar central engine may dominate the emission, GRB 060218 (Mirabal & Gotthelf 2010) and GRB 090709A (Mirabal & Gotthelf 2009; de Luca et al. 2010). The prompt emission of GRB 090709A possibly showed evidence of a periodic signal (Golenetskii et al. 2009; Gotz et al. 2009; Markwardt et al. 2009; Ohno et al. 2009), however this was ruled out with a more careful analysis of the prompt data from BAT, X-ray Telescope (XRT) and X-ray Multi-mirror Mission (XMM) observations of the X-ray afterglow (Cenko et al. 2010; de Luca et al. 2010). However, in the majority of these studies, the authors have targeted a constant spin period whereas a magnetar central engine is expected to have a rapidly decelerating spin period which would be very difficult to detect in standard searches for periodic signals. Dichiara et al. (2013) did conduct a deceleration search, however they were targeting signals in the prompt emission where we do not expect the signal from a spinning down magnetar.


For the TRAPPIST-1 system, data obtained by HST provide initial constraints on the extent and composition of the planet’s atmospheres, suggesting that the four innermost planets do not have a cloud/haze-free H2-dominated atmosphere (de Wit et al. 2016, 2018). However, follow-up work by Moran et al. (2018) have shown that HST data can also be fit to a cloudy/hazy H2-dominated atmosphere. Complementary to HST, NASA’s Spitzer Space Telescope—which played a major role in the discovery and orbital determination of TRAPPIST-1d, e, f, and g (Gillon et al. 2017)—has also allowed us to put additional constraints on the atmospheric composition of TRAPPIST-1b. Transit observations with Spitzer (Delrez et al. 2018) have found a +208 ± 110 ppm difference between the 3.6 and 4.2 μm bands, suggesting CO2 absorption. Spitzer also showed that transit depth measurements do not show any hint of significant stellar contamination in the 4.5 μm spectral range. Morris et al. (2018) reached the same conclusion using a “self-contamination” approach based on the Spitzer data set. Spitzer's “Red Worlds” Program encompassed over 1000 hours of observations of the TRAPPIST-1 system, whose global results have been presented (Ducrot et al. 2020). HST and Spitzer measurements have also been combined with transit light curves obtained from space with K2 (Luger et al. 2017) and from the ground with the SPECULOOS-South Observatory (Burdanov et al. 2018; Gillon 2018) and Liverpool Telescope (Steele et al. 2004) where Ducrot et al. (2018) produced featureless transmission spectra for the planets in the 0.8–4.5 μm wavelength range, showing an absence of significant temporal variations of the transit depths in the visible. Additional ground-based observations with the United Kingdom Infra-Red Telescope, Anglo-Australian Telescope, and Very Large Telescope also show no substantial temporal variations of transit depths for TRAPPIST-1 b, c, e, and g (Burdanov et al. 2019). While the K2 optical data set detected a 3.3 day periodic 1% photometric modulation, it is not present in the Spitzer observations (Delrez et al. 2018). Further constraints on the molecular weight and presence/absence of atmospheres on the TRAPPIST-1 planets will require additional observations with future facilities.


Elsässer variables (Elsässer 1950) are usually employed in solar wind studies to separate the outward-propagating waves (denoted by z+) and the reflected or inward propagating waves (denoted by z−). The separation is exact for even fully nonlinear, unidirectionally propagating waves in homogeneous and incompressible plasma, i.e., Alfvén waves, and it even holds for radially inhomogeneous (along a purely radial magnetic field) but otherwise homogeneous plasma without nonlinear interactions (Hollweg & Isenberg 2007; Magyar et al. 2019b). However, beyond pure Alfvén wave dynamics, it is often overlooked that transverse inhomogeneities, compressibility, and the nonlinear interaction of waves renders the separation of fluctuations into inward and outward-propagating waves inexact. For example, inhomogeneity and compressibility allows for waves (e.g., fast, slow MHD waves, surface Alfvén waves, kink waves, etc.) that are mostly described by both Elsässer variables as they propagate (Magyar et al. 2019a, 2019b). In fact, waves other than pure Alfvén waves generally perturb both Elsässer variables as they propagate. While kink waves, both propagating and standing, are routinely observed in the corona (e.g., Nakariakov et al. 1999; Tomczyk et al. 2007; Anfinogentov et al. 2015; Wang 2016; Nechaeva et al. 2019), evidence of surface Alfvén waves in the solar wind is as of yet inconclusive (e.g., Horbury et al. 2001; Vasquez et al. 2001; Paschmann et al. 2013). Besides waves that are not pure Alfvén waves, structures (inhomogeneities) advected by the solar wind also perturb both Elsässer variables (Tu & Marsch 1990, 1995; Zank et al. 2012; Adhikari et al. 2015). The nonlinear interaction of Alfvén waves can generate purely magnetic fluctuations, 2D modes (k∥ = 0) or condensates which as well perturb both Elsässer variables (Boldyrev & Perez 2009; Howes & Nielson 2013). Indeed, the nature of the inward z− Elsässer variable is often not clear (Wang et al. 2018). Previous studies on Alfvén wave dynamics in radially inhomogeneous models often mention the existence of an “anomalous” z− component that is co-propagating with z+ (Velli et al. 1989; Verdini et al. 2009; Perez & Chandran 2013). The issue of anomalous waves is solved by Hollweg & Isenberg (2007), who showed that, while the continuously generated, reflected z− components might show up as co-propagating in a harmonic analysis, their impulse response analysis shows that these reflected Alfvén waves still follow sunward characteristics, i.e., that there are no truly co-propagating Elsässer variables in these studies. Nevertheless, the coherence of the Elsässer variables resulting from this linear coupling of Alfvén waves seems to influence their spectrum (Verdini et al. 2009).


The use of N-body simulations that include fragmentation allows us to perform a more detailed study of the final composition of the planets formed. In particular, we can study the water loss and/or accretion of the final planets more realistically than in the classic models of accretion. Marcus et al. (2010) presented two empirical models for the mantle stripping in differentiated planetary embryos after a collision. The authors set a simple planet structure of two layers, assuming differentiation in core and mantle, where the mantle could be composed by silicate or ice. In this work, the authors concluded that the more energetic the collision, the more mass from the mantle is lost. Therefore, for violent collisions, water could be more easily removed. Dvorak et al. (2015) performed SPH (smoothed particle hydrodynamics) simulations and studied water loss in planetary embryos and water retained in significant fragments after a collision. They concluded that the impact velocity and the impact angle play a key role in the water loss of a planetary embryo after a collision. The investigations developed by Marcus et al. (2010) and Dvorak et al. (2015) suggest that incorporating a realistic model of volatile transport and removal in an N-body code, may lead to reduced water contents on the resulting terrestrial-like planets, in comparison with those derived from classical models that assume perfect mergers. Burger et al. (2018) studied the volatile loss and transfer. The authors focused on hit-and-run encounters using SPH simulations. They concluded that the cumulative effect of several hit-and-run collisions could efficiently strip off volatile layers of protoplanets. Driven by this, Dugaro et al. (2019) studied the water delivery in planets formed in the habitable zone (HZ), using the mantle stripping models derived by Marcus et al. (2010) in their N-body simulationswith fragmentation. The authors showed that fragmentation is not a barrier for the surviving of water worlds in the HZ, and fragments may be important in the final water content of the potentially habitable terrestrial planets formed in situ.


Here we outline our sample of strongly lensed Planck-selected, dusty star-forming galaxies, hereafter “LPs” (Table 1). Our sample of 24 LPs began with a Planck and Herschel cross-match identification of eight objects (8/24) with continuum detections at 857 GHz (Harrington et al. 2016) greater than 100 mJy. The remaining 16/24 LPs were selected based on continuum detections by Planck, at 857, 545, and/or 353 GHz in the maps of all the available, clean extragalactic sky. These bright Planck point sources were then analyzed through a filtering process using a WISE color selection for the four WISE bands (3.4, 4.6, 12, 22 μm; Yun et al. 2008; D. Berman et al. 2021, in preparation). Other methods to identify strong gravitational lenses using (sub)millimeter data were independently verified by other teams using Planck and Herschel color criteria (Cañameras et al. 2015). The 24 LPs presented in these analyses include eight systems identified by Cañameras et al. (2015). The use of Planck and WISE data resulted in the discovery of the brightest known, dusty starburst galaxy at z > 1, the “Cosmic Eyebrow” (Díaz-Sánchez et al. 2017; Dannerbauer et al. 2019), which has also been independently recovered as one of the LPs presented in this survey work. Note that LPsJ1329 corresponds to the location on the sky associated with the Cosmic Eyebrow-A lens component (Dannerbauer et al. 2019). Table 1 shows the size of the lensed emission for each of the LPs, in which there are 21/24 with lens sizes ≤10″. Half of the LPs are galaxy–galaxy lenses, while the other half are a mix of cluster or group lensing. The foreground lens galaxies have a negligible contribution to the observed FIR emission of the lensed galaxy (Harrington et al. 2016). The LPs have CO-based spectroscopic redshifts ranging from zCO ∼ 1.1 to 3.6 (Harrington et al. 2016, 2018; Cañameras et al. 2018b; this work). They are comparable or brighter in CO and FIR luminosity than other strongly lensed SPT- (Weiß et al 2013; Strandet et al. 2016, 2017) or Herschel-selected; dusty star-forming galaxies (Harris et al. 2012; Bussmann et al. 2013, 2015; Yang et al. 2017). The Planck and Herschel wavelength selections preferentially target z ∼ 2–3 galaxies, versus the millimeter-selected SPT sources with a median closer to z ∼ 4, although with a wide range between z ∼ 2 and 7 (Weiß et al 2013; Spilker et al. 2016; Strandet et al. 2016; Reuter et al. 2020).


Classical bulges (hereafter ClBs) are the central building blocks in many early-type spiral galaxies. Classical bulges might have formed as a result of major mergers during the early phase of cosmic evolution (Kauffmann et al. 1993; Baugh et al. 1996; Hopkins et al. 2009; Naab et al. 2014), or through a number of other mechanisms such as the monolithic collapse of primordial gas clouds (Eggen et al. 1962), the coalescence of giant clumps in gas-rich primordial galaxies (Noguchi 1999; Immeli et al. 2004; Elmegreen et al. 2008), violent disk instability at high-redshift (Ceverino et al. 2015), multiple minor mergers (Bournaud et al. 2007; Hopkins et al. 2010), and accretion of small companions or satellites (Aguerri et al. 2001). Although most of these studies do not provide quantitative predictions for the bulge kinematics, it is generally believed that ClBs formed through these processes have low rotation compared to the random motion. For example, Naab et al. (2014) showed that spheroids produced by minor and major mergers (which include ClBs) in full cosmological hydrodynamical simulations have a wide range of rotational properties; the massive ones have V/σ less than 0.5. Elmegreen et al. (2008) reported dispersion dominated clump-origin ClBs with upper limit on V/σ ~ 0.4−0.5, where V is the rotation velocity and σ is the central velocity dispersion. A similar study by Inoue & Saitoh (2012) suggests that clump-origin bulges have exponential surface density profiles and rotate rapidly with V/σ ~ 0.9, resembling pseudobulges (Kormendy & Kennicutt 2004). However, using cosmological hydrodynamical simulations with continuous gas accretion, Ceverino et al. (2015) showed that massive classical bulges with non-zero angular momenta are produced at high redshift, but provided no estimate on the bulge V/σ. Overall, there is no clear quantitative picture of the rotational motion induced during the formation of classical bulges in numerical simulations.


Another important consistency check regards the angular size distribution of the sources. Figure 6 shows the cumulative size distributions of the final catalogs combined together, in four flux density bins (yellow solid lines). Such distributions can be considered reliable only down to a flux-dependent minimum intrinsic size (see vertical gray lines), below which most of the sources cannot be reliably deconvolved and they are conventionally assigned Θ = 0. The observed distributions are compared with various realizations of the cumulative distribution function described by Eq. (6), obtained by varying either the function exponent q (left and right columns respectively) or the assumed median size – flux relations (see various black lines).The original function proposed by Windhorst et al. (1990) (Eq. (6) with q = 0.62, see left column) does provide a good approximation of the observed distributions, when assuming the original Θmed − S relation described by Eq. (7), only at flux densities S150 MHz≳10 mJy (see long-dashed lines). This is perhaps not surprising considering that this relation was calibrated at 1.4 GHz down to a few mJy fluxes. At the lowest flux densities (S150 MHz≲1 mJy) we need to assume a steepening of the parameter m (see Eq. (8)), to get a good match with observations (dotted line in the top left panel). This is consistent with what proposed for higher frequency deep surveys (as discussed earlier in this section). At intermediate fluxes (S150 MHz ~ 1−10) mJy, on the other hand, none of the discussed median size – flux relations can reproduce the observed size distribution (see second-row panel on the left). It is interesting to note, however, that if we assume a steeper exponent for the distribution function described by Eq. (7) (i.e., q = 0.80), we get a very good match with observations at all fluxes, when assuming a flux-dependent scaling factor (k = k(S); see Eq. (9)) for the Windhorst et al. (1990) median size – flux relation (black solid lines on the right). The median sizes derived from the T-RECS simulated catalogs (Bonaldi et al. 2019) also provide good results for q = 0.80 (dot-dashed lines on the right), except again at intermediate fluxes (S150 MHz ~ 1−10), where they show strong discrepancies with observations also in Fig. 5. This seems to indicate that the number density of extended radio galaxies in this flux density range is over-estimated in the T-RECS simulated catalogs.


Until recently, only β Pic and 49 Cet were known as CO-bearing debris disks (Vidal-Madjar et al. 1994; Zuckerman et al. 1995; Roberge et al. 2000). Later, more such objects were discovered, and the first statistical studies could be done (Greaves et al. 2016; Lieman-Sifry et al. 2016; Péricaud et al. 2017). Our present list partly incorporates these earlier samples and extends them to a full list of 17 dust-rich debris disks in the solar neighborhood (Section 2). In Figure 3(a), we display the 12CO 2–1 (or 12CO 3–2 for HR 4796) line luminosities (or upper limits), as a function of the fractional luminosity for our sample. Line luminosities of detected CO-bearing disks span almost two orders of magnitude, in which the brightest disks have luminosities that are comparable to those of fainter Herbig Ae and T Tauri disks (Ansdell et al. 2016; Péricaud et al. 2017). Since the sensitivity of the HR 4796 observation is nearly two orders of magnitude worse than that of the other measurements, we discard this object from the following analysis, reducing the size of our statistical sample to 16. For the other objects, by adopting the highest upper limit (source No. 7) in Figure 3(a), we derived a detectability threshold of ∼1.4 ×104 Jy km s−1 pc2 for the 12CO 2–1 line luminosity. With our 3 new discoveries, we found 11 disks in this sample that harbor CO gas, resulting in a very high detection rate of 68.8

−
13.1

+
8.9

%

. Because of the small sample, we computed the uncertainties (corresponding to 68% confidence interval) using the binomial distribution approach proposed by Burgasser et al. (2003). Our result indicates that the presence of CO gas in dust-rich debris disks around young A-type stars is more likely the rule than the exception. The obtained incidence rate of 11/16 is valid above our detectability threshold for the 12CO J = 2–1 line luminosities. Nevertheless, we cannot rule out that all of our targets harbor CO gas at some level. Remarkably, as Figure 3(a) shows, apart from HR 4796, all of the disks with

L

disk

/

L

bol

>
2
×

10

−
3

 contain detectable levels of CO gas.


Sudden outbursts of NH3 simultaneously with H2S detected with the ROSINA-DFMS instrument on the Rosetta S/C point to the presence of abundant ammonium hydrosulphide in or on carbonaceous grains from comet 67P/Churyumov-Gerasimenko. There seems to be a clear distinction between the nucleus ice, where H2S and NH3 exist independently and grains, where they desorb together. S2 is much more abundant on grains compared to water than in the ice of the comet, while S3 is found only in grain impacts. This higher abundance points to radiolysis in these grains, which means they must have been exposed to energetic particles over an extended time. While for operational reasons, S4 could not be measured close to the dust impacts, S4 was clearly identified in periods where the coma was very dusty (Calmonte et al. 2016). Longer sulphur chains very likely are refractory, not sublimating at temperatures reached in the instrument or on grains in the coma. While Sn can also be formed from pure H2S ice by photo processing (Cazaux et al. 2022), the fact that S3 is clearly related to dust and is not found in the normal nucleus ice, where H2S is quite abundant, indicates that S3 is a product of radiolysis of the ammonium salt. In addition, photo processing of H2S results not only in Sn, but also in H2S2 (Cazaux et al. 2022), a species not detected in the DFMS m/z 66 and m/z 65 (HS2) spectra. This exposure rules out a contemporary formation of the salt on the surface or in the interior of the comet or a formation of the salt in the mid-plane of the protoplanetary disc, while the comet accreted. A pre-stellar formation is therefore likely. The salt is semivolatile, less volatile than water and could probably have survived quite high temperatures. It seems that on these grains, acids and ammonia are all locked in salts, be it sulphur, halogens, or carbon bearing acids like HOCN. If indeed, a relatively large part of sulphur and nitrogen is therefore in a semivolatile state in these grains, then the depletion of nitrogen in comets and of sulphur in star-forming regions could probably be explained, primarily because salts escape detection unless they experience temperatures above water sublimation. With the JWST S/C in orbit, there is hopefully the possibility to detect salts, or at least several of the acids in ices, which are supposed to be part of ammonium salt, like HOCN, H2CO, and formamide while looking for ammonium salts in star-forming regions and possibly comets.


The orbital decay of BSBHs may slow down or stall at ∼pc scales (e.g. Begelman et al. 1980; Milosavljević & Merritt 2001; Zier & Biermann 2001; Yu 2002; Vasiliev, Antonini & Merritt 2014; Dvorkin & Barausse 2017; Tamburello et al. 2017), or the barrier may be overcome in gaseous environments (e.g. Gould & Rix 2000; Escala et al. 2004; Hayasaki, Mineshige & Sudou 2007; Hayasaki 2009; Cuadra et al. 2009; Lodato et al. 2009; Chapon, Mayer & Teyssier 2013; Rafikov 2013; del Valle et al. 2015), in triaxial or axisymmetric galaxies (e.g. Yu 2002; Berczik et al. 2006; Preto et al. 2011; Khan et al. 2013, 2016; Vasiliev, Antonini & Merritt 2015; Gualandris et al. 2017; Kelley, Blecha & Hernquist 2017a), and/or by interacting with a third SMBH in hierarchical mergers (e.g. Valtonen 1996; Blaes, Lee & Socrates 2002; Hoffman & Loeb 2007; Kulkarni & Loeb 2012; Tanikawa & Umemura 2014; Bonetti et al. 2018). The accretion of gas and the dynamical evolution of BSBHs are likely to be coupled (Ivanov, Papaloizou & Polnarev 1999; Armitage & Natarajan 2002; Haiman, Kocsis & Menou 2009; Bode et al. 2010, 2012; Farris, Liu & Shapiro 2010, 2011; Kocsis, Haiman & Loeb 2012; Shi et al. 2012; D’Orazio, Haiman & MacFadyen 2013; Shapiro 2013; Farris et al. 2014, 2015) such that the occurrence rate of BSBHs depends on the initial conditions and gaseous environments at earlier phases (e.g. thermodynamics of the host galaxy interstellar medium; Dotti et al. 2007, 2009; Dotti, Sesana & Decarli 2012; Fiacconi et al. 2013; Mayer 2013; Tremmel et al. 2018). Quantifying the occurrence rate of BSBHs at various merger phases is therefore important for understanding the associated gas and stellar dynamical processes. This is a challenging problem for three main reasons. First, BSBHs are expected to be rare (e.g. Foreman, Volonteri & Dotti 2009; Volonteri, Miller & Dotti 2009), and only a fraction of them accrete enough gas to be ‘seen’. Secondly, the physical separations of BSBHs that are gravitationally bound to each other (≲a few pc) are too small for direct imaging. Even VLBI cannot resolve BSBHs except for in the local universe (Burke-Spolaor 2011). CSO 0402+379 (discovered by VLBI as a double flat-spectrum radio source separated by 7 pc) remains the only secure case known (Rodriguez et al. 2006; Bansal et al. 2017, see Kharb, Lal & Merritt 2017; however, for a possible 0.35-pc BSBH candidate in NGC 7674). Thirdly, various astrophysical processes complicate their identification such as bright hot spots in radio jets (e.g. Wrobel, Walker & Fu 2014b). Until recently, only a handful cases of dual active galactic nuclei (AGNs) – galactic-scale progenitors of BSBHs – were known (Owen et al. 1985; Junkkarinen et al. 2001; Komossa et al. 2003; Ballo et al. 2004; Hudson et al. 2006; Max, Canalizo & de Vries 2007; Bianchi et al. 2008; Guidetti et al. 2008). While great strides have been made in identifying dual AGNs at kpc scales (e.g. Gerke et al. 2007; Comerford et al. 2009, 2012, 2015; Green et al. 2010; Liu et al. 2010, 2013, 2018; Fabbiano et al. 2011; Fu et al. 2011, 2012, 2015a,b; Koss et al. 2011, 2012, 2016; Rosario et al. 2011; Teng et al. 2012; Woo et al. 2014; Wrobel, Comerford & Middelberg 2014a; McGurk et al. 2015; Müller-Sánchez et al. 2015; Shangguan et al. 2016; Ellison et al. 2017; Satyapal et al. 2017), there is no confirmed BSBH at sub-pc scales (for recent reviews, see e.g. Popović 2012; Burke-Spolaor 2013; Bogdanović 2015; Komossa & Zensus 2016).


For the clustering measurements, we use the sample of galaxies compiled in Data Release 11 (DR11) of the SDSS-III project. The SDSS-III is a spectroscopic investigation of galaxies and quasars selected from the imaging data obtained by the SDSS (York et al. 2000) I/II covering about 11,000 deg2 (Abazajian et al. 2009) using the dedicated 2.5 m SDSS Telescope (Gunn et al. 2006). The imaging employed a drift-scan mosaic CCD camera (Gunn et al. 1998) with five photometric bands (

 and z; Fukugita et al. 1996; Smith et al. 2002; Doi et al. 2010). The SDSS-III (Eisenstein et al. 2011) BOSS project (Ahn et al. 2012; Dawson et al. 2013) obtained additional imaging data of about 3000 deg2 (Aihara et al. 2011). The imaging data was processed by a series of pipelines (Lupton et al. 2001; Pier et al. 2003; Padmanabhan et al. 2008) and corrected for Galactic extinction (Schlegel et al. 1998) to obtain a reliable photometric catalog. This catalog was used as an input to select targets for spectroscopy (Dawson et al. 2013) for conducting the BOSS survey (Ahn et al. 2012) with the SDSS spectrographs (Smee et al. 2013). Targets are assigned to tiles of diameter 3° using an adaptive tiling algorithm designed to maximize the number of targets that can be successfully observed (Blanton et al. 2003). The resulting data were processed by an automated pipeline which performs spectral classification, redshift determination, and various parameter measurements, e.g., the stellar-mass measurements from a number of different stellar population synthesis codes which utilize the photometry and redshifts of the individual galaxies (Bolton et al. 2012). In addition to the galaxies targeted by the BOSS project, we also use galaxies that pass the target selection but have already been observed as part of the SDSS-I/II project (legacy galaxies). These legacy galaxies are subsampled in each sector so that they obey the same completeness as that of the CMASS sample (Anderson et al. 2014).


Collisions with charged particles, namely with electrons and heavier ions are not considered in the radiative-transfer calculations presented in this paper, but may play an important role in the excitation of the CH Λ-doublet, particularlyin regions with high electron fractions,

$x_{\textrm{e}} = n_{\textrm{e}}/(n_{\textrm{H}}  2n_{\textrm{H}_{2}}) >10^{-5}$xe=ne/(nH+2nH2)>10−5
–10−4. Such high electron fractions are ubiquitous in the diffuse molecular clouds present along the lines of sight studied here and may even be prevalent in the clouds surrounding the observed HII regions themselves, making electrons an important collision partner at low gas temperatures T ≤ 100 K. Bouloy & Omont (1977, 1979) studied the impact of collisional excitation by electrons on Λ-doublet transitions with particular emphasis on the ground-state Λ-doublet transitions of OH. These authors compute the collisional rate coefficients for collisions with electrons, either using perturbation methods such as those used by Goss & Field (1968) or using the Born approximation, both of which yield comparable results. These authors concluded that collisions with electrons, while incapable of inducing level inversion in the ground-state lines of OH at 18 cm, are responsible for thermalising them. Bouloy et al. (1984) further studied the excitation conditions of the ground-state lines of CH, and model the excitation by considering the radiative and collisional (de-) excitation of CH with H, H2, and electrons. Their results once again point to the role played by the collisions with electrons in thermalising the CH lines rather than inverting them. However, the excitation temperature of OH is found to be a 1–2 K above TCMB as derived from the resolved optical spectra of the OH A − X band transitions (Felenbok & Roueff 1996) or when measured by comparing the emission and absorption profiles of the radio L-band transitions of OH (Liszt & Lucas 1996, and references therein). This implies that densities much higher than the critical density are needed for thermalisation, which might also be the case for CH. More recently, Goldsmith & Kauffmann (2017) examined the impact of electron excitation on high-dipole-moment molecules like HCN, HCO+, CS, and CN in various interstellar environments. As long-range forces dominate the collisional cross-sections for electron excitation, the cross-sections and, in turn, the collisional rate coefficients scale with the square of the electric dipole moment, μe. Hence, the electron collisional rate coefficients for CH with μe = 1.46 D (Phelps & Dalby 1966) are ≃ 25 % of those of HCN with μe = 2.985 D (Ebenstein & Muenter 1984). Scaling the value of the HCN–e− collisional rate coefficient at T ≤ 100 K from Faure et al. (2007), we find the CH–e− collisional rates to be of the order of ~ 2 × 10−5 cm3 s−1. From this we can compute the critical electron fractional abundance, x* (e−), which defines the fractional abundance of electrons required for the collisional rate coefficients with electrons to be the same as that with H2 such that x* (e−) = ncrit(e−)∕ncrit(H2). Under the validity of these assumptions, x*(e−) for CH is ~10−6, making CH likely to be affected by electron excitation. Therefore, a complete treatment of the radiative–collisional (de-)excitation of the CH ground state would still require the availability of accurate collisional rate coefficients for collisional excitation by electrons. However, to our knowledge, these are currently not available.


Given the number densities within the mass-dissociation index plane of Fig. 8, we now ask ourselves whether known dissociated clusters, such as the Bullet cluster, are expected in L210N1024NR? The Bullet Cluster has a mass of $\sim 1.5 \times 10^{15} \, {\rm M}_{\odot }$ (e.g. Clowe et al. 2004; Bradač et al. 2006; Clowe et al. 2006) and we estimated a dissociation index of SBullet ∼ 0.335 ± 0.06. As seen in Fig. 8 there are no Bullet cluster analogues (structures of approximate mass and dissociation) in L210N1024NR, this is unsurprising as a simulation requires a significantly larger volume than that of L210N1024NR ((210cMpc h−1)3) to expect such an object (e.g. Lee & Komatsu 2010; Thompson & Nagamine 2012; Bouillot et al. 2015; Kraljic & Sarkar 2015; Thompson et al. 2015). From the distribution presented in Fig. 8, it is trivial to estimate the required cosmological volume (the effective volume, Veff) to expect structures of a given mass and dissociation index. By separating the 2D distribution on the mass-dissociation index planes into the component 1D distributions of mass and dissociation the effective volume is computed as
(12)$$\begin{eqnarray}
V_\text{eff}~^{-1} &=&\int \int \,{\rm{ d}} S \, {\rm{ d}} M \phi (S, M) \\
&=& \int _{S_\text{a}}^{S_\text{b}} \, {\rm{ d}} S \phi _S(S) \int _{M_\text{a}}^{M_\text{b}} \, {\rm{ d}} M \phi _M(M)~,
\end{eqnarray}$$where ϕS(S) is the number density function associated with S and $\phi _\mathit {M}(\mathit {M})$ is the mass function presented in Fig. 7. Assuming a probable range of S = 0.335 ± 0.06 and $1 \lt M \lt 2 \times 10^{15} \, {\rm M}_{\odot }$ we estimate a number density ∼4.92 × 10−10 Mpc−3 or that an effective volume of ∼2.03 Gpc3 would be required to observe a single Bullet-like cluster. This result is inline with the number density estimate of the order of ∼10−10 Mpc−3 by Thompson et al. (2015), which improves on previous estimates (e.g. Lee & Komatsu 2010; Thompson & Nagamine 2012; Bouillot et al. 2015) due to more sophisticated halo finding methods (e.g. Behroozi, Wechsler & Wu 2013). Conversely, it was estimated by Kraljic & Sarkar (2015) (utilizing the same halo finder as Thompson et al. 2015) that given an effective volume of ∼14.6 Gpc3, no Bullet cluster analogues are expected, however as indicated by a pairwise velocity distribution it would be expected that present binary halo–halo orbits have the potential to form a Bullet-like object.


In this scenario, the present γ-ray emission from the Fermi bubbles arises predominantly through inverse Compton scattering of an energetic non-thermal cosmic ray (CR) electron population in the remnant structures with ambient radiation supplied by the interstellar radiation field (ISRF) and the cosmological microwave background (CMB). A subdominant component due to non-thermal bremsstrahlung may also be present, emitted primarily from regions of high gas and non-thermal electron density within the bubble (OY22). Models of this nature are broadly referred to as ‘leptonic’ models (see also Su et al. 2010; Zubovas, King & Nayakshin 2011; Su & Finkbeiner 2012; Fujita, Ohira & Yamazaki 2013 for similar configurations). Other similar approaches invoking Sgr A* activity, but where the CR composition is not specifically required to be leptonic, invoke a pair of jet-driven outflowing bubbles assuming constant AGN activity or continuous energy injection over Myr time-scales (e.g. Zhang & Guo 2020). Notably, these models have been able to account for the bi-conical X-ray structures observed near the GC as part of the same phenomenon as the Fermi bubbles. Alternative proposals have also been discussed, where the bubbles arise from the confluence of a number of processes operating more gradually within the inner part of the Milky Way (Thoudam 2013). These could include tidal disruption events (TDEs) occurring at regular intervals of 10s to 100s kyr (Cheng et al. 2011; Ko et al. 2020), or the action of a bipolar galactic outflow driven by the ongoing intense GC star-formation activity and/or the processes associated with Sgr A* (Lacki 2014), with the resulting γ-ray glow instead arising from a hadronic CR population interacting with an advected supply of entrained gas in the wind (the ‘hadronic’ models – see  Crocker & Aharonian 2011; Cheng et al. 2014, 2015; Crocker et al. 2014, 2015; Mou et al. 2014, 2015; Razzaque & Yang 2018).