For the first time, we detected [C ii] 158-μm emission from a GRB host galaxy at z > 2. This is the second detection of [C ii] 158-μm emission among known GRB host galaxies, following GRB 980425 (Michałowski et al. 2016). The [C ii] 158-μm fine structure line is the dominant cooling line of the cool interstellar medium, arising from photodissociation regions (PDR) on molecular cloud surfaces. It is one of the brightest emission lines from star-forming galaxies from FIR to metre wavelengths, almost unaffected by dust extinction. [C ii] 158-μm luminosity, L[C II], has been discussed as an indicator of SFR (e.g. Stacey et al. 2010). If L[C II] scales linearly with SFR, the ratio to FIR luminosity, L[C II]/LFIR, is expected to be constant, since LFIR is a linear function of SFR (e.g. Kennicutt 1998a). However, LC II/LFIR is not constant, but declines with increasing LFIR, known as the ‘[C ii] deficit’ (e.g. Luhman et al. 1998, 2003; Malhotra et al. 2001; Sargsyan et al. 2012; Díaz-Santos et al. 2013, 2017; Spilker et al. 2016). The [C ii] deficit persists when including high-z galaxies (e.g. Stacey et al. 2010; Wang et al. 2013; Rawle et al. 2014). In Fig. 5, we compare the [C ii] deficit in the GRB 080207 host and other star-forming galaxies. Two GRB hosts are shown by stars: GRB 080207 (orange star) and 980425 (blue star). The comparison sample is compiled from the literature up to z ∼ 3 (Malhotra et al. 2001; Cormier et al. 2010, 2014; Ivison et al. 2010; Stacey et al. 2010; Sargsyan et al. 2012; Farrah et al. 2013; Magdis et al. 2014; Brisbin et al. 2015; Gullberg et al. 2015; Schaerer et al. 2015). Active galactic nuclei are separated from star-forming galaxies based on either (i) the explicit description in the literature or (ii) EWPAH 6.2μm  0.1 (Sargsyan et al. 2012). As reported by previous studies (e.g. Maiolino et al. 2009; Stacey et al. 2010), high-z galaxies are located at a different place from local galaxies in the L[C II]/LFIR–LFIR plane.


Jiang et al. (1975) estimated the infrared band strength of CO ice deposited at high pressure compared to modern setups: their deposition rate of 0.5 to 2 μm requires a pressure about 16 to 66 times higher than typical experiments performed at 1 × 10−6 mbar during deposition. To obtain this band strength, Jiang et al. (1975) also adopted a literature value of the CO ice density measured at a relatively high temperature, 30 K (Vegard 1930). According to Luna et al. (2022) and this work, the CO ice density depends on the temperature and pressure during deposition, and therefore, the value of the CO infrared band strength in Jiang et al. (1975) needs a revision. We propose to use $\mathcal {A}(\rm CO)$  = (8.7 ± 0.5) × 10−18 cm  molecule−1 for future column density estimations. This value is valid in the experimental range from 11 to 28 K deposition temperatures investigated in this work, for which no variations of the integrated absorbance area and sticking probability were found (Cazaux et al. 2017). The CO ice column density values reported in previous experimental and observational papers might thus be underestimated, they would be about 23  per cent lower than the actual value. Most of the CO ice column densities reported in the literature adopted a band strength of $\mathcal {A}(\rm CO)$  = 1.1 × 10−17 cm  molecule−1 (Jiang et al. 1975), they would need to be multiplied by a factor of 1.3 for correction. An example is the number of monolayers on the surface, or just beneath the surface of the ice, involved in the photodesorption of CO, i.e. N  = 5 × 1015 molecules cm−2 or about 5 monolayers (ML) where 1 ML  = 1 × 1015 molecules cm−2 (Muñoz Caro et al. 2010; Fayolle et al. 2011; Chen et al. 2014). After correction, this becomes 6.5 ML. Considering our average density value of CO ice deposited at 11 K, 0.837 g cm−3, and equation (24), it is found that the thickness of one monolayer is 0.56 nm, and 3.36 nm for the top 6.5 ML of ice. The UV photons emitted by the MDHL that are absorbed deeper than 3.36 nm would not lead us to photodesorption of CO molecules.


SFR and

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

 can be derived from observations, but the molecular gas mass, MH2 = αCO ×

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

, is the key quantity of interest. The observed SFR/

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

 ratio is related to both the depletion time (τdep) and the CO-to-H2 conversion factor (αCO) as SFR/

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

 = SFR/(MH2/αCO) = αCO/τdep. Many studies have shown that αCO increases rapidly at low metallicity theoretically and observationally (Israel 1997; Glover & Mac Low 2011; Leroy et al. 2011; Narayanan et al. 2012; Elmegreen et al. 2013; Shi et al. 2016). If τdep is constant, as is found for the nearby disc galaxies (Leroy et al. 2008; Bigiel et al. 2011), then the dependence of the SFR/

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

 ratio on metallicity is the direct consequence of the variation of αCO. However, τdep in dwarf galaxies remains uncertain and tends to be shorter at a lower mass and at a higher specific SFR (sSFR; Saintonge et al. 2011; Shi et al. 2014; Hunt et al. 2020). IZw18, as a dwarf galaxy with a relatively high sSFR and low mass, is likely to have low τdep. We speculates that as αCO = τdep ×

SFR

L

CO
(
1
−
0
)

′

$ \frac{\mathrm{SFR}}{L^\prime_{\mathrm{CO(1{-}0)}}} $

, the high SFR/

L

CO(1-0)
′

$ L^\prime_{\rm CO(1{-}0)} $

 ratio of IZw18 would overwhelm the possibly low τdep, and this would result in a high αCO. Moreover, considering the dependence of αCO on both sSFR and metallicity for the metal poor galaxies, IZw18 follows the empirical relation found in Hunt et al. (2020) within the uncertainty of the stellar mass, as shown in the bottom right corner of Fig. 5. This relation was derived based on a recent compilation of ∼400 metal poor galaxies (Ginolfi et al. 2020, MAGMA). We note that the individual star-forming regions of the metal-poor galaxies from Shi et al. (2015, 2016) all fall slightly above the relation, but well within the scatter of the global measurements of the galaxies. This indicates that αCO changes continuously with metallicity and sSFR. A further constraint on the conversion factor (αCO) of IZw18 is beyond the scope of this paper.


The behaviour of the radio emission from the early-type magnetic stars has been definitively quantified by the empirical scaling relationship between the radio spectral luminosity and the magnetic flux rate: $L_{\nu ,\mathrm{rad}} {\propto } (\Phi / P_{\mathrm{rot}})^2 = {B_{\mathrm{p}}^2 R_{\ast }^{4}} / {P_{\mathrm{rot}}^2}$, found by Leto et al. (2021) and confirmed by Shultz et al. (2022). This empirical relationship is the consequence of the physical mechanism supporting the non-thermal acceleration able to produce the electrons responsible for the magnetospheric radio emission of early-type magnetic stars. It was recently demonstrated by Owocki et al. (2022) that the power provided by centrifugal breakout events, continuously occurring within the stellar magnetosphere, is directly related to the power of the radio emission. From this point of view, the CBOs are non-random events. Breakouts occur continuously in a well-constrained magnetospheric region, where the resulting reconnection of the magnetic fields drives the acceleration of the local electrons. The CBO luminosity is related to the stellar parameters by the relation (Owocki et al. 2022):
$$\begin{eqnarray*}
L_{\mathrm{CBO}} = \frac{B_{\mathrm{p}}^2 R_{\ast }^3}{P_{\mathrm{rot}}} \times W {\mathrm{~~(erg \, s^{-1}),}}
\end{eqnarray*}$$with Bp in gauss, R* in centimetres, and Prot in seconds. In the above relation $W=2 \pi R_{\ast } / P_{\mathrm{rot}} \sqrt{G M_{\ast }/R_{\ast }}$ is the dimensionless critical rotation parameter calculated as the ratio between the equatorial stellar velocity and the corresponding orbital velocity, defined by the gravitational law at the stellar equator (G = 6.67408 × 10−8 cm3 g−1 s−2 is the gravitational constant). Once the explicit relation of W is substituted within the LCBO definition, we find that $L_{\mathrm{CBO}} \propto {B_{\mathrm{p}}^2 R_{\ast }^{4.5} M_{\ast }^{-0.5}} / {P_{\mathrm{rot}}^2} \propto L_{\nu ,\mathrm{rad}}$, which in practice is almost the same relation empirically found, except for the term related to the mass, that in any case has negligible effect. The maximum possible value of W is W = 1, which, due to magnetic braking, would have to be at the beginning of the star’s life; however no magnetic star has ever been found with W⪆0.5 (Shultz et al. 2019b). In fact, this parameter progressively decreases as the star loses angular momentum throughout its life (Keszthelyi et al. 2019, 2020).


The presence of neutral lines in the IRIS spectra allow us an accurate absolute wavelength calibration, and therefore a more accurate determination of line Doppler shifts, as compared to some other spectral observations; e.g., with EIS. For the post-flare loops, we find that the Fe xii emission is largely unshifted at the loop tops and predominantly redshifted at both loop footpoints, as expected for the draining of plasma in the later phases of the flare, when the heating has ceased. For the moss observations, we find that the distribution of Doppler shifts is peaked at a redshift of a few km s−1, but with significant wings on both the red and the blue sides. In our interpretation, guided by the Bifrost 3D models, Fe xii appears redshifted for hotter coronal temperatures (≳4 MK), while blueshifted when the corona is at cooler temperatures. The distribution of Doppler shift we observe shows a wide range of blue- and redshifts, and it suggests that there may be a continuous mix of heating and cooling in the moss we observed. For cooler TR lines (

[K]

), the dominance of redshift is well established (e.g., Doschek et al. 1976; Dere et al. 1984; Achour et al. 1995; Chae et al. 1998b; Peter & Judge 1999). The physical processes causing the observed redshifts are still debated, and several models have been proposed (e.g., Hansteen 1993; Patsourakos & Klimchuk 2006; Peter et al. 2006; Hansteen et al. 2010). Hansteen et al. (2010) use 3D radiative MHD models of the solar atmosphere to investigate the TR redshifts, and find that in their model the heating is dominated by rapid intermittent events at low heights which heat the plasma locally to coronal temperature and produces downflows at TR temperature due to the local overpressure. The model analyzed by Hansteen et al. (2010) reached modest coronal temperatures (≲2.5 MK), therefore with significant Fe xii emission in loop structures rather than at the loop footpoints, as in the case for moss. Here we used a different Bifrost 3D MHD simulation, which reaches higher coronal temperatures and represents a better model for moss observations. We find that the moss Fe xii (mostly) redshifted emission we observe with IRIS is reproduced in the model when the corona gets hot enough and most of the Fe xii is confined in the TR loops. The Fe xii then behaves like a TR line, and is redshifted, in the model, as a consequence of the local heating at low heights which causes upflows in the hotter lines and downflows in the cooler (TR) lines.


After crossing the bow shock, neutral atoms can interact with the shocked protons. The interaction length is given by

(2)

\begin{equation}
\lambda = \frac{V_{\rm NS}}{X_{\rm ion} n_{\rm ISM} r_{\rm c} \langle \sigma (v_{\rm rel}) v_{\rm rel} \rangle },
\end{equation}

where Xion is the ionization fraction of the ISM, rc is the compression ratio of the bow shock, σ(vrel) is the relevant cross-section of the process under consideration and 〈σvrel〉 is the collision rate averaged over the ion distribution function. When the ion distribution is a Maxwellian, 〈σvrel〉 is well approximated (within 20 per cent) by the expression (Zank et al. 1996; Blasi et al. 2012)

(3)

\begin{equation}
\langle \sigma v_{{\rm rel}} \rangle \approx \sigma (U^*) U^*,
\end{equation}

where

(4)

\begin{equation}
U^*= \sqrt{\frac{8}{\pi } \frac{2 k_{\rm B} T}{m_{\rm p}} }
\end{equation}

is the average, relative speed between the incoming hydrogen atom and ions (T is the temperature of the shocked ISM determined by the Rankine–Hugoniot jump conditions, assumed to be ≫TISM). Using the fiducial values nISM = 0.1 cm−3 and VNS = 300 km s−1, together with an ionization fraction of 90 per cent and rc = 4 (the typical value for strong shocks), leads to the following estimates for the mean free paths:

(5)

\begin{eqnarray}
\lambda _{{\rm ion},{\rm p}} & \approx 3.0 \times 10^{20} \, \rm cm,
\end{eqnarray}

(6)

\begin{eqnarray}
\lambda _{{\rm ion},{\rm e}} & \approx 2.2 \times 10^{16} \, \rm cm,
\end{eqnarray}

(7)

\begin{eqnarray}
\lambda _{\rm CE} & \approx 1.5 \times 10^{15} \, \rm cm
\end{eqnarray}

for the ionization due to collisions with protons, electrons and CE, respectively. Note that λion, e has been calculated under the assumption that the electrons downstream of the shock equilibrate rapidly with protons, thereby acquiring the same temperature. If this assumption does not hold, the collisional length scale for ionization due to electrons can become much larger than the value reported in equation (6). In addition, the values (5)–(7) are to be taken as lower limits as they are valid just ahead of the nebula, where the compression ratio and the temperature obtain their maximum values.


The driving mechanism and physical nature of large-scale EUV waves have been hotly debated over the past 20 years (Chen 2017). Because chromospheric Moreton waves were believed to be driven by the flare pressure pulses for a long time after Uchida (1968), solar physicists naturally thought at the beginning that EUV waves also have the same driving mechanism (e.g., Khan & Aurass 2002; Hudson et al. 2003; Warmuth et al. 2004). However, more and more recent high-resolution observational studies have shown that EUV waves are in fact bow shocks driven by CMEs instead of flare pressure pulses (e.g., Chen 2006; Kienreich et al. 2009; Patsourakos & Vourlidas 2009; Patsourakos et al. 2010; Liu et al. 2011b; Shen & Liu 2012a, 2012c; Xue et al. 2013; Long et al. 2017a; Shen et al. 2017a; Cunha-Silva et al. 2018). Further investigations indicated that EUV waves can also be excited by other kinds of slightly different processes, such as newly formed expanding loops, coronal jets, and mini-filament eruptions on small scales (e.g., Podladchikova et al. 2010; Zheng et al. 2012, 2013; Su et al. 2015; Shen et al. 2017a), and the sudden expansion of transequatorial loops driven by coronal jets on large scales (Shen et al. 2018e, 2018d). In these scenarios, those authors found that such EUV waves often have a shorter lifetime (a few minutes) compared to those driven by CMEs (tens of minutes). Moreover, Shen et al. (2018a) reported an interesting non-CME-associated homologous EUV wave event in which the initial speeds of the waves are about 1000 km s−1, and the authors proposed that the observed EUV waves were large-amplitude nonlinear fast-mode magnetosonic waves or shocks directly driven by the associated recurrent coronal jets, resembling the generation of a piston shock in a tube. These new observations suggest that EUV waves can be excited by different kinds of coronal disturbances, e.g., CMEs, coronal jets, and mini-filament eruptions.


The HD 189733 system offers the possibility to study these various interactions (Table 1). It is a binary system with a K2V dwarf (HD 189733 A, hereafter HD 189733) and a M4V dwarf (HD 189733 B; Bakos et al. 2006) at a mean separation of ∼220 au. The bright primary (V = 7.7) hosts a transiting hot Jupiter at 0.03 au (Bouchy et al. 2005), whose strong irradiation and large occultation area make it particularly favourable for atmospheric characterization (see Pino et al. 2018 and references inside for observations of the lower atmospheric layers). Recent transit observations in the near-infrared (Salz et al. 2018) revealed absorption by helium in an extended but compact thermosphere. Transit observations in the far-ultraviolet (FUV) previously revealed absorption by a dense and hot layer of neutral oxygen at higher altitudes (Ben-Jaffel & Ballester 2013). The close distance of HD 189733 to the Sun (19.8 pc) enables observations of the stellar Lyman α line with the Hubble Space Telescope (HST). Atmospheric escape of neutral hydrogen was first detected in the unresolved line with the HST Advanced Camera for Surveys (ACS) in two out of three epochs (Lecavelier des Etangs et al. 2010), and in the line resolved with the HST Space Telescope Imaging Spectrograph (STIS) in one out of two epochs (Lecavelier des Etangs et al. 2012; Bourrier et al. 2013). These observations provided the first indication of temporal variations in the physical conditions of an evaporating planetary atmosphere. Bourrier & Lecavelier des Etangs (2013) attributed the high-velocity, blueshifted absorption signature detected by Lecavelier des Etangs et al. (2012) to intense charge-exchange between the planet exosphere and the stellar wind, proposing that an X-ray flare observed before the transit increased the atmospheric mass-loss and/or increased the density of the stellar wind. The latter scenario is favoured by thermal escape simulations from Chadney et al. (2017), who found that the energy input from a flare would not increase sufficiently the mass-loss. Interestingly the tentative detection of absorption by ionized carbon (Ben-Jaffel & Ballester 2013) and excited hydrogen (Jensen et al. 2012; Cauley et al. 2015, 2016; Cauley, Redfield & Jensen 2017; Kohl et al. 2018) before and during the transit of HD 189733b could be explained by the interaction of the stellar wind with the planetary magnetosphere or escaping material. The observed temporal variability in the atmospheric escape of neutral and excited hydrogen is likely linked to the high-level of activity from the host star, which results in a fast-changing radiation, particle, and magnetic environment for the planet. Enhanced activity in the stellar chromosphere and transition region have also been observed after the planetary eclipse, and attributed to signatures of magnetic star–planet interactions (Pillitteri et al. 2010, 2011, 2014, 2015). Evidence for modulation in the Ca ii lines at the orbital period of the planet, during an epoch of strong stellar magnetic field, further supports this scenario (Cauley et al. 2018). The detectability of star–planet interactions in the HD 189733 system however remain uncertain, and intrinsic stellar variability and inadequate sampling has been proposed to explain the observed variations (Route 2019). These results show the need to study contemporaneously and in different epochs the upper atmosphere of HD 189733b and its high-energy environment.


Due to recent major advances in the observational studies of active galactic nuclei (AGN), evidence is growing that massive black holes (MBHs) heavier than $10^5\, \rm {M_\odot }$ form in nature and power AGN activity at the centres of galaxies through gas accretion (Schmidt 1963; Hopkins, Richards & Hernquist 2007; Merloni & Heinz 2008; Ueda et al. 2014; Aird et al. 2015). The demographic study of AGN and the dynamics of stars and gas around the centre of nearby galaxies, further provided evidence that most (if not all) massive galaxies in the Universe host MBHs in their nuclei (Genzel & Townes 1987; Dressler & Richstone 1988; Kormendy 1988; Kormendy & Richstone 1992; Genzel, Hollenbach & Townes 1994; Salucci et al. 1999; Peterson et al. 2004; Vestergaard & Peterson 2006). Even more, the existing correlations between the mass of MBHs and key properties of their host galaxies hint for their co-evolution (Haehnelt & Rees 1993; Faber 1999; O’Dowd, Urry & Scarpa 2002; Häring & Rix 2004; Kormendy & Ho 2013; Savorgnan et al. 2016). Even though these findings sharpened our knowledge on the role of MBHs in the formation and evolution of galaxies, there is a need to contextualize galaxies and MBHs within the broad cosmological context. It is commonly accepted that the Universe behaves in a hierarchical way. Cosmic structure formed through the hierarchical assembly of dark matter (DM) haloes, and the galaxies observed nowadays assembled through mergers with smaller companions and accretion of matter from the cosmic filaments (White & Rees 1978; White & Frenk 1991; Haehnelt & Rees 1993; Kauffmann et al. 1999; Guo et al. 2011; Vogelsberger et al. 2014a,b; Schaye et al. 2015; Nelson et al. 2018; Pillepich et al. 2018). Consequently, the existence of MBHs at the centre of galaxies and the main role of mergers in the Universe, hint for the existence of massive black hole binary systems (MBHBs) that might have formed and coalesced throughout the whole Universe lifetime.

Kormendy 1988
O’Dowd, Urry & Scarpa 2002


The two components in V593 Cen are early-type stars that contain the convective core and the radiative envelope. This suggests that the period oscillation cannot be explained by the magnetic activity cycle mechanism, which is usually adopted to explain the cycle period change of later-type binary stars (e.g., Applegate 1992; Lanza et al. 1998). Therefore, the light-travel-time effect via the presence of a tertiary body is adopted to explain the cyclic change of the orbital period (e.g., Irwin 1952; Chambliss 1992; Borkovits & Hegedüs 1996; Liao & Qian 2010). Using this method, tertiary bodies have been detected orbiting massive close binary stars. Some examples are V701 Sco (Qian et al. 2006), V382 Cyg, TU Mus (Qian et al. 2007), and AI Cru (Zhao et al. 2010). Though it seems that binaries favor a more uniform distribution of eccentricities (Tokovinin 1997, 2008; Raghavan et al. 2010; Moe & Di Stefano 2017), we still assume that the tertiary wanders in the circular orbit. This is because the period change (in Figure 2) displays a periodic oscillation, and this oscillation is fitted either by a sine curve without eccentricities, or by using the value of the outer orbital period and the inner binary period (Pouter and Pinner). The period ratio for this multiple system is about Pouter/Pinner = 2.5 × 104. For this kind of multiple star system with a very large period ratio, the outer orbit usually has very low or no eccentricities (Shatsky 2001; Tokovinin 2004, 2008). Moreover, in large Pouter/Pinner ratio multiple system (commonly found in triple systems) the Kozai effect becomes too weak (Tokovinin 2008). For brevity, considering that the third body in the V593 Cen system is moving in a circular orbit, the projected radius of the orbit of the eclipsing pair rotating around the mass central of the triple system, the mass function, and the masses of the third component could be computed with these equations as follows:
5

6

where A3 is the amplitude of the O − C oscillation, c is the light speed, P3 is the period of the O − C oscillation, and i′ is the inclination of the tertiary orbital plane. Finally, when the inclination i′ = 90°, it is calculated that the lowest mass of the tertiary companion is 4.3 (0.3) M⊙ and the separation between the binary and the tertiary companion is about 25.5(2.2) au. For the other inclinations, the third body should have a larger mass and a shorter separation. When the inclination i′ = 826, which shows that the third body is in the same plane with the inner binary, its mass is calculated to be 4.7 (0.4) M⊙ and the separation is 25.2(2.8) au.


The binary characteristics of early-type magnetic stars may provide crucial clues, allowing one to test alternative fossil field hypotheses. The non-magnetic chemically peculiar stars of Am (A-type stars with enhanced lines of Fe-peak elements) and HgMn (late-B stars identified by strong lines of Hg and/or Mn) types are frequently found in close binaries (Gerbaldi, Floquet & Hauck 1985; Ryabchikova 1998; Carquillat & Prieur 2007), including eclipsing systems (Nordstrom & Johansen 1994; Strassmeier et al. 2017; Takeda et al. 2019). In contrast, only about ten close (Porb  20 d) spectroscopic binaries containing at least one magnetic ApBp star are known (Landstreet et al. 2017). The overall incidence rate of magnetic upper main sequence stars in close binaries is less than 2 per cent (Alecian et al. 2015), although this fraction is significantly higher if one includes wide long-period systems (Mathys 2017). This low incidence of magnetic ApBp stars in close binaries is frequently considered as an argument in favour of the stellar merger origin of fossil fields (de Mink et al. 2014; Schneider et al. 2016). In this context, confirmation of magnetic ApBp stars in short-period binary systems gives support to alternative theories or, at least, demonstrates that early-type stars may acquire magnetic fields through different channels. In addition, detached close binary stars, particularly those showing eclipses, are valuable astrophysical laboratories that provide model-independent stellar parameters and allow one to study pairs of co-evolving stars formed in the same environment. Until recently, no early-type magnetic stars in eclipsing binaries were known. The first such system, HD 66051, was identified by Kochukhov et al. (2018). The second system, HD 62658 containing twin components of which only one is magnetic, was found by Shultz et al. (2019). Several other eclipsing binaries containing candidate ApBp stars were proposed (Hensberge et al. 2007; González, Hubrig & Castelli 2010; Skarka et al. 2019), but the magnetic nature of these stars has not been verified by direct detections of their fields using the Zeeman effect. In this paper, we put a spotlight on another candidate eclipsing magnetic Bp star, which received little attention prior to our work despite being significantly brighter than the confirmed magnetic eclipsing systems HD 62658 and HD 66051.


In our model atom, singlet and triplet terms are often separated, and thus we would
            like to estimate how this total cross-section is distributed among final spin states. We
            considered a representation where the spin of the Rydberg electron is coupled to the
            spin of the other electrons in the target atom Sc (assuming
            this to be a good quantum number) to give the total electronic spin of the Rydberg atom,
              S.
            Recoupling of angular momenta gives the required spin-changing scattering amplitude in
            terms of the singlet and triplet scattering amplitudes (i.e. in the representation in
            which the Rydberg electron spin and the hydrogen atom spin are coupled), and the
            differential cross-section for change of spin S → S′, S ≠ S′,
            is found to be \begin{eqnarray} |f_{\rm e}(p,\theta,S \rightarrow S')|^2 & = & \frac{(2S'+1)}{8(2S_{\rm c}+1)} |f^+ - f^-|^2 \\[3mm]  & = &\frac{(2S'+1)}{2(2S_{\rm c}+1)} \sigma_{\rm exch} \end{eqnarray}

|

f

e

(
p,θ,S
→

S

′

)

|

2

=

(
2

S

′

+
1
)

8
(
2

S

c

+
1
)

|

f

+

−

f

−

|

2

=

(
2

S

′

+
1
)

2
(
2

S

c

+
1
)

σ

exch

where σexch is the
            differential cross-section for electron exchange in elastic e−+H  collisions (e.g. Field 1958; Burke
              & Schey 1962). The derivation essentially follows Dalgarno & Rudge (1965), except that in our case the electron
            spin is coupled to the core electronic spin and not to the nuclear spin. We note that
              S′ =
                Sc ± 1/2 and the factor
              (2S′ + 1) /
                2(2Sc + 1) is always in the range 0 to
            1 and represents the probability that an electron exchange leads to a change in the
            total electronic spin of the Rydberg atom. In the limit p → 0, − f± →
                a±, where a± are the
            scattering lengths, which have values of a+ = 5.965 and a− = 1.769
            atomic units (Schwartz 1961). In this limit, for
            the case of Sc =
              1/2, via comparison of the expressions for the cross sections we
            find that \begin{eqnarray} \sigma_{nl,S=0 \rightarrow n'l',S'=0} = 0.706 \: \sigma_{nl \rightarrow n'l'}, \\ \sigma_{nl,S=0 \rightarrow n'l',S'=1} = 0.294 \: \sigma_{nl \rightarrow n'l'}, \\ \sigma_{nl,S=1 \rightarrow n'l',S'=0} = 0.098 \: \sigma_{nl \rightarrow n'l'}, \\ \sigma_{nl,S=1 \rightarrow n'l',S'=1} = 0.902 \: \sigma_{nl \rightarrow n'l'}. \end{eqnarray}

σ

nl,S
=
0
→

n

′

l

′

,

S

′

=
0

=
0.706

σ

nl
→

n

′

l

′

,

σ

nl,S
=
0
→

n

′

l

′

,

S

′

=
1

=
0.294

σ

nl
→

n

′

l

′

,

σ

nl,S
=
1
→

n

′

l

′

,

S

′

=
0

=
0.098

σ

nl
→

n

′

l

′

,

σ

nl,S
=
1
→

n

′

l

′

,

S

′

=
1

=
0.902

σ

nl
→

n

′

l

′

.


However, two recent pieces of evidence may suggest that standard disk accretion is not the only process at work in growing a BH to the (super)massive regime. The first is the discovery of an increasing number of active BHs with masses M
• ≳ 109
M
⊙ at very high redshifts, z ≳ 7 (e.g., Mortlock et al. 2011; Wu et al. 2015; Venemans et al. 2017, 2018; Reed et al. 2019; Banados et al. 2018, 2021; Wang et al. 2019, 2021), when the age of the universe was shorter than 0.8 Gyr. The second is the robust measurements of extreme BH masses M
• ≳109−10
M
⊙ at the center of early-type galaxies with stellar mass M
⋆ ≳ 1011
M
⊙ (e.g., McConnell et al. 2011; Ferre-Mateu et al. 2015; Thomas et al. 2016; Mehrgan et al. 2019; Dullo et al. 2021) that formed most of their old stellar component during a star formation episode lasting some 108 yr at z ≳ 1, as demonstrated by astroarcheological measurements of their stellar ages and α-enhanced metal content (e.g., Thomas et al. 2005, 2010; Gallazzi et al. 2006, 2014; Johansson et al. 2012; Maiolino & Mannucci 2019; Morishita et al. 2019; Saracco et al.2020). These observations concur to raise the issue of how billion-solar-mass BHs may have grown in less than a gigayear. In fact, this is somewhat challenging if standard disk accretion starts from a light seed of ∼102
M
⊙ of stellar origin and proceeds with the typical Eddington ratios λ ≲ 1 as estimated out to z ∼ 6 in active BHs (see Vestergaard & Osmer 2009; Nobuta et al. 2012; Kelly & Shen 2013; Dai et al. 2014; Kim & Im 2019; Duras et al. 2020; Ananna et al. 2022), which would require an overall time ≳0.8/λ Gyr to attain ∼109
M
⊙. Solutions may invoke mechanisms able to rapidly produce heavy BH seeds of 103−5
M
⊙, thus reducing the time required to attain the final masses by standard disk accretion (see Natarajan 2014, Mayer & Bonoli 2019, Inayoshi et al. 2020, and Volonteri et al. 2021 for recent and exhaustive reviews). Viable possibilities comprise direct collapse of gas clouds within a (proto)galaxy, possibly induced by galaxy mergers or enhanced matter inflow along cosmic filaments (e.g., Lodato & Natarajan 2007; Mayer et al. 2010, 2015; Di Matteo et al. 2012, 2017); merging of stars inside globular or nuclear star clusters (e.g., Portegies Zwart et al. 2004; Devecchi et al. 2012; Latif & Ferrara 2016; Kroupa et al. 2020); and migration of stellar BHs toward the nuclear galaxy regions via dynamical friction against the dense gas-rich environment in strongly star-forming progenitors of local massive galaxies (e.g., Boco et al. 2020, 2021).
11

11
When referring to the dynamical friction mechanism, the term “seeds” is used in a broader sense with respect to the classic meaning in the literature. A seed is usually referred to as the first compact object on which subsequent disk accretion occurs, eventually leading to the formation of a supermassive BH. The heavy seeds formed with the dynamical friction mechanism are by-products of multiple mergers of already-existing stellar mass BHs (that in turn could be referred to as light seeds) forming across a wide redshift range.


In contrast to previous studies, the MFP expressions employed here are defined to include independent rigidity and nonlinear radial dependences. Furthermore, this approach does not make the simplifying assumption that λ
⊥ is proportional to λ
∥ (see, e.g., Ferreira et al. 2001). In order to study the radial and rigidity dependences of MFPs required to fit observations, as well as the values of these quantities at 1 au so as to compare these with theoretical predictions of the same, this study still implements simplified parametric expressions for these MFPs, motivated by the fact that the transport code here employed is written to perform on graphic processing units GPUs in CUDA, which allows for the vast performance increase required of such a study, but limits the number of definable variables due to memory constraints (see Dunzlaff et al. 2015, for additional information). Subsequently, λ
∥ and λ
⊥ are defined as
5

λ∥=λ∥,0PP0αrreβ,

6

λ⊥=λ⊥,0PP0γrreδ,

where r

e
 = 1 au, and P
0 = 1 GV, in an approach similar to that employed by Engelbrecht & Di Felice (2020). Here, parameters α and β specify the respective rigidity and radial dependences of the parallel MFP, which assumes a value of λ
0,∥ (in astronomical units) at Earth, with γ, δ, and λ
0,⊥ playing the same roles with respect to the perpendicular MFP. In the parameter studies employed here, these quantities are varied over a broad range of values, chosen so as to include, as well as extend beyond, various theoretical predictions for these quantities. In terms of MFP values at 1 au, the ranges λ
0,∥ ∈ [0.05, 1.0]  au and λ
0,⊥ ∈ [0.001, 0.5]  au are chosen, so as to include values reported in previous such studies for the parallel MFP (e.g., Palmer 1982; Bieber et al. 1994; Dröge et al. 2014) as well as for the perpendicular MFP (e.g., Ferrando et al. 1993; Giacalone 1998; Burger et al. 2000; Zhang et al. 2007; Vogt et al. 2020), while taking into account various theoretical predictions and expressions previously used in modeling studies for these MFPs (e.g., Ferreira et al. 2003; Engelbrecht & Burger 2013b; Nndanganeni & Potgieter 2018; Engelbrecht 2019). For the rigidity dependences, it is assumed that α ∈ [−0.1, 0.67] and γ ∈ [−0.1, 0.33], incorporating the flat rigidity dependence expected of λ
∥ for low-energy electrons (e.g., Dröge 1994; Potgieter 1996; Evenson 1998) within a range that includes dependences expected at higher rigidities as well as from different scattering theories (e.g., Dröge 2000; Teufel & Schlickeiser 2002; Shalchi & Schlickeiser 2004; Burger et al. 2008; Engelbrecht & Burger 2010). The range of rigidity dependences for λ
⊥ was chosen following similar reasoning, with the aim of incorporating a flat rigidity dependence within a range that incorporates moderate rigidity dependences expected from various theories (e.g., Shalchi et al. 2004a; Gammon & Shalchi 2017; Dempers & Engelbrecht 2020) as well as prior particle transport studies (e.g., Zhang et al. 2007; Dröge & Kartavykh 2009) and numerical test particle simulations of diffusion coefficients (e.g., Minnie et al. 2007a; Dundovic et al. 2020). Lastly, radial dependences were varied such that β ∈ [0.5, 1.5] and δ ∈ [0.5, 1.5], taking into account various possible radial dependences for these MFPs reported by studies employing turbulence quantities yielded by various turbulence transport models as inputs for several theoretical expressions (see, e.g., Engelbrecht & Burger 2013b; Wiengarten et al. 2016; Chhiber et al. 2017; Zhao et al. 2017; Adhikari et al. 2021). In order to study the various dependencies in Equations (5) and (6), four initial electron energies are chosen, motivated by the fact that for these energies spacecraft data is available for periods of good and bad magnetic connection to the Jovian source as well as at the Jovian source itself. The three higher energies furthermore correspond to the three lowest-energy channels of the ISEE 3 data by Moses (1987). As the results of the parameter studies have to be compared to spectral fits (given by the dotted lines in Figure 1), this choice allows for the most transparent analysis of the deviation of the simulation results from in situ data. The six parameters governing the MFP expressions used in this study, along with the relevant initial energies, are
7

E=[0.0025,0.006,0.0085,0.0125]GeVλ∥,0=[0.05,0.106,0.224,0.473,1.0]auλ⊥,0=[0.001,0.00473,0.02236,0.1057,0.5]auα=[−0.1,−0.01,0.01,0.04,0.165,0.67]γ=[−0.1,−0.01,0.01,0.03,0.1,0.33]β=[0.5,0.658,0.866,1.397,1.5]δ=[0.5,0.658,0.866,1.397,1.5],

which form a parameter space of consisting of 90,000 grid points, resulting in 180,000 simulations for comparison with observational points both well and badly connected to the Jovian source as shown in Figure 1.


Torques caused by a misalignment between the spin axis of a black hole and the angular momentum of the accretion disc can also induce jet precession. In an overall misaligned system the inner part of the accretion disc is aligned with the black hole spin axis by the combined effect of Lense–Thirring precession and internal viscosity of the disc. This process is known as the Bardeen–Petterson effect (Bardeen & Petterson 1975). The outer disc will maintain its angular momentum direction. Through the accretion of material of the outer disc on to the black hole, the black hole spin axis will precess and align with the outer disc. The precession time-scale is of the same order as the alignment time-scale (Scheuer & Feiler 1996). Two aspects of the disc-induced precession, namely the relatively slow precession rate and the single possible precession cycle in the alignment lifetime (Scheuer & Feiler 1996; Lodato & Pringle 2006), make this process less likely to be a cause of jet precession for the given radio sources. For example, following Lodato & Pringle (2006), adopting their numerical values for accretion efficiency ε and disc viscosity parameters α1, 2 and assuming black hole spin a ≈ 1, we obtain a precession period of
(3)
\begin{eqnarray*}
\begin{split} t_\mathrm{prec} &\approx t_\mathrm{align} \approx 7\, \, \mathrm{Myr} \\
&\times \, a^{11/16} \left(\frac{\epsilon }{0.1}\right)^{7/8} \left(\frac{L}{0.1 L_\mathrm{Edd}}\right)^{-7/8} M_9^{-1/16} \\
&\times \left(\frac{\alpha _1}{0.03}\right)^{15/16} \left(\frac{\alpha _2}{0.3}\right)^{-11/16} \, .\\
\end{split}
\end{eqnarray*}
This is larger than the precession period we estimated for Cygnus A. Simulations show that if the precession time-scale becomes comparable to the source age, very asymmetric lobe morphologies result (Donohoe & Smith 2016). As discussed in Section 3.3, the sources in our sample do not show such asymmetries, but often show smooth, ring-like hotspot features that suggest a local impact time-scale short compared to the source ages and thus many precession cycles. This is not consistent with precession induced by a misaligned accretion disc. A process that produces one precession cycle only, where the period of the cycle can be much shorter than the source age (equation 3), would have only a limited probability to be detected at the time a given source is observed. However, we also find precession frequently in comparatively old sources, e.g. 3C 33.1, 3C 303, and 3C 436. These are three of the seven sources with modelled ages of log(age/yr) ≥ 7.9. This makes disc-induced precession less likely for this subpopulation.


We end by studying the BH mass versus stellar velocity dispersion, σ, in Fig. 7. This is also a benchmark scaling, and it is considered to be one of the best predictors for BH masses, i.e. to have a small intrinsic scatter (0.3 dex). This is perhaps the most problematic for Horizon-AGN. If we fit a single logarithmic slope to the BH mass versus stellar velocity dispersion, we find a value of 4.02 for galaxies with σ > 60 km s−1, compared to 4.38 found by Kormendy & Ho on a culled subsample of the galaxies shown in Fig. 7. The main reason is that velocity dispersions in the simulation are underestimated at Mbulge  1011 M⊙ (σ  200 km s−1), and overestimated above, when comparing to the observed Faber–Jackson relation (Bernardi et al. 2003). The dearth of BHs with MBH  5 × 107  M⊙ in intermediate-mass galaxies highlighted in the discussion of Fig. 5 also contributes, but given the good agreement in the MBH versus Mbulge relation, it is a minor contribution, for this specific comparison. One contribution to the discrepancy is that galaxy sizes in the simulation are larger than for observed galaxies with mass ≲ 1011 M⊙. In fact, the galaxy sizes are in good agreement with observed low-mass late-type galaxies, but are larger, by a factor of ∼ 3–4, than observed low-mass early-type galaxies (Dubois et al., in preparation). Since the complete sample is a mix of late- and early-type galaxies, there appears to be an overall mismatch at σ  200 km s−1 of a factor of ${\sim } \sqrt{2}$ in the velocity dispersion (roughly corresponding to an average overestimate of the galaxy size of a factor of 2). The velocity dispersion is also sensitive to the DM halo central profile, which depends not only on DM halo mass but also on a subtle interplay with the baryons and AGN feedback. Indeed, the central DM densities in Horizon-AGN are lower than for the Horizon-noAGN simulation, which differs only in that AGN are not included in the latter. We defer a more detailed study quantifying the exact contribution of this effect as a function of galaxy mass and assembly history to another paper (Peirani et al., in preparation).


However, our understanding of the scaling relations in lower-mass galaxies is lacking. This includes galaxies near the break of the galaxy stellar-mass function

 (e.g., Baldry et al. 2012) like the Milky Way (MW), as well as galaxies at lower masses. Measurements of black hole (BH18

18
In this article, we use the abbreviations SMBH and BH interchangeably.
) masses are lacking in these galaxies. Despite the MW’s precisely measured central BH mass of

 (Boehle et al. 2016; Gillessen et al. 2017), we actually know very little about the demographics of BHs in galaxies of this mass. In particular, there are only 11 published

 in spiral galaxies within ∼1 dex in stellar mass of the MW, but there are strong indications that the relations between

 and galaxy properties that hold at higher mass (Kormendy & Ho 2013; McConnell et al. 2013; Saglia et al. 2016; van den Bosch et al. 2016) break down for M⋆ spiral galaxies (Greene et al. 2010, 2016; Läsker et al. 2016). For instance, Andromeda is only ≲2 times more massive than the MW, but its SMBH is two orders of magnitude higher,

 ∼1.4 × 108M⊙ (Bender et al. 2005). At even lower galaxy masses, the changes in the inferred

 are twofold: not only does the scatter increase enormously at low mass (e.g., Reines et al. 2013), but it also seems that the

 are lower at fixed stellar mass or σ of the lower-mass systems (e.g., Kormendy & Ho 2013; Scott et al. 2013; Greene et al. 2016; Saglia et al. 2016; Nguyen et al. 2017, 2018, 2019, hereafter N17, N18, N19). Despite the incomplete sample of

 at lower masses, there are already two intriguing findings from the limited data available: (1) a factor of 2 difference in normalization seen in the

–σ correlations for early-type (ETGs) and late-type (LTGs) galaxies (McConnell et al. 2013) and (2) a large scatter up to two orders of magnitude of MBH seen around the global scaling relations for low-mass systems (Greene et al. 2010; Scott et al. 2013; Graham & Scott 2015; Greene et al. 2016; Läsker et al. 2016; Chilingarian et al. 2018).


The 21-cm line is the transition line of atomic hydrogen between two hyperfine states that are split from the ground state due to the spin coupling of the nucleus and electron. The 21-cm brightness temperature from a bulk of hydrogen gas depends on the matter density, the neutral fraction, the velocity gradient, and the spin temperature. During the EoR, it is reasonable to assume that X-ray heating and Ly α-pumping effects are so efficient that the H i spin temperature is much greater than the CMB temperature. In this limit, the power spectrum of 21-cm brightness temperature fluctuations can be written as (Datta et al. 2012; Mao et al. 2013)
(3)
\begin{eqnarray*}
P_{\Delta T}({\boldsymbol k},{\boldsymbol n},z) = \widetilde{\delta T}_b^2 \bar{x}_{\rm H\,{\small I}}^2\, \left[ b_{\rho _{\rm H\,{\small I}}}(z) + \mu _{\boldsymbol k}^2 \right]^2\, P_{\zeta }(k,{\boldsymbol n},z),
\end{eqnarray*}
at large scales, with
(4)
\begin{eqnarray*}
\widetilde{\delta T}_b(z) = (23.88\, {\rm mK}) (\frac{\Omega _b h^2}{0.02}) \sqrt{\frac{0.15}{\Omega _M h^2} \frac{1+z}{10}}.
\end{eqnarray*}
Note that $\bar{x}_{\rm H\,{\small I}}$ is the global neutral fraction that implicitly depends on the redshift, and μk ≡ k · n/|k| (cosine of angle between line of sight n and wave vector k of a given Fourier mode). We can relate the neutral hydrogen density fluctuations to the total matter density fluctuations with a simple bias parameter, the neutral density bias, $b_{\rho _{\rm H\,{\small I}}}(k,z) \equiv \tilde{\delta }_{\rho _{\rm H\,{\small I}}}({\boldsymbol k},z)/\tilde{\delta }_{\rho }({\boldsymbol k},z)$. At large scales, the bias approaches to a scale-independent value, $b_{\rho _{\rm H\,{\small I}}}(z)$. Here, we assume that the baryonic distribution traces the cold dark matter distribution at large scales, so $\delta _{\rho _{\rm H}} = \delta _{\rho }$. Also, we assume that no additional hemispherical power asymmetry was generated during the evolution of the Universe, so the dipolar modulation of the curvature perturbation is transferred to the 21-cm power spectrum with the same form.


The sensitivity of dwarfs to galaxy formation physics, like the gas density at which stars form (e.g. Kravtsov 2003; Saitoh et al. 2008) or the details of galactic outflows (e.g. Read et al. 2006b), makes them a natural ‘rosetta stone’ for constraining galaxy formation models. Early work simulating dwarf galaxies focused on high resolution small box simulations, stopping at high redshift (z ∼ 5–10) to avoid gravitational collapse on the scale of the box (e.g. Abel et al. 1998; Gnedin & Kravtsov 2006; Read et al. 2006b; Mashchenko, Wadsley & Couchman 2008). These simulations demonstrated that stellar winds, SN feedback, and ionizing radiation combine to prevent star formation in halo masses below ∼107–8 M⊙ (Read et al. 2006b; Bland-Hawthorn, Sutherland & Webster 2015). Furthermore, once cooling below 104 K is permitted and gas is allowed to reach high densities $n_{\rm max} {\gtrsim}10\,{\rm cm}^{-3}$ (requiring a spatial and mass resolution better than $\Delta x {\lesssim}100$ pc and mmin  103 M⊙; Pontzen & Governato 2012; Benitez-Llambay et al. 2019; Bose et al. 2019; Dutton et al. 2019), star formation becomes much more stochastic and violent (Mashchenko et al. 2008; Pontzen & Governato 2012; Dutton et al. 2019). The repeated action of gas cooling and blow-out due to feedback expels DM from the galaxy centre, transforming an initially dense DM cusp to a core (e.g. Read & Gilmore 2005; Pontzen & Governato 2012). Finally, independent of any internal sources of feedback energy, cosmic reionization can halt star formation in low-mass galaxies (e.g. Gnedin & Kravtsov 2006). Dwarfs that have not reached a mass of $M_{200} {\gtrsim}10^8$ M⊙ (see Section 2.4 for a definition of M200) by the redshift that reionization begins (z ∼ 8–10; Gnedin & Kaurov 2014; Ocvirk et al. 2018) are gradually starved of fresh cold gas, causing their star formation to shut down by a redshift of z ∼ 4 (Oñorbe et al. 2015). This is similar to the age of nearby ‘ultra-faint’ dwarf galaxies (UFDs) that have $M_* {\lesssim}10^5$ M⊙, suggesting that at least some of these are likely to be relics from reionization, inhabiting pre-infall halo masses in the range M200 ∼ 108–9 M⊙ (Gnedin & Kravtsov 2006; Bovill & Ricotti 2009, 2011; Brown et al. 2014; Weisz et al. 2014; Jethwa, Erkal & Belokurov 2018; Read & Erkal 2019).


Moreover, there could also be some contribution to the detected temperature asymmetry from high-latitude gas clouds in our Galaxy along the line of sight toward M 81. In this respect we note that M 81 is at about 40.9° north of the Galactic disk, where contamination from the Milky Way is expected to be low. However, interpretation of astronomical observations is often hampered by the lack of direct distance information. Indeed, it is often not easy to judge whether objects on the same line of sight are physically related or not. Since the discovery of the Arp’s Loop (Arp 1965) the nature of the interstellar clouds in this region has been debated; in particular whether they are related to the tidal arms around the galaxy triplet (Sun et al. 2005; de Mello et al. 2008) or to Galactic foreground cirrus (Sollima et al. 2010; Davies et al. 2010). Already Sandage et al. (1976) presented evidence showing that we are observing the M 81 triplet through widespread Galactic foreground cirrus clouds and de Vries et al. 1987 built large-scale HI, CO, and dust maps that showed Galactic cirrus emission toward the M 81 region with NH ≃ 1−2  ×  1020 cm-2. The technique used to distinguish between the emission from extragalactic or Galactic gas and dust relies on spectral measurements and on the identification of the line of sight velocities, which are expected to be different in each case. Unfortunately, in the case of the M 81 Group, this technique appears hardly applicable since the radial velocities of extragalactic and Galactic clouds share a similar LSR (local standard of rest) velocity range (Heithausen 2012). Several small-area molecular clouds (SAMS), that is, tiny molecular clouds in a region where the shielding of the interstellar radiation field is too low (so that these clouds cannot survive for a long time), have been detected by Heithausen (2002) toward the M 81 Group. More recently, data from the Spectral and Photometric Imaging Receiver (SPIRE) instrument onboard Herschel ESA space observatory and Multiband Imaging Photometer for Spitzer (MIPS) onboard Spitzer allowed the identification of several dust clouds north of the M 81 galaxy with a total hydrogen column density in the range 1.5–5 × 1020 cm-2 and dust temperatures between 13 and 17 K (Heithausen 2012). However, since there is no obvious difference among the individual clouds, there was no way to distinguish between Galactic or extragalactic origin although it is likely that some of the IR emission both toward M 81 and NGC 3077 is of Galactic origin. Temperature asymmetry studies in Planck data may be indicative of the bulk dynamics in the observed region provided that other Local (Galactic) contamination in the data is identified and subtracted. This is not always possible, as in the case of the M 81 Group, and therefore it would be important to identify and study other examples of dust clouds where their origin, either Galactic or extragalactic, is not clear. One such example might be provided by the interacting system toward NGC 4435/4438 (Cortese et al. 2010) where the SAMS found appear more consistent with Galactic cirrus clouds than with extragalactic molecular complexes. Incidentally, the region A1 within R0.50 has been studied by Barker et al. (2009), who found evidence for the presence of an extended structural component beyond the M 81 optical disk, with a much flatter surface brightness profile, which might contain ≃10–15% of the M 81 total V-band luminosity. However, the lack of both a similar analysis in the other quadrants (and at larger distances from the M 81 center) and the study of the gas and dust component associated to this evolved stellar population, hamper our understanding of whether this component may explain the observed temperature asymmetry toward the M 81 halo.


Using a single-fluid approach Chian & Oliveira (1994, 1996), and Oliveira & Chian (1996) studied the effects of parametric instabilities on standing waves with applications to the magnetosphere. These authors developed a theory of MHD parametric instabilities driven by a left- or right-hand circularly polarised standing Alfvén wave in a low plasma-β under very specific initial conditions. Israelevich & Ofman (2004) and Israelevich & Ofman (2011) using both the MHD approach and hybrid simulations investigated numerically the generation of an electric field parallel to the magnetic field, due to non-linear induced ion-acoustic waves produced by the presence of a standing Alfvénic pump. In a different context, Mottez (2015) also explored the generation of a parallel electric field using PIC (particle in cell) simulations when two counter-propagating Alfvén waves are excited. However, the mechanism proposed by this author is not related to the ponderomotive force that it is usually operating under such conditions. A ponderomotive force (see the review of Lundin & Guglielmi 2006) generates density enhancements periodically in time and space which are normally called cavities and depend on gradients in density or electric fields. The periodicity depends on the value of gas pressure of the system and the amplitude of these enhancements is related quadratically with the amplitude of the initial transverse excitation (see also Rankin et al. 1994; Tikhonchuk et al. 1995). Interestingly, ponderomotive forces can lead to separation of species, although this topic is not addressed in the present work since the quasi-neutrality condition is imposed for the plasma composed of kinetic protons and the treatment of electrons is that of a fluid. Recently, Martínez-Gómez et al. (2018) used the multi-fluid approach to model high-frequency waves in partially ionised plasmas similar to those in quiescent solar prominences. These authors show that non-linear Alfvén waves generate density and pressure perturbations and that the friction due to collisions dissipates a fraction of the wave energy, which is transformed into heat that raises the temperature of the plasma.


The X-ray transient MAXI J1820+070 was first detected on 2018 March 11 (Kawamuro et al. 2018) by the Monitor of All-sky X-ray Image (MAXI, Matsuoka et al. 2009) and was associated with the optical transient ASASSN-18ey (Denisenko 2018; Tucker et al. 2018). In the X-rays, the source flux exceeded 3 Crabs (Bozzo et al. 2018; Mereminskiy et al. 2018), and in the optical, the source reached a magnitude of mV = 12 − 13 (Littlefield 2018; Russell et al. 2018). The parallax of the source π = 0.3 ± 0.1 mas was presented in the Gaia DR2 catalogue (Gaia Collaboration 2018). This corresponds to a distance of

3
.

9

−
1.3

+
3.3

$ 3.9^{+3.3}_{-1.3} $

 kpc (Gandhi et al. 2018a). This unusually bright event allows a detailed investigation of multi-wavelength spectral and timing properties. The course of the outburst was monitored in radio (Trushkin et al. 2018; Polisensky et al. 2018), sub-millimeter (Tetarenko et al. 2018a), optical (Baglio et al. 2018), X-rays (Uttley et al. 2018), and γ-rays (Bozzo et al. 2018; Kuulkers et al. 2018). Because the object was sufficiently bright even for small telescopes, the target was almost continuously monitored, and a rich variety of phenomena was observed. Fast variability and powerful flares in the optical and infrared (Littlefield 2018; Sako et al. 2018; Gandhi et al. 2018b; Casella et al. 2018), optical, and X-ray quasi-periodic oscillations (Mereminskiy et al. 2018; Yu et al. 2018a,b; Buisson et al. 2018; Zampieri et al. 2018) as well as low linear polarisation (Berdyugin et al. 2018) were detected in the source. A 17 h photometric period was recently reported (Patterson et al. 2018) and was tentatively associated with the orbital or superhump period (previously, the source showed a 3.4 h periodicity, Richmond 2018). The X-ray spectral and timing properties as well as the optical-to-X-ray flux ratio suggests that the source is a black hole binary (Baglio et al. 2018; Mereminskiy et al. 2018).


We study the impact of secondary outgassing associated with partial melting and volcanism following accretion and a possible magma ocean phase. Raymond et al. (2007) showed that the accretion of planets around M-dwarf stars is faster than around more massive stars, which has also been calculated by Lissauer (2007) and Ida & Lin (2005). We therefore assumed that the accretion of a planet around an M dwarf is very fast (1 Myr), while for the other stellar cases we assumed an accretion time of 20 Myr. This corresponds to the lower limit in accretion times of Raymond et al. (2007) for planets around stars with masses greater than 0.8 MSun. After accretion, the planet may reside in a magma ocean stage, which we assume to last 1 Myr, following the results by Spohn & Schubert (1991) and Lebrun et al. (2013), among others. For high stellar irradiation, the planet may be trapped in a long-term magma ocean stage, which may last from several millions of years to billions of years (see e.g. Hamano et al. 2013, Nikolaou et al. 2019). For planets in a long-term magma ocean stage, Hamano et al. (2013) suggested very low water concentrations in the mantle at the end of the magma ocean stage because the planet can only exit this stage via severe atmospheric water loss because thick, water-rich atmospheres show a thermal blanketing effect suppressing the cooling of the planet via the Planck feedback. Their estimates of interior water reservoirs for such cases correspond to our lower limit of the initial mantle water concentration of 34 ppm. We nevertheless assumed in our calculations that the magma ocean has a short duration. This allows the high-luminosity pre-main sequence phase of the M dwarfs to have a larger impact on the HZ evolution. Any CO2 and H2O outgassed during the magma ocean phase is neglected in our calculations, i.e. we started our outgassing calculations with a 1 bar N2 atmosphere. Neglecting any atmospheric H2O and CO2 from the magma ocean phase results in a very narrow HZ at the beginning of our calculations, which is then only extended by secondary outgassing. This facilitates the evaluation of the potential of secondary outgassing from a stagnant-lid planet to form a HZ.


The theory of LK migration with STs involves two important time-scales. The first is the time-scale for quadrupole LK eccentricity oscillations, tLK, given by
(40)
\begin{eqnarray*}
t_{\rm LK} &=& \left(\frac{10^6}{2\pi }\,\text{yr}\right)\left(\frac{M_\mathrm{ b}}{\rm{ M}_\odot }\right)^{-1}\left(\frac{M_*}{\mathrm{ M}_\odot }\right)^{1/2}\left(\frac{a_0}{1\, \text{au}}\right)^{-3/2}\nonumber \\
&&\times \, \left(\frac{a_{\rm b,eff}}{100\, \text{au}}\right)^{3} ,
\end{eqnarray*}
where a0 is the initial semimajor axis of the planet’s orbit, Mb is the mass of the (stellar) companion, and
(41)
\begin{eqnarray*}
a_{\rm b,eff} \equiv a_{\rm b} (1-e_{\rm b}^2)^{1/2} ,
\end{eqnarray*}
with ab and eb the semimajor axis and eccentricity of the companion’s orbit, respectively. The time that the planet spends near the maximum eccentricity, emax, is of the order of (e.g. Anderson et al. 2016)
(42)
\begin{eqnarray*}
\Delta t(e_{\rm max}) \sim (1-e_{\rm max}^2)^{1/2} t_{\rm LK}.
\end{eqnarray*}
The second important time-scale is that of orbital decay due to STs given by (Alexander 1973; Hut 1981).
(43)
\begin{eqnarray*}
t^{-1}_{\rm ST} &=& \left|\frac{\skew4\dot{a}}{a}\right|_{\rm ST} = (6 k_{2\mathrm{ p}} \Delta t_{\mathrm{ L}}) \frac{M_*}{M_\mathrm{ p}} \left(\frac{R_\mathrm{ p}}{a}\right)^{5}\frac{n^{2}}{(1-e^2)^{15/2}} \nonumber \\
&&\times \, \left[f_1(e)\ - \frac{f_2^2(e)}{f_5(e)}\right],
\end{eqnarray*}
where k2p is the tidal Love number of the planet, ΔtL is the lag time, f1(e) = 1 + 31e2/2 + 255e4/8 + 185e6/16 + 25e8/64, and f2(e) and f5(e) are given in equations (30) and (31), respectively. (We have assumed that the planet has a pseudo-synchronous spin rate.) Tidal dissipation is most efficient near the maximum eccentricity. Since the planet only spends a fraction (${\sim } \sqrt{1-e_{\rm max}^2}$) of the time near emax, the effective orbital decay rate during LK migration is
(44)
\begin{eqnarray*}
t^{-1}_{\rm ST,LK} &=& \left(\left|\frac{\skew4\dot{a}}{a}\right|_{\rm ST} \sqrt{1-e^2}\right)_{e_{\rm max}} \nonumber \\
&\approx& \frac{1.27}{{\rm Gyr}} \left(\frac{k_{\rm 2p}}{0.37}\right) \left(\frac{\Delta t_{\rm L}}{1 {\rm s}}\right)\left(\frac{M_\mathrm{ p}}{\mathrm{ M}_{\odot }}\right)^2 \left(\frac{M_\mathrm{ p}}{M_\mathrm{ J}}\right)^{-1} \nonumber \\
&&\times \; \left(\frac{R_\mathrm{ p}}{R_\mathrm{ J}}\right)^5\left(\frac{a_0}{1\, {\rm au}}\right)^{-1} \left(\frac{r_{\rm p, min}}{0.025\, {\rm au}}\right)^{-7} \,,
\end{eqnarray*}
where rp, min = a0(1 − emax) is the minimum pericentre distance (in the second of the above equalities, we have used emax = 0.96). Successful migration within a few Gyr requires rp, min ≲ 0.025 au, corresponding to a final (circularized) planet semimajor axis aF ≲ 0.05 au. A planet would need to be more dissipative than Jupiter (larger ΔtL)5 to become an HJ with a larger aF.


Several optical/near-UV surveys have targeted the CDF-S and its flanking fields, with the main purpose of photometrically and spectroscopically calculating the redshifts of the detectable sources. In the context of the ESO imaging survey (EIS, Arnouts et al. 2001) project, multi-colour (six broad band filters, U′UBVRI) photometric measurements of about 75 000 sources have been provided to observe the Extended-CDF-S (E-CDF-S) through the Wide Field Imager (WFI, Baade et al. 1998, 1999) at the MPG/ESO 2.2 m telescope on La Silla. By means of the same telescope and instrument, a nearly coincident area of 31.5 × 30 arcmin2 has been observed in 17 passbands (350–930 nm) within the COMBO-17 survey, obtaining a catalogue of more than 60 000 objects including astrometry, classification, and photometric redshifts, with around 10 000 galaxies identified at RVega 24 (Wolf et al. 2004, 2008). Another optical and near-IR multi-wavelength survey has been developed by Cardamone et al. (2010) as part of the Multi-Wavelength Survey by Yale-Chile (MUSYC, Gawiser et al. 2006), combining the Subaru 18-bands imaging (400–900 nm) to 14 other available ground-based and Spitzer images: about 80 000 galaxies are catalogued at R  ≤  27, and about 30 000 redshifts are computed. The Great Observatories Origins Deep Survey (GOODS, Giavalisco et al. 2004), instead, covers a 320 arcmin2 region in the CDF-S (GOODS-S) by using the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS, Ford et al. 2003), through the filters F435W,F606W,F775W, and F850LP; it catalogued about 29 600 objects, at m ≤ 27. Moreover, an 11 arcmin2 region within the GOODS-S (Hubble Ultra Deep Field, UDF, Beckwith et al. 2006) has been deeply observed by the ACS in the same bands, reaching a uniform limiting magnitude m ~ 29 for point sources. Voyer et al. (2009) also presented the U-band observations of the same area, achieved through the HST Wide-Field Planetary Camera 2 (WFPC2)/F300W filter (λmax ~ 300 nm) and providing a catalogue of 96 objects. The Arizona CDFS Environment Survey (ACES, Cooper et al. 2012) spectroscopically targeted the E-CDF-S by means of the Inamori-Magellan Areal Camera and Spectrograph on the Magellan-Baade telescope (565–920 nm), yielding more than 5000 redshifts for R ≤ 24.1 sources; it widens therefore previous sets of spectroscopic redshift estimates of CDF-S sources (e.g. Szokoly et al. 2004; Le Fèvre et al. 2005; Ravikumar et al. 2007; Popesso et al. 2009; Balestra et al. 2010; Silverman et al. 2010), for a total of about 1300 ACES objects with comparison published redshifts. All the publicly available spectroscopic redshift estimates in the CDF-S have been compiled in the ESO CDF-S Master Catalogue, version 3.01.


Figure 5 shows that, toward the Galactic plane,
            ⟨ B⊥
            ⟩ is mostly oriented along the plane, corresponding to a polarization
          angle close to 0°. This is
          especially the case toward the outer MW regions. There are a few exceptions, in particular
          toward lines of sight that are tangent to spiral arms (Cygnus X, ℓII ≃ 81°,
            bII ≃
            0°; Carina, ℓII ≃ 277°, bII ≃ −9°),
          where the polarization signal is actually the smallest in the plane because in those
          regions the large-scale magnetic field is parallel to the LOS. This was already noted by
            Heiles (1996) (and references therein) and Benoît et al. (2004). We also note that the homogeneity
          of the field orientation being parallel to the plane extends away from the plane and up to
            | bII | ≃
            10° in many regions (in particular the Fan). At intermediate latitudes,
          the field orientation follows a few of the well known filamentary intensity structures of
          the local ISM. In particular, this is the case for the Aquila Rift and most of Loop I
          (outside the latitude range bII ≃ 50°−60°), where the structure of
            ⟨ B⊥
            ⟩ follows the intensity flare and loop elongation. As addressed
          earlier, this orientation of ⟨
                B⊥ ⟩ in those regions was
          already noted in the synchrotron polarized maps of WMAP (Gold et al. 2011). Other regions, however, show a variety of relative
          orientations between the field projection and intensity structures, which can also be
          orthogonal in some instances. Thus studies with Planck submillimetre
          polarization (Planck Collaboration Int. XXXII 2015;
            Planck Collaboration Int. XXXIII 2015) hold
          promise as a valuable complement to optical and near infrared polarization studies of the
          relative orientation of the magnetic field and structure (e.g., Goodman et al. 1990; Chapman et al.
            2011).


Blazars are the subclass of active galactic nuclei (AGNs) characterized by a relativistic jet that is aligned close (≤10°) to the observer's line of sight (Urry & Padovani 1995). The Doppler-boosted non-thermal emission from the relativistic jets is highly variable at all observed timescales over essentially the entire electromagnetic spectrum. Variability seen over a timescale of less than a day is called intraday variability (IDV), or microvariability (Wagner & Witzel 1995), variations over a few days to months are often called short-term variability (STV), and fluctuations observed over several months to years, or even decades, are known as long-term variability (LTV; Gupta et al. 2004). The two classical subclasses of blazars are BL Lacertae objects (BL Lacs), which have no detectable, or very weak (EW  5Å), optical emission lines (Marcha et al. 1996), and flat spectrum radio quasars (FSRQs), which have the usual strong quasar emission lines in their optical spectra. The two broad bumps seen in the broadband spectral energy distributions (SEDs) of blazars indicate two different emission mechanisms. The low-energy peak is well understood to be caused by the synchrotron emission from relativistic electrons in the jet. However, the origin of the high-energy peak is still under debate. In the leptonic model, the high-energy component is interpreted as the inverse Compton (IC) scattering of synchrotron photons themselves (synchrotron-self Compton, SSC; e.g., Bloom & Marscher 1996), or external photons (external Compton, EC; e.g., Blandford & Levinson 1995) by the same electrons responsible for the synchrotron emission. In the alternative hadronic models, processes such as proton and muon synchrotron emission are thought to be responsible for the high-energy bump (e.g., Böttcher 2007). Blazars are also classified through the value of the peak frequency of the synchrotron component. It typically lies in the infrared to optical region in the low-frequency peak blazars (LBL), while in the high-frequency peaked blazars (HBL) it is located at FUV to X-ray energies (Padovani & Giommi 1995). The high-energy components of blazar SEDs peak at GeV energies in LBLs and at TeV energies in HBLs, but some LBLs and the intermediate-peaked blazars (IBLs) have still been detected at TeV energies.


Now we estimate the average number of strong lensing systems within the region of possible GW sky location ($\mathbf {\delta \Omega }$), and capable of reproducing the observed ΔtGW and RGW. The probability for an object at redshift zs being strongly lensed by a foreground galaxy with redshift in [zl, zl + dzl] and velocity dispersion in [σv, σv + dσv] is
(2)$$\begin{eqnarray*}
{\rm d}\tau = \frac{{\rm d}n(\sigma _v,z_l)}{{\rm d}\sigma _v}S_{cr}(\sigma _v,z_l,z_s)\frac{{\rm d}V(z_l)}{{\rm d}z_l}{\rm d}\sigma _v{\rm d}z_l.
\end{eqnarray*}$$Here n(σv, zl) is the comoving number density of the lens galaxies per unit velocity dispersion σv at redshift zl. $S_{cr}(\sigma _v,z_l,z_s)=\pi \theta _E^2$ is the cross-section for a lens of velocity dispersion σv at zl and a source at zs. V(zl) is the comoving volume within zl. For simplicity and consistency with previous research on detection rate of SLGWs for ET (Piórkowska et al. 2013; Biesiada et al. 2014; Li et al. 2018), we ignore the redshift dependence of n(σv, zl), and adopt the following Schechter distribution function
(3)$$\begin{eqnarray*}
\frac{{\rm d}n(\sigma _v,z_l)}{{\rm d}\sigma _v} = n_*\left(\frac{\sigma _v}{\sigma _{v*}}\right)^\alpha \exp \left[-\left(\frac{\sigma _v}{\sigma _{v*}}\right)^\beta \right]\frac{\beta }{\Gamma (\alpha /\beta)}\frac{1}{\sigma _v},
\end{eqnarray*}$$where $(n_*,\sigma _{v*},\alpha ,\beta)=(8.0\times 10^{-3}\, h^3\, {\rm Mpc^{-3}},161\, {\rm km\, s}^{-1}, 2.32$, $2.67)$ comes from the galaxy sample of SDSS DR5 (Choi, Park & Vogeley 2007). Γ(α/β) is the Gamma function. Integrating equation (2), one obtains the probability for an object at redshift zs being strongly lensed. It is (Piórkowska et al. 2013)
(4)$$\begin{eqnarray*}
\tau (z_s)=\frac{8}{15}\pi ^3(1+z_s)^3d_s^3\left(\frac{\sigma _{v*}}{c}\right)^4n_*\frac{\Gamma (\frac{4+\alpha }{\beta })}{\Gamma (\alpha /\beta)}.
\end{eqnarray*}$$To estimate the expected number of strongly lensed galaxies in redshift range (zs, zs + δzs), we need the redshift distribution of source galaxies. Here we approximate it by (Holz & Hughes 2005)
(5)$$\begin{eqnarray*}
\frac{{\rm d}N}{{\rm d}r}=N_0r^a\exp (-(r/r_*)^b),
\end{eqnarray*}$$where (a, b, r*) = (1, 4, c/H0) and r is the comoving distance (Kaiser 1992; Hu 1999). The normalization factor N0 is chosen to satisfy $\int \frac{{\rm d}N}{{\rm d}r} {\rm d}r=40 \, \rm galaxies\, arcmin^{-2}$, corresponding to several important survey projects in the future like LSST, WFIRST, and Euclid (Yao et al. 2017). Then the expected number of strongly lensed galaxies in redshift range (zs, zs + δzs) will be
(6)$$\begin{eqnarray*}
N_{\rm lg} = \int _{z_s}^{z_s+\delta z_s} \tau (z)\frac{{\rm d}N}{{\rm d}r}\frac{{\rm d}r(z)}{{\rm d}z}{\rm d}z.
\end{eqnarray*}$$The normalized differential distribution of the expected number of the lensed galaxies is shown as Fig. 1. It shows that most of lensed galaxies distribute in the range of redshift (0.5, 2.5) and the distribution peaks at z ≈ 1.5.


The PWN morphology depends on the pulsar’s Mach number

, where v is the pulsar’s velocity and cs is the speed of sound in the ambient medium. Because even the transverse velocity of B0656+14,

 km s−1 (Brisken et al. 2003)—where i is the angle between the velocity vector and the line of sight—exceeds the typical speed of sound in the ISM,

–30 km s−1, the pulsar moves supersonically and the interaction of its wind with the ISM should produce a bow-shock PWN. X-ray emission from such a PWN is due to synchrotron radiation from the shocked pulsar wind outside the termination shock (TS). For large

 and small values of the magnetization parameter σ of the preshock isotropic wind (Kennel & Coroniti 1984), the TS acquires a bullet-like shape (Bucciantini et al. 2005) with a distance

 between the pulsar and bullet head, where

 dyn cm−2 is the ram pressure,

 is the mass density of the ambient medium,

 (in units of cm−3), and

. The bullet’s cylindrical radius is

 and the distance of its back surface from the pulsar is

. The shocked wind outside the TS is confined by the contact discontinuity (CD) surface, which has an approximately cylindrical shape behind the bullet, with a radius

. The shocked wind flows along and past the bullet, reaching very high flow velocities: ∼(0.1–0.3)c in the “inner channel” (

) and up to (0.8–0.9)c in the “outer channel” (

) of the collimated outflow. The appearance of such a PWN depends on the velocity orientation angle i. For i close to 90°, the PWN would look like a bow-shock structure accompanied by a tail of collimated wind outflow. However, for small

 the PWN would look nearly round because of the projection effect. Because the emission from the nearly relativistic flow is predominantly directed along the flow velocity, the apparent PWN radius

 is smaller,

, in the case of approaching pulsar than in the case of receding pulsar, when

.


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).


The linearized equation of motion (see derivation of Equation (A12)) for the unmagnetized case is given by (also refer to Christensen-Dalsgaard 2003):
9

where cs(r), ρ0(r), and g(r) are the sound speed, density, and gravity (directed radially inward), respectively, and

 denotes the covariant spatial derivative operator. For all ensuing calculations and derivations, we write Equation (9) in the form

, where the magnetically unperturbed wave operator

 is self-adjoint (Goedbloed & Poedts 2004). To solve for the eigenmodes of the unperturbed model S, the boundary conditions employed are: (a)

 and the Eulerian pressure perturbation at r = 0 are finite, and (b) the Lagrangian pressure perturbation at r = R⊙ vanishes (see Section 17.6 in Cox 1980). As already mentioned earlier, the Cowling approximation is used, and hence, the gravitational Poisson equation is not needed while finding the eigenmodes. We suppress the subscript “0” in the unperturbed eigenfunctions

 and eigenfrequencies ωk,0 for the rest of this paper. Unless specified otherwise, any instance of ωk or

 should be assumed to imply eigenfrequencies and eigenfunctions of Equation (9), respectively. The Sun is treated as a fluid body with a vanishing shear modulus and, hence, is unable to sustain shear waves (although the presence of magnetic fields complicates this assumption). Thus, the eigenfunctions of the background model contain no toroidal components (see Chapter 8 of Dahlen & Tromp 1998), rendering them purely spheroidal. We write the displacement field

 in the basis of vector spherical harmonics (and thereafter GSH) as follows:
10

11

Here,

 denote spherical polar coordinates, with basis vectors

 and

 where n is the radial order, ℓ is the angular degree, and m is the azimuthal order. The dimensionless lateral covariant derivative operator is denoted by

. The basis vectors in spherical polar coordinates are related to those in the GSH basis via
12


Equation (1) describes the departure of the system from a ‘background’, time-independent state of exact corotation, where this departure is driven by $\dot{\boldsymbol {\Omega }}$. As a consequence, all quantities appearing in (1) are perturbations with respect to the corotating state. For instance, the magnetic field can be decomposed into a fixed background part, $\boldsymbol {B}_0$, plus a small, spin-down-induced perturbation, $\boldsymbol {b} \equiv \delta \boldsymbol {B}$, i.e.

(4)

\begin{equation}
\boldsymbol {B} \approx \boldsymbol {B}_0 + \boldsymbol {b}.
\end{equation}

The magnetic force can therefore be expressed to leading order

(5)

\begin{equation}
\boldsymbol {F}_{\rm mag} \approx - \nabla \left(\frac{\boldsymbol {B}_0 \cdot \boldsymbol {b} }{4\pi } \right) + \frac{1}{4\pi } \left[ (\boldsymbol {b} \cdot \nabla ) \boldsymbol {B}_0 + (\boldsymbol {B}_0 \cdot \nabla ) \boldsymbol {b} \right].
\end{equation}

The nature of the coupling force, $\boldsymbol {F}_{\rm cpl}$, is determined by the interaction between the superfluid's quantized vortices and the other components of the star (e.g. protons, electrons, the magnetic field). The simplest form for this force is provided by the Hall & Vinen mutual friction force (Hall & Vinen 1956a,b)

(6)

\begin{equation}
\boldsymbol {F}_{\rm cpl} = 2\Omega _{\rm n}\rho _{\rm n}\left\lbrace {\cal B}\left[ \hat{\boldsymbol {z}} \times (\hat{\boldsymbol {z}} \times \boldsymbol {w}) \right]+ {\cal B}^\prime (\hat{\boldsymbol {z}} \times \boldsymbol {w}) \right\rbrace ,
\end{equation}

where $\boldsymbol {w}= \boldsymbol {v}_{\rm n}-\boldsymbol {v}_{\rm p}$ is the velocity lag between the neutrons and the charged particles, ${\cal B}$ and ${\cal B}^\prime$ are mutual friction coefficients (which we take to be uniform throughout the star) and a ‘hat’ denotes a unit vector. This form of the force assumes a vortex array aligned with the common spin axis. According to our setup of the problem we have,

(7)

\begin{equation}
\boldsymbol {w} \approx \varpi (\Omega _{\rm n}-\Omega _{\rm p}) \hat{\varphi } \equiv \varpi \Omega _{\rm np}\hat{\bf {\varphi }},
\end{equation}

where ϖ = rsin θ is the usual cylindrical radius. The mutual friction coupling force becomes

(8)

\begin{equation}
\boldsymbol {F}_{\rm cpl} = -2\varpi \rho _{\rm n}\Omega _{\rm n}\Omega _{\rm np}\left({\cal B}\hat{\varphi } + {\cal B}^\prime \hat{\varpi}\right).
\end{equation}

The most commonly considered mutual friction mechanism in neutron star cores is the scattering of electrons by the magnetic field of individual vortices, in which case ${\cal B}^\prime \ll {\cal B}\sim 10^{-4}$ (Alpar et al. 1984; Andersson, Sidery & Comer 2006).


The first equation describes the time evolution of f(p), the momentum distribution of the non-thermal particles, from here on supposed to be relativistic electrons, since we are considering a leptonic model:
(1)$$\begin{eqnarray*}
\frac{\partial f}{\partial t} = \frac{1}{p^2} \frac{\partial }{\partial p} \left[ p^2 D_p \frac{\partial f}{\partial p} + p^2 \left(\frac{\partial p}{\partial t} \right)_{\text{rad}} f \right] + \frac{f}{t_\text{esc}} +I_f ,
\end{eqnarray*}$$where p is the particle momentum, Dp the momentum diffusion coefficient, (∂p/∂t)rad the cooling coefficient due to radiative emission, and If is the particle injection rate. Starting from momentum distribution, it is possible to calculate the electron energy density, in fact n(γ) = 4πp2f(p)mec, where me is the electron mass. Supposing only parallel and antiparallel propagating magnetohydrodynamic (MHD) waves interacting with electrons, the acceleration time can be estimated as (Miller & Roberts 1995; Kakuwa 2016):
(2)$$\begin{eqnarray*}
t_\text{acc} = \left[ \frac{2 \beta _w^2 c}{U_B r_g}\int _{k\gt k_\text{res}} \frac{W_B(k)}{k} dk \right]^{-1} ,
\end{eqnarray*}$$where βw = va/c is the wave velocity in unit of c, UB = B2/8π is the magnetic energy density, rg = γmec2/eB is the Larmor radius, WB(k) ≈ W(k)/2 is the magnetic component of the energy density of turbulence fields per unit of k [indicated with W(k)] and kres = 2π/rg is the resonant wavenumber. Note that an electron with a fixed γ can interact only with Alfvén waves whose wavelength is smaller than its Larmor radius. From the time acceleration, the momentum diffusion coefficient can be obtained using Dp = p2/2tacc. Neglecting inverse Compton and adiabatic cooling for simplicity, the cooling time is equal to:
(3)$$\begin{eqnarray*}
t_\text{cool} = \frac{6 \pi m_e c}{\sigma _T B^2 \gamma } ,
\end{eqnarray*}$$where B is the mean magnetic field and σT is the Thomson cross-section. From the cooling time, it is possible to calculate the cooling coefficient, in fact $t_\text{cool} = p/\dot{p}$. Another effect influencing the electron spectra is the spatial escape from the acceleration site:
(4)$$\begin{eqnarray*}
t_\text{esc} = \frac{R}{c} + \frac{R^2}{k_\parallel } ,
\end{eqnarray*}$$where k∥ = crg/9ζ(kres) is the spatial diffusion coefficient along the mean magnetic field, while ζ(k) = kWB(k)/UB is the relative amplitude of the turbulent magnetic field energy density for a given k. Note that if the mean free path is large (which implies a small diffusion coefficient), the escape time is simply equal to the geometric escape time, that is the ratio between the dimension of the acceleration region and the particles velocity. Finally, the last term of equation (1) describes the particle injection, identified here as the electrons accelerated at the recollimation shock and advected downstream. Supposing a strong shock, the injected electron density per unit time is equal to:
(5)$$\begin{eqnarray*}
I_n(\gamma) = I_{n,0} \, \gamma ^{-2}\, e^{-\frac{\gamma }{\gamma _\text{cut}}} \quad \text{with} \quad \gamma _\text{inj}\lt \gamma \lt 100\, \gamma _\text{cut} .
\end{eqnarray*}$$The extremes of the injection range are fixed and they are estimated thanks to recent simulations of diffusive shock acceleration: γinj = 103 and γcut = 105 (Zech & Lemoine 2021). From the injected electron density per unit time, it is possible to obtain the injection distribution in the momentum space, in fact In(γ) = 4πp2mecIf(p).


In addition to the selection using

, we filter our sample of 1083 double-component galaxies. Our aim is to select S0 galaxies for which we can reliably measure the colors of both the bulge and disk components. To this end, we exclude 68 double-component galaxies with unphysical fits:

1.
nbulge pegged at the allowed fit limits (37 galaxies) and

2.
re,bulge or re,disk pegged at the allowed fit limits (31 galaxies).

We also exclude 404 double-component galaxies with unresolved radial parameters:

1.
re,disk is less than 5% larger than re,bulge, since the bulge is defined as the most compact component (134 galaxies), and

2.
re,bulge is smaller than 80% of the PSF half-width at half-maximum in order to recover reliable bulge properties (Gadotti 2009; Meert et al. 2015; 270 galaxies).

Finally, we exclude 142 double-component galaxies for which the separation of the bulge and disk measurements is unreliable:

1.
nbulge  0.6 to exclude bars, which typically have n ∼ 0.5 (Gadotti 2008a, 2008b; 33 galaxies), and

2.
bulge-to-total flux ratio (B/T) lower than 0.2 and larger than 0.8 to exclude galaxies dominated by one component (Meert et al. 2015; 109 galaxies).

The filter causing the exclusion of the largest number of double-component fits is due to the impact of the PSF on the geometrical properties of the bulge, which tends to make the bulge rounder and harder to constrain. The effects of the PSF on the bulge properties have been detected in several previous works (Gadotti 2008a, 2009; Bernardi et al. 2014; Meert et al. 2015). For galaxies with double-component unphysical fits or unreliable bulge/disk separation, the single-component fit is preferable. Galaxies with unresolved radial parameters are not useful for investigating separate bulge and disk properties, but they are still better characterized by a double-component fit. In conclusion, the number of SDSS+ATLAS single-component galaxies is 782 (47%), the number of SDSS+ATLAS reliable double-component galaxies is 469 (28%), and the number of SDSS+ATLAS double-component galaxies with unresolved radial parameters is 404 (25%). For the single-component sample, 388 galaxies are bulge-dominated with Sérsic index n > 2, and 394 are disk-dominated with Sérsic index n  2. For the reliable double-component sample, 418 galaxies are better described by the most complex Sérsic + exponential model, 37 by the simple Sérsic + exponential model, and 14 by the de Vaucouleurs + exponential model. The range in stellar mass of the 469 double-component galaxies is

. The galaxies with unresolved bulge and disk radial parameters are characterized by a lower galaxy stellar mass and smaller size compared to the double-component galaxies with reliable bulge and disk measurements. The median

 difference is 0.145 ± 0.037 dex, and the median galaxy size difference, measured as the single-component re, is 0469 ± 0091. Thus, the exclusion of these galaxies limits our study to double-component galaxies that tend to have a large stellar mass and size for which the bulge and disk colors are reliably estimated.


Besides sunspots, filaments are one of the most important features in studying solar magnetic fields. They are usually formed above the polarity inversion line (PIL), supported by either helically coiled or dipped magnetic fields (Gilbert et al. 2001). Therefore, filaments are representative signatures of discrete magnetic fields at large scales. Scientific observations of filaments started in the 1880s at the Arcetri Astrophysical Observatory in Florence, Italy (McIntosh et al. 2019). Thus, the long time span of the archive of filaments becomes crucial in studying the long-term variations of solar magnetic fields. Poleward and equatorward migrations of filaments have already been studied extensively, e.g., Cliver (2014), Xu et al. (2018), McIntosh et al. (2019), and Tlatova et al. (2020). Among different types of filaments (Hansen & Hansen 1975), polar crown filaments (PCFs) locate at the highest latitudes and usually reside near the polar crown cavities (Leroy et al. 1983). Their underlying PILs separate the predominate polar field of the previous cycle and the dispersed field of the current cycle, drifting poleward from the trailing polarities of sunspots or sunspot groups in lower latitudes (Tang 1987; Panesar et al. 2014). Also known as “rush-to-the-pole,” the poleward migration is an important property of PCFs and a good indicator of the polarity reversal of the solar magnetic field and the occurrence of the solar maximum (e.g., Ananthakrishnan 1952; Cliver 2014). Moreover, polar magnetic fields are difficult to measure because of the projection effect, and they are much weaker than the fields in low-latitude and active regions. Recently, Xu et al. (2018) studied the PCFs observed by the Kanzelhöhe Solar Observatory (KSO) and Big Bear Solar Observatory (BBSO) between 1973 and 2018 (cycles 21–24). Their results show the following. The poleward migration speeds of PCFs are between 0.4 and 0.7 per Carrington rotation (CR), which is about 5°–9° yr–1. This result is consistent with the speed found by Ulrich (2010) and Altrock (2014; for coronal Fe xiv emission). The tilt angles of the filaments were studied in Mazumder et al. (2018) and Diercke & Denker (2019). The authors found that the tilt angles of the filaments have opposite signs in the northern and southern solar hemispheres, as compared to the expected interactive region formation. Similar results are found in a much longer time span by Hao et al. (2015) and Tlatov et al. (2016). In addition, the asymmetric distribution of filaments in the northern and southern solar hemispheres is found to be correlated with the asymmetry of sunspot areas. More recently, McIntosh et al. (2019) studied the activity cycles near the solar equator, using a comprehensive data set going back to 1880. The authors showed that the timing of PCFs reaching the polar caps is correlated with cycle terminators, which are defined as the rapid intensity changes of extreme-UV brightenings occurring between cycles. They also suggested that PCFs are representative features in studying the polar magnetic reversal quantitatively. Tlatova et al. (2020) analyzed prominences observed in the Ca ii K line at the solar limb, starting as early as 1897 with hand drawings and digitized records after 1910. Their results show systematically higher migration speeds of PCFs than those found by Xu et al. (2018), Ulrich (2010), and Altrock (2014) but with similar variation trends. The discrepancy is likely due to several factors, such as the selection of prominences over filaments and the latitude range. Chatterjee et al. (2020) performed an automated detection of prominence latitudes from Ca ii K disk blocked images from Kodaikanal Solar Observatory (since 1906) and quantified the nonlinearity in poleward migration with its variation over 10 solar cycles combining data from KSO and Meudon Observatory in Meudon, France (Demarcq et al. 1985).


The importance of obtaining high-fidelity constraints on ΩK cannot be overstated. The sign and value of ΩK play an important role in determining the future evolution of the universe. From the model-building side, constraints on ΩK have important consequences for models of inflation, most of which predict an universe that is spatially flat to the level of

 (see, e.g., Kazanas 1980; Starobinsky 1980; Guth 1981; Mukhanov & Chibisov 1981; Sato 1981; Albrecht & Steinhardt 1982; Linde 1982). Conversely, detecting ∣ΩK∣ ≠ 0 at the

 level or larger could be a problem for most models of inflation (Linde 2008; Guth & Nomura 2012; Kleban & Schillo 2012), although others (see, e.g., Bull & Kamionkowski 2013) have argued that this might not be as problematic. It is generally simpler to construct inflationary models in a spatially open universe (see, e.g., Coleman & De Luccia 1980; Gott 1982; Ratra 1994; Ratra & Peebles 1995, 1994; Bucher et al. 1995; Linde 1995; Yamamoto et al. 1995; Linde 2008; Guth & Nomura 2012; Kleban & Schillo 2012), whereas achieving the same result in a spatially closed universe might require more fine-tuning (see, e.g., Ratra 1985; Hartle & Hawking 1987; Linde 2003; Ratra 2017). In any case, the importance of spatial curvature in modern cosmology is the reason why a huge number of works have been devoted to providing and forecasting constraints on ΩK from current and future cosmological observations.6

6
For an inevitably incomplete list of such works, see, e.g., Vardanyan et al. (2009), Carbone et al. (2011), Li & Zhang (2012), Bull et al. (2015), Takada & Dore (2015), Di Dio et al. (2016), Leonard et al. (2016), Rana et al. (2017), Ooba et al. (2018a), Jimenez et al. (2018), Ooba et al. (2018b), Park & Ratra (2018, 2019a, 2019b, 2019c, 2020), Denissenya et al. (2018), Bernal et al. (2019), Li et al. (2020), Wang et al. (2020), Zhai et al. (2020), Geng et al. (2020), Heinesen & Buchert (2020), Gao et al. (2020), Khadka & Ratra (2020), Nunes & Bernui (2020), Liu et al. (2020), Chudaykin et al. (2021), Benisty & Staicova (2020), Shimon & Rephaeli (2020), Tröster et al. (2020), Di Valentino et al. (2021), and Qi et al. (2020).

Kleban & Schillo 2012
Kleban & Schillo 2012


Any massive stellar source from ~8 M⊙ can burn hot enough to completely ionise the surrounding molecular material to form an H II region (Wood & Churchwell 1989; Churchwell 1990; Kurtz 2005) and destroy any complex chemical tracers that are traditionally used to understand the kinematics of their natal environments and their accretion discs. Initial arguments by Walmsley (1995) argued that high accretion rates will cause a very dense H II region that is optically thick to radio emission and thus the H II would not be seen. However, for heavily accreting massive YSOs the onset of the H II region could be delayed via stellar bloating (Palla & Stahler 1992; Hosokawa & Omukai 2009; Hosokawa et al. 2010; Kuiper & Yorke 2013) where the effective temperature of the star is much cooler than it would be considering a main sequence star of the same luminosity. In these models, a halt or considerable reduction in accretion (≪10−3 M⊙ yr−1), or growth beyond ~30−40 M⊙, will result in the YSO contracting to a “main-sequence” configuration, heating significantly, and being able to create an H II region. The fine details of this transition are still somewhat unclear since they depend on the assumed accretion law and the initial conditions chosen for the stellar evolution calculations (Haemmerlé & Peters 2016). Observations in the infra-red (IR) have been made in search of cool stellar atmospheres that point to bloated stars, however these studies remain inconclusive (Linz et al. 2009; Testi et al. 2010). Alternative scenarios are that H II regions can be gravitationally trapped at very early ionisation stages (see Keto 2003, 2007), or flicker due to chaotic shielding of the ionising radiation by an accretion flow, leading to a non-monotonous expansion (Peters et al. 2010b). Hyper Compact (HC) H II regions (0.03 pc) are thought to be the earliest ionisation stage and therefore could relate to the halt of accretion, be a marker of a transition phase.


Abundances are estimated by fitting the different ions in the observed spectrum. Microturbulence is again fixed at 20 $\rm km\, s^{-1}$  for all elements, the same value adopted for the Si ionization balance. All abundances are given by number and were initially adopted to be solar and then modified to reach a better fit to the observed spectrum. Most He  lines are well fitted with a solar abundance, but the strong He i 4471 Å  line points to a slightly enriched He abundance, which is consistent with the detection of He ii 4686. Thus, we adopt $\epsilon = N({\rm He})/[N({\rm H})+N({\rm He})] =0.12\pm 0.04$, a mildly enriched He  abundance, but still consistent with the solar one. The N abundance is poorly constrained, as the N iii lines around 4630–4640 are blended with O and C and are highly sensitive to wind details (see Rivero-González et al. 2011), whereas the N ii lines from 5666 to 5710, clear in this spectral type, are outside the spectral range of our observations. Thus, we have to rely on the very weak N ii 5005.15 and 5007.33 Å  lines (a blend in our spectrum). These lines are consistent with a solar N abundance. The absence of N ii 4630.54 and N iii 4634.13 (which should be present if the N abundance is higher than solar) is also consistent with a solar N abundance. Taking into account the uncertainties in stellar parameters, we determine a value of log(N/H) + 12 = 7.83 ± 0.15 (with 7.83 ± 0.05 being the solar value from Asplund et al. 2009). The abundance of C  relies on the C ii λ4267 line, which is weak. Unfortunately, the C ii 6578-83 lines are not well reproduced by our model. We estimate a C  abundance of log(C/H) + 12 = 8.13$^{+0.20}_{-0.25}$, a factor of 2 below solar, where errors have again been obtained by fitting the line with varying stellar parameters. However, given the uncertainties, the line weakness and the moderate S/N in the region, we cannot completely rule out a solar abundance. For Mg we have only the doublet line at λ4481, which fits at a slightly lower than solar abundance, but fully compatible with solar – we obtain log(Mg/H) + 12 = 7.53 ± 0.15 – where the relatively low uncertainty is probably an underestimate as we have only one line available. On the contrary, for O and Si we have several lines and thus the same uncertainty of 0.15 dex is more representative. We obtain in both cases solar abundances, with uncertainties determined by varying the stellar parameters (except the microturbulence, which we kept fixed through the whole analysis). Abundances are given in Table 3.


A gravitational lens deflects light from background sources depending on their projected distance R in the lens plane. The deflection angle $\hat{\alpha }(R)$ of a thin axially symmetric lens where the distances between the source, the lens, and the observer are much larger than the size of the lens is directly related to its cumulative mass $\mathcal {M}(R)$ through
(35)$$\begin{eqnarray*}
\hat{\alpha }(R) = \frac{4 G\mathcal {M}(R)}{c^2 R}
\end{eqnarray*}$$(Schneider, Ehlers & Falco 1992, equation 8.5), c being here the speed of light. Introducing DL, DS, and DLS the angular distances respectively between the observer and the lens, between the observer and the source, and between the lens and the source, one can express the scaled deflection angle
(36)$$\begin{eqnarray*}
\alpha (R)\equiv \frac{D_{\rm L}D_{\rm LS}}{r_\mathrm{ c} D_{\rm S}} \hat{\alpha }(R)
\end{eqnarray*}$$and the convergence
(37)$$\begin{eqnarray*}
\kappa (R) \equiv \frac{{\Sigma }(R)}{\Sigma _{\rm crit}}
\end{eqnarray*}$$where distances in the lens plane are scaled in units of rc and Σcrit = c2DS/4πGDLDLS is the lensing critical surface density. Introducing $\widetilde{\kappa _0} \equiv \widetilde{\Sigma }(0)/\Sigma _{\rm crit}$ with equation (29) and $\mathcal {B}(a,b)=\Gamma (a)\Gamma (b)/\Gamma (a+b)$, the scaled deflection angle for an untruncated DZ profile yields
(38)$$\begin{eqnarray*}
\widetilde{\alpha }(X) = \frac{\sqrt{\pi } ~\widetilde{\kappa _0} }{\Gamma (2-2a)\Gamma (5)} X^2~ H_{3,3}^{2,2} \left[ \left. \begin{array}{c}(-6,4), (-\frac{1}{2},1), (0,1)\\
(-\frac{1}{2},1), (-2a,4), (-\frac{3}{2}, 1) \end{array} \right| X^2 \right], \nonumber \\
\end{eqnarray*}$$which has a series expansion analogous to that of $\widetilde{\mathcal {M}}(X)$. For a DZ profile truncated at the virial radius, $\alpha (X) = \widetilde{\alpha }(X)-X \widetilde{\Sigma }(c)/ \Sigma _{\rm crit}$. We give analytic expressions for the lensing potential in Appendix F (Supporting Information).