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.


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.


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.


Integrating equation (2), one obtains the probability for an object at redshift zs being strongly lensed. It is

(4)$$\begin{eqnarray*}
\tau (z_s)=\frac{8}{15}\pi ^3(1+z_s)^3d_s^3\left(\frac{\sigma _{v*}}{c}\right)^4n_*\frac{\Gamma (\frac{4+\alpha }{\beta })}{\Gamma (\alpha /\beta)}.
\end{eqnarray*}$$


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.


The effects of the PSF on the bulge properties have been detected in several previous works


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).


Scientific observations of filaments started in the 1880s at the Arcetri Astrophysical Observatory in Florence, Italy


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).


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)


Poleward and equatorward migrations of filaments have already been studied extensively, e.g.,


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.


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


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


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).


Stars with masses 130 M
⊙ ≲ M
ZAMS ≲ 250 M
⊙ are subject to the pair instability, which disrupts them completely, and hence no BH forms. Stars with M
ZAMS ≳ 250 M
⊙ can collapse directly to IMBHs with a mass ≳135 M
⊙. Thus, in the standard picture, there should be an upper stellar-mass BH gap in the range [65, 135] M
⊙, and any BHs observed in this range (e.g., the primary BH of GW190521) have to form via other formation channels—for example, through hierarchical coalescence of smaller BHs or direct collapse of a stellar merger between an evolved star and a main-sequence companion (e.g., Quinlan & Shapiro 1989; Portegies Zwart & McMillan 2000; Ebisuzaki et al. 2001; Miller & Hamilton 2002; O’Leary et al. 2006; Gerosa & Berti 2017; Antonini et al. 2019; Di Carlo et al. 2019, 2020; Rodriguez et al. 2019; Gayathri et al. 2020a; Kimball et al. 2021; Mapelli et al. 2021). However, the exact mass boundaries of the gap depend on parameters that are uncertain. For example, the 12C(α, γ)16O nuclear reaction rate, which converts carbon to oxygen in the core, can affect the boundary significantly (Takahashi 2018; Farmer et al. 2020; Costa et al. 2021; Woosley & Heger 2021). Here, to better determine whether future GW observations will be able to observe BHs in the mass gap, we recompute the mass-gap boundaries with updated 12C(α, γ)16O reaction rates and increased mass and temporal resolution. The complexity of this system has made a reliable analysis of the reaction a decades-old challenge.
7

7
The reaction rate for 12C(α, γ)16O is determined by the quantum structure of the compound nucleus 16O as an α cluster system. It is characterized by the interfering ℓ = 1 waves of the J

π
 = 1− resonances and sub-threshold levels defining an E1 component for the reaction cross section as well as by the ℓ = 2 components and interference from broad J

π
 = 2+ resonances and the nonresonant E2 external capture to the ground state of 16O. In addition to these two main E1 and E2 ground-state components, transitions to higher lying excited states occur that also add to the total cross section (see, e.g., Buchmann & Barnes 2006; deBoer et al. 2017). The rapidly declining cross section at low energies has prohibited a direct measurement of the reaction at stellar temperatures, and the reaction rate is entirely based on the theoretical analysis and extrapolation of the experimental data toward lower energies. The rate newly derived by deBoer et al. (2017), using a multichannel analysis approach, derives for the first time a reliable prediction for the interference patterns within the reaction components by taking into account all available experimental data sets that cover the near-threshold energy range of the 12C(α, γ)16O process.


We gathered four independent measurements of

 from the literature: 2.312 ± 0.095 (Carter et al. 1988), 2.237 ± 0.013 (Beuing et al. 2002), 2.292 ± 0.0429

9
Wegner et al. (2003) listed a value within a 0.595

 kpc radius aperture, and this value is converted to the value for an aperture with 1/8 of

 using Equation (1) of Cappellari et al. (2006).
 (Wegner et al. 2003), and 2.212 ± 0.045 (Ogando et al. 2008). We adopt a weighted mean value,

, of the four independent measurements. Using the

 values from Table 1 after applying the Galactic extinction correction and the FP coefficients from various sources, we derive distances to NGC 4993 in seven different bands. Table 3 summarizes the result. In total, 17 different estimates are derived. Two of the FP relations are based on

, and z (Bernardi et al. 2003; La Barbera et al. 2010), and for these cases, the

, and I values are converted to

, and i values while keeping the structural parameters in the corresponding bands. Uncertainties in the distances are dominated by the intrinsic dispersion in the FP relation, which has errors of about 7 to 12 Mpc (or 23% of the derived value; e.g., Bernardi et al. 2003). The errors from the observed quantities amount to only ≲10% of the FP distance, and we ignore them. The uncertainty in the Hubble constant is only a few percent or less according to recent estimates (e.g., Riess et al. 2016) and also can be neglected. We note that the rms dispersion of the 17 estimates is 5.3 Mpc, smaller than the FP distance error of each estimate. This suggests that the 17 estimates are not independent quantities (e.g., they share a

 value and are based on an identical object), and the rms dispersion of 5.3 Mpc is an uncertainty related to the wavelength and the adopted FP relation parameters for each measurement. Therefore, we consider a typical intrinsic scatter in the FP relation along

 of 0.09 dex (23% of the value) as our distance error. Considering these factors, we adopt 37.7 ± 8.7

 Mpc as the FP-based distance. Note that this is an angular diameter distance, and at z = 0.009783 of NGC 4993 (Levan et al. 2017), the luminosity distance is 2% larger at 38.4 ± 8.9

 Mpc. We also note that the mean angular diameter distance of 37.7 Mpc is accurate to about

 Mpc in regard to uncertainties due to wavelengths and FP parameter sets.


Uncertainties in the distances are dominated by the intrinsic dispersion in the FP relation, which has errors of about 7 to 12 Mpc (or 23% of the derived value; e.g.,


For the transition range around ion scales, the turbulent energy is believed to further cascade and the kinetic physics begins (Alexandrova et al. 2013; Kiyani et al. 2015). Various forms of coherent structures (i.e., current sheet, vortex filaments) are usually found to reside near this range (i.e., between the end of the MHD range and proton scales in Lion et al. 2016; Perrone et al. 2016, 2017). Hence, in the following analysis we focus on fluctuations of similar scales, where the frequency ranges is chosen as [0.4, 3] Hz, and the timescale is [0.3, 3] s. It has been shown recently that, in case of a collisionless turbulent system as the solar wind, the intermittency, non-Gaussian fluctuations, and phase coherence of magnetic field components are interrelated (Perrone et al. 2017). We expect this relation to be present here and thus take similar procedures as in Perrone et al. (2017) to search for the intermittent events. The first step is to reconstruct the fluctuations using the bandpass filter based on wavelet transforms (Torrence & Compo 1998; He et al. 2012; Wang et al. 2014; Perrone et al. 2016). The magnetic field fluctuations are thus defined as
1

where i represent the magnetic field components, j represent the scale index,

 is the constant scales step,

 is the real part of the wavelet coefficient

, Cδ = 0.776.

 is the Morlet mother function and

 at time t = 0 (Torrence & Compo 1998),

 and

 are taken to be 0.3 and 3 s, respectively. The second step is to determine the threshold energy as defined by

, where

 is the standard deviation of the Gaussian fit for each magnetic field fluctuation component.

 are fitted to be 0.36, 0.32, and 0.49 nT, leading to a

 ∼  4.3 nT2. From a statistical point of view, 99.7% of all the values in Gaussian distribution are within

 from the mean. In other words, the events whose total energy

 are larger than

 could contribute to the non-Gaussian part of the distributions. Figure 1(e) presents the PDFs of the normalized magnetic field fluctuations

 together with their Gaussian fits. The presence of clear non-Gaussian tails suggest the abundance of intermittent events during the whole interval. The last step is to locate these events. As seen in Figure 1(b), there are approximately 14 events (with at least 10 s in duration) with

 larger than 5 nT2. We have picked one interval during 10:59:41.5–10:59:48.5 UT for further study. This event is characterized by large magnetic energy ∼150  nT2 and strong flow vorticity up to ∼1.4 /s as compared to the mean value of ∼0.4 /s. Furthermore, the local intermittency measure (LIM) exhibits an extension of temporal scale from several tens of seconds to sub-seconds, as seen in the LIMtotal of Figure 1(c). The LIM spectrogram, as a function of time and scales, is computed as the instantaneous energy of fluctuations normalized to its mean value over the studied time interval (Farge 1992):
2

where

 is the total fluctuation energy. For the coherent structures (space or time localized energetic events), one of its intrinsic properties is the energy coupling over many scales (Farge 1992; Frisch 1995), which is usually manifested in the spanning of LIM at a wide range of spatial scales (or temporal scales if the Taylor frozen-in hypothesis is assumed; Lion et al. 2016; Perrone et al. 2017). While for the wave phenomenon, the energy distribution is typically focused around a certain frequency, i.e., Alfvén ion cyclotron (Alexandrova et al. 2004) or electron cyclotron waves (Lacombe et al. 2014). Therefore, the extension of LIM provides evidence of coupling from MHD to sub-ion scales and hence implies the presence of a coherent structure (Farge 1992; Frisch 1995; Lion et al. 2016; Perrone et al. 2017). Note that because

,

, and LIMtotal exhibit nearly the same features in this event, only LIMtotal is presented here.


The magnetic field fluctuations are thus defined as
1

where i represent the magnetic field components, j represent the scale index,

 is the constant scales step,

 is the real part of the wavelet coefficient

, Cδ = 0.776.

 is the Morlet mother function and

 at time t = 0


A triple system consists of an inner binary with masses m1 and m2 (defined as m1 ≥ m2), and a third object with mass m3 (in our case, the third object is always the central SMBH). It is convenient to describe the orbit using Jacobi coordinates (see e.g. Murray & Dermott 2000, p. 441). In this treatment, the dominant motion of the triple system can be divided into two separate Keplerian orbits: one is the motion of objects 1 and 2 relative to each other, and the other one is the motion of object 3 relative to the centre of mass of the inner binary. We denote the relative position vectors, semimajor axes, and eccentricities of these systems by $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, a1 and a2, and e1 and e2, respectively. If the third object is sufficiently distant from the inner binary, a perturbative approach can be used to describe the evolution of the system. In the usual secular approximation (e.g. Marchal 1990), the two orbits exchange angular momentum, but not energy. Hence, the eccentricities and orientations of the orbits can change, but the semimajor axes cannot. The coupling term in the Hamiltonian can be written in a power series expansion of parameter α = a1/a2 (see e.g. Harrington 1968), where the Hamiltonian is
(16)
\begin{eqnarray*}
\mathcal {H} &=&\, \frac{m_1 m_2}{2a_1} + \frac{(m_1+m_2)m_3}{2a_2} \nonumber \\
&&+\, \frac{1}{a_2} \sum \limits _{j=2}^{\infty } \alpha ^j M_\mathrm{ j} \left(\frac{r_1}{a_1} \right)^j \left(\frac{a_2}{r_2} \right)^{j+1} P_\mathrm{ j}(\mathrm{cos}\Phi) ,
\end{eqnarray*}
Pj is the jth-degree Legendre polynomial, Φ is the angle between $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, and
(17)
\begin{eqnarray*}
M_\mathrm{ j} = m_1 m_2 m_3 \frac{m_1^{j-1}+(-1)^j(m_2)^{j-1}}{(m_1+m_2)^j} .
\end{eqnarray*}
When analysing these hierarchical triple systems, it is particularly convenient to adopt the canonical coordinates known as Delaunay’s elements (e.g. Valtonen & Karttunen 2006). The coordinates are the mean anomalies, l1 and l2, the longitudes of ascending nodes, h1 and h2, and the arguments of periastron, g1 and g2. One can then eliminate the short-period terms by a canonical transformation to get equations that describe the long-term evolution of the triple system. This technique is known as the Von Zeipel transformation (see e.g. Brouwer 1959). Naoz et al. (2013a) showed that going beyond the quadrupole-order approximation is necessary in order to acquire highly eccentric Kozai excitations. They also pointed out a common mistake in the Hamiltonian treatment of these hierarchical systems in previous studies, which could lead to erroneous conclusions in the test particle limit. The full octupole-order equations of motion can be found in Naoz et al. (2013a).


A detailed description of the dynamo model can be found in our previous paper (Pipin & Kosovichev 2019, hereafter PK19). The model describes the dynamo generation of large-scale magnetic fields (LSMFs) in the bulk of the solar convective zone (CZ). The model is based on the mean-field induction equation (Krause & Rädler 1980):
1

where the induction vector of the LSMF,

, is represented as the sum of the toroidal and poloidal components:

where r is the radial distance, θ is the polar angle, and

 is the unit vector in the azimuthal direction. The mean electromotive force

 describes the turbulent generation effects, pumping, and diffusion:
2

where the symmetric tensor αij stands for the turbulent generation of the LSMF by kinetic and magnetic helicities, the antisymmetric tensor γij describes the turbulent pumping effect, and the anisotropic (in the general case) tensor ηijk is the eddy diffusivity of the LSMF (Pipin 2018). The large-scale (LS) flow field,

, produces the LS toroidal magnetic field from the LS poloidal field by means of the differential rotation,

. The meridional circulation,

, advects the LSMF in the convection zone. The angular momentum conservation and the equation for the azimuthal component of large-scale vorticity,

, determine distributions of the differential rotation and meridional circulation:
3

4

where

 is the turbulent stress tensor:
5

(see detailed description in PK19). Also,

 is the mean density,

 is the mean entropy, and

 is the gradient along the axis of rotation. The mean heat transport equation determines the mean entropy variations from the reference state due to the generation and dissipation of LSMF and large-scale flows (Pipin & Kitchatinov 2000):
6

where

 is the mean temperature, Fr is the radiative heat flux, and Fc is the anisotropic convective flux. An analytical mean-field expression for Fc takes into account the effect of the Coriolis force, and the influence of the LSMF on the turbulent convection (see PK19). The last two terms in Equation (6) take into account the convective energy gain and sink caused by the generation and dissipation of LSMF and large-scale flows. The reference profiles of mean thermodynamic parameters, such as entropy, density, and temperature are determined from the stellar interior model MESA (Paxton et al. 2011, 2013). The radial profile of the typical convective turnover time, τc, is determined from the MESA code, as well. We assume that τc does not depend on the magnetic field and global flows. The convective rms velocity is determined from the mixing-length approximation
7

where

 is the mixing length, αMLT = 1.9 is the mixing-length parameter, and Hp is the pressure height scale. Equation (7) determines the reference profiles for the eddy heat conductivity,

, eddy viscosity, νT, and eddy diffusivity, ηT, as follows:
8

9

10

The model gives the best agreement of the angular velocity profile with helioseismology results for

 (PK19). Also, the dynamo model reproduces the solar magnetic cycle period, ∼22 years, if

.


Let the data be denoted $\mathscr {D}=\lbrace \boldsymbol {\hat{z}},\boldsymbol {\hat{\theta }},\boldsymbol {\hat{s}},\boldsymbol {\hat{\imath }}\rbrace$ (and possibly the experimental covariance matrix), and $\mathcal {S}$ be the proposition that we have observed some data thus passing the selection criteria. The probability that a given data set is observed depends on the selection criteria. The likelihood of observing $\mathscr {D}$ given ϑ and that we have observed some data is (Loredo 2004; Mandel, Farr & Gair 2019)
(54)$$\begin{eqnarray*}
p(\mathscr {D}\mid \mathcal {S},\vartheta) =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\mathscr {D}\mid \vartheta)}{p(\mathcal {S}\mid \vartheta)} =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\vartheta \mid \mathscr {D})}{\int \mathrm{d}\mathscr {D}^{\prime }\, p(\mathcal {S}\mid \mathscr {D}^{\prime },\vartheta)\, p(\vartheta \mid \mathscr {D}^{\prime })},
\end{eqnarray*}$$where $p(\mathcal {S}|\mathscr {D},\vartheta)$ is the selection function, and we used Bayes’ theorem in the second equality so $p(\vartheta |\mathscr {D}) \propto p(\mathscr {D}|\vartheta)p(\vartheta)$. If we assume that all objects that exceed some threshold are successfully observed then $p(\mathcal {S}|\mathscr {D},\vartheta)=1$. In the ideal scenario of no selection effects (see Section 3) clearly all possible data sets are observable. In the case of a cut-off, equivalent to replacing with a truncated distribution, as in (5), we have $p(\mathcal {S}|\mathscr {D},\vartheta)=1$ if it exceeds some threshold, and $p(\mathcal {S}|\mathscr {D},\vartheta)=0$ otherwise. Thus, $p(\mathcal {S}|\vartheta)$ is the fraction of all possible data sets that are observable given parameters ϑ. From (54) we can see that the form of the posterior distribution subject to selection effects, $p(\vartheta |\mathscr {D},\mathcal {S})$, is simply that of $p(\vartheta |\mathscr {D})$ given by (44) but with a different normalization.


The gas inside the filaments may be turbulent and the dissipation of this turbulence could, in principle, also contribute to the filament powering. In order to obtain a very rough estimate of the turbulent dissipation rate, we assume that the velocity dispersion σ in the filaments is at best comparable to the sound speed in the Hα-emitting filament gas. Otherwise, there should be evidence for shock heating but that is not observed. In general, shock heating should lead to a correlation between [N ii]/Hα emission line ratios and the velocity dispersion of the gas, but such correlations have not been detected in the Perseus and Centaurus clusters where this issue was studied (Hatch et al. 2006; Canning et al. 2011; albeit the caveat that these models did not incorporate non-thermal pressure which could affect the nature of the shocks). The absence of shocks implies that
19

where fg is fraction of thermal pressure support in the filament and cs,0 is the sound speed in the absence of any non-thermal pressure in the filament for 104 K. For fg = fcr = fB = 1/3, we get σ ∼ 24 km s−1. The turbulent power

. Typical masses of filaments are in the range from Mfil ∼ 104 to 106 M⊙ (Conselice et al. 2001). Using a filament mass of 106 M⊙ and assuming a turbulence injection scale comparable to the filament width lturb = 60 pc (c.f. Canning et al. 2016 who use smaller value), we get Lturb ∼ 2×1038 erg s−1, which is a few percent of the Hα luminosity of the resolved Horseshoe or Northern filaments in Perseus. This is not a strict upper limit on the contribution of turbulent dissipation as the filament may consist of a number of subfilaments. However, inside the filaments the Hα phase, while possibly not completely volume filling, may be more volume filling than the phase corresponding to the dense molecular gas. Furthermore, the power contributed by turbulent dissipation depends on the uncertain mass in the Hα phase in the filament, and that mass is smaller than the total Mfil mass of the filament adopted above. Most importantly however, the measurements of internal turbulence in the filaments are very difficult because of limited spatial resolution. Current measurements of the velocity dispersion inside the filaments are very likely to be significantly overestimated due to filament or subfilament confusion (Canning et al. 2016) and consequently velocity dispersions could be consistent with values lower than those adopted above. We thus conclude that it is at least plausible that turbulent heating is not the dominant powering mechanism.


On 2017 August 17, a gravitational wave (GW) event (GW170817; Abbott et al. 2017a, 2017b) was linked with a faint gamma-ray prompt emission of GRB 170817A (Goldstein et al. 2017; Savchenko et al. 2017). Immediately, GRB 170817A was followed by an extensive observational campaign covering radio, optical, and X-ray bands (see, e.g., Troja et al. 2017; Abbott et al. 2017b; Kisaka et al. 2018; D’Avanzo et al. 2018a, and references therein). The observations of the nonthermal spectrum of GRB 170817A gathered during the first ≈ 900 days after the initial merger were analyzed by several authors. It was shown that they were consistent with synchrotron forward-shock emission generated by the deceleration of an off-axis structured jet with an opening angle θ

j
 ≈ 5° that was observed from a viewing angle in the range of 15° ≤ θ
obs ≤ 25° (Troja et al. 2017; Kasliwal et al. 2017b; Lamb & Kobayashi 2017, 2018; Resmi et al. 2018; Margutti et al. 2017a; Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b; Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c). In some proposed models, the off-axis structured jet is formed with an off-axis jet with a cocoon (Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b) and a shock breakout (Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c; Urrutia et al. 2021). Recently, Hajela et al. (2022) analyzed the latest X-ray and radio observations of GRB 170817A collected with the Chandra X-ray Observatory, the Very Large Array (VLA), and the MeerKAT radio interferometer about 3.3 yr after the initial merger. These new observations did not agree with the best-fit synchrotron curves from the off-axis jet model, thus reporting evidence of a new X-ray emission component. Given these contrasting observations, the authors offered the solution to explain this phenomenon in the context of either radiation from accretion processes on the compact-object remnant or a KN afterglow.


No studies have yet provided an extensive and conclusive kinematics analysis of populations of simulated star-forming galaxies that can be broadly contrasted to the results from slit-like and IFS-like current observations. So far, large uniform-volume simulations have been analysed in order to shed light on the dichotomy and formation of fast and slow rotators (e.g. Penoyre et al. 2017; Li et al. 2018; Schulze et al. 2018, with Illustris and Magneticum) and to quantify the connection between galaxy morphology and spin on the one side and AGN feedback, environment and/or mergers on the other (Dubois et al. 2016; Rodriguez-Gomez et al. 2017; Choi et al. 2018; Lagos et al. 2018a,b, with Illustris, Horizon-AGN and EAGLE). Recently, van de Sande et al. (2019) have compared structural and kinematic properties of simulated galaxies from EAGLE, Hydrangea (Bahé et al. 2017), Horizon-AGN (Dubois et al. 2016), and Magneticum (Dolag et al., in preparation) with observed galaxies in the SAMI, ATLAS3D, CALIFA, and MASSIVE surveys. However, in most cases these works have principally focused on massive early-type galaxies at $z$ ∼ 0, which are expected to be mostly quenched. This choice has been dictated by the limited numerical resolution of such models, which is generally thought to be insufficient to properly capture even the kiloparsec-scale kinematics of intermediate and low-mass galaxies that are observationally accessible, especially in the low-redshift Universe. Other analyses have investigated the relation between stellar shapes and stellar kinematics with EAGLE and across galaxy types (Thob et al. 2019). Furthermore, there is a rich body of works on the problem of the evolution and conservation of galaxies angular momentum, also informed by the results of large uniform-volume simulations and across galaxy types (e.g. more recently Genel et al. 2015; Lagos et al. 2017, with Illustris and EAGLE). However, in most cases so far, the focus has been placed on the final $z$ = 0 outcome, or on the stellar components, or the analyses have been carried out via theoretical characterizations of galaxies properties (e.g. resolution element-based angular momenta) that cannot be easily contrasted to observations. In fact, so far, no quantitative analysis of the spatially averaged or map-based internal kinematics of star-forming galaxies within large uniform-volume simulations exists.


have compared structural and kinematic properties of simulated galaxies from EAGLE, Hydrangea (Bahé et al. 2017), Horizon-AGN (Dubois et al. 2016), and Magneticum (Dolag et al., in preparation) with observed galaxies in the SAMI, ATLAS3D, CALIFA, and MASSIVE surveys.


Other analyses have investigated the relation between stellar shapes and stellar kinematics with EAGLE and across galaxy types


The primary focus of this Letter is the spatial distribution of galaxies and their orientations projected onto the sky. While the former is characterized by the fluctuations of number density, denoted by

, the latter is quantified by the two-component ellipticity, (γ+ , γ× ), defined with the minor-to-major-axis ratio q on the celestial sphere:
1

with ϕx being the misalignment angle relative to the reference axis. We will below set q to zero for simplicity, which corresponds to the galaxy being assumed to be a line along its major axis (Okumura et al. 2009). In the weak-lensing measurements, a more convenient characterization of the ellipticity distribution is the rotation-invariant decomposition called E-/B-modes,

 (Kamionkowski et al. 1998; Crittenden et al. 2002), and these are defined, in Fourier space, by

, where

 are the Fourier counterpart of the ellipticity fields, and ϕk is the azimuthal angle of the wavevector projected on the celestial sphere, measured from the x-axis. Then, we consider the two-point statistics among

 and

. To quantify the cosmological information encoded in these statistics, we adopt the LA model as mentioned above. In Fourier space, it is given by
2

with

 being the redshift-dependent coefficient (Okumura & Taruya 2020). Here we used the Poisson equation to relate the gravitational potential to the mass density field,

. Note that the observable ellipticities are density-weighted, i.e.,

, but at large scales, the term

 is higher order and can be ignored. Then Equation (2) leads to γB = 0, and the nonvanishing two-point statistics in Fourier space become the auto-power spectra of the galaxy density and E-mode ellipticity, and their cross power spectrum, which we respectively denote by Pgg, PEE, and PgE. In redshift space, where the line-of-sight position of galaxies is determined by the redshift, the observed galaxy density field is affected by the effect of RSD. Furthermore, the ellipticity of galaxies is measured on the celestial sphere normal to the line of sight. Thus, all the power spectra considered here exhibit anisotropies along the line-of-sight direction, and denoting the directional cosine between the wavevector and line-of-sight direction by μ, they are expressed as a function of k and μ. In the linear theory limit, we have (see Okumura & Taruya 2020, for their configuration-space counterparts)
3

4

5

Here, we assume the linear bias relation between the galaxy and matter density fields, and b1 is the coefficient. The quantity f is the linear growth rate defined by

 with a and D being, respectively, the scale factor of the universe and linear growth factor, and

 is the linear-order matter power spectrum at the redshift z.


Figures 4 and 5 display two enlarged regions from Figure 3, Region 1 being quiet Sun and Region 2 being plage. We have plotted a variety of statistics in these plots that measure the components of the profiles. The emission at h1v generally originates in the upper photosphere. Based on inspection of this and additional higher spatial resolution IRIS data sets, we determine that the structures are a mix of reverse granulation and grains. At h2v, much of the small granular structure disappears and a diffuse halo of bright structures overlay and surround magnetic concentrations. The brightest plage is almost twice as bright as the brightest network, and the maximum h2v emission occurs near the center of the magnetic flux concentration. Single-peaked profiles are relativity common (∼10%) in plage, as depicted by the contiguous red regions in Figure 4. Figure 6(a) shows an example of a single-peaked plage profile. While our algorithm separates between single-peaked and double-peaked profiles to aid in statistical analysis, there is in fact a smooth continuum of weakly separated, weakly reversed profiles that bridge those two profile categories (Figure 6(c) shows an example). In the semi-empirical models of the solar atmosphere, the Mg ii h source function above the temperature minimum has a local maximum at an altitude of 1.2 Mm (see Figure 7 in Leenaarts et al. 2013a), while the line core forms near a height of 2 Mm. To produce a single-peaked profile the atmosphere must be structured such that source function does not vary significantly between

 at h2 and h3. At high densities, the source function closely adheres to the Planck function, which rises through the chromosphere. These profiles are consistent with the model that hot high-density loops are rooted in plage. Most sunspot profiles are single peaked, and an example is displayed in Figure 6(e). Single-peaked umbral profiles were previously reported in Lites & Skumanich (1982) and Morrill et al. (2001). Sunspot profiles are half as bright at h2 and 20% narrower at h1 than plage. The far wing intensity (beyond h1) is lower in sunspots than plage or internetwork. Single-peaked profiles also occur in internetwork regions, albeit at much lower frequency (∼1%). Figure 6(b) shows a typical internetwork single-peaked profile. Although this profile contains a single maximum, it still requires a substantial contribution from the negative Gaussian amplitude (h in Equation (1)) to achieve a low

 Figures 6(d) and (f) show profiles that occur at low frequencies in the internetwork (

). These profiles represent some of the extreme variations from the mean profile shape that we observe in this large data set. The profile of Figure 6(d) could be produced by extreme upflows in the upper chromosphere that completely mask the photons in the nominal position of the h2v peak and wing. Figure 6(f) illustrates Mg ii h in absorption, which requires that the source function monotonically decrease from the photosphere precluding the mid-chromospheric temperature rise.


For case 1, we find that most of the stellar values of the MW halo ([Fe/H]  −1) reside between the evolutionary tracks explored here and also in the coloured region. Note that the carbon stars are excluded in Fig. 5, since they might have been contaminated by the binary mass transfer from former asymptotic-giant-branch stars. The smaller SFE for a less-massive galaxy leads to an increase of [Eu/Fe] at a lower [Fe/H]. It is notable that our model predicts that almost all highly r-process-enhanced stars ([Eu/Fe] > 1 at [Fe/H] ∼ −3) originate from UFD-sized (M*  105) building blocks [but see a caution in Section 4 for the applicability of the mass–metallicity relation of Kirby et al. (2013) to UFD-sized systems], as also suggested in Ishimaru et al. (2015) and Ojima et al. (2018). For instance, our model suggests that J1521−3538 (star, the highest measured Eu/Fe, Cain et al. 2020) and CS 31082−001 (diamond, Hill et al. 2002) were born in building-block galaxies of M* ∼ 103 and 5 × 103 (in M⊙), respectively. This is a consequence of the fact that a less-massive galaxy contains a smaller amount of gas to be mixed with Eu from an NSM. In fact, the highly eccentric orbit of the former indicates that J1521−3538 originates from a building-block galaxy subsequently accreted by the MW (Cain et al. 2020). In contrast, the r-process-deficient stars may have been born in the most-massive building-block galaxies of M* ∼ 108 M⊙, like HD 4306 (circle, the lowest measured Eu/Fe, Ishimaru et al. 2004) and HD 122563 (triangle, Honda et al. 2006). Note that our model predicts the presence of even further r-process-deficient stars with [Eu/Fe]  −1 at [Fe/H] ≲ −3. The absence of such stars with measured Eu is likely due to the current detection limit for Eu (Ishimaru et al. 2004). Overall, the result for case 1 (a mean delay of 0.1 Gyr in the lognormal term for NSMs) appears in qualitative agreement with those in Ishimaru et al. (2015) and Ojima et al. (2018) with a fixed delay of 0.1 Gyr (see Appendix A for additional tests).


It is notable that our model predicts that almost all highly r-process-enhanced stars ([Eu/Fe] > 1 at [Fe/H] ∼ −3) originate from UFD-sized (M*  105) building blocks

as also suggested in Ishimaru et al. (2015) and


Overall, the result for case 1 (a mean delay of 0.1 Gyr in the lognormal term for NSMs) appears in qualitative agreement with those in Ishimaru et al. (2015) and

with a fixed delay of 0.1 Gyr (see Appendix A for additional tests).


In contrast, the r-process-deficient stars may have been born in the most-massive building-block galaxies of M* ∼ 108 M⊙, like HD 4306


As to the chemical enrichment history of individual galaxies, we have exploited the standard code che-evo incorporated into GRASIL (Silva et al. 1998, 2011; Bressan et al. 2002; Panuzzo et al. 2003; Vega et al. 2005). For spheroidal galaxies, it reproduces the observed local relationship between stellar metallicity and stellar mass and its weak evolution with redshift (see Arrigoni et al. 2010; Spolaor et al. 2010; Gallazzi et al. 2014). For disk galaxies at z ≲ 2, it reproduces the observed relationship between gas metallicity and stellar mass, including its appreciable redshift dependence (see Andrews & Martini 2013; Zahid et al. 2014; de la Rosa et al. 2016; Onodera et al. 2016). In both cases, within an individual galaxy the metallicity behavior is closely approximated by
4

i.e., it increases from Z = 0 almost linearly with the galactic age, and then after a time τ = Δτψ it saturates to the value Zsat. The dependence of Zsat and Δ on SFR/stellar mass can be described with an expression inspired by analytic chemical evolution models (see Cai et al. 2013; Feldmann 2015); this yields
5

here s ≈ 3 is the ratio between the dynamical timescale of the infalling gas and the star formation timescale,

 is the recycling stellar mass fraction,

 is the metal yield (assuming the Romano et al. 2010 stellar yields), and out is the mass-loading factor of galactic outflows from stellar winds and SN explosions. In the above equation we have provided a fit for out as a function of the final stellar mass M⋆ from the results of the che-evo code; a similar expression concurrently describes the time-averaged outcome from hydrodynamical simulations of stellar feedback (e.g., Hopkins et al. 2012). As a result, typical values

 are obtained for galaxies with final stellar masses in the range M⋆ ∼ 109–1011 M⊙, respectively (see, e.g., Chruslinska et al. 2019); the related quantity Δ ∼ 0.1–0.3 specifies how quickly the metallicity saturates to such values as a consequence of the interplay between cooling, dilution, and feedback processes. Note that several chemical evolution codes present in the literature, reproducing comparably well observations on the chemical abundances in galaxies of different stellar masses, also share a similar age-dependent metallicity behavior.


The dependence of Zsat and Δ on SFR/stellar mass can be described with an expression inspired by analytic chemical evolution models (see


As to the chemical enrichment history of individual galaxies, we have exploited the standard code che-evo incorporated into GRASIL


In the above equation we have provided a fit for out as a function of the final stellar mass M⋆ from the results of the che-evo code; a similar expression concurrently describes the time-averaged outcome from hydrodynamical simulations of stellar feedback (e.g.,


The unsmoothed normalized higher order cumulants or $S_N = \langle \delta ^N \rangle _c/\langle \delta ^2 \rangle ^{N-1}_c$ can be expressed in terms of the tree-level vertices denoted as νN using the following expressions (Bernardeau et al. 2002a):
(20)$$\begin{eqnarray*}
&& S_3 = 3\nu _2; \quad S_4 = 4\nu _3+ 12\nu _2^2; \quad S_5 = 5\nu _4+ 60\nu _3\nu _2 + 60\nu _2^3. \quad
\end{eqnarray*}$$The vertices νN are the angular averages of the mode-coupling kernels FN defined in equation (1c) i.e. νN = N!〈 FN〉 introduced in Section   2.2 in the Fourier domain.
(21)$$\begin{eqnarray*}
&& \nu _N = N!\langle F_N \rangle = N! \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_1} \over 4\pi } \cdots \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_N} \over 4\pi } F_N({\bf k}_1.\cdots , {\bf k}_N);\quad \mathrm{ d}{{\bf \widehat{\Omega }}}_k = \sin \theta _k \mathrm{ d}\theta _k \mathrm{ d}\varphi _k.
\end{eqnarray*}$$The following generating function approach was introduced in Bernardeau (1992, 1994b). The generating functions ${\cal G}_{\delta }(\tau)$ are solved using the equations of gravitational dynamics encapsulated in Euler–Continuity–Poisson equations. Here, τ plays the role of a dummy variable. In the perturbative regime, the νN parameter can be computed for an arbitrary N(22)$$\begin{eqnarray*}
&& {\cal G}_{\delta }(\tau) = \sum _{n}{\nu _N \over N!}\tau ^N = -\tau + {12 \over 14} \tau ^2 - {29 \over 42} \tau ^3 + {79 \over 147} \tau ^4 - {2085 \over 5096} \tau ^5 + \cdots
\end{eqnarray*}$$Next, using equation (20), the one-point cumulants in 2D (Munshi et al. 1999b), denoted as ΣN as opposed to SN parameters which represent the cumulants in 3D, can be used to compute the cumulants to arbitrary order in 2D (Munshi et al. 1999b)
(23)$$\begin{eqnarray*}
&& \Sigma _3 = {36\over 7}; \quad \Sigma _4={2540 \over 49}; \quad \Sigma _5={793}; \quad \Sigma _6=16370;
\end{eqnarray*}$$The generalization of the one-point cumulants i.e the SN parameters to the two-point cumulant correlators $C_{pq} = \langle \delta ^p_1\delta _2^q\rangle _c/\langle \delta ^2\rangle _c^{p+q-1}\langle \delta _1\delta _2\rangle _c$ or Cpq parameters was introduced in Bernardeau (1996a). The lower order normalized cumulant correlators can also be expressed in terms of the tree-level vertices νN just as the one-point cumulants introduced in equation (20)
(24)$$\begin{eqnarray*}
&& C_{21} = 2\nu _2; \quad C_{31} = 3\nu _3+ 6\nu _2^2; \quad C_{41} = 4\nu _4 + 36 \nu _3\nu _2 + 24 \nu _2^3.
\end{eqnarray*}$$The corresponding quantities for convergence maps can be defined in an analogous manner and are denoted with a superscript κ i.e. $C^{\kappa }_{pq}$. To compare with observed or simulated data smoothing of the field is necessary. The smoothed generating function ${\cal G}^s_{\delta }$ can be computed from the unsmoothed generating function ${\cal G}_{\delta }$. The generating functions ${\cal G}^s_{\delta }$ and ${\cal G}_{\delta }$ are related by the following implicit relation (Bernardeau 1995):
(25)$$\begin{eqnarray*}
&& {\cal G}_{\delta }^s(\tau)= {\cal G}_{\delta }(\tau [1+ {\cal G}_{\delta }^s]^{-(2+n)/4}).
\end{eqnarray*}$$A top hat smoothing window is assumed and the power spectrum is approximated locally as a power law P(k) ∝ kn (Bernardeau 1995; Munshi et al. 1999b). For other window functions, e.g. Gaussian window generic results are not possible for arbitrary N. However, an order-by-order approach can be adopted to obtain the lower order cumulants (Matsubara 2003). Notice that the smoothed power law depends on the spectral index while unsmoothed vertices depend solely on the gravitational collapse in 3D spherical or cylindrical in 2D. The smoothed vertices can be recovered by Taylor-expanding the smoothed generating function ${\cal G}^s$. Using these vertices, it is possible to now compute the 2D skewness Σ3 and kurtosis Σ4 can be computed (Munshi et al. 1999b)
(26a)$$\begin{eqnarray*}
&& \Sigma _3 = {36 \over 7} -{3\over 2} (n+2);
\end{eqnarray*}$$(26b)$$\begin{eqnarray*}
&& \Sigma _4 = {2540 \over 49} - 33(n+2) + {21 \over 4}(n+2)^2.
\end{eqnarray*}$$


denoted as ΣN as opposed to SN parameters which represent the cumulants in 3D, can be used to compute the cumulants to arbitrary order in 2D


A top hat smoothing window is assumed and the power spectrum is approximated locally as a power law P(k) ∝ kn


Using these vertices, it is possible to now compute the 2D skewness Σ3 and kurtosis Σ4 can be computed


Interestingly, the presence of such an underdense librating region at the leading side of the planet has also been observed in simulations of giant planets which undergo inward runaway migration (Artymowicz 2004; D'angelo & Lubow 2008). Here, it is worth noting that the low-density region does not result from the gap opening process but is rather due to radiative effects. Never the less, using the terminology employed in the framework of Type III migration, Fig. 3 shows that radiative effects can give rise to a coorbital mass deficit given by (Masset & Papaloizou 2003)

(8)

\begin{equation}
\delta \mathrm{m}=4\pi a x_\mathrm{s}\left(\Sigma _\mathrm{s}-\Sigma _h\right),
\end{equation}

where Σs is the surface density at the upstream separatrix and Σh the mean surface density in the librating region. In order to provide an estimation for the coorbital mass deficit in the librating region, we make in the following the assumption that the only contribution to the density perturbation in the librating region results from entropy advection. In reality, however, the density perturbation features an additional contribution linked to the production of vortensity at the outgoing separatrix (Paardekooper et al. 2011). Ignoring this contribution gives the following expression for the surface density Σ− in the librating region (Baruteau & Masset 2008; Paardekooper et al. 2011):

(9)

\begin{equation}
\Sigma _-=\Sigma _\mathrm{s}\left(1-2\frac{\mid \xi \mid x}{\gamma a}\right) \quad \mathrm{for} \quad 0<x<x_\mathrm{s}.
\end{equation}

Here, we have also ignored the radial density gradient and assumed that the corotation torque was initially unsaturated. This gives a mean surface density in the librating region:

(10)

\begin{equation}
\Sigma _h=\Sigma _\mathrm{s}(1-\frac{\mid \xi \mid x_\mathrm{s}}{\gamma a}).
\end{equation}

The coorbital mass deficit then becomes:

(11)

\begin{equation}
\delta \mathrm{m} = 4\pi \mid \xi \mid x_\mathrm{s}^2 \Sigma _\mathrm{s}/\gamma ,
\end{equation}

which shows that the coorbital mass deficit scales with the entropy gradient and the width of the horseshoe region. We note in passing that at the boundary corresponding to $\dot{a} \,{\sim }\, \dot{a_f}$, we expect the corotation torque exerted on the planet to be given by (e.g. Papaloizou et al. 2007):

(12)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{1}{2}\dot{a_f}\Omega _\mathrm{p}a \delta \mathrm{m}.
\end{equation}

Using the expression for the coorbital mass deficit given by equation (11), the previous equation then becomes:

(13)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{3}{4}\frac{\xi }{\gamma }\Sigma _\mathrm{s} x_\mathrm{s}^4\Omega _\mathrm{p}^2,
\end{equation}

so that we recover the standard expression for the entropy-related horseshoe drag (Baruteau & Masset 2008). For $\dot{a} > \dot{a_f}$, however, the expression for the unsaturated corotation torque given by equation (12) is no longer valid and should be modified (Papaloizou et al. 2007). In that case, we expect the corotation torque to be prevented from saturation since the horseshoe region does not longer extend to the full 2π in azimuth, so that phase-mixing cannot occur. In isothermal discs with a large-scale vortensity gradient, Ogilvie & Lubow (2006) found that drift rates with $\dot{a} > \dot{a_f}$ can even lead to torques that are much larger than the unsaturated value in absence of migration. They showed that this arises due to corotational torques caused by a vortensity asymmetry in the coorbital region between gas on the leading and trailing sides of the planet. In the limit of low viscosity, the trapped material tends to conserve its initial vortensity as the planet migrates, leading to a possibly significant vortensity contrast between the librating region and the local disc. In that case, the impact on migration depends in fact on the sign of the vortensity deficit (Paardekooper 2014) which is defined as

(14)

\begin{equation}
\delta (\omega /\Sigma )=1-\frac{\omega _a/\Sigma _a}{\omega _{0}/\Sigma _{0}},
\end{equation}

where ωa/Σa is the local disc vortensity at the position of the planet and ω0/Σ0 the vortensity at the initial location of the planet. In the case where the planet migrates in the direction set by the vortensity-related horseshoe drag, the vortensity deficit is positive and this yields a positive feedback on migration, whereas if the planet migrates in the opposite direction to that set by the corotation torque, the vortensity deficit is negative and this gives rise to a negative feedback on migration (Paardekooper 2014).


Using the expression for the coorbital mass deficit given by equation (11), the previous equation then becomes:

(13)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{3}{4}\frac{\xi }{\gamma }\Sigma _\mathrm{s} x_\mathrm{s}^4\Omega _\mathrm{p}^2,
\end{equation}

so that we recover the standard expression for the entropy-related horseshoe drag


Standard data reduction was applied to the data taken with HIPO (red and blue) and FPI+. This includes bias and dark subtraction, and in the case of FPI+ also flat field correction. For FLITECAM we did not acquire bias frames, as bias contributions are generally very low for this type of NIR detector array. Furthermore, it is complicated and time consuming to obtain a reliable flat field on such a narrow band filter as the 1.9 μm Paschen-α continuum filter. Since it was not possible to take long enough exposures during this campaign, we used K-band flat fields taken on the same flight before our observation run, which, however, did not improve the photometric precision significantly. Similarly, dark subtraction did not show any improvement in photometric precision. Additionally, we corrected for the sky background using dithered images taken during the observation run. Stars were detected with the Source Extractor by Bertin & Arnouts (1996). Aperture photometry was applied using Image Reduction and Analysis Facility’s (IRAF Tody 1993) DAOPHOT (Stetson 1987) using circular apertures. The optimal aperture radius with the lowest noise level was found to be six pixels for FLITECAM and 12 pixel for HIPO and FPI+. As part of the DAOPHOT routine, an annulus around the target was used to estimate the sky background in each exposure. Next to GJ 1214, we extracted the light curves of two additional bright stars within our field of view (see Table 1 and Fig. 1). To identify stars in the images we calculated a rough astrometric solution for each image using data provided by Astrometry.net (Barron et al. 2008). In the FLIPO setup SOFIA does not provide an image rotator to compensate field rotation during long integrations. This introduces a rotation of the images over time. Due to SOFIA’s unique setup, the telescope must periodically undergo so-called “line-of-sight (LOS) rewinds”. The required frequency of LOS rewinds depends on the rate of field rotation experienced by the target, which is a complex function of the position of the target in the sky relative to that of the aircraft heading. These need to be carefully timed with regard to the transit observation, to not interfere with, for example, ingress or egress. While we kept the target star, GJ 1214, in boresight, the comparison stars moved over the CCD due to this field rotation. This is one of the main factors introducing systematic noise and limiting the photometric precision of the instrument and is another reason why reliable flat fields are crucial for this kind of time series observation. After comparing the light curves, the brighter star was selected as comparison star to correct for first-order systematic effects present in all light curves.


To identify stars in the images we calculated a rough astrometric solution for each image using data provided by Astrometry.net


Dust grains are modelled using Lagrangian tracer particles, where each simulated particle represents a collection of dust grains with similar properties and motions. Trajectories of the particles are integrated using the fully implicit method of Bai & Stone (2010), which we have incorporated into the VL integrator in Athena. In a Cartesian coordinate system, Athena solves an equation of motion for each particle given by

(12)

\begin{equation}
\frac{{\rm d} {{\boldsymbol v}_i}}{{\rm d}t} = -\frac{{{\boldsymbol v}_i} - {\boldsymbol u}}{t_{\rm stop}},
\end{equation}

with ${{\boldsymbol v}_i}$ the velocity vector of particle i, ${\boldsymbol u}$ the local gas velocity vector, and tstop the particle stopping time due to gas drag. Neglecting grain charges and assuming only pure hydrogen gas, the (collisional) drag law is given by (Draine & Salpeter 1979)

(13)

\begin{equation}
\frac{{\rm d} {\boldsymbol v}_i}{{\rm d}t} \approx -\frac{2 \pi a^2 n k_{\rm B} T G_0(s)}{(4/3)\pi \rho _{\rm d} a^3},
\end{equation}

with

(14)

\begin{equation}
G_0(s) \approx \frac{8 s}{3 \sqrt{\pi }} \left(1 + \frac{9\pi }{64} s^2\right)^{1/2}
\end{equation}

and

(15)

\begin{equation}
s \equiv \left(\frac{m_{\rm H} {\boldsymbol v}_{\rm rel}^2}{2 k_{\rm B} T}\right)^{1/2},
\end{equation}

where a is the dust grain radius, kB is the Boltzmann constant, T is the temperature of the gas, n is the gas number density, ρd is the internal density of the dust (which we treat as constant at ρd = 3.0 g cm−3), mH is the mass of hydrogen, and ${\boldsymbol v}_{\rm rel} \equiv {{\boldsymbol v}_i} - {\boldsymbol u}$ is the relative velocity difference between the dust and gas. The stopping distance is evaluated as

(16)

\begin{equation}
t_{\rm stop} = \frac{\sqrt{\pi }}{2\sqrt{2}} \frac{a \rho _{\rm d}}{n \sqrt{m_{\rm H} k_{\rm B} T}} \left(1 + \frac{9\pi m_{\rm H}}{128 k_{\rm B} T} {\boldsymbol v}^2_{\rm rel}\right)^{-1/2}.
\end{equation}

The gas properties (n, T, ${\boldsymbol u}$) at each particle's location are calculated from nearby grid points using a triangular-shaped cloud (TSC) interpolation scheme (Hockney & Eastwood 1988). There is no momentum feedback from the particles on the gas.


The bottom panel of Fig. 9 shows the mass accretion histories for Z = 10−2 (-2Sol), 10−1, (-1Sol_from-2Sol), and Z⊙ (Sol_from-2Sol), starting from the envelope structure of Z = 10−2 Z⊙. In the early phase of M* ≲ 100 M⊙, while radiative feedback is still weak, the accretion histories are mostly determined by the initial condition and very similar to each other. At later time, radiative feedback starts operating and the accretion history depends on the metallicity in a way that the final mass is lower, or the feedback is stronger, at lower metallicities. This metallicity dependence comes from the dust attenuation of ionizing photons, which can be seen in the following analytic argument. The photoionization feedback quenches the accretion by the photoevaporation of an accretion disc, which proceeds most efficiently near the gravitational radius
(19)$$\begin{eqnarray*}
r_{\rm g} &=& \frac{GM_{*}}{c_{\rm i}^2} \nonumber \\
&\simeq & 1.1 \times 10^3 ~{\rm au} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)
\end{eqnarray*}$$(e.g. Hollenbach et al. 1994; McKee & Tan 2008), where ci is the sound speed of the ionized gas for which we assume T = 3 × 104 K. Equations (15) and (19) indicate that the gravitational radius is located outside the dust destruction front and thus the ionizing photons around the gravitational radius is attenuated by dust grains. Using the disc surface density at r ∼ rg set by the ionization balance,
(20)$$\begin{eqnarray*}
n_0 &=& \left(\frac{3 S_*}{4 \pi r_{\rm g}^3 \alpha _{\rm B}} \right)^{1/2} \nonumber \\
&\simeq & 2 \times 10^{7} ~{\rm cm^{-3}} \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)^{-3/2},
\end{eqnarray*}$$where S* is the stellar emissivity of ionizing photons, we find the optical depth from the star to the gravitational radius,
(21)$$\begin{eqnarray*}
\tau _{\rm d} &=& \rho \kappa _{\rm d} r_{\rm g} \nonumber \\
&=& 3.0 \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{200~M_{\odot }} \right)^{-1/2} \left(\frac{\kappa _{\rm d}}{350~{\rm cm^2 g^{-1}}} \right) \nonumber\\
&&\times \,\left(\frac{Z}{10^{-2}~Z_{\odot }} \right),
\end{eqnarray*}$$larger than unity for Z ≳ 10−2 Z⊙, where κd is the dust opacity for Lyman-limit frequency at the solar metallicity, meaning that the photoionizing radiation is substantially attenuated by the dust, especially at higher metallicity. This explains the trend of weaker feedback at higher metallicity seen in Fig. 9. According to recent studies (Tanaka, Tan & Zhang 2017; Kuiper & Hosokawa 2018; Tanaka et al. 2018), the impact of photoionization is significantly diminished by the dust attenuation at the solar metallicity. In fact, in our cases with Z = 0.1 (-1Sol_from-2Sol) and 1 Z⊙ (Sol_from-2Sol), the photoionization effect being disabled by this process, the accretion continues until the star becomes very massive with >300 M⊙ and the radiation-pressure feedback finally shuts off the inflow. In this process, the radiation force by the infrared dust re-emission also contributes in terminating the accretion (see equation 17). In particular, it can also repel the flow coming from the shade of the disc, for which the radiation force by the stellar direct light is ineffective (e.g. Yorke & Bodenheimer 1999; Kuiper et al. 2010b; Kuiper & Yorke 2013). Our results suggest that the metallicity dependence of the radiation-pressure effect is weaker than that of the photoionization owing to the dust attenuation in metal-enriched environments. As a result, the final stellar mass increases with increasing metallicity for Z > 10−2 Z⊙.


We scale the maps of NGC 4710 to a system that can be directly compared with the Milky Way bulge and compare the resulting maps with those constructed by Zoccali et al. (2014) based on interpolation of the GIBS survey rotational curves. In this comparison we found that the rotation map of NGC 4710 and the rotation map of the Milky Way bulge show a remarkable similarity. Although the velocity dispersion map of the bulge of NGC 4710 appears noisier than that of the Milky Way bulge, owing to the way the maps are constructed, we see that the velocity dispersion map of the Milky Way bulge is more vertically elongated in the centre than in the velocity dispersion map of NGC 4710. We see a similar change in the velocity dispersion maps constructed using different bar orientation angles in the simulation from Cole et al. (2014). This is also in good agreement with the features identified in the LOS kinematic maps of BP bulges modelled by Iannuzzi & Athanassoula (2015). We thus suggest that central increase in the velocity dispersion profile of the Milky Way bulge can be partially the consequence of the Milky Way bar and BP bulge viewing angle of 27° with respect to the Sun-Galactic centre LOS. However, a comparison of the rotation and velocity dispersion curves at different heights from the plane, used by Zoccali et al. (2014) to obtain the rotation maps, are very similar to those of the bulge of NGC 4710. These 1D rotation profiles are also in good agreement with the simulation of a galaxy hosting a pure BP bulge. The differences identified between the inner regions of the σ maps of the Milky Way and NGC 4710 are not seen in the 1D curves. This suggests that the vertical elongation observed in the central region of the σ map of the Milky Way bulge could be an artefact from the plane interpolation method to account for the σ peak observed at b = −2. Instead, the σ peak in the Milky Way bulge would be limited to Galactic latitudes | b |   2 (~0.28 kpc).


We scale the maps of NGC 4710 to a system that can be directly compared with the Milky Way bulge and compare the resulting maps with those constructed by

based on interpolation of the GIBS survey rotational curves.

In this comparison we found that the rotation map of NGC 4710 and the rotation map of the Milky Way bulge show a remarkable similarity.

Although the velocity dispersion map of the bulge of NGC 4710 appears noisier than that of the Milky Way bulge, owing to the way the maps are constructed, we see that the velocity dispersion map of the Milky Way bulge is more vertically elongated in the centre than in the velocity dispersion map of NGC 4710.


However, a comparison of the rotation and velocity dispersion curves at different heights from the plane, used by

to obtain the rotation maps, are very similar to those of the bulge of NGC 4710.