If CO intensity profile is available, we follow the procedure described in


For all galaxies the extended profiles of $\Sigma _\mathrm{H\,\small {I}}$ obtained from observations at the WSRT telescope were found in Yim & van der Hulst (2016) and Noordermeer et al. (2005). The surface density of $\Sigma _\mathrm{H_2}$ is usually calculated from observations of CO molecule lines, but the exact value of the conversion factor XCO remains debatable. If CO intensity profile is available, we follow the procedure described in Yim & van der Hulst (2016) to obtain $\Sigma _\mathrm{H_2}$. For NGC 4258, NGC 4725 (Yim & van der Hulst 2016), and NGC 2985 (Young et al. 1995), the CO(J = 1 → 0) line was used to calculate $\Sigma _\mathrm{H_2}$:

\begin{equation*}
\Sigma _\mathrm{H_2} (\text{M}_{\odot }\,\mathrm{pc}^{-2}) = 3.2\times I_\mathrm{CO}(\mathrm{K\,km\,s^{-1}})\,.
\end{equation*}
However, for three of four remaining galaxies only the total mass of molecular hydrogen $M_\mathrm{H_2}$ was found. Regan et al. (2001) make a conclusion based on the BIMA survey data that $\Sigma _\mathrm{H_2}(R)$ profile decreases exponentially with a scale equal to the scale length of the disc h on the average. Therefore for these remaining galaxies we found central surface densities from the formula of the total mass of the molecular disc $\displaystyle \Sigma _\mathrm{H_2}(0) = \frac{M_\mathrm{H_2}}{2 {\pi} \, h^2}$ and then extended them exponentially with the appropriate h scale length. If more than one photometry is used for an examined galaxy this approach gives several possible profiles of $\Sigma _\mathrm{H_2}(R)$ and several estimates of Q, respectively both for one-fluid and two-component cases. We approximate the $\Sigma _\mathrm{H_2}(R)$ and $\Sigma _\mathrm{H\, \small {I}}(R)$ profiles using linear interpolation between points. The total gas surface density was corrected for the presence of helium and other heavy elements using $\Sigma _\mathrm{g} = 1.36\,(\Sigma _\mathrm{H\,\small {I}} + \Sigma _\mathrm{H_2})$ formula (see for example Leroy et al. 2008).


We have also plotted the scaling relation of S and

 as suggested by the Schmidt SF relation

 (green dashed line). The relation is normalized such that the line passes through the MW average

 For data points that lie around this relation,

 is smaller for higher densities. For

 the ISM around the Schmidt relation has a fV,hot of only a few percent. Surprisingly, for ISM in star-burst regimes, little hot gas exists and the medium is thermally stable. Observationally, however, powerful winds are ubiquitously associated with such high SF rates, and the mass loading (

) is on the order of unity or even higher (e.g., Heckman et al. 2000; Steidel et al. 2004). Note that we have found the PEH rate to be a very important factor in determining the thermal state of the ISM. If the PEH rates we have adopted are reasonable, then the data seem to suggest that one needs other mechanism(s) to drive a wind for the high density regions. For example, “clustering” of SNe may work, although a factor of 10 or more increase in S is needed for

 above the average value to have a thermally driven wind. Another factor is the pre-SN feedback, such as photo-ionization, radiation pressure and winds from massive stars, which can create low density tunnels, facilitating SN energy to leak out (e.g., Rogers & Pittard 2013). Alternatively, runaway OB stars, which can easily migrate more than a hundred parsecs, may lead to a significant fraction of core collapse SNe exploding outside the immediate high density SF clouds. For a low density medium

 the critical SN rate for the ISM to have a thermal runaway is much less stringent. Another possible mechanism for wind launching is by the cosmic rays. As the cosmic rays diffuse out, the pressure gradient exerts accelerating force on the baryonic gas. Recent simulations have shown that this mechanism is promising to drive winds with a reasonable mass loading (Uhlig et al. 2012; Hanasz et al. 2013; Salem & Bryan 2014).


The momentum equations for the neutron and proton–electron fluids in the MHD approximation are (Mendell 1991a, 1991b; Glampedakis et al. 2011)
5

6

where

 is the smooth-averaged velocity of the proton–electron fluid, ρn,p are the mass densities of the fluids, pn,p are scalar potentials related to thermodynamic variables in Equation (146), and

 is the external driving force associated with the magnetic dipole torque on the star. The neutron fluid is inviscid, while the proton–electron fluid has kinematic viscosity νee arising from electron–electron scattering. The two fluids are coupled by the mutual friction force

, which arises from electron scattering from magnetized neutron vortices and pinning interactions. The force acts equally and oppositely on the two fluids and is given by (see, e.g., Hall 1960; Khalatnikov 1965; Hills & Roberts 1977; Barenghi et al. 1983; Chandler & Baym 1986; Mendell 1991b; Peralta 2007; Glampedakis et al. 2011)
7

where

 and

 are the mutual friction coefficients; the first term is dissipative and the second term is nondissipative. The mutual friction coefficients are related to scattering and pinning parameters in Section 3. Electron scattering from flux tubes is connected with the evolution of the magnetic field and describes processes analogous to ohmic and Hall diffusion; see, for example, Graber et al. (2015). These effects are small compared with the inertial modes studied in this paper; see Appendix B for further discussion. The restoring force due to tension of the vortex lines is (see, e.g., Hall 1960; Khalatnikov 1965; Hills & Roberts 1977; Baym & Chandler 1983; Mendell 1991a; Peralta 2007; Glampedakis et al. 2011)
8

where νn is the vortex line tension parameter, defined in (124). The vortex line tension is negligible compared with other terms in (5) and (6); see Equation (168). We set the vortex tension to zero everywhere in this paper except in the analysis of the Donnelly–Glaberson instability in Section 4.2.1, where it determines the instability condition. In a type II superconductor, the magnetic stresses arise from the tension of the array of the quantized flux tubes and is given by (Easson & Pethick 1977)
9

where

 is the lower critical field for type II superconductivity. The evolution of the magnetic field is determined by the induction equation
10


The momentum equations for the neutron and proton–electron fluids in the MHD approximation are

5

6

where

 is the smooth-averaged velocity of the proton–electron fluid, ρn,p are the mass densities of the fluids, pn,p are scalar potentials related to thermodynamic variables in Equation (146), and

 is the external driving force associated with the magnetic dipole torque on the star.


The restoring force due to tension of the vortex lines is (see, e.g.,

8

where νn is the vortex line tension parameter, defined in (124).


In a type II superconductor, the magnetic stresses arise from the tension of the array of the quantized flux tubes and is given by

9

where

 is the lower critical field for type II superconductivity.


As seen in the previous section, the mass of the PNS increases with the accretion, which eventually approaches the maximum mass allowed with the adopted EOSs. Then, the PNS would gravitationally collapse to a black hole. The moment when the PNS approaches its maximum mass corresponds to the onset of the instability. In order to determine the onset of instability in the evolution of the PNSs, we make a linear analysis with the radial perturbation on the PNS models at each time-step after corebounce. For this purpose, one can derive the perturbation equations as
(3)$$\begin{eqnarray*}
\frac{\mathrm{ d}\xi }{\mathrm{ d}r} &=& -\left[\frac{3}{r} + \frac{p^{\prime }}{p+\varepsilon }\right]\xi - \frac{\eta }{r\Gamma },
\end{eqnarray*}$$(4)$$\begin{eqnarray*}
\frac{\mathrm{ d}\eta }{\mathrm{ d}r} &=& \left[r(p+\varepsilon) e^{2\Lambda }\left(\frac{\omega ^2}{p} e^{-2\Phi } - 8\pi \right) - \frac{4p^{\prime }}{p} + \frac{r(p^{\prime })^2}{p(p+\varepsilon)}\right]\xi \nonumber\\
&&-\, \left[\frac{\varepsilon p^{\prime }}{p(p+\varepsilon)} + 4\pi r(p+\varepsilon)e^{2\Lambda }\right]\eta ,
\end{eqnarray*}$$where p, ε, and Γ denote the pressure, energy density, and adiabatic index for the background PNS models, while ξ and η are perturbative variables given by ξ ≡ Δr/r and η ≡ Δp/p with the radial displacement, Δr, and the Lagrangian perturbation of pressure, Δp (Chandrasekhar 1964; Chanmugan 1977; Gondek, Haensel & Zdunik 1997).3 The prime in the equations denotes the radial derivative and the adiabatic index is given by
(5)$$\begin{eqnarray*}
\Gamma \equiv \left(\frac{\partial \ln p}{\partial \ln n_{\rm b}}\right)_s = \frac{p+\varepsilon }{p}c_\mathrm{ s}^2,
\end{eqnarray*}$$where nb, s, and cs denote the baryon number density, entropy per baryon, and sound velocity, respectively. We remark that one can derive the Sturm–Liouville type second-order differential equation with respect to ξ from equations (3) and (4). To solve the eigenvalue problem with respect to the eigenvalue ω2, one should impose the appropriate boundary conditions. The boundary condition at the stellar surface comes from the condition to remove the singularity in equation (4) (Chanmugan 1977), i.e.
(6)$$\begin{eqnarray*}
\eta = -\left[\left(\frac{\omega ^2R_{\rm PNS}^3}{M_{\rm PNS}} + \frac{M_{\rm PNS}}{R_{\rm PNS}}\right) \left(1-\frac{2M_{\rm PNS}}{R_{\rm PNS}}\right)^{-1}+ 4\right]\xi ,
\end{eqnarray*}$$while the boundary condition at the centre is the regularity condition, i.e.
(7)$$\begin{eqnarray*}
3\Gamma \xi + \eta = 0.
\end{eqnarray*}$$In addition, as normalization of the eigenfunction, we set ξ = 1 at the stellar centre. Then, with the resultant eigenvalue ω2, the frequency of radial oscillations are given by
(8)$$\begin{eqnarray*}
f_{\xi } = {\rm sgn}(\omega ^2)\sqrt{|\omega ^2|}/2\pi ,
\end{eqnarray*}$$where the system is unstable when ω2 becomes negative.


To solve the eigenvalue problem with respect to the eigenvalue ω2, one should impose the appropriate boundary conditions. The boundary condition at the stellar surface comes from the condition to remove the singularity in equation (4)

i.e.
(6)$$\begin{eqnarray*}
\eta = -\left[\left(\frac{\omega ^2R_{\rm PNS}^3}{M_{\rm PNS}} + \frac{M_{\rm PNS}}{R_{\rm PNS}}\right) \left(1-\frac{2M_{\rm PNS}}{R_{\rm PNS}}\right)^{-1}+ 4\right]\xi ,
\end{eqnarray*}$$while the boundary condition at the centre is the regularity condition, i.e.
(7)$$\begin{eqnarray*}
3\Gamma \xi + \eta = 0.
\end{eqnarray*}$$In addition, as normalization of the eigenfunction, we set ξ = 1 at the stellar centre. Then, with the resultant eigenvalue ω2, the frequency of radial oscillations are given by
(8)$$\begin{eqnarray*}
f_{\xi } = {\rm sgn}(\omega ^2)\sqrt{|\omega ^2|}/2\pi ,
\end{eqnarray*}$$where the system is unstable when ω2 becomes negative.


To calculate the synchrotron emission from the previous RMHD jet models, we need to establish some assumptions. While the radio continuum emission we are interested in is being produced by a population of non-thermal electrons (and maybe positrons), the RMHD simulations discussed previously account only for the evolution of the thermal electrons present in the jet. Establishing a relationship between the thermal and non-thermal populations requires a detailed prescription for the particle acceleration processes that connect both populations, presumably taking place in strong shocks or in magnetic reconnection events (see e.g., Sironi et al. 2015). A proper treatment of particle acceleration/injection in shocks (e.g., Kirk et al. 2000) or magnetic reconnection (e.g., Lyubarsky 2005) requires a microscopic description of the fluid, such as in particle-in-cell (PIC) simulations (e.g., Nishikawa et al. 2016), and its implementation in macroscopic RMHD models, such as the one used here, still falls outside current computing capabilities, given the vastly different scales involved. Nevertheless, as a first-order approximation, we consider that the internal energy of the non-thermal population is a constant fraction of the thermal electrons considered in the RMHD simulations (e.g., Gómez et al. 1995, 1997; Komissarov & Falle 1997; Broderick & McKinney 2010; Porth et al. 2011). Alternatively, the non-thermal population can also be considered to be proportional to the magnetic energy density (e.g., Porth et al. 2011), which determines the particle acceleration efficiency in shocks and magnetic reconnection events. No significant differences are found in our emission calculations when considering the latter approach for particle acceleration, given the similarities between the gas pressure and magnetic energy density distributions in our RMHD simulations, except for the particular case of jet spine brightening discussed in more detail in Section 3.3. On the other hand, we note that particle acceleration at shock fronts is probably the most important ingredient for computing the expected non-thermal emission from our RMHD simulations. Our results should therefore be considered in these cases as a first-order approximation, which could be used as a base model to test different prescriptions for in situ particle acceleration in future modeling.

Background
Background
Background
Future Work


Establishing a relationship between the thermal and non-thermal populations requires a detailed prescription for the particle acceleration processes that connect both populations, presumably taking place in strong shocks or in magnetic reconnection events (see e.g.,


Nevertheless, as a first-order approximation, we consider that the internal energy of the non-thermal population is a constant fraction of the thermal electrons considered in the RMHD simulations (e.g.,


The relative escape fraction of galaxies brighter than M1500 ~ −20.2 (L = 0.5L∗) should be ~16% to provide a photoionization rate Γ-12 ~ 0.95, in agreement with the observed value at z ~ 3, shown in Fig. 7. Alternatively, the escape fraction can remain at approximately a few percent, but the LF must be integrated down to very faint magnitudes, at M1500 = −14.2 (Fontanot et al. 2014; Alavi et al. 2014) to yield Γ-12 ~ 0.9. The calculations above are quite conservative, since we are assuming a rather steep LF (α = −1.73). At z ~ 3 the estimated slopes of the galaxy LF are generally flatter, with α ~  [−1.4,−1.5] (Sawiki & Thompson 2006; Finkelstein et al. 2015; Parsa et al. 2015), with the exception of the LF of LAEs by Cassata et al. (2011), which has α ~ −1.8 at z ≥ 3. Adopting the parametrization of Sawiki & Thompson (2006), with α = −1.43, and integrating down to M1500 = −14.2, the ionizing emissivity is Γ-12 ≤ 0.31, which is insufficient to keep the Universe reionized. Similarly, if we adopt the LF by Cucciati et al. (2012) with a slightly steeper slope (α = −1.50), but brighter M∗, we find an emissivity of Γ-12 ≤ 0.26 at M1500 = −14.2, far from providing enough ionizing photons to maintain the reionization of the Universe. The latter LF is based on the VVDS (Le Fèvre et al. 2013) spectroscopic survey, so it should be robust against interlopers in the z ~ 3 LF. At z ≥ 3, Cassata et al. (2011) find that the LF of LAEs is relatively steep, with a slope of ~− 1.8. However, these galaxies are much fainter than the typical UV selected galaxies discovered at the same redshifts. They measured a star formation rate density, which is a factor of 5 lower than that by Reddy & Steidel (2009), with the implication of a lesser contribution of the LAEs to the HI ionizing background at z ~ 3. Table 3 summarizes the different HI photoionization rates derived for different parametrization of the z ~ 3 galaxy LF and at different luminosities.


There is not yet a clear consensus for observational efforts to detect primordial galaxy alignments in clusters or superclusters. Rood & Sastry (1972) were the first to claim that satellite galaxies in Abell 2199 tend to point in the direction of the major axis of the BCG. Subsequently, primordial galaxy alignment was found in more clusters, such as Abell 521 (Plionis et al. 2003), Abell 1689 (Hung et al. 2010), Coma (Djorgovski 1983; Kitzbichler & Saurer 2003), Virgo (West & Blakeslee 2000), Abell 999 and Abell 2197 (Adams, Storm & Strom 1980; Thompson 1976), and in some cluster samples [Yang et al. 2006 (0.01  z  0.2); Plionis 1994; Plionis et al. 2003 (z  0.15); West, Jones & Forman 1995 (z  0.2); Agustsson & Brainerd 2006 (median redshift z = 0.058); Faltenbacher et al. 2007 (0.01  z  0.2)]. However, some other measurements have been consistent with random orientations of satellite galaxies in clusters [e.g. Hawley & Peebles 1975; Dekel 1985; van Kampen & Rhee 1990; Godłowski & Ostrowski 1999; Strazzullo et al. 2005 (z  0.27); Torlina, De Propris & West 2007; Trevese, Cirimele & Flin 1992; Panko, Juszczyk & Flin 2009 (z  0.18); Sifón et al. 2015 (0.05  z  0.55)]. On one hand, the primordial alignment is primarily found in unrelaxed clusters, such as Coma (Sanders et al. 2013; Simionescu et al. 2013), Abell 1689 (Kawaharada et al. 2010), Virgo (Urban et al. 2011) and Abell 521 (Maurogordato et al. 2000; Ferrari et al. 2003, 2006), which agrees with stronger alignment in dynamically young clusters. On the other hand, at higher redshifts, the primordial alignment is denied in many clusters (Sifón et al. 2015) other than Abell 521 (z ∼ 0.25). This is unexpected because the fraction of the dynamically young clusters at high redshifts is larger (Hashimoto et al. 2007; Maughan, Forman & Van Speybroeck 2008; Mann & Ebeling 2012; Weißmann, Böhringer & Chon 2013), according to the hierarchical clustering models of structure formation.


We performed our spectral fit in the XSPEC v12.8.2 and 12.9.0 (Arnaud 1996) spectral analysis environments. We verified that the resulting fits are consistent across both versions of XSPEC. Twenty-three parameters describe the spectral continuum and emission lines that NuSTAR observes in the 4–79 keV range. Six parameters describe two iron emission lines (we use the Gaussian function for both lines), three describe the cyclotron resonant scattering absorption feature (CRSF), one parameter describes the interstellar absorption (

), and one parameter describes the cross-normalization of the FPMA and FPMB modules (

). The absorbing column is frozen at

 cm−2, a value that approximates the galactic absorption to Her X-1  (Fürst et al. 2013). We utilized the tbabs absorption model with the “WILM” abundance table and the “BCMC” cross-sections (Wilms et al. 2000). This leaves 12 parameters to describe the RDRS model. One of these is an RDRS model normalization that we freeze at 1.0. One is a numerical switch that allows us to turn on or off any of the three principal seed-photon processes for the Comptonization (see Section 2). We freeze the distance to the Her X-1 system at D = 6.6 kpc (Reynolds et al. 1997), and the NS mass and radius are set to their standard values (

 and 10 km). The distance to Her X-1 obtained by Reynolds et al. of

 kpc is sufficiently consistent, for our purposes, with a more recently determined distance of

 kpc obtained by Leahy & Abdallah (2014). Thus, we adopt the Reynolds et al. distance for these calculations. The value of the input magnetic field strength to the RDRS model is tied to the centroid of the fitted cyclotron absorption line (

) via

 where z* is the gravitational redshift to the NS surface. The six remaining parameters describing the RDRS model (BW) are: mass accretion rate (

), plasma Comptonizing temperature (Te), accretion cap radius (r0), scattering cross-section perpendicular to the magnetic field (

), scattering cross-section parallel to the magnetic field (

), and average scattering cross-section (

). The scattering cross-section perpendicular to the magnetic field (

) is frozen at the Thomson electron scattering cross-section (

). This leaves five free parameters,

, Te, r0,

, and

, to describe the X-ray continuum.


The extent of the evolution of the faint red-sequence population, however, is still under debate. As opposed to local clusters that exhibit a flat faint end, or even an upturn at the faint end of their red-sequence LFs (e.g., Popesso et al. 2006; Agulli et al. 2014; Moretti et al. 2015; Lan et al. 2016), various studies have revealed that clusters at intermediate and high redshifts show a continual decrease in the fraction of the faint red-sequence population with redshift, which indicates a gradual buildup of the faint red-sequence population over time since z ∼ 1.5 (e.g., Dressler et al. 1997; Smail et al. 1998; De Lucia et al. 2004, 2007; Kodama et al. 2004; Tanaka et al. 2007; Gilbank et al. 2008; Rudnick et al. 2009, 2012; Stott et al. 2009; Martinet et al. 2015; Zenteno et al. 2016; Sarron et al. 2018; Zhang et al. 2019). This is also supported by findings that cluster galaxies on the high-mass end of the red sequence are on average older than those on the low-mass end (e.g., Nelan et al. 2005; Sánchez-Blázquez et al. 2009; Demarco et al. 2010a; Smith et al. 2012). Contrary to the above-mentioned studies, a number of studies have reported that there is little or no evolution of the faint end of the red-sequence cluster LF up to z ∼ 1.5 (e.g., Andreon 2006; Crawford et al. 2009; De Propris et al. 2013, 2015; Andreon et al. 2014; Cerulo et al. 2016), which in turn suggests the early formation of the faint end, similar to bright red-sequence galaxies. De Propris et al. (2013) proposed that the discrepancy is primarily caused by surface brightness selection effects, which lowers the detectability of faint galaxies at high redshift. Nevertheless, a recent study by Martinet et al. (2017) extensively investigated the effect of surface brightness dimming with 16 CLASH clusters in the redshift range of 0.2  z  0.6. They concluded that surface brightness dimming alone could not explain the observed redshift evolution of the faint end. Other possible explanations of the discrepancy invoke the radial and mass dependence of the faint red-sequence population, both of which are also debated in local cluster LF studies (see, e.g., Popesso et al. 2006; Barkhouse et al. 2007; Lan et al. 2016). While there may be a (weak) dependence of the red-sequence LF on cluster mass (or cluster properties that are mass proxies) at intermediate redshift (e.g., De Lucia et al. 2007; Muzzin et al. 2007; Rudnick et al. 2009; Martinet et al. 2015), it remains unclear whether this effect exists at higher redshift. It is also possible that the disagreements in the literature are driven by the large cluster-to-cluster variations, sample selections, or methods used to derive the LF, as observed in most of the above-mentioned works.


As opposed to local clusters that exhibit a flat faint end, or even an upturn at the faint end of their red-sequence LFs


Other possible explanations of the discrepancy invoke the radial and mass dependence of the faint red-sequence population, both of which are also debated in local cluster LF studies (see, e.g.,


Contrary to the above-mentioned studies, a number of studies have reported that there is little or no evolution of the faint end of the red-sequence cluster LF up to z ∼ 1.5 (e.g.,


proposed that the discrepancy is primarily caused by surface brightness selection effects, which lowers the detectability of faint galaxies at high redshift.


In our axisymmetric simulations without magnetohydrodynamical (MHD) effects, angular momentum transport in the accreting flow is given by imposing explicit viscosity. The viscous stress tensor is given by
11

σij=ρν˜∂vj∂xi+∂vi∂xj−23(∇·v)δij,

where

ν˜

 is the shear viscosity, and the bulk viscosity is neglected. To mimic angular momentum transport associated with MHD turbulence driven by the magnetorotational instability (MRI) in a sufficiently ionized disk (e.g., Balbus & Hawley 1998; Stone & Pringle 2001; McKinney & Gammie 2004; Bai 2011; Narayan et al. 2012), we assume the azimuthal components of the shear tensor are nonzero and, in spherical polar coordinates, are given by
12

σrϕ=ρν˜∂∂rvϕr,

13

σθϕ=ρν˜sinθr∂∂θvϕsinθ

(e.g., Stone et al. 1999; Fernández & Metzger 2013; Inayoshi et al. 2019). The strength of anomalous shear viscosity is calculated with the α-prescription (Shakura & Sunyaev 1973),
14

ν˜=αcs2ΩK·exp−∣z∣H,

where α is the viscous parameter, c
s is the sound speed,

ΩK≡(GM•/r3)1/2

, and H( ≡ c
s/ΩK) is the disk scale height. Note that the exponential factor imposes that the viscous process is active near the mid-plane. The strength of viscosity is set to
15

α=α0+αmaxexp−ρcritρ2.

The first term corresponds to the strength of MRI turbulence, and the value is set to α
0 = 0.01 (e.g., Zhu & Stone 2018; Takasao et al. 2018). The second term characterizes the torque caused by non-axisymmetric structure (e.g., spiral arms) excited in a marginally unstable disk against its self-gravity. The density threshold, above which viscosity turns on, is assumed to be

ρcrit≡ΩK2/(πG)≃2.2×10−21

(M•/105M⊙)(r/10pc)−3gcm−3

. This choice is motivated by the following reasons. The local gravitational instability of a rotating disk is described by Toomre’s Q parameter (Toomre 1964) defined by
16

Q≡csκΩπGΣ,

where

κΩ2=4Ω2+dΩ2/dlnr

 is the epicyclic frequency, and Σ is the disk surface density. For a geometrically thin cold disk around a point mass, the Q-value is approximated to

Q≃ΩK2/(2πGρ)=0.5(ρcrit/ρ)

. Thus, the second term of the right-hand side of Equation (15) is written as

∝exp(−4Q2)

, which is a commonly used parameterization of the effective viscosity adopted in semi-analytical models of a self-gravitating disk;

∝exp(−QA/B)

 where A > 0 and B > 0 (Zhu et al. 2009; Takahashi et al. 2013; see also Kratter & Lodato 2016). The value of

αmax

 depends on the level of non-axisymmetric structures in a disk. A previous three-dimensional RHD simulation study of a dusty circumnuclear disk around a BH shows that the mass inflow velocity is as high as a substantial fraction of the freefall velocity due to strong torque caused by spiral arms in the disk, indicating

αmax≃O(1)

 (Toyouchi et al.2021). We here adopt

αmax=2

 (Hirano et al. 2014; Fukushima et al. 2020). The critical Q-value for the onset of gravitational torque caused by spiral arms in a disk is considered to be Q ∼ 1, but the exact value depends on various properties of the disk (e.g., cooling, heating, and disk irradiation). We note that in our viscous model, the second term in Equation (15) becomes larger than α
0 when Q ≤ 1.15.


Bright arc-like structures extending from the Sun, when seen in extreme-ultraviolet (EUV) wavelengths, are called coronal loops. Being a characteristic feature of the corona, it is important to assimilate high-resolution information on these structures in order to understand how plasma in the outer atmosphere is heated to multi-million Kelvin temperatures. The plasma temperatures and pressures are known to vary along the lengths of the loops, whose gradients provide some insight on the underlying heating mechanisms (Rosner et al. 1978), but the main attribute of critical importance is the cross-field morphology (Patsourakos & Klimchuk 2007). It has been suggested that coronal loops consist of multiple unresolved thin strands (Reale 2014; Klimchuk 2015). Observational evidence demonstrating low filling factors (Di Matteo et al. 1999), different degrees of “fuzziness” observed in loops at different temperatures (Tripathi et al. 2009; Reale et al. 2011), and plasma dynamics at small spatial scales (Antolin & Rouppe van der Voort 2012) indicate that this scenario is likely to be true. The typical cross-sections of the strands were estimated to be a few tens to a few hundreds of kilometers (DeForest 2007; Antolin & Rouppe van der Voort 2012; Brooks et al. 2013; Klimchuk 2015). Possible braiding of these strands can instigate magnetic reconnection, thus releasing significant energy to directly heat the plasma inside the loop (Cirtain et al. 2013; Klimchuk 2015). Each strand is believed to be impulsively and independently heated, leading to a multithermal configuration across the loop (Cargill 1994; Reale et al. 2005; Klimchuk et al. 2008; Tripathi et al. 2011). However, not all studies are consistent with this scenario since some loops demonstrate isothermal structuring (Del Zanna & Mason 2003; Noglik et al. 2008; Schmelz et al. 2009). One of the main issues plaguing observations of coronal loops is the intrinsic optically thin emission (Del Zanna & Mason 2003; Terzo & Reale 2010). Since the observed intensities are integrated along the line of sight, it becomes difficult to exclude emission from overlapping structures, which can be formed at different temperatures and inadvertently imply a multithermal structure. In this letter, we use a unique set of observations to reveal the multi-stranded and multithermal nature of a coronal loop, and isolate two of its components through the application of MHD seismology. More importantly, our results are free from the adverse effects arising from line of sight integrations.


Each strand is believed to be impulsively and independently heated, leading to a multithermal configuration across the loop


One of the main issues plaguing observations of coronal loops is the intrinsic optically thin emission

Since the observed intensities are integrated along the line of sight, it becomes difficult to exclude emission from overlapping structures, which can be formed at different temperatures and inadvertently imply a multithermal structure.


It has been suggested that coronal loops consist of multiple unresolved thin strands


In this section we show that the growth of the m = 1 mode in the disc results in the formation of a spiral density wave with a constant global pattern speed ΩP. As the disc is differentially rotating this means that there is a location in the disc, the so-called corotation radius rco, where the spiral density wave has the same orbital velocity as the fluid in the disc. Inside rco the wave travels slower than the fluid, which means that it has negative angular momentum with respect to the fluid. While the wave amplitude is linear, it does not interact with the fluid, but once non-linear amplitude effects come into play, the spiral density wave can couple to the fluid via dissipation (Papaloizou & Lin 1995; Goodman & Rafikov 2001; Heinemann & Papaloizou 2012). When this happens, the wave begins to transport angular momentum outwards as the fluid loses angular momentum to the wave inside the corotation radius. The development and persistence of the spiral density wave are shown in Fig. 3. The figure shows the fractional change in density between two successive snapshots of the tracer particles Δρ/ρ at nine different times of the disc evolution, as well as the tracer particles that are located at rco (grey circles). The location of the corotation radius is defined as the radius where the m = 1 pattern speed and the fluid orbital velocity are equal. As usual, the non-axisymmetric modes in the disc are analysed by means of an azimuthal Fourier transform of the rest-mass density ρ (Zurek & Benz 1986; Heemskerk et al. 1992):

(5)

\begin{equation}
D_m = \int \, \alpha \, \sqrt{\gamma } \,\rho \, {\rm e}^{-i\,m\,\phi } \, d^3x \,,
\end{equation}

with α and γ being the space–time lapse function and the determinant of the three-dimensional metric, respectively. Similarly, the mode analysis performed using the tracers is accomplished by means of the following sum

(6)

\begin{equation}
D_m = \sum _j^N \, m_j \, {\rm e}^{-i\,m\,\phi },
\end{equation}

where mj is the mass of each tracer particle. From the mode amplitudes, the pattern speed of an azimuthal mode with mode number m is defined as (see e.g. Heemskerk et al. 1992)

(7)

\begin{equation}
\Omega _{\rm P} = \frac{1}{m}\frac{d \phi _m}{dt},
\end{equation}

where the phase angle ϕm is given by

(8)

\begin{equation}
\phi _m = \mathrm{tan}^{-1} \left(\frac{\mathrm{Im}(-D_m)}{\mathrm{Re}(D_m)} \right).
\end{equation}

For our simulation, we obtain an orbital period of the m = 1 pattern of PP ∼ 1.96 ms, which is slightly shorter than twice the initial orbital period of the disc at the location of the density maximum (see Table 1). We note that we have chosen the same range of the fractional change in the rest-mass density in all snapshots shown in Fig. 3, which corresponds to the interval [−0.1, 0.1]. At these wave amplitudes, non-linear effects are negligible compared to the linear effects (Masset & Tagger 1997). This plot range has, however, been chosen for visualization purposes and the wave amplitudes are actually much larger than 0.1 after the saturation of the m = 1 growth, reaching almost unity. At these wave amplitudes non-linear effects are important and the spiral density wave can couple to the fluid via dissipation. To show the development of weak shocks, we plot the fractional change in the entropy S = p/ρΓ for tracers with Δρ/ρ > 0.1 in Fig. 4. From this plot, we clearly see an increase in the fluid entropy in the inner and outer regions of the spiral density wave. In some snapshots of Fig. 3, we can see fractional changes in ρ in the form of two spiral arms in the inner regions of the disc. The dominant m = 1 mode developing in the disc should not produce these, which means there could be a different mechanism at play. Standing shocks in the inner regions of tilted accretion discs in the form of two spiral arms have been observed and analysed in Henisey, Blaes & Fragile (2012) and Generozov et al. (2014) and are a distinctive feature of tilted accretion discs. Finally, we note that the mesh refinement boundaries of our computational grid also produce entropy changes due to numerical dissipation. However, these changes are about an order of magnitude smaller than the physical increase of entropy due to the development of shocks in the spiral density wave.


In the context of a standard flare and coronal mass ejection (CME) model (e.g., Shibata et al. 1995; Lin et al. 2004), there are two parallel flare ribbons where nonthermal electrons collide and heat the chromosphere, which are observed in the Ca ii H, Hα, UV, and EUV wavelengths. Apart from the two ribbons, a particular type of flare ribbons, i.e., circular ribbons, exist (Masson et al. 2009; Reid et al. 2012; Wang & Liu 2012; Jiang et al. 2013; Liu et al. 2013; Sun et al. 2013; Yang et al. 2015; Zhang et al. 2015). They are always associated with the spine-fan configuration in the presence of the magnetic null point, which is a singular point where the magnetic field vanishes (

 = 0; Lau & Finn 1990). The magnetic field

 near the null point can be expressed as the linear term

, where

 is a Jacobian matrix with elements

 and

 is the position vector

 centered at the null point (Parnell et al. 1996). The divergence-free condition (

) requires that the sum of the three eigenvalues equals zero (Zhang et al. 2012). The two eigenvectors corresponding to the two eigenvalues of the same sign determine the fan surface, which divides the space into two regions having a distinct connectivity. The third eigenvector corresponding to the third eigenvalue of the opposite sign determines the direction of spines passing through the null point. Magnetic reconnection and particle acceleration in null point reconnection regions have been explored in analytical studies (Priest & Titov 1996; Litvinenko 2004) and three-dimensional (3D) numerical simulations (Rosdahl & Galsgaard 2010; Baumann et al. 2013a, 2013b). The circular ribbons are believed to be intersections of the fan surfaces and the chromosphere. The central or inner ribbons within the circular ribbons are thought to be intersections of the inner spines and the chromosphere (Reid et al. 2012; Wang & Liu 2012). Sometimes, there are multiple flare ribbons owing to the extraordinarily complex magnetic topology of the active regions (ARs; Joshi et al. 2015; Liu et al. 2015).


Our motivation for this work is manifold. First, we have incorporated the gas-accretion models of Papaloizou  Nelson (2005) into the combined N-body and protoplanetary disc evolution code presented in Coleman  Nelson (2014, 2016a), in order to increase the realism of the planetary system formation simulations that can be computed with this numerical tool. The purpose of this paper is to provide a simplified examination of gas accretion on to isolated, non-interacting planetary cores at a variety of orbital locations using the evolving disc models from Coleman  Nelson (2016a), as a precursor to presenting simulations that consider the contemporaneous accretion of solids and gas on to protoplanets under the influence of migration within the evolving disc models. A second motivation is to examine how accreting planets with different initial core masses at different locations in the disc evolve within disc lifetimes. Observations indicate the presence of exoplanets (super-Earths, Neptunes and Jupiters) that are likely to be gas bearing with a broad range of orbital radii, covering the interval 0.04au  ap  100au (e.g. Marois etal. 2008; Borucki etal. 2011). Analysis of the occurrence rates as a function of planet radius in the Kepler data suggests that planets with radii in the range 0.85  Rp  4R are present around 50percent of Sun-like stars (e.g. Fressin etal. 2013), which is consonant with the results from earlier radial velocity surveys that focused on low-mass planets (Mayor etal. 2011). Transit timing variations (Wu  Lithwick 2013; Hadden  Lithwick 2016) and radial velocity measurements (Marcy etal. 2014; Weiss etal. 2016) indicate that the super-Earths and Neptunes in the Kepler data with radii Rp4R have masses in the range 2  Mp  15M, and show a tendency for the bulk density to increase for Rp1.5R and decrease for 1.5Rp4R, indicating that the smaller planets are composed almost entirely of solids, whereas the larger planets have larger radii due to the presence of significant gaseous envelopes that have been accreted during formation. Comparing theoretical models with the data suggests that the gaseous envelopes contribute little to the total masses of the super-Earths and Neptunes (Lopez  Fortney 2014), with recent estimates indicating that planets with Rp  1.2R contain 1percent gas by mass, rising to 5percent for planets with Rp3R (Wolfgang  Lopez 2015). Considering the gas giant exoplanets, numerous attempts have been made to constrain the core masses and total heavy element abundances of those transiting planets that have mass estimates from radial velocity measurements using evolutionary models of the internal structure (Guillot etal. 2006; Burrows etal. 2007; Miller  Fortney 2011; Thorngren etal. 2016). These studies provide a strong indication that the majority of gas giant exoplanets are substantially enriched with heavy elements relative to their host stars, with heavy element masses typically being 20M or larger. These results are consistent with the idea that giant planets form when a sufficiently massive (i.e. in the range 1015M) core forms that is able to accrete gas and undergo runaway gas accretion sufficiently early in the disc lifetime, in basic agreement with the theoretical core accretion models. In the absence of a sufficiently massive core that forms early enough, a more common outcome is the formation of super-Earths or Neptunes that fail to become gas giants because of the relatively slow rate of gas accretion until the envelope mass becomes comparable to the core mass. It is clearly of interest to address the above observational results through theoretical modelling of in situ planet formation, and to compare the outcomes of our calculations with recent work that has considered gas accretion on to planets with short (Lee, Chiang  Ormel 2014; Batygin, Bodenheimer  Laughlin 2016) and long orbital periods (Piso  Youdin 2014; Piso, Youdin  Murray-Clay 2015). Our third motivation is to consider the longer term cooling and contraction of the gaseous envelopes for those planets that remain in a state of quasi-static equilibrium throughout the period of gas accretion when the disc is present. Since the observations are typically relevant to exoplanets that have existed around their stars for Gyrs, they will have evolved for time periods of this duration after the epoch of formation. This is the first time that our models have been used to examine this longer term evolution, and our approach is similar to that used in recent population synthesis studies (see the recent review by Mordasini etal. 2015).


To study the effect that assuming light traces mass has, we use a model of a cluster based on real data that contains 174 small scale galaxy-scale haloes (spectroscopically identified by Grillo et al. 2015) embedded in two large, cluster-scale DM haloes (Jauzac et al. 2014; Grillo et al. 2015). In this study, we act only to study the effect of changes to the galaxy-scale haloes on the positions of multiple images. We use the position of multiple images in the Hubble Frontier Field (HFF) galaxy cluster, MACS J0416 and the best-fitting cluster model that fits these multiple images as derived by the parametric strong lensing algorithm lenstool (Jullo et al. 2007; Jauzac et al. 2014). This equates to 140 multiple images and 174 cluster members. Each potential was originally fitted with a pseudo-isothermal elliptical mass distribution (PIEMD; Kneib et al. 1996; Natarajan & Kneib 1997; Elíasdóttir et al. 2007; Jullo et al. 2007), which follows the analytical density profile,

(1)

\begin{equation}
\frac{\rho (r)}{\rho _0} = \frac{1}{\sqrt{r^2+r_{\rm core}^2}} - \frac{1}{\sqrt{r^2+r_{\rm cut}^2}},
\end{equation}

where r is the radial distance from the centre of the halo, and the profile is parametrized by the core radius, rcore, and the cut radius, rcut. The cluster itself has a measured mass within a 200 kpc aperture of (1.6 ± 0.01) × 1014 M⊙ (Jauzac et al. 2014), and the mass of each galaxy has a lognormal distribution centred around log(M/M⊙) = 10.1 ± 0.6. The distribution of ellipticities peaks at 0.1 and decreases towards larger ellipticities. The galaxies span the entire range between 0 and 0.9. We split each galaxy halo potential into two components: a baryonic core and a DM halo. We separate the halo into two PIEMDs with the baryonic $r^{\rm B}_{\rm core} = r_{\rm core}$. For the baryonic cut radius, we conservatively cut it to a quarter of the original cut radius $r^{\rm B}_{\rm cut} = \frac {r_{\rm cut}}{4}$ (Velander et al. 2014), although this could be smaller. We also force the dark matter core radius to the baryonic cut radius, $r^{\rm DM}_{\rm core} = r^{\rm B}_{\rm cut}$ and the dark matter cut radius as the original cut radius $r^{\rm DM}_{\rm cut} = r_{\rm cut}$. Separating the halo like this means that the mass of each halo is conserved with respect to the pre-split halo. Given this two-component galaxy model, we project the images back to the source plane to create a list of background sources for the cluster. To do this, we use the same redshift information for the sources as that in Jauzac et al. (2014).


From the scattering experiments, we have obtained the HVB distribution d3Nhvb/(dmchirpdvejdτ). Note that this distribution varies with M1 and q, thus it is a function of mchirp, τ and veject but also of log M1 and log q. After the ejection, a given HVB will escape from the galactic potential with initial velocity vej and merge in the dark matter halo or in the IGM at a distance R away from the host galactic centre. This distance depends on the flight time τ, and can be computed explicitly by assuming a specific galactic potential. In this work, we adopt the characteristic galactic potential of elliptical galaxies (e.g. Kenyon et al. 2008, 2014),
(19)$$\begin{eqnarray*}
\phi (r)& =& \phi _\mathrm{bh}(r)+\phi _b(r) + \phi _h(r)\nonumber\\
&=& -\frac{G(M_1+M_2)}{r} -\frac{GM_\mathrm{ b}}{r+r_\mathrm{ b}} -\frac{GM_\mathrm{ h}}{r}\ln \left(1+\frac{r}{r_\mathrm{ h}}\right),
\end{eqnarray*}$$where Mb is the mass of the bulge, Mh is the mass of the dark matter halo, rb is the scale radius of the bulge, and rh is the scale radius of the dark matter halo. The choice of the spherical potential in equation (19) is dictated by computational convenience, as it allows us to perform 1D calculations to compute the ejected CO binary trajectory. The addition of a disc component would produce a deflection of the ejected trajectories, without significantly affecting the overall distribution of distances travelled. We scale the variables Mb, Mh, rb, and rh with the total mass of the SMBHB M1(1 + q) by means of the relations (e.g. Kormendy & Ho 2013; Kulier & Ostriker 2015)
(20)$$\begin{eqnarray*}
\frac{M_1(q+1)}{10^9\mathrm{ M}\mathrm{ }_\odot } &=& 0.49\left(\frac{M_b}{10^{11}\mathrm{ M}_\odot }\right)^{1.16},\nonumber\\
\frac{M_h}{10^{13}\mathrm{ M}_\odot } & = & 0.30\left(\frac{M_b}{10^{11}\mathrm{ M}_\odot}\right),\nonumber\\
\frac{M_b}{10^6 \mathrm{ M}_\odot }\bigg (\frac{\mathrm{kpc}}{r_b}\bigg)^3 &=& 4000, \nonumber\\
\frac{M_h}{10^6 \mathrm{ M}_\odot }\bigg (\frac{ \mathrm{kpc}}{r_h}\bigg)^3 &=& 125.
\end{eqnarray*}$$The ejection properties are recorded at a distance of $r\, \sim \, 50a_{\mathrm{bh}}$, given the condition that the total energy of the compact binary must be positive. Thus, we can calculate the travelled distance R = D(M1(1 + q), τ, vej)
(21)$$\begin{eqnarray*}
\int _0^D\frac{\mathrm{ d}r}{\sqrt{2E_0-2\phi (r)- 2l_0^2/r^2}} =\int _0^{\tau _\mathrm{GW}} \mathrm{ d}t,
\end{eqnarray*}$$where E0 and l0 are, respectively, the total energy and total angular momentum per unit mass at the time of ejection. The distribution of HVBs as a function of mass, travelling time τ, and distance from the galaxy centre at a merger distance R can then be calculated as
(22)$$\begin{eqnarray*}
&& {\frac{\mathrm{d}^3 N_\mathrm{hvb}}{\mathrm{d}m_\mathrm{chirp}\mathrm{d}\tau \mathrm{d}R}(\log q,\log M_1,m_\mathrm{chirp},\tau ,R)}\nonumber\\
&& {\quad=\frac{\mathrm{d}}{\mathrm{d}R}\oint _{D(M_1,q,\tau ,v_\mathrm{ej}) \lt R}\frac{\mathrm{d}^3 N_\mathrm{hvb}}{\mathrm{d}m_\mathrm{chirp}\mathrm{d}v_\mathrm{ej}\mathrm{d}\tau }\mathrm{d}v_\mathrm{ej}.}
\end{eqnarray*}$$Fig. 10 shows the 2D-PDF of the distance R and the corresponding velocity at that distance of the simulated HVBs. Note that not all ejected mergers can be observed within the age of the universe.


Observations of the lowest-lying rotational levels of water—those able to probe cold gas—have required space-based observatories. The Infrared Space Observatory-Short Wavelength Spectrometer (ISO-SWS; de Graauw et al. 1996; Kessler et al. 1996) covered the

 ro-vibrational band (symmetric bending mode) of H2O centered near 6 μm, and absorption out of the lowest-lying levels of the ortho and para nuclear spin modifications (1

 and 0

, respectively) was detected toward several massive protostars (Boonman  van Dishoeck 2003). Due to the low spectral resolution of the observing configuration though (

 using SWS in AOT6 grating mode), these lines were significantly blended with absorption from other nearby H2O lines, making the determination of level-specific column densities impossible. Instead, the entire

 band was fit simultaneously assuming a single temperature to determine the total water column density,

. The Submillimeter Wave Astronomy Satellite (SWAS; Melnick et al. 2000b) provided much higher spectral resolution (

 km s−1) and covered the 1

–1

 pure rotational transition of H2O at 557 GHz. This line was observed in both emission and absorption in multiple sources (e.g., Melnick et al. 2000a; Snell et al. 2000), demonstrating the ability to probe cold water. More recently, the study of low-lying rotational levels at high spectral resolution (∼0.5 km s−1) has been facilitated by the Heterodyne Instrument for the Far-Infrared (HIFI; de Graauw et al. 2010) on board the Herschel Space Observatory (Pilbratt et al. 2010). Water has been detected in both emission and absorption out of levels with

 in several protostars (e.g., van Dishoeck et al. 2011; van der Tak et al. 2013), and in absorption out of the 0

 and 1

 levels in the molecular ISM (e.g., Sonnentrucker et al. 2010; Flagey et al. 2013). Observations that resolve the velocity structure of absorption lines are vital to both determining level-specific column densities and understanding the dynamics of the absorbing/emitting regions. This is especially important for protostars as such objects contain multiple dynamical components (e.g., disk, envelope, jets, outflows, shocks).


Within this framework, Lovric et al. (2017) introduced a spectral color index in the solar UV that is linked to the ratio between the flux integrate over the Far-UV (FUV) and Middle-UV (MUV) spectral broadbands. Such a descriptor can be used to characterize UV stellar emission, which modulates the photochemistry of molecular species, e.g., oxygen, in the atmospheres of planets (e.g., Tian et al. 2014). By using solar irradiance measurements obtained with radiometers on board the Solar Radiation and Climate Experiment (SORCE) satellite (McClintock et al. 2005) for almost a solar cycle, the authors showed that the color index so defined is linearly correlated with the Bremen Magnesium II index, which is an excellent proxy of the magnetic activity (Viereck et al. 2004). Lovirc et a. showed that the correlation coefficient is slightly different for the descending phase of Cycle 23 and the ascending phase of the subsequent cycle. Such a difference was ascribed to residual instrumental effects, which, when compensated for, lead to a correlation coefficient constant with time. On the other hand, several solar photospheric and chromospheric indices present a clear asymmetry during different phases of a cycle (namely, an hysteresis pattern, e.g., Bachmann & White 1994; Criscuoli 2016; Salabert et al. 2017), whereas correlation coefficients between indices may vary from cycle to cycle (e.g., Bruevich & Yakunina 2014; Tapping & Morgan 2017), so that the question arises whether, and to what extent, the data corrections applied by Lovric et al. (2017) might include a physical variation of solar emission. The purposes of this paper are to answer this question and to investigate the physical mechanisms determining the high linear relation between the UV spectral color index and solar activity measured with the Mg ii index. To this aim, we compare the results of Lovric et al. (2017) with synthetic indices obtained with an irradiance reconstruction technique based on the use of semi-empirical atmosphere models and full-disk observations.


Lovric et al. (2017)
Lovric et al. (2017)

introduced a spectral color index in the solar UV that is linked to the ratio between the flux integrate over the Far-UV (FUV) and Middle-UV (MUV) spectral broadbands.

On the other hand, several solar photospheric and chromospheric indices present a clear asymmetry during different phases of a cycle

whereas correlation coefficients between indices may vary from cycle to cycle

so that the question arises whether, and to what extent, the data corrections applied by

might include a physical variation of solar emission.

Background Background Compare/Contrast Compare/Contrast Compare/Contrast Compare/Contrast


The purposes of this paper are to answer this question and to investigate the physical mechanisms determining the high linear relation between the UV spectral color index and solar activity measured with the Mg ii index. To this aim, we compare the results of

with synthetic indices obtained with an irradiance reconstruction technique based on the use of semi-empirical atmosphere models and full-disk observations.


The MF is an important statistical tool to understand the formation of stars (Sharma et al. 2017 and references therein). The MF is often expressed by a power law,

 and the slope of the MF is given as

, where N (log m) is the number of stars per unit logarithmic mass interval. We have used our deep optical data to generate the MF of the Sh 2-305 region. For this, we have utilized the optical V0 versus

 CMDs of the sources in the target region and that of the nearby field region of equal area and decontaminated the previous sources of foreground/background stars and corrected for data incompleteness using a statistical subtraction method already described in detail in our previous papers (see Sharma et al. 2007, 2012, 2017; Pandey et al. 2008, 2013; Chauhan et al. 2011; Jose et al. 2013). As an example, in Figure 10 (left panel), we have shown the V0 versus

 CMDs for the stars lying within the central cluster “Mayer 3” in panel (a) and for those in the reference field region selected as an annular region outside the boundary of the Sh 2-305 region (see Section 3.1) in panel (b). The magnitudes were corrected for the AV values derived from the reddening map (see Section 3.7). In panel (c), we have plotted the statistically cleaned V0 versus

 CMD for the central cluster “Mayer 3,” which is showing the presence of PMS stars in the region. The ages and masses of the stars in this statistically cleaned CMD have been derived by applying the procedure described earlier in our previous papers (Chauhan et al. 2009; Sharma et al. 2017). For reference, the post-MS isochrone for 2 Myr calculated by Marigo et al. (2008; thick blue curve) along with the PMS isochrones of 0.1 and 3.5 Myr (purple curves) and evolutionary tracks of different masses (red curves) by Siess et al. (2000) are also shown in panel (c). These isochrones are corrected for the distance of Sh 2-305 (3.7 kpc; see Section 3.3). The corresponding MF has subsequently been plotted in Figure 10 (right panel) for the central cluster “Mayer 3.” For this, we have used only those sources that have ages equivalent to the average age of the optically identified YSOs combined with error (i.e.,

; see Table 6). Our photometry is more than 90% complete up to V = 21.5 mag, which corresponds to the detection limit of a 0.8 M⊙ (see Figure 10(c)) PMS star of ≃1.8 Myr age embedded in the nebulosity of AV ≃ 3.0 mag (i.e., the average values for the optically detected YSOs; see Table 6). We have applied a similar approach to derive the MF of the southern clustering and the whole region of Sh 2-305 (see Section 3.1), and the corresponding values of their MF slopes in the mass range 1.5  M⊙  6.6 are given in Table 7. The MF of the northern clustering cannot be determined due to an insignificant number of optically detected stars.


The characteristic frequency of curvature radiation is
8

ν

curv

=

3
c

γ

3

4
π
ρ

=
2.4
×

10

3

γ

3

ρ

30

−
1

Hz
,

where

ρ
=
30

ρ

30

 km is the curvature radius and

γ

 is the Lorentz factor of an emitting electron. For a typical FRB, we have
9

γ
=
75

ρ

30

1

/

3

ν

curv,9

1

/

3

,

where

ν

curv,9

=

ν

curv

/

10

9

Hz

. We then discuss whether or not a photon can propagate through the plasma in the magnetosphere by considering three effects. First, the plasma frequency in the emission region is
10

ν

p

=

1

2
π

4
π

n

e

e

2

m

e

1

/

2

.

In a highly magnetized magnetosphere, the plasma effect is negligible if

ν

p

γ

1

/

2

ν

curv

 (Lyubarskii & Petrova 1998), which is further written as
11

a

30

>
1.0

B

*
,
12

2

/

9

γ

2

−
2

/

9

ν

curv,9

−
4

/

9

,

where

γ

2

=
γ

/

10

2

 and we have taken the electron density of the emission region to be roughly that of the acceleration region. Second, the cyclotron absorption in the magnetosphere is generally considered in pulsar physics, whose optical depth is given by Lyubarskii & Petrova (1998):
12

τ

cyc

≃
2
×

10

−
3

B

*
,
12

3

/

5

P

*

−
9

/

5

ν

curv,9

−
3

/

5

γ

2

−
3

/

5

,

where

P

*

=
2
π

/

Ω

*

 and the electron number density outside of the flux tube (see Figure 1) has been assumed to be described by the Goldreich–Julian density. Thus, the cyclotron absorption can be ignored if

P

*

>
0.03

B

*
,
12

1

/

3

ν

curv,9

−
1

/

3

γ

2

−
1

/

3

 s. Third, the characteristic distance (l) of a photon due to the Thomson scattering is (Lyubarskii & Petrova 1998)
13

l
=

1

(
1
−

β

e

cos
θ
)

n

¯

e

σ

T

=

1.5
×

10

14

cm

n

¯

e
,
10

(
1
−

β

e

cos
θ
)

,

where

β

e

c

 is the velocity of an electron, θ is the angle between the motion directions of the electron and photon, and

n

¯

e
,
10

=

n

¯

e

/

10

10

cm

−
3

 has been assumed to be the mean electron number density of the magnetosphere. The light cylinder RL for a pulsar with period P* is

R

L

=

cP

*

/

2
π
=
4.88
×

10

9

(

P

*

/

1

s
)

cm
≪
l

. Therefore, we conclude that a photon with frequency of order 1 GHz can propagate freely through the magnetosphere.


Second, the cyclotron absorption in the magnetosphere is generally considered in pulsar physics, whose optical depth is given by

:
12

τ

cyc

≃
2
×

10

−
3

B

*
,
12

3

/

5

P

*

−
9

/

5

ν

curv,9

−
3

/

5

γ

2

−
3

/

5

,

where

P

*

=
2
π

/

Ω

*

 and the electron number density outside of the flux tube (see Figure 1) has been assumed to be described by the Goldreich–Julian density.


Third, the characteristic distance (l) of a photon due to the Thomson scattering is

13

l
=

1

(
1
−

β

e

cos
θ
)

n

¯

e

σ

T

=

1.5
×

10

14

cm

n

¯

e
,
10

(
1
−

β

e

cos
θ
)

,

where

β

e

c

 is the velocity of an electron, θ is the angle between the motion directions of the electron and photon, and

n

¯

e
,
10

=

n

¯

e

/

10

10

cm

−
3

 has been assumed to be the mean electron number density of the magnetosphere.


Our next goal is to estimate the density, spatial extent, and total baryonic mass contained in the CGM of NGC 3221, given that the detected signal is truly from NGC 3221. We calculate the average EM profile from the cumulative EI profile. As the EI of the larger annulus encompasses the EI of the smaller annulus, the uncertainties are not independent. We assume that the correlation coefficient of the consecutive measurements is 0.5, i.e., the covariance is half the geometric mean of their own variance: σi,i−1 = 0.5σiσi−1. We fit the EM profile with a β-model. The β-model has a density profile
3

where no is the central density and rc is the core radius, resulting in an EM profile similar to the surface brightness profile (Sarazin 1986)
4

where EMo is the central EM. Because of the small number of data points, large error bars, and the degeneracy between the parameters, we could not fit the three parameters of the β-model simultaneously. Therefore, as we did for the surface brightness profile in Section 3.1, we fit the EM profile in two different ways:

(1)
A β-model: We first fixed β = 0.5 and fit the radial profile for EMo and rc. Then, we fixed rc to the best-fit value (152.4 kpc) and fit for EMo and β. The resulting fit (χ2/dof = 3.3/4) yielded parameter values EMo =(2.8 ± 1.1) × 10−5 cm−6 kpc and β = 0.56 ± 0.36. We could not determine the range of rc from the fit, but we obtain the range of rc empirically for the best-fitted values of EMo and β that is consistent with the 1σ error span of the EM profile, with minimum rc = 110 kpc and maximum rc = 225 kpc. Interestingly, the best-fitted values and the confidence intervals are consistent with those obtained by fitting the radial profile of surface brightness in Section 3.1. Additionally, we fit the EM profile with Model B, a constant-density model.

(2)
A constant-density model: The truncated constant-density homogeneous medium (n = no(constant)) has an elliptical profile of projected EM,
5

where Rout is the spatial extent of the gaseous medium. We obtain best-fitted (χ2/dof = 1.4/4) values of n =

 = (2.7 ± 0.2) × 10−4 cm−3 and Rout = 175 ± 2 kpc (Figure 8).


The solution will be a retarded integral over the past light cone of the field point. The domain of the integration can be conveniently partitioned into a near-zone domain and a wave-zone domain. In the near zone, the field point distance from the source

 is small compared with a characteristic wavelength

, where tc is the characteristic timescale of the dynamics of the source and c is the speed of light. The boundary of the near and wave zones is situated roughly at an arbitrary radius

. This radius is of the same order of magnitude as λc. The integrals can vanish at this boundary. In fact, by partitioning the domain of integration into these two zones, all integrations will be finite, and one can show that the contributions from the near-zone spatial integrals containing

 (which diverge at

) will be canceled by corresponding terms from the wave-zone integrals (Will & Wiseman 1996). As a result, this calculation does not depend on the arbitrary boundary radius

. Keeping these facts in mind, one can find the PN metric as (Poisson & Will 2014)
1

where Ψ is defined as
2

Potentials that appear in the metric components are given by
3

where quantities in integrands are evaluated at time t and position

. The matter variables in the PN approach are

, where Σ* is the conserved surface mass density defined as
4

in which Σ is the proper surface density and g is the determinant of the metric tensor. Here u0 is the zeroth component of the velocity four-vector uα. It should be noted that in the Newtonian limit, there is no difference between Σ* and the proper surface density Σ. Moreover, p is the pressure, Π = /Σ is the internal energy per unit mass, and vj is the fluid’s velocity field defined with respect to the time coordinate t. Note that  is the proper surface density of internal energy and U is the gravitational potential for surface density Σ*. In the following subsection, we will introduce the Newtonian potential in terms of Σ. Using the PN metric and the conservation equation of the energy-momentum tensor, i.e.,

, one can find the PN equations of hydrodynamics (Poisson & Will 2014). In fact, in the 1pn approximation, the relativistic corrections appear as terms proportional to order c−2 in the equations. The continuity equation in the PN limit takes the following form:
5

In the cylindrical coordinate system (R, φ, z), where the velocity is

, Equation (5) is given by
6

Moreover, the Euler equation in the PN approximation can be written as follows:
7

The terms inside the braces are PN corrections. In the 1pn approximation, we ignore terms proportional to order c−4 and higher. The left-hand side of Equation (7) can be written as

. After some algebraic manipulations and using vector analysis in the cylindrical coordinate system, one can decompose Equation (7) to components R and φ. The R component of the Euler equation reads
8

Also, the φ component of the Euler equation can be written as
9

In order to have a complete set of differential equations, the abovementioned equations should be joined with the first law of thermodynamics and an equation of state (EOS). The first law of thermodynamics for a perfect fluid can be written as
10

On the other hand, the pressure and density of the fluid can be related to each other via an EOS. A simple case suitable for astrophysical aims is the barotropic equation, given as
11

where the conserved surface density is
12


We study here the Sco OB1 association (Figs. 1 and 2), using this and other techniques. The general properties of this large OB association, which spans almost 5° on the sky, and is surrounded by a ring-shaped HII region called Gum 55, are reviewed by Reipurth (2008). Its central cluster NGC 6231 contains several tens of OB stars, which have been extensively studied. On the other hand, many fewer studies, all recent, were devoted to the full mass spectrum, using optical photometry (Sung et al. 1998, 2013) and X-rays (Sana et al. 2006, 2007; Damiani et al. 2016; Kuhn et al. 2017a,b). The currently accepted distance of NGC 6231 is approximately 1580 pc, and its age is between 2and 8 Myr, with a significant intrinsic spread (Sung et al. 2013; Damiani et al. 2016). No ongoing star formation is known to occur therein, however. Approximately one degree North of the cluster, the loose cluster Trumpler 24 (Tr 24) also belongs to the association. There is little literature on this cluster (Seggewiss 1968; Heske & Wendker 1984, 1985; Fu et al. 2003, 2005) which unlike NGC 6231 lacks a well-defined center and covers about one square degree on the sky. Its age is  10 Myr according toHeske & Wendker (1984, 1985), who find several PMS stars, and its distance is 1570–1630 pc according to Seggewiss (1968). Other studies of the entire Sco OB1 association include MacConnell & Perry (1969 – Hα-emission stars), Schild et al. (1969 – spectroscopy), Crawford et al. (1971 – photometry), Laval (Laval 1972a,b – gas and star kinematics, respectively), van Genderen et al. (1984 – Walraven photometry), and Perry et al.(1991 – photometry). At the northern extreme of Sco OB1, the partially obscured HII region G345.45+1.50 and its less obscured neighbor IC4628 were studied by Laval (1972a), Caswell & Haynes (1987), López et al. (2011), and López-Calderón et al. (2016). They contain massive young stellar objects (YSOs; Mottram et al. 2007), maser sources (Avison et al. 2016), and the IRAS source 16562-3959 with its radio jet (Guzmán et al. 2010), outflow (Guzmán et al. 2011), and ionized wind (Guzmán et al. 2014), and are therefore extremely young (1 Myr or less). The distance of G345.45+1.50 was estimated as 1.9 kpc by Caswell & Haynes (1987), and 1.7 kpc by López et al. (2011), in fair agreement with distances of Sco OB1 stars. In Fig. 1 of Reipurth (2008) a strip of blue stars is visible, connecting NGC 6231 to the region of IC4628.


Its age is  10 Myr according to
who find several PMS stars


H α velocity dispersion. for Hα, the mass of ionized hydrogen is not a good estimate of the Hα intensity, since the emission decreases with increasing temperature. Here, we compute the Hα flux from two contributions, namely the recombination of ionized hydrogen and collisional excitation from the ground state to level n = 3. We have to neglect the contribution from H ii regions around young massive stars since these are not treated in the simulations. Collisional excitations to n > 3 represent a negligible contribution to the total Hα emission since transitions to these levels are significantly less likely (Anderson et al. 2000, 2002, but see also Péquignot & Tsamis 2005 and references therein). We compute mHα (dLHα) and Mtot,Hα (Ltot,Hα), required in equation (9), following the emissivity calculations for recombination and collisional excitation of Draine (2011), Dong & Draine (2011), and Kim et al. (2013b) (but see also Aggarwal 1983):

(11)

\begin{equation}
{\rm d}L_{\mathrm{H}\alpha ,\mathrm{R}}\propto T_4^{-0.942-0.031\ln (T_4)}n_\mathrm{e} n_{\mathrm{H\small {II}}} {\rm d}V\,,
\end{equation}

(12)

\begin{equation}
{\rm d}L_{\mathrm{H}\alpha ,\mathrm{C}}\propto \frac{\Gamma _{13}(T_\mathrm{e})}{\sqrt{T_\mathrm{e}}}{\rm e}^{\frac{-12.1 \mathrm{eV}}{k_\mathrm{B} T_\mathrm{e}}} n_\mathrm{e} n_{\mathrm{H\small {I}}} {\rm d}V\,,
\end{equation}

where T4 = T/104 K, Te is the electron temperature, nj is the number density of species j in cm−3, dV is the zone volume, and

(13)

\begin{eqnarray}
\Gamma _{13}(T_\mathrm{e})&=&0.35-2.62\times 10^{-7}T_\mathrm{e}-8.15\times 10^{-11}T_\mathrm{e}^2\nonumber\\
&&+\;6.19\times 10^{-15}T_\mathrm{e}^3.
\end{eqnarray}

We apply equation (12) only for cells with temperatures 4000  T  25 000 K assuming Te = T. We only consider this temperature range because emission at T  4000 K will be negligible, while we expect to find very little atomic hydrogen at T > 25 000 K. We also assume $n_\mathrm{e}=n_{\mathrm{H\,\small {II}}}$, which results in an ∼10 per cent error at most in regions where helium is ionized.


SNRs are thought to be the dominant contributors to the Galactic CR flux up to the ‘knee’ at E = 1015 eV. The Galactic production rate of CRs can be explained if ηCR ∼ 10 per cent of the SN energy (ηCR is the cosmic ray acceleration efficiency and ESN ∼ 1051 erg) goes into accelerating CRs at SN blast waves (e.g. Aharonian 2004). Many middle-age SNRs (tSNR ∼ 105 yr) found in dense environments hosting GMCs have been observed as bright GeV and TeV sources. In fact, SNRs interacting with GMCs, as inferred from the detection of OH masers (e.g. Frail et al. 2013), constitute the dominant fraction of Galactic GeV SNRs (e.g. Thompson, Baldini & Uchiyama 2012). An estimate of the GeV/TeV flux from the hadronic component can be obtained with simple arguments. At high energies (Eγ > 1 GeV), the γ-ray spectrum is spectrally similar to the parent distribution of CR protons (e.g. Aharonian & Atoyan 1996). Then, for a power-law distribution of CR protons

(96)

\begin{equation}
n_{\rm p}(E_{\rm p}) = K_{\rm p}E_{\rm p}^{-s_{\rm p}}
\end{equation}

where the normalization Kp is obtained by assuming that the CR energy density $U_{\rm CR}\sim 3\eta _{\rm CR}E_{\rm SN}/4\pi R_{\rm SNR}^3$ (e.g. Drury, Aharonian & Völk 1994), the γ-ray photon emissivity [phs−1 cm−3 erg−1] is (e.g. Aharonian 2004)

(97)

\begin{equation}
J_{\gamma} (E_\gamma ) = \int _{E_{\rm p, min}}^\infty \frac{2cn_{\rm GMC}\sigma _{\rm pp}(E_{\rm p})\eta _An_{\rm p}(E_{\rm p})}{\sqrt{(E_{\rm p}-m_{\rm p}c^2)^2\kappa _{\pi} ^2-m_{\pi} ^2c^4}}{\rm d}E_{\rm p}
\end{equation}

where the threshold proton energy for π0 production is $E_{\rm p, min} = m_{\rm p}c^2+ \kappa _{\pi} ^{-1}(E_\gamma + m_{\pi} ^2c^4/4E_\gamma )$, mp and mπ are the proton and π0 masses, ηA ≃ 1.5 includes the contribution of nuclei other than protons towards the production of γ-rays (Dermer 1986), κπ = 0.17 is the mean fraction of proton's kinetic energy transferred to π0-meson per collision, and nGMC is the target proton number density (assumed uniform) in the GMC. The p–p inelastic collision cross-section is given by (e.g. Cheng & Romero 2004)

(98)

\begin{equation}
\sigma _{\rm pp}(E_{\rm p}) \approx 30 \left[0.95+0.06\ln \left(\frac{E_{\rm p}}{\rm GeV}\right)\right]\,{\rm mb}
\end{equation}

which is assumed to vanish below proton kinetic energy Ep − mpc2  1 GeV. Since σpp only has a weak logarithmic dependence on proton energy, the γ-ray spectrum is expected to reproduce the spectrum of the parent proton population. The integrated γ-ray photon flux in the GeV-region for sp = 2.15, E1 = 1 GeV and E2 = 100 GeV is

(99)

\begin{equation}
\Phi _\gamma = \int _{E_1}^{E_2}\frac{J_\gamma (E^{\prime }_\gamma )V_{\rm TeV}}{4\pi d^2} {\rm d}E_\gamma ^{\prime } \simeq 8.16\times 10^{-10} A\,{\rm cm}^{-2}\,{\rm s}^{-1}
\end{equation}

where $A = \eta _{\rm CR} E_{\rm SN,51}n_{\rm GMC}d_4^{-2}$. This agrees with the photon flux measured by Fermi (H.E.S.S. Collaboration 2015) in the (1–100) GeV region Φ ≈ 8.93 × 10−9 cm−2 s−1 for ηCR ∼ 0.1 and nGMC ∼ 109 cm−3. This is a reasonable estimate of the target proton number density in light of the fact that Tian et al. (2007) find a number density of ∼103 cm−3 for the GMC associated with W41 from 13CO observations.


Keeping these facts in mind, one can find the PN metric as


Using the PN metric and the conservation equation of the energy-momentum tensor, i.e.,

, one can find the PN equations of hydrodynamics


There are many types of nonstationarity for a shock front, e.g., self-reformation (Lembège & Dawson 1987; Hada et al. 2003; Chapman et al. 2005; Matsukiyo & Scholer 2014), self-excited ripples (Winske & Quest 1988; Savoini & Lembège 1994; Lembège et al. 2004; Burgess & Scholer 2007), and pre-existing waves or turbulence (Giacalone 2005; Guo & Giacalone 2010), some of which likely cause the unexpected motions of the TS. These nonstationarities are predicted and observed by numerical simulations and satellite observations, respectively. The term “self-reformation” describes a process wherein the particles reflected by the shock ramp accumulate ahead of the shock and form a shock foot, which then grows and becomes the new ramp. The new ramp starts to reflect incident particles, and the process repeats. Self-reformation of the shock front was predicted by both hybrid simulations (Tiu et al. 2011; Hellinger & Trávnícek 2002; Lembège et al. 2009; Yuan et al. 2009; Su et al. 2012) and particle-in-cell (PIC) simulations (Lembège & Dawson 1987; Hada et al. 2003; Nishimura et al. 2003; Scholer et al. 2003; Lee et al. 2005a; Yang et al. 2009, 2013) for large Mach number and low

 shocks, where

 is the ratio of the thermal pressure of ions to the magnetic pressure. Even in the presence of PUIs, self-reformation occurs under some conditions (Chapman et al. 2005; Lee et al. 2005b; Oka et al. 2011; Yang et al. 2012a; Matsukiyo & Scholer 2014). For the heliospheric TS, the values of

 and the Mach number are relatively low and high, respectively. The TS is generally believed to be in the supercritical regime and probably undergoes self-reformation (Burlaga et al. 2008). Self-excited ripples are usually found in two-dimensional (2D) hybrid simulations (Thomas 1989; Burgess & Scholer 2007) and PIC simulations (Savoini & Lembège 1994; Lembège et al. 2009; Yang et al. 2012b). Shock front ripples are robust in three-dimensional (3D) hybrid simulations (Thomas 1989; Hellinger et al. 1996). The filament instability of the shock front found in high-dimensional simulations can also contribute to the shock front nonstationarity (Spitkovsky 2005; Guo & Giacalone 2013; Caprioli & Spitkovsky 2014). Not all of the simulations above include PUIs. By using the 2D Los Alamos hybrid simulation code with PUIs, Liu et al. (2010) studied the Alfvén-cyclotron and mirror modes excited in the near-TS heliosheath. The impact of the PUIs on the shock front ripples and self-reformation is not mentioned. It is expected that the density of PUIs at the TS is of the order of 20%–30% of the solar wind density (Richardson et al. 2008; Wu et al. 2009; Matsukiyo & Scholer 2014). The impact of such a high percentage of PUIs on the shock front ripples has not been studied yet. The relevance of the shock front ripples to the multiple TS crossings still remains unclear. A precise description of this influence would require at least a 2D full particle model.


However, in the case of stationary collisionless stellar systems with gravitomagnetic forces, the situation is not so simple, because while E is still an integral of motion, J

z
 is not conserved along orbits and so it cannot be used as an isolating integral in the DF. Fortunately, the problem is solved as follows. As is well known, the Lagrangian (per unit mass) associated with Equation (30) is
34

=∥v∥22−(ϕ+A·v),

where, at variance with the EM case (e.g., see Landau & Lifshitz 1971), the + sign in the generalized potential derives from the – sign in the gravitomagnetic Lorentz force. Suppose now that
v
 = v

φ
(R, z)
e

φ
, so that
A
 = A

φ
(R, z)
e

φ
 from the results in Section 2.1. The Euler–Lagrange equations show immediately that J

z
 is not conserved, but a second integral of motion exists, I
2 = J

z
 − RA

φ
. The Jeans theorem then in principle allows for a two-integral DF, f(E, I
2), and an interesting question arises whether such a DF is consistent with the assumption of streaming motions with
v
 = v

φ
(R, z)
e

φ
 used to prove the existence of I
2. The answer is in the affirmative, and it is easy to prove that properties (1) and (2) mentioned above for systems supported by a f(E, J

z
) are preserved by the generalized f(E, I
2), in particular the fact that v

R
 = v

z
 = 0. Therefore, the vertical and radial Jeans Equations (B2) for a stationary, axisymmetric two-integral system with gravitomagnetic forces, supported by a DF of the family f(E, I
2), become
35

∂ρσ2∂z=−ρ∂ϕ∂z+jBR[j],j=ρvφ,∂ρσ2∂R−ρΔR=−ρ∂ϕ∂R−jBz[j],

where

Δ=vφ2¯−σ2

, and in the isotropic rotator case

Δ=vφ2

. Finally, the azimuthal Jeans equation vanishes identically also in the gravitomagnetic case, as obvious from the last of Equations (B2). Notice how, at variance with the classical Newtonian case, now the vertical and radial velocity dispersions depend on the ordered rotational field v

φ
(R, z), yet to be determined.


We performed an accurate estimation of the physical properties derived from SED fitting (e.g., absolute magnitude, stellar mass, and star formation rate) and characterized each sample as a whole with the median of each of these properties. Figure 6 shows the median of the probability distribution of stellar mass, absolute magnitude, and star formation rate for the BJGKn 6 (red squares) samples as a function of redshift. In the stellar mass panel, the dashed line shows the evolution of the stellar mass for galaxies with present-day stellar masses of log (M∗) ≈ 10.7, such as our Milky Way galaxy. The data are taken from the analysis of the CANDELS survey by van Dokkum et al. (2013). The observed mass growth determined by van Dokkum et al. (2013), which slightly increase by a factor of 0.1 dex from z = 1 to z = 0.5,  agrees well with the flat behavior found in this work. In the star formation rate panel, the dashed line shows the evolution of the implied star formation rate that is caused by the evolution of the stellar mass of galaxies with present-day stellar masses of log (M∗) ≈ 10.7, such as our Milky Way, also taken from van Dokkum et al. (2013). Our results are slightly lower, suggesting that the SFR is sufficient to account for the increment in stellar mass between z = 1 and z = 0.5  and that major mergers play a minor role in this redshift range. The evolution of the SFR and sSFR resembles the expected main-sequence behavior of star-forming galaxies (Rodighiero et al. 2011). The BJGK evolution from z ~ 1 to z = 0 of the B-band absolute luminosity agrees with the evolution measured by Tasca et al. (2014). They found that the contribution of disks to the total B-band luminosity decreases by 30%  from z ~ 1 to z = 0, while for the BJGs the median B-band luminosity decreases by a factor of 0.5 dex. The distribution of galaxy ages obtained by iSEDfit has a large dispersion, but even so, its increment between two epochs corresponds to the Universe age increment indicated by the redshift. According to the models of Lagos et al. (2014), galaxies of properties similar to the BJGs host most of the neutral gas at 0.5 z 1.5. Hence the BJG samples contain tentative targets to sample the neutral gas with submillimeter surveys.


, which slightly increase by a factor of 0.1 dex from z = 1 to z = 0.5,  agrees well with the flat behavior found in this work.


In the star formation rate panel, the dashed line shows the evolution of the implied star formation rate that is caused by the evolution of the stellar mass of galaxies with present-day stellar masses of log (M∗) ≈ 10.7, such as our Milky Way, also taken from

Our results are slightly lower, suggesting that the SFR is sufficient to account for the increment in stellar mass between z = 1 and z = 0.5  and that major mergers play a minor role in this redshift range.


We revisit the total (star + circumstellar) mass estimate for the M1701117 system, and the likelihood of it evolving into a brown dwarf. From ALMA continuum observations, the total (dust+gas) mass arising from the circumstellar material in the M1701117 system is 20.98 ± 1.24 MJup. To estimate the intrinsic stellar mass for M1701117, we have used the infrared photometry for this object and the evolutionary models by Baraffe et al. (2003). The J band, in particular, is considered to be least affected by the potential effects of veiling at bluer wavelengths, and circumstellar disc emission further into the infrared (e.g. Hartigan et al. 1995; White & Hillenbrand 2004; Cieza et al. 2005). Fig. 10 shows the UKIDSS (J − K, J) colour–magnitude diagram (cmd). The filled circles represent known members from previous studies in the σ Orionis cluster (Béjar et al. 1999, 2001; Zapatero et al. 2000; Caballero et al. 2007; Caballero 2008). The cluster suffers from negligible reddening (AV ≤ 1 mag; Béjar et al. 1999) and the photometric data has been corrected for the interstellar reddening. M1701117 appears much redder than the main cluster member sequence, which is due to the large K-band excess emission from the circumstellar material. We have overplotted in Fig. 10 the 1, 3, 5, and 8 Myr isochrones from the Baraffe et al. (2003) models. The isochrones at ages of 1 Myr are not provided in these models. The mass scale shown on the right-hand side in Fig. 10 is from the 1 Myr model isochrone at a distance of 410 pc for the cluster. Assuming an age of 1 Myr, we can estimate a stellar mass of ∼40 MJup and a stellar luminosity of 0.012 L$\odot$ for M1701117. The location of M1701117 to the red of the 1 Myr isochrone in the cmd indicates an age younger than 1 Myr. The typical age of the embedded phase or the age of the protostar formation is considered to be ∼0.01–0.05 Myr, based on the statistics of embedded-to-Class II sources (e.g. Evans et al. 2009). On the other hand, the embedded phase may be shorter for very low-mass protostars and brown dwarfs, as indicated by numerical simulations. The estimates on the stellar mass and luminosity for M1701117 will be lower if the object is younger than 1 Myr.


The filled circles represent known members from previous studies in the σ Orionis cluster


The cluster suffers from negligible reddening (AV ≤ 1 mag;


Baraffe et al. (2003)
Baraffe et al. (2003)

We have overplotted in Fig. 10 the 1, 3, 5, and 8 Myr isochrones from the

To estimate the intrinsic stellar mass for M1701117, we have used the infrared photometry for this object and the evolutionary models by


The typical age of the embedded phase or the age of the protostar formation is considered to be ∼0.01–0.05 Myr, based on the statistics of embedded-to-Class II sources (e.g.


For MS high-velocity stars seemingly ejected from the Galactic disc, on the other hand, two main processes are typically blamed. In the dynamical ejection scenario (DES; e.g. Poveda, Ruiz & Allen 1967; Leonard & Duncan 1990; Leonard 1991; Perets & Šubr 2012; Oh & Kroupa 2016), exchange encounters in dense stellar systems (Aarseth 1974) may eject stars at high velocities. In the binary supernova scenario (BSS; e.g. Blaauw 1961; Boersma 1961; Tauris & Takens 1998; Portegies Zwart 2000; Tauris 2015; Renzo et al. 2019b), the massive primary in a binary system explodes in a core-collapse (CC) supernova, disrupting the binary and ejecting its MS companion with a velocity comparable to its pre-CC orbital velocity. Both processes are known to occur in the Milky Way (Hoogerwerf, de Bruijne & de Zeeuw 2001; Jilinski et al. 2010) and are generally thought to be responsible for the known sample of ‘runaway stars’ with ejection velocities of ${\ge}30\!-\!40\, \mathrm{km\ s^{-1}}$ (Blaauw 1961), though their relative contribution is not yet well constrained (see Hoogerwerf et al. 2001; Renzo et al. 2019b). With characteristic ejection speeds of the order of a few tens of km s−1, it is not yet known whether these mechanisms can eject stars of the order of hundreds of km s−1 with sufficient frequency to explain the current known sample of ‘hyper-runaway stars’ (HRSs) – runaway stars ejected near to or above the Galactic escape velocity at their location. While ejection velocities in the neighbourhood of ∼1000 km s−1 are possible in both the DES (Leonard 1991) and BSS (e.g. Tauris & Takens 1998; Tauris 2015) scenarios, these situations are thought to be rare. Recent N-body simulations of young star clusters have found that ejections in excess of 200 km s−1 are very rare (Perets & Šubr 2012; Oh & Kroupa 2016). Binary population synthesis models simulating a large number of binary systems show that ejections above 200 km s−1 are vastly outnumbered by ejections of the order of ∼10 km s−1 (Portegies Zwart 2000; Eldridge, Langer & Tout 2011; Renzo et al. 2019b).


though their relative contribution is not yet well constrained

Both processes are known to occur in the Milky Way

and are generally thought to be responsible for the known sample of ‘runaway stars’ with ejection velocities of ${\ge}30\!-\!40\, \mathrm{km\ s^{-1}}$


Binary population synthesis models simulating a large number of binary systems show that ejections above 200 km s−1 are vastly outnumbered by ejections of the order of ∼10 km s−1


CCSNe are also of interest for GW astronomy as targets in their own right. As the sensitivity of GW detectors increases, they will begin to detect not only binary mergers but also other lower amplitude sources of GWs such as CCSNe. Accurate knowledge of the GW emission from CCSNe will be essential for detection and parameter estimation. The GW signal from rotational core bounce has already been well covered in the literature (e.g. Dimmelmeier et al. 2008; Abdikamalov et al. 2014; Fuller et al. 2015; Richers et al. 2017). In the non-rotating case, the GW emission from the post-bounce phase has been studied using self-consistent 3D simulations by many groups (Kuroda, Kotake & Takiwaki 2016; Andresen et al. 2017, 2019; Kuroda et al. 2017, 2018; Powell & Müller 2019, 2020; Radice et al. 2019; Andresen, Glas & Janka 2020; Mezzacappa et al. 2020; Pan et al. 2020). The structure of the GW emission has shown common features in different simulations from recent years. The dominant emission feature in the GW emission is due to the quadrupolar surface f/g mode 1 of the proto-neutron star (PNS), which produces GW frequencies rising in time from a few hundred Hz up to a few kHz (Müller, Janka & Wongwathanarat 2012; Sotani et al. 2017; Kuroda et al. 2018; Morozova et al. 2018; Torres-Forné et al. 2018, 2019). In addition, some models (Kuroda et al. 2016, 2017; Andresen et al. 2017; Mezzacappa et al. 2020; Powell & Müller 2020) exhibit low-frequency GW emission due to the standing accretion shock instability (SASI; Blondin, Mezzacappa & DeMarino 2003; Blondin & Mezzacappa 2006; Foglizzo et al. 2007). In rapidly rotating models, very strong GW emission can also occur during the post-bounce phase due to a corotation instability (Takiwaki & Kotake 2018). The emerging understanding of the GW emission features has led to the formulation of universal relations for the GW emission (Torres-Forné et al. 2019) and paved the way for phenomenological modelling for CCSN signals (Astone et al. 2018). Further work is still needed, however, to extend these models to fully explore CCSN GW signals from across the progenitor parameter space. The majority of 3D simulations that include GW emission are for progenitor stars below $30\, \mathrm{M}_{\odot }$. In this paper, we perform simulations of high-mass Population III (Pop-III) stars in the pulsational pair instability regime to expand the parameter space coverage of 3D simulations and to provide further insights into the massive and very massive star remnant BH population.


The GW signal from rotational core bounce has already been well covered in the literature


The dominant emission feature in the GW emission is due to the quadrupolar surface f/g mode 1 of the proto-neutron star (PNS), which produces GW frequencies rising in time from a few hundred Hz up to a few kHz


In rapidly rotating models, very strong GW emission can also occur during the post-bounce phase due to a corotation instability


The emerging understanding of the GW emission features has

and paved the way for phenomenological modelling for CCSN signals


Several faint and sparse knots are detected over all the field of observations, especially around the systemic velocity of IC443 vLSR = −4.5 km s−1 (Hewitt et al. 2006). These structures, noticeable between v = −7.5 km s−1 and v = −1.5 km s−1, might either correspond to a slice of turbulent medium driven by the SN shockwave and/or belong to the ambient gas associated with the NW-SE molecular cloud in which IC443G is embedded (Lee et al. 2012). Other than that, the description of the region probed by our observations can be divided into the following six distinct structures:

1.Cloudlet: in the upper part of the field we observe a large (~ 5′ × 2′, i.e., ~ 2.8 × 1.1 pc) elongated cloudlet detected between v = −7.0 km s−1 and v = −5.5 km s−1 (indicated by the letter “A” in Fig. 2), which is also detected in 12CO(3–2). This structure was labeled G1 by Zhang et al. (2010), as part of the double-peaked morphology of the extended G region. The 13CO J = 1–0 counterpart of this structure is much brighter than the other main structures in the field, and it is also detected in the transitions J = 2–1 and J = 3–2, and in C18O J = 1–0 and J = 2–1. This structure was also presented and characterized by Lee et al. (2012) who proposed the label SC 03, among a total of 12 SCs (of size ~1′) found in IC443.
2.Ring-like structure: a ring-like structure seemingly lying in the center of the field (indicated by the letter “B” in Fig. 2), appearing between v = −5.5 km s−1 and v = −4.5 km s−1 and also detected in 12CO(3–2). It has a semimajor axis of 1.5′, or 0.8 pc. This structure might be spurious and is likely to be physically connected to the elongated cloudlet as both are spatially contiguous and their emission lines are spectrally close. It is partially detected as well in our observations of 13CO J = 1–0, J = 2–1 and J = 3–2, and also has a faint, partial counterpart in C18O J = 1–0 and J = 2–1. To understand the nature of this region we searched for counterparts in Spitzer Multiband Imaging Photometer (MIPS), WISE, DSS and XMM-Newton data; and in near-infrared and optical point source catalogs (Sect. 5), without success. Owing to projection effects, this apparent circular shape could also be explained by an unresolved and clumpy distribution of gas.
3.Shocked clump: in the lower part of the field we identify a very bright clump emitting between v = −31.0 km s−1 and v = 16 km s−1. This structure of size ~2′ × 0.75′ (~ 1.1 × 0.4 pc), which is detected in the 12CO(3–2) transition as well, belongs to the southwestern ridge of the molecular shell of the SNR and has been described as a shocked molecular structure by several studies (van Dishoeck et al. 1993; Cesarsky et al. 1999; Snell et al. 2005; Shinn et al. 2011; Zhang et al. 2010). The core of the shocked clump (indicated by the letter “C” in Fig. 2) is also detected in 13CO in the transitions J = 1–0, J = 2–1, and J = 3–2, and in C18O J = 1–0 and J = 2–1.
4.Shocked knot: an additional shocked knot (indicated by the letter “D” in Fig. 2) is also detected to the west of the previously described structure. This fainter and smaller structure is spatially separated from the main shocked clump.
5.At the same position as the shocked clump and extending southward and westward, we find a faint, elongated clump emitting between v = 5.0 km s−1 and v = 7.5 km s−1. This structure (indicated by the symbol “*” in Fig. 2) spatially coincides with the shocked clump, yet the peak velocity is not exactly the same (see developments on kinematics of the region in Sect. 3.3). It has a faint counterpart in 13CO(1–0). Observations of the ambient molecular cloud by Lee et al. (2012) indicate this structure as part of a faint NE-SW complex of molecular gas in the velocity range +3 km s−1 vLSR  +10 km s−1.
6.Finally, the 13CO(1–0) map (Fig. 4, right panel and Fig. D.2) indicates a large clump of gas extending from the bottom center to the right end of the field, with a bright knot in the bottom right corner of the field. However, this structure has no bright, well-defined counterpart in any of the 12CO transitions maps. It is spatially and kinematically correlated with the faint and diffuse 12CO J = 1–0 and J = 2–1 emission seen in the velocity range −5.5 km s−1 vLSR  −2 km s−1. From the comparison with the 12CO observations of Lee et al. (2012) and 13CO observations of Su et al. (2014) toward the SNR, we conclude that this structure is part of the western molecular complex observed in the velocity range −10 km s−1 vLSR  0 km s−1.


This structure was also presented and characterized by

who proposed the label SC 03, among a total of 12 SCs (of size ~1′) found in IC443.


Observations of the ambient molecular cloud by

indicate this structure as part of a faint NE-SW complex of molecular gas in the velocity range +3 km s−1 vLSR  +10 km s−1.


From the comparison with the 12CO observations of

and 13CO observations of Su et al. (2014) toward the SNR, we conclude that this structure is part of the western molecular complex observed in the velocity range −10 km s−1 vLSR  0 km s−1.


The total energy at each point is defined as a sum of four separate components; the CCSD(T) energy extrapolated to the complete basis set (CBS) limit and three energy increments. Each increment represents a source of electron correlation omitted from the above CBS energy: core-correlation, scalar-relativistic correlation, and higher order dynamic correlation. A nomenclature has been used in the past to describe this composite theory by its components, where C represents CBS-extrapolation, cC represents core-correlation, R represents scalar-relativity, and E represents higher order correlation (Fortenberry et al. 2011). For example, the CcCRE potential surface (equation 2) includes all of the corrections, but the CcCR omits the higher order correlation correction.
(2)
\begin{equation*}
E_\text{CcCRE} = E_\text{CBS} + \Delta E_\text{core} + \Delta E_\text{rel} + \Delta E_\text{hlc}
\end{equation*}
Two different procedures are used to extrapolate the CCSD(T) energy to the CBS limit. The first scheme used is the three-point total energy extrapolation shown in equation (3, Martin & Lee 1996).
(3)
\begin{equation*}
E(X) = E_\text{CBS}+ A(X+1/2)^{-4} + B(X+1/2)^{-6}
\end{equation*}
The second scheme extrapolates the Hartree–Fock energy and the correlation energy, represented here as ε, separately using equations (4) and (5), respectively (Feller 1993; Helgaker et al. 1997).
(4)
\begin{equation*}
E^\text{HF}(X) = E^\text{HF}_\text{CBS} + A e^{BX}
\end{equation*}
(5)
\begin{equation*}
\varepsilon ^\text{(T)}(X) = \varepsilon ^\text{(T)}_\text{CBS} + A X^{-3}
\end{equation*}
The resulting CCSD(T)/CBS energy in this latter scheme is $E^{\prime }_\text{CBS} = E^\text{HF}_\text{CBS} + \varepsilon ^\text{(T)}_\text{CBS}$, where the prime is simply used to distinguish the two CBS extrapolations. Each energy correction (increment) is defined as the difference between the energy with the correlation effect considered and the energy without it when using the same basis set. The core-correlation correction, ΔEcore, is EMTc − EMT as defined with the geometry correction. Scalar relativity is incorporated through ΔErel, which is taken as the difference in the all-electron CCSD(T)/cc-pCV(T+d)Z-DK energy with and without the Douglas–Kroll–Hess (DKH) corrections to second-order (Reiher & Wolf 2004). The energy contribution from non-iterative quadruple excitations in the coupled cluster wavefunction, ΔEhlc, is defined as the difference between CCSDT(Q)/cc-pV(T+d)Z and CCSD(T)/cc-pV(T+d)Z at each point.


However, in our model, the interactions between strangeons are dominated by strong force. The detailed calculations of the shear modulus is still not performed, but it should be much larger than that of the NS’s crust. If the burst oscillation frequencies observed in low-mass X-ray binaries correspond to the first few torsional modes of SSs, the shear modulus should be about one thousand times of the NS’s crust, say $\mu \simeq 4\times 10^{32}\, \rm erg\, cm^{-3}$ (Xu 2003; Owen 2005). We do not know how large the tidal deformability will deviate from the fluid case for SSs, but we can obtain some key insights of this problem from the calculations of tidal deformabilities for crystalline colour superconducting phase (Lau, Leung & Lin 2017, 2019). QSs composed of crystalline colour superconducting phase are rigid with extremely high shear modulus (Alford, Bowers & Rajagopal 2001; Mannarelli, Rajagopal & Sharma 2007). The shear modulus is approximately given by (Mannarelli et al. 2007)
(77)$$\begin{eqnarray*}
\mu =2.47 \, \mathrm{MeV} / \mathrm{fm}^{3}\left(\frac{\Delta }{10 \, \mathrm{MeV}}\right)^{2}\left(\frac{\mu _{\rm q}}{400 \, \mathrm{MeV}}\right)^{2}\, ,
\end{eqnarray*}$$where μq is the average quark chemical potential and Δ is the gap parameter which is in the range of 5–$25\, \rm MeV$. If the shear modulus is taken to be $\mu =4\times 10^{32}\, \rm erg\, cm^{-3}$ ($\Delta \simeq 5\, \rm MeV$), according to the results in Lau et al. (2017, 2019), the deviations of the tidal deformabilities from the fluid case are in the order of $1{{\ \rm per\ cent}}$. For larger gap parameters, the deviations can be very large and more interestingly, they found that the universal relations between the moment of inertia and tidal deformability will deviate from the ones for fluid stars. If we take the shear modulus to be about $\mu \simeq 4\times 10^{32}\, \rm erg\, cm^{-3}$, the corrections may not be very large and the results discussed in our work is a good approximation. But if the shear modulus is actually larger, then we need to consider the corrections due to the elasticity.


We derive leftover planetesimal populations directly from our simulations sets that incorporate planetesimals (see Table 1 and the left panel of Figure 1), and perform an additional batch of ∼105 simulations of the final few giant impacts in our systems to derive collisional fragment distributions. This additional suite of computations utilizes a modified version of the Mercury integration package described in Chambers (2013) that incorporates collisional parameter space mapped in Stewart & Leinhardt (2012) to track hit-and-run and fragmenting impacts. Aside from the fragmentation algorithm, the integration parameters (i.e., time step, ejection radius, etc.) are identical to those utilized in Section 2.1. To prevent the calculation from becoming intractable, the user must establish a minimum fragment mass (MFM; 0.005 M
⊕ in our simulations based on previous experience with the code; Quintana et al. 2016; Clement et al. 2019b). When the algorithm detects a fragmenting collision, the total remnant mass is divided into a number of equal-mass fragments with M > M
MFM that are ejected in uniformly spaced directions in the collisional plane at v ≃ 1.05v
esc. If the total eroded mass is less than the MFM, the collision is considered totally accretionary. While this treatment of fragmentation is obviously somewhat contrived (see Section 4 for a discussion of some of these caveats), a first-order dissection of the distinctive populations of collisionally generated debris and unaltered planetesimals is important for our study as the fragment material might posses enhanced volatile and light element inventories if it is ejected from near the surface of partially or fully differentiated proto-planets. Moreover, since we utilize these simulations to infer the orbital distribution of debris given a supposed size frequency distribution (SFD), our particular selection of MFM is not carried forward into our study of late bombardment, and is therefore only important for keeping our experiments computationally feasible.