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Any massive stellar source from ~8 M⊙ can burn hot enough to completely ionise the surrounding molecular material to form an H II region (Wood & Churchwell 1989; Churchwell 1990; Kurtz 2005) and destroy any complex chemical tracers that are traditionally used to understand the kinematics of their natal environments and their accretion discs. Initial arguments by Walmsley (1995) argued that high accretion rates will cause a very dense H II region that is optically thick to radio emission and thus the H II would not be seen. However, for heavily accreting massive YSOs the onset of the H II region could be delayed via stellar bloating (Palla & Stahler 1992; Hosokawa & Omukai 2009; Hosokawa et al. 2010; Kuiper & Yorke 2013) where the effective temperature of the star is much cooler than it would be considering a main sequence star of the same luminosity. In these models, a halt or considerable reduction in accretion (≪10−3 M⊙ yr−1), or growth beyond ~30−40 M⊙, will result in the YSO contracting to a “main-sequence” configuration, heating significantly, and being able to create an H II region. The fine details of this transition are still somewhat unclear since they depend on the assumed accretion law and the initial conditions chosen for the stellar evolution calculations (Haemmerlé & Peters 2016). Observations in the infra-red (IR) have been made in search of cool stellar atmospheres that point to bloated stars, however these studies remain inconclusive (Linz et al. 2009; Testi et al. 2010). Alternative scenarios are that H II regions can be gravitationally trapped at very early ionisation stages (see Keto 2003, 2007), or flicker due to chaotic shielding of the ionising radiation by an accretion flow, leading to a non-monotonous expansion (Peters et al. 2010b). Hyper Compact (HC) H II regions (0.03 pc) are thought to be the earliest ionisation stage and therefore could relate to the halt of accretion, be a marker of a transition phase.
Citation: Kuiper & Yorke 2013
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Citation: Haemmerlé & Peters 2016
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Citation: Testi et al. 2010
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Abundances are estimated by fitting the different ions in the observed spectrum. Microturbulence is again fixed at 20 $\rm km\, s^{-1}$  for all elements, the same value adopted for the Si ionization balance. All abundances are given by number and were initially adopted to be solar and then modified to reach a better fit to the observed spectrum. Most He  lines are well fitted with a solar abundance, but the strong He i 4471 Å  line points to a slightly enriched He abundance, which is consistent with the detection of He ii 4686. Thus, we adopt $\epsilon = N({\rm He})/[N({\rm H})+N({\rm He})] =0.12\pm 0.04$, a mildly enriched He  abundance, but still consistent with the solar one. The N abundance is poorly constrained, as the N iii lines around 4630–4640 are blended with O and C and are highly sensitive to wind details (see Rivero-González et al. 2011), whereas the N ii lines from 5666 to 5710, clear in this spectral type, are outside the spectral range of our observations. Thus, we have to rely on the very weak N ii 5005.15 and 5007.33 Å  lines (a blend in our spectrum). These lines are consistent with a solar N abundance. The absence of N ii 4630.54 and N iii 4634.13 (which should be present if the N abundance is higher than solar) is also consistent with a solar N abundance. Taking into account the uncertainties in stellar parameters, we determine a value of log(N/H) + 12 = 7.83 ± 0.15 (with 7.83 ± 0.05 being the solar value from Asplund et al. 2009). The abundance of C  relies on the C ii λ4267 line, which is weak. Unfortunately, the C ii 6578-83 lines are not well reproduced by our model. We estimate a C  abundance of log(C/H) + 12 = 8.13$^{+0.20}_{-0.25}$, a factor of 2 below solar, where errors have again been obtained by fitting the line with varying stellar parameters. However, given the uncertainties, the line weakness and the moderate S/N in the region, we cannot completely rule out a solar abundance. For Mg we have only the doublet line at λ4481, which fits at a slightly lower than solar abundance, but fully compatible with solar – we obtain log(Mg/H) + 12 = 7.53 ± 0.15 – where the relatively low uncertainty is probably an underestimate as we have only one line available. On the contrary, for O and Si we have several lines and thus the same uncertainty of 0.15 dex is more representative. We obtain in both cases solar abundances, with uncertainties determined by varying the stellar parameters (except the microturbulence, which we kept fixed through the whole analysis). Abundances are given in Table 3.
Citation: Rivero-González et al. 2011
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A gravitational lens deflects light from background sources depending on their projected distance R in the lens plane. The deflection angle $\hat{\alpha }(R)$ of a thin axially symmetric lens where the distances between the source, the lens, and the observer are much larger than the size of the lens is directly related to its cumulative mass $\mathcal {M}(R)$ through
(35)$$\begin{eqnarray*}
\hat{\alpha }(R) = \frac{4 G\mathcal {M}(R)}{c^2 R}
\end{eqnarray*}$$(Schneider, Ehlers & Falco 1992, equation 8.5), c being here the speed of light. Introducing DL, DS, and DLS the angular distances respectively between the observer and the lens, between the observer and the source, and between the lens and the source, one can express the scaled deflection angle
(36)$$\begin{eqnarray*}
\alpha (R)\equiv \frac{D_{\rm L}D_{\rm LS}}{r_\mathrm{ c} D_{\rm S}} \hat{\alpha }(R)
\end{eqnarray*}$$and the convergence
(37)$$\begin{eqnarray*}
\kappa (R) \equiv \frac{{\Sigma }(R)}{\Sigma _{\rm crit}}
\end{eqnarray*}$$where distances in the lens plane are scaled in units of rc and Σcrit = c2DS/4πGDLDLS is the lensing critical surface density. Introducing $\widetilde{\kappa _0} \equiv \widetilde{\Sigma }(0)/\Sigma _{\rm crit}$ with equation (29) and $\mathcal {B}(a,b)=\Gamma (a)\Gamma (b)/\Gamma (a+b)$, the scaled deflection angle for an untruncated DZ profile yields
(38)$$\begin{eqnarray*}
\widetilde{\alpha }(X) = \frac{\sqrt{\pi } ~\widetilde{\kappa _0} }{\Gamma (2-2a)\Gamma (5)} X^2~ H_{3,3}^{2,2} \left[ \left. \begin{array}{c}(-6,4), (-\frac{1}{2},1), (0,1)\\
(-\frac{1}{2},1), (-2a,4), (-\frac{3}{2}, 1) \end{array} \right| X^2 \right], \nonumber \\
\end{eqnarray*}$$which has a series expansion analogous to that of $\widetilde{\mathcal {M}}(X)$. For a DZ profile truncated at the virial radius, $\alpha (X) = \widetilde{\alpha }(X)-X \widetilde{\Sigma }(c)/ \Sigma _{\rm crit}$. We give analytic expressions for the lensing potential in Appendix F (Supporting Information).
Citation: Schneider, Ehlers & Falco 1992
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We gathered four independent measurements of 





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

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





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





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





, of the four independent measurements. Using the 





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





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





, and I values are converted to 





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





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





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





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





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





 Mpc in regard to uncertainties due to wavelengths and FP parameter sets.
Citation: Bernardi et al. 2003
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Citation: Bernardi et al. 2003
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A triple system consists of an inner binary with masses m1 and m2 (defined as m1 ≥ m2), and a third object with mass m3 (in our case, the third object is always the central SMBH). It is convenient to describe the orbit using Jacobi coordinates (see e.g. Murray & Dermott 2000, p. 441). In this treatment, the dominant motion of the triple system can be divided into two separate Keplerian orbits: one is the motion of objects 1 and 2 relative to each other, and the other one is the motion of object 3 relative to the centre of mass of the inner binary. We denote the relative position vectors, semimajor axes, and eccentricities of these systems by $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, a1 and a2, and e1 and e2, respectively. If the third object is sufficiently distant from the inner binary, a perturbative approach can be used to describe the evolution of the system. In the usual secular approximation (e.g. Marchal 1990), the two orbits exchange angular momentum, but not energy. Hence, the eccentricities and orientations of the orbits can change, but the semimajor axes cannot. The coupling term in the Hamiltonian can be written in a power series expansion of parameter α = a1/a2 (see e.g. Harrington 1968), where the Hamiltonian is
(16)
\begin{eqnarray*}
\mathcal {H} &amp;=&amp;\, \frac{m_1 m_2}{2a_1} + \frac{(m_1+m_2)m_3}{2a_2} \nonumber \\
&amp;&amp;+\, \frac{1}{a_2} \sum \limits _{j=2}^{\infty } \alpha ^j M_\mathrm{ j} \left(\frac{r_1}{a_1} \right)^j \left(\frac{a_2}{r_2} \right)^{j+1} P_\mathrm{ j}(\mathrm{cos}\Phi) ,
\end{eqnarray*}
Pj is the jth-degree Legendre polynomial, Φ is the angle between $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$, and
(17)
\begin{eqnarray*}
M_\mathrm{ j} = m_1 m_2 m_3 \frac{m_1^{j-1}+(-1)^j(m_2)^{j-1}}{(m_1+m_2)^j} .
\end{eqnarray*}
When analysing these hierarchical triple systems, it is particularly convenient to adopt the canonical coordinates known as Delaunay’s elements (e.g. Valtonen & Karttunen 2006). The coordinates are the mean anomalies, l1 and l2, the longitudes of ascending nodes, h1 and h2, and the arguments of periastron, g1 and g2. One can then eliminate the short-period terms by a canonical transformation to get equations that describe the long-term evolution of the triple system. This technique is known as the Von Zeipel transformation (see e.g. Brouwer 1959). Naoz et al. (2013a) showed that going beyond the quadrupole-order approximation is necessary in order to acquire highly eccentric Kozai excitations. They also pointed out a common mistake in the Hamiltonian treatment of these hierarchical systems in previous studies, which could lead to erroneous conclusions in the test particle limit. The full octupole-order equations of motion can be found in Naoz et al. (2013a).
Citation: Harrington 1968
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Let the data be denoted $\mathscr {D}=\lbrace \boldsymbol {\hat{z}},\boldsymbol {\hat{\theta }},\boldsymbol {\hat{s}},\boldsymbol {\hat{\imath }}\rbrace$ (and possibly the experimental covariance matrix), and $\mathcal {S}$ be the proposition that we have observed some data thus passing the selection criteria. The probability that a given data set is observed depends on the selection criteria. The likelihood of observing $\mathscr {D}$ given ϑ and that we have observed some data is (Loredo 2004; Mandel, Farr & Gair 2019)
(54)$$\begin{eqnarray*}
p(\mathscr {D}\mid \mathcal {S},\vartheta) =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\mathscr {D}\mid \vartheta)}{p(\mathcal {S}\mid \vartheta)} =\frac{p(\mathcal {S}\mid \mathscr {D},\vartheta)\, p(\vartheta \mid \mathscr {D})}{\int \mathrm{d}\mathscr {D}^{\prime }\, p(\mathcal {S}\mid \mathscr {D}^{\prime },\vartheta)\, p(\vartheta \mid \mathscr {D}^{\prime })},
\end{eqnarray*}$$where $p(\mathcal {S}|\mathscr {D},\vartheta)$ is the selection function, and we used Bayes’ theorem in the second equality so $p(\vartheta |\mathscr {D}) \propto p(\mathscr {D}|\vartheta)p(\vartheta)$. If we assume that all objects that exceed some threshold are successfully observed then $p(\mathcal {S}|\mathscr {D},\vartheta)=1$. In the ideal scenario of no selection effects (see Section 3) clearly all possible data sets are observable. In the case of a cut-off, equivalent to replacing with a truncated distribution, as in (5), we have $p(\mathcal {S}|\mathscr {D},\vartheta)=1$ if it exceeds some threshold, and $p(\mathcal {S}|\mathscr {D},\vartheta)=0$ otherwise. Thus, $p(\mathcal {S}|\vartheta)$ is the fraction of all possible data sets that are observable given parameters ϑ. From (54) we can see that the form of the posterior distribution subject to selection effects, $p(\vartheta |\mathscr {D},\mathcal {S})$, is simply that of $p(\vartheta |\mathscr {D})$ given by (44) but with a different normalization.
Citation: Mandel, Farr & Gair 2019
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The gas inside the filaments may be turbulent and the dissipation of this turbulence could, in principle, also contribute to the filament powering. In order to obtain a very rough estimate of the turbulent dissipation rate, we assume that the velocity dispersion σ in the filaments is at best comparable to the sound speed in the Hα-emitting filament gas. Otherwise, there should be evidence for shock heating but that is not observed. In general, shock heating should lead to a correlation between [N ii]/Hα emission line ratios and the velocity dispersion of the gas, but such correlations have not been detected in the Perseus and Centaurus clusters where this issue was studied (Hatch et al. 2006; Canning et al. 2011; albeit the caveat that these models did not incorporate non-thermal pressure which could affect the nature of the shocks). The absence of shocks implies that
19





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





. Typical masses of filaments are in the range from Mfil ∼ 104 to 106 M⊙ (Conselice et al. 2001). Using a filament mass of 106 M⊙ and assuming a turbulence injection scale comparable to the filament width lturb = 60 pc (c.f. Canning et al. 2016 who use smaller value), we get Lturb ∼ 2×1038 erg s−1, which is a few percent of the Hα luminosity of the resolved Horseshoe or Northern filaments in Perseus. This is not a strict upper limit on the contribution of turbulent dissipation as the filament may consist of a number of subfilaments. However, inside the filaments the Hα phase, while possibly not completely volume filling, may be more volume filling than the phase corresponding to the dense molecular gas. Furthermore, the power contributed by turbulent dissipation depends on the uncertain mass in the Hα phase in the filament, and that mass is smaller than the total Mfil mass of the filament adopted above. Most importantly however, the measurements of internal turbulence in the filaments are very difficult because of limited spatial resolution. Current measurements of the velocity dispersion inside the filaments are very likely to be significantly overestimated due to filament or subfilament confusion (Canning et al. 2016) and consequently velocity dispersions could be consistent with values lower than those adopted above. We thus conclude that it is at least plausible that turbulent heating is not the dominant powering mechanism.
Citation: Canning et al. 2011
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On 2017 August 17, a gravitational wave (GW) event (GW170817; Abbott et al. 2017a, 2017b) was linked with a faint gamma-ray prompt emission of GRB 170817A (Goldstein et al. 2017; Savchenko et al. 2017). Immediately, GRB 170817A was followed by an extensive observational campaign covering radio, optical, and X-ray bands (see, e.g., Troja et al. 2017; Abbott et al. 2017b; Kisaka et al. 2018; D’Avanzo et al. 2018a, and references therein). The observations of the nonthermal spectrum of GRB 170817A gathered during the first ≈ 900 days after the initial merger were analyzed by several authors. It was shown that they were consistent with synchrotron forward-shock emission generated by the deceleration of an off-axis structured jet with an opening angle θ

j
 ≈ 5° that was observed from a viewing angle in the range of 15° ≤ θ
obs ≤ 25° (Troja et al. 2017; Kasliwal et al. 2017b; Lamb & Kobayashi 2017, 2018; Resmi et al. 2018; Margutti et al. 2017a; Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b; Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c). In some proposed models, the off-axis structured jet is formed with an off-axis jet with a cocoon (Lazzati et al. 2017; Gottlieb et al. 2018b; Fraija et al. 2019b) and a shock breakout (Gottlieb et al. 2018b; Hotokezaka et al. 2018; Fraija et al. 2019c; Urrutia et al. 2021). Recently, Hajela et al. (2022) analyzed the latest X-ray and radio observations of GRB 170817A collected with the Chandra X-ray Observatory, the Very Large Array (VLA), and the MeerKAT radio interferometer about 3.3 yr after the initial merger. These new observations did not agree with the best-fit synchrotron curves from the off-axis jet model, thus reporting evidence of a new X-ray emission component. Given these contrasting observations, the authors offered the solution to explain this phenomenon in the context of either radiation from accretion processes on the compact-object remnant or a KN afterglow.
Citation: Margutti et al. 2017a
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No studies have yet provided an extensive and conclusive kinematics analysis of populations of simulated star-forming galaxies that can be broadly contrasted to the results from slit-like and IFS-like current observations. So far, large uniform-volume simulations have been analysed in order to shed light on the dichotomy and formation of fast and slow rotators (e.g. Penoyre et al. 2017; Li et al. 2018; Schulze et al. 2018, with Illustris and Magneticum) and to quantify the connection between galaxy morphology and spin on the one side and AGN feedback, environment and/or mergers on the other (Dubois et al. 2016; Rodriguez-Gomez et al. 2017; Choi et al. 2018; Lagos et al. 2018a,b, with Illustris, Horizon-AGN and EAGLE). Recently, van de Sande et al. (2019) have compared structural and kinematic properties of simulated galaxies from EAGLE, Hydrangea (Bahé et al. 2017), Horizon-AGN (Dubois et al. 2016), and Magneticum (Dolag et al., in preparation) with observed galaxies in the SAMI, ATLAS3D, CALIFA, and MASSIVE surveys. However, in most cases these works have principally focused on massive early-type galaxies at $z$ ∼ 0, which are expected to be mostly quenched. This choice has been dictated by the limited numerical resolution of such models, which is generally thought to be insufficient to properly capture even the kiloparsec-scale kinematics of intermediate and low-mass galaxies that are observationally accessible, especially in the low-redshift Universe. Other analyses have investigated the relation between stellar shapes and stellar kinematics with EAGLE and across galaxy types (Thob et al. 2019). Furthermore, there is a rich body of works on the problem of the evolution and conservation of galaxies angular momentum, also informed by the results of large uniform-volume simulations and across galaxy types (e.g. more recently Genel et al. 2015; Lagos et al. 2017, with Illustris and EAGLE). However, in most cases so far, the focus has been placed on the final $z$ = 0 outcome, or on the stellar components, or the analyses have been carried out via theoretical characterizations of galaxies properties (e.g. resolution element-based angular momenta) that cannot be easily contrasted to observations. In fact, so far, no quantitative analysis of the spatially averaged or map-based internal kinematics of star-forming galaxies within large uniform-volume simulations exists.
Citation: Lagos et al. 2018a
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Citation: van de Sande et al. (2019)
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Citation: Thob et al. 2019
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The primary focus of this Letter is the spatial distribution of galaxies and their orientations projected onto the sky. While the former is characterized by the fluctuations of number density, denoted by 





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





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





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





, where 





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





 and 





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





with 





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





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





, but at large scales, the term 





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







4







5





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





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





 is the linear-order matter power spectrum at the redshift z.
Citation: Kamionkowski et al. 1998
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