adsabs/FOCAL huggingface dataset:
Column 11.—Adopted distance DH and its uncertainty σD, both in Mpc. For objects with cz⊙ > 6000 





, the distance is simply estimated as czcmb/H, where czcmb is the recessional velocity measured in the cosmic microwave background reference frame (Lineweaver et al. 1996) and H is the Hubble constant, adopted to be 70 





 Mpc−1. For objects with czcmb  6000 





, we use the local peculiar velocity model of Masters (2005), which is based in large part on the SFI++ catalog of galaxies (Springob et al. 2007) and results from analysis of the peculiar motions of galaxies, groups, and clusters using a combination of primary distances from the literature and secondary distances from the Tully–Fisher relation. The resulting model includes two attractors with infall onto the Virgo cluster and the Hydra-Centaurus supercluster, as well as a quadrupole and a dipole component. The transition from one distance estimation method to the other is selected to be at cz⊙ = 6000 





  because the uncertainties in each method become comparable at that distance. Where available, primary distances in the published literature are adopted; we also use secondary distances, mainly from Tully et al. (2013), for galaxies with czcmb  6000 





. When the galaxy is a known member of a group (Springob et al. 2007), the group systemic recessional velocity czcmb is used to determine the distance estimate according to the general prescription just described. Where primary distances are not available, objects in the Virgo region are assigned to likely Virgo substructures (Mei et al. 2007), and then the distances to those subclusters are adopted following Hallenbeck et al. (2012). Errors on the distance are generated by running 1000 Monte Carlo iterations where peculiar velocities are drawn from a normal distribution each time, as described in Section 4.1 of Jones et al. (2018). Such errors are likely underestimates in the vicinity of major attractors such as Virgo. It should be noted that the values quoted here are Hubble distances, not comoving or luminosity distances.
Citation: Springob et al. 2007
Citation function: Uses
Citation: Springob et al. 2007
Citation function: Uses; Uses
We stress the uncertainty of our present interpretation, given that the density and temperature structures of our target sources are not yet constrained in a sufficiently detailed way on our resolved spatial scales. In addition, our 44 GHz observationsare still subject to missing short-spacing, which nevertheless should not impact regions with high intensity. To provide a quantitative sense of how missing short-spacing biased the observed Stokes I intensity fractionally, we assume that exterior to the disks (or disk-like structures) around IRAS 16293–2422 A and B there is ~2 M⊙ of gas on ≳10′′ scales (Mundy et al. 1990), and assume that the gas to dust mass ratio is 100. The averaged dust mass column density is Σdust ~ 0.1 g cm−2. Assuming the dust opacity at 230 GHz is κ230 GHz = 0.5

$^{0.5}_{-0.4}$−0.4+0.5
 cm2 g−1 (c.f., Draine 2006), and the dust opacity spectral index is β ~1−1.75 (Li et al. 2017), the derived averaged dust optical depth at 44 GHz is 

$\tau_{\mbox{\tiny 44 GHz}}=\kappa_{\mbox{\tiny 230 GHz}}(44/230)^{\beta}\mathrm{\Sigma}_{\mbox{\tiny dust}} = $τ 44 GHz=κ 230 GHz(44/230)βΣ dust=
2.8

$^{16}_{-2.2}\,\times$−2.2+16 ×
 10−3. Assuming a nominal 20–30 K dust temperature on 10′′ scales (van Dishoeck et al. 1995; Ceccarelli et al. 2000; Chandler et al. 2005), the brightness temperature of the extended dust emission is ~ (25 ± 5) × (1 − e−τ44 GHz) = 0.069

$^{0.57}_{-0.058}$−0.058+0.57
 K. This is approximately how much missing short-spacing can bias the Stokes I intensity. If we take regions where the observed brightness temperature in the Robust = 0 weighted 44 GHz image is above 1.4 K (i.e., 6σ; the second contour from bottom in the right panel of Fig. 1) for example, missing flux can bias Stokes I intensity by at most ~(0.64 K)/(1.4 K + 0.64 K) ~ 31%. For structures detected at the 6σ significance, clean errors and thermal noise (e.g., 3σ) may dominate the errors in the measured polarization percentages. The brightness temperature of the missing flux can be comparable with the 6σ detection limit of our Robust = 2 weighted 44 GHz image (~0.6 K), and therefore the observed polarization percentages can be biased by over ~50% in the low intensity (3–6σ) area in that image (Table B.1). Our assessment is that for our present 44 GHz images, missing short-spacing biased the observed polarization percentages of the detected, spatially-extended, dust emission component. Since the brightness temperature of the extended component is low (Fig. 1) such that dust is likely to be optically thin (i.e., Tbg = TCMB), missing short-spacing does not dramatically flip the observed polarization position angles by 90° (Sect. 4.1; Figs. 2–4).
Citation: Chandler et al. 2005
Citation function: Uses; Uses
To definitively assess whether or not the growth of the vertical kinetic energy observed in our simulations is associated with the disc eccentricity, we performed three additional simulations using the parameters of the fiducial model, except that: (i) In the first simulation, the disc orbits a single star and the cooling time-scale is set to tc = 0.1Ω−1; (ii) In a second run, the disc orbits a single star and the cooling time-scale is set to tc = 0.001Ω−1; and (iii) In the third run, we restarted the fiducial model at time t0 = 955 but with the system slowly transitioning from a central binary system into a single star located at the centre of mass. This is done by treating the gravitational potential as the sum of two terms, one corresponding to the binary plus one corresponding to a single star, with weighting factors that change with time. More precisely, the gravitational potential for this simulation is given by (Mutter et al. 2017)
(23)$$\begin{eqnarray}
\Phi _{\rm trans}=(1-W (t))\Phi _{\rm bin}-\frac{GM_\star }{r} W(t)
\end{eqnarray}$$with
(24)$$\begin{eqnarray}
W(t) &amp;= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1 &amp; \text{if } \,\, t-t_0 \ge 100T_{\rm bin}\\
\frac{1}{2}\left[1-\cos \left(\frac{\pi (t-t_0)}{100T_{\rm bin}}\right) \right] &amp; \text{if} \,\, t-t_0 \lt 100T_{\rm bin}\end{array}.\right.
\end{eqnarray}$$The aim of simulations (i) and (ii) is to check that our fiducial disc is stable to the VSI whereas run (iii) is useful to examine whether the turbulent flow is maintained under unforced conditions. For these three calculations, the time evolution of the vertical kinetic energy is shown in Fig. 5. Considering simulations (i) and (ii), exponential growth of the vertical kinetic energy due to growth of the VSI only occurs for the case with tc = 0.001 Ω−1. This is consistent with the results of Richard, Nelson & Umurhan (2016), who found that for h = 0.05 the disc remains stable to the VSI provided tc > 0.05 Ω−1. It also demonstrates that our fiducial model, which has tc = Ω−1, is stable to the VSI. For simulation (iii), the saturation level of the vertical kinetic energy remains essentially unchanged after the switch (t = 1055). Contours of the normalized density perturbation one scale height above the mid-plane, and of the meridional velocity in a [R, Z] plane located at azimuth φ = π, are presented in Fig. 6 at t = 1500. They clearly reveal that both the eccentric mode and the turbulence can be maintained under unforced conditions. The continued existence of the eccentric mode at that time is not surprising since m = 1 free eccentric modes can be long lived (Papaloizou 2005b). The persistence of the turbulent flow is a strong indication that the instability originates from the presence of an eccentric mode in the disc.
Citation: Mutter et al. 2017
Citation function: Uses; Uses
It has been clear since many years now (Matthews et al. 1964) that the majority of powerful radio-AGN is hosted within the most massive galaxies (stellar masses M∗∼1011M⊙\documentclass[12pt]{minimal}
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				\begin{document}$$M_* \sim 10^{11}\, M_\odot$$\end{document} and above) known in the universe. This result has been confirmed at least up to z∼1\documentclass[12pt]{minimal}
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				\begin{document}$$z\sim 1$$\end{document} (and in some cases beyond) both by direct estimates (e.g., Heeschen 1970; Ekers and Ekers 1973; Auriemma et al. 1977; Jenkins 1982; Best et al. 2005a; Mauch and Sadler 2007; Smolčić et al. 2009; Magliocchetti et al. 2016; Sabater et al. 2019; Capetti et al. 2022) and also by the extremely tight correlation in the MK-z\documentclass[12pt]{minimal}
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				\begin{document}$$M_K-z$$\end{document} plane exhibited by the hosts of radio-AGN (e.g., Lilly and Longair 1984; Eales et al. 1997; Jarvis et al. 2001; De Breuck et al. 2002; Willott et al. 2003) that can be explained by assuming that the observed K-band light is dominated by emission from the old stellar population and that radio-AGN have very similar (large) host masses all across half of the age of the universe. More recent results based on rest-frame 1.6 and 4.5 μ\documentclass[12pt]{minimal}
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				\begin{document}$$\upmu$$\end{document}m photometry also report very high masses for the hosts of radio-AGN, at least up to z∼3\documentclass[12pt]{minimal}
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				\begin{document}$$z\sim 3$$\end{document} if not beyond (Seymour et al. 2007; De Breuck et al. 2010; Gürkan et al. 2014; Drouart et al. 2016).
Citation: Mauch and Sadler 2007
Citation function: Background
On a ∼30000 au scale, a bipolar outflow associated with this source was detected in the east-west direction by FCRAO single-dish observations in 12CO 




J
=
1
−
0


 molecular line emission (Narayanan et al. 2012) and by James Clerk Maxwell Telescope (JCMT) single-dish observations in 12CO 




J
=
3
−
2


 molecular line emission (Hogerheijde et al. 1998). Their results show that the blue and red robes of the outflows are on the eastern and western sides, respectively. On the other hand, inner parts (∼8000 au scale) of the outflow mapped with the Nobeyama Millimeter Array (NMA) in 12CO 




J
=
1
−
0


 show the opposite distribution, i.e., stronger blueshifted emission on the western side and stronger redshifted emission on the eastern side (Tamura et al. 1996). Mid-infrared observations toward L1527 IRS with Spitzer Space Telescope shows bright bipolar scattered light nebulae along the outflow axis in the ∼20000 au scale (Tobin et al. 2008). They fitted a protostellar envelope model to near- and mid-infrared scattered light images and spectral energy distribution (SED). As a result, the inclination angle of the envelope around L1527 IRS was estimated to be 




i
=
85
°


, where 




i
=
90
°


 is the edge-on configuration. In addition, Oya et al. (2015) found that the western side is closer to observers. This inclination angle is consistent with a disk-like structure of dust highly elongated along the north-south direction, which was spatially resolved for the first time in 7-mm continuum emission by the Very Large Array (VLA; Loinard et al. 2002). By expanding the studies by Tobin et al. (2008, 2013), we fitted a model composed of an envelope and a disk to (sub)millimeter continuum emissions and visibilities observed with the Submillimeter Array (SMA) and the Combined Array for Research in Millimeter Astronomy (CARMA), as well as infrared images and SED. Their best-fitting model suggests a highly flared disk structure (




H
∝


R


1.3




, H = 48 au at R = 100 au) with a radius of 125 au. This study has geometrically distinguished the protostellar disk and envelope around L1527 IRS, although they were not kinematically distinguished from one another.
Citation: Tamura et al. 1996
Citation function: Compare/Contrast
The error budget associated with Δtij was estimated by considering three sources of uncertainties. Firstly, we propagated the statistical error inferred from the MCMC sampling. This was simply done by constructing histograms for Δtij h−1 from the independent samples of SIEg model parameters reported in Figs. 3 and 4. An example of these histograms is illustrated in Fig. 7 for the case of zl = 0.7. Secondly, we considered the impact of hypothetical massive objects lying on the line-of-sight by scaling the theoretical time delay distance with an external convergence term κext such that 




D

Δ
t


=

D

Δ
t



theory


/

(
1
−

κ


ext


)



$ D_{\Delta t} = D_{\Delta t}^{\text{ theory}} / (1 - \kappa_{\text{ ext}}) $


 (see, e.g., Keeton 2003). We applied a different scaling for each set of model parameters used to construct the histograms (see Fig. 7). The κext values were randomly drawn from a zero mean normal distribution and characterized by a conservative standard deviation σκ = 0.03 (see, e.g., Wong et al. 2017). When combined, the typical errors are ∼7.7% for ΔtCB, ∼7.8% for ΔtCA, and ∼7.6% for ΔtCD, compared to the median values. Thirdly, we considered the impact of the Source Position Transformation (SPT), a degeneracy existing between different lens density profiles that reproduce equally well the lensing observables, except for the product Δt H0 (Schneider & Sluse 2013, 2014). A short summary is given in Appendix A. We SPT-transformed the SIEg model using the modified deflection law 





α

̂

=


α



SIEg


+

β

−


β

̂


(

β

)



$ {\hat{\boldsymbol{\alpha}}} = \boldsymbol{\alpha}_{\text{ SIEg}} + \boldsymbol{\beta} - {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) $


 along with a radial stretching of the source plane defined by 





β

̂


(

β

)

=

[
1
+
f
(
|

β

|
)
]


β




$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = [1 + f(|\boldsymbol{\beta}|)]\ \boldsymbol{\beta} $


. In particular, we considered the special case where the deformation function f(|β|) is the lowest-order expansion of more general functions, f(|β|) = f0 + f2|β|2/(2θE), where the constants f0 = f(0) and 



f
2

=
θ
E
2


f

′′

(0)

$ f_2 = \theta_{{\rm E}}^2 f^{\prime\prime}(0) $


 quantify the magnitude of the deformation. This choice explores a large variety of degeneracies characterized by an isotropic transformation of the source plane. In the end, this at least defines a lower limit on the impact of the SPT on the time delays. In the particular case of f2 = 0, the SPT reduces to the well-known mass-sheet degeneracy (MSD) characterized by 





β

̂


(

β

)

=
λ

β

≡

(
1
+

f
0

)


β



$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = \lambda \boldsymbol{\beta} \equiv (1 + f_0) \boldsymbol{\beta} $


. In our analysis, we only explored degeneracies that cannot be explained by a MSD, hence f0 = 0.
Citation: Keeton 2003
Citation function: Uses
For completeness and ease of our discussions, here we summarize briefly the main features of the NS EOS-metamodel we use in solving the NS inverse-structure problem. More details can be found in our previous publications (Zhang et al. 2018; Xie & Li 2019, 2020; Zhang & Li 2019a, 2019b, 2019c, 2020). We assume the cores of NSs are made of totally charged neutral neutrons, protons, electrons, and muons (the npeμ model) at β-equilibrium. Unlike the widely used composition-degenerate spectral functions and/or piecewise polytropes that directly parameterize the pressure as a function of energy or baryon density, to probe the high-density symmetry energy, we have to keep the isospin dependence of the EOS and retain explicitly the composition information at all densities. For this reason, we metamodel NS EOS by starting at the single-nucleon energy level with explicitly isospin dependence. Specifically, we parameterize the E0(ρ) and Esym(ρ) according to
2






E


0


(
ρ
)
=


E


0


(


ρ


0


)
+





K


0




2










ρ
−


ρ


0




3


ρ


0









2


+





J


0




6










ρ
−


ρ


0




3


ρ


0









3


,




3









E


sym


(
ρ
)


=




E


sym






ρ


0




+
L





ρ
−


ρ


0




3


ρ


0













+






K


sym




2










ρ
−


ρ


0




3


ρ


0









2


+





J


sym




6










ρ
−


ρ


0




3


ρ


0









3







where E0(ρ0) = −15.9 ± 0.4 MeV is the binding energy and K0 ≈ 230 ± 20 MeV (Shlomo et al. 2006; Piekarewicz 2010; Garg & Colò 2018) is the incompressibility at the saturation density ρ0 of SNM, while Esym(ρ0) = 31.7 ± 3.2 MeV is the magnitude and L ≈ 58.7 ± 28.1 MeV is the slope of symmetry energy at ρ0 (Li & Han 2013; Oertel et al. 2017) based on earlier surveys of over 50 analyses of both terrestrial experiments and astrophysical observations, respectively. A very recent survey of 24 new analyses of NS observations since GW179817 indicates that L ≈ 57.7 ± 19 MeV and Ksym ≈ −107 ± 88 MeV at 68% confidence level (Li et al. 2021). In this study, we use L = 58.7 as its most probable value and vary it within ±20 MeV. We keep the E0(ρ0), K0, and Esym(ρ0) at their most probable values given above as they have been relatively well determined and their variations within their remaining uncertain ranges have been shown to have little effects on the masses, radii, and tidal deformabilities of NSs Fattoyev et al. (2013, 2014), Xie & Li (2019, 2020). As a reference, we notice that very recent calculations based on a set of commonly used Hamiltonian operators including two- and three-nucleon forces derived from chiral effective field theory predicted that L = 47 ± 3 MeV, Ksym = −146 ± 43 MeV, Jsym = 90 ± 334 MeV (Somasundaram et al. 2021). While in the same calculations, with a prior J0 = −300 ± 400 MeV, the posterior ranges between J0 =−573 ± 133 MeV and J0 = −172 ± 243 MeV depending on the scaling schemes used. The EOS parameter ranges we used are consistent with these calculations.
Citation: Shlomo et al. 2006
Citation function: Uses; Uses
Next, we estimated the distance and reddening. As in Brown et al. (2014), we compared the Eri II main sequence to a synthetic Victoria-Regina theoretical isochrone from VandenBerg et al. (2014) and the horizontal branch of Eri II to the horizontal branch of M92, which was observed with the same ACS filters by Brown et al. (2005). The synthetic main-sequence isochrone was constructed assuming an age of 13 Gyr, the metallicity distribution displayed in Figure 3, and a binary fraction of 0.48 (Geha et al. 2013). For M92, we assumed a distance modulus of 14.62 mag (the mean of literature measurements by Del Principe et al. 2005, Sollima et al. 2006, and Paust et al. 2007) and reddening of E(B − V) = 0.023 mag (Schlegel et al. 1998). We included main-sequence stars more than 0.5 mag below the main-sequence turnoff (m814 > 27.67) in order to avoid portions that are age-sensitive, and we used horizontal branch stars that are bluer than m606 − m814 = −0.68 in order to avoid the RR Lyrae instability strip. Fitting the main sequence and the horizontal branch simultaneously, we found m − M = 22.65 mag (d = 339 kpc) and E(B − V) = 0.03 mag, as shown in Figure 4. This distance modulus is consistent, within the uncertainties, with the distance derived from the Eri II RR Lyrae stars by A. Garofalo et al. (2021, in preparation). The statistical uncertainties on the distance fit are very small (0.015 mag in distance modulus, 0.003 mag in reddening), and they are certainly dominated by systematics in the choice of comparison cluster/isochrone and the magnitude and color range of stars to include. By analogy to Brown et al. (2014), we assume overall uncertainties of 0.07 mag in m − M and 0.01 mag in E(B − V). Similar to a few of the ultra-faint dwarfs studied by Brown et al. (2012, 2014), the distance modulus determined with this approach is in modest disagreement with some literature results (Crnojević et al. 2016 measured m − M = 22.8 ± 0.1 mag and Koposov et al. 2015 estimated m − M = 22.9 mag, although Bechtol et al. 2015 found a comparable value of m − M = 22.6 mag) and the derived reddening is larger than the E(B − V) = 0.009 mag value obtained by Schlafly & Finkbeiner (2011) (also see Mutlu-Pakdil et al. 2019). However, for consistency with previous star formation history analyses, we adopt the derived values for the remainder of this study.
Citation: Del Principe et al. 2005
Citation function: Uses; Uses
Citation: Geha et al. 2013
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Citation: A. Garofalo et al. (2021, in preparation)
Citation function: Similarities
Citation: Crnojević et al. 2016
Citation function: Background; Differences