implications for the proposed loose correlation between explosion energy and kick velocity (Bray Eldridge 2016 ; Janka 2017 ; Vigna-Gómez et al. 2018 ). Our results constitute further evidence that this correlation is not a tight one. More likely, the explosion energy only determines the maximum attainable kick velocity, below which one expects a distribution of kicks depending on the degree of unipolarity or bipolarity (Müller et al. 2019 ). This can be accounted for in a more general formulation than in Bray Eldridge ( 2016 ) by a factor that measures the explosion asymmetry (Janka 2017 ). The evolution of the spin-kick angle α in model m39 is noteworthy. Some observations have suggested that the birth spins and kicks of neutron stars tend to be aligned (Johnston et al. 2005 ; Ng Romani 2007 ; Noutsos et al. 2013 ). However, even though hydrodynamic mechanisms for spin-kick alignment have been proposed (Janka 2017 ), no trend towards spin-kick alignment has been observed so far in 3D CCSN simulations of non-rotating progenitors (Wongwathanarat et al. 2013 ; Müller et al. 2019 ). For our non-rotating explosion model y20, we also find a large angle α ≈ 70° between the spin and kick. For the rapidly rotating model m39, however, the spin-kick angle stays at about α ≈ 30°. In this case, spin-kick alignment comes about because the bipolar outflows are aligned with the initial rotation axis, which is not changed appreciably by asymmetric accretion on to the PNS. This suggests that progenitor rotation could play a role in explaining spin-kick alignment (provided that the observational evidence remains robust). However, even for model m39, the spin-kick alignment is not perfect, and is not clear whether the polar alignment of the outflows is a robust feature for rotating progenitors. In fact, some previous 3D supernova simulations of rotating progenitors rather find that the explosion develops preferably in the equatorial plane (Summa et al. 2018 ; Takiwaki Kotake 2018 ). Clearly, more simulations are required to determine the implications of rotation for the explosion morphology and spin-kick alignment. 6 GRAVITATIONAL WAVE EMISSION The gravitational-wave emission for our models, as measured by an observer at the pole and the equator, is shown in Fig. 6 . The high mass and rotation of model m39 results in high gravitational-wave amplitudes of up to 40 cm at the pole and over 100 cm at the equator. Model y20 reaches amplitudes of ∼10 cm at the pole, and ∼18 cm at the equator. Model s18np has a much lower amplitude of ∼5 cm at the equator, due to the lack of shock revival in this model, however its prompt convection signal in the direction of the equator reaches amplitudes as high as model y20. Figure 6. The gravitational-wave emission for our models as measured at the pole (left-hand panel) and the equator (right-hand panel). Top is model m39, middle is model s18np, and bottom is model y20. All models show the rise of gravitational-wave frequency

O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Citation I-Citation O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O B-Model I-Model I-Model O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Model I-Model O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O B-Model I-Model O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Model I-Model O O O B-Model I-Model O O O O O O B-EntityOfFutureInterest O


the total magnetic fluctuations and of the parallel electric fluctuations, and (top and bottom rows, respectively) for (left column) and β i = 0.2 (right column). A classical intermittency analysis has been performed on both simulations at about the peak of the nonlinear activity. In order to define the large-scale limit of the inertial range, we evaluated the perpendicular and parallel autocorrelation functions, respectively defined as and (Frisch 1995 ). We assumed isotropy in the perpendicular xy -plane, with the parallel direction along B 0 , i.e., along z . The e -folding length approximately gives the integral scale, which is about in the perpendicular direction (corresponding to ), for both regimes. The situation is different in the parallel direction, where the parallel correlation length is for β p = 0.2, while it is for β p = 1 (corresponding to and ∼0.5, respectively). This is in qualitative agreement with the features spotted in Figure 1 and quantitatively with the corresponding spectra (see Figure 4 ), indicating differences already in the large-scale properties of the fluctuations possibly due to a different decorrelation mechanism along the mean field. The level of intermittency can be better quantified by the probability distribution functions (PDFs) of the magnetic field increments at a given scale r , defined asWe show here the statistics of the perpendicular increments, namely, , spanning this increment from lengths larger than the correlation scale , down to the smallest available scale (Δ x ∼ 0.08 d p ). These distributions are reported in Figure 3 (a) for the β p = 0.2 regime for three cases, namely, and 0.08. These PDFs, as expected, become increasingly intermittent going toward smaller scales. In order to compare among cases, and among different times, we measured the scale-dependent kurtosis χ —the fourth-order moment of the increments in Equation ( 4 )—that can be measured asThis quantity is reported in Figure 3 (b), as a function of the perpendicular scale r , for the two values of β , at two distinct times. At large scales, for r > 5 d p , the distribution becomes Gaussian, where χ ∼ 3, in agreement with the computation of the correlation lengths. At small scales, in the inertial range of turbulence, there is an enhancement due to the intermittent nature of the cascade, due to the presence of coherent structures and nonlinear waves. At the smallest scales, a saturation of the multifractality is observed, in agreement with observations in the SW. In fact, the study of high-order structure functions up to the sixth moment and of their exponents, shows deviation from monofractality (not shown here). Here, this process of saturation might also be slightly affected by the presence of artificial dissipation. It is important to note that at scales in the inertial-dispersive range, the case with β p = 0.2 is more intermittent than the β p = 1 regime, indicating a higher degree of coherency in the small-scale fluctuations.

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before the analyses pipelines were finalized. Our decision to focus on the NGC exclusively did come after a comparison with S16 had occurred, though we believe that we are well justified in our choice. S16 provide the current best weak lensing mass calibration of redMaPPer clusters, and they discuss in detail how their mass calibration compares to others in the literature. While the constraints on the slope of the mass–richness relation are in good agreement between the two works, there is mild tension on the recovered amplitude assuming our fiducial cosmology, with the clustering-derived constraints preferring a somewhat higher amplitude. At the richness pivot point used in this work, λ 0 = 35, S16 predict log 10 ( M /M ⊙ ) = 14.43 ± 0.04, while our constraints yield log 10 ( M /M ⊙ ) = 14.62 ± 0.08; this discrepancy corresponds to roughly 2.2σ tension. Because the clustering analysis prefers a slightly lower value of α than the weak lensing analysis, this tension is reduced for λ > 35. For λ 35, the tension remains roughly the same since the error bars increase as one moves away from the λ pivot point. We postpone a discussion of the origin of this tension, and a more detailed comparison of the two results, to future work. That is, our current results reflect our best understanding of both the weak lensing and clustering analysis prior to the comparison of the two works. We have also considered a power-law bias–richness relation as per equation ( 10 ). This model has five parameters: A λ , α λ , β λ and two bias parameters for 5 λ 20 clusters in the two redshift bins. Fitting this model to the data, we find A λ = 3.54 ± 0.13, α λ = 0.57 ± 0.08 and β λ = 2.0 ± 1.1 with χ 2 /d.o.f. = 57.7/75. 7 CHARACTERIZING THE UNCERTAINTY IN THE MASS–RICHNESS RELATION The uncertainties on the mass–richness parameters presented in Fig. 6 and Table 1 receive contributions from several sources: statistical noise in the measurement of the angular correlation functions, uncertainty on the parameters of the line-of-sight projection model, uncertainty on the parameters of the assembly bias model and scatter around the b ( M ) relation from Tinker et al. ( 2010 ). In addition to these sources of scatter that are explicitly included in our model, it is also possible that our measurement is affected by systematic biases that we have not modelled. Below, we determine the approximate contributions of both modelled sources of scatter and unmodelled systematic errors to the uncertainty budget of our parameter constraints. 7.1 Statistical noise in the correlation function measurement We first investigate how statistical noise in the measurement of w αβ (θ) contributes to our parameter uncertainties. The contribution of statistical noise to our parameter uncertainties can be determined by setting the parameters of the line-of-sight projection model, the parameters of the assembly bias model and b scaling to constant values rather than allowing these parameters to vary.

O O O O O O O O O O O O B-Dataset O O O O O O O B-Citation O O O O O O O O O O O O O B-Citation O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula B-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


however, need to find more stars with similar travel times as S5-HVS1. It is remarkable that the age of S5-HVS1 ejection is close to both the age of the disc of young stars around the GC (Lu etal. 2013 ) and the age of the Fermi bubbles (Bland-Hawthorn Cohen 2003 ; Su, Slatyer Finkbeiner 2010 ) which have been potentially associated with the recent accretion event in the GC (Zubovas, King Nayakshin 2011 ; Guo Mathews 2012 ), thus potentially linking these different astrophysical objects. An alternative scenario that would naturally produce a time-variable HVS spectrum is that involving an Intermediate Mass Black Hole (IMBH) orbiting the GC (Yu Tremaine 2003 ; Levin 2006 ). In this mechanism, during the inspiral of the IMBH, the HVS production rate peaks and then subsides due to dynamical friction around the SMBH (Baumgardt, Gualandris Portegies Zwart 2006 ; Darbha etal. 2019 ), with the fastest HVS being ejected in the final phase of the in-spiral. This mechanism produces a strongly anisotropic distribution with the fastest stars in the orbital plane of the IMBH (Rasskazov etal. 2019 ). There is also some indication that the HVSs produced by this mechanism tend to have higher velocities and a flatter velocity spectrum than the classical Hills mechanism (Sesana, Haardt Madau 2007 ). While there is currently not much evidence for the presence of an IMBH in the GC (Gualandris Merritt 2009 ) other than a shallow stellar density slope that can be produced by an IMBH scattering (Baumgardt etal. 2006 ), if an IMBH inspiral happened a few Myr ago, then it would produce an excess of nearby and fast HVSs with a narrow range of ejection times. To test for this possibility we need to search for other nearby HVS and see if there is an excess of stars that were ejected at roughly the same time as S5-HVS1 (5Myr ago), are strongly anisotropic and that have a velocity spectrum inconsistent with the Hills mechanism. An interesting consequence of the fact that S5-HVS1 has lower mass than most other HVS in the halo is that its expected lifetime given the stellar mass of 2.3M is quite long around a 1Gyr. By the end of its life the star would have travelled a distance of 2Mpc, traversing a large fraction of the Local Group. This suggests that searching for such ejected stars at large distances from the MW or Andromeda (Sherwin, Loeb OLeary 2008 ) is quite promising. On top of being well separated in colourmagnitude space from other contaminants, an S5-HVS1-like star would eventually evolve on to the red giant branch and thus be detectable much more easily. Searches for S5-HVS1-like stars within the whole Local Group will be possible with upcoming deep imaging surveys like LSST (LSST Science Collaboration etal. 2009 ). In this paper, we tried to use the positionvelocity information on S5-HVS1 to constrain the distance from the Sun to the GC and the Galactic Solar velocity. We have not been able to constrain those simultaneously, mainly due to the precision of

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of this procedure at z = 6 (red) and z = 7 (blue) in Figure 4 as obtained by changing only the outflows (left) or changing only the column density (right). While both scenarios produce stacks that have consistent FWHM values with observations at z = 6, the changing outflows scenario yields also a consistent FWHM with z = 7 measurements. This also confirms that changing the column density predicts very high FWHM >400 km s −1 at z = 7. As quoted in the legend for changing the outflows scenario (left), z = 6 lines are broader than those at z = 7, and consistent with Pentericci et al. ( 2018 ). The lines are highly asymmetric with extended red wings. The blue peaks disappear due to the poor resolution. Figure 4. Example for stacks of randomly selected 52 mock spectra at z = 6 (red) and 19 mock spectra at z = 7 (blue) with f esc ( z = 7) ≈ 0.5 f esc ( z = 6) in agreement with the measured drop of X Ly α . We assembled the stacks following the method outlined in Pentericci et al. ( 2018 ) with spectral resolution R = 1390 (corresponding to Δ = 216 km s −1 ). All these spectra are either obtained by changing only outflows while keeping the column density N H I fixed (left) or changing only the column density N H I while keeping outflows fixed. In all cases, dust is kept fixed. In either scenario, the FWHM of stacks at z = 6 is consistent with values reported in Pentericci et al. ( 2018 , FWHM( z ∼ 6) = 300 ± 30). The FWHM of stacks obtained by changing outflows at z = 7 is also consistent with Pentericci et al. ( 2018 ) measurements (FWHM( z ∼ 7) = 220 ± 25) while those produced by changing the column density N H I are not, and hence indicating that changing the outflow can naturally lead to a higher Ly α escape as well as broader lines at lower redshifts as observations indicate. To quantify the width using the above recipe, we generate randomly 10,000 combination of stacks at z = 6 and z = 7 from the total number of model combinations (i.e., 143 and 201) in each scenario and compute their widths. We show the resulting width distribution at these redshifts in Figure 5 . Results from changing outflows and changing column density scenarios are presented by dashed and solid lines. Shaded red and blue areas show Pentericci et al. ( 2018 ) measurements at z ∼ 6 and z ∼ 7, respectively. From this exercise alone, we constrain the FWHM, using changing column density scenario, to 261.7 ± 14.3 km s −1 at z = 6 and 434.9 ± 47.8 km s −1 at z = 7, and using changing outflows scenario, to 291.5 ± 7.9 km s −1 z = 6 and 224.6 ± 22.8 km s −1 at z = 7. It is evident that, over all possible combinations and the prior range assumed, the changing outflows scenario produces consistent width distribution within the 1 σ level of Pentericci et al. ( 2018 ) measurements. This confirms that changing the column density predicts inconsistent width distributions. It might be noteworthy that while we use the Pentericci et al. ( 2018 ) results in this section, other studies of

O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O


( 1 ) returned best-fitting parameters of α = 2.29 ± 0.08 and log ( N ) = −7.3 ± 0.1. The total number of degrees of freedom was 10 970. These values along with the measured Whittle statistic are provided in Table 4 . Allowing both N and C to vary between data sets did improve the fit by Δ S = 90, but there was no change to the measured slope. Additionally, allowing the fit to be driven by the low-frequency ‘red-noise slope’ by truncating the PSDs at 10 −4 Hz returns best-fitting parameters that are consistent with those listed in Table 4 . We also fitted equation ( 1 ) to PSDs generated in ‘soft’ and ‘hard’ energy bands (0.3–1 and 1–10 keV, respectively). One may expect to observe differences in the PSD at different energies based on the energy dependence of the rms variability at low and high frequencies, as shown in Fig. 7 . For example, it is clear that larger variations are observed between the slow and fast variations in the soft band compared to the hard. Subsequently, we may expect the soft band to have a steeper PSD. However, we found that the PSD does not exhibit any significant energy-dependence with best-fitting values typically remaining consistent within the uncertainties. As such, these variations are most likely contained within the uncertainties of the PSD fitting. These values are reported in Table 4 . Table 4. The best-fitting parameters of the two models (power-law and bending-power-law) fitted to the PSDs of NGC 3227 in three energy bands: 0.3–10, 0.3–1, and 1–10 keV. The fits are applied to all six XMM–Newton EPIC-pn observations from 2016. See Section 3.5 for details. We then searched for evidence of a break (or bend) in the PSD by fitting a bending power law (plus a constant): where N is the normalization, ν b is the break frequency, and and are the spectral indices below and above the bend, respectively (see González-Martín Vaughan 2012 for further details, who find that a significant bend is detected in the PSDs of 15 out of their sample of 104 AGN observed with XMM–Newton ). Fitting equation ( 3 ) to the 0.3–10 keV PSD improves the fit by Δ S = 10 for two additional free parameters, suggesting the marginal detection of a bend. The low-frequency slope, , is consistent with a value of 1, consistent with the X-ray PSD slopes observed in other Seyfert galaxies. Meanwhile, the high-frequency slope above the bend is steeper, with , where the best-fitting bend frequency is Hz, largely consistent with previous results (Uttley McHardy 2005 ; González-Martín Vaughan 2012 ; Arévalo Markowitz 2014 ). Extending this fit to the soft and hard bands, we find consistent results (Δ S = 9 and Δ S = 10, respectively) with break frequencies of and Hz and high-frequency slopes of and , respectively. Again, the bend detection is marginal and we find no evidence of any energy-dependence of the PSD. The best-fitting values are summarized in Table 4 and the bending-power-law fit is shown in Fig. 9 . In Fourier-based PSD analysis, it is

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panels. In the labels we show the average cross-power above 20′′. Figure 2. Stacked CIB–CXB cross-power spectra: between 3.6 μ m and [0.5–1], [1–2], [2–4], and [4–7] keV (top panels); between 4.5 μ m and CXB (bottom panels). Open circles with dashed error bars denote the absolute values of negative results. Note that the shown spectra are subtracted by the cross-power between IR and the stowed X-ray map, that is, the potential contamination from the X-ray particle background is removed from our final results. The final mean powers (used for further SED analysis) are summarized in Table 1 . As demonstrated in Cappelluti et al. ( 2013 , 2017 ), Mitchell-Wynne et al. ( 2016 ), and Li et al. ( 2018 ) the CIB versus CXB cross-power is in excess over shot noise at angular scales larger than 20′′. Since at those scales known foregrounds are significantly weaker than our signal (Helgason et al. 2014 ; Cappelluti et al. 2017 ; Kashlinsky et al. 2018 ), we therefore calculate the weighted mean cross-power above 20′′ asbetween each IR and X-ray band, for each field and for the stacked cross-power spectrum. As shown in Table 2 , the [1–2] keV band is significantly correlated with all IR wavelengths at ∼5 σ significance level, while the [0.5–1] and [2–4] keV bands are correlated with the IR wavelengths at ∼1–3 σ levels. There is no significant cross-power signal between the [4–7] keV band and any IR wavelengths so we cannot exclude an intrinsic correlation. By including the COSMOS field with a larger area, our new analysis extends the CIB–CXB cross-power analysis to ∼3000′′, doubling the largest scale reached in Cappelluti et al. ( 2017 ). The reason the highest significance is in the [1–2] keV band is due the higher S/N due to the peak in effective area of the Chandra mirror in that spectral region. Stacked results in the hard X-ray band (i.e., [2–4] keV + [4+7] keV) are not as significant as in Li et al. ( 2018 ) since deep fields add data affected by strong cosmic variance on very large scales. In the hard band most of the events are particles, so the low astrophysical hard X-ray photon statistics is another source of noise. As a cross-check we compared the significance of our stacks with those shown in Figure 17 of Li et al. ( 2018 ) for the [2–7] keV band versus 3.6 and 4.5 μ m and found consistent results. Table 2 Cross-power Spectrum Amplitude (>20′′), Calculated as a Note. a In units of 10 −11 photon s −1 cm −2 nW m −2 sr −1 . 4.2. Cross-power Hardness Ratios Similar to the X-ray color–color diagram, we look into three hardness ratios for the mean cross -power, which are the equivalent of photometric color indices, following the typical definition (Brunner et al. 2008 ) The statistical 1 σ hardness ratio errors are calculated from the mean power (Table 2 ) by error propagation. The mean 1 σ errors for the hardness ratios are ∼0.3 for HR1 and HR2 and ∼0.8 for HR3. Due to the larger errors in HR3, resulting from the uncertainties from harder

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


region that is magnetically connected to the bow shock and is filled with ions or electrons backstreaming from the bow shock (Eastwood et al. 2005 ). The backstreaming particles can excite many instabilities, generating various waves (Johlander et al. 2018 ; Shan et al. 2018 ; Shi et al. 2020 ) and transient structures (Schwartz et al. 2018 ; Bai et al. 2020 ; Turner et al. 2020 ). In recent years, a new type of magnetic structure, electron-scale magnetic holes, has been revealed to exist upstream of the terrestrial bow shock (Yao et al. 2019a ; Wang et al. 2020b , 2020c ) as well as in the magnetosheath (Huang et al. 2017a ; Yao et al. 2017 ) and current sheet (Ge et al. 2011 ; Sun et al. 2012 ). The sizes of these holes are within 2 ρ i (ion gyroradii) (Sun et al. 2012 ; Wang et al. 2020d ), and their cross sections have complicated shapes (Liu et al. 2020b )—some are circular (Liu et al. 2019 ) or like a two-dimensional sheet (Wang et al. 2020d ). Bipolar electron velocities are often observed in the cross section, suggesting that there are electron vortices inside the structures (Huang et al. 2017a ; Wang et al. 2020c ). In contrast, electron-scale magnetic holes have little effect on the ion population (Balikhin et al. 2012 ; Sundberg et al. 2015 ). These structures can trap electrons, and have rich wave activity (Huang et al. 2017b , 2019 ; Yao et al. 2019b ). Several mechanisms have been provided to explain their generation, such as electron mirror instabilities (Yao et al. 2019a ), tearing-mode instabilities (Balikhin et al. 2012 ), solitary waves (Ji et al. 2014 ; Li et al. 2016 ), and electron vortex magnetic holes (Haynes et al. 2015 ). Since they can survive up to 100 electron gyroperiods (Haynes et al. 2015 ) and they can propagate with the ambient flow (Wang et al. 2020d ), it is difficult to identify whether an electron-scale magnetic hole in the terrestrial foreshock is locally generated or originates from the upstream undisturbed solar wind. Thus, the generation mechanism of electron-scale magnetic holes upstream of the bow shock is still not fully understood. Magnetic peaks are another kind of magnetic structure, characterized by an enhancement of the magnetic field strength (Balikhin et al. 2010 ). They widely exist in the planetary magnetosheath (Horbury et al. 2004 ; Balikhin et al. 2010 ) and in the heliosheath (Burlaga et al. 2006 ). In the heliosheath, their sizes are of the order of 20 ρ i (Burlaga et al. 2006 ). They mainly appear in the mirror-unstable region, and they are suggested to be generated by ion mirror instabilities (Soucek et al. 2008 ). Using observations of the four Cluster spacecraft, Horbury et al. ( 2004 ) studied the propagation of magnetic peaks based on timing analysis (Harvey 1998 ). The authors found that their propagation velocities are scattered around zero within the timing errors in the plasma frame, and suggested that they are stationary in the plasma frame. The authors also found that their

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amplitude than a denser sample (gray region). Second, if the galaxy sample is sufficiently sparse, then the genus amplitude becomes sensitive to galaxy bias (or more precisely the combination b 2 A s )—the width of the filled regions indicates the sensitivity of to bias, for given number density and fixing all other cosmological parameters. For our fiducial number density a ∼10% variation in the galaxy bias generates a ∼0.5% uncertainty in the genus amplitude (see red shaded area). One can also observe a faint redshift evolution in the red shaded region—this is due to the amplitude of the matter power spectrum decreasing with redshift relative to the shot-noise term, which we have fixed to be constant at each redshift. However, if we increase the number density of the galaxy sample by an order of magnitude (gray shaded region), the sensitivity of the genus amplitude to the bias becomes negligible—as the shot noise decreases, the genus amplitude becomes insensitive to the amplitude of the power spectrum. The redshift evolution is also suppressed by selecting a denser sample. A more general word of caution is required: the shot-noise contribution is not perfectly represented by a white-noise power spectrum with . In fact, as we stray into scales at which shot noise affects our results, the expansion of the genus in Hermite polynomials will not provide a good representation of the Poissonian signal, as the Poisson distribution has a different moment-generating function. , where is the mean galaxy separation of the catalog, is an important condition on our analysis. Shot noise has been discussed in Appleby et al. ( 2017 ) and further in S. Appleby et al. (2020, in preparation). We note that both the redshift-space distortion effect and shot noise introduce a redshift dependence to the genus amplitude. For the smoothing scales and redshifts used in this work, the redshift-space distortion effect systematically decreases the genus amplitude with increasing redshift. Conversely, the shot-noise contribution increases the genus amplitude with redshift. Both effects are and will act to practically cancel one another. To completely suppress the shot noise and bias effects, the condition is required. For the smoothing scales and number densities considered in this work, the genus amplitude is only weakly sensitive to the combination of b , A s , and P SN . Nevertheless, we must include the P SN contribution to the power spectrum to avoid systematic bias in our cosmological parameter reconstruction. Appendix B Effect of Variation of Cosmological Parameters on Measured Genus Amplitudes At numerous points in our analysis when measuring the genus curves from the BOSS galaxy catalog, we have been forced to fix the distance–redshift relation. Specifically, we smoothed the field with constant comoving scale R G = 20 Mpc, which corresponds to angular smoothing on the unit sphere. We measured the genus per unit area, and so divided the genus by the total area

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O B-Model I-Model O O B-Model I-Model O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O


to evolve only a slice spanning ( r , θ), and with constant φ, thus assuming that the system will remain rotationally symmetric during the evolution. In order to justify to what extent and under which conditions this assumption is valid the comparison with 3D computation was needed. We will comment on that in Section 4.3 . However, within the 2.5-D approach, the φ component of the four-velocity is considered. We prescribe this component in Boyer–Lindquist coordinates according to relations given by equations ( 4 ) and ( 5 ). We chose several exemplary simulations and show their results in the form of snapshots. Every snapshot contains four panels with the slices of and its equatorial profile, ρ in arbitrary units, and λ in geometrized units, and it is labelled by the time t in geometrized units, where M is the mass of central black hole. The axes show the position in geometrized units. The red colour in the slice of radial Mach number corresponds to the supersonic motion, and blue regions indicate the subsonic accretion. The shock front is located at the place where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre. The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre. On top of the radial Mach number map the velocity streamlines are plotted in blue. These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel. 4.2.1 Bondi solution equipped with angular momentum At the beginning we check that for pure Bondi solution with λ = 0 we get the stationary solution of the flow with the properties consistent with the analytical solution. After that, we start to study the slowly rotating flows. For the first simulation, we pick the parameters such that λ eq belongs to the multicritical region (the region in the parameter space where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular , λ eq = 3.6M and γ = 4/3. In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data.). In other words, we expect that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way that it gets very close to 1, but does not touch it. In Fig. 2 , the initial conditions at t = 0 M are shown and the final state at t f = 10 4 M of the gas is depicted in Fig. 7 . At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch. It is interesting to note that for t = t f the Mach number actually crosses the line, however only on a very short radial range. Usually, the presence of the

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power-law template used by Wik et al. ( 2014b ) to actually fit the harder high-mass binary dominated sources) gives essentially the same results. Figure 2. NuSTAR spectrum (black) within a circular region of 30′′ centered on the X-ray peak joint-fitted with the Chandra ICM (blue) and LMXB (red) spectra. The dashed lines are the individual components discussed in the text. The vertical dashed blue lines below 3 keV are the Gaussian models, to take into account the line emission. Beyond ∼5 keV, the X-ray emission is dominated by the AGN-related activities (core, knots, and jet) and is completely dominated by these activities beyond ∼15 keV. Error bars are at 1 σ . The 3–40 keV NuSTAR spectrum was joint-fitted with the Chandra ICM and LMXB spectra by including these two component models as well as a single power-law model to represent the combined emission from the core, knots, and the jet. The same Galactic absorption model was also used for all the components. We tied all the parameters for the ICM and LMXB models, except for the ICM component. The ratio of the two APEC normalizations is tied to that of the Chandra model but not the overall normalizations to allow for the cross-calibration uncertainty of NuSTAR and Chandra and the difference in response due to the much larger PSF of NuSTAR . The difference in best-fit normalizations turns out to be small, with less than a 1% difference when constraining the overall hard X-ray emission from the AGN activities (this section) and about 6% when decomposing the core emission from the knots/jet (Section 4 ). The best-fit spectra with all the model components are shown in Figure 2 . The best-fit power-law index of the overall AGN-related emission is and the associated 20–40 keV flux, which completely dominates the spectrum, is erg cm −2 s −1 . The spectrum is well characterized by a single power law with no evidence of cut-off in energy below ∼100 keV (3 σ ). 11 11 The lower limit of the cut-off energy is determined by the fitting of a cut-off power-law model ( cutoffpl). The 20–40 keV flux measured is an order of magnitude lower than that determined by de Jong et al. ( 2015 ) with the Suzaku observation taken in 2006. It is believed that the X-ray emission above 20 keV detected near the end of 2006 was due to the HST -1 flare (de Jong et al. 2015 ; Prieto et al. 2016 ). The Chandra observations in 2017 show that the soft emission (≲10 keV) from HST -1 has become fainter by a factor of ∼40 since 2006 November. By simple scaling, the contribution from HST -1 in 20–40 keV should be a factor of a few lower than that from the core. This is indeed the case where HST -1 is a factor of six fainter than the core in 2017 measured in this band (Section 4 ). Therefore, for the first time, we have detected hard X-ray emission above ∼10 keV dominated by the core, which also gives the upper limit of its quiescent emission in hard X-rays. 4. Decomposing the Hard X-Ray Emission of the Core,

O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O B-Wavelength O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-Model I-Model O O O O O O O O B-Wavelength I-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O B-Wavelength I-Wavelength B-Telescope O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion B-CelestialObjectRegion O O B-CelestialObjectRegion O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-Telescope O O O O O O O O O O O O O B-Telescope O B-Telescope O O O O O O O O O O B-Model O B-Telescope O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O B-Model O B-Wavelength I-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Telescope O O O O O O O O O B-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O B-Telescope O O O O O O O O B-Wavelength I-Wavelength O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-Wavelength I-Wavelength O O O O O O O O O O O O B-CelestialObjectRegion O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O B-CelestialObjectRegion O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O B-CelestialObjectRegion O O O O O O O O O O O B-Wavelength I-Wavelength O O O B-Wavelength I-Wavelength O O O B-CelestialObjectRegion


We perform this analysis for the outer and the inner (if present) spirals. On the one hand, we observe that all the outer spirals have similar pitch angles (|φ| ∼ 16°), but the simulation with T s = 20 kyr (|φ| ∼ 20°). On the other hand, we find that the inner spirals for the moving shadows (|φ| ∼ 23°) are bit larger than the planetary ones (|φ| ∼ 14°). Again, we see that the pitch angle increases with T s (see Table 1 ). In Fig. 3 , we show a synthetic image of the emission of the disc at a wavelength equal to 1 μm, where the intensity has been normalized and scaled by r 2 . This calculation is done by means of the radmc-3d code (Dullemond et al. 2012 ), modifying it to mimic the shadows by blocking the photons sent from the star in the shadowed regions. It is based on the density field for the simulation with prograde shadows with T s = 10 kyr, assuming a dust-to-gas ratio of 0.01. Figure 3. Radiative transfer calculation at 1 μm for the simulation of prograde shadows with T s = 10 kyr, shown in the third column of Fig. 2 . We assume a dust-to-gas ratio equal to 0.01. 4 DISCUSSION Moving shadows dramatically changes the disc structure and evolution. For retrograde rotations, we do not observe any spirals, despite the fact that there are two shadows cast in the disc. Because the shadows are always confronting the gas motion, there is not enough time for it to cool down in order to create a gradient pressure to act as a symmetry breaking. This could be different if a more efficient cooling mechanism is taken into account (e.g. β cooling prescription, Gammie 2001 ). Realistically, it is expected that upper layers of the gas cool down faster, resulting in a colder shadowed region able to produce a gradient pressure, promoting sharper spiral features for this mode. For static shadows, we obtain similar results as in Montesinos et al. ( 2016 ). However, in the prograde rotation case, a gas portion under the shadow moving at the same shadow's speed will always be in the shade. This corotating position can be obtained by equating the azimuthal velocity with the shadow speed v s = (2π/ T s ) r . For a Keplerian disc, where , we obtain R co = ( GM ⋆ ( T s /2π) 2 ) 1/3 . In Fig. 4 , we plot R co as a function of the shadow's period and the azimuthal velocity along with the shadow speed for the different periods T s . The intersection between both curves gives the corotating location for each model (red dots). These points are in excellent agreement with the location of the local perturbations observed in the density fields shown in Fig. 2 (see also Table 1 ). We also found that the perturbation development is independent of the stellar luminosity or the gas density. Figure 4. Left-hand panel: Shadow rotation speed ( v s = 2π/ T s r ) for different periods T s (colored dotted lines) along with the azimuthal velocity (solid curve). The intersections (red dots) give the corotating locations, in remarkable agreement with the location of the perturbations

O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O B-Software I-Software B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Model I-Model O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


consuming and may suffer from a low S/N (Grier et al. 2013 ) caused by the combination of weak absorption lines, surface brightness dimming, instrument sensitivity and atmospheric transmission. In the rest-frame optical regime, the QSO and host galaxy contribution can be separated in high S/N single-aperture spectra by directly modelling them with template spectra for each component (Shen et al. 2008a , 2015 ). While a large sample can be studied with this technique, the fixed aperture size prevents such studies from measuring σ * at a physical size matched to the spheroid component. Furthermore, the reliability of the measurements decreases with increasing AGN luminosity and contrast ratio, thus limiting this technique to AGN with L 5100 ≲ 10 45 erg s −1 . To probe more luminous AGN, deep off-axis spectroscopy a few arcsec away from the QSO has been obtained to reduce the contrast between QSO and host galaxy emission (Wolf Sheinis 2008 ; Hughes et al. 2000 ; Wold et al. 2010 ). While σ * can be reliably derived from these off-axis spectra (Wolf Sheinis 2008 ), an uncertain aperture correction needs to be applied to infer comparable measurements representative for the spheroidal component. A direct decomposition the stellar continuum of the host galaxy underneath very luminous QSOs with deep optical integral-field unit observations was previously attempted with VIMOS for HE 1029−1401 (Husemann et al. 2010 ). Although HE 1029−1401 is much closer with a redshift of z = 0.086 but has a similar luminosity as PG 1307+085, the achieved S/N in the host spectrum was significantly lower than what we obtained with MUSE for PG 1307+085. This can be clearly attributed to the superior sensitivity, spatial resolution/sampling, and image quality of MUSE compared to VIMOS. These capabilities allow direct measurements of σ * with MUSE even in a high contrast regime that has not been accessible to any other techniques so far. 4.2 The nature of the large extended emission line region 4.2.1 Ionized gas shells driven by AGN outflows? A number of recent studies report kpc-scale high-velocity outflows to be common around luminous AGN with line widths of about 1000 km s −1 (e.g. Cano-Díaz et al. 2012 ; Liu et al. 2013 ; Harrison et al. 2014 ; Liu, Zakamska Greene 2014 ). Those outflows have been interpreted as spherically symmetric radiation pressure-dominated winds (Liu et al. 2013 ) and are possibly common for AGN with L bol > 2 × 10 45 erg s −1 (Zakamska Greene 2014 ). However, not all QSOs above this luminosity threshold exhibit such powerful superbubble-like outflows (e.g. Husemann et al. 2010 , 2013 ). The ionized gas kinematics of the ENLR around PG 1307+085 exhibits only small line widths with a velocity dispersion of 50 km s −1 in all the bright extended knots without any asymmetric component. This confidently rules out the AGN-driven superbubble scenario at least for this particular QSO. Although PG 1307+085 is classified as radio-quiet QSO with R 0.1

O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O B-Instrument O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O B-CelestialObject I-CelestialObject O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O B-Instrument O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O B-Instrument O O B-Instrument O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O B-Wavelength O O O O


RTE) applied to the spectra and polarized spectra emitted by the object. The RTE characterizes the interaction of the matter with the light that traverses through it. This interaction leaves its “fingerprints” in many different ways, for instance in the shape of the continuum of the spectra coming from the object, or in the presence or absence and the shape of spectral lines. Properly modeled, these features allow approximate characterization of the studied object. In recent decades, the advent of polarimetric observations has allowed the generalization of the RTE to polarized light ( Unno 1956 ; Rachkovsky 1967 ; Landi Degl’Innocenti Landi Degl’Innocenti 1972 ). This step was of great significance, as the polarization of the light encodes information on the magnetic field configuration as well as the geometry of the system, which allows a deeper insight into the system being studied. In solar physics, there are different approaches to solving the polarized RTE depending on the magnetic field regime adopted: atomic, Hanle, Zeeman, and/or Paschen (see e.g., Landi Degl’Innocenti Landolfi 2004 ). In this work, we focus on the solution of the polarimetric RTE under the Zeeman magnetic regime, which is usually the dominant one at this atmosphere layer. There are already several methods to address the solution of the RTE, each with its respective accuracy (see, e.g., del Toro Iniesta Ruiz Cobo 2016 ). There are several computational codes that allow one to infer the optical depth ( τ ) dependence of the physical properties in the solar atmosphere; for example, SIR ( Ruiz Cobo del Toro Iniesta 1992 ), SPINOR ( Frutiger et al. 2000 ), NICOLE ( Socas-Navarro et al. 2015 ), SNAPI ( Milić van Noort 2018 ), and STIC ( de la Cruz Rodríguez et al. 2019 ). These are of particular relevance here as they allow the furthest insight into the properties of the solar atmosphere in a three-dimensional volume. In these cases, the conversion between optical depth τ and geometrical height z is then performed by considering the relation d τ = − ρ κ c d z , where ρ is the density and κ c the continuum opacity, with the latter being a nonlinear function of the temperature T and gas pressure P g . Since the proper calculation of the gas pressure (and density) involves the whole three-dimensional volume, for which there is no account in the aforementioned codes, the gas pressure and density inferred by them are unrealistic, and hence also the conversion between τ and z . Deviations from hydrostatic equilibrium involve, among others, time-variation of the velocity, advection and viscous terms and, the most relevant at this atmospheric layer, the Lorentz force. The latter is responsible for the Wilson depression, a depression of the height at which photospheric spectral lines are formed with respect to the quiet-Sun because of the presence of magnetic field. Because the geometrical scale z is then inaccurate, so is the calculation of the spatial derivatives, such as those

B-Model O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-Model O O O O O O O B-Model I-Model I-Model I-Model I-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Model O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O B-Software O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Software O B-Citation I-Citation I-Citation I-Citation O B-Software O B-Citation I-Citation I-Citation I-Citation O B-Software O B-Citation I-Citation I-Citation I-Citation O O B-Software O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O


et al. 2009 ; Rosario et al. 2013 , 2016 , 2018 , and references therein) or mid-infrared neon ([Ne ii ] 12.81 μ m and [Ne iii ] 15.56 μ m) emission lines (Zhuang, Ho Shangguan 2019 ). The infrared observations used in these works are obtained using large apertures, including the whole galaxy emission. As discussed in Rosario et al. ( 2016 ), despite finding a good correlation between far-infrared emission and SFR (which can be attributed to leaking of the SF regions to the whole galaxy or that the SF is constant over several hundreds of Myr) a substantial component of the cold dust luminosity that is associated with a diffuse interstellar radiation field can come from evolved stars. Even if the contribution from an old age stellar population is not dominant, the far-infrared may not be a good tracer of SF since dust can be heated by stars that are older than a few hundred Myr (Kennicutt et al. 2009 ; Hao et al. 2011 ; Rosario et al. 2016 ). Other recent effort on the bluer part of the spectrum has been made using a re-calibration [O ii ] λ3727 emission line (Zhuang et al. 2019 ; Zhuang, Ho Shangguan 2020 ), which is metallicity and electron density dependent. From the above, it is clear that it is thus necessary to find an independent way to obtain SFRs in AGN hosts. A powerful technique to disentangle the components summing up to a galaxy spectral energy distribution (SED) is stellar populations synthesis (e.g. Cid Fernandes et al. 2004 , 2005 ; Riffel et al. 2009 , 2008 ; Walcher et al. 2011 ; Baldwin et al. 2018; Cid Fernandes 2018 ; Salim, Boquien Lee 2018 ; Peterken et al. 2020 , and references therein). The synthesis is based on the simultaneous fit of different proportions of composite or simple stellar populations (SSPs) templates from a base of such spectra to the observed spectrum. In the case of the starlight fitting code (Cid Fernandes et al. 2004 , 2005 ; Cid Fernandes 2018 ), it returns the values of the gas mass rate that has been converted into stars throughout the galaxy life as well as this rate for each single SSP included in the base. These values can therefore be used to compute the SFR via stellar population synthesis, SFR ⋆ . In fact, this was already applied in Asari et al. (2007) using single fiber Sloan Digital Sky Survey (SDSS) data and older stellar population models generations (they used Bruzual Charlot 2003 models). From the discussion above, it is clear that it is of utmost importance to characterize the SFR of AGN hosts – in particular in the narrow-line region (NLR) and extended narrow-line region (ENLR), which are photoionized by the AGN. And in order to verify the actual role of the AGN on the SFR, it is necessary to compare the SFR values obtained for the NLR and ENLR with those obtained for a matched control sample of non-active galaxies at similar distances from the nucleus, as well as with predictions from simulations. In Rembold et al. ( 2017 ), we described the method we have used to select

B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation O O O O O B-Wavelength O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Survey I-Survey I-Survey I-Survey B-Survey O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O


polarized) radiation. Then, the radio spectral index S ν ∝ ν α can provide information as well, but it will often be ambiguous. Third, variability can be an indication, with rapid variability a sign of nonthermal emission (on timescales as short as a few minutes, e.g., Forbrich et al. 2017 ), but there is again some ambiguity on intermediate timescales. Fourth, radio brightness temperature is a rarely employed but powerful indicator, since this technique allows us to select sources with brightness temperatures that are above that of thermal emission. This can naturally be accomplished by using Very Long Baseline Interferometry (VLBI), with which it is thus possible to filter out any thermal emission, enabling a census of nonthermal radio emission that remains largely unaffected by thermal radio emission. Apparent brightness is inversely proportional to the synthesized beam size for an unresolved source, and for milli-arcsecond resolution VLBI observations in C -band (4–8 GHz), even faint detections have corresponding brightness temperatures of several 10 6 K, well above possible values of free–free emission ( ≈ 10 4 –10 5 K), even more so when considering that this is a lower limit. Even at this high resolution, the stellar radio emission remains unresolved, with a synthesized beam size corresponding to a linear scale of order 1 au in our case. Overall, the two most useful indicators thus are polarization and brightness temperature. In recent years, it has become possible to study populations of radio stars at unprecedented sensitivity, with a range of radio observatories. Our focus has been on observing YSOs in the Orion Nebula Cluster, the nearest young rich open cluster containing high-mass stars ( d ∼ 400 pc; Menten et al. 2007 ; Kounkel et al. 2017 ). Altogether, the ONC has 3500 stellar members that formed less that 2 Myr ago (Hillenbrand et al. 2013 ), many of which emit X-rays (e.g., Getman et al. 2005a ). Radio emission from YSOs in the ONC and the nearby Kleinmann–Low nebula was first observed when the NRAO Very Large Array became available (Churchwell et al. 1987 ; Garay et al. 1987 ), and prior to the VLA expansion, 77 radio sources had been observed in this cluster (Zapata et al. 2004 ). After the VLA expansion was completed, we obtained a deep pointing with what is now called the Karl G. Jansky Very Large Array, which allowed us to identify 556 radio sources in the ONC (Forbrich et al. 2016 ). Our analysis revealed extreme variability for many objects, strongly suggesting nonthermal emission (Forbrich et al. 2017 ). While this analysis included the derivation of in-band spectral indices for the brightest sources, it has been impossible so far to extract reliable wideband polarization information across the entire primary beam of this deep pointing. To obtain a more reliable census of the incidence of nonthermal emission, and to then use this information for a proper motion census of the inner ONC, we have utilized the

O O O O B-Wavelength O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Telescope O O B-Wavelength I-Wavelength B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O B-Wavelength O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength O B-Citation I-Citation I-Citation I-Citation O B-Wavelength O O O O O B-CelestialObject O O O B-CelestialObject I-CelestialObject O O O O O B-Observatory B-Telescope I-Telescope I-Telescope O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Telescope O O B-Wavelength O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Telescope O O O O O O O O O O O O O O B-Telescope I-Telescope I-Telescope I-Telescope I-Telescope I-Telescope O O O O O O B-Wavelength O O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O


T icb = 6.36 × 10 4 K; the density ρ icb = 1.47 g cm −3 . The composition is uniform with 100% silicates out to a temperature T = 2.1 × 10 4 K and radius 2.72 × 10 9 cm, decreasing to 50% at T = 5.42 × 10 3 K and radius 3.39 × 10 9 cm, and to 1% at T = 2960 K and radius 4.84 × 10 9 cm. During the isolated phase, the parameter in Equation ( 7 ) is set to 0.04. During the first 10 8 yr, an additional 0.25 M ⊕ is lost, giving M XY = 0.42 M ⊕ , R p = 7.43 R ⊕ , L = 2.0 × 10 −10 L ⊙ , and a cooling time of 8 × 10 8 yr. At this time, M ⊕ yr −1 . As in Run 0.128 (Rev), a radiative zone extends inward to T = 1500 K. Later, the luminosity declines to ≈10 −11 L ⊙ , the cooling time increases by an order of magnitude and the rate of mass loss declines significantly, by a factor of 400, to 2 × 10 −12 M ⊕ yr −1 , by the final time. Between t = 10 8 yr and t = 7 × 10 9 yr, a further 0.13 M ⊕ is lost from the H/He envelope. The radius of the final model planet agrees quite well with that of the actual planet, as does the total mass. The total heavy-element mass (including the inner and outer cores) is M Z = 7.52 M ⊕ , and the H/He mass M XY = 0.29 M ⊕ . The inner core of 1.87 M ⊕ has R icore = 1.075 R ⊕ and mean density 8.35 g cm −3 . The region of almost 100% heavy elements has radius 1.84 R ⊕ , and the mean density of the inner plus outer cores is 5.86 g cm −3 . The mass of this final model is very close to that of the in situ model 0.128 (Rev). The temperature at the boundary between the inner and outer cores ( T icb ) is similar (1.54 × 10 4 K versus 1.75 × 10 4 K), and the corresponding density is slightly lower (7.59 versus 8.0 g cm −3 ). These differences are presumably caused primarily by the different masses of the inner cores. Both final models have outer radiative zones, extending inward to T = 1850 K in the in situ case and to 1500 K in the present case; they include less than 1% of the envelope mass. The structure of the final model for Run 1.00 (Rev) is plotted in Figure 12 . Figure 12. Structure of the model in Run 1.00 (Rev) at time 7.03 × 10 9 yr. Symbols and curves: as in Figure 11 . The mean density of the inner core is 8.35 g cm −3 . 3.4. Migration Model: Run 1.00(Old) Run 1.00(Old) starts at 1 au with σ init = 190 g cm −2 , slightly lower than that in Run 1.00(Rev). Outer densities and temperatures are the same in the two runs. The masses and radius as a function of time are given in Figure 13 . At first, M core increases very rapidly until, at t ≈ 7 × 10 3 yr, it reaches 6.78 M ⊕ , close to the isolation mass of 6.81 M ⊕ . At this point, M env = 0.084 M ⊕ . The temperature T cb is 9.91 × 10 3 K, much lower than the value of T icb = 6.1 × 10 4 K reached at a comparable evolutionary phase in Run 1.00 (Rev). The lower mean molecular weight in the H/He envelope accounts for much of this difference. The luminosity during the solid accretion phase averages about 10 −4 L ⊙ , corresponding to a mass accretion rate onto the core of 1 to 1.5 ×

B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


8 . It can be seen from Fig. 8 that the age of red giants estimated by our model is usually younger than the age given by Pinsonneault et al. ( 2018 ). This is determined by the sample (Sanders Das 2018 ; Yu et al. 2018 ) we used to train the red giant age model. Pinsonneault et al. ( 2018 ) used the revised scaling relation when inferring stellar mass from seismic data, and they took the influence of the [ α /Fe] into account when estimating the age of red giants. Wu et al. ( 2018 ) also mentioned these two points when comparing the red giant ages of Sanders Das ( 2018 ). It should be mentioned that, in principle, using asteroseismology to estimate the age of red giants should be much more precise. In this work, we used the stellar age given by Sanders Das ( 2018 ) as a standard, established different stellar age models by using different stellar parameters, and compared the model results. By comparison, we infer which parameters are used to establish the stellar age model that is more accurate, that is, to filter out which stellar parameters are more relevant to the stellar age. Figure 8. The age of the red giant estimated using our model is compared with the age given by Pinsonneault et al. ( 2018 ). 4.2 EW of the magnetic activity lines and the age of red giants In Table 3 , by comparing the input parameter combinations ‘b’ and ‘j’, ‘d’ and ‘f’, ‘g’, ‘h’, ‘i’, we know that no matter adding all or part of the EW of the magnetic activity lines to the input parameters, the red giants age estimation cannot be improved. This suggest that in the red giants, the EW of the magnetic activity lines has no direct correlation with the age of the red giants. When studying the relationship between stellar chromospheric activity and age in the open clusters, Zhang et al. ( 2019 ) found that the chromospheric activity indices (log and log ) of low-mass stars with strong surface magnetic field are strongly correlated with the stellar age, while for those stars with relatively large mass and weak surface magnetic field, their stellar chromospheric activity indices are weakly correlated with the age. In other words, the relationships between stellar age and chromosphere activity indices are affected by the intensity of the stellar surface magnetic field. Red giants are formed when low- or intermediate-mass stars have finished burning the hydrogen in their cores. Stello et al. ( 2016 ) mentioned that in the red giants with asteroseismic signals, the outer layer dynamo has shut off at the end of core-hydrogen-burning, and most of these red giants have core-dynamo-generated fields. In Section 2 , by analysing the EW of the magnetic activity lines, we know that there is no strong magnetic activity in the chromosphere and photosphere of all red giants in this study sample. This is consistent with the conclusion of Stello et al. ( 2016 ). Moreover, the results of Chaplin et al. ( 2000 ) showed that the amplitude of the solar p mode decreases with the increase of

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here but contains also a 1/ r dependence. Recently, Vigren Eriksson ( 2017 ) developed a 1D model to test the ability of H 2 O molecules to interrupt ion acceleration along weak electric fields in the radial direction. From the model results, they raised concerns about the correctness in assuming u I, flow = u for r r IN as given by Equation ( 1 ) or ( 2 ). As an example, for perihelion conditions, r IN from equation (1 ) is at ∼1000 km, while the estimated effective ion flow speed at 200 km was calculated as several km s −1 , assuming a rather weak electric field strength of the order of ∼0.03–0.1 mV m -1 , which may commence due to the electron pressure gradient force. Their finding would support the observations of H 3 O + /H 2 O + number density ratios observed at times to be lower than predicted by models running with the assumption of collisionally coupled ions (see fig. 7 of Fuselier et al. 2016 ). The observed ratios were observed to be variable and at times also at the level predicted by collisionally coupled models (see in particular, Fig. 6 of Fuselier et al. 2016 ), particularly so when the RPC-Ion Electron Sensor (RPC-IES, see Mandt et al. 2016 ) did not see clear signs of ion acceleration. Vigren Eriksson ( 2017 ) referred also to preliminary values of effective ion speeds (not restricted to ion flow speeds) of several km s −1 as calculated from combined measurements by the Mutual Impedance Probe (MIP, Trotignon et al. 2007 ) and the Langmuir Probe (LAP, Eriksson et al. 2007 ), both being subsystems of the Rosetta Plasma Consortium (RPC, Carr et al. 2007 ). In this work, we will first (Section 2 ) discuss the MIP and LAP measurements and how they can serve to estimate the mean (or rather the effective) ion speed. The selected time interval for the study, 2015 August 2–4, near perihelion, is discussed in Sec-tion 3 . Results are presented and discussed in Section 4 . The derived effective ion speeds are markedly higher than the neutral outflow velocity and also shown to at least periodically be in good agreement with effective ion drift speeds derived from an independent method; a simple flux conservation model relying on MIP and COPS measurements. The work is briefly summarized in Section 5 . 2 METHOD Our method utilized to derive effective ion speeds is based on combining electron number densities measured by the MIP with voltage–current characteristics from the ion side (negative voltages) of Langmuir probe bias voltage sweeps (we used the LAP1 sensor, see Eriksson et al. 2007 for details). The MIP measurements consist of determining the mutual impedance between two electric antennas – a transmitting and a receiving one. The transmitting electrode injects an oscillating current in the plasma, at a given frequency, while the receiving one measures the potential difference induced in the plasma at the same frequency. The frequency is increased step by step, in a frequency range that ideally contains the plasma frequency. The

O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Instrument I-Instrument I-Instrument B-Instrument O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument I-Instrument I-Instrument B-Instrument B-Citation I-Citation I-Citation I-Citation O O O B-Instrument I-Instrument B-Instrument B-Citation I-Citation I-Citation I-Citation O O O O O O B-Collaboration I-Collaboration I-Collaboration B-Collaboration B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Instrument O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O B-Instrument I-Instrument O O O O O O B-Instrument O O B-Citation I-Citation I-Citation I-Citation O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


magnification, which depends on the microlensing model and blending. Our method conserves the variability and noise in the original light curves, as well as information on the quality of individual measurements. Let be the flux of the source and the unmagnified flux from possible blended stars and/or the lens itself. The flux of the baseline object () is magnified during a microlensing event by a factorwhere A ( t ) is the model magnification and is the blending parameter. If there is no blending ( f s = 1), then A ′( t ) = A ( t ); if the blending is very strong (), the observed magnification . To inject a microlensing event into the database light curve, we needed to transform the observed flux and its uncertainty to . The naive transformation is incorrect because it preserves the original photon noise during the magnified part of the event. To illustrate this, let us consider a constant I = 19.5 star with a typical rms light-curve scatter of 0.1 mag (left panel of Figure 9 ). The transformation A ′( t ) = 100 would shift the mean magnitude to I ′ = 14.5, but it would preserve the original scatter, whereas the observed scatter of constant stars of that magnitude is much smaller (Figure 9 ). Figure 9. The rms light-curve scatter as a function of the mean magnitude of a star in the I band in the OGLE-IV field BLG521.12. Left panel: real data. Right panel: constant stars injected on top of real light curves (, where 0 ≤ Δ m ≤ 0.75 is drawn from a uniform distribution). The orange line is the best-fit model from Skowron et al. ( 2016 ). Assuming that every observed fluctuation in brightness of the initial light curve from its mean brightness ( F i − F 0 ) is due to the observational noise, the following transformation ensures that the photon noise is properly scaled:where F 0 and are the mean flux in the baseline and the magnified mean flux, respectively (the corresponding magnitudes are m 0 and ). The ratio can be calculated by assuming the photometric noise model of Skowron et al. ( 2016 ):In this simple model, the observed scatter is the sum of the Poisson noise contributions from the object and background (parameterized by m B ). To verify the proposed model, we injected “constant” stars on top of real light curves from the database (which correspond to the transformation ). The right panel of Figure 9 shows the rms scatter of simulated light curves, which is consistent with that of real data. Our simple model underpredicts the scatter of the brightest stars ( I ≲ 15), likely because the accuracy of their photometry is limited by the accuracy of modeling the PSF and thus the quality of image subtractions, not by the photon noise. This effect does not influence our detection efficiency simulations, because the vast majority of events are fainter than I = 15. Similarly, we used the noise model of Skowron et al. ( 2016 ) to transform the uncertainties:When the predicted uncertainties were equal to or below 0.003 mag, we assumed mag, as in the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey B-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O


: pseudo-ADI image of the inner regions obtained by differentiating the Q Φ image (see Ligi et al. 2018 , for more details). The white cross marks the position of the star, and the cyan circle shows the size of the coronagraphic mask. The other labels refer to the blobs we discuss in this paper that are also visible in these images. Thecolor scale of the differential image is five times less extended to show the faint structures better. In both panels, N is up and E to the left; a segment represents 1 and 0.5 arcsec in the left and right panel, respectively. Fig. 2 Signal-to-noise ratio (S/N) maps for the individual epochs for IFS. The individual S/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median over the wavelength (see Mesa et al. 2015 ). Upper row, left panel : JD = 57180.17, middle panel : JD = 57201.12, and right panel : JD = 57499.34. Lower row, left panel : JD = 57566.15, middle panel : JD = 57873.30, and right panel : JD = 58288.19. In all panels, the central 0.1 arcsec is masked, the solid white line at the bottom represents 1 arcsec, the white cross represents the position of the star. N is up and E to the left. The upper panels of Fig. 3 show images of HD 169142 obtained by combining the six individual images, assuming the distance given by Gaia DR2, a mass of 1.7 M ⊙ (see below), and circular orbits. 3.1 Coincidence images To improve our ability of discerning faint signals, we combined data from different epochs using a coincidence map (see the lower panel of Fig. 3 ). The principle of this coincidence map is to start with S/N maps for individual epochs. We used S/N maps after correcting for the small-number-statistics effect using the formula by Mawet et al. (2014) . The maps were then multiplied by each other. The S/N maps average to zero, with both positive and negative values for individual pixels. Of course, this may result in a false-positive signal for an even number of negative signals in the individual S/N maps. To avoid this problem, we arbitrarily set to negative the result for a given pixel when the signal for that pixel was negative for at least one epoch. Of course, this is not a realistic flux map: the aim is merely to identify consistent signals throughout all individual images. To consider the orbital motion around the star over the three years covered by our observations, we divided the field into 65 rings, each one 2 IFS pixels wide (15 mas, i.e., about 1.8 au at the distance of HD 169142). For each ring, we rotated the S/N maps obtained at different epochs with respect to the first reference image according to Kepler’s third law. When the distance to the star was fixed, the only free parameter remaining in this model is the (dynamical) stellar mass. If there is a companion orbiting the star, the signal is maximized for a value of the mass that, if the assumptions made (circular motion seen face on) are correct, is the dynamical mass of the star. If these hypotheses are not

O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O B-Software I-Software I-Software O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O