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-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 B-Wavelength O O O O O O B-Wavelength 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-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 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 B-Wavelength 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 B-Wavelength 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 B-Wavelength O O B-CelestialObject I-CelestialObject 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 B-Wavelength 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 O O O O O O O O O O O O O O O O B-Wavelength 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 O O 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 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 B-Software I-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 O O O O O O O O O O O O O O O O O

then actually expect less active nuclear star formation and/or inflows in NGC 1097, which will lead to the deficiency of a geometrically thick torus in this LLAGN as compared to luminous AGNs such as NGC 1068 due to the deficiency of the energy source(s): this is an expected trend by the evolutionary torus models (Section 1 ). Another remarkable thing is that the envelope of the CND traced by the CO(3–2) emission and that traced by the NIR H 2 emission (Davies et al. 2009 ; Hicks et al. 2009 ) are almost identical in NGC 1097 at our available resolutions (Figure 3 (d)). This indicates that the outer radial extent of the molecular CND is unrelated to gas temperature, although the vertical height of the CND could differ (see below), which is consistent with hydrodynamic simulations of CND-scale gas distribution (e.g., Wada et al. 2009 ). However, we also found that their internal distributions differ significantly: while the two knots stand out in the CO(3–2) map, the NIR H 2 emission is clearly concentrated toward the nucleus, likely reflecting complex excitation conditions including fluorescence around the AGN. Thus, we should be careful to simply assume a linear conversion from NIR H 2 line luminosity to CO-based (Müller-Sánchez et al. 2006 ) when using that luminosity as a tracer of molecular mass. Gas dynamics is also key for studying the nature of the torus. Figures 4 (a) and (b) show the observed line-of-sight velocity and velocity dispersion maps of the CO(3–2) line in NGC 1097, respectively. It is evident that rotation (kinematic P.A. ≃ 118°) dominates the global gas dynamics, while streaming motions as well as high-velocity dispersions along the gas spirals/arc are detected. Figure 4. Intensity-weighted (a) mean velocity and (b) velocity dispersion maps of the CO(3–2) emission in the central 400 pc of NGC 1097. The stars mark the AGN location. The systemic velocity is 1254 km s −1 (thick line). Thin contours indicate 1080–1400 km s −1 in steps of 20 km s −1 for (a), and 10–40 km s −1 in steps of 10 km s −1 for (b), respectively. (c) and (d) Position–velocity diagrams (PVDs) of the CO(3–2) line along the adopted major axis (P.A. = 118°) and minor axis (P.A. = 208°) shown in the blue contours. The overlaid red contours indicate the PVDs produced by our dynamical model (MODEL-1). Both contours are plotted at 5, 10, ..., 40 σ , where 1 σ = 0.84 mJy beam −1 . Due to the inclined geometry of the concentric rings, no model component exists at offset ≳11 along the minor axis. To extract basic beam-deconvolved dynamical information, particularly the rotation velocity () and the velocity dispersion ( σ ), we fitted tilted rings to these observed maps with the 3D Barolo code (Di Teodoro Fraternali 2015 ). The main parameters are dynamical center, systemic velocity (), , σ , galactic inclination ( i ), and P.A., all of which can be varied for each ring. Our initial fits indicated a dynamical center as the AGN position, as 1254 km s −1 . Thus , σ ,


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We thank A. Strong and T. Porter for advice regarding the details of the GALPROP ISRF model used in this paper. We also thank A. M. Stutz, Y. L. Shirley, J. Steinacker, and E. Keto for helpful discussions and comments on the paper draft. The work of J.K. and S.E.R. was supported by the Deutsche Forschungsgemeinschaft priority program 1573 (“Physics of the Interstellar Medium”). H.L. and M.N. were funded by the Deutsches Zentrum für Luft- und Raumfahrt (DLR). PACS has been developed by a consortium of institutes led by MPE (Germany), including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).


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Acknowledgments This research was conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. SB acknowledges funding support from the Australian Research Council through a Future Fellowship (FT140101166). GAMA is a joint European-Australasian project based around a spectroscopic campaign using the AAT. The GAMA input catalogue is based on data taken from the SDSS and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KIDS, VISTA VIKING, WISE , Herschel -ATLAS, GMRT and ASKAP providing ultraviolet to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO and the participating institutions. The GAMA web site is http://www.gama-survey.org/ .


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J.C.L. and K.M.H. acknowledge support from the INSU-CNRS national programs PCMI and PNP. V.W. is funded by the ERC Starting Grant (3DICE, grant agreement 336474). Acknowledgement is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research (PRF#53105-DN16 for FG summer support). The Rennes team acknowledges support from the Agence Nationale de la Recherche, contract ANR-11-BS04-024-CRESUSOL-01, the INSU-CNRS national program PCMI, the Institut National de Physique (INP CNRS), the Région Bretagne and the Université de Rennes 1. S.D.L.P. acknowledges financial support from the Institut Universitaire de France.


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 O O O B-Wavelength I-Wavelength 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 B-Formula I-Formula I-Formula O O B-CelestialRegion I-CelestialRegion O B-Wavelength I-Wavelength O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation 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 O O O O O O O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Model I-Model 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 O O O O O O O O O O B-Model I-Model O B-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula 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 B-Model I-Model O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Model I-Model O O O O O O B-Formula I-Formula I-Formula O O O O B-Wavelength I-Wavelength O O B-Survey B-Wavelength 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-Wavelength I-Wavelength O O O B-Formula I-Formula I-Formula O B-CelestialRegion I-CelestialRegion O O O O B-Model O O B-Citation I-Citation I-Citation I-Citation 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 B-Formula I-Formula I-Formula 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 B-Formula I-Formula I-Formula O O O O O O O O O O B-Wavelength I-Wavelength O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Wavelength O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O B-Telescope B-Instrument 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 B-Telescope B-Instrument B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O B-Wavelength I-Wavelength O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength 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 B-Wavelength I-Wavelength O O O O O O O O B-Formula I-Formula I-Formula O O O B-Wavelength I-Wavelength O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Wavelength I-Wavelength O O

is listed in Table 1 , and plotted in Figure 2 together with LAE LFs at z ∼ 5.7–7.3 from the literature. The LF shows dramatic evolution not only in normalization, but also in the shape. Figure 2. Top: the Ly α LFs of LAE surveys at z ∼ [5.7, 6.6, 7, 7.3]. The red filled circles show our results at z ∼ 7 from the LAGER-COSMOS field. References: Ly α LFs at z = 5.7 from Ouchi et al. ( 2008 ; blue filled squares), Konno et al. ( 2017 ; blue filled stars), and Santos et al. ( 2016 ; light-blue filled hourglass), at z = 6.6 from Ouchi et al. ( 2010 ; green filled squares), Konno et al. ( 2017 ; green filled stars), and Santos et al. ( 2016 ; dark-green filled hourglass), at z ∼ 7 from this work (red filled circles) and Ota et al. ( 2010 ; pink empty circles with plus), and at z = 7.3 from Shibuya et al. ( 2012 ; purple empty circles), and Konno et al. ( 2014 ; purple empty circles with cross). The colored regions in blue and green show the best Schechter fit with ±1 σ error in L * at z ∼ 5.7 and 6.6, respectively, from Konno et al. ( 2017 ). The solid and dashed lines in red (gray) show the best Schechter fits at z ∼ 6.9 with L range 42.65 ≤ log 10 L ≤ 43.25 (42.65 ≤ log 10 L ≤ 43.65) (see Table 2 ). A truncated Schechter function is also plotted (red solid line, see the text for details) to measure the bright-end LF bump. The purple line shows the best Schechter fit at z ∼ 7.3 from Konno et al. ( 2014 ). All these Schechter fits have a fixed faint-end slope of α = −2.5. Bottom: the histogram of Ly α luminosities of LAGER LAEs. The average completeness fraction for our LAEs interpolated from Section 2.2 is also plotted as a red solid line with red triangles. Table 1 Ly α Luminosity Function at z ∼ 7 from LAGER-COSMOS Field Note. a 1 σ Poisson errors from Gehrels ( 1986 ). The relatively fainter end of our LF (42.65 ≤ log 10 L Ly α ≤ 43.25) at z ∼ 7 can be fitted with a a single Schechter function in the form ofBecause our survey does not go far below L *, we fix α = −2.5 in our fitting, which is suggested from recent studies on Ly α LFs at z = 5.7 and z = 6.6 by Konno et al. ( 2017 ). Konno et al. ( 2017 ) report the largest narrowband surveys to date for LAEs at z = 5.7 and z = 6.6 taken with Subaru HSC on the 14–21 deg 2 sky, and their best-fit Schechter function analyses also include the smaller but much deeper narrowband surveys taken with Subaru Suprime-Cam (Ouchi et al. 2008 , 2010 ) at the corresponding redshifts. Considering that the Ly α LFs by Konno et al. ( 2017 ) are derived from the largest LAE samples at z = 5.7 and z = 6.6 with the widest luminosity range, and we have used consistent selection and analysis methods, we choose their Ly α LFs for comparison in the following sections. At z > 7, we choose the Ly α LF at z = 7.3 by Konno et al. ( 2014 ) for comparison. The best-fit Schechter function parameters of these LFs are listed in Table 2 and plotted in the upper panel of Figure 3 . We find a large drop in Φ of Ly α LF, compared


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M.A. and J.P.W. were supported by NSF and NASA grants AST-1208911 NNX15AC92G, respectively. M.A. also acknowledges support from NSF AST-1518332, NASA NNX15AC89G, NNX15AD95G/NEXSS, and the Center for Integrative Planetary Science at UC Berkeley. Leiden is supported by the European Union A-ERC grant 291141 CHEMPLAN, by the Netherlands Research School for Astronomy (NOVA), and by grant 614.001.352 from the Netherlands Organization for Scientific Research (NWO). A.M. and C.F.M. are supported by ESO Fellowships. M.R.H. is supported by NWO grant 614.001.352. M.T. has been supported by the DISCSIM project, grant agreement 341137 funded by the European Research Council under ERC-2013-ADG. This paper makes use of the following ALMA data: 2015.1.00222.S, 2016.1.01239.S, 2013.1.00226.S, 2013.1.01020.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work benefited from NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate.


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

of Fe xxi 1354.09 Å. On the other hand, the shorter period cannot be detected in these detrended fluxes, which further confirms that the 3.1 minute oscillations can only be observed in the Doppler shift of Fe xxi 1354.09 Å. Figure 9. Normalized Fourier power spectra of the detrended time series from IRIS Fe xxi 1354.09 Å (a, b), GOES SXR fluxes (c, d), and AIA EUV light curves (e, f). A horizontal purple line in each panel indicates the 99.99% confidence level. As mentioned above, the Doppler shift oscillations within a period of ∼3.1 minutes ( P 1 ) can be interpreted as the standing kink oscillations in the hot flare loop. On the other hand, the longer period ( P 2 ) is different from the shorter one ( P 1 ), because it can be observed simultaneously in SXR and EUV fluxes, as well as in the time series of Doppler shift and line-integrated intensity of Fe xxi 1354.09 Å. Those facts imply the process of periodic energy release in this flare (Tian et al. 2015 ; Li et al. 2017b ). The longer period is most likely to be linked to the quasi-periodic pulsations during the solar flare, which seems to be caused by an MHD wave in the slow mode (Fang et al. 2015 ; Van Doorsselaere et al. 2016 ). Based on the formation temperature ( 11 MK) of Fe xxi 1354.09 Å, the local sound speed ( v s ≈ ) in the flare loop is estimated to be ∼500 km s −1 (e.g., Nakariakov Ofman 2001 ; Kumar et al. 2013 , 2015 ). While the deprojected length ( L ) and longer period ( P 2 ) of this flare loop have been identified as ∼81 Mm and ∼10 ± 4 minutes, then the phase speed of this flare loop ( v h = 2L/ P 2 ) can be estimated as about 200−450 km s −1 . This phase speed is slower than the local sound speed, implying it might be a slow MHD wave (Mandal et al. 2016 ). In other words, the longer period ( P 2 ) might be explained as the MHD wave in a slow mode. To compare the Doppler shift oscillations, we also use the sine function with a damping amplitude (Equation ( 3 )) to fit the time series of Doppler shift during peaks “5” and “6” (e.g., Nakariakov et al. 1999 ; Wang et al. 2002 ; Anfinogentov et al. 2015 ), because the Doppler shift oscillations are most pronounced during this time interval. Figure 8 (upper) shows the fitting result, the overlaid turquoise profile gives the best fitting, and it appears to match well with the Doppler velocities (black).Here, v 0 is the initial Doppler velocity, v m is the initial oscillatory amplitude, and P 1 , ψ , and τ are the shorter dominant period, initial phase, and decay time of the oscillations, respectively. 5. Conclusions Using observations from multi-instruments, i.e., IRIS , SDO /AIA, SDO /HMI, Hinode /XRT, and NoRH, we investigate the Doppler shift oscillations in a hot loop during a GOES M7.1 flare on 2014 October 27. The main results are summarized as follows. (I) The Doppler shift oscillations are detected in a hot flare line, i.e., Fe xxi 1354.09 Å from IRIS spectroscopic observations. The time series of Doppler shift


O O O O O B-Software 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O 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 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 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 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 O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

noise random number generator within GILDAS . However, the data cube has undergone many transformations and the true noise may not be precisely Gaussian. Thus, we extract contiguous channels from signal-free regions of the cube which we use as noise. In fact, as any transformations were designed to preserve particularly the region where signal is present, this is a sort of worst-case noise. We first examine what we obtain from mock clouds with no velocity gradient, that is to say with v xy constant. Figure 13 shows the results for T max = 200 mK using both purely random noise and noise taken from signal-free but unoptimized regions of the cube. The gradients, for equivalent clouds and noise levels, are clearly more dispersed with the “real” cube noise. The dispersions, as measured by the full width at half power of the distribution (divided by 2.35 to give the equivalent for a Gaussian), are approximately 0.008 and 0.015 km s −1 pc −1 for the random and cube noise injection. T max = 200 mK corresponds roughly to the median signal in the cloud sample. Fig. 13 Histogram of gradients from noise. The black line shows the distribution of the gradients for purely random noise. The red line shows the distribution when noise is from signal-free regions of the cube. No velocity gradient has been injected here. We have also tested with higher and lower S/N levels. For T max = 100 mK, the dispersion increases to 0.014 and 0.029 km s −1 pc −1 for respectively random and cube noise. For T max = 400 mK, the dispersion decreases to 0.005 and 0.0085 km s −1 pc −1 for respectively random and cube noise. 5 Evaluating uncertainties on cloud rotation The same operations were done with the observed gradients, creating the same clouds but injecting the velocity gradient deduced from the observations for each cloud. Should there be a link between size or shape and the velocity gradient deduced from calculating the first moment, the link would be preserved in these tests. Figure 14 shows the distribution of the retrieved gradients. The clouds were created with the observed gradients and shapes and then noise was added and the gradients remeasured. This can be directly compared with Fig. 10 . The distribution in Fig. 14 is of course wider because the gradients from Fig. 10 have been injected and then noise added. The process was repeated with T max = 100 mK and the distribution is significantly wider but the prograde-retrograde asymmetry is nonetheless preserved. We are thus confident that the gradients retrieved reflect the true distribution of velocity gradients, although the distribution is likely broadened by the presence of the noise in the datacube. Fig. 14 Like Fig. 13 except that the observed gradients (see Fig. 10 ) have been injected, noise added, and the gradients recovered. Black line shows the distribution of the gradients for purely random noise. The red line shows the distribution when noise from signal-free regions of the cube. As in Fig. 13 , T


O O O O O O O O O O O O O O O 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 I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-Citation I-Citation I-Citation I-Citation O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength 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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength 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 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 B-Tag O O B-Tag O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength 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 B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-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 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 B-Citation I-Citation I-Citation I-Citation I-Citation 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 O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula 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 B-Model O O O O O O O O O O O O O

) resolved the molecular line emission from above and below the disk, we could not detect the emission in the southwest of the disk, possibly due to the lower sensitivity and resolution. Figure 3. (a) Peak positions of the SiO v = 1 and 2 J = 1–0 maser emission (gray circle) and the SiO v = 4 J = 11–10 emission (colored circle). (b) Peak positions of the SiO v = 1 and 2 J = 1–0 maser emission (gray circle) and 29 SiO v = 2 J = 11–10 emission (colored cross). The color of the SiO v = 4 and 29 SiO v = 2 emission represents the line-of-sight velocity. The cross at (0, 0) corresponds to the position of Source I (Goddi et al. 2011 ). The 29 SiO v = 2 J = 11–10 emission is mainly found between the eastern and northern arms of the X-shape and also in the vertical structures from the disk, such as blueshifted emission along the eastern arm and high-velocity components near the disk midplane. It confirms that the 29 SiO v = 2 J = 11–10 emission traces the rotating outflow as other SiO line emission (e.g., Plambeck Wright 2016 ; Hirota et al. 2017 ; Ginsburg et al. 2018 ). On the other hand, the v = 4 J = 11–10 emission is only found in a line connecting the base of eastern and northern arms of 43 GHz SiO masers, which resembles the H 2 O emission in Ginsburg et al. ( 2018 ). The morphology and velocity gradient of this line suggests that it traces both the surface of the disk and the base of the outflow. 4.2. The Kinematics of <italic>v</italic> = 4 <italic>J</italic> = 11–10 Line Emission Figure 4 shows the position–velocity diagrams of the SiO v = 4 J = 11–10 and 29 SiO v = 2 J = 11–10 emissions along the axis with the position angle of 140° (Hirota et al. 2014 ). The spatial distribution and the velocity gradients along the major axis of the continuum emission indicate that the SiO v = 4 J = 11–10 emission is related to the rotating disk. However, the absence of the high-velocity components at the central position suggests that the high-frequency SiO line emissions are associated with a rotating ring structure rather than a filled disk. To find the parameters of the ring structure, we fit the observed rotation curve with a simulated rotation curve using a simple model similar to those of Hirota et al. ( 2014 ) and Plambeck Wright ( 2016 ). We assumed a thin, edge-on ring structure in Keplerian rotation around a central star. In each position, the gas with uniform temperature and density emit the optically thin emission. The thermal width of 3 km s −1 was assumed. We set the systemic velocity of 6 km s −1 , which is the central value of the observed velocity range. We simulated the rotation curve for a range of inner radii r in and outer radii r out with central masses ranges of 5 to 15 M ⊙ . As a result, we assess that r in = 12 au, r out = 26 au, and a central mass of 7 M ⊙ for the best parameter. Figure 4. (a) Position–velocity diagram of the SiO v = 4 J = 11–10 emission. The Keplerian velocity curves for 10 M ⊙ (gray) and 15 M ⊙ (black) are


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A.G.E. was supported by grant NNX10AT78G from NASA’s Goddard Space Flight Center. N.H.B. was supported by a STFC grant. We thank Gary Zank for useful comments.


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A.J.D. is supported by a Porat Fellowship at Stanford University. We thank Evan Kirby and Andrew Wetzel for stimulating science discussions, and for providing very useful comments on an earlier version of this manuscript. We also thank an anonymous referee for providing valuable comments that improved the quality of our paper. The simulations used were run using computational resources at SLAC; we gratefully acknowledge the support of the SLAC computational team. This research also used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02–05CH11231. A.J.D. thanks J.B.C. for his impeccable timing and invaluable insight.


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 B-Model I-Model 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 I-Citation 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

strength of the magnetic field (Gehman, Adams Watkins 1996b ). The purpose of this paper is to add the effect of the ambipolar diffusion in the gravitational instability of filamentary MCs with magnetic field using non-ideal MHD. The outline of this article is as follows: in Section 2 , we present the basic MHD equations, which include the induction equation with an ambipolar diffusion term. We also describe the physical assumptions that are applied to the system. The unperturbed state of the system is introduced in Section 3 . In Section 4 , we perform linear perturbation analysis and then describe briefly the boundary conditions (BCs) and numerical method which are used to solve the perturbed equations in Sections 5 and 6 , respectively. We express the results in Section 7 . In Section 8 , we discuss our findings and finally, we summarize our conclusions in Section 9 . 2 MHD EQUATIONS WITH AD If the degree of ionization in a medium is very low, the force associated with the pressure gradient of the ions and their gravitational force could be neglected in comparison with the Lorentz and ion-neutral drag force. Accordingly, the Lorentz force and the drag force exerted by the neutrals on the ions could virtually balance each other. The equations of MHD in this so-called strong coupling approximation (Shu 1983 ) with self-gravity and AD term are as follows: where is the neutral gas density, is the fluid velocity, p is the pressure, is the gravitational potential and is the magnetic field strength, whereas A is the AD coefficient indicated by (Shu, Adams Lizano 1970 ), where i and are the ions density and the drag coefficient, respectively. Draine, Roberge Dalgarno ( 1983 ) proposed the value of 3.510 13 cm 3 g 1 s 1 . Due to the cosmic radiation, MC are partially ionized. Elmegreen ( 1979 ) showed that one could approximate the relation between ions and neutrals density as where C is a constant equals to C 310 16 cm 3/2 g 1/2 . So the equation ( 5 ) can be recast as where is a substitution for C and equals to 10.510 3 cm 3/2 g 1/2 s 1 . These assumptions lead to a fractional ionization of n i / n n 10 7 for n n 10 4 cm 3 , that is completely centred within 10 6 to 10 8 of observation of MC cores. By making the equations dimensionless, and A become 11.465 and 0.007 3/2 , respectively (see Section A ). The aforementioned fluid equations, are closed by addition of one equation of state, which is considered here to be isothermal, i.e. and, where c s is the isothermal sound speed which is about 0.2kms 1 for an MC with temperature of 10K and mean molecular weight of 2.36. The above values for the number density and sound speed, will give the value of 0.7 for a typical MC threaded by a magnetic field of 10G. 3 UNPERTURBED STATE The unperturbed fluid is supposed to be in the hydrostatic equilibrium state ( = 0). We make the assumption that the initial magnetic field does not have any role in stability of the filament. Therefore the density and


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Acknowledgements C-HL is indebted to the referee, whose comments greatly improved the manuscript. Based on observations (GN-2017B-FT-4) obtained at the Gemini Observatory and processed using the Gemini iraf package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). The author wishes to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community and is very fortunate to have the opportunity to conduct observations from this mountain.


O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O 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 B-CelestialObject O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObject B-CelestialObject O B-CelestialObject 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation 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 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 B-Citation I-Citation O B-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 O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation 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 O O O O O O O O O O O O O O O O O O O O O O O O O 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-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

was larger than the current discovered value, increasing the critical migration rate. The rate of migration can slow as the planets approach the inner edge of the disc due to the decrease in surface density (Kley Haghighipour 2014 , 2015 ). This slowing makes the likelihood of scattering more probable since it increases the time of interaction with the N:1 resonances. Additionally, if there exists a lack of circumbinary planets and especially multiplanet systems, this might be indicative of slow migration rates. 6.3 Application to the Pluto and Charon binary With a mass ratio of 0.1184, the dynamics around Pluto and its largest moon, Charon, behaves similar to the binary star systems described above. The binary dwarf planet has four discovered circumbinary moons: Styx, Nix, Kerberos, and Hydra. These moons are near to the 3:1, 4:1, 5:1, and 6:1 resonances with Charon and are therefore close to resonances with each other. None of these resonances are currently active according to the best estimates for the orbits and masses of the moons (Showalter Hamilton 2015 ). The very low eccentricities of all the bodies in the system makes active resonances unlikely due to the small resonance widths. Showalter Hamilton ( 2015 ) suggests the masses of the circumbinary moons are larger than predicted by albedo alone (Brozović et al. 2015 ) due to a librating three-body resonant angle for higher masses. While moon–moon resonances may help constrain the mass of the moons, the location of the moons near the N:1 resonances remains unexplained. The very low eccentricity of the moons suggests that in situ formation is likely (Brozović et al. 2015 ) but the collision from which the binary formed is not thought to have created enough leftover material at large enough distance to form the moons at their current location (Canup 2005 , 2011 ). Bromley Kenyon ( 2015 ) demonstrate that processes like physical collisions and collective gravitational scattering can expand the debris in the system to larger distances, facilitating formation. Resonant transport cannot explain the migration of all moons simultaneously since different N:1 resonances require different binary eccentricity for migration (Lithwick Wu 2008 ; Cheng et al. 2014 ). Resonant transport due to an eccentric Charon would also excite the eccentricity of the moons. Even outside of a resonance, the secular effects can excite the eccentricity of the moons above their current values. Formation models that predict a high initial eccentricity for the binary need to explain the current low eccentricity of the moons and overcome the likelihood of instabilities from the number of resonances in an eccentric system. Previous efforts have used stability criteria to constrain the maximum eccentricity of Charon as well as the maximum mass of the moons (Youdin, Kratter Kenyon 2012 ; Smullen Kratter 2017 ). The instabilities described here can also limit the possible masses and formation history of the moons. The potential


O O 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 I-Survey B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

more massive galaxies, LTGs have the same negative gradient regardless of mass. Metallicity gradients steepening for more massive ETGs is consistent with other studies from the MaNGA survey, Oyarzún et al. ( 2019 ) (after noting that they probe larger radii) and Lacerna et al. ( 2020 ). For C and Mg, at low masses the radial gradients are slightly positive, and they become shallower or slightly negative as galaxy mass increases. The gradients are independent of the type of galaxy. For the heavier elements, ETGs display similar gradients at all masses for [Ca/Fe], and become slightly negative as mass increases for [Ti/Fe]. High-mass spirals show larger variation in gradients with mass. The overall trend for Ca is unclear, but for Ti, the gradients become more negative as galaxy mass increases, like ETGs. N gradients go from slightly positive or shallow at low masses, to negative at high masses for both types. Finally, for Na, ETGs display a slight steepening in the gradient for the most massive galaxies while gradients are consistent for both types at intermediate masses. This suggests that the mass or some other property drives Na abundance variation rather than galaxy type. Briefly, we find several interesting features while studying these radial gradients in these parameters for different galaxy types. The most striking difference is in the age gradients. While ETGs display relatively flat age gradients, at all masses, LTGs show negative age gradients, which become steeper with mass. Metallicity behaves exactly the opposite way, with negative gradients steepening with mass for ETGs and negative gradients overall for LTGs. These properties could provide clues into how these different galaxy types and their stellar populations assembled and evolved. C and Mg abundances and radial gradients are remarkably similar for both types, suggesting that star formation time-scales are independent of galaxy type but rather depend on some other physical property i.e. galaxy mass or velocity dispersion. Na behaves in the same way as metallicity, showing negative gradients for LTGs, and negative gradients that steepen with mass for ETGs. We explore relations with velocity dispersion in the following section. 3.2.4 Local versus global relationships Along with radial gradients, it is informative to look at trends with velocity dispersion. We plot the spatially resolved parameters for both galaxy types, in different mass bins, so that we can identify local deviations (radial) from the global relation with velocity dispersion. The increased sample size and inclusion of different morphologies considerably expands our parameter space compared to P19, section 3.3. Fig. 18 shows all our derived stellar population parameters against the velocity dispersion. These velocity dispersion profiles shown in Fig. 5 have been calculated by combining the dispersions from the individual spectra belonging to each bin. The symbols represent three key pieces of information: (i)


O O O O O O O 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 B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation 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 B-Software O O O O O 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 O O O O O O O O O O O O O O O O O O O O O B-Database I-Database I-Database O O O B-URL O O O O O O O O O O O O O O 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 O B-Model I-Model I-Model I-Model 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O 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-Citation I-Citation I-Citation I-Citation 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

and discuss the future work needed to improve our understanding of r -process enrichment at high redshifts. 2. Method To model the formation of DNSs, we use published results from the StarTrack (Belczynski et al. 2002 , 2006 , 2008 ) population synthesis code. These simulations are described in detail in Dominik et al. ( 2012 ) and include three major improvements over previous StarTrack versions with regards to stellar winds (Belczynski et al. 2010 ), common envelope (CE) formulation, and compact object formation (Fryer et al. 2012 ). We have analyzed the intermediate data from these simulations that are publicly available. 7 7 The public data from the Synthetic Universe Project are available at: www.syntheticuniverse.org . We briefly summarize the method in this section. Each of the models we analyze has 2 × 10 6 binaries, initialized by four different parameters: (i) the primary star’s mass M 1 ; (ii) the mass ratio q = M 2 / M 1 , with M 2 being the mass of the secondary star; (iii) the semimajor axis a of the binary; and (iv) the eccentricity e . The masses of the primary are drawn from the Kroupa initial mass function (IMF) from 5 to 150 M ⊙ . The mass ratio is assumed to have a flat distribution between q = 0–1 with the minimum mass of the secondary considered to be 3 M ⊙ . The distribution of initial binary separation is assumed to be flat in log( a ), and the eccentricity is drawn from thermal equilibrium distribution Ξ( e ) = 2 e , with e ranging from 0 to 1. 2.1. Model Variations A number of parameters are varied among the models, which are meant to parameterize the uncertainties in our understanding of binary neutron star formation. The first parameter that is varied deals with the behavior of binaries when a star enters into a CE with a Hertzsprung gap (HG) donor star; during a CE, a star enters the envelope of its companion, exchanging orbital energy to unbind the donor’s envelope. For giant stars, with a clear core–envelope boundary, the end result of this process (so long as there is enough orbital energy available to keep the system from merging) is a closely bound binary comprised of the accretor star and the giant star’s core. However, HG stars lack well-defined cores, and studies are inconclusive as to whether binaries entering into a CE during this phase can survive without merging (Deloye Taam 2010 ). We note that allowing HG stars to survive a CE significantly increases the merger rates of double black hole binaries to a level that exceeds current LIGO constraints (Belczynski et al. 2007 ), although it matches NSM rate estimates (Chruslinska et al. 2018 ). Two different submodels, A and B, have been analyzed, and treat CE events with HG donor stars differently. Submodel A treats HG stars such that a core could be distinguished from an envelope in their evolutionary phase, hence a successful CE ejection is possible. Submodel B assumes any system entering into a CE with a HG donor will merge. As a second parameter, the


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The Swedish 1-m Solar Telescope is operated by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. We made use of resources provided by the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Centre North at Umeå University. JdlCR is supported by grants from the Swedish Research Council (2015–03994), the Swedish National Space Board (128/15) and the Swedish Civil Contingencies Agency (MSB). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (SUNMAG, grant agreement 759548). This research was supported by the CHROMOBS and CHROMATIC grants of the Knut och Alice Wallenberg foundation. This research has made use of NASA’s Astrophysical Data System.


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

from CO line emission ( Olofsson et al. 2002 ; González Delgado et al. 2003 ), as well as a linear fit to the observed values. These observed wind properties can be compared with wind properties derived from the model grid in order to evaluate what combinations of seed particle abundance and piston amplitude best reproduce the observational data. Constraining the pulsation period in the model data to less than 800 days to fit the periods of the observed sample, we found that models with log( n d ∕ n H ) = − 14.5 and u p = 2.0 give the best agreement with observations. The bottom panel of Fig. 12 shows the distribution of the differences between model mass-loss rates and the expected mass-loss rates from the linear fit to the observed values. The middle panel of Fig. 12 shows mass-loss rate as a function of pulsation period for the same observational sample as shown in the top panel. One way to asses how sensitive the mass-loss rates are to the pulsation mode is to look at the group of stars indicated by blue symbols in Fig. 12 . These stars are semiregular/irregular variables (SRVs), probably pulsating in the first overtone as indicated by their short pulsation periods. Even though these stars presumably pulsate in first overtone, they follow the same correlation between mass-loss rates and wind velocities as the fundamental mode pulsators (red symbols). This is clearly seen in the top panel of Fig. 12 , which shows the mass-loss rates and wind velocities for stars pulsating in different modes. Assuming that our reasoning about the stars with shorter periods being first overtone pulsators is correct, it seems that differences in the pulsation modes may not notably affect the mass-loss rates in M-type AGB stars. 5.2 Trends at the longest pulsation periods Figure 13 shows the mass-loss rate as a function of pulsation period for all models in the grid with a stellar wind. Also shown are observed mass-loss rates estimated from CO-line emission against pulsation periods for M-type AGB stars from Olofsson et al. (2002) and González Delgado et al. (2003) , as well as a sample of OH/IR stars taken from De Beck et al. (2010) and He Chen (2001) , with mass-loss rates estimated from CO-line and dust emission, respectively. In model data and in observations the mass-loss rates level off at longer pulsation periods, even if the models do not reach quite as high as the observed mass-loss rates. However, it is important to remember that the grid is currently biased against models with very high mass-loss rate, where dust formation and wind acceleration is very abrupt, resulting in non-converging dynamical structures (see Sect. 4.1 ). The exact level where the mass-loss rate flattens in the model values might also depend on the inner boundary conditions of the models, for example the value of f L or the phase-shift of the luminosity variation with respect to the radial variation of the surface layers ( Liljegren et al. 2016 , 2017 ). It might also be


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O O O 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-Location I-Location B-Grant I-Grant I-Grant O O B-Grant I-Grant O O O B-Grant B-Person O B-Person O O O O B-Organization I-Organization B-Grant I-Grant I-Grant I-Grant I-Grant O O O B-Person O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Person O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person O O O O B-Fellowship I-Fellowship I-Fellowship I-Fellowship B-Organization I-Organization O O B-Organization B-Telescope B-Grant O B-Grant B-Person O O O O O O B-Grant I-Grant O O O B-Person I-Person O O O B-Person O B-Person I-Person O B-Person I-Person I-Person O O O

The authors thank the anonymous referee for suggestions that led to important improvements in the paper. The research leading to these results has received funding from the European Union’s Horizon 2020 Programme under the AHEAD project (grant agreement n. 654215). A.T. and E.N.E. would like to thank Boğaziçi University Research Fund Grant Number 13760 for financial support. A.T. acknowledges support by the Study in Italy Grant, Research Under Academic Supervision. F.T. acknowledges support by the Programma per Giovani Ricercatori – anno 2014 “Rita Levi Montalcini”. M.G. is supported by the Lyman Spitzer Jr. Fellowship (Princeton University) and by NASA Chandra GO7-18121X and GO8-19104X. R.S. acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.0. The authors thank Massimo Cappi for useful discussions. A.T. thanks François Mernier and Jelle de Plaa for useful discussions.


O O O O O O O O O O O O O O O O O O B-Person I-Person B-Person I-Person B-Person I-Person O B-Person I-Person O O O B-Person O B-Person O O O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant O O O O O O B-Organization O B-Person O O B-Organization O B-Grant B-Person B-Person O B-Person O O O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant O O O O O O B-Grant O B-Person O B-Person O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant B-Person O B-Person O O O B-Organization O O B-Grant B-Person O O O B-Observatory O B-Grant O B-Organization O O B-Grant B-Person O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person O O O B-Grant I-Grant I-Grant I-Grant O O B-Location B-Organization I-Organization I-Organization I-Organization I-Organization O O B-Grant B-Person O B-Person O O O O O B-Grant I-Grant I-Grant I-Grant O O O B-Grant I-Grant O B-Grant B-Person O O O O B-Grant I-Grant I-Grant O B-Grant B-Person O O O O B-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant I-Grant B-Person O O O O O B-Organization I-Organization I-Organization I-Organization I-Organization B-Organization O O B-Grant B-Person O O O O B-Organization I-Organization O B-Grant O B-Grant O O O O O O O B-Observatory O B-Identifier O B-Identifier B-Observatory O O O O B-Observatory O O O O B-Organization O O B-Organization O O O B-Organization O B-Organization O B-Organization B-Location O B-Organization B-Location I-Location I-Location O O O O B-Location I-Location I-Location O B-Observatory I-Observatory I-Observatory O O O B-Observatory B-Organization O B-Observatory O O O O O O B-Organization B-Telescope I-Telescope I-Telescope O O O O O O O B-Organization I-Organization I-Organization I-Organization B-Organization O O O O B-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization I-Organization O B-Organization O B-Grant I-Grant

We are grateful to the anonymous referee for constructive comments that helped us improve the manuscript. We thank Luis Colina, Alvaro Labiano, Leonard Burtscher and Daniel Asmus for useful discussions. A.A.-H. and I.G.-B. acknowledge support from the Spanish Ministry of Science, Innovation and Universities through grant AYA2015-64346-C2-1-P, which was party funded by the FEDER program. A.A.-H. also acknowledges CSIC grant PIE201650E36. A.A.-H., S.G.-B. and A.U. acknowledge support from the Spanish Ministry of Science, Innovation and Universities through grant PGC2018-094671-B-I00, which was party funded by the FEDER program. S.G.-B. and C.R.A. acknowledge the Spanish Plan Nacional de Astronomía y Astrofísica under grant AYA2016-76682-C3-2-P. M.P.-S. and D.R. acknowledge support from STFC through grant ST/N000919/1. T.D.-S. acknowledges support from ALMA-CONICYT project 31130005 and FONDECYT regular project 1151239. S.F.H. is supported by European Research Council Starting Grant ERC-StG-677117 DUST-IN-THE-WIND. C.R.A. also acknowledges the Ramón y Cajal Program of the Spanish Ministry of Science and Technology through project RYC-2014-15779. S.R. and M.V. gratefully acknowledge support from the Independent Research Fund Denmark via grant numbers DFF 4002-00275 and 8021-00130. L.K.H. acknowledges funding from the INAF PRIN-SKA 2017 program 1.05.01.88.04. C.R. acknowledges support from the CONICYT+PAI Convocatoria Nacional subvencion a instalacion en la academia convocatoria año 2017 PAI77170080. D.R. acknowledges the support of the Science and Technology Facilities Council (STFC) through grant ST/P000541/1. AU acknowledges support from the Spanish MINECO grants ESP2015-68964 and AYA2016-79006. This paper makes use of the following ALMA data: ADS/JAO.ALMA.2016.1.00254.S and JAO.ALMA.2016.1.01236.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.


O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 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 O O O B-Citation I-Citation I-Citation I-Citation 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 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 B-Software I-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula 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 B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation 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 B-Formula I-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

the density and dynamic range of the frequency sampling may be computationally attainable via offloading this part of the workload to multicore architectures within the MHD code in the future, even though the attainable speedup may in fact be limited by restrictions in the memory bandwidth. Even without these technological advancements, we have demonstrated that the spectral TF method of Brandenburg et al. ( 2008 ) and its time-dependent counterpart introduced by Hubbard Brandenburg ( 2009 ) are highly useful tools and indeed produce interesting results for advanced applications, such as our investigation of realistic high-Mach-number turbulence driven by SNe in the multiphase ISM, where mean-field effects occur naturally as an outcome of the combined action of gravity and the Coriolis force. Notwithstanding the challenges of understanding the quenching in the strong-field limit (Gressel, Bendre Elstner 2013a ) and covering the parameter space required by genuinely global models (Gressel et al. 2013b ), combining these approaches (as pioneered by Rheinhardt Brandenburg 2012 ) will allow to derive the most comprehensive mean-field closures achievable. APPENDIX A: BENCHMARK USING HELICAL FORCING IN THE KINEMATIC LIMIT We verify our implementation of the spectral TF method for the simple test case of helically forced (mildly compressible) hydrodynamic turbulence, that is, in the strict kinematic limit of the mean-field dynamo, where the back reaction via the Lorentz force is ignored. To this end, we perform a hydrodynamic DNS in a cubic domain, solving along with the TF induction equation ( 11 ), and using as the equation of state. We closely follow Brandenburg ( 2001 ) and adopt a helical forcing function , with the normalization with respect to the numerical time-step, δ t , and f 0 ≲ 1 a dimensionless forcing amplitude. The forcing obtains its helical character via with an isotropically sampled arbitrary unit vector required for constructing a vector . Note that , and – being an eigenfunction of the curl operator – we have , such that the helicity of the forcing, , is positive definite. The wave vector is chosen from a list of 66 (158) pre-computed discrete wave vectors obeying the periodicity of the box, and that have , with k f = 3 ( k f = 5), and Δ k = 1/4. Note that both and the random phase φ( t ) are updated after each time-step, δ t , of the DNS. A1 Non-locality in space In Fig. A1 , we show the spectral dependence of and , adopting units of and . Both curves are very well approximated by a Lorentzian with a characteristic length scale . This is in contrast to the results of Brandenburg et al. ( 2008 ), who find k c = k f for , and for , respectively, for a very similar test case with k f = 5 and Rm = 10. While the qualitative trend is the same, the difference is much less pronounced in our case. Figure A1. Spectral amplitudes for the α effect (top panel) and turbulent diffusion, η T (lower panel) for helically forced isothermal


O B-Citation I-Citation O B-Citation I-Citation 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 B-Citation I-Citation I-Citation I-Citation 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 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 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 O O O 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 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 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 B-Model O O O O O O O O O O O O O O B-Wavelength O B-Wavelength 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 B-Model 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-Wavelength O O B-Wavelength I-Wavelength I-Wavelength 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 O O O O O O O O B-Model O O O O O O O O O O B-Model O O O O B-Wavelength 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 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 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 B-Model 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 B-Wavelength O O O O O O O O O O O O B-Wavelength O O O O

spacetime (Mattingly 2005 ; Amelino-Camelia 2013 ). Any possible violation of Lorentz invariance would have far-reaching consequences for our understanding of the nature, i.e., the pillars of modern physics will be shocked and new physics is needed (Cao et al. 2018a ). Therefore, the pursuit of testing the possible LIV at much higher precision has continued in recent decades, concerning various astrophysical and cosmological observations. Formulating and quantitatively interpreting the test of LIV is another question: an interesting proposal, in this respect, has been formulated in the frameworks of many QG: at small spatial scales, a foamy structure of spacetime predicted by QG theory will interact only with the high-energy photons (Amelino-Camelia et al. 1998 ; Amelino-Camelia Piran 2001 ). In this case, the speed of light is varying at a different energy range in vacuum, i.e., the high and low-energy photons will not reach us at the same time. More specifically, the deformed velocity of light usually takes the form , where E QG is effective QG energy scale, s ± , is a dimensionless parameter depending on the particular QG model, and c is the limiting speed of light on low-energy scales, respectively. On low-energy scales, E ≪ E QG , the effect of LIV will be more obvious and the high-energy photons propagate slower than low-energy photons. Although the QG effect is expected to be very weak (since E QG is typically close to the Planck energy scale), some effects of LIV are expected to increase with energy and over very long distances due to cumulative processes in photon propagation. Therefore, astrophysical searches provide sensitive probes of LIV and its potential signatures, such as the energy-dependent time delay and many other phenomena. Gamma-ray bursts (GRBs), the most luminous astrophysical events observable proposed as distance indicators at high redshift (Pan et al. 2013 ; Cao Zhu 2014 ), create such an opportunity to test the LIV, through the precise measurement of unprecedented very-high-energy photons. Compared with commonly used supernova Ia, the advantages of GRBs lies in their high-energy photons in the gamma-ray band (from KeV to GeV), short spectral lags, and the propagation distance at cosmological scales. Such a natural laboratory provides a possibility of testing LIV, through the well-measured time delays between light curves in different energy bands caused by the LIV. Recently, some advances have been made concerning the limits on LIV using different samples of GRBs (Ellis et al. 2003 , 2006 ; Rodriguez Martínez Piran 2006 ; Jacob Piran 2007 ; Pan et al. 2015 ; Zhang Ma 2015 ; Wei et al. 2017 ; Zhang et al. 2018b ; Zou et al. 2018 ). The original idea of such studies of possible LIV constraints can be traced back to the papers of Ellis et al. ( 2003 ), which developed a method to analyze samples of GRBs with different redshifts and energy bands, by extracting time-dependent features from the GRB signals. It is worth


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This investigation is based on observations obtained with XMM-Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This research made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. We acknowledge support from NASA ADAP grant NNX15AE63G. We thank Eric Gotthelf for valuable discussions and the anonymous referee for several helpful comments.


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

ways. They facilitate the comparative analysis of measurements in a pairwise fashion. One may use them with n measurements of a particular variable to generate n ( n −1)/2 comparisons for each pair of data. The benefits are twofold: (1) one can test how well each pair of data fits the models, and (2) assess how closely the published error bars fit a normal distribution, thereby providing some indication of possible contamination by correlated systematic uncertainties. Zheng et al. ( 2016 ) recently used such an approach to conclude that the stated errors in cosmic chronometer data are strongly non-Gaussian, suggesting that the quoted measurement uncertainties are almost certainly not based exclusively on statistical randomness (see also Leaf Melia 2017 ). As we shall see, the diagnostic Δμ( z i , z j ) is expected to be zero if the model being tested is the correct cosmology. To allow for possible non-Gaussianity in the published errors, we shall use both weighted-mean and median statistics to determine the degree to which each model's distribution of Δμ( z i , z j ) values is consistent with this null result. So while Wei, Wu Melia ( 2016 ) optimized the overall ΛCDM and R h = ct parametric fits to the H ii galaxy Hubble diagram, here we will test the consistency of each fit with individual pairs of data. We will begin with a brief description of the data in Section 2 , and then define and apply the diagnostic Δμ( z i , z j ) in Section 3 . The outcome of our analysis will be discussed in Section 4 , followed by our conclusions in Section 5 . 2 OBSERVATIONAL DATA AND METHODOLOGY We base our analysis on the methodology described in Chávez et al. ( 2012 , 2014 ) and Terlevich et al. ( 2015 ), using their total sample of 156 sources, including 107 local H ii galaxies, 24 GEHRs, and 25 high- z H ii galaxies. The correlation between the emission-line luminosity and the ionized gas velocity dispersion may be written as (Chávez et al. 2012 ; Chávez et al. 2014 ; Terlevich et al. 2015 ) where α is the slope and the constant κ represents the logarithmic luminosity at log σ(Hβ) = 0. As noted, previous applications of this relation have produced a very small scatter in the correlation for L (Hβ), making it a viable luminosity indicator for cosmology. But one cannot completely avoid its cosmology dependence because the Hβ luminosity is calculated using the expression where D L is the model-dependent luminosity distance at redshift z and F (Hβ) is the reddening corrected Hβ flux. From equation (1), we may then obtain the distance modulus of an H ii galaxy according to with an associated error in terms of σ log σ and σ log F , these being the 1σ uncertainties in log σ(Hβ) and log F (Hβ), respectively. This is to be compared with the theoretical distance modulus as a function of the cosmology-dependent luminosity distance D L . In ΛCDM, the luminosity distance may be written where p de = w de ρ de is the dark-energy equation of state; radiation is


We are grateful to the referee for constructive and useful comments that improved this paper.


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transformation back to the lab frame then gives an electric field and an associated electrostatic potential in the lab frame. We assume here that the jet velocity greatly exceeds the difference J / ne between electron and ion drift velocities, so the jet velocity is nearly the same as the Harris-frame velocity. This means that in the jet frame the electric field and associated electrostatic potential are zero. The nonrelativistic Lorentz transformation relates the jet-frame electric field E ′ to the lab-frame electric field E byBecause the Harris construction in the jet frame gives E ′ = 0, and assuming a jet velocity Equation ( 20 ) gives the lab-frame radial electric field asUsing Equation ( 14 ), the lab-frame radial electric field can then be expressed asSince a gauge with has been assumed, the radial electric field reduces toCombining Equations ( 22 ) and ( 23 ) and integrating gives the lab-frame electrostatic potential asso using Equation ( 16 ) we have 5. Construction of the Electric and Magnetic Field Vectors We now have sufficient information to construct the complete lab-frame electric field. Because of axisymmetry, the electrostatic potential V is independent of φ , so the toroidal electric field isthis electric field component is missing in the static models used by Mestel ( 1961 ), Blandford Payne ( 1982 ), Lynden-Bell ( 2003 ), and Pudritz et al. ( 2012 ); it also differs from the singular E φ proposed in Contopoulos ( 1994 ). The axial electric field in the jet region iswhere Equation ( 24 ) has been used and the last line comes from Equation ( 17 ). The radial electric field using A r = 0 is given by Equation ( 21 ) asThe lab-frame magnetic field is There is no lab-frame electric field parallel to the magnetic field sincewhere Equation ( 9 ) has been used. 6. Demonstration That the Time-dependent Induction Equation Is Satisfied Because ψ is time dependent, both the electric and magnetic lab-frame fields are time dependent. This contrasts with the previous analytic models (e.g., Mestel 1961 ; Sauty 1994 ; Beskin 2010 ; Contopoulos 1994 ) where the jet fields were assumed to be independent of time and the lengthening of the jet was not taken into account. These previous analytic models assumed that the magnetic field is independent of time, so the induction equationis reduced to , which forces U × B to be the gradient of a potential. In contrast, time dependence of the magnetic field is retained here, so U × B is not the gradient of a potential. Using , we see thatsoDirect calculation of the time derivative of Equation ( 29 ) giveswhich on invoking Equation ( 9 ) becomesComparison of Equations ( 33 ) and ( 35 ) shows that the time-dependent induction equation, Equation ( 31 ), is exactly satisfied. Various points of view have been expressed on how the detailed flow moves with respect to the magnetic field. According to one point of view (e.g., Blandford Payne 1982 ), the predominant flow is along magnetic field lines (beads on a


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This work was supported by the São Paulo State Science Foundation, FAPESP, and the Brazilian National Research Council, CNPq. This work has made use of the facilities of the Computation Center of the University of São Paulo (LCCA-USP) and of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase was made possible by the Brazilian agency FAPESP (grant 2009/54006-4) and the INCT-A. The authors are grateful to the anonymous referee for numerous suggestions and corrections to this paper.


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field, S , exists near the pole. This result is due to spatial symmetry. Figure 4 shows the time-averaged toroidal flux S in . The toroidal magnetic field S decreases gradually by the radius from the star. The maximum value of S falls inside the light cylinder radius. The maximum value does not match the radius of the light cylinder. The concentrated poloidal current area is the light cylinder interior. Figure 4. Time-averaged toroidal magnetic flux function S for the center of the star. . The toroidal magnetic field, S , increases due to large electrical conductivity . In our model, the value increases with the increase of r Equation ( 49 ). The toroidal magnetic field increase proportionally to electrical conductivity. Figure 3 (c) shows the electromagnetic field for , five times bigger than the case. In addition, this field area spreads out. This is in contrast to the poloidal magnetic field G , which is not changed by the electrical conductivity. 4.1.3. Electrostatic Potential Φ Figure 5 (a) shows the electrostatic potential contour of . From to 0.08, contours are equally spaced for each 0.01. In the numerical calculation, the electrostatic potential on the star’s surface does not change with time because of the boundary conditions Equation ( 36 ). Blue lines in Figure 5 (a) spread out from the surface boundary and extend in the radial direction. Blue lines are connected vertically toward the equatorial plane because the field’s boundary condition has only an R component in this plane. Figure 5 (b) shows the electrostatic potential of the spatial region from Z = 0 to 30. Here, the contour line is drawn in increments of 0.005 from . Figure 5. Electrical potential Φ for . The red line is the light cylinder radius R = 5. The black line is the equal value of the poloidal-magnetic flux function G . The black dotted line shows a dipole-magnetic flux equal to the value of the function. The bottom panels show zoomed-out views. The densities of the equipotential lines are proportional to the strength of the electric field, which is strong on the star surface and decreases as one goes radially outward. No electric field dominant () area exist in . The magnetic field in the obtained results is always stronger than the electric field. Figure 5 (c) is the drawn contours of the electrostatic potential cases. Electrical conductivity increases along with the electrostatic potential of the equatorial plane. However, the difference of electrostatic potential of the equatorial plane between the and is about 10%. When the electrical conductivity is different from this case, the overall structure of the solution remains similar. For example, electrical conductivity is increased in a toroidal magnetic field, but little shape change was observed about the potential. 4.1.4. Poloidal Current jp Current density is divided into poloidal and toroidal components. The poloidal component of current density j p consists of j r and : Figure 6 (a) plots


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ACKNOWLEDGEMENTS ADM holds an Australian Research Council (ARC) Future Fellowship (FT160100206), and acknowledges support from ARC Discovery Project DP150103294. We would like to thank the anonymous referee for their thorough reading of this manuscript and for their helpful suggestions for improvement. This paper is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada–France–Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. This paper is also based, in part, on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). These observations were obtained via programmes GN-2014B-Q-26 and GN-2015B-Q-17.