a year. The second scenario pointed out above implies the presence of an unseen massive perturber, i.e. a trans-Plutonian planet. Close encounters between minor bodies and planets can induce fragmentation directly via tidal forces (e.g. Sharma, Jenkins Burns 2006 ) or indirectly by exciting rapid rotation (e.g. Scheeres etal. 2000 ; Ortiz etal. 2012 ). Alternatively, wide binary asteroids can be easily disrupted during close encounters with planets. The existence of wide binaries among the populations of minor bodies orbiting beyond Neptune is well documented (e.g. Parker etal. 2011 ). Wide binary asteroids have very low binding energies and can be easily dissociated during close encounters with planets (e.g. Agnor Hamilton 2006 ; Parker Kavelaars 2010 ). Binary asteroid dissociation may be able to explain the properties of this pair of ETNOs, but only if there is a massive perturber well beyond Pluto. In order to test the viability of this hypothesis, we have performed thousands of numerical experiments following the prescriptions discussed by de la Fuente Marcos etal. ( 2016 ) and aimed at finding the most probable orbital properties of a putative perturber able to tilt the orbital plane of the pair 4746402013 RF 98 from an initial angular separation close to zero at dissociation to the current value of nearly 4. These simulations involve N -body integrations backwards in time under the influence of an unseen perturber with varying orbital and physical parameters (per numerical experiment). Our preliminary results indicate that a planet with mass, m , in the range 1020M moving in an eccentric (0.10.4) and inclined (2050) orbit with semimajor axis of 300600au, may be able to induce the observed tilt on a time-scale of 510Myr. Perturbers with m 10M or a 600au are unable to produce the desired effect. Fig. 4 illustrates the typical outcome of these calculations. A detailed account of these numerical experiments will be presented in an accompanying paper (de la Fuente Marcos, de la Fuente Marcos Aarseth, in preparation). The orbital parameters of this putative planet are somewhat consistent with those of the object discussed by Holman Payne ( 2016 ). Super-Earths may form at 125750au from the Sun (Kenyon Bromley 2015 , 2016 ). Our analysis favours a scenario in which 4746402013 RF 98 were once a binary asteroid that became unbound after a relatively recent gravitational encounter with a trans-Plutonian planet at hundreds of astronomical units from the Sun. An alternative explanation involving an asteroid break-up near aphelion, also after a close encounter with a planet, is possible but less probable because it requires an approach at closer range, i.e. 20 planetary radii versus 0.8au, for binary dissociation. Figure 4. Evolution of the angular separation between the orbital poles of the pair (474640) 2004 VN 112 2013 RF 98 for three representative test calculations with different perturbers. Red: a = 549au, e = 0.21, i = 47 and m = 16M . Blue: a

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 B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-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 B-CelestialObject 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 B-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 B-EntityOfFutureInterest 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 B-Citation I-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 B-CelestialObject I-CelestialObject I-CelestialObject 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 O O O O O O O O O O O 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 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 B-Citation I-Citation I-Citation I-Citation I-Citation B-EntityOfFutureInterest O O O O O O B-CelestialObject B-Citation I-Citation I-Citation O B-Citation 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 O O B-CelestialObjectRegion I-CelestialObjectRegion 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 B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O


with metallicity as [O/Fe], or other [ α /Fe] ratios do. Relevant discussion in previous comparison studies should therefore be revisited. In particular, Cooke et al. (2015) showed [O/Fe] as a function of [Fe/H] for [Fe/H] ≲−2, and [S/Zn] as a function of [Zn/H] for [Fe/H] ≳−2, assuming that both abundance ratios were tracing [ α /Fe] versus [Fe/H] in DLAs. By doing this, they observed a ‘knee’ in these abundance ratios where they started to decrease at [Fe/H] ≈−2, analogous to that observed in [ α /Fe] as a function of [Fe/H] in dwarf galaxies. However, it is clear from Fig. 8 that this assumption does not hold (see also Sect. 4 ). Therefore, although a ‘knee’ in [ α /Fe] abundance ratios in DLAs cannot be excluded at this point, further confirmation is needed, preferably using the same probe throughout the entire metallicity range. Fig. 8 Schematic view of the mean trends of [O/Fe] (magenta solid lines) and [S/Zn] (green dashed lines) in the Sculptor dSph as a function of [Fe/H] ( top ) and [Zn/H] ( bottom ). References: [O/Fe]: Hill et al. in prep. [S/Zn]: Skúladóttir et al. (2015a , 2017 ). 6 Conclusions For the first time we have made a detailed chemical comparison between DLAs and stars in dwarf galaxies by using the volatile elements S and Zn without the help of any other element as proxy. This was made possible by recent measurements of S and Zn abundances in around 80 RGB stars in the Sculptor dwarf spheroidal galaxy ( Skúladóttir et al. 2015a , 2017 ). Most of the stars in this galaxy formed >10 Gyr ago, which correspond to the redshifts at which the majority of DLAs are observed (i.e., z ≳ 1.8). Our analysis of the Sculptor data shows that S abundances behave as expected of an α -element, primarily created by core-collapse SN. However, there is also a non-negligible contribution to S from the yields of type Ia SN. This makes [S/H] a reasonabletracer of the metallicity evolution in galaxies. In low-metallicity stars, S measurements are challenging, so when not available, other α -elements can be used as proxy, preferably: Ca, Ti or the average of the two (see Appendix A ). The lighter α -elements, O and Mg, should not be used as proxies for S since, unlike S, they have negligible contribution from type Ia SN. In light of the available measurements in the Milky Way and its satellite dwarf galaxies, it is clear that [Zn/Fe] is neither the same at a given metallicity in all galaxies probed here, nor constant in a given galaxy throughout its entire evolutionaryhistory. The nucleosynthetic channels that lead to the formation of Zn remain unclear, but the following statement isin agreement with both theory and observations: Type Ia SN are not significant producers of Zn. The formation channels of Zn are therefore not the same as Fe, and the [Zn/Fe] ratio is highly dependent on the star formation history of each system. Thus, using [Zn/H] as a proxy for [Fe/H] is fraught with potential pitfalls. Furthermore, [S/Zn] should not be used as a

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


2013 ), ∼50 days in OJ 287 (Pihajoki et al. 2013 ); on an LTV timescale, QPOs of ∼12 yr, and ∼400 days in OJ 287 (Sillanpää et al. 1988 ; Valtonen et al. 2008 ; Bhatta et al. 2016 ), ∼3 yr in S5 0716+714 (Raiteri et al. 2003 ), ∼8 yr in BL Lacertae (Villata et al. 2004 ), ∼585 days in PKS B0048–097 (Kadler et al. 2006 ), ∼0.92 yr in PKS 1510–089 (Xie et al. 2008 ), 5–6 yr in AO 0235+164 (Raiteri et al. 2008 ), ∼420 days in 1ES 2321+419 (Rani et al. 2009 ), ∼6.3 yr in GB6 J1058+5628 (Nesci 2010 ), ∼10 and ∼6 yr in NRAO 530 (An et al. 2013 ), ∼315 days in PKS 2155–304 (Sandrinelli et al. 2014 ), 2.18 ± 0.08 yr in PG 1553+113 (Ackermann et al. 2015 ), and 5.2 yr in PG 1302–102 (Graham et al. 2015 ). However, the physical mechanism causing these possible periodicities is still unclear and requires further research . The physical mechanism of the quasi-periodic behavior in blazars is still in general controversial and not understood well. Several models have been proposed to interpret this behavior, although a few evidences for QPOs were detected. However, many in the scientific community are deeply skeptical of QPO claims where very few cycles are observed and/or one fails to account for random red noise variability (e.g., Tagliaferri et al. 1996 ; Benlloch et al. 2001 ; Vaughan 2005 ; Vaughan Uttley 2006 ; Espaillat et al. 2008 ; Gaur et al. 2010 ; Press 2013 ). Recently, Vaughan et al. ( 2016 ) reexamined the detection of periodic modulations of 5.2 yr in PG 1302−102 reported by Graham et al. ( 2015 ) and found that a simple stochastic process is preferred over a sinusoidal variation. They also emphasized that such a red noise stochastic process can occasionally mimic the sinusoid-like periodicity in a few-cycle light curve while the periodic candidate can be distinguished from a stochastic process by observing it for ∼5 cycles. Furthermore, we also note that many of the QPO claims mentioned above suffer from incomplete statistical analysis, e.g., periodicity claims in Sandrinelli et al. ( 2014 ), Ackermann et al. ( 2015 ), and Bhatta et al. ( 2016 ) only persist for a small number of cycles ; the baseline PSD in Ackermann et al. ( 2015 ) was difficult to measure and their light curve can be explained by a pure red noise process; Valtonen et al. ( 2008 ) and Wang Su ( 2016 ) failed to properly account for the random red noise variations; and Sandrinelli et al. ( 2014 ) seem to discuss only single-trial confidence limits instead of global significance limits. In our QPO analysis, the temporary sinusoid-like trends visually appearing for a few periods in 2012–2014 might cause a certain test of LSP and PDM analyses to spuriously indicate a QPO detection, and the stochastic red noise variability might dominate the total variance of the light curve and lead to the emergence of spurious peaks/dips appearing in the DACF. Although the significance of the candidate QPO is relatively low when accounting for the red noise processes, some discussions on

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so that we could compare the value of the dimensionless growth rate directly to the predicted value. However, there does not appear to be an accurate way to measure this directly from the simulation data. Future simulations that insert a growing mode with a single, well-defined value for k would be desirable. In the dynamic simulations we present here, it is likely that there is significant power at a range of wavenumbers. In this case, we can expect growth to be slower initially, until sufficient power is in the fastest-growing modes. For the mode to fit in the disk (recall that we cannot bend the disk on scales ≲ H ), we must have Hk ≲ 1 (Ogilvie 2000 ), so we can expect the fastest-growing modes to appear with Hk less than, but not much less than, unity. However, to be consistent, we report the values of that can be measured from the simulation data. For simulation A with α = 0.03, H / R = 0.02, and θ = 30°, we find that the warp amplitude in the inner disk regions rapidly grows to greater than the critical value for R ≲ 15 R g (which for this viscosity is ψ c ≈ 0.5). This inner region breaks into multiple rings, starting from the inner parts and ending at ≈ 30 R g . Outside of this region, ψ remains below the critical value at all times, and no breaks are observed in this region. For this simulation, we analyze two clear breaks. The first break occurs between R ≈ 12–18 R g , with a peak in the warp amplitude at R p ≈ 15 R g while the warp amplitude is growing from ψ 0 = 2.5 to ψ 1 = 4. The second break forms an initial peak in the warp amplitude at R ≈ 20 R g , and by the time the peak warp amplitude is ≈ 2, the peak has settled at a radius of R p ≈ 27 R g . For the first break, we measure Δ t ≈ 1500 (in units of GM / c 3 ), and with the peak at R p ≈ 15 R g we have Ω ≈ 0.02. This yields χ ≈ 0.015. For the second peak, we have Δ t ≈ 4000, R p ≈ 27 R g , Ω ≈ 8 × 10 −3 , and therefore χ ≈ 0.015. This means that, when scaled to the local dynamical time, the growth rate was approximately the same in both cases, as expected. In this simulation, we expect the dimensionless growth rate to be in the range of 0.2–0.5 (see Figures 2 and 3 ). If , this would imply k −1 ≈ (3–5) H , which seems reasonable, but we also note that it is likely that the total level of viscosity in the simulation is slightly higher than α = 0.03 (see discussion in Section 2 ) and thus we would expect the growth rate to be reduced somewhat (i.e., if α ≈ 0.05, then the predicted dimensionless growth rate implies k −1 ≈ (2–3) H ). For simulation B, with α = 0.03, H / R = 0.02, and θ = 60°, we analyze two breaks in the disk, with the first occurring at R p ≈ 24 R g and the second beginning at R ≈ 30 R g and settling into a strong peak at R p ≈ 40 R g . For the first break, we measure Δ t ≈ 1650, and with the peak at R p ≈ 24 R g we have Ω ≈ 0.01. This yields χ ≈ 0.03. For the second peak, we have Δ t ≈ 5400, R p ≈ 40 R g , Ω ≈ 4 × 10 −3 , and therefore χ ≈ 0.02. As before, when

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


resolved temperature profiles up to . The fifth cluster, PLCK G266.627.3, had already been observed to similar depth with Chandra ( ks, proposal , PI P. Mazzotta). In addition to our deep XMM-Newton data, the four SPT clusters had previously been observed as part of the Chandra X-Ray Visionary Project proposal (P.I. B. Benson). Exposure times for these observations (typically 80 ks), were tuned so as to obtain counts per cluster to allow for global properties to be measured ( McDonald et al. 2014 ). The original ks snapshot observation of PLCK G266.627.3, obtained as part of the Planck - XMM-Newton validation programme, is described in Planck Collaboration XXVI (2011) . Observation details are given in Table 1 . All the XMM-Newton observations were taken using the European Photon Imaging Camera (EPIC), combining the data taken with the MOS, MOS, and PN cameras. Chandra observations were taken using the Advanced CCD imaging spectrometer (ACIS, Garmire et al. 2003 ). XMM-Newton and Chandra images of all five objects are shown in Appendix A . Table 2 Global properties. Notes. (a) Spectroscopic temperature measured in the region. (b) is the product of and ( Kravtsov et al. 2006 ). (d) The gas mass within , derived using the combined density profiles. (c,e) and are determined iteratively using the relation of Arnaud et al. (2010) , assuming self-similar evolution. 2.2. Data preparation We processed Chandra observations using the Chandra Interactive Analysis of Observations (CIAO, Fruscione et al. 2006 ) ver. 4.7 and the calibration database 3 ver. 4.6.5. The latest version of calibration files were applied following the prescriptions detailed in cxc.harvard.edu/ciao/ guides/acis_data.html? using the chandra_repro tool. To process the XMM-Newton datasets, we used the Science Analysis System 4 pipeline ver. and calibration files as available in December . New event files with the latest calibration applied were produced using the emchain and epchain tools. 2.3. Data filtering We reduced the contamination from high-energy particles using the Very Faint Mode status bit 5 and the keyword PATTERN for the Chandra and XMM-Newton datasets, respectively. In particular, we removed from the analysis all the events for which the keyword PATTERN is and for MOS, and PN cameras. To remove periods of anomalous count rate, i.e. flares, we followed the prescriptions described in Markevitch’s COOKBOOK 6 and Pratt et al. (2007) for Chandra and XMM-Newton , respectively. For both datasets, we extracted a light curve for each observation and removed from the analysis the time intervals where the count rate exceeds times the mean value. If there were multiple observations of the same objects, these were merged after the processing and cleaning procedures. We list in Table 1 the effective exposure times, after cleaning, for each instrument. Point sources were identified using the CIAO wavdetect tool ( Freeman et al. 2002 ) on and exposure-corrected images. For

O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O B-Telescope O O O O O B-Person I-Person O O O O O B-Telescope O O O B-Telescope O O O O O O O O O B-Survey I-Survey I-Survey I-Survey O O B-Person I-Person O O 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 B-CelestialObject I-CelestialObject O O O O O B-Collaboration I-Collaboration I-Collaboration I-Collaboration I-Collaboration O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Telescope O O O O O B-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O O O O O B-Instrument B-Instrument O B-Instrument O B-Telescope O O O O O B-Instrument I-Instrument I-Instrument I-Instrument B-Instrument B-Citation I-Citation I-Citation I-Citation O B-Telescope 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 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 B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Telescope O O O B-Software I-Software I-Software I-Software I-Software B-Software B-Citation I-Citation I-Citation I-Citation O B-Identifier I-Identifier O O B-Database I-Database I-Database B-Identifier I-Identifier O O O O O O O O O O O O O B-URL I-URL O O B-Software O O O O B-Telescope O O O O B-Software I-Software I-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 B-Software O B-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 B-Telescope O B-Telescope 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 B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Telescope 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 O O O O O O O O O O O O O O O O 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-Software O O B-Citation I-Citation I-Citation I-Citation O O O O O O


reasons for these two regimes goes far beyond the scope of this paper. Compared to ETGs, there is a higher dispersion and a less inclined slope for the LTG relation, which is more noticeable in the massive galaxy range, as was already pointed out by Georgiev et al. ( 2016 ). This indicates that the nuclei in LTGs are on average less massive at a fixed host galaxy mass than nuclei in ETGs. The weighted linear and polynomial least-squares fits are shown in Table 3 . Based on considerations that the mechanisms that are responsible for the build-up of the central massive objects are similar for nuclei and massive BHs, previous studies have discussed their possible evolutionary connection (e.g., Ferrarese et al. 2006 ; Neumayer Walcher 2012 ). For the purpose of comparison, we illustrate the black hole–galaxy mass relation from McConnell Ma ( 2013 ) with its 1 σ and 3 σ uncertainty limits. This relation was obtained from fitting the compilation of 35 ETGs with dynamical measurements of the bulge stellar mass, with mass range of 10 9 to 10 12 M ⊙ . Their sample is well populated for bulges more massive than 2 × 10 10 M ⊙ (see their Figure 3). Assuming that we can extrapolate to lower bulge masses, we see in Figure 10 that the scaling relation for BHs and their ETG hosts is similar to the relation between nuclei and their host galaxy mass down to 10 9 M ⊙ . For galaxies with lower masses, the nucleus-to-galaxy mass relation becomes flatter. Table 3 Scaling Relations Note. For the nucleus versus galaxy mass relation (), we set and . For the mass ratio ( η n ) relation as a function of galaxy mass (), we define y ≡ log( η n ) and . The mass ratio between nucleus and its host galaxy () as a function of galaxy mass is shown at the bottom panel of Figure 10 . We find a clear anticorrelation of η n versus over the entire galaxy mass range; that is, the lower the galaxy mass, the more significant becomes the nucleus. In the massive galaxy regime (), we note a large scatter in the relation, which is the main reason why previous studies assumed a constant equivalent luminosity ratio for their samples, like for instance the ACS Virgo nucleated galaxies with (Côté et al. 2006 ), for the ACS Fornax nucleated galaxies (Turner et al. 2012 ), and for LTGs (Georgiev et al. 2016 ). However, in the faint dwarf galaxy regime (), a clear and strong trend appears, reaching to η n ≃ 10% for a dwarf galaxy with a stellar mass of 10 7 M ⊙ . The four low-mass ETGs studied by Nguyen et al. ( 2017 , 2018 ) have η n values up to 1.7%, which are in agreement with the general trend. The extension of the η n versus anticorrelation toward the faint dwarf galaxy population appears to be similar for ETGs and LTGs. This, in turn, suggests that nuclei at the smallest masses are subject to localized processes that work on parsec scales within the galaxy core regions independent of galaxy type. 5.3. Comparison with Theoretical Predictions The two proposed formation scenarios for

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cases for toroidal magnetic field configuration combined with the rotation. Fig. 2(a) shows the density contour with the uniform angular frequency Ω = 0.0628 rad s −1 with kinetic to gravitational energy ratio KE/GE ∼3.58 × 10 −6 . Since the angular frequency is small, it does not affect the star considerably, resulting in a prolate star. On the other hand, Fig. 2(b) illustrates a star with angular frequency Ω = 3.6537 rad s −1 with KE/GE ∼ 1.33 × 10 −2 , and due to this high angular velocity, the low-density region is affected by the rotation more significantly than the high-density region, resulting in an overall oblate-shaped white dwarf. Here, in both the cases, ρ c ∼ 2.2 × 10 10 g cm −3 , and magnetic field at the centre of the white dwarf B max ∼ 2.7 × 10 14 G. Indeed, a few white dwarfs are observed with the surface magnetic field of ∼10 9 G (Heyl 2000 ; Vanlandingham et al. 2005 ; Brinkworth et al. 2013 ); hence, central field might be much larger than 10 9 G. In fact, it has already been argued in the literature that the central field could be as large as 10 14 –10 15 G (Franzon Schramm 2015 ; Shah Sebastian 2017 ). This is similar to the magnetic field in the case of the Sun, where it is known from observations that the surface field is ∼1 G. However, the central field could be ∼10 4 –10 7 G, indicating an increase of four to seven orders of magnitude (Parker 1979 ). Now, in time, Sun-like stars will become white dwarfs and they may possess similar field profile as earlier, as argued by Das Mukhopadhyay ( 2014 ). Hence, a white dwarf with B max considered here might have surface field of 10 9 G, which observed data have already confirmed. As a result, magnetic to gravitational energy ratio ME/GE is ∼0.1. The surface of a white dwarf is determined, when the density decreases at least up to seven to eight orders of magnitude compared to its central density, i.e. ρ c ∼ 10 7–8 ρ surf (ideally zero), where ρ surf is the density at the surface. The barcode shows the density of the white dwarf at different radii. The typical isocontours of magnetic field strength are shown in Fig. 3 . It is confirmed herein that the surface magnetic field can decrease up to ∼10 9 G even if the central field is ∼10 14 G. However, GW astronomy may also help in identifying magnetized white dwarfs with surface fields higher than 10 9 G, as argued later. Figure 2. Density isocontours of uniformly rotating white dwarf with toroidal magnetic field. Figure 3. Isocontours of magnetic field strength for uniformly rotating white dwarf with toroidal magnetic field. The black solid line represents the surface of the white dwarf. All the parameters are same as in Fig. 2(a) . Table 1 shows different h 0 for various ρ c . In the table, M represents the mass of compact object, R E is the equatorial radius, R P is the polar radius, and ν is the linear frequency defined as ν = Ω/2. It is observed that for a given ρ c , h 0 increases with the increase in rotation because h

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 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 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 O O O O O O O O O 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 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 B-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 B-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 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 B-Citation I-Citation O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest 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 O O O O O O O O O O O O O O O O O O O O O O O O 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 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-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O


and started observations in two polarization filters 60 s after notice time and 79 s after trigger time at 2015-03-01 19:39:23 ut . The very first frame showed a bright (15.3 m) optical source with the coordinates and an error of 1 arcsecond (Buckley et al. 2015 ). The Swift XRT X-ray telescope and UVOT ultraviolet telescope also detected X-ray and optical emission, respectively, from GRB 150301B 82 and 85 s after the BAT trigger (Lien et al. 2015 ). Later, 5.12 h after the burst, the Very Large Large Telescope (Paranal Observatory, Chile) and X-shooter measured the redshift of this burst, which was found to be z = 1.5169 (de Ugarte Postigo et al. 2015 ). MASTER observations. MASTER robotic telescope observed this gamma-ray burst in the alert mode in two polarization planes with variable exposures (see Section 2.2 ). As a result of our observations, the following light curve was obtained (see Fig. 1 and Table 1 ). Figure 1. GRB 150301B MASTER-SAAO light curve in two telescope tubes with different polaroid orientations (45° and 135° ). Table 1. Photometry data for GRB 150301B by MASTER in the unfiltered band with two orthogonal polarizing filters. MASTER-SAAO telescope observations of GRB 150301B lasted from 2015-03-01 19:39:23 ut through 2015-03-01 20:48:11, i.e. for about 1 h beginning with the 79th second after the trigger. Observations were then interrupted because of heavy clouds. We observed well-defined optical emission from GRB 150301B up until 2000 s after the trigger while the source remained brighter than the limiting magnitude on our frames. Unfavorable weather conditions prevented us from extending the light curve by obtaining deep stacked frames. Throughout the entire time interval from 79 to 2000 s the source faded in accordance with the power law F ∼ t −1.16 ± 0.05 which, in general, is typical of GRB that time (Mészáros 2006 ). 3.1 Spectrum from the optical to X-ray Observations of the gamma-ray burst with MASTER telescopes were performed synchronously with observations on the Swift /XRT X-ray telescope (Fig. 2 ). Optical data corrected for absorption N H = 6.2 × 10 20 cm −2 (Maselli et al. 2015 ) which is consistent with the Galactic value (Willingale et al. 2013 ). Note that variability observed in X-rays in the form of three local maxima in the 88, 191 and 282 s does not occur in the optical wavelength, that has already been observed previously (for example, Ziaeepour et al. 2008 ; Li et al. 2012 ). Figure 2. Optical, X-ray and gamma observation light curve of GRB 150301B. The Swift /XRT 0.3–10 keV and Swift /BAT 15–150 keV light curves are available from the Swift Burst Analyser (Evans et al. 2010 , http://www.swift.ac.uk/burst_analyser/ ). 3.2 Polarimetric measurements The technique of polarimetric measurements used on MASTER network telescopes allows detecting the optical emission polarization at a level of 5 per cent or higher for objects brighter than 16 m or estimating a lower limit for polarization (if the object

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continuum from decay of the metastable 2s level of H i , since that would peak at around 1450 Å and drop toward zero as the wavelength approaches Ly α , which is not observed. Rather, it is dust-scattered galactic background starlight, as also seen with HUT (Raymond et al. 1995 ). That continuum was attributed to dust-scattered light from B stars based on a private communication from J. Murthy, whose UV background observations showed that this area of the sky is unusually bright. 4. Analysis 4.1. Proper Motion We have used H α images for six epochs from 1998 to 2016, as listed in Table 1 , to measure the proper motion of the Balmer filament that delineates the bow shock. All the images were first placed on a common world coordinate system using a few hundred stars from the Gaia First Data Release (Gaia Collaboration et al. 2016 ). We then extracted a 1D profile for each epoch from a section 8′′ long, oriented normal to the shock front at the bow shock position. For each epoch pair separated by ∼4 yr or more, we carried out a fitting procedure to determine the proper motion over the intervening interval—a total of 14 independent epoch pairs. Since the 2015 and 2016 images had significantly better seeing than those at the earlier epochs, it was necessary to smooth the sharper images with an appropriate Gaussian to obtain acceptable fits. Figure 2 shows the bow shock region at three of the epochs, displayed in different colors. All the epoch pairs gave reasonably good fits and results ranging from 0314 ± 0019 yr −1 to 0357 ± 0023 yr −1 , with most of the pairs ranging from 0336 yr −1 to 0349 yr −1 . We took a simple average of all 14 measurements to obtain our final result for the proper motion of bow shock filament: 0341 ± 0010 yr −1 , where the uncertainty should include both statistical errors and possible systematic ones due to misalignment of images from different epochs. This corresponds to −0245 ± 0008 yr −1 in R.A. and +0238 ± 0008 yr −1 in decl. This value is 22% faster than the proper motion of 0280 yr −1 obtained by Winkler et al. ( 2003 ) for the brighter main portion of the NW filament in SN 1006, located a few arcminutes to the NE. It is hardly surprising that the bow shock region is moving faster, since the shock is apparently being driven ahead here by an ejecta fragment. Indeed, it was for this reason that we selected it for study with COS—to study the fastest shock that was also bright enough to enable us to obtain an ultraviolet spectrum in a reasonable amount of time. Combining our proper motion with the distance of 1.84 ± 0.14 kpc from the proper motion and shock speed of the main NW filament, we obtain a shock speed of 3030 ± 250 for the bow shock. Our long sequence of images will enable a more complete study of the proper motions of optical filaments in SN 1006, which will be reported in a separate publication. 4.2. Optical Spectrum We extracted the spectrum of a 15′′ section of the IMACS slit centered on the HST /COS

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 O O O O O O O O O O O O O O O O B-Telescope 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-Person I-Person O B-Wavelength 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 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-Dataset I-Dataset I-Dataset I-Dataset 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 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 O O O O O O O O O O O O O O O O O O O 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 B-CelestialObjectRegion I-CelestialObjectRegion 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-Instrument 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 B-CelestialObjectRegion I-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 O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O B-Wavelength O O O O O O O O O O O B-Instrument O O O O B-Telescope B-Instrument


between the two populations is clear: RQ AGNs occupy only the top-right region of the plot, meaning that they have a weak and steep X-ray emission. On the other hand, most blazars have a stronger and flat emission and they are located in the bottom left-hand region. Based on this distinction, we set two thresholds to differentiate blazars and non-blazar AGNs. In particular, as shown in Fig. 5 , we adopt the values = 1.8 and = 1.355 as thresholds to separate the two populations. These limits include all the confirmed high-z blazars. We then apply these criteria to the high-z AGN in the CLASS sample (Fig. 6 ). In particular, we plot here the 14 objects with a reasonable estimate of the photon index (error 0.4). The remaining 10 will be discussed further below. The sources of our sample which have been already confirmed as blazars in the literature are highlighted by a black circle. In addition, in the plot it is also included the classification as blazar/non-blazars (flat/peaked) based on the radio spectrum and discussed in C19 (blue points = peaked, red squares = flat, and purple diamonds = uncertain). Figure 5. Photon index as a function of the index for the comparison samples. In orange we report the BZCAT blazars observed by Swift-XRT , with red squares the few confirmed blazars at high-z and with blue points the RQ AGNs at high-z. The red star represent the z = 5 blazar DESJ014132.4-542749.9, Belladitta et al. ( 2019 ). The continuous black line represents the dependence of the photon index with a variation of the X-ray intensity () of the jet due to beaming effects considering a coronal emission with = 1.55 and = 2, while for the jet emission we assume = 1.6. The black cross represents the critical angle assuming that the jet seen at θ = 0 is 50 times more intense than the corona at 10 keV. The plot is divided in four areas by a vertical line at = 1.355 and a horizontal one at = 1.8, which correspond to the thresholds assumed for the classification. Figure 6. Photon index as a function of the for the high-z objects in the CLASS sample with an error on the photon index lower than 0.4. The objects are plotted with different colours and markers depending on the radio spectral classification reported in C19 (‘flat’ i.e. good blazar candidates, ‘peaked’ i.e. probably non-blazars and ‘uncertain’) and the black circles represent the few confirmed blazars at z > 4 in the literature. As in the previous figure, the plot is divided in four areas by a vertical line at = 1.355 and a horizontal one at = 1.8. Considering the objects plotted in Fig. 6 , there is only one source whose and are not consistent with a blazar nature (GB6J012126+034646), meaning that its X-ray spectrum is too steep and too weak to be produced by an oriented jet (grey region). On the other hand, as expected, the majority of the sources (13) are in the bottom left-hand region (green), meaning that they are strong and flat enough to suggest that they are bona fide blazars.

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 O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B-Survey O O O B-Observatory 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-CelestialObject 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-Wavelength I-Wavelength B-TextGarbage O O O O O O O O O O O O O O O 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 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 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 B-Citation 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 O O O O O O O O O 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 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


impossible by using an encryption algorithm and allowing the specification of side constraints to be met. Section 5.6 computes the shift of parameters to determine whether blinding succeeded. Finally, the physical units are restored, and the output is a purposefully biased covariance matrix which leads to the requested bias in physical parameters, with the origin of the bias being untraceable due to the intermediate encryption and side constraints. 3.3 Finding sensible target parameters for blinding At the outset of the blinding strategy, the blinder has to pick an origin of the posterior shift, and target parameters which are preferred after shifting. Typically, shifting the yet unknown posterior by two to three standard deviations is the sought aim. Having never computed the full posterior, its standard deviations are however not known yet, so we effectively wish to shift by the multiple of a yet unknown quantity. We thus require an abstract prediction of the shift’s magnitude. To this aim, we exploit scaling relations: For a multivariate Gaussian posterior, the best-fit will occur at the minimal chisquared, . The associated 90 per cent credibility contour can then be chosen to run along an isocontour which lies by Δχ 2 above this minimum. The more parameters are estimated, the larger Δχ 2 has to be, since more parameters will widen up the joint posterior. The values of Δχ 2 where the 90 per cent credibility contour lies above the minimum are known (they are integrals of χ 2 -distributions), and we tabulate them as a function of free parameters in Table 1 . Table 1. Δχ 2 -values where the 90 percent credibility contour lies above the minimum χ 2 , as a function of number of parameters. If the blinder chooses target parameters which differ from the initial parameters by more than Δχ 2 shown here, then a posterior shift of more than three posterior standard deviations will ensue. This is not advised. The blinder thus has to know how many parameters are to be inferred, and has to pick initial parameters and target parameters which differ approximately by Δχ 2 -values as given in Table 1 . If the blinder has no physical intuition for which parameter values are likely to be preferred by the data, then the linearized best-fitting estimator of equation ( 7 ) can be evaluated, which will yield parameter values close to the full non-linear best fit . We found that the values do not need to be picked with too much care, as long as they are within about two posterior standard deviation from the true best-fit . We provide a public code 2 for the algorithm, which assists the blinder via equation ( 7 ) and Table 1 in finding sensible parameters for the shift origin, and for the shift destination . 3.4 Data transformation The general astronomical data set will come in ‘natural’ units, which might depend on estimator choice or the adopted (physical) units of measurements, such as parsec or mega-parsec. To gain generality, we thus standardize the data set.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 adds it to the envelope. We see that even with the large amount of energy that we add to the envelope as it expands to twice its original radius in a short time, the basic structure stays similar. For example, there is a density inversion in the outer part of the envelope and the convection speed stays below the sound speed. This implies that convection can carry even more energy than it carries now. Figure 3. Profiles of four quantities of our AGB stellar model in the outer layers of the envelope at t = 0 (the beginning of the plunge-in phase). The right edge of both panels is at the photosphere of the star. Upper panel: the temperature (black) and the density (magenta) in a logarithmic scale. Lower panel: the sound speed (blue) and the convective velocity (red). The orange, green and purple dotted vertical lines mark the zones of 90 per cent, 50 per cent and 10 per cent ionization of hydrogen, respectively. Figure 4. Like Fig. 3 but at from the beginning of the plunge-in phase. Figure 5. Like Fig. 3 but at the end of the plunge-in phase at . We next calculate the energy transport time from radius r to the surface of the star, R , by photon diffusion, as given in equation ( 1 ), and by convection, as given in equation ( 4 ). We present these energy transport times at the three evolutionary stages in Figs 6 – 8 , respectively. The convection velocity drops to very low values in the very outer part of the envelope, and so we calculate the convection time to the radius of 1 per cent hydrogen ionization fraction instead of the photosphere. This change is of no significance since in the outer region photon diffusion carries most of the energy. Figure 6. Energy transport times from radius r to the surface of the star at radius R for t = 0 when we start the energy injection. Since the convection velocity drops to very low values in the very outer part of the envelope, we calculate the convection time to the radius of 1 per cent hydrogen ionization fraction instead of the photosphere. This radius is also the right boundary of the graph. We present the photon diffusion time according to equation ( 1 ) by the red line, and the convection time according to equation ( 4 ) with the blue line. The diffusion time from R ion, 50 is . Note that the photon diffusion time becomes much shorter than the convection time in the outskirts of the envelope. Figure 7. Like Fig. 6 but at from the beginning of the plunge-in phase. At this time . Figure 8. Like Fig. 6 but at the end of the plunge-in phase at . At this time . From the figures we learn the following about the energy transport times in the inflated envelope. Convection can carry energy out from the entire hydrogen recombination zone very efficiently, as the convection transport time from 90 per cent hydrogen ionization fraction to the surface is shorter than the plunge-in time, in our case. The convection cells can move even faster, up to the sound speed that is larger than the convective speed of 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 B-Model I-Model I-Model 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


FRB repeater exhibits several features commonly found in earthquakes. In fact, nonlinear dissipative systems always show self-organized criticality (SOC) behaviors. With a solid crust or stiff equation of state (EOS) of a pulsar, a star can build up stresses that make the crust crack and adjust the stellar shape in order to reduce its deformation. The characters for these processes of statistical independence, nonlinear coherent growth, and random rise times are consistent with an SOC system (Aschwanden 2011 ). According to the SOC theory, NS quakes are giant catastrophic events like earthquakes, and are probably accompanied by global seismic vibrations or oscillations. Hence, starquakes in a pulsar share similar statistical distributions with earthquakes. Here we present two possible scenarios. One scenario involves a normal NS with a solid crust and a superfluid core, and with a strong toroidal magnetic field (i.e., magnetar). An NS can form a solid crust quickly after its birth. The stellar shape changes from oblate to spherical configuration, and thermal and dynamic responses will induce stresses in the crusts. When stress buildup in the dense solid crust extends beyond a yield point a starquake occurs, with a sudden energy release that supports FRBs. This process also brings a slight change in the moment of inertia, with an abrupt jump of angular frequency that is the so-called pulsar glitch (Ruderman 1969 ; Link Epstein 1996 ). For a highly magnetized NS, the stellar crust couples with the magnetosphere. In this scenario, starquakes induce the magnetic curl or twist ejected into the magnetosphere from the crust in a few milliseconds (Thompson et al. 2002 , 2017 ). Electrons in the magnetosphere are suddenly accelerated to ultra-relativistic velocity by magnetic reconnection (Zhang Yan 2011 ) and move along the magnetic field lines, resulting in the production of curvature radiation. The characteristic frequency of the curvature radiation iswhere R c is the curvature radius with a typical value of ∼10 km, and γ is the Lorentz factor of electrons. With a detected FRB frequency ν c = 4–8 GHz, a γ ≈ 50–100 is required. If generated by such stress-induced reconnections (e.g., Lyutikov 2015 ), the FRB has an estimated duration timescalewhere the scale of the reconnection-unstable zone L ∼ 1–10 km, and the Alfvén velocity is v A ≃ B /(4 πρ c ) 1/2 ∼ 0.01 c , in which ρ c ≃ 10 14 g cm −3 is the average mass density of the crust. The sudden elastic and magnetic energy release in the crustal stress is estimated to bewhere h c ≃ R /30 ≃ 0.3 km is the crustal thickness (Thompson et al. 2017 ), ε is the shear strain, and the total stress including crustal shear stress and Maxwell stress in magnetosphere isin which μ is the shear modulus, B is the surface magnetic field, and B z is the component of magnetic field perpendicular to the direction of plastic flow. Within the crust, the force balance με ≃ BB z /(4 π ) implies that is smaller than ∼10 −2

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


must be taken into account when analysing gas observations and comparing it with dust levels. However, if CO is unshielded, its short lifetime (∼120 yr) means that its mass will be set by the present mass-loss rate. Conversely, atomic carbon has a longer lifetime as it will viscously evolve on longer time-scales which will depend on the kinematic viscosity. We used this new numerical code to produce the first population synthesis model for exocometary gas around A stars. Population synthesis studies are ideal to deal with degenerated problems and provide constraints on population properties rather than individual systems. Informed by previous studies that fit the dust evolution, we generated samples of 10 4 stars hosting planetesimal discs with random radius, mass, and stellar properties. We evolved these systems up to a random age between 3–100 Myr and compare the final distribution to the observed distribution by applying the same selection filters to our model population. We found that our model can reproduce the distribution of CO masses well when viscosities are high (α ≈ 0.1), even producing a bimodal distribution with a large population of unshielded CO discs and a small fraction of shielded ones. The shielded population is nevertheless common in our models among systems with bright debris discs, which explains why shielded discs are common among the sample of 17 A stars with bright and cold debris discs. The high-α value implies that gas needs to be lost on time-scales of ∼1–10 Myr. Otherwise, carbon readily accumulates shielding CO and producing a large population of shielded and massive CO discs, even around systems with low planetesimal disc masses, which is inconsistent with observations. Although, we focused on A stars, we also tested our model against observations of FGK stars. While massive and shielded CO gas discs are found around A stars, observations of FGK stars have showed that these have low CO gas masses, with only a few detections. Using our model, we showed that this is consistent with the collisional evolution studies of discs around FGK stars compared to A stars. Their lower initial disc masses means that gas release rates are lower. Our model quantitatively shows that massive CO gas discs should be very rare around FGK stars if they have similar viscosity levels compared to A stars. An important test of viscous evolution models of gas in debris discs, is whether gas is observed to extend inwards and accrete on to the star. In Section 5 , we discussed recent observations of neutral carbon around a few systems which seem to indicate that carbon has not spread inwards forming an accretion disc as expected in a viscous evolution scenario. This might suggest that either gas release started recently or is lost through another mechanism, e.g. radiation pressure. Nevertheless, our modelling constrains the gas loss time-scale to be ∼1–10 Myr (viscous time-scale), which could provide insights into alternative

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


could be directly understood from the second term on the RHS of Eq. ( 34 ). Also the damping due to the plasma resistivity is clearly shown by the presence of the third term on the RHS of Eq. ( 34 ). Since the viscosity is small for coronal conditions, the deceleration is not very strong in the long wavelength limit, but in the long run would gradually become significant. 3. Discussions and conclusions In the solar atmosphere MHD waves interact with the plasma. When a gradient in the density is present, a wave guide is created that guides waves leading to the appearance of collective modes. In this work a magnetized cylindrical structure whose density is higher than its surroundings has been taken into consideration. The interplay of the equilibrium twist and rotation exhibits interesting features of the nature of the torsional wave that are of interest for coronal seismology. We considered a twisted and rotating magnetic flux tube, which resembles a solar spiral or tornado, and studied the influence of its helical properties on other dissipating parameters linked to the oscillations of the torsional waves. The analytic expressions obtained for the dependence of the phase speed of the torsional wave on the equilibrium twist and rotation enabled a quantitative study of the interplay of various actors on its propagation. The comparison of the explicit expressions obtained for the phase speed of the torsional wave in the present study with the implicit version, in all cases proved adequate for the robustness of the explicit expressions. This is reflected in the figures where the numerical results are compared with analytic solutions. The full dispersion relation for axisymmetric oscillations in the long wavelength limit has been obtained by showing the joint effects of the equilibrium twist and rotation with the viscosity and gravity (Eq. ( 19 )). An explicit expression for the dependency of the torsional wave frequency in the presence of viscosity, equilibrium, and environmental conditions on the wave number has been obtained (Eq. ( 20 )). This explicit dispersion relation also gives the damping and deceleration of the torsional wave guided by solar plasma structures. The fine consistency between the full dispersion relation and the explicit expressions for the frequency, damping, and speed of the torsional wave justifies the model and results. The magnetic twist owes its efficiency on the torsional wave frequency to not only the flux-tube rotation and coronal conditions, but also to the viscosity and gravity. In a zero plasma- β structure that is non-rotating, the magnetic twist acts very weakly on the torsional wave frequency (Fig. 3 ). If in such conditions the viscosity and solar gravity become negligible, the magnetic twist, no matter how large, would be completely neutral without any effect on the frequency. The damping of the torsional wave due to viscosity in solar structures is proportional to the strength of the magnetic twist and

O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O B-CelestialObject O O O O O O O O O O O


refer to φ w = 6 V, hν = 10.29 V (for λ ν = 121.57 nm), T d ≈ 205 K (for latitude θ = 70°), and non-thermal photoelectron distribution with A = 0.1. The parameter q is observed to increase with φ o . It is also noticed that the effect of dust charge fluctuation on the wave dispersion is pronounced for small z o values; for instance, q > 1 for z o = 30 and φ o > 0.2 (Fig. 5a ) and for φ o = 1.0 and z o 104 (Fig. 5b ). The positive plasma potential φ o infers enhanced population density of the photoelectrons in the vicinity compared to the case φ o = 0; this may induce additional shielding to the floating dust particles and effectively reduces the coupling between charged dust particles. It may result in a decrease in the phase speed and frequency of the propagating DA wave. This nature has been depicted in Figs 5(c) and (d) . The calculations corresponding to non-zero φ o are of significance to the lunar exospheric plasma, which exhibits a photoelectron sheath with a positive potential and varies with the altitude. Another interesting point evident from the wave dispersion (Figs 2 – 5 ) is the occurrence of a cut off with z o , Δ, and φ o for which ω r takes imaginary values, and it refers to a damped wave solution for the excited DA modes. Figure 5. (a) and (b) ( top panel ): Parameter q A = [(Λ ch /Ω ch ) n de ] A and (c) and (d) ( bottom panel ): wave dispersion ( ω r / k ) in long-wavelength regime ( kλ e 1) for the non-thermal distribution of photoelectrons. Left-hand and right-hand columns refer to parameters’ variation with φ o and z o , respectively. The calculations refer to the parameters φ w = 6 V, hν = 10.29 eV, T d ≈ 205 K, and A = 0.1. The black (solid), red (dash), green (dot), blue (dash–dotted), magenta (dash–dot–dotted), and brown (short dash) in the left -hand and right-hand columns correspond to the magnitude of the varying parameter z o = 30, 50, 100, 150, 200, and 250, and φ o = 0.1, 0.5, 0.8, 1.0, and 1.2, respectively. As understood from the discussion, the significant coupling between floating dust and electrons within the photoelectron sheath is a favourable condition for generating such low-frequency DA modes on the sunlit locations, and these physical conditions strongly depend on dust charge z o and constituent sheath plasma parameters (i.e. density and potential). It is of interest to comment on the influence of the solar wind on the excitation of these DA modes. The solar wind is an essential constituent configuring the electrostatic environment over Moon. Due to larger photoemission current (∼5 μA m −2 ), the surface charging over the sunlit location and sheath population (∼500 cm −3 ) is dominated by emitted photoelectrons, and the solar wind plasma (density ∼5 cm −3 and current ∼10 nA m −2 ) essentially helps in attaining dynamic equilibrium of the constituent charging currents. In the wave generation processes, the relative motion of solar wind to the photoelectron sheath plasma contributes in exciting

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Thus, while large-scale structure for different types of DM can appear identical, careful attention to structure on subgalactic scales can be a powerful tool to distinguish DM models. A powerful probe of the gravitationally bound structures of dark matter is strong gravitational lensing. Given a matter over or under density, the deflection angle along the line of sight is given by an integral over the induced gravitational potential (Narayan Bartelmann 1997 ),where χ is the distance along the line of sight, χ s is the distance to the source, and is the gravitational potential. In the thin lens approximation (Narayan Bartelmann 1997 ), valid in the limit that the thickness of the lensing galaxy is small compared to the distance to the lens, this takes a simplified form,where D LS , D L , and D S are the angular diameter distances from the lens to the source, from the observer to the lens, and from the observer to the source, respectively, and z is the distance along the line of sight. From this expression one can straightforwardly compute the lensing due to any gravitational potential Ψ. The gravitational potential is in turn determined by matter density via the Poisson equation, ∇ 2 Ψ ∝ ρ . The linearity of this equation implies that the total lensing due to the separate contributions, e.g., of a halo and halo substructure, is simply the sum of the individual contributions. That is,where is the external shear due to large-scale structure, and are the lensing due to the halo and halo substructure respectively. The well studied case is the lensing due to the spherical substructures expected from hierarchical structure formation in the context of noninteracting particle dark matter. However, as mentioned in the 1 , other types of substructure can exist in models of dark matter outside the WIMP paradigm. As a prototypical example, we will consider dark matter condensates, namely superfluids, which exhibit substructure in the form of vortices. We now proceed to develop this in detail. 3. Case Study: Dark Matter Superfluidity The canonical example of a condensate dark matter model is axion dark matter. Axions were introduced as a solution to the strong-CP problem of the SM (Peccei Quinn 1977 ; Weinberg 1978 ; Wilczek 1978 ), and soon thereafter proposed as a dark matter candidate (Abbott Sikivie 1983 ; Dine Fischler 1983 ; Preskill et al. 1983 ). It was later argued that axions could form a Bose–Einstein condensate and exhibit superfluidity (Sin 1994 ; Hu et al. 2000 ; Silverman Mallett 2002 ; Hui et al. 2017 ; Sikivie Yang 2009 ; Berezhiani Khoury 2015 ; Ferreira et al. 2019 ). To emphasize the superfluid nature of axions, we can rewrite the field equations in terms of fluid equations. Being comprised of extremely light particles at an incredibly high number density, axion dark matter is well described by a coherent scalar field. Moreover, because dark matter as we observe it is cold, the system is well described by a nonrelativistic

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


polarization fraction. Note also that the magnetic pole and equator are the two main contributors to the surface emissions in the current configuration (see the explanations given later). We next discuss the phase-averaged quantities. The semi-amplitudes and polarization fractions are shown in the upper and lower panels of Figure 6 , respectively. Note that, rigorously speaking, the semi-amplitude is not a phase-averaged quantity, but we consider it here just for comparison. In this paper, the semi-amplitude is defined as the quantity related to the total variation of the polarization angle during the rotational period divided by four, which is expressed aswhere the supremum is taken over all possible partitions of the range for ψ . This definition coincides with that given in Taverna et al. ( 2015 ) in most cases but not always (see below). The photon energy is set to following Taverna et al. ( 2015 ). In the left column, the mode conversion is neglected on purpose for comparison with the previous work, whereas it is incorporated in the right column. The results without mode conversion are consistent with those in Taverna et al. ( 2015 ). The discontinuous suppression of the semi-amplitude on the diagonal line observed in our result (see the top left panel of Figure 6 ) but absent in their result is mainly due to the fact that our definition of the semi-amplitude is not completely the same as theirs. Figure 6. Semi-amplitudes of the polarization angle (upper panels) and phase-averaged polarization fractions (lower panels) for different configurations of a rotating neutron star. The left and right columns correspond to the cases without and with the mode conversion, respectively. The photon energy is set to . We explain this in more detail using Figure 7 , which shows how the polarization angle changes with the rotational phase for different combinations of η and γ . Figure 7. (a) Top views of the observed patches (orange circles) on the polarization-limiting surface in the middle row and the polarization directions (red arrows) at different rotational phases in the bottom row for (left), (center), and (right). The trajectories of the (extended) magnetic north pole are drawn in green. Blue arrows and black crosses in the bottom panels indicate the directions of the average magnetic fields at the different rotational phases and the position of the coordinate origin, respectively. (b) Polarization angles as functions of the rotational phase. The angle η is fixed to , whereas the angle γ is changed: . In the middle panels of Figure 7 (a), we schematically draw the top views of the observed patch on the polarization-limiting surface for , , and . The green circles indicate the trajectories of the (extended) magnetic north pole on this surface. Note that they are not exact circles in general. In the bottom panels, we give the corresponding average magnetic fields (blue arrows) and polarization directions (red arrows), which are estimated from 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 O O O O O O O O O O O O O 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O


is obtained by taking the sum of the contributions from all dark matter halos We have made use of the python code colossus (Diemer 2018 ) to generate the dark matter halo mass function using the parameterized formula fitting to an N -body simulation proposed by Tinker et al. ( 2008 ), and the concentration–dark matter halo mass relation proposed by Prada et al. ( 2012 ). We have tested systematics by using Raeymaekers ( 2007 ) for N -body simulation fits for the dark matter halo mass function, and Dutton Macciò ( 2014 ) for the concentration–halo mass relation. The results show a different choice of dark matter halo population modeling will not change the final event rate by orders of magnitude. Assuming f PBH = 100%, the result of Equation ( 10 ) is shown in Figure 1 . The main contribution to the integral in Equation ( 10 ) is from low-mass dark matter halos, therefore the result depends sensitively on the lower end for halo mass. To illustrate the dependence on lower mass cutoff , we plot as a function of in Figure 1 . Also note that the integrand of Equation ( 10 ) does not depend on M PBH due to compensation between the primordial black hole number density and the cross section. To determine the lower limit of the dark matter halo mass for M PBH = 0.01/0.1/1 M ⊙ , we follow the criterion by Equation (11) of Bird et al. ( 2016 ) requiring that the dark matter halo evaporation timescale due to dynamical relaxation exceeds the dark energy domination timescale (∼3 × 10 9 yr). This choice results in the lower integral limit of Equation ( 10 ) being 0.3/3/21 M ⊙ for M PBH = 0.01/0.1/1 M ⊙ . Choosing a lighter dark matter halo cutoff would contribute more binary primordial black hole merger events if the halos would not have evaporated at present and the NFW profile is still preserved. High resolution dark matter N -body simulations are needed to precisely resolve the smallest structure of subsolar mass primordial black hole halos but this is beyond the scope of this paper. Figure 1. Event rate for binary primordial black hole merger as a function of the lower cutoff of dark matter halo mass. The markers are the event rate for M PBH = 0.01/0.1/1 M ⊙ , which are 528/140/45 Gpc −3 yr −1 , respectively. With the above choices of lower integral cutoff, as shown in Figure 1 , the event rate for M PBH = 0.01/0.1/1 M ⊙ is 528/140/45 Gpc −3 yr −1 . As a comparison, for M PBH = 30 M ⊙ , the event rate is Gpc −3 yr −1 , as computed in Bird et al. ( 2016 ). Therefore, the merger rate of subsolar primordial black holes increases by one to two orders of magnitude compared to that of 30 M ⊙ . To compare with observations, the sensitive volume, which is defined to be the orientation-averaged volume assuming a match-filtering signal-to-noise ratio (S/N) > 8 (Abbott et al. 2018b ), is Gpc 3 for M PBH = 0.01/0.1/1 M ⊙ for Advanced LIGO at its design sensitivity. Given the above estimated event rate, under our assumption of delta function mass distribution, the

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


spherical target and the stripped mass from the envelope as functions of time. Time is scaled with the mean pusher transit time (≈19.4 ns). The stripped mass is normalized with the total envelope mass and is identified as forward ( v z > 0) traveling mass above the mass center (MC, blue line). Also plotted is the stripped mass obtained when its velocity becomes larger than 4 km s −1 (green line). The evolution of the stripped mass and core velocity obtained with this criterion is shown by the green line in Figure 11 . Our second procedure simply counts as stripped any envelope material with coordinates above the center of mass of the target. This second method leads to the blue line in Figure 11 . As can be seen, the precise criterion to decide when the mass element has been stripped mainly affects the rising part of the curves, which is steeper with the second criterion. Nevertheless, both criteria lead to a similar asymptotic behavior. The curves show a similar overall trend to the astrophysical case presented in Figure 7 . Furthermore, by taking the ratio between the asymptotic core velocity (AACV) and the average ejecta velocity (AEV) this results in values of AACV/AEV ≈ and 1.6 × 10 –2 for the astrophysical and the laboratory simulations respectively, which fulfills Equation ( 9 ). Figure 12 shows a plot of the angular distribution of stripped mass from the two-layer companion/target. The baseline curve labeled t = 0 depicts the mass distribution function of the Cu envelope of the target before the impact (the shell in gray in the sketch) as a function of the axial angle θ (). Figure 12 shows the laboratory and astrophysics distributions at the common time . The lines for the laboratory and astrophysical simulations lie above the reference line, indicating that much of the stripped material is coming from the lower hemisphere of the target. The consequences of using planar or spherical blasts in the laboratory setting are quite evident from the figure, and support the conclusions from the simple toy model stated in Section 2 above. For curved fronts the agreement between the astrophysical and laboratory simulations is excellent. Figure 12. Distribution of stripped mass of the companion star/two-layer target envelope as a function the azimuth measured from its center of mass (CM). The curve labeled “ t = 0” is the initial mass distribution before the collision. The curves labeled “Companion Star” and “Hemispherical” refer to the astrophysical and experimental simulations respectively. For comparison, the simulation of the experiment using a planar blast wave generated with the same laser conditions (labeled “Planar”) is also presented. 5. Discussion and Conclusions We presented the results of a proposed design for a laboratory experiment at high energy density in which a spherically expanding blast wave collides with a spherical target composed of a dense core surrounded by a low-density shell. The experiment was optimized to match

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


because, first, the amount of early injected electrons, which has already been well cooled, will be larger, and secondly, the cooling break will appear at lower energy for a larger t age . Figure 5. Influence of pulsar’s initial rotation period P 0 on the predicted nebula extent. See Section 3.1.2 for details. 3.1.3 Braking index The braking index n determines the injection profile of electrons and its influence on the energy-dependent morphology is complex. For a large braking index, e.g. n = 3, the age of the pulsar t age is much longer than the initial spin-down time-scale τ 0 (see Fig. 1 ). The spin-down luminosity in this case decreases as t −2 except at very early time (equation 3 ). For a smaller braking index, the spin-down luminosity decreases more quickly with time for t > τ 0 . However, τ 0 will become comparable or even larger than the age of the pulsar. As a result, the global decline slope of the spin-down luminosity from the birth of the pulsar to the present day is flatter for a smaller n . Note that radiation at smaller (larger) radius mainly arises from electrons injected at later (earlier) time, so a smaller n results in a steeper decrease of the electron density profile and hence leads to a less extended morphology and vice versa. Such a tendency can be seen from Fig. 6 except at the low-energy end and the high-energy end. At the low-energy end, the extent of the PWN is mainly determined by the age of the system, so it is larger for a smaller n . At the high-energy end, the cooling time-scale of electrons is very short (within a few thousand years) so the radiating electrons are just recently injected. In other word, the spatial distribution of high-energy electrons is not sensitive to the long-term temporal behaviour of the spin-down luminosity and hence the curves for all three values of n converge at high-energy end. Figure 6. Influence of pulsar’s braking index n on the predicted nebula extent. See Section 3.1.3 for details. 3.1.4 Spectral index We assume a single power law of index p for the injection spectrum (equation 5 ). The SED shows a peak around GeV that is naturally formed due to cooling of electrons and superposition of electrons injected at different epoch, without invoking a broken power-law function for the injection spectrum. The peak in the SED roughly corresponds to the transition in the slope of the present-day electron spectrum from −3 to >−3, so the injection spectral index p can influence the peak energy in the SED. We show the present-day electron spectrum ( E e versus ) with decomposing the contributions from different injection epoch in Fig. 7 for different p . We note that the spectrum shows a hump structure around the cooling break instead of a simple broken power law. This is because the broken power-law spectrum can be formed via synchrotron or IC cooling only when the injection rate of electrons is constant, while the latter decreases with time as the pulsar spins down. Figure 7. Electron

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