Balmer filaments in Tycho's supernova remnant: an interplay between cosmic-ray and broad-neutral precursors
Sladjana Knežević, Ronald Läsker, Glenn van de Ven, Joan Font, John C. Raymond, Coryn A.L. Bailer-Jones, John Beckman, Giovanni Morlino, Parviz Ghavamian, John P. Hughes, Kevin Heng
DD RAFT VERSION S EPTEMBER
15, 2018
Preprint typeset using L A TEX style AASTeX6 v. 1.0
BALMER FILAMENTS IN TYCHO’S SUPERNOVA REMNANT: AN INTERPLAY BETWEEN COSMIC-RAY ANDBROAD-NEUTRAL PRECURSORS S LADJANA K NE ˇ ZEVI ´ C , R ONALD
L ¨
ASKER , G LENN VAN DE V EN , J OAN F ONT , J OHN
C. R
AYMOND , C ORYN
A. L.B
AILER -J ONES , J OHN B ECKMAN , G IOVANNI M ORLINO , P ARVIZ G HAVAMIAN , J OHN
P. H
UGHES , AND K EVIN H ENG Department of Particle Physics and Astrophysics, Faculty of Physics, The Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel Benoziyo Fellow, Email: [email protected], ORCID: 0000-0003-1416-8069 Finnish Centre for Astronomy with ESO (FINCA), University of Turku, V¨ais¨al¨antie 20, FI-21500 Kaarina, Finland Max Planck Institute for Astronomy, K¨onigstuhl 17, D-69117, Heidelberg, Germany Instituto de Astrof´ısica de Canarias, V´ıa L´actea, La Laguna, Tenerife, Spain Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, U.S.A. INFN Gran Sasso Science Institute, viale F. Crispi 7, 67100 L’Aquila, Italy Department of Physics, Astronomy and Geosciences Towson University, Towson, MD 21252, U.S.A. Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854, U.S.A. University of Bern, Center for Space and Habitability, Sidlerstrasse 5, CH-3012, Bern, Switzerland
ABSTRACTWe present H α spectroscopic observations and detailed modelling of the Balmer filaments in the supernovaremnant Tycho (SN 1572). We used GH α FaS (Galaxy H α Fabry-P´erot Spectrometer) on the William HerschelTelescope with a 3.4 (cid:48) × (cid:48) field-of-view, 0.2 (cid:48)(cid:48) pixel scale and σ instr = 8 . kms − resolution at 1 (cid:48)(cid:48) seeing for ∼ hours, resulting in 82 spatial-spectral bins that resolve the narrow H α line in the entire SN 1572 north-eastern rim. For the first time, we can therefore mitigate artificial line broadening from unresolved differentialmotion, and probe H α emission parameters in varying shock and ambient medium conditions. Broad H α lineremains unresolved within spectral coverage of 392 kms − . We employed Bayesian inference to obtain reli-able parameter confidence intervals, and quantify the evidence for models with multiple line components. Themedian H α narrow-line full-width at half-maximum of all bins and models is W NL = (54 . ± . kms − atthe 95% confidence level, varying within [35, 72] kms − between bins and clearly broadened compared to theintrinsic (thermal) ≈
20 kms − . Possible line splits are accounted for, significant in ≈ of the filament,and presumably due to remaining projection effects. We also find wide-spread evidence for intermediate-lineemission of a broad-neutral precursor, with median W IL = (180 ± kms − (95% confidence). Finally, wepresent a measurement of the remnant’s systemic velocity, V LSR = − kms − , and map differential line-of-sight motions. Our results confirm the existence and interplay of shock precursors in Tycho’s remnant. Inparticular, we show that suprathermal narrow-line emission is near-universal in SN 1572 and that, in absence ofan alternative explanation, collisionless supernova remnant shocks constitute a viable acceleration source forGalactic TeV Cosmic-Ray protons. Keywords: cosmic rays – ISM: individual objects (Tycho) – ISM: supernova remnants – shock waves INTRODUCTIONSupernova remnant (SNR) shocks are suspected to be thelong-sought Galactic cosmic-ray (CR) sources. Observa-tional evidence for a particle acceleration at work in SNRshas been seen throughout synchrotron emission from elec-trons, gamma radiation, signatures of an amplified magneticfield, higher compression ratio than predicted by jump condi-tions and through optical line profiles from the shocks prop-agating in a partially ionized plasma (for a review see Helderet al. (2012)). Since most of the CRs that we detect on Earthconsist of protons, finding evidence for accelerated CR pro- tons in SNRs with the energies up to the ’knee’ ( ∼ ) isthus of a particular importance. Insight in CR protons is pos-sible either through gamma ray spectrum as a result of neutralpion decay (Ackermann et al. 2013), or throughout opticalwavelength window by carefully analyzing the H α -line pro-files (Helder et al. 2009, 2010; Nikoli´c et al. 2013). Thelatter shock emission, usually also referred to optical emis-sion from Balmer-dominated shocks (BDSs), is the subjectof study of this paper.The spectra of BDSs, typically observed around SNRs thatoriginate from Type Ia supernova explosions, show the pres-ence of strong two-component hydrogen lines Heng (2010). a r X i v : . [ a s t r o - ph . H E ] A ug When a shock wave encounters partly ionized interstellarmedium (ISM), the cold pre-shock hydrogen atoms over-run by the shock can either be excited by hot post-shockgas resulting in the narrow H α -component emission, orenter a charge exchange (CE) process with the hot post-shock plasma producing hot neutrals whose collisional ex-citation then give rise to the broad H α component. The two-component H α -line parameters provide valuable informationon the CR precursor existence in the shocks. A narrow linebroadened beyond 10–20 km s − gives direct evidence of thenon-thermal particle presence in the shock precursor (Mor-lino et al. 2013). This is due to the fact that hydrogen atomsare ionized at temperatures larger than ≈
10 000 K, but alsobecause the lifetime of the neutral hydrogen in the post-shockregion is too short for the collisional interaction to broadenthe line profiles (Smith et al. 1994). The CRs will heat thecold neutrals in the interstellar medium, resulting in broad-ening of the narrow H α -line, but the CRs will also reduce thebroad H α -line width by removing energy from the protons inthe post-shock region.Several authors proposed that narrow line broadening canalso arise from a broad-neutral (BN) precursor: hot neutralscreated in CE processes between hot protons and slow neu-trals streaming to the pre-shock region (Hester et al. 1994;Smith et al. 1994). Recent theoretical studies (Blasi et al.2012; Morlino et al. 2012) show that a BN precursor doesnot broaden the narrow component, but rather introduces athird intermediate component with the FWHM (full widthat half maximum) of around ∼
150 km s − and depends onthe shock speed. The reason is that only a small number ofincoming neutrals does interact with ions in the BN precur-sor because its extent, which corresponds to the interactionlength of the returning neutrals, is much smaller than theCE interaction length of the incoming neutrals. Therefore,Balmer lines can be used to study the microphysics of colli-sionless shocks and are currently the only mean that give aninsight into the collisionless shocks.We have observed Tycho’s SNR, which has already beenwell studied across all wavelength ranges (Reynolds & Elli-son 1992; Stroman et al. 2009; Bamba et al. 2005; Katsudaet al. 2010; Lee et al. 2004; Tian et al. 2011; Acciari et al.2011; Giordano et al. 2012) and the evidence for particle ac-celeration in shocks of Tycho’s SNR including accelerationup to the knee in the CR spectrum was found (Warren et al.2005; Cassam-Chena¨ı et al. 2008; Eriksen et al. 2011; Slaneet al. 2014). In 1572, the star exploded as a Type Ia super-nova, leaving a remnant at an estimated heliocentric distanceof 2.3 ± . At that distance,the remnant’s diameter of 8 (cid:48) corresponds to ≈ Several optical studies based on modelling the observed H α -line spectragive a distance 2–3 kpc. Radio, X-ray and gamma-ray observations preferlarger distances 3–5 kpc (for a review see Hayato et al. 2010). sity gradients in the medium around the remnant (Williamset al. 2013) modified the evolution of the shock, which in turnresulted in the asymmetric remnant. The lower shock veloc-ity inferred in the northeastern (NE) part suggests the shockinteraction with a dense ambient medium, namely a diffusecloud (Reynolds & Keohane 1999; Lee et al. 2004).Previous optical studies of Tycho’s SNR have shown in-dications for CRs (e.g. Ghavamian et al. 2000; Lee et al.2007; Lee et al. 2010). However, these studies focused onthe H α -bright, but very complex ’knot g’, where multiple ordistorted shock fronts can contribute to the measured narrow-line broadening and thus partially mimic the effect that CRacceleration would have. Using the Fabry-P´erot instrument GH α FaS (Galaxy H α Fabry-P´erot System) on the WilliamHerschel Telescope (WHT), we observed a great portion ofthe shock front in the NE region of the remnant. The highspatial and spectral resolution together with the large field-of-view (FOV) of the instrument, allow us to measure thenarrow H α -line width across individual parts of the shockssimultaneously, and thereby study the indicators of CR pres-ence in a large variety of shock front conditions. In partic-ular, the spatial resolution allows us to distinguish intrinsicline broadening from line broadening originating in geomet-ric distortions and differential kinematics. Moreover, our ob-servational setup provides us with a unique insight into theintermediate component existence along the entire filamentpreviously only reported for the bright ’knot g’ (Ghavamianet al. 2000; Lee et al. 2007). Apart from vastly enhancingthe amount and quality of the spectroscopic data availablefor the NE filament, our study also improves the analysis: in-stead of fitting line models, we employ Bayesian inference toobtain full information and realistic uncertainties on the lineparameters, as well as quantitative, reliable evidence for thepresence of an intermediate line (IL) originating in a BN pre-cursor and multiple shock fronts. The interpretation of theresults is based on predictions of the state-of-the-art shockmodels that include the effects of BN and CR precursor onthe observed H α profiles (Morlino et al. 2012, 2013). Withthe GH α FaS spectral coverage of approximately 400 km s − we are not able to resolve the broad H α -component that wasfound to be about ≈ − in previous studies (Cheva-lier et al. 1980; Ghavamian et al. 2001). OBSERVATIONS & DATA REDUCTIONIn order to resolve the narrow H α lines along the rim of Ty-cho’s SNR we have used the instrument GH α FaS mountedon the Nasmyth focus of the 4.2 m WHT (Hernandez et.al. 2008), which operates at the Observatory del Roquede Los Muchachos in La Palma, Canary Islands. GH α FaS is a Fabry-P´erot interferometer-spectrometer with a FOV of3 (cid:48) .4 × (cid:48) .4. Its detector is an Image Photon Counting System(IPCS) for which the absence of readout noise is an advan-tage for observations of diffuse emission from extended ob-jects. IPCS cameras are almost insensitive to cosmic rays Figure 1 . The left panel shows a composite image of the remnant ( ∼ (cid:48) in diameter) of Tycho Brahe’s 1572 supernova, combining data from theChandra X-ray Observatory (yellow, green, blue; NASA/CXC/SAO), Spitzer Space Telescope (red; NASA/JPL-Caltech), and the Calar AltoObservatory (white stars; Krause et al.). The transparent magenta box indicates the pointing of the ACAM (Auxiliary-port CAMera) on theCassegrain focus of the WHT with a FOV of 4 (cid:48) × (cid:48) . The center panel shows a zoom-in on the ACAM FOV. Using the same pointing as withACAM, we have covered the same region with the GH α FaS
Fabry-P´erot interferometer with a FOV of 3 (cid:48) .4 × (cid:48) .4. The green box marks theregion which is zoomed-in in the right panel to show our reduced and integrated GH α FaS H α image. and thus do not require cosmic ray rejection. We used ahigh resolution mode, acquiring data on 1024 × with R ∼
21 000 resolving power and a pixel scale of nearly0.2 (cid:48)(cid:48) /pixel. The free spectral range (FSR) of the etalon was8.56 ˚A or 392 km s − centered at 6561 ˚A and split into 48 channels that differ in their central wavelength, leading to asampling velocity resolution of 8.16 km s − . The instrumentresponse function is well approximated by a Gaussian withFWHM of 19 km s − (Blasco et al. 2010).The observations were conducted on 15-19 November2012 under FWHM (cid:39) (cid:48)(cid:48) seeing conditions. Successive ex-posures differ by one channel, and thus 48 successive expo-sures complete one cycle that covers the full spectral range.Total integration time was ≈
10 s -exposures. Observations conducted in several cy-cles provide homogeneous airmass and atmospheric condi-tions for all channels. We reduced the data (see Figure 1) byfirst applying the phase correction to all exposures individ-ually, where we follow the standard procedure for GH α FaS data described in (Hernandez et. al. 2008). The phase cor-rection is a process of designating the photons’ positions forthe interference rings and assigning the corresponding wave-length to each position ( x, y ) on the image, λ i = λ i ( x, y ) .As a result, from each exposure D i ( x, y ) , we build a data- subcube D i ( x, y, λ ) with 48 monochromatic images. In or-der to use the largest possible GH α FaS
FOV, we did not usethe optical derotator. Therefore, we have to align and dero-tate the observed data-subcubes before co-adding them. Todetermine the exposures’ relative pointing and orientation,we measure the centroid positions of bright point sources(stars) on a stack that combines each exposure with the × exposures that precede and suceed it. Thus, we assure thatat least three bright point sources are detected with enoughflux for a < pixel centroid precision. This requirementmeans that we have to discard the first four and the last four D i of each ”run” of consecutive observations, but this con-cerns only 57 exposures (2%). Along with removing 13 cy-cles that suffer from reflected light or other defects that wenoticed in visual inspection of the data, and also excluding7 cycles that lack reference sources for derotation, we retain2439 exposures in 52 cycles. The final datacube D ( x, y, λ ) results from summing all data-subcubes and consists of 48calibrated constant-wavelength slices.Besides the datacube, that is, the stack of all aligned data-subcubes, we produce a background-cube and a flatfield-cube. To this end, we model the background flux inindividual exposures as well as the flatfield image (theposition-dependent throughput of the optical system), andsubsequently process the individual background and flatfieldframes in exactly the same manner as the correspondingdata frames. By constructing the co-added background- andflatfield-cube and including them in our parametric models ofthe observed shock emission, as opposed to subtracting thebackground from the individual exposures and dividing themby the flatfield image before calibration and co-addition, wepreserve the photon (Poisson) statistics in the data-cube andsimultaneously account for the variable effective exposuretime in the stack of aligned subcubes. Details on modellingof the flatfield and background frames are given in AppendixA. ANALYSISOur analysis is guided by the following goals and princi-ples:1. to determine the H α narrow-line (NL) width across amaximal area of the observed shock2. to achieve maximum spatial resolution, implying min-imal binning (bin size) and signal per bin3. to still extract line parameters reliably and, in particu-lar, characterize their uncertainties accurately4. to do so even when including up to two additional lines(10 model parameters)5. to compare single-NL and multi-line models and toquantify their relative evidenceIn order to achieve these aims we decide to perform param-eter estimation and model comparison using Bayesian infer-ence instead of traditional (maximum-likelihood, minimum- χ ) fitting routines. We also account for the Poisson statisticsof the data, as opposed to the often tacitly applied Gaussianapproximation. Details on our method can be found in sub-section 3.4, Appendices E and F.3.1. Motivation for a multi-line analysis
Narrow H α lines in non-radiative shocks around SNRs areconventionally modeled by a single Gaussian, which theoret-ically has a width (FWHM) of W NL ∼ km s − in accor-dance with the pre-shock temperature of ∼ K expectedfor the warm interstellar medium. However, visual inspec-tion of our data indicates that this basic model may not be de-scriptive; theoretical considerations also justify investigationof more complex models for the shock emission spectrum.Since the spatial elements (bins) cover a small but finite partof the shock, and moreover the observed filament is the pro-jection of an extended shock section along the line-of-sight(LOS), a single-NL model is only suitable if one assumes thatwithin each projected resolution element (bin), the followingconditions are realized: • the pre-shock ambient medium is homogeneous withconstant temperature; • the velocity distribution is uniform, without differen-tial bulk velocity components along the LOS; and • there are no precursors (classical BDS).If the three conditions above are all satisfied, the projectioneffects cannot modify the NL width, because both the widthand the centroid are the same everywhere, corresponding tothe upstream plasma temperature and bulk speed, respec-tively. Unresolved or projected inhomogeneous pre-shock temperature causes the superposition of Gaussians of dif-ferent width. The presence of a CR precursor can alter theNL in two different ways: the cold neutrals in the interstellarmedium will be heated, resulting in the NL being broadenedbeyond the normal 10–20 km s − gas dispersion (Morlino etal. 2013) and they acquire a bulk speed up to few % of theshock speed. Therefore, inhomogeneous CR emission canbe one reason for a non-Gaussian NL. CRs also transfer mo-mentum to the pre-shock neutrals and potentially introduce aDoppler shift between the gas in the precursor and the pre-shock gas not affected by the precursor if the shock is notmoving strictly perpendicular to the LOS (Lee et al. 2007),in addition to shifts from any pre-existing differential bulkmotion in the ambient medium. These projected or spatiallyunresolved shifts will also alter the line shape, and if the ef-fect is pronounced enough, even lead to a split in the NL.One way to account for those distortions of the Gaussian lineshape is by allowing generalized Gaussians with non-zerothird- and fourth-order moments, for example using a Gauss-Hermite-polynomial. However, we visually identify splits inthe narrow line, and choose to represent a non-Gaussian NLby a sum of two Gaussian components.Apart from CRs there is another possible precursor – a BNprecursor, which introduces a new line component: the so-called intermediate line (IL) that can be described by a Gaus-sian with FWHM in the range 100–350 km s − for a typicalshock velocity in the range of [1500, 3500] km s − (Morlinoet al. 2012). We therefore also investigate models that includethis IL.Previous observations focused on the brightest H α knot,where the spectrum was measured by combining all pixelsacross the so-called ‘knot g’ (Ghavamian et al. 2000; Leeet al. 2007) which, as HST imaging shows, has a complexstructure (Lee et al. 2010). It was then used to estimate theNL and IL line widths of ’knot g’, with best-fit 44 ± − and 150 km s − (Ghavamian et al. 2000), or 45.3 ± − and 108 ± − (Lee et al. 2007).We set out to check if these results hold when ‘knot g’is spatially resolved and its parts analyzed individually, orif spatially averaging the spectrum introduced an artificiallylarge W NL and an IL that may have mimicked the effectsof a CR or BN precursor. We also vastly extend the arealcoverage of those earlier studies to include the lower-surfacebrightness parts of the filament, and there, too, exploit thehigh spatial resolution provided by the GH α FaS instrument.In this way we can investigate whether signature of the exis-tence of CR and BN precursor are present also in regions lesscomplex than the ’knot g’ and looking for possible differencein the physical properties of the shock. Strictly speaking the IL is not a perfect Gaussian, because it results fromthe population of neutrals undergoing CE in the BN precursor and they havenot enough time to thermalize to a single temperature (Morlino et al. 2012).
We find that the contribution of the emission arising in thephoto-ionization precursor (PIP), previously suggested andmeasured by Ghavamian et al. (2000) and Lee et al. (2007),is negligible (see Appendix B), so that we do not need toaccount for it in our filament flux models.3.2.
Definition of models
For each location (bin), we consider several parametrizedmodels ( S ) to characterize the shock H α emission. Regard-less of the parametrization (type of model) or specific param-eter values ( θ ), we factor in the local flatfield spectrum ( F )and add to it the observed background spectrum ( B ) beforecomparing the model with the data. This has the advantage ofpreserving the correct photon statistics, and contrasts with thecommon approach of subtracting the background from thedata and dividing by the flatfield before modelling. Hence,for a given location (bin), the full model is represented by M ( θ ) = S ( θ ) × F + B , while by model we mostly referjust to S , the intrinsic, or ”source” component. Note that inthis expression, F and B are the result of binning the flatfieldcube and the background cube in the same way as the data,and therefore they only depend on wavelength ( λ ) for the binthat M describes. F and B are constructed separately fromthe data and inserted into the model without free parameters.More information on how we established F and B can befound in Section 2 and Appendix A.Every S consists of a NL with Gaussian profile as wellas a constant component ( c ) that accounts for the sum of thecontinuum level and the broad H α line (BL), which is severaltimes wider than our spectral range. Depending on its type, S may further include one or two Gaussian components thatrepresent an IL or an additional NL. Overall, in each bin wetherefore have four different models S to compare with thedata spectrum:1. NL - constant plus single narrow line2. NLNL - constant plus two narrow lines3.
NLIL - constant plus one narrow and one intermediateline4.
NLNLIL - constant plus two narrow lines and one in-termediate lineIn connection with calculating the models’ relative evi-dence, we also consider a ”no-line model” (0L, i.e. withthe constant-spectrum as the only component), which givesan auxilary baseline for the relevance of at least the NL be-ing present in the data. A possible choice of model param-eters are the component fluxes ( f c = f constant and f i = f { NL , NL1 , NL2 , IL } ), line centroids ( µ i ), and lines’ FWHM( W i ). The general form of this model is S ( λ ) = f c + (2 π ) − / (cid:88) i f i exp (cid:0) − ( λ − µ i ) / (2 σ i ) (cid:1) /σ i , (1) where σ i = (cid:112) W i + W / √ is the observed (in-strumentally broadened) Gaussian dispersion, and W instr theFWHM of the instrumental response.In practice, our models employ a transformed version ofthose parameters, which has the advantage of their more di-rect interpretation, and a simpler functional form of the de-sired parameter priors (see next paragraph). For example,instead of the two NL centroids, µ NL1 and µ NL2 , the NLNLand NLNLIL models use the NL centroid mean ( < µ NL > )and the separation between the NLs ( ∆ µ NL ). Similarly, theIL centroid is specified by its offset from the NL centroidor NLNL centroid mean, ∆ µ IL = µ IL − µ NL . We replacethe component fluxes by the total flux and the components’flux fractions, and use the logarithm of the total flux and linewidths as they are strictly positive quantities. A more de-tailed account of the definition of model parameters can befound in Appendix D.The parameter priors, P ( θ ) , are an integral part of themodel definition: they encapsulate what we know (or as-sume) about the relative probabilities of parameter values apriori , before considering the data. In particular, they canimpose parameter boundaries by way of being zero outsideof those. The line centroid parameters are effectively re-stricted to our spectral window. In our models the FWHMparameters W NL and W IL are limited to [15, 100] km s − and [100, 350] km s − respectively. The lower boundary of W NL reflects the lower limit of the pre-shock temperature( ≈ W IL range is the theoretical expectation forshock velocities around 2000 km s − and a range of shockparameters (see Figure 10 in Morlino et al. 2012).Inside of the parameter boundaries, we desire to notstrongly favor any particular parameter values a priori. How-ever, on physical grounds we prefer a smooth transition ofthe prior to zero for parameter values approaching the bound-aries (see Figure 2). We therefore employ shifted and scaledBeta distributions with α = β = 1 . for the centroid param-eters and logarithmic line widths, and a Dirichlet distributionwith α = 1 . for the component flux fractions which havethe constraint of being summed up to unity. For compari-son, the special case of ”flat” Beta and Dirichlet distributionswould be realized by setting α (= β ) = 1 . . Thus, our choiceslightly favors the centre of the allowed parameter range. Thelogarithm of the total flux has a flat unbounded prior. Modelparameters and their priors are summarized in the Table S1,and detailed definitions presented in Appendix D. f c f NL f IL W NL W IL F tot [counts] μ NL Δμ IL [kms -1 ][kms -1 ] [kms -1 ][kms -1 ] Figure 2 . Parameter estimation via Bayesian inference for a bin in the NE filament of Tycho’s SNR. The top-right blue panel shows the observedspectrum (solid-black line), the background model (dashed-black line) and components of the intrinsic median NLIL model (dashed-red lines).The median model is overplotted with the solid red line. The remaining 8 panels are 1D-marginalized posteriors over model parameters (solid-black lines): total flux ( F tot in counts), flux fractions in the continuum ( f c ) and lines ( f NL and f IL ), NL centroid ( µ NL ), IL offset from theNL centroid ( ∆ µ IL ), and intrinsic line widths ( W NL and W IL ) with the latter three quantities given in km s − . Dashed-black lines are priordistributions and vertical red lines are the estimated parameters of the median model, i.e. the median values (solid red), and the boundariesenclosing the highest density 95% confidence intervals (dashed red). Binning
We analyze two shock filaments, one in the more easternpart of the NE rim which contains ’knot g’, and the otherin the more northern part (Figure 3). We use the WeightedVoronoi Tessellation (Diehl & Statler 2006) with the adap-tive bin size to spatially bin the pixels and obtain a signal-to-noise ratio (
S/N ) of (cid:38) (Appendix C) in the wavelength-integrated signal that remains after subtraction of the back-ground. This implies an average minimum S/N of 1.4 perspectral element. Due to the seeing of 1 (cid:48)(cid:48) we require at least 5 pixels across, so that for a round bin this implies a minimumof 19 pixels. We exclude bins that would require > pix-els for our target S/N , so that unaccounted-for residual back-ground variations, which we estimate to be at most ∼ ofthe background level, do not significantly effect our measure-ments. Following these criteria, we study 73 Voronoi bins inthe eastern and 9 Voronoi bins in the northern filament.3.4. Parameter estimation and model comparison
For each Voronoi bin, we want to find which of the fourmodels, M (see Sec. 3.2), and which vector of model param- n o r t h e r n fi l a m e n t e a s t e r n fi l a m e n t W NL [kms -1 ] f IL f IL W NL [kms -1 ] Figure 3 . NE filament of Tycho’s SNR. The two boxes on the GH α FaS H α image show northern and eastern shock filaments. The four panels ineach box represent: the bin contours overplotted on the background subtracted cube, spatial variation of the median values of evidence-weightedNL width (in km s − ) and IL flux fraction posteriors, and bins that show necessity of IL (red), second NL (blue), and both the second NL andIL (green). In this last (rightmost) panel, white indicates either bins with too little flux, or those where no line is required in addition to thesingle NL to describe the data. eters θ , best explain its data (spectrum). Since data are noisy,model comparison and parameter estimation are inherentlyprobabilistic. For both tasks and the reasons discussed be-low, we use Bayesian inference.The ”standard” approach to estimate parameters are themaximum-likelihood (or minimum- χ ) method. It relies on awell-defined likelihood maximum (mode), and convergenceof the optimizer deteriorates when high noise or multiplemodes are present. Both conditions are met in our study:we desire high spatial resolution and therefore small bin size,implying low S/N . Our models are non-linear and compriseup to 10 parameters, implying generally multiple modes. Wewish to characterize all those modes, not just the ”main”(global) maximum, towards reliable, high-confidence levelparameter uncertainties (”errors”), instead of the minimalbut common 68%-confidence (” − σ ” ) error. Often, andin our models with their non-trivial likelihood function, er-ror propagation over a large parameter range is cumbersomeor impossible. Finally, the maximum of the likelihood doesnot provide a quantitative, well-defined measure for relative probability of different models with different parametrization(model comparison). These circumstances make maximum-likelihood or other fitting methods insufficient for our pur-poses.Bayesian inference, by contrast, provides the full, mul-tivariate parameter probability distribution function (PDF),the so-called posterior P ( θ | D, M ) , as well as its integral(marginalization) over all parameters, P ( D | M ) – the evi-dence . Evidences of models are the relative model proba-bilities, without reference to any specific (e.g., best-fit) pa-rameter values. Bayes’ theorem states that the posterior as afunction of θ is P ( θ | D, M ) = P ( D | θ, M ) P ( θ | M ) P ( D | M ) . (2)It is proportional to the product of likelihood, L = P ( D | θ, M ) , and prior , P ( θ | M ) . L reflects the model andis the probability of the data for given model parameters andmeasurement errors. Our IPCS instrument counts photons,which are described by a Poisson distribution that thereforerepresents the measurement error, with expectation value andvariance equal to the flux predicted by the spectral model.The prior is the parameter PDF that we know or assume be-fore taking into account the data at hand. Apart from makingthese assumptions or knowledge explicit (fitting methods im-plicitly assume a flat prior in the chosen parameters), it hasthe advantage of naturally facilitating self-consistent param-eter changes.We represent the N -dimensional posterior PDF of the N model parameters as a sample, which we obtain using aMarkov chain Monte Carlo (MCMC) method. For detailson the sampling algorithm, see Appendix E. The posteriorcan be summarized in many ways. One is the posterior max-imum; we do not emphasize as it is a relatively noisy estima-tor and its computation is not unique. Instead, we provide themedian and the highest-density (shortest) 95%-confidenceintervals of the one-dimensional marginalized distributionsthat result from integrating P ( θ | D ) over all but one parame-ter. We deem 95% to be the minimal confidence level worthquoting, and more reliable than the frequently employed 68%(” − σ ”) level, which carries a high probability (32%) of notincluding the optimal parameter values.For the parameter estimation, we are interested only inthe posterior’s shape (relative parameter probabilities)whichdoes not necessitate normalization by the evidence. How-ever, the Bayes factor, i.e. the evidence ratio of two models,is a probabilistically well-defined, quantitative measure forcomparing models. For a given M , the evidence is definedas P ( D | M ) = (cid:82) θ P ( D | θ, M ) P ( θ | M )d θ . It is the probabil-ity of observing the data when assuming that the model is”true” but parameters are not specified. In practice, the evi-dence integral is often high-dimensional and therefore com-putationally intensive. In order to approximate it numeri-cally, we use the cross validation (CV) likelihood (Bailer-Jones, C.A.L. 2012), particularly the leave-one-out CV like-lihood: P ( D | M ) = L LOO − CV ( D | M ) , see also Appendix F.To compute it, samples are drawn from the data partition pos-teriors instead of the prior as in ”standard” evidence integrals,and it has the advantage that it depends on the prior only tosecond order. Each bin spectrum has 48 elements and theLOO-CV is applied to these elements by predicting each ofthem from the remaining 47 elements under the model M (marginalized over the parameters).Since P ( D | M ) can be a very large (or small) number, itsabsolute value is meaningless, and only relative values areneeded for different models, we express it as the base-10logarithm of the ratio with some reference model. As a mat-ter of choice, we consider 0.5 dex log-evidence differencesas ”significant” to clearly prefer one model over another.This choice is somewhat conservative; testing on simulateddata reveals that we start to distinguish the correct models at0.2 dex, while our numerical precision is around 0.05 dex.With 0.5 dex criterion per bin we do not rule out the respec-tive other models, but rather indicate that significant evidence exists that an IL or double-NL (or both) is present in the dataas a ”population” in the filament overall, or conversely, thatsuch an additional line emission is most likely not presentif the evidence for the NL model relative to other models islarger than 0.5 dex.The Bayes factors, and in turn the fraction of bins that showsignificant evidence for a double-NL or an additional IL, de-pend on the line width used to distinguish an NL from an IL.Our choice of FWHM=100 km s − as lower IL width limitcorresponds to shock speeds as low as 1500 km s − whileprevious results show the shock speed of Tycho’s SNR tobe at least 2000 km s − (Ghavamian et al. 2001). W IL <
100 km s − requires unrealistic full electron-proton tempera-ture equilibrium (Caprioli 2015), or shock speeds lower than1500 km s − and zero equilibration, i.e. T e /T p = m e /m p .The latter case was already debated by Morlino et al. 2012,and furthermore T e /T p (cid:28) . was never measured in anyof the remnants (Ghavamian et al. 2013). On the other hand,NL widths larger than 60 km s − have not been observed be-fore, and > km s − would require cosmic ray accelera-tion efficiency >
40% (Morlino et al. 2013), compared to themore realistic 10–20% efficiency in SNR shocks (as was alsofound in Tycho’s SNR by Morlino & Caprioli 2012). There-fore, our NL-IL separating line width limit robustly distin-guishes NL and IL that arise from different processes (coldneutral excitation and broad-neutral precursor). RESULTSFollowing analysis described above, we calculated pos-terior parameter distributions for every model and Voronoibin, which we then summarized by the median of the 1D-marginalized posteriors as a central parameter estimator, andthe boundaries of the shortest (highest-density) 95% confi-dence interval. We also compared models using the Bayesfactors (evidence ratios) calculated by the CV likelihoodmethod. Our focus is on the measured NL width, evidencefor and magnitude of a split in the NL, evidence for an IL,its strength and width, the variation of LOS velocities acrossthe filament, and possible correlations among the line param-eters.In Figure 2, we illustrate the application of our analysis toone of the bins from our data set. The observed spectrum(solid-black line) is shown in the top blue panel, and themedian NLIL model in solid red, while the model compo-nents are shown with dashed lines: the nonparametric back-ground model (dashed black), and the parametrized ”source”(SNR emission) model components – a constant component,one NL and one IL – in dashed red. The background spec-trum, which has been derived and fixed independently (seeAppendix A) shows geocoronal and Galactic H α emission,including the Galactic [NII] line at around -130 km s − LOSvelocity. The 1D-marginalized posterior parameter distribu-tions (solid black) and prior parameter distributions (dashedblack) are shown in the remaining 8 panels. The prior pa-rameter distributions are overplotted for comparison with theposterior. The priors are chosen to not strongly prefer anyparameter values over the allowed parameter range, but taperoff smoothly towards zero at the range limits. Most poste-riors are significantly different from the priors, thus all theparameters are well constrained by the data. The parame-ters of the median model are denoted with solid-red verti-cal lines. In this example, the median NL FWHM width is W NL ≈ km s − . The IL is W IL ≈
210 km s − wide andcomprises ≈
40% of the total flux. With dashed-red verticallines we marked the boundaries of the 95%-confidence inter-val. Compared to all other model parameters, the posteriorshape of W IL is more sensitive to the choice of prior (Ap-pendix F). Still, the median and 95%-confidence intervals of W IL of a flat and Beta prior agree within ≈ evidence-weighted • We consider the distribution of all 82 evidence-weighted posteriors’ medians, as well as their × lower and upper 95%-confidence interval boundaries(Figure 6, top panels). This yields information on thevariability of line parameters across the filament. • Second, we combine (average) the evidence-weightedposteriors of all bins, providing a representation of theinformation that we typically find in one individual bin(Figure 6, middle row of panels, black curves). Wealso combine model-specific posteriors separately, toillustrate the relative contribution of different modelsand how parameters estimates depend on the modelchoice (coloured curves). • Finally, we evaluate the parameter constraints as im-posed by all data (bins) combined. For that purpose,we sample from all bins’ evidence-weighted posteri-ors, each time computing the median value over allbins. That is, we define one new parameter for eachmodel parameter: the cross-bin median . Its posterioris plotted in the bottom row of Figure 6, and againsummarized by its median and 95% confidence inter-val boundaries. We emphasize that this measure isdecidedly distinct from a modelling the spectrum ofall bins combined, because it is still based on models constrained by all bins individually and independently,and, in particular, allows for local shifts in the line cen-troids to avoid artificial line broadening.Apart from estimating model parameters, we compare theprobability of models (different number of emission lines)against each other. In each bin, we define a model as favoredif it has the highest evidence and a > . dex logarithmic ev-idence ratio ( > probability) over the NL model (seeFigure 3, right panel). This threshold is a matter of choice; itis twice the 0.2 dex by which the correct model is typically fa-vored in our tests on simulated data, and reflects our approx-imate notion of the minimum for a ”significant” probability.If no model satisfies the 0.5 dex requirement, we considerthe fiducial single-NL model as favored. In this scheme, thedata in one bin may favor an IL (NLIL or NLNLIL model), adouble-NL (NLNL or NLNLIL model), both (NLNLIL), ornone of them (NL). Using this criterion for each bin sepa-rately, we ascertain the fraction of bins in which the evidenceindicates an IL, as well as the fraction of double-NL occur-rence. 4.1. Narrow-Line width
The example in Figure 2 is by no means the only one wherethe Narrow Line (NL) width, W NL , is much larger thanthe maximally allowed thermal NL broadening (20 km s − ).On the contrary: our central estimator, the median ofthe evidence-weighted W NL posterior, is never lower than35 km s − , in any bin. The spatial distribution of estimated W NL is shown in panel 2 of Figure 3. We do not recognizea strong spatial pattern of W NL ; most of the bin-to-bin vari-ations appear to be randomly distributed. However, in thenorthern part of the eastern filament there appears to be atrend of lower (higher) W NL on the pre-shock (post-shock)side, and a generally higher W NL in the southern part. Thespatial variation of W NL (Figure 3) probably indicates varia-tions in the amount of neutrals in the ambient medium: moreneutrals imply a more efficient ion-neutral damping of mag-netic waves excited by CRs, thus resulting in a pre-shock gasheated to larger temperatures. The histogram of median and95%-confidence interval boundaries can be found in the top-left panel of Figure 6. As measured by the cross-bin median,the global W NL is (54 . ± .
8) km s − with 95% confidence.Suprathermal NL widths are required even when only mod-els with double-NL or an additional IL are considered; theiraverage W NL is only a few km s − lower (see next two sub-sections). 4.2. Evidence for a split in the NL
In 18% of the Voronoi bins (15 of 82) we find significantevidence for an NLNL(IL) model, i.e. for a split in the NL.For one of these bins (bin 5 in Table S3), we show posteriorsin Figure 5. In this example, NL centroids and widths arewell determined, while the flux of the second (right, high-velocity) NL closely follows the prior distribution. The two0
NLIL vs NL NLNL vs NL NLIL vs NLNLNLNLIL vs NL NLNLIL vs NLNL NLNLIL vs NLIL NLIL vs NL NLNL vs NL NLIL vs NLNLNLNLIL vs NL NLNLIL vs NLNL NLNLIL vs NLIL
Figure 4 . Spatial variation and histogram representation of logarithmic (base 10) evidence ratios across the NE rim in Tycho’s remnant. Thelogarithmic evidence ratios are shown in the range ± > NLs are separated by ≈
40 km s − and have median widthsof ≈
52 km s − and 70 km s − . That is, despite using twoGaussians, both are much wider than the thermal 20 km s − ,indicating that the broadening is mostly not an artifact of un-resolved differential line-of-sight motion. Not only is the logevidence ratio of NLNL to NL model larger than 1 dex infavor of NLNL model, but it is also more than 3 times ( (cid:38) against an IL.The example in Figure 5 is not untypical: even in locations where double-NL models are favored , the average measured W NL is 49 km s − , again much larger than the upper limit ofthe intrinsic thermal NL width. The median parameters and95%-confidence intervals for the favored model of each binare listed in Tables S4 and S5. The cross-bin median NL-centroid separation constrained from all the bins in the NErim is ∆ µ NL = 38 . ± . km s − (95% confidence).In case of a perfectly spherical shock and a homogeneousambient ISM, the parts of the filament closest to the upstreamare seen edge-on and we would expect to see a single NL,while the parts of the filament closer to the downstream areinclined to the LOS and should exhibit a split in the NL.1 f c f NL1 f NL2 W NL1 [kms -1 ] W NL2 [kms -1 ] F tot [counts]< μ NL > [kms -1 ] Δμ NL [kms -1 ] Figure 5 . Parameter estimation via Bayesian inference for a bin in the NE filament of Tycho’s SNR that requires a second NL. The top-right bluepanel shows the observed spectrum (solid-black line), the background model (dashed-black line) and components of the intrinsic median NLNLmodel (dashed-red lines). The median model is overplotted with the solid red line. The remaining 8 panels are 1D-marginalized posteriors overmodel parameters (solid-black lines): total flux ( F tot in counts), flux fractions in the continuum ( f c ) and two narrow lines ( f NL1 , f NL2 ), NLcentroid mean (cid:104) µ NL (cid:105) , the separation between the two NLs ∆ µ NL , and intrinsic NL widths ( W NL1 , W NL2 ), the latter four all given in km s − .Dashed-black lines are prior distributions and vertical red lines are the estimated parameters of the median model, i.e. the median values (solidred), and the boundaries enclosing the highest density 95% confidence intervals (dashed red). Evidence for a double NL in the eastern filament is foundin a few inner bins (blue bins in Figure 3), but also in someouter (green) bins. Given the small number of double-NLoccurrence and its scattered locations, any determination ofthe shell geometry would be vague. However, detection ofdouble NLs with W NL (cid:29)
20 km s − clearly points towardheating and momentum transfer in the CR precursor. 4.3. Intermediate-Line evidence and parameters
As quantified by the Bayes factors, we find that 34% of theVoronoi bins (28 bins out of 82) are significantly better ex-plained when a line is added to the fiducial NL-only model.In 74% of those, the NLIL or NLNLIL model is also pre-ferred over an NLNL model, which means 24% of the binsoverall. This is illustrated in Figure 4, where logarithmic(base 10) ratios of all models versus one another are shownin separate panels and each panel shows the spatial variationof the corresponding ratio. These values are visualized by the2
Figure 6 . Summary of results for narrow-line width ( W NL , left panels), intermediate-to-narrow line flux fraction ( f IL / f NL , centre-left), IL width( W IL , centre-right) and NL-centroid separation ( ∆ µ NL , right-most panels). All figures are based on the marginalized posteriors of all 82 binsand × models, weighted within individual bins by relative model evidence. Top row:
Distribution of median (green) and highest-density95%-confidence interval boundaries (blue, red), quantifying the variation across the filament. For the f IL /f NL , we adopted a log-scale inorder to make the low-end confidence interval boundary histogram more visible. Middle:
Sum of all posteriors (solid black), illustrating theaverage posterior an individual bin and the typical relative contributions of single-NL (orange), NLNL (green), NLIL (blue) and NLNLIL (red)models.
Bottom:
Posterior of the median across all bins (solid), in effect using all data to constrain the respective parameter values. These aresignificantly narrower than the individual bin posterior. The prior is shown by the dashed curve in the middle and bottom panels, whereas theprior of the cross-bin median is given in the bottom panels. histograms, but also tabulated in the Tables S2 and S3. Apartfrom just a few bins, there is no clear favor of NLNLIL modelover NLIL model (panel 6). The situation changes when itcomes to comparison between NLNLIL and NLNL (panel 5), further confirming the necessity of an IL component. Oneexample (bin 17 in Table S2) is illustrated in Figure 2, wherethe logarithmic evidence ratio of the NLIL relative to the NLand NLNL models is larger than 1 dex. This means that an3IL in addition to a single NL explains our data more than 10times better than a simple NL model or NLNL model, irre-spective of any particular parameter values. Both NLIL andNLNLIL models are equally likely for this bin which impliesno need for the second NL component. For this particularbin, we present posteriors for all other models in Figures A.4,A.5, A.6.Prominent ILs are seen across the entire filament (panel3 in Figure 3), with IL flux fraction of up to 42%. If weconsider only bins that favor NLIL or NLNLIL model, we getan IL flux fraction of 28% on average, and the intermediate-to-narrow flux fraction f IL /f NL is estimated at 0.61 with a95% confidence interval of (0.01–1.87).In contrast to the IL flux, W IL is not well constrained inindividual bins, as indicated by the similarity of prior and av-erage bin-specific posterior (Figure 6, middle panels). How-ever, several bins with high S/N and/or strong IL emission(as the bin in Figure 2) have good constraints on IL width.Moreover, the combination of the data in all bins provideseven better information: the cross-bin median is W IL ∈ [166 . , . km s − (95% confidence, see bottom of Fig-ure 6). The global 95%-confidence interval of f IL /f NL is[0.34, 0.47]; however, we caution that it varies considerablybetween bins, with median log( f IL /f NL ) ∈ [ − . , . .4.4. Line-of-sight velocity
We present the observed line-of-sight (LOS) velocity, i.e.the NL centroid µ NL , in Figure 7. The map of the bin-specific µ NL is shown in the left panels. In the top-rightpanel, we show the corresponding histogram (green), as wellas the distribution of 95%-confidence interval boundaries.Across the entire NE rim, we find median LOS velocitiesin the range [ − . , − . km s − . Notably, the northernfilament moves with respect to the eastern filament; µ NL = − . km s − and − . km s − on average. The bin-to-binvariations in µ NL are most noticeable in the eastern filament,indicating inhomogeneities in the shock and, in turn, in theISM density. The bulk (median) LOS velocity of the filamentas a whole is µ NL = − . ± . ) km s − at the 95% confi-dence level (see middle-right panel). Converting to the localstandard of rest (LSR), we obtain V LSR ≈ − km s − . Wealso check for the correlation between LOS velocities andsurface brightness (bottom-right panel), but find them to beuncorrelated. 4.5. ’knot g’ This location in the eastern filament has been the targetof previous observational H α studies, but it was not spa-tially resolved with spectroscopic data. We spatially resolvethis ’knot g’, and the entire NE filament, with our spectro-scopic data for the first time. More precisely, the centroidsof 6 of our bins drop within the Lee et al. 2007 slit. Av-eraging over these bins, we find W NL = 49 ±
15 km s − (68% confidence), which is in agreement with Lee et al.’s e a s t e r n fi l a m e n t n o r t h e r n fi l a m e n t -150 -100 -50 0 -150 -100 -50 0 ρ =0 ρ =-0.2 -150 -100 -50 0 μ NL [kms-1]
20 15 10 5
Figure 7 . NL centroid (line-of-sight velocity) variation in the NErim (left panels), distribution of median µ NL (green) and highest-density 95%-confidence interval boundaries (blue, red) in the top-right panel, and posterior of the median across all bins (solid) usingall data to constrain µ NL (middle-right panel). The prior is over-plotted with the dashed line. The northern (top-left) filament movessystematically with respect to the eastern filament, and the latterexhibits significant internal differential motion. The bottom-rightpanel shows that NL centroid does not correlate with the surfacebrightness. Symbols and notation are the same as for Figure 8. reported values of 44 ± − . The mean LOS veloc-ity is − . km s − enclosed by 68% confidence interval of[ − − − . The mean LSR velocity in ’knot g’is found to be ≈ − km s − in agreement with previ-ous estimate by Lee et al., but with a blue-shifted offsetof nearly 6 km s − . Typical measured parameters and their68% confidence intervals in ’knot g’ are W IL = 188 [105,229] km s − , f IL / f NL = 0.35 [0.05, 0.54], and ∆ µ NL = 32.4[8.1, 46.7] km s − .4.6. Correlation in parameters
We investigate the correlation between various line param-eters and surface brightness, as well as correlation amongmodel parameters, using Spearman’s rank correlation coeffi-cient ( ρ ). It is sensitive to any monotonic relationship, evenif it is not linear as for Pearson’s correlation coefficient, anddoes not require normally distributed data. The result indi-cates strong correlation (or anti-correlation) for values close4 ρ =0.05 ρ =0.38 ρ =-0.02 ρ =0.15 ρ =0.01 ρ =0.57 ρ =-0.11 ρ =-0.03 ρ =0.05 ρ =-0.32 ρ =-0.38 ρ =-0.32 ρ =0.49 ρ =0.47 single-NL double-NL eastern northern (1) ρ =0.05 ρ =0.32 (2)(5) (6) (7) (8)(3) (4) Figure 8 . Parameter correlation: top row shows median of evidence-weighted W NL , f IL /f NL , W IL and ∆ µ NL posteriors and their dependenceon the surface brightness, i.e. total model S flux over bin area (panels 1-4). In the bottom-row panels 5-8 correlation among model parametersis tested. Squares (circles) refer to single (double) NL models in the eastern filament (black/magenta points) and northern filament (gray/pinkpoints) based on the highest CV. Wherever we have NLNL(IL) model as the one with the highest CV, W NL is set to the average of the two NLswidths. In panel 8 for bins favoring single-NL models we use the posterior results for their NLNL(IL) models. Labeled are Spearman’s rankcorrelation coefficient ρ that correspond to the color-relevant points, where black label refers to black and gray points together, magenta labelto magenta and pink points. We indicate typical error bars on the surface brightness and parameters (average shortest 68% confidence intervals)in the right corner of every panel. to ± W NL , f IL /f NL , W IL and ∆ µ NL posteriors againstthe surface brightness (SB, total flux of the intrinsic modeldivided by the bin area). We separately test correlation forbins for which a single-NL model (NL or NLIL, marked withsquares) has the highest evidence, and those bins for which adouble-NL model (NLNL or NLNLIL, marked with circles)has the highest evidence of all models. We do not find anystrong correlation: parameters of single-NL models are un-correlated with surface brightness, while double-NL modelsshow weak correlations with SB. The highest correlation iswith ∆ µ NL ( ρ =0.57), meaning that with increasing surfacebrightness, the two NL centroids tend to be more separated.In the bottom-row panels of the same Figure we estimatedcorrelation among the parameters. Although we see a hintat positive correlation between NL width and NL separation ( ρ =0.47 and ρ =0.49, see panel 8), we do not find a clear cor-relation among any of the parameters. We get very similarresults when we apply Pearson’s correlation: its coefficient iseither very close to the Spearman’s rank coefficient or closerto zero. In addition, the typical uncertainty of all these points(plotted in the top-right corner of each panel and given asan average shortest 68% confidence interval) propagates intocorrelation rank coefficient uncertainty. DISCUSSIONPrevious H α observations of Tycho’s ‘knot g’ (Lee et al.2007) were modeled by Wagner et al. (2009) computinga series of time-dependent numerical simulations of CR-modified shocks. Assuming a distance of 2.1 kpc to theremnant, they found the CR diffusion coefficient of κ =2 × cm s − and the lower limit of the injection param-eter ξ inj = 4.2 × − to be in good agreement with the obser-5vations, suggesting that CR acceleration in the shock is effi-cient. Diffuse emission 1 (cid:48)(cid:48) ( ∼ cm) ahead of the easternfilament was also detected by Lee et al. (2010) and inter-preted as emission from the CR precursor with T ∼
80 000–100 000 K.In what follows we will summarize the main results of ourpaper and look for theoretical explanations for our findings.
Presence of a CR precursor : The NL width is muchbroader than 20 km s − ( ≈
55 km s − on average) in the en-tire NE rim regardless of describing the shock emission witha single or double NL. In other words, even when differen-tial velocities (double NL) are present and accounted for, theNL is still significantly broadened. This clearly points to-ward gas heating in a CR precursor (Morlino et al. 2013).Furthermore, momentum transfer in a CR precursor mightresult in a split in the NL if we have two inclined shocksprojected on our LOS. This is something that we observe,more precisely we find significant Bayesian evidence for itin 18% of the data. Apart from the separation between thetwo NLs being 38 km s − on average, at the same time wefind their intrinsic widths are around 49 km s − . If we as-sume that we have contribution of two shocks inclined withthe same angle to the LOS, and that neutrals in the CR pre-cursor acquired 10% of the previously estimated shock ve-locity of km s − (Ghavamian et al. 2001), we find thatthe shock normal inclination of 85 ◦ –86 ◦ explains the cen-troid separation of 38 km s − . Obviously, for the same shockspeed and the smaller acquired bulk velocity, we would needthe shocks slightly more inclined, i.e. with angles < ◦ .However, since only few bins show evidence for a split inthe NL, we do not have enough information to construct thegeometry of the shock. Presence of a BN precursor : The main signature of theBN precursor is an IL (Morlino et al. 2012); 24% of the binsdemand an additional line being specifically IL. We find amedian value for the intrinsic IL width of 180 km s − , com-prising on average a 41% intermediate-to-narrow flux ratio.Moreover, the observed high pre-shock neutral fraction of0.9 (Ghavamian et al. 2001) in combination with the shockvelocity of ∼ km s − supports the BN presence sincea large neutral fraction and the specified shock velocity con-tribute to efficient charge-exchange and larger number of cre-ated broad-neutrals that take essential part in the formationof and heating in the BN precursor. Broad-neutrals are ion-ized inside the BN precursor almost immediately when theycross the shock. The newly created protons move with a bulkspeed larger than the Alfv´en speed, hence they can trigger thestreaming instability (and possibly other kind of instabilities)resulting in an increase of the ion temperature in the BN pre-cursor. In turn, the CE between pre-shock neutrals and warmions creates warm neutrals that produce the IL.For illustrative purpose in Figure 9 we report the tempera-ture profile of the pre-shock gas for specific values of shockparameters as calculated in Morlino et al. (2013) where the B N Figure 9 . Heating and length scales of CR and BN precursor for aspecific shock and CR properties (Morlino et al. 2013). Zero marksthe location of the shock front, while the negative distance from theshock front represents distance in the pre-shock region. As labeled,different lines present the extent and the level of heating in the pre-cursor for different amount of turbulent heating η TH . The blacksolid line shows the case without CRs and clearly shows the extendof the BN precursor alone. Immediate pre-shock region is affectedby both BN and CR precursors where the temperature reaches sev-eral 10 K. CR precursor extends much further from the shock front( > cm) where the gas is heated to several 10 K. CR and the BN precursor can be clearly distinguished. BNprecursor acts on scales 10 –10 cm in the immediate pre-shock region and can heat the gas to temperature of ∼ K.CR precursor length is much larger, depends of the maximumenergy of the accelerated particles and extends over several10 cm. The expected gas temperature in the CR precursoris ∼ K. Collesionless shock model prediction in partially ionizedmedium : Following results of Morlino et al. (2013) and theirFigure 9, 40–50 km s − NL widths require efficient turbu-lent heating η TH in the CR precursor, but also acceleratedparticles with the maximum momentum of p max = 40 TeV/cor higher. However, this result is dependent on the shockvelocity assumption which is 4000 km s − in the cited pa-per. Furthermore, Figure 10 in the same paper shows the ILwidth as a function of the CR acceleration efficiency (cid:15) for afixed shock speed V sh = 4000 km s − , η TH = 0.5 and p max = 50 TeV/c and two values of downstream electron-to-protontemperature ratios β down being either 0.01 or 1. Interestingly,our measured IL width of 180 km s − on average can be ex-plained with the mentioned shock parameters and the acceler-ation efficiency of 15-25%, although any further constraintsare difficult since we do not know β down . The latter value,as well as the shock velocity, is something that can be con-strained from the observations of the broad H α component.In order to obtain the measured W NL ≈ km s − , onerequires a combination of shock parameters (and CR accel-eration included): p max = 10 TeV/c, V sh =2500 km s − and β down of up to 0.1 (Morlino et al. 2013). At the same time6this configuration predicts IL width of around 300 km s − and intermediate-to-narrow flux ratio of 1.6 on average forvarious values of η TH and (cid:15) . Both these IL parameters aremuch higher than the median values we infer from our dataand analysis. We speculate that somewhat higher β down than0.1 (see Figure 13 in Morlino et al. 2012) and a shock ve-locity of around 3000 km s − would possibly be able to ex-plain the observed W IL ≈ km s − , f IL /f NL ≈ . and W NL ≈ km s − . Finally, we also notice that keeping η TH constant and increasing (cid:15) the model of Morlino et al. (2013)predicts a simultaneous increase in W NL and decrease in W IL and f IL /f NL . We find a hint for such an anti-correlation be-tween W NL and W IL (panel 7 in our Figure 8). Average and variation of velocities:
Our result for the me-dian LSR-corrected line-of-sight velocity, V LSR = ( − ± km s − is in agreement with earlier study by Lee etal. (2007), who reported narrow-line H α LSR velocity − . km s − of ’knot g’. Furthermore, Ghavamian et al.(2017) reported a value of − . km s − , that falls withinour range [ − −
25] km s − of observed LSR velocities inthe NE rim. In HI 21-cm observations toward Tycho’s NErim Reynoso et al. (1999) found V LSR = − . km s − andassociated the location of Tycho’s SNR and the HI cloudto Perseus arm. Similarly, CO emission was found at V LSR = − . km s − (Lee et al. 2004, see also Zhou etal. 2016), and might be associated with SN1572’s pre-shockgas. However, as pointed out by Tian et al. (2011), thereis no clear evidence that either HI cloud or CO cloud arephysically associated with Tycho’s NE rim. We leave furtherdiscussion of the V LSR result and its possible interpretationto future work, where we will also investigate shock and CRproperties in more detail (see Section 6). SUMMARY AND CONCLUSIONSWe present H α spectroscopic observations of Tycho’s NEBalmer filaments. This study provides spectroscopic datathat for the first time are spatially resolved (spectro-imagery),with large coverage that comprises and resolves the entire NEfilament. Our analysis is based on Bayesian inference thatenables a quantitative, probabilistic and well-defined modelcomparison, and a comprehensive, complete characterizationof the parameter probabilities.We find that the broadening of the NL beyond 20 km s − that was noted in previous studies was not an artifact of thespatial integration, and that it extends across the whole fil-ament, not only the previously covered ’knot g’. NL widthin the NE rim is typically found around 55 km s − . Such alarge width cannot be due to superposition of multiple lines.In fact, our data analysis allows us to take projection effectsinto account when interpreting the data. We are able to distin-guish between single-NL and double-NL models where wefind significant evidence for split in NL in 18% of the Voronoibins. The widths of the two NLs are around 49 km s − andtheir centroid separations 38 km s − on average. A NL width of 55 km s − implies a temperature of the up-stream gas of ≈
68 000 K. If this were the temperature ofthe unperturbed ISM where the SNR is expanding, no neu-tral hydrogen would exist in the first place, contradicting thepresence of H α emission. Hence our finding is the signatureof the existence of a mechanism able to heat the upstreamplasma in a region ahead of the shock much smaller than thecollisional ionization length-scale. As shown by the previousstudy of Morlino et al. (2013), a CR precursor is the bestcandidate to explain the widening of the NL, opening thepossibility to study particle acceleration at shock using H α emission. The fact that the NL width ranges from 35 km s − to 72 km s − across the NE rim suggests that the amount ofneutrals in the ambient medium varies which imposes dif-ferent degree of damping of magnetic waves excited by CRstreaming.Likewise, we confirm the suspected presence of an IL, andshow it to be widespread (24% of the bins). Typical IL widthsand intermediate-to-narrow flux ratios are 180 km s − and0.41, respectively.Our model parameters also comprise the line-of-sight ve-locity centroids. After correction to the local standard of rest,their median is V LSR = ( − ± km s − , in agreement withthe Lee et al. (2007) investigation of ’knot g’.Overall, our results reveal an interplay between two pre-cursors in the Tycho’s NE rim: broaden NL widths point to-ward the evidence for the presence of a CR precursor, whiledetected IL reveals the presence of a BN precursor.From the knowledge of the NL width only it is not pos-sible to determine the CR acceleration efficiency, becausesuch width depends on many parameters (shock speed, max-imum energy of accelerated particles, electron-ion equilibra-tion and turbulent heating). Nevertheless, we can concludethat, assuming a shock speed between 2500 and 3000 km s − ,our result is compatible with having a maximum CR energy >
10 TeV, a turbulent heating >
10% and an acceleration ef-ficiency > few %. The degeneracy between these parame-ters could be broken using other information coming fromthe broad-line width and intensity (giving more precise in-formation on shock speed and electron/ion equilibration) andX-ray/gamma-ray observations (determining the maximumenergy of accelerated particles). The difficulties in perform-ing such calculations rely in the fact that the broad line is notknown with the same accuracy as the NL and IL and that thegamma-ray emission do not have enough spatial and spectralresolution to fix unambiguously the maximum energy (Park2015; Morlino & Blasi 2016). Improvements in this regardswill surely come from the Cherenkov Telescope Array. Fur-thermore, parallel to the study of the GH α FaS narrow H α -line profiles, we have conducted an investigation on the samepart of the Tycho’s remnant using OSIRIS (Optical Systemfor Imaging and low-intermediate Resolution Spectroscopy)on the GTC (Gran Telescopio Canarias) to observe the broadH α -line profiles. In a forthcoming paper we will present re-7sults of broad-H α components of the same spatial locations(bins) along the filaments as presented in this paper, whichcombined with narrow-H α components and applied shockmodels will give a better handle on the overall conditions inthe shock, and will enable us to quantify CR properties.ACKNOWLEDGEMENTSWe would like to thank the anonymous referee for the con-structive report and a valuable contribution towards the com-pleteness of this paper. We would also like to thank Ren´eAndrae (MPIA, Heidelberg) abd Joonas N¨attil¨a (Tuorla Ob-servatory, University of Turku) for fruitful discussions andhelpful suggestions on Bayesian inference.8 APPENDIXA. Data, Flatfield and Background
In order to compare our models of Tycho’s SNR spec-tra to the data, we need to account for the variable sen-sitivity (”flatfield”) and the background flux. We eschewthe standard method of applying background subtractionand flatfield correction to compensate for both effects inthe individual exposures. Instead, we reconstruct the flat-field and background in the final product of the data reduc-tion pipeline, which is a cube (position, wavelength) of co-added, ”stacked” individual observations. Notably, the flat-field will also have a wavelength-dependence in addition tobegin position-dependent. We eventually include flatfieldand background in our models. The advantage of our ap-proach is the preservation of photon statistics and thereforeaccurate uncertainties of model parameters and evidence.Before describing the construction of individual as well asthe co-added flatfield and background, we formalize the pro-cess by which the data are generated and how it is propagatedby the data reduction pipeline.
MEASUREMENT PROCESS
Each datum (measurement) D xy,i with pixel indices ( x, y ) and exposure index i is the response of the telescope andinstrument to the incoming, seeing-convolved flux d = d ( α, δ, λ ) , which varies with sky coordinates ( α, δ ) andwavelength λ : D xy,i = (cid:90) ∞−∞ d ,i ( x, y, λ ) F xy R xy,i ( λ ) d λ , (1)where d ,i ( x, y, λ ) ≡ d ( α = α i ( x, y ) , δ = δ i ( x, y ) , λ ) has the astrometric solution, α = α i ( x, y ) , δ = δ i ( x, y ) , andthe integral over the area of pixel ( x, y ) implicitly applied. F xy is the spatially varying sensitivity of the detector andoptical system – the flatfield image . As expected, and as ver-ified by us (see below), it is the same for all exposures. R xy,i ( λ ) is the pixel- and exposure-specific wavelength fil-ter imposed by the Fabry-P´erot interferometer (etalon). Inour case, it is accurately described by a universal line-spreadfunction (LSF), r ( λ ) : R xy,i ( λ ) = r (Λ xy,i − λ ) , (2)where the wavelength calibration Λ xy,i returns the central,maximum-throughput wavelength for each pixel and expo-sure (tuning of the etalon). By definition, the LSF is centeredon (peaks at) the origin. In our case it is a Gaussian withdispersion measured from calibration spectra.Inserting (2) in (1), one sees that the datum D xy,i is the LSF-convolved spectral flux d i ( x, y, λ ), evaluated at Λ xy,i : D xy,i = (cid:90) ∞−∞ d ,i ( x, y, λ ) F xy r (Λ xy,i − λ ) d λ = F xy · (cid:16) d ,i ∗ λ r (cid:1) ( x, y, λ = Λ xy,i ) (3) ≡ F xy · d i ( x, y, Λ xy,i ) . DATA PROCESSING
By way of converting the observed images into data sub-cubes , D xy,i → D xyl,i , we make the wavelength informationcontained in them explicit and obtain a format that allowsdirect co-addition. Each of the 48 ”slices” (third index) ofa sub-cube corresponds to a wavelength, λ l . Each observed Λ xy,i is bracketed by two slices, and the corresponding flux D xy,i is assigned to them via linear interpolation: D xyl,i = D xy,i · T xyl,i T xyl,i = t l (Λ xy,i ) (4) ≡ − | Λ xy,i − λ l | / ∆ λ : | Λ xy,i − λ l | ≤ ∆ λ | Λ xy,i − λ l | > ∆ λ , where ∆ λ is the ”size” of each slice, i.e. the distance be-tween the slices’ central wavelengths, λ l . Another way todescribe this assignment is that each slice imposes a trian-gle filter t l ( λ ) = max (0 , − | λ − λ l | / ∆ λ ) . We shall useit again to construct the co-added flatfield and backgroundcubes.As D xy,i itself is a filtered version of the flux, the sub-cube D xyl,i at a given pixel ( x, y ) is nearly ”empty”, exceptfor two slices l . Adequate coverage of the spectrum there-fore necessitates multiple tunings of the etalon, or imagingthe source in varying locations on the detector, since evenfor unchanged tuning, Λ depends on x and y . The result-ing multiple sub-cubes are then coadded, albeit after project-ing and spatially resampling them onto a common astromet-rically calibrated frame ( x (cid:48) , y (cid:48) ) : D xyl,i α, δ −→ D x (cid:48) y (cid:48) l,i (5) D x (cid:48) y (cid:48) l = (cid:88) i D x (cid:48) y (cid:48) l,i . (6) D x (cid:48) y (cid:48) l is the final, co-added data cube which, apart from spa-tial binning, directly constrains the SNR shock models. Weaccount for the spatial sensitivity variations and any residualnon-constant spectral sampling rate by including them in themodels, in form of the co-added flatfield cube, F x (cid:48) y (cid:48) l , whichwe derive below along with the background cube, B x (cid:48) y (cid:48) l . PROPAGATION OF FLATFIELD AND BACKGROUND
We represent the incoming flux d ,i as the sum of ”source”flux ( s ,i ) that comprises celestial objects, in particular theSNR filament that we are interested in, and background flux( b ,i ). Because of the linearity of the integral in (3), the same9applies to the LSF-convolved flux d i : d ,i ( x, y, λ ) = s ,i ( x, y, λ ) + b ,i ( x, y, λ ) s i ( x, y, λ ) = (cid:0) s ,i ∗ λ r (cid:1) ( x, y, λ ) b i ( x, y, λ ) = (cid:0) b ,i ∗ λ r (cid:1) ( x, y, λ ) ( ) = ⇒ d i ( x, y, λ ) = s i ( x, y, λ ) + b i ( x, y, λ ) D xy,i = F xy · [ s i ( x, y, Λ xy,i ) + b i ( x, y, Λ xy,i )] ≡ F xy · S xy,i + B xy,i . Note that in contrast to the source component of the data, S xy,i = s i ( x, y, Λ xy,i ) , (7)we have absorbed the flatfield in the definition of the data’sbackground component, B xy,i = F xy · b i ( x, y, Λ xy,i ) . (8)Due to linearity of the wavelength assignment (4), D xyl,i = T xyl,i F xy S xy,i + T xyl,i B xy,i ≡ F xyl,i S xy,i + B xyl,i . Here we have defined the background subcube , B xyl,i = T xyl,i B xy,i and the flatfield sub-cube F xyl,i = T xyl,i F xy ,which are the background component of the data and theflatfield image (the response to a flat spectrum), respectivelytransformed to a sub-cube using the same slice-assignment(4) that was used for the data itself. Also, as for the data,background and flatfield sub-cubes are projected onto thesame frame and co-added: F xyl,i α, δ → F x (cid:48) y (cid:48) l,i S xy,i α, δ → S x (cid:48) y (cid:48) ,i B xyl,i α, δ → B x (cid:48) y (cid:48) l,i D x (cid:48) y (cid:48) l = (cid:88) i ( F x (cid:48) y (cid:48) l,i S x (cid:48) y (cid:48) ,i + B x (cid:48) y (cid:48) l,i ) S x (cid:48) y (cid:48) ,i = s i ( x (cid:48) , y (cid:48) , Λ x (cid:48) y (cid:48) ,i ) is s i ( x, y, Λ xy,i ) with x = x i ( x (cid:48) ) , y = y i ( y (cid:48) ) given by the astrometric solution. Thesame transformation applies to b . Since the co-added frame’s ( x (cid:48) , y (cid:48) ) corresponds to a unique sky position and we as-sume the filament emission (but not necessarily the back-ground!) to be constant between exposures, s i ( x (cid:48) , y (cid:48) , λ ) = s ( x (cid:48) , y (cid:48) , λ ) . Then, D x (cid:48) y (cid:48) l = (cid:88) i F x (cid:48) y (cid:48) l,i s ( x (cid:48) , y (cid:48) , Λ x (cid:48) y (cid:48) ,i ) + F x (cid:48) y (cid:48) l,i b i ( x (cid:48) , y (cid:48) , Λ x (cid:48) y (cid:48) ,i ) ≈ (cid:32)(cid:88) i F x (cid:48) y (cid:48) l,i (cid:33) s ( x (cid:48) , y (cid:48) , λ l ) + (cid:88) i F x (cid:48) y (cid:48) l,i b i ( x (cid:48) , y (cid:48) , λ l ) (9) ≡ F x (cid:48) y (cid:48) l · s ( x (cid:48) , y (cid:48) , λ l ) + B x (cid:48) y (cid:48) l . In the second line, we have used that F x (cid:48) y (cid:48) l,i is non-zeroonly for l nearest to the sampled wavelength, λ l ≈ Λ x (cid:48) y (cid:48) ,i . Hence, F x (cid:48) y (cid:48) l,i · s ( x (cid:48) , y (cid:48) , Λ x (cid:48) y (cid:48) ,i ) ≈ F x (cid:48) y (cid:48) l,i · s ( x (cid:48) , y (cid:48) , λ l ) for all l = 1 , . . . , , and analogous for b . However, thebackground does not factor out of the co-addition, as it maychange between exposures. In the last step, we have definedthe flatfield and background cubes F x (cid:48) y (cid:48) l and B x (cid:48) y (cid:48) l as thesum of the flatfield and background sub-cubes. BACKGROUND AND FLATFIELD MODEL
In order to isolate the source flux component of the data,we still need to model the flatfield and background.We begin by investigating the background on the coad-ded data (6,9). Using S
EXTRACTOR we mask sources(stars, galaxies, and the filament itself) on the wavelength-integrated frame D x (cid:48) y (cid:48) = (cid:80) l =1 D x (cid:48) y (cid:48) l . This frame is thedeepest, highest signal-to-noise ratio ( S/N ) data product wehave available, hence the mask is as complete as possible.We obtain the set of unmasked pixels, ( x (cid:48) u , y (cid:48) u ) which we canuse to measure the background without ”contamination” bycelestial sources: D x (cid:48) u ,y (cid:48) u ,l = (cid:88) i B x (cid:48) u y (cid:48) u l,i = B x (cid:48) u y (cid:48) u l . Next, in order to ameliorate potential undetected low surface-brightness source flux incursion, to measure variability of B ,and to average over spectral sensitivity variations in D x (cid:48) y (cid:48) l ,we select 32 boxes of (25 pix) ≈ (5 (cid:48)(cid:48) ) in particularlyobject-poor locations. For each box B j = { ( x (cid:48) u , y (cid:48) u ) j } , wemeasure the spatially integrated spectrum: B ( j ) l = (cid:88) ( x (cid:48) u ,y (cid:48) u ) ∈ B j B x (cid:48) u y (cid:48) u l ( ) = (cid:88) ( x (cid:48) u ,y (cid:48) u ) ∈ B j (cid:88) i F x (cid:48) u y (cid:48) u l,i b i ( x (cid:48) u , y (cid:48) u , λ l ) . It turns out that, apart from flux normalization, all B ( j ) l are nearly the same , with differences at the percent level (seeFigure A.1). This allows us to conclude thati) the spectral part of the co-added flatfield is spatiallyinvariant, as desired and expected after co-adding ∼ exposures with different pointing, orientation andwavelength tuning plus averaging over a ∼ (cid:48)(cid:48) boxii) the spectral shape of the background component is spa-tially invariant: b i ( x (cid:48) , y (cid:48) , λ ) = a i ( x (cid:48) , y (cid:48) ) · b ( λ ) , (10)where a i is the amplitude of the background.Proof comes by contradiction: if the spectral shape of the5 (cid:48)(cid:48) box-averaged flatfield or background varied with location, B ( j ) l would vary. Notably, we also see no systematic changeof the background between the pre- and post-shock regions.Variations of the background flux on smaller ( < (cid:48)(cid:48) ) spatialscales are physically unlikely and in any case must be small0as the observed variability can entirely be accounted for byPoisson (measurement) noise (middle and bottom panels inFigure A.1). In the bottom panel of Figure A.1 we presentdifferences between the actual background and the modeledbackground. There is no systematic effect, and the spatialvariation is within the measurement uncertainty, which inturn is much smaller than the signal. The variation betweenbackground boxes, as measured by the 32-element samplestandard deviation normalized by the photon noise, is . on average across the spectrum (solid-purple line), with max-imum value of . . The average absolute deviation (dashed-purple line) is 0.77 on average, close to the theoretical valueof 0.8 expected for the absolute value of a standard-normallydistributed variable. Therefore, any differences between theactual background (“Data”) and model (“Bkg”) are fully ex-plained by measurement uncertainties.We can therefore measure b ( λ ) on the co-added data cube.By combining all 32 background boxes, we additionally min-imize noise and residual systematics: b ( λ ) = (cid:88) j B ( j ) l ≡ ˜ b ( λ ) . The tilde indicates that ˜ b is a model of the background spec-trum. Normalization of b ( λ ) is absorbed in the a i . We now assume that a i is spatially invariant, i.e. that the backgroundamplitude changes only between exposures but not across thefield-of-view: a i ( x (cid:48) , y (cid:48) ) = a i Therefore, also a i ( x, y ) = a i is spatially constant, and wehave D x u y u ,i = B x u y u ,i = F x u y u · a i ˜ b (Λ x u y u ,i ) . Here, ( x u , y u ) are the non-masked pixels of the co-addedframe, reprojected onto the individual exposures. We do notknow the background amplitudes a i yet, but can already usethe knowledge of b ( λ ) to model F xy : (cid:32)(cid:88) i D ˜ x u ˜ y u ,i ˜ b (Λ ˜ x u ˜ y u ,i ) (cid:33) fit+norm. −→ ˜ F xy . (11)This way, we ”divide out” the non-constant background spec-trum, which is imprinted on the measurement via Λ xy,i . Thesum on the left side is only carried out for pixels (˜ x u , ˜ y u ) that are not masked in any of the exposures. The result of thesum is a non-normalized flatfield image, which is then mod-eled by a fourth-order polynomial. The fit eliminates pixelnoise and interpolates over masked pixels. It is followed bynormalization, such that (cid:80) x,y ˜ F xy = N pix .Now we justify that the background amplitude is spatiallyconstant, and even that the flatfield is indeed constant as nor-mally expected. We cannot prove this for individual expo-sures, as the fluxes are impractically small. However, wecan restrict the sum in (11) to different subsets of exposures.This way, we derived ˜ F xy for exposures of only one specific ( D a t a - B k g ) / √ D a t a individual Bkg boxes mean -- σ Figure A.1 . The top panel shows the wavelength-integrated datacube D x (cid:48) y (cid:48) . The effect of the flatfield is clearly visible as aradial surface-brightness pattern of the background flux in addi-tion to inhomogeneous coverage (empty corners). Overlaid are (25 pix) ≈ (5 (cid:48)(cid:48) ) boxes in the pre-shock/post-shock (blue/red) re-gion, which we used to estimate the incident background spectrum b ( λ ) and, in turn, the flatfield. The middle panel presents the nor-malized spectra of the boxes (light blue/red lines), while thick blueand red lines are the mean profiles of the pre- and post-shock boxes.The shapes of the background spectra are similar across the en-tire FOV. The bottom panel indicates differences between the actualbackground (”Data”) and the background model (”Bkg”), normal-ized by the Poisson noise. Dashed-black lines represent the differ-ences in the 32 background boxes separately, dashed-purple line isthe mean absolute difference, and solid-purple line is the standarddeviation between the boxes. ˜ F , but we do not observe such variations.With ˜ F xy at hand, we measure a i as the flux scaling re-quired to match the data, a i = (cid:80) x u ,y u D x u y u ,i (cid:80) x u ,y u ˜ F xy · ˜ b (Λ x u y u ,i ) , and reconstruct the background in each exposure: ˜ B xy,i = ˜ F xy · a i ˜ b (Λ xy,i ) . (12)We then transform the flatfield and background models ˜ F xy and ˜ B xy,i to sub-cubes in the common astrometric frame andco-add them, in the same way the data (4, 5,6). As for the”real” flatfield and background, (9) ensures that the resultingcoadded cubes correspond to the actual flatfield and back-ground components in the data cube.We use ˜ B x (cid:48) y (cid:48) l and ˜ F x (cid:48) y (cid:48) l to check once more whetherour assumption of a uniform background and invariant flat-field are fulfilled: the wavelength-integrated ( B/F ) x (cid:48) y (cid:48) = (cid:80) l B x (cid:48) y (cid:48) l /F x (cid:48) y (cid:48) l is indeed flat, and background residuals ( D − B ) x (cid:48) y (cid:48) are zero apart from Poisson noise and objects(see Figure A.2).B. Photo-ionization precursor (PIP)
When gas starts to cool and recombine downstream, theproduced photons escape to the pre-shock region and forma PIP (Raymond 1979). Although non-radiative shocks lackrecombination zones, PIP can still be created, where the mainsources of the photons produced downstream are HeI λ
584 ˚Aand HeII λ
304 ˚A. Ghavamian et al. (2000) reported on dif-fuse H α , but also [NII] and [SII] emission extending over 1 (cid:48) in front of the Tycho’s NE rim, and suggested that it arisesin a PIP. They predicted that the pre-shock gas was heatedin the PIP to ∼
12 000 K. Subsequently, Lee et al. (2007)measured PIP spectrum in front of ’knot g’: narrow H α with W NL ≈
34 km s − , and [NII] λ − .Similarity of the spectra in the pre- and post-shock back-ground boxes (middle panel in Figure A.1), where pre-shockboxes partially cover the region of the suspected PIP, unam-biguously shows that the PIP emission in our data is negligi-ble. Furthermore, since the signal of a PIP increases towardsthe shock front, we searched for its signature in 9 pre-shock, -pixel regions (magenta boxes in Figure A.3) that weretaken to be closer to the filament than the background boxes( ≈ (cid:48)(cid:48) away from the filament), but still far enough so that wedo not pick up on projected filament emission and emission Figure A.2 . Wavelength-integrated background residuals ( D − B ) x (cid:48) y (cid:48) . The grey scale is linear from -10 to +10 counts/pixel. Thesmooth gray regions have low signal or are even zero due low ef-fective exposure time and hence lower flux (the flatfield was notdivided out). The sources (white) were masked in the flatfield andbackground construction procedure and additionally avoided by thebackground-probing boxes. This applies in particular also to theapparent faint but extended brightness around the eastern (left) andnorthern (top) filaments. in the CR precursor. The top-right panel shows the compar-ison of the putative PIP signal and background model, nor-malized by measurement noise. The mean normalized PIPlevel is indicated by the solid-magenta line, and is 0.13 onaverage. Therefore, the possible PIP signal is consistent withzero. We also show the comparison between the mean PIPand background flux, including their 1-sigma uncertainties(bottom-left panel). Finally, in the bottom-right panel wedemonstrate that PIP signal is negligible in comparison tothe filament flux. We chose here the location (bin) with thesmallest Signal per area; the signal is larger in all other bins,and the putative PIP even smaller in relation to the signal.We therefore conclude that there is none or negligible PIPcontribution to our filament flux models.C. Spatial binning of the data
The Voronoi binning (Cappellari & Copin 2003) was per-formed setting two criteria: the targeted S/N and the mini-mum bin size. The minimum bin size has to be set in order toaccount for the seeing of (cid:39) (cid:48)(cid:48) , which for the spatial scale of0.2 (cid:48)(cid:48) /pixel gives the minimal bin size of (cid:39)
19 pixels. We setthe targeted S/N=10. The code of Cappellari & Copin (2003)in its standard version tends to create elongated-shaped binsin the direction perpendicular to the shock filament. The HSTimage of Tycho’s SNR seen in Figures 1 & 2 in Lee et al.(2010) shows complex filamentary structure including sev-eral very bright knots as a result of different shock emission2 databackground model Voronoi bin PIP
Voronoi bin with minimal Signal (= Data-Bkg) per Area ( D a t a - B k g ) / √ D a t a ( D a t a - B k g ) / A r e a C o un t s V [kms -1 ] V [kms -1 ] V [kms -1 ] mean PIP spectrum across 9 boxes of 169 pixels individual PIP boxes mean PIP PIP boxes in front of eastern fi lament PIP emission versus noise Figure A.3 . Magenta boxes in the top-left panel indicate the region where we extracted spectra in search for a potential PIP signal. Thedifference between data and background model in individual pre-shock boxes (black-dashed) and mean PIP level (solid-magenta), divided bythe measurement errors, shows that residual emission is consistent with Poisson noise. The mean flux (solid-black) and the background model(solid-red) in the regions of putative PIP emission are directly compared in bottom-left panel, with their − σ uncertainties are indicated withthe corresponding color-shaded regions. Putative PIP flux and the background are entirely consistent with one another. In bottom-right panel,we compare filament emission (black line and grey-shaded region) in the bin with smallest surface brightness to the PIP-region signal (magentaline and pink-shaded region), again illustrating that any possible PIP signal is negligible. projected along the LOS. Small-scale differential gradientsare very notable in the direction of the shock normal, the rea-son why we used the Weighted Voronoi Tessellation adap-tation of the Cappellari & Copin (2003) (Diehl & Statler2006) that created a couple of rounder bins in the same di-rection instead. This way we created 85 spatial Voronoi binsin the eastern and 15 in the northern part of the rim (Figure 3).Although we see only 2%-level variations between boxes of25 pixel size, these residual background variations becomeimportant relative to the S/N for bin sizes of 400 pixels andlarger. This yields the upper size limit of 400 pixels for thebins used in our analysis. 73 bins (out of 85) in the easternand 9 bins (out of 15) in the northern filament fulfill the abovecriteria and are further considered while the bins with morethan 400 pixels are excluded from the analysis.D. Model parameters and prior PDFs
As already mentioned in Section 3.2, we define the totalflux, continuum and line flux fractions, line centroids andwidths as models parameters. In case of the NLNL and NLNLIL models, we define the NL centroid mean (cid:104) µ NL (cid:105) andthe separation between the two NLs ∆ µ NL as parameters. ILcentroid is introduced with its offset from the NL centroid(mean) ∆ µ IL .The parameters that we actually sample from are slightlydifferent from the parameters that we use in Section 3;the difference enables direct application of the prior PDFs(Dirichlet or Beta distributions) on the parameters. Theseprior distributions require parameter sets defined in the range(0, 1), except for the naturally based logarithm of the to-tal flux ln ( F tot ) for which we use flat unbound prior. Thecontinuum and line flux fractions are by definition in (0, 1)range, where we set the continuum flux fraction to be de-pendent on flux fractions in the lines f c = 1 − (cid:80) f i . Thefact that flux fractions sum up to 1 and that they are in therange (0, 1) makes the Dirichlet distribution (cid:81) i f α i − i , i =[ N L, N L , N L , IL, c ] perfect choice for their prior PDF.Since we do not favor any of the flux components, weuse symmetric Dirichlet distribution with the same index α which we set to α = 1 . to disfavor zero fluxes.3 Table S1 . Model parameters and their prior PDFs. All parameters apart from ln ( F tot ) are defined in (0,1) range.parameters meaning priorln ( F tot ) natural log-based total flux flat prior f i , i = [c , NL , NL1 , NL2 , IL] flux fractions Dirichlet prior: (cid:81) i f α − i , α = 1 . ; (cid:80) i f i = 1 , f i ∈ (0 , µ (cid:48) NL , (cid:104) µ (cid:48) NL (cid:105) NL centroid, NL centroid mean Beta prior: x α − (1 − x ) β − , α = β = 1 . ; x ∈ (0, 1) ∆ µ (cid:48) NL separation between the two NLs ∆ µ (cid:48) IL IL centroid offset from NL centroid (mean) w (cid:48) NL(IL)
NL (IL) natural log-widthmodel model parametersNL ln ( F tot ) , f NL , f c , µ (cid:48) NL , w (cid:48) NL NLNL ln ( F tot ) , f NL1 , f
NL2 , f c , (cid:104) µ (cid:48) NL (cid:105) , ∆ µ (cid:48) NL , w (cid:48) NL1 , w (cid:48)
NL2
NLIL ln ( F tot ) , f NL , f IL , f c , µ (cid:48) NL , ∆ µ (cid:48) IL , w (cid:48) NL , w (cid:48) IL NLNLIL ln ( F tot ) , f NL1 , f
NL2 , f IL , f c , (cid:104) µ (cid:48) NL (cid:105) , ∆ µ (cid:48) NL , ∆ µ (cid:48) IL , w (cid:48) NL1 , w (cid:48)
NL2 , w (cid:48) IL The line parameters µ NL , (cid:104) µ NL (cid:105) , ∆ µ NL , ∆ µ IL are alldefined with the following functional form: x (cid:48) = ( x − x min ) / ( x max − x min ) . µ NL or (cid:104) µ NL (cid:105) are defined in the range[ V cen − V FSR / , V cen + V FSR / ], where V cen is the centerof the free velocity range V FSR (FSR in velocity units) and ∆ µ NL in the range [0, V FSR / ] so that in the most extremecase the two NL centroids are at the edges of the spectral cov-erage. ∆ µ IL is within ± V FSR / having the absolute upperboundary set to the upper (lower) boundary of the NL (IL)width ≈
100 km s − . Instead of line widths W NL and W IL defined in the range [15, 100] km s − and [100, 350] km s − respectively (Morlino et al. 2012, 2013), we used their log-widths (natural logarithm) denoted as w NL(IL) = ln W NL(IL) in the same functional form as for the centroids and sep-arations. For all these parameters we define a symmetricBeta distribution prior x α − (1 − x ) β − with the arguments α = β = 1 . - slightly favoring the central values of the de-fined parameters ranges. Model parameters and their priorsare summarized in the Table S1.In addition to Figure 2 where we plotted posteriors for thefavored NLIL model for one of the bins, in Figures A.4, A.5,A.6 we show the posteriors for the NL, NLNL and NLNLILmodels, where the latter two figures show the parameters thatwe actually sample from.E. MCMC sampling
Posterior samples were drawn ensemble MCMC sampler(Goodman & Weare 2010), an implementation of which hasbeen popularized as ”emcee” (Foreman-Mackey et al. 2013).Among other advantages, this method provides for (near-)optimal tuning at every stage of the sampling, which wouldotherwise be a substantial challenge and obstacle in the wayof efficient sampling considering 82 different data sets, fourdifferent models for each of them, up to 10 model parameters,and our intent to test the sampler and results based on hun-dreds of additional simulated data sets. We draw the initialparameters for 128 parallel chains (walkers) uniformly be-tween the prior boundaries. The unnormalized log-posterior of a model is computed as a sum of unnormalized log-priorand log-likelihood for the proposed walker position. Sinceour data (fluxes in the spectral bins) result from Poisson pro-cesses, the likelihood is the product of each datum’s probabil-ity under a Poisson distribution with expectation value equalto the model prediction. After taking the logarithm, ln L = N=48 (cid:88) i =1 d i ln( m i ) − m i − ln ( Γ ( d i + 1)) , (13)where m i are the model predictions at each spectral bin i ,and d i are the corresponding data. The last term in eq. 13(the factorial term) was left out of the posterior sampling(but not the evidence calculation) because it is modelindependent. We refer to Foreman-Mackey et al. (2013)and Goodman & Weare (2010) for details of the samplingalgorithm. Before checking the chains’ convergence, wedisregard the first 25%, but at least 512, of the samples ofeach walker (”burn-in”), and further thin the chains until theautocorrelation time of the thinned sample is smaller that5. In order to achieve low noise and to set a first minimumthreshold for convergence of the chains, we require at least2 =8096 total samples (all walkers combined) to be keptafter thinning. Once this minimum number of samples isreached, we additionally impose the following convergence(stopping) criterion: we split the sample in subsamples andcompute the desired estimators (maximum-posterior sample,median and 95%-confidence interval boundaries) for eachsubsample. The variances of the subsample estimators arethen required to be smaller than 5% of the mean parametervalue. In order to reduce the probability of coincidentallyfavorable (small) variances from possible ”modes” in thechains, we repeat the process twice, first taking each ofthe 128 chains as one subsample, and second each ofthe 128-element walker states as subsamples, and use thearithmetic mean of the resulting eight relative estimatorstandard deviations towards our 5%-criterion.4To ensure that the applied procedure gives the cor-rect results, we performed tests of the posterior samplingroutine, using simulated data with known model parameters.We tested models with S/N in the range [5 , and varyingbackground level (0%, 50% and 90% of the total flux). First,we checked if the posterior distribution reproduces the priorin the S/N = 0 limit, but also if the posterior approached adelta function in the infinity limit, i.e. for very large
S/N .Second, we check if the model parameters of the inputmodel are reproduced statistically. We ran the algorithm for200 different realizations of the same model, i.e. each timedrawing the data from the same model prediction with itsspecified uncertainty included (each a set of 48 Poisson dis-tributions). The mean of the distribution of median valuesis always consistent with the input model parameter and thetypical scatter of this mean is 10-20% for the range of
S/N in our real data.Finally, we vary model parameters by randomly choosingthem 200 times from within the prior boundaries. Again, foreach of the resulting simulated data we sample the posterioras described above, and evaluate it in form of the median ofthe marginalized posteriors. We find that the measured me-dian values scatter symmetrically around the 1:1 relation withthe input model parameters, with the scatter being roughlyequal to the individual posteriors’ standard deviation, as de-sired. The distribution of measured values become biased asthe input parameters approach the parameter range bound-aries, as expected for our priors.F.
Evidence calculation via LOO-CV likelihood
We use the cross-validation (CV) likelihood, specificallyits ”leave-one-out” (LOO) variant, to compute model evi-dences and to compare models (Bailer-Jones, C.A.L. 2012).We prefer it over the standard numeric (”Bayesian”) integral,because it draws samples from the posterior instead of theprior. It is hence more efficient, less dependent on the choiceof prior, and in some cases numerically more stable. The ideaof CV likelihood is to evaluate the likelihood of part of thedata, given the model and the rest of the data. In our casewe have 48 data points and measure how well any 47 datapoints (the complement, D − k ) under the model M predictthe 48th data point (the partition, D k ), as quantified by thecomplement’s posterior P ( θ | D − k , M ) and its prediction for D k . The process is repeated for all possible (48) partitions.Each time we leave out one datum ( D k ), its partition likeli-hood L k is given by L k = P ( D k | D − k , M ) = (cid:90) θ P ( D k | θ, M ) P ( θ | D − k , M ) d θ . (14)The first term in the integrand is the likelihood of D k , whilethe second term is the posterior PDF after considering theinformation contained in D − k . We can numerically (Monte-Carlo) integrate by drawing a number of samples N from P ( θ | D − k , M ) : L k ≈ N N (cid:88) n =1 P ( D k | θ n , M ) . (15)Assuming that the data points are independent, the LOO-CV likelihood is the product of all partition likelihoods: L LOO − CV = (cid:81) k =1 L k , or ln L LOO − CV = (cid:88) k =1 ln L k . (16)It can be shown that L LOO − CV for model M is equal to theBayesian evidence ( E ( M ) ), and we henceforth use it to com-pute the Bayes factors, E ( M ) /E ( M ) to compare models.We tested the ability of the Bayes factors to indicate thecorrect model by employing simulated data. We find thatthe typical numerical precision of ln L LOO − CV is better than0.05 dex. At the same time, for data generated from param-eters and with S/N that are typical for the actual data, the”right” model’s ln L LOO − CV is (cid:38) . ( (cid:38) ) betterthan any of the alternative models. Nevertheless, on heuris-tic grounds (what probability is considered statistically ”sig-nificant enough” ?), and in order to bracket the practicallylimited scope of such tests, we adopt a more conservative +0 . threshold ( probability) before we consider amodel preferable over another.We have also tested the dependence of the evidence ratioson the choice of prior. We used the same functional form forthe priors (Table S1), but different α , β distribution parame-ters. Specifically, we tested α D values for the Dirichlet distri-bution, and α B = β B for the Beta distribution with values of ( α D , α B ) = { (1 . , . , (1 , . , (1 . , , (1 , } , where (1 . , . was used for the results presented in the main partof the paper. We found that for all bins, the mean standarddeviation of the appropriate evidence ratios is (cid:46) ( α D , α B ) = (1 , instead of the fiducial (1 . , . , areapplied. We present results for the same two Voronoi binsshown in Figures 2 & 5. The shape of the posteriors and theirmedians are very similar to those obtained with the adoptedpriors, demonstrating that our results do not depend stronglyon the choice of prior. W IL posteriors for the adopted andflat prior are different, but similar to the degree that medianand confidence interval boundaries change only by 30 km s − ( ≈ ). Even with the flat prior, 210 km s − is clearly moreprobable than other W IL parameter values, in particular pre-ferred over values close to the W IL limits. This shows thatthe preference of central W IL values is not just borne out ofthe prior shape or range, but genuinely reflects constraintsprovided by the data, even if they are not as strong as forother parameters.5 f c f NL W NL F tot [counts] μ NL [kms -1 ] [kms -1 ] Figure A.4 . Parameter estimation of an NL model via Bayesian inference for the bin in the NE filament of Tycho’s SNR for which we presentedposterior of NLIL model parameters in Figure 2 (see its caption for explanation). ln(F tot )f NL1 f NL1 +f NL2 < μ NL > [kms -1 ] Δμ NL [kms -1 ] lnW NL1 lnW
NL2
Figure A.5 . Parameter estimation of an NLNL model via Bayesian inference for the bin in the NE filament of Tycho’s SNR for which wepresented posterior of NLIL model parameters in Figure 2 (see its caption for explanation). Instead of F tot , W NL1 , W NL2 , f NL1 , f NL2 , wepresent log (natural logarithm) of total intrinsic flux and intrinsic line widths (ln ( F tot ) , ln W NL1 , ln W NL2 ), and cumulative flux fractions ( f NL1 , f NL1 + f NL2 ). ln(F tot )f NL1 f NL1 +f NL2 f NL1 +f NL2 +f IL lnW NL1 lnW
NL2 lnW IL < μ NL > [kms -1 ] Δμ NL [kms -1 ] Δμ IL [kms -1 ] Figure A.6 . Parameter estimation of an NLNLIL model via Bayesian inference for the bin in the NE filament of Tycho’s SNR for which wepresented posterior of NLIL model parameters in Figure 2 (see its caption for explanation). Instead of F tot , W NL1 , W NL2 , W IL , f NL1 , f NL2 , f IL , we present log (natural logarithm) of total intrinsic flux and intrinsic line widths (ln ( F tot ) , ln W NL1 , ln W NL2 , ln W IL ), and cumulative fluxfractions ( f NL1 , f NL1 + f NL2 , f NL1 + f NL2 + f IL ). f c f NL f IL W NL W IL F tot [counts] μ NL Δμ IL [kms -1 ][kms -1 ] [kms -1 ][kms -1 ] Figure A.7 . Parameter estimation of an NLIL model via Bayesian inference for the bin in the NE filament of Tychos SNR for which we presentedposterior in Figure 2 (see its caption for explanation). Posteriors in black are calculated for flat Dirichlet and Beta priors for all parameters.Posteriors from Figure 2 are overplotted in grey. Among all parameters, the W IL posterior is the most sensitive to the prior choice. Even so, themedian and 95%-confidence intervals agree within ≈ f c f NL1 f NL2 W NL1 [kms -1 ] W NL2 [kms -1 ] F tot [counts]< μ NL > [kms -1 ] Δμ NL [kms -1 ] Figure A.8 . Parameter estimation of an NLNL model via Bayesian inference for the bin in the NE filament of Tychos SNR for which wepresented posterior in Figure 5 (see its caption for explanation). Posteriors in black are calculated for flat Dirichlet and Beta priors for allparameters. Posteriors from Figure 5 are overplotted in grey. Table S2 . Model comparison for the 73 spatial bins in the eastern shock filamentof Tycho’s SNR. Columns 1–5: number of the (Voronoi) bin, x and y coordi-nates of the bin centroid, number of combined pixels, and signal-to-noise ratio.Columns 6–10: relative log CV likelihoods of the favored model (denoted with0) to other models.Bin x [ (cid:48)(cid:48) ] y [ (cid:48)(cid:48) ] Pix S/N 0L NL NLNL NLIL NLNLIL(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1 22.0 59.2 20 14.40 27.073 0.108 0 0.077 0.2342 21.4 58.4 26 14.21 31.122 0.307 0.273 0 0.0983 22.4 58.2 21 14.96 40.522 0.245 0 0.236 0.4474 22.8 59.7 26 12.52 13.405 0 0.318 0.220 0.4765 21.6 57.4 22 12.33 27.723 0.435 0.343 0 0.1106 21.0 60.7 46 13.08 31.293 0.459 0.391 0 0.0917 19.6 64.2 188 12.04 39.630 0.483 0.325 0.270 08 21.1 59.5 26 15.19 30.855 0.176 0.113 0 0.1609 18.3 61.1 378 12.84 37.771 0.376 0.348 0 0.18510 20.5 67.3 234 11.43 33.046 0.410 0.146 0.473 011 20.4 57.1 51 12.56 25.860 0.236 0 0.370 0.29512 21.3 56.3 32 11.07 25.137 0 0.217 0.074 0.26613 22.4 57.2 22 11.97 17.165 0 0.161 0.246 0.35614 19.7 58.5 104 13.81 42.871 0.594 0.366 0.088 015 22.5 55.9 48 10.68 14.121 0 0.182 0 0.14116 21.3 55.0 40 10.47 19.664 0.188 0.178 0.272 017 19.5 55.3 137 13.05 30.044 1.456 1.307 0 0.02718 22.4 54.5 27 9.69 15.533 0 0.203 0.013 0.12119 21.4 53.8 37 10.44 12.876 1.387 0 0.475 0.20320 20.2 53.4 51 11.33 17.289 0.094 0.360 0 0.12621 23.4 55.0 40 8.49 9.724 0.436 0.084 0 0.09222 21.3 52.6 31 10.51 20.602 0 0.109 0.050 0.20723 22.5 53.3 30 10.98 19.092 0.518 0.212 0.232 024 18.5 52.7 138 11.15 23.991 0.107 0 0.333 0.30325 22.3 52.1 27 9.17 4.052 0 0.249 0.132 0.33326 20.4 51.8 38 11.57 10.662 0 0.267 0.281 0.43427 21.6 51.4 23 9.86 18.144 0.016 0 0.081 0.11928 19.4 51.0 48 11.10 21.513 0.018 0 0.456 0.44429 20.8 50.8 19 10.63 20.281 0.166 0.223 0 0.14630 23.9 53.6 67 9.73 9.985 0.524 0.488 0 0.02431 19.3 49.6 36 10.09 23.207 0 0.131 0.041 0.25432 20.3 50.0 28 11.16 18.678 0.459 0.628 0 0.15233 20.9 49.0 32 10.58 16.697 0.218 0.105 0.010 034 19.9 48.7 26 11.08 21.270 0.081 0.098 0 0.09935 18.7 48.1 47 12.01 23.668 0 0.085 0.084 0.26536 20.2 47.5 48 11.38 21.533 0 0.008 0.052 0.16537 17.5 49.6 165 11.17 19.644 0.316 0.352 0 0.11238 19.0 46.9 37 11.92 25.193 0.730 0.622 0 0.07039 18.6 46.0 21 12.10 23.762 0.603 0 0.834 0.45240 20.3 45.5 115 12.72 21.540 1.005 0 0.258 0.03341 17.3 44.3 59 11.28 23.175 0.503 0.713 0 0.07242 18.9 44.9 35 11.55 23.385 0.731 0.432 0 0.08443 18.5 44.2 16 9.10 11.179 1.116 0.790 0.065 0 Table S2 – continued from previous pageBin x [ (cid:48)(cid:48) ] y [ (cid:48)(cid:48) ] Pix S/N 0L NL NLNL NLIL NLNLIL(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)44 16.4 46.4 279 12.59 27.058 0.125 0.066 0 0.08245 18.4 43.4 31 11.30 21.451 0.033 0.072 0 0.17346 16.8 42.6 98 12.74 22.116 1.082 0.673 0 0.02047 18.7 42.3 52 11.63 20.054 0 0.303 0.172 0.40448 16.9 40.6 58 12.11 26.149 0.207 0 0.051 0.04149 17.9 41.3 29 11.62 18.944 0.195 0.282 0 0.11850 18.0 39.7 42 12.24 20.337 0.324 0.289 0 0.11251 15.5 35.6 58 11.36 28.214 0.685 0.432 0.459 052 17.4 38.1 21 10.47 17.933 0.324 0 0.113 0.04553 16.9 39.0 38 12.27 22.999 0.265 0 0.404 0.10754 18.5 38.0 58 10.56 12.943 0 0.022 0.292 0.26855 16.9 37.3 29 12.04 18.049 0.466 0.152 0.195 056 17.9 36.7 29 11.74 18.795 0.198 0.316 0 0.15157 16.8 36.2 23 12.53 16.550 0.589 0.619 0 0.08358 19.5 33.6 344 11.89 16.533 0.187 0.395 0 0.11759 17.8 35.7 34 12.40 30.850 0.695 0 0.361 0.15060 16.6 35.2 19 11.81 16.934 0.091 0 0.096 0.11961 17.2 33.5 25 11.15 17.640 0.033 0.203 0 0.11862 17.3 34.7 28 10.86 19.534 0 0.038 0.172 0.26463 16.4 34.2 25 11.63 26.490 0.648 0.649 0 0.11064 17.6 32.1 42 12.86 26.277 2.415 1.670 0 0.09865 16.5 32.8 29 13.09 22.555 0 0.123 0.128 0.27866 16.5 31.7 34 12.12 27.519 0.086 0.126 0 0.15467 16.0 30.7 32 12.02 17.226 0 0.131 0.088 0.20368 14.4 31.7 155 12.61 32.667 0 0.148 0.165 0.39369 13.2 22.1 59 9.33 18.851 0.095 0.241 0 0.17170 12.3 18.4 285 10.04 15.999 0.047 0.152 0 0.11271 13.8 26.8 47 8.78 8.551 0 0.239 0.030 0.19472 15.3 33.6 61 11.76 19.728 0.090 0.377 0 0.18573 13.3 20.6 59 9.43 11.914 0 0.208 0.138 0.300
Table S3 . Same as Table S2 just for the 9 bins in the northern filament.Bin x [ (cid:48)(cid:48) ] y [ (cid:48)(cid:48) ] Pix S/N 0L NL NLNL NLIL NLNLIL(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1 13.0 23.7 351 14.30 33.932 0.846 0 0.667 0.3572 10.4 20.6 62 15.31 43.600 0.483 0.479 0 0.2643 8.9 22.1 169 15.36 53.524 1.645 0.423 0 0.0094 12.3 20.8 169 14.77 41.332 0.003 0 0.068 0.2055 8.6 19.7 129 15.50 49.913 1.194 0 0.644 0.5176 11.7 17.3 355 16.72 34.586 0.470 0 0.186 0.0777 9.0 17.8 75 16.31 42.528 0.959 0 0.741 0.2698 9.1 15.0 190 14.65 40.034 1.090 0.479 0.329 09 7.4 16.5 104 14.73 37.961 0.086 0.246 0.025 0 Table S4 . Median and highest density 95%-confidence interval of parameter posteriors in the favored modelof Voronoi bins. Shown here are the spatial bins in the eastern filament of Tycho’s SNR. If none of multi-line models is at least ≈ W NL1 W NL2 W IL f IL /f < NL > ∆ µ NL [kms − ] [kms − ] [kms − ] [kms − ]1 NL 60.53 - - - -[43.84, 80.68]2 NL 65.12 - - - -[48.10, 84.66]3 NL 59.23 - - - -[44.61, 76.77]4 NL 72.07 - - - -[49.95, 94.80]5 NL 60.90 - - - -[44.14, 81.94]6 NLIL 40.63 - 191.02 0.51 -[24.78, 56.31] [100.15, 308.96] [0.01, 1.55]7 NLNLIL 35.23 31.68 195.54 0.26 21.65[15.02, 63.35] [15.11, 79.41] [102.05, 310.43] [0, 0.64] [0.04, 108.74]8 NL 60.78 - - - -[45.34, 78.82]9 NL 46.85 - - - -[32.28, 61.53]10 NL 40.87 - - - -[28.68, 54.37]11 NL 54.04 - - - -[35.29, 73.60]12 NL 78.70 - - - -[62.54, 95.65]13 NL 70.71 - - - -[49.50, 92.40]14 NLNLIL 40.34 35.18 191.06 0.33 22.74[15.76, 81.33] [15.50, 72.99] [102.59, 311.14] [0, 0.80] [0.16, 101.29]15 NL 71.88 - - - -[43.78, 99.09]16 NL 75.86 - - - -[56.59, 96.57]17 NLIL 39.13 - 212.36 0.88 -[22.30, 56.71] [100.18, 307.88] [0.01, 6.15]18 NL 82.43 - - - -[64.85, 98.87]19 NLNL 45.04 65.68 - - 91.34[15.39, 90.16] [39.58, 90.64] [41.02, 115.24]20 NL 74.45 - - - -[54.71, 96.58]21 NL 73.00 - - - -[47.82, 98.64] Table S4 – continued from previous pageBin Model W NL1 W NL2 W IL f IL /f < NL > ∆ µ NL [kms − ] [kms − ] [kms − ] [kms − ]22 NL 68.76 - - - -[50.01, 88.42]23 NLNLIL 48.04 49.68 168.44 0.42 51.99[15.86, 88.08] [19.25, 84.92] [100.05, 295.69] [0, 1.78] [11.42, 153.20]24 NL 59.97 - - - -[43.21, 77.48]25 NL 66.63 - - - -[36.35, 96.93]26 NL 70.97 - - - -[47.59, 95.35]27 NL 58.66 - - - -[39.30, 81.32]28 NL 56.87 - - - -[41.50, 74.12]29 NL 56.07 - - - -[36.44, 79.31]30 NLIL 59.03 - 181.53 1.44 -[21.68, 93.56] [103.90, 288.14] [0, 7.08]31 NL 56.38 - - - -[37.83, 74.36]32 NLIL 41.53 - 150.76 1.02 -[15.31, 76.64] [100.17, 278.80] [0, 3.29]33 NL 38.17 - - - -[22.16, 56.19]34 NL 47.04 - - - -[30.20, 65.05]35 NL 72.60 - - - -[56.45, 89.57]36 NL 59.26 - - - -[40.97, 80.03]37 NL 42.92 - - - -[23.91, 62.21]38 NLIL 41.82 - 170.06 0.74 -[21.28, 66.79] [100.12, 294.73] [0, 2.26]39 NLNL 51.41 56.03 - - 60.15[16.46, 90.57] [29.12, 80.18] [9.50, 84.17]40 NLNL 38.06 63.21 - - 39.90[15.61, 61.48] [26.59, 96.32] [1.32, 144.16]41 NLIL 48.34 - 201.48 0.62 -[24.99, 70.31] [103.46, 310.50] [0.03, 1.67]42 NLIL 48.09 - 162.60 0.61 -[27.07, 72.44] [100.39, 291.90] [0.01, 1.82]43 NLNLIL 44.57 33.42 171.73 0.75 55.59[15.43, 87.20] [15.03, 77.23] [100.31, 295.94] [0.01, 2.23] [1.57, 97.03] Table S4 – continued from previous pageBin Model W NL1 W NL2 W IL f IL /f < NL > ∆ µ NL [kms − ] [kms − ] [kms − ] [kms − ]44 NL 65.49 - - - -[48.42, 85.13]45 NL 55.89 - - - -[38.00, 76.15]46 NLIL 35.29 - 221.61 0.89 -[17.55, 53.39] [100.39, 317.73] [0, 10.90]47 NL 71.18 - - - -[51.66, 91.45]48 NL 49.41 - - - -[32.16, 66.58]49 NL 43.21 - - - -[27.87, 63.95]50 NL 61.30 - - - -[42.56, 83.35]51 NLNLIL 50.84 37.90 201.87 0.31 37.67[16.21, 81.76] [16.10, 66.80] [104.16, 313.03] [0, 0.89] [1.03, 90.84]52 NL 61.42 - - - -[39.35, 85.21]53 NL 54.00 - - - -[38.21, 71.73]54 NL 47.73 - - - -[23.04, 68.53]55 NLNLIL 33.67 50.83 187.38 0.50 34.31[15.24, 72.78] [17.70, 88.41] [101.07, 312.58] [0, 1.49] [1.00, 82.17]56 NL 58.08 - - - -[38.88, 80.69]57 NLIL 45.30 - 199.89 1.12 -[17.24, 72.11] [107.28, 312.43] [0.02, 3.11]58 NL 74.99 - - - -[54.48, 97.77]59 NLNL 57.49 58.12 - - 63.14[19.54, 93.95] [36.03, 82.85] [11.53, 93.23]60 NL 66.49 - - - -[46.53, 89.49]61 NL 73.42 - - - -[52.04, 96.65]62 NL 55.28 - - - -[37.46, 75.62]63 NLIL 49.93 - 202.95 0.49 -[32.62, 69.52] [106.47, 316.80] [0, 1.34]64 NLIL 38.86 - 191.52 1.12 -[19.78, 60.12] [107.16, 298.62] [0.12, 2.53]65 NL 68.07 - - - -[47.10, 92.23] Table S4 – continued from previous pagebin model W NL1 W NL2 W IL f IL /f < NL > ∆ µ NL [kms − ] [kms − ] [kms − ] [kms − ]66 NL 57.62 - - - -[42.10, 77.46]67 NL 65.35 - - - -[46.28, 87.53]68 NL 72.89 - - - -[57.12, 89.85]69 NL 60.22 - - - -[41.25, 82.71]70 NL 46.09 - - - -[27.58, 67.10]71 NL 50.96 - - - -[29.85, 76.82]72 NL 78.28 - - - -[61.10, 96.81]73 NL 64.24 - - - -[42.79, 89.21] Table S5 . Same as Table S4 just for Voronoi bins in the Tycho’s northern filament.Bin Model W NL1 W NL2 W IL f IL /f < NL > ∆ µ NL [kms − ] [kms − ] [kms − ] [kms − ]1 NLNL 33.85 53.16 - - 28.04[15.04, 61.64] [21.03, 87.55] [0.29, 156.41]2 NLIL 58.50 - 195.28 0.39 -[41.77, 75.74] [102.92, 316.25] [0, 1.13]3 NLIL 54.32 - 156.07 0.41 -[37.83, 69.52] [100.12, 274.97] [0.02, 1.08]4 NL 59.63 - - - -[37.09, 78.26]5 NLNL 51.95 69.86 - - 39.81[25.10, 73.85] [34.12, 97.51] [2.97, 173.42]6 NLNL 42.30 56.58 - - 20.37[15.65, 83.53] [20.46, 91.93] [0.05, 67.38]7 NLNL 37.89 60.23 - - 42.00[15.09, 66.94] [26.97, 93.47] [6.73, 184.78]8 NLNLIL 44.66 34.55 149.45 0.36 29.63[15.68, 70.78] [15.22, 80.12] [100.36, 291.76] [0, 0.89] [0.23, 86.09]9 NL 56.38 - - - -[37.58, 74.56]6 REFERENCES