3D mapping of the Crab Nebula with SITELLE. I. Deconvolution and kinematic reconstruction
MMNRAS , 1–15 (2019) Preprint 11 January 2021 Compiled using MNRAS L A TEX style file v3.0
3D mapping of the Crab Nebula with SITELLE. I. Deconvolutionand kinematic reconstruction
T. Martin , (cid:63) , D. Milisavljevic , L. Drissen , D´epartement de physique, de g´enie physique et d’optique, Universit´e Laval, 2325, rue de l’universit´e, Qu´ebec (Qu´ebec), G1V 0A6, Canada Centre de Recherche en Astrophysique du Qu´ebec (CRAQ) Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present a hyperspectral cube of the Crab Nebula obtained with the imagingFourier transform spectrometer SITELLE on the Canada-France-Hawaii telescope. Wedescribe our techniques used to deconvolve the 310 000 individual spectra ( R = )containing H α , [N ii ] λ λ ii ] λ λ Ni-rich ejecta. These characteristics reflect criticaldetails of the original supernova of 1054 CE and its progenitor star, and may favoura low-energy explosion of an iron-core progenitor. We demonstrate that our mainfindings are robust despite regions of non-homologous expansion driven by accelerationof material by the pulsar wind nebula.
Key words: instrumentation: interferometers – methods: data analysis – techniques:imaging spectroscopy – supernovae: general – ISM: supernova remnants
Young supernova (SN) remnants ( < yr) in the MilkyWay and Magellanic Clouds provide rare opportunities toprobe the explosion mechanisms and progenitor systems ofSNe with observations capable of producing three dimen-sional reconstructions. Their proximity (from a few to lessthan a hundred kpc) and high expansion velocity (a fewthousand km s − ) allow not only to disentangle the differentDoppler components with relatively modest spectral resolu-tions, but also to determine the tangential velocity of theirfilaments using images obtained decades apart (or sometimesa few years apart using the Hubble Space Telescope ; HST ).Asymmetries in chemical or ionization structure, expansion (cid:63)
E-mail: [email protected] velocity or density can then be probed with much greaterprecision than what is possible for unresolved extragalacticobjects (Milisavljevic & Fesen 2017). Such reconstructions,made for SNRs including Cassiopeia A (Cas A) (DeLaneyet al. 2010; Milisavljevic & Fesen 2013; Alarie et al. 2014;Milisavljevic & Fesen 2015; Grefenstette et al. 2017), 1E0102.2-7219 (Vogt & Dopita 2010; Vogt et al. 2018), andN132D (Law et al. 2020), are critical for establishing strongempirical links between SNe and SNRs that can be comparedto state-of-the-art simulations evolving from core collapse toremnant (Orlando et al. 2015, 2016, 2020; Ono et al. 2020).Among the most studied yet still enigmatic remnantsdeserving of 3D reconstruction is the Crab Nebula (SN 1054,NGC 1952). Despite decades of investigation, the progeni-tor star’s initial mass and the properties governing the SNexplosion remain uncertain (Davidson & Fesen 1985; Hes- © a r X i v : . [ a s t r o - ph . H E ] J a n T. Martin et al. ter 2008). The total mass of its ejecta (2-5 M (cid:12) ; Fesen et al.1997) is much less than the plausible mass of the progen-itor (8-13 M (cid:12) ; Nomoto 1987), and although SN 1054 wasmore luminous than a normal Type II SN ( − mag vs. − . mag; Clark & Stephenson 1977), its kinetic energy( ≈ × erg) is surprisingly low compared to the canoni-cal ∼ erg. The standard explanation is that most of themass and 90% of the kinetic energy of SN 1054 reside inan invisible freely expanding envelope of cold and neutralejecta traveling ∼ km s − far outside the Crab (Cheva-lier 1977). However, this theorized outer envelope has neverbeen robustly detected to remarkably low upper limits (Fe-sen et al. 1997; Lundqvist & Tziamtzis 2012). A weak C IV λ HST spectrum of the Crab pulsar is suggestive of an ionized outerenvelope (Sollerman et al. 2000; Hester 2008), but only ex-tending out to ∼ km s − and tracing a relatively smallamount of material ( ∼ . M (cid:12) ).Models and observations generally support a low en-ergy SN origin potentially associated with an O-Ne-Mg corethat collapses and explodes as electron-capture supernova(ECSN) (Nomoto et al. 1982; Hillebrandt 1982; Kitauraet al. 2006). However, such explosions are generally faint( M V > − mag) and thus inconsistent with the brightnessof SN 1054 estimated from historical records. Fesen et al.(1997) and Chugai & Utrobin (2000) suggested that SN 1054was a low energy SN with additional luminosity provided bycircumstellar interaction. Smith (2013) supports this viewand identified potential Crab-like analogs in many recentType IIn events. Tominaga et al. (2013) found that the highpeak luminosity could instead be related to the large extentof the progenitor star and not necessarily associated withstrong circumstellar interaction. Gessner & Janka (2018)questioned the appropriateness of an ECSN origin for theCrab, as their simulations found that hydrodynamic neu-tron star kicks associated with O-Ne-Mg core progenitorsare much below the ∼ km s − measured for the Crabpulsar (Kaplan et al. 2008). Yang & Chevalier (2015) foundthe Crab’s overall properties to be consistent with expec-tations from a pulsar wind nebula evolving inside a freelyexpanding low energy supernova.There have been multiple attempts at mapping thethree dimensional structure of the Crab. However, the largeangular size of the Crab ( ∼ (cid:48) ) and complexity of its nu-merous overlapping filamentary structures presents manychallenges. Lawrence et al. (1995) created three-dimensionalspatial models of the line-emitting [O III] λ λ Figure 1.
Atmospheric transmission as a function of the step index.The date when each part of the cube was obtained is shown asan orange dotted line.
Figure 2.
Interferogram of the M1 pulsar before and after the cor-rection for the sky transmission and the varying background (resp.blue and orange line). Atmospheric transmission is reported indotted black. The large oscillation seen at step ∼ correspondsto the Zero path difference of the interferometer, and is charac-teristic of a continuum source, while the low-amplitude beatingpatterns observed along the interferogram are caused by the mul-tiple emission lines of the nebula included within the . (cid:48)(cid:48) radiusof the aperture. lar disk left behind from pre-SN mass loss (Fesen et al. 1992).To date there does not exist a complete mapping of individ-ual emission lines throughout the remnant sensitive to faintemission on fine scales.In this paper we introduce a hyperspectral cube of theCrab obtained with the imaging Fourier transform spec-trometer SITELLE (Drissen et al. 2019). SITELLE has afield of view of 11 (cid:48) × (cid:48) , high sensitivity down to 350 nm,and is especially powerful for observing emission line sourcesabove a low continuum background (Bennett 2000; Maillardet al. 2013). Together, these characteristics make SITELLEuniquely suited to meet the challenges of observing the Crab.In this paper we describe preliminary SITELLE obser-vations and associated analysis of the Crab obtained in apassband covering 647-685 nm with spectral resolution 9 600.These data are the highest resolution ever obtained withSITELLE on an astrophysical target and a spectacular op-portunity to demonstrate the full potential of this new tech-nology. We also describe the techniques used to deconvolvethe spectra and produce a 3D reconstruction of the super-nova remnant. A more detailed analysis of these data incombination with planned complementary observations atother wavelengths will follow in a subsequent paper. MNRAS000
Interferogram of the M1 pulsar before and after the cor-rection for the sky transmission and the varying background (resp.blue and orange line). Atmospheric transmission is reported indotted black. The large oscillation seen at step ∼ correspondsto the Zero path difference of the interferometer, and is charac-teristic of a continuum source, while the low-amplitude beatingpatterns observed along the interferogram are caused by the mul-tiple emission lines of the nebula included within the . (cid:48)(cid:48) radiusof the aperture. lar disk left behind from pre-SN mass loss (Fesen et al. 1992).To date there does not exist a complete mapping of individ-ual emission lines throughout the remnant sensitive to faintemission on fine scales.In this paper we introduce a hyperspectral cube of theCrab obtained with the imaging Fourier transform spec-trometer SITELLE (Drissen et al. 2019). SITELLE has afield of view of 11 (cid:48) × (cid:48) , high sensitivity down to 350 nm,and is especially powerful for observing emission line sourcesabove a low continuum background (Bennett 2000; Maillardet al. 2013). Together, these characteristics make SITELLEuniquely suited to meet the challenges of observing the Crab.In this paper we describe preliminary SITELLE obser-vations and associated analysis of the Crab obtained in apassband covering 647-685 nm with spectral resolution 9 600.These data are the highest resolution ever obtained withSITELLE on an astrophysical target and a spectacular op-portunity to demonstrate the full potential of this new tech-nology. We also describe the techniques used to deconvolvethe spectra and produce a 3D reconstruction of the super-nova remnant. A more detailed analysis of these data incombination with planned complementary observations atother wavelengths will follow in a subsequent paper. MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. The data were obtained using the SITELLE instrumentmounted to Canada-France-Hawaii telescope (CFHT) dur-ing the course of an engineering run spanning the nightsof November 22, 25 and 26, 2016. SITELLE combines a2D imaging detector with a Michelson interferometer. Twocomplementary interferometric data cubes are obtained byrecording images, on two 2k ×
2k CCD detectors, at differentpositions of the moving mirror inside the Michelson. Fouriertransforms are then used to convert these cubes into a singlespectral data cube. Spectral resolution is set by the maxi-mum path difference between the two arms of the interfer-ometer, reached by displacing its moving mirror through aseries of steps of several hundred nanometers each. The spec-tral range is selected by using interference filters; SITELLEcovers the 350 - 850 nm range with a series of 8 filters, tai-lored to specific needs. Spatial sampling is . (cid:48)(cid:48) per pixel,leading to a field of view of (cid:48) × (cid:48) and over 4 million spec-tra. Our raw data for the Crab consist in an interferometriccube of 1682 steps (with a step size of 2843 nm) with anexposure time of 5.3 s per step (followed by an overhead of3.8 s for CCD readout and concurrent mirror movement andstabilization), leading to an integration time of 2.48 hoursover a 4.25 hours total data aquisition time. The medianseeing, measured on the image obtained from the combi-nation of all detrended and aligned interferometric images,was 1.17 ± . (cid:48)(cid:48) . This engineering data was aimed at testingSITELLE’s high resolution capabilities with a target reso-lution of 10 000 in the SN3 filter (647 - 685 nm passband:H α , [N ii ] λ λ ii ] λ λ ORBS (Martin et al. 2012; Martin 2015) withoutany special treatment with respect to the rest of the dataobtained during the same run. As an example of the qual-ity of the reduction, we present in Figure 2 the raw inter-ferogram of the pulsar, obtained with an aperture of . (cid:48)(cid:48) radius, before and after the correction for the sky transmis-sion. Any error on this correction on the interferograms cansignificantly impact the quality of the calculated spectrumand especially its instrumental line shape (ILS) (Martin &Drissen 2016).We have measured the effective resolution by fitting amodel on a spectrum of the sky (dominated by OH lines inthis spectral range) integrated over a small region of the cube(a circular aperture with a 100 pixels radius) with the anal-ysis software ORCS (Martin et al. 2015). In order to take intoaccount the modulation efficiency loss at high OPD, whichmay broaden the observed ILS, we have modelized the ILSas the convolution of a sinc (the natural ILS of a Fouriertransformed spectrum) with a Gaussian (resulting from thebroadening of the observed sky lines by the modulation ef-
Figure 3.
Sky spectrum used to measure the effective resolution( R = ± ) (blue line) along with the fitted model (orangeline). The bottom panel shows an enlarged portion of the toppanel around H α (15237 cm − ) ficiency loss) and used the sincgauss model described inMartin et al. (2016) (see Figure 3). The measured full-widthat half maximum of the sky lines was 1.718 ± . cm − whichleads to a resolving power R = ± @ H α (Martin et al.2016). From the parameters of the cube, the theoretical res-olution which we should have been measured at the sameposition is 9640. We can conclude that the effect of the mod-ulation efficiency loss at high OPD is at most of the orderof 10 % at a resolution of 9640. α , [N II] AND[S II] The Crab Nebula is long known to display a very complexfilamentary structure that remains interpreted as the resultof Rayleigh-Taylor instabilities at the interface of the syn-chrotron nebula and the thermal ejecta (Hester et al. 1996;Hester 2008). This filamentary structure, ionized by theshock of the expanding synchrotron nebula, shows’s partic-ularly strong emission in H α , [N ii ] and [S ii ]. Multiple com-ponents of filamentary emission are visible along any lineof sight with velocities ranging from -1500 to 1500 km s − (see Figure 4) that must be separated in order to computethe correct mapping of the flux and velocity of the observedemission lines. The emission covers a circular surface of ∼ (cid:48) in diameter, which spans approximately one million of thefour million spectra contained in the data cube.Inspired by the algorithms developed by ˇCadeˇz et al.(2004) and Charlebois et al. (2010) on similar data sets,we have written an algorithm to automate detection andfitting of overlapping emission components observed in thespectra. This algorithm analyses each spectrum individuallyin 3 steps (see Figure 5):(i) evaluation of the probability, as a score, of having onecomponent at a given velocity;(ii) enumeration of all the individual velocity componentsalong the line of sight; MNRAS , 1–15 (2019)
T. Martin et al.
Figure 4.
Two examples of spectra containing multiple compo-nents and their fit.
A spectrum with 2 non-overlapping components, obtained near the center of the nebula.
A spectrum with 4 overlapping componentsobtained in a complex filamentary region of the nebula. The plotis split in two parts to help distinguish all 4 components. Note thatif only 4 of the components have been found by the algorithm, ad-ditional dim components may exist. This example demonstratesat the same time the quality and the limitations of our algorithm. (iii) fit of the spectrum with a model combining all thevelocity components at the same time.
The first step is based on the convolution of the analyzedspectrum S ( λ ) with a comb-like spectrum K ( v , λ ) made ofa subset of the emitting lines of each component: H α , the[N ii ] doublet and the [S ii ] doublet (see bottom-left panel ofFigure 5). All lines of the comb have the same amplitude.The calculated score C ( v ) is simply: C ( v ) = (cid:90) S ( λ ) K ( v , λ ) d λ . (1) C ( v ) is maximum when the position of the modeled emission-lines of K ( v , λ ) coincides with the position of the emission-lines of the spectrum.Ideally, if the explored velocity range is not too largeand no line of the comb is matched with another emission-line, each velocity component of the spectrum will produce one peak with an approximately Gaussian shape. The cen-troid of the peak gives the component velocity and its ampli-tude scales with the integral of the flux in the lines presentin K .However, the biggest challenge with this approach ap-pears when the comb and the analyzed spectrum containsmultiple lines within the range of velocity scanned. For ex-ample, one line of the comb (e.g., H α ) may coincide with theposition of a neighbouring line at a different velocity (e.g.,[N ii ]). In this case, even with only one component along theline of sight, C ( v ) will show multiple peaks; the highest be-ing the real one because it reflects the velocity at which thelargest number of lines are coincident. With multiple compo-nents however, if one component is much brighter than theothers, the secondary peaks in C ( v ) created by the brightestcomponent may be even higher than the primary peak of thesecond components, in which case the correct enumerationof the components is compromised.Figure 6 reproduces this issue by showing a syntheticspectrum made of two velocity components, one being 5times brighter than the other. The comb used for the analy-sis contains 5 emission lines (H α , [N ii ], [S ii ]) and is shownin the bottom-left quadrant of Figure 5.Using equation 1 without any special treatment leads toa score C ( v ) with multiple false peaks (black line) where thepeak related to the dimmest component cannot be retrieved.We mitigate complicating factors with use of equation 1by adding a number of physically-based conditions that mustbe respected in order to get a non-zero value of C ( v ) . One isto force the presence of all the lines by computing indepen-dent scores for each line and compute the product of theirprobability. Let K i ( v , λ ) be the kernel for the line i , i ∈ { H α , [ N ii ] λ , [ N ii ] λ , [ N ii ] λ , [ S ii ] λ , [ S ii ] λ } .If we want all 5 lines to be present in order to have a non-zeroscore we would rewrite equation 1 as C i ( v ) = (cid:90) S ( λ ) K i ( v , λ ) d λ , (2) C ( v ) = Π i C i ( v ) . (3)However, all 5 lines are not always detectable. Some-times only the [ S ii ] or [ N ii ] + H α lines are visible. Thuswe further separate these two groups and put additionalphysics-based conditions to finally write the mitigated ver-sion of the score: C [ N ii ] = (cid:113) C [ N ii ] λ × C [ N ii ] λ , (4) C [ S ii ] = (cid:113) C [ S ii ] λ × C [ S ii ] λ , (5) C ( v ) = C H α × C [ N ii ] + C [ S ii ] ; (6)with the following additional conditions that constrain the[S ii ] and [N ii ] line ratios to be realistic enough, C [ N ii ] = if C [ N ii ] λ < C [ N ii ] λ , (7) C [ N ii ] = if C [ N ii ] λ < . × C [ N ii ] λ , (8) C [ S ii ] = if C [ S ii ] λ < . × C [ S ii ] λ , (9) C [ S ii ] = if C [ S ii ] λ < . × C [ S ii ] λ . (10)The value of 0.14 ( (cid:39) / ) has been manually optimized tohelp reject obviously wrong scores while keeping lower SNRcomponents. This value is necessarily kept lower than thetheoretical ratios of [S ii ] lines (between 0.5 and 1.5) and MNRAS000
The first step is based on the convolution of the analyzedspectrum S ( λ ) with a comb-like spectrum K ( v , λ ) made ofa subset of the emitting lines of each component: H α , the[N ii ] doublet and the [S ii ] doublet (see bottom-left panel ofFigure 5). All lines of the comb have the same amplitude.The calculated score C ( v ) is simply: C ( v ) = (cid:90) S ( λ ) K ( v , λ ) d λ . (1) C ( v ) is maximum when the position of the modeled emission-lines of K ( v , λ ) coincides with the position of the emission-lines of the spectrum.Ideally, if the explored velocity range is not too largeand no line of the comb is matched with another emission-line, each velocity component of the spectrum will produce one peak with an approximately Gaussian shape. The cen-troid of the peak gives the component velocity and its ampli-tude scales with the integral of the flux in the lines presentin K .However, the biggest challenge with this approach ap-pears when the comb and the analyzed spectrum containsmultiple lines within the range of velocity scanned. For ex-ample, one line of the comb (e.g., H α ) may coincide with theposition of a neighbouring line at a different velocity (e.g.,[N ii ]). In this case, even with only one component along theline of sight, C ( v ) will show multiple peaks; the highest be-ing the real one because it reflects the velocity at which thelargest number of lines are coincident. With multiple compo-nents however, if one component is much brighter than theothers, the secondary peaks in C ( v ) created by the brightestcomponent may be even higher than the primary peak of thesecond components, in which case the correct enumerationof the components is compromised.Figure 6 reproduces this issue by showing a syntheticspectrum made of two velocity components, one being 5times brighter than the other. The comb used for the analy-sis contains 5 emission lines (H α , [N ii ], [S ii ]) and is shownin the bottom-left quadrant of Figure 5.Using equation 1 without any special treatment leads toa score C ( v ) with multiple false peaks (black line) where thepeak related to the dimmest component cannot be retrieved.We mitigate complicating factors with use of equation 1by adding a number of physically-based conditions that mustbe respected in order to get a non-zero value of C ( v ) . One isto force the presence of all the lines by computing indepen-dent scores for each line and compute the product of theirprobability. Let K i ( v , λ ) be the kernel for the line i , i ∈ { H α , [ N ii ] λ , [ N ii ] λ , [ N ii ] λ , [ S ii ] λ , [ S ii ] λ } .If we want all 5 lines to be present in order to have a non-zeroscore we would rewrite equation 1 as C i ( v ) = (cid:90) S ( λ ) K i ( v , λ ) d λ , (2) C ( v ) = Π i C i ( v ) . (3)However, all 5 lines are not always detectable. Some-times only the [ S ii ] or [ N ii ] + H α lines are visible. Thuswe further separate these two groups and put additionalphysics-based conditions to finally write the mitigated ver-sion of the score: C [ N ii ] = (cid:113) C [ N ii ] λ × C [ N ii ] λ , (4) C [ S ii ] = (cid:113) C [ S ii ] λ × C [ S ii ] λ , (5) C ( v ) = C H α × C [ N ii ] + C [ S ii ] ; (6)with the following additional conditions that constrain the[S ii ] and [N ii ] line ratios to be realistic enough, C [ N ii ] = if C [ N ii ] λ < C [ N ii ] λ , (7) C [ N ii ] = if C [ N ii ] λ < . × C [ N ii ] λ , (8) C [ S ii ] = if C [ S ii ] λ < . × C [ S ii ] λ , (9) C [ S ii ] = if C [ S ii ] λ < . × C [ S ii ] λ . (10)The value of 0.14 ( (cid:39) / ) has been manually optimized tohelp reject obviously wrong scores while keeping lower SNRcomponents. This value is necessarily kept lower than thetheoretical ratios of [S ii ] lines (between 0.5 and 1.5) and MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. Figure 5.
The fitting process.
Top-left:
Example of a raw spectrum extracted from a complex filamentary region of the nebula containingat least 4 components.
Bottom-left:
Example of a 5 emission-lines comb-like spectrum which would be convolved with the analyzedspectrum. In fact, the spectrum is convolved with the individual lines of the comb. All lines of the comb have the same intensity. Theapparent difference comes from the function sampling.
Top-right:
Mitigated correlation score obtained after the first step of the analysis(black) with the 4 fitted components detected at the second step. The detection threshold used for the enumeration of the brightestcomponents of the score (see section 3.2) is shown in dotted red.
Bottom-right:
Fit realized on the spectrum shown in the top-left panel;the positions of the 5 fitted emission-lines are shown for each of the 4 fitted components. basicmitigated
Figure 6.
Top:
Two-components noiseless model spectrum.
Bot-tom:
Basic and mitigated scores computed from the top-panelspectrum. The numerous lines appearing in the basic score (blackline) do not reflect real components or noise. Instead, they cor-respond to velocities where some lines of the comb coincide withanother emission line (e.g., where the H α line of the comb co-incides, at a given velocity, with an [N ii ] line of the spectrum).Applying physical constraints when computing a mitigated score(in red) reduces these aliases, which permits the detection of dim-mer velocity components. The addition of noise only contributessmall amplitude peaks that are discarded (see top-right panel ofFigure 5). [N ii ] lines (2.94 in the low-density regime) (Osterbrock &Ferland 2006) to take into account noise associated with themeasured flux.The results of this mitigated score is drawn on Figure 6with a red line. The real peaks are both present and all thefalse peaks have been removed. A more realistic example ofthis score is also shown in the top-right quadrant of Figure 5,where the false peaks, if not completely removed, have beenattenuated enough so that the 4 brightest components areclearly visible. Once the score is obtained, we must go through a peak de-tection process to enumerate the brightest components andevaluate their velocity, which is done by measuring the cen-troid of each detected peak. As the components may havesimilar velocities, their peaks may overlap, which compli-cates the detection.We have used an iterative detection procedure not un-like the CLEAN algorithm (H¨ogbom 1974). At each itera-tion, only the brightest peak is detected, fitted and removedbefore moving on to the next iteration until no peaks canbe detected above a threshold. The threshold was manuallyadjusted to keep the number of false detections negligible atthe expense of loosing some of the dimmest components. Anexample of the resulting detection is shown in the top-rightquadrant of Figure 5, where only 4 of the possibly 5 com-ponents are bright enough to be considered. Since the noiseof an FTS spectrum is distributed over all the channels andproportional to the total flux of the source, using a variablethreshold based on our knowledge of the noise level for eachspectrum may help in detecting components in dimmer re-
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Figure 7.
Integral of the emission in all five emission lines ( top ) and velocity ( bottom ) maps obtained. When components are overlapping,only the front component is displayed. The color mapping has been discretized for the sake of clarity, but it does not reflect the actualprecision of the values. The red cross indicates the center of expansion computed by Kaplan et al. (2008). The velocity uncertainty isshown in Figure A1. The background is the integral of the flux measured (emission lines + continuum) in the whole filter.MNRAS000
Integral of the emission in all five emission lines ( top ) and velocity ( bottom ) maps obtained. When components are overlapping,only the front component is displayed. The color mapping has been discretized for the sake of clarity, but it does not reflect the actualprecision of the values. The red cross indicates the center of expansion computed by Kaplan et al. (2008). The velocity uncertainty isshown in Figure A1. The background is the integral of the flux measured (emission lines + continuum) in the whole filter.MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. gions of the Crab. This possibility will be explored in futureversions of our algorithm.At the end of this step we detected emission in 310 000pixels (of the ∼ Once all the components are enumerated, a fit of the wholespectrum is attempted. This fit is done with
ORCS , a Pythonmodule designed especially to fit the spectra obtained withSITELLE (Martin et al. 2015). Given that the effective res-olution is 10 % smaller than the theoretical resolution, thesecondary lobes of the sinc ILS are small enough that asimple Gaussian model can be used. Five emission-lines arefitted for each component: H α , the [N ii ] λ λ ii ] λ λ ii ] lines isalso fixed at 3. Consequently, only 5 parameters are fittedfor each component: 1 for the amplitude of the [N ii ] doublet,3 for the amplitudes of the other lines, and 1 for the veloc-ity of all 5 lines. An example of the resulting fit is shownin the bottom-right quadrant of Figure 5. We can see thatmost lines are well-fitted except for a possible 5 th compo-nent which was neglected at the enumeration step becausethe threshold has been kept high enough to minimize therisks of false detections.The data obtained after the automatic fitting procedureis a set of 25 flux maps (5 emission lines × The remarkable work of Trimble (1968) demonstrated thatproper motion velocity vectors of filaments all share thesame origin both in position and time, and that the ex-pansion velocity is approximately proportional to the ra-dius. Subsequent analyses have come to similar conclusions(Wyckoff & Murray 1977; Nugent 1998; Kaplan et al. 2008;Bietenholz & Nugent 2015), though their results on the lo-cation of the explosion center or the mean expansion ve-locity differ by a few arcseconds (Kaplan et al. 2008) (seeTable 3.4). The expansion model we adopt is based on 3 pa-rameters: the right ascension and declination ( α c , δ c ) of theexpansion center and the expansion factor e (e.g., Bietenholz& Nugent 2015): µ α = e ( α − α c ) , (11) µ δ = e ( δ − δ c ) , (12) Reference ∆ α ∆ δ Outburst(arcsec) (arcsec) Date (CE)Trimble (1968) 7.6(1.7) -8.5(1.4) 1140(15)Wyckoff & Murray (1977) 8.2(2.7) -8.6(3.6) 1120(7)Nugent (1998) 9.4(1.7) -8.0(1.3) 1130(16)Kaplan et al. (2008) 8.4(0.4) -8.1(0.4)
Table 1.
Comparison of the expansion parameters computed byseveral authors. Most of the data comes from Table 4 of Nugent(1998) and Table 3 of Kaplan et al. (2008). Following Kaplan et al.(2008), ∆ α and ∆ δ are the offset in right ascension and declinationbetween the center of the explosion and the star 5 (cid:48)(cid:48) to the northeastof the pulsar (which was first used by Trimble 1968 as a referencegiven its proximity to the center and its small proper motion).The coordinates of the reference star α = h m . s , δ =+ ◦ (cid:48) . (cid:48)(cid:48) come from the GAIA DR2 catalog (Brown et al.2018). where α and δ are the coordinates of the filament and µ α , µ δ denote the proper motion along the right ascension anddeclination axes.It has long been known that when the measured ex-pansion velocity is projected back to the origin, the com-puted outburst date lies around 1130 CE, which is nearly ahundred years after the recorded outburst date of 1054 CE(Stephenson & Green 2002). This is attributed to materialhaving been accelerated by the Crab’s pulsar wind nebula.Thus, we can expect some sort of signature of this acceler-ation preferentially near the center of the explosion. Fromthe preliminary results of a new analysis of the proper mo-tion of the Crab (Martin et al., in preparation), we believethat there indeed might be an accelerated expansion nearthe center, which means that the expansion factor is highernear the center than it would be if following a purely lin-ear model. However, as a first approximation we choose toconsider a simple linear model and use the expansion fac-tor e = . ( ) × − year − computed by Nugent (1998)along with the expansion center α = h m . ( ) s , δ =+ ◦ (cid:48) . ( . ) (cid:48)(cid:48) determined by Kaplan et al. (2008) fromtheir study of the pulsar proper motion. The validity of thishypothesis is discussed in more detail in section 4.1.1.To construct a 3D mapping of the Crab Nebula we re-quire an estimate of its distance. We adopt 2 kpc, computedby Trimble (1973), who estimates it to lie between 1.7 kpcand 2.4 kpc from prior morphological considerations. Thisdistance has not been improved since then and is still in usein recent articles (see e.g., Hester 2008; Kaplan et al. 2008).At this distance 1 (cid:48)(cid:48) =9.696 × − pc and the radial distance d to the expansion center can be computed from the radialvelocity v r via the expansion factor (Ng & Romani 2004): d = v r e (13)Knowing the expansion factor and the distance to theCrab makes it possible to obtain a mapping of our datain the Euclidean space (in parsecs) as shown in Figure 8and appendix B. Given the complexity of our data wehave created an interactive visualization in Python acces-sible through a Jupyter Notebook. It may be found at https://github.com/thomasorb/M1_paper . The 3D visu-alization program can also be run directly in any html
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T. Martin et al. O b s e r v e r li n e o f s i g h t NorthEast F r o n t L e f t To p Figure 8.
Front (as seen from Earth) and top view of the H α emission. The right panel displays an isometric representation of the nebulashowing the different viewpoints considered in the paper. The central part is filled with the inner envelope (see section 4.1.2) so that therear-facing components are obscured. The red arrow represents the pulsar torus axis and the red line shows the intersection of the innersurface with the pulsar torus plane as fitted by Ng & Romani (2004). All axes are in parsecs and the spatial grid of the left and centralpanels has a 1 pc stepping. The orientation symbol is color coded: East is green, North is blue and the line-of-sight direction is red. browser at https://mybinder.org/v2/gh/thomasorb/M1_paper/master and does not require any particular comput-ing knowledge.Two movies have been created with the help of panda3d ,an open-source framework for 3D rendering (Goslin & Mine2004). They are available online as supplementary material.Both show the total flux emitted in all 5 emission lines.Each data point is represented as a small cube and cor-responds to one velocity component at a given pixel. Nodata points have been added by interpolation. The thick-ness of some of the brightest filaments along the line of sightcomes from the fact that they could be resolved and fittedwith two components instead of one. The Milky Way back-ground has been adequately positioned to simulate whatwould be the typical perspective of someone moving aroundthe nebula. The Milky Way map has been created by theNASA/Goddard Space Flight Center Scientific VisualizationStudio ( https://svs.gsfc.nasa.gov/4851 ) based in parton the data obtained with Gaia (Brown et al. 2018). One ofthe movies shows a glowing sphere at the center to simulatethe blue continuum emitted by the pulsar wind nebula withan intensity to roughly match that observed in the composite HST image presented by Loll et al. (2013). The soundtrackis a sonification of the data set. Using the interferogramsdirectly as a sound wave, we have mixed multiple samplesplayed at different rates. The volume of the samples is re-lated to the square of the distance to the nebula and theplaying speed is related to the velocity of the observer withrespect to the nebula. A second movie highlights the geome-try of the Crab Nebula. The obtained data is shown with theinner and outer envelopes described in the next section. Thepulsar axis and the plane of the pulsar torus are indicated.
Each of the H α , [N ii ] λ λ ii ] λ λ (cid:48)(cid:48) at 2 kpc i.e., . × − pc) and an elementof spectral resolution along the line of sight (35 km s − i.e. . × − pc), which yields a voxel volume of . × − pc .Representing our observations as voxels in this way pro-vides a detailed representation of the Crab’s complex distri-bution of material and permits an investigation of its mor-phology at small and large scales. For example, it is worthtesting whether or not the Crab is an ellipsoid, as long sus-pected (e.g., Hester 2008). To this end we can analyse theradial distribution of the emitting material contained in asolid angle originating from the explosion center and obtainthe radial extent of the nebula by computing the outer limitof this distribution in all directions. To illustrate, we show inFigure 9 the distribution of material integrated over all di-rections along the pulsar torus axis. We see that no materialextends beyond 2 pc.Because the emission I ( H α ) is very roughly proportionalto the square of the ionized hydrogen density n p (at a con-stant temperature along the line of sight), I ( H α ) ∝ n p , wechoose to weight this distribution by (cid:112) I ( H α ) in order toapproximately sample the distribution of the material den-sity. We define the outer limit of the nebula in one particulardirection as the radius that encompasses 97% of the materialemitting in H α which is contained in a solid angle of 22.5 ◦ .We repeat this procedure over all directions, using the coor-dinates of the vertices of a 320-faced icosphere which ensuresan homogeneous distribution of the probing directions (seeappendix C1 for more details). Consequently, we obtain the MNRAS000
Each of the H α , [N ii ] λ λ ii ] λ λ (cid:48)(cid:48) at 2 kpc i.e., . × − pc) and an elementof spectral resolution along the line of sight (35 km s − i.e. . × − pc), which yields a voxel volume of . × − pc .Representing our observations as voxels in this way pro-vides a detailed representation of the Crab’s complex distri-bution of material and permits an investigation of its mor-phology at small and large scales. For example, it is worthtesting whether or not the Crab is an ellipsoid, as long sus-pected (e.g., Hester 2008). To this end we can analyse theradial distribution of the emitting material contained in asolid angle originating from the explosion center and obtainthe radial extent of the nebula by computing the outer limitof this distribution in all directions. To illustrate, we show inFigure 9 the distribution of material integrated over all di-rections along the pulsar torus axis. We see that no materialextends beyond 2 pc.Because the emission I ( H α ) is very roughly proportionalto the square of the ionized hydrogen density n p (at a con-stant temperature along the line of sight), I ( H α ) ∝ n p , wechoose to weight this distribution by (cid:112) I ( H α ) in order toapproximately sample the distribution of the material den-sity. We define the outer limit of the nebula in one particulardirection as the radius that encompasses 97% of the materialemitting in H α which is contained in a solid angle of 22.5 ◦ .We repeat this procedure over all directions, using the coor-dinates of the vertices of a 320-faced icosphere which ensuresan homogeneous distribution of the probing directions (seeappendix C1 for more details). Consequently, we obtain the MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. Figure 9.
Representation of the emitting material contained ina solid angle originating from the explosion center and alignedalong the pulsar axis. All axes are in parsecs and the spatial gridhas a 1 pc stepping. outer envelope of the emitting material, i.e., the dominantshape of the Crab as made up by its visible gas. Two per-spectives of this outer limit surface are shown in Figure 10,and all six perspectives are shown in Figure B1.Note that, doing this in the plane of the sky and con-sidering only the red part of the visible spectrum would re-veal the well known elliptical outer envelope of the Crab.But the 3D outer envelope is most surprising since it differsnotably from the generally assumed ellipsoidal shape (e.g.,Hester 2008). As viewed from the top , which we define asbeing along the axis perpendicular to the observer’s line ofsight looking from the north toward the south, a conspicu-ous heart-shaped morphology oriented along the pulsar axisis visible. The most rapidly expanding NW and SE lobesare separated by 120 ◦ of each other. The NW lobe is nearlyaligned with the pulsar torus axis, but the SE lobe is not.The potential effects of inhomogeneities of the expan-sion factor must be considered since this spatial reconstruc-tion is mostly based on a homologous expansion hypothesis(i.e., a constant expansion factor in all directions) whichis not absolutely true. Using the data obtained by Trim-ble (1968), we find that the expansion is indeed acceler-ated towards the north-west direction by a factor of 15 %with respect to the median value of the expansion factor( e = . ( ) × − year − ) computed by Nugent (1998) Figure 10.
Top:
Outer envelope enclosing 97% of the materialemitting in H α . The binning angle covers 22.5 ◦ . Bottom:
Innerenvelope enclosing 8% of the material emitting in H α . The binningangle covers 45 ◦ . The positions of the dark bays (Fesen et al. 1992)and the arcade region (Dubner et al. 2017) are shown as well asthe High-Helium bands first observed by Uomoto & MacAlpine(1987b) (in gray). We have also reported the topologic featuresreferenced in the text. Other details are the same as Figure 8. and used in this article to derive the 3D model of the neb-ula (see Figure 11). Of course, this accelerated expansion isassociated with a dilatation of the ellipsoidal shape of thenebula in the same direction. Knowing only the proper mo-tion of the nebula and using a homologous expansion modelto compute its shape, one would have exaggerated this di-latation effect, resulting in a nebula with an envelope evenmore extended towards the north-east direction. We havefitted an ellipsoid to the computed outer envelope and ob-tained the dilatation ratio. The center of the ellipsoid coin-cides with the center of expansion and its major axis followsthe axis of the pulsar torus. As shown in Figure 11, the di-latation ratio of the SE-Lobe (which gives its heart shapeto the nebula) is around 1.3. It would require an expansioninhomogeneity of the same order to keep the model compat-ible with an ellipsoid, which is two times larger than whatis observed in the celestial plane. Moreover, this expansioninhomogeneity should not be related with any extension ofthe nebula in this direction, which is in contradiction withthe correlation we observed based on Trimble (1968) data.We are thus confident that, if the computed 3D model ofthe nebula might indeed be exaggerated along the line-of-sight, the general shape should not differ enough to make itcompletely compatible with an ellipsoid. The Crab shows an intricate complex of filamentary struc-tures going deep under its outer envelope, and it can be
MNRAS , 1–15 (2019) T. Martin et al.
Figure 11.
Top:
Ellipsoid (wireframe) fitted to the outer envelope.The ellipsoid represents the surface of dilatation factor unity. Itscenter coincides with the center of expansion and its major axisfollows the axis of the pulsar torus. The color of the outer enveloperepresents its dilatation factor. Regions above the ellipsoid arered while regions below are blue. All axes are in parsecs andthe spatial grid has a 1 pc stepping. The orientation symbol iscolor coded: East is green, North is blue and the line-of-sightdirection is red.
Bottom-left:
Histogram of the dilatation of theouter envelope with respect to the fitted ellipsoid.
Bottom-right: median of the expansion factor measured by Trimble (1968) alongdifferent directions (north is up and east to the left). The circlesof homologous expansion are represented with dotted lines. Thenorth-west acceleration appears clearly and can be related to thedilatation along the north-west lobe but there is no accelerationas high as the 1.3 dilatation factor observed in the SE-Lobe. Evenif we consider the largest expansion factor measured in the planeof the sky (1.15), this is not sufficient to explain the factor 1.3dilatation observed in the south-west lobe. difficult to easily identify 3D locations of material. Thus, itis of interest to define the inner extent of this material, i.e.the size and the morphology of the central void of ionized gasaround the center of expansion, to distinguish front-facingfrom rear-facing ejecta. If we look at the distribution of ma-terial and integrate over all directions (Figure 9) we see thatno material is located below 0.5 pc. However, we take thisa step further by examining the 3D extent of this void inall directions. Using the same procedure as the one used toobtain the outer envelope, we obtained the inner envelopeconsidering a radius enclosing only 8% of the material emit-ting in H α in a solid angle of 45 ◦ . The limit of 8 % mayseem high but is explained by the relatively high numberof spurious detections near the center. We have thus slowlyincreased the limit up to the point where the 3D shape ofthe inner surface was not changing anymore (except for itsscale). Two perspectives of this inner limit surface are shownin Figure 10, and all six perspectives are shown in Figure B2. Comparing the inner and outer envelopes reveals cleartrends in the overall morphology of gas. Material aroundthe plane defined by the pulsar torus mapped by Ng & Ro-mani (2004) is much closer to the explosion center than thematerial distributed along the pulsar axis. This can be seenas a circular pinched valley running along the pulsar torusplane (labeled equatorial valley ). On the front of the nebulatwo small depressions are seen (labeled DB1 and DB2 onthe figure) that coincide with the pinched velocity regionsobserved by MacAlpine et al. (1989a) and the helium-richbands observed by Uomoto & MacAlpine (1987b). Anotherindentation is seen in the general region of the “arcade ofloops” described by Dubner et al. (2017). The two depres-sions DB1 and DB2 also coincide with the positions of thedark bays, (Fesen et al. 1992), also observed in the X-ray(Seward et al. 2006), and UV (see e.g., Dubner et al. 2017).Interestingly, the locations of the dark bays also coincidewith a conspicuous restriction oriented along the east-westplane running along the perimeter of the inner envelope (la-beled inner ring ). Together, these shared features betweenthe inner and outer envelopes strongly suggest a constrainedexpansion of the nebular material, potentially due to inter-action with a pre-existing circumstellar disk left by the pro-genitor star (Fesen et al. 1992; Smith 2013).
High resolution images of the Crab show that its filamentsexhibit a complex and fine structure (see, e.g., Hester et al.1996, Blair et al. 1997, and Sankrit et al. 1998). Many fila-ments are less than an arcsecond in width and point inwardinto the center of the nebula, with lengths ranging from ≈ (cid:48)(cid:48) − (cid:48)(cid:48) . Filaments are also often connected by arc-likebridges of emission with a “bubble-and-spike” morphology(Hester 2008). Numerous studies have associated this mor-phology with Rayleigh-Taylor (RT) instabilities (see for ex-ample Chevalier & Gull 1975; Hester et al. 1996; Bucciantiniet al. 2004; Stone & Gardiner 2007; Porth et al. 2014). Theseinstabilities are generally characterized by 2 parameters. (1)Angular size, i.e., the wavelength of the perturbations, whichis strongly related to the stability of the shell. NumericalMHD simulations (Bucciantini et al. 2004) show that onlyperturbations at a scale smaller than ∼ π / should give riseto RT instabilities and to the observed structures with pro-truding fingers. (2) The size of the filaments, which shouldalso be related to the wavelength of these perturbations.In Figure 12 we show the whole structure as if all vox-els were at the same radius in a classical Mercator projec-tion, and in Figure D1 as orthographic projections. All mapsare gridded with triangles having sides covering an angle of16.6 ◦ ( (cid:39) π / ), which is helpful to measure the relative sizesof the different structures.Crab material is largely distributed along boundariesresembling a honeycomb. This structure is hierarchical; i.e.,larger regions ( ∼ π / . ) have thicker filaments, while otherregions, in particular in the high-velocity lobes, exhibit amuch smaller and thinner distribution ( ∼ π / ). The inte-rior of the largest structures is generally divided into smallerregions that are also at a larger radii. The size of the struc-tures appear to be anti-correlated with the radius; i.e., the MNRAS000
High resolution images of the Crab show that its filamentsexhibit a complex and fine structure (see, e.g., Hester et al.1996, Blair et al. 1997, and Sankrit et al. 1998). Many fila-ments are less than an arcsecond in width and point inwardinto the center of the nebula, with lengths ranging from ≈ (cid:48)(cid:48) − (cid:48)(cid:48) . Filaments are also often connected by arc-likebridges of emission with a “bubble-and-spike” morphology(Hester 2008). Numerous studies have associated this mor-phology with Rayleigh-Taylor (RT) instabilities (see for ex-ample Chevalier & Gull 1975; Hester et al. 1996; Bucciantiniet al. 2004; Stone & Gardiner 2007; Porth et al. 2014). Theseinstabilities are generally characterized by 2 parameters. (1)Angular size, i.e., the wavelength of the perturbations, whichis strongly related to the stability of the shell. NumericalMHD simulations (Bucciantini et al. 2004) show that onlyperturbations at a scale smaller than ∼ π / should give riseto RT instabilities and to the observed structures with pro-truding fingers. (2) The size of the filaments, which shouldalso be related to the wavelength of these perturbations.In Figure 12 we show the whole structure as if all vox-els were at the same radius in a classical Mercator projec-tion, and in Figure D1 as orthographic projections. All mapsare gridded with triangles having sides covering an angle of16.6 ◦ ( (cid:39) π / ), which is helpful to measure the relative sizesof the different structures.Crab material is largely distributed along boundariesresembling a honeycomb. This structure is hierarchical; i.e.,larger regions ( ∼ π / . ) have thicker filaments, while otherregions, in particular in the high-velocity lobes, exhibit amuch smaller and thinner distribution ( ∼ π / ). The inte-rior of the largest structures is generally divided into smallerregions that are also at a larger radii. The size of the struc-tures appear to be anti-correlated with the radius; i.e., the MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. Figure 12.
Top : Mercator projection of total flux of the emitting material in the Crab. The densest and deepest buried material (at lessthan 1.1 pc of the center) is represented in black while the faster expanding material is in gray. As the angles are not conserved, weoverplot a grid of triangles which vertices sides cover an angle of 16.6 ◦ ( (cid:39) π / ). The red line shows the pulsar torus plane as fitted by Ng& Romani (2004). The center of the latitude and longitude axes corresponds to the center of expansion in the face-on view. Longitudeincreases towards the West. Middle and bottom : Mercator projections of 3C 58 (Lopez & Fesen 2018), and Cas A (Milisavljevic & Fesen2013), respectively.MNRAS , 1–15 (2019) T. Martin et al.
Figure 13.
Slices in 3 different directions with respect to the line of sight direction. The slice at 90 ◦ beeing the canonical point of viewperpendicular to the line of sight. All points at ± ◦ are merged together so that the angular size of a slice is 30 ◦ . A top view of theselected pixels is shown at the bottom. It shows the slice extent used to generate the plots in the corresponding upper panel. Axes arein pc. The orientation symbol is color coded: East is green, North is blue and the line-of-sight direction is red. largest and most conspicuous structures are at smallest radiiwith smallest expansion velocities. Some of the largest struc-tures, seen clearly on the top and back views of Figure 12,exhibit a polygonal shape and are found all along the pulsartorus equatorial plane. One also sees these structures whenviewing the Crab through narrow slices along different ori-entations (Figure 13). The largest regions are localized inthe interior with diameters of 0.5 to 1 pc. Above these atlarger radii lie numerous smaller bubbles. There exist very few kinematic maps of supernova remnantsdetailed enough that a comparison of the shape of their fila-mentary structure can be studied. Figure 12 shows a compar-ison of the mercator projections of the Crab, 3C 58 (Lopez &Fesen 2018) and Cas A (Milisavljevic & Fesen 2013). Large-scale ejecta rings may be a common phenomenon of young,core-collapse SNRs (Milisavljevic & Fesen 2017). The largestand deepest structures of the Crab are similar both in sizeand shape as those seen in 3C 58. However, both the Craband Cas A share small scale circular formations.3C 58 has many overlapping properties with the Crab. The Crab Nebula is far brighter and more luminous than3C 58, but both remnants are bright in both the radio and X-rays in their center and harbor young, rapidly spinning cen-tral pulsars that provide the magnetic field and relativisticparticles that generate the observed center-filled synchrotronradiation. 3C 58 may be connected to the historical event of1181 CE (Stephenson & Green 2002; Kothes 2013), whichwould make it only 127 years younger than the Crab. How-ever, this relatively young age is inconsistent with its overallangular size ( . (cid:48) × . (cid:48) ) and proper motion measurementsof its expanding ejecta (Fesen et al. 1988; van den Bergh1990) that suggest a much older remnant, potentially as oldas ± yr.Cas A is the youngest known core-collapse remnant withan estimated explosion date of 1681 CE (Fesen et al. 2006).Milisavljevic & Fesen (2013) demonstrated that the bulk ofthe remnant’s main shell ejecta are arranged in several well-defined complete or broken ring-like structures. These ringstructures have diameters that can be comparable to the ra-dius of the remnant ( ∼ pc). Some rings show considerableradial extensions giving them a crown-like appearance, whileother rings exhibit a frothy, ring-like substructure on scalesof ∼ . pc. A subsequent three-dimensional map of its inte-rior unshocked ejecta made from near-infrared observations MNRAS000
Slices in 3 different directions with respect to the line of sight direction. The slice at 90 ◦ beeing the canonical point of viewperpendicular to the line of sight. All points at ± ◦ are merged together so that the angular size of a slice is 30 ◦ . A top view of theselected pixels is shown at the bottom. It shows the slice extent used to generate the plots in the corresponding upper panel. Axes arein pc. The orientation symbol is color coded: East is green, North is blue and the line-of-sight direction is red. largest and most conspicuous structures are at smallest radiiwith smallest expansion velocities. Some of the largest struc-tures, seen clearly on the top and back views of Figure 12,exhibit a polygonal shape and are found all along the pulsartorus equatorial plane. One also sees these structures whenviewing the Crab through narrow slices along different ori-entations (Figure 13). The largest regions are localized inthe interior with diameters of 0.5 to 1 pc. Above these atlarger radii lie numerous smaller bubbles. There exist very few kinematic maps of supernova remnantsdetailed enough that a comparison of the shape of their fila-mentary structure can be studied. Figure 12 shows a compar-ison of the mercator projections of the Crab, 3C 58 (Lopez &Fesen 2018) and Cas A (Milisavljevic & Fesen 2013). Large-scale ejecta rings may be a common phenomenon of young,core-collapse SNRs (Milisavljevic & Fesen 2017). The largestand deepest structures of the Crab are similar both in sizeand shape as those seen in 3C 58. However, both the Craband Cas A share small scale circular formations.3C 58 has many overlapping properties with the Crab. The Crab Nebula is far brighter and more luminous than3C 58, but both remnants are bright in both the radio and X-rays in their center and harbor young, rapidly spinning cen-tral pulsars that provide the magnetic field and relativisticparticles that generate the observed center-filled synchrotronradiation. 3C 58 may be connected to the historical event of1181 CE (Stephenson & Green 2002; Kothes 2013), whichwould make it only 127 years younger than the Crab. How-ever, this relatively young age is inconsistent with its overallangular size ( . (cid:48) × . (cid:48) ) and proper motion measurementsof its expanding ejecta (Fesen et al. 1988; van den Bergh1990) that suggest a much older remnant, potentially as oldas ± yr.Cas A is the youngest known core-collapse remnant withan estimated explosion date of 1681 CE (Fesen et al. 2006).Milisavljevic & Fesen (2013) demonstrated that the bulk ofthe remnant’s main shell ejecta are arranged in several well-defined complete or broken ring-like structures. These ringstructures have diameters that can be comparable to the ra-dius of the remnant ( ∼ pc). Some rings show considerableradial extensions giving them a crown-like appearance, whileother rings exhibit a frothy, ring-like substructure on scalesof ∼ . pc. A subsequent three-dimensional map of its inte-rior unshocked ejecta made from near-infrared observations MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. sensitive to [S III] λ λ Ni-rich ejecta(Li et al. 1993; Blondin et al. 2001). Such plumes can pushthe nuclear burning zones located around the Fe core out-ward, creating dense shells separating zones rich in O, S, andSi from the Ni-rich material. After the SN shock breakoutadditional energy input from the radioactive decay of Nicontinues to drive inflation of Ni-rich structures and fa-cilitates mixing between ejecta components. This late timeexpansion can modify the overall SN ejecta morphology ontimescales of weeks or months. Compression of surroundingnonradioactive material by hot expanding plumes of radioac-tive Ni-rich ejecta generates a “Swiss cheese”-like structurethat is frozen into the homologous expansion when the radio-active power of Ni is strongest. Gabler et al. (2020) foundthat this “Ni-bubble effect” accelerates the bulk of the nickelin their 3D models and causes an inflation of the initiallyoverdense Ni-rich clumps, which leads to underdense, ex-tended fingers, enveloped by overdense skins of compressedsurrounding matter.Stockinger et al. (2020) recently performed 3D full-sphere simulations of supernovae originating from non-rotating progenitors similar to those anticipated to be as-sociated with the SN 1054. Their low energy explosions( ∼ . − . × erg) are compared in two contrasting sce-narios: (1) iron-core progenitors at the low-mass end of thecore-collapse supernova domain ( ≈ M (cid:12) ), and (2) a super-AGB progenitor with an oxygen-neon-magnesium core thatcollapses and explodes as electron-capture supernova. Theydisfavor associating SN 1054 with an electron capture su-pernova because the kick experienced by the neutron star isnegligible and inconsistent with the observed ≈ km s − transverse velocity of Crab pulsar. Instead, they favour sim-ulations with iron-core progenitors with less 2nd dredge-upthat result in highly asymmetric explosions with hydrody-namic and neutrino-induced NS kick of > km s − and aNS spin period of ∼ ms, not unlike the Crab pulsar. Theresulting distribution of Ni-rich material from these ex-plosions, which enable efficient mixing and dramatic shockdeceleration in the extended hydrogen envelope, is poten-tially consistent with our mapping of Crab ejecta. However,simulations extending the evolution from ∼ days to ∼ We have presented a 3D kinematic reconstruction of theCrab Nebula that has been created from a hyperspectralcube obtained with SITELLE. The data is comprised of310 000 high resolution ( R = ) spectra containing H α ,[N ii ], and [S ii ] line emission, and represent the most de-tailed homogeneous spectral data set ever obtained of theCrab Nebula. Our findings can be summarized as follows:(i) The general shape of the Crab, as measured by 97%of the material emitting in H α , occupies a “heart-shaped”volume and is symmetrical about the plane of the pulsarwind torus. This morphology runs counter to the generallyassumed ellipsoidal volume and is not an artifact of assuminga uniform global expansion. The most rapidly expandingNW and SE lobes are separated by 120 ◦ of each other. TheNW lobe is nearly aligned with the pulsar torus axis, butthe SE lobe is not.(ii) Conspicuous restrictions in the distribution of ma-terial as mapped by the inner and outer limits of emis-sion is seen along the band of He-rich filaments (Uomoto& MacAlpine 1987b; MacAlpine et al. 1989b). Notable de-pressions are also coincident with the east and west darkbays (Fesen et al. 1992). Together these features are con-sistent with constrained expansion of Crab ejecta, possiblyassociated with interaction between the supernova and a pre-existing cirumstellar disk.(iii) The filaments follow a honeycomb-like distributiondefined by a combination of straight and rounded boundariesat large and small scales. The scale size is anti-correlatedwith distance from the center of expansion; i.e., largest fea-tures are found at smallest radii. The structures are not un-like those seen in other SNRs, including 3C 58 and Cas A,where they have been attributed to turbulent mixing pro-cesses that encouraged outwardly expanding plumes of ra-dioactive Ni-rich ejecta.The observed kinematic characteristics reflect criticaldetails concerning the original supernova of 1054 CE andits progenitor star, and may favour a low-energy explosionof an iron-core progenitor as opposed to an oxygen-neon-magnesium core that collapses and explodes as an electron-capture supernova. Planned future observations will provideadditional hyperspectral cubes spanning more wavelengthwindows that include the emission lines of [O II] λ λ β , [O III] λ λ λ λ DATA AVAILABILITY
The data underlying this article will be shared on reasonablerequest to the corresponding author.
MNRAS , 1–15 (2019) T. Martin et al.
ACKNOWLEDGEMENTS
We thank the referee for comments that greatly improvedthe paper. We thank R. Fesen for many helpful discussionsthat guided interpretation of our data, and Hans-ThomasJanka who commented on an earlier draft of the manuscript.We are also thankful to the python (Van Rossum & Drake2009) community and the free softwares that made the anal-ysis of this data possible: numpy (Oliphant 2006), scipy (Vir-tanen et al. 2020), pandas (McKinney et al. 2010), panda3d (Goslin & Mine 2004), pyvista (Sullivan & Kaszynski 2019), matplotlib (Hunter 2007) and astropy (Price-Whelan et al.2018). This paper is based on observations obtained withSITELLE, a joint project of Universit´e Laval, ABB, Univer-sit´e de Montr´eal and the Canada-France-Hawaii Telescope(CFHT) which is operated by the National Research Coun-cil (NRC) of Canada, the Institut National des Science del’Univers of the Centre National de la Recherche Scientifique(CNRS) of France, and the University of Hawaii. The au-thors wish to recognize and acknowledge the very significantcultural role that the summit of Mauna Kea has always hadwithin the indigenous Hawaiian community. LD is gratefulto the Natural Sciences and Engineering Research Coun-cil of Canada, the Fonds de Recherche du Qu´ebec, and theCanadian Foundation for Innovation for funding. DM ac-knowledges support from the National Science Foundationfrom grants PHY-1914448 and AST-2037297.
REFERENCES
Alarie A., Bilodeau A., Drissen L., 2014, MNRAS, 441, 2996Andresen H., M¨uller B., M¨uller E., Janka H. T., 2017, MNRAS,468, 2032Baril M. R., et al., 2016, in Evans C. J., Simard L., Takami H.,eds, Proceedings of SPIE. International Society for Optics andPhotonics, p. 990829Bennett C. L., 2000, Imaging the Universe in Three Dimensions.Proceedings from ASP Conference Vol. 195. Edited by W. vanBreugel and J. Bland-Hawthorn. ISBN: 1-58381-022-6 (2000),195Bietenholz M. F., Nugent R. L., 2015, Monthly Notices of theRoyal Astronomical Society, 454, 2416Black C. S., Fesen R. A., 2015, MNRAS, 447, 2540Blair W. P., Davidson K., Fesen R. A., Uomoto A., MacAlpineG. M., Henry R. B. C., 1997, ApJS, 109, 473Blondin J. M., Borkowski K. J., Reynolds S. P., 2001, ApJ, 557,782Brown A. G. A., et al., 2018, Astronomy & Astrophysics, 616, A1Bucciantini N., Amato E., Bandiera R., Blondin J. M., Del ZannaL., 2004, Astronomy & Astrophysics, 423, 253Burrows A., Radice D., Vartanyan D., 2019, MNRAS, 485, 3153Charlebois M., Drissen L., Bernier A. P., Grandmont F., BinetteL., 2010, AJ, 139, 2083Chevalier R. A., 1977, in Schramm D. N., ed., Astrophysicsand Space Science Library, Vol. 66, Supernovae. p. 53,doi:10.1007/978-94-010-1229-4 5Chevalier R. A., Gull T. R., 1975, ApJ, 200, 399Chugai N. N., Utrobin V. P., 2000, A&A, 354, 557Clark D. H., Stephenson F. R., 1977, The historical supernovaeCouch S. M., Chatzopoulos E., Arnett W. D., Timmes F. X.,2015, ApJ, 808, L21Davidson K., Fesen R. A., 1985, ARA&A, 23, 119DeLaney T., et al., 2010, ApJ, 725, 2038Drissen L., et al., 2019, MNRAS, 485, 3930 Dubner G., Castelletti G., Kargaltsev O., Pavlov G. G., Bieten-holz M., Talavera A., 2017, The Astrophysical Journal, 840,82Fesen R. A., Kirshner R. P., Becker R. H., 1988, in Roger R. S.,Landecker T. L., eds, IAU Colloq. 101: Supernova Remnantsand the Interstellar Medium. p. 55Fesen R. A., Martin C. L., Shull J. M., 1992, The AstrophysicalJournal, 399, 599Fesen R. A., Shull J. M., Hurford A. P., 1997, The AstronomicalJournal, 113, 354Fesen R. A., et al., 2006, The Astrophysical Journal, 645, 283Gabler M., Wongwathanarat A., Janka H.-T., 2020, arXiv e-prints, p. arXiv:2008.01763Gessner A., Janka H.-T., 2018, ApJ, 865, 61Goslin M., Mine M. R., 2004, Computer, 37, 112Grefenstette B. W., et al., 2017, ApJ, 834, 19Hester J. J., 2008, Annual Review of Astronomy and Astro-physics, 46, 127Hester J. J., et al., 1996, The Astrophysical Journal, 456, 225Hillebrandt W., 1982, A&A, 110, L3Hunter J. D., 2007, Computing in science & engineering, 9, 90H¨ogbom J. A., 1974, Astronomy and Astrophysics SupplementSeries, 15, 417Kaplan D. L., Chatterjee S., Gaensler B. M., Anderson J., 2008,The Astrophysical Journal, 677, 1201Kitaura F. S., Janka H. T., Hillebrandt W., 2006, A&A, 450, 345Kothes R., 2013, Astronomy & Astrophysics, 560, A18Kuroda T., Kotake K., Hayama K., Takiwaki T., 2017, ApJ, 851,62Law C. J., et al., 2020, ApJ, 894, 73Lawrence S. S., MacAlpine G. M., Uomoto A., Woodgate B. E.,Brown L. W., Oliversen R. J., Lowenthal J. D., Liu C., 1995,AJ, 109, 2635Levenberg K., 1944, Quarterly of Applied Mathematics, 2, 164Li H., McCray R., Sunyaev R. A., 1993, ApJ, 419, 824Loll A. M., Desch S. J., Scowen P. A., Foy J. P., 2013, The As-trophysical Journal, 765, 152Lopez L. A., Fesen R. A., 2018, The Morphologies andKinematics of Supernova Remnants ( arXiv:1804.00024 ),doi:10.1007/s11214-018-0481-x, https://link.springer.com/article/10.1007/s11214-018-0481-x
Lundqvist P., Tziamtzis A., 2012, MNRAS, 423, 1571MacAlpine G. M., McGaugh S. S., Mazzarella J. M., Uomoto A.,1989a, ApJ, 342, 364MacAlpine G. M., McGaugh S. S., Mazzarella J. M., Uomoto A.,1989b, The Astrophysical Journal, 342, 364Maillard J. P., Drissen L., Grandmont F., Thibault S., 2013, Ex-perimental Astronomy, 35, 527Marquardt D. W., 1963, Journal of the Society for Industrial andApplied Mathematics, 11, 431Martin T., 2015, Phd thesis, Universit´e LavalMartin T., Drissen L., 2016, in Reyl´e C., Richard J., Cambr´esy L.,Deleuil M., P´econtal E., Tresse L., Vauglin I., eds, Proceedingsof the annual meeting of the French Society of Astronomy &Astrophysics Lyon, June 14-17, 2016. pp 23–28Martin T., Drissen L., Joncas G., 2012, in Radziwill N. M.,Chiozzi G., eds, Vol. 2, SPIE - Software and Cyberinfras-tructure for Astronomy II. pp 84513K–84513K–9Martin T., Drissen L., Joncas G., 2015, Astronomical Data Anal-ysis Software an Systems XXIV (ADASS XXIV), 495Martin T. B., Prunet S., Drissen L., 2016, Monthly Notices of theRoyal Astronomical Society, 463, 4223McKinney W., et al., 2010, in Proceedings of the 9th Python inScience Conference. pp 51–56Mezzacappa A., et al., 2020, Phys. Rev. D, 102, 023027Milisavljevic D., Fesen R. A., 2013, ApJ, 772, 134Milisavljevic D., Fesen R. A., 2015, Science, 347, 526Milisavljevic D., Fesen R. A., 2017, in Alsabti A. W., Murdin P.,MNRAS000
Lundqvist P., Tziamtzis A., 2012, MNRAS, 423, 1571MacAlpine G. M., McGaugh S. S., Mazzarella J. M., Uomoto A.,1989a, ApJ, 342, 364MacAlpine G. M., McGaugh S. S., Mazzarella J. M., Uomoto A.,1989b, The Astrophysical Journal, 342, 364Maillard J. P., Drissen L., Grandmont F., Thibault S., 2013, Ex-perimental Astronomy, 35, 527Marquardt D. W., 1963, Journal of the Society for Industrial andApplied Mathematics, 11, 431Martin T., 2015, Phd thesis, Universit´e LavalMartin T., Drissen L., 2016, in Reyl´e C., Richard J., Cambr´esy L.,Deleuil M., P´econtal E., Tresse L., Vauglin I., eds, Proceedingsof the annual meeting of the French Society of Astronomy &Astrophysics Lyon, June 14-17, 2016. pp 23–28Martin T., Drissen L., Joncas G., 2012, in Radziwill N. M.,Chiozzi G., eds, Vol. 2, SPIE - Software and Cyberinfras-tructure for Astronomy II. pp 84513K–84513K–9Martin T., Drissen L., Joncas G., 2015, Astronomical Data Anal-ysis Software an Systems XXIV (ADASS XXIV), 495Martin T. B., Prunet S., Drissen L., 2016, Monthly Notices of theRoyal Astronomical Society, 463, 4223McKinney W., et al., 2010, in Proceedings of the 9th Python inScience Conference. pp 51–56Mezzacappa A., et al., 2020, Phys. Rev. D, 102, 023027Milisavljevic D., Fesen R. A., 2013, ApJ, 772, 134Milisavljevic D., Fesen R. A., 2015, Science, 347, 526Milisavljevic D., Fesen R. A., 2017, in Alsabti A. W., Murdin P.,MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. eds, , Handbook of Supernovae. p. 2211, doi:10.1007/978-3-319-21846-5 97Ng C., Romani R. W., 2004, The Astrophysical Journal, 601, 479Nomoto K., 1987, ApJ, 322, 206Nomoto K., Sparks W. M., Fesen R. A., Gull T. R., Miyaji S.,Sugimoto D., 1982, Nature, 299, 803Nugent R. L., 1998, Publications of the Astronomical Society ofthe Pacific, 110, 831Oliphant T. E., 2006, A guide to NumPy. Vol. 1, Trelgol Pub-lishing USAOno M., Nagataki S., Ferrand G., Takahashi K., Umeda H.,Yoshida T., Orland o S., Miceli M., 2020, ApJ, 888, 111Orlando S., Miceli M., Pumo M. L., Bocchino F., 2015, ApJ, 810,168Orlando S., Miceli M., Pumo M. L., Bocchino F., 2016, ApJ, 822,22Orlando S., Wongwathanarat A., Janka H. T., Miceli M., OnoM., Nagataki S., Bocchino F., Peres G., 2020, arXiv e-prints,p. arXiv:2009.01789Osterbrock D., Ferland G., 2006, Astrophysics of gaseous nebulaeand active galactic nuclei. University Science Books, SausalitoPorth O., Komissarov S. S., Keppens R., 2014, Monthly Noticesof the Royal Astronomical Society, 443, 547Price-Whelan A. M., et al., 2018, The Astronomical Journal, 156,123Sankrit R., et al., 1998, ApJ, 504, 344Seward F. D., Tucker W. H., Fesen R. A., 2006, The AstrophysicalJournal, 652, 1277Smith N., 2013, MNRAS, 434, 102Sollerman J., Lundqvist P., Lindler D., Chevalier R. A., FranssonC., Gull T. R., Pun C. S. J., Sonneborn G., 2000, ApJ, 537,861Stephenson F. R., Green D. A., 2002, in Historical supernovaeand their remnants.Stockinger G., et al., 2020, MNRAS, 496, 2039Stone J. M., Gardiner T., 2007, The Astrophysical Journal, 671,1726Sullivan C. B., Kaszynski A., 2019, Journal of Open Source Soft-ware, 4, 1450Tominaga N., Blinnikov S. I., Nomoto K., 2013, ApJ, 771, L12Trimble V., 1968, The Astronomical Journal, 73, 535Trimble V., 1973, Publications of the Astronomical Society of thePacific, 85, 579Uomoto A., MacAlpine G. M., 1987b, The Astronomical Journal,93, 1511Uomoto A., MacAlpine G. M., 1987a, AJ, 93, 1511Van Rossum G., Drake F. L., 2009, Python 3 Reference Manual.CreateSpace, Scotts Valley, CAVirtanen P., et al., 2020, Nature Methods,Vogt F., Dopita M. A., 2010, ApJ, 721, 597Vogt F. P. A., Bartlett E. S., Seitenzahl I. R., Dopita M. A.,Ghavamian P., Ruiter A. J., Terry J. P., 2018, Nature As-tronomy, 2, 465Westernacher-Schneider J. R., O’Connor E., O’Sullivan E., Tam-borra I., Wu M.-R., Couch S. M., Malmenbeck F., 2019, Phys.Rev. D, 100, 123009Wongwathanarat A., Janka H.-T., M¨uller E., Pllumbi E., WanajoS., 2017, ApJ, 842, 13Wyckoff S., Murray C. A., 1977, Monthly Notices of the RoyalAstronomical Society, 180, 717Yang H., Chevalier R. A., 2015, ApJ, 806, 153ˇCadeˇz A., Carrami˜nana A., Vidrih S., 2004, ApJ, 609, 797van den Bergh S., 1990, ApJ, 357, 138 APPENDIX A: VELOCITY UNCERTAINTYAPPENDIX B: 3D MAPSAPPENDIX C: ICOSPHEREAPPENDIX D: FILAMENTARY STRUCTURE
This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS , 1–15 (2019) T. Martin et al.
Figure A1.
Map of the velocity uncertainty related to Figure 7. The uncertainty is an output of the fit realised with
ORCS (Martin et al.2015). It is calculated from the covariance matrix returned by the Levenberg-Marquardt minimization process (Levenberg 1944; Marquardt1963). As all 5 emission-lines for each velocity component share the same velocity parameter, the uncertainty is generally smaller thatthe uncertainty that would have been obtained by fitting the emission-lines independently. As other constraints are implemented inthe spectrum model (see section 3.3) the emission-lines velocities and fluxes are the parameters of one spectrum model. As such theiruncertainty are only loosely related to the SNR of each emission-line (see Martin et al. 2015 for more details).MNRAS000
ORCS (Martin et al.2015). It is calculated from the covariance matrix returned by the Levenberg-Marquardt minimization process (Levenberg 1944; Marquardt1963). As all 5 emission-lines for each velocity component share the same velocity parameter, the uncertainty is generally smaller thatthe uncertainty that would have been obtained by fitting the emission-lines independently. As other constraints are implemented inthe spectrum model (see section 3.3) the emission-lines velocities and fluxes are the parameters of one spectrum model. As such theiruncertainty are only loosely related to the SNR of each emission-line (see Martin et al. 2015 for more details).MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. Figure B1.
Outer envelope related to Figure 10. All 6 viewing angles are shown. Details are the same as Figure 8 and 10. The orientationsymbol is color coded: East is green, North is blue and the line-of-sight direction is red.MNRAS , 1–15 (2019) T. Martin et al.
Figure B2.
Inner envelope related to Figure 10. All 6 viewing angles are shown. Details are the same as Figure 8 and 10. The orientationsymbol is color coded: East is green, North is blue and the line-of-sight direction is red. MNRAS000
Inner envelope related to Figure 10. All 6 viewing angles are shown. Details are the same as Figure 8 and 10. The orientationsymbol is color coded: East is green, North is blue and the line-of-sight direction is red. MNRAS000 , 1–15 (2019)
D mapping of the Crab Nebula. I. Figure C1.
Icosahedron ( a , 20 faces), subdivided 2 times along its edges to make the 80 ( b ) and 320 faces ( c ) icospheres. This type ofsphere approximation is interesting since its vertices are homogeneously distributed. We used the 320 icosphere to construct the innerand outer envelope as well as to make a homogeneous grid on the projections of Figures 12 and D1.MNRAS , 1–15 (2019) T. Martin et al.
Figure D1.
Orthographic projections of all the voxels at the same unit radius to reveal the structuration of the filamentary envelopeof the nebula. This is presented as an alternative to Mercator projections shown in Figure 12 which produce important distortions farfrom the equator. A grid of triangles with 16.6 ◦ ( (cid:39) π / ) sides helps to measure the size of the structures. The red arrow indicates thedirection of the pulsar torus and the red line shows its equator. The orientation symbol is color coded: East is green, North is blue andthe line-of-sight direction is red. MNRAS000