Evidence For Particle Acceleration to the Knee of the Cosmic Ray Spectrum in Tycho's Supernova Remnant

2.1 MB·English

by  Kristoffer A. Eriksen

Accepted to ApJ Letters
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
EVIDENCE FOR PARTICLE ACCELERATION TO THE KNEE OF THE COSMIC RAY SPECTRUM IN
TYCHO’S SUPERNOVA REMNANT
Kristoffer A. Eriksen1, John P. Hughes1, Carles Badenes2,3, Robert Fesen4, Parviz Ghavamian5, David
Moffett6, Paul P. Plucinksy7, Cara E. Rakowski8, Estela M. Reynoso9,10,Patrick Slane7
Accepted to ApJ Letters
ABSTRACT
Supernova remnants (SNRs) have long been assumed to be the source of cosmic rays (CRs) up to
the “knee” of the CR spectrum at 1015 eV, accelerating particles to relativistic energies in their blast
waves by the process of diffusive shock acceleration (DSA). Since cosmic ray nuclei do not radiate
1
1 efficiently, their presence must be inferred indirectly. Previous theoretical calculations and X-ray
0 observations show that CR acceleration modifies significantly the structure of the SNR and greatly
2 amplifies the interstellar magnetic field. We present new, deep X-ray observations of the remnant
of Tycho’s supernova (SN 1572, henceforth Tycho), which reveal a previously unknown, strikingly
n
ordered pattern of non-thermal high-emissivity stripes in the projected interior of the remnant, with
a
spacingthatcorrespondstothegyroradiiof1014−1015 eVprotons. Spectroscopyofthestripesshows
J
the plasma to be highly turbulent on the (smaller) scale of the Larmor radii of TeV energy electrons.
7
Models of the shock amplification of magnetic fields produce structure on the scale of the gyroradius

ofthehighestenergyCRspresent,buttheydonotpredictthehighly-orderedpatternweobserve. We
]
E interpret the stripes as evidence for acceleration of particles to near the knee of the CR spectrum in
regions of enhanced magnetic turbulence, while the observed highly ordered pattern of these features
H
provides a new challenge to models of DSA.
.
h Subjectheadings: accelerationofparticles—Cosmicrays—ISM:individualobjects(SN1572,Tycho)
p — ISM: supernova remnants
-
o
tr 1. INTRODUCTION chrotronemissionobservedfromthelimbsofyoungSNRs
s (Koyamaetal.1995;Hughesetal.2000a;Gotthelfetal.
a Energetic arguments have long-favored the hypothe-
2001; Hwang et al. 2002) indicates the presence of elec-
[ sis that supernova shocks are responsible for the bulk
of the cosmic ray (CR) flux at Earth from 108 eV to trons to tens of TeVs. Though CR nuclei comprise the
1 vast majority of mass in SNR relativistic particles, they
the “knee”, a flattening of the CR spectrum at approx-
v imately 1015 eV (Blandford & Eichler 1987). This re- do not radiate efficiently, and their presence must be in-
4 ferred indirectly, either by the modification of the shock
quires that ∼ 10% of the mechanical energy of Galactic
5 dynamics (Warren et al. 2005) and thermal state of the
supernova remnants (SNRs) be converted into relativis-
4 plasma (Hughes et al. 2000b; Decourchelle et al. 2000;
tic particles over the remnants’ lifetime. In the favored
1 Patnaudeetal.2009),orbytheproductionofsecondary
modelofthisprocess,diffusiveshockacceleration(DSA),
1. self-generated magnetic turbulence provides scattering pions (Butt et al. 2001; Aharonian et al. 2006).
0 centers from which suprathermal particles may diffuse Bell&Lucek(2001)predictedsignificantamplification
1 between the up- and downstream fluid, gaining energy ofthemagneticfieldinsupernovashocksbyCR-induced
1 turbulentprocesses,allowingforaccelerationtotheknee
(on average) with each shock crossing (Bell 1978). The
: withinaSNRlifetime. Thispictureissupportedbyhigh
v ubiquitous non-thermal radio emission in SNRs is ev-
spatial resolution observations from the Chandra X-ray
i idence of GeV electrons in SNRs, and the X-ray syn-
X Observatory that require high magnetic fields to explain
the short synchrotron cooling time implied by the geo-
ar 1Department of Physics and Astronomy, Rutgers, the State metrically thin rims of young remnants (Vink & Laming
UniversityofNewJersey,Piscataway,NJ08854,USA
2School of Physics and Astronomy, Tel-Aviv University, 2003; Ballet 2006). An extensive theoretical literature
Tel-Aviv69978,Israel has developed detailing a range of plasma instabilities
3Benoziyo Center for Astrophysics, Weizmann Institure of that might generate these high fields. Of these, the cos-
Science,Rehovot76100,Israel mic ray current driven instability (CRCD, often referred
4Department of Physics and Astronomy, Dartmouth Col-
toasBell’snon-resonantmechanism,Bell2004),inwhich
lege,Hanover,NH03755,USA
5Space Telescope Science Institute, Baltimore, MD 21218, the magnetic field is amplified by turbulence generated
USA byCRspropagatingaheadoftheshock,hasreceivedthe
6Department of Physics, Furman University, Greenville, SC
most study, though other processes (e.g., acoustic insta-
29613,USA
7Harvard-Smithsonian Center for Astrophysics,Cambridge, bilities, resonant instabilities, etc.) have been suggested.
MA02138,USA MHD (Bell 2005) and particle-in-cell (PIC) simulations
8Space Science Division, Naval Research Labora- (Riquelme & Spitkovsky 2009) show that the CRCD in-
tory,Washington,DC20375,USA
stabilitydrivesplasmamotionswhichsuppressgrowthon
9Instituto de Astronomia´ y Fi´sica del Espacio,Buenos Aires,
Argentina small scales, until the dominant scale of the turbulence
10Faculty of Exact and Natural Sciences, University of iscomparabletotheLarmorradiusofthehighestenergy
BuenosAires,Argentina
2 Eriksen et al.
Fig. 1.— Chandra X-ray 4.0–6.0 keV image of the Tycho supernova remnant, smoothed with a ∼ 0.75(cid:48)(cid:48) Gaussian and displayed with
anarcsinhscaling,showingvariousregionsofstripinginthenonthermalemission. Clockwisefromtheupperright: a)Themainwestern
stripesdiscussedinthisLetter;b)Afainterensembleofstripes;c)apreviously-knownbrightarcofnon-thermalemission,withournewly
discoveredstreamers;d)filamentsof“rippledsheet”morphologycommoninopticalobservationsofmiddle-agedSNRs.
cosmic rays. This results in a non-uniform medium with brightened shell, a number of bright regions are seen to-
structured, filamentary density and field enhancements ward the projected interior of the remnant. While the
∼ 10 times greater than the mean quantities, surround- strongest of these features are visible in earlier Chandra
ing cavities of size approximately equal to the Larmor data (Warren et al. 2005) our deeper observation reveals
radius of the highest energy particles present. Indeed, astrikingpatternofnearly-regularlyspacedstripes. The
Bell’s analysis and simulations show that once the insta- brightest group, centered ∼ 30(cid:48)(cid:48) interior to the west-
bilityhassaturated,themediumconsistsofanetworkof ern limb (see Figure 2), has a peak surface brightness
holes evacuated of plasma and field, enclosed by “wan- twice that of the brightest sections of the rims, and is
deringfilaments”ofhighdensityandfrozen-infield. One the primary subject of this communication. A second,
goal of observations is to discover evidence for structure fainter pattern extends east-west, 55 − 75(cid:48)(cid:48) inside the
in SNR shocks on this spatial scale. southern rim, and there is evidence for several other re-
gions of striping near the detection limit. Comparison
2. OBSERVATIONS
with a shallower 2004 Chandra observation reveals no
We observed Tycho in April 2009 with the Chandra
statistically significant change in the brightness of the
X-ray Observatory Advanced CCD Imaging Spectrome-
stripes, ruling out any dramatic flux variability like that
ter imaging array, as part of a Cycle 10 Large Program
observed in the non-thermal X-ray filaments of the SNR
(LP), using the 4 ACIS-I front-side illuminated CCDs,
RX J1713.7-3946 (Uchiyama et al. 2007). Our prelim-
operatedinFaintmode. Thetotalprogramwassplitinto
inary proper motion measurements for the stripes are
nine individual obsids, which we reprocessed with CIAO
consistent with the overall expansion of the blast wave
version 4.1, using the gain tables and CTI correction in
and, in particular, show no evidence for non-radial flow.
CALDB 4.1.3. Examination of the lightcurves revealed
There are no obvious counterparts to these features in
no significant background flares. The registration of the
the radio (Reynoso et al. 1997), nor in the mid-IR.
individual pointings was improved using the measured
relative positions of background point sources, though
3. ANALYSIS
the necessary shifts were typically (cid:28)1(cid:48)(cid:48). For imaging
3.1. Location of the Stripes
analyses,theregisteredeventlistsweremergedusingthe
standard CIAO tools, and have a total average livetime Before investigating the nature of the stripes, we first
of ∼ 734.1ks. For spectroscopy, counts were extracted must locate them within the three-dimensional volume
andRMFsandARFsweregeneratedforeachindividual of the remnant. The canonical picture of a young SNR
obsid, which we fit jointly in XSPEC version 12.6.0. consists of three distinct fluid discontinuities: the blast
InFigure1weshowtheimagefromthe4–6keVband, wave, which marks the shock propagating into the am-
which is dominated by the synchrotron component of bient medium, a Rayleigh-Taylor (R-T) unstable con-
the X-ray spectrum. Apart from the well-known limb- tactdiscontinuity(CD)attheejecta-interstellarmaterial
Cosmic Rays in Tycho 3
288
00 213
Fig. 2.—(Left)Theprojectedintensityprofilefortworegionscontainingthewesternstripes. Thebrightstripeistheredpeakarounda
positionof35(cid:48)(cid:48). Thelocationsofprominentstripesareindicatedbythehorizontallineswithtickmarks. Thepeaksinthenorthernprofile
arespacedby9.9(cid:48)(cid:48),11.3(cid:48)(cid:48),6.9(cid:48)(cid:48),8.3(cid:48)(cid:48),and13.3(cid:48)(cid:48);thoseinthesouthernprofilearespacedby4.4(cid:48)(cid:48),9.4(cid:48)(cid:48),6.8(cid:48)(cid:48),and7.4(cid:48)(cid:48). (Right)Grayscale
imageoftheChandradatainaline-freeregionofthespectrum(4.2–6.0keV)showingtheextractionregionsfortheprofilesplottedinthe
left panel. The profiles run from the lower left to the upper right and the tick marks on the boxes correspond to the same tick marks as
in the left panel. The scale bar in the lower right is 10(cid:48)(cid:48) long. The green boxes denote the spectral regions analysed in the Letter. The
brightestregionisthebrightstripe;thecoadditionofthefainterregionsconstitutesthefaintstripes.
boundary, and a reverse shock that propagates into the The observed peak surface brightness of the stripes is,
stellar remains. Warren et al. (2005) set an upper-limit however, 5 times that of the faint limb region. Thus
fortheazimuthally-dependentprojectedradiusofthere- the peak intrinsic emissivity of the synchrotron-emitting
verse shock in Tycho using the location of the Fe Kα plasma in the stripes must be a factor of 25 higher than
emission. Adopting their center of expansion, the west- is typical for the blast wave. In principle, this brighten-
ernstripespeakataradiusof220(cid:48)(cid:48),welloutsidethe190(cid:48)(cid:48) ing could be due to a local enhancement in the ambient
position of the reverse shock at that azimuth. While the density and/or magnetic field. While we can not en-
position of the stripes does coincide with the Warren et tirely exclude this possibility, we consider the existence
al. estimate of the CD, the regularly-spaced, linear mor- of a pre-existing structure around Tycho with the right
phology of the non-thermal stripes does not correspond combination of increased density, magnetic field, and, as
to any features in the R-T plumes of thermal emission we show below, turbulence necessary to produce the ob-
tracing the ejecta boundary, nor is the CD a prominent served correlations between the ordered structure and
feature elsewhere in the 4–6 keV band. Conversely, the spectral variations improbable. In contrast, models of
blastwaveisabrightsourceof4–6keVemission,andwe CR-drivenmagneticfieldamplificationproducestructure
identify the stripes as projected features of this forward in the precursor density and field, and we find it more
shock. likelythatthestripesmarkaregionextensivelymodified
Tycho’s blast wave is traced by a very thin shell of X- by the acceleration process. This assumption is implicit
ray emission, with a typical thickness only 1-2% of its in the analysis in this Letter.
radius (Warren et al. 2005). Since the stripes are seen
inprojectionawayfromtherim, theirline-of-sightpath- 3.2. Spectroscopy
length through the shell is small. Thus their intrinsic The spatial position of the stripes, seen in projection
emissivity must be high relative to the limb regions. We against the bright thermal X-rays of the shocked ejecta,
examinedtworegionsofdifferentbrightnessonthewest- greatly complicates spectroscopy. This can be mitigated
ernrimwithminimalejectacontamination,andassumed by restricting analysis to the 4.2-10 keV range, where
a spherical shell model with a radius of 214(cid:48)(cid:48), chosen to theemissionisdominatedbythesynchrotroncontinuum
match the curvature of the western limb. The radial and the Fe Kα line complex at ∼6.45 keV. Throughout
brightness profile of the brighter rim region is too nar- this work we set the absorbing column to N = 7 ×
H
row and peaked to be consistent with a simple projected 1021 cm−2, which has negligible effect at these energies.
shell geometry, meaning that its high surface brightness The spectrum of the brightest stripe is well-fit with a
likely arises from a local enhancement in emissivity, per- power-lawofphotonindexofΓ=2.11+0.08,significantly
hapssimilartothestripes. Incontrast, theprofileofthe −0.10
harder than the spectrum of the entire SNR at these
fainter region is well-fit by a shell with thickness 2% of
energies (Γ = 2.72 ± 0.02; this fit includes faint lines
its radius, and is likely more typical of the blast wave
identified by Tamagawa et al. 2009), and a coaddition
as a whole. This simple model predicts that features at
of several of the lower surface brightness stripes (Γ =
80% of the shock radius (where the stripes are located)
2.52±0.05; this fit includes an Fe K line).
would have a surface brightness 20% that of the limb.
While photon indices are useful for characterization
4 Eriksen et al.
Fig. 3.—(Left)BroadbandspectraofTychofromtheradiototheX-rayfortheentireSNR(top),anensembleoffaintstripes(middle)
and the bright stripe (bottom). The radio points for the latter two came from our own estimates based on the VLA radio data, taking
account of the remnant’s expansion. The integrated radio flux density values for the entire SNR are taken from Klein et al. (1979). The
ChandradataareplottedinredandforthetoptwospectratheFeKlineisvisible. Theoverplottedcurvesarethebest-fitphenomenological
model for the continuum emission of a power law (extrapolated from the radio band) with an exponential cut-off in the X-ray regime.
(Right) Chandra spectra of the three spatial regions analyzed. The top part shows the observed spectral intensity versus photon energy
(datapointswitherrorbars)alongwiththebestfitmodel. Thebottompartshowstheresiduals(data−model)inunitsofthefluxerror.
Thedatapointsandmodelcurveshereincludeinstrumentaleffects,whichhavebeenremovedfortheX-raydatapointsplottedontheleft
panel.
of the non-thermal spectrum, the broadband spectrum differentcut-offfrequenciesorradiospectralindices. Our
gradually departs from a simple power law over several spectral analysis (detailed in Table 1) allows us to dis-
decades in energy. Detailed DSA models of synchrotron criminate between these two options. When the spec-
spectrashowasmalldegreeofcurvaturethroughthera- tra for the Bright and Faint regions are fit with radio
dio to UV, with a cut-off in the X-rays, set (in young flux densities fixed at their nominal values, both regions
SNRs like Tycho) by the short cooling time of TeV-scale prefer an index very close to the average for the entire
electrons (Ellison et al. 2000). Absent detailed spectral remnant. Conversely, if we instead require that both re-
calculations for Tycho, we adopt a common approxima- gions have the same cut-off, the fitted radio indices are
tion and assume a power-law electron momentum dis- α = 0.565 (bright) and α = 0.623 (faint). We cannot
tribution with an exponential cut-off, which produces a rejectthislevelofvariationoutrightsincemeasurements
photonspectrumthatisnearlyapower-lawfromthera- ofthespatialvariationoftheradiospectralindexarenot
dio through the UV and is curved in X-rays (Reynolds available for Tycho, though in Cas A (a plausible com-
& Keohane 1999). For this model, the low energy index parison object) the radio index does vary spatially by
(α ) and the flux normalization are set by radio ob- much as ±0.1 (Wright et al. 1999). On the other hand,
radio
servations,whilethelocationofthehighenergycut-offis the best fit for a single cut-off frequency is clearly worse
constrained with the X-ray spectrum. For the spectrum (∆χ2 = 13.8) than for independent cut-off values. An
oftheentireremnant,assumingaradiospectralindexof F-test shows that the fit including independent cut-off
α =0.65, afluxdensityat1GHzof∼50Jy(Kothes frequencies is an improvement at a significance level of
radio
etal.2006),andagainincludingthefaintX-rayemission more than 3σ. Thus we conclude that the cut-off fre-
lines from Tamagawa et al. (2009), the fit to the Chan- quenciesaresignificantlydifferentinthebrightandfaint
dra data gives ν = 1.9×1017 Hz (hν ∼ 0.79keV). spectral regions analysed.
cut cut
This provides a remarkably good fit (χ2ν = 1.05), and is 4. INTERPRETATION
consistent with measurements from Suzaku. (The statis-
4.1. The Diffusion Coefficient
tical error on this number is so small as to be irrelevant;
the precision to which we measure it is far greater than DSA requires magnetic field inhomogeneities strong
the efficacy of our emission model or the accuracy of the enough that particles can scatter multiple times across
radio flux and spectral index measurements.) If we in- the shock front. This diffusionof CRs in the magnetized
stead assume α = 0.61 (Klein et al. 1979), the fit turbulence is generally assumed to be close to the Bohm
radio
is slightly worse (χ2 = 1.10), and the cut-off moves to regime, where the mean-free path is the particle gyrora-
ν
lower energies (ν =0.9×1017 Hz;hν ∼0.37keV). dius. Departure from this limit can be parameterized as
cut cut
k = D /D , where D is the diffusion coefficient
For the stripes, we estimated the radio flux in these 0 0 0,Bohm 0
at the electron cut-off energy. Parizot et al. (2006) show
regions from a VLA map. The fits favor slightly shal-
that for an electron distribution limited by synchrotron
lower indices (α ∼0.60) and higher cut-off energies,
radio
losses, the photon cut-off energy (E = hν ) is set
ν =19+13×1017 Hz(7.9+5.4keV)fortheBrightStripe γ,cut cut
cut −9 −3.7 by the diffusion coefficient, shock velocity (V , in units
and ν = 2.8±0.2×1017 Hz (1.16±0.04keV) for the 3
cut of 1000 km s−1), and compression ratio, r:
ensemble of fainter stripes. We plot the broadband syn-
chrotron spectra of these regions in Figure 3. 16(r−1)
k =0.14E−1 V2 (1)
Although it is clear that the Bright Stripe and Faint 0 γ,cut,keV 3,shock 3r2
Ensemblespectrarequiredifferentbest-fitparameterval-
At a distance of 4.0±1.0kpc (Hayato et al. 2010), with
ues, in principle these differences might be due to either
a blast wave proper motion of 0.3(cid:48)(cid:48) yr−1 (Katsuda et al.
Cosmic Rays in Tycho 5
2010), v = 5700 kms−1. For an unmodified shock (r = locity of 5700 kms−1, fits to models of Tycho’s syn-
s
4) and our fit cut-off energies, k = 5.8 (Whole SNR), chrotron emission (Cassam-Chena¨ı et al. 2007) predict
0
3.9±0.1(faintstripes),and0.6+0.4(BrightStripe). With an upstream field strength of B ∼ 30 µG, indicating
−0.3 E = 2×1015 eV — just at the knee of the Galactic
a higher compression more applicable to Tycho (r = 7), CR
CR spectrum.
these numbers become 3.8, 2.6±0.1, and 0.4+0.3. (The
−0.2
error bars reflect only the statistical error in ν , and
cut 5. CONCLUSION
not the ∼ 25% uncertainty in the distance to Tycho.)
Our value for the whole SNR is near to those found by If the Bell mechanism is active in the magnetic field
Parizotet al.(2006)forTycho(k =4.9−10), whilethe amplificationinSNRshocks,andifthepatternofstripes
0
very small numbers for the brightest stripe are similar weobserveinTychoisanobservableconsequenceofthis
to their results for G347.3-0.5 and SN 1006 (k = 0.2− process, the stripes are direct evidence of particles ac-
0
0.9). Since k > 1 for isotropic turbulence, this may celerated to the knee. There are several caveats to con-
0
be due to our oversimplified emission model, and could sider before accepting this conclusion. Most obviously,
indicatethatthephotonspectrumfallsmoregraduallyto while MHD and PIC simulations of the CRCD generate
high energies, as seen in some DSA calculations (Ellison structureintheamplifiedmedium,theydonotshowthe
etal.2010), ormaybeindicativeofanisotropicdiffusion regular and nearly periodic pattern that we observe in
(Reville et al. 2008). Regardless, our measured photon Tycho. Secondly,whentheCRback-reactionisincluded
cut-off energies require an order of magnitude decrease (Riquelme&Spitkovsky2009),PICsimulationsindicate
in the diffusion coefficient in the stripes relative to the that the Bell instability alone may be neither fast nor
average value in Tycho, and indicate that diffusion in efficient enough to provide the observed amplification,
the stripes is very near the Bohm limit. Furthermore, particularly for Tycho’s low ambient density. Additional
since Bohm diffusion requires δB/B ∼1, k ∼1 implies instabilities are required, and seem to be indicated by
0
fully developed turbulence, at least on the scale of the the degree of turbulence in the stripes implied by our
gyroradius of the highest energy (∼ TeV) electrons in observations. Riquelme & Spitkovsky (2010) propose a
the brightest stripes. new instability (the Parallel Current Driven Instability,
PCDI) that acts on smaller scales (i.e., lower energies)
4.2. The Origin of the Stripes and provides significant field amplification over that es-
tablished by the CRCD. Further theoretical work on the
Might the spatial structure and enhanced turbulence
degree of turbulence generated by proposed mechanisms
inthestripesbeamanifestationoftheplasmainstability
onTeVparticlescalesseemsparticularlywarranted. Re-
drivingthecosmicrayacceleration? Thelargestpossible
gardless, the match in scales between the observed spac-
characteristic scale of the acceleration process is set by
ing of the Tycho stripes and the Larmor radius of knee-
the gyroradius of the highest energy particles present. If
region CRs — a scale intrinsic to the CRCD — is tan-
weidentifythegapsbetweenthestripes(l ∼8(cid:48)(cid:48))with
gap talizing, whiletheorderedstructurepresentsachallenge
twicetheprotongyroradius,anestimateofthemagnetic
to current models of diffusive shock acceleration.
field yields a measurement of the energy, given by Equa-
tion 2.
(cid:16)l (cid:17)(cid:16) D (cid:17)(cid:16) B (cid:17) This work was partially supported by Chandra award
ECR =9 1ga(cid:48)(cid:48)p 4.0kpc µG ×1012 eV (2) (grantnumberGO9-0078X)toRutgersUniversity. EMR
is member of the CIC (CONICET) and is funded by the
The choice of B is somewhat uncertain. If the stripes CONICET,UBAandANPCyTprojects. JPHwouldlike
are a consequence of the highest energy CRs interact- toacknowledgehelpfuldiscussionswithMarioRiquelme,
ing with the non-amplified ISM field (B ∼ 3 µG), for Anatoly Spitkovsky, Tom Jones, Roger Blandford, and
a gap spacing of 8(cid:48)(cid:48), E = 2 × 1014 eV. However, Martin Laming. We also thank Christopher McKee for
CR
these features may arise from a region where the field contributing to the original Chandra proposal.
has already been somewhat amplified. For a shock ve- Facilities: Chandra.
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TABLE 1
Stripe Spectroscopic Fits
Region α Fν (1GHz) νcut χ2/d.o.f.
(Jy) (Hz)
BrightStripe 0.61* 3.33×10−2* 2.60±0.09×1018 194.89/205
BrightStripe 0.603+0.010 3.33×10−2* 1.88+1.25×1018 194.30/204
−0.025 −0.89
BrightStripe 0.637+0.014 6.66×10−2* 1.93+2.09×1018 194.26/204
−0.012 −0.74
BrightStripe 0.65* 8.0+2.3×10−2 2.37+1.64×1018 194.27/204
−1.5 −1.00
BrightStripe 0.61* 4.2+0.8×10−2 1.56+1.53×1018 194.27/204
−1.1 −0.44
FaintEnsemble 0.61* 0.39* 3.52±0.03×1017 667.29/687
FaintEnsemble 0.601+0.009 0.39* 2.84±0.10×1017 666.31/686
−0.004
FaintEnsemble 0.635+0.011 0.78* 2.88±0.11×1017 666.55/686
−0.004
FaintEnsemble 0.65* 0.899+0.032 3.53±0.23×1017 666.37/686
−0.026
FaintEnsemble 0.61* 0.394±0.012 3.47+0.22×1017 667.25/686
−0.36
* Fixedparameter
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