text: 8 Out of interest, let us take the small k limit of this the second instability band we need to look at the leading − result. This leaves a result for µ that is linear in k, even harmonics ω = 2m, 0, +2m. The ω = 0 mode is k as we proved it should be in Part 1 [1] where we did easily solved for, leaving just two unknown coeﬃcients. − a general long wavelength analysis, and the Goldstone We note that these couple to ω = 4m, +4m, giving ﬁ- theorem ensured a gapless spectrum. If we are both at nite corrections. However, a reasonable approximation small wavenumber and small amplitude (the lower left for the central results, arises from just focussing on the − × corner of the stability charts) we obtain from eqs. (38, 39) ω =(cid:18) 2m, +2(cid:19)m harmonics. We ﬁnd that the 2 2 matrix the following relationship among the speeds problem for the second instability band takes the form − c2 d δ(cid:32)φ (cid:33)(cid:18) (cid:19) c2 = −S (44) +2m I 2q 1 dτ δφ−2m − − − whose sign governs the stab−ility structure. Indeed for the = i A 4m−2 BA2 C BA2− δφ+2m (51) case q = 2, we obtain c2I = c2S/3. 4m BA2 C 4m2 + BA2 A δφ−2m C. Second Instability Band The eigenval(cid:115)ues of this matrix are the Floquet exponents (cid:18) (cid:19) (cid:18) (cid:19) In the previous subsection we discussed the (cid:15) expan- of the second band sion that governs the behavior at small amplitudes, and applied it to the ﬁrst instability band. In principle one 1 − B2 2 − − − B2 2 µ = C A 4m2 (52) can go further and study any band to any desired order k 4m A A in perturbation theory. Here we mention the key leading order results of the second instability band. We continue The full expression for µ after substituting in the above k to study m2 > 0, but now we specialize to q = 2, i.e., values for A, B, C is somewhat complicated, but it suf- dimension 4 potentials. ﬁces to discuss its features. In the previous subsection we found the solution to Firstly, lets discuss at what k value the band starts at O → → ((cid:15)) in eqs. (21, 25) (by taking q 2). For the next in the small amplitude limit. If we take φ 0, then O → → → a band, we need the background solution to ((cid:15)3). With- B 0, C 0, and A k2 + m2, and µ becomes k out going through the full details, we ﬁnd that the result → ± i − to this order is µ (k2 3m2) (53) k 4m λ φ3 φ0 = ((cid:15) φa1 + (cid:15)3φa3) cos(m τ) + (cid:15)3 32 ma12 cos(3 m τ) (45) Hence the value of k∗ that sets the imaginary part of µ to zero, and hence corresponds to the start of the k where (cid:112) (cid:112) instability band, is √ 2m ∓ m φa1 = | |, φa3 = | | (46) k∗ = 3 m (54) 3 λ 24 3 λ In Section we will explain this wavenumber as arising Substitution into the perturbation equations leads to a → from 4φ 2φ particle annihilations. Hill’s function h that is more complicated than the stan- Next, we discuss the shape and width of the instability dard Mathieu equation, namely O band. If we work only to ((cid:15)2) we ﬁnd that both the h(τ) = A + 2 B cos(2 m τ) + 2 C cos(4 m τ) (47) left hand and right hand edge of the instability band coincide. As a function of amplitude this merely provides where we have gone to the requirOed number of harmonics. the overall bending of the band. If we eOxpress this in The coeﬃcients a(cid:0)re required to(cid:1) ((cid:15)4), and we ﬁnd them terms of the amplitude φ , we ﬁnd that to ((cid:15)2) we have a to be √ − B = γ7m22 ±24(cid:15)2 − 23(cid:15)4 (48) kedge = 3 m + (64√3γ) λmφ2a (55) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19) γ m2 C = (cid:15)4 (49) On the other hand, weOﬁnd a splitting between the left 36 and right hand edges at ((cid:15)4). This splitting, which gives A = m2 (cid:0)1 ± 2γ − 1(cid:1) (cid:15)2 + 1 − 25γ (cid:15)4 the width of the band, is found to be 3 36 | − | ∓ − γ γ √ 1 λ2φ4 +k2 1 (cid:15)2 + (cid:15)4 (50) ∆k = k k = a (56) r,edge l,edge 64 3 m2 where we again use γ from eq. (28) with q = 2; so γ = 3 for δφ(cid:107), and γ = 1 for δφ⊥. So for the isocurvature modes (γ = 1) the width is zero. We then substitute into the harmonic expansion (34). This means there is no instability band. This is in ac- Previously we studied the ﬁrst instability band by track- cord with the bottom panel of Fig. 1. On the other hand, − ing the leading odd harmonics ω = m, +m. To study for the adiabatic modes (γ = 3) there does exist a ﬁnite ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: anSdUeBxMteITnTsEivDeTOreAsoPJluOtNio2n01st4uAdUieGsUwSTil6l be needed to further as- certain the role of turbulence in the core-collapse supernova Preprint typeset using LATEX style emulateapj v. 04/17/13 mechanism. THE ROLE OF TURBULENCE IN NEUTRINO-DRIVEN CORE-COLLAPSE SUPERNOVA EXPLOSIONS SEAN M. COUCH1,2,4 AND CHRISTIAN D. OTT2,3,5 Keywords: hydrodynamics – neutrinos – Stars: supernovae: general ACKNOWLEDGEMENTS Couch, S. M., Graziani, C., & Flocke, N. 2013, ApJ, 778, 181 Couch, S. M., & O’Connor, E. P. 2014, ApJ, 785, 123 The authors thank David Radice and Ernazar Abdikamalov Couch, S. M., & Ott, C. D. 2013, ApJ, 778, L7 for help with the computation of the numerical Reynolds Couch, S. M., et al. 2014, in preparation number. The authors acknowledge further helpful discus- Dessart, L., Burrows, A., Livne, E., & Ott, C. D. 2006, ApJ, 645, 534 Dolence, J. C., Burrows, A., Murphy, J. W., & Nordhaus, J. 2013, ApJ, 765, sions with W. 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label: section

text: be

label: paragraph

text: needed

text: to

text: further

text: as-

text: certain

text: the

text: role

text: of

text: turbulence

text: in

text: the

text: core-collapse

text: supernova

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: mechanism.

text: THE

text: ROLE

text: OF

text: TURBULENCE

text: IN

text: NEUTRINO-DRIVEN

text: CORE-COLLAPSE

text: SUPERNOVA

text: EXPLOSIONS

text: SEAN

label: author

text: M.

text: AND

text: CHRISTIAN

text: D.

text: Keywords:

text: hydrodynamics

label: abstract

text: neutrinos

text: Stars:

text: supernovae:

text: general

text: ACKNOWLEDGEMENTS

text: Couch,

label: reference

text: S.

text: M.,

text: Graziani,

text: C.,

text: Flocke,

text: N.

text: ApJ,

text: Couch,

text: S.

text: M.,

text: O’Connor,

text: E.

text: P.

text: ApJ,

text: The

text: authors

text: thank

text: David

text: Radice

text: and

text: Ernazar

text: Abdikamalov

text: Couch,

text: S.

text: M.,

text: Ott,

text: C.

text: D.

text: ApJ,

text: for

text: help

text: with

text: the

text: computation

text: of

text: the

text: numerical

text: Reynolds

text: Couch,

text: S.

text: M.,

text: et

text: al.

text: in

text: preparation

text: number.

text: The

text: authors

text: acknowledge

text: further

text: helpful

text: discus-

text: Dessart,

text: L.,

text: Burrows,

text: A.,

text: Livne,

text: E.,

text: Ott,

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text: hanced absorption of neutrino energy is insufﬁcient to explain than in 3D in Section . We present our analysis of the strength the relative ease of explosion in 2D and 3D simulations as of turbulence across our wide range of 3D simulations in Sec- compared with 1D simulations. We show that the turbulent tion and show that turbulence plays a central role in pushing SUBMITTED TO APJ ON 2014 AUGUST 6 ramPrepprrinetstsyuperseetiussinsgyLsAtTeEXmsatytlieceamlluylat(eaapnjdv. a04r/t1i7ﬁ/1c3ially!) higher in 2D 3D models over the critical threshold to explosion. We discuss our results and present our conclusions in Section . THE ROLE OF TURBULENCE IN NEUTRINO-DRIVEN CORE-COLLAPSE SUPERNOVA EXPLOSIONS SEAN M. COUCH1,2,4 AND CHRISTIAN D. OTT2,3,5 Keywords: hydrodynamics – neutrinos – Stars: supernovae: general 2. SIMULATIONS τ are the neutrino optical depths. See Couch & O’Connor (2ν0i 14) for detailed deﬁnitions of these quantities and a com- 2.1. Methods plete description of how they are calculated in our leakage Our numerical method is the same as that described in scheme. Equation (1) is a solution to the spherically sym- Couch & Ott (2013) and Couch & O’Connor (2014). Our metric neutrino transport equation, save for the addition of core-collapse supernova simulation application is built in the the fheat factor. This is included as a means of increasing (or FLASH framework (Dubey et al. 2009; Lee et al. 2014). We decreasing) the local (i.e. microscopic) efﬁciency of the neu- utilize a directional-unsplit PPM hydrodynamics scheme (c.f., trino heating without changing other details of the neutrino Colella & Woodward 1984) with an approximate HLLC Rie- radiation as would result from, e.g., a simple change in the mann solver in smooth ﬂow and a HLLE Riemann solver near opacities. The nominal value of fheat is unity. Due to the shocks. Self-gravity is included via a fast and accurate mul- strong non-linearities of the core-collapse supernova prob- tipole approximation (Couch et al. 2013) including Legendre lem, small changes in the local heating efﬁciency, fheat, can orders up to 16. We employ adaptive mesh reﬁnement (AMR) result in large changes in the global heating efﬁciency (e.g., on a Cartesian grid with a minimum grid spacing of 0.49 km. Ott et al. 2013). The maximum allowed reﬁnement level is limited as a func- All simulations described here make use of the 15-M pro- tion of radius from the origin, yielding a pseudo-logarithmic genitor of Woosley & Heger (2007). Such a star, if i(cid:12)t were grid spacing with radius. The ﬁrst forced decrement in re- to occur in Nature, would be likely to result in a success- ﬁnement in our ﬁducial models occurs around 100 km, the ful explosion and the formation of a neutron star based on second at 200 km, the third at 400 km, and so on. This gives both its compactness (O’Connor & Ott 2011; Ugliano et al. an average effective resolution of α = (cid:104)∆x /r(cid:105) ∼ 0.75%, 2012; Kochanek 2014) and its low helium core mass (Clausen i or ∼ 0◦.43, in the angular directions, where ∆x is the linear et al. 2014). As in Couch & Ott (2013), we apply non-radial, i zone size and r is the distance from the origin. momentum-preserving velocity perturbations to the otherwise We include the effects of heating, cooling, and electron spherically-symmetric initial model. These perturbations are fraction evolution due to neutrinos via an approximate, ray- applied only to the θ-direction velocity component and are by-ray, multi-species leakage scheme (O’Connor & Ott 2010; described by, Ott et al. 2013). This scheme accounts for the cooling due to heavy lepton neutrinos and antineutrinos and cooling and electron fraction changes due to electron neutrinos and anti- δv = M c sin[(n − 1)θ] sin[(n − 1)ζ] cos(nφ) , (2) θ pert S neutrinos. Charged-current heating is included via (Janka 2001), (cid:28) (cid:29) where M is the peak perturbation Mach number, c is pert S the local adiabatic sound speed, n is the number of nodes in L (r) 1 ρX the interval θ = [0, π], and ζ = π(r −r )/(r − Qνi = fheat 4νπir2 Fνi σνi ma(nm/up) e−2τνi, (1) r(rpoeurtg,mhliyn)r. ∼Th1e0p0e0rtkumrba−tio5n0s0a0rekmap)p, ltiheepdeiwrntn,imtehriinnirothnecpSoeirr/teO,missahxleefltl where i runs over both electron neutrinos and anti-neutrinos, completely spherical. L (r) are the neutrino luminosities as a function of radius, 2.2. Overview of Models r,νi(cid:104)1/F (cid:105) is the energy-averaged mean inverse ﬂux factor, We run a total of 19 high-resolution, 3D core-collapse su- σ are νtihe charged-current cross sections, ρ is the matter pernova simulations in which we vary the scale and amplitude deνni sity, X(n/p) are the neutron/proton mass fractions, and of the initial perturbations, fheat, and the resolution.11. We re- fer to the simulations using the scheme n[node count]m[initial perturbation Mach number, times ten] [heat factor]. We con- 1 Flash Center for Computational Science, Department of Astron- omy & Astrophysics, University of Chicago, Chicago, IL, 60637, sider 7 node counts: 0, 3, 5, 9, 13, 17, 33, and 4 amplitudes: smc@ﬂash.uchichago.edu 0 M, 0.1 M, 0.15 M, and 0.2 M. These values yield veloc- 2 TAPIR, Walter Burke Institute for Theoretical Physics, MC 350- ity perturbations in the Si/O shell not unlike the ﬂuctuations 17, California Institute of Technology, Pasadena, CA 91125, USA, found in realistic, multi-dimensional simulations of convec- cott@tapir.caltech.edu tive nuclear burning in massive stars in both scale and strength 3 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU WPI), The University of Tokyo, Kashiwa, Japan 4 Hubble Fellow 11 Some data for unperturbed models are taken from Couch & O’Connor 5 Alfred P. Sloan Research Fellow (2014). ##LTLine## ##LTLine## ##LTLine##

text: hanced

text: absorption

text: of

text: neutrino

text: energy

text: is

text: insufﬁcient

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text: explain

text: than

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text: Section

text: We

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text: of

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text: strength

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text: relative

text: ease

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text: across

text: our

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text: compared

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text: turbulence

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text: SUBMITTED

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text: over

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text: critical

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text: explosion.

text: We

text: discuss

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text: present

text: our

text: conclusions

text: in

text: Section

text: THE

text: ROLE

text: OF

text: TURBULENCE

text: IN

text: NEUTRINO-DRIVEN

text: CORE-COLLAPSE

text: SUPERNOVA

text: EXPLOSIONS

text: SEAN

text: M.

text: AND

text: CHRISTIAN

text: D.

text: Keywords:

text: hydrodynamics

text: neutrinos

text: Stars:

text: supernovae:

text: general

text: SIMULATIONS

text: τ

text: are

text: the

text: neutrino

text: optical

text: depths.

text: See

text: Couch

text: O’Connor

text: for

text: detailed

text: deﬁnitions

text: of

text: these

text: quantities

text: and

text: a

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text: Methods

text: plete

text: description

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text: how

text: they

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text: Couch

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text: metric

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text: equation,

text: save

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text: core-collapse

text: supernova

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text: application

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text: built

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text: the

text: fheat

text: factor.

text: This

text: is

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text: et

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text: decreasing)

text: the

text: local

text: (i.e.

text: microscopic)

text: efﬁciency

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text: the

text: neu-

text: utilize

text: a

text: directional-unsplit

text: PPM

text: hydrodynamics

text: scheme

text: (c.f.,

text: trino

text: heating

text: without

text: changing

text: other

text: details

text: of

text: the

text: neutrino

text: Colella

text: Woodward

text: with

text: an

text: approximate

text: HLLC

text: Rie-

text: radiation

text: as

text: would

text: result

text: from,

text: e.g.,

text: a

text: simple

text: change

text: in

text: the

text: mann

text: solver

text: in

text: smooth

text: ﬂow

text: and

text: a

text: HLLE

text: Riemann

text: solver

text: near

text: opacities.

text: The

text: nominal

text: value

text: of

text: fheat

text: is

text: unity.

text: Due

text: to

text: the

text: shocks.

text: Self-gravity

text: is

text: included

text: via

text: a

text: fast

text: and

text: accurate

text: mul-

text: strong

text: non-linearities

text: of

text: the

text: core-collapse

text: supernova

text: prob-

text: tipole

text: approximation

text: (Couch

text: et

text: al.

text: including

text: Legendre

text: lem,

text: small

text: changes

text: in

text: the

text: local

text: heating

text: efﬁciency,

text: fheat,

text: can

text: orders

text: up

text: to

text: We

text: employ

text: adaptive

text: mesh

text: reﬁnement

text: (AMR)

text: result

text: in

text: large

text: changes

text: in

text: the

text: global

text: heating

text: efﬁciency

text: (e.g.,

text: on

text: a

text: Cartesian

text: grid

text: with

text: a

text: minimum

text: grid

text: spacing

text: of

text: km.

text: Ott

text: et

text: al.

text: The

text: maximum

text: allowed

text: reﬁnement

text: level

text: is

text: limited

text: as

text: a

text: func-

text: All

text: simulations

text: described

text: here

text: make

text: use

text: of

text: the

text: pro-

text: tion

text: of

text: radius

text: from

text: the

text: origin,

text: yielding

text: a

text: pseudo-logarithmic

text: genitor

text: of

text: Woosley

text: Heger

text: Such

text: a

text: star,

text: if

text: were

text: grid

text: spacing

text: with

text: radius.

text: The

text: ﬁrst

text: forced

text: decrement

text: in

text: re-

text: to

text: occur

text: in

text: Nature,

text: would

text: be

text: likely

text: to

text: result

text: in

text: a

text: success-

text: ﬁnement

text: in

text: our

text: ﬁducial

text: models

text: occurs

text: around

text: km,

text: the

text: ful

text: explosion

text: and

text: the

text: formation

text: of

text: a

text: neutron

text: star

text: based

text: on

text: second

text: at

text: km,

text: the

text: third

text: at

text: km,

text: and

text: so

text: on.

text: This

text: gives

text: both

text: its

text: compactness

text: (O’Connor

text: Ott

text: Ugliano

text: et

text: al.

text: an

text: average

text: effective

text: resolution

text: of

text: α

text: Kochanek

text: and

text: its

text: low

text: helium

text: core

text: mass

text: (Clausen

text: i

text: or

text: in

text: the

text: angular

text: directions,

text: where

text: ∆x

text: is

text: the

text: linear

text: et

text: al.

text: As

text: in

text: Couch

text: Ott

text: we

text: apply

text: non-radial,

text: i

text: zone

text: size

text: and

text: r

text: is

text: the

text: distance

text: from

text: the

text: origin.

text: momentum-preserving

text: velocity

text: perturbations

text: to

text: the

text: otherwise

text: We

text: include

text: the

text: effects

text: of

text: heating,

text: cooling,

text: and

text: electron

text: spherically-symmetric

text: initial

text: model.

text: These

text: perturbations

text: are

text: fraction

text: evolution

text: due

text: to

text: neutrinos

text: via

text: an

text: approximate,

text: ray-

text: applied

text: only

text: to

text: the

text: θ-direction

text: velocity

text: component

text: and

text: are

text: by-ray,

text: multi-species

text: leakage

text: scheme

text: (O’Connor

text: Ott

text: described

text: by,

text: Ott

text: et

text: al.

text: This

text: scheme

text: accounts

text: for

text: the

text: cooling

text: due

text: to

text: heavy

text: lepton

text: neutrinos

text: and

text: antineutrinos

text: and

text: cooling

text: and

text: electron

text: fraction

text: changes

text: due

text: to

text: electron

text: neutrinos

text: and

text: anti-

text: δv

text: M

text: c

text: sin[(n

text: sin[(n

text: cos(nφ)

text: θ

text: pert

text: S

text: neutrinos.

text: Charged-current

text: heating

text: is

text: included

text: via

text: (Janka

text: where

text: M

text: is

text: the

text: peak

text: perturbation

text: Mach

text: number,

text: c

text: is

text: pert

text: S

text: the

text: local

text: adiabatic

text: sound

text: speed,

text: n

text: is

text: the

text: number

text: of

text: nodes

text: in

text: L

text: (r)

text: ρX

text: the

text: interval

text: θ

text: π],

text: and

text: ζ

text: π(r

text: −r

text: )/(r

text: Qνi

text: fheat

text: Fνi

text: σνi

text: ma(nm/up)

text: r(rpoeurtg,mhliyn)r.

text: where

text: i

text: runs

text: over

text: both

text: electron

text: neutrinos

text: and

text: anti-neutrinos,

text: completely

text: spherical.

text: L

text: (r)

text: are

text: the

text: neutrino

text: luminosities

text: as

text: a

text: function

text: of

text: radius,

text: Overview

text: of

text: Models

text: is

text: the

text: energy-averaged

text: mean

text: inverse

text: ﬂux

text: factor,

text: We

text: run

text: a

text: total

text: of

text: high-resolution,

text: core-collapse

text: su-

text: σ

text: are

text: νtihe

text: charged-current

text: cross

text: sections,

text: ρ

text: is

text: the

text: matter

text: pernova

text: simulations

text: in

text: which

text: we

text: vary

text: the

text: scale

text: and

text: amplitude

text: deνni

text: sity,

text: X(n/p)

text: are

text: the

text: neutron/proton

text: mass

text: fractions,

text: and

text: of

text: the

text: initial

text: perturbations,

text: fheat,

text: and

text: the

text: We

text: re-

text: fer

text: to

text: the

text: simulations

text: using

text: the

text: scheme

text: n[node

text: count]m[initial

text: perturbation

text: Mach

text: number,

text: times

text: ten]

text: [heat

text: factor].

text: We

text: con-

text: Flash

text: Center

text: for

text: Computational

text: Science,

text: Department

text: of

text: Astron-

text: omy

text: Astrophysics,

text: University

text: of

text: Chicago,

text: Chicago,

text: IL,

text: sider

text: node

text: counts:

text: and

text: amplitudes:

text: smc@ﬂash.uchichago.edu

text: M,

text: M,

text: M,

text: and

text: M.

text: These

text: values

text: yield

text: veloc-

text: TAPIR,

text: Walter

text: Burke

text: Institute

text: for

text: Theoretical

text: Physics,

text: MC

text: ity

text: perturbations

text: in

text: the

text: Si/O

text: shell

text: not

text: unlike

text: the

text: ﬂuctuations

text: California

text: Institute

text: of

text: Technology,

text: Pasadena,

text: CA

text: USA,

text: found

text: in

text: realistic,

text: multi-dimensional

text: simulations

text: of

text: convec-

text: cott@tapir.caltech.edu

text: tive

text: nuclear

text: burning

text: in

text: massive

text: stars

text: in

text: both

text: scale

text: and

text: strength

text: Kavli

text: Institute

text: for

text: the

text: Physics

text: and

text: Mathematics

text: of

text: the

text: Universe

text: (Kavli

text: IPMU

text: WPI),

text: The

text: University

text: of

text: Tokyo,

text: Kashiwa,

text: Japan

text: Hubble

text: Fellow

text: Some

text: data

text: for

text: unperturbed

text: models

text: are

text: taken

text: from

text: Couch

text: O’Connor

text: Alfred

text: P.

text: Sloan

text: Research

text: Fellow

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1. Volume renderings of speciﬁc entropy for several of the 3D simulations at 150ms after bounce. Darker, red colors correspond to speciﬁc entropies − − of ∼14kB baryon 1 while lighter, yellow colors correspond to entropies of ∼18kB baryon 1. The blu−e colors, which highlight the shock surface and the lower-entropy cooling region near the protoneutron star, correspond to speciﬁc entropies of ∼5kB baryon 1. Models with stronger perturbations show higher speciﬁc entropies in the gain layer and a greater shock extension. This is a result of the stronger turbulence, and concomitant higher neutrino heating efﬁciency (Meakin & Arnett 2007; Arnett & Meakin 2011). The per- sake of the present discussion, we shall consider a model to and turbulent pressure, in these models. turbations we apply also result in total added kinetic ener- have “exploded” if the shock radius exceeds 400 km with no gies very similar to the convective kinetic energies found in sign of returning. This is, of course, only an approximate def- the above-cited works: 2.2 × 1047 erg, 4.9 × 1047 erg, and inition of a successful explosion. The other perturbed models 8.7 × 1047 erg for M= 0.1, 0.15, and 0.2, respectively. These with fheat > 1.00 (i.e., n9m2, n13m2) are near the explosion energies are orders of magnitude below both the thermal and threshold, but as we show below, an analysis of the strength gravitational binding energies of the progenitor’s Si/O shell. of turbulence in these models implies that conditions are less Most simulations were carried out with fheat = 1 but for a few favorable for runaway shock expansion than in model n5m2. combinations of node count and amplitude we have increased In Figure 1, we show entropy volume renderings from 12 of fheat. our 19 3D simulations. These visualizations illustrate that tur- In Table 1, we list all the simulations along with key param- bulence is stronger, and the shock radii larger, for models with SUBMITTED TO APJ ON 2014 AUGUST 6 etePrrseprainntdtypreessetuulstisn.g LAOTEnXlystymle eomduelaltenap5jmv.204/117.0/123, at both high- and stronger precollapse perturbations and/or higher fheat. low-resolution, explodes within the simulated time. For the THE ROLE OF TURBULENCE IN NEUTRINO-DRIVEN CORE-COLLAPSE SUPERNOVA EXPLOSIONS SEAN M. COUCH1,2,4 AND CHRISTIAN D. OTT2,3,5 Keywords: hydrodynamics – neutrinos – Stars: supernovae: general 1 Flash Center for Computational Science, Department of Astron- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Volume

label: caption

text: renderings

text: of

text: speciﬁc

text: entropy

text: for

text: several

text: of

text: the

text: simulations

text: at

text: after

text: bounce.

text: Darker,

text: red

text: colors

text: correspond

text: to

text: speciﬁc

text: entropies

text: of

text: baryon

text: while

text: lighter,

text: yellow

text: colors

text: correspond

text: to

text: entropies

text: of

text: baryon

text: The

text: blu−e

text: colors,

text: which

text: highlight

text: the

text: shock

text: surface

text: and

text: the

text: lower-entropy

text: cooling

text: region

text: near

text: the

text: protoneutron

text: star,

text: correspond

text: to

text: speciﬁc

text: entropies

text: of

text: baryon

text: Models

text: with

text: stronger

text: perturbations

text: show

text: higher

text: speciﬁc

text: entropies

text: in

text: the

text: gain

text: layer

text: and

text: a

text: greater

text: shock

text: extension.

text: This

text: is

text: a

text: result

text: of

text: the

text: stronger

text: turbulence,

text: and

text: concomitant

text: higher

text: neutrino

text: heating

text: efﬁciency

text: (Meakin

text: Arnett

text: Arnett

text: Meakin

text: The

text: per-

text: sake

text: of

text: the

text: present

text: discussion,

text: we

text: shall

text: consider

text: a

text: model

text: to

text: and

text: turbulent

text: pressure,

text: in

text: these

text: models.

text: turbations

text: we

text: apply

text: also

text: result

text: in

text: total

text: added

text: kinetic

text: ener-

text: have

text: “exploded”

text: if

text: the

text: shock

text: radius

text: exceeds

text: km

text: with

text: no

text: gies

text: very

text: similar

text: to

text: the

text: convective

text: kinetic

text: energies

text: found

text: in

text: sign

text: of

text: returning.

text: This

text: is,

text: of

text: course,

text: only

text: an

text: approximate

text: def-

text: the

text: above-cited

text: works:

text: erg,

text: erg,

text: and

text: inition

text: of

text: a

text: successful

text: explosion.

text: The

text: other

text: perturbed

text: models

text: erg

text: for

text: M=

text: and

text: respectively.

text: These

text: with

text: fheat

text: (i.e.,

text: are

text: near

text: the

text: explosion

text: energies

text: are

text: orders

text: of

text: magnitude

text: below

text: both

text: the

text: thermal

text: and

text: threshold,

text: but

text: as

text: we

text: show

text: below,

text: an

text: analysis

text: of

text: the

text: strength

text: gravitational

text: binding

text: energies

text: of

text: the

text: progenitor’s

text: Si/O

text: shell.

text: of

text: turbulence

text: in

text: these

text: models

text: implies

text: that

text: conditions

text: are

text: less

text: Most

text: simulations

text: were

text: carried

text: out

text: with

text: fheat

text: but

text: for

text: a

text: few

text: favorable

text: for

text: runaway

text: shock

text: expansion

text: than

text: in

text: model

text: combinations

text: of

text: node

text: count

text: and

text: amplitude

text: we

text: have

text: increased

text: In

text: Figure

text: we

text: show

text: entropy

text: volume

text: renderings

text: from

text: of

text: fheat.

text: our

text: simulations.

text: These

text: visualizations

text: illustrate

text: that

text: tur-

text: In

text: Table

text: we

text: list

text: all

text: the

text: simulations

text: along

text: with

text: key

text: param-

text: bulence

text: is

text: stronger,

text: and

text: the

text: shock

text: radii

text: larger,

text: for

text: models

text: with

text: SUBMITTED

text: TO

text: APJ

text: ON

text: AUGUST

text: etePrrseprainntdtypreessetuulstisn.g

text: LAOTEnXlystymle

text: at

text: both

text: high-

text: and

text: stronger

text: precollapse

text: perturbations

text: and/or

text: higher

text: fheat.

text: low-resolution,

text: explodes

text: within

text: the

text: simulated

text: time.

text: For

text: the

text: THE

text: ROLE

text: OF

text: TURBULENCE

text: IN

text: NEUTRINO-DRIVEN

text: CORE-COLLAPSE

text: SUPERNOVA

text: EXPLOSIONS

text: SEAN

text: M.

text: AND

text: CHRISTIAN

text: D.

text: Keywords:

text: hydrodynamics

text: neutrinos

text: Stars:

text: supernovae:

text: general

text: Flash

text: Center

text: for

text: Computational

text: Science,

text: Department

text: of

text: Astron-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: −πδF π − 0 δM = + (3φ δφ φ δφ ) , (33) κ 2l2 1 0 0 1 − E δQ = 2πδ . (34) e 0 The minus sign in (34) comes from the sign diﬀerence between the electric ﬁeld density and the canonical momentum of the gauge ﬁeld. The electric charge can be immediately integrated, and is given by the leading term of the canonical momentum of the gauge ﬁeld: − E Q = 2π . (35) e 0 It is clear that the second term in (33), which takes into account the contribution of (cid:54) (cid:54) the scalar ﬁeld to the mass, provided φ = 0 and φ = 0, needs a boundary condition 0 1 for integrating it, i. e., a functional relation φ = φ (φ ). In simple words, the mass is 1 1 0 determined after imposing boundary condit(cid:90)ion(cid:18)s, and is given(cid:19)by2 −πF π − dφ 0 1 M = + 3φ φ dφ . (36) κ 2l2 1 0 dφ 0 0 Apart from the boundary conditions φ = 0 or φ = 0, there is only one additional 0 1 case which also leads a vanishing contribution from scalar ﬁeld to the mass: the functional relation φ = γφ3, (37) 1 0 where γ is a constant without variation. These three boundary conditions share a same feature: they do not spoil the conformal invariance of a scalar ﬁeld approaching to inﬁnity in the form (28), as pointed out in [7] (for a recent related discussion in four dimensions see [31]). Any other functional relation φ = φ (φ ), in the way of Designer Gravity [32], breaks 1 1 0 2 Two arbitrary additive constants (but ﬁxed, i. e. without variation) appear in the integration of (33) and (34). They will be set to zero in order that the massless BTZ has a vanishing mass, and in absence of the gauge ﬁeld, the solution be electrically uncharged. 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −πδF

label: equation

text: π

text: δM

text: δφ

text: φ

text: δφ

text: κ

text: E

text: δQ

text: 2πδ

text: e

text: The

text: minus

text: sign

text: in

text: comes

text: from

text: the

text: sign

text: diﬀerence

text: between

text: the

text: electric

text: ﬁeld

text: density

text: and

text: the

text: canonical

text: momentum

text: of

text: the

text: gauge

text: ﬁeld.

text: The

text: electric

text: charge

text: can

text: be

text: immediately

text: integrated,

text: and

text: is

text: given

text: by

text: the

text: leading

text: term

text: of

text: the

text: canonical

text: momentum

text: of

text: the

text: gauge

text: ﬁeld:

text: E

text: Q

text: 2π

text: e

text: It

text: is

text: clear

text: that

text: the

text: second

text: term

text: in

text: which

text: takes

text: into

text: account

text: the

text: contribution

text: of

text: the

text: scalar

text: ﬁeld

text: to

text: the

text: mass,

text: provided

text: φ

text: and

text: φ

text: needs

text: a

text: boundary

text: condition

text: for

text: integrating

text: it,

text: i.

text: e.,

text: a

text: functional

text: relation

text: φ

text: φ

text: (φ

text: In

text: simple

text: words,

text: the

text: mass

text: is

text: determined

text: after

text: imposing

text: boundary

text: and

text: is

text: −πF

text: π

text: dφ

text: M

text: 3φ

text: φ

text: dφ

text: κ

text: dφ

text: Apart

text: from

text: the

text: boundary

text: conditions

text: φ

text: or

text: φ

text: there

text: is

text: only

text: one

text: additional

text: case

text: which

text: also

text: leads

text: a

text: vanishing

text: contribution

text: from

text: scalar

text: ﬁeld

text: to

text: the

text: mass:

text: the

text: functional

text: relation

text: φ

text: γφ3,

text: where

text: γ

text: is

text: a

text: constant

text: without

text: variation.

text: These

text: three

text: boundary

text: conditions

text: share

text: a

text: same

text: feature:

text: they

text: do

text: not

text: spoil

text: the

text: conformal

text: invariance

text: of

text: a

text: scalar

text: ﬁeld

text: approaching

text: to

text: inﬁnity

text: in

text: the

text: form

text: as

text: pointed

text: out

text: in

text: (for

text: a

text: recent

text: related

text: discussion

text: in

text: four

text: dimensions

text: see

text: Any

text: other

text: functional

text: relation

text: φ

text: φ

text: (φ

text: in

text: the

text: way

text: of

text: Designer

text: Gravity

text: breaks

text: Two

text: arbitrary

text: additive

text: constants

text: (but

text: ﬁxed,

text: i.

text: e.

text: without

text: variation)

text: appear

text: in

text: the

text: integration

text: of

text: and

text: They

text: will

text: be

text: set

text: to

text: zero

text: in

text: order

text: that

text: the

text: massless

text: BTZ

text: has

text: a

text: vanishing

text: mass,

text: and

text: in

text: absence

text: of

text: the

text: gauge

text: ﬁeld,

text: the

text: solution

text: be

text: electrically

text: uncharged.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the mass and electric charge of the black hole without self-interaction potential described in Section . IV. THERMODYNAMICS This section is devoted to study thermodynamic properties of the charged hairy black holes shown in Sec. . The conjugate variables associated to the conserved charges, mass and electric charge, are the temperature and the electric potential, respectively. The temperature can be obtained by means of the surface gravity κ H κ H T = , (44) 2π − ∇ ∇ which is given by κ2 = 1/2 µχ χν with χµ = (1, 0, 0). Additionally, the electric H ν µ potential is deﬁned as − ∞ − q Φ := A (r ) A ( ) = . (45) 0 + 0 r + The entropy can be found using the modiﬁed Bekenstein-Hawking area formula, 4π2r + S = Ω(r ) , (46) + κ − where the factor Ω(r ) = 1 κφ(r )2/8 comes from the nonminimally coupling term + + in the action [33, 34]. Since this factor is not positive deﬁnite, the entropy could become negative. In order to avoid such a non-well-behaved thermodynamic situation, solutions in which Ω(r ) is negative must be discarded as black holes. For this reason, it is necessary + to impose additional constrains on the integration constants and α as discussed in detail below. We start examining the validity of the ﬁrst law for the black holes introduced in Sec.. Using the variation of the global charges (33) and (34), and the expressions for the temper- ature (44), entropy (46) and electric potential (45), evaluated on each particular black hole, it is possible to prove that the ﬁrst law of black hole thermodynamics δM = T δS + ΦδQ (47) 15 ##LTLine## ##LTLine## ##LTLine##

text: the

text: mass

text: and

text: electric

text: charge

text: of

text: the

text: black

text: hole

text: without

text: self-interaction

text: potential

text: described

text: in

text: Section

text: IV.

text: THERMODYNAMICS

text: This

text: section

text: is

text: devoted

text: to

text: study

text: thermodynamic

text: properties

text: of

text: the

text: charged

text: hairy

text: black

text: holes

text: shown

text: in

text: Sec.

text: The

text: conjugate

text: variables

text: associated

text: to

text: the

text: conserved

text: charges,

text: mass

text: and

text: electric

text: charge,

text: are

text: the

text: temperature

text: and

text: the

text: electric

text: potential,

text: respectively.

text: The

text: temperature

text: can

text: be

text: obtained

text: by

text: means

text: of

text: the

text: surface

text: gravity

text: κ

text: H

text: κ

text: H

text: T

text: 2π

text: which

text: is

text: given

text: by

text: κ2

text: µχ

text: χν

text: with

text: χµ

text: Additionally,

text: the

text: electric

text: H

text: ν

text: µ

text: potential

text: is

text: deﬁned

text: as

text: q

text: Φ

text: A

text: (r

text: A

text: r

text: The

text: entropy

text: can

text: be

text: found

text: using

text: the

text: modiﬁed

text: Bekenstein-Hawking

text: area

text: formula,

text: S

text: Ω(r

text: κ

text: where

text: the

text: factor

text: Ω(r

text: κφ(r

text: comes

text: from

text: the

text: nonminimally

text: coupling

text: term

text: in

text: the

text: action

text: Since

text: this

text: factor

text: is

text: not

text: positive

text: deﬁnite,

text: the

text: entropy

text: could

text: become

text: negative.

text: In

text: order

text: to

text: avoid

text: such

text: a

text: non-well-behaved

text: thermodynamic

text: situation,

text: solutions

text: in

text: which

text: Ω(r

text: is

text: negative

text: must

text: be

text: discarded

text: as

text: black

text: holes.

text: For

text: this

text: reason,

text: it

text: is

text: necessary

text: to

text: impose

text: additional

text: constrains

text: on

text: the

text: integration

text: constants

text: and

text: α

text: as

text: discussed

text: in

text: detail

text: below.

text: We

text: start

text: examining

text: the

text: validity

text: of

text: the

text: ﬁrst

text: law

text: for

text: the

text: black

text: holes

text: introduced

text: in

text: Sec..

text: Using

text: the

text: variation

text: of

text: the

text: global

text: charges

text: and

text: and

text: the

text: expressions

text: for

text: the

text: temper-

text: ature

text: entropy

text: and

text: electric

text: potential

text: evaluated

text: on

text: each

text: particular

text: black

text: hole,

text: it

text: is

text: possible

text: to

text: prove

text: that

text: the

text: ﬁrst

text: law

text: of

text: black

text: hole

text: thermodynamics

text: δM

text: T

text: δS

text: ΦδQ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: instabilities as in asymptotically ﬂat spacetimes, provided the mass of the scalar ﬁeld is bounded from below by the Breitenlohner-Freedman one m2 [19–21]. In three dimensions, BF − − − m2 = l 2, and because of (5), in our case the mass of the scalar ﬁeld is 3/4l 2, which BF satisﬁes the mentioned bound. We look for static and circularly symmetric conﬁgurations described by the line element − − ds2 = F (r)dt2 + F 1(r)dr2 + r2dθ2, (6) a scalar ﬁeld depending just on the radial coordinate, and a gauge ﬁeld of the form −∞ A = A (r)dt, which generates a purely radial electric ﬁeld. The coordinates range as < t ∞ ≤ ∞ ≤ t < , 0 r < , 0 θ < 2π. − From the subtraction Et Er in (2a), a second-order diﬀerential equation for the scalar t r (cid:114) ﬁeld is obtained, whose integration yields b φ(r) = , (7) r + c where b and c are integration constants. Moreover, from the nonlinear Maxwell equation (2c) the gauge ﬁeld is easily obtained (modulo gauge transformations) −q A = dt. (8) r The constant q is related with the electric charge as we will show below in Sec. . Finally, the metric function F (r) can be directly obtained from equation (5), which gives r2 a 2 F (r) = + a + , (9) l2 1 r where a and a are integration constants. It is clear, from the line element (6) and the 1 2 radial dependence of F shown in (9), that these solutions are asymptotically anti-de Sitter spacetimes whose asymptotic behaviors match the well-known Brown-Henneaux conditions [22]. However, as is discussed in Sec. , boundary conditions on the matter ﬁelds could spoil the conformal invariance of the full conﬁguration. 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: instabilities

text: as

text: in

text: asymptotically

text: ﬂat

text: spacetimes,

text: provided

text: the

text: mass

text: of

text: the

text: scalar

text: ﬁeld

text: is

text: bounded

text: from

text: below

text: by

text: the

text: Breitenlohner-Freedman

text: one

text: In

text: three

text: dimensions,

text: BF

text: l

text: and

text: because

text: of

text: in

text: our

text: case

text: the

text: mass

text: of

text: the

text: scalar

text: ﬁeld

text: is

text: which

text: BF

text: satisﬁes

text: the

text: mentioned

text: bound.

text: We

text: look

text: for

text: static

text: and

text: circularly

text: symmetric

text: conﬁgurations

text: described

text: by

text: the

text: line

text: element

text: F

text: F

text: a

text: scalar

text: ﬁeld

text: depending

text: just

text: on

text: the

text: radial

text: coordinate,

text: and

text: a

text: gauge

text: ﬁeld

text: of

text: the

text: form

text: A

text: A

text: (r)dt,

text: which

text: generates

text: a

text: purely

text: radial

text: electric

text: ﬁeld.

text: The

text: coordinates

text: range

text: as

text: t

text: t

text: r

text: θ

text: From

text: the

text: subtraction

text: Et

text: Er

text: in

text: a

text: second-order

text: diﬀerential

text: equation

text: for

text: the

text: scalar

text: t

text: r

text: ﬁeld

text: is

text: obtained,

text: whose

text: integration

text: yields

text: b

text: φ(r)

text: r

text: c

text: where

text: b

text: and

text: c

text: are

text: integration

text: constants.

text: Moreover,

text: from

text: the

text: nonlinear

text: Maxwell

text: equation

text: the

text: gauge

text: ﬁeld

text: is

text: easily

text: obtained

text: (modulo

text: gauge

text: transformations)

text: −q

text: A

text: dt.

text: r

text: The

text: constant

text: q

text: is

text: related

text: with

text: the

text: electric

text: charge

text: as

text: we

text: will

text: show

text: below

text: in

text: Sec.

text: Finally,

text: the

text: metric

text: function

text: F

text: (r)

text: can

text: be

text: directly

text: obtained

text: from

text: equation

text: which

text: gives

text: a

text: F

text: (r)

text: a

text: r

text: where

text: a

text: and

text: a

text: are

text: integration

text: constants.

text: It

text: is

text: clear,

text: from

text: the

text: line

text: element

text: and

text: the

text: radial

text: dependence

text: of

text: F

text: shown

text: in

text: that

text: these

text: solutions

text: are

text: asymptotically

text: anti-de

text: Sitter

text: spacetimes

text: whose

text: asymptotic

text: behaviors

text: match

text: the

text: well-known

text: Brown-Henneaux

text: conditions

text: However,

text: as

text: is

text: discussed

text: in

text: Sec.

text: boundary

text: conditions

text: on

text: the

text: matter

text: ﬁelds

text: could

text: spoil

text: the

text: conformal

text: invariance

text: of

text: the

text: full

text: conﬁguration.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 13 – The value of R and L determines the temperature at the inner boundary (T ), which is in SN in the primary regulator of the ionization structure of the illuminated progenitor wind. Our main diagnostic for the ionization structure is the ratio of He I and He II lines, which are very sensitive to R and L in this parameter range. To minimize the confounding eﬀect of the uncertainties in in SN the ﬂux calibration and dust reddening, we use neighboring lines when setting these parameters: He I λλ4471, 4713 and He II λ4686, all of which are ﬁt reasonably well by the ﬁnal model. Our best-ﬁt values for R and L are in extremely good agreement with predictions for this epoch in SN by Chugai (2001, see their Fig. 2), which are based upon light-curve model ﬁts to bolometric ﬂux estimates from March 14 and onward (Fassia et al. 2000). Additional constraints on the ionization structure come from the absence of N IV λ4057 and N IV λ7123. To ﬁt the absolute-ﬂux level, § we assume a distance of d = 17.00 Mpc (Tully 1994) and, as explained in , a standard Milky Way dust-extinction law (R = 3.1, Cardelli et al. 1989), ﬁnding a best-ﬁt total dust reddening of − V A = 0.15 mag or E(B V ) = 0.05 mag. V Given the high spectral resolution, we are able to fully resolve both the narrow and broad components of the features. This allows us to independently determine v and M˙ . We ﬁnd ± − wind a progenitor wind speed of v = 40 5 km s 1. To constrain this value, we use the width of wind the narrow component of the emission lines and the P-Cygni absorption component of N III λ4097 ± × − − and He I λ5876. We obtain a progenitor mass-loss rate of M˙ = (6 1) 10 3 M(cid:12) yr 1, using the strength of Hγ, Hβ, and He II λ4686 lines as diagnostics. The advantage of using these lines rather than the stronger Hα line is that they are formed in the inner parts of the outﬂow (Fig. 4) and thus are less aﬀected by time-dependent eﬀects such as a variable progenitor M˙ , the changing radiation ﬁeld from the SN, or light-travel time. Our model overestimates Hα, which may indicate either that part of the extended region responsible for the Hα emission (Fig. 4) has not yet been illuminated − by the SN, or that the density structure is steeper than r 2, or both. Our best-ﬁt value for M˙ is within the range of previously-published estimates for the mass loss rate from SN 1998S’s progenitor ≈ − − − (M˙ 10 4–10 2 M(cid:12) yr 1, e.g., Lentz et al. 2001; Chugai 2001; Pooley et al. 2002; Moriya et al. 2014). × Figure 4 indicates that (except for the Hα line) material between approximately 6 1014 cm × − and 2 1015 cm dominates the CMFGEN spectrum. Assuming a constant v of 40 km s 1, this ∼ wind corresponds to material launched from the progenitor 5–15 yr before core collapse. As shown in Table 2, our model indicates that the progenitor wind presents material mildly enriched by the CNO cycle, with He and N being enhanced and H, C, and O being depleted. The He mass fraction Y is relatively well constrained since the spectrum of SN 1998S has strong H, He I, and He II lines. ± We use the relative strengths of the He and H lines as diagnostics and ﬁnd Y = 0.49 0.1. The N abundance is constrained using N III λλ4634, 4640 and the O abundance is determined from [O III] λλ4363, 4959, 5007. We employ C III λλ4647, 4750, 5696 to constrain the C abundance, but the absolute value there should be taken with care given the well-known dependence of these C lines on details of the model atoms (Martins & Hillier 2012). We estimate uncertainties of 50%, 30%, and 50% on the C, N, and O abundances, respectively. We assume solar abundances for the

text: The

text: value

text: of

text: R

text: and

text: L

text: determines

text: the

text: temperature

text: at

text: the

text: inner

text: boundary

text: (T

text: which

text: is

text: in

text: SN

text: in

text: the

text: primary

text: regulator

text: of

text: the

text: ionization

text: structure

text: of

text: the

text: illuminated

text: progenitor

text: wind.

text: Our

text: main

text: diagnostic

text: for

text: the

text: ionization

text: structure

text: is

text: the

text: ratio

text: of

text: He

text: I

text: and

text: He

text: II

text: lines,

text: which

text: are

text: very

text: sensitive

text: to

text: R

text: and

text: L

text: in

text: this

text: parameter

text: range.

text: To

text: minimize

text: the

text: confounding

text: eﬀect

text: of

text: the

text: uncertainties

text: in

text: in

text: SN

text: the

text: ﬂux

text: calibration

text: and

text: dust

text: reddening,

text: we

text: use

text: neighboring

text: lines

text: when

text: setting

text: these

text: parameters:

text: He

text: I

text: and

text: He

text: II

text: all

text: of

text: which

text: are

text: ﬁt

text: reasonably

text: well

text: by

text: the

text: ﬁnal

text: model.

text: Our

text: best-ﬁt

text: values

text: for

text: R

text: and

text: L

text: are

text: in

text: extremely

text: good

text: agreement

text: with

text: predictions

text: for

text: this

text: epoch

text: in

text: SN

text: by

text: Chugai

text: see

text: their

text: Fig.

text: which

text: are

text: based

text: upon

text: light-curve

text: model

text: ﬁts

text: to

text: bolometric

text: ﬂux

text: estimates

text: from

text: March

text: and

text: onward

text: (Fassia

text: et

text: al.

text: Additional

text: constraints

text: on

text: the

text: ionization

text: structure

text: come

text: from

text: the

text: absence

text: of

text: N

text: IV

text: and

text: N

text: IV

text: To

text: ﬁt

text: the

text: absolute-ﬂux

text: level,

text: we

text: assume

text: a

text: distance

text: of

text: d

text: Mpc

text: (Tully

text: and,

text: as

text: explained

text: in

text: a

text: standard

text: Milky

text: Way

text: dust-extinction

text: law

text: (R

text: Cardelli

text: et

text: al.

text: ﬁnding

text: a

text: best-ﬁt

text: total

text: dust

text: reddening

text: of

text: V

text: A

text: mag

text: or

text: E(B

text: V

text: mag.

text: V

text: Given

text: the

text: high

text: spectral

text: resolution,

text: we

text: are

text: able

text: to

text: fully

text: resolve

text: both

text: the

text: narrow

text: and

text: broad

text: components

text: of

text: the

text: features.

text: This

text: allows

text: us

text: to

text: independently

text: determine

text: v

text: and

text: M˙

text: We

text: ﬁnd

text: wind

text: a

text: progenitor

text: wind

text: speed

text: of

text: v

text: km

text: s

text: To

text: constrain

text: this

text: value,

text: we

text: use

text: the

text: width

text: of

text: wind

text: the

text: narrow

text: component

text: of

text: the

text: emission

text: lines

text: and

text: the

text: P-Cygni

text: absorption

text: component

text: of

text: N

text: III

text: and

text: He

text: I

text: We

text: obtain

text: a

text: progenitor

text: mass-loss

text: rate

text: of

text: M˙

text: yr

text: using

text: the

text: strength

text: of

text: Hγ,

text: Hβ,

text: and

text: He

text: II

text: lines

text: as

text: diagnostics.

text: The

text: advantage

text: of

text: using

text: these

text: lines

text: rather

text: than

text: the

text: stronger

text: Hα

text: line

text: is

text: that

text: they

text: are

text: formed

text: in

text: the

text: inner

text: parts

text: of

text: the

text: outﬂow

text: (Fig.

text: and

text: thus

text: are

text: less

text: aﬀected

text: by

text: time-dependent

text: eﬀects

text: such

text: as

text: a

text: variable

text: progenitor

text: M˙

text: the

text: changing

text: radiation

text: ﬁeld

text: from

text: the

text: SN,

text: or

text: light-travel

text: time.

text: Our

text: model

text: overestimates

text: Hα,

text: which

text: may

text: indicate

text: either

text: that

text: part

text: of

text: the

text: extended

text: region

text: responsible

text: for

text: the

text: Hα

text: emission

text: (Fig.

text: has

text: not

text: yet

text: been

text: illuminated

text: by

text: the

text: SN,

text: or

text: that

text: the

text: density

text: structure

text: is

text: steeper

text: than

text: r

text: or

text: both.

text: Our

text: best-ﬁt

text: value

text: for

text: M˙

text: is

text: within

text: the

text: range

text: of

text: previously-published

text: estimates

text: for

text: the

text: mass

text: loss

text: rate

text: from

text: SN

text: progenitor

text: (M˙

text: yr

text: e.g.,

text: Lentz

text: et

text: al.

text: Chugai

text: Pooley

text: et

text: al.

text: Moriya

text: et

text: al.

text: Figure

text: indicates

text: that

text: (except

text: for

text: the

text: Hα

text: line)

text: material

text: between

text: approximately

text: cm

text: and

text: cm

text: dominates

text: the

text: CMFGEN

text: spectrum.

text: Assuming

text: a

text: constant

text: v

text: of

text: km

text: s

text: this

text: wind

text: corresponds

text: to

text: material

text: launched

text: from

text: the

text: progenitor

text: yr

text: before

text: core

text: collapse.

text: As

text: shown

text: in

text: Table

text: our

text: model

text: indicates

text: that

text: the

text: progenitor

text: wind

text: presents

text: material

text: mildly

text: enriched

text: by

text: the

text: CNO

text: cycle,

text: with

text: He

text: and

text: N

text: being

text: enhanced

text: and

text: H,

text: C,

text: and

text: O

text: being

text: depleted.

text: The

text: He

text: mass

text: fraction

text: Y

text: is

text: relatively

text: well

text: constrained

text: since

text: the

text: spectrum

text: of

text: SN

text: has

text: strong

text: H,

text: He

text: I,

text: and

text: He

text: II

text: lines.

text: We

text: use

text: the

text: relative

text: strengths

text: of

text: the

text: He

text: and

text: H

text: lines

text: as

text: diagnostics

text: and

text: ﬁnd

text: Y

text: The

text: N

text: abundance

text: is

text: constrained

text: using

text: N

text: III

text: and

text: the

text: O

text: abundance

text: is

text: determined

text: from

text: [O

text: III]

text: We

text: employ

text: C

text: III

text: to

text: constrain

text: the

text: C

text: abundance,

text: but

text: the

text: absolute

text: value

text: there

text: should

text: be

text: taken

text: with

text: care

text: given

text: the

text: well-known

text: dependence

text: of

text: these

text: C

text: lines

text: on

text: details

text: of

text: the

text: model

text: atoms

text: (Martins

text: Hillier

text: We

text: estimate

text: uncertainties

text: of

text: and

text: on

text: the

text: C,

text: N,

text: and

text: O

text: abundances,

text: respectively.

text: We

text: assume

text: solar

text: abundances

text: for

text: the


text: – 14 – other elements, noting that several Fe IV lines that are predicted by the model are not observed. The absence of these lines could be reconciled with the observations either by decreasing R (i.e. in increasing T ) or by decreasing the Fe abundance to 1/3–1/2 of the solar value. in 4. Discussion 4.1. A Cartoon Outer CSM Inner CSM Photosphere SN Ejecta Obscured or Not Illuminated Fig. 5.— A cartoon representation of SN 1998S and its circumstellar environment at the time of observation. For the purposes of this discussion, we deﬁne three zones: the SN ejecta (shown in black), the inner CSM region (blue), and the outer CSM region (orange). The photosphere of the system, which falls in the CSM, is shown in red. A fraction of the CSM is obscured by the photosphere and some regions have not yet received any SN radiation owing to nonzero light-travel time; these eﬀects are illustrated by the gray mask, assuming the observer is to the right (ﬁgure not to scale). Figure 5 displays a simple cartoon of SN 1998S and its CSM at the time of our HIRES observation. SN 1998S was a hydrogen-rich core-collapse SN, and the expanding SN ejecta exhibited ≈ − radial velocities of v 7000 km s 1 (Leonard et al. 2000). As indicated by our CMFGEN modeling ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: other

text: elements,

text: noting

text: that

text: several

text: Fe

text: IV

text: lines

text: that

text: are

text: predicted

text: by

text: the

text: model

text: are

text: not

text: observed.

text: The

text: absence

text: of

text: these

text: lines

text: could

text: be

text: reconciled

text: with

text: the

text: observations

text: either

text: by

text: decreasing

text: R

text: (i.e.

text: in

text: increasing

text: T

text: or

text: by

text: decreasing

text: the

text: Fe

text: abundance

text: to

text: of

text: the

text: solar

text: value.

text: in

text: Discussion

text: A

text: Cartoon

text: Outer

font: default

text: CSM

text: Inner

text: CSM

text: Photosphere

text: SN

text: Ejecta

text: Obscured

text: or

text: Not

text: Illuminated

text: Fig.

text: A

text: cartoon

text: representation

text: of

text: SN

text: and

text: its

text: circumstellar

text: environment

text: at

text: the

text: time

text: of

text: observation.

text: For

text: the

text: purposes

text: of

text: this

text: discussion,

text: we

text: deﬁne

text: three

text: zones:

text: the

text: SN

text: ejecta

text: (shown

text: in

text: black),

text: the

text: inner

text: CSM

text: region

text: (blue),

text: and

text: the

text: outer

text: CSM

text: region

text: (orange).

text: The

text: photosphere

text: of

text: the

text: system,

text: which

text: falls

text: in

text: the

text: CSM,

text: is

text: shown

text: in

text: red.

text: A

text: fraction

text: of

text: the

text: CSM

text: is

text: obscured

text: by

text: the

text: photosphere

text: and

text: some

text: regions

text: have

text: not

text: yet

text: received

text: any

text: SN

text: radiation

text: owing

text: to

text: nonzero

text: light-travel

text: time;

text: these

text: eﬀects

text: are

text: illustrated

text: by

text: the

text: gray

text: mask,

text: assuming

text: the

text: observer

text: is

text: to

text: the

text: right

text: (ﬁgure

text: not

text: to

text: scale).

text: Figure

text: displays

text: a

text: simple

text: cartoon

text: of

text: SN

text: and

text: its

text: CSM

text: at

text: the

text: time

text: of

text: our

text: HIRES

text: observation.

text: SN

text: was

text: a

text: hydrogen-rich

text: core-collapse

text: SN,

text: and

text: the

text: expanding

text: SN

text: ejecta

text: exhibited

text: radial

text: velocities

text: of

text: v

text: km

text: s

text: (Leonard

text: et

text: al.

text: As

text: indicated

text: by

text: our

text: CMFGEN

text: modeling

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: – 22 – Fig. 9.— Velocity structure of helium lines in the early HIRES spectrum of SN 1998S. He I lines are shown in the top eight panels while He II lines are in the bottom two. The HIRES spectrum is shown in black, red lines illustrate simple Gaussian ﬁts to the data, and the vertical ashed lines indicate the rest frame. The He II lines are signiﬁcantly broader and blueshifted compared to the He I lines. A complete list of lines and their properties is in Table 1. a factor of 2 greater than we observe on March 4, Fassia et al. 2001). To explore this further, we adopt an approximation from Chugai et al. (2002): the maximum velocity to which the SN could ≈ − − accelerate a slow wind through Thomson scattering alone is v 80 E r 2 km s 1, where E is the 49 15 49 time-integrated energy released through radiation in units of 1049 erg and r is the radius in units 15 of 1015 cm. We estimate E by assuming that the luminosity of the young SN roughly follows a ≈49 − − ∼ power law in time: L α(t t )n. We assume the time since explosion (t t ) is 5 d and ≈ SN 0 0 we determine n 3.25 from early photometry reported by Li et al. (1998) and Qiu et al. (1998). Adopting our modeled values for the luminosity and radius of the photosphere, we calculate that ≈ ∼ − E 0.6 at the time of our HIRES spectrum. This yields a maximum v of 130 km s 1. As 49 Chugai et al. (2002) notes, including UV line-absorption eﬀects increases the maximum velocity by ∼ ∼ − 50 % in SN 1998S, and including the original wind velocity adds another 40 km s 1. This more ∼ − than accounts for the full He II velocity of 140–180 km s 1 measured from our HIRES spectrum. § ∝ − In we adopt a ρ r 2 wind proﬁle for the CSM beyond the CDS. This provides a good ##LTFigure##

text: Fig.

text: Velocity

text: structure

text: of

text: helium

text: lines

text: in

text: the

text: early

text: HIRES

text: spectrum

text: of

text: SN

text: He

text: I

text: lines

text: are

text: shown

text: in

text: the

text: top

text: eight

text: panels

text: while

text: He

text: II

text: lines

text: are

text: in

text: the

text: bottom

text: two.

text: The

text: HIRES

text: spectrum

text: is

text: shown

text: in

text: black,

text: red

text: lines

text: illustrate

text: simple

text: Gaussian

text: ﬁts

text: to

text: the

text: data,

text: and

text: the

text: vertical

text: ashed

text: lines

text: indicate

text: the

text: rest

text: frame.

text: The

text: He

text: II

text: lines

text: are

text: signiﬁcantly

text: broader

text: and

text: blueshifted

text: compared

text: to

text: the

text: He

text: I

text: lines.

text: A

text: complete

text: list

text: of

text: lines

text: and

text: their

text: properties

text: is

text: in

text: Table

text: a

text: factor

text: of

text: greater

text: than

text: we

text: observe

text: on

text: March

text: Fassia

text: et

text: al.

text: To

text: explore

text: this

text: further,

text: we

text: adopt

text: an

text: approximation

text: from

text: Chugai

text: et

text: al.

text: the

text: maximum

text: velocity

text: to

text: which

text: the

text: SN

text: could

text: accelerate

text: a

text: slow

text: wind

text: through

text: Thomson

text: scattering

text: alone

text: is

text: v

text: E

text: r

text: km

text: s

text: where

text: E

text: is

text: the

text: time-integrated

text: energy

text: released

text: through

text: radiation

text: in

text: units

text: of

text: erg

text: and

text: r

text: is

text: the

text: radius

text: in

text: units

text: of

text: cm.

text: We

text: estimate

text: E

text: by

text: assuming

text: that

text: the

text: luminosity

text: of

text: the

text: young

text: SN

text: roughly

text: follows

text: a

text: power

text: law

text: in

text: time:

text: L

text: α(t

text: t

text: )n.

text: We

text: assume

text: the

text: time

text: since

text: explosion

text: (t

text: t

text: is

text: d

text: and

text: SN

text: we

text: determine

text: n

text: from

text: early

text: photometry

text: reported

text: by

text: Li

text: et

text: al.

text: and

text: Qiu

text: et

text: al.

text: Adopting

text: our

text: modeled

text: values

text: for

text: the

text: luminosity

text: and

text: radius

text: of

text: the

text: photosphere,

text: we

text: calculate

text: that

text: E

text: at

text: the

text: time

text: of

text: our

text: HIRES

text: spectrum.

text: This

text: yields

text: a

text: maximum

text: v

text: of

text: km

text: s

text: As

text: Chugai

text: et

text: al.

text: notes,

text: including

text: UV

text: line-absorption

text: eﬀects

text: increases

text: the

text: maximum

text: velocity

text: by

text: in

text: SN

text: and

text: including

text: the

text: original

text: wind

text: velocity

text: adds

text: another

text: km

text: s

text: This

text: more

text: than

text: accounts

text: for

text: the

text: full

text: He

text: II

text: velocity

text: of

text: km

text: s

text: measured

text: from

text: our

text: HIRES

text: spectrum.

text: In

text: we

text: adopt

text: a

text: ρ

text: r

text: wind

text: proﬁle

text: for

text: the

text: CSM

text: beyond

text: the

text: CDS.

text: This

text: provides

text: a

text: good

text: ##LTFigure##


text: – 30 – Acknowledgments We thank Avishay Gal-Yam and Nathan Smith for useful discussions and comments, and we are grateful to the organizers of the 2014 CAASTRO Annual Scientiﬁc Conference at which several helpful discussions of this work occurred. We thank Tom Matheson for assistance with an early reduction of the data. The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientiﬁc partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration (NASA); the observatory was made possible by the generous ﬁnancial support of the W. M. Keck Foundation. We wish to recognize and acknowledge the very signiﬁcant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community; we are most fortunate to have the opportunity to conduct observations from this mountain. This research has made use of the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with NASA. This research has also made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. A.V.F.’s supernova group at UC Berkeley is supported through NSF grant AST–1211916, the TABASGO Foundation, and the Christopher R. Redlich Fund. Research by D.C.L. at San Diego State University is supported by NSF grants AST–1009571 and AST–1210311. This work was supported in part by the NSF under grant No. PHYS–1066293. A.V.F. thanks the Aspen Center for Physics for its hospitality during the “Fast and Furious” workshop in June 2014. J.H.G. is supported by an Ambizione Fellowship of the Swiss National Science Foundation. Facilities: Keck:HIRES REFERENCES Anupama, G. C., Sivarani, T., & Pandey, G. 2001, A&A, 367, 506 Ben-Ami, S., Konidaris, N., Quimby, R., Davis, J. T., Ngeow, C. C., Ritter, A., & Rudy, A. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Bjorkman, J. E., & Cassinelli, J. P. 1993, ApJ, 409, 429 Bowen, D. V., Roth, K. C., Meyer, D. M., & Blades, J. C. 2000, ApJ, 536, 225 Branch, D., Jeﬀery, D. J., Blaylock, M., & Hatano, K. 2000, PASP, 112, 217 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 Chevalier, R. A., & Irwin, C. M. 2011, ApJ, 729, L6 Chugai, N. N. 1991, MNRAS, 250, 513

text: Acknowledgments

text: We

text: thank

text: Avishay

text: Gal-Yam

text: and

text: Nathan

text: Smith

text: for

text: useful

text: discussions

text: and

text: comments,

text: and

text: we

text: are

text: grateful

text: to

text: the

text: organizers

text: of

text: the

text: CAASTRO

text: Annual

text: Scientiﬁc

text: Conference

text: at

text: which

text: several

text: helpful

text: discussions

text: of

text: this

text: work

text: occurred.

text: We

text: thank

text: Tom

text: Matheson

text: for

text: assistance

text: with

text: an

text: early

text: reduction

text: of

text: the

text: data.

text: The

text: data

text: presented

text: herein

text: were

text: obtained

text: at

text: the

text: W.

text: M.

text: Keck

text: Observatory,

text: which

text: is

text: operated

text: as

text: a

text: scientiﬁc

text: partnership

text: among

text: the

text: California

text: Institute

text: of

text: Technology,

text: the

text: University

text: of

text: California,

text: and

text: the

text: National

text: Aeronautics

text: and

text: Space

text: Administration

text: (NASA);

text: the

text: observatory

text: was

text: made

text: possible

text: by

text: the

text: generous

text: ﬁnancial

text: support

text: of

text: the

text: W.

text: M.

text: Keck

text: Foundation.

text: We

text: wish

text: to

text: recognize

text: and

text: acknowledge

text: the

text: very

text: signiﬁcant

text: cultural

text: role

text: and

text: reverence

text: that

text: the

text: summit

text: of

text: Mauna

text: Kea

text: has

text: always

text: had

text: within

text: the

text: indigenous

text: Hawaiian

text: community;

text: we

text: are

text: most

text: fortunate

text: to

text: have

text: the

text: opportunity

text: to

text: conduct

text: observations

text: from

text: this

text: mountain.

text: This

text: research

text: has

text: made

text: use

text: of

text: the

text: Keck

text: Observatory

text: Archive

text: (KOA),

text: which

text: is

text: operated

text: by

text: the

text: W.

text: M.

text: Keck

text: Observatory

text: and

text: the

text: NASA

text: Exoplanet

text: Science

text: Institute

text: (NExScI),

text: under

text: contract

text: with

text: NASA.

text: This

text: research

text: has

text: also

text: made

text: use

text: of

text: the

text: NASA/IPAC

text: Extragalactic

text: Database

text: (NED)

text: which

text: is

text: operated

text: by

text: the

text: Jet

text: Propulsion

text: Laboratory,

text: California

text: Institute

text: of

text: Technology,

text: under

text: contract

text: with

text: NASA.

text: A.V.F.’s

text: supernova

text: group

text: at

text: UC

text: Berkeley

text: is

text: supported

text: through

text: NSF

text: grant

text: the

text: TABASGO

text: Foundation,

text: and

text: the

text: Christopher

text: R.

text: Redlich

text: Fund.

text: Research

text: by

text: D.C.L.

text: at

text: San

text: Diego

text: State

text: University

text: is

text: supported

text: by

text: NSF

text: grants

text: and

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: the

text: NSF

text: under

text: grant

text: No.

text: A.V.F.

text: thanks

text: the

text: Aspen

text: Center

text: for

text: Physics

text: for

text: its

text: hospitality

text: during

text: the

text: “Fast

text: and

text: Furious”

text: workshop

text: in

text: June

text: J.H.G.

text: is

text: supported

text: by

text: an

text: Ambizione

text: Fellowship

text: of

text: the

text: Swiss

text: National

text: Science

text: Foundation.

text: Facilities:

text: Keck:HIRES

text: REFERENCES

text: Anupama,

text: G.

text: C.,

text: Sivarani,

text: T.,

text: Pandey,

text: G.

text: A&A,

text: Ben-Ami,

text: S.,

text: Konidaris,

text: N.,

text: Quimby,

text: R.,

text: Davis,

text: J.

text: T.,

text: Ngeow,

text: C.

text: C.,

text: Ritter,

text: A.,

text: Rudy,

text: A.

text: in

text: Society

text: of

text: Photo-Optical

text: Instrumentation

text: Engineers

text: (SPIE)

text: Conference

text: Series,

text: Vol.

text: Society

text: of

text: Photo-Optical

text: Instrumentation

text: Engineers

text: (SPIE)

text: Conference

text: Series

text: Bjorkman,

text: J.

text: E.,

text: Cassinelli,

text: J.

text: P.

text: ApJ,

text: Bowen,

text: D.

text: V.,

text: Roth,

text: K.

text: C.,

text: Meyer,

text: D.

text: M.,

text: Blades,

text: J.

text: C.

text: ApJ,

text: Branch,

text: D.,

text: Jeﬀery,

text: D.

text: J.,

text: Blaylock,

text: M.,

text: Hatano,

text: K.

text: PASP,

text: Cardelli,

text: J.

text: A.,

text: Clayton,

text: G.

text: C.,

text: Mathis,

text: J.

text: S.

text: ApJ,

text: Chevalier,

text: R.

text: A.,

text: Irwin,

text: C.

text: M.

text: ApJ,

text: Chugai,

text: N.

text: N.

text: MNRAS,