Sec$on 5. Radia$on from explosive transients (2)

5.1 Light curves of supernovae

5.2 Applica$ons to neutron star mergers

Light curves

-20

-18

-16

-14

-12

-10 0 50 100 150 200 250 300 350

Abso

lute

mag

nitu

de

Days after the explosion

Type IaType Ib and Ic

Type II Type I - Peak - L(Ia) > L(Ib, Ic)

Type II - plateau - L(Ia) > L(II)

1043 erg s-1

1042

1041

Various types of explosive transients

What determines their luminosity and $mescale?

Sec$on 4 Sec$on 5

103 104

10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

bound-free

free-free

e-

lines

wavelength [A]

opac

ity [c

m-1]

758 PINTO & EASTMAN Vol. 530

of this section will then be applied to an examination offrequency-averaged mean opacities.

A simple application of the analytic model of Paper Ishows that, for explosion models with 0.7 M

_\ M \ 1.4

and 0.35 the central tem-M_

M_

\ M(56Ni) \ 1.4 M_

,perature near maximum luminosity is 1.5 ] 104 K [ T [

K and the density is g cm~3.2.5 ] 104 10~14 [ o [ 10~12Under these conditions, the continuum opacity at opticalwavelengths is dominated by electron scattering. Forcentral temperatures K, the peak of theT

c[ 1.5 ] 104

Planck function is in the UV where the(jBB [ 1900 A! )opacity is dominated by bound-bound transitions. Theopacity from a thick forest of lines is greatly increased byvelocity shear Doppler broadening (Karp et al. 1977).

Figure 1 displays the various sources of opacity for amixture of 56Ni (20%), 56Co (70%), and 56Fe (10%), at adensity of 10~13 g cm~3 and temperature of 2.5 ] 104 K,typical (perhaps) of maximum light in a Chandrasekhar-mass explosion. The excitation and ionization were com-puted from the Saha-Boltzmann equation. The opacityapproximation of Eastman & Pinto (1993Èalso see below)was used for the line opacity, which greatly exceeds thatfrom electron scattering. Bound-free and free-free tran-sitions contribute negligibly to the overall opacity, but theyare important contributors to the coupling between theradiation Ðeld and the thermal energy of the gas. Theopacity is very strongly concentrated in the UV and falls o†steeply toward optical wavelengths. As was noted byMontes & Wagoner (1995), the line opacity between 2000and 4000 falls o† roughly as We willA! d ln ij/d ln j D [10.show that the steepness of this decline toward the optical

has important implications for the e†ective opacity in SNeIa and the way in which energy escapes.

2.2. Di†usionNot only is the opacity from lines greater than that of

electron scattering, it is also fundamentally di†erent in char-acter from a continuous opacity. In a medium where theopacity varies slowly with wavelength, photons have anexponential distribution of path lengths. Their progressthrough an optically thick medium is a random walk with amean path length given by (oi)~1. In a supersonicallyexpanding medium dominated by line opacity, however,there is a bimodal distribution of path lengths. The lineopacity is concentrated in a Ðnite number of isolated reso-nance regions. Within these regions, where a photon hasDoppler shifted into resonance with a line transition, themean free path is very small. Outside these regions, the pathlength is determined either by the much smaller continuousopacity or by the distance the photon must travel to haveDoppler shifted into resonance with the next transition oflonger wavelength. For the physical conditions of Figure 1,the mean free path of a UV photon goes from approx-imately 5 ] 1014 cm in the continuum (because of electronscattering) to less than D106 cm when in a line. The usualrandom walk description of continuum transport must bemodiÐed to take this bimodal distribution into account.

Within a line, a photon scatters on average N D 1/ptimes, where p is the probability per scattering for escape,which is accomplished by Doppler shifting out of reso-nance. In spite of the possibly large number of scatteringsneeded for escape, a photon spends only a small fraction of

FIG. 1.ÈMonochromatic opacity sources at maximum light for a Chandrasekhar-mass model of a Type Ia supernova from a time-dependent, multi-frequency, LTE calculation. The physical conditions are o \ 10~13 g cm~3, T \ 2.5 ] 104 K and t \ 14 days. The line opacity shown here is the frequencybinÈ averaged expansion opacity, as given by equation (9) (see text) and not the much larger monochromatic Sobolev opacity. The line opacity has not beenplotted below 2 ] 10~14 cm~1, so as not to obscure the contributions from continuum opacity sources.

Pinto & Eastman 2000

Opacity in supernova ejecta (Type Ia SN, ρ = 10-13 g cm-3)

κ (cm2 g-1)

1.0

10

102

103

10-1

10-2

10-3

束縛遷移

電子散乱

-20

-18

-16

-14

-12

-10 0 50 100 150 200 250 300 350

Abso

lute

mag

nitu

de

Days after the explosion

Type IaType Ib and Ic

Type II

~20 days

~100 days

56Ni

Shock hea$ng

1043 erg s-1

1042 erg s-1

56Ni

Type Ia SNe eject more 56Ni

Light curves

Observa$ons <=> physical quan$$es

E,2Mej,2and2M(56Ni)

spectrum

velocity~2E1/22MejG1/2

M(56Ni)

M(56Ni)

t

L

~2202d

t2~2κ1/22EG1/42Mej3/4

Stripped2envelope2SN2(Type2Ib/c,2Ia)

3

Light curves Spectra

0

1

2

5600 5800 6000 6200 6400 6600Fl

ux

Wavelength (A)

v ~ E1/2 Mej-1/2

+ chemical composi$on

E, Mej, M(56Ni), X (element)

Applica$ons to Type Ib/Ic supernovae

Lyman+16

344 J. D. Lyman et al.

a progenitor, it follows that the ZAMS mass of an SN Ib would beexpected to be lower than that of a Ic, for equal Mej. The appar-ent similarity in the Mej distributions shown here would then hinttowards lower CO core masses (and thus lower ZAMS mass) forSNe Ib compared to Ic. However, as mentioned, the helium massrequired to produce an SN Ib spectrum may be only ∼ 0.1 M⊙(Hachinger et al. 2012). Considering this, alongside the compara-tively large uncertainties on our Mej values, the distributions cannotbe used to rule conclusively on any potential mass sequence (or lackof) in the ZAMS masses of SNe Ib and Ic.

Despite the modest sample sizes (from a statistical viewpoint),the SNe Ic-BL manifest themselves as very different in two of thethree explosion parameters determined here. Their MNi and EK val-ues are much larger on average than the distributions of any of theother SN types. However, unlike the EK distributions, where eventhe least energetic SN Ic-BL is more energetic than the majority ofSNe IIb, Ib and Ic, the largest MNi masses of SNe Ic-BL are matchedby those of SNe Ib and Ic. This indicates the presence of BL is nota certainty when a large amount of MNi is synthesized (since wesee SNe Ib and Ic with comparable MNi), and thus the peak bright-ness is not a uniquely determining factor. Additionally, although thehigh-velocity nature of SNe Ic-BL naturally implies a large EK/Mej

ratio (equation 2), these large EK/Mej ratios are occurring at verysimilar Mej values of the other SN subtypes (SNe Ic-BL are indistin-guishable from the individual or combined IIb/Ib/Ic distribution),favouring an energy source that is decoupled from a dependence onthe mass of the exploding core.

4.4 Explosion parameter correlations

The parameters derived from the modelling are plotted against eachother in Fig. 8. The bulk of SNe IIb, Ib and Ic appear to form afairly tight correlation in the Mej–EK plot, this is a result of thesimilar vph values they exhibit (which, in turn, gives the Mej/EK

ratio). Conversely, SNe Ic-BL, which can have very high velocities(Table 4), are found at larger EK/Mej ratios, as dictated by equa-tion (2). Some splitting of SNe Ic-BL occurs with the ‘hypernovabranch’ (i.e. very high Mej and EK values; e.g. Mazzali et al. 2013)being populated by SNe 1998bw and 2005kz, whereas other SNeIc-BL sit at similar Mej values to other SN types, but with higherEK values. SN 2011bm appears as an intermediate member of thehypernova branch in these plots, despite displaying very modestvelocity, with vph = 9000 km s−1. In this case, the huge explosionparameter values found were due to the extremely slow evolution ofthe SN (Valenti et al. 2012, Fig. 2), and could point to an alternativesignature of the explosion of a very massive star, perhaps withoutthe angular momentum to produce an accretion-disc powered jet.Although SN Ic-BL Mej values have a similar distribution to thoseof other SE SN types, their MNi values (barring SN 2002ap) aremuch higher than the bulk of SE SNe, indicating the production ofMNi is much more efficient in these explosions.

SNe IIb appear to be the most homogeneous subtype of SE SNe asevident from the clustering of their bolometric light-curve properties(Fig. 1) and explosion parameters (Fig. 8). This may be a result of amuch more restrictive progenitor range for SNe IIb (e.g. Yoon et al.2010).

Hamuy (2003) show a correlation of increasing EK with MNi forSNe IIP and to a weaker extent this is also found for SE SNe.Although there is a large amount of scatter and SN 2005hg is aprominent outlier, a correlation between MNi and EK can be seenin Fig. 8 – no highly energetic and MNi deficient events are seen.Similarly, a correlation between MNi and v50 (vph at 50 d after

Figure 8. Derived explosion parameters (MNi, Mej and EK) of literatureSE SNe are plotted against each other. Data are given in Table 5. SNe arecolour-coded according to their type.

MNRAS 457, 328–350 (2016)

at National A

stronomical O

bservatory of Japan on September 15, 2016

http://mnras.oxfordjournals.org/

Dow

nloaded from

Mej ~ 1-3 Msun Ek ~ (0.5-5) x 1051 erg

Ia型

Type II

Core-collapse SNe and their progenitors (親星)

Fe

He

Type Ib

He

Type Ic

C

Mej >~ 6-8 Msun (single stars)Wolf-Rayet stars

Inconsistent with observa$ons => importance of binary evolu$on?

Sec$on 5. Radia$on from explosive transients (2)

5.1 Light curves of supernovae

5.2 Applica$ons to neutron star mergers

Sekiguchi+15, 16M ~ 10-3 - 10-2 Msun

v ~ 0.1 - 0.2 c

Top view Side view

NS merger => mass ejec$on

Supernova vs NS merger

Supernova NS merger

Power source 56Ni r-process elements

Ejecta mass 1-10 Msun 0.01 Msun

Ejecta velocity 5,000-10,000 km/s 30,000-60,000 km/s (0.1c-0.2c)

Kinetic energy 1051 erg 1-5 x 1050 erg

Composition H, He, C, O, Ca, Fe-group r-process elements

r-process nucleosynthesis in NS merger

(C) Nobuya Nishimura

Transients from compact object mergers 2653

3 R A D I OAC T I V E H E AT I N G

3.1 Network calculations

In this section we present calculations of the radioactive heating ofthe ejecta. We use a dynamical r-process network (Martınez-Pinedo2008; Petermann et al. 2008) that includes neutron captures, pho-todissociations, β-decays, α-decays and fission reactions. The latterincludes contributions from neutron-induced fission, β delayed fis-sion and spontaneous fission. The neutron capture rates for nucleiwith Z ≤ 83 are obtained from the work of Rauscher & Thielemann(2000) and are based on two different nuclear mass models: theFinite Range Droplet Model (FRDM; Moller et al. 1995) and theQuenched version of the Extended Thomas–Fermi with StrutinskyIntegral (ETFSI-Q) model (Pearson, Nayak & Goriely 1996). Fornuclei with Z > 83 the neutron capture rates and neutron-inducedfission rates are obtained from Panov et al. (2010). β-decay ratesincluding emission of up to three neutrons after β-decay are fromMoller, Pfeiffer & Kratz (2003). β-delayed fission and spontaneousfission rates are determined as explained by Martınez-Pinedo et al.(2007). Experimental rates for α and β decay have been obtainedfrom the NUDAT data base.1 Fission yields for all fission processesare determined using the statistical code ABLA (Gaimard & Schmidt1991; Benlliure et al. 1998). All heating is self-consistently addedto the entropy of the fluid following the procedure of Freiburghauset al. (1999). The change of temperature during the initial expan-sion is determined using the Timmes equation of state (Timmes &Arnett 1999), which is valid below the density ρ ∼ 3 × 1011 g cm−3

at which our calculation begins.As in the r-process calculations performed by Freiburghaus et al.

(1999), we use a Lagrangian density ρ(t) taken from the NS–NSmerger simulations of Rosswog et al. (1999). In addition to ρ(t), theinitial temperature T , electron fraction Ye and seed nuclei properties(A, Z) are specified for a given calculation. We assume an initialtemperature T = 6 × 109 K, although the subsequent r-process heat-ing is not particularly sensitive to this choice because any initial ther-mal energy is rapidly lost to P dV work during the initial expansionbefore the r-process begins (Meyer 1989; Freiburghaus et al. 1999).For our fiducial model we also assume Ye = 0.1, Z ≃ 36, A ≃ 118(e.g. Freiburghaus et al. 1999).

Our results for the total radioactive power E with time are shownin Fig. 1. On time-scales of interest the radioactive power can bedivided into two contributions: fission and β-decays, which aredenoted by dashed and dotted lines, respectively. The large heatingrate at very early times is due to the r-process, which ends whenneutrons are exhausted at t ∼ 1 s ∼10−5 d. The heating on longertime-scales results from the synthesized isotopes decaying back tostability. On the time-scales of interest for powering EM emission(tpeak ∼ hours–days; equations3), most of the fission results fromthe spontaneous fission of nuclei with A ∼ 230–280. This releasesenergy in the form of the kinetic energy of the daughter nuclei andfast neutrons, with a modest contribution from γ -rays. The othersource of radioactive heating is β-decays of r-process product nucleiand fission daughters (see Table 1 for examples corresponding toour fiducial model). In Fig. 1 we also show for comparison theradioactive power resulting from an identical mass of 56Ni and itsdaughter 56Co. Note that (coincidentally) the radioactive power ofthe r-process ejecta and 56Ni/56Co are comparable on time-scales∼1 d.

1http://www.nndc.bnl.gov/nudat2/

Figure 1. Radioactive heating rate per unit mass E in NS merger ejectadue to the decay of r-process material, calculated for the Ye = 0.1 ejectatrajectory from Rosswog et al. (1999) and Freiburghaus et al. (1999). Thetotal heating rate is shown with a solid line and is divided into contributionsfrom β-decays (dotted line) and fission (dashed line). For comparison wealso show the heating rate per unit mass produced by the decay chain56Ni → 56Co → 56Fe (dot–dashed line). Note that on the ∼day time-scalesof interest for merger transients (t ∼ tpeak; equation 3) fission and β-decaysmake similar contributions to the total r-process heating, and that the r-process and 56Ni heating rates are similar.

Table 1. Properties of the dominant β-decay nuclei at t ∼ 1 d.

Isotope t1/2 Qa ϵbe ϵc

ν ϵdγ Eavg e

γ

(h) (MeV) (MeV)

135I 6.57 2.65 0.18 0.18 0.64 1.17129Sb 4.4 2.38 0.22 0.22 0.55 0.86128Sb 9.0 4.39 0.14 0.14 0.73 0.66129Te 1.16 1.47 0.48 0.48 0.04 0.22132I 2.30 3.58 0.19 0.19 0.62 0.77135Xe 9.14 1.15 0.38 0.40 0.22 0.26127Sn 2.1 3.2 0.24 0.23 0.53 0.92134I 0.88 4.2 0.20 0.19 0.61 0.8656Nif 146 2.14 0.10 0.10 0.80 0.53

aTotal energy released in the decay.b,c,dFraction of the decay energy released in electrons, neutrinos and γ -rays.eAverage photon energy produced in the decay.f Note: 56Ni is not produced by the r-process and is only shown for compar-ison [although a small abundance of 56Ni may be produced in accretion discoutflows from NS–NS/NS–BH mergers (Metzger et al. 2008b)].

In Fig. 2 we show the final abundance distribution from ourfiducial model, which shows the expected strong second and thirdr-process peaks at A ∼ 130 and ∼195, respectively. For comparison,we show the measured Solar system r-process abundances withpoints. The computed abundances are rather different to the oneobtained by Freiburghaus et al. (1999) due to an improved treatmentof fission yields and freeze-out effects.

Although we assume Ye = 0.1 in our fiducial model, the ejectafrom NS mergers will possess a range of electron fractions (seeSection 2.1). To explore the sensitivity of our results to the ejectacomposition we have run identical calculations of the radioactiveheating, but varying the electron fraction in the range Ye = 0.05–0.35. Although in reality portions of the ejecta with different compo-sitions will undergo different expansion histories, in order to makea direct comparison we use the same density trajectory ρ(t) as wasdescribed earlier for the Ye = 0.1 case. Fig. 3 shows the heating rate

C⃝ 2010 The Authors. Journal compilation C⃝ 2010 RAS, MNRAS 406, 2650–2662

Metzger+10

NS merger ~ t -1.3

Power source

1 day1 hr 10 days

supernova

r-process and kilonovae 9

Figure 4. Heating rates (black) from β-decay (top), α-decay (middle), and fission (bottom) for mFE-a (left) and mFE-b (right)with the top 11 isotopes (in different colors) that have more than 10% contributions at the maxima.

Wanajo 18

Radioac$ve decay luminosity

1 day

10 day

The Astrophysical Journal, 775:113 (16pp), 2013 October 1 Tanaka & Hotokezaka

1040

1041

1042

1 10

UV

OIR

Lum

inos

ity (

erg

s-1)

Days after the merger

NSM-allκ=0.1 cm2 g-1

κ=1.0 cm2 g-1

κ=10 cm2 g-1

Figure 2. Bolometric light curve of the model NSM-all (black, multi-frequencysimulations). This light curve is compared with the light curves for the samemodel but with the gray approximation of UVOIR transfer (κ = 0.1, 1, and10 cm2 g−1 for the blue, purple, and red lines, respectively). The result ofmulti-frequency transfer is most similar to that of gray transfer withκ = 10 cm2 g−1.(A color version of this figure is available in the online journal.)

compared with the light curves for the same model but with thegray approximation of the UVOIR transfer. The blue, purple, andred lines show the cases with gray mass absorption coefficientsof κ = 0.1, 1.0, and 10 cm2 g−1, respectively. The result ofmulti-frequency transfer closely follows the light curve withthe gray opacity of κ = 10 cm2 g−1. This result indicates thatr-process element-rich NS merger ejecta are more opaque thanpreviously assumed (κ ≃ 0.1 cm2 g−1; e.g., Li & Paczynski1998; Metzger et al. 2010), by a factor of about 100. As a result,the bolometric light curve becomes fainter and the timescale

becomes longer.7 This result is consistent with the findings ofKasen et al. (2013) and Barnes & Kasen (2013).

Figure 3 shows the mass absorption coefficient as a functionof wavelength at t = 3 days in the model NSM-all at v = 0.1c.The mass absorption coefficient is as high as 1–100 cm2 g−1 atoptical wavelengths. The resulting Planck mean mass absorptioncoefficient is about κ = 10 cm2 g−1 (Figure 15). As a result,the bolometric light curve of multi-frequency transfer mostclosely follows that of gray opacity of κ = 10 cm2 g−1 inFigure 2.

The high opacity in r-process element-rich ejecta is alsoconfirmed by a comparison with other simple models. Figure 4shows the comparison of the bolometric light curve from themodels NSM-all, NSM-dynamical, NSM-wind, and NSM-Fe.Compared with the NSM-Fe model, the other models showfainter light curves. This finding indicates that elements heavierthan Fe contribute to the high opacity. The opacity in the modelNSM-Fe is also shown in Figure 3. The opacity in the NSM-allmodel is higher than that in the NSM-Fe model by a factor ofabout 100 at the center of optical wavelengths (∼5000 Å).

As inferred from Figure 4, the NSM-dynamical model (55 !Z ! 92) has a higher opacity than that of the NSM-windmodel (31 ! Z ! 54). This finding arises because lanthanoidelements (57 ! Z ! 71) make the largest contribution to thebound–bound opacity, as demonstrated by Kasen et al. (2013).Note, however, that even with the elements with 31 ! Z ! 54,the opacity is higher than that of Fe.

Figure 5 shows the multi-color light curves of the modelNSM-all. In general, the emission from NS merger ejecta is redbecause of (1) a lower temperature than SNe and (2) a higheroptical opacity than in SNe. In particular, the optical light curves

7 We show the results of our multi-frequency transfer simulations at t ! 1day. Because of the lack of bound–bound transition data for triply ionized ionsin our line list (Figure 1), the opacities at earlier epochs are not correctlyevaluated. We hereafter show the results when the temperature at thecharacteristic velocity is below 10,000 K, when the dominant ionization statesare no longer triply ionized ions. A detailed discussion is presented inAppendix B.

0.001

0.01

0.1

1

10

100

1000

5000 10000 15000

κ (c

m2 g

-1)

Wavelength (A)

NSM-allNSM-Fe

Figure 3. Mass absorption coefficient κ at v = 0.1c in the models NSM-all and NSM-Fe as a function of wavelength (t = 3 days after the merger). In r-processelement-rich ejecta, the opacity is higher than in Fe-rich ejecta by a factor of about 100 around the center of optical wavelengths (∼5000 Å).(A color version of this figure is available in the online journal.)

6

Higher opacity by factor of 100

bound-bound opacity

r-process Fe

(Kasen+13, Tanaka & Hotokezaka 13)

Opacity

Nh Mc Ts Og

open s shell

open p-shell

open d-shell

open f shell

0

1

2

3

4

5

E (e

V)

Fe II

0

1

2

3

4

5

E (e

V)

Nd II

� =hc

�E

Radia$on from NS merger

E,2Mej,2and2M(56Ni)

spectrum

velocity~2E1/22MejG1/2

M(56Ni)

M(56Ni)

t

L

~2202d

t2~2κ1/22EG1/42Mej3/4

Stripped2envelope2SN2(Type2Ib/c,2Ia)

3

Light curves Spectra

0

1

2

5600 5800 6000 6200 6400 6600Fl

ux

Wavelength (A)

v ~ E1/2 Mej-1/2

+ Chemical composi$onM(r) = Mej

x 10 x 1/100

Fainter and faster than supernovae Higher veloci$es than supernovae

Summary: Radia$on from explosive transients

• Timescale of radia$on

• Expansion and photon diffusion

• Timescale t ~ κ1/2 Mej3/4 Ek-1/4

• Opacity: Bound-bound transiXons and electron scaYering

• Applica$ons to NS merger

• Lower ejecta mass (x 1/100), Faster expansion (x 5)

• Higher opacity (x 100)

• Kilonova: Fainter and faster than supernovae