A B C Y TP024014-12 質量大降着円盤質量（太陽質量単位）放出質量（太陽質量単位） C C • スピンの傾きは実効的にスピンの効果を弱めるため

重力波テンプレート作成に向けたブラックホール中性子星連星合体の高解像度シミュレーション

川口恭平A

,

木内建太A

, 久徳浩太郎B

, 中野寛之C

,

大川博督D

, 柴田大A

, 田中雅臣E

, 谷口敬介F

A:基礎物理学研究所 B:理化学研究所 C:京都大学 D:早稲田大学 E:国立天文台 F:琉球大学

Y TPYUKAWA INSTITUTE FOR THEORETICAL PHYSICS

第29回理論懇シンポジウム＠東北大学

本発表で話すこと• ブラックホール中性子星連星合体の研究意義

• ブラックホール中性子星連星合体のシミュレーション研究

• これまでの研究

• スピンの傾いたブラックホール中性子星連星合体KK, K. Kyutoku, H. Nakano, H. Okawa, M. Shibata, and K. Taniguchi, PRD (2015)

• ブラックホール中性子連星星合体からのkilonova/macronovaKK, K. Kyutoku, M. Shibata, and M. Tanaka, ApJ (2016)

• これからの研究

• ブラックホール中性子星連星合体の重力波波形の高精度計算

Ref: B. Metzger and E. Berger 2012

ブラックホール中性子星連星合体• ブラックホール中性子星連星合体は地上重力波干渉計の有望な重力波源であり、その波形からは様々な物理の情報が得られると期待されている・ブラックホール時空の情報→重力理論・中性子星の半径の情報→原子核物理学

• 中性子星を含むため、合体には様々な電磁波対応天体が付随すると期待されている・ショートガンマ線バースト（←降着円盤）・Kilonova/Macronova (←質量放出）・Radio Flare, etc.重力波と電磁波対応天体の同時観測によりより多くの連星合体の情報を抜き出せる（ex. 合体の位置情報）

ref) http://atc.mtk.nao.ac.jp/KAGRA/images/KAGRA.png

KAGRA

Ref: K. Kyutoku et al. 2015

シミュレーション研究

Ref: M. Tanaka et al. 2015

The Astrophysical Journal, 780:31 (9pp), 2014 January 1 Tanaka et al.

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Figure 9. Expected observed ugrizJHK-band light curves (in AB magnitudes) for the BH–NS merger models APR4Q3a75 (red solid lines) and H4Q3a75 (blue solidlines) and the NS–NS merger models APR4-1215 (red dashed lines) and H4-1215 (blue dashed lines). The light curves are those averaged over all solid angles. Thedistances to the events are set to be 400 Mpc (BH–NS) and 200 Mpc (NS–NS). K correction is taken into account. Horizontal lines show typical limiting magnitudesfor wide-field telescopes (5σ with 10 minute exposure). For optical wavelengths (ugriz bands), “1 m,” “4 m,” and “8 m” limits are taken or deduced from those ofPTF (Law et al. 2009), CFHT/Megacam, and Subaru/HSC (Miyazaki et al. 2006), respectively. For NIR wavelengths (JHK bands), “4 m” and “space” limits aretaken or deduced from those of Vista/VIRCAM and the planned limits of WFIRST (Green et al. 2012) and WISH (Yamada et al. 2012), respectively.(A color version of this figure is available in the online journal.)

6

• ブラックホール中性子星連星の合体過程は連星の質量、ブラックホールスピン、中性子星の半径などに依存する

• 特に合体直前～合体の重力波波形、質量放出、降着円盤形成などの依存性を定量的に明らかにするためには数値相対論によるシミュレーション研究が必要→これまでも多くの数値相対論シミュレーションが　行われ、合体の定量的依存性が解明されてきている　(e.g., K. Kyutoku et al. 2015, F. Foucart et al. 2016)

• 電磁波対応天体の同時観測のためには周波数依存性を含めた光度曲線の定量的予測とパラメータ依存性の解明が肝要→数値相対論の結果に基づいた輻射輸送計算による　kilonova/maronovaの光度曲線の予測が行われている　(M. Tanaka and K. Hotokezaka 2014)

これまでの研究

• これまで行われてきた多くのブラックホール中性子星連星合体の数値相対論シミュレーションでは、ブラックホールスピンと軌道運動の軸が揃った場合の着目して、そのパラメータ依存性が調べられてきた

• 連星進化計算の研究はブラックホールスピンの向きが軌道運動の軸と揃っていない（傾いた）ブラックホール中性子星連星が形成されうる事を示唆(K. Belczynski et al. 2008)

• ブラックホールスピンが傾いた系では慣性系のひきずりの効果により連星が歳差運動する→スピンの傾きは重力波波形や放出される物質、　形成される降着円盤などの性質に大きな影響を与え得る

ブラックホールスピンの傾きがブラックホール中性子連星合体からの重力波波形や降着円盤形成、質量放出に与える影響を数値相対論シミュレーションによって

（中性子星の状態方程式依存性込みで）系統的に調べた

スピンが傾いた系

SBH

MBH MNS

L

スピンが揃った系

ブラックホールスピンの傾きの影響

歳差運動

• KK, K. Kyutoku, H. Nakano, H. Okawa, M. Shibata, and K. Taniguchi, PRD (2015)

MS1i60のモデル表面:~1010g/cm3

ブラックホール

中性子星

結果：質量放出・降着円盤形成• 数値相対論シミュレーションにより降着円盤と放出される物質の質量のスピンの傾きと状態方程式依存性を定量的に明らかにした

• 大きな質量を持った物質の放出や降着円盤の形成が起こるためには、中性子星がブラックホールの最内安定軌道（ISCO)よりも十分外側で潮汐破壊される必要がある

compactness parameter C of the NS in Fig. 6. Thedependence of Mdisk and Meje is clear: Both of themdecrease monotonically with the increases of C and itilt;0.For a moderate value of compactness C ¼ 0.160, itilt;0should be smaller than 50° for Mdisk to be larger than0.1M⊙. On the other hand, Mdisk ≳ 0.1M⊙ even if itilt;0 ≈70° for a stiff EOS that realizes C ¼ 0.140. For a soft EOSwith which C ≳ 0.175,Mdisk is smaller than 0.1M⊙ for anyvalue of itilt;0. Meje > 0.01M⊙ is possible for itilt;0 < 85°with C ¼ 0.140, for itilt;0 < 65° with C ¼ 0.160, and foritilt;0 < 30° with C ¼ 0.175.

Figure 7 comparesMdisk andMeje obtained by numericalsimulations for aligned-spin BH-NS mergers [17] withthose for the misaligned-spin cases. Each line describes theresults of Mdisk and Meje for the misaligned-spin BH-NSmergers interpolated linearly for χeff. Each point in Fig. 7shows the results of the aligned-spin BH-NS mergers withthe same mass ratio ðQ ¼ 5Þ and the same EOS as weemployed in this paper, but with smaller BH spin χ ¼ 0.5.We also plot a new result for the model with Q ¼ 5, H4EOS, and χ ¼ 0.375. For both Mdisk and Meje, the resultsof the aligned-spin case agree approximately with the

TABLE V. The list of M>AH, Mdisk, Meje, vave, and veje. The subscripts 10 ms and 20 ms denote the values evaluated at ≈10 ms and≈20 ms after the onset of merger, respectively. The results for the aligned-spin case are obtained from [17]. The center dots imply thatwe were not able to take the data for them. For the model APR4i0, the simulation was stopped before t − tmerge ≈ 20 ms. For the modelAPR4i90, the masses of the disk and ejecta are so small that accurate values cannot be derived for them. For the i90 models, the data forPeje;i were not output.

Model M>AH;10 msðM⊙Þ Mdisk;10 msðM⊙Þ Mdisk;20 msðM⊙Þ Meje;10 msðM⊙Þ vave;10 msðcÞ veje;10 msðcÞAPR4i0 0.068 0.059 $ $ $ 8 × 10−3 0.26 0.099APR4i30 0.022 0.017 0.014 5 × 10−3 0.30 0.057APR4i60 4 × 10−3 2 × 10−3 2 × 10−4 1 × 10−4 0.27 0.078APR4i90 $ $ $ $ $ $ $ $ $ <10−4 0.24 $ $ $ALF2i0 0.24 0.20 0.16 0.046 0.21 0.15ALF2i30 0.16 0.13 0.10 0.033 0.27 0.17ALF2i60 0.026 0.016 0.013 0.010 0.28 0.048ALF2i90 2 × 10−4 1 × 10−4 <10−4 <10−4 0.26 $ $ $H4i0 0.32 0.27 0.21 0.050 0.22 0.18H4i30 0.25 0.21 0.19 0.042 0.27 0.21H4i60 0.084 0.072 0.061 0.012 0.25 0.14H4i90 3 × 10−3 2 × 10−3 2 × 10−3 1 × 10−3 0.28 $ $ $MS1i0 0.36 0.28 0.22 0.079 0.24 0.19MS1i30 0.30 0.23 0.19 0.070 0.28 0.23MS1i60 0.18 0.14 0.11 0.041 0.27 0.21MS1i90 0.022 0.012 0.011 0.010 0.27 $ $ $

FIG. 6 (color online). The contour for Mdisk (left panel) and Meje (right panel) evaluated at ≈10 ms after the onset of merger in theplane of the NS compactness C and initial value of itilt.

KYOHEI KAWAGUCHI et al. PHYSICAL REVIEW D 92, 024014 (2015)

024014-12

質量　大

質量　大

放出質量（太陽質量単位）降着円盤質量（太陽質量単位）

CC

• スピンの傾きは実効的にスピンの効果を弱めるためスピンの傾きは降着円盤質量や放出質量を減少させる

�eff = �cositilt

Q :=MBH

MNS

質量比

C :=MNS

RNS

コンパクトネス

� :=SBH

M2BH

スピンパラメータスピンの傾き

itilt := cos

�1

✓SBH · LSBHL

◆

rISCO . rtidal潮汐半径

単調減少関数

() rISCO

MBH(�) . C�1Q�2/3

ISCO半径

itilt

Lorb

SBH

Kilonova/macronovaのパラメータ依存性

• 先行研究(M. Tanaka and K. Hotokezaka 2014)による数値相対論の結果に基づいた輻射輸送計算ではいくつかの限られたパラメータのブラックホール中性子星連星合体について、kilonova/maronovaの光度曲線の予測が行われている。数値相対論シミュレーションや輻射輸送計算は計算コストが高く、ブラックホール中性子星連星合体のパラメータ空間全てをカバーするのは難しい（大変）。そこで

• i) 連星合体から放出される物質の質量と速度のフィッティング式

• ii) Kilonova/Macronovaの光度曲線の半解析モデルをこれまでのシミュレーション結果を用いて導出し、組合わせることで広いパラメータ領域でのkilonova/macronovaのパラメータ依存性を調べた

Light curve modelFitting formula

NR simulation

Parameter study

RT simulation

Calibration /Validity check

Mej,fit (Q,�, itilt,MNS,EOS)

L (t;Mej, vave)

• KK, K. Kyutoku, M. Shibata, and M. Tanaka, ApJ (2016)

放出物質のフィッティング式3

bration, we only apply the fitting formulas for the casethat �e↵ 0.9.

We determine the fitting parameters from the simula-tion data using the least squares method. The best–fitvalues for the parameters were obtained as follows:

a1 =4.464 ⇥ 10�2,

a2 =2.269 ⇥ 10�3,

a3 =2.431,

a4 =�0.4159,

n1 =0.2497,

n2 =1.352 (4)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Mej

,fit[M

sun]

Mej[Msun]

0 20 40 60 80

100

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

6M

ej/M

ej[%

]

Mej[Msun]

6Mej,NR/Mej26Mej,NR/Mej

Fig. 1.— The comparison of the ejecta mass fitting formula withthe results of numerical simulations. Each point in the top panelshows the ejecta mass derived by simulations listed in Table 2(horizontal axis) and the fitting model using corresponding binaryparameters (vertical axis). The errors of the data are esti-mated using equation (5). The bottom panel shows com-parison of the estimated relative error of the data and therelative residual error of the ejecta mass fitting model with thebest–fit parameters.

Figure 1 plots the comparison of the ejecta mass fittingformula with the results of numerical simulations andthe relative fitting error of the data as a function of theejecta mass. The data for Mej � 0.05M� is fitted within⇠ 20%, while the data with 0.04M� and 0.02M� are fit-ted within ⇠ 30% and ⇠ 40%, respectively. Because theresults obtained by the simulations include errors due tonumerical discretization, some dispersion is unavoidableeven if the fitting model is appropriate. Since only lim-ited data in Table 2 were published with expliciterror measurement, we assume the estimated nu-merical error of the simulation data by

�Mej,NR =q

(0.1 Mej)2 + (0.005M�)2 (5)

referring to the estimated numerical error discussed inthe Appendix of Kyutoku et al. (2015) and Kawaguchiet al. (2015). Figure 1 shows that the errors of thefitting are consistent with these estimated errors.The error of the ejecta mass induces the relative error in

the peak luminosity only by about a half of its relativeerror because the luminosity of the kilonova/macronova

is approximately proportional to M1/2ej in the early phase.

This error in the peak luminosity is comparable to oreven smaller than the systematic error of the model ofkilonova/macronova lightcurves as we see below. Thereduced �2 for this fit is defined as,

�2 =1

NNR � Np � 1

NNR

X

i=1

✓

Mej � Mej,fit

�Mej,NR

◆2

, (6)

where NNR = 45 and Np = 6 are the number of thedata and the parameter. �2 for the model of equa-tion (1) is 0.85, and �2 become larger than 1 whenwe reduced the number of the parameter, such asa3, a4, n1 and n2. Thus, we refrain from increasing thenumber of parameters in this paper to improve the fit-ting accuracy, in order to avoid the simulation data tobe over–fitted. We note that this fitting formulacould have systematic errors due to the choice ofthe NS EOSs that are used for the fitting. For ex-ample, we should check whether our fitting for-mula can appropriately predict the ejecta massfor two EOSs that give the same NS compactnessbut give di↵erent MNS,⇤ with the same MNS. Thiscan only be checked by testing the fitting modelwith the data using EOS which is not used in thispaper. We keep it as a future task.

We also derive a fitting model for the averaged velocityof the ejecta as a simple linear model of Q:

vave = (0.01533 Q + 0.1907) c. (7)

The relative error of the fitting is always within 10%.However, as is discussed in Kyutoku et al. (2015),we expect that the ejecta velocity measured inthe numerical–relativity simulation can be over-estimated by ⇠ 20%, and thus, the relative errorin the velocity fitting formula can be ⇠ 30%. Thiserror in ejecta velocity can cause ⇠ 15% relativeerror in tc and the bolometric luminosity in thelightcurve model introduced below, which onlyweakly a↵ects the following discussions.

2.2. Kilonova/Macronova

Here, we derive a model for the kilonova/macronovafrom the anisotropic ejecta with velocity distribution re-sulting from BH-NS mergers in reference to the kilo-nova/macronova model introduced in Piran et al.(2013); Kisaka et al. (2015). Rosswog et al. (2013);Kyutoku et al. (2013, 2015); Kawaguchi et al.(2015) showed that, the ejecta expands homologouslyand exhibits a crescent–like shape in most cases for BH-NS mergers (see Figure 2). We describe this morphologyof the ejecta by modeling the ejecta shape as a partialsphere in the longitudinal and latitudinal directions asshown in Figure 3. We employ spherical coordinates set-ting the ejecta on the equatorial plane. The latitudinalcoordinate ✓ is measured from the equatorial plane. Dueto the homologous expansion, each shell with the sameradius has a velocity v = r/t, where t is the elapsed timeafter the merger.

Kyutoku et al. (2013) and Kyutoku et al. (2015)showed that, the opening angle of its arc is typically

12

TABLE 2The list of results obtained by recent numerical–relativity simulations: Q, �, i

tilt

, Mej

, and vave

are the mass ratio of thebinary, the dimensionless spin parameter of the BH, the initial misalignment angle between the BH spin and the orbital

angular momentum, the ejected mass, and the mass–weighted root mean square of the ejecta velocity distribution,respectively. “New” in the “Ref” column denotes the data obtained by our new numerical simulations.

ID Q � itilt

[�] EOS Mej

vave

[c] Ref

1 3 0 0 APR4 < 1⇥ 10�3 0.20 [1]2 3 0 0 ALF2 0.003 0.22 [1]3 3 0 0 H4 0.006 0.22 [1]4 3 0 0 MS1 0.02 0.23 [1]5 3 0.5 0 APR4 0.002 0.21 [1]6 3 0.5 0 ALF2 0.02 0.24 [1]7 3 0.5 0 H4 0.03 0.23 [1]8 3 0.5 0 MS1 0.05 0.24 [1]9 3 0.75 0 APR4 0.01 0.23 [1]10 3 0.75 0 ALF2 0.05 0.25 [1]11 3 0.75 0 H4 0.05 0.24 [1]12 3 0.75 0 MS1 0.07 0.25 [1]13 3 0.75 31 H4 0.03 0.22 New14 3 0.75 62 H4 0.02 0.24 New15 3 0.75 93 APR4 < 1⇥ 10�3 0.21 New16 3 0.75 93 H4 0.006 0.22 New17 5 0.5 0 APR4 < 1⇥ 10�3 0.23 [1]18 5 0.5 0 ALF2 0.01 0.27 [1]19 5 0.5 0 H4 0.02 0.26 [1]20 5 0.5 0 MS1 0.05 0.27 [1]21 5 0.75 0 APR4 0.008 0.25 [1]22 5 0.75 0 ALF2 0.05 0.28 [1]23 5 0.75 0 H4 0.05 0.27 [1]24 5 0.75 0 MS1 0.08 0.28 [1]25 5 0.75 33 APR4 0.005 0.30 [2]26 5 0.75 33 ALF2 0.03 0.27 [2]27 5 0.75 33 H4 0.04 0.27 [2]28 5 0.75 32 MS1 0.07 0.28 [2]29 5 0.75 63 APR4 0.001 0.27 [2]30 5 0.75 63 ALF2 0.007 0.28 [2]31 5 0.75 63 H4 0.01 0.25 [2]32 5 0.75 63 MS1 0.01 0.27 [2]33 5 0.75 94 APR4 < 1⇥ 10�3 0.24 [2]34 5 0.75 94 ALF2 < 1⇥ 10�3 0.26 [2]35 5 0.75 94 H4 0.001 0.28 [2]36 5 0.75 93 MS1 0.01 0.27 [2]37 7 0.5 0 APR4 < 1⇥ 10�3 0.23 [1]38 7 0.5 0 ALF2 < 1⇥ 10�3 0.27 [1]39 7 0.5 0 H4 0.003 0.29 [1]40 7 0.5 0 MS1 0.02 0.30 [1]41 7 0.75 0 APR4 < 1⇥ 10�3 0.27 [1]42 7 0.75 0 ALF2 0.02 0.29 [1]43 7 0.75 0 H4 0.04 0.29 [1]44 7 0.75 0 MS1 0.07 0.30 [1]45 7 0.75 33 H4 0.03 0.28 New

—. 2014b, Phys. Rev. D, 89, 102005Piran, T., Nakar, E., & Rosswog, S. 2013, MNRAS, 430, 2121Read, J. S., Lackey, B. D., Owen, B. J., & Friedman, J. L. 2009,

Phys. Rev. D, 79, 124032Roberts, L. F., Kasen, D., Lee, W. H., & Ramirez-Ruiz, E. 2011,

ApJ, 736, L21Rosswog, S. 2005, ApJ, 634, 1202Rosswog, S., Piran, T., & Nakar, E. 2013, MNRAS, 430, 2585Somiya, K. 2012, Classical and Quantum Gravity, 29, 124007Takami, H., Kyutoku, K., & Ioka, K. 2014, Phys. Rev. D, 89,

063006

Tanaka, M., & Hotokezaka, K. 2013, ApJ, 775, 113Tanaka, M., Hotokezaka, K., Kyutoku, K., et al. 2014, ApJ, 780,

31Tanvir, N. R., Levan, A. J., Fruchter, A. S., et al. 2013, Nature,

500, 547Wanajo, S., Sekiguchi, Y., Nishimura, N., et al. 2014, ApJ, 789,

L39Yang, B., Jin, Z.-P., Li, X., et al. 2015, Nature Communications,

6, 7323

qh�M2

ejei ⇡ 0.005M�p

h�v2avei ⇡ 0.01c

最新のブラックホール中性子星連星合体の数値相対論シミュレーション結果（45モデル）(K. Kyutoku et al. 2015 & KK et al. 2015)をもとに、連星合体のパラメータから放出物質の質量と速度を与えるフィッティング式を導出した。

vej,fit (Q)Mej,fit (Q,�, itilt,MNS,EOS)

光度曲線モデル

'ej = ⇡, ✓ej = 1/5, ✏th = 0.5

= 10 cm2 g�1

(ref. S. Wanajo et.al 2014)

• ブラックホール中性子星連星合体の数値相対論シミュレーションから得られた放出物質の形状、速度分布の知見を取り入れ、bolometric luminosityの現象論的モデルを導出した

✏̇(t) = 1.6⇥ 1010erg g�1 s�1

✓t

day

◆�1.2

L(t) = (1 + ✓ej

) ✏th

✏̇(t)Mobs

(t)

Mobs

(t) =

⇢M

eje

ttobs

t tobs

,M

eje

t > tobs

.

simulations in [Tanaka et al. 2014] that the emis-sion from the radial edge is smaller than the theone from the latitudinal truncation by an order(See Fig. 8 in [Tanaka et. al 2014].). Consideringthe random walk of the photon, the depth of thevisible mass is determined by the condition thatthe distance to the latitudinal edge is compara-ble to the distance that a photon di↵uses, namelyvt(✓ej � ✓) ⇡ ct/⌧ . Here, ⌧ ⇡ ⇢vt(✓

ej

� ✓) isan optical depth measured from the cross sectionof latitudinal truncation. From this condition, weobtain the depth of the visible mass ✓obs as,

✓obs(t) = ✓ej �✓

2'

ej

✓ej(vmax � vmin)c

Meje

◆1/2

t,

=: ✓ej

✓

1 � t

tobs

◆

, (8)

where we set tobs =⇣

✓

ej

M

eje

2'ej(vmax

�v

min

)c

⌘1/2

. Using

this result, the mass being visible is given as,

Mobs(t) = Meje �Z

'

ej

0

d'

Z

✓

obs

(t)

�✓

obs

(t)

sin✓d✓

Z

v

max

v

min

dvv

2t

3⇢,

= Mejet

tobs. (9)

At t = tobs, the whole ejecta becomes visible.Thus at the latter time (t > tobs), Mobs(t) = Meje.

As is treated in [Piran et. al 2013], we as-sume that the observed luminosity is dominatedby the energy release via radioactive decay [Ko-robkin et. al 2012]. According to [Korobkinet. al 2012, Wanajo et. al 2014], the spe-cific heating rate is approximately given by apower law ✏̇(t) ⇡ ✏̇0 (t/day)�↵, and we set ✏̇0 =1.58 ⇥ 1010erg g�1 s�1 and ↵ = 1.2 by referringto [Tanaka et al. 2014]. The resulting bolometricluminosity is given as,

L(t) =

8

<

:

(1 + ✓ej) ✏th✏̇0Mejet

t

obs

⇣

t

day

⌘�↵

t tobs,

(1 + ✓ej) ✏th✏̇0Meje

⇣

t

day

⌘�↵

t > tobs.

(10)Here, ✏th is the e�ciency of thermalization in-troduced in [Korobkin et. al 2012]. The factor(1 + ✓ej) is put to describe the contribution fromthe radial truncation.

In order to check the validity of the analyticmodel obtained above, we compare this modelwith the results obtained by radiation transfer

1e+39

1e+40

1e+41

1e+42

1 10

Bolo

met

ric L

umin

osity

[erg

/s]


APR4Q3a75H4Q3a75

MS1Q3a75APR4Q3a75-model

H4Q3a75-modelMS1Q3a75-model

Fig. 2.— The dashed lines denote light curvesof Macronova/Kilonova predicted by the radia-tion transfer simulation performed in[Tanaka et al.2014] . The solid lines denote the light curves pre-dicted by the analytic model obtained in Sec. 2.2with the corresponding model parameters.

simulations in [Tanaka et. al 2014]. Particu-larly, we focus on the cases for “APR4Q3a75”,“H4Q3a75” and “MS1Q3a75” in [Tanaka et. al2014]. We plot the light curves predicted by theanalytic model in Fig. 2. Here, we set 'ej = ⇡,✓ej = 1/5, vmin = 0.02c and ✏

th

= 0.5 accord-ing to [Kyutoku et. al 2015, Piran et. al 2013].(Meje, vmax) are set to be (1 ⇥ 10�2

M�, 0.41c),(5 ⇥ 10�2

M�, 0.35c) and (5 ⇥ 10�2M�, 0.42c) for

“APR4Q3a75”, “H4Q3a75” and “MS1Q3a75”, re-spectively. These values of vmax are chosen so thatmass weighted mean root squares of the velocitydistribution agree with the values of vch in [Tanakaet. al 2014].

Comparing Fig. 2 with Fig. 3 in [Tanaka et. al2014], we found that the light curves of the ana-lytic model agree well with the results of radiationtransfer simulations within a factor of ⇠ 1.2.

Because we can not know the spectra from thisanalytic model, we need to make additional treat-ment to predict the magnitude in each band fre-quency, which can be obtained more directly bythe observation. However, the spectra of the kilo-

4

• 輻射輸送計算の計算結果を用いて各バンド等級(u,g,r,i,z,J,H,K ̶バンド)の時間発展についても半解析的なモデルを導出

MX (t;Mej, vej) (X = u, g , r , i , z , J ,H ,K )

of artificial atmosphere, which is inevitable in conserva-tive hydrodynamics schemes. The atmospheric density isat most 103 g cm!3, and negligible for ejecta (see Fig. 1).Indeed, we confirmed that ejecta properties depend veryweakly on the atmospheric density as far as it is suffi-ciently low. Specifically, varying the atmospheric densityby an order of magnitude changes the ejecta propertiesonly by 10%–20%. Finally, we improve grid resolutionsby "20% so that the NS radius is covered by #50 gridpoints in the highest resolution runs. The radius of theejecta is always covered by #50–60 grid points in theequatorial plane, and #10 grid points in the perpendicu-lar direction (see the next section). We perform simula-tions with 3 different resolutions for selected models, andestimate that ejecta properties are accurate within #10%for many cases and within a factor of 2 for the worstcases in which the ejecta mass is small. This accuracy issufficient for the purpose of this article, which mainlydiscuss qualitative signatures. A convergence study willbe presented in a separate paper with detailed discussionsof our systematic simulations.

III. MASS EJECTION

When the NS is disrupted by the BH tidal field, a one-armed spiral structure called the tidal tail is formed aroundthe BH. Although a large part of the tail eventually fallsback onto the remnant disk and BH, its outermost partobtains a sufficient angular momentum and kinetic energyto become unbound via hydrodynamic angular momentumtransport processes. Dynamical mass ejection from the BH-NSmerger is driven dominantly by this tidal effect. We alsofind that some material in the vicinity of the BH becomesunboundwhen the tidal tail hits itself as it spirals around theBH. This ejection may be ascribed to the shock heating, butthis shock-driven component is always subdominant.Ejecta exhibits a crescentlike shape as depicted in Fig. 1

in most cases. Specifically, a typical opening angle of theejecta in the equatorial plane is ’ej # !. Such a nonax-isymmetric shape arises because the sound-crossing timescale is shorter than the orbital period at the onset of tidaldisruption. Furthermore, ejecta spreads dominantly in theequatorial plane and expands only slowly in the directionperpendicular to the equatorial plane (hereafter, thez-direction). The reason for this is that the ejection isdriven mainly by the tidal effect, which is most efficientin the equatorial plane. Thus, a portion of circumferentialmaterial will be subsequently swept by the ejecta. A typicalhalf opening angle of the ejecta around the equatorial planeis "ej # 1=5 radian; and, this implies that the ejecta veloc-ity in the equatorial plane vk is larger by a factor of1="ej # 5 than that in the z-direction v?. Here, vk maybe identified with the radial velocity, and the azimuthalvelocity should become negligible soon after the ejectiondue to the angular momentum conservation. Indeed,azimuthal velocity is very small in Fig. 1. Aside from theejecta itself, the region above the remnant BH is muchclearer than that for the NS-NS merger, and thus thebaryon-loading problem of GRB jets may be less severe.The ejecta mass Mej depends on binary parameters and

NS EOSs and are typically in the range of "0:01–0:1M$when the tidal disruption occurs and the disk mass Mdisk

exceeds"0:1M$. Important values are shown in Table I forrepresentative models in which Mdisk * 0:1M$. The valueofMej is generally largewhen theNSEOS is stiff and theNSradius is large for fixed values ofQ and#, because the massejection is driven primarily by the tidal effect. This depen-dence on the NS EOS and radius is opposite to the case oftheNS-NSmerger in general relativity, where rapid rotationand oscillation of a remnant massive NS drive ejection[18,19]. We speculate that ejecta from the BH-NS mergermight account for a substantial portion of the r-processnucleosynthesis (see also below) compared to that fromthe NS-NS merger if the realistic NS EOS is stiff, andvice versa.When theNS is not disrupted prior to themerger,the ejecta mass is negligible for current astrophysical inter-est. The ejecta mass is always smaller than the disk mass,and a relation Mej # 0:05–0:25Mdisk approximately holds

-1000 0 1000x (km)

y (k

m)

5

v=0.5c

6

7

8

9

10

-1500 -1000 -500 0 500 1000 1500x (km)

0

500

1000

1500

z (k

m)

5

6

7

8

9

10 v=0.5c

-1000

0

1000

FIG. 1 (color online). The rest-mass density ($) profiles ofejecta overplotted with the velocity at #10 ms after the mergerfor Q ¼ 5, # ¼ 0:75, and H4 EOS model. We only show un-bound material to elucidate geometry of the ejecta, and the blankregion between the ejecta and BH is filled with unshown boundmaterial. The top and bottom panels are for the xy- andxz-planes, respectively. The color panel on the right of eachplot is log 10$ in g cm!3. The region above "500 km in thebottom panel is much clearer than that in a typical NS-NSmerger (see the corresponding panels of Figs. 3–5 in [18]).

KOUTAROU KYUTOKU, KUNIHITO IOKA, AND MASARU SHIBATA PHYSICAL REVIEW D 88, 041503(R) (2013)

RAPID COMMUNICATIONS

041503-2

Ref: K. Kyutoku et al. 2013

加熱率

フィッティング式+光度曲線モデル

今回導出したモデルは各バンドについて～１(AB)等級程度の範囲でM. Tanaka and K. Hotokezaka 2014の射輸送輸送計算の結果を再現できた

rate may be sources of error in the lightcurve model and theradiation-transfer simulations, as we discuss in Section 5.

3. IMPLICATIONS FOR OBSERVING STRATEGIES

Once the detection of gravitational waves from a compactbinary merger is achieved, the chirp mass

[ ( )]% � �Q Q M1chirp3 1 5

NS, the symmetric mass ratio

( )O � �Q Q1 2, and the effective spin parameterχeff = χ cos itilt of the binary will be estimated (e.g.,O’Shaughnessy et al. 2014a, 2014b). Assuming a BH–NSmerger event, the analytic model we constructed in this paperpredicts the brightness and the duration of the kilonova/macronova for detected events. Tanaka et al. (2014) pointedout that observations in the i-band filter of wide-field 8 m class

Table 3The Bolometric Corrections for ugrizJHK-band Magnitudes Calculated from the Results of the Model APR4Q3a75 in Tanaka et al. (2014)

Rescaled Time Bolometric Correction ΔMX (mag) (X: Band Filter)

( )( ) :t MMday

0.01 1 3.2

ej X = u g r i z J H K

1.5 −0.28 −0.45 −0.47 −0.71 −0.97 −1.61 −2.37 −4.552.0 −0.19 −0.57 −0.59 −0.32 −0.48 −0.07 1.18 0.412.5 0.25 −0.13 −0.20 0.05 −0.28 −1.50 0.86 2.223.0 0.93 0.93 0.60 0.56 0.37 −1.98 −4.29 −3.783.5 0.47 1.25 1.36 1.26 1.04 −1.38 −4.65 −6.364.0 −0.34 1.29 1.20 1.60 1.44 −0.74 −3.31 −6.124.5 0.03 0.74 1.15 1.51 1.65 −0.29 −2.33 −5.255.0 −0.39 0.37 0.89 0.99 1.76 0.42 −1.73 −4.105.5 −0.69 0.68 0.74 0.44 1.13 0.94 −0.69 −3.576.0 −1.21 0.41 0.74 0.50 1.15 1.05 −0.66 −3.556.5 −3.65 −0.58 1.05 0.94 1.28 1.66 −0.67 −3.767.0 −4.40 −0.87 −0.16 1.04 1.82 1.46 −0.07 −3.687.5 −1.74 −2.55 0.14 1.30 2.09 1.39 0.01 −3.248.0 −1.36 −2.87 0.15 1.31 2.11 1.38 0.04 −3.118.5 −3.75 −1.90 −0.35 1.16 2.26 1.53 0.15 −2.409.0 −4.70 −1.57 −0.57 1.08 2.29 1.59 0.21 −2.109.5 −3.93 −0.36 0.62 1.24 2.50 2.31 0.31 −1.3410.0 −4.00 −0.44 0.56 1.20 2.47 2.29 0.32 −1.3010.5 3.18 −0.52 0.84 1.34 2.74 2.43 0.30 −1.1111.0 L −0.60 1.97 1.92 3.79 2.93 0.20 −0.5411.5 L −0.68 1.89 1.87 3.73 2.90 0.21 −0.5112.0 L 0.29 1.59 2.14 3.74 2.93 0.43 0.3312.5 L 0.53 1.44 2.18 3.70 2.91 0.50 0.6113.0 L 0.43 1.36 2.12 3.64 2.89 0.51 0.6313.5 L L 2.05 3.59 3.00 3.00 1.27 1.0414.0 L L 2.19 3.97 2.77 3.02 1.49 1.17

Figure 6. Comparison of the ugriz-band (left) and the JHK-band (right) AB magnitudes of the results obtained in Tanaka et al. (2014) (dashed lines) and the lightcurvemodels that are derived from the ejecta mass model and the bolometric correction model (solid lines) for H4Q3a75.

7

The Astrophysical Journal, 825:52 (12pp), 2016 July 1 Kawaguchi et al.

• 重力波によって制限されると期待される連星合体のパラメータ対して、どういった場合にkilonova/macronovaが観測され得るかを明らかにした。

Kilonova/macronovaのパラメータ依存性

�e↵ & 0.67 (Mch/M�)� 1.45

9

Effe

ctiv

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Mass Ratio Q

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Mass Ratio Q

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0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

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Mass Ratio Q

0

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0.2

0.3

0.4

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0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

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in P

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Mass Ratio Q

0

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0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

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0

0.1

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0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

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3 4 5 6 7 8 9

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3 4 5 6 7 8 9

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3 4 5 6 7 8 9

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0.9

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3 4 5 6 7 8 9

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3 4 5 6 7 8 9

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3 4 5 6 7 8 9

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0.9

2 2.2 2.4 2.6 2.8 3

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t= 3 days t= 7 days t= 14 days

AP

R4 H

-ban

d D

=400

Mpc

[M�][M�][M�]

AP

R4 i-b

and

D=4

00 M

pc

[M�] [M�] [M�]

H4 i-b

and

D=4

00 M

pc

[M�] [M�] [M�]

H4 H

-ban

d D

=400

Mpc

[M�][M�][M�]

22–23

23–24

24–25

24–25

25–26

26–27

23–24

25–26

26–27

27–28

>28

>2827–2825–26

26–2723–24

24–25

25–2624–25

26–2725–26

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

Fig. 8.— Expected i and H–band AB magnitudes as functions of the chirp mass Mch

(lower horizontal axis) or the mass ratio Q (upperhorizontal axis) and the e↵ective dimensionless spin parameter �

e↵

. The mass of the NS is set to be MNS

= 1.35M� and the distance tothe BH–NS binary is set to be 400 Mpc.

9

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ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

Mass Ratio Q

t= 3 days t= 7 days t= 14 days

AP

R4 H

-ban

d D

=400

Mpc

[M�][M�][M�]

AP

R4 i-b

and

D=4

00 M

pc

[M�] [M�] [M�]

H4 i-b

and

D=4

00 M

pc

[M�] [M�] [M�]

H4 H

-ban

d D

=400

Mpc

[M�][M�][M�]

22–23

23–24

24–25

24–25

25–26

26–27

23–24

25–26

26–27

27–28

>28

>2827–2825–26

26–2723–24

24–25

25–2624–25

26–2725–26

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

(Mej

. 10�3 M�

)

Fig. 8.— Expected i and H–band AB magnitudes as functions of the chirp mass Mch

(lower horizontal axis) or the mass ratio Q (upperhorizontal axis) and the e↵ective dimensionless spin parameter �

e↵

. The mass of the NS is set to be MNS

= 1.35M� and the distance tothe BH–NS binary is set to be 400 Mpc.

MNS = 1.35M�D = 400Mpc.

kilonova/macronova magnitudes and its dependence on thebinary parameters by using the model derived in Section 2. InSection 4, we apply our model to GRB 130603B as anillustration. Finally, a summary and remarks on this work arepresented in Section 5. Our notation for physically importantquantities is as follows: the BH mass, MBH; the NS mass, MNS;the ejecta mass,Mej; the mass-weighted root mean square of theejecta velocity, vave; the mass ratio, Q = MBH/MNS; thedimensionless spin parameter of the BH, D � cS GMBH BH

2 ; theangle between the BH spin and the orbital angular momentum,itilt; and the compactness of the NS, � � GM c RNS

2NS. G and c

denote the gravitational constant and the speed of light,respectively. The BH and the NS masses are the Arnowitt-Deser-Misner (ADM) masses at infinite separation.

2. MODELS

In this section we derive fitting formulas for the mass andvelocity of dynamical ejecta from BH–NS mergers, and ananalytic model for the lightcurve of a kilonova/macronova. Weonly consider the dynamical ejecta which are ejected during themerger process. The ejecta from the BH-accretion disk system,which could be formed in the post-merger phase, are not takeninto account. We remark on the effect of this additional ejectacomponent in Section 5.

2.1. Ejecta Mass

Foucart (2012) introduced a fitting formula for the massremaining outside the remnant BH after BH–NS mergers. Thisremaining mass includes both the remnant disk mass and ejectamass. As the ejecta mass is defined as the gravitationallyunbound component of the remaining mass, we expect that asimilar form of fitting model can also be useful for the ejectamass. Therefore, by referring to Foucart (2012), we propose afitting model for the ejecta mass as

{ ( )

˜ ( )

( )

� �

D

� �

�

� � �

�

�

�⎪

⎛⎝⎜

⎞⎠⎟

⎫⎬⎭

M

Ma Q

a Q r

aM

Ma

Max 1 2

1 , 0 , 1

n

n

ej

NS,1

1

2 ISCO eff

3NS

NS,4

1

2

( )D D� icos , 2eff tilt

and

˜ ( )( ) ( )( )

( ){( ) ( ) }

( )

D

DDD D

D

� �

� � � �

� � �q � � �

� �

r Z

Z Z Z

Z

Z Z

3

sign 3 3 2 ,

1 11 1 ,

3 . 3

ISCO 2

1 1 2

12 1 3

1 3 1 3

22

12

We generalize the model of Foucart (2012) as follows: (i) weset the exponents of Q, i.e., n1 and n2, as fitting parameters and(ii) we add a term proportional to the specific binding energy ofthe NS, � �M M1 NS NS, , where �MNS, is the total baryon massof the NS.

We use the results of recent numerical-relativity simulationsperformed by the Kyoto group (Kawaguchi et al. 2015;Kyutoku et al. 2015) to determine the fitting parameters, whichare summarized in Table 2. We use the data for the BH–NS

mergers with various mass ratios, Q, BH spin magnitudes, χ,BH spin orientations, itilt, and NS equations of state (EOSs). Inthe simulations, four phenomenological EOSs described inRead et al. (2009) are employed. We list the key quantities of aNS with the EOS models which are employed in the numerical-relativity simulations in Table 1. Table 2 lists the new results ofnumerical-relativity simulations. These new simulations andthe computations for initial data are performed using the samemethod as in Kawaguchi et al. (2015). We note that the NSmass is fixed to be 1.35Me for all the models. Lovelace et al.(2013) show that very massive ejecta (Mej 0.2Me) can beproduced in the case where the BH spin is extremely high (χeff∼ 0.97). However, because Lovelace et al. (2013) is, so far, theonly study for such a rapidly spinning case, and we do not useits data for the calibration, we only apply the fitting formulasfor the case where χeff � 0.9.We determine the fitting parameters from the simulation data

using the least squares method. The best-fit values for theparameters were obtained as follows:

( )

� q� q��

�

�

aaaann

4.464 10 ,2.269 10 ,2.431,

0.4159,0.2497,1.352. 4

12

23

3

4

1

2

Figure 1 plots the comparison of the ejecta mass fittingformula with the results of numerical simulations and therelative fitting error of the data as a function of the ejecta mass.The data for Mej � 0.05Me are fitted within ∼20%, while thedata with 0.04Me and 0.02Me are fitted within ∼30% and∼40%, respectively. Because the results obtained by thesimulations include errors due to numerical discretization,some dispersion is unavoidable even if the fitting model isappropriate. Since only limited data in Table 2 were publishedwith explicit error measurements, we assume the estimatednumerical error of the simulation data using

( ) ( ) ( ):% � �M M M0.1 0.005 5ej,NR ej2 2

referring to the estimated numerical error discussed in theappendix of Kyutoku et al. (2015) and Kawaguchi et al. (2015).Figure 1 shows that the errors of the fitting are consistent withthese estimated errors. The error of the ejecta mass induces therelative error in the peak luminosity only by about a half of itsrelative error because the luminosity of the kilonova/macro-nova is approximately proportional to Mej

1 2 in the early phase.This error in the peak luminosity is comparable to or evensmaller than the systematic error of model of kilonova/

Table 1The Key Quantities of a NS with the Piecewise Polytropic EOSs (Read

et al. 2009) Employed in the Numerical-relativity Simulations

Model R1.35(km) M∗,1.35(Me) �1.35

APR4 11.1 1.50 0.180ALF2 12.4 1.49 0.161H4 13.6 1.47 0.147MS1 14.4 1.46 0.138

Note. R1.35, M∗,1.35, and �1.35 are the radius, the Baryon rest mass, and thecompactness parameter for the isolated NS with MNS = 1.35 Me, respectively.

2

The Astrophysical Journal, 825:52 (12pp), 2016 July 1 Kawaguchi et al.

Mchirp =⇥Q3/(1 +Q)

⇤1/5MNS

RNS=

13.6

km

Application to GRB130603B10

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

MS1

Mass Ratio Q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

MS1

Mass Ratio Q

[M�] [M�]

[M�] [M�]

�MJ < 1 mag

�MJ<

2 mag

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

H4

Mass Ratio Q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

H4

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

ALF2

Mass Ratio Q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

ALF2

Mass Ratio Q

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

APR4

Mass Ratio Q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 2.2 2.4 2.6 2.8 3

3 4 5 6 7 8 9

Effe

ctiv

e Sp

in P

aram

eter

ref

f

Chirp Mass Mchirp

APR4

Mass Ratio Q

Fig. 9.— Allowed parameter region for explaining the observation of GRB130603B. The results for four EOSs are shown. The blueand light–blue colored regions show the parameter space in which the di↵erence of J–band magnitude between the prediction and theobservation of GRB130603B obtained by HST is within 1 mag and 2 mag, respectively.

t ⇠ 7 daysMJ ,abs = �15.7 mag

• red shift: z=0.356

• An excess in H-band from after glow model is observed.-25.3 mag, t=9.4 days by HST→

NOTE: 違う電磁波放射源（e.g. disk wind)の存在が結果を変え得る

�MJ = MJ ,abs �MJ ,model

• GRB130603B: Kilonova association candidate(Tanvir et al. 2013)

• GRB130603B に BHNS連星合体からの kilonova/macronova 付随しているとし、今回導出したモデルが~1等級の範囲で正しいとすると連星のパラメータは青い領域に制限される。

@ GRB rest frame

11.1 km

13.6 km

12.4 km

14.4 km

中性子星半径

• スピンが傾いたブラックホール中性子星連星合体を数値相対論シミュレーションにより系統的に調べ、連星合体のパラメータ依存性の定量的理解を、歳差運動するような系までひろげた

• kilonova/macronovaの源となる質量放出について、スピンが傾いた系を含めた広いパラメータ空間で有効なフィッティング式を導出した。また、これを輻射輸送計算をよく再現する半解析的モデルと組み合わせる事でkilonova/macronovaのパラメータ依存性を明らかにした※（現在、放出物質の加熱率や不透明度には不定性が存在する）

• スピンパラメータが1に近いようなブラックホール中性子連星合体については、まだ定量的（定性的にも？）な理解が不十分である

• 合体時に放出される物質以外にも、合体後降着円盤から放出されると期待されるアウトフローもkilonova/macronovaの源になりえる。これらの成分はこの研究では考慮されておらず、今後の課題である

これまでの研究のまとめ

これからの研究

重力波テンプレート作成に向けた高精度シミュレーション

• 中性子星はブラックホールからの潮汐力によって潮汐変形を起こし、重力波の位相進化に影響をあたえる。潮汐変形に伴う位相の変化から中性子星の変形度合いを読み取る事によって中性子星の半径や状態方程式に制限をかけることができる

• 実際の観測でノイズに埋もれた重力波のシグナルの中から中性子星の潮汐変形の効果を読み取るためには、高精度長時間の波形モデル（テンプレート）が必要。特に潮汐変形が最も強くあらわれる合体直前の波形については数値相対論による予測が必要不可欠→高解像度シミュレーションによるブラックホール中性子連星合体の重力波波形の作成

潮汐変形

��TidalGW (t)

⇤ ⇠✓c2RNS

GMNS

◆5

（無次元）潮汐変形度位相の変化

Preliminary Result• 現在、スピンしていないブラックホール中性子星連星合体の高解像度(Δx=160m)、長時間（~35周期）のシミュレーションをはじめている。系統誤差をコントロールするために解像度を変えたいくつも計算を行いその収束性を調べている段階である。

-0.2

-0.1

0

0.1

0.2

0 20 40 60 80 100

r A/m

0 h

(l=2,

m=2

)

tret [ms]

N=72 Extrapolated

+x

Prelimin

ary

MBH = 4.05 M�, MNS = 1.35 M�, � = 0, EOS = H4(RNS = 13.6 km)

今後の方針

19

• 実際の観測では数値相対論だけでは計算時間の観点から必要とされる長時間の波形モデルを作るのは難しい。そこで重力波波形の解析的モデルとの組み合わせによりこれを実現する

• ハブリッド化の系統誤差をおさえるためには、可能な限り合体直前まで正確な解析モデルを使用する必要がある。そこで現在、最も有望な重力波波形の現象論的モデルであるEffective-one body モデルとのハブリッド化を考えている

• 来春から、ポツダムのAlbert Einstein Institute (マックスプランク)でEOBの研究者と共同研究を行う予定

数値相対論波形（合体直前～合体）

EOB波形（合体前）

ハイブリッド波形

重力波観測に向けた長時間、高精度な波形を実現

長時間○精度△

長時間×精度○

重力波波形

-8e-23

-6e-23

-4e-23

-2e-23

0

2e-23

4e-23

6e-23

8e-23

5 10 15 20 25 30

h[D

=100

Mpc

]

t[ms]

H4, i=0, theta=0

+x

J

Litilt = 0�, ✓ = 0�

-6e-23

-4e-23

-2e-23

0

2e-23

4e-23

6e-23

5 10 15 20 25 30

h[D

=100

Mpc

]

t[ms]

H4, i=90, theta=0

+x

J

L

itilt = 90�, ✓ = 0�

-4e-23

-3e-23

-2e-23

-1e-23

0

1e-23

2e-23

3e-23

4e-23

5 10 15 20 25 30

h[D

=100

Mpc

]

t[ms]

H4, i=90, theta=90

+x

J

L

itilt = 90�, ✓ = 90�

-4e-23

-3e-23

-2e-23

-1e-23

0

1e-23

2e-23

3e-23

4e-23

5 10 15 20 25 30

H4, i=0, theta=90

+x

J

L

itilt = 0�, ✓ = 90�

7

u

gr

iz

JHK

-4

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14

6M

X

(t/1day)(0.01Msun/Meje)1/3.2

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 2 4 6 8 10 12 14

6M

X-6

MX(APR4

)


-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 2 4 6 8 10 12 146

MX-6

MX(APR4

)(t/1day)(0.01Msun/Meje)1/3.2

-8

-6

-4

-2

0

2

0 2 4 6 8 10 12 14

6M

X


u

gr

iz

JHK

-4

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14

6M

X


-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 2 4 6 8 10 12 14

6M

X-6

MX(APR4

)


-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 2 4 6 8 10 12 14

6M

X-6

MX(APR4

)(t/1day)(0.01Msun/Meje)1/3.2

-8

-6

-4

-2

0

2

0 2 4 6 8 10 12 14

6M

X


Fig. 5.— The upper panel: the evolution of time-rescaled bolometric corrections calculated from the results from Tanaka et al. (2014). Thesolid, dashed, and dotted curves denote the bolometric corrections for APR4Q3a75, H4Q3a75, MS1Q3a75, and MS1Q7a75, respectively.The lower panel: the di↵erence of the time-rescaled bolometric correction of H4Q3a75 (the dashed curves), MS1Q3a75 (the dotted curves),MS1Q7a75 (the dot–dashed curves) from APR4Q3a75.

21

22

23

24

25

26

27

28 5 10 15 20 25 30

Mag

nitu

de [D

=400

Mpc

]


u-modelg-modelr-modeli-modelz-model

u-THg-THr-THi-THz-TH

21

22

23

24

25

26

27

28 5 10 15 20 25 30

Mag

nitu

de [D

=400

Mpc

]


J-modelH-modelK-model

J-THH-THK-TH

Fig. 6.— The comparison of ugriz–band (left figure) and the JHK–band (right figure) AB magnitudes between the results obtained inTanaka et al. (2014) (dashed lines) and the light-curve models that are derived from the ejecta mass model and the bolometric correctionmodel (solid lines) for H4Q3a75.

pendence of the ejecta mass has been shown by previ-ous numerical simulations (e.g., Kyutoku et al. 2015;Kawaguchi et al. 2015).

In Figure 8, we show the contour plots of i and H–band magnitudes at t = 3, 7, and 14 days after themerger for APR4 and H4 as a function of chirp massMchirp = [Q3/(1 + Q)]1/5MNS and �e↵ = � cos itilt. Themagnitudes and the chirp mass are calculated assum-ing MNS = 1.35M�. The distance is set to be 400Mpc.We cut the regions from the plot in which theejecta mass is below 10�3 M� because the dataof the radiation–transfer simulation is not longenough to calculate the bolometric correction at14 days after the merger for the case that the

ejecta mass is below 10�3 M�. We note that iand H–band magnitude are always dimmer than26 mag in those regions for both APR4 and H4.

For the case that the NS EOS is H4, i–band mag-nitude can reach ⇠ 23 mag for rapidly spinning BH,�e↵ � 0.5 for a wide range of Q. The emission in i–band with 26 mag can last for ⇠ 7 days after themerger. On the other hand, a BH–NS merger withsmall BH spin and large chirp mass, particularly with�e↵ . 0.67 (Mch/M�) � 1.45, produces no ejecta orejecta with mass below 10�3 M� and leads to a kilo-nova/macronova dimmer than ⇠ 26 mag in i–band. Incontrast to the i–band magnitude, the H–band magni-tude becomes brighter as the time elapses. This time

Bolometric Correction• 各バンドフィルター(X=u,g,r,z,i,J,H,K) のを次のように定義する。

nova are determined by very complex frequency-dependent radiative processes of lanthanoids, so itis di�cult to model them analytically. Thus, in-stead, we introduce a semi-analytical approach toreproduce the spectra of kilonovae.

We define bolometric correction of X–band�M

X

as

�M

X

(t) := Mbol (L(t)) � M

X

(t), (11)

where M

X

and Mbol are the X–band magnitudeand the bolometric magnitude, respectively. Asis shown in Fig. 3, we found that the evolutionsof the bolometric correction of each band filter ob-tained by radiation transfer simulations of [Tanakaet al. 2014] almost agree with each other by rescal-

ing the elapsed time by factor of M

�1/4eje , particu-

larly for riz–bands.

!Physical Reasons!

Though physical reason for this is not clear yet,this result shows that this scaling rule can be fol-lowed in the mass range of Meje = 0.01–0.07M�.This fact allows us to calculate the bolometric cor-rection for kilonova with arbitrary ejected massfrom the result of some specific mass. Thus, inthis paper, we use the bolometric correction ob-tained by rescaling the result of radiation transfersimulation, particularly the result for APR4Q3a75in [Tanaka et al. 2014] to predict each band mag-nitude of a kilonova.

Figure 4 shows a comparison of the ugriz-bandmagnitudes between the model obtained in Sec.2.2(we call this model “the analytic model” in thefollowing) and the radiation transfer simulationin [Tanaka et al. 2014]. The band magnitudesare calculated with the combination of the massand the velocity fitting formula, the bolometricluminosity model, and the bolometric correctionmodel. We set a hypothetical distance as 400 Mpc.For each band magnitude, we found that the ana-lytic model always agrees with the result of radi-ation transfer simulation within ⇠ 1[mag.].

3. Implications for Observing Strategies

!Say something about GW parameter estima-tion!

In fig.5, we plot evolutions of i-band magnitudefor several binary parameters. The label for themodel denotes the EOS name, the mass ratio Q

21

22

23

24

25

26 2 4 6 8 10 12 14

Mag

nitu

de [D

=400

Mpc

]


u-modelg-modelr-modeli-modelz-model

u-THg-THr-THi-THz-TH

Fig. 4.— The comparison of ugriz–band magni-tudes between the result obtained in [Tanaka etal. 2014] (dashed lines) and the light-curve modelwith combination of the ejecta mass model andthe bolometric correction model (solid lines) forH4Q3a75.

and the e↵ective spin parameter �e↵ of the bi-nary. Specifically, “Q3”, “Q5”, and “Q7” denotethe models with Q = 3, 5, and 7, respectively.“a0”, “a5”, and “a75” denote the models with�e↵ = 0, 0.5, and 0.75, respectively. We can seethat a kilonova becomes brighter as the chirp massgets smaller and �e↵ get larger. This dependencereflects the fact that the ejecta mass becomes largewhen the mass ratio of the binary gets small andthe e↵ective spin of the BH gets large. Similarly, akilonova is always dimmer for APR4 than for H4,which also reflects the fact that the NS with largerradius produces more massive ejecta. (These de-pendence of ejecta mass which have been shownby many previous studies of numerical simulations[e.g. Kyutoku et al. 2015, Kawaguchi et al. 2015],and thus, please refer those papers for details.)

In fig.6, we show the contour plot of i and zband magnitude at t = 2, 3, 5, 7 days after themerger for APR4 and H4 as a function of chirpmass Mchirp = {Q3

/(1 + Q)}1/5MNS and �e↵ =� cos itilt. The magnitudes and the chirp mass arecalculated assuming MNS = 1.35M�. A hypothet-ical distance is set to be 400Mpc.

5

• 我々は様々な質量の放出物質のモデルについて輻射輸送シミュレーションから得られる各バンドのbolometric correctionがという時間リスケーリングのもとで、どれも似た振る舞いをすることに着目。

Time rescaled BC @ M. Tanaka et al. 2014

, ここでは X バンドのAB絶対等級、　は bolometric magnitudeを表す。M

bol

MX

t ! t0 = t/M1/3.2ej

この性質を利用すれば、時間リスケールされたbolometric correctionを輻射輸送シミュレーションの結果からモデル化する事で任意の質量のモデルについてのbolometric correction、すなわちband magnitudeを計算できる

�M̃X (t0) = �MX

⇣t0 M1/3.2

ej

⌘! �MX (t) ⇡ �M̃X

⇣t/M1/3.2

ej

⌘

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A B C Y TP024014-12 質量大降着円盤質量（太陽質量単位）放出質量（太陽質量単位） C C • スピンの傾きは実効的にスピンの効果を弱めるため