GRB放射を再現する相対論的流体-輻射輸送シミュレーション

大西 直文 (東北大工)長倉 洋樹 (Caltech)伊藤 裕貴 (理研)山田 章一 (早稲田大理工)

1

第29回 理論懇シンポジウム＠東北大学青葉山キャンパス 2016/12/21

http://www.astroarts.co.jp/news/2013/08/08grb/index-j.shtml

石井 彩子 (東北大工学研究科 D3)

http://www.astroarts.co.jp/news/2013/08/08grb/index-j.shtml

ガンマ線バースト(GRB)

• 宇宙最大級の爆発現象• ガンマ線放射時間により分類

• 非熱的スペクトルが観測されているが, 詳細な放射機構は未解明2

No. 1, 1999 CGRO OBSERVATIONS OF GRB 990123 85

FIG. 2.ÈDeconvolved spectra from the CGRO detectors, shown both as photon Ñux and in units. The spectra have been rebinned intoNE

E2NE

\ lflwider bins for clarity. Each spectra is calculated using the actual accumulation times (Table 1), except for the EGRET TASC spectrum, which uses a shortertime interval during which the emission was intense (see text).

MER data is summed from LADs 0 and 4, which had anglesto the burst of and respectively. (For GRB27¡.5 46¡.0,990123, high time resolution data from the energy channelfrom 230 to 320 keV is missing because of a telemetry gap ;however, all channels are available at 2.048 s resolution viathe CONT data type.) These MER rates show the burstÏstemporal morphology (see Fig. 1) and are particularly usefulfor studying spectral evolution.

Figure 1 (lower panels) shows the evolution of usingEpÐts to 16 channel MER spectra from LADs 0 and 4

rebinned in time to provide S/N of at least 100. In order toimprove the reliability of the Ðts and because there is littleevidence for temporal variations in b, the GRB function wasused with b Ðxed at [3.11. This value of b was obtainedfrom the joint Ðt to the BATSE data shown in Figure 2 andis consistent with the values obtained from the other instru-ments (see Table 1). As can be seen, increases by a largeE

pfactor every time there is a spike in the light curve, as istypical of ““ hardness-intensity ÏÏ spectral evolution. Addi-

tionally, the maximum is greater in the Ðrst spike than inEpthe second, decreases more rapidly than the count rateE

pand has an overall decreasing trend, behaviors that aretypical of ““ hard-to-soft ÏÏ evolution (Ford et al. 1995). Thesmall maximum for the second spike is consistent withE

pthe absence of that spike in the 4È8 MeV light curve (Fig. 1).Two intervals during the Ðrst spike have values ofE

p1470 ^ 110 keV. Such values are exceptional : only threebursts of the 156 studied by Preece et al. (1999) have spectrawith values above 1000 keV.E

pTo investigate the burst spectrum over the broadestenergy range possible, we extend the LAD spectra by alsoÐtting the SD data. The high-energy resolution SHERBdata can be Ðtted satisfactorily by the GRB function dis-cussed above ; the Ðts are consistent with the Ðts to otherdata types. SD 4 provides detections of burst Ñux to at leastthe 4.0È8.0 MeV band (Fig. 1).

For the multi-instrument Ðt shown in Figure 2, theBATSE data from LAD 0, SD 4, and SD discriminators 0

Briggs et al. 1999

∝Eα ∝Eβ

Internal Shock Model

Photospheric Emission Model

photosphere Internal shock External shock

Low efficiency for gamma-ray production

Clustering of peak enegy ~ 1MeV

(e.g., Rees & Meszaros 2005,

flaw

Natural consequence of fireball model

High radiation efficiency

Difficult to model hard

Collapsing massive star

Relativistic jet

Photosphere

Long GRBの起源

α ~ −1 β ~ −2.5

✓ Short GRB ( 2 s) → Compact star binary merger ✓ Long GRB ( 2 s)　→大質量星の重力崩壊

.

&

GRBの数値的研究

• ジェットをモデル化した定常　　流体場中での輻射輸送計算　　　　　(Pe’er & Ryde 2011, Ito+ 2013, Shibata+ 2014)

• ジェット内部構造の時間変化を　再現する相対論的流体計算　　　(Aloy+ 2000, Mizuta+ 2006, Nagakura+ 2011)

3

相対論的流体と輻射輸送のカップリング計算

より詳細に再現するには, 非定常流体中での輻射輸送計算が必要

D(#4)2A"1$8+(@&$-,$K4+$L)-0"F"B-?*M$')4"H-(+*$"?@$$L6-+-*064)2A$.#2**2-?*$,)-#$%-+"B?F$E-11"0*2?F$8+")*

Jet

Ito et al. 2013

The Astrophysical Journal, 777:62 (17pp), 2013 November 1 Ito et al.

where νin (νsc) and θin (θsc) are the frequency and anglebetween the fluid-velocity and photon-propagation directionbefore (after) the scattering, respectively. Hence, if θsc < θin,photons gain energy and vice versa. Photons that have crossedthe boundary layer from the sheath region to the spine regiontend to gain energy when they are scattered there (upscatter).This is simply because the photons in the sheath region tend tohave larger angle between their propagation direction and fluidvelocity than do those in the spine region. In contrast, photonsthat have crossed the boundary layer from the spine region tothe sheath region tend to lose energy (downscatter) for the samereason. Consequently, some fraction of photons that cross theboundary layer multiple times can gain because the energy gainby the upscattering overcomes the downscattering, on average.This mechanism can give rise to a non-thermal spectrum at thehigh frequencies.

To obtain a rough estimation of the average energy gainand loss rate (νsc/νin) for each process, we approximate theradially expanding spine and sheath regions as a plane parallelflow. Under the aforementioned consideration, the typical anglebetween the photon-propagation direction and the fluid-velocitydirection for the photons in the spine (sheath) region can beestimated as roughly ⟨θv⟩0 ∼ Γ−10 (⟨θv⟩1 ∼ Γ

−11 ). Because the

angle θv is conserved along the photon’s path in the case ofa plane parallel flow, the typical energy gain rate by theupscattering in the spine region can be evaluated by substitutingθin = ⟨θv⟩1 ∼ Γ−11 and θsc = ⟨θv⟩0 ∼ Γ

−10 in Equation (11) and

is given as follows:

〈νsc

νin

〉

up∼ 1 − β0cos ⟨θv⟩1

1 − β0cos ⟨θv⟩0∼ 1

2

{

1 +(

Γ0Γ1

)2}

. (12)

Similarly, the typical energy-loss rate by the downscattering inthe sheath region is given as follows:

〈νsc

νin

〉

down∼ 1 − β1cos⟨θv⟩0

1 − β1cos⟨θv⟩1∼ 1

2

{

1 +(

Γ1Γ0

)2}

. (13)

From Equations (12) and (13), it is clear that the energy gain bythe upscattering overcomes the energy loss by the downscatter-ing (⟨νsc/νin⟩up⟨νsc/νin⟩down ∼ (1/4)[2+(Γ0/Γ1)2 +(Γ1/Γ0)2] >1). It is also clear that the efficiency of the acceleration per eachcycle of crossing ⟨νsc/νin⟩up⟨νsc/νin⟩down ∼ (1/4)[2+(Γ0/Γ1)2 +(Γ1/Γ0)2] is controlled by the ratio between the bulk Lorentzfactor of the two regions Γ0/Γ1, and the acceleration increasesas the ratio becomes larger.

Although the average value of the energy ratio νsc/νin roughlyobeys Equations (12) and (13), the dispersion around theaverage value is large, given that it depends sensitively onthe scattering angles (θin and θsc; see Equation (11)). Whena photon from the sheath region that has an angle θin = f1Γ−11is scattered in the spine region with an angle θsc = f0Γ−10 ,the energy gain by the scattering can be written as νsc/νin ∼(1 + f 20 )

−1[1 + f 21 (Γ0/Γ1)2]. It is clear from the Equation (11)that a small change in the scattering angles (θin = ⟨θv⟩0 ∼ Γ−10and θsc = ⟨θv⟩1 ∼ Γ−11 ) leads to a large change in the energyratio. For example, in the case of f0 = 0 and f1 = 2, theenergy ratio resulting from the upscattering is larger than thetypical value by a factor of (νsc/νin)(⟨νsc/νin⟩up)−1 ∼ 2[1 +4(Γ0/Γ1)2][1 + (Γ0/Γ1)2]−1 ∼ 8.

Note also that, once the photon energy (evaluated in theelectron rest-frame) approaches the electron rest-mass energy

Figure 3. Observed luminosity spectrum in the case of spine-sheath jet inwhich the spine jet with half-opening angle of θ0 = 0.◦5 is embedded in awider sheath outflow with half-opening angle of θ1 = 1◦. The values usedfor dimensionless entropy (terminal Lorentz factor) and kinetic luminosity arechosen as η0 = 400 and L0 = 1053 erg s−1 for the spine and η1 = 200 andL1 = (η1/η0)L0 = 5 × 1052 erg s−1 for the sheath, respectively. The initialradius of fireball is chosen as ri = 108 cm in both regions. The various linesshow the cases where the observer angle with respect to the jet axis is θobs = 0◦(red), 0.◦25 (green), 0.◦4 (blue), 0.◦5 (purple), 0.◦6 (light blue) and 0.◦75 (black).(A color version of this figure is available in the online journal.)

hνcmf ∼ mec2, where me is the electron rest mass, the scatteringcan no longer be approximated as elastic because recoil effectbecomes non-negligible (Klein–Nishina effect). In this case, theacceleration efficiency is significantly reduced.

In Figure 3, we display the obtained result for the case ofa stratified jet with η0 = 400 and L0 = 1053 erg s−1 for thespine and η1 = 200 and L1 = (η1/η0)L0 = 5 × 1052 erg s−1for the sheath. As mentioned in Section 2, the injection radiusis set at a position where a velocity shear between the tworegions develops (rinj = rs1). The corresponding optical depthis τ (rinj) ∼ 100 for the spine and τ (rinj) ∼ 180 for thesheath. The various lines in the figure show the cases for theobserver angle with respect to the jet axis being θobs = 0◦(red), 0.◦25 (green), 0.◦4 (blue), 0.◦5 (purple), 0.◦6 (light blue) and0.◦75 (black). The spectrum varies sensitively with the observerangle. The spectrum for θobs = 0◦ is thermal-like and nearlyidentical to that obtained in the case of a uniform jet (Figure 2).The reason for this is simple. Because most of the scatteredphotons propagate in a direction within a cone of half-openingangle ∼1/Γ ∼ 0.◦14(Γ/400)−1, the majority of the observedphotons are from a region of θ ! 0.◦14. Hence, only a smallfraction of photons from the sheath region and the boundary(θ " θ0 = 0.◦5) can reach the observer, so that the spectrum doesnot deviate largely from the case of uniform jet. In contrast, if theobserver angle is larger, photons from the sheath and boundarylayer become observable. As a result, a non-thermal componentappears above the peak energy of the thermal spectrum as aresult of the photon acceleration in the boundary layer. The non-thermal component is hardest when the observer angle is alignedto the boundary layer θobs = θ0 = 0.◦5 and becomes softer asthe deviation between θobs and θ0 becomes larger, because theboundary layer corresponds to the site of photon acceleration. Asmentioned earlier, the photon acceleration becomes inefficientwhen the photon energy becomes large enough so that the recoilof electrons cannot be neglected (Klein–Nishina effect). Hence,in all cases, the spectrum does not extend up to energies higherthan hν ∼ Γ0mec2 ∼ 200(Γ0/400) MeV.

7

非熱的スペクトルを生成

ジェット構造の多次元的な時間変化

を再現

Nagakura 2010

カップリング計算に向けて

4

輻射輸送計算

相対論的流体計算

輻射計算の背景場輻射と物質の相互作用によるフィードバック

先行研究

GRB輻射流体計算に必要な要素

�✓ 超相対論的流速 (ローレンツファクター Γ 100)✓ 輻射の非等方性✓ 輻射媒介衝撃波 (A. Levinson and O. Bromberg 2008, R. Budnik+ 2010)

✓ モンテカルロ(MC)輻射計算と流体のカップリング (N. Roth and D. Kasen 2015, A. M. Beloborodov 2016)

✓ 超相対論的流体場中でのMC輻射計算の検証および適切な計算条件の調査 (Ishii+ 2015, Ishii+ 2016 (submitted))

研究の目的

• 衝撃波面が不連続的な場合となまっている場合のモデル流体場中で輻射計算を行い, 計算結果を比較

• 衝撃波速度の違いによる影響を調査• 光子の初期放射位置によるスペクトルへの影響を評価

5

輻射-相対論的流体カップリング計算を目指し衝撃波構造が放射スペクトルへ与える影響を調べる

輻射輸送の計算手法

6

輻射輸送方程式�

1c

⇤⇤t + � ·⇥

�I = j + ⇥4�

� �⇤I⌅d��d�� � [k + ⇤] ⇥I

:光速　 :時刻 :光の入射方向 :光の散乱方向 :光の強度 :放射率 :入射光の振動数 :散乱光の振動数 :散乱断面積 :散乱カーネル :吸収断面積

c t ��

�I j

� k

����

吸収

散乱

CMF 電子

光子

OBF

自由行程を計算

ローレンツ変換

輸送CMF

輸送

OBF

自由行程, 角度を再計算

放射 CMF : comoving frameOBF : observer frame

輻射輸送計算→モンテカルロ法吸収は除き, コンプトン散乱を導入

流体静止系で評価

計算条件

7

zi=0 1 2 3 4 imax=105

Sampling photons

1012 cm5 6 ......

Right boundary

grid number

Discontinuousshock wave

• 不連続な衝撃波および波面がなまっている衝撃波を与え, 　　衝撃波背後で光子を放射させ, 衝撃波前方でサンプリングする

• ρ は衝撃波静止系にて関数を用いて与える　　　　　　　　　　　　　　 (ρmax, ρmin の値はRankine-Hugoniot の関係式より導出)

• v は衝撃波静止系において連続の式を用いて算出する

Every photons are put behind shock wave initially (106 sample particles)

計算条件

8

ρ

zi=0 1 2 3 4 imax=105

Sampling photons

1012 cm

ρmax

ρmin5 6 ......

Right boundary

grid number

Smearedshock wave

• 不連続な衝撃波および波面がなまっている衝撃波を与え, 　　衝撃波背後で光子を放射させ, 衝撃波前方でサンプリングする

• ρ は衝撃波静止系にて関数を用いて与える　　　　　　　　　　　　　　 (ρmax, ρmin の値はRankine-Hugoniot の関係式より導出)

• v は衝撃波静止系において連続の式を用いて算出する

Every photons are put behind shock wave initially (106 sample particles)

計算条件

9

ρ

zi=0 1 2 3 4 imax=105

Sampling photons

1012 cm

ρmax

ρmin5 6 ......

Right boundary

grid number

Smearedshock wave

• 不連続な衝撃波および波面がなまっている衝撃波を与え, 　　衝撃波背後で光子を放射させ, 衝撃波前方でサンプリングする

• ρ は衝撃波静止系にて関数を用いて与える　　　　　　　　　　　　　　 (ρmax, ρmin の値はRankine-Hugoniot の関係式より導出)

• v は衝撃波静止系において連続の式を用いて算出する

Every photons are put behind shock wave initially (106 sample particles)

10−7

10−5

10−3

10−1

101

103

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

emissiondiscontinuous shock

δ τsh ≈ 6.3 × 10−5 (2 cells)

δ τsh ≈ 1.8 × 10−4 (6 cells)

δ τsh ≈ 3.1 × 10−4 (10 cells)

異なる衝撃波幅での比較(Γs=10)

10

• 衝撃波幅の変化はスペクトルに影響を及ぼさない

コンプトン散乱によるカットオフ

�⌧sh =wshsini

wsh : 衝撃波幅sini : 初期の平均自由行程

流体とのカップリング計算

11

• 流体計算(Γs=10)とバックリアクションなしでカップリング(nagakura et al. 2011)• 光子が衝撃波面を跨ぐ際, 逆コンプトン散乱によってエネルギーが 変化している

Samplingescaped photons

z

逆コンプトン散乱によるエネルギー変化

x [c

m]

dens

ity [g

/cm

3 ]

z [cm]

モデル計算とのスペクトル比較

12

• Γs=10 のモデル計算とよく似たスペクトル形状を示す• モデル計算で, 流体計算において衝撃波がなまる様子をうまく再現できていたことが確認された

10−610−510−410−310−210−1100101

10−2 100 102 104

∆ N

(E) /

N

photon energy, E [keV]

modeled shock (10−cell width)coupling

10−7

10−5

10−3

10−1

101

103

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

emissiondiscontinuous shock

δ τsh ≈ 5.5 × 10−5 (2 cells)

δ τsh ≈ 1.6 × 10−4 (6 cells)

δ τsh ≈ 2.7 × 10−4 (10 cells)

異なる衝撃波幅での比較(Γs=100)

13

• 衝撃波幅の増加に伴い, スペクトルの高エネルギー成分が減少• 流速分布がなまることにより逆コンプトン散乱により増加するエネルギーが抑えられるため

z

v ~ 0.999 c v ~ 0

High-energy gain

shock wave

zLower-energy gain

v

不連続衝撃波のとき

衝撃波がなまっているとき

コンプトン散乱によるカットオフ

�⌧sh =wshsini

wsh : 衝撃波幅sini : 初期の平均自由行程

期待されるピーク位置

10−7

10−5

10−3

10−1

101

103

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

emissiondiscontinuous shock

δ τsh ≈ 6.3 × 10−5 (2 cells)

δ τsh ≈ 1.8 × 10−4 (6 cells)

δ τsh ≈ 3.1 × 10−4 (10 cells)

衝撃波速度による自由行程の違い

14

• 衝撃波速度が速いほど衝撃波幅の取り扱いを気をつけなければいけない

wsh : 衝撃波幅δs : 平均自由行程

�⌧sh =wsh�s

z

flow

z

flow

δτsh (Γs =10)

δτsh (Γs =100)

2cell 6.3×10-5 5.5×10-5

6cell 1.8×10-4 1.6×10-410cell 3.1×10-4 2.7×10-4

δτsh (Γs =10)

δτsh (Γs =100)

2cell 6.2×10-3 2.1×10-1

6cell 1.8×10-2 6.4×10-110cell 3.1×10-2 1.1

流速方向光子について

流速逆方向光子について

10−7

10−5

10−3

10−1

101

103

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

emissiondiscontinuous shock

δ τsh ≈ 5.5 × 10−5 (2 cells)

δ τsh ≈ 1.6 × 10−4 (6 cells)

δ τsh ≈ 2.7 × 10−4 (10 cells)

Γs = 10

Γs = 100

10−7

10−5

10−3

10−1

101

103

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

emissionτ ≈ 2

τ = 2.02τ = 2.04τ = 2.06

異なる放射位置のスペクトル比較

15

• 衝撃波幅が δτsh ~ 2.7 × 10-4 のときの計算結果• 放射位置の τ の増加に伴い, スペクトルの高エネルギー成分が減少• 衝撃波面に到達し逆コンプトン散乱を経験する光子が減少するため

z

v

放射位置の設定

zzzτ ~22.06

スペクトルの重ね合わせ

16

• 放射位置 τ = 2 - 2.06 のスペクトルを重ね合わせる (Δτ = 0.01)• 衝撃波幅の広がりにより, 観測から得られているβの値が再現される• 数値拡散により衝撃波幅が広がる可能性もある　　　　　　　　　　　　　　　　　→流体計算を行い適切な計算条件を調査する必要がある

10−8

10−6

10−4

10−2

100

102

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

discontinuous shockδ τsh ≈ 2.7 × 10

−4 (10 cell)δ τsh ≈ 1.3 × 10

−3 (50 cells)δ τsh ≈ 2.7 × 10

−3 (100 cells)

スペクトルの重ね合わせ

17

∝ E-2

10−8

10−6

10−4

10−2

100

102

10−2 100 102 104 106

∆ N

(E) /

N

photon energy, E [keV]

discontinuous shockδ τsh ≈ 2.7 × 10

−4 (10 cell)δ τsh ≈ 1.3 × 10

−3 (50 cells)δ τsh ≈ 2.7 × 10

−3 (100 cells)

• 放射位置 τ = 2 - 2.06 のスペクトルを重ね合わせる (Δτ = 0.01)• 衝撃波幅の広がりにより, 観測から得られているβの値が再現される• 数値拡散により衝撃波幅が広がる可能性もある　　　　　　　　　　　　　　　　　→流体計算を行い適切な計算条件を調査する必要がある

まとめ

• 衝撃波速度が速いほど衝撃波幅の広がりによるスペクトルへの影響が顕著に現れた

• Γs=100では, 衝撃波幅の増加および放射位置τの増加に伴い　　高エネルギー成分が減少した

• 衝撃波幅の広がりにより観測から得られているβの値を再現　できる可能性が示された

18

輻射-相対論的流体カップリング計算を目指し衝撃波構造が放射スペクトルへ与える影響を調べた

今後の予定• 一次元相対論的流体と輻射のバックリアクション入りカップリング• 輻射の影響による衝撃波構造の変化を検証• 輻射媒介衝撃波の構造が放射スペクトルに与える影響について調査

計算条件

19

ρ

zi=0 1 2 3 4 imax=105

Sampling photons

1012 cm

ρmax

ρmin5 6 ......

Right boundary

grid number

Smearedshock wave

• ρ は衝撃波静止系にて関数を用いて与える　　　　　　　　　　　　　　 (ρmax, ρmin の値はRankine-Hugoniot の関係式より導出)

• v は衝撃波静止系において連続の式を用いて算出する

Every photons are put behind shock wave initially (106 sample particles)

⇢ =

8><

>:

⇢max

(z zsh

(t)� �/2)12 (⇢max � ⇢min)

h1 + sin⇡(zsh(t)�z)

�

i+ ⇢

min

(zsh

(t)� �/2 < z zsh

(t) + �/2)

⇢min

(z > zsh

(t) + �/2),

フィードバック入りのカップリング計算

• 輻射フィードバックを考慮した　　　　カップリング計算コードを開発する

20

相対論的ラグランジュ流体コード開発(J. R. Wilson & G. J. Mathews 2003)

モーメント的手法によるモンテカルロ輻射計算と流体計算のカップリング

(N. Roth & D. Kasen 2015)

輻射流体衝撃波形成のテスト計算(A. M. Beloborodov 2016)

ラグランジュ流体のメリット

• 密度が高く自由行程が短い領域で自動的にセル幅が小さくなる

• 輻射とのカップリングの際エネルギー保存を保ちやすい

輻射とのカップリング手法

• モーメント式の E0, Fi0 に相当する量をモンテカルロ法を用いて評価する

• 輻射力G0を観測者系へとローレンツ変換し, 流体のソース項へと入れる

21

E0 =1

cV�t⌃pEp�⌫0⌫

�2lp

F i0 =1

cV�t⌃pEp�⌫0⌫

�2lpni0

G00 = c(�0EE0 � �0ParT 4)Gi0 = �

i0FF

i0/c

MC法によるE0, Fi0の評価

全光子が滞在した分のlpを合計

輻射衝撃波のテスト計算

• 非相対論的流体場において, 領域中心で反射衝撃波を生成する• 先行研究よるテスト計算結果と類似の構造が得られる• より高速流でのテスト計算が必要

22

−0.5

0

0.5

1.0

−2 0 2

arbi

trary

uni

ts

x/L

initial velocitydensityvelocity

カップリング計算結果 Beloborodov (2016) の計算結果