中性子星連星合体
シミュレーションに向けた
モンテカルロニュートリノ輻射
流体コードの開発
川口 恭平
東京大学 宇宙線研究所
共同研究：柴田大、藤林翔（AEI）

研究会「高エネルギー現象で探る宇宙の多様性Ⅰ」
1

Neutron star mergers
InspiralMergerPost-Merger

NS-NS~10-100ms

G
EOS
ν
B

PostNewtonian

Full GR, dynamical
zero temperature

1s~

Full GR, staAonaryﬁnite temperatureν cooling & heaAng
MHD

Remnants / outﬂow formed in the merger/post-merger phase2
will be the source of the electromagnetic
counterpartsElectromagne,c Counterparts
to NS binary mergers•

Various transient EM counterparts areproposed for NS binary mergers•

for example,•

short-hard gamma-ray-burst•

AQerglow•

cocoon emission•

kilonovae/macronovae•

radio ﬂare, etc.

•

Host galaxy idenAﬁcaAon, remnant
properAes, environment•

Possible synthesis site of r-process nuclei
3

Ref: B. Metzger and E. Berger 2012

Kilonova lightcurve prediction
GR-viscous-νR-HD BNS merger simulaAon(Merger:3D; Post-merger:2D)

Density evoluAon

axisymmetrize

Bolometric Luminosity [erg/s]

Fujibayashi et al. 2020

Kilonova
RadiaAve transer simulaAon(KK et al. 2021)

0.02 s
0.1 s

1s

rext
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5s

AT2017gfo
dynamical(Waxman+ ν drivenet al. 2018)
Total
Total heating rate
0.02 s0°°<20°°ν driven
41 <460.1 s
86°viscous<90°& ν driven

1x10421s

1x10415 s dynamical
1x1040~10s

1Ejecta property(density, velocity, pressure)-19

Long-term
Hydrodynamical simula7on~10000s

dynamical + ν driven
ν driven

1s

0.02 s1s0.02 s0.1 sviscous (ν-irradia4on enhanced)
viscosity driven

0.1 s
5 s dynamicalrin = rext
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x
4Z

300 d

g
z
K

-18
-17

10

t [day]Z 300 d
1 °0°<20f= 2✏(t)dt˙
c w heating
0.1 d

w/o heating

15
16

-16

17

-15

18

-14

19

-13

20

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21

-11

5s

0

AB absolute magnitudezviscous driven

22
0.5

1

2 3 4 5 7 10 16
t [day]AB apparent magnitude [40 Mpc]

3D
~100msPHYSICAL REVIEW D 9
PHYSICAL REVIEW D 91, 064059 (2015)

Y. Sekiguchi et al. 2015

SEKIGUCHI et al.

Neutrino-matter interaction plays
an important role in the merger/post-merger
phase of a BNS merger:

0.35

average Ye

0.30
0.25
0.20
0.15

SFHo
no heatDD2
no heat

0.10
0.05

•

Determines the thermodynamical propertyof the remnant NS and disk

100

0

5

10
15
20
t - tM-6 [ms]

25

30

F
b
t
a
i
d
[
i

Fraction of mass

o
o
10-1
(
h
-2
10
f
p
10-3
E
•
s
0
0.1
0.2
0.3
0.4
0.5
E
Ye
s
The Astrophysical Journal, 901:122 (27pp), 2020 October 1
FIG. 5 (color online). Same as Fig. 4 but for simulations with anddwithout (denoted as no heat) the neutrino heating for SFHo [red andbmagenta (no heat)] and DD2 [blue and light blue (no heat)].
FIG. 2fraction,(color online).
Contoursof entropy
the electron
fraction,s Y(righthalf),inand
entropy
FIG. 2 (color online). Contours of the electron
Y e (left half),
and theper baryon,
x-ythe(lower)andperx-zbaryon, s (right half), in xee (lefthalf),(upper)planes.
Leftpanel:(27pp),
for DD2
atSFHo8.5
mstheaftermerger.
Middle Rightpanel:panel:for SFHo
at 5.0 ms after the merger. Rigi
(upper) planes. Left panel: for DD2
at 8.5 ms
after themerger.
Middlepanel:
forat after
5.0
the merger.
for SFHo•
The AstrophysicalJournal,901:1222020
October
1 msaThemass-distribution
histograms
also shift towards theat 15.0 ms after the merger.
at 15.0 ms after the merger.
higher Y e side due to the neutrino heating. However, thecdistributions still show a broad feature even without
neutrino
heating.
This suggeststhat theinpositron
captureartificial
grid-based
simulatib
For TM1,
the forresultsartificial
are basicallysimilarthe
those foratmosphere
into
grid-based
simulations,inatmosphereparticular
For TM1, the results are basically similar
to those−3 M ) A
from−3thestrongheating
generalexceptfor the fact whenthat theDD2 except for the fact that the tidallyDD2
ejectedcomponent
whentheshockejectamass
lowto(≲10
thetidally
ejectaejectedmassresulting
iscomponentlow (≲10M⊙) as inthecaseof isdue⊙þþ
relativistic
gravityis
primarily
responsibleformakingthewouli−4
more
dominant
and thee captureis less efficient.inAlso,
is more dominant and the e capture isislessefficient.
Also,
TM1. beThecontaminationin mass
TM1.
The contamination
mass would∼10M⊙e
Y e distribution
broad for DD2 and SFHo. For much stiffer
the aneutrino-driven
appears
to playexpand
a majorforkm
the neutrino-driven wind appears to playmajor role for windwhen
whenejectato ∼2000 km in o
the ejecta
to role∼2000
inour thesettingof expand
the
e
EOS
like
TM1,
the
neutrino
heating
would
play
a
relatively
3 of 3 atmosphere with density ∼103 g=cm3, whi
the curve fort −densitytM−6 >∼10
5 ms
the mass ejection (see the curve for thet − tmass
5 ms of(see atmosphereM−6 >ejection
with
g=cm
, whileit would be
of
f
major
role.
Althoughourtreatmentfortheneutrinotransfer−2 if the ejecta are as massiv
becausethe totalorderejectaofmass
for thisEOS
is rather
Fig. 1) because the total ejecta mass forFig.
this1)EOS
is rather
orderofresults
percent
percent
if theejectaare as massive∼10M⊙ . that the neutrinow
is anapproximate
one,
ouras
indicate
small. Here,note ofthat it The
is notcontaminationeasy to exclude
theeffect ofsmall. Here, note that it is not easy to exclude
the effectcontamination
would
be at a sp
inheatingY e would
at Thea similar
For in Y e theplays bean importantrolelevel.
in determining
chemicalthisconsider
reason, intheas
following, we basically nthis reason, in the following,DD2
propertiesweofbasicallythe ejecta.
arepresentativeofa stiff (or moderately barepresentative
of
a
stiff
(or
moderately
stiff)
EOS.
Electron νElectron ν
4
4
TM1
TM1
Electron anti-ν
Electron anti-ν
t
For
SFHo,
shock
waves are formed seve
For
SFHo,
shock
waves
are
formedseveraltimes
duringIV.
SUMMARYAND
DISCUSSION(K. Thorne 1981, M. Shibata et al. 2011, Y. Sekiguchi et al. 2015, 2016
3
3
Heavy νHeavy νhthemerger
phase
as
the
MNS
oscillatthemerger
phase
as
the
MNS
oscillates
with
a
high
2
2
We have reported the first numerical results of radiation, F. Foucart et al. 2015, D. Radice et al. 2016,S
amplitude,
and,hence,
acertainfraction
amplitude,
and,hence,
acertainfractionofmatterorigihydrodynamics simulations in general relativity focusing on
1
1
a
nally
ejected
by
the
tidal
interactionissubnally ejected by the tidal
interaction
is subsequently
heatedsee also McKinney et al. 2014, Sadowski et al. 2014, Takahashi et 0al. 2016
the propertiesof thedynamical ejectaof the equal-mass BNS
0
byincrease
shocks(sneutronincreases),resulting).in th
up by shocks (s increases),
inup
the
the
4
4
DD2 resulting
mergerwith typicalmassof eachof
star (1.35M
DD2⊙
for GR-RMHD)S.
Fujibayashi
et
al.
2020
values
of
Y
via
weak
interactions.
values
of
Y
via
weak
interactions.
On
the
other
hand,
other
e
e
3
Three modern finite-temperature EOS are employed On
to the3parts
lessinfluenced
by
the
shock
parts
less
influenced
by
the
shock
heatingpreservetheclarify the dependence of the ejecta properties on the heati
22
Figure 1. Snapshots for model DD2-125M at t=0, 1, 2, and 4 s. Each panel with the color
neutron-richof theoriginal
neutront
neutron-richnatureoftheneutronstars.
Asmassanatureresult
EOS.
We
that
the
total
of (bottom
theejectais larger(toporiginalright),foundentropy(bottomleft),
andelectronfractionright).
1
1
of
these
two
facts,
theejectacanhavecof these two facts, theforejectacanhavehighersmaller-radiusvalues of s andsofterEOS(giving
neutron stars), and ithighe00Controls the nucleosynthesisin the outﬂow

Possible mechanism for launchinga relativistic outﬂow / jet (pair-annihilation)

5

Luminosity [1053erg/s]

erg/s]The moment formalism M1(M0) method isoften used for the latest merger simulations totake the eﬀ ect of neutrino transport into account53•

SEKIGUCHI et al.

SEKIGUCHI et al.

Luminosity [10•Neutrino-matter interaction

SFHo
no heatDD2
no heatLimitation of M1 method(truncated-moment formalism)
M1-method

Full Boltzmann (grid-based)
The Astrophysical Journal, 901:96 (21pp), 2020 October 1

Y. Asahina et al. 2020

Figure 3. (Top panel) Distribution of the radiation energy density at t=10
using our method. The two radiation beams cross each other. (Bottom panel)Same as the top panel but using the M1 method. Unphysical radiation collisionoccurs.

Figure 4. Proﬁles of the radiation energy density (a) for our method and (b) for
the M1 method. In our method, the radiation reaches the origin. A low radiationenergy density region is formed in the M1 method due to the unphysicalcollision.

Shadow
M1 method is not always guaranteed to provide 4.4.
physicallycorrect results.

In this test, we show the propagation of radiation around gas(see, e.g., H. Nagakura et al. 2017 & Y. Asahina et clouds,
al. 2020
in GR)
whichforaregrid-based
optically thickfull-Boltzmann
for the absorption.method
The
6
and lose their directivity. Such an unphysical collision works tosimulation region is set to be 0x7 and 0y7, and aMonte Carlo methods
Neutron star merger simulation

F. Foucart et al. 2020

(GRHD+MCRadiation)
MC transort in NSNS mergers
MC transort in NSNS mergers

7

5

Figure 4. Left: Angular distribution of the electron antineutrinos leaving the grid (3 5) ms after merger; ✓ is the angle withrespect to the angular momentum vector, i.e. polar neutrinos are on the left, equatorial neutrinos on the right, and each bincovers the same surface area. Center: Neutrino luminosity in the high resolution MC simulation and the M1 simulation. Right:
Average (number-weighted) energy of neutrinos leaving the computational grid in the same simulations.

has Ye ⇠ 0.2, is close to the orbital plane (✓ > 60 ), and is observed (1 2) ms post-merger and in the final
GRejecta
Monte-Carlo
RHD:
snapshot. It is thus most likely ejected during core bounces, from the hottest regions of the remnant, where neutrino
is thesignificant. 2015
This would most naturally explain the properties of the excess ejecta, and the fact
N.pressureRoth& mostD. Kasenthat the neutrino luminosity surprisingly varies less with MC sampling rate than the ejected mass (see below). Fig. 3
also shows
is so difficult to accurately estimate the amountof unboundmaterial in aaccurate,
simulation: the steepness
3D:
physically
Ryan
etwhy
al.it 2015of the Lorentz factor distributions implies that a very small error in the estimated location of the boundary between
Milleretunboundal. 2019,
bound andmaterial2020
has large e↵ects on the predicted but
mass ofcomputationallyunbound material. The second,
more minor
expensivedi↵erence is that the cuto↵ of the velocity distribution is slightly lower in both MC simulations (vmax ⇠ 0.5) than with
F.M1Foucart
et al. 2017, 2018, 2020
(vmax ⇠ 0.55).
The properties of neutrinos escaping the system are summarized in Fig. 4. After the first 2 ms, the average energy

Figure 1. Snapshot of the MC simulation with the largestofnumberof packets,
after ⌫¯merger.
We show(Left),
neutrinos
and the5 ms⌫e andinthe
thedensity
M1 andMC simulations agree within ⇠ 20%. The ⌫x luminosity,
e luminosities
temperature (Center), and electron fraction (Right) in the equatorial
plane
(Top)
and
a
verticalslice
passing
throughthecenteron the other hand, is nearly twice as large in the MC simulation. Di↵erences between the two MC simulations areof the remnant (Bottom). We can see the post-merger hypermassive neutron star at the center of the figures, surrounded by anegligible compared to di↵erences between MC and M1. The luminosity of heavy-lepton neutrinos is one of the mostdense torus. Low-density outflows are launched mostly along the edges of the torus.

our useful
M17scheme,for
becausea large region
close to the neutronstar systematic
surface has negligible
a butDeveloping an axisymmetric (2d)uncertain
codeobservable
wouldinbe
longterm
simulations
andstudies!

Monte-Carlo method:
Procedure
photon/ν packet
absorption

Random event

propagation

emission(geodesic)

scattering

photon/ν production

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Radiation feedback

Axisymmetric GR-MCRHD code•

Ray transfer (log radiation energy density)

Geodesic:
4th order spatial interpolation

8

0

7
-1

6

•

•

z [MBH]

Hydrodynamics:
GR hydro (ﬁxed metric)
3rd Order MUSCL+ Kurganov-Tadmor (central)scheme

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Time integration:
SSP-RK3 (3rd order)
for hydro & geodesic solver

z [MBH]•5

isotropic scattering(as a ﬁrst step)

6

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Case 1 (gas dom.)abs = 1 g/cm3<latexit sha1_base64="otF2fMLLnZ4eG0QRqGkS/hQwmpk=">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</latexit><latexit sha1_base64="qIaAB8e/42VSRZyZRx2i5GH1dq4=">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</latexit>

1abs ⇢c= 1s<latexit sha1_base64="cnH2gUz1478UHpLLXE7JQLFuONg=">AAACiXichVG7ThtBFD0shIchYKBBolnFgEgR6y5YxiAhISigNA8DEous2WUwI/al3bUlsPgBfoCCCiQURTS0SUuTH6DgE1BKIqVJwd21I5SC5I5m5syZe+6cmbECR0Ux0WOH1tn1rrunty/TP/B+cCg7PLId+fXQlhXbd/xw1xKRdJQnK7GKHbkbhFK4liN3rOOVZH+nIcNI+d5WfBLIfVfUPHWobBEzVc1OlKtNM3T1mojOFqfNVeG6Qv9kfDTDI183ZRApJ0nLUX6+aMxQUU9AcdaYT0CJCswYeUojh3aU/exnmDiADxt1uJDwEDN2IBBx24MBQsDcPprMhYxUui9xhgxr65wlOUMwe8xjjVd7bdbjdVIzStU2n+JwD1mpY5Ie6As903e6pSf6/WatZloj8XLCs9XSyqA6dD62+eu/KpfnGEevqn96jnGIUupVsfcgZZJb2C194/TieXNhY7I5Rdf0g/1f0SPd8w28xk/7Zl1uXCLDH/DnlfW3wfZM3ijmC+uF3NJy+yt6MY4PmOb3nsMS1lBGhc89xx2+4pvWrxlaSVtopWodbc0o/gpt5QVSz5cR</latexit>

Pgas = (1)⇢✏<latexit sha1_base64="pIFqKFZ8xMmY/YCxVG2FbDurybY=">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</latexit>

Tgas =

(

1)mp
kB

✏

Radiation temperature

10

Case 2 (rad dom.)

Multi-energy (neutrino)<latexit sha1_base64="4XnOXlNDj3LG/qaKfYFP9JgJzBw=">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</latexit>11⇢ = 10 g/cm

3

Tgas,ini = 12.4 MeV<latexit sha1_base64="squ9v0JlRMm8V+3y/aeD+/WGFuw=">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</latexit><latexit sha1_base64="w56RikkxnLzjA1bSET1Wan2Esq8=">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</latexit><latexit sha1_base64="bqpP0m1urGNolvj9Q61wH0A+jF8=">AAAClHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvoB9gGxFenxunWaoNpbTAdU5SrUJSYUmsbk1aUmFxtaFhbbWhUmxgSZxIvoGygZ2lpYGZqqABmWBibQxhmBqYKhnoGYKDMAAUB+QLLGWIYUhjyGZIZShlyGVIZ8hhKgOwchkSGYiCMZjBkMGAoAIrFMlQDxYqArEywfCpDLQMXUG8pUFUqUEUiUDQbSKYDedFQ0TwgH2RmMVh3MtCWHCAuAupUYFA1uGqw0uCzwQmD1QYvDf7gNKsabAbILZVAOgmiN7Ugnr9LIvg7QV25QLqEIQOhC6+bSxjSGCzAbs0Eur0ALALyRTJEf1nV9M/BVkGq1WoGiwxeA92/0OCmwWGgD/LKviQvDUwNms3ABYwAWCgr4GaEGekZmumZBJooOzhBo4KDQZpBiUEDGN7mDA4MHgwBDKFAe6cy7GY4wnCUSYzJhsmZyRWilIkRqkeYAQUw+QEAUrKbXw==</latexit>

P = PeTeq = 11.0 MeV<latexit sha1_base64="eA9dId3gPKZbSyCnkD1eWPZRM0E=">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</latexit>

Teq

11 4
+ Pe+ + Prad =
aT
12

✏3⌫
f✏⌫ (T ) /exp (✏⌫ /kB T ) + 1<latexit sha1_base64="W3AS3mPpRiyPiFz1kqcw1a1/EZA=">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</latexit>2(1 + 3gA)G2F 2
abs ⇡✏⌫⇡/mp<latexit sha1_base64="ziaz6nwdrJskUWeo/mRFXSnbrGY=">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</latexit>⇣ ✏ ⌘2⌫⇡ 5.7 ⇥ 10 20 cm2 /g
1MeV

✏5⌫
⌘✏⌫ (T ) /exp (✏⌫ /kB T ) + 1<latexit sha1_base64="RfLBbkuIV1OND1HDNT+e7RMMJDc=">AAAC0HichVHPTxNBFH67+APrDwpcTLxsbDAlJvWt0gKeCF48QmmBhMHN7jItk87uTmanDbDZEK/8Ax48aWKM8c/w4tWDB/4E4xETLh583dYYSdC32Zlvvvd9b97MBEqK1CCeWvbElavXrk/eKN28dfvOVHl6ZjNN+jrk7TCRid4O/JRLEfO2EUbybaW5HwWSbwW9Z8P81oDrVCRxyxwqvhv53Vh0ROgborxyj3HjexnjKhWSiIzF/TyvtuaZ0okyCetoP7yQflHPs4zpyOEHKmeSd0z1guJRzysEq3krZ1p09838Qzf3yhWsLS9jo+46BVh6sjgCDaw7bg2LqMA41pLye2CwBwmE0IcIOMRgCEvwIaVvB1xAUMTtQkacJiSKPIccSuTtk4qTwie2R2OXVjtjNqb1sGZauEPaRdKvyenAHH7FD3iGn/EjfsOfl9bKihrDXg5pDkZerrypk7sb5/91RTQb2P/j+mfPBjqwVPQqqHdVMMNThCP/4OjV2cbT5lz2AN/id+r/DZ7iJzpBPPgRvlvnzddQogf4fcvO5WDzcc1t1BbWFyorq+OnmIR7cB+qdN+LsALPYQ3atO8XOLfAsuymfWAf2y9HUtsae2bhr7BPfgFqULW5</latexit>

11

Time integration
Radiation ﬁeld•

Operator splitting method is often
employed for the coupling between
radiation ﬁeld and ﬂuid part:→ time integration is 1st order for
entire simulation

photon/ν packet<latexit sha1_base64="4eICWUJVPBeH1Yp2paVrke95rM4=">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</latexit><latexit sha1_base64="qet4r6UBtk2g7Ab7jNpY10aQAP4=">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</latexit>

y(t)

y(t +

<latexit sha1_base64="HZJPqT/GQTmmMy47msEdycw/kDQ=">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</latexit>

N. Roth & D. Kasen 2015, Ryan et al. 2015, Miller et al. 2019, 2020,
F. Foucart et al. 2017, 2018, 2020•y = y(t +How can we implement higher-ordertime integration scheme?
•

•

t)

y(t)?

t)

x

“Addition” of radiation ﬁeld

Usual iterative higher-order time
integration schemes are notapplicable for radiation ﬂied
described by MC packets

y1<latexit sha1_base64="zgeKf2y4kQyiQnnkCeu7B/qzrMI=">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</latexit>

Algebraic addition of radiation ﬁeld
can be deﬁned by appropriate
thinning and joint of MC packets

y2<latexit sha1_base64="K4Zxzme4qzdh/8MMyBD78ms3sfc=">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</latexit><latexit sha1_base64="PZhjwX22f/2iMrJBisBFPW4DjXw=">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</latexit>↵N1<latexit sha1_base64="zHrT7o5FEA8DrvhMOJIwWNjdTv4=">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</latexit><latexit sha1_base64="mEwwFBV/fplWzzwGcEev9L7uxT8=">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</latexit>

12y⇤ = ↵y1 + (1

(1

↵)N2

↵)y2 (↵ 2 [0, 1])

Higher-order scheme
un , yn : matter and radiation ﬁeld at n-th time step
<latexit sha1_base64="+cQQDBO9fUBM8xgmORAYxoEmlQA=">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</latexit><latexit sha1_base64="H0AtbVFQUClF8WLGlP3YBFvSUuc=">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</latexit><latexit sha1_base64="Cudpn8vOKDo8No3mVMiy96CJaYE=">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</latexit>

u1 = un +y(t +F (un , yn )y1 = G (yn , up )u2 = un +<latexit sha1_base64="hEOe8vLGhqXWyDgBNmXZYoEDgEU=">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</latexit>

F (u1 , yn )t)|u = G [y(t), u]

du
t=
dt

F [u, y(t)]*including feed back fromradiation ﬁeld during t~t+Δty2 = G (yn , up )1
1
2
un+1 = u1 + u2 + u3
6
6
3
1
1
2
yn+1 = y1 + y2 + y3
663
<latexit sha1_base64="BzXZz+8CqrCj72yGJIgpYq3KHoo=">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</latexit>

u3 = un +F (up , yn )y3 = G (yn , up )11
up = un + (u1 + u2 )24Guarantees 2nd order accuracy for time integration*hydro scheme reduces to SSP-RK3 for13the case that radiation ﬁeld is negligible

Convergence test

14

Eddington Limit
Linj = 1.25LEdd<latexit sha1_base64="2pEz2wkdrP4xGc/qE+D812ropaI=">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</latexit>

4⇡GMBH c
LEdd =
log10 ⇢ [g/cm3 ]<latexit sha1_base64="neUX03Rko7c5gk/aqwhteFPi9Jk=">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</latexit>

Linj = LEdd

log10 ⇢ [g/cm3 ]<latexit sha1_base64="kyLl6k3jeFRWmmXhvfnjdZr6U30=">AAACiXichVHLLgRBFD3ae7wGG4nNxCAWMm4jiJWwsfQaJNNj0t1qRke/0t0zCZ35AT9gYUUiIja2bG38gIVPEEsSGwt3ejoRBLdSVadO3XPrVJXmmoYfED02SI1NzS2tbe2Jjs6u7p5kb9+m75Q9XWR1x3S8bU31hWnYIhsYgSm2XU+olmaKLW1/qba/VRGebzj2RnDgiryllmyjaOhqwFQhORwqnpUynVK1EMpUVcYVb89RxnMRXZrQrZ2par6QTFOGokj9BHIM0ohjxUleQMEuHOgow4KAjYCxCRU+txxkEFzm8giZ8xgZ0b5AFQnWljlLcIbK7D6PJV7lYtbmda2mH6l1PsXk7rEyhRF6oEt6oXu6oid6/7VWGNWoeTngWatrhVvoORpYf/tXZfEcYO9T9afnAEXMRV4N9u5GTO0Wel1fOTx+WZ9fGwlH6Yye2f8pPdId38CuvOrnq2LtBAn+APn7c/8Em5MZeSYzvTqdXliMv6INgxjCGL/3LBawjBVk+dwjXOMGt1KHJEtz0nw9VWqINf34EtLSB38nlx0=</latexit><latexit sha1_base64="tUPSl+wlHLPopKKQTwQC8iDvC2w=">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</latexit>

Linj = 0.75LEdd<latexit sha1_base64="kyLl6k3jeFRWmmXhvfnjdZr6U30=">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</latexit><latexit sha1_base64="1zMSmT2IoaSvxFHOjRlpotviB9Y=">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</latexit><latexit sha1_base64="kyLl6k3jeFRWmmXhvfnjdZr6U30=">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</latexit>

3

BH

MBH = 1 M ,<latexit sha1_base64="FwKP1OOG+PZkBk6Q8X/kYdE8JN8=">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</latexit>

abs = 1 cm2 /g

log10 ⇢ [g/cm ]radiaAon=0<latexit sha1_base64="Dfu39XDhYlStEWJc8c1USij7qLM=">AAAChXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsoxGQnFhQkxlfHFOUqJCYV19oaxuiAOcm5cUb66bXxAsoGepaWZqbGBgpghrmhOYRhYmGiYKhnAAbKDFAQkC+wnCGGIYUhnyGZoZQhlyGVIY+hBMjOYUhkKAbCaAZDBgOGAqBYLEM1UKwIyMoEy6cy1DJwAfWWAlWlAlUkAkWzgWQ6kBcNFc0D8kFmFoN1JwNtyQHiIqBOBQZVg6sGKw0+G5wwWG3w0uAPTrOqwWaA3FIJpJMgelML4vm7JIK/E9SVC6RLGDIQuvC6uYQhjcEC7NZMoNsLwCIgXyRD9JdVTf8cbBWkWq1msMjgNdD9Cw1uGhwG+iCv7Evy0sDUoNkMXMAIgIWyAm5GmJGeoZmeSaCJsoMTNCo4GKQZlBg0gOFtzuDA4MEQwBAKtLeNYQ3DVoZtTOxMukwmTGYQpUyMUD3CDCiAyR4Ap12V8A==</latexit>

15

Prescription for optically thick region
Dynamical Amescale(resoluAon in the simulaAon)<latexit sha1_base64="kPH7Xdht+f6VFwNT/KoMSzfWhak=">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</latexit>

x
tLC =
c

<latexit sha1_base64="YOguxevZdJc/UlGQMPYSqlns080=">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</latexit><latexit sha1_base64="V6Xv9aG71NrkvMUK444NQEu2RLM=">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</latexit>

Thermal Amescaleugas
tems =
uems
ugas
=
uems
1 ugas=⇢c urad

tems
1
ugas
=
tLC
abs ⇢ x urad
1 ugas=⌧ 1 (for⌧ urad⌧<latexit sha1_base64="6yp7nLYDJZn24ZHyaSyfCPmNdLY=">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</latexit>

tems ⌧

tabsorption

reemission

1)

PrescripAon:(Foucart et al. 2020, Fleck & Cummings 1971)⇡<latexit sha1_base64="OK4GQUNmOD8PiSt5oCxSY32eooQ=">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</latexit>

abs ! 0abs = (1<latexit sha1_base64="ixB2PJbzpZyEx4gg6sYQASLQxXc=">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</latexit>

sct ! 0sct = sct +

)abs
abs

JusAﬁed if the state in the cell isclose to thermal equilibrium
16

scattering

Shock tube
4
efl = aTfl
<latexit sha1_base64="nQ8qzj+GMdzFyuAKcutxiuOH/kU=">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</latexit>⌧<latexit sha1_base64="KXEIKKRKcnmUOdMB+/0t8mK6Asg=">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</latexit>: radiation pressure dom. gas

1

efl ⇡ erad @ thermal equilibrium<latexit sha1_base64="hlzWs8MQ8KABiJWJFbMzMfnmXGE=">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</latexit>

17

Shock tube
radiation pressure dom. gas(the packet number was not suﬃcient...)18

Next steps/tasks
Radiation energy density

Rest-mass density

19Summary•

We develop an axisymmetric (2d) GR-MCRHD code
as a tool to study NS binary mergers in the the post-merger phase•

Most of the infrastructures are implemented:••

photon/ν packet transfer, hydrodynamics, appropriate matter-radiation interaction•

Higher-order time integration MC scheme
for matter-radiation interaction is implemented and demonstrated(for the ﬁrst time, as far as I know)•

prescription for optically thick region (needs further tests)

Tasks and problems to be solved:•

more simple/robust treatment of optically thick / thermalized region•

more realistic microphysical process(Compton scattering, energy-dependent cross-section)•

implementing dissipation/ang. mom. transport process in hydrodynamics(viscosity, magnetic ﬁelds)•

parallelization / optimization

20