低光度GRBの星周物質相互作用モデル
Circumstellar medium (CSM)- interaction
model for low-luminosity GRBs
Akihiro Suzuki (NAOJ)

Suzuki, Maeda, & Shigeyama (2017), ApJ, 832,32
Suzuki, Maeda, & Shigeyama (2018), in press

理論懇シンポジウム2018, 京都大学基礎物理学研究所, 2018/12/19

Gamma-ray bursts➡ bursts of gamma-ray photons: ~1 event/x10

2day, isotropic

2.5

8–13 keV
BATSE-SD0

2
1.5

x10

4

x10

2

x10

➡ classiﬁcation: long-soft/short-hardRate (counts/s)function” (Band et al. 1993)4➡ spectrum well represented by a ”Band

24–120 keV
BATSE-LAD0

3
2
1
1.5
1
0.5
0
4

320–1090 keV
BATSE-LAD0

1–2 MeVCOMPTEL
Burst Mode2
2–4 MeVCOMPTEL
Burst Mode

1.5
1
3

4–8 MeVBATSE-SD4

x10

1

x10

2

2

2

Ep (MeV)1– 25 –

0

50

100

0

50

100

1.5
1
0.5

α

0
-1
-2

Time Since BATSE Trigger (s)Fig. 4.— Sky distribution of GBM triggered GRBs in celestial coordinates. Crosses indicate long GRBs (T90 > 2 s); asterisks
indicate short GRBs. Also shown are the GBM GRBs simultaneously detected by Swift (red squares)

distribution of Fermi GRBs on the celestial sphere(3rd Fermi GBM catalog, 2016)

Briggs+ (1999)

Gamma-ray bursts➡ bursts of gamma-ray photons: ~1 event/day, isotropic
3

10
–1

Flux (photons · cm · s · MeV )➡ spectrum well represented by a ”Band

1

10

–2

–1

function” (Band et al. 1993)

E2 NE (erg · cm–2· s–1)

➡ classiﬁcation: long-soft/short-hard

GRB 990123

2

10

0

10

-1

10

BATSE SD0
BATSE SD1
BATSE LAD0BATSE SD4OSSE
COMPTEL Telescope
COMPTEL Burst Mode
EGRET TASC

-2

10

-3

10

-4

10

-6

10

-7

10

– 25 –

-8

10

0.01

0.1

1

10

100

Photon Energy (MeV)

Briggs+ (1999)
Fig. 4.— Sky distribution of GBM triggered GRBs in celestial coordinates. Crosses indicate long GRBs (T90 > 2 s); asterisks
indicate short GRBs. Also shown are the GBM GRBs simultaneously detected by Swift (red squares)

distribution of Fermi GRBs on the celestial sphere(3rd Fermi GBM catalog, 2016)

−1

0

1

2

3

Log T50

HR vs T90 plot(3rd Fermi GBM catalog, 2016)

Gamma-ray bursts

Hardness
0.5

ratio

➡ bursts of gamma-ray photons: ~1 event/function” (Band et al. 1993)“short-hard” GRBs

– 26 –−0.5➡ spectrum well represented by a ”Band

Log HR

0.0

day, isotropic➡ classiﬁcation: long-soft/short-hard−1.0Long-soft GRBs
−1

0

1
Log T90

synchrotron

23

logT90Fig. 8.— Classification based on the hardness-duration diagram. Here we show only G~2 sechardness errors less than the hardness itself. Colors indicate their group membershipaverage short/hard, blue: on average long/soft). Ellipses show the best fitting multivariate
models. In the T90-HR case (bottom) the best model has components with equal volume a(the major and minor axes of the ellipses are equal) but their orientation is not constra
T50-HR (top) the best model has similar properties as for T90-HR, only the orientati
components is constrained to be the same (see Figure 9 for BIC values of diﬀerent mode

Blast wave
ジェット
中心核

重い星

ロング ガンマ線バースト

中性子星

ショート ガンマ線バースト

Fig. 5.— Distribution of GRB durations in the 50–300 keV energy range. The upper plot
T50 and the lower plot shows T90 . Also shown are the lognormal fits separately to long and
GRBs (see text for details).

low-luminosity GRBs➡

sub-energetic class of long GRBs➡

only nearby events are detected, but event rate is highe.g., 230+490-190 Gpc-3 yr-1 (Soderberg+ 2006 ), 100-1800 Gpc-3 yr-1 (Guetta&Della Valle 2007)➡

They accompany broad-lined Ic SNe➡

Ex. GRB 980425/SN 1998bw, GRB 060218/SN 2006aj, GRB100316D/ SN2010bh

Luminosity Lγ,iso

Isotropic energy Eiso

Duration T90

peak energy Ep

GRB 980425
SN 1998bw

6×1046 erg/s

9×1047 erg

35 s

122 keVGRB 060218
SN 2006aj

2×1046 erg/s

4×1049 erg

2100 s

4.7 keVGRB 100316D
SN 2010bh

5×1046 erg/s

6×1049 erg

1300 s18 keVcf. Liso~1051 erg/s, Eiso~1052-53 erg for standard GRBs

New LLGRB 171205A @ 168Mpc➡

Swift detection on 2017/12/05(D'Elia+2017, GCN circular 22177)➡

Obs. Data provided by Swift UK Data Centre

Eiso~2.2x1049[erg], T90~190[s](DʼElia+2018)➡

follow-up optical, radio observation➡

SN bump after a few days(de Ugarte Postigo+2017, GCN circular 22207)

GRB 980425
SN 1998bw
GRB 060218
SN 2006aj
GRB 100316D
SN 2010bh
GRB 171205A
SN 2017iuk

Luminosity Lγ,iso

Isotropic energy Eiso

Duration T90

peak energy Ep

6 x 1046 erg/s

9 x 1047 erg

35 s

122 keV2 x 1046 erg/s

4 x 1049 erg

2100 s

4.7 keV5 x 1046 erg/s

6 x 1049 erg

1300 s18 keV1 x 1047erg/s

2.2 x 1049 erg

190 s

125keVcf. Liso~1051 erg/s, Eiso~1052-53 erg for standard GRBs

New LLGRB 171205A @ 168Mpc➡

Swift detection on 2017/12/05(D'Elia+2017, GCN circular 22177)

Eiso~2.2x1049[erg], T90~190[s](DʼElia+2018)➡

follow-up optical, radio observation➡

SN bump after a few days

B 171205A: a low luminosity GRB

ctrum(de Ugarte Postigo+2017, GCN circular 22207)(1+z)Epeak

GRB 980425
SN 1998bw
GRB 060218
SN 2006aj
GRB 100316D
SN 2010bh
GRB 171205A
SN 2017iuk

isotropic radiated energy➡

high L

7

Luminosity Lγ,iso

Isotropic energy Eiso

Duration T90

peak energy Ep

6 x 1046 erg/s

9 x 1047 erg

35 s

122lowkeVL

2 x 1046 erg/s

4 x 1049 erg
2100 sburst duration
6 x 1049 erg
1300 s5 x 1046 erg/s
6 x 1046erg/s
Eiso

1.2 x 1049 erg

Fig. 9.DʼElia+2018
E
in rest-frame versus E . The Swift/BAT GRB sam-

190 s

4.7 keV
18 keV
N/A (single PL)low-luminosity GRBs➡

GRB 980425/SN 1998bw, GRB 060218/SN 2006aj, GRB 100316D/SN 2010bh➡

low Liso, low Epeak➡

luminosity function

log(L/1051erg/s)

Liang+(2007)
LOW-LUMINOSITY
GAMM

log(z)—Jointed contours of the logarithmic GRB detection rate [log (dN /dt)] distribution in a two-dimensional (2D) [ log L, log (z)]-plane, as compared with

New LLGRB 171205A @ 168Mpc➡

Swift detection on 2017/12/05(D'Elia+2017, GCN circular 22177)
synchrotron

Eiso~1.2x1049[erg], T90~190[s]Blast wave
(Lien+2017,
GCN circular 22184)
ジェット

➡

中心核

follow-up optical, radio observation

重い星

中性子星

ガンマ線バースト
ガンマ線バースト➡ ロング
SN bumpafter aショート
few days

B 171205A: a low luminosity GRB

ctrum(de Ugarte Postigo+2017, GCN circular 22207)(1+z)Epeak

GRB 980425
SN 1998bw
GRB 060218
SN 2006aj
GRB 100316D
SN 2010bh
GRB 171205A
SN 2017iuk

isotropic radiated energy➡

high L

7

Luminosity Lγ,iso

Isotropic energy Eiso

Duration T90

peak energy Ep

6 x 1046 erg/s

9 x 1047 erg

35 s

122lowkeVL

2 x 1046 erg/s

4 x 1049 erg
2100 sburst duration
6 x 1049 erg
1300 s5 x 1046 erg/s
6 x 1046erg/s
Eiso

1.2 x 1049 erg

Fig. 9.DʼElia+2018
E
in rest-frame versus E . The Swift/BAT GRB sam-

190 s

4.7 keV
18 keV
N/A (single PL)

Origin of low-luminosity GRBs➡

central engine was a neutron star? (e.g., Mazzali+2006)➡

oﬀ-axis GRB?

➡

relativistic shock breakout with dense CSM, oﬀ-axis/weak/failed jet, cocoonCSM interaction(Kulkarni+1998, Tan+2001, Campana+2006, Li 2007;
Toma+2007; Wang+ 2007; Waxman+ 2007,Suzuki&Shigeyama 2012, Nakar
2015, Irwin&Chevalier 2016 )Suzuki&Shigeyama (2013)
Nakar (2015)

3

Low-Luminosity GRBLong GRB

Gamma-rays/Hard X-rays(low-luminosity, un-collimated, soft, non-variable)

Gamma-rays(luminous, collimated, hard, variable)

A mildly relativistic
shock

Extended
mass

Shock
breakout

Core
Core

Core
~1011 cm

1013-1014 cm

An Ultra-Relativistic jet
Penetrates the core – choked in the extended material~1011 cm
An Ultra-Relativistic jet
Penetrates the coreFig. 2.— A schematic sketch illustrating the similarity and dif-dense CSM around massive stars?
➡

“Non-detection” of fast, bright shock breakout signals from type II SNe➡

shock breakout emission can be smeared out or “delayed” by optically thick gas.

➡

SN progenitor surrounded by dense CSMs?

➡

we do not know if such CSMs are ubiquitously present for all types of CCSNe

Forster, Moriya + (2018)

credit: NAOJ

dense CSM around massive stars?
➡

“ﬂash spectroscopy”, right after SN discovery➡

CSM illuminated by the SN shock breakout light➡

centrally conﬁned CSM with a density much larger than normal stellar winddensitydM/dt=10-3M◉/yr?

1014cm1015cmradius

SN 2013fs(=iPTF13dqy) Yaron + (2017)

Ejecta with mildly relativistic speeds and CSM➡

SN ejecta (with max Γ~2-10) colliding with circum-stellar medium (CSM),
leading to the dissipation of the kinetic energy into the thermal energy of theshocked gas.

➡

the thermal energy diﬀusing out through the shell is responsible for the
A. Suzuki, K. Maeda, and T. Shigeyama
prompt emission

pana et al. 2006; Starling et al. 2011). The unusually
gamma-ray emission and the large blackbody radii inby theHow
thermal muchcomponent energy
indicate that acan
hot ejecta
beonent with a photospheric radius larger than typicalof compact
stars play a vitalin producing highreleased
in roleejecta-CSM
y emission.
Radio observations of
SNe are another tool for probinteraction?
ighly energetic explosions through the presence ofhock waves propagating in the circumstellar medium) of the•exploding
star.
Follow-up ofradioSNobservationsofkinetic
energyejectaSNe have also been conducted in great detail. GRB• density
structureare indeed
known to bea bright radio emitter, e.g.,
98bw (Kulkarni
al. 1998; Weiler et al. 2002), which• CSMetdensity
ly caused by synchrotron emission produced by therd shock sweeping the ambient gas. Furthermore, thevery of radio-loud SNe Ic-BL without any gamma-ray
ure, e.g., SN 2009bb and SN 2012ap, have revealed a
Suzuki, Maeda, Shigeyama (2018)ation of relativistic SNe, whose radio emission strongly
te the presence of ejecta traveling at (mildly) relativis-

density

Optically thick stage
RSejectaFS

gamma-ray

shell
CSM

CD

densityradius

Optically thin stageRSFS

X-ray, radio
opt-UV

photospherephotosphereCDradiusOne-zone semi-analytical model➡

steady wind: ρ=Ar-2, A★=A/(5x1011 [g/cm])(dM/dt=10-4M◉/yr, for vwind=1000km/s)➡

freely expanding trans-relativistic ejecta: cβ=r/t,

Ejecta distribution in velocity space

Ejecta distribution in physical spaceρρkinetic energy=Erelρej∝(Γβ)0ρ∝(Γβ)

0ρej∝(Γβ)-nρ∝(Γβ)-nρcsm=Ar-2

relativistic

Erelcomponent
Γβ=1

ΓmaxβmaxΓβ

r=3x1013cm radiusinternalp/⇢⌧ 1. ofThevalidity(13)of these
hocks, pfs,dand penergy,
R.H.S.
Equation
rep- approxirs,d . Theuemations
be checkedin Section
esentsthise↵ect.willThe
quantities
Ffs and3, where
Frs in approximated
the
es,
with
results offluxes
numerical
calcu.H.S. ofsolutions
Equationare(13)compareddenote themomentumofralations.
as flowing
into the shocked gas through the forward ander- reverse shocks. They also contribute to the temporalhe➡ we approximate
the shocked region as a thin shell and solve the EOM.
ere
volution of the momentumof the Evolution
shell and expressedin2.3. Temporalof the Shellpeerms of thevelocities,
and rs , oftheforwardandfs evolutionThe
dynamical
of
the
shell
is
determinedbyand so on
al.
➡
shock
radii
R
,R
,
shell
mass
Ms,
shell
momentumSr,fs
rs
everse shocks
and the post-shockdensities⇢fs (1)and&⇢rs
at
SUZUKI,
MAEDA,SHIGEYAMAthefollowingtwocompetinge↵ects,thedeceleration
ed forward and the reverse shocks as follows,
heby loading mass and (2) the acceleration by the pressure
cal
TABLE
1
2
gradient.
the
mass,
the
Lorentz
factor, theexFfs We
=Total⇢fsdenote
(
),
(14)
s fs
s
s massof trans-relativistic
ejecta for Erel,51 = 1
momentuminﬂux
fromejectavelocity
andtheposition
of
the
shell
by
M
andpressure gradient forces , s , s , bydecelerationanndintegration ofn Equation(2) overthe volumen ofon maxRs . The
n=1
=
2
n
=
3
=4
n=5
2
shell
leads
=4 M⇢rsto s5.47
n- 10 the3.78
rss ( ⇥rs10 4 M
s ).
⇥F10
6.98 ⇥ 10 4(15)M
8.14 ⇥ 10 4 M
9.00 ⇥ 10 4 M
4M
4M
4M
4M
4M
2, 5 dS
4.64
⇥
10
5.90
⇥
10
7.13
⇥
10
8.19
⇥
10
9.02
⇥
10
r
2 Equation 2(1)2
Next, theintegrationofgives
the2⇥pfollowing
4M
4M
4 M4⇡R8.46
+ 10
4⇡R
F
4⇡R
F
=
4⇡R
6.06 ⇥
6.87
⇥
10
7.69
10
⇥ 10 4 M
9.14 ⇥ 10 4 M
fs
rs
rs,d
um 3
fs
rs
fs
rs pfs,ddt
quation governing
the temporal evolution of the mass
he(13)
f the shell,
ese
Since theshockedofgas
in between
forwardρthe of the shellmomentums the temporal
evolution
theis velocityof the where
theandvelocitytoSrs is assumeddMsreverseshocks,the di↵erence
2 the shell is2 accelerated bypost-shock
velocity. Using the shock
velocity,
one
mass
Ms
+
4⇡R
F
4⇡R
F
=
0
(16)
fs m,fs
rs m,rsty
theshellpost-shockat tothetheforwardtheandthe reverse
dt
Rspost-shejectan Rs ofofthe
evolves pressure
accordingforwardshock
Lorentz factor radius
fs , theshocks, p
and prs,d . The R.H.S. of Equation
ation,
sitythe⇢fs,d(13), andrepthe post-shockshellpressure pfs,d , as fo
he mass fluxes fs,dFm,fs andFm,rs flowing into
resents
this thee↵ect.
The andquantities
Fshocks
fs andareFrs in the1)hockedgas
throughforwardreversedRs of Equation shell1
increases
its mass fluxes of
L.H.S.
(13)
denotethemomentum= s,(20)
xpressed as follows,,
fs = p2
ssgasdt
flowing into theby
shockedgas throughthe forward andsweeping
CSM/ejecta
1fs
ut
the velocity
reversealso
contribute to(17)the temporalFm,fsshocks.
⇢fs shell.
( fs ThissTheys ),equation of theof=the
Erel
2
CSM
(1
)
evolutionofthemomentum
of
the
shell
and
expressed
in
nd
s
fs
ated alongforward/reverse
with Equations (16)
and
(19)
to
shock velocity⇢fs,d = ⇢fs,u
,
nd
2
2
terms
of
the
velocities,
and
,
of
the
forwardandArs
e position of the shell at time t. fs
)
u( s
fs
r=Rs
radiusdeterminedbyjump⇢shockreverse F
shocksandpost-shockdensities(18)⇢fs and ⇢rs at
m,rs =
rs thes ( rss ).condition
and2.4. theShockconditions
forwardand the reverse shocks as follows,

One-zone semi-analytical model

dMs
2
2
cked gas internalcannot+ 4⇡R
Fpm,fs and
4⇡Rrs
Fm,rs
=these0the (15)energy,p/⇢⌧1.
validity
ofatapproxifsThe
nce
in
the
post-shockpressure,
p
,
fs
rs
dt
time unlike
ressedasnonfollows,uemationswill
be checkedin SectionF3, where
which can be solved with respdes,and thereverseshocks.
We denotemass, approximatedtheDespitediffiThe mass fluxes
Fm,fs and them,rs flowing into theafactor,geometricallysolutionsare comparedwithresultsof
of(16)
numerical
shockedgasthroughthe ,forwardandreverseshocks.calcufor a given set the momentumzas
thevelocity
andtheposition
the
for=
⇢
(
)
=
⇢
m,fs
a
a
a
fs
a
fs
raion as we will
see
They are expressed as follows,
lations.
sndreverseshocks by Ms , s , s Rfs , and Rrs . Weof theshell to be
erFm,fs = ⇢a a ( a⇢a fs ,
(16)fs ) =
shelltemporalapproximatheevolutionofthemomentumSofthe
we approximate the shocked region ras a thin shell and solve the EOM.
ere
olve the equation
2.3. Temporal Evolution of the Shell
1+
s =
and
Fm,rs
= ⇢ej direction). = ⇢ej ej ( ej rs ).(17)ong
theasFrsm,rs
follows,
peme thatthe radialrestej ( ej
(17)
ral.
the internalTheen- dynamical evolution of the shell is determined byshockHereradii⇢ R, fs,R, rsand, shelldenotemasstheMs,
shellthemomentumSr, and so on
proximations
will
density,
Lorentzfacaaafollowingtwocompetinge↵ects,the deceleration
anddenotethedensity,
Lorentz
faced
a the
2solutions
2 and
2the
2(1)
The radius of the shell evolve
imatedtor,
thevelocityoftheunshocked
ambient
gas at the
+cal
4⇡R
F
4⇡R
F
=
4⇡R
p
4⇡R
p
,
(13)
fs
rs
rs
fs
fs by
rs
fsthe
rs
massshock.
and (2)
byCOLLIDING
thegaspressure
MILDLYRELATIVISTIC
WITH AN AMBIENT GAS
calculations.
elocityofloadingthe unshocked
ambient
gasofEJECTA
at
forwardSince
the acceleration
velocity
the theambient
is ingequation,
gradient.
We denotethevelocity
mass,is thefactor,negligibly
small, theset toLorentz
zero, a andthen the
exk.
Since
the
velocityoftheambientgas
is
be calculated
a givensetof theshock
radii and the
The reverse shock velocity is determined indRa similarws
the for
R.H.S
of Equation(16)
is Equation
obtained.
hemomentuminﬂux
fromejectavelocity
andtheposition
of
the
shell
by
M
,
,
,
and
tanShelland 2nd
terms
of
the
R.H.S.
of
(13)
s
s
s
decelerationbypressure
gradient force =velocityof theshell, Lorentz
or zero,the corresponding
Lorentz
tos theforwardshock,
sto
Defining
thefactorandthevelocity
of
s
all,
the
velocity
is
set
and
then
MILDLYRELATIVISTICEJECTACOLLIDING
WITH
ANAMBIENTGAS
5ell is determineda fromdt
Rfactor
integrationofEquation
(2)
over
the
volumeofThesevariablesareobtainedtheshock
s . Thes . the
nt
the
force
exertedby
thepost-shockpressure
atonshell,
thefollowing
relation
betweenthese
variables,
=
(
,
).
(3
the deceleration
rs
sh ej
s
jump
conditions
at
the
shock
fronts.
The
derivationofEquation(16)
isobtained.
theforward
shell
leads
toradii andthemomentum,andthethequantities
mass
obtained,
ed
aand
given
set
of theshock
Theis reverseshock
velocity is determinedin a similar way
n-forthrough
tum
the
erse
fs andThe
temporalevolution
theshock jumpshocks.
condition isThedescribed
in elsewhereF(see,Oncethereverseshock velocity
is obtained, ofthe the
reve
ρ
the
shell
,
or
the
corresponding
Lorentz
totheforwardshock,e
Lorentz
factorandthevelocityofsrationdue
tothe
spreviouss
2,theEJECTA
Sdenote= Ms GAS(18)
Appendixof theWITH
paper,r Suzukis&s Shigeyamashock
factorpost-shockdensity⇢rs ,ganTheseL.H.S.
ofEquation(13)
themomendSTIC
COLLIDINGAN
AMBIENT
52 Lorentz
rs , thershock
radii,
R
and
R
are
2
2
2
variablesareobtained
from
the
shock
fs
rs
the
fs and prs , at(2014),
some
textbooks,orreview
papers,suchfsas,
Lan- 4⇡R+fronts.
4⇡R
Fbetween
4⇡R
Frs
= 4⇡R
pthroughp(fs,d=(33)
thersshpost-shock
evaluated,
rs factor
ej , s ). pressure
fsderivationrs,dum
following
relationthesevariables,
fsintors
ejecta prs are
which
can
be
solvedwithrespect
totheLorentz
tions
thegas
shock
The
of
uxes
flowing
theshockedgas
tetheatofmass,
thedtis&describedtions,
dau
Lifshitz(1987);Martı́
& Müller(2003)).inThus,
khe
radii
and theThe
reverseshock
velocity
is determineda similar
way
for
a
given
set
the
momentum
and
the
mass,
ump
conditioninelsewhere(see,
1 shellOncethereverseshock
velocity
is obtained, the reverse
osition
of
the
form,
and
the
mass
is
obtained,
(13)
we
will
not
repeatthederivationin
thispaper.
ward
andthereverseshocks,reflects
themo-rs , the post-shock density,(3
ondingLorentz
to theforwardshock,whichrs =⇢p, anddRs shockofprevious
paper,
Suzuki& Shigeyama
fs
Lorentz
factor
Rfsthe, andRrs . We2
rsese
1
The theshock velocity sh
is generally
expressed
as a funcfrom
the shock
rs
2 the
=
gas
is
in
betweenforwardandthee textbooks,orreview shocked
papers,such
as,ambientLanSmconservation.
Since
the
gas
is
movingthepost-shockpressurepareevaluated,
mentumSrSince
of
the
=
(
,
).
(33)
rs
r
rs
sh
ej
s
tion
theM
velocity and the Lorentz
gas in1 + of2 .the (18)(19)
e derivationof r of=dt
s = factor
S
hitz(1987);Martı́
&shocks,Müller(2003)).
Thus,
(
)
s
s
s
reverse
the
shell
is
acceleratedbythedi↵erence
ws,
ej
ej
rs
M
the upstream,in thedownstream,s we neglect
1
locitysmallerthantheejecta,u andu , andd andelsewhere(see,⇢rs = ⇢ej
,
(3
Once
the
reverseshock
velocity
is obtained,
the reverse
repeat themuch
derivationin thispaper.
ty
p
=
,
(34)
rs
(
)
E
rel
of thepressure
attheforwardandfollows,
u s
rs
d , as post-shock& Shigeyama
2
shock
Lorentz
factorthe
post-shockdensity⇢rsthe,followandreverse
and
CSM
WITH
GAS
5
rs , to1
ki velocity
isgenerally
expressedasafunc2ANshAMBIENT
ntributionoftheambientgas
themomentumTheradiusoftheshell
evolvesaccording
to
the
rs
4⇡R
p
,
(13)
solvedwithrespect
totheLorentz
factorrs fs, such as,
Lanthepost-shockpressurep R.H.S.
are evaluated,
shocks,pfs,dandu ofp2d (rs,d. The
Equation(13) repanddRrsvelocity
andthe Lorentz
factorthe1)( ofing
equation,u gasd )in drs (
u
d)
2
the
shell
and
thus
set
the
flux
to
zero,
F
=
0.
(
)
(2003)).
Thus,
( uin
, this
,
ej ej
rs
fs
⇢ej ej ( ej (35))( s ej
d) =
= (3
rs
s ) radiusm, theandthedownstream,
tpaper.
momentumandmass,
2 ( the
dR
r=R
u and
u , shd andresentse↵ect.
The
quantities
F
and
F
in
the
1
s
⇢
=
⇢
,
fs
rs
)
(
1)(
)
rs
ej
u d u
d a
u
u
d
d
p
=
.
e
shock
velocityisdeterminedinsimilarwayofEquation(13)rs1)(20)u ( sp = s, ,(34)dt
rs = momentum
ws,
rs )
other
hand,
the
supply
of
the
of
the
1
s
rs
2
dt
(23)
L.H.S.
ofEquation(13)
denotethemomentum
fluxesofshock aspressure
1
essed
a func-at s
rs
ward
shock,
2
and
when
the
pressure
inthe
upstreamisnegligible.
Oncethethethroughshock
plays
athedominantrole
uantities
Fand(fs the) dTheinto( temporal1)(
orof theu gasinudreverseu shockedd)evolutionofthe forwardandreversess2
2.5. andNon-relativistic
and Ultra-relativistic
Limitsflowingthegasthroughforwarddgas
These
equationscanbeintegra
2
=
,
(
)
S
⇢
(
)(
)
ej
ej
rs
ej
ej
rs
ej
s
shock
velocity
isobtained,thepost-shockdensity⇢ejequa2momennote
the
sh
d
r).
stream,andradii,
governedbysimilarfs
rs are(dthe(shock1)()⇢R⇢rsdand=dshell.
, (19)
(35)
=
(
,
(33)
easing
the
The
flux
F
is
u dtheu rsmomentumd ) shus ofuR
p=.of the(36)is much smaller ,than t
ej
ut
ej
rs
reverseshocks.
They
also
contributetothe
temporalWhen
the velocity
the shell
rs
=
1
+
.
way,
once
velocities,
andpressure
p
are
found
from
the
following
relations,s()
cked gas through
s
fs
d
1
u
s
rs
tions,
s rs
2
(23)
speed
of
light
(“non-relativistic
regime”)
or
the
Lore
M
ed
asinfollows,
evolution
ofnegligible.
theis momentumofshell and expressed inhreflects
themo- velocitynds OncedRfs) theessuretheupstreamisthe(reverseshock
obtained,thereverseanduushfactorof the shell is Limits
much larger than unity (“ult1)(gas= 2.5.
(21)
Non-relativistic
and Ultra-relativistic
u is moving
d)
fs ,
nt
⇢d = ⇢density,(24)
u ⇢ej 2⇢ (fsej
terms
of
the
velocities,
and
,
of
the
forward
and
,
)(
)
ity
is
obtained,thepost-shockA
dtsh ) ⇢rs , rsejsshrelativistic regime”), self-similarsolutions2.4.
Shockdescribing
Jumpt
ntz
factor,the
post-shockandd (density
dejd
rs
rs
)
jecta,
weneglect
2
u
u
d
d
p
=
.
(36)
When
the
velocity
of
the
shell
is
much
smaller
than
the
rs
e
p
are
found
from
the
following
relations,dthe reverse
F
=
⇢
(
),
(14)
shocksandthepost-shockdensities⇢fs and⇢rsknown
at (Chevalierflow are1982; Nakamura & Shigeya
shell
evolvesaccordingtothefollow1
and
rs
ej
ej
ej
rs
s
rs
ej
oof
the
momentum(23)
hockpressurepareevaluated,
speed
of
light
(“non-relativistic
regime”)
ortheLorentzandrsdRrsshocks as follows, 2006). TheIn
time
dependenceof physical variables
of t
2
order
to
integratetheequforwardthereverseto zero,Fthe= 0.
gible.
Onceu( ush )and⇢()()
fs thefactor
of
the
shell
is
much
largerthanunity(“ultra=
,
(22)
u
u
d
u
sh
u
rs
2.5., Non-relativistic
Limits flow can be obtained by imposing the condition that t⇢ =⇢(24)and Ultra-relativistic

One-zone semi-analytical model➡
➡

Bolometric LC for prompt emission➡

ﬁducial model: Erel,51=0.5,A★=25,n=5 (dM/dt=2.5x10-4M◉/yr for vw=103km/s)➡

theoretical emission model is consistent with observed prompt gamma-ray and
X-ray light curves➡

note: theoretical model produce bolometric light curves➡

spectral evolution is the next step

ρ

5x1050erg

ρej∝(Γβ)0~10-4M◉n=5
ρej∝(Γβ)-nρcsm=Ar-2

Erel
r=3x1013cm radius

Suzuki, Maeda, & Shigeyama (2018)

Erad vs Tburst➡

Erad-Tburst diagram➡

longer bursts show larger radiatedenergies➡

in ejecta-CSM interaction model,
this trend can be explained byincreasing CSM density (or mass)➡

GRB 171205A is consistent withthe trend.

➡

GRB171205A: Erel~1051[erg], A★~several 10

Suzuki, Maeda, & Shigeyama (2018)

Non-thermal X-ray and radio emission
6➡

D’Elia et al.: GRB 171205A: a low luminosity GRB

Afterglow emission following theprompt gamma-ray

10

2

101

Swift XRT observations + radio(NOEMA, ALMA, VLA )

10

-1

10

-2

10-3

a

lly
incta
cal
gh-

obof
um
s of
RB.g.,
ich
the
the
ray
da
gly
vis15;

10

Optically thick stage

106s
↓

-4

4

10

5

10

V filter
B filter
U filter
UVW1 filter
UVM2 filter
UVW2 filter
XRT flux density (@ 1 keV)
10density

105s
↓

100

Flux density (mJy)

➡

104s
↓

2

10

3

10
Time since T0 (s)

6

10

7

XRT & UVOT obs. (DʼElia+2018)

RS
ejectaFS

gamma-ray

Fig. 6. Swift/XRT and /UVOT light curve of the GRB 171205A afterglow. UVOT data are corrected for the host galaxy and extinction
contribution. Information on the UVOT filter passbands can be found at: http://www.swift.ac.uk/analysis/uvot/filters.php.

shell
CSM

CD

densityradius

Optically thin stageRSFS

X-ray, radio
opt-UV

photospherephotosphereCDradiusthe bolometric rest-frame band to take this e↵ect into account,increases just slightly the Eiso error (less than 10%). Concerning
the spectral model uncertainties, if we use a Band function in49
stead of a CPL, we obtain Eiso = 3.6+0.9
0.9 ⇥ 10 , a slightly higher
value.
The Eiso derived for this burst is similar to those of otherlow-luminosity GRBs. For example, the Eiso (in the observed
15-150 keV band) for GRB 060218 is 2.57 ⇥ 1049 erg (assuming T 90 = 2100 s, see Campana et al. 2006) and GRB 100316D
has Eiso3.70 ⇥ 1049 erg (assuming T 90
1300 s, seeStarling et al. 2011). However, besides the low Eiso , other properties of GRB 171205A seem to be di↵erent from GRB 060218
and GRB 100316D. The BAT spectrum of GRB 171205A is
harder than those of GRB 060218 and GRB 100316D, whichhave power-law indices of 2.18 and 2.36, respectively (Lien etal. 2016). In addition, both GRB060218 and GRB 100316D
show long-lasting burst emission of more than ⇠ 1000 s, while
GRB171205A has no obvious emission after ⇠ T0 +200 s, though
we note that GRB171205A went out of the BAT field of view at
T0 + 479 s. As shown in Fig. 9, GRB 171205A lies outside theAmati relation from the BAT long GRB (Krimm et al. 2009) andthe Konus-Wind (Tsvetkova et al. 2017) samples, even adopting
the more conservative values for Epeak (see Section 3.1.3) andEiso .
Other low-luminosity GRBs such as 171205A are not consistent with the Amati relation. As discussed by Amati et al. (2007),
this lack of consistency strongly indicates that these LL-GRBs
are not just normal, ”cosmological” events seen o↵-axis. GRBs
are supposed to be jetted sources seen on-axis (Sari et al. 1999)
or very close to it. Theoretically, if a GRB event were seen o↵axis, the prompt emission detected would be weaker, since theradiation is strongly beamed toward the direction of motion byrelativistic e↵ects. It is found that Ep / and Eiso / 1 ↵ (Amati
et al. 2007, Yamazaki et al. 2003), where ↵ is the low-energy
photon index and is the relativistic Dopper factor which, inturn, depends on the half-opening angle ✓ jet of the GRB jets
and the angle between the GRB jet axis and the observer ✓obs := ( (1 v/c cos(✓obs ✓ jet ))) 1 . This parameter decreases as✓obs ✓ jet increases.
Since Ep and Eiso depend on in di↵erent ways, GRBs seeno↵-axis cannot follow the Amati relation as GRBs seen on-axis.
We find that, in principle, this might be the case for 171205A.

assumed
non-thermal
emission ✏ismodelisotropic.
Thus,
their
momentum
distribution
dN/dp
pein
). many& Sariwe introduce
a parameterande (t, 2002),
e exprasand
function
t and
theof
norm
of theSNe(e.g.,
et
al.
1998;
Sari
Esin
The
temporalevolution
theelectron
momen
aafraction✏ofe timeofSaritheinternal
energyof&momentu
the
gas20
dN/dp
pof
e (t, 2002),
eis).thetribution
obtainedsolvingthefollowingaa&Sari
we introduce
a parameter✏ e andof
streamshock
isbyconverted
to theenergyThe temporal
evolutionofspacetheelectronmomentumequationin themomentumfor isthe
range
from
aelectrons,
fraction✏ueeleoftheinternal
energy
of
the
gas
in
=
✏
u
,
where
u
the
internae
int
int
3 m c to6bytributionis theobtainedsolving thefollowing
adve
10stream
p
=
10
m
c,
of
shock
is
convertedtotheenergy
of
n
e
max
e
sity
at
the
shock
front.
The
averageenergyofequation
the momentum
space
for therange fromp
Suzuki,(2018)!inMaeda," &✏ Shigeyama#!
electrons,
u
=
u
,
where
u
is
the
internal
int
ele
3m
6 mintelectrongiven
by the@thermaldN@ = 10eis
dNelectrond Ṅintern
10c,e c to=pmaxe( ṗsyn+ ṗicThe+ ṗad)+energy of. a
sity
at
the
shock
front.
averagedividedbytheelectronnumberdensity!
"#! e nin@t dpe
@pe
dpe
dp
ele i
@thermaldN@
dN
d Ṅ interna
given
stream,=electronuele /nele.is +( ṗsynṗic + byṗad )the electron+.
We
injectelectrons
withapower-law
momentum@tdivided
dpe by@pthe
dpe density
dpe innele indeelectron number

Non-thermal X-ray and radio emission➡

electron distribution in momentum
space

➡

parameters: p, εe, εB

The momentum loss rates, ṗ

and ṗ , for

syn
ic
tion
with uan exponentp,stream,/n.
We
injectelectrons
momentumdist
ele Compton
ele with a power-law
andcooling canbe calculat! inverse(pThe
momentumloss
rates,
ṗsynandareṗic ,given
for
tion
with
an
exponent
p,
d
Ṅ
p
for
p

p

p
e
max
corresponding
energyloss
rates.
Theyine➡ synchrotron, inverse Compton, and/(. calculate!
andinverseComptoncoolingcanbep0otherwiseq
ddpṄe in 4 pue
for pin  pe  pmax
adiabatic cooling/.
T
B
corresponding
energyrates.
They
are given
2 c2loss
2,
ṗ
=
p
m
+
p
0
otherwise
dp
synee the minimumeThee normalization
injection moin
2 c2 q and
3m
euB
4
T
p
are
determinedbydissipation
at
a
2the
2,
in
2 +energy
The
and
the
minimum
injectionratemomeṗsynnormalization
=
p
m
c
p
e
e
e
2front
2 and
and
ward
shock
the
averageelectron energy.
3m
c
p
are
determinedbytheenergydissipation
rate at AsthinSuzuki,K.
Maeda,
andT.
Shigeyamaelly
Optically thick stage
qnon-thermal
density
assumedinTfront
many
emissionmodels
fo
ward
shock
and
the average electronenergy.
As us
in4
u
rad
2
2
2+p ,
and
ṗic SNe= in(e.g.,
p
m
c
cta
e
and
Sari
et
al.
1998;
Sari &models
Esin 2001;
assumedmany
non-thermalfor Gee emission
RS22qnverse Compton
emission. We calculate the nonmomentum
rate is
cal
3m
closs
ejecta
e
4
u
and
SNe
(e.g.,
Sari
et2 al.
1998;
Sari & Esin
2001;
G
FS
&
Sari
2002),
we
introduce
a2 parameter✏ e andassu
Tradgamma-ray2+
ghṗic = pe dV
p
m
c
p
,
emission following
the
early
high-energy
emission
e
e
e
2
shell
&
Sari= 2002),
weintroduceaparameter
assume2
a ṗwhere
fraction✏oftheinternal
energyof ✏the
gas
th
e and
is
the
Thomsoncross
section,
u
the
e
3m
c
T
B in
.
e
ad
the
methoddeveloped
by
SM18.
We
focus
on
the
obCSM
a
fraction✏ edtoftheinternal
energygas ind3Vthestreamofshock
is uconverted
to ofthetheenergyofthe
nonergydensity,andtheenergydensity
of
s
of
rad
CD
mal emission from the forward shock,becausethewhereof thetheThomsoncross
section,uBnon-ththemradius
Tu isshock
streamisconvertedtotheenergyofumelectrons,
=
✏
u
,
where
u
is
the
internalenee
int equation,
intEquation
advectionele
The Themagnetic
energydensity
is given (23),by uBis =n
rate at the forward shock
front
dominates
sissipationofergy
density,andutheenergydensityofseelectrons,uele= ✏ e ufront.
uaveragethe energy
internalofenergyradint , whereint is
Optically thin stage
sity
at
the
shock
The
a
sin
solvedbya
1st-order
implicit
upwindscheme.
density✏ B is another microphysics parameter specifying
RBt of the reverse shock (Section
2.1).
sity
at magnetic
theshock front.
Theaverageof auB
singleThe
energydensityis energy
giveninternal
by
=en✏
.g.,
thermalelectron
is
given
by
the
electronofthemagnetic
energydensity
to internal
the internalRSu:thermalphotons+
synchrotronrad➡
thermalelectron
is
given
by
the
electron
energ
ich
✏
is
anothermicrophysics
parameter
specifying
B
dividedbytheelectronnumberdensity
n
in
th
FS
ele
X-ray, radio
the
sity.
As
we
will
see
below,
thephotosphericanddividedbythe electron
numberdensityneleinternalin the dofthemagnetic
energydensity
to
the
2.4.2
Synchrotron
emission
photonsstream,u/n.
theele contributeele
lectron momentum distribution
opt-UVemissionto seedphotons for inverstream,u/n. see below,
ele
ele
ray
sity.TheAs momentum
we willtheṗsynphotosphericandloss
rates,
and
ṗ
,
for
sync
ic
da
The
momentumloss
rates,
ṗsyn andṗ energy, for synchrFora givenelectronmomentum
distribution,caemission.
Thus,
we
use
the
den
photospherephotosphere
non-thermal electrons
producedat theCD shockemissiontoseedradiation
photons
for invers
radius front
andinversecontribute
Compton cooling
can
be ic
calculated
f
gly
and
inverseComptoncooling
canbecalculated
fromofsynchrotron
emissivity
j
and
the
self-abso
tospheric Thus,andsynchrotron
photons
forgiven
urad
.byT
⌫,syn
vis- approximation. In other words, we assume thatoneemission.
we
use
the
radiationenergydens
corresponding
energyloss
rates.
Theyare
corresponding
loss rates. They areby w
15;
efficient ↵ energy
are straightforward.
Wegivenuse theNon-thermal X-ray and radio emission
10

A. Suzuki, K. Maeda, and T. Shigeyama➡ p=3.0, εe=0.08, εB=3x10-3➡

Suzuki, Maeda, & Shigeyama (2018)

Swift XRT observations + radio(NOEMA, ALMA, VLA )
γ-ray

➡

A★,out=0.5 (normal stellar wind)

X-ray(inverse Compton)

ρOpticalρej∝(Γβ)(photospheric)0ρej∝(Γβ)-nρcsm=Ar-2
Erel

Radio
(synchrotron)

r=3x1013cm radiusNon-thermal X-ray and radio emission➡

p=3.0, εe=0.08, εB=3x10-3➡

Swift XRT observations + radio(NOEMA, ALMA, VLA )➡

A★,out=0.5 (normal stellar wind)

14

Suzuki, Maeda,

Suzuki, Maeda, & Shigeyama (2018)ρρej∝(Γβ)0ρej∝(Γβ)-nρcsm=Ar-2

Erel
r=3x1013cm radiusFigure 10. Spectral energy distributions at tobs = 3 ⇥ 103 ,
104 , and 105 s from top to bottom. The luminosity per unit
frequency (or equivalently the flux density) is plotted for theSummary➡

low-luminosity GRBs: a sub-energetic
population of long GRBs➡

A part of them are really a distinct
population powered by CSM interaction.

➡

origin of (sub-)relativistic ejecta?

➡

There should be more llGRBs, but how
to distinguish between oﬀ-axis GRBs
and CSM-powered transients.

➡

Future deep and/or wide X-ray surveys
will unveil the hidden population.

Suzuki, Maeda, & Shigeyama (2018)