EHTによるSgr A*への落下ガス雲の直接撮像を用いたブラックホールスピン測定
Kotaro Moriyama1, Shin Mineshige2,
Mareki Honma3, Kazu Akiyama1, 4

1: MIT Haystack observatory,
2: Kyoto university,
3: NAOJ,
4: 2017 Jansky Fellow

Moriyama et al. in press, arXiv:1910.10713

Goal and Background
Goal： Observational proof of the black hole spacetime to test of general relativity theory

Spacetime is uniquely determined by the black hole mass and spin: (M, a)
M ：Relatively accurately estimated by observing the motions of stars and gas(e.g., Ozel et al. 2010; Gravity Collaboration et al. 2018; EHT collaboration et al. 2019)
EHT collaboration et al. (2019)a： NOT easy to measure↑ Important to consider the relativistic effectnear the black hole(e.g., Abbott et al. 2017; Broderick et al. 2016; EHT collaboration et al. 2019)

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!2

Sgr A* Imaging: Advantages and Difficulties
Sgr A*

Quantities

EHT

50μas

Black hole mass [M⊙ ]Distance [kpc]Apparent size [µas]Period at ISCO (a/M = 0)

Credit: H. Shiokawa

Sgr A*
4 × 106

850
31 minM87*
6.5 × 1091.68 × 10

440

37 day(e.g., Gravity Collaboration et al. 2018; EHT collaboration et al. 2019)

Advantages:

・Largest apparent diameter among all black hole candidates (~50 μas)
・Relativistic flux variation (~30 min-1hr) → Black hole spin indicator?

Lu et al. 2018

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!3

Gas Cloud Falling Toward the Black Hole
ObserverRed shifted photon (De-beaming)

Innermost stable orbit (rms)Blue shifted photon (Beaming)

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!4′

4.05050.40

0.5316

1/3
exp(−1.8899xM3 ),

1/6
1/2
OF
NUMERICAL
Gas cloud model
xM
xM
xM
IONS
chord angle,
δφ,
and
it
emits
radiation
with
the
power
Emissivity
model
where
K
is
the
modified
Besselfunctionofthesecond2
ud modellaw spectrum. The
emissivity
intheinertial
frameofDistance fromconstant,the central axis
kind, k is the Boltzmann’s
B is the magnetic
the gas cloud is expressedbyc ray-tracing method
the T
gas cloudphoton indexdensity,
field, and ne , e, me , ofande are the number!
"
Mineshige (2015), we
charge, mass, and temperature of the 2electron,
respecΓ+1j(νem ) = j0 exp − (R/R0 (φ)) νem ,of a gas cloud and tratively.
itational field (see also
R0 (φ) = Rcloud
[1
−
(φ
−
φ
)/δφ]
,
(1)
cloud
Photonof the gas
2004). We postulate
2.3. MethodsThicknessof numericalcalculation
frequency
cloud (ring or arc)
ccretion disk is tidally
where
R isthetheemissivity
radial distancethe central
the
Using
profilefrom[Equations(1) axis
and of(2)],
ue to the azimuthal vearc-shaped
gastransfer
cloud, R
thicknessthe radiativeequation
written as
cloud is aischaracteristic
Radiation transfer equationof the gas cloud. Then,of the cross section of thegas+∆ν
cloud, νem is the photon% νobse, keeping its original
frequency,∆I(νj0 is a )numericalconstant,Γisthephoton3=g
j(ν/g)∆ℓdν,(3)
obs
stulate that the initial
index of the power-law spectrum,
the region of [φcloud −νobss set to be 0.98 ri , andδφ, φcloud + δφ] corresponds to the azimuthal region ofning
the accretion
motion of disk.
theof the
where
I(ν
)
is
the
intensityofraysreaching
theobobs
relativistic
eﬀect
the arc-shaped gas cloud, and φ is the azimuthal angle
plane.
Here,
xobsanand
oud
model
with
arc
server, νobs is the observed photon frequency, ∆ν is theof the emission point.

e observer’s plane, theI (xM ) =

1+

+
1/4

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!5

Gas clouds falling onto the black hole∘a/M = 0.9, viewing angle = 75 inner edge of the disk = rISCO

rg = GM/c 2

rg /c = 20sec (SgrA*) = 0.5day (M87)

rg = 5μas (SgrA*) = 3.8μas (M87)32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!6

Spin Indicator: Photon Rotation and Time Lag

Simulated movie

rg /c = 20 sec (SgrA*) = 0.4 day (M87*)
rg = 5 μas (SgrA*) = 3.8 μas (M87*)

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan

!7

⟨

Norma

he
0.8
0.4
Light
curve
of a gas ring
l0.6
BI
0.2

rg /c = 20sec (SgrA*) = 0.5day (M87)
rg = 5μas (SgrA*) = 3.8μas (M87)0rocedures are as
0 follows:
10 20 30 40 50
ct/rg’s
cedures in Moriyama, et al.
ch

Total radiation
DirectSecondary

0.6
2.0
0.4
1.8
0.2
1.6
0.0
0
10
20
30
40
50
3
(c)

2

1.2

1.0

⟨g⟩

0.4
(a)
1.0
2.4 (d)
0.8
2.2

π⟩

⟨φray /2π⟩⟨rray /rg ⟩ flux
Normalized

0.0
or observations, ∆ℓ is the in3of a ray in Boyer-Lindquist(c)energy-shift
factor, g, is writ.
µwhereucloud is2the four velocnnatthe gas cloud, and pµ is theitted photon. 1We fix the valbdamental quantities of VLBIg-Light curve of a ring

0.8

0.6

0.4
2.4

ct/rg
g⟩

Simulated movie

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan

2.2!8

Light curve of an arc-shaped gas model

Figure 3. Snapshots of the direct radiation [panels (a) and (b)] and secondary component [panel (c)] of a gas ring, where
the parameters are the same as those in Figure 2. The abscissa is xobs /rg , while the ordinate is yobs /rg , and the color contours
represent the relative brightness of the gas-cloud image, ∆f (xobs , yobs , t)/∆fnorm , where the normalized brightness, ∆fnorm , isat the brightest position of the direct component, (xobs /rg , yobs /rg ) = (−3.1, 0.7). Here, the spacial regions of panels (b) and(c) correspond to the dotted rectangular frame region in panel (a).

Normalized flux

Single arc gas cloud(a)
1.0

Total (Ring)

0.8

Direct (arc)Secondary

Various initial phase(b)

Align maximum peaks(c)

φ0 /π = 0.00.6

φ0 /π = 0.0

φ0 /π = 0.5

φ0 /π = 0.5

φ0 /π = 1.0

φ0 /π = 1.0

φ0 /π = 1.5

φ0 /π = 1.5

0.4
0.2
0.0

0

10

20

30

ct/rg

40

50 0

10

20

30

ct/rg

40

50 0

10

20

30

40

50

ct/rg

Figure 4. Flux variation of the arc-shaped gas clouds where the arc chord angle is δφ = 2π/3, φo is the initial azimuthal angle,

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan!9

Spin Dependency of the Echo of Radiation
i=75°

45

(a)

2.5

40

a/M = 0.98

2.0

0.9

1.5
1.0

0.6

Time interval
Photon period

30
25
20

0.5
0.0

(b)

35
cδt/rgNormalized flux

3.0

0
0

10 20 30 40 50 60
ct/rg

15

0.0 0.2 0.4 0.6 0.8 1.0
a/M
Moriyama et al. in press

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !10

Depency of Echo on Other Parameters
Thickness

Spectrum index

Arc-chord angle

Viewing angle

Infalling velocity

Rotation velocity

a/M=0.9i=75°

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !11reases,andSynthetic
so the period of photon
EHT
Observation (2017-2020)
ases. Therefore, the estimated spiner than Gas
theclouds
corresponding
true
ones
with various initial
parameters(u, v) coverage for SgrA*
velocities are based on GRMHD/RIAF )in Figure(initial
7b).
Expected EHT in 2017-2020

loud has no angular momentum
the estimated spin value is larger
nce the beaming eﬀect outside rph
the light bending eﬀect near the
r orbit is dominant (see blue square
Here, we cannot estimate aset in the
M = 0 and 0.2, since the peaks due
g eﬀect do not appear and each flux
decreases. We can find the peaks
atrue /M = 0.6 − 0.9, since the gas
Moriyama et al. in press
ational velocity near the black holeme-dragging eﬀect.
32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !12
rg /c = 20 sec (SgrA*) = 0.4 day (M87*)
rg = 5 μas (SgrA*) = 3.8 μas (M87*)

MK
SMT
AZ
SPT
SP

4700

Spin Dependenceof Superposed Light
Curve
SP
9000
AZ

11000

11000

9000

Baseline: ALMA/APEX-JCMT/SMA
Synthetic light curve

Align by maximum peaks

Superposed light curve

Unique dependenceof the spin∘a/M = 0.9, viewing angle = 75 inner edge of the disk = rISCO

Moriyama et al. in press

◦
◦
ation
for
infallinggas
clouds
for
the
case
with
(a/M,
i)
=
(0.9,
75
):
(a)
spatialemissivity
9. Synthetic EHT observation for infalling gas clouds for the case with (a/M, i) = (0.9, 75 ): (a) spatial emissivity
32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !13

Estimated Spin Values and Detectability

Detectability (%)

1.0 (a)Baseline: ALMA-JCMT/SMA
100 (b)aest/M

0.8
0.6
0.4
0.2
0.0

0.0 0.2 0.4 0.6 0.8 1.0
atrue/M

80
60
40
20
0.0 0.2 0.4 0.6 0.8 1.0
atrue/M

Other baselines also provides similar estimation for spin values
Moriyama et al. in press

Figure 10. The accuracy of the estimated black hole spin

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !14

Table 3. Estimated spin value of each baseline
baselines

aest /M for atrue /M = 0.2

atrue /M = 0.6

atrue /M = 0.9

(1)

(2)

(3)

(4)

ALMA-JCMT/SMA

0.39 (0.10-0.66)

0.56 (0.33-0.77)

0.86 (0.82-0.90)

ALMA-SMT

0.33 (-0.00-0.62)

0.56 (0.31-0.76)

0.87 (0.82-0.90)

JCMT/SMA-SMT

0.34 (0.04-0.62)

0.62 (0.37-0.80)

0.89 (0.83-0.94)

ALMA- PV 30 m

0.27 (-0.03-0.59)

0.57 (0.34-0.75)

0.86 (0.82-0.90)

ALMA-LMT

0.30 (0.01-0.61)

0.55 (0.31-0.75)

0.86 (0.82-0.90)

ALMA-SPT

0.31 (0.01-0.61)

0.54 (0.28-0.73)

0.86 (0.82-0.90)

ALMA-NOEMA

0.31 (-0.01-0.62)

0.56 (0.34-0.76)

0.87 (0.82-0.90)

LMT-JCMT/SMA

0.33 (0.02-0.63)

0.56 (0.34-0.74)

0.87 (0.84-0.92)

LMT- PV 30 m

0.29 (-0.01-0.62)

0.56 (0.34-0.75)

0.86 (0.82-0.90)

LMT-SPT

0.33 (0.04-0.62)

0.57 (0.33-0.77)

0.87 (0.84-0.92)

LMT-SMT

0.31 (0.02-0.60)

0.57 (0.31-0.75)

0.87 (0.84-0.92)

NOEMA-SPT

0.35 (0.03-0.63)

0.58 (0.32-0.77)

0.88 (0.84-0.92)

PV 30 m-SPT

0.35 (0.05-0.67)

0.58 (0.34-0.77)

0.88 (0.84-0.92)

PV 30 m-NOEMA

0.31 (0.02-0.60)

0.56 (0.31-0.75)

0.87 (0.84-0.92)

JCMT/SMA-SPT

0.38 (0.05-0.68)

0.62 (0.36-0.81)

0.89 (0.84-0.94)

SMT-SPT

0.39 (0.05-0.69)

0.65 (0.40-0.84)

0.91 (0.85-0.97)

Note— Top three lines show the estimated spin values for redundant site, and bottomlines corresponds to other ones.

EHT Movie for Infalling Gas Clouds
Rotating gas cloud around the black hole
(1 period = 30 min, Broderick & Loeb (2006)

Synthetic movie using full complex visibility(EHT imaging software is SMILI)

Basic idea:
Johnson et al. 2017

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !16

Summary and Future Issues
3DGRMHD simulation

Summary:• Method for black hole spin measurement with EHT
observations• Investigate validity of the spin measurement based on
2017-2020 EHT synthetic observation

Next issues for GRMHD + Real observations:(Used in EHT Collaboration et al. 2019)• Investigate properties of infalling gas clouds• Investigate the accuracy of the black hole spin estimations

Moriyama et al. in press, arXiv:1910.10713
32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !17

Uncertainty in the inner edge of the accretion disk

32th Rironkon Symposium, 12/25, 2019, NAOJ Mitaka, Tokyo, Japan !18