原始惑星系円盤におけるダスト偏光と磁場
Grain alignment in protoplanetary disks
Ryo Tazaki

Astronomical Institute, Tohoku University
Collaborators: A. Lazarian and H. Nomura
1. Introduction
2. Grain alignment in the ISM
3. Grain alignment in protoplanetary disks
4. Discussion : Alignment eﬃciency
5. Summary

1

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

原始惑星系円盤におけるダスト偏光と磁場
Grain alignment in protoplanetary disks
Ryo Tazaki

Astronomical Institute, Tohoku University
Collaborators: A. Lazarian and H. Nomura
1. Introduction
2. Grain alignment
the ISM
Takeinhome
message:torque inalignment
theory disks
predicts that
3. Radiative
Grain alignment
protoplanetaryalignment :axis
is not necessary
4. Discussion
Alignmenteﬃciency to be B-field!
5. Summary

2

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Grain alignment and polarization
ARTICLEIN PRESS
Lazarian(2007)
A. Lazarian / Journal of Quantitative Spectroscopy & Radiative Transfer 106 (2007) 225–256

FIR/mmopticalpolarization E ⊥ Bpolarization
E||
B
Polarization of starlight passing through a cloud of aligned dust grains. The direction of polarization (E) is parallel to the p(emission)
direction of magnetic
field. (b) Polarization of radiation from a optically thin cloudof aligned dust grains. The directio(absorption)n (E) is perpendicular to the plane of the sky direction of magnetic field.

3

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

RESEARCH
Our group is studying different aspects of the first stages of star formation. The main topics that our researchcovers are: (1) The study of thea magnetic fields in star forming regions from mm and sub-mmb6
spectro-polarimetric observations;(2) The study of the interaction process between the powerful protosetllar
winds and the dense molecular environment in star forming regions. In particular we are carrying out adetailed observational and theoretical study of the molecular regions illuminated by Herbig-haro objects; (3)The study of the properties 5of molecular clouds with and without star formation at high angular resolution. The
research is mainly done using interferometers (VLA, VLBA, SMA), and single-dish radio telescopes. The
research is being doing in collaboration with a large group of theoretical and observational astronomers: Drs.
4
R.Estalella, M.T. Beltrán (Universitat
de Barcelona), Prof. G. Anglada, Dr. Mayra Osorio, J.F Gómez (IAA,
CSIC; Spain), Dr. S.Viti, Prof. D.A.Williams (University College London), , O.Morata (LAEFF) and P.T.P.Ho(SMA). P. Palmeirim et al.: Herschel view of the Taurus B211/3 filament and striations: evidence of filamentary growth?

Tracing tool of B-ﬁeld structure

ISM
NGC 891

Annu. Rev. Astron. Astrophys. 2015.53:501-539. Downloaded from www.annualreviews.org
Access provided by Tokyo Institute of Technology on 01/19/16. For personal use only.

p (%)

molecular cloud

GRAPHICS

3

cloud core(~1000 au)
2

Klare et al. (1972)

High-angular-resolution measurements ofpolarized dust emission toward theAndersson& Potter(2005) system NGC 1333
low-mass
protostellar
H-bandIRAS 4A shows that the observed
6.2 cm0012
34
magnetic
field morphology
is in5agreement
AV (mag.)with the standard theoretical models of theformation of Sun-like stars in magnetized
Figure 1
molecular clouds at scales of a few
astronomical
units; gravity(a) The amount of polarization p as a functionhundredof extinction
AV is boundedby thehasrelation p/AV < 3% mag−1 (solid line), indicating acorrelation between polarization and dust. However,themagnetic
broad distribution
indicates
overcome
support, andthe that not all environments produce the sameamount of polarization. Reproduced from Andersson
with permissionof Springer Science+Business Media. (b) The orientation
magnetic(2015)
field traces
a clear hourglassof the optical/near-IR polarization (blue bars) is
systematicallyorthogonalthat of synchrotron radiation ( yellow bars), which itself isshape.
The magnetic
field isto
substantiallyknown to be orthogonal to the projected magnetic(courtesy
Jones). in the
more fieldimportant
thanT.J.
turbulenceFig. 3. a) Herschel/SPIRE 250 µm image of the B211/B213/L1495 region in Taurus. The light blue and purple curves show the crests of theevolutionof theand the vectors
initial from
B213 and B211 segments of the whole filament discussed in this paper, respectively. b) Displayof opticaland system,
infrared polarization
Heyer et al. (2008), Heiles (2000), and Chapman et al. (2011) tracing the magnetic field orientationinthe
B211/L1495
region,overlaid
on ourmisalignment of the magnetic and spin
Herschel/SPIRE 250 µm image. The plane-of-the-sky projection of the magnetic field appears to be oriented perpendicular to the B211/B213
axes
may
have
importantin lower
the right
filament and roughly aligned with the general direction of the striations overlaid in blue. The
green,
blue,and been
black segmentsin thecorner represent the average position angles of the polarization vectors, low-density striations,and B211offilament,
respectively.
formation
the binary
system.
Girart et al. 2006, Science, Vol. 313. no.
POLARIZATION REDUCTION
FACTORS
pp. 812- 814 images of star-forming
and most of the compact cores. Figure 2 shows that the B211 the filamentary5788,
structure
in HerschelProtoplanetarydisks(~ 100 au)

ALMA!
4

1

Andersson et al. 2015

Palmeirin+ (2013)

Girart+ (2006)

filament is surrounded by a large number of lower-density fila- clouds (e.g., Arzoumanian et al. 2011; Hill et al. 2011; Peretto
ments or striations which are oriented
roughly
perpendicularal. 2012; Schneider
et al. 2012).
with a rel- from theory. One way to capture thisSeveraleffects
reduce totheetobserved
polarization
fromUsingthe DisPerSE
ideal estimatedthe main filament. It is important to stress that the striations can ative “persistence” threshold of 1021 cm 2 (⇠5 in the map –information is the polarization reduction factor (i.e., Lee & Draine 1985),
also be seen in the original maps (cf. Figs. 1a and 3a) and that see Sousbie 2011 for the formal definition of “persistence”) andthe curvelet transform was merely used to enhance their con- an absolute column density threshold of 1–22⇥ 1021 cm 2 , we
γ , lower density! =filament
RF costrast. Some of these striations are also visible in the 12 CO(1–0) could trace the crests of the B211
and 441300
1/6
and CO(1–0) maps of Goldsmith et al. (2008) at 45 resolu- filamentary structures (see Fig. 3). Due to di↵ering backgroundwhere
γ is theinclinationof theonmagnetic
the lineof sight,
R is1a),theweRayleigh reduction factor (a function oftion. Following André et al. (2010),
the column
densitymap of levels
either side fieldof theto
B211/3filament(see Fig.
Fig.R.
2 was
also
convertedtoanapproximate
map
of
mass
per
adopteddi↵erent
columndensity
thresholdsonthenorth-eastern
Tazaki星形成と銀河構造における磁場の役割
the angle between the magnetic
field and the precession axis of the grain; Greenberg鹿児島大学,
1968), and F is2017.12.22
the reduction
unit length along the filaments by multiplying the local column side (2 ⇥ 1021 cm 2 ) and south-western side (1021 cm 2 ). The re-Linear polarization of HL Tau by ALMA
＊ E-vectors are shown!

Scattering polarization
HL Tau (λ=870 µm)

Kataoka et al.

alignment…?

AASTEX wavelength-dependent polarization

Model (Kataoka+15’)

HL Tau (λ=3.1 mm)

Stephens+17’

Kataoka+17’

100 AU.
ALMA polarimetric observations at 3 mm (top,Figure2. Comparisonof the polarization images between
1.3 mm=2 ]. (Center)2 ], (CARMA Stephens et al. 2014) andetal.
2017),of1.3(middle),and
870 µm[Jymmarcsec
s in units
The(bottom),polarized
intensity map in units of [mJy =arcsec
red vectors show the >3
polarization(i.e.,The ALMA image is smoothed to have the same beam size of CARMA, where the beam
(ALMA,
this morphology
observation).
ndegree
in
units
of
%,
overlaidwiththepolarization
vectors, which is the same as the middle
ave not been rotated). Vector00lengths are
00 linearly propor0.65
⇥
0.56
with
the
PA
of
79.5
degrees. The color scale represents the polarized intensity while the grey contours r
P . The color scale shows the polarized intensity, which isonly show 3 detections.
I contours
in each panel
theStokes
continuum
emission.
The levels of the grey contours are (10, 20, 40, 80, 160, 320, 640, 1280) ⇥ I where I = 2.1 mJ
for [3, 10, 25, 50, 100, 200, 325, 500, 750, 1000] ⇥ I , where
for the CARMA data and= 0.017 mJy/beam ALMA data.
154, and 460 µJy bm 1 for 3 mm, 1.3 mm, and 870 µm,I
ly. 5
鹿児島大学, 2017.12.22
R. Tazaki
星形成と銀河構造における磁場の役割

Does grain alignment with B-field really happen?

anics

reproduces the multiple-ring structure observed with ALMA.

Outline
1. Introduction
2. Grain alignment in the ISM
3. Grain alignment in protoplanetary disks
4. Discussion : Alignment eﬃciency
5. Summary

6

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Classical theory: Paramagnetic dissipation
Davis & Greenstein (1951), see also Spitzer (1978)•

Dissipation results in non-zero magnetic torque (M B), which acts
to reduce the component of angular momentum perpendicular to B.

Initial stateJ

Final state

J
J

J||

※ assume angular momentum vector || minor axisMJ(so-called “internal alignment”)•

But, this alignment process takes place slowly…

tDG
7

1.5

a
10 yr
0.1 µmR. Tazaki

62>> gas collision(damping) timescale
星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

4.3 DDSCAT calculations

Fig. 4 presents the test grain shapes that we hfor using DDSCAT. Parameters for calculations
Shapes 1, 2 and 3 have been used in DW97
shapes 4 and 5. In addition, we created a mirro
shape 1, namely, shape 1* and provided the DD
a spheroidal grain (see more details in Table 1)
functions for astronomical silicate in which aviolet is removed (see DW97; Weingartner &Lazarian 2005).
We discussed for AMO that the sign of hel
by taking the mirror image of the grain. We
procedure to the irregular grains and obtained
ones obtained for AMO (see Figs 7 and 8).
Note that we observe that Qe1 and Qe2 chwhen we calculate torques for a mirror image othe shape 1 has one type of helicity, while shsymmetric image of shape 1, that is, shape 1*,
helicity.

Fast alignment mechanism? → radiative torque (RAT)•

Dolginov & Mitrophanov (1976):
Spin-up of helical grains by left- and right-handed photons•

Draine & Weingartner (1996, 1997)

RAT is important! RATs from anisotropic
radiation field result in rapid grain alignment
with B-field even in the absence of theparamagnetic dissipation!
https://web.astro.princeton.edu/people/bruce-draine

Lazarian & Hoang (2007)•

AA53CH13-Andersson

ARI

23 July 2015

16:20

a

RAT alignment can be understood by asimple helical grain model! The role of RAT
is spin-up (down), alignment, and induce
precession. RATs often tend to spin-down
the grains (see also Weingartner & Draine 2003).

b

A helical grain–AMO
e^ 2

^

e2

β

a^2

^

a1

8

http://www.astro.wisc.edu/~lazarian/R. Tazaki

om www.annualreviews.org
6. For personal use only.

e^3

Figure 3

a^ 1
l1

e^1

k

l2
α

Radiation beam

Φ

Θ

N

a^ 2

e^ 1
e^ 3

Figure 4. Geometry of grains under study: shapes 1
an ellipsoidal shape.

(a) The orientation of a grain, described by three principal axes â1 , â2 , and â3 , in the laboratory coordinate system (scattering referenc
system with axes ê1 , ê2 , and ê3 ) is defined by three angles, !, β, and ", in which the direction of the incident photon beam k is alongê1 . The symmetry axis of the grain, â1 , is at an angle ! with respect to the radiation, and " is the precession angle of the grain axis⃝⃝
CC
The Authors.
Journalcompilation
R
about the radiation direction. (b) A model of a helical grain used2007
to analyticallycompute radiativealignment
torques induced 2007
by

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

alignment torque F is perpendicular to J, the fast precession of the grain causes the average valueof F to approach zero as J approaches B (i.e., as ξ decreases). Thus the positions corresponding
to J parallel with B (ξ = 0 and ξ = π) are stationary points, regardless of the functional forms of
the radiative torques. The detailed dynamics of the grain, including whether the stationary points
Lazarian & Hoang 2007
are stable, attractor points, require the full solution of the equations of motion (cf. Draine &Weingartner 1997, Lazarian & Hoang 2007a). Addressing the question of whether the alignment
Helicalgrainto B requires use of the actual AMO expressions
can occur bwith
J perpendicular
for the effect”
torque“Barnett
A helical grain–AMOaeefficiencies with respect
tothel radiation direction. The narrow range of Magnetization
radiation-to-magneticdue
N

Overview of RAT alignment

July 2015

16:20

^

^

2

2

2

α

Radiation beamβΦF is alignment torque (⊥ to J )e
H is spin-up torque (II to J ) e

a^1

l1

^

Θ

to grain spinning

^
a
1

1

k

^

1

Grain (© M Matsuo)e^ 3Magnetic field
Magnetic
fieldradiation with respect to the magnetic field. The numerical calculations forlar grain shapes in Draine & Weingartner (1997) indicated that the alignment
the long axes perpendicular to the magnetic field, for most directions of the, similar to DG alignment. However, this study did not consider grain flipping
Additionally, because the radiative torques for the three grain shapes were sonclear what exactly caused the tendency to produce the correct alignment
ndency was a universal feature of RATs. A subsequent numerical study byaine (2003) completed the description of the alignment process in Draine &), establishing that RATs not only accelerate grains but can also decelerate
m into states of slow, near-thermal, rotation rates.
lations provide evidence that RATs are important in grain dynamics and yieldwith observations for a limited number of specific cases. However, they do notselves to identifying the underlying physical mechanisms of the grain behavior
pendence on grain and radiation field characteristics. Resolving this problem
merical approach, by calculating the interaction between hundreds of grainn of varying wavelength, would be prohibitively difficult.
as resolved by Lazarian & Hoang (2007a), who circumvented the need forof the grain shape by identifying the helicity of an irregular grain, interactings the key property for evaluating the effects of RATs. Because neither sphericalate/oblate grains have helicity, the Lazarian & Hoang (2007a) model uses anweightless mirror attached at an angle to its body (Figure 3b), a shape thatIn this analytical model (AMO), changing the angle at which the mirror isboth left-handed and right-handed model grains. It is obviously not possibleroperties of irregular grains with such a simple model. In particular the model

am
e
tb
h
Lig

ISRF

Einstein-de Haas effect)Ba grain, described by three principal axes â1 , â2 , and â3 , in the laboratory coordinate system (scattering reference

2 , and ê3 ) is defined by three angles, !, β, and ", in which the direction of the incident photon beam k is alongs of the grain, â1 , is at an angle ! with respect to the radiation, and " is the precession angle of the grain axis
rection. (b) A model of a helical grain used to analytically compute radiative alignment torques induced bycattering coordinate system, in which ! = 0 for simplicity only. A perfectly reflecting oblate spheroid isless mirror of length l2 by a rod l1 . The vector normal to the mirror, N, lies in the plane â1 â2 with angle α to â2 .
duced from Lazarian & Hoang (2007a). Abbreviation: AMO, analytical model.

Stationary
points:(Inverse
effect of
F = 0 for ξ = 0 or π

Ψ

J||a1

ξ

Magnetic moment

Fast precession
Larmorprecession
about B

Falignment
torque FH

spin-up (down)
torque H

dF/dξ < 0 is an attractor point
If spin-up torques H > 0, we have high-J attractor
Larmor precession + radiative torqueIf spin-up torques H < 0, we have zero-J attractor→ alignment with respect to B
＊ precession axis = alignment axis

Andersson et al. 2015

9

Figure 5

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Test of RAT alignment by observations
RadiativeAndersson
torques: analytical model et915al. 2015
see e.g.,•! = 90◦ , we have

Table 1. Grain shapes and parameters for calculation of RATs.

In the ISM, larger grains are better aligned.
Q abs · e2 ∼ k cos βRe!

[ p j E inc ]eikx j .

(22)Grain shapes

Dipole #Size (µm)

Shape 1
Shape 1∗
Shape 2
Shape 3
Shape 4
Shape 5
Hollow 1

83200053248
45056
10257015000
11000
8320000.05–0.2
0.2
0.2
0.2
0.2
0.2
1.0

Wavelength (µm)

Helicityj

Since x j = r sin αsin β, it follows

ISRF
1.2
ISRF
ISRF
1.2
1.2
0.1

Right-handed
Left-handed
Left-handed
Left-handed
Left-handed
Left-handed
Right-handed- minimum aligned grain size: a ~ 0.045 µm (Kim & Martin 1995)
Q abs · e2 ∼•!

cos β cos(A sin β)[ p j E inc ],

(23)

j

where the term [p j Einc ] is a function that is independent of β. It is
obvious that Qabs · e2 is a function of β which is zero when averaging
is performed for β over [0, 2π ]. We calculated Qabs · e2 for different
λ/a and grain shapes, and found that Qe2 is indeed close to zero atcos ! = 0, which is consistent with our analytical expectation.

In RAT alignment, small grains do not align.
AA53CH13-Andersson

ARI

23 July 2015

For all calculations here, we adopt the dielectric function for astronomicalsilicate.

16:20- minimum aligned grain size : ~ λ(rad ﬁeld)/2 (Lazarian & Hoang 2007).
- short wavelength end of ISRF: λ=912 Å (Lyman limit)
Fig. 5 provides a comparison of normalized RATs between AMO
and DDSCAT calculations performed for two irregular grains inducedby monochromatic radiation field of λ = 0.2 µ m (see more in SecFig. 4 presents the test grain shapes that we have calculated RATs
tion 4.4). It can be seen that they possess the same symmetric properfor using DDSCAT. Parameters for calculations are given in Table 1.
ties as well as zero-points. Also, the functional form of normalized
Shapes 1, 2 and 3 have been used in DW97. We added to themQ
e1 and Qe2 calculated for the irregular grains and AMO are reshapes 4 and 5. In addition, we created a mirror symmetric shape of
markably similar, in particular for the Qe2 component. Typically,
shape 1, namely, shape 1* and provided the DDSCAT calculations for
the RAT components for shape 2 are similar to those of AMO witha spheroidal grain (see more details in Table 1). We adopt dielectric
max
max
Qe1
/Qe2
= 1 ratio, while RATs of shape 4 are similar to thosefunctions for astronomical silicate in which a feature in the ultramax
max
of
AMO
with Qe1/Qe2= 0.78. Hence, by changing the ratio of
violet is removed (see DW97; Weingartner & Draine 2001; Cho &2
amplitudes
of
the
RAT
components
for AMO, we can obtain an1
10Lazarian 2005).
alytical expressions of RATs for a number of irregular Total
grains.grain
To size
We discussed for AMO that the sign of helicity can be changed
have RATs appropriate to irregular grains, it is necessary
touseby taking the mirror image of the grain. We performed a similar
distribution
Θ = 0˚Shape 1, astronomicalsilicate
1 functional
DDSCAT to estimate the magnitude of RATs. Combining
procedure to the irregular grains and obtained results similar to the(scaled)forms from AMO and magnitude from DDSCAT, we can obtain anones obtained for AMO (see0.9
Figs 7 and 8).
alytical approximate expressions for RATs components of irreguNote that we observe that Qe1 and Qe2 change synchronously0
lar grains. Note that in Figs 3 and 5, we normalized RATs over
10whenwe calculate torques for a mirror image of a grain. We see thatmax
|Qe1
| = |Qe1 (! = 0)|, so Qmax
the shape 1 has one type of helicity, while shapes 4, 5 and mirror
e1 is the same for all realizations
0.8
of AMO and the irregular grains. It is easy to see that with thissymmetric image of shape 1, that is, shape 1*, have another type of
0.1grains, but getschoice, AMO reproduces very well Qe2 for irregularhelicity.
4.3 DDSCAT calculations

10–6
0.1

10

0.7

aeff = 0.32 µm
0.6µm
aeff = 0.5
aeff = 1.0 µm
aeff = 3.0 µm
Fit with0.5α = 3
Fit with α = 4
0.4

AV = 0A V = 0.4

A V = 2.4
A V = 210 1.0 10.01
a ≈ 0.045 µm
10–3

2

3

10.04

λ/aeff AV (mag)Figure 11

R. Tazaki

5

6

7
100.0

0.01

0.1

Andersson et al. 2015references thereinfrom www.annualreviews.org
9/16. For personal use only.

10–4

b

m (a)a

10–2

a

λ**
max (µm)QΓb

1<a> (µm)

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22(a) The wavelength of maximum polarization, λmax , is correlated with the extinction, AV . Different color points correspond to data

Outline
1. Introduction
2. Grain alignment in the ISM
3. Grain alignment in protoplanetary disks
4. Discussion : Alignment eﬃciency
5. Summary

11

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Grain alignment in PPDs (Previous works)
Cho & Lazarian (2007)

Bertrang+ (2016)

Based on the old RAT alignment theory

MHD simulation+ perfectly aligned grains

B

Figure 1. The ring state: Self-consistent radiative tra• InFig.
PPDs,
radiationcanbestrongis because
thepresence
10.—Simulated
observations.
Degreeof polarization
calculated1.3
formm
theof
total
radiation(i.e.,
interiorþof
surface)from theobservations
disk. The disk in
(top
row),
and
simulated
ALMA
angle between disk symmetry axis and the line of sight.
minimal displayedpolarization
degree is 1%, i.e., the 3
the central star. → RAT alignment is expected!
(Cho &
Lazarian 2007)
right edge-on). The color map shows the total (unpolariwith the spatial• Too optimistic conditions for grain alignment
withresolution
B-ﬁeldindicated by the beam siz
characteristic polarization structure in all of the three oare used in previous studies.

e.g., grain precession is NOT included in Cho & Lazarian 07 model.
12

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

taken from Miyake & Nakagawa (1993), which is a mixture of silicate andH2 O ice.

Grain alignment in PPDs (Our work)

Surface density profile of dust grain in disk is assumed toRT,
LazarianandNomura(2017)
be self-similar solution, such that⎡ %& Hoang• We apply RAT alignment theory(Lazarian% &−γ&2−γ ⎤ 2007)
⎢
⎥
M
R
R
⎢
⎥
d expected polarization⎢⎥⎥⎥
to PPDs, andcalculate
the
map
⎢⎢⎣−
Σd = (2 − γ)
exp
(30)
⎦
2 R
Rc
2πR
c
c
to be compared with ALMA.

・ whereRadiative
transfer
calculation
Dullemond
Md isthe disk
dust mass,(RADMC-3D,
rc is the cut-oﬀ
radius,etRal.is2012)
the

disk
radius.
For simplicity,
vertical
distribution
of
dust
grain
Centralstar：T-Tauri
star (4000K,
2Rsun)√
2
2-4
isDiskassumedtobegaussian,
ρ
=
Σ
/(
2πH
)
exp[−(z/2H
mass: 10 Msun
d
d
d
d )],
where Hd isEstimate
the dust scaleheight.
The
parametersaresumthe magnitude of radiative torquemerized in Table 2. Note that the temperature structure andat eachof self-consistently.
the disk
density structure
doeslocation
not treated
In this pawe assume
・ per,
Strengthof toroidal magnetic ﬁeld (Okuzumi et al. 2014)- R .−2
B(R) = 10 µG(31)
100 AU(Okuzumi
et al. 2014). We assume
magnetic field is鹿児島大学,
constant
13
2017.12.22
R. Tazaki
星形成と銀河構造における磁場の役割

Can grains align with B-ﬁeld in PPDs?
••

Smaller grains do not aligndue to ineﬃcient RAT.
At r=50 au and midplane,
grain size > 20 µm can align.

Alignmentis possiblea > ~20 µm
Radiative alignment

o
e
s
Ga

~ λ(rad. ﬁeld)/2

d
us

a

in
mp

g

•

For such grain sizes, Larmor
precession is suppressed by
the gaseous damping.

La
rm
or
pr
ec
es
si
on

~FIR@midplane

us
o
e
s
Ga pingmdaGrain alignment
with B-ﬁeld
hardly occur!
14

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

We discussed for AMO that the sign of hel
by taking the mirror image of the grain. We
procedure to the irregular grains and obtained
ones obtained for AMO (see Figs 7 and 8).
Note that we observe that Qe1 and Qe2 chwhen we calculate torques for a mirror image othe shape 1 has one type of helicity, while shsymmetric image of shape 1, that is, shape 1*,
helicity.

RAT alignment in the absence of B-ﬁeld
Lazarian & Hoang 2007•

RAT alignment with B → 0

Normalized angular momentum- No Larmor precession = No B-ﬁeld alignment- RAT induces precession about radiative ﬂux.
torques: analytical model→ alignment with respect to radiative ﬂuxRadiative
happens.

where $B is the Larmor precession rate of J around thefield B. Here F, H, G are RAT components projected to tξˆ, Ĵ , φ̂, which are given by (see DW97)
F(ξ, ψ, φ) = Q e1 (%, 0)[− sin ψ cos ξ cos φ − cos ψ sin+ Q e2 (%, 0)[cos &(cos ψ cos ξ cos φ

Initial state− sin ψ sin ξ ) + sinFinal& cos ξstatesin φ]+ Q e3 (%, 0)[cos & cos ξ sin φ + sin &(sin− cos ψ cos ξ cos φ)],

-1.0

1.0

0.0

H (ξ, ψ, φ) = Q e1 (%,Figure0)[−4.sin
ψ sin ξ cos φ + cos ψ cos
Geometry of grains under study: shapes 1
radiative flux || minor axis
radiative flux
an ellipsoidal shape.
+ Q e2 (%, 0)[cos &(sin ψ cos ξ

Lazarian & Hoang 2007
cos
Figure 15. Phase trajectory map for grain shape 1 for ψ = 0 shows the◦+ cos ψ⃝C 2007
sin ξThecosφ) +Journal
sin &compilationsin ξ sin⃝C φ],
Authors.
2007 R

G(ξ, ψ, φ) = Q e1 (%, 0)[sin ψ鹿児島大学,
sin φ]
152017.12.22
R. Tazaki
alignment with one high-J
attractor point C, two low-J attractor 星形成と銀河構造における磁場の役割
points A

Alignment with radiation direction•

RAT induce precession

Alignmentis possiblea > ~20 µm

around radiation direction.
(Lazarian & Hoang 2007)•

Radiative alignment

For grains > 20 µm,
rad. precession is faster

o
e
s
Ga

than Larmor precession and

d
us

a

in
mp

g

gaseous damping timescales.

Initial stateradiative flux
16

R. Tazaki

La
rm
or
pr
ec
es
si
on

→ “radiation alignment”
Final state

us
o
e
s
Ga pingmdaRadiative
precession

radiative flux || minor axis
星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Millimeter wave polarization of disks
Previous
study
14(assumed B-field alignment)

Bertrang et al. 2016••

Tazaki et al.

Our study(Based on the alignment theory)

RT+2017

Figure 1. The ring state: Self-consistent radiative transfer simulations of the polarized emission of
Alignment
theorypredicts
withrow;
radiation
direction
1.3 mm (top row),and simulated
ALMAalignment
observations (bottom
CASA v5.4.2,
band 6, configura
minimal displayed polarization degree is 1%, i.e., the 3 level of ALMA) at three di↵erent disk orien
happens,
which results in azimuthal pol. vectors.
right edge-on). The color map shows the total (unpolarized) intensity which is overplotted by polarizat
with the spatial resolution indicated by the beam size given as white ellipses. The toroidal magne
Polarization
vectors at (sub-)mm wavelengths do notcharacteristic polarization structure in all of the three orientations in the simulated ALMA observatio

trace the magnetic ﬁeld structure in PPDs!
17

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Millimeter wave polarization of disks
Previous
study
14(assumed B-field alignment)

Bertrang et al. 2016

Tazaki et al.

Our study(Based on the alignment theory)

RT+2017

Figure 1. The ring state: Self-consistent radiative transfer simulations of the polarized emission of
1.3Short
mm (topsummary:row), and simulated
ALMA
observations(bottom row; CASA
v5.4.2, band 6, configura
Whyis B-fieldalignment
so inefficient?
minimal displayed polarization degree is 1%, i.e., the 3 level of ALMA) at three di↵erent disk orien
Alinedgrainsize~
λ(rad.
field)/2@
disk
midplaneright edge-on). The color map shows the total (unpolarized) intensity which is overplotted by polarizat~ a fewtens
of size
micronISMellipses.
dust ~The
0.1µm)
with the spatial resolution indicatedby thebeam
given (>>
as white
toroidal magne
characteristic
polarization
structurein all of Larmor
the three orientationsin thesimulated- Larger
grains show
slowerprecession,
andthen ALMA observatio

the Larmor precession is suppressed by gaseous collisions.
18

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Radiative grain alignment in
In MIR, we 8may observe magnetically aligned
grains!
Tazaki et
al.

λ[µm]

Larmor precession timescaletLa
1.3 year
0.1 µm

2

B
5 µG

1

(0)
10 4

1

λ~µm
λ~100µm

grain size being aligned 〜 λ(rad field)/2

When does B-field alignment happen?
- Fast Larmor precession
forfieldgrainsbeing
Figure 4. The radiationin the disk.
(a) Energy density of the radiation urad [erg cm3 ] (b) the spectrum
averaged anisotropy parameter γ are shown.
aligned.
B- Less gaseous parameter
damping(lowdensity)in the disk. Above the disk photosphere, the enand the gaseous dam
Disk ergy
surfaceis favorable
densitylayerof radiation
is dominated by the stellar radiar = 50 AU, the radia
4tion.
Hence,
atthesurfacelayer,u≈AT∆Ω∗ , whereradfield emitted from afor B-field alignment!
−15
−3
−4
A = 7.57 × 10
and then19

erg cmK

is the radiation constant,and then for smaller
diative torques beco
! R "−2
ure 5 shows that at−6
We can expect toroidaluradB-fieldalignment
grain size is ≈ 27
≃ 5.2 × 10
erg cm−3(51)
10 AU
a millimeter sized gin MIR wavelength!
plane, although thesfor T ∗ = 4000 K and R∗ = 2R⊙ . The anisotropy parameter(see Figure 2).
At tunity, and ⃗k ispointing
Figure9. Same radiallyas Figure 7, but
for diﬀerent magnetic
properties
鹿児島大学,
2017.12.22
R. Tazaki of the radiation field is almost
星形成と銀河構造における磁場の役割
lower
that
paramagnetic inclusions model (f =0.1, φ than= 0), theand (ii)
supmm-wave polarization of PPDs
Alignment with
ALMA
Alignment
with
AASTEX wavelength-dependent
polarization
Tazaki et al.
(Band 3, λ=3.1 mm)radiation direction
magnetic field
RT+2017

Kataoka+2017

Bertrang et al. 2016

100 AUigure 1. The ring state: Self-consistent radiative transfer simulations of the polarized emission of aligned aspherical d
3 mm (top row), andsimulated
ALMA observations(bottom
row;between
CASA v5.4.2,
band
6, configuration
1.5handinte
Figure2. Comparisonof the polarization
images= 1.3 mm
(CARMA
Stephens etC40-5,
al. 2014)
inimal displayed polarization
is 1%, The
i.e., ALMA
the 3 image
level isofsmoothed
ALMA) toat havethreethedi↵erent
disksizeorientations
face-on(ALMA, this degree
observation).
same beamof CARMA,(left:
where
the b
00
•
0.6500 map⇥ 0.56showswiththethe total
PA of(unpolarized)
79.5 degrees. The
color scale
represents
the polarized
intensity while
the grey contou
ght edge-on).
The color
intensitywhich
is overplottedby polarization
vectors.The
vecto
the continuum
emission.
Thebeamlevels sizeof thegiven
grey contours(10, 20, 40,
80,toroidal
160, 320, 640,1280) ⇥field
ith the spatial resolution
indicatedby theas whiteare
ellipses.
The
magnetic
topologyisI whereI = 2.1- seems
to CARMA
be consistent
with
grain
alignment
withobservation.
radiative ﬂux.
for thedata
andof Ithe
= 0.017
mJy/beam
ALMA
data.
haracteristic
polarization
structurein allthreeorientations
inthesimulated ALMAHL Tau in Band 3 shows azimuthal polarization vectors.

20

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

k¼
wn)

perpendicular to both axes. Then, the cross sections Cx 0 and Cy 0 are"!
"!
Cx0 ¼ Cavg þ 13 R C? ! Ck 1 ! 3 cos2 & ;!
"Cy0 ¼ Cavg þ 13 R C? ! Ck ;

for
gral

MIR polarization observations

where Cavg ¼ (2C? þ Ck )/3, C? and Ck are cross sections with respect to themagnetic field, and & is the angle between the magnetic field and the plane of the sky.
We assume the Rayleigh reduction factor, R, is 1. For the x- and y-axes that also lie
in the plane of the sky,Cx ' Cx 0 cos2 $ þ Cy 0 sin2 $;Cy ' Cx 0 sin2 $ þ Cy 0 cos2 $;19Þ

Cho & Lazarian 2007

where $ is the angle between the x- and x0-axis.

D. Li et al. Li et al. 201712

5

Polarization at 10.3 µm
Absorptive component
Emissive component

4

3
CQ TauVV Ser2

1
MWC 29700

r the total radiation (i.e., interior þ surface) from the disk. The disk inclination angle $ is the•21

CL07 Model

Fractional polarization (per cent)620

AB AurHD 179218

40
60
Disk inclination (degree)80

MIR polarization obs. traces
B-ﬁeldofof
Figuretoroidal
4. Mid-IR polarization
thesurface
sample. The layer
dash line ispolarized emission at 10 µm predicted by Cho & Lazarian (2007)of PPDs?
for a flared disc, assuming a toroidal B-field and RAT alignment.
R. Tazaki

The horizontal
line is the mean 10-µm polarization
of YSOsin鹿児島大学,
2017.12.22星形成と銀河構造における磁場の役割

Outline
1. Introduction
2. Grain alignment in the ISM
3. Grain alignment in protoplanetary disks
4. Discussion : Alignment eﬃciency
5. Summary

22

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

due to evaporation is
ignment (Hoang & Lazarian 2009b).
evaporated molecules e
nal alignment, or internal relaxation, is a process where
ning dust grains. The
n axis for the maximum
inertiaalignswiththeangular
Coupling of J and minormomentum
axis becomes ze
tum vector (Purcell 1979). The rotational kinetic energy
dy 2,•canbe given by(see Lazarian& Roberge
1997)
No.
2009
RADIATIVETORQUES ALIGNMENT
Rotational
kinetic
energyHoang & Lazarian 2009 t 1321
gas =2of the grain at the gas temperature
Tgas , T̂d = Td /15 K with Td
Jbeing the dust
andE (qtemperature,)=(1 + sin2 q (h - 1)) ,(1 )!
!
" ωI∣∣τ #2 $−2" ω τ #2 $−2
1 n
1 el
f (J ) = 1.6105 1 ++ 0.47 1 +.
2
2»8.1 ´ 10• (Internal) Energy dissipation leads(16) with respect toh = I∣∣ I^, I∣∣ and I⊥ represent the inertia
5
to
θ→
0
(“Internalalignment”)
5
fortheJ = angular
Jth . InFor
grainslarger
than
1
µm,
f
(J
)
≈
1.6
×
10
nhe minor
axis
and
major
axis,
J
is
momentum
t
~10
where
align,int
sec rg is the gas deabove equations ω1 = J cos θ/I∥ and τn is given byminor% −1 axis&â1 and(17)rain, and θ is τthe= 1/
angle
betweenJ . The rotational−1
2kB Tg mmH , where m
τ
+
τ
,
n
ne
nn
%
&
energy
of
a
body
can
into
heat
by
internal
2 be22 converted−4−3
is a geometrical coefﬁcwhere τne = 3 × 10 (2.7/gn ) 10 cm /ne s and τnn =h̄/(3.8g
0.58τne (nthes, with electron
and nucleus of a body J is 0.1 µm
dissipation.
momentumn nn µn ) ≈ Sincee /nn )angulardensity ne and nn , are the relaxation timesJof interaction of3
ed when weandconsider
torque-freemotionnucleus–electron
nucleus–nucleus
spins, respectively.
Also,of a rigid body,G∣∣ ={3 + 4 (1 - e−11τdissipation
sleadsis the spin–spin
relaxation
timeofelectronicel = 2.9 × 10
Figuregreatest
4. Timescales as functions of grain sizefor internal relaxation (t )to alignment of the axis of the16
spins, and cos θ = 1/2 is chosen (see LD99; HL09).
radiative (t ), and gas damping (t ) calculated for J = J and for the diffuse•
Largergrainsdonotshowinternalalignment.
ISM
withstandard parameters: gas density n = 30 cm , gas temperatureof the grainswith
its
angularmomentum
vector.
Various3.3. Radiative Alignment Timescale
T = 100 K, and dust temperature T = 20 K. Filled circles denote wheret &= tLazarian.
dissipation
processes
soaccelerate
far been
proposed:
Barnett• characteristic
Without
internalalignment2009):
The
timescaleforhaveRATstothe(Hoanggrain
ω is estimatedbyfrom J(Purcell= 0 to J = I11979),
onnuclearrelaxation&- alignment eﬃciency:~ 10% @(Lazarian
High-J
attractor(spin-up state)
For
λ
<
1.8a,
⟨H
⟩
∼
0.4,
φ
=
g
˜(
e
)
I1 ω
I1 ω
m
1999a), superparamagnetic(Lazariantrad ==,Barnett relaxation
(18)
Γrad
M⟨H ⟩φ' (3 state)~ 100% @ Low-J
attractor (spin-down
1.5
T̂
ω
1
t
rad
−1.5 −6.5 gas
ine
1999a;
Hoang&
Lazarian
2008),
andinelastic= 70ρ̂ŝ â−5f (J ).
(23)
where ⟨H ⟩φ is the spin-up component of RATs that is averagedtintωT̂d ûrad λ̂2 T
2. W
over23the torque-freemotionandtheprecession
angle
φ
using
e
=
1
s
where
on
(Purcell1979;
Lazarian&Efroimsky1999).
鹿児島大学,
2017.12.22
R. Tazaki
星形成と銀河構造における磁場の役割
m
int

rad

gas

int

gas

th
−3

d

rad

gular momentum smaller than the value of high-J attractor point. Filed circles show initial position of grains. Filled arrow shows the time interval of
0.5tgas . Some grains get aligned at low-J attractor point over a short time t< 0.5tgas , some others get to the high-J attractor points over t ∼ 3tgas .

We discussed for AMO that the sign of helicity can be cha
by taking the mirror image of the grain. We performed a sim
procedure to the irregular grains and obtained results similar tones obtained for AMO (see Figs 7 and 8).
Note that we observe that Qe1 and Qe2 change synchrono
when we calculate torques for a mirror image of a grain. We see
the shape 1 has one type of helicity, while shapes 4, 5 and msymmetric image of shape 1, that is, shape 1*, have another tyhelicity.

Alignment eﬃciency : High-J or Low-J?
Lazarian & Hoang 2007

Helicity

λ=ISRF
B-ﬁeldλ=1.2 µm

Figure 25. Upper panel: The shift ofmax /Qmax for different ψfunction of Qe1
e2
low-J attractor point with respect to ψ
max /Qmax
Radiation
Qe1
e2 and for irregular grains. T
degree of alignment R = 1.5 cos2 ξ −Figure 4. Geometry of grains under study: shapes 1–3 are similar to tanalignment.
ellipsoidal shape.

ψmax /Qmax for which• For⃝ 2007 The Authors.
⃝ 2007 RAS, MNRAS 378
grainalignment
with
rad.
see
ψ=0.
Journal compilation
Figure24. Ratioof Qe1the phase
map direction,
has high- or low-Jwe shoulde2attractor points as a function of ψ. Curved solid lines show predictions by• AboveamountAMO,
solid lines condition(maps with low-Jdepends
attractor pointson
only),thedottedlines (maps of magnetic inclusions,attractor point coincides to cos ξwith high-J
points) show the result for irregulargrains.
suchattractor
as superparamagnetic
inclusions(Lazarian & Hoang 2008, Hoang & Lazarian 2016).
C

24

R. TazakiCwith long axes perpendicular to ma
鹿児島大学, 2017.12.22星形成と銀河構造における磁場の役割
tendency is consistent
with our ear

ones obtained for AMO (see Figs 7 and 8).
Note that we observe that Qe1 and Qe2 change
when we calculate torques for a mirror image of a gr
the shape 1 has one type of helicity, while shapes
symmetric image of shape 1, that is, shape 1*, havehelicity.

Example: RAT alignment calculation

Low-J attractor (partial alignment ~ 20%)

Downloaded from http://mnras.oxfordjournals.org/ at (D) To

J/J_thermal

Hoang & Lazarian 2008

High-J attractor(perfect alignment~ 100%)

Figure 4. Geometry of grains under study: shapes 1–3 aran ellipsoidal shape.

cos( J, B)⃝
C 2007 The Authors. Journal compilation ⃝C 2007 RAS, M

Figure 16. Phase trajectory maps for λ = 1.2 µm and a = 0.2 µm, and• Most grains
evolve into Low-J attractor …
ψ = 70◦ corresponding to no thermal fluctuations (upper panel) and with→alignmenteﬃciencyof ~20%thermalfluctuations(lower panel).
For the case in which J is fixed, whengrains are driven to the crossover J = 0, a 1 flips over. So grains frequently
25

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

J / I1ωT

15

Toward the higher alignment eﬃciency…
10

Hoang & Lazarian
2016, (see also Hoang & Lazarian 2008)
5

Low-J attractor → High-J attractor
0

(c)

(d)

15

J / I1ωT

J/J_thermal

20

10

5

0

−1.0

−0.5

0.0

cosξ

0.5

−0.5

1.0

cos( J, B)

•

0.0

cosξ

0.5

Fig. 4.— Same as Figure 3, but for the case stochastic excitations by collisions and magnetic fluctuations are co
represents grains heading to the high-J attractor point (red circles), while blue arrows represent grains follow cyc
presence of the high-J attractor, grains at the low-J attractor are strongly disturbed and excited to the high-J attra
whereas grains follows perturbed cyclic trajectories in the absence of high-J attractors (see (a) and (b) panels).

If high-J
attractors
arepresent,thestochastic
perturbation,suchastimestep as follows:
and purple lines in (a)), we can seealignment
is lower,atlevel of R̃ ∼gas bombardment, bringsthegrainsat
Low-J
attractors
tohigh-JN! (3 cos2 ξi − 1)lowing disturbed trajectories, the gr
Q̃J =perfect alignment,(29)
attractor! → almost
occurs!
fraction of time in the vicinity of tw
Ngr
gr

i

26

1

R. Tazaki
R̃ =

Ngr
!
(3 cos2 ξi − 1)Q̃X (Ji )of cos ξ = ±1,w which results in analignment.

星形成と銀河構造における磁場の役割(30)

鹿児島大学, 2017.12.22

Outline
1. Introduction
2. Grain alignment in the ISM
3. Grain alignment in protoplanetary disks
4. Discussion : Alignment eﬃciency
5. Summary

27

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22

Summary•

Now ALMA starts to observe linear polarizationof protoplanetary disks in (sub-)mm-wavelengths.

•

In disks, dust grains in midplane do not align with B-ﬁeld.
- Mainly because large grains show slow Larmor precession.
→ALMA observations may not provide B-ﬁeld structure of PPDs.

•

RAT alignment theory predicts that large grains in the disk
midplane may align with radiation direction instead of B-ﬁeld.

•

Magnetically aligned grains can present at surface layer of thedisks, which can be veriﬁed by the MIR polarimetric obs.

28

R. Tazaki

星形成と銀河構造における磁場の役割

鹿児島大学, 2017.12.22