星形成銀河における円盤不安定性と星形成クランプの形成プロセス
Disc instability in star-forming galaxies andformation mechanisms of their giant clumps

井上 茂樹
カブリ数物連携宇宙研究機構 / 東京大学
Shigeki Inoue
Kavli IPMU / U TokyoI am going to talk about “giant clumps” in galaxies

“clump”
 is a very ambiguous word.
 e.g. gas clouds, star cluster, dwarf galaxies etc.

 In this talk, “clump” I refer to as is “giant clumps”
 found in gas-rich galaxies (mainly in high-z galaxies)
 massive star clusters containing much gas
 ～108 M at the largest, ～1 kpc in size.
 actively star-forming

Star formation in galaxies
 Colour – stellar mass relation
Quiescent/
Quenched

Star-forming

Schawinski et al. (2014)

Star formation in galaxies
 Colour – stellar mass relation
Quiescent/
Quenched

Red sequence

Schawinski et al. (2014)

Green valley

blue cloudStar-forming

Star formation in galaxies
 Colour – stellar mass relation

Ellipticals

Discs

Schawinski et al. (2014)

Star-formation “main sequence”
 Tight correlations between stellar mass and SFR
 Evolving with redshifts
 The origin is still a mystery

Whitaker et al. (2012)

Low-redshift

high-redshift

What do the “main-sequence” galaxies look like?
Wisnioski et al. (2015)

SFR [M / yr]

Stellar mass
distributions

Line-of-sight velocity distributions

Wisnioski et al. (2015)

Stellar mass [M]
Most of star-forming galaxies
are rotating discs.

Most of star formation takes
place in disc galaxies.

Many of SF galaxies are clumpy Clumpy galaxies• observed mainly at highredshifts z~2.
• Quite rare at low-redshifts.
Guo et al. (2014)

 The clumpy structures are detected with Hα and UV light.
 They are star-forming clumps
 ～108 M at the largest, ～1 kpc in sizeMany of SF galaxies are clumpy Clumpy galaxies

Guo et al. (2012)

each

total

Guo et al. (2014)

 The clumpy structures are detected with Hα and UV light.
 They are star-forming clumps
 ～108 M at the largest, ～1 kpc in size

 Each clump accounts for ～10% of the galactic total SFR.
 The total SFR in the clumps accounts for ～50% of the galactic SFR.
 although the SFR fraction depends on spatial resolutions
 see Dessauges-Zavadsky et al. 2016, Cava et al. 2017

Giant clumps are the main SF sites in gas-rich clumpy galaxies.

Clumpy fraction and cosmic SFR
Shibuya et al. (2016)
Clumpy fractions
Cosmic SFRD

Clumpy fraction and cosmic SFR
Shibuya et al. (2016)
Clumpy fractions
Cosmic SFRD

Chicken or Egg?
••

Formation of clump increases SFR?
Gas accretion (increase of SFR) triggers clump formation?

Fate of giant clumps: longevity or ephemerality?
 The long-lived scenario
 Noguchi 98, 99; Immeli+ 04; Bournaud+ 06, 08, 14;Ceverino+ 10, 12; Inoue+ 11, 12, 14; Forbs+ 13; Moody+ 14
 Observed giant clumps are gravitationally bound structures.
 migrate to the galactic centres by dynamical friction and finally form bulges.

 Possible if feedback processes are weak and/or clump masses are large.

 The short-lived scenario
 Genel+ 12; Hopkins+ 11; Buck+ 17; Oklopčić+ 17.
 Observed giant clumps are less-bound and/or tidally disrupted soon.
 cannot migrate to the galactic centres and are transient structures.

 Possible if feedback processes are strong and/or clump masses are small.

Still under debate

Clump formation and star formation
 Thus, formation mechanisms of giant clumps are closely
related to (global) physics of star formation in galaxies.
 What drives giant-clump formation?
 Gravitational instability (GI) Cosmological gas accretion
 Galactic mergers
 What suppress giant-clump formation? (why clumps disappear?) Disc stabilization by gas consumption and/or heating Growth of a massive bulge
 Cessation of galactic mergers
Shibuya et al. (2016)

Guo et al. (2012)

“Non-linear violent disc instability with highToomre’s Q in high-redshift clumpy discgalaxies”
MNRAS, 456, 2052 (2016)

SI, Avishai Dekel, Nir Mandelker, Daniel Ceverino,Frederic Bournaud, Joel PrimackWhy are they clumpy?
• It has been proposed;
 Galaxies are highly gas-rich (stream-fed) in their early formation stages.
 Cold gas discs in the galaxies are Toomre unstable (Noguchi 1998, 1999).

Clump formation is caused by ‘Toomre instability’
Stars

Gas

Inoue & Saitoh (2012)

Toomre instability• From a local and linear perturbation theory foraxisymmetric perturbations,• In observations,

Qgas
Genzel et al. (2011)
with SINFONI

𝐐 ≤ 𝟏 in observed clumpy discs.

Qgas

Toomre instability• However, dynamically unstable states cannot belong-lasting.
 The unstable regions will collapse soon, then their disc(inter-clump) regions turn over to stable states within
the local dynamical time-scales.

The observation might beunderestimating Toomre Q…
QToomre analysis in cosmological sims.

•Cosmological simulations
 Ceverino et al. (2010, 2013) using ART code 10pc-order resolution with radiation pressure.

How to measure 𝑄2𝑐𝑜𝑚𝑝• 2-component model (Romeo & Wiegert 2011)
 𝑄𝑔𝑎𝑠 =𝜅𝑔𝑎𝑠 𝜎𝑔𝑎𝑠
𝜋𝐺Σ𝑔𝑎𝑠,𝑄𝑠𝑡𝑎𝑟 =

𝜅𝑠𝑡𝑎𝑟 𝜎𝑠𝑡𝑎𝑟
3.36𝐺Σ𝑠𝑡𝑎𝑟

𝑄2𝑐𝑜𝑚𝑝 −1 = 𝑊𝑄𝑔𝑎𝑠 −1 + 𝑄𝑠𝑡𝑎𝑟 −1
ቐ𝑄2𝑐𝑜𝑚𝑝 −1 = 𝑄𝑔𝑎𝑠 −1 + 𝑊𝑄𝑠𝑡𝑎𝑟 −1

𝑊≡(𝑖𝑓 𝑄𝑔𝑎𝑠 > 𝑄𝑠𝑡𝑎𝑟 )(𝑖𝑓 𝑄𝑔𝑎𝑠 < 𝑄𝑠𝑡𝑎𝑟 )𝜎𝑔𝑎𝑠 𝜎𝑠𝑡𝑎𝑟

𝜎𝑔𝑎𝑠 2 +𝜎𝑠𝑡𝑎𝑟 2

 𝜎 is velocity dispersion (not sound speed).
 𝜅 is calculated from mean velocity fields of gas/star.
𝜅≡

2<𝑣𝜙 > 𝑑<𝑣𝜙 >
<𝑣𝜙 >
+
𝑅
𝑑𝑅
𝑅

 Young stars (age<100 Myr) are considered to be “gas “
 Bulge stars are removed ; 𝑗𝑧 /𝑗𝑚𝑎𝑥 < 0.7
 Gaussian smoothing with FWHM=1.2 kpc
 to focus on 𝑀𝑐𝑙𝑢𝑚𝑝 = 108−9 M A razor-thin disc model (which gives lower limits)

Cosmological simulations
• V07
•
•
•
•

𝑧 = 2.13
𝑀𝑣𝑖𝑟 = 8.8 × 1011 M
𝑀𝑠𝑡𝑎𝑟 = 5.6 × 1010 M
𝑓𝑔𝑎𝑠 = 0.18

• 𝐵Τ𝑇 = 0.37 (kinematic)• 𝑆𝐹𝑅 = 27.5 M yr-1

log Σgas [M pc-2]

log Σstars [M pc-2]

gas
stars
log Σ [M pc-2]

σ [km s-1]

log κ [km s-1 kpc-1]

log Q

Purple Q<1: linear instability
Blue Q=1-1.8: non-linear instability
Green Q=1.8-3: dissipative instability
Yellow, Red, Black: Q>3: stable state
White: imaginary κ (Q cannot be defined)log Σgas+stars [M pc-2]

log ΣSFR [M yr-1 kpc-2]

log Q2comp• Instability (Q<1) can only be seen in/around the clumps.
• Disc (inter-clump) regions seem to be stable (Q>2).
Purple Q<1: linear instability
Blue Q=1-1.8: non-linear instability
Green Q=1.8-3: dissipative instability
Yellow, Red, Black: Q>3: stable state
White: imaginary κ (Q cannot be defined)Cosmological simulations•V08 𝑧 = 1.33
 𝑀𝑣𝑖𝑟 = 6.0 × 1011 M 𝑀𝑠𝑡𝑎𝑟 = 1.9 × 1010 M 𝑓𝑔𝑎𝑠 = 0.42
 𝐵Τ𝑇 = 0.45 (kinematic)
 𝑆𝐹𝑅 = 33.1 M yr-1

log Σgas [M pc-2]

log Σstars [M pc-2]

log Σgas+stars [M pc-2]

log ΣSFR [M yr-1 kpc-2]

log Q2comp• Instability (Q<1) can only be seen in/around the clumps.
• Disc (inter-clump) regions seem to be stable (Q>2).
Purple Q<1: linear instability
Blue Q=1-1.8: non-linear instability
Green Q=1.8-3: dissipative instability
Yellow, Red, Black: Q>3: stable state
White: imaginary κ (Q cannot be defined)Cosmological simulations•V13
 𝑧 = 1.78
 𝑀𝑣𝑖𝑟 = 3.4 × 1011 M 𝑀𝑠𝑡𝑎𝑟 = 1.2 × 1010 M 𝑓𝑔𝑎𝑠 = 0.40
 𝐵Τ𝑇 = 0.46 (kinematic)
 𝑆𝐹𝑅 = 11.2 M yr-1

log Σgas [M pc-2]

log Σstars [M pc-2]

log Q2comp• Instability (Q<1) can only be seen in/around the clumps.
• Disc (inter-clump) regions seem to be stable (Q>2).
Purple Q<1: linear instability
Blue Q=1-1.8: non-linear instability
Green Q=1.8-3: dissipative instability
Yellow, Red, Black: Q>3: stable state
White: imaginary κ (Q cannot be defined)Non-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumpsNon-linear formation of clumps• Distributions of Q on proto-clumps.
 The initial masses Mclump > 108 M

Clump detection scheme(Mandelker+ 2014)
We trace clumps back intime and space, and thenwe look into proto-clumps
which are detected forthe first time.

Non-linear formation of clumps• Distributions of Q on proto-clumps.
 The initial masses Mclump > 108 M

Clump detection scheme(Mandelker+ 2014)
We trace clumps back intime and space, and thenwe look into proto-clumps
which are detected forthe first time.

Significant fractions of clumps start forming with Q>1.8

How do giant clumps form?
• Non-perturbative scenarios
Gas dissipation
𝑄𝑐𝑟𝑖𝑡 = 2 − 3 if gas cooling is rapid.(Elmegreen 2011)

How do giant clumps form?
• Non-perturbative scenarios
Gas dissipation
𝑄𝑐𝑟𝑖𝑡 = 2 − 3 if gas cooling is rapid.(Elmegreen 2011)
 Small-scale formation and merger/accretion growthQ can be <1 on small scales (e.g. Romeo et al 2010)
 Q-measurement can depend on physical scales, e.g. Larson low

 We applied the Gaussian smoothing with FWHM=1.2 kpc

A giant clump may form bymergers of small clumps.
(Behrendt et al. 2015)

How do giant clumps form?
• Non-perturbative scenarios
Gas dissipation
𝑄𝑐𝑟𝑖𝑡 = 2 − 3 if gas cooling is rapid.(Elmegreen 2011)
 Small-scale formation and merger/accretion growthQ can be <1 on small scales (e.g. Romeo et al 2010)
Non-axisymmetric perturbation
𝑚 ≠ 0 perturbations (Griv & Gedalin 2012) unstable up to 𝑄 ≅ 2 for finite thickness.

How do giant clumps form?
• Perturbative scenarios
Minor mergers
 Satellite accretion can disturb a disc.

Pre-existing clumps Clumps also disturb a disc and stimulate formation ofother clumps.

Cold stream flowing in a disc
 Streams can join a disc with slow or counter rotation.
 Slow rotation leads to low 𝜅How do giant clumps form?
• Perturbative scenarios
Minor mergers
 Satellite accretion can disturb a disc.

Pre-existing clumps Clumps also disturb a disc and stimulate formation ofother clumps.

Cold stream flowing in a disc
 Streams can join a disc with slow or counter rotation.

Compressive turbulence Compressing gas can indicate a high 𝜎 (i.e. high Q)
 But a clump will form there

High σ and high Q

But, stars will form

Toomre instability• In observations,

Qgas
Genzel et al. (2011)
with SINFONI

𝐐 ≤ 𝟏 in the observations.
𝐐 > 𝟏 in our simulations.

Qgas

Obs. vs. sims.
• Toomre Q of gas component.
• 𝐐 ≤ 𝟏 in the observations.

Genzel et al. (2011)～3-5 times lower in the obs.

Obs. vs. sims.
• Toomre Q of gas component.
• 𝐐 ≤ 𝟏 in the observations.
 The observation might be underestimating Q.
 Perhaps, it might be die to overestimation of gas density.
 The gas density in obs. is measured from Hα luminosity.

Hα(observable)

conversion factor

SFR

KS relation

Σ𝑔𝑎𝑠

 e.g.
 KS relation may be different in high-z galaxies…?
 Star-formation efficiency may be higher in high-z galaxies…?

Summary• Q>2-3 in disc (inter-clump) regions,• Q<1 inside/around giant clumps.
• Formation of new clumps can start with Q>2-3.

• Clump formation is NOT NECESSARILY
due to the (standard) Toomre instability.
• Maybe induced by other mechanisms.
 minor mergers, pre-existing clumps, cold streams, etc..

• There is tension between our sims and IFS obs.
 Gas density may be overestimated in IFS?

Transition of giant clump formation
mechanisms at z~2:from Toomre to spiral-arm instabilities
Beyond Toomre’s Q
MNRAS in press
arXiv: 1706.01895

Shigeki Inoue (Kavli IPMU / U Tokyo)
w/ Naoki Yoshida, Takatoshi Shibuya

Spiral or Clumpy?

• Clumpy galaxies• Giant clumps• Gas-rich (fgas>30%)

Guo et al. (2014)

Spiral-arm fragmentation?

Law et al. (2012)

Spiral galaxies emerge at z<2 (Elmegeen+14)

Elmegreen& Elmegreen (2014)

Spiral or Clumpy?
• Isolated disc galaxy simulations• Gas + stellar discs• Isothermal gas (no star formation, no feedback)• Moving-mesh code: Arepo

𝒇𝒈𝒂𝒔 = 𝟎. 𝟐

𝒇𝒈𝒂𝒔 = 𝟎. 𝟐𝟓

Sp i r a l - arm fr a g me ntati o n a s a c l u m p fo r m a ti o n m e c h a ni sm• Spiral arms can fragment into clumps,• if the gas fraction is high and/or the disc is kinematically cold.

• Spiral-arm fragmentation is not Toomre instability!
• Spiral-arm fragmentation could be a possible mechanism of giant clump formation.

Beyond Toomre’s Q• The aims of this study:• How spiral-arm fragmentation occurs?

• Derive an instability parameter and its criterion

linear perturbation theory for a spiral arm

Takahashi, Tsukamoto & Inutsuka (2016)
 Proto-planetary discs
 Spiral-arm fragmentation takes place only if 𝑄 < 0.6 on the spiral arm.

 Their linear perturbation analysis also supports this criterion.

Set-up for the linear perturbation theory• Now considering…
• Gravitational instability for azimuthal perturbations on an axisymmetric spiral (ring).

yperturbations
𝛿𝐴 exp 𝑖 𝑘𝑅𝜙 − 𝜔𝑡Rspiral arm

𝜙xcontinuity:

R-momentum:
φ-momentum:

Assuming:• The spiral has a rigid rotation since selfgravitating.
Ω = −𝐵• Replace surface density Σ with line-massΥ = 1.4𝑊Σ (Gaussian).

Set-up for the linear perturbation theory• Now considering…
• Gravitational instability for azimuthal perturbations on an axisymmetric spiral (ring).

yperturbations
𝛿𝐴 exp 𝑖 𝑘𝑅𝜙 − 𝜔𝑡Rspiral arm

𝜙xcontinuity:
R-momentum:

φ-momentum:

Assuming:• The spiral has a rigid rotation since selfgravitating.
Ω = −𝐵• Replace surface density Σ with line-massΥ = 1.4𝑊Σ (Gaussian).

A dispersion relation for a single -component model• One can obtain the dispersion relation for the perturbations,• The Poisson equation for the perturbations is
𝐾 : Bessel function
𝐿 : Struve function𝑓(𝑘𝑊)𝑊 : half width of arm

A dispersion relation for a single -component model• One can obtain the dispersion relation for the perturbations,(cf. Takahashi, Tsukamoto & Inutsuka 2016)• This can be transformed as• When 𝜔2 < 0, the spiral is unstable.
• Considering this in the boundary case 𝜔2 = 0, the new instability parameter andits criterion can be defined as

A dispersion relation for a two -component model• A galaxy usually has gas and stars. The dispersion relations of gas and stars are,gas:
stars:• Because gas and stars interact only through gravity, they are connected in the
Poisson eq.,

• Then, one can obtain the two-component dispersion relation,• Finally, I obtain the new instability condition for 2-comp. models,2

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The fragmenting case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable case

Demonstration
The stable caseClump mass estimation• If the unstable wavelength is long, λ ≫ 𝑊,
• An unstable perturbation is expected to collapse along
the arm (1D collapse)

Clump mass estimation• If the unstable wavelength is long, λ ≫ 𝑊,
• An unstable perturbation is expected to collapse along
the arm (1D collapse)•𝑀𝑐𝑙 ~Σ𝑊𝜆Clump-mass prediction•𝑀𝑐𝑙 ~Σ𝑊𝜆• 𝑴𝒄𝒍 ~𝟏𝟎𝟖 − 𝟏𝟎𝟖.𝟓 M (predicted)

Clump-mass prediction•𝑀𝑐𝑙 ~Σ𝑊𝜆• 𝑴𝒄𝒍 ~𝟏𝟎𝟖 − 𝟏𝟎𝟖.𝟓 M (predicted)

Toomre Instability (TI)
(e.g. Dekel et al. 2009)v.s.
Spiral-Arm Instability (SAI)(Inoue & Yoshida 2017)For low-z clumpy galaxies

TI vs SAI• Toomre Instability
Disc formation

Toomre instability
𝑄<1• Spiral-Arm Instability
Disc formation
Toomre instability and/or
swing amplification

Spiral arm formation
Spiral-Arm Instability
𝑆<1

Giant clump formation

Giant clump formation

TI vs SAI• Toomre Instability• Spiral-Arm Instability

TI vs SAI• Toomre Instability• 2D collapse• 𝑀𝑐𝑙 ~𝜋Σ 𝜆Τ2 2• Spiral-Arm Instability

TI vs SAI• Toomre Instability• Spiral-Arm Instability• 2D collapse• 1D collapse• 𝑀𝑐𝑙 ~𝜋Σ 𝜆Τ2 2• 𝑀𝑐𝑙 ~Σ𝑊𝜆

TI vs SAI• Toomre Instability• Spiral-Arm Instability• 2D collapse• 1D collapse• 𝑀𝑐𝑙 ~𝜋Σ 𝜆Τ2 2• 𝑀𝑐𝑙 ~Σ𝑊𝜆

When 𝑄 ≅ 1,

When 𝑆 ≅ 1,

Scaling relations of high -z clumps• From our analysis, we can obtain scaling relations of properties of giant clumps.

Spiral-arm instability
expected scaling relation:

1.3
𝜎cl
𝑅cl ∝
𝑅d
𝑉

Toomre instability
expected scaling relation:

𝜎cl
𝑅cl ∝
𝑅d
𝑉

𝑅cl : clump radius,𝜎cl :vel. disp. with in clump,𝑅d : disc radius,

𝑉: disc rot. vel.

Scaling relations of high -z clumps

redshift z~0
Data from Fisher+17: DYNAMO• Neither model is rejected by the observations.

Scaling relations of high -z clumps• From our analysis, we can obtain scaling relations of properties of giant clumps.

Spiral-arm instability
expected scaling relation:𝜂 ≈ 𝑓𝑔 : gas fraction including DM

Toomre instability
expected scaling relation:
𝑅cl : clump radius,𝜎cl :vel. disp. with in clump,𝑅d : disc radius,

𝑉: disc rot. vel.

Scaling relations of high -z clumps• Toomre Instability• Spiral-Arm Instability
Shibuya & SI (in prep.)redshiftredshiftz=0-1
z=0-1

redshift z=0-1

Data from Shibuya+16: HST @ z=0-1• Our SAI model appears better consistent withthe observations of 𝑀𝑐𝑙 Τ𝑀𝑑 ~10% clumps.

Transition of the clump formation mechanisms

high zToomre instability: Q < 1time

Possibly non-linear: Q > 2 − 3(Inoue+ 2016)z~2

The onset of spiral galaxies
Low+12 @ z=2.1
Elmegreen+14 @ z~1.8

z~1
Spiral-arm instability: 𝑆 < 1

z=0

Clumpy fraction and cosmic SFR
Clumpy fractions
Cosmic SFRD

• Shibuya et al. (2016)

Clumpy fraction and cosmic SFR
Onset of spiral galaxies
Clumpy fractions
Cosmic SFRD

• Shibuya et al. (2016)

Clumpy fraction and cosmic SFR
Onset of spiral galaxies
Clumpy fractions
Cosmic SFRD

Toomre instability: Q < 1

Spiral-arm instability: 𝑆 < 1• Shibuya et al. (2016)

Summary• Our SAI model appears better consistent with the
low-z observations.
• The TI model cannot reproduce the scaling relation of the observations
despite that the TI model relays on fewer assumptions than our SAI
model.

• There could be transition of clump formation mechanisms• @ z=2~1, from Toomre instability to spiral-arm instability

See arXiv: 1706.01895

A little about magnetic fields
for Toomre and spiral-arm instabilities

Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).

B

B

𝑅

𝜃

Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations

B

𝑅

𝜃

Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations
 The B-fields work against the perturbations.

B

Azimuthal B-fields can stabilize azimuthal
perturbations by magnetic pressure.

𝑅𝜃Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations
 The B-fields work against the perturbations.

B

Azimuthal B-fields can stabilize azimuthal
perturbations by magnetic pressure.

 Azimuthal perturbations
 The B-fields do nothing in azimuthal direction.

𝑅𝜃Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations
 The B-fields work against the perturbations.

B𝜃

Azimuthal B-fields can stabilize azimuthal
perturbations by magnetic pressure.

 Azimuthal perturbations
Coriolis force

 The B-fields do nothing in azimuthal direction.
 But, work against Coriolis force.
𝑅Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations
 The B-fields work against the perturbations.

B𝜃

Azimuthal B-fields can stabilize azimuthal
perturbations by magnetic pressure.

 Azimuthal perturbations
Coriolis force

 The B-fields do nothing in azimuthal direction.
 But, work against Coriolis force.
Azimuthal B-fields can destabilize azimuthal
perturbations by cancelling Coriolis force.

𝑅Toroidal magnetic fields in a disc galaxy
 The Galactic B-fields are approximately toroidal (Rand & Kulkarni 1989).
 𝐵𝜃 ～1 𝜇G (e.g. Inoue & Tabara 1981, Mouschovias 1983, ).
 Radial perturbations: Toomre instability
 The B-fields work against the perturbations.
Dispersion relation:(modified) Toomre Q:

 Azimuthal perturbations: Spiral-arm instability
 The B-fields do nothing in azimuthal direction.
 But, work against Coriolis force.
Dispersion relation:
cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

Ideal MHD simulation
 Gas + stellar discs (𝑓𝑔𝑎𝑠 = 0.2, isothermal gas)
 Initial conditions
No magnetic fields

Summary Assuming toroidal magnetic fields,
 The magnetic fields can suppress growth of radial
perturbations.
 Toomre instability can be mitigated.

 The magnetic fields can enhance growth of azimuthal
perturbations.
 By cancelling Coriolis force
 Therefore, spiral-arms can be destabilized by the magnetic fields.

 Giant-clump formation by spiral-arm fragmentation can beenhanced in toroidal magnetic fields in a spiral galaxy.