Spiral-arm instability:
Magnetic destabilization

Shigeki Inoue
Naoki Yoshida(Kavli IPMU / U. Tokyo)

Clumpy galaxies• Observed in the high-z universe （z > 1）
– clump clusters / chain galaxies
in the high-z

in the local universe～109 yr?
with HST

Guo et al. (2014)

Elmegreen et al. (2013)• `Clumpy’ galaxies can be formative stages of spiral galaxies.
– `Giant clumps’ (~ 108-9 M at the largest)– Clumpy galaxies account for ~ 30-50 % at z=1-3• Tadaki+14, Livermore+15, Guo+15

In this workshop last year…
• Isolated disc galaxy simulations• Gas + stellar discs• Isothermal gas (no star formation, no feedback)• Moving-mesh code: Arepo

𝒇𝒈𝒂𝒔 = 𝟎. 𝟐

𝒇𝒈𝒂𝒔 = 𝟎. 𝟐𝟓

In this workshop last year…
• One can obtain the dispersion relation for the perturbations,

PressureSelf-gravity

Coriolis force(cf. Takahashi et al. 2016)• When 𝜔2 < 0, the spiral is unstable.
• Considering this in the boundary case 𝜔2 = 0, the instability
criterion can be defined as• Spiral-arm instability(Inoue & Yoshida 2018)
Line-mass:

Half-width of spiral arm:

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

Demonstration
The fragmenting caseInoue & Yoshida (2018)

How does magnetic field
affect spiral arm?

Does magnetic field stabilize ordestabilize the arm?

If it destabilizes, it may drive clump
formation with high SFR.

Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.
 𝐵𝜃 ～1 𝜇G around the sun (e.g. Inoue & Tabara 1981, Mouschovias 1983).

Han (2017)

Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

BB𝑅

𝜙Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

 Radial perturbations

B𝑅
cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

𝜙Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

 Radial perturbations
 The magnetic pressure work against the perturbations.

B

Toroidal B-fields can stabilize radialperturbations by magnetic pressure.

𝑅
cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

𝜙Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

 Radial perturbations
 The magnetic pressure work against the perturbations.

B

Toroidal B-fields can stabilize radial
perturbations by magnetic pressure.

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

𝑅cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

𝜙Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

 Radial perturbations
 The magnetic pressure work against the perturbations.

B

𝜙Toroidal B-fields can stabilize radialperturbations by magnetic pressure.

 Azimuthal perturbations
Coriolis force

 The B-fields do nothing in φ-direction.
 But, work against Coriolis force.

𝑅cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

Toroidal magnetic fields in a disc galaxy
 Galactic B-fields are approximately toroidal and/or following spiral arms.

 Radial perturbations
 The magnetic pressure work against the perturbations.

B

𝜙Toroidal B-fields can stabilize radial
perturbations by magnetic pressure.

 Azimuthal perturbations
Coriolis force

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

𝑅cf. Elmegreen (1987, 1991), Kim & Ostriker (2001)

Set-up for the linear perturbation theory
 Now considering…
 Gravitational instability for azimuthal perturbations on an axisymmetricspiral (ring).
yR
𝜙spiral arm• Linear perturbation equationsx• 𝐴 → 𝐴0 + 𝛿𝐴
• consider the first-order terms

continuity:
R- and φ-momenta:(ideal) Faraday's law:Set-up for the linear perturbation theory
 Now considering…
 Gravitational instability for azimuthal perturbations on an axisymmetricspiral (ring).
yperturbations
𝛿𝐴 exp 𝑖 𝑘𝑅𝜙 − 𝜔𝑡R
𝜙spiral armx
continuity:

R- and φ-momenta:(ideal) Faraday's law:

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 axisymmetricspiral (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).

The dispersion relation of MHD
 One can obtain the dispersion relation for the perturbations,
Magnetics
Thermalpressure

Self-gravity
Coriolis forceBasically, a high 𝜔2 means a stable state.

 Strong magnetic field (large 𝑣𝐴 ) cancels Coriolis force.
 Coriolis force has a stabilizing effect.
 Canceling the stabilizing effect is a destabilizing effect.

 Toroidal magnetic fields can drive spiral-arm fragmentation.

Ideal MHD simulationsat t=400 Myr

How can we predict the spiral-arm instability One can obtain the dispersion relation for the perturbations,
Magnetics
PressureSelf-gravity
Coriolis force

 If 𝜔2 > 0 for all wavenumbers 𝑘, the arm is stable.
 If 𝜔2 < 0 for some wavenumbers 𝑘, the arm is unstable.
 If 𝜔2 is complex for some wavenumbers 𝑘, the arm is unstable.
 Growth rates of perturbations can be computed as 𝜔grow = 0 for stable perturbations.
 𝜔grow > 0 for unstable perturbations.

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-fieldIgnoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The fragmenting case𝛽𝑖𝑛𝑖 = 5

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Demonstration

Including B-field

Ignoring B-field

The stable case𝛽𝑖𝑛𝑖 = 100

Two-component model: gas and stars
 This analysis can be applied to a multi-component model Stars are not affected by magnetic fields.

 However, the dispersion relation becomes very complicated…
 The two-component dispersion relation isGasStars This is reduced to a sixth-order (bi-cubic) equation of 𝜔

Clump mass estimate
 A clump mass can be estimated from our linear analysis.
 Assuming typical values of clumpy disc galaxies
 Disc rotation velocity 𝑉𝜙 = 200 km/s ・Clump formation radius 𝑅 = 5 kpc
 Spiral arm width 𝑊 = 0.5 kpc
・High turbulence 𝜎~50 km/s
Typical𝜎~50 km/s

Magnetic case  The estimated mass 𝑴𝒄𝒍 ~𝟏𝟎𝟖 Mwith 𝛽 = 0.1
 Consistent with observed clumps

Clump mass The estimated mass is almost
independent from magnetic fields.

Non-magnetic casewith 𝛽 = ∞Velocity dispersion

Clump mass estimate
 A clump mass can be estimated from our linear analysis.
 Assuming typical values of clumpy disc galaxies
 Disc rotation velocity 𝑉𝜙 = 200 km/s ・Clump formation radius 𝑅 = 5 kpc
 Spiral arm width 𝑊 = 0.5 kpc

Velocity dispersion

𝛽 = 0.1

Surface density

Non-magnetic casewith 𝛽 = ∞Unstable
wavelength

Magnetic case  The estimated mass 𝑴𝒄𝒍 ~𝟏𝟎𝟖 Mwith 𝛽 = 0.1
 Consistent with observed clumps

Clump mass

Typical𝜎~50 km/s

𝛽=∞

𝛽=∞

𝛽 = 0.1

Summary Toroidal magnetic fields can destabilize spiral arms
and drive the formation of giant clumps.
 By canceling Coriolis force.
 Presence of magnetic fields may induce clump formation and stimulate starformation in spiral galaxies.

 In MHD simulations, our linear analysis can characterizefragmentation of spiral arms and predict giant clump formation.
 This implies that spiral-arm fragmentation is basically a linear process.
 The analysis becomes inaccurate if the magnetic field is ignored.
 Our analysis can also be applied to multi-component models.

 From our analysis, a typical clump mass is almost independent fromstrength of magnetic fields.
 The toroidal fields causes a wide and low-density region to collapse.
 The long wavelength and the low density compensate the dependence on Bfields, and result in formation of clumps with similar masses.