渦状腕の線形摂動理論：
渦状腕分裂によるクランプ形成.
Beyond Toomre’s Q
MNRAS submitted
arXiv: 1706.01895 （←昨日！！）

井上 茂樹 (CREST/SPPEXA fellow)
カブリ数物連携宇宙研究機構 / 東京大学
w/ 吉田 直紀 (東京大学)

Spiral or Clumpy?
at low-z• Spiral galaxies• Low redshifts• Gas-rich (fgas~10%)

Toomre instability

at high-z• Clumpy galaxies• Giant clumps• High redshifts (mainly)• Gas-rich (fgas~30%)

Toomre instability
with HST

Guo et al. (2014)

really…?

What was Toomre instability???

• From local and linear perturbation theory foraxisymmetric perturbations in smooth discs,
Velocity dispersion
or sound speed(pressure)

Epicyclic frequency(Coriolis force)#$
The stability condition: 𝑄 ≡>1%&'
Surface density(self-gravity)

If Q<1, the local region isgravitationally unstable,
going to collapse.

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

S pi ral -a rm fra g me n ta ti o n a s a c l u mp fo rma ti o n me c h a n i s m• Spiral arms can fragment into clumps,• if a gas fraction is high and/or a 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• Discuss if the fragmentation can form high-z clumps

Let’s go to linear perturbation theory for a spiral arm!!
(線形摂動理論)

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

yperturbation
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).

yperturbation
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 𝜔K < 0, the spiral is unstable.
• Hence, the new instability parameter and its 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 thePoisson eq.,
• Then, one can obtain the two-component dispersion relation,• Finally, I obtain the new instability condition for 2-comp. models,Demonstration

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Clump mass estimation• An unstable perturbation is expected to collapse
along the arm (1D collapse)

Clump mass estimation• An unstable perturbation is expected to collapse
along the arm (1D collapse)•𝑀OP ~Σ𝑊𝜆Clump-mass prediction• An unstable perturbation is expected to collapse
along the arm (1D collapse)•𝑀OP ~Σ𝑊𝜆• 𝑴𝒄𝒍 ~𝟏𝟎𝟖 − 𝟏𝟎𝟖.𝟓 M8 (predicted)

Limitations of our models• Our simulations are ideal• Isolated models• Isothermal gas• No star formation or stellar feedback• Very thin discs• Our analytic model is also ideal• Razor-thin spiral arms
• Tight-winding approximation• Equilibrium for spiral arms
• Gaussian density distribution assumed• We also find a few non-linear fragmentation• Clump formation can occur even if the instability condition is not satisfied.
• But, non-linear fragmentation is quite rare in our simulations.

Violent Disc Instability (VDI)(based on Toomre instability; Dekel et al. 2009)v.s.
Spiral-Arm Instability (SAI)VDI vs SAI• Violent Disc 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

VDI vs SAI• Violent Disc Instability• Spiral-Arm Instability

VDI vs SAI• Violent Disc Instability• 2D collapse• 𝑀OP ~𝜋Σ 𝜆⁄2 K• Spiral-Arm Instability

VDI vs SAI• Violent Disc Instability• Spiral-Arm Instability• 2D collapse• 1D collapse• 𝑀OP ~𝜋Σ 𝜆⁄2 K• 𝑀OP ~Σ𝑊𝜆

Scaling relations of high-z clumps• From our analysis, we can obtain scaling relations of properties of giant clumps.
the most unstable wavelength:
expected clump mass:
expected velocity dispersion
within a clump:
Spiral-arm instability
expected scaling relation:b.c𝜎\]𝑅\] ∝𝑅a𝑉Toomre instability
expected scaling relation:𝜎\]𝑅\] ∝𝑅a𝑉𝑅\] : clump radius,𝜎\] :vel. disp. with in clump,𝑅a : disc radius,

𝑉: disc rot. vel.

Scaling relations of high-z clumps• Both models of the spiral-arm and Toomre instability can explain the observations.
• Neither is rejected by the observations.

• Our spiral-arm instability theory can be a possible
mechanism of giant clump formation in high-z galaxies.

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:

Toomre instability
expected scaling relation:
𝑅\] : clump radius,𝜎\] :vel. disp. with in clump,𝑅a : disc radius,

𝑉: disc rot. vel.

Scaling relations of high-z clumps• Violent Disc Instability• Spiral-Arm Instability• The observations may prefer our SAI model.
• But, the sample size is too small.
• If the most gas-poor one is excluded, the fittings become comparable in both models.

Summary• We analytically derived an instability parameter and its criterion for spiralarm fragmentation.

• Our novel instability parameter can characterize remarkably well
fragmentation of spiral arms and clump formation following.
• Neither model of our spiral-arm nor Toomre instability is inconsistent withcurrent observations.
• The spiral-arm instability could be a possible mechanism of giant
clump formation in high-redshift galaxies.

• It is also interesting to adopt our instability analysis tolocal spiral galaxies.