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Contents §0 Introduction: Importance of Galactic Bars §1 General Observed Properties of Barred Galaxies §2 Dynamical Effects of Bars

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§3 Dynamics of Bars

Numerical simulations have extensively been used to clarify bar dynamics.

3.1 Bar Instability
◆ Ostriker & Peebles (1973, ApJ, 186, 467)

'Rotating massive disks are generally unstable and produce bars spontaneously'

Ostriker-Peeble criterion : Trot/|W| > 0.14  unstable

where Trot = kinetic energy, W = potential energy

How to suppress instability

1)Increase random motion (and reduce rotation) in disk so that Trot

2)Increase halo mass so that $W ↗
◆ Hohl (1976,AJ,81,30)

Stabilizing effect of halos

Disk: (Q=1)

Halo: rigid sphere of uniform density


1)point-mass like halo needs 60% of total mass to suppress bar instability.

2)more extended halo is more effective, requiring 40% of total mass.

◆ Athanassoula & Sellwood (1986, MN,221,213)

Stabilizing effects of random motions as well as massive halos


Disk : Σ(r)=(+1)-3/2 (cutt off at 〜5a)

(M: total mass, f: disk mass fraction, a: disk scale length)

Bulge (rigid): ρ(r)=( +1)-5/2 [M(1-f): bulge mass, b: bulge scale length = a]

Distribution function (equilibrium solution of collisionless Boltzmann equation)

1) Miyamoto models (1971, PASJ, 23,21)

2) Kalnajs models (1976, ApJ, 205, 751)
Bar was extracted by Fourier analysis for early evolutionary states

Exponential growth rate was determined for linear growth phases

 growth rate = 0.169q-0.092-0.048

Conclusions: pressure support (stellar random motions) reduces the halo mass required to inhibit bar instability in real galaxies.
◆ Berman & Mark (1979,AA,77,31)

‘Small but compact’ bulges are sufficient to stabilize disks

(i.e., massive haloes not necessary)
- pure disk simulation (Kuzmin disk with Q=1)  strong bar

- inner disk part was replaced by rigid Plummer sphere (i.e. bulge) with mass 0.3 and scale length 1/ of the disk (i.e., disk mass = 0.7) no bar

3.2 Parameter dependence of bar properties

◆ Sellwood (1981,AA,99,362)

Dependence on shape of rotation curve

Disk : Σ(r)=(+1)-3/2 (cut off at 2a)

(M: total mass, f: disk mass fraction, a: disk scale length)

Bulge (rigid): ρ(r)=( +1)-5/2 (M(1-f): bulge mass, b: bulge scale length)

[This bulge is highly concentrated to the center, and not intended to represent massive halo]
f=0.75, Q=1


Bar length LB = the radius at which m=2 Fourier component drops sharply and begins to change phase with radius.

 LB is larger for more slowly rising rotation curve

 LB growth is larger for initially shorter bar (because more matter is available outside bar, which absorbs bar angular momentum and helps bar growth)

Gravitational softening stabilizes disk considerably.

Stable model by Berman & Mark (1979,AA,77,31) is affected by too large gravitational softening.

1) No theoretical interpretation is possible for the correlation between rmax and bar length


2) Observational implications:

Models correspond to early-type galaxies (with 25% mass in bulge)

Obs: Athanassoula & Martinet (1980, AA, 87, L10) bar length and bulge size

Galaxies (earlier than Sbc) from Kormendy (1979,ApJ,227,714)

- Bulge radius measured at a constasnt surface brightness by Kormendy.

- Deprojection of bar length using inclination and line-of-nodes obtained assuming that inner rings or lenses are flat circles. (outer disk may be warped)

- D0 ≡diameter at 25 mag/arcsec2 in B band

Good correlation  What determines bar-size?

Lynden-Bell (1979, MN, 187,101), Contopoulos (1980, AA, 81,198)

Bars and Hubble types

◆ Combes & Elmegreen (1993,AA,271,391)

Claims realization of early and late type bars

Disk+bulge numerical simulations

- Stellar disk : Toomre disk (mass Md, scale length ad)

- Gas disk : exponential disk (scale length 8kpc)

(Both disks truncated at Rd)

- Bulge : Plummer sphere (mass Mb, scale length ab)

Model parameters

ab ad Mb Md Rd Q

Late-type models

CS1: 3 10 2 23 15 1.5 (small low-mass bulge)

CSE: 30 10 20 20 30 1.5 (massive halo)

PSG1: 3 10 2 23 25 Q(r)(polar grid model)
Early-type models

CS2: 1.5 6 16.67 8.33 15 0.0 (massive bulge and concentrated disk)

CSG2: 1.5 6 16.67 8.33 15 1.5

PSG2: 0.5 6 5 30 25 Q(r)(polar grid model)

Late-type models generally characterized by slowly rising rotation curve (Fig.2),

and early-type models by rapidly rising rotation curve (Fig.7)

Fig.9 Bar difference

Early-type: bar length = 6.2 kpc, flat density profile within 3 kpc

Late-type: bar length = 9 kpc, exponential profile

Gas response


-no halo component was included  early-type models have smaller disk mass fraction and more stable (inclusion of halo may invert this situation)

- bar length relative to disk size is not shown to match the observed trend for different Hubble types

Effects of halo concentration

◆ Athanassoula & Misiriotis (2002,MN,330,35)

Numerical simulations intended for comparison with observations

Model parameters have been widely varied




Disk: Md=1, h=1 (disk cutoff at 7.5), Q=0.9

Halo: Mh=5, rc=10

Bulger: Mb=0.6, a=0.4

(all components were modeled by particles)
Model MH (‘massive halo model’): γ=0.5 (no bulge)

Model MD (‘massive disk model’): γ=5 (no bulge)

Model MDB (‘massive disk with bulge’): MD + bulge


1) Bar length decreases 2) Bar becomes fatter 3) Edge-on shape: X boxy
Bar shape analysis

◆ Athanassoula (2003,MN,341,1179)

Bars in halo/bulge-dominated systems stronger than bars in disk-dominated systems.


Isolated disk galaxy evolution

- Inner disk emit angular momentum by resonant stars

- Emitted angular momentum is absorbed by resonant stars in outer disk, halo, and bulge.

This prediction by linear theory (Lynden-Bell & Kalnajs, 1972, MN, 157,1) is confirmed by numerical simulations for non-linear cases.

 Fig.1 : disk+halo model

(here, resonant stars are found by calculating Ω and κ by spectral analysis,

so strictly speaking, existence of resonant orbits does not necessarily mean the existence of resonance itself)

- Colder and massive components are more effective in emitting and absorbing angular momentum

- As angular momentum is lost, bar becomes longer, thinner and pattern

speed decreases.

 bar becomes stronger (longer and thinner) and slows down more quickly,

as halo and/or bulge mass increases

§4 Origins (Formation) of Bars
4.1 Bar instability
4.2 Tidal Triggering

◆ Noguchi (1987, MN, 228, 635; 1988, A&A, 203, 259)

Close Encounter simulations for disk+halo galaxy models

'Even stable galactic disks make bars under the influence of tidal

forces from another galaxy'
◆ Byrd & Valtonen (1990, ApJ, 350, 89)

Bar formation by galaxy cluster tidal field ( S0 and AGN in galaxy cluster)

4.3 Bar incidence in various environments
◆van den Bergh (2002, AJ, 124, 782)}

Sample: Northern 930 Shapley-Ames galaxies divided into

(1) field, (2) group, or (3) cluster environments.

based mainly on inspection of Palomar Sky Survey prints.

Barred or non-barred classified by two competent morphologists, using blue images.
Results on bar fraction:

- field : 25 ± 3 %

- group : 19 ± 4 %

- cluster : 28 ± 3 %

Caution : Morphology-Density Relation (Dressler 1980, ApJ, 236, 351)

Not affected by different morphological mix in different environments.

For Sc and SBc,

- field : 17 ± 3 %

- group : 12 ± 5 %

- cluster : 25 ± 5 %

 Bar frequency not dependent on environments

 Bar formation determined by intrinsic properties of parent galaxy.

◆ Thompson (1981, ApJ, 244, L43)

Sample: Coma Cluster galaxies with diameter > 16”

1) Bars occur twice as often in Core (r<28') as in Annulus (mean r=50')

or Outer Area (mean r=98')

2) No difference in luminosity distribution between bars and nonbars.
◆ Elmegreen, Elmegreen & Bellin (EEB, 1990,ApJ, 364, 415)


1) Binary sample :

- Turner binary(1976, ApJ, 208, 20)(87 galaxies) : selected by angular separation

- Peterson binary(1979, ApJS, 40, 527)(227 galaxies) : selected by angular separation and magnitude difference

2) Group sample :

- Geller & Huchra group (1983, ApJS, 52,61)(460 galaxies) : selected by velocity difference and projected separation (using CfA Redshift Survey)
D12 = 2 sin (Θ/2) V/H0 < DL, where V = (v1 + v2)/2

V12 = |v1 - v2| < VL

- Turner & Gott group (1976, ApJS, 32, 409)(221 galaxies): procedure complicated but based on position alone.

3) Field sample : Turner & Gott (1976, ApJS, 32,409)(119 galaxies)

4) All spirals : RC2
Morphology : RC2 and UGC

1) Higher bar fraction in binary sample only for early morphological type (Fig.1)

2) Smaller galaxies in binary tend to be barred.

3) Excess of early-type galaxies in binary systems. (Fig.2)


Interactions turn perturbed galaxies into barred-and-earlier-type (Table 4)

◆Giuricin et al. (1993, ApJ, 407 22)}

improvement of EEB's analysis (introduce objective environmental parameters)
Sample: Nearby Galaxies Catalogue (Tully 1988)

= Shapley-Ames Catalogue (magnitude-limited, Sandage & Tammann 1981)

+ all-sky HI survey (diameter-limited, Fisher & Tully 1981, ApJS, 47,139 ;

Reif et al. 1982, A&AS, 50, 451)

2367 galaxies (Table 2)

Morphology : RC3 morphology

Local Galaxy Density Parameters

Distances :

- non-cluster galaxies = radial velocity

- cluster galaxies = mean velocity of the cluster

1) ρσ

ρi = C exp[ -ri2/2(F1/3σ)2 ] (C = 1/(2π)3/2^3 and F is incompleteness)

ρσ = ∑iρi,σ (∑ is taken over galaxies with MB ≤ -16)

2) CR : number of galaxies with MB ≤ -16 and distance less than R, divided by


3) distances d1 (d2,d3,...)$ of the first (second, third,...) nearest galaxy


1) Early types (Sa-Sab) : bar fraction larger in denser environments (in terms of ρ0.25, C0.5, and d1) (Fig.5, Fig.9)

2) Late types (Sb-Sm) : bar fraction not dependent on environments (Fig.6, 7, 8)

3) Result 1) is due to less luminous (MB > -20) early type galaxies (Fig.11)

4) For 1Mpc< scale, no dependence is observed for any morphological types







These results agree with interaction scenario, because

1) Less massive galaxies tend to have more massive companion.

2) Less massive galaxies tend to have more gently rising rotation curve.

And these two characteristics are favorable to tidally-triggered bar formation (Noguchi, 1987, MN, 228, 635).

§5 Evolution of Bars
5.1 bending  boxy/peanuts bulges

Predicted by numerical simulatins

◆ Combes & Sanders (1981, AA, 96, 164)

垂直方向の resonance (特に Inner Lindblad Resonance) によって、 粒子が垂直方向の速度成分を増加させるため, バーはbox/peanut shape に。

Supporting observations

◆ Lütticke et al. (A&AS, 145, 405 ,2000)

Statistics of box- and peanut-shaped bulges

Previous studies: incomplete (or inhomogeneous) samples

>1000 edge-on galaxies from RC3

- D25>2’, -3.5 < T < 9.5

- Nearly edge-on galaxies (logR25>0.3 for T≤-0.5, logR25>0.35 for others)

(limitation by diameter is better than by magnitude, because RC3 claims

completeness for D25>1’, also magnitude limit will bias for early-type galaxies)
Inspection of Digitized Sky Survey images (B,V,R)

Classification into box/peanut or elliptical bulges

 45% of all bulges are box/peanut-shaped (>40% from S0 to Sd)

 box/peanut-shaped bulges (so numerous)are edge-on bar

◆ Lütticke et al. (A&A, 362, 435 , 2000)

JHK (K’) observation of 60 edge-on galaxies
bar/bulge length ratio ↗

as elliptical (4)  box-shaped  peanut(1) (Fig.4)

 N-body simulations (i.e, difference in viewing angle)

(inner region = bulge + thin bar + b/p structure)

Another signature of bars in edge-on galaxies

◆Athanassoula & Bureau (1999, ApJ, 522, 699)

Isothermal gas

Potential:Athanassoula (1992,MN,259,328;MN,259,345)

Characteristics of PV (position-velocity) diagram

1) Model with ILR

- Steep feature  nuclear spiral

Peak velocity is maximum when ψ=90°(bar seen side-on), because gas nearly follows x2 orbits

- Less steep feature  circular rotation of gas outside bar

Feature not dependent on ψ

- Bar region (e.g., offset shock and spiral arms just outside bar) do not contribute to PV diagram because of small associated mass

2) Model without ILR (i.e., without x2 orbits)

- Steep feature is lacking (absence of x2 orbits leads to no nuclear spirals)

Observed bar signatures

- Gap between steep and less steep features ( paucity of gas in bar region)

- Viewing angle of bar  (max velocity of nuclear component / velocity in the outer component)

[In case of no ILR  very difficult]

◆ Bureau & Freeman (AJ, 118, 126, 1999)

17 boxy/peanut bulges

Hα long slit spectroscopy (along the major axis) to investigate gas kinematics

14 out of 17 B/P galaxies have a bar

None of spheroidal bulges is associated with a bar

IC 4937 : peanut  ψ=90°, large nuclear/outer velocity ratio

NGC 1886 : boxy  ψ=0° , small velocity ratio

5.2 Dissolution of bars by growth of central mass

(注意: bending instability と区別せよ!)

◆Hasan & Norman (1990, ApJ, 361,69)

2D orbit surface of section


fraction of phase space occupied by regular x-1 orbits decreases as the central

core mass increases.

Physical insight:

epicyclic theory  x-1 orbit exist only between outer ILR and corotation.

 As the central mass increases, ILR appears and moves outward finally to the bar end (=CR), so that the region where x1 family exists shrinks.

◆Hasan & Norman (1993, ApJ, 409,91)

orbital study in a given 3D potential

characteristic diagram and surface of section


from 2D orbit study: as the central core mass increases, radial ILR appears

 x1 orbits become unstable (stochastic)  bar dissolution

(具体的なメカニズムは言及せず。ILRの出現と instability を関連させているだけ)

from 3D orbit study: radial ILR is close to vertical ILR  help heating of disk

stars into a bulge.

◆Norman, Sellwood, Hasan (1996,ApJ,462,114)

銀河の総質量の 5% の central mass によって短時間(1 bar rotation) に bar が壊れる。

N-body simulations

Mass ratio: Disk 75%, bulge+core(rigid components) 25%

After bar has fully formed and settled into boxy shape due to bending instability,

radius of core decreases gradually (over about 4 bar rotation periods) with mass fixed

 bar vanish suddenly (Fig.2)

 new axisymmetric ‘bulge’ is not boxy but somewhat peaky (Fig.8)
Reason: (revealed by ‘Surface of section’)

- Boxy/peanut bars are supported by 2:2:1 orbit family, which is analog of x1 family

in 3 dimensions (these bars are essentially the same as 2D bars)

- As the core shrinks, bar-supporting stellar orbits (x1 orbits) become stochastic

(resonance との関係は言及なし)

Surface of Section:

Core mass = 0%

Core mass = 3%

Comment on other ‘explanation’

‘ Orbit scattering by the central core’ (Norman, May, van Albada, 1985,ApJ,296,20)

 This explanation applies only to slowly rotating case, in which ‘box’ orbits

are predominant. Box orbits pass near the center.

But in rapidly rotating case, loop orbits (like x1 orbits) are dominant.

Loop orbits avoid the center.
Supporting observation?

◆ Carollo et al. (2001, ApJ, 546, 216)

HST V,H,J observation of R1/4 and exponential bulges
結果: late-type spiral に見つかった全ての Exp bulges は central nuclei という compact な component を持つ。luminosity から mass を推定してみると,これらのnuclei は 'massive enough to dissolve progenitor bars'である。 また、 nuclei の color からそれらは母体の Exp bulge と似た stellar population を持つと推測される。
5.3 Nested bars or 'Bars-within-bars'
◆ Shlosman, Frank, & Begelman (1989, Nature, 338, 45 )

'Bars-within-bars' scenario for AGN fueling

large-scale bar  gas inflow  central massive gas disk  central (secondary)

bar  and so on..... (= bar instability cascades to successively smaller scales)

◆ Kormendy (1993,IAUS,153,209)

'Kinematics of extragalactic bulges: evidence that some bulges are really disks'

- many bulges of barred galaxies well above 'the oblate line' in

V/σ-ε diagram. (Fig.3)

- barred galaxy bulges tend to be located below Faber-Jackson relation. (Fig.6)
disk-like or triaxial bulges
◆ Elmegreen et al.(1996, AJ, 111,1880)

Near-Infrared Observations of Isophotal Twists in Barred Spirals

1) Sample 1: JHK photometry of 12 barred galaxies (also B,I for 9 galaxies)

ellipse fit

twist ≡ position angle change of >10° from ‘inner region’ to bar end

(Bar end is determined by image and contour map)

2 types of twists

continuous change of PA = triaxial bulge

discontinuous jump between two values = bars-within-bars
2) Sample 2: NIR samples from literature

3) Sample 3: B photograph of barred (SB) galaxies in Sandage & Bedke Atlas

eye inspection
Results (以下では nested bars と triaxial bulge は区別せず)

- Twists are seen only in early-type(-Sbc) spirals

- Twists are seen in some of flat bars, but not in exponential bars

Twist is related to ILR, where stellar orbits change from x2 to x1.

(consistent with other resonance indications: nuclear rings supposed to be

related to ILR are found in SBbc and earlier but not in later-types)

- Flat bars have ellipticity increasing with radius, but exponential bars

have a constant ellipticity

◆ Friedli & Martinet (1993、AA, 277, 27)

Double-bar numerical experiments

Some models include rigid massive halos

No star formation included
Model(I): small unstable stellar disk + large unstable stellar disk

(expectation: small bar appears  after that (or simultaneously) large bar appears. the small disk creates ILR for the primary bar)

 only small bar forms (large bar never develops)


small bar  large spiral arms  large disk heats up  no large scale bar instability
Model(II): Model(I)+large gas disk

large and small bars form simultaneously in 1Gyr (10 bar


double-barred state lasts for a few bar rotation

 small bar dissolves (due to gas accretion to the center) but large bar remains
Model(III): similar to Model(II) but with different parameters

 large stellar bar forms (in 1 bar rotation)

 small stellar bar forms (after a few bar rotation)

double-bar state lasts for a few bar rotation

 both bars weaken because of gas accumulation
1) two bars rotate with different pattern speeds in Models (II) and (III).

2) double-barred state is transient.

3) dissipative component and moderate ILR are essential in forming double bar systems.

◆Wozniak et al (1995,AAS,111,115)

BVRI photometry of (not necessarily barred) 36 galaxies (=candidates for having misaligned secondary bar or triaxial bulge)  strongly biased sample

Analysis = ellipse fitting

Define three types of structures based on radial behavior of e & PA (in projected image)

1) Bar: emin emax emin, PA=const

2) Bar-within-bar : emin esmax emin, PA=const (secondary bar), emin epmax emin, PA=const (primary bar)

3) Barred plus twisted isophotes : emin es e’, PA changes (twisted structure), e’ emax emin, PA=const (bar)
Bar length (lp, ls) = distance (semi-major axis) from center to emin

(emin length > visual estimate > emax length)

Bar luminosity (Lp, Ls) = luminosity inside isophote of semi-major axis (lp,ls)
Ratios: β ≡lp/ls, γ ≡ Lp/Ls
Uncertainty arises from

- High inclination of the galaxy

- Pixel size

- Seeing

- Dust

B+B 3.7<β<18.0 2.0<γ<7.5

B+B+B, B+T+B 4.8<β13<13.6 2.3<γ13<7.7 (1.8<β12<3.0 1.4<γ12<1.9)

 θ shows no preference (paucity at small values will be artificial)

 β1212 in B+B similar to β1313 in B+B+B,B+T+B

 β1212 in B+B+B are different


 esmax smaller than epmax (could be seeing or pixel size effects on the secondary bars)

High percentage of Seyfert nuclei among B+B (46% compared with a few % in disk galaxies) (consistent with Bars-in-bars scenario for nuclear activity)
Friedli & Martinet model : consistent with β-γ relation (Fig.6), but primary bar is too round (Fig.7) (probably because poorer spatial resolution at large radii in their simulation)

◆ Friedli et al. (1996, A&AS, 118, 461)

JHK photometry of 13 galaxies from Wozniak et al (1995,AAS,111,115)

Same analysis as Wozniak et al.

- Results are similar to I-band results exceptγ tend to be smaller for K band (Fig.3) probably because K better traces stellar mass

- θ is distributed evenly (deprojection does not change the result)

Hubble type dependence

- θ shows no dependence

- β andγ larger for late types (for γ, probably because of bulge effect)

- esmax shows no dependence

- epmax probably larger for late-types

Numerical model

1) Bno : mass ratios = disk 0.5, bulge 0.05, gas 0.055

2) Bsf : same as Bno but includes star formation

Two models evolve qualitatively similar.

Primary bar forms  gas inflow  secondary bar forms in the center

Two bars rotate with different pattern speeds

◆ Laine, Shlosman, Knapen, & Peletier (2002,ApJ,567,97)

Theoretically motivated by Shlosman et al. (1989, Nature, 338, 45 )

  1. Seyfert sample (56): vhel < 6000 kms-1 HST/NICMOS F160W(H) images

+ a few well-known Seyferts

R>0.45 (RC3)

2) Control sample (56): similar distribution to Seyfert sample in

- absolute B magnitude

- distance

- axis ratio (R)

- morphological type

NICMOS(H-band) + 2MASS(H-band) [5”-30”] + DSS (optical)[30”-]

- ellipse fit (IRAF/GALPHOT)

- deprojection (assuming galaxy outer parts are flat and circular)

Bar detection:

1) ellipticity (1-b/a) rise and fall by >0.1

2) position angle constant (<20°)

Bar length ≡ radius where ellipticity peaks

Bar ellipticity ≡ maximum ellipticity

1) critical length = 0.06 ×D25 or 1.6 Kpc ( Fig.3 Fig.4)

Fig.7 (secondary bar length ⇔ ILR)

2) Twists are predominant in late-type spirals (Fig.5)

◆Erwin & Sparke (2002, AJ, 124,65)

Motivation: Past observational study compared incidence of double bars in Seyfert and normal galaxies. But no complete sample was used.

This study = survey of a complete sample of early-type optically barred galaxies
Sample: barred (after RC2) S0-Sa galaxies from UGC

δ>-10°, vhel< 2000 km/s, D25>2’, axial ratio a/b <2

Virgo cluster galaxies excluded to avoid possible environmental effects

 38 galaxies (20 S0, 10 S0/a, 8 Sa) (25 SB, 13 SAB)

Imaging in B and R bands (WYIN telescope)

Archival HST data (WFPC2 with F606W, F814W, NICMOS with F160W)


Stellar structures: secondary bars, inner disks, stellar nuclear rings

Gaseous structures: dusty and star-forming nuclear rings, nuclear spirals, off-plane dust structures (small polar rings, inclined dust disks)
Detection: ellipse fit to R band, F814W, F160W images  candidate features was examined by unsharp masking and color maps (Fig.2)  discriminate between bars, rings, spirals (can have similar ellipse-fit features)

Result: (Table2)

- at least 1/4 of galaxies have inner bars (in all Hubble types)

- inner stellar disks are more common in earlier types

- dusty/star-forming rings and nuclear spirals are more common in late types

- none of these features show any preference for bar strength (SB or SAB)

- off-plane gas are most common in S0


Angle between outer and inner bars (Fig.3)  no significant preference for either leading or trailing  two bars rotate independently
Sizes of inner structures (Fig.4)

 Inner bars have 0.05 – 0.14 primary bar length (240-1000pc)

 bar length ratio in simulation : 0.26-0.5 (Friedli & Martinet), 0.21 (Friedli et al. 1996), 0.15 (Rautiainen & Salo 1999)
Ellipticity: smaller than primary bars (Fig.6) (intrinsic roundness and/or bulge contamination)
Inner disks have a wider size distribution (Fig.4)
nuclear ring:

- dusty/star-forming rings in late types

- Stellar rings in S0 (evolved from star-forming ring / or product of double bar forcing) (the latter scenario cannot explain two stellar rings without inner bar in the sample)

- Co-existence with inner bar: 6 out of 10 double barred galaxies (vs. only 27% in single bar galaxies) For these galaxies, inner bar length is only a bit smaller than ring sizes

 consistent with [inner bar’s CR = outer bar’s ILR]-hypothesis because nuclear rings are believed to mark ILR
[inner bars do not require nuclear rings]

Nuclear activity:

31 galaxies have classification in the literature

- AGN and HII nuclei incidences agree with Ho et al.

- inner bar / inner disks do not influence nuclear activity

- AGN activity is closely related to nuclear dusty/star-forming ring, nuclear spirals, off-plane dust

- SAB may have a higher AGN incidence than SB

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