Deconstructing double-barred galaxies in 2D and 3D. I. Classical nature of the dominant bulges

TL;DR

This study uses advanced 2D and 3D photometric techniques to analyze double-barred galaxies, revealing that most host classical bulges, which may be essential for inner bar formation.

Contribution

First combined application of 2D photometric decompositions and 3D shape analysis to double-barred galaxies for bulge characterization.

Findings

Almost all bulges are classical in nature.

Only 2 galaxies show potential disc-like bulges.

Presence of a hot central structure may be necessary for inner bar formation.

Abstract

We present here a thorough photometric analysis of double-barred galaxies, consisting of i) two-dimensional photometric decompositions including a bulge, inner bar, outer bar, and (truncated) disc; and ii) three-dimensional statistical deprojections to derive the intrinsic shape of bulges, inner bars, and outer bars. This is the first time the combination of both techniques is applied to a sample of double-barred galaxies. It represents a step forward with respect to previous works, which are based on properties of the integrated light through ellipse fitting and unsharp masking. In this first paper of a series of two, we analyse the nature of the dominant bulges within double-barred systems by using several photometric diagnostics, namely S\'ersic index, Kormendy relation, colours, and the better suited intrinsic flattening. Our results indicate that almost all bulges in our sample are classical, whereas only 2 out of the 17 galaxies under study appear as potential candidates to host secularly-formed disc-like bulges. Such result poses the possibility that having a central hot structure may be a requirement for inner bar formation.

Tables

Table 1. Table 1: DOUBLE-BARRED GALAXIES SAMPLE.

Name	Morphological type	Distance (Mpc)	$z$
(1)	(2)	(3)	(4)
NGC 357	SB(r)0/a	31.6	0.008
NGC 718	SAB(s)a	22.6	0.006
NGC 2642	SB(r)bc	56.8	0.014
NGC 2681	(R’)SAB(rs)0/a	17.2	0.002
NGC 2859	(R)SB(r)0+	24.3	0.006
NGC 2950	(R)SB(r)00	14.9	0.004
NGC 2962	(R)SAB(rs)0+	30.0	0.007
NGC 3368	SAB(rs)ab	10.5	0.003
NGC 3941	SB(s)00	12.2	0.003
NGC 3945	(R)SB(rs)0+	19.3	0.004
NGC 4314	SB(rs)a	12.0	0.003
NGC 4340	SB(r)0+	15.3	0.003
NGC 4503	SB0-:	15.3	0.004
NGC 4725	SAB(r)ab	12.4	0.004
NGC 5850	SB(r)b	35.2	0.009
NGC 7280	SAB(r)0+	24.3	0.006
NGC 7716	SAB(r)b:	34.1	0.009

Table 2. Table 2: Intrinsic 3D shape of the different structures in our sample.

Galaxy	Structure	B/A	C/A
NGC 357	Bulge	0.94${}_{}{}^{0.96}{}_{0.84}{}^{}$	0.76${}_{}{}^{1.01}{}_{0.34}{}^{}$
NGC 357	Outer bar	0.46${}_{}{}^{0.51}{}_{0.39}{}^{}$	1.96${}_{}{}^{2.00}{}_{0.69}{}^{}$
NGC 357	Inner bar	0.91${}_{}{}^{1.00}{}_{0.74}{}^{}$	0.96${}_{}{}^{1.26}{}_{0.36}{}^{}$
NGC 718	Bulge	0.81${}_{}{}^{0.86}{}_{0.69}{}^{}$	1.21${}_{}{}^{1.89}{}_{0.34}{}^{}$
NGC 718	Outer bar	0.36${}_{}{}^{0.41}{}_{0.16}{}^{}$	1.04${}_{}{}^{1.79}{}_{0.24}{}^{}$
NGC 718	Inner bar	0.56${}_{}{}^{0.69}{}_{0.24}{}^{}$	0.99${}_{}{}^{1.84}{}_{0.24}{}^{}$
NGC 2642	Bulge	0.84${}_{}{}^{0.89}{}_{0.69}{}^{}$	0.91${}_{}{}^{2.00}{}_{0.26}{}^{}$
NGC 2642	Outer bar	0.16${}_{}{}^{0.26}{}_{0.06}{}^{}$	0.69${}_{}{}^{1.24}{}_{0.16}{}^{}$
NGC 2642	Inner bar	0.14${}_{}{}^{0.21}{}_{0.04}{}^{}$	0.34${}_{}{}^{0.79}{}_{0.11}{}^{}$
NGC 2681	Bulge	0.91${}_{}{}^{0.94}{}_{0.79}{}^{}$	0.56${}_{}{}^{1.09}{}_{0.09}{}^{}$
NGC 2681	Outer bar	0.76${}_{}{}^{0.76}{}_{0.69}{}^{}$	1.66${}_{}{}^{2.00}{}_{0.89}{}^{}$
NGC 2681	Inner bar	0.89${}_{}{}^{0.91}{}_{0.76}{}^{}$	1.31${}_{}{}^{1.71}{}_{0.69}{}^{}$
NGC 2859	Bulge	0.96${}_{}{}^{1.00}{}_{0.91}{}^{}$	0.86${}_{}{}^{1.01}{}_{0.64}{}^{}$
NGC 2859	Outer bar	0.64${}_{}{}^{0.69}{}_{0.54}{}^{}$	1.79${}_{}{}^{2.00}{}_{0.74}{}^{}$
NGC 2859	Inner bar	0.34${}_{}{}^{0.41}{}_{0.11}{}^{}$	0.41${}_{}{}^{0.66}{}_{0.14}{}^{}$
NGC 2950	Bulge	0.66${}_{}{}^{0.71}{}_{0.41}{}^{}$	0.59${}_{}{}^{0.81}{}_{0.19}{}^{}$
NGC 2950	Outer bar	0.49${}_{}{}^{0.54}{}_{0.24}{}^{}$	0.36${}_{}{}^{0.64}{}_{0.11}{}^{}$
NGC 2950	Inner bar	0.74${}_{}{}^{0.76}{}_{0.49}{}^{}$	0.36${}_{}{}^{0.69}{}_{0.09}{}^{}$
NGC 2962	Bulge	1.00${}_{}{}^{1.00}{}_{0.96}{}^{}$	0.96${}_{}{}^{0.99}{}_{0.91}{}^{}$
NGC 2962	Outer bar	0.76${}_{}{}^{0.79}{}_{0.44}{}^{}$	0.19${}_{}{}^{0.46}{}_{0.06}{}^{}$
NGC 2962	Inner bar	1.00${}_{}{}^{1.00}{}_{0.96}{}^{}$	0.64${}_{}{}^{0.74}{}_{0.49}{}^{}$
NGC 3368	Bulge	0.91${}_{}{}^{0.94}{}_{0.74}{}^{}$	0.41${}_{}{}^{0.64}{}_{0.19}{}^{}$
NGC 3368	Outer bar	0.54${}_{}{}^{0.59}{}_{0.31}{}^{}$	0.46${}_{}{}^{0.74}{}_{0.11}{}^{}$
NGC 3368	Inner bar	0.44${}_{}{}^{0.51}{}_{0.19}{}^{}$	0.31${}_{}{}^{0.54}{}_{0.09}{}^{}$
NGC 3941	Bulge	0.91${}_{}{}^{0.94}{}_{0.76}{}^{}$	0.61${}_{}{}^{0.79}{}_{0.39}{}^{}$
NGC 3941	Outer bar	0.51${}_{}{}^{0.59}{}_{0.26}{}^{}$	0.29${}_{}{}^{0.51}{}_{0.09}{}^{}$
NGC 3941	Inner bar	0.71${}_{}{}^{0.76}{}_{0.51}{}^{}$	0.56${}_{}{}^{0.81}{}_{0.16}{}^{}$
NGC 3945	Bulge	0.94${}_{}{}^{0.96}{}_{0.86}{}^{}$	0.66${}_{}{}^{0.79}{}_{0.49}{}^{}$
NGC 3945	Outer bar	0.81${}_{}{}^{0.86}{}_{0.66}{}^{}$	1.66${}_{}{}^{1.86}{}_{1.01}{}^{}$
NGC 3945	Inner bar	0.51${}_{}{}^{0.64}{}_{0.21}{}^{}$	0.29${}_{}{}^{0.51}{}_{0.09}{}^{}$
NGC 4314	Bulge	0.84${}_{}{}^{0.91}{}_{0.64}{}^{}$	0.66${}_{}{}^{1.39}{}_{0.09}{}^{}$
NGC 4314	Outer bar	0.21${}_{}{}^{0.26}{}_{0.09}{}^{}$	0.36${}_{}{}^{0.61}{}_{0.11}{}^{}$
NGC 4314	Inner bar	0.84${}_{}{}^{0.89}{}_{0.66}{}^{}$	0.76${}_{}{}^{1.49}{}_{0.14}{}^{}$
NGC 4340	Bulge	0.91${}_{}{}^{0.96}{}_{0.79}{}^{}$	0.54${}_{}{}^{0.74}{}_{0.34}{}^{}$
NGC 4340	Outer bar	0.29${}_{}{}^{0.31}{}_{0.11}{}^{}$	0.51${}_{}{}^{0.84}{}_{0.14}{}^{}$
NGC 4340	Inner bar	0.24${}_{}{}^{0.26}{}_{0.11}{}^{}$	0.61${}_{}{}^{1.01}{}_{0.16}{}^{}$
NGC 4503	Bulge	0.84${}_{}{}^{0.91}{}_{0.64}{}^{}$	0.64${}_{}{}^{0.76}{}_{0.41}{}^{}$
NGC 4503	Outer bar	0.61${}_{}{}^{0.66}{}_{0.31}{}^{}$	0.29${}_{}{}^{0.44}{}_{0.06}{}^{}$
NGC 4503	Inner bar	0.44${}_{}{}^{0.51}{}_{0.24}{}^{}$	0.41${}_{}{}^{0.76}{}_{0.16}{}^{}$
NGC 4725	Bulge	0.81${}_{}{}^{0.86}{}_{0.59}{}^{}$	0.71${}_{}{}^{0.86}{}_{0.46}{}^{}$
NGC 4725	Outer bar	0.74${}_{}{}^{0.76}{}_{0.46}{}^{}$	0.44${}_{}{}^{0.59}{}_{0.19}{}^{}$
NGC 4725	Inner bar	0.21${}_{}{}^{0.49}{}_{0.11}{}^{}$	0.79${}_{}{}^{2.00}{}_{0.31}{}^{}$
NGC 5850	Bulge	1.00${}_{}{}^{1.00}{}_{1.00}{}^{}$	0.94${}_{}{}^{0.99}{}_{0.91}{}^{}$
NGC 5850	Outer bar	0.24${}_{}{}^{0.31}{}_{0.09}{}^{}$	0.24${}_{}{}^{0.36}{}_{0.09}{}^{}$
NGC 5850	Inner bar	0.39${}_{}{}^{0.44}{}_{0.21}{}^{}$	1.86${}_{}{}^{2.00}{}_{0.61}{}^{}$
NGC 7280	Bulge	0.89${}_{}{}^{0.96}{}_{0.64}{}^{}$	0.31${}_{}{}^{0.64}{}_{0.11}{}^{}$
NGC 7280	Outer bar	0.61${}_{}{}^{0.69}{}_{0.34}{}^{}$	0.31${}_{}{}^{0.59}{}_{0.09}{}^{}$
NGC 7280	Inner bar	0.11${}_{}{}^{0.16}{}_{0.04}{}^{}$	0.11${}_{}{}^{0.24}{}_{0.04}{}^{}$
NGC 7716	Bulge	0.64${}_{}{}^{0.74}{}_{0.29}{}^{}$	0.49${}_{}{}^{0.94}{}_{0.11}{}^{}$
NGC 7716	Outer bar	0.46${}_{}{}^{0.54}{}_{0.21}{}^{}$	0.46${}_{}{}^{0.79}{}_{0.14}{}^{}$
NGC 7716	Inner bar	0.91${}_{}{}^{0.94}{}_{0.79}{}^{}$	0.49${}_{}{}^{0.91}{}_{0.11}{}^{}$

Table 3. Table 3: Bulge colours (within one bulge effective radius), bulge masses, and galaxy masses.

Galaxy	(g’-i’)	(r’-i’)	log(M⋆,bulge (M⊙))	log(Mgal (M⊙))
NGC 357	1.52	0.49	10.27	10.88
NGC 718	1.07	0.34	9.83	10.33
NGC 2642	1.53	0.46	10.45	10.62
NGC 2681	-0.04	-0.10	7.91	9.51
NGC 2859	1.20	0.41	10.40	10.64
NGC 2950	1.18	0.36	9.97	10.38
NGC 2962	1.46	0.47	10.00	10.61
NGC 3368	1.47	0.50	10.44	**
NGC 3941	1.20	0.39	9.80	10.41
NGC 3945	1.25	0.38	9.32	10.78
NGC 4314	1.10	0.39	9.76	10.19
NGC 4340	1.14	0.40	9.38	9.99
NGC 4503	1.29	0.43	9.92	10.47
NGC 4725	1.28	0.49	10.24	**
NGC 5850	1.38	0.47	10.66	11.02
NGC 7280	1.40	0.34	10.10	10.34
NGC 7716	1.38	0.46	10.53	10.67

Table 4. Table 4: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN r’-BAND. First set of 9/17 galaxies.

		NGC 357	NGC 718	NGC 2642	NGC 2681	NGC 2859	NGC 2950	NGC 2962	NGC 3368	NGC 3941
Bulge	${\mu }_e$	18.52 $\pm$ 0.26	18.42 $\pm$ 0.15	19.58 $\pm$ 0.33	15.96 $\pm$ 0.13	18.52 $\pm$ 0.16	16.86 $\pm$ 0.16	18.32 $\pm$ 0.15	18.91 $\pm$ 0.16	17.53 $\pm$ 0.13
	Re	2.68 $\pm$ 0.43	3.15 $\pm$ 0.32	2.65 $\pm$ 0.53	1.65 $\pm$ 0.17	5.63 $\pm$ 0.47	2.65 $\pm$ 0.22	2.22 $\pm$ 0.23	14.59 $\pm$ 1.22	3.84 $\pm$ 0.38
	n	1.62 $\pm$ 0.18	2.24 $\pm$ 0.27	1.57 $\pm$ 0.29	0.69 $\pm$ 0.05	1.66 $\pm$ 0.13	1.88 $\pm$ 0.15	1.65 $\pm$ 0.20	1.44 $\pm$ 0.11	2.28 $\pm$ 0.17
	b/a	0.90 $\pm$ 0.04	0.84 $\pm$ 0.04	0.84 $\pm$ 0.05	0.86 $\pm$ 0.02	0.95 $\pm$ 0.02	0.69 $\pm$ 0.02	0.97 $\pm$ 0.04	0.69 $\pm$ 0.02	0.78 $\pm$ 0.02
	PA	38.76 $\pm$ 5.05	163.89 $\pm$ 5.20	127.90 $\pm$ 7.76	25.71 $\pm$ 3.31	103.53 $\pm$ 2.97	97.36 $\pm$ 2.97	176.86 $\pm$ 5.20	154.56 $\pm$ 2.97	18.42 $\pm$ 3.31
	B/T	0.170	0.214	0.092	0.125	0.259	0.257	0.135	0.214	0.224
Disc	${\mu }_o$	20.81 $\pm$ 0.07	20.52 $\pm$ 0.07	21.11 $\pm$ 0.14	20.15 $\pm$ 0.14	21.92 $\pm$ 0.14	20.17 $\pm$ 0.14	21.12 $\pm$ 0.07	20.71 $\pm$ 0.07	19.29 $\pm$ 0.07
	hinner	29.56 $\pm$ 1.38	22.38 $\pm$ 0.85	27.61 $\pm$ 1.88	33.38 $\pm$ 0.81	63.41 $\pm$ 1.05	26.33 $\pm$ 0.4	33.73 $\pm$ 1.28	80.36 $\pm$ 1.33	31.17 $\pm$ 0.74
	b/a	0.82 $\pm$ 0.01	0.95 $\pm$ 0.01	0.97 $\pm$ 0.02	0.89 $\pm$ 0.01	0.84 $\pm$ 0.01	0.64 $\pm$ 0.01	0.57 $\pm$ 0.01	0.65 $\pm$ 0.01	0.65 $\pm$ 0.01
	PA	18.78 $\pm$ 0.37	19.70 $\pm$ 0.58	151.41 $\pm$ 0.90	12.40 $\pm$ 0.25	84.73 $\pm$ 0.45	123.80 $\pm$ 0.45	2.46 $\pm$ 0.58	160.7 $\pm$ 0.45	9.15 $\pm$ 0.25
	houter	13.93 $\pm$ 0.49	N/A	12.01 $\pm$ 0.42	18.70 $\pm$ 0.65	N/A	N/A	N/A	N/A	19.99 $\pm$ 0.70
	Rbreak	48.62 $\pm$ 1.65	N/A	45.55 $\pm$ 1.54	88.20 $\pm$ 2.99	N/A	N/A	N/A	N/A	39.40 $\pm$ 1.33
	D/T	0.667	0.645	0.806	0.594	0.537	0.444	0.587	0.527	0.614
Outer bar	${\mu }_o$	20.73 $\pm$ 0.30	20.42 $\pm$ 0.32	21.37 $\pm$ 0.45	19.43 $\pm$ 0.18	20.70 $\pm$ 0.20	19.63 $\pm$ 0.20	20.69 $\pm$ 0.32	20.16 $\pm$ 0.20	19.03 $\pm$ 0.18
	a	38.42 $\pm$ 0.63	46.90 $\pm$ 0.78	53.08 $\pm$ 1.53	39.64 $\pm$ 0.39	72.93 $\pm$ 0.79	45.59 $\pm$ 0.49	49.52 $\pm$ 0.82	135.14 $\pm$ 1.46	34.98 $\pm$ 0.34
	n	3	4	3	3	3	3	2	3	2
	b/a	0.44 $\pm$ 0.04	0.42 $\pm$ 0.02	0.24 $\pm$ 0.06	0.73 $\pm$ 0.02	0.63 $\pm$ 0.03	0.53 $\pm$ 0.03	0.47 $\pm$ 0.02	0.59 $\pm$ 0.03	0.44 $\pm$ 0.02
	PA	119.74 $\pm$ 0.28	150.89 $\pm$ 0.33	117.11 $\pm$ 0.58	83.57 $\pm$ 0.16	159.32 $\pm$ 0.13	156.05 $\pm$ 0.13	174.39 $\pm$ 0.33	128.19 $\pm$ 0.13	171.29 $\pm$ 0.16
	c	3	2	4	2	2	2	2	2	2
	Bar/T	0.106	0.127	0.097	0.170	0.182	0.204	0.224	0.247	0.136
Inner bar	${\mu }_o$	19.93 $\pm$ 0.98	18.50 $\pm$ 0.24	18.55 $\pm$ 2.04	17.81 $\pm$ 1.09	18.33 $\pm$ 1.06	18.32 $\pm$ 1.06	19.88 $\pm$ 0.24	17.55 $\pm$ 1.06	19.01 $\pm$ 1.09
	a	13.61 $\pm$ 2.46	4.11 $\pm$ 0.70	3.26 $\pm$ 1.10	13.75 $\pm$ 1.16	11.21 $\pm$ 1.15	17.18 $\pm$ 1.76	14.99 $\pm$ 2.54	10.47 $\pm$ 1.07	13.42 $\pm$ 1.14
	n	3	2	2	3	3	4	3	3	3
	b/a	0.90 $\pm$ 0.06	0.63 $\pm$ 0.06	0.18 $\pm$ 0.13	0.89 $\pm$ 0.03	0.37 $\pm$ 0.04	0.63 $\pm$ 0.04	0.74 $\pm$ 0.06	0.44 $\pm$ 0.04	0.71 $\pm$ 0.03
	PA	157.75 $\pm$ 2.85	31.99 $\pm$ 2.21	160.04 $\pm$ 3.64	75.77 $\pm$ 2.09	62.32 $\pm$ 1.30	140.56 $\pm$ 1.30	0.96 $\pm$ 2.21	133.83 $\pm$ 1.30	165.58 $\pm$ 2.09
	c	2	4	2	2	4	2	2	4	2
	Bar/T	0.057	0.013	0.005	0.111	0.022	0.094	0.053	0.012	0.026

Table 5. Table 5: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN r’-BAND. Last set of 8/17 galaxies

		NGC 3945	NGC 4314	NGC 4340	NGC 4503	NGC 4725	NGC 5850	NGC 7280	NGC 7716
Bulge	${\mu }_e$	18.39 $\pm$ 0.16	19.57 $\pm$ 0.13	19.16 $\pm$ 0.16	18.93 $\pm$ 0.16	18.50 $\pm$ 0.16	20.00 $\pm$ 0.16	18.93 $\pm$ 0.26	17.97 $\pm$ 0.15
	Re	5.72 $\pm$ 0.48	10.94 $\pm$ 1.09	6.01 $\pm$ 0.50	5.05 $\pm$ 0.42	6.80 $\pm$ 0.57	7.77 $\pm$ 0.65	3.71 $\pm$ 0.59	2.04 $\pm$ 0.21
	n	1.57 $\pm$ 0.12	0.94 $\pm$ 0.07	2.15 $\pm$ 0.17	2.64 $\pm$ 0.20	1.68 $\pm$ 0.13	2.14 $\pm$ 0.17	3.37 $\pm$ 0.36	1.70 $\pm$ 0.21
	b/a	0.83 $\pm$ 0.02	0.81 $\pm$ 0.02	0.76 $\pm$ 0.02	0.75 $\pm$ 0.03	0.83 $\pm$ 0.03	0.97 $\pm$ 0.03	0.65 $\pm$ 0.04	0.60 $\pm$ 0.04
	PA	150.63 $\pm$ 2.97	129.41 $\pm$ 3.31	98.85 $\pm$ 2.97	18.90 $\pm$ 2.97	29.97 $\pm$ 2.97	158.06 $\pm$ 2.97	66.56 $\pm$ 5.05	55.52 $\pm$ 5.20
	B/T	0.258	0.214	0.250	0.205	0.091	0.182	0.280	0.144
Disc	${\mu }_o$	21.20 $\pm$ 0.07	21.49 $\pm$ 0.07	20.73 $\pm$ 0.07	19.45 $\pm$ 0.07	20.06 $\pm$ 0.07	21.67 $\pm$ 0.07	20.59 $\pm$ 0.07	20.11 $\pm$ 0.07
	hinner	46.59 $\pm$ 0.77	62.51 $\pm$ 1.51	34.17 $\pm$ 0.57	26.40 $\pm$ 0.44	85.43 $\pm$ 1.42	61.49 $\pm$ 1.02	26.74 $\pm$ 1.25	16.34 $\pm$ 0.62
	b/a	0.70 $\pm$ 0.01	0.92 $\pm$ 0.01	0.67 $\pm$ 0.01	0.46 $\pm$ 0.01	0.50 $\pm$ 0.01	0.75 $\pm$ 0.01	0.65 $\pm$ 0.01	0.82 $\pm$ 0.01
	PA	157.45 $\pm$ 0.45	135.48 $\pm$ 0.25	91.94 $\pm$ 0.45	9.83 $\pm$ 0.45	43.10 $\pm$ 0.45	137.26 $\pm$ 0.45	74.19 $\pm$ 0.37	41.49 $\pm$ 0.58
	houter	N/A	25.56 $\pm$ 0.89	N/A	N/A	N/A	N/A	11.12 $\pm$ 0.39	N/A
	Rbreak	N/A	106.49 $\pm$ 3.6	N/A	N/A	N/A	N/A	44.65 $\pm$ 1.51	N/A
	D/T	0.466	0.486	0.624	0.713	0.863	0.698	0.629	0.735
Outer bar	${\mu }_o$	20.27 $\pm$ 0.20	20.28 $\pm$ 0.18	21.72 $\pm$ 0.20	20.41 $\pm$ 0.20	20.43 $\pm$ 0.20	21.52 $\pm$ 0.20	20.48 $\pm$ 0.30	21.28 $\pm$ 0.32
	a	57.00 $\pm$ 0.62	126.16 $\pm$ 1.24	62.90 $\pm$ 0.68	32.60 $\pm$ 0.35	59.09 $\pm$ 0.64	130.87 $\pm$ 1.42	20.24 $\pm$ 0.33	30.38 $\pm$ 0.50
	n	2	3	1	2	3	4	2	2
	b/a	0.66 $\pm$ 0.03	0.25 $\pm$ 0.02	0.43 $\pm$ 0.03	0.46 $\pm$ 0.02	0.61 $\pm$ 0.03	0.25 $\pm$ 0.03	0.51 $\pm$ 0.04	0.48 $\pm$ 0.02
	PA	73.54 $\pm$ 0.13	146.77 $\pm$ 0.16	30.98 $\pm$ 0.13	176.76 $\pm$ 0.13	35.81 $\pm$ 0.13	116.04 $\pm$ 0.13	57.25 $\pm$ 0.28	17.82 $\pm$ 0.33
	c	2	2	4	2	3	2	2	3
	Bar/T	0.219	0.276	0.109	0.065	0.040	0.110	0.067	0.073
Inner bar	${\mu }_o$	18.88 $\pm$ 1.06	16.95 $\pm$ 1.09	18.41 $\pm$ 1.06	19.63 $\pm$ 1.06	18.28 $\pm$ 1.06	19.21 $\pm$ 1.06	16.87 $\pm$ 0.98	19.27 $\pm$ 0.24
	a	21.90 $\pm$ 2.24	4.92 $\pm$ 0.42	8.83 $\pm$ 0.90	12.03 $\pm$ 1.23	11.54 $\pm$ 1.18	11.00 $\pm$ 1.13	4.00 $\pm$ 0.72	8.38 $\pm$ 1.42
	n	3	4	4	4	3	2	2	3
	b/a	0.41 $\pm$ 0.04	0.83 $\pm$ 0.03	0.36 $\pm$ 0.04	0.69 $\pm$ 0.04	0.34 $\pm$ 0.04	0.25 $\pm$ 0.04	0.17 $\pm$ 0.06	0.84 $\pm$ 0.06
	PA	159.97 $\pm$ 1.30	158.37 $\pm$ 2.09	19.99 $\pm$ 1.30	146.14 $\pm$ 1.30	137.33 $\pm$ 1.30	46.83 $\pm$ 1.30	114.24 $\pm$ 2.85	56.17 $\pm$ 2.21
	c	3	2	4	2	5	4	1	2
	Bar/T	0.056	0.024	0.017	0.017	0.006	0.010	0.024	0.047

Table 6. Table 6: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN g’-BAND. First set of 9/17 galaxies.

		NGC 357	NGC 718	NGC 2642	NGC 2681	NGC 2859	NGC 2950	NGC 2962	NGC 3368	NGC 3941
Bulge	${\mu }_e$	19.55 $\pm$ 0.33	19.04 $\pm$ 0.31	20.42 $\pm$ 0.33	16.21 $\pm$ 0.13	19.16 $\pm$ 0.15	17.74 $\pm$ 0.15	19.25 $\pm$ 0.31	19.82 $\pm$ 0.16	18.44 $\pm$ 0.13
	Re	2.76 $\pm$ 0.55	3.02 $\pm$ 0.53	2.62 $\pm$ 0.52	1.48 $\pm$ 0.15	5.29 $\pm$ 0.54	2.77 $\pm$ 0.28	2.28 $\pm$ 0.40	15.36 $\pm$ 1.29	3.97 $\pm$ 0.40
	n	1.60 $\pm$ 0.29	2.02 $\pm$ 0.40	1.19 $\pm$ 0.22	2.02 $\pm$ 0.15	1.50 $\pm$ 0.18	1.67 $\pm$ 0.20	1.46 $\pm$ 0.29	1.37 $\pm$ 0.11	2.59 $\pm$ 0.19
	b/a	0.87 $\pm$ 0.05	0.83 $\pm$ 0.05	0.78 $\pm$ 0.05	0.86 $\pm$ 0.02	0.92 $\pm$ 0.04	0.71 $\pm$ 0.04	0.93 $\pm$ 0.05	0.64 $\pm$ 0.03	0.78 $\pm$ 0.02
	PA	33.72 $\pm$ 7.76	164.48 $\pm$ 7.79	131.33 $\pm$ 7.76	29.70 $\pm$ 3.31	92.96 $\pm$ 5.20	100.99 $\pm$ 5.20	7.11 $\pm$ 7.79	152.84 $\pm$ 2.97	19.79 $\pm$ 3.31
	B/T	0.164	0.203	0.053	0.166	0.204	0.244	0.110	0.173	0.214
Disc	${\mu }_o$	21.76 $\pm$ 0.14	21.25 $\pm$ 0.09	21.82 $\pm$ 0.14	20.47 $\pm$ 0.14	22.53 $\pm$ 0.07	20.99 $\pm$ 0.07	22.12 $\pm$ 0.09	21.47 $\pm$ 0.09	20.08 $\pm$ 0.14
	hinner	29.70 $\pm$ 2.03	22.70 $\pm$ 1.77	37.83 $\pm$ 2.58	33.94 $\pm$ 0.82	72.72 $\pm$ 2.77	27.69 $\pm$ 1.05	38.96 $\pm$ 3.03	84.96 $\pm$ 1.41	32.34 $\pm$ 0.78
	b/a	0.83 $\pm$ 0.02	0.94 $\pm$ 0.01	0.97 $\pm$ 0.02	0.88 $\pm$ 0.01	0.76 $\pm$ 0.01	0.64 $\pm$ 0.01	0.57 $\pm$ 0.01	0.67 $\pm$ 0.01	0.65 $\pm$ 0.01
	PA	19.24 $\pm$ 0.90	24.29 $\pm$ 0.73	155.77 $\pm$ 0.90	13.76 $\pm$ 0.25	78.24 $\pm$ 0.58	124.60 $\pm$ 0.58	4.41 $\pm$ 0.73	161.45 $\pm$ 0.45	10.05 $\pm$ 0.25
	houter	14.38 $\pm$ 0.50	N/A	14.56 $\pm$ 0.51	21.22 $\pm$ 0.74	N/A	N/A	N/A	N/A	20.89 $\pm$ 0.73
	Rbreak	47.57 $\pm$ 1.61	N/A	39.77 $\pm$ 1.35	87.60 $\pm$ 2.96	N/A	N/A	N/A	N/A	37.81 $\pm$ 1.28
	D/T	0.681	0.649	0.874	0.595	0.628	0.453	0.624	0.562	0.621
Outer bar	${\mu }_o$	21.72 $\pm$ 0.45	21.22 $\pm$ 0.42	22.32 $\pm$ 0.45	19.88 $\pm$ 0.18	21.35 $\pm$ 0.32	20.38 $\pm$ 0.32	21.49 $\pm$ 0.42	20.87 $\pm$ 0.20	19.76 $\pm$ 0.18
	a	37.64 $\pm$ 1.08	48.77 $\pm$ 1.46	54.07 $\pm$ 1.55	41.34 $\pm$ 0.41	70.69 $\pm$ 1.17	46.05 $\pm$ 0.76	49.66 $\pm$ 1.48	136.42 $\pm$ 1.48	34.86 $\pm$ 0.34
	n	3	4	3	3	3	3	2	3	2
	b/a	0.44 $\pm$ 0.06	0.43 $\pm$ 0.09	0.24 $\pm$ 0.06	0.71 $\pm$ 0.02	0.60 $\pm$ 0.02	0.55 $\pm$ 0.02	0.48 $\pm$ 0.09	0.61 $\pm$ 0.03	0.44 $\pm$ 0.02
	PA	119.74$\pm$ 0.57	150.08 $\pm$ 0.50	118.46 $\pm$ 0.57	85.55 $\pm$ 0.16	160.20$\pm$ 0.33	156.86 $\pm$ 0.33	174.30 $\pm$ 0.50	130.34 $\pm$ 0.13	172.29$\pm$ 0.16
	c	3	2	4	2	2	2	2	2	2
	Bar/T	0.099	0.130	0.067	0.154	0.154	0.211	0.217	0.254	0.136
Inner bar	${\mu }_o$	20.91 $\pm$ 2.04	19.02 $\pm$ 1.77	18.21 $\pm$ 2.04	18.39 $\pm$ 1.09	19.18 $\pm$ 0.24	19.17 $\pm$ 0.24	20.72 $\pm$ 1.77	18.60 $\pm$ 1.06	19.60 $\pm$ 1.09
	a	13.99 $\pm$ 4.73	4.11 $\pm$ 1.33	2.15 $\pm$ 0.73	14.12 $\pm$ 1.20	10.71 $\pm$ 1.81	17.69 $\pm$ 2.99	14.98 $\pm$ 4.84	11.74 $\pm$ 1.20	13.21 $\pm$ 1.12
	n	3	2	2	3	3	4	3	3	3
	b/a	0.87 $\pm$ 0.13	0.69 $\pm$ 0.95	0.26 $\pm$ 0.13	0.85 $\pm$ 0.03	0.30 $\pm$ 0.06	0.65 $\pm$ 0.06	0.75 $\pm$ 0.95	0.42 $\pm$ 0.04	0.72 $\pm$ 0.03
	PA	151.62$\pm$ 3.64	35.37 $\pm$ 0.38	166.23 $\pm$ 3.64	91.29 $\pm$ 2.09	61.67 $\pm$ 2.21	142.39 $\pm$ 2.21	179.84 $\pm$ 0.38	145.63 $\pm$ 1.30	168.06$\pm$ 2.09
	c	2	4	2	2	4	2	2	4	2
	Bar/T	0.057	0.018	0.006	0.085	0.013	0.092	0.049	0.011	0.029

Table 7. Table 7: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN g’-BAND. Last set of 8/17 galaxies

		NGC 3945	NGC 4314	NGC 4340	NGC 4503	NGC 4725	NGC 5850	NGC 7280	NGC 7716
Bulge	${\mu }_e$	20.71 $\pm$ 0.15	20.16 $\pm$ 0.26	20.05 $\pm$ 0.15	19.81 $\pm$ 0.15	19.22 $\pm$ 0.16	20.85 $\pm$ 0.15	19.87 $\pm$ 0.33	18.08 $\pm$ 0.31
	Re	11.50 $\pm$ 1.17	10.32 $\pm$ 1.65	6.32 $\pm$ 0.64	5.14 $\pm$ 0.52	6.14 $\pm$ 0.51	7.45 $\pm$ 0.76	4.26 $\pm$ 0.85	1.37 $\pm$ 0.24
	n	3.41 $\pm$ 0.42	0.80 $\pm$ 0.09	2.58 $\pm$ 0.32	2.60 $\pm$ 0.32	1.86 $\pm$ 0.14	2.12 $\pm$ 0.26	2.16 $\pm$ 0.40	1.93 $\pm$ 0.38
	b/a	0.88 $\pm$ 0.04	0.82 $\pm$ 0.04	0.75 $\pm$ 0.04	0.75 $\pm$ 0.04	0.85 $\pm$ 0.03	0.97 $\pm$ 0.04	0.60 $\pm$ 0.05	0.45 $\pm$ 0.05
	PA	149.56 $\pm$ 5.20	123.02 $\pm$ 5.05	98.14 $\pm$ 5.20	20.03 $\pm$ 5.20	34.43 $\pm$ 2.97	148.30 $\pm$ 5.20	67.54 $\pm$ 7.76	53.90 $\pm$ 7.79
	B/T	0.340	0.207	0.256	0.197	0.070	0.138	0.232	0.082
Disc	${\mu }_o$	22.67 $\pm$ 0.07	22.17 $\pm$ 0.07	21.52 $\pm$ 0.07	20.27 $\pm$ 0.07	20.93 $\pm$ 0.07	22.37 $\pm$ 0.07	21.31 $\pm$ 0.14	20.88 $\pm$ 0.09
	hinner	69.07 $\pm$ 2.63	59.84 $\pm$ 2.79	34.96 $\pm$ 1.33	26.45 $\pm$ 1.01	99.59 $\pm$ 1.65	65.99 $\pm$ 2.51	26.90 $\pm$ 1.84	18.18 $\pm$ 1.42
	b/a	0.67 $\pm$ 0.01	0.92 $\pm$ 0.01	0.67 $\pm$ 0.01	0.46 $\pm$ 0.01	0.50 $\pm$ 0.01	0.74 $\pm$ 0.01	0.65 $\pm$ 0.02	0.82 $\pm$ 0.01
	PA	166.07 $\pm$ 0.58	137.31 $\pm$ 0.37	91.68 $\pm$ 0.58	9.97 $\pm$ 0.58	42.99 $\pm$ 0.45	138.64 $\pm$ 0.58	74.48 $\pm$ 0.90	37.85 $\pm$ 0.73
	houter	N/A	26.52 $\pm$ 0.93	N/A	N/A	N/A	N/A	10.90 $\pm$ 0.38	N/A
	Rbreak	N/A	108.86 $\pm$ 3.68	N/A	N/A	N/A	N/A	44.51 $\pm$ 1.51	N/A
	D/T	0.455	0.500	0.620	0.721	0.891	0.762	0.644	0.781
Outer bar	${\mu }_o$	21.42 $\pm$ 0.32	21.04 $\pm$ 0.30	22.47 $\pm$ 0.32	21.22 $\pm$ 0.32	21.00 $\pm$ 0.20	22.40 $\pm$ 0.32	21.50 $\pm$ 0.45	21.92 $\pm$ 0.42
	a	58.16 $\pm$ 0.96	125.89 $\pm$2.05	62.42 $\pm$ 1.03	32.01 $\pm$ 0.53	54.26 $\pm$ 0.59	135.53 $\pm$ 2.24	21.19 $\pm$ 0.61	30.48 $\pm$ 0.91
	n	2	3	1	2	3	4	2	2
	b/a	0.63 $\pm$ 0.02	0.26 $\pm$ 0.04	0.44 $\pm$ 0.02	0.48 $\pm$ 0.02	0.66 $\pm$ 0.03	0.24 $\pm$ 0.02	0.53 $\pm$ 0.06	0.48 $\pm$ 0.09
	PA	74.99 $\pm$ 0.33	147.29 $\pm$ 0.28	30.81 $\pm$ 0.33	176.82$\pm$ 0.33	36.25 $\pm$0.13	114.98 $\pm$ 0.33	55.43 $\pm$ 0.57	17.19 $\pm$ 0.50
	c	2	2	4	2	3	2	2	3
	Bar/T	0.138	0.275	0.108	0.065	0.035	0.091	0.060	0.071
Inner bar	${\mu }_o$	19.23 $\pm$ 0.24	17.63 $\pm$ 0.98	18.79 $\pm$ 0.24	20.62 $\pm$ 0.24	19.21 $\pm$ 1.06	19.68 $\pm$ 0.24	16.44 $\pm$ 2.04	19.47 $\pm$ 1.77
	a	19.44 $\pm$ 3.29	4.54 $\pm$ 0.82	8.56 $\pm$ 1.45	13.29 $\pm$ 2.25	12.84 $\pm$ 1.32	10.52 $\pm$ 1.78	2.41 $\pm$ 0.81	8.57 $\pm$ 2.77
	n	3	4	4	4	3	2	2	3
	b/a	0.47 $\pm$ 0.06	0.69 $\pm$ 0.06	0.25 $\pm$ 0.06	0.64 $\pm$ 0.06	0.26 $\pm$ 0.04	0.19 $\pm$ 0.06	0.41 $\pm$ 0.13	0.76 $\pm$ 0.95
	PA	161.04 $\pm$ 2.21	165.23 $\pm$ 2.85	21.76 $\pm$ 2.21	148.87$\pm$ 2.21	140.93 $\pm$ 1.30	45.91 $\pm$ 2.21	106.65$\pm$ 3.64	56.82 $\pm$ 0.38
	c	3	2	4	2	5	4	1	2
	Bar/T	0.067	0.018	0.016	0.017	0.004	0.009	0.064	0.066

Table 8. Table 8: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN i’-BAND. First set of 9/17 galaxies.

		NGC 357	NGC 718	NGC 2642	NGC 2681	NGC 2859	NGC 2950	NGC 2962	NGC 3368	NGC 3941
Bulge	${\mu }_e$	18.03 $\pm$ 0.26	18.21 $\pm$ 0.16	19.12 $\pm$ 0.26	16.68 $\pm$ 0.13	18.11 $\pm$ 0.16	16.66 $\pm$ 0.16	17.93 $\pm$ 0.15	18.30 $\pm$ 0.16	17.26 $\pm$ 0.13
	Re	2.70 $\pm$ 0.43	3.38 $\pm$ 0.28	2.52 $\pm$ 0.40	2.59 $\pm$ 0.26	5.59 $\pm$ 0.47	2.90 $\pm$ 0.24	2.35 $\pm$ 0.24	13.75 $\pm$ 1.15	3.98 $\pm$ 0.40
	n	1.63 $\pm$ 0.18	2.36 $\pm$ 0.18	1.62 $\pm$ 0.18	0.69 $\pm$ 0.05	1.66 $\pm$ 0.13	1.84 $\pm$ 0.14	1.61 $\pm$ 0.20	1.29 $\pm$ 0.10	2.62 $\pm$ 0.19
	b/a	0.89 $\pm$ 0.04	0.84 $\pm$ 0.03	0.87 $\pm$ 0.04	0.96 $\pm$ 0.02	0.95 $\pm$ 0.03	0.72 $\pm$ 0.03	0.98 $\pm$ 0.04	0.70 $\pm$ 0.03	0.78 $\pm$ 0.02
	PA	41.14 $\pm$ 5.05	165.64 $\pm$ 2.97	125.17 $\pm$ 5.05	0.00 $\pm$ 3.31	97.41 $\pm$ 2.97	97.97 $\pm$ 2.97	4.64 $\pm$ 5.20	156.86 $\pm$ 2.97	16.72 $\pm$ 3.31
	B/T	0.164	0.209	0.085	0.129	0.237	0.267	0.135	0.197	0.226
Disc	${\mu }_o$	20.30 $\pm$ 0.07	20.13 $\pm$ 0.07	20.71 $\pm$ 0.07	19.77 $\pm$ 0.07	21.44 $\pm$ 0.07	19.73 $\pm$ 0.07	20.78 $\pm$ 0.07	20.22 $\pm$ 0.07	18.93 $\pm$ 0.07
	hinner	30.02 $\pm$ 1.40	22.83 $\pm$ 0.38	30.91 $\pm$ 1.44	33.65 $\pm$ 0.81	67.91 $\pm$ 1.13	25.83 $\pm$ 0.43	37.46 $\pm$ 1.43	82.42 $\pm$ 1.37	32.08 $\pm$ 0.78
	b/a	0.84 $\pm$ 0.01	0.95 $\pm$ 0.01	0.97 $\pm$ 0.01	0.89 $\pm$ 0.01	0.80 $\pm$ 0.01	0.65 $\pm$ 0.01	0.55 $\pm$ 0.01	0.67 $\pm$ 0.01	0.65 $\pm$ 0.01
	PA	20.91 $\pm$ 0.37	16.23 $\pm$ 0.45	153.14 $\pm$ 0.37	12.22 $\pm$ 0.25	80.90 $\pm$ 0.45	124.19 $\pm$ 0.45	5.98 $\pm$ 0.58	158.78 $\pm$ 0.45	9.15 $\pm$ 0.25
	houter	15.39 $\pm$ 0.54	N/A	14.20 $\pm$ 0.50	22.13 $\pm$ 0.77	N/A	N/A	N/A	N/A	20.62 $\pm$ 0.72
	Rbreak	46.09 $\pm$ 1.56	N/A	40.09 $\pm$ 1.36	88.81 $\pm$ 3.01	N/A	N/A	N/A	N/A	38.39 $\pm$ 1.30
	D/T	0.681	0.658	0.822	0.637	0.575	0.457	0.594	0.539	0.616
Outer bar	${\mu }_o$	20.22 $\pm$ 0.30	20.02 $\pm$ 0.20	20.86 $\pm$ 0.30	19.21 $\pm$ 0.18	20.25 $\pm$ 0.20	19.28 $\pm$ 0.20	20.20 $\pm$ 0.32	19.56 $\pm$ 0.20	18.67 $\pm$ 0.18
	a	38.09 $\pm$ 0.62	45.97 $\pm$ 0.50	48.35 $\pm$ 0.79	40.37 $\pm$ 0.40	72.16 $\pm$ 0.78	45.74 $\pm$ 0.50	49.62 $\pm$ 0.82	130.17 $\pm$ 1.41	35.13 $\pm$ 0.34
	n	3	4	3	3	3	3	2	3	2
	b/a	0.44 $\pm$ 0.04	0.43 $\pm$ 0.03	0.26 $\pm$ 0.04	0.74 $\pm$ 0.02	0.63 $\pm$ 0.03	0.53 $\pm$ 0.03	0.49 $\pm$ 0.02	0.59 $\pm$ 0.03	0.44 $\pm$ 0.02
	PA	119.74$\pm$ 0.28	150.81 $\pm$ 0.13	118.57 $\pm$ 0.28	84.79 $\pm$ 0.16	159.38$\pm$ 0.13	156.76 $\pm$ 0.13	173.62 $\pm$ 0.33	128.00 $\pm$ 0.13	171.50 $\pm$ 0.16
	c	3	2	4	2	2	2	2	2	2
	Bar/T	0.101	0.120	0.089	0.159	0.169	0.195	0.225	0.245	0.131
Inner bar	${\mu }_o$	19.40 $\pm$ 0.98	18.11 $\pm$ 1.06	18.01 $\pm$ 0.98	18.35 $\pm$ 1.09	17.92 $\pm$ 1.06	18.05 $\pm$ 1.06	19.55 $\pm$ 0.24	16.15 $\pm$ 1.06	18.58 $\pm$ 1.09
	a	13.40 $\pm$ 2.42	4.16 $\pm$ 0.43	3.19 $\pm$ 0.58	18.52 $\pm$ 1.57	11.15 $\pm$ 1.14	17.16 $\pm$ 1.76	15.48 $\pm$ 2.62	8.88 $\pm$ 0.91	13.23 $\pm$ 1.12
	n	3	2	2	3	3	4	3	3	3
	b/a	0.89 $\pm$ 0.06	0.63 $\pm$ 0.04	0.16 $\pm$ 0.06	0.75 $\pm$ 0.03	0.34 $\pm$ 0.04	0.61 $\pm$ 0.04	0.72 $\pm$ 0.06	0.41 $\pm$ 0.04	0.76 $\pm$ 0.03
	PA	157.67$\pm$ 2.85	35.03 $\pm$ 1.30	159.41 $\pm$ 2.85	71.40 $\pm$ 2.09	61.78 $\pm$ 1.30	141.49 $\pm$ 1.30	0.80 $\pm$ 2.21	132.61 $\pm$ 1.30	170.13 $\pm$ 2.09
	c	2	4	2	2	4	2	2	4	2
	Bar/T	0.054	0.013	0.004	0.075	0.019	0.081	0.046	0.018	0.027

Table 9. Table 9: ALL PARAMETERS FOR THE DOUBLE-BARRED FITS IN i’-BAND. Last set of 8/17 galaxies

		NGC 3945	NGC 4314	NGC 4340	NGC 4503	NGC 4725	NGC 5850	NGC 7280	NGC 7716
Bulge	${\mu }_e$	19.23 $\pm$ 0.16	19.23 $\pm$ 0.13	18.90 $\pm$ 0.16	18.62 $\pm$ 0.16	18.25 $\pm$ 0.16	19.77 $\pm$ 0.16	18.69 $\pm$ 0.26	17.54 $\pm$ 0.15
	Re	10.06 $\pm$ 0.84	11.25 $\pm$ 1.13	6.45 $\pm$ 0.51	5.39 $\pm$ 0.45	7.22 $\pm$ 0.61	8.79 $\pm$ 0.74	4.04 $\pm$ 0.65	1.97 $\pm$ 0.20
	n	3.30 $\pm$ 0.26	1.00 $\pm$ 0.07	2.34 $\pm$ 0.17	2.70 $\pm$ 0.21	2.27 $\pm$ 0.18	2.46 $\pm$ 0.19	3.07 $\pm$ 0.33	2.19 $\pm$ 0.27
	b/a	0.87 $\pm$ 0.03	0.81 $\pm$ 0.02	0.76 $\pm$ 0.03	0.76 $\pm$ 0.03	0.84 $\pm$ 0.03	0.97 $\pm$ 0.03	0.66 $\pm$ 0.04	0.57 $\pm$ 0.04
	PA	147.98 $\pm$ 2.97	132.45 $\pm$ 3.31	98.75 $\pm$ 2.97	19.61 $\pm$ 2.97	28.52 $\pm$ 2.97	160.29 $\pm$ 2.97	68.06 $\pm$ 5.05	56.16 $\pm$ 5.20
	B/T	0.329	0.210	0.241	0.207	0.101	0.195	0.269	0.154
Disc	${\mu }_o$	21.12 $\pm$ 0.07	21.05 $\pm$ 0.07	20.33 $\pm$ 0.07	19.02 $\pm$ 0.07	19.59 $\pm$ 0.07	21.40 $\pm$ 0.07	20.19 $\pm$ 0.07	19.73 $\pm$ 0.07
	hinner	57.57 $\pm$ 0.95	63.47 $\pm$ 1.53	35.24 $\pm$ 0.58	26.62 $\pm$ 0.44	85.69 $\pm$ 1.42	66.96 $\pm$ 1.11	27.34 $\pm$ 1.27	15.95 $\pm$ 0.61
	b/a	0.68 $\pm$ 0.01	0.92 $\pm$ 0.01	0.69 $\pm$ 0.01	0.46 $\pm$ 0.01	0.47 $\pm$ 0.01	0.75 $\pm$ 0.01	0.65 $\pm$ 0.01	0.83 $\pm$ 0.01
	PA	163.30 $\pm$ 0.45	136.98 $\pm$ 0.25	91.44 $\pm$ 0.45	9.79 $\pm$ 0.45	45.33 $\pm$ 0.45	139.03 $\pm$ 0.45	74.00 $\pm$ 0.37	40.91 $\pm$ 0.58
	houter	N/A	28.52 $\pm$ 1.00	N/A	N/A	N/A	N/A	11.92 $\pm$ 0.42	N/A
	Rbreak	N/A	102.74 $\pm$ 3.48	N/A	N/A	N/A	N/A	44.19 $\pm$ 1.50	N/A
	D/T	0.450	0.498	0.621	0.711	0.854	0.683	0.636	0.719
Outer bar	${\mu }_o$	20.11 $\pm$ 0.20	19.87 $\pm$ 0.18	21.31 $\pm$ 0.20	19.98 $\pm$ 0.20	19.87 $\pm$ 0.20	21.08 $\pm$ 0.20	20.23 $\pm$ 0.30	20.89 $\pm$ 0.32
	a	58.81 $\pm$ 0.64	126.15 $\pm$ 1.24	62.71 $\pm$ 0.83	32.83 $\pm$ 0.36	56.18 $\pm$ 0.61	134.14 $\pm$ 1.45	20.69 $\pm$ 0.34	28.85 $\pm$ 0.48
	n	2	3	1	2	3	4	2	2
	b/a	0.61 $\pm$ 0.03	0.25 $\pm$ 0.02	0.44 $\pm$ 0.03	0.45 $\pm$ 0.03	0.60 $\pm$ 0.03	0.25 $\pm$ 0.03	0.49 $\pm$ 0.04	0.51 $\pm$ 0.02
	PA	72.26 $\pm$ 0.13	146.89 $\pm$ 0.16	30.93 $\pm$ 0.13	176.77 $\pm$ 0.13	34.57 $\pm$ 0.13	115.75 $\pm$ 0.13	55.17 $\pm$ 0.28	16.61 $\pm$ 0.33
	c	2	2	4	2	3	2	2	3
	Bar/T	0.153	0.265	0.120	0.063	0.040	0.112	0.056	0.072
Inner bar	${\mu }_o$	18.05 $\pm$ 1.06	16.35 $\pm$ 1.09	18.03 $\pm$ 1.06	18.88 $\pm$ 1.06	17.98 $\pm$ 1.06	18.75 $\pm$ 1.06	14.61 $\pm$ 0.98	18.69 $\pm$ 0.24
	a	19.78 $\pm$ 2.03	4.80 $\pm$ 0.41	8.88 $\pm$ 0.91	10.54 $\pm$ 1.08	12.30 $\pm$ 1.26	11.15 $\pm$ 1.14	2.23 $\pm$ 0.40	8.12 $\pm$ 1.37
	n	3	4	4	4	3	2	2	3
	b/a	0.46 $\pm$ 0.04	0.82 $\pm$ 0.03	0.36 $\pm$ 0.04	0.75 $\pm$ 0.04	0.28 $\pm$ 0.04	0.24 $\pm$ 0.04	0.17 $\pm$ 0.06	0.84 $\pm$ 0.06
	PA	159.84 $\pm$ 1.30	152.92 $\pm$ 2.09	19.82 $\pm$ 1.30	150.13 $\pm$ 1.30	137.08 $\pm$ 1.30	47.49 $\pm$ 1.30	112.04 $\pm$ 2.85	53.15 $\pm$ 2.21
	c	3	2	4	2	5	4	1	2
	Bar/T	0.067	0.026	0.019	0.019	0.005	0.010	0.040	0.055

Table 10. Table 10: ALL PARAMETERS FOR THE SINGLE-BARRED FITS IN r’-BAND. First set of 9/17 galaxies.

		NGC 357	NGC 718	NGC 2642	NGC 2681	NGC 2859	NGC 2950	NGC 2962	NGC 3368	NGC 3941
Bulge	${\mu }_e$	18.72 $\pm$ 0.10	17.98 $\pm$ 0.17	19.45 $\pm$ 0.19	16.96 $\pm$ 0.07	18.10 $\pm$ 0.09	17.26 $\pm$ 0.09	18.70 $\pm$ 0.17	18.95 $\pm$ 0.09	17.66 $\pm$ 0.07
	Re	3.46 $\pm$ 0.23	2.62 $\pm$ 0.33	2.53 $\pm$ 0.30	2.99 $\pm$ 0.13	5.01 $\pm$ 0.26	3.68 $\pm$ 0.19	3.32 $\pm$ 0.42	15.18 $\pm$ 0.79	4.55 $\pm$ 0.20
	n	1.63 $\pm$ 0.09	1.93 $\pm$ 0.14	1.71 $\pm$ 0.19	1.65 $\pm$ 0.06	1.45 $\pm$ 0.06	2.03 $\pm$ 0.09	1.80 $\pm$ 0.13	1.94 $\pm$ 0.09	2.27 $\pm$ 0.08
	b/a	0.93 $\pm$ 0.02	0.86 $\pm$ 0.03	0.82 $\pm$ 0.04	0.92 $\pm$ 0.01	0.87 $\pm$ 0.02	0.74 $\pm$ 0.02	0.90 $\pm$ 0.03	0.68 $\pm$ 0.02	0.79 $\pm$ 0.01
	PA	27.46 $\pm$ 3.06	172.88 $\pm$ 2.80	134.82 $\pm$ 5.23	52.81 $\pm$ 1.87	72.40 $\pm$ 3.05	110.61 $\pm$ 3.05	179.38 $\pm$ 2.80	151.07 $\pm$ 3.05	12.81 $\pm$ 1.87
	B/T	0.245	0.211	0.097	0.255	0.268	0.389	0.207	0.251	0.282
Disc	${\mu }_o$	20.96 $\pm$ 0.05	20.49 $\pm$ 0.05	21.11 $\pm$ 0.11	20.14 $\pm$ 0.11	21.73 $\pm$ 0.11	20.08 $\pm$ 0.11	21.10 $\pm$ 0.05	20.77 $\pm$ 0.05	19.42 $\pm$ 0.05
	hinner	33.25 $\pm$ 1.15	22.06 $\pm$ 0.86	27.61 $\pm$ 2.04	33.26 $\pm$ 0.68	56.99 $\pm$ 0.84	25.33 $\pm$ 0.37	33.42 $\pm$ 1.30	82.14 $\pm$ 1.21	34.14 $\pm$ 0.69
	b/a	0.82 $\pm$ 0.01	0.95 $\pm$ 0.02	0.97 $\pm$ 0.02	0.89 $\pm$ 0.01	0.85 $\pm$ 0.01	0.64 $\pm$ 0.01	0.57 $\pm$ 0.02	0.65 $\pm$ 0.01	0.64 $\pm$ 0.01
	PA	18.58 $\pm$ 3.61	16.97 $\pm$ 3.92	151.36 $\pm$ 7.19	12.33 $\pm$ 1.31	85.66 $\pm$ 1.98	125.91 $\pm$ 1.98	2.38 $\pm$ 3.92	161.32 $\pm$ 1.98	9.21 $\pm$ 1.31
	houter	14.14 $\pm$ 0.71	N/A	12.01 $\pm$ 0.60	18.66 $\pm$ 0.93	N/A	N/A	N/A	N/A	20.15 $\pm$ 1.01
	Rbreak	47.01 $\pm$ 1.35	N/A	45.54 $\pm$ 1.31	88.63 $\pm$ 2.54	N/A	N/A	N/A	N/A	39.49 $\pm$ 1.13
	D/T	0.652	0.651	0.806	0.595	0.537	0.453	0.588	0.516	0.595
Outer bar	${\mu }_o$	20.70 $\pm$ 0.31	20.29 $\pm$ 0.25	21.36 $\pm$ 0.57	19.59 $\pm$ 0.24	20.53 $\pm$ 0.16	19.75 $\pm$ 0.16	20.79 $\pm$ 0.25	20.23 $\pm$ 0.16	19.11 $\pm$ 0.24
	a	37.34 $\pm$ 0.74	45.33 $\pm$ 1.12	52.93 $\pm$ 2.51	40.96 $\pm$ 0.46	69.60 $\pm$ 0.80	46.36 $\pm$ 0.53	50.83 $\pm$ 1.26	136.20 $\pm$ 1.56	35.93 $\pm$ 0.41
	n	3	4	3	3	3	3	2	3	2
	b/a	0.44 $\pm$ 0.03	0.44 $\pm$ 0.06	0.24 $\pm$ 0.07	0.69 $\pm$ 0.02	0.62 $\pm$ 0.03	0.44 $\pm$ 0.03	0.45 $\pm$ 0.06	0.59 $\pm$ 0.03	0.41 $\pm$ 0.02
	PA	119.74 $\pm$ 3.02	150.94 $\pm$ 2.00	116.85 $\pm$ 4.79	84.81 $\pm$ 1.64	159.22 $\pm$ 1.89	158.60 $\pm$ 1.89	174.36 $\pm$ 2.00	127.74 $\pm$ 1.89	171.24 $\pm$ 1.64
	c	3	2	4	2	2	2	2	2	2
	Bar/T	0.103	0.138	0.097	0.149	0.195	0.159	0.205	0.233	0.123

Table 11. Table 11: ALL PARAMETERS FOR THE SINGLE-BARRED FITS IN r’-BAND. Last set of 8/17 galaxies

		NGC 3945	NGC 4314	NGC 4340	NGC 4503	NGC 4725	NGC 5850	NGC 7280	NGC 7716
Bulge	${\mu }_e$	18.08 $\pm$ 0.09	19.72 $\pm$ 0.07	18.81 $\pm$ 0.09	18.64 $\pm$ 0.09	18.12 $\pm$ 0.09	19.61 $\pm$ 0.09	18.64 $\pm$ 0.10	18.07 $\pm$ 0.17
	Re	6.33 $\pm$ 0.33	10.95 $\pm$ 0.48	4.97 $\pm$ 0.26	4.64 $\pm$ 0.24	5.43 $\pm$ 0.28	6.82 $\pm$ 0.35	3.19 $\pm$ 0.21	2.45 $\pm$ 0.31
	n	1.28 $\pm$ 0.06	2.07 $\pm$ 0.07	2.04 $\pm$ 0.09	2.23 $\pm$ 0.10	1.54 $\pm$ 0.07	1.99 $\pm$ 0.09	3.49 $\pm$ 0.19	1.52 $\pm$ 0.11
	b/a	0.69 $\pm$ 0.02	0.81 $\pm$ 0.01	0.86 $\pm$ 0.02	0.79 $\pm$ 0.02	0.92 $\pm$ 0.02	0.89 $\pm$ 0.02	0.67 $\pm$ 0.02	0.67 $\pm$ 0.03
	PA	156.67 $\pm$ 3.05	134.37 $\pm$ 1.87	91.60 $\pm$ 3.05	12.92 $\pm$ 3.05	5.76 $\pm$ 3.05	41.03 $\pm$ 3.05	72.10 $\pm$ 3.06	55.53 $\pm$ 2.80
	B/T	0.317	0.261	0.259	0.219	0.088	0.179	0.288	0.201
Disc	${\mu }_o$	21.12 $\pm$ 0.05	21.49 $\pm$ 0.05	20.68 $\pm$ 0.05	19.43 $\pm$ 0.05	20.05 $\pm$ 0.05	21.58 $\pm$ 0.05	20.59 $\pm$ 0.05	20.13 $\pm$ 0.05
	hinner	44.60 $\pm$ 0.66	62.51 $\pm$ 1.27	33.76 $\pm$ 0.50	26.20 $\pm$ 0.39	85.06 $\pm$ 1.25	59.17 $\pm$ 0.87	26.95 $\pm$ 0.94	16.44 $\pm$ 0.64
	b/a	0.72 $\pm$ 0.01	0.92 $\pm$ 0.01	0.66 $\pm$ 0.01	0.46 $\pm$ 0.01	0.50 $\pm$ 0.01	0.74 $\pm$ 0.01	0.65 $\pm$ 0.01	0.82 $\pm$ 0.02
	PA	156.77 $\pm$ 1.98	137.39 $\pm$ 1.31	92.96 $\pm$ 1.98	9.99 $\pm$ 1.98	43.07 $\pm$ 1.98	136.37 $\pm$ 1.98	73.83 $\pm$ 3.60	41.36 $\pm$ 3.92
	houter	N/A	25.72 $\pm$ 1.29	N/A	N/A	N/A	N/A	11.19 $\pm$ 0.56	N/A
	Rbreak	N/A	105.84 $\pm$ 3.03	N/A	N/A	N/A	N/A	44.40 $\pm$ 1.27	N/A
	D/T	0.468	0.480	0.628	0.722	0.864	0.706	0.639	0.732
Outer bar	${\mu }_o$	20.25 $\pm$ 0.16	20.25 $\pm$ 0.24	21.75 $\pm$ 0.16	20.51 $\pm$ 0.16	20.11 $\pm$ 0.16	21.41 $\pm$ 0.16	20.40 $\pm$ 0.31	21.43 $\pm$ 0.25
	a	57.05 $\pm$ 0.65	123.78 $\pm$ 1.40	63.33 $\pm$ 0.73	33.19 $\pm$ 0.38	56.17 $\pm$ 0.64	123.71 $\pm$ 1.42	20.90 $\pm$ 0.42	32.07 $\pm$ 0.79
	n	2	3	1	2	3	4	2	2
	b/a	0.63 $\pm$ 0.03	0.24 $\pm$ 0.02	0.45 $\pm$ 0.03	0.45 $\pm$ 0.03	0.60 $\pm$ 0.03	0.26 $\pm$ 0.03	0.48 $\pm$ 0.03	0.46 $\pm$ 0.06
	PA	73.25 $\pm$ 1.89	146.78 $\pm$ 1.64	31.14 $\pm$ 1.89	177.80 $\pm$ 1.89	36.86 $\pm$ 1.89	116.54 $\pm$ 1.89	55.13 $\pm$ 3.02	16.95 $\pm$ 2.00
	c	2	2	4	2	3	2	2	3
	Bar/T	0.215	0.258	0.112	0.060	0.048	0.114	0.073	0.067

Equations

$$I_{\mathrm{b}}(r_{\mathrm{b}})=I_{\mathrm{e}}10^{-b_n\left[{\left(\frac{r_{\mathrm{b}}}{R_{\mathrm{e}}}\right)}^{\frac{1}{n}}-1\right]},$$

$$I_{\mathrm{d}}(r_{\mathrm{d}})=I_0\left[e^{\frac{-r_{\mathrm{d}}}{h}}\theta +e^{\frac{-r_{\mathrm{break}}(h_{\mathrm{out}}-h)}{h_{\mathrm{out}}h}}{e}^{\frac{-r_{\mathrm{d}}}{h_{\mathrm{out}}}}(1-\theta )\right],$$

$$\theta =\{\begin{array}{ll}0& \mathrm{if}{r}_{\mathrm{d}}>r_{\mathrm{break}}\\ 1& \mathrm{if}{r}_{\mathrm{d}}<r_{\mathrm{break}},\end{array}$$

$$I_{\mathrm{bar}}(r_{\mathrm{bar}})=I_{0,\mathrm{bar}}{\left[1-{\left(\frac{r_{\mathrm{bar}}}{a_{\mathrm{bar}}}\right)}^2\right]}^{n_{\mathrm{bar}}+0.5};\mathrm{for}{r}_{\mathrm{bar}}\le a_{\mathrm{bar}},$$

$$r={\left(|x{|}^c+{\left|\frac{y}{(1-{ϵ}_{\mathrm{bar}})}\right|}^c\right)}^{1/c},$$

Full text

††pagerange: Deconstructing double-barred galaxies in 2D and 3D. I. Classical nature of the dominant bulges–A††pubyear: 2018

1 Introduction

Double-barred galaxies are disc-like galaxies hosting two stellar structures in the shape of a bar: an outer bar alike to the single bars observed in $\sim$60-70% of all disc galaxies (Aguerri et al., 2009; Erwin, 2018), and the so-called inner bar that refers to an additional, smaller barred structure that has been found in at least $\sim$30% of all barred galaxies (Laine et al., 2002; Erwin, 2004).

Observations of galaxies with two nested stellar bars date back to the second half of the twentieth century (see de Vaucouleurs, 1975, for a first identification of NGC 1291 as a double-barred galaxy). After serendipitous detections, such as those of Sandage & Brucato (1979) and Kormendy (1979), Erwin (2004) was the first to make a thorough compilation of photometric data from various facilities in a catalog of 50 double-barred galaxies. More individuals have been later detected by large photometric surveys such as the Spitzer Survey of Stellar Structures in Galaxies (S4G; Sheth et al., 2010). Moreover, inner bars have been observed up to a redshift $\sim$0.15 (Lisker et al., 2006), thus supporting an scenario in which they are not occasional but common stellar structures. This is a surprising result as theoretically it is not possible to have two sets of closed barred orbits coexisting within a rotating disc; the problem of the orbital support of double-barred systems requires the complex definition of loops introduced by Maciejewski & Sparke (1997) and Maciejewski & Sparke (2000). Simulating the formation and evolution of double-barred galaxies has indeed been proved a hard task (e.g. Du et al., 2015; Wozniak, 2015).

After almost 40 years of studies, many important questions remain open about double-barred galaxies. This work aims at providing photometric constraints to some of them through the most complete photometric study of double-barred galaxies ever performed. Our analysis combines two-dimensional (2D) photometric decompositions of double-barred galaxies in their multiple structural components (bulges, bars, and discs), and a three-dimensional (3D) statistical deprojection of their bulges and (inner and outer) bars. We remark this is the first time either 2D decompositions or 3D deprojections of a sample of double-barred galaxies are presented.

In a series of two papers, we explore the following unanswered questions about double-barred galaxies: i) whether there exists a major incidence of disc-like bulges within double-barred galaxies, where secular evolution is assumed to take place in a very efficient way; ii) whether inner bars form secularly out of disc-like bulges already present in barred galaxies; iii) whether inner bars are transient or long-lived structures; and iv) whether all barred galaxies will develop an inner bar at some stage of their lives. This first paper is devoted to the presentation of the analysis and the photometric properties of bulges within double-barred systems (questions i and ii), while the properties of inner and outer bars (questions iii and iv) are studied in de Lorenzo-Cáceres et al. (2018c, in preparation; hereafter Paper II).

1.1 Frequency of disc-like bulges in double-barred galaxies

Galactic bulges are properly defined as the central excess of light found in the surface brightness distribution of a disc galaxy, i.e., they imply a central concentration of stars. According to their photometric and kinematic properties, these structures are classified into two main groups: classical and disc-like bulges. Both types are believed to be linked to the different evolutionary processes forming them. Classical bulges are considered the result of fast actions, such as monolithic collapse or mergers (Eggen et al., 1962; Aguerri et al., 2001; Bournaud et al., 2007; Hopkins et al., 2010). These are pressure-supported systems rather analogous to elliptical galaxies. On the other hand, disc-like bulges show more ordered motions dominated by rotation. Sometimes referred to as pseudobulges, they are thought to be formed through secular evolution mainly driven by bars (Kormendy & Kennicutt, 2004, although see Eliche-Moral et al. 2006 for a numerical evidence of minor mergers causing similar bulge growth). Indeed, bars promote an angular momentum exchange between dark and baryonic matter or even between baryons and baryons. As a consequence, stars are dragged towards the outer galactic regions or gas is brought into the centre, where it concentrates and may eventually form new stars and stellar structures such as bulges (Combes et al., 1990; Friedli & Benz, 1995; Muñoz-Tuñón et al., 2004; Boone et al., 2007).

Shlosman et al. (1989, 1990) show that a double-barred system may theoretically be more efficient than a single bar in the transportation of gas towards the central galactic regions, being even able to reach the sphere of influence of the central black hole. This is the reason why double bars have been proposed as the gas channel for triggering active galactic nuclei, although no conclusive observational evidence has been found so far (Márquez et al., 2000). It is therefore sensible to state that a major incidence of secularly-formed structures should be found in double-barred galaxies, where gas inflow is very efficient. Disc-like bulges are commonly considered the clearest consequence of secular evolution, and inner bars themselves are considered as evidence for the presence of secularly-promoted bulges by many authors (Fisher & Drory, 2016). Whether disc-like bulges are frequent in double-barred galaxies has not been purposedly studied in the literature so far, with the exception of de Lorenzo-Cáceres et al. (2012), who present evidence of the existence of a disc-like bulge in the centre of the double-barred galaxy NGC 357.

During the last decades, great effort has been put into the search of photometric diagnostics that allow to discern the nature of bulges (i.e. classical vs. disc-like). The Sérsic index, $n$, appeared at first to be the best candidate. For example, Fisher & Drory (2008) found that classical bulges tend to have $n>$2, whereas disc-like bulges show $n<$2. The Sérsic index is a mathematical parameter describing the shape of the surface-brightness profile of a bulge, once isolated from the rest of galaxy components through a photometric decomposition. Another diagnostic involves the projection of the fundamental plane onto the integrated surface brightness of the bulge within one effective radius versus the bulge effective radius, also known as the Kormendy (1977) relation. This has been argued to be a good discriminator between classical and disc-like bulges, as it shows a correlation that creates a top sequence mostly populated by classical bulges, while the existence of bulges with lower effective surface brightness for a given radius (i.e., outside the classical relation) is explained by their disc-like nature (e.g. Fisher & Drory, 2008; Gadotti, 2009; Fisher & Drory, 2016; Neumann et al., 2017).

More recently, Costantin et al. (2018) tested the photometric diagnostics commonly used to discern between classical and disc-like bulges, and compared them with some kinematic diagnostics. Their results indicate that projected quantities such as the Sérsic index alone do not provide conclusive results and they propose the use of the intrinsic 3D shape of bulges as a better photometric discriminator, based on the intrinsic flattening of the bulge component with respect to the other galaxy structures (see also Méndez-Abreu et al., 2018). Indeed, classical bulges are expected to be pretty spherical structures. On the other hand, rotating disc-like bulges should appear flatter, also depending on the disc thickening.

1.2 2D photometric decompositions

All published photometric studies about double-barred galaxies so far rely on analyses of the integrated galaxy light (e.g. Erwin, 2004). Inner bars are usually detected through isophotal analysis as bumps in the ellipticity and position angle profiles (Friedli & Martinet, 1993; Laine et al., 2002), and/or through a careful visual inspection of unsharp-masked images (Erwin & Sparke, 2003). Better suited photometric decompositions, which model the galaxy light as a combination of individual structures instead of studying the integrated properties, have not been systematically applied for double-barred cases. The only notable exceptions are the recent works by Méndez-Abreu et al. (2017), who performed 2D photometric decompositions of all galaxies of the Calar Alto Legacy Integral-Field Area survey (CALIFA; Sánchez et al., 2012), including two double-barred cases, and de Lorenzo-Cáceres et al. (2018a, submitted; see also Méndez-Abreu et al. 2019), who decomposed the S4G images of the two double-barred galaxies present in the spectroscopic TIMER project (Gadotti et al., 2018).

The rapid progress in the field of photometric decompositions has resulted in several sophisticated codes that dissect the galaxy into various components by making 2D models of each structure (e.g. Peng et al., 2002; de Souza et al., 2004; Méndez-Abreu et al., 2008; Erwin, 2015). However, most of the published works include up to two components, namely disc and bulge, while other structures such as stellar bars are not taken into account (e.g., Simard et al., 2011). Bulge and disc certainly account for most of the galaxy light but not including bars has an effect on the measured parameters, as shown by Laurikainen et al. (2006), Gadotti (2008), Fisher & Drory (2008), and Méndez-Abreu et al. (2017), among others. Dismissing the bar leads to large uncertainties and overestimation of the bulge parameters ($n$ and effective radius) and bulge contribution to total light.

Here we present the first consistent 2D multi-component photometric decomposition of a homogenous set of images of 17 double-barred galaxies, fitting a bulge, a (truncated) disc, and the two bars with the code GASP2D (Méndez-Abreu et al., 2008). The analysis of the bulge properties is therefore performed with no contamination of the inner and outer bars.

1.3 3D deprojection: intrinsic shapes

While photometric decompositions provide the individual parameters for the structural components shaping galaxies, such analysis is based on projected images of the galaxies on the sky. Therefore, and even after deprojecting bar lengths and ellipticities, only the two dimensions in the galaxy plane are explored. Recent development in mathematical and statistical techniques has shown that it is possible to retrieve information about the missing third direction and actually derive the intrinsic 3D shape of galaxy structures (bulges and bars), as reviewed in Méndez-Abreu (2016). Studies so far have focused on the case of non-barred and single-barred galaxies. In the set of papers by Costantin et al. (2018) and Méndez-Abreu et al. (2018), the intrinsic shapes of bulges and bars in the CALIFA sample are analysed. Among other conclusions, the authors find that the intrinsic flattening of bulges with respect to bars holds important clues for understanding the formation path of these structures.

In this project we derive the 3D shape of bulges and (inner and outer) bars with the code galaXYZ (Méndez-Abreu et al., 2010; Costantin et al., 2018). We emphasise that this is the first time the intrinsic shape of inner bars within double-barred systems are studied.

The paper is organised as follows: the parent sample and images are described in Sect. 2, while a brief outline of GASP2D and the procedure followed in this work to perform 2D multi-component photometric decompositions of double-barred galaxies are provided in Sect. 3. Section 4 describes galaXYZ and the procedure used to derive the 3D intrinsic shape of bars and bulges. In Sect. 5 we quantify the possible bias introduced in the bulge parameters by dismissing the presence of an inner bar. Section 6 shows an analysis of the photometric properties of bulges in double-barred galaxies and the discussion about their nature is provided in Sect. 7. Conclusions are wrapped up in Sect. 8. A flat cosmology with $\Omega_{m}=$0.3, $\Omega_{\Lambda}=$0.7, and $H_{0}=$75 km s*-1* Mpc*-1* is assumed throughout the paper. These are the same parameters adopted by Gadotti (2009) and Méndez-Abreu et al. (2017), whose works are used for comparison throughout this paper.

2 The sample of double-barred galaxies

The sample of double-barred galaxies is extracted from a catalog of 67 barred galaxies with inner structures presented in Erwin (2004). Among these, 50 galaxies are double-barred, as classified by two photometric diagnostics, namely ellipse fitting and unsharp masking. Bar properties such as sizes and position angles (both measured from ellipse fitting) are also provided by Erwin (2004), who uses a compilation of results from the literature and new measurements made on a variety of images in the optical and near infrared.

For the present analysis we first selected all double-barred galaxies from the catalog of Erwin (2004) with available Sloan Digital Sky Survey (SDSS; York et al., 2000) imaging, thus obtaining a preliminary sample of 23 objects. SDSS provides a homogeneous set of $g^{\prime}$-, $r^{\prime}$-, and $i^{\prime}$-band images with medium spatial resolution, suitable for inner bar detection at the redshift of our galaxies ($z<0.015$), and a field-of-view large enough to reach the outermost regions of the galaxy, as required for a proper modelling of the disc component. In particular, we use the images from the SDSS Data Release 9 (Ahn et al., 2012). All images are already soft-bias subtracted, flat-field corrected, sky subtracted, and flux calibrated using the standard SDSS pipelines. Some further treatment of the data is necessary to both re-calibrate the images from nanomaggies to counts (required for the fitting procedure with GASP2D) and to refine the sky subtraction. Details on this process are described in Pagotto et al. (2017) and Costantin et al. (2018). The point spread function (PSF) is measured on each image using a circular Moffat function (Moffat, 1969). The mean values of the Full Width at Half Maximum (FWHM) for the $g^{\prime}$-, $r^{\prime}$-, and $i^{\prime}$-images are 1.32, 1.15, and 1.17 arcsec, respectively.

Notwithstanding the careful inspection made by Erwin (2004), our 2D multi-component photometric decompositions reveal that some of these galaxies had been either misclassified as double barred or the SDSS spatial resolution is not enough to readily distinguish the inner bar. In the former case, we found this is mainly due to the presence of other components (e.g., stellar inner rings or complex dust structures). Six galaxies were finally removed from our preliminary sample, namely Mrk 573, UGC 524, NGC 1068, NGC 4303, NGC 4321, and NGC 4736. Our definitive sample is therefore composed of 17 double-barred galaxies. Table 1 shows the galaxy sample together with some relevant parameters.

All our double-barred galaxies are nearby, with $z<0.015$ as indicated in Table 1. Fisher & Drory (2016) establish that, given the SDSS spatial resolution and a typical bulge effective radius of 2 kpc, photometric decompositions aiming at accurately deriving bulge properties with SDSS data should restrict to galaxies up to $z$=0.03. Costantin et al. (2017) explore the possible errors on the bulge structural parameters when their angular sizes are close to the size of the image PSF. They find that even for bulges with effective radius 1.2$\times\sigma$ of the PSF ($\sigma\sim\rm FWHM/2.35$), the bulge parameters can be recovered within a 10% error. We conclude that our photometric decomposition analysis is therefore not hampered by resolution effects affecting small bulges or inner bars.

2.1 Control samples of single-barred galaxies

Two control samples of single-barred galaxies are used throughout this paper: those from Méndez-Abreu et al. (2017) and Gadotti (2009).

Méndez-Abreu et al. (2017) present the 2D photometric decompositions of the SDSS $g^{\prime}-$, $r^{\prime}-$, and $i^{\prime}-$images for 404 galaxies from the CALIFA survey (Sánchez et al., 2012). They make use of the GASP2D code. The methodology used in Méndez-Abreu et al. (2017) is very similar to the one followed in this work and described in Sect. 3. For this reason, the photometric parameters for their 162 single-barred galaxies are used as a control sample throughout this paper. Note that the CALIFA sample expands a wider redshift range than our double-barred sample, covering from $z$=0.005 up to $z$=0.03. In addition, the analysis of Méndez-Abreu et al. (2017) includes two double-bar hosts as well: NGC 7716 and NGC 23. While NGC 7716 is included in the sample presented here, NGC 23 does not belong to the parent sample from Erwin (2004) and it was therefore not initially considered for this work. Results from Méndez-Abreu et al. (2017) for these two galaxies are shown together with our measurements whenever it is possible, in order to demonstrate the good consistency between both works and particularly for the one galaxy in common.

Gadotti (2009) also performs 2D multi-component photometric decomposition of SDSS images in all the bands. His analysis includes nearly 1000 galaxies, among which he finds 287 bar+bulge hosts. The sample is more distant (0.02$<z<$0.07) than the sample presented here, and no search for possible double bars is performed; some level of contamination is therefore expected. Note that the code used for the analysis of Gadotti (2009) is BUDDA (de Souza et al., 2004), which describes bars with Sérsic profiles instead of the Ferrers profiles used in this work (see Sect. 3). By using the works of both Méndez-Abreu et al. (2017) and Gadotti (2009) as control single-barred samples for our analysis, we further assess possible biases due to the different approaches considered by the fitting codes.

3 Two-dimensional multi-component photometric decompositions with GASP2D

The 2D multi-component photometric decompositions are performed with the code GASP2D  (Méndez-Abreu et al., 2008, 2014, 2017). GASP2D  fits the galaxy 2D surface-brightness distribution with a combination of structural components, parameterised by known mathematical functions. A Levenberg-Marquardt algorithm based on a $\chi^{2}$ minimisation is used to find the most suitable set of parameters describing the galaxy light.

GASP2D  has already been tested in a number of works to find the structural composition of a variety of objects, such as AGN hosts (Benítez et al., 2013), galaxies with decoupled polar bulges (Corsini et al., 2012), and isolated galaxies (Morelli et al., 2016). For the present study, the ability of simultaneously fitting two bar structures was added to the code, as already introduced in Méndez-Abreu et al. (2017). The available components for the fitting are therefore a bulge, up to two bars, and a disc that might show none, positive, or negative bending (e.g. Erwin et al., 2005; Pohlen & Trujillo, 2006).

For the sake of completeness, in the following we will present the analytical functions describing each component. More details on the fitting procedure can be found in Méndez-Abreu et al. (2017).

The surface brightness distribution of the bulge component is parameterised with a Sérsic profile (Sérsic, 1968) of the form

[TABLE]

where $r_{\rm b}$ is the radius measured in the reference system of the bulge. $R_{\rm e}$, $I_{\rm e}$, and $n$ are the effective (or half-light) radius, the surface brightness at $R_{\rm e}$, and the Sérsic index describing the curvature of the profile, respectively, and $b_{n}\simeq 0.868\,n-0.142$ (Caon et al., 1993).

The surface brightness distribution of a galaxy disc is allowed to take three different shapes, namely: (i) Type I profile, a single exponential profile, (ii) Type II profile, a double exponential law with a down-bending beyond the so-called break radius, and (iii) Type III profile, a double exponential law with an up-bending in the outer parts of the disc. To account for these possibilities we adopt the following parameterisation:

[TABLE]

where

[TABLE]

and $r_{\rm d}$ is the radius measured in the reference system of the disc. $I_{0}$, $h$, $h_{\rm out}$, and $r_{\rm break}$ are the central surface brightness, inner scale-length, outer scale-length, and break radius of the disc, respectively.

The projected surface density of a 3D Ferrers ellipsoid (Ferrers 1877, see also Aguerri et al. 2009) is used to describe the surface-brightness distribution of both the inner and outer bar components:

[TABLE]

where $r_{\rm bar}$ is the radius in the reference system of each bar. The inner and outer bars are allowed to have different ellipticities and position angles. $I_{\rm 0,bar}$, $a_{\rm bar}$, and $n_{\rm bar}$ represent the central surface brightness, length, and shape parameter of the bar, respectively. It is worth noting that $a_{\rm bar}$ is not an effective radius, but the radius where the bar intensity drops to zero.

The bar surface-brightness distribution is assumed to be axially symmetric with respect to a generalised ellipse (Athanassoula, 1990). Therefore, the radial coordinate is defined as

[TABLE]

where $\epsilon_{\rm bar}$ is the bar ellipticity and $c$ controls the shape of the isophotes. A bar with pure elliptical isophotes has $c$=2. It is $c>$2 if the isophotes are boxy, and $c<$2 if they are discy. The parameters $\epsilon_{\rm bar}$ and $c$, as well as the position angle, are assumed to be constant as a function of radius.

3.1 Selection of the $n$ and $c$ bar parameters

Recovering all possible bar parameters previously described as free variables in the fitting process is rather difficult even for single-barred galaxies, due to the high number of degeneracies among the parameters. The most commonly used procedure in photometric decompositions involving Ferrers profiles is to keep fixed the two shape parameters to their default $n_{\rm bar}$=2 and $c$=2 values (Laurikainen et al., 2005; Méndez-Abreu et al., 2017).

Since we are particularly interested in studying the structure of inner and outer bars with great accuracy, we have investigated which values of $n_{\rm bar}$ and $c$ provide the best fits. For this purpose, we first perform the $r^{\prime}$-band double-barred fit in the usual way, i.e., with fixed values of $n_{\rm bar}$=2 and $c$=2. Variations of the profile with $n_{\rm bar}$ are explored first: the results from the usual fits are introduced as fixed initial conditions for all the inner and outer bar parameters ($I_{\rm 0,bar}$, $a_{\rm bar}$, $b/a$, and $PA$) except for the outer bar length $a_{\rm OB}$, which is allowed to vary. We remark that the correlation between bar length and $n_{\rm bar}$ makes it mandatory to keep the bar length as a free variable when studying variations of $n_{\rm bar}$. GASP2D  is then run again with fixed integer values for $n_{\rm OB}\in[1,4]$. The trends $n_{\rm OB}$ vs $\chi^{2}$ are inspected so the minimum providing the best $n_{\rm OB}$ parameter for the outer bar is found. The process is then repeated for the inner bar case, fixing both $n_{\rm OB}$ and $a_{\rm OB}$ parameters to the newly recovered values.

A similar procedure is carried out to derive the best $c$ values, varying the integer values of $c$ within the range $c\in[1,5]$. For this case, bar length as well as the rest of bar parameters (including the updated $n_{\rm bar}$ values just obtained in the previous step) can be kept fixed and therefore this procedure is just a $\chi^{2}$ computation rather than a fitting. Again, the outer bar $c$ is explored first.

As the inner bar is small and located at the galaxy centre, a slight modification of its global shape does not affect the large scale of the outer bar fitting. However, the extrapolation of the outer bar profile towards the central regions accounts for a certain amount of light that may significantly affect the inner bar contribution. Determining the outer bar shape in first place, before exploring $n_{\rm bar}$ and $c$ variations for the inner bar, is therefore a sensible procedure.

The final parameters for the 2D photometric decompositions of double-barred galaxies in $r^{\prime}$-band, including $n_{\rm bar}$ and $c$ values for the Ferrers bar profiles, are included in Tables 4 and 5 and shown in Fig. 1. We find that two thirds of the outer bars are well reproduced with the default value $c$=2, while this percentage slightly decreases down to 53% for the inner bar case. The remaining inner and outer bars tend to be boxier than the nominal elliptical case. $n_{\rm bar}$ values differ even more from the standard case, as only 6 and 4 out of 17 outer and inner bars, respectively, show $n_{\rm bar}$=2. Except for one outer bar, bars in general show steeper profiles with $n_{\rm bar}>$2, being $n_{\rm bar}$=3 the preferred value for most cases.

3.2 Double- and single-bar fits

All the sample galaxies were found to host a bulge component, two bars, and a disc. The $r^{\prime}$-band images are used as benchmarks to perform the first fit. Figure 2 shows an example of the final $r^{\prime}$-band fit for the galaxy NGC 357, which is composed of a Type-II disc, a bulge, and the inner and outer bars.

Following the prescription given in Méndez-Abreu et al. (2017), the final parameters for the $r^{\prime}$-band are used as initial guesses for fitting the $g^{\prime}$- and $i^{\prime}$-band images, fixing the $n_{\rm bar}$ and $c$ values to the best estimates obtained as described in Sect. 3.1. Although very similar, slight band-dependent differences in the measured parameters are expected. Best-fitting parameters for all the structures in each galaxy and band are shown in Tables 4 to 9.

The results from the double-barred fits in the $r^{\prime}$-band are also used as initial guesses for single-barred $r^{\prime}$-band fits. The outcomes are found in Tables 10 and 11. These single-barred fits are done for comparison purposes as discussed in Sect. 5.

We remark here that all parameters included in Tables 4 to 11 correspond to direct measurements from the images, i.e., lengths, radii, and scalelengths are provided in arcsec and projected onto the plane of the sky. For the analysis presented throughout the paper, bar parameters (lengths, ellipticities, and position angles) have been deprojected following the recipes given by Gadotti et al. (2007) and angular sizes have been transformed into physical scales. As explained in Zou et al. (2014), bar parameters for galaxies with inclinations $i>$60*∘* have not been deprojected due to the high uncertinties introduced in the process.

3.3 Error computation

The errors on the individual parameters involved in the fit were computed using a set of tailor-made mock galaxies in a Monte Carlo fashion. The full description of the methodology is presented in Méndez-Abreu et al. (2017) and we refer the reader to this paper for a complete description. Here we provide a brief summary for the sake of clarity.

A sample of 500 mock double-barred galaxies was simulated using a combination of structural parameters constrained within the limits of our real galaxy sample in the $r^{\prime}$-band (see Tables 4 and 5). Each model was built up using the equations provided in Sect. 3 for each distinct structure on a 2D grid with the SDSS pixel scale (0.396 arcsec/px). The total galaxy model was then convolved with a circular Moffat PSF with the typical FWHM of our SDSS images in $r^{\prime}$-band (Sect. 3) in order to reproduce the observed spatial resolution. We also adopted the typical values of CCD gain (4.86 e-/ADU) and read-out noise (5.76 e-), and added the background and photon noise from the galaxy to yield a signal-to-noise ratio similar to that of the observations.

Finally, the mock images were analysed using GASP2D in the same way as real images. The difference between the input and output values provides us with a systematic (mean value) and statistical (standard deviation) error on the individual parameters. Both errors are added in quadrature to obtain the final values shown in Tables 4 to 11.

4 Three-dimensional shapes with galaXYZ

We derive the intrinsic 3D shape for the bulges, outer bars, and inner bars of our sample using the galaXYZ code, which follows the procedure described in Méndez-Abreu et al. (2010) and Costantin et al. (2018). This method has been previously applied to different samples of galactic bulges and outer bars, but it is used here for the first time in inner bars. Méndez-Abreu et al. (2018) and Costantin et al. (2018) have already demonstrated that the statistical approach to derive the intrinsic 3D shape is applicable to any galactic structure if the initial assumptions are fulfilled, namely: i) all structures under study can be modelled by a triaxial ellipsoid in the same equilibrium plane as the disc; ii) the galaxy disc is considered to be an oblate spheroid. We allow for the disc to have a intrinsic thickness according to a normal distribution with mean intrinsic axial ratio $\langle q_{0,d}\rangle$=0.267 and standard deviation $\sigma_{q_{0,{\rm d}}}$=0.102 (Rodríguez & Padilla, 2013); and iii) all structures share the same centre, which is adopted as the centre of the galaxy.

All previous conditions are met by the three structures studied here. A caveat is that stellar bars can develop vertically-extended components during the so-called buckling phase. During this time, the bar creates what is called a box/peanut (B/P) structure in its central regions (Combes & Sanders, 1981; Athanassoula et al., 1983; Martinez-Valpuesta et al., 2006). These structures do not comply with our first hypothesis. However, in Méndez-Abreu et al. (2018) it is demonstrated that the parameters obtained from our photometric decomposition represent the ’thin’ part of the bar. Possible B/P structures present in a bar therefore have a small impact in the derived intrinsic shape.

The galaXYZ code needs as input the measured values of the projected geometric parameters (ellipticity and position angle) of the disc (representative of the galaxy inclination and position of the lines-of-nodes), and the structures under study, i.e. bulges, outer bars, and inner bars in our case. All these parameters, and their corresponding errors, are provided by the 2D photometric decomposition carried out in Sect. 3. Then, galaXYZ randomly generates 1000 geometric configurations by adopting for each parameter a Gaussian distribution centred on its measured value and with a standard deviation equal to its uncertainty. For each geometric configuration, the code evaluates equations 5 and 6 in Costantin et al. (2018). This is carried out in a Monte Carlo fashion with 5000 simulations of the intrinsic semiaxis ratios $B/A$ and $C/A$.

The result of this analysis is a joined probability distribution function (PDF) of $B/A$ and $C/A$, i.e., the PDF of the intrinsic 3D shape of each structure. An example is shown in Fig. 3 for the inner bar of NGC 2859. Our approach is entirely based on the projected geometric properties of each structure and the PDF is calculated independently for each structure in a galaxy. The most probable $B/A$ and $C/A$ values and their corresponding 1$\sigma$ uncertainties are shown in Table 2. We notice here that the width of the PDF in either $B/A$ or $C/A$ does not only depend on the photometric decomposition errors, but mostly on the lack of knowledge of the Euler angle ($\phi$), i.e., the angle describing the position of the intrinsic major axis of the structure with respect to the line-of-nodes in the plane of the disc. Therefore, there are some projected configurations (combination of the disc and structure geometry) that are less suited to derive the 3D shape and they provide large uncertainties. Values with 1$\sigma$ errors in any intrinsic semiaxis ratio larger than 0.5 are not included in the final analysis.

We remark that GASP2D provides the size of each structure as projected on the sky; those values are then deprojected following the methodology described in Gadotti (2009), thus sizes in the galaxy plane are finally estimated. On the other hand, galaXYZ performs a statistical recovery of the size of each structure in both the galaxy plane and perpendicular to it. Since galaXYZ uses the outcomes from GASP2D as input values, the results from both analyses are not fully independent. However, the PDF delivered by galaXYZ include statistical as well as methodology-intrinsic uncertainties, and we select as final value the one single point with the maximun likelihood. A good agreement between the in-plane sizes retrieved by GASP2D and galaXYZ therefore indicates the goodness of our combined analysis. Such analysis is shown in Fig. 4. In this case, all 17 galaxies from the sample are included in the plot regardless of their error bars: those finally removed from the analysis with error bars greater than 0.5 are shown in grey, while the coloured symbols correspond to the remaining 8 galaxies. The excellent correspondence between the measurements from both techniques, even for the dismissed galaxies, argues in favor of this analysis and shows how conservative the uncertainties provided by the PDF are.

5 Influence of inner bars in bulge parameters

Photometric decompositions have been widely used in the literature for studying the central regions of galaxies and, particularly, the nature of bulges (e.g., Gadotti, 2009). Current estimates establish that $\sim$20% of disc galaxies are double-barred (e.g., Erwin & Sparke, 2002; Laine et al., 2002; Erwin, 2011), and this is most likely a lower limit due to the small sizes of inner bars ($\sim$23% of the outer bar size as shown in Paper II). Despite their frequent presence, inner bars have never been taken into account when decomposing galaxies and not including bars in the photometric decompositions may significantly affect the derived bulge parameters (e.g., Laurikainen et al., 2006; Méndez-Abreu et al., 2017), hampering the conclusions obtained in many works. In Fig. 5 we compare the double- and single-barred $r^{\prime}$-band fits of the double-barred galaxies in order to quantify how much bulge measurements (effective radius, Sérsic index, ellipticity, and position angle) are affected when inner bars are not included in the analysis.

The effective radii measurements are in good agreement, with a mean difference of 0.07 kpc between the single- and double-bars fits. This value is lower than the typical errors introduced by the adopted methodology (see Tables 10 and 11) for all the galaxies, and therefore negligible. The agreement between Sérsic indices is also remarkably good except for the two galaxies with $n<$1, one of which hosts a extremely small bulge (NGC 2681). For the remaining galaxies we obtain a mean difference of 0.2, slightly higher than the errors from the decompositions. The ellipticity shows the largest discrepancy among all parameters, with a mean difference of 0.05, i.e., larger than 10% of the typical values. We note however that such measurement is of the order or only slightly larger than the typical errors derived from GASP2D. Finally, the position angle is the best behaved measurement, with only one discrepant galaxy corresponding to a very spherical bulge ($\epsilon=$0.03), where the position angle is irrelevant.

We can therefore conclude that the bulge parameters are rather insensitive to disregarding the inner bar in the fits. However, small differences are observed. With the aim of identifying the main contributor to such discrepancies, measurements in Fig. 5 are colour-coded attending to the length of the (dismissed) inner bars. A subtle trend pointing at largest inner bars causing the largest differences in the derived ellipticity and effective radius of the bulges is observed, although inner bar size does not account for discrepancies in the Sérsic indices. The same test has been performed by colour-coding the values with other quantities such as Sérsic index, bulge effective radius, and Re/aIB ratio. No clear correlations have been found, apart from the expected fact that galaxies with large inner bars with respect to the bulge size (i.e., small Re/aIB ratios) tend to compensate the dismissed inner bar by increasing the size of the bulge (Re). We therefore conclude that most likely a combination of all those parameters (inner bar parameters with respect to bulge parameters) is responsible for the differences found between the single- and double-barred fits. We emphasise again that deviations are small and generally within the error bars.

We note that the inner bar contribution to the total galaxy light can be as low as 0.5% with a mean value of 4%, while the outer bar accounts for [4%, 28%] of the total light. It is therefore reasonable to find that bulge measurements do not vary in a significant way when inner bars are not accounted for and therefore their light is included as bulge light.

For the sake of completeness, we have also investigated whether dismissing the inner bar has any effect over the outer bar parameters. While the length and position angle of the outer bar are absolutely independent from including the inner bar or not, very subtle differences are found for the outer bar ellipticity. However, such differences do not depend on the inner bar size and they are well within the error bars, thus supporting the idea that outer bar parameters are not affected by the inner bar.

6 The nature of bulges within double-barred galaxies

6.1 Traditional photometric diagnostics

Figure 6 shows the distribution of Sérsic indices for the bulges of our double-barred sample. Following the traditional demarcation n$$=2, we find that 6 out of the 17 double-barred galaxies host bulges with $n>$2 and they can therefore be considered classical bulges. The remaining 11 galaxies, i.e. the majority (65%) of the sample, host disc-like bulges attending to this pure Sérsic index diagnostics. Since our conclusions might be hampered by the limited size of the sample, we compare with the 162 single-barred galaxies of Méndez-Abreu et al. (2017), also shown in Fig. 6. This comparison sample does show a higher incidence of bulges with $n<$2 (77%), thus complying with the expectation that single/double-barred galaxies should show a large fraction of disc-like bulges. In particular, the two double-barred galaxies in Méndez-Abreu et al. (2017) host bulges with $n<$2. For the sake of completeness, Fig. 6 shows the results for the 287 barred galaxies included in the sample of Gadotti (2009) as well. We remind the reader that this represents a more distant sample where bars are modelled with a Sérsic profile instead of a Ferrers profile; these differences may account for the higher abundance of bulges with $n>$2 found in this case (59%).

The Kormendy (1977) relation is drawn in Fig. 7. As in the previous plot, the samples of bulges within barred galaxies from Gadotti (2009) and Méndez-Abreu et al. (2017) are also shown. The dashed line represents the demarcation found by Gadotti (2009) to separate between the classical and disc-like nature of the bulges. We remark that the CALIFA barred galaxies in Méndez-Abreu et al. (2017) are analysed with GASP2D; many bulges in this sample are compatible with being classical but there is also a less populated cloud of disc-like-compatible bulges which extends up to the regime of large bulges. Surprisingly, all bulges in double-barred galaxies lay at the top sequence of classical bulges regardless of their Sérsic indices. In fact, no trend with $n$ shows up in Fig. 7. We must note that the well-known dependence of the Kormendy relation with the stellar mass (Nigoche-Netro et al., 2008) makes this diagnostic not fully reliable. Such problem is shown in Costantin et al. (2017), where it is also claimed that a single line division might not be the correct way to separate classical and disc-like bulges using this diagnostic.

In summary, the traditional Sérsic index and Kormendy-relation discriminators provide contradictory results, with a majority of disc-like bulges in the first case and all classical bulges in the second case for the double-barred galaxies under study.

6.2 3D intrinsic shape of bulges

In Costantin et al. (2018), different diagnostics to constrain the nature of bulges are put at test. The main conclusion of that work is that the traditional pure-photometric methods based on projected quantities such as the Sérsic index do not provide clear separations between classical and disc-like bulges (see also Méndez-Abreu et al., 2018). The probabilistic retrieval of the 3D intrinsic shape of bulges represents a better suited diagnostic, as it allows the study of the intrinsic flattening of these structures, once isolated from the remaining components of the galaxy.

Disc-like bulges are expected to be flattened structures with a close-to-circular projection in the galaxy plane, due to their rotating nature. Only two bulges in our sample comply with these requirements, as indicated by the results shown in Table 2: NGC 3368 ($B/A=$0.91 and $C/A=$0.41) and NGC 7280 ($B/A=$0.89 and $C/A=$0.31). They therefore represent the best candidates to host disc-like bulges. Note that NGC 3368 does show a bulge Sérsic index $n<$2, while NGC 7280 would be classified as a classical-bulge host according to the Sérsic index discriminator.

Méndez-Abreu et al. (2018) discuss why bulges which are intrinsically flatter than their coexistent single bars represent even stronger candidates to have a disc-like nature. Such conclusion is based on the fact that the vertical extension of bars is closely related to that of galaxy discs, so if bulges are as flats as bars, they are also at least as flat as discs, a result which is hard to reconcile with a classical nature. The left panel of Fig. 8 shows the intrinsic semiaxis ratios of bulges and outer bars. Only the 10 out of 17 double-barred galaxies with uncertainties lower than 0.5 in the derived 3D parameters of bulges and outer bars are plotted. Four galaxies show close values of the flattening between bulges and outer bars. These are NGC 3368, NGC 4340, NGC 7280, and NGC 7716, being NGC 3368 the only one with an intrinsic flattening for the bulge ($C/A=$0.41) estrictly lower than that for the outer bar ($C/A=$0.46). NGC 3368 and NGC 7280 therefore remain as the two best candidates for hosting disc-like bulges. Note that the bulge of NGC 7716 has an axis ratio $B/A=$0.64, which does not fulfill the axisymmetric projection requirement.

For the sake of completeness, the right panel of Fig. 8 shows similar results but comparing the intrinsic flattening of bulges and inner bars. A different set of 10 galaxies with low uncertainties affecting bulge and inner bar measurements are plotted in this case. NGC 3368 and NGC 7280 are included in this right panel: they host thicker bulges than the corresponding inner bars, while NGC 4340 hosts a flatter and almost circular bulge, and NGC 3941 and NGC 7716 show similar intrinsic flattenings for bulges and inner bars. The intrinsic shape of inner bars has never been studied in the literature and no relationship between it and the nature of bulges is a-priori expected. A detailed comparison between the 3D shapes of inner and outer bars is presented in Paper II.

7 Discussion

We discuss here our results within the context of the two major questions brought up in Sect. 1 about bulges within double-barred systems: i) do double-barred galaxies host a larger fraction of disc-like bulges than non-barred galaxies due to the efficient secular evolution that is expected to take place in them?; and ii) are bars within bars formed due to the presence of a disc-like component resulting from secular evolution due to the large-scale bar?

Following both the Kormendy relation (Fig. 7) and the 3D intrinsic shape (Fig. 8) as diagnostics for the bulge nature, our results agree with double-barred galaxies mostly hosting classical bulges. Only the bulge Sérsic indices point towards a larger fraction of disc-like bulges.

Two galaxies among the sample of 17 individuals, NGC 3368 and NGC 7280, stand out as the best candidates for hosting disc-like bulges as suggested by our preferred diagnostics of intrinsic flattening. With the aim of understanding better the origin of bulges within double-barred galaxies, we compute colours for the isolated bulges by using the results from the 2D photometric decompositions in the different bandpasses (Table 3). Results among galaxies can be directly compared as we use colours integrated within one bulge effective radius, so no bias due to different galaxy sizes affects the results. All our bulges show rather red colours, with median values of ($g^{\prime}-i^{\prime}$)$=$1.28 and ($r^{\prime}-i^{\prime}$)$=$0.41. Note these values agree with the results for classical bulges in, e.g., Gadotti (2009, although take into account they integrate inside a fibre instead of bulge effective radius). If disc-like bulges are secularly formed after a star-forming process triggered by gas which has inflowed along the bar to the central regions, they should show bluer colours due to a younger stellar content with respect to the older surrounding structures. NGC 3368 and NGC 7280 show colours which are even redder than the median, with ($g^{\prime}-i^{\prime}$) values of 1.47 and 1.40, respectively. Bulge colours do indeed point towards an old, classical nature for all bulges in the sample.

Figure 3 in Fisher & Drory (2011) shows that disc-like bulges are more frequent in less massive galaxies, and therefore any mass bias in our sample needs to be considered as well (see also Fig. 12 in Gadotti, 2009). We calculate the galaxy masses by using the colours and the recipe given by Zibetti et al. (2009). Results are shown in Table 3. Our galaxies span a wide range of masses between 3$\times$109 and 1$\times$1011 M*⊙, with a median value of 2.6$\times$1010* M*⊙*. Actually, a larger fraction of disc-like bulges is expected in galaxies at these masses, so no mass bias appears to be affecting our analysis.

In summary, most of the diagnostics indicate our double-barred galaxies host classical bulges. At first glance, this finding may look like against the expectations: secular processes should have played a major role in these kind of galaxies and therefore a significant presence of disc-like bulges is expected. However, we must note here that the traditional paradigm of a single, either classical or disc-like, bulge in a galaxy has recently been challenged by many authors: the coexistence of several bulges of different nature has been observationally demonstrated by Méndez-Abreu et al. (2014) and Erwin et al. (2015). An exhaustive analysis combining photometric with kinematic information is claimed to be required in order to clearly distinguish the nature and possible presence of more than one bulge within a galaxy. If our double-barred galaxies were hosting composite bulges, the analysis presented here would correspond to the dominant structure in light, which would be the one revealed in the images and the photometric decompositions. A previously-formed classical bulge could therefore be hiding the presence of a secularly-formed disc-like bulge in our galaxies.

In de Lorenzo-Cáceres et al. (2018a, submitted), we investigate the kinematics and stellar populations of two double-barred galaxies for which MUSE TIMER integral-field spectroscopic data is available: NGC 1291 and NGC 5850 (also included in this sample). For those galaxies we perform 2D photometric decompositions of 3.6 $\mu$m images including a bulge, two bars, (truncated) disc and also an inner disc. Whether we can call these inner discs as disc-like bulges (and therefore composite bulges as both galaxies do host an additional bulge component as well) is a matter of semantics out of the scope of this discussion. However, we do find a strong connection between the size of the inner discs and the inner bars, suggestive of a dynamical origin of those inner bars from instabilities in the small-scale discs. The goal of the photometric study of double-barred galaxies presented here is to study the nature of the dominant bulges, as we do not have the support of high resolution spectroscopic data which may reveal the presence of rotating inner discs. Therefore we cannot discard the possibility of having a faint inner disc coexisting with the classical bulges in several (if not all) double-barred galaxies, as it is indeed the case for NGC 5850 (and also NGC 357, NGC 4725 and NGC 2859, see the kinematic evidences in de Lorenzo-Cáceres et al., 2008, de Lorenzo-Cáceres et al. 2012, and de Lorenzo-Cáceres et al. 2013).

The B/P structures developed at the inner regions of large stellar bars are considered also as ’bulges’ by several authors, attending to the fact that they are central components which cause an excess of light with respect to the pure disc and bar. Although we emphasise that B/P are not isolated structures but they rather belong to the bars, we have performed here a visual inspection of our double-barred galaxies with the aim at looking for signatures of the presence of a B/P component coexisting with the dominant bulge. In particular, we search for a barlens: an oval component seen in moderately inclined galaxies which is supposed to be the face-on counterpart of the X-shaped B/P structure seen edge-on (Athanassoula et al., 2015; Laurikainen & Salo, 2017). Other indicators of the presence of a B/P are isophotal twists in the innermost bar regions, which tend to show boxy isophotes accompanied of narrow outermost isophotes (the so-called spurs, see Erwin & Debattista, 2013, 2017). Note that the bulge component found in our analysis and studied throughout this paper does not correspond to the possible B/P revealed by this visual inspection. 7 out of 17 galaxies do not show signatures of the presence of a B/P (NGC 718, NGC 2681, NGC 2962, NGC 3368, NGC 4503, NGC 7280, and NGC 7716), while 4 out of 17 galaxies host clear barlenses (NGC 2950, NGC 3945, NGC 4314, NGC 4340). For the remaining 6 galaxies, the classification of B/P structures is unclear: NGC 357, NGC 2642, NGC 2859, NGC 3941, and NGC 4725 show weak elliptical structures, spurs, or isophotal twists that may be due to B/P. Finally, NGC 5850 shows a component resembling a barlens but the kinematic study presented in de Lorenzo-Cáceres et al. (2018a, submitted) reveals that it is indeed an inner disc.

In order to constrain whether there is any further relation between our dominant bulges and the double bars, correlations between the bar parameters (ellipticity and length) and bulge parameters (Sérsic index and effective radius) have been searched for, with no particular results found. As a final note, the fact that all our galaxies are better fitted by including a bulge component, i.e, all double-barred galaxies do host a bulge, might be relevant for understanding their formation. Our sample includes some late Hubble types up to Sbc but no bulgeless galaxy is found. This may point towards a connection between the presence of inner bars and formation of bulges, and it even poses the question of whether the presence of a classical bulge is required for forming an inner bar through, e.g., dynamical stabilization of the central regions.

8 Conclusions

We present a complete photometric study of a sample of 17 double-barred galaxies consisting of two analyses which are for the first time applied to these kind of objects: i) 2D photometric decompositions including a bulge, inner bar, outer bar, and (truncated) disc, and ii) 3D deprojections of bulges, inner bars, and outer bars thus retrieving their actual intrinsic shape. While the photometric properties of bars are explored in the companion Paper II, the current work focuses on the properties of bulges. In particular, we constrain the classical vs. disc-like nature of bulges in double-barred systems. Our galaxy sample spans a wide range in galaxy masses. The main results are:

• •

All double-barred galaxies under study host a bulge component.

• •

The bulge properties derived through photometric decompositions are not significantly affected by dismissing the presence of the inner bar.

• •

65% of the galaxies host bulges with Sérsic index $n<$2.

• •

All bulges lay at the top sequence of the Kormendy (1979) relation, in the region supposedly populated by classical bulges.

• •

No correlations are found between inner bar properties (length and ellipticity) and bulge properties (Sérsic index and effective radius).

• •

Only 2 out of 17 bulges show an intrinsic shape compatible with a disc-like nature, i.e., almost circular in-plane projection and flattened off-plane profile: NGC 3368 and NGC 7280.

• •

3 out of 17 bulges (including NGC 3368 and NGC 7280) are flatter than their corresponding outer bars. This has been argued to be an indication of their disc-like nature.

• •

Inner bars are either flatter or with similar flattening than their coexisting bulges.

• •

All bulges show rather red colours.

Most previous results support a classical nature for bulges within double-barred galaxies. A major incidence of secularly-formed structures such as disc-like bulges is expected in these galaxies, where two non-axisymmetric structures may help to transport gas to the central regions. We note that composite bulges are not studied in this work and therefore we cannot rule out the possibility of a faint disc-like component, or even other kind of secularly-formed substructure, coexisting within these galaxies; we refer to the dominant bulge in this analysis. The presence of a dominant classical bulge in all double-barred galaxies under study suggests that hosting a hot component may be necessary for the dynamical development of an inner bar within a barred galaxy.

Acknowledgments

We are very grateful to the anonymous referee, whose comments helped improving the content and presentation of this paper. AdLC acknowledges support from grants ST/J001651/1 (UK Science and Technology Facilities Council - STFC) and AYA2016-77237-C3-1-P (Spanish Ministry of Economy and Competitiveness - MINECO). JMA acknowledges support from the Spanish Ministry of Economy and Competitiveness (MINECO) by grant AYA2017-83204-P.

Appendix A GASP2D structural parameters of 18 double-barred galaxies