COLOR–MAGNITUDE RELATION AND MORPHOLOGY OF LOW-REDSHIFT ULIRGs IN SLOAN DIGITAL SKY SURVEY

In Figure 2, we plot the ^0.1g − ^0.1r color against k-corrected ^0.1r-band absolute magnitude $M_{^{0.1}r}$ for all comparison sample galaxies, which consists of the 436,762 galaxies from the NYU SDSS Value Added Catalog (DR4) within the same redshift range. Individual galaxies are shown as small dots while the contours represent the galaxy number density. The bimodal distribution of the SDSS comparison galaxies is clearly shown in the color–magnitude diagram where the red sequence and the blue cloud form two separate groups. We adopt the empirical definition of the red sequence from Weinmann et al. (2006), such that the red-sequence galaxies follow the color–magnitude relations,

$$\begin{equation} ^{0.1}g-^{0.1}r > 0.7 - 0.033(M_{^{0.1}r}-5\log h+16.5). \end{equation} \tag{ 1 }$$

This relation is shown in Figure 2 where the red sequence lies above the upper straight line. The centroids of the red sequence and the blue cloud are separated by ∼0.3 in ^0.1g − ^0.1r, and between the two populations there is a relatively low-density region called the "green valley." We define the green valley as a strip below the red sequence with a width of 0.1 in ^0.1g − ^0.1r, shown as the region between the two straight lines in Figure 2. The fiducial width of 0.1 is chosen to be one-third of the color difference between the centroids of the red sequence and the blue cloud, and is also roughly equal to the color difference between the centroid and the blue limit of the red sequence (the upper straight line). The low-density region between the red sequence and the blue cloud is wider at fainter magnitudes and narrower at brighter magnitudes, and our simple cut may miss some green-valley galaxies at fainter magnitudes but include more blue-cloud galaxies at brighter magnitudes. However, this will not affect our main results as we discuss later.

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Table 2 lists the statistics of the colors and magnitudes of the ULIRGs and SDSS comparison sample and their subsamples. We summarize below those optical characteristics of the ULIRGs and their subsamples (AGN and non-AGN) and compare them with those of the SDSS comparison sample.

Table 2. Photometry Statistics of the ULIRGs and SDSS Comparison Sample

Sample Property	ULIRGs		SDSS Comparison Sample
Number of sources	52a		436762
$\langle M_{^{0.1}r}\rangle \pm \sigma _{M_{^{0.1}r}}$	−21.4 ± 0.72		−20.2 ± 1.17
$\langle ^{0.1}g-^{0.1}r\rangle \pm \sigma _{^{0.1}g-^{0.1}r}$	0.57 ± 0.22		0.77 ± 0.25
	AGN ULIRGs	Non-AGN ULIRGs	Red sequence	Green valley	Blue cloud
Number of sources	12a	40	216648	51904	168119
$\langle M_{^{0.1}r}\rangle \pm \sigma _{M_{^{0.1}r}}$	−22.1 ± 0.60	−21.1 ± 0.62	−20.6 ± 1.01	−20.4 ± 1.01	−19.8 ± 1.26
$\langle ^{0.1}g-^{0.1}r\rangle \pm \sigma _{^{0.1}g-^{0.1}r}$	0.61 ± 0.30	0.58 ± 0.17	0.95 ± 0.15	0.76 ± 0.04	0.52 ± 0.18

Notes. ^aThe ULIRGs sample excludes the two saturated sources FSC12540+5708 and FSC12265+0219, and these two sources are also AGN ULIRGs.

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The first obvious trend is that the ULIRGs are very luminous in the optical. The median k-corrected absolute magnitude is $M_{^{0.1}r} = -21.4$, or equivalently 2.5 L_* given $M_{^{0.1}r}=-20.44$ for an L_* galaxy (Blanton et al. 2003b). This is consistent with the median of R-band absolute magnitudes for the whole 1 Jy sample (∼2L_*; KVS02). For comparison the median k-corrected absolute magnitude of the SDSS galaxies is $M_{^{0.1}r}=-20.4$, which is 1 mag fainter than that of the ULIRGs. The absolute magnitudes of the ULIRGs range from $M_{^{0.1}r} = -19.78$ to $M_{^{0.1}r} = -23.23$ (excluding the two saturated sources), a luminosity range of a factor of 25. Fifty out of the 54 ULIRGs (∼93%; including the two saturated sources) are more luminous than the median k-corrected r-band absolute magnitude of the SDSS comparison sample. The difference in magnitude distribution between the ULIRGs and the SDSS comparison sample is further illustrated in Figure 3. The ULIRG sample has a much narrower distribution than that of the comparison sample. The ULIRGs are on average 1 mag brighter than that of the entire comparison sample (see panel(a)) as well as the individual subsample of the red sequence, the green valley, and the blue cloud (panel (c)). At the highest luminosity bins, the ULIRG sample outnumbers all three SDSS subsamples in fraction ($M_{^{0.1}r}-5\log h<-21$; panel (c)).

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Another striking trend is that the ULIRGs are very blue, with only a few exceptions. Their ^0.1g − ^0.1r colors span a wide range, from 0.04 to 1.5, and the majority (87%) of the ULIRGs have optical colors as blue as those of the blue-cloud galaxies. The median ^0.1g − ^0.1r color of the ULIRGs is 0.58 and very close to that of the blue cloud (0.55). The median ^0.1g − ^0.1r color of the red sequence and green valley are 0.95 and 0.78, respectively. The color distributions of the ULIRGs and the SDSS galaxies are further illustrated in Figures 4(a) and (c). Considering galaxy color a function of galaxy luminosity, we limit the r-band absolute magnitudes of the SDSS comparison sample to the same magnitude range as that of the ULIRGs ($-19.8<M_{^{0.1}r}-5\log h<-23.2$). The peak of the ULIRG distribution is ∼0.3 mag bluer than that of the SDSS sample within the same r-band magnitude range, but very similar to that of the blue cloud within the same r-band magnitude range (^0.1g − ^0.1r ∼ 0.6). The ULIRGs have a much larger fraction in the bluer bins (0.5 < ^0.1g − ^0.1r < 0.7) than the comparison sample, but a much lower fraction in the redder bins (0.7 < ^0.1g − ^0.1r < 1.0), except for the reddest bins (^0.1g − ^0.1r>1.2) and the bluest bins (^0.1g − ^0.1r < 0.1). The ULIRG sample outnumbers the blue-cloud galaxies in fraction at blue colors (^0.1g − ^0.1r < 0.7), has a similar fraction compared to the green-valley galaxies at green colors (0.7 < ^0.1g − ^0.1r < 0.9), and has a much lower fraction than the red-sequence galaxies at red colors (^0.1g − ^0.1r>0.9).

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The blue colors of the ULIRGs seem inconsistent with the working hypothesis that ULIRGs are dusty, massive systems. However, the patchy mix of blue and red regions in the optical color images (see Figure 1) suggests that the dust extinction is patchy, and the extended distribution of young, blue stars dominate the overall color of these galaxies. Veilleux et al. (1999) have discussed the blue optical continuum colors of the ULIRGs and suggested that the stellar light suffer less dust extinction than the emission-line gas. Our ULIRGs would appear even more luminous and bluer in the optical if corrected for the internal extinction.

The AGN ULIRGs are among the most luminous sources in the ULIRG sample, with a median $M_{^{0.1}r} = -21.9$, 0.8 mag more luminous than the median of the non-AGN ULIRGs. As shown in Figure 3(b), the AGN ULIRGs are more concentrated toward high luminosities and outnumber the non-AGN ULIRGs at the highest luminosity bin ($M_{^{0.1}r}-5\log h<-23$). They also have a much flatter color distribution than the non-AGN ULIRGs (see Figure 4(b)). There is little overlap between the AGN ULIRGs and the green-valley galaxies in both magnitude distribution (see Figure 3(d)) and color distribution (see Figure 4(d)).

We overlay the colors and magnitudes of the ULIRG sample on top of the SDSS contours in Figure 2, with different symbols representing ULIRGs known to host an AGN (AGN ULIRGs, circles) and H ii-region like and LINER galaxies (non-AGN ULIRGs, stars). The images of FSC12540+5708 and FSC12265+0219 are saturated in the g and r band, and their magnitudes are shown as upper limits. The error bar shows σ(g_P,g − g_P,r), the standard deviation of difference in g-band magnitude measured between g- and r-band apertures (Section 3).

The ULIRGs form a distinct group in the color–magnitude diagram: 24 out of the 52 unsaturated ULIRGs (∼46%) lie outside the 90% level contour of the SDSS galaxies in the same redshift range. Strikingly, only three (6⁺⁶₋₄%³) lie in the green valley, and none hosts an AGN. The majority of the ULIRGs (47 out of 54, or 87⁺⁷₋₁₁%) are located below the bottom straight line that defines the red limit of the blue cloud, and four (7⁺⁵₋₆%) are located above the upper straight line that defines the blue limit of the red sequence.

In the local universe, most ULIRGs are major mergers of late-type galaxies, and are proposed to be the precursors of massive ellipticals (e.g., Toomre & Toomre 1972; Toomre 1977; Mihos & Hernquist 1994, 1996; Barnes & Hernquist 1996). Given the favorable scenario in which the red sequence is built up by "wet mergers" of late-type galaxies (e.g., Bell et al. 2004b; Faber et al. 2007), one may expect a concentration of merging galaxies in the green valley, the area connecting the blue cloud and the red sequence in the color–magnitude diagram. The low fraction (6%) of our ULIRGs in the green valley, however, suggests that most of these ULIRGs are not in the transition phase from the blue cloud to the red sequence. Instead, the majority (87%) of these ULIRGs have typical blue-cloud colors, suggesting that they are still undergoing rapid star formation, and that significant quenching of star formation has yet to start.

AGNs may play a crucial role in quenching star formation and accelerate the transition of blue-cloud galaxies to the red sequence (e.g., Hopkins et al. 2006; Croton et al. 2006). Indeed, some studies have found an excess of X-ray-selected AGNs in the green valley (e.g., Georgakakis et al. 2008; Silverman et al. 2008; Hickox et al. 2009; Schawinski et al. 2009). None of the AGN ULIRGs in our sample, however, is located in the green valley, and only two (17%) are redder than the blue limit of the red sequence. The remaining 10 (83%) unsaturated sources are bluer than the red limit of the blue cloud.

Hickox et al. (2009) suggested an evolutionary sequence for massive galaxies with halo masses between 10¹²and10¹³ M_☉, in which a high accretion rate optical- and infrared-bright SMG/ULIRG/quasar phase evolves quickly (≲10⁸ yr), and is followed by a slightly lower accretion rate "green-valley" phase (lasting ∼ 1 Gyr) when the objects are detected as X-ray AGNs. This phase is followed by an intermittent radio AGN phase in which the galaxies reside in the red sequence with lowest accretion rates. In the context of the Hickox et al. study, the majority of our AGN ULIRGs are still in the early optical-bright phase and have yet to evolve to the green valley.

In the optical bands, the AGN ULIRGs are among the most luminous in the ULIRG sample, and most have blue colors (see Section 4). AGNs are point-like sources residing in the nucleus region and typically have blue optical colors. Thus, one may expect the presence of an AGN would affect the color and magnitude of the host galaxy (e.g., Hickox et al. 2009). However, our AGN ULIRGs are low-luminosity Seyfert galaxies and the AGN emission at optical wavelengths may be highly attenuated. In order to test whether their high luminosity and blue color are due to a central AGN, we estimate the AGN color and luminosity contribution by a simple model fitting.

4.4.1. Methodology

4.4.2. AGN Host Galaxies on the Color–Magnitude Diagram

On average the subtracted point sources contribute only a small amount to the total luminosity of the AGN ULIRGs in both g and r bands. This suggests that most of their optical luminosity is associated with the stellar hosts. The maximum, median, and mean ratio of the subtracted point source to the total flux, using the maximum subtraction method, are 67%, 28%, 31% in g band, and 74%, 27%, and 31% in r band, respectively. Using the smooth host subtraction method, these values are 61%, 11%, and 21% in g band, 74%, 8%, and 21% in r band, respectively. The maximum, median, and mean magnitude differences between the hosts and the unsubtracted sources are 1.19, 0.36, and 0.45 (1.48, 0.34, and 0.47) in g band (r band), using the maximum subtraction method, and the corresponding values are 1.02, 0.12, and 0.31 (1.03, 0.09, and 0.31) in g band (r band), respectively, using the smooth host subtraction method.

We overlay the ^0.1g − ^0.1r versus $M_{^{0.1}r}$ color–magnitude diagram for the 12 AGN ULIRGs (filled circles) and their AGN-subtracted hosts (open circles) on the SDSS color–magnitude contours in Figure 5. Some hosts are much bluer after the subtraction (e.g., FSC11119+3257, bluer by 0.42 using both methods), while some are much redder (e.g., FSC01572+0009, redder by 0.33 using the maximum subtraction method). However, the median color difference between the host and the original source is only 0.007 and 0.005 mag, using the maximum subtraction method and the smooth host method, respectively. Even after the maximum subtraction, only one (FSC11119+3257) source moved closer to the green valley, and the remaining 11 sources are located close to their original positions in the color–magnitude diagram and barely overlapped with the SDSS comparison galaxies. This distinction is clearer using the smooth host subtraction method, after which the hosts are much closer to their unsubtracted counterparts. In either case, most hosts (10 out of 12) remain luminous and blue in the color–magnitude diagram. With or without the point source subtraction, these AGN-host ULIRGs do not appear in the green valley, the hypothesized transitional stage between the blue cloud and the red sequence that was associated with X-ray- and IR-selected AGNs (e.g., Hickox et al. 2009). The blue colors of our AGN host galaxies are broadly consistent with other studies of optically selected AGNs (e.g., Kauffmann et al. 2003; Jahnke et al. 2004; Silverman et al. 2008). Kaviraj (2008) studied a local LIRG (L>10¹¹ L_☉) sample and also found that virtually all LIRGs are in the blue cloud and the AGN has no impact on the star formation in their host galaxies. In a broad agreement with our findings, Cowie & Barger (2008) also found that the bulk of extinction-corrected 24 μm-selected sources in the GOODS-North field lie in the blue cloud.

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Our study is limited by the small sample size, and the point source decomposition is limited by the spatial resolution of SDSS. A more complete sample observed with Hubble Space Telescope (HST) or the next generation James Webb Space Telescope (JWST; Gardner et al. 2006) is necessary in order to fully characterize the optical properties of the host galaxies in AGN ULIRGs.

Motivated by the high infrared and bolometric luminosity, it has been proposed that ULIRGs are the early stage of dusty quasars, and a ULIRG-QSO evolutionary scenario has been suggested (Sanders et al. 1988). Although the sources of the bolometric luminosity for ULIRGs are nearly completely obscured by dust in the visible bands, a comparison of their host properties with those of QSOs should provide an important test for the proposed evolutionary scenario. For example, if the ULIRGs and QSOs are identical except for a geometrical difference such as the viewing angle and dust geometry, then the optical properties of their hosts should be largely identical. If a purely temporal evolution and an AGN feedback process removing gas and dust are the key difference (e.g., Narayanan et al. 2009), then a significant overlap with a systematic shift is expected in the host color–magnitude relation.

The QSO sample we selected is composed of 1070 objects in the SDSS DR5 quasar catalog (Schneider et al. 2007) within the same redshift range (0.018 < z < 0.265) as the ULIRGs. The SDSS QSOs are type 1 quasars, selected based on their broadband colors, with M_i < −22 mag and spectroscopically confirmed to have broad lines (FWHM greater than 1000 km s⁻¹). We also selected a type 2 quasar sample for comparison, which consists of 402 objects in the SDSS type 2 quasar catalog (Reyes et al. 2008) within the same redshift range, selected based on their optical emission lines. Systematic photometric difference should be minimal among the SDSS QSOs, type 2 quasars, and the ULIRGs due to the shared origin of the data. For SDSS QSOs and type 2 quasars, we apply the k-correction and extinction correction using the same procedure as for the ULIRGs to obtain the absolute magnitudes. We plot in Figure 6 the ^0.1g − ^0.1r colors versus $M_{^{0.1}r}$ of the QSOs and the type 2 quasars with dots and contours, on top of the contours of the SDSS field galaxies and the ULIRG symbols.

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Strikingly, the distribution of the type 2 quasars peaks in the green valley where the X-ray- and IR-selected AGNs are found (e.g., Hickox et al. 2009). In comparison, the SDSS QSOs overlap only slightly with the blue cloud and extend toward very blue colors. The difference in their distributions in the color–magnitude diagram is at least partly due to selection effects. The SDSS QSOs are selected unobscured type 1 quasars where the blue continuum and high luminosity of the central AGNs are visible, while the type 2 quasars are selected obscured AGNs and thus the dust and gas block the view to the central blue and luminous AGNs. Studying whether these green type 2 quasars represent galaxies in a transitional stage from the blue cloud to the red sequence by AGN quenching, or their green colors are simply color combinations of reddened obscured AGNs and the host galaxies, is beyond the scope of this paper.

The ULIRGs are distributed very differently from the SDSS QSOs and the type 2 quasars in the color–magnitude diagram. The mean absolute magnitude of the ULIRGs ($\langle M_{^{0.1}r}\rangle =-21.4$) is very close to that of the SDSS QSOs ($\langle M_{^{0.1}r}\rangle =-21.3$), but their mean color (〈^0.1g − ^0.1r〉 = 0.57) is 0.35 mag redder than that of the SDSS QSOs (0.22). The SDSS QSOs occupy a narrow range in $M_{^{0.1}r}$, overlap only slightly with the ULIRGs in the color–magnitude diagram, and extend farther to bluer colors than the ULIRGs. In comparison, the $\langle M_{^{0.1}r}\rangle$ of ULIRGs is 1.1 mag more luminous than that of the type 2 quasar (−20.3), and their 〈^0.1g − ^0.1r〉 is 0.14 mag bluer (0.57 compared to 0.74). The ULIRGs overlap with the type 2 quasars only at faint magnitudes ($M_{^{0.1}r}>-21$), and all but one AGN ULIRGs are more luminous than 90% of the type 2 quasars. We also implement two-dimensional K–S tests (Peacock 1983; Fasano & Franceschini 1987) between different samples in the color–magnitude two-dimensional parameter space, and the results are listed in Table 3. The test results indicate that neither the ULIRG sample nor the AGN-ULIRG subsample is drawn from the same underlying distribution with the SDSS QSO sample or with the type 2 quasar sample.

We model the color–magnitude evolutionary tracks for the ULIRGs using the stellar synthesis model BC03 (Bruzual & Charlot 2003) in order to gain some insight into the distribution and evolution scenarios of ULIRGs on the color–magnitude diagram. Rather than providing a full solution to the SEDs with best-fit parameters, the primary goal of our models is to demonstrate qualitatively the possible evolutionary paths these galaxies may follow on the color–magnitude diagram and to establish the associated timescales.

We start with the simplest cases in which the star formation histories are (1) a passively evolved single stellar population (SSP) burst and (2) a constant star formation rate (constant). Figure 7 shows the evolutionary tracks for these two models with a solid line (SSP) and a dashed line (constant). Both tracks are k-corrected to z = 0.1 in the same manner as the ULIRGs. The SSP track is normalized so that it passes through the median color (^0.1g − ^0.1r = 0.58) at the median absolute magnitude of the ULIRGs ($M_{^{0.1}r} = -21.43$, or L_r = 2.5L_r,*). The constant track is normalized to reach 2.5L_* at t = 20 Gyr. We use solar metallicity in our models for simplicity and ignore intrinsic extinction as we are primarily concerned with the unobscured stellar population. Three symbols are plotted on each track to mark three ages, 1 Gyr, 2 Gyr, and 5 Gyr. In the SSP model, we find that the stellar populations of ULIRGs have ages spanning from 200 Myr to 8 Gyr based on their optical colors, with a median age of ∼1 Gyr. One exception is the extremely red source FSC11119+3257 whose color is too red to be reproduced by the SSP model. Our modeling suggests that a galaxy with a median age takes 1.5 Gyr to reach the red sequence. In the constant star formation case, the reddest color of the stellar population (at t = 20 Gyr) is ^0.1g − ^0.1r = 0.43, which is bluer than 80% of the ULIRGs. This implies that in the constant star formation case the star formation has to be quenched in order to move the ULIRGs to the red sequence.

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Next we construct a model with a more complex star formation history and apply different metallicity and dust treatments to try to understand their effects on the color and magnitude. Our model assumes that (1) a ULIRG is a merger system consisting of two equal-mass late-type galaxies and that the merger triggers a starburst, (2) the star formation history consists of an SSP component associated with each progenitor and an exponentially decaying starburst triggered by the merger, (3) the SSP components are identical with the same age and the starburst is triggered when the SSP components evolve to the median color and magnitude of the ULIRGs, and (4) the starburst forms as much as 10% of its progenitors' stellar mass with an e-folding time of 10⁷ yr. The timescale of 10⁷ yr is close to the dynamical timescale of ULIRGs such as Arp 220 (6 × 10⁶ yr; Mauersberger et al. 1996) and 10% is a fiducial fraction of mass formed during a merger that is consistent with the results in the numerical simulations of disk mergers (e.g., Mihos & Hernquist 1996). Given that the progenitors are two L_* galaxies with M_* ∼ 10¹¹ M_☉, the calculated star formation rate starts with a maximum value of ∼2000 M_☉ yr⁻¹ and decreases to several hundred M_☉ yr⁻¹ after an e-folding time. Although these star formation rates seem extremely high, they are required by the observed high FIR luminosities. We calculate the star formation rate of the ULIRGs from their FIR luminosities using the FIR–SFR relation (Kennicutt 1998). The star formation rates range from 170 M_☉ yr⁻¹ to 1370 M_☉ yr⁻¹, with a median of 270 M_☉ yr⁻¹, which are consistent with our model predicted values.

We examine three different metallicities, Z = 0.008, Z = 0.02, and Z = 0.05, corresponding to a sub-solar, solar, and super-solar metallicity, respectively, and construct three base models using this star formation history. We assume that the starburst has the same metallicity as the passively evolved progenitor populations. We set up a grid of different dust extinctions following Charlot & Fall (2000) for each metallicity, but apply the dust extinction only to the starburst population. There are two parameters in the dust treatment method: the total effective V-band optical depth τ_V that affects stars younger than 10⁷ yr, and the fraction μ of extinction that comes from the diffuse interstellar medium (ISM) and thus affects stars of all ages. We choose τ_V between 1 and 10 with a step size of 1. Bruzual & Charlot (2003) noted that the average value of μ is 0.3, and we adopt this value for our modeling. Our choice of optical depth is conservative, based on the common concept that these galaxies are dusty starbursts and/or AGNs. However, we reiterate that the ULIRGs on average are blue and bright, and in order to produce the blue colors the dust attenuation associated with the escaping optical emission has to be modest. As discussed earlier, it is also possible that the geometry of the dust distribution in the system is patchy so that the high infrared luminosity originates from the most dust-attenuated regions, while the optical emission comes from relatively less attenuated regions, e.g., tidal structures at large distances.

We plot nine modeled tracks along with the ULIRGs on the color–magnitude diagram in Figure 8. The tracks start from the time when the starburst is induced. The three rows of panels show sub-solar, solar, and super-solar metallicity, from top to bottom, respectively. For each metallicity, there are three tracks with τ_V = 1,  5,  9, from left to right, respectively. The tracks are k-corrected to z = 0.1 and normalized to the median $M_{^{0.1}r}$ which is roughly 2.5 L_*. Symbols are plotted on the tracks to mark the age of 1, 2, and 5 Gyr after the starburst starts. These tracks clearly suggest that both metallicity and dust extinction play important roles in our understanding of the evolution of these systems. Dust extinction mostly affects the blue optical colors of the systems younger than several hundred million years. For all metallicities, the stellar populations reach their bluest colors in less than 20 Myr, but the bluest color a stellar population can reach becomes increasingly red with increasing dust extinction. The blue colors of many ULIRGs can be produced only with the lowest dust extinctions. However, there is almost no color difference between different dust extinctions when the populations are older than 1 Gyr and their ^0.1g − ^0.1r colors are redder than 0.7, and any systems with the same metallicity will redden to the red sequence at roughly the same time. Metallicity affects the range of colors and the timescale of the color evolution of the stellar populations, and higher metallicity systems span wider ranges in the color space, and evolve more quickly. As shown in Figure 8, the bluest ULIRGs can be reproduced only by the highest metallicity track. After 20 Gyr, the super-solar track is 0.25 mag redder than the sub-solar case. It takes a sub-solar population 2 Gyr to evolve to the red sequence, 1.5 Gyr for the solar population, and only 1 Gyr for the super-solar one.

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In reality, the star formation history is almost certainly more complicated than the simple models we discuss here. In the third model, we use a simplified version of star formation history produced from numerical simulations of gaseous disk mergers by Springel et al. (2005). Springel et al. used GADGET-2 (Springel 2005) to trace star formation in a merger of two spiral galaxies and studied the effects of black hole feedback on the quenching of star formation. The two spiral galaxies have equal dynamical masses of 3.85 × 10¹² M_☉, and form stars at an exponentially decaying rate before the merger, with the pre-merger star formation rate ∼200 M_☉ yr⁻¹. The merger-triggered starburst happens at t = 1.5 Gyr and the peak star formation rate is ∼2000 M_☉ yr⁻¹. In the black hole feedback scenario, the star formation is quenched right after the starburst and the star formation rate decreases to less than 1 M_☉ yr⁻¹ at t = 2.5 Gyr, while in the no feedback scenario the star formation rate decays slowly and remains at several solar masses per year for several Gyr. The intrinsic extinction is ignored in this model. We plot the evolutionary tracks in Figure 7 with the dotted line representing the track without AGN feedback and the dashed line representing the one with AGN feedback. Three symbols are plotted on each track to mark the ages of 1, 2, and 5 Gyr. We find that some of our most luminous ULIRGs are located very close to these tracks in the color–magnitude diagram. Both tracks evolve more slowly than the SSP track on the same plot because the starburst produces new young blue stars. Between the two tracks, the one with AGN feedback clearly reddens more quickly than the one without AGN feedback: it takes 8.5 Gyr (or 7 Gyr after the starburst) for the galaxies without AGN feedback to evolve to the red sequence, which is 1.5 Gyr longer than the one with AGN feedback. The difference in the timescales is consistent with the scenario in which AGN feedback quenches the star formation effectively, and consequently helps form the color bimodality of galaxies.

In summary, these simple models suggest possible evolutionary paths of the ULIRGs on the color–magnitude diagram. We conclude that constant star-forming galaxies cannot evolve to the green valley without the star formation quenched. Galaxies with lower star formation activity evolve to the red sequence more quickly and spend shorter time in the green valley, as seen in the SSP model and the numerical simulation model. Dust affects mostly the blue colors of the stellar population, and has almost no influences on the red colors and the timescale of evolution to the red sequence. A stellar population with higher metallicity explores a wider range in the color space, and evolves more quickly to the red sequence.

The color evolution of the ULIRGs may be represented by any one of the models with different masses and dust extinctions. For example, the evolutionary track of the most luminous ULIRGs in our sample may be represented by either the numerical simulation models with slightly different mass, or by the SSP track in Figure 7 with 10 times more mass; the evolutionary track of a ULIRG with the median color and magnitude may be represented by the numerical simulation models with 90% of the optical emission being attenuated. It is also possible that the color evolution of a galaxy is the result of composite star formation histories, so that the galaxy may leave the blue cloud and enter the red sequence more than once. We will not discuss these more complicated scenarios because they are out of the scope of this paper.

We use the code developed by Lotz et al. (LPM04) to calculate the morphology coefficients G and M₂₀ for both the 54 ULIRGs and the comparison sample, galaxies appearing on the same SDSS images of the ULIRGs. The Gini coefficient, G, is defined as

$$\begin{equation} G = \frac{1}{{\mid}{\bar{X}}{\mid} n(n-1)}\sum _{i}^{n}(2i-n-1){\mid} X_i{\mid}, \end{equation} \tag{ 2 }$$

where n is the number of pixels and X_i's are the sorted pixel intensities in increasing order. The total second-order moment M_tot of the pixels assigned to the galaxy is defined as

$$\begin{equation} M_{\rm tot} = \sum _{i}^{n}f_i[(x_i-x_c)^2+(y_i-y_c)^2], \end{equation} \tag{ 3 }$$

where f_i is the flux in each pixel in decreasing order, x_i and y_i are the coordinates of the pixel, and x_c, y_c is the center of the galaxy, computed by minimizing M_tot. M₂₀ is the normalized second-order moment of the brightest 20% of the galaxy's flux, such that

$$\begin{equation} M_{20} = {\rm log}_{10}\left(\frac{\sum _i M_i}{M_{\rm tot}}\right)\ {\rm while}\ \sum _i f_i < 0.2f_{\rm tot}, \end{equation} \tag{ 4 }$$

where f_tot is the total flux of the galaxy.

As noted by LPM04 and Lotz et al. (2008b), the quantity G is very sensitive to the ratio of low-surface-brightness pixels to high-surface-brightness pixels, so a well-defined segmentation map is essential to measure G and M₂₀ accurately. Also, as pointed out by Lisker (2008), measurement uncertainties of G are minimized when using the Petrosian radius as the aperture size. We thus adopt the same approach as LPM04 and assign the pixels brighter than the surface brightness at the Petrosian radius (semimajor axis as measured in an elliptical aperture) to the galaxy. The measured coefficients are not robust for a source with the signal-to-noise ratio per pixel within the Petrosian radius (〈S/N〉) less than 2.5, or with a Petrosian radius (r_P) less than twice the size of FWHM of the PSF (∼13 in SDSS) (LPM04 and Lotz et al. 2008b), where the signal-to-noise ratio per pixel is defined as the ratio of the mean flux and the mean rms noise within an aperture. We therefore exclude all sources with 〈S/N〉 < 2.5, and/or r_P < 26. The u-band images are noisy in general: almost 60% of the ULIRGs have 〈S/N〉 less than 2.5, and thus we do not include the u-band results further in our analysis. We also exclude possible candidates for stars in the comparison sample by rejecting sources with the SExtractor star/galaxy separation parameter CLASS_STAR greater than 0.9. The final selection yields the measured morphology parameters for 50, 50, 51, and 42 ULIRGs, and 1215, 2059, 2019, and 484 comparison galaxies, in the g, r, i, and z band, respectively.

In Figure 9, we plot the g-, r-, i-, and z-band G against M₂₀ of the comparison galaxies using contours and dots, where the contours are 10 equally spaced levels between 10% and 90% of the peak number density, and dots show galaxies located outside the lowest 10% contour level. On top of them, we overlay the morphology parameters of ULIRGs with circles (AGN ULIRGs) and stars (H ii-region like/LINERs). We also plot the empirical lines that divide the parameter space into four morphology regions: mergers, elliptical galaxies, irregular galaxies, and spiral galaxies, according to the classifications for z ∼ 0 galaxies described in LPM04 and Lotz et al. (2008a):

$$\begin{eqnarray} \rm {Merger:}\, & G > -0.115M_{20}+0.384 & \nonumber \\ \rm {Elliptical:}\, & G < -0.115M_{20}+0.384 &\nonumber\\ &{\rm and}\ G > 0.115M_{20}+0.769& \nonumber \\ \rm {Irregular:}\, & G > 0.115M_{20}+0.697, &\nonumber\\ &G < 0.115M_{20}+0.769,&\nonumber \\ & {\rm and}\ G < -0.115M_{20}+0.384 & \nonumber \\ \rm {Spiral:}\, &G < -0.115M_{20}+0.384 & \nonumber\\ &{\rm and}\ G < 0.115M_{20}+0.697.& \nonumber \end{eqnarray} \tag{ 5 }$$

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The distributions of the ULIRGs at all wavelengths are consistent in that they form a similarly heterogeneous group in this parameter space. The centroids of the distribution in G–M₂₀ are [0.55,−1.42], [0.55,−1.43], [0.57,−1.47], [0.54,−1.45], in g, r, i, and z band, respectively, all of which are located very close (ΔG < ±0.02 for any given M₂₀) to the separation line between merger and non-merger galaxies. There are only 21(42⁺¹⁸₋₂₄%), 21(42⁺²⁰₋₂₂%), 25(49⁺¹⁹₋₂₇%), and 15(36⁺²⁴₋₂₆%) ULIRGs located in the merger region, in g, r, i, and z band, respectively. These are very low fractions compared to the z ∼ 0 results where the majority (80%) of ULIRGs are located in the merger region (LPM04). The distributions of the ULIRGs in the parameter space vary slightly with wavelengths: the merger fraction is the highest (49%) using the i-band derived parameters, and lowest (36%) using the z-band derived ones. Although the morphology parameters of the ULIRGs span a wide range in the parameter space, their G–M₂₀ distribution is clearly different from that of the comparison sample galaxies. The distribution centroids of the comparison galaxies in the four bands are all located in the empirical region of spiral galaxies, and only 56(5⁺¹⁴₋₄%), 117(6⁺¹⁹₋₅%), 146(7⁺²¹₋₆%), and 26(5⁺²¹₅%) galaxies are found in the empirical region of mergers, in g, r, i, and z band, respectively.

The poor physical resolution of the SDSS images (700 pc pixel⁻¹ at z ∼ 0.1) may lead to higher random and systematic uncertainties in G and M₂₀ (see LPM04), and thus careful estimation of the uncertainties in G and M₂₀ is necessary. Using image and noise simulations (described below in Section 5.3), we estimate the uncertainties of morphology measurements (ΔG ≲ 0.05 and M₂₀ ≲ 0.3–0.4) in SDSS images and plot the largest uncertainties as error bars in Figure 9. We find that a galaxy seen against higher background noise tends to move to the left lower corner in the G–M₂₀ plot, and we plot a vector to illustrate the typical changes in G and M₂₀ resulting from adding background noise until the source only barely satisfies the signal-to-noise ratio and Petrosian radius criteria. The vector shows that a source selected as a merger at high signal-to-noise ratio could move away from the merger region and stay in the late-type galaxy region if it is immersed in higher background, and consequently, the merger fraction selected by G and M₂₀ is likely to be a lower limit. Although less than half of the ULIRGs have empirical merger G–M₂₀ relations, the heterogeneous distributions are qualitatively consistent with what was found in the recent numerical simulations of merging galaxies of Lotz et al. (2008b). We will discuss these noise simulations further, together with the uncertainties in measuring the morphologies, in Section 5.3.

AGN ULIRGs are slightly more concentrated in the merger region than non-AGN ULIRGs: there are 45%, 50%, 60%, and 50% of AGN ULIRGs in the merger region, in g, r, i, and z band, respectively, compared to 41%, 40%, 48%, and 30% for non-AGN ULIRGs. However the difference is not statistically significant given the small number of AGN ULIRGs (11 in g band and 10 in other bands based on the signal-to-noise ratio and size selection criteria). There is no significant difference between the mean G (〈G〉) and M₂₀ (〈M₂₀〉) for the AGN ULIRGs and non-AGN ULIRGs: for AGN ULIRGs, 〈G〉 is 0.58, 0.57, 0.59, 0.58, and 〈M₂₀〉 is −1.48, −1.48, −1.36, −1.48, in g, r, i, and z band, respectively; in comparison, 〈G〉 is 0.54, 0.54, 0.56, 0.53, and 〈M₂₀〉 is −1.42, −1.45, −1.49, −1.42 for non-AGN ULIRGs in the same bands.

Our visual inspection shows that almost all the ULIRGs are disturbed systems (Figure 2), and this is consistent with the visual classifications by VKS02, who found that all but one of the 118 IRAS 1 Jy sample show visual merger features. We perform the G–M₂₀ analysis and find that more disturbed galaxies do have higher G and M₂₀ coefficients. This is illustrated in Figure 1 where more disturbed galaxies tend to occupy the upper left corner of the mosaic with higher values of both G and M₂₀. However, our studies also show that ULIRGs are a heterogeneous group in G–M₂₀ space: less than half of the sources lie above the solid line in each panel of Figure 9, the region where most local mergers and ULIRGs were found by LPM04. Interestingly, there is only one source located in the early-type region, and all the remaining sources fall in the region where irregular and late-type galaxies are located. This heterogeneous distribution seems to be inconsistent with the result of LPM04 that 80% of local ULIRGs are located in the typical merger regions in the parameter space. However, the spread in the parameter space we found is supported by numerical simulations. Lotz et al. (2008b) analyzed the morphological parameters in SDSS g-band images of mergers of equal mass gas-rich spirals by using the N-body/hydrodynamic simulation code GADGET (Springel et al. 2001) and Monte Carlo radiative transfer code SUNRISE (Jonsson 2006; Jonsson et al. 2006), and find that the mergers are most disturbed in G–M₂₀ at the first pass, but they can have normal galaxies morphologies at other merger stages. They also noted that two-thirds of the ULIRGs in LPM04 exhibit double or multiple nuclei, and therefore are more effectively selected by G–M₂₀. The timescale is only 0.2–0.6 Gyr for a merger to appear in the empirical merger region defined by LPM04, and the range in this timescale depends on dust, orbit parameters, and observing orientations. Thus at face value, our results imply that about half of our ULIRGs have been captured within this time window. The fact that the other half of our ULIRGs do not appear in the empirical merger region does not mean that they are not mergers. Instead, they might be in other merger stages when their morphological parameters are consistent with those of normal galaxies. The heterogeneous picture of our ULIRGs thus may represent different evolutionary epochs of the ULIRGs.

In order to gain some insights into the G–M₂₀ morphology, we compare our g-band G–M₂₀ classification with the interaction classification for the same sources made by Veilleux et al. (2002). Veilleux et al. classified each object into one of the six sequential merging stages according to its morphology. The comparison of two different classifications is illustrated in Figure 10. We find that all triple mergers by the classification of Veilleux et al. are also classified as mergers by G–M₂₀. Sources in their type IIIa group, i.e., wide binary pre-merger systems with apparent separations greater than 10 kpc, are also very likely to be recognized as G–M₂₀ mergers: ∼80% of such sources are recognized as mergers using G–M₂₀. Sources in their other groups are less likely to be recognized by G–M₂₀ as mergers, and the fraction of G–M₂₀ mergers appears to decrease toward later merging stages. We find that G–M₂₀ mergers make up ∼31% of the close binary pre mergers (type IIIb), ∼50% of the compact mergers (type IVa), ∼46% of the diffuse mergers (type IVb), and only ∼11% of the old mergers (type V) are G–M₂₀ mergers, respectively. Sources classified as close binary pre-merger systems (type IIIb) and old mergers (type V) have very large fractions to be recognized as spiral galaxies by G–M₂₀ (∼54% and ∼67%, respectively). These comparison results are qualitatively consistent with the simulation results by Lotz et al. (2008b) in which the most disturbed G–M₂₀ morphology happens during the first pass and maximum separation. However, since our measured morphology is limited by relatively small number statistics and relatively large uncertainties, we do not conclude strongly that the observed morphology can be well explained by the simulations.

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We also find that uncertainties in measuring the morphologies contribute significantly to the distribution of the ULIRGs in the parameter space. LPM04 reported that the measurements of G and M₂₀ are robust to within 10% when the source has signal-to-noise ratio per pixel 〈S/N〉 greater than 2.5, the Petrosian radius larger than 2.5 times the PSF FWHM, and the physical resolution (parsec per pixel) higher than 500 pc pixel⁻¹. However, when the physical resolution is lower the uncertainties of morphologies largely increase (Figure 6 in their paper) since small structures are washed out. At z ∼ 0.1–0.2 the spatial resolution of the SDSS ULIRGs is roughly 0.7–1.3 kpc pixel⁻¹ and the uncertainties need to be calibrated carefully.

We perform simple simulations to estimate the uncertainties of G and M₂₀ for the SDSS ULIRGs at different noise levels. We use four model elliptical galaxies, four spiral galaxies selected from SDSS images, and three merging galaxies selected from our ULIRG sample. The four elliptical galaxies are modeled using de Vaucoulers profiles and are added to real SDSS images to mimic the observational conditions. The three spiral galaxies are selected with bright g-band magnitudes of 13.9, 15.2, and 16.4, respectively, much brighter than the median g-band magnitudes of the SDSS sample (18.0 mag). The three ULIRGs (FSC08572+3915, FSC14060+2919, FSC14121−0126) are selected with bright g-band magnitudes and unmistakable merger morphology. The g-band magnitudes are 16.4, 16.7 and 17.8, respectively, all brighter than the median g-band magnitudes of the ULIRGs (17.8 mag). For each image, we generate random noise at a series of different levels, and at each level we generate 20 noise maps independently and add them to the original image. Morphologies, Petrosian radius, and 〈S/N〉 are measured for each noise-added source and compared to the original measurements. Although our simulation is limited by the sample size, ∼90% of the noise-added sources selected by the signal-to-noise and size criteria (〈S/N〉>2.5 and R_p>26) have g-band magnitudes between 16 and 19, and g-band Petrosian radii between 3 and 12 arcsec (∼ 6–22 kpc at z ∼ 0.1), ranges similar to those of the ULIRGs.

We find that the measured Petrosian radius and the 〈S/N〉 do not always decrease monotonically with the amount of noise added, and they are broadly distributed at low signal-to-noise levels. We also find that, rather than remaining fixed for a given source, the measured G decreases and M₂₀ increases systematically with increasing noise. Therefore, when seen against higher background noise, a galaxy tends to move to the lower left corner on the G–M₂₀ plot. This effect will decrease the fraction of the mergers observed in the merger region. The Gini coefficient G decreases by as much as 0.2 with increasing background noise until the galaxy is no longer detected (either r_p < 2× FWHM, or 〈S/N〉 < 2.5), and M₂₀ increases by as much as 0.5. The standard deviations of G at every noise level are all less than 0.05, and the spiral galaxies have the smallest dispersions. The standard deviations of M₂₀ are all less than 0.3–0.4, and the elliptical galaxies have the smallest dispersions. There is no systematic difference for the uncertainties between different bands. These uncertainties (ΔG = 0.05 and ΔM = 0.5) are shown in Figure 9 as error bars, and the systematic trend with increasing noise in G and M₂₀ is shown as an arrow.

The aperture within which the morphologies are measured is also an important source of uncertainty. As noted by Lisker (2008), G value measured within larger apertures are systematically larger, and the measured uncertainties are minimized when the Petrosian radius is used as the aperture size. The reason behind this is that more low-surface-brightness pixels are included within a larger aperture size, which steepens the intensity distribution of the pixels and consequently increases the value of G. Some of the low-surface-brightness features of the source, often at larger distances from the center, are mistakenly excluded when the aperture size is chosen too small, which flattens the pixel intensity distribution and consequently decreases the value of G. When assigning pixels to the sources we adopt the surface brightness at the Petrosian radius as the threshold, and assign pixels brighter than the threshold to the source. Therefore, according to their study our measured morphologies should have minimized uncertainties, because our apertures should be very close to the Petrosian radius and not significantly larger or smaller.

The morphology, color–magnitude relation, and the infrared luminosity of ULIRGs present a rather complicated picture of these systems. The huge infrared luminosity in our ULIRGs suggests that their dominant powering sources should be obscured by dust. However, the majority of them are optically bright and blue. This further implies that dust is not uniformly distributed and that significant amount of the unobscured stellar light is seen directly. Indeed, as shown in Figure 1, there are obvious color gradients in the ULIRGs, with their tidal features at large distances appearing blue and their central regions appearing red. Lotz et al. (2008b) suggested that merging systems are most disturbed in G–M₂₀ space and are located in the merger region in the G–M₂₀ plot during the first pass when the tidal features are prominent. After the first pass they will move to the late-type and irregular region (see Figure 5 in their paper). Thus, we would expect the integrated colors of those disturbed systems selected by G–M₂₀ to be relatively bluer than the ones with normal G–M₂₀ relations, due to the blue colors of the tidal features. Recent hydrodynamic simulations of disk–disk mergers also suggested that the infrared luminosity of ULIRGs will peak during the final merger when enough metals and dust have been accumulated (e.g., Jonsson et al. 2006), so one would also expect sources with higher infrared luminosity to be preferentially found in the late-type and irregular region.

We test these hypotheses by plotting the g-band Gini versus M₂₀ for the ULIRGs in Figure 11, with different symbol sizes representing their FIR luminosities and different symbol colors representing their ^0.1g − ^0.1r colors. There are no strong relations between the optical colors and G or M₂₀, except that optically redder sources are slightly more preferentially located in the non-merger regions with lower G values. This is further illustrated in the small panel in the same plot, where we plot the color distributions of sources with G ⩾ 0.55, the median G of the whole sample, and of sources with G < 0.55. The sources with lower G values tend to distribute toward redder optical colors, although the reddest source has a higher-than-median G value (0.62). All AGN ULIRGs have G values greater than 0.5. There are no obvious correlations between the FIR luminosity and G or M₂₀.

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These results are broadly consistent with the scenario that the blue optical colors originate from more disturbed systems, and these systems are not seen during the phase when the FIR emission peaks. However, as mentioned earlier, the uncertainties in measuring G and M₂₀ could be substantial, and the measured morphologies are affected by orientation, orbital parameters, viewing angle, and dust content (Lotz et al. 2008b). We thus do not conclude any strong relations among the morphologies, the FIR luminosity, and the optical color of these system.