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Mon. Not. R. Astron. Soc. 000, 1­13 (2010)

Printed 21 September 2011

A (MN L TEX style file v2.2)

A universal ultraviolet-optical colour­colour­magnitude relation of galaxies
I1gor V. Chilingarian1,2, and Ivan Yu. Zolotukhin
3 ,4 ,2
Centre de Donn´es astronomiques de Strasbourg, Observatoire astronomique de Strasbourg, Universit´ de Strasbourg, e e CNRS UMR 7550, 11 rue de l'Universit´, 67000 Strasbourg, France e 2 Sternberg Astronomical Institute, Moscow State University, 13 Universitetsky prospect, Moscow, 119992, Russia 3 Observatoire de Paris, LERMA, UMR 8112, 61 Av. de l'Observatoire, 75014 Paris, France 4 Observatoire de Paris, VO Paris Data Centre, 61 Av. de l'Observatoire, 75014 Paris, France

arXiv:1102.1159v2 [astro-ph.CO] 19 Sep 2011

Accepted 2011 Sep 15. Received 2011 Sep 15; in original form 2011 Feb 6

ABSTRACT

The bimodal galaxy distribution in the optical colour­magnitude diagram (CMD) comprises a narrow "red sequence" populated mostly by early-type galaxies and a broad "blue cloud" dominated by star-forming systems. Although the optical CMD allows one to select red sequence ob jects, neither can it be used for galaxy classification without additional observational data such as spectra or high-resolution images, nor to identify blue galaxies at unknown redshifts. We show that adding the near ultraviolet colour (GALEX NUV eff = 227 nm) to the optical (g - r vs Mr ) CMD reveals a tight relation in the three-dimensional colour­colour­magnitude space smoothly continuing from the "blue cloud" to the "red sequence". We found that 98 per cent of 225 000 low-redshift (Z < 0.27) galaxies follow a smooth surface g - r = F (Mr , N U V - r) with a standard deviation of 0.03­0.07 mag making it the tightest known galaxy photometric relation given the 0.9 mag range of k -corrected g - r colours. Similar relations exist in other NUV­optical colours. There is a strong correlation between morphological types and integrated N U V - r colours of galaxies, while the connection with g - r is ambiguous. Rare galaxy classes such as E+A or tidally stripped systems become outliers that occupy distinct regions in the 3D parameter space. Using stellar population models for galaxies with different star formation histories, we show that (a) the (N U V - r, g - r) distribution at a given luminosity is formed by ob jects having constant and exponentially declining star formation rates with different characteristic timescales with the red sequence part consistent also with simple stellar population; (b) colour evolution for exponentially declining models goes along the relation suggesting a weak evolution of its shape up-to a redshift of 0.9; (c) galaxies with truncated star formation histories have very short transition phase offset from the relation thus explaining the rareness of E+A galaxies. This relation can be used as a powerful galaxy classification tool when morphology remains unresolved. Its mathematical consequence is the possibility of precise and simple redshift estimates from only three broad-band photometric points. We show that this simple approach being applied to SDSS and GALEX data works better than most existing photometric redshift techniques applied to multi-colour datasets. Therefore, the relation can be used as an efficient search technique for galaxies at intermediate redshifts (0.3 < Z < 0.8) using optical imaging surveys. Key words: galaxies: (classification, colours, luminosities, masses, radii, etc.) ­ galaxies: photometry ­ galaxies: stellar content ­ galaxies: distances and redshifts

1
The best-fitting photometric relation coefficients and other supporting technical information are available at the pro ject website: http://specphot.sai.msu.ru/galaxies/ E-mail: chil@sai.msu.ru (IC); iz@sai.msu.ru (IZ) c 2010 RAS

INTRODUCTION

Understanding observational asp ects of galaxy evolution requires to classify them in regard to various prop erties such as morphology, luminosity, stellar p opulation characteristics,


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lowed us to reject quasars and prominent broad-line active galactic nuclei. This list contains 377 923 sources. After that we app ended b oth GALEX GR5 and UKIDSS DR8 data by joining our reference list of ob jects with these surveys using the spatial match criterion, namely the b est match within the angular separation of 3 arcsec. For the GALEX data we made use of the pre-calculated crossmatch b etween SDSS and GALEX surveys (Budavari et al. ´ 2009) accessible through GALEX CasJobs as the xsdssdr7 table restricting the angular separation to < 3 arcsec. For the UKIDSS LAS we queried multi-cone search programmatic access interface with effectively the same parameters for every ob ject from our initial list. The SDSS ­GALEX join returned 223 646 galaxies detected in N U V , 144 639 in F U V (eff = 155 nm), and 136 781 in b oth filters. The SDSS ­UKIDSS match contains 176 868 galaxies detected in the Y band, 178 806 in J , 187 789 in H , 188 221 in K , among them 158 578 in all four NIR bands. For 96 939 of those galaxies we had photometric data from GALEX N U V , including 59 994 with GALEX F U V measurements. All the technical op erations on tables were p erformed with the stilts software (Taylor 2006). We used SDSS Petrosian magnitudes (PetroMag * ), GALEX extended source calibrated magnitudes (nuv mag and fuv mag ), and UKIDSS Petrosian magnitudes (PetroMag * ) to construct multi-wavelength sp ectral energy distributions (SED). Here we notice, that even though SDSS model magnitudes (modelMag * ) generally have lower formally computed statistical uncertainties than Petrosian magnitudes, esp ecially in blue photometric bands, in case of "blue cloud" galaxies they are often hamp ered by the differences b etween the observed light distribution and those assumed (axisymmetrical exp onential or de Vaucouleurs) for the computation of model magnitudes. Petrosian magnitudes may underestimate the total galaxy flux by 15­ 20 p er cent in case of face-on de Vaucouleurs profiles (Yasuda et al. 2001). However, in our case this offset is similar for SDSS and UKIDSS data while for the GALEX measurements it is not imp ortant b ecause of high photometric uncertainties significantly exceeding 15 p er cent for red galaxies having the light profile shap e affected by this effect. As long as GALEX and SDSS contain photometric measurements in the AB system, but UKIDSS magnitudes are in the Vega system, we applied zero-p oint transformations available in the literature (Hewett et al. 2006) to the NIR magnitudes. We are using integrated photometry of galaxies, therefore ap erture effects have little imp ortance in the present study and we do not need to apply ap erture corrections. Then, all magnitudes were corrected for the effects of Galactic extinction. The UKIDSS and SDSS catalogues provide selective extinction values in all photometric bands, while for GALEX we used the provided E (B - V ) value (Schlegel et al. 1998) and computed extinctions in UV bands assuming AN U V = 8.87 · E (B - V ), AF U V = 8.29 · E (B - V ). At this p oint we created a calibrated photometric sample of low-redshift galaxies in 11 bands, from far-UV to NIR. It is easily reproducible at any workstation with the Internet access, however, due to the data access p olicy, the DR4 latest public release of UKIDSS catalogues has to b e used instead of DR8. Systematic uncertainties of SDSS p oint source photomc 2010 RAS, MNRAS 000, 1­13

internal dynamics. In the present-day era of deep wide-field imaging surveys, there is a need in efficient mechanisms of galaxy classification and selection using minimal available information. Colour­colour and colour­magnitude diagrams (CMD) have b een traditionally used for this purp ose. In the optical CMD (g - r, Mr ) (Strateva et al. 2001; Baldry et al. 2004; Blanton et al. 2003a), the very narrow "red sequence" (Visvanathan & Sandage 1977) ( (g - r ) 0.04 mag) formed mostly by elliptical and lenticular galaxies is used to identify early-typ e memb ers of galaxy clusters b ecause at low redshift it moves as a whole remaining tight in the colour space. However, optical CMDs cannot b e used for detailed classification of galaxies, either for the selection of other than red galaxies b ecause of several degeneracies: (1) there is no unambiguous connection b etween the galaxy morphological typ e and its p osition on the CMD; (2) the red part of the CMD is contaminated by 25 p er cent with late-typ e galaxies having weak ongoing star formation (SF) attenuated by the dust; (3) the blue cloud overlaps with the loci of "E+A" p oststarburst galaxies (Dressler & Gunn 1983) (PSG) having blue colours, early-typ e morphology, often disky kinematics (Chilingarian et al. 2009b) but no or weak ongoing SF. In the (N U V - r, Mr ) space, b oth the red sequence and the blue cloud b ecome pronounced but quite broad ( (N U V - r ) 2 mag) sequences (Wyder et al. 2007). Such a width of the red sequence is due to the UV flux sensitivity to even small fractions of young stars that was shown to b e connected to the environment of early-typ e galaxies (Kavira j et al. 2007). At the same time, (1) the sequences are too broad to use them for the efficient photometric selection of galaxies; (2) there is still an ambiguity b etween the N U V - r colour and a galaxy morphological class as well as the presence of ongoing SF: PSGs still reside in the blue cloud.

2 2.1

THE UV­OPTICAL GALAXY PHOTOMETRIC SAMPLE Catalogue construction

Using Virtual Observatory data mining, we constructed a photometric sample of 225 000 galaxies excluding quasars and bright active galactic nuclei (AGN) based on their sp ectral classification by the Sloan Digital Sky Survey Data Release 7 (Abaza jian et al. 2009) in the absolute magnitude range -25 < Mz < -15 mag at low redshifts (0.007 < Z < 0.27). We cross-identified the sp ectral sample of SDSS DR7 galaxies with the UV Galaxy Evolution Explorer satellite (Martin et al. 2005) Release 5 (GALEX GR5) catalogue in the CASJobs (Szalay et al. 2002) catalogue access systems of SDSS and GALEX and rejected the matches separated by more than 3 arcsec on the sky. First, we employed the SDSS DR7 CasJobs service to select galaxies in the redshift range from 0.007 to 0.27 from the SDSS DR7 sp ectroscopic sample in the strip es covered (already or in the survey plan) by the United Kingdom Infrared Telescop e Deep Imaging Sky Survey (Lawrence et al. 2007) Large Area Survey Data Release 8 (UKIDSS LAS DR8). We selected only the ob jects classified as galaxies by the SDSS sp ectroscopic pip eline (SpecClass = 2 ), that al-


A universal galaxy photometric relation
etry do not exceed 1 p er cent (Ivezi´ et al. 2004) in g r , c whereas for extended sources they may b e a few times larger. However, since the red sequence in the optical CMD of our sample constructed from galaxies p opulating a large area on the sky is as tight as 0.03 mag, we conclude that either the systematic errors on optical magnitudes are within this range, or they are strongly correlated b etween g and r bands so that they cannot hamp er the results of our analysis. Statistical uncertainties of SDSS photometric measurements are generally b etter than 0.015 mag reflecting the sp ectroscopic target selection of SDSS : galaxies from the sp ectral sample are at least a few magnitudes brighter than the limiting magnitude of the photometric survey. At the same time, the median value of N U V magnitude uncertainties is as large as 0.15 mag across the whole sample. However, the range of N U V - r galaxy colours is 7.5 mag compared to 0.9 mag in g - r . Therefore, the relative "resolution" of our analysis p er colour range is very similar in b oth colours.

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Second, we used the topcat1 table manipulation software (Taylor 2005) to visualise obtained k-corrections as functions of redshifts and various observed colours, as we did for optical and NIR bands (Chilingarian et al. 2010). Similarly, we found that UV k-corrections could b e precisely approximated by low-order p olynomial functions of redshifts and certain colours. The b est filter combinations are N U V - g and F U V - u for the N U V and F U V bands resp ectively with standard deviations of the surface fitting residuals of ab out 0.08 mag and 0.15 mag. Then, we used these approximations to compute k-corrections for all galaxies in our sample. The newly obtained approximations of k-corrections are available from the new version of the "kcorrections calculator" service2 .

3

THE COLOUR­COLOUR­MAGNITUDE RELATION AND ITS PROPERTIES

2.2

Computation of k-corrections

As we compare photometric measurements for galaxies at different redshifts, we have to correct them for the changes of effective rest-frame wavelengths of filter bandpasses known as k-corrections (Oke & Sandage 1968; Hogg et al. 2002; Blanton & Roweis 2007). Their computation is an imp ortant step for obtaining the fully calibrated homogeneous dataset. Here we provide some details regarding the k-correction computation in GALEX UV bands, while the procedure for optical and NIR filters was exhaustively describ ed earlier (Chilingarian et al. 2010). The imp ortance of accurate kcorrection computation is illustrated by the fact that earlier studies of galaxies in the (N U V - r, g - r ) colour­colour diagram (Yi et al. 2005) did not rep ort a sequence of galaxies which would b e obvious if one took a galaxy sample in a narrow redshift range. Due to high sensitivity of UV fluxes to the recent SF and mass fractions of young stars as little as 1 p er cent, the UV-to-NIR SED of a galaxy usually cannot b e precisely represented by a single simple stellar p opulation, that is, a p opulation of stars of the same age and metallicity. Therefore, kcorrections cannot b e computed by the single SSP fitting as it can b e done in optical and NIR bands (Chilingarian et al. 2010). Other effects, such as a non-thermal emission from a moderate-luminosity active galaxy nucleus, or emission in certain sp ectral lines, create additional difficulties in the UV bandpasses. Most of these effects were tackled and successfully taken into account using the nonnegative matrix factorization (Blanton & Roweis 2007). In the same work, the authors computed 5 synthetic template sp ectra representative of different galaxies and galaxy comp onents. Here we used the 2-step process to compute kcorrections for our galaxies. First, we constructed a subsample including some 25 000 galaxies detected in all 11 bands with high signal-to-noise ratios in the UV bands. We then fitted their SEDs using a non-negative linear combination of 5 representative templates (Blanton & Roweis 2007) attenuated using the Cardelli extinction law (Cardelli et al. 1989) leaving the colour excess E (B - V ) a free parameter.
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We insp ected the combined GALEX ­SDSS dataset visually using topcat in three dimensions (Mz , N U V - r , g - r ) and detected a very thin continuous distribution of b oth, blue and red galaxies along a smooth surface with very few outliers. Then we approximated it with a low order twodimensional p olynomial (see App endix B for details). Given the range of observed g - r colour of 0.9 mag, the disp ersion of the g - r residuals that decreases from 0.07 to 0.03 mag going from blue to red N U V - r colours without significant dep endence on the luminosity at Mz < -17.5 makes it the tightest known photometric relation of galaxies. At lower luminosities, the disp ersion of the residuals increases. In our case this can b e explained by significantly lower numb er of ob jects due to the sp ectroscopic target selection algorithm used in SDSS and also by the p oor quality of photometric measurements, b ecause dwarf galaxies have lower mean surface brightness values than giants and, consequently, their magnitudes cannot b e precisely measured in relatively shallow wide-field surveys that we used to construct our catalogue. An additional factor increasing the scatter is the p eculiar motions of galaxies inside clusters and groups not taken into account which hamp er the Hubble law distance estimates. The 3D distribution of galaxies and its pro jection onto the (N U V - r , g - r ) plane are shown in Fig. 1, 2. The choice of the z band for the M axis is not that imp ortant: the relation b ehaves similarly when using r iz or NIR absolute magnitudes. We selected the SDSS z band for presentation purp oses as the z luminosity range is slightly higher than in the r band. Yi et al. (2005) presented a (N U V - r , g - r ) colour­ colour plot for galaxies and stars in their fig. 1. However, the galaxy photometric measurements were not prop erly kcorrected as authors did not p ossess multi-band photometry, therefore no tight colour relation was revealed. Rather tight relation in the (N U V - r , g - r ) colour­ colour diagram for normal galaxies was mentioned in Schiminovich et al. (2007). Then Salim et al. (2005) and Haines et al. (2008) attempted to use b oth, optical and nearUV information to study the star formation histories (SFHs)
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http://www.star.bris.ac.uk/~ mbt/topcat/ http://kcor.sai.msu.ru/UVtoNIR.html


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Figure 2. The 3-dimensional distribution of galaxies in the (N U V - r , Mz , g - r ) space. The 3D plot presents the density distribution of 225 000 galaxies in the colour­colour­magnitude space with the increasing density going from yellow to red, the best-fitting polynomial surface as a mesh grid immersed in it, and standard deviations of the fitting residuals shown as bars in the (N U V - r , Mz ) plane with their colours corresponding to the magnitude ranges in Fig. 1. PSGs (Goto 2007), and compact elliptical galaxies (cE) are shown as tetrahedra and cubes. The top face of the plot demonstrates their pro jected positions on-to the (N U V - r , Mz ) plane.

3.1

Connection to morphology

Figure 1. The pro jection of the colour­colour­magnitude relation on-to the colour­colour plane. The upper panel demonstrates the logarithm of the number density plot in grayscale with solid lines showing the relations for galaxies of constant luminosity derived from the best-fitting polynomial surface equation. Four bottom plots show fitting residuals in different magnitude ranges as density plots with dashed lines indicating their ±1 levels, which are normalised to the maximum value in every N U V - r bin. Residuals for the two low-luminosity intervals are computed with coarser binning compared to the brighter galaxies in order to account for lower ob ject counts at those luminosities. Red sequence, blue cloud, and the loci of certain types of outliers are identified. The direction of internal extinction is shown with a vector.

To explore the connection of the colour­colour­magnitude relation to galaxy morphology, we employed the morphological catalogue of SDSS galaxies (Fukugita et al. 2007) available through the Vizier service3 , which contains morphological typ es for 1 465 intermediate luminosity and giant galaxies at Z > 0.03 from our sample. We see a continuous change of morphological typ es along the surface in the N U V - r direction with a typical disp ersion of 0.7. . .0.8 Hubble typ e. At the same time, the optical g - r colour turns to b e a very bad morphological indicator: the red sequence region contains galaxies of all morphologies from ellipticals to S c late-typ e spirals. The two-dimensional histogram of morphologies vs N U V - r colours is displayed in Fig. 3. One can see that the S0a and Sa galaxy typ es span a very broad range of colours demonstrating the difficulties of the visual classification of early-typ e disc galaxies. It corresp onds to a simple linear correlation b etween the N U V - r and a Hubble typ e which can b e expressed as: TYPE = 6.6 - 1.1 · (N U V - r ),

and dust effects and the effects of environment on the evolution of galaxies. However, the residual scatter of ob jects from this relation still remains high (an order of 0.15 mag in g - r ) due to the dep endence of b oth galaxy colours on luminosity. Adding the absolute magnitude as the third dimension decreases the scatter by a factor of 3 in the absolute magnitude range (-25 < Mr < -15 mag). It is remarkable, that the relation in the 3D space is followed by star-forming as well as passively evolving galaxies.

where the "TYPE" values will corresp ond to the Hubble typ es as < 0 for E, 1 for S0, 2 for Sa, 3 for Sb, 4 for Sc, higher values for Irr. Red outliers (N U V - r > 4 mag) ab ove the surface (yellow and red p oints in Fig. 3) mostly have later typ es, i.e. spiral galaxies, while outliers b elow it (blue and violet p oints in Fig. 3) have earlier typ es compared to galaxies on the sequence with similar N U V - r colours.
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Figure 3. Connection between visually determined galaxy morphology (Fukugita et al. 2007) and a N U V - r colour. Black contours correspond to galaxies following the relation (numbers are for the counts), while individual outliers beyond 1 are shown as crosses with the colours representative of the deviation from the relation in the g - r colour: yellow to red for galaxies above the relation, blue to violet for ob jects below it.

The connection b etween the morphology and the luminosity suggesting that more luminous galaxies have earlier morphological typ es is much looser and may b e affected by the selection effects in our sample. 3.2 Outliers from the relation

We identify several classes of outliers from the relation comprising 2 p er cent of the total sample (see Fig. 1, 2). (i) Early-typ e PSGs selected from the catalogue of H strong galaxies (Goto 2007) (359 matches with our sample) p opulate a region 0.15 mag b elow the surface in g - r spanning 3 < N U V - r < 5 mag colours explaining the nature of "blue early-typ e outliers" from the morphology­ (N U V - r ) relation describ ed ab ove. These are galaxies with truncated or multi-modal SFH where the last strong star formation episode has just b een finished. The passively evolving newly formed stellar p opulation reddens much faster in the N U V - r colour than in the g - r one so that a PSG at first departs from the blue part of the relation (right in Fig. 1) and notably later (after 2.5­3 Gyr) moves up increasing g - r towards the locus of red sequence galaxies. (ii) Compact elliptical galaxies (Chilingarian et al. 2009a; Price et al. 2009) are residing ab ove the red sequence region of the colour­colour­magnitude relation at the low luminosity part. A few examples of new cE galaxies are shown in Fig 2. However, their colours are never redder than those of the most massive galaxies at the bright red sequence end. This fact is explained by their formation via severe tidal stripping of more massive progenitors, most likely early-typ e disc galaxies, by massive elliptical or cluster/group dominant galaxies. Progenitors of cEs are stripp ed in the innermost regions of galaxy clusters, when the SF is ceased b ecause their interstellar medium is already removed by the ram pressure stripping created by the hot intergalactic gas
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(Gunn & Gott 1972). Dep ending on the previous SFH, these ob jects must reside either on the colour­colour­magnitude relation, or slightly b elow it, in the PSG locus. During the relatively fast tidal stripping process lasting ab out 1 Gyr (Chilingarian et al. 2009a), their stellar p opulation prop erties and colours change insignificantly while the mass and, consequently, the luminosity may decrease by a factor of 10 or more, hence moving a galaxy off the relation if it was sitting on it. Thus, red cE colours are explained by high stellar metallicities inherited from their progenitors which is confirmed by detailed studies of nearby cEs (Rose et al. 2005; Chilingarian & Bergond 2010). There may b e some very rare intermediate-age cEs originating from PSGs whose colours will b e bluer, however their passive evolution will quickly move them ab ove the colour­colour­magnitude relation. (iii) Dusty starforming galaxies such as edge-on spirals are sometimes found ab ove the flattened red part of the relation (N U V - r > 4 mag) also b eing consistent with the locus of late-typ e morphological outliers. Their p ositions are explained by the extinction vector direction shown in Fig. 1. If internal extinction is very strong then a galaxy is moved up-right in the diagram and may end up ab ove the locus of red sequence galaxies. (iv) Galaxies with strong ongoing SF but yet small mass fractions of newly formed stars including ongoing and recent mergers may have very p eculiar colours b ecause of strong nebular emission lines and/or large quantities of dust. (v) Narrow-line AGNs having low contribution of their nuclei to the total light in the optical band and hence classified as normal galaxies by the SDSS pip eline may have strong UV excess. We did not apply any particular filtering to our data to exclude these ob jects, therefore our sample may b e slightly contaminated by them at a sub-p er cent level. (vi) "Non-physical" ob jects: galaxies casually overlapping with either foreground stars or galaxies at different redshifts create outliers which may b e located in almost any part of the parameter space except the region very red in (N U V - r ) and very blue in (g - r ).

4 4.1

DISCUSSION Effects of stellar population evolution and internal extinction

The distribution of galaxies in the colour­colour­magnitude space is governed by three factors: (1) stellar mass, (2) star formation and chemical enrichment histories including the ongoing SF, and (3) internal extinction. Therefore, there must b e a connection b etween galaxy p ositions on the diagram and their stellar p opulation prop erties. We fitted a sub-sample of 133 000 SDSS DR7 sp ectra with simple stellar p opulation (SSP) models using the nbursts technique (Chilingarian et al. 2007b,a) and hence obtained their SSPequivalent ages and metallicities. The nbursts technique includes the multiplicative p olynomial continuum that absorbs flux calibration errors and makes the fitting insensitive to the internal extinction in a galaxy. We also notice that SDSS sp ectra obtained in 3 arcsec wide circular ap ertures may not b e representative of entire galaxies in case of strongly extended ob jects with notable gradients of the


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Figure 4. Connection between galaxy positions on the colour­colour­magnitude relation and their SSP-equivalent stellar populations. The slice of the relation in a narrow luminosity range (-23 < Mz < -21.5 mag) is displayed in the colour­colour pro jection. Mean ages of stellar populations of 40,000 galaxies obtained from the fitting of their SDSS DR7 optical spectra are colour­coded. The evolutionary tracks of stellar population models without internal extinction for the solar metallicity and various SFH are overplotted (see text). The colours of the tracks correspond to the ages of synthetic galaxies formed 12 Gyr ago, the ticks on the tracks are given every 1 Gyr. The N U V - r colours in the models are empirically corrected by -0.7 mag (see the text).

stellar p opulation prop erties. However, we can draw some qualitative conclusions. Even in the over-simplified case of SSP-equivalent parameters, we observe a strong connection b etween average stellar p opulation prop erties of galaxies and their p osition in the (Mz , N U V - r, g - r ) parameter space. Galaxies sitting close to the b est-fitting surface exhibit moderate metallicity gradient as a function of luminosity and almost no variations in the colour­colour plane except the blue end of the sequence (N U V - r < 2.5; g - r < 0.5) where the metallicity quickly decreases. The luminosity­ metallicity relation of early-typ e galaxies known to b e resp onsible for the tilt of their optical colour­magnitude relation (Kodama & Arimoto 1997) similarly causes the tilt of the colour­colour­magnitude surface in its red part. The observed age eff tion are more imp ortant. nosities, the age smoothly 500 Myr at N U V - r ects in the colour­colour pro jecAt intermediate and high lumiincreases along the sequence from 1.5, g - r 0.3 to 13 Gyr

at the red end. For low luminosity galaxies, the oldest SSP-equivalent ages of red galaxies decrease to 10 Gyr at Mz = -18 mag b eing in accordance with the known anticorrelation of mean stellar p opulation ages with luminosities of dwarf elliptical galaxies in clusters (van Zee et al. 2004; Michielsen et al. 2008; Chilingarian et al. 2008; Chilingarian 2009; Smith et al. 2009a,b). Because of the same effect, the upp er "edge" of the broad red sequence in the (N U V - r ; Mz ) CMD is strongly tilted at Mz > -19 mag. At all luminosities, there is a notable age gradient across the sequence, that is, higher g - r colours corresp ond to older p opulations. Also, the disp ersion of age estimates (log t) increases while moving towards blue colours with values totally uncorrelated with colours at the very blue end of the sequence corresp onding to mean stellar ages t < 500 Myr. In Fig. -21.5 mag) metallicity the median 4 we present a luminosity slice (-23.0 < Mz < of the sample having the median SSP-equivalent [Fe/H] = -0.02 dex ( [Fe/H] = 0.13 dex) and redshift Z = 0.11 ( Z = 0.03) corresp onding to
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the light travel time of 1.4 Gyr. At this redshift range, the 3 arcsec wide ap ertures enclose a large fraction of light from galaxies. The qualitative b ehaviour of mean stellar p opulation ages at other luminosities is similar. We also present the evolutionary tracks for galaxies having various SFH typ es. The models were constructed from the pegase.2 (Fioc & Rocca-Volmerange 1997) models computed using the synthetic low-resolution BaSeL stellar library (Lejeune et al. 1997) for N U V - r colours, and g - r colours predicted by another family of stellar p opulation models (Vazdekis et al. 2010) computed using a large Medium-resolution Isaac Newton Telescop e library of empirical sp ectra (MILES, S´nchez-Blazquez et al. 2006). The a ´ combination of the two families of stellar p opulation models was essential, as Maraston et al. (2009) demonstrated that the offsets b etween predicted and observed colours of red galaxies in the SDSS photometric system was due to the nature of synthetic stellar sp ectra used to construct stellar p opulation models. The prop osed solution was to use models based on empirical stellar sp ectra. The MILES stellar library used to construct models presented here, has the b est coverage of the stellar atmosphere parameters compared to all other existing published sources except the high-resolution ELODIE library which, however, has too narrow wavelength coverage making the computation of g and r colours imp ossible. The N U V - r colours for all models displayed in Fig. 4 were empirically corrected by -0.7 mag. This offset is probably of the same nature as that describ ed by Maraston et al. (2009), however no models based on empirical stellar sp ectra are available in the NUV yet. The tracks shown in Fig. 4 were computed as follows. First, we computed colours and luminosities of simple stellar p opulations having [Fe/H] =0 dex using the pegase.2 code for the ages of 30, 50, 100 Myr and further till 17 Gyr with a step of 50 Myr. Second, we computed g - r colours from the MILES-based models and interp olated them to the same age grid. The resulting SSP track is shown in Fig. 4 ­ its young part is strongly offset from the observed distribution of galaxies towards red N U V - r then joining the main red sequence concentration at ages t > 5 Gyr. Third, we integrated the computed SSP luminosities in different photometric bands up-to 12 Gyr using two families of an SFH: (a) exp onentially declining star formation rate (SFR) with the three characteristic timescales texp = 1, 2, 4 Gyr; and (b) truncated SFHs: constant SFR until a given moment of time from the galaxy formation ep och (ttr = 4, 6, 10 Gyr) followed by the immediate SF cessation and passive evolution afterwards. Our galaxy evolutionary tracks are simpler than real galaxies b ecause they do not include the intrinsic metallicity evolution and other processes such as gas infall from filaments or satellites, mergers, etc., however, we can use them for some qualitative conclusions: (i) SSP models at low and intermediate ages (t < 5 Gyr) to not have any corresp onding galaxies observed that suggests that (obviously) none of the massive galaxies in our sample was formed recently and quickly. However, it matches quite well the locus of the oldest red sequence galaxies. (ii) The main colour sequence can b e explained by galaxies formed immediately after the Big Bang (ab out 12 Gyr taking into account the median redshift of our sample) and
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having various typ es of a SFH. In its main part, the slop e of the colour­colour relation and the direction of the internal extinction match each other very well. (iii) A galaxy having a constant SFR will end up near the low blue end of the sequence and can b e moved up­right along it by the internal extinction. A family of exp onentially declining SFH with different characteristic timescales form a curved sequence well corresp onding to the observed relation. The N U V - r colour evolution pace at 3 < N U V - r < 5 mag anticorrelates with the SFR characteristic timescale (see also Wyder et al. (2007) for a similar plot in the (N U V - r, u - r ) colour space without any selection on the luminosity). Models for lower metallicities well reproduce the colour­colour relation at lower luminosities suggesting the universality of exp onentially declining SFHs. We stress that this SFH typ e is not the only one that is able to explain the observed galaxy distribution in the colour­colour­magnitude space, however, it is the simplest model with the smallest numb er of free parameters compared to other alternatives (e.g. multiple starbursts, an exp onentially declined law with an additional burst). (iv) Galaxies having truncated SFHs have a very short transition phase on their way to the red sequence region lasting ab out 1 Gyr after the SF cessation when their N U V - r colour reddens radically, by 3 mag while the g - r change remains ab out 0.2 mag. The PSG locus b elow the main colour sequence is well matched by this transition phase and their rareness is consistent with a short duration of the transition. (v) Dusty star-forming galaxies ab ove the sequence at red N U V - r colours are also explained: they are moved up­right from the sequence following the direction of the extinction vector. (vi) The shap es of evolutionary tracks for galaxies with exp onentially declining SFH clearly shows that the evolution of a ma jority of galaxies goes along the relation during 6­8 Gyr. Therefore, we would expect a weak evolution of the presented colour­colour­magnitude relation shape at least up-to a redshift Z 0.9, although the distribution of galaxies on it wil l evolve. This suggests it to be a unique search instrument for distant galaxy clusters using broad band g iJ images.

4.2

Colour­colour­magnitude relations in other colour pairs

We found similar photometric relations in other colour pairs, however they are more strongly affected by observational biases and galaxy evolutionary phenomena. Colour­colour­ magnitude relations involving only optical colours are very tight b ecause of strong degeneracies b etween the colours but for the same reason have very limited astrophysical applications. For example, in the (u - r, g - r, Mr ) colour space the u - r is close to 2 â (g - r ) at all luminosities, i.e. they are linearly dep endent for most galaxies, hence virtually no information is added by the third dimension.

4.2.1

Other near-UV­optical colour combinations

The colour­colour­magnitude relation remains in place when other optical colour combinations together with the


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brightness of galaxies decreases. The computed k-corrections also have higher uncertainties in F U V as well as internal extinction effects introducing additional scatter. The red sequence region in the colour­colour pro jection extends from 2.5 mag in N U V - r to 4 mag in F U V - r due to even higher sensitivity of F U V colours to small fractions of young stars. However, all combinations involving F U V magnitudes are sensitive to the UV upturn in old early-typ e galaxies (Code 1969; Bertola et al. 1982) likely caused by the stellar evolution (Yi et al. 1997), which results in the ambiguity of the relation in the red sequence region. That is, after some "turning p oint" (7­8 Gyr), the F U V - r colour b ecomes bluer when the stars are getting older. We used NIR UKIDSS photometry to test the existence of photometric relations in the combinations involving optical-NIR colours. None of the combinations except (N U V - Y , g - Y , MY ) provides a relation having similar tightness to what we detected in the optical colours: the fitting residuals are of an order of 0.2 mag or larger. It is known (see e.g. Maraston 2005) that J H K colours are sensitive to AGB stars presenting in intermediate-age stellar p opulations, and that at certain ages (1­2 Gyr) the optical­NIR colours (g - H or r - H ) are dominated by them b eing redder than the colours of old stellar p opulations by a few tenths of a magnitude. Then, given a much larger range of e.g. N U V - H than that of r - H , this excess will b e significant, and it will strongly dep end on the SFH of a given galaxy, so that galaxies with intermediate N U V - H colours having different SFH families may have significantly different g - H or r - H colours smearing out the intermediate-to-red part of the relation except its very red end. In addition, optical-NIR colours are more sensitive to the metallicity than the optical ones. Hence, the natural relatively low metallicity spread of galaxies at a given luminosity will introduce high scatter of their g - H or r - H colours. Because of the AGB phase, for truncated SFHs, the colour evolution in the (N U V - H , g - H ) plane will also b e more complex than in the optical colour and it will strongly dep end on the truncation time.

N U V are used provided that there is enough wavelength lever in the optical colour to distinguish b etween red and blue galaxies. That is, colours like u - r , g - i, g - z , r - z but not r - i and i - z . In App endix C we provide figures similar to Fig. 1 constructed for different colour pairs. Photometric measurements in the u band have relatively p oor quality compared to g and r , therefore the residuals of the (N U V - r, u - r, Mr ) relation are ab out four times larger than those of (N U V - r, g - r, Mr ). An additional factor increasing the scatter is an imp ortant difference b etween the extinction vector direction and the blue slop e of the relation in the (N U V - r, u - r ) plane compared to (N U V - r, g - r ) increasing the scatter at N U V - r < 4 mag. The (N U V - i, g - i, Mi ) relation has fitting residuals ab out 50 p er cent higher than the (N U V - r, g - r, Mr ) one, although one would exp ect them to b e similar given very high quality of i and r band photometry and similar dep endence of these colours on the stellar p opulation evolution. We explain this by higher uncertainties of the k-correction computation in the i band connected to a broad range of the H + [Ni i] emission line strength in our galaxies. In our sample, except the very low-redshift ob jects (Z < 0.03), the observed-frame i band may b e contaminated by the H + [Ni i] emission in a galaxy which may vary a lot from ob ject to ob ject. However, during the k-correction computation we rely on some average line strength provided by the template sp ectra of Blanton & Roweis (2007). Therefore, ob jects with very weak or very strong emission lines will have their observed r - i colours redder or bluer than what is predicted by the templates and what was included in the 2D-p olynomial approximations of k-corrections. Hence, the i band k-correction values may b e biased. Even though, given a 50 p er cent larger range of g - i colours compared to g - r , the relation in this colour space may b e used in the same way as the one presented in Section 3. The (N U V - z , g - z ) colour combination provides another very good alternative to (N U V - r, g - r, Mr ), however with 80 p er cent larger scatter b ecause of lower photometric quality in the z band compared to r in the SDSS. The remarkable features of this relation are: (a) a notably higher luminosity tilt in the red sequence region and (b) low residuals in the blue part of the relation caused by even b etter coincidence of the colour change direction due to the stellar p opulation evolution and internal extinction than in the (N U V - r , g - r ) plane. The (N U V - z , r - z ) colour pair starts to suffer from very similar b ehaviour of r and z burdened by relatively high photometric errors in the z band. Therefore, although the r - z colour range is nearly the same as that in g - r , the disp ersion of the residuals is notably higher that complicates the usage of the (N U V - z , r - z , Mz ) colour­colour­ magnitude relation. 4.2.2 Combinations including Far-UV and NIR colours

4.3

Redshifts from three photometric points

GALEX F U V measurements have on average much worse quality than the N U V ones b ecause of the lower detector sensitivity and also lower fluxes for intermediate-age and old galaxies. However, we still can see similar colour­colour­ magnitude relations if we use F U V magnitudes instead of N U V although with higher residuals esp ecially in the low-luminosity part of the relation where the mean surface

This section of the pap er aims at an indep endent mathematical proof of the existence of the tight UV­optical colour­ colour­magnitude relation of galaxies. The detailed discussion of the technique and its practial applications to the existing photometric survey data will b e provided in a separate pap er. The mathematical consequence of the relation and smooth dep endencies of k-corrections on observed colours is the p ossibility of the existence of a univocal functional dep endence of a redshift on observed colours and magnitudes of galaxies. Such a dep endence, if found, would confirm the existence of the universal colour­colour­magnitude relation. Imp ortantly, it arises from a non-zero curvature of the colour­colour­magnitude surface and significantly different colour­magnitude distributions for the two colours used. In a degenerated case, e.g. (u - r, g - r, Mr ) where the two colour­magnitude distributions are very close to the linear dep endence and, therefore, galaxies reside on a surface very
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A universal galaxy photometric relation
similar to a plane, the photometric redshift determination b ecomes imp ossible. For the following computations we define the two subsamples from the main galaxy sample extended to the redshift z = 0.52 (270 016 ob jects) by using their k-corrected g - r colours, red galaxies (g - r >0.73 mag, 77 070 ob jects) and blue galaxies (g - r <0.7 mag, 167 157 ob jects) excluding a small fraction of ob jects in the "green valley". These samples were again separated into a low- (z 0.25) and high-redshift (z > 0.25) parts containing 214 770 (32 317 red + 160 310 blue ) and 56 275 (45 365 red + 7 227 blue ) galaxies corresp ondingly. Here, most of blue galaxies in the high-redshift sample come from the deep SDSS Stripe 82 imaging (Adelman-McCarthy et al. 2006) and the fraction b etween blue and red galaxies clearly demonstrates the target selection algorithm of SDSS biased towards luminous red galaxies at intermediate and high redshifts. We approximated the sp ectroscopic redshifts of galaxies in our low-redshift subsample as a three-dimensional p olynomial function of observed r , N U V - r , and g - r , and attempted to recover the photometric redshifts zphot (Fig 5). The disp ersion ((z )) of the residuals (z ) = (zphot - zsp ec )/(1+ zsp ec ) of 0.025 together with catastrophic failure rate (defined as fraction of ob jects with (z ) > 3 ((z ))) of = 0.8 p er cent is comparable to the b est available photometric redshift techniques exploiting multi-band FUVto-NIR photometry, sophisticated mathematical and statistical algorithms (Way & Srivastava 2006) and additional morphological information (Wray & Gunn 2008). For red and blue galaxies, the residuals and the catastrophic failure rates were: ((z ))red = 0.021, red = 2.2 p er cent and ((z ))blue = 0.024, blue = 0.7 p er cent. Consequently, at higher redshifts when the restframe N U V photometric band shifts to the optical domain, it should b e p ossible to determine photometric redshifts precisely using u - r - z , g - r - z , or g - i - Y broadband photometry. We tested this hyp othesis with our high-redshift subsample by fitting their redshifts as a function of observed SDSS (z , u - z , r - z ) and obtained the residuals having a disp ersion ((z )) = 0.036 and the rate of catastrophic failures = 1.1 p er cent. For red and blue galaxies, the residuals and the catastrophic failure rates were: ((z ))red = 0.034, red = 1.0 p er cent and ((z ))blue = 0.047, blue = 1.2 p er cent. These relatively large errors are mostly due to the very p oor quality of u-band Petrosian magnitudes having typical uncertainties of an order of 0.3 mag for highredshift galaxies. We notice here, that if one uses model magnitudes instead of Petrosian ones, the relation b ecomes much tighter for the red subsample ( ((z ))red = 0.027), however it nearly disapp ears for blue galaxies whose u-band model magnitudes do not corresp ond to their real photometric prop erties b ecause of light distributions b eing very far from regular exp onential or de Vaucouleurs profiles. The demonstrated p ossibility of the precise photometric redshift computation for b oth, red and blue galaxies with a small fraction of outliers from three photometric p oints involving a near-UV and optical colours proves the existence of the tight unversal colour­colour­magnitude relation for normal galaxies of all typ es, not only red sequence ob jects. We compare these metrics to the existing photometric redshifts techniques. Using the "le phare" photometric
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Figure 5. Recovery of galaxy distances from three photometric points. Density histograms of photometric redshift (Zphot ) determination by fitting their spectroscopic redshifts (Zspec ) as a polynomial function of three parameters: an observed magnitude and two observed colours corrected for the Galactic extinction. The blue plot displays a sample of 214 000 low-redshift galaxies (0.03 < Z < 0.25) with NUV, g , and r photometric measurements from GALEX and SDSS. The green plot is sample of 56 000 intermediate-redshift galaxies (0.2 < Z < 0.52) whose redshifts Zphot were determined using u, r , and z SDSS DR7 photometric data. Red and blue solid lines denote median of Zphot and the standard deviations of (Zphot - Zspec )/(1 + Zspec ) residuals for blue and red galaxy subsamples respectively.

redshift code combined with a template optimisation procedure and the application of a Bayesian approach, based on the sample of galaxies with 9 individual photometric measurements of a quality similar to ours, Ilb ert et al. (2006) find the disp ersion ((z )) to b e 0.025 and = 1.9 p er cent (though we stress that the authors defined the catastrophic failure limit at the fixed level of 0.15 which corresp onded to 6 in their statistics). Mobasher et al. (2007) analyse the p erformance of different photometric redshift codes on a dataset that comprises 16 photometric p oints for every SED in question. The b est result achieved in their study with own method reaches the disp ersion of residuals as low as ((z )) = 0.033 and = 2.2 p er cent. The SDSS database provides several photometric redshift estimates obtained as describ ed in Adelman-McCarthy et al. (2007). We have extracted three of them: (1) those obtained from the comparison of the observed colors of galaxies to a semi-empirical reference set (hereafter photoz) from the photoz table, and (2) neural network estimators derived from galaxy magnitudes ("D1") and (3) colours ("CC2") from the photoz2 table. In general, they p erform quite well for red galaxies in our high-redshift subsample ( ((z ))red,photoz = 0.020 and red,photoz = 1.7 p er cent, ((z ))red,D1 = 0.020 and red,D1 = 1.6 p er cent, ((z ))red,CC2 = 0.023 and red,CC2 = 1.7 p er cent). However, similarly to our approach,


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mated by a low-order p olynomial surface with the residuals of 0.03­0.07 mag in the entire covered luminosity (-23.5 < Mr < -15.5 mag) and colour (0 < N U V - r < 7.5 mag) ranges. We have demonstrated that there is a strong correlation b etween the N U V - r colour and the galaxy morphology while for the optical g - r colour this correlation is much weaker. We identified several classes of the outliers constituting ab out 2 p er cent of the total galaxy p opulation and explained their nature, the most imp ortant b eing rare compact elliptical and p oststarburst E+A galaxies. We have p erformed stellar p opulation modelling and shown that the relation can b e explained by galaxies having constant and exp onentially declining SFHs, while truncated SFHs with very rapid colour evolution after the SF cessation explain the prop erties of E+A outliers. The most imp ortant conclusion is the predicted weak dep endence of the colour­ colour­magnitude relation shap e on a redshift up-to Z 0.9 that will allow to use it for search of distant galaxy clusters in 3-colour broad band images. The existence of such a tight photometric relation suggests the p ossibility of the high-precision photometric redshift estimates using only three photometric p oints that we have also illustrated. Our empirical photometric redshift technique using a minimalistic input dataset has a quality comparable to or b etter than most published Zphot techniques, while it is much simpler to implement. The detailed astrophysical interpretation of the describ ed photometric relation still has to b e done. However, it can b e used already as a p owerful galaxy classification and selection instrument based only on their integrated photometry as well as a tool to search for representatives of rare galaxy typ es in photometric galaxy samples.

the quality is worse for blue galaxies: ((z ))blue,photoz = 0.042 and blue,photoz = 3.4 p er cent, ((z ))blue,D1 = 0.039 and blue,D1 = 2.0 p er cent, ((z ))blue,CC2 = 0.044 and CC2 = 1.7 p er cent. One has to keep in mind that (1) these techinques use a lot of additional information (e.g. morphology, size, etc.) and (2) all the galaxies in the sp ectroscopic SDSS sample in fact constitute a training sample of these methods, so one has to check those numb ers against massive third-party redshift surveys. We also note that since the redshift determination by three photometric p oints is a mere mathematical consequence from the colour-colour-magnitude relation, photometric redshift outliers are themselves ob jects that fall aside from the relation, namely PSGs, dusty starbursts, and AGN. To summarise, compared to existing photometric redshift techniques, presented method requires a factor of 3 to 5 more modest investment in observing time (due to the fact that individual galaxy measurements do not need to b e made in as many photometric bands), b eing able to provide redshifts for large samples of galaxies at the same or b etter level of accuracy. Moreover, prop osed p olynomial evaluation is significantly simpler from the methodological p oint of view than 2 minimization with Bayesian priors used in mainstream photometric redshift codes. There are two main disadvantages of our approach: (1) it is not precise for non-typical galaxies, i.e. outliers from the colour­colour­magnitude relation; (2) it works only for those regions of the parameter space, that are well sampled with sp ectral redshift measurements. Latter means e. g. that it is p ossible to go b eyond the SDSS sp ectral sample magnitude limit retaining the declared precision of our method if one uses an external source of sp ectral redshifts to calibrate the functional relation to work with the SDSS photometry at fainter magnitudes. But without such a calibration, photometric redshifts estimates for faint galaxies will b e wrong. Conceptually, the presented multi-dimensional p olynomial fit resembles the training of artificial neural networks sometimes used for the photometric redshift determination (e.g. D'Abrusco et al. 2007), though the underlying machinery is different. In b oth cases, there is a non-linear transformation (a 3D-p olynomial function in our case or consequent multi-level sigmoid transformations in case of neural networks) of some input measurements into the output redshift estimate. And the coefficients of a transformation are tuned ("trained") in a way to work as good as p ossible for the reference ("training") dataset. Hence, b oth methods unlike 2 template fitting family are insensitive to systematic errors of the input data. That is, the "templates" are constructed from the dataset itself and are tolerant to its problems by construction.

ACKNOWLEDGMENTS We acknowledge th e ADASS conference series (http://www.adass.org/ ), b ecause this result emerged while preparing the tutorial for the ADASS-xx meeting. This work would have b een imp ossible without technologies develop ed by the International Virtual Observatory Alliance (http://www.ivoa.net/ ) and tools and services maintained by the participating national VO pro jects: Euro-VO, VAO, Astrogrid. This research has made use of TOPCAT, develop ed by Mark Taylor at the University of Bristol; Aladin develop ed by the Centre de Donn´es Astronomiques de e Strasb ourg (CDS); the "exploresdss" script by G. Mamon (IAP); the VizieR catalogue access tool (CDS). Funding for the SDSS and SDSS -I I has b een provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/ . GALEX (Galaxy Evolution Explorer) is a NASA Small Explorer, launched in April 2003.We gratefully acknowledge NASA's supp ort for construction, op eration, and science analysis for the GALEX mission, develop ed in coop eration with the Centre National d'Etudes Spatiales of France and the Korean Ministry of Science and Technology. Authors acc 2010 RAS, MNRAS 000, 1­13

5

SUMMARY

We presented the universal very tight colour­colour­ magnitude relation in optical and near-UV filters followed by the vast ma jority of non-active galaxies of all morphological typ es covering at least 8 magnitudes in luminosity from the sample including 225 000 low-redshift (Z < 0.27) galaxies observed by SDSS and GALEX surveys. A sp ecial case is the connection of the optical g - r colour to the N U V - r colour and Mr luminosity which we approxi-


A universal galaxy photometric relation
knowledge the Oversun-Scalaxy (http://www.scalaxy.ru/ ) cloud computing provider for resources used to p erform a part of this study. Sp ecial thanks to F. Comb es, A. Graham, and R. Ibata for useful discussions and suggestions.

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APPENDIX A: VALIDATION OF THE RESULT Two factors may in principle lead to the spurious creation of the colour­colour­magnitude relation presented in this work: (a) sample selection biased toward sp ecific colours corresp onding to the describ ed surface, (b) serious faults in the k-correction computation artificially bringing most galaxies to that relation. As far as we apply no selection to the SDSS DR7


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sp ectroscopic sample of galaxies based on their morphology, colours, sizes etc., and GALEX is a full-sky survey providing photometric information for all detected ob jects, the only source of selection effects may b e the sp ectroscopic target selection of SDSS. The corresp onding algorithms are exhaustively describ ed (Eisenstein et al. 2001; Strauss et al. 2002), and we find no evidences for them to introduce any biases on galaxy selection that may lead to the creation of "empty" regions in the colour­colour­ magnitude parameter space. This is also well illustrated by the broad and very well filled colour distribution of SDSS galaxies (Strateva et al. 2001; Blanton et al. 2003a; Baldry et al. 2004) and studies of galaxy luminosity functions based on SDSS (Blanton et al. 2003b). In order to test the existence of p otential problems in the k-correction computations, we have conducted two sp ecific tests. For this, we selected two sub-samples of galaxies in narrow redshift ranges, z1 : 0.03 < z1 < 0.05 (24 319 galaxies) and z2 : 0.08 < z2 < 0.10 (37 303 galaxies). We chose relatively low-redshift samples b ecause the SDSS targeting algorithm has a limiting magnitude r = 17.77 mag in a 3 arcsec ap erture so that galaxies at higher redshifts do not sample well the luminosity axis of the parameter space. The first test was to compare the colour­colour­ magnitude distributions of galaxies in the two sub-samples. In case of k-correction computation problems, one would exp ect systematic differences b etween the two sub-samples, which we have not detected (see Fig. A1a for an "edge-on" view of the relation). The second test was to entirely disable the kcorrection computation for these two sub-samples. Because k-corrections do not change significantly within the narrow redshift sub-samples, but do change b etween the subsamples, one would exp ect to get two tight colour­colour­ magnitude sequences qualitatively resembling the relation for k-corrected magnitudes, but with some quantitative differences such as the blue cloud slop es and red sequence p ositions. We obtained the result exactly as predicted by this intuitive assumption. The edge-on views of the non kcorrected relations are presented in Fig. A1b. However, the strongest argument supp orting the existence of a universal colour­colour­magnitude relation in UV­optical colours is the p ossibility of computing photometric redshifts using only three observed colours as demonstrated in the Discussion section.

Figure A1. Tests with the two sub-samples of galaxies in narrow redshift ranges. The top panel demonstrates the number density plots for galaxies in two narrow redshift ranges in the pro jection shown in green and blue, corresponding to the edge-on view of the colour­colour­magnitude relation. The bottom panel is the same as the top one, but without applying k -corrections to the measurements.

APPENDIX B: FITTING SURFACES INTO STRONGLY NON-UNIFORM THREE-DIMENSIONAL SCATTERED DATASETS A surface fitting procedure is an essential mathematical comp onent required to obtain results presented in the pap er. The observational photometric datasets for galaxies from wide-field survey have strongly non-uniform distribution in the colour­colour­magnitude space due to the sup erp osition of the complex distribution of galaxies connected to their physics and various selection effects and observational biases. Using visual insp ection, we revealed a distribution of p oints in the colour­colour­magnitude space close to a

smooth surface, but applying standard 2 -based linear surface fitting techniques did not yield the results of a reasonable quality b ecause: (a) the density of p oints in the NUVcolour­magnitude plane varies by several orders of magnitude while the individual measurements have comparable quality; (b) distribution of p oints around the surface is sometimes significantly asymmetric and non-Gaussian. The former prop erty of the distribution leads to the fact that the scarcely p opulated regions can deviate significantly from the surface without notable change of the goodness-of-fit as the b est-fitting surface tries to minimize the deviation in the densest regions of the parameter space. The latter prop erty leads to the biased fitting results as the 2 technique assumes the Gaussian distribution. The -sigma clipping technique will not solve the problem here b ecause we are dealing with a large numb er of p oints deviating from the symmetric distribution and not with individual outliers.
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To tackle these issues in a simple way, we decided to use a two­step technique for the surface fitting. First, we defined a fine grid (cell size of ab out 0.25â0.25 mag) in the NUV-colour­magnitude plane and in every bin computed median values of the optical colour b eing fitted. This allowed us to pick up the maxima of the (e.g. g - r ) colour distributions in every bin and not the mean values, that was critical in order to account for the asymmetrical distribution of p oints around the surface. Then we filtered out the values where the 2D-histogram counts in a given bin were b elow some threshold (usually, 5 or 7 galaxies). At the second step, we fitted a low-order p olynomial surface into these median values using a standard routine fitting linearly the p olynomial coefficients and assigning equal weights to all p oints remained after the filtering at the first stage. This way we took into account the strongly non-uniform distribution of galaxies in the NUV-colour­magnitude plane. This two­step approach resulted in almost flat distribution of residuals displayed in Fig. 1 and in the next App endix.

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APPENDIX C: COLOUR­COLOUR­MAGNITUDE RELATIONS IN DIFFERENT COLOUR COMBINATIONS In Fig. C1­C4 we show the colour­colour pro jections of the colour­colour­magnitude relation in different near-UV­ optical colours describ ed in Section 4.2. All the fitting results including the coefficients of the b est-fitting surfaces, fitting residuals in colour­magnitude bins and other essential information for the usage of the relations are provided in the electronic form4 .

Figure C1. The same as Fig. 1 but for the (N U V - r , u - r , Mr ) space.

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http://specphot.sai.msu.ru/galaxies/

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Figure C2. The same as Fig. 1 but for the (N U V - i, g - i, Mi ) space.

Figure C3. The same as Fig. 1 but for the (N U V - g , g - z , Mz ) space.

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Figure C4. The same as Fig. 1 but for the (N U V - z , r - z , Mz ) space.

c 2010 RAS, MNRAS 000, 1­13