Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://kcor.sai.msu.ru/arXiv1002.2360.pdf Äàòà èçìåíåíèÿ: Fri Feb 12 03:23:23 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:23:38 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: transit

Mon. Not. R. Astron. Soc. 000, 1­12 (2009)

Printed 11 February 2010

A (MN L TEX style file v2.2)

Analytical approximations of K -corrections in optical and near-infrared bands
I1gor V. Chilingarian1
2

,2,3

, Anne-Laure Melchior

1,4

and Ivan Yu. Zolotukhin2

Observatoire de Paris, LERMA, CNRS UMR 8112, 61 Av. de l'Observatoire, 75014 Paris, France Sternberg Astronomical Institute, Moscow State University, 13 Universitetskij prospect, 119992, Moscow, Russia 3 Centre de donn´es astronomiques de Strasbourg, Observatoire astronomique de Strasbourg, UMR 7550, e Universit´ de Strasbourg / CNRS, 11 rue de l'Universit´, 67000 Strasbourg, France e e 4 Universit´ Pierre et Marie Curie - Paris 6, 4 Place Jussieu, 75252 Paris Cedex 5, France e

Accepted 2010 February 10. Received 2010 February 01; in original form 2009 August 14

ABSTRACT

To compare photometric properties of galaxies at different redshifts, the fluxes need to be corrected for the changes of effective rest-frame wavelengths of filter bandpasses, called K -corrections. Usual approaches to compute them are based on the template fitting of observed spectral energy distributions (SED) and, thus, require multi-colour photometry. Here, we demonstrate that, in cases of widely used optical and nearinfrared filters, K -corrections can be precisely approximated as two-dimensional loworder polynomials of only two parameters: redshift and one observed colour. With this minimalist approach, we present the polynomial fitting functions for K -corrections in SDSS ug riz , UKIRT WFCAM Y J H K , Johnson-Cousins U B V Rc Ic , and 2MASS J H Ks bands for galaxies at redshifts Z < 0.5 based on empirically-computed values obtained by fitting combined optical-NIR SEDs of a set of 105 galaxies constructed from SDSS DR7 and UKIDSS DR5 photometry using the Virtual Observatory. For luminous red galaxies we provide K -corrections as functions of their redshifts only. In two filters, g and r, we validate our solutions by computing K -corrections directly from SDSS DR7 spectra. We also present a K -corrections calculator, a web-based service for computing K -corrections on-line. Key words: galaxies: (classification, colours, luminosities, masses, radii, etc.) ­ galaxies: photometry ­ galaxies: evolution ­ galaxies: stellar content ­ galaxies: fundamental parameters

1

INTRODUCTION

Extragalactic studies usually require comparison b etween photometric data for different galaxy samples, in particular, comparing measurements obtained for distant galaxies to the local Universe, where prop erties of galaxies are studied in a much greater detail. Generally, any differences in observable parameters arise from: (1) astrophysical properties of galaxies and (2) observational biases. The former ones include the galaxy evolution effects: due to the light travel time, we see distant galaxies as they were looking several Gyr ago, so the evolution during the last p eriod of their lifetime simply cannot b e observed. On the other hand, observational biases arise from the process how the observations are carried out and, therefore, change drastically from one facility to another, even assuming the data are p erfectly reduced and calibrated. These include effects of ap erture,

spatial resolution, and a family of effects connected to the photometric bandpasses. Photometric data often originate from different observational studies exploiting different photometric systems and, thus require colour transformations to b e applied (e.g. Fukugita et al. 1995). But even if the data are obtained in the course of one given pro ject, a galaxy sample may contain ob jects at different redshifts. In a broad wavelength range, from ultra-violet to nearinfrared, the sp ectral energy distributions (SED) of nonactive galaxies are mostly determined by their stellar p opulation prop erties, i.e. age and chemical comp osition, and effects of internal dust attenuation increasing dramatically at short wavelengths (Calzetti et al. 1994; Fitzpatrick 1999). Stellar p opulation SEDs are very far from flat distributions and exhibit prominent features (e.g. Fioc & RoccaVolmerange 1997; Bruzual & Charlot 2003). At the same time, redshifting the galaxy sp ectrum is equivalent to shifting the corresp onding filter transmission curve. This ex-



E-mail: Igor.Chilingarian@astro.unistra.fr; chil@sai.msu.ru

c 2009 RAS


2

I. Chilingarian et al.
cross-matching of the SDSS DR7 sp ectroscopic sample in strip es 9 to 16 with the Large Area Survey catalogue of UKIDSS DR5. We selected 190,275 galaxies having sp ectroscopic redshifts in a range 0.03 < Z < 0.6 provided by SDSS DR7 using the SDSS CASJobs Service1 . The spatial cross-identification with UKIDSS DR5 with a search radius of 3 arcsec selecting the b est p ositional matches in case of multiple ob jects within this radii resulted in 170,533 ob jects, 87,161 of which were detected in all four UKIDSS Large Area Survey photometric bands (Y , J , H , and K ). In order to compute K -corrections, the photometric measurements from the two data sources have to b e homogeneous. We use SDSS fibre magnitudes corresp onding to d = 3 arcsec circular ap ertures (fiberMags ), and computed the corresp onding 3 arcsec ap erture magnitudes for UKIDSS ob jects by linearly interp olating b etween the values provided for three ap ertures (2.0, 2.8, and 5.7 arcsec) applying zero-p oint corrections (Hewett et al. 2006) for converting UKIDSS magnitudes from the Vega into the AB system. The lower redshift limit, Z = 0.03, is selected in order to minimise the ap erture effects: at a distance of 120 Mp c, corresp onding to this redshift, the 3 arcsec ap erture encloses ab out 1.75 kp c, i.e. significant part of the bulge even for giant galaxies, thus stellar p opulations in galactic nuclei would not dominate the light. Beyond the selected upp er redshift limit, Z = 0.6, the fraction of normal galaxies in SDSS significantly decreases, b ecause of the magnitude-limited selection of SDSS sp ectroscopic targets, and at the same time the quality of absorption-line sp ectra b ecomes quite p oor. We use magnitudes in 3 arcsec ap ertures and not the Petrosian magnitudes to b e able to compare them directly with the SDSS DR7 sp ectra obtained within the same ap ertures. The median 3 arcsec ap erture magnitude uncertainties are 0.01 mag and b etter for g , r , i, Y , J , H , and K , 0.017 mag for z , and 0.07 mag for the u band resp ectively. All magnitudes are corrected for the foreground Galactic extinction according to Schlegel et al. (1998). There are imp ortant systematic offsets of unknown origin b etween SDSS DR7 fibre magnitudes and UKIDSS DR5 photometry in 3 arcsec ap ertures. We fit 3rd order p olynomial using 5 colours starting from r - i and redder except z - Y and compute the offset b etween the exp ected z - Y value from the b est-fitting p olynomial and the observed one. The offset has a mean value of 0.22 mag and a standard deviation of 0.13 mag indep endent from other parameters (e.g. observed colours and a redshift). We therefore subtract it from all UKIDSS magnitudes. This effect is illustrated in Fig. 1. In the top panel, we display the combined SDSS-UKIDSS SED of some galaxy and its b est-matching pegase.2 template obtained by the K -correction determination procedure describ ed b elow. The middle and b ottom panels show the residuals b etween the observed SED and its model, and colours in consequent sp ectral bands with the b est-fitting 3rd order p olynomial. From the middle panel, it is clear that the NIR part of the SED is offset by a constant value from the optical part, which is evident as the measured z - Y colour strongly deviates from the fitting p olynomial.

plains the difference in fluxes in the same bandpass for two hyp othetical galaxies having exactly identical SEDs but b eing at different redshifts. Historically, this difference is called K -correction (Oke & Sandage 1968). The K -correction formalism is presented in detail and thoroughly discussed in Hogg et al. (2002); Blanton & Roweis (2007). Nowadays, in the era of large wide-area photometric and sp ectroscopic extragalactic surveys, the precise, fast and simple computation of K -corrections has b ecome a crucial p oint for the successful astrophysical interpretation of data. Several approaches were presented in the literature (Fukugita et al. 1995; Mannucci et al. 2001; Blanton & Roweis 2007; Roche et al. 2009). Blanton & Roweis (2007) provide a software package to compute K -corrections for datasets in any photometric system. However, since their method is based on the SED fitting technique, the results critically dep end on the availability of multi-colour photometric data. Fukugita et al. (1995) and Mannucci et al. (2001) provide only qualitative dep endence of K -corrections on redshift and galaxy morphological typ e; the latter is very difficult to assess in an automatic way and these methods therefore require availability of original galaxy images in addition to photometric measurements. The aim of our study is to explore the parameter space of typical observed galaxy prop erties and to provide simple and precise analytical approximations of K -corrections in widely used optical and NIR photometric bandpasses, based on the minimal set of observables. To achieve this goal, we exploit a large homogeneous database of optical-to-NIR galaxy SEDs compiled from modern wide-area photometric surveys. In the next section, we describ e our galaxy sample, details on the computation of K -corrections using pegase.2 stellar p opulation models (Fioc & Rocca-Volmerange 1997) and comparison of obtained values with those computed using the kcorrect code (Blanton & Roweis 2007). In Section 3, we describ e analytical approximations, and the validation of our results using sp ectral-based K -corrections. In Section 4, we compare our results with the literature and briefly discuss some astrophysical interpretation of our technique. App endices provide tables with coefficients of b estfitting p olynomials and present a "K-corrections calculator" service.

2 2.1

EMPIRICAL COMPUTATION OF K -CORRECTIONS Galaxy sample

We compute K -corrections using a large sample of opticalto-NIR SEDs of nearby galaxies constructed using Virtual Observatory technologies to retrieve and combine photometric measurements from Sloan Digital Sky Survey Data Release 7 (SDSS DR7, Abaza jian et al. 2009) and the UKIRT Infrared Deep Survey Data Release 5 (UKIDSS DR5, Lawrence et al. 2007). Comprehensive description of this multi-colour photometric catalogue will b e provided in a separate pap er (Chilingarian et al. in prep), here we give a brief summary essential for understanding the empirical K -correction computations. We constructed a sample of galaxies excluding broadline active galactic nuclei (AGN) by p erforming the spatial

1

http://cas.sdss.org/CasJobs c 2009 RAS, MNRAS 000, 1­12


Analytical approximations of K -corrections

3

Figure 2. Comparison of empirically calculated K -corrections obtained by fitting the photometric data using pegase.2 SSP models (kP 2 ) and by the kcorrect (kBR07 ) code. Each panel displays the difference between the two approaches as a function of redshift. Solid lines denote the median differences and their standard deviations are shown with dotted lines.

Figure 1. Top panel presents the optical-to-NIR SED of SDSS J155023.03-000023.8 in a 3-arcsec aperture as provided in the catalogue (black diamonds), its NIR part empirically corrected (blue diamonds), its best-matching pegase.2 template used for K -correction computation (solid red line). Fitting residuals are shown in middle panel (same symbols). Bottom panel displays colours as a function of wavelength (black crosses), the best-fitting 3rd order polynomial for 5 colours (r - i, i - z , Y - J , J - H , and H - K ) is shown as a solid green line; the corrected value of z - Y is denoted by the blue cross.

2.2

Computation of K -corrections

We used two approaches to compute K -corrections: (1) the kcorrect software package by Blanton & Roweis (2007) and (2) pegase.2-based computations describ ed hereafter. The latter technique allows us to roughly estimate mean stellar p opulation prop erties as well as internal extinction in galaxies. The analytical fitting is p erformed for b oth methods indep endently. We first ran the kcorrect software package to compute the K -corrections for our sample of galaxies. This package is based on a mathematical algorithm, namely non-negative matrix factorisation, which creates model-based template sets. The initial set of hundreds of templates is reduced to a basis of five, which in principle can b e used to interpret the galaxy SED in terms of stellar p opulations. The linear combination of these templates is then fitted into the set of broadband fluxes available for each galaxy to derive the K -corrections in all bands. Second, we computed a grid of Simple Stellar Populations (SSP) using the pegase.2 evolutionary synthesis code (Fioc & Rocca-Volmerange 1997) for a set of 75 ages nearly logarithmically spaced b etween 25 Myr and 16.5 Gyr and 10
c 2009 RAS, MNRAS 000, 1­12

metallicities b etween -2.5 < [Fe/H] < +1.0 dex. Such a grid was computed separately for redshifts b etween 0 < Z < 0.6 with a step of 0.05. We apply the Fitzpatrick (1999) extinction law for each of 750 SSPs at each redshift varying the AV b etween 0 and 2.25 mag with a step of 0.15 mag ending up with 11250 template SEDs p er redshift. In order to compute K -corrections for a given galaxy, we p erform a linear interp olation of the SSP grid to its redshift, then pick up the b est-matching template SED in terms of 2 normalising b oth data and templates by the mean fluxes in all filters. Since our photometric uncertainties are quite small, this approach would not result in significant biases. Once the b est-matching template has b een found, the K -corrections in all photometric bands are computed as Kf (Z ) = -2.5 log(F (0)/F (Z )), where F (Z ) and F (0) are fluxes in a given filter at redshift Z and in the restframe. We provide an example of an observed SED and its b estmatching template in the upp er panel of Fig 1. Since we can calculate model fluxes in any photometric band, we also used the same technique to compute K -corrections in Johnson-Cousins U B V Rc Ic and 2MASS J H Ks bands. We compare the values computed in this fashion with the value derived by the kcorrect code from the same photometric dataset. The comparison for all 9 SDSS-UKIDSS bands is provided in Fig 2. In general, the results obtained by using the two approaches in ug r iz Y J H K bands are quite similar. However, in certain sp ectral bands, some statistically significant differences are evident. The worst situation is observed in the SDSS u band mainly for two reasons: relatively p oor quality of the u band photometry esp ecially for ob jects at higher redshifts and very high sensitivity of UV colours to even low mass fractions of recently formed stars, which affect fluxes at longer wavelengths much weaker. Therefore, the SSP fitting


4

I. Chilingarian et al.
Since b oth K -correction computation techniques used in our study are based on the stellar p opulation models, and our analytical approximations exploit simple p olynomial fitting technique without any clipping of outliers, we have to exclude strong active galactic nuclei (AGN) and quasars, as well as SDSS targeting artifacts (e.g. aircrafts, satellites, minor planets) and ob jects with wrong redshift determinations, which may affect our b est-fitting solutions. We have fitted all 190,275 SDSS DR7 sp ectra using the nbursts full sp ectral fitting technique (Chilingarian et al. 2007b,a) in the restframe wavelength range b etween 3900 and 6750°, A and selected for our further analysis only 164,108 ob jects having reduced 2 /DOF < 0.9 (median 2 /DOF = 0.67, which is smaller than unity b ecause of slight sp ectral oversampling of SDSS data). The interpretation of the sp ectral fitting results will b e provided together with the presentation of a sp ectrophotometric catalogue in Chilingarian et al. (in prep). We computed and fitted K -corrections in this fashion for a sample of 164,108 SDSS DR7 galaxies with well-fitted sp ectra using only SDSS 5-bands photometry, and to the merged SDSS­UKIDSS sample containing photometric information in all 9 bands for 74,254 galaxies. While the results from the two approaches remain nearly statistically identical in u, g , and r bands, the difference b ecomes significant in i and esp ecially in z . Two distinct sequences b ecome clear in the redshift vs K -correction plots in i and z in case of the 5-bands based computation. Only one of the two sequences remains in each case if full 9-band SEDs are used, suggesting that the second sequence is created by wrong stellar p opulation templates picked up from the template grid. This clearly demonstrates the imp ortance of the NIR part to compute K -corrections if using a fully empirical approach. Consequently, r.m.s of the p olynomial surface fitting residuals in case of 9-bands based K -corrections in the z band is four times smaller compared to the 5 bands. In Fig 3­4, we display the analytical approximations of K -corrections in 9 bands for the values computed using pegase.2-based matching and the kcorrect code. Upp er panels demonstrate computed values as a function of redshift and an observed colour, colour-coded according to the scale bars presented in the plots. The lower panels display the mean residuals and their r.m.s. as a function of redshift. Since we use a fixed grid of pegase.2 templates and do not p erform interp olation on the age and metallicity axes, the individual template age sequences b ecome visible at high redshifts range in some filters (e.g. Y ). However, the computational errors due to this discretisation do not exceed 0.03 mag. Black lines in the plots denote the b ehaviour of K -corrections for galaxies having fixed restframe (i.e. K -corrected) g - r colours. Solid lines are for 0.73 < g - r < 0.81 mag, dashed for 0.58 < g - r < 0.70 mag, dot-dashed for 0.4 < g - r < 0.6 mag, and triple-dot-dashed for g - r < 0.15 mag, roughly corresp onding to luminous red galaxies (LRG), early-typ e spirals (S-early), late-typ e spirals (S-late), and actively star-forming galaxies (SF). These lines are constructed by connecting median values within 0.03-wide redshift bins in the colour bins given ab ove. We notice that the u-band K -corrections computed with the kcorrect code at higher redshifts turn to b e higher than exp ected for LRGs defined by g - r restframe colours, which is probably indicative of a template mismatch.
c 2009 RAS, MNRAS 000, 1­12

approach presumably should not work well. At the same time, the linear combination of 5 templates used by the kcorrect package may also produce significant biases due to age and/or metallicity mismatch b etween the templates and real galaxies. Finally, we provide the K -corrections for the u band, emphasising that since no indep endent verification is p ossible in our case, one has to use these results with caution. The same statement applies to K band K corrections, although the results of the two approaches well match each other, b ecause the computation relies on the extrap olation of galaxy SED in the NIR part, where stellar p opulation models and, corresp ondingly, the template sp ectra are of much lower quality than in the optical wavelength domain. Computations in the r , i, and J bands agree remarkably well except the high-redshift (Z > 0.5) end of the J band. There are some systematic discrepancies b etween the two techniques in the g , z , Y , and H bands. They may originate from the fact that pegase.2 SSPs are built using the theoretical stellar library, shown to introduce colour differences b etween synthetic sp ectra and observed ones at least in the SDSS photometric system (Maraston et al. 2009). We will analyse the g and r band results b elow using direct sp ectral-based K -corrections, while for the remaining bandpasses no indep endent test can b e p erformed since no large samples of galaxy sp ectra are available in those wavelength domains. However, we note that the discrepancies are an order of 0.05 mag, hence, b oth approaches may b e used in the photometric studies. Therefore, we will proceed with the rest of our analysis using b oth techniques of the K correction computation, addressing them as BR07 and SSP for the kcorrect and pegase.2 SSP-based approaches resp ectively.

3 3.1

RESULTS Analytical approximations

We observe a large scatter of K -corrections as functions of redshift reaching 2 mag in all SDSS-UKIDSS bands except H and K . However, exploring the data with the topcat software2 , we found that adding just one observed colour as a second parameter and approximating the K -correction as a surface in the three-dimensional space, significantly reduces the residual scatter bringing it to the order of K -correction computation uncertainties. We fit K -correction values in every filter q as a p olynomial surface of a form: Kq (Z, m
f1

- m f2 ) =

NZ Nc XX x= 0 y = 0

a

x ,y

Z x (m

f1

- m f2 ) y ,

(1)

where ax,y are p olynomial coefficients, Z is a sp ectroscopic redshift, mf1 and mf2 are observed magnitudes in filters f1 and f2 chosen for every filter q , NZ and Nc are empirically selected p olynomial p owers in the redshift and colour dimensions resp ectively. Given that K -corrections are zero by definition at Z = 0, no constant term is needed and all a0,y = 0.

2

http://www.star.bris.ac.uk/mbt/topcat/


Analytical approximations of K -corrections

5

Figure 3. K -corrections in 9 photometric SDSS-UKIDSS bands computed using pegase.2 SSP matching and the residuals of the analytical fitting. Upper panel for each filter presents computed K -corrections vs redshift with a colour-coded observed colour used to perform the fitting. Sequences of galaxies with constant restframe g - r colours roughly corresponding to different morphological types are overplotted (see the text for details). Lower panels display the residual between the analytical approximation and measured values (bold black line) and r.m.s. of the fitting residuals (dotted lines).

The 4 morphological typ es are well separated in the u, g , and r bands; only very blue ob jects exhibit a distinguished b ehaviour in i, while in NIR bands the K -corrections b ecome nearly indep endent of a restframe colour of a galaxy (or its morphological typ e). Some tables containing the coefficients of p olynomial approximations are provided in App endix A. Additional tables for K -correction approximations using different observed colours are provided on the "K-corrections calculator" web-site describ ed in App endix B. Classical Johnson-Cousins photometric system is still widely used for extragalactic research, as well as NIR bands of the 2MASS survey (Skrutskie et al. 2006), therefore computation and analytical approximation of K -corrections in
c 2009 RAS, MNRAS 000, 1­12

these bands are of a great practical imp ortance. However, we do not have photometric measurements for galaxies in our sample obtained in these bands. We used the photometric transformations defined in Jordi et al. (2006) for stars and those provided on the web-pages of SDSS in order to convert available SDSS ug r iz magnitudes into Johnson-Cousins U B V Rc Ic , and UKIDSS Y J H K to 2MASS J H Ks transformations presented in Hewett et al. (2006). Then, we used these magnitudes to fit K -corrections in the U B V Rc Ic and 2MASS J H Ks bands as functions of "computed" observed colours defined in the same photometric system. The coefficients of these p olynomial approximations are available at the "K-corrections calculator" web-site. We note that the approximations for the Johnson­Cousins and 2MASS filters


6

I. Chilingarian et al.

Figure 4. The same as Fig 3, but the K -corrections are computed using the kcorrect code.

are provided as functions of colours in the Vega system, whereas for SDSS and UKIRT WFCAM bands the colours are expressed in AB magnitudes. 3.2 Validation using SDSS DR7 spectra

We fetched optical SDSS sp ectra in the wavelength range b etween 3800 and 9200° for all galaxies from our sample, A therefore we were able to p erform indep endent direct verification of the analytical approximations of K -corrections presented ab ove. The available sp ectral coverage allows us to compute fluxes directly from the sp ectra using b oth real and redshifted filter transmission curves, i.e. to obtain real restframe fluxes, for g , r , and i filters. The highest redshifts, where reliable computation is still p ossible for the r and i bands are 0.28 and 0.08 resp ectively due to the upp er wavelength limit of the SDSS sp ectral coverage. We use a similar technique to Roche et al. (2009) for the computa-

tion of sp ectral-based K -corrections, with two main differences: (1) we redshift the filter transmission curves instead of blueshifting the sp ectra in order to prevent interp olation of fluxes and smoothing defects in the sp ectra; (2) in case of missing data within the wavelength range, we interp olate fluxes linearly or extrap olate them using constant level in F if the red tail of the redshifted filter transmission curve goes b eyond 9200°. The maximal adopted truncation of the A filter transmission curve was at a level of 1 p er cent of the maximal transmission. In Fig 5, we present the differences b etween sp ectralbased K -corrections in g and r bands and the analytical approximations of those computed empirically from SDSS fiberMag using pegase.2 SSP templates and the kcorrect code as functions of redshift and observed g - r colour. In the r band, the agreement b etween sp ectral-based K -corrections and analytical fitting derived ab ove is as good as 0.02 mag. For the low-redshift (Z < 0.3) part of the sample, the situc 2009 RAS, MNRAS 000, 1­12


Analytical approximations of K -corrections
ation is similar in the g band, although the systematic difference reaches 0.06 mag. However, for the higher redshifts (0.3 < Z < 0.5), the discrepancies b ecome as large as 0.15­ 0.2 mag. This inconsistency p ossibly originates from underestimated g synthetic magnitudes due to a very low signal in the blue part of the sp ectra for ob jects having Z > 0.3. Formally computed photometric errors of synthetic g magnitudes in this redshift range reach 0.2 mag.

7

4 4.1

DISCUSSION AND CONCLUSIONS Comparison with literature

We compare K -corrections computed in this study with the results describ ed in literature. Fukugita et al. (1995) presented K -corrections of the SDSS photometric system in their fig. 20. Since no data are provided in the numerical form, we are able to compare the results only qualitatively. The b ehaviour of the values as functions of redshifts agree well at Z < 0.5, different morphological typ es of galaxies in Fukugita et al. (1995) are related to the reconstructed restframe colours. The elliptical galaxy template of Fukugita et al. (1995) b ehaves similarly to the sequences denoted as "LRG" in Fig. 3­4 corresp onding to luminous red galaxies. We notice that the filter transmission curves used in Fukugita et al. (1995) are somewhat different from those obtained from the telescop e and presented at the SDSS DR7 web pages, esp ecially for the g and r bands, explaining why the exact match b etween the two approaches cannot b e achieved. Mannucci et al. (2001) present in their fig. 7 NIR K corrections computed from the templates sp ectra of galaxies having different morphological typ es computed with the pegase code. Although their J H K bands are slightly different from those of the UKIDSS, the b ehaviour of K -corrections as functions of redshift is very similar. Authors mention that K -corrections in H and K are virtually insensitive to the galaxy morphological typ es, which we observe as their very weak dep endence on galaxy colours. We compare our results with the sp ectral-based K corrections presented in Roche et al. (2009). The authors deal only with red sequence galaxies, therefore we are able to test only a small although very imp ortant part of the parameter space. The values of K -corrections provided in tables 1­2 of Roche et al. (2009) are overplotted in Fig 5. The agreement is remarkably good, which is, however, exp ected b ecause our sp ectral-based K -corrections computed in the same manner agree well with the b est-fitting solutions.

4.2

Recovered restframe magnitudes and LRGs
Figure 5. Differences between analytically approximated and directly measured spectral-based K -corrections in the g (top) and r (bottom) bands. Upper panels display computed values of K corrections. Dashed lines denote spectral-based K -corrections for early-type galaxies presented in Roche et al. (2009). Two bottom panels for each spectral band display the differences between spectral-based K -corrections and those computed using pegase.2 SSP matching and the kcorrect code, respectively.

It is well known that the restframe colours correlate with the galaxy morphology, and there is a pronounced bimodality in the colour distribution of galaxies (see e.g. Strateva et al. 2001; Baldry et al. 2004; Balogh et al. 2004). The socalled "red sequence" includes elliptical and lenticular galaxies with some fraction of early-typ e spirals demonstrating a lack of ongoing star formation, whereas the "blue cloud" contains actively star-forming ob jects represented mostly by late-typ e spiral and irregular galaxies. Some morphologically
c 2009 RAS, MNRAS 000, 1­12


8

I. Chilingarian et al.
Table 1. Coefficients of the best-fitting polynomials for the redshift-dependence of K -corrections for luminous red galaxies. Initial K -correction values were computed from the pegase.2 SSPs. Z K K K K K K K K K
u g r i z Y J H K 1

Z

2

Z

3

Z

4

Z

5

5.93938 2.61617 0.312233 0.234538 0.897075 0.402992 -0.076704 0.382926 -1.75997

-30.5247 -4.44391 14.3325 14.3162 3.60112 5.30858 -4.48411 -1.81590 5.48023

179.473 93.0132 -68.2493 -97.2754 -34.7890 -18.3172 27.1585 -13.1657 -56.4175

-380.488 -284.582 136.254 246.775 93.0266 17.6760 -45.6481 57.5486 175.939

282.011 252.245 -87.3360 -207.028 -79.5246 -0.31400 22.6928 -59.0677 -160.754

Figure 6. Colour-magnitude diagram presenting data for 74,254 galaxies using the SDSS DR7 and UKIDSS DR5 Petrosian magnitudes in the g , r , and H bands respectively K -corrected using the analytical approximations presented in this study. Black and blue contours are for the pegase.2-based and kcorrect derived values respectively. Levels of the two-dimensional density plot correspond to the powers of two from 2 (outermost) to 1024 (innermost) galaxies per bin of 0.25 magâ0.025 mag.

Table 2. Same as in Table 1 but the kcorrect computed K corrections. Z K K K K K K K K K
u g r i z Y J H K 1

Z

2

Z

3

Z

4

Z

5

classified early-typ e galaxies indeed sit b elow the red sequence in the "green valley" or even "blue cloud" regions. In Fig 6, we provide the colour-magnitude plot of 74,254 galaxies from SDSS DR7 and UKIDSS, where their total Petrosian magnitudes were K -corrected using the presented analytical approximations. Some of the ob jects residing b elow the red sequence referred as E+A galaxies (Dressler & Gunn 1983; Couch & Sharples 1987; Dressler et al. 1999) often exhibit weak if any emission lines in their sp ectra ruling out ma jor star formation. However, Balmer absorptions are remarkably deep, which is a good evidence for the presence of young stars that was confirmed by the recent study where authors used full-sp ectral fitting (Chilingarian et al. 2009b). Despite the early-typ e morphologies of E+A, these galaxies are indistinguishable from spirals if one uses only integrated colours and therefore, colour transformations and K -corrections b ehave very similarly for these two morphologically different classes of galaxies. Hence, the restframe colours are not tightly connected to the galaxy morphology, but to the internal prop erties such as ongoing star formation and, therefore, they are sp ecific of stellar p opulations. Another example would b e dwarf elliiptical (dE) and compact elliptical (cE) galaxies. The former ones usually have intermediate-age stellar p opulations with metallicities lower than those of LRGs (see e.g. Chilingarian et al. 2008; Chilingarian 2009), therefore their global optical colours are bluer, and their K -corrections in the corresp onding bands are lower than those of more massive early-typ e galaxies, which results in quite a strong tilt of the red sequence at MH,AB > -20 mag (see Fig 6) reproducing its b ehaviour for nearby dE galaxies (Janz & Lisker 2009). On the other hand, cE galaxies with luminosities similar to dEs, originate from the tidal stripping of more massive early-typ e progenitors and, therefore, exhibit old metal-rich stellar p opulations (Chilingarian et al. 2009a). For this reason, cEs have significantly redder colours compared to dEs and, therefore, have b ehaviour of K -corrections very similar to LRGs.

4.20000 2.17470 0.710579 0.702681 0.643953 -0.245996 0.106358 0.268479 -2.80894

-24.5015 10.3810 10.1949 4.27115 -1.88400 21.8772 -5.06024 3.03488 15.6923

229.149 1.49141 -57.0378 -37.2060 13.6952 -137.019 18.4707 -35.8994 -96.8401

-574.272 -76.6656 133.141 112.054 -35.0960 322.051 -3.73196 98.6524 256.235

434.901 88.6641 -99.9271 -105.976 29.9249 -257.136 -23.9595 -83.9401 -220.691

There is a weak dep endence of the colour of red sequence galaxies on the galaxy luminosity reflecting, in particular, a mass-metallicity relation of early-typ e galaxies. Massive (and luminous) red sequence galaxies compared to lower-mass systems contain more metal-rich stellar p opulations having intrinsically redder colours than the metal-p oor ones. The sp ecial case of LRGs is very imp ortant for understanding galaxy formation and evolution, therefore we provide sp ecific approximations for these ob jects. Since the intrinsic spread of LRGs restframe colours is very low, ab out 0.04 mag, their K -corrections can b e well approximated as a function of a single parameter, the redshift. We selected a sample of LRGs based on their restframe colours, and fitted the K -corrections as p olynomial functions of their redshifts in all 9 photometric bands. The coefficients of b est-fitting 5th order p olynomials with no constant term are presented in Table 1 and Table 2 for the pegase.2 SSP-based and kcorrect-computed values corresp ondingly. Every row corresp onds to a given photometric band and contains coefficients from the 1st to the 5th p ower of redshift. Rest-frame colours of LRGs provide a natural test for the quality of K -corrections. The vast ma jority of stars in these ob jects is b elieved to form on short timescales in the early Universe and then evolve passively. The redshift z = 0.5 corresp onds to the lookback time ab out 5 Gyr. Thus, if one assumes LRGs to b e as old as 12 Gyr in the local Universe and contain no younger stars, then at z = 0.5 their restframe optical colours would b e slightly bluer with the differences (u - r ) 0.25 mag, (g - r ) 0.08 mag, and redder colours different by less than 0.03 mag (values estimated from the colour evolution of the solar metallicity
c 2009 RAS, MNRAS 000, 1­12


Analytical approximations of K -corrections

9

the corresp onding panels in Fig. 2). The pegase.2 SSPbased K -band K -corrections at Z > 0.3 look slightly underestimated, conversely to the kcorrect-computed ones, which are overestimated. In the H band, b oth methods seem to underestimate K -corrections, but SSP-based approach is stronger affected. The decisive answer to the questions ab out the K correction computation in NIR bands will b e given only when the next generation NIR stellar p opulation models based on empirical stellar libraries b ecome available. The H and K band solutions presented in this pap er have to b e used with caution keeping in mind that some systematic errors may b e introduced to the final results.

4.3

Summary

We present p olynomial approximations of K -corrections in commonly used optical (SDSS ug r iz and Johnson-Cousins U B V Rc Ic ) and near-infrared (UKIRT WFCAM Y J H K and 2MASS J H Ks ) broad-band filters as functions of redshift and observed colours for galaxies at redshifts Z < 0.5. The traditional K -correction computation techniques based on the SED fitting require multi-colour photometry, which is not always available. Our approach allows one to compute restframe galaxy magnitudes of the same quality using a minimal set of observables including only two photometric p oints and a redshift. For luminous red galaxies, we provide the fitting solutions of K -corrections as 1-dimensional p olynomial functions of a redshift.

ACKNOWLEDGMENTS
Figure 7. Dependence of recovered restframe colours of LRGs on redshift. Solid and dashed lines are for the pegase.2 SSP-based and kcorrect computed K -corrections respectively.

pegase.2 SSP). In Fig 7, we present the b ehaviour of recovered restframe colours of LRGs as functions of redshift. All presented colours have SDSS r as one of the bands, b ecause the b est consistency of b oth K -correction computation algorithms is reached in this band. The bluest u - r colour exhibits significant changes over redshift exceeding by a factor of 3 the exp ectations from the passive evolution of a SSP. This, however, can b e explained by a "tail" of the star formation history since even small mass fractions of intermediate mass stars strongly affect the u band photometry. On the other hand, we cannot exclude the template mismatch b etween the model and real galaxies to b e partly resp onsible for this effect. Surprisingly, the g - r colour does not evolve at all if we use the kcorrect values of g -band K -corrections, but evolves slightly stronger than exp ected from the SSP evolution when using the pegase.2 based values, b eing somewhat consistent with the u - r colour b ehaviour. While r - i and r - z colours b ehave similarly in b oth approaches exhibiting, as exp ected, virtually no evolution, in NIR bands the situation is different. Both techniques are consistent in the J band, although demonstrating rather unexp ected evolution by ab out 0.08 mag, whereas in H and K they are significantly different (as also can b e seen from
c 2009 RAS, MNRAS 000, 1­12

In this study, we used the UKIDSS DR5 survey catalogues available through the WFCAM science archive and SDSS DR7 data. 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/. We acknowledge the usage of the topcat software by M.Taylor. IC and IZ acknowledge the supp ort from the RFBR grants 07-02-0029 and 09-02-00032. Sp ecial thanks to A. Sergeev for the "K-corrections calculator" web-site design. This research is supp orted by the VO Paris Data Centre. We thank our anonymous referee for useful suggestions.

REFERENCES Abaza jian, K. N., et al. 2009, ApJS, 182, 543 Baldry, I. K., Glazebrook, K., Brinkmann, J., Ivezi´, Z., c Lupton, R. H., Nichol, R. C., & Szalay, A. S. 2004, ApJ, 600, 681 Balogh, M. L., Baldry, I. K., Nichol, R., Miller, C., Bower, R., & Glazebrook, K. 2004, ApJ, 615, L101 Blanton, M. R. & Roweis, S. 2007, AJ, 133, 734 Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000


10

I. Chilingarian et al.
Table A1. The coefficients ax,y of the two-dimensional polynomial approximation of K -corrections Ku (Z, u - r ) in the SDSS u band (see Equation 1). In order to derive a value for Ku , one needs to sum the polynomial terms of a form Z x and (u - r )y from the table heading and its first column multiplied by the coefficients in corresponding table cells. (u - r )0 Z Z Z Z Z Z
0 1 2 3 4 5

Calzetti, D., Kinney, A. L., & Storchi-Bergmann, T. 1994, ApJ, 429, 582 Chilingarian, I., Cayatte, V., Revaz, Y., Dodonov, S., Durand, D., Durret, F., Micol, A., & Slezak, E. 2009a, Science, 326, 1379 Chilingarian, I., Prugniel, P., Sil'chenko, O., & Koleva, M. 2007a, in IAU Symp osium, Vol. 241, Stellar Populations as Building Blocks of Galaxies, ed. A. Vazdekis & R. R. Peletier (Cambridge, UK: Cambridge University Press), 175­176, arXiv:0709.3047 Chilingarian, I. V. 2009, MNRAS, 394, 1229 Chilingarian, I. V., Cayatte, V., Durret, F., Adami, C., Balkowski, C., Chemin, L., Lagan´, T. F., & Prugniel, P. a 2008, A&A, 486, 85 Chilingarian, I. V., De Rijcke, S., & Buyle, P. 2009b, ApJ, 697, L111 Chilingarian, I. V., Prugniel, P., Sil'chenko, O. K., & Afanasiev, V. L. 2007b, MNRAS, 376, 1033 Couch, W. J. & Sharples, R. M. 1987, MNRAS, 229, 423 Dressler, A. & Gunn, J. E. 1983, ApJ, 270, 7 Dressler, A., Smail, I., Poggianti, B. M., Butcher, H., Couch, W. J., Ellis, R. S., & Oemler, A. J. 1999, ApJS, 122, 51 Fioc, M. & Rocca-Volmerange, B. 1997, A&A, 326, 950 Fitzpatrick, E. L. 1999, PASP, 111, 63 Fukugita, M., Shimasaku, K., & Ichikawa, T. 1995, PASP, 107, 945 Hewett, P. C., Warren, S. J., Leggett, S. K., & Hodgkin, S. T. 2006, MNRAS, 367, 454 Hogg, D. W., Baldry, I. K., Blanton, M. R., & Eisenstein, D. J. 2002, arXiv:astro-ph/0210394 Janz, J. & Lisker, T. 2009, ApJ, 696, L102 Jordi, K., Greb el, E. K., & Ammon, K. 2006, A&A, 460, 339 Lawrence, A., et al. 2007, MNRAS, 379, 1599 Mannucci, F., Basile, F., Poggianti, B. M., Cimatti, A., Daddi, E., Pozzetti, L., & Vanzi, L. 2001, MNRAS, 326, 745 Maraston, C., Str¨mb¨ck, G., Thomas, D., Wake, D. A., & oa Nichol, R. C. 2009, MNRAS, 394, L107 Oke, J. B. & Sandage, A. 1968, ApJ, 154, 21 Roche, N., Bernardi, M., & Hyde, J. 2009, MNRAS, 398, 1549 Schlegel, D. J., Finkb einer, D. P., & Davis, M. 1998, ApJ, 500, 525 Skrutskie, M. F., et al. 2006, AJ, 131, 1163 Strateva, I., et al. 2001, AJ, 122, 1861

(u - r )1 0 2.24658 20.4939 -42.3042 63.0036 0

(u - r )2 0 0.141845 -3.82771 -4.05721 0 0

(u - r )3 0 -0.13441 0.789867 0 0 0

0 1.63349 -71.84 257.509 -308.573 42.8572

Table A2. Same as in Table A1 but for the SDSS g band as a function of redshift and the g - r colour. (g - r )0 Z Z Z Z Z Z
0 1 2 3 4 5

(g - r )1 0 3.97338 -8.04213 -31.1241 71.5801 0

(g - r )2 0 0.774394 11.0321 -17.5553 0 0

(g - r )3 0 -1.09389 0.781176 0 0 0

0 -0.900332 3.65877 -16.7457 87.3565 -123.671

UKIDSS Y J H K bands. Tables A19­A23 contain the coefficients for approximations of pegase.2-based K -corrections in the Johnson­Cousins U B V Rc Ic bands. The coefficients for the 2MASS J H Ks bands as well as for the other colour combinations of SDSS and UKIDSS bands are provided online through the "K-corrections calculator" service.

APPENDIX A: 2-DIMENSIONAL APPROXIMATIONS OF K -CORRECTIONS Here we provide tables containing coefficients of the analytical approximations of K -corrections. Columns in every table contain colour terms of the fitting solution, from constant to the 3rd order resp ectively. Rows are for the redshift dep endence. The maximal degree of the p olynomial surface is set to 5 or 7 (for the H band), therefore the coefficients for the higher degrees, in b oth colour and redshift, are set to zero. Tables A1­A9 and A10­A18 present the coefficients for approximations of pegase.2-based and kcorrect-computed K -corrections in the SDSS ug r iz and
Table A3. Same as in Table A1 but for the SDSS r band as a function of redshift and the g - r colour. (g - r )0 Z Z Z Z Z Z
0 1 2 3 4 5

(g - r )1 0 3.81378 9.85141 -30.3631 -25.0159 0

(g - r )2 0 -3.56114 -5.1432 38.5052 0 0

(g - r )3 0 2.47133 -7.02213 0 0 0

0 -1.61294 9.13285 -81.8341 250.732 -215.377

c 2009 RAS, MNRAS 000, 1­12


Analytical approximations of K -corrections
Table A4. Same as in Table A1 but for the SDSS i band as a function of redshift and the g - i colour. (g - i)0 Z Z Z Z Z Z
0 1 2 3 4 5

11

Table A9. Same as in Table A1 but for the UKIDSS K band as a function of redshift and the J - K colour. (J - K )0 Z Z Z Z Z Z
0 1 2 3 4 5

(g - i)1 0 4.68318 5.14198 -14.154 -51.6662 0

(g - i)2 0 -3.70678 -2.64767 27.2864 0 0

(g - i)3 0 1.5155 -3.63215 0 0 0

(J - K )1 0 4.14968 -39.5749 94.0769 -44.0291 0

(J - K )2 0 1.15579 11.7 -35.1023 0 0

(J - K )3 0 -1.94003 4.7809 0 0 0

0 -2.41799 11.2598 -94.7387 285.775 -222.641

0 -2.80374 13.4077 -69.7725 157.649 -132.317

Table A5. Same as in Table A1 but for the SDSS z band as a function of redshift and the r - z colour. (r - z )0 Z Z Z Z Z Z
0 1 2 3 4 5

Table A10. Same as in Table A1 but using the kcorrect code. (u - r )0 Z Z Z Z Z Z Z Z
0 1 2 3 4 5 6 7

(u - r )1 0 -12.0027 39.7152 236.944 -659.764 395.301 -67.5999 0

(u - r )2 0 5.57928 -18.7077 -52.6404 162.85 -83.6382 0 0

(u - r )3 0 -0.825005 4.0901 -3.43017 -1.62112 0 0 0

(r - z )1 0 3.35566 -23.1956 75.2648 -48.3299 0

(r - z )2 0 0.469411 -3.32427 -10.2986 0 0

(r - z )3 0 0.350873 1.78842 0 0 0

0 -1.7252 14.9772 -41.1269 11.3667 23.5438

0 8.81624 33.0392 -1223.73 6304.63 -15288.5 18533.6 -8719.48

Table A6. Same as in Table A1 but for the UKIDSS Y band as a function of redshift and the Y - H colour. (Y - H Z Z Z Z Z Z
0 1 2 3 4 5

)0

(Y - H

)1

(Y - H

)2

(Y - H

)3

Table A11. Same as in Table A10 but for the SDSS g band as a function of redshift and the g - r colour. (g - r )0 Z Z Z Z Z Z
0 1 2 3 4 5

0 -2.01575 14.3112 -46.4835 80.2341 -66.1958

0 2.70429 -13.2354 12.7464 27.9842 0

0 6.01384 -4.03961 -26.3263 0 0

0 -5.17119 13.2633 0 0 0

(g - r )1 0 2.2796 -14.8073 -49.2478 131.339 0

(g - r )2 0 4.16029 19.261 -40.9139 0 0

(g - r )3 0 -3.27579 4.28022 0 0 0

0 -0.962084 15.6602 -82.9388 273.308 -312.677

Table A7. Same as in Table A1 but for the UKIDSS J band as a function of redshift and the J - K colour. (J - K Z Z Z Z Z Z
0 1 2 3 4 5

)0

(J - K

)1

(J - K

)2

(J - K

)3

Table A12. Same as in Table A10 but for the SDSS r band as a function of redshift and the g - r colour. (g - r )0 Z Z Z Z Z Z
0 1 2 3 4 5

0 -0.765217 1.59864 -4.02136 18.5608 -40.3567

0 2.43055 -14.646 18.077 25.2691 0

0 -0.427304 12.0911 -26.1137 0 0

0 0.277662 -1.2131 0 0 0

(g - r )1 0 2.61848 16.0682 -49.337 12.0421 0

(g - r )2 0 -2.99032 -2.16736 22.9267 0 0

(g - r )3 0 1.59058 -4.24709 0 0 0

0 -0.351251 1.93312 -69.9339 253.373 -235.32

Table A8. Same as in Table A1 but for the UKIDSS H band as a function of redshift and the H - K colour. (H - K )0 Z Z Z Z Z Z Z Z
0 1 2 3 4 5 6 7

(H - K )1 0 -1.05192 80.0291 -564.952 1569.47 -1893.45 869.396 0

(H - K )2 0 -15.5123 192.688 -848.543 1741.31 -1488.4 0 0

(H - K )3 0 -18.1957 179.956 -646.653 761.264 0 0 0

Table A13. Same as in Table A10 but for the SDSS i band as a function of redshift and the g - i colour. (g - i)0 Z Z Z Z Z Z
0 1 2 3 4 5

0 -0.642942 26.3667 -274.11 1081 -1938.48 1448.38 -249.952

(g - i)1 0 8.51897 -31.8699 40.9251 -44.6466 0

(g - i)2 0 -3.44266 7.13812 6.77963 0 0

(g - i)3 0 0.823935 -2.26748 0 0 0

0 -5.58619 22.398 -16.5911 -12.0117 21.0947

c 2009 RAS, MNRAS 000, 1­12


12

I. Chilingarian et al.
Table A17. Same as in Table A10 but for the UKIDSS H band as a function of redshift and the H - K colour. (H - K )0 Z Z Z Z Z Z Z Z
0 1 2 3 4 5 6 7

Table A14. Same as in Table A10 but for the SDSS z band as a function of redshift and the r - z colour. (r - z )0 Z Z Z Z Z Z
0 1 2 3 4 5

(r - z )1 0 3.08833 -8.48028 53.5534 -77.1975 0

(r - z )2 0 -0.726039 -8.18852 13.6829 0 0

(r - z )3 0 1.06364 -1.35281 0 0 0

(H - K )1 0 1.93083 -17.3496 267.725 -1657.94 4005.55 -3298.45 0

(H - K )2 0 -4.7581 28.7714 127.193 -851.199 1071.17 0 0

(H - K )3 0 -6.67018 8.05742 109.678 -217.543 0 0 0

0 -1.426 2.9386 8.08986 -93.2991 133.298

0 0.132484 8.61784 -83.7003 134.398 567.048 -2000.79 1697.02

Table A15. Same as in Table A10 but for the UKIDSS Y band as a function of redshift and the Y - H colour. (Y - H )0 Z Z Z Z Z Z
0 1 2 3 4 5

(Y - H )1 0 4.7578 -26.5297 34.7785 26.8742 0

(Y - H )2 0 2.28856 10.1083 -41.2725 0 0

(Y - H )3 0 -4.02782 10.5582 0 0 0

Table A18. Same as in Table A10 but for the UKIDSS K band as a function of redshift and the J - K colour. (J - K )0 Z Z Z Z Z Z
0 1 2 3 4 5

0 -2.62137 29.4209 -141.372 311.42 -264.997

(J - K )1 0 1.66876 -7.83528 -17.793 70.0829 0

(J - K )2 0 1.45967 13.3436 -42.0747 0 0

(J - K )3 0 -3.40684 9.32974 0 0 0

0 -3.1771 17.9897 -114.067 318.424 -299.557

APPENDIX B: K-CORRECTIONS CALCULATOR To facilitate the usage of describ ed analytical approximations and calculation of necessary K -corrections for user's data, we provide a dedicated web-site entitled "Kcorrections calculator"3 offering several ways of op eration. Firstly, one can determine K -corrections for ob jects of interest one by one by manually filling interactive web form with the data available. This is the most straightforward method to determine K -corrections and it works in any modern web browser. As a second option, the ready-to-use code snipp ets written in p opular data languages (C, IDL, Python) together with the corresp onding analytical expressions are provided to simplify the integration of K -correction functionality into the user's code. Both ways allow a user to calculate K -corrections for SDSS ug r iz and UKIRT WFCAM Y J H K (in AB magnitudes), but also for Johnson-Cousins U B V Rc Ic and 2MASS J H Ks bands (in Vega magnitudes), choosing the most convenient colour as an input argument from several options. Based on user demands and feedback, we will consider the development of a web service for bulk calculation of K -corrections and publication of additional filter-colour combinations in the calculator.
Table A19. Same as in Table A1 but for the Johnson U band as a function of redshift and the U - Rc colour. (U - Rc )0 Z Z Z Z Z Z Z Z
0 1 2 3 4 5 6 7

(U - Rc )1 0 2.31564 13.2852 -124.303 428.811 -39.2842 -641.309 0

(U - Rc )2 0 -0.411492 6.74212 -9.92117 -124.492 197.445 0 0

(U - Rc )3 0 -0.0362256 -2.16222 12.7453 -14.3232 0 0 0

0 2.84791 -18.8238 -307.885 3040.57 -10677.7 16022.4 -8586.18

Table A20. Same as in Table A1 but for the Johnson B band as a function of redshift and the B - Rc colour. (B - Rc )0 Z Z Z Z Z Z
0 1 2 3 4 5

(B - Rc )1 0 3.45377 -3.99873 -44.4243 86.789 0

(B - Rc )2 0 0.818214 6.44175 -12.6224 0 0

(B - Rc )3 0 -0.630543 0.828667 0 0 0

0 -1.99412 15.9592 -101.876 299.29 -304.526

3

http://kcor.sai.msu.ru/

Table A21. Same as in Table A1 but for the Johnson V band as a function of redshift and the V - Ic colour. (V - Ic )0 Z Z Z Z Z Z
0 1 2 3 4 5

Table A16. Same as in Table A10 but for the UKIDSS J band as a function of redshift and the J - K colour. (J - K Z Z Z Z Z Z
0 1 2 3 4 5

(V - Ic )1 0 -1.3982 -17.9194 -13.6809 39.4246 0

(V - Ic )2 0 4.76093 8.32856 -9.25747 0 0

(V - Ic )3 0 -1.59598 0.622176 0 0 0

)0

(J - K

)1

(J - K

)2

(J - K

)3

0 -0.472236 -0.107502 -18.2002 111.89 -162.057

0 2.1536 -8.00546 -27.5709 99.1294 0

0 0.811858 16.6955 -46.9334 0 0

0 -1.87211 4.85177 0 0 0

0 -1.37734 19.0533 -86.9899 305.09 -324.357

c 2009 RAS, MNRAS 000, 1­12


Analytical approximations of K -corrections
Table A22. Same as in Table A1 but for the Cousins Rc band as a function of redshift and the B - Rc colour. (B - Rc )0 Z Z Z Z Z Z
0 1 2 3 4 5

13

(B - Rc )1 0 4.64989 5.34587 -30.3302 -19.3575 0

(B - Rc )2 0 -2.86494 0.408024 18.4741 0 0

(B - Rc )3 0 0.90422 -2.47204 0 0 0

0 -2.83216 4.97464 -57.3361 224.219 -194.829

Table A23. Same as in Table A1 but for the Cousins Ic band as a function of redshift and the V - Ic colour. (V - Ic )0 Z Z Z Z Z Z
0 1 2 3 4 5

(V - Ic )1 0 17.6389 -1.99263 -42.9575 -67.5785 0

(V - Ic )2 0 -15.2414 10.663 46.7401 0 0

(V - Ic )3 0 5.12562 -10.8329 0 0 0

0 -7.92467 15.7555 -88.0145 266.377 -164.217

c 2009 RAS, MNRAS 000, 1­12