Mon. Not. R. Astron. Soc. 000, 1–10 (2002) Printed 24 October 2018 (MN LATEX style file v2.2)
Colour Pairs for Constraining the Age and Metallicity of
Stellar Populations ⋆
Zhongmu Li1,2 † and Zhanwen Han1
1National Astronomical Observatories/Yunnan Observatory, the Chinese Academy of Sciences, Kunming, 650011, China
2Graduate School of the Chinese Academy of Sciences
Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11
ABSTRACT
Using a widely used stellar population synthesis model, we study the ability of using
pairs of AB system colours to break the well-known stellar age–metallicity degeneracy
and give constraints on two luminosity-weighted stellar-population parameters (age
and metallicity). The relative age and metallicity sensitivities of AB system colours
that relate to u, B, g, V , r, R, i, I, z, J ,H , andK bands are presented, and the abilities
of various colour pairs for breaking the age–metallicity degeneracy are quantified.
Our results suggest that a few pairs of colours can be used to constrain the two
above stellar-population parameters. This will be very useful for exploring the stellar
populations of distant galaxies. In detail, colour pairs [(r−K), (u−R)] and [(r−K), (u−
r)] are shown to be the best pairs for estimating the luminosity-weighted stellar ages
and metallicities of galaxies. They can constrain two stellar-population parameters on
average with age uncertainties less than 3.89 Gyr and metallicity uncertainties less
than 0.34 dex for typical colour uncertainties. The typical age uncertainties for young
(Age < 4.6 Gyr) populations and metal-rich (Z > 0.001) populations are small (about
2.26 Gyr) while those for old populations (Age > 4.6 Gyr) and metal-poor (Z < 0.001)
populations are much larger (about 6.88 Gyr). However, the metallicity uncertainties
for metal-poor populations (about 0.0024) are much smaller than for other populations
(about 0.015). Some other colour pairs can possibly be used for constraining the two
parameters, too.
As a whole, the estimation of stellar-population parameters is likely to reliable
only for early-type galaxies with small colour errors and globular clusters, because
such objects contain less dust. In fact, no galaxy is totally dust-free and early-type
galaxies are also likely have some dust, e.g., E(B-V)∼ 0.05, which can change the
stellar ages about 2.5 Gyr and metallicities (Z) about 0.015. When we compare the
photometric estimates with previous spectroscopic estimates, we are shown some dif-
ferences, especially when comparing the stellar ages determined by two methods. The
differences mainly result from the young populations of galaxies. Therefore, it is dif-
ficult to get the absolute values of stellar ages and metallicities, but the results are
useful for getting some relative values.
In addition, our results suggest that colours relating to both UBV RIJHK and
ugriz magnitudes are much better than either UBV RIJHK colours or ugriz colours
for breaking the well-known degeneracy. The results also show that the stellar ages
and metallicities of galaxies observed by the Sloan Digital Sky Survey (SDSS) and the
Two-Micron All-Sky Survey (2MASS) can be estimated via photometry data.
Key words: galaxies: stellar content — galaxies: photometry — galaxies: elliptical
and lenticular, cD.
⋆ The data are available at the CDS or on request to the authors.
† E-mail: zhongmu.li@gmail.com
1 INTRODUCTION
The formation and evolution of galaxies is one of the
hottest topics in astronomy and astrophysics. Great progress
in the field has been had (see, e.g., Thomsen & Baum
c© 2002 RAS
http://arxiv.org/abs/0704.1202v4
2 Zhongmu Li and Zhanwen Han
1989, Kormendy & Djorgovski 1989, Kodama et al. 1998,
van Dokkum & Stanford 2003, Kauffmann & Charlot 1998,
Baugh et al. 1996, Baugh et al. 1998, Cole et al. 2000,
De Lucia et al. 2006, and Thomas 1999), and two
luminosity-weighted stellar-population parameters (age and
metallicity) of galaxies are crucial. A lot of stellar
population synthesis models such as Bruzual & Charlot
(2003)(hereafter BC03), Worthey (1994), Vazdekis (1999)
, Fioc & Rocca-Volmerange (1997), and Zhang et al. (2005)
have been brought forward for stellar population studies and
a great deal of observational data have been supplied by
big surveys such as the Sloan Digital Sky Survey (SDSS)
and the Two-Micron All-Sky Survey (2MASS). However, it
is difficult to measure the stellar ages and metallicities of
some distant galaxies (e.g., galaxies with redshifts greater
than 0.3) via the spectra-like methods. The reason is that
reliable spectra or line-strength indices (see, e.g., Worthey
1994, Gallazzi et al. 2005, Li et al. 2006) are usually avail-
able for some nearby galaxies. If we can use one of important
observational results, i.e., photometry, we will be able to ex-
plore the stellar populations of some distant galaxies. Be-
cause colours are easier to be obtained than spectra and are
independent of the distances of objects, they are good can-
didates for studying the stellar populations of such galaxies.
Many works have been tried in this way, e.g., Dorman et al.
(2003), Yi (2003), Wu et al. (2005), James et al. (2006),
Li et al. (2007); Li & Han (2007), and Kaviraj et al. (2006).
It seems possible to break the stellar age–metallicity de-
generacy (Worthey 1994) by colours. Many pairs of colours
(hereafter colour pairs) are used in previous works, e.g.,
(U −R) and (R−K) by Peletier & Balcells (1996), (B−R)
and (R−K) by Bell & de Jong (2000), (V −I) and (V −K)
by Puzia et al. (2002), (B−K) and (J−K) by James et al.
(2006), (B−V ) and (B−K) by Li et al. (2007), (B−V ) and
(V −K) by Lee et al. (2007), and most colours used are only
in UBV RIJHK bands. As advocated by, e.g., Kong et al.
(2006), colours relating to both UBV RIJHK and ugriz
bands, i.e., (B − z) and (z − Ks), can be used to select
galaxies with various redshifts. Similar colours can possibly
be used to break the stellar age–metallicity degeneracy. We
intend to give some investigations in this work, using the
BC03 stellar population synthesis model.
The paper is organized as follows. In Sect. 2, we briefly
introduce the BC03 model and the calculation of colours. In
Sect. 3, we present the age and metallicity sensitivities of
colours. In Sect. 4, we try to search for colour pairs that can
be used to constrain stellar age and metallicity. In Sect. 5,
we give our discussions and conclusions.
2 THE BC03 MODEL AND CALCULATION OF
COLOURS
The BC03 model is a widely used model in stellar popula-
tion synthesis study. Its standard model takes the Chabrier
(2003) initial mass function (IMF) and uses Padova 1994
library tracks to calculate integrated colours. A few alter-
native stellar spectral libraries are considered to give the
spectral predictions of simple stellar populations (SSPs).
The model provides us magnitudes and colours on both
Johnson-Cousins-Glass (UBV RIJHKLM bands) and AB
(ugriz bands) systems. More detailed information about
Table 1. Relative metallicity sensitivities of 66 AB system
colours.
colour rms colour rms colour rms
(B −K) 2.0294 (I − z) 0.8218 (B −H) 0.5758
(R−K) 1.8610 (i−K) 0.8094 (u− g) 0.5654
(u−K) 1.6904 (g − i) 0.7863 (g − V ) 0.5400
(I − J) 1.6125 (B − r) 0.7554 (i−H) 0.5178
(V −K) 1.5774 (B −R) 0.7335 (H −K) 0.5105
(r −K) 1.4114 (V − I) 0.7237 (R− I) 0.5074
(i− J) 1.4031 (g − r) 0.7027 (r −R) 0.5012
(I −K) 1.2654 (g − R) 0.6952 (r −H) 0.4834
(R− J) 1.2160 (V −R) 0.6896 (J −K) 0.4707
(g − J) 1.1094 (V − r) 0.6877 (R− i) 0.4675
(B − z) 1.0914 (u− I) 0.6725 (u− V ) 0.4440
(u− J) 1.0474 (r − i) 0.6722 (z −H) 0.4180
(B − J) 1.0455 (R − z) 0.6675 (g −H) 0.4116
(V − J) 1.0145 (u−B) 0.6632 (u− r) 0.4000
(V − z) 1.0069 (r − z) 0.6534 (B − g) 0.3984
(u− z) 0.9985 (g − z) 0.6404 (B − V ) 0.3908
(z − J) 0.9805 (r − I) 0.6382 (u−R) 0.3887
(g −K) 0.9719 (u−H) 0.6320 (V −H) 0.3835
(r − J) 0.9247 (i− z) 0.6151 (z −K) 0.3637
(B − i) 0.8788 (u− i) 0.5909 (R−H) 0.3344
(V − i) 0.8719 (I −H) 0.5867 (i− I) 0.2262
(B − I) 0.8414 (g − I) 0.5856 (J −H) 0.1670
the model please refer to Bruzual & Charlot (2003). In this
work, BV RIJHK magnitudes on AB system are recalcu-
lated from those given on Johnson-Cousins-Glass system, by
taking -0.1, 0.0, 0.2, 0.45, 0.9, 1.4, and 1.9 as the differences
between the zero points of AB system and Johnson-Cousins-
Glass system magnitudes, in B, V , R, I , J , H and K bands,
respectively 1.
3 AGE AND METALLICITY SENSITIVITIES
OF COLOURS
We study the age and metallicity sensitivities by a rela-
tive metallicity sensitivity (rms) technique, which was used
by Worthey (1994) and Li et al. (2007), and find age- and
metallicity-sensitive colours based on the rms of colours.
The rms method estimates the rms of each colour by the
ratio of percentage change of age to that of metallicity
when they lead to the same change in a colour respectively.
Colours with large rms (>1.0) are more sensitive to metallic-
ity and those with small rms (<1.0) to stellar age. Following
Li et al. (2007), we calculated the rms of each colour in this
work.
The rms of 66 AB system colours are calculated in the
work. The detailed data are listed in Table 1. As we see,
(B − K), (R − K), (u − K), (I − J), (V − K), (r − K),
(i− J), (I −K), (R− J), and (g − J) are more sensitive to
stellar metallicity while (J −H), (i− I), (R−H), (z −K),
(V −H), (u−R), (B−V ), (B− g), (u− r), and (g−H) to
stellar age. We select the former ten colours as age-sensitive
colours while the latter ten as metallicity-sensitive colours.
1 http://www.astro.livjm.ac.uk/∼ikb/convert-units/node1.html
c© 2002 RAS, MNRAS 000, 1–10
http://www.astro.livjm.ac.uk/$^\sim $ikb/convert-units/node1.html
Colour Pairs for Constraining Stellar-Population Parameters 3
4 COLOUR PAIRS FOR BREAKING THE
AGE–METALLICITY DEGENERACY
4.1 Colour pairs for general studies
Using the 10 metallicity-sensitive colours and the 10 age-
sensitive colours presented in the last paragraph of Sect. 3,
we buildup 100 colour pairs. Each colour pair includes a
metallicity-sensitive colour and an age-sensitive colour. Us-
ing the colour pairs one by one, we fit the stellar ages and
metallicities of 500 testing stellar populations. To make the
results useful for estimating the parameters of all kinds of
stellar populations, the ages and metallicities of the test-
ing populations are generated randomly within the ranges
of 0.1–15 Gyr and 0.0001–0.05, respectively. The averages of
uncertainties in age and metallicity, ∆t and ∆Z, are then
calculated by taking typical uncertainties for input colours.
The typical uncertainties for U , B, V , R, I , J , H , K, u, g,
r, i, and z magnitudes are taken as 0.109, 0.116, 0.059, 0.03,
0.07, 0.08, 0.09, 0.126, 0.11, 0.01, 0.007, 0.007 and 0.012 mag,
respectively. These values are estimated using the data sup-
plied by the NASA/IPAC Extragalactic Database (NED),
the 2MASS and SDSS surveys. Then we investigate the abil-
ity of each colour pair for breaking the well-known stellar
age–metallicity degeneracy by comparing the ∆t and ∆Z.
A least-square method is used to fit the ages and metal-
licities of stellar populations in the work. The uncertain-
ties in stellar-population parameters are estimated by tak-
ing the maximum uncertainties in the two parameters when
considering the uncertainties of colours. One can refer to
Denicoló et al. (2005) or Li et al. (2006) for more details.
The main results are listed in Table 2.
Because colour pairs that can well break the stellar age–
metallicity degeneracy lead to small uncertainties in stel-
lar age and metallicity, pairs with small ∆t and ∆Z are
better for constraining two stellar-population parameters.
However, as we see, some colour pairs have only small ∆t or
small ∆Z. In this case, it is difficult to compare the abilities
of various colour pairs. We defined a parameter, uncertainty
parameter (UP ), to solve this problem. The UP is calcu-
lated by taking the average of the relative uncertainties of
stellar ages and metallicities of the 500 testing stellar pop-
ulations. According to the calculation of UP , colour pairs
with small UP s are more suitable for breaking the stellar
age–metallicity degeneracy. The UP s of colours are shown
in Table 2, together with ∆ts and ∆Zs. Considering actual
applications, we only list the results of colour pairs that
have UP s smaller than 2.0 when taking typical uncertain-
ties for colours. In the table, colours are sorted by an in-
creasing order of UP . Note that we also list the average
metallicity uncertainty in dex. As we see, [(r−K), (u−R)]
and [(r −K), (u− r)] are the best colour pairs for breaking
the well-known degeneracy. Given typical uncertainties for
colours, the two colour pairs can constrain stellar-population
parameters with relative uncertainties smaller than about
96%, which corresponds to average uncertainties in stel-
lar age and metallicity smaller than 3.89 Gyr and 0.34
dex, respectively. Some other pairs, i.e., [(R−K), (u−R)],
[(I −K), (u−R)], [(R −K)], (u− r)] and [(i− J), (u−R)]
can possibly be used to give constraints on the stellar ages
and metallicities of galaxies as they have small UP s (6 1.13)
for typical uncertainties of colours. In addition, colour pairs
relating to both UBV RIIHJK bands and ugriz bands are
shown to be much better than those only relating to one of
the two kinds of bands. To see the abilities of colour pairs
for breaking the age–metallicity degeneracy, we show the
colour–colour grids of four pairs in Fig. 1.
However, as we note from Table 2, the uncertainties in
two stellar-population parameters are very big. This results
from the large observational uncertainties. Because the ob-
servational uncertainties depend on surveys and they will
possibly be decreased in future surveys, we tried to find the
best pairs for various uncertainties (0.02, 0.05 and 0.10) in
colours. We assumed that all colours have the same uncer-
tainty in each test. The results are shown in Table 2. As we
see, if the uncertainties of colours are less than 0.05 mag, the
stellar ages and metallicities of galaxies can be constrained
with uncertainty less than 1 Gyr and 0.0076, respectively,
via (I −K) and (u−R).
4.2 Colour pairs for special studies
Because we often need to estimate the ages and metallic-
ities of some special stellar populations, e.g., the old pop-
ulations of globular clusters, we tried to find some colour
pairs that are suitable for estimating the two luminosity-
weighted stellar-population parameters of such populations.
In detail, the best colour pairs for estimating the stellar ages
and metallicities of young (t < 4.6 Gyr), old (t > 4.6 Gyr),
metal poor (Z < 0.001), and metal rich (Z > 0.001) stellar
populations are found by taking the above typical uncer-
tainties of colours. The best ten colour pairs for studying
each kind of special population are listed in Table 3. As we
see, some colour pairs for studying different kinds of stel-
lar populations are various. However, [(r −K), (u− r)] can
be used to constrain the age and metallicity of all kinds of
stellar populations. We can also find that when colour pair
[(r−K), (u− r)] is used to study the stellar-population pa-
rameters of various populations, the uncertainties of results
are different. The age uncertainties of old or metal-poor pop-
ulations are usually larger than those of young or metal-rich
populations.
4.3 Composite colour pairs
In practice, we can also use colour pairs including magni-
tudes on different systems, as colours relating to the same
bands but on different systems usually have similar proper-
ties for breaking the well-known stellar age–metallicity de-
generacy (Worthey 1994). For example, we can use colour
pair [(r−Ks), (u− r)], in which ur magnitudes are on AB-
system and Ks magnitude on 2MASS system instead of
[(r−K), (u−r)], in which all magnitudes are on AB-system.
Of course, we can use Johnson-Cousins-Glass system colours
together with AB system colours.
In the work, we analyzed colour pairs relating to five
AB-system bands (ugriz) and three 2MASS bands (JHKs).
According to the results, colour pairs [(u − r), (r − Ks)],
[(u−r), (i−J)] and [(u−Ks), (z−Ks)] are more suitable for
constraining stellar-population parameters than others. The
UP s of the three colour pairs can refer to Table 2. Because
these pairs have colour-colour grids similar to those shown
in Fig. 1, we do not show them here.
c© 2002 RAS, MNRAS 000, 1–10
4 Zhongmu Li and Zhanwen Han
Table 2. The abilities of various colour pairs for breaking the age–metallicity degeneracy. Here ∆t and ∆Z (or ∆[Z/H]) are the average
uncertainties of stellar ages and metallicities, while UP is the uncertainty parameter, i.e., the average of relative uncertainties of ages
and metallicities of 500 testing stellar populations.
Uncertainty (mag) typical 0.02 0.05 0.10
Colour pair ∆t ∆Z ∆[Z/H] UP ∆t ∆Z UP ∆t ∆Z UP ∆t ∆Z UP
(Gyr) (dex) (Gyr) (Gyr) (Gyr)
[(r −K), (u− R)] 3.89 0.0176 0.33 0.95 0.37 0.0047 0.17 1.01 0.0085 0.35 2.13 0.0145 0.66
[(r −K), (u− r)] 3.57 0.0176 0.34 0.96 0.40 0.0045 0.17 1.14 0.0093 0.40 2.58 0.0146 0.78
[(R−K), (u−R)] 4.15 0.0191 0.37 1.06 0.33 0.0044 0.16 0.96 0.0084 0.35 1.98 0.0142 0.63
[(I −K), (u− R)] 3.89 0.0217 0.44 1.09 0.27 0.0032 0.12 0.89 0.0076 0.31 2.33 0.0131 0.58
[(R−K), (u− r)] 4.33 0.0192 0.38 1.11 0.32 0.0040 0.15 1.04 0.0087 0.37 2.19 0.0142 0.67
[(i− J), (u−R)] 3.16 0.0222 0.40 1.13 0.45 0.0078 0.25 1.28 0.0148 0.55 3.54 0.0228 1.19
[(i− J), (u− r)] 3.18 0.0224 0.41 1.16 0.50 0.0089 0.29 1.44 0.0148 0.63 3.71 0.0206 1.19
[(I −K), (u− r)] 4.12 0.0218 0.45 1.19 0.28 0.0034 0.12 0.92 0.0078 0.32 2.34 0.0133 0.61
[(u−K), (z −K)] 5.21 0.0188 0.37 1.34 0.34 0.0041 0.16 1.48 0.0081 0.39 4.01 0.0144 0.90
[(g − J), (z −K)] 6.21 0.0176 0.35 1.37 0.48 0.0073 0.31 2.27 0.0082 0.44 4.56 0.0132 0.98
[(r −K), (R −H)] 4.20 0.0255 0.39 1.54 0.48 0.0073 0.31 3.54 0.0181 1.08 3.79 0.0211 1.26
[(B −K), (z −K)] 6.81 0.0187 0.37 1.55 0.32 0.0072 0.28 2.13 0.0084 0.43 4.61 0.0146 0.96
[(i− J), (g −H)] 5.42 0.0243 0.47 1.56 0.42 0.0096 0.34 2.61 0.0168 0.77 7.28 0.0257 2.26
[(i− J), (z −K)] 5.09 0.0175 0.30 1.59 0.42 0.0096 0.34 2.47 0.0158 0.83 5.65 0.0146 1.71
[(V −K), (i− I)] 5.61 0.0249 0.49 1.59 0.42 0.0096 0.34 2.80 0.0201 1.05 3.08 0.0203 1.10
[(V −K), (u− R)] 5.38 0.0230 0.51 1.62 0.38 0.0050 0.18 1.19 0.0101 0.44 2.82 0.0152 0.80
[(V −K), (u− r)] 5.56 0.0227 0.50 1.72 0.57 0.0052 0.21 1.32 0.0107 0.48 3.13 0.0151 0.98
[(I − J), (z −K)] 4.72 0.0160 0.31 1.76 2.43 0.0110 0.93 2.72 0.0110 0.74 5.54 0.0153 2.05
[(r −K), (g −H)] 7.77 0.0226 0.43 1.86 0.66 0.0064 0.29 2.99 0.0096 0.55 6.85 0.0176 1.44
[(g − J), (u− r)] 5.58 0.0262 0.53 1.94 0.91 0.0140 0.53 2.36 0.0182 1.02 6.45 0.0264 2.74
[(I − J), (J −H)] 5.23 0.0280 0.59 1.95 0.91 0.0140 0.53 2.36 0.0182 1.02 6.06 0.0215 2.06
[(R− J), (u− r)] 5.53 0.0224 0.55 1.97 0.40 0.0074 0.23 1.95 0.0178 0.94 5.09 0.0233 1.88
[(R−K), (g −H)] 8.02 0.0227 0.47 1.98 0.60 0.0077 0.35 2.79 0.0094 0.54 6.13 0.0167 1.27
Table 3. Best colour pairs for estimating the ages and metallicities of old (Age > 4.6 Gyr), young (Age < 4.6 Gyr), metal-rich (Z >
0.001), and metal-poor (Z < 0.001) stellar populations. Symbols have the same meanings as in Table 2. The values are calculated using
the typical colour uncertainties. Note that the age uncertainties of metal-poor populations are not always right because some testing
populations are out of the colour-colour grid when taking their colour uncertainties into account.
Old Population Young or Metal-rich Population Metal-poor Population
Colour pair ∆t ∆Z UP Colour pair ∆t ∆Z UP Colour pair ∆t ∆Z UP
[(r −K), (u− r)] 6.88 0.0141 0.73 [(r −K), (u− r)] 2.26 0.0174 0.76 [(r −K), (u− r)] 3.64 0.0024 4.96
[(i− J), (u− r)] 7.17 0.0146 0.73 [(i− J), (u− r)] 1.34 0.0222 0.78 [(r −K), (u− R)] 3.98 0.0024 4.96
[(i− J), (u−R)] 7.58 0.0149 0.75 [(i− J), (u−R)] 1.43 0.0220 0.81 [(R −K), (u− R)] 4.10 0.0026 5.52
[(r −K), (u− R)] 7.51 0.0144 0.76 [(r −K), (u− R)] 2.54 0.0173 0.81 [(g − J), (z −K)] 1.85 0.0032 5.69
[(R−K), (u− r)] 8.17 0.0170 0.89 [(R−K), (u− r)] 2.70 0.0195 0.85 [(R −K), (u− r)] 3.89 0.0030 5.79
[(R−K), (u−R)] 8.47 0.0174 0.91 [(R−K), (u−R)] 2.64 0.0195 0.86 [(B −K), (z −K)] 3.67 0.0032 6.18
[(V −K), (u− R)] 9.57 0.0188 1.07 [(R− J), (u− r)] 2.23 0.0216 0.93 [(i− J), (B − V )] 3.42 0.0031 6.23
[(i− J), (i− I)] 11.29 0.0177 1.10 [(I −K), (u− R)] 2.46 0.0235 0.95 [(u−K), (z −K)] 6.33 0.0029 6.26
[(V −K), (u− r)] 9.19 0.0187 1.13 [(I −K), (u− r)] 2.50 0.0235 0.96 [(I −K), (B − V )] 3.87 0.0030 6.47
[(I −K), (z −K)] 9.79 0.0170 1.16 [(R− J), (u− R)] 2.29 0.0250 1.06 [(r −K), (B − V )] 3.44 0.0031 6.67
5 APPLICATION OF COLOURS AND
COLOUR PAIRS
5.1 Using colour pairs to constrain
stellar-population parameters
To test the application of colour pairs to estimate the ages
and metallicities of stellar populations, we select 1 646 lu-
minous (absolute magnitude Mr < −22 and r-band Pet-
rosian magnitude < 17.77) early-type (concentration index
C > 2.8) galaxies observed by both 2MASS and the sec-
ond release of SDSS (SDSS-DR2). All the sample galaxies
have small magnitude uncertainties (< 0.15 mag). Note that
only the galaxies with colour-fitted stellar ages smaller than
15 Gyr and stellar metallicities richer than 0.008 are se-
lected for our sample galaxies, because the age of the uni-
verse was shown to be smaller than about 15 Gyr (e.g.,
Shafieloo et al. 2006) and the results for populations with
c© 2002 RAS, MNRAS 000, 1–10
Colour Pairs for Constraining Stellar-Population Parameters 5
Figure 1. Colour–colour grids of four colour pairs that are suitable for constraining stellar age and metallicity. Colours are on AB system.
Solid and dashed lines represent constant age and metallicity, respectively. Note that we did not mark for constant ages of 6, 8, 10, 12
Gyr, as the limitation of the space. The four panels are for [(r−K), (u−R)], [(r−K), (u− r)], [(R−K), (u− r)], and [(I−K), (u−R)],
respectively.
metallicities poorer than 0.008 seems unreliable as large un-
certainties. Then we use (r −K) and (u− r) colours to es-
timate two stellar-population parameters of these galaxies,
according to the results shown in Tables 2 and 3. The K-
band magnitudes of galaxies are calculated from Ks-band
magnitudes supplied by 2MASS, by using the same method
as Bessell (2005). The k-corrections of Ks-band magnitudes
are estimated as −6log(1 + z), as used by Girardi et al.
(2003), where z is the redshift. The galactic extinctions
of K-band magnitudes are calculated using the model of
Burstein & Heiles (1982). The u and z bands magnitudes,
and their k-corrections and galactic extinctions are taken
from SDSS. The differences between SDSS magnitudes and
AB magnitudes are taken into account according to the val-
ues supplied by SDSS. In Fig. 2, the sample galaxies are
shown on the (u − r) versus (r − K) grid. For clearly, we
only plotted the error bars for the first 10 galaxies of our
sample, which are marked in black. We see that a rough
estimation for the two parameters of galaxies can be ob-
tained, although the uncertainties are somewhat big. As ex-
amples, we list the stellar-population parameters of a few
galaxies in Table 4, in which the data of three subsets of
galaxies are listed. In fact, even if we take Lick indices for
such work, the uncertainties in stellar ages and metallicities
of these galaxies are very big because of the large obser-
vational uncertainties. For our sample galaxies, the uncer-
tainties in Hβ and [MgFe] indices are typically about 0.3 Å,
which will lead to large uncertainties in stellar-population
parameters. The Lick indices and uncertainties of our sam-
ple galaxies are taken from the Garching SDSS catalogs
(http://www.mpa-garching.mpg.de/SDSS/DR4/). Further-
more, we found that 24 galaxies are out of the theoreti-
cal grid of BC03 models. This possibly results from the ef-
fects of young stellar populations of galaxies, the limitation
of theoretical models, and large observational uncertainties
of colours. In special, we found that most galaxies can fall
inside the theoretical colour-colour grid when taking their
colour uncertainties into account. This means that most of
the 24 galaxies could have physical ages, but as pointed out
by Li et al. (2007), the presence of young populations in
such early-type galaxies can also make metal-rich popula-
tions outside the theoretical colour-colour grids.
c© 2002 RAS, MNRAS 000, 1–10
http://www.mpa-garching.mpg.de/SDSS/DR4/
6 Zhongmu Li and Zhanwen Han
Table 4. Stellar ages and metallicities of old (Age > 4.6 Gyr), young metal-rich (Age < 4.6 Gyr and Z > 0.02), and young metal-
poor (Age < 4.6 Gyr and Z < 0.02) galaxies. The results are measured by two kinds of methods (photometric and spectroscopic).
The photometric results are determined via (u − K) and (z − K) colours in the work, and the spectroscopic results are obtained by
Gallazzi et al. (2005). The symbol “objID” is the unique SDSS identifier of each galaxy. Note that the definition of metal-poor galaxy
here is different from that in other places of the paper, as the limitation of our sample galaxies.
Photometric results Spectroscopic results
Population type objID Age Error Z Error Age Error Z Error
(Gyr) (Gyr) (Gyr) (Gyr)
587727177921921120 4.6
0.0293
+0.0110
−0.0125
0.0411
+0.0077
−0.0171
Old 587727225693143113 5.7
+10.7
0.0300
+0.0143
−0.0176
0.0291
+0.0207
−0.0088
587727225692160083 9.1
0.0416
+0.0082
−0.0108
0.0350
+0.0141
−0.0127
587727177921986670 2.7
0.0478
+0.0022
−0.0121
0.0241
+0.0220
−0.0133
Young and metal-rich 587727225160466548 2.2
0.0385
+0.0089
−0.0081
0.0301
+0.0162
−0.0114
587727225689538771 2.7
0.0432
+0.0068
−0.0124
0.0305
+0.0146
−0.0109
587727230524063879 3.7 +8.0
0.0189 +0.0128
−0.0121
7.4 +2.9
0.0226 +0.0184
−0.0097
Young and metal-poor 587727227302707220 3.8
0.0166
+0.0038
−0.0085
0.0251
+0.0179
−0.0073
588848898845114541 2.9 +2.7
0.0192 +0.0097
−0.0044
3.5 +2.5
0.0135 +0.0304
−0.0054
In the work, we fitted the stellar ages and metallicities
of our sample galaxies via BC03 SSPs with ages within 0.1–
19.96 Gyr and metallicities within 0.0001–0.05. In Fig. 3, we
compare the parameters determined by colours with those
by a few Lick indices (Gallazzi et al. 2005). The results of
Gallazzi et al. (2005) were obtained by comparing D4000,
Hβ, HδA+HγA, [Mg2Fe], and [MgFe]
′ indices of galaxies to
the values of theoretical stellar populations and have taken
the effects of young populations (YSPs) into account. The
detailed data about the sample galaxies can be obtained on
request to the authors or via the CDS in the future. From
panels (a) and (c) of Fig. 3, we find that the stellar ages and
metallicities determined by colours are respectively smaller
and somewhat richer than those determined by Lick indices.
The reason is that there are composite stellar populations
(CSPs) in galaxies and the YSPs make the results derived
from colours bias younger and richer in metal compared
to those of the dominant populations (DSPs) of galaxies
(Li & Han 2007). In detail, because the age of a YSP affects
the colours of a star system more stronger than the mass
fraction of the YSP, YSPs with only a few percent stellar
mass can make the colour-fitted stellar populations younger
and metal-richer than DSPs of galaxies if the YSPs are not
too old. Then in panels (b) and (d) Fig. 3, we compare the
stellar-population parameters measured by colours and cor-
rected for the effects of YSPs to those determined by Lick in-
dices. The correction is accomplished using the possible dis-
tributions of the differences between the stellar-population
parameters of the DSPs of CSPs and the parameters derived
from two colours of CSPs (Li & Han 2007). The above distri-
butions were obtained by a statistical method, in which the
fractions of YSPs were assumed to depend on the ages of the
DSPs and YSPs, and the fraction of a YSP was assumed to
decline exponentially with decreasing age of the YSP (see,
e.g., Thomas et al. 2005). In special, the distributions can
be used to give a rough correction for the effects of YSPs.
Here, we get the corrected stellar age of a galaxy by submit-
ting from the colour-fitted result a random value that fits to
the distribution of the difference between the stellar ages of
DSPs and those derived from two colours. A similar method
is used to get the corrected stellar metallicities. For more
clearly, we show the distributions of the uncorrected and
corrected parameters in Fig. 4. We are shown that after the
correction, the distributions of stellar ages obtained by two
methods, especially the peaks, become more similar after the
correction (Fig. 4). However, it is also shown that the distri-
butions of stellar metallicities obtained by various methods
are similar. Therefore, it seems that without any correction,
we can get correct distributions of stellar metallicities of lu-
minous early-type galaxies via colours. Note that because we
used a least-square fitting method similar to Denicoló et al.
(2005) and Li et al. (2006), the effects of the uncertainties
of colours were not taken into account here. The uncertain-
ties may effect the above distributions slightly. Furthermore,
the dust may contribute to the difference between the two
kinds of results. In fact, a small amount of dust can make
(u−r) colours of galaxies redder, and then lead to additional
uncertainties in stellar ages. In detail, a dust of E(B − V )
= 0.05 will change the (u − r) and (r − K) colours about
0.12 mag, and then lead to about 2.5 Gyr age uncertainty
and 0.015 metallicity uncertainty. In addition, the average
stellar-population parameters obtained by photometry and
spectroscopy are found to be similar (about 7 Gyr for age
and 0.02 for metallicity). Therefore, although it is difficult
to get the accurate stellar-population parameters of each
galaxy via colours, we can get reliable values for the aver-
age age and metallicity of a sample of galaxies, as can the
distributions of the two parameters.
c© 2002 RAS, MNRAS 000, 1–10
Colour Pairs for Constraining Stellar-Population Parameters 7
Figure 2. Our sample galaxies (1 646 galaxies) in the (u− r) versus (r−K) grid. Dark points with error bars show the first 10 galaxies
of our sample. The colour uncertainties of other galaxies can refer to those shown. Lines have the same meanings as in Fig. 1. Here we
did not mark for the constant age of 8 and 16 Gyr.
5.2 Using colours in conjunction with
spectroscopic indices
Because some colours are shown to be sensitive to stellar
age or metallicity, we can possibly use colours in conjunc-
tion with Lick indices to study stellar-population parame-
ters. For example, we can use a metallicity-sensitive colour
together with an age-sensitive Lick index to estimate the
stellar ages and metallicities of galaxies. We have a try in this
work. The above galaxy sample are used here. The stellar-
population parameters are fitted via Hβ and (r − K) in-
dices using the same method as Sect 5.1. The results are
compared to those determined by Gallazzi et al. (2005) in
panels (a) and (b) of Fig. 5. Note that the results did NOT
correct for the effects of young populations. We see that
using Hβ instead of (u − r), the stellar ages obtained are
closer to those of Gallazzi et al. (2005). This means that
line indices are affected by young populations of galaxies
more slightly. However, the fitted stellar metallicities of some
galaxies are poorer than those obtained by Gallazzi et al.
(2005). It should mainly result from the effects of YSPs in
galaxies. In fact, YSPs make the (r −K) colour of galaxies
obviously bluer than those of the DSPs, but they affect the
Hβ index more slightly. When (r − K) is used in conjunc-
tion with Hβ to estimate the stellar-population parameters
of galaxies, the two indices will lead to more poor metallic-
ities compared those determined by a colour pair. One can
see a Hβ versus [MgFe] grid for comparison, as the shape
of Hβ versus (u − r) grid is similar to that of Hβ versus
[MgFe]. Therefore, it is difficult to use photometry in con-
junction with spectroscopy to estimate the metallicities of
galaxies. Furthermore, we tried to give some final results for
the stellar-population parameters of our sample galaxies us-
ing Hβ, [MgFe], (u−r) and (r−K) indices together. Because
the uncertainties of colours and Lick indices are different,
we use a χ2 fit (see, e.g., Press et al. 1992) to estimate the
stellar-population parameters. The stellar metallicities are
shown to be different significantly from those determined
by previous work. The comparison can be seen in panels
(c) and (d) of Fig. 5. The results did not correct for the
effects of young populations, either. Because the results of
(Gallazzi et al. 2005) have taken the effects of YSPs into ac-
count, the above results are certainly affected by the YSPs
in galaxies. The results may also effected by the large ob-
servational uncertainties. However, it seems that the YSPs
effect the results stronger, because when we tried to esti-
mate the stellar-population parameters via Hβ and [MgFe],
we got poorer metallicities than those of the previous work.
As a whole, our results suggest that colour pairs can be used
c© 2002 RAS, MNRAS 000, 1–10
8 Zhongmu Li and Zhanwen Han
Figure 3. Comparison of stellar ages and metallicities derived from colours with those from Lick indices. The results derived from
colours are obtained in this work and those derived from Lick indices were obtained by Gallazzi et al. (2005). The typical errors of
stellar ages obtained by colours and Lick indices are about 3.80 Gyr and 4.17 Gyr, respectively. The symbol “lick” denotes the results
of Gallazzi et al. (2005) while “colour” means the results derived from (r − K) and (u − r). The suffix “corrected” means the results
corrected for the effects of young stellar populations. Dashed lines show a ± 3.5 Gyr spread about the unity (solid) line for stellar ages
in panels a) and b) while show a ± 0.015 spread in stellar metallicities in panels c) and d).
more conveniently for estimating the stellar-population pa-
rameters of distant galaxies, as colours can be obtained more
easily than spectra and can constrain the stellar-population
parameters with uncertainties similar to Lick indices (typi-
cally 4 Gyr for age and 0.015 for metallicity).
6 DISCUSSIONS AND CONCLUSIONS
We investigated the relative metallicity sensitivities of AB
system colours relating to u, B, g, V , r, R, i, I , z, J , H , and
K bands in the first place. Then we studied the abilities of
colour pairs for constraining luminosity-weighed stellar ages
and metallicities. The results showed that [(r−K), (u−R)]
and [(r − K), (u − r)] are the best colour pairs for break-
ing the stellar age–metallicity degeneracy while colour pairs
such as (R−K), (u−R)], [(I−K), (u−R)], [(R−K)], (u−r)]
and [(i − J), (u − R)] can also be used. Colour pairs [(r −
K), (u−R)] and [(r −K), (u− r)] can measure two stellar-
population parameters with small uncertainties (∆t 6 3.89
Gyr, ∆[Z/H] 6 0.34 dex for typical uncertainties in colours).
However, the age uncertainties for old populations (Age >
4.6 Gyrs) and metal-poor populations (Z < 0.001) are al-
ways significantly larger than for young populations (Age <
4.6 Gyrs) and metal-rich populations (Z > 0.001). The rea-
son is that the colours of some old populations or metal-poor
populations are largely indistinguishable within present typ-
ical errors. One can see Fig. 3 for comparison. However,
the metallicity uncertainty of metal-poor populations (about
0.0024) is much less than that of other populations (about
0.015). In the work, we did not take the uncertainties in the-
oretical stellar population models into account. A detailed
study about it please refer to the work of Yi (2003). Further-
c© 2002 RAS, MNRAS 000, 1–10
Colour Pairs for Constraining Stellar-Population Parameters 9
Figure 5. Stellar-population parameters measured using colours in conjunction with line indices are plotted versus those measured by a
few line indices. The suffix “comp” represents the results derived from Hβ and (r−K) while “all” for the results measured by two colours
in conjunction with two line indices. Lines have meanings similar to Fig. 4. The typical uncertainties for stellar ages and metallicities
are 4 Gyr and 0.015, respectively.
more, colours are usually affected by the dust of galaxies,
even if early-type ones. A typical effect of dust in early-type
galaxies, E(B-V)∼ 0.05, will change the (r−K) and (u− r)
colours of galaxies about 0.12 mag, and then lead to addi-
tional uncertainties in stellar-population parameters (about
2.5 Gyr for age and 0.015 for metallicity). Thus it is more
suitable to use colours to measure the stellar-population pa-
rameters of galaxies with poor dust and gas, e.g., luminous
early-type ones (with small colour uncertainties). Actually,
to quantify the age and metallicity errors induced by a typ-
ical dust is in study. When we compare the results deter-
mined by photometry with those determined by Lick in-
dices, we found that the stellar ages determined by colours
are less than those determined by Lick indices. The differ-
ence mainly results from the effects of young populations in
these galaxies. We also found that the average uncertainties
of stellar-population parameters determined by colours and
Lick indices are similar (typically 4 Gyr for age and 0.015 for
metallicity). Therefore, it is actually difficult to get the ab-
solute values of stellar ages and metallicities via colours, but
we can get some relative values. This will be useful for some
statistical studies of the stellar populations of galaxies. In
addition, our results suggest that it is better to use colours
relating to both UBV RIJHK and ugriz bands than to use
those only relating to one of the two kinds of bands for esti-
mating the two stellar-population parameters. According to
the results, we can estimate the stellar-population parame-
ters of some distant galaxies, via the photometry data sup-
plied by, e.g., SDSS and 2MASS. The possible uncertainties
of using various colour pairs can be estimated. The results
presented in the paper can also help us to choose suitable
bands for the observation of stellar population studies.
We also tried to find some colour pairs that are suitable
for estimating the luminosity-weighted ages and metallicities
of some special stellar populations. The results show that
some colour pairs for estimating the two stellar-population
c© 2002 RAS, MNRAS 000, 1–10
10 Zhongmu Li and Zhanwen Han
Figure 4. Comparison of the distributions of stellar-population
parameters obtained by two kinds of methods. Solid lines are for
the results obtained by Lick indices (Gallazzi et al. 2005). Dotted
lines in panels a) and c) show the results obtained by [(r − K),
(u−r)] and without any correction, while those in in panels b) and
d) show the results corrected for the effects of young populations
using the results of Li & Han (2007).
parameters of young (Age < 4.6 Gyr), old (Age > 4.6 Gyr),
metal-poor (Z < 0.001), and metal-rich (Z > 0.001) popula-
tions are different. However, [(r−K), (u−r)] can be used to
estimate the ages and metallicities of all stellar populations.
Although we took the BC03 model for our work, the re-
sults can be used for other models as most stellar population
synthesis models predict similar colours for the same popu-
lation. Furthermore, we suggest to choose a colour pair for
estimating the two stellar-population parameters of galax-
ies, although a few colours can give smaller range for the
two parameters. The reason is that galaxies usually contain
more than two populations, and via comparing observational
colours with predictions of simple stellar populations, we can
only measure less stellar ages and richer stellar metallicities
for galaxies compared to the dominant stellar populations
(DSPs) of galaxies. For a sample of galaxies, the effects of
young populations on the ages and metallicities of dominant
stellar populations can be roughly corrected and some reli-
able estimations for the averages and distributions of two
stellar-population parameters can be obtained (Li & Han
2007).
Note that the BC03 model is a single stellar popula-
tion model, which does not take the binary interactions into
account. This is actually different from the real stellar popu-
lations of galaxies. If stars of a population evolve as binaries
rather than single stars, the colours of the population will be
different with those of a single stellar population. Typically,
the (u− r) and (r −K) colours predicted by binary stellar
populations will be bluer about 0.05 mag than those pre-
dicted by single populations. Using binary populations in-
stead of single populations, the stellar ages and metallicities
will be about 1.14 Gyr older and 0.0093 richer, respectively.
A detailed study about this will be given in the future.
ACKNOWLEDGMENTS
We greatly acknowledge the anonymous referee for some use-
ful comments. We also thank Profs. Gang Zhao, Xu Kong,
and Dr. Fenghui Zhang for useful discussions, Dr. Anna Gal-
lazzi for some useful discussions and her group for the line
indices, stellar ages and metallicities of our sample galaxies,
the Sloan Digital Sky Survey (SDSS), the Two-Micron All-
Sky Survey (2MASS) and the NASA/IPAC Extragalactic
Database (NED) for the photometry data of galaxies. This
work is supported by the Chinese National Science Foun-
dation (Grant Nos 10433030, 10521001), and the Chinese
Academy of Science (No. KJX2-SW-T06).
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c© 2002 RAS, MNRAS 000, 1–10
	Introduction
	The BC03 model and calculation of colours
	age and metallicity sensitivities of colours
	colour pairs for breaking the age–metallicity degeneracy
	Colour pairs for general studies
	Colour pairs for special studies
	Composite colour pairs
	Application of colours and colour pairs
	Using colour pairs to constrain stellar-population parameters
	Using colours in conjunction with spectroscopic indices
	discussions and conclusions

  Using a widely used stellar population synthesis model, we study the ability
of using pairs of AB system colours to break the well-known stellar
age--metallicity degeneracy and give constraints on the two stellar-population
parameters (age and metallicity). The relative age and metallicity
sensitivities of AB system colours that relate to u, B, g, V, r, R, i, I, z, J,
H, and K bands are presented, and the abilities of various colour pairs for
breaking the age--metallicity degeneracy are quantified by an uncertainty
parameter (UP) method. Our results suggest that a few pairs of colours can be
used to constrain the two above stellar-population parameters. This will be
very useful for exploring the stellar populations of distant galaxies. In
detail, colour pairs [(r-K), (u-R)] and [(r-K), (u-r)] are shown to be the best
pairs for estimating stellar ages and metallicities. They can constrain two
stellar-population parameters on average with age uncertainties less than 3.89
Gyr and metallicity uncertainties less than 0.34 dex for typical uncertainties
in colours. Some other colour pairs, such as [(R-K), (u-R)], [(I-K), (u-R)],
[(R-K), (u-r)] and [(i-J), (u-R)], can possibly be used for constraining the
two parameters, too. As a whole, our results suggest that colours relating to
both UBVRIJHK and ugriz magnitudes are much better than either UBVRIJHK colours
or ugriz colours for breaking the well-known degeneracy. The results also show
that the stellar ages and metallicities of galaxies observed by the Sloan
Digital Sky Survey (SDSS) and the Two-Micron All-Sky Survey (2MASS) can be
estimated via photometry data. It is also shown that the colours can be used in
conjunction with line indices to measure stellar-population parameters.

Introduction
	The BC03 model and calculation of colours
	age and metallicity sensitivities of colours
	colour pairs for breaking the age–metallicity degeneracy
	Colour pairs for general studies
	Colour pairs for special studies
	Composite colour pairs
	Application of colours and colour pairs
	Using colour pairs to constrain stellar-population parameters
	Using colours in conjunction with spectroscopic indices
	discussions and conclusions
