Exploring the Variable Sky with the Sloan Digital Sky Survey
Branimir Sesar1, Željko Ivezić1, Robert H. Lupton2, Mario Jurić3, James E. Gunn2,
Gillian R. Knapp2, Nathan De Lee4, J. Allyn Smith5, Gajus Miknaitis6, Huan Lin6,
Douglas Tucker6, Mamoru Doi7, Masayuki Tanaka8, Masataka Fukugita9, Jon
Holtzman10, Steve Kent6, Brian Yanny6, David Schlegel10, Douglas Finkbeiner11, Nikhil
Padmanabhan10, Constance M. Rockosi12, Nicholas Bond2, Brian Lee10, Chris
Stoughton6, Sebastian Jester13, Hugh Harris14, Paul Harding15, Jon Brinkmann16,
Donald P. Schneider17, Donald York18, Michael W. Richmond19, Daniel Vanden Berk17
ABSTRACT
1University of Washington, Dept. of Astronomy, Box 351580, Seattle, WA 98195-1580
2Princeton University Observatory, Princeton, NJ 08544-1001
3Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
4Dept. of Physics & Astronomy, Michigan State University, East Lansing, MI 48824-2320
5Dept. of Physics & Astronomy, Austin Peay State University, Box 4608, Clarksville, TN 37044
6Fermi National Accelerator Laboratory, Box 500, Batavia, IL 60510
7Institute of Astronomy, University of Tokyo, 2-21-1 Osawa,Mitaka, Tokyo 181-0015, Japan
8Dept. of Astronomy, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo,
113-0033, Japan
9Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba, Japan
10New Mexico State University, Box 30001, 1320 Frenger St., Las Cruces, NM 88003
10Lawrence Berkeley National Laboratory, One Cyclotron Road, MS 50R5032, Berkeley, CA, 94720
11Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
12University of California–Santa Cruz, 1156 High St., Santa Cruz, CA 95060
13School of Physics and Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK
14U.S. Naval Observatory, Flagstaff Station, Box 1149, Flagstaff, AZ 86002
15Department of Astronomy, Case Western Reserve University, Cleveland, Ohio 44106
16Apache Point Observatory, 2001 Apache Point Road, Box 59, Sunspot, NM 88349-0059
17Department of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802
18University of Chicago, Astronomy & Astrophysics Center, 5640 S. Ellis Ave., Chicago, IL 60637
19Rochester Institute of Technology, Department of Physics, 84 Lomb Memorial Dr., Rochester, NY 14623-
http://arxiv.org/abs/0704.0655v1
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We quantify the variability of faint unresolved optical sources using a catalog
based on multiple SDSS imaging observations. The catalog covers SDSS Stripe
82, which lies along the celestial equator in the Southern Galactic Hemisphere
(22h 24m < αJ2000 < 04h 08m, −1.27
◦ < δJ2000 < +1.27
◦, ∼ 290 deg2), and
contains 58 million photometric observations in the SDSS ugriz system for 1.4
million unresolved sources that were observed at least 4 times in each of the
gri bands (with a median of 10 observations obtained over ∼5 years). In each
photometric bandpass we compute various low-order lightcurve statistics such as
root-mean-square scatter (rms), χ2 per degree of freedom, skewness, minimum
and maximum magnitude, and use them to select and study variable sources.
We find that 2% of unresolved optical sources brighter than g = 20.5 appear
variable at the 0.05 mag level (rms) simultaneously in the g and r bands. The
majority (2/3) of these variable sources are low-redshift (< 2) quasars, although
they represent only 2% of all sources in the adopted flux-limited sample. We
find that at least 90% of quasars are variable at the 0.03 mag level (rms) and
confirm that variability is as good a method for finding low-redshift quasars as
is the UV excess color selection (at high Galactic latitudes). We analyze the
distribution of lightcurve skewness for quasars and find that is centered on zero.
We find that about 1/4 of the variable stars are RR Lyrae stars, and that only
0.5% of stars from the main stellar locus are variable at the 0.05 mag level. The
distribution of lightcurve skewness in the g − r vs. u − g color-color diagram
on the main stellar locus is found to be bimodal (with one mode consistent with
Algol-like behavior). Using over six hundred RR Lyrae stars, we demonstrate rich
halo substructure out to distances of 100 kpc. We extrapolate these results to
expected performance by the Large Synoptic Survey Telescope and estimate that
it will obtain well-sampled 2% accurate, multi-color lightcurves for ∼ 2 million
low-redshift quasars, and will discover at least 50 million variable stars.
Subject headings: Galaxy: halo — Galaxy: stellar content — quasars: general
— RR Lyrae
1. Introduction
Variability is an important phenomenon in astrophysical studies of structure and evolu-
tion, both stellar and galactic. Some variable stars, such as RR Lyrae, are an excellent tool
for studying the Galaxy. Being nearly standard candles (thus making distance determina-
tion relatively straightforward) and being intrinsically bright, they are a particularly suitable
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tracer of Galactic structure. In extragalactic astronomy, the optical continuum variability of
quasars is utilized as an efficient method for their discovery (van den Bergh, Herbst & Pritchet
1973; Hawkins 1983; Koo, Kron & Cudworth 1986; Hawkins & Veron 1995), and is also fre-
quently used to constrain the origin of their emission (Kawaguchi et al. 1998; Trevese et al.
2001; Martini & Schneider 2003).
Despite the importance of variability, the variable optical sky remains largely unex-
plored and poorly quantified, especially at the faint end. To what degree different variable
populations contribute to the overall variability, how they are distributed in magnitude and
color, what the characteristic time-scales and the dominant mechanisms of variability are,
are just some of the questions that still remain to be answered. To address these questions,
several contemporary projects aimed at regular monitoring of the optical sky were started.
Some of the more prominent surveys in terms of the sky coverage, depth, and cadence are:
• The Faint Sky Variability Survey (Groot et al. 2003) is a very deep (17 < V < 24)
BV I survey of 23 deg2 of sky, containing about 80,000 sources sampled at timescales
ranging from minutes to years.
• The QUEST Survey (Vivas et al. 2001) monitors 700 deg2 of sky from V = 13.5 to a
limit of V = 21.
• ROTSE-I (Akerlof et al. 2000) monitors the entire observable sky twice a night from
V = 10 to a limit of V = 15.5. The Northern Sky Variability Survey (Woźniak et al.
2004) is based on ROTSE-I data.
• OGLE (most recently OGLE III; Udalski et al. 2002) monitors ∼ 100 deg2 towards the
Galactic bulge from I = 11.5 to a limit of I = 20. Due to the very high stellar density
towards the bulge, OGLE II has detected about 270,000 variable stars (Woźniak et al.
2002; Żebruń et al. 2002).
• The MACHO Project monitored the brightness of ∼ 60 million stars in ∼ 90 deg2 of
sky toward the Magellanic Clouds and the Galactic bulge for ∼ 7 years to a limit of
V ∼ 24 (Alcock et al. 2001).
A comprehensive review of past and ongoing variability surveys can be found in Becker et al.
(2004).
Recognizing the outstanding importance of variable objects, the last Decadal Survey Re-
port (Astronomy and Astrophysics Survey Committee, Board on Physics and Astronomy, Space Studies Board, National Research Council
2001) highly recommended a major new initiative for studying the variable sky, the Large
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Synoptic Survey Telescope (LSST; Tyson et al. 2002; Walker 2003). The LSST1 will offer an
unprecedented view of the faint variable sky: according to the current designs it will scan
the entire accessible sky every three nights to a limit of V ∼ 25 with two observations per
night in two different bands (selected from a set of six). One of the LSST science goals2 will
be the exploration of the transient optical sky: the discovery and analysis of rare and exotic
objects (e.g. neutron star and black hole binaries), gamma-ray bursts, X-ray flashes, and of
new classes of transients, such as binary mergers and stellar disruptions by black holes. The
observed volume of space, and the requirement to recognize and monitor these events — in
real time — on a “normally” variable sky, will present a challenge to the project.
Since LSST will utilize3 the Sloan Digital Sky Survey (SDSS; York et al. 2000) photo-
metric system (ugriz, Fukugita et al. 1996), multiple photometric observations obtained by
the SDSS represent an excellent dataset for a pre-LSST study that characterizes the faint
variable sky and quantifies the variable population and its distribution in magnitude-color-
variability space. Here we present such a study of unresolved sources in a region that has
been imaged multiple times by the SDSS.
In Section 2 we give a brief overview of the SDSS imaging survey and repeated scans
of a ∼ 290 deg2 region called Stripe 82. In Section 3, we describe methods used to select
candidate variable sources from the SDSS Stripe 82 data assembled, averaged and recal-
ibrated by Ivezić et al. (2007), and present tests that show the robustness of the adopted
selection criteria. In the same section, we discuss the distribution of selected variable sources
in magnitude-color-variability space. The Milky Way halo structure traced by selected can-
didate RR Lyrae stars is discussed in Section 4, and in Section 5 we estimate the fraction of
variable quasars. Implications for surveys such as the LSST are discussed in Section 6, and
our main results are summarized in Section 7.
2. Overview of the SDSS Imaging and Stripe 82 Data
The quality of photometry and astrometry, as well as the large area covered by the
survey, make the SDSS stand out among available optical sky surveys (Sesar et al. 2006).
The SDSS is providing homogeneous and deep (r < 22.5) photometry in five bandpasses
1See [HREF]http://www.lsst.org
2For more details see [HREF]http://www.lsst.org/Science/science goals.shtml
3LSST will also use the Y band at ∼ 1 µm. For more details see the LSST Science Requirement Document
at [HREF]http://www.lsst.org/Science/lsst baseline.shtml
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(u, g, r, i, and z, Gunn et al. 1998; Hogg et al. 2002; Smith et al. 2002; Gunn et al. 2006;
Tucker et al. 2006) accurate to 0.02 mag (root-mean-square scatter, hereafter rms) for un-
resolved sources not limited by photon statistics (Scranton et al. 2002; Ivezić et al. 2003a),
and with a zeropoint uncertainty of 0.02 mag (Ivezić et al. 2004a). The survey sky coverage
of 10,000 deg2 in the northern Galactic cap, and 300 deg2 in the southern Galactic cap will
result in photometric measurements for well over 100 million stars and a similar number
of galaxies (Stoughton et al. 2002). The recent Data Release 5 (Adelman-McCarthy et al.
2007)4 lists photometric data for 215 million unique objects observed in 8000 deg2 of sky as
part of the “SDSS-I” phase that ran through June 2005. Astrometric positions are accu-
rate to better than 0.1′′ per coordinate (rms) for sources with r < 20.5 (Pier et al. 2003),
and the morphological information from the images allows reliable star-galaxy separation to
r ∼ 21.5 (Lupton et al. 2002). In addition, the 5-band SDSS photometry can be used for
very detailed source classification; e.g. separation of quasars and stars (Richards et al. 2002),
spectral classification of stars to within 1-2 spectral subtypes (Lenz et al. 1998; Finlator 2000;
Hawley et al. 2002), and even remarkably efficient color selection of the horizontal branch and
RR Lyrae stars (Yanny et al. 2000; Sirko et al. 2004; Ivezić et al. 2005) and low-metallicity
G and K giants (Helmi et al. 2003).
The equatorial Stripe 82 region (22h 24m < αJ2000 < 04h 08m, −1.27
◦ < δJ2000 <
+1.27◦, ∼ 290 deg2), observed in the southern Galactic cap, presents a valuable data source
for variability studies. The region was repeatedly observed (65 imaging runs by July 2005,
but not all cover the entire region), and it is the largest source of multi-epoch data in the
SDSS-I phase. Another source of the large number of scans is the SDSS-II Supernova Survey
(Frieman et al. 2007). By averaging the repeated observations of Stripe 82 sources, more
accurate photometry than the nominal 0.02 mag single-scan accuracy can be achieved. This
motivated Ivezić et al. (2007) to produce a catalog of recalibrated Stripe 82 observations.
The catalog lists 58 million photometric observations for 1.4 million unresolved sources that
were observed at least 4 times in each of the gri bands (with a median of 10 observations
obtained over ∼ 5 years). The random photometric errors for PSF (point spread function)
magnitudes are below 0.01 mag for stars brighter than 19.5, 20.5, 20.5, 20, 18.5 in ugriz,
respectively (about twice as accurate for individual SDSS runs), and the spatial variation of
photometric zeropoints is not larger than ∼0.01 mag (rms). Following Ivezić et al. (2007),
we use PSF magnitudes because they go deeper at a given signal-to-noise ratio than aperture
magnitudes, and have more accurate photometric error estimates than model magnitudes.
In addition, various low-order statistics such as root-mean-square scatter (Σ), χ2 per degree
of freedom (χ2), lightcurve skewness (γ), minimum and maximum PSF magnitude, were
4Please see [HREF]http://www.sdss.org/dr5
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computed for each ugriz band and each source. We compute χ2 per degree of freedom as
(xi − 〈x〉)
and lightcurve skewness γ as5
(n− 1)(n− 2)
(xi − 〈x〉)
3 (3)
(xi − 〈x〉)2 (4)
where n is the number of detections, xi is the magnitude, 〈x〉 is the mean magnitude, and
ξi is the photometric error.
Separation of quasars and stars, as well as efficient color selection of horizontal branch
and RR Lyrae stars, depend on accurate u band photometry. To ensure this, we select
748,084 unresolved sources from the Ivezić et al. (2007) catalog with at least 4 detections in
the u band. A catalog of variable sources selected from this sample is analyzed in Section 3
below.
3. Analysis of Stripe 82 Catalog of Variable Sources
In this section we describe methods for selecting candidate variable sources, and present
tests that show the robustness of the adopted selection criteria. The distribution of selected
variable sources in magnitude-color-variability space is also presented.
3.1. Methods and Selection Criteria
Due to a relatively small number of observations per source and random sampling, we
do not perform lightcurve fitting, but instead use low order statistics to select candidate
variables and study their properties. There are four parameters (median PSF magnitude,
root-mean-square scatter Σ, χ2, and lightcurve skewness γ) measured in five photometric
5We use equations from [HREF]http://www.xycoon.com/skewness small sample test 1.htm.
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bands (u, g, r, i, and z), for a total of 20 parameters. In the analysis presented here, we
utilize eight of them:
• median PSF magnitudes in the ugr bands (corrected for interstellar extinction using
the map from Schlegel, Finkbeiner & Davis 1998) because the g − r vs. u − g color-
color diagram has the most classification power (e.g. Smolčić et al. 2004 and references
therein).
• Σ and χ2 in the g and r bands, and
• lightcurve skewness γ(g) (the g band combines a high signal-to-noise ratio and large
variability amplitude for the majority of variable sources).
The observed root-mean-square scatter Σ includes both the intrinsic variability σ and
the mean photometric error 〈ξ(m)〉 as a function of magnitude. The dependence of Σ on
magnitude in the ugriz bands, is shown in Figure 1. For sources brighter than 18, 19.5, 19.5,
19, and 17.5 mag in the ugriz, respectively, the SDSS delivers 2% photometry with little or
no dependence on magnitude. We determine 〈ξ(m)〉 by fitting a fourth degree polynomial
to median Σ values in 0.5 mag wide bins (here we assume that the majority of sources are
not variable). The theoretically expected 〈ξ(m)〉 function (Strateva et al. 2001)
〈ξ(m)〉 = a+ b100.4m + c100.8m (5)
provides equally good fits. We define the intrinsic variability σ (hereafter rms scatter σ) as
σ = (Σ2 − 〈ξ(m)〉2)1/2 (6)
for Σ > 〈ξ(m)〉, and σ = 0 otherwise.
As the first variability selection criterion, we adopt σ(g) > 0.05 mag and σ(r) > 0.05
mag (hereafter written as σ(g, r) > 0.05 mag). At the bright end, this criterion is equivalent
to selecting sources with rms scatter greater than 2.5σ0, where σ0 = 0.02 mag is the mea-
surement noise. Selection cuts are applied simultaneously in the g and r bands to reduce
the number of “false positives” (intrinsically non-variable sources selected as candidate vari-
able sources due to measurement noise). About 6% of sources pass the σ cut in each band
separately, and ∼ 3% of sources pass the cut in both bands simultaneously. By selecting
sources with σ(g, r) > 0.05 mag, we also select faint sources that have large σ due to large
photometric errors at the faint end. To only select faint sources with statistically significant
rms scatter, we apply the χ2 test as the second selection cut.
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In the χ2 test, the value of χ2 per degree of freedom (calculated with respect to a
weighted mean magnitude and using errors computed by the photometric pipeline) deter-
mines whether the observed lightcurve is consistent with the Gaussian distribution of er-
rors. Large χ2 values show that the rms scatter is inconsistent with random fluctuations.
Ivezić et al. (2003a, 2007) used multi-epoch SDSS observations to show that the photometric
error distribution in the SDSS roughly follows a Gaussian distribution. A comparison of χ2
distributions in the g and r bands with a reference Gaussian χ2 distribution is shown in
Figure 2. As evident, χ2 distributions in both bands roughly follow the reference Gaussian
χ2 distribution for χ2 < 1, demonstrating that median photometric errors are correctly de-
termined. The discrepancy for larger χ2 is due to variable sources rather than non-Gaussian
error distributions, as we demonstrate below.
The second selection cut, χ2(g) > 3 and χ2(r) > 3 (hereafter written as χ2(g, r) > 3),
selects ∼ 90% of σ(g, r) > 0.05 mag sources, as shown in Figure 2 (middle panels). The
effectiveness of the χ2 test is demonstrated in the bottom panel of Figure 2. For magnitudes
fainter than g = 20.5, the fraction of candidate variables decreases as photometric errors
increase. The selection is relatively uniform for sources brighter than g = 20.5, and we
adopt this value as the flux limit for the selected variable sample.
There are 662,195 sources brighter than g = 20.5 in the full sample. Using σ(g, r) > 0.05
mag and χ2(g, r) > 3 as the selection criteria, we select 13,051 candidate variable sources6.
Therefore, at least 2% of unresolved optical sources brighter than g = 20.5 appear variable
at the > 0.05 mag level (rms) simultaneously in the g and r bands. The fraction of selected
variable sources is not a strong function of the minimum required number of observations,
but it does depend on the stellar density because the number of stars increases at lower
Galactic latitudes (see Fig. 5 in Ivezić et al. 2007) while the quasar count remains the same.
3.2. The Counts of Variable Sources
In this section we estimate the completeness and efficiency of the candidate variable
sample, and discuss the dependence of counts, rms scatter, σ(g)/σ(r) ratio, and the lightcurve
skewness γ(g) on the position in the g − r vs. u− g color-color diagram.
6A list of candidate variable sources and their data from Ivezić et al. (2007) are publicly available from
[HREF]http://www.sdss.org/dr5/products/value added/index.html
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3.2.1. Completeness
The selection completeness, defined as the fraction of true variable sources recovered by
the algorithm, depends on the lightcurve shape and amplitudes. Due to a fairly large number
of observations (median of 10), and small σ(g, r) cutoff compared to typical amplitudes of
variable sources (e.g. most RR Lyrae stars and quasars have peak-to-peak amplitudes ∼ 1
mag), we expect the completeness to be fairly high for RR Lyrae stars (& 95%, see Section 4)
and quasars (∼ 90%, see Section 5). The completeness for other types of variable sources,
such as flares and eclipsing binaries, is hard to estimate, but is probably low due to sparse
sampling.
3.2.2. Efficiency
The selection efficiency, defined as the fraction of true variable sources in the candidate
variable sample, determines the robustness of the selection algorithm. The main diagnostic
for the robustness of the adopted selection criteria is the distribution of selected candidates
in the SDSS color-magnitude and color-color diagrams. The position of a source in these
diagrams is a good proxy for its spectral classification (Lenz et al. 1998; Fan 1999; Finlator
2000; Smolčić et al. 2004).
Figure 3 compares the distribution of candidate variable sources to that of all sources
in the g − r vs. u − g color-color diagram. Were the selection a random process, the se-
lected candidates would have the same distribution as the full sample. The distributions of
candidate variables and of the full sample are remarkably different, demonstrating that the
candidate variables are not randomly selected from the parent sample.
The three dominant classes of variable objects are quasars, RR Lyrae stars, and stars
from the main stellar locus. The most obvious difference between the variable and the full
sample distributions is a much higher fraction of low-redshift quasars (< 2.2, recognized by
their UV excess, u − g < 0.7, see Richards et al. 2002) and RR Lyrae stars (u − g ∼ 1.15,
g − r < 0.3, see Ivezić et al. 2005) in the variable sample, and vividly shown in the bottom
panel of Figure 3.
Another interesting feature visible in this panel is a gradient in the fraction of variable
main stellar locus stars (perpendicular to the main stellar locus). We investigate this gradient
by first defining principal colors
P1 = 0.91u− 0.495g − 0.415r − 1.28 (7)
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s = −0.249u+ 0.794g − 0.555r + 0.234 (8)
where P1 and s are principal axis parallel and perpendicular to the main stellar locus,
respectively (Ivezić et al. 2004a). The s color is a measure of metallicity (Lenz et al. 1998),
and s > 0.05 stars are expected to be metal poor (Helmi et al. 2003). Sources with r < 19
and 0 < P1 < 0.9 are selected and binned in four s bins. For each bin we calculate the
fraction of source with σ(g) > 0.05 mag, the fraction of variable sources (selected with
σ(g, r) > 0.05 mag and χ2(g, r) > 3), median σ(g), and the total number of sources in the
bin (see Table 2). A greater fraction of variable sources in the last bin (s > 0.06) indicates
that, on average, metal-poor main stellar locus stars are more variable than the metal-rich
stars. This could be because this sample of metal-poor stars is expected to have a high
fraction of giants.
In order to quantify the differences between the full and the variable sample, we follow
Sesar et al. (2006) and divide the g − r vs. u− g color-color diagram into six characteristic
regions, each dominated by a particular type of source, as shown in Figure 4. The fractions
and counts of variable and all sources in each region are listed in Table 1 for g < 19, g < 20.5,
and g < 22 flux-limited samples. Notably, in the adopted g < 20.5 flux limit, the fraction
of Region II sources (dominated by low-redshift quasars) in the variable sample is 63%, or
∼ 30 times greater than the fraction of Region II sources in the full sample (∼ 2%). The
fraction of Region IV sources (which include RR Lyrae stars) in the variable sample is also
high when compared to the full sample (∼ 6 times higher).
As shown in Table 1, in the g = 20.5 flux-limited sample, we find that low-redshift
quasars and RR Lyrae stars (i.e. Regions II and IV) make 70% of the variable population,
while representing only 3% of all sources. Quasars alone account for 63% of the variable
population. Stars from the main stellar locus represent 95% of all sources and 25% of the
variable sample: about 0.5% of the stars from the locus are variable at the > 0.05 mag level.
3.3. The Properties of Variable Sources
Various lightcurve properties, such as shape and amplitude, are expected to be correlated
with stellar types. In this section we study the distribution of the rms scatter in the u and g
bands, and σ(g)/σ(r) ratio as a function of the u− g and g− r colors. To emphasize trends,
we bin sources and present median values for each bin.
The distribution of the median σ(u) and σ(g) values in the g − r vs. u − g color-color
diagram is shown in the top two panels of Figure 5. RR Lyrae stars show larger rms scatter
– 11 –
(& 0.3 mag) in the u and g bands, than low-redshift quasars or stars from the main stellar
locus. Quasars also show slightly larger rms scatter in the u band (∼ 0.1 mag) than in
the g band (∼ 0.07 mag), as discussed by Kinney et al. (1991),Ivezić et al. (2004b), and
Vanden Berk et al. (2004). If we define the degree of variability as the root-mean-square
scatter in the g band, then on average RR Lyrae stars show the greatest variability, followed
by quasars and the main stellar locus stars.
Another distinctive characteristic of variable sources is the ratio of flux changes in
different bandpasses. This property can be used to select different types of variable sources.
For example, RR Lyrae stars are bluer when brighter, a behavior used by Ivezić et al. (2000)
to select RR Lyrae using 2-epoch SDSS data. Here we define a new parameter, σ(g)/σ(r), to
express the ratio of flux changes in the g and r bands, and study its distribution in the g− r
vs. u − g color-color diagram. In particular, we examine this distribution and its median
values for three dominant classes of variable sources: quasars, RR Lyrae stars, and stars
from the main stellar locus.
The bottom left panel in Figure 5 shows the distribution of median σ(g)/σ(r) values as
a function of u− g and g − r colors. Using Fig. 5 we note that on average:
• RR Lyrae stars have σ(g)/σ(r) ∼ 1.4
• Main stellar locus stars have σ(g)/σ(r) ∼ 1, and
• Quasars show a σ(g)/σ(r) gradient in the g − r vs. u− g color-color diagram.
The average value of σ(g)/σ(r) ∼ 1.4 in Region IV indicates that RR Lyrae stars
dominate the variable source count in this region. The ratio of 1.4 for RR Lyrae stars was
also previously found by Ivezić et al. (2000). While Figure 5 only presents median values of
the rms scatter, Figure 6 shows how the rms scatter in the g and r bands correlates with the
u − g color for individual sources. Variable sources that follow the σ(g) = 1.4σ(r) relation
also correlate with the u− g color, and have u− g ∼ 1, as expected for RR Lyrae stars.
The average ratio of σ(g)/σ(r) ∼ 1 (i.e. gray flux variations) for stars in the main stellar
locus suggests that the variability could be caused by eclipsing systems. The distribution of
γ(g) for main stellar locus stars further strengthens this possibility, as discussed in Section 3.4
below.
The gradient in the σ(g)/σ(r) ratio observed for low-redshift quasars in the g−r vs. u−g
color-color diagram suggests that the variability correlation between the g and r bands is
more complex than in the case of RR Lyrae or main stellar locus stars. Wilhite et al. (2006)
– 12 –
show that the photometric color changes for quasars depend on the combined effects of con-
tinuum changes, emission-line changes, redshift, and the selection of photometric bandpasses.
They note that due to the lack of variability of the lines, measured photometric color is not
always bluer in brighter phases, but depends on redshift and the filters used. To verify the
dependence of broad-band photometric variability on redshift, we plot σ(g)/σ(r) vs. redshift
for all spectroscopically confirmed unresolved quasars from Schneider et al. (2005) which are
in Stripe 82, as shown in Figure 7. We confirm that the broad-band photometric variability
depends on the redshift, and that the σ(g)/σ(r) gradient in the g − r vs. u − g color-color
diagram can be explained by the increase in σ(g)/σ(r) from ∼ 1 to ∼ 1.6 in the 1.0 to 1.6
redshift range. This effect is due to the Mg II emission line (more stable in flux than the con-
tinuum) moving through the r band filter over this redshift range. The implied correlation
of the u−g and g− r colors with redshift is consistent with the discussion by Richards et al.
(2002). The lack of noticeable correlation of σ(g) with redshift is due to the combined effects
of the dependence of σ(g) on the rest-frame wavelength and time which cancel out (for a
detailed model see Ivezić et al. 2004b).
3.4. Skewness as a Proxy for Dominant Variability Mechanism
Lightcurve skewness, a measure of the lightcurve asymmetry, provides additional in-
formation on the type of variability. Negatively skewed, asymmetric lightcurves indicate
variable sources that spend more time fainter than (mmin+mmax)/2, where mmin and mmax
are magnitudes at the minimum and maximum. Type ab RR Lyrae stars, for example,
have negatively skewed lightcurves (γ ∼ −0.5, Wils, Lloyd & Bernhard 2006). Positively
skewed, asymmetric lightcurves indicate variable sources that spend more time brighter
than (mmin + mmax)/2 (e.g. eclipsing systems). Sources with symmetric lightcurves will
have γ ∼ 0.
The bottom right panel in Figure 5 shows the distribution of the median γ(g) as a
function of the position in the g − r vs. u − g color-color diagram. On average, quasars
and c type RR Lyrae stars (u − g ∼ 1.15, g − r < 0.15) have γ(g) ∼ 0, ab type RR Lyrae
(u − g ∼ 1.15, g − r > 0.15) have negative skewness (γ(g) ∼ −0.5), and stars in the main
stellar locus have positive skewness.
Figure 8 shows the distribution of the lightcurve skewness in the ugi bands for spec-
troscopically confirmed unresolved quasars from Schneider et al. (2005) which are in Stripe
82, candidate RR Lyrae stars (selection details are discussed in Section 4 below), and main
stellar locus stars from our variable sample. Stars in the main stellar locus show a bimodal
γ(g) distribution. This distribution suggests at least two, and perhaps more, different popu-
– 13 –
lations of variables. Indeed, when spectroscopically confirmed M dwarfs are selected, a third
peak appears at γ(g) −2.5, possibly associated with flaring M dwarfs (Kowalski et al. 2007).
The bimodality similar to the one in the g band is also discernible in the r band, while it is
less pronounced in the i band and not detected in the u and z bands (the r and z data are
not shown).
A comparison of the u− g and g − r color distributions for variable main stellar locus
stars brighter than g = 19 and a subset with highly asymmetric lightcurves (γ(g) > 2.5)
is shown in Figure 9. The subset with asymmetric lightcurves has an increased fraction of
stars with colors u− g ∼ 2.5 and g− r ∼ 1.4, that correspond to M stars. This may indicate
that M stars have a higher probability of being associated with an eclipsing companion than
stars with earlier spectral types. However, the selection effects are probably important since
a companion is easier to detect (due to the low luminosity of M dwarfs). Kowalski et al.
(2007) examine these issues using lightcurve data on a sample of spectroscopically confirmed
M dwarfs. Finally, quasars have symmetric lightcurves (γ ∼ 0) and their distribution of
skewness does not change between bands.
4. The Milky Way Halo Structure Traced by Candidate RR Lyrae Stars
Studies of substructures in the Galactic halo, such as clumps and streams, can constrain
the formation history of the Milky Way. One of the best tracers to study the outer halo
are RR Lyrae stars because they are nearly standard candles, are sufficiently bright to be
detected at large distances (5− 100 kpc for 14 < r < 20.7), and are sufficiently numerous to
trace the halo substructure with a high spatial resolution. The General Catalog of Variable
Stars (GCVS; Kholopov et al. 1988) lists7 RR Lyrae stars as RR Lyrae type ab (RRab)
and type c (RRc) stars. RRab stars have asymmetric lightcurves, periods from 0.3 to 1.2
days, and amplitudes from V ∼ 0.5 to V ∼ 2. RRc stars have nearly symmetric, sometimes
sinusoidal, lightcurves, with periods from 0.2 to 0.5 days, and amplitudes not greater than
V ∼ 0.8. In this work we assume MV = 0.7 as the absolute V band magnitude of RRab and
RRc stars. A comprehensive review of RR Lyrae stars can be found in Smith (1995).
In this section we fine tune criteria for selecting candidate RR Lyrae stars, and estimate
the selection completeness and efficiency. Using selected candidate RR Lyrae stars, we
recover a known halo clump associated with the Sgr dwarf tidal stream, and find several new
halo substructures.
7A list of GCVS variability types can be found at [HREF]http://www.sai.msu.su/groups/cluster/gcvs/gcvs/iii/vartype.txt
– 14 –
4.1. Criteria for Selecting RR Lyrae Stars
Figures 3, 4, and 5 show that RR Lyrae stars occupy a well-defined region (Region IV)
in the g− r vs. u−g color-color diagram, and Figure 6 shows how RR Lyrae stars follow the
σ(g) = 1.4σ(r) relation. Motivated by these results, we introduce color and σ(g)/σ(r) cuts
to specifically select candidate RR Lyrae stars from the variable sample, and study their
distribution in the rms scatter-color-lightcurve skewness parameter space.
RR Lyrae stars have distinctive colors and can be selected with the following criteria
(Ivezić et al. 2005):
0.98 < u− g < 1.30 (9)
− 0.05 < Dug < 0.35 (10)
0.06 < Dgr < 0.55 (11)
− 0.15 < r − i < 0.22 (12)
− 0.21 < i− z < 0.25 (13)
where
Dug = (u− g) + 0.67(g − r)− 1.07 (14)
Dgr = 0.45(u− g)− (g − r)− 0.12. (15)
We apply these cuts to our sample of candidate variables and select 846 sources. It is
implied by Ivezić et al. (2005) that RR Lyrae should always stay within these color bound-
aries, even though their colors change as a function of phase. Their distribution in the g− r
vs. u − g color-color diagram and rms scatter in the g band are shown in Figure 10 (top
left panel). The distribution of sources in the RR Lyrae region is inhomogeneous. Sources
with large rms scatter in the g band (& 0.2 mag) are centered around u− g ∼ 1.15, and are
separated by g − r ∼ 0.12 into two groups. A comparison with Figure 3 from Ivezić et al.
(2005) suggests that these large rms scatter sources might be RR Lyrae type ab (RRab,
g − r > 0.12) and type c stars (RRc, g − r < 0.12). Small rms scatter sources (. 0.1 mag)
have a fairly uniform distribution, and are slightly bluer with u− g . 1.1.
The distribution of sources from the RR Lyrae region in the σ(r) vs. σ(g) diagram is
presented in the top right panel of Figure 10. The majority of large rms scatter sources
follow the σ(g) = 1.4σ(r) relation, as expected for RR Lyrae stars. Since RR Lyrae stars
are bluer when brighter, or equivalently, have greater rms scatter in the g band than in the
r band, we require 1 < σ(g)/σ(r) 6 2.5 and select 683 candidate RR Lyrae stars.
– 15 –
A comparison of u − g color distributions for candidate RR Lyrae stars and of sources
with RR Lyrae colors, but not tagged as RR Lyrae stars, presented in the bottom left panel
of Figure 10, demonstrates the robustness of the RR Lyrae selection. The two distributions
are very different (the probability that they are the same is 10−4, as given by the KS test),
with the candidate RR Lyrae distribution peaking at u−g ∼ 1.15, as expected for RR Lyrae
stars.
One property that distinguishes RRab from RRc stars is the shape (or skewness) of their
lightcurves (in addition to lightcurve amplitude and period). RRab stars have asymmetric
lightcurves, while RRc lightcurves are symmetric. In the top left panel of Figure 10, we
noted that g − r ∼ 0.12 seemingly separates high rms scatter sources into two groups. If
g − r ∼ 0.12 is the boundary between the RRab and RRc stars, then the same boundary
should show up in the distribution of lightcurve skewness as a function of the g − r color.
As shown in Figure 10 (bottom left panel), this is indeed the case. On average, sources with
g− r < 0.12 have γ(g) ∼ 0 (symmetric lightcurves), as RRc stars, while g− r > 0.12 sources
have γ(g) ∼ −0.5 (asymmetric lightcurves) typical of RRab stars.
We show in Section 4.2 that candidate RR Lyrae stars with γ(g) > 1 are contaminated
by eclipsing variables. Therefore, to reduce the contamination by eclipsing variables, we also
require γ(g) 6 1, and select 634 sources as our final sample of candidate RR Lyrae stars.
4.2. Completeness and efficiency
The selection completeness, defined as the fraction of recovered RR Lyrae stars, will
depend on the color cuts, σ(g, r) cutoff, and the number of observations. The color cuts
(Eqs. 9 to 15) applied in Section 4.1 were chosen to minimize contamination by sources
other than RR Lyrae stars while maintaining an almost 100% completeness (Ivezić et al.
2005). With the σ(g, r) cutoff at 0.05 mag (small compared to the ∼ 1 mag typical peak-
to-peak amplitudes of RR Lyrae stars), and a fairly large number of observations per source
(median of 10), we estimate the RR Lyrae selection completeness to be & 95% (see Appendix
in Ivezić et al. 2000).
To determine the selection efficiency, defined as the fraction of true RR Lyrae stars in the
RR Lyrae candidate sample, we positionally match 683 candidate RR Lyrae stars selected
by 1 < σ(g)/σ(r) 6 2.5 to a sample of RR Lyrae sources selected from the SDSS Light-
Motion-Curve Catalog (LMCC; Bramich et al. 2007). This catalog covers the same region of
the sky as the one discussed here, but includes more recent SDSS-II observations that allow
the construction of lightcurves. We match 613 candidates, while 70 candidate RR Lyrae
– 16 –
stars from our sample, for some reason, do not have a match in the LMCC (De Lee, private
communication). Following the classification based on phased lightcurves by De Lee et al.
(2007), we find that 71% of sources in our candidate RR Lyrae sample are classified as RRab
and RRc, 28% are classified as variable non-RR Lyrae stars, and only 1% are spurious, non-
variable sources. The most significant contamination comes from a population of variable
sources bluer than u−g ∼ 1.1 (dotted line, bottom left panel Figure 11), possibly Population
II δ Scuti stars, also known as SX Phoenicis stars (Hoffmeister, Richter & Wenzel 1985).
The top left and the bottom right panels in Figure 11, show that RRab and RRc-
dominated regions are separated by g−r ∼ 0.12, as already hinted in Figure 10. Also, variable
non-RR Lyrae sources with γ(g) > 1 are classified by De Lee et al. (2007) as eclipsing
variables, justifying our γ(g) 6 1 cut.
To summarize, using color criteria and criteria based on σ(g), σ(r), and γ(g) RR Lyrae
stars are selected with & 95% completeness and ∼ 70% efficiency.
4.3. The Spatial Distribution of Candidate RR Lyrae Stars
Using the selection criteria from Section 4.1 we isolate 634 RR Lyrae candidates. The
magnitude-position diagram for these candidates within 2.5◦ from the Celestial Equator is
shown in Figure 12.
As discussed by Ivezić et al. (2005), an advantage of the data representation utilized
in Figure 12 (magnitude–right ascension diagram) is its simplicity – only “raw” data are
shown, without any post-processing. However, the magnitude scale is logarithmic and thus
the spatial extent of structures is heavily distorted. In order to avoid these shortcomings,
we have applied a Bayesian method for estimating continuous spatial density distribution
developed by Ivezić et al. (2005) (see their Appendix B). The resulting density map is shown
in the right panel in Figure 13. The advantage of that representation is that it better conveys
the significance of various local overdensities. For comparison, we also show a map of the
northern part of the equatorial strip constructed using 2-epoch data discussed by Ivezić et al.
(2000).
We detect several new halo substructures at & 3σ significance (compared to expected
Poissonian fluctuations) and present their approximate locations and properties in Table 3.
The most distant clump is at 100 kpc from the Galactic center. The strongest clump in
the left wedge belongs to the Sgr dwarf tidal stream as does the clump marked by C in
the right wedge (Ivezić et al. 2003a). We note that the apparent “clumpiness” of the candi-
date RR Lyrae distribution increases with increasing radius, similar to CDM predictions by
– 17 –
Bullock, Kravtsov & Weinberg (2001). A detailed comparison of their models with the data
presented here will be discussed elsewhere (Sesar et al., in prep).
5. Are All Quasars Variable?
The optical continuum variability of quasars has been recognized since their first op-
tical identification (Matthews & Sandage 1963), and it has been proposed and utilized as
an efficient method for their discovery (van den Bergh, Herbst & Pritchet 1973; Hawkins
1983; Koo, Kron & Cudworth 1986; Hawkins & Veron 1995). The observed characteristics
of the variability of quasars are frequently used to constrain the origin of their emission
(e.g. Kawaguchi et al. 1998 and references therein; Martini & Schneider 2003; Pereyra et al.
2006). Recently, significant progress in the description of quasar variability has been made by
employing the SDSS data (de Vries, Becker & White 2003; Ivezić et al. 2004b; Vanden Berk et al.
2004; de Vries et al. 2005; Sesar et al. 2006). Here we expand these studies by quantifying
the efficiency of quasar discovery using variability.
A preliminary comparison of color and variability based methods for selecting quasars
using SDSS data was presented by Ivezić et al. (2003b). They found that 47% of spectroscop-
ically confirmed unresolved quasars with UV excess have the g band magnitude difference
between two observations obtained two years apart larger than 0.15 mag. We can improve on
their analysis because now there are significantly more observations obtained over a longer
time period. Since quasars vary erratically and the rms scatter of their variability (the so-
called structure function) increases with time (e.g. Vanden Berk et al. 2004 and references
therein), the variability selection completeness is expected to be higher than ∼ 50% obtained
by Ivezić et al. (2003b).
First, although the adopted variability selection criteria discussed above are fairly con-
servative, we find that at least 63% of low-redshift quasars are variable at the > 0.05 mag
level (simultaneously in the g and r bands over observer’s time scales of several years) in
the g < 20.5 flux-limited sample. Second, even this estimate is only a lower limit: given the
spectroscopic confirmation for a large flux-limited sample of quasars, it is possible to relax
the adopted variability selection cutoff without a prohibitive contamination by non-variable
sources.
There are 2,492 unresolved quasars in the catalog of spectroscopically confirmed SDSS
quasars (Schneider et al. 2005) from Stripe 82. The fraction of these objects that vary more
than σ in the g and r bands, as a function of σ, is shown in Figure 14. We also show the
analogous fraction for stars from the stellar locus. About 93% of quasars vary with σ > 0.03
– 18 –
mag. For a small fraction of these objects the measured rms scatter is due to photometric
noise, and the stellar data limit this fraction to be at most 3%. Conservatively assuming that
none of these 3% of stars is intrinsically variable, we estimate that at least 90% of quasars
are variable at the 0.03 mag level on time scales up to several years.
6. Implications for Surveys such as LSST
The Large Synoptic Survey Telescope (LSST) is a proposed imaging survey that aims
to obtain repeated multi-band imaging to faint limiting magnitudes over a large fraction of
the sky. The LSST Science Requirement Document8 calls for ∼ 1000 repeated observations
of a solid angle of ∼ 20, 000 deg2 distributed over the six ugrizY photometric bandpasses
and over 10 years. The results presented here can be extrapolated to estimate the lower
limit on the number of variable sources that the LSST would discover.
The single-epoch LSST images will have a 5σ detection limit9 at r ∼ 24.7. Hence, 2%
accurate photometry, comparable to the subsample with g < 20.5 discussed here, will be
available for stars with r . 22. The USNO-B catalog (Monet et al. 2003) shows that there
are about 109 stars with r < 21 across the entire sky. About half of these stars are in the
parts of the sky to be surveyed by the LSST. The simulations based on contemporary Milky
Way models, such as those developed by Robin et al. (2003) and Jurić et al. (2007), predict
that there are about twice as many stars with r < 22 than with r < 21 across the whole
sky. Hence, it is expected that the LSST will detect about a billion stars with r < 22. This
estimate is uncertain to within a factor of two or so due to unknown details in the spatial
distribution of dust in the Galactic plane and towards the Galactic center.
We found that at least 0.5% of stars from the main stellar locus can be detected as
variable with photometry accurate to ∼ 2%. This is only a lower limit because a much
larger number of LSST observations obtained over a longer timespan than the SDSS data
discussed here would increase this fraction. Hence, our results imply that the LSST will
discover at least 50 million variable stars (without accounting for the fact that stellar counts
greatly increase closer to the Galactic plane). Unlike the SDSS sample, where RR Lyrae
stars account for ∼ 25% of all variable stars, the number of RR Lyrae stars in the LSST
sample will be negligible compared to other types of variable stars.
As estimated by Jurić et al. (2007) using deeper coadded SDSS photometry, there are
8 Available at [HREF]http://www.lsst.org/Science/lsst baseline.shtml
9An LSST Exposure Time Calculator is available at [HREF]www.lsst.org
– 19 –
about 100 deg−2 low-redshift quasars with r < 22 (see also Beck-Winchatz & Anderson 2007
and references therein). Therefore, with a sky coverage of ∼ 20, 000 deg2, the LSST will
obtain well-sampled accurate multi-color lightcurves for ∼ 2 million low-redshift quasars.
Even at the redshift limit of ∼ 2, this sample will be complete to Mr ∼ −24, that is,
almost to the formal quasar luminosity cutoff, and will represent an unprecedented sample
for studying quasar physics.
7. Conclusions and Discussion
We have designed and tested algorithms for selecting candidate variable sources from
a catalog based on multiple SDSS imaging observations. Using a sample of 13,051 selected
candidate variable sources in the adopted g < 20.5 flux-limited sample, we find that at least
2% of unresolved optical sources appear variable at the > 0.05 mag level, simultaneously
in the g and r bands. A similar fraction of variable sources (∼ 1%) was also found by
Sesar et al. (2006) using recalibrated photometric POSS and SDSS measurements, and by
Morales-Rueda et al. (2006) using the Faint Sky Variability Survey data (∼ 1%).
Thanks to the multi-color nature of the SDSS photometry, and especially to the u band
data, we can obtain robust classification of selected variable sources. The majority (2/3)
of variable sources are low-redshift (< 2) quasars, although they represent only 2% of all
sources in the adopted g < 20.5 flux-limited sample. We find that about 1/4 of variable stars
are RR Lyrae stars, and that only 0.5% of stars from the main stellar locus are variable at
the 0.05 mag level.
The distribution of γ(g) for main stellar locus stars is bimodal, suggesting at least two,
and perhaps more, different populations of variables. About a third of variable stars from
the stellar locus show gray flux variations in the g and r bands (σ(g)/σ(r) ∼ 1), and positive
lightcurve skewness, suggesting variability caused by eclipsing systems. This population has
an increased fraction of M type stars.
RR Lyrae stars show the largest rms scatter in the u and g bands, followed by low-
redshift quasars. The ratio of rms scatter in the g and r bands for RR Lyrae is ∼ 1.4, in agree-
ment with Ivezić et al. (2000) results based on 2-epoch photometry. The mean lightcurve
skewness for RR Lyrae stars is ∼ −0.5, in agreement with Wils, Lloyd & Bernhard (2006).
We selected a sample of 634 candidate RR Lyrae stars, with an estimated & 95% complete-
ness and ∼ 70% efficiency. Using these stars, we detected rich halo substructure out to
distances of 100 kpc. The apparent “clumpiness” of the candidate RR Lyrae distribution in-
creases with increasing radius, similar to CDM predictions by Bullock, Kravtsov & Weinberg
– 20 –
(2001).
Low-redshift quasars show a dependence of σ(g)/σ(r) on redshift, consistent with dis-
cussions in Richards et al. (2002) and Wilhite et al. (2006). The lightcurve skewness dis-
tribution for quasars is centered on zero in all photometric bands. We find that at least
90% of quasars are variable at the 0.03 mag level (rms) on time scales up to several years.
This confirms that variability is as a good a method for finding low-redshift quasars at high
(|b| > 30) Galactic latitudes as is the UV excess color selection. The fraction of variable
quasars at the > 0.1 mag level obtained here (30%, see Figure 14) is comparable to 36%
found by Rengstorf, Brunner & Wilhite (2006).
The multiple photometric observations obtained by the SDSS represent an excellent
dataset for estimating the impact of surveys such as the LSST on studies of the variable
sky. Our results indicate that the LSST will obtain well-sampled 2% accurate multi-color
lightcurves for ∼ 2 million low-redshift quasars, and will discover at least 50 million variable
stars. The number of variable stars discovered by the LSST will be of the same order as the
number of all stars detected by the SDSS. With about 1000 data points in six photometric
bands, it will be possible to recognize and classify variable objects using lightcurve mo-
ments of higher order than skewness discussed here, including lightcurve folding for periodic
variables.
We acknowledge support by NSF grant AST-0551161 to the LSST for design and de-
velopment activity.
Funding for the SDSS and SDSS-II has been 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/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions. The Participating Institutions are the American Museum of Natural History,
Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western
Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Ad-
vanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute
for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the
Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National
Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), New Mexico State University, Ohio State University, University of
Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Obser-
http://www.sdss.org/
– 21 –
vatory, and the University of Washington.
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This preprint was prepared with the AAS LATEX macros v5.2.
http://arxiv.org/abs/astro-ph/0303012
Table 1. The distribution of candidate variable sources in the g − r vs u− g diagram
g < 19 g < 20.5 g < 22
Regiona Nameb % allc % vard var/alle Nvar/N
% allc % vard var/alle Nvar/N
% allc % vard var/alle Nvar/N
I white dwarfs 0.14 0.59 4.25 3.50 0.24 0.40 1.69 3.34 0.28 0.45 1.64 4.51
II low-redshift QSOs 0.45 30.88 68.83 56.58 1.90 62.90 33.03 65.10 4.07 70.01 17.22 47.30
III dM/WD pairs 0.08 0.53 6.54 5.37 0.83 2.08 2.50 4.92 1.21 3.79 3.13 8.61
IV RR Lyrae stars 1.28 16.81 13.11 10.78 1.33 7.95 5.99 11.81 1.48 6.41 4.33 11.90
V stellar locus stars 96.27 48.77 0.51 0.42 94.49 25.15 0.27 0.52 91.89 18.33 0.20 0.55
VI high-redshift QSOs 1.78 2.42 1.36 1.12 1.21 1.52 1.26 2.48 1.07 1.01 0.95 2.60
total count 411,667 3,384 662,195 13,051 748,067 20,553
aThese regions are defined in the g − r vs. u− g color-color diagram, with their boundaries shown in Fig. 4
bAn approximate description of the dominant source type found in the region
cThe fraction of all sources in a magnitude-limited sample found in this color region, with the magnitude limits listed on top.
dThe number of candidate variables from the region, expressed as a fraction of all variable sources
eThe ratio of values listed in columns d) and c)
fThe number of candidate variables from the region, expressed as a fraction of all sources in that region
Table 2. The fraction of variable main stellar locus stars as a function of the s color
Bin % σ(g) > 0.05a % varb 〈σ(g)〉c Countsd
s < −0.02 3.23 0.36 0.017 46,836
−0.02 < s < 0.02 2.92 0.28 0.017 136,910
0.02 < s < 0.06 4.61 1.18 0.019 29,106
s > 0.06 11.50 4.10 0.027 4,547
aFraction of sources with σ(g) > 0.05 mag
bFraction of variable sources (selected using σ(g, r) > 0.05 mag and
χ2(g, r) > 3)
cMedian σ(g)
dNumber of sources in the bin
Table 3. Approximate locations and properties of detected overdensities
Labela Nb 〈R.A.〉c 〈d〉d 〈r〉e 〈u− g〉f 〈g − rg〉 〈γ(g)〉h Nb/N
A 84 330.95 21 17.02 1.14 0.18 -0.50 0.62
B 144 309.47 22 16.76 1.12 0.16 -0.57 0.64
C 54 33.69 25 17.61 1.13 0.20 -0.68 0.29
D 8 347.91 29 18.02 1.14 0.23 -0.50 0.38
E 11 314.06 43 18.84 1.09 0.20 -0.41 0.75
F 11 330.26 48 19.16 1.07 0.20 -0.46 0.38
G 10 354.81 55 19.46 1.10 0.22 -0.69 0.38
H 7 43.57 57 19.32 1.05 0.04 0.06 1.34
I 4 311.34 72 19.98 1.08 0.11 -0.10 2.0
J 26 353.58 81 20.21 1.11 0.20 -0.27 0.58
K 8 28.39 84 20.35 1.10 0.20 0.14 0.44
L 3 339.01 92 20.45 1.06 0.16 0.08 0.67
M 5 39.45 102 20.73 1.07 0.11 0.36 1.67
aOverdensity’s label from Fig. 13
bNumber of candidate RR Lyrae in the overdensity
cMedian Right Ascension
dMedian distance (in kpc)
eMedian r band magnitude
fMedian u− g color
gMedian g − r color
hMedian γ(g)
iThe number ratio of candidate RR Lyrae with g − r < 0.12 and g − r > 0.12
– 28 –
Fig. 1.— The dependence of the median root-mean-square (rms) scatter Σ in SDSS ugrz
bands on magnitude (symbols). The vertical bars show the rms scatter of Σ in each bin
(not the error of the median). The dependence of Σ in the i band is similar to the r band
dependence. In each band, a fourth-degree polynomial is fitted through medians and shown
by the solid line.
– 29 –
Fig. 2.— Top: The cumulative distribution of χ2 g and r values for all sources (solid line)
and a reference Gaussian χ2 distribution with 9 degrees of freedom (dashed line). Vertical
dashed lines show adopted selection cuts on χ2(g) and χ2(r) values. Middle: The fraction
of σ(g, r) > 0.05 mag sources with χ2 per degree of freedom greater than χ2 (only in the
g or r band: solid line, both in the g and r bands: dashed line). Bottom: The fraction of
σ(g, r) > 0.05 mag sources with χ2(m) > 2 (dashed line) or χ2(m) > 3 (solid line) as a
function of magnitude for m = g, r bands, respectively.
– 30 –
Fig. 3.— The distribution of counts for the full sample (top), candidate variable sample
(middle), and the ratio of two counts (bottom) in the g− r vs. u− g color-color diagram for
sources brighter than g = 20.5, binned in 0.05 mag bins. Contours outline distributions of
unbinned counts. Note the remarkable difference between the distribution of all sources and
that of the variable sample, which demonstrates that the latter are robustly selected.
– 31 –
Fig. 4.— The distribution of 18,329 candidate variable sources brighter than g = 21 in
representative SDSS color-magnitude and color-color diagrams. Candidate variables are
color-coded by their rms scatter in the g band (0.05-0.2, see the legend, red if larger or equal
than 0.2). Only sources brighter than g = 20 are plotted in color-color diagrams. Note
how RR Lyrae stars (u − g ∼ 1.15, red dots, σ(g) & 0.2 mag) and low-redshift quasars
(u− g 6 0.7, green dots, σ(g) & 0.1 mag) stand out as highly variable sources. The regions
marked in the top right panel are used for quantitative comparison of the overall and variable
source distributions (see Table 1).
– 32 –
Fig. 5.— The distribution of the rms scatter σ(u) (top left), rms scatter σ(g) (top right),
σ(g)/σ(r) ratio (bottom left), and γ(g) (bottom right) for the variable sample in the g − r
vs. u − g color-color diagram. Sources are binned in 0.05 mag wide bins and the median
values are color-coded. Color ranges are given at the top of each panel, going from blue to
red, where the green color is in the mid-range. Values outside the range saturate in blue or
red. Contours outline the count distributions on a linear scale in steps of 15%. Flux limit is
g < 20.5, with an additional u < 20.5 limit in the top left panel. Bottom left: On average,
RR Lyrae stars have σ(g)/σ(r) ∼ 1.4, main stellar locus stars have σ(g)/σ(r) ∼ 1, while
low-redshift quasars show a gradient of σ(g)/σ(r) values. Bottom right: On average, quasars
and c type RR Lyrae stars (u − g ∼ 1.15, g − r < 0.15) have γ(g) ∼ 0, ab type RR Lyrae
(u− g ∼ 1.15, g− r > 0.15) have negative skewness, and stars in the main stellar locus have
positive skewness.
– 33 –
Fig. 6.— The distribution of candidate variable sources in the g < 20.5 flux-limited sample
is shown by linearly-spaced contours, and by symbols color-coded by the u − g color for
sources with σ(g) > 0.05 mag and σ(r) > 0.05 mag. The dotted lines show the adopted
σ(g, r) selection cut. The thick solid line shows σ(g) = σ(r), while the dashed line shows
σ(g) = 1.4σ(r) relation representative of RR Lyrae stars. Note that sources following the
σ(g) = 1.4σ(r) relation tend to have u−g ∼ 1, as expected for RR Lyrae stars. The greyscale
background shows the fraction of χ2(g, r) > 3 sources which also have σ(g) > x and σ(r) > y
and demonstrates that large χ2 sources also have large σ.
– 34 –
0 0.5 1 1.5 2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3 3.5
Fig. 7.— The dependence of σ(g)/σ(r) (top), g − r (middle), and σ(g) on redshift for
a sample of spectroscopically confirmed unresolved quasars from Schneider et al. (2005).
The σ(g)/σ(r) gradient shown in Fig. 5 (bottom left panel) can be explained by the local
maximum of σ(g)/σ(r) in the 1.0 to 1.6 redshift range.
– 35 –
Fig. 8.— The lightcurve skewness distribution in the ugi bands for spectroscopically con-
firmed unresolved quasars (dotted line), candidate RR Lyrae stars (dashed line), and variable
main stellar locus stars (solid line, Region V, see Fig. 4 for the definition). The distribution
of the skewness in the r band is similar to the g band distribution, and the distribution of
skewness in the z band is similar to the u band distribution (therefore the r and z data
are not shown). Stars in the main stellar locus show bimodality in the γ(g) suggesting at
least two, and perhaps more, different populations of variables. Similar bimodality is also
discernible in the r band, while it is less pronounced in the i band and not detected in the u
and z bands. Quasars have symmetric lightcurves (γ ∼ 0) and their distribution of skewness
does not change between bands.
– 36 –
0.5 1 1.5 2 2.5 3 3.5
0.5 1 1.5 2
Fig. 9.— A comparison of the u−g (left) and g−r (right) color distributions for variable main
stellar locus stars brighter than g = 19 (dashed lines), and a subset with highly asymmetric
lightcurves (γ(g) > 2.5, solid lines). The subset with highly asymmetric lightcurves has an
increased fraction of stars with colors u− g ∼ 2.5 and g− r ∼ 1.5, characteristic of M stars.
– 37 –
0.9 1 1.1 1.2 1.3
0 0.1 0.2 0.3 0.4 0.5
0.9 1 1.1 1.2 1.3
-0.1 0 0.1 0.2 0.3
Fig. 10.— Top left: The distribution of 846 candidate variable sources from the RR Lyrae
region (dashed lines, see Fig. 3 in Ivezić et al. 2005) in the g−r vs. u−g color-color diagram.
The symbols mark the time-averaged values and are color-coded by σ(g) (0.05 to 0.2, blue to
red). The dotted horizontal line shows the boundary between the RRab and RRc-dominated
regions. Top right: Sources from the top left panel divided into 3 groups according to their
σ(g)/σ(r) values: candidate RR Lyrae stars with 1 < σ(g)/σ(r) 6 2.5 (large dots), sources
with σ(g)/σ(r) 6 1 (triangles), and sources with σ(g)/σ(r) > 2.5 (squares). Small dots show
sources with RR Lyrae colors that fail the variability criteria. The dashed lines show the
σ(g) = σ(r) and σ(g) = 2.5σ(r) relations, while the dotted line shows the σ(g) = 1.4σ(r)
relation. Bottom left: A comparison of the u− g color distributions for candidate RR Lyrae
stars (solid line) and sources with RR Lyrae colors but not tagged as RR Lyrae stars (dashed
line). Bottom right: The dependence of γ(g) on the g−r color for candidate RR Lyrae stars.
The boundary g − r = 0.12 (vertical dotted line) separates candidate RR Lyrae stars into
those with asymmetric (γ(g) ∼ −0.5) and symmetric (γ(g) ∼ 0) lightcurves, corresponding
to RRab and RRc stars, respectively. The condition γ(g) 6 1 (horizontal dashed line) is
used to reduce the contamination of the RR Lyrae sample by eclipsing variables.
– 38 –
0.9 1 1.1 1.2 1.3
0 0.1 0.2 0.3 0.4 0.5
0.9 1 1.1 1.2 1.3
-0.1 0 0.1 0.2 0.3
Fig. 11.— The distribution of candidate RR Lyrae stars selected with 1 < σ(g)/σ(r) 6 2.5
and classified by De Lee et al. (2007), shown in diagrams similar to Fig. 10. Symbols show
RRab stars (red dots), RRc stars (blue dots), variable non-RR Lyrae stars (green dots),
and non-variable sources (open squares, only four sources). A comparison of the u − g
color distribution for RRab (solid line), RRc (dashed line), and variable non-RR Lyrae stars
(dotted line) is shown in the bottom left panel.
– 39 –
60 50 40 30 20 10 0 -10 -20 -30 -40 -50
634 Stripe 82 RR Lyrae candidates
25 kpc
50 kpc
75 kpc
100 kpc
JK LM
Fig. 12.— The magnitude-position distribution of 634 Stripe 82 RR Lyrae candidates within
−55◦ < R.A. < 60◦ and |Dec| 6 1.27◦. Approximate distance (shown on the right y-axis)
is calculated assuming Mr = 0.7 mag for RR Lyrae stars. Dashed lines show where sample
completeness decreases from approximately 99% to 60% due to the χ2 cut (see the bottom
right panel in Fig. 2). Closed curves are remapped ellipses and circles from Fig. 13 that
mark halo substructure.
– 40 –
25       
50       
75       
100            kpc
Fig. 13.— Left: The spatial distribution of candidate RR Lyrae stars discovered by SDSS
along the Celestial Equator. Distance is calculated assuming Eq. 3 from Ivezić et al. (2005)
and MV = 0.7 mag as the absolute magnitude of RR Lyrae in the V band. The right
wedge corresponds to candidate RR Lyrae selected in this work (634 candidates, shown in
Fig. 12) and the left wedge is based on the sample from Ivezić et al. (2000) (296 candidates).
Right: The number density distribution of candidate RR Lyrae stars shown in the left panel,
computed using an adaptive Bayesian density estimator developed by Ivezić et al. (2005).
The color scheme represents the number density multiplied by the cube of the galactocentric
radius, and displayed on a logarithmic scale with a dynamic range of 300 (from light blue to
red). The green color corresponds to the mean density – both wedges with the data would
have this color if the halo number density distribution followed a perfectly smooth r−3 power-
law. The purple color marks the regions with no data. The yellow regions are formally ∼ 3σ
significant overdensities, and orange/red regions have an even higher significance (using
only the counts variance). The strongest clump in the left wedge belongs to the Sgr dwarf
tidal stream as does the clump marked by C in the right wedge (Ivezić et al. 2003a). An
approximate location and properties of labeled overdensities are listed in Table 3. The
Ivezić et al. (2000) sample is based on only 2 epochs and thus has a much lower completeness
(∼ 56%) resulting in a lower density contrast.
– 41 –
Fig. 14.— The fraction of spectroscopically confirmed unresolved QSOs (fQSO, solid line)
and the fraction of sources from the stellar locus (floc, dashed line) brighter than g = 19.5 and
r = 19.5 that have rms scatter larger than σ in the g and r bands. The ratio fQSO/(1+ floc)
(dotted line), which corresponds to the implied fraction of variable QSOs, peaks at a level
of 90% for σ = 0.03 mag.
	Introduction
	Overview of the SDSS Imaging and Stripe 82 Data
	Analysis of Stripe 82 Catalog of Variable Sources
	Methods and Selection Criteria
	The Counts of Variable Sources
	Completeness
	Efficiency
	The Properties of Variable Sources
	Skewness as a Proxy for Dominant Variability Mechanism
	The Milky Way Halo Structure Traced by Candidate RR Lyrae Stars 
	Criteria for Selecting RR Lyrae Stars
	Completeness and efficiency
	The Spatial Distribution of Candidate RR Lyrae Stars
	Are All Quasars Variable?
	Implications for Surveys such as LSST
	Conclusions and Discussion

  We quantify the variability of faint unresolved optical sources using a
catalog based on multiple SDSS imaging observations. The catalog covers SDSS
Stripe 82, and contains 58 million photometric observations in the SDSS ugriz
system for 1.4 million unresolved sources. In each photometric bandpass we
compute various low-order lightcurve statistics and use them to select and
study variable sources. We find that 2% of unresolved optical sources brighter
than g=20.5 appear variable at the 0.05 mag level (rms) simultaneously in the g
and r bands. The majority (2/3) of these variable sources are low-redshift (<2)
quasars, although they represent only 2% of all sources in the adopted
flux-limited sample. We find that at least 90% of quasars are variable at the
0.03 mag level (rms) and confirm that variability is as good a method for
finding low-redshift quasars as is the UV excess color selection (at high
Galactic latitudes). We analyze the distribution of lightcurve skewness for
quasars and find that is centered on zero. We find that about 1/4 of the
variable stars are RR Lyrae stars, and that only 0.5% of stars from the main
stellar locus are variable at the 0.05 mag level. The distribution of
lightcurve skewness in the g-r vs. u-g color-color diagram on the main stellar
locus is found to be bimodal (with one mode consistent with Algol-like
behavior). Using over six hundred RR Lyrae stars, we demonstrate rich halo
substructure out to distances of 100 kpc. We extrapolate these results to
expected performance by the Large Synoptic Survey Telescope and estimate that
it will obtain well-sampled 2% accurate, multi-color lightcurves for ~2 million
low-redshift quasars, and will discover at least 50 million variable stars.

Introduction
Variability is an important phenomenon in astrophysical studies of structure and evolu-
tion, both stellar and galactic. Some variable stars, such as RR Lyrae, are an excellent tool
for studying the Galaxy. Being nearly standard candles (thus making distance determina-
tion relatively straightforward) and being intrinsically bright, they are a particularly suitable
– 3 –
tracer of Galactic structure. In extragalactic astronomy, the optical continuum variability of
quasars is utilized as an efficient method for their discovery (van den Bergh, Herbst & Pritchet
1973; Hawkins 1983; Koo, Kron & Cudworth 1986; Hawkins & Veron 1995), and is also fre-
quently used to constrain the origin of their emission (Kawaguchi et al. 1998; Trevese et al.
2001; Martini & Schneider 2003).
Despite the importance of variability, the variable optical sky remains largely unex-
plored and poorly quantified, especially at the faint end. To what degree different variable
populations contribute to the overall variability, how they are distributed in magnitude and
color, what the characteristic time-scales and the dominant mechanisms of variability are,
are just some of the questions that still remain to be answered. To address these questions,
several contemporary projects aimed at regular monitoring of the optical sky were started.
Some of the more prominent surveys in terms of the sky coverage, depth, and cadence are:
• The Faint Sky Variability Survey (Groot et al. 2003) is a very deep (17 < V < 24)
BV I survey of 23 deg2 of sky, containing about 80,000 sources sampled at timescales
ranging from minutes to years.
• The QUEST Survey (Vivas et al. 2001) monitors 700 deg2 of sky from V = 13.5 to a
limit of V = 21.
• ROTSE-I (Akerlof et al. 2000) monitors the entire observable sky twice a night from
V = 10 to a limit of V = 15.5. The Northern Sky Variability Survey (Woźniak et al.
2004) is based on ROTSE-I data.
• OGLE (most recently OGLE III; Udalski et al. 2002) monitors ∼ 100 deg2 towards the
Galactic bulge from I = 11.5 to a limit of I = 20. Due to the very high stellar density
towards the bulge, OGLE II has detected about 270,000 variable stars (Woźniak et al.
2002; Żebruń et al. 2002).
• The MACHO Project monitored the brightness of ∼ 60 million stars in ∼ 90 deg2 of
sky toward the Magellanic Clouds and the Galactic bulge for ∼ 7 years to a limit of
V ∼ 24 (Alcock et al. 2001).
A comprehensive review of past and ongoing variability surveys can be found in Becker et al.
(2004).
Recognizing the outstanding importance of variable objects, the last Decadal Survey Re-
port (Astronomy and Astrophysics Survey Committee, Board on Physics and Astronomy, Space Studies Board, National Research Council
2001) highly recommended a major new initiative for studying the variable sky, the Large
– 4 –
Synoptic Survey Telescope (LSST; Tyson et al. 2002; Walker 2003). The LSST1 will offer an
unprecedented view of the faint variable sky: according to the current designs it will scan
the entire accessible sky every three nights to a limit of V ∼ 25 with two observations per
night in two different bands (selected from a set of six). One of the LSST science goals2 will
be the exploration of the transient optical sky: the discovery and analysis of rare and exotic
objects (e.g. neutron star and black hole binaries), gamma-ray bursts, X-ray flashes, and of
new classes of transients, such as binary mergers and stellar disruptions by black holes. The
observed volume of space, and the requirement to recognize and monitor these events — in
real time — on a “normally” variable sky, will present a challenge to the project.
Since LSST will utilize3 the Sloan Digital Sky Survey (SDSS; York et al. 2000) photo-
metric system (ugriz, Fukugita et al. 1996), multiple photometric observations obtained by
the SDSS represent an excellent dataset for a pre-LSST study that characterizes the faint
variable sky and quantifies the variable population and its distribution in magnitude-color-
variability space. Here we present such a study of unresolved sources in a region that has
been imaged multiple times by the SDSS.
In Section 2 we give a brief overview of the SDSS imaging survey and repeated scans
of a ∼ 290 deg2 region called Stripe 82. In Section 3, we describe methods used to select
candidate variable sources from the SDSS Stripe 82 data assembled, averaged and recal-
ibrated by Ivezić et al. (2007), and present tests that show the robustness of the adopted
selection criteria. In the same section, we discuss the distribution of selected variable sources
in magnitude-color-variability space. The Milky Way halo structure traced by selected can-
didate RR Lyrae stars is discussed in Section 4, and in Section 5 we estimate the fraction of
variable quasars. Implications for surveys such as the LSST are discussed in Section 6, and
our main results are summarized in Section 7.
2. Overview of the SDSS Imaging and Stripe 82 Data
The quality of photometry and astrometry, as well as the large area covered by the
survey, make the SDSS stand out among available optical sky surveys (Sesar et al. 2006).
The SDSS is providing homogeneous and deep (r < 22.5) photometry in five bandpasses
1See [HREF]http://www.lsst.org
2For more details see [HREF]http://www.lsst.org/Science/science goals.shtml
3LSST will also use the Y band at ∼ 1 µm. For more details see the LSST Science Requirement Document
at [HREF]http://www.lsst.org/Science/lsst baseline.shtml
– 5 –
(u, g, r, i, and z, Gunn et al. 1998; Hogg et al. 2002; Smith et al. 2002; Gunn et al. 2006;
Tucker et al. 2006) accurate to 0.02 mag (root-mean-square scatter, hereafter rms) for un-
resolved sources not limited by photon statistics (Scranton et al. 2002; Ivezić et al. 2003a),
and with a zeropoint uncertainty of 0.02 mag (Ivezić et al. 2004a). The survey sky coverage
of 10,000 deg2 in the northern Galactic cap, and 300 deg2 in the southern Galactic cap will
result in photometric measurements for well over 100 million stars and a similar number
of galaxies (Stoughton et al. 2002). The recent Data Release 5 (Adelman-McCarthy et al.
2007)4 lists photometric data for 215 million unique objects observed in 8000 deg2 of sky as
part of the “SDSS-I” phase that ran through June 2005. Astrometric positions are accu-
rate to better than 0.1′′ per coordinate (rms) for sources with r < 20.5 (Pier et al. 2003),
and the morphological information from the images allows reliable star-galaxy separation to
r ∼ 21.5 (Lupton et al. 2002). In addition, the 5-band SDSS photometry can be used for
very detailed source classification; e.g. separation of quasars and stars (Richards et al. 2002),
spectral classification of stars to within 1-2 spectral subtypes (Lenz et al. 1998; Finlator 2000;
Hawley et al. 2002), and even remarkably efficient color selection of the horizontal branch and
RR Lyrae stars (Yanny et al. 2000; Sirko et al. 2004; Ivezić et al. 2005) and low-metallicity
G and K giants (Helmi et al. 2003).
The equatorial Stripe 82 region (22h 24m < αJ2000 < 04h 08m, −1.27
◦ < δJ2000 <
+1.27◦, ∼ 290 deg2), observed in the southern Galactic cap, presents a valuable data source
for variability studies. The region was repeatedly observed (65 imaging runs by July 2005,
but not all cover the entire region), and it is the largest source of multi-epoch data in the
SDSS-I phase. Another source of the large number of scans is the SDSS-II Supernova Survey
(Frieman et al. 2007). By averaging the repeated observations of Stripe 82 sources, more
accurate photometry than the nominal 0.02 mag single-scan accuracy can be achieved. This
motivated Ivezić et al. (2007) to produce a catalog of recalibrated Stripe 82 observations.
The catalog lists 58 million photometric observations for 1.4 million unresolved sources that
were observed at least 4 times in each of the gri bands (with a median of 10 observations
obtained over ∼ 5 years). The random photometric errors for PSF (point spread function)
magnitudes are below 0.01 mag for stars brighter than 19.5, 20.5, 20.5, 20, 18.5 in ugriz,
respectively (about twice as accurate for individual SDSS runs), and the spatial variation of
photometric zeropoints is not larger than ∼0.01 mag (rms). Following Ivezić et al. (2007),
we use PSF magnitudes because they go deeper at a given signal-to-noise ratio than aperture
magnitudes, and have more accurate photometric error estimates than model magnitudes.
In addition, various low-order statistics such as root-mean-square scatter (Σ), χ2 per degree
of freedom (χ2), lightcurve skewness (γ), minimum and maximum PSF magnitude, were
4Please see [HREF]http://www.sdss.org/dr5
– 6 –
computed for each ugriz band and each source. We compute χ2 per degree of freedom as
(xi − 〈x〉)
and lightcurve skewness γ as5
(n− 1)(n− 2)
(xi − 〈x〉)
3 (3)
(xi − 〈x〉)2 (4)
where n is the number of detections, xi is the magnitude, 〈x〉 is the mean magnitude, and
ξi is the photometric error.
Separation of quasars and stars, as well as efficient color selection of horizontal branch
and RR Lyrae stars, depend on accurate u band photometry. To ensure this, we select
748,084 unresolved sources from the Ivezić et al. (2007) catalog with at least 4 detections in
the u band. A catalog of variable sources selected from this sample is analyzed in Section 3
below.
3. Analysis of Stripe 82 Catalog of Variable Sources
In this section we describe methods for selecting candidate variable sources, and present
tests that show the robustness of the adopted selection criteria. The distribution of selected
variable sources in magnitude-color-variability space is also presented.
3.1. Methods and Selection Criteria
Due to a relatively small number of observations per source and random sampling, we
do not perform lightcurve fitting, but instead use low order statistics to select candidate
variables and study their properties. There are four parameters (median PSF magnitude,
root-mean-square scatter Σ, χ2, and lightcurve skewness γ) measured in five photometric
5We use equations from [HREF]http://www.xycoon.com/skewness small sample test 1.htm.
– 7 –
bands (u, g, r, i, and z), for a total of 20 parameters. In the analysis presented here, we
utilize eight of them:
• median PSF magnitudes in the ugr bands (corrected for interstellar extinction using
the map from Schlegel, Finkbeiner & Davis 1998) because the g − r vs. u − g color-
color diagram has the most classification power (e.g. Smolčić et al. 2004 and references
therein).
• Σ and χ2 in the g and r bands, and
• lightcurve skewness γ(g) (the g band combines a high signal-to-noise ratio and large
variability amplitude for the majority of variable sources).
The observed root-mean-square scatter Σ includes both the intrinsic variability σ and
the mean photometric error 〈ξ(m)〉 as a function of magnitude. The dependence of Σ on
magnitude in the ugriz bands, is shown in Figure 1. For sources brighter than 18, 19.5, 19.5,
19, and 17.5 mag in the ugriz, respectively, the SDSS delivers 2% photometry with little or
no dependence on magnitude. We determine 〈ξ(m)〉 by fitting a fourth degree polynomial
to median Σ values in 0.5 mag wide bins (here we assume that the majority of sources are
not variable). The theoretically expected 〈ξ(m)〉 function (Strateva et al. 2001)
〈ξ(m)〉 = a+ b100.4m + c100.8m (5)
provides equally good fits. We define the intrinsic variability σ (hereafter rms scatter σ) as
σ = (Σ2 − 〈ξ(m)〉2)1/2 (6)
for Σ > 〈ξ(m)〉, and σ = 0 otherwise.
As the first variability selection criterion, we adopt σ(g) > 0.05 mag and σ(r) > 0.05
mag (hereafter written as σ(g, r) > 0.05 mag). At the bright end, this criterion is equivalent
to selecting sources with rms scatter greater than 2.5σ0, where σ0 = 0.02 mag is the mea-
surement noise. Selection cuts are applied simultaneously in the g and r bands to reduce
the number of “false positives” (intrinsically non-variable sources selected as candidate vari-
able sources due to measurement noise). About 6% of sources pass the σ cut in each band
separately, and ∼ 3% of sources pass the cut in both bands simultaneously. By selecting
sources with σ(g, r) > 0.05 mag, we also select faint sources that have large σ due to large
photometric errors at the faint end. To only select faint sources with statistically significant
rms scatter, we apply the χ2 test as the second selection cut.
– 8 –
In the χ2 test, the value of χ2 per degree of freedom (calculated with respect to a
weighted mean magnitude and using errors computed by the photometric pipeline) deter-
mines whether the observed lightcurve is consistent with the Gaussian distribution of er-
rors. Large χ2 values show that the rms scatter is inconsistent with random fluctuations.
Ivezić et al. (2003a, 2007) used multi-epoch SDSS observations to show that the photometric
error distribution in the SDSS roughly follows a Gaussian distribution. A comparison of χ2
distributions in the g and r bands with a reference Gaussian χ2 distribution is shown in
Figure 2. As evident, χ2 distributions in both bands roughly follow the reference Gaussian
χ2 distribution for χ2 < 1, demonstrating that median photometric errors are correctly de-
termined. The discrepancy for larger χ2 is due to variable sources rather than non-Gaussian
error distributions, as we demonstrate below.
The second selection cut, χ2(g) > 3 and χ2(r) > 3 (hereafter written as χ2(g, r) > 3),
selects ∼ 90% of σ(g, r) > 0.05 mag sources, as shown in Figure 2 (middle panels). The
effectiveness of the χ2 test is demonstrated in the bottom panel of Figure 2. For magnitudes
fainter than g = 20.5, the fraction of candidate variables decreases as photometric errors
increase. The selection is relatively uniform for sources brighter than g = 20.5, and we
adopt this value as the flux limit for the selected variable sample.
There are 662,195 sources brighter than g = 20.5 in the full sample. Using σ(g, r) > 0.05
mag and χ2(g, r) > 3 as the selection criteria, we select 13,051 candidate variable sources6.
Therefore, at least 2% of unresolved optical sources brighter than g = 20.5 appear variable
at the > 0.05 mag level (rms) simultaneously in the g and r bands. The fraction of selected
variable sources is not a strong function of the minimum required number of observations,
but it does depend on the stellar density because the number of stars increases at lower
Galactic latitudes (see Fig. 5 in Ivezić et al. 2007) while the quasar count remains the same.
3.2. The Counts of Variable Sources
In this section we estimate the completeness and efficiency of the candidate variable
sample, and discuss the dependence of counts, rms scatter, σ(g)/σ(r) ratio, and the lightcurve
skewness γ(g) on the position in the g − r vs. u− g color-color diagram.
6A list of candidate variable sources and their data from Ivezić et al. (2007) are publicly available from
[HREF]http://www.sdss.org/dr5/products/value added/index.html
– 9 –
3.2.1. Completeness
The selection completeness, defined as the fraction of true variable sources recovered by
the algorithm, depends on the lightcurve shape and amplitudes. Due to a fairly large number
of observations (median of 10), and small σ(g, r) cutoff compared to typical amplitudes of
variable sources (e.g. most RR Lyrae stars and quasars have peak-to-peak amplitudes ∼ 1
mag), we expect the completeness to be fairly high for RR Lyrae stars (& 95%, see Section 4)
and quasars (∼ 90%, see Section 5). The completeness for other types of variable sources,
such as flares and eclipsing binaries, is hard to estimate, but is probably low due to sparse
sampling.
3.2.2. Efficiency
The selection efficiency, defined as the fraction of true variable sources in the candidate
variable sample, determines the robustness of the selection algorithm. The main diagnostic
for the robustness of the adopted selection criteria is the distribution of selected candidates
in the SDSS color-magnitude and color-color diagrams. The position of a source in these
diagrams is a good proxy for its spectral classification (Lenz et al. 1998; Fan 1999; Finlator
2000; Smolčić et al. 2004).
Figure 3 compares the distribution of candidate variable sources to that of all sources
in the g − r vs. u − g color-color diagram. Were the selection a random process, the se-
lected candidates would have the same distribution as the full sample. The distributions of
candidate variables and of the full sample are remarkably different, demonstrating that the
candidate variables are not randomly selected from the parent sample.
The three dominant classes of variable objects are quasars, RR Lyrae stars, and stars
from the main stellar locus. The most obvious difference between the variable and the full
sample distributions is a much higher fraction of low-redshift quasars (< 2.2, recognized by
their UV excess, u − g < 0.7, see Richards et al. 2002) and RR Lyrae stars (u − g ∼ 1.15,
g − r < 0.3, see Ivezić et al. 2005) in the variable sample, and vividly shown in the bottom
panel of Figure 3.
Another interesting feature visible in this panel is a gradient in the fraction of variable
main stellar locus stars (perpendicular to the main stellar locus). We investigate this gradient
by first defining principal colors
P1 = 0.91u− 0.495g − 0.415r − 1.28 (7)
– 10 –
s = −0.249u+ 0.794g − 0.555r + 0.234 (8)
where P1 and s are principal axis parallel and perpendicular to the main stellar locus,
respectively (Ivezić et al. 2004a). The s color is a measure of metallicity (Lenz et al. 1998),
and s > 0.05 stars are expected to be metal poor (Helmi et al. 2003). Sources with r < 19
and 0 < P1 < 0.9 are selected and binned in four s bins. For each bin we calculate the
fraction of source with σ(g) > 0.05 mag, the fraction of variable sources (selected with
σ(g, r) > 0.05 mag and χ2(g, r) > 3), median σ(g), and the total number of sources in the
bin (see Table 2). A greater fraction of variable sources in the last bin (s > 0.06) indicates
that, on average, metal-poor main stellar locus stars are more variable than the metal-rich
stars. This could be because this sample of metal-poor stars is expected to have a high
fraction of giants.
In order to quantify the differences between the full and the variable sample, we follow
Sesar et al. (2006) and divide the g − r vs. u− g color-color diagram into six characteristic
regions, each dominated by a particular type of source, as shown in Figure 4. The fractions
and counts of variable and all sources in each region are listed in Table 1 for g < 19, g < 20.5,
and g < 22 flux-limited samples. Notably, in the adopted g < 20.5 flux limit, the fraction
of Region II sources (dominated by low-redshift quasars) in the variable sample is 63%, or
∼ 30 times greater than the fraction of Region II sources in the full sample (∼ 2%). The
fraction of Region IV sources (which include RR Lyrae stars) in the variable sample is also
high when compared to the full sample (∼ 6 times higher).
As shown in Table 1, in the g = 20.5 flux-limited sample, we find that low-redshift
quasars and RR Lyrae stars (i.e. Regions II and IV) make 70% of the variable population,
while representing only 3% of all sources. Quasars alone account for 63% of the variable
population. Stars from the main stellar locus represent 95% of all sources and 25% of the
variable sample: about 0.5% of the stars from the locus are variable at the > 0.05 mag level.
3.3. The Properties of Variable Sources
Various lightcurve properties, such as shape and amplitude, are expected to be correlated
with stellar types. In this section we study the distribution of the rms scatter in the u and g
bands, and σ(g)/σ(r) ratio as a function of the u− g and g− r colors. To emphasize trends,
we bin sources and present median values for each bin.
The distribution of the median σ(u) and σ(g) values in the g − r vs. u − g color-color
diagram is shown in the top two panels of Figure 5. RR Lyrae stars show larger rms scatter
– 11 –
(& 0.3 mag) in the u and g bands, than low-redshift quasars or stars from the main stellar
locus. Quasars also show slightly larger rms scatter in the u band (∼ 0.1 mag) than in
the g band (∼ 0.07 mag), as discussed by Kinney et al. (1991),Ivezić et al. (2004b), and
Vanden Berk et al. (2004). If we define the degree of variability as the root-mean-square
scatter in the g band, then on average RR Lyrae stars show the greatest variability, followed
by quasars and the main stellar locus stars.
Another distinctive characteristic of variable sources is the ratio of flux changes in
different bandpasses. This property can be used to select different types of variable sources.
For example, RR Lyrae stars are bluer when brighter, a behavior used by Ivezić et al. (2000)
to select RR Lyrae using 2-epoch SDSS data. Here we define a new parameter, σ(g)/σ(r), to
express the ratio of flux changes in the g and r bands, and study its distribution in the g− r
vs. u − g color-color diagram. In particular, we examine this distribution and its median
values for three dominant classes of variable sources: quasars, RR Lyrae stars, and stars
from the main stellar locus.
The bottom left panel in Figure 5 shows the distribution of median σ(g)/σ(r) values as
a function of u− g and g − r colors. Using Fig. 5 we note that on average:
• RR Lyrae stars have σ(g)/σ(r) ∼ 1.4
• Main stellar locus stars have σ(g)/σ(r) ∼ 1, and
• Quasars show a σ(g)/σ(r) gradient in the g − r vs. u− g color-color diagram.
The average value of σ(g)/σ(r) ∼ 1.4 in Region IV indicates that RR Lyrae stars
dominate the variable source count in this region. The ratio of 1.4 for RR Lyrae stars was
also previously found by Ivezić et al. (2000). While Figure 5 only presents median values of
the rms scatter, Figure 6 shows how the rms scatter in the g and r bands correlates with the
u − g color for individual sources. Variable sources that follow the σ(g) = 1.4σ(r) relation
also correlate with the u− g color, and have u− g ∼ 1, as expected for RR Lyrae stars.
The average ratio of σ(g)/σ(r) ∼ 1 (i.e. gray flux variations) for stars in the main stellar
locus suggests that the variability could be caused by eclipsing systems. The distribution of
γ(g) for main stellar locus stars further strengthens this possibility, as discussed in Section 3.4
below.
The gradient in the σ(g)/σ(r) ratio observed for low-redshift quasars in the g−r vs. u−g
color-color diagram suggests that the variability correlation between the g and r bands is
more complex than in the case of RR Lyrae or main stellar locus stars. Wilhite et al. (2006)
– 12 –
show that the photometric color changes for quasars depend on the combined effects of con-
tinuum changes, emission-line changes, redshift, and the selection of photometric bandpasses.
They note that due to the lack of variability of the lines, measured photometric color is not
always bluer in brighter phases, but depends on redshift and the filters used. To verify the
dependence of broad-band photometric variability on redshift, we plot σ(g)/σ(r) vs. redshift
for all spectroscopically confirmed unresolved quasars from Schneider et al. (2005) which are
in Stripe 82, as shown in Figure 7. We confirm that the broad-band photometric variability
depends on the redshift, and that the σ(g)/σ(r) gradient in the g − r vs. u − g color-color
diagram can be explained by the increase in σ(g)/σ(r) from ∼ 1 to ∼ 1.6 in the 1.0 to 1.6
redshift range. This effect is due to the Mg II emission line (more stable in flux than the con-
tinuum) moving through the r band filter over this redshift range. The implied correlation
of the u−g and g− r colors with redshift is consistent with the discussion by Richards et al.
(2002). The lack of noticeable correlation of σ(g) with redshift is due to the combined effects
of the dependence of σ(g) on the rest-frame wavelength and time which cancel out (for a
detailed model see Ivezić et al. 2004b).
3.4. Skewness as a Proxy for Dominant Variability Mechanism
Lightcurve skewness, a measure of the lightcurve asymmetry, provides additional in-
formation on the type of variability. Negatively skewed, asymmetric lightcurves indicate
variable sources that spend more time fainter than (mmin+mmax)/2, where mmin and mmax
are magnitudes at the minimum and maximum. Type ab RR Lyrae stars, for example,
have negatively skewed lightcurves (γ ∼ −0.5, Wils, Lloyd & Bernhard 2006). Positively
skewed, asymmetric lightcurves indicate variable sources that spend more time brighter
than (mmin + mmax)/2 (e.g. eclipsing systems). Sources with symmetric lightcurves will
have γ ∼ 0.
The bottom right panel in Figure 5 shows the distribution of the median γ(g) as a
function of the position in the g − r vs. u − g color-color diagram. On average, quasars
and c type RR Lyrae stars (u − g ∼ 1.15, g − r < 0.15) have γ(g) ∼ 0, ab type RR Lyrae
(u − g ∼ 1.15, g − r > 0.15) have negative skewness (γ(g) ∼ −0.5), and stars in the main
stellar locus have positive skewness.
Figure 8 shows the distribution of the lightcurve skewness in the ugi bands for spec-
troscopically confirmed unresolved quasars from Schneider et al. (2005) which are in Stripe
82, candidate RR Lyrae stars (selection details are discussed in Section 4 below), and main
stellar locus stars from our variable sample. Stars in the main stellar locus show a bimodal
γ(g) distribution. This distribution suggests at least two, and perhaps more, different popu-
– 13 –
lations of variables. Indeed, when spectroscopically confirmed M dwarfs are selected, a third
peak appears at γ(g) −2.5, possibly associated with flaring M dwarfs (Kowalski et al. 2007).
The bimodality similar to the one in the g band is also discernible in the r band, while it is
less pronounced in the i band and not detected in the u and z bands (the r and z data are
not shown).
A comparison of the u− g and g − r color distributions for variable main stellar locus
stars brighter than g = 19 and a subset with highly asymmetric lightcurves (γ(g) > 2.5)
is shown in Figure 9. The subset with asymmetric lightcurves has an increased fraction of
stars with colors u− g ∼ 2.5 and g− r ∼ 1.4, that correspond to M stars. This may indicate
that M stars have a higher probability of being associated with an eclipsing companion than
stars with earlier spectral types. However, the selection effects are probably important since
a companion is easier to detect (due to the low luminosity of M dwarfs). Kowalski et al.
(2007) examine these issues using lightcurve data on a sample of spectroscopically confirmed
M dwarfs. Finally, quasars have symmetric lightcurves (γ ∼ 0) and their distribution of
skewness does not change between bands.
4. The Milky Way Halo Structure Traced by Candidate RR Lyrae Stars
Studies of substructures in the Galactic halo, such as clumps and streams, can constrain
the formation history of the Milky Way. One of the best tracers to study the outer halo
are RR Lyrae stars because they are nearly standard candles, are sufficiently bright to be
detected at large distances (5− 100 kpc for 14 < r < 20.7), and are sufficiently numerous to
trace the halo substructure with a high spatial resolution. The General Catalog of Variable
Stars (GCVS; Kholopov et al. 1988) lists7 RR Lyrae stars as RR Lyrae type ab (RRab)
and type c (RRc) stars. RRab stars have asymmetric lightcurves, periods from 0.3 to 1.2
days, and amplitudes from V ∼ 0.5 to V ∼ 2. RRc stars have nearly symmetric, sometimes
sinusoidal, lightcurves, with periods from 0.2 to 0.5 days, and amplitudes not greater than
V ∼ 0.8. In this work we assume MV = 0.7 as the absolute V band magnitude of RRab and
RRc stars. A comprehensive review of RR Lyrae stars can be found in Smith (1995).
In this section we fine tune criteria for selecting candidate RR Lyrae stars, and estimate
the selection completeness and efficiency. Using selected candidate RR Lyrae stars, we
recover a known halo clump associated with the Sgr dwarf tidal stream, and find several new
halo substructures.
7A list of GCVS variability types can be found at [HREF]http://www.sai.msu.su/groups/cluster/gcvs/gcvs/iii/vartype.txt
– 14 –
4.1. Criteria for Selecting RR Lyrae Stars
Figures 3, 4, and 5 show that RR Lyrae stars occupy a well-defined region (Region IV)
in the g− r vs. u−g color-color diagram, and Figure 6 shows how RR Lyrae stars follow the
σ(g) = 1.4σ(r) relation. Motivated by these results, we introduce color and σ(g)/σ(r) cuts
to specifically select candidate RR Lyrae stars from the variable sample, and study their
distribution in the rms scatter-color-lightcurve skewness parameter space.
RR Lyrae stars have distinctive colors and can be selected with the following criteria
(Ivezić et al. 2005):
0.98 < u− g < 1.30 (9)
− 0.05 < Dug < 0.35 (10)
0.06 < Dgr < 0.55 (11)
− 0.15 < r − i < 0.22 (12)
− 0.21 < i− z < 0.25 (13)
where
Dug = (u− g) + 0.67(g − r)− 1.07 (14)
Dgr = 0.45(u− g)− (g − r)− 0.12. (15)
We apply these cuts to our sample of candidate variables and select 846 sources. It is
implied by Ivezić et al. (2005) that RR Lyrae should always stay within these color bound-
aries, even though their colors change as a function of phase. Their distribution in the g− r
vs. u − g color-color diagram and rms scatter in the g band are shown in Figure 10 (top
left panel). The distribution of sources in the RR Lyrae region is inhomogeneous. Sources
with large rms scatter in the g band (& 0.2 mag) are centered around u− g ∼ 1.15, and are
separated by g − r ∼ 0.12 into two groups. A comparison with Figure 3 from Ivezić et al.
(2005) suggests that these large rms scatter sources might be RR Lyrae type ab (RRab,
g − r > 0.12) and type c stars (RRc, g − r < 0.12). Small rms scatter sources (. 0.1 mag)
have a fairly uniform distribution, and are slightly bluer with u− g . 1.1.
The distribution of sources from the RR Lyrae region in the σ(r) vs. σ(g) diagram is
presented in the top right panel of Figure 10. The majority of large rms scatter sources
follow the σ(g) = 1.4σ(r) relation, as expected for RR Lyrae stars. Since RR Lyrae stars
are bluer when brighter, or equivalently, have greater rms scatter in the g band than in the
r band, we require 1 < σ(g)/σ(r) 6 2.5 and select 683 candidate RR Lyrae stars.
– 15 –
A comparison of u − g color distributions for candidate RR Lyrae stars and of sources
with RR Lyrae colors, but not tagged as RR Lyrae stars, presented in the bottom left panel
of Figure 10, demonstrates the robustness of the RR Lyrae selection. The two distributions
are very different (the probability that they are the same is 10−4, as given by the KS test),
with the candidate RR Lyrae distribution peaking at u−g ∼ 1.15, as expected for RR Lyrae
stars.
One property that distinguishes RRab from RRc stars is the shape (or skewness) of their
lightcurves (in addition to lightcurve amplitude and period). RRab stars have asymmetric
lightcurves, while RRc lightcurves are symmetric. In the top left panel of Figure 10, we
noted that g − r ∼ 0.12 seemingly separates high rms scatter sources into two groups. If
g − r ∼ 0.12 is the boundary between the RRab and RRc stars, then the same boundary
should show up in the distribution of lightcurve skewness as a function of the g − r color.
As shown in Figure 10 (bottom left panel), this is indeed the case. On average, sources with
g− r < 0.12 have γ(g) ∼ 0 (symmetric lightcurves), as RRc stars, while g− r > 0.12 sources
have γ(g) ∼ −0.5 (asymmetric lightcurves) typical of RRab stars.
We show in Section 4.2 that candidate RR Lyrae stars with γ(g) > 1 are contaminated
by eclipsing variables. Therefore, to reduce the contamination by eclipsing variables, we also
require γ(g) 6 1, and select 634 sources as our final sample of candidate RR Lyrae stars.
4.2. Completeness and efficiency
The selection completeness, defined as the fraction of recovered RR Lyrae stars, will
depend on the color cuts, σ(g, r) cutoff, and the number of observations. The color cuts
(Eqs. 9 to 15) applied in Section 4.1 were chosen to minimize contamination by sources
other than RR Lyrae stars while maintaining an almost 100% completeness (Ivezić et al.
2005). With the σ(g, r) cutoff at 0.05 mag (small compared to the ∼ 1 mag typical peak-
to-peak amplitudes of RR Lyrae stars), and a fairly large number of observations per source
(median of 10), we estimate the RR Lyrae selection completeness to be & 95% (see Appendix
in Ivezić et al. 2000).
To determine the selection efficiency, defined as the fraction of true RR Lyrae stars in the
RR Lyrae candidate sample, we positionally match 683 candidate RR Lyrae stars selected
by 1 < σ(g)/σ(r) 6 2.5 to a sample of RR Lyrae sources selected from the SDSS Light-
Motion-Curve Catalog (LMCC; Bramich et al. 2007). This catalog covers the same region of
the sky as the one discussed here, but includes more recent SDSS-II observations that allow
the construction of lightcurves. We match 613 candidates, while 70 candidate RR Lyrae
– 16 –
stars from our sample, for some reason, do not have a match in the LMCC (De Lee, private
communication). Following the classification based on phased lightcurves by De Lee et al.
(2007), we find that 71% of sources in our candidate RR Lyrae sample are classified as RRab
and RRc, 28% are classified as variable non-RR Lyrae stars, and only 1% are spurious, non-
variable sources. The most significant contamination comes from a population of variable
sources bluer than u−g ∼ 1.1 (dotted line, bottom left panel Figure 11), possibly Population
II δ Scuti stars, also known as SX Phoenicis stars (Hoffmeister, Richter & Wenzel 1985).
The top left and the bottom right panels in Figure 11, show that RRab and RRc-
dominated regions are separated by g−r ∼ 0.12, as already hinted in Figure 10. Also, variable
non-RR Lyrae sources with γ(g) > 1 are classified by De Lee et al. (2007) as eclipsing
variables, justifying our γ(g) 6 1 cut.
To summarize, using color criteria and criteria based on σ(g), σ(r), and γ(g) RR Lyrae
stars are selected with & 95% completeness and ∼ 70% efficiency.
4.3. The Spatial Distribution of Candidate RR Lyrae Stars
Using the selection criteria from Section 4.1 we isolate 634 RR Lyrae candidates. The
magnitude-position diagram for these candidates within 2.5◦ from the Celestial Equator is
shown in Figure 12.
As discussed by Ivezić et al. (2005), an advantage of the data representation utilized
in Figure 12 (magnitude–right ascension diagram) is its simplicity – only “raw” data are
shown, without any post-processing. However, the magnitude scale is logarithmic and thus
the spatial extent of structures is heavily distorted. In order to avoid these shortcomings,
we have applied a Bayesian method for estimating continuous spatial density distribution
developed by Ivezić et al. (2005) (see their Appendix B). The resulting density map is shown
in the right panel in Figure 13. The advantage of that representation is that it better conveys
the significance of various local overdensities. For comparison, we also show a map of the
northern part of the equatorial strip constructed using 2-epoch data discussed by Ivezić et al.
(2000).
We detect several new halo substructures at & 3σ significance (compared to expected
Poissonian fluctuations) and present their approximate locations and properties in Table 3.
The most distant clump is at 100 kpc from the Galactic center. The strongest clump in
the left wedge belongs to the Sgr dwarf tidal stream as does the clump marked by C in
the right wedge (Ivezić et al. 2003a). We note that the apparent “clumpiness” of the candi-
date RR Lyrae distribution increases with increasing radius, similar to CDM predictions by
– 17 –
Bullock, Kravtsov & Weinberg (2001). A detailed comparison of their models with the data
presented here will be discussed elsewhere (Sesar et al., in prep).
5. Are All Quasars Variable?
The optical continuum variability of quasars has been recognized since their first op-
tical identification (Matthews & Sandage 1963), and it has been proposed and utilized as
an efficient method for their discovery (van den Bergh, Herbst & Pritchet 1973; Hawkins
1983; Koo, Kron & Cudworth 1986; Hawkins & Veron 1995). The observed characteristics
of the variability of quasars are frequently used to constrain the origin of their emission
(e.g. Kawaguchi et al. 1998 and references therein; Martini & Schneider 2003; Pereyra et al.
2006). Recently, significant progress in the description of quasar variability has been made by
employing the SDSS data (de Vries, Becker & White 2003; Ivezić et al. 2004b; Vanden Berk et al.
2004; de Vries et al. 2005; Sesar et al. 2006). Here we expand these studies by quantifying
the efficiency of quasar discovery using variability.
A preliminary comparison of color and variability based methods for selecting quasars
using SDSS data was presented by Ivezić et al. (2003b). They found that 47% of spectroscop-
ically confirmed unresolved quasars with UV excess have the g band magnitude difference
between two observations obtained two years apart larger than 0.15 mag. We can improve on
their analysis because now there are significantly more observations obtained over a longer
time period. Since quasars vary erratically and the rms scatter of their variability (the so-
called structure function) increases with time (e.g. Vanden Berk et al. 2004 and references
therein), the variability selection completeness is expected to be higher than ∼ 50% obtained
by Ivezić et al. (2003b).
First, although the adopted variability selection criteria discussed above are fairly con-
servative, we find that at least 63% of low-redshift quasars are variable at the > 0.05 mag
level (simultaneously in the g and r bands over observer’s time scales of several years) in
the g < 20.5 flux-limited sample. Second, even this estimate is only a lower limit: given the
spectroscopic confirmation for a large flux-limited sample of quasars, it is possible to relax
the adopted variability selection cutoff without a prohibitive contamination by non-variable
sources.
There are 2,492 unresolved quasars in the catalog of spectroscopically confirmed SDSS
quasars (Schneider et al. 2005) from Stripe 82. The fraction of these objects that vary more
than σ in the g and r bands, as a function of σ, is shown in Figure 14. We also show the
analogous fraction for stars from the stellar locus. About 93% of quasars vary with σ > 0.03
– 18 –
mag. For a small fraction of these objects the measured rms scatter is due to photometric
noise, and the stellar data limit this fraction to be at most 3%. Conservatively assuming that
none of these 3% of stars is intrinsically variable, we estimate that at least 90% of quasars
are variable at the 0.03 mag level on time scales up to several years.
6. Implications for Surveys such as LSST
The Large Synoptic Survey Telescope (LSST) is a proposed imaging survey that aims
to obtain repeated multi-band imaging to faint limiting magnitudes over a large fraction of
the sky. The LSST Science Requirement Document8 calls for ∼ 1000 repeated observations
of a solid angle of ∼ 20, 000 deg2 distributed over the six ugrizY photometric bandpasses
and over 10 years. The results presented here can be extrapolated to estimate the lower
limit on the number of variable sources that the LSST would discover.
The single-epoch LSST images will have a 5σ detection limit9 at r ∼ 24.7. Hence, 2%
accurate photometry, comparable to the subsample with g < 20.5 discussed here, will be
available for stars with r . 22. The USNO-B catalog (Monet et al. 2003) shows that there
are about 109 stars with r < 21 across the entire sky. About half of these stars are in the
parts of the sky to be surveyed by the LSST. The simulations based on contemporary Milky
Way models, such as those developed by Robin et al. (2003) and Jurić et al. (2007), predict
that there are about twice as many stars with r < 22 than with r < 21 across the whole
sky. Hence, it is expected that the LSST will detect about a billion stars with r < 22. This
estimate is uncertain to within a factor of two or so due to unknown details in the spatial
distribution of dust in the Galactic plane and towards the Galactic center.
We found that at least 0.5% of stars from the main stellar locus can be detected as
variable with photometry accurate to ∼ 2%. This is only a lower limit because a much
larger number of LSST observations obtained over a longer timespan than the SDSS data
discussed here would increase this fraction. Hence, our results imply that the LSST will
discover at least 50 million variable stars (without accounting for the fact that stellar counts
greatly increase closer to the Galactic plane). Unlike the SDSS sample, where RR Lyrae
stars account for ∼ 25% of all variable stars, the number of RR Lyrae stars in the LSST
sample will be negligible compared to other types of variable stars.
As estimated by Jurić et al. (2007) using deeper coadded SDSS photometry, there are
8 Available at [HREF]http://www.lsst.org/Science/lsst baseline.shtml
9An LSST Exposure Time Calculator is available at [HREF]www.lsst.org
– 19 –
about 100 deg−2 low-redshift quasars with r < 22 (see also Beck-Winchatz & Anderson 2007
and references therein). Therefore, with a sky coverage of ∼ 20, 000 deg2, the LSST will
obtain well-sampled accurate multi-color lightcurves for ∼ 2 million low-redshift quasars.
Even at the redshift limit of ∼ 2, this sample will be complete to Mr ∼ −24, that is,
almost to the formal quasar luminosity cutoff, and will represent an unprecedented sample
for studying quasar physics.
7. Conclusions and Discussion
We have designed and tested algorithms for selecting candidate variable sources from
a catalog based on multiple SDSS imaging observations. Using a sample of 13,051 selected
candidate variable sources in the adopted g < 20.5 flux-limited sample, we find that at least
2% of unresolved optical sources appear variable at the > 0.05 mag level, simultaneously
in the g and r bands. A similar fraction of variable sources (∼ 1%) was also found by
Sesar et al. (2006) using recalibrated photometric POSS and SDSS measurements, and by
Morales-Rueda et al. (2006) using the Faint Sky Variability Survey data (∼ 1%).
Thanks to the multi-color nature of the SDSS photometry, and especially to the u band
data, we can obtain robust classification of selected variable sources. The majority (2/3)
of variable sources are low-redshift (< 2) quasars, although they represent only 2% of all
sources in the adopted g < 20.5 flux-limited sample. We find that about 1/4 of variable stars
are RR Lyrae stars, and that only 0.5% of stars from the main stellar locus are variable at
the 0.05 mag level.
The distribution of γ(g) for main stellar locus stars is bimodal, suggesting at least two,
and perhaps more, different populations of variables. About a third of variable stars from
the stellar locus show gray flux variations in the g and r bands (σ(g)/σ(r) ∼ 1), and positive
lightcurve skewness, suggesting variability caused by eclipsing systems. This population has
an increased fraction of M type stars.
RR Lyrae stars show the largest rms scatter in the u and g bands, followed by low-
redshift quasars. The ratio of rms scatter in the g and r bands for RR Lyrae is ∼ 1.4, in agree-
ment with Ivezić et al. (2000) results based on 2-epoch photometry. The mean lightcurve
skewness for RR Lyrae stars is ∼ −0.5, in agreement with Wils, Lloyd & Bernhard (2006).
We selected a sample of 634 candidate RR Lyrae stars, with an estimated & 95% complete-
ness and ∼ 70% efficiency. Using these stars, we detected rich halo substructure out to
distances of 100 kpc. The apparent “clumpiness” of the candidate RR Lyrae distribution in-
creases with increasing radius, similar to CDM predictions by Bullock, Kravtsov & Weinberg
– 20 –
(2001).
Low-redshift quasars show a dependence of σ(g)/σ(r) on redshift, consistent with dis-
cussions in Richards et al. (2002) and Wilhite et al. (2006). The lightcurve skewness dis-
tribution for quasars is centered on zero in all photometric bands. We find that at least
90% of quasars are variable at the 0.03 mag level (rms) on time scales up to several years.
This confirms that variability is as a good a method for finding low-redshift quasars at high
(|b| > 30) Galactic latitudes as is the UV excess color selection. The fraction of variable
quasars at the > 0.1 mag level obtained here (30%, see Figure 14) is comparable to 36%
found by Rengstorf, Brunner & Wilhite (2006).
The multiple photometric observations obtained by the SDSS represent an excellent
dataset for estimating the impact of surveys such as the LSST on studies of the variable
sky. Our results indicate that the LSST will obtain well-sampled 2% accurate multi-color
lightcurves for ∼ 2 million low-redshift quasars, and will discover at least 50 million variable
stars. The number of variable stars discovered by the LSST will be of the same order as the
number of all stars detected by the SDSS. With about 1000 data points in six photometric
bands, it will be possible to recognize and classify variable objects using lightcurve mo-
ments of higher order than skewness discussed here, including lightcurve folding for periodic
variables.
We acknowledge support by NSF grant AST-0551161 to the LSST for design and de-
velopment activity.
Funding for the SDSS and SDSS-II has been 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/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions. The Participating Institutions are the American Museum of Natural History,
Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western
Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Ad-
vanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute
for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the
Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National
Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), New Mexico State University, Ohio State University, University of
Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Obser-
http://www.sdss.org/
– 21 –
vatory, and the University of Washington.
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Table 1. The distribution of candidate variable sources in the g − r vs u− g diagram
g < 19 g < 20.5 g < 22
Regiona Nameb % allc % vard var/alle Nvar/N
% allc % vard var/alle Nvar/N
% allc % vard var/alle Nvar/N
I white dwarfs 0.14 0.59 4.25 3.50 0.24 0.40 1.69 3.34 0.28 0.45 1.64 4.51
II low-redshift QSOs 0.45 30.88 68.83 56.58 1.90 62.90 33.03 65.10 4.07 70.01 17.22 47.30
III dM/WD pairs 0.08 0.53 6.54 5.37 0.83 2.08 2.50 4.92 1.21 3.79 3.13 8.61
IV RR Lyrae stars 1.28 16.81 13.11 10.78 1.33 7.95 5.99 11.81 1.48 6.41 4.33 11.90
V stellar locus stars 96.27 48.77 0.51 0.42 94.49 25.15 0.27 0.52 91.89 18.33 0.20 0.55
VI high-redshift QSOs 1.78 2.42 1.36 1.12 1.21 1.52 1.26 2.48 1.07 1.01 0.95 2.60
total count 411,667 3,384 662,195 13,051 748,067 20,553
aThese regions are defined in the g − r vs. u− g color-color diagram, with their boundaries shown in Fig. 4
bAn approximate description of the dominant source type found in the region
cThe fraction of all sources in a magnitude-limited sample found in this color region, with the magnitude limits listed on top.
dThe number of candidate variables from the region, expressed as a fraction of all variable sources
eThe ratio of values listed in columns d) and c)
fThe number of candidate variables from the region, expressed as a fraction of all sources in that region
Table 2. The fraction of variable main stellar locus stars as a function of the s color
Bin % σ(g) > 0.05a % varb 〈σ(g)〉c Countsd
s < −0.02 3.23 0.36 0.017 46,836
−0.02 < s < 0.02 2.92 0.28 0.017 136,910
0.02 < s < 0.06 4.61 1.18 0.019 29,106
s > 0.06 11.50 4.10 0.027 4,547
aFraction of sources with σ(g) > 0.05 mag
bFraction of variable sources (selected using σ(g, r) > 0.05 mag and
χ2(g, r) > 3)
cMedian σ(g)
dNumber of sources in the bin
Table 3. Approximate locations and properties of detected overdensities
Labela Nb 〈R.A.〉c 〈d〉d 〈r〉e 〈u− g〉f 〈g − rg〉 〈γ(g)〉h Nb/N
A 84 330.95 21 17.02 1.14 0.18 -0.50 0.62
B 144 309.47 22 16.76 1.12 0.16 -0.57 0.64
C 54 33.69 25 17.61 1.13 0.20 -0.68 0.29
D 8 347.91 29 18.02 1.14 0.23 -0.50 0.38
E 11 314.06 43 18.84 1.09 0.20 -0.41 0.75
F 11 330.26 48 19.16 1.07 0.20 -0.46 0.38
G 10 354.81 55 19.46 1.10 0.22 -0.69 0.38
H 7 43.57 57 19.32 1.05 0.04 0.06 1.34
I 4 311.34 72 19.98 1.08 0.11 -0.10 2.0
J 26 353.58 81 20.21 1.11 0.20 -0.27 0.58
K 8 28.39 84 20.35 1.10 0.20 0.14 0.44
L 3 339.01 92 20.45 1.06 0.16 0.08 0.67
M 5 39.45 102 20.73 1.07 0.11 0.36 1.67
aOverdensity’s label from Fig. 13
bNumber of candidate RR Lyrae in the overdensity
cMedian Right Ascension
dMedian distance (in kpc)
eMedian r band magnitude
fMedian u− g color
gMedian g − r color
hMedian γ(g)
iThe number ratio of candidate RR Lyrae with g − r < 0.12 and g − r > 0.12
– 28 –
Fig. 1.— The dependence of the median root-mean-square (rms) scatter Σ in SDSS ugrz
bands on magnitude (symbols). The vertical bars show the rms scatter of Σ in each bin
(not the error of the median). The dependence of Σ in the i band is similar to the r band
dependence. In each band, a fourth-degree polynomial is fitted through medians and shown
by the solid line.
– 29 –
Fig. 2.— Top: The cumulative distribution of χ2 g and r values for all sources (solid line)
and a reference Gaussian χ2 distribution with 9 degrees of freedom (dashed line). Vertical
dashed lines show adopted selection cuts on χ2(g) and χ2(r) values. Middle: The fraction
of σ(g, r) > 0.05 mag sources with χ2 per degree of freedom greater than χ2 (only in the
g or r band: solid line, both in the g and r bands: dashed line). Bottom: The fraction of
σ(g, r) > 0.05 mag sources with χ2(m) > 2 (dashed line) or χ2(m) > 3 (solid line) as a
function of magnitude for m = g, r bands, respectively.
– 30 –
Fig. 3.— The distribution of counts for the full sample (top), candidate variable sample
(middle), and the ratio of two counts (bottom) in the g− r vs. u− g color-color diagram for
sources brighter than g = 20.5, binned in 0.05 mag bins. Contours outline distributions of
unbinned counts. Note the remarkable difference between the distribution of all sources and
that of the variable sample, which demonstrates that the latter are robustly selected.
– 31 –
Fig. 4.— The distribution of 18,329 candidate variable sources brighter than g = 21 in
representative SDSS color-magnitude and color-color diagrams. Candidate variables are
color-coded by their rms scatter in the g band (0.05-0.2, see the legend, red if larger or equal
than 0.2). Only sources brighter than g = 20 are plotted in color-color diagrams. Note
how RR Lyrae stars (u − g ∼ 1.15, red dots, σ(g) & 0.2 mag) and low-redshift quasars
(u− g 6 0.7, green dots, σ(g) & 0.1 mag) stand out as highly variable sources. The regions
marked in the top right panel are used for quantitative comparison of the overall and variable
source distributions (see Table 1).
– 32 –
Fig. 5.— The distribution of the rms scatter σ(u) (top left), rms scatter σ(g) (top right),
σ(g)/σ(r) ratio (bottom left), and γ(g) (bottom right) for the variable sample in the g − r
vs. u − g color-color diagram. Sources are binned in 0.05 mag wide bins and the median
values are color-coded. Color ranges are given at the top of each panel, going from blue to
red, where the green color is in the mid-range. Values outside the range saturate in blue or
red. Contours outline the count distributions on a linear scale in steps of 15%. Flux limit is
g < 20.5, with an additional u < 20.5 limit in the top left panel. Bottom left: On average,
RR Lyrae stars have σ(g)/σ(r) ∼ 1.4, main stellar locus stars have σ(g)/σ(r) ∼ 1, while
low-redshift quasars show a gradient of σ(g)/σ(r) values. Bottom right: On average, quasars
and c type RR Lyrae stars (u − g ∼ 1.15, g − r < 0.15) have γ(g) ∼ 0, ab type RR Lyrae
(u− g ∼ 1.15, g− r > 0.15) have negative skewness, and stars in the main stellar locus have
positive skewness.
– 33 –
Fig. 6.— The distribution of candidate variable sources in the g < 20.5 flux-limited sample
is shown by linearly-spaced contours, and by symbols color-coded by the u − g color for
sources with σ(g) > 0.05 mag and σ(r) > 0.05 mag. The dotted lines show the adopted
σ(g, r) selection cut. The thick solid line shows σ(g) = σ(r), while the dashed line shows
σ(g) = 1.4σ(r) relation representative of RR Lyrae stars. Note that sources following the
σ(g) = 1.4σ(r) relation tend to have u−g ∼ 1, as expected for RR Lyrae stars. The greyscale
background shows the fraction of χ2(g, r) > 3 sources which also have σ(g) > x and σ(r) > y
and demonstrates that large χ2 sources also have large σ.
– 34 –
0 0.5 1 1.5 2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3 3.5
0 0.5 1 1.5 2 2.5 3 3.5
Fig. 7.— The dependence of σ(g)/σ(r) (top), g − r (middle), and σ(g) on redshift for
a sample of spectroscopically confirmed unresolved quasars from Schneider et al. (2005).
The σ(g)/σ(r) gradient shown in Fig. 5 (bottom left panel) can be explained by the local
maximum of σ(g)/σ(r) in the 1.0 to 1.6 redshift range.
– 35 –
Fig. 8.— The lightcurve skewness distribution in the ugi bands for spectroscopically con-
firmed unresolved quasars (dotted line), candidate RR Lyrae stars (dashed line), and variable
main stellar locus stars (solid line, Region V, see Fig. 4 for the definition). The distribution
of the skewness in the r band is similar to the g band distribution, and the distribution of
skewness in the z band is similar to the u band distribution (therefore the r and z data
are not shown). Stars in the main stellar locus show bimodality in the γ(g) suggesting at
least two, and perhaps more, different populations of variables. Similar bimodality is also
discernible in the r band, while it is less pronounced in the i band and not detected in the u
and z bands. Quasars have symmetric lightcurves (γ ∼ 0) and their distribution of skewness
does not change between bands.
– 36 –
0.5 1 1.5 2 2.5 3 3.5
0.5 1 1.5 2
Fig. 9.— A comparison of the u−g (left) and g−r (right) color distributions for variable main
stellar locus stars brighter than g = 19 (dashed lines), and a subset with highly asymmetric
lightcurves (γ(g) > 2.5, solid lines). The subset with highly asymmetric lightcurves has an
increased fraction of stars with colors u− g ∼ 2.5 and g− r ∼ 1.5, characteristic of M stars.
– 37 –
0.9 1 1.1 1.2 1.3
0 0.1 0.2 0.3 0.4 0.5
0.9 1 1.1 1.2 1.3
-0.1 0 0.1 0.2 0.3
Fig. 10.— Top left: The distribution of 846 candidate variable sources from the RR Lyrae
region (dashed lines, see Fig. 3 in Ivezić et al. 2005) in the g−r vs. u−g color-color diagram.
The symbols mark the time-averaged values and are color-coded by σ(g) (0.05 to 0.2, blue to
red). The dotted horizontal line shows the boundary between the RRab and RRc-dominated
regions. Top right: Sources from the top left panel divided into 3 groups according to their
σ(g)/σ(r) values: candidate RR Lyrae stars with 1 < σ(g)/σ(r) 6 2.5 (large dots), sources
with σ(g)/σ(r) 6 1 (triangles), and sources with σ(g)/σ(r) > 2.5 (squares). Small dots show
sources with RR Lyrae colors that fail the variability criteria. The dashed lines show the
σ(g) = σ(r) and σ(g) = 2.5σ(r) relations, while the dotted line shows the σ(g) = 1.4σ(r)
relation. Bottom left: A comparison of the u− g color distributions for candidate RR Lyrae
stars (solid line) and sources with RR Lyrae colors but not tagged as RR Lyrae stars (dashed
line). Bottom right: The dependence of γ(g) on the g−r color for candidate RR Lyrae stars.
The boundary g − r = 0.12 (vertical dotted line) separates candidate RR Lyrae stars into
those with asymmetric (γ(g) ∼ −0.5) and symmetric (γ(g) ∼ 0) lightcurves, corresponding
to RRab and RRc stars, respectively. The condition γ(g) 6 1 (horizontal dashed line) is
used to reduce the contamination of the RR Lyrae sample by eclipsing variables.
– 38 –
0.9 1 1.1 1.2 1.3
0 0.1 0.2 0.3 0.4 0.5
0.9 1 1.1 1.2 1.3
-0.1 0 0.1 0.2 0.3
Fig. 11.— The distribution of candidate RR Lyrae stars selected with 1 < σ(g)/σ(r) 6 2.5
and classified by De Lee et al. (2007), shown in diagrams similar to Fig. 10. Symbols show
RRab stars (red dots), RRc stars (blue dots), variable non-RR Lyrae stars (green dots),
and non-variable sources (open squares, only four sources). A comparison of the u − g
color distribution for RRab (solid line), RRc (dashed line), and variable non-RR Lyrae stars
(dotted line) is shown in the bottom left panel.
– 39 –
60 50 40 30 20 10 0 -10 -20 -30 -40 -50
634 Stripe 82 RR Lyrae candidates
25 kpc
50 kpc
75 kpc
100 kpc
JK LM
Fig. 12.— The magnitude-position distribution of 634 Stripe 82 RR Lyrae candidates within
−55◦ < R.A. < 60◦ and |Dec| 6 1.27◦. Approximate distance (shown on the right y-axis)
is calculated assuming Mr = 0.7 mag for RR Lyrae stars. Dashed lines show where sample
completeness decreases from approximately 99% to 60% due to the χ2 cut (see the bottom
right panel in Fig. 2). Closed curves are remapped ellipses and circles from Fig. 13 that
mark halo substructure.
– 40 –
25       
50       
75       
100            kpc
Fig. 13.— Left: The spatial distribution of candidate RR Lyrae stars discovered by SDSS
along the Celestial Equator. Distance is calculated assuming Eq. 3 from Ivezić et al. (2005)
and MV = 0.7 mag as the absolute magnitude of RR Lyrae in the V band. The right
wedge corresponds to candidate RR Lyrae selected in this work (634 candidates, shown in
Fig. 12) and the left wedge is based on the sample from Ivezić et al. (2000) (296 candidates).
Right: The number density distribution of candidate RR Lyrae stars shown in the left panel,
computed using an adaptive Bayesian density estimator developed by Ivezić et al. (2005).
The color scheme represents the number density multiplied by the cube of the galactocentric
radius, and displayed on a logarithmic scale with a dynamic range of 300 (from light blue to
red). The green color corresponds to the mean density – both wedges with the data would
have this color if the halo number density distribution followed a perfectly smooth r−3 power-
law. The purple color marks the regions with no data. The yellow regions are formally ∼ 3σ
significant overdensities, and orange/red regions have an even higher significance (using
only the counts variance). The strongest clump in the left wedge belongs to the Sgr dwarf
tidal stream as does the clump marked by C in the right wedge (Ivezić et al. 2003a). An
approximate location and properties of labeled overdensities are listed in Table 3. The
Ivezić et al. (2000) sample is based on only 2 epochs and thus has a much lower completeness
(∼ 56%) resulting in a lower density contrast.
– 41 –
Fig. 14.— The fraction of spectroscopically confirmed unresolved QSOs (fQSO, solid line)
and the fraction of sources from the stellar locus (floc, dashed line) brighter than g = 19.5 and
r = 19.5 that have rms scatter larger than σ in the g and r bands. The ratio fQSO/(1+ floc)
(dotted line), which corresponds to the implied fraction of variable QSOs, peaks at a level
of 90% for σ = 0.03 mag.
	Introduction
	Overview of the SDSS Imaging and Stripe 82 Data
	Analysis of Stripe 82 Catalog of Variable Sources
	Methods and Selection Criteria
	The Counts of Variable Sources
	Completeness
	Efficiency
	The Properties of Variable Sources
	Skewness as a Proxy for Dominant Variability Mechanism
	The Milky Way Halo Structure Traced by Candidate RR Lyrae Stars 
	Criteria for Selecting RR Lyrae Stars
	Completeness and efficiency
	The Spatial Distribution of Candidate RR Lyrae Stars
	Are All Quasars Variable?
	Implications for Surveys such as LSST
	Conclusions and Discussion
