# STRUCTURE FUNCTION ANALYSIS OF LONG-TERM QUASAR VARIABILITY Page: 7 of 18

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Long Term Quasar Variability. II

the gSDSS-band.

The actual data for the gSDSS-band have been plotted

in Fig. 5. The bottom panel shows the magnitude differ-

ences as function of intrinsic time-lag. Since the quasars

are quite spread out in redshift space (cf. Fig.1), we have

to bring the actual time separation between the obser-

vations onto the reference frame of the quasar itself (by

dividing it by a (1 + z) factor). This has the additional

advantage of smoothing out the time-lag distribution. So

even though the observing campaigns were well separated

in time (1950's, 1990's, and ~ 2000), resulting in time-

lags clustering around a few, ~ 10, and ~ 50 years, the

(1 + z) redistributing factor results in a pretty smooth

distribution up to time-lags of ~ 40 years (cf. Fig. 5,

bottom panel).

The top four panels of Fig. 5 provide a direct picture

of the increase in FWHM (and hence the rms) of the

magnitude difference distribution as time-lags increase.

This is actually the definition of the SF (see next sec-

tion). Since the total number of time-lag measurements

decreases with increasing time-lag, this is not immedi-

ately obvious looking at the point-cloud in the lower

panel. The numbers of data-points for the current 10-

year time-lag binning are: 86 795, 50 974, 22 627, and

8 520 permutations respectively. While the numbers do

decline, they are still large enough to assess the FWHM

of the distribution very accurately. The last bin alone al-

ready contains 30% of the total number of permutations

used for Paper I.

3. STRUCTURE FUNCTION

Our analysis of Paper I, and the current paper, will

utilize the SF as the tool to characterize the quasar vari-

ability. SF's are not very sensitive to aliasing problems

due to discrete and/or sparse time sampling (e.g., Hughes

et al. 1992), which make them well suited for our pur-

pose. As before, we define the SF as:5(r) =n (N KmKG) -mWj]2)

N~r)(1)

with the summation over all the combinations of mea-

surements for which Tr - - t2. In our case we group

all the n(n - 1)/2 permutations into bins which contain

at least 200 measurements. The SF value for each bin is

then given by the rms of the magnitude permutations.

3.1. Error estimates

This results in 1500 bins, which are then banned again

onto a fixed grid in log time-lag space (running from

-0.97 to 1.55 in 0.06 dex bins for a total of 42). This

facilitates easy comparison between model and actual

SF curves. It also allows us to approximate the error

on a particular SF point by calculating the rms of the

1500/42 ~ 36 values inside each bin. This is basically

the same method as we employed in Paper I. The pre-

sented error-bars reflect therefore accurately the actual

local SF uncertainties. It should be stressed that, unlike

a well monitored SF of a single source for which all of the

bins are cross-correlated with each other and an objec-

tive error estimate is hard to give, our bins are essentially

independent. Out of the 150 000 or so time-lag measure-

ments (per band) only measurements for a single quasar(about 4) are correlated with each other. In other words,

each of the SF bins contains a virtually completely differ-

ent set of quasars. This bin-independence also allows us

to quantify SF similarities in terms of their offset distri-

butions. Assuming two SF curves, labeled A and B, both

of which are binned to the same N 42 bins specified

above, we can define:iN

O N KZSA )

AOzzzN l (SA(i)SB Q)

SB(i) - 2)

(2)

(3)after substituting N ~ N N - 1. The quantities O

and AO represent the mean SF offset and its lo- uncer-

tainty, respectively. We will use this metric in particular

for our SF asymmetry part of the paper.

3.2. Stellar Structure Function

The SF for the calibration stars serves multiple pur-

poses. If we assume that stars, on average, are not vari-

able, then the SF derived from it should not exhibit any

correlation with time-lag. In other words, it should be

parallel to the i-axis (in plots like Fig. 6). This was in-

deed found to be the case for the stars in Paper I, which

clearly illustrated the significant differences between the

SF behavior of stars and quasars. However, given the

much smaller sample sizes for Paper I (note that the

number of calibration stars is linked to the number of

quasars), the overall stellar SF was rather noisy. It just

served to make the point that constructing an SF from

a random sample of stars resulted in a non-variable SF

curve, but it clearly was not good enough to go beyond

that. The current sample, however, is large enough. In

the next few sections, we will discuss the stellar SF in

more detail.

3.2.1. Stellar Type Dependencies

In the same way spectral differences between the av-

erage stellar spectrum and a quasar spectrum lead to

slightly different passband corrections, and therefore, ad-

ditional noise to the variability measure, spectral differ-

ences among stars themselves will inflate its SF variabil-

ity signal as well. This has to be considered in the con-

struction of the stellar SF. The reason we can use stars

to calibrate the quasars at all, is that the mean of the

stellar color distribution does not change that much go-

ing from one sightline to another. The stellar population

therefore does not change a lot across the sky covered by

DR22.

In order to limit the stellar spectral range allowed for

our template SF, we only included stars within a magni-

tude range (17 < r < 21), and an (r-g) color within 0.2

magnitudes of the typical stellar color of (r-g) 0.4 (cf.

Stoughton et al. 2002). This color cut effectively limits

the allowed range of stellar colors, and improves the pass-

band calibrations accordingly. The resulting time-lag

permutation database contains 2.1 million data-points

2 It should be noted in this respect that the DR2 does not cover

the galactic plane.

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de Vries, W; Becker, R; White, R & Loomis, C. STRUCTURE FUNCTION ANALYSIS OF LONG-TERM QUASAR VARIABILITY, article, November 15, 2004; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc1410080/m1/7/: accessed May 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.