<figure><img src="image_35.png" /><figcaption>**Figure 33.** Timescale vs. rms, separated into [quasi-]periodic and aperiodic behavior, in both bands. We also display the distributions of each set of points.</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>**Figure 34.** Left: correspondence between optical and infrared rms values for all objects in our data set. Right: optical and infrared aperiodic timescales in each band. These are quantized, because the aperiodic timescale involves division of the total time baseline (the same for all sources) by the number of peaks larger than a particular amplitude. Here the points at timescales of ~40 days are lower limits. Timescales are not shown for objects that have a quasi-periodic optical or infrared light curve, since we find that the detected periods, if present in both bands, are always similar.</figcaption></figure>

less correlated or varying with a different color slope in other parts. This makes it difficult to quantify correlation via a single parameter.

As an alternative, we have computed a time-dependent Stetson index by comparing 20 point sections (1.7 days) of the IRAC and interpolated *CoRoT* light curves. The 20 point step size was motivated by the typical duration of short-timescale light-curve fluctuations, which is a few days. Well-correlated

light curves should display a positive Stetson index with little dependence on time. Light curves that are correlated at some times but not others will instead display a fluctuating Stetson index, reaching large values at some times and dropping to zero or below at others. To differentiate this behavior from chance correlation between sections of two light curves, we have performed simulations using time series from different stars. We match optical and infrared light curves from different stars,
<figure><img src="image_37.png" /><figcaption>**Figure 35.** Distributions of median running Stetson index for pairs of *CoRoT* and IRAC light curves in our disk-bearing sample. Here the values from the 3.6 and 4.5 µm bands are averaged. The hashed region shows the distribution for randomly selected light-curve pairs, and dotted lines mark the 95% confidence interval for random Stetson indices. Values greater than 0.6 indicate well-correlated optical and infrared light curves.</figcaption></figure>

thereby guaranteeing that there should be no true correlation. Only light curves with at least 250 points are allowed, since the more sparsely sampled HDR mode light curves will have a broader Stetson index distribution. One thousand simulations of randomly matched light-curve pairs revealed that 95% of uncorrelated light curves have a median running Stetson index between -0.5 and 0.5 when sampled at every 20 points. Repeating the simulations on HDR light curves for every five points, the value increased to 1.0. We also ran the simulations for a sampling step size of 50 points, but found that the running Stetson index showed more systematics, and its median was less consistent between the two IRAC channels, despite flux behavior being nearly the same in these bands. The distribution of median running Stetson index for *CoRoT* light curves paired with IRAC light curves is shown in Figure 35.

We report the median running Stetson index for all objects in Table 4. Stars with large negative running Stetson indices exhibit anticorrelated behavior in the optical and infrared.

The fractions of correlated versus uncorrelated and anticorrelated light curves may indicate the percentage of sources in which variability is disk dominated. To determine this number, we assemble the distribution of median running Stetson indices in Figure 35, along with the distribution expected for uncorrelated light curves, as derived in our simulations. Although we have drawn a cutoff of 95% confidence, it is clear that the set of Stetson indices is skewed with respect to the distribution defined by the simulations. We conclude that there is low-level optical/infrared correlation in many of the objects; this is not surprising since stellar variability will generally have a small infrared contribution due to emission from the long-wavelength tail of the star's SED. In general, however, the larger infrared contribution from the dust means that variability originating in the disk will dominate the light curve.

Overall, 38% of our sample shows evidence of correlation at the 2σ level, and 58% shows correlation at the 1σ level. There is generally a correlation between the shapes in the optical and those in the infrared, but in many cases it is quite weak (i.e., it consists of transient dips or bursts matching up). To further illuminate the relationship between optical and infrared behavior, we have plotted a correlation matrix comparing the

<figure><img src="image_38.png" /><figcaption>**Figure 36.** Correlation matrix between assigned optical and infrared variability types. For each optical morphology class, this diagram shows the fractions of objects that occupy each infrared class. It should be read along rows. For example, just over 10% of optical variables classified as aperiodic dippers are also aperiodic dippers in the infrared; the reverse (i.e., 10% of aperiodic infrared dippers being aperiodic dippers in the optical) is not necessarily true. Continuing along the same row, we also see that just over 50% of aperiodic optical dippers are long-timescale variables in the infrared. The remaining 34% of optical variables are a mixture of periodic dipper (~6%), stochastic (~11%), quasi-periodic (~6%), and unclassifiable (11%) in the infrared.</figcaption></figure>

morphologies assigned in each band. Figure 36 confirms a low-level optical/infrared correlation at best in most sources.

## 7.2. Well-correlated Optical and Infrared Variability

We have identified light curves that are correlated in the optical and infrared by selecting those with a large median running Stetson index. Some of these light curves display partial correlation in that the infrared flux trends mimic those in the optical over part of the time series, but at other points there is no resemblance between the two bands.

There are two scenarios for correlation of behavior in the optical and infrared. First, if there is only one variability process affecting the star, then the light curves should exhibit a relatively simple wavelength dependence. Second, if the disk flux dominates in the infrared bands but variability reflects mainly reprocessed stellar light, then we would also expect to see correlated behavior, either in or out of phase. We test these scenarios in Section 8 by comparing the Stetson index to measures of disk to star flux.

We have identified three broad categories of morphological behavior among the well-correlated light curves: optical dippers, bursters and stochastic stars with similar infrared amplitude, and optical stochastic stars with lower amplitude infrared light curves. A selection of these is presented in Figure 37, while optical and infrared light curves for the entire 162 member disk-bearing data set are provided in the Appendix, Figure 49. Some of these have behavior that makes sense in the context of the extinction-dominated variability model. Here, fluctuations in the light curve are caused by changing amounts of dust blocking the star, and the optical/infrared correlation simply reflects the wavelength dependence of extinction, diluted by any flux from the inner disk. Confronting this model are the two objects Mon-000183 and Mon-000566, which display *deeper* dips in the infrared than in the optical. This behavior is not expected from
<figure><img src="image_39.png" /><figcaption>**Figure 37.** Light curves with correlated optical and infrared behavior. Small black points are *CoRoT* data, light gray points are 3.6 µm data, and dark gray points are 4.5 µm data (sometimes hidden behind the 3.6 µm points). Labels show the Mon ID along with the optical and infrared morphologies, respectively; morphology abbreviations are the same as in Table 4.</figcaption></figure>

any standard extinction law and may require unique geometry, such as occultations of the disk by itself (i.e., the front blocks of flux from the back wall).

The remaining set of well-correlated optical and infrared light curves displays very similar morphology and amplitude in each band. Half of these are members of the optical bursting class, and the rest are stochastic or unknown morphology. Strong optical/infrared correlation for these would be expected if flux changes are reflective of increases in accretion, the radiation from which is absorbed and reemitted by the inner disk.

### 7.3. Uncorrelated Optical and Infrared Variability

Some 62% of the variable light curves in our sample display little correlation between the optical and infrared bands. In these cases, there are likely two or more variability mechanisms at work, connected separately with the star and the disk. Alternatively, a single variability mechanism could be at work if it only affects one wavelength region. Since the infrared flux is in most cases not dominated by stellar or accretion emission, disk-driven mechanisms are necessary to explain the often large-amplitude (~0.1 mag) infrared modulation on week or longer timescales. One possible origin is magnetic turbulence, proposed by Turner (2013) to modulate the inner disk scale height and thereby alter the observed mid-infrared emission.

We have assembled a set of the most prominent uncorrelated optical and infrared behavior in Figure 38. The wide range of morphologies in both bands is evident. Particularly interesting examples of uncorrelated behavior occur in Mon-000185, Mon-000273, Mon-000357, Mon-000876, and Mon-000928, for which there is almost no optical variability, but high-amplitude

(0.1–0.3 mag) infrared light-curve excursions on 5–10 day timescales. We have checked the individual images of these objects for erroneous cross matching of *CoRoT* and *Spitzer* sources, but in all cases there is no nearby star that could be a better match. This type of behavior occurs in a few percent of our disk-bearing stars.

The selection of objects with uncorrelated behavior is particularly helpful for investigating the distribution of infrared variability timescales, since we do not have to worry about much contamination from optical processes. We have measured the aperiodic infrared timescales of stars with median running Stetson indices less than 0.6. As illustrated in Figure 39, the distribution displays a clump around 5–15 days, although there are many additional objects with infrared timescales at 20 days and longer. The individual examples of rapid infrared changes, as well as the overall distribution, lend support to the idea that disk structural changes may be occurring close to or even faster than the local dynamical timescale in some cases. The latter is between a few days and a few weeks, depending on stellar mass and dust properties.

### 7.4. Inverse Correlation: Optical and Infrared Phase Shifts

A prediction of YSO variability models for edge-on systems (e.g., Kesseli et al. 2013) is that inner disk regions may receive non-uniform illumination from the central star, thereby reradiating infrared flux in a time-variable manner. An example is an accretion hot spot on the stellar surface, radiation from which interacts only with the region of the inner disk on that hemisphere of the star. In this case, we expect to observe the infrared emission 180 degrees out of phase with the hot spot
<figure><img src="image_40.png" /><figcaption>**Figure 38.** Light curves with uncorrelated optical and infrared behavior. Small black points are *CoRot* data, light gray points are 3.6 µm data, and dark gray points are 4.5 µm data (sometimes hidden behind the 3.6 µm points).</figcaption></figure>

signature, since we do not view the heated region of the disk wall until it has rotated behind the star.

Surprisingly, we encounter very few such examples of optical/infrared phase shifts in our time series. The only object with a clear inverse relationship between the two bands is Mon-001031. Mon-001132 displays a more subtle phase lag, but this shift does not appear to be constant across the entire observation time. We present the two light curves in Figure 40.

## 8. VARIABILITY IN THE CONTEXT OF STELLAR AND DISK PROPERTIES

For most of the *CoRot/Spitzer* data set, we have available spectral types, classification of the infrared excess, and optical through infrared photometry enabling determination of position on the H-R diagram. In many cases, we also have measurements of the equivalent width of prominent emission lines, such as Hα. We have correlated the variability properties measured in Section 6 with a number of the above-mentioned parameters and searched for combinations that may offer physical insight.

### 8.1. Relationship of Variability to Stellar Parameters

The main stellar parameter of interest is the effective temperature, which serves as a proxy for mass. We have estimated this

for the 112 stars in our sample that have available spectral types in Table 3. We then plotted the variability properties involving timescale and amplitude against temperature. One might expect infrared variability timescale to scale with temperature since it could reflect dynamics near the sublimation radius, which is in turn dependent on stellar mass. For completeness, we have separated the timescale measurements into quasi-periodic and aperiodic sets, as well as divided them into the two wavelength bands before comparing with temperatures. The correlation diagrams resulting from this exercise are shown in Figures 41. Timescale, whether periodic or aperiodic, does not show any prominent trends as a function of temperature. However, there is a lower envelope to the distribution of aperiodic infrared timescales versus temperature; this could be consistent with the orbital period at the sublimation setting the minimum timescale for disk variability. In addition, the implied lack of a global dependence of the variability timescale on mass does not necessarily mean that there is no relationship between these two parameters. We suspect that there are subclasses of variability that exhibit different behavior as a function of stellar mass. For example, the dipper objects show larger infrared amplitudes in comparison to the optical for preferentially later spectral types. Finally, there may be currently unobservable factors, such as disk inclination, that influence the variability properties.
<figure><img src="image_41.png" /><figcaption>**Figure 38.** (Continued)</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>**Figure 39.** Distribution of timescales measured with the PeakFind algorithm for infrared light curves that are not well correlated with their optical counterparts.</figcaption></figure>

<figure><img src="image_43.png" /><figcaption>**Figure 40.** Light curves with potential phased shifts between their optical and infrared behavior. Small black points are *CoRot* data, light gray points are 3.6 µm data, and dark gray points are 4.5 µm data (sometimes hidden behind the 3.6 µm points).</figcaption></figure>

We have also carried out a similar comparison of variability amplitude versus temperature, as shown in Figure 42. Here, we find the surprising result that aperiodic variability amplitudes are significantly lower around the cool stars in our sample. This

trend is the opposite of what one would expect from a detection bias, in which it is harder to detect low-amplitude variability around fainter stars. The effect is stronger in the optical, but also seems to appear in the infrared. This is in contrast to the periodic variability, which shows no appreciable mass dependence. The
<figure><img src="image_44.png" /><figcaption>**Figure 41.** We plot variability timescale against effective temperature (where spectral types are available), as a proxy for mass. We do not see correlations between these two parameters, although the lower envelope of aperiodic infrared timescales is roughly consistent with the orbital period at the disk sublimation radius.</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>**Figure 42.** rms vs. stellar effective temperature, for [quasi-]periodic and aperiodic variables in both bands. The amplitudes of optical variability appear to grow with mass, while no such trend is seen in the infrared.</figcaption></figure>

rms/temperature trend in aperiodic variables may reflect a correlation of magnetic field strength and/or configuration with mass. However, this may be contradicted by recent results from Zeeman-Doppler imaging that suggest that the dipole field component *weakens* with increasing mass (e.g., Gregory et al. 2012).

Perhaps the most telling correlation we have noted is the connection between light-curve flux asymmetry (*M*) and $H\alpha$ equivalent width. We plot these two parameters in Figure 43. While there is not a one-to-one correspondence between them, it is clear that the more negative *M* values correspond to stronger $H\alpha$ emission. This finding supports the idea that the burster
disk obscuration model may be invalid in part. A further puzzle came in the form of two stars (Mon-000183 and Mon-000566) displaying dips that are *deeper* in the infrared (nearly 10%) than in the optical (~2%–3%). A very non-standard extinction law or geometric peculiarities will have to be invoked to explain these two enigmatic objects.

Pure periodicities are very rare in the disk-bearing sample. We argue that each of these classes represents a distinct physical variability mechanism on the star or in the disk. While follow-up studies including spectroscopy are needed to confirm physical mechanisms, the data are consistent with variable circumstellar obscuration, unsteady accretion, rotating starspots, and rapid structural changes in the disk—and likely multiple of these processes happening simultaneously. Some of the variability may be due directly to the effects of these phenomena on physical properties, while some may be due to radiative transfer effects on the SED.

The clear asymmetries in the dipper- and burster-type light curves motivated us to develop a statistic measure of asymmetry, “*M*.” In combination with the quasi-periodicity, “*Q*,” we can uniquely identify the variability type of each star in each band without relying on subjective evaluation. Although we set out to understand the wavelength dependence of variability in our sample, we have found that in fact optical and infrared variability behavior in young disk-bearing stars is not well correlated in over 50% of cases. Most of our stars have different optical and infrared classes, with dipping and bursting behavior becoming less common at longer wavelengths and long-timescale variability becoming more common.

We highlight the set of high-amplitude infrared variables for which there is little to no corresponding variability in the optical. We have measured timescales of order ~5–10 days for these variables, providing evidence that they represent structural rearrangements in the disk. Among the entire sample, we find that high-amplitude (>0.1 mag) infrared variability appears in objects with SED class of type II and earlier, whereas the majority of type II/III transitional disk systems have rms infrared amplitudes less than 0.05 mag. Other correlations identified between variability properties and star/disk parameters include increases in aperiodic amplitude with stellar effective temperature, as well as increased flux asymmetry with $H\alpha$ equivalent width. These correlations hint at a connection between variability properties and magnetospheric structure.

Overall, we have developed a process for automatically classifying variability that can be applied to other data sets and clusters as well. We have focused exclusively on time series photometry from *CoRoT* and *Spitzer* here, but in future papers we will explore variability at the full range of available wavelengths.

This work is based in part on observations made with the *Spitzer Space Telescope*, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. S.H.P.A. acknowledges support from CNPq, CAPES, and Fapemig. R.G. gratefully acknowledges funding support from NASA ADAP grants NNX11AD14G and NNX13AF08G and Caltech/JPL awards 1373081, 1424329, and 1440160 in support of *Spitzer Space Telescope* observing programs. M.M.G. acknowledges support from INCT-A/CNPq. K.Z. received a Pegasus Marie Curie Fellowship of the Research Foundation Flanders (FWO) during part of this work and received funding from the Eu-

ropean Research Council under the European Community's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement No. 227224 (PROSPERITY).

# APPENDIX

## A.1. NGC 2264 Membership Criteria

NGC 2264 has been the subject of many young cluster studies, from photometric and $H\alpha$ censuses (e.g., Rebull et al. 2002; Lamm et al. 2004) to X-ray (e.g., Ramírez et al. 2004a; Flaccomio et al. 2006; Feigelson et al. 2013) and radial velocity surveys (e.g., Fűrész et al. 2006). To cull a reliable membership list from candidates reported in the literature, we required that objects meet at least two of six criteria: (1) photometric data consistent with the $V-I$ or $R-I$ cluster locus defined by Flaccomio et al. (2006) (see their Section 3.2); (2) strong photometric $H\alpha$ emission, according to the criteria of Sung et al. (2008), or spectroscopic $H\alpha$ EW larger than 10 Å; (3) X-ray detection at a flux greater than $10^{-4} L_{bol}$ (Ramírez et al. 2004a, 2004b; Flaccomio et al. 2006); (4) radial velocity consistent with NGC 2264 membership, as classified by Fűrész et al. (2006); (5) mid-infrared excess indicating a disk (i.e., Class I, II, or flat SED, according to selection methods described in Section 4.2); and (6) spatial location coinciding with an $A_V > 7$ region of NGC 2264, if an object displays an infrared excess or X-ray emission but is not detected in the optical.

We identified highly embedded objects for criterion (6) by plotting X-ray and mid-infrared source locations on the extinction map produced by Teixeira et al. (2012). These obscured stars were added to our overall membership list for further study, but not included in this paper since we are focusing specifically on targets with *CoRoT* detections.

The above requirements eliminate most field dwarf and extragalactic contaminants from our membership sample. We have avoided selecting members based on variability detection so as not to bias our statistical analysis of the light-curve types.

## A.2. IRAC Staring Data

While not used extensively in this paper, the high-cadence staring data from *Spitzer/IRAC* offer an important window into short-timescale infrared variability in YSOs. In preparing it for analysis, we performed a series of procedures to remove systematics. The steps described below may be of general interest to other *Spitzer/IRAC* users.

### A.2.1. Staring Data Quality and Correction of Systematics

The pixel-phase effect, or variation of measured flux as a function of sub-pixel position, is a known issue affecting the IRAC detector (see the IRAC instrument handbook, v2.0.3). Intrapixel sensitivity variations introduce two systematic effects to the staring (i.e., continuous, non-dithered) light curves: first, the flux may vary up to 6% as the pointing shifts gradually from one side of a pixel to the other; second, the flux exhibits oscillations at the percent level on <1 hr timescales. The first effect reflects the pointing stability of *Spitzer*, while the second was determined by the engineering team to be associated with the cycling of a battery heater on board the satellite, causing periodic flexure between the star tracker and the cold focal plane. The result is that the pointing center on the detector oscillates within individual pixels, convolving their sensitivity variations with the intrinsic flux. In 2010 October, the *Spitzer* engineering team was able to reduce both the amplitude and period of the
<figure><img src="image_46.png" /><figcaption>**Figure 46.** Normalized flux values as a function of pixel position for a non-variable star in our sample (top). The coordinate 0,0 is the pixel center. The pointing varied primarily in the Y direction, as shown in the flux vs. Y and flux vs. X plots at middle and bottom. The red line indicates the slope in flux vs. Y that minimizes the pixel-phase peak in the periodogram shown in Figure 47; this is *not* a direct linear fit to the data.

(A color version of this figure is available in the online journal.)</figcaption></figure>

heater-associated temperature fluctuations, the latter from ~60 minutes to ~40 minutes. However, the effect remains prominent at the 0.5% level in both channels, comparable to the white-noise contribution for bright stars in the sample.

Our goal of assessing infrared variability of young stars on short timescales and at low amplitudes necessitates the removal of pixel-phase variations to the extent possible from the data. A pixel-phase correction exists for Warm *Spitzer* observations, but tests on our light curves revealed that it works for stars on some pixels, but not others. Several techniques have been developed to provide a more consistent correction, many of which involve fitting a polynomial to the flux as a function of detector X and Y position (e.g., Bonfils et al. 2011). Unfortunately, this approach does not work well for many of our targets, as the erratic variability that we seek to study cannot be modeled analytically and thus prevents a robust fit. In Cody & Hillenbrand (2011), we adopted a different technique, involving a Gaussian model of the intra-pixel sensitivity variation, for which free parameters were fit by minimizing the height of the pixel-phase oscillation peak in the periodogram. Maps of the pixel sensitivity were determined independently for each star, and the inferred flux

variation as a function of X and Y position was then subtracted from the light curves.

While this method provided satisfactory corrections without compromising intrinsic stellar variability, it was computationally intensive. To apply a similar technique to our ~500 staring targets, we require an algorithm with fewer free parameters. Fortunately, during our 2011 observations of NGC 2264, object centroids were stable to ~0.08 pixel (i.e., 0.1) in the X detector position over the course of each 16–26 hr staring mode observation. Flux variations are much better correlated with the Y centroids, which vary by 0.3–0.4 pixels (0.3–0.5). Plots of the normalized flux versus Y for stars lacking obvious variability by eye revealed nearly linear trends, as shown in Figure 46. We therefore chose a basic pixel-phase correction that fits a single slope characterizing the pixel sensitivity as a function of Y position. To determine this slope, A, we express the sensitivity, s, as a linear function of Y position on the detector:

$$s(y) = A(y - y_{\text{med}}) + 1. \quad (A1)$$

Here we have normalized the sensitivity to a median value of 1.0 in the area of the pixel where the data fall. Applying
<figure><img src="image_47.png" /><figcaption>**Figure 47.** Top: uncorrected light curve, showing the pixel-phase oscillation, as well as longer timescale systematics. Middle: light curve processed with the standard Warm Spitzer pixel-phase correction; this algorithm does not fully remove the systematics. Bottom: light curve corrected by the procedure described in this paper. Below these light curves, we display the periodograms of the corrected light curves (dashed, dotted curves), as compared with the periodogram of the raw light curve (solid curve).</figcaption></figure>

this sensitivity function, the true flux data, $d_{\text{true}}(y)$, will be observed as

$$d_{\text{obs}}(y) = d_{\text{true}}(A(y - y_{\text{med}}) + 1). \quad (\text{A2})$$

Note that we have specified the observed and true data to have the same median. To correct the observed data, we then invert this equation, exploiting the fact that $y - y_{\text{med}} \ll 1$:

$$d_{\text{true}}(y) = d_{\text{obs}}/(A(y - y_{\text{med}}) + 1) \sim d_{\text{obs}}(1 - A(y - y_{\text{med}})). \quad (\text{A3})$$

We wish to determine the value of A so that the corrected light curve is devoid of pixel-related systematics. To do this, we take advantage of the pixel-phase effect, which causes $y(t)$, the detector position, to oscillate rapidly as a function of time. The period of oscillation is approximately 40 minutes. To determine this value more accurately, we compute a Fourier transform periodogram for each chunk of staring data, $y(t)$, first subtracting out its median value $y_{\text{med}}$ to remove low-frequency

systematics:

$$FT(f, y(t_k)) = \frac{2}{N} \left[ \left( \sum_{k=0}^{N} \sin(-2\pi f t_k)(y(t_k) - y_{\text{med}}) \right)^2 + \left( \sum_{k=0}^{N} \cos(-2\pi f t_k)(y(t_k) - y_{\text{med}}) \right)^2 \right]^{1/2}, \quad (\text{A4})$$

where f is frequency in the periodogram, k is the index of points in the time series, and N is the total number of points. We numerically locate the highest peak in this periodogram and note its frequency, $f_0$. We find a value of ~37.5 cycles per day; this can be seen with the solid curve in the bottom panels of Figure 47. The corresponding pixel-phase period is $1/f_0 \sim 38.4$ minutes.
We will now use the oscillatory behavior of $y(t)$ to determine the best value of the slope A in the pixel sensitivity function $s(y)$. We do this by minimizing the value of the Fourier transform periodogram of the data themselves, $d(y(t_k))$, at $f_0$. The value of A resulting from this process can then be used to remove the effects of pixel sensitivity variation from the data without compromising variability. We begin by writing out the Fourier transform periodogram for the true data, again subtracting out the median value, $d_{\text{med}}$, to eliminate periodogram systematics:

$$ 
\begin{aligned}
FT(f, d(y(t_k))) &= \frac{2}{N} \left[ \left( \sum_{k=0}^{N} \sin(-2\pi f t_k) (d_{\text{true}}(t_k) - d_{\text{med}})^2 \right. \right. \\
&\qquad \left. \left. + \left( \sum_{k=0}^{N} \cos(-2\pi f t_k) (d_{\text{true}}(t_k) - d_{\text{med}}) \right)^2 \right) \right]^{1/2} \\
&= \frac{2}{N} \left[ \left( \sum_{k=0}^{N} \sin(-2\pi f t_k) (d_{\text{obs}}(t_k) \cdot (1 - A(y_k - y_{\text{med}})) - d_{\text{med}}) \right)^2 \right. \\
&\qquad \left. + \left( \sum_{k=0}^{N} \cos(-2\pi f t_k) (d_{\text{obs}}(t_k) \cdot (1 - A(y_k - y_{\text{med}})) - d_{\text{med}}) \right)^2 \right]^{1/2}.
\end{aligned}
\quad (A5) $$

We next minimize the periodogram with respect to the slope, A, at the peak frequency $f_0$:

$$ \frac{dFT(f_0, d(y(t_k)))}{dA} = 0. \quad (A6) $$

It can be shown that this equation takes the form

$$ 0 = AC_1 + C_2 + AC_3 + C_4, \quad (A7) $$

where $C_1$, $C_2$, $C_3$, and $C_4$ are constants, as follows:

$$ C_1 = - \left[ \sum_{k=0}^{N} \sin(-2\pi f_0 t_k) \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} \cdot (y_k - y_{\text{med}}) \right]^2, \quad (A8) $$

$$ C_2 = \left[ \sum_{k=0}^{N} \sin(-2\pi f_0 t_k) \left( \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} - 1 \right) \right] \times \left[ \sum_{k=0}^{N} \sin(-2\pi f_0 t_k) \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} \cdot (y_k - y_{\text{med}}) \right], \quad (A9) $$

$$ C_3 = - \left[ \sum_{k=0}^{N} \cos(-2\pi f_0 t_k) \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} \cdot (y_k - y_{\text{med}}) \right]^2, \quad (A10) $$

$$ C_4 = \left[ \sum_{k=0}^{N} \cos(-2\pi f_0 t_k) \left( \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} - 1 \right) \right] \times \left[ \sum_{k=0}^{N} \cos(-2\pi f_0 t_k) \frac{d_{\text{obs}}(t_k)}{d_{\text{med}}} \cdot (y_k - y_{\text{med}}) \right]. \quad (A11) $$

Operating on the four sections of by-BCD staring light curves, we determine analytically the slope A that minimizes the pixel-phase oscillation signal by computing

$$ A = \frac{-(C_2 + C_4)}{C_1 + C_3}. \quad (A12) $$

<figure><img src="image_48.png" /><figcaption>**Figure 48.** Standard deviations for the first section of IRAC staring light curves in channel 1 (top) and channel 2 (bottom), as a function of magnitude. The lower envelope of values closely matches the uncertainties predicted by Poisson noise and sky background, whereas the higher values are due to stellar variability.</figcaption></figure>

Finally, we divide out the pixel response from the light-curve sections by calculating $d_{\text{true}}$ as in Equation (3). Because the X and Y positions are well correlated in time, elimination of the variation associated with shifts in Y only should effectively remove flux trends in X as well (see Figure 46).

Application of our algorithm resulted in a substantial reduction in the level of systematic variability in our light curves, as indicated by both visual inspection and measurement of the height of the periodogram peak at the 40 minute pixel-phase oscillation period. An example comparing our modified correction with the standard correction and a raw light curve is shown in Figure 47. For approximately 1% of light curves, the pixel sensitivity distribution was not well modeled by a linear fit and our correction introduced oscillatory features that were not present in the original light curve. In these cases, we retained the raw light curve or the version corrected with the standard Warm Spitzer pixel-phase prescription, depending on which one displayed lower levels of systematics.

The rms values of the corrected 3.6 µm staring light curves ranged from 0.003 mag (at a magnitude of 7.5) to 0.1 mag (at a magnitude of 15.5). The 4.5 µm light curves had similar but slightly lower precision (by a factor of ~1.5). The rms values as a function of magnitude for one of the staring photometry sections are presented in Figure 48; these are consistent with the predicted uncertainties based on Poisson noise and sky background.
<figure><img src="image_49.png" /><figcaption>**Figure 49.** Light curves for all stars in the 162 member disk-bearing sample of this paper. Small black points are *CoRot* data, light gray points are IRAC 3.6 µm data, and dark gray points are IRAC 4.5 µm data (sometimes hidden behind the 3.6 µm points). More information is available in Tables 3 and 4; objects are presented in the same order here.</figcaption></figure>

(An extended version of this figure is available in the online journal.)
### A.2.2. Merging the Mapping and Staring Data

Each section of staring photometry was surrounded by lower cadence mapping photometry. While both data types contain the same systematics, those taken in mapping mode are much less correlated in time, since the dithering and other pointing changes sample the varying pixel sensitivity in a random manner. We therefore cannot remove the mapping systematics but have accounted for them (e.g., Section 3.3) for statistical purposes. Much of the staring photometry, on the other hand, displays significant and correctable zero-point offsets from the surrounding mapping points. This effect occurs when the sensitivity of a single pixel containing the object centroid during a staring observation differs from the mean sensitivity across the many different pixels occupied during mapping observations. Since discontinuities in the light curves create challenges for variability analysis, we have introduced a further set of corrections to ensure smooth transitions between staring and mapping photometry. Following pixel-phase mitigation, we selected the set of ~280 by-BCD staring points lying within 1.2 hr of the beginning or end of each staring data section. Likewise, we select the set of ~6 mapping points (from by-AOR photometry) lying within 9.6 hr of the beginning or end of each mapping section. For each set of photometry, we computed a linear fit to the magnitudes as a function of time, to account for short-timescale variability. Offsets between adjacent staring and mapping light-curve sections were then determined by subtracting the fit values at the point midway between them. Each staring section had an offset at both ends, from which we determined the mean. This final offset was then applied to the entire staring section, thereby knitting it to the surrounding mapping data.

The typical offset between staring and mapping data was ±0.01–0.05 mag. It is unclear as to why it is often larger than the random scatter in the mapping data. Since the offset fits are undoubtedly affected by observational noise, we use the combined light curves with caution and provide statistics separately for mapping data and individual staring data sections. To produce staring light curves with lower uncertainties, we further binned sets of 10 points for a final cadence of 2.5 minutes. In all, there were twenty-six stars in our disk-bearing sample with both IRAC staring data and CoRoT observations.

### A.3. Complete Combined Optical/Infrared Data Set

We assemble in Figure 49 the entire 162 member disk-bearing data set observed by CoRoT and Spitzer. Small black dots are optical data, light gray points are 3.6 µm data, and dark gray points are 4.5 µm data.

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