Extinction and the Dimming of KIC 8462852

The Ultraviolet/Optical Telescope (UVOT) is one of three instruments on board the Swift mission. It is a modified Ritchey-Chrétien 30 cm telescope with a wide ($17^{\prime} \times 17^{\prime}$) field of view and a microchannel plate intensified CCD detector operating in photon-counting mode (see details in Roming et al. 2000, 2004, 2005). UVOT provides images at 23 resolution and includes a clear white filter, u, b, and v optical filters, uvw1, uvm2, and uvw2 UV filters, a magnifier, two grisms, and a blocking filter. The uvw2 and uvw1 filters have substantial red leaks, which have been characterized to high precision by Breeveld et al. (2010) and are included in the current UVOT filter curves. Calibration of the UVOT is discussed in depth by Poole et al. (2008) and Breeveld et al. (2011).

In full-frame mode, the CCD is read every 11 ms, which creates a problem of coincidence loss (similar to pile-up in the X-ray) for stars with count rates greater than 10 cts s⁻¹ (∼15 mag, depending on filter). The camera can be used in a windowed mode, in which a subset of the pixels is read. Given the brightness of KIC 8462852, to reduce the coincidence corrections we observed in a $5^{\prime} \times 5^{\prime}$ window (70 × 70 pixels), resulting in a 3.6 ms readout time. The observations were generally 1 ks in duration, utilizing a mode that acquired data in five filters (v, u, uvw1, uvm2, and uvw2, from 5900 to 1600 Å). To improve the precision of the photometric measurements and ensure that the target star and comparison stars all landed in the readout window, observations were performed with a "slew in place," in which Swift observed the field briefly in the "filter of the day"—one of the four UV filters—before slewing a second time for more precise positioning. The slew-in-place images were used for additional data points in the UV. KIC 8462852 was first observed on 2015 October 22 and then approximately every three days from 2015 December 4 to 2016 March 27. It was later observed in coordination with the Spitzer campaign starting in 2016 August.

X-ray data were obtained simultaneously with the Swift/XRT (X-ray Telescope, Burrows et al. 2005). No X-ray emission within the passband from 0.2 to 10 keV is seen from KIC 8462852 in 52 ks of exposure time down to a limit of $5\times {10}^{-15}$ erg s⁻¹ cm⁻² using the online analysis tools of Evans et al. (2009).

Optical observations in B, V, and R bands were taken with the 684 mm aperture Keller F4.1 Newtonian New Multi-Purpose Telescope (NMPT) of the public observatory AstroLAB IRIS, Zillebeke, Belgium. The CCD detector assembly is a Santa Barbara Instrument Group (SBIG) STL 6303E operating at −20 °C. A 4 inch Wynne corrector feeds the CCD at a final focal ratio of 4.39, providing a nominal field of view of $20^{\prime} \times 30^{\prime}$. The 9 μm physical pixels project to 062 and are read out binned to 3 × 3 pixels, i.e., 186 per combined pixel. The B, V, and R filters are from Astrodon Photometrics, and have been shown to reproduce the Johnson/Cousins system closely (Henden 2009). The earliest observation was made on 2015 September 29. There is a gap in time coverage from 2016 January 8 to June 8. We report the observations through 2016 December.

Spitzer observations are made with the IRAC with a uniform exposure design, which uses a cycling dither at 10 positions on the full array with 12 s frame time. Using multiple dither positions tends to even out the intra- and inter-pixel response variations of the detector; repeating the same dither pattern at every epoch puts KIC 8462852 roughly on the same pixels of the detector, further reducing potential instrumental bias on the photometry.¹⁰ Our first Spitzer observation was executed on 2016 January 16. There is a gap in the time baseline of the monitoring in the period from 2016 April to July (from MJD 57475 to 57605) when KIC 8462852 was out of the visibility window of Spitzer.¹¹ The Swift–Spitzer coordinated monitoring is still underway at the time this paper was written.

Swift/UVOT data were obtained directly from the HEASARC archive.¹² We then used the HEASARC FTOOLS software¹³ program UVOTSOURCE on the transformed sky images to generate point-source photometry.

The Swift/UVOT photometry includes a filter-dependent correction that accounts for the decline in instrument sensitivity (Breeveld et al. 2011). However, the potential fading of KIC 8462852 pushes the boundaries of the UVOT calibration, which is specified to within 1%. To check for any residual sensitivity changes, we made photometric measurements for four additional field stars in the UVOT field. The reference stars selected are KIC 8462934, KIC 8462763, KIC 8462843, and KIC 8462736, which all lie 14 to 20 from KIC 8462852. Since these reference stars are all fainter than KIC 8462852 by 2–5 mag in the UV and have larger photometric errors individually, we took their weighted average as our photometric reference to minimize the noise. Their linear fits indicate that the average reference star appears to get brighter in the UVOT data over the full range of our time coverage, especially when the post-gap data are considered (−22.2 ± 5.2 mmag yr⁻¹). We examined the reference stars individually and found that all four of them follow similar and consistent brightening rates, eliminating the possibility of "bad" star contamination. This suggests a small residual instrumental trend in the UVOT data, which could reflect a small overestimate of the sensitivity loss or a small residual in the coincidence loss correction.

To remove the instrumental trend of Swift/UVOT, we subtract the normalized magnitude of the average reference star from the absolutely calibrated magnitude of KIC 8462852. This inevitably propagates the photometric uncertainties of the average reference star into the KIC 8462852 light curve. In all following discussions, we only use the corrected calibration data of Swift/UVOT. All the Swift photometry is given in Table 1 and is displayed in Figure 1. For a first search for trends in the brightness of the star, we performed a linear fit to the photometry. As shown in Table 2, with the corrected calibration the fading of KIC 8462852 is seen at a 3σ significance level in the pre-gap data and by >2σ in the full data set.

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Table 2.  Trends of the Brightness of KIC 8462852

		Pre-Gap				Full Data
Waveband	${\lambda }_{\mathrm{eff}}$	dm/dt	σ(dm/dt)	${\chi }_{\mathrm{red}}^{2}$	siga	dm/dt	σ(dm/dt)	${\chi }_{\mathrm{red}}^{2}$	siga
	(μm)	(mmag yr−1)	(mmag yr−1)			(mmag yr−1)	(mmag yr−1)
uvw2	0.2030	83.7	66.6	0.81	⋯	71.8	31.5	0.92	⋯
uvm2	0.2231	210.5	203.9	1.03	⋯	−60.3	86.5	1.07	⋯
uvw1	0.2634	108.5	61.5	0.96	⋯	17.4	24.5	1.04	⋯
u	0.3501	69.0	44.0	0.29	⋯	42.7	17.8	0.30	⋯
v	0.5402	54.9	32.7	0.27	⋯	1.9	14.5	0.34	⋯
Swift Averageb		71.0	22.6	⋯	Y	22.1	9.7	⋯	⋯
B	0.435	20.2	11.4	⋯	⋯	26.3	1.5	⋯	Y
V	0.548	16.2	8.1	⋯	⋯	21.6	1.5d	⋯	Y
R	0.635	1.6	6.7	⋯	⋯	13.1	1.0	⋯	Y
[3.6]c	3.550	⋯	⋯	⋯	⋯	5.1	1.5	1.99	Y
[4.5]c	4.493	⋯	⋯	⋯	⋯	4.8	2.0	1.84	⋯
Spitzer Averageb		⋯	⋯	⋯	⋯	5.0	1.2	⋯	Y

Notes.

^aSignificance flag. "Y" means that the magnitude changing rate is significantly different from zero at the 3σ level or higher. Otherwise this is left blank. ^bAveraging the data, ignoring the wavelength information, implicitly assumes a gray color of the fading. ^cWe only have two epochs of Spitzer observations before the gap, and the gap is offset relative to the ground-based data, so we do not quote values pre-gap. ^dComputed omitting the first night, which is systematically high and had a large number of measurements that drive the slope inappropriately.

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The Swift data suggest that the star faded during our observations, and such behavior would be consistent with the long-term secular fading of KIC 8462852 observed previously (Montet & Simon 2016; Schaefer 2016). To probe this behavior further, we turn to the ground-based BVR photometry obtained at AstroLAB IRIS and available from AAVSO. We did not detect any major dips in the flux from the monitoring up to 2016 December, but our measurements do indicate a slight long-term secular dimming.

We used differential photometry relative to four stars in the field, selected to be similar in brightness and color to KIC 8462852 (see Table 3). These observations were obtained simultaneously with those of KIC 8462852, typically in a series in V followed by B and then R. The reductions utilized the LesvePhotometry reduction package (de Ponthière 2013), which is optimized for time-series observations of variable stars. It automates reduction of the data in a series of observations, providing a homogeneous database of photometry for KIC 8462852. Errors from photon noise on the source, scintillation noise and background noise are determined for each observation, using the methodology in Newberry (1991). We eliminated data from a single night when the source was at large (>2) airmass. We also eliminated data from nights with larger than normal estimated errors (>0.025 magnitudes rms) for KIC 8462852, as identified either by the reduction package or by large rms errors for the data obtained within that night. The remaining photometry in B, V, and R bands is displayed in Figure 1. The scatter is somewhat larger than implied by the internal error estimates; therefore, we will base our error estimates on the scatter. Linear fits show a dimming in all three colors over the full data set, and probable dimming but not at signficant levels pre-gap (see Table 2). No significant dimming is seen post-gap. In estimating the uncertainties of these slopes, we found that the reduced ${\chi }^{2}$ using the reported internal errors was large (2 to 4 depending on the band), consistent with our finding that the rms scatter of the measurements is larger than the reported errors. We brought the reduced ${\chi }^{2}$ of the fit to ∼1 by adding an additional error of 0.01 mag in quadrature to each measurement.

There are two issues with the linear fits. The first is that, given that there is no evidence for fading in the data after the interruption due to solar viewing constraints (see Figure 1), the fits tend to be high toward the beginning and low toward the end of the post-gap sequence. This is particularly prominent in the v and V fits as shown in Figure 1. The linear fits are meant as the simplest way to quantify the dimming that would include all the data in each band, but they do not appear to be exactly the correct dependence.

The second issue is that, for the ground-based data, our discussion above does not include the possibility of systematic errors affecting the comparison pre- and post-gap, effects that are not included in the LesvePhotometry package. (We have already eliminated such errors for the ${\text{}}{Swift}$/UVOT data and, as discussed below, they should be negligible for the Spitzer measurements.) To test for such effects, we turned to the photometry of the four reference stars to estimate night-to-night and longer-term errors and to select the nights with the most consistent observations to see if the dimming was apparent just using this subset of best measurements. Since these further steps made no reference to the photometry of KIC 8462852, they should introduce no bias in its measurements.

We first computed the standard deviations of running sets of 45 measurements for each of these four stars. When this value exceeded an average of 0.015 per star, we investigated the photometry involved and eliminated nights contributing disproportionately to the value. Following this step, we examined the consistency of the remaining measurements of the reference stars. Since we do not know the "true" magnitudes of the stars, we instead tested for the consistency of the measurements of each star across the gap in time coverage. To establish a baseline, we identified two long consecutive sets of measurements for each star and each color, one on each side of the gap, that agreed well. We then tested each of the additional nights of data to see if they were consistent with this baseline, and added in the data for the nights that did not degrade the agreement across the gap. We carried out this procedure individually for each of the three colors, but found that the same nights were identified as having the highest quality photometry in each case. The final typical mismatch in photometry across the gap was 0.0032 magnitudes, showing that this vetting was effective in identifying nights with consistent results for the four reference stars. The photometry of KIC 8462852 on these nights is listed in Table 4 and plotted in Figure 1.

The photometry selected to be of highest internal consistency is generally consistent with the rest of the measurements. The final averages for KIC 8462852 just based on these nights before and after the gap in the time series are shown in Table 5. Each band shows a small but statistically significant dimming from pre-gap to post-gap, both in the initial photometry (Table 2) and in that selected to be of highest quality.

Table 5.  Summary of Ground-based Monitoring of KIC 8462852

MJD Range	$\langle B\rangle$ a	err(B)b	$\langle V\rangle$ a	err(V)b	$\langle R\rangle$ a	err(R)b
57322-57396	12.374	0.0032	11.845	0.0032	11.453	0.0032
57549-57693	12.395	0.0032	11.859	0.0032	11.466	0.0032
Differences	0.021	0.0045	0.014	0.0045	0.013	0.0045

Notes.

^aAverage magnitude. ^bCombined rms errors of the mean, i.e., rms scatter divided by the square root of (n − 1), where n is the number of measurements.

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There is a hint of fading in the pre-gap data, but averaged over the three bands the net change is 0.010 ± 0.005 magnitudes, i.e., small and potentially insignificant. The post-gap data indicate no significant long-term secular changes beyond the errors of ∼0.003 magnitudes in any of the bands. These results suggest that most of the change in brightness occurred while the star was in the gap for the ground-based photometry. This behavior would be consistent with that observed for long-term dimming using Kepler data, where most of the change is a drop in brightness by ∼2% over a period of 300 days (Montet & Simon 2016). The gap in our data is intriguingly about 1400 days (approximately twice the interval between the two large dips in the light curve) past the time of the similar dimming seen in the Kepler data. The amplitude in the V and R bands (which together approximate the Kepler spectral response) is about 1.4 ± 0.3% (Table 5), and the duration of the gap in our highest quality BVR data is ∼150 days, values that are also reminiscent of the Kepler observations.

For the purpose of probing for a long-term secular trend, we do not consider the earlier Spitzer photometry (Marengo et al. 2015). That observation was made under the SpiKeS program (Program ID 10067, PI M. Werner) in January 2015, too far from the epochs of our new data. In addition, the SpiKeS observation was executed with an AOR design different from ours for the dedicated monitoring of KIC 8462852, which may lead to different instrumental systematics in the photometry.

The photometry we did use is from AORs 58782208, 58781696, 58781184, 58780928, 58780672, 58780416, and 58780160 (PID 11093, PI K. Y. L. Su) and 58564096 and 58564352 (PID 12124, PI Huan Meng). Photometric measurements were made on cBCD (artifact-corrected basic calibrated data) images with an aperture radius of 3 pixels and sky annulus inner and outer radii of 3 and 7 pixels, with the pixel phase effect and array location dependent response functions corrected. Aperture correction factors are 0.12856 and 0.12556 magnitudes at 3.6 and 4.5 μm, respectively (Carey et al. 2012). Individual measurements were averaged for each epoch. The photometry is summarized in Table 6. During the period when the KIC 8462852 data were obtained, the photometric performance of IRAC is expected to have varied by less than 0.1% per year (i.e., <1 mmag yr⁻¹).¹⁴ There is a suggestion of a fading at the ∼3σ level of significance in the 3.6 μm band; there is also a fading in the 4.5 μm band at lower significance.

The three independent sets of observations presented in this paper all show evidence of dimming in KIC 8462852. Measurements reported by the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee et al. 2014; Kochanek et al. 2017) do not completely overlap with ours, but a preliminary analysis shows that they show a fading of about 8 mmag at V, comparing their data between MJD of 57200 and 57334 with that between MJD of 57550 and 57740; their measurements may indicate a further small fading after that. The ASAS-SN photometry has not been tested thoroughly for systematic errors at this small level (K. Stanek, private communication, and see warning at https://asas-sn.osu.edu/); nonetheless, the results agree within the mutual errors with ours. The dimming is also corroborated in measurements by B. Gary.¹⁵

However, we find that the amount of dimming is significantly less in the infrared than in the optical and ultraviolet, as shown in Figure 1. We will investigate in Section 4 what constraints the wavelength dependence lets us put on this event. To do so, we need to focus on periods when measures are available in all the relevant bands (see Figure 1), since otherwise wavelength-independent and -dependent brightness changes are degenerate.

The first sequence of Swift measurements extends well beyond the end date for the first sequence of BVR measurements. For the purposes of Section 4, we compute the dimming using the average of the measurements through MJD 57396. For the ground-based measurements, we take only the measurements on nights that passed our test for consistency of the comparison star measurements. We average all the data from such nights pre-gap and, separately, post-gap, and base the errors on the rms scatter of the individual measurements. For Spitzer, there is only a single measurement close in time to the first ground-based sequence, namely at JD 2457404. To analyze these data, we first confirm the errors by computing the scatter in both bands (combined). This calculation indicates an error of 0.002, slightly larger than the quoted errors of 0.0012–0.0016. Using our more conservative error estimate, the change from the first point to the average of the post-gap ones at 3.6 μm is 0.0049 ± 0.0021 mag, and at 4.5 μm it is 0.0033 ± 0.0021 mag, or an average of 0.0041 ± 0.0015. We compare this value with the change between the average of the two sets obtained from the ground closest in time to the first Spitzer one, namely at MJD 57395 and 57396 (both of which passed our tests for high-quality data), versus the later (post-gap) ground-based measurements. We averaged the measurements in B and V together into a single higher-weight point for these two nights and then compared with the similar average of the post-gap measurements. The net change is 0.012 ± 0.0023 magnitudes, i.e., significantly larger than the change in the infrared.

Table 7 summarizes the measurements we will use to examine the color dependence of the dimming of KIC 8462852. It emphasizes conservative error estimation (including systematic ones) and homogeneous data across the three observatories, at the cost of nominal signal to noise. The dimming is apparent, at varying levels of statistical significance, in every band. The values in the table also agree with the slopes we computed previously (and noting that the time interval for the differences is about 74% of a year), as shown in the table for the cases with relatively high weight values so comparisons are meaningful.

Many of the hypotheses to explain the variability of KIC 8462852 depend on the presence of a substantial amount of circumstellar material. Excess emission from circumstellar dust is therefore an interesting diagnostic. Such an excess has not been found at a significant level (Lisse et al. 2015; Marengo et al. 2015; Boyajian et al. 2016; Thompson et al. 2016). However, previous searches have used data from 2MASS (Skrutskie et al. 2006), GALEX (Morrissey et al. 2007), warm Spitzer (Fazio et al. 2004), WISE (Wright et al. 2010), and new optical observations, which were taken more than 10 years apart, to constrain the stellar atmospheric models that are used to look for excess. The variability of the star, particularly if it has been fading for years (Montet & Simon 2016), could undermine the searches for excesses.

Now we can test if KIC 8462852 has had a significant excess at 3.6 and 4.5 μm by fitting stellar models with the Swift data taken at specific epochs and compare the model-predicted stellar IR flux with the simultaneous Spitzer measurements. By 2016 October, there have been five epochs at which we have Swift and Spitzer observations taken within 24 hr: MJD 57403, 57454, 57621, 57635, and 57673. The first two are before March 2016, whereas the last three epochs are observed after the gap. To analyze these results, we adopt the ATLAS9 model (Castelli & Kurucz 2004) with the stellar parameters obtained from spectroscopic observations, $\mathrm{log}(g)=4.0$, and [M/H] = 0.0. We allow T_eff to vary between 6700 and 7300 K with an increment of 100 K. We find that the stellar models with ${T}_{\mathrm{eff}}=7000$ and 7100 K provide the best fits to the Swift photometry at all five epochs, with a minimum reduced ${\chi }^{2}$ from 0.08 to 1.5 (see Figure 2). We did not use the Spitzer/IRAC measurements to further constrain the fit, since we did not want to bias any evidence for an infrared excess. Although these temperatures are slightly higher than originally estimated by Boyajian et al. (2016), they are in reasonable agreement with the SED model based on the 2MASS photometry (${T}_{\mathrm{eff}}=6950$ K, Marengo et al. 2015) and the recent IRTF/SpeX spectrum leading to a classification of F1V–F2V, i.e., ∼6970 K (Lisse et al. 2015). All of these values can be somewhat degenerate with changes in the assumed logg and metallicity. For our purposes, however, having a good empirical fit into the ultraviolet allows placing constraints on the extinction. All five best-fit stellar models, one for each epoch, have A_V in the range of 0.68 to 0.78, 2.0 to 2.3 times higher than A_V = 0.341 found by Boyajian et al. (2016) with photometric measurements years apart. In the Spitzer/IRAC wavebands at 3.6 and 4.5 μm, the observed flux densities match the model-predicted stellar output fairly well at all five epochs. The average excess is −0.39 ± 0.30 mJy at 3.6 μm and −0.29 ± 0.21 mJy at 4.5 μm. We conclude that we do not detect any significant excess of KIC 8462852 with near-simultaneous Swift and Spitzer observations. Our conclusion is consistent with that of Boyajian et al. (2016), showing that it is independent of the uncertainties in fitting the stellar SED.

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We have tested these conclusions using the average post-gap B, V, R, [3.6], and [4.5] measurements (Table 7) rather than the UVOT UV ones, standard stellar colors (Mamajek 2017), and a standard extinction law (Rieke & Lebofsky 1985; Chapman et al. 2009). The best fit was obtained assuming the star is of F1V spectral type (nominal temperature of 7030 K), with A_V = 0.61. Given the uncertainties in the extinction law, the stellar models, and intrinsic stellar colors (e.g., the effects of metallicity), this agreement is excellent. The assigned extinction level also agrees roughly with the relatively red color of the star relative to an F2V comparison star in infrared spectra (C. M. Lisse, private communication). The conclusion about the absence of any infrared excess is unmodified with this calculation.

Upper limits to the level of circumstellar dust were also determined at 850 μm with JCMT/SCUBA-2 (Thompson et al. 2016), and from WISE at 12 and 22 μm, in all cases where the stellar variations are relatively unimportant. Assuming that the grains emit as blockbodies and are distributed in a narrow, optically thin ring at various radii from the star, we place upper limits of ∼4 × 10⁻⁴ for the fractional luminosity, ${L}_{\mathrm{dust}}/{L}_{* }$, for warm rings of radii between 0.1 and 10 au and an order of magnitude higher for cold rings lying between 40 and 100 au, which would be a typical cold-ring size for a star of this luminosity (see also Boyajian et al. 2016). These limits are consistent with the presence of a prominent debris disk, since even around young stars these systems usually have ${L}_{\mathrm{dust}}/{L}_{* }\lesssim {10}^{-3}$ (Wyatt et al. 2007; Kenyon & Bromley 2008).

We carried out a second calculation to place upper limits on the possible dust masses. We again assumed that the dust is distributed in an optically thin ring (0.1 au wide in the ring plane). We took the optical constants derived for debris disk material (Ballering et al. 2016) and used the Debris Disk Radiative Transfer Simulator¹⁶ (Wolf & Hillenbrand 2005). The minimum grain radius was set to 1 μm, roughly the blowout size for spherical particles around an F1/2 V star, and the power-law index of the particle distribution was taken as 3.65 (Gáspár et al. 2012). We take the upper limit at 850 μm to be 4.76 mJy, or 5.6σ above zero, i.e., 3σ added to the 2.6σ "signal" at the position of the star (rather than the 3σ above zero as in Thompson et al. 2016). Similarly, we take a 3σ upper limit of 0.63 mJy at 4.5 μm. We took the cataloged upper limits for the two WISE bands, which are computed in a similar way but at a 2σ level. Stellar parameters are assumed to be ${T}_{\mathrm{eff}}=7000$ K and ${L}_{* }=5\,{L}_{\odot }$ (Pecaut & Mamajek 2013; Mamajek 2017).

The resulting limits are shown in Figure 3. The upper limit at 100 au is slightly higher than the mass of 0.017 ${M}_{\oplus }$ for a similar range of dust sizes (i.e., up to 2 mm) in the Fomalhaut debris ring (Boley et al. 2012). Again, a prominent but not extraordinary debris disk is allowed, corresponding for example to two to three orders of magnitude more dust than orbits the Sun. The upper limits also permit sufficient dust mass to yield significant extinction. To demonstrate, we assume that the dust is in a ring at radius R with a thickness perpendicular to the orbital plane of 0.1 R; for simplicity we take an ISM-like dust particle size distribution with constant density within this ring. The resulting upper limit on the extinction is ${A}_{V}\sim 0.1$ (Güver & Özel 2009). This value is independent of the radius assumed for the ring, since as shown in Figure 3, the upper limits for the mass scale roughly as R², which is also the scaling of the ring area under our assumptions. Of course, this is only a rough estimate, but it is sufficient to demonstrate that detectable levels of extinction can be consistent with the upper limits on the thermal emission of any material surrounding the star. That is, current measurements allow enough material to orbit KIC 8462852 to account for a number of the hypotheses for its behavior, such as the inspiralling and disintegration of massive comets. The minimum mass required to account for the long-term secular dimming through extinction is also within these mass constraints.

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We now explore the hypothesis that the long-term secular dimming of KIC 8462852 is due to variable extinction by dust in the line of sight. The absence of excess emission at 3.6 and 4.5 μm means that the photometry at these wavelengths is a measure of the stellar photospheric emission. Under the assumption that the fading of the star indicated in Table 7 is due to dust passing in front of the star, the relative amounts of dimming at the different wavelengths can therefore be used to constrain the wavelength dependence of the extinction from the UV to 4.5 μm in the IR.

Under this hypothesis, the dimming of KIC 8462852 may arise either from the ISM or circumstellar material. For convenience, we describe the color of the fading in the terminology of interstellar extinction, although circumstellar material might have different behavior if the color were measured to high accuracy. The Galactic ISM extinction curve from 0.1 to 3 μm can be well characterized by only one free parameter, the total-to-selective extinction ratio, defined as ${R}_{V}={A}_{V}/E(B-V)$ (Cardelli et al. 1989). Longward of 3 μm, measurements toward diffuse ISM in the Galactic plane (Indebetouw et al. 2005), toward the Galactic center (Fritz et al. 2011), and toward dense molecular clouds in nearby star-forming regions (Chapman et al. 2009) reveal consistent shallow wavelength dependence of the ISM extinction in the Spitzer/IRAC bands. We find that extrapolating the analytical formula in Cardelli et al. (1989; CCM89, hereafter) to the 3.6 and 4.5 μm bands of IRAC yields ${A}_{[3.6]}/{A}_{{K}_{s}}$ from 0.42 to 0.54 and${A}_{[4.5]}/{A}_{{K}_{s}}$ from 0.28 to 0.37 for R_V values from 2.5 to 5.0, a range suitable for most sight lines in the Milky Way. Although the CCM89 extinction law does not claim to apply to these wavelengths, the 3.6 and 4.5 μm extrapolations are in good agreement with the IRAC observations (see Table 3 in Chapman et al. 2009). Therefore, for simplicity, we adopt the CCM89 extinction law for all the seven bands monitored.

We have fitted the wavelength-dependent dimming in Table 7 with extinction curves using the formulation in CCM89, parameterized by R_V and extended to 4.5 μm. Figure 4 shows the results. Because of the relatively small level of dimming in the ultraviolet, the best-fitting extinction curves are relatively "gray," i.e., have large values of R_V. The vertical dashed lines show confidence levels corresponding to 1, 2, and 3σ. Values of ${R}_{V}\sim 5$ are favored; the general value for the ISM, R_V = 3.1 is disfavored at a confidence level >90%. Extinction even more gray than that given by R_V = 5 (or a form differing more fundamentally from the interstellar law) is a definite possibility. However, a completely neutral extinction law is excluded because of the small variations at [3.6] and [4.5].

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So far we have conducted simple fits via ${\chi }^{2}$ minimization to the fading and extinction curves individually. However, they are interrelated. Therefore, we now fit them simultaneously using the Feldman and Cousins method (F&C, Feldman & Cousins 1998; Sanchez et al. 2003). The method differs from a regular ${\chi }^{2}$ minimization by allowing setting of physical boundaries in the fitting parameters (e.g., $2\lt {R}_{V}\lt 6$), and by adjusting the ${\chi }^{2}$ statistics accordingly, via a Monte Carlo approach. Figure 5 shows the resulting confidence intervals. Those for ${\chi }^{2}$ minimization agree excellently with the simple single-parameter results in Figure 4. The F&C formalism suggests a similar conclusion, i.e., ${R}_{V}\gt 3.1$, but at somewhat higher confidence, $\gt 98$%. The derived best-fit dimming rate also agrees with the simple ${\chi }^{2}$ analysis. We also tried to fit the pre- and post-gap data separately. The result was not successful: the F&C method fails to constrain reasonable values of R_V and ${{dA}}_{V}/{dt}$. This suggests that there is no measurable dimming in the pre- and post-gap data sets separately (e.g., the dimming happened during the gap), which is also in agreement with the findings from simple ${\chi }^{2}$ minimization.

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