Isabel L Colman

Daniel Huber

Timothy R Bedding

James D Kuszlewicz

Jie D Yu

Paul R Beck

Yvonne D Elsworth

Rafael R García

Steven D Kawaler

Savita A Mathur

James S Kuszlewicz

Paul G Beck

Rafael A García

Dennis Stello

Timothy R White

Evidence for compact binary systems around Kepler red giants

Keywords: stars, oscillations (including pulsations), (stars, ) binaries (including multiple), close

come

INTRODUCTION

This paper sets out to solve a long-standing problem in the study of oscillating Kepler red giants. The analysis of red giants has been an area of rapid growth with the advent of data from the Kepler mission (Borucki et al. 2010). In particular, asteroseismology has allowed unprecedented insights into their core fusion (Bedding et al. 2011), internal rotation (Mosser et al. 2012;Beck et al. 2012), and internal magnetic fields (Stello et al. 2016). Red giants have also contributed to the field of galactic archaeology, where the study of red giant populations is used to map the formation Figure 1. Top panel: Amplitude spectrum of KIC 4350501, the first noted red giant with an anomalous peak, using only 43 days of data (Q0 and Q1), as in Bedding et al. (2010). Bottom panel: Amplitude spectrum calculated using all four years of Kepler data. The oscillations are centred at 140 µHz; the anomalous peak is at ∼86 µHz. lope with g modes propagating in the radiative core. Solarlike oscillations are stochastically excited and damped, with narrower peaks indicating longer mode lifetimes, as expected for mixed modes. For a recent review of red giants and their oscillations, see Hekker & Christensen-Dalsgaard (2016). In the case of KIC 4350501, more data showed the anomalous peak to be intrinsically narrow and revealed a subharmonic at half the frequency of the peak, suggesting that the peak was not a mixed mode but rather the signature of tidal interactions in a binary system. We have subsequently found many other red giants that show this type of behaviour, including the presence of harmonics and subharmonics. However, these peaks are present at such short periods that any binary companion would have to be orbiting within the convective envelope of the red giant.

This raises the possibility that we are observing common-envelope systems [START_REF] Paczynski | Structure and Evolution of Close Binary Systems[END_REF]. This is a phase of binary evolution that has been extensively studied with modelling and population synthesis. Evidence to confirm the existence of common-envelope systems is hard to come by; the closest method we have to direct detection is studying observational phenomena indicative of a past commonenvelope phase. Recent studies have used the shaping of planetary nebulae with binary central stars to better understand common-envelope interaction (Hillwig et al. 2016), and jets in planetary nebulae to constrain the magnetic fields of common-envelope binaries (Tocknell et al. 2014). It has also been postulated that the common-envelope phase could be integral to the evolution of red giants into sdB stars and cataclysmic variables (Beck et al. 2014). The observation of a common-envelope system would provide important confirmation for these theories of binary evolution. For a recent review of our understanding of common-envelope systems, see Ivanova et al. (2013).

Another possibility is that these objects may be examples of hierarchical triple systems, where a compact binary orbits a red giant, e.g. "Trinity" (Derekas et al. 2011;Fuller et al. 2013). These anomalous peaks could arise from a background or foreground compact binary that has contaminated the light collected from the red giant. There is still a limit to our knowledge of hierarchical triple systems and common-envelope binaries. The Trinity system is the best-studied observational example of a hierarchical triple involving a red giant. The red giant in Trinity does not exhibit any oscillations, and it is expected that red giant oscillations would be suppressed by binarity, but we found no such trend in the stars in this study. Ultimately, we cannot extrapolate from the case of Trinity to the many possible cases in this study, so the population of triple systems remains a hypothesis.

This study examines a sample of 168 light curves that exhibit both red giant oscillations and an anomalous peak, often with harmonics or a subharmonic. In this paper, we outline the method used to identify chance alignments, and comment on the statistics of possible physically associated systems.

METHODS AND ANALYSIS

Data preparation

The 168 stars studied were all discovered among Kepler red giants by visual inspection of power spectra. Many were included in Huber et al. (2010), Huber et al. (2011), or Stello et al. (2013). Additional stars were taken from Yu et al. (2016) or found in the APOKASC sample [START_REF] Pinsonneault | [END_REF].

We began by downloading and preparing Kepler simple aperture photometry (SAP) light curves from MAST1 . We processed the light curves following García et al. (2011), initially performing a high-pass filter using a Gaussian of width 100 days. We followed this by clipping all outliers further than 3σ from the mean. Finally, we took a Fourier transform to produce the amplitude spectrum.

We first located the comb-like pattern of solar-like oscillations, typical of red giant stars. Then, we were able to identify anomalous peaks. These have no particular position in relation to the solar-like oscillations. The majority of anomalous peaks had amplitudes higher than or comparable to the oscillations. Some anomalous peaks were found at similar frequencies to the oscillations, which led to a degree of confusion in stars that were identified previously with fewer quarters of data. A subset of the stars that we initially considered to fit this pattern were discarded from this study due to the anomalous peak representing an oscillatory ℓ = 0 or ℓ = 2 mode with a relatively broad peak, implying a shorter mode lifetime.

Using the frequency of the high-amplitude anomalous peak, we phase-folded each star's time series. The majority of the resulting phase curves displayed ellipsoidal variation, lending weight to the theory that these peaks are due to binarity. None of the anomalous peaks included in this study displayed the phase variation expected of a red giant oscillation. Examples are given in Figure 2. In some cases there were also subharmonics present in the Fourier spectra, as with KIC 4350501 (Figure 1), or a series of peaks indicative of an eclipse.

In two cases, we found the anomalous peaks to come from γ-Doradus oscillations and five stars where the source of the anomalous peaks are δ-Scuti pulsators, indicating possible binary systems. An example of red giants contaminated by γ-Dor and δ-Scuti oscillations is shown in Figure 3. Although these stars do not conform to the typical pattern of red giant oscillations with one anomalous peak and possible harmonics and subharmonics, we in-clude them in this paper as they were studied with the same processes as the remainder of the sample and provide confirmation that the method works independently of what type of target it is used to analyse.

Pixel power spectrum analysis

In the previous section we covered the process of identifying stars for this sample. For this, we used SAP light curves, which are a composite of several 4 ′′ Kepler pixels comprising the so-called 'optimal aperture.' To locate the true sources of these anomalous peaks, it was necessary to examine the area around each of the targets. For this, we used Kepler target pixel files (TPFs), which are available for download from MAST. TPFs provide a 'postage stamp' image of pixels around Kepler targets. We employed the same methods outlined in Section 2.1 to process light curves from individual pixels in each TPF.

During the Kepler mission, the orientation of the telescope changed by 90 • every quarter. Because of this, we examined each quarter of pixel data separately. In cases where oscillations were not visible with only one quarter of data, we stitched together the light curves of quarters with the same orientation, which occurred every fourth quarter.

By taking a Fourier transform of each pixel time series, we could more accurately locate the source of the anomalous peaks in the Fourier spectra. We identified these by inspection, based on the pixels included in the optimal aperture around the target star. In many cases, the source of the anomalous peak was obviously separated from the source of the solar-like oscillations. Figure 4 shows an example of such a TPF for KIC 7461601, where the optimal aperture is indicated by red shading. In this case, the anomalous peak is primarily located outside the optimal aperture. It is evident that its source is separate to the source of the oscillations. Of the 168 stars analysed, we found 87 to display this type c ⃝ 0000 RAS, MNRAS 000, 000-000 of clear separation. We interpret these as chance alignments of red giants with background or foreground binaries. The other 81 stars did not show this sort of clear separation. Figure 5 shows the TPF for KIC 3736251, a case where there is no clear distinction between the pixel source of the red giant oscillations and the anomalous peak. We interpret these as possibly physically associated systems.

Difference imaging

We performed a more detailed study of the TPFs with difference imaging, which has been successfully applied to the identification of false positive exoplanet transits (Bryson et al. 2013). We selected postage stamp images that fell in time within 10% bands centred on the maximum and minimum points of the phased light curve. To create the difference image, we took the average of both sets of images and subtracted the average about the minima from the average about the maxima. This new image retained the dimensions of a TPF postage stamp and could easily be compared to the average images, as shown in Figure 6. From this, we could see which pixels were the source of the flux variations at the period of the anomalous peak.

There remain some caveats for the use of difference imaging. The method we used was designed for ellipsoidal variation, and so it was less useful for the few stars in the sample where the phased light curve showed an eclipse, or where the identified anomalous peak belonged to δ-Scuti oscillations with multiple high-amplitude peaks. There were several other issues with using the phased light curves, particularly in stars with a low signal-to-noise ratio where the periodicity was hard to discern by looking at the phased light curve, due to scatter. Additionally, difference imaging was less successful for cases where the contaminant was at an angular distance greater than 30 ′′ from the target star. Some stars with clear contamination in the TPF did not show any variation in the difference image, which suggested that the contaminant was located outside the optimal aperture. This tended to coincide with low-amplitude anomalous c ⃝ 0000 RAS, MNRAS 000, 000-000 peaks. In such cases, it was clear simply from the TPFs that there was contamination. Despite this, we were still able to gain valuable information from difference imaging. For stars with no evident contamination in the TPFs, the difference images tended not to show variation when compared to the average im-ages. Difference imaging also helped to confirm the status of stars with low signal in the TPF Fourier spectra. Conversely, the difference images reinforced the status of stars with more tentative classification as spatially separated. It follows that many of the cases where difference imaging did not confirm contamination correspond to widely separated chance alignments.

To identify the true source of chance alignments, widelyseparated or otherwise, we next compared both the average and difference images to higher-resolution images of the same area of sky, using 1 ′ cutouts from the UKIRT WF-CAM (the UK Infrared Telescope Wide Field Camera) survey (Lawrence et al. 2007). An example is shown in Figure 7. Kepler TPFs contain world coordinate system (WCS) information, which allowed us to calculate the orientation of the postage stamp. We displayed coordinates on both types of images in the form of a compass rose, from which we could see whether there were any possible contaminant stars from the same position as the anomalous source as shown in the TPF Fourier spectra. Looking for matches in both the KIC and the UKIRT object catalogue, we were able to use a Kepler light curve to confirm the source of contamination in 18 cases (see Table 2 in the appendix chance alignments (Table 4), we noted one or more possible stars that could be the contaminant, especially closer to the Galactic plane, which is where most of the chance alignments were found.

DISCUSSION

Spatial distribution

We found that, of the 168 red giants with anomalous peaks, 87 could be identified as chance alignments. For 18 of these chance alignments, we confirmed their status with the analysis of Kepler light curves of nearby stars which we identified as the sources of contamination. We could not spatially resolve the 81 other stars, and we refer to these as possibly associated systems. We identified four of the five δ-Scuti anomalous peaks as chance alignments. The two γ-Dor anomalous peaks remain possible physical associations.

Figure 8 shows the distribution of our sample over the Kepler field of view (FOV), with chance alignments in panel (a) and possibly associated systems in panel (b). We present these populations in galactic coordinates, and note that the bottom of the field at lower galactic latitudes is closer to the Galactic plane and has a higher density of stars. At higher galactic latitudes we observe a marked paucity of stars as expected, both in the field itself and in the sample considered in this study. Similarly, this pattern presents itself in the distribution of chance alignments. It is therefore noteworthy that the possibly associated systems seen in panel (b) seem to be spread quite evenly across the FOV.

We further analysed these populations by examining their cumulative distributions as a function of galactic latitude, shown in Figure 9. We compare this to a distribution of 1,000 red giants drawn randomly from a list of all oscillating Kepler red giants provided from Yu et al. (in preparation).

The distribution of the possible physical associations closely matches the distribution of random red giants, which implies that they are not chance alignments. It is also noteworthy that these distributions visibly differ from the distribution of chance alignments, which increases sharply at low galactic latitudes, reflecting the higher density of stars closer to the Galactic plane. The probability of finding a chance alignment between two populations is expected to scale as the square of surface density. This gives us an important insight into the nature of this population and suggests that we may be observing a distinct population of systems, possibly hierarchical triples or common envelope binaries.

Amplitude distribution

We searched for a possible correlation between the intrinsic luminosities of the red giants and the amplitudes of their anomalous peaks. It might be expected that if a compact binary is physically associated with a red giant, the amplitude of variations from the binary might correlate inversely with the luminosity of the red giant, due to dilution. We observed no correlation, which led us to compare our stars to a sample drawn from the Kepler Eclipsing Binary Catalog (Prša et al. 2011) 2 . We selected the sample of eclipsing binaries (EBs) by their morphology, which is a measure of the ellipticity of their phased light curves calculated by locally linear embedding (Matijevič et al. 2012). The cut off for EB selection was a morphology number > 0.7, chosen by visual inspection of stars in the catalog to match those with light curves similar to those in our sample. In Figure 10, we plot the amplitudes of the anomalous peaks in our sample and of the Kepler EBs against Kepler magnitude. These data show that more ellipsoidal variation tends to have a lower amplitude of variation, a trend which is also present in our sample. The measure of ellipticity in our data was based on a ranking of the shape of phased light curves and on a different scale to the Catalog's morphology number, so we do not display it in Figure 10.

From this exercise, we can explain the lack of correlation between the intrinsic luminosities of red giants and the amplitudes of their anomalous peaks by the broad range of amplitudes present across ellipsoidal variables, as exhibited by the Catalog sample. We also note that while our sample is overall brighter than the Catalog sample, the distribution of amplitudes is what would be expected for a sample primarily exhibiting ellipsoidal variation. The histogram in the right panel of Figure 10 shows that the distribution of the possible physical associations closely matches the distribution of the Catalog EBs. The anomalous peaks of both the possible physical associations and the chance alignments are more present at lower amplitudes, but this is markedly noticeable for the latter. This effect can be explained by the wide angular separation between the target stars and their contaminants, so less of the contaminating light enters the optimal aperture. This leads to systematically lower apertures. In the case of the possible physical associations, this dilution could be caused by a compact binary companion. This strengthens the conclusion that the possible physical associations comprise a distinct population.

Modelling of chance alignments

While the majority of chance alignments found in this study involved contaminants further than 4 ′′ from the target star, it is possible that there could be contaminants within 4 ′′ of the target that our methods do not have the sensitivity to detect. To test whether we could expect to find more chance alignments within the remaining 81 stars, we analysed a model of a stellar population in the Kepler FOV. This also helped us in understanding the underlying statistics around chance alignments of red giants and background or foreground binary systems.

We used the modelling software Galaxia (Sharma et al. 2011), which allows the user to synthesise an artificial population of stars within a given area of sky, here chosen to match the Kepler FOV. We defined a chance alignment as any two stars found within a Kepler pixel of each other, namely 4 ′′ . Galaxia does not take into account the existence of binary systems, and hence any chance alignments that are detected in the simulation are true chance alignments.

Our model goes down to an apparent J magnitude of 20. Once the synthetic FOV had been simulated, we then searched for stars analogous to those in the sample by minimising over apparent J magnitude within a set range of galactic coordinates b and ℓ, and stellar parameters T eff , log(g) and [Fe/H] ( Mathur et al. 2016). The nearest match to each target star was designated a blend if we located another star within 4 ′′ of it. This process is illustrated in Figure 11.

We found the occurrence of 4 ′′ chance alignments in the model to be rare, with only 18 of 168 these stars fitting the criterion, or 10.7%. This is much lower than the observed fraction (as discussed in Section 3.1) because here we are only looking at matches within 4 ′′ , which cannot be discerned by the techniques covered in Section 2. This can be compared with the figure quoted within 0.15 ′′ -4 ′′ with a probability of 12.6% ± 0.9. Despite the fact that Ziegler et al. were not focused on red giants in the same way as our study, our value is just over 2σ from the result given by Ziegler et al., which places the two samples in good agreement. This suggests that a similar proportion of our sample will contain chance alignments within 4 ′′ . By inspection of UKIRT images, 9 of the identified chance alignments appear to be close to or within 4 ′′ . This implies that we should expect to find roughly 9 more chance alignments within 4 ′′ among the 81 stars that have not been identified as chance alignments. This is a strong result, and leaves us with a sizeable population of possible physical associations involving an oscillating red giant.

CONCLUSIONS

From a sample of 168 red giant stars with anomalous highamplitude peaks, we found 87 could be discounted as chance alignments, with the remaining 81 exhibiting no contamination outside a Kepler pixel. This leaves the opportunity for these stars to be physically associated systems such as a common-envelope binary or hierarchical triple systems. We observe that this population appears to follow the distribution of randomly-selected stars from the Kepler FOV. This distinguishes them from the population of chance alignments, which appear with a greater density towards the galactic plane. We have constructed and examined a model of a synthetic population in the Kepler field which suggests that such close chance alignments are rare, which would imply that most of these stars are more likely to be physically associated systems. This may point to hierarchical triple systems, or to common-envelope binaries.

Future work includes an observation of all remaining targets by Robo-AO (Baranec et al. 2011), an adaptive optics system which has been used previously to examine Kepler exoplanet host candidates. We will also look to spectroscopic follow-up observations, with the possibility of identifying the spectral lines of companion stars or radial velocity variations. In addition to this, further opportunities will arise to search for these unusual red giants in data from K2 and TESS. A larger sample from different areas of the sky would aid our understanding of these unusual cases and aid more detailed analysis of a possible new population of systems.

The fundamental parameters of the stars in this study are listed in tables in the appendix. All code used for analysis is available online at https://github.com/astrobel/ chancealignments. (Mathur et al. 2016). In cases where we found the anomalous peak to be part of γ-Dor oscillations, the peak frequency (ν peak ) is marked with an asterisk. We mark δ-Scuti anomalous peaks with two asterisks.

Figure 3 .

 3 Figure 3. Top row: KIC 6707691, showing both γ-Dor (left) and red giant (right) oscillations. Bottom row: KIC 9771905, showing both red giant (left) and δ-Scuti (right) oscillations. Each set of oscillations is isolated to better display its features, as in each star the red giant oscillations have significantly lower amplitudes than the classical pulsator oscillations.

Figure 4 .

 4 Figure 4. The Kepler TPF aperture of KIC 7461601, a red giant showing contamination from a chance alignment with a binary. Each panel represents a pixel, showing an amplitude spectrum calculated using the same methods as in Figure 1 with frequencies up to the Kepler long cadence Nyquist frequency, 283.21µHz. Amplitudes in each pixel are auto-scaled in order to better display qualitative features. More compact tick marks indicate higher overall amplitudes. Shaded pixels indicate the optimal aperture.

Figure 5 .Figure 6 .

 56 Figure 5. The Kepler TPF aperture of KIC 3736251, showing no contamination. Each panel represents a pixel, as in Figure 4.

  Figure 7. A UKIRT image showing a 1 ′ field of view around KIC 7461601. To indicate scale, the compass arms are 6 ′′ , as in Figure 6. The solid line points north, and the dashed line points east.

Figure 8 .

 8 Figure8. The sample of stars in this study is shown across the Kepler field of view in galactic coordinates. Panel (a) shows red giants with anomalous peaks that we classified as chance alignments, shown as green circles. In the case where a Kepler light curve was available to confirm this contamination, the star is shown by a pink square. Panel (b) shows the population of red giants exhibiting an anomalous peak and where a physical separation cannot be discerned.

Figure 9 .

 9 Figure 9. Cumulative distributions in galactic latitude of the populations shown in panels (a) and (b) of Figure 8. The solid blue line is taken from a random sample of 1,000 Kepler red giants.

Figure 10 .

 10 Figure 10. Left: Relationship between the amplitude of the anomalous peaks in our sample and Kepler magnitude. Chance alignments are shown with green circles, and possible physical associations with red triangles. For comparison, we also display the amplitudes of a population selected from the Kepler Eclipsing Binary Catalog with morphology number > 0.7, shown with blue squares. Right: Histogram of each population distributed across amplitude. Not pictured: two outliers with amplitudes above 700ppm, one chance alignment (KIC 4071950) and one possible physical association (KIC 6526377).

  A variety of light curves phased on the period of the anomalous peak and binned. The top row shows stars that could not be discounted as chance alignments in this study. The bottom row shows chance alignments.

		1.00012		6124426			1.0006		6468112			1.0010		6526377			1.0006		11192141
	Fractional intensity	0.99998 1.00000 1.00002 1.00004 1.00006 1.00008 1.00010					0.9998 1.0000 1.0002 1.0004					0.9990 0.9995 1.0000 1.0005					0.9994 0.9996 0.9998 1.0000 1.0002 1.0004
		0.0 0.99996	0.5	1.0	1.5	2.0	0.0 0.9996	0.5	1.0	1.5	2.0	0.0 0.9985	0.5	1.0	1.5	2.0	0.0 0.9992	0.5	1.0	1.5	2.0
			Period: 0.30 days			Period: 0.40 days			Period: 0.70 days			Period: 0.68 days
		1.0003		2018906					4072864			1.00012		5556726			1.00012		7198587
	Fractional intensity	0.9997 0.9998 0.9999 1.0000 1.0001 1.0002					0.9999 1.0000 1.0001 1.0002					0.99996 0.99998 1.00000 1.00002 1.00004 1.00006 1.00008 1.00010					1.00002 1.00004 1.00006 1.00008 1.00010
		0.0 0.9996	0.5	1.0	1.5	2.0	0.0 0.9998	0.5	1.0	1.5	2.0	0.0 0.99994	0.5	1.0	1.5	2.0	0.0 1.00000	0.5	1.0	1.5	2.0
			Period: 0.26 days			Period: 0.95 days			Period: 0.48 days			Period: 0.57 days
	Figure 2.

  in a study of false positive KOIs (Kepler Objects of Interest) by Ziegler et al. (2016), who found that planet host candidates have a nearby star

		1.00		700						1000
													chance alignments
													possible associations
		0.96		600									Kepler EBs
										800
		0.92		500
	Kepler EB morph number	0.84 0.88	Amplitude (ppm)	300 400						400 600
		0.80
				200
		0.76								200
				100
		0.72
				6 0	8	10	12	14	16	0.000 0	0.002	0.004	0.006	0.008	0.010
						Kp						Normalised density
	c ⃝ 0000 RAS, MNRAS 000, 000-000

Table 1 .

 1 An illustration of the process used to search for chance alignments in the Galaxia model, described in Section 3.3. The concentric rings have radius 4 ′′ , intended to represent the maximum distance between stars that could fall on the same Kepler pixel. The colour scale represents J magnitude. Details of the 81 possible physical associations. Stellar parameters are taken from the NASA Exoplanet Archive, data release 25

		4681356
		6.492
			12.8
		6.490
			13.6
		6.488	14.4
	b	6.484 6.486	15.2 16.0	J Magnitude
		6.482	16.8
		6.480	17.6
		6.478	18.4
		74.574 74.576 74.578 74.580 74.582 74.584 74.586 74.588
		ℓ
	Figure 11.

c ⃝ 0000 RAS, MNRAS 000, 000-000

Table 2 .

 2 Details of the 18 confirmed chance alignments with a known entry in the KIC. In cases where we found the anomalous peak to be part of δ-Scuti oscillations, the peak frequency (ν peak ) is marked with two asterisks.

	KIC RA (deg)	DEC (deg)	Kp	log(g)	T eff (K) [Fe/H] νmax (µHz) ν peak (µHz)	P peak (days)
	1726211 292.50447	37.29278	10.93	2.39	4981	-0.74	31.56	66.01	0.18
	1726245 292.51086	37.25521	11.57	2.59	4837	0.21	53.66	43.78	0.26
	2160901 291.5993895 37.53966	12.06	2.67	4676	0.24	56.54	2.97	3.89
	2449020 292.68245	37.75239	11.91	2.84	5007	0.07	66.76	13.98	0.83
	2573092 290.85716	37.87599	11.58	2.46	4723	0.07	31.68	44.69	0.26
	3356438 294.92994	38.47866	11.97	2.85	4999	0.07	58.40	5.94	1.95
	3530823 287.101181	38.60348	11.74	2.56	5036	0.07	44.55	21.27	0.54
	3546046 291.8050695 38.64798	11.96	3.18	4845	0.16	186.12	7.71	1.50
	3736251 288.20211	38.8702	13.59	3.38	5148	-0.72	25.97	85.71	0.14
	3858714 293.65017	38.95757	11.94	2.61	4852	0.21	48.27	61.02	0.19
	3973137 296.026121	39.0659	13.65	2.41	4926	-0.58	36.93	2.56	4.53
	4043436 287.40072	39.10863	12.77	2.42	4653	0.10	31.36	15.61	0.74
	4149966 289.5747495 39.25492	10.08	2.79	4934	0.07	72.19	5.02	2.30
	4164236 293.17628	39.24902	13.97	2.47	4738	-0.04	35.29	61.29	0.19
	4279165 295.51473	39.36227	12.38	2.61	4868	-0.16	46.14	222.12	0.05
	4374169 293.93991	39.41256	11.67	2.67	4891	0.07	40.07	7.83	1.48
	4456739 289.362849	39.54505	12.02	2.44	4658	0.36	41.78	61.63	0.19
	4555699 289.94031	39.69181	12.80	2.57	4768	-0.08	26.92	2.63	4.40
	4681356 297.897023	39.709683	13.45	2.57	4710	-0.36	47.16	63.68	0.18
	4830095 290.10768	39.96584	13.10	2.45	5166	-0.50	30.92	53.11	0.22
	5112950 295.37307	40.20586	12.77	2.53	4753	0.00	41.51	91.43	0.13
	5462460 295.65894	40.62042	12.40	2.41	4999	-0.50	32.70	40.99	0.28
	5793628 292.67057	41.06844	11.10	2.48	4878	-0.48	36.04	49.55	0.23
	5985252 298.17206	41.23463	11.00	2.31	4936	-0.50	27.47	37.37	0.31
	6124426 292.1016	41.46219	13.88	3.68	5406	-0.28	205.92	39.16	0.30
	6185964 284.57787	41.55982	12.98	2.65	4852	-0.42	27.75	39.86	0.29
	6382801 296.9842395 41.73708	13.72	2.68	4724	0.28	38.10	69.47	0.17
	6451664 294.57057	41.89605	12.56	2.45	4996	0.07	35.19	50.16	0.23
	6462755 297.30729	41.84648	10.44	2.53	4785	-0.16	27.53	32.96	0.35
	6468112 298.50825	41.8638	9.70	3.06	5089	-0.04	64.40	*29.22	0.40
	6526377 293.07981	41.94772	11.81	2.59	4789	0.00	32.30	16.59	0.70
	6610354 293.0651805 42.04949	9.31	2.61	4883	-0.20	45.46	7.74	1.50
	6707691 295.9976805 42.17442	11.91	2.89	5122	0.07	85.82	*27.31	0.42
	6716840 298.00289	42.11263	11.91	2.55	4841	0.36	27.57	40.50	0.29
	6753216 282.487031	42.22581	11.49	3.34	5149	-0.74	46.87	**245.20	0.05
	6929104 284.55087	42.46167	13.80	2.96	5054	-0.22	23.77	41.97	0.28
	6948654 291.84099	42.43724	13.96	3.08	5048	0.10	32.78	36.40	0.32
	6952430 293.019909	42.47953	11.83	2.48	4784	0.07	36.22	61.48	0.19
	7267370 286.715829	42.88117	12.32	2.56	4835	0.07	43.74	61.69	0.19
	7272332 288.70806	42.86038	13.26	2.63	4779	-0.14	46.89	58.60	0.20
	7418275 281.62991	43.00333	13.37	3.41	5369	-0.10	221.62	57.55	0.20
	7447072 292.4606	43.05733	13.26	2.93	5059	-0.16	33.26	23.22	0.50
	7511777 286.22405	43.12891	13.72	3.53	5140	0.08	221.24	12.05	0.96
	7596350 287.81166	43.25247	11.10	2.58	5153	-0.50	39.24	43.92	0.26
	7816294 289.78314	43.52288	11.47	2.62	4644	0.18	48.01	25.45	0.45
	8092097 289.94732	43.93543	12.80	2.43	4781	-0.20	24.50	33.33	0.35
	8095225 290.95461	43.9071	13.43	3.32	5268	-0.04	79.96	76.07	0.15
	8462775 301.53743	44.40842	10.89	2.68	4828	0.02	33.92	52.81	0.22
	8870432 285.3189	45.1689	9.78	2.53	4733	0.56	35.00	45.39	0.25
	9008090 286.0825605 45.3942	12.79	2.49	4785	0.07	38.46	67.39	0.17
	9029195 294.38823	45.33999	10.93	2.52	4848	0.21	40.43	11.68	0.99
	9146423 288.12324	45.55873	10.95	2.15	4499	-0.02	18.73	54.12	0.21
	9210116 288.06693	45.68367	10.11	2.75	4912	0.07	53.87	43.65	0.27
	9541892 297.49509	46.18827	12.67	2.54	4809	0.07	34.31	35.88	0.32
	9605626 297.981431	46.26839	13.83	3.06	5122	-0.38	33.35	48.30	0.24
	9763419 288.4587	46.50697	11.55	2.41	4934	0.07	32.20	49.54	0.23
	9777198 294.5120895 46.59932	12.63	2.48	4775	0.21	36.39	40.76	0.28
	9851743 298.97724	46.61877	10.46	2.54	4869	0.07	44.81	60.31	0.19
	9908646 298.160649	46.73501	13.68	2.97	5021	-0.14	23.58	40.08	0.29
	9969574 298.46844	46.88932	12.09	2.96	4812	0.30	108.63	4.00	2.90
	c ⃝ 0000 RAS, MNRAS 000, 000-000

Table 3 .

 3 Details of the 69 presumed chance alignments. In cases where we found the anomalous peak to be part of δ-Scuti oscillations, the peak frequency (ν peak ) is marked with two asterisks.

	KIC RA (deg)	DEC (deg) log(g)	Kp	T eff (K) [Fe/H] νmax (µHz) ν peak (µHz)	P peak (days)
	1870196	291.8805	37.34748	12.65 3.20	4895	0.10	191.59	60.77	0.19
	2018906 292.42326	37.428	13.20 3.30	5046	-0.54	155.70	44.07	0.26
	2163856 292.22694	37.55744	11.70 2.78	5010	0.07	72.14	144.29	0.08
	2301349 291.09195	37.64004	13.43 2.72	4615	0.36	64.62	33.95	0.34
	2569650 290.19609	37.81097	15.88 3.62	4986	0.22	187.60	74.04	0.16
	2569935 290.222441	37.80783	13.12 1.55	4082	0.36	5.21	71.10	0.16
	2696115 286.8690795 37.95218	11.85 2.19	4619	0.21	20.41	42.07	0.28
	2710194 290.79134	37.92911	12.15 2.68	4580	0.24	54.74	33.60	0.34
									c ⃝ 0000 RAS, MNRAS 000, 000-000

c ⃝ 0000 RAS, MNRAS 000, 000-000

http://archive.stsci.edu/kepler/ c ⃝ 0000 RAS, MNRAS 000, 000-000

c ⃝ 0000 RAS, MNRAS 000, 000-000

http://keplerebs.villanova.edu/ c ⃝ 0000 RAS, MNRAS 000, 000-000

ACKNOWLEDGMENTS

This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. The UKIDSS project is defined in Lawrence et al. (2007). UKIDSS uses the UKIRT Wide Field Camera (WF-CAM; Casali et al. (2007)) and a photometric system described in Hewett et al. (2006). The science archive is described in Hambly et al. (2008). We have used data from the 2nd data release, which is described in detail in Warren et al. (2007). IC acknowledges scholarship support from the University of Sydney. DH acknowledges support by the Australian Research Council's Discovery Projects funding scheme (project number DE140101364) and support by the National Aeronautics and Space Administration under Grant NNX14AB92G issued through the Kepler Participating Scientist Program. PGB, and RAG acknowledge the ANR (Agence Nationale de la Recherche, France) program IDEE (n • ANR-12-BS05-0008) "Interaction Des Etoiles et des Exoplanetes". PGB, and RAG also received funding from the CNES grants at CEA. YE acknowledges the support of the UK Science and Technology Facilities Council (STFC). Funding for the Stellar Astrophysics Centre (SAC) is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). SM would like to acknowledge support from NASA grants NNX12AE17G and NNX15AF13G and NSF grant AST-1411685. DS is the recipient of an Australian Research Council Future Fellowship (project number FT1400147).


Comment on the paper by Rovelli & al. about the compatibility of various "gauge conditions"

Germain Rousseaux

Université de Nice Sophia-Antipolis, Institut Non-Linéaire de Nice, UMR 6618 CNRS-UNSA, 1361, route des Lucioles. 06560 Valbonne, France.

(Germain.Rousseaux@inln.cnrs.fr)

The compatibility "demonstrated" by Rovelli & al. in [START_REF] Magliaro | Compatibility of radial, Lorenz and harmonic gauges[END_REF] between various "gauge conditions" both in Classical Electromagnetism and General Relativity can be better understood if one distinguishes "gauge conditions" of the solution type and "gauge conditions" of the constraint type.

Recently, we showed that the well-known Coulomb and Lorenz "gauge conditions" (see [START_REF] Jackson | Historical roots of gauge invariance[END_REF] for an introduction to the various "gauge conditions" encountered in the literature) were, in fact, not equivalent because they must be interpreted as physical constraints that is electromagnetic continuity equations [START_REF] Rousseaux | On the physical meaning of the gauge conditions of Classical Electromagnetism : the hydrodynamics analogue viewpoint[END_REF]. In addition, we were able to demonstrate that the Coulomb "gauge condition" is the galilean approximation of the Lorenz "gauge condition" within the magnetic limit of Lévy-Leblond & Le Bellac [START_REF] Bellac | Galilean Electromagnetism[END_REF][START_REF] Rousseaux | Lorenz or Coulomb in Galilean Electromagnetism ?[END_REF][START_REF] De Montigny | On the electrodynamics of moving bodies at low velocities[END_REF][START_REF] De Montigny | On some applications of the galilean electrodynamics of moving bodies[END_REF]. So, to "make a gauge choice" that is choosing a gauge condition is, as a consequence of our findings, not related to the fact of fixing a special couple of potentials. Gauge conditions are completely uncorrelated to the supposed indeterminacy of the potentials. Hence, we propose to rename the "gauge conditions" like the ones of Lorenz or Coulomb by physical "constraints".

The purpose of our comment is to show that a "demonstration" of the compatibility between the Fock-Schwinger "gauge condition" and the Lorenz "gauge conditions" by Rovelli & al. in [START_REF] Magliaro | Compatibility of radial, Lorenz and harmonic gauges[END_REF] is blurred by a slip in terminology. As a matter of fact, the Lorenz equation ∂ µ A µ = 0 is according to us a constraint whereas the Fock-Schwinger equation x µ A µ = 0 is what we called a "solution" submitted to the Lorenz constraint. If so, the "compatibility" is obvious... First of all, let us illustrate what we mean by a "solution" under a physical constraint. One often finds in textbooks that we can describe a uniform magnetic field B = Be z by either the so-called symmetric "gauge" A 1 = 1/2B × r or by the so-called Landau "gauge" [START_REF] Jackson | Historical roots of gauge invariance[END_REF]. This two "gauges" are related by a gauge transformation :

A 1 = 1 2 B × r = 1 2 [-By, Bx, 0] (1)
becomes either :

A 2 = [0, Bx, 0] or A 3 = [-By, 0, 0] (2)
with the gauge functions ±f = ±xy/2. However, there is no discussion in the litterature of the following issue. As a matter of fact, if we consider a solenoid with a current along e θ , the magnetic field is uniform (along e z ) and could be described by the symmetric "gauge" or the Landau "gauge". Yet, the vector potential in the Landau "gauge" A 2 is along e y whereas the vector potential in the symmetric gauge is along e θ . We advocate that only the symmetric "gauge" is valid in this case because it does respect the symmetry of the currents (J = Je θ ) whereas the Landau "gauge" does not. Moreover, the symmetric "gauge" (or the Landau "gauge") is not, in fact, a gauge condition but a solution describing a uniform magnetic field under the Coulomb constraint (∇.A 1 = 0). In order to understand this last point, one can picture an analogy between Fluid Mechanics and Classical Electromagnetism. Indeed, the solenoid is analogous to a cylindrical vortex core with vorticity w and we know that the velocity inside the core is given by u = 1/2w × r which is analogous to the symmetric gauge for an incompressible flow (∇.u = 0). Outside the vortex core, the velocity is given by [START_REF] Guyon | Physical Hydrodynamics[END_REF] :

u = Γ∇θ 2π = Γ 2πr e θ ( 3
)
where Γ is the flux of vorticity inside the vortex or the circulation of the velocity outside the vortex. One recovers the analogue formula for the vector potential outside a solenoid... Of course, if the problem we are considering does not feature the cylindrical geometry (two horizontal plates with opposite surface currents for example, analogous to a plane Couette flow [START_REF] Guyon | Physical Hydrodynamics[END_REF]), one of the Landau "gauges" A 2 or A 3 submitted to the constraints ∇.A 2 = 0 or ∇.A 3 = 0 must be used instead of the symmetric "gauge" A 1 according to the necessity of respecting the underlying distribution/symmetry of the currents which is at the origin of both the vector potential and the magnetic field. To give a magnetic vector field without specifying its current source is an ill-posed problem which was interpreted so far by attributing an indeterminacy to the vector potential which is wrong. Now, how can we test experimentally this argument based on symmetry ? If the current of the solenoid varies with time, it will create an electric field which is along e θ as the vector potential because the electric field is minus the time derivative of the vector potential. If the currents in the horizontal plates change with time, a horizontal electric field will appear for the same reason.

We come back now to the main claim of Rovelli & al. [START_REF] Magliaro | Compatibility of radial, Lorenz and harmonic gauges[END_REF]. We will argue that the Fock-Schwinger equation is a solution submitted to the Lorenz constraint. Let us write the Fock-Schwinger "gauge" x µ A µ = 0 in projection for cartesian coordinates x 0 A 0 + x 1 A 1 + x 2 A 2 + x 3 A 3 = 0. The four-position and four-potential have components x µ = (-ct, x) and A µ = (V /c, A). For example, the Fock-Schwinger equation becomes xA x + yA y + zA z = tV in cartesian coordinates.

The Lorenz "gauge" ∇.A + 1/c 2 ∂V /∂t = 0 becomes after a Fourier transformation ik.A -iω/c 2 V = 0. But a light wave is such that |k| = ω/c.

A plane or spherical light wave has a null space-time interval (ds 2 = dx µ .dx µ = 0) :

x 2 + y 2 + z 2 = c 2 t 2 or r 2 = c 2 t 2 .
The Lorenz constraint for a plane or spherical light wave becomes : V = cA x or V = cA r where A x or A r is the component of the light wave which is parallel to the wave-vector k. With x = ct (y = z = 0) or r = ct for outgoing waves, one gets V = cA x = x/tA x or V = cA r = r/tA r that is the Fock-Schwinger solution : xA x = tV or rA r = tV .

As a conclusion, the Fock-Schwinger solution describes the propagation of a light wave submitted to the Lorentzcovariant Lorenz constraint which is the continuity equation allowing the propagation. It is interesting to notice that V. A. Fock himself derived the Fock-Schwinger equation starting with the Lorenz equation and using a gauge transformation [START_REF] Fock | Proper Time in Classical and Quantum Mechanics in Selected Works[END_REF]...