TESS Observations of Cepheid Stars: First Light Results

To produce light curves from the FFIs we used the FITSH package developed by Pál (2012). This tool provided us with differential photometric solutions for the target stars. The data reduction process as implemented in the FITSH pipeline has been split into two parts, namely the global full-frame analysis and the target object photometry. As the first step, the calibrated FFIs are analyzed in terms of derivation of the plate solution. We perform the astrometric cross-matching with the Gaia DR2 source catalog (Gaia Collaboration et al. 2016, 2018). In this step, we also derive the flux zero-point with respect to the Gaia G_RP magnitudes due to the similarities between the TESS and the Gaia photometric systems. Finalizing the plate solutions, corrections were applied to account for all other optical aberrations as well as to treat the differential velocity aberration.

In the second step, we trim small, 64 × 64 pixel frames centered on our target stars and execute the differential image analysis only on those sub-frames. Any type of differential image analysis requires a so-called reference frame, from which the deviations are expected to be minimal and signal-to-noise (S/N) ratio to be maximal. To create this reference frame, we select 9 or 11 individual and subsequent sub-frames for each sector around the mid-times of the observation series when differential velocity aberration, and thus the apparent shifts in the stellar images are minimal. We calculated the median of these frames to maximize the S/N.

We needed to obtain a set of reference fluxes for the targets of interest in parallel with the derivation of the image convolution coefficients. The image convolution is able to correct for many of the instrumentation and/or intrinsic differences between the target frames and the reference frame, including the slight drift caused by the differential velocity aberration, the spacecraft jitter, background, and stray light variations. It also helps to eliminate or significantly decrease the effects on the frames acquired during momentum wheel dumps. Since reference fluxes are difficult to accurately obtain even at moderately confused stellar fields for TESS due to the large pixel size, we simply used the Gaia DR2 G_RP magnitudes for this purpose. However, the final fluxes (i.e., the sums of the reference fluxes and the respective residual fluxes on the convolved and subtracted images) needed to be adjusted because the phase of the light variations could be significantly different in the reference frames assembled for the subsequent TESS sectors. The aperture size was set to be 2.5 pixels (525), which worked well for most of our targets. We tested smaller aperture sizes (1.5 px and 2.0 px) and found that they have no effect on the frequency content but can reduce the measured pulsation amplitude by up to 30%. Therefore we used 2.5 px apertures whenever possible, except for very crowded fields where we used smaller radii to minimize contamination from nearby variable stars. In Figure 1, we display example frames for WX Tuc, including a Digitized Sky Survey (DSS) image, the reference image used for the differential photometry process, as well as some typical differential frames that show up during the process.

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The light-curve outputs of the FITSH pipeline contained a small number of outliers, which are mostly due to momentum dumps of the reaction wheels at every 2.5 days when TESS is not in fine pointing for a few minutes. Low-quality segments can also be identified in some light curves that are caused by other issues, such as stray light or glints. To eliminate these data points, we used a 3σ-clipping cleaning method: we deleted points that differed by more than three times the standard deviation of the residual compared to a fitted signal composed of the five highest-amplitude frequencies. Finally, we applied barycentric correction to the time stamps of each target. The light curves are provided in the Appendix and are also available online for further analysis. ³²

Three of our stars were included in the 2 minutes cadence target list assembled by the TESS Asteroseismic Science Consortium: β Dor, XZ Cet, and UY Eri. In the cases of XZ Cet and UY Eri, we downloaded the SPOC-generated light curves from MAST, and used the Simple Aperture Photometry, SAP_FLUX (Twicken et al. 2010; Morris et al. 2020), which we converted to magnitude units using 20.54 mag as the zero-point. We applied outlier removal to them, and proceeded with the analysis. However, we found that the SPOC light curve for β Dor produced very different flux values from one sector to the other. As β Dor is bright enough to saturate the TESS detectors and it is a large-amplitude variable, the length of the bleed column (i.e., the set of pixels that stores the excess flux from the star) changes considerably over the pulsation cycle. Closer inspection revealed that the photometric aperture set by the pipeline was not always capturing the bleed columns entirely. This problem cannot be solved with scaling the light curves of different sectors to the same amplitude, because flux loss occurs only in the brightest pulsation phases (i.e., around the maxima), and therefore the light curve gets periodically distorted. We note that the same variable bleed column length problem affected the observations of the star RR Lyr during the early stages of the Kepler mission (Kolenberg et al. 2011).

In order to capture the flux from the star properly, we generated custom apertures that were optimized to the bleed columns at maximum light with the lightkurve tool (Lightkurve Collaboration et al. 2018; Barentsen et al. 2019). Fortunately, the target pixel files (TPFs) were large enough to contain the bleed column entirely in four out of five sectors, albeit barely. The comparison of the SPOC and the custom masks of Sector 1 is presented in Figure 2. Even with the custom apertures, small differences between the successive sectors still remained. We were able to minimize these by scaling and shifting the sectors to the same median flux and amplitude, except for Sector 4 where we detected flux loss caused by the bleed column extending beyond the TPF edge at maximum light. The sector-stitched SPOC and custom-aperture light curves can be seen in Figure 3, as well as the phased light curves of each sector that we discuss in Section 3.1. Our custom light curve of β Dor is also available in the Appendix and online, together with the FITSH reduced ones.

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The star β Dor is among the brightest classical Cepheids in the sky; therefore, it has been the main subject of several earlier studies (see, e.g., Bell & Rodgers 1967; Lub et al. 1979; Turner 1980). It is also an important contributor to the Cepheid-based investigations concerning the Galactic structure and rotation (Kraft & Schmidt 1963), its abundance gradient (Andrievsky et al. 2002), the P–L relation (Benedict et al. 2007), and the pulsation velocity projection factor (Parsons 1972; Nardetto et al. 2006, 2007; Breitfelder et al. 2016), to mention just a few. The TESS data of β Dor provide new and precise input for models and future analysis.

Here we focus on only one phenomenon that is clearly visible in the light curve in Figure 3: the cycle-to-cycle variation of the amplitudes. This is seen not only between sectors but during a particular sector, too, ruling out incorrect data stitching as a cause. The variation reaches ∼5 × 10⁴ e⁻ s⁻¹ flux (∼5 mmag, ∼1.3 percent of the peak-to-peak amplitude), which is two orders of magnitude higher than the photometric precision of the data points. Blends cannot be responsible for these differences either: only one neighboring bright star shows high-amplitude variation, ASAS J053422-6225.4 (G_RP = 7.95 mag), but on a much longer timescale than the observed phenomenon. Moreover it contaminates only in Sector 1, when it falls in the saturation column (Figure 2), but not in later sectors. Thus we conclude that the cycle-to-cycle variations must be intrinsic, and β Dor is another example where precise space-based photometry reveals low-amplitude fluctuations in the pulsation, after SZ Tau, observed by the MOST satellite (Evans et al. 2015b), and V1154 Cyg, measured with Kepler (Derekas et al. 2017). Cycle-to-cycle variation in the RV curves has been detected in the long-period fundamental-mode Cepheids RS Pup and ℓCar, on the order of a few kilometers per second (Anderson 2014; Anderson et al. 2016; Anderson 2016). Evidence for cycle-to-cycle differences in the angular diameter variability has also been found in the case of the latter star, which may lead to ∼10 mmag photometric fluctuations. For β Dor, an upper limit of <0.57 km s⁻¹ was found for cycle-to-cycle variations in the RV curve by Taylor & Booth (1998).

The investigation of RV and photometric fluctuations in Cepheid pulsations is a new and promising area of research. As we showed in the data of the early sectors of β Dor, TESS makes it possible to combine accurate photometric observations with spectroscopic measurements of modulation and fluctuations in the pulsation for the brightest Cepheids in the sky and thus has the potential to provide important contributions to solve the question of (in)stability of Cepheid pulsation.

Anomalous Cepheids (ACs), also known as BL Boo stars form the rarest and least understood type of Cepheids. They are ∼0.5–2 mag brighter than RR Lyrae stars of similar pulsation periods. This property led to their discovery in the Sculptor dwarf galaxy (Baade & Hubble 1939). Since then, dozens of anomalous Cepheids have been found in the Local Group dwarf galaxies (see Stetson et al. 2014; Martínez-Vázquez et al. 2017 and references in Fiorentino & Monelli 2012) and hundreds in the Magellanic Systems (Soszyński et al. 2008b, 2015a). More than 50 ACs are known in the Galactic bulge (Soszyński et al. 2017; Udalski et al. 2018), but only a few are known in the Galactic globular clusters (Zinn & Dahn 1976; Navarrete et al. 2015). McCarthy & Nemec (1997) presented the only detailed high-dispersion spectroscopic study of an anomalous Cepheid (BL Boo) so far.

Single-star evolution models indicate that ACs are intermediate-age (1–6 Gyr), metal-poor (Z = ∼0.0001), He-burning stars with masses of 1.3–2M_⊙, but they do not explain their existence in old and/or high metal abundance (Z > 0.008) environments. That suggests that at least a fraction of these variable stars must have evolved via binary channels, where mass transfer or merger may play an important role (Bono et al. 1997; Gautschy & Saio 2017).

ACs have their own P–L relation that differs from that of the other Cepheid classes. Their pulsation periods (∼0.6–3 days) overlap with that of the RR Lyrae stars, BL Her, and classical Cepheid stars. However, their distinction from the other Cepheid types is possible, since their light-curve shapes significantly differ at a given period value. Despite these factors, the classification of these stars is often unreliable, mostly because of the uncertainties in absolute brightness and the lack of fine details in the light-curve shape caused by noise and the sparseness of data. This is especially true for the Milky Way galaxy, where, for example, a recent revision of classifications resulted in finding several new ACs previously assumed to be type II Cepheids (Jurkovic 2018). Hopefully the Gaia and TESS surveys will solve this situation and help us to assign the proper type of variability.

ACs pulsate either in the fundamental or first-overtone radial mode. The existence of the latter was revealed by the discovery of the two distinct linear P–L relationships of ACs (Nemec et al. 1988, 1994), which was also confirmed by the OGLE (Soszyński et al. 2008b, 2015a) and VMC Surveys (Ripepi et al. 2014) of the Magellanic Clouds. We now believe that nearly one-third of ACs pulsate in the first-overtone mode. In this paper we present the TESS measurements of nine ACs, two of which belong to the overtone group (we marked them as ACEP-F and ACEP-1O in Table 1).

Here we briefly present the BL Her stars (BLHs) as well, since our sample contains one member of this class. BLHs are the shortest-period type II Cepheids (with pulsation periods between 1–5 days). They are low-mass stars (0.5–0.6 M_⊙) evolving on the blue horizontal branch, and pulsating almost exclusively in the fundamental radial mode. The first double-mode pulsators among them in the bulge (Smolec et al. 2018b; Udalski et al. 2018) and two purely first-overtone candidates in the LMC (Soszyński et al. 2019) have been discovered only very recently. Theoretical calculations suggest overtone pulsation to be very rare among BLHs, in agreement with the observations.

Figure 4 shows the TESS data of 10 short-period Cepheid candidates in the Milky Way we have analyzed: light curves, phase curves, residual phase curves, and residual Fourier spectra (after pre-whitening with the main frequency and its harmonic series to the order indicated in Table 2). Our 11th short-period Galactic target, XZ Cet, to which we dedicate a separate section (Section 3.2.1), is excluded here.

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In order to analyze the light-curve quality, we measured the noise level by calculating the standard deviation of the residual light curves. We found it to be ∼4 mmag for stars brighter than 14th mag, ∼10 mmag for stars between 14th–15th mag and ∼63 mmag for the faintest star, AV Gru (16.84 mag). Almost all stars suffer from a slight trend that also appears as a low-frequency excess in the residual spectra. We suspect the origin of these trends is instrumental. In the case of WX Tuc, we identified contamination from scattered light being responsible for the amplitude change at the beginning of the TESS orbits (BJD 2458326 and 2458340). The cause of the transient decrease in the pulsation amplitude at UY Eri (marked with gray) is also due to instrumental effects. We discuss the various potential instrumental issues in more detail in Section 3.3.6.

We noticed cycle-to-cycle variation on the order of 10 mmag in the only BLH of the sample, AA Gru. This star is easily recognizable by its differing light-curve shape. The same is true for the overtone anomalous Cepheid AK PsA, which has the most sinusoidal light curve in Figure 4.

The fundamental-mode ACs often have a characteristic small bump on the ascending branch of their light curves. Here such features are only noticeable in the longest-period ones, UY Eri and OGLE-GAL-ACEP-006. The latter shows remnant peaks in the residual Fourier spectrum at the pulsation frequency and its harmonics. These may originate from problems with stitching the data obtained in different sectors: contamination from neighboring stars may differ from sector to sector, which can affect the measured pulsation amplitudes. Small changes in the pulsation amplitude over time can generate side-peaks in the frequency spectrum. We experienced that with OGLE-GAL-ACEP-006, as well as at targets in the Magellanic Clouds (see Section 3.3.). A brighter star can be found at 76.18 arcsec separation (TYC 9368-421-1), which shows ∼0.4 mag variability according to Gaia measurements, and therefore may cause the instability of the light curve of OGLE-GAL-ACEP-006. Another star, the overtone pulsator AK PsA, also shows incoherent frequencies and displays a weak amplitude modulation, similar to the Blazhko effect commonly seen in the fundamental-mode RR Lyrae stars (RRab), and less commonly in their overtone mode siblings (RRc). Unfortunately, the full modulation cycle is not covered here, if there is any. One AC with possible Blazhko modulation (or multimode pulsation) has been reported by Stetson et al. (2014) in the galaxy Leo IV, and other Blazhko AC candidates have recently been discovered from the K2 mission data by Plachy et al. (2019), all pulsating in the fundamental mode.

The last star in Figure 4, RV Men was previously classified as first-overtone classical Cepheid in the ASAS database (Pojmanski 1997), RRc in VSX, and as RRab by Maintz (2005) as well as by ASAS-SN (Jayasinghe et al. 2019) and Gaia DR2 (Gaia Collaboration et al. 2017). The discrepancy of classifications of this 10th magnitude star sparked our interest. TESS uncovered the fine details of its light-curve shape, which is unusual both for Cepheids or RR Lyrae stars, as well as its instability in the form of irregular phase and amplitude changes (causing a forest of remnant peaks in the residual spectra) that are unlike the Blazhko effect. Therefore, we suspect that this star is a rotational variable with a stable stellar spot pattern that falsified the classification based on sparsely sampled photometric data. Stable light curves of rotational variables have recently been presented by Iwanek et al. (2019).

3.2.1. XZ Ceti

The variability of XZ Ceti was discovered by Hoffmeister (1934), and the star was initially classified as RR Lyrae type by Dean et al. (1977). Later, Teays & Simon (1985) fitted linear nonadiabatic pulsation models to it and concluded that XZ Ceti must be a first-overtone AC, or a first-overtone BLH. A detailed investigation concerning the type and pulsation properties of XZ Ceti was carried out by Szabados et al. (2007). Based on RV measurements, the authors ruled out XZ Cet as a classical Cepheid and proposed the type AC, so XZ Cet became the first AC identified in the Galactic field. Their analysis showed systematic shifts in the RV at certain phases, possibly indicating binarity. Instabilities in the pulsation were also reported, as well as a secular period change in the O−C diagram from historical data dating back to the Harvard photographic plate archive containing information from 1888 on.

The short-cadence TESS light curve of XZ Cet uncovered further interesting features in this star (Figure 5). After pre-whitening with the main frequency and its 13 harmonics, a group of low-amplitude frequencies appeared, of which we could clearly identify the ∼0.61 mode (f₂) and its linear combination with the overtone mode (Table 3). We found 12 other frequencies above our 4.0 S/N amplitude threshold. In the past, no additional modes had yet been reported in any anomalous Cepheid, let alone this many. We discovered a familiar pattern in the frequencies, the subharmonic structure connected to f₂, observed previously in Kepler RRc stars (Moskalik et al. 2015). Between 0.5f₂ and 1.5f₂, the former has a higher amplitude while the latter has combinations with the main pulsation frequency (f₁+1.5f₂, 2f₁+1.5f₂, 3f₁+1.5f₂). The 0.5f₂ subharmonic was also detected in OGLE overtone RR Lyrae and classical Cepheid stars that show f₂, but mostly in the form of power excess rather than a single peak (Smolec & Śniegowska 2016; Netzel & Smolec 2019). Further analysis of the low-amplitude peaks uncovered a frequency (f₃) with ∼0.63 period ratio, and its higher-amplitude subharmonic (0.5f₃). We note, however, that the separation between 0.5f₂ and 0.5f₃ is smaller than the frequency resolution (∼0.0365 days⁻¹) and for f₂ and f₃ is just above that. So we treat f₃ with caution. Low-amplitude oscillation near 0.61P₁ was first found in the RRd star, AQ Leo, measured by the MOST satellite (Gruberbauer et al. 2007). Since then it has become evident that a small fraction (∼8%–9%) of overtone pulsators exhibit weakly excited modes with a 0.6–0.64 period ratio, as this has been shown for RR Lyrae stars in the Galactic bulge (Netzel & Smolec 2019) and classical Cepheids in the SMC (Soszyński et al. 2010). In addition, a significant fraction (39% for the SMC and 47% for the LMC) of overtone Cepheids of the Magellanic Clouds show power excess with these period ratios (Śniegowska & Smolec 2018).

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Table 3. Fourier Solution of XZ Cet

	Frequency	Amplitude	Phase
	(c/d)	(m)
f1	1.215105(8)	0.15298(6)	0.13958(6)
2f1	2.43021(5)	0.02638(6)	0.8554(3)
3f1	3.64532(9)	0.01327(6)	0.5430(6)
4f1	4.8604(2)	0.00519(6)	0.183(2)
5f1	6.0755(10)	0.00120(6)	0.831(8)
6f1	7.291(3)	0.00041(6)	0.12(2)
7f1	8.5057(10)	0.00092(6)	0.734(10)
8f1	9.7202(10)	0.00104(6)	0.387(9)
9f1	10.9359(10)	0.00097(6)	0.031(10)
10f1	12.1511(10)	0.00078(6)	0.68(12)
11f1	13.366(2)	0.00060(6)	0.34(15)
12f1	14.581(3)	0.00043(6)	0.96(2)
13f1	15.796(4)	0.00028(6)	0.61(3)
f2	1.97717(10)	0.00930(6)	0.8511(10)
f1 + f2	3.1922(2)	0.00462(6)	0.332(2)
2f1 + f2	4.4074(5)	0.00265(6)	0.883(3)
3f1 + f2	5.622(10)	0.00117(6)	0.531(8)
4f1 + f2	6.838(10)	0.00073(6)	0.192(12)
f2 − f1	0.7621(4)	0.00336(6)	0.839(3)
0.5f2	0.9886(3)	0.00353(6)	0.3348(11)
1.5f2	2.9657(7)	0.00152(6)	0.819(5)
f1 + 1.5f2	4.1808(10)	0.00107(6)	0.361(10)
2f1 + 1.5f2	5.3960(10)	0.00110(6)	0.845(12)
3f1 + 1.5f2	6.611(2)	0.00064(6)	0.474(15)
f3	1.9296(10)	0.00097(6)	0.980(11)
0.5f3	0.9648(3)	0.00428(6)	0.321(13)

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These modes, which are sometimes referred to as f_x (or f_0.61) modes, are hypothesized to be nonradial modes (Dziembowski 2016), where the actual pulsation frequency is the 0.5f_x peak that typically appears with a smaller amplitude (or does not appear at all) due to geometric cancellation. We note, however, that it can appear with a larger amplitude, or is the only peak detected (i.e., harmonics are not detected, Smolec & Śniegowska 2016; Süveges & Anderson 2018; Netzel & Smolec 2019). The nonradial modes of spherical-harmonic degree ℓ = 7, 8, and 9 form sequences in the Petersen diagram (from top to bottom, respectively), which slightly differ for RR Lyrae and classical Cepheid stars (Figure 6). In RR Lyrae stars, only the ℓ = 8 and ℓ = 9 modes seem to be present with a smaller middle sequence caused by their combination peaks. The f₂ and f₃ modes of XZ Cet are placed just between the variability classes in period, and clearly match the ℓ = 8 and ℓ = 9 sequences.

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New multicolor photometric observations were obtained for XZ Cet at the Remote Observatory Atacama Desert (ROAD; Hambsch 2012) in Chile between 16 October 2019 and 13 January 2020. Data are available in the AAVSO database (with HMB observer code). We have analyzed the data taken in BVRI passbands and discovered that the f₂ mode is visible in all four bands, and the f₁+f₂ combination, too, in the B band. We plotted the phase curves and the residual phase curves after removing the main periodicity with four harmonics and folded with P_0.61 (see Figure 7). We also calculated the Fourier amplitudes and ratios in different colors (Table 4). The extensive study of RRc stars in the globular cluster M3 by Jurcsik et al. (2015) reported color measurements of the 0.61 mode for the first time. The authors showed that the RRc stars with the 0.61 mode are bluer than the regular ones and redder than the RRd stars, as they are exactly between these two groups in the (B − V) – (V − I) plane.

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Table 4. Fourier Amplitudes for XZ Cet Light Curves in Various Photometric Bands

Filter	A1O	A0.61	A0.61/A1O
I	0.1413(5)	0.0077(5)	0.054(4)
R	0.1831(6)	0.0086(6)	0.047(3)
V	0.2227(6)	0.0114(6)	0.052(3)
B	0.2921(6)	0.0128(6)	0.044(2)
TESS	0.15298(6)	0.00930(6)	0.0608(4)

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The TESS light curve of XZ Cet provided the first detection of nonradial pulsation in an AC-type star, and hopefully more will follow in the future.

TESS observed both Magellanic Clouds in Sectors 1 and 2. Moreover, the LMC was continuously observed during the entire first year of the mission. It is our misfortune, however, that as a consequence of the P–L relation, the shortest-period Cepheids are the faintest ones, seriously limiting our target selection in the Magellanic System. To avoid significant confusion, we selected our targets at lower-density stellar fields in the outskirts of the LMC and the SMC, and from the Magellanic Bridge. We also added two stars from the inner region of SMC to test the limits of TESS. Targets in TESS CCD frames of Sectors 1 and 2 are displayed in Figure 8. This region of the sky has been extensively studied by the OGLE Survey; thus, we have the opportunity to compare all of our findings with OGLE data. The FITSH light curves of the selected Cepheids in the Magellanic System are plotted in Figure 9.

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3.3.1. Bump Cepheids

3.3.2. pWVir Stars

W Vir stars represent the intermediate-period subtype of type II Cepheids: they are 1.5-2 mag fainter than the classical Cepheids at similar pulsation periods, making their TESS photometry more challenging. Here we present two members from the so-called "peculiar W Vir" group (OGLE LMC-T2CEP-023 and OGLE LMC-T2CEP-280; Figure 9). These stars are on average brighter and bluer than the regular ones, and can also be recognized from light-curve morphology, since shapes are closer to those of the DCEP-F type. Only 34 members of this group are known in the Magellanic Clouds (Soszyński et al. 2018) and the same amount in the Galactic bulge (Soszyński et al. 2017). Half of the group shows signs of binarity in the form of ellipsoidal or eclipsing variation superimposed on the pulsation. The same was detected in OGLE LMC-T2CEP-023, where an 88.61 day orbital period could be determined (Soszyński et al. 2010). The TESS light curve of this star clearly shows the eclipses beside cycle-to-cycle changes of the pulsation and a long-period additional variability. We removed the pulsation signal and plotted the residual light curves folded with the orbital period for both the TESS and OGLE data in Figure 10. These residual light curves reveal ellipsoidal variability due to tidal distortion effects, as well as a difference between the widths of eclipses. The latter phenomenon could be an indication of a disk being present in the system, suggesting that mass transfer was happening in the past, as in the case of OGLE-LMC-T2CEP-211, a similar eclipsing system with a pWVir star analyzed by Pilecki et al. (2018a). We also found the combination frequencies of 1/P_puls + 1/P_orbit and 1/P_puls − 2/P_orbit, the latter of which is also seen in the OGLE data. The combination frequencies are evidence for phase modulation caused by binary motion, and they are often detected in type II Cepheid light curves if the Cepheid is a member of an eclipsing or an ellipsoidal binary system.

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OGLE LMC-T2CEP-023 is a strongly blended star, with several variables within and nearby the aperture, with periodicities known from the OGLE Survey. We found excess power near to the pulsation frequency and harmonics that may originate from the neighboring variable star OGLE LMC-CEP-0615, which has a pulsation period of ∼5.1 days, very close to the 5.23 day period of T2CEP-023. We could not identify the origin of the mean brightness decrease: it might be of instrumental origin or an intrinsic feature of the system.

The other pWVir star, OGLE LMC-T2CEP-280 is located in a less-dense stellar environment, away from the center of the LMC. We observe cycle-to-cycle changes in this star with amplitudes up to 10 mmag, which was not seen in the OGLE data. The scatter of the light curve is also significant (∼5 mmag), and the timescale of the sectors causes a prominent peak in the Fourier spectrum, highlighting the difficulty of accurate data stitching, which is especially challenging when cycle-to-cycle variations occur. Due to the instrumental issues, we cannot be certain that the observed cycle-to-cycle variation is intrinsic, although it is characteristic of type II Cepheids.

3.3.3. OGLE LMC-CEP-227

OGLE LMC-CEP-227 is the famous Cepheid variable in a double-lined eclipsing binary system, which was the first classical Cepheid star with a precisely determined mass (Soszynski et al. 2008a; Pietrzyński et al. 2010; Pilecki et al. 2018b). Binary systems with a Cepheid component are of great importance because dynamical masses and radii can be directly determined and compared to pulsation masses and radii, providing a link between theoretical models of stellar evolution, atmospheres, and pulsation. OGLE LMC-CEP-227 has been the subject of many detailed studies (e.g., Prada Moroni et al. 2012; Marconi et al. 2013) including various arguments concerning the p-factor, the projection factor that relates observed RVs to intrinsic pulsation velocities (Pilecki et al. 2013). The TESS data of this star have the potential to further increase the precision of the derived stellar properties and the p-factor itself.

Here we focus only on the detectability of the light-time effect. OGLE LMC-CEP-227 is the primary component of the system with an orbital period of 309.404 ± 0.002 d. In Sectors 1–5, the stars were in the orbital phase when the Cepheid component was approaching us but did not reach the eclipse. We calculated the light-time effect with the parameters given by Pilecki et al. (2013), and we constructed the O−C diagram, displayed in Figure 11, where "O" is defined by third-order polynomial fits to individual maxima. The pulsation period was determined to be the best fit to the light-time effect: C = JD2454896.285+3.7970782E. The O−C follows the slope of the curve, but it is doubtful if we could recognize the light-time effect only from that. The success depends on the amplitude of the light-time effect, the duration of observational interval, and the uncertainty of O−C residuals. The latter is related to both the brightness and crowding.

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New calculations based on Gaia DR2 put the binary fraction of Galactic Cepheids above 80% (Kervella et al. 2019). This is a significantly higher fraction than the 29 ± 8% value found by Evans et al. (2015a) based on radial velocity data, although that was considered to be a lower limit by the authors too. A list of binary Cepheid systems is maintained by Szabados (2003). ³³ On the whole, we have a good chance of discovering binarity if we can separate the light-time effect from evolutionary period changes in the O−C data. Such analysis would benefit enormously from repeated visits of the continuous viewing zones over multiple mission extensions. In contrast, we have the possibility to catch eclipses, as we do in the case of OGLE LMC-CEP-227 in the later TESS sectors. However, the light-time effect is not sensitive to all types of binaries, as demonstrated by Groenewegen & Jurkovic (2017), who could not identify the known binary OGLE LMC-T2CEP-023 this way.

We note that we did not use any detrending algorithm on any of our light curves to demonstrate the effect of contamination. In the case of OGLE LMC-CEP-227, we noticed a long-term variation in the mean brightness, not observed in the OGLE data (Figure 9). After investigating the possible blend stars, we concluded that OGLE-LMC-LPV-5076 is the most probable source of this trend.

3.3.4. Detectability of Additional Modes

We mentioned in Section 3.2.1 that nonradial modes may be excited in overtone pulsators. Here we report a case where the nonradial mode was not detected in OGLE, but is clearly seen in the TESS data. OGLE LMC-CEP-3377, located in the Magellanic Bridge, was identified in OGLE-IV (Soszyński et al. 2015b) and was classified as first-overtone classical Cepheid (DCEP-1O). The additional mode we found in the TESS light curve belongs to the middle sequence of f_x modes (Figure 6). The amplitude of this mode is 2 mmag, which is below the detection limit of OGLE-IV (3 mmag). It is very likely that this frequency comes from the Cepheid star, as there are no known variables in its close neighborhood. However, we observed an additional slow, sinusoidal variation in the light curve (with ∼20 mmag full amplitude) that may originate either from a blended source, from a companion star, or from instrumental effects. O−C calculations were not helpful in this case; thus, more data and investigations are needed.

We have to be careful when interpreting signals as extra modes in Cepheids. Here we describe another interesting case of a spurious additional frequency found in the double-mode Cepheid OGLE LMC-CEP-4419. Table 5 summarizes the results of our Fourier analysis. We detected the fundamental mode (f₀, 2f₀) and the overtone mode (f₁), as well as their linear combinations (f₀+f₁, f₁–f₀, 2f₀+f₁). In addition, we detected a low-amplitude frequency series that can be explained either as combination frequencies 2f₀–f₁, 4f₀–2f₁, 6f₀–3f₁ or as a third mode f₂, 2f₂, 3f₂. But these combinations would be very unusual, and f₂ would not fit to the known frequency ratio groups in the Petersen diagram as an additional mode (Süveges & Anderson 2018). Therefore we investigated the neighboring stars and found that f₂ is the pulsation frequency of the fundamental-mode classical Cepheid, OGLE LMC-CEP-4414, located 6337 away (Figure 12). To check the contamination from this star we created light curves with smaller apertures using radii of 2, 1.5, and 1 pixels. The smaller the radius, the weaker the contamination, as is visible in Table 5, and it drops below the detection limit at the 1.0 px aperture. Nevertheless, we do not consider the tight apertures to be better in general, even if we can avoid a large fraction of the contaminations. The intrinsic low-amplitude modes may also remain undetectable if only a small fraction of the flux is collected.

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Table 5. Frequency Content and Fourier Amplitudes of OGLE LMC-CEP-4419 Light Curves Constructed with 2.5, 2.0, 1.5, and 1.0 Pixel Radius Apertures, Respectively

ID	Frequency	A(2.5 px)	A(2.0 px)	A(1.5 px)	A(1.0 px)
f0	0.28744(4)	0.1030(8)	0.0828(7)	0.0745(5)	0.0546(4)
f1	0.40093(4)	0.0972(8)	0.0802(7)	0.0711(5)	0.0514(4)
f2 = 2f0–f1	0.17613(6)	0.0658(8)	0.0249(7)	0.0109(5)	⋯
f0+f1	0.68837(10)	0.0346(8)	0.0287(7)	0.0249(5)	0.0178(4)
2f2 = 4f0 − 2f1	0.35225(11)	0.0312(8)	0.0092(7)	⋯	⋯
f1 − f0	0.1135(2)	0.0192(8)	0.0192(7)	0.0165(5)	0.0119(4)
2f0	0.5749(2)	0.0186(8)	0.0154(7)	0.0136(5)	0.0100(4)
3f2 = 6f0–3f1	0.5284(3)	0.0125(8)	⋯	⋯	⋯
2f0+f1	0.9758(4)	0.0103(8)	0.0075(7)	⋯	⋯

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3.3.5. Cepheids in the SMC

We identified another case of strong contamination between Cepheids in the SMC as well. Our DCEP-F-type target star, SX Tuc, lies away from the dense center of the galaxy, but its surroundings are still crowded with variable stars. Even with 2-px apertures, we could only generate a light curve that is a composite, which also contains variation from a neighboring Cepheid at 2141 distance, OGLE SMC-CEP-3650. This latter star is brighter, and its variability dominates the composite Fourier spectrum displayed in Figure 13. Moreover, the first harmonic frequency of OGLE SMC-CEP-3650 is very close to the pulsation frequency of SX Tuc. Despite all of these complications, we were able to determine the harmonic series of SX Tuc up to the seventh harmonic. The Fourier parameters were compared with the OGLE values, and we found only a few percent difference. This implies that contamination does not make Fourier parameter calculations necessarily impossible, but may cause some uncertainties.

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We examined three DCEP-F stars in the SMC (see Figures 8 and 9). SW Tuc shows a changing light-curve shape. We found frequencies in the residual spectra at ∼0.87, ∼1.74, and ∼0.62 c/d. Although there are no known variables in the neighborhood, contamination is the most likely explanation for these signals. TT Tuc and AT Tuc are located in the outskirts of the SMC. In the light curve of TT Tuc, we detected the pulsation frequency of the neighbor OGLE-SMC-RRLYR-1868 at ∼1.42 c/d. AT Tuc displays cycle-to-cycle changes, possibly caused by contamination from an unidentified source.

We selected two DCEP-1O type stars from the Magellanic Bridge: OGLE SMC-CEP-4952 (Jacyszyn-Dobrzeniecka et al. 2016) and OGLE SMC-CEP-4955 (Soszyński et al. 2015b). The former is the farthest known Cepheid in the Magellanic Bridge, and the faintest star in our sample. Its type is also problematic, since it has recently been reclassified as DCEP-F based on distance calculations using optical Period–Wesenheit relations (Jacyszyn-Dobrzeniecka et al. 2020). However, the R₂₁ and R₃₁ Fourier parameters based on the OGLE light curves do not support the reclassification. We could reliably detect the first harmonic from the TESS light curve, but there is no clear peak at the second harmonic frequency above the noise level. Therefore we can only give an upper limit for the R₃₁ value, at R₃₁ < 0.065. That puts this star very far from the DCEP-F domain in the $\mathrm{log}P-{R}_{31}$ plane, but agrees with the region where classical overtone and type II Cepheids are located (Figure 15). The proper classification of OGLE SMC-CEP-4952, as well as its origin in that far out region of the Magellanic Bridge, remains uncertain.

The overtone OGLE SMC-CEP-4955 is a blended star, which might be responsible for the trend visible in its light curve. This trend also shows the periodicity of the TESS orbits. No additional mode is detected beyond the main frequency and its two harmonics.

Finally, we present the only known double-mode Cepheid in the Magellanic Bridge, OGLE-SMC-CEP-4951. Both the O1 and O2 modes could be detected in TESS data, the former along with two harmonics, and one combination frequency is also visible. The orbital period of TESS causes weak peaks at ∼0.07 and ∼0.14 c/d.

3.3.6. On the Instrumental Issues

Cepheids are one of the primary distance indicators in the local universe (Riess et al. 2018a); therefore, the accuracy of our distance measurements can depend on our ability to identify them properly. The OGLE Catalogue is a good source of classifications for the Magellanic System, where both the light-curve shape and the brightness information are used. The latter is applied through a reddening-free brightness quantity, the Wesenheit magnitude. However, the accuracy of Cepheid distances may also depend on blending corrections for remote targets in crowded fields, as pointed out by Majaess (2020).

The identification of Cepheids in the Galactic field is more uncertain. Except for the brightest, most well-studied Cepheids, we detect a significant fraction of misclassifications among Cepheid candidates (Szabó et al. 2011; Poretti et al. 2015). A considerable number of the Cepheid candidates identified in Gaia DR2 had low absolute magnitudes, well below the expected P–L relation, indicating that they are either other variables or that their parallaxes are inaccurate (Gaia Collaboration et al. 2017). These were subsequently reclassified in a follow-up work that produced a cleaner sample (Ripepi et al. 2019). The Galactic Cepheids, in particular, were analyzed in detail, employing both numerical parameters and visual inspections of the light curves along with period-Fourier parameter and period-absolute Wesenheit diagrams plus literature sources.

We adopted the same formulae used in that paper to calculate the Gaia Wesenheit magnitudes (W = G − 1.90(G_BP − G_RP)) and calculated the absolute Wesenheit magnitudes of those TESS stars whose offset-corrected parallaxes were above zero $({W}_{A}=W+5\,\mathrm{log}{\varpi }_{\mathrm{corr}}+5)$. We used the same 0.046 milliarcsecond (mas) parallax zero-point offset calculated by Riess et al. (2018b) for consistency. We note that the Gaia DR2 parallax zero-point offset was estimated by various authors, and published values span a wide range from 0.0 to –0.1 mas (e.g., Groenewegen 2018; Stassun & Torres 2018; Marconi et al. 2021). Corrected parallaxes remained negative for AV Gru and AK PsA. Due to the lack of G_BP and G_RP values, we could not calculate W_A for β Dor either. We plotted the original and cleaned DR2 Cepheid samples and the P–L relations obtained from the cleaned DR2 sample in Figure 14. We then overplotted our TESS Milky Way stars to see if they line up with the PLRs or the loci defined by the Gaia classifications. As Figure 14 demonstrates, the absolute Wesenheit magnitudes are too uncertain for these stars for a conclusive result, indicating that the underlying parallaxes themselves are too uncertain (see Table 6). This is not unexpected: Ripepi et al. (2019) themselves state that especially at the short-period end of the Cepheid populations, the different classes overlap. Even with the Fourier parameters of the Gaia light curves at hand, they could not classify all objects unambiguously, concluding that collecting well-sampled, high-precision light curves remains the best way to solve this issue.

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Unambiguous classification of pulsating stars based exclusively on light-curve shape and period would be a significant achievement in stellar astrophysics. There are many clear cases where reliability of the classification depends exclusively on the quality of the photometric data. It was demonstrated nearly 40 yr ago that the light-curve structure in its quantitative form can describe the Hertzsprung progression (Simon & Lee 1981), and it is a sensitive tool for discriminating the pulsation modes in RR Lyrae stars (Simon & Teays 1982) and classical Cepheids. Since then, Fourier decomposition has become a widely used classifier for pulsating stars. However, as with all techniques, it has its own limitations that mostly arise from close similarities between the light curves of pulsators and other variable stars (rotational variables and eclipsing/ellipsoidal binaries). The diverse light-curve shapes that appear within the subtypes of Cepheids further complicate the problem.

The Fourier parameters R₂₁, R₃₁, ϕ₂₁, and ϕ₃₁ form clusters for the various types of Cepheids when plotted against the pulsation period. In Figure 15 we plot the OGLE-IV I-band values for LMC and Galactic bulge Cepheids (Soszyński et al. 2014, 2015b, 2015a, 2017, 2018), where we separated the classical Classical Cepheids (left panels) from type II and anomalous Cepheids (right panels) for better visibility. Since the passband of TESS is similar to the OGLE I band, we could overlay the Fourier parameters of our target stars listed in Table 2. Fourier parameters calculated from the OGLE data are available for many of our targets, and we found them to differ from TESS values only by a few percent, which changes the position of the stars on the plots only slightly. The clusters in Figure 15 seemingly overlap on the individual plots, but if we take all four plots into account, class membership becomes more clear. We note, however, that the two peculiar W Vir stars (marked with large red diamonds at $\mathrm{log}P\sim 0.7$) cannot be distinguished from DCEP-F stars based only on their Fourier parameters. The Cepheid impostor, RV Men, is also indicated (with brown). Based on its location, this star can belong to the RR Lyrae stars, demonstrating the uncertainty of this classifier method, in which the Fourier coefficients represent average values for the light curves, and no fast changes are taken into account.

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Many Galactic Cepheid candidates observed by large sky surveys have under-sampled light curves from which Fourier parameters can be determined only with large errors. The precise and continuous light curves of TESS will certainly solve this problem, and availability of fine details of the light-curve structure allows us to filter out the nonpulsating variables. We note, however, that the low-order Fourier parameters are only simplified quantitative measures of the average light curve, and this fact is independent of the photometric accuracy. Using more information embedded in the fine details of brightness variation can lead to significant progress in stellar classification, and in this way, we could better utilize the TESS precision. A recent attempt in this direction has been made by Szklenár et al. (2020) who used phased light curves of diverse variable stars to test an image-based classification attained with machine-learning methods. TESS may provide a new basis for such efforts.

3.4.1. Comparison with Models

The light-curve shape of RR Lyrae and Cepheids stars depend on global stellar parameters. The pioneering study of Jurcsik & Kovacs (1996) presented a formula for RRab metallicity as a linear function of the period and the ϕ₃₁ coefficient. An analogous correlation has been found for the RRc stars as well (Morgan et al. 2007). Such a clear relation between the light-curve shape and the metallicity is not known for Cepheid stars. Although there are indications that metallicity affects the pulsation amplitudes of Cepheids, the nature of the dependence has been debated (Szabados & Klagyivik 2012; Majaess et al. 2013). Recent efforts to estimate fundamental parameters such as mass, radius, luminosity, and effective temperature from photometry have also demonstrated that light-curve structure is statistically important both in RR Lyrae and Cepheid types (Bellinger et al. 2020).

The efficacy of the Fourier decomposition technique to compare observed and theoretical Cepheid light curves has been recognized and widely used since the 1980s (Simon & Davis 1983; Petersen & Hansen 1984; Stellingwerf & Donohoe 1986). A recent comparative study explores the variation of Fourier parameters at different compositions and mass–luminosity levels as a function of the wavelength and the period (Bhardwaj et al. 2017) to provide stringent constraints for the stellar pulsation models (Bono et al. 2000, 2010; Marconi et al. 2013 and references therein).

We adopted the I-band models from Bhardwaj et al. (2017), computed with metallicities representative for the LMC (Y = 0.25, Z = 0.008), as described by Marconi et al. (2013), and compared them with our parameters. Our sample contains only three fundamental and one overtone Cepheids whose Fourier parameters can be derived from both the TESS data and OGLE I -band photometry, and we can directly compare them to those of the models in Figure 16, for example as a function of stellar mass. The offset between the parameters of the two surveys is in the range of model grid resolution; thus, a precise fit can be obtained only after recomputing models and generating light curves in the TESS passband. Extending the grid toward shorter periods, where TESS observations are the most efficient, would also be welcome. Given the models covering a range of chemical composition, the combination of TESS and OGLE observations and theoretical models may be the basis of promising future investigations on physical parameters of Cepheids in the Galaxy and the Magellanic Clouds.

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3.4.2. High-order Fourier Parameters

Although the light-curve shape can be approximately described with the Fourier parameters of the main frequency and its first two harmonics, further harmonics may provide considerably more details. The number of detectable harmonics is a function of data sampling (cf. the Nyquist frequency) and the noise level. With space-based photometric missions, a high number of harmonics (10–20) could be detected for a large number of Galactic field pulsating stars, providing a basis for statistical studies of high-order harmonic structures that could be useful for stellar classification, too. We plotted the high-order Fourier parameters for the amplitude (R_n1) in Figure 17. A similar plot was presented for some CoRoT RRab stars by Benkő et al. (2016) who discovered that a deep local minimum exists in most amplitude ratio curves, and the position of the minimum is dependent on the pulsation period.

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Here we have only a small sample, but we can still declare that for the four ACEP-F stars that show significant minima in the amplitude ratio series, the longer the period, the lower the harmonic order where the minimum occurs. Moreover, both overtone stars show minima but at different harmonic orders, despite their periods being almost equal. The impostor star, RV Men has the steepest slope. The BLH star has a similar R₂₁ value as the overtone ACs, but then goes on a different path. We intend to analyze more TESS light curves to figure out what these characteristics can be used for.

Several stars in our sample show light-curve fluctuations, but in the dense stellar fields, it may be impossible to unambiguously determine whether these are intrinsic features or only systematics and instrumental signals. Instead of disentangling the photometry, we can use the O−C technique to detect fluctuations in the time domain. We used this method to recover the light-time effect caused by a companion, as we presented in Section 3.3.3. In this section we discuss our results for the Galactic Cepheids.

There are various methods to measure the onset and observed length of a pulsation cycle, depending on which characteristic points of the light curve are used and how they are fitted. Derekas et al. (2012) showed that the median brightness on the rising branch provides the most precise way to construct an O−C diagram for V1154 Cyg, the Kepler Cepheid. For TESS, however, flux loss and contamination may affect the moment of the median brightness; thus, O−C values have been calculated from the timing of the sharp light-curve maxima, too.

In Figure 18 we present O−C diagrams for four stars: β Dor, XZ Cet, AK PsA, and OGLE GAL-ACEP-006. We used two different methods to calculate the O−C values where the C values are always based on the mean pulsation period of the star. We determined the times of maxima using a third-order polynomial for β Dor, and fifth-order polynomials for the rest of the stars to locally fit the maxima (blue points). Polynomials with different orders were tested before selecting these values. In the second method, we calculated the times of the median of the ascending branch (pink). Ephemerides are given in Table 7.

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Table 7. Ephemerides Used to Plot O−C Graphs

β Dor	${C}_{\mathrm{MAX}}$ = JD2458326.7988+9.84318E
	CMED = JD2458334.4241+9.84318E
XZ Cet	${C}_{\mathrm{MAX}}$ = JD2445285.4260+0.822974E
	CMED = JD2445285.2760+0.822974E
AK PsA	${C}_{\mathrm{MAX}}$ = JD2458325.9529+0.82210E
	CMED = JD2458325.8086+0.82210E
GAL-ACEP-006	${C}_{\mathrm{MAX}}$ = JD2458326.1237+1.883636E
	CMED = JD2458325.9206+1.883636E

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A trend is clearly visible in the O−C curves of AK PsA. This O−C variation along with the detected amplitude modulation at the millimagnitude level (see Section 3.2) suggests that AK PsA may be a Blazhko ACEP-FO, where the modulation is so weak that only precise and continuous observations from space could recover it. Since the proposed modulation cycle was not fully covered, we will need more observations to confirm this detection and determine the modulation period.

In the O−C diagram of the ACEP-F-type star OGLE GAL-ACEP-006, we see strong irregular period jitter on the order of 5–10 minutes (depending on the technique) instead of a coherent trend. In comparison, most of our targets show scatter of a few minutes in their O−C values. It is not clear yet if the larger scatter in the pulsation period of OGLE GAL-ACEP-006 is caused by systematics an/or elevated noise in the detectors and processing, or if it is a physical fluctuation in the pulsation, which was observed in other Cepheids before. We can only answer this question after more TESS Cepheid observations have been processed.

The O−C values of β Dor are consistent with the historical O−C diagram, considering the scatter present in the ground-based data (Berdnikov et al. 2003). We also examined the cycle-to-cycle variation and detected slight changes, but the curve is dominated by a strong drop in the times of the median of the ascending branch in Sector 4. These data points coincide with the maxima that suffered flux loss in the 2 minutes observations, indicating that even seemingly small photometric inaccuracies can cause noticeable problems during data analysis.

Finally, the O−C curve of XZ Cet clearly shows fast variations. The frequency of the variations is 0.76 d⁻¹, which is equal to the f₂ − f₁ difference between the two strongest modes present in the star. The same signal is present in the median of the ascending branch but with a smaller amplitude. Clearly, the O−C variations here are primarily driven by mode beating instead of variations in either of the modes. Similar beating between the overtone mode and the f_X mode was also described in RRc stars (Hippke et al. 2015). These findings demonstrate that intrinsic variations of the first overtone cannot be always detected via conventional O−C methods: in those cases, temporal variations in Fourier amplitudes and phases are more reliable indicators.

Accurate O−C data from the TESS observations can also be used to compound existing long-term data sets to study evolutionary period changes. These in turn can then indicate in what direction certain Cepheids are crossing the instability strip (Turner et al. 2006). However, such studies require decades-long baselines and thus coverage beyond the capabilities of TESS, for now.