Discovery of Two Pulsating Extremely Low-mass Pre-white Dwarf Candidates in the TESS Eclipsing Binaries

As the most common stellar remnants in the Galaxy, white dwarf (WD) stars are thought to be the final evolutionary stage of stars less massive than about 11 M_☉ (Siess 2007). The vast majority (more than 95%) of all stars, including our Sun, will end their lives as WDs (Althaus et al. 2010). Their mass distribution is one of the most important properties of the WD population. For DA WDs comprising about 85% of all known WDs (Eisenstein et al. 2006), the mass distribution suggested the existence of a small population of extremely low-mass white dwarf stars (ELM WDs) with masses below 0.3 M_☉ (Kepler et al. 2007, 2019; Kleinman et al. 2013). They are most likely to be born in interactive binary systems through either common-envelope evolution or stable Roche-lobe overflow mass transfer (Althaus et al. 2013; Istrate et al. 2016b; Chen et al. 2017; Li et al. 2019), since the universe is not old enough for a single star to evolve into a helium-core WD. More than 100 ELM WDs and their precursors have been found in binary systems through multiple survey campaigns such as ELM, WASP, PTF, and Kepler (e.g., Maxted et al. 2014a; Brown et al. 2016; van Roestel et al. 2018; Wang et al. 2019). Recently, Pelisoli et al. (2019) identified 50 new high-probability pre-ELM WDs (or pre-He WDs, i.e., helium WD precursors) and Pelisoli & Vos (2019) compiled a catalog of 5762 ELM WD candidates through analysis of the second data release of Gaia (DR2) and the Sloan Digital Sky Survey (SDSS) spectral database. Among the known ELM WDs and pre-ELM WDs, their companions are carb-oxygen WDs (Li et al. 2019), the millisecond pulsars (Istrate et al. 2014), or A/F-type dwarfs in EL CVn-type binaries (Maxted et al. 2014a; Wang et al. 2018).

EL CVn-type stars are a small group of eclipsing binaries composed of an A/F-type dwarf star and a hotter pre-ELM WD. They are in a rarely observed state in which the detached pre-ELM WD component is going through a phase of contraction and evolving to higher effective temperatures at an almost constant luminosity (Chen et al. 2017), the so-called (bloated) pre-WD phase. Before reaching the WD cooling track, the pre-ELM WDs may spend up to several billion years in the transition phase through stable hydrogen shell burning (Istrate et al. 2014), so it can explain a number of such objects discovered. Maxted et al. (2014a) discovered 17 bright EL CVn-type binaries from ground-based SuperWASP light curves and this new class of eclipsing binaries was named after the brightest variable star, EL CVn. Using the Palomar Transient Factory photometric database, van Roestel et al. (2018) found 36 EL CVn stars with machine-learning techniques. There are a total of 13 such samples discovered by the Kepler mission so far (Zhang et al. 2019). Observationally, the light curves of EL CVn-type binaries are featured by the boxy-shaped primary eclipse, i.e., steep ingress and egress as well as a well-defined flat bottom due to the occultation of the dwarf star (Maxted et al. 2014a; van Roestel et al. 2018). Besides, their light curves outside eclipses exhibit sinusoidal modulation, in most cases caused by the reflection of the hotter pre-ELM WD. From the point of view of binary evolution theory, EL CVn stars can be produced by stable mass transfer in low-mass binaries, but not rapid common envelop evolution (Chen et al. 2017).

Another interesting aspect is that several EL CVn stars exhibit multiperiodic photometric variations. Multiperiodic pulsations of six EL CVn stars that are attributed to δ Sct-type pulsators have been observed in the recent few years (Maxted et al. 2014b; Faigler et al. 2015; Guo et al. 2017; Zhang et al. 2017; Wang et al. 2018). δ Scuti variables are main-sequence or immediate post-main-sequence stars with A–F spectral types. They pulsate in low to intermediate radial order p, g, and mixed modes, which are driven by the κ mechanism acting mainly at the He ii partial ionization zone (Aerts et al. 2010; Xiong et al. 2016). The pulsation frequencies for most δ Sct stars are in the range 4–60 c day⁻¹. As of now, there are about 200 δ Scuti stars in eclipsing binaries (Liakos & Niarchos 2017). Apart from the dwarf star, the pre-ELM WD among the EL CVn-type systems can also show pulsations. The pre-ELM WDs in three EL CVn-type eclipsing binaries WASP 0247–25 (Maxted et al. 2013), WASP 1628+10 (Maxted et al. 2014b), and KIC 9164561 (Zhang et al. 2016) are supposed to be pulsators. In the T_eff−log g diagram, the pulsating extremely low-mass pre-white dwarf stars (pre-ELMV) are located outside the boundaries of the extended ELM WDs and ZZ Ceti instability strips, suggesting that they constitute a new class of pulsating stars (Córsico et al. 2019). The seismic properties of pre-ELM WDs had been explored by several authors (e.g., Jeffery & Saio 2013; Córsico et al. 2016). These authors found that pre-ELM WDs can pulsate in radial and nonradial p and g modes, which are excited by the κ–γ mechanism operating mainly in the He⁺–He⁺⁺ ionization zone. It is possible to investigate their internal structure by means of the tools of asteroseismology (e.g., Istrate et al. 2017). As such, pulsating EL CVn-type binaries play an important and unique role for our understanding of the formation and evolution of pre-ELM WDs, since the binarity of these stars provides the possibility to directly measure their physical parameters, such as mass and radius. However, the exact nature of pulsating EL CVn stars is still unclear because of a few objects.

The Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2014) plans to observe nearly the whole sky during its 2 yr mission with four wide-field cameras, providing a combined field of view of 24° × 96°. Full frame images will be collected every 30 minutes while a subset of preselected target stars (>200,000) will be recorded with a two-minute cadence. Depending mainly on the ecliptic latitude, each star observed by TESS will have temporal baselines ranging from 27 days to about 1 yr. With such high-precision space photometry from TESS, the number of pulsating EL CVn-type binaries is expected to increase. We performed a search and study of EL CVn stars using the short-cadence data from Sector 1 through Sector 15 and discovered two new pre-ELMV candidates in the TESS eclipsing binaries, TIC 149160359 and TIC 416264037. The light variability of TIC 149160359 (=HD 36276; T_mag = 9.993) was first recorded by the Wide-field Infrared Survey Explorer and Chen et al. (2018) classified it as an EA-type eclipsing binary with an orbital period of 1.1207645 days. This star was also observed by the Radial Velocity Experiment, which gives the following atmospheric parameters: T_eff = 7652 ± 83 K, log g = 3.98 ± 0.16, and [M/H] = 0.56 ± 0.12 (Kunder et al. 2017). TIC 416264037 (=UCAC4 695-065574; T_mag = 13.445) is 1 out of 36 eclipsing EL CVn binaries discovered by the Palomar Transient Factory (van Roestel et al. 2018). With dozens of sparse brightness measurements, the authors estimated the orbital period of the binary system to be about 1.1599993 days, but did not see any light variations apart from the eclipsing light changes. A low-resolution spectrum for TIC 416264037 was obtained by the LAMOST project (Luo et al. 2018) and its atmospheric parameters are as follows: T_eff = 7393 ± 55 K, log g = 4.13 ± 0.09, and [Fe/H] = −0.053 ± 0.0.053. The Gaia DR2 parallaxes for TIC 149160359 and TIC 416264037 are 1.8301 ± 0.0445 and 0.3482 ± 0.0189 mas (Gaia Collaboration et al. 2018), respectively.

The target TIC 149160359 was observed by TESS during Sectors 11–13 in two-minute cadence mode. The short-cadence TESS photometry for TIC 416264037 was collected in Sectors 14–15. Their light curves that are provided by the TESS Science team (Jenkins et al. 2016) were downloaded from the Mikulski Archive for Space Telescopes (MAST).⁴ Only the Simple Aperture Photometry data was used for the present work, since the Pre-search Data Conditioning Simple Aperture Photometry (PDCSAP) is optimized for planet transits and the PDCSAP detrending may have adverse effects on eclipsing binary data (Slawson et al. 2011). We detrended the raw light curve of each 13.7 day orbit of the TESS spacecraft, separately with different global trend. First of all, we removed the data points with dispersions larger than 5σ by doing a local weighted linear regression smoothing (LOWESS; Cleveland 1979) fit. Following Hambleton et al. (2013), a first- or second-order Legendre polynomial was used to fit the out-of-eclipse portions of the light curve. The long-term trends were then removed from the original data. At last, the flux measurements were converted to magnitudes and the offsets between observation orbits were corrected. As an example, a short section of the resultant light curves for both stars is displayed in Figure 1. Their light curves show a boxy-shaped primary eclipse and sinusoidal modulation outside eclipses, the typical features of EL CVn-type binaries. It indicates that TIC 149160359 is probably a new EL CVn system with multiperiodic pulsations. TIC 416264037 is confirmed to be an EL CVn-type binary and the relatively large scatter of TESS measurements may imply the existence of intrinsic pulsations. We will discuss this in the following sections.

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In order to derive the orbital ephemeris of the two binary systems, the times of minimum light were calculated using a visual program (Peranso; Paunzen & Vanmunster 2016), which is mainly based on the K–W method (Kwee & van Woerden 1956). In total, we collect 132 and 87 minimum light times for TIC 149160359 and TIC 416264037, respectively, which are compiled in Table 4 of the Appendix. With these data, we obtained their linear ephemeris using the least-squares method. The new linear ephemeris for TIC 149160359 is:

$$\begin{eqnarray}&&{T}_{\min }=2458600.65490(6)+1.120738(1)\times E\end{eqnarray} \tag{ 1 }$$

for TIC 416264037, it is:

$$\begin{eqnarray}&&{T}_{\min }=2458684.7231(4)+1.15991(2)\times E.\end{eqnarray} \tag{ 2 }$$

The phases of all TESS observations for TIC 149160359 and TIC 416264037 were computed by using the newly derived linear ephemerides. We plotted their phased light curves in Figure 2. It should be noted that the timing of the shallower eclipse, when the more massive A-type dwarf star is eclipsed, was chosen as the reference epoch in this work.

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We made use of the 2013 version of the Wilson–Devinney (W–D) binary modeling code (Wilson & Devinney 1971; Wilson 1979, 1990, 2012) to analyze the TESS light curves of the two eclipsing binaries. Since the light curves of TIC 149160359 display different systematic effects between Sectors 11–12 and Sector 13 duo to imperfect de-trending, we fitted the two sets of light curves separately. In this paper, we define the luminous, more massive, primary star as star 1 and the hotter less-massive component as star 2. The effective temperatures (T₁) of star 1 were kept constant during the whole operation using the results from spectroscopic observations (Kunder et al. 2017; Luo et al. 2018). Circular orbit (e = 0) and synchronous rotation (F₁ = F₂) assumptions for both binary systems are acceptable in view of their short orbital period (P_orb < 1.16 days). The TESS-band limb-darkening coefficients (x_T, y_T) and the bolometric ones (X_bolo, Y_bolo) were adopted from Claret (2017) and van Hamme (1993), respectively. The initial value of the mass ratio was assumed to be 0.1, since almost all known EL CVn-type binaries have very low-mass ratios of ∼0.1. At first, we tried to use the canonical values of the gravity brightening coefficients (g₁ = g₂ = 1; Lucy 1967) and the bolometric albedos (A₁ = A₂ = 1; Ruciński 1969) for radiative atmospheres. However, the light-curve residuals still present obvious variations. We found that the light-curve fit is much better by setting g₁ and A₁ as free parameters. Therefore, we let the gravity brightening coefficient and the bolometric albedo of star 1 vary, while g₂ and A₂ were fixed at the canonical values, since they have a negligible impact for the hotter secondary star with T_eff > 8000 K. In general, the adjustable parameters included the phase shift, the orbital inclination (i), the gravity brightening coefficient of star 1 (g₁), the surface temperature of star 2 (T₂), the bolometric albedo of star 1 (A₁), the surface potential of both components (Ω_1,2), the mass ratio (q), and the relative monochromatic luminosity of star 1 (L₁). The fitted parameters for the best-fit model and their uncertainties are given in Table 1. As mentioned above, we modelled the two sets of light curves of TIC 149160359 separately. In Table 1, we just give the fitting results from Sectors 11–12. The theoretical light curves of TIC 149160359 and TIC 416264037, generated by the W–D binary model, are plotted as red solid lines in Figures 1 and 2. The corresponding light residuals, calculated by using the observational data minus the final binary model, are displayed in the bottom of each panel.

Table 1.  Photometric Solutions and Physical Parameters for All Studied Systems

Parameter	TIC 149160359		TIC 416264037
	Star 1	Star 2	Star 1	Star 2
Porb (days)	1.120738 ± 0.000001		1.15991 ± 0.00002
i (deg)	84.45 ± 0.04		80.31 ± 0.16
q = M2/M1	0.0906 ± 0.0004		0.1086 ± 0.0027
Teff(K)	7652 ± 83a	8675 ± 100b	7393 ± 55c	9545 ± 100b
Ω	2.648 ± 0.001	2.541 ± 0.004	2.703 ± 0.008	3.033 ± 0.041
${L}_{i}/{({L}_{1}+{L}_{2})}_{T}$	0.947 ± 0.001	0.053 ± 0.001	0.945 ± 0.007	0.055 ± 0.007
Gravity Brightening, g	0.521 ± 0.012	1.000b	0.467 ± 0.048	1.000b
Bolometric Albedo, A	0.705 ± 0.007	1.000b	0.924 ± 0.026	1.000b
Xbolo	0.643	0.659	0.637	0.668
Ybolo	0.249	0.139	0.261	0.074
xT	0.522	0.483	0.531	0.445
yT	0.306	0.250	0.315	0.238
r (pole)	0.3900 ± 0.0002	0.0831 ± 0.0006	0.3844 ± 0.0011	0.0683 ± 0.0028
r (point)	0.4097 ± 0.0002	0.0843 ± 0.0007	0.4042 ± 0.0013	0.0688 ± 0.0029
r (side)	0.4039 ± 0.0002	0.0834 ± 0.0007	0.3977 ± 0.0012	0.0684 ± 0.0028
r (back)	0.4070 ± 0.0002	0.0842 ± 0.0007	0.4011 ± 0.0013	0.0687 ± 0.0029
Absolute Parameters
a (R☉)	5.6 ± 0.2		5.7 ± 0.2
M (M☉)	1.75 ± 0.10	0.16 ± 0.01	1.70 ± 0.10	0.18 ± 0.01
R (R☉)	2.27 ± 0.06	0.47 ± 0.01	2.28 ± 0.07	0.39 ± 0.02
L (L☉)	15.8 ± 1.1	1.1 ± 0.1	13.9 ± 0.9	1.2 ± 0.1
log g (cgs)	3.97 ± 0.09	4.29 ± 0.10	3.95 ± 0.11	4.51 ± 0.17

Notes. r stands for the equivalent radii (r₁ = R₁/a, r₂ = R₂/a);

^aResult from Kunder et al. (2017); ^cCited from Luo et al. (2018); ^bAssumed values.

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In the absence of radial velocity measurements for the two binary systems, it is impossible to directly derive their physical parameters through the photometric solutions. Following Zhang et al. (2013) and Rappaport et al. (2015), we used two accessible parameters, i.e., the mean stellar density (${\bar{\rho }}_{1}\,\equiv$ M₁/(4πR₁³/3)) and the effective temperatures (T₁) of the primary star, to estimate their mass. Combining with the Kepler's third law (i.e., ${P}_{\mathrm{orb}}^{2}=\tfrac{4{\pi }^{2}{a}^{3}}{G({M}_{1}+{M}_{2})}$), we can obtain the following relation:

$$\begin{eqnarray}&&{\left\{\displaystyle \frac{a}{{R}_{1}}\right\}}^{3}=\displaystyle \frac{1}{3\pi }{{GP}}_{\mathrm{orb}}^{2}{\bar{\rho }}_{1}(1+q).\end{eqnarray} \tag{ 3 }$$

As discussed by Rappaport et al. (2015), a rather accurate value of ${\bar{\rho }}_{1}$ is expectable for EL CVn-type binaries that have a very low-mass ratio of ∼0.1. Based on our photometric solutions and Equation (3), we find that ${\bar{\rho }}_{1}=0.217\pm 0.001$ g cm⁻³ for TIC 149160359 and ${\bar{\rho }}_{1}=0.211\pm 0.008$ g cm⁻³ for TIC 416264037. Figure 3 displays the evolution tracks of the Yonsei–Yale stellar isochrones (Demarque et al. 2004) in the ${\bar{\rho }}_{1}$–T_eff plane, wherein the locations of the two primary stars are shown as the red dots. Based on which, the masses of the primary stars for TIC 149160359 and TIC 416264037 were estimated to be 1.75 ± 0.10 and 1.70 ± 0.10 M_☉, respectively. The masses of the hotter secondary stars were then obtained with the mass ratios (q = M₂/M₁) from the photometric solutions. Other parameters, such as the orbital separation (a), the radii, and luminosities of the components, were subsequently deduced, as summarized in Table 1.

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As shown in Figure 1, the residual light curve of TIC 149160359 exhibits obvious short-period light pulsations. The large deviations displaying among the light-curve residuals of TIC 416160359 also hints probable intrinsic pulsations of the binary star. To probe into the pulsation nature of the two EL CVn-type binaries, we have made the Fourier analysis of their light-curve residuals with the software package Period04 v1.2 (Lenz & Breger 2005). The standard procedure of iterative pre-whitening was used to extract significant frequencies from the residual light curves and the searching frequency was extended to the Nyquist limit of the short-cadence TESS data, f_Ny ≃ 359 day⁻¹. We stopped this process until none of the remaining peaks had a signal-to-noise amplitude ratio (S/N) larger than 4.0, suggested by Breger et al. (1993). As a result, we extracted 46 significant frequencies (S/N ≥ 4) for TIC 149160359 and 10 frequencies for TIC 416264037. These detected frequencies and their amplitudes, phases, errors, and S/N values are summarized in Tables 2 and 3. The uncertainties presented here were computed via analytical simulations in Period04 following the treatment of Montgomery & Odonoghue (1999).

All the detected frequencies have been checked for possible orbital frequency harmonics (f_i = kf_orb) or linear combination terms (f_i = f_j ± mf_orb or f_i = mf_j ± nf_k). Following Kurtz et al. (2015), we restricted our search to the second-order combinations and m, n are integers less than 3. The frequency resolution was set as 1.5/ΔT, where ΔT is the observation time span in days (Loumos & Deeming 1978). A combined frequency shall satisfy two criteria: the difference between the predicted and the observed frequency was within the frequency resolution, and the amplitude of the presumed combination term was lower than that of both parent frequencies. We found four orbital harmonic peaks and 21 other combination terms for TIC 149160359, while only one such frequency for TIC 416264037. These harmonic and combined frequencies are marked in Tables 2 and 3. The remaining frequencies are treated as independent modes for further analysis.

4.1. TIC 149160359

The Fourier amplitude spectrum of TIC 149160359 is displayed in the upper panel of Figure 4, wherein the independent frequencies are marked by their ID number. A zoom-in of the frequency range of 20–35 day⁻¹ is shown in the upper right inset to make it clearer. As can be clearly seen from the picture, TIC 149160359 presents two frequency concentration ranges: one is between 21 and 35 day⁻¹, and the other between 63 and 77 day⁻¹. Generally, the amplitudes of the first group are higher than the second region. Out of 21 independent frequencies, 17 are located in the first region, and the highest peak is the f₁ ≃ 27.5227 day⁻¹ with an amplitude of 1.22 mmag and an S/N of 64.2. The remaining four independent frequencies, f₁₃, f₁₄, f₂₃, and f₃₅, belong to the second group and the dominant frequency is the f₁₃ ≃ 67.4009 day⁻¹ with an amplitude of 0.31 mmag and an S/N of 23.6.

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As we mentioned earlier, the two components of EL CVn binaries can be shown to pulsate. To diagnose which star gives rise to the light pulsations, we performed a Fourier analysis of its residual light curves only from two specific orbital phases (−0.05 ≤ ϕ ≤ 0.05, −0.45 ≤ ϕ ≤ 0.55). Figure 5 displays their amplitude spectra between 0 and 100 day⁻¹. The spectra appear as a comb-like structure due to the strong aliasing from the selected data only during the two eclipses. The main pulsating signals in both cases are located around 28.414 day⁻¹ and none of the significant differences are visible in the frequency range of 21–35 day⁻¹, even if when the hotter but less-luminous pre-ELM WD was totally occulted by the A-type primary star. Since the pre-ELM WD companion is quite faint and contributes only ∼5% of the luminosity, this phenomenon is reasonable even though TIC 149160359 is a totally eclipsing binary with the orbital inclination of about 84°. Besides, the short-period light variations during the two eclipses are clear, as shown in Figure 1. All of these indicate that those independent frequencies between 21 and 35 day⁻¹ are very likely to come from the A-type primary star. The derived physical parameters suggest that the primary star of TIC 149160359 is a somewhat evolved A-type main-sequence star and located within the δ Sct instability strip in the Hertzsprung–Russell (H–R) diagram (Xiong et al. 2016), implying that it is a δ Sct pulsator.

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Interestingly, we could hardly see any pulsating signals in the frequency range of 63–77 day⁻¹ when the pre-ELM WD component completely disappeared behind the A-type primary star around orbital phases 0.45–0.55. However, when it was passing in front of the primary star (−0.05 ≤ ϕ ≤ 0.05), there seemed to be pulsating signals between 63 and 77 day⁻¹. We therefore conclude that the four independent frequencies between 63 and 77 day⁻¹ may be attributed to the intrinsic pulsations of the pre-ELM WD. If so, it is the fourth pre-ELMV candidate detected in the EL CVn-type eclipsing binaries.

4.2. TIC 416264037

This paper is the first contribution to the ongoing project for the search and study of pre-ELMVs in the TESS eclipsing binaries. Based on the short-cadence TESS data from Sector 1 through Sector 15, we found two new pre-ELMV candidates, TIC 149160359 and TIC 416264037. Their light curves are shown as a boxy-shaped primary eclipse and sinusoidal modulation outside eclipses, similar to those of well known EL CVn-type binaries. We have used the W–D binary modeling code to analyze their light curves. Light-curve modeling produces very low-mass ratios (q ≃ 0.1) for both binary systems with a detached configuration. By comparing the derived physical parameters with the mass–radius, mass–luminosity, and H–R diagrams (Ibanoǧlu et al. 2006; Xiong et al. 2016), the more massive A-type primary stars are somewhat evolved, but still stay within the δ Sct instability strip on the main-sequence band. The less-massive components of TIC 149160359 and TIC 416264037 are probably stripped red giants or pre-ELM WDs in view of their absolute physical parameters, as shown in Table 1. After a comparison of their cold degenerate radius (Nelson & Rappaport 2003), one can find that they are quite thermally bloated, closely resembling those of pre-ELM WDs discovered by the Kepler mission (e.g., Rappaport et al. 2015; Zhang et al. 2017). We therefore conclude that TIC 149160359 and TIC 416264037 are probably dA+pre-ELM WD eclipsing binaries, i.e., EL CVn-type binaries. In addition, TIC 149160359 is the first EL CV-type binary candidate discovered by the TESS satellite.

Apart from the eclipsing light changes, short-period light variations are shown in the residual light curves of TIC 149160359 and TIC 416264037. We have made the Fourier analysis of their light-curve residuals with the Period04 program. The Fourier amplitude spectrum of TIC 416264037 presents two frequency concentration ranges: one is between 21 and 35 day⁻¹, and the other between 63 and 77 day⁻¹. The pulsating signals among the low-frequency region can be detected during the two eclipses, even though the pre-ELM WD companion completely disappeared behind the A-type primary star around orbital phases 0.45–0.55. This suggest that the 17 independent frequencies between 21 and 35 day⁻¹ are probably due to the intrinsic pulsations of the δ Sct-type primary star. Using the equation $Q={P}_{\mathrm{pul}}{({\bar{\rho }}_{1}/{\rho }_{\odot })}^{1/2}$ (Breger 2000), the pulsation constants of the 17 independent frequencies were calculated to be 0.0114–0.0184 days, indicating that they all belong to low-order p modes of δ Scuti stars (Fitch 1981). For the high-frequency signals, the situation is different. When the pre-ELM WD component was totally occulted by the primary star, we could hardly see any pulsating signals among the second frequency region. However, when it was passing in front of the primary star, there seemed to be pulsating signals between 63 and 77 day⁻¹. We therefore conclude that the remaining four pulsating signals (f₁₃, f₁₄, f₂₃, f₃₅), with periods of 1281.88 ± 0.01, 1359.89 ± 0.01, 1269.17 ± 0.01, and 1131.67 ± 0.01 s, probably come from the pre-ELM WD component. Pre-ELMV stars can pulsate in p, g, and mixed modes (Córsico et al. 2016; Istrate et al. 2016a). By comparing with the theoretical results of linear pulsation stability analysis of pre-ELMVs (see the Figure 3 in Istrate et al. 2016a), the four pulsating signals detected in the pre-ELMV component of TIC 149160359 may arise from g modes.

The Fourier amplitude spectrum of TIC 416264037 also shows two frequency concentration ranges. Out of nine independent frequencies, seven reside within the low-frequency range of 12.5–19.9 day⁻¹. Two pulsating signals, f₄ = 122.2698 day⁻¹ and f₁₀ = 112.3603 day⁻¹, were detected in the high-frequency region. The seven independent frequencies between 12.5 and 19.9 day⁻¹, which are often observed in δ Sct stars, could be attributed to the intrinsic pulsations of the A-type primary star located in the δ Sct instability strip. However, the two high-frequency signals are much higher than the frequencies normally detected in δ Scuti stars. We therefore suggest that they are probably due to pulsations of the pre-ELM WD component. In a similar way, we calculated the pulsation constants of the 7 independent frequencies exhibited by the δ Sct-type component. The Q values of 0.0195–0.0309 days correspond to low-order p modes of δ Scuti stars (Fitch 1981). The pre-ELM WD component of TIC 416264037 was found to pulsate in two significant modes with periods of 706.63 ± 0.01 and 768.95 ± 0.01 s, respectively. They are likely due to low-order g modes based on the theoretical results of linear pulsation stability analysis of pre-ELMVs (Istrate et al. 2016a).

As summarized in Table 8 of Córsico et al. (2019), there are only 10 pre-ELMVs, including 5 candidates and 5 confirmed cases. With the two new candidates, TIC 149160359 and TIC 416264037, the number of pre-ELMV stars increases to 12. As a new class of pulsating stars, pre-ELM WDs can pulsate in radial and nonradial p and g modes, which allow us to diagnose their inner structures and fundamental properties using the tools of asteroseismology. These pre-ELMVs among the eclipsing binaries play an important and unique role for understanding their formation and evolution, since they provide the possibility to directly determine their absolute physical parameters, which can be utilized to improve asteroseismic modeling. In addition, TIC 149160359 and TIC 416264037 are bright enough for high-resolution spectroscopic observations and high-cadence multicolor photometry. We therefore appeal to pay more attention to the two pulsating EL CVn-type binaries.

We thank the anonymous referee for valuable comments. This paper includes data collected by the TESS mission. Funding for the TESS mission is provided by the NASA Explorer Program. K.W. acknowledges funding by the Meritocracy Research Funds of China West Normal University (grant No. 17YC515) and the Fundamental Research Funds of China West Normal University (grant Nos. 17C051, 16E016). X.B.Z. acknowledges support from the National Natural Science Foundation of China (NSFC, grant Nos. 11973053, 11833002, U1731111).

Software: Period04 v.1.2⁵ (Lenz & Breger 2005), Wilson–Devinney (W-D) binary code⁶ (Wilson & Devinney 1971; Wilson 1979, 1990, 2012), Matplotlib (Hunter 2007).

In Table 4, we give the eclipse timings of the two eclipsing binaries, TIC 149160359 and TIC 416264037. The epochs and the (O-C)₁ residuals for all the times of minimum light were calculated with respect to the newly derived linear ephemerides.