A Long-period Pre-ELM System Discovered from the LAMOST Medium-resolution Survey

2.1.1. LAMOST Spectra

LAMOST is a uniquely designed 4 m reflecting Schmidt telescope that is able to observe 4000 spectra simultaneously in a field of view of 5° (Cui et al. 2012; Zhao et al. 2012). The wavelength coverage of LAMOST low-resolution (R ∼ 1800) spectra ranges from 3690 to 9100 Å (Luo et al. 2016). The LAMOST medium-resolution (R ∼ 7500; see Liu et al. 2005) spectra have two arms: the blue arm covers the wavelength range 4950–5350 Å, and the red arm covers the range 6300–6800 Å (Zong et al. 2018). For both low- and medium-resolution spectra, LAMOST's observation strategy is to perform 2–4 consecutive short exposures for 10–20 minutes each (see Table 1). In the study of close binaries with a period of less than one day, the RV changes significantly between two single LAMOST exposures. Hence, the RVs of the short-exposure LAMOST spectra are crucial in our study (Mu et al. 2022).

LAMOST MRS conducted the first observation of J0419 on 2019 November 8, with six consecutive exposures. The RVs (see Section 3.4) span from 166 to −46 km s⁻¹ in 2.4 hr. LAMOST LRS made another six consecutive exposures on 2019 December 20, each with an exposure time of 600 s, and the resulting RVs are from 105 to −13 km s⁻¹. On 2020 December 30, 2021 January 26, and 2021 January 28, LAMOST MRS performed follow-up observations on J0419 and obtained a total of 12 single-exposure spectra. Owing to a bright Moon, the LAMOST follow-up spectra have very low signal-to-noise ratio (S/N). Nevertheless, we still use them to measure the corresponding RVs. The details of LAMOST spectroscopic data are summarized in Table 1.

We combine the LAMOST spectra observed on the same night after correcting the wavelength of each spectrum to the rest frame and plot them in Figure 1. The spectra show evident emission lines of the Balmer series and He i with significant double-peak characteristics in most LAMOST observations, suggesting that the emission lines are not produced by the visible star. We discuss the emission lines in Section 4.1.

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2.1.2. The 2.16 m Telescope Spectra

2.1.3. The Lijiang 2.4 m Telescope Spectra

We collect the light curves of J0419 from several publicly available photometric surveys. The light curves are used to determine the orbital period and analyze the variability (Figure 2). We introduce the photometric data below.

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2.2.1. TESS

TESS observed J0419 in two episodes in 2018 and 2020, respectively, using the full frame image mode. The first episode was taken from 2018 November 15 to December 11, and the second from 2020 November 20 to December 16, with exposure times of 1426 s and 475 s, respectively. Totals of 1176 and 3589 points were obtained in the two episodes. We use the Python package lightkurve ¹² (Lightkurve Collaboration et al. 2018) to reduce the data and get the TESS light curves of J0419. Images with background counts higher than 150 have been eliminated before extracting the light curve, because we cannot obtain reliable flux from these seriously contaminated images. After the visual inspection, we retain 1111 and 3347 points for the first and second observations, respectively. We use a pixel level decorrelation (PLD, see Deming et al. 2015; Luger et al. 2016, 2018) method to remove systematic instrumental trends. The TESS light curves are shown in Figure 3.

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2.2.2. ASAS-SN

2.2.3. CRTS

2.2.4. ZTF

The Gaia DR3 ID of J0419 is 3298897073626626048. We collect the astrometric information of J0419 from Gaia early data release 3 (EDR3; see Gaia Collaboration et al. 2021), which provides a parallax of ϖ = 1.45 ± 0.03 mas with proper motions of μ_α = 2.17 ± 0.03 mas yr⁻¹ and μ_δ = −0.68 ± 0.02 mas yr⁻¹. Based on a parallax zero-point correction zpt = −0.043908 mas from Lindegren et al. (2021), we obtain a distance from J0419 to the Sun, d = 671.3 ± 12.5 pc.

We use the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982) to determine the photometric period of J0419. To improve the accuracy of the period value, we use a time series as long as possible from 2005 to 2021 to calculate the Lomb–Scargle power spectrum. We reject the ASAS-SN data for the concerns that the long trend might interfere with the measurement results. The light curves of CRTS, TESS, and ZTF are combined after the flux normalization. We estimate the uncertainty of the period by using a bootstrap method (Efron et al. 1979) that we repeat for 10,000 measurements, randomly removing some points from each measurement. The Lomb–Scargle periodogram gives a period with an error of P_orb = 0.6071890(3) days. Note that for the ellipsoidal variation, the real orbital period is twice the peak period on the Lomb–Scargle power spectrum.

In order to determine the zero-point of ephemeris T₀, we use a three-term Fourier model (Morris & Naftilan 1993),

$$\begin{eqnarray}\begin{array}{rcl}f(t) & = & {a}_{0}\cos [\omega (t-{T}_{0})]+{a}_{1}\cos [2\omega (t-{T}_{0})]\\ & & +{a}_{2}\cos [3\omega (t-{T}_{0})],\end{array}\end{eqnarray} \tag{ 1 }$$

to fit the normalized light curve, where ω = 2π/P_orb, a₀, a₁, and a₂ are the parameters used to fit the light curve profile. We find the best-fitting parameters by minimizing the χ² statistics, which yields the zero-point of ephemeris of T₀ = 2,453,644.8439(5), where T₀ corresponds to the superior conjunction. We list P_orb and T₀ in Table 2. The folded light curves from different surveys or filters using P_orb and T₀ are shown in Figure 4.

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Table 2. Orbital and Stellar Parameters of J0419

Parameter	Unit	Value	Note
Astronomical parameters
R.A.	h:m:s (J2000)	04:19:20.07	R.A.
Decl.	d:m:s (J2000)	+07:25:45.4	Decl.
Gaia parallax	mas	1.45 ± 0.03	The parallax measured by Gaia EDR3
d (Gaia)	pc	671.3 ± 12.5	Distance derived from Gaia EDR3
μα	mas yr−1	2.17 ± 0.03	Proper motion in R.A. direction
μδ	mas yr−1	–0.68 ± 0.02	Proper motion in decl. direction
G-band magnitude	mag	14.70 ± 0.01	The G-band magnitude measured by Gaia EDR3
Orbital parameters
Porb	days	0.6071890(3)	Orbital period
T0	HJD	2,453,644.8439(5)	Ephemeris zero-point
K1	km s−1	216 ± 3	RV semi-amplitude of the visible star
γ	km s−1	86 ± 3	The systemic RV of J0419
f(M2)	M⊙	0.63 ± 0.03	Mass function of the compact star
Parameters of the pre-ELM
Teff	K	${5793}_{-133}^{+124}$	Effective temperature derived from SED fitting
Teff (spectral fit)	K	5776 ± 168	Effective temperature derived from spectral fitting
$\mathrm{log}g$ (spectral fit)	dex	3.95 ± 0.45	Surface gravity from spectral fitting
$\mathrm{log}g$	dex	3.90 ± 0.01	Surface gravity from SED fitting
Metallicity	[M/H]	–0.86 ± 0.24	Metallicity from spectral fitting
M1	M⊙	0.176 ± 0.014	Mass of the visible star
R1	R⊙	${0.782}_{-0.019}^{+0.021}$	Effective radius of the visible star
Lbol	L⊙	${0.62}_{-0.10}^{+0.11}$	Bolometric luminosity of the visible star
A(V)SED	mag	${0.34}_{-0.10}^{+0.07}$	The extinction value obtained from the SED fitting

Note. The astronomical parameters are from Gaia EDR3. The stellar parameters from SED fitting and spectral fitting are all listed in this table.

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The folded light curves (Figure 4) show ellipsoidal variability with amplitudes of about 0.3 mag, together with evidence of mass transfer (the obvious emission lines in the spectra), indicating that the visible star already fills its Roche lobe. We did not find any outburst event of this source in the 15 years of photometric monitoring, suggesting that the mass transfer rate is very low. The result is similar to El-Badry et al. (2021a, 2021b) and different to normal CVs.

The high-cadence TESS observation can be used to show the light-curve profile at each period. The light curve observed in 2020 (bottom panel of Figure 3) exhibits a typical ellipsoidal variation, while the light curve observed in 2018 (top panel of Figure 3) shows a long-term evolution of the peaks and valleys beyond the period. The timescale of the evolution seems to be several tens of days, which is consistent with the timescale of spot activity (Hussain 2002; Reinhold et al. 2013). We suspect that the long-term evolution of the light curve observed in 2018 may result from spot activity.

The folded light curves (except for the TESS data observed in 2020) show larger scatters than the measurement errors (see the ZTF light curves in Figure 4). The extra scatter could be for many reasons. The spot activity we mentioned above may bring about an additional scatter. The flux from the accretion disk may also contribute to the dispersion of the light curves, although the mass transfer rate of J0419 is very low. The temperature and $\mathrm{log}g$ (see Section 3.6) suggest that J0419 falls in the pre-ELM WD instability strip (Córsico et al. 2016; Wang et al. 2020), in which the pulsation can be driven by the κ–γ mechanism (Unno et al. 1989) and the "convective driving" mechanism (Brickhill 1991) acting in the H-ionization and He-ionization zones, and the scatter may be partly from the pulsation.

We obtain the template used to measure the RV of each single-epoch spectrum of J0419 through the Python package PyHammar, ¹³ and the best-fitting stellar type is G0. The RVs are then measured by using the cross-correlation function. The uncertainties of the RVs are estimated using the "flux randomization random subset sampling" (FR/RSS) method (Peterson et al. 1998). Only the spectral wavelength from 4910 to 5375 Å is used to measure the RVs to avoid disruption of Balmer and He i emission lines and telluric lines (see Figure 1). Because of the low resolution, the spectra observed by the 2.16 m telescope using the G4 grism are excluded from the RV measurements.

We fold the RVs in one phase using the period P_orb = 0.607189 days and T₀ = 2,453,644.8439 derived from Section 3.2 and display the result in Figure 5. Since the visible star fills its Roche lobe, the orbital circularization is effective, i.e., the binary moves along a circular orbit (Zahn 1977). Therefore, we fit the RVs with a circular orbit model following the equation

$$\begin{eqnarray}&&V(t)=-{K}_{1}\sin \left[\omega (t+{\rm{\Delta }}t)\right]+\gamma ,\end{eqnarray} \tag{ 2 }$$

where K₁ is the semi-amplitude of RVs of the visible star, ω = 2π/P_orb, γ is the systemic velocity of J0419 toward the Sun, and Δt represents the possible zero-point shift caused by the limited period accuracy when folding the RVs. The fitting results are K₁ = 216 ± 3 km s⁻¹, γ = 86 ± 3 km s⁻¹, and Δt = 12 ± 3 minutes. We display the RV model curve in Figure 5. The best-fitting RV model matches the measured RVs well.

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The mass function of a binary is defined as

$$\begin{eqnarray}&&f({M}_{2})=\displaystyle \frac{{M}_{2}^{3}{\sin }^{3}i}{{\left({M}_{1}+{M}_{2}\right)}^{2}}=\displaystyle \frac{{K}_{1}^{3}{P}_{\mathrm{orb}}}{2\pi G},\end{eqnarray} \tag{ 3 }$$

where M₁ and M₂ are the masses of the visible star and the compact star, respectively, K₁ is the semi-amplitude of the RVs of the visible star, P_orb is the orbital period, and i is the inclination angle of the binary to the observer. The mass function gives the minimum possible mass of the compact star. Using P_orb = 0.6071890(3) days and K₁ = 216 ± 3 km s⁻¹, we get the mass function of the compact star, f(M₂) = 0.63 ± 0.03 M_⊙.

Our spectra were observed in different telescopes or instruments with very different wavelength coverage and resolution. It is difficult to generate a mean spectrum by including all the spectra. Another problem is that most of the spectra have obvious emission lines that may interfere with the measurements of stellar parameters. Therefore, we only use the Lijiang spectrum to make the measurement; it has a good S/N (40.5) and the weakest emission lines.

We use the Python package The_Payne ¹⁴ to interpolate the model spectra. The_Payne is a spectral interpolation tool that is able to return a template spectrum when we provide a group of stellar parameters. Based on a neural net and spectral interpolation algorithm (Ting et al. 2019), The_Payne can interpolate the spectral grid with flexible labels efficiently.

We adopt the BOSZ grid of Kurucz model spectra (Bohlin et al. 2017) with five labels (T_eff, $\mathrm{log}g$, [M/H], [C/M], [α/M]) to train the template model. The stellar label intervals provided by BOSZ are ΔT_eff = 250 K, ${\rm{\Delta }}\mathrm{log}g=0.5$ dex, and Δ[Fe/H] = 0.25 dex. Before the training, we have reduced the resolution of the template to match the resolution of the observed spectrum, and the fluxes of the template spectra are also normalized using pseudo-continua that were generated by convolving the template spectra with a Gaussian kernel (σ_width = 50 Å).

We construct a likelihood function considering both χ² and a systematic error,

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}p(f| \lambda ,\mathrm{pms},{\sigma }_{\mathrm{sys}})\\ \ =\,-\displaystyle \frac{1}{2}\sum _{\lambda ={\lambda }_{0}}^{{\lambda }_{n}}\left[\displaystyle \frac{{\left({f}_{\lambda }-{\mathrm{model}}_{\lambda }\right)}^{2}}{{s}_{\lambda }^{2}}+\mathrm{ln}(2\pi {s}_{\lambda }^{2})\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{s}_{\lambda }^{2}={\sigma }_{\lambda }^{2}+{\sigma }_{\mathrm{sys}}^{2},\end{eqnarray} \tag{ 5 }$$

f represents the normalized spectrum, pms represents the stellar labels of the template spectrum, and σ_sys is the systematic error. The posteriors of pms and σ_sys are sampled by the software emcee (Foreman-Mackey et al. 2013) based on the Markov Chain Monte Carlo method.

The emcee sampling yields the result T_eff = 5776 K, $\mathrm{log}g=3.95$ dex, and [M/H] = −0.86 dex. Similar to Xiang et al. (2019) and El-Badry et al. (2021b), our fitting also underestimates the uncertainties of the stellar parameters with small ΔT_eff = 56 K, ${\rm{\Delta }}\mathrm{log}g=0.19$ dex, and Δ[M/H] = 0.08 dex. For simplicity, we directly use 3σ of the posterior sample as our fitting uncertainties. The fitting results are listed in Table 2.

Figure 6 shows the fitting result of the Lijiang spectrum. The gray spectrum in the top panel is the observation data, and the red spectrum is the best-fitting template. The residual spectrum of f_obs − f_model is shown in the bottom panel. The shaded region in Figure 6 is masked when we perform the fit. We find the model spectrum agrees well with the observed spectrum, except for Hβ, Hα, and several weak He i emission lines clearly showing on the residual spectrum. These emission lines are common in CV spectra (e.g., Sheets et al. 2007). The LAMOST spectra with higher resolution show clear double-peak emission lines, which suggests that these lines should be produced by the accretion disk rather than the visible star. The equivalent widths of the emission lines vary greatly in the different observations. If the EWs of the emission lines reflect the mass transfer rate in the binary, the EW variations indicate that the mass transfer process is intermittent. We discuss the emission lines more in Section 4.1.

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Our spectra show strong sodium "D" absorption lines beyond the template spectrum at wavelengths of 5890 Å and 5896 Å, which is similar to the result of El-Badry et al. (2021b). Sodium is thought to originate from the CNO processing that can only reach the surface of a star after most of its envelope has been stripped off. Therefore, sodium enhancement is generally observed in evolved CV donors. The presence of strong sodium absorption lines suggests that the visible star of J0419 is an evolved star and has lost most of its hydrogen envelope.

We use the broadband SED to constrain the stellar parameters and the extinction of J0419. In the SED fitting, the peak wavelength from UV to optical can be used to constrain the effective temperature, T_eff. The deviation of the SED slope from the Rayleigh–Jeans law in the mid-infrared band is generally considered to be caused by extinction, and can be used to estimate the extinction value (Majewski et al. 2011). The difference in color between the uband and another band represents the metal abundance, [Fe/H] (Huang et al. 2021). If we know the distance of a source, we can get its effective radius from the SED fitting.

We use the Python package astroARIADNE ¹⁵ to fit the SED of J0419. astroARIADNE is designed to fit broadband photometry automatically based on a list of stellar atmosphere models by using the nested sampling algorithm. The fitting parameters in astroARIADNE include T_eff, $\mathrm{log}g$, [Fe/H], distance, stellar radius, and extinction parameter A(V).

We collect multiband photometric data of J0419. Because Galaxy Evolution Explorer (Martin et al. 2005) and Swift (Gehrels et al. 2004) did not observe J0419, we do not have the UV points. The photometric data flag of J0419 in the Sloan Digital Sky Survey (SDSS; York & Adelman 2000) suggests that its magnitudes may have problems. We therefore use the APASS (B, g, V, r, and i bands; Henden et al. 2015) data instead. Our photometric data also include Pan-STARRS (g, r, i, z, and y bands; Chambers et al. 2016), the Two Micron All Sky Survey (2MASS; J, H, and K_s bands; Skrutskie et al. 2006), and the Wide-field Infrared Survey Explorer (WISE; W₁, W₂ bands; Wright et al. 2010).

For the APASS survey, the DR9 data include six detections, and DR10 includes four detections. We merge these two data sets using the inverse of the error as the weight. For the Pan-STARRS survey, the numbers of single-epoch detections in each band are 8, 10, 22, 16, and 13, respectively. Considering that J0419 shows significant variations, we add systematic uncertainties caused by random sampling of the light curves to the errors of the photometric data. The systematic uncertainties in each band are estimated as ${\sigma }_{\mathrm{sys}}=\mathrm{std}/\sqrt{N}$, where std is the mean standard deviation of the light curves of J0419 and N is the number of observations in a band. The 2MASS survey only observed J0419 once on 1999 December 7, 08:27:34.51, and the corresponding phase is 0.28. We assume that the IR-band light curve has the same amplitude of variation as the optical band to calculate the deviation between the observed magnitude and the mean magnitude and we add 0.1 mag to 2MASS data to correct the deviation. Considering the time interval between 2MASS and other surveys (10–20 years) and a possible trend in the light curve, we increase the magnitude uncertainties of 2MASS to 0.1 mag. All the photometric data are summarized in Table 3.

Table 3. The SED of J0419

Survey	Filter	Nobs	λeffective	AB mag	Vega mag	$\mathrm{log}\lambda {f}_{\lambda }$
			(μm)	(mag)	(mag)	$\mathrm{log}(\mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{cm}}^{-2})$
APASS	Johnson B	10	0.435		15.55 ± 0.07	−10.787 ± 0.026
	SDSS g	10	0.472	15.13 ± 0.05		−10.689 ± 0.020
	Johnson V	10	0.550		14.83 ± 0.07	−10.633 ± 0.027
	SDSS r	10	0.619	14.61 ± 0.07		−10.597 ± 0.026
	SDSS i	10	0.750	14.42 ± 0.07		−10.606 ± 0.028
Pan-STARSS	PS1 g	8	0.487	14.98 ± 0.03		−10.643 ± 0.014
	PS1 r	10	0.621	14.69 ± 0.03		−10.633 ± 0.012
	PS1 i	22	0.754	14.49 ± 0.02		−10.637 ± 0.008
	PS1 z	16	0.868	14.35 ± 0.02		−10.641 ± 0.009
	PS1 y	13	0.963	14.30 ± 0.03		−10.669 ± 0.010
2MASS	2MASS J	1	1.241		13.43 ± 0.1	−10.787 ± 0.040
	2MASS H	1	1.651		13.10 ± 0.1	−10.968 ± 0.040
	2MASS Ks	1	2.166		13.03 ± 0.1	−11.243 ± 0.040
WISE	W1	26	3.379		12.97 ± 0.02	−11.745 ± 0.010
	W2	26	4.629		12.91 ± 0.03	−12.115 ± 0.011

Note. The number of observations is shown in the third column, where the APASS data are a combination of DR9 and DR10. Both APASS and Pan-STARRS data have added additional systematic errors caused by sampling. The magnitudes of 2MASS have been increased by 0.1 mag to correct the phase offset, and the errors have been increased to 0.1 mag. The AB and Vega magnitude systems are displayed in separate columns.

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The visible star of J0419 has filled its Roche lobe (Section 3.3). In this case, the mean density is given by

$$\begin{eqnarray}&&\bar{\rho }=\displaystyle \frac{3{M}_{1}}{4\pi {R}_{1}^{3}}\cong 110{P}_{\mathrm{hr}}^{-2}\,{\rm{g}}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 6 }$$

where M₁ is the mass, R₁ is the equivalent radius of the Roche lobe-filling star, and the period, P_hr, is in units of hours (Frank et al. 2002). Equation (6) shows that the mean density of the Roche lobe depends only on the orbital period. Combining the radius derived from astroARIADNE fitting and the mean density of the Roche lobe, we can calculate the mass and log g of the visible star.

We fit the SED in the following way. First, we use the distance measured by Gaia EDR3 as the prior of the distance parameter, and other parameters are set to default values. Then we use the radius from the fitting result to calculate the mass and $\mathrm{log}g$ of the visible star (Equation (6)). Second, we update the $\mathrm{log}g\,$ prior by using our calculation and refit the SED. $\mathrm{log}g\,$ and [Fe/H] have only a little effect on the SED, so the fitting parameters converge quickly. The radius obtained from the SED fitting is ${R}_{1}={0.782}_{-0.019}^{+0.021}\,{R}_{\odot }$, and the corresponding mass of the visible star is M₁ = 0.176 ± 0.014 M_⊙. The effective temperature of the visible star is ${T}_{\mathrm{eff}}={5793}_{-133}^{+124}\,{\rm{K}}$. The bolometric luminosity derived from the SED fitting is ${L}_{{\rm{bol}}}=4\pi {R}_{1}^{2}\sigma {T}_{{\rm{eff}}}^{4}={0.62}_{-0.10}^{+0.11}\,{L}_{\odot }$. We summarize the SED fitting result in Table 2, and show the best-fit model in Figure 7.

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According to the mass function, f(M₂) = 0.63 M_⊙ and the mass of the visible star M₁ = 0.176 M_⊙, we obtain the minimum mass of the compact star, M₂ ≥ 0.9 M_⊙. If we assume the compact star is a WD and estimate its radius using the mass–radius relation of Bédard & Bergeron (2020), the corresponding WD radius is ${R}_{2,\max }\lt 0.01{R}_{\odot }$. Even if the temperature of the compact star is 20,000 K, its flux contribution to the total luminosity is less than 3% and is negligible. The flux contribution from the accretion disk is more complex and will be discussed in Section 4.2. Our analysis in that section demonstrates that the flux contribution in the optical band is dominated by the visible star.

J0419 exhibits ellipsoidal variability. The main parameters modulating the ellipsoidal light curves are the inclination angle i, the mass ratio q = M₂/M₁, the filling factor f_fill, the limb darkening factors, and the gravity darkening exponent β₁ (von Zeipel 1924). To estimate the inclination of the binary, we use Phoebe 2.3 ¹⁶ (Prša & Zwitter 2005; Prša et al. 2016; Conroy et al. 2020) to model the light curve and RV curve of J0419, and inversely solve the orbital parameters. Phoebe is an open-source software package, which is based on the Wilson–Devinney software (Wilson & Devinney 1971), for computing light and RV curves of binaries using a superior surface discretization algorithm. Main physical effects in a binary system have been considered, including eclipse, the distortion of stellar shape due to the Roche potential, radiative properties (atmosphere intensities, gravity darkening, limb darkening, mutual irradiation), and spots.

In fitting the light curve, we set the temperature of the donor to 5793 K as derived from the SED fitting (Section 3.6) with the Phoenix atmosphere model. We adopt the logarithmic limb darkening law and obtain the coefficient values self-consistently from the Phoebe atmosphere model. The gravity darkening law states that ${T}_{\mathrm{eff}}^{4}\propto {g}_{\mathrm{eff}}^{{\beta }_{1}}$, where β₁ is the gravity darkening exponent. It is generally assumed that, for stars in hydrostatic and radiative equilibrium (T_eff ≳ 8000 K), β₁ = 1 (von Zeipel 1924), and for stars with convective envelopes (T_eff ≲ 6300 K), β₁ = 0.32 (Lucy 1967). The theoretical dependence of β₁ upon T_eff is obtained by Claret & Bloemen (2011, see their Figure 2). J0419 appears to be in the temperature range where the transition from convection to equilibrium occurs, therefore we set β₁ as a free parameter and adopt a common normal distribution, ${\beta }_{1}\sim { \mathcal N }(0.32,0.1)$, as its prior, just similar to El-Badry et al. (2021b).

Because the visible star of J0419 is filling its Roche lobe, we set the model to be semidetached (f_fill = 1). We use an equivalent radius of ${R}_{1}\sim { \mathcal N }(0.78,0.02){R}_{\odot }$ obtained from the SED fitting as the prior of the radius parameter. The free parameters in our fit are i, q, β₁, γ, and sma, where sma is the semimajor axis of a binary orbit.

Except for data from the second TESS observation, the folded light curves show larger scatters than the measurement errors. We therefore fit only the second TESS data. To simplify the calculation of the model, we rebin the light curve of TESS to 40 points. The errors of the rebinned light curve include both the measurement errors and a systematic error that is estimated using a median filter method (Zhang et al. 2019).

We find that a pure ellipsoidal model cannot explain the observed data well. Indeed, the residuals between the observed and model fluxes depend upon the phases (see Figure 8). Thus, we add a spot to the model to compensate for the phase-dependent residuals. The spot component is defined with four parameters—relteff, radius, colatitude, and longitude. The relteff parameter is the ratio of the spot temperature to the local intrinsic temperature of the star. The radius parameter represents the angular radius of the spot. The remaining two parameters, colatitude and longitude, indicate the colatitude and longitude of the spot on the stellar surface, respectively. We only set relteff, radius, and longitude to be free parameters. The colatitude is fixed to be 90°. We use the Akaike information criterion (AIC) and Bayesian information criterion (BIC) to compare the pure ellipsoidal model with the one with a spot. For the model without a spot, the AIC and BIC of the best-fitting result are −105 and −85, respectively. For the model with a spot, the AIC and BIC of the best-fitting result are −282 and −255, respectively. Hence, adding a spot improves the fitting result greatly. Figure 8 illustrates the best-fit model with a spot, which is in good agreement with the observed light curve. The residuals between the TESS light curve and the model with a spot are much smaller than the variability amplitude. The light-curve residuals in Figure 8 show a weak periodic structure. Considering that the reduced χ² is 1.1 (close to 1.0), the above structure is statistically insignificant. The fitting yields an inclination angle of $i={66.5}_{-1.7}^{+1.4}$ degrees and a mass of the compact object of ${M}_{2}={1.09}_{-0.05}^{+0.05}\,{M}_{\odot }$. Figure 9 illustrates the distributions of the model parameters (see also Table 4). We stress that the inferred inclination angle would not significantly change if we omit the spot component.

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Table 4. Parameters from the Joint Fit of Light and RV Curves

Parameter	Value	Parameter	Value
i (deg)	${66.5}_{-1.7}^{+1.4}$	q (M2/M1)	${6.2}_{-0.5}^{+0.6}$
sma (R⊙)	${3.27}_{-0.05}^{+0.05}$	γ (km s−1)	${87.5}_{-3.6}^{+3.6}$
β1	${0.22}_{-0.04}^{+0.04}$	spot relteff	${1.018}_{-0.001}^{+0.001}$
Spot radius (deg)	${78}_{-9}^{+8}$	spot long. (deg)	${186}_{-1}^{+1}$
t0_rv (minutes)	${11.7}_{-2.8}^{+2.7}$	R1 (R⊙)	${0.771}_{-0.022}^{+0.020}$
M1(M⊙)	${0.177}_{-0.015}^{+0.014}$	M2 (M⊙)	${1.094}_{-0.049}^{+0.053}$
K1 (km s−1)	${214.8}_{-3.4}^{+3.4}$

Note. The spot long. is the longitude of the spot, where 0 means that the spot faces the companion of the binary. The t0_rv parameter accounts for the small phase offset.

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J0419 has had nine nights of observations, and multiple spectra have been taken on most of them. We normalize each single-exposure spectrum and measure the EW of the Hα emission line after subtracting the continuum component, where we generate the continuum template by using The_Payne with the stellar parameters obtained in Section 3.5. We obtain the EW of the Hα emission line by integrating each residual spectrum from 6520 to 6610 Å. The EWs of Hα are listed in Table 1. As can be seen from Figure 1, the EWs of the Hα emission line have changed greatly from night to night. But on the same night, or on two observation nights close together, the EWs change little. These show that the timescale of the variations of emission lines is from several days to tens of days. Some works found that the flickering timescale of the emission lines is similar to the that of the continuum in CVs (e.g., Ribeiro & Diaz 2009), ranging from minutes to hours. The mechanism causing the flicker could be condensations in the matter stream (Stockman & Sargent 1979), nonuniform mass accretion, or turbulence in the accretion disk (Elsworth & James 1982). The longer variability timescale of the emission lines of J0419 may be due to the low mass transfer rate.

According to the mass function (Equation (3)) and the mass of the visible star, the mass ratio is q = M₂/M₁ > 5.3. If we assume that the emission lines originate from the accretion disk, the RV semi-amplitude of the emission lines will be K_em < 41.3 km s⁻¹. The resolution and S/N of the J0419 spectra are not enough to measure the RVs of the emission lines. However, we find that the wavelength shift of the emission lines is much smaller than that of the continuum component, which disfavors the stellar origin of the emission lines.

The radiation from the companion star or disk for an accreting binary system will lead to an overestimation of the radius and mass of the donor. For J0419, as we mentioned in Section 3.6, due to the large mass of the compact star, its flux contribution to the total luminosity is negligible. The radiation from the accretion disk is more complicated. Most normal CVs have strong radiation from disks in the optical band. However, for evolved donors with higher temperatures, their mass transfer rate is very low (El-Badry et al. 2021a, 2021b) and the donors dominate the luminosities. The spectra of J0419 are also clearly dominated by the donor. We list the reasons below:

• 1.  
  the SED is well fitted by a pure stellar model;
• 2.  
  the template spectrum matches the observation spectra well;
• 3.  
  the light curves of J0419 show ellipsoidal variability, and no CV characteristics were found, such as violent variability on a short timescale or outburst events.

Most CVs have low-mass donors whose orbital periods are several hours (Section 1). The donors show characteristics similar to main-sequence stars with the same mass because they have the same chemical composition and structure. Unlike normal CVs, pre-ELMs in the CV stage have evolved and do not follow the donor sequence. Evolved donors generally have higher temperatures and possibly more bloated radii than normal CVs with similar donor masses.

We collect the mass and radius data of normal CVs from Patterson et al. (2005), and pre-ELM data from El-Badry et al. (2021a), and plot them in Figure 10. The red and black points in Figure 10 are normal CVs; the orange diamond points are pre-ELMs. The solid line is the empirical mass–radius relation of normal CVs from Knigge et al. (2011). J0419 is labeled by the red star for comparison. We can see that J0419 deviates hugely from the empirical mass–radius relation of normal CVs. Objects from El-Badry et al. (2021a) either fall on the mass–radius relation or deviate slightly from it, although their temperatures are significantly higher than those of normal CV donors with the same mass. These show that the visible star of J0419 is very bloated and more evolved than the objects of El-Badry et al. (2021a).

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Compared with normal CVs, the SED of J0419 is dominated by the donor star, and emission lines in the spectra are weaker. No outburst events were detected in 15 years of monitoring. We do not find random variability on a short timescale from hours to days in the high-cadence TESS light curves, which is different from the light curves of most normal CVs (Bruch 2021). The objects of El-Badry et al. (2021a) exhibit similar properties. These indicate that the mass transfer rate of pre-ELMs is very low compared with that of normal CVs.

Several works have reported the pre-ELMs. These objects are stripped stars with burning hydrogen envelopes that are more bloated and cooler than ELMs. Most of the reported pre-ELMs are found in EL CVns. Their companions are main-sequence A or F stars. We collect ELMs/pre-ELMs to compare their properties. The EL CVn-type stars are from Maxted et al. (2013, 2014a), Córsico et al. (2016), Corti et al. (2016), Gianninas et al. (2016), Zhang et al. (2017), Wang et al. (2020), and Lee et al. (2020), the pre-ELMs in DD binaries are from El-Badry et al. (2021a), and the ELMs are from Brown et al. (2020).

The pre-ELMs in EL CVns are hotter than pre-ELMs in DD binaries (see Figure 11), which should be due to the selection effect. Most of the reported EL CVns are detached binaries selected from eclipsing systems. Their radii have been shrinking after the detachment, accompanied by an increase in surface temperatures. The reported pre-ELMs (including J0419 and the sources of El-Badry et al. 2021a) in DD binaries all have distinct ellipsoidal variability. They are filling or close to filling their Roche lobes with lower temperatures. Hence, the pre-ELMs in EL CVns are at a later stage of evolution than the reported pre-ELMs in DD binaries.

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The mass and temperature of J0419 are similar to those of the sources of El-Badry et al. (2021a). They are all in the transition from mass transfer to detached. Compared with the sources in El-Badry et al. (2021a), J0419 has the smallest surface gravity (see Figure 11), which is due to its far longer orbital period. According to the evolutionary model of ELMs (Sun & Arras 2018; Li et al. 2019), pre-ELMs with longer initial periods will be more evolved before the mass transfer begins, resulting in smaller $\mathrm{log}g$ and longer periods. Long-period pre-ELMs like J0419 have rarely been reported in previous works.

Similar to El-Badry et al. (2021a, 2021b), we show the position of J0419 on the H-R diagram in Figure 12. For comparison, we plot the ELMs obtained from Brown et al. (2020) and the pre-ELM objects obtained from El-Badry et al. (2021a). We also show the CV sample using the data from Ritter & Kolb (2003) and Knigge (2006). The solid line in Figure 12 is the main sequence obtained from isochrones (Morton 2015) with an age of log(age/yr) = 8.3. Most CVs in Figure 12 fall on the main sequence. Both the ELMs and pre-ELMs are well below the main sequence, showing that they are evolved stripped stars.

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While the pre-ELMs of El-Badry et al. (2021a) deviate significantly from the main sequence, J0419 almost falls on it, which makes the method of selecting J0419 analogs via the H-R diagram ineffective. We can only search for J0419 analogs by combining the variations in features along with the SED fitting and spectroscopic information, which significantly limits their sample size. The multiple exposure strategy of LAMOST is beneficial in searching for such long-period pre-ELMs and binary systems consisting of a visible star and a compact star (Yi et al. 2019).

According to Sun & Arras (2018) and Li et al. (2019), the orbit of J0419 will continue to shrink with angular momentum loss due to magnetic braking until the convective envelope becomes too thin. After the orbital contraction ends, the accretion process will stop, and the radius of the visible star will begin to shrink with increasing temperature and $\mathrm{log}g$. The visible star will gradually evolve into an ELM WD.

The typical temperature of the transition from mass-transferring CVs to detached ELMs is about 6500 K (Sun & Arras 2018), which is related to the Kraft break (Kraft 1967). When the temperature is higher than this value, stars will lack the convective envelopes to generate a magnetic field, so that magnetic braking is no longer effective. The sample of El-Badry et al. (2021a) verifies this statement well. In their sample, emission lines only occur in sources with T_eff < 6600 K, and sources with higher temperatures have no emission lines found. J0419 with the donor temperature of T_eff = 5793 K seems also to obey the law.

The evolution of ELMs mainly depends on the initial mass of the visible star and the initial orbital period. The stars in WD + MS binaries with sufficiently long initial periods will ascend to the red giant branch when the mass transfer begins. The orbits of such systems will expand with the mass transfer (above the bifurcation period, see Figure 6 of Li et al. 2019). For donors just leaving the main sequence when mass transfer begins, their orbits will shrink due to magnetic braking. Both the mass and period of J0419 appear to be in between these two cases.

For WDs in binaries beyond the bifurcation period, there is a tight relationship between the core mass of a low-mass giant and the radius of its envelope, resulting in a good correlation between the period and the core mass at the termination of mass transfer (Rappaport et al. 1995). For systems below the bifurcation period (P_orb ≲ 16–22 hr, ${M}_{{\rm{core}}}\lesssim 0.18\,{M}_{\odot }$), the correlation between the radius and mass of the donors is unclear.

Figure 13 shows the mass–period distribution of helium stars. The long-period systems are radio pulsar binaries or EL CVn-type systems, and most of the short-period objects are ELM/pre-ELMs reported in recent years. J0419 is just at the junction of the upper and lower systems. With a mass of M₁ = 0.176 ± 0.014 M_⊙ and a period of P_orb = 0.607189 days, J0419 appears to follow the M_WD–P_orb relation. The proprieties of closing to the bifurcation period and ongoing mass transfer make J0419 a unique source to connect ELM/pre-ELM systems, wide binaries, and CVs. Systems with periods longer than 14 hr follow the M_WD–P_orb relation well. But for short-period targets, the correlation becomes diffuse. Sources with short periods and relatively large mass at the bottom of Figure 13 are thought to be generated through the common-envelope channel (Li et al. 2019).

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