Fundamental Parameters of the Eclipsing Binary DD CMa and Evidence for Mass Exchange

The first spectroscopic binary to be discovered was one of the stellar components of Mizar (ζ Ursae Majoris; Pickering 1890). These spectroscopic binaries, defined as those showing two sets of lines moving in antiphase in their spectra, have high astrophysical value, as the use of well-established methods including the analysis of the eclipse depth and its duration allows us to derive the whole set of orbital and stellar parameters. Some of the current studies of binary star populations show that the observed distributions of binary separation extend from 10⁻² to 10 ⁵ au. The explanation for this wide range of separations must lie in the details of the star formation process (Hubber & Whitworth 2005). Among this huge variety of binaries, there is a group whose separation is close enough to allow interaction through stellar wind accretion or Roche lobe overflow (Eggleton 2006). In close binary systems, mass transfer is always accompanied by an exchange of angular momentum. In cases where the transferred mass directly impacts the accretor, the excess kinetic energy of the gas stream is radiated in an optically thick hot spot or in a hot equatorial band (Ulrich & Burger 1976). In some interacting binaries, the existence of additional light sources as an accretion disk, gas stream, and shock regions complicates the task of obtaining stellar parameters from the light-curve analysis only. Combined spectroscopic and photometric information is needed to constrain the physical scenarios and help us to understand these stellar systems. On the other hand, the evolution of close binaries of intermediate mass is not yet well understood, as we lack knowledge of the details of the process of systemic mass loss and stellar mass exchange throughout the interaction stage (Sana et al. 2012; de Mink et al. 2014). In order to have a global picture of the binary evolution, understanding these evolutionary stages is essential, because it has been shown that binary interaction dominates the evolution of massive stars, and more than 70% of all massive stars will exchange mass with a companion during its lifetimes (Sana et al. 2012).

In this paper, we present a study of DD CMa, an interesting short orbital period eclipsing binary showing signatures of mass exchange, which could be in an advanced evolutionary stage. DD CMa was discovered as a variable star of Algol type after inspecting photographic plates in the Laboratoire d'Astronomie et de Géodésie de l' Université de Louvain (Deurinck 1949). According to SIMBAD, ¹⁰ this object is an Eclipsing Algol Semidetached (EA/SD) binary; it is also named ASAS ID 072409–1910.8 and is characterized by α₂₀₀₀ = 07:24:09, δ₂₀₀₀ = −19:10:48, V = 11.56 ± 0.11 mag, and B − V = 0.1 ± 0.19 mag. The orbital period has been successively reported as P_o = 2.0083807 ± 0.0000027 days, P_o = 2.0083 days, 2.008452 days, and 2.0084 ± 0.0001 days by Deurinck (1949), ASAS (Pojmanski 1997), ¹¹ the International Variable Star Index (VSX), ¹² and Rosales & Mennickent (2017). The distance based on the Gaia ¹³ DR2 parallax is 2632 [+309, −251] pc (Bailer-Jones et al. 2018). The object has not been studied in detail until now. In this study, we expect to determine for the first time the fundamental stellar and orbital parameters of this system and to contribute to the knowledge of its evolutionary stage.

In Section 2, we present a new photometric analysis of DD CMa. In Section 3, we present new spectroscopic data acquired by us along with our methods of data reduction. In the same section, we calculate the orbital parameters of the system and the physical parameters of the brighter star. In Section 4, we model the V-band light curve of the binary and derive a complete set of parameters including the system inclination, the stellar separation, and the absolute magnitudes, temperatures, masses, and radii for both stars. In Section 5, we fit the spectral energy distribution, obtaining the distance to the system. In Section 6, we discuss the Hα residual emission. In Section 7, we provide a discussion of our results. Finally, in Section 8, we summarize the main results of our investigation.

In this section, we present the analysis of the ASAS and ASAS-SN ¹⁴ (Shappee et al. 2014; Kochanek et al. 2017; Jayasinghe et al. 2019) photometric databases. For our analysis, we have considered only the best-quality data, excluding lower-quality magnitudes available in the database. These photometric time series present a skewed distribution with a tail toward faint magnitudes (Figure 1). Those observed skewed data in both distributions basically correspond to the magnitudes measured during the primary and secondary eclipses that range from approximately 11.6–12.6 mag. Then, if the primary and secondary eclipses are less deep, the skews would be much smaller. We also present the analysis of eclipse timings and survey infrared photometry.

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2.1. A Search for Periodicities

2.2. Analysis of the Main Eclipse Timing

In order to improve the accuracy of the orbital period, we selected the photometric data points with phases close to the primary minimum and performed an analysis of observed (O) minus calculated (C) eclipse times using a test period of 2.00844 days and following Sterken (2005). In this analysis, the O − C deviations can be represented as a function of the number of cycles with a straight line whose zero point and slope are the corrections needed for the (linear) ephemeris zero point and test period, respectively. We include 6 times of primary minimum studied by Deurinck (1949) and 16 new times of minima measured by us from the ASAS and ASAS-SN databases. The data set of the minima covers 15,289 cycles, i.e., 84 yr, and is presented in Table 1. Because some times of minima are published without errors, we use a simple least-squares fit for our analysis. Our result displayed in Figure 3 shows that the fit can be performed with a straight line, indicating a constant period. The new ephemeris is given by

$$\begin{eqnarray}&&{\mathrm{HJD}}_{\min }=2427537.311(7)+E\times 2\buildrel{\rm{d}}\over{.} 0084530(6).\end{eqnarray} \tag{ 1 }$$

Table 1. Times of Main Eclipse Minima Studied in This Paper. Data from Deurinck (1949) are in JD and Others in HJD

JD/HJD	Source	JD/HJD	Source
2427537.336	DEURINCK	2454470.68677	ASAS
2427539.334	DEURINCK	2454482.68797	ASAS
2427810.453	DEURINCK	2454757.84982	ASAS
2428240.276	DEURINCK	2457392.95175	ASAS-SN
2428246.29	DEURINCK	2457774.58377	ASAS-SN
2428507.367	DEURINCK	2457774.58384	ASAS-SN
2451889.76435	ASAS	2458053.74451	ASAS-SN
2452763.46515	ASAS	2458053.74487	ASAS-SN
2453030.61594	ASAS	2458055.74683	ASAS-SN
2453048.65734	ASAS	2458055.74717	ASAS-SN
2454205.53838	ASAS	2458244.55000	ASAS-SN

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Due to the much longer time baseline considered in this paper, we find that this period is more reliable than P_o = 2.0083807 ± 0.0000027 days, provided by Deurinck (1949). Actually, the same period given by Equation (1) is found when doing the O − C analysis using as test Deurinck's period. The ephemeris given in Equation (1) will be used for the photometric and spectroscopic analysis in the rest of the paper.

The light curve phased with the ephemerides given in Equation (1) is shown in Figure 2. Three points suggest a deeper main eclipse during the epochs covered by ASAS, but the larger scatter shown by ASAS data places doubt on the significance of this finding. Actually, in other epochs, we also observe—few—fainter-than-average ASAS magnitudes around the secondary eclipse and around phases 0.2 and 0.35, for instance.

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2.3. Multiband WISE and 2MASS Photometry

We performed a search for photometric data in the database of the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and find the mean colors of W1 − W2 = −0.006 ± 0.031 mag and W2 − W3 = 0.297 ± 0.095 mag. In addition, using data of the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), we obtained mean colors of J − H = 0.015 ± 0.034 mag and H − K = −0.025 ± 0.034 mag. We did not correct the colors for interstellar extinction because it should be insignificant at these wavelengths. We compared these infrared colors with those of systems with circumstellar envelopes, namely, Be stars—rapidly rotating B-type stars with circumstellar disks; double periodic variables (DPVs)—close binaries similar to β Lyrae showing super-orbital photometric cycles; and W Serpentis—close interacting binaries showing large variability. The comparison data used in this work are from Mennickent et al. (2016). We observe that DD CMa does not show a color excess in JHK photometry—it is located close to a single star of temperature 10,000 K in the color–color diagram—but in the WISE color–color diagram, the system is located in the area of objects with circumstellar envelopes showing an excess in W2 − W3. This can be considered as evidence for circumstellar material (see Figure 4).

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In this section, we describe our spectroscopic observations and provide results of our study of radial velocities and spectral decomposition.

3.1. Spectroscopic Observations

3.2. Radial Velocities

We find sets of lines of both stellar components in the spectra, i.e., the system is an SB2 binary. Therefore, we were able to trace the movement of each stellar component individually, by measuring their radial velocities (RVs), using different absorption lines. We used for the primary (more massive star) He i 4713, Hβ λ 4861.33, He i 4921.9, He i 5875, Hα λ 6562.817, and He i 6678.149, and for the secondary (less massive star), Hβ λ 4861.33 and Hα λ 6562.817.

In order to determine the line-profile center, peak strength, equivalent width, and radial velocity for these lines, we used the deblending routine included in the splot IRAF task. This allowed us to define the continuum region and the initial positions for each blended line, which were fitted with Gaussian-type functions of the adjustable position and broadening. The measured radial velocities for the secondary and primary components are given in Table 3.

Table 3. Radial Velocities and Their Errors for the Less Massive Star (Hα and Hβ) and the More Massive Star (He i 4713 Å, Hβ, and He i 4921.9 Å)

HJD'	Hα	Error	Hβ	Error	He i 4713	Error	Hβ	Error	He i 4921.9	Error
	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)	( km s−1)
8148.70016	241.3	5.8	234.6	5.8	−68.7	1.2	−71.8	1.2	−72.3	1.2
8149.64998	−237.0	5.8	−307.5	5.8	71.6	1.2	65.0	1.2	64.6	1.2
8162.67824	271.5	5.8	274.5	5.8	−71.8	1.2	−69.0	1.2	−74.3	1.2
8163.58789	−274.0	5.8	⋯	⋯	72.8	1.2	66.6	1.2	54.5	1.2
8176.57953	224.1	5.8	242.7	5.8	−60.6	1.2	−61.2	1.2	−70.8	1.2
8177.57591	−236.8	5.8	−228.3	5.8	⋯	⋯	⋯	1.2	61.0	1.2
8190.53966	178.8	5.8	178.6	5.8	−35.4	1.2	−48.5	1.2	−41.1	1.2
8191.52936	−175.8	5.8	−187.5	5.8	48.8	1.2	40.4	1.2	37.8	1.2
8204.50431	188.2	5.8	⋯	⋯	−22.3	1.2	−16.7	1.2	−26.8	1.2
8205.48963	−101.6	5.8	⋯	⋯	34.8	1.2	17.6	1.2	25.0	1.2

Note. We give HJD' = HJD–2,450,000.

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The RVs for the secondary were fitted using a sine function through a Marquart–Levenberg method (Marquardt 1963) of a nonlinear least-squares fit, wherein the solutions and respective errors are obtained through an iterative succession of local linearization. We obtained an amplitude K_c = 265.1 ± 7.3 km s⁻¹ and a zero point 0.5 ± 5.5 km s⁻¹. In the case of the primary, the RVs were fitted using a sine function with K_h = 70.9 ± 1.8 km s⁻¹ and zero point 0.0 ± 1.1 km s⁻¹. Both solutions assume a circular orbit (Figure 5).

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In order to obtain the orbital parameters of DD CMa, we used a public subroutine based on a genetic algorithm called pikaia (Charbonneau 1995). The code determines the single parameter set that minimizes the difference between the model's predictions and the data, producing a series of theoretical velocities and finding the best parameters through a minimization of the chi-square:

$$\begin{eqnarray}&&\begin{array}{l}{\chi }_{6}^{2}({P}_{{\rm{o}}},\tau ,\omega ,e,K,\gamma )=\displaystyle \frac{1}{N-6}\\ \ \times \ \displaystyle \sum _{{\rm{j}}=1}^{N}{\left(\displaystyle \frac{{V}_{{\rm{j}}}^{\mathrm{obs}}-V({t}_{{\rm{j}}};{P}_{{\rm{o}}},\tau ,\omega ,e,K,\gamma )}{{\sigma }_{{\rm{j}}}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 2 }$$

wherein (N − 6) corresponds to the number of degrees of freedom of the fit to six parameters, N is the number of observations, ${V}_{{\rm{j}}}^{\mathrm{obs}}$ is the observed radial velocity in the data set, and V(t_j; P_o, ω, e, K, γ) is the radial velocity at the time t_j. The other parameters such as P_o represent the orbital period, τ the time of passage through the periastron, ω the periastron longitude, e the orbital eccentricity, K the half-amplitude of the radial velocity, and γ the velocity of the system center of mass. The theoretical radial velocities are computed through Equation (2.45), given by Hilditch (2001):

$$\begin{eqnarray}&&V(t)=\gamma +K(\cos (\omega +\theta (t))+e\cos (\omega )).\end{eqnarray} \tag{ 3 }$$

In order to solve Equation (3), it is necessary to compute the true anomaly θ. This parameter is obtained after solving the equation relating the true and the eccentric anomaly:

$$\begin{eqnarray}&&\tan \left(\displaystyle \frac{\theta }{2}\right)=\sqrt{\displaystyle \frac{1+e}{1-e}}\tan \left(\displaystyle \frac{E}{2}\right),\end{eqnarray} \tag{ 4 }$$

but to obtain the value of the eccentric anomaly E, it is necessary to solve Equation (2.35) of Hilditch (2001):

$$\begin{eqnarray}&&E-e\sin (E)=\displaystyle \frac{2\pi }{{P}_{{\rm{o}}}}(t-\tau ).\end{eqnarray} \tag{ 5 }$$

Because Equation (5) has no analytic solution, it is necessary to use an iterative method to solve it. Then, after computing the solution, we can solve Equation (3) by obtaining the radial velocity solution. Following this procedure, we obtained the orbital parameters shown in Table 4.

Table 4. Orbital Parameters for the Stellar Components of DD CMa Obtained through the Minimization of χ² Given by Equation (4)

	Parameter	Best Value	Lower limit	Upper limit
	τ*	150.709	150.694	150.722
	e	0.04419	0.02000	0.07000
	ω(rad)	3.31212	3.27151	3.25651
Sh	Kh ( km s−1)	70.6	68.4	72.9
	γ ( km s−1)	0.4	−1.2	2.0
	N	28
	χ2	25.1513
	τ*	150.709	150.686	150.730
	e	0.00000	0.00000	0.02000
	ω(rad)	0.31410	0.24846	0.38346
Sc	Kc ( km s−1)	264.2	253.7	275.2
	γ ( km s−1)	0.0	−8.3	8.2
	N	17
	χ2	14.3765

Note. The value τ^* = τ − 2450000 is given and the maximum and minimum are one isophote of 1σ. S_h and S_c stand for solutions for the more massive and less massive star, respectively.

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In addition, we implemented a test for the significance of the eccentricity using the following equation (Lucy & Sweeney 1971):

$$\begin{eqnarray}&&{p}_{1}={\left(\displaystyle \frac{\sum {\left(O-C\right)}_{\mathrm{ecc}}^{2}}{\sum {\left(O-C\right)}_{\mathrm{circ}}^{2}}\right)}^{(n-m)/2}.\end{eqnarray} \tag{ 6 }$$

Lucy's test condition says that if p₁ < 0.05, an elliptic orbit is accepted, while if p₁ ≥ 0.05, a circular orbit is statistically preferred. The subindex ecc indicates the residuals of an eccentric fit, circ corresponds to the residuals of a circular orbit fit, n is the number of observations, and m the number of free parameters used in the eccentric fit. We obtained p₁ = 0.2226, i.e., DD CMa has an orbit compatible with a circular orbit (Figure 6). We notice consistent results for the orbital solutions obtained from the radial velocities of the primary and secondary stars. In the next sections, we use the orbital period derived from the photometric analysis and the orbital parameters of Table 4.

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3.3. Spectral Decomposition

3.4. Physical Parameters of the More Massive Star and Interstellar Reddening

We have compared the decomposed average spectrum of the hotter star with a grid of synthetic spectra constructed with the stellar spectral synthesis program spectrum, ¹⁵ which uses atmospheric models computed by ATLAS9 ¹⁶ (Castelli & Kurucz 2003) in local thermodynamic equilibrium. We looked for the best-fit synthetic spectrum by minimizing residuals among the observed and the theoretical spectra, considering the veiling factor. Briefly, this factor is a correction incorporated on the spectrum of the hotter star that takes into account the additional light that is contributed by the companion. This veiling produces an artificial weakness of the absorption lines that can be removed using the aforementioned factor.

The theoretical models were computed using two groups of effective temperatures: the first considers 9000 ≤ T_h ≤ 11,000 K with steps of 250 K and the second group considers 11,000 ≤ T_h ≤ 36,000 K with steps of 1000 K. This difference between the resolution of both groups matches the grid resolution. The surface gravity varies from 1.5 to 5.0 dex with steps of 0.5 dex, the macroturbulent velocity varies from 0 to 10 km s⁻¹ with steps of 1 km s⁻¹, the $v\sin i$ varies from 10 to 140 km s⁻¹ with steps of 10 km s⁻¹, and we considered two values for the microturbulent velocity of 0.0 and 2.0 km s⁻¹. The following parameters are kept fixed in the grid: the stellar metallicity M/H = 0 and the mixing length parameter l/H = 1.25. In addition, we constructed spectra with veiling factor η running 0.0–0.9 with step 0.1.

The implemented method converged successfully to a minimum chi-square at T_h = 20,000 ± 500 K, $\mathrm{log}{g}_{{\rm{h}}}=4.0\,\pm 0.25\,\mathrm{dex}$, v_mac = 1.0 ± 0.5 km s⁻¹, v_mic = 0.0, ${v}_{1{\rm{r}}}\sin i=80\,\pm 5\,\mathrm{km}\,{{\rm{s}}}^{-1},$ and η = 0.4 ± 0.05 with six degrees of freedom and the best value of ${\chi }_{6}^{2}=0.174$ (Figure 6, right, and Figure 7).

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In addition, we measured the equivalent width (EW) of the diffuse interstellar bands of every spectrum by fitting simple Gaussians to the 5780 and 5797 Å lines to determine the average reddening in the system's direction, which will be used later to determine the distance of the system. For that, we have used the formula given by Herbig (1993) computing an average reddening of E(B − V) = 0.46 ± 0.03 and an interstellar absorption A_V = 1.42 ± 0.09.

3.5. Mass Ratio, Roche Lobe Radius, and Structure

After exploring the spectrum of the hotter star, we noticed that it reveals emission in Hα and Hβ that are characteristic of interacting binaries of Algol type. Therefore, we assume a semidetached nature of the system, i.e., the less massive star fills its Roche lobe. From the radial velocity half-amplitudes, we get a system mass ratio q = K_h/K_c = 0.268 ± 0.01. On the other hand, we obtain the ratio between the effective Roche lobe radius of the less massive star R_L and the orbital separation a, using the following relation by Eggleton (1983):

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{\rm{L}}}}{a}=\displaystyle \frac{0.49{q}^{2/3}}{0.6{q}^{2/3}+\mathrm{ln}(1+{q}^{1/3})}.\end{eqnarray} \tag{ 7 }$$

Assuming R_c = R_L, the above equation gives us R_c/a =0.274 ± 0.008. Now, if a star fills its Roche lobe, its mean density $\bar{\rho }$ is directly related to its orbital period (e.g., Eggleton 1983, 2006), and using $\bar{{\rho }_{{\rm{c}}}}=3{M}_{{\rm{c}}}/4\pi {R}_{{\rm{c}}}^{3}$, M_c = Mq/(1 + q) along with Kepler's third law, it is possible to obtain

$$\begin{eqnarray}&&\bar{{\rho }_{{\rm{c}}}}=\displaystyle \frac{3q}{(1+q)}\displaystyle \frac{1}{{\left({R}_{2}/a\right)}^{3}}\displaystyle \frac{\pi }{{{GP}}^{2}}=0.048\pm 0.004\,{\rm{g}}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 8 }$$

Thus, the obtained value is close to that expected for a giant of type B9. The derivation of the spectral types of the binary stellar components is given in Section 7.

In order to extract the physical information hidden in the orbital light curve, we use an algorithm that fits the data solving the inverse problem (Djurasevic 1992; Djurašević 1996; Djurašević et al. 2013). Briefly, this program is based on the Roche model and the principles described by Wilson & Devinney (1971). The program determines the optimal stellar and physical parameters that best fit the observed and theoretical light curve through an iterative cycle of corrections, solving the inverse problem with a method based on the one described by Marquardt (1963). We assume synchronous rotation for both components. This is justified because the stars are quite close, so tidal forces are expected to have synchronized the stellar spins rapidly. In addition, q and T_h are fixed to the values determined in our previous spectroscopic study.

The results of the model of the ASAS V-band light curve are given in Table 5, first two columns. The binary consists of a hot star of mass M_h = 6.3 ± 0.1 M_⊙, temperature T_h = 20,000 ± 500 K, and radius R_h = 3.7 ± 0.1 R_⊙ while the most evolved star has a mass M_c = 1.7 ± 0.1 M_⊙, a temperature T_c = 11,320 ± 200 K, and radius R_c = 3.6 ± 0.1 R_⊙. Both stars are separated by 13.4 ± 0.2 R_⊙. The orbital inclination is 840 ± 03.

Table 5. Results of the Analysis of DD CMa ASAS (First Two Columns) and ASAS-SN (Last Two Columns) V-band Light Curve Obtained by Solving the Inverse Problem for the Roche Model

ASAS		ASAS-SN
Quantity		Quantity
n	448	n	562
Σ(O − C)2	0.7996	Σ(O − C)2	0.2230
σrms	0.0423	σrms	0.0199
i [°]	84.0 ± 0.3	i [°]	81.8 ± 0.2
Fh	0.584 ± 0.01	Fh	0.510 ± 0.009
Fc	1.000 ± 0.003	Fc	1.000 ± 0.002
Th [K]	20000	Th [K]	20000
Tc [K]	11320 ± 200	Tc [K]	11350 ± 100
Ωh	3.944	Ωh	4.478
Ωc	2.394	Ωc	2.394
Rh [D = 1]	0.271	Rh [D = 1]	0.237
Rc [D = 1]	0.253	Rc [D = 1]	0.253
${{ \mathcal M }}_{{\rm{h}}}[{{ \mathcal M }}_{\odot }]$	6.3 ± 0.1	${{ \mathcal M }}_{{\rm{h}}}[{{ \mathcal M }}_{\odot }]$	6.4 ± 0.1
${{ \mathcal M }}_{{\rm{c}}}[{{ \mathcal M }}_{\odot }]$	1.7 ± 0.1	${{ \mathcal M }}_{{\rm{c}}}[{{ \mathcal M }}_{\odot }]$	1.7 ± 0.1
${{ \mathcal R }}_{{\rm{h}}}[{R}_{\odot }]$	3.7 ± 0.1	${{ \mathcal R }}_{{\rm{h}}}[{R}_{\odot }]$	3.2 ± 0.1
${{ \mathcal R }}_{{\rm{c}}}[{R}_{\odot }]$	3.6 ± 0.1	${{ \mathcal R }}_{{\rm{c}}}[{R}_{\odot }]$	3.7 ± 0.1
$\mathrm{log}\ {g}_{{\rm{h}}}$	4.11 ± 0.02	$\mathrm{log}\ {g}_{{\rm{h}}}$	4.23 ± 0.02
$\mathrm{log}\ {g}_{{\rm{c}}}$	3.55 ± 0.02	$\mathrm{log}\ {g}_{{\rm{c}}}$	3.55 ± 0.02
${M}_{\mathrm{bol}}^{{\rm{h}}}$	−3.44 ± 0.1	${M}_{\mathrm{bol}}^{{\rm{h}}}$	−3.14 ± 0.1
${M}_{\mathrm{bol}}^{{\rm{c}}}$	−0.94 ± 0.1	${M}_{\mathrm{bol}}^{{\rm{c}}}$	−0.97 ± 0.1
aorb[R⊙]	13.4 ± 0.2	aorb[R⊙]	13.5 ± 0.2

Note. Fixed parameters: $q={{ \mathcal M }}_{{\rm{c}}}/{{ \mathcal M }}_{{\rm{h}}}=0.268$—mass ratio of the components; T_h = 20,000 K—temperature of the more massive hotter component (h); F_c = 1.0—filling factor for the critical Roche lobe of the less massive cooler component (c); f_h = 1.0, f_c = 1.00—nonsynchronous rotation coefficients of the hotter and colder star, respectively; β_h = 0.25, β_c = 0.25—gravity-darkening coefficients of the hotter and colder star; A_h = 1.0, A_c = 1.0—albedo coefficients of the hotter and colder star. Quantities: n—number of observations; ${\rm{\Sigma }}{\left(O-C\right)}^{2}$—final sum of squares of residuals between observed (LCO) and synthetic (LCC) light curves; σ_rms—rms of the residuals; i—orbit inclination (in arc degrees); F_h = R_h/R_zc —filling factor for the critical Roche lobe of the hotter, more massive star (ratio of the stellar polar radius to the critical Roche lobe radius along z-axis); T_c—temperature of the less massive (cooler) star; Ω_h,c—dimensionless surface potentials of the hotter and cooler star, respectively; R_h,c—polar radii of the components in units of the distance between their centers; L_h/(L_h + L_c)—luminosity (V-band) of the more massive, hotter component; ${{ \mathcal M }}_{{\rm{h}},{\rm{c}}}[{{ \mathcal M }}_{\odot }]$, ${{ \mathcal R }}_{{\rm{h}},{\rm{c}}}[{R}_{\odot }]$—stellar masses and mean radii of stars in solar units; $\mathrm{log}\ {g}_{{\rm{h}},{\rm{c}}}$—logarithm (base 10) of the system components' effective gravity; ${M}_{\mathrm{bol}}^{{\rm{h}},{\rm{c}}}$—absolute stellar bolometric magnitudes; a_orb [R_⊙]—orbital semimajor axis given in solar radius units.

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The results of the model of the ASAS-SN V-band light curve are given in Table 5, last two columns, and they are in general in close agreement with those obtained with ASAS data. We notice the following differences: ASAS-SN R_h = 3.2 ± 0.1 R_⊙, i.e., it is 0.5 ± 0.1 R_⊙ smaller than the value obtained with ASAS and the ASAS-SN orbital inclination of 810 ± 02, which has a variation of 22 ± 04 from the ASAS value. These variations are larger than the formal errors; however, the observed changes are probably influenced by the different qualities of the data (the ASAS light curve is noisier), and it is possible that some of the errors are underestimated. Geometrical views along with the observed and calculated light curves for the ASAS and ASAS-SN model are given in Figure 8. The interesting fact that the scatter of the residuals is larger on the primary eclipse might indicate that our simple model of two stars is not able to represent a more complex system, with the eventual addition of emitting/absorbing light sources, as expected in a mass-transferring system, as discussed in Section 7. At present, we have no information about the structure of the circumstellar matter or gas stream; hence, it is not possible to include these additional emission/absorption structures in the light-curve model. Therefore, we caution about a possible bias in the orbital and stellar parameters derived from the light-curve model because of the modeling limitations.

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We compiled the photometric fluxes available for DD CMa to build the spectral energy distribution (SED), extracting the information from the Vizier Photometric viewer ¹⁷ (Table 6). We selected single-star theoretical models from the Virtual Observatory SED Analyzer ¹⁸ (Bayo et al. 2008) to perform an SED fit determining the system distance. Specifically, we use those models with T_c = 11,000 K, $\mathrm{log}{g}_{{\rm{c}}}=4.0\,\mathrm{dex}$, T_h = 19,000 K, $\mathrm{log}{g}_{{\rm{h}}}=3.5\,\mathrm{dex}$. In addition, we considered E(B − V) = 0.458 as derived in Section 3.3. The best-fitting model is the one that minimizes the reduced χ², considering the composite flux described in the following equation (Fitzpatrick & Massa 2005):

$$\begin{eqnarray}&&{f}_{\lambda }={f}_{\lambda ,0}{10}^{-0.4E(B-V)[k(\lambda -V)+R(V)]},\end{eqnarray} \tag{ 9 }$$

where E(B − V) is the color excess, k(λ − V) ≡ E(λ − V)/E(B − V) is the normalized extinction curve, R(V) ≡ A(λ)/E(B − V) is the ratio of total extinction to reddening at V, and f_λ,0 corresponds to the sum of intrinsic surface fluxes of both stars at the wavelength λ:

$$\begin{eqnarray}&&{f}_{\lambda ,0}={\left(\displaystyle \frac{{R}_{{\rm{c}}}}{d}\right)}^{2}\left[{\left(\displaystyle \frac{{R}_{{\rm{h}}}}{{R}_{{\rm{c}}}}\right)}^{2}{f}_{{\rm{h}},\lambda }+{f}_{{\rm{c}},\lambda }\right],\end{eqnarray} \tag{ 10 }$$

where the parameters f_h,λ and f_c,λ represent the fluxes of the hotter and colder star, respectively, and the normalized extinction curve was calculated using the prescriptions described in Martin & Whittet (1990) and Cardelli et al. (1989). The best fit resulted in a distance d = 2300 ± 50 pc with a χ² = 0.462, close to the Gaia value mentioned in Section 1, namely, 2632 [+309 −251] pc (Figure 9).

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We have subtracted the theoretical contribution of the hotter and cooler stars from the average spectrum. While the parameters used for the hotter star are those calculated in Section 3.4, those for the secondary star arise from the light-curve model. We model the cooler star with an ATLAS9 atmospheric model with effective temperature T_c = 11,000 K, surface gravity $\mathrm{log}g=3.5\,\mathrm{dex}$, macroturbulent velocity v_mac = 0.0 km s⁻¹ according to Smalley (2004), a fixed microturbulent velocity v_mic = 0.0 km s⁻¹, $v\sin i=93.2\,\pm 3.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$, a mixing length parameter of strong convection l/H = 1.25, and a fixed stellar metallicity M/H = 0.0.

The spectra resulting from the subtraction of the stellar theoretical spectra reveal broad emission peaks in Hα and Hβ (Figure 10). A study of the Hα residual emission through the orbital cycle shows a complex and variable structure—a double, sometimes single, usually asymmetrical emission whose maximum intensity closely follows the velocity of the donor star (Figure 11).

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The temperature of the hotter star, radius, and mass are consistent with typical properties of main-sequence stars, while the cold star is clearly evolved, showing a larger radius and smaller mass than expected for main-sequence stars of the same temperature (Harmanec 1988). We determine the stellar spectral types B9 and B2.5 following the calibration of Harmanec (1988). Therefore, the system consists of an early B-type dwarf and a slightly evolved late B-type giant. We notice that the fractional radius of the hotter star R_h/a = 0.24 ± 0.01 and the mass ratio q = 0.268 place the system into the area of "direct-impact" semidetached binaries (Figure 12). This means that if mass transfer occurs due to Roche lobe overflow, the gas stream hits directly the more massive star.

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The line emission shoulders observed in Balmer lines and the clear emissions observed after removing the theoretical underlying spectra, along with the infrared color excess, indicate the presence of circumstellar matter. The complex emission profiles observed in Hα and Hβ residual spectra do not correspond to the typical double-peaked emission of an optically thin accretion disk. We notice that the broad emission profiles have high velocities in their wings, probably indicating an origin in a region of rapidly rotating material. The half-width at half-maximum of the Hα emission is ∼700 km s⁻¹. This velocity is compatible with the critical velocity of early B-type stars (Ekström et al. 2008). It is in principle possible that the impact of the stream in the stellar surface speed it up to near-critical velocity (Packet 1981). However, the $v\sin \,i$ value determined from the fit to the helium lines—80 km s⁻¹—suggests that the hotter star rotates almost synchronously with the orbit; the expected projected synchronous velocity is 92 km s⁻¹. We notice that other broadening mechanisms besides rotation could be acting in the Balmer lines. For instance, in Be star disks, Thompson scattering of Balmer line photons by free envelope electrons produces extended Hα emission wings (Dachs & Rohe 1990 and references therein).

Quite interesting is the behavior of the velocity of the Hα emission peak, which roughly follows the donor path. We conjecture that material escaping from the donor through the inner Lagrangian point in the form of a gas stream is responsible for part of this emission. The fact that part of the material is observed receding and the other part approaching the observer at almost all orbital phases covered by our spectra reveals the complex structure of the gaseous envelope. This view is consistent with the semidetached nature of the binary and its direct-impact condition; it is also possible that part of the emission comes from the impact point of the stream on the gainer.

We propose that DD CMa is a semidetached direct-impact binary consisting of two B-type stars. The cooler star overfills its Roche lobe and transfers matter onto the hotter star through an accretion gas stream that directly hits the star's surface. The observed emission in Hα and Hβ are probably in a gas stream between both stars, reflecting the process of mass exchange between the stars. The circumstellar matter is also revealed by the infrared color excess found in the WISE photometry.

The error in the orbital period = 0.0000006 days, given by Equation (2), might be interpreted as a possible drift of the orbital period between P_o − and P_o + in Δt = 18 yr. This implies a possible change in the orbital period of less than roughly 2_P /Δt = 6.6 × 10⁻⁸ day yr⁻¹. For conservative mass transfer, this imposes an upper limit for the mass transfer rate. The expected period change in the conservative case is (Huang 1963)

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{P}}_{o}}{{P}_{o}}=-3\dot{{{ \mathcal M }}_{{\rm{c}}}}\left(\displaystyle \frac{1}{{{ \mathcal M }}_{{\rm{c}}}}-\displaystyle \frac{1}{{{ \mathcal M }}_{{\rm{h}}}}\right).\end{eqnarray} \tag{ 11 }$$

Using the aforementioned numbers and the derived stellar masses, we determine $\dot{{{ \mathcal M }}_{{\rm{c}}}}\,\lt$ 2.55 × 10⁻⁸ M_⊙ yr⁻¹. We have not found evidence for a change in the orbital period, but in case it does change, the upper limit we have found imposes a restriction for the mass transfer rate, in the conservative case, reflected in the above figure.

When comparing the stellar luminosities and temperatures with those predicted by the PARSEC ¹⁹ isochrones for solar-composition stellar models (Bressan et al. 2012), we find that the hotter star is located near the main sequence, whereas the cooler star is slightly evolved. Both stars cannot fit the same age simultaneously, when considering single-star evolution models (Figure 13, top panel). However, the position of the stars in the luminosity–temperature diagram can be understood in terms of mass exchange in a close interacting binary. We searched for the best binary model among those described in Van Rensbergen & De Greve (2016) following the method described in Mennickent & Djurašević (2013), which consists of a χ² minimization of modeled and observed luminosities, masses, temperatures, radii, and periods.

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The best fit is obtained with a 2.026 day orbital period binary of initial masses 6 and 2.4 M_⊙ and initial orbital period 1.5 days, whose less massive star is filling its Roche lobe, igniting the hydrogen in its core. The binary has an age of 50 Myr, and the cooler star is currently transferring matter onto the more massive star at a rate $\dot{{{ \mathcal M }}_{{\rm{c}}}}$ = 1.1 × 10⁻⁶ M_⊙ yr⁻¹ (Figure 13; lower panel). As the grid of models is rather coarse—not including all possible initial parameters—and the model is simplified, because it does not consider the effect of circumstellar matter on the stellar luminosities and temperatures, we cannot expect a perfect match. In this sense, our result is qualitative rather quantitative. This explains the discrepancy among the mass transfer rates, for instance. We include the best-fit evolutionary track just to show that the current stellar and orbital parameters can in principle be reproduced by binary evolution considering mass exchange between the stellar components. In this view, the initially less massive star evolved to higher masses and luminosities following a track almost parallel to the main sequence, while acquiring mass from the initially more massive star. The initially more massive star followed a completely different track, decaying in mass, temperature, and luminosity. We observe that the above changes might explain the position of the binary components in the luminosity–temperature diagram, corroborating our general picture of mass exchange described in previous paragraphs.

In this work, we have presented a detailed spectroscopic and photometric study of the eclipsing binary DD CMa. This is the first spectroscopic study of this system and our main results are the following:

• 1.  
  We analyzed main eclipse times spanning 84 yr and find an improved orbital period of 2.0084530(6) days.
• 2.  
  From the analysis of the spectral energy distribution, we find a distance of 2300 pc with a formal error of 50 pc, close to the reported value based on the Gaia DR2, viz. 2632 [+309 −251] pc.
• 3.  
  From the analysis of the radial velocities, we found a circular orbit, q = 0.268 ± 0.01, K_h = 70.6 ± 1.2 km s⁻¹ and γ = 0.4 ± 0.6 km s⁻¹. In addition, its companion showed a semiamplitude K_c = 264.2 ± 5.4 km s⁻¹.
• 4.  
  We find Hα absorption flanked by weak emission shoulders. This finding, along with the clear detection of Balmer emission in residual spectra and the infrared excess observed in WISE photometry, suggests the presence of circumstellar emitting gas. The radial velocities of the residual emission suggest a partial origin in the gas stream. The above suggests that DD CMa is in a mass-exchange evolutionary stage.
• 5.  
  We solved the inverse problem to derive the parameters producing the best match between a theoretical light curve and the observed one. The best model of the V-band ASAS-SN light curve includes a slightly evolved late B-type star of M_c = 1.7 ± 0.1 M_⊙, T_c = 11,350 ± 100 K, and R_c = 3.7 ± 0.1 R_⊙. It also includes an early B-type star with M_h = 6.4 ± 0.1 M_⊙, T_h = 20,000 ± 500 K, and R_h = 3.2 ± 0.1 R_⊙. The inclination of the orbit is 818.
• 6.  
  When comparing the obtained orbital and stellar parameters with those of published evolutionary tracks for intermediate-mass binaries, we find that they can be reproduced considering the mass exchange occurring when the initially more massive star fills its Roche lobe as a result of its nuclear evolution.

We thank the anonymous referee, who provided useful insights on the first version of this paper. J.R., R.E.M., and D.S. acknowledge support by Fondecyt 1190621, Fondecyt 1201280, and the BASAL (CATA) PFB-06/2007. G.D. acknowledges the financial support of the Ministry of Education and Science of the Republic of Serbia through project 176004 "Stellar Physics" and the financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia through contract No. 451-03-68/2020-14/200002. M. Cabezas acknowledges support by Astronomical Institute of the Czech Academy of Sciences through the project RVO 67985815. I.A. and M. Curé acknowledge support from Fondecyt 1190485. I.A. is also grateful for the support from Fondecyt Iniciación 11190147.