TV UMi: A Shallowly Eclipsing Marginal-Contact Binary

Abstract

Using twenty sectors of TESS observations and the hitherto unutilized radial velocities from the David Dunlap Observatory survey, we fully characterize the close binary TV UMi. Its nearly sinusoidal light curves are well explained by a low-inclination, shallowly-eclipsing model in marginal contact, with a dark spot whose longitudinal migration is strongly correlated with the eclipse time variations. We derive the orbital parameters of the binary and determine the masses and radii of the components with a precision of a few percent. The estimated age and the position of TV UMi on the theoretical HR diagram indicate it’s a relatively young late-type contact binary of the W subtype.

1. Introduction

A large fraction (20–30%) of eclipsing binaries are contact systems [1,2], where the component stars are close enough to form a stable and long-lived co-rotating common envelope. Because the common envelope fills a Roche surface, the ratio of the component radii is proportional to the ratio of the component masses [3]; that is, the more massive star must necessarily be the larger. This unique property of contact binaries allows, in principle, the determination of the mass ratio from photometric light curves. Absolute parameters of the stars and the orbital parameters of the system can then be estimated even in the absence of velocity curves, which are necessary for full characterization of non-contact binaries.

In the era of large survey astronomy, photometric light curves are abundant, but velocity curves remain a rare commodity; only a small fraction (about two hundred) of contact binaries have been characterized using radial velocities as well as light curves [4]. In this modest sample, a major contribution was made by the David Dunlap Observatory (hereafter DDO) radial velocity survey [5], which remains incompletely utilized.

One of the stars observed in the DDO survey for which no combined photometric and spectroscopic solution has been published to date is TV UMi ($\mathrm{RA}={15}^{\mathrm{h}}{00}^{\mathrm{m}}{59}^{\mathrm{s}}.7$, $\mathrm{DEC}={73}^{\circ }{03}^{\prime }{11}^{″}.5$, ${\mathrm{V}}_{\mathrm{mag}}=8.9$, $P\approx 0.4\mathrm{d}$): a close binary with an almost sinusoidal light curve and a visual companion that complicates both photometric and spectroscopic measurements. The companion is itself an eccentric, spectroscopic binary with an orbital period of about 31 days [6], making TV UMi a quadruple system and all the more astrophysically interesting. The components of the wide system were resolved in 2018 with the separation of $0^{″}.3$ [7,8]. It has been matched with a ROSAT X-ray source by Haakonsen and Rutledge [9].

The optical variability of TV UMi was examined by Selam [10], who used the HIPPARCOS photometry to classify the system as a contact binary and obtain an estimate of its orbital parameters with the simplified light curve synthesis method of Rucinski [11]. The photometric mass ratio resulting from this analysis ($q_{PH}=0.1$) is much smaller than the spectroscopic value obtained by Pribulla et al. [6] ($q_{SP}=0.739$), which is not surprising given the very low inclination and the absence of obvious eclipses.

In this work, we use the abundant TESS [12] observations in combination with the DDO radial velocities to completely characterize TV UMi. The TESS data and their reduction are described in Section 2. The radial velocities are taken from Pribulla et al. [6], who measured them using the broadening function technique [13]. As our analysis shows, TV UMi is a shallowly eclipsing, marginal-contact W UMa binary of the W subtype with a dark spot of likely magnetic origin.

Apart from standard simultaneous modeling of the phase-binned light curve and the velocity curve (detailed in Section 3), we also model each individual orbital cycle present in the TESS observations and establish the correlation between longitudinal spot migration and eclipse time variations. Finally, we place the components of TV UMi on the HR diagram in Section 4, estimate its age and compare it to a selection of similar contact binaries.

2. TESS Light Curves

The TESS data were obtained from the MAST Portal1 using the lightkurve 2.52 Python package [14]. We selected the observations made at the two-minute cadence throughout 20 TESS sectors, making for about 323,000 individual observations over 1138 orbital cycles, of which 1027 have complete phase coverage.

Orbital phases were calculated with the ephemeris given in the next subsection. Data from different TESS sectors were stitched into a single light curve as follows: in every sector, the PDCSAP flux and its corresponding error were divided by the maximal value of a parabola fitted through the phase-folded light curve maximum between phases 0.24 and 0.26, resulting in normalized flux values. For the purpose of modeling, we phase-binned the resulting light curve to 1000 normal points.

Eclipse Times and the Updated Ephemeris

We measured the eclipse times from the TESS data by fitting a parabola through each minimum. Using these measurements and the initial period reported by Pribulla et al. [6], we updated the ephemeris for TV UMi as follows:

$$Mi{n}_I\left[HJD\right]=2458683.739173\left(37\right)+0.4155480\left(1\right)·E$$

(1)

The eclipse time measurements are given in

Table 1. Eclipse times from TESS observations.

Eclipse Time [HJD]	Error	Type
2,458,683.739173	3.7 × ${10}^{-4}$	1
2,458,683.945526	4.4 × ${10}^{-5}$	2
2,458,684.154674	1.5 × ${10}^{-5}$	1
2,458,684.361434	5.8 × ${10}^{-5}$	2
2,458,684.569953	1.0 × ${10}^{-6}$	1
2,458,684.776637	1.3 × ${10}^{-4}$	2
2,458,684.986067	6.8 × ${10}^{-5}$	1
2,458,685.192522	5.4 × ${10}^{-5}$	2
2,458,685.401022	3.0 × ${10}^{-6}$	1
2,458,685.607896	6.4 × ${10}^{-5}$	2
…	…	…

This table is available in its entirety as supplemental data available at https://doi.org/10.5281/zenodo.18411807 (accessed on 28 January 2026). Only a portion is shown here for guidance regarding its form and content.

.

Note that the radial velocities in Pribulla et al. [6] were phased with a different ephemeris, where the shallower eclipse was taken as the point of reference corresponding to phase 0. In the current work, the deeper minimum corresponds to phase 0. Thus, there is a phase shift of 0.5 between the radial velocities shown in Figure 3 of Pribulla et al. [6] and those shown in Figure 1 of the present work.

Table 2. Model parameters and derived quantities. The range pertains to the upper and lower limits of free parameters during model optimization.

Quantity	Value	Error	Range
$a_{orb}\left[{\mathrm{R}}_{\odot }\right]$	3.037	0.030	1–10
q	0.7411	0.0030	0.7–0.8
$\gamma \left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	−9.55	0.50	−30–30
$i{[}^{\circ }]$	48.28	0.59	30–60
$T_1\left[\mathrm{K}\right]$	6152	200
$T_2\left[\mathrm{K}\right]$	6336	365	5500–7500
$f_1$	1.0033	0.0029	1.0–1.1
${\mathit{\ell }}_3$	0.4405	0.0043	0.0–0.5
Phase Shift (LC)	0.4984	0.0013	0.4–0.6
Phase Shift (RV)	0.5486	0.0010	0.4–0.6
Dimensionless surface potential ${\mathrm{\Omega }}_{1,2}$	3.3063	0.0088
Degree of overcontact [%]	2.1	1.8
Spot on the Primary Star
${\displaystyle \frac{T_{spot}}{T_{star}}}$	0.9770	0.0037	0.9–1.0
Angular Radius ${[}^{\circ }]$	27.9	2.5	10–45
Longitude ${[}^{\circ }]$	150	35	0–360
Latitude ${[}^{\circ }]$	18	15	−90–90
Absolute Parameters
$M_1\left[{\mathrm{M}}_{\odot }\right]$	1.254	0.038
$M_2\left[{\mathrm{M}}_{\odot }\right]$	0.929	0.028
$R_1\left[{\mathrm{R}}_{\odot }\right]$	1.240	0.016
$R_2\left[{\mathrm{R}}_{\odot }\right]$	1.080	0.015
$L_1\left[{\mathrm{L}}_{\odot }\right]$	1.95	0.51
$L_2\left[{\mathrm{L}}_{\odot }\right]$	1.68	0.48
$log{g}_1$	4.3493	0.0028
$log{g}_2$	4.3393	0.0029
$K_1\left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	117.43	0.40
$K_2\left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	158.46	0.44

Table 2. Model parameters and derived quantities. The range pertains to the upper and lower limits of free parameters during model optimization.

Quantity	Value	Error	Range
$a_{orb}\left[{\mathrm{R}}_{\odot }\right]$	3.037	0.030	1–10
q	0.7411	0.0030	0.7–0.8
$\gamma \left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	−9.55	0.50	−30–30
$i{[}^{\circ }]$	48.28	0.59	30–60
$T_1\left[\mathrm{K}\right]$	6152	200
$T_2\left[\mathrm{K}\right]$	6336	365	5500–7500
$f_1$	1.0033	0.0029	1.0–1.1
${\mathit{\ell }}_3$	0.4405	0.0043	0.0–0.5
Phase Shift (LC)	0.4984	0.0013	0.4–0.6
Phase Shift (RV)	0.5486	0.0010	0.4–0.6
Dimensionless surface potential ${\mathrm{\Omega }}_{1,2}$	3.3063	0.0088
Degree of overcontact [%]	2.1	1.8
Spot on the Primary Star
${\displaystyle \frac{T_{spot}}{T_{star}}}$	0.9770	0.0037	0.9–1.0
Angular Radius ${[}^{\circ }]$	27.9	2.5	10–45
Longitude ${[}^{\circ }]$	150	35	0–360
Latitude ${[}^{\circ }]$	18	15	−90–90
Absolute Parameters
$M_1\left[{\mathrm{M}}_{\odot }\right]$	1.254	0.038
$M_2\left[{\mathrm{M}}_{\odot }\right]$	0.929	0.028
$R_1\left[{\mathrm{R}}_{\odot }\right]$	1.240	0.016
$R_2\left[{\mathrm{R}}_{\odot }\right]$	1.080	0.015
$L_1\left[{\mathrm{L}}_{\odot }\right]$	1.95	0.51
$L_2\left[{\mathrm{L}}_{\odot }\right]$	1.68	0.48
$log{g}_1$	4.3493	0.0028
$log{g}_2$	4.3393	0.0029
$K_1\left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	117.43	0.40
$K_2\left[{\displaystyle \frac{\mathrm{km}}{\mathrm{s}}}\right]$	158.46	0.44

3. Modeling

The analysis of the light and velocity curves was done with the modernized version of the program described in Djurasevic [15] and Djurasevic et al. [16], that calculates a numerical model of a close binary system where the stars are represented as Roche surfaces. The model outputs synthetic light curves by summing the flux contributions from elementary surfaces whose visibility at each phase of the orbital motion depends on the orbit inclination and the relatives sizes of the stars. The flux contributions are calculated in the blackbody approximation, taking in consideration various second-order effects, such as the limb-darkening, gravity darkening and mutual irradiation. The original code was extensively modified for automated computing of large numbers of models in parallel (see e.g., [17]), and expanded to allow the synthesis and fitting of velocity curves.

One or more spots may be present on each star. In this model, the spots are circular and parametrized with angular coordinates, radius and the temperature ratio of the affected elementary surfaces with and without the spot. The inclusion of spots in the models of eclipsing binaries is often necessary to account for the asymmetry of the light curve knows as the O’Connell effect [18], where one of the maxima is notably higher than the other, which implies surface brightness inhomogeneity. Spots may be hotter (bright) or cooler (dark) than the unaffected star surface.

In the case of TV UMi, the inclusion of one spot is sufficient to account for the light curve asymmetry. We place it on the cooler of the two components, as it is more likely to exhibit magnetic activity. Alternate models, with multiple spots and with a single spot on the hotter star, were considered and rejected because they did not produce superior fits to the observations.

We fit the light and velocity curves simultaneously. The following parameters are free during model optimization: the mass ratio (q); the orbital separation (a); the center of mass velocity ($\gamma$); the orbital inclination (i); the filling factor (the ratio of the star’s polar radius to the polar radius of the critical Roche lobe) of the primary star ($f_1$); the effective temperature of the secondary star ($T_2$); the third light, ${\mathit{\ell }}_3=F_3/(F_1+F_2+F_3)$; the phase shifts of the light curve and the velocity curve; and the spot parameters: the longitude ($\lambda$), latitude ($\phi$), angular size ($\sigma$) and the temperature contrast of affected elementary surfaces, $T_{spot}/T_{star}$. The parameter space of spot locations, sizes and temperatures was explored using additional model grids.

The temperature of the primary star was fixed to the value from the tabulations in Eker et al. [19] for a star of spectral type F8 ($T_1=6151.8 K$), as classified by Pribulla et al. [6]. The gravity-darkening exponents and the albedos were fixed to the theoretical values appropriate for stars with convective envelopes: ${\beta }_{1,2}=0.08$ and $A_{1,2}=0.5$ [20,21,22]. The limb-darkening was calculated using a four-term law with filter-specific coefficients from Claret [23]. We consider the rotation of the components synchronized with the orbital motion.

The final model parameters obtained by fitting the phase-binned light curve and the velocity curves are given in Table 2. The errors are the standard deviations from the ensemble of models for individual cycles (discussed in detail in the next section), with the exception of $T_1$, for which we assume an uncertainty of 200 K roughly corresponding to the range between spectral classes F7 and G0 in the tabulations of Eker et al. [19]. The observations and the model are shown in Figure 1 and Figure 2.

Models of Individual Orbital Cycles

Apart from modeling the phase-binned light curve of TV UMi, we also modeled each of the 1027 orbital cycles with complete phase coverage present in the TESS data as separate light curves, using the phase-binned model as the starting point, but otherwise in the same manner (the same parameters were allowed to vary or were fixed). This procedure provides an ensemble of models that can be used to estimate the precision of the model parameters and the derived quantities, similarly to the bootstrap method. At the same time, it provides a time series of model parameters that can be analyzed for evidence of significant variation.

As can be seen from Table 2, most of the model parameters change only slightly from cycle to cycle, except for the spot parameters, whose locations vary over a wide range, and the secondary temperature. This can also be seen in the corner plot constructed from the ensemble of individual cycle models, given in Figure 3, which shows that the parameters determining the configuration of the system are very well constrained.

Plotting the spot longitude as a function of time together with the O-C residuals (OCR) in Figure 4 demonstrates a remarkable agreement between the longitude and primary eclipse OCR. Meanwhile, the secondary eclipse OCR appear anti-correlated with both the spot longitude and the primary eclipse OCR. This is an expected observational signature of spot migration according to Tran et al. [24], who found anti-correlated primary and secondary OCR in a sample of contact binaries observed by the Kepler space telescope and explained it with a simplified model of a spotted binary. This phenomenon was further analyzed and confirmed in our previous works [17,25].

The apparent (anti-)correlation with the (secondary) primary OCR in the case of TV UMi is further demonstrated in Figure 5.

4. Discussion

The interpretation of the results presented above is complicated by the fact that TV UMi is a quadruple system. However, the analysis of the contact binary is sound regardless of the signature of the non-eclipsing member in the observations:

The radial velocities we use were measured with the broadening function technique, where the non-eclipsing member was clearly distinguished from the components of the contact binary [6], and therefore, does not affect the radial velocities and the orbital parameters of the contact pair.

The contamination of the photometric observations by the non-eclipsing member is accounted for by the third light parameter (${\mathit{\ell }}_3$), which represents constant additional flux. Our estimate of the third light, ${\mathit{\ell }}_3\approx 0.44$, is in excellent agreement with the spectroscopic estimate from Pribulla et al. [6] ($L_{34}/(L_1+L_2)\approx 0.9$, which would correspond to ${\mathit{\ell }}_3\approx 0.47$). As the companion binary is non-eclipsing and has a period two orders of magnitude longer than the contact binary, its contribution to the light curve of TV UMi could be considered as constant; but even if it has some variability of its own, that too would be accounted for by the third light in the individual cycle models. While we do not find significant variation of the third light across cycles, we cannot rule out that such variability exists.

Based on the arguments above, we conclude that the characterization of the contact binary is reliable despite its membership in a multiple system.

Although the light curve of TV UMi appears to be entirely sinusoidal, the best-fitting model suggests a low-inclination system with shallow eclipses and spots causing the O’Connell effect.

Throughout this work, we observe the convention that the larger and more massive star is denoted with the index 1 and referred to as the primary, and the smaller, less massive star is denoted with the index 2 and referred to as the secondary. The mass ratio is defined as $q=M_2/M_1$ and is always less than unity.

According to these definitions and the ephemeris given in Equation (1), in TV UMi, the secondary star is eclipsed in the deeper minimum at phase 0. This situation corresponds to the W subtype of W UMa binaries defined by Binnendijk [26], where the smaller, less massive secondary appears to be hotter than the larger, more massive primary.

With the mass ratio of $q=0.74$, TV UMi belongs to the H subtype of W UMa stars defined by Csizmadia and Klagyivik [27]. Latković et al. [28] found that H stars have lower temperatures, shorter periods, and smaller fillouts than non-H stars. Compared to the summary statistics given in the same paper, TV UMi has a temperature higher than average, a period that’s also slightly longer than the average, but its fillout is smaller than 90% of the sample presented by Latković et al. [28] (hereafter, WUMaCat).

Following the method of age calculation for W UMa binaries developed by Yildiz and Doğan [29] and Yıldız [30], as detailed in Appendix A of Latković et al. [28], we estimate the age of TV UMi at $\tau =2.02\pm 0.82$ Gy, which makes it younger than 90% stars with age estimates in WUMaCat.

Figure 6 shows TV UMi on the theoretical Hertzsprung-Russell diagram (HRD) with a selection of similar systems found in WUMaCat (listed in

Table 3. Contact binaries similar to TV UMi in order of increasing period. All the quantities are taken from Latković et al. [28]. The masses, radii and luminosities are in solar units.

Name	$\mathit{P}\left[\mathbf{d}\right]$	q	$\mathit{f}[\%]$	$\mathit{i}{[}^{\circ }]$	${\mathit{T}}_1\left[\mathbf{K}\right]$	${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{R}}_1$	${\mathit{R}}_2$	${\mathit{L}}_1$	${\mathit{L}}_2$	$\mathit{\tau }\left[\mathbf{Gy}\right]$	Ref.
NSVS 2700153	0.228456	0.77	0.07	47.8	4785	0.65	0.50	0.67	0.60	0.22	0.16		[34]
V336 TrA	0.266768	0.72	0.16	80.8	4840	0.91	0.65	0.85	0.73	0.35	0.30		[35]
44 Boo *	0.267812	0.56	0.00	72.8	5300	0.98	0.55	0.87	0.66	0.51	0.24		[36]
V704 Mon	0.278000	0.85	0.35	66.5	5399	1.06	0.90	0.87	0.81	0.52	0.46		[37]
XY Leo *	0.284101	0.73	0.21	68.6	4742	0.85	0.62	0.88	0.76	0.35	0.37	5.04	[38]
V1139 Cas	0.297103	0.63	0.04	68.3	5933	1.05	0.66	0.94	0.76	0.98	0.79	3.43	[39]
GZ And *	0.305003	0.53	0.12	87.0	5810	1.12	0.59	1.00	0.74	1.03	0.45		[40]
V599 Aur	0.316536	0.62	0.04	60.3	5230	1.07	0.66	0.99	0.80	0.66	0.54	4.27	[41]
TW Cet *	0.316851	0.75	0.06	81.2	5753	0.96	0.72	0.95	0.83	0.89	0.73	2.85	[38]
SW Lac *	0.320716	0.79	0.35	79.8	5515	1.24	0.96	1.09	0.98	0.97	0.95		[42]
V369 Cep	0.328192	0.53	0.17	74.7	5088	0.93	0.49	1.01	0.76	0.77	0.61	5.51	[43]
GW Leo	0.336149	0.88	0.03	54.1	5832	1.08	0.95	1.00	0.94	1.04	1.27		[44]
V692 Pup	0.339999	1.00	0.23	62.1	5600	1.06	1.06	1.00	1.00	0.88	0.76		[45]
V479 Lac	0.345759	0.80	0.03	80.4	5652	1.00	0.80	1.01	0.91	0.94	0.80		[46]
TIC 428534573	0.376532	0.42	0.25	60.3	6494	0.65	0.28	1.01	0.69	1.63	0.77	3.20	[47]
TX Cnc *	0.382883	0.45	0.17	62.2	6121	1.32	0.60	1.27	0.89	2.03	1.09	4.06	[38]
NSVS 908513	0.399592	0.71	0.15	75.2	5923	0.61	0.43	0.98	0.84	1.03	0.65	6.55	[48]
BG Vul	0.403236	0.88	0.45	74.5	5868	1.01	0.89	1.23	1.17	1.63	1.15		[49]
TIC 185058337	0.415110	0.50	0.48	68.4	4805	1.20	0.47	1.22	0.81	1.62	0.72	4.60	[50]
V899 Her *	0.421172	0.57	0.24	68.7	5700	2.10	1.19	1.57	1.22	2.32	1.40		[51]
V542 Aur	0.422480	0.78	0.12	62.6	6386	1.61	1.26	1.37	1.23	1.75	1.42		[52]
AV Pup	0.435014	0.90	0.10	80.8	6255	1.27	1.14	1.29	1.23	2.29	1.94		[53]
AA UMa *	0.468135	0.55	0.14	79.6	5920	1.61	0.89	1.53	1.17	2.57	1.55	2.75	[54]
DN Cam *	0.498309	0.44	0.49	73.1	6530	1.85	0.82	1.77	1.22	5.14	2.71	2.43	[40]

* Systems with a spectroscopically determined mass ratio.

). The figure also shows the zero-age main sequence (ZAMS), the terminal-age main sequence (TAMS), and selected isochrones from the MIST grid of evolutionary models [31,32,33] for rotating stars (rot=vvcrit0.4) and solar metallicity (feh=p0.00). The isochrone corresponding to the age of the system as estimated above ($log\left(\tau \right[yr\left]\right)=9.3$) lies below the primary of TV UMi, which is instead best described by the isochrone corresponding to $log\left(\tau \right[yr\left]\right)=9.6$, or $\tau \approx 4.0$ Gy. The discrepancy is significant, even when considering the upper limit of the age estimate from the absolute parameters, of $\tau \approx 2.84$ Gy. However, this is not surprising, because the MIST models were made for single stars, and as a contact binary, TV UMi can be expected to deviate from single-star evolution.

We assumed that a dark spot on the larger (and cooler) star is responsible for the visible O’Connell effect. The usual interpretation of such spots in close binaries is that they are of magnetic origin, because the tidal spin-orbit synchronization compels the stars to rotate much faster than they would if they were single, and fast rotation produces stronger magnetic fields. Our cycle-by-cycle analysis showed that the spot migrates, and that its longitudinal motion is strongly correlated with the eclipse timing variations.

5. Conclusions

The combined analysis of TESS light curves and velocity curves from the DDO survey shows that TV UMi is a young ($\tau =2.02$ Gy) shallowly eclipsing ($i\approx {48}^{\circ })$ marginal contact binary (degree of contact of 2.1%) of the W subtype, with a large mass ratio ($q=0.74$), an F7-G1 primary and a G6-K0 secondary. The masses and radii of the components are determined with a precision of a few percent. Eclipse time variations are found to be strongly correlated with the longitudinal migration of a large, dark spot on the primary star. Although we could detect no period change over the five years of TESS observations, continued monitoring of eclipse times might reveal whether the system is evolving towards deeper contact or a detached configuration, and help characterize the orbital motion of the quadruple system it belongs to.