Absolute-magnitude Calibration for W UMa-type Systems Based on Gaia Data^*

The motivation for this study is the availability of newly measured, high-precision parallaxes from the Gaia mission as well as standardized UBV photometry for W UMa binaries; the details of the photometric database are described in the next sub-section. Cross-matching of the catalogs led to the selection of EW binaries from Rucinski & Duerbeck (1997), Pojmanski et al. (2005), Gettel et al. (2006), and Terrell et al. (2012), provided that they have corresponding Gaia parallaxes. After removing the poorer data points, the final sample consists of 318 EW binaries with absolute-magnitude errors $\sigma ({M}_{V})\,\lt 0.5$ mag.³ We used the standard formula for absolute magnitudes: ${M}_{V}=5\mathrm{log}(\pi )+{V}_{\max }-3.1\times {E}_{(B-V)}-10$ with $\sigma ({M}_{V})$ estimated through propagation of errors contributed by the parallax and the photometry.

The parallax determination via TGAS is a substantial improvement to that of the Hipparcos sample (HIP; Rucinski & Duerbeck 1997). The median parallax error for the HIP sample was 1.30 mas (milliarcseconds) whereas it is 0.28 mas for the TGAS sample. Similarly, the median relative error $\sigma (\pi )/\pi$ has been reduced from 0.13 to 0.038. Because the magnitude scale M_V is logarithmic, the latter value is of greater importance. From the HIP sample of 40 systems, 30 have data in the TGAS database.⁴ This rather high fraction of HIP systems absent from the TGAS database may indicate frequent occurrences of perturbing companions to these binaries (Rucinski et al. 2007), which would have introduced difficulties in the astrometric solutions.

The HIP and TGAS results for common systems appear to be in mutual agreement as shown in Figure 1; the graph presents the properties of the input parallaxes for both data sets. The TGAS relative errors $\sigma (\pi )/\pi$ are approximately five times smaller than those of HIP, which results in a five-fold increase in accessible distances $d=1000/\pi$ (with π in mas or milliarcseconds units) and thus a larger volume by about 125 times. The apparent scaling of the minimum relative errors with the distance (see Figure 1) implies a typical minimum parallax error of about 0.22 mas for the TGAS sample.

Zoom In Zoom Out Reset image size

As pointed by Lutz & Kelker (1973), relative errors of $\sigma (\pi )/\pi \gt 0.05$ may result in biased determinations of distances. As seen in Figure 1, this may have happened to a large fraction of the TGAS sample at distances greater than about 200 pc. Thus, while the new sample includes more distant binaries than the HIP sample, the quality of the parallax data in terms of $\sigma (\pi )/\pi$ for distant objects in both samples is comparable and moderate. However, thanks to its much larger sample volume, the TGAS data set includes more stars within a wider range of parameters. We made sure that the binaries observed with lower accuracy contributed less to the final M_V calibration solutions using a weighting scheme with weights $\propto \,1/{{\sigma }^{2}}_{({M}_{V})}$.

The nearest system 44 Boo (i Boo, HIP 73695) does not have a Gaia parallax but its Hipparcos measurement has been included in the current sample because of the importance of its small relative parallax error: ${\pi }_{\mathrm{HIP}}=79.95\pm 1.56$ mas. The value $(B-V)=0.94\pm 0.02$ has been taken from Fabricius & Makarov (2000) while the value ${V}_{\max }=5.87\pm 0.02$ has been taken from Hill et al. (1989). This object is the only one in our sample with an HIP number in place of a TYC number.

In selecting the photometric data, ${V}_{\max }$ and B − V, we decided not to use scattered photometric information published in numerous, single-object papers. Recently, discoveries have become more organized due to large-scale, systematic photometric surveys. We intentionally chose precision over potentially improved accuracy by preferentially using these survey data.

The sky has been uniformly surveyed for stellar variability with photometric errors $\lt 0.01$ mag (enabling discoveries of variability with amplitudes of about 0.05 mag) by the Hipparcos satellite down to $V\lt 7.5$ (Rucinski 2002). Tycho-2 magnitude and color data (European Space Agency 1997) are reliable to about $V\simeq 10$ (see Figure 2 in Raghavan et al. 2010), but the survey may be incomplete in terms of variability detections. Calibrated photometry for large parts of the sky reaching down to $V\lt 15$ has been undertaken by a few wide-sky surveys. In addition to the Hipparcos and the Tycho-2 catalogs, data for ${V}_{\max }$ have been taken from three other sources that we judged to be large enough to provide sufficient uniformity: Gettel et al. (2006), Pojmanski et al. (2005),⁵ and Terrell et al. (2012). We re-evaluated the errors for ${V}_{\max }$ and assumed that they are not smaller than 0.01 mag due to residual calibration errors.

Uniformly calibrated $B-V$ data for EW binaries have been taken primarily from Terrell et al. (2012), which analyzed 606 binaries with ${V}_{\max }\lt 14$. No directly measured colors exist for about a third of the binaries with measured parallaxes and readily available ${V}_{\max }$ values. For such cases, we used the infrared indices $J-H$ and $H-K$ from the 2MASS survey (Skrutskie et al. 2006) to estimate the $B-V$ colors. While this may be a risky conversion, as the color indices for EW binaries do not necessarily obey the MS relations, this approach assures a high degree of uniformity within our data. Using the extensive tabulation of Pecaut & Mamajek (2013),⁶ the agreement between the independent color transformations from $J-H$ and $H-K$ to $B-V$ gave estimates for the uncertainties of the derived $B-V$ values.

The interstellar extinction (A_V) and reddening (${E}_{B-V}$) corrections must be included, because their neglect would create systematic biases to ${V}_{\max }$ and $B-V$. For EW binaries, these corrections are expected to be relatively small for distances of tens to a hundred parsecs but become larger and very uncertain beyond a few hundred parsecs due to the very non-uniform distribution of interstellar matter in the local Galactic Disk.

For uniform treatment of the interstellar corrections, we utilized the distances derived from the new Gaia parallaxes and assumed an exponential density decay with galactic height to approximate the interstellar matter distribution. For this, we closely followed Nataf et al. (2013) but modified the paper's approach (developed for fields at the center of the Galaxy) in the following way: instead of assuming that interstellar extinction "hits the wall" at the Galactic Bulge, we assumed that the maximum values are determined by integrating to infinity as in Schlegel et al. (1998). In general, for any distance R and galactic latitude b:

$$\begin{eqnarray}&&{A}_{V}={\int }_{0}^{R}\exp (-r| \sin (b)| /H){dr},\end{eqnarray} \tag{ 1 }$$

with the scale height $H=164$ pc (Nataf et al. 2013). Here, the upper limit can be the actual distance to the binary ($R=d$) or it can be infinity ($R=\infty$), with the latter giving A_V^max as in Schlegel et al. (1998). From the ratio of the definite integrals to both values of R:

$$\begin{eqnarray}&&{A}_{V}={A}_{V}^{\max }\times (1-\exp (-d| \sin (b)| /H)).\end{eqnarray} \tag{ 2 }$$

Finally, we assumed that the relation between the reddening and absorption corrections is the standard law, ${A}_{V}=3.1{E}_{B-V}$.

All data used in this paper are listed in Table 1. The table is arranged by increasing orbital period.

Table 1.  Observational Data

Name	TYC	P (days)	${V}_{\max }$	$\sigma ({V}_{\max })$	$B-V$	$\sigma (B-V)$	${E}_{B-V}$	$\sigma (E(B-V))$	π (mas)	$\sigma (\pi )$	MV	$\sigma ({M}_{V})$
CC Com	1986-2106-1	0.220686	11.30	0.09	1.20	0.03	0.010	0.001	14.22	0.54	7.03	0.12
V523 Cas	3257-167-1	0.233692	10.62	0.05	1.08	0.02	0.002	0.005	15.84	0.53	6.61	0.09
RV Gru	8446-662-1	0.259516	10.94	0.03	0.93	0.03	0.004	0.000	7.78	0.65	5.38	0.18
EI CVn	2548-936-1	0.260770	11.86	0.02	0.97	0.02	0.009	0.001	6.83	0.33	6.00	0.11
VZ Psc	581-259-1	0.261259	10.20	0.05	1.15	0.01	0.020	0.001	18.80	0.46	6.51	0.07
15867853	5507-705-1	0.263549	10.83	0.01	0.84	0.01	0.027	0.002	9.20	0.79	5.57	0.19
772054	4375-2406-1	0.264667	11.31	0.02	0.69	0.01	0.017	0.001	6.33	0.28	5.26	0.10
44 Boo	HIP 73695	0.267818	5.87	0.02	0.94	0.02	0.001	0.002	79.95	1.56	5.38	0.05
V384 Ser	2035-175-1	0.268739	11.87	0.02	1.01	0.03	0.018	0.004	5.43	0.23	5.49	0.09
MU Aqr	5178-1376-1	0.272218	11.12	0.02	1.06	0.07	0.031	0.002	7.45	0.74	5.38	0.22

Note. The star names, when fully numerical, correspond to the numbers in Gettel et al. (2006); in catalogs, the numbers are referred to as GGM2006-nnnn. TYC are the numbers in the Tycho Catalog. All photometric data are in magnitude units, while the parallax is in milliarcseconds (mas) = 0.001 arcsec. The full version of the table is available electronically.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

The current sample has some limitations that impact the present attempts to obtain a luminosity calibration: (1) the decreased precision of the parallax data for the most distant objects (Figure 1); (2) the loss of about one fourth of the binaries due to apparently more stringent criteria (than those of the HIP sample) when the Tycho and the Gaia data were combined into TGAS; and (3) our neglect of the potential presence of companions to the EW binaries. We comment on them in turn, bypassing #1, as it is a rather obvious one.

Due to limitation #2, the current sample cannot be used to estimate the spatial density of EW binaries. For the faintest systems, the volume currently explored is particularly small. These are the binaries with the shortest orbital periods, which, similar to previous studies, remain under-represented in spatial counts. This is illustrated in Figure 2 in a plot of accessible luminosities (expressed in M_V) versus the distance. The limits for the apparent magnitudes $V=12$ and $V=13$ are shown; they roughly correspond to the depth of the Tycho data, which defined the TGAS data selection via the initial-epoch positions for the proper-motion eliminations. The decrease in the accessible range of M_V magnitudes may combine with (or possibly modify) the apparent sharp drop in spatial density of EW binaries taking place at about ${M}_{V}\gt 5$ (Rucinski 2006, 2007). As shown in Figure 1, the healthy increase in binary numbers versus ${V}_{\max }$ stops beyond $V\simeq 12$. Thus, the definition of the short-period cut-off for EW binaries remains an open question due to the lack of data for the faintest systems.

Zoom In Zoom Out Reset image size

We are very much aware of the fact that many EW binaries have companions whose presence may affect the M_V and the B − V photometric data. Targeted searches (D'Angelo et al. 2006; Pribulla & Rucinski 2006; Rucinski et al. 2007) found that additional components are very common with a frequency approaching one hundred percent. Unfortunately, the searches had limited depths, with only the brightest EW binaries (usually $V\lt 10$) having been successfully investigated. With this in mind, we decided not to correct for the brightness contributions of these companions, as adjusting only the apparently brightest systems could potentially introduce a systematic luminosity bias in the sample; we simply assumed that the presence of individual binary companions would show up in the increased $\sigma ({M}_{V})$ errors.

Finally, in evaluating the ${V}_{\max }$ values, we entirely neglected slow trends in mean luminosity (Rucinski & Paczynski 2002) observed in about one-third of the EW binaries (Marsh et al. 2017). These very recently discovered trends will no doubt be studied in the coming years, but at this moment, we again had to assume that they will simply increase the $\sigma ({M}_{V})$ errors.

The previous absolute-magnitude calibrations for EW binaries were developed as linear, three-parameter relations of the form ${M}_{V}={a}_{0}+{a}_{P}\mathrm{log}P+{a}_{C}{(B-V)}_{0}$. Rucinski (1994) simplistically argued that the expected calibration may have such a shape, possibly with a contribution from the mass ratio (q), which is usually unknown without spectroscopic studies. Although it was recognized that the period–color correlation discovered by Eggen (1967) possibly plays a role through the correlation of $\mathrm{log}P$ and B − V, it was not explicitly included in these considerations. The improved data from the HIP sample seemed to confirm the need to include both the period and the color, but the significance of a solution with two independent parameters was low though fully explainable by the moderate quality of the parallax data.

The two terms in the previous calibrations, a_P and a_C, relate to the quantities that are not only mutually correlated but are also determined with very different uncertainties. Thanks to the repetitive nature of photometric observations, the orbital period is usually known with very high accuracy compared to the color, which is usually relatively inaccurate due to a large fraction of EW binaries never having been observed in standard photometric systems. In conjunction, the individual calibrations to standard systems of many binaries give results with discrepancies at the level of a few hundredths of a magnitude, indicating the presence of systematic errors in the color calibrations. Two additional physical causes produce further uncertainties: (1) EW binaries do change their colors with the orbital phase (for $B-V$, it is typically by 0.02–0.05 mag)—a variation interpreted within the Lucy model as due to gravity distributions over the common equipotential; and (2) interstellar extinction systematically reddens the colors but is poorly determined in individual cases. As a result, the colors are seldom known to better than a few hundredths of a magnitude, which makes color a much weaker independent variable than the period.

Our least-squares fits have been done in succession starting from (1) the simplest linear fit utilizing $\mathrm{log}P$ as the independent variable, (2) by splitting the full $\mathrm{log}P$ range into three linear sub-sections, (3) by considering a quadratic dependence, and finally (4) by supplementing the relation with the de-reddened color index ${(B-V)}_{0}$ as the second independent variable. Figure 3 shows the M_V versus $\mathrm{log}P$ dependence while the coefficients obtained for the fits are given in Table 2. We used the standard least-squares regression to evaluate the values of ${\chi }^{2}$, the reduced ${\chi }_{r}^{2}$ (per degree of freedom), the mean weighted standard deviation and the mean weighted absolute deviation δ, with weights for the individual stars $\propto \,1/{{\sigma }^{2}}_{({M}_{V})}$. Because the intrinsic scatter in M_V is unknown, we used bootstrap experiments to estimate the coefficient errors. The experiments consisted of 10,000 samples with repetitions from the population of n stars (for the whole sample, $n=318$). The coefficient uncertainties were estimated as the 15.8th and the 84.1st percentiles of the resulting coefficient distributions relative to the median (50th percentile); such scatter levels correspond to the mean standard errors of a Gaussian distribution. To improve the quality of the least-squares fits, we shifted the independent variables X and Y to the approximate centers of their respective ranges, as given in Table 2. We measured the quality of each fit by calculating the coefficient of determination, R², which is defined as the fraction of variance reproduced by the model, ${R}^{2}=1-{{SS}}_{\mathrm{res}}/{{SS}}_{\mathrm{tot}}$, where ${{SS}}_{\mathrm{res}}$ is the sum of the residual (unexplained) variances computed by using local, non-parametric smoothing, while ${{SS}}_{\mathrm{tot}}$ is the sum of the total variances. Thus $0\leqslant {R}^{2}\leqslant 1$, where 1 indicates the best model fit and 0 indicates the fit not represented by the assumed model.

Zoom In Zoom Out Reset image size

Table 2.  Calibration

Fit	Details	a0	a1	a2	n	${\chi }^{2}$	${\chi }_{r}^{2}$		δ	R2
	${{\boldsymbol{M}}}_{{\boldsymbol{V}}}={{\boldsymbol{a}}}_{{\bf{0}}}+{{\boldsymbol{a}}}_{{\bf{1}}}{\boldsymbol{X}}$
#1	$0.22\lt P\lt 0.90$ days				318
	$X=\mathrm{log}P+0.35$	3.21	−9.89			1962.2	6.2	0.40	0.31	0.71
	P0 = 0.4467 days	${3.21}_{-0.08}^{+0.08}$	$-{9.85}_{-0.67}^{+0.61}$
#2	$P\leqslant 0.275$ days				12
	$X=\mathrm{log}P+0.60$	6.11	−23.27			174.0	19.3	0.36	0.26	(1.0)
	P0 = 0.2512 days	${6.15}_{-0.19}^{+0.29}$	$-{22.37}_{-7.73}^{+2.44}$
#3	$P\geqslant 0.575\,{\rm{d}}$				9
	$X=\mathrm{log}P+0.15$	1.90	−4.85			6.13	1.0	0.21	0.15	(1.0)
	P0 = 0.7079 days	${1.91}_{-0.09}^{+0.08}$	$-{4.54}_{-0.86}^{+1.40}$
#4	$0.275\lt P\lt 0.575$ days				297
	$X=\mathrm{log}P+0.40$	3.73	−8.67			1453.3	4.9	0.37	0.29	0.84
	P0 = 0.3981 days	${3.73}_{-0.06}^{+0.06}$	$-{8.67}_{-0.64}^{+0.65}$
	${{\boldsymbol{M}}}_{{\boldsymbol{V}}}={{\boldsymbol{a}}}_{{\bf{0}}}+{{\boldsymbol{a}}}_{{\bf{1}}}{\boldsymbol{X}}+{{\boldsymbol{a}}}_{{\bf{2}}}{{\boldsymbol{X}}}^{{\bf{2}}}$
#5	$0.22\lt P\lt 0.90$ days				318
	$X=\mathrm{log}P+0.35$	3.28	−7.54	+9.96		1806.8	5.8	0.38	0.30	0.82
	P0 = 0.4467 days	${3.29}_{-0.08}^{+0.08}$	$-{7.52}_{-1.18}^{+1.31}$	$+{10.19}_{-5.51}^{+5.72}$
	${{\boldsymbol{M}}}_{{\boldsymbol{V}}}={{\boldsymbol{a}}}_{{\bf{0}}}+{{\boldsymbol{a}}}_{{\bf{1}}}{\boldsymbol{X}}+{{\boldsymbol{a}}}_{{\bf{2}}}{\boldsymbol{Y}}$
#6	$0.22\lt P\lt 0.90$ days				318
	$X=\mathrm{log}P+0.35$	3.21	−5.70	$+2.49$		812.4	2.6	0.26	0.20	0.79
	$Y={(B-V)}_{0}-0.5$	${3.20}_{-0.05}^{+0.05}$	$-{5.77}_{-0.38}^{+0.44}$	$+{2.54}_{-0.32}^{+0.29}$
	P0 = 0.4467 days
#7	$0.275\lt P\lt 0.575$ days				297
	$X=\mathrm{log}P+0.40$	3.82	−9.04	$+0.05$		1638.4	5.6	0.39	0.31	0.83
	$Y={(B-V)}_{0}-0.5$	${3.81}_{-0.05}^{+0.05}$	$-{8.92}_{-0.81}^{+0.81}$	$+{0.12}_{-0.45}^{+0.56}$
	P0 = 0.3981 days

Note. For each solution, the results are given twice: (1) for a standard least-squares determination for n stars, with the corresponding ${\chi }^{2}$, the reduced ${\chi }_{r}^{2}$, the weighted mean-square deviation , the weighted mean absolute deviation δ (weights were $\propto \,1/{\sigma }_{i}^{2}$), and the coefficient of determination R² (see the text), and (2) as a result of a bootstrap experiment (10,000 repetitions) with the median (50th percentile) and the asymmetric errors given as differences from the median for the 15.8th and the 84.1st percentiles of the coefficient distributions. In the column "Details," the orbital period P is specified by its range. Note that shifts to mid-ranges of $\mathrm{log}P$ and ${(B-V)}_{0}$ have been applied in the definition of the independent variables X and Y to improve the least-squares solutions. P₀ is the period corresponding to the $\mathrm{log}P$ shift in the definition of X.

Download table as:  ASCIITypeset image

We discuss below the calibration fits in relation to Table 2, which gives the numerical details. Figure 3 illustrates the fits, as explained in its caption.

• Solution #1: The linear fit over the whole period range, $0.22\lt P\lt 0.88$ days, is obviously an oversimplification. The fit shows systematic trends in the residuals (see Figure 3) for the short-period and the long-period ends where EW binaries are fainter than what the linear regression predicts. The trends are reflected in the value of ${R}^{2}=0.71$. The very steep dependence at the short-period end and the shallow dependence at the long-period end were noted before (Rucinski 2006; Pawlak 2016) and may at least partially be explained by the shift of the dominant flux away from the V-band filter transmission.
• Solutions #2 and #3: The separate linear fits for the short-period end, $0.275\lt P$ days, and the long-period end, $P\lt 0.575$ days, are based on very few stars. Thus, no systematic trends in the solutions are visible and R² could not be reliably determined. We note that the values of ${\chi }_{r}^{2}$ are very different for the two solutions: while ${\chi }_{r}^{2}=1.0$ for the long-period segment (#3), ${\chi }_{r}^{2}=19.3$ for the short-period segment (#2). The large value of ${\chi }_{r}^{2}$ may be explained either by our under-estimation of the $\sigma ({M}_{V})$ errors, by an inappropriate linear model (oversimplified description of the functional dependence of M_V on binary parameters), or by an intrinsic scatter in M_V (e.g., due to the neglected companions or the slow luminosity trends, Section 2.4). We return to this subject in Section 4. We note that the particularly steep relation for the short-period systems is driven mostly by the new, high-precision parallax determinations for CC Com and V523 Cas; the steepness of the dependence may have important implications on the detectability of short-period systems.
• Solution #4: This is the best defined among the single-parameter, $\mathrm{log}P$-dependent solutions. It covers the central range $0.275\lt P\lt 0.575$ days and is based on a large number of stars. Its better definition, relative to #1, is reflected in the decreased value of ${\chi }_{r}^{2}$ from 6.2 to 4.9 and the increased value of R² from 0.71 to 0.84.
• Solution #5: A quadratic fit for the whole period range is only slightly better than the linear fit #1. It does not reproduce the ends very well, and it only leads to a marginal improvement in ${\chi }^{2}$ and R². The added quadratic term was found to be insignificant through the F-test: the change from ${\chi }_{2}^{2}=1962$ to ${\chi }_{3}^{2}=1807$ results in $F=0.0003$.

Before discussing the two solutions utilizing the de-reddened color index ${(B-V)}_{0}$, #6 and #7, we present the familiar color–magnitude diagram (CMD) for the solar neighborhood EW binaries (Figure 4). The EW binaries are located mostly above and to the right of the Zero Age Main Sequence. While evolved cores of one or both components may produce an increase in luminosity, the Lucy model predicts the main shift to occur in the color. With the observed tendency of unequal masses in EW binaries ($q\ne 1$), the secondary component should add almost no light but a lot of surface area, which must result in a lower effective temperature. For details on the expected shifts in magnitude and color versus the mass ratio q, see Figure 8 in Rucinski & Duerbeck (1997).

• Solution #6: This fit is basically solution #1 with the added color term. It has the same form as the calibrations in Rucinski (1994) and Rucinski & Duerbeck (1997). Because the color term has been added to improve the fit over the whole period range, we attempted to visualize a possible coupling of the M_V deviations with the color by marking in Figure 3 the binaries that deviate from the linear period dependence #1. An obvious correlation is visible at the short-period end (for ${M}_{V}\gt 5.5$) where all of the binaries are red and fainter than what the linear fit predicts. While fit #6 is definitely better than fit #1 and shows an expected reduction in ${\chi }^{2}$, the formal F-test gives $F=0.004$ and does not support the need for two variables; the deviating ends are too poorly populated to influence the solution. The slope coefficients are somewhat similar to those in Rucinski & Duerbeck (1997), but not identical, which probably reflects the different proportion of the included short-period systems in the respective samples.
• Solution #7: The fit for the central part of the period distribution ($0.275\lt P\lt 0.575$ day) with the color term included is particularly informative. With the short-period and the long-period ends removed, the color term becomes entirely redundant. Thus, the inclusion of the short-period, red binaries that deviate in luminosity from the linear $\mathrm{log}P$ dependence resulted in the bias in the previous calibration (Rucinski & Duerbeck 1997) as well as in solution #6.

Zoom In Zoom Out Reset image size

In conclusion, we feel that solution #4 is the best for representing the majority of detected EW binaries. For short-period and long-period binaries, the best solutions are currently #2 and #3, but future studies may result in a better functional form that could cover the entire period range. Finally, the color term is apparently not needed at all.