ANALYSIS OF DETACHED ECLIPSING BINARIES NEAR THE TURNOFF OF THE OPEN CLUSTER NGC 7142

Radial velocities were determined from cross correlation using the IRAF task fxcor with a solar spectrum (Hinkle et al. 2000) over the region 4880 to 5750 Å. On nearly every night that a cluster star was observed, we also observed a radial velocity standard, and the difference between the measurement of the standard and the literature value determined a zero point that was applied to the final star velocity. The radial velocity corrections only vary slightly (by a few tenths of a km s⁻¹) from night to night, but change more significantly with the instrument configuration and season. If a standard was not observed on the night of a cluster observation, the correction was taken by averaging ones from nearby nights. The radial velocities for the three binaries under consideration here are given in Table 2, while the phased velocity curves are shown in Figures 1, 2, and 3.

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Rotational velocities were determined from clean spectra for both components. This involved creating a first attempt at a clean spectrum by shifting all of the spectra to the rest frame for one of the components and then combining the spectra using a median with a fairly aggressive sigma clip to remove the spectral features of the other component. With preliminary A and B component spectra, we could divide them back into the original spectra (shifted to the correct velocity) and then repeat the process on these residual spectra files to generate a second set of cleaned A and B component spectra. The cleaned component spectra were then cross-correlated against the solar spectrum, and the width of the cross-correlation peak measured. We then generated synthetic spectra with different rotational velocities and cross-correlated them against the same solar template spectrum. Finally, we estimated the rotational velocity by interpolating in the grid of results for cross-correlation peak widths.

Abundance analysis for binaries is a fairly specialized and complicated endeavor. To accomplish this, we computed synthetic spectra using the 2010 version of MOOG (Sneden 1973) with a line list based largely on the Kurucz line list⁵ but with some lines adjusted to fit a solar spectrum (Hinkle et al. 2000). MOOG is able to account for continuum light contributed by a companion star in the binary. To minimize the number of free parameters, we used surface gravities and flux ratios from the binary analysis (because the uncertainties on these parameters are far smaller than could be obtained with any spectroscopic analysis) while we let the effective temperature and metallicity of the model atmospheres vary.

We calculated synthetic spectra on a grid of effective temperatures and metallicities with steps of 250 K and 0.07 dex, respectively. We then divided the observed spectrum into the synthetic spectrum, and in several small wavelength regions calculated the residuals about a fitted constant value, where the constant was allowed to change from region to region to compensate for errors in setting the continuum. The regions we chose were: 4866–4995 Å, 4995–5220 Å, 5220–5390 Å, 5300–5330 Å, and 5390–5620 Å. The 5300–5330 Å region is given extra weight by being used twice because it contains a mix of strong lines that increase and decrease with changes in temperature, making it particularly sensitive to T_eff. We interpolated the results between grid points to determine the best parameters for each region. The results from the regions were then averaged together to give a final effective temperature and metallicity, along with an estimate of the random uncertainties. For V2 we derived T_A = 6238 ± 52 K and [Fe/H]_A = −0.03 ± 0.06, while T_B = 6276 ± 63 K and [Fe/H]_B = −0.12 ± 0.02. Because these two stars are found to have identical characteristics within the uncertainties in the later analysis of the binary, it is unlikely that the inputs to the spectral modeling (log g and/or the flux ratios) are responsible for the difference in the metallicities derived. It is more likely that systematic errors dominate the internal uncertainties. For V375 Cep we derive T_A = 6230 ± 50 and [Fe/H]_A = +0.09 ± 0.02 (where the quoted uncertainties are errors for the mean), while the B component was too weak to yield useful results. If we consider systematic uncertainties due to inputs for the spectroscopic analysis (such as oscillator strengths and microturbulence), the uncertainties are larger. We estimate that the overall uncertainties are 100 K for T_A and 0.05 dex for [Fe/H]_A. The abundance derived by Jacobson et al. (2008) for NGC 7142 giants is +0.14 ± 0.01. Our metallicity for V375 Cep is thus within 2σ of the Jacobson et al. value while our metallicity for the V2 system suggests that it may be a non-member.

Membership determinations for a poorly-studied cluster like NGC 7142 can be fairly difficult. To date, proper motions have only been published for a few stars in the cluster field (e.g., Baumgardt et al. 2000). In addition, there have only been a relatively small number of high precision radial velocity measurements. Jacobson et al. (2007) identified 6 cluster stars out of a sample of 17 and found an average radial velocity of −48.6 ± 1.1 km s⁻¹, while Jacobson et al. (2008) found −50.3 ± 0.3 km s⁻¹ from higher resolution spectra of 4 of the same candidate members. Sandquist et al. (2011) observed three red clump star candidates, and found that one had a velocity consistent with these averages, but two had velocities of −43.9 and −44.0 km s⁻¹. Looking more carefully at the photometry for these stars, the two stars with the higher velocities are fainter than other candidates in the Two Micron All Sky Survey (2MASS) K_s band by more than 0.2 mag, but bluer in the (J − K_S) color. This could indicate that these are foreground giant stars.

From the binary modeling discussed later, we find a system velocity γ = −17.2 km s⁻¹ for V1, which unambiguously rules out cluster membership. By comparison, the system velocity for V375 Cep was found to be γ = −49.86 ± 0.05 km s⁻¹, in very good agreement with the mean values found by the two high-resolution spectroscopic studies of the cluster. We therefore judge V375 Cep to be a very likely cluster member.

The fit for V2 returns a system velocity (γ = −42.57 ± 0.02 km s⁻¹) that falls near the mean cluster value, but about 7–8 km s⁻¹ higher. Although there has not been extensive enough proper motion or radial velocity survey of cluster stars to determine a reliable velocity dispersion for NGC 7142, the dispersion is expected to be ≲1 km s⁻¹ for bound clusters with typical masses and radii (Piskunov et al. 2008), and most old open clusters do seem to have radial velocity dispersions of that size (NGC 188, Geller et al. 2008; NGC 6819, Hole et al. 2009; Berkeley 32, Randich et al. 2009). Thus, V2 is unlikely to simply be in the wing of the cluster radial velocity distribution. The effects of a tertiary on a long period orbit could potentially produce this difference between the presently measured system velocity and the cluster mean, and we examine that possibility in more detail in the next subsection. A strong gravitational interaction within the cluster could also give a cluster member enough energy to escape.

We can examine other information (such as projected sky position and CMD position) that provides circumstantial evidence. Janes & Hoq (2011) found an "effective" radius of 4' for the cluster, which roughly corresponds to 1.5 times the σ-width of a Gaussian fitted to the stellar distribution. V375 Cep is projected 29 from the cluster center, nonmember V1 is 39 from center, and V2 is approximately 48 from center. Once again V2 has a lower likelihood of cluster membership, but this could also be related to its large velocity relative to other cluster members.

As can be seen in Figure 4, the system photometry (see Table 3) and decomposed optical photometry of V375 places it firmly within the main sequence band. The colors of V1 are very similar to those of the other eclipsing binaries despite indications that both component masses are lower than the primary masses of the other binaries (see Section 5.1), which is consistent with the smaller reddening of a foreground object.

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The system photometry for V2 is brighter and slightly bluer than cluster turnoff stars, but when the photometry is decomposed, only the color is slightly discrepant—the stars are at the blue edge of the distribution at the turnoff in the CMD, which could be explained by lower-than-average reddening. They are slightly brighter than the primary star in V375 Cep in optical bands. In the 2MASS bandpasses, they are approximately the same brightness, assuming that the individual stars are about 0.75 mag fainter than the combined photometry of V2 and the primary of V375 Cep is about 0.15 mag fainter than the combined photometry of that binary (roughly consistent with the I-band secondary eclipse depth). If V2 is a member of the cluster having lower reddening than V375 Cep, it is consistent that the primary of V375 Cep is brighter relative to the two stars of V2 in the 2MASS bands.

To summarize, we judge V1 to be nonmember based on its radial velocity. For V375, the argument for cluster membership is much stronger than it is for V2. None of the information we have available unambiguously supports V2 membership. The age determination for V2 more definitively argues against cluster membership, however, and that will be discussed in Section 5.1.2.

Because there are now numerous examples of known triple systems in open clusters (a partial list includes Mermilliod & Mayor 1989; Mermilliod et al. 1994; Alencar et al. 1997; Sandquist et al. 2003; Liu et al. 2011; Jeffries et al. 2013), it is worth checking whether the influence of tertiary stars can be detected. If a tertiary star is massive or bright enough, it can affect the models of an eclipsing binary enough to produce significant systematic errors in the measured characteristics of the eclipsing stars. None of the binaries we discuss here had a third set of detectable lines in our spectra, but with a long enough baseline of observations, photometric methods (such as eclipse timing) or spectroscopic methods (such as center-of-mass motion) can reveal tertiaries via their effects on the eclipsing binary. Table 4 gives our measurements of the times of eclipse minimum for the binaries. Due to the relatively small number of radial velocity and eclipse minimum observations for V1, the detection of a tertiary star's effects is unlikely, so we do not discuss it here.

The radial velocity results for V2 are shown in Figure 2, assuming the best fit mass ratio q = 1.001. This binary shows more than a 7 km s⁻¹ offset from the cluster mean velocity (−50.3 ± 0.3 km s⁻¹ from 4 stars; Jacobson et al. 2008), which could potentially result from the action of a tertiary star. However, the center-of-mass velocities did not vary significantly over three seasons and more than a 700 day interval of observations, and there is no sign of variations in eclipse timing.

The lower panel of Figure 1 shows the measured center-of-mass velocities for V375 Cep, assuming the best fit mass ratio q = 0.676 from the binary models. There does not appear to be evidence of significant motion during the three seasons (covering more than 1100 days) that we observed the system. Our own eclipse observations for V375 Cep cover a period of almost 1800 days. For the finely sampled light curves from our study, we used the method of Kwee & van Woerden (1956) to determine times of minima and the errors, and we show a comparison of those times with a best-fitting linear ephemeris in Figure 5. We include in Table 4 our best estimates of eclipse minima from the published observations of Crinklaw & Talbert (1991) and Seeberger et al. (1991) to improve the accuracy of the ephemeris and test for the possibility of a nonlinear ephemeris over the 27 yr baseline.

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The earlier observations were discussed in Sandquist et al. (2011). Most of the observations by Crinklaw & Talbert (1991) agree well with our phased light curve and they had observations in and out of eclipse on the night of one eclipse. Using our model light curves in BV, we fit their data in order to derive an approximate time of minimum. One additional observation in V on a different date also appears to have fallen near an eclipse minimum. Crinklaw & Talbert shifted the photometric zero points of each of their frames to be consistent, so the relative photometry of the single observation should be approximately correct. The interval between this and the nearest primary eclipse is about 14 orbital cycles, but implies a period of about 1.9164 days, which is significantly different than we find in our more recent observations.

Sandquist et al. (2011) concluded that data from Seeberger et al. (1991) was not of sufficient quality to test for nonlinearities in the ephemeris. Seeberger et al. quoted fairly large uncertainties (0.05 mag) on their measurements, and the shape and depth of the observations on one night (HJD 2446650) that appeared to contain a primary eclipse egress were inconsistent with our model light curves. This may be due to their use of photographic plates as the recording medium.

We conclude that there is a possibility of eclipse timing variations for the V375 Cep system, but the fact that it has been well-behaved during the time covered by our own eclipse observations and radial velocity measurements makes the existence of a tertiary less probable.

In order to use the photometry for temperature estimates for the stars, we need to have a measurement of the reddening. Sandquist et al. (2011) derived a reddening value [E(B − V) = 0.32 ± 0.06] via a comparison of the red clump stars in NGC 7142 with those of M67. We revisit that estimate here by examining how the difference in median clump magnitudes between the two clusters changed with filter. M67 and NGC 7142 have similar ages and metallicities, and have values in ranges where small differences have minimal effects on the photometry of the red clump (Girardi & Salaris 2001; Grocholski & Sarajedini 2002).

We have, however, calculated theoretical corrections for intrinsic differences in clump magnitude from Girardi & Salaris (2001) models in order to make the reddening determination more precise. At constant age, the higher metallicity of NGC 7142 (Δ[Fe/H] =0.14) is theoretically expected to make the clump magnitude brighter in 2MASS infrared filters (by 0.05 mag in K_s), but increasingly fainter at bluer wavelengths, reaching almost 0.13 mag in B. At constant metallicity, the larger age of M67 is expected to make the red clump fainter in all filters by approximately 0.03 mag (Girardi & Salaris 2001; Grocholski & Sarajedini 2002).

We made use of our own optical photometry along with 2MASS (Skrutskie et al. 2006) and WISE (Wright et al. 2010) infrared photometry to simultaneously derive the differences in true distance moduli [$\Delta (m-M)_0 = 2.45^{+0.11}_{-0.07}$] and optical depths (Δτ₁ = 0.278 ± 0.053) between the two clusters, assuming an extinction law based on the study by McCall (2004) with Cardelli et al. (1989) used to extend predictions to the WISE filters. Δ(m − M)₀ is primarily determined by observations in the infrared where the extinction is small, while Δτ₁ is constrained by the variation in the extinction from filter to filter. The fit is shown in Figure 6. The uncertainties on each measurement are based on uncertainties on the medians of each clump (dominated by the uncertainties for NGC 7142), and the goodness of fit was calculated using a χ² algorithm. The uncertainties in Δ(m − M)₀ and Δτ₁ were derived from the ranges covered by fits that were within 1 of the minimum value. Using the well-determined distance modulus ((m − M)₀ = 9.60 ± 0.03; Sandquist 2004) and reddening (E(B − V) = 0.041 ± 0.004; Taylor 2007) for M67, we find $(m-M)_0 = 12.05^{+0.11}_{-0.09}$ and E(B − V) = 0.29 ± 0.05 for NGC 7142. Because there appears to be a significant amount of differential reddening in the cluster, infrared colors should be employed when possible. We discuss below the characteristics of each system that we are able to exploit to produce temperature estimates from the photometry of the binary stars.

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The characteristics of the binary systems make it possible to obtain good estimates of the colors of the component stars. Table 3 lists the photometry for the binary systems and their components, and Figure 4 shows their positions in the CMD.

Table 4. Photometric Minima

Eclipse	Filter	mJDa
V1
P	R	53639.9356 ± 0.0006
P	I	54657.7878 ± 0.0005
S	V	54678.7786 ± 0.0008
S	B	55355.7864 ± 0.0008
S	R	55383.7955 ± 0.0007
P	I	55390.8320 ± 0.0009
S	I	55411.8168 ± 0.0014
V2
P	R	53639.6889 ± 0.0002
P	I	54656.9770 ± 0.0011
P	V	55423.8558 ± 0.0011
P	V	55517.7581 ± 0.0002
P	I	55736.8673 ± 0.0003
S	I	55755.9707 ± 0.0004
P	B	55783.8178 ± 0.0003
S	V	55802.9238 ± 0.0026
S	V	55849.8720 ± 0.0011
V375 Cep
P	V	$46650.484^{+0.005}_{0.020}$
P	V	47442.81 ± 0.025
P	BV	47469.659 ± 0.004
S	R	53594.9749 ± 0.0066
S	R	53596.8816 ± 0.0003
S	R	53598.7923 ± 0.0005
P	R	53599.7467 ± 0.0002
S	R	53600.7011 ± 0.0006
P	R	53637.9396 ± 0.0003
P	R	53639.8496 ± 0.0002
S	R	53640.8054 ± 0.0004
P	R	53641.7594 ± 0.0002
P	B	54630.9733 ± 0.0024
S	B	54633.8450 ± 0.0016
P	B	54634.7949 ± 0.0007
P	I	54655.8013 ± 0.0003
S	I	54656.7542 ± 0.0008
P	I	54657.7114 ± 0.0002
S	V	54675.8491 ± 0.0012
P	V	54676.8071 ± 0.0005
S	V	54677.7630 ± 0.0007
P	V	54678.7170 ± 0.0004
P	V	55121.7629 ± 0.0005
P	V	55146.5934 ± 0.0009
S	V	55372.8853 ± 0.0012

Note. ^amJD = HJD − 2400000.

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For V2, the two eclipses are deep and very nearly the same in depth, which supports our later results (Section 4.2.1) that the two stars are nearly identical in all of their major characteristics. As a result, the system color is an excellent representation of the star colors. (The components are therefore about 0.75 mag fainter than the combined photometry.) Later results also indicate that the binary is probably not a member of the cluster, and is slightly behind the cluster. To get a temperature estimate, we therefore use the infrared colors of the binary and assume that the reddening and metallicity of the binary are close to that of NGC 7142. Using these assumptions, we find T_eff = 6150 ± 200 K from the (J − K_S) color using the color–temperature calibration of Casagrande et al. (2010). The temperatures derived from optical colors are consistent with this estimate, but are much more uncertain due to reddening uncertainties.

The main difficulty for V1 is that it appears to be a foreground system, and so the cluster metallicity and reddening do not apply. However, we can derive fairly accurate temperature estimates if we note that the stellar temperatures only differ by a little under 2% according to later models (see Section 4.3), so that the system color will be a fair representation of the colors of the components. (The luminosities of the stars differ, however, so we have used the results of the binary models to determine the fraction of the flux contributed by each star. The estimates of the component magnitudes are given in Table 3, and the results are plotted in Figure 4.) We can minimize the effects of reddening uncertainties by using infrared colors. A reddening approximately equal to the mean cluster reddening gives an upper limit to the average temperature of about 6120 K. Given the slightly super-solar masses of the stars (again, see Section 4.3), the Sun's temperature provides us with a lower temperature limit. Temperature uncertainty due to the unknown metallicity is likely to be small (a few 10 s of K) as long as the stars have near-solar abundances. Based on these arguments we constrain the primary (hotter) star temperature to be between 5850 K and 6250 K.

In the case of V375 Cep, we can see a period of totality in the secondary eclipse, so that the light from the secondary star can be precisely disentangled from that of the primary. The errors on the secondary star photometry in this case are calculated from

$$\sigma ^2(m_2) = \sigma ^2(m_{12}) + \frac{\sigma ^2(\Delta m_2)}{(10^{(\Delta m_2 / 2.5)} -1)^2}$$

where m₂ is the secondary magnitude, m₁₂ is the binary magnitude, and Δm₂ is the secondary eclipse depth. For shallow eclipse depths, the factor in the denominator of the second term amplifies the uncertainty considerably.

We do not have measurements of the secondary eclipse depths in infrared filters for V375 Cep, so we resort to optical/near-infrared colors. These imply T_A = 6080 ± 170 K for the primary star, again using the Casagrande et al. (2010) calibration. For comparison, we obtained a temperature 6230 ± 100 K, [M/H] =0.09 ± 0.05, and [α/Fe] =0.0 ± 0.15 from our spectroscopic analysis in Section 3.2. There is greater uncertainty for the secondary star resulting from the uncertainties in the photometric deconvolution, but the most certain determination using the (V − I) color puts its temperature at about 5050 ± 180 K.

The light curves (with primary and secondary eclipses of different depths, as seen in Figures 7 and 8) and radial velocities for V375 both implied from the start that the mass ratio for this system was likely to be significantly different from 1. This led us to believe that the system's light was dominated by one of the stars, and that star therefore resided close to the cluster turnoff. The decomposed photometry shown in Figure 4 confirms this, and makes the primary star an excellent candidate for constraining the cluster age if it is a member.

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Another notable feature of the light curves for this system is the small amount of out-of-eclipse light variation. This can be seen especially in the R_C light curve, which generally had the images with the highest signal-to-noise ratio and also had the best coverage of out-of-eclipse phases. In spite of the rather short orbital period, effects due to the distortion of the stellar surfaces by the other star are barely discernable. Most of the scatter in other filter bands comes from observations made during poor weather conditions. However, there is a hint that stellar activity might be producing some night-to-night variations in B. More observations would be needed to confirm this.

4.1.1. Radial Velocity Modeling

Because this binary appears to be circularized and there is little or no out-of-eclipse light variation, the light curves do not effectively constrain the mass ratio of the stars. Therefore we modeled the radial velocities separately from the light curves. In the radial velocity models, we fit for the velocity semi-amplitude of the primary star K_A, the mass ratio q = M_B/M_A, and the system velocity γ as parameters. We did an experiment where we allowed the system velocities to differ for the two stars (to allow for differences in gravitational redshift or convective blueshift resulting from their differing evolutionary states), but found a difference of only 0.1 km s⁻¹. This negligibly affected the derived masses.

The initial estimates of the velocity uncertainties for the two stars were scaled separately to return reduced χ² values of 1. After scaling, the typical uncertainties were 0.5–2 km s⁻¹ for the primary and 2–4 km s⁻¹ for the secondary. Once the light curves were modeled, the orbital inclination i was used in a final modeling run to derive the stellar masses. The results are given in Table 5.

Table 5. Characteristics of the Eclipsing Binaries

Parameter	V375 Cep		V2		V1
	Limb Darkening	Atmospheres	Limb Darkening	Atmospheres
γ (km s−1)	−49.86 ± 0.05		−42.64 ± 0.02	−42.66	$-17.16^{+0.03}_{-0.09}$
q	0.676 ± 0.004		1.0001 ± 0.0033	0.9992	0.951 ± 0.004
KA (km s−1)	89.13 ± 0.24		69.92 ± 0.20	69.91	80.81 ± 0.30
vrot(A) (km s−1)	$42^{+2}_{-4}$ (set)
vrot(B) (km s−1)	$20^{+10}_{-5}$ (set)
t0 − 2450000	3599.74650	3599.74651	5502.85043	5502.8507
σ(t0)	±0.00007	±0.00008	±0.00008
tc − 2450000					3639.9062 ± 0.0003
P (days)	1.90968257	1.90968252	15.6505950	15.6505954	4.6690576
σ(P) (days)	±0.00000016	±0.00000018	±0.0000016		±0.0000007
i (°)	$85.34^{+0.02}_{-0.05}$	$85.39^{+0.03}_{-0.02}$	89.703 ± 0.008	89.650	83.38 ± 0.02
e	0 (set)		0.52165 ± 0.00002	0.52158	$0.0379^{+0.0031}_{-0.0004}$
ω (°)	90 (set)		326.692 ± 0.003	326.696	$285.0^{+0.2}_{-1.2}$
RA/a	0.1937 ± 0.0003	0.1958 ± 0.0004	0.04381 ± 0.00011	0.04350	0.0877 ± 0.0009
RA/RB	1.8098 ± 0.0018	1.840 ± 0.002	1.0022 ± 0.0030	0.9847	$1.140^{+0.010}_{-0.021}$
RB/a	0.10704 ± 0.00020	$0.10644^{+0.00014}_{-0.00022}$	0.04372 ± 0.00012	0.04417	0.0769 ± 0.0007
(RA + RB)/a	0.3008 ± 0.0005	0.3023 ± 0.0006	0.08753 ± 0.00012	0.08767	0.1647 ± 0.0003
TB/TA	0.8268 ± 0.0008	0.8122 ± 0.0012	0.9969 ± 0.0005	0.9944	0.983 ± 0.002
MA (M☉)	1.288 ± 0.017	1.288 ± 0.017	1.377 ± 0.009	1.379	1.147 ± 0.012
MB (M☉)	0.871 ± 0.008	0.871 ± 0.008	1.377 ± 0.009	1.378	1.090 ± 0.013
RA (R☉)	1.623 ± 0.006	1.642 ± 0.007	1.616 ± 0.005	1.605	1.338 ± 0.008
RB (R☉)	0.897 ± 0.003	0.893 ± 0.004	1.613 ± 0.005	1.630	1.190 ± 0.008
log gA (cgs)	4.129 ± 0.003	4.120 ± 0.003	4.160 ± 0.003	4.166	4.244 ± 0.007
log gB (cgs)	4.473 ± 0.002	4.478 ± 0.002	4.162 ± 0.003	4.152	4.324 ± 0.007

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4.1.2. Light Curve Modeling

The light curves of V2 show two very deep (0.7–0.8 mag) eclipses per cycle (see Figure 9), and two components are very clearly seen in spectra of the system. The separation of the eclipses in phase (Δϕ = 0.2206) and the much longer duration of the shallower eclipse conclusively show that the system has a substantial eccentricity.

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4.2.1. Combined Radial Velocity and Light Curve Modeling

For eccentric binaries, both the radial velocities and the light curves contain information on the orbits of the two stars, so it is more important to model the two datasets simultaneously. As we did with V375 Cep, we scaled the errors for each dataset (photometry by filters, radial velocities for each component) separately to produce a reduced $\chi ^2_\nu$ value near 1.

For a combined run with model atmospheres, we fitted the binary with a set of 12 parameters: orbital period P, time of periastron t₀, velocity semi-amplitude of the primary star K_A, mass ratio q, system velocity γ, eccentricity e, argument of periastron ω, inclination i, ratio of the stellar radii to average orbital separation R_A/a and R_B/a, primary star temperature T_A, and temperature ratio T_B/T_A. When using a quadratic limb darkening law, we forced the fitted limb darkening coefficients to be the same for both stars due to the indications that the star temperatures, masses, and radii were nearly identical. Generally when limb darkening coefficients are fitted, they are not tied to the stellar temperatures. In our case, when we allowed the coefficients to vary independently, the fits converged on values that were significantly different for the two stars. The most likely reason is that systematic trends in the eclipse light curves were presenting χ² incentives for the coefficients to differ.

We trimmed the light curve data down to observations in and near eclipse (see Figure 10) because of the lack of significant variation at other phases. As expected for a fairly long period binary, there is no sign of variation associated with nonsphericity of the stars. We did make zero point adjustments to the photometry (as we did for V375 Cep), but in all cases the shifts were less than 0.011 mag.

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The main result of the analysis is that the two stars have very similar characteristics. In particular, the mass ratio q is consistent with 1 to within the 1σ uncertainty. As a result, we cannot definitively state which star is the more massive one, and so for the purposes of this paper, we will define the primary star to be the one eclipsed during the deeper eclipse. According to the binary star modeling, the primary star is slightly larger and hotter at about 2σ and 4σ levels of significance, respectively. However, the radius and temperature ratios only differ from 1 by less than a percent. These results are supported observationally by the very long eclipse ingresses and egresses with no sign of totality (in spite of an inclination found to be within 05 of 90°), and by the very similar depths of the eclipses.

The combined photometry of this system puts it in the blue straggler portion of the cluster CMD, and when the components are decomposed, they fall at the blue end of the distribution of likely cluster stars at the turnoff. However, our radial velocities clearly identify it as a nonmember. Although we have only three radial velocity observations, we conducted trial model runs to get preliminary estimates of the star characteristics. As can be seen in Figure 11, the light curves show relatively shallow eclipses (∼0.08 and 0.12 mag), and the secondary eclipse is found at phase ϕ = 0.492, indicating a slight eccentricity. Binaries with periods shorter than 5 days are typically found to be circularized even in young populations (Meibom & Mathieu 2005), so it is worth trying to establish how large the eccentricity is.

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In a relatively short period binary like V1, it is not surprising to find some evidence of spot activity. During most nights of observation, there were few deviations in the light curve that could be identified with spot activity. However, on the night of one secondary eclipse (HJD 2455383.8), we found that the out-of-eclipse level was fainter than was typical in R observations, and there was a difference in the pre- and post-eclipse levels as well. To correct for this to first order, we applied a zero point shift to observations from that night to bring the average out-of-eclipse level for that night into agreement with others.

We then followed a procedure similar to that of binary V2, modeling the radial velocities and photometry simultaneously. We used the same binary model parameters with the exception of substituting time of conjunction (primary eclipse) t_c for time of periastron t₀. t_c is more directly constrained by observations in our combined dataset for V1. The model fit indicates that the binary orbit has a very small but significant eccentricity (the radial velocities are nearly consistent with a circular orbit), and the long axes of the orbits are almost in the plane of the sky.

The masses and radii for the six stars are plotted in Figure 12. Comparing the results of limb darkening law and model atmosphere runs in Table 5, there are relatively small (∼1%) but significant differences in radius. In the discussion below, we use the results from limb darkening law runs for their greater ability to fit eclipse ingresses and egresses. However, it should be remembered that we have not identified the root cause of the differences.

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5.1.1. V375 Cep

When the components of the two stars in V375 Cep are compared with isochrones in the M–R plane, we are immediately confronted with several issues. The most striking one involves the radius of the lower mass secondary star in the V375 Cep system. A star of mass 0.87 M_☉ should have not have evolved significantly during the lifetime of a cluster like NGC 7142, but we find that the star is more than 10% larger than expected from models.

This kind of behavior has been seen before: Clausen et al. (2009) discuss well-studied eclipsing binaries in the field containing stars with masses of 0.80–1.10 M_☉, finding that stars in binaries with short periods (0.6–2.8 days) tend to have larger radii and lower temperatures than predicted. Indicators such as spot-induced photometric variations and X-ray emission support the idea that stellar activity is related to the radius discrepancies (Torres et al. 2006). Stellar activity is thought to produce magnetic flux tubes that can inhibit the flow of the convective gas blobs that transport energy to the surface, forcing the star to grow in size to compensate for the lost transport capability. Stellar models have been produced that can reproduce such anomalously large radii via an ad hoc decrease in the mixing length parameter (Chabrier et al. 2007), or recently via a self-consistent (although one-dimensional) treatment of the magnetic field (Feiden & Chaboyer 2012).

Because the primary star has a larger mass, its convective envelope is predicted to be about an order of magnitude smaller in mass than that of the secondary star. As such, magnetic activity should play a less important role in influencing the energy transport in the outer layers of the star, and the radius should be closer to predictions for the cluster age (although it may still be inflated to a smaller degree). The decomposed photometry of the primary star places it toward the blue edge of the main sequence band for NGC 7142 (see Figure 13), supporting the idea that its temperature has not been affected significantly.

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FL Lyr (Popper et al. 1986), V1061 Cyg (Torres et al. 2006), and EF Aqr (Vos et al. 2012) are three other short-period binaries (P = 2.1782, 2.3467, and 2.8536 days, respectively) containing inflated secondary stars and primary star masses (1.218 ± 0.016 M_☉, 1.282 ± 0.015 M_☉, and 1.244 ± 0.008 M_☉, respectively) similar to that of V375 Cep (1.288 ± 0.017 M_☉). In the cases of V1061 Cyg and FL Lyr, the primary star radius can be matched with standard models for reasonable ages of 2.4 and 3.4 Gyr. It should be understood that this is not conclusive evidence that the primary stars are free of influences that modify the radius. A modest increase in radius could be camouflaged as a larger age in both cases. For EF Aqr (which has the largest orbital period of the three systems), the primary star shows some signs of being affected. However, the secondary stars can be definitely tagged as unusual because their radii are significantly larger than could possibly be expected for reasonable ages at their lower masses.

V375 Cep is a potentially more interesting test case than those field binaries because the cluster age can be constrained independently using the CMD, and its orbital period is even shorter. Taking the mass and radius of the primary star at face value, an age of about 3.3–3.6 Gyr is indicated, depending on the model. This is consistent with results from isochrone fitting in the CMD (Sandquist et al. 2011). We also measured the rotational velocities of the two stars from broadening of the spectral lines. Because the binary has a short period and has circularized, the stars should have synchronized their spins with the orbit, and the measured rotational velocities are indeed consistent with synchronous rotation for the measured stellar radii. In order to check the possibility that magnetic activity is responsible for the unusually large radius of the secondary, we looked at several indicators. We see very little evidence of spot-induced light curve variations unless the variations occur preferentially in the B bandpass. We looked for signs of X-ray emission in archival data from space-based missions. Although the cluster was observed by XMM-Newton (PI: Verbunt), no source was detected at the position of V375 Cep during a pointing of more than 10600 s. So we do not have corroborating evidence that magnetic activity is responsible for the unusual characteristics of the secondary. A search for emission in the core of the Ca ii H and K lines is probably one of the more promising ways remaining for proving the presence of such activity.

Unfortunately, the primary star in V375 Cep is so far the only "normal" star that can be used to derive the age of NGC 7142 using the eclipsing binary technique. Other groups (Clausen et al. 2009; Torres et al. 2006) have attempted to derive age constraints from inflated stars like V375 Cep B using models with reduced convective mixing length, but this is beyond the scope of this study. Ideally a full analysis for this cluster would make use of three or more stars so that composition questions could be addressed. Helium, for example, is one of the more substantial unknowns affecting an age analysis, and its abundance for stars of super-solar metallicity is still somewhat uncertain. However, the helium abundance can be inferred from the shape of the mass-radius isochrones if the observational data is sufficiently precise (Brogaard et al. 2011, 2012). Brogaard et al. discussed the old, very metal-rich cluster NGC 6791, which has a helium abundance significantly above the solar value. NGC 7142 stars have metal content a little less than halfway between the Sun and NGC 6791. Until we have additional cluster stars for analysis, we will implicitly be using helium enrichment laws assumed by the different model isochrones, meaning that the helium abundance will be super-solar. This enrichment will have an effect on the age determination if the assumed value is significantly in error.

Figures 13 and 14 show a CMD of the cluster with isochrones pinned to the position of V375 Cep A at the mass measured here. Generally speaking, isochrones of the age implied by the binary star analysis are consistent with cluster photometry as well. The details differ between isochrone sets due to differences in physics. NGC 7142 is in a range of ages where the physics of the convective core (including core overshooting and CNO cycle reaction rates) is important.

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The photometry of V375 Cep B is consistent with that of a cluster main sequence star, although the position predicted for a star of its mass from isochrones is only marginally consistent with the photometry. Indications from field binaries (Stassun et al. 2012) are that chromospheric activity tends to increase the stellar radius and decrease effective temperature in a way that leaves the luminosity unchanged. In short period binaries such as V1061 Cyg (Torres et al. 2006), activity induced by forced synchronous rotation also appears to drive similar changes that keep the luminosity approximately constant.

Because of its position in the CMD, the determination of the mass of V375 Cep A is essentially a direct measurement of the turnoff mass for the cluster. Single stars appear to reach slightly bluer colors just before starting their subgiant branch evolution toward the red giant branch, but V375 Cep A is quite close to bluest point on the main sequence, which is the traditional definition of the turnoff. Subsequent evolution is comparatively rapid, and this places a strong upper limit on the age of the cluster. Isochrones with ages of 4 Gyr or above would require a star of V375 Cep A's mass to have evolved significantly to the blue (and then red). We can therefore rule out the much greater age (6.9 ± 0.9 Gyr) determined by Janes & Hoq (2011) in their study of the cluster CMD.

As we discussed in a study of eclipsing binaries in the somewhat younger cluster NGC 6819 (Sandquist et al. 2013), the Dartmouth isochrones (Dotter et al. 2008) are to be preferred among current publicly available isochrones because they include the most up-to-date inputs for physics that affects the evolution of turnoff-mass stars. Since that paper, the PARSEC models (Bressan et al. 2012) have been revised, and contain similar input physics. One important difference between the Dartmouth and PARSEC models and most others is the inclusion of an improved nuclear reaction rate for the CNO cycle reaction ¹⁴N(p, γ)¹⁵O. In addition, stellar model calculations typically include a varying amount of convective core overshooting for stars with masses around that of V375 Cep A. The amount (expressed in units of the pressure scale height H_P) is typically ramped up from zero at a lower mass limit (1.1 M_☉ for the Dartmouth models) to a maximum value at a high mass limit (0.2H_P at 1.3 M_☉). Both physics effects have minimal effects on the CMD except near the cluster turnoff and subgiant branch, where the details of central hydrogen exhaustion in the stars significantly influence the shape of the isochrones. With the possibility of nailing down the isochrones at the position of one or more binary stars, we therefore have leverage to test the physics of the stellar cores. This test would be stronger in NGC 7142 if the effects of differential reddening and field star contamination could be reduced. Until then, the indications are that different isochrone sets can reproduce the cluster turnoff in a qualitative sense.

Before leaving the discussion of this binary star, we use the stars to calculate a distance modulus for the cluster. From the measured radius and effective temperature, we can calculate the bolometric luminosity. For the temperature, we have used the spectroscopic estimate in order to avoid uncertainties associated with the cluster reddening. After applying a theoretical bolometric correction (VandenBerg & Clem 2003), we derive M_V and the distance modulus. The primary star provides the best estimate [(m − M)_V = 12.86 ± 0.07] because its photometry and its effective temperature are better constrained, but the measurement from the secondary star is completely consistent [(m − M)_V = 12.86 ± 0.15] if the star's temperature is derived from the effective temperature of the primary and the temperature ratio from the light curve fits using the analytic limb darkening law. These measurements are in nice agreement with our previous determination using the CMD (12.96 ± 0.24; Sandquist et al. 2011), but are of higher precision and effectively independent of any need for reddening estimates.

5.1.2. V2

5.1.3. V1