DS Andromedae: A Detached Eclipsing Double-lined Spectroscopic Binary in the Galactic Cluster NGC 752^*

The BVR_CI_C data described previously in SM88, and a much smaller number of U observations, constitute the photometric data we used for the current analyses. We now summarize the acquisition and treatment of these data and provide additional detail not included in SM88. All are photoelectric data. The photometry was obtained by S.J.S. in the 1982–83 observing season at the Rothney Astrophysical Observatory (RAO) in Alberta, the McDonald Observatory (McDO) in Texas, and the Table Mountain Observatory (TMO) in California, over the HJD intervals 2445243.8 to 2445384.8, 2445600.8 to 2445604.0, and 2445683.6 to 2445683.8, respectively. The RAO data are differential, obtained with the chopped, gated pulse-counting photoelectric setup designated RADS, for Rapid Alternate Detection System. RADS is described and discussed in Milone et al. (1982), Milone & Robb (1983), Schiller (1986), and Milone & Pel (2011), the last in the context of the historical development of differential photometry. The inconvenient 1.01 day period of DS And necessitated that RADS be used to gather as much data as possible within a season, even on nights that were not quite photometric or when low-grade aurora and air glow were present, although pains were taken to avoid nights with strong auroral activity. Variable and comparison star data were measured alternately, with background sky near the variable and near the comparison stars sampled prior to stellar measurements. The chopping distance between sky and star was kept small to minimize ringing by the secondary mirror as it momentarily came to a stop at each position. The effect on photon counting was minimized further by a selected dead time, during which no pulses could be counted, inserted into the mirror driving function via a potentiometer on the RADS control box. Tests of accuracy and precision achieved with RADS are described by Schiller (1986). The m.s.e. (mean standard error, or rms error) of a single RAO observation was found to be in the range 0.010 to 0.020 mag in V. The number of RADS observations made in each passband were 150, 193, 354, and 197 in I_C, R_C, V, and B, respectively.

The five-filter McDO and TMO data were obtained on larger telescopes (90 and 60 cm, respectively) in darker and more photometric conditions, yielding a typical m.s.e. of a single observation of about 0.005 mag There are 145 observations in each of the I_C, R_C, V, and B passbands, and 94 in U. The reduced, systemic data were standardized to the Johnson–Cousins system through observations at each site of stars in the standard star lists of Landolt (1983). The comparison and check stars for all differential observations were BD +37°425 (F6) and BD +37°450 (G8). The standard deviation of a single observational difference was found to be 0.015 mag in V, and both stars were considered constant within this level of precision. Along with the lists of absolute photometry data, a full list of the extinction coefficients, transformation coefficients, and their uncertainties, the procedure used to obtain them, as well as checks and tests of their reliability, are given in Schiller (1986, pp. 36–43, 142–162ff). These steps fulfill the requirement of the Direct Distance Estimation (DDE) method incorporated in recent versions of the WD program that the photometric data be in a well-calibrated system.

The radial velocity (RV) data acquired at the Dominion Astrophysical Observatory (DAO) and listed in SM88 were obtained from image-tube-enhanced large-grained photographic IIa-O emulsion plates. The spectra have low signal-to-noise ratio (S/N) and display a distorted (S-shaped) cross-dispersion distribution on the plates. The adopted measurement procedure was designed to avoid the most significant curvature at the edges of the spectral lines. The scanner digitized the spectra in 5 μm steps, and care was taken to align the scanner slit perpendicular to the spectrometer dispersion direction projected onto the photographic plate. The length of the scanner slit was narrowed to about one-third the width of the stellar spectrum on the photographic plate. Two separate scans were made of each stellar spectrum, one above and one below the center of the stellar spectrum. The slit length was short enough that the two scans digitized the central, most perpendicular/linear portion of the spectrum and avoided the edges. The lamp comparison spectra, both above and below the stellar spectra, were scanned similarly. The two stellar spectral scans were wavelength-calibrated separately, based on lamp calibration spectra, with the use of a fourth- or fifth-order polynomial fit to remove residual nonlinearities (to less than 1 μm, rms) in the wavelength calibration that still remained from the S-shape effects. After wavelength calibration, the two spectra were combined to improve the S/N ratio. This was carried out for each plate for both the variable and RV standard star spectra. Application of the VCROSS cross-correlation analysis package (Hill 1982) between pairs of velocity standards revealed a precision for a single measurement of ±1.5 km s⁻¹, setting a lower limit for this technique; the upper limit was estimated to be 10–15 km s⁻¹. Further details on the DAO data and analyses can be found in SM88 and Schiller (1986, pp. 43–60).

Initially, analyses were carried out only on the photometric and RV data presented in Schiller (1986) and SM88. New RV data (Mellergaard Amby 2011) were used first to enhance and later to replace the older RV data to obtain more precise results for the dynamic parameters. The Mellergaard Amby (2011) data are listed in Table 1. The spectra were obtained with the Nordic Optical Telescope and the FIES spectrograph in the medium-resolution (R = 46,000) mode (http://www.not.iac.es/instruments/fies/). Higher resolution was precluded by the large rotational velocities of V sin i = 106(3) and 62(2) km s⁻¹ for the two components, in agreement with rotational velocities of stars synchronously rotating with the orbital period, 104 and 63 km s⁻¹, respectively. Exposure lengths of 20 minutes provided sufficient S/N and limited cosmic-ray hits. The data were reduced using the FIEStool package and wavelength calibrations were obtained with ThAr spectra framing the target exposures. The wavelength calibration is good to roughly 100 ms⁻¹. The RVs were calculated from the wavelength-calibrated reduced spectra by the method of broadening functions developed by Rucinski (1999a, 1999b) and employed as described by Lu et al. (2001). In the broadening function, we included the rotation velocity as a free parameter as well as the RVs, as the rotation is the major broadening component and the rotational velocity V sin i is different for the two components. The technique fails during eclipses, where the two spectra overlap heavily; three spectra obtained at these phases were not used in the analyses. Mellergaard Amby (2011) analyzed these RV data (see Section 5). RV data and varied photometry suites were used simultaneously in all runs.

The 2013 version of the WD program (Wilson & Van Hamme 2013 and references therein) was used exclusively. In WD, there are general level weights, more specific curve weights, and, for each datum, an individual weight. The level weight switch, labeled "NOISE" in WD, was set at 1, appropriate for photon statistics, for all runs. The WD Differential Corrections routine (hereafter DC), output file provides two tables to measure the goodness of the fitting: "Standard Deviations for Computation of Curve Dependent Weights" for each curve, and the "Input–Output in F and D formats." The first table provides sigmas for each curve, in specified VUNITS (here, 100 km s⁻¹) for RV data and units of erg cm⁻³ s⁻¹ for photometric data; they must be inserted into the DC input file for the following run. The weights are computed internally. These weights are critical to the adequate relative weighting of the RV and photometric data. A software switch (KSD = 1) when set in the input file automatically updates the curve weights after each iteration during a run. Multiple iterated corrections of the main set, and subsets of uncorrelated or weakly correlated parameters, are always obtained. The second table lists input and output parameters and their standard errors, and, at the foot of this table, "the mean residual for input values," a mean weighted residual of all  the curves, in which weights are applied so that both light and RV curves contribute appropriately to the resulting, mixed-unit, weighted mean residual. This $\langle {r}_{\mathrm{in}}\rangle$ is the fitting datum that we used to assess the overall goodness of fit of each converged solution. Parameter correlations were dealt with in two ways. First, the Marquardt (1963) damping constant, λ, was set to 10⁻⁶ to lessen the effects of parameter correlations in the full set on the damped least-squares results. Second, several subsets of weakly correlated parameters were adjusted in each run. The subsets were rarely needed, as the multiple iteration operation usually produced full convergence within six runs. Each run consisted of 30 or 40 iterations. We considered full  convergence to be achieved when adjustments became smaller than the probable errors, i.e., less than two-thirds of the standard deviations listed in the DC output files for all adjusted parameters.

In all cases, we adjusted the semimajor axis, a; the systemic or gamma velocity, V_sys; the orbital inclination, i; the temperature of the cooler star, T₂; the modified Kopal potentials, Ω_1,2; the mass ratio, q = M₂/M₁ (where component 1 is the primary star, taken here as that eclipsed at primary minimum, thus the hotter component); the epoch, t₀, specifying the instant of a particular conjunction and primary eclipse minimum; and the orbital period, P. In many runs, T₁, was adjusted also. In some runs, the passband luminosity parameters, ${{L}^{\lambda }}_{1}$, were adjusted; otherwise, the logarithm of the distance, log d, was adjusted. In most runs, the third-light parameter, ${{{\ell }}_{3}}^{\lambda }$, was adjusted. The initial values of the limb-darkening coefficients were taken from Van Hamme's (1993) table via a desk-top GUI devised by D. Terrell (1995, private communication). For all trials, we used logarithmic limb-darkening coefficients (LD_1,2 = −2), internally computed beyond the first iteration. These values closely matched the flux-weighted limb-darkening coefficients produced by the LC routine. Usually the albedo parameters, ALB1, 2, were fixed at 1.000, appropriate for the radiative envelopes of stars earlier than the Sun in spectral type. In Section 3.8, we discuss trials where these were set to values appropriate for convective envelopes.

Modeling began with Model 0, the BVR_CI_C and RV data listed in Schiller (1986), with initial values for the parameters from SM88. Note that by "model" we mean any WD data configuration run to convergence. As work progressed, S.J.S. and E.F.M. became aware of the work of Th.M.A. and obtained permission to make use of his RV data (Section 2.2). Both RV sets were incorporated in the modeling, and these data were used in the preliminary results reported by us in 2014 (Milone et al. 2015), but it soon became clear that Th.M.A.'s RVs alone would provide greater precision in the dynamically sensitive parameters (a, V_sys, i, q, Ω_1,2, t₀, P). From Model 21 on, the Mellergaard Amby (2011) RVs alone were used in the modeling. In the earliest runs, the temperature of the hotter star was fixed at 6775 K, the temperature adopted by Schiller (1986) and Schiller & Milone (1988) from the early-F spectral classification and apparent color index of the system according to Table 1 of Popper (1980). Other fixed values assumed for T₁ were 6795 K, to test the effects of a small change, and 6964 K, found in the newer tables and formulae of Flower (1996), as corrected by Torres (2010). Later, when ${{L}^{\lambda }}_{1}$ was adjusted in four-passband runs, T₁ values were fixed at the means of two- or three-passband runs in which log d and both temperatures had been adjusted. We call the fixed T₁ runs the "1T" models. Modeling thus consists of two types of runs: those involving all four passbands, in which the passband luminosities, T₂, and the other parameters are adjusted but not the log d parameter; and those involving only two or three passbands, in which both T₁ and T₂ and the log d parameter, along with other parameters except  the passband luminosities, are adjusted. The reason for the latter scheme is the temperature–distance (T–d) theorem as discussed by Wilson (2007) and Wilson (2008, their Sections 2–4), which, to avoid under- or overconditioned circumstances, specifies the number of passbands run simultaneously if log d is one of the adjusted parameters. With this operational dichotomy, several groups of models were explored. We label models in which both temperatures and log d were adjusted "2T" Models.

Models 1–27 tested the effects of different interstellar extinction coefficients, A_V, and metallicity, [M/H], across the ranges reported in studies of the cluster NGC 752 (see Section 6). Three values of both quantities were tested in early trials: A_V = 0.075, 0.100, 0.125 mag and [M/H] = −0.1, 0, +0.1, in all combinations. The resulting effects on the parameters, including the distance, were found to be slight. Models with A_V = 0.125 and [M/H] > 0 yielded larger mean residuals and so were slightly less favored. The trials suggested that 0.075 ≤ A_V ≤ 0.100 and −0.1 ≤ [M/H] ≤ 0.0, in accordance with the trends of the cluster studies. It should be noted that Models 0 to 27 did not include adjustments for third light. In most of the later runs, A_V was fixed at 0.100, and [M/H] was fixed at 0, the solar value. We also carried out grid determinations for A_V, adjusting all parameters, including third light, for the suite of VB and RV data, where each individual RADS datum was weighted relative to non-RADS data (see Section 3.4). Fixed A_V values were stepped from 0.05 to 0.15 in units of 0.0125. A fitting of the squares of the mean residuals to a parabola yielded a minimum at A_V = 0.1059, justifying the use of the values 0.1 for the bulk of the previous trials. The conclusions about [M/H] were tested further (see Section 3.6), along with the effects of different calibration constants (see Table 3 of Wilson & Van Hamme 2013). With an adopted solar abundance and A_V = 0.100 interstellar extinction values, iterations between the four-passband, ${{L}^{\lambda }}_{1}$- adjusted runs of Model 41 and the three-passband, log d adjusted runs were performed in the Model 28 series to obtain fully consistent values of T₁ and ${{L}^{\lambda }}_{1}$ in the final averages of the runs where log d was adjusted. This was not strictly necessary, as not having fully consistent ${{L}^{\lambda }}_{1}$ and ${{L}^{\lambda }}_{2}$ input values does not affect other parameters, but the input values of ${{L}^{\lambda }}_{1}$ as well as the computed values of ${{L}^{\lambda }}_{2}$ are listed in the LC output, so it is desirable for them to be seen consistent with other quantities.

As no previous work had produced evidence of third light in the DS And system, we did not immediately explore that possibility in the present study. However, in checking over a problem we had with data input, R. E. Wilson ran one of our input files adjusting third light and found it to be significant. Subsequently, from Model 28 onward, with a few exceptions mentioned later, third light was adjusted, and, in nearly every case where this was done, it was found to be significant at about the 10% level in each passband, including U. This result implies that DS And appears slightly dimmer and farther than we at first reported: an average detached model distance of 436(1) pc (Milone et al. 2015), where the uncertainty is the formal m.s.e. of the mean, signifying mainly the lack of dispersion among models. The estimated systematic error is an order of magnitude greater. Nevertheless, most of the adjusted parameters were not significantly affected by the adjustment of third light.

The RADS and non-RADS data (hereafter, R & n-R in this section and in tables)  were obtained at slightly different epochs over the 1982–83 observing season (see Section 2.1), and these contribute unevenly to the folded light curves. Accordingly, experiments were undertaken to see if separating the R & n-R data or weighting one set relative to the other would improve the fittings. R & n-R data in each passband were entered as separate bands in Models 28A and 37–40. For all runs, curve weights (see Section 3.1) were applied to each band of all runs. In Models 40a–f (the I_CR_C, VB, I_CV, I_CB, R_CV, and R_CB runs, respectively), the R & n-R data were divided into separate bands, and the sigma of each band was inserted into the next run's DC input file, with unit individual weights in all bands. The parameter means of these models, which we call Model 40, produced the smallest mean residual, $\langle {r}_{\mathrm{in}}\rangle$, of any of the models, with extremes of 2.77 ×10⁻⁸ for Model 40a and 4.02 × 10⁻⁸ for Model 40b, and an average of 3.53(20) × 10⁻⁸ for Model 40. The ratio of the inverse square of the mean of the full curve sigmas in each passband of the R & n-R light curves in Models 37 (RADS) and 38 (non-RADS) was used as the basis to compute and apply a relative weighting to the RADS data for some runs. These were found to be 0.2391, 0.1569, 0.0933, and 0.0884, for the I_C, R_C, V, and B passbands, respectively. In Model 28B, three-passband suites with combined R & n-R data, these weights were applied to each individual RADS datum; non-RADS data had unit weight. The Model 28B fitting precision, $\langle {r}_{\mathrm{in}}\rangle$ = 4.396 × 10⁻⁸, was better than that for most but not the best models. To test the robustness of the Model 40 solutions, we ran all four passbands, again divided into R & n-R bands (for a total of eight bands, in formal violation of the T–d theorem): Model 40''. Model 40'' fittings are not quite as good as those with fewer bands, possibly due to overconditioning, but the parameters are consistent. The adjusted parameters for Models 40a–f, 40'', and 40 are given in Table 2. The m.s.e.'s are below each weighted mean, and the errors computed from the inverse square root of the sums of the squares of the inverse standard deviations of each parameter follow in brackets. The adopted uncertainty is the larger of the two in each case; systematic error is not included. The values for third light normalized by the sum of the light of both stars and third light are listed for both R & n-R bands in the lower part of the table, and the weighted means of the combined R & n-R sets are given in footnote a. The light curves and RV curves of Models 40a and 40b are plotted in Figures 1(A)–(C) with their residuals. VB passbands with the weighted RADS data and the non-RADS data combined were used in the A_V grid search (see Section 3.2). The A_V grid-fitting curve is shown in Figure 2. Figures 1(b) and (c) show that the R & n-R data are not fully consistent. Modeling them separately, or together but applying the derived individual weights to the RADS data, produces fitting curves that overall do not appear to match the data perfectly. The non-RADS data have less scatter, but less phase coverage at maximum light and none at secondary minimum. The RADS data are relatively sparse in the first maximum of the I_C curve, and although suggestive of an asymmetry in the maxima (the O'Connell effect—see Davidge & Milone 1984), the enhancement at 0.25^p is absent in the R_C light curve. SM88 suggested that this and mid-eclipse discrepancies were due to variability in the system on timescales shorter than the two months required to observe the full light curve. Attempts to model the I_C anomaly have not been satisfactory, given that at least two light curves are required to model the base temperatures of the two stars and log d. A new complete light curve in this passband would be desirable. Nevertheless, both R & n-R data treatments yielded the smallest mean residuals of all models, and parameter values are consistent with those of group means.

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Table 2.  DS Andromedae Model 40 Adjusted Parameters^a

Model	40a	40b	40c	40d	40e	40f	40''	40
(Passbands)/	(IcRc)	(VB)	(IcV)	(IcB)	(RcV)	(RcB)	(IcRcVB)	$\langle$IcRcVB $\rangle$
Parameter							(eight bands)	(columns 2–7)
a	5.941	5.923	5.945	5.918	5.938	5.910	5.877	5.930
(RSun)	0.036	0.035	0.034	0.037	0.034	0.037	0.039	0.006[0.015]
Vsys	+8.09	+8.21	+8.16	+8.09	+8.17	+8.08	+8.05	+8.13
(km s−1)	0.56	0.63	0.58	0.57	0.59	0.58	0.60	0.02[0.24]
i	89.47	88.86	89.01	88.71	89.56	89.81	89.44	89.35
(deg)	0.53	0.37	0.46	0.45	0.40	0.26	0.28	0.19[0.16]
T1	6884	7052	7014	7027	7162	7091	7059	7056
(K)	81	30	38	21	51	20	16	21[12]
T2	5752	6019	5983	5894	6087	5968	5987	5971
(K)	66	31	38	36	46	29	21	33[15]
Ω1	3.566	3.577	3.580	3.572	3.579	3.567	3.551	3.574
	0.017	0.015	0.016	0.017	0.015	0.017	0.016	0.003[0.007]
Ω2	4.274	4.321	4.304	4.319	4.314	4.271	4.238	4.302
	0.068	0.051	0.063	0.072	0.054	0.059	0.052	0.009[0.024]
q	0.659	0.656	0.663	0.653	0.661	0.650	0.642	0.657
	0.009	0.009	0.009	0.010	0.009	0.010	0.010	0.002[0.004]
t0	.403315	.401926	.402397	.403477	.402252	.403277	.402898	.402813
HJD	.001093	.001232	.001150	.001119	.001149	.001112	.001130	.000265
fractn								[.000465]
P(day)	188143	189627	189052	188180	189142	188299	188576	188697
1.0105+	001137	001301	001206	001171	001211	001171	001196	000247[000488]
log d	2.666	2.675	2.676	2.668	2.692	2.682	2.675	2.678
(pc)	0.011	0.006	0.008	0.008	0.008	0.006	0.004	0.004[0.003]
(m0 – M)	8.329	8.373	8.380	8.338	8.462	8.412	8.376	8.390
(mag)	0.056	0.030	0.040	0.042	0.039	0.028	0.020	0.018[0.015]
ℓ3(Ic)R	0.113	⋯	0.100	0.086	⋯	⋯	0.103	0.100
(ℓ1+2+3)	0.021	⋯	0.019	0.023	⋯	⋯	0.008	0.007[0.012]
ℓ3(Rc)R	0.112	⋯	⋯	⋯	0.099	0.101	0.090	0.103
(ℓ1+2+3)	0.019	⋯	⋯	⋯	0.015	0.013	0.007	0.004[0.009]
ℓ3(V)R	⋯	0.099	0.112	⋯	0.106	⋯	0.105	0.103
(ℓ1+2+3)	⋯	0.010	0.018	⋯	0.014	⋯	0.006	0.004[0.007]
ℓ3(B)R	⋯	0.097	⋯	0.097	⋯	0.109	0.104	0.101
(ℓ1+2+3)	⋯	0.009	⋯	0.022	⋯	0.012	0.006	0.004[0.007]
ℓ3(Ic)n	0.110	⋯	0.097	0.083	⋯	⋯	0.097	0.098
(ℓ1+2+3)	0.021	⋯	0.019	0.023	⋯	⋯	0.007	0.008[0.012]
ℓ3(Rc)n	0.116	⋯	⋯	⋯	0.100	0.104	0.092	0.105
(ℓ1+2+3)	0.020	⋯	⋯	⋯	0.015	0.013	0.007	0.004[0.009]
ℓ3(V)n	⋯	.0.084	0.098	⋯	0.092	⋯	0.092	0.088
(ℓ1+2+3)	⋯	0.009	0.018	⋯	0.015	⋯	0.006	0.004[0.007]
ℓ3(B)n	⋯	0.083	⋯	0.086	⋯	0.097	0.090	0.088
(ℓ1+2+3)	⋯	0.009	⋯	0.022	⋯	0.012	0.006	0.005[0.007]
$\langle \mathrm{res}.\rangle$ × 10−8	2.7720	4.0207	3.3160	3.8900	3.2863	3.8837	3.5380	3.5281

Note.

^aThe Model 40 entries in the last column are the weighted means of Models 40a through 40f; the m.s.e.'s of the means are below (followed by those from the combination of parameter s.d.'s, in brackets). They indicate only the dispersion among the runs and contain no systematic error estimates (see Sections 3.4 and 4). The third-light parameter values have been normalized by the total system light (at phases 0.25 and 0.75). The first four ℓ₃(pb) rows are the RADS bands values, the last four are non-RADS bands values. The weighted means of both bands are I_C: 0.099(1); R_C: 0.104(1); V: 0.095(7); B: 0.094(7).

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Spots were introduced to model the largest RV1 residuals in Figure 1(a) as well as the slight light-curve enhancements in Figures 1(b) and (c) and described in Section 3.4. In spot modeling, it is typical to place cool spots on the cooler star, but in this system, that model did not converge. Thus, to improve the fitting of the RV₁ data in the first quadrature by weighting some disk grid elements to simulate a velocity distortion, a cool spot was placed on star 1 (in Models 42A, 42B, and 43r), and for this spot, start and stop times were set for the RV data interval only. For the most noticeable enhancement in maximum I (that following the primary minimum) seen in the I_C light curve (Figure 1(b)), a hot spot was placed on star 1 (Models 42–50). Usually all four parameters of a spot were adjusted. In two-spot models, usually only one parameter was adjusted per spot, and the parameters cyclically adjusted until consistency was obtained. The moments of onset and end of the spots were varied prior to runs to test the effects of spots on each of the observing segments, but no improvements were found. The latter spot parameters were relatively poorly determined, and the RV fitting not noticeably improved, so the two-spot models were abandoned. In Model 51, a hot spot was placed on star 2's opposite hemisphere. In Model 52, a single cool spot was placed on star 1 but on the opposite hemisphere from that assumed for models with a hot spot on star 1. Star 2 could be expected to have a deeper convection zone, if present, than star 1, but model runs with a cool spot on star 2 failed to converge. Third light was removed for Models 47 and 48 to see if spots alone could provide a better fit. None of the spot models, with or without third light, improved the fit significantly. The mean parameters of this group of models are listed in Column 7 of Table 3; note the large $\langle {r}_{\mathrm{in}}\rangle$ value. A pdf of the full spreadsheet Table 3 in the repository shows the spot parameters of all models. The adopted model does not include spots.

Table 3.  DS Andromedae Adjusted Parameters Summary^a

Group/	2-T	ARV	Av = .1	Solar	Third	Spot	Non-sp	R/nR	Wtd Means
Parameter	Models	Models	Models	[M/H]	Light	Models	Models	Cases
a	5.862	5.925	5.933	5.933	5.937	5.948	5.934	5.917	5.940
(RSun)	0.015	0.006	0.007	0.007	0.005	0.003	0.007	0.011	0.004
Vsys	+7.34	+8.13	+8.14	+8.14	+8.15	+8.16	+8.15	+8.11	+8.15
(km s−1)	0.22	0.01	0.01	0.01	0.01	0.01	0.02	0.02	0.01
i	87.43	88.49	88.92	88.87	89.55	89.08	89.23	89.23	89.32
(deg)	0.30	0.26	0.24	0.25	0.07	0.33	0.22	0.08	0.17
T1	7059	7063	7067	7064	7070	7053	7071	7046	7064
(K)	6	8	9	9	8	11	12	21	3
T2	5931	5929	5928	5921	5962	5946	5926	5954	5950
(K)	13	17	23	25	08	14	34	18	6
Ω1	3.542	3.593	3.595	3.595	3.592	3.608	3.587	3.575	3.595
	0.012	0.004	0.004	0.005	0.006	0.005	0.005	0.012	0.005
Ω2	4.252	4.355	4.334	4.338	4.286	4.352	4.299	4.282	4.305
	0.034	0.017	0.018	0.019	0.008	0.033	0.014	0.015	0.010
q	0.624	0.656	0.658	0.658	0.659	0.656	0.658	0.654	0.657
	0.008	0.002	0.002	0.002	0.002	0.001	0.002	0.003	0.001
log d	2.657	2.664	2.670	2.669	2.690	2.670	2.677	2.675	2.673
(pc)	0.004	0.004	0.004	0.005	0.008	0.009	0.003	0.004	0.003
t0	.399622	.402897	.402789	.402793	.402663	.402679	.402751	.402719	.402756
2436142+	.001099	.000071	.000071	.000071	.000058	.000098	.000070	.000083	.000056
P (day)	92165	88605	88693	88703	88776	88791	88733	88711	88729
1.010514+	1189	58	57	57	33	60	62	61	40
$\langle r\rangle$ × 10−8	7.907	9.469	8.858	8.789	9.337	10.511	8.623	5.198	8.587

Note.

^aBelow each group mean is the m.s.e. of the mean, signifying mainly the dispersion among models of that group but containing no estimate of the significant systematic error. Column 10 shows the grand means of the weighted means of columns 3 through 9, and the same qualification for its m.s.e.'s applies. The square root of the sum of the inverse squares of the m.s.e.'s of the means is usually smaller; for parameters a to log d these are, respectively, a, 0.002; V_sys, 0.004; i, 0.05; T₁, 4; T₂, 6; Ω₁, 0.002; Ω₂, 0.005; q, 0.001, log d, 0.002, t₀, .000028; and P, 19 × 10⁻⁸. A pdf copy of the complete spreadsheet version of Table 3 can be found in the Zenodo repository 10.5281/zenodo.2553042. See the text for details.

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The DDE algorithm deals with absolute units and requires a calibration constant (CALIB in Wilson & Van Hamme 2013) for each passband. The calibration constants are used when log d is adjusted. The constants for V and B used for all previous modeling were 0.36949 for V and 0.62350 for B (Wilson 2015, private communication), 0.122 for I_C and 0.225 for R_C (Bessell 1979), all in units of erg s⁻¹ cm⁻³. The effects of changing both metallicity and passband calibration constants were explored in Models 28C and 28D. A_V = 0.1059 (see Sections 3.2 and 3.4, and Figure 2) was adopted for these trials. The test series made use of only two passbands, ensuring strict adherence to the T–d theorem (Section 3.1). The designations, calibration constants, and the sources from which they were derived were taken from Wilson & Van Hamme (2013, Table 3, reproduced in the detailed Section 3, in the repository). For each of the listed models, runs were made for each of the three metallicities −0.1, 0, 1, and +0.1, signified by suffixes a, b, and c, respectively, in Tables 4 and 5. The last two columns of Table 4 contain $\langle {r}_{\mathrm{in}}\rangle$ means for the calibration set, i.e., averaged across the metallicities, and the m.s.e. of the means. The mean residuals of the three metallicity runs for each calibration constant are not significantly different. There is no significant difference in $\langle {r}_{\mathrm{in}}\rangle$ between the runs with the R. E. Wilson (2015, private communication) and the Wilson et al. (2010) calibration constants, but those of the former are consistently smaller. Those for the other pairs of calibration constants are significantly larger.

Table 4.  Models 28Cn and 28Dn Metallicity and Calibration Constant Trial Results^a

Model	[M/H]	$\langle {r}_{\mathrm{in}}\rangle$ × 10−8	Mean cal n $\langle {r}_{\mathrm{in}}\rangle$ × 10−8	eMean × 10−8
28C1a	−0.1	4.71649
28C1b	0.0	4.71141	4.70260	0.01145
28C1c	+0.1	4.67989
28C2a	−0.1	5.18844
28C2b	0.0	5.20415	5.20268	0.00784
28C2c	+0.1	5.21546
28C3a	−0.1	4.98328
28C3b	0.0	4.99198	4.98632	0.00283
28C3c	+0.1	4.98370
28C4a	−0.1	4.76459
28C4b	0.0	4.77593	4.76320	0.00779
28C4c	+0.1	4.74907
28C5a	−0.1	4.73450
28C5b	0.0	4.73586	4.72445	0.01073
28C5c	+0.1	4.70300
28D1a	−0.1	2.98538
28D1b	0.0	2.98791	2.99006	0.00349
28D1c	+0.1	2.99689
28D2a	−0.1	2.87757
28D2b	0.0	2.87112	2.87095	0.00387
28D2c	+0.1	2.86416
$\langle 28\mathrm{Cb}\rangle$ runs	0.0	⋯	4.88387	0.09422
$\langle 28\mathrm{Db}\rangle$ runs	0.0	⋯	2.92951	0.08258

Note.

^aColumns 4 and 5 show the mean residuals and their uncertainties for the three metallicity models (suffixes a, b, and c, respectively) of each calibration constant. The mean residuals for the Model 28Ca,c and 28Da,c runs are 4.87746(9138), 4.86622(10268) × 10⁻⁸; and 2.93146(7623), 2.93052(9385) × 10⁻⁸, respectively.

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Table 5.  DS Andromedae [M/H] and Calibration Test Models 28Cn,Dn Adjusted Parameter Means^a

Model/	28C1	28C2	28C3	28C4	28C5	28D1	28D2	[M/H]	[M/H]	[M/H]
Parameter								−0.1	+0.0	+0.1
								$\langle \mathrm{Cn}\rangle$ b	$\langle \mathrm{Cn}\rangle$ b	$\langle \mathrm{Cn}\rangle$ b
a	5.956	5.956	5.958	5.956	5.957	5.939	5.944	5.956	5.957	5.957
(RSun)	0.002	0.001	0.001	0.001	0.000	0.002	0.002	0.002	0.001	0.002
Vsys	+8.35	+8.35	+8.37	+8.36	+8.34	+8.07	+8.10	+8.35	+8.35	+8.37
(km s−1)	0.01	0.00	0.01	0.02	0.06	0.01	0.01	0.00	0.01	0.01
i	89.46	89.78	89.42	89.12	89.15	88.93	89.09	89.88	89.11	89.27
(deg)	0.10	0.19	0.25	0.33	0.10	0.06	0.20	0.10	0.10	0.24
T1	7052	7510	7539	7327	7088	6817	7275	7318	7287	7305
(K)	13	13	14	74	13	6	14	114	100	105
T2	5876	6210	6230	6079	5903	5731	6081	6061	6039	6080
(K)	24	10	10	52	24	8	3	88	77	68
Ω1	3.570	3.569	3.575	3.567	3.570	3.569	3.568	3.572	3.571	3.567
	0.002	0.001	0.001	0.003	0.001	0.001	0.002	0.001	0.002	0.002
Ω2	4.289	4.271	4.288	4.286	4.295	4.285	4.281	4.279	4.286	4.293
	0.005	0.003	0.005	0.005	0.007	0.002	0.004	0.002	0.004	0.008
q	0.664	0.665	0.666	0.665	0.662	0.658	0.663	0.665	0.664	0.663
(M2/M1)	0.001	0.000	0.001	0.000	0.002	0.001	0.002	0.000	0.001	0.002
t0	.4014	.4013	.4013	.4013	.4015	.4033	.4032	.4014	.4014	.4013
HJDfr.	.0001	.0000	.0000	.0001	.0001	.0000	.0000	.0000	.0000	.0001
P (day)	19025	19031	19035	19039	19018	18813	18822	19079	19025	19037
1.0105+	00005	00002	00003	00013	00006	00002	00002	00002	00004	00011
log d	2.681	2.733	2.746	2.706	2.685	2.658	2.715	2.706	2.710	2.715
(pc)	0.003	0.003	0.003	0.001	0.002	0.002	0.000	0.013	0.012	0.013

Notes.

^aMean standard errors are given for each model below the parameters. The inverse square root of the sums of the inverse squares of the individual-run errors is usually similar but sometimes may exceed the m.s.e. of the mean. The robustness of most parameters is striking but T_1,2 are particularly sensitive to the choice of calibration constant, as the m.s.e.'s of their means indicate. ^bThe mean parameters for the same [M/H] in columns 9–11 are for 28C models only.

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Some VB runs yielded T_1,2 that were too high for the early-F observed spectral type of the system, so their use here can be rejected as unphysical, further supporting the selection of the R. E. Wilson (2015, private communication) V and B calibration constants. For the I_C and R_C passbands, the constants of Bessell et al. (1998) produced consistently smaller mean residuals compared with the Bessell (1979) calibration constants, but yielded temperatures that were, again, too high. The resulting high temperatures skew the radiative parameter means across the models as can be seen in the last three columns of Table 5. The m.s.e. of T_1,2 are ∼100 and 80 K, and those of log d ∼ 0.012, setting upper limits for systematic error due to calibration constant selection.

Suspecting that the secondary star may exhibit a large reflection effect, due to the proximity in this system of a larger and hotter companion, we ran the template model 28C1b with enhanced reflection (Wilson et al. 2010), mref  = 3, keeping the number of reflections at (nref =) 2. This model (28C1bR) showed significant changes to some of the parameters, but the overall fit was no better. The mean residual for this model was 4.758 × 10⁻⁸, whereas $\langle {r}_{\mathrm{in}}\rangle$ = 4.711 × 10⁻⁸ for Model 28C1b. As the temperature of star 2 had now slipped below that of the Sun (in WD 2013, T_Sun = 5779 K is assumed), Model 28C1bR was rerun with convective envelope coefficients A₂ = 0.500 and g₂ = 0.32, and designated Model 28C1bRC. This did not improve the fitting ($\langle {r}_{\mathrm{in}}\rangle$ = 4.964 × 10⁻⁸). The insertion of nonsynchronous rotation factors did not make significant changes to the parameters and to their uncertainties: we set F₁ = P_orb/P_rot,1 = 1.047 and F₂ = P_orb/P_rot,2 = 1.006, appropriate for v₁ = 103 and v₂ = 63 km s⁻¹, slightly off synchronism. Running this last model with an additional reflection, nref  = 3, again produced no significant changes and only a slight improvement in $\langle {r}_{\mathrm{in}}\rangle$. The means of the adjusted parameters of two trials are included in Table 6 (column 2).

Star 1 had been consistently the larger component in all detached model run results. Thus, we initially tested semidetached (SD) models in mode 4, in which the potential of Star 1 is not adjusted but internally set to that of its inner Lagrangian surface; they failed to converge.

After the runs described in Sections 3.1–3.7 were completed, we sought a model that would not only provide better fittings to the data but would also help with the secondary luminosity problem, to be discussed in Section 6. We used a template with detailed reflection, convective envelope parameters for A₂ and g₂ and ran tests in mode 5, in which Ω₁ is adjusted while Ω₂ is internally computed by the program. After many runs, convergence was achieved with ${{{\ell }}_{3}}^{V,B}$ fixed at zero: Model 53_SD5a, the parameters of which are listed in column 3 of Table 6 under header "53a." The errors, beneath the parameters, are their standard deviations, from the DC output files. With the I_CR_C suite and ${{{\ell }}_{3}}^{{Ic},{Rc}}$ fixed at 0, convergence was achieved after 15 runs. This is Model 53_SD5b (header "53b" in Table 6). The two sets of SD models differ significantly from each other and from the detached solutions. To check the robustness of the SD solutions, third light was again adjusted and after four runs, convergence was achieved for the I_CR_C passband suite, Model 53_SD5b1 (header "53b1" in Table 6). For the VB passband suite, convergence was also found, Model 53_SD5a1 (header "53a1" in Table 6), but in this case, the ${{{\ell }}_{3}}^{V}$ value was not significant and the ${{{\ell }}_{3}}^{B}$ value only marginally so. In all the SD5 models studied thus far, T_1,2 were larger than those in the detached models, but had been run with convective envelope parameters A₂ and g₂ Setting both to 1.00, appropriate for radiative envelopes (leaving aside the issue of the appropriateness of such an envelope for a lobe-filling star), we obtained convergence for the I_CR_C passband suite in the next run; this is Model 53_SD5b2 ("53b2" in the Table 6 header). No convergence was found for the VB passband suite for the radiative envelope case. We had previously found that restricting the range of phases around quadrature from 0.10^P to 0.06^P for the calculation of the standard errors at maximum light improved $\langle {r}_{\mathrm{in}}\rangle$ for VB suite runs. I_CR_C suite runs showed no such improvement, so the range was left at 0.1^P. Model 53_SD5a1 rerun with the range reset to 0.10^P is Model 53_SD5a2 (Table 6 header "53a2"). The results are not significantly different. Although parameter errors may differ, in all cases, $\langle {r}_{\mathrm{in}}\rangle$ for the converged semidetached models exceeded those of Model 40 detached models. These results confirm the impression given by the rounded, transit-like primary minimum and the occultation-like secondary minimum already seen in the SM88 plots. We therefore favor the detached mode models.

Given the presence of third light, Models 28C1b and 28D1b were rerun with the period variation, or P-dot (for dP/dt), parameter adjusted along with all the other parameters. The two sets of trials show significant dP/dt terms, but one only marginally, and they disagree. Only the period and epoch, among the other parameters, were significantly changed. For Model 28C1bP, involving VB passbands, $\langle {r}_{\mathrm{in}}\rangle$ = 4.462 × 10⁻⁸ and critical parameters t₀ = 2436142.2819(183), P₀ = 1.01053838(295), and dP/dt = −1.393 (211) × 10⁻⁹ days/day. With detailed reflection, nref  = 3, Model 28C1bPR yielded $\langle {r}_{\mathrm{in}}\rangle$ = 4.481 × 10⁻⁸, t₀ =2436142.2798(185), P₀ = 1.01053864(299), and dP/dt =−1.405(214) × 10⁻⁹. For Model 28D1bPR, with I_CR_C passbands, $\langle {r}_{\mathrm{in}}\rangle$ = 2.879 × 10⁻⁸, t₀ = 2436142.4493(204), P₀ = 1.01051138(329), and dP/dt = +5.330(2356) × 10⁻⁹. The results for the VB and I_CR_C suites are significantly different, casting doubt on the reality of the dP/dt term and making determination of parameters of any third body a matter for future investigation. A recent major O − C study is not supportive: an examination of all the times of minimum over the interval 1932 to 2015 by R. H. Nelson (2018, private communication; data available at http://www.aavso.org/bob-nelsons-o-c-files). With weightings of 0.1 applied to visual and photographic data and 1 for photoelectric and CCD times of minima, he found a marginally significant but very small, quadratic term, corresponding to dP/dt = +2.715(1151) × 10⁻¹¹ day/day.

Modeling was conducted in two stages. All five passbands as well as the two RV bands were run with L₁, but not log d, adjusted in a Model 41 template file. The final converged values of the parameters were then used as starting parameters for subsequent runs of suites of two-band photometric and the RV curves, namely Models 55a (I_CR_C), 55b (I_CV), 55c (I_CB), 55d (I_CU), 55e (R_CV), 55f (R_CB), 55g (R_CU), 55h (VB), 55i (VU), and 55j (BU), in which the passband luminosities were not adjusted but log d was. The U bands were run with calibration constant 0.4221 (R. E. Wilson 2015, private communication). The weighted means of the adjusted and absolute parameters define Model 55. Excluding Models 55e and 55j, with parameters T_1,2 and ℓ₃, which exceeded Chauvenet's criterion, we define Model 55'. Its $\langle {r}_{\mathrm{in}}\rangle$ exceeded that of Model 40 but the parameters agree within errors. We conclude that the exclusion of the U data for the bulk of the trials has had minimal impact on the adjusted parameters, and the increased uncertainty in many parameters when U data are included justifies their exclusion from the bulk of the modeling trials. Still, the exercise provided U curve parameters. The relative U passband luminosity [L₁/(L₁₊L₂)]^U = 0.912(3), and(ℓ₃/ℓ₁₊₂₊₃)^U = 0.122(9), where the errors are the m.s.e.'s of the means. For further modeling details, see the document "DS ANDROMEDAE CURVES ANALYSES: EXTENDED DISCUSSION VERSION" with references and a pdf copy of the expanded online spreadsheet version of Table 3 in Zenodo 10.5281/zenodo.2553042.

Table 10 shows a small effect of the tested range of metallicities and a relatively large effect of different calibration constants on the radiative parameters but no effect on the masses and radii. Columns 9, 10, and 11 indicate a trend for luminosity and distance to increase, and a very slight trend for third light to decrease, with increasing [M/H]. The absolute parameters derived from the detailed reflection and semidetached models (see Section 3.8) are listed in Table 11. Note the apparent absence and only marginally significant amount of third light in V and B, respectively, in the 53a models. The large range of values for several of the parameters of the 53b (I_CR_C) models is characteristic. Semidetached model parameters are excluded from group and grand means.

Table 11.  DS Andromedae Absolute Parameters of Detailed Reflection and Semidetached Models^a

Model/Parameter	28CR3	53a	53a1	53a2	53b	53b1	53b2
M1/MSun	1.651	1.966	1.961	1.963	2.025	1.705	1.664
M2/MSun	1.117	1.331	1.328	1.329	1.463	1.160	1.133
R1/RSun	2.119	1.979	1.969	1.970	1.669	1.320	1.364
R2/RSun	1.237	2.179	2.178	2.178	2.365	2.182	2.068
Mbol1	2.26	2.22	2.37	2.37	2.27	3.08	3.04
Mbol2	4.34	2.46	3.10	3.10	2.27	2.73	2.74
log g1	4.00	4.14	4.14	4.14	4.30	4.43	4.39
log g2	4.30	3.89	3.88	3.89	3.86	3.82	3.86
${{\mathscr{L}}}_{1}$/${{\mathscr{L}}}_{\mathrm{Sun}}$	9.91	10.28	8.95	8.95	9.82	4.66	4.83
	0.02	0.05	0.04	0.04	0.05	0.02	0.02
${{\mathscr{L}}}_{2}$/${{\mathscr{L}}}_{\mathrm{Sun}}$	1.46	8.24	4.57	4.57	9.82	6.43	6.37
	0.00	0.04	0.02	0.02	0.05	0.03	0.03
ℓ3(p1)	0.067	⋯	0.001	0.001	⋯	0.278	0.336
(ℓ1+2+3)	0.010		0.012	0.012		0.021	0.015
ℓ3(p2)	0.068	⋯	0.032	0.032	⋯	0.311	0.362
(ℓ1+2+3)	0.009		0.013	0.013		0.020	0.014
d (pc)	465	529	492	481	605	562	581
	8	4	9	8	6	12	10

Note.

^aAll of the unaveraged model runs shown here have detailed reflection, and all except 53b2 have convective envelopes. All models designated "53" are semidetached. Normalized third-light passbands p₁ and p₂ are, respectively, V and B for Models 28CR3 and 53a1,a2, and I_C and R_C for Models 53b1,2; third light was not applied in Models 53a,b. Mean errors are given for the d and ℓ₃ parameters; errors for ${{\mathscr{L}}}_{\mathrm{1,2}}$ are based on an assumed error e(M_bol) = ±0.005. Note the range of parameter values among the semidetached models.

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The polar, component-facing ("point"), side, and back radii of the components for a selection of detached and semidetached models are listed in Table 12. The units are in each model's semimajor axis, a. The run results are from the DC output files for all but Model 40. We define the quantity δ as the difference between the equatorial radius and polar radius, divided by the equatorial radius. As this quantity is neither an ellipticity in the traditional Russell–Merrill modeling usage nor oblateness, which is properly applied only to the effect of rotational distortion, we refer to δ as a departure-from-sphericity index. Taking the mean of the point and back radii as the equatorial radius of the Model 40 data, we find r_eq,1 = 0.3710(2), r_eq,2 = 0.2159(6), and δ₁ = 0.0870(8) and δ₂ = 0.0346(40).

The weighted means of the relative passband luminosities are shown in column 10 of Table 7 for Model 40 and Table 8 for the group means. These quantities and the observed magnitudes at maximum light (from the "Standard Deviations for Computation of Curve Dependent Weights" table produced by the DC-routine) permit the computation of system and component magnitudes and color indices. In the absence of third light, the results can be obtained directly. However, if the apparent system magnitude includes a third-light contribution, computed system and component magnitudes, and colors, ignoring it cannot be correct. We now suggest two ways to find third-light-free magnitudes and colors of the components and system.

5.3.1. Technique 1

The recommended procedure involves passband luminosities, ${{\mathscr{L}}}_{\lambda 1}$ and ${{\mathscr{L}}}_{\lambda 2}$, from LC, the log d parameter, and solar data from the literature (R. E. Wilson 2018, private communication). Note that ${{\mathscr{L}}}_{\lambda \mathrm{1,2}}$ are free of third light. This is Technique 1, hereafter Tech. 1. The absolute visual magnitude of a star is given by

$$\begin{eqnarray}&&{M}_{{\rm{V}}}={M}_{\mathrm{VSun}}-2.5\,\mathrm{log}{{\mathscr{L}}}_{V},\end{eqnarray} \tag{ 1 }$$

with similar expressions for other passbands. M_VSun is the Sun's absolute visual magnitude; ${{\mathscr{L}}}_{V}$ is the V passband luminosity in solar units, ${{\mathscr{L}}}_{\mathrm{VSun}}$. We assume that the scaling from bolometric to passband luminosities is dealt with satisfactorily in the LC routine (Wilson 2008, Section 4.3). Via Equation (1), we obtain M_V1 and M_V2, also free of third light. M_V,1+2 follows:

$$\begin{eqnarray}&&{M}_{{\rm{V}},1+2}={M}_{{\rm{V}}1}-2.5\,\mathrm{log}(1+{{\mathscr{L}}}_{{\rm{V}}2}/{{\mathscr{L}}}_{{\rm{V}}1}).\end{eqnarray} \tag{ 2 }$$

The intrinsic magnitude V₀ for each component and combined light is then computed from

$$\begin{eqnarray}&&{V}_{0}={M}_{{\rm{V}}}+5\,\mathrm{log}\,d-5.\end{eqnarray} \tag{ 3 }$$

The addition of A_V to Equation (3) gives the apparent magnitudes. Substitution of λ for V generalizes the procedure for other passbands. For solar values, Allen (1973, p. 162) lists M_Bol,Sun = 4.75 and M_VSun = 4.83(3) (see Torres 2010 for error estimates), (B – V)_Sun = 0.65 and (U – B)_Sun = 0.13. From Ramirez et al. (2012), (V – R_C)_Sun = 0.356(3) and (V – I_C)_Sun = 0.701(3). Wilson & Van Hamme (2013, Table 5) adopted M_Bol,Sun = 4.75 and ${{\mathscr{L}}}_{\mathrm{Sun}}$ = 3.846 10²⁶ W, per Harmanec & Prsa (2011), and consistent with the bolometric scale zero point (i.e., the luminosity for which M_Bol = 0.00), ${{\mathscr{L}}}_{0}$ = 3.055 10²⁸ W (Andersen 1999). (See also 2015 IAU Resolutions B2 and B3, available from: http://astronomy2015.org/resolutions and elaborated in Prsa et al. 2016.)

Returning to DS And, we use Model 40 means: [${{\mathscr{L}}}_{\mathrm{Ic}1}$ = 8.23(12), ${{\mathscr{L}}}_{\mathrm{Ic}2}$ = 1.77(7)]${{\mathscr{L}}}_{\mathrm{Ic},\mathrm{Sun}}$; [${{\mathscr{L}}}_{\mathrm{Rc}1}$ = 9.27(34), ${{\mathscr{L}}}_{\mathrm{Rc}2}$ = 1.81(10)]${{\mathscr{L}}}_{\mathrm{Rc},\mathrm{Sun}}$; [${{\mathscr{L}}}_{{\rm{V}}1}$ = 10.55(28), ${{\mathscr{L}}}_{{\rm{V}}2}$ = 1.90(5)]${{\mathscr{L}}}_{\mathrm{Vsun}}$; [${{\mathscr{L}}}_{{\rm{B}}1}$ = 13.17(18), ${{\mathscr{L}}}_{{\rm{B}}2}$ = 1.76(7)]${{\mathscr{L}}}_{\mathrm{Bsun}}$; log d = 2.678(4) (pc). With A_V = 0.100, and polynomial representations for A_λ/A_V from Cardelli et al. (1989, Equations 3(a), (b)) we get the A_λ =  0.132, 0.157, 0.084, and 0.061 mag for the B, U, R_C, and I_C passbands, respectively. Color indices are then computed. The results are shown in columns 2, 3, and 4 of Table 13 under "Allen (1973)." However, if we use M_VSun = 4.81(3) as advised by Torres (2010), with (B – V)_Sun = 0.653(3) and (U – B) = 0.158(9) additionally from Ramirez et al. (2012), magnitudes are 0.02 mag brighter than those computed with the Allen (1973) solar values; the color indices are similar, if not identical (see Table 13 columns under "Torres 2010"). Finally, with the Mamajek (2012) and Pecaut & Mamajek (2013) values M_Bolsun = 4.7554(4) and M_VSun = 4.862(20), results are 0.03 and 0.05 mag fainter than those obtained with the Allen (1973) and Torres (2010) values, respectively (see Table 13, columns 8–10). With the Allen (1973) values, the intrinsic color indices imply spectral types of ∼F4, G8, and ∼F6, ±2–3 subclasses, for stars 1, 2, and the system, respectively, with correspondingly cool temperatures, in Table 5 of Pecaut & Mamajek (2013). Model 40 temperatures, on the other hand, imply spectral types ∼F1 ± 1 and G0 ± 2 for stars 1 and 2, respectively, in that table.

Table 13.  DS Andromedae Computed Magnitudes and Color Indices Free of Third Light^a

Method	Technique 1									Technique 2
Sun Data	VSun = −26.74			VSun = −26.76			VSun = −26.71			...
Source	Allen (1973)			Torres (2010)			Pecaut & Mamajek (2013)			...
Comp/λ, CI	1	2	1+2	1	2	1+2	1	2	1+2	1	2	1+2
V	10.76	12.62	10.58	10.74	12.60	10.56	10.79	12.65	10.61	10.73	12.59	10.55
e(V)	0.04	0.04	0.04	0.03	0.04	0.03	0.04	0.03	0.04	0.02	0.02	0.01
V0	10.66	12.52	10.48	10.64	12.50	10.46	10.69	12.55	10.51	10.63	12.49	10.45
e(V0)	0.04	0.04	0.04	0.03	0.04	0.03	0.04	0.03	0.04	0.02	0.02	0.01
MV	2.27(4)	4.13(4)	2.09(4)	2.25(3)	4.11(4)	2.09(3)	2.30(4)	4.16(3)	2.12(4)	2.24(2)	4.10(3)	2.06(2)
(B – V)	0.44(5)	0.77(7)	0.49(5)	0.44(5)	0.77(7)	0.49(5)	0.44(4)	0.77(6)	0.49(4)	0.35(2)	0.67(2)	0.39(2)
(B – V)0	0.41(4)	0.73(7)	0.45(5)	0.41(5)	0.74(7)	0.46(5)	0.41(4)	0.74(6)	0.46(4)	0.32(2)	0.64(2)	0.36(2)
(V – IC)	0.47(5)	0.66(7)	0.50(6)	0.47(5)	0.66(7)	0.50(5)	0.47(5)	0.66(6)	0.50(5)	0.44(2)	0.64(3)	0.48(2)
(V – IC)0	0.43(5)	0.62(7)	0.46(5)	0.43(5)	0.62(6)	0.46(5)	0.43(5)	0.62(6)	0.46(5)	0.41(2)	0.60(3)	0.44(2)
(V – RC)	0.23(5)	0.32(8)	0.24(6)	0.23(5)	0.32(6)	0.25(5)	0.23(5)	0.32(8)	0.24(5)	0.19(2)	0.28(2)	0.21(1)
(V – RC)0	0.22(5)	0.30(8)	0.24(6)	0.22(5)	0.30(8)	0.23(5)	0.22(5)	0.30(8)	0.23(5)	0.18(2)	0.26(2)	0.19(1)
(U – B)	0.11(5)	0.52(6)	0.15(5)	0.13(5)	0.54(6)	0.18(5)	0.13(5)	0.54(6)	0.18(5)	0.01(2)	0.42(2)	0.05(2)
(U – B)0	0.08(5)	0.49(6)	0.12(5)	0.11(5)	0.52(6)	0.15(5)	0.11(4)	0.52(6)	0.16(4)	−0.01(2)	0.40(2)	0.03(2)

Note.

^aBased on Model 40 determinations. The "1+2" values differ from the observed magnitudes and color indices of the DS And system because of the presence of third light in the system. Technique 1 depends on solar data and the derived absolute passband luminosities of each component and the determined distance from the best-fitting model, whereas Technique 2 involves the removal of normalized third light from the observed system light. Both are model-dependent. As for other tables, the uncertainties reported for the magnitudes and color indices are formal internal errors. Note the significant agreement between Tech. 1 and 2 computed magnitudes but the systematic differences in the B – V and U – B color indices; nevertheless, all agree within 2σ. See Section 5.3.2 for details.

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5.3.2. Technique 2

Another way involves the extraction of third light from the observed magnitudes at maximum light. In this technique, hereafter Tech. 2, we start with the observed system magnitudes and proceed to find third-light-free observed, intrinsic, and absolute magnitudes. If the third-light source is a star, we can use the relation

$$\begin{eqnarray}&&{m}_{\lambda ,1+2+3}={m}_{\lambda ,1+2}-2.5\,\mathrm{log}\,[1/(1-{r}_{\lambda })],\end{eqnarray} \tag{ 4 }$$

where m_λ,1+2 is the magnitude of stars 1 and 2 combined, m_λ,1+2+3 is that of the total system, i.e., including third light, and r_λ = (ℓ₃/ℓ₁₊₂₊₃)^λ, the fraction of the total light that ℓ₃ contributes to maximum light. From Models 40 and 55', r_Ic =0.0989(13), r_Rc = 0.1037(10), r_V = 0.0954(75), r_B = 0.0941(66), and r_U = 0.1221(88). The factor ℓ₁₊₂₊₃ is selected for a designated phase of maximum (0.25^P, 0.75^P) in LC output files. Magnitudes at maximum are I_C,1+2+3 = 9.959(12), R_C,1+2+3 =10.222(5), V₁₊₂₊₃ = 10.439(4), B₁₊₂₊₃ = 10.833(9), and U₁₊₂₊₃ = 10.852(5).

Solving Equation (4) for m₁₊₂ and Equation (2) for m₁, we then find m₂ from

$$\begin{eqnarray}\begin{array}{rcl}{m}_{\lambda 2} & = & {m}_{\lambda 1}-2.5\,\mathrm{log}({{\mathscr{L}}}_{\lambda 2}/{{\mathscr{L}}}_{\lambda 1})\\ & = & \ {m}_{\lambda 1}-2.5\,\mathrm{log}[(1/k)-1)],\end{array}\end{eqnarray} \tag{ 5 }$$

where k = ${{\mathscr{L}}}_{\lambda 1}$/(${{\mathscr{L}}}_{\lambda 1}$ + ${{\mathscr{L}}}_{\lambda 2}$). The Tech. 2 computed, third-light-free, V, V₀, and M_V magnitudes and color indices are shown in the last three columns of Table 13 and in footnote c of Table 14. Those listed in Table 14 itself include third light to allow comparison with previous analyses that did not adjust third light. The system and component apparent magnitudes of Tech. 2 agree within formal errors with those of Tech. 1 but agree only to within ∼2σ with Tech. 1 B – V color indices. Other color index differences are smaller but all are bluer than those found by Tech. 1. The spectral type implied by Tech. 2 ((B – V)₁₊₂ in Popper (1980, Table 1) and (B – V)_0,1+2 in Pecaut & Mamajek (2013, Table 5) is F2 ± 1, in agreement with published spectral types F2–3; they imply F1 ± 1, G2 ± 2 for stars 1 and 2, respectively, and T₁ =7100(100) K and T₂ = 5800(100) K.

Table 14.  DS Andromedae Properties Determinations

Property	Schiller & Milone (1988)			Mellergaard Amby (2011)			Present Adopted Model
Comp.	1	2	System	1	2	System	1	2	System
P (days)1.01051+	⋯	⋯	87 ± 2	⋯	⋯	870a	⋯	⋯	8870 ± 0025 ± 0120
t0 (HJD)2436142+	⋯	⋯	.405 ± .002	⋯	⋯	cf fna	⋯	⋯	.40281 ± .00027 ± 00114
a/RSun	⋯	⋯	5.77 ± 0.14	⋯	⋯	5.92 ± 0.01	⋯	⋯	5.930 ± 0.006 ± .015
Vsystem (km s−1)	⋯	⋯	2.5 ± 2.0	⋯	⋯	8.44 ± 0.60	⋯	⋯	8.13 ± 0.02 ± .50
i (deg)	⋯	⋯	84.3 ± 0.5	⋯	⋯	89.72 ± 0.02	⋯	⋯	89.35 ± 0.19 ± .45
M/MSun	1.58 ± 0.17	0.94 ± 0.10	⋯	1.625 ± 0.018	1.105 ± 0.014	⋯	1.655 ± 0.003 ± .030	1.087 ± 0.005 ± .040	⋯
R/RSun	2.10 ± 0.08	1.19 ± 0.05	⋯	2.146 ± 0.015	1.159 ± 0.014	⋯	2.086 ± 0.003 ± .013	1.255 ± 0.005 ± .012	⋯
Teff (K)	6775a	5997 ± 17	⋯	6775a	6144 ± 4	⋯	7056 ± 21 ± 140	5971 ± 33 ± 130	⋯
${\mathscr{L}}$ (${{\mathscr{L}}}_{\mathrm{Sun}}$)	8.7b ± 1.2	1.6 ± 0.3	⋯	8.75 ± 0.40	1.73 ± 0.12	⋯	9.58 ± 0.12 ± .20	1.77 ± 0.03 ± .06	⋯
V	10.62 ± 0.02	12.47 ± 0.03	10.44	10.70 ± 0.10	12.49 ± 0.12	⋯	10.619c ± 0.012	12.479c ± 0.017	10.439c ± 0.004
B – V	0.38 ± 0.03	0.63 ± 0.05	0.40	0.38 ± 0.04	0.55 ± 0.06	⋯	0.350c ± 0.016	0.675c ± 0.020	0.394c ± 0.009
V – RC	0.20 ± 0.03	0.31 ± 0.04	0.22	⋯	⋯	⋯	0.204c ± 0.014	0.287c ± 0.018	0.217c ± 0.007
V – IC	0.46 ± 0.03	0.64 ± 0.04	0.49	⋯	⋯	⋯	0.448c ± 0.018	0.640c ± 0.022	0.480c ± 0.013
(m – M)0	⋯	⋯	8.17 ± 0.15	8.30 ± 0.12	8.28 ± 0.15	⋯	⋯	⋯	8.390 ± 0.018 ± .060
d (pc)	⋯	⋯	431 ± 30	464 ± 16	459 ± 18	⋯	⋯	⋯	477 ± 4 ± 12

Notes.

^aAssumed and unadjusted. The Mellergaard Amby (2011) unadjusted t₀ = 245128.68930 from his RV curve. ^bCorrected to agree with M_bol (Schiller & Milone 1988, Table 7). ^cMeans of RADS and non-RADS reference magnitudes from curve weights tables in column 10 include both stars and third light. Third-light-free values are V: 10.73(2), 12.59(2), 10.55(1); (B – V): 0.35(2), 0.67(2), 0.39(2); (V – R_C): 0.19(2), 0.28(2), 0.21(1); (V – I_C): 0.44(2), 0.64(3), 0.48(2); (U – B): 0.01(2), 0.42(2), 0.05(2), for stars 1, 2, and 1+2, respectively; a similar systematic error may be present in all color indices. Uncertainties of other values in the last three columns are m.s.e.'s followed by conservative systematic error estimates. See the discussion of errors in Sections 4 and 5. The inverse square root of the sums of the inverse squares of the individual-run standard deviations are e_P = 49 × 10⁻⁹ days; e_t0 = 47 × 10⁻⁵; e_a = 0.015 R_Sun; e_Vsys = 0.24 km s⁻¹; e_i = 016; e_T1 = 12 K; e_T2 = 15 K; ${e}_{{{\mathscr{L}}}_{1}}$ = 0.007; ${e}_{{{\mathscr{L}}}_{2}}$ = 0.04; e_(m−M)0 = 0.015; and e_d = 3 pc.

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5.3.3. Adopted Magnitudes and Color Indices

The determination of membership of a star in a cluster is based on its location on the plane of the sky, its position on the cluster's CMD, its distance, RV, and proper motion relative to other candidates for membership. Figure 3, the CMD of NGC 752 with the derived third-light-free apparent magnitudes and color indices of DS And, demonstrates that the primary component is at or near the main-sequence turnoff, whereas the secondary is brighter than expected for a single main-sequence star with its color index. The Model 40 adopted mean distance is d = 476.5 ± 3.9 ± 12 pc, corresponding to a corrected distance modulus (m – M)₀ = 8.390 ± 0.018 ± 0.080 mag, with the formal internal error followed by a conservative estimate of systematic error, in agreement with that of the best-fitting single run, Model 40a (Table 2), d = 463(12) and (m – M)₀ = 8.329(56) and that of the grand means (Table 3), d = 470.8(35) pc, and $\langle$(m – M)₀$\rangle$ = 8.364(17). In DR2, the second data release of the Gaia mission (Gaia Collaboration et al. 2018), π = 2.19676(4041) mas, which implies d = 455.2(+8.5, −8.2) pc, and (m – M)₀ = 8.291(40), within ~2σ of the current results. The revised cluster distances obtained by Twarog et al. (2015), (m – M)₀ = 8.20(5), and Agüeros et al. (2018) differ by just less than 3σ, however. The latter determines combinations of properties through Bayesian analysis, finding d = 438(+8, −6) pc and (m – M)₀ = 8.21(+4, −3), age of 1.34(6) Gyr, metallicity [Fe/H] =+0.02(1), and A_V = 0.198(+8, −9). Most previous work found A_V closer to 0.10 mag.

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Our adopted systemic velocity, 8.13 ± 0.02 ± 0.50 km s⁻¹, agrees with the mean cluster velocity, +9 km s⁻¹, found by Rebeirot (1970), that of Friel et al. (1989), +8.5(30) km s⁻¹, excluding a star with variable RV, and the mean velocity found by Pilachowski et al. (1988), 5.9(24) km s⁻¹, from six stars that meet the proper motion criterion of Ebbighausen (1939) classes 1 and 2 (but without the exclusion of several stars for various reasons, leaving a mean of +4.9(7) km s⁻¹). It is inconsistent with the adopted cluster velocity of Daniel et al. (1994), +5.5(6) km s⁻¹, based on 33 stars but excluding stars differing by 3σ from the mean. On the other hand, it should be noted that the distribution of high-precision radial velocities found by Maderak et al. (2013) peaks at +5.4(1) km s⁻¹ but the core histogram spans the range 2 to 8 km s⁻¹, so it may be possible to set the RV criteria too narrowly. Moreover, our FIES RVs are highly precise and accurate but it has been suggested that if all RV results were reduced to the IAU or other uniformly accurate system, agreement might be closer. DS And's proper motion is also supportive of membership. Ebbighausen (1939) assigned it to class 2, with a reasonable chance of being a cluster member; the probability of membership was rated at >99% by Francic (1989), and at 99% by Platais (1991), whose catalog indicates the values μ_x = 7.9 and μ_y = −6.4 mas yr⁻¹, where (x, y) are closely equatorial, with an estimated uncertainty of about 1.5 mas yr⁻¹. The Gaia DR2 lists the proper motion components as μ_α = 10.72(8) and μ_δ = −11.78(8) mas yr⁻¹. Thus, the preponderance of sky location, CMD locations, distance, extinction and metallicity, RV, and proper motion evidence is supportive of cluster membership.

The primary and secondary stars, free of third light, are placed on the CMDs in Figures 3 and 4. The primary is seen to be just at the turnoff point of the cluster and the secondary to be sitting roughly on the "binary main sequence," i.e., with brightness about twice that of a single star of its color index. Figure 4 displays the WEBDA V versus (V – R_C) CMD, with three sets of Padova isochrones and binary sequences scaled to log age = 9.20 or 1.58 Gyr, (m–M)₀ = 8.39, and for a metallicity of −0.08 (slightly different than the bulk of our models but within the range of the metallicities tested in the trials described in Section 3). The third-light-free V and (V – R_C) of the system and components are superimposed as crosses. Sets of curves are shown for models with metals fractions Z = 0.004, 0.008, and 0.019. The lighter curves are the "binary sequences." A log age 9.20 model seems to provide a better fitting than the 9.15 or 9.25 models provided by this online source. If these models are appropriate for NGC 752, Z would need to lie closer to 0.008 than 0.019 to fit (if crudely) both the main-sequence and the giant branch, but a younger cluster implies a larger Z. The primary component is seen to be near or slightly below the main-sequence turnoff region, and on the Z = 0.008 isochrone curve. The secondary, just to the blue of this isochrone, does not seem to deviate very much from other later type main-sequence stars in the cluster, but neither does it appear to be on the same isochrone as the primary or at least not one with the same metals fraction. A similar result is found with Geneva isochrones, which are ∼0.02–0.04 mag bluer. Figure 5 places the components, labeled 1 and 2, on a log ${\mathscr{L}}$ versus log T plot amid isochrones for a range of ages. The isochrones are plotted from data in the online tables provided at http://stellar.dartmouth.edu/models/grid.html incorporating the empirical BVR_CI_C photometry of VandenBerg & Clem (2003) and, if we understand correctly, the VandenBerg et al. (2006) values of the He fraction Y = 0.2696, metals fraction Z = 0.016115, and α-element abundance α/Fe = 0.00. If these convective overshoot models with solar composition, metallicity, and helium fraction theoretical models are appropriate for the DS And system, the age of the primary is about 1.55(5) Gyr, significantly different from that found by Agüeros et al. (2018), 1.34(6) Gyr, but not significantly more than that found for NGC 752 by Twarog et al. (2015), 1.45(5) Gyr; not significantly less than that found for convective overshoot isochrone fitting to the cluster's main sequence by Daniel et al. (1994), 1.7(1) Gyr; and effectively the same as that found by Girardi et al. (2002), 1.55(10) Gyr. In this plot, the secondary appears to be noncoeval with the primary, a circumstance that appears to be confirmed in Figures 6 and 7, adapted from plots kindly provided by D. A. VandenBerg (2017, private communication). The evolutionary tracks plotted in Figures 6 and 7 are for overshoot models with Z = 0.013, and Z = 0.0188, respectively. The large dots on the tracks mark equivalent evolutionary phase (eep) points. The placements of the components among the tracks suggest that although both components are approximately in the expected regions for their masses, the secondary appears to be slightly older than the primary, as gauged by the evolutionary phase points. We note that the components appear to be closer to the appropriate track for their masses in Figure 6 than in Figure 7.

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A vertical shift with respect to the isochrones is explained if the secondary is overluminous compared to single star models (see Figure 5) perhaps because of its proximity to a hotter, larger, and much more luminous companion. A lower luminosity would bring it much closer to the evolutionary phase point of the primary star in Figures 6 and 7. Following a suggestion by G. Torres, the components are placed in an R versus M plane in Figure 8, with a selection of VandenBerg et al. (2006) isochrones for models with the same properties as in Figure 5. The radii are not listed in the tables but were computed from log g and mass values. The secondary appears to be too large for its mass, but the primary appears to be smaller than expected for its mass, if the models are appropriate. In Figure 9, the mass and radius of the primary and secondary stars are compared with the well-established masses and radii of the components of selected binary stars of similar spectral types (filled symbols), listed by Torres et al. (2010, Table 2). The Sun is also shown for reference. The secondary's mass and radius are similar to those of Alpha Cen A [G2V, M = 1.105(7) M_Sun, R = 1.224 R_Sun, in a wide binary with P = 79.91 yr]. This star is cooler (5824(26) versus 5971(33) K) and more metal-rich ([Fe/H] = 0.24(4)) than the DS And secondary. The primary's mass and radius are similar to those of FS Mon A (F2V, M = 1.632(10) M_Sun, R = 2.052(12) R_Sun, in a 1.91 day binary). This star is significantly cooler than the DS And primary [6715(100) versus 7056(21) K], but its metallicity is not given and was assumed to be 0.00. The temperature differences show that the stars are not exactly analogs and thus offer only weak support for the idea that the DS And components are anomalous, given systematic error estimates in T_1,2 of ∼150 K, but neither can the idea be ruled out. The masses and radii calculated with the polynomial representation of log T, log g, and [Fe/H] terms (Torres 2010, Tables 1 and 3) are depicted by the unfilled and starred symbols. The differences between the determined and calculated values for the DS And components are in the range of those shown for the other stars (marked by red lines). The (lighter) regression line for the calculated points closely matches that of the stars themselves. Note that both DS And components fall close to the regression line for the plotted stars. We conclude that the primary may be somewhat radiatively anomalous compared with generic solar models and with stars of similar properties, but the secondary is certainly anomalous (already clear from its location on the CMD), but specific theoretical modeling needs to be done to check this conclusion. Another example of an oversized if cooler than expected star in a close binary of solar composition is EF Aqr B (M = 0.946(6) M_Sun, R = 0.956(12) R_Sun in a 2.85 day binary; Vos et al. 2012. It has a high rotational velocity (18(4) km s⁻¹) due to synchronicity, but far less than DS And B's 63 km s⁻¹. Strikingly (see below), EF Aqr is an X-ray source, and strong magnetic fields have been invoked to suppress its envelope convection. There are many other cases (see Feiden & Chaboyer 2012, 2014) with similar anomalies. Fekel's (1997) Table 1 shows that early FV and early GV stars' v sin i measurements typically fall within ranges of 10–20 and 1–10 km s⁻¹, respectively. Thus, rotational speed-up may be key to DS And's anomalies. There are at least two other anomalous objects with very similar proper motions and high probability of membership in NGC 752 (Platais 1991): QX And (H235), the only overcontact system in the cluster, and the blue straggler H209, so prominently visible on the extreme left in Figures 3 and 4. Both are among eight NGC 752 objects detected in X-rays (Belloni & Verbunt 1996), with DS And, lying just outside the inner region of the detector, as a possible detection (Van den Berg & Verbunt 2001 describe this as a probable detection). The X-ray luminosities of QX And, H209, and DS And are given, respectively, as 7.4 × 10²⁹, 6.7 × 10²⁹, and 2.9 × 10³⁰ erg s⁻¹. Like DS And, QX And is a close binary with a variable light curve and with at least one component that is likely magnetically active, but the circumstance for H209 is different. It is a very wide spectroscopic binary with a period of 1574 days (Van den Berg & Verbunt 2001, citing a private communication from D. Latham), and no very close companion has been detected spectroscopically by Van den Berg & Verbunt (2001). Those authors conclude that the cause of the X-ray emission and blue straggler nature of H209 is unknown. Twarog et al. (2015) noted that its age with respect to the Victoria–Regina isochrones is less than 0.1 Gyr. It is possible that H209 is a merger product of a former close binary, and so "reborn"; in any case, it is an anomalous object. Considering that such evolved binaries are not to be expected in an "intermediate-age" cluster unless they have lost angular momentum to wider components, the three anomalous objects would suggest that a dedicated search for wider components in these binary systems is in order.

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