3 Light curve analysis

The fits to the light curves were performed using an improved version of the Wilson-Devinney program (Wilson & Devinney 1971; hereafter W-D) that includes a model atmosphere routine developed by Milone et al. (1992) for the computation of the stellar radiative parameters. A detached configuration was chosen for all solutions. Both reflection and proximity effects were taken into account, even though the light curves do not show strong evidence for these. The bolometric albedo was set at the canonical value of 0.5 for stars with convective envelopes. The gravity-brightening coefficient was adopted as 0.2 following the theoretical results of Claret (2000a). A mass ratio of $q=M_{\rm B}/M_{\rm A}=0.9184$ was adopted from D99 and the temperature of the primary component (eclipsed at phase 0.0) was set to 3160 K (see Sect. 4 for discussion). For the limb darkening we used a logarithmic law as defined in Klinglesmith & Sobieski (1970). In our W-D implementation, first- and second-order limb darkening coefficients are interpolated at each iteration from a set of tables computed in advance using a grid of Kurucz model atmospheres. However, CU Cnc has a temperature somewhat below the coolest model and a extrapolation had to be made. We also ran some tests by fixing the coefficients at the values computed by Claret (2000b) for the appropriate effective temperatures. The change had negligible effects but we finally adopted this latter prescription to avoid extrapolations. Finally, third light fractions were set to the values provided in Sect. 2.

The iterations with the W-D code were carried out automatically until convergence, and a solution was defined as the set of parameters for which the differential corrections suggested by the program were smaller than the internal probable errors on three consecutive iterations. As a general rule, several runs with different starting parameters are used to make realistic estimates of the uncertainties and to test the uniqueness of the solution.

Simultaneous W-D fits to the RI light curves were carried out using the mean error of a single measurement as the relative weight, $w_{\lambda}$, of each passband ("curve-dependent'' weighting scheme). Note that the number of photometric measurements we have is so large that 4-point normals had to be computed at the densest phases in or around the eclipses when running the W-D fits. We initially solved for the following light curve parameters: the orbital inclination (i), the temperature of the secondary ( ${T_{\rm eff}}_{\rm B}$), the gravitational potentials ( $\Omega_{\rm A}$ and $\Omega_{\rm B}$), the luminosity of the primary at each passband ($L_{\rm A}$), a phase offset ( $\Delta\phi$), and the spot parameters. The orbital eccentricity was set to zero as the light curves do not show any evidence for eclipses of different width or a secondary eclipse at an orbital phase other than 0.5. Some tests were run to check for the possibility of a small non-zero orbital eccentricity. The W-D solutions converged towards a value of $e=0.001\pm0.003$, thus consistent with our adoption of a null eccentricity.

As expected for a system with partial eclipses and similar components, numerous test solutions revealed that the ratio of the radii ( $k\equiv r_{\rm B}/r_{\rm A}$, where $r_{\rm A}$ and $r_{\rm B}$ are the fractional radii in units of the separation) is poorly constrained. The problem is, in principle, severe because it implies that the light curves are not sufficiently sensitive to discriminate between the sizes of the components. Thus, some source of external information has to be used to infer a value for k. With the ratio of temperatures well determined from the light curves, a spectroscopic estimation of the luminosity ratio would provide the necessary information.

Two high resolution and S/N echelle spectra were obtained with the UES instrument at the William Herschel Telescope (La Palma) as part of the Service Programme. The observations were carried on March 8-9, 2001 (HJD 2451977.499) at orbital phase 0.287. The raw images were reduced using standard NOAO/IRAF tasks (including bias subtraction, flat field correction, sky-background subtraction, cosmic ray removal, extraction of the orders, dispersion correction, merging, and continuum normalization). Equivalent width ratios for the components were measured for 10 relatively clean and isolated Ca  I and Fe  I absorption features. The average of the individual values yielded $EW_{\rm B}/EW_{\rm A}=0.73\pm0.05$ at a mean wavelength of $\lambda=5900$ Å. Since the effective temperatures of both components are very similar, as shown below, the equivalent width ratio is analogous to luminosity ratio. From this we derive values of $[L_{\rm B}/L_{\rm A}]_{V}=0.72\pm0.05$, $[L_{\rm B}/L_{\rm A}]_{R}=0.74\pm0.05$ and $[L_{\rm B}/L_{\rm A}]_{I}=0.76\pm0.05$, for the luminosity ratios at the V, R and I passbands, respectively. Our value for the V-band luminosity ratio is indeed in good agreement with that obtained by D99. Although not explicitly given in the paper, Delfosse (2000, priv. comm.) kindly computed the luminosity ratio from their spectra and obtained a value of 0.68, which lies within one sigma of our determination.

In practice, the luminosity ratio or the ratio of radii is not one of the parameters considered explicitly in the W-D code. Instead, are the surface potentials of the stars (which, along with q, control the sizes) that can be fixed or left free. We thus ran a number of solutions by adopting different values for the potential of the primary star until reaching agreement between the resulting luminosity ratio and the observed one. This occurred for $\Omega_{\rm A}=19.0$. All further solutions were ran by fixing $\Omega_{\rm A}$ at this value.

The shape of the out-of-eclipse region in the light curve clearly indicates that one or both components have surface inhomogeneities. A feature incorporated by the W-D program is the capability of fitting a simple spot model. The spots, assumed to be circular in shape, are described by W-D through four parameters, namely, the longitude, latitude, angular radius, and temperature ratio (spot relative to photosphere). Some of the spot parameters must be fixed when running solutions to prevent lack of convergence. For example, the latitude of the spots can be inferred theoretically. Granzer et al. (2000) find that a star such as the components of CU Cnc (0.4-$M_{\odot }$ star in the ZAMS with an angular velocity 10 times the solar value) favours strongly the formation of spots at latitudes of about 60$^\circ$. Tests ran with W-D indeed indicated a marginally better fit for spots at high latitudes.

As it seems obvious, the radius of the spot and the temperature contrast are strongly correlated. No empirical information is available on the temperature ratio between the spots and the surrounding photosphere for CU Cnc. However, Hatzes (1995) analysed Doppler tomography of the M-type eclipsing binary YY Gem and found the presence of dark areas cooler than the photosphere with $T_{\rm spot}/T_{\rm phot}=0.87{-}0.92$. For CU Cnc, we expect similar values. We thus ran a series of solutions and tested a variety of spot combinations (number of spots, locations, sizes, temperature ratios, etc.) to achieve the best possible fit to the light curves (see a detailed discussion of a similar procedure in TR02). Our preferred solution calls for two spots - one on each component -. The spot on the primary component, which transits the central meridian at orbital phase 0.438, appears to be relatively small, with a radius of 9$^\circ$, and $\sim$450 K cooler than the photosphere. In contrast, the secondary component has a much larger spot with a radius of 31$^\circ$ and a temperature difference with the surrounding photosphere of $\sim$200 K. The center of the dark area transits the central meridian at orbital phase 0.997. Its large size and relatively small temperature difference seems to indicate that this is more likely a spot complex rather than a homogeneous spot. Thus, the parameters determined represent an average over an extended photospheric area probably covered with patchy dark spots.

The photometric effect of the spot configuration adopted here is illustrated in Fig. 2. Note that the spot solution has only a mild effect on the rest of the light curve parameters. Furthermore, the spot solution is not unique and other configurations may be equally valid from a strictly numerical point of view.

Figure 2: Photometric effects of the spots on the R and I light curves. The spike during the phase of the primary minimum arises when the primary component is partially eclipsed and the spot on the secondary component is in front thus enhancing its relative contribution.

The best-fitting two-spot model described above yields rms residuals of 0.008 mag and 0.005 mag for the R and I light curves, respectively. These small values reflect both the quality of the photometry and the excellent performance of the spot model. The synthetic light curve superimposed to the observations can be seen in Fig. 1, together with the residuals of the fit for each light curve. Also, for illustrative purposes, we show a detail of the eclipse phases in Fig. 3. The solution indicates that the eclipses are partial and that $\sim$25% of each component's flux is blocked during the corresponding eclipses. The parameters resulting from the light curve fit are listed in Table 3. The uncertainties given in this table were not adopted from the formal probable errors provided by the W-D code, but instead from numerical simulations and other considerations. Several sets of starting parameters were tried in order to explore the full extent of the parameter space. In addition, the W-D iterations were not stopped after a solution was found, instead, the program was kept running to test the stability of the solution and the geometry of the $\chi^2$ function near the minimum. The scatter in the resulting parameters from numerous additional solutions yielded estimated uncertainties that we consider to be more realistic, and are generally several times larger than the internal statistical errors. For CU Cnc, however, an additional error source must be considered and this is the uncertainty in the measurement of the luminosity ratio. The empirical value obtained from the spectra carries a small but significant measurement error that must be properly accounted for. We did this by running new W-D solutions for values of $\Omega_A$ set to yield the observed luminosity ratio plus and minus 1$\sigma$. The average differences between the adopted and the new parameters were combined quadratically with the uncertainties described above to yield the values listed in Table 3. As a consequence of the uncertainty in the luminosity ratio, the radii of the components have larger errors ($\sim$2%) than would be expected from the quality of the light curve

Figure 3: Details of the fits to the primary and secondary eclipses. Note the high density of observations and the excellent agreement with the synthetic light curve.

 

 

Table 3: Results from the light curve analysis for CU Cnc.

Parameter	Value
Geometric and radiative parameters
P (days) (fixed)	2.771468
e (fixed)	0
i (deg)	86.34 $\pm$ 0.03
$q\equiv M_{\rm B}/M_{\rm A}$ (fixed)	0.9184
$\Omega_{\rm A}$ (fixed)	19.00 $\pm$ 0.22
$\Omega_{\rm B}$	19.39 $\pm$ 0.40
$r_{\rm A}$ (volume)	0.0553 $\pm$ 0.0007
$r_{\rm B}$ (volume)	0.0501 $\pm$ 0.0012
$k\equiv r_{\rm B}/r_{\rm A}$	0.906 $\pm$ 0.025
$T_{\rm eff}^{\rm A}$ (K) (fixed)	3160
$T_{\rm eff}^{\rm B}/T_{\rm eff}^{\rm A}$	0.9892 $\pm$ 0.0019
$L_{\rm B}/L_{\rm A}$ (R band, phase 0.287)	0.74 $\pm$ 0.05
$L_{\rm B}/L_{\rm A}$ (I band, phase 0.287)	0.76 $\pm$ 0.05
F3 (R band, phase 0.215) (fixed)	0.23
F3 (I band, phase 0.215) (fixed)	0.25
Albedo (fixed)	0.5
Gravity brightening (fixed)	0.2
Limb darkening coefficients (Logarithmic law)
$x_{\rm A}$ and $y_{\rm A}$ (R band)	0.875, 0.390
$x_{\rm B}$ and $y_{\rm B}$ (R band)	0.875, 0.390
$x_{\rm A}$ and $y_{\rm A}$ (I band)	0.844, 0.530
$x_{\rm B}$ and $y_{\rm B}$ (I band)	0.844, 0.530
Spot parameters
Phase for spot #1	0.438
Radius for spot #1 (deg)	9
$\Delta T$ for spot #1 (K)	450
Location of spot #1	Primary
Phase for spot #2	0.997
Radius for spot #2 (deg)	31
$\Delta T$ for spot #2 (K)	200
Location of spot #2	Secondary
rms residuals from the fits
$\sigma_R$ (mag)	0.008
$\sigma_I$ (mag)	0.005

The only independent determination of the light curve parameters of CU Cnc comes from D99. As mentioned earlier, the authors only had sparse observations and the quality of the solution is not very high. Importantly, the effect of star spots were taken into account in the analysis. The final radii published by D99 have errors of 12-16%. Even so, the agreement between D99's solution and the parameters listed in Table 3 is mostly within their large quoted uncertainties. The most significant difference is in the relative radii of the components, where D99 report very unequal components (k=0.67). Recall that the ratio of radii cannot be determined from the light curve alone so it is not surprising that D99 found rather inconsistent values. The sum of the relative radii, however, is tightly constrained by the duration of the eclipses and our value ( $0.1054\pm0.0006$) is extremely close to that found by D99 (0.1052).