4 Absolute dimensions and effective temperatures

The absolute dimensions of the CU Cnc components can be readily determined by combining the photometric parameters in Table 3 with the spectroscopic solution from D99. When doing so, we obtain the physical properties listed in Table 4.

 

 

Table 4: Absolute dimensions and radiative properties for the components of CU Cnc.

Property	Primary	Secondary
Spectral type	M3.5 Ve	M3.5 Ve
Mass ($M_{\odot }$)	0.4333 $\pm$ 0.0017	0.3980 $\pm$ 0.0014
Radius ($R_{\odot}$)	0.4317 $\pm$ 0.0052	0.3908 $\pm$ 0.0094
$\log g$ (cgs)	4.804 $\pm$ 0.011	4.854 $\pm$ 0.021
$T_{\rm eff}$ (K)	3160 $\pm$ 150	3125 $\pm$ 150
$\log (L/L_{\odot})$	-1.778 $\pm$ 0.083	-1.884 $\pm$ 0.086
$M_{\rm bol}$ (mag)	9.19 $\pm$ 0.21	9.45 $\pm$ 0.21
$\pi$ (mas)	78.05 $\pm$ 5.69
MV (mag)	11.95 $\pm$ 0.16	12.31 $\pm$ 0.16
BCV (mag)	-2.76 $\pm$ 0.26	-2.86 $\pm$ 0.26

The absolute value of the effective temperature for eclipsing binaries cannot be determined from the light curves alone and independent methods must be used. Temperature determination is usually quite problematic when one has to deal with very cool stars such as CU Cnc. Two independent approaches present themselves to estimate the effective temperature of CU Cnc: one based on photometric calibrations and one based on the known absolute dimensions and distance.

Synthetic colors, which are the base of photometric calibrations, are computed by modelling the stellar atmospheres. Albeit some discrepancies remain, M-star model atmospheres have made significant progress over the past decade (Allard et al. 1997). Line opacities from a large number of molecular species are now included in the calculations and substantial improvements have been made in the input physics, particularly the equation of state. For cool stars, the infrared colors are especially suited to estimate the effective temperature because most of the stellar flux is emitted in that wavelength range. Thus, we collected from the literature the available magnitudes and colors for CU Cnc, which are listed in Table 5. Simple linear transformations were applied to refer the observed magnitudes and colors to the photometric systems most widely used.

 

 

Table 5: Joint observed magnitudes and color indices for CU Cnc.

Mag. or color	Value	Notes
V	$11.89\pm0.01$	From Weis (1991)
$(R-I)_{\rm C}^{{\rm a}}$	1.60	From Weis (1991) and transformations in Bessell (1979)
$(V-I)_{\rm C}^{{\rm a}}$	2.80	From Weis (1991) and transformations in Bessell (1979)
$J_{\rm 2MASS}$	$7.536\pm0.018$	2MASS Second Incremental Data Release $^{{\rm b}}$
$H_{\rm 2MASS}$	$6.912\pm0.030$	2MASS Second Incremental Data Release $^{{\rm b}}$
$K_{\rm 2MASS}$	$6.626\pm0.029$	2MASS Second Incremental Data Release $^{{\rm b}}$
$K_{\rm BB}^{{\rm c}}$	$6.670\pm0.031$	Using transformations from Carpenter (2001)
$K_{\rm CIT}^{{\rm d}}$	$6.650\pm0.031$	Using transformations from Carpenter (2001)
$(V-K)_{\rm BB}^{{\rm c}}$	$5.22\pm0.03$
$(V-K)_{\rm CIT}^{{\rm d}}$	$5.24\pm0.03$

$^{{\rm a}}$

Cousins (1976) photometric system.

$^{{\rm b}}$

http://www.ipac.caltech.edu/2mass/releases/second/doc/explsup.html

$^{{\rm c}}$

Bessell & Brett (1988) homogenized system.

$^{{\rm d}}$

Caltech photometric system (Elias et al. 1982, 1983).

The magnitudes and color indices in Table 5 correspond to the combined light of the two components of CU Cnc. Since the effective temperatures of the two components are very similar, it is justified to use the joint color indices as those representing the "mean'' component of CU Cnc. To estimate the mean effective temperature we employed a number of modern photometric calibrations. We assumed no interstellar reddening in our calculations as expected for a system that lies less than 20 pc away. The resulting temperatures, together with the references, atmosphere model names, and comments, are presented in Table 6. Most of the calibrations are based upon synthetic colors and model atmosphere calculations, except for Leggett (1992), who provides an empirical calibration. There is remarkably good agreement between all the temperature estimates. A plain average of the independent values (i.e., excluding one of the redundant temperatures from the BaSeL model) yields $T_{\rm eff}=3140\pm40$ K. The formal error, however, does not reflect any possible systematics that could be present in the calibration. We have thus adopted a more realistic error value of $T_{\rm eff}=3140\pm150$ K based on the results of Leggett et al. (1996).

 

 

Table 6: Effective temperature determinations for the mean component of CU Cnc.

Reference	Model	$T_{\rm eff}$ (K)	Colors used
Allard et al. (2000)	STARDusty2000	3130	$(V-I)_{\rm C}$, $(V-K)_{\rm CIT}$
Hauschildt et al. (1999)	NextGen	3170	$(V-K)_{\rm CIT}$
Lejeune et al. (1998)	BaSeL 3.1	3170	$(V-I)_{\rm C}$, $(V-K)_{\rm BB}$
Lejeune et al. (1998)	BaSeL 2.2	3170	$(V-I)_{\rm C}$, $(V-K)_{\rm BB}$
Bessell et al. (1998)	NMARCS	3070	$(V-I)_{\rm C}$, $(V-K)_{\rm BB}$
Leggett (1992)	-	3145	$(R-I)_{\rm C}$, $(V-I)_{\rm C}$, $(V-K)_{\rm CIT}$

As mentioned above, there is an alternative approach to compute the temperature of the components based on the method described in Ribas et al. (1998). This is a simple method that relates the observed radius and luminosity with the temperature. In the case of CU Cnc, the absolute radii of the components are given in Table 4. The empirical luminosities were computed from the apparent magnitudes, the parallax in the Hipparcos catalogue (ESA 1997) and a bolometric correction. The latter is especially delicate because bolometric correction calibrations are known to be the source of systematic errors. To minimise these, we used the bolometric correction in the K band, which was computed from the models of Allard et al. (2000) and resulted in ${BC_{\rm K}}_{\rm BB}=+2.75\pm0.08$ mag. This leads to an apparent bolometric luminosity of $m_{\rm bol}=9.42\pm0.08$. From the luminosity ratio, the individual bolometric magnitudes can be computed and we find ${m_{\rm bol}}_{\rm A}=10.04\pm0.08$ and ${m_{\rm bol}}_{\rm B}=10.31\pm0.08$. The absolute bolometric magnitudes follow from introducing the Hipparcos distance: ${M_{\rm bol}}_{\rm A}=9.50\pm0.18$and ${M_{\rm bol}}_{\rm B}=9.77\pm0.18$. The effective temperature of each component can be computed in a straightforward manner and we obtain ${T_{\rm eff}}_{\rm A}=2950\pm120$ K and ${T_{\rm eff}}_{\rm B}=2910\pm120$ K.

The agreement with the photometric determinations described above is not perfect but nevertheless fairly good given the uncertainties. The values agree within one sigma of their respective error bars. However, a caveat is due at this point. As it turns out, there are reasonable concerns about the accuracy of the Hipparcos parallax, even taking into account its relatively large error (see Table 4). CU Cnc is a faint object near the threshold for detection by Hipparcos. Thus, the individual transit data are of poor quality. A reanalysis of the transit data (considering also the visual component) has not provided any better results (Arenou 1999, priv. comm.). As a reference, we have computed the temperature of CU Cnc as a function of the distance:

$${T_{\rm eff}}_{\rm A}=2610 \sqrt{\frac{d\mbox{(pc)}}{10}} \;\... ...ff}}_{\rm B}=2575 \sqrt{\frac{d\mbox{(pc)}}{10}} \;\;\mbox{K}.$$

Given the concerns with the Hipparcos distance raised above, we prefer to adopt the temperatures derived from photometric calibrations. The final adopted individual temperatures for CU Cnc (computed from the average temperature and the temperature ratio derived from the light curves) are listed in Table 4 along with their associated errors. Also included in Table 4 are the absolute magnitudes of the components in the V band, which have been computed from the parallax listed in the Hipparcos catalogue. Note that an empirical determination of the bolometric correction for each component follows from the comparison of the V-band absolute magnitude with the bolometric magnitudes (resulting from the radii, effective temperatures and ${M_{\rm bol}}_{\odot}=4.74$).