Passband Luminosities Phoebe

where the second column applies to main sequence stars (V), sub-giants (IV), giants (III), and bright giants (II), the third one (I) to supergiants. To make use of this scheme, CIs from other passbands, e.g., Johnson V and Cousins I, need to be trans formed into B - V in order to use Flower (1996), e.g., by exploiting Caldwell et al. (1993) who provide different color index dependencies on B - V .As these calibrations only serve to obtain an initial value of TB(t), there is no need to worry about accuracy. As we see below the individual temperatures reproducing the color index are derived in an iterative scheme exploiting the synthetic energy distribution of the binary.

The Wilson-Devinney (WD) model used in the following discussion computes the observable flux scaled to an arbitrary level. The model adapts to this level by determining for each passband i the corresponding WD passband luminosity, L 1i, not considering any color information that might have been present in the data. Because Tb is observationally revealed by its B - V (or any other suitable) color index,1 some of the relevant temperature information is lost. One solution to this problem is to couple the passband luminosity by exploiting the observed color index (the method proposed by Prsa & Zwitter 2005b), another solution is to resort to Wilson's (2007a, b, 2008) temperature-distance theorem matching discussed in Sect. 5.1.2.3. Both approaches depend on reliable photometric calibrations [cf. Landolt (1992) covering celestial equator regions, Henden & Honeycutt (1997) and Bryja & Sandtorf (1999) covering fields around cataclysmic variables, Henden & Munari (2000) covering fields around symbiotic binaries].

In Prsa & Zwitter's (2005b) light curve program PHOEBE, described in Sect. 8.2, the parameters L 1i are initially regarded as simple level-setting quantities - physical context comes in only after the color index relationship is exploited. For the sake of simplicity, consider that input observational data are supplied in magnitudes rather than fluxes, without any arbitrary scaling of the data: Colors must be preserved and the photometry must be absolute, i.e., not relative, and fully transformed to a standard system, for which the zero magnitude flux is known accurately. PHOEBE, built on WD, inherently works with flux using a single, passband-independent parameter m0 to transform all light curves from magnitudes to fluxes. The value of m0 is chosen so that the fluxes of the dimmest light curve are of the order of unity. It is a single quantity for all passbands, which immediately implies that the magnitude difference, now the flux ratio, is preserved; hence, the color index is preserved. If the distance to the binary is also known (e.g., from astrometry), m0 immediately yields observed luminosities in physical units; intrinsic luminosities in physical units are obtained if the color excess E(B — V) is also known. This is where physics comes in: From such a set of observations, the calculated L 1i are indeed passband luminosities, the ratios of which are the constraints we need. Passband luminosities of light curves are now connected by the corresponding color indices. Note that the temperatures can be derived without the distances.

Once the color index relationship is set, only a single passband luminosity L 1i is adjusted, while the remaining L 1i are computed from the color index constraints. This way color indices are preserved and effective temperatures of the binary may

1 Useful relations among color indices are given in Caldwell et al. (1993). Although color indices depend also on log g/g0, metallicity, and rotational velocity, their effect is much smaller than the temperature.

be obtained. For a given model solution, a synthetic spectrum of the binary may be computed from the current values of Teff, log(g), vrot, and metallicity, [M/H]. As the stars are not spherical, this spectrum is constructed by integrating over local emergent intensities computed for each surface element. That spectrum is then "corrected" for interstellar extinction (by multiplying it with a wavelength-dependent interstellar extinction curve), Rayleigh scattering due to Earth's atmosphere, and any other intrinsic wavelength-dependent corrections (circumbinary attenuation, clouds, etc.) that are included in the model (Prsa 2009). The yielded spectral energy distribution is then multiplied by the CCD, optics and filter response functions (this combination is usually referred to as the passband transmission function). By this one obtains the theoretical spectrum as it would be detected. To obtain the theoretical flux for the given passband, we need to integrate this spectrum over wavelength. By repeating the same process for different passbands, we can obtain theoretical fluxes for all passbands at all phases, the ratios of which - as one might have guessed from the discussion in the preceding section - are the color indices and these need to be constrained by observations.

A historical definition of color indices, dating back to the time of photometers and differential photometry, requires that their values are set to 0.0 for Vega (spectral type A0V). Stellar energy distribution functions are not as accommodating as to provide us with that a priori, so one needs to offset the theoretical color indices. This is achieved easily by computing passband fluxes for Vega (by the same procedure sketched above), deriving color indices and setting them to 0. This provides color index offsets that are then applied to binary star color indices.

Applying (5.1.15) yields TB(t) at some observational time t orphase. For binaries with well-determined Roche surface potentials, T1 and T2 follow from the least-squares analysis. For wide detached binaries, this is somewhat hindered by the fact that the Roche potentials and temperatures are fully correlated via the surface brightness ratio (the eclipse depth ratio is directly proportional to the surface brightness ratio, which is in turn a function of [(T1, R1), (T2, R2)]). This renders individual T1 and T2 uncertain to the extent of the degeneracy between surface brightness parameters. Regardless, color constraining projects out only those combinations of parameters that preserve TB(t) and hence the color index. Because the relation between effective temperatures of individual components is, in cases where parameter correlations are weaker, fully determined by the light curve shape (predominantly by the primary-to-secondary eclipse depth ratio) and because the sum of both components' contributions must match the effective temperature of the binary, the color-constrained least-squares method yields effective temperatures of individual components.

5.1.2.3 Both Temperatures from Absolute Light Curves

The essential requirement for direct distance estimation described in Sect. 5.1.4 and temperature determination follows from Wilson's (2007a, b, 2008) EB temperature-distance theorem: Eclipsing binary light curves can yield temperatures of both stars and distance if and only if the data are standardized, the absolute geometry is determined,, and two or more substantially different photometric passbands are fitted. Here temperature means effective mean surface temperature excluding the reflection effect.

In this context, the "standardization" requirement can be fulfilled by calibration from standard2 magnitudes to physical units as described by Wilson (2007a, b) and discussion in Wilson (2008, Sect. 5). Absolute flux calibrations for the Stromgren uvby photometric system have been derived by Fabregat & Reig (1996) and Gray (1998). Two calibrations suitable for converting standard magnitudes in the Johnson system to standard physical units are Johnson (1965, 1966) and Bessell (1979), designated by subscripts J and B, respectively, below. They have been converted by Wilson (2007a, b) and lead to the following table:

Band Aeff(«) f

fB

fJ/fB

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