\journalname
Astronomy&AstrophysicsReview

Analysis techniques

\label{analysis}
Results of the highest accuracy require complete, high-quality data, analysed with appropriate techniques and with a critical assessment of formal and – especially – systematic errors. Criteria for and examples of suitable observational data were discussed in A91 (chiefly, accuracy and phase coverage of the light and radial-velocity curves). We have inspected all the original data used in the determinations listed in Table \ref{tableMR} to satisfy ourselves that they are adequate to support the published accuracies.
A detailed discussion of state-of-the-art analysis techniques as of 1991 for both spectroscopic and photometric data was given in A91. This need not be repeated here, but in the following we briefly review the main developments in observational and computational techniques since that time.
\label{massdet}Mass determination.
The most critical requirement for obtaining accurate masses is an accurate determination of the orbital velocities from the observed double-lined spectra, both for eclipsing and non-eclipsing binaries, because the derived masses are proportional to the third power of these velocities. This requires spectra of good spectral resolution and S/N ratio, properly analysed.
The most significant progress in the intervening two decades is the great advance in digital spectroscopy, chiefly through the use of modern échelle spectrographs and CCD detectors, coupled with the perhaps even greater advances in numerical analysis techniques. Today’s binary star observers employ spectra of much higher resolution and S/N ratio than the vast majority of the studies reported in A91, and accurate velocities can be derived with sophisticated mathematical techniques. These include two-dimensional cross-correlation techniques (\citealt{todcor}, extended to systems with three or four components by \citealt{tricor, quadcor}), the broadening function technique \citep{1992AJ....104.1968R}, and several variants of the ‘disentangling’ technique \citep[e.g.,][]{1991ApJ...376..266B, simon, hadrava, gonzalez} operating either in wavelength space or Fourier space.
The disentangling method takes advantage of the fact that a set of spectra distributed over the orbital cycle of a double-lined binary displays the same two spectra, only shifted by different relative velocities. Best-estimate values for the two spectra and the orbital elements are then extracted from the observations by a statistical analysis technique. The individual spectra can then be further analysed by standard single-star procedures to derive effective temperatures and chemical compositions for the two stars – a significant additional advantage. The determination of individual radial velocities is optional in this technique, since the orbital elements can be fit directly to the spectra with no need for an intermediate stage of measuring actual Doppler shifts \citep{1998A+A...331..167H}. Similarly, the orbital elements can be fit directly to an ensemble of one- or two-dimensional cross-correlation functions, as done, e.g., by \cite{1997ApJ...485..167T} and \cite{1999A+A...351..619F}. A minor inconvenience is that the orbital fit cannot easily be visualized, since there are no velocities to display.
The critical point, in line-by-line measurements as well as in the above more powerful, but less transparent methods, is to ascertain that the resulting velocities and orbital elements are free of systematic error. The most straightforward test today is to generate a set of synthetic binary spectra from the two component spectra and the orbital elements as determined from a preliminary analysis, computed for the observed phases and with a realistic amount of noise added. The synthetic data set is then analysed exactly as the real data, and the input and output parameters are compared. The differences, if significant, are a measure of the systematic errors of the procedure, and can be added to the real observations to correct for them \citep[see, e.g.,][]{popper94, dmvir, hshya}.
The least-squares determination of spectroscopic orbital elements from the observed radial velocities is in principle straightforward. However, subtle differences exist between various implementations, which may lead to somewhat different results from the same data sets. We have therefore systematically recomputed orbital elements from the original observations.
A point of minor importance, but still significant in mass determinations of the highest accuracy possible today, is the value of the physical constants used to compute the stellar masses and orbital semi-axis major from the observed orbital parameters. The recommended formulae are:
\begin{eqnarray} M_{1,2}\sin^{3}i & = & 1.036149\times 10^{-7}(1-e^{2})^{3/2}(K_{1}+K_{2})^{2}K_{2,1}P\nonumber \\ a\sin i & = & 1.976682\times 10^{-2}(1-e^{2})^{1/2}(K_{1}+K_{2})P\nonumber \\ \end{eqnarray}
where \(M_{1,2}\sin^{3}i\) are in units of solar masses, the orbital velocity amplitudes \(K_{2,1}\) are in km s\({}^{-1}\), the orbital period \(P\) in days, and the orbital semi-axis major in solar radii; \(i\) and \(e\) are the orbital inclination and eccentricity, respectively. The numerical constants given above correspond to the currently accepted values, in SI units, for the heliocentric gravitational constant, \(GM_{\odot}=1.3271244\times 10^{20}\) m\({}^{3}\) s\({}^{-2}\) \citep[see][]{standish} and the solar radius, \(R_{\odot}=6.9566\times 10^{8}\) m \citep{2008ApJ...675L..53H}. However, some authors still use the old value of \(1.0385\times 10^{-7}\) for the constant in the mass formula – a difference of 0.23%, which is not entirely negligible by today’s standards. Uncertainties in the solar radius itself also affect the stellar radii when expressed in that unit, but at a level \(<0.1\)%, at which the very definition of the radius of a star comes into play.
Accordingly, we have recomputed all the masses and radii listed in Table \ref{tableMR} from our own solutions of the original observations, using the constants listed above.
\label{lcanalysis}Light curve analysis.
A variety of codes exists to analyse the light curves of eclipsing binaries and derive the orbital parameters (\(i\), \(e\), and \(\omega\)) and stellar radii in units of the orbital semi-axis major. The most frequent obstacle to an accurate radius determination from such codes, notably in partially eclipsing systems, is the fact that a wide range of combinations of stellar radii, \(i\), \(e\), and \(\omega\), may yield light curves that are essentially indistinguishable.
Whether or not convergence is easy, the results must therefore always be checked against the spectroscopic determination of \(e\) and \(\omega\) and the temperature and luminosity ratio of the two stars. Because the luminosity ratio is proportional to the square of the ratio of the radii, it is a particularly sensitive – often indispensable – constraint on the latter \cite[see][for an extreme example]{tzfor}.
The relative depths of the light curve minima and the colour changes during eclipse are usually robustly determined from the light curve solution and are good indicators of the surface flux ratio between the two stars. Whenever possible, we have used these data to check the published temperature differences between the two stars.
Formal error estimates from the codes rarely include the contribution of systematic effects. Comparing separate solutions of light curves in several passbands is one way to assess the reliability of the results; computing light curves for several parameter combinations and evaluating the quality of the fit to the data is another.
\label{consistency}Consistency checks.
In all cases, it is important to verify the consistency of the different types of information on a given system as thoroughly as possible. The values for the period (rarely a problem except in systems with apsidal motion), \(e\) and \(\omega\) of the orbit must be internally consistent, as must the luminosity ratio of the components as measured from the light curves and seen in the spectra.
Certain light-curve codes, e.g., the widely-used WD code \citep{WD}, allow one to input a set of light curves in several colours as well as the radial-velocity observations, and return a single set of results for the stellar and orbital parameters. From a physical point of view this is clearly the preferable procedure, and the resulting single sets of masses, radii, etc. often have impressively small formal errors.
However, such a procedure tends to obscure the effects of flaws and inconsistencies in the data and/or the binary model, and consistency checks such as those described above become difficult or impossible. If consistency has been verified independently, a combined, definitive solution can be performed with confidence, but the basic philosophy should always be that consistency is a condition to be verified, not assumed.
In compiling the data presented in Table \ref{tableMR}, we have verified that the conditions described in this section are satisfied in every system, usually by recomputing the stellar parameters from the original data. The numbers presented here will therefore not always be strictly identical to those found in the original analyses.