6 The mass luminosity relation

6.1 Computation

The masses and parallaxes derived from the previous study and the Hipparcos magnitudes have been used to determine the relation between the mass and the luminosity of the stars in this sample. This aims firstly to detect anomalous results from the residuals of the fit and secondly to compare this fit to other investigations of the mass-luminosity relation for the early type stars. The absolute magnitudes have been computed by using our parallax solution listed in Tables 7-8. The individual magnitudes of the components came either directly from the Hipparcos results (when $\Delta m$'s computation was possible), or from the combination of a ground-based $\Delta m$ with the Hipparcos composite magnitude $H\rm _p$.In any case the relation is,

$$H^{\rm abs}_{1,2} = H_{1,2} + 5 \log\pi -10$$ (2)

where $H^{\rm abs}_1$ and $H^{\rm abs}_2$ are respectively the Hipparcos absolute magnitudes of the primary and secondary components, $\pi$ is the Hipparcos parallax in mas, and H₁, H₂ are the apparent magnitudes, derived by,

$$H_1=H_{\rm p}+2.5\log(1+10^{-0.4\Delta m});\ H_2=H_1+\Delta m\ .$$ (3)

The ground-based $\Delta m$ has been expressed in the Hipparcos photometric system when possible (see Sect. 4). The results are listed in Table 12.

In the following, two sets of data are considered: the first one (Fig. 2) concerns the objects studied in the present paper (excluding the 6 revised stars taken from Paper II), while the second one includes also all the results obtained in our previous work (Figs. 3, 4 and 5). For the first three plots the components with relative error of the mass larger than 25% have been discarded. This limit is reduced to 13% in the case of the last plot, Fig. 5. There are 30 components belonging to 17 systems in the first set and 99 components belonging to 52 systems in the second group, including the non-main sequence objects and the stars lying outside of the main distribution, called "outliers".

The size of each dot in Figs. 2, 3, 4 and 5 is a visual indication of the relative quality of the mass: the bigger dots correspond to the better results ($\sigma_M/M < 10\%$) and so on to the smaller dots by steps of 5%, up to 25%. The error bars in both $\log(M/M_\odot)$ and absolute magnitude are only represented on the last two plots.

6.2 The different fits

We have used a very simple and robust procedure to fit a weighted polynomial model to the data. The obvious outliers have been first excluded from the fit by assigning a null weight to their contribution. Then, two dashed lines parallel to the fitted straight line have been drawn to highlight the main distribution of the scatter, arbitrarily taken at a distance of $\pm 0.15$ in $\log(M/M_\odot)$. Eventually we could identify additional outliers by this way and iterate the procedure. The rejected systems, shown in Fig. 3, have been discussed on a case by case basis in Sect. 5.1.

The fits are based on a cleaned sample obtained after the removal of every non-main sequence object, identified by their assumed spectral class (SIMBAD database, Hipparcos catalogue, literature), an exception being made for the intermediate class IV-V objects. The rejected systems are listed in Table 13. When the composite spectral class of the pair is the only available information, we assume the components belong to the same class, and thus probably reject more stars than needed. The outliers are naturally also removed and we are left with 75 components of 42 systems when $\sigma_M/M < 25\%$(Fig. 4), or just 32 components of 18 systems if the restriction is pushed to 13% (Fig. 5).

  

Table 13: List of the non-main sequence components excluded from the fit in Figs. 4 and 5

  

Figure 2: Mass Luminosity diagram for 30 components belonging to 17 systems studied in this paper, with $\sigma_M/M < 25\%$. The dot's size depends on the quality of the mass (see Sect. 6.1). Open circles stand for the components outside the main sequence and for the outliers, the latter being identified by their Hipparcos ID. The diamond represents the position of the Sun, assuming its absolute mag is 4.9. Note: the parameter in ordinate is not the luminosity but the absolute magnitude in the Hipparcos band; this distinction is significant for the stars outside the range $0.6-2.0\ M_{\odot}$

  

Figure 3: Same as Fig. 2 for the whole set of binaries: 99 components belonging to 52 systems (compilation of the stars contained in this paper and the previous one)

It is well known that a single linear relation cannot fit all mass ranges at a time. The M-L relations for the very high or very low mass stars differ from that concerning the central part of the diagram (see for example Cester et al. 1983, or Söderhjelm et al. 1997). In the present case, the components with masses larger than 2 $M_\odot$ or smaller than 0.6 $M_\odot$ were not taken into account in the fit of the central part. These limits are materialized on the plots by two vertical dotted lines. An additional restriction was applied to the absolute magnitude, namely $H^{\rm abs}<8$, in order to rule out any ambiguous object. Due to the very small number of objects in the extreme ranges of the diagram, we did not try to fit a relation in these two areas. Thus, the number of stars used in the fit is respectively 54 and 23 in Figs. 4 and 5.

The linear regression was computed in a very classic way, already used by Cester (1983). Briefly, It consists of fitting the data with two different straight lines by minimizing a $\chi^2$ function estimated in two orthogonal directions, and by taking the line which intercepts the two previous ones and whose slope is the average of the two slopes. In each case the weights of the data points have been set to $1/\sigma_i^2$, $\sigma_i$ representing the standard deviation of the $i^{\hbox{ th}}$ point in $\log(M)$ or in absolute magnitude, depending on the direction considered. The errors of each coefficient of the solution and the correlation coefficient between the two parameters $\log(M/M_\odot)$ and $H^{\rm abs}$are also determined.

This gives for the first case (Fig. 4),

$$\begin{eqnarray} \log(M/M_\odot) &= & 0.625 - 0.1292 H^{\rm abs} \\ & & \pm.011 \phantom{-}\pm.0024. \nonumber\end{eqnarray}$$ (4)

For this set of early type main sequence stars, the bolometric correction with respect to the Hipparcos absolute magnitude has been determined by Cayrel (1997) and can be rounded to $BC_{\rm Hp} \approx -0.2$ (or $m_{\rm bol} = Hp\, -\, 0.2$), which is accurate enough in the present context. Using $m_{b_\odot} = 4.72$, the above fit can be expressed with the luminosity as,

$$\begin{eqnarray} \log(L/L_\odot) &= & 0.033 + 3.096 \log(M/M_\odot) \\ & & \pm.066 \phantom{+}\pm.057. \nonumber\end{eqnarray}$$ (5)

In the second case limited to the best solutions (Fig. 5) we get,

$$\begin{eqnarray} \log(M/M_\odot) &= & 0.537 - 0.1074 H^{\rm abs} \\ & & \pm.010 \phantom{-}\pm.0020 \nonumber\end{eqnarray}$$ (6)

or equivalently for the luminosity,

$$\begin{eqnarray} \log(L/L_\odot) &= & -0.032 + 3.724 \log(M/M_\odot) \\ & & \phantom{0} \pm.079 \phantom{+}\pm.070. \nonumber\end{eqnarray}$$ (7)

The correlation coefficient between the "X" and "Y" variables of the diagram equals 0.62 in the first case (Eqs. (4) and (5)) and 0.70 in the second one (Eqs. (6) and (7)). According to the number of data points in each case, these values show that the linear dependance is highly significant (the probability to produce such correlation values by chance if the variables were independant is less than 0.001).

  

Figure 4: Mass Luminosity relation fitted over 54 main sequence components with $\sigma_M/M < 25\%$ (75 components visible in total). Non-main sequence stars and outliers have been removed. The dot's size depends on the quality of the mass. The two vertical dotted lines indicate the limits of the mass range considered for the fit, represented by the plain straight line. See details in Sect. 6.2. As bolometric corrections were not computed, the scatter's behaviour in the two extreme mass ranges of the diagram is not physically significant. The Sun is at the centre of the diagram (diamond)

  

Figure 5: Same as Fig. 4 (the previous remarks still hold here) with a stronger restriction in mass quality: $\sigma_M/M <$ 13%. Over the 32 visible components, 23 are effectively participating to the fit

6.3 Discussion

We have processed the same data as above with a similar filtering in mass, including this time the non-main sequence objects. It turns out that their influence to the fit is negligible, since each numerical coefficient of Eqs. (4)-(7) changes only by about 3%. The restriction in mass is on the other hand very important, even for such a sample with a fairly large scatter.

The slopes of the mass-luminosity relation are signifcantly different according to the selection threshold. This may indicate that the formal errors in the linear fit are underestimated. One must notice that the procedure to allow for the absence of an independent and perfectly controlled variable in the model fitting, is rather ad-hoc and lacks statistical rigor. Given the precision of the masses, it is however an acceptable approach. The value K = 3.7 found in the second and more reliable solution agrees well with recent determinations on similar material (Lampens et al. 1997). A more refined solution and thoughtful discussion would not only require an improved knowledge of the masses, but that of the spectral type and luminosity class for every component and was beyond the main scope of this paper.