Andersen (1991) provides a compilation of data on all eclipsing binaries (EB) known at the time - a total of 90 stars, most of which are on the main sequence. Section 4 in Andersen
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSGC07 (MSol) 0 1 2 3 4 RCHARA (R Sol )
Figure 7.24: Radius-Mass: The CHARA radii and model masses are plotted for stars in common with the Holmberg et al. (2007) survey (GC07). The 1-σerrors on radius and mass are also shown.
(1991) argues that the motivation for compiling the EB data is to aid in the prediction of single star properties where masses and radii are unobtainable by direct measurements for a large number of stars. We use these data on eclipsing binaries to compare with our results for single stars in this section.
Effective temperatures of EB stars are not able to be determined directly because the distances to the systems are not known to great accuracy. Due to the fact that the stars are in binaries, their parallaxes could be difficult to determine because the orbital motion of the binary in the sky around the center of mass of the system is particularly difficult to deconvolve from the parallactic displacement. In addition, interstellar reddening is also a
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSGC07 (MSol) 0 1 2 3 4 RCHARA (R Sol ) [Fe/H] -1.5 +0.4
Figure 7.25: Radius-Mass-Metallicity: The CHARA radii and model masses are plotted for stars in common with the Holmberg et al. (2007) survey (GC07). The grayscale color corresponds to the metallicity [Fe/H] of the star.
factor in the distant systems when converting observed photometry to absolute magnitudes. Thus, a primary advantage of measuring the angular diameters of single stars for which we know the distances with great accuracy is that reddening can be ignored. Nearby stars will provide the means to calibrate the temperature relations for EB’s and can also be applied to a large number of stars. Also, in Andersen (1991) the luminosities are derived via the Stefan-Boltzmann equation, using the measured EB radii and model derived TEFF. In the
discussions to follow, keep in mind that these EB luminosities and temperatures might have systematic offsets due to the indirect determination of these quantities.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSTak07 (MSol)
0 1 2 3 4 RCHARA (R Sol )
Figure 7.26: Radius-Mass: The CHARA radii and model masses are plotted for stars in common with the Takeda (2007) survey (Tak07). Each point is represented by a circle, and the 1-σ errors in radius are shown (Takeda 2007 does not provide mass errors).
Eclipsing binary star and single star radii versus (B −V) color index are compared in Figure 7.33. The general direction of evolution off the main sequence is marked in the top right of the plot. One can see that for stars even on the main sequence there is quite a spread in radius for a given (B−V). It is interesting to note that for stars redder thanB−V ≈0.5, EB stars are more evolved than CHARA stars (although the data are sparse in this region for EBs). For stars bluer than B −V ≈ 0.5, the CHARA stars are more evolved than the EB stars. This might be from a selection effect that all nearby stars observed with CHARA are field stars, and hence older than EBs found in dense young clusters. The important conclusion here is that there is no systematic offset seen when comparing the radii from
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSTak07 (MSol)
0 1 2 3 4 RCHARA (R Sol ) [Fe/H] -1.5 +0.4
Figure 7.27: Radius-Mass-Metallicity: The CHARA radii and model masses are plotted for stars in common with the Takeda (2007) survey (Tak07). The grayscale color corresponds to the metallicity [Fe/H] of the star.
eclipsing binary and single stars. This supports the conclusion that models are doing a poor job of predicted radii for single stars (§7.1).
Exploring the mass-radius relations in single versus binary stars, we find a similar re- lationship. Figure 7.34 shows that there is still much scatter in the mass-radius relation for main sequence stars and that there is no systematic offset when comparing values from binary to single stars. The masses used here are the masses derived from our measured CHARA radii and logg estimates. In the previous section, Figure 7.31 and Figure 7.32 showed that for stars of larger masses, there were increasingly larger differences between the model masses from the references, and our derived CHARA masses. In the region of higher
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSAP99 (MSol)
4000 5000 6000 7000 8000 9000 10000 TCHARA (K)
Figure 7.28: Temperature-Mass: The CHARA temperatures and model masses are plotted for stars in common with the Allende Prieto & Lambert (1999) survey (AP99). The 1-σerrors on temperature and mass are also shown.
masses in Figure 7.34, the derived CHARA masses are very consistent with the EB values, so perhaps the errors in gravity are not as large as previously thought, and the techniques for determining masses from the models need to be tweaked. The mass-radius relation of R ∝M0.8 is shown as the dotted line, which holds for both binary and single main sequence
stars of less than ≈3.5M⊙.
Figure 7.35 is the radius-luminosity relation for both the EB stars and the single CHARA stars. The larger the radii, the more spread in luminosity is found in these stars. For the main sequence stars observed with CHARA this spread is minimal. For the eclipsing binary stars, whose spectral types range from O8−M1, the spread on the luminosity-radius plane
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSGC07 (MSol) 4000 5000 6000 7000 8000 9000 10000 TCHARA (K)
Figure 7.29: Temperature-Mass: The CHARA temperatures and model masses are plotted for stars in common with the Holmberg et al. (2007) survey (GC07). The 1-σ errors on temperature and mass are shown.
is significant. Within the range of radii measured with CHARA (logR/R⊙ ≈ −0.2 to 0.6), there is a tight relation of binary stars to single stars up to logR/R⊙ ≈ 0.15. For stars larger than this radius, there is a minimum luminosity for a given radius consistent within each data set, but the spread to higher luminosities of the EB sample increases significantly more than the single stars.
Figure 7.36 shows the mass to color index (B −V) relation for EB and CHARA stars with masses derived from logg estimates. For the sample of EBs, Andersen (1991) points out that stellar evolution on the main sequence can be seen by the fact that for a certain color index, there is a range of masses (EB mass error is typically ≈ 1.4%). This effect is
0.0 0.5 1.0 1.5 2.0 2.5 3.0 MASSTak07 (MSol)
4000 5000 6000 7000 8000 9000 10000 TCHARA (K)
Figure 7.30: Temperature-Mass: The CHARA temperatures and model masses are plotted for stars in common with the Takeda (2007) survey (Tak07). Each point is represented by a circle, and the 1-σerrors in temperatures are shown (Takeda 2007 does not provide mass errors).
most apparent in spectral types A-F (0.0 . B −V . 0.5), where for the EB data points, there is a spread in the right direction of the plot (the direction of stellar evolution). For the CHARA stars, the error in mass is much larger. However, the same trend seen in Figure 7.33 (radius versus color index) is seen with respect to stellar mass versus color index, where the stars bluer than B−V .0.45 are more evolved than the stars in the EB sample.
There does seem to be a systematic offset between EB masses and CHARA masses derived from gravity when plotted against luminosity, as seen in Figure 7.37. Although the scatter is large, the systematics appear for stars with M ≥ 1.5M⊙, the same position as in Figure 7.31, where the CHARA masses are larger than they should be if logg estimates are
0 1 2 3 4 MCHARA (MSol) 0 1 2 3 4 MAP99 (M Sol )
Figure 7.31: CHARA Masses Versus Model Masses: The CHARA masses derived from measured radii and logg estimates from Allende Prieto & Lambert (1999) (AP99) compared to model masses of the same stars included in Allende Prieto & Lambert (1999). The dotted line shows the 1:1 relation. Errors are not shown, however the errors for the CHARA derived masses are ≈20% due to uncertainty in gravity estimates.
overestimated. However, the errors in CHARA derived masses may diminish the significance of this effect. An equally likely contributor to this effect is that this could be a problem with the derived EB luminosities.
0 1 2 3 4 MCHARA (MSol) 0 1 2 3 4 MTak07 (M Sol )
Figure 7.32: CHARA Masses Versus Model Masses: The CHARA masses derived from measured radii and loggestimates from Takeda (2007) (Tak07) compared to model masses of the same stars included in Allende Prieto & Lambert (1999). The dotted line shows the 1:1 relation. Errors are not shown, however the errors for the CHARA derived masses are≈20% due to uncertainty in gravity estimates.
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 COLOR INDEX (B-V) 0.0 0.5 1.0 LOG Radius (R Sol ) = EB = CHARA
Figure 7.33: Eclipsing Binary and CHARA Radii Versus (B-V):The CHARA radii (filled circles) and eclipsing binary radii (open circles) are plotted against color index (B−V). In most cases, the errors in radii are smaller than the data points. The arrow in the top right side of the plot indicates the direction of evolution off the main sequence.
1.5 1.0 0.5 0.0 -0.5 LOG Mass (MSol)
-0.5 0.0 0.5 1.0 1.5 LOG Radius (R Sol ) = EB = CHARA / AP99 = CHARA / Tak07
Figure 7.34: Eclipsing Binary and CHARA Masses Versus Radius: The EB radii and masses (open circles) are from Andersen (1991). CHARA data from this work are plotted, where the mass is derived from the logg estimates combined with CHARA radii for stars in Allende Prieto & Lambert (1999) (AP99) and Takeda (2007) (Tak07). In most cases, the errors in radii are smaller than the data points. Mass errors for EB’s are typically smaller than the data point. A representative error in CHARA mass is plotted on the bottom left of the plot window. The dotted line is the mass-radius relation for main sequence stars
0.0 0.5 1.0 LOG Radius (RSol)
-2 0 2 4 6 LOG Luminosity (L Sol )
Figure 7.35: Eclipsing Binary and CHARA Luminosities Versus Radii: The EB data are from Andersen (1991) and are plotted as open circles. CHARA data from this work are plotted as closed circles. In most cases, the errors in radii and luminosities are smaller than the data points.
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 COLOR INDEX (B-V) -0.5 0.0 0.5 1.0 1.5 LOG Mass (M Sol ) = EB = CHARA / AP99 = CHARA / Tak07
Figure 7.36: Eclipsing Binary and CHARA Mass Versus (B−V): The EB data are from Andersen (1991) and are plotted as open circles. The mass is derived from the loggestimates combined with CHARA radii for stars in Allende Prieto & Lambert (1999) (AP99) and Takeda (2007) (Tak07) are plotted as green and blue filled circles, respectively. In most cases, the errors in color index (B−V) are smaller than the data point. Mass errors for EB’s are typically smaller than the data point. A representative error in CHARA mass is plotted on the bottom left of the plot window. The arrow in the upper right position of the plot points in the direction of stellar evolution.
1.5 1.0 0.5 0.0 -0.5 LOG Mass (MSol)
-2 0 2 4 6 LOG Luminosity (L Sol ) = EB = CHARA / AP99 = CHARA / Tak07
Figure 7.37: Eclipsing Binary and CHARA Mass Versus Luminosity: The EB data are from Andersen (1991) and are plotted as open circles. The mass is derived from the loggestimates combined with CHARA radii for stars in Allende Prieto & Lambert (1999) (AP99) and Takeda (2007) (Tak07) are plotted as filled green and blue circles, respectively. In most cases, the error in luminosity is smaller than the data point. Mass errors for EB’s are typically smaller than the data point, whereas the error in CHARA masses are much larger (representative CHARA mass error shown in the bottom left position of the plot window). The dotted line is the relation: M ∝L3.8
. The arrow in the upper right position of the plot points in the direction of stellar evolution.