Mass evaluation for red giant stars based on the spectroscopically determined atmospheric parameters

Abstract

The mass ($M$) of a star can be evaluated from its spectroscopically determined effective temperature ($T_{\mathrm{eff}}$) and metallicity ([Fe/H]) along with the luminosity ($L$; derived from parallax), while comparing them with grids of theoretical evolutionary tracks. It has been argued, however, that such a track-based mass ($M_{\mathrm{trk}}$) may tend to be overestimated for the case of red giants. Meanwhile, there is an alternative approach of evaluating mass ($M_{\mathit{gLT}}$) directly from surface gravity ($g$), $L$, and $T_{\mathrm{eff}}$. The practical reliability of $M_{\mathit{gLT}}$ was examined for $\sim 100$ benchmark giants in the Kepler field, for which atmospheric parameters are already determined and the reliable mass ($M_{\mathrm{seis}}$) along with the evolutionary status are known from asteroseismology. In addition, similar check was also made for the accuracy of $M_{\mathrm{trk}}$ for comparison. It turned out that, while a reasonable correlation is seen between $M_{\mathit{gLT}}$ and $M_{\mathrm{seis}}$ almost irrespective of the stellar property, its precision is rather insufficient because $\log (M_{\mathit{gLT}}/M_{\mathrm{seis}})$ distributes rather widely within $\sim \pm 0.2$–0.3 dex. In contrast, the reliability of $M_{\mathrm{trk}}$ was found to depend on the evolutionary status. Although $M_{\mathrm{trk}}$ and $M_{\mathrm{seis}}$ are satisfactorily consistent with each other (typical dispersion of $\log (M_{\mathrm{trk}}/M_{\mathrm{seis}})$ is within ∼± 0.1 dex) for H-burning red giants as well as He-burning 2nd clump giants of higher mass, $M_{\mathrm{trk}}$ tends to be considerably overestimated as compared to $M_{\mathrm{seis}}$ by up to ≲ 0.4 dex for He-burning 1st clump giants of lower mass. Accordingly, $M_{\mathit{gLT}}$ and $M_{\mathrm{trk}}$ are complementary with each other in terms of their characteristic merit and demerit.

Appendix: effect of parallax correction on mass determination

Regarding the parallax data necessary for evaluating $L$ (luminosity) by using Eq. (2), we invoked Gaia DR2 values ($\pi _{\mathit{Gaia}}$) for all of the 103 giants in the Kepler field. After the release of these Gaia data, several studies argued the possibility of small zero-point offset to be applied to $\pi _{\mathit{Gaia}}$. It is interesting to examine the impact of such small $\pi$ corrections in mass determination, given that evaluation of $M_{\mathrm{trk}}$ from intricate evolutionary tracks may be sensitive to a slight change in $L$.

Zinn et al. (2019) reported based on asteroseismic data of red giants in the Kepler field that zero-point offset exists in the Gaia DR2 parallaxes, which depends on Gaia astrometric pseudocolour ($\nu _{\mathrm{eff}}$) and Gaia $G$-band magnitude. According to the formula derived by them, the values of this offset ($\pi _{\mathrm{Zinn}} - \pi _{\mathit{Gaia}}$) for each of our 103 Kepler giants ($\nu _{\mathrm{eff}} \sim$ 1.5–1.6, $G \sim$ 6.7–12.1; cf. Fig. 8a) turned out ∼ 0.05–0.07 m.a.s. as shown in Fig. 8b. This amount of slight increase in $\pi$ leads to a decrease in $\log L$ by ≲ 0.01–0.02 dex ($-0.4 \log (\pi _{ \mathrm{Zinn}}/\pi _{\mathit{Gaia}})$). Although such a small change in $\log L$ has a negligible effect in $M_{\mathit{gLT}}$ (only a few per cent at most), it might have an appreciable impact on $M_{\mathrm{trk}}$. We thus conducted a test determination of $M_{\mathrm{trk}}$ again based on PARAM but adopting $\pi _{\mathrm{Zinn}}$ instead of $\pi _{\mathit{Gaia}}$ (other input parameters were kept unchanged). The resulting mass values ($M_{\mathrm{trk}}^{\mathrm{Z}}$) are compared with those of original $\pi _{\mathit{Gaia}}$-based $M_{\mathrm{trk}}$ in Fig. 8c, where appreciable deviations (up to ≲ 20–30%) are observed for some cases despite a reasonable correlation overall. A closer inspection revealed that such problematic stars are seen in the luminosity range of $\log (L/L_{\odot}) \sim$ 1.5–2.0 (cf. upper inset of Fig. 8d), which almost coincides with the region of large discrepancies between $M_{\mathrm{trk}}$ and $M_{\mathrm{seis}}$ (cf. Sect. 5.1). Accordingly, this effect of $M_{\mathrm{trk}}$ changes due to $\pi$ correction is eventually inconspicuous in the $\log (M_{\mathrm{trk}}/M_{\mathrm{seis}})$ vs. $\log L$ diagram (Fig. 8d), because stars showing appreciable differences (up to $\log (M_{\mathrm{trk}}^{\mathrm{Z}}/M_{\mathrm{trk}}) \lesssim 0.1$ dex) just fall in the luminosity region where the dispersion is already large, as can be confirmed by comparing Fig. 8d with Fig. 5h (both appear almost similar). As such, we may state that the essential conclusion is not affected at all by using $\pi _{\mathrm{Zinn}}$ instead of $\pi _{\mathit{Gaia}}$.

Fig. 8

(a) Correlation between $G$ (magnitude in the $G$-band) and $\nu _{\mathrm{eff}}$ (astrometric pseudocolour) for the 103 Kepler giants (taken from Gaia DR2 catalogue). (b) Zero-point offset corrections ($\pi _{\mathrm{Zinn}} - \pi _{\mathit{Gaia}}$), which were evaluated from $G$ and $\nu _{\mathrm{eff}}$ by using Zinn et al.’s (2019) relation, plotted against $\pi _{\mathit{Gaia}}$. (c) Correlation between $M_{\mathrm{trk}}^{\mathrm{Z}}$ (derived by using PARAM but with $\pi _{\mathrm{Zinn}}$) and $\pi _{\mathit{Gaia}}$-based $M_{\mathrm{trk}}$ derived in Sect. 4, where the meanings of the symbols are the same as in Fig. 3. (d) The logarithmic ratios of $\log (M_{\mathrm{trk}}^{Z}/M_{\mathrm{seis}})$ plotted against $\log L^{\mathrm{Z}}$ ($\pi _{\mathrm{Zinn}}$-based luminosity), which is arranged similarly to Fig. 5h for the sake of comparison. Note that $\log L^{\mathrm{Z}}$-dependence of $\log (M_{\mathrm{trk}}^{\mathrm{Z}}/M_{\mathrm{trk}})$ is also depicted in the upper inset of this panel. (e) Distance differences for 103 Kepler giants between Bailer-Jones et al.’s (2018) values ($d_{\mathrm{Bailer}}$) and those derived from Gaia DR2 parallaxes ($d_{\mathit{Gaia}} \equiv \pi _{\mathit{Gaia}}^{-1}$), plotted against $d_{\mathit{Gaia}}$. (f) Difference between the “pseudo”-parallax (formally derived as $\pi ^{*}_{\mathrm{Bailer}} \equiv d_{\mathrm{Bailer}}^{-1}$) and $\pi _{\mathit{Gaia}}$, plotted against $\pi _{\mathit{Gaia}}$

Throughout this study, the distance ($d$) was derived as the reciprocal of the parallax ($1/\pi$) as usually done. However, Bailer-Jones et al. (2018) argued that distances obtained by simply inverting the Gaia DR2 parallaxes may suffer appreciable errors. Since this effect becomes progressively more significant with an increase in $\sigma _{\pi}/\pi$ (see Fig. 6 in Bailer-Jones et al. 2018; $\sigma _{\pi}$ is the error of $\pi$), it may be less significant in the present case where $\sigma _{\pi}/\pi$ is only several per cent at most. Anyway, we also checked how much correction is expected for our Kepler-field giants, because the revised distance data ($d_{\mathrm{Bailer}}$) are available in Bailer-Jones et al.’s (2018) catalogue. Figure 8e shows that the difference between $d_{\mathrm{Bailer}}$ and $d_{\mathit{Gaia}}$ ($\equiv 1/\pi _{\mathit{Gaia}}$; adopted in this study) is almost zero at small $d$ but systematically grows towards larger $d$. Interestingly, in terms of the “pseudo”-parallax (formally derived as $\pi ^{*}_{\mathrm{Bailer}} \equiv d_{\mathrm{Bailer}}^{-1}$), the difference between these two ($\pi ^{*}_{\mathrm{Bailer}} - \pi _{\mathit{Gaia}}$) is almost constant at $\simeq +0.03$ m.a.s. (Fig. 8f), which means that this effect is equivalent to the zero-point offset of $\pi$ as long as our program stars are concerned. Considering that this offset is half as small as that of Zinn et al. (2019) discussed above, we may similarly state that it does not have any significant impact on our final conclusion.