# 7.7.3 Calibration models

• Identification of red supergiants. Long period variables consist of both red giant (mainly on the asymptotic giant branch – AGB) and supergiant stars. According to Wood et al. (1983) red supergiants can be identified based on their absolute bolometric magnitude $M_{bol}$ and main period $P$. If the LPVs are plotted in a $M_{bol}$-$P$ diagram, two distinct regions are found: one occupied by AGB stars and one occupied by supergiants. AGB stars never exceed a certain value $\max(M_{bol,P})$ that depends on the period. Therefore the following relation can be seen as upper luminosity limits:

 $\max(M_{bol,P})=-5.62787-0.00383*P+1.875*10^{-6}*P^{2}$ (7.12)

All stars brighter than $\max(M_{bol,P})$ have been classified as supergiant.

• Computation of the bolometric correction $BC(\mbox{G})$ for the $G$ band using the $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ colour (which is computed as mean($G_{\rm BP}$) - mean($G_{\rm RP}$)).

The procedure distinguishes three different cases:

1. (a)

Supergiants. Based on the compilation of M${}_{bol}$ values for red supergiants by Levesque et al. (2005) a mean value of $\mathrm{BC}_{G}$ =-0.71$\pm$0.3 mag was chosen for all red supergiants, where the bolometric error represents the standard deviation around the mean value.

2. (b)

LPVs with $G$ amplitude larger than 3 mag (Mira-like). The $G$ amplitude is computed as the 5-95% trimmed range, using the LEGACY strategy of commons-math to compute the percentiles In Gaia DR2, no distinction was made between M/S- and C-stars. The correction function is based on synthetic spectra computed with MARCS. A fixed value of $\mathrm{BC}_{G}$ =-2.2$\pm$0.005 mag was used in Gaia DR2, based on Kerschbaum et al. (2010).

3. (c)

Other LPVs. The following relation is used, based on synthetic spectra of hydrostatic M-star models (Aringer et al. 2016):

 $\displaystyle\mathrm{BC}_{G}$ $\displaystyle=$ $\displaystyle 0.2438-0.25155*(G_{\mathrm{BP}}-G_{\mathrm{RP}})-0.11433*(G_{% \mathrm{BP}}-G_{\mathrm{RP}})^{2}$ (7.13) $\displaystyle+0.00154*(G_{\mathrm{BP}}-G_{\mathrm{RP}})^{3}$

However, if the uncertainty in $G_{\rm BP}$ or $G_{\rm RP}$ is larger than 4 mag, the bolometric correction is computed as if $G_{\mathrm{BP}}-G_{\mathrm{RP}}=3.25$ mag, i.e. $\mathrm{BC}_{G}$ = -1.729$\pm$1.892 mag.

The colour has not been corrected for extinction since no reddening correction was available during our processing.

It is important to note that these bolometric corrections are preliminary results. Improved values will be provided in Gaia DR3.